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# The matrices of argumentation frameworks
Xu Yuming
$School\ of\ Mathematics,Shandong\ University,Jinan,China$ Corresponding
author. E-mail: xuyuming@sdu.edu.cn
Abstract
We introduce matrix and its block to the Dung’s theory of argumentation
frameworks. It is showed that each argumentation framework has a matrix
representation, and the common extension-based semantics of argumentation
framework can be characterized by blocks of matrix and their relations. In
contrast with traditional method of directed graph, the matrix way has the
advantage of computability. Therefore, it has an extensive perspective to
bring the theory of matrix into the research of argumentation frameworks and
related areas.
Keywords: Argumentation framework; extension semantics; matrix; block
1\. Introduction
In recent years, the area of argumentation begins to become increasingly
central as a core study within Artificial Intelligence. A number of papers
investigated and compared the properties of different semantics which have
been proposed for argumentation frameworks (AFs, for short) as introduced by
Dung [8, 4, 3, 9, 6]. In early time, many of the analysis of arguments are
expressed in natural language. Later on, a tradition of using diagrams has
been developed to explicate the relations between the components of the
arguments. Now, argumentation frameworks are usually represented as directed
graphs, which play a significant role in modeling and analyzing the extension-
based semantics of AFs. For further notations and techniques of argumentation,
we refer the reader to [8, 2, 15, 1].
Our aim is to introduce matrix as a new mathematic tool to the research of
argumentation frameworks. First, we assign a matrix of order $n$ for each
argumentation framework with $n$ arguments. Each element of the matrix has
only two possible values: one and zero, where one represents the attack
relation and zero represents the non-attack relation between two arguments
(they can be the same one). Under this circumstance, the matrix can be thought
to be a representation of the argumentation framework. Secondly, we analysis
the internal structure of the matrix corresponding to various extension-based
semantics of the argumentation framework, and obtain the matrix approaches to
determine the stable extension, admissible extension and complete extension,
which can be easily realized on computer.
As will be seen in later, the matrix of an argumentation framework is not only
visualized as the directed graph, but also has another significant advantage
on the aspect of computation. We shall study various extension-based semantics
of the argumentation framework by comparing and computing the matrix of the AF
and its blocks.
2\. Dung’s theory of argumentation
Argumentation is a general approach to model defeasible reasoning and
justification in Artificial Intelligence. So far, many theories of
argumentation have been established. Among them, Dung’s theory of
argumentation framework is quite influence. In fact, it is abstract enough to
manage without any assumption on the nature of arguments and the attack
relation between arguments. Let us first recall some basic notion in Dung’s
theory of argumentation framework. We restrict them to finite argumentation
frameworks.
An argumentation framework is a pair $F=(A,R)$, where $A$ is a finite set of
arguments and $R\subset A\times A$ represents the attack-relation. For
$S\subset A$, we say that
(1) $S$ is conflict-free in $(A,R)$ if there are no $a,b\in S$ such that
$(a,b)\in R$;
(2) $a\in A$ is defeated by $S$ in $(A,R)$ if there is $b\in S$ such that
$(b,a)\in R$;
(3) $a\in A$ is defended by $S$ in $(A,R)$ if for each $b\in A$ with $(b,a)\in
R$, we have $b$ is
defeated by $S$ in $(A,R)$.
(4) $a\in A$ is acceptable with respect to $S$ if for each $b\in A$ with
$(b,a)\in R$, there is some
$c\in S$ such that $(c,b)\in R$.
The conflict-freeness, as observed by Baroni and Giacomin[1] in their study of
evaluative criteria for extension-based semantics, is viewed as a minimal
requirement to be satisfied within any computationally sensible notion of
”collection of justified arguments”. However, it is too weak a condition to be
applied as a reasonable guarantor that a set of arguments is ”collectively
acceptable”.
Semantics for argumentation frameworks can be given by a function $\sigma$
which assigns each AF $F=(A,R)$ a collection $\mathcal{S}\subset 2^{A}$ of
extensions. Here, we mainly focus on the semantic
$\sigma\in\\{s,a,p,c,g,i,ss,e\\}$ for stable, admissible, preferred, complete,
grounded, ideal, semi-stable and eager extensions, respectively.
Definition 1[14] Let $F=(A,R)$ be an argumentation framework and $S\in A$.
(1) $S$ is a stable extension of $F$, $i.e.$, $S\in s(F)$, if $S$ is conflict-
free in $F$ and each
$a\in A\setminus S$ is defeated by $S$ in $F$.
(2) $S$ is an admissible extension of $F$, $i.e.$, $S\in a(F)$, if $S$ is
conflict-free in $F$ and each
$a\in A\setminus S$ is defended by $S$ in $F$.
(3) $S$ is a preferred extension of $F$, $i.e.$, $S\in p(F)$, if $S\in a(F)$
and for each $T\in a(F)$,
we have $S\not\subset T$.
(4) $S$ is a complete extension of $F$, $i.e.$, $S\in c(F)$, if $S\in a(F)$
and for each $a\in A$
defended by $S$ in $F$, we have $a\in S$.
(5) $S$ is a grounded extension of $F$, $i.e.$, $S\in g(F)$, if $S\in c(F)$
and for each $T\in c(F)$,
we have $T\not\subset S$.
(6) $S$ is an ideal extension of $F$, $i.e.$, $S\in i(F)$, if $S\in a(F)$,
$S\subset\cap\\{T:T\in p(F)\\}$ and
for each $U\in a(F)$ such that $U\subset\cap\\{T:T\in p(F)\\}$, we have
$S\not\subset U$.
(7) $S$ is a semi-stable extension of $F$, $i.e.$, $S\in ss(F)$, if $S\in
a(F)$ and for each $T\in a(F)$,
we have $R^{+}(S)\not\subset R^{+}(T)$, where $R^{+}(U)=\\{U\cap\\{b:(a,b)\in
R,A\in U\\}\\}$.
(8) $S$ is a eager extension of $F$, $i.e.$, $S\in e(F)$, if $S\in c(F)$,
$S\subset\cap\\{T:T\in ss(F)\\}$ and
for each $U\in a(F)$ such that $U\subset\cap\\{T:T\in ss(F)\\}$, we have
$S\not\subset T$.
Note that, there are some elementary properties for any argumentation
framework $F=(A,R)$ and semantic $\sigma$. If $\sigma\in\\{a,p,c,g\\}$, then
we have $\sigma(F)\neq\emptyset$. And if $\sigma\in\\{g,i,e\\}$, then
$\sigma(F)$ contains exactly one extension. Furthermore, the following
relations hold for each argumentation framework $F=(A,R)$:
$s(F)\subseteq p(F)\subseteq c(F)\subseteq a(F)$.
Since every extension of an AF under the standard semantics (stable,
preferred, complete and grounded extensions) introduced by Dung is an
admissible set, the concept of admissible extensions plays an important role
in the study of argumentation frameworks.
3\. The matrix of an argumentation framework
We know that the directed graph is a traditional tool in the research of
argumentation frameworks, and has the feature of visualization [7, 10, 11]. It
is widely used for modeling and analyzing argumentation frameworks. In this
section, we shall introduce the matrix representation of argumentation
frameworks. Except for the visualization, the matrix also has the advantage of
computability in analyzing argumentation frameworks and computing various
extension semantics.
An $m\times n$ matrix $A$ is a rectangular array of numbers, consisting of $m$
rows and $n$ columns, denoted by
$A=\left(\begin{array}[]{cccccc}a_{1,1}&a_{1,2}&.&.&.&a_{1,n}\\\
a_{2,1}&a_{2,2}&.&.&.&a_{2,n}\\\ .&.&.&.&.&.\\\
a_{m,1}&a_{m,2}&.&.&.&a_{m,n}\end{array}\right).$
The $m\times n$ numbers $a_{1,1},a_{1,2},...,a_{m,n}$ are the elements of the
matrix $A$. We often called $a_{i,j}$ the $(i,j)$th element, and write
$A=(a_{i,j})$ for short. It is important to remember that the first suffix of
$a_{i,j}$ indicates the row and the second the column of $a_{i,j}$.
A column matrix is an $n\times 1$ matrix, and a row matrix is an $1\times n$
matrix, denoted by
$\left(\begin{array}[]{c}x_{1}\\\ x_{2}\\\ .\\\ .\\\ .\\\
x_{n}\end{array}\right),\left(\begin{array}[]{cccccc}x_{1}&x_{2}&.&.&.&x_{n}\end{array}\right)$
respectively. Matrices of both these types can be regarded as vectors and
referred to respectively as column vectors and row vectors. Usually, the $i$th
row of a matrix $A$ is denoted by $A_{i,*}$, and the $j$th column of $A$ is
denoted by $A_{*,j}$.
Definition 2 In an $n\times m$ matrix $A=(a_{i,j})$, we specify any $k(\leq
min\\{n,m\\})$ different rows $i_{1},i_{2},...,i_{k}$ and the same number of
different columns $i_{1},i_{2},...,i_{k}$. The elements appearing at the
intersections of these rows and columns form a square matrix of order $k$. We
call this matrix a principal block of order $k$ of the original matrix $A$; it
is denoted by
$M=\left(\begin{array}[]{cccccc}a_{i_{1},i_{1}}&a_{i_{1},i_{2}}&.&.&.&a_{i_{1},i_{k}}\\\
a_{i_{2},i_{1}}&a_{i_{2},i_{2}}&.&.&.&a_{i_{2},i_{k}}\\\ .&.&.&.&.&.\\\
a_{i_{k},i_{1}}&a_{i_{k},i_{2}}&.&.&.&a_{i_{k},i_{k}}\end{array}\right),$
or $M=M_{i_{1},i_{2},...,i_{k}}^{i_{1},i_{2},...,i_{k}}$ for short.
Definition 3 If in the original $n\times m$ matrix $A=(a_{i,j})$, we delete
the rows and columns which make up the block
$M=M^{i_{1},i_{2},...,i_{k}}_{i_{1},i_{2},...,i_{k}}$, then the remaining
elements form an $(n-k)\times(m-k)$ matrix. We call this matrix the
complementary block of $M$, and is denoted by the symbol
$\overline{M}=\overline{M_{i_{1},i_{2},...,i_{k}}^{i_{1},i_{2},...,i_{k}}}$.
Definition 4 In an $n\times m$ matrix $A$, we specify any $k(\leq n)$
different rows $i_{1},i_{2},...,i_{k}$ and $h(\leq m)$ different columns
$j_{1},j_{2},...,j_{h}$. The elements appearing at the intersections of these
rows and columns form a $k\times h$ matrix. We call this matrix a $k\times h$
block of the original matrix $A$; it is denoted by
$M=\left(\begin{array}[]{cccccc}a_{i_{1},j_{1}}&a_{i_{1},j_{2}}&.&.&.&a_{i_{1},j_{h}}\\\
a_{i_{2},j_{1}}&a_{i_{2},j_{2}}&.&.&.&a_{i_{2},j_{h}}\\\ .&.&.&.&.&.\\\
a_{i_{k},j_{1}}&a_{i_{k},j_{2}}&.&.&.&a_{i_{k},j_{h}}\end{array}\right),$
or $M=M_{i_{1},i_{2},...,i_{k}}^{j_{1},j_{2},...,j_{h}}$ for short.
For the underlying set $A$ of an argumentation framework $F=(A,R)$, there is
no ordering in nature. But, in many cases the ordering set can benefit us a
lot. Contrasting with the form $A=\\{a,b,...\\}$, it is more convenience to
put $A=\\{1,2,...,n\\}$ while the cardinality of $A$ is large. In particular,
we can map each argument to the corresponding row and column of a matrix. We
will follow this arrangement in the below discussion.
Definition 5 Let $F=(A,R)$ be an argumentation framework with
$A=\\{1,2,...,n\\}$. The matrix of $F$, denoted by $M(F)$, is a Boolean matrix
of order $n$, its element is determined by the following rules:
(1) $a_{i,j}=1$ iff $(i,j)\in R$;
(2) $a_{i,j}=0$ iff $(i,j)\notin R$.
Example 6 Considering the argumentation framework $F=(A,R)$, where
$A=\\{1,2,3\\}$ and $R=\\{(1,2),(2,3),(3,1)\\}$. By the definition, we have
the following matrix of $F$:
$M(F)=\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\ 1&0&0\end{array}\right)$
Example 7 Given an argumentation framework $F=(A,R)$, where $A=\\{1,2,3,4\\}$
and $R=\\{(1,2),(1,3),(2,1),(2,3),(3,4)\\}$. The matrix of $F$ is as follows:
$M(F)=\left(\begin{array}[]{cccc}0&1&1&0\\\ 1&0&1&0\\\ 0&0&0&1\\\
0&0&0&0\end{array}\right)$
In comparison with graph-theoretic way and mathematical logic way, the matrix
of an argumentation framework has many excellent features. First, it possess a
concise mathematical format. Secondly, it contains all information of the $AF$
by combining the arguments with attack relation in a specific manner in the
matrix $M(F)$. Also, it can be deal with by program on computer. The most
important is that we can import the knowledge of matrix to the research of
argumentation frameworks.
4\. Determination of the conflict-free sets
As we know, there is no efficient method for us to decide a conflict set in an
argumentation framework, even we can draw up the directed graph of the AF.
After we introduce the matrix of the AF, the situation will be changed
completely. By checking the matrix of the argumentation framework, we can
easily find out all the conflict-free sets of the AF. Let us see an example,
firstly.
Example 8 Given an argumentation framework $F=(A,R)$, where
$A=\\{1,2,3,4,5\\}$ and $R=\\{(1,2),(2,3),(2,5),(4,3),(5,4)\\}$. Then, we can
easily to show that the collection of conflict-free sets of $F$ is
$\\{\emptyset,\\{1\\},\\{2\\},\\{3\\},\\{4\\},\\{5\\}\\{1,3\\},\\{1,4\\},\\{1,5\\},\\{2,4\\},\\{3,5\\},\\{1,3,5\\}\\}$,
by the routine method of directed graph.
On the other hand, we consider the matrix of $F=(A,R)$ and study its structure
from the level of blocks. First, we write out the matrix of $F$:
$M(F)=\left(\begin{array}[]{ccccc}0&1&0&0&0\\\ 0&0&1&0&1\\\ 0&0&0&0&0\\\
0&0&1&0&0\\\ 0&0&0&1&0\end{array}\right).$
By observing the principal blocks of the above matrix, we find that there are
five zero principal blocks of order 1
$M^{1}_{1}=\left(\begin{array}[]{c}0\end{array}\right),M^{2}_{2}=\left(\begin{array}[]{c}0\end{array}\right),M^{3}_{3}=\left(\begin{array}[]{c}0\end{array}\right),M^{4}_{4}=\left(\begin{array}[]{c}0\end{array}\right),M^{5}_{5}=\left(\begin{array}[]{c}0\end{array}\right)$
corresponding to the conflict-free sets $\\{1\\}$, $\\{2\\}$, $\\{3\\}$,
$\\{4\\}$, $\\{5\\}$, respectively. There are five zero principal blocks of
order 2
$M^{1,3}_{1,3}=\left(\begin{array}[]{cc}0&0\\\
0&0\end{array}\right),M^{1,4}_{1,4}=\left(\begin{array}[]{cc}0&0\\\
0&0\end{array}\right),M^{1,5}_{1,5}=\left(\begin{array}[]{cc}0&0\\\
0&0\end{array}\right),M^{2,4}_{2,4}=\left(\begin{array}[]{cc}0&0\\\
0&0\end{array}\right),M^{3,5}_{3,5}=\left(\begin{array}[]{cc}0&0\\\
0&0\end{array}\right)$
corresponding to the conflict-free sets $\\{1,3\\}$, $\\{1,4\\}$, $\\{1,5\\}$,
$\\{2,4\\}$, $\\{3,5\\}$, respectively. Also, there is a zero principal block
of order 3
$M^{1,3,5}_{1,3,5}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\
0&0&0\end{array}\right)$
corresponding to the conflict-free sets $\\{1,3,5\\}$.
Note that, the above blocks are all principal blocks which are zero in the
matrix $M(F)$, and there is a one to one correspond between the collection of
all conflict-free sets of $F$ and the set of all zero principal blocks of
$M(F)$. In fact, for any argumentation framework $F$ there exists such
corresponding relation between the collection of all conflict-free sets of $F$
and the set of all zero principal blocks of $M(F)$.
Since it is easy to find out the zero principal blocks in the matrix of an
argumentation framework, we obtain a good way to decide the conflict-free sets
of the $AF$ through its matrix. Certainly, this way can be carried out on the
computer readily.
Definition 9 Let $F=(A,R)$ be an argumentation framework with
$A=\\{1,2,...,n\\}$, and $S=\\{i_{1},i_{2},...,i_{k}\\}\subset A$. The
principal block
$M^{i_{1},i_{2},...,i_{k}}_{i_{1},i_{2},...,i_{k}}=\left(\begin{array}[]{cccccc}a_{i_{1},i_{1}}&a_{i_{1},i_{2}}&.&.&.&a_{i_{1},i_{k}}\\\
a_{i_{2},i_{1}}&a_{i_{2},i_{2}}&.&.&.&a_{i_{2},i_{k}}\\\ .&.&.&.&.&.\\\
a_{i_{k},i_{1}}&a_{i_{k},i_{2}}&.&.&.&a_{i_{k},i_{k}}\end{array}\right)$
of order $k$ in the matrix $M(F)$ is called the $cf$-block of $S$, and denoted
by $M^{cf}$.
Theorem 10 Given an argumentation framework $F=(A,R)$ with
$A=\\{1,2,...,n\\}$, then $S=\\{i_{1},i_{2},...,i_{k}\\}\subset A$ is a
conflict-free set in $F$ iff the $cf$-block
$M^{i_{1},i_{2},...,i_{k}}_{i_{1},i_{2},...,i_{k}}$ of $S$ is zero.
Proof Assume that $M^{i_{1},i_{2},...,i_{k}}_{i_{1},i_{2},...,i_{k}}=0$, then
for arbitrary $1\leq s,t\leq k$ we have $a_{i_{s},i_{t}}=0$, $i.e.$,
$(i_{s},i_{t})\notin R$. Thus, $S=\\{i_{1},i_{2},...,i_{k}\\}$ is a conflict-
free set in $F$.
Suppose $S=\\{i_{1},i_{2},...,i_{k}\\}\subset A$ is a conflict-free set in
$F$, then for arbitrary $1\leq s,t\leq k$ we have that $(i_{s},i_{t})\notin
R$, $i.e.$, $a_{i_{s},i_{t}}=0$. Therefore, we have
$M^{i_{1},i_{2},...,i_{k}}_{i_{1},i_{2},...,i_{k}}=0$.
5\. Determination of the stable extensions
Example 11 We continuous to study the argumentation framework $F=(A,R)$, where
$A=\\{1,2,3,4,5\\}$ and $R=\\{(1,2),(2,3),(2,5),(4,3),(5,4)\\}$. Since the
stable extension is firstly a conflict-free set, we can look for the stable
extension from the collection
$\\{\emptyset,\\{1\\},\\{2\\},\\{3\\},\\{4\\},\\{5\\}\\{1,3\\},\\{1,4\\},\\{1,5\\},\\{2,4\\},\\{3,5\\},\\{1,3,5\\}\\}$
of conflict-free sets. In fact, the set $S=\\{1,3,5\\}$ is the only stable
extension in $F$ by a simple discussion.
Again, we turn our attention to the matrix of the $F=(A,R)$:
$M(F)=\left(\begin{array}[]{ccccc}0&1&0&0&0\\\ 0&0&1&0&1\\\ 0&0&0&0&0\\\
0&0&1&0&0\\\ 0&0&0&1&0\end{array}\right).$
Since $S=\\{1,3,5\\}$ is a stable extension of $F$, the arguments $2$ and $4$
are defeated by $\\{1,3,5\\}$. This fact is reflected in the matrix $M(F)$ of
$F$ as follows.
In the column vector $F_{*,2}$ (column 2), $a_{1,2}=1$ means that $(1,2)\in
R$, and thus the argument $1$ attacks the argument $2$. In the column vector
$F_{*,4}$ (column 4), $a_{5,4}=1$ means that $(5,4)\in R$, and thus the
argument $5$ attacks the argument $4$.
From the behavior of the elements $a_{1,2}=1$ and $a_{5,4}=1$ in the matrix
$M(F)$, we can extract a matrix approach to decide that the conflict-free set
$S=\\{1,3,5\\}$ is a stable extension: Corresponding to the arguments $2,4\in
A\setminus S$, we firstly pick out the column vectors $F_{*,2}$ and $F_{*,4}$
in the matrix $M(F)$, then check the elements $a_{1,2},a_{3,2},a_{5,2}$ of
$F_{*,2}$, and the elements $a_{1,4},a_{3,4},a_{5,4}$ of $F_{*,4}$. If there
is one element of $\\{a_{1,2},a_{3,2},a_{5,2}\\}$ which is non-zero, then the
argument $2$ is defeated by $S$. Similar result is hold for the argument $4$.
This process leads to a block of the matrix $M(F)$ at the intersection of
columns $2,4$ and rows $1,3,5$.
To sum up, we can decide that the conflict set $S=\\{1,3,5\\}$ is a stable
extension by the fact that the two column vectors of the above block of the
matrix $M(F)$ are all non-zero. Further analysis indicates that the converse
is also true. This motivation makes us to give the following definition.
Definition 12 Let $F=(A,R)$ be an argumentation framework with
$A=\\{1,2,...,n\\}$, and $S=\\{i_{1},i_{2},...,i_{k}\\}\subset A$ is a stable
extension of $F$. The $k\times h$ block
$M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}=\left(\begin{array}[]{cccccc}a_{i_{1},j_{1}}&a_{i_{1},j_{2}}&.&.&.&a_{i_{1},j_{h}}\\\
a_{i_{2},j_{1}}&a_{i_{2},j_{2}}&.&.&.&a_{i_{2},j_{h}}\\\ .&.&.&.&.&.\\\
a_{i_{k},j_{1}}&a_{i_{k},j_{2}}&.&.&.&a_{i_{k},j_{h}}\end{array}\right)$
in the matrix $M(F)$ is called the $s$-block of $S$ and denoted by $M^{s}$,
where $\\{j_{1},j_{2},...,j_{h}\\}=A\setminus S$.
In other words, the elements appearing at the intersections of rows
$i_{1},i_{2},...,i_{k}$ and columns $j_{1},j_{2},...,j_{h}$ in the matrix
$M(F)$ form the $s$-block $M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}$
of $S$.
Theorem 13 Given an argumentation framework $F=(A,R)$ with
$A=\\{1,2,...,n\\}$, then $S=\\{i_{1},i_{2},...,i_{k}\\}\subset A$ is a stable
extension in $F$ iff the following conditions hold:
(1) The $cf$-block $M^{i_{1},i_{2},...,i_{k}}_{i_{1},i_{2},...,i_{k}}$ of $S$
is zero,
(2) Every column vector of the $s$-block
$M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}$ of $S$ is non-zero, where
$A\setminus S$
$=\\{j_{1},j_{2},...,j_{h}\\}$.
Proof Let $S$ be a conflict-free set and $A\setminus
S=\\{j_{1},j_{2},...,j_{h}\\}$, then we need only to prove that every element
of $A\setminus S(1\leq t\leq h)$ is defeated by $S$ in $F$ iff all column
vectors of the $s$-block $M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}$
of $S$ are non-zero.
Assume that every element of $A\setminus S(1\leq t\leq h)$ is defeated by $S$
in $F$. Take any column vector $A_{*,j_{t}}(1\leq t\leq h)$ of the $s$-block
$M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}$ of $S$, then we have
$j_{t}\in A\setminus S$. By the assumption, there is some element $i_{r}\in
S(1\leq r\leq k)$ such that the argument $i_{r}$ attacks the argument $j_{t}$,
$i.e.$, $(i_{r},j_{t})\in R$. It follows that $a_{i_{r},j_{t}}=1$ in the
matrix $M(F)$ and the $s$-block
$M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}$ of $S$, and thus the
column vector $A_{*,j_{t}}$ is non-zero.
Conversely, suppose that all column vectors of the $s$-block
$M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}=M^{s}$ of $S$ are non-zero.
Take any element $j_{t}\in A\setminus S(1\leq t\leq h)$, then
$M^{s}_{*,j_{t}}$ is a column vector of the $s$-block
$M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}=M^{s}$ of $S$. By the
hypothesis, we know that $A_{*,j_{t}}$ is non-zero. Therefore, there is some
$i_{r}\in S(1\leq r\leq k)$ such that $a_{i_{r},j_{t}}=1$, $i.e.$,
$(i_{r},j_{t})\in R$. This means that the argument $i_{r}$ attacks the
argument $j_{t}$ of $S$ in $F$, and thus we claim that $j_{t}$ is defeated by
$S$ in $F$.
6\. Determination of the admissible extensions
Example 14 Let us return to the argumentation framework $F=(A,R)$, where
$A=\\{1,2,3,4,5\\}$ and $R=\\{(1,2),(2,3),(2,5),(4,3),(5,4)\\}$. Since an
admissible extension is necessarily a conflict-free set, we can look for the
admissible extension from the collection
$\\{\emptyset,\\{1\\},\\{2\\},\\{3\\},\\{4\\},\\{5\\}\\{1,3\\},\\{1,4\\},\\{1,5\\},\\{2,4\\},\\{3,5\\},\\{1,3,5\\}\\}$
of conflict-free sets. By definition, it is easy to check that $\\{1\\}$,
$\\{1,5\\}$ and $\\{1,3,5\\}$ are all the admissible extensions in $F$.
Since $\\{1,3,5\\}$ is also a stable extension and $\\{1\\}$ is not typical
enough as an admissible extension in $F$, we will mainly concentrate on the
admissible extension $S=\\{1,5\\}$ which is not a stable extension in $F$.
First, we write out the matrix of argumentation framework $F=(A,R)$:
$M(F)=\left(\begin{array}[]{ccccc}0&1&0&0&0\\\ 0&0&1&0&1\\\ 0&0&0&0&0\\\
0&0&1&0&0\\\ 0&0&0&1&0\end{array}\right).$
Secondly, we study the structure of the matrix $M(F)$ of $F$ to find out the
internal properties which can reflect the fact that $S=\\{1,5\\}$ is an
admissible extension.
In the column vector $M(F)_{*,5}$ of the matrix $M(F)$, $a_{2,5}=1$ means that
$(2,5)\in R$, $i.e.$, the argument $2$ attacks the argument $5$. Under this
circumstance, the element $a_{1,2}=1$ in the row vector $M(F)_{*,2}$ of the
matrix $M(F)$ implies that $(1,2)\in R$, $i.e.$, the argument $1$ attacks the
argument $2$. This illustrates that the argument $5$ is defended by
$\\{1,5\\}$ in $F$. In the column vector $M(F)_{*,1}$ of the matrix $M(F)$, we
have $a_{i,1}=0$ for each $1\leq i\leq 5$. It follows that the argument $1$ is
defended by $\\{1,5\\}$ in $F$.
In the above analysis, the behavior of $a_{2,5}=1$ and $a_{1,2}=1$ in the
matrix $M(F)$ is intrinsic for the fact that the argument $5$ is defended by
$\\{1,5\\}$ in $F$. This inspires us a general idea to decide the conflict-
free set $S=\\{1,5\\}$ to be admissible through the structure of the matrix
$M(F)$ of $F$.
(1) In order to decide whether the arguments of $\\{1,5\\}=S$ are defended by
$S$, we should firstly find the attackers of the argument $1$ and $5$. So, we
must pick out the column vectors $M(F)_{*,1}$ and $M(F)_{*,5}$ of the matrix
$M(F)$ corresponding to the arguments $1$ and $5$ respectively. Since the set
$S$ is conflict-free, there is no attack relation between $1$ and $5$, $i.e.$,
$a_{1,1}=0,a_{5,1}=0,a_{1,5}=0,a_{5,5}=0$. Therefore, we only need to check
the elements $a_{2,1},a_{3,1},a_{4,1}$ of the column vector $M(F)_{*,1}$, and
the elements $a_{2,5},a_{3,5},a_{4,5}$ of the column vector $M(F)_{*,5}$. Each
non-zero element of the set $\\{a_{2,1},a_{3,1},a_{4,1}\\}$ tells us an
attacker of the argument $1$, and each non-zero element of the set
$\\{a_{2,5},a_{3,5},a_{4,5}\\}$ tells us an attacker of the argument $5$. This
leads to a block of the matrix $M(F)$ at the intersection of column $1,5$ and
row $2,3,4$, which is exactly the $s$-block of $S$.
(2) After having determined the attackers $(\in\\{2,3,4\\})$ of the argument
$1$ and $5$, we should secondly to check whether these attackers are defeated
by $S=\\{1,5\\}$. For example, $a_{2,5}=1$ means that the argument $2$ is an
attacker of the argument $5$. So, we should check the element $a_{1,2}$ and
$a_{5,2}$ to see whether the attacker $2$ of the argument $5$ is defeated by
$\\{1,5\\}$. Similar situation holds for any other attackers of the argument
$1$ and $5$. Namely, we need also to check the elements $a_{1,3},a_{5,3}$ ( if
the argument $3$ is an attacker of the argument $1$ or $5$ ) and elements
$a_{1,4},a_{5,4}$ ( if the argument $4$ is an attacker of the argument $1$ or
$5$). This process leads to a block of the matrix $M(F)$ at the intersection
of columns $2,3,4$ and rows $1,5$.
In summary, we need to check two blocks (related to $S=\\{1,5\\}$) of the
matrix $M(F)$ in order to decide that the conflict-free set $S=\\{1,5\\}$ is
an admissible extension. This motivate us to give the following definition.
Definition 15 Let $F=(A,R)$ be an argumentation framework with
$A=\\{1,2,...,n\\}$, and $S=\\{i_{1},i_{2},...,i_{k}\\}\subset A$ is an
admissible extension of $F$. The $h\times k$ block
$M^{j_{1},j_{2},...,j_{h}}_{i_{1},i_{2},...,i_{k}}=\left(\begin{array}[]{cccccc}a_{j_{1},i_{1}}&a_{j_{1},i_{2}}&.&.&.&a_{j_{1},i_{k}}\\\
a_{j_{2},i_{1}}&a_{j_{2},i_{2}}&.&.&.&a_{j_{2},i_{k}}\\\ .&.&.&.&.&.\\\
a_{j_{h},i_{1}}&a_{j_{h},i_{2}}&.&.&.&a_{j_{h},i_{k}}\end{array}\right)$
of the matrix $M(F)$ is called the $a$-block of $S$ and denoted by $M^{a}$,
where $\\{j_{1},j_{2},...,j_{h}\\}=A\setminus S$.
In other words, the elements appearing at the intersection of rows
$j_{1},j_{2},...,j_{h}$ and columns $i_{1},i_{2},...,i_{k}$ in the matrix
$M(F)$ form the $a$-block $M^{j_{1},j_{2},...,j_{h}}_{i_{1},i_{2},...,i_{k}}$
of $S$.
Note that, there is a natural relation between the $a$-block
$M^{j_{1},j_{2},...,j_{h}}_{i_{1},i_{2},...,i_{k}}$ and the $s$-block
$M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}$ in matrix theory. Namely,
the $a$-block $M^{j_{1},j_{2},...,j_{h}}_{i_{1},i_{2},...,i_{k}}$ of $S$ is
precisely the complementary block of the $s$-block
$M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}$ of $S$ in the matrix
$M(F)$.
For convenience, in this section we may assume that the sequences
$i_{1},i_{2},...,i_{k}$ and $j_{1},j_{2},...,j_{h}$ are all increasing.
Theorem 16 Given an argumentation framework $F=(A,R)$ with
$A=\\{1,2,...,n\\}$, then $S=\\{i_{1},i_{2},...,i_{k}\\}\subset A$ is an
admissible extension in $F$ iff the following conditions hold:
(1) The $cf$-block $M^{i_{1},i_{2},...,i_{k}}_{i_{1},i_{2},...,i_{k}}$ of $S$
is zero,
(2) The column vector of $s$-block
$M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}$ of $S$ corresponding to
the non-zero row
vector of the $a$-block $M^{j_{1},j_{2},...,j_{h}}_{i_{1},i_{2},...,i_{k}}$ of
$S$ is non-zero, where $A\setminus S=\\{j_{1},j_{2},...,j_{h}\\}$.
Proof Let $S$ be a conflict-free set and $A\setminus
S=\\{j_{1},j_{2},...,j_{h}\\}$. We need only to prove that every $i_{r}\in
S(1\leq r\leq k)$ is defended by $S$ in $F$ iff the column vector of $s$-block
$M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}$ of $S$ corresponding to
the non-zero row vector of the $a$-block
$M^{j_{1},j_{2},...,j_{h}}_{i_{1},i_{2},...,i_{k}}$ of $S$ is non-zero
Assume that every $i_{r}\in S(1\leq r\leq k)$ is defended by $S$ in $F$. If
the row vector $M^{a}_{t,*}(1\leq t\leq h)$ of the $a$-block
$M^{j_{1},j_{2},...,j_{h}}_{i_{1},i_{2},...,i_{k}}=M^{a}$ of $S$ is non-zero,
then there is some $i_{r}(1\leq r\leq k)$ such that $a_{j_{t},i_{r}}=1$. Note
that $a_{j_{t},i_{r}}$ is at the intersection of row $t$ and column $r$ of the
$a$-block $M^{a}$ of $S$, and at the intersection of row $j_{t}$ and column
$i_{r}$ of the matrix $M(F)$. This implies that $(j_{t},i_{r})\in R$, $i.e.$,
the argument $j_{t}$ attacks the argument $i_{r}$. By the assumption, there is
some $i_{q}\in S(1\leq q\leq k)$ such that the argument $i_{q}$ attacks the
argument $j_{t}$, $i.e.$, $(i_{q},j_{t})\in R$. It follows that
$a_{i_{q},j_{t}}=1$ in the matrix $M(F)$. But, $a_{i_{q},j_{t}}$ is also an
element of the $s$-block $M^{s}$, which is at the intersection of row $q$ and
column $t$ of $M^{s}$. Namely, $a_{i_{q},j_{t}}$ is an element of the column
vector $M^{s}_{*,t}$ of $M_{s}$. Therefore, we conclude that the column vector
$M^{s}_{*,t}$ of $s$-block
$M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}=M^{s}$ of $S$ is non-zero.
Conversely, suppose that the column vector of $s$-block
$M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}$ of $S$ corresponding to
the non-zero row vector of the $a$-block
$M^{j_{1},j_{2},...,j_{h}}_{i_{1},i_{2},...,i_{k}}$ of $S$ is non-zero. For
any fixed $i_{r}\in S(1\leq r\leq k)$, if there is no $j_{t}\in A\setminus
S(1\leq t\leq h)$ such that the argument $j_{t}$ attacks the argument $i_{r}$,
then by the fact that $S$ is a conflict-free set we claim that there is no
$i\in A$ such that the argument $i$ attacks the argument $i_{r}$. It follows
that argument $i_{r}\in S$ is defended by $S$ in $F$.
Otherwise, there is some $j_{t}\in A\setminus S(1\leq t\leq h)$ such that the
argument $j_{t}$ attacks the argument $i_{r}$. It follows that
$(j_{t},i_{r})\in R$, $i.e.$, $a_{j_{t},i_{r}}=1$. Since the element
$a_{j_{t},i_{r}}$ is at the intersection of row $t$ and column $r$ of the
$a$-block $M^{j_{1},j_{2},...,j_{h}}_{i_{1},i_{2},...,i_{k}}=M^{a}$ of $S$,
the row vector $M^{a}_{t,*}$ of the $a$-block $M^{a}$ of $S$ is non-zero. By
the assumption, we conclude that the corresponding column vector $M^{s}_{*,t}$
of the $s$-block $M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}=M^{s}$ of
$S$ is non-zero. Therefore, there is some $i_{q}\in S(1\leq q\leq k)$ such
that $a_{i_{q},j_{t}}=1$. Note that, the element $a_{i_{q},j_{t}}$ is at the
intersection of row $q$ and column $t$ of the $s$-block
$M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}$ and at the intersection of
row $i_{q}$ and column $j_{t}$ of the matrix $M(F)$. Consequently, we have
that $(i_{q},j_{t})\in R$, $i.e.$, the argument $i_{q}\in S$ attacks the
argument $j_{t}$. Now, we have proved that the argument $i_{r}\in S$ is also
defended by $S$ in $F$.
Remark: The fact that any stable extension must be admissible is clearly
expressed by the properties of $s$-blocks in the matrix. In other words, the
condition every column vector of the $s$-block
$M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}$ of $S$ are non-zero is
stronger than that the column vector of the $s$-block
$M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}$ of $S$ corresponding to
the non-zero row vector of the $a$-block
$M^{j_{1},j_{2},...,j_{h}}_{i_{1},i_{2},...,i_{k}}$ of $S$ is non-zero.
7\. Determination of the complete extensions
Example 17 Consider the argumentation framework $F=(A,R)$, where
$A=\\{1,2,3,4,5\\}$ and $R=\\{(1,2),(2,3),\\{2,4\\},(2,5),(4,3),(5,4)\\}$.
Since the admissible extension is necessarily a conflict-free set, we can find
out the admissible extension from the collection of conflict-free sets
$\\{\emptyset,\\{1\\},\\{2\\},\\{3\\},\\{4\\},\\{5\\}\\{1,3\\},\\{1,4\\},\\{1,5\\},\\{3,5\\},\\{1,3,5\\}\\}$.
By the directed graph of $F$, it is easy to check that $\\{1,5\\}$ and
$\\{1,3,5\\}$ are all the admissible extensions in $F$. Furthermore, one can
verify that $S_{1}=\\{1,3,5\\}$ is the only complete extension in $F$, while
$S_{2}=\\{1,5\\}$ is not.
Next, we will analysis the different expressions in the matrix $M(F)$ of $F$
between $\\{1,3,5\\}$ (as a complete extension but not an admissible
extension) and $\\{1,5\\}$ (as an admissible extension). By comparing them, we
extract the matrix method to decide that an admissible extension is complete.
Let us firstly write out the matrix of the argumentation framework $F$:
$M(F)=\left(\begin{array}[]{ccccc}0&1&0&0&0\\\ 0&0&1&1&1\\\ 0&0&0&0&0\\\
0&0&1&0&0\\\ 0&0&0&1&0\end{array}\right).$
In the column vector $M(F)_{*,2}$ of the matrix $M(F)$, $a_{1,2}=1$ means that
$(1,2)\in R$, $i.e.$, the argument $1$ attacks the argument $2$. Since
$S_{1}=\\{1,3,5\\}$ is a conflict-free set, there is no element of $S_{1}$
which attacks the argument $1$. It follows that the arguments $2$ is not
defended by $S_{1}$ in $F$. In the column vector $M(F)_{*,4}$ of the matrix
$M(F)$, $a_{5,4}=1$ means that $(5,4)\in R$, $i.e.$, the argument $5$ attacks
the argument $4$. Also because that $S_{1}=\\{1,3,5\\}$ is a conflict-free
set, there is no element of $S_{1}$ which attacks the argument $5$. Thus, we
have that the arguments $4$ is not defended by $S_{1}$ in $F$. These are
exactly the reasons for the admissible extension $S_{1}=\\{1,3,5\\}$ to be a
complete extension.
Next, we will mainly focus our attention on the argument $3$ with respect to
$S_{2}=\\{1,5\\}$.
In the column vector $M(F)_{*,3}$ of the matrix $M(F)$, $a_{2,3}=1$ means that
$(2,3)\in R$, and $a_{4,3}=1$ means that $(4,3)\in R$. Therefore, both
arguments $2$ and $4$ attack the argument $3$. On the other hand, in the
column vector $M(F)_{*,2}$ of the matrix $M(F)$, $a_{1,2}=1$ means that
$(1,2)\in R$, $i.e.$, the argument $1$ attacks the argument $2$. In the column
vector $M(F)_{*,4}$ of the matrix $M(F)$, $a_{5,4}=1$ means that $(5,4)\in R$,
$i.e.$, the argument $5$ attacks the argument $4$. Consequently, we have that
the argument $3$ is defended by $S_{2}=\\{1,5\\}$ in $F$. It is precisely that
the argument $3$ is not included in $S_{2}$ which leads to the fact that
$S_{2}=\\{1,5\\}$ is not a complete extension.
From the above analysis, we find a simple fact: In an argumentation framework
$F=(A,R)$ with $A=\\{1,2,...,n\\}$, an admissible extension
$S=\\{i_{1},i_{2},...,i_{k}\\}$ is complete iff each argument of $A\setminus
S=\\{j_{1},j_{2},...,j_{h}\\}$ is not defended by $S$ in $F$. And, we can
summarize the process to decide an admissible extension $S$ to be complete by
the blocks of matrix $M(F)$ of $F$ as follows:
(1) First, we pick out the column vectors
$M(F)_{\ast,j_{1}},M(F)_{\ast,j_{2}},...,M(F)_{\ast,j_{h}}$ of the matrix
$M(F)$ corresponding to the arguments of $A\setminus
S=\\{j_{1},j_{2},...,j_{h}\\}$. For each argument $j_{t}\in A\setminus S(1\leq
t\leq h)$, we check the elements $a_{1,j_{t}},a_{2,j_{t}},...,a_{n,j_{t}}$ in
the column vector $M(F)_{\ast,j_{t}}$ of the matrix $M(F)$ to find all the
attackers of the argument $j_{t}$.
(2) For each argument $j_{t}(1\leq t\leq h)$, we consider two cases with
respect to its attackers.
(a) There is some $j_{p}\in A\setminus S(1\leq p\leq h)$ such that
$a_{j_{p},j_{t}}=1$ in the column vector $M(F)_{\ast,j_{t}}$ of the matrix
$M(F)$, $i.e.$, $(j_{p},j_{t})\in R$, then the argument $j_{p}$ attacks the
argument $j_{t}$ in $F$. In order that the argument $j_{t}$ is not defended by
$S$, any argument $i_{r}\in S(1\leq r\leq k)$ should not attack the argument
$j_{p}$. Thus, we have $(i_{r},j_{p})\notin R$, $i.e.$, $a_{i_{r},j_{p}}=0$
for all $1\leq r\leq k$.
(b) There is no $j_{p}\in A\setminus S(1\leq p\leq h)$ such that
$a_{j_{p},j_{t}}=1$ in the column vector $M(F)_{\ast,j_{t}}$ of the matrix
$M(F)$, then there must be some $i_{r}\in S(1\leq r\leq k)$ such that
$a_{i_{r},j_{t}}=1$ in the column vector $M(F)_{\ast,j_{t}}$. Otherwise, there
is no $i\in A$ such that $a_{i,j_{t}}=1$, $i.e.$, there is no $i\in A$ such
that $(i,j_{t})\in R$. It follows that there is no argument $i\in A$ which
attacks the argument $j_{t}$ in $F$. This implies that the argument $j_{t}$ is
defended by $S$ in $F$, and thus $S$ is not a complete extension.
In case $(a)$, the elements $"a_{j_{p},j_{t}}"(1\leq p\leq h,1\leq t\leq h)$
form a block of the matrix $M(F)$ at the intersection of row
$j_{1},j_{2},...,j_{h}$ and the same number of columns. The elements
$"a_{i_{r},j_{t}}"(1\leq r\leq k,1\leq t\leq h)$ form anther block of the
matrix $M(F)$ at the intersection of row $i_{1},i_{2},...,i_{k}$ and the
column $j_{1},j_{2},...,j_{h}$, which is exactly the $s$-block of $S$. In case
$(b)$, one can find that the elements considered form the same blocks as in
case $(a)$. This motivation makes us to give the following definition.
Definition 18 Let $F=(A,R)$ be an argumentation framework with
$A=\\{1,2,...,n\\}$, and $S=\\{i_{1},i_{2},...,i_{k}\\}\subset A$ is a
complete extension of $F$. The block
$M^{j_{1},j_{2},...,j_{h}}_{j_{1},j_{2},...,j_{h}}=\left(\begin{array}[]{cccccc}a_{j_{1},i_{1}}&a_{j_{1},i_{2}}&.&.&.&a_{j_{1},i_{k}}\\\
a_{j_{2},i_{1}}&a_{j_{2},i_{2}}&.&.&.&a_{j_{2},i_{k}}\\\ .&.&.&.&.&.\\\
a_{j_{h},i_{1}}&a_{j_{h},i_{2}}&.&.&.&a_{j_{h},i_{k}}\end{array}\right)$
of order $h$ in the matrix of $M(F)$ is called the $c$-block of $S$ and
denoted by $M^{c}$, where $\\{j_{1},j_{2},...,j_{h}\\}=A\setminus S$.
In other words, the elements appearing at the intersection of rows
$j_{1},j_{2},...,j_{h}$ and the same number of columns in the matrix $M(F)$
form the $c$-block $M^{j_{1},j_{2},...,j_{h}}_{j_{1},j_{2},...,j_{h}}$ of $S$.
Note that, the $c$-block
$M^{c}=M^{j_{1},j_{2},...,j_{h}}_{j_{1},j_{2},...,j_{h}}$ of $S$ is exactly
the complementary block of the $s$-block
$M^{s}=M^{i_{1},i_{2},...,i_{k}}_{i_{1},i_{2},...,i_{k}}$ of $S$, in the
matrix $M(F)$ of $F$.
Now, the fact that $S_{1}=\\{1,3,5\\}$ is a complete extension in the above
example can be verified by the following conditions:
(1) The column vector of $s$-block $M^{1,3,5}_{2,4}$ of $S_{1}$ corresponding
to the non-zero row vector
of $c$-block $M^{2,4}_{2,4}$ of $S_{1}$ is zero;
(2) The column vector of $s$-block $M^{1,3,5}_{2,4}$ of $S_{1}$ corresponding
to the zero column vector
of $c$-block $M^{2,4}_{2,4}$ of $S_{1}$ is non-zero.
For convenience, in this section we also assume that the sequences
$i_{1},i_{2},...,i_{k}$ and $j_{1},j_{2},...,j_{h}$ are all increasing.
Lemma 19 Let $F=(A,R)$ be an argumentation framework with $A=\\{1,2,...,n\\}$,
then $S=\\{i_{1},i_{2},...,i_{k}\\}\subset A$ is a complete extension of $F$
iff $S$ is an admissible extension and each argument $j_{t}\in S(1\leq t\leq
h)$ is not defended by $S$ in $F$.
Theorem 20 Given an argumentation framework $F=(A,R)$ with
$A=\\{1,2,...,n\\}$, then the admissible extension
$S=\\{i_{1},i_{2},...,i_{k}\\}\subset A$ is a complete extension in $F$ iff
the following conditions hold:
(1) the column vector of $s$-block
$M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}$ of $S$ corresponding to
the non-zero row vector
of the $c$-block $M^{j_{1},j_{2},...,j_{h}}_{j_{1},j_{2},...,j_{h}}$ of $S$ is
zero,
(2) the column vector of $s$-block
$M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}$ of $S$ corresponding to
the zero column vector
of the $c$-block $M^{j_{1},j_{2},...,j_{h}}_{j_{1},j_{2},...,j_{h}}$ of $S$ is
non-zero,
where $A\setminus S=\\{j_{1},j_{2},...,j_{h}\\}$.
Proof Let $S$ be an admissible extension and $A\setminus
S=\\{j_{1},j_{2},...,j_{h}\\}$, we need only to prove that every $j_{t}\in
S(1\leq t\leq h)$ is not defended by $S$ in $F$ iff the condition $(1)$ and
$(2)$ are hold.
Assume that every $j_{t}\in A\setminus S(1\leq t\leq h)$ is not defended by
$S$ in $F$. If the row vector $M^{{}^{c}}_{r,*}(1\leq r\leq h)$ of the
$c$-block $M^{j_{1},j_{2},...,j_{h}}_{j_{1},j_{2},...,j_{h}}$ of $S$ is non-
zero, then there is some $1\leq t\leq h$ such that $a_{j_{r},j_{t}}=1$,
$i.e.$, $(j_{r},j_{t})\in R$. It follows that the argument $a_{j_{r}}$ attacks
the argument $a_{j_{t}}$. By the assumption, there is no argument in $S$ which
attacks the argument $a_{j_{r}}$. Therefore, for each $i_{q}\in S(1\leq q\leq
k)$ we have $(i_{q},j_{r})\notin R$, $i.e.$, $a_{i_{q},j_{r}}=0$. This means
that the column vector $M^{{}^{s}}_{*,r}$ of the $s$-block
$M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}$ of $S$ is zero.
If the column vector $M^{{}^{c}}_{*,t}(1\leq t\leq h)$ of the $c$-block
$M^{j_{1},j_{2},...,j_{h}}_{j_{1},j_{2},...,j_{h}}$ of $S$ is zero, then for
each $1\leq p\leq h$ we have that $a_{j_{p},j_{t}}=0$, $i.e.$,
$(j_{p},j_{t})\notin R$. Therefore, there is no argument in $A\setminus S$
which attacks the argument $j_{t}$. If there is no argument in $S$ which
attacks the argument $j_{t}$, then there is no argument in $A$ which attacks
the argument $j_{t}$. It follows that the argument $j_{t}$ is defended by $S$
in $F$, a contradiction with the assumption. Thus, there is some argument
$i_{r}\in S(1\leq r\leq k)$ which attacks the argument $j_{t}$, $i.e.$,
$(i_{r},j_{t})\in R$. This implies that $a_{i_{r},j_{t}}=1$, and thus the
column vector $M^{{}^{s}}_{*,t}$ of the $s$-block
$M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}$ of $S$ is non-zero.
Conversely, suppose that the conditions $(1)$ and $(2)$ are hold. Let
$j_{t}\in A\setminus S(1\leq t\leq h)$, we consider the column vector
$M^{c}_{*,t}$ of the $c$-block
$M^{j_{1},j_{2},...,j_{h}}_{j_{1},j_{2},...,j_{h}}$ of $S$. If the column
vector $M^{c}_{*,t}$ is zero, then by condition $(2)$ we have that the column
vector $M^{s}_{*,t}$ of the $s$-block
$M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}$ of $S$ is non-zero. It
follows that there is some $i_{q}\in S(1\leq q\leq k)$ such that
$a_{i_{q},j_{t}}=1$, $i.e.$, $(i_{q},j_{t})\in R$. This means that the
argument $i_{q}$ attacks the argument $j_{t}$ in $F$. Considering that $S$ is
a conflict-free set, there is no argument $i_{r}\in S(1\leq r\leq k)$ which
attacks the argument $i_{q}$ in $F$.
If the column vector $M^{c}_{*,t}$ is non-zero, then the row vector
$M^{c}_{t,*}$ is also non-zero. By condition $(1)$, the column vector
$M^{s}_{*,t}$ of the $s$-block
$M^{i_{1},i_{2},...,i_{k}}_{j_{1},j_{2},...,j_{h}}=M^{s}$ of $S$ is zero. It
follows that $a_{i_{r},j_{t}}=0$, $i.e.$, $(i_{r},j_{t})\notin R$ for each
$1\leq r\leq k$. This implies that there is no argument $i_{r}\in S(1\leq
r\leq k)$ which attacks the argument $j_{t}$ in $F$.
To sum up, we conclude that the argument $j_{t}\in A\setminus S(1\leq t\leq
h)$ is not defended by $S$.
8\. Conclusions and perspectives
In this paper, we introduced the matrix $M(F)$ of an argumentation framework
$F=(A,R)$, and the $cf$-block $M^{cf}$, $s$-block $M^{s}$, $a$-block $M^{a}$
and $c$-block $M^{c}$ of a set $S\subset A$, presented several theorems to
decide various extensions (stable, admissible, complete) of the AF, by blocks
of the matrix $M(F)$ of $F$ and relations between these blocks.
Interestingly, the $s$-block $M^{s}$ ($a$-block $M^{a}$, $c$-block $M^{c}$) of
$S$ corresponds to the determination for $S$ to be a stable extension
(admissible extension, complete extension respectively). And, the $c$-block of
$S$ is exactly the complementary block of the $cf$-block of $S$, the $a$-block
of $S$ is exactly the complementary block of the $s$-block of $S$.
Furthermore, we can decide basic extensions of an argumentation framework by
the special feature of blocks and relations between these blocks. These facts
indicate that there is indeed a corresponding relation between the
argumentation framework and its matrix. So, we can investigate the structure
and properties of an argumentation framework by using the theory and method of
matrix.
For the other common extension semantics (preferred, grounded, ideal, semi-
stable and eager) of Dung’s argumentation framework not discussed in the above
sections, we can also provide the matrix method to describe them, by combining
the obtained results. For example, if we want to decide that a complete
extension $S\subset A$ is grounded in $F=(A,R)$, we could first find out all
the complete extensions by theorem 20. Then, we compare the $cf$-blocks of
these complete extensions. If the $cf$-block of $S$ is the minimal one in the
collection of $cf$-blocks of all complete extensions, then we claim that $S$
is a grounded extension.
The prospectives are that, we can find out the internal pattern of AFs and the
relations between different objects which we concerned in AFs, by studying
blocks of the matrix of AFs. Our future goal is to develop the matrix method
in the related areas, such as argument acceptability, dialogue games,
algorithmic and complexity and so on [7, 11, 8, 13, 16, 12].
References
## References
* [1] P. Baroni, M. Giacomin, On principle-based evaluation of extension-based argumentation semantics, Artificial Intelligence 171 (2007), 675-700.
* [2] T. J. M. Bench-Capon, Paul E. Dunne, Argumentation in artificial intelligence, Artificial intelligence 171(2007)619-641
* [3] M.Caminada, Semi-stable semantics, in: Frontiers in Artificial Intelligence and its Applications, vol. 144, IOS Press, 2006, pp. 121-130.
* [4] C.Cayrol, M.C.Lagasquie-Schiex, Graduality in argumentation, J. AI Res. 23 (2005)245-297.
* [5] S.Coste-Marquis, C.Devred, C.Devred, Symmetric argumentation frameworks, in: Lecture Notes in Artificial Intelligence, vol. 3571, Springer-Verlag, 2005, pp. 317-328.
* [6] S.Coste-Marquis, C.Devred, P. Marquis, Prudent semantics for argumentation frameworks, in: Proc. 17th ICTAI, 2005, pp. 568-572.
* [7] Y.Dimopoulos, A. Torres, Graph theoretical structures in logic programs and default theories, Teoret. Comput. Sci. 170(1996)209-244.
* [8] P.M. Dung, On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and $n$-person games, Artificial Intelligence 77 (1995), 321-357.
* [9] P.M.Dung, P. Mancarella, F. Toni, A dialectic procedure for sceptical assumption-based argumentation, in: Frontiers in Artificial Intelligence and its Applications, vol. 144, IOS Press, 2006, pp. 145-156.
* [10] P.E.Dunne, Computational properties of argument systems satisfying graph-theoretic constrains, Artificial Intelligence 171 (2007), 701-729.
* [11] P.E.Dunne, T. J. M. Bench-Capon, Coherence in finite argument systems, Artificial intelligence 141(2002)187-203.
* [12] P.E.Dunne, T. J. M. Bench-Capon, Two party immediate response disputes: properties and efficiency, Artificial Intelligence 149 (2003), 221-250.
* [13] H.Jakobovits, D.Vermeir, Dialectic semantics for argumentation frameworks, in: Proc. 7th ICAIL, 1999, pp. 53-62.
* [14] E. Oikarinen, S.Woltron, Characterizing strong equivalence for argumentation frameworks, Artificial intelligence(2011), doi:10.1016/j.artint.2011.06.003.
* [15] G. Vreeswijk, Abstract argumentation system, Artificial intelligence 90(1997)225-279.
* [16] G. Vreeswijk, H.Pakken, Credulous and sceptical argument games for preferred semantics, in: Proceedings of JELIA’2000, the 7th European Workshop on Logic for Artificial Intelligence, Berlin, 2000, pp. 224-238.
|
arxiv-papers
| 2011-10-07T00:28:58 |
2024-09-04T02:49:22.903895
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xu Yuming",
"submitter": "Yuming Xu",
"url": "https://arxiv.org/abs/1110.1416"
}
|
1110.1608
|
††thanks: Contribution of an agency of the U.S. government; not subject to
copyright
# Advanced code-division multiplexers for superconducting detector arrays
K. D. Irwin irwin@nist.gov H. M. Cho W. B. Doriese J. W. Fowler G. C.
Hilton M. D. Niemack C. D. Reintsema D. R. Schmidt J. N. Ullom L. R. Vale
National Institute of Standards and Technology, Boulder, CO 80305
###### Abstract
Multiplexers based on the modulation of superconducting quantum interference
devices are now regularly used in multi-kilopixel arrays of superconducting
detectors for astrophysics, cosmology, and materials analysis. Over the next
decade, much larger arrays will be needed. These larger arrays require new
modulation techniques and compact multiplexer elements that fit within each
pixel. We present a new in-focal-plane code-division multiplexer that provides
multiplexing elements with the required scalability. This code-division
multiplexer uses compact lithographic modulation elements that simultaneously
multiplex both signal outputs and superconducting transition-edge sensor (TES)
detector bias voltages. It eliminates the shunt resistor used to voltage bias
TES detectors, greatly reduces power dissipation, allows different dc bias
voltages for each TES, and makes all elements sufficiently compact to fit
inside the detector pixel area. These in-focal-plane code-division
multiplexers can be combined with multi-gigahertz readout based on
superconducting microresonators to scale to even larger arrays.
## I Introduction
Arrays of superconducting transition-edge sensors1 (TES) are widely used to
detect millimeter-wave, submillimeter, and x-ray signals. The development of
kilopixel arrays has required cryogenic signal multiplexing techniques. To
date, all deployed arrays use either time-division multiplexing (TDM)2 or
frequency-division multiplexing (FDM)3. The potential advantages of
multiplexing TES devices with Walsh codes have been anticipated4, 5, and code-
division multiplexing (CDM) circuits are now emerging6 that can significantly
increase the number of pixels multiplexed in each output channel, with more
compact modulation elements.
Code-division multiplexing (CDM) shares many of the advantages of both TDM and
FDM. In CDM, the signals from all TESs are summed with different Walsh-matrix
polarity patterns. In the simplest case of two-channel CDM, the sum of the
signals from TESs 1 and 2 would first be measured, followed by their
difference. The original signals can be reconstructed from the reverse
process. CDM can use the same room-temperature electronics as TDM. Unlike TDM,
CDM does not suffer from the aliasing of SQUID noise by $\sqrt{N}$, where $N$
is the number of pixels multiplexed. CDM uses smaller filter elements and
simpler room-temperature electronics than FDM, and it allows dc biasing of the
TES sensor, making it easier to achieve optimal energy resolution7.
Several CDM circuits have been demonstrated. One implementation, flux-summed
CDM6 ($\rm{\Phi}$-CDM), has been used to achieve average multiplexed energy
resolution of 2.78 eV $\pm$ 0.04 eV FWHM at 6 keV with a small array of TES
x-ray microcalorimeters8. Here we present a more advanced CDM multiplexing
circuit topology that allows scaling to much larger multiplexing factors than
$\rm{\Phi}$-CDM. In current-summed CDM (I-CDM), only one SQUID amplifier is
required for each column of detectors. The current from all TES calorimeters
in a column flows out in one pair of wires, with a coupling polarity that is
switched at each pixel by compact, ultra-low-power switches in the focal plane
itself. These switches can be placed underneath an overhanging x-ray absorber,
so that separate wires need not be extracted from each pixel. In I-CDM, the
voltage bias source for the TES calorimeters does not dissipate power at the
cold stage, making the power dissipation in even megapixel arrays manageable.
Because the dc voltage bias source for the TES calorimeters is naturally
multiplexed, different bias voltages can be chosen for each pixel. Finally,
the number of address wires required scales only logarithmically with the
number of rows multiplexed. Logarithmic encoding of the address lines is made
possible by the periodic nature of the response of the superconducting
interferometer switches to address flux6.
## II Superconducting polarity modulation switches
I-CDM requires a circuit that can modulate the polarity of the current
coupling from a TES to its SQUID amplifier. The modulation has unity gain;
amplification occurs only after the signal from many TES pixels is summed with
different polarities. After the signal bandwidth is limited by a one-pole
$L/R$ low-pass filter formed by a Nyquist inductor $L_{\rm{nyq}}$ and the TES
resistance, superconducting switches steer the current from the TES into one
of two pathways that couple to the SQUID with opposite polarity. Because the
polarity modulation occurs at much higher frequency than the bandwidth of the
signal, there is no degradation in performance from detector noise aliasing.
We have already fabricated and tested appopriate polarity modulators9 as part
of a previous CDM circuit implementation. Fig. 1a shows the current coupled to
the SQUID ($I_{\rm{squid}}$) vs. the input current ($I_{\rm{in}}$) for the two
settings of the modulator.
Figure 1: Polarity modulation. (a) Experimental measurements of a polarity
modulator. Current into the SQUID ($I_{\rm{squid}}$) vs. the input current
($I_{\rm{}_{in}}$) in two different states: positive (solid, positive slope)
data are for no applied address flux; negative (dashed, negative slope) data
are for address flux $\Phi_{\rm{add}}=\Phi_{0}/2$. Data are shown for three
different pixels summed into one SQUID (black, red, and green — the data for
all three curves are nearly identical). The inflection points near $\pm
7\rm{\mu A}$ are indicative of the transition to the normal state above the
current-carrying capacity of the modulator used in this experiment. (b) A
photograph of the new generation of polarity-modulation switches that are used
in I-CDM. The switch contains four Josephson junctions. Junction 1 and 2 are
separated by a serial gradiometer, as are junctions 3 and 4. Junctions 2 and 3
are adjacent, and behave as a single junction with twice the critical current.
This design provides larger operating margins and higher current-carrying
capacity. An expanded view is shown for the serial gradiometer separating
junctions 3 and 4.
The polarity modulator contains superconducting switches10, 11 that are based
on compact, low-inductance SQUIDs controlled by an applied flux. These
switches are designed so that their critical currents modulate from a maximum
value at zero flux (zero applied address current) to very near to zero at a
flux of $\Phi_{0}/2$. At zero applied address flux, the switch is closed, and
the TES current flows in parallel through its Josephson junctions, which are
in a zero-voltage state. At a flux of $\Phi_{0}/2$, the combined current flow
through the parallel Josephson junctions in each switch drops near zero, and
the switch acts as a normal resistor with a value orders of magnitude larger
than the TES resistance. The ‘open’ resistance is large enough that it
introduces no significant Johnson-Nyquist current noise. The switches used in
I-CDM consist of four Josephson junctions rather than two (Fig. 1b), which
allows both larger operating margins and higher current-carrying capacity than
two-junction switches10. The new switches work well, have high yield, and have
wide operating margins.
## III Current-Summed (I-CDM) Array Architecture
The I-CDM array architecture presented here uses the polarity modulators shown
in Fig. 1. Figure 2a shows the I-CDM circuit diagram for a small 4-pixel
array. In this circuit, each TES is wired in series with a Nyquist inductor
that is large enough that the current through the TES is approximately
constant during a multiplexed frame. Each TES (and its Nyquist inductor) is
coupled to a polarity modulator, schematically represented in the figure as
two single-pole double-throw (SPDT) switches that connect the two electrodes
of the TES and Nyquist inductor alternately to the two wires coming from the
SQUID coil. The current from all four TESs is summed with different polarities
into one pair of wires, with the polarity of each summation determined by the
state of the associated pair of SPDT switches. The column is read out with a
single SQUID amplifier on the same silicon chip.
Figure 2: The I-CDM multiplexer. (a) A schematic of a four-pixel I-CDM
multiplexer. The currents through all TES pixels (variable resistors in the
figure) and their Nyquist inductors $L_{\rm{nyq}}$ are modulated in a Walsh
pattern and summed in parallel into the input coil of a SQUID. In the example
state shown in the schematic, TES pixels 1 and 4 are coupled to the SQUID with
positive polarity, while TES pixels 2 and 3 are coupled with negative
polarity. A voltage bias is applied to the detectors by means of a current
$I_{\rm{bias}}$, which is injected into the primary of a coupled inductor.
$I_{\rm{bias}}$ induces a voltage $U(t)$ on the secondary of the inductor. (b)
A periodic bias current ramp $I_{\rm{bias}}$ chosen to bias TES 1 at zero
voltage, and TES 2-4 at $\approx$ 38 nV. (c) The induced voltage levels $U(t)$
on the secondary. (d) The induced voltage $V(t)$ on the series combination of
TES 3 and its Nyquist inductor $L_{\rm{nyq}}$ (e) The voltage across TES 3,
which is approximately constant except for a 0.16 % rms ripple.
In I-CDM, the TES detectors are dc biased. The average voltage bias level
$\overline{V}$ on each pixel is set by applying a repeating linear current
ramp $I_{\rm{bias}}$(t) to the coupled inductor in Fig. 2a. One example of an
$I_{\rm{bias}}$(t) pattern is shown in Fig. 2b. The current ramp
$I_{\rm{bias}}$(t) induces a repeating series of voltage levels $U(t)$ on the
secondary of the coupled coil (Fig. 2c). The polarity of the coupling between
each pixel and the bias signal $U(t)$ is modulated in a Walsh code. Figure 2d
shows the modulated voltage bias across the series combination of TES 3 and
its Nyquist inductor, $V(\rm{TES3}+L_{\rm{nyq}})$, for pixel 3. In the four-
pixel example in Fig. 2, the vector of average voltages $\overline{V}$ seen by
the four TES pixels is $\overline{V}_{i}=\sum W_{ij}U_{j}/4$ (summed over
j=1..4), where $W_{ij}$ is the 4$\times$4 Walsh matrix and $U_{j}$ is the
value of the voltage $U(t)$ for each pixel during the four Walsh periods. The
voltage across the TES itself stays approximately constant because the
impedance of $L_{\rm{nyq}}$ is large compared to the TES resistance at the
modulation frequency. The voltage across pixel 3, V(TES3), is shown as an
example in Fig. 2e.
All Walsh matrices are non-singular, thus any combination of average TES bias
voltages $\overline{V}$ can be established by multiplying the desired values
of $\overline{V}$ by the inverse of the Walsh matrix,
$W_{ij}^{-1}=(1/4)W_{ij}$. The first TES is typically disconnected because its
output is not modulated in the Walsh code, so it doesn’t share the same
benefits from nulling common-mode pickup. For large multiplex factors, this
results in only a small loss in the number of pixels. Thus, we set
$\overline{V_{1}}=0$, which has the added benefit that $\rm{I_{bias}}$ will
return to the same level after each frame. The four voltages on the secondary
$U_{i}$ must be $U_{i}=\sum W_{ij}\overline{V}_{i}$ (summed over j=1..4), or
$\vspace{-3 pt}\begin{pmatrix}U_{1}\\\ U_{2}\\\ U_{3}\\\ U_{4}\\\
\end{pmatrix}=\begin{pmatrix}1&1&1&1\\\ 1&1&-1&-1\\\ 1&-1&-1&1\\\ 1&-1&1&-1\\\
\end{pmatrix}\begin{pmatrix}0\\\ \overline{V_{2}}\\\ \overline{V_{3}}\\\
\overline{V_{4}}\\\ \end{pmatrix}.$ (1)
The example of Fig. 2 is for the case in which all of the TES voltages are
chosen to have the same value $V_{0}$. In this case, $U_{1}=3V_{0}$, and
$U_{2}=U_{3}=U_{4}=-V_{0}$. The numerical values chosen in Fig. 2 are for a
particular TES detector design, with $V_{0}$=38 nV, row periods of 250 ns, and
$L_{nyq}$=100 nH. In this example, the voltage across the TES is approximately
constant (Fig. 2e) with an rms ripple of only 0.16 %. Since this ripple occurs
over periods much shorter than the response time of the TES, it does not
measurably degrade the achievable energy resolution.
As the polarity of each TES is switched, it generates a back-action voltage.
This would first appear to be a source of crosstalk, since it also acts on
other TES pixels. However, over a full frame, the crosstalk back-action is
null due to the orthogonality of the Walsh vectors. The back-action of the
switching on the TES itself, averaging over multiple frames, appears as an
additional source of resistance $R_{\rm{s}}=L_{\rm{sw}}/\tau_{\rm{dwell}}$ in
series with the voltage bias 6, where $\tau_{\rm{dwell}}$ is the average time
between switching. The source resistance, $R_{\rm{s}}$, must be kept small
compared to the TES bias resistance $R_{0}$ to maintain a voltage bias.
Another constraint is placed by Josephson-frequency oscillations: the voltage
applied to the ‘open’ switch will cause a small ac leakage current to
oscillate at the frequency $V/\Phi_{0}$. The switch circuit must be designed
so that the Josephson oscillations are out of band.
Figure 3: (a) A photograph of part of a 32$\times$32 TES x-ray calorimeter
array that was fabricated as a geometric test. This array uses Mo-Cu TES
thermometers, and will later be integrated with x-ray absorbers cantilevered
over the multiplexer components. Inset: a magnified view of four pixels. The
area of relieved silicon nitride membranes is the darker outline around the
‘I’-shaped Cu x-ray absorber attachment structures. (b) A full lithographic
layout of the I-CDM multiplexer, which fits beneath overhanging x-ray
absorbers. Each two-lobed blue coil is a Nyquist inductor. The symmetry of the
lobes ensures that the magnetic field on the adjacent TESs is close to zero.
The polarity switches are the circuit elements running horizontally between
the I-shaped posts.
I-CDM has great potential to increase the scalability of both TES bolometer
instruments for far-infrared / millimeter-wave measurements, and for TES x-ray
detectors. In order to demonstrate the potential to incorporate I-CDM
multiplexer components within an x-ray detector pixel, we have fabricated a
32$\times$32 TES x-ray calorimeter test array with pixels on a 300 $\mu$m
pitch, and room for the I-CDM multiplexer components (Fig. 3a). In a full
implementation, x-ray absorbers will be cantilevered over the multiplexer
components, and connected to the ‘I’-shaped absorber attachment structures. We
have also developed a full lithographic layout of an I-CDM multiplexer
integrated in this test array (Fig. 3b).
## IV Conclusions
I-CDM uses compact multiplexing elements that fit beneath an x-ray absorber in
a TES array with a 300 $\mu$m pixel pitch. I-CDM modulation elements are much
smaller than the LC filters used in FDM and the microwave resonators used in
MKIDs and microwave SQUIDs. I-CDM does not use shunt resistors to voltage bias
TES detectors, greatly reducing the power dissipation, and making it possible
to scale to larger arrays.
The output bandwidth provided by dc SQUID amplifiers is typically a few
megahertz, which limits the total multiplexing factor. Greater output
bandwidth and much higher multiplexing factors can be achieved by using
microwave SQUID multiplexers12 as the readout SQUIDs in I-CDM instead of
traditional dc SQUIDs.
###### Acknowledgements.
We acknowledge support from NASA under grant NNG09WF27I.
## References
* 1 K.D. Irwin, Appl. Phys. Lett. 66, 1998, (1995).
* 2 J.A. Chervenak, K.D. Irwin, E.N. Grossman, J.M. Martinis, C.D. Reintsema, and M.E. Huber, Appl. Phys. Lett. 74, 4043, (1999).
* 3 J. Yoon, J. Clarke, J.M. Gildemeister, A.T. Lee, M.J. Myers, P.L. Richards, and J.T. Skidmore, Appl. Phys. Lett. 78, 371, (2001).
* 4 B. Karasik, and W. McGrath, Proc. of 12th Int’l Symp. on Space Terahertz Technology, 436, (2001).
* 5 M. Podt, J. Weenink, J. Flokstra, and H. Rogalla, Physica C 368, 218, (2002).
* 6 K.D. Irwin, M.D. Niemack, J. Beyer, H.M. Cho, W.B. Doriese, G.C. Hilton, C.D. Reintsema, D.R. Schmidt, J.N. Ullom, and L.R. Vale, Supercond. Sci. Technol. 23, 034004, (2010).
* 7 L. Gottardi, J. van der Kuur, P.A.J. de Korte, R. Den Hartog, B. Dirks, M. Popescu, and H.F.C. Hoevers, AIP Conference Proceedings 1185, 538, (2009).
* 8 J.W. Fowler, W.B. Doriese, G.C. Hilton, K.D. Irwin, D.R. Schmidt, G. Stiehl, J.N. Ullom, and L.R. Vale, Proc. of 14th Int’l Workshop on Low Temp. Detectors, Heidelberg, Germany, Aug. 1-5, 2011, submitted.
* 9 M.D. Niemack, J. Beyer, H.M. Cho, W.B. Doriese, G.C. Hilton, K.D. Irwin, C.D. Reintsema, D.R. Schmidt, J.N. Ullom, and L.R. Vale, Appl. Phys. Lett. 96, 1635093, (2010).
* 10 H.H. Zappe, IEEE Trans. on Magnetics 13, 41, (1977).
* 11 J. Beyer, and D. Drung, Supercond. Sci. Technol. 21, 105022, (2008).
* 12 J.A.B. Mates, G.C. Hilton, K.D. Irwin, L.R. Vale, and K.W. Lehnert, Appl. Phys. Lett. 92, 023514, (2008).
|
arxiv-papers
| 2011-10-07T18:52:34 |
2024-09-04T02:49:22.920544
|
{
"license": "Public Domain",
"authors": "K. D. Irwin, H. M. Cho, W. B. Doriese, J. W. Fowler, G. C. Hilton, M.\n D. Niemack, C. D. Reintsema, D. R. Schmidt, J. N. Ullom, and L. R. Vale",
"submitter": "Kent Irwin",
"url": "https://arxiv.org/abs/1110.1608"
}
|
1110.1804
|
arxiv-papers
| 2011-10-09T07:55:42 |
2024-09-04T02:49:22.937253
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhi-Feng Pang, Li-Lian Wang, and Yu-Fei Yang",
"submitter": "Zhifeng Pang",
"url": "https://arxiv.org/abs/1110.1804"
}
|
|
1110.1923
|
# Decompositions of the Automorphism Group of a Locally Compact Abelian Group
Iian B. Smythe ibs24@cornell.edu Department of Mathematics, Cornell
University, Ithaca, NY 14853-4201
(Date: October 9, 2011.)
###### Abstract.
It is well known that every locally compact abelian group $L$ can be
decomposed as $L_{1}\oplus\mathbb{R}^{n}$, where $L_{1}$ contains a compact-
open subgroup. In this paper, we use this decomposition to study the
topological group $\operatorname{Aut}(L)$ of automorphisms of $L$, equipped
with the $g$-topology. We show that $\operatorname{Aut}(L)$ is topologically
isomorphic to a matrix group with entries from $\operatorname{Aut}(L_{1})$,
$\operatorname{Hom}(L_{1},\mathbb{R}^{n})$,
$\operatorname{Hom}(\mathbb{R}^{n},L_{1})$, and
$\operatorname{GL}_{n}(\mathbb{R})$, respectively. It is also shown that the
algebraic portion of the decomposition is not specific to locally compact
abelian groups, but is also true for objects with a well-behaved decomposition
in an additive category with kernels.
###### 2010 Mathematics Subject Classification:
Primary: 22D45, 22B05. Secondary: 54H11, 18E05, 20K30.
I would like to thank the Natural Sciences and Engineering Research Council of
Canada, the University of Manitoba, and Cornell University for financial
support which has enabled this research.
## 1\. Introduction
Given a collection of mathematical objects with a notion of isomorphism, it is
often of interest to study the self-isomorphisms, or automorphisms, of those
objects. In particular, the set of all such automorphisms is a group under
composition, and there is an interplay between the structure of this group of
automorphisms and the underlying object. Classical examples include
permutation groups of sets, which encompasses the whole of group theory,
automorphism groups of fields in the context of Galois theory, and groups of
diffeomorphisms of smooth manifolds. In the setting of topological spaces,
where automorphisms are self-homeomorphisms of a space $X$, it is natural to
consider endowing this automorphism group $\operatorname{Homeo}(X)$ with a
topology related to that of $X$. If $X$ is locally compact, then
$\operatorname{Homeo}(X)$ and its subgroups can be made into topological
groups, via the so-called _$g$ -topology_, or _Birkhoff topology_ , generated
by the subbasis consisting of sets of the form
$(C,U)=\\{f\in\operatorname{Homeo}(X):f(C)\subseteq U\text{ and
}f^{-1}(C)\subseteq U\\},$
where $C$ is a compact subset of $X$, and $U$ an open subset of $X$ [1]. This
is the coarsest refinement of the compact-open topology wherein both
composition and inversion are continuous.
When $L$ is a Hausdorff locally compact group, denote by
$\operatorname{Aut}(L)$ the group of topological automorphisms of $L$, a
closed subgroup of $\operatorname{Homeo}(L)$, endowed with the $g$-topology.
In general, $\operatorname{Aut}(L)$ is not locally compact, even in the case
where $L$ is a compact abelian group [9, 26.18 (k)], which has led many to
study conditions under which local compactness holds. For example, if $L$ is
compact, totally disconnected, and nilpotent, then local compactness, and in
fact, compactness, of $\operatorname{Aut}(L)$ are equivalent to all Sylow
subgroups having finitely many topological generators [16]. Recent work of
Caprace and Monod has shown that if $L$ is totally disconnected, compactly
generated and locally finitely generated, then $\operatorname{Aut}(L)$ is
locally compact [5, I.6]. It is also known that $\operatorname{Aut}(L)$ is a
Lie group provided $L$ is connected and finite dimensional [12]. It has been
shown that automorphism groups of compact abelian groups are universal for the
class of non-archimedean groups in the sense that every non-archimedean group
embeds as a topological subgroup of $\operatorname{Aut}(K)$, for some compact
abelian $K$; see [15] and [14].
In the case where $L$ is a locally compact abelian (LCA) group, Levin [10]
gave criterion for local compactness of $\operatorname{Aut}(L)$, provided $L$
contained a discrete subgroup such that the quotient was compact. Levin’s
analysis utilizes the additional structure of LCA groups afforded to us by
their duality theory, and in particular, the following canonical decomposition
of such groups.
###### Theorem 1.1. ([9, 24.30], [3, Cor. 1])
If $L$ is an LCA group, then $L\cong L_{1}\oplus\mathbb{R}^{n}$, where $L_{1}$
is an LCA group containing a compact-open subgroup. Further, $n$ is uniquely
determined, and $L_{1}$ is determined up to isomorphism.
The main result of this paper is a structural decomposition of the
automorphism group of an LCA group, using the decomposition in Theorem 1.1:
###### Theorem A.
Let $L$ be an LCA group with $L=L_{1}\oplus\mathbb{R}^{n}$, where $L_{1}$
contains a compact-open subgroup. Then, as topological groups,
$\operatorname{Aut}(L)\cong\begin{pmatrix}\operatorname{Aut}(L_{1})&\operatorname{Hom}(\mathbb{R}^{n},L_{1})\\\
\operatorname{Hom}(L_{1},\mathbb{R}^{n})&\operatorname{GL}_{n}(\mathbb{R})\end{pmatrix},$
where the latter is equipped with the product topology.
The algebraic portion of Theorem A can be extracted and established in a more
general setting.
###### Theorem B.
Let $\mathcal{C}$ be an additive category with kernels, and $A=B\oplus C$ an
object in $\mathcal{C}$ such that:
1. (I)
$\delta\in\operatorname{End}(C)$ is an automorphism of $C$ if and only if the
zero morphism $\mathbf{0}$ is a kernel of $\delta$; and
2. (II)
For every pair of morphisms $\gamma:B\to C$ and $\beta:C\to B$, one has that
$\gamma\beta=\mathbf{0}$.
Then, as groups,
$\operatorname{Aut}(A)\cong\begin{pmatrix}\operatorname{Aut}(B)&\mathcal{C}(C,B)\\\
\mathcal{C}(B,C)&\operatorname{Aut}(C)\end{pmatrix}.$
The paper is structured as follows: In §2, we provide topological
preliminaries regarding the compact-open and $g$-topologies. §3 is a
discussion of an abstract categoral setting wherein we prove Theorem B. In §4,
we present the proof of Theorem A.
## 2\. Preliminaries
Throughout this paper, all spaces are assumed to be Hausdorff, and in
particular, all topological groups are Tychonoff [11, 1.21]. Recall that if
$X$ and $Y$ are topological spaces and $\mathcal{F}$ a collection of
continuous functions from $X$ to $Y$, the _compact-open topology_ on
$\mathcal{F}$ is the topology generated by the subbasis consisting of sets of
the form
$[C,U]=\\{f\in\mathcal{F}:f(C)\subseteq U\\},$
where $C$ is a compact subset of $X$, and $U$ an open subset of $Y$ (see [7],
[18, §43]). For locally compact $X$, composition of maps is continuous in
$\operatorname{Homeo}(X)$ when endowed with the compact-open topology, a
consequence of the following property:
###### Theorem 2.1. ([6, 3.4.2])
If $X$, $Y$ and $Z$ are topological spaces, with $Y$ locally compact, then the
composition map $C(Y,Z)\times C(X,Y)\to C(X,Z)$ is continuous with respect to
the compact-open topology.
However, inversion may fail to be continuous in $\operatorname{Homeo}(X)$ with
respect to the compact-open topology [4, p. 57-58]; this shortcoming is
remedied by the $g$-topology. The two topologies coincide when $X$ is compact,
discrete, or locally connected, but not in general [1]. One can characterize
convergence in the $g$-topology in terms of the compact-open topology as in
the following proposition.
###### Proposition 2.2. ([1, 5. (ii)])
Let $X$ be a locally compact space. A net $(f_{\lambda})$ in
$\operatorname{Homeo}(X)$ converges to $f\in\operatorname{Homeo}(X)$ in the
g-topology, written $f_{\lambda}\xrightarrow{g}f$, if and only if
$(f_{\lambda})$ converges to $f$ and $(f_{\lambda}^{-1})$ converges to
$f^{-1}$ in the compact-open topology, written
$f_{\lambda}\xrightarrow{c.o.}f$ and
$f_{\lambda}^{-1}\xrightarrow{c.o.}f^{-1}$.
Given an LCA group $L$, $\operatorname{Aut}(L)$ is a closed subgroup of
$\operatorname{Homeo}(L)$, endowed with the $g$-topology. Theorem 1.1 implies
a decomposition of $\operatorname{End}(L)$, the (additive) group of
topological endomorphisms of $L$, endowed with the compact-open topology, into
a topological ring of $2\times 2$ matrices. In particular, every element of
$\operatorname{Aut}(L)$ can be algebraically represented in this way, but we
caution that since $\operatorname{Aut}(L)$ carries the $g$-topology, it is
_not_ a subspace of $\operatorname{End}(L)$. We note for future reference that
if $L=\mathbb{R}^{n}$, then its ring of endomorphisms and group of
automorphisms are familiar objects:
###### Remark 2.3. ([9, 26.18 (i)])
1. (a)
Taken with the compact-open topology,
$\operatorname{End}(\mathbb{R}^{n})=M_{n}(\mathbb{R})$, where the latter
carries its standard topology as a subspace of $\mathbb{R}^{n^{2}}$.
2. (b)
Taken with the $g$-topology,
$\operatorname{Aut}(R^{n})=\operatorname{GL}_{n}(\mathbb{R})$, where the
latter carries its standard topology. In particular, the compact-open and
$g$-topologies on $\operatorname{Aut}(\mathbb{R}^{n})$ coincide.
## 3\. A Categorical Setting
In this section, we prove Theorem B. First, we recall the following
terminology from category theory:
###### Definition 3.1 ([13, VIII.2]).
Let $\mathcal{C}$ be a category.
1. (a)
An object $\mathbf{0}$ in $\mathcal{C}$ is a _zero object_ if for every object
$A$ of $\mathcal{C}$, there are unique morphisms $\mathbf{0}\to A$ and
$A\to\mathbf{0}$.
2. (b)
If $\mathcal{C}$ has a zero object and $A$ and $B$ are objects in
$\mathcal{C}$, then the _zero morphism_ $\mathbf{0}:A\to B$ is the composite
of the morphism $A\to\mathbf{0}$ and $\mathbf{0}\to B$.
3. (c)
A _kernel_ of a morphism $f:A\to B$ is a morphism $k:K\to A$ such that:
1. (i)
$fk=\mathbf{0}$; and
2. (ii)
every morphism $h\colon C\to A$ such that $fh=\mathbf{0}$ factors uniquely
through $k$, that is, there is a unique morphism $k^{\prime}\colon C\to A$
making the following diagram commutative:
$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k}$$\scriptstyle{\mathbf{0}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{B}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exists!k^{\prime}}$$\scriptstyle{h}$$\scriptstyle{\mathbf{0}}$
4. (d)
$\mathcal{C}$ is _preadditive_ if for every pair of objects $A$ and $B$ in
$\mathcal{C}$, the set $\mathcal{C}(A,B)$ of morphisms from $A$ to $B$ is an
abelian group, and the composition
$\circ\colon\mathcal{C}(B,C)\times\mathcal{C}(A,B)\to\mathcal{C}(A,C)$ is
bilinear for every $A$, $B$, and $C$.
5. (e)
$\mathcal{C}$ is _additive_ if it has a zero object, and every two objects in
$\mathcal{C}$ have a biproduct.
###### Examples 3.2.
1. (a)
The category $\mathsf{Ab}$ of abelian groups and their homomorphisms is
additive, with the zero object being the trivial group, and biproducts being
direct sums.
2. (b)
The category $\mathsf{LCA}$ of locally compact abelian groups and their
continuous homomorphisms is additive, with the zero object being the trivial
group, and biproducts being direct products with the product topology.
In an additive category $\mathcal{C}$, the abelian group
$\operatorname{End}(A):=\mathcal{C}(A,A)$ is a ring with respect to
composition for every object $A$ in $\mathcal{C}$.
###### Proposition 3.3. ([13, p. 192])
Let $\mathcal{C}$ be an additive category and $A=A_{1}\oplus
A_{2}\oplus\cdots\oplus A_{n}$ an object of $\mathcal{C}$. Then,
$\operatorname{End}(A)\cong\begin{pmatrix}\operatorname{End}(A_{1})&\mathcal{C}(A_{2},A_{1})&\cdots&\mathcal{C}(A_{n},A_{1})\\\
\mathcal{C}(A_{1},A_{2})&\operatorname{End}(A_{2})&\cdots&\mathcal{C}(A_{n},A_{2})\\\
\vdots&\vdots&&\vdots\\\
\mathcal{C}(A_{1},A_{n})&\mathcal{C}(A_{2},A_{n})&\cdots&\operatorname{End}(A_{n})\\\
\end{pmatrix}$
as rings, where composition is given by matrix multiplication.
For an object $A$ in $\mathcal{C}$, we denote the set of all automorphisms
(self-isomorphisms) of $A$ by $\operatorname{Aut}(A)$; it is a group under
composition.
###### Remark 3.4.
Let $\mathcal{C}$ be an additive category, and suppose that $A=B\oplus C$ is
an object of $\mathcal{C}$ such that $\mathcal{C}(B,C)=\\{\mathbf{0}\\}$.
Then,
$\operatorname{Aut}(A)\cong\begin{pmatrix}\operatorname{Aut}(B)&\mathcal{C}(C,B)\\\
\mathbf{0}&\operatorname{Aut}(C)\end{pmatrix}$
as groups.
Theorem B is an analogue of the aforementioned decomposition of
$\operatorname{Aut}(A)$ when $\mathcal{C}(B,C)$ is not necessarily trivial. To
this end, for the remainder of this section, we fix an additive category
$\mathcal{C}$ such that every morphism has a kernel, and an object $A$ in
$\mathcal{C}$ such that $A=B\oplus C$, with $B$ and $C$ objects of
$\mathcal{C}$ satisfying the following conditions:
1. (I)
$\delta\in\operatorname{End}(C)$ is an automorphism of $C$ if and only if the
zero morphism $\mathbf{0}$ is a kernel of $\delta$.
2. (II)
For every pair of morphisms $\gamma:B\to C$ and $\beta:C\to B$, one has that
$\gamma\beta=\mathbf{0}$.
Theorem B is a consequence of Proposition 3.3, and the equivalence of (i) and
(iii) in Theorem 3.5 below.
###### Theorem 3.5.
Let $\varphi\in\operatorname{End}(A)$, where
$\varphi=\begin{pmatrix}\alpha&\beta\\\
\gamma&\delta\end{pmatrix}\in\begin{pmatrix}\operatorname{End}(B)&\mathcal{C}(C,B)\\\
\mathcal{C}(B,C)&\operatorname{End}(C)\end{pmatrix}.$
Then, the following statements are equivalent:
1. (i)
$\varphi$ is an automorphism of $A$;
2. (ii)
$\delta$ is an automorphism of $C$, and the _quasi-determinant_
$\det(\varphi):=\alpha-\beta\delta^{-1}\gamma$ is an automorphism of $B$;
3. (iii)
$\delta$ is an automorphism of $C$, and $\alpha$ is an automorphism of $B$.
We rely on the following elementary fact from ring theory in the proof of
Theorem 3.5.
###### Remark 3.6.
Let $R$ be a (unital) ring, and $n\in R$ a nilpotent element such that
$n^{2}=0$. Then, $(\mathbf{1}+n)^{-1}=(\mathbf{1}-n)$, and in particular,
$(\mathbf{1}+n)$ is invertible.
###### Proof.
(i)$\Longrightarrow$(ii): In order to show that $\delta$ is automorphism, let
$k:K\to C$ be a kernel of $\delta$. Denote the canonical projections
$\pi_{1}:B\oplus C\to B$ and $\pi_{2}:B\oplus C\to C$, $\iota_{K}\colon K\to
B\oplus K$ and $\iota_{B}\colon B\to B\oplus C$ canonical injections, and set
$\psi=\mathbf{0}\oplus k:B\oplus K\to B\oplus C$. Then, one has
$\psi\iota_{K}=(0,k)$ written componentwise as a morphism into $B\oplus C$,
and so $\varphi\psi\iota_{K}=(\beta k,0)$. Thus,
$\displaystyle\pi_{1}\varphi\psi\iota_{K}$ $\displaystyle=\beta k.$ Put
$g:=\pi_{2}\varphi^{-1}\iota_{B}:B\to C$. Since $\displaystyle\iota_{B}\beta
k$ $\displaystyle=(\beta k,0)=\varphi\psi\iota_{K},$ one obtains that
$\displaystyle g\beta k=\pi_{2}\varphi^{-1}\iota_{B}\beta k$
$\displaystyle=\pi_{2}\varphi^{-1}\varphi\psi\iota_{K}=\pi_{2}\psi\iota_{K}=k.$
However, $g\colon B\to C$ and $\beta\colon C\to B$, and so by condition (II),
$g\beta=\mathbf{0}$. Therefore, $k=\mathbf{0}$, and it follows from condition
(I) that $\delta$ is an automorphism.
To establish the second condition, observe that $\varphi$ can be expressed as
follows:
$\displaystyle\varphi=\begin{pmatrix}\mathbf{1}_{B}&\mathbf{0}\\\
\mathbf{0}&\delta\end{pmatrix}\begin{pmatrix}\mathbf{1}_{B}&\beta\\\
\mathbf{0}&\mathbf{1}_{C}\end{pmatrix}\begin{pmatrix}\det(\varphi)&\mathbf{0}\\\
\mathbf{0}&\mathbf{1}_{C}\end{pmatrix}\begin{pmatrix}\mathbf{1}_{B}&\mathbf{0}\\\
\delta^{-1}\gamma&\mathbf{1}_{C}\end{pmatrix}.$ (1)
Since the matrices $\begin{pmatrix}\mathbf{1}_{B}&\mathbf{0}\\\
\mathbf{0}&\delta\end{pmatrix}$, $\begin{pmatrix}\mathbf{1}_{B}&\beta\\\
\mathbf{0}&\mathbf{1}_{C}\end{pmatrix}$ and
$\begin{pmatrix}\mathbf{1}_{B}&\mathbf{0}\\\
\delta^{-1}\gamma&\mathbf{1}_{C}\end{pmatrix}$ are invertible, the remaining
matrix is also invertible. The latter occurs if and only if
$\det(\varphi)\in\operatorname{Aut}(B)$.
(ii)$\Longrightarrow$(i) is an immediate consequence of (1).
(ii)$\Longrightarrow$(iii): It is given that $\delta$ is an automorphism of
$C$. It follows from the definition of $\det(\varphi)$ that
$\alpha=\det(\varphi)+\beta\delta^{-1}\gamma$. By multiplying both sides by
$\det(\varphi)^{-1}$, one obtains
$\alpha\det(\varphi)^{-1}=\mathbf{1}_{B}+\beta\delta^{-1}\gamma\det(\varphi)^{-1}.$
By condition (II), $\gamma\det(\varphi)^{-1}\beta=\mathbf{0}$, because
$\det(\varphi)^{-1}\beta\in\mathcal{C}(C,B)$, and thus
$(\beta\delta^{-1}\gamma\det(\varphi)^{-1})^{2}=\mathbf{0}$. Therefore, by
Remark 3.6, $\alpha\det(\varphi)^{-1}$ is invertible, and its inverse is
$\mathbf{1}_{B}-\beta\delta^{-1}\gamma\det(\varphi)^{-1}$. Hence, $\alpha$ is
invertible, and
$\displaystyle\alpha^{-1}=\det(\varphi)^{-1}(\mathbf{1}_{B}-\beta\delta^{-1}\gamma\det(\varphi)^{-1}).$
(2)
(iii)$\Longrightarrow$(ii): It is given that $\delta$ is an automorphism of
$C$. One can express $\det(\varphi)\alpha^{-1}$ as
$\det(\varphi)\alpha^{-1}=\mathbf{1}_{B}-\beta\delta^{-1}\gamma\alpha^{-1}.$
By condition (II), $\gamma\alpha^{-1}\beta=\mathbf{0}$, because
$\alpha^{-1}\beta\in\mathcal{C}(C,B)$, and thus
$(-\beta\delta^{-1}\gamma\alpha^{-1})^{2}=0$. Therefore, by Remark 3.6,
$\det(\varphi)\alpha^{-1}$ is invertible, and its inverse is
$\mathbf{1}_{B}+\beta\delta^{-1}\gamma\alpha^{-1}$. Hence, $\det(\varphi)$ is
invertible, and
$\displaystyle\det(\varphi)^{-1}=\alpha^{-1}(\mathbf{1}_{B}+\beta\delta^{-1}\gamma\alpha^{-1}).$
(3)
This completes the proof. ∎
The proof of Theorem 3.5 also enables us to provide an explicit formula for
the inverse of an element in $\operatorname{Aut}(A)$.
###### Corollary 3.7.
If $\varphi\in\operatorname{Aut}(A)$ with
$\varphi=\begin{pmatrix}\alpha&\beta\\\ \delta&\gamma\end{pmatrix}$, then
$\varphi^{-1}=\begin{pmatrix}(\det(\varphi))^{-1}&-(\det(\varphi))^{-1}(\beta\delta^{-1})\\\
-\delta^{-1}\gamma(\det(\varphi))^{-1}&\delta^{-1}\end{pmatrix}.$
###### Proof.
The inverse $\varphi^{-1}$ can be obtained by expressing $\varphi$ in the form
provided in (1), and calculating the inverse of each of the factors as
follows:
$\varphi^{-1}=\begin{pmatrix}\mathbf{1}&\mathbf{0}\\\
-\delta^{-1}\gamma&\mathbf{1}_{C}\end{pmatrix}\begin{pmatrix}\det(\varphi)^{-1}&\mathbf{0}\\\
\mathbf{0}&\mathbf{1}_{C}\end{pmatrix}\begin{pmatrix}\mathbf{1}_{B}&-\beta\\\
\mathbf{0}&\mathbf{1}_{C}\end{pmatrix}\begin{pmatrix}\mathbf{1}_{B}&\mathbf{0}\\\
\mathbf{0}&\delta^{-1}\end{pmatrix}.$
Therefore,
$\varphi^{-1}=\begin{pmatrix}\det(\varphi)^{-1}&-(\det(\varphi))^{-1}(\beta\delta^{-1})\\\
-\delta^{-1}\gamma(\det(\varphi))^{-1}&\delta^{-1}\gamma(\det(\varphi))^{-1}\beta\delta^{-1}+\delta^{-1}\end{pmatrix}.$
However, $(\det(\varphi))^{-1}\beta\in\mathcal{C}(C,B)$, so
$\gamma(\det(\varphi))^{-1}\beta=\mathbf{0}$ by condition (II), and
$\delta^{-1}\gamma(\det(\varphi))^{-1}\beta\delta^{-1}=\mathbf{0}$. ∎
We now apply these general results to the category $\mathsf{LCA}$.
###### Proposition 3.8.
$\mathsf{LCA}$ is an additive category with kernels, and the decomposition of
an LCA group given in Theorem 1.1 satisfies conditions (I) and (II). That is,
given $L=L_{1}\oplus\mathbb{R}^{n}$, where $L_{1}$ contains a compact-open
subgroup, then:
1. (a)
$\delta\in\operatorname{End}(\mathbb{R}^{n})$ is an automorphism if and only
if it has trivial kernel;
2. (b)
for every pair of continuous homomorphisms $\gamma:L_{1}\to\mathbb{R}^{n}$ and
$\beta:\mathbb{R}^{n}\to L_{1}$, one has $\gamma\beta=\mathbf{0}$.
###### Proof.
By Example 3.2(b), $\mathsf{LCA}$ is an additive category. If $f\colon L\to H$
is a continuous homomorphism of LCA groups, then the inclusion map
$k\colon\ker f\to L$ is a kernel of $f$ in the sense of Definition 3.1 (c).
(a) follows from Proposition 2.3.
(b) Since $\mathbb{R}^{n}$ is connected, $\beta(\mathbb{R}^{n})$ is contained
in the connected component $c(L_{1})$ of $L_{1}$, which is compact. One has
$\gamma(c(L_{1}))=\\{0\\}$, because the only compact subgroup of
$\mathbb{R}^{n}$ is the trivial one. Hence, $\gamma\beta=\mathbf{0}$. ∎
###### Corollary 3.9.
Let $L=L_{1}\oplus\mathbb{R}^{n}$ be an LCA group, where $L_{1}$ contains a
compact-open subgroup, and $\varphi\in\operatorname{End}(L)$, with
$\varphi=\begin{pmatrix}\alpha&\beta\\\
\gamma&\delta\end{pmatrix}\in\begin{pmatrix}\operatorname{End}(L_{1})&\operatorname{Hom}(\mathbb{R}^{n},L_{1})\\\
\operatorname{Hom}(L_{1},\mathbb{R}^{n})&M_{n}(\mathbb{R})\end{pmatrix}.$
Then, the following statements are equivalent:
1. (i)
$\varphi$ is an automorphism of $L$;
2. (ii)
$\delta$ is an automorphism of $\mathbb{R}^{n}$ (i.e., in
$\operatorname{GL}_{n}(\mathbb{R})$), and the quasi-determinant of $\varphi$
is an automorphism of $L_{1}$;
3. (iii)
$\delta$ is an automorphism of $\mathbb{R}^{n}$, and $\alpha$ is an
automorphism of $L_{1}$. ∎
## 4\. $\operatorname{Aut}(L)$ and Decompositions of $L$
In this section, whenever $L$ and $H$ are LCA groups, the group
$\operatorname{Hom}(L,H)$ of continuous homomorphisms from $L$ to $H$, and the
ring $\operatorname{End}(L)$ of continuous endomorphisms of $L$, are endowed
with the compact-open topology, while the group $\operatorname{Aut}(L)$ of
topological automorphisms, will have the $g$-topology. We show that the
results of §3 remain true for LCA groups with a topological enrichment in the
sense that the algebraic isomorphisms from §3 become topological isomorphisms
in the presence of the aforementioned topologies. The culmination of this work
is Theorem A, a topological enrichment of Theorem B. We begin with the
following enrichment of Proposition 3.3:
###### Proposition 4.1.
Let $L=L_{1}\oplus L_{2}\oplus\cdots\oplus L_{n}$, where each $L_{i}$ is an
LCA group. Then,
$\operatorname{End}(L)\cong\begin{pmatrix}\operatorname{End}(L_{1})&\operatorname{Hom}(L_{2},L_{1})&\cdots&\operatorname{Hom}(L_{n},L_{1})\\\
\operatorname{Hom}(L_{1},L_{2})&\operatorname{End}(L_{2})&\cdots&\operatorname{Hom}(L_{n},L_{2})\\\
\vdots&\vdots&&\vdots\\\
\operatorname{Hom}(L_{1},L_{n})&\operatorname{Hom}(L_{2},L_{n})&\cdots&\operatorname{End}(L_{n})\\\
\end{pmatrix}$
as topological rings, where the latter is equipped with the product topology.
###### Proof.
Let $[A_{(i,j)}]$ denote the matrix ring on the right-hand side, where
$A_{(i,j)}=\operatorname{End}(L_{i})$ if $i=j$, and
$A_{(i,j)}=\operatorname{Hom}(L_{j},L_{i})$ otherwise. We define the map
$F:\operatorname{End}(L)\to[A_{(i,j)}]$ by
$F(\varphi)=[\pi_{i}\varphi\iota_{j}],$
where $\pi_{i}$ is the canonical projection of $L$ onto $L_{i}$, and
$\iota_{j}$ the inclusion of $L_{j}$ into $L$. By Proposition 3.3, $F$ is a
ring homomorphism from $\operatorname{End}(L)$ onto $[A_{(i,j)}]$. This map is
continuous, since all of the spaces involved are given the compact-open
topology, and the map $\varphi\mapsto\pi_{i}\varphi\iota_{j}$ is continuous by
Proposition 2.1. The inverse of $F$ is given by
$F^{-1}([\alpha_{i,j}])=\sum_{(i,j)}{\iota_{i}\alpha_{i,j}\pi_{j}},$
which is continuous by Proposition 2.1 and the continuity of addition in
$\operatorname{End}(L)$. ∎
From now on, if $L$ is a direct sum of (finitely many) LCA groups, we identify
$\operatorname{End}(L)$ with the aforesaid matrix decomposition. Recall that
$\operatorname{Aut}(L)$ is equipped with the $g$-topology, which need not
coincide with the topology inherited from $\operatorname{End}(L)$. Therefore,
decomposition results concerning $\operatorname{End}(L)$ do not automatically
give rise to those for $\operatorname{Aut}(L)$. Nevertheless, in the simplest
case, a topological enrichment of Remark 3.4 holds.
###### Proposition 4.2.
Suppose that $L=A\oplus B$, where $A$ and $B$ are LCA groups with
$\operatorname{Hom}(A,B)=\mathbf{0}$. Then,
$\operatorname{Aut}(L)\cong\begin{pmatrix}\operatorname{Aut}(A)&\operatorname{Hom}(B,A)\\\
\mathbf{0}&\operatorname{Aut}(B)\end{pmatrix},$
as topological groups, where the right-hand side is equipped with the product
topology.
###### Proof.
Define
$F\colon\operatorname{Aut}(L)\to\begin{pmatrix}\operatorname{Aut}(A)&\operatorname{Hom}(B,A)\\\
\mathbf{0}&\operatorname{Aut}(B)\end{pmatrix},\text{ by
}F(\varphi)=\begin{pmatrix}\alpha&\beta\\\ 0&\delta\end{pmatrix}.$
By Remark 3.4, $F$ is well-defined and it is a group isomomorphism, and in
particular, $\alpha$ and $\delta$ are automorphisms of $A$ and $B$,
respectively. Thus, it remains to be seen that $F$ is also a homeomorphism.
Let $(\varphi_{\lambda})$ be a net converging (in the $g$-topology) to
$\varphi\in\operatorname{Aut}(L)$, where
$F(\varphi_{\lambda})=\begin{pmatrix}\alpha_{\lambda}&\beta_{\lambda}\\\
0&\delta_{\lambda}\end{pmatrix},\text{ and
}F(\varphi)=\begin{pmatrix}\alpha&\beta\\\ 0&\delta\end{pmatrix}.$
One may show by direct computation that
$\begin{pmatrix}\alpha_{\lambda}&\beta_{\lambda}\\\
0&\delta_{\lambda}\end{pmatrix}^{-1}=\begin{pmatrix}\alpha_{\lambda}^{-1}&-\beta_{\lambda}\\\
0&\delta_{\lambda}^{-1}\end{pmatrix},\text{ and
}\begin{pmatrix}\alpha&\beta\\\
0&\delta\end{pmatrix}^{-1}=\begin{pmatrix}\alpha^{-1}&-\beta\\\
0&\delta^{-1}\end{pmatrix}.$
Since $\varphi_{\lambda}\xrightarrow{g}\varphi$, by Proposition 2.2,
$\begin{pmatrix}\alpha_{\lambda}&\beta_{\lambda}\\\
0&\delta_{\lambda}\end{pmatrix}\xrightarrow{c.o.}\begin{pmatrix}\alpha&\beta\\\
0&\delta\end{pmatrix},\text{ and
}\begin{pmatrix}\alpha_{\lambda}^{-1}&-\beta_{\lambda}\\\
0&\delta_{\lambda}^{-1}\end{pmatrix}\xrightarrow{c.o.}\begin{pmatrix}\alpha^{-1}&-\beta\\\
0&\delta^{-1}\end{pmatrix}.$
In particular, $\alpha_{\lambda}\xrightarrow{c.o.}\alpha$ and
$\alpha_{\lambda}^{-1}\xrightarrow{c.o.}\alpha^{-1}$, and by Proposition 2.2
applied to $\operatorname{Aut}(A)$, we have that
$\alpha_{\lambda}\xrightarrow{g}\alpha$. Similarly,
$\delta_{\lambda}\xrightarrow{g}\delta$ in $\operatorname{Aut}(C)$. It is
clear that $\beta_{\lambda}\to\beta$ in $\operatorname{Hom}(B,A)$, and thus,
$F(\varphi_{\lambda})\to F(\varphi)$. Therefore $F$ is continuous.
To see that $F^{-1}$ is continuous, suppose that
$\begin{pmatrix}\alpha_{\lambda}&\beta_{\lambda}\\\
0&\delta_{\lambda}\end{pmatrix}\to\begin{pmatrix}\alpha&\beta\\\
0&\delta\end{pmatrix}\text{ in
}\begin{pmatrix}\operatorname{Aut}(A)&\operatorname{Hom}(B,A)\\\
\mathbf{0}&\operatorname{Aut}(B)\end{pmatrix}.$
One has that $\beta_{\lambda}\to\beta$ in $\operatorname{Hom}(B,A)$,
$\alpha_{\lambda}\xrightarrow{g}\alpha$ in $\operatorname{Aut}(A)$, and
$\delta_{\lambda}\xrightarrow{g}\delta$ in $\operatorname{Aut}(B)$. By
Proposition 2.2, $\alpha_{\lambda}\xrightarrow{c.o.}\alpha$ and
$\alpha_{\lambda}^{-1}\xrightarrow{c.o.}\alpha$, and
$\delta_{\lambda}\xrightarrow{c.o.}\delta$ and
$\delta_{\lambda}^{-1}\xrightarrow{c.o.}\delta^{-1}$. The compact-open
topology on $\operatorname{End}(L)$ coincides with the product topology where
each of the component spaces have the compact-open topology, as given in
Proposition 4.1. Therefore,
$\begin{pmatrix}\alpha_{\lambda}&\beta_{\lambda}\\\
0&\delta_{\lambda}\end{pmatrix}\xrightarrow{c.o.}\begin{pmatrix}\alpha&\beta\\\
0&\delta\end{pmatrix},\text{ and
}\begin{pmatrix}\alpha_{\lambda}^{-1}&-\beta_{\lambda}\\\
0&\delta_{\lambda}^{-1}\end{pmatrix}\xrightarrow{c.o.}\begin{pmatrix}\alpha^{-1}&-\beta\\\
0&\delta^{-1}\end{pmatrix}.$
Hence, $F$ is a topological isomorphism. ∎
If $L$ is a compactly generated LCA group, then $L\cong
K\oplus\mathbb{R}^{n}\oplus\mathbb{Z}^{m}$, where $K$ is the maximal compact
subgroup of $L$ [9, 9.8], while if $L$ is a connected LCA group, then $L\cong
K\oplus\mathbb{R}^{n}$ where $K$ is the maximal compact connected subgroup of
$L$ [9, 9.14]. Combining these facts with Proposition 4.2, we have the
following:
###### Corollary 4.3.
1. (a)
Let $L\cong K\oplus\mathbb{R}^{n}\oplus\mathbb{Z}^{m}$ be a compactly
generated LCA group, where $K$ is the maximal compact subgroup of $L$. Then,
$\operatorname{End}(L)\cong\begin{pmatrix}\operatorname{End}(K)&\operatorname{Hom}(\mathbb{R}^{n},K)&K^{m}\\\
\mathbf{0}&M_{n}(\mathbb{R})&\mathbb{R}^{mn}\\\
\mathbf{0}&\mathbf{0}&M_{m}(\mathbb{Z})\\\ \end{pmatrix}\text{ and
}\operatorname{Aut}(L)\cong\begin{pmatrix}\operatorname{Aut}(K)&\operatorname{Hom}(\mathbb{R}^{n},K)&K^{m}\\\
\mathbf{0}&\operatorname{GL}_{n}(\mathbb{R})&\mathbb{R}^{mn}\\\
\mathbf{0}&\mathbf{0}&\operatorname{GL}_{m}(\mathbb{Z})\\\ \end{pmatrix}.$
2. (b)
Let $L\cong K\oplus\mathbb{R}^{n}$ be a connected LCA group, where $K$ is the
maximal compact connected subgroup of $L$. Then,
$\operatorname{End}(L)\cong\begin{pmatrix}\operatorname{End}(K)&\operatorname{Hom}(\mathbb{R}^{n},K)\\\
\mathbf{0}&M_{n}(\mathbb{R})\end{pmatrix},\text{ and
}\operatorname{Aut}(L)\cong\begin{pmatrix}\operatorname{Aut}(K)&\operatorname{Hom}(\mathbb{R}^{n},K)\\\
\mathbf{0}&\operatorname{GL}_{n}(\mathbb{R})\end{pmatrix}.$
###### Proof.
(a) The only compact subgroup of $\mathbb{R}^{n}\oplus\mathbb{Z}^{m}$ is the
trivial one, and so
$\operatorname{Hom}(K,\mathbb{R}^{n}\oplus\mathbb{Z}^{m})=\mathbf{0}$. Thus,
by Proposition 4.2, we have that
$\operatorname{Aut}(L)\cong\begin{pmatrix}\operatorname{Aut}(K)&\operatorname{Hom}(\mathbb{R}^{n}\oplus\mathbb{Z}^{m},K)\\\
\mathbf{0}&\operatorname{Aut}(\mathbb{R}^{n}\oplus\mathbb{Z}^{m})\end{pmatrix}.$
The only connected subgroup of $\mathbb{Z}^{m}$ is trivial, so
$\operatorname{Hom}(\mathbb{R}^{n},\mathbb{Z}^{m})=\mathbf{0}$, and so
$\operatorname{Aut}(\mathbb{R}^{n}\oplus\mathbb{Z}^{m})=\begin{pmatrix}\operatorname{Aut}(\mathbb{R}^{n})&\operatorname{Hom}(\mathbb{Z}^{m},\mathbb{R}^{n})\\\
\mathbf{0}&\operatorname{Aut}(\mathbb{Z}^{m})\end{pmatrix}.$
One can easily show that
$\operatorname{Hom}(\mathbb{R}^{n}\oplus\mathbb{Z}^{m},K)\cong\operatorname{Hom}(\mathbb{R}^{n},K)\times\operatorname{Hom}(\mathbb{Z}^{m},K)$.
$\operatorname{Aut}(\mathbb{R}^{n})=\operatorname{GL}_{n}(\mathbb{R})$ by
Remark 2.3,
$\operatorname{Aut}(\mathbb{Z}^{m})=\operatorname{GL}_{m}(\mathbb{Z})$ by [9,
26.18(g)], and it is elementary that
$\operatorname{Hom}(\mathbb{Z}^{m},G)\cong G^{m}$ for any topological group
$G$. Hence,
$\operatorname{Aut}(L)\cong\begin{pmatrix}\operatorname{Aut}(K)&\operatorname{Hom}(\mathbb{R}^{n},K)&\operatorname{Hom}(\mathbb{Z}^{m},K)\\\
\mathbf{0}&\operatorname{Aut}(\mathbb{R}^{n})&\operatorname{Hom}(\mathbb{Z}^{m},\mathbb{R}^{n})\\\
\mathbf{0}&\mathbf{0}&\operatorname{Aut}(\mathbb{Z})\\\
\end{pmatrix}\cong\begin{pmatrix}\operatorname{Aut}(K)&\operatorname{Hom}(\mathbb{R}^{n},K)&K^{m}\\\
\mathbf{0}&\operatorname{GL}_{n}(\mathbb{R})&\mathbb{R}^{mn}\\\
\mathbf{0}&\mathbf{0}&\operatorname{GL}_{m}(\mathbb{Z})\\\ \end{pmatrix}.$
(b) The only compact subgroup of $\mathbb{R}^{n}$ is the trivial one, and so
$\operatorname{Hom}(K,\mathbb{R}^{n})=\mathbf{0}$. The remainder of the result
follows from Proposition 4.2. ∎
A few additional results of this flavour are found in [17, §25].
Fix an LCA group $L=L_{1}\oplus\mathbb{R}^{n}$, where $L_{1}$ contains a
compact-open subgroup. Corollaries 3.7 and 3.9 imply that
$\displaystyle\operatorname{Aut}(L)\cong\begin{pmatrix}\operatorname{Aut}(L_{1})&\operatorname{Hom}(\mathbb{R}^{n},L_{1})\\\
\operatorname{Hom}(L_{1},\mathbb{R}^{n})&\operatorname{GL}_{n}(\mathbb{R})\end{pmatrix}$
(4)
as (abstract) groups. Theorem A is established once we show that this
isomorphism is topological, a result that follows from the equivalence of (i)
and (iv) in Theorem 4.4 below.
###### Theorem 4.4.
Let $(\varphi_{\lambda})$ be a net in $\operatorname{Aut}(L)$, and
$\varphi\in\operatorname{Aut}(L)$. The following statements are equivalent:
1. (i)
$\varphi_{\lambda}\xrightarrow{g}\varphi$ in $\operatorname{Aut}(L)$;
2. (ii)
$\varphi_{\lambda}\xrightarrow{c.o.}\varphi$ and
$\det(\varphi_{\lambda})\xrightarrow{g}\det(\varphi)$;
3. (iii)
$\varphi_{\lambda}\xrightarrow{c.o.}\varphi$ and
$(\det(\varphi_{\lambda}))^{-1}\xrightarrow{c.o.}(\det(\varphi))^{-1}$;
4. (iv)
$\varphi_{\lambda}\xrightarrow{c.o.}\varphi$ and
$\alpha_{\lambda}\xrightarrow{g}\alpha$;
5. (v)
$\varphi_{\lambda}\xrightarrow{c.o.}\varphi$ and
$\alpha_{\lambda}^{-1}\xrightarrow{c.o.}\alpha^{-1}$.
###### Proof.
Throughout the proof, we identify automorphisms in $\operatorname{Aut}(L)$
with their matrix representations as provided in (4), and use the following
convention to denote components:
$\varphi_{\lambda}=\begin{pmatrix}\alpha_{\lambda}&\beta_{\lambda}\\\
\gamma_{\lambda}&\delta_{\lambda}\end{pmatrix}$ and
$\varphi=\begin{pmatrix}\alpha&\beta\\\ \gamma&\delta\end{pmatrix}$.
Furthermore, by Remark 2.3, the compact-open and $g$-topologies coincide on
$\operatorname{Aut}(\mathbb{R}^{n})$. So $\delta_{\lambda}\to\delta$ if and
only if $\delta_{\lambda}^{-1}\to\delta^{-1}$, and given one, we need not
verify the other.
(i)$\Longrightarrow$(ii): Since the $g$-topology is finer than the compact-
open one, it follows that $\varphi_{\lambda}\xrightarrow{c.o.}\varphi$. By
Proposition 2.2 applied to $\operatorname{Aut}(L_{1})$, it suffices to show
that $\det(\varphi_{\lambda})\xrightarrow{c.o.}\det(A)$ and
$\det(\varphi_{\lambda})^{-1}\xrightarrow{c.o.}\det(\varphi)^{-1}$. Since
$\alpha_{\lambda}\xrightarrow{c.o.}\alpha$, $\beta_{\lambda}\to\beta$,
$\gamma_{\lambda}\to\gamma$ and $\delta_{\lambda}\to\delta$, where each of the
spaces involved carries the compact-open topology, it follows by Proposition
2.1 that
$\det(\varphi_{\lambda})=\alpha_{\lambda}-\beta_{\lambda}\delta_{\lambda}^{-1}\gamma_{\lambda}\xrightarrow{c.o.}\alpha-\beta\delta^{-1}\gamma=\det(\varphi).$
By Corollary 3.7,
$\displaystyle\varphi_{\lambda}^{-1}$
$\displaystyle=\begin{pmatrix}(\det(\varphi_{\lambda}))^{-1}&-(\det(\varphi_{\lambda}))^{-1}(\beta_{\lambda}\delta_{\lambda}^{-1})\\\
-\delta_{\lambda}^{-1}\gamma_{\lambda}(\det(\varphi_{\lambda}))^{-1}&\delta_{\lambda}^{-1}\end{pmatrix},\text{
and }$ $\displaystyle\varphi^{-1}$
$\displaystyle=\begin{pmatrix}(\det(\varphi))^{-1}&-(\det(\varphi))^{-1}(\beta\delta^{-1})\\\
-\delta^{-1}\gamma(\det(\varphi))^{-1}&\delta^{-1}\end{pmatrix}.$
Since $\varphi_{\lambda}^{-1}\xrightarrow{c.o.}\varphi^{-1}$, in particular,
the $(1,1)$-entry of $\varphi_{\lambda}^{-1}$ converges to the $(1,1)$-entry
of $\varphi^{-1}$. Hence,
$(\det(\varphi_{\lambda}))^{-1}\xrightarrow{c.o.}(\det(\varphi))^{-1}$.
(ii)$\Longrightarrow$(iii) follows from Proposition 2.2 applied to
$\operatorname{Aut}(L_{1})$.
(iii)$\Longrightarrow$(i): Since $\varphi_{\lambda}\xrightarrow{c.o.}\varphi$,
by Proposition 2.2, it suffices to show that
$\varphi_{\lambda}^{-1}\xrightarrow{c.o.}\varphi^{-1}$. We know that
$(\det(\varphi_{\lambda}))^{-1}\xrightarrow{c.o.}(\det(\varphi))^{-1}$, so by
Proposition 2.1,
$\displaystyle-(\det(\varphi_{\lambda}))^{-1}(\beta_{\lambda}\delta_{\lambda}^{-1})$
$\displaystyle\to-(\det(\varphi))^{-1}(\beta\delta^{-1})\text{ in
$\operatorname{Hom}(\mathbb{R}^{n},L_{1})$, and}$
$\displaystyle-\delta_{\lambda}^{-1}\gamma_{\lambda}(\det(\varphi_{\lambda}))^{-1}$
$\displaystyle\to-\delta^{-1}\gamma(\det(\varphi))^{-1}\text{ in
$\operatorname{Hom}(L_{1},R^{n})$.}$
Thus, one has
$\begin{pmatrix}(\det(\varphi_{\lambda}))^{-1}&-(\det(\varphi_{\lambda}))^{-1}(\beta_{\lambda}\delta_{\lambda}^{-1})\\\
-\delta_{\lambda}^{-1}\gamma_{\lambda}(\det(\varphi_{\lambda}))^{-1}&\delta_{\lambda}^{-1}\end{pmatrix}\xrightarrow{c.o.}\begin{pmatrix}(\det(\varphi))^{-1}&-(\det(\varphi))^{-1}(\beta\delta^{-1})\\\
-\delta^{-1}\gamma(\det(\varphi))^{-1}&\delta^{-1}\end{pmatrix}.$
That is, $\varphi_{\lambda}^{-1}\xrightarrow{c.o.}\varphi^{-1}$, and hence,
$\varphi_{\lambda}\xrightarrow{g}\varphi$.
(iii)$\Longrightarrow$(iv): Since
$\varphi_{\lambda}\xrightarrow{c.o.}\varphi$, one has
$\alpha_{\lambda}\xrightarrow{c.o.}\alpha$. Thus, by Proposition 2.2, it
suffices to show that $\alpha_{\lambda}^{-1}\to\alpha^{-1}$. By (2),
$\displaystyle\alpha_{\lambda}^{-1}$
$\displaystyle=(\det(\varphi_{\lambda}))^{-1}(\mathbf{1}_{B}-\beta_{\lambda}\delta_{\lambda}^{-1}\gamma_{\lambda}(\det(\varphi_{\lambda}))^{-1})$
$\displaystyle\alpha^{-1}$
$\displaystyle=(\det(\varphi))^{-1}(\mathbf{1}_{B}-\beta\delta^{-1}\gamma(\det(\varphi))^{-1}).$
Therefore, by Proposition 2.1,
$\alpha_{\lambda}^{-1}=(\det(\varphi_{\lambda}))^{-1}(\mathbf{1}_{B}-\beta_{\lambda}\delta_{\lambda}^{-1}\gamma_{\lambda}(\det(\varphi_{\lambda}))^{-1})\xrightarrow{c.o.}(\det(\varphi))^{-1}(\mathbf{1}_{B}-\beta\delta^{-1}\gamma(\det(\varphi))^{-1})=\alpha^{-1}.$
(iv)$\Longrightarrow$(v) follows from Proposition 2.2 applied to
$\operatorname{Aut}(L_{1})$.
(v)$\Longrightarrow$(iii): By (3),
$\displaystyle(\det(\varphi_{\lambda}))^{-1}$
$\displaystyle=\alpha_{\lambda}^{-1}(\mathbf{1}_{B}+\beta_{\lambda}\delta_{\lambda}^{-1}\gamma_{\lambda}\alpha_{\lambda}^{-1})$
$\displaystyle(\det(\varphi))^{-1}$
$\displaystyle=\alpha^{-1}(\mathbf{1}_{B}+\beta\delta^{-1}\gamma\alpha^{-1})$
Therefore, by Proposition 2.1,
$(\det(\varphi_{\lambda}))^{-1}=\alpha_{\lambda}^{-1}(\mathbf{1}_{B}+\beta_{\lambda}\delta_{\lambda}^{-1}\gamma_{\lambda}\alpha_{\lambda}^{-1})\xrightarrow{c.o.}\alpha^{-1}(\mathbf{1}_{B}+\beta\delta^{-1}\gamma\alpha^{-1})=(\det(\varphi))^{-1}.$
This establishes the remaining equivalence. ∎
We remark that the previous theorem has a striking similarity to the purely
algebraic Theorem 3.5. In both cases, we have reduced a question regarding
elements of $\operatorname{Aut}(L)$ to a question regarding only its diagonal
components, utilizing the quasi-determinant as an intermediate step. Also,
observe that (i)$\Longrightarrow$(ii) in Theorem 4.4 implies that the quasi-
determinant $\det:\operatorname{Aut}(L)\to\operatorname{Aut}(L_{1})$ is
continuous.
## Acknowledgments
This research was conducted as part of an NSERC Undergraduate Summer Research
Award under the supervision of Gábor Lukács. I thank Dr. Lukács for his
wisdom, guidance, attention to detail, and understanding; without him, this
work would simply have not been possible.
I would also like to thank Karen Kipper for carefully proof-reading this paper
for grammar and spelling.
## References
* [1] Arens, Richard. Topologies for homeomorphism groups. Amer. J. Math. 68 (4) (1946), 593-610.
* [2] Armacost, David L. _The structure of locally compact abelian groups_ , Marcel Dekker Inc., New York, 1981.
* [3] Armacost D. L. and Armacost W. L. Uniqueness in structure theorems for lca groups. Can. J. Math. 30 (3) (1978), 593-599.
* [4] Braconnier, J. Sur les groupes topologiques localement compacts, J. Math. Pures Appl. 27 (9) (1948), 1-85.
* [5] Caprace, P. and Monod, N. Decomposing locally compact groups into simple pieces. Math. Proc. Cambridge Philos. Soc. 150 (2011), 97-128.
* [6] Engelking, Ryszard. _General topology_ , Sigma Series in Pure Math., 6, Heldermann Verlag, Berlin, 2e, 1989. Translated from Polish by the author.
* [7] Fox, Ralph H. On topologies for function spaces. Bull. Amer. Math. Soc. 51 (1945), 429-432.
* [8] Gleason, A. M. and Palais, R. S. On a class of transformation groups. Amer. J. Math. 79 (3) (1957), 631-648.
* [9] Hewitt, E. and Ross, K. A. _Abstract harmonic analysis Vol. I_. Springer, Berlin, 1963.
* [10] Levin, Martin D. The automorphism group of a locally compact abelian group. Acta. Math. 127 (1971), 259-278.
* [11] Lukács, Gábor. _Compact-like topological groups_. Research and Exposition in Math., 31, Heldermann Verlag, Berlin, 2009.
* [12] Lee, D. H. and Wu, T.-S. The group of automorphisms of a finite-dimensional topological group. 15 (3) (1968), 321-324.
* [13] Mac Lean, Saunders. _Categories for the working mathematician_. Graduate Texts in Mathematics. Springer, New York, 2e, 1998.
* [14] Megrelishvili, M. and Shlossberg, M. Notes on non-archimedean topological groups. Topology Appl. To appear. (2011)
* [15] Mel’nikov, O. V. Compactness conditions for groups of automorphisms of topological groups. Matematicheskie Zametki. 19 (5) (1976) 735-743.
* [16] Moskalenko, Z. I. Automorphism groups of compact, totally disconnected, nilpotent groups. Ukrainskii Matematicheskii Zhurnal. 32 (1) (1980), 46 52.
* [17] Stroppel, Markus. _Locally compact groups_. EMS Textbooks in Mathematics. European Mathematical Society, Zurich, 2006.
* [18] Willard, Stephen. _General topology_. Addison-Wesley, Reading, Mass., 1970. (reprinted by Dover)
|
arxiv-papers
| 2011-10-10T04:32:07 |
2024-09-04T02:49:22.945201
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Iian B. Smythe",
"submitter": "Iian Smythe",
"url": "https://arxiv.org/abs/1110.1923"
}
|
1110.1963
|
# Depth of factors of square free monomial ideals
Dorin Popescu Dorin Popescu, ”Simion Stoilow” Institute of Mathematics of
Romanian Academy, Research unit 5, P.O.Box 1-764, Bucharest 014700, Romania
dorin.popescu@imar.ro
###### Abstract.
Let $I$ be an ideal of a polynomial algebra over a field, generated by $r$
square free monomials of degree $d$. If $r$ is bigger (or equal, if $I$ is not
principal) than the number of square free monomials of $I$ of degree $d+1$,
then $\operatorname{depth}_{S}I=d$. Let $J\subsetneq I$, $J\not=0$ be
generated by square free monomials of degree $\geq d+1$. If $r$ is bigger than
the number of square free monomials of $I\setminus J$ of degree $d+1$, or more
generally the Stanley depth of $I/J$ is $d$, then
$\operatorname{depth}_{S}I/J=d$. In particular, Stanley’s Conjecture holds in
theses cases.
Key words : Monomial Ideals, Depth, Stanley depth.
2000 Mathematics Subject Classification: Primary 13C15, Secondary 13F20,
13F55, 13P10.
The support from grant ID-PCE-2011-1023 of Romanian Ministry of Education,
Research and Innovation is gratefully acknowledged.
## Introduction
Let $S=K[x_{1},\ldots,x_{n}]$ be the polynomial algebra in $n$ variables over
a field $K$, $d$ a positive integer and $I\supsetneq J$, be two square free
monomial ideals of $S$ such that $I$ is generated in degrees $\geq d$,
respectively $J$ in degrees $d+1$. Let $\rho_{d}(I)$ be the number of all
square free monomials of degree $d$ of $I$. It is easy to note (see Lemma 1.1)
that $\operatorname{depth}_{S}I/J\geq d$. Our Theorem 2.2 gives a sufficient
condition which implies $\operatorname{depth}_{S}I/J=d$, namely this happens
when
$\rho_{d}(I)>\rho_{d+1}(I)-\rho_{d+1}(J).$
Suppose that this condition holds. Then the Stanley depth of $I/J$ (see [11],
[2], or here Remark 2.6) is $d$ and if Stanley’s Conjecture holds then
$\operatorname{depth}_{S}I/J\leq d$, that is the missing inequality. Thus to
test Stanley’s Conjecture means to test the equality
$\operatorname{depth}_{S}I/J=d$, which is much easier since there exist very
good algorithms to compute $\operatorname{depth}_{S}I/J$ but not so good to
compute the Stanley depth of $I/J$. After a lot of examples computed with the
computer algebra system SINGULAR we understood that a result as Theorem 2.2 is
believable. The above condition is not necessary as Example 2.4 shows.
Necessary and sufficient conditions could be possible found classifying some
posets (see Remark 2.5) but this is not the subject of this paper.
The proof of Theorem 2.2 uses the Koszul homology and can be read without
other preparation. Our first section gives easy proofs in special cases, but
they are mainly an introduction in the subject. Remarks 1.7, 1.9 show that the
Koszul homology seems to be the best tool in our problems. Section $2$ starts
with one example, where we give the idea of the proof of Theorem 2.2.
If $I$ is generated by more (or equal, if $I$ is not principal) square free
monomials of degree $d$ than ${n\choose d+1}$, or more general than
$\rho_{d+1}(I)$, then $\operatorname{depth}_{S}I=d$ as shows our Corollary
3.4. This extends [9, Corollary 3], which was the starting point of our
research, the proof there being easier. Remark 3.5 says that the condition of
Corollary 3.4 is tight.
The conditions above are consequences of the fact that
$\operatorname{sdepth}I/J=d$, as we explained in Remark 2.6, and we saw that
they imply $\operatorname{depth}I/J=d$. But what happens if we just suppose
that $\operatorname{sdepth}I/J=d$? Then there exists a monomial square free
ideal $I^{\prime}\subset I$ such that
$\rho_{d}(I^{\prime})>\rho_{d+1}(I^{\prime})-\rho_{d+1}(I^{\prime}\cap J)$
using our Theorem 4.1 (somehow an extension of [10, Lemma 3.3]) and it follows
also $\operatorname{depth}_{S}I/J=d$ by our Theorem 4.3.
We owe thanks to a Referee who found gaps in a preliminary version of our
paper.
## 1\. Factors of square free monomial ideals
Let $J\subsetneq I$, be two nonzero square free monomial ideals of $S$ and $d$
a positive integer. Let $\rho_{d}(I)$ be the number of all square free
monomials of degree $d$ of $I$. Suppose that $I$ is generated by square free
monomials $f_{1},\ldots,f_{r}$, $r>0$, of degrees $\geq d$ and $J$ is
generated by square free monomials of degree $\geq d+1$. Set
$s:=\rho_{d+1}(I)-\rho_{d+1}(J)$ and let $b_{1},\ldots,b_{s}$ be the square
free monomials of $I\setminus J$ of degree $d+1$.
###### Lemma 1.1.
$\operatorname{depth}_{S}I,\ \operatorname{depth}_{S}I/J\geq d$.
###### Proof.
By [2, Proposition 3.1] we have $\operatorname{depth}_{S}I\geq d$,
$\operatorname{depth}_{S}J\geq d+1$. The conclusion follows from applying the
Depth Lemma in the exact sequence $0\rightarrow J\rightarrow I\rightarrow
I/J\rightarrow 0$.
###### Lemma 1.2.
Suppose that $J=E+F$, $F\not\subset E$, where $E,F$ are ideals generated by
square free monomials of degree $d+1$, respectively $>d+1$. Then
$\operatorname{depth}_{S}I/J=d$ if and only if
$\operatorname{depth}_{S}I/E=d$.
###### Proof.
We may suppose that in $E$ there exist no monomial generator of $F$. In the
exact sequence
$0\rightarrow J/E\rightarrow I/E\rightarrow I/J\rightarrow 0$
we see that the first end is isomorphic with $F/(F\cap E)$ and has depth $\geq
d+2$ by Lemma 1.1. Apply the Depth Lemma and we are done.
Before trying to extend the above lemma it is useful to see the next example.
###### Example 1.3.
Let $n=4$, $d=1$, $I=(x_{2})$, $E=(x_{2}x_{4})$, $F=(x_{1}x_{2}x_{3})$. Then
$\operatorname{depth}_{S}I/E=3$ and $\operatorname{depth}_{S}I/(E+F)=2$.
###### Lemma 1.4.
Let $H$ be an ideal generated by square free monomials of degrees $d+1$. Then
$\operatorname{depth}_{S}I/J=d$ if and only if
$\operatorname{depth}_{S}(I+H)/J=d$.
###### Proof.
By induction on the number of the generators of $H$ it is enough to consider
the case $H=(u)$ for some square free monomial $u\not\in I$ of degrees $d+1$.
In the exact sequence
$0\rightarrow I/J\rightarrow(I+(u))/J\rightarrow(I+(u))/I\rightarrow 0$
we see that the last term is isomorphic with $(u)/I\cap(u)$ and has depth
$\geq d+1$ by Lemma 1.1, since $I\cap(u)$ has only monomials of degrees
$>d+1$. Using the Depth Lemma the first term has depth $d$ if and only if the
middle has depth $d$, which is enough.
Using Lemmas 1.2, 1.4 we may suppose always in our consideration that $I$, $J$
are generated in degree $d$, respectively $d+1$, in particular $f_{i}$ have
degree $d$.
###### Lemma 1.5.
Let $e\leq r$ be a positive integer and $I^{\prime}=(f_{1},\ldots,f_{e})$,
$J^{\prime}=J\cap I^{\prime}$. If
$\operatorname{depth}_{S}I^{\prime}/J^{\prime}=d$ then
$\operatorname{depth}_{S}I/J=d$ .
###### Proof.
In the exact sequence
$0\rightarrow I^{\prime}/J^{\prime}\rightarrow I/J\rightarrow
I/(I^{\prime}+J)\rightarrow 0$
the right end has depth $\geq d$ by Lemma 1.1 because
$I/(I^{\prime}+J)\cong(f_{e+1},\ldots,f_{r})/((J+I^{\prime})\cap(f_{e+1},\ldots,f_{r})))$
and $(J+I^{\prime})\cap(f_{e+1},\ldots,f_{r})$ is generated by monomials of
degree $>d$. If the left end has depth $d$ then the middle has the same depth
by the Depth Lemma.
###### Lemma 1.6.
Suppose that there exists $i\in[r]$ such that $f_{i}$ has in $J$ all square
free multiples of degree $d+1$. Then $\operatorname{depth}_{S}I/J=d$.
###### Proof.
We may suppose $i=1$. By our hypothesis $J:f_{1}$ is generated by $(n-d)$
variables. If $r=1$ then the depth of $I/J\cong S/(J:f_{1})$ is $d$. If $r>1$
apply the above lemma for $e=1$.
###### Remark 1.7.
Suppose in the proof of the above lemma that $f_{1}=x_{1}\cdots x_{d}$. Then
the hypothesis says that $(x_{d+1},\ldots,x_{n})f_{1}\subset J$. It follows
that $z=f_{1}e_{\sigma_{1}}$, $e_{\sigma_{1}}=e_{d+1}\wedge\ldots\wedge e_{n}$
induces a nonzero element in the Koszul homology module $H_{n-d}(x;I/J)$ of
$I/J$ (some details from Koszul homology theory are given in Example 2.1).
Thus $\operatorname{depth}_{S}I/J\leq d$ by [1, Theorem 1.6.17], the other
inequality follows from Lemma 1.1. This gives a different proof of the above
lemma using stronger tools, which will be very useful in the next section. We
also remind that
$H_{n-d}(x;I/J)\cong\operatorname{Tor}_{n-d}^{S}(K,I/J)\not=(0)$ gives
$\operatorname{pd}_{S}I/J\geq n-2$, which means
$\operatorname{depth}_{S}I/J\leq 2$ by Auslander-Buchsbaum Theorem [1, Theorem
1.3.3].
###### Lemma 1.8.
Suppose that $r\geq 2$ and the least common multiple $b=[f_{1},f_{2}]$ has
degree $d+1$ and it is the only monomial of degree $d+1$ which is in
$(f_{1},f_{2})\setminus J$. Then $\operatorname{depth}_{S}I/J=d$.
###### Proof.
Apply induction on $r\geq 2$. Suppose that $r=2$. By hypothesis the greatest
common divisor $u=(f_{1},f_{2})$ have degree $d-1$ and after renumbering the
variables we may suppose that $f_{i}=x_{i}u$ for $i=1,2$. By hypothesis the
square free multiples of $f_{1},f_{2}$ by variables $x_{i}$, $i>2$ belongs to
$J$. Thus we see that $I/J$ is a finite module over a polynomial ring in
$(d+1)$-variables and we get $\operatorname{depth}_{S}I/J\leq d$ since $I/J$
it is not free. Now it is enough to apply Lemma 1.1. If $r>2$ then apply Lemma
1.5 for $e=2$.
###### Remark 1.9.
We see in the proof of the above lemma (similarly as in Remark 1.7) that if
$u=x_{n-d+2}\cdots x_{n}$ then $z=f_{1}e_{\sigma_{1}}-f_{2}e_{\sigma_{2}}$,
$e_{\sigma_{1}}=e_{2}\wedge\ldots\wedge e_{n-d+1}$,
$e_{\sigma_{2}}=e_{1}\wedge e_{3}\wedge\ldots\wedge e_{n-d+1}$ induces a
nonzero element in $H_{n-d}(x;I/J)$. Thus $\operatorname{depth}_{S}I/J\leq d$
again by [1, Theorem 1.6.17].
###### Proposition 1.10.
Let $b_{1},\ldots,b_{s}$ be the monomials of degree $d+1$ from $I\setminus J$.
Suppose that $r>s$ and for each $i\in[r]$ there exists at most one $j\in[s]$
with $f_{i}|b_{j}$. Then $\operatorname{depth}_{S}I/J=d$.
###### Proof.
If there exists $i\in[r]$ such that $f_{i}$ has in $J$ all square free
multiples of degree $d+1$, then we apply Lemma 1.6. Otherwise, each $f_{i}$
has a square free multiple of degree $d+1$ which is not in $J$. By hypothesis,
there exist $i,j\in[r]$, $i\not=j$ such that $f_{i},f_{j}$ have the same
multiple $b$ of degree $d+1$ in $I\setminus J$. Now apply the above lemma.
###### Corollary 1.11.
Suppose that $r>s\leq 1$. Then $\operatorname{depth}_{S}I/J=d$.
###### Proposition 1.12.
Suppose that $r>s=2$. Then $\operatorname{depth}_{S}I/J=d$.
###### Proof.
Using Lemma 1.5 for $e=3$ we reduce to the case $r=3$. By Lemma 1.6 we may
suppose that each $f_{i}$ divides $b_{1}$, or $b_{2}$. By Proposition 1.10 we
may suppose that $f_{1}|b_{1}$, $f_{1}|b_{2}$, that is $f_{1}$ is the greatest
common divisor $(b_{1},b_{2})$. Assume that $f_{2}|b_{1}$. If $f_{2}|b_{2}$
then we get $f_{2}=(b_{1},b_{2})=f_{1}$, which is false. Similarly, if
$f_{3}|b_{1}$ then $f_{3}\not|b_{2}$ and we may apply Lemma 1.8 to
$f_{2},f_{3}$. Thus we reduce to the case when $f_{3}|b_{2}$ and
$f_{3}\not|b_{1}$. We may suppose that $b_{1}=x_{1}f_{1}$, $b_{2}=x_{2}f_{1}$
and $x_{1},x_{2}$ do not divide $f_{1}$ because $b_{i}$ are square free. It
follows that $b_{1}=x_{i}f_{2}$, $b_{2}=x_{j}f_{3}$ for some $i,j>2$ with
$x_{i},x_{j}|f_{1}$.
Case $i=j$
Then we may suppose $i=j=3$ and $f_{1}=x_{3}u$ for a square free monomial $u$
of degree $d-1$. It follows that $f_{2}=x_{2}u$, $f_{3}=x_{1}u$. Let
$S^{\prime}$ be the polynomial subring of $S$ in the variables
$x_{1},x_{2},x_{3}$ and those dividing $u$. Then for each variable
$x_{k}\not\in S^{\prime}$ we have $f_{i}x_{k}\in J$ and so $I/J\cong
I^{\prime}/J^{\prime}$, where $I^{\prime}=I\cap S^{\prime}$, $J^{\prime}=J\cap
S^{\prime}$. Changing from $I,J,S$ to $I^{\prime},J^{\prime},S^{\prime}$ we
may suppose that $n=d+2$ and $u=\Pi_{i>3}^{n}x_{i}$. Then
$I/J\cong(I:u)/(J:u)\cong(x_{1},x_{2},x_{3})S/(x_{1}x_{2})S$. Then
$\operatorname{depth}_{S}I/J=d-1+\operatorname{depth}_{T}(x_{1},x_{2},x_{3})T/(x_{1}x_{2})$.
By Lemma 1.2 it is enough to see that
$\operatorname{depth}_{T}(x_{1},x_{2},x_{3})T/(x_{1}x_{2})T=1$.
Case $i\not=j$
Then we may suppose $i=3$, $j=4$ and $f_{1}=x_{3}x_{4}v$ for a square free
monomial $v$ of degree $d-2$. It follows that
$f_{2}=x_{1}f_{1}/x_{3}=x_{1}x_{4}v$, $f_{3}=x_{2}f_{1}/x_{4}=x_{2}x_{3}v$.
Let $S^{\prime\prime}$ be the polynomial subring of $S$ in the variables
$x_{1},x_{2},x_{3},x_{4}$ and those dividing $v$. As above $I/J\cong
I^{\prime\prime}/J^{\prime\prime}$, where $I^{\prime\prime}=I\cap
S^{\prime\prime}$, $J^{\prime\prime}=J\cap S^{\prime\prime}$. Changing from
$I,J,S$ to $I^{\prime\prime},J^{\prime\prime},S^{\prime\prime}$ we may suppose
that $n=d+2$ and $v=\Pi_{i>4}^{n}x_{i}$. Then
$I/J\cong(I:v)/(J:v)\cong(x_{1}x_{4},x_{2}x_{3},x_{3}x_{4})S/(x_{1}x_{2}x_{3},x_{1}x_{2}x_{4})S.$
Then
$\operatorname{depth}_{S}I/J=d-2+\operatorname{depth}_{T^{\prime}}(x_{1}x_{4},x_{2}x_{3},x_{3}x_{4})T^{\prime}/(x_{1}x_{2}x_{3},x_{1}x_{2}x_{4})T^{\prime}.$
By Lemma 1.2 it is enough to see that
$\operatorname{depth}_{T^{\prime}}(x_{1}x_{4},x_{2}x_{3},x_{3}x_{4})T^{\prime}/(x_{1}x_{2}x_{3},x_{1}x_{2}x_{4})T^{\prime}=2.$
###### Lemma 1.13.
Suppose that $d=1$, $f_{i}=x_{i}$, $i\in[r]$ and $b_{j}\in
S^{\prime}=K[x_{1},\ldots,x_{r}]$ for all $j\in[s]$. Then
$\operatorname{depth}_{S}I/J=1$ independently of $r,s$ ($s$ may be greater
than $r$).
###### Proof.
Set $I^{\prime}=I\cap S^{\prime}$ and $J^{\prime}=J\cap S^{\prime}$. Then
$\operatorname{depth}_{S^{\prime}}S^{\prime}/I^{\prime}=0$ and
$\operatorname{depth}_{S^{\prime}}S^{\prime}/J^{\prime}>0$ by Lemma 1.1. From
the following exact sequence
$0\rightarrow I^{\prime}/J^{\prime}\rightarrow
S^{\prime}/J^{\prime}\rightarrow S^{\prime}/I^{\prime}\rightarrow 0$
it follows that $\operatorname{depth}_{S^{\prime}}I^{\prime}/J^{\prime}=1$ by
the Depth Lemma. If $r<n$ then note that $(x_{r+1},\ldots,x_{n})I\subset J$
and so
$\operatorname{depth}_{S}I/J=\operatorname{depth}_{S}(I^{\prime}S/J^{\prime}S)-(n-r)=\operatorname{depth}_{S^{\prime}}I^{\prime}/J^{\prime}=1$.
###### Proposition 1.14.
Suppose that $d=1$ and $r>s$. Then $\operatorname{depth}_{S}I/J=1$.
###### Proof.
By Lemma 1.13 we may suppose that $I=(x_{1},\ldots,x_{r})$ with $r<n$. Using
Lemma 1.6 we may suppose that each $x_{i}$, $i\in[r]$ divides a certain
$b_{k}$. Apply induction on $s$, the case $s\leq 2$ being done in Proposition
1.12. Assume that $s>2$. We may suppose that each $b_{k}$ is a product of two
different $x_{i}$, $i\in[r]$ because if let us say $b_{s}$ is just a multiple
of one $x_{i}$, $i\in[r]$, for example $x_{r}$, then we may take
$I^{\prime}=(x_{1},\ldots,x_{r-1})$, $J^{\prime}=J\cap I^{\prime}$ and we get
$\operatorname{depth}_{S}I^{\prime}/J^{\prime}=1$ by induction hypothesis on
$s$ since $r-1>s-1$, that is $\operatorname{depth}_{S}I/J=1$ by Lemma 1.5. But
if each $b_{k}$ is a product of two different $x_{i}$, $i\in[r]$ we see that
$b_{j}\in S^{\prime}=K[x_{1},\ldots,x_{r}]$ for all $j\in[s]$ and we may apply
again Lemma 1.13.
## 2\. Main result
We want to extend Proposition 1.14 for the case $d>1$. Next example is an
illustration of our method.
###### Example 2.1.
Let $n=6$, $d=2$, $f_{1}=x_{1}x_{6}$, $f_{2}=x_{1}x_{5}$, $f_{3}=x_{1}x_{3}$,
$f_{4}=x_{3}x_{4}$,
$f_{5}=x_{2}x_{4}$,
$J=(x_{1}x_{2}x_{4},x_{1}x_{2}x_{5},x_{1}x_{2}x_{3},x_{1}x_{2}x_{6},x_{1}x_{3}x_{6},x_{1}x_{4}x_{5},x_{1}x_{4}x_{6},$
$x_{2}x_{4}x_{5},x_{2}x_{4}x_{6},x_{3}x_{4}x_{5},x_{3}x_{4}x_{6})$
and $I=(f_{1},f_{2},f_{3},f_{4},f_{5})$. We have $s=4$,
$b_{1}=x_{5}f_{1}=x_{6}f_{2}$, $b_{2}=x_{3}f_{2}=x_{5}f_{3}$,
$b_{3}=x_{4}f_{3}=x_{1}f_{4}$, $b_{4}=x_{2}f_{4}=x_{3}f_{5}$. Let
$\partial_{i}:K_{i}(x;I/J)\rightarrow K_{i-1}(x;I/J)$, $K_{i}(x;I/J)\cong
S^{{6\choose i}}$, $i\in[6]$ be the Koszul derivation given by
$\partial_{i}(e_{j_{1}}\wedge\ldots\wedge
e_{j_{i}})=\sum_{k=1}^{i}(-1)^{k+1}x_{j_{k}}e_{j_{1}}\wedge\ldots\wedge
e_{j_{k-1}}\wedge e_{j_{k+1}}\wedge\ldots\wedge e_{j_{i}}.$
We consider the following elements of $K_{4}(x;I/J)$
$e_{\sigma_{1}}=e_{2}\wedge\ldots\wedge e_{5},\
e_{\sigma_{2}}=e_{2}\wedge\ldots\wedge e_{4}\wedge e_{6},\
e_{\sigma_{3}}=e_{2}\wedge e_{4}\wedge\ldots\wedge e_{6},$
$e_{\sigma_{4}}=e_{1}\wedge e_{2}\wedge e_{5}\wedge e_{6},\
e_{\sigma_{5}}=e_{1}\wedge e_{3}\wedge e_{5}\wedge e_{6}.$
Then the element
$z=f_{1}e_{\sigma_{1}}-f_{2}e_{\sigma_{2}}-f_{3}e_{\sigma_{3}}-f_{4}e_{\sigma_{4}}+f_{5}e_{\sigma_{5}}$
satisfies
$\partial_{4}(z)=(-b_{1}+b_{1}))e_{2}\wedge e_{3}\wedge
e_{4}+(b_{2}-b_{2})e_{2}\wedge e_{4}\wedge e_{6}+$ $(b_{3}-b_{3})e_{2}\wedge
e_{5}\wedge e_{6}+(b_{4}-b_{4})e_{1}\wedge e_{5}\wedge e_{6}=0,$
since $(b_{k})$ are the only monomials of degree $3$ which are not in $J$.
Note that in a term $ue_{\sigma}$ of an element from
$\operatorname{Im}\partial_{5}$ we have $u$ of degree $\geq 3$ because $I$ is
generated in degree $2$. Thus $z\not\in\operatorname{Im}\partial_{5}$ induces
a nonzero element in $H_{4}(x;I/J)$. By [1, Theorem 1.6.17] we get
$\operatorname{depth}_{S}I/J\leq 2$, which is enough.
###### Theorem 2.2.
If $r>s$ then $\operatorname{depth}_{S}I/J=d$, independently of the
characteristic of $K$.
###### Proof.
Let $\operatorname{supp}f_{i}=\\{j\in[n]:x_{j}|f_{i}\\}$,
$e_{\sigma_{i}}=\wedge_{j\in([n]\setminus\operatorname{supp}f_{i})}\ e_{j}$
and
$e_{\tau_{k}}=\wedge_{j\in([n]\setminus\operatorname{supp}b_{k})}\ e_{j}$. By
[1, Theorem 1.6.17] it is enough to show, as in the above example, that there
exist $y_{i}\in K$, $i\in[r]$ such that
$z=\sum_{i=1}^{r}y_{i}f_{i}e_{\sigma_{i}}$ induces a nonzero element of
$H_{n-d}(x;I/J)$. Let $\partial_{i}$ be the Koszul derivation as above. Then
$\partial_{n-d}(z)=\sum_{k=1}^{s}(\sum_{i\in[r],f_{i}|b_{k}}\varepsilon_{ki}y_{i})b_{k}$
for some $\varepsilon_{ki}\in\\{1,-1\\}$. Thus $\partial_{n-d}(z)=0$ if and
only if $\sum_{i\in[r],f_{i}|b_{k}}\varepsilon_{ki}y_{i}=0$ for all $k\in[s]$.
This is a system of $s$ homogeneous linear equations in $r$ variables $Y$,
which must have a nonzero solution in $K$ because $r>s$. As in the above
example $z\not\in\operatorname{Im}\partial_{n-d+1}$ if $I$ is generated in
degree $d$ (this may be supposed by Lemmas 1.2, 1.4).
The condition given in Theorem 2.2 is tight as shows the following two
examples.
###### Example 2.3.
Let $n=4$, $d=2$, $f_{1}=x_{1}x_{3}$, $f_{2}=x_{2}x_{4}$, $f_{3}=x_{1}x_{4}$
and $I=(f_{1},\ldots,f_{3})$, $J=(x_{2}x_{3}x_{4})$ be ideals of $S$. We have
$r=s=3$, $b_{1}=x_{1}x_{2}x_{3}$, $b_{2}=x_{1}x_{2}x_{4}$,
$b_{3}=x_{1}x_{3}x_{4}$, and $\operatorname{depth}_{S}I/J=d+1$.
###### Example 2.4.
Let $n=6$, $d=2$, $f_{1}=x_{1}x_{5}$, $f_{2}=x_{2}x_{3}$, $f_{3}=x_{3}x_{4}$,
$f_{4}=x_{1}x_{6}$, $f_{5}=x_{1}x_{4}$, $f_{6}=x_{1}x_{2}$, and
$I=(f_{1},\ldots,f_{6})$,
$J=(x_{1}x_{2}x_{4},x_{1}x_{2}x_{5},x_{1}x_{3}x_{5},x_{1}x_{3}x_{6},x_{1}x_{4}x_{6},x_{2}x_{3}x_{5},x_{2}x_{3}x_{6},x_{3}x_{4}x_{5},x_{3}x_{4}x_{6}).$
We have $r=s=6$ and $b_{1}=x_{1}x_{4}x_{5}$, $b_{2}=x_{2}x_{3}x_{4}$,
$b_{3}=x_{1}x_{2}x_{3}$, $b_{4}=x_{1}x_{5}x_{6}$, $b_{5}=x_{1}x_{3}x_{4}$,
$b_{6}=x_{1}x_{2}x_{6}$ but $\operatorname{depth}_{S}I/J=2$ (although $d=2$).
###### Remark 2.5.
The above example 2.4 shows that one could find a nice class of factors of
square free monomial ideals with $r=s$ but $\operatorname{depth}_{S}I/J=d$
similarly as in [9, Lemma 6]. An important tool seems to be a classification
of the possible posets given on $f_{1},\ldots,f_{r},b_{1},\ldots,b_{s}$ by the
divisibility.
###### Remark 2.6.
Given $J\subsetneq I$ two square free monomial ideals of $S$ as above one can
consider the poset $P_{I\setminus J}$ of all square free monomials of
$I\setminus J$ (a finite set) with the order given by the divisibility. Let
${\mathcal{P}}$ be a partition of ${\mathcal{P}}\ \ P_{I\setminus J}$ in
intervals $[u,v]=\\{w\in P_{I\setminus J}:u|w,w|v\\}$, let us say
$P_{I\setminus J}=\cup_{i}[u_{i},v_{i}]$, the union being disjoint. Define
$\operatorname{sdepth}{\mathcal{P}}=\operatorname{min}_{i}\operatorname{deg}v_{i}$
and
$\operatorname{sdepth}_{S}I/J=\operatorname{max}_{\mathcal{P}}\operatorname{sdepth}{\mathcal{P}}$,
where ${\mathcal{P}}$ runs in the set of all partitions of $P_{I\setminus J}$.
This is the Stanley depth of $I/J$, in fact this is an equivalent definition
(see [11], [2]). If $r>s$ then it is obvious that
$\operatorname{sdepth}_{S}I/J=d$ and so Theorem 2.2 says that Stanley’s
Conjecture holds, that is
$\operatorname{sdepth}_{S}I/J\geq\operatorname{depth}_{S}I/J$. In general the
Stanley depth of a monomial ideal $I$ is greater than or equal with the
Lyubeznik’ size of $I$ increased by one (see [3]). Stanley’s Conjecture holds
for intersections of four monomial prime ideals of $S$ by [5] and [7] and for
square free monomial ideals of $K[x_{1},\ldots,x_{5}]$ by [6] (a short
exposition on this subject is given in [8]). Also Stanley’s Conjecture holds
for intersections of three monomial primary ideals by [13]. In the case of a
non square free monomial ideal $I$ a useful inequality is
$\operatorname{sdepth}I\leq\operatorname{sdepth}\sqrt{I}$ (see [4, Theorem
2.1]).
## 3\. Around Theorem 2.2
Let $S^{\prime}=K[x_{1},\ldots,x_{n-1}]$ be a polynomial ring in $n-1$
variables over a field $K$, $S=S^{\prime}[x_{n}]$ and $U,V\subset S^{\prime}$,
$V\subset U$ be two square free monomial ideals. Set $W=(V+x_{n}U)S$.
Actually, every monomial square free ideal $T$ of $S$ has this form because
then $(T:x_{n})$ is generated by an ideal $U\subset S^{\prime}$ and
$T=(V+x_{n}U)S$ for $V=T\cap S^{\prime}$.
###### Lemma 3.1.
([6]) Suppose that $U\not=V$ and
$\operatorname{depth}_{S^{\prime}}S^{\prime}/U=\operatorname{depth}_{S^{\prime}}S^{\prime}/V=\operatorname{depth}_{S^{\prime}}U/V$.
Then
$\operatorname{depth}_{S}S/W=\operatorname{depth}_{S^{\prime}}S^{\prime}/U$.
###### Lemma 3.2.
Suppose that $U\not=V$ and
$d:=\operatorname{depth}_{S^{\prime}}S^{\prime}/U=\operatorname{depth}_{S^{\prime}}S^{\prime}/V$.
Then $d=\operatorname{depth}_{S^{\prime}}U/V$ if and only if
$d=\operatorname{depth}_{S}S/W$.
###### Proof.
The necessity follows from the above lemma. For sufficiency note that in the
exact sequence
$0\rightarrow VS\rightarrow W\rightarrow US/VS\rightarrow 0$
the depth of the left end is $d+2$ and the middle term has depth $d+1$. It
follows that $\operatorname{depth}_{S}US/VS=d+1$ by the Depth Lemma, which is
enough.
Let $I$ be an ideal of $S$ generated by square free monomials of degree $\geq
d$ and $x_{n}f_{1},\ldots,x_{n}f_{r}$, $r>0$ be the square free monomials of
$I\cap(x_{n})$ of degree $d$. Set $U=(f_{1},\ldots,f_{r})$, $V=I\cap
S^{\prime}$.
###### Theorem 3.3.
If $r>\rho_{d}(U)-\rho_{d}(U\cap V)$ then
$\operatorname{depth}_{S}S/I=\operatorname{depth}_{S^{\prime}}(U+V)/V=d-1$.
###### Proof.
By Theorem 2.2 we have
$\operatorname{depth}_{S^{\prime}}(U+V)/V=\operatorname{depth}_{S^{\prime}}U/(U\cap
V)=d-1$. Using Lemmas 1.2, 1.4 we get
$\operatorname{depth}_{S^{\prime}}(U+V)/V=\operatorname{depth}_{S^{\prime}}((I:x_{n})\cap
S^{\prime})/(I\cap S^{\prime})=d-1.$
If $\operatorname{depth}_{S^{\prime}}S^{\prime}/(I\cap
S^{\prime})=\operatorname{depth}_{S^{\prime}}S^{\prime}/((I:x_{n})\cap
S^{\prime})=d-1$ then $\operatorname{depth}_{S}S/I=d-1$ by Lemma 3.2. If
$\operatorname{depth}_{S^{\prime}}S^{\prime}/((I:x_{n})\cap S^{\prime})=d-2$
then in the exact sequence
$0\rightarrow S/(I:x_{n})\xrightarrow{x_{n}}S/I\rightarrow S^{\prime}/(I\cap
S^{\prime})\rightarrow 0$
the first term has depth $d-1$ and the other two have depth $\geq d-1$ by
Lemma 1.1. By the Depth Lemma it follows that
$\operatorname{depth}_{S}S/I=d-1$.
It remains to consider the case when at least one of
$\operatorname{depth}_{S^{\prime}}S^{\prime}/((I:x_{n})\cap S^{\prime})$ and
$\operatorname{depth}_{S^{\prime}}S^{\prime}/(I\cap S^{\prime})$ is $\geq d$.
Using the Depth Lemma in the exact sequence
$0\rightarrow((I:x_{n})\cap S^{\prime})/(I\cap S^{\prime})\rightarrow
S^{\prime}/(I\cap S^{\prime})\rightarrow S^{\prime}/((I:x_{n})\cap
S^{\prime})\rightarrow 0$
we see that necessarily the depth of the last term is $\geq d$ and the depth
of the middle term is $d-1$. But then the Depth Lemma applied to the previous
exact sequence gives $\operatorname{depth}_{S}S/I=d-1$ too.
The following corollary extends [9, Corollary 3].
###### Corollary 3.4.
Let $I$ be an ideal generated by $\mu(I)>1$ square free monomials of degree
$d$. If $\mu(I)\geq\rho_{d+1}(I)$, in particular if $\mu(I)\geq{n\choose
d+1}$, then $\operatorname{depth}_{S}I=d$.
###### Proof.
We have $I=(V+x_{n}(U+V))S$ as above. Renumbering the variables we may suppose
that $U,V\not=0$. Note that $\mu(I)=r+\rho_{d}(V)$ and
$\rho_{d+1}(I)=\rho_{d+1}(V)+\rho_{d}(U+V)>\rho_{d}(V)+\rho_{d}(U)-\rho_{d}(U\cap
V)$. By hypothesis, $\mu(I)\geq\rho_{d+1}(I)$ and so
$r>\rho_{d}(U)-\rho_{d}(U\cap V)$. Applying Theorem 3.3 we get
$\operatorname{depth}_{S}S/I=d-1$, which is enough.
###### Remark 3.5.
Take in Example 2.3 $S^{\prime}=K[x_{1},\ldots,x_{5}]$ and
$L=(J+x_{5}I)S^{\prime}$. We have $\mu(L)=4<{5\choose 3+1}$, that is the
hypothesis of the above corollary are not fulfilled. This is the reason that
$\operatorname{depth}_{S^{\prime}}L=4$ by Lemma 3.2 since
$\operatorname{depth}_{S}I/J=3$. Thus the condition of the above corollary is
tight.
## 4\. Minimal Stanley depth
Let $S=K[x_{1},\ldots,x_{n}]$ be the polynomial algebra in $n$-variables over
a field $K$, $d$ a positive integer and $J\subsetneq I$, be two square free
monomial ideals of $S$. Let $\rho_{d}(I)$ be the number of all square free
monomials of degree $d$ of $I$. Suppose that $\rho_{d}(I)>0$ and $I$ is
generated in degree $\geq d$. It follows that
$\operatorname{sdepth}_{S}I/J\geq d$.
###### Theorem 4.1.
The following statements are equivalent:
1. (1)
$\operatorname{sdepth}_{S}I/J=d$
2. (2)
there exist some square free monomials of degree $d$ in $I$, which generate an
ideal $I^{\prime}$ such that
$\rho_{d}(I^{\prime})>\rho_{d+1}(I^{\prime})-\rho_{d+1}(I^{\prime}\cap J)$.
###### Proof.
If $J\not=0$ then $J$ is generated in degree $\geq d$, even we may suppose
that $J$ is generated in degree $\geq d+1$ using an easy isomorphism. Let
${\mathcal{M}}_{d}(I)$ be the set of all square free monomials of $I$ of
degree $d$ and ${\mathcal{B}}={\mathcal{M}}_{d+1}\setminus J$. We consider the
bipartite graph $G$ defined by $V(G)={\mathcal{M}}_{d}(I)\cup{\mathcal{B}}$,
an edge of $G$ can have only endpoints $f\in{\mathcal{M}}_{d}(I)$ and
$b\in{\mathcal{B}}$ with $f|b$. Given $f\in{\mathcal{M}}_{d}(I)$ let
$\Gamma(f)$ be the set of all vertices $b$ adjacent to $f$ and for
$A\subset{\mathcal{M}}_{d}(I)$ set $\Gamma(A)=\cup_{f\in A}\Gamma(f)$. By P.
Hall’s marriage theorem [12] there is a complete matching from
${\mathcal{M}}_{d}(I)$ to ${\mathcal{B}}$ if and only if $|\Gamma(A)|\geq|A|$
for every subset $A\subset{\mathcal{M}}_{d}(I)$. Thus
$\operatorname{sdepth}_{S}I/J=d$ if and only if there exists no complete
matching above and so there exists a subset $A\subset{\mathcal{M}}_{d}(I)$
such that $|\Gamma(A)|<|A|$, that is $I^{\prime}=(A)$ satisfies the second
statement.
For $J=0$ we get the following corollary, which is closed to [10, Lemma 3.3]
###### Corollary 4.2.
The following statements are equivalent:
1. (1)
$\operatorname{sdepth}_{S}I=d$
2. (2)
there exist some square free monomials of degree $d$ in $I$, which generate an
ideal $I^{\prime}$ such that $\rho_{d}(I^{\prime})>\rho_{d+1}(I^{\prime})$.
###### Theorem 4.3.
If $\operatorname{sdepth}_{S}I/J=d$ then $\operatorname{depth}_{S}I/J=d$, that
is Stanley’s conjecture holds in this case.
###### Proof.
By Theorem 4.1 there exists a monomial square free ideal $I^{\prime}\subset I$
such that
$\rho_{d}(I^{\prime})>\rho_{d+1}(I^{\prime})-\rho_{d+1}(I^{\prime}\cap J)$.
Then $\operatorname{depth}_{S}I^{\prime}/(I^{\prime}\cap J)=2$ by Theorem 2.2
(if $J=0$ we apply Corollary 3.4). Now it is enough to apply Lemma 1.5.
## References
* [1] W. Bruns and J. Herzog, Cohen-Macaulay rings, Revised edition. Cambridge University Press (1998).
* [2] J. Herzog, M. Vladoiu, X. Zheng, How to compute the Stanley depth of a monomial ideal, J. Algebra, 322 (2009), 3151-3169.
* [3] J. Herzog, D. Popescu, M. Vladoiu, Stanley depth and size of a monomial ideal, Proc. Amer. Math. Soc., 140 (2012), 493-504, arXiv:AC/1011.6462v1.
* [4] M. Ishaq, Upper bounds for the Stanley depth, Comm. Algebra, 40(2012), 87-97.
* [5] A. Popescu, Special Stanley Decompositions, Bull. Math. Soc. Sc. Math. Roumanie, 53(101), no 4 (2010), arXiv:AC/1008.3680.
* [6] D. Popescu, An inequality between depth and Stanley depth, Bull. Math. Soc. Sc. Math. Roumanie 52(100), (2009), 377-382, arXiv:AC/0905.4597v2.
* [7] D. Popescu, Stanley conjecture on intersections of four monomial prime ideals, to appear in Communications in Algebra, arXiv:AC/1009.5646.
* [8] D. Popescu, Bounds of Stanley depth, An. St. Univ. Ovidius. Constanta, 19(2),(2011), 187-194.
* [9] D. Popescu, Depth and minimal number of generators of square free monomial ideals, An. St. Univ. Ovidius, Constanta, 19(3), (2011), 163-166, arXiv:AC/1107.2621.
* [10] Y.H. Shen, When will the Stanley depth increase, arxiv:AC/1110.3182v1.
* [11] R. P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math. 68 (1982) 175-193.
* [12] J. H. van Lint, R.M. Wilson, A course in combinatorics, Cambridge Univ. Press, Cambridge, 2001.
* [13] A. Zarojanu, Stanley Conjecture on three monomial primary ideals, to appear in Bull. Math. Soc. Sc. Math. Roumanie, arXiv:AC/11073211.
|
arxiv-papers
| 2011-10-10T08:54:42 |
2024-09-04T02:49:22.954894
|
{
"license": "Public Domain",
"authors": "Dorin Popescu",
"submitter": "Dorin Popescu",
"url": "https://arxiv.org/abs/1110.1963"
}
|
1110.2046
|
# Fitting in a complex $\chi^{2}$ landscape using an optimized hypersurface
sampling
L. C. Pardo1, M. Rovira-Esteva1, S. Busch2, J.-F. Moulin3, J. Ll. Tamarit1
1Grup de Caracterització de Materials, Departament de Física i Enginyieria
Nuclear, ETSEIB, Universitat Politècnica de Catalunya, Diagonal 647, 08028
Barcelona, Catalonia, Spain 2Physik Department E13 and Forschungs-
Neutronenquelle Heinz Maier-Leibnitz (FRM II), Technische Universität München,
Lichtenbergstr. 1, 85748 Garching, Germany 3Helmholtz-Zentrum Geesthacht,
Institut für Werkstoffforschung, Abteilung WPN, Instrument REFSANS,
Forschungs-Neutronenquelle Heinz Maier-Leibnitz (FRM II), Lichtenbergstr. 1,
85748 Garching, Germany
###### Abstract
Fitting a data set with a parametrized model can be seen geometrically as
finding the global minimum of the $\chi^{2}$ hypersurface, depending on a set
of parameters $\\{P_{i}\\}$. This is usually done using the Levenberg-
Marquardt algorithm. The main drawback of this algorithm is that despite of
its fast convergence, it can get stuck if the parameters are not initialized
close to the final solution. We propose a modification of the Metropolis
algorithm introducing a parameter step tuning that optimizes the sampling of
parameter space. The ability of the parameter tuning algorithm together with
simulated annealing to find the global $\chi^{2}$ hypersurface minimum,
jumping across $\chi^{2}\\{P_{i}\\}$ barriers when necessary, is demonstrated
with synthetic functions and with real data.
###### pacs:
02.50.Cw,02.50.Ng,02.60.Pn,02.50.Tt
## I Introduction
Fitting a parametrized model to experimental results is the most usual way to
obtain the physics hidden behind data. However, as nicely reported by
Transtrum et al. PRLfit , this can be quite challenging and it usually takes
“weeks of human guidance to find a good starting point”. Geometrically, the
problem of finding a best fit corresponds to finding the global minimum of the
$\chi^{2}$ hypersurface. As this hypersurface is often full of fissures, local
minima prohibit an efficient search. The human guidance consists usually of a
set of tricks (depending on every particular problem) that allow to choose the
starting point in this landscape such that the first minimum found is indeed
the global minimum.
This problem is usually due to the mechanism that is behind classical fit
algorithms such as Levenberg-Marquardt (LM) numrecipes : a set of parameters
$\\{P_{i}\\}$ is optimized by varying the parameters and accepting the
modified parameter set as a starting point for the next iteration only if this
new set reduces the value of a cost or merit function such as $\chi^{2}$. From
a geometrical point of view, those algorithms allow only downhill movements in
the $\chi^{2}\\{P_{i}\\}$ hypersurface. Therefore they can get stuck in local
minima or get lost in flat regions of the $\chi^{2}$ landscape PRLfit . This
means that they are only able to find an optimal solution if they are
initialized around the absolute minimum of the $\chi^{2}$ hypersurface.
The challenge of finding the global minimum can be alternatively tackled by
Bayesian methods Bayes ; Sivia_book as demonstrated in different fields such
as astronomy or biology bayesapp , solid state physics bayesappCM ,
quasielastic neutron scattering data analysis Sivia_QENS , and Reverse Monte
Carlo methods RMC . We follow a Bayesian approach to the fit problem in this
contribution. This method is based on another mechanism to wander around in
parameter space: instead of allowing only downhill movements, parameter
changes that increase $\chi^{2}$ can also be accepted if the change in
$\chi^{2}$ is compatible with the data errors.
To do that, a Markov Chain Monte Carlo (MCMC) method is used, where the Markov
Chains are generated by the Metropolis algorithm hastings . However, while in
the case of the LM algorithm the initialization of parameters is critical to
the convergence of the algorithm, it is here the tuning of the maximum
parameter change allowed at each step (called parameter jumps hereafter) that
will decide the success of the algorithm to find the global
$\chi^{2}\\{P_{i}\\}$ minimum in an efficient way.
If the parameter jumps are chosen too small, the algorithm will always accept
any parameter change, getting lost in irrelevant details of the
$\chi^{2}\\{P_{i}\\}$ landscape. If chosen too large, the parameters will
hardly be accepted and the algorithm will get stuck every now and then.
Moreover, in the case of models defined by more than one parameter, when
parameter jumps are not properly chosen, the parameter space can be over-
explored in the direction of those parameters with too small jump lengths, in
other words, the model would be insensitive to the proposed change of these
parameters. On the other hand, some other parameters can be associated to a
jump so big that changes are hardly ever accepted.
Different schemes have been proposed in order to change parameter jumps to
explore the target distribution efficiently using Markov Chains under the
generic name of adaptive MCMC Andrieu2008 . Using the framework of the
Stochastic Approximation Benveniste1990 we present in this work an algorithm
belonging to the group of “Controlled Markov Chains” Borkar1990 ; Andrieu2001
where the calculation of new parameter jumps takes the history of the Markov
Chain and previous parameter jumps into account.
Two main approaches are known which take the Markov Chain history into
account: Adaptive Metropolis (AM) algorithmsHaario2001 (implemented for
example in PyMC PYMC ) and algorithms that use rules following Robbins-Monro
update Robbins1951 ; Gilks1998 ; Andrieu2001 . In the first case, parameter
jumps are tuned using the covariance matrix at every step, so that once the
adaptation is finished the algorithm should be wandering with a parameter jump
close to the “error” of the parameter (defined as the variance of the
posterior parameter PDF). In some cases, this kind of algorithm Andrieu2008
can get stuck if the acceptance ratio of a parameter is too high or too low.
In this case the Markov Chain stops learning from the past history, thus the
optimization is stopped with suboptimal parameter jumps. This problem is
overcome by Robbins-Monro update rules that change parameter jumps so that
they are accepted with an optimal ratio.
The main danger of optimized Metropolis algorithms is that adaptation might
cause the Markov Chain to not converge to the target distribution anymore. In
other words, the Markov Chain might lose its ergodicity. For example in the
case of AM algorithms, the generated chain is not Markovian since it depends
on the history of the chain. However, as demonstrated by Haario et al.
Haario2001 , the chain is able to reproduce the target distribution, i.e. is
ergodic. In the second type of algorithms, the Robbins-Monro type, ergodicity
properties must be assured by updating only at regeneration times Gilks1998 .
In any case, as pointed out by Andrieu et al. Andrieu2008 the convergence to
the target distribution is assured if optimization vanishes. In other words,
if parameter jumps oscillate around a fixed value the ergodic property of the
Markov Chain is assured.
The presented algorithm is based on the stochastic approach of Robbins-Monro
with an updating rule inspired by the one of Gilks et al. Gilks1998 .
Optimization of parameter jumps is therefore performed with two goals in mind:
* •
To calculate them in such a way that all parameters are accepted with the same
ratio. Adjusting parameter jumps so that all parameter changes will have the
same acceptance ratio is important to explore the $\chi^{2}\\{P_{i}\\}$
landscape with the same efficiency in all parameter directions.
* •
To adjust parameter jumps to a value tailored to the stage of the fit. This
will turn out to be important when exploring the $\chi^{2}\\{P_{i}\\}$
hypersurface using the simulated annealing technique Kirkpatrick1984 , since
this allows the parameter jumps to be optimized to explore
$\chi^{2}\\{P_{i}\\}$ (see subsection fitting in a complex $\chi^{2}$
landscape): at the beginning of the fit process the algorithm will set
parameter jumps to a large value to explore large portions of the $\chi^{2}$
landscape, and at the final stages these parameter jumps will be set to small
values by the same algorithm in order to find its absolute minimum.
Geometrically, we can interpret the algorithm as setting the parameter step
sizes to a value related to the hypersurface landscape. First, it modifies the
parameter jump to take into account the shape of the hypersurface along a
parameter direction. If $\chi^{2}\\{P_{k}\\}$ (the cut along a parameter $k$)
is flat (the parameter direction is “sloppy” following Sethna’s nomenclature
sloppy ), the parameter step size is set to a larger value, and parameters
will move faster in this sloppy direction. On the contrary, in the directions
where the $\chi^{2}\\{P_{k}\\}$ has a larger slope (the “stiff” direction
following Sethna’s nomenclature), parameter steps will be set to a smaller
value so that they are accepted with the same as the previous ones. Second, it
modifies the parameter jumps to take the shape of the global $\chi^{2}$
landscape into account when the simulated annealing is used. At the beginning
of the fit parameter jumps will be set to a large value so that details of
$\chi^{2}\\{P_{k}\\}$, i.e. local minima, will be smeared out, making it
easier to find the global minimum. However, during the last steps of the
fitting process, parameter steps will be set to a small value by the algorithm
so that the system will be allowed to relax inside the minimum.
The present work gives a detailed description on how the algorithm works, and
will be organized as follows: We first recall briefly on the Metropolis method
applied to generate Markov Chains. In the next section, the proposed algorithm
to optimize the parameter step size is introduced. Afterwards, we check its
robustness to find optimized parameter jumps using a simple test function; and
finally we test the ability of the regenerative algorithm combined with the
simulated annealing technique to find the global minimum of $\chi^{2}$, even
with poor initialization values, using a simple function with a complex
$\chi^{2}\\{P_{i}\\}$ landscape. The algorithm presented in this work has been
implemented in the program FABADA fabada .
## II The fit method
### II.1 Fitting with the Bayesian ansatz
Fitting data using the Metropolis algorithm is based on an iterative process
where successively proposed parameter sets are accepted according to the
probability that these parameters describe the actual data, given all
available evidence. Hence this method makes use of our knowledge of the error
bars of the data.
We now briefly recall how this can be done using a Metropolis algorithm, to
proceed in the next section with the algorithm to adjust parameter jumps.
We should first start with the probabilistic bases behind the $\chi^{2}$
definition. The probability $\mathbb{P}(H\mid D)$ that an hypothesis $H$ is
correctly describing an experimental result $D$ is related to the likelihood
$\mathbb{P}(D\mid H)$ that experimental data $D_{k}$ ($k=1,\ldots,n$) are
correctly described by a model or hypothesis $H_{k}$ ($k=1,\ldots,n$); using
Bayes theorem Sivia_book ; Bayes ,
$\mathbb{P}(H_{k}\mid D_{k})=\frac{\mathbb{P}(D_{k}\mid
H_{k})\cdot\mathbb{P}(H_{k})}{\mathbb{P}(D_{k})}$ (1)
where $\mathbb{P}(H_{k}\mid D_{k})$ is called the _posterior_ , the
probability that the hypothesis is in fact describing the data.
$\mathbb{P}(D_{k}\mid H_{k})$ is the _likelihood_ , the probability that the
description of the data by the hypothesis is good. $\mathbb{P}(H_{k})$ is
called the _prior_ , the probability density function (PDF) summarizing the
knowledge we have about the hypothesis before looking at the data.
$\mathbb{P}(D_{k})$ is a normalization factor to assure that the integrated
posterior probability is unity.
In the following we will assume no prior knowledge (maximum ignorance prior
Sivia_book ), in this special case Bayes theorem takes the simple form
$\mathbb{P}(H_{k}\mid D_{k})\propto\mathbb{P}(D_{k}\mid H_{k})\equiv L$ (2)
where $L$ is a short notation for likelihood.
Although this is by no means a prerequisite, we will assume in the following
that the likelihood that every single data point $D_{k}$ described by the
model or hypothesis $H_{k}$ follows a Gaussian distribution. The case of a
Poisson distribution was discussed previously fabada_paper . For data with a
Gaussian distributed uncertainty with width $\sigma$, the likelihood for each
individual data point takes the form
$\mathbb{P}(D_{k}\mid
H_{k})=\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{1}{2}\left(\frac{H_{k}-D_{k}}{\sigma_{k}}\right)^{2}\right]$
(3)
and correspondingly, the likelihood that _the whole_ data set is described by
this hypothesis is
$\displaystyle\mathbb{P}(D_{k}\mid H_{k})$ $\displaystyle\propto$
$\displaystyle\prod_{k=1}^{n}\exp\left[-\frac{1}{2}\left(\frac{H_{k}-D_{k}}{\sigma_{k}}\right)^{2}\right]$
(4) $\displaystyle=$
$\displaystyle\exp\left[{-\frac{1}{2}\sum_{k=1}^{n}\left(\frac{H_{k}-D_{k}}{\sigma_{k}}\right)^{2}}\right]$
$\displaystyle=$ $\displaystyle\exp\left(-\frac{\chi^{2}}{2}\right)\quad.$
The Metropolis algorithm will in this special case consist on the proposition
of successive sets of parameters $\\{P_{i}\\}$. A new set of parameters is
generated changing one parameter at a time using the rule
$P_{i}^{\mathrm{new}}=P_{i}^{\mathrm{old}}+r\cdot\Delta P_{i}^{\mathrm{max}}$
(5)
where $\Delta P_{i}^{\mathrm{max}}$ is the maximum change allowed to the
parameter or parameter jump and $r$ is a random number between -1.0 and 1.0.
The new set of parameters will always be accepted if it lowers the value of
$\chi^{2}$, or, if the opposite happens it will be accepted with a probability
$\frac{\mathbb{P}(H\\{P_{i}^{\mathrm{l+1}}\\}\mid
D_{k})}{\mathbb{P}(H\\{P_{i}^{\mathrm{l}}\\}\mid
D_{k})}=\exp\left(-\frac{\chi^{2}_{\mathrm{l+1}}-\chi^{2}_{\mathrm{l}}}{2}\right)$
(6)
where $\chi^{2}_{\mathrm{l+1}}$ and $\chi^{2}_{\mathrm{l}}$ correspond to the
$\chi^{2}$ for the proposed new set of parameters and the old one,
respectively. Otherwise, this new parameter value will be rejected and the fit
function does not change during this step.
The Metropolis algorithm described here is very similar to the one used in
statistical physics to find the possible molecular configurations
(microstates) at a given temperature. In that case the algorithm minimizes the
energy of the system while allowing changes in molecular positions that yield
an increase of the energy if it is compatible with the temperature.
Inspired by the similarities between fitting data using a Bayesian approach
and molecular modeling using Monte Carlo methods, a simulated annealing
procedure proposed by Kirkpatrick Kirkpatrick1984 might optionally be used
(see for example Mortensen2005 ; Schulte1996 ). Following the idea of that
work, the $\chi^{2}$ landscape might be compared with an energy landscape used
to describe glassy phenomena Debenedetti2001 . What we do is to start at high
temperatures, i.e. in the liquid phase, where details of the energy landscape
are not so important. By lowering the temperature fast enough the system might
fall into a local minima, i.e. in the glassy phase. In that case the system is
quenched as it is normally done by standard fitting methods. The presented
algorithm aims to avoid being trapped in local minima using an ”annealing
schedule” as suggested by Kirkpatrick. This is done by artificially increasing
the errors of the data to be fitted and letting the errors slowly relax until
they reach their true values. Because this is very similar to what is
performed in molecular modeling, the parameter favoring the uphill movements
in equation 7 is usually called _temperature_ , yielding the acceptance rule
$\frac{\mathbb{P}(H(P_{i}^{\mathrm{l+1}})\mid
D_{k})}{\mathbb{P}(H(P_{i}^{\mathrm{l}})\mid
D_{k})}=\exp\left(-\frac{\chi^{2}_{\mathrm{l+1}}-\chi^{2}_{\mathrm{l}}}{2\cdot
T}\right)\quad.$ (7)
As it happens with Monte Carlo simulations, increasing the temperature will
increase the acceptance of parameter sets that increase $\chi^{2}$, thus
making the jump over $\chi^{2}$ barriers between minima easier.
### II.2 Adjusting the parameter step size
The objective of tuning the parameter step size is to choose a proper value
for $\Delta P_{i}^{\mathrm{max}}$ in equation 5 to optimize the parameter
space exploration.
Given the total number of algorithm steps $N$ and the number of steps that
yield a change in $\chi^{2}$, i. e. the number of successful attempts, $K$,
the ratio $R$ of steps yielding a $\chi^{2}$ change is $R=K/N$.
$R_{\mathrm{desired}}$ is defined as the ratio with which _some parameter_
should be accepted in a step. As we want every parameter to be changed with
the same ratio, $R_{i,\mathrm{desired}}=R_{\mathrm{desired}}/m$ where $m$ is
the number of parameters.
The algorithm is initialized with a first guess for the parameter step sizes.
This first guess, as will be seen shortly, is not important due to the fast
convergence of the algorithm to the optimized values. The calculation of a new
$\Delta P_{i}^{\mathrm{max}}$, i.e. the regeneration of the Markov Chain, is
done after $N$ steps, i.e. at regeneration times, through the equation
$\Delta P_{i}^{\mathrm{max,new}}=\Delta
P_{i}^{\mathrm{max,old}}\cdot\frac{R_{i}}{R_{i,\mathrm{desired}}}$ (8)
where $R_{i}$ is the actual acceptance ratio of parameter $i$. Following the
previous equation, if the calculated ratio $R_{i}/R_{i,\mathrm{desired}}$ is
equal to one, i. e. if all parameters are changing with the same predefined
ratio, $\Delta P_{i}^{\mathrm{max}}$ will not be changed.
If during the fit process a change of parameter $P_{i}$ is too often accepted,
the parameter space is being over explored with regard to parameter $i$. The
algorithm will then make $\Delta P_{i}^{\mathrm{max}}$ larger in order to
reduce its acceptance. The contrary happens if the acceptance is too low for a
parameter: the algorithm makes $\Delta P_{i}^{\mathrm{max}}$ smaller to
increase its acceptance ratio. This will set different step sizes for each
parameter, making the exploration of all of them equally efficient.
## III Demonstrations of fitting functions
### III.1 Fitting in a well-behaved $\chi^{2}$ landscape
The optimization of the parameter step size is shown using the Gaussian
function
$y(x)=\frac{A}{W\sqrt{2\pi}}\exp\left[-\frac{(x-C)^{2}}{2W^{2}}\right]$ (9)
where $A$ is the amplitude, $W$ is the width and $C$ is the center of the
Gaussian. A function has been generated with the parameter set
$\\{A,W,C\\}=\\{10,1,5\\}$ and a normally distributed error with $\sigma=0.1$
was added. A series of tests with different initial values for parameter jumps
and different desired acceptance ratios have been carried out (see below for
details). The initial parameters for the fit were $\\{A,W,C\\}=\\{2,2,2\\}$.
In all cases the algorithm was able to fit the data as can be seen in 1.
Figure 1: Circles: Generated Gaussian function to test the algorithm with the
parameters $\\{A,W,C\\}=\\{10,1,5\\}$. Dashed line: starting point for all
performed tests ($\\{A,W,C\\}=\\{2,2,2\\}$). Solid line: best fit, i. e.
minimum $\chi^{2}$ fit, of the Gaussian function.
The parameter step size was adjusted every 1000 steps. Three cases are shown
in figure 2: an initial $\Delta P_{i}^{\mathrm{max}}$ of 10 (a very large jump
compared to the parameter values, nearly always resulting in a rejection of
the new parameters) and an $R_{\mathrm{desired}}$ of 66%, the same $\Delta
P_{i}^{\mathrm{max}}$ with an $R_{\mathrm{desired}}$ of 9% and finally a
$\Delta P_{i}^{\mathrm{max}}$ of $10^{-4}$ (a very small jump compared to the
parameter values, resulting in a slow exploration of the parameter space) and
an $R_{\mathrm{desired}}$ of 9%. It can be seen that the algorithm manages in
all these extreme cases to adapt the jump size quickly and reliably in order
to make $R$ equal to $R_{\mathrm{desired}}$.
Figure 2: Total acceptance ratio $R$ as a function of the number of steps when
$R_{\mathrm{desired}}$ is set to 66% and 9% (solid and dashed or dotted
lines). In the second case ($R_{\mathrm{desired}}=9\%$), dashed and dotted
lines represent the values of $R$ as a function of algorithm step for two
different parameter step size initializations ($\Delta
P_{i}^{\mathrm{max}}=10$ and $\Delta P_{i}^{\mathrm{max}}=10^{-4}$
respectively)
In figure 3 we show the three individual acceptance ratios $R_{i}$ for the
different parameters as a function of the fit steps for different
initialization values of the parameter jumps $\Delta P_{i}$, for different
values of $R_{\mathrm{desired}}$, and setting the number of steps to
recalculate parameter jumps $N$ to 1000. When the total acceptance ratio is
set to $R_{\mathrm{desired}}=66\%$ (solid line), the algorithm is able to
change all parameter jumps (see figure 3(b)), making the acceptance ratio
$R_{i}$ of every parameter equal to $R_{\mathrm{desired}}/m=22\%$ and thus the
total acceptance ratio $R$ to 66%. The same happens if the acceptance is set
to 9%: the algorithm finds the parameter step sizes (see dashed line in Fig.
3(b)) which yield a total acceptance ratio of 9% within the first 5000 steps,
no matter how the parameter step sizes were initialized.
Figure 3: (color online) a) Acceptance ratio $R_{i}$ for parameters $A$, $W$,
$C$ involved in the fit of the Gaussian following equation 9 ( red triangles,
green squares and blue circles respectively) when $R_{\mathrm{desired}}$ is
set to 66% and 9% (solid and dashed lines). b) Parameter step size as a
function of the number of steps (line and symbols code as in figure a). The
inset shows a cut through the $\chi^{2}$ hypersurface along $A$ and $C$
directions fixing W to the best fit value.
To explicitly show how this is linked with the geometrical features of the
$\chi^{2}$ landscape, the inset of figure 3(b) shows a cut of the $\chi^{2}$
hypersurface along parameters $A$ and $C$, leaving parameter $W$ fixed to its
best fit value $W_{\mathrm{BF}}$. As can readily be seen, the
$\chi^{2}\\{A,C,W=W_{\mathrm{BF}}\\}$ hypersurface is sloppy in the direction
of parameter $A$ and stiff in the direction of parameter $C$. The algorithm
has thus correctly calculated a parameter step size which is larger for $A$
than for $C$, along whose direction the $\chi^{2}$ well is narrower. This fact
makes the final parameter step sizes proportional to the errors of each
parameter – if the global minimum is not multimodal, is quadratic in all
parameters, and those are not correlated.
In order to show the robustness of the algorithm, we have also made disparate
initial guesses for parameter step sizes $\Delta P_{i}^{\mathrm{max}}$ about
three decades below the correct acceptance ratio, setting
$R_{\mathrm{desired}}=9\%$. As displayed in figure 3, after about 5000 steps
the acceptance ratio $R$ ($N$ is again 1000 steps) has already reached the
desired value. It can be seen in figure 4(a) that the acceptance ratio for
each parameter reaches again the value $R_{\mathrm{desired}}/m=3\%$ and
parameter step sizes are virtually equal to those obtained previously as shown
in figure 4(b).
Figure 4: (color online) a) Acceptance ratio $R_{i}$ for parameters $A$
(triangles), $W$ (squares), $C$ (circles) involved in the fit of the Gaussian
following equation 9 when initial parameter step sizes are set to $\Delta
P_{i}=10$ (dashed line) and $\Delta P_{i}=10^{-4}$ (dotted line). b) Parameter
step size as a function of the number of steps (lines and symbols as in figure
a).
To stress the relevance of the aforementioned algorithm to explore the
parameter space correctly, thus assuring its convergence, we have calculated
the normalized $\Delta\chi^{2}$PDF in all tested cases. As can be seen in
figure 5, the $\Delta\chi^{2}$ PDF after $10^{5}$ steps matches the chi-square
distribution
$\mathbb{P}(\Delta\chi^{2})\propto\left(\Delta\chi^{2}\right)^{\left(\frac{m}{2}-1\right)}\exp\left(-\frac{\Delta\chi^{2}}{2}\right)$
(10)
with $m=3$ as expected numrecipes . In figure 5 we show the $\Delta\chi^{2}$
PDF obtained after $10^{4}$ steps for different cases: first setting $\Delta
P_{i}^{\mathrm{max}}$ equal to the value calculated by the algorithm and
second setting $\Delta P_{i}^{\mathrm{max}}$ equal to the initial guess and
finally to a value, calculated a posteriori, which is proportional to the best
fit parameters $\Delta P_{i}^{\mathrm{max}}=0.1P_{i}$ (inset of figure 5)
As can be seen in figure 5, when $\Delta P_{i}^{\mathrm{max}}$ is set much
higher than the optimal step sizes, the Metropolis algorithm scans the whole
parameter space $\\{P_{i}\\}$, but jumping between disparate regions with very
different values of $\chi^{2}$, therefore with a low acceptance rate of new
parameter sets (dashed line in figure 5). This causes a poor exploration of
parameter space. In contrast, a small value over-explores only a restricted
portion of $\\{P_{i}\\}$, falling very often in local minima of the parameter
space (dotted line in the same figure). Also choosing parameter jumps
proportional to the final parameters leads to a poor exploration of parameter
space (solid line in the same figure). Finally, after the same number of
steps, when using the optimized parameter step sizes obtained by the algorithm
the $\chi^{2}$ PDF follows the theoretical expectation, meaning that the
parameter space is correctly sampled.
Figure 5: The dashed line represents a chi-square distribution for three
parameters, i. e. $m=3$ (see text for details). Solid line is the obtained PDF
associated to $\Delta\chi^{2}$ when calculated for $10^{5}$ steps. Circles
represent the same distribution when calculated using only $10^{4}$ steps. The
inset shows the $\chi^{2}$ PDFs when calculated with parameters allowed to
change with $\Delta P_{i}=10^{-4}$, $\Delta P_{i}=10$, $\Delta P=0.1P_{i}$.
Successive PDFs are displaced on the ordinate axis for clarity of the figure.
### III.2 Fitting in a complex $\chi^{2}$ landscape
As pointed out before, one of the main problems when dealing with data fitting
using the LM algorithm is to find a proper set of initial parameters close
enough to the global minimum of the $\chi^{2}\\{P_{i}\\}$ hypersurface. As an
example we show in figure 6 the function $\sin(x/W)$ for $W=5$ affected by a
normal distributed error with $\sigma=0.1$. In figure 7(a) we show the
$\chi^{2}\\{W\\}$ landscape associated to the generated function. As it can be
seen, the $\chi^{2}\\{W\\}$ landscape for this function has a great number of
local minima and a global minimum at $W=5$. We have fitted the function using
the LM algorithm and initializing the parameter at $W_{i}=2$ and $W_{i}=15$
(see figure 6). As expected, both fits were not able to find the global
minimum that fits the function. In fact only if the LM algorithm is
initialized between $W=3.6$ and $W=9.0$ it is able to succeed in fitting the
data.
Figure 6: (color online) Synthetic $\sin(x/5)$ function (circles) together
with the best fit using parameter step sizes tuning together with simulated
annealing (line). Dashed lines are the fits using the LM algorithm with
starting parameters $W_{i}=2$ and $W_{i}=15$.
We now test the ability of our algorithm to jump across $\chi^{2}$ barriers
delimiting successive local minima to find the global one. For this task we
have used the simulated annealing method, decreasing the temperature one
decade every 3000 steps from $T=1000$ to $T=1$. The parameter jump calculation
has been performed every $N=1000$ steps. While the initial temperature allows
to explore wide regions of the parameter space, the last temperature will let
the acceptance be determined only by the real errors of the data.
In figure 7(b) we show the parameter $W$ as a function of algorithm step for
the two aforementioned initializations together with the $\chi^{2}$ landscape
(a). Parameter step sizes were initialized after a first run of optimization
of 2000 steps. As can be seen in this figure, after 3000 steps both runs have
already reached the absolute $\chi^{2}$ minimum. Successive steps just relax
the system to the final temperature $T=1$.
As it can be seen in figure 7, the way the minimum is reached depends on the
parameter initialization. Parameter step sizes are larger for the run started
with $W_{i}=15$ with a flat local minimum. The contrary happens with the run
initialized at $W_{i}=2$, parameter step sizes are set small due to the narrow
wells of the $\chi^{2}$ landscape in this region. However, both runs are able
to avoid getting stuck in local minima, jumping over rather high $\chi^{2}$
barriers and successfully reaching the best fit.
Figure 7: (color online) (a) $\chi^{2}\\{W\\}$ landscape obtained for the
function $\sin(x/W)$ with a normal error associated of $\sigma=0.1$ (see
figure 6). (b) Algorithm steps for two different initializations , black solid
line for $W_{i}=2$ and red dashed line for $W_{i}=15$, as a function of
parameter $W$
## IV Conclusion
Classical fit schemes are known to fail when the parameters are not
initialized close enough to the final solution. We have proposed in this work
to use an Adaptive Markov Chain Monte Carlo Through Regeneration scheme,
adapted from that of Gilks et al. Gilks1998 , combined with a simulated
annealing procedure to avoid this problem.
The proposed algorithm tunes the parameter step size in order to assure that
all of them are accepted in the same proportion. Geometrically the parameter
step size is set large when a cut of $\chi^{2}\\{P_{i}\\}$ along this
parameter is flat, i. e. when the change of the $\chi^{2}\\{P_{i}\\}$
hypersurface along this parameter is sloppy. Similarly the parameter step size
is set small if $\chi^{2}\\{P_{i}\\}$ wells are narrow.
Moreover, the step sizes can be modulated by a temperature added to the
acceptance equation that makes jumps across $\chi^{2}$ barriers easier, i. e.
using a simulated annealing method Kirkpatrick1984 . From a geometric point of
view, a high temperature makes the $\chi^{2}\\{P_{i}\\}$ wells artificially
broader, smearing out details of local minima. This is important at the first
stages of a fit process. At final stages of the fitting, temperature is
decreased, making parameter jumps smaller, and thus allowing the system to
relax, once it is inside the global minimum.
By fitting simulated data including statistical errors we verified that our
algorithm actually fulfills the requirements of ergodicity (it converges to
the target distribution), robustness (the ability to reach the $\chi^{2}$
minimum independent of the choice of starting parameters), ability to escape
local minima and to explore efficiently the $\chi^{2}$ landscape, and
guarantee that it will self tune to converge to the global minimum avoiding an
infinite search with large steps.
More complex problems have already successfully been studied with this
algorithm such as model selection using Quasielastic Neutron Scattering data
QENS , non-functional fits in the case of dielectric spectroscopy dielectric
or finding the molecular structure from diffraction data with a model defined
by as many as 27 parameters freon . In the last case, the proper
initialization of parameters to use a LM algorithm would have been a difficult
task, made easy by the use of the presented algorithm.
## V Acknowledgments
This work was supported by the Spanish Ministry of Science and Technology
(FIS2008-00837) and by the Catalonia government (2009SGR-1251). We would also
like to thank helpful comments and discussions on the manuscript made from K.
Kretschmer,Anand Patil and Christopher Fonnesbeck and A. Font.
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* (19) J. J. Waterfall, F. P. Casey, R. N. Gutenkunst, K. S. Brown, C. R. Myers, P. W. Brouwer, V. E. and J. P. Sethna, Phys. Rev. Lett. 104 060201 (2010)
* (20) FABADA program (Fit Algortihm for Bayesian Analysis of DAta) can be found in http//fisicaetseib.upc.es/gcm/members/lcpardo/software
* (21) L. C. Pardo, M. Rovira-Esteva, S. Busch, M. D. Ruiz-Martín, J. Ll. Tamarit, FABADA: a Fitting Algorithm for Bayesian Analysis of DAta. J. Phys.: Conf. Ser. Conference proceeding of the Spanish Neutron Scattering Society Meeting 2010. Accepted.; L. C. Pardo, M. Rovira-Esteva, S. Busch, M. D. Ruiz-Martín, J. Ll. Tamarit, and T. Unruh, arXiv:0907.3711
* (22) J. J. Mortensen, K. Kaasbjerg, S. L. Frederiksen, J. K. Nørskov, J. P. Sethna, and K.W. Jacobsen Phys. Rev. Lett. 95 216401 (2005)
* (23) Schulte J, Phys. Rev. E 53 R1348 (1996)
* (24) P. G. Debenedetti, F. H. Stillinger, Nature, 410 (2001)
* (25) M. Rovira-Esteva, A. Murugan, L. C. Pardo, S. Busch, M. D. Ruiz-Martín, M.-S. Appavou, J. Ll. Tamarit, C. Smuda, T. Unruh, F. J. Bermejo, G. J. Cuello, and S. J. Rzoska, Phys. Rev. B 81 092202 (2010); S. Busch, C. Smuda, L. C. Pardo, T. Unruh, J. Am. Chem. Soc., 132 3232 (2010)
* (26) J. C. Martinez-Garcia, J. Ll. Tamarit, L. C. Pardo, M. Barrio, S. J. Rzoska and A. Droz-Rzoska, J. Phys. Chem. B 114 6099 (2010)
* (27) M. Rovira-Esteva, N. A. Murugan, L. C. Pardo, S. Busch, J. Ll. Tamarit, Sz. Pothoczki, G. J. Cuello, and F. J. Bermejo, Phys. Rev. B 84, 064202 (2011)
|
arxiv-papers
| 2011-10-10T14:05:55 |
2024-09-04T02:49:22.968732
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "L.C. Pardo, M. Rovira-Esteva, S. Busch, J.-F. Moulin, J.Ll. Tamarit",
"submitter": "Luis Carlos Pardo",
"url": "https://arxiv.org/abs/1110.2046"
}
|
1110.2255
|
# The fluctuating $\alpha$-effect and Waldmeier relations in the nonlinear
dynamo models
V.V. Pipin1-3 and D.D. Sokoloff3,4 1Institute of Solar-Terrestrial Physics,
Russian Academy of Sciences,
2 Institute of Geophysics and Planetary Physics, UCLA, Los Angeles, CA 90065,
USA
3NORDITA, Roslagstullsbacken 23, 106 91 Stockholm, Sweden
4Department of Physics, Moscow State University, Moscow, 119991, Russia
###### Abstract
We study the possibility to reproduce the statistical relations of the sunspot
activity cycle, like the so-called Waldmeier relations, the cycle period -
amplitude and the cycle rise rate - amplitude relations, by means of the mean
field dynamo models with the fluctuating $\alpha$-effect. The dynamo model
includes the long-term fluctuations of the $\alpha$-effect and two types of
the nonlinear feedback of the mean-field on the $\alpha$-effect including the
algebraic quenching and the dynamic quenching due to the magnetic helicity
generation. We found that the models are able to reproduce qualitatively and
quantitatively the inclination and dispersion across the Waldmeier relations
with the 20% fluctuations of the $\alpha$-effect. The models with the dynamic
quenching are in a better agreement with observations than the models with the
algebraic $\alpha$-quenching. We compare the statistical distributions of the
modeled parameters, like the amplitude, period, the rise and decay rates of
the sunspot cycles, with observations.
## 1 Introduction
It is observed that the sunspot’s activity is organized in time and latitude
and forms the large scales patterns which are called the Maunder butterfly
diagram. This pattern is believed to be produced by the large-scale toroidal
magnetic field generated in the convection zone. Another component of the
solar activity is represented by the global poloidal magnetic field extending
outside the Sun and shaping the solar corona. Both components synchronously
evolve as the solar 11-year cycle progresses. The global poloidal field
reverses the sign in the polar regions near the time of maximum of sunspot
activity.
A remarkable feature of cyclic solar activity is that it is far to be just a
cycle. Cycle amplitude and shape varies from one cycle to the other and
prognostic abilities of any study of solar activity looks as its very
attractive destination. Solar activity observations give various hints that
various tracers of solar activity which are exploited to quantify the
phenomenon demonstrate some relation one to the other what opens a possibility
to predict future evolution of solar activity basing on available observations
of other indices. Waldmeier [36] pointed out at first this option (an inverse
correlation between the length of the ascending phase of a cycle, or its "rise
time", and the peak sunspot number of that cycle) and applied it, [37], to
give a prediction for the following cycle. The latter paper is in practice the
first accessible (at least for German speaking readers) paper in the area.
Later other relation of this this type was suggested and summarized as
Waldmeier relations. This development was clearly summarized by [35] and
recently by [12]. The nature of the physical processes, that are manifested in
the Waldmeier relations, is not clear, see discussion, e.g., in [6, 10, 8]. It
seems to be remarkable, however, that these statistical properties of magnetic
activity are also existed for the other tracers related with the sunspot
activity (e.g., sunspot group and squares of sunspot groups, see [35, 12, 8]),
and even for the other kind of the solar and stellar activity indices, e.g.,
for the Ca II index [31]. The Waldmeier relations are considered as a valuable
test of the dynamo models [15, 8, 29].
A natural way to push the understanding of the problem forward is to clarify
the physics underlying Waldmeier relations. It is more or less accepted that
cyclic solar activity is driven by a dynamo, i.e. a mechanism which transforms
kinetic energy of hydrodynamical motions into magnetic one. Most of the
current solar dynamo models suggest that the toroidal magnetic field that
emerges on the surface and forms sunspots is generated near the bottom of the
convection zone, in the tachocline or just beneath it in a convection
overshoot layer (see, e.g., [32]). This kind of dynamo can be approximated by
the Parker’s surface dynamo waves [26]. The direction of the dynamo waves
propagation is defined by the Parker-Yoshimura rule [38]. It states that for
the $\alpha\Omega$ kind dynamo the waves propagates along iso-surfaces of the
angular velocity. The propagation process can be modified by the turbulent
transport (associated with the mean drift of magnetic activity in the
turbulent media by means turbulent mechanisms), by the anisotropic turbulent
diffusivity (see, [14]), and by meridional circulation [7]. A viewpoint, which
is an alternative to the Parker’s surface dynamo waves is presented by the
distributed dynamo with subsurface shear, e.g. [3]. The dynamo waves here
propagates along the radius in the main part of the solar convection zone,
[14]. The near surface activity is shaped by the subsurface shear. One more
option is the flux-transport dynamo, e.g. [7, 9].
In the context of dynamo theory, the Waldmeier relations have to be explained
by some mechanism which varies amplitude and shape of activity cycle and
fluctuations $\alpha$-effect are considered below as such mechanism. This idea
extend the approach proposed in [29] to explain these relations by changing
the magnitude of the $\alpha$-effect.
The physical idea underlying this mechanism can be presented as follows.
$\alpha$-coefficient is a mean quantity taken over ensemble of convective
vortexes. Number $N$ of the vortexes in solar convective shell is large
however much smaller then, say, the Avogardo number, so fluctuations being
proportional to $N^{-1/2}$ may be not negligible. Particular choice of $N$ is
obviously model dependent however if we take just for orientation $N=10^{4}$
then $N^{-1/2}=0.01$. Taking into account that $\alpha$ is usually about 1/10
of turbulent velocity we consider a dozen percent of $\alpha$-fluctuations as
a comfortable estimate. From the other hand, governing equations for large-
scale solar magnetic field deal with spatial averaging and have to include a
contribution of $\alpha$-fluctuations, [13].
A straightforward application of the idea with vortex turnover time and vortex
size as correlation time and length for $\alpha$-fluctuations needs
fluctuations much larger then mean $\alpha$. [24], [34] based on experiences
in direct numerical simulations, e.g. [4], and results of current helicity
(related to $\alpha$) observation in solar active regions, e.g. [39]
considered $\alpha$-fluctuations with correlation time comparable with cycle
length and correlation length comparable with the extent of the latitudinal
belts where sunspots occur to conclude that a reasonable $\alpha$-noise of
order of few dozen percents is sufficient to explain Grand minima of solar
activity. The aim of this paper is to apply this idea to explain Waldmeier
relations.
## 2 Basic equations
### 2.1 2D model
The dynamo model is based on the standard mean-field induction equation in
perfectly conductive media [19]:
$\frac{\partial\mathbf{B}}{\partial
t}=\boldsymbol{\nabla}\times\left(\mathbf{\boldsymbol{\mathcal{E}}+}\mathbf{U}\times\mathbf{B}\right)$
where $\boldsymbol{\mathcal{E}}=\overline{\mathbf{u\times b}}$ is the mean
electromotive force, with $\mathbf{u,\,b}$ being the turbulent fluctuating
velocity and magnetic field respectively; $\mathbf{U}$ is the mean velocity
(differential rotation). The axisymmetric magnetic filed:
$\mathbf{B}=\mathbf{e}_{\phi}B+\nabla\times\frac{A\mathbf{e}_{\phi}}{r\sin\theta}$
$\theta$ \- polar angle. We have used the expression for
$\boldsymbol{\mathcal{E}}$ obtained by [27] (hereafter P08) and write it as
follows:
$\mathcal{E}_{i}=\left(\alpha_{ij}+\gamma_{ij}\right)\overline{B}_{j}-\eta_{ijk}\nabla_{j}\overline{B}_{k}.$
(1)
Tensor $\alpha_{i,j}$ represents the alpha effect, including the hydrodynamic
and magnetic helicity contributions,
$\alpha_{ij}=C_{\alpha}\left(1+\xi\right)\psi_{\alpha}(\beta)\sin^{2}\theta\alpha_{ij}^{(H)}+\alpha_{ij}^{(M)},$
(2)
where the hydrodynamical part of the $\alpha$-effect, $\alpha_{ij}^{(H)}$,
$\xi$ is the noise, and the quenching function, $\psi_{\alpha}$, are given in
Appendix (see also in [28]). The hydrodynamic $\alpha$-effect term is
multiplied by $\sin^{2}\theta$ ($\theta$ is co-latitude) to prevent the
turbulent generation of magnetic field at the poles. The contribution of the
small-scale magnetic helicity
$\overline{\chi}=\overline{\mathbf{a\cdot}\mathbf{b}}$ ($\mathbf{a}$ is a
fluctuating vector-potential of magnetic field) to the $\alpha$-effect is
defined as $\alpha_{ij}^{(M)}=C_{ij}^{(\chi)}\overline{\chi}$, where
coefficient $C_{ij}^{(\chi)}$ depends on the turbulent properties and
rotation, and is given in Appendix. The other parts of Eq.(1) represent the
effects of turbulent pumping, $\gamma_{ij}$, and turbulent diffusion,
$\eta_{ijk}$. They are the same as in PK11. We describe them in Appendix.
Figure 1: Parameters of the solar convection zone: a) the contours of the
constant angular velocity plotted for the levels $(0.75-1.05)\Omega_{0}$ with
a step of $0.025\Omega_{0}$, $\Omega_{0}=2.86\cdot 10^{-7}s^{-1}$; b) turnover
convection time $\tau_{c}$, and the RMS convective velocity $u^{\prime}_{c}$
and the background turbulent diffusivity $\eta_{T}^{(0)}$ profiles; c) the
radial profiles of the $\alpha$-effect tensor components.
The nonlinear feedback of the large-scale magnetic field to the
$\alpha$-effect is described as a combination of an “algebraic” quenching by
function $\psi_{\alpha}\left(\beta\right)$ (see Appendix and [29]), and a
dynamical quenching due to the magnetic helicity conservation constraint. The
magnetic helicity, $\overline{\chi}$ , subject to a conservation law, is
described by the following equation [18, 5, 33]:
$\displaystyle\frac{\partial\overline{\chi}}{\partial t}$ $\displaystyle=$
$\displaystyle-2\left(\boldsymbol{\mathcal{E}\cdot}\overline{\mathbf{B}}\right)-\frac{\overline{\chi}}{R_{\chi}\tau_{c}}+\boldsymbol{\nabla}\cdot\left(\eta_{\chi}\boldsymbol{\nabla}\bar{\chi}\right),$
(3)
where $\tau_{c}$ is a typical convection turnover time. Parameter $R_{\chi}$
controls the helicity dissipation rate without specifying the nature of the
loss. It seems to be reasonable that the helicity dissipation is most
efficient in the near surface layers because of the strong decrease of
$\tau_{c}$ (see Figure 1b). The last term in Eq.(3) describes the diffusive
flux of magnetic helicity [20]. We use the solar convection zone model
computed by [32], in which the mixing-length is defined as
$\ell=\alpha_{MLT}\left|\Lambda^{(p)}\right|^{-1}$, where
$\mathbf{\boldsymbol{\Lambda}}^{(p)}=\boldsymbol{\nabla}\log\overline{p}\,$ is
the pressure variation scale, and $\alpha_{MLT}=2$. The turbulent diffusivity
is parametrized in the form, $\eta_{T}=C_{\eta}\eta_{T}^{(0)}$, where
$\eta_{T}^{(0)}={\displaystyle\frac{u^{\prime}\ell}{3}}$ is the characteristic
mixing-length turbulent diffusivity, $\ell$ and $u^{\prime}$ are the typical
correlation length and RMS convective velocity of turbulent flows,
respectively and $C_{\eta}$ is a constant to control the intensity of
turbulent mixing. In the paper we use $C_{\eta}=0.05$. The differential
rotation profile, $\Omega=\Omega_{0}f_{\Omega}\left(x,\mu\right)$,
$x=r/R_{\odot}$, $\mu=\cos\theta$ is a modified version of an analytical
approximation to helioseismology data, proposed by [2], see Fig. 1a.
Figure 2: The typical time-latitude and the time-radius (at the $30^{\circ}$
latitude) diagrams of the toroidal field (grey scale), the radial field
(contours at left panel) and the poloidal magnetic field (contours at the
right panel) evolution in 2D1 model (see Table 1). The toroidal field averaged
over over the subsurface layers in the range of $0.9-0.99R_{\odot}$ , the
radial field is taken at the top of the convection zone.
We use the standard boundary conditions to match the potential field outside
and the perfect conductivity at the bottom boundary. As discussed above, the
penetration of the toroidal magnetic field in to the near surface layers is
controlled by the turbulent diffusivity and pumping effect. For magnetic
helicity, similar to [11] and [21], we use the time dependent conditions
provided be Eq.3 and the helicity flux conservation the condition
$\boldsymbol{\nabla_{r}}\bar{\chi}=0$ is applied at the bottom and at the top
of domain. The latter gives a smooth transfer for solutions with and without
the diffusive helicity flux.
The left panel on the Fig. 2 shows the typical the time-latitude diagram for
the toroidal magnetic averaged over the subsurface layers $0.9-0.99R_{\odot}$
and the radial magnetic at the top of the integration domain. The right panel
shows the time-radius the time-radius diagram for the toroidal an poloidal
magnetic field evolution at $30^{\circ}$ latitude.
We demonstrate it by Fig. 3 which shows the time-latitude diagrams for
toroidal and radial magnetic field evolution for the models 1D1, 1D3 and 2D1.
For the latter model we show the toroidal magnetic averaged over the
subsurface layers $0.9-0.99R_{\odot}$ and the radial magnetic field is given
for the top of the integration domain. For the model 2D1 we show the time-
radius diagram for the toroidal an poloidal magnetic field evolution at
$30^{\circ}$ latitude. The other models listed in Table 1, having the same
general patterns of the magnetic field evolution, are differed from the models
shown on the Fig. 3 in some details (mostly associated with magnetic helicity
evolution).
### 2.2 1D model
For comparison with the previous studies and also to study how the additional
dimension affect the statistical properties of the dynamo we consider the $1D$
model similar to that studied by [24]:
$\displaystyle\frac{\partial A}{\partial t}$ $\displaystyle=$
$\displaystyle\sin\theta\left(\left(1+\xi\right)\cos\theta\psi_{\alpha}(B)+\chi\right)B+\sin\theta\frac{\partial}{\partial\theta}\left(\frac{1}{\sin\theta}\frac{\partial
A}{\partial\theta}\right)-\eta_{CZ}A,$ (4) $\displaystyle\frac{\partial
B}{\partial t}$ $\displaystyle=$
$\displaystyle-\mathcal{D}\tilde{\Omega}\left(\theta\right)\frac{\partial
A}{\partial\theta}+\frac{\partial}{\partial\theta}\left(\frac{1}{\sin\theta}\frac{\partial\sin\theta
B}{\partial\theta}\right)-\eta_{CZ}B,$ (5)
where the large-scale radial shear
$\tilde{\Omega}\left(\theta\right)=\partial\Omega/\partial r$. The $1D$ model
employs two possibilities for the shear profile. In one case we put
$\tilde{\Omega}\left(\theta\right)=1$, that give us the model explored by
[24]. In another case we use
$\tilde{\Omega}=\frac{1}{10}\left(5\sin^{2}\theta-4\right),$ (6)
which is suggested by [16]. In agreement with the helioseismology results for
the bottom of the convection zone, this profile is positive in equatorial
regions and negative near the poles. The magnetic field strength in Eq.(5) is
measured in the units of the equipartition magnetic field strength and the
time is normalized to the typical diffusive time,
$R_{\odot}^{2}/\eta_{T}^{(0)}$. The evolution of the magnetic helicity for the
1D model is governed by equation:
$\displaystyle\frac{\partial\overline{\chi}}{\partial t}$ $\displaystyle=$
$\displaystyle-2\left((1+\xi)\cos\theta\psi_{\alpha}(B)+\chi\right)B^{2}-2B\frac{\partial}{\partial\theta}\left(\frac{1}{\sin\theta}\frac{\partial
A}{\partial\theta}\right)$ $\displaystyle+$
$\displaystyle\frac{2}{\sin^{2}\theta}\frac{\partial
A}{\partial\theta}\frac{\partial\sin\theta
B}{\partial\theta}-\frac{\overline{\chi}}{R_{\chi}}+\frac{\eta_{\chi}}{\sin\theta}\frac{\partial}{\partial\theta}\left(\frac{1}{\sin\theta}\frac{\partial\bar{\chi}}{\partial\theta}\right).$
In what follows we will discuss the 1D models with the constant shear, because
they are more relevant to compare with observations. The differences in
results for the 1D models with the variable shear given by Eq.(6) will be
briefly mentioned in subsequent sections.
Figure 3: The left panel shows the time-latitude diagrams of the toroidal
field (grey scale) and the radial field (contours) for the 1D1 model (see
Table 1). The right panel shows the estimated sunspot number in the the
separated cycles in the 1D1 model (see Section 2.4).
Summarizing, we exploit much more detailed and realistic dynamo models then
[24], [34]. Our point is that Waldmeier relations are a much more delicate
phenomena rather Grand minima and the bulk of our knowledge concerning recent
solar cycles is much more rich then that one for remote past when Grand minima
took place.
### 2.3 Noise model
The noise, $\xi$, contributes in the hydrodynamic part of the $\alpha$-effect
(see, Eqs.(2,4)). Following to [34] the models employ the long-term Gaussian
fluctuating $\xi$ of the small amplitude with RMS deviation given in the Table
1 (last column). The time of the renewal of the $\xi$ is equal to the period
of the model. The random numbers were generated with help of the standard F90
subroutine quality of contemporary standard noise generator subroutine is
shown to be sufficient for such kind of modelling, see e.g. [1]). It would be
more realistic to consider the renewal time as the fluctuating quantity as
well, but we would like to separate this effect for the different study. Also,
we found that the models which employ the magnetic helicity effect show the
very intermittent long term behaviour. This makes the analysis procedure
(e.g., division to subsequent cycles) more complicated. We isolate ourselves
from these phenomena by considering the noise models with the lower RMS in
case if the magnetic helicity is employed. 111In part, the given problem is
likely due to the very rough model for the Wolf number, see Eq.(8).
Model | $\eta_{CZ}$ | $\overline{\chi}$ | $\eta_{\chi}/\eta_{T}$ | $R_{\chi}$ | $B_{0}$ | $C_{W}$ | $\sigma$
---|---|---|---|---|---|---|---
1D1 | 1 | 0 | 0 | 10 | 3 | 1200 | 0.15
2D1 | | Eq.(2) | $10^{-5}$ | 200 | 800 | 1 | 0.15
2D2 | | Eq.(2) | $0.3$ | $10^{6}$ | 200 | 1 | 0.15
Table 1: Parameters the dynamo models: the type of the nonlinear quenching of
the $\alpha$-effect, if the magnetic helicity is $\overline{\chi}=0$ then the
model employ only the algebraic quenching which is described by
$\psi_{\alpha}$and otherwise by the dynamic quenching due to magnetic helicity
described by Eq.(3) or Eq.(2.2); $\eta_{\chi}/\eta_{T}$ is the ratio between
the turbulent magnetic helicity diffusivity and the turbulent magnetic
diffusivities; the profile of the shear in the 1D models; the $\alpha$-effect
parameter in the 2D models; the parameter $R_{\chi}$ controls the helicity
dissipation rate; the parameter $B_{0}/B_{{\rm eq}}$ controls the sunspot
number parameter in the 1D models. It is the ratio between the typical
strength of the toroidal magnetic field producing the sunspots and the
equipartition magnetic field strength; $B_{0}$ is the typical strength of the
toroidal magnetic field controlling the sunspots number parameter in the 2D
models; $C_{W}$ is the parameter to calibrate the modeled sunspot number
relative to observations; $\sigma$ is the standard deviation of the Gaussian
noise in the model
### 2.4 The sunspot cycle model and the Waldmeier relations
In the paper we define the Waldmeier relations as the set of the mean
properties of the sunspot cycle. We will deal with the following properties of
the Wolf sunspot number (which is taken either from observational database or
simulated from the model): the relation between period and amplitude of the
same cycle, the relation between rise rate and amplitude of the cycle and the
shape of the sunspot cycle, characterized by the ratio between the decay rate
and the rise rate in the cycle. The other kind of relations, like the link
between the rise time and amplitude of the cycle, can be considered as the
derivative from the above relation. For comparison with other analysis of the
observational data and also with the results of the dynamo models presented by
Karak and Choudhuri[8] we show the results for the rise time of the cycles as
well, the relation between the rise time and amplitude of the cycle and the
relation between the cycle amplitude and period of the preceding cycle (see,
[35] and [12]). The amplitude of the cycles is defined by difference between
the maximum sunspot number and the sunspot number in the preceding minimum.
Even for the harmonic cycles the latter differs from zero due to the spatial
overlap in subsequent cycles. The period of the cycle is equal to the time
between the subsequent minima, the rise time of the cycle is defined by the
difference between the moment of the cycle maximum and the moment of the
preceding minimum of the cycle. The rise rate is defined as the ratio between
the difference of the sunspot number amplitude during maximum and minimum of
the cycle and the rise time of the cycle. The similar definition is for the
decay rate of the cycle.
Remind that sunspots are not directly presented in dynamo models and we have
to relate its number to a quantity involved in a dynamo model under
consideration. We assume that the sunspots are produced from the toroidal
magnetic fields by means of the nonlinear instability and avoid to consider
the instability in details. To model the sunspot number $W$ produced by the
dynamo we use the following anzatz
$W\left(t\right)=C_{W}\left\langle B_{{\rm
max}}\right\rangle_{SL}\exp\left(-\frac{B_{0}}{\left\langle B_{{\rm
max}}\right\rangle_{SL}}\right),$ (8)
where for the 2D models $\left\langle B_{{\rm max}}\right\rangle_{SL}$ is the
maximum of the toroidal magnetic field strength over latitudes averaged over
the subsurface layers in the range of $0.9-0.99R_{\odot}$ and for the 1D
models $\left\langle B_{{\rm max}}\right\rangle_{SL}$ is simply the maximum of
the toroidal magnetic field strength over latitudes; $B_{0}$ is the typical
strength of the toroidal magnetic field that is enough to produce the sunspot;
$C_{W}$ is the parameter to calibrate the modeled sunspot number relative to
observations. The all parameters which were employed in the different models
are listed in the Table 1.
In the dynamo models we explore the effect of the Gaussian fluctuations of the
$\alpha$-effect, or parameter $C_{\alpha}$ with the typical time equal to the
period of the cycle and the standard deviations less than $0.2C_{\alpha}$. In
the models presented here we fix the standard deviation to $0.15C_{\alpha}$.
| 1D1 | 2D1 | 2D2 | SIDC | NIMV (2004)
---|---|---|---|---|---
Period | 11.02$\pm$0.66 | 11.07$\pm$1.08 | 10.97$\pm$0.92 | 11.01$\pm$1.12 | 11.02$\pm$1.49
Amplitude | 115.7$\pm$33.6 | 103.3$\pm$40.5 | 96.3$\pm$25.7 | 108.2$\pm$38.1 | 87.6$\pm$43.9
Rise Rate | 18.62$\pm$6.14 | 25.39$\pm$11.95 | 19.91$\pm$5.95 | 25.81$\pm$12.74 | 19.48$\pm$13.38
Rise Time | 6.11$\pm$.33 | 4.06$\pm$.77 | 4.73$\pm$.36 | 4.32$\pm$1.07 | 4.82$\pm$1.32
Shape | 1.27$\pm$0.2 | .59$\pm$0.15 | .77$\pm$0.08 | .69$\pm$0.31 | .83$\pm$0.34
Rise Rate - Amplitude | 5.4x+14.2$\pm$3.0 0.99 | 3.3x+18.8$\pm$7.6 0.98 | 4.2x+12.4$\pm$5.6 0.98 | 2.9x+33.2$\pm$8.9 0.97 | 3.1x+27.8$\pm$15.7 0.93
Period - Amplitude(a) | -31.7x+463.9 $\pm$26.2 -0.63 | -17.5x+298.0 $\pm$34.0 -0.54 | -17.25x+2856 $\pm$20.3 -0.62 | -23.6x+368.5 $\pm$28.0 -0.68 | -8.4x+179.9 $\pm$42.0 -0.29
Period - Amplitude(b) | -17.9x+312.3 $\pm$31.4 -0.35 | -8.9x+202.9 $\pm$38.9 -0.28 | -6.3x+165.4 $\pm$25.0 -0.22 | -11.2x+231.7 $\pm$35.9 -0.33 | -6.9x+163.4 $\pm$42.7 -0.23
Rise Time - Amplitude | -82.1x+617.4 $\pm$18.3 -0.84 | -25.6x+207.5 $\pm$35.3 -0.49 | -33.0x+252.8 $\pm$22.7 -0.47 | -26.7x+234. $\pm$24. -0.75 | -16.1x+165.4 $\pm$38.5 -0.48
Rise Rate - Decay Rate | 1.0x+4.0$\pm$3.1 0.9 | 0.43x+3.3$\pm$2.2 0.92 | 0.68x+1.6$\pm$1.6 0.93 | 0.34x+6.4$\pm$2.6 0.85 | 0.42x+5.3$\pm$4.1 0.81
Table 2: First five rows contain information for the mean and variance
(standard deviation) for the parameters of the sunspot cycles in the different
data set. The shape of the cycle is defined as ratio between the decay rate
and the rise rate of the cycle. Last five rows show the linear fits with the
mean-square error bar and the correlation coefficient. In the relation Period-
Amplitude (a) we compare the cycle amplitudes to period of the _preceding_
cycle (see [12, 35]), and in the relation Period-Amplitude (b) we compare
these parameters for the _same_ cycle.
For comparison with simulation we use the smoothed data set from [30] which
starts at 1750. Choosing this data set we appreciate that in principle
Waldmeier relations can be valid for normal cycle only and their applicability
to epochs of Grand minima of solar activity must be addressed separately.
Available instrumental data concerning solar activity in XVII - early XVIII
centuries gives a limited possibility only to address this important point
which obviously is out of the scope of this paper. From the other hand, there
are various indirect (mainly isotopic) tracers of solar activity which give a
limited information concerning its shape over much longer time interval rather
instrumental data. Our point is that Waldmeier relations and the regularities
of such long-term time series (see, e.g.,[23, 22]) have to be discussed in a
separate paper and here use as an illustrative example the extended time
series of the sunspot data proposed by [25] (referred hereafter as NIMV).
These data sets are shown on Fig. 4. The Table 2 contains the linear fits and
correlations between the different parameters of the cycles for observational
data sets and for the dynamo models as well. In particular, the parameters of
the relation between rise time and amplitude and parameters of the Amplitude-
Period effect (a) and (b) (associated with period of the _preceding_ and the
_same_ cycle) for SIDC data set are in a good agreement with the results by
Vitinskij et al [35] and Hathaway et al[12]. The similar conclusion can be
done if we compare our analysis for SIDC data set for the relation between
rise rate and amplitude of the cycles with analysis given by Vitinskij et al
[35].
Figure 4: The sunspot data sets. Upper raw: left \- SIDC and right -
[25](NIMV), lower raw - corresponding cycles distributions
## 3 Results
The typical time-latitude diagrams for the dynamo models were shown in Figures
2 and 3. The shape of the simulated sunspot cycles in 1D1 model can be seen on
the right panel Figure 3. The simulated sunspot cycles for the 2D1 and 2D2
models are shown on the the Figure 5. We can conclude that the shape of the
simulated sunspot cycles (and, perhaps, the associated Waldmeier relations) is
directly related with the spatial shape of the toroidal magnetic field
evaluational patterns. For example, in the 1D1 model the maximum of the
butterfly diagram is very close to equator and butterfly wing is elongated
toward the pole. In such a pattern of the toroidal magnetic field evolution
the decay phase of the sunspot activity is shorter than the rise phase. The
opposite situation is in the models 2D1 and 2D2. The physical mechanisms which
produce the short rise and the long decay of the toroidal magnetic field
activity were discussed recently by Pipin and Kosovichev [29].
Figure 5: Left panel shows cycles distributions for the model 2D1 and the
right panel - model 2D2.
To proceed further we would like to discuss the statistical properties of the
cycle parameters those involved in the Waldmeier relations. The 1D models have
the much less cycle period than diffusive time of the system. Therefore, we
scale the periods of these models by factor $\sim 50$ . The Table 2 show the
results for the mean and the variance (standard deviations) for the period,
amplitude, rise rate and the shape of the sunspot cycles in the different data
sets. From that Table we see that the 1D1 model has the smaller variance in
the period, amplitude and rise rate of the cycles as compared to the others
data sets. The shape asymmetry of the cycles in 1D1 is opposite to the others
cases as well. Also we can see that the mechanism of the helicity loss in the
dynamo model influences the mean and variance of the sunspot cycles
parameters. In particular, the model 2D2 with the increased diffusive loss of
the magnetic helicity has the lower variance of the period and amplitude of
the sunspot cycles and has the more symmetric shape of the cycle as compared
to the model 2D1. The difference in the synthetic data set of the sunspot
cycles provided by NIMV as compared with the SIDC is likely due to the fact
that the SIDC data set does not cover the periods with low magnetic activity.
This argument is also applied if we compared NIMV and, e.g., 2D1 model. The
parameters of the 2D1 model does not allow to have the extended periods of
time with very low sunspot cycles.
Figure 6: CDF distributions, red line - SIDC the data set, blue line - the
data set from [25].
The difference of the the statistical properties of the given data set can be
seen in further detail using the cumulative distribution probability
functions. The cumulative distributions are constructed as follows. At the
beginning, we sort each distribution for each parameter and each model in
increasing order. After this we compute the following
$\mathrm{CDF}(P_{i})=\frac{\sum_{k=1}^{i}k}{N},$ (9)
where $P_{i}$ is the parameter under consideration (say, the cycle period)
having the order number $i$ (after sorting the set in increasing order) and
$N$ is the total number of the instances of the given parameter in the set.
Equation (9) approximate the probability for the parameter $P$ to have the
values in interval between $P_{\rm min}$ and $P_{i}$. The accuracy of the
approximation improves under $N\rightarrow\infty$. We will use the log-normal
cumulative distribution constructed on the base of the SIDC data set as the
reference distribution. The SIDC data set has only 23 instances of the sunspot
cycles. To construct the reference log-normal distribution we use the standard
mean and variance of the cycles parameters (period, amplitude, rise rate and
asymmetry) given in the Table 2. Then take the natural logarithm of them and
construct the log-normal distribution of the length 1000 using those mean and
variance. The results are shown on the Fig. 6.
It is clearly seen that log-normal distribution is a good fit for the
distributions of the sunspot cycles period in the SIDC data set and also for
model 2D2. The difference of the SIDC data set from the log-normal
distribution is seen in the probabilities distributions for the rise rates and
the shape of the cycles. It is, however, unclear if these differences is due
to the limited data set of cycles covered by SIDC. The data set produced by
the models and the NIMV data set can be equally well approximated by the log-
normal distributions (with different mean and variance). For the dynamo
models, the difference between the distributions computed by Eq.(9) and the
log-normal approximations for them is less visible than for SIDC and NIMV
sets.
Figure 7: The Waldmeier relations for 1D1 (left) and 2D1 (right) models. The
linear fits are shown the solid lines, the dashed lines shows the fits for the
SIDC data and the dash-dot line - for the NIMV data.
Fig. 7 shows the Waldmeier relations for the 1D1 and 2D1 models together with
their linear fits and also fits for the SIDC and NIMV data sets. The
parameters of the linear fits are summarised in the Table 2. It is seen that
the model 2D1 is well to reproduce the SIDC data set, and the difference to
the NIMV data is not very large. The correspondence of the 2D2 model to the
SIDC and the NIMV is not as good as for the 2D1 model. This is also can be
expected by results presented in Fig. 6 and by Table 2. Finally, we can
conclude that 1D1 model has only qualitative agreement for the relations
between the rise rate - amplitude, and the period - amplitude of the sunspot
cycles.
## 4 Discussion and Conclusions
In the paper we have studied the possibility to reproduce the statistical
relations of the sunspot activity cycle, like the so-called Waldmeier
relations, by means of the mean field dynamo model with the fluctuating
$\alpha$-effect. The dynamo model includes the long-term fluctuations of the
$\alpha$-effect. The dynamo models employ two types of the nonlinear feedback
of the mean-field on the $\alpha$-effect including the algebraic quenching and
the dynamic quenching due to the magnetic helicity generation. The paper
presents the results for three particular dynamo models.
The presented 1D model is similar to model discussed by Moss et al. [24]. It
uses the constant shear and the algebraic quenching of the $\alpha$-effect.
The results for this model disagree with observations (SIDC data set) about
the shape of the simulated sunspot (decay rate is higher than rise rate) even
though it is qualitatively reproduce the basic Waldmeier relations for the
Rise Rate-amplitude and the cycle Period-amplitude (see left column in Fig.
7). It was found that the variance of the cycle parameters in the long-term
evolution is less than in 2D models. It is interesting, that under the level
of noise the 1D models involving the magnetic helicity show the smaller mean
even though having the stronger variances of the simulated sunspot parameters.
Although we could scale the mean parameters of those models to the
observational values, we did not present the results for these models because
they have the Waldmeier relations which are quantitatively the same as those
presented for 1D1 model in Table 2 and Fig. 7.
We checked the 1D models with the spatially variable shear like that suggested
by Kitchatinov et al. [16]. In agreement with the helioseismology results, the
given 1D models have the realistic latitudinal profile of the shear (see
Eq.(6)). Although, these models qualitatively reproduce the relation between
the rise rate and amplitude of the cycle, they fail with the other kind of
relations, having the positive correlation between the period and amplitude of
the cycle and the equal rate for the rise and decay phase of the simulated
sunspot cycles.
Similar to the 1D cases the magnetic helicity contribution to the
$\alpha$-effect results to decrease of the toroidal magnetic field strength
and to growth the variance of the cycle parameters in the long-term evolution
of the magnetic activity. The strong variance of the cycle parameters is
expected from SIDC data set and from NIMV as well. For this reason in the
paper we discuss the 2D model which involves the effect of the magnetic
helicity. The 2D models employ two different description for the magnetic
helicity loss, to overcome the problem of the $\alpha$-effect catastrophic
quenching. The term $-\overline{\chi}/R_{\chi}\tau_{c}$ in Eq.(3) describes
the magnetic helicity loss with the dissipation rate $(\tau_{c}R_{\chi})^{-1}$
without specifying the nature of the loss. Note, that $\tau_{c}$ is varied
from about 2 months near bottom of the convection zone to a few hours at the
top of the integration domain (which is $0.99R_{\odot}$). Thus, for the
$R_{\chi}=200$ used in the model 2D1, the typical decay time for the magnetic
helicity is varied from about 4 solar cycles at the bottom of the convection
zone to a time which is less than one month at the top of the convection zone.
It is not clear if this simple description is satisfactory approximation for
the magnetic helicity loss. Therefore we checked the alternative possibility
using the diffusive helicity flux. Although, the model that employ the
diffusive helicity flux is in satisfactory agreement with SIDC data, the
correspondence to observation in this model is not as good as for the model
2D1. We find the the variance of the cycle parameters in the model 2D2 is less
than in the model 2D1 while the SIDC and NIMV data sets show higher variances
than the model 2D1.
The detailed comparison the results of our models with those given by Karak
and Choudhuri [8] is not possible, because we have used a different definition
for the amplitude of the cycle and the rise time. They did not give the
results for the linear fits coefficients and only provide the correlation
coefficients in the Waldmeier relations involving the Rise Rate-amplitude and
the Rise Time-Amplitude of the cycle. Bearing in mind the differences in
definition that their “high diffusivity model” with fluctuating meridional
circulation is comparable with our 2D1 and 2D2 models. It is not clear however
what is the typical shape of the cycle in their model. This is an important
issue as we have seen in example given by model 1D1. It has qualitative
agreement with SIDC data about the period - amplitude and the rise rate -
amplitude relations even-though having the rise time of the cycle greater than
the decay time.
In the models under consideration, the asymmetry between the ascent and decent
phase of the sunspot cycle is inherent from the pattern of the toroidal
magnetic field activity. In particular, the 1D1 model has the toroidal
magnetic field butterfly diagram with maximum located very close to equator.
Therefore, applying the definition Eq.(8) for this type of the toroidal
magnetic field evolutional pattern we obtain the decent phase of the sunspot
activity shorter than the ascent phase. The opposite situation is in 2D
models. There, we relate the sunspot activity with the toroidal field in the
subsurface layers. The turbulent diffusivity in the model decrease outward
this leads to increases the decay time when the toroidal field gets closer to
the surface (see [29]). We find that the effect of the magnetic helicity on
the $\alpha$-effect can amplify or saturate the asymmetry of the cycle shape
depending on the mechanisms of the helicity loss employed in the model.
It is expected that the nonlinear dynamo mechanisms affect both the magnetic
cycle profile and the statistical properties of the cycles. The paper
illustrates the impact of the non-linear $\alpha$-effect for the algebraic and
the dynamic non-linearities. Recently, Kitchatinov & Olemskoy [17] suggested
that the non-linear diffusion could promotes the events similar to the Maunder
minimum provided there are the small fluctuations in the $\alpha$-effect. This
mechanism does not work in our models, because on the rise phase of the cycle,
the growing toroidal magnetic fields results to the turbulent diffusivity
quenching and this effect makes the rise phase of the cycle longer, i.e., the
smaller turbulent diffusion, the longer evolutionary time scale. The opposite
situation is expected for the decay phase of the magnetic cycle.
The comparison of the SIDC data set and the synthetic data set provided by
Nagovizyn et al. [25](NIMV) reveals the significant difference in the
statistical properties of the cycle parameters. This seems to be a result of
the wider cycle variations range covering by the NIMV data set. The model
presented in the paper don’t cover the variations seen in NIMV because the
selected models almost have no the extended events with low cycles like the
so-called Maunder minimum which were observed during the 16-th century. This
motivated us to extend our study and explore the models which have the more
intermittent variations of the sunspot cycle. This work is planned for the
future papers.
Summarizing the main findings of the paper we conclude as follows. We found
that the dynamo models, having the reasonably good the time-latitude diagram
of the toroidal magnetic field evolution, are able to reproduce qualitatively
the inclination and dispersion across the Waldmeier relations with less than
20% Gaussian fluctuations of the $\alpha$-effect. The 2D models have better
agreement with observations than 1D. In particular, 1D models fail to
reproduce the asymmetric shape of the sunspot cycle with short rise and long
decay phases. The statistical distributions of the cycle parameters show the
log-normal probability distributions for the all data sets analysed in the
paper. The parameters of these distributions are different for all data sets.
Again the 1D model is significantly different from others in this sense. The
2D model that employs the simplest form of the helicity loss via the term
$-\overline{\chi}/R_{\chi}\tau_{c}$ agrees well with the SIDC, even-though the
long-term variations in this model is not intermittent enough, and this seems
to be a reason for its difference to the NIMV data set in some aspects. The
employ of the diffusive loss in the magnetic helicity evolution equation
results to decreasing in the variations of the cycle parameters. The further
study of the magnetic helicity transport mechanisms should clarify the likely
candidates which are responsible for the magnetic helicity loss from the
dynamo region. We have seen that the analysis of the statistical relations of
the sunspot cycle may provide the valuable diagnostic tool for this study.
## Acknowledgements
The authors thanks the Nordita program "Dynamo, Dynamical Systems and
Topology" for the financial support. D.S. is grateful to RFBR for financial
support under grant 09-05-00076-a and V.P. thanks for the financial support
from NASA LWS NNX09AJ85G grant and for the partial support under RFBR grant
10-02-00148-a.
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## 5 Appendix
We describe some parts of the mean-electromotive force. The basic formulation
is given in P08. For this paper we reformulate tensor $\alpha_{i,j}^{(H)}$,
which represents the hydrodynamical part of the $\alpha$-effect, by using
Eq.(23) from P08 in the following form,
$\displaystyle\alpha_{ij}^{(H)}$ $\displaystyle=$
$\displaystyle\delta_{ij}\left\\{3\eta_{T}\left(f_{10}^{(a)}\left(\mathbf{e}\cdot\boldsymbol{\Lambda}^{(\rho)}\right)+f_{11}^{(a)}\left(\mathbf{e}\cdot\boldsymbol{\Lambda}^{(u)}\right)\right)\right\\}+$
$\displaystyle+$ $\displaystyle
e_{i}e_{j}\left\\{3\eta_{T}\left(f_{5}^{(a)}\left(\mathbf{e}\cdot\boldsymbol{\Lambda}^{(\rho)}\right)+f_{4}^{(a)}\left(\mathbf{e}\cdot\boldsymbol{\Lambda}^{(u)}\right)\right)\right\\}$
$\displaystyle+$ $\displaystyle
3\eta_{T}\left\\{\left(e_{i}\Lambda_{j}^{(\rho)}+e_{j}\Lambda_{i}^{(\rho)}\right)f_{6}^{(a)}+\left(e_{i}\Lambda_{j}^{(u)}+e_{j}\Lambda_{i}^{(u)}\right)f_{8}^{(a)}\right\\}.$
The contribution of magnetic helicity
$\overline{\chi}=\overline{\mathbf{a\cdot}\mathbf{b}}$ ($\mathbf{a}$ is a
fluctuating vector magnetic field potential) to the $\alpha$-effect is defined
as $\alpha_{ij}^{(M)}=C_{ij}^{(\chi)}\overline{\chi}$, where
$C_{ij}^{(\chi)}=2f_{2}^{(a)}\delta_{ij}\frac{\tau_{c}}{\mu_{0}\overline{\rho}\ell^{2}}-2f_{1}^{(a)}e_{i}e_{j}\frac{\tau_{c}}{\mu_{0}\overline{\rho}\ell^{2}}.$
(11)
The turbulent pumping, $\gamma_{i,j}$, is also part of the mean electromotive
force in Eq.(23)(P08). Here we rewrite it in a more traditional form (cf,
e.g., ),
$\gamma_{ij}=3\eta_{T}\left\\{f_{3}^{(a)}\Lambda_{n}^{(\rho)}+f_{1}^{(a)}\left(\mathbf{e}\cdot\boldsymbol{\Lambda}^{(\rho)}\right)e_{n}\right\\}\varepsilon_{inj}-3\eta_{T}f_{1}^{(a)}e_{j}\varepsilon_{inm}e_{n}\Lambda_{m}^{(\rho)},$
(12)
The effect of turbulent diffusivity, which is anisotropic due to the Coriolis
force, is given by:
$\eta_{ijk}=3\eta_{T}\left\\{\left(2f_{1}^{(a)}-f_{1}^{(d)}\right)\varepsilon_{ijk}-2f_{1}^{(a)}e_{i}e_{n}\varepsilon_{njk}\right\\}.$
(13)
Functions $f_{\\{1-11\\}}^{(a,d)}$ depend on the Coriolis number
$\Omega^{*}=2\tau_{c}\Omega_{0}$ and the typical convective turnover time in
the mixing-length approximation: $\tau_{c}=\ell/u^{\prime}$. They can be found
in P08. The turbulent diffusivity is parametrized in the form,
$\eta_{T}=C_{\eta}\eta_{T}^{(0)}$, where
$\eta_{T}^{(0)}={\displaystyle\frac{u^{\prime}\ell}{3}}$ is the characteristic
mixing-length turbulent diffusivity, $u^{\prime}$ is the RMS convective
velocity, $\ell$ is the mixing length, $C_{\eta}$ is a constant to control the
intensity of turbulent mixing. The others quantities in Eqs.(5,12,13) are:
$\mathbf{\boldsymbol{\Lambda}}^{(\rho)}=\boldsymbol{\nabla}\log\overline{\rho}$
is the density stratification scale,
$\mathbf{\boldsymbol{\Lambda}}^{(u)}=\boldsymbol{\nabla}\log\left(\eta_{T}^{(0)}\right)$
is the scale of turbulent diffusivity,
$\mathbf{e}=\boldsymbol{\Omega}/\left|\Omega\right|$ is a unit vector along
the axis of rotation. Equations (5,12,13) take into account the influence of
the fluctuating small-scale magnetic fields, which can be present in the
background turbulence and stem from the small-scale dynamo (see discussions
in). In our paper, the parameter
$\varepsilon={\displaystyle\frac{\overline{\mathbf{b}^{2}}}{\mu_{0}\overline{\rho}\overline{\mathbf{u}^{2}}}}$,
which measures the ratio between the magnetic and kinetic energies of
fluctuations in the background turbulence, is assumed equal to 1\. This
corresponds to the energy equipartition. The quenching function of the
hydrodynamical part of $\alpha$-effect is defined by
$\psi_{\alpha}=\frac{5}{128\beta^{4}}\left(16\beta^{2}-3-3\left(4\beta^{2}-1\right)\frac{\arctan\left(2\beta\right)}{2\beta}\right).$
(14)
Note, in notation of P08 $\psi_{\alpha}=-3/4\phi_{6}^{(a)}$, and
$\beta={\displaystyle\frac{\left|\overline{B}\right|}{u^{\prime}\sqrt{\mu_{0}\overline{\rho}}}}$.
|
arxiv-papers
| 2011-10-11T02:54:11 |
2024-09-04T02:49:22.981740
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V. V. Pipin and D. D. Sokoloff",
"submitter": "Valery Pipin",
"url": "https://arxiv.org/abs/1110.2255"
}
|
1110.2268
|
# Search for Chargino-Neutralino Associated Production in Dilepton Final
States with Tau Leptons
R. Forrest Department of Physics, UC Davis, Davis, CA, USA M. Chertok
Department of Physics, UC Davis, Davis, CA, USA (on behalf of the CDF
Collaboration)
###### Abstract
We present a search for chargino and neutralino supersymmetric particles
yielding same signed dilepton final states including one hadronically decaying
tau lepton using 6.0 $fb^{-1}$ of data collected by the the CDF II detector.
This signature is important in SUSY models where, at high $\tan{\beta}$, the
branching ratio of charginos and neutralinos to tau leptons becomes dominant.
We study event acceptance, lepton identification cuts, and efficiencies. We
set limits on the production cross section as a function of SUSY particle mass
for certain generic models.
## I Introduction
In the search for new phenomena, one well-motivated extension to the Standard
Model (SM) is supersymmetry (SUSY). One very promising mode for SUSY discovery
at hadron colliders is that of chargino-neutralino associated production with
decay into three leptons. Charginos decay into a single lepton through a
slepton
$\tilde{\chi}_{1}^{\pm}\rightarrow~{}\tilde{l}^{(*)}~{}\nu_{l}\rightarrow\tilde{\chi}_{1}^{0}~{}l^{\pm}~{}\nu_{l}$
and neutralinos similarly decay into two detectable leptons
$\tilde{\chi}_{2}^{0}\rightarrow~{}\tilde{l}^{\pm(*)}~{}l^{\mp}\rightarrow\tilde{\chi}_{1}^{0}~{}l^{\pm}~{}l^{\mp}$
. The detector signature is thus three SM leptons with associated missing
energy from the undetected neutrinos and lightest neutralinos,
$\tilde{\chi}_{1}^{0}$ (LSP), in the event. Many previous searches have used
all three leptons for detection rut_note ; Forrest, R. for the CDF
Collaboration (2009).
The most generic form of SUSY is the MSSM model which, in many parameter
spaces, gives the lepton signature that interests us susy_primer .
Unfortunately there are far too many free parameters in this model to test
generically. In the past it has been tradition to use a specific gravity
mediated SUSY breaking model called mSugra. For this analysis we adopt a more
generic method, in which we present results in terms of exclusions in
sparticle masses as opposed to mSugra parameter space.
We construct simplified models of SUSY wherein we do not hope to develop a
full model of SUSY, but an effective theory that can be easily translated to
describe kinematics of arbitrary models. We set the masses at the electroweak
scale and include the minimal suite of particles necessary to describe the
model and effectively decouple all other particles, by setting their masses
$>$ TeV range. We also tune the couplings of the particles to mimic models
that preferentially decay to taus.
Specific models will determine permitted decay modes Ruderman:2010kj .
Different models’ SUSY breaking method will determine allowed decay modes in
broad categories. In this analysis we present two types of generic models. The
first is a simplified gravity breaking model similar to mSugra; the second is
a simplified gauge model, which encompasses a broad suite of theories with
gauge mediated SUSY breaking (GMSB).
The simplified gravity model we generally have electroweak ($W^{\pm}$)
production of $\tilde{\chi}_{1}^{\pm},\tilde{\chi}_{2}^{0}$ pairs.
$\tilde{\chi}_{1}^{\pm}$ then decays to $\tilde{l}^{\pm},\nu_{l}$ and
$\tilde{\chi}_{2}^{0}$ goes to $\tilde{l}^{\pm}l^{\mp}$. All the sleptons
decay as normal $\tilde{l}^{\pm}\rightarrow l^{\pm},\tilde{\chi}_{1}^{0}$. We
can tune the branching ratio to slepton flavors. For each SUSY point, we
choose two branching ratios
BR($\tilde{\chi}_{2}^{0},\tilde{\chi}_{1}^{\pm}\rightarrow\tilde{\tau}+X)=1,1/3$.
We choose the masses of the $\tilde{\chi}_{1}^{\pm}$ and
$\tilde{\chi}_{2}^{0}$ to be equal.
The simplified gauge model is motivated by gauge mediated SUSY breaking
scenarios. Generally, the LSP is the gravitino which is very light: in the
sub-keV range. Also, charginos do not couple to right handed sleptons in these
models, therefore all chargino decays are to taus, so
BR($\tilde{\chi}_{1}^{\pm}\rightarrow\tilde{\tau}_{1}\nu_{\tau})=1$ always.
The $\tilde{\chi}_{2}^{0}$ can decay to all lepton flavors. The final feature
of this model is that $\tilde{\chi}_{1}^{\pm}$ or $\tilde{\chi}_{2}^{0}$ don’t
decay through SM bosons.
## II Analysis Overview
Our approach is to look for the two same signed leptons from trilepton events
since the opposite signed pair has the disadvantage of large standard model
backgrounds from electroweak Z decay.
We select one electron or muon and one hadronically decaying tau. Requiring a
hadronicaly decaying tau adds sensitivity to high tan$\beta$ SUSY space. Our
main backgrounds therefore will be SM W + Jets where the W boson decays to an
electron or muon and the jet fakes a hadronic tau in our detector.
Our background model is comprised of two distinct types. We use Monte Carlo to
account for common SM processes naturally entering the background as well as
processes with real taus that might contain a fake lepton. Any process
involving a jet faking a tau is covered in our tau fake rate method, these
processes would be W + Jet, conversion+Jet and QCD. In all these processes,
the jet fakes a tau and a lepton comes from the other leg of the event.
Our fake rate is measured in a sample of pure QCD jets Aaltonen et al. (2009).
We validate the measurement by applying it to three distinct orthogonal
regions to our signal.
We select our dilepton events and first understand the opposite signed lepton-
tau region. After applying an $H_{T}$ cut, we develop confidence that we
understand the primary and secondary backgrounds, $Z\rightarrow\tau\tau$ and W
+ jets respectively. We then look at the same signed signal region, where we
expect to be dominated by our fake rate background.
To set limits in the M(Chargino) vs. M(Slepton) plane, a grid of signal points
is generated. We optimize a $\rm\,/\\!\\!\\!\\!{\it E_{T}}$ cut as a function
of model parameters for each point to increase our sensitivity to signal.
Limits are then found at each point, and iso-contours are interpolated to form
our final limits on SUSY process cross section.
## III Dataset And Selection
We use 1.96 TeV $p\bar{p}$ collision data from the Fermilab Tevatron
corresponding to 6.0 $fb^{-1}$ of integrated luminosity from the CDF II
detector. The data is triggered by requiring one lepton object, and electron
or muon; as well as a cone isolated tau like object. We then apply standard
CDF selection cuts to the objects. Electrons and muons are required to have an
$E_{T}$ ($P_{T}$) cut of 10 GeV. One pronged taus have a $P_{t}$ cut of 15
GeV/c and three pronged taus have a 20 GeV/c cut. The $P_{T}$ for a tau is
considered to be the visible momentum: the sum of the tracks and $\pi^{0}$’s
in the isolation cone.
To reduce considerable QCD backgrounds we apply a cut on $H_{T}$ defined as
the sum of the tau, lepton and $\rm\,/\\!\\!\\!\\!{\it E_{T}}$ in the event.
The $H_{T}$ cut is 45,50,55 GeV/c for the $\tau_{1}-\mu$, $\tau_{1}-e$ and
$\tau_{3}-\ell$ channels. We cut events were $d\phi(l,\tau)<0.5$ as well as
events with OS leptons within 10 GeV of the Z boson mass.
$\rm\,/\\!\\!\\!\\!{\it E_{T}}$ is corrected for all selected objects and any
jets observed in the event.
Monte Carlo is scaled to reflect trigger inefficiencies as well as
inefficiencies from lepton and tau reconstruction.
## IV Backgrounds
Our background model is comprised of two distinct types. We use Monte Carlo to
simulate detector response to Diboson, $t\bar{t}$, Z boson processes as well
as real taus from W decay. These processes are normalized to their SM cross
section and weighted by scale factors to account for inefficiencies in
trigger, ID and reconstruction. Any process involving a jet faking a tau is
covered in our tau fake rate method, these processes would be W + Jet,
conversion+Jet and QCD. In all these processes, the jet fakes a tau and a
lepton comes from the other leg of the event.
We measure the fake rate in a sample of QCD jets. Our rate is defined as the
ratio of tau objects to loose taus where loose taus are tau like objects that
pass our trigger. Because the trigger has very decent tau discriminating
ability, this relative fake rate is fairly high. In terms of applying the fake
rate to fakeable objects, in order to not overestimate our fake contributions
we have a subtraction procedure for the preponderance of real taus that pass
through our trigger. The measurement of the fake rate in the leading jet and
sub leading QCD jet constitutes the systematic on the measurement.
We validate our tau fake rates in three different orthogonal regions to our
signal. These regions reflect the three processes the fake rate will account
for in the analysis.
## V OS Validation
Before we look at signal data in out blind analysis, a major validation step
is to confirm agreement in the OS region. This region is dominated by
$Z\rightarrow\tau\tau$ decays, which gives us confidence in our scale factor
application. The secondary background in this region is W+ Jets, which serves
as an additional check on our fake rate background. As can be seen in Table 1
as well as in Figure 1 and we have good confidence in our background model.
CDF Run II Preliminary $6.0\ \textrm{fb}^{-1}$ OS $\ell-\tau$ Process Events
$\pm$ stat $\pm$ syst Z$\rightarrow\tau\tau$ $6967.3\pm 56.4\pm 557.4$
Jet$\rightarrow\tau$ $4526.5\pm 26.8\pm 1064.5$ Z$\rightarrow\mu\mu$ $262.5\pm
20.1\pm 21.0$ Z$\rightarrow ee$ $82.5\pm 8.6\pm 6.6$ W$\rightarrow\tau\nu$
$371.5\pm 12.4\pm 36.4$ t$\bar{\textrm{t}}$ $36.3\pm 0.3\pm 5.1$ Diboson
$61.3\pm 0.9\pm 6.0$ Total 12308.0 $\pm\ 67.3\pm 1202.3$ Data 12268
Table 1: Total OS control region. |
---|---
Figure 1: Plots of the OS Control Region, Electron $E_{T}$ (left) and Muon
$P_{T}$ (right).
### V.1 Observed Data and Limit Setting
After gaining confidence in the OS control region, we unblind the analysis and
set limits on our models. For each signal point, we choose a
$\rm\,/\\!\\!\\!\\!{\it E_{T}}$ cut that optimizes the $s/\sqrt{b}$ at that
point. To allow simple interpretation, we form an analytical expression for
the $\rm\,/\\!\\!\\!\\!{\it E_{T}}$ cut as a function of model parameters.
Because of large QCD and conversion backgrounds at low $\rm\,/\\!\\!\\!\\!{\it
E_{T}}$ all limit setting is done above $\rm\,/\\!\\!\\!\\!{\it E_{T}}=20\
GeV$. The results are below in table 2. Kinematic plots of the SS region are
in Figure 5.
CDF Run II Preliminary $6.0\ \textnormal{fb}^{-1}$ SS $\ell-\tau$ Process
Events $\pm$ stat $\pm$ syst Z$\rightarrow\tau\tau$ $10.2\pm 2.2\pm 0.8$
Jet$\rightarrow\tau$ $1152.7\pm 15.2\pm 283.1$ Z$\rightarrow\mu\mu$ $0.0\pm
0.0\pm 0.0$ Z$\rightarrow ee$ $0.0\pm 0.0\pm 0.0$ W$\rightarrow\tau\nu$
$96.9\pm 6.4\pm 9.5$ t$\bar{\textnormal{t}}$ $0.7\pm 0.0\pm 0.1$ Diboson
$4.3\pm 0.2\pm 0.4$ Total 1264.8 $\pm\ 16.6\pm 283.3$ Data 1116
Table 2: SS signal region used in limit setting, $\rm\,/\\!\\!\\!\\!{\it E_{T}}>20\ GeV$. Both Electron and Muon Channels. |
---|---
Figure 2: Plots of the SS Signal Region, Electron $E_{t}$ (left) and a log version (right). |
---|---
Figure 3: Plots of the SS Signal Region, Electron $H_{t}$ (left) and a electron $\rm\,/\\!\\!\\!\\!{\it E_{T}}$ (right). |
---|---
Figure 4: Plots of the SS Signal Region, Muon $P_{t}$ (left) and a log version (right). |
---|---
Figure 5: Plots of the SS Signal Region, Muon $\rm\,/\\!\\!\\!\\!{\it E_{T}}$
(left) and tau cluster $E_{T}$ (right).
After the $\rm\,/\\!\\!\\!\\!{\it E_{T}}$ cut is applied at each point, we
find SUSY production cross section limits and interpolate these contours in
the M(Chargino) vs. M(Slepton) plane. The final results can be found in
Figures 6 through 8.
|
---|---
Figure 6: Expected limits (pb) for Simplified Gauge Model for BR to taus of 100% ( left), and 33%(right) |
---|---
Figure 7: Expected limits (pb) for Simplified Gravity Model with LSP = 120 GeV for BR to taus of 100% (left), 33% (right). |
---|---
Figure 8: Expected limits (pb) for Simplified Gravity Model with LSP = 220 GeV
for BR to taus of 100% (left), 33% (right).
## References
* (1) A Supersymmetry Primer, Stephen P. Martin, hep-ph/9709356.
* Forrest, R. for the CDF Collaboration (2009) Forrest, R., for the CDF Collaboration 2009, arXiv:0910.1931
* (3) J. T. Ruderman, D. Shih, JHEP 1011, 046 (2010). [arXiv:1009.1665 [hep-ph]].
* Aaltonen et al. (2009) Aaltonen, T., Adelman, J., Akimoto, T., et al. 2009, Physical Review Letters, 103, 201801
* (5) Search for Supersymmetry in $p\bar{p}$ Collisions at $\sqrt{s}$ = 1.96 TeV Using the Trilepton Signature for Chargino-Neutralino Production, CDF Collaboration, Phys. Rev. Lett. 101, 251801 (2008), DOI:10.1103/PhysRevLett.101.251801
|
arxiv-papers
| 2011-10-11T04:31:44 |
2024-09-04T02:49:22.992280
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "R. Forrest, M. Chertok (for the CDF Collaboration)",
"submitter": "Robert Forrest",
"url": "https://arxiv.org/abs/1110.2268"
}
|
1110.2416
|
# Supervised learning of short and high-dimensional temporal sequences for
life science measurements
F.-M. Schleif 1 A. Gisbrecht 1 B. Hammer 1
(1Univ. of Bielefeld, CITEC Center of Excellence,
Universitätsstrasse 21-23, 33615 Bielefeld, Germany
01\. September 2011
Technical report follow of the Dagstuhl Seminar 11341
Learning in the context of very high dimensional data
21.08.11 - 26.08.11
Organizer: Michael Biehl (Univ. of Groningen, NL), Barbara Hammer (Univ.
Bielefeld, DE), Erzsébet Merényi (Rice Univ., US),
Alessandro Sperduti (Univ. of Padova, IT), Thomas Villmann (Univ. of Applied
Sc. Mittweida, DE))
###### Abstract
Motivation: The analysis of physiological processes over time is becoming
increasingly important. The measurements are often given by spectrometric or
gene expression profiles over time with only few time points but a large
number of measured variables. The analysis of such temporal sequences is
challenging and only few methods have been proposed. The information can be
encoded time independent, by means of classical expression differences for a
single time point or in expression profiles over time. Available methods are
limited to unsupervised and semi-supervised settings. The predictive variables
can be identified only by means of wrapper or post-processing techniques. This
is complicated due to the small number of samples for such studies. Here, we
present a supervised learning approach, termed _Supervised Topographic Mapping
Through Time_ (SGTM-TT). It learns a supervised mapping of the temporal
sequences onto a low dimensional grid. We utilize a hidden markov model (HMM)
to account for the time domain and relevance learning to identify the relevant
feature dimensions most predictive over time. The learned mapping can be used
to visualize the temporal sequences and to predict the class of a new
sequence. The relevance learning permits the identification of discriminating
masses or gen expressions and prunes dimensions which are unnecessary for the
classification task or encode mainly noise. In this way we obtain a very
efficient learning system for temporal sequences.
Results: The results indicate that using simultaneous supervised learning and
metric adaptation significantly improves the prediction accuracy for
synthetically and real life data in comparison to the standard techniques. The
discriminating features, identified by relevance learning, compare favorably
with the results of alternative methods. Our method permits the visualization
of the data on a low dimensional grid, highlighting the observed temporal
structure.
Contact: fschleif@techfak.uni-bielefeld.de
Keywords: high-dimensional time series, short time series, prototype learning,
relevance learning, topographic mapping
## 1 Introduction
The analysis of high-dimensional, short time series, or temporal sequences is
a challenging task. On the one hand side the data are not any longer identical
and independent distributed (i.i.d) due to the time constraint, on the other
hand the dimensionality of the data is large, complicating the learning of a
predictive model. Standard time series methods like auto-regressive moving
average (ARMA) or extensions thereof (see e.g. [9]) are in general not
applicable due to the limited number of time points and the large
dimensionality of the data. Some methods have been proposed to model this type
of data. In [20] an unsupervised projection techniques was proposed employing
a so called temporal context. The temporal data have been processed by a kind
of Self Organizing Map (SOM) [11] but the learning was modified such that it
depends on the the current temporal context. A further unsupervised proposal
has been made in [14] using the Generative Topographic Mapping Through Time
(GTM-TT) ([3]). Some new hidden variables were introduced to account for the
relevance of the different feature dimensions, to accounts, in a non-
discriminative manner, for the explained variance in the data over time. A
supervised two-class method solely based on hidden markov models was proposed
in [13]. It models the two different data distribution by independent HMMs and
evaluates the generated models to obtain a ranking of the input dimensions.
Subsequently the model was improved by selecting a set of features using a
wrapper strategy. In [6] a similar approach was proposed but in a semi-
supervised scenario, introducing classwise constraints in the hidden markov
model. The importance of the individual features was determined using a
complex post processing procedure. Another supervised method using all
features, based on Support Vector Machine (SVM) and a Kalman filter was
proposed in [5].
While the first two approaches have been found to be very effective for
unsupervised analysis, the last mentioned methods focus on supervised and
semi-supervised analysis. The results in [13] are very promising, with $85\%$
prediction accuracy on a real life multiple sclerosis data (MS) set, but make
strong pre-assumptions about the underlying HMM structure. Also, it is
proposed for two class scenarios, only. The approach in [5] improved this
result by $2-5\%$ but in a black box scenario, without additional feature
selection. The approach in [6] is evaluated also with respect to the results
of [13] achieving improved performance for the same MS data sets. There is
still ongoing work of research in this field, reflecting the high demand for
effective methods dealing with this type of data. The application field is not
limited to the bio-medical domain as considered in [13, 6, 8] but covers a
broader field of applications also in industry and geo-science as reflected in
[14, 20].
The identification of the relevant input dimensions of a temporal sequence is
very important as outlined in [14, 13] to obtain better understanding of the
data, to reduce the processing complexity and to improve the overall
prediction accuracy. As already motivated by some of the prior references,
prototype methods (see e.g. [11]) have been found to be very effective for the
analysis of high dimensional data also to analyze temporal sequences. In [3],
the _Generative Topographic Mapping - through time_ (GTM-TT), an unsupervised
prototype based method for the topographic projection of high-dimensional,
temporal sequences was proposed. GTM-TT learns a hidden markov model (HMM) of
a data generating process and represents the data by a prototype based
representation in time and space. Like in ordinary prototype methods the GTM-
TT approximates the data distribution by a vector quantization of the data
space. The temporal dependence between the prototype is modeled by an
appropriate HMM. Additionally the prototypes are assigned to a fixed grid
representation or lattice, which permits, provided the topology is preserved
(see [22]), the easy visualization and interpretation of the data trajectory
in a low dimensional space. In this contribution we extend the GTM-TT to a
supervised method and integrate relevance learning to identify the relevant
dimensions over time. Then we will briefly review Generative Topographic
Mapping (GTM) and Generative Topographic Mapping Through Time. Subsequently,
we outline our method and apply and discuss it for different experimental
data. The paper is closed with links to further extensions and open questions.
## 2 Approach and Methods
### 2.1 Generative Topographic Mapping
The Generative Topographic Mapping (GTM) as introduced in [4] models data
$\mathbf{x}\in\mathbb{R}^{D}$ by means of a mixture of Gaussians which is
induced by a lattice of points $\mathbf{w}$ in a low dimensional latent space
which can be used for visualization.
Figure 1: GTM-TT consisting of a HMM in which the hidden states are given by
the latent points of the GTM model. The emission probabilities are governed by
the GTM mixture distribution [3]. The different data distributions,
exemplified in 3D (bottom) and indicated by the color/shading are mapped to
the 2D grid (top). Here we consider $9$ hidden states on a $3\times 3$ grid.
The data distribution may change over time and hence also the mapping of the
GTM is effected over time, assuming smooth transitions.
The lattice points are mapped via
$\mathbf{w}\mapsto\mathbf{t}=y(\mathbf{w},\mathbf{W})$ to the data space,
where the function is parametrized by $\mathbf{W}$; one can, for example, pick
a generalized linear regression model based on Gaussian base functions
$y:\mathbf{w}\mapsto\Phi(\mathbf{w})\cdot\mathbf{W}$ (1)
where the base functions $\Phi$ are equally spaced Gaussians.The high-
dimensional points $y$ are so called prototypes of the original data space,
representing a larger set of points, they are responsible for, as measured by
Eq. (5). They can be directly inspected and permit to summarize the data.
Every latent point induces a Gaussian
$p(\mathbf{x}|\mathbf{w},\mathbf{W},\beta)=\left(\frac{\beta}{2\pi}\right)^{D/2}\exp\left(-\frac{\beta}{2}\|\mathbf{x}-y(\mathbf{w},\mathbf{W})\|^{2}\right)$
(2)
with variance $\beta^{-1}$, which gives the data distribution as mixture of
$K$ modes
$p(\mathbf{x}|\mathbf{W},\beta)=\sum_{k=1}^{K}p(\mathbf{w}^{k})p(\mathbf{x}|\mathbf{w}^{k},\mathbf{W},\beta)$
(3)
where, usually, $p(\mathbf{w}^{k})$ is taken as Dirac distributions of the
prototypes. Training of GTM optimizes the data log-likelihood
$\ln\left(\prod_{n=1}^{N}\left(\sum_{k=1}^{K}p(\mathbf{w}^{k})p(\mathbf{x}^{n}|\mathbf{w}^{k},\mathbf{W},\beta)\right)\right)$
(4)
by means of an expectation maximization (EM) approach with respect to the
parameters $\mathbf{W}$ and $\beta$. In the E step, the responsibility of
mixture component $k$ for point $n$ is determined as
$r^{kn}=p(\mathbf{w}^{k}|\mathbf{x}^{n},\mathbf{W},\beta)=\frac{p(\mathbf{x}^{n}|\mathbf{w}^{k},\mathbf{W},\beta)p(\mathbf{w}^{k})}{\sum_{k^{\prime}}p(\mathbf{x}^{n}|\mathbf{w}^{k^{\prime}},\mathbf{W},\beta)p(\mathbf{w}^{k^{\prime}})}$
(5)
In the M step, the weights $\mathbf{W}$ are determined solving the equality
$\mathbf{\Phi}^{T}\mathbf{G}_{\mathrm{old}}\mathbf{\Phi}\mathbf{W}_{\mathrm{new}}^{T}=\mathbf{\Phi}^{T}\mathbf{R}_{\mathrm{old}}\mathbf{X}$
(6)
where $\mathbf{\Phi}$ refers to the matrix of base functions $\Phi$ evaluated
at points $\mathbf{w}^{k}$, $\mathbf{X}$ to the data points, $\mathbf{R}$ to
the responsibilities, and $\mathbf{G}$ is a diagonal matrix with accumulated
responsibilities $G_{nn}=\sum_{k}r^{kn}(\mathbf{W},\beta)$. The variance can
be computed by solving
$\frac{1}{\beta_{\mathrm{new}}}=\frac{1}{ND}\sum_{k,n}r^{kn}(\mathbf{W}_{\mathrm{old}},\beta_{\mathrm{old}})\|{\Phi}(\mathbf{w}^{k})\mathbf{W}_{\mathrm{new}}-\mathbf{x}^{n}\|^{2}$
(7)
where $D$ is the data dimensionality and $N$ the number of data points.
### 2.2 Relevance learning
The principle of relevance learning has been introduced in [10] as a
particularly simple and efficient method to adapt the metric of prototype
based classifiers according to the given situation at hand. It takes into
account a relevance scheme of the data dimensions by substituting the squared
Euclidean metric by the weighted form
$d_{\boldsymbol{\lambda}}(\mathbf{x},\mathbf{t})=\sum_{d=1}^{D}\lambda_{d}^{2}(x_{d}-t_{d})^{2}\,.$
(8)
The principle is extended in [18, 17] to the more general metric form
$d_{\boldsymbol{\Omega}}(\mathbf{x},\mathbf{t})=(\mathbf{x}-\mathbf{t})^{T}\boldsymbol{\Omega}^{T}\boldsymbol{\Omega}(\mathbf{x}-\mathbf{t})$
(9)
Using a square matrix $\boldsymbol{\Omega}$, a positive semi-definite matrix
which gives rise to a valid pseudo-metric is achieved this way. In [18, 17],
these metrics are considered in local and global form, i.e. the adaptive
metric parameters can be identical for the full model, or they can be attached
to every prototype present in the model. Relevance learning for GTM has been
already introduced in [7] for i.i.d. data. In case of temporal sequences some
modification of the original principle are necessary and also the supervision
will be handled differently as pointed out subsequently. First however we
review the GTM through time as described in [3, 15] which is the basic method
to process i.i.d. data in our approach.
### 2.3 Generative Topographic Mapping Through-Time
The GTM through time (GTM-TT) has been introduced in [3]. For data vectors
$\mathbf{x}$ which have the form of a time series the vectors are no longer
independent. Nearby timepoints are likely to be correlated. As pointed out in
[3] such effects can be captured using Hidden Markov Models (HMM). Accordingly
in [3] the GTM is equipped by a HMM, constructing a kind of a topology-
constrained HMM
The structure of the GTM-TT is shown in Figure 1. Assuming a sequence length
$T$, of hidden states $Z=\\{z_{1},\ldots,z_{n},\ldots z_{T}\\}$ and the
observed multidimensional time series $X=\\{x^{1},x^{2},\ldots,x^{n},\ldots
x^{T}\\}$, the probability of the observations is given by
$p(X)=\sum_{\text{all sequences }Z}p(Z,X)$ (10)
where $p(Z,X)$ defines the complete data likelihood as in HMM models [4]
taking the following form:
$p(Z,X)=p(z_{1})\prod_{n=2}^{T}p(z_{n}|z_{n-1})\prod_{n=1}^{T}p(x^{n}|z_{n})$
(11)
So it consists of the initial state probability, the transition probability
between two hidden states, capturing the temporal dependence, and the
probability to observe a specific sequence in a given state also known as
emission probability (covered by Eq. (2)). The model parameters are
$\Theta=(\pi,A,W,\beta)$ where $\pi=\\{\pi_{j}\\}:\pi_{j}:=p(z_{1}=j)$ are the
initial state probabilities. $A=\\{a_{ij}\\}:a_{ij}=p(z_{n}=j|z_{n-1}=i)$ are
the transition state probabilities, and $\\{W,\beta\\}$ are given by Eq. (6).
Again we control the gaussians by a common invariance $\beta$. As in HMM the
above likelihood can be efficiently calculated using the _forward backward
procedure_ [23]. The probability being in state $\mathbf{w}_{k}$ at time $n$,
given the observation sequence and the model, also known as responsibility
$r^{kn}$ is calculated as:
$r^{kn}=p(z_{n}=\mathbf{w}^{k}|X,\Theta)=\frac{A_{kn}B_{kn}}{p(X|\Theta)}$
(12)
The forward variable $A_{kn}$ is the joint probability of the past sequences
$\\{x^{1},\ldots,x^{n}\\}$ and the state $z_{n}=\mathbf{w}^{k}$, i.e.
$A_{kn}=p(\\{x^{1},\ldots,x^{n}\\},z_{n}=\mathbf{w}^{k}|\Theta)$, given by the
following recursive equation:
$A_{kn}=\left(\sum_{i=1}^{K}A_{i,n-1}p_{i,k}\right)p_{k}(x^{n})$ (13)
where $A_{k,1}=\pi_{k}p_{k}(x^{1})$. The backward variable $B_{kn}$ which is
the probability of the future sequence $x^{n+1},x^{n+2},\ldots,x^{N}$ given
the hidden state $z_{n}=\mathbf{w}^{k}$, i.e.
$B_{kn}=p(\\{x^{n+1},x^{n+2},\ldots,x^{N}\\}|z_{n}=\mathbf{w}^{k},\Theta)$ is
calculated using the following recursive equation:
$B_{kn}=\sum_{i=1}^{K}p_{i,k}p_{i}(x^{n+1})B_{i,n+1}$ (14)
where $B_{k,T}=1$. The whole parameter estimation can be accomplished by a
maximum likelihood optimization using the EM algorithm as sketched above.
Details can be found in [19].
### 2.4 Supervised GTM-TT
Assume that data point $X$ is equipped with label information $l$ which is
element of a finite set of different labels $L$, e.g. $L=\\{0,1\\}$. Lets
assume we have only two labels 111An extension to multiple labels is straight
forward.. The data are divided into two groups, according to the labeling and
we train one separate GTM-TT per group. To keep the models comparable, the
$\beta$ update for the models is linked to each other and optimized in the
outer loop. The parameters $\mathbf{W}$ are determined for each model
individually leading to $\mathbf{W}_{0}$ and $\mathbf{W}_{1}$. We will further
assume that the grid structure is common for both models. The learning
procedure is thus similar to GTM-TT and depicted in Figure 1.
1:function Supervised GTM-TT($X$,$L$,$K$)
2: [Xn,Pars ] = normalize(X)
3: [X1,X2,L1,L2] = splitdata(Xn,L)
4: [$M_{0}$, $M_{1}$] = init both GTM-TT models
5: repeat
6: call train_single_step for $M_{0}$, $M_{1}$
7: call convergence_check for $M_{0}$, $M_{1}$
8: call optimize_beta for $M_{0}$, $M_{1}$
9: $\bar{\beta}=$ calculate mean of the $\beta$
10: call update_beta($M_{0}$, $M_{1}$,$\bar{\beta}$)
11: until convergence is true for both models
12:end function
Algorithm 1 Pseudocode of supervised GTM-TT
We denote the obtained model as Supervised GTM-TT (SGTM-TT) and the submodels
as $M_{0}$ and $M_{1}$. The concept of the SGTM-TT is depicted schematically
in Figure 2.
Figure 2: Illustration of the SGTM-TT. It consists of multiple GTM-TT models.
It behaves similar to the regular GTM-TT but the training is classwise and the
$\beta$ parameter is common over the different models. The different classwise
models (top) are used to represent the data distribution (bottom) over time
(from left to right).
### 2.5 Classification using SGTM-TT
To classify new data points with the SGTM-TT model different approaches can be
taken. The simplest one is to make direct use of the samplewise likelihoods
considering the class wise models. In that case a new point is assigned to the
model with maximal likelihood considering one model against the rest. A more
interesting approach is to combine the performance of the generative SGTM-TT
model with a discriminative approach like the SVM [21]. Again we use the
likelihood values from the forward procedure (13) of the SGTM-TT and define a
kernel as follows:
$\displaystyle Lik_{j}^{l}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{K}A_{i,j}\text{ for a series $j$ and a sub-model }l$
(15) $\displaystyle K(X_{j},X_{k})$ $\displaystyle=$
$\displaystyle\sum_{l=1:\\#L}Lik_{j}^{l}\cdot Lik_{k}^{l}\;\text{ with equal
prior}$ (16)
Hence the kernel $K$ is based on a kernel function of inner-products in a one
dimensional feature space of the likelihood-values. In the following we will
make use of this approach employing a standard SVM implementation.
### 2.6 Relevance learning for SGTM-TT
Relevance learning for GTM has been introduced in [7], as the Relevance GTM
(R-GTM). The basic idea for Relevance GTM is to introduce an adaptive metric
for the GTM. The original Euclidean metric is replaced by a parametric
distance like the weighted Euclidean metric (8). After each GTM training step
the prototypes are post-labeled according to its responsibilities, employing
the labeling $L$ of the datapoints. Subsequently the metric parameters of the
distance are adapted according to an optimization criterion. In the article of
[7] different cost functions E where suggested.
The data of the GTM-TT are not any longer i.i.d. and, as mentioned before we
observe a sequence of states $Z$ for a given time series $X$. In the SGTM-TT
we know the labeling of the prototypes, assuming constant labels over time,
due to the split of the learning problem according to the data labeling.
Further, using a common metric and common $\beta$ parameters the prototypes
$Y$ exist still in the same common dataspace. Relevance learning can now be
done in the same way as for R-GTM. This however is often not useful because
the original relevance learning ignores the time domain. If data separation is
observed over time and not for a single time point the R-GTM approach will
fail. For temporal sequences we may also be interested on two views of
relevance, namely relevant, or separating input _dimensions_ $x_{i}$ but also
relevant _time points_ in a temporal sequence $x$. Taking this problem into
account we consider two distance measures, one for the time domain, denoted as
$d^{t}$ and one for the time-independent data space $d^{\lambda}$. A
parametrization of $d^{t}$ can be used to account for the relevance of
specific time points, e.g to prune out time points which are irrelevant for
the representation of the data in a discriminative manner. Parameters on
$d^{\lambda}$ can be used to identify discriminating feature dimensions, e.g.
to prune out noisy dimensions. Subsequently, we provide a distance measure
which can be used for $d^{t}$ and a specific form for $d^{\lambda}$. For
simplicity we will use a simple _global_ , _diagonal_ metric learning scheme
in the experiments.
SGTM-TT provides a probabilistic prediction of the internal representation
$\hat{\mathbf{x}}$ of a time series $\mathbf{x}$ considering the two GTM-TT
models, we obtain one reconstruction each:
$\displaystyle\hat{\mathbf{x}^{n}}^{l}_{i}$ $\displaystyle=$ $\displaystyle
Y_{l}(\underset{k}{\arg\max}\left(r^{kn}\right),i)\;\forall i\in[1,D]$
$\displaystyle\text{with }l\in\\{0,1\\}$
Now, two distances are calculated over time for each point and each dimension
$i$: $d^{t}(\hat{\mathbf{x}^{n}}^{0}_{i},\mathbf{x}^{n}_{i})$,
$d^{t}(\hat{\mathbf{x}^{n}}^{1}_{i},\mathbf{x}^{n}_{i})$. Using one of the
suggested cost functions in the paper of [7] we can calculate the relevance of
the individual dimensions for the separation between the two reconstructions
per point and hence between the different models.
Like for R-GTM the metric adaptation is done by an appropriate optimization
scheme on the cost functions, here we will use stochastic gradient descend,
with a fixed learning rate $\epsilon=0.1$. To avoid convergence to trivial
optima such as zero we pose constraints on the metric parameters of the form
$\|\boldsymbol{\lambda}\|=1$ or
$\mathrm{trace}(\boldsymbol{\Omega}^{T}\boldsymbol{\Omega})^{2}=1$, for matrix
learning. This is achieved by normalization of the values, i.e. after every
gradient step, $\boldsymbol{\lambda}$ is divided by its length, and
$\boldsymbol{\Omega}$ is divided by the square root of
$\mathrm{trace}(\boldsymbol{\Omega}^{T}\boldsymbol{\Omega})$.
A pseudo code of the SGTM-TT with relevance learning is depicted in 2.
1:function SGTM-TT-R($X$,$L$,$K$)
2: [Xn,Pars ] = normalize(X)
3: [X1,X2,L1,L2] = splitdata(Xn,L)
4: initialize the common metric
5: [$M_{0}$,$M_{1}$] = init both GTM-TT models
6: repeat
7: call train_single_step for each GTM-TT model
8: call convergence_check for each GTM-TT model
9: if $cycle>10$ then
10: $\forall X,\forall i=1:D$ call reconstruct($X_{i},M_{0},M_{1}$)
11: $\forall X,\forall i=1:D$ call $d^{t}(M_{0},\hat{x_{i}}_{0},x_{i})$
12: $\forall X,\forall i=1:D$ call $d^{t}(M_{1},\hat{x_{i}}_{1},x_{i})$
13: $\forall X$ call calculate_metric_update
14: average the metric updates and normalize
15: update the metric parameter annealed by $\epsilon$
16: end if
17: call optimize_beta for each GTM-TT model
18: $\bar{\beta}=$ calculate mean of the $\beta$
19: call update_beta(M1,M2,$\bar{\beta}$)
20: until convergence is true for both models
21:end function
Algorithm 2 Pseudocode of supervised GTM-TT with relevance learning
Usually, we alternate between one EM step, one epoch of gradient descent, and
normalization in our experiments and start the metric learning after $10$
epochs of EM learning to allow a reasonable pre-positioning of the GTM-TT in
the dataspace. The metric learning is annealed by $\epsilon$. Since EM
optimization is much faster than gradient descent, this way, we can enforce
that the metric parameters are adapted on a slower time scale. Hence we can
assume an approximately constant metric for the EM optimization, i.e. the EM
scheme optimizes the likelihood as before. Metric adaptation takes place
considering quasi stationary states of the GTM solution due to the slower time
scale. The call of train_single_step is a regular EM optimization step of the
GTM-TT but without the adaptation of the parameter $\beta$ which is postponed
to allow a linking between the two GTM-TT models included in the SGTM-TT.
Now, we briefly review a concrete cost function $E$ of the relevance GTM for
the metric adaptation as already introduced in [7] but account for the
alternative distance calculations mentioned before.
#### Cost function - Generalized Relevance GTM (GRGTM)
Metric parameters have the form $\boldsymbol{\lambda}$ or
$\boldsymbol{\lambda}^{k}$ for a diagonal metric (8) and $\boldsymbol{\Omega}$
or $\boldsymbol{\Omega}^{k}$ for a full matrix (9), depending on whether a
local or global scheme is considered. In the following, we define the general
parameter $\Theta^{k}$ which can be chosen as one of these four possibilities
depending on the given setting. Thereby, we can assume that $\Theta^{k}$ can
be realized by a matrix which has diagonal form (for relevance learning) or
full matrix form (for matrix updates).
The cost function of generalized relevance GTM is taken from generalized
relevance learning vector quantization (GRLVQ), which can be interpreted as
maximizing the hypothesis margin of a prototype based classification scheme
[10, 18]. The cost function has the form
$E(\Theta)=\sum_{n}E_{n}(\Theta)=\sum_{n}\operatorname{sgd}\left(\frac{d_{\Theta^{+}}(\mathbf{x}^{n},\hat{\mathbf{x}^{n}}^{+})-d_{\Theta^{-}}(\mathbf{x}^{n},\hat{\mathbf{x}^{n}}^{-})}{d_{\Theta^{+}}(\mathbf{x}^{n},\hat{\mathbf{x}^{n}}^{+})+d_{\Theta^{-}}(\mathbf{x}^{n},\hat{\mathbf{x}^{n}}^{-})}\right)$
(17)
where $\operatorname{sgd}(x)=(1+\exp(-x))^{-1}$, $\hat{\mathbf{x}^{n}}^{\pm}$
is the reconstruction of $\mathbf{x}^{n}$ over time using the model $M_{0}$ or
$M_{1}$ depending on the label of $\mathbf{x}$, $+$ indicates the model with
the same level $-$ the model with a different label or the model for the
remaining data.
The adaptation formulas can be derived thereof by taking the derivatives with
respect to the metric parameter. Depending on the form of the metric, the
derivative of the metric is simple
$\frac{\partial
d_{\boldsymbol{\lambda}}(\mathbf{x},\hat{\mathbf{x}^{n}})}{\partial\lambda_{i}}=2\lambda_{i}d^{t}(x_{i},\hat{x^{n}}_{i})^{2}$
(18)
for a diagonal metric and
$\frac{\partial
d_{\boldsymbol{\Omega}}(\mathbf{x},\hat{\mathbf{x}^{n}})}{\partial\Omega_{ij}}=2d^{t}(x_{j},\hat{x^{n}}_{j})\sum_{d}\Omega_{id}d^{t}(x_{d},\hat{x^{n}}_{d})$
(19)
for a full matrix.
For simplicity, we denote the respective squared distances to the closest
correct and wrong model, respectively, by
$d^{+}=d_{\Theta^{+}}(\mathbf{x}^{n},\hat{\mathbf{x}}^{+})$ and
$d^{-}=d_{\Theta^{-}}(\mathbf{x}^{n},\hat{\mathbf{x}}^{-})$. The term
$\operatorname{sgd}^{\prime}$ is a shorthand notation for
$\operatorname{sgd}^{\prime}((d^{+}-d^{-})/(d^{+}+d^{-}))$. Given a data point
$\mathbf{x}^{n}$ the derivative of the corresponding summand of cost function
$E$ with respect to metric parameters yields
$\frac{\partial{E_{n}}}{\partial\Theta^{+}}=2\operatorname{sgd}^{\prime}\cdot\frac{d^{-}}{(d^{+}+d^{-})^{2}}\cdot\frac{\partial
d^{+}}{\partial\Theta^{+}}$ (20)
for the parameters of the closest correct prototype and
$\frac{\partial{E_{n}}}{\partial\Theta^{-}}=-2\operatorname{sgd}^{\prime}\cdot\frac{d^{+}}{(d^{+}+d^{-})^{2}}\cdot\frac{\partial
d^{-}}{\partial\Theta^{-}}$ (21)
for the parameters attached to the closest wrong model. All other parameters
are not affected. As pointed out before we choose only a global metric such
that the update corresponds to the sum of these two derivatives.
#### Distance measure for functional data
(a) Two functions: Euc = $L^{p}$-norm (b) Two functions: Euc $\neq$
$L^{p}$-norm
Figure 3: Illustration of the $L^{p}$-norm. Plot (a) indicates the case in
which the distance between two functions is equal, both for Euclidean or
$L^{p}$-norm. In plot (b) parts of the functions are interchanging (crossing).
The distance using Euc is still the same as in plot (a) but for the
$L^{p}$-norm the distance is changed, giving a more realistic measure of the
distance of the two functions.
Here we consider a functional distance measure as an extension of the $L^{p}$
norm proposed in ([12]) subsequently denoted as (FUNC). The functional
distance measure has the advantage of taking the functional nature of the data
into account, or in our case the dependence over time, which also constitutes
a function $f(t)$, with potentially discrete arguments $t$. It has been
already successfully used for the analysis of biomedical data as shown in
[16]. The standard Euclidean distance considers the individual features of a
signal independent, so that a change in the order of the positions does not
affect the calculated distance. However, the measurement points over time are
not independent, so that a distance taking this aspect into account can be
considered to be more appropriate for this type of data. Lee proposed a
distance measure taking the functional structure into account by involving the
previous and next values of a signal $v_{i}$ in the $i$-th term of the sum,
instead of $v_{i}$ alone. Assuming a constant sampling period $\tau$, the
proposed norm (FUNC) is:
$\mathcal{L}_{p}^{fc}\left(\mathbf{v}\right)=\left(\sum_{k=1}^{D}\left(A_{k}\left(\mathbf{v}\right)+B_{k}\left(\mathbf{v}\right)\right)^{p}\right)^{\frac{1}{p}}$
(22)
with
$\displaystyle
A_{k}\left(\mathbf{v}\right)=\begin{cases}\frac{\tau}{2}|v_{k}|&\text{if
}0\leq v_{k}v_{k-1}\\\
\frac{\tau}{2}\frac{v_{k}^{2}}{|v_{k}|+|v_{k-1}|}&\text{if
}0>v_{k}v_{k-1}\end{cases}$ (23) $\displaystyle
B_{k}\left(\mathbf{v}\right)=\begin{cases}\frac{\tau}{2}|v_{k}|&\text{if
}0\leq v_{k}v_{k+1}\\\
\frac{\tau}{2}\frac{v_{k}^{2}}{|v_{k}|+|v_{k+1}|}&\text{if
}0>v_{k}v_{k+1}\end{cases}$ (24)
representing the triangles on the left and right sides of $v_{i}$ and $D$
being the data dimensionality. For the data considered in this paper $v$ is a
time series or a prototype reconstruction. As for $L_{p}$, the value of $p$ is
assumed to be a positive integer. At the left and right extremes of the
sequence, $v_{0}$ and $v_{D}$ are assumed to be equal to zero. The concept of
the $L^{p}$-norm is shown in Figure 3. The calculation of this norm is
slightly more complex than that of the standard Euclidean.
### 2.7 Data set description
Subsequently we consider two data sets to evaluate our approach.
#### 2.7.1 Simulated data sets
The first one is a simulated two class scenario, proposed in the paper of
[13]. It consists of $100$ samples divided into two classes of $50$ samples
each. For each sample $100$ features have been generated with $8$ time points.
Out of the $100$ features, only $10$ where substantially differentiating
between the classes. The generation mechanism behind the simulated data is to
sample the time series from a piecewise linear function. At a later step,
sample-specific variation is included by shrinking and expanding the curves.
#### 2.7.2 Multiple sclerosis data
The second data set is taken from [2] (IBIS) in the prepared form, given in
[6]. The data are taken from a clinical study analyzing the response of
multiple sclerosis (MS) patients to the treatment. Blood sample entrenched
with mono-nuclear cells from $52$ relapsing-remitting MS patients were
obtained $0,3,6,7,12,18$ and $24$ months after initiation of IFN$\beta$
therapy. This resulted on an average of $7$ measurements across the $2$ years.
Expression profiles were obtained using one-step kinetic reverse-transcription
PCR over $70$ genes selected by the specialists to be potentially related to
IFN$\beta$ treatment. Overall, $8\%$ of the measurements were missing due to
patients missing the appointments. After the two year endpoint, patients were
classified as either good or bad responders, depending on strict clinical
criteria. Bad responders were defined as having suffered two or more relapses
or having a confirmed increase of at least one point on the expanded
disability status scale (EDSS). A good responder was to have a suppression of
relapses and not allowed to have an increase on the EDSS. From the $52$
patients, $33$ were classified as good and $19$ as bad responders. A more
detailed description of the data set is available in the paper of [2] and the
supplemented material, therein.
## 3 Results and Discussion
For the simulated and the MS data set, we reanalyzed the classification
accuracy of the SGTM-TT with $9$ hidden states and $4$ basis functions. The
analysis was done within a $4$ fold cross-validation with $5$ repetitions. We
compared it with the general HMM classifier (HMM-Lin) and the discriminative
HMM classifier (HHM-Disc-Lin) proposed in [13]. We also included the results
of [2] who originally proposed the MS study, the analysis of [1], employing a
Kalman Filter combined with an SVM approach and [6] proposing a semi-
supervised analysis coupled with a wrapper and cut-off technique to identify
discriminating features.
Figure 4: Relevance profile as obtained using SGTM-TT with relevance
learning. The plot shows the average relevance (blue/dark), minimal relevance
(green/bright) and the standard deviation of the relevance, flipped to the
negative part of the relevance axis. We observe that the standard-deviation is
relatively small, hence the relevance profiles of different runs are very
stable. The most discriminative features (high-relevance), can in parts also
be found in [6] but some additional features appear to be relevant, and our
proposed set consists of $7$ genes rather $17$ like in [6]
### 3.1 Simulated data
We applied SGTM-TT with relevance learning for the simulated data set of [13].
We observed an overall prediction accuracy of $94\pm 4$. The relevance profile
identified all known $10$ features and effectively pruned out the remaining
irrelevant data dimensions. Our results are slightly better than those
reported in [13] $(90\%)$ and by [6] $(92\%)$.
### 3.2 Multiple sclerosis experiment
Method | Number of genes | Test accuracy (%)
---|---|---
SGTM-TT | $70$ | $85.66\pm 8.3$
SGTM-TT-R | $7$ | $93.43\pm 5.8$
IBIS | $3$ | $74.20$
Kalman-SVM | - | $87.80$
Lin-Best | $7$ | $85.00$
Costa-Best | $17$ | $92.70\pm 6.1$
Table 1: Prediction accuracies on the test data for different models using
the MS data set. We observe improved predicition accuracy employing feature
selection. This is also true for SGTM-TT which improved by $\approx 6\%$ using
relevance learning and the SVM classifier. Interestingly also the prediction
accuracy on the full data set, including all features and without relevance
learning is quite good with nearly $84\%$ and hence close to the best result
proposed in [13].
In Table 1 we have summarized the prediction (test-set) results for the
classification of the MS data set in comparison to the results given in [2].
The obtained mappings of the SGTM-TT are topology preserving222In our
observations the topographic error was reasonable small. and we analyzed the
mapping of the points to its prototypes and the neighborhoods. The map for the
first class is depicted for two temporal sequences in Figure 5.
Figure 5: Illustration of the $3\times 3$ SGTM-TT mapping for the responder
class. Plots in the first row show a typical state sequences. Also if the
state sequences $Z$ are not identical we can expect that the underlying
signals $X$ are similar due to its close neighborhood on the map. This is
reflected by such clustered signals at the bottom. The start of a sequence is
indicated by $\square$ and the termination state by a $\circ$.
As expected, results improved by integration of feature selection or relevance
learning compared to the full feature set. Overall the SGTM-TT with relevance
learning performed very well and achieved good results of $92.5\%$ with
respect to the best reported model and also a smaller number of necessary
features. 333We would like to stress that due to the small sample size and the
$4$ fold cross-validation a missclassification of $1$ point, accounts an error
of $8\%$.. Further the integrated relevance learning avoids multiple, time
consuming runs within a wrapper approach like for the techniques used in [13,
6]. The obtained relevance profile is depicted in Figure 4 and provides direct
access to an interpretation of the relevant features, or marker-candidates,
pruning irrelevant or noise dimensions. The values of the relevance profile
are roughly gaussian distributed with $\mu=0.1$. We calculate a threshold
$\zeta$ for the most relevant features using $\zeta=\mu+\sigma$ and obtain $7$
most relevant features, summarized in Table 2.
Genes | Relevance | found by Lin (7) | found by Costa (17)
---|---|---|---
MAP3K1 | 0.3014 | X | X
NFkBIB | 0.2609 | - | -
IRF8 | 0.2584 | - | X
Caspase 10 | 0.2471 | X | X
Jak2 | 0.1869 | X | X
FLIP | 0.1842 | - | -
RIP | 0.1647 | - | -
Table 2: Most relevant genes using SGTM-TT with relevance learning.
The SGTM-TT also inherently models different subgroups by the probabilistic
regularizing model of the GTM and GTM-TT [4, 19]. Hence the model complexity
is not so critical provided the map is reasonable large. This is a plus with
respect to the approach presented in [6] which has the number of groups as an
additional meta parameter.
## 4 Conclusion
We have presented a theoretically sound approach for the analysis of short
temporal sequences. It is based on the novel idea to introduce supervision and
relevance learning into Generalized Topographic Mapping through time. Our
results show that we are able to achieve improved or similar performance to
alternative methods for the simulated and the MS data set. Further the
prototype concept of the underlying method permits a better understanding of
the model and extended visualization performance. We also obtain a direct
ranking of the individual features employing the relevance profile, rather by
use of wrapper techniques. In future work we will explore more advance metric
adaptation schemes and alternative functional distance measures. Further we
would like to apply our approach to non-clinical data and make it more
flexible with respect to missing values.
## Acknowledgment
The authors thank: Peter Tino, University of Birmingham for interesting
discussions about probabilistic modeling and support during the early stage of
this project and Falk Altheide, University of Bielefeld and Tien-ho Lin,
Carnegie Mellon University, USA for support with the simulation data. We would
also give extra thanks to Ivan Olier, University of Manchaster, UK; Iain
Strachan, AEA Technology, Harwell, UK and Markus Svensen, Microsoft Research,
Cambridge, UK for providing code and support with the GTM and GTM-TT.
##### Funding:
This work was supported by the German Res. Fund. (DFG), HA2719/4-1 (Relevance
Learning for Temporal Neural Maps) and by the Cluster of Excellence 277
Cognitive Interaction Technology funded in the framework of the German
Excellence Initiative.
## References
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* [2] Sergio E Baranzini, Parvin Mousavi, Jordi Rio, Stacy J Caillier, Althea Stillman, Pablo Villoslada, Matthew M Wyatt, Manuel Comabella, Larry D Greller, Roland Somogyi, Xavier Montalban, and Jorge R Oksenberg. Transcription-based prediction of response to ifnβ using supervised computational methods. PLoS Biol, 3(1):e2, 12 2004.
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* [5] Karsten M. Borgwardt, S. V. N. Vishwanathan, and Hans-Peter Kriegel. Class prediction from time series gene expression profiles using dynamical systems kernels. In Altman et al. [1], pages 547–558.
* [6] Ivan G. Costa, Alexander Schönhuth, Christoph Hafemeister, and Alexander Schliep. Constrained mixture estimation for analysis and robust classification of clinical time series. Bioinformatics, 25(12), 2009.
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* [8] Christoph Hafemeister, Ivan G. Costa, Alexander Schönhuth, and Alexander Schliep. Classifying short gene expression time-courses with bayesian estimation of piecewise constant functions. Bioinformatics, in press, 2011.
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* [13] Tien-ho Lin, Naftali Kaminski, and Ziv Bar-Joseph. Alignment and classification of time series gene expression in clinical studies. In ISMB, pages 147–155, 2008.
* [14] Iván Olier and Alfredo Vellido. Advances in clustering and visualization of time series using gtm through time. Neural Networks, 21(7):904–913, 2008.
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* [16] F.-M. Schleif, T. Riemer, U. Börner, and L. Schnapka-Hille M. Cross. Genetic algorithm for shift-uncertainty correction in 1-D NMR based metabolite identifications and quantifications. Bioinformatics, 27(4):524–533, 2011.
* [17] P. Schneider, M. Biehl, and B. Hammer. Distance learning in discriminative vector quantization. Neural Computation, 21:2942–2969, 2009.
* [18] P. Schneider, K. Bunte, H. Stiekema, B. Hammer, T. Villmann, and M. Biehl. Regularization in matrix relevance learning. IEEE Transactions on Neural Networks, 21:831–840, 2010.
* [19] I. G. D. Strachan. Latent Variable Methods for Visualization Through Time. PhD thesis, University of Edinburgh, Edinburgh, UK, 2002.
* [20] M. Strickert and B. Hammer. Merge SOM for temporal data. Neurocomputing, 64:39–72, 2005.
* [21] Vladimir N. Vapnik. The nature of statistical learning theory. Springer New York, Inc., New York, NY, USA, 1995.
* [22] Thomas Villmann, Ralf Der, Michael Herrmann, and Thomas M Martinetz. Topology preservation in self-organizing feature maps: exact definition and measurement. IEEE Transactions on Neural Networks, 8(2):256–266, 1997.
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|
arxiv-papers
| 2011-10-11T16:19:06 |
2024-09-04T02:49:23.005123
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "F.-M. Schleif, A. Gisbrecht, B. Hammer",
"submitter": "Frank-Michael Schleif",
"url": "https://arxiv.org/abs/1110.2416"
}
|
1110.2427
|
# Thermodynamics of elementary excitations in artificial magnetic square ice
R.C. Silva1, F.S. Nascimento1, L.A.S. Mól1, W.A. Moura-Melo1 and A.R. Pereira1
apereira@ufv.br 1Departamento de Física, Universidade Federal de Viçosa,
Viçosa, 36570-000, Minas Gerais, Brazil
###### Abstract
We investigate the thermodynamics of artificial square spin ice systems
assuming only dipolar interactions among the islands that compose the array.
The emphasis is given on the effects of the temperature on the elementary
excitations (magnetic monopoles and their strings). By using Monte Carlo
techniques we calculate the specific heat, the density of poles and their
average separation as functions of temperature. The specific heat and average
separation between monopoles with opposite charges exhibit a sharp peak and a
local maximum, respectively, at the same temperature, $T_{p}\approx
7.2D/k_{B}$ (here, $D$ is the strength of the dipolar interaction and $k_{B}$
is the Boltzmann constant). As the lattice size is increased, the amplitude of
these features also increases but very slowly. Really, the specific heat and
the maximum in the average separation $d_{max}$ between oppositely charged
monopoles increase logarithmically with the system size, indicating that
completely isolated charges could be found only at the thermodynamic limit. In
general, the results obtained here suggest that, for temperatures $T\geq
T_{p}$, these systems may exhibit a phase with separated monopoles, although
the quantity $d_{max}$ should not be larger than a few lattice spacings for
viable artificial materials.
###### pacs:
75.75.-c, 75.60.Ch, 75.60.Jk
††: NJP
## 1 Introduction
New methods for exploring geometric frustrations in magnetic systems have been
developed recently. Such methods consist in creating arrays of nanomagnets
designed to resemble the disordered magnetic state known as spin ice. They are
essentially composed of lithographically defined two-dimensional ($2d$)
ferromagnetic nanostructures (elongated permalloy nanoparticles) with single-
domain elements organized in diverse types of geometries (square lattice [1],
hexagonal, brickwork [2], kagome [3, 4] etc). Since their geometries are
determined lithographically, lattice symmetry and topology can be directly
controlled, allowing experimental investigation of a vast set of important
theoretical models of statistical physics [5]. These artificial magnetic
compounds have the potential of increasing our understanding of disordered
matter and may also lead to new technologies. Therefore, artificial spin ices
are object of intense theoretical and experimental investigations [1, 2, 3, 4,
6, 7, 8, 9, 10, 11, 12, 13, 14].
The trouble is that, in artificial spin ice patterns, the magnetization is
unaffected by thermal fluctuations because the magnetic islands contain a
large number of spins. Despite the fact that the moment configuration is
athermal, these artificial materials can be described through an effective
thermodynamics formalism [15, 16]; in addition, some works have introduced a
predictive notion of effective temperature [7, 16]. For instance, an external
drive, in the form of an agitating magnetic field behaves as a thermal bath
and controls the temperature [7, 16]. Alternatively, this problem was
addressed very recently by using a material with an ordering temperature near
room temperature [17]; such experimental work on a square lattice in an
external magnetic field confirms a dynamical ”pre-melting” of the artificial
spin ice structure at a temperature well below the intrinsic ordering
temperature of the island material, creating a spin ice array that has real
thermal dynamics of the artificial spins over an extended temperature range
[17]. These findings and also other future possibilities make evident that a
more detailed analysis of the effects of thermal fluctuations on a lower
dimensional spin ice material should be of large interest for a better
understanding of these frustrated systems. In particular, it would also be
important to know the roles of elementary excitations in the thermodynamic
properties of artificial magnetic ices.
The main aim of this work is exactly this investigation. We are interested in
the temperature effects on the excitations (“magnetic monopole defects” and
their strings). Actually, since the prediction of monopoles in the usual
three-dimensional ($3d$) spin ice materials [18] and their experimental
detection [19, 20, 21, 22, 23], the search for these objects in artificial
compounds has become an important issue [8, 11, 10, 3]. The possible existence
of these excitations in artificial and controllable systems is of great
interest because they could be studied at room temperature and, more
important, they could be directly observed with modern experimental
techniques. Curiously, in the case of artificial systems, while the square
lattice was the first to be produced [1], the direct observation of magnetic
monopole defects and their motion was firstly accomplished in a kagome
geometry [3]. Still, in this kagome lattice, a direct, real-space observation
of the interplay of Dirac strings and monopoles was reported by Mengotti et.al
[4]. For a square lattice, the direct observation of such excitations came
only afterward because there was a primary experimental problem: until last
year, none of such systems had achieved its ground state through thermodynamic
equilibrium [13]. Despite predictions[6, 8, 9], the studies till recently do
not have shown a long-range ordered configuration, perhaps because the
researchers have used only non-thermal methods to randomize the array. This
problem was experimentally solved by Morgan et. al. [10]. These authors have
reported that by allowing the magnetic islands to interact as they are
gradually formed at room temperature, the artificial square spin ice can be
effectively thermalized, allowing it to find its predicted ground state very
closely; thus, they could also identify the small departures from the ground
state as elementary excitations of the system, at frequencies that follow a
Boltzmann law. Subsequently, Magnetic Force Microscopy (MFM) images of a large
number of isolated excitations with their string shapes and corresponding
moment flip maps were described in square lattices [10]. Therefore, the
experimental results considering magnetic artificial square ices obtained in
Ref.[10] (which demonstrates the thermal ground-state ordering and the
elementary excitations) and Ref.[17] (which achieves a thermodynamic melting
transition by using a material with ordering temperature near room
temperature) lead us to think that more progress on the development of such
arrays may become available in the near future, establishing opportunities to
experimentally elucidate their real thermodynamics.
Figure 1: (Color online) Specific heat as a function of temperature. It
exhibits a sharp peak, at a temperature $T_{p}\sim 7.2D/k_{B}$, which the
amplitude increases very slowly with the system size $L$. Inset: the specific
heat peak diverges logarithmically with the system size $L$.
## 2 The model and outlook
Here, we consider an arrangement of dipoles similar to that experimentally
investigated in Ref. [1]. In our approach, however, the magnetic moment
(“spin”) of the island is replaced by an Ising-like point dipole at its
center. In this approach, the internal degrees of freedom of each island are
not being considered, as well as higher order interactions. We expect that
this simplification does not change significantly the main physical properties
of the system. As shown in Ref. [24], if the lattice spacing is about two
times larger than the island’s longest axis, the effect of higher order
interactions is negligible. For smaller lattice spacings the effect of higher
order interactions is to give more stability for the lowest energy states. In
this way one may expect that as the island size increases, approaching the
lattice spacing, the ground-state should be more robust and the appearance of
excitations would cost more energy. While the consideration of the internal
degrees of freedom would reduce the energy scale, the consideration of higher
order interactions would increase it, but none of them are expected to change
the physical picture discussed here. Thus, in our approach, at each site
$(x_{i},y_{i})$ of the square lattice two spin variables are defined:
$\vec{S}_{x(i)}$ with components $S_{x}=\pm 1$, $S_{y}=0,S_{z}=0$ located at
$\vec{r}_{x}=(x_{i}+1/2,y_{i})$, and $\vec{S}_{y(i)}$ with components
$S_{x}=0$, $S_{y}=\pm 1,S_{z}=0$ at $\vec{r}_{y}=(x_{i},y_{i}+1/2)$.
Therefore, in a lattice of volume $L^{2}=l^{2}a^{2}$ ($a$ is the lattice
spacing) one gets $2\times l^{2}$ spins. Representing the spins of the islands
by $\vec{S}_{i}$, which can assume either $\vec{S}_{x(i)}$ or
$\vec{S}_{y(i)}$, then the artificial spin ice is described by the following
Hamiltonian
$\displaystyle H_{SI}$ $\displaystyle=$ $\displaystyle Da^{3}\sum_{i\neq
j}\left[\frac{\vec{S}_{i}\cdot\vec{S}_{j}}{r_{ij}^{3}}-\frac{3(\vec{S}_{i}\cdot\vec{r}_{ij})(\vec{S}_{j}\cdot\vec{r}_{ij})}{r_{ij}^{5}}\right],$
(1)
where $D=\mu_{0}\mu^{2}/4\pi a^{3}$ is the coupling constant of the dipolar
interaction. We perform standard Monte Carlo techniques to obtain
thermodynamic averages of the system defined by Hamiltonian (1). Periodic
boundary conditions were implemented by means of the Ewald Summation [25, 26],
used here to avoid spurious results brought about by the use of a cut-off
radius[27]. Our Monte Carlo procedure comprises a combination of single spin
flips and loop moves [28], where all spins contained in a closed random loop
are flipped according to the Metropolis prescription. In our scheme one Monte
Carlo step (MCS) consists of $2\times l^{2}$ single spin flips and $0.7\times
l^{2}$ worm moves. Usually, $10^{4}$ MCS were shown to be sufficient to reach
equilibrium configurations and we have used $10^{5}$ configurations to get
thermodynamic averages.
Figure 2: (Color online) Density of pairs of unit-charged monopoles as a
function of temperature. Inset: density of doubly charged monopole pairs.
Figure 3: (Color online) The average separation between charges exhibits a
maximum around the same temperature $T_{p}$ in which the specific heat has a
sharp feature. The inset shows, in more details, the region around the
maximum.
Before presenting the Monte Carlo calculations, it would be interesting to
remark on some previous results [8, 11, 9] and some expectations for these
arrays. The ground state configuration of the system in a square lattice is
twofold degenerate. If one considers the vorticity in each plaquette,
assigning a variable $\sigma=+1$ and $-1$ to clockwise and anticlockwise
vorticities respectively, the ground state looks like a checkerboard, with an
antiferromagnetic arrangement of the $\sigma$ variable [8, 10]. Of course, the
ground state clearly obeys the ice rule (two spins point inward and two point
outward in each vertex), but with configurations of topology $1$ (in $2d$,
there are two topologies that obey the ice rule. However, they are not
degenerate and topology $2$ is more energetic than topology $1$; see Refs.[1,
8] for more details). The most elementary excitation is related to the
inversion of a single spin (dipole) to generate a localized pair of defects.
This is the $3-in$, $1-out$ state in a particular vertex and the $3-out$,
$1-in$ state in its adjacent vertex. In principle, these defects could be
separated without further violation of the ice rule. Indeed, in our previous
papers [8, 9], we have numerically shown that these defects behave as a
monopoles pair since their interaction follows a $d=3$ Coulomb law $q/R$,
where $q$ measures the strength of the interaction and $R$ is the distance
between the poles. However, we have also pointed out that an isolated monopole
should be hard to see as effective low-energy degrees of freedom in the $2d$
square spin ice because the background antiferromagnetic order in the ground
state confines them [8], since the ice rule is not degenerate in two
dimensions. Actually, in $2d$, there are additional excitations not present in
the usual $3d$ spin ice [18], namely, energetic one-dimensional strings of
dipoles (resultant spins at each vertex along a line of adjacent vertices)
that terminate in monopoles with opposite charges. Such string excitations
could be seen as lines which pass by adjacent vertices that obey the ice rule
but sustaining topology $2$ (instead of topology $1$) and hence they cost an
energy equal to $b$ times their length $X$, where $b$ is the string tension.
When the temperature $T$ of the system is near absolute zero, the shortest
path length connecting the monopoles gives the potential energy. The most
general expression for the total cost of a pair of monopoles separated by a
distance $R$ is the sum of the usual Coulombic term roughly equal to $q/R$,
and a term roughly equal to $bX$ resulting from the string joining the
monopoles (there is, of course, also a constant term associated with the
creation energy of a pair). Note that there is not a unique identification of
a given path connecting the ends (monopoles) of the excitation. It is
explicitly considered in the fact that the energy is proportional to $X$,
which can assume different values for a given $R$. For a sufficiently long
string, the string energy is completely dominant; for a short string the
Coulomb interaction may have some importance if the size of the end-point
monopoles is even smaller (as always occur for these systems). With the above
features, these excitations are, to some extent, more similar to Nambu
monopoles [29] than Dirac monopoles. Really, as Nambu suggested, for a
modified Dirac monopole theory, the string connecting monopoles has energy and
is oriented, having a sense of polarization[29].
In the artificial square ices, the ordering causes an anisotropy in the system
making the monopoles interaction highly dependent on the direction in which
the monopoles are separated in the crystal plane [9]. This anisotropy is
manifested in both the Coulomb and linear terms of the potential in such a way
that we explicitly write [9]
$\displaystyle V(R)=q(\phi)/R+b(\phi)X+c$ (2)
where $\phi$ is the angle that the line joining the monopole defects makes
with the $x$-axis of the array. Numerically, for instance,
$q(0)\approx-3.88Da$, $b(0)\approx 9.8D/a$ while $q(\pi/3)\approx-4.1Da$,
$b(\pi/3)\approx 10.1D/a$. The constant $c\approx 23D$, associated with the
pair creation energy [9] ($E_{c}\approx 29D$) is independent of $\phi$.
Similar results can be found in the experimental work for the square lattice.
Indeed, in Ref.[10], the authors have found that, at a temperature $T$, these
excitations arise in the system according to the Boltzmann law
$\sim\exp(-\beta V(R))$ with $b\approx 10D/a$, $V(a)=E_{c}\approx 30D$ and
$\beta=1/k_{B}T$, where $k_{B}$ is the Boltzmann constant. They have also
classified the elementary excitations by the number of flipped spins (given by
$n$) and a mnemonic character for shape. The three most observed defects are
represented by $1$ (a single pair with charges separated by only one lattice
spacing) followed by $2L$ (a pair with $n=2$ with the shape of $L$) and $4O$
(an isolated string loop with no charges and having $n=4$ flipped spins) [10].
Curiously, the second excited state should be $4O$ since its energy is smaller
than the energy of $2L$ defect.
Figure 4: (Color online) The maximum of the average separation $d_{max}$
between opposite charges increases logarithmically with the system size $L$.
Figure 5: (Color online) Snapshot of a particular configuration of
excitations for a temperature $T=6.0D/k_{B}$ in a lattice with $L=10a$. Red
and black circles are positive and negative charges respectively. In general,
for all temperatures below $T_{p}$, each monopole is clearly confined to its
counterpart by a string (see the blue arrow indicating the direction of the
string for the larger pair. Small pairs (i.e., monopoles bound together
tightly in pairs) are indicated by a green arrow. Figure 6: (Color online)
Snapshot of a particular configuration of excitations for a temperature
$T=7.6D/k_{B}$ in a lattice with $L=10a$. Red and black circles are positive
and negative charges respectively. For a temperature above $T_{p}$, a small
amount of monopoles does not have a string connecting them to their
counterparts and, therefore, they seem to be isolated. There are also some
pieces of strings (i.e., one-dimensional regions obeying topology $2$, as
indicated by blue paths) that do not connect monopoles. Small pairs are
indicated by a green arrow.
In principle, for the thermodynamics of these systems, the following argument
should be valid: at low temperatures, there is insufficient thermal energy to
create long strings (with length $X$ larger than one lattice spacing) and so,
the monopoles (with opposite charges) are bound together tightly in pairs. On
the other hand, as the temperature is increased, the average separation
between the constituents of a pair should also increase, which means that
larger strings may become present in the system. Of course, there are several
ways of connecting two monopoles by a string of length $X$. Therefore,
considering states with $X>>R$, we remember then that the number of
configurations for the $m$-step self-avoiding random walk is $N=\delta^{m}$,
where $\delta$ is a constant and equal to $3$ for a $2d$ square lattice. For
the string with sufficient large $X$, $N$ is well approximated by the random
walk result and one obtains $N\simeq\delta^{X/a}$. So the entropy of strings
is proportional to $X$, i.e., the many possible ways of connecting two
monopoles with a string give rise to a string configurational entropy
proportional to $X$. Crudely speaking, then, the string free energy $F=[b-(\ln
3)k_{B}T/a]X$ will imply in an effective string tension $[b-(\ln 3)k_{B}T/a]$
which is positive in the low temperature region and the monopoles are
completely confined. Above a certain temperature, it becomes negative, namely,
the string looses its tension. The tension decreases like $[b-(\ln
3)k_{B}T/a]$ with increasing $T$, vanishing at some critical temperature
$k_{B}T_{c}\approx ba/\ln 3$. Using the average value for the string tension
in Eq.(2), i.e., $b\approx 10D/a$, we then estimate $k_{B}T_{c}\approx 9.1D$.
Of course, these theoretical arguments always overestimate the critical
temperature. Although this picture leads to a rich physics for this system,
predicting free magnetic monopoles and a phase transition, things may be a
little more complicated. Really, additionally to the entropic effect discussed
just above, there is another entropic contribution which manifests against
monopole separation; the monopoles should become close together because it
would provide more ways to arrange the surrounding dipoles in the lattice.
Such effect introduces a $2d$ Coulombic interaction between the poles, which
is proportional to $T$ (i.e., $V_{s}=T\ln(R/a)$). If the temperature in which
the string looses its tension is high enough, on the order of $9.1D$ as
estimated, then, around this value of $T$, the confining potential $V_{s}$
must be very strong, possibly preventing the freedom for the poles. With all
these expectations, it would be important to investigate how the elementary
excitations behave as a function of temperature. Our calculations is a first
step in this direction.
## 3 Results
Now we present the results from Monte Carlo Simulations. The calculations
shown here are for lattices with sizes $10,20,30,40,50,60$ and $70$ lattice
spacings but in all figures we present only the results for lattice sizes
$40,60,70$. We start by presenting the results for the specific heat (see
Fig.1). We notice that, for all lattice sizes studied, the specific heat
exhibits a sharp feature at a temperature $T_{p}$ approximately equal to
$7.2D/k_{B}$. Indeed, the position of this peak does not seem to move as the
lattice size $L$ is varied. On the other hand, its amplitude $C_{max}$
increases much slowly as $L$ increases. In the inset of Fig.1, we show how
$C_{max}$ behaves with $L$. Therefore, with the obtained data we expect a
logarithmic divergence of the specific heat in the thermodynamic limit. We
also analyzed the pair density and the average separation between monopoles
with opposite charges as a function of $T$. It is useful here to distinguish
two types of monopoles: the less energetic ones in which the spins (in a
vertex) are in the $3-in$, $1-out$ or $3-out$, $1-in$ states (here referred to
as unit-charged monopoles) and the most energetic ones in which the spins are
in the $4-in$ or $4-out$ states (doubly charged monopoles). Figure 2 shows the
density of pairs containing monopoles with unitary charge ($\rho_{S}$) and
also the density of pairs containing doubly charged monopoles ($\rho_{D}$, see
the inset). They are calculated as the one-half of the thermodynamic average
of the absolute value of the charge ($\pm 1$) and ($\pm 2$) respectively,
summed over the lattice. For both cases, the density increases monotonously up
to a maximum value achieved in the high-temperature limit.
Figure 7: (Color online) The density of string loops $4O$ ($\rho_{O}$) also
exhibits a maximum around the temperature $T_{p}\simeq 7.2D$ (green balls).
This defect carries no charge and is the second excited state. Just for
comparison, the density of pairs with opposite charges ($\rho_{s}$) is also
shown (red balls).
The size of the monopole pairs constitutes an internal degree of freedom,
since the energy of a pair depends on the distance between the members of the
pair. Here we would like to know the average distance $r_{M}$ between two
opposite poles as a function of temperature. Such a thermodynamic quantity may
contain information about the possibility of monopoles separation and how they
are organized into the system. For this calculation we consider only defects
with unitary charges. The grouping of monopoles into pairs is unique as long
as the distances between them are smaller than the average distance between
the monopoles $r_{M}=1/\sqrt{\rho_{S}}$. As the size of the monopole pairs
becomes larger than $r_{M}$, one would simply have to redefine the monopole
pairs. The average size $r_{M}$ of the monopole pairs is calculated by using
the method of assignment problems; it deals with the question of how to assign
$n$ items (jobs, students) to $n$ other items (machines, tasks) [30]. In our
case, we would like to assign $n$ positive charges to $n$ negative charges for
a given configuration in such a way that the sum of distances of all possible
pairing be a minimum. The results are shown in Fig.3. The average separation
has a local maximum at the same temperature $T_{p}$ in which the specific heat
exhibits a peak ($\sim 7.2D/k_{B}$). We notice that the amplitude of this
maximum increases slowly as the system size increases. Indeed, like the
specific heat peak, the maximum in the average separation $d_{max}$ also
increases logarithmically with the system size $L$ ($d_{max}\propto\ln L$, see
Fig.4) and hence, one could expect that a certain quantity of monopoles may be
almost isolated for very large arrays. Indeed, in our simulations for
temperatures $T\geq T_{p}$ considering lattices with $L\leq 80a$, we could
observe some charges relatively distant from their respective counterparts
(separated by distances of the order of $5a$). For instance, we show in Fig. 5
a distribution of positive (red circles) and negative (black circles)
monopoles in a small lattice with $L=10a$ observed in our simulations for a
temperature $T=6.0D/k_{B}$ (i.e., below $T_{p}$). Note that there are few
excitations and all monopoles with opposite charges are coupled by a string,
forming pairs. On the other hand, Fig. 6 shows the same system for a
temperature above $T_{p}$ ($T=7.6D/k_{B}$). In this case, we see that a small
quantity of monopoles are not connected by strings. In principle, they are
free although some of them are not completely isolated (i.e., far away from
other opposite poles). Furthermore, we also notice that some strings seem to
be detached, not terminating in monopoles; there are few pieces of strings
dispersed along the system (as said before, strings could be seen as lines
which pass by adjacent vertices that obey the ice rule but sustaining topology
$2$ rather than topology $1$). Of course, these figures exhibit only samples
from a large number of data, but most of the data should be similar to the
features of Fig.5 for the regime of low temperatures and the features of Fig.6
for the regime of high temperatures. Things must be clearer in the
thermodynamic limit; in this case, some monopoles should become infinitely
separated from their counterpart for temperatures $T\geq 7.2D/k_{B}$. However,
as the temperature is increased from zero, the monopole pair density grows
simultaneously with an increase of the pair size (see also Fig. 2). As the
pairs become denser, there is less space to put in new pairs and hence the
average pair size $r_{M}$ decreases for high temperatures. Really, we observe
that, for $T<T_{p}$ the average separation $r_{M}$ does not depends on the
lattice size $L$, while for $T\geq T_{p}$, this quantity has a tiny dependence
on $L$ (at least in the range $7.2D/k_{B}<T<12D/k_{B}$). In this case, it is
possible that monopoles may become completely isolated even for high
temperatures ($T>T_{p}$) when $L\rightarrow\infty$. This picture for infinite
systems corroborates the theoretical expectations for the existence of a phase
with free monopoles [8] in large $2d$ artificial square ices, but the
transition temperature ($\sim 7.2D/k_{B}$) should be little smaller than the
estimated value $\sim 9.1D/k_{B}$ discussed earlier (remembering that the
arguments of energy-entropy, in general, overestimate the correct quantity).
We have also calculated the density of string loops $4O$, which is the defect
with no charge but having the second lower energy (second excited state). Like
the specific heat and the average separation, the density of defects $4O$ also
displays a feature at $T_{p}$ (see Fig. 7). Note that the string loops $4O$
almost do not appear in the system for temperatures smaller than $T_{p}$.
Indeed, they surge suddenly at $T_{p}$ and then, for temperatures above
$T_{p}$, their number starts to decrease while the density of monopole pairs
starts to increase more appreciable. Figures ( 8) and ( 9) show typical
distributions of defects $4O$ in the system for temperatures below and above
$T_{p}$, respectively.
Figure 8: (Color online) A typical configuration of string loops of the type
$4O$ for a temperature below $T_{p}$ (here, $T=6D/k_{B}$). At $T_{p}$, the
number of $4O$ excitations proliferate in such way that a percolated cluster
seems to be formed. The figure also shows the pairs of monopoles. Figure 9:
(Color online) A typical configuration of string loops of the type $4O$ for a
temperature above $T_{p}$ (here, $T=8D/k_{B}$). The figure also shows the
pairs of monopoles.
## 4 Discussion
In summary, assuming the spin-spin interaction to be purely dipole-dipole, we
notice that, at a temperature $T_{p}$, there is a maximum in the mean
separation of opposite monopoles that increases logarithmically with the
system size $L$ ($d_{max}\propto\ln L$). Hence, the distance between monopoles
with opposite charges in the thermodynamic limit ($L\to\infty$) should diverge
weakly, suggesting a possible unbinding of monopole pairs ($T<T_{p}$) into
”free” monopoles ($T>T_{p}$). However, to the authors knowledge, for a finite
monopole density there is no diagnostic for (de)confinement based on a pair
distribution function, for reasons analogous to the failure of the Wilson loop
(which only knows perimeter laws in the presence of dynamical matter) to
diagnose deconfinement in gauge theories. Indeed, from the three approaches
that have been used to measure the static potential associated with the
breaking of long flux tube between two quarks in QCD (i.e., correlation of
Polyakov loops, variational ansatz and Wilson loops), string breaking has been
seen only using the first two methods. On the other hand, the divergence found
in $r_{M}$ could be understood in two different ways. It may be associated
with either a vanishing string tension (which would lead to effectively free
poles) or simply by the fact that in an order-disorder transition the
correlation length (which is the only characteristic length of the system)
diverges at the critical temperature. In this case, since the mean distance
should be given in terms of the correlation length, then, it should also
diverge. Of course, these two distinct ways to describe the system are closely
related. We are faced thus with the question of the existence or not of a
phase transition in this system. If there is a phase transition, other
question arises: what is its nature? It is worthy to note at this point [31]
that although this system is closely related to the 16-vertex model, for which
an exact solution is known, the range and symmetry of the interactions differ
and thus we do not expect to observe the same critical behavior. Nevertheless,
one point that deserves remark is the possible similarities between this
system and the Ising model. In the two degenerated ground states, the $\sigma$
variables, related to the vorticity of each plaquette, can be seem as the
spins of an antiferromagnetic (AF) Ising model. In the AF Ising model, as the
temperature raises, clusters of flipped spins are found in the system and at
the critical temperature one can find percolated clusters of spins. If there
is some similarities between these systems one may expect thus that the $4O$
excitations, which can be viewed as flipped $\sigma$ variables, form clusters
at low temperature that percolates at the critical temperature, justifying
thus the increasing number of these excitations at the transition temperature.
This picture is corroborated by the logarithmic divergence of the specific
heat. Unfortunately, our results are not conclusive about the possibility of a
phase transition, and much more work has to be done in order to answer this
question. To try to put some extra light on the topic, we have also done some
calculations restricting the islands interaction to nearest neighbors
converging in the same vertex, which would lead to a kind of generalized $2d$
Ising system with the same ground state. Nevertheless, we have obtained that
the vertices with topology $3$, in the $3$-in/$1$-out and $3$-out/$1$-in
states, remain connected by strings (but now, there is no Coulomb interaction
anymore). The interaction energy between two opposite vertices in topology $3$
(type $III$ vertices) is given by $b_{I}X+c_{I}$, where $b_{I}=26D/a$ and
$c_{I}=34D$, much bigger than the usual results obtained for the long ranged
dipolar interaction. Since the string tension remains, the arguments
associated with the string configurational entropy should maintain valid and
we have again the same problem as before (but with different energetics; for
instance, the value of the temperature in which the quantities show a maximum
changes to $16D/k_{B}$). Indeed, the specific heat, the average separation
between opposite type $III$ vertices etc, have the same behavior found for the
system with long-range dipolar interaction (not shown here).
From a practical point of view, the divergence in $r_{M}$ in the thermodynamic
limit, and thus the phase of large separation among monopoles should not be
expected in finite systems. Due to the slow logarithmic divergence, the
extrapolation of our results to a $2d$ lattice containing the Avogadro’s
number ($N_{a}^{2/3}=10^{16}=10^{8}\times 10^{8}$) of islands will imply in
$d_{max}\sim 2.5a$ only. On the other hand, even with small values for
$d_{max}$, some monopoles may become isolated for temperatures near $7.2D$
(see Fig. 6). The challenge of building arrays using new materials (with an
ordering temperature near room temperature ) and/or with reduced island volume
and moment (and possibly with larger $L$) should be then an important issue
for technological applications. Indeed, it concerns with the excitations
evolution in these artificial compounds. These developments may experimentally
determine the possibility of monopole dynamics, their lifetimes and so on. For
instance, based only on the average separation results, we speculate that,
near the temperature $T_{p}$, the annihilation process of monopoles (without
strings) should be more probable to occur in small arrays than in large arrays
due to the fact that the mean separation between such opposite charges
increases with the system size.
The authors thank CNPq, FAPEMIG, CAPES and FUNARBE (Brazilian agencies) for
financial support. We would like to thank Professors R. Moessner and G.M.
Wysin for a careful reading of the manuscript and for helpful comments.
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|
arxiv-papers
| 2011-10-11T16:51:48 |
2024-09-04T02:49:23.015139
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "R. C. Silva, F. S. Nascimento, L. A. S. M\\'ol, W. A. Moura-Melo, A. R.\n Pereira",
"submitter": "Rodrigo Silva",
"url": "https://arxiv.org/abs/1110.2427"
}
|
1110.2440
|
# Electromagnetic emission from hot medium measured by the PHENIX experiment
at RHIC
Takao Sakaguchi for the PHENIX collaboration Brookhaven National Laboratory,
Physics Department, Upton, NY 11973, USA takao@bnl.gov
###### Abstract
Electromagnetic radiation has been of interest in heavy ion collisions because
they shed light on early stages of the collisions where hadronic probes do not
provide direct information since hadronization and hadronic interactions occur
later. The latest results on photon measurement from the PHENIX experiment at
RHIC reflect thermodynamic properties of the matter produced in the heavy ion
collisions. An unexpectedly large positive elliptic flow measured for direct
photons are hard to be explained by many models.
## 1 Introduction
The experiments utilizing relativistic heavy ion collisions have been aiming
to find a new state of matter, quark-gluon plasma (QGP), that should have
existed in the early stage of the Universe (Fig. 2).
Figure 1: Phase diagram of the nuclear matter.
Figure 2: Photon emission in relativistic heavy ion collisions.
The QGP is an interesting state in the sense that it is not only a discovery
subject, but also a unique place to understand the nature of QCD matter, such
as quark confinement or the chiral symmetry restoration. The unique feature of
the study at the Relativistic Heavy Ion Collider (RHIC) at the Brookhaven
National Laboratory is that one can utilize the probe with high $Q^{2}$
(perturbative probe) to investigate the QCD matter in thermal region (low
$Q^{2}$, non perturbative matter).
Many intriguing phenomena have been observed at RHIC since it started of
running in 2000. The high transverse momentum ($p_{T}$) hadron production from
the initial hard scattering was observed, and the large suppression of their
yields suggested that the matter is sufficiently dense to cause parton-energy
loss prior to hadronization [1]. The large elliptic flow of particles and its
scaling in terms of particle kinetic energy suggests that the system is
locally in equilibrium as early as 0.3 fm/c, and the flow occurs already on
the partonic level.
Because they interact with the medium and other particles only
electromagnetically and are largely unaffected by final state interactions,
photons serve as a direct and penetrating probe of the early stages at high
temperature and high density [2]. At leading order, the production processes
of photons are annihilation ($q\bar{q}\rightarrow\gamma g$) and Compton
scattering ($qg\rightarrow\gamma q$) (Figure 4). Their yields are proportional
to $\alpha\alpha_{s}$, which are $\sim$40 times lower than hadrons from strong
interactions.
Figure 3: Photon production process.
Figure 4: Sources of photons from various stages of collisions.
A calculation predicts that the photon contribution from the QGP state is
predominant in the $p_{T}$ range of 1$<p_{T}<$3 GeV/$c$ [3]. For $p_{T}>$3
GeV/$c$, the signal is dominated by a contribution from initial hard
scattering, and $p_{T}<$1 GeV, the signal is from hadron gas through processes
of $\pi\pi(\rho)\rightarrow\gamma\rho(\pi)$, $\pi K^{*}\rightarrow K\gamma$
and etc. Figure 4 shows a landscape of photon sources as a function of the
time they are produced. The vertical axis corresponds to transverse momenta of
photons. We have one another degree of freedom, virtual mass, in photon
measurement, which will be explained in detail in a later section. These
photons can be measured after a huge amount of background photons coming from
hadron decays ($\pi^{0}$, $\eta$, $\eta^{\prime}$ and $\omega$, etc.) are
subtracted off from inclusive photon distributions. The typical signal to
background ratio is $\sim$1 % at 2 GeV, and $\sim$10 % at 5 GeV in case of p+p
collisions. The signal from QGP is predicted to be $\sim$10 % of the inclusive
photons. For Au+Au collisions, thanks to a large suppression of high $p_{T}$
hadrons, the ratio is enhanced by the same degree. PHENIX [4] has measured
photons throughout the first decade of RHIC operations. We present here a
review of the results.
## 2 Measurement of initial hard scattering photons in heavy ion collisions
One of the big successes by now in electro-magnetic radiation measurements is
the observation of high $p_{T}$ direct photons that are produced in initial
hard scattering [5] in relativistic heavy ion collisions. Figure 6 shows the
direct photon spectra in Au+Au collisions at $\sqrt{s_{NN}}$=200 GeV for
different centralities.
Figure 5: Direct photon spectra in Au+Au collisions at $\sqrt{s_{NN}}$=200
GeV.
Figure 6: Nuclear modification factors ($R_{AA}$ for photons, $\pi^{0}$ and
$\eta$ in 10 % central Au+Au collisions at $\sqrt{s_{NN}}$=200 GeV.
The lines show the NLO pQCD calculations [6] scaled by the nuclear thickness
function ($T_{AA}$). The fact that the data are well described by the lines
show that the yields are following the $T_{AA}$ scaling and suggest that the
source is the initial hard scattering. Figure 6 shows the nuclear modification
factors ($R_{AA}$) for direct photons, $\pi^{0}$ and $\eta$ for 0-10 % central
Au+Au collisions at the same center-of-mass (cms) energy. $R_{AA}$ is defined
as the ratio of the yield in nucleus-nucleus collisions divided by that in p+p
collisions scaled by $T_{AA}$. The high $p_{T}$ hadron suppression is
interpreted as a consequence of an energy loss of hard-scattered partons in
the hot and dense medium. It was strongly supported by the fact that the high
$p_{T}$ direct photons are not suppressed and well described by a NLO pQCD
calculation. The small suppression seen in the highest $p_{T}$ is likely due
to the fact that the ratio of the yields in Au+Au to p+p was computed without
taking the isospin dependence of direct photon yields into account [7].
## 3 Measurement of direct photons through its internal conversion
There is a huge background arising from $\pi^{0}$ decaying into two photons,
which makes it very difficult to look at the direct photon signal at low
$p_{T}$, where thermal photons from QGP manifest, with traditional calorimetry
of (real) photons. However, if we look at photons with a small mass (virtual
photons) instead, we can select the mass region where $\pi^{0}$ contribution
ceases (Fig 7). For the case of $p_{T}>>M$, the yield of virtual photons is
expected to be dominated by internal conversion of real photons [8, 9]. For
obtaining direct photon yield, we fit the measured invariant mass distribution
with the function:
$F=(1-r)f_{c}+rf_{d},$
where $f_{c}$ is the cocktail calculation (photons from various hadron
decays), $f_{d}$ is the mass distribution for direct photons, and $r$ is the
free parameter in the fit. Next, using the Kroll-Wada formula [10] to account
for the Dalitz decays of $\pi^{0}$, $\eta$ and direct photons, $r$ is defined
as the ratio of direct photons to inclusive photons:
$r=\frac{\gamma^{*}_{\rm dir}(m_{ee}>0.15)}{\gamma^{*}_{\rm
inc}(m_{ee}>0.15)}\propto\ \frac{\gamma^{*}_{\rm dir}(m_{ee}\approx
0)}{\gamma^{*}_{\rm inc}(m_{ee}\approx 0)}\ =\frac{\gamma_{\rm
dir}}{\gamma_{\rm inc}}\equiv r_{\gamma}$
Then, the invariant yield of direct photons is calculated as $\gamma_{\rm
inc}\times r_{\gamma}$. As described in [9], the procedure is demonstrated in
Fig 7 for 1.0$<p_{T}<$1.5 GeV/$c$.
Figure 7: Invariant mass distributions of electron-pairs and comparison with
possible hadron sources of electron-pairs.
The dotted lines show the contributions from various hadrons, the solid blue
is the sum of these contributions, and the solid red line shows the
distribution from direct photons converted internally. The $r$ value is
determined by the fit to the data. The error of the fit corresponds to the
statistical error. We applied the procedure as a function of $p_{T}$ for
various centrality selections in p+p and Au+Au collisions, and obtained the
$p_{T}$ spectra, as shown in Fig 9.
Figure 8: Direct photon spectra obtained from the measurement of internal
conversion of photons in Au+Au collisions.
Figure 9: Direct photon yield in Au$+$Au and $d$$+$Au collisions scaled by the
difference of $N_{\rm coll}$.
The distributions are for 0–20 %, 20–40 % centrality and MB events for Au$+$Au
collisions. For $p_{T}<$2.5 GeV/$c$ the Au+Au yield are visibly higher than
the scaled p+p yield. The distributions were then fitted with the p+p fit plus
exponential function to obtain slopes and dN/dy for three centralities. The
slopes are estimated to be $\sim$220 MeV. The lines show the theoretical
expectation from a literature [3]. One may question whether or not the excess
arises from a source that exists only in Au$+$Au collisions. For example, an
effect that could increase the yield is cold-nuclear-matter (CNM) effect such
as $k_{T}$ broadening (Cronin effect). To quantify the contribution we
analyzed 2008 $d$$+$Au data with the same procedure [11]. Figure 9 shows the
Au$+$Au yield compared to the $d$$+$Au yield scaled by $N_{\rm coll}$. It
clearly shows that there is an enhancement over CNM effects in Au+Au
collisions.
## 4 Exploring new degree of freedom in direct photon measurement
On exploring the matter produced, one wants to explore a new degree of freedom
of the observables. The angular dependence of the photon yield with respect to
the plane defined by impact parameter (event plane) is one of the degrees that
can be investigated. Rapidity dependence will be another degree of freedom,
which may shed light to the pre-equilibrium state of the collisions.
Figure 10: Source dependence of elliptic flow ($v_{2}$) of direct photons.
Figure 11: Rapidity dependence of direct photons.
It is predicted that the second order of the Fourier transfer coefficient
($v_{2}$, elliptic flow) of angular distributions of photons show the
different sign and/or magnitude, depending on the production processes [12]
(Fig. 11). The observable is powerful to disentangle the contributions from
various photon sources in the $p_{T}$ region where they intermix. The photons
from hadron-gas interaction and thermal radiation may follow the collective
expansion of a system, and give a positive $v_{2}$. The amount of photons
produced by jet-photon conversion or in-medium bremsstrahlung increases as the
medium to traverse increases. Therefore these photons show a negative $v_{2}$.
The fragmentation photons will give positive $v_{2}$ since larger energy loss
of jets is expected orthogonal to the event plane.
PHENIX has measured the $v_{2}$ of direct photons by subtracting the $v_{2}$
of hadron decay photons off from that of the inclusive photons, following the
formula below:
${v_{2}}^{dir.}=(R\times{v_{2}}^{incl.}-{v_{2}}^{bkgd.})/(R-1),\ \ \
R=(\gamma/\pi^{0})_{meas}/(\gamma/\pi^{0})_{bkgd}$
The elliptic flow of $\pi^{0}$ and inclusive photons are shown in Fig. 12(a),
and the one for direct photons is shown in Fig. 12(b).
Figure 12: Elliptic flow of (a, left) $\pi^{0}$ and inclusive photons and (b,
right) direct photons.
The $v_{2}$ of direct photons is large and positive, and comparable to the
flow of hadrons for $p_{T}<$3 GeV/$c$. This result is hard to be explained by
many models. Several models qualitatively predicted the positive flow of the
photons assuming the photons are boosted with hydrodynamic expansion of the
system, but the amount is significantly lower than the measurement [13]. There
is one model that gives relatively large flow by including hadron-gas
interaction [14].
## 5 Summary
Direct photons are a powerful tool to investigate the collision dynamics.
PHENIX has measured direct photons over wide $p_{T}$ ranges, including hard
scattering and thermal photons, and extracted quantities, such as slope
parameters, that reflect thermodynamic properties of the matter. An
unexpectedly large positive elliptic measured for direct photons are hard to
be explained by many models.
## References
## References
* [1] K. Adcox, et al. (PHENIX Collaboration), Nucl. Phys. A757 (2005) 184–283.
* [2] P. Stankus, Ann. Rev. Nucl. Part. Sci. 55 (2005) 517–554.
* [3] S. Turbide, R. Rapp, C. Gale, Phys. Rev. C 69 (2004) 014903.
* [4] K. Adcox, et al., Nucl. Instrum. Meth. A499 (2003) 469–479.
* [5] S. S. Adler et al. (PHENIX Collaboration), Phys. Rev. Lett. 94, 232301 (2005).
* [6] L. E. Gordon, W. Vogelsang, Phys. Rev. D 48 (1993) 3136–3159.
* [7] F. Arleo, JHEP 0609, 015 (2006).
* [8] A. Adare, et al. (PHENIX Collaboration), Phys. Rev. Lett. 104 (2010) 132301.
* [9] A. Adare, et al. (PHENIX Collaboration), Phys. Rev. C 81 (2010) 034911.
* [10] N. M. Kroll, W. Wada, Phys. Rev. 98 (1955) 1355–1359.
* [11] T. Sakaguchi, Nucl. Phys. A855, 141-148 (2011).
* [12] S. Turbide, C, Gale and R.J. Fries, Phys. Rev. Lett. 96, 032303 (2006); R. Chatterjee et al., Phys. Rev. Lett. 96, 202302 (2006)
* [13] R. Chatterjee, D. Srivastava, Phys. Rev. C 79 (2009) 021901.
* [14] H. van Hees, C. Gale and R. Rapp, arXiv:1108.2131 [hep-ph].
|
arxiv-papers
| 2011-10-11T17:16:58 |
2024-09-04T02:49:23.025071
|
{
"license": "Public Domain",
"authors": "Takao Sakaguchi (for the PHENIX Collaboration)",
"submitter": "Takao Sakaguchi",
"url": "https://arxiv.org/abs/1110.2440"
}
|
1110.2515
|
# Normalized Mutual Information to evaluate overlapping community finding
algorithms
Aaron F. McDaid, Derek Greene, Neil Hurley
Clique Reseach Cluster, University College Dublin, Ireland.
aaronmcdaid@gmail.com
###### Abstract
Given the increasing popularity of algorithms for overlapping clustering, in
particular in social network analysis, quantitative measures are needed to
measure the accuracy of a method. Given a set of true clusters, and the set of
clusters found by an algorithm, these sets of clusters must be compared to see
how similar or different the sets are. A normalized measure is desirable in
many contexts, for example assigning a value of 0 where the two sets are
totally dissimilar, and 1 where they are identical.
A measure based on normalized mutual information, [1], has recently become
popular. We demonstrate unintuitive behaviour of this measure, and show how
this can be corrected by using a more conventional normalization. We compare
the results to that of other measures, such as the Omega index [2].
A C++ implementation is available online.
111https://github.com/aaronmcdaid/Overlapping-NMI
In a non-overlapping scenario, each node belongs to exactly one cluster. We
are looking at overlapping, where a node could belong to many communities, or
indeed to no clusters. Such a set of clusters has been referred to as a
_cover_ in the literature, and this is the terminology that we will use.
For a good introduction to our problem of comparing covers of overlapping
clusters, see [2]. They describe the Rand index, which is defined only for
disjoint (non-overlapping) clusters, and then show how to extend it to
overlapping clusters. Each pair of nodes is considered and the number of
clusters in common between the pair is counted. Even if a typical node is in
many clusters, it’s likely that a randomly chosen pair of nodes will have zero
clusters in common. These counts are calculated for both covers and the Omega
index is defined as the proportion of pairs for which the shared-cluster-count
is identical, subject to a correction for chance.
## I Mutual information
Meila [3] defined a measure based on mutual information for comparing disjoint
clusterings. Lancichinetti et al. [1] proposed a measure also based on mutual
information, extended for covers. This measure has become quite popular for
comparing community finding algorithms in social network analysis. It is this
measure we are primarily concerned with there, and we will refer to it as
$\mbox{NMI}_{LFK}$after the authors’ initials.
We are proposing to use a different normalization to that used in
$\mbox{NMI}_{LFK}$, but first we will define the non-normalized measure which
is based very closely on that in $\mbox{NMI}_{LFK}$. You may want to compare
this to the final section of Lancichinetti et al. [1].
Given two covers, $X$ and $Y$, we must first see how to measure the similarity
between a pair of clusters. $X$ and $Y$ are matrices of cluster membership.
There are $n$ objects. The first cover has $K_{X}$ clusters, and hence $X$ is
an $n\times K_{X}$ matrix. $Y$ is an $n\times K_{Y}$ matrix. $X_{im}$ tells us
whether node $m$ is in cluster $i$ in cover $X$.
To compare cluster $i$ of the first cover to cluster $j$ of the second cover,
we compare the vectors $X_{i}$ and $Y_{j}$. These are vectors of ones and
zeroes denoting which clusters the node is in.
* •
${a=\sum_{m=1}^{n}[X_{im}=0\wedge Y_{jm}=0]}$
* •
${b=\sum_{m=1}^{n}[X_{im}=0\wedge Y_{jm}=1]}$
* •
${c=\sum_{m=1}^{n}[X_{im}=1\wedge Y_{jm}=0]}$
* •
${d=\sum_{m=1}^{n}[X_{im}=1\wedge Y_{jm}=1]}$
If $a+d=n$, and therefore $b=c=0$, then the two vectors are in complete
agreement.
The lack of information between two vectors is defined:
$\displaystyle H(X_{i}|Y_{j})=$ $\displaystyle{}H(X_{i},Y_{j})-H(Y_{j})$
$\displaystyle=$ $\displaystyle{}h(a,n)+h(b,n)+h(c,n)+h(d,n)$
$\displaystyle{}-h(b+d,n)-h(a+c,n)$ (1)
where $h(w,n)=-w\log_{2}\frac{w}{n}$.
There is an interesting technicality here. Imagine a pair of clusters but
where the memberships have been defined randomly. There is a possibility that
there will be a small amount of mutual information, even in the situation
where the two vectors are negatively correlated with each other. In extremis,
if the two vectors are near complements of each other, mutual information will
be very high. We wish to override this and define that there is zero mutual
information in this case. This is defined in equation (B.14) of [1]. We also
use this restriction in our proposal.
$\begin{split}H^{*}&(X_{i}|Y_{j})=\\\
&\left\\{\begin{split}H(X_{i}|Y_{j})\;&\mbox{~{}if}\;h(a,n)+h(d,n)\geq
h(b,n)+h(c,n)\\\
h(c+d,n)+h(a+b,n)\;&\mbox{~{}otherwise}\end{split}\right.\end{split}$ (2)
This allows us to compare vectors $X_{i}$ and $Y_{j}$, but we want to compare
the entire matrices $X$ and $Y$ to each other. We will follow the
approximation used by [1] here and match each vector in $X$ to its best match
in $Y$,
$H(X_{i}|Y)=\underset{j\in\\{1,\dots K_{Y}\\}}{\min}H^{*}(X_{i}|Y_{j})$ (3)
then summing across all the vectors in $X$,
$H(X|Y)=\sum_{i\in\\{1,\dots K_{X}\\}}H(X_{i}|Y)$ (4)
$H(Y|X)$ is defined in a similar way to $H(X|Y)$, but with the roles reversed.
We will also need to define the (unconditional) entropy of a cover,
$\displaystyle H(X)$ $\displaystyle=\sum_{i=1}^{K_{X}}H(X_{i})$
$\displaystyle=\sum_{i=1}^{K_{X}}\left(h\left(\sum_{m=1}^{n}[X_{im}=1],n\right)+h\left(\sum_{m=1}^{n}[X_{im}=0],n\right)\right)\;,$
where $\sum_{m=1}^{n}[X_{im}=1]$ counts the number of nodes in cluster $i$,
end $\sum_{m=1}^{n}[X_{im}=0]$ counts the number of nodes not in cluster $i$,
## II Useful identities
$I(X:Y)$$H(Y|X)$$H(X|Y)$$H(Y)$$H(X)$ Figure 1: Mutual information and
variation of information. The total information $H(X,Y)=H(X|Y)+I(X:Y)+H(Y|X)$.
fig. 1 gives us an easy way to remember the following useful identities, which
apply to any mutual information context.
$\displaystyle H(X)=$ $\displaystyle I(X:Y)+H(X|Y)$ $\displaystyle H(Y)=$
$\displaystyle I(X:Y)+H(Y|X)$ $\displaystyle H(X,Y)=$ $\displaystyle
H(X)+H(Y|X)$ $\displaystyle H(X,Y)=$ $\displaystyle H(Y)+H(X|Y)$
$\displaystyle H(X,Y)=$ $\displaystyle\overbrace{I(X:Y)}^{\text{mutual
information}}+\overbrace{H(X|Y)+H(Y|X)}^{\text{variation of information}}$
The first two equalities give us two definitions for the mutual information,
$I(X:Y)$. In theory, these should be identical, but due to the approximation
used in eq. 3 they may be different. Therefore, we will use the average of the
two.
$I(X:Y):=\frac{1}{2}\left[H(X)-H(X|Y)+H(Y)-H(Y|X)\right]$ (5)
We are now ready to discuss normalization, contrasting the method of [1] with
our alternative.
Lancichinetti et al. [1] define their own normalization of the _variation of
information_ ,
$\frac{1}{2}\left(\frac{H(X|Y)}{H(X)}+\frac{H(Y|X)}{H(Y)}\right)$ (6)
and hence their normalized mutual information is
$\mbox{NMI}_{LFK}=1-\frac{1}{2}\left(\frac{H(X|Y)}{H(X)}+\frac{H(Y|X)}{H(Y)}\right)$
(7)
There are of course many ways to normalize a quantity such as the _variation
of information_. Normalization typically involves division by a quantity $c$,
$\frac{H(X|Y)+H(Y|X)}{c(X,Y)}$ (8)
where $c$ is a function of $X$ and $Y$ which is guaranteed to be greater than
or equal to the numerator. But $\mbox{NMI}_{LFK}$does not use a normalization
of this standard form, instead using eq. 6.
There is another aspect to the non-standard normalization used in
$\mbox{NMI}_{LFK}$; they insert an extra normalization factor into their
definition of $H(X_{i}|Y_{j})$. But this is not the root cause of the problems
we will describe, hence we will not dwell on it. Our change is to remove all
the normalization steps from their analysis and instead use a more
conventional normalization of the form of eq. 8.
## III Unintuitive behaviour
There are circumstances where $\mbox{NMI}_{LFK}$overestimates the similarity
of two clusters. We will show how an alternative normalization will fix these
problems.
Imagine a cover $X$, and we are comparing it to a cover $Y$. Further, imagine
$Y$ has only one cluster ($K_{Y}=1$) and this cluster is identical to one of
the clusters in $X$. For large $K_{X}$, we would expect the normalized mutual
information to be quite low. An intuitive result would be approximately
$\frac{1}{K_{X}}$.
However, $\mbox{NMI}_{LFK}(X,Y)$ will be at least $0.5$ in cases like this.
This is because $H(Y|X)$ will be zero bits (the single cluster in $Y$ can be
encoded with zero bits because it has a perfect match among the clusters of
$X$) and this will result in a contribution of $0.5$ to the
$\mbox{NMI}_{LFK}$.
The other problematic example involves the power set. There are $n$ objects in
total. A cover involving every subset of the $n$ objects will create $2^{n}-1$
clusters; we will ignore the empty subset. This is the power set, which we
denote as $p(n)$.
$\mbox{$\mbox{NMI}_{LFK}$}(X,p(n))$ will again be slightly greater than $0.5$.
This is because every cluster in $X$ will have a perfect match in $p(n)$ and
this will result in $H(X|p(n))=0$.
In both these examples $\mbox{NMI}_{LFK}$ gives a score slightly above $0.5$.
The intuitive behaviour in these cases would be for a similarity score close
to $0$. We will demonstrate this behaviour in our experiments in section V
When we remove the normalization from $\mbox{NMI}_{LFK}$, and instead use a
more conventional normalization strategy eq. 8, we will find more intuitive
behaviour.
## IV normalization
Figure 2: As more communities are found, the scores of $\mbox{NMI}_{LFK}$and
$\text{NMI}_{max}$ increase. For a small number of communities found, the
intuitive result is a small value, and this is the behaviour of our proposed
measure.
Typically a normalization will involve a simple division of the absolute
quantity by a quantity which is gauranteed to be an upper bound, giving us a
number between zero and one.
The following sequence of inequalities from Vinh et al. [4] provide
possibilities for normalization.
$\begin{split}I(X:Y)\leq&\min(H(X),H(Y))\\\ \leq&\sqrt{H(X),H(Y)}\\\
\leq&\frac{1}{2}\left(H(X)+H(Y)\right)\\\ \leq&\max(H(X),H(Y))\\\
\leq&H(X,Y)\end{split}$ (9)
Any of the five expressions on the right can be used, and [4] suggest a
measure based on $\max(H(X),H(Y))$. The Normalized Information Distance is
recommended
$d_{max}=1-\frac{I(X,Y)}{\max(H(X),H(Y))}$
where zero means perfect similarity and one means dissimilarity. We want a
measure with the opposite behaviour, so we’ll use the corresponding normalized
mutual information
$NMI_{max}=\frac{I(X:Y)}{\max(H(X),H(Y))}$ (10)
where $I(X:Y)$ is as defined in eqs. 2, 3, 4 and 5
This can also be understood with reference to fig. 1. The problem with
$\mbox{NMI}_{LFK}$ arises when one cover is more complicated than the other,
for example if one cover has many more clusters than the other cover. This
corresponds to one circle in fig. 1 being much larger than the other. In both
the unintuitive examples mentioned in section III, we will find that one of
the circles will be much larger than the other and that the overlap between
the two circles will be quite large, almost the full size of the smaller
circle. As a result, one of the terms inside the brackets in eq. 7 will be
small and will bring the $\mbox{NMI}_{LFK}$to 0.5.
## V evaluation
See fig. 2. There are 200 nodes, divided into 20 communities. Each community
has 10 nodes and they do not overlap. We fix one of our covers, $X$, to be the
full set of twenty communities. $Y$ contains a subset of these communities. As
we go from left to right, the number of communities in $Y$ increases from 1 to
20.
The communities in $Y$ are perfect copies of communities in $X$. Therefore,
$X=Y$ when all 20 communities are used. We see this in fig. 2 at the right,
where both measures report an NMI of $1.0$.
This plot confirms the unintuitive behaviour of $\mbox{NMI}_{LFK}$when few
communities are found. On the left of the plot, when $Y$ has only one
community, the score is $0.5$.
The linear relationship of our NMImax, going from 0 to 1 as the number of
communities in $Y$ increases, is intuitive.
## VI conclusion
We have identified unintuitive behaviour in the version of NMI proposed by [1]
. We have identified the root cause of the behaviour and shown how the use of
a conventional normalization can lead to more intuitive behaviour.
A simple experiment was performed to confirm the existence of the unintuitive
behaviour and demonstrate the more intuitive behaviour.
There are a variety of normalized measures to measure the similarity of
covers. There is no unique set of evaluation criteria to decide on the best,
but we suggest that our measure is the most intuitive definition based on
normalized mutual information.
## VII Acknowledgements
This work is supported by Science Foundation Ireland under grant 08/SRC/I1407:
Clique: Graph and Network Analysis Cluster.
## References
* Lancichinetti et al. [2009] Andrea Lancichinetti, Santo Fortunato, and Janos Kertesz. Detecting the overlapping and hierarchical community structure in complex networks. _New J. Phys._ , 11(3):033015+, March 2009. ISSN 1367-2630. doi: 10.1088/1367-2630/11/3/033015. URL http://dx.doi.org/10.1088/1367-2630/11/3/033015.
* Collins and Dent [1988] L.M. Collins and C.W. Dent. Omega: A general formulation of the rand index of cluster recovery suitable for non-disjoint solutions. _Multivariate Behavioral Research_ , 23(2):231–242, 1988. ISSN 0027-3171.
* Meila [2007] M. Meila. Comparing clusterings—an information based distance. _Journal of Multivariate Analysis_ , 98(5):873–895, May 2007. ISSN 0047259X. doi: 10.1016/j.jmva.2006.11.013. URL http://dx.doi.org/10.1016/j.jmva.2006.11.013.
* [4] Nguyen X. Vinh, Julien Epps, and James Bailey. Information Theoretic Measures for Clusterings Comparison: Variants, Properties, Normalization and Correction for Chance. _Journal of Machine Learning Research_. URL http://www.jmlr.org/papers/volume11/vinh10a/vinh10a.pdf.
|
arxiv-papers
| 2011-10-11T21:45:31 |
2024-09-04T02:49:23.033409
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Aaron F. McDaid, Derek Greene, Neil Hurley",
"submitter": "Aaron Francis McDaid",
"url": "https://arxiv.org/abs/1110.2515"
}
|
1110.2812
|
# Direct Numerical Simulation of Single-mode Rayleigh-Taylor Instability
Tie Wei Daniel Livescu Los Alamos National Laboratory, Los Alamos, NM, 87544
twei@lanl.gov, livescu@lanl.gov
Rayleigh-Taylor instability (RTI) is an interfacial instability that occurs
when a high density fluid is accelerated or supported against gravity by a low
density fluid. This instability is of fundamental importance in a multitude of
applications, from fluidized beds, oceans and atmosphere, to inertial or
magnetic confinement fusion, and to astrophysics. The interface between the
two fluids is unstable to any perturbation with a wavelength larger than the
cutoff due to surface tension (for the immiscible case) or mass diffusion (for
the miscible case).
The video shows the evolution of density and vorticity field from our Direct
Numerical Simulation (DNS) of high perturbation Reynolds number single-mode
RTI. The development of single-mode RTI can be divided into a number of
stages, depending on which physical effect dominates the instability growth.
At early times, if the initial perturbations amplitudes are small compared to
their wavelength and the growth is not dominated by diffusive effects, the
flow can be described by linearized equations and the perturbation amplitude
grows exponentially with time (exponential growth stage-EG). With increasing
bubble and spike speed, the differential velocity on the two sides of the
interfaces leads to the development of the Kelvin-Helmholtz instability on the
edges of the bubbles and spikes. However, not long after the non-linear
effects become important, the vortical motions generated by the Kelvin-
Helmholtz instability are weak, and the flow at the tip of the bubble is still
potential. This potential flow regime is characterized by a “quasi-constant”
bubble front speed, and this staged is called ‘potential flow stage’ (PFG).
As the fluid accelerates due to the buoyancy forces, the initial vortices grow
larger and start interacting. One of the first consequences of this
interaction is that the vortices split and form pairs of counter-rotating
vortices (one for each bubble and spike) which start self-propelling towards
the tips of the bubbles and spikes. The motions become more complicated due to
the further break-up, however, the first vortex pair still moves on an
accelerating trajectory such that the induced velocity at the tips of the
bubble/spike continues to increase. The consequence is that the velocity no
longer follows the potential flow theory and the tips of the bubble/spike
undergo a ‘re-acceleration stage’(RA). A new stage, chaotic development (CD),
was revealed in our DNS after the re-acceleration stage. The chaotic
development is caused by the complex vortical motions and interactions, which
can be clearly in the later part of the movie. Since such complex motions have
non-integrable dynamics, the bubble/spike velocities present chaotic temporal
behavior.
The parameters used in the 2D simulation is shown in table 1.
$L_{h}\times L_{v}$ | $N_{h}\times N_{v}$ | $g$ | $\nu$ | $Sc$
---|---|---|---|---
$2048\times 10240$ | $2048\times 12800$ | $11.0$ | $1.0$ | $1.0$
Table 1: Simulation parameters. $L_{h}$: domain size in the horizontal
direction; $L_{v}$: domain size in the vertical direction; $N_{h},N_{v}$: grid
numbers in the horizontal and vertical, respectively; $g$: gravity; $\nu$:
kinematic viscosity; $Sc$: Schmidt number.
|
arxiv-papers
| 2011-10-12T23:22:26 |
2024-09-04T02:49:23.068184
|
{
"license": "Public Domain",
"authors": "Tie Wei and Daniel Livescu",
"submitter": "Tie Wei",
"url": "https://arxiv.org/abs/1110.2812"
}
|
1110.2877
|
# Search for narrow resonances in the lepton final state at CMS
G. Kukartsev Department of Physics and Astronomy, Brown University,
Providence, RI, USA
###### Abstract
We discuss the results of searches for high-mass narrow resonances decaying
into pairs of leptons using pp collisions at 7 TeV delivered by LHC and
collected with the CMS detector in 2010 and 2011. These include searches for
the ${Z^{0}}^{\prime}$ bosons and RS gravitons.
## I Introduction
Several theoretical models predict new $\mathrm{\,Te\kern-1.00006ptV}$-scale
resonances decaying into a pair of leptons. Models of particular interest for
the presented analysis include the Sequential Standard Model (SSM) with
standard-model-like couplings, and certain grand-unification-motivated models
($\Psi$) Leike:1998wr . Both predict narrow $Z^{0}$-boson-like states
(${Z^{0}}^{\prime}$). We also consider Kaluza-Klein excitations in the
Randall-Sundrum (RS) model of extra dimensions ($G_{\mathtt{KK}}$)
Randall:1999vf ; Randall:1999ee . We use the four listed models as benchmarks
while we search for a narrow resonance, which is similar to the SSM
${Z^{0}}^{\prime}$, in the dimuon and the dielectron channels. We perform a
likelihood-based shape analysis of the reconstructed dilepton invariant mass
($m_{ll}$) spectra. The approach provides robustness against uncertainties in
the absolute background rate.
The recent searches for ${Z^{0}}^{\prime}\to l^{+}l^{-}$ and
$G_{\mathtt{KK}}\to l^{+}l^{-}$ were published by the Tevatron experiments
D0_RS ; D0_Zp ; CDF_RS ; CDF_Zp . There are indirect constraints from LEP-II
delphi ; aleph ; opal ; l3 .
## II Detector and Experiment
CMS is a general-purpose particle detector located at the LHC proton-proton
collider at CERN. A prominent feature of the detector is a superconducting
solenoid with the internal diameter of 6$\rm\,m$ and an axial field of 3.8 T.
The solenoid encloses the pixel detector, the silicon tracker, the crystal
electromagnetic calorimeter (ECAL) and the brass and scintillator hadron
calorimeter (HCAL). Outside the solenoid there is a steel flux return yoke
instrumented with the gas ionization detectors, which constitute the CMS muon
system. A diagram of the detector is shown in Figure 1. Further details can be
found elsewhere JINST . For the presented results, 1.1$\mbox{\,fb}^{-1}$ of
integrated luminosity were used.
Figure 1: The CMS detector.
## III Data and Monte Carlo
The presented results were obtained using the data recorded by the CMS
experiment in 2011. The data were taken using proton-proton colliding beams
with the center-of-mass energy of $7\mathrm{\,Te\kern-1.00006ptV}$. The size
of the dataset corresponds to an integrated luminosity of approximately
$1.1\mbox{\,fb}^{-1}$. The size of the data sample used in the dielectron
analysis is $25\mbox{\,pb}^{-1}$ smaller due to different quality requirements
for the data.
The signal and background processes were modeled using Monte Carlo
simulations. Depending on the process, PYTHIA v6.424 Sjostrand:2006za ,
MADGRAPH MADGRAPH and POWHEG v1.1 Alioli:2008gx ; Nason:2004rx ;
Frixione:2007vw event generators together with the CTEQ6L1 Pumplin:2002vw
parton distribution function (PDF) set were used. The full CMS detector
simulation was done with GEANT4 GEANT4 . The generated events were passed
through the CMS trigger simulation and full reconstruction sequence.
## IV Event Selection
We developed dedicated selection criteria for each of the two dilepton
channels under consideration. Even though the underlying physics processes
under study are similar, reconstruction of different lepton flavors in the
detector differs substantially. For electrons, we reconstruct the transverse
energy using calorimeter information, while the muon reconstruction is based
on the tracking and the muon systems for the measurement of the transverse
momenta. Dilepton invariant mass reconstruction deteriorates for higher values
in the dimuon channel and improves in the dielectron channel. We require the
muons to be reconstructed with the opposite charge, and do not impose this
restriction on dielectron pairs. For the dimuon pairs reconstruction, we
reduce systematic uncertainty by performing data-driven studies with cosmic-
ray muons. The dielectron channel entails higher background rates from
misreconstructed strong scattering signal, and requires tighter selection,
which leads to lower efficiency and acceptance.
### IV.1 Trigger
For the dimuon pair event candidates, we used a single muon trigger with
sufficiently high minimum transverse momentum requirement
($p_{T}>30\mathrm{\,Ge\kern-1.00006ptV}$). The muon was firstly required to be
detected in the muon system, and then matched to a track in the silicon
tracker. For dielectron pairs, the trigger requires two sufficiently energetic
deposits ($33\mathrm{\,Ge\kern-1.00006ptV}$) in ECAL, with at least one of the
deposits matched to level-one deposit. The corresponding deposit in HCAL must
be small (less than $15\%$). In later portions of the dataset, a match to the
activity in the silicon pixel tracker was required.
### IV.2 Lepton Reconstruction and Pile-up
Standard CMS techniques apply to the reconstruction, calibration and
identification of the leptons MUO-10-004-PAS ; EGMPAS ; EWK-10-002-PAS . For
all leptons, the reconstructed track was required to be consistent with the
beam interaction point, to be topologically isolated from the hadronic
signatures, and to be sufficiently energetic in the plane transverse to the
beam axis ($p_{T}>35\mathrm{\,Ge\kern-1.00006ptV}$ for muons and electrons in
the ECAL barrel, $p_{T}>40\mathrm{\,Ge\kern-1.00006ptV}$ for endcap
electrons). The muons are then reconstructed via a global fit of the tracker
and the muon system information with proper quality requirements met: there
should be enough hits (more than 10) in the silicon tracker, at least 1 hit in
the pixel detector, and a track reconstructed in the tracker and extrapolated
to the muon system must be compatible with the hits in the muon system, with
hits in at least 2 of the muon stations. The transverse impact parameter
relative to the beam interaction point is required to be less than
$0.2\rm\,cm$. The electrons are reconstructed as an ECAL cluster matched to a
track in the silicon tracker. The ECAL cluster seeds the track in the pixel
detector, which in turn seeds the track in the tracker. Each track must have
at least five hits, and a hit in each of the three pixel layers. The
reconstructed electron candidate must be within either barrel ($|\eta|<1.442$)
or endcap ($1.56<\eta<2.5$) ECAL acceptance regions, and less than $5\%$ of
the energy must be deposited in HCAL.
Leptons are required to be isolated from other activity in the tracker, in
order to suppress background from jets misreconstructed as leptons, and from
non-prompt leptons. The isolation is defined using a cone $\delta
R=\sqrt{(\delta\eta)^{2}+(\delta\phi)^{2}}$ centered on the lepton axis where
$\eta$ is pseudo-rapidity and $\phi$ is the azimuthal angle relative to the
beam axis.
For the muon, the sum of transverse momenta of all other tracks, consistent
with the primary vertex, in the cone of $0.3$ must be less than $10\%$ of the
muon $p_{T}$. The efficiency of this isolation requirement was shown to be
stable with the number of primary vertexes as indication of robustness against
pile-up in a higher instantaneous luminosity regime.
For the electron, the sum of all track $p_{T}$ in the cone of $0.04$ is
required to be less than $7\mathrm{\,Ge\kern-1.00006ptV}$ in the barrel of
ECAL, and less than $15\mathrm{\,Ge\kern-1.00006ptV}$ for the endcap. The
tracks are required to be consistent with the reconstructed primary vertex.
The calorimeter isolation for the electrons requires that the sum of $E_{T}$
of all deposits in the ECAL and the HCAL to be less than
$0.03E_{T}+2\mathrm{\,Ge\kern-1.00006ptV}$ relative to the the electron
$E_{T}$. For the electrons in endcap, we exploit the HCAL segmentation along
the beam axis. The isolation energy is required to be less than
$0.03\cdot\max{(0,E_{T}-50\mathrm{\,Ge\kern-1.00006ptV})}+2.5\mathrm{\,Ge\kern-1.00006ptV}$
where $E_{T}$ is determined from ECAL and the first layer of HCAL. In the
second layer, the HCAL $E_{T}$ must be less than
$0.5\mathrm{\,Ge\kern-1.00006ptV}$. Additionally, the shape of the transverse
energy deposit is required to be compatible with the expected electron signal,
and a good match in $\eta$ and $\phi$ with the corresponding track is
required.
### IV.3 Lepton pair selection
We select events with two reconstructed leptons: either muons or electrons,
originating from a well-reconstructed primary vertex. The vertex must be
within $2\rm\,cm$ from the beam interaction point in the transverse plane, and
within $24\rm\,cm$ along the beam axis, to suppress cosmic ray background. For
the muon pair event candidates, an additional protection against cosmic muons
is required as an opening angle between the two muons being less than
$(\pi-0.02)$.
For the dimuon events, we require opposite charges for the two muons as it
reduces the fraction of events with a large mismeasurement of the momentum. We
suppress events with many poorly reconstructed tracks in order to reduce beam
background. At least one muon has to match a high-level trigger (HLT) muon. As
an additional quality requirement, the muon pair is required to be consistent
with a common vertex.
For the electron pair events, at least one of the electrons is required to be
reconstructed in the barrel part of the detector. In order to suppress
background from photon conversions, we impose requirement on the distance to
the nearest track and an opening angle with it.
### IV.4 Efficiency and Acceptance
We measure efficiency of triggering, lepton reconstruction and identification
with “tag-and-probe” method MUO-10-004-PAS ; EWK-10-002-PAS . We use a pure
sample of dimuon pairs requiring that their invariant mass is consistent with
the Z boson mass
($60\mathrm{\,Ge\kern-1.00006ptV}<m_{\mathtt{ll}}<120\mathrm{\,Ge\kern-1.00006ptV}$).
One of the muons in the pair is reconstructed with stringent quality
requirements (tag), and the other is used as a probe for efficiency estimates.
Contributing factors also include track reconstruction and electron
clustering. We measure the single muon trigger efficiency to be $95.0\%\pm
0.3\%$ in the barrel and $89.9\%\pm 0.4\%$ in the endcap. The efficiency of
the muon identification is measured to be $96.4\%\pm 0.2\%$ in the barrel and
$96.0\%\pm 0.3\%$ in the endcap. The efficiency of the track reconstruction in
the internal tracker is found to be above $99\%$ in the whole acceptance
range. Figure 2 represents the overall acceptance and efficiency values for
the dielectron channel, as a function of the dilepton invariant mass. Similar
behavior with higher overall acceptance and efficiency values is observed in
the dimuon channel.
Figure 2: Acceptance and efficiency (left) and invariant mass resolution
(right), dielectron channel.
## V Resolution
We study detector performance using Standard Model processes with W and Z
mesons and their leptonic final states. We also use cosmic muons. The muon
momentum resolution ranges from $1\%$ at few tens of
$\mathrm{\,Ge\kern-1.00006ptV}$ (Z boson peak scale) to approximately $10\%$
above $1\mathrm{\,Te\kern-1.00006ptV}$. Tracker-based reconstruction yields
better performance at low momenta, while the muons reconstructed in the muon
system have better resolution at high momenta. However, energy loss in the
steel yoke and showers in the muon chambers can spoil the global fit. We find
that adding muon system hits to the tracker-based fit improves resolution for
muons with $p_{T}$ greater than approximately
$200\mathrm{\,Ge\kern-1.00006ptV}$ PTDR2 . The most comprehensive algorithm
(”Tune P”) makes track-by-track decisions about which hits in which subsystems
to use. The resolution is also sensitive to the alignment of the muon and the
tracker systems.
Unlike for muons, the electron energy resolution improves with energy. The
ECAL resolution is better than $0.5\%$ for unconverted photons with transverse
energies above $100\mathrm{\,Ge\kern-1.00006ptV}$. The invariant mass
resolution of dielectron pairs is modeled with a Crystal Ball function and
obtained from Monte Carlo simulation, with additional smearing applied. The
smearing is obtained from comparisons of the Z-boson peak fits in data and
Monte Carlo simulation of the $Z\to ee$ process. At
$1\mathrm{\,Te\kern-1.00006ptV}$, the dielectron invariant mass resolution is
approximately $1.3\%$ when both electrons are in the barrel acceptance region,
and approximately $2.4\%$ when one of the electrons is in the endcap region.
For the electrons in the barrel section of the detector, energy scale is
established using neutral pions and checked using the Z peak.
## VI Background
The Drell-Yan process produces the dominant irreducible background, with the
next biggest contribution from the top pair and other top-like processes (tW,
diboson and $Z\to\tau\tau$). The remaining background comes from jet
misidentification as leptons ($1\%-5\%$ depending on the channel), and from
cosmic muons in the dimuon channel. We found that the contribution from the
latter, and from diphoton processes misreconstructed as dielectrons are
negligible. Figures 3 and 4 depict the observed dilepton data overlaid with
the background components. The individual components are normalized to next-
to-leading order, and then to the Z-boson peak in data.
Figure 3: Dimuon invariant mass (left) and the corresponding cumulative
spectrum (right). Individual components are normalized to NLO and then
together to the Z-boson peak.
Figure 4: Dielectron invariant mass (left) and the corresponding cumulative
spectrum (right). Individual components are normalized to NLO and then
together to the Z-boson peak.
The overall background rate and the shape of the dilepton invariant mass
distribution are taken from the Drell-Yan Monte Carlo corrected to next-to-
next-to-leading-order with FEWZz v1.X FEWZ , PYTHIA v6.409 and CTEQ6.1 PDF
Stump:2003yu . For the purposes of setting the limits on the dilepton
resonance cross section, the variation in the shape due to added top-like and
other background sources ($5\%-10\%$), the uncertainties in k-factor,
generator choice and PDF sets are covered conservatively by a background rate
uncertainty of $20\%$($15\%$) in the dimuon (dielectron) channel.
As a cross check of the top-like background model, we compare data and Monte
Carlo distributions of the dilepton invariant mass where the flavor and
electric charge of the two leptons are required to be different (“e$\mu$”
method). The reasoning is that if the two leptons do not originate from a
resonance, there is no special reason for them to be of the same flavor. For
each dielectron and dimuon event, we expect to observe nearly two $e\mu$
events (the actual ratio is slightly different due to different efficiencies
for electrons and muons). Figure 5 demonstrates the comparison between data
and Mote Carlo for the $e\mu$ events, which we find satisfactory.
Figure 5: Invariant mass of an electron and a muon of the opposite charge.
## VII Statistical Inference
We set $95\%$ C.L. upper limits on the cross section ratio as defined in
Equation 5, assuming uniform prior on the parameter of interest and Lognormal
likelihood constraint terms on the nuisance parameters in order to model
systematic uncertainties. We use the likelihood formalism to estimate the
model parameters (via maximum likelihood, ML), and to build a likelihood ratio
to be used as a test statistic. In the Bayesian methods the likelihood is
further multiplied by priors to obtain the posterior pdf. We define the
unbinned likelihood for a data set as
$\L({\bm{x}}|{\bm{\theta}},{\bm{\nu}})=\prod_{i=1}^{N}f(x_{i}|{\bm{\theta}},{\bm{\nu}}),$
(1)
where the product is over the events in the data set ${\bm{x}}$,
$f(x|{\bm{\theta}},{\bm{\nu}})$ is the probability density function of the
observable $x$, $x_{i}$ is the value of the observable in the $i-$th event,
${\bm{\theta}}$ is a vector of the model parameters of interest, ${\bm{\nu}}$
is a vector of nuisance parameters.
It is often convenient and advantageous to define an extended likelihood by
adding the Poisson term. It provides the normalization of the data in terms of
the event yield:
$\L({\bm{x}}|\mu,{\bm{\theta}},{\bm{\nu}})=\frac{\mu^{N}e^{-\mu}}{N!}\prod_{i=1}^{N}f(x_{i}|{\bm{\theta}},{\bm{\nu}}),$
(2)
where $N$ is the number of events in the data sample $\\{x_{i}\\}$, $\mu$ is
the Poisson mean number of events. In the following we will use extended
likelihoods everywhere.
It is useful to define the profile likelihood ratio test statistic
$t_{\theta}=-2\ln\lambda({\bm{\theta}})=-2\ln{\frac{\L_{\mathtt{B}}({\bm{\theta}},\hat{\hat{{\bm{\nu}}}}_{\mathtt{B}})}{\L_{\mathtt{S+B}}({\bm{\theta}},\hat{\hat{{\bm{\nu}}}}_{\mathtt{SB}})}},$
(3)
where $\L_{\mathtt{B}}$ and $\L_{\mathtt{S+B}}$ are the likelihood values for
the background-only and for the signal-plus-background models, and
${\bm{\nu}}_{\mathtt{B}}$ is a subset of ${\bm{\nu}}_{\mathtt{SB}}$. The _hat_
notation ($\hat{\,\,\,}$) symbolizes the ML estimator, i.e.
$\hat{{\bm{\theta}}}$ is the value of ${\bm{\theta}}$ that maximizes the
likelihood for a given model. The double hat notation ($\hat{\hat{\,\,\,}}$)
stands for a conditional ML estimator for a given value of a parameter of
interest, i.e. $\hat{\hat{{\bm{\nu}}}}$ is the value of ${\bm{\nu}}$ that
maximizes the likelihood for a given model, for a given value of
${\bm{\theta}}$.
There are several ${Z^{0}}^{\prime}$ models that we consider in the analysis.
In the Sequential Standard Model (SSM), the ${Z^{0}}^{\prime}$ has the same
couplings as the standard model $Z$ boson. The $\Psi$ model is based on an
$E_{6}$ gauge symmetry. For the models overview see Z-Boson searches in
Nakamura:2010zzi . We use the SSM model by default everywhere unless
explicitly stated otherwise.
For the ${Z^{0}}^{\prime}$ search, we define the signal-plus-background
likelihood as
$\L_{\mathtt{S+B}}({\bm{m}}|\theta,{\bm{\nu}})=\frac{\mu^{N}e^{-\mu}}{N!}\prod_{i=1}^{N}\left(\frac{\mu_{\mathtt{S}}(\theta)}{\mu}f_{\mathtt{S}}(m_{i}|{\bm{\nu}}_{\mathtt{S}})+\frac{\mu_{\mathtt{B}}}{\mu}f_{\mathtt{B}}(m_{i}|{\bm{\nu}}_{\mathtt{B}})\right),$
(4)
where ${\bm{m}}$ is the dataset in which $m_{i}$ is the value of the
observable $m$ (the invariant mass of the lepton pair) in $i$-th event,
$\theta$ denotes the parameter of interest, either the cross section or the
cross section ratio, as defined further, ${\bm{\nu}}$ is the vector of the
nuisance parameters, $f_{\mathtt{S}}$ and $f_{\mathtt{B}}$ are the PDFs for
the signal and the background (specific shapes are defined later in the
document), $N$ is the total number of events observed, $\mu_{\mathtt{S}}$ and
$\mu_{\mathtt{B}}$ are the expected signal and the background event yields,
respectively, and $\mu=\mu_{\mathtt{S}}+\mu_{\mathtt{B}}$ is the total number
of events expected. Note that the expected signal yield $\mu_{\mathtt{S}}$ is
a function of the parameter of interest. The parameter of interest is the
cross section ratio
$R_{\sigma}=\frac{\sigma_{{Z^{0}}^{\prime}\to\ell^{+}\ell^{-}}}{\sigma_{Z^{0}\to\ell^{+}\ell^{-}}},$
(5)
where $\sigma_{Z^{0}\to\ell^{+}\ell^{-}}$ is the cross section multiplied by
the branching ratio for $pp\to Z^{0}\to\ell^{+}\ell^{-}$. Such an approach
allows to exclude the uncertainty on the integrated luminosity from the
measurement. In this case, we parameterize the expected signal yield as
$\mu_{\mathtt{S}}=N_{Z}\cdot
R_{\sigma}\cdot\frac{\epsilon_{\mathtt{sel}}({Z^{0}}^{\prime})\cdot{\cal
A}({Z^{0}}^{\prime})\cdot}{\epsilon_{\mathtt{sel}}(Z^{0})\cdot{\cal
A}(Z)}\equiv N_{Z}\cdot R_{\sigma}\cdot R_{\epsilon}\cdot R_{{\cal A}}.$ (6)
Here $N_{Z}$ is the observed number of $Z^{0}$ events, and $R_{\epsilon}\cdot
R_{{\cal A}}$ denotes the fraction in Equation 6. The PDF, which represents
the ${Z^{0}}^{\prime}$ signal model, is
$f_{\mathtt{S}}(m_{ll}|M,\Gamma,w)=\mathtt{BW}(m_{ll}|M,\Gamma)\otimes\mathtt{Gaussian}(0,w),$
(7)
where $m_{ll}$ is the invariant mass of the two leptons (the observable),
$\mathtt{BW}$ stands for the resonant Breit-Wigner shape, $\Gamma$ is its
width, $w$ is the width of the Gaussian resolution function. For combining
multiple analysis channels, the corresponding likelihoods are multiplied
together in order to build the combined likelihood.
## VIII Systematic uncertainty
We combine all systematic uncertainties into three components that we treat
independently: an uncertainty on signal sensitivity (includes uncertainties on
signal and Z acceptances and efficiencies and on the Z event count), the
background rate uncertainty, which is described in Section VI, and the mass
scale uncertainty in the dielectron channel ($1\%$).
## IX Results
We present reconstructed dilepton invariant mass distributions in the CMS data
in Figures 3 and 4 superimposed with the individual background components from
Monte Carlo. We use the mass spectra to set $95\%$ C.L. Bayesian upper limits
on the dilepton resonance cross section ratio (Equation 5). We use several
popular theoretical models to set lower limits on the corresponding resonance
masses, including the Sequential Standard Model ${Z^{0}}^{\prime}$. Figure 6
displays the observed limits overlaid with the median expected limits and 1-
and 2-standard deviation quantile bands. Theoretical estimates for four
popular theoretical models are overlaid as well. Figure 7 displays similar
limit plots for the combination of the dimuon and dielectron channels. the
likelihood ratio in Equation 3 is asymptotically distributed as a $\chi^{2}$
distribution with number of degrees of freedom equal to the difference in the
numbers of free parameters between the two models.
By combining the two channels, we exclude ${Z^{0}}^{\prime}$ masses for the
SSM and E6-motivated $\Psi$ model below 1940$\mathrm{\,Ge\kern-1.00006ptV}$
and 1620$\mathrm{\,Ge\kern-1.00006ptV}$, respectively. The corresponding
limits in the individual dimuon(dielectron) channels are
1780(1730)$\mathrm{\,Ge\kern-1.00006ptV}$ and
1440(1440)$\mathrm{\,Ge\kern-1.00006ptV}$. Combined analysis excludes masses
of the RS Kaluza-Klein gravitons for couplings of 0.05 and 0.10 at
1450$\mathrm{\,Ge\kern-1.00006ptV}$ and 1780$\mathrm{\,Ge\kern-1.00006ptV}$.
The corresponding dimuon(dielectron) numbers are
1240(1300)$\mathrm{\,Ge\kern-1.00006ptV}$ and
1640(1590)$\mathrm{\,Ge\kern-1.00006ptV}$.
Figure 6: Exclusion limits on the dilepton resonance cross section times
branching fraction relative to the Z-boson standard model production, dimuon
channel (left) and dielectron channel (right). Figure 7: Combined dimuon and
dielectron channel exclusion limits on the dilepton resonance cross section
times branching fraction relative to the Z-boson standard model production.
We identify the most signal-like patterns in the data. They correspond to a
dimuon resonance at 1080$\mathrm{\,Ge\kern-1.00006ptV}$ and a dielectron
resonance at 950$\mathrm{\,Ge\kern-1.00006ptV}$, with local significances of
1.7 and 2.2 standard deviations, respectively. Corrected for the “trials
factor” (a consideration that a signal-like fluctuation can happen at an
arbitrary mass value), the significances become 0.3 and 0.2 standard
deviations, respectively. Combined analysis suggest a dilepton resonance-like
signature at 970$\mathrm{\,Ge\kern-1.00006ptV}$ with local significance of 2.1
and significance corrected for the trials factor of 0.2 standard deviations.
Figures 8 and 9 display the sampling distributions of the likelihood ratio
test statistic (3) obtained from ensembles of background-only
pseudoexperiments, used for estimating significances including the trials
factor corrections, overlaid with the value from data.
Figure 8: Significance in the dimuon channel (left) and in the dielectron
channel (right). A histogram corresponds to an ensemble of background-only
pseudoexperiments. The red line is a value observed in data. A plotted value
corresponds to the most signal-like pattern in a dataset, in a fine scan of
the spectrum over the allowed invariant mass values. Figure 9: Combined
significance in the two channels. A histogram corresponds to an ensemble of
background-only pseudoexperiments. The red line is a value observed in data. A
plotted value corresponds to the most signal-like pattern in a dataset, in a
fine scan of the spectrum over the allowed invariant mass values.
## X Conclusion
The CMS Collaboration has searched for high-mass narrow resonances in the
dilepton invariant mass spectra in the dimuon and the dielectron channels,
using 1.1$\mbox{\,fb}^{-1}$ of integrated luminosity recorded by the CMS
detector operating at the LHC proton-proton collider at CERN, with the center-
of-mass energy of 7$\mathrm{\,Te\kern-1.00006ptV}$. The individual channel and
combined spectra are consistent with the Standard Model expectations. The 95%
C.L. upper limits have been set on the product of the new resonance production
cross section and the corresponding branching fraction relative to the
Standard Model Z boson production. Mass limits have been set on the resonances
predicted by the SSM and $\Psi$ ${Z^{0}}^{\prime}$ models, and on the RS
Kaluza-Klein gravitons for couplings of 0.05 and 0.1.
###### Acknowledgements.
We wish to congratulate our colleagues in the CERN accelerator departments for
the excellent performance of the LHC machine. We thank the technical and
administrative staff at CERN and other CMS institutes, and acknowledge support
from: FMSR (Austria); FNRS and FWO (Belgium); CNPq, CAPES, FAPERJ, and FAPESP
(Brazil); MES (Bulgaria); CERN; CAS, MoST, and NSFC (China); COLCIENCIAS
(Colombia); MSES (Croatia); RPF (Cyprus); Academy of Sciences and NICPB
(Estonia); Academy of Finland, ME, and HIP (Finland); CEA and CNRS/IN2P3
(France); BMBF, DFG, and HGF (Germany); GSRT (Greece); OTKA and NKTH
(Hungary); DAE and DST (India); IPM (Iran); SFI (Ireland); INFN (Italy); NRF
and WCU (Korea); LAS (Lithuania); CINVESTAV, CONACYT, SEP, and UASLP-FAI
(Mexico); PAEC (Pakistan); SCSR (Poland); FCT (Portugal); JINR (Armenia,
Belarus, Georgia, Ukraine, Uzbekistan); MST and MAE (Russia); MSTD (Serbia);
MICINN and CPAN (Spain); Swiss Funding Agencies (Switzerland); NSC (Taipei);
TUBITAK and TAEK (Turkey); STFC (United Kingdom); DOE and NSF (USA).
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|
arxiv-papers
| 2011-10-13T09:27:43 |
2024-09-04T02:49:23.078529
|
{
"license": "Public Domain",
"authors": "Gennadiy Kukartsev (for the CMS Collaboration)",
"submitter": "Gennadiy Kukartsev",
"url": "https://arxiv.org/abs/1110.2877"
}
|
1110.3066
|
# Mass-Radius Relationships for Exoplanets II:
Grüneisen Equation of State for Ammonia
Damian C. Swift dswift@llnl.gov Condensed Matter and Materials Division,
Lawrence Livermore National Laboratory, 7000 East Avenue, Livermore,
California 94550, USA
(September 18, 2011; modified October 13, 2011 – LLNL-JRNL-505357)
###### Abstract
We describe a mechanical equation of state for NH3, based on shock wave
measurements for liquid ammonia. The shock measurements, for an initial
temperature of 203 K, extended to 1.54 g/cm3 and 38.6 GPa. The shock and
particle speeds were fitted well with a straight line, so extrapolations to
higher compressions are numerically stable, but the accuracy is undetermined
outside the range of the data. The isentrope through the same initial state
was estimated, along with its sensitivity to the Grüneisen parameter. Mass-
radius relations were calculated for self-gravitating bodies of pure ammonia,
and for differentiated ammonia-rock bodies. The relations were insensitive to
variations in the Grüneisen parameter, indicating that they should be accurate
for studies of planetary structure.
ammonia, shock, equation of state, planetary structure
## 1 Introduction
Ammonia, NH3, is a common molecule in ice giant planets, which appear to be
found widely throughout the galaxy (Schneider, 2011). The equation of state
(EOS) of ammonia is therefore important for our understanding of planetary
structures and their evolution, potentially to pressures of order 1 TPa for
the base of the ice-rock interface in icy exoplanets. An accurate EOS for
ammonia is also needed for studies of hypervelocity impacts, such as meteoroid
collisions with ice giants. Furthermore, ammonia is a simple prototype for
bonds occurring in chemical explosives, for which densities from up to around
twice that of zero-pressure solids are of interest for shock initiation and
detonation.
Although shock compression experiments have been performed on ammonia to
pressures of several tens of gigapascals (Marsh, 1980), the only equation of
state readily available is SESAME 5520 (Holian, 1984), based on National
Bureau of Standards gas phase data (Haar & Gallagher, 1978), and is tabulated
to a maximum density of 0.765 g/cm3, which is barely greater than the zero-
pressure density for liquid ammonia. Quasistatic compression experiments have
been performed in which the density and sound speed were measured along
isotherms (Abramson, 2008; Li et al, 2009), but the highest pressures reported
have reached only a few gigapascals.
## 2 Empirical Grüneisen equation of state
Shock experiments have been reported previously in which the shock and
particle speeds $u_{s}$ and $u_{p}$ were measured for a range of shock
pressures, for liquid ammonia at an initial temperature of 203 K (Marsh, 1980)
(Fig. 1). The uncertainties in $u_{s}$ and $u_{p}$ were approximately 1%.
These data can be fitted by a straight line fit
$u_{s}=c_{0}+s_{1}u_{p}$ (1)
with
$\displaystyle c_{0}$ $\displaystyle=$ $\displaystyle 2.00\pm
0.13\quad(6.7\%)$ $\displaystyle s_{1}$ $\displaystyle=$ $\displaystyle
1.511\pm 0.039\quad(2.6\%),$
where the standard errors shown are from the residual fitting error,
neglecting the uncertainty in measurement. The experimental measurements
exhibited a slightly curved trend, but the number of points was not great
enough to justify a higher-order fit. Solving the Rankine-Hugoniot equations
for a steady shock (Zel’dovich & Raizer, 1966) using the fitted parameters
rather than the individual shock measurements, the highest shock pressure was
38.6 GPa, giving a mass density of 1.54 g/cm3. The observed sound speed in
liquid ammonia at 203 K is 1.9535 km/s (Bowen & Thompson, 1968), which is
consistent with the extrapolated Hugoniot data. (Fig. 1)
Figure 1: Principal shock Hugoniot of liquid ammonia (initial temperature 203
K): experimental measurements and least-squares fit. The point at zero
particle speed is the observed sound speed, which was not included in the fit.
The curve labelled WCS is the Woolfolk-Cowperthwaite-Shaw universal liquid
equation of state, whose sole fitting parameter is the sound speed at zero
pressure.
Various universal EOS have been proposed for different classes of material. It
is interesting to compare with the ‘universal liquid EOS’ of Woolfolk et al
(1973), whose only material-specific parameter is the sound speed at zero
pressure. This EOS does not reproduce the shock data for ammonia, which is
softer and more linear than the universal EOS (Fig. 1).
The fit to the shock Hugoniot can be used to predict the mechanical equation
of state, using the Hugoniot as a reference curve (McQueen et al, 1970)
$\displaystyle p(\rho,e)$ $\displaystyle=$ $\displaystyle
p_{r}(\rho)+\Gamma(\rho)\left[e-e_{r}(\rho)\right]$ (2) $\displaystyle
p_{r}(\rho)$ $\displaystyle=$
$\displaystyle\displaystyle\frac{c_{0}^{2}\rho_{r}\rho(\rho-\rho_{r})}{\left[\rho+s_{1}(\rho-\rho_{r})\right]^{2}}$
(3) $\displaystyle e_{r}(\rho)$ $\displaystyle=$ $\displaystyle
e_{0}+\frac{1}{2}p_{r}(\rho)\left(\frac{1}{\rho_{r}}-\frac{1}{\rho}\right),$
(4)
where $\rho_{r}$ is the initial density on the reference curve, and
$p_{r}(\rho)$ was derived for zero initial pressure, as here. Other
experiments are required to determine $\Gamma(\rho)$, such as sound speed
measurements on the Hugoniot, a shock Hugoniot from a different initial state,
or ramp compression. However, $\Gamma$ can be estimated from the slope of the
shock Hugoniot as $2s_{1}-1$, which is accurate for cubic crystals (Skidmore,
1965). Thus $\rho_{r}=0.725$ g/cm3 and $\Gamma\simeq 2.022$.
Given the mechanical EOS, the isentrope through any state can be calculated by
integrating the $-p\,dv$ work numerically (Swift, 2008). Isentropes calculated
from Grünseisen EOS fitted to shock data typically behave unphysically at high
compression, where the assumption that the Grüneisen parameter is a function
of density only breaks down. For the ammonia fit, the breakdown occurred at a
mass density of 2.145 g/cm3. The isentrope was well-behaved to several
terapascals, though its accuracy was undetermined. The isentrope should be
reasonably accurate at least up to the peak compression in the shock data,
which equates to 22.1 GPa on the isentrope. To investigate the sensitivity to
the Grüneisen parameter, isentropes were calculated for the nominal value
above, and for values 10% lower and higher. With this variation in $\Gamma$,
the pressure varied by 10% at 20 GPa, rising to 25% at 500 GPa. (Fig. 2)
Figure 2: Principal isentrope of liquid ammonia (initial temperature 203 K)
deduced from mechanical equation of state fitted to principal shock Hugoniot,
showing sensitivity to assumed Grünseisen parameter.
## 3 Mass-radius relationships
Mass-radius relationships were calculated using the deduced EOS, for a self-
gravitating body comprising pure ammonia and also for differentiated bodies
consisting of a rocky core and an ammonia mantle, using the numerical methods
described previously (Swift et al, 2011). Separate mass-radius curves were
constructed for the nominal and perturbed values of the Grüneisen parameter.
In all cases, the temperature at the surface was taken to be 203 K, to match
the initial state in the shock experiments. The rocky core was modeled using
an EOS for basalt, SESAME 7530 (Barnes & Lyon, 1988), as was done previously
(Swift et al, 2011).
The variations in $\Gamma$ made a negligible difference to the mass-radius
relations. At high masses, the mass-radius relation for pure ammonia
asymptoted to a power-law behavior $R=\alpha M^{\beta}$ with $\alpha=1.4395\pm
0.0005$ and $\beta=0.32889\pm 0.00004$. For an incompressible material,
$M=\frac{4}{3}\pi r^{3}\rho_{0},$ (5)
giving $\beta=1/3$. The difference in the fitted value is small but
significant; $\alpha$ is considerably less than the incompressible value of
$\left(\frac{4}{3}\pi\rho_{0}\right)^{-1/3}$. The mass-radius relation was
also deduced using the SESAME EOS, which matched that from the Grüneisen EOS
up to 0.1 M${}_{\mbox{E}}$, above which point the extrapolation beyond the
bounds of the table gave unphysical behavior. (Figs 3 to 5.)
Figure 3: Mass-radius relation deduced from equation of state for liquid
ammonia (surface temperature 203 K), also showing relation for incompressible
material and least-squares fit to the relation, which is dominated by high
pressure behavior. Figure 4: Variation of central pressure with mass, for
liquid ammonia (surface temperature 203 K). Figure 5: Mass-radius relations
for differentiated ammonia-rock bodies. The percentages are the mass fraction
of ammonia in the body.
The planetary radius for pure ammonia did not exhibit a maximum within the
range of masses investigated. The range of compressions explored by the shock
experiments was equivalent to the central pressure in bodies of pure ammonia
up to around 2/3 M${}_{\mbox{E}}$(1.5 R${}_{\mbox{E}}$). However, the mass-
radius relation is accurate for significantly larger bodies, because the mass
and volume are dominated by matter at much lower pressures until the average
density exceeds 1.5 g/cm3 or so: approximately 4 M${}_{\mbox{E}}$, 2.5
R${}_{\mbox{E}}$, and a core pressure of 100 GPa. The relation may be accurate
for even larger bodies, but it has not been validated by EOS experiments.
## 4 Conclusions
The relation between shock and particle speeds in liquid ammonia appears
linear to within the scatter in the data up to pressures of at least 39 GPa. A
Grüneisen mechanical equation of state was constructed using the principal
Hugoniot of initial state zero pressure and 203 K as a reference, and
estimating the Grüneisen parameter $\Gamma$ from the slope of the Hugoniot.
Isentropes were calculated through the same state, the sensitivity to $\Gamma$
rising with pressure.
Mass-radius relations were calculated for self-gravitating bodies consisting
of ammonia, and differentiated ammonia-rock mixtures. The mass-radius
relations were insensitive to variations in $\Gamma$, indicating that the
relations should be reliable for comparison to planetary measurements, to
central pressures substantially above those reached in the shock experiments.
## Acknowledgements
This work was performed under the auspices of the U.S. Department of Energy
under contract DE-AC52-07NA27344.
## References
* Schneider (2011) Schneider, J. 2011, The Extrasolar Planets Encyclopaedia (version 2.06), http://exoplanet.eu and references therein
* Marsh (1980) Marsh, S.P. (Ed) 1980, LASL Shock Hugoniot Data, University of California, Berkeley
* Holian (1984) Holian, K.S. (Ed.) 1984, Los Alamos National Laboratory report LA-10160-MS Vol 1c
* Haar & Gallagher (1978) Haar, L. & Gallagher, J.S. 1978, J. Phys. Chem. Ref. Data, 7, 3, 635
* Abramson (2008) Abramson, E.H. 2008, J. Chem. Eng. Data, 53, 1986
* Li et al (2009) Li, F., Li, M., Cui, Q., Cui, T., He, Z., Zhou, Q., & Zou, G. 2009, J. Chem. Phys., 131, 134502
* Zel’dovich & Raizer (1966) Zel’dovich, Ya.B. & Raizer, Yu.P. 1966, Physics of shock waves and high temperature hydrodynamic phenomena, Academic Press, New York
* Bowen & Thompson (1968) Bowen, D.E. and Thompson, J.C. 1968, J. Chem. Eng. Data, 13, 2, 206–208
* Woolfolk et al (1973) Woolfolk, R.W., Cowperthwaite, M., & Shaw, R. 1973, Thermochimica Acta, 5, 409-414
* McQueen et al (1970) McQueen, R.G., Marsh, S.P., Taylor, T.W., Fritz, J.N., & Carter, W.J., in Kinslow, R. (Ed) 1970, High Velocity Impact Phenomena, Academic Press, New York
* Skidmore (1965) Skidmore, I.C. 1965, App. Materials Res. 131–147
* Swift (2008) Swift, D.C. 2008 J. Appl. Phys., 104, 7, 073536
* Swift et al (2011) Swift, D.C. et al 2011, ApJ manuscript 83565 (accepted), preprint arXiv:1001.4851.
* Barnes & Lyon (1988) Barnes, J.F. & Lyon, S.P. 1988, SESAME Equation of State Number 7530, Basalt, Los Alamos National Laboratory report LA-11253-MS
|
arxiv-papers
| 2011-10-13T20:51:14 |
2024-09-04T02:49:23.093718
|
{
"license": "Public Domain",
"authors": "Damian C. Swift",
"submitter": "Damian Swift",
"url": "https://arxiv.org/abs/1110.3066"
}
|
1110.3094
|
Nigel Collier, Son Doan 11institutetext: National Institute of Informatics,
Tokyo, Japan
collier@nii.ac.jp
WWW home page: http://born.nii.ac.jp/dizie
# Syndromic classification of Twitter messages
Nigel Collier 11 Son Doan 11
###### Abstract
Recent studies have shown strong correlation between social networking data
and national influenza rates. We expanded upon this success to develop an
automated text mining system that classifies Twitter messages in real time
into six syndromic categories based on key terms from a public health
ontology. 10-fold cross validation tests were used to compare Naive Bayes (NB)
and Support Vector Machine (SVM) models on a corpus of 7431 Twitter messages.
SVM performed better than NB on 4 out of 6 syndromes. The best performing
classifiers showed moderately strong F1 scores: respiratory = 86.2 (NB);
gastrointestinal = 85.4 (SVM polynomial kernel degree 2); neurological = 88.6
(SVM polynomial kernel degree 1); rash = 86.0 (SVM polynomial kernel degree
1); constitutional = 89.3 (SVM polynomial kernel degree 1); hemorrhagic = 89.9
(NB). The resulting classifiers were deployed together with an EARS C2
aberration detection algorithm in an experimental online system.
###### Keywords:
epidemic intelligence, social networking, machine learning, natural language
processing
## 1 Introduction
Twitter is a social networking service that allows users throughout the world
to communicate their personal experiences, opinions and questions to each
other using micro messages (‘tweets’). The short message style reduces thought
investment java:2007 and encourages a rapid ‘on the go’ style of messaging
from mobile devices. Statistics show that Twitter had over 200 million
users111http://www.bbc.co.uk/news/business-12889048 in March 2011,
representing a small but significant fraction of the international population
across both age and
gender222http://sustainablecitiescollective.com/urbantickurbantick/20462/twitter-
usage-view-america with a bias towards the urban population in their 20s and
30s. Our recent studies into novel health applications collier:2011c have
shown progress in identifying free-text signals from tweets that allow
influenza-like illness (ILI) to be tracked in real time. Similar studies have
shown strong correlation with national weekly influenza data from the Centers
for Disease Control and Prevention and the United Kingdom’s Health Protection
Agency. Approaches like these hold out the hope that low cost sensor networks
could be deployed as early warning systems to supplement more expensive
traditional approaches. Web-based sensor networks might prove to be
particularly effective for diseases that have a narrow window for effective
intervention such as pandemic influenza.
Despite such progress, studies into deriving linguistic signals that
correspond to other major syndromes have been lacking. Unlike ILI, publicly
available gold standard data for other classes of conditions such as
gastrointestinal or neurological illnesses are not so readily available.
Nevertheless, the previous studies suggest that a more comprehensive early
warning system based on the same principles and approaches should prove
effective. Within the context of the DIZIE project, the contribution of this
paper is (a) to present our data classification and collection approaches for
building syndromic classifiers; (b) to evaluate machine learning approaches
for predicting the classes of unseen Twitter messages; and (c) to show how we
deployed the classifiers for detecting disease activity. A further goal of our
work is to test the effectiveness of outbreak detection through geo-temporal
aberration detection on aggregations of the classified messages. This work is
now ongoing and will be reported elsewhere in a separate study.
### 1.1 Automated Web-sensing
In this section we make a brief survey of recent health surveillance systems
that use the Web as a sensor source to detect infectious disease outbreaks.
Web reports from news media, blogs, microblogs, discussion forums, digital
radio, user search queries etc. are considered useful because of their wide
availability, low cost and real time nature. Although we will focus on
infectious disease detection it is worth noting that similar approaches can be
applied to other public health hazards such as earthquakes and typhoons
earle:2010 ; sakaki:2010 .
Current systems fall into two distinct categories: (a) event-based systems
that look for direct reports of interest in the news media (see hartley:2010
for a review), and (b) systems that exploit the human sensor network in sites
like Twitter, Jaiku and Prownce by sampling reports of symptoms/GP visits/drug
usage etc. from the population at risk szomszor:2009 ; lampos:2010 ;
signorini:2011 . Early alerts from such systems are typically used by public
health analysts to initiate a risk analysis process involving many other
sources such as human networks of expertise.
Work on the analysis of tweets, whilst still a relatively novel information
source, is related to a tradition of syndromic surveillance based on analysis
of triage chief complaint (TCC) reports, i.e. the initial triage report
outlining the reasons for the patient visit to a hospital emergency room. Like
tweets they report the patient’s symptoms, are usually very brief, often just
a few keywords and can be heavily abbreviated. Major technical challenges
though do exist: unlike TCC reports tweets contain a very high degree of noise
(e.g. spam, opinion, re-tweeting etc.) as well as slang (e.g. itcy for itchy)
and emoticons which makes them particularly challenging. Social media is
inherently an informal medium of communication and lacks a standard vocabulary
although Twitter users do make use of an evolving semantic tag set. Both TCC
and tweets often consist of short telegraphic statements or ungrammatical
sentences which are difficult for uncustomised syntactic parsers to handle.
In the area of TCC reports we note work done by the RODS project wagner:2004
that developed automatic techniques for classifying reports into a list of
syndromic categories based on natural language features. The chief complaint
categories used in RODS were respiratory, gastrointestinal, botulinic,
constitutional, neurologic, rash, hemorrhagic and none. Further processes took
aggregated data and issued alerts using time series aberration detection
algorithms. The DIZIE project which we report here takes a broadly similar
approach but applies it to user generated content in the form of Twitter
messages.
## 2 Method
DIZIE currently consists of the following components: (1) a list of latitudes
and longitudes for target world cities based on Twitter usage; (2) a lexicon
of syndromic keywords used as an initial filter, (3) a supervised machine
learning model that converts tweets to a word vector representation and then
classifies them according to six syndromes, (4) a post-processing list of stop
words and phrases that blocks undesired contexts, (5) a MySQL database holding
historic counts of positive messages by time and city location, used to
calculate alerting baselines, (6) an aberation detection algorithm, and (7) a
graphical user interface for displaying alerts and supporting evidence.
After an initial survey of high frequency Twitter sources by city location we
selected 40 world cities as candidates for our surveillance system. Sampling
in the runtime system is done using the Twitter API by searching for tweets
originating within a 30km radius of a city’s latitude and longitude, i.e. a
typical commuting/shopping distance from the city centre. The sampling rate is
once every hour although this can be shortened when the system is in full
operation. In this initial study we focussed only on English language tweets
and how to classify them into 6 syndromic categories which we describe below.
Key assumptions in our approach are that: (a) each user is considered to be a
sensor in the environment and as such no sensor should have the capacity to
over report. We controlled over reporting by simply restricting the maximum
number of messages per day to be 5 per user; (b) each user reports on personal
observations about themselves or those directly known to them. To control (a)
and (b) and prevent over-reporting we had to build in filtering controls to
mitigate the effects of information diffusion through re-reporting,
particularly for public personalities and mass events. Re-tweets, i.e.
repeated messages, and tweets involving external links were automatically
removed.
### 2.1 Schema development
A syndrome is a collection of symptoms (both specific and non-specific) agreed
by the medical community that are indicative of a class of diseases. We chose
six syndrome classes as the targets of our classifier: constitutional,
respiratory, gastrointestinal, hemorrhagic, rash (i.e. dermatological) and
neurological. These were based on an openly available public health ontology
developed as part of the BioCaster project collier:2008a by a team of experts
in computational linguists, public health, anthropology and genetics.
Syndromes within the ontology were based on RODS syndrome definitions and are
linked to symptom terms - both technical and laymen’s terms - through typed
relations. We use these symptoms (syndromic keywords) as the basis for
searching Twitter and expanded them using held out Twitter data.
### 2.2 Twitter Data
After defining our syndromes we examined a sample of tweets and wrote
guidelines outlining positive and negative case definitions. These guidelines
were then used by three student annotators to classify a sample of 2000 tweets
per syndrome into positive or negative for each of the syndrome classes. Data
for training was collected by automatically searching Twitter using the
syndromic keywords over the period 9th to 24th July 2010. No city filtering
was applied when we collected the training data. Typical positive example
messages are: “Woke up with a stomach ache!”, “Every bone in my body hurts”,
and “Fever, back pain, headache… ugh!”. Examples of negative messages are:
“I’m exercising till I feel dizzy”, “Cabin fever is severe right now”,
“Utterly exhausted after days of housework”. Such negative examples include a
variety of polysemous symptom words such as fever in its senses of raised
temperature and excitement and headache in its senses of a pain in the head or
an inconvenience. The negative examples also include cases where the context
indicates that the cause of the syptom is unlikely to be an infection, e.g.
headache caused by working or exercising. The training corpus is characterised
using the top 7 terms calculated by mutual association score in Table 1. This
includes several spurious associations such as ‘rt’ standing for ‘repeat
tweet’, ‘botox’ which is discussed extensively as a treatment for several
symptoms and ‘charice’ who is a new pop idol.
The final corpus was constructed from messages where there was total agreement
between all three annotators. This data set was used to develop and evaluate
supervised learning classifiers in cross-fold validation experiments. A
summary of the data set is shown in Table 2. Inter-annotator agreement scores
between the three annotators are given as Kappa showing agreement between the
two highest agreeing annotators. Kappa indicates strong agreement on most
syndromic classes with the noteable exception of gastrointestina and
neurological.
Table 1: Top 7 terms by syndrome calculated by mutual information score. * indicates a spurious association. Resp | Gastro | Const | Hemor | Rash | Neuro
---|---|---|---|---|---
throat | stomach | botox∗ | pain | road | headache
sore | ache | body | hemorrhage | heat | coma
cough | gib | charice∗ | muscle | arm | worst
flu | feel | jaw | tired | tired | gave
nose | rt∗ | hurts | pray | rash | giving
rt∗ | bad | stomach | brain | itcy | vertigo
cold | worst | sweating | guiliechelon∗ | face | pulpo∗
Table 2: Structure of the annotated syndrome corpus of Twitter messages. Syndrome | Positives (P) | Negatives (N) | P/N | Kappa
---|---|---|---|---
Respiratory | 627 | 738 | 0.85 | 0.67%
Gastrointestinal | 489 | 676 | 0.72 | 0.49%
Neurological | 549 | 434 | 1.26 | 0.42%
Rash | 914 | 592 | 1.54 | 0.86%
Hemorrhagic | 320 | 711 | 0.45 | 0.92%
Constitutional | 1043 | 338 | 3.09 | 0.78%
### 2.3 Classifier models
DIZIE employs a two stage filtering process. Since Twitter many topics
unrelated to disease outbreaks, DIZIE firstly requests Twitter to send it
messages that correspond to a set of core syndromic keywords, i.e. the same
sampling strategy used to collect training/testing data. These keywords are
defined in the BioCaster public health ontology collier:2008a . In the second
stage messages which are putatively on topic are filtered more rigorously
using a machine learning approach. This stage of filtering aims to identify
messages containing ambiguous words whose senses are not relevant to
infectious diseases and messages where the cause of the symptoms are not
likely to be infectious diseases. About 70% of messages are removed at this
second stage.
To aid in model selection our experiments used two widely known machine
learning models to classify Twitter messages into a fixed set of syndromic
classes: Naive Bayes (NB) and support vector machines (SVM) joachims:98 using
a variety of kernel functions. Both models were trained with binary feature
vectors representing a dictionary index of words in the training corpus. i.e.
a feature for the test message was marked 1 if a word was present in the test
message which had been seen previously in the training corpus otherwise 0. No
normalisation of the surface words was done, e.g. using stemming, because of
the high out of vocabulary rate with tools trained on general language texts.
Despite the implausibility of its strong statistical independence assumption
between words, NB tends to perform strongly. The choice to explore keywords as
features rather than more sophisticated parsing and conceptual analysis such
as MPLUS christensen:2002 was taken from a desire to evaluate less expensive
approaches before resorting to time consuming knowledge engineering.
The NB classifier exploits an estimation of the Bayes Rule:
$P(c_{k}|d)=\frac{P(c_{k})\times\prod_{i=1}^{m}P(f_{i}|c_{k})^{f_{i}(d)}}{P(d)}$
(1)
where the objective is to assign a given feature vector for a document $d$
consisting of $m$ features to the highest probability class $c_{k}$.
$f_{i}(d)$ denotes the frequency count of feature $i$ in document $d$.
Typically the denominator $P(d)$ is not computed explicitly as it remains
constant for all $c_{k}$. In order to compute the highest value numerator NB
makes an assumption that features are conditionally independent given the set
of classes. Right hand side values of the equation are estimates based on
counts observed in the training corpus of classified Twitter messages. We used
the freely available Rainbow toolkit333http://www.cs.cmu.edu/
mccallum/bow/rainbow/ from CMU as the software package.
SVMs have been widely used in text classification achieving state of the art
predictive accuracy. The major distinction between the two approaches are that
whereas NB is a generative classifier which forms a statistical model of each
class, SVM is a large-margin binary classifier. SVM operates as a two stage
process. Firstly the feature vectors are projected into a high dimensional
space using a kernel function. The second stage finds a maximum margin
hyperplane within this space that separates the positive from the negative
instances of the syndromic class. In practice it is not necessary to perfectly
classify all instances with the level of tolerance for misclassification being
controlled by the C parameter in the model. A series of binary classifiers
were constructed (one for each syndrome) using the SVMLight software package
444http://svmlight.joachims.org/. We explored polynomial degree 1, 2, 3 and
radial basis function kernels.
### 2.4 Temporal model
In order to detect unexpected rises in the stream of messages for each
syndrome we implemented a widely used change point detection algorithm called
the Early Aberration and Reporting System (EARS) C2 hutwagner:2003 . C2
reports an alert when its test value $S_{t}$ exceeds a number $k$ of standard
deviations above a historic mean:
$S_{t}=max(0,(C_{t}-(\mu_{t}+k\sigma_{t}))/\sigma_{t})$ (2)
where $C_{t}$ is the count of classified tweets for the day, $\mu_{t}$ and
$\sigma_{t}$ are the mean and standard deviation of the counts during the
history period, set as the previous two weeks. $k$ controls the number of
standard deviations above the mean where an alert is triggered, set to 1 in
our system. The output of C2 is a numeric score indicating the degree of
abnormality but this by itself is not so meaningful to ordinary users. We
constructed 5 banding groups for the score and showed this in the graphical
user interface.
## 3 Results
### 3.1 Classifying Twitter Messages
Results for 10-fold cross validation experiments on the classification models
are shown in Table 3. Overall the SVM with polynomial degree 1 kernel
outperformed all other kernels with other kernels generally offering better
precision at a higher cost to recall. Precision (Positive predictive) values
ranged from 82.0 to 93.8 for SVM (polynomial degree 1) and from 83.3 to 99.0
for NB. Recall (sensitivity) values ranged from 58.3 to 96.2 for SVM
(polynomial degree 1) and from 74.7 to 90.3 for NB. SVM tended to offer a
reduced level of precision but better recall. In the case of one syndrome
(Hemorrhagic) we noticed an unusually low level of recall for SVM but not for
NB.
SVM’s performance seemed moderately correlated to the positive/negative ratio
in the training corpus and also showed weakness for the two classes
(Hemorrhagic and Gastrointestinal) with the smallest positive counts. Naive
Bayes performed robustly across classes with no obvious correlation either to
positive/negative ratio or the volume of training data. Low performance was
seen in both models for the gastrointestinal syndrome. This was probably due
to the low number of training examples resulting from the low inter-annotator
agreement on this class and the requirement for complete agreement between all
three annotators.
Table 3: Evaluation of automated syndrome classification using naive Bayes and Support Vector Machine models on 10-fold cross validation. P - Precision, R - Recall, F1 - F1 score. 1 SVM using a linear kernel, 2 SVM using a polynomial kernal degree 2, 3 SVM using a polynomial kernal degree 3, R SVM using a radial basis function kernel. | Naive Bayes | SVM1 | SVM2 | SVM3 | SVMR
---|---|---|---|---|---
Synd. | P | R | F1 | P | R | F1 | P | R | F1 | P | R | F1 | P | R | F1
Resp. | 90.3 | 82.4 | 86.2 | 85.4 | 82.5 | 83.8 | 83.0 | 71.0 | 76.5 | 86.4 | 61.3 | 71.7 | 66.7 | 3.2 | 6.2
Gast. | 83.3 | 75.5 | 79.2 | 85.9 | 78.4 | 81.8 | 92.7 | 79.2 | 85.4 | 91.4 | 66.7 | 77.1 | 73.1 | 39.6 | 51.3
Neur. | 98.2 | 74.7 | 84.8 | 83.2 | 95.0 | 88.6 | 77.9 | 98.2 | 86.9 | 62.4 | 98.2 | 76.3 | 90.0 | 63.0 | 74.1
Rash | 94.5 | 76.1 | 84.3 | 82.0 | 90.6 | 86.0 | 76.9 | 91.2 | 83.4 | 67.7 | 94.5 | 78.9 | 60.7 | 100.0 | 75.5
Hem. | 89.4 | 90.3 | 89.9 | 93.8 | 58.3 | 71.7 | 100.0 | 50.0 | 66.7 | 100.0 | 50 | 66.7 | 87.5 | 43.8 | 58.3
Con. | 99.0 | 79.8 | 88.4 | 83.6 | 96.2 | 89.3 | 83.6 | 93.3 | 88.2 | 78.6 | 99.0 | 87.7 | 76.5 | 100 | 86.7
### 3.2 Technology dissemination
An experimental service for syndromic surveillance called DIZIE has been
implemented based on the best of our classifier models and we are now
observing its performance. The service is freely available from an online
portal at http://born.nii.ac.jp/dizie. As shown in Figure 1 the graphical user
interface (GUI) for DIZIE shows a series of radial charts for each major world
city with each band of the chart indicating the current level of alert for one
of the six syndromes. Alerting level scores are calculated using the Temporal
Model presented above. Each band is colour coded for easy recognition.
Alerting levels are calculated on the classified twitter messages using the
EARS C2 algorithm described above. Data selection is by city and time with
drill down to a selection of user messages that contributed to the current
level. Trend bars show the level of alert and whether the trend is upwards,
downwards or sideways. Charting is also provided over an hourly, daily, weekly
and monthly period. The number of positively classified messages by city is
indicated in Figure 2 for a selection of cities.
Figure 1: Radial graphs showing syndromic alert levels for major world cities.
Colour coding on the radial segments indicates the alerting degree
automatically assigned to a syndrome in a city based on the previous hour’s
Twitter counts and the previous 2 weeks as a baseline. The page is updated
every hour. Clicking on the graph for a city displays the frequency graph and
also the matching tweets for the current hour. Figure 2: Number of Tweets by a
sample of major world cities classified by DIZIE during the period 2nd March
2011 to 31st August 2011.
Navigation links are provided to and from BioCaster, a news event alerting
system, and we expect in the future to integrate the two systems more closely
to promote greater situation awareness across media sources. Access to the GUI
is via regular Web browser or mobile device with the page adjusting
automatically to fit smaller screens.
## 4 Conclusion
Twitter offers unique challenges and opportunities for syndromic surveillance.
Approaches based on machine learning need to be able (a) to handle biased
data, and (b) to adjust to the rapidly changing vocabulary to prevent a flood
of false positives when new topics trend. Future work will compare keyword
classifiers against more conceptual approaches such as christensen:2002 and
also compare the performance characteristics of change point detection
algorithms.
Based on the experiments reported here we have built an experimental
application called DIZIE that samples Twitter messages originating in major
world cities and automatically classifies them according to syndromes. Access
to the system is openly available. Based on the outcome of our follow up study
we intend to integrate DIZIE’s output with our event-based surveillance system
BioCaster which is currently used by the international public health
community.
## Acknowledgements
This work was in part supported by grant in aid support from the National
Institute of Informatics’ Grand Challenge Project (PI:NC). We are grateful to
Reiko Matsuda Goodwin for commenting on the user interface in the early stages
of this study and helping in data collection for the final system.
## References
* [1] A. Java, X. Song, T. Finin, and B. Tseng. Why we twitter: Understanding microblogging usage and communities. In Proc. 9th WebKDD and 1st SNA-KDD Workshop on Web Mining and Social Network Analysis, ACM, 12th August 2007.
* [2] N. Collier, S. T. Nguyen, and M.T.N. Nguyen. OMG U got flu? analysis of shared health messages for bio-surveillance. Biomedical Semantics, 2(Suppl 5):S10, September 2011.
* [3] P. Earle. Earthquake twitter. Nature Geoscience, 3(4):221–222, 2010. doi:10.1038/ngeo832.
* [4] T. Sakaki, M. Okazaki, and Y. Matsuo. Earthquake shakes twitter users: real-time event detection by social sensors. In Proc. of the 19th International World Wide Web Conference, Raleigh, NC, USA, pages 851–860, 2010.
* [5] D. Hartley, N. Nelson, R. Walters, R. Arthur, R. Yangarber, L. Madoff, J. Linge, A. Mawudeku, N. Collier, J. Brownstein, G. Thinus, and N. Lightfoot. The landscape of international biosurveillance. Emerging Health Threats J., 3(e3), January 2010. doi:10.1093/bioinformatics/btn534.
* [6] Martin Szomszor, Patty Kostkova, and Ed De Quincey. swineflu : Twitter predicts swine flu outbreak in 2009. Number December. 2009.
* [7] V. Lampos, T. De Bie, and N. Cristianini. Flu detector - tracking epidemics on twitter. In Machine Learning and Knowledge Discovery in Databases, volume 6223/2010, pages 599–602. Lecture Notes in Computer Science, 2010.
* [8] A. Signorini, A. M. Segre, and P. M. Polgreen. The use of twitter to track levels of disease activity and public concern in the U.S. during the influenza a h1n1 pandemic. PLoS One, 6(5):e19467, 2011.
* [9] M. M. Wagner, J. Espino, F.C. Tsui, P. Gesteland, W. Chapman, W. Ivanov, A. Moore, W. Wong, J. Dowling, and J. Hutman. Syndrome and outbreak detection using chief-complaint data - experience of the real-time outbreak and disease surveillance project. Morbidity and Mortality Weekly Report (MMWR), 53 (Suppl):28–31, 2004.
* [10] N. Collier, S. Doan, A. Kawazoe, R. Matsuda Goodwin, M. Conway, Y. Tateno, Q. Ngo, D. Dien, A. Kawtrakul, K. Takeuchi, M. Shigematsu, and K. Taniguchi. BioCaster:detecting public health rumors with a web-based text mining system. Bioinformatics, 24(24):2940–1, December 2008. doi:10.1093/bioinformatics/btn534.
* [11] T. Joachims. Text categorization with support vector machines: Learning with many relevant features. In Proceedings of the European Conference on Machine Learning, 1998\.
* [12] L. M. Christensen, P. J. Haug, and M. Fiszmann. Mplus: A probabilistic medical language understanding model. In Proceedings of the Workshop on Natural Language Processing in the Biomedical Domain, Philadelphia, USA, July 2002.
* [13] L. Hutwagner, W. Thompson, M. G. Seeman, and T. Treadwell. The bioterrorism preparedness and response early aberration reporting system (EARS). J. Urban Health, 80(2):i89–i96, 2003.
|
arxiv-papers
| 2011-10-13T23:42:32 |
2024-09-04T02:49:23.109982
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nigel Collier, Son Doan",
"submitter": "Nigel Collier",
"url": "https://arxiv.org/abs/1110.3094"
}
|
1110.3151
|
# Minimum penalized Hellinger distance for model selection in small samples
Papa Ngoma, Bertrand Ntepb a,bLMA - Laboratoire de Mathématiques Appliquées
Université Cheikh Anta Diop
BP 5005 Dakar-Fann Sénégal
a e-mail : papa.ngom@ucad.edu.sn
b ntepjojo@yahoo.fr
###### Abstract
In statistical modeling area, the Akaike information criterion AIC, is a
widely known and extensively used tool for model choice. The $\phi$-divergence
test statistic is a recently developed tool for statistical model selection.
The popularity of the divergence criterion is however tempered by their known
lack of robustness in small sample. In this paper the penalized minimum
Hellinger distance type statistics are considered and some properties are
established. The limit laws of the estimates and test statistics are given
under both the null and the alternative hypotheses, and approximations of the
power functions are deduced. A model selection criterion relative to these
divergence measures are developed for parametric inference. Our interest is in
the problem to testing for choosing between two models using some
informational type statistics, when independent sample are drawn from a
discrete population. Here, we discuss the asymptotic properties and the
performance of new procedure tests and investigate their small sample
behavior.
###### keywords:
Generalized information, estimation, hypothesis test, Monte Carlo simulation.
AMS Subject Classification : 62F03, 62F05, 60F40,94A17.
††thanks: This research was supported, in part, by grants from AIMS(African
Institute for Mathematical Sciences) 6 Melrose Road, Muizenberg-Cape Town 7945
South Africa
## 1 Introduction
A comprehensive surveys on Pearson Chi-square type statistics has been
provided by many authors as Cochran (1952), Watson (1956) and Moore
(1978,1986), in particular on quadratics forms in the cell frequencies.
Recently, Andrews(1988a, 1988b) has extended the Pearson chi-square testing
method to non-dynamic parametric models, i.e., to models with covariates.
Because Pearson chi-square statistics provide natural measures for the
discrepancy between the observed data and a specific parametric model, they
have also been used for discriminating among competing models. Such a
situation is frequent in Social Sciences where many competing models are
proposed to fit a given sample. A well know difficulty is that each chi-square
statistic tends to become large without an increase in its degrees of freedom
as the sample size increases. As a consequence goodness-of-fit tests based on
Pearson type chi-square statistics will generally reject the correct
specification of every competing model.
To circumvent such a difficulty, a popular method for model selection, which
is similar to use of Akaike (1973) Information Criterion (AIC), consists in
considering that the lower the chi-square statistic, the better is the model.
The preceding selection rule, however, does not take into account random
variations inherent in the values of the statistics.
We propose here a procedure for taking into account the stochastic nature of
these differences so as to assess their significance. The main propose of this
paper is to address this issue. We shall propose some convenient
asymptotically standard normal tests for model selection based on
$\phi-$divergence type statistics. Following Vuong (1989, 1993), the
procedures considered here are testing the null hypothesis that the competing
models are equally close to the data generating process (DGP) versus the
alternative hypothesis that one model is closer to the DGP where closeness of
a model is measured according to the discrepancy implicit in the
$\phi-$divergence type statistic used. Thus the outcomes of our tests provide
information on the strength of the statistical evidence for the choice of a
model based on its goodness-of-fit. The model selection approach proposed here
differs from those of Cox (1961, 1962) and Akaike (1974) for non nested
hypotheses. This difference is that the present approach is based on the
discrepancy implicit in the divergence type statistics used, while these other
approaches as Vuong’s (1989) tests for model selection rely on the Kullback-
Leibler (1951) information criterion (KLIC).
Beran (1977) showed that by using the minimum Hellinger distance estimator,
one can simultaneously obtain asymptotic efficiency and robustness properties
in the presence of outliers. The works of Simpson (1989) and Lindsay (1994)
have shown that, in the tests hypotheses, robust alternatives to the
likelihood ratio test can be generated by using the Hellinger distance. We
consider a general class of estimators that is very broad and contains most of
estimators currently used in practice when forming divergence type statistics.
This covers the case studies in Harris and Basu (1994); Basu et al. (1996);
Basu and Basu (1998) where the penalized Hellinger distance is used.
The remainder of this paper is organized as follows. Section 2 introduces the
basic notations and definitions. Section 3 gives a short overview of
divergence measures. Section 4 investigates the asymptotic distribution of the
penalized Hellinger distance. In section 5, some applications for testing
hypotheses are proposed. Section 6 presents some simulation results. Section 7
concludes the paper.
## 2 Definitions and notation
In this section, we briefly present the basic assumptions on the model and
parameters estimators, and we define our generalized divergence type
statistics.
We consider a discrete statistical model, i.e $X_{1},X_{2},\ldots X_{n}$ an
independent random sample from a discrete population with support
$\mathcal{X}=\\{1,\ldots,m\\}$. Let $P=\left(p_{1},\ldots,p_{m}\right)^{T}$ be
a probability vector i.e $P\in\Omega_{m}$ where $\Omega_{m}$ is the simplex of
probability m-vectors,
$\Omega_{m}=\big{\\{}\left(p_{1},p_{2},\ldots,p_{m}\right)\in\mathbb{R}^{m}\
;\displaystyle\sum_{i=1}^{m}p_{i}=1,\ p_{i}\geq 0,i=1,\dots,m\big{\\}}.$
We consider a parameter model
$\mathcal{P}=\\{P_{\theta}=\left(p_{1}(\theta),\ldots,p_{m}(\theta)\right)^{T}:\
\theta\in\Theta\\}$
which may or may not contain the true distribution $P$, where $\Theta$ is a
compact subset of k-dimensional Euclidean space (with $k<m-1$). If
$\mathcal{P}$ cointains $P$, then there exists a $\theta_{0}\in\Theta$ such
that $P_{\theta_{0}}=P$ and the model $\mathcal{P}$ is said to be correctly
specified.
We are interested in testing
$H_{0}:P\in{\cal{P}}\ (\hbox{ with true parameter}\ {\theta_{0}})\ \hbox{
versus }H_{1}:P\in\Omega_{m}-\cal{P}.$
By $\parallel\cdot\parallel$ we denote the usual Euclidean norm and we
interpret probability distributions on $\mathcal{X}$ as row vectors from
$\mathbb{R}^{m}$. For simplicity we restrict ourselves to unknown true
parameters $\theta_{0}$ satisfying the classical regularity conditions given
by Birch (1964):
1\. True $\theta_{0}$ is an interior point of $\Theta$ and $p_{i\theta_{0}}>0$
for $i=1,\ldots,m$. Thus
$P_{\theta_{0}}=\left(p_{1\theta_{0}},\ldots,p_{m_{\theta_{0}}}\right)^{T}$ is
an interior point of the set $\Omega_{m}$.
2\. The mapping $P:\Theta\longrightarrow\Omega_{m}$ is totally differentiable
at $\theta_{0}$ so that the partial derivatives of $p_{i}$ with respect to
each $\theta_{j}$ exist at $\theta_{0}$ and $p_{i}(\theta)$ has a linear
approximation at $\theta_{0}$ given by
$p_{i}(\theta)=p_{i}(\theta_{0})+\sum_{j=1}^{k}(\theta_{j}-\theta_{0j})\frac{\partial
p_{i}(\theta_{0})}{\partial\theta_{j}}+o(\parallel\theta-\theta_{0}\parallel)$
where $o(\parallel\theta-\theta_{0}\parallel)\hbox{ denotes a function
verifying
}\displaystyle\lim_{\theta\longrightarrow\theta_{0}}\frac{o(\parallel\theta-\theta_{0}\parallel)}{\parallel\theta-\theta_{0}\parallel}=0.$
3\. The Jacobian matrix $\displaystyle J(\theta_{0})=\left(\dfrac{\partial
P_{\theta}}{\partial\theta}\right)_{\theta=\theta_{0}}=\left(\frac{\partial
p_{i}(\theta_{0})}{\partial\theta_{j}}\right)_{\begin{subarray}{c}1\leq i\leq
m\\\ 1\leq j\leq k\end{subarray}}$ is of full rank (i.e. of rank k and $k<m$).
4\. The inverse mapping $P^{-1}:{\cal{P}}\longrightarrow\Theta$ is continuous
at $P_{\theta_{0}}.$
5\. The mapping $P:\Theta\longrightarrow\Omega_{m}$ is continuous at every
point $\theta\in\Theta$.
Under the hypothesis that $P\in\mathcal{P}$, there exists an unknown parameter
$\theta_{0}$ such that $P=P_{\theta_{0}}$ and the problem of point estimation
appears in a natural way. Let $n$ be sample size. We can estimate the
distribution
$P_{\theta_{0}}=\left(p_{1}(\theta),p_{2}(\theta),\ldots,p_{m}(\theta)\right)^{T}$
by the vector of observed frequencies
$\widehat{P}=(\hat{p}_{1},\ldots,\hat{p}_{m})$ on $\mathcal{X}$ ie of
measurable mapping $\mathcal{X}^{n}\longrightarrow\Omega_{m}$.
This non parametric estimator $\widehat{P}=(\hat{p}_{1},\ldots,\hat{p}_{m})$
is defined by
$\hat{p}_{j}=\frac{N_{j}}{n},\quad
N_{j}=\displaystyle{\sum_{i=1}^{n}}T^{i}_{j}(X_{i})\hbox{ \ where \
}T^{i}_{j}(X_{i})=\left\\{\begin{array}[]{lll}1&&\hbox{if }X_{i}=j\\\
0&&\hbox{otherwise}\end{array}\right.$ (2.1)
We can now define the class of $\phi$-divergence type statistics considered in
this paper.
## 3 A brief review of $\phi$-divergences
Many different measures quantifying the degree of discrimination between two
probability distributions have been studied in the past. They are frequently
called distance measures, although some of them are not strictly metrics. They
have been applied to different areas, such as medical image registration
(Josien P.W. Pluim, 2001), classification and retrieval, among others. This
class of distances is referred, in the literature, as the class of $\phi$, f
or g-divergences (Csisz$\grave{a}$r (1967); Vajda (1989); Morales et al.
(1995); Pardo (2006); Bassetti et al. (2007)) or the class of disparities
(Lindsay (1994)). The divergence measures play an important role in
statistical theory, specially in large theories of estimation and testing.
Later many papers have appeared in the literature, where divergence or entropy
type measures of information have been used in testing statistical hypotheses.
Among others we refer to McCulloch (1988), Read and Cressie (1988), Zografos
et al. (1990), Salicr$\grave{u}$ et al. (1994), Bar-Hen and Daudin (1995),
Men$\grave{e}$ndez et al. (1995, 1996, 1997), Pardo et al. (1995), Morales et
al. (1997, 1998), Zografos (1994, 1998), Bar-Hen (1996) and the references
therein. A measure of discrimination between two probability distributions
called $\phi$-divergence, was introduced by Csisz$\acute{a}$r (1967).
Recently, Broniatowski et al. (2009) presented a new dual representation for
divergences. Their aim was to introduce estimation and test procedures through
divergence optimization for discrete or continuous parametric models. In the
problem where independent samples are drawn from two different discrete
populations, Basu et al. (2010) developped some tests based on the Hellinger
distance and penalized versions of it.
Consider two populations $X$ and $Y$, according to classifications criteria
can be grouped into $m$ classes species $x_{1},x_{2},\ldots,x_{m}$ and
$y_{1},y_{2},\ldots,y_{m}$ with probabilities $P=(p_{1},p_{2},\ldots,p_{m})$
and $Q=(q_{1},q_{2},\ldots,q_{m})$ respectively. Then
$D_{\phi}(P,Q)=\sum_{i=1}^{m}q_{i}\phi(\frac{p_{i}}{q_{i}})$ (3.2)
is the $\phi-$divergence between $P$ and $Q$ (see Csisz$\acute{a}$r, 1967) for
every $\phi$ in the set $\Phi$ of real convex functions defined on
$[0,\infty[$. The function $\phi(t)$ is assumed to verify the following
regularity condition :
$\phi:[0,+\infty[\longrightarrow\mathbb{R}\cup\\{\infty\\}$ is convex and
continuous, where $0\phi(\frac{0}{0})=0$ and
$0\phi(\frac{p}{0})=\lim_{u\longrightarrow\infty}\left(\phi(u)/u\right)$. Its
restriction on $]0,+\infty[$ is finite, twice continuously differentiable in a
neighborhood of $u=1$, with $\phi(1)=\phi^{\prime}(1)=0$ and
$\phi^{\prime\prime}(1)=1$ (cf. Liese and Vajda (1987)).
We shall be interested also in parametric estimators
$\widehat{Q}=\widehat{Q}_{n}=P_{\hat{\theta}}$ (3.3)
of $P_{\theta_{0}}$ which can be obtained by means of various point estimators
$\hat{\theta}=\hat{\theta}^{(n)}:\mathcal{X}^{(n)}\longrightarrow\Theta$
of the unknown parameter $\theta_{0}$.
It is convenient to measure the difference between observed $\widehat{P}$ and
expected frequencies $P_{\theta_{0}}$. A minimum Divergence estimator of
$\theta$ is a minimizer of $D_{\phi}(\widehat{P},P_{\theta_{0}})$ where
$\widehat{P}$ is a nonparametric distribution estimate. In our case, where
data come from a discrete distribution, the empirical distribution defined in
(2.1) can be used.
In particular if we replace $\phi_{1}(x)=-4[\sqrt{x}-\frac{1}{2}(x+1)]$ in
(3.2) we get the Hellinger distance between distribution $\widehat{P}$ and
$P_{\theta}$ given by
$D_{\phi_{1}}(\widehat{P},P_{\theta})=HD_{\phi_{1}}(\widehat{P},P_{\theta})=\displaystyle
2\sum_{i=1}^{m}\big{(}\hat{p}_{i}^{1/2}-p_{i}^{1/2}(\theta)\big{)}^{2}\quad;\quad\phi_{1}\in\Phi.$
(3.4)
Liese and Vajda (1987), Lindsay (1994) and Morales et al. (1995) introduced
the so-called minimum $\phi$-divergence estimate defined by
$D_{\phi}(\widehat{P},P_{\widehat{\theta}})=\displaystyle\min_{\theta\in\Theta}D_{\phi}(\widehat{P},P_{\theta})\quad;\quad\phi\in\Phi.$
(3.5)
$\hat{\theta}_{\phi}=\displaystyle
arg\min_{\theta\in\Theta}D_{\phi}(\widehat{P},P_{\theta})\quad;\quad\phi\in\Phi.$
(3.6)
###### Remark 3.1
The class of estimates (3.4) contains the maximum likelihood estimator (MLE).
In particular if we replace $\phi=-\log x+x-1$ we get
$\hat{\theta}_{KL_{m}}=\displaystyle
arg\min_{\theta\in\Theta}KL_{m}(P_{\theta},\widehat{P})=\displaystyle
arg\min_{\theta\in\Theta}\sum_{i=1}^{m}-\log p_{i}(\theta)\hat{p}_{i}=MLE$
where $KL_{m}$ is the modified Kullback-Leibler divergence.
Beran (1977) first pointed out that the minimum Hellinger distance estimator
(MHDE) of $\theta$, defined by
$\hat{\theta}_{H}=\displaystyle
arg\min_{\theta\in\Theta}HD_{\phi}(\widehat{P},P_{\theta})$ (3.7)
has robustness proprieties.
Further results were given by Tamura and Boos (1986), Simpson (1987), and
Donoho and Liu (1988), Simpson (1987, 1989) and Basu et al. (1997) for more
details on this method of estimation. Simpson, however, noted that the small
sample performance of the Hellinger deviance test at some discrete models such
as the Poisson is somewhat unsatisfactory, in the sense that the test requires
a very large sample size for the chi-square approximation to be useful
(Simpson (1989), Table 3). In order to avoid this problem, one possibility is
to use the penalized Hellinger distance (see Harris and Basu, (1994); Basu et
al., (1996); Basu and Basu, (1998) ; Basu et al. (2010)). The penalized
Hellinger distance family between the probability vectors $\widehat{P}$ and
$P_{\theta}$ is defined by :
$PHD^{h}(\widehat{P},P_{\theta})=2\left[\displaystyle\sum_{i\in\varpi}^{m}\big{(}\hat{p}_{i}^{1/2}-p_{i}^{1/2}(\theta)\big{)}^{2}+h\displaystyle\sum_{i\not\in\varpi^{c}}^{m}p_{i}(\theta)\right]$
(3.8)
where $h$ is a real positive number with $\varpi=\\{i:\hat{p}_{i}\neq
0\\}\hbox{ and }\varpi^{c}=\\{i:\hat{p}_{i}=0\\}$. Note that when $h=1$, this
generates the ordinary Hellinger distance (Simpson, 1989).
Hence (3.7) can be written as follows
$\hat{\theta}_{PH}=\displaystyle
arg\min_{\theta\in\Theta}PHD^{h}_{\phi}(\widehat{P},P_{\theta})$ (3.9)
One of the suggestions to use the penalized Hellinger is motivated by the fact
that this suitable choice may lead to an estimate more robust than the MLE.
A model selection criterion can be designed to estimate an expected overall
discrepancy, a quantity which reflects the degree of similarity between a
fitted approximating model and the generating or true model. Estimation of
Kullback’s information (see Kullback-Leibler (1951)) is the key to deriving
the Akaike Information criterion AIC (Akaike (1974)).
Motivated by the above developments, we propose by analogy with the approach
introduced by Vuong (1993), a new information criterion relating to the
$\phi$-divergences. In our test, the null hypothesis is that the competing
models are as close to the data generating process (DGP) where closeness of a
model is measured according to the discrepancy implicit in the penalized
Hellinger divergence.
## 4 Asymptotic distribution of the penalized Hellinger distance
Hereafter, we focus on asymptotic results. We assume that the true parameter
$\theta_{0}$ and mapping $P:\Theta\longrightarrow\Omega_{m}$ satisfy
conditions 1-6 of Birch (1964).
We consider the m-vector $P_{\theta}=(p_{1\theta},\ldots,p_{m\theta})^{T}$,
the $m\times k$ Jacobian matrix
$J_{\theta}=\left(J_{jl}(\theta)\right)_{j=1,\ldots,m;\ l=1,\ldots,k}$ with
$J_{jl}(\theta)=\displaystyle\frac{\partial}{\partial\theta_{l}}p_{j\theta},$
the $m\times k$ matrix
$D_{\theta}=diag\left(P_{\theta}^{-1/2}\right)J_{\theta}$ and the $k\times k$
Fisher information matrix
$I_{\theta}=\left(\sum_{j=1}^{m}\frac{1}{p_{j\theta}}\frac{\partial
p_{j\theta}}{\partial\theta_{r}}\frac{\partial
p_{j\theta}}{\partial\theta_{s}}\right)_{r,s=1,\ldots,k}=D_{\theta}(\theta)^{T}D_{\theta}$
where
$diag\left(P_{\theta}^{-1/2}\right)=diag\left(\frac{1}{\sqrt{p_{1}(\theta)}},\ldots,\frac{1}{\sqrt{p_{m}(\theta)}}\right)$
The above defined matrices are considered at the point $\theta\in\Theta$ where
the derivatives exist and all the coordinates $p_{j}(\theta)$ are positive.
The stochastic convergences of random vectors $X_{n}$ to a random vector $X$
are denoted by $X_{n}\stackrel{{\scriptstyle P}}{{\longmapsto}}X$ and
$X_{n}\stackrel{{\scriptstyle\mathcal{L}}}{{\longmapsto}}X$ (convergences in
probability and in law, respectively). Instead
$c_{n}X_{n}\stackrel{{\scriptstyle P}}{{\longmapsto}}0$ for a sequence of
positive numbers $c_{n}$, we can write $\|X\|=o_{p}(c_{n}^{-1})$.
(This relation means
$\lim_{x\rightarrow\infty}\lim\sup_{x\rightarrow\infty}\mathbb{P}(\|c_{n}X_{n}\|>x)=0$)
An estimator $\widehat{P}$ of $P_{\theta_{0}}$ is consistent if for every
$\theta_{0}\in\Theta$ the random vector
$\left(\widehat{p}_{1},\ldots,\widehat{p}_{m}\right)$ tends in probability to
$\left(p_{1\theta_{0}}\ldots,p_{m\theta_{0}}\right)$, i.e. if
$\lim_{n\longrightarrow\infty}\mathbb{P}(\parallel\widehat{P}-P_{\theta_{0}}\parallel>\varepsilon)=0\hbox{
for all }\varepsilon>0.$
We need the following result to prove Theorem (4.3).
###### Proposition 4.1
(Mandal et al. 2008)
Let $\phi\in\Phi$, let $p:\Theta\rightarrow\Omega_{m}$ be twice continuously
differentiable in a neighborhood of $\theta_{0}$ and assume that conditions
1-5 of Section 2 hold. Suppose that $I_{\theta_{0}}$ is the $k\times k$ Fisher
Information matrix and $\widehat{\theta}_{PH}$ satisfying (3.7) then the
limiting distribution of $\sqrt{n}(\widehat{\theta}_{PH}-\theta_{0})$ as
$n\longrightarrow+\infty$ is $N[0,I^{-1}_{\theta_{0}}]$
###### Lemma 4.2
We have
$\sqrt{n}(\widehat{P}-P_{\theta_{0}})\stackrel{{\scriptstyle\mathcal{L}}}{{\longmapsto}}\mathcal{N}\left[0,\Sigma_{P_{\theta_{0}}}\right]$
where
$\widehat{P}(\theta_{0})=(\widehat{p}_{1\theta_{0}},\ldots,\widehat{p}_{m\theta_{0}})$
an estimator of $P_{\theta_{0}}=(p_{1\theta_{0}},\ldots,p_{m\theta_{0}})$
defined in (2.1) with
$\Sigma_{P_{\theta_{0}}}=diag(P_{\theta_{0}})-P_{\theta_{0}}P_{\theta_{0}}^{T}.$
proof. Denote $\displaystyle
V=\left[\frac{N_{1}-np_{1\theta_{0}}}{\sqrt{n}},\ldots,\frac{N_{m}-np_{m\theta_{0}}}{\sqrt{n}}\right]$
and $N_{j}=\displaystyle\sum_{1}^{n}T^{i}_{j}$ where
$T^{i}_{j}(X_{i})=\left\\{\begin{array}[]{lll}1&&\hbox{si }X_{i}=j\\\
0&&\hbox{otherwise}\end{array}\right.$
$\displaystyle V$ $\displaystyle=$
$\displaystyle\left\\{\frac{1}{\sqrt{n}}\left(\sum_{i=1}^{n}T^{i}_{1}-np_{1\theta_{0}}\right);\ldots;\frac{1}{\sqrt{n}}\left(\sum_{i=1}^{n}T^{i}_{m}-np_{m\theta_{0}}\right)\right\\}$
$\displaystyle=$
$\displaystyle\left\\{\sqrt{n}\left(\frac{1}{n}\sum_{i=1}^{n}T^{i}_{1}-p_{1\theta_{0}}\right);\ldots;\frac{1}{\sqrt{n}}\left(\frac{1}{n}\sum_{i=1}^{n}T^{i}_{m}-p_{m\theta_{0}}\right)\right\\}$
and applying the Central Limit Theorem we have
$\left(\frac{N_{1}-np_{1\theta_{0}}}{\sqrt{n}},\ldots,\frac{N_{m}-np_{m\theta_{0}}}{\sqrt{n}}\right)\stackrel{{\scriptstyle\mathcal{L}}}{{\longmapsto}}\mathcal{N}\left[0,\Sigma_{P_{\theta_{0}}}\right]$
where
$\Sigma_{P_{\theta_{0}}}=diag(P_{\theta_{0}})-P_{\theta_{0}}P_{\theta_{0}}^{T}.$
(4.10)
$\square$
For simplicity, we write $D_{H}^{h}(\widehat{P},P_{\widehat{\theta}_{PH}})$
instead of $PHD^{h}(\widehat{P},P_{\widehat{\theta}_{PH}})$.
###### Theorem 4.3
Under the assumptions of Proposition (4.1), we have
$\sqrt{n}(\widehat{P}-P_{\widehat{\theta}_{PH}})\stackrel{{\scriptstyle\mathcal{L}}}{{\longmapsto}}\mathcal{N}\left[0,\Lambda_{\theta_{0}}\right]$
where
$\displaystyle\Lambda_{\theta_{0}}$ $\displaystyle=$
$\displaystyle\Sigma_{\theta_{0}}-\Sigma_{\theta_{0}}M_{\theta_{0}}^{T}-M_{\theta_{0}}\Sigma_{\theta_{0}}+M_{\theta_{0}}\Sigma_{\theta_{0}}M_{\theta_{0}}^{T}$
$\displaystyle M_{\theta_{0}}$ $\displaystyle=$ $\displaystyle
J_{\theta}I^{-1}_{\theta_{0}}(\theta_{0})^{T}diag\big{(}P_{\theta_{0}}^{1/2}\big{)}$
$\displaystyle\Sigma_{\theta_{0}}$ $\displaystyle=$
$\displaystyle\Sigma_{P_{\theta_{0}}}$ (4.11)
proof. A first order Taylor expansion gives
$P_{\widehat{\theta}_{PH}}=P_{\theta_{0}}+J_{\theta_{0}}(\widehat{\theta}_{PH}-\theta_{0})^{T}+o(||\widehat{\theta}_{PH}-\theta_{0}||)$
(4.12)
In the same way as in Morales et al. (1995), it can be established that :
$\widehat{\theta}_{PH}=\theta_{0}+I^{-1}_{\theta_{0}}D_{\theta_{0}}^{T}diag\left[P_{\theta_{0}}^{-1/2}\right]\left(\widehat{P}-P_{\theta_{0}}\right)^{T}+o(||\widehat{P}-P_{\theta_{0}}||)$
(4.13)
From (4.12) and (4.13) we obtain
$P_{\widehat{\theta}_{PH}}=P_{\theta_{0}}+J_{\theta_{0}}I^{-1}(\theta_{0})D_{\theta_{0}}^{T}diag\left[P_{\theta_{0}}^{-1/2}\right]\left(\widehat{P}-P_{\theta_{0}}\right)^{T}+o(||\widehat{P}-P_{\theta_{0}}||)$
therefore the random vectors
$\left[\begin{array}[]{c}\widehat{P}-P_{\theta_{0}}\\\
P_{\widehat{\theta}_{PH}}-P_{\theta_{0}}\end{array}\right]_{2m\times 1}{\hbox{
and }}\left[\begin{array}[]{c}I\\\ M_{\theta_{0}}\end{array}\right]_{2m\times
m}\times(\widehat{P}-P_{\theta_{0}})_{m\times 1}$
Where $I$ is the $m\times m$ unity matrix, have the same asymptotic
distribution.
Furthermore it is clear (applying TCL) that
$\sqrt{n}(\widehat{P}-P_{\theta_{0}})\stackrel{{\scriptstyle\mathcal{L}}}{{\longmapsto}}\mathcal{N}\left[0,\Sigma_{\theta_{0}}\right]$
being $\Sigma_{\theta_{0}}$ the $m\times m$ matrix
$diag\left[P_{\theta_{0}}\right]-P_{\theta_{0}}P_{\theta_{0}}^{T}$ implies
$\sqrt{n}\left[\begin{array}[]{c}\widehat{P}-P_{\theta_{0}}\\\
P_{\widehat{\theta}_{PH}}-P_{\theta_{0}}\end{array}\right]_{2m\times
1}\stackrel{{\scriptstyle\mathcal{L}}}{{\longmapsto}}\mathcal{N}\left[0,\
\left(\begin{array}[]{c}I\\\
M_{\theta_{0}}\end{array}\right)\Sigma_{\theta_{0}}(I,M_{\theta_{0}}^{T})\right]$
therefore, we get
$\sqrt{n}(\widehat{P}-P_{\widehat{\theta}_{PH}})=\sqrt{n}(\widehat{P}-P_{{\theta_{0}}})+\sqrt{n}(P_{\theta_{0}}-P_{\widehat{\theta}_{PH}})\stackrel{{\scriptstyle\mathcal{L}}}{{\longmapsto}}\mathcal{N}\left[0,\Lambda(\theta_{0})\right]$
(4.14)
$\Lambda_{\theta_{0}}=\Sigma_{\theta_{0}}-\Sigma_{\theta_{0}}M_{\theta_{0}}^{T}-M_{\theta_{0}}\Sigma_{\theta_{0}}+M_{\theta_{0}}\Sigma_{\theta_{0}}M_{\theta_{0}}^{T}$
$\square$
The case which is interest to us here is to test the hypothesis
$H_{0}:P\in\mathcal{P}$. Our proposal is based on the following penalized
divergence test statistic $D_{H}^{h}(\widehat{P},P_{\widehat{\theta}_{PH}})$
where $\widehat{P}$ and $\widehat{\theta}_{PH}$ have been introduced in
Theorem (4.3) and (3.7) respectively.
Using arguments similar to those developed by Basu (1996), under the
assumptions of (4.3) and the hypothesis $H_{0}:P=P_{\theta}$, the asymptotic
distribution of $2nD_{H}^{h}(\widehat{P},P_{\widehat{\theta}_{PH}})$ is a chi-
square when $h=1$ with $m-k-1$ degrees of freedom. Since the others members of
penalized Hellinger distance tests differ from the ordinary Hellinger distance
test only at the empty cells, they too have the same asymptotic distribution.
(See Simpson 1989, Basu, Harris and Basu 1996 among others).
Considering now the case when the model is wrong i.e $H_{1}:P\neq P_{\theta}$.
We introduce the following regularity assumptions
$(A_{1})$
There exists $\theta_{1}=\displaystyle arg\ inf_{\theta\in\Theta}PHD^{h}(P,\
P_{\theta})$ such that :
$P_{\widehat{\theta}_{PH}}\stackrel{{\scriptstyle
as}}{{\longmapsto}}P_{\theta_{1}}$
when $n\rightarrow+\infty$
$(A_{2})$
There exists $\theta_{1}\in\Theta$ ;
${\Lambda^{\ast}}=\left(\begin{array}[]{lll}\Lambda_{11}&&\Lambda_{12}\\\
\Lambda_{21}&&\Lambda_{22}\end{array}\right)$, with $\Lambda_{11}=\Sigma_{p}$
in (4.10) and $\Lambda_{12}=\Lambda_{21}$ such that
$\sqrt{n}\left(\begin{array}[]{rll}\widehat{P}&-&P\\\
P_{\widehat{\theta}_{PH}}&-&P_{\theta_{1}}\end{array}\right)\stackrel{{\scriptstyle\mathcal{L}}}{{\longmapsto}}\mathcal{N}\left[0,\Lambda^{\ast}\right]$
###### Theorem 4.4
Under $H_{1}:P\neq P_{\theta}$ and assume that conditions $(A_{1})$ and
$(A_{2})$ hold, we have :
$\sqrt{n}\left(D_{H}^{h}(\widehat{P},P_{\widehat{\theta}_{PH}})-D_{H}^{h}(P,P_{{\theta}_{1}})\right)\stackrel{{\scriptstyle\mathcal{L}}}{{\longmapsto}}\mathcal{N}\left[0,\Omega^{2}_{(\theta,P)}\right]$
where
$\Omega^{2}_{(\theta,P)}=H^{T}\Lambda_{11}H+H^{T}\Lambda_{12}J+J^{T}\Lambda_{21}H+J^{T}\Lambda_{22}J$
(4.15)
$H^{T}=(h_{1},\ldots,h_{m})$ with $h_{i}=\left(\dfrac{\partial}{\partial
p_{i}^{1}}D_{H}^{h}(p^{1},p^{2})\right)_{p^{1}=p,p^{2}=p(\theta_{1})}$ ,
$i=1,\ldots,m$
and
$J^{T}=(j_{1},\ldots,j_{m})$ with $j_{i}=\left(\dfrac{\partial}{\partial
p_{i}^{2}}D_{H}^{h}(p^{1},p^{2})\right)_{p^{1}=p,p^{2}=p(\theta_{1})}$ ,
$i=1,\ldots,m$
proof. A first order Taylor expansion gives
$\displaystyle D_{H}^{h}(\widehat{P},P_{\widehat{\theta}_{PH}})$
$\displaystyle=$ $\displaystyle
D_{H}^{h}(P,P_{{\theta}_{1}})+H^{T}(\widehat{P}-P)+J^{T}(P_{\widehat{\theta}_{PH}}-P_{\theta_{1}})$
(4.16) $\displaystyle+$ $\displaystyle
o(||\widehat{P}-P||+||P_{\widehat{\theta}_{PH}}-P_{\theta_{1}}||)$
From the assumed assumptions $(A_{1})$ and $(A_{2})$, the result follows.
$\square$
## 5 Applications for testing hypothesis
The estimate $D_{H}^{h}(\widehat{P},P_{\widehat{\theta}_{PH}})$ can be used to
perform statistical tests.
### 5.1 Test of goodness-fit
For completeness, we look at
$D_{H}^{h}(\widehat{P},P_{\widehat{\theta}_{PH}})$ in the usual way, i.e., as
a goodness-of-fit statistic. Recall that here $\theta_{PH}$ is the minimum
penalized Hellinger distance estimator of $\theta$. Since
$D_{H}^{h}(\widehat{P},P_{\widehat{\theta}_{PH}})$ is a consistent estimator
of $D_{H}^{h}(P,P_{\theta})$, the null hypothesis when using the statistic
$D_{H}^{h}(\widehat{P},P_{\widehat{\theta}_{PH}})$ is
$H_{0}:\ D_{H}^{h}(P,P_{\theta})=0\quad\hbox{ or equivalently, }\quad H_{0}:\
P=P_{\theta}$
Hence, if $H_{0}$ is rejected so that one can infer that the parametric model
$P_{\theta}$ is misspecified. Since $D_{H}^{h}(P,P_{\theta})$ is non-negative
and takes value zero only when $P=P_{\theta}$, the tests are defined through
the critical region
$C_{\theta_{PH}}=\left\\{2nD_{H}^{h}(\widehat{P},P_{\widehat{\theta}_{PH}})>q_{\alpha,k}\right\\}$
where $q_{\alpha,k}$ is the $(1-\alpha)-$quantile of the
$\chi^{2}-$distribution with $m-k-1$ degrees of freedom.
###### Remark 5.1
Theorem (4.4) can be used to give the following approximation to the power of
test $H_{0}:\ D_{H}^{h}(P,P_{\theta})=0$.
Approximated power function is
$\beta_{(P)}=\mathbb{P}\left[2nD_{H}^{h}(\widehat{P},P_{\widehat{\theta}_{PH}})>q_{\alpha,k}\right]\approx
1-{\cal{F}}_{n}\left(\frac{q_{\alpha,k}-2nD_{H}^{h}(P,P_{\theta})}{2\sqrt{n}\Omega_{(\theta,P)}}\right)$
(5.17)
where $q_{\alpha,k}$ is the $(1-\alpha)$-quantile of the $\chi^{2}$
distribution with $m-k-1$ degrees of freedom and ${\cal{F}}_{n}$ is a sequence
of distribution functions tending uniformly to the standard normal
distribution ${\cal{F}}(x)$. Note that if $H_{0}:\ D_{H}^{h}(P,P_{\theta})\neq
0$, then for any fixed size $\alpha$ the probability of rejection $H_{0}:\
D_{H}^{h}(P,P_{\theta})=0$ with the rejection rule
$2nD_{H}^{h}(\widehat{P},P_{\widehat{\theta}_{PH}})>q_{\alpha,k}$ tends to one
as $n\rightarrow\infty$.
Obtaining the approximate sample $n$, guaranteeing a power $\beta$ for a give
alternative $P$, is an interesting application of formula (5.17). If we wish
the power to be equal to $\beta^{\ast}$, we must solve the equation
$\beta^{\ast}=1-{\cal{F}}\left[\frac{\sqrt{n}}{\Omega_{(\theta,P)}}\left(\frac{1}{2n}q_{\alpha,k}-D_{H}^{h}(P,P_{\theta})\right)\right].$
It is not difficult to check that the sample size $n^{\ast}$, is the solution
of the following equation
$n^{2}D_{H}^{h}(P,P_{\theta})^{2}-nD_{H}^{h}(P,P_{\theta})q_{\alpha,k}+\left(\frac{q_{\alpha,k}}{2}\right)^{2}-n\Omega^{2}_{(\theta,P)}\left[{\cal{F}}^{-1}(1-\beta^{\ast})\right]^{2}.$
The solution is given by
$n^{\ast}=\frac{(a+b)-\sqrt{a(a+2b)}}{2D_{H}^{h}(P,P_{\theta})^{2}}$
with $a=\Omega^{2}_{(\theta,P)}\left[{\cal{F}}^{-1}(1-\beta)\right]^{2}$ and
$b=q_{\alpha,k}D_{H}^{h}(P,P_{\theta})$ and the required size is
$n_{0}=[n^{\ast}]+1$ , where $[\cdot]$ denotes “integer part of”.
### 5.2 Test for model selection
As we mentioned above, when one chooses a particular $\phi-$divergence type
statistic
$D_{H}^{h}(\widehat{P},P_{\widehat{\theta}_{PH}})=PHD_{H}^{h}(\widehat{P},P_{\widehat{\theta}_{PH}})$
with $\widehat{\theta}_{PH}$ the corresponding minimum penalized Hellinger
distance estimator of $\theta$, one actually evaluates the goodness-of-fit of
the parametric model $P_{\theta}$ according to the discrepancy
$D_{H}^{h}(P,P_{\theta})$ between the true distribution $P$ and the specified
model $P_{\theta}$. Thus it is naturel to define the best model among a
collection of competing models to be the model that is closest to the true
distribution according to the discepancy $D_{H}^{h}(P,P_{\theta})$.
In this paper we consider the problem of selecting between two models. Let
$G_{\mu}=\left\\{G(.\mid\mu);\mu\in\Gamma\right\\}$ be another model, where
$\Gamma$ is a $q-$dimensional parametric space in $\mathrm{R^{q}}$. In a
similar way, we can define the minimum penalized Hellinger distance estimator
of $\mu$ and the corresponding discrepancy $D_{H}^{h}(P,G_{\mu})$ for the
model $G_{\mu}$.
Our special interest is the situation in which a researcher has two competing
parametric models $P_{\theta}$ and $G_{\mu}$, and he wishes to select the
better of two models based on their discrimination statistic between the
observations and models $P_{\theta}$ and $G_{\mu}$, defined respectively by
$D_{H}^{h}(\widehat{P},P_{\widehat{\theta}_{PH}})$ and
$D_{H}^{h}(\widehat{P},G_{\widehat{\mu}_{PH}})$.
Let the two competing parametric models $P_{\theta}$ and $G_{\mu}$ with the
given discrepancy $D_{H}^{h}(P,\cdot)$.
###### Definition 5.2
$\displaystyle H_{0}^{eq}:$ $\displaystyle
D_{H}^{h}(P,P_{\theta})=D_{H}^{h}(P,G_{\mu})$ means that the two models are
equivalent, $\displaystyle H_{P_{\theta}}:$ $\displaystyle
D_{H}^{h}(P,P_{\theta})<D_{H}^{h}(P,G_{\mu})$ means that $P_{\theta}$ is
better than $G_{\mu}$, $\displaystyle H_{G_{\mu}}:$ $\displaystyle
D_{H}^{h}(P,P_{\theta})>D_{H}^{h}(P,G_{\mu})$ means that $P_{\theta}$ is worse
than $G_{\mu}$,
###### Remark 5.3
1) It does not require that the same divergence type statistics be used in
forming $D_{H}^{h}(\widehat{P},P_{\widehat{\theta}_{PH}})$ and
$D_{H}^{h}(\widehat{P},G_{\widehat{\mu}_{PH}})$. Choosing, however, different
discrepancy for evaluating competing models is hardly justified.
2) This definition does not require that either of the competing models be
correctly specified. On the other hand, a correctly specified model must be at
least as good as any other model.
The following expression of the indicator
$D_{H}^{h}(P,P_{\theta})-D_{H}^{h}(P,G_{\mu})$ is unknown, but from the
previous section, it can be estimated by the the difference
$\sqrt{n}\left[D_{H}^{h}(\widehat{P},P_{\widehat{\theta}_{PH}})-D_{H}^{h}(\widehat{P},G_{\widehat{\mu}_{PH}})\right]$
This difference converges to zero under the null hypothesis $H_{0}^{eq}$, but
converges to a strictly negative or positive constant when $H_{P_{\theta}}$ or
$H_{G_{\mu}}$ holds.
These properties actually justify the use of
$D_{H}^{h}(\widehat{P},P_{\widehat{\theta}_{PH}})-D_{H}^{h}(\widehat{P},G_{\widehat{\mu}_{PH}})$
as a model selection indicator and common procedure of selecting the model
with highest goodness-of-fit.
As argued in the introduction, however, it is important to take into account
the random nature of the difference
$D_{H}^{h}(\widehat{P},P_{\widehat{\theta}_{PH}})-D_{H}^{h}(\widehat{P},G_{\widehat{\mu}_{PH}})$
so as to assess its significance. To do so we consider the asymptotic
distribution of
$\sqrt{n}\left[D_{H}^{h}(\widehat{P},P_{\widehat{\theta}_{PH}})-D_{H}^{h}(\widehat{P},G_{\widehat{\mu}_{PH}})\right]$
under $H_{0}^{eq}$.
Our major task is to to propose some tests for model selection, i.e., for the
null hypothesis $H_{0}^{eq}$ against the alternative $H_{P_{\theta}}$ or
$H_{G_{\mu}}$. We use the next lemma with $\widehat{\theta}_{PH}$ and
$\widehat{\mu}_{PH}$ as the corresponding minimum penalized Hellinger distance
estimator of $\theta$ and $\mu$.
Using $P$ and $P_{\theta}$ defined earlier, we consider the vector
$K_{\theta}^{T}=(k_{1},\ldots,k_{m})\hbox{ where
}k_{i}=\left(\dfrac{\partial}{\partial
p_{i}^{1}}D_{H}^{h}(P^{1},P^{2})\right)_{P^{1}=P,P^{2}=P_{\theta}}\hbox{ with
}i=1,\dots,m$ $Q_{\theta}^{T}=(q_{1},\ldots,q_{m})\hbox{ where
}q_{i}=\left(\dfrac{\partial}{\partial
p_{i}^{2}}D_{H}^{h}(P^{1},P^{2})\right)_{P^{1}=P,P^{2}=P_{\theta}}\hbox{ with
}i=1,\dots,m$
###### Lemma 5.4
Under the assumptions of the Theorem (4.4), we have
1. (i)
for the model $P_{\theta}$,
$D_{H}^{h}(\widehat{P},P_{\widehat{\theta}_{PH}})=D_{H}^{h}(P,P_{\theta})+K_{\theta}^{T}(\widehat{P}-P)+Q_{\theta}^{T}(P_{\widehat{\theta}_{PH}}-P_{\theta})+o_{P}(1)$
2. (ii)
for model $G_{\mu}$,
$D_{H}^{h}(\widehat{P},G_{\widehat{\mu}_{PH}})=D_{H}^{h}(P,G_{\mu})+K_{\mu}^{T}(\widehat{P}-P)+Q_{\mu}^{T}(G_{\widehat{\mu}_{PH}}-G_{\mu})+o_{P}(1)$
proof.
The results follows from a first order Taylor expansion. $\square$
We define
$\Gamma^{2}=(K_{\theta}-K_{\mu};Q_{\theta}-Q_{\mu})^{T}\Lambda^{\ast}(K_{\theta}-K_{\mu};Q_{\theta}-Q_{\mu})$
which is the variance of
$(K_{\theta}-K_{\mu};Q_{\theta}-Q_{\mu})^{T}\left(\begin{array}[]{rll}\widehat{P}&-&P\\\
P_{\widehat{\theta}_{PH}}&-&P_{\theta_{1}}\end{array}\right)$. Since
$K_{\theta}$, $K_{\mu}$, $Q_{\theta}$, $Q_{\mu}$ and $\Lambda^{\ast}$ are
consistently estimated by their sample analogues $K_{\widehat{\theta}}$,
$K_{\widehat{\mu}}$, $Q_{\widehat{\theta}}$, $Q_{\widehat{\mu}}$ and
${\widehat{\Lambda}}^{\ast}$, hence $\Gamma^{2}$ is consistently estimated by
$\widehat{\Gamma}^{2}=(K_{\widehat{\theta}}-K_{\widehat{\mu}};Q_{\widehat{\theta}}-Q_{\widehat{\mu}})^{T}\widehat{\Lambda}^{\ast}(K_{\widehat{\theta}}-K_{\widehat{\mu}};Q_{\widehat{\theta}}-Q_{\widehat{\mu}})$
Next we define the model selection statistic and its asymptotic distribution
under the null and alternatives hypothesis.
Let
$\mathcal{HI}^{h}=\frac{\sqrt{n}}{\widehat{\Gamma}}\left\\{D_{H}^{h}(\widehat{P},P_{\widehat{\theta}_{PH}})-D_{H}^{h}(\widehat{P},G_{\widehat{\mu}_{PH}})\right\\}\\\
$
where $\mathcal{HI}^{h}$ stands for the penalized Hellinger Indicator.
The following theorem provides the limit distribution of $\mathcal{HI}^{h}$
under the null and alternatives hypothesis.
###### Theorem 5.5
Under the assumptions of theorem (4.4), suppose that
$\Gamma\neq 0$, then:
1. (i)
Under the null hypothesis $H_{0}^{eq}$,
$\mathcal{HI}^{h}\stackrel{{\scriptstyle\mathcal{L}}}{{\longmapsto}}\mathcal{N}(0,1)$
2. (ii)
Under the null hypothesis $H_{P_{\theta}}$,
$\mathcal{HI}^{h}\longrightarrow-\infty$ in probability
3. (iii)
Under the null hypothesis $H_{G_{\mu}}$,
$\mathcal{HI}^{h}\longrightarrow+\infty$ in probability
proof.
From the lemma (5.4), it follows that
$\displaystyle
D_{H}^{h}(\widehat{P},P_{\widehat{\theta}_{PH}})-D_{H}^{h}(\widehat{P},G_{\widehat{\mu}_{PH}})$
$\displaystyle=$ $\displaystyle
D_{H}^{h}(P,P_{\theta})-D_{H}^{h}(P,G_{\mu})+K_{\theta}^{T}(\widehat{P}-P)-K_{\mu}^{T}(\widehat{P}-P)$
$\displaystyle+$ $\displaystyle
Q_{\theta}^{T}(P_{\widehat{\theta}_{PH}}-P_{\theta})-Q_{\mu}^{T}(G_{\widehat{\mu}_{PH}}-G_{\mu})+o_{P}(1)$
Under $H_{0}^{eq}$ : $P_{\theta}=G_{\mu}$ and
$P_{\widehat{\theta}_{PH}}=G_{\widehat{\mu}_{PH}}$ we get :
$\displaystyle
D_{H}^{h}(\widehat{P},P_{\widehat{\theta}_{PH}})-D_{H}^{h}(\widehat{P},G_{\widehat{\mu}_{PH}})$
$\displaystyle=$ $\displaystyle
K_{\theta}^{T}(\widehat{P}-P)-K_{\mu}^{T}(\widehat{P}-P)$ $\displaystyle+$
$\displaystyle
Q_{\theta}^{T}(P_{\widehat{\theta}_{PH}}-P_{\theta})-Q_{\mu}^{T}(P_{\widehat{\theta}_{PH}}-P_{\theta})+o_{P}(1)$
$\displaystyle=$
$\displaystyle\left(K_{\theta}-K_{\mu},Q_{\theta}-Q_{\mu}\right)^{T}\left(\begin{array}[]{c}\widehat{P}-P\\\
P_{\widehat{\theta}_{PH}}-P_{\theta}\\\ \end{array}\right)+o_{P}(1)$
Finally, applying the Central Limit Theorem and assumptions (A1)-(A2), we can
now immediately obtain
$\mathcal{HI}^{h}\stackrel{{\scriptstyle\mathcal{L}}}{{\longmapsto}}\mathcal{N}(0,1).$
$\square$
## 6 Computational results
### 6.1 Example
To illustrate the model procedure discussed in the preceding section,we
consider an example. we need to define the competing models, the estimation
method used for each competing model and the Hellinger penalized type
statistic to measure the departure of each proposed parametric model from the
true data generating process.
For our competing models, we consider the problem of choosing between the
family of poisson distribution and the family of geometric distribution. The
poisson distribution $P(\lambda)$ is parameterized by $\lambda$ and has
density
$f(x,\lambda)=\frac{\exp({-\lambda})\times{\lambda}^{x}}{x!}\quad\hbox{for
$x\,\in\mathbf{N}$ and zero otherwise}.$
The geometric distribution $G(p)$ is parameterized by $p$ and has density
$g(x,p)=(1-p)^{x-1}\times p\quad\hbox{for $x\,\in\mathbf{N^{*}}$ and zero
otherwise}.$
We use the minimum penalized Hellinger distance statistic to evaluate the
discrepancy of the proposed model from the true data generating process. We
partition the real line into $m$ intervals
$\\{[C_{i-1},C_{i}[,\,i=1,\cdots,m\\}$ where $C_{0}=0$ and $C_{m}=+\infty$.
The choice of the cells is discussed below.
The corresponding minimum penalized Hellinger distance estimator of $\lambda$
et $p$ are :
$\displaystyle\hat{\lambda}_{PH}=\displaystyle
arg\min_{\lambda\in\Theta}D_{H}^{h}(\widehat{P},P_{\lambda})$ $\displaystyle=$
$\displaystyle
arg\min_{\lambda\in\Theta}\left[\sum_{i\in\varpi}^{m}({f}_{i}^{1/2}-p^{1/2}_{i\lambda})^{2}+h\sum_{i\in\varpi^{c}}^{m}p_{i\lambda}\right]$
$\displaystyle\hat{p}_{PH}=\displaystyle
arg\min_{p\in\Theta}D_{H}^{h}(\widehat{P},P_{p})$ $\displaystyle=$
$\displaystyle
arg\min_{p\in\Theta}\left[\sum_{i\in\varpi}^{m}({f}_{i}^{1/2}-p^{1/2}_{ip})^{2}+h\sum_{i\in\varpi^{c}}^{m}p_{ip}\right]$
$p_{i\lambda}$ and $p_{ip}$ and are probabilities of the cells
$[C_{i-1},C_{i}[$ under the poisson and geometric true distribution
respectively.
We consider various sets of experiments in which data are generated from the
mixture of a poisson and geometric distribution. These two distributions are
calibrated so that their two means are close (4 and 5 respectively). Hence the
DGP (Data Generating Process) is generated from $M(\pi)$ with the density
$m(\pi)=\pi\ Pois(4)+(1-\pi)\ Geom(0.2)$
where $\pi(\pi\in[0,1])$ is specific value to each set of experiments.
Figure 1 : Histogram of DGP=Pois(4) with n=50 Figure 2 : Comparative barplot
of $HI_{n}$ depending $n$
In each set of experiment several random sample are drawn from this mixture of
distributions. The sample size varies from $20$ to $300$, and for each sample
size the number of replication is $1000$. In each set of experiment, we choose
two values of the parameter $h=1$ and $h=1/2$, where $h=1$ corresponds to the
classic Hellinger distance. The aim is to compare the accuracy of the
selection model depending on the parameter setting chosen.
n | 20 | 30 | 40 | 50 | 300
---|---|---|---|---|---
$\widehat{p}$ | 0.210(0.03) | 0.195(0.03) | 0\. 197(0.02) | 0.205(0.02) | 0.201(0.01)
$\widehat{\lambda}$ | 3.950(0.46) | 4.090(0.4) | 4.015(0.31) | 4.015(0.28) | 4.011(0.13)
DHP(Pois) | h=1 | 0.133(0.07) | 0.081(0.05) | 0.059(0.03) | 0.042(0.03) | 0.037(0.01)
| h=1/2 | 0.096(0.04) | 0.064(0.03) | 0.048(0.02) | 0.034(0.02) | 0.03(0.01)
DHP(Geom) | h=1 | 0.391(0.28) | 0.348(0.12) | 0.298(0.09) | 0.282(0.10) | 0.271(0.05)
| h=1/2 | 0.278(0.07) | 0.262(0.08) | 0.242(0.06) | 0.236(0.06) | 0.231(0.03)
$\mathcal{HI}^{h}$ | $h=1/2$ | -3.67(2.14) | -4.32(2.69) | -4.34(2.38) | -4.83(2.52) | -4.97(2.18)
| Correct | 77% | 87% | 92% | 96% | 100%
| Indecisive | 23% | 13% | 08% | 04% | 00%
| Incorrect | 00% | 00% | 00% | 00% | 00%
$\mathcal{HI}^{h}$ | $h=1$ | -3.61(3.03) | -3.98(2.48) | -3.73(2.29) | -4.16(2.35) | -4.25(1.87)
| Correct | 70% | 79% | 83% | 86% | 93%
| Indecisive | 30% | 21% | 17% | 14% | 07%
| Incorrect | 00% | 00% | 00% | 00% | 00%
Table 1 : DGP=Pois(4)
In order a perfect fit by the proposed method, for the chosen parameters of
these two distributions, we note that most of the mass is concentrated between
0 and 10. Therefore, the chosen partition has eight cells defined by
$\\{[C_{i-1},C_{i}[=[i-1,i[,\,i=1,\cdots,7\\}$ and $[C_{7},C_{8}[=[7,+\infty[$
represents the last cell. We choose different values of $\pi$ which are
$0.00,\,0.25,0.535,\,0.75,\,1.00$. Although our proposed model selection
procedure does not require that the data generating process belong to either
of the competing models, we consider the two limiting cases $\pi=1.00$ and
$\pi=0.00$ for they correspond to the correctly specified cases. To
investigate the case where both competing models are misspecified but not at
equal distance from the DGP, we consider the case $\pi=0.25$, $\pi=0.75$ and
$\pi=0.535$. The former case correspond to a DGP which is poisson but slightly
contaminated by a geometric distribution. The second case is interpreted
similarly as a geometric slightly contaminated by a poisson distribution. In
the last case, $\pi=0.535$ is the value for which the poisson
$D_{H}^{h}(\widehat{P},P_{\widehat{\lambda}_{PH}})$ and the geometric
$D_{H}^{h}(\widehat{P},G_{\widehat{p}_{PH}})$ families are approximatively at
equal distance to the mixture $m(\pi)$ according to the penalized Hellinger
distance with the above cells. Thus this set of experiments corresponds
approximatively to the null hypothesis of our proposed model selection test
$\mathcal{HI}^{h}$.
Figure 3 : Histogram of DGP=Geom(0.2) with n=50 Figure 4 : Comparative barplot
of $HI_{n}$ depending,$n$
The results of our different sets of experiments are presented in table 1-5.
The first half of each table gives the average values of the the minimum
penalized Hellinger distance estimator $\widehat{\lambda}_{PH}$ and
$\widehat{p}_{PH}$, the penalized Hellinger goodness-of-fit statistics
$D_{H}^{h}(\widehat{P},P_{\widehat{\lambda}_{PH}})$ and
$D_{H}^{h}(\widehat{P},G_{\widehat{p}_{PH}})$, and the Hellinger indicator
statistic $\mathcal{HI}^{h}$. The values in parentheses are standard errors.
The second half of each table gives in percentage the number of times our
proposed model selection procedure based on $\mathcal{HI}^{h}$ favors the
poisson model, the geometric model, and indecisive. The tests are conducted at
$5\%$ nominal significance level.
n | 20 | 30 | 40 | 50 | 300
---|---|---|---|---|---
$\widehat{p}$ | 0.196(0.04) | 0.213(0.03) | 0.203(0.02) | 0.203(0.02) | 0.201(0,01)
$\widehat{\lambda}$ | 3.920(1.0) | 4.206(0.89) | 4.021(0.67) | 4.109(0.58) | 4.03(0.34)
DHP(Pois) | h=1.0 | 0.356(0.14) | 0.309(0.10) | 0.271(0.09) | 0.253(0.08) | 0.244(0.07)
| h=0.5 | 0.281(0.1) | 0.273(0.07) | 0.254(0.07) | 0.246(0.07) | 0.237(0.02)
DHP(Geom) | h=1 | 0.150(0.06) | 0.089(0.05) | 0.053(0.03) | 0.039(0.02) | 0.033(0.01)
| h=1/2 | 0.103(0.04) | 0.067(0.03) | 0.044(0.02) | 0.035(0.02) | 0.027(0.98)
$\mathcal{HI}^{h}$ | $h=1/2$ | 1.880(1.43) | 2.560(1.37) | 3.020(1.25) | 3.340(1.14) | 3.40(1.03)
| Correct | 42% | 72% | 81% | 90% | 97%
| Indecisive | 58% | 28% | 19% | 10% | 03%
| Incorrect | 00% | 00% | 00% | 00% | 00%
$\mathcal{HI}^{h}$ | $h=1$ | 1.710(1.07) | 2.260(1.05) | 2.760(0.96) | 3.01(0.65) | 4.19(0.32)
| Correct | 36% | 62% | 77% | 84% | 92%
| Indecisive | 64% | 38% | 23% | 16% | 08%
| Incorrect | 00% | 00% | 00% | 00% | 00%
Table 2 : DGP=Geom(0.2) |
In the first two sets of experiments ($\pi=0.00\hbox{ and }\pi=1.00$) where
one model is correctly specified, we use the labels ‘correct’, ‘incorrect’ and
‘indecisive’ when a choice is made.
Figure 5 : Histogram of
DGP=0.75$\times$Geom+0.25$\times$Pois with n=50 Figure 6 : Comparative barplot
of $HI_{n}$ depending $n$
The first halves of tables 1-5 confirm our asymptotic results. They all show
that the minimum penalized Hellinger estimators $\widehat{\lambda}_{PH}$ and
$\widehat{p}_{{}_{PH}}$ converge to their pseudo-true values in the
misspecified cases and to their true values in the correctly specified cases
as the sample size increases . With respect to our $\mathcal{HI}^{h}$, it
diverges to $-\infty$ or $+\infty$ at the approximate rate of $\sqrt{n}$
except in the table 5. In the latter case the $\mathcal{HI}^{h}$ statistic
converges, as expected, to zero which is the mean of the asymptotic
$\mathcal{N}(0,1)$ distribution under our null hypothesis of equivalence.
n | 20 | 30 | 40 | 50 | 300
---|---|---|---|---|---
$\widehat{p}$ | 0.213(0.13) | 0.197(0.12) | 0.208(0.08) | 0.202(0.05) | 0.202(0.01)
$\widehat{\lambda}$ | 4.160(0.72) | 3.910(0.55) | 4.180(0.55) | 3.970(0.43) | 4.022(0.21)
DHP(Pois) | h=1 | 0.546(0.13) | 0.472(0.1) | 0.412(0.09) | 0.402(0.08) | 0.367(0.06)
| h=1/2 | 0.344(0.07) | 0.340(0.05) | 0.320(0.05) | 0.311(0.05) | 0.304(0.03)
DHP(Geom) | h=1 | 0.150(0.06) | 0.089(0.05) | 0.053(0.03) | 0.039(0.02) | 0.021(0.01)
| h=1/2 | -3.67(2.62) | -4.32(2.53) | -4.34(2.47) | -4.83(2.27) | -5.37(2.01)
$\mathcal{HI}^{h}$ | $h=1/2$ | 1.220(1.02) | 1.820(0.89) | 2.080(1.12) | 2.370(0.99) | 3.102(0.84)
| Geom | 23% | 40% | 50% | 64% | 81%
| Indecisive | 77% | 60% | 50% | 36% | 19%
| Pois | 00% | 00% | 00% | 00% | 00%
$\mathcal{HI}^{h}$ | $h=1$ | 0.840(1.29) | 0.831(1.27) | 0.845(1.16) | 0.967(1.05) | 1.131(0.78)
| Geom | 17% | 15% | 19% | 22% | 33%
| Indecisive | 80% | 83% | 89% | 77% | 66%
| Pois | 03% | 02% | 02% | 01% | 01%
Table 3 : DGP=0.75$\times$Geom(0.2)+0.25$\times$Pois(4) |
With the exception of table 1 and 2, we observed a large percentage of
incorrect decisions. This is because both models are now incorrectly
specified. In contrast, turning to the second halves of the tables 1-2, we
first note that the percentage of correct choices using $\mathcal{HI}^{h}$
statistic steadily increases and ultimately converges to $100\%$.
Figure 7 : Histogram of
DGP=0.25$\times$Geom+0.75$\times$Pois with n=50 Figure 8 : Comparative barplot
of $HI_{n}$ depending $n$
The preceding comments for the second halves of tables 1 and 2 also apply to
the second halves of tables 3 and 4. In all tables (1,2,3 and 4), the results
confirm, in small samples, the relative domination of the model selection
procedure based on the penalized Hellinger statistic test ($h=1/2$) than the
other corresponding to the choice of classical Hellinger statistic test
($h=1$), in percentages of correct decisions. Table 5 also confirms our
asymptotics results : as sample size incerases, the percentage of rejection of
both models converges, as it should, to 100%.
n | 20 | 30 | 40 | 50 | 300
---|---|---|---|---|---
$\widehat{p}$ | 0.213(0.03) | 0.212(0.03) | 0.210(0.02) | 0.206(0.02) | 0.203(0.01)
$\widehat{\lambda}$ | 4.110(0.43) | 4.090(0.31) | 3.970(0.28) | 4.020(0.26) | 4.019(0.17)
DHP(Pois) | h=1 | 1.779(0.45) | 1.634(0.30) | 1.650(0.28) | 1.570(0.24) | 1.520(0.21)
| h=1/2 | 1.443(0.24) | 1.473(0.21) | 1.520(0.20) | 1.500(0.18) | 1.483(0.14)
DHP(Geom) | h=1 | 2.055(0.35) | 1.870(0.25) | 1.860(0.21) | 1.790(0.19) | 1.704(0.11)
| h=1/2 | 1.640(0.15) | 1.660(0.15) | 1.700(0.14) | 1.690(0.13) | 1.632(0.10)
$\mathcal{HI}^{h}$ | $h=1/2$ | -2.40(1.27) | -2.44(1.1) | -2.49(1.08) | -2.77(1.01) | -2.89(0.92)
| Geom | 00% | 00% | 00% | 00% | 00%
| Indecisive | 38% | 37% | 32% | 27% | 21%
| Pois | 62% | 63% | 68% | 83% | 79%
$\mathcal{HI}^{h}$ | $h=1$ | -2.18(1.37) | -2.37(1.33) | -2.31(1.36) | -2.66(1.18) | -2.83(1.06)
| Geom | 00% | 00% | 00% | 00% | 00%
| Indecisive | 48% | 45% | 46% | 30% | 24%
| Pois | 52% | 55% | 54% | 70% | 76%
Table 4 : DGP=0.75$\times$Pois(4)+0.25$\times$Geom(0.2) |
In figures 1, 3, 5, 7 and 9 we plot the histogramm of datasets and overlay the
curves for Geometric and poisson distribution. When the DGP is correctly
specified figure 1, the poisson distribution has a reasonable chance of being
distinguished from geometric distribution.
Figure 9 : Histogram of
DGP=0.465$\times$Geom+0.535$\times$Pois with n=50 Figure 10 : Comparative
barplot of $HI_{n}$ depending $n$
Similarly, in figure 3, as can be seen, the geometric distribution closely
approximates the data sets. In figures 5 and 7 the two distributions are close
but the geometric (figure 5) and the poisson distributions (figure 7) does
appear to be much closer to the data sets. When $\pi=0.535$, the distributions
for both (figure 9) poisson distribution and geometric distribution are
similar, while being slightly symmetrical about the axis that passes through
the mode of data distribution. This follows from the fact that these two
distributions are equidistant from the DGP.
and would be difficult to distinguish from data in practice.
n | 20 | 30 | 40 | 50 | 300
---|---|---|---|---|---
$\widehat{p}$ | 0.196(0.06) | 0.204(0.05) | 0.211(0.03) | 0.213(0.207) | 0.204(0.01)
$\widehat{\lambda}$ | 3.968(0.61) | 3.962(0.46) | 3.981(0.374) | 4.023(0.309) | 4.011(0.11)
DHP(Pois) | h=1 | 2.869(0.63) | 2.600(0.46) | 2.582(0.36) | 2.525(0.38) | 2.311(0.25)
| h=1/2 | 2.633(0.30) | 2.492(0.28) | 2.369(0.27) | 2.302(0.26) | 2.142(0.17)
DHP(Geom) | h=1 | 2.867(0.52) | 2.682(0.37) | 2.553(0.30) | 2.495(0.20) | 2.237(0.12)
| h=1/2 | 2.157(0.21) | 2.200(0.20) | 2.263(0.20) | 2.287(0.19) | 2.291(0.15)
$\mathcal{HI}^{h}$ | $h=1/2$ | -0.079(1.04) | 0.038(1.05) | 0.182(0.99) | 0.334(1.10) | 0.442(0.67)
| Geom | 03% | 04% | 05% | 10% | 13%
| Indecisive | 92% | 92% | 93% | 88% | 88%
| Pois | 05% | 04% | 02% | 02% | 01%
$\mathcal{HI}^{h}$ | $h=1$ | 0.186(1.14) | 0.248(1.64) | 0.378(0.90) | 0.452(0.86) | 0.617(0.73)
| Geom | 05% | 06% | 04% | 09% | 11%
| Indecisive | 92% | 90% | 95% | 90% | 88%
| Pois | 03% | 04% | 01% | 01% | 01%
Table 5 : DGP=0.535$\times$Pois(4)+0.465$\times$Geom(0.2) |
The preceding results in tables and the theorem (5.5) confirm, in figures 2,
4, 6 and 8, that the Hellinger indicator for the model selection procedure
based on penalized hellinger divergence statistic with $h=0.5$ (light bars)
dominates the procedure obtained with $h=1$ (dark bars) corresponding to the
ordinary Hellinger distance. As expected, our statistic divergence
$\mathcal{HI}^{h}$ diverges to $-\infty$ (figure 2, 8) and to $+\infty$
(figure 4, figure 8) more rapidly when we use the penalized Hellinger distance
test than the classical Hellinger distance test.
Hence, Figure 10 allows a comparison with the asymptotic $\mathcal{N}(0,1)$
approximation under our null hypothesis of of equivalence. Hence the indicator
$\mathcal{HI}^{1/2}$, based on the penaliezd Hellinger distance is closer to
the mean of $\mathcal{N}(0,1)$ than is the indicator $\mathcal{HI}^{1}$.
## 7 Conclusion
In this paper we investigated the problems of model selection using divergence
type statistics. Specifically, we proposed some asymptotically standard normal
and chi-square tests for model selection based on divergence type statistics
that use the corresponding minimum penalized Hellinger estimator. Our tests
are based on testing whether the competing models are equally close to the
true distribution against the alternative hypotheses that one model is closer
than the other where closeness of a model is measured according to the
discrepancy implicit in the divergence type statistics used. The penalized
Hellinger divergence criterion outperforms classical criteria for model
selection based on the ordinary Hellinger distance, especially in small
sample, the difference is expected to be minimal for large sample size. Our
work can be extended in several directions. One extension is to use random
instead of fixed cells. Random cells arise when the boundaries of each cell
$c_{i}$ depend on some unknown parameter vector $\gamma$, which are estimated.
For various examples, see e.g., Andrews (1988b). For instance, with
appropriate random cells, the asymptotic distribution of a Pearson type
statistic may become independent of the true parameter $\theta_{0}$ under
correct specification. In view of this latter result, it is expected that our
model selection test based on penalized Hellinger divergence measures will
remain asymptotically normally or chi-square distributed.
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|
arxiv-papers
| 2011-10-14T08:56:59 |
2024-09-04T02:49:23.119695
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Papa Ngom and Bertrand Ntep",
"submitter": "Ngom Papa",
"url": "https://arxiv.org/abs/1110.3151"
}
|
1110.3153
|
# Approximated $l$-states of the Manning-Rosen potential by Nikiforov-Uvarov
method
Sameer M. Ikhdair sikhdair@neu.edu.tr Department of Physics, Near East
University, Nicosia, North Cyprus, Turkey
###### Abstract
The approximately analytical bound state solutions of the $l$-wave Schrödinger
equation for the Manning-Rosen (MR) potential are carried out by a proper
approximation to the centrifugal term. The energy spectrum formula and
normalized wave functions expressed in terms of the Jacobi polynomials are
both obtained for the application of the Nikiforov-Uvarov (NU) method to the
Manning-Rosen potential. To show the accuracy of our results, we calculate the
eigenvalues numerically for arbitrary quantum numbers $n$ and $l$ with two
different values of the potential parameter $\alpha.$ It is found that our
results are in good agreement with the those obtained by other methods for
short potential range, small $l$ and $\alpha.$ Two special cases are
investigated like the $s$-wave case and Hulthén potential case.
Keywords: Bound states; Manning-Rosen potential; NU method.
###### pacs:
03.65.-w; 02.30.Gp; 03.65.Ge; 34.20.Cf
## I Introduction
One of the important tasks of quantum mechanics is to find exact solutions of
the wave equations (nonrelativistic and relativistic) for certain type of
potentials of physical interest since they contain all the necessary
information regarding the quantum system under consideration. For example, the
exact solutions of these wave equations are only possible in a few simple
cases such as the Coulomb, the harmonic oscillator, pseudoharmonic and Mie-
type potentials [1-8]. For an arbitrary $l$-state, most quantum systems could
be only treated by approximation methods. For the rotating Morse potential
some semiclassical and/or numerical solutions have been obtained by using
Pekeris approximation [9-13]. In recent years, many authors have studied the
nonrelativistic and relativistic wave equations with certain potentials for
the $s$\- and $l$-waves. The exact and approximate solutions of these models
have been obtained analytically [10-14].
Many exponential-type potentials have been solved like the Morse potential
[12,13,15], the Hulthén potential [16-19], the Pöschl-Teller [20], the Woods-
Saxon potential [21-23], the Kratzer-type potentials [12,14,24-27], the Rosen-
Morse-type potentials [28,29], the Manning-Rosen potential [30-33],
generalized Morse potential [34] and other multiparameter exponential-type
potentials [35]. Various methods are used to obtain the exact solutions of the
wave equations for this type of exponential potentials. These methods include
the supersymmetric (SUSY) and shape invariant method [19,36], the variational
[37], the path integral approach [31], the standard methods [32,33], the
asymptotic iteration method (AIM) [38], the exact quantization rule (EQR)
[13,39,40], the hypervirial perturbation [41], the shifted $1/N$ expansion
(SE) [42] and the modified shifted $1/N$ expansion (MSE) [43], series method
[44], smooth transformation [45], the algebraic approach [46], the
perturbative treatment [47,48] and the Nikiforov-Uvarov (NU) method
[16,17,20–26,49-51] and others. The NU method [51] is based on solving the
second-order linear differential equation by reducing to a generalized
equation of hypergeometric type. It has been used to solve the Schrödinger
[14,16,20,22,48,49], Dirac [17,28,34,50], Klein-Gordon [21,24,25,50] wave
equations for such kinds of exponential potentials.
The NU method has shown its power in calculating the exact energy levels of
all bound states for some solvable quantum systems. Motivated by the
considerable interest in exponential-type potentials [12-35], we attempt to
study the quantum properties of another exponential-type potential proposed by
Manning and Rosen (MR) [29-33]
$V(r)=\frac{\hbar^{2}}{2\mu
b^{2}}\left(\frac{\alpha(\alpha-1)e^{-2r/b}}{\left(1-e^{-r/b}\right)^{2}}-\frac{Ae^{-r/b}}{1-e^{-r/b}}\right),$
(1)
where $A$ and $\alpha$ are two-dimensionless parameters but the screening
parameter $b$ has dimension of length and corresponds to the potential range
[33]. This potential is used as a methematical model in the description of
diatomic molecular vibrations [52,53] and it constitutes a convenient model
for other physical situations. Figure 1 plots the Manning-Rosen potential (1)
versus $r$ for various screening distances $b=0.025,$ $0.050,$ and $0.100$
considering the cases (a) $\alpha=0.75$ and (b) $\alpha=1.50.$ It is known
that for this potential the Schrödinger equation can be solved exactly for
$s$-wave (i.e., $l=0$) [32]. Unfortunately, for an arbitrary $l$-states
($l\neq 0),$ in which the Schrödinger equation does not admit an exact
analytic solution. In such a case, the Schrödinger equation is solved
numerically [54] or approximately using approximation schemes
[18,50,55,56,57]. Some authors used the approximation scheme proposed by
Greene and Aldrich [18] to study analytically the $l\neq 0$ bound states or
scattering states of the Schrödinger or even relativistic wave equations for
MR potential [13,21]. We calculate and find its $l\neq 0$ bound state energy
spectrum and normalized wave functions [29-33]. The potential (1) may be
further put in the following simple form
$V(r)=-\frac{Ce^{-r/b}+De^{-2r/b}}{\left(1-e^{-r/b}\right)^{2}},\text{
}C=A,\text{ }D=-A-\alpha\text{(}\alpha-1)\text{,}$ (2)
It is also used in several branches of physics for their bound states and
scattering properties. Its spectra have already been calculated via
Schrödinger formulation [30]. In our analysis, we find that the potential (1)
remains invariant by mapping $\alpha\rightarrow 1-\alpha.$ Further, it has a
relative minimum value $V(r_{0})=-\frac{A^{2}}{4\kappa b^{2}\alpha(\alpha-1)}$
at $r_{0}=b\ln\left[1+\frac{2\alpha(\alpha-1)}{A}\right]$ for
$A/2+\alpha(\alpha-1)>0$ which provides $2\alpha>1+\sqrt{1-2A}$ as a result of
the first derivative $\left.\frac{dV}{dr}\right|_{r=r_{0}}=0.$ For the case
$\alpha=0.75,$ we have the criteria imposed on the value of $A$ is
$A>\alpha/2=3/8.$ For example, in $\hbar=\mu=1,$ the minimum of the potential
is $V(r_{0})=-\alpha/16b^{2}(\alpha-1).$ The second derivative which
determines the force constants at $r=r_{0}$ is given by
$\left.\frac{d^{2}V}{dr^{2}}\right|_{r=r_{0}}=\frac{A^{2}\left[A+2\alpha(\alpha-1)\right]^{2}}{8b^{4}\alpha^{3}(\alpha-1)^{3}}.$
(3)
The purpose of this paper is to investigate the $l$-state solution of the
Schrödinger-MR problem within the Nikiforov-Uvarov method to generate accurate
energy spectrum. The solution is mainly depends on replacing the orbital
centrifugal term of singularity $\sim 1/r^{2}$ [17] with Greene-Aldrich
approximation scheme. consisting of the exponential form [16]. Figure 2 shows
the behaviour of the singular term $r^{-2}$ and various approximation schemes
recently used in Refs. [18,34,55,56].
The paper is organized as follows: In Section II we present the shortcuts of
the NU method. In Section III, we derive $l\neq 0$ bound state eigensolutions
(energy spectrum and wave functions) of the MR potential by means of the NU
method. In Section IV, we give numerical calculations for various diatomic
molecules. Section V, is devoted to for two special cases, namely, $l=0$ and
the Hulthén potential. The concluding remarks are given in Section VI.
## II Method
The Nikiforov-Uvarov (NU) method is based on solving the hypergeometric type
second order differential equation [51]. Employing an appropriate coordinate
transformation $z=z(r),$ we may rewrite the Schrödinger equation in the
following form:
$\psi_{n}^{\prime\prime}(z)+\frac{\widetilde{\tau}(z)}{\sigma(z)}\psi_{n}^{\prime}(z)+\frac{\widetilde{\sigma}(z)}{\sigma^{2}(z)}\psi_{n}(z)=0,$
(4)
where $\sigma(z)$ and $\widetilde{\sigma}(z)$ are the polynomials with at most
of second-degree, and $\widetilde{\tau}(s)$ is a first-degree polynomial.
Further, using $\psi_{n}(z)=\phi_{n}(z)y_{n}(z),$ Eq. (4) reduces into an
equation of the following hypergeometric type:
$\sigma(z)y_{n}^{\prime\prime}(z)+\tau(z)y_{n}^{\prime}(z)+\lambda
y_{n}(z)=0,$ (5)
where $\tau(z)=\widetilde{\tau}(z)+2\pi(z)$ (its derivative must be negative)
and $\lambda$ is a constant given in the form
$\lambda=\lambda_{n}=-n\tau^{\prime}(z)-\frac{n\left(n-1\right)}{2}\sigma^{\prime\prime}(z),\text{\
\ \ }n=0,1,2,...$ (6)
It is worthwhile to note that $\lambda$ or $\lambda_{n}$ are obtained from a
particular solution of the form $y(z)=y_{n}(z)$ which is a polynomial of
degree $n.$ Further, $\ y_{n}(z)$ is the hypergeometric-type function whose
polynomial solutions are given by Rodrigues relation
$y_{n}(z)=\frac{B_{n}}{\rho(z)}\frac{d^{n}}{dz^{n}}\left[\sigma^{n}(z)\rho(z)\right],$
(7)
where $B_{n}$ is the normalization constant and the weight function $\rho(z)$
must satisfy the condition [51]
$w^{\prime}(z)-\left(\frac{\tau(z)}{\sigma(z)}\right)w(z)=0,\text{
}w(z)=\sigma(z)\rho(z).$ (8)
In order to determine the weight function given in Eq. (8), we must obtain the
following polynomial:
$\pi(z)=\frac{\sigma^{\prime}(z)-\widetilde{\tau}(z)}{2}\pm\sqrt{\left(\frac{\sigma^{\prime}(z)-\widetilde{\tau}(z)}{2}\right)^{2}-\widetilde{\sigma}(z)+k\sigma(z)}.$
(9)
In principle, the expression under the square root sign in Eq. (9) can be
arranged as the square of a polynomial. This is possible only if its
discriminant is zero. In this case, an equation for $k$ is obtained. After
solving this equation, the obtained values of $k$ are included in the NU
method and here there is a relationship between $\lambda$ and $k$ by
$k=\lambda-\pi^{\prime}(z).$ After this point an appropriate $\phi_{n}(z)$ can
be calculated as the solution of the differential equation:
$\phi^{\prime}(z)-\left(\frac{\pi(z)}{\sigma(z)}\right)\phi(z)=0.$ (10)
## III Bound-state solutions for arbitrary $l$-states
To study any quantum physical system characterized by the empirical potential
given in Eq. (1), we solve the original $\mathrm{SE}$ which is given in the
well known textbooks [1,2]
$\left(\frac{p^{2}}{2m}+V(r)\right)\psi(\mathbf{r,}\theta,\phi)=E\psi(\mathbf{r,}\theta,\phi),$
(11)
where the potential $V(r)$ is taken as the MR form in (1). Using the
separation method with the wavefunction
$\psi(\mathbf{r,}\theta,\phi)=r^{-1}R(r)Y_{lm}(\theta,\phi),$ we obtain the
following radial Schrödinger eqauation as
$\frac{d^{2}R_{nl}(r)}{dr^{2}}+\left\\{\frac{2\mu
E_{nl}}{\hbar^{2}}-\frac{1}{b^{2}}\left[\frac{\alpha(\alpha-1)e^{-2r/b}}{\left(1-e^{-r/b}\right)^{2}}-\frac{Ae^{-r/b}}{1-e^{-r/b}}\right]-\frac{l(l+1)}{r^{2}}\right\\}R_{nl}(r)=0,$
(12)
Since the Schrödinger equation with above MR effective potential has no
analytical solution for $l\neq 0$ states, an approximation to the centrifugal
term has to be made. The good approximation for the too singular kinetic
energy term $l(l+1)r^{-2}$ in the centrifugal barrier is taken as [18,33]
$\frac{1}{r^{2}}\approx\frac{1}{b^{2}}\frac{e^{-r/b}}{\left(1-e^{-r/b}\right)^{2}},$
(13)
in a short potential range. To solve it by the present method, we need to
recast Eq. (12) with Eq. (13) into the form of Eq. (4) by making change of the
variables $r\rightarrow z$ through the mapping function $r=f(z)$ and energy
transformation:
$z=e^{-r/b},\text{ }\varepsilon=\sqrt{-\frac{2\mu
b^{2}E_{nl}}{\hbar^{2}}},\text{ }E_{nl}<0,$ (14)
to obtain the following hypergeometric equation:
$\frac{d^{2}R(z)}{dz^{2}}+\frac{(1-z)}{z(1-z)}\frac{dR(z)}{dz}$
$+\frac{1}{\left[z(1-z)\right]^{2}}\left\\{-\varepsilon^{2}+\left[A+2\varepsilon^{2}-l(l+1)\right]z-\left[A+\varepsilon^{2}+\alpha(\alpha-1)\right]z^{2}\right\\}R(z)=0.$
(15)
It is noted that the bound state (real) solutions of the last equation demands
that
$z=\left\\{\begin{array}[]{ccc}0,&\text{when}&r\rightarrow\infty,\\\
1,&\text{when}&r\rightarrow 0,\end{array}\right.$ (16)
and thus provide the finite radial wave functions $R_{nl}(z)\rightarrow 0.$ To
apply the hypergeometric method (NU), it is necessary to compare Eq. (15) with
Eq. (4). Subsequently, the following value for the parameters in Eq. (4) are
obtained as
$\widetilde{\tau}(z)=1-z,\text{\ }\sigma(z)=z-z^{2},\text{\
}\widetilde{\sigma}(z)=-\left[A+\varepsilon^{2}+\alpha(\alpha-1)\right]z^{2}+\left[A+2\varepsilon^{2}-l(l+1)\right]z-\varepsilon^{2}.$
(17)
If one inserts these values of parameters into Eq. (9), with
$\sigma^{\prime}(z)=1-2z,$ the following linear function is achieved
$\pi(z)=-\frac{z}{2}\pm\frac{1}{2}\sqrt{a_{1}z^{2}+a_{2}z+a_{3}},$ (18)
where $a_{1}=1+4\left[A+\varepsilon^{2}+\alpha(\alpha-1)-k\right],$
$a_{2}=4\left\\{k-\left[A+2\varepsilon^{2}-l(l+1)\right]\right\\}$ and
$a_{3}=4\varepsilon^{2}.$ According to this method the expression in the
square root has to be set equal to zero, that is,
$\Delta=a_{1}z^{2}+a_{2}z+a_{3}=0.$ Thus the constant $k$ can be determined as
$k=A-l(l+1)\pm a\varepsilon,\text{ \ }a=\sqrt{(1-2\alpha)^{2}+4l(l+1)}.$ (19)
In view of that, we can find four possible functions for $\pi(z)$ as
$\pi(z)=-\frac{z}{2}\pm\left\\{\begin{array}[]{c}\varepsilon-\left(\varepsilon-\frac{a}{2}\right)z,\text{
\ \ \ for \ \ }k=A-l(l+1)+a\varepsilon,\\\
\varepsilon-\left(\varepsilon+\frac{a}{2}\right)z;\text{ \ \ \ for \ \
}k=A-l(l+1)-a\varepsilon.\end{array}\right.$ (20)
We must select
$\text{\ }k=A-l(l+1)-a\varepsilon,\text{
}\pi(z)=-\frac{z}{2}+\varepsilon-\left(\varepsilon+\frac{a}{2}\right)z,$ (21)
in order to obtain the polynomial, $\tau(z)=\widetilde{\tau}(z)+2\pi(z)$
having negative derivative as
$\tau(z)=1+2\varepsilon-\left(2+2\varepsilon+a\right)z,\text{
}\tau^{\prime}(z)=-(2+2\varepsilon+a).$ (22)
We can also write the values of $\lambda=k+\pi^{\prime}(z)$ and
$\lambda_{n}=-n\tau^{\prime}(z)-\frac{n\left(n-1\right)}{2}\sigma^{\prime\prime}(z),$
$n=0,1,2,...$ as
$\lambda=A-l(l+1)-(1+a)\left[\frac{1}{2}+\varepsilon\right],$ (23)
$\lambda_{n}=n(1+n+a+2\varepsilon),\text{ }n=0,1,2,...$ (24)
respectively. Letting $\lambda=\lambda_{n}$ and solving the resulting equation
for $\varepsilon$ leads to the energy equation
$\varepsilon=\frac{(n+1)^{2}+l(l+1)+(2n+1)\Lambda-A}{2(n+1+\Lambda)},\text{
}\Lambda=\frac{-1+a}{2},$ (25)
from which we obtain the discrete energy spectrum formula:
$E_{nl}=-\frac{\hbar^{2}}{2\mu
b^{2}}\left[\frac{(n+1)^{2}+l(l+1)+(2n+1)\Lambda-A}{2(n+1+\Lambda)}\right]^{2},\text{
\ }0\leq n,l<\infty$ (26)
where $n$ denotes the radial quantum number. It is found that $\Lambda$
remains invariant by mapping $\alpha\rightarrow 1-\alpha,$ so do the bound
state energies $E_{nl}.$ An important quantity of interest for the MR
potential is the critical coupling constant $A_{c},$ which is that value of
$A$ for which the binding energy of the level in question becomes zero.
Furthermore, from Eq. (26), we have (in atomic units $\hbar=\mu=Z=e=1),$
$A_{c}=(n+1+\Lambda)^{2}-\Lambda(\Lambda+1)+l(l+1).$ (27)
Next, we turn to the radial wave function calculations. We use $\sigma(z)$ and
$\pi(z)$ in Eq (17) and Eq. (21) to obtain
$\phi(z)=z^{\varepsilon}(1-z)^{\Lambda+1},$ (28)
and weight function
$\rho(z)=z^{2\varepsilon}(1-z)^{2\Lambda+1},$ (29)
$y_{nl}(z)=C_{n}z^{-2\varepsilon}(1-z)^{-(2\Lambda+1)}\frac{d^{n}}{dz^{n}}\left[z^{n+2\varepsilon}(1-z)^{n+2\Lambda+1}\right].$
(30)
The functions $\ y_{nl}(z)$, up to a numerical factor, are in the form of
Jacobi polynomials, i.e., $\ y_{nl}(z)\simeq
P_{n}^{(2\varepsilon,2\Lambda+1)}(1-2z),$ and physically holds in the interval
$(0\leq r<\infty$ $\rightarrow$ $0\leq z\leq 1)$ [58]. Therefore, the radial
part of the wave functions can be found by substituting Eq. (28) and Eq. (30)
into $R_{nl}(z)=\phi(z)y_{nl}(z)$ as
$R_{nl}(z)=N_{nl}z^{\varepsilon}(1-z)^{1+\Lambda}P_{n}^{(2\varepsilon,2\Lambda+1)}(1-2z),$
(31)
where $\varepsilon$ and $\Lambda$ are given in Eqs. (14) and (19) and $N_{nl}$
is a normalization constant. This equation satisfies the requirements;
$R_{nl}(z)=0$ as $z=0$ $(r\rightarrow\infty)$ and $R_{nl}(z)=0$ as $z=1$
$(r=0).$ Therefore, the wave functions, $R_{nl}(z)$ in Eq. (31) is valid
physically in the closed interval $z\in[0,1]$ or $r\in(0,\infty).$ Further,
the wave functions satisfy the normalization condition:
$\int\limits_{0}^{\infty}\left|R_{nl}(r)\right|^{2}dr=1=b\int\limits_{0}^{1}z^{-1}\left|R_{nl}(z)\right|^{2}dz,$
(32)
where $N_{nl}$ can be determined via
$1=bN_{nl}^{2}\int\limits_{0}^{1}z^{2\varepsilon-1}(1-z)^{2\Lambda+2}\left[P_{n}^{(2\varepsilon,2\Lambda+1)}(1-2z)\right]^{2}dz.$
(33)
The Jacobi polynomials, $P_{n}^{(\rho,\nu)}(\xi),$ can be explicitly written
in two different ways [59,60]::
$P_{n}^{(\rho,\nu)}(\xi)=2^{-n}\sum\limits_{p=0}^{n}(-1)^{n-p}\binom{n+\rho}{p}\binom{n+\nu}{n-p}\left(1-\xi\right)^{n-p}\left(1+\xi\right)^{p},$
(34)
$P_{n}^{(\rho,\nu)}(\xi)=\frac{\Gamma(n+\rho+1)}{n!\Gamma(n+\rho+\nu+1)}\sum\limits_{r=0}^{n}\binom{n}{r}\frac{\Gamma(n+\rho+\nu+r+1)}{\Gamma(r+\rho+1)}\left(\frac{\xi-1}{2}\right)^{r},$
(35)
where
$\binom{n}{r}=\frac{n!}{r!(n-r)!}=\frac{\Gamma(n+1)}{\Gamma(r+1)\Gamma(n-r+1)}.$
After using Eqs. (34) and (35), we obtain the explicit expressions for
$P_{n}^{(2\varepsilon,2\Lambda+1)}(1-2z):$
$P_{n}^{(2\varepsilon,2\Lambda+1)}(1-2z)=(-1)^{n}\Gamma(n+2\varepsilon+1)\Gamma(n+2\Lambda+2)$
$\times\sum\limits_{p=0}^{n}\frac{(-1)^{p}}{p!(n-p)!\Gamma(p+2\Lambda+2)\Gamma(n+2\varepsilon-p+1)}z^{n-p}(1-z)^{p},$
(36)
$P_{n}^{(2\varepsilon,2\Lambda+1)}(1-2z)=\frac{\Gamma(n+2\varepsilon+1)}{\Gamma(n+2\varepsilon+2\Lambda+2)}\sum\limits_{r=0}^{n}\frac{(-1)^{r}\Gamma(n+2\varepsilon+2\Lambda+r+2)}{r!(n-r)!\Gamma(2\varepsilon+r+1)}z^{r}.$
(37)
Inserting Eqs. (36) and (37) into Eq. (33), one obtains
$1=bN_{nl}^{2}(-1)^{n}\frac{\Gamma(n+2\Lambda+2)\Gamma(n+2\varepsilon+1)^{2}}{\Gamma(n+2\varepsilon+2\Lambda+2)}$
$\times\sum\limits_{p,r=0}^{n}\frac{(-1)^{p+r}\Gamma(n+2\varepsilon+2\Lambda+r+2)}{p!r!(n-p)!(n-r)!\Gamma(p+2\Lambda+2)\Gamma(n+2\varepsilon-p+1)\Gamma(2\varepsilon+r+1)}I_{nl}(p,r),$
(38)
where
$I_{nl}(p,r)=\int\limits_{0}^{1}z^{n+2\varepsilon+r-p-1}(1-z)^{p+2\Lambda+2}dz.$
(39)
Using the following integral representation of the hypergeometric function
[59.60]
${}_{2}F_{1}(\alpha_{0},\beta_{0}:\gamma_{0};1)\frac{\Gamma(\alpha_{0})\Gamma(\gamma_{0}-\alpha_{0})}{\Gamma(\gamma_{0})}=\int\limits_{0}^{1}z^{\alpha_{0}-1}(1-z)^{\gamma_{0}-\alpha_{0}-1}(1-z)^{-\beta_{0}}dz,$
$\mathop{\mathrm{R}e}(\gamma_{0})>\mathop{\mathrm{R}e}(\alpha_{0})>0,$ (40)
which gives
${}_{2}F_{1}(\alpha_{0},\beta_{0}:\alpha_{0}+1;1)/\alpha_{0}=\int\limits_{0}^{1}z^{\alpha_{0}-1}(1-z)^{-\beta_{0}}dz,$
(41)
where
${}_{2}F_{1}(\alpha_{0},\beta_{0}:\gamma_{0};1)=\frac{\Gamma(\gamma_{0})\Gamma(\gamma_{0}-\alpha_{0}-\beta_{0})}{\Gamma(\gamma_{0}-\alpha_{0})\Gamma(\gamma_{0}-\beta_{0})},$
$(\mathop{\mathrm{R}e}(\gamma_{0}-\alpha_{0}-\beta_{0})>0,\text{
}\mathop{\mathrm{R}e}(\gamma_{0})>\mathop{\mathrm{R}e}(\beta_{0})>0).$ (42)
For the present case, with the aid of Eq. (40), when
$\alpha_{0}=n+2\varepsilon+r-p,$ $\beta_{0}=-p-2\Lambda-2,$ and
$\gamma_{0}=\alpha_{0}+1$ are substituted into Eq. (41), we obtain
$I_{nl}(p,r)=\frac{{}_{2}F_{1}(\alpha_{0},\beta_{0}:\gamma_{0};1)}{\alpha_{0}}=\frac{\Gamma(n+2\varepsilon+r-p+1)\Gamma(p+2\Lambda+3)}{(n+2\varepsilon+r-p)\Gamma(n+2\varepsilon+r+2\Lambda+3)}.$
(43)
Finally, we obtain
$1=bN_{nl}^{2}(-1)^{n}\frac{\Gamma(n+2\Lambda+2)\Gamma(n+2\varepsilon+1)^{2}}{\Gamma(n+2\varepsilon+2\Lambda+2)}$
$\times\sum\limits_{p,r=0}^{n}\frac{(-1)^{p+r}\Gamma(n+2\varepsilon+r-p+1)(p+2\Lambda+2)}{p!r!(n-p)!(n-r)!\Gamma(n+2\varepsilon-p+1)\Gamma(2\varepsilon+r+1)(n+2\varepsilon+r+2\Lambda+2)},$
(44)
which gives
$N_{nl}=\frac{1}{\sqrt{s(n)}},$ (45)
where
$s(n)=b(-1)^{n}\frac{\Gamma(n+2\Lambda+2)\Gamma(n+2\varepsilon+1)^{2}}{\Gamma(n+2\varepsilon+2\Lambda+2)}$
$\times\sum\limits_{p,r=0}^{n}\frac{(-1)^{p+r}\Gamma(n+2\varepsilon+r-p+1)(p+2\Lambda+2)}{p!r!(n-p)!(n-r)!\Gamma(n+2\varepsilon-p+1)\Gamma(2\varepsilon+r+1)(n+2\varepsilon+r+2\Lambda+2)}.$
(46)
## IV Numerical Results
To show the accuracy of our results, we calculate the energy eigenvalues for
various $n$ and $l$ quantum numbers with two different values of the
parameters $\alpha.$ Its shown in Table 1, the present approximately numerical
results are not in a good agreement when long potential range (small values of
parameter $b$). The energy eigenvalues for short potential range (large values
of parameter $b$) are in agreement with the other authors. The energy spectra
for various diatomic molecules like $HCl,CH,LiH$ and $CO$ are presented in
Tables 2 and 3. These results are relevant to atomic physics [61-64],
molecular physics [65,66] and chemical physics [67,68], etc.
## V Discussions
In this work, we have utilized the hypergeometric method and solved the radial
$\mathrm{SE}$ for the M-R model potential with the angular momentum $l\neq 0$
states. We have derived the binding energy spectra in Eq. (26) and their
corresponding wave functions in Eq. (31).
Let us study special cases. We have shown that for $\alpha=0$ $(1)$, the
present solution reduces to the one of the Hulthén potential [16,19,57]:
$V^{(H)}(r)=-V_{0}\frac{e^{-\delta r}}{1-e^{-\delta r}},\text{
}V_{0}=Ze^{2}\delta,\text{ }\delta=b^{-1}$ (47)
where $Ze^{2}$ is the potential strength parameter and $\delta$ is the
screening parameter and $b$ is the range of potential. We note also that it is
possible to recover the Yukawa potential by letting $b\rightarrow\infty$ and
$V_{0}=Ze^{2}/b.$ If the potential is used for atoms, the $Z$ is identified
with the atomic number. This can be achieved by setting $\Lambda=l,$ hence,
the energy for $l\neq 0$ states
$E_{nl}=-\frac{\left[A-(n+l+1)^{2}\right]^{2}\hbar^{2}}{8\mu
b^{2}(n+l+1)^{2}},\text{ \ }0\leq n,l<\infty.$ (48)
and for $s$-wave ($l=0)$ states
$E_{n}=-\frac{\left[A-(n+1)^{2}\right]^{2}\hbar^{2}}{8\mu
b^{2}(n+1)^{2}},\text{ \ }0\leq n<\infty$ (49)
Essentially, these results coincide with those obtained by the Feynman
integral method [31,56] and the standard way [32,33], respectively.
Furthermore, if taking $b=1/\delta$ and identifying $\frac{A\hbar^{2}}{2\mu
b^{2}}$ as $Ze^{2}\delta,$ we are able to obtain
$E_{nl}=-\frac{\mu\left(Ze^{2}\right)^{2}}{2\hbar^{2}}\left[\frac{1}{n+l+1}-\frac{\hbar^{2}\delta}{2Ze^{2}\mu}(n+l+1)\right]^{2},$
(50)
which coincides with those of Refs. [16,19]. Further, we have (in atomic units
$\hbar=\mu=Z=e=1)$
$E_{nl}=-\frac{1}{2}\left[\frac{1}{n+l+1}-\frac{(n+l+1)}{2}\delta\right]^{2},$
(51)
which coincides with Refs. [16,33].
The corresponding radial wave functions are expressed as
$R_{nl}(r)=N_{nl}e^{-\delta\varepsilon r}(1-e^{-\delta
r})^{l+1}P_{n}^{(2\varepsilon,2l+1)}(1-2e^{-\delta r}),$ (52)
where
$\varepsilon=\frac{\mu
Ze^{2}}{\hbar^{2}\delta}\left[\frac{1}{n+l+1}-\frac{\hbar^{2}\delta}{2Ze^{2}\mu}(n+l+1)\right],\text{
}0\leq n,l<\infty,$ (53)
which coincides for the ground state with that given in Eq. (6) by Gönül et
al. [18]. In addition, for $\delta r\ll 1$ (i.e., $r/b\ll 1),$ the Hulthén
potential turns to become a Coulomb potential: $V(r)=-Ze^{2}/r$ with energy
levels and wave functions:
$E_{nl}=-\frac{\varepsilon_{0}}{(n+l+1)^{2}},\text{ }n=0,1,2,..$
$.\varepsilon_{0}=\frac{Z^{2}\hbar^{2}}{2\mu a_{0}^{2}},\text{
}a_{0}=\frac{\hbar^{2}}{\mu e^{2}}$ (54)
where $\varepsilon_{0}=13.6$ $eV$ and $a_{0}$ is Bohr radius for the Hydrogen
atom. The wave functions are
$R_{nl}=N_{nl}\exp\left[-\frac{\mu
Ze^{2}}{\hbar^{2}}\frac{r}{\left(n+l+1\right)}\right]r^{l+1}P_{n}^{\left(\frac{2\mu
Ze^{2}}{\hbar^{2}\delta(n+l+1)},2l+1\right)}(1+2\delta r)$
which coincide with Refs. [3,16,22].
## VI Conclusions and Outlook
In this work approximately analytical bound states for the $l$-wave
Schrödinger equationwith the MR potential have been presented by making a
proper approximation to the too singular orbital centrifugal term $\sim
r^{-2}.$ The normalized radial wave functions of $l$-wave bound states
associated with the MR potential are obtained. The approach enables one to
find the $l$-dependent solutions and the corresponding energy eigenvalues for
different screening parameters of the MR potential.
We have shown that for $\alpha=0,1,$ the present solution reduces to the one
of the Hulthén potential. We note that it is possible to recover the Yukawa
potential by letting $b\rightarrow\infty$ and $V_{0}=Ze^{2}/b.$ The Hulthén
potential behaves like the Coulomb potential near the origin (i.e.,
$r\rightarrow 0$) $V_{C}(r)=-Ze^{2}/r$ but decreases exponentially in the
asymptotic region when $r\gg 0,$ so its capacity for bound states is smaller
than the Coulomb potential [16]. Obviously, the results are in good agreement
with those obtained by other methods for short potential range, small $\alpha$
and $l.$ We have also studied two special cases for $l=0,$ $l\neq 0$ and
Hulthén potential. The results we have ended up show that the NU method
constitute a reliable alternative way in solving the exponential potentials.
We have also found that the criteria for the choice of parameter $A$ requires
that $A$ satisfies the inequality $\sqrt{1-2A}<2\alpha-1.$ This means that for
real bound state solutions $A$ should be chosen properly in our numerical
calculations.
A slight difference in the approximations of the numerical energy spectrum of
Schrödinger-MR problem is found in Refs. [55,56] and present work since the
approximation schemes are different by a small shift $\delta^{2}/12.$ In our
recent work [17], we have found that the physical quantities like the energy
spectrum are critically dependent on the behavior of the system near the
singularity ($r=0$). That is why, for example, the energy spectrum depends
strongly on the angular momentum $l$, which results from the $r^{-2}$
singularity of the orbital term, even for high excited states. It is found
that the $r^{-2\text{ }}$ orbital term is too singular, then the validity of
all such approximations is limited only to very few of the lowest energy
states. In this case, to extend accuracy to higher energy states one may
attempt to utilize the full advantage of the unique features of Schrödinger
equation. Therefore, it is more fruitful to perform the analytic approximation
of the less singularity $r^{-1}$ rather than the too singular term $r^{-2}$
which makes it possible to extend the validity of the results to higher
excitation levels giving better analytic approximation for a wider energy
spectrum [69].
###### Acknowledgements.
Work partially supported by the Scientific and Technological Research Council
of Turkey (TÜBİTAK).
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Figure 1: Variation of MR potential as function of separation distance $r$
taking various values for the screening parameter $b$ when (a) $\alpha=0.75$
and (b) $\alpha=1.50.$
Figure 2: A plot of the variation of the singular orbital term $1/r^{2}$
(dotted-solid line) with the approximations of (a) Ref. 34 (dash line), the
conventional Greene-Aldrich of Ref. 18 (dash-dot line) and improved [55,56]
(solid line) replacing the term $1/r^{2}$ with respect to $r$ where
$\delta=0.1$ $fm^{-1},$ and (b) the improved approximation [55] with various
shifting constants.
Table 1: Energies (in atomic units) of different $n$ and $l$ states and for $\alpha=0.75$ and $\alpha=1.5,$ $A=2b.$ | | $\alpha=0.75$ | | | $\alpha=1.5$ | |
---|---|---|---|---|---|---|---
states | $1/b$ | Present | QD [33] | LSl [54] | Present | QD [33] | LS [54]
$2p$ | $0.025$ | $-0.1205793$ | $-0.1205793$ | $-0.1205271$ | $-0.0900228$ | $-0.0900229$ | $-0.0899708$
| $0.050$ | $-0.1084228$ | $-0.1084228$ | $-0.1082151$ | $-0.0802472$ | $-0.0802472$ | $-0.0800400$
| $0.075$ | $-0.0969120$ | $-0.0969120$ | $-0.0964469$ | $-0.0710332$ | $-0.0710332$ | $-0.0705701$
| $0.100$ | $-0.0860740$ | | | $-0.0577157$ | |
$3p$ | $0.025$ | $-0.0459296$ | $-0.0459297$ | $-0.0458779$ | $-0.0369650$ | $-0.0369651$ | $-0.0369134$
| $0.050$ | $-0.0352672$ | $-0.0352672$ | $-0.0350633$ | $-0.0274719$ | $-0.0274719$ | $-0.0272696$
| $0.075$ | $-0.0260109$ | $-0.0260110$ | $-0.0255654$ | $-0.0193850$ | $-0.0193850$ | $-0.0189474$
| $0.100$ | $-0.0181609$ | | | $-0.0127043$ | |
$3d$ | $0.025$ | $-0.0449299$ | $-0.0449299$ | $-0.0447743$ | $-0.0396344$ | $-0.0396345$ | $-0.0394789$
| $0.050$ | $-0.0343082$ | $-0.0343082$ | $-0.0336930$ | $-0.0300629$ | $-0.0300629$ | $-0.0294496$
| $0.075$ | $-0.0251168$ | $-0.0251168$ | $-0.0237621$ | $-0.0218120$ | $-0.0218121$ | $-0.0204663$
$4p$ | $0.025$ | $-0.0208608$ | $-0.0208608$ | $-0.0208097$ | $-0.0172249$ | $-0.0172249$ | $-0.0171740$
| $0.050$ | $-0.0119291$ | $-0.0119292$ | $-0.0117365$ | $-0.0091019$ | $-0.0091019$ | $-0.0089134$
| $0.075$ | $-0.0054773$ | $-0.0054773$ | $-0.0050945$ | $-0.0035478$ | $-0.0035478$ | $-0.0031884$
$4d$ | $0.025$ | $-0.0204555$ | $-0.0204555$ | $-0.0203017$ | $-0.0183649$ | $-0.0183649$ | $-0.0182115$
| $0.050$ | $-0.0115741$ | $-0.0115742$ | $-0.0109904$ | $-0.0100947$ | $-0.0100947$ | $-0.0095167$
| $0.075$ | $-0.0052047$ | $-0.0052047$ | $-0.0040331$ | $-0.0042808$ | $-0.0042808$ | $-0.0031399$
$4f$ | $0.025$ | $-0.0202886$ | $-0.0202887$ | $-0.0199797$ | $-0.0189222$ | $-0.0189223$ | $-0.0186137$
| $0.050$ | $-0.0114283$ | $-0.0114284$ | $-0.0102393$ | $-0.0105852$ | $-0.0105852$ | $-0.0094015$
| $0.075$ | $-0.0050935$ | $-0.0050935$ | $-0.0026443$ | $-0.0046527$ | $-0.0046527$ | $-0.0022307$
$5p$ | $0.025$ | $-0.0098576$ | $-0.0098576$ | $-0.0098079$ | $-0.0081308$ | $-0.0081308$ | $-0.0080816$
$5d$ | $0.025$ | $-0.0096637$ | $-0.0096637$ | $-0.0095141$ | $-0.0086902$ | $-0.0086902$ | $-0.0085415$
$5f$ | $0.025$ | $-0.0095837$ | $-0.0095837$ | $-0.0092825$ | $-0.0089622$ | $-0.0089622$ | $-0.0086619$
$5g$ | $0.025$ | $-0.0095398$ | $-0.0095398$ | $-0.0090330$ | $-0.0091210$ | $-0.0091210$ | $-0.0086150$
$6p$ | $0.025$ | $-0.0044051$ | $-0.0044051$ | $-0.0043583$ | $-0.0035334$ | $-0.0035334$ | $-0.0034876$
$6d$ | $0.025$ | $-0.0043061$ | $-0.0043061$ | $-0.0041650$ | $-0.0038209$ | $-0.0038209$ | $-0.0036813$
$6f$ | $0.025$ | $-0.0042652$ | $-0.0042652$ | $-0.0039803$ | $-0.0039606$ | $-0.0039606$ | $-0.0036774$
$6g$ | $0.025$ | $-0.0042428$ | $-0.0042428$ | $-0.0037611$ | $-0.0040422$ | $-0.0040422$ | $-0.0035623$
Table 2: Energy spectrum of $HCl$ and $CH$ (in $eV$) for different states where $\hbar c=1973.29$ $eV$ $A^{\circ},$ $\mu_{HCl}=0.9801045$ $amu,$ $\mu_{CH}=0.929931$ $amu$ and $A=2b.$ states | $1/b$111$b$ is in $pm$. | $HCl/$ $\alpha=0,1$ | $\alpha=0.75$ | $\alpha=1.5$ | $CH/$ $\alpha=0,1$ | $\alpha=0.75$ | $\alpha=1.5$
---|---|---|---|---|---|---|---
$2p$ | $0.025$ | $-4.81152646$ | $-5.14278553$ | $-3.83953094$ | $-5.07112758$ | $-5.42025940$ | $-4.04668901$
| $0.050$ | $-4.31837832$ | $-4.62430290$ | $-3.42259525$ | $-4.55137212$ | $-4.87380256$ | $-3.60725796$
| $0.075$ | $-3.85188684$ | $-4.13335980$ | $-3.02961216$ | $-4.05971155$ | $-4.35637111$ | $-3.19307186$
| $0.100$ | $-3.41205201$ | $-3.66996049$ | $-2.46161213$ | $-3.59614587$ | $-3.86796955$ | $-2.59442595$
$3p$ | $0.025$ | $-1.86633700$ | $-1.95892730$ | $-1.57658128$ | $-1.96703335$ | $-2.06461927$ | $-1.66164415$
| $0.050$ | $-1.42316902$ | $-1.50416901$ | $-1.17169439$ | $-1.49995469$ | $-1.58532495$ | $-1.23491200$
| $0.075$ | $-1.03998066$ | $-1.10938179$ | $-0.82678285$ | $-1.09609178$ | $-1.16923738$ | $-0.87139110$
| $0.100$ | $-0.71676763$ | $-0.77457419$ | $-0.54184665$ | $-0.75544012$ | $-0.81636557$ | $-0.57108145$
$3d$ | $0.025$ | $-1.86633700$ | $-1.91628944$ | $-1.69043293$ | $-1.96703335$ | $-2.01968093$ | $-1.78163855$
| $0.050$ | $-1.42316902$ | $-1.46326703$ | $-1.28220223$ | $-1.49995469$ | $-1.54221615$ | $-1.35138217$
| $0.075$ | $-1.03998066$ | $-1.07124785$ | $-0.93029598$ | $-1.09609178$ | $-1.12904596$ | $-0.98048917$
| $0.100$ | $-0.71676763$ | $-0.74022762$ | $-0.63472271$ | $-0.75544012$ | $-0.78016587$ | $-0.66896854$
$4p$ | $0.025$ | $-0.85301300$ | $-0.88972668$ | $-0.73465318$ | $-0.89903647$ | $-0.93773100$ | $-0.77429066$
| $0.050$ | $-0.47981981$ | $-0.50878387$ | $-0.38820195$ | $-0.50570801$ | $-0.53623480$ | $-0.40914700$
| $0.075$ | $-0.21325325$ | $-0.23361041$ | $-0.15131598$ | $-0.22475912$ | $-0.24621462$ | $-0.15948008$
$4d$ | $0.025$ | $-0.85301300$ | $-0.87244037$ | $-0.78327492$ | $-0.89903647$ | $-0.91951202$ | $-0.82553574$
| $0.050$ | $-0.47981981$ | $-0.49364289$ | $-0.43054552$ | $-0.50570801$ | $-0.52027690$ | $-0.45377517$
| $0.075$ | $-0.21325325$ | $-0.22198384$ | $-0.18257890$ | $-0.22475912$ | $-0.23396076$ | $-0.19242977$
$4f$ | $0.025$ | $-0.85301300$ | $-0.86532198$ | $-0.80704413$ | $-0.89903647$ | $-0.91200956$ | $-0.85058739$
| $0.050$ | $-0.47981981$ | $-0.48742442$ | $-0.45146566$ | $-0.50570801$ | $-0.51372292$ | $-0.47582404$
| $0.075$ | $-0.21325325$ | $-0.21724109$ | $-0.19844068$ | $-0.22475912$ | $-0.22896211$ | $-0.20914735$
$5p$ | $0.025$ | $-0.40318193$ | $-0.42043305$ | $-0.34678391$ | $-0.42493521$ | $-0.44311709$ | $-0.36549429$
$5d$ | $0.025$ | $-0.40318193$ | $-0.41216309$ | $-0.37064268$ | $-0.42493521$ | $-0.43440094$ | $-0.39064034$
$5f$ | $0.025$ | $-0.40318193$ | $-0.40875104$ | $-0.38224366$ | $-0.42493521$ | $-0.43080479$ | $-0.40286723$
$5g$ | $0.025$ | $-0.40318193$ | $-0.40687867$ | $-0.38901658$ | $-0.42493521$ | $-0.42883140$ | $-0.41000558$
$6p$ | $0.025$ | $-0.17919244$ | $-0.18788038$ | $-0.15070181$ | $-0.18886059$ | $-0.19801728$ | $-0.15883277$
$6d$ | $0.025$ | $-0.17919244$ | $-0.18365796$ | $-0.16296387$ | $-0.18886059$ | $-0.19356705$ | $-0.17175642$
$6f$ | $0.025$ | $-0.17919244$ | $-0.18191355$ | $-0.16892216$ | $-0.18886059$ | $-0.19172852$ | $-0.17803620$
$6g$ | $0.025$ | $-0.17919244$ | $-0.18095818$ | $-0.17240246$ | $-0.18886059$ | $-0.19072160$ | $-0.18170426$
Table 3: Energy spectrum of $LiH$ and $CO$ (in $eV$) for different states where $\hbar c=1973.29$ $eV$ $A^{\circ},$ $\mu_{LiH}=0.8801221$ $amu,$ $\mu_{CO}=6.8606719$ $amu$ and $A=2b.$ states | $1/b$111$b$ is in $pm$. | $LiH/$ $\alpha=0,1$ | $\alpha=0.75$ | $\alpha=1.5$ | $CO/$ $\alpha=0,1$ | $\alpha=0.75$ | $\alpha=1.5$
---|---|---|---|---|---|---|---
$2p$ | $0.025$ | $-5.35811876$ | $-5.72700906$ | $-4.27570397$ | $-1.374733789$ | $-0.734690030$ | $-0.548509185$
| $0.050$ | $-4.80894870$ | $-5.14962650$ | $-3.81140413$ | $-1.233833096$ | $-0.660620439$ | $-0.488946426$
| $0.075$ | $-4.28946350$ | $-4.60291196$ | $-3.37377792$ | $-1.100548657$ | $-0.590485101$ | $-0.432805497$
| $0.100$ | $-3.79966317$ | $-4.08687021$ | $-2.74125274$ | $-0.974880471$ | $-0.524284624$ | $-0.351661930$
$3p$ | $0.025$ | $-2.07835401$ | $-2.18146262$ | $-1.75568186$ | $-0.533243776$ | $-0.279849188$ | $-0.225227854$
| $0.050$ | $-1.58484188$ | $-1.67504351$ | $-1.30479958$ | $-0.406623254$ | $-0.214883153$ | $-0.167386368$
| $0.075$ | $-1.15812308$ | $-1.23540823$ | $-0.92070588$ | $-0.297139912$ | $-0.158484490$ | $-0.118112862$
| $0.100$ | $-0.79819287$ | $-0.86256629$ | $-0.60340076$ | $-0.204792531$ | $-0.110654417$ | $-0.077407337$
$3d$ | $0.025$ | $-2.07835401$ | $-2.13398108$ | $-1.88246712$ | $-0.533243776$ | $-0.273758013$ | $-0.241492516$
| $0.050$ | $-1.58484188$ | $-1.62949505$ | $-1.42786117$ | $-0.406623254$ | $-0.209039964$ | $-0.183173338$
| $0.075$ | $-1.15812308$ | $-1.19294225$ | $-1.03597816$ | $-0.299139912$ | $-0.153036736$ | $-0.132900580$
| $0.100$ | $-0.79819287$ | $-0.82431793$ | $-0.70682759$ | $-0.204792531$ | $-0.105747722$ | $-0.090675460$
$4p$ | $0.025$ | $-0.94991579$ | $-0.99080017$ | $-0.81811023$ | $-0.243720118$ | $-0.127104916$ | $-0.104951366$
| $0.050$ | $-0.53432763$ | $-0.56658202$ | $-0.43230193$ | $-0.137092566$ | $-0.072684041$ | $-0.055457903$
| $0.075$ | $-0.23747895$ | $-0.26014869$ | $-0.16850556$ | $-0.060930029$ | $-0.033373205$ | $-0.021616756$
$4d$ | $0.025$ | $-0.94991579$ | $-0.97155012$ | $-0.87225543$ | $-0.243720118$ | $-0.124635422$ | $-0.111897390$
| $0.050$ | $-0.53432763$ | $-0.54972102$ | $-0.47945575$ | $-0.137092566$ | $-0.070521025$ | $-0.061507037$
| $0.075$ | $-0.23747895$ | $-0.24720134$ | $-0.20331998$ | $-0.060930029$ | $-0.031712252$ | $-0.026082927$
$4f$ | $0.025$ | $-0.94991579$ | $-0.96362308$ | $-0.89872483$ | $-0.243720118$ | $-0.123618500$ | $-0.115293020$
| $0.050$ | $-0.53432763$ | $-0.54279613$ | $-0.50275243$ | $-0.137092566$ | $-0.069632666$ | $-0.064495655$
| $0.075$ | $-0.23747895$ | $-0.24191980$ | $-0.22098366$ | $-0.060930029$ | $-0.031034710$ | $-0.028348915$
$5p$ | $0.025$ | $-0.44898364$ | $-0.46819450$ | $-0.38617877$ | $-0.115195837$ | $-0.060062386$ | $-0.049540988$
$5d$ | $0.025$ | $-0.44898364$ | $-0.45898506$ | $-0.41274791$ | $-0.115195837$ | $-0.058880953$ | $-0.052949414$
$5f$ | $0.025$ | $-0.44898364$ | $-0.45518540$ | $-0.42566677$ | $-0.115195837$ | $-0.058393512$ | $-0.054606711$
$5g$ | $0.025$ | $-0.44898364$ | $-0.45310033$ | $-0.43320910$ | $-0.115195837$ | $-0.058126029$ | $-0.055574280$
$6p$ | $0.025$ | $-0.19954881$ | $-0.20922370$ | $-0.16782162$ | $-0.051198285$ | $-0.026840287$ | $-0.021529017$
$6d$ | $0.025$ | $-0.19954881$ | $-0.20452162$ | $-0.18147666$ | $-0.051198285$ | $-0.026237080$ | $-0.023280755$
$6f$ | $0.025$ | $-0.19954881$ | $-0.20257904$ | $-0.18811182$ | $-0.051198285$ | $-0.025987876$ | $-0.024131947$
$6g$ | $0.025$ | $-0.19954881$ | $-0.20151514$ | $-0.19198748$ | $-0.051198285$ | $-0.025851393$ | $-0.024629136$
|
arxiv-papers
| 2011-10-14T09:08:23 |
2024-09-04T02:49:23.130786
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sameer M. Ikhdair",
"submitter": "Sameer Ikhdair",
"url": "https://arxiv.org/abs/1110.3153"
}
|
1110.3249
|
# Studies of $b$-hadron decays to charming final states at LHCb
S. Ricciardi (on behalf of the LHCb Collaboration) STFC Rutherford Appleton
Laboratory, Chilton, Didcot, Oxfordshire, OX11 0QX, UK
###### Abstract
We present studies from the LHCb experiment of decays of the type $H_{b}\to
H_{c}X$, where $H_{b}$ represents a beauty hadron ($B^{\pm}$, $B^{0}$ or
$\Lambda_{b}^{0}$) and $H_{c}$ a charmed hadron ($D^{0}$, $D^{(*)+}$,
$D_{s}^{+}$ or $\Lambda_{c}^{+}$). Such decays are important for the
determination of the CKM angle $\gamma$, a key goal of the LHCb physics
programme. We exploit the data accumulated in 2010, and in the early months of
the 2011 run. We report on the observation of new decay modes, and first
measurements on the road to a precise determination of $\gamma$.
## I Introduction
Decays of $b$-hadrons (($H_{b}$) to open charm are of great interest both in
the context of $CP$ violation studies and of QCD studies of heavy-quark
dynamics. In particular, the angle $\gamma$ of the CKM Unitarity Triangle can
be determined from $H_{b}\to(D^{0},\bar{D^{0}})X_{s}$, where $D^{0}$ and
$\bar{D^{0}}$ decay to a common final state, thanks to the interference of
$b\to u$ and $b\to c$ tree-level transitions. In addition, the abundant and
predictable relative rate of suitable decays to open-charm can be used to
measure the production fractions of different $b$ meson and baryon species.
Since all $b$-hadron species are produced in $pp$ collisions at the LHC, these
measurements are needed to normalise measurements of $B^{0}_{s}$ and
$\Lambda^{0}_{b}$ branching fractions to those of known $B^{+}$ or $B^{0}$
decays, and to determine absolute branching fractions.
In these proceedings, we report on the most recent results from LHCb using
data accumulated in 2010, and in the early months of the 2011 run. In Section
III, we describe preliminary measurements on the road of a precise
determination of $\gamma$ with charged $B$ decays. In Section IV, we present
two different measurements of the $B_{s}$ production fraction, with
semileptonic and fully hadronic decays, which combined give the most accurate
determination of $f_{s}/f_{d}$. Finally, in Section V, we present the first
observation of the $\Lambda^{0}_{b}\to D^{0}pK^{-}$ decay and a hint of the
neutral beauty strange baryon $\Xi^{0}_{b}$, also reconstructed in the
$D^{0}pK^{-}$ final state.
## II The LHCb experiment
The LHCb experiment has been designed to study decays of $b$-hadrons from $pp$
collisions at the LHC. The detector has been described elsewhere bib:LHCb .
Here, we just mention the salient experimental features which are critical for
the measurement of $\gamma$ and are common to many hadronic decays to open-
charm. Above all, since sensitivity to $\gamma$ arises from the interference
of the $b\to c$ with the suppressed $b\to u$ amplitude, a large data sample is
mandatory. It is ensured by: the high integrated luminosity that the LHC
delivers, the large $b\bar{b}$ cross-section within the LHCb detector
acceptance, and a dedicated and flexible trigger, which can select efficiently
$b$-hadron decays. The vertex detector and the tracking system also play a
crucial role: a momentum resolution smaller than 1% and clear separation of
secondary vertices from the primary vertex enable the separation of $b$-hadron
decays from different sources of background components, both prompt, from the
primary vertex, and non-prompt, due to long-lived hadron decays other than
signal. In addition, the excellent pion-kaon separation over a wide momentum
range, provided by two RICH detectors, is vital to distinguish the different
$H_{b}$ and $D$ decays of interest. For example, it is necessary to separate
$B^{+}\to DK^{+}$, from the about ten times more abundant $B^{+}\to D\pi^{+}$
in the measurements of $\gamma$ using charged $B^{+}\to DK^{+}$ decays.111In
the following, $D$ indicates a superposition of $D^{0}$ and $\bar{D^{0}}$.
## III $CP$ violation studies with charged $B$ decays
Despite the impressive achievements by experiments at $B$-factories and the
Tevatron, the CKM angle $\gamma$ is still the least well-determined angle of
the Unitarity Triangle. The current average of direct measurements of $\gamma$
has an uncertainty of about $10^{\circ}$ bib:CKMUTfitters . This precision can
be significantly improved at LHCb in the near future.
Already now, the 2010 and early 2011 LHCb data-sets are sufficient to set
constraints on the $CP$ asymmetries and measure the ratio of branching
fractions of the most sensitive $B^{+}\to DK^{+}$ decay over the favoured
$B^{+}\to D\pi^{+}$ mode. These measurements demonstrate the capability of
LHCb in three well-established methods for extracting $\gamma$: the GLW method
bib:GLW , which uses $D$ decays to $CP$-eigenstates (e.g., $K^{+}K^{-}$), the
ADS method bib:ADS , where the $D^{0}$ or $\bar{D^{0}}$ is reconstructed in a
final state accessible to both Cabibbo-favoured (CF) and doubly-Cabibbo-
suppressed (DCS) transitions (e.g., $K^{\pm}\pi^{\mp}$), and the GGSZ method
bib:GGSZ , which exploits the interference over the Dalitz plot of $D$ decays
to three-body final states (e.g., $K^{0}_{S}\pi^{+}\pi^{-}$).
### III.1 Measurements of GLW observables
As first step towards a measurement of $\gamma$ with the GLW method, the ratio
of the $B^{+}\to DK^{+}$ branching fraction to that of $B^{+}\to D\pi^{+}$ is
measured using the 2010 LHCb dataset, corresponding to 36.5 pb-1 bib:LHCbGLW .
The measurement is performed, simultaneously, for the Cabibbo-favoured (CF)
$D\to K^{+}\pi^{-}$ and the $D\to K^{+}\pi^{-}\pi^{+}\pi^{-}$ decay, and,
separately, for the $CP$-even $D\to K^{+}K^{-}$ (CP+) decay (a difference
between the two can be expected due to the relative larger interference
between the $b\to u$ and the $b\to c$ transitions in the $CP$-even case).
The $B^{+}\to DK^{+}$ and $B^{+}\to D\pi^{+}$ signal yields are extracted with
an unbinned extended maximum-likelihood fit to the $B$-mass distributions. The
fit is performed simultaneously to four different mass distributions, which
are obtained by separating the sample according to the charge of the bachelor
hadron, and the value of a particle identification discriminant for the
bachelor. The used PID discriminant is the difference of the log-likelihood
between the kaon and pion hypotheses, DLLKπ. The results of the fit are shown
in Fig. 1 for the $D\to KK$ mode.
Figure 1: Reconstructed $B$ mass distributions for $B^{\pm}\to(KK)_{D}K^{\pm}$
(top) and $B^{\pm}\to(KK)_{D}\pi^{\pm}$ (bottom) candidates. Sensitivity to
charge asymmetry is obtained by splitting $B^{-}$ (left) and $B^{+}$ (right).
The solid red (green) curve is the fitted $B^{\pm}\to DK^{\pm}$ ($B^{\pm}\to
D\pi^{\pm}$) signal. The dashed lines indicate the different background
components from: charmless (red and green, if present), combinatoric,
partially reconstructed, and semileptonic decays (blue)bib:LHCbGLW .
The ratio of branching fractions is computed from the fitted yields and the
ratio of efficiencies. The PID efficiency determination uses a data
calibration sample of pions and kaons from $D^{*+}\to
D^{0}(K^{-}\pi^{+})\pi^{+}$ decay and a re-weighting technique to take into
account small difference in the kinematics between the calibration sample and
the signal samples. Other efficiencies (geometric acceptance, trigger and
reconstruction) are very similar for the two decay modes, and their ratio is
derived from Monte Carlo simulations.
The results are: ${\cal{R}}_{CF}^{K/\pi}=(6.30\pm 0.38\pm 0.40)\%,$ and
${\cal{R}}_{CP+}^{K/\pi}=(9.31\pm 1.89\pm 0.53)\%.$ From the ratio of the CP+
over CF measurements, the following $\gamma$-sensitive observable is computed:
${\cal{R}}_{CP+}=1.48\pm 0.31\pm 0.12.$
In addition, three $CP$ asymmetries are measured between the $B^{-}$ and the
$B^{+}$ decay rates:
$\displaystyle{\cal{A}}_{CF~{}}^{DK}$ $\displaystyle=$ $\displaystyle(-0.08\pm
0.06\pm 0.02)\%,$ $\displaystyle{\cal{A}}_{CP+}^{DK}$ $\displaystyle=$
$\displaystyle(0.07\pm 0.18\pm 0.07)\%,$ $\displaystyle{\cal{A}}_{CP+}^{D\pi}$
$\displaystyle=$ $\displaystyle(0.01\pm 0.04\pm 0.01)\%.$
None of the measured asymmetries significantly deviates from zero, but all
results agree with existing measurements within their uncertainties. The main
systematic uncertainties are associated to possible differences in the trigger
response, to the PID calibration procedure, and to the parameterisation of the
background. With larger data samples, we will be able to extract $\gamma$ and
all hadronic unknowns by combining ${\cal{A}}_{CP+}^{DK}$ and
${\cal{R}}_{CP+}$ with measurements of additional $\gamma$-sensitive
observables from other methods.
### III.2 Towards a GGSZ measurement
The ratio of the $B^{+}\to DK^{+}$ and $B^{+}\to D\pi^{+}$ branching fractions
is also measured in the $D\to K_{S}^{0}\pi^{+}\pi^{-}$ final state, as it is
the first step towards the measurement of $\gamma$ with the GGSZ method
bib:LHCbGLW . As for the GLW analysis, the $B^{+}\to DK^{+}$ and $B^{+}\to
D\pi^{+}$ samples are separated by the value of DLLKπ for the bachelor hadron.
Yields are extracted with a simultaneous fit to the $B$ invariant mass
distributions for the two samples. The results are shown in Figure 2. In 36.5
pb-1, the fitted signal yield for $B^{+}\to D\pi^{+}$ is 95${}^{+14}_{-12}$
events, and the ratio of branching fraction of $B^{+}\to DK^{+}$ and $B^{+}\to
D\pi^{+}$ is
${\cal{R}}_{K_{S}^{0}\pi\pi}^{K/\pi}=(12.0^{+6.0}_{-5.0}\pm 1.0)\%,$
where the largest systematic uncertainty is due to the appropriateness of the
fit model (7%).
Figure 2: The recosntructed $B$ mass distributions for
$B^{\pm}\to(K^{0}_{S}\pi\pi)_{D}K^{\pm}$ (left) and
$B^{\pm}\to(K^{0}_{S}\pi\pi)_{D}\pi^{\pm}$ (right). The dashed lines indicate
the fitted signal contribution (dark yellow), the background from
combinatorial (red) and partially reconstructed decays (light green), and the
cross-feed between $B\to DK$ and $B\to D\pi$ (dark green)bib:LHCbGLW .
### III.3 Hunting for the ADS suppressed modes
The $CP$ asymmetry in the decay rates of the suppressed ADS mode $B^{+}\to
D(K^{-}\pi^{+})K^{+}$ is expected to be enhanced by the fact that the two
interfering amplitudes in these modes have similar size. However, the
branching fraction of this mode is small, ${\cal{O}}(10^{-7})$, hence its
observation is difficult. The most competitive result to date is from the
Belle Collaboration bib:BelleADS , who observed $56.0^{+15.1}_{-14.2}$ events
($4.1~{}\sigma$ significance) in their full data set of $772\times 10^{6}$
$B\bar{B}$ pairs collected at the $\Upsilon(4S)$.
The LHCb collaboration has performed a search for these modes using the early
2011 data-set, corresponding to 343 pb-1 of data bib:LHCbADS . The analysis is
similar to the GLW analysis just described. An improved event selection, based
on a “Boosted Decision Tree” algorithm, is used in this case to isolate the
$B^{+}\to D(K\pi)h^{+},h={K,\pi}$ candidates from the background. As for the
previously described analyses, the value of the particle identification
variable DLLKπ for the $B$-meson bachelor track is used to effectively
separate $B\to D\pi$ from $B\to DK$. Eight signal yields are extracted with an
unbinned maximum-likelihood fit, corresponding to the two $B$-meson charges,
the two product of kaon charges (opposite-sign kaons are suppressed and same-
sign kaons are favoured), and the two fail/pass slices according to the PID
requirement on the bachelor, DLL${}_{K\pi}>$4\. Particular attention has been
paid to model the signal and the different background components in the fit.
The results of the fit to the suppressed modes, summed over both $B$ charges,
are shown in Figure 3.
Figure 3: The reconstructed $B$ mass distribution for the ADS suppressed
candidates, summed over both $B$ charges. The red line indicates the signal
component. The background components are from combinatorial and partial
reconstruction (dashed blue), charmless sources (magenta), and $B^{\pm}\to
D\pi^{\pm}$ (green)bib:LHCbADS .
The charge asymmetry between the $B^{-}$ and $B^{+}$ suppressed modes is
measured to be
$A_{ADS}^{DK}=-0.39\pm 0.17\pm 0.02,$
and the average partial rate $R_{ADS}^{DK}$ of the suppressed over favoured
mode
$R_{ADS}^{DK}=(1.66\pm 0.39\pm 0.24)\times 10^{-2},$
which corresponds to $4.0~{}\sigma$ significance for the evidence of the
suppressed decay. The main sources of systematic uncertainties are the PID
calibration procedure and the background model. All these preliminary results
are highly competitive with existing measurements and consistent with world
averages.
## IV Measurements of the $B^{0}_{s}$ production fraction
Strange $B$ mesons offer a still largely unexplored window on $CP$ violation
studies and searches of physics beyond the Standard Model. For example, the
determination of the absolute branching fraction for the rare decay
$B^{0}_{s}\to\mu^{+}\mu^{-}$ provides an important constraint to different new
physics models. The LHCb measurement of
${\cal{B}}(B^{0}_{s}\to\mu^{+}\mu^{-})$ bib:Olivier and of other $B^{0}_{s}$
decay branching fractions relies on the knowledge of $f_{s}/f_{d}$, the ratio
of $B^{0}_{s}$ production to $B^{0}$ production.
We have performed two measurements of the ratio $f_{s}/f_{d}$ using the
relative abundance of $B^{0}_{s}\to D^{-}_{s}\pi^{+}$ to $B^{0}\to
D^{-}K^{+}$, and to $B^{0}\to D^{-}\pi^{+}$ decays, and a measurement of the
ratio $f_{s}/(f_{u}+f_{d})$, where $f_{u}$ is the $B^{+}$ production fraction,
using $H_{b}$ semileptonic decays, identified by the detection of a muon and a
charmed hadron. In this section, we report on the hadronic bib:hadronic and
semileptonic bib:semileptonic measurements, and on their combination
bib:average .
### IV.1 $f_{s}/f_{d}$ from hadronic decays
The reconstruction of $B^{0}_{s}\to D_{s}^{-}\pi^{+}$ decays is the first step
towards the time-dependent analysis of $B^{0}_{s}\to D_{s}^{-}K^{+}$, which is
sensitive to $\gamma$. In addition, the ratio of its branching fraction to
U-spin related $B^{0}$ decay modes can be used to measure $f_{s}/f_{d}$
bib:FST . Here, we report on the latter measurement, which has been performed
with a sample of 35 pb-1 collected in 2010. Two normalisation modes are used:
$B^{0}\to D^{-}K^{+}$ and $B^{0}\to D^{-}\pi^{+}$. The first is dominated by
contributions from colour-allowed tree-diagram amplitudes, and is therefore
theoretically well-understood. The second leads to a smaller statistical
uncertainty due to its greater yield, but suffers from an additional
theoretical uncertainty due to the contribution from a $W$-exchange diagram.
The relative yields of the three decay modes are extracted from unbinned
maximum likelihood fits to the mass distributions, which are shown in Fig. 4.
Figure 4: Mass distributions of the $B^{0}_{s}\to D_{s}^{-}\pi^{+}$, $B^{0}\to
D^{-}K^{+}$, and $B^{0}\to D^{-}\pi^{+}$ candidates (left to
right)bib:hadronic .
The value of $f_{s}/f_{d}$ is found to be
$f_{s}/f_{d}=0.250\pm 0.024{\mathrm{(stat.)}}\pm 0.017{\mathrm{(syst.)}}\pm
0.017{\mathrm{(theor.)}}$
from the relative yields of $B^{0}_{s}\to D_{s}^{-}\pi^{+}$ with respect to
$B^{0}\to D^{-}K^{+}$, and
$f_{s}/f_{d}=0.256\pm 0.014{\mathrm{(stat.)}}\pm 0.019{\mathrm{(syst.)}}\pm
0.026{\mathrm{(theor.)}}$
from $B^{0}_{s}\to D_{s}^{-}\pi^{+}$ with respect to $B^{0}\to D^{-}\pi^{+}$.
### IV.2 $b$-hadron production fraction measurements from semileptonic decays
The semileptonic measurement of the $b$-hadron production fractions is based
on 3 pb-1 of LHCb data collected in 2010. We measure two production ratios:
that of $\bar{B^{0}_{s}}$ and that of $\Lambda_{b}^{0}$ relative to the sum of
$B^{-}$ and $\bar{B^{0}}$. The relative fractions are extracted from the
yields in four different final states: $D^{0}\mu^{-}\bar{\nu}X$,
$D^{+}\mu^{-}\bar{\nu}X$, $D_{s}\mu^{-}\bar{\nu}X$, and
$\Lambda_{c}\mu^{-}\bar{\nu}X$. We do not attempt to separate $f_{u}$ and
$f_{d}$, but we measure their sum from $D^{0}$ and $D^{+}$ channels, taking
into account corrections due to cross-feed from $\bar{B^{0}_{s}}$ and
$\Lambda_{b}^{0}$ decays.
The $H_{b}$ signals are separated from various sources of background yields by
studying the two-dimensional distributions of the charm candidate invariant
mass and impact parameter (IP) with regard to the primary $pp$ collision
vertex. This approach allows us to determine the background coming from false
combinations and from prompt charm production. As an example, the results of
the fit for the $D^{+}_{s}\mu^{-}\bar{\nu}X$ candidates are shown in Fig. 5.
Figure 5: The logarithm of the IP distributions for (a) right sign and (c)
wrong sign $D^{0}$ candidate combinations with a muon. The dotted curves show
the combinatorial backgrounds, the small red-solid curve the prompt-charm
contributions, the dashed curves the signal, the purple-dashed curves
represent a background originating from $\Lambda_{c}$ reflection, and the
green-solid curves the total. The invariant $K^{-}K^{+}\pi^{+}$ mass spectra
are also shown for right sign (b) and wrong-sign (d)
combinationsbib:semileptonic .
The fractions $f_{s}/(f_{u}+f_{d})$ and $f_{\Lambda_{b}^{0}}/(f_{u}+f_{d})$
are determined as function of the pseudo-rapidity $\eta$, and the charmed
hadron-muon pair transverse momentum, $p_{t}$.
Figure 6: Ratio between $B^{0}_{s}$ and light $B$ meson production fractions
as a function of the transverse momentum of the $D_{s}\mu$ pair in two bins of
$\eta$. The errors shown are statistical only bib:semileptonic .
We find
$f_{s}/(f_{u}+f_{d})=0.134\pm 0.004^{+0.011}_{-0.010},$
with no significant dependence on $\eta$, nor $p_{t}$, as shown in Fig. 6. The
main systematic uncertainty is from limited knowledge of the charm-hadron
branching fractions. A dependence on $p_{t}$ is instead found for the
$\Lambda_{b}^{0}$ fragmentation function with regard to the sum of $B^{-}$ and
$\bar{B^{0}}$. Assuming a linear dependence, we get
$f_{\Lambda_{b}}/(f_{u}+f_{d})=(0.404\pm 0.017\pm 0.027\pm
0.105)\times[1-(0.031\pm 0.004\pm 0.003)\times p_{t}({\mathrm{GeV}})]$, where
the errors are statistical, systematic and (for the constant term) an absolute
scale uncertainty due to the error in ${\cal{B}}(\Lambda_{c}\to pK\pi)$,
respectively. No $\eta$ dependence is found. More details on this measurement
can be found in Ref. bib:semileptonic .
### IV.3 Average
If we use isospin symmetry to set $f_{u}=f_{d}$, the LHCb measurements of the
ratio of strange $B$ meson to light neutral $B$ meson, obtained using $H_{b}$
semileptonic decays, is in good agreement with the two measurements obtained
with hadronic decays. Therefore, we combine them to derive
$f_{s}/f_{d}=0.267^{+0.021}_{-0.020}.$
Since we do not observe a dependence upon properties such as the transverse
momentum or rapidity of the $B$ meson, although our measurement is obtained
from data within the LHCb acceptance, it is reasonable to assume that it is
valid in other phase-space regions. We also note that, despite the fact that
this ratio is not a-priori universal, our result is in remarkable agreement
with the average between results from LEP and Tevatron experiments
($f_{s}/f_{d}=0.271\pm 0.027$ bib:HFAG ).
## V Studies of beauty baryon decays to charm
The study of $b$-baryons is a largely unexplored area where LHCb has great
potential for measurements of spectroscopy and $CP$ violation. In particular,
we look for $\Lambda_{b}^{0}\to DpK$, which is an unobserved $\Lambda_{b}^{0}$
decay mode. This channel is sensitive to the angle $\gamma$ bib:DpK ,
similarly to $\Lambda_{b}^{0}\to D\Lambda$, as originally proposed in Ref.
bib:DLambda , but it presents some advantages compared to $D\Lambda$, because
the $pK$ pair originates from the $\Lambda_{b}^{0}$ decay vertex, rather than
from a long-lived intermediate particle, therefore a larger reconstruction
efficiency is expected at LHCb. In addition, the use of the full phase-space
of the three-body decay may enhance the sensitivity compared to the two-body
process.
The $D^{0}pK^{-}$ final state is studied together with the $\Lambda_{b}^{0}\to
D^{0}p\pi^{-}$, and $\Lambda_{b}^{0}\to\Lambda_{c}^{+}\pi^{-}$ decays, which
have similar kinematics and can be used as normalisation channels. With 333
pb-1 taken by LHCb in early 2011, we measure the ratio of branching fractions
$\frac{{\cal{B}}(\Lambda_{b}^{0}\to D^{0}p\pi^{-})\times{\cal{B}}(D^{0}\to
K^{-}\pi^{+})}{{\cal{B}}(\Lambda_{b}^{0}\to\Lambda_{c}^{+}\pi^{-})\times{\cal{B}}(\Lambda_{c}^{+}\to
K^{-}p\pi^{+})}=0.119\pm 0.006\pm 0.013.$
We also present the first observation of the $\Lambda_{b}^{0}\to DpK$ decay
and measure the ratio of branching fractions
$\frac{{\cal{B}}(\Lambda_{b}^{0}\to D^{0}pK^{-})}{{\cal{B}}(\Lambda_{b}^{0}\to
D^{0}p\pi^{-})}=0.112\pm 0.019^{+0.011}_{-0.014}.$
The significance of the $\Lambda_{b}^{0}\to DpK$ signal is 6.3 $\sigma$. As
seen in Fig. 7, in the $DpK^{-}$ final state we find a hint of production of
the neutral beauty-strange baryon $\Xi^{0}_{b}$ with significance of 2.6
$\sigma$, and we measure the ratio of branching fraction times production
ratio with respect to those for the $\Lambda_{b}^{0}$
$\frac{f_{b\to\Xi^{0}_{b}}\times{\cal{B}}(\Xi_{b}^{0}\to
D^{0}pK^{-})}{f_{b\to\Lambda^{0}_{b}}\times{\cal{B}}(\Lambda_{b}^{0}\to
D^{0}pK^{-})}=0.29\pm 0.12\pm 0.18.$
Figure 7: Invariant mass spectrum of $DpK^{-}$. Results of the fit are
overlaid to the data pointsbib:DpK .
We measure the difference of the $\Xi^{0}_{b}$ and $\Lambda_{b}^{0}$ masses to
be equal to (181.8 $\pm$ 5.5 $\pm$ 0.5) MeV/c2, which is in good agreement
with the recent measurement of the CDF Collaboration bib:XiCDF .The main
sources of systematic uncertainty in these results are the description of the
signal and background lineshapes, and, for the branching fraction
measurements, the determination of reconstruction and PID efficiency ratios.
## VI Conclusions
LHCb is on track for a precise measurement of the CKM angle $\gamma$ using
$b$-hadron decays to open charm, with both well-established modes (GLW, ADS,
GGSZ) and unique ways (e.g., using $B^{0}_{s}\to D^{\pm}_{s}K^{\mp}$, and
$\Lambda_{b}^{0}\to DpK$ decays). In particular, the 4.0 $\sigma$ evidence of
the ADS suppressed mode is highly competitive with the previous measurements.
Another important by-product of the study of $b$-hadron decays to open charm
is the most precise measurement of $f_{s}/f_{d}$, the $B^{0}_{s}$ production
fraction with respect to that of $B^{0}_{d}$, from the combination of LHCb
results obtained with hadronic and semileptonic decays.
All these results have been obtained with data samples from 2010 or early
2011, which are just a fraction of the total expected by the end of this year
(1 fb-1). Hence, an improved precision in the measurement of $\gamma$ and many
other measurements with charming tree-level final states can be expected very
soon from LHCb.
###### Acknowledgements.
I am grateful to Steven Blusk, Tim Gershon, Vava Gligorov, and Olaf Steinkamp
for the help in the preparation of this contribution. I would like also to
thank the organisers of DPF2011 for inviting us to such an exquisite
conference.
## References
* (1) A. Augusto Alves et al., LHCb Collaboration, JINST 3 (2008) S080005.
* (2) CKMfitter Group, http://ckmfitter.in2p3.fr/; UTFit Collaboration, http://www.utfit.org/.
* (3) M. Gronau and D. London, Phys. Lett. B253 (1991) 483; M. Gronau and D. Wyler, Phys. Lett. B265 (1991) 172.
* (4) D. Atwood, I. Dunietz and A. Soni, Phys. Rev. Lett. 78 (1997) 3257; D. Atwood, I. Dunietz and A. Soni, Phys. Rev. D63 (2001) 036005.
* (5) A. Giri, Yu. Grossman, A. Soffer, and J. Zupan, Phys. Rev. D68 (2003) 054018.
* (6) LHCb Collaboration, LHCb-CONF-2011-031 (2011).
* (7) Y. Horii et al., Belle Collaboration, Phys. Rev. Lett. 106 (2011) 231803.
* (8) LHCb Collaboration, LHCb-CONF-2011-044 (in preparation).
* (9) R. Aaij et al., LHCb Collaboration, Phys. Lett. B669 (2011) 330.
* (10) R. Aaij et al., LHCb Collaboration, arXiv:1106.4435 [hep-ex].
* (11) LHCb Collaboration, LHCb-CONF-2011-028 (2011).
* (12) LHCb Collaboration, LHCb-CONF-2011-034 (2011).
* (13) R. Fleischer, N. Serra, and N. Tuning, Phys. Rev. D82 (2010) 034038;R. Fleischer, N. Serra, and N. Tuning, Phys. Rev. D83 (2011) 014017.
* (14) Heavy Flavour Averaging Group, http://www.slac.stanford.edu/xorg/hfag/
* (15) LHCb Collaboration, LHCb-CONF-2011-036 (2011).
* (16) I. Dunietz, Z. Phys. C56 (1992) 129.
* (17) T. Aaltonen et al., Phys. Rev. Lett. 107 (2011) 102001.
|
arxiv-papers
| 2011-10-14T15:32:41 |
2024-09-04T02:49:23.141853
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Stefania Ricciardi",
"submitter": "Stefania Ricciardi",
"url": "https://arxiv.org/abs/1110.3249"
}
|
1110.3278
|
# Curved geometry and Graphs111Based on a talk given at Loops ’11, Madrid, on
24 May 2011.
Francesco Caravelli Perimeter Institute for Theoretical Physics,
Waterloo, Ontario N2L 2Y5 Canada,
and
University of Waterloo, Waterloo, Ontario N2L 3G1, Canada,
and
Max Planck Institute for Gravitational Physics, Albert Einstein Institute,
Am Mühlenberg 1, Golm, D-14476 Golm, Germany fcaravelli@perimeterinstitute.ca
###### Abstract
Quantum Graphity is an approach to quantum gravity based on a background
independent formulation of condensed matter systems on graphs. We summarize
recent results obtained on the notion of emergent geometry from the point of
view of a particle hopping on the graph. We discuss the role of connectivity
in emergent Lorentzian perturbations in a curved background and the
Bose–Hubbard (BH) model defined on graphs with particular symmetries.
## 1 Introduction
The gravitational interaction may be an effective description of an underlying
theory which does not suffer from the well-known problems plaguing Einstein’s
gravity, as for instance perturbative non-renormalizability 222However, there
is still hope that gravity is non-perturbativly renormalizable. This is the
starting point of Asymptotic Safety and Loop Quantum Gravity[2, 3].. These
ideas are the starting point of Quantum Graphity [1]. The target per se is
understanding the emergence of gravity through model building of discrete
quantum systems. A similar approach is Analogue Models [4], although in the
continuum and considering Quantum Field Theory as the main tool. The
motivation for the introduction of simplified models for Quantum Gravity is
understanding the features of background independence in contexts in which not
all the obstructions typical of gravity are present and, in particular, better
understanding the phenomenon of emergence in background independent contexts.
The first Quantum Graphity model was introduced in [5, 6]. The basic
motivation in [5] was to construct a Hamiltonian on a Hilbert space associated
to the degrees of freedom of a graph (a set of nodes $V$ with cardinality
$\mathscr{N}$ and of links $\mathscr{E}$ of cardinality
$\mathscr{N}(\mathscr{N}-1)/2$) such that the ground state of the Hamiltonian
is a graph which has geometrical properties. The initial hope, in [5], was
that by fixing the parameters of the reduced model (the one with on/off
links), the ground state of the Hamiltonian would have been a graph of average
degree $d$ and with geometrical properties similar to simplicial
complexes333However the results obtained numerically in [6] suggested that the
low-energy structure of the graph is string-like: the ground state is a one of
a 1-dimensional object. This result is compatible with the mean field theory
analysis performed in the reduced model[7], in which the model was mapped to
an Ising-type Hamiltonian. This mapping allowed a straightforward use of the
mean field theory techniques well known from the study of Ising-type models,
after having identified the average degree of the graph as an order parameter.
In [8] a different Hamiltonian based on graphs and close in spirit to [5]
found, instead, a 2-dimensional complex in a low-energy phase. This last
result gives some hope that a generic mechanism to obtain low energy
$d$-dimensional simplicial complexes exists. .
A second model, which we now describe, was introduced in [9]. The main
motivation for the introduction of [9] was the interpretational issue of the
external bath (i.e. the temperature of the system): how can it be interpreted
in a closed system (i.e. the Universe)? The same problem arises in closed
quantum systems, where one could ask why thermalization is such a general
phenomenon despite the unitary time evolution on the Hilbert space. In general
the solution to the problem is strongly dependent on the observability of full
Hilbert space, i.e. what part of it should be traced out in order to observe
decoherence in the local observables. From this perspective, [9] is a
simplified model in which these questions can be answered in a background
independent context. While the graph degrees of freedom are the same as the
model introduced in [5], additional degrees of freedom associated to bosonic
particles with a Bose–Hubbard (BH) interaction are present. The result is a
dynamical graph with local interactions.
## 2 Matter coupling: a graph-dynamical Bose–Hubbard model
The model introduced in [9] and recently further studied in [10], focuses on
the study of interaction between matter (in the specific case, bosonic degrees
of freedom on the vertices of the graph) and the graph. The energy terms for
links and particles are of the form
$\widehat{H}_{0}=\sum_{i}\mu\
\widehat{a}^{\dagger}_{i}\widehat{a}_{i}+\sum_{ij}\nu\
\widehat{b}^{\dagger}_{ij}\widehat{b}_{ij},$ (1)
with $\widehat{a}_{i}$, $\widehat{b}_{ij}$ are hard core bosonic ladder
operators on the space of vertices and links respectively and $\mu$,$\nu$
coupling constants. There are two other terms, namely a BH interaction,
$\widehat{H}_{BH}=-E\sum_{ij}(\widehat{a}^{\dagger}_{i}\widehat{a}_{j}+h.c.)\otimes\widehat{P}_{ij},$
(2)
where $\widehat{P}_{ij}$ is a projector on the on links $ij$, and an
interaction between the bosonic particles and the links,
$\widehat{H}_{int}=-E\sum_{ij}\widehat{P}^{L}_{ij}(\widehat{a}^{\dagger}_{i}\widehat{a}^{\dagger}_{j}\otimes\widehat{b}_{ij}+h.c.).$
(3)
$\widehat{P}^{L}_{ij}$ is a nonlocal projector which annihilates graph states
unless the link $ij$ is not on a triangle. This keeps the dynamics local.
Thermalization. Simulations[9] showed that, in the long time regime, the
classical model evolves into random graphs. In the quantum case, instead, the
damping of oscillations of graphs observables, e.g. the degree of the graph,
indicates that the model thermalizes to a metastable state. We studied the
case of a 4-vertices graph with a link turned off and the others on. We
observed dumping for the vertex degree observables. For the studied case, the
asymptotic state turned out to be a mixed state of the graph with the link
turned on and the link turned off.
## 3 Graphs and curved geometry
Trapped surfaces. An interesting feature of this model, first noticed in [9],
is that regions of high connectivity are able to trap particles of low energy.
This was studied in detail in [10] on a fixed graph. The graph chosen is the
one of Fig. 1LABEL:sub@2dkn, which has a rotational symmetry: single particle
states can be labeled by two quantum numbers, the shell position $n$ and the
internal position, $\theta$, thus states of the form $|i,\theta\rangle$. In
this case, the problem can be simplified by noticing that the Hamiltonian is
diagonal in blocks of constant angular momentum. We can thus introduce, in the
1-particle sector, the _delocalized_ states, defined as:
$|n\rangle=\frac{1}{\sqrt{K_{n}}}\sum_{\theta=1}^{K_{n}}|n,\theta\rangle,$ (4)
where $K_{n}$ is the number of sites inside a shell $n$ (see Fig.
1LABEL:sub@2dnt) The full quantum evolution of a delocalized state can be
_fully_ reduced to a 1-dimensional BH model if the initial state is
delocalized, where the coefficients of the Hamiltonian are opportunely chosen.
This allowed to prove that the ground state, for the graph in Fig.
1LABEL:sub@2dkn, is $|n=0\rangle$.
In fact, this reduction showed that particles inside the highly connected
central region of Fig. 1a feel a potential which is proportional to the
degree. In particular, between particle states inside the central region and
the outside regions there is an energy gap proportional the relative degree.
This prevents low energy particles from escaping the trap, thus confirming the
argument of [9].
Emergent curved space. Another interesting finding of [10] is that, in the
reduced models, expectation values of number operators on the graph exhibit an
emergent Lorentz symmetry. The reduced effective 1-dimensional BH Hamiltonian,
for a generic rotationally invariant graph, takes the form,
$\widehat{H}_{BH}=-\sum_{i}f_{i\
i-1}(\widehat{b}^{\dagger}_{i}\widehat{b}_{i-1}+h.c.),$ (5)
with $f_{ij}=d_{ij}E$ and $d_{ij}$ is the relative degree between the shell
$i$ and the shell $j$, $\widehat{b}^{\dagger}_{i}$,$\widehat{b}_{i}$ is a
symmetry reduced ladder operators. The idea then is to study the following
expectation values (e.v.)
$\langle\widehat{b}^{\dagger}_{i}\widehat{b}_{i}\rangle=\Psi_{i}(t)$. The
interesting result is that these e.v. satisfy the following (closed) relation
on the manifold of classical states:
$\displaystyle\frac{\hbar^{2}}{2}\partial_{t}^{2}\Psi_{n}(t)=$ $\displaystyle
f_{n-1,n}^{2}\left(\Psi_{n+1}(t)+\Psi_{n-1}(t)-2\Psi_{n}(t)\right)$
$\displaystyle+\left(f_{n+1,n}^{2}-f_{n-1,n}^{2}\right)\left(\Psi_{n+1}(t)-\Psi_{n}(t)\right).$
(6)
This fully describes the time evolution of the probability density. In the
continuum limit, eqn. (6) becomes
$\Big{[}\partial_{t}^{2}-\partial_{x}\Big{(}n(x)\partial_{x}\Big{)}\Big{]}\Psi(x,t)=0,$
so it has a site-dependent speed of propagation for the density. This equation
is everywhere locally Lorentz-invariant, with a local speed of propagation
given by $c_{x}=1/\sqrt{n(x)}$. The higher-dimensional version of this
equation is known to be related to the Gordon background[4]. In fact, there is
no obstruction to repeating the same procedure for graphs which can be
foliated in more than one dimension. Thus, curved space is encoded in the
connectivity of the graph (i.e. the local degree).
(a) The $\mathscr{K}_{N}$ configuration.
(b) A non-trivial graph in which the coupling constants of the 1d-reduced
model are site-dependent.
Figure 1: Two graphs on which the wavefunction can be symmetry-reduced.
## 4 Conclusions
The emergence of gravity in a background independent contexts is a complete
new research direction within the field of Analogue Models. In the case of
graph-based models as ours it is not always obvious what are the right
questions to ask. Indeed, we have shown in [10] that by focusing on a
particular set of graphs and by asking the right question, “How does the
particle probability density evolve?”, the phenomenon of emergence (in our
case of a Lorentz symmetry) can be identified. This happens because the graph
we considered can be foliated, and then the states classified, according to
this foliation. More recently, the same technique has been used to study the
effects of disordered locality on the Lorentz symmetry, showing that a mass
term appears [11]. We have identified graph configurations which, in a
appropriate limit, can be considered as trapping surfaces. By considering
delocalized states (i.e. symmetry-reduced wavefunctions), we showed that high
connectivity implies a high energy gap between particles being inside the
region or outside.
It is not clear, at this point, how general these models are and how many of
the emergent phenomena of Analogue Models can be reproduced. We believe that
many interesting questions related to background independence (and emergent
gravity [12]) can be addressed in such simplified models. The full connection
between a graph and the low energy manifold associated to it is an open
problem to address. It is tempting, given the large literature on Regge
calculus, to associate a simplicial complex to the graph when possible. On the
other hand, it may be more physical to require that only the scaling
properties - for instance looking at the Heat Kernel of the Laplacian of the
graph[13]\- match those of manifolds of a specific dimension.
Aknowledgements
We would like to thank A. Hamma, F. Markopoulou and A. Riera for several
stimulating discussions and for the collaboration in the main work that we
summarized here. Also, we thank L. Sindoni for many fruitful conversations
about emergent gravity. Research at Perimeter Institute is supported by the
Government of Canada through Industry Canada and by the Province of Ontario
through the Ministry of Research & Innovation.
## References
* [1] F. Markopoulou, A. Hamma, New J. Phys. 13:095006 (2011) [arXiv:1011.5754]
* [2] C. Rovelli, (Lectures given at Zakopane 2011) [arXiv:1102.3660]
* [3] M. Reuter, F. Saueressig, (Lectures given at Zakopane 2007) [arXiv:0708.1317v1]
* [4] C. Barcelo, S. Liberati, M. Visser, Living Rev. Rel. 8 12 (2005) [arXiv:gr-qc/0505065]
* [5] T. Konopka, F. Markopoulou, L. Smolin, [arXiv:hep-th/0611197]; T. Konopka, F. Markopoulou, S. Severini, Phys. Rev. D 77, 104029 (2008) [arXiv:0801.0861]
* [6] T. Konopka, Phys. Rev. D 78,044032 (2008) [arXiv:0805.2283]
* [7] F. Caravelli, F. Markopoulou, Phys. Rev. D 84, 024002 (2011), [arXiv:1008.1340]
* [8] F. Conrady, J. Statist. Phys. 142:898 (2011) [arXiv:1009.3195]
* [9] A. Hamma, F. Markopoulou, S. Lloyd, F. Caravelli, S. Severini, K. Markstrom, Phys. Rev. D 81, 104032 (2010) [arXiv:0911.5075]
* [10] F. Caravelli, A. Hamma, F. Markopoulou, A. Riera, Phys. Rev. D 85, 044046 (2012) [arXiv:1108.2013]
* [11] F. Caravelli, F. Markopoulou, [arXiv:1201.3206]
* [12] L. Sindoni, Contribution to SIGMA Special Issue ”Loop Quantum Gravity and Cosmology” (to appear) [arXiv:1110.0686]
* [13] T. Filk, Class.Quant.Grav.17:4841-4854 (2000) [arXiv:hep-th/0010126]
|
arxiv-papers
| 2011-10-14T17:41:36 |
2024-09-04T02:49:23.150052
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Francesco Caravelli",
"submitter": "Francesco Caravelli",
"url": "https://arxiv.org/abs/1110.3278"
}
|
1110.3410
|
# Effects of nuclear deformation and neutron transfer in capture process, and
origin of fusion hindrance at deep sub-barrier energies
V.V.Sargsyan1,2, G.G.Adamian1, N.V.Antonenko1, W. Scheid3, and H.Q.Zhang4
1Joint Institute for Nuclear Research, 141980 Dubna, Russia
2International Center for Advanced Studies, Yerevan State University, M.
Manougian 1, 0025, Yerevan, Armenia
3Institut für Theoretische Physik der Justus–Liebig–Universität, D–35392
Giessen, Germany
4China Institute of Atomic Energy, Post Office Box 275, Beijing 102413, China
###### Abstract
The roles of nuclear deformation and neutron transfer in sub-barrier capture
process are studied within the quantum diffusion approach. The change of the
deformations of colliding nuclei with neutron exchange can crucially influence
the sub-barrier fusion. The comparison of the calculated capture cross section
and the measured fusion cross section in various reactions at extreme sub-
barrier energies gives us information about the fusion and quasifission.
###### pacs:
25.70.Jj, 24.10.-i, 24.60.-k
Key words: sub-barrier capture, fusion hindrance, quasifission
## I Introduction
The nuclear deformation and neutron-transfer process have been identified as
playing a major role in the magnitude of the sub-barrier capture and fusion
cross sections Gomes . There are a several experimental evidences which
confirm the importance of nuclear deformation on the capture and fusion. The
influence of nuclear deformation is straightforward. If the target nucleus is
prolate in the ground state, the Coulomb field on its tips is lower than on
its sides, that then increases the capture or fusion probability at energies
below the barrier corresponding to the spherical nuclei. The role of neutron
transfer reactions is less clear. A correlation between the overall transfer
strength and fusion enhancement was firstly noticed in Ref. Henning . The
importance of neutron transfer with positive $Q$-values on nuclear fusion
(capture) originates from the fact that neutrons are insensitive to the
Coulomb barrier and therefore they can start being transferred at larger
separations before the projectile is captured by target-nucleus Stelson .
Therefore, it is generally thought that the sub-barrier fusion cross section
will increase Pengo ; Roberts ; Stefanini3236s110pd ; Acker ; Sonzogni
because of the neutron transfer. As suggested in Ref. Broglia , the
enhancements in fusion yields may be due to the transfer of a neutron pair
with a positive $Q$-value. However, as shown recently in Ref. Scarlassara ,
the two-neutron transfer channel with large positive $Q$-value weakly
influences the fusion (capture) cross section in the 60Ni + 100Mo reaction at
sub-barrier energies. So, from the present data an unambiguous signature of
the role of neutron transfer channel could not be inferred.
The experiments with various medium-light and heavy systems have shown that
the experimental slopes of the complete fusion excitation function keep
increasing at low sub-barrier energies and may become much larger than the
predictions of standard coupled-channel calculations. This was identified as
the fusion hindrance Jiang . More experimental and theoretical studies of sub-
barrier fusion hindrance are needed to improve our understanding of its
physical reason, which may be especially important in astrophysical fusion
reactions Zvezda .
It is worth remembering that the first evidences of hindrance for compound
nucleus formation in the reactions with massive nuclei ($Z_{1}\times
Z_{2}>1600$) at energies near the Coulomb barrier were observed at GSI already
long time ago GSI . The theoretical investigations showed that the probability
of complete fusion depends on the competition between the complete fusion and
quasifission after the capture stage Volkov ; nasha ; Avaz . As known, this
competition can strongly reduce the value of the fusion cross section and,
respectively, the value of the evaporation residue cross section in the
reactions producing superheavy nuclei. Although the quasifission was
originally ascribed to the reactions with massive nuclei, it is the general
phenomenon which is related to the binary decay of nuclear system after the
capture, but before the compound nucleus formation which could exist at
angular momenta treated. The mass and angular distributions of the
quasifission products depend on the entrance channel and bombarding energy
Volkov . Because the capture cross section is the sum of the fusion and
quasifission cross sections, from the comparison of calculated capture cross
sections and measured fusion cross sections one can extract the hindrance
factor and show a role of the quasifission channel in the reactions with
various medium-mass and heavy nuclei at extreme sub-barrier energies.
In the present paper the quantum diffusion approach EPJSub ; EPJSub1 is
applied to study the fusion hindrance and the roles of nuclear deformation and
neutron transfer in sub-barrier capture process. With this approach many
heavy-ion capture reactions at energies above and well below the Coulomb
barrier have been successfully described EPJSub ; EPJSub1 ; Conf . Since the
details of our theoretical treatment were already published in Refs. EPJSub ;
EPJSub1 , the model will be shortly described in Sec. II. The calculated
results will be presented in Sec. III.
## II Model
In the quantum diffusion approach the collisions of nuclei are treated in
terms of a single collective variable: the relative distance between the
colliding nuclei. The nuclear deformation effects are taken into consideration
through the dependence of the nucleus-nucleus potential on the deformations
and orientations of colliding nuclei. Our approach takes into consideration
the fluctuation and dissipation effects in collisions of heavy ions which
model the coupling with various channels. We have to mention that many
quantum-mechanical and non-Markovian effects accompanying the passage through
the potential barrier are taken into consideration in our formalism EPJSub ;
our ; VAZ . The details of used formalism are presented in our previous
articles EPJSub ; EPJSub1 . All parameters of the model are set as in Ref.
EPJSub . All calculated results are obtained with the same set of parameters
and are rather insensitive to the reasonable variation of them EPJSub ;
EPJSub1 . The heights of the calculated Coulomb barriers $V_{b}=V(R_{b})$
($R_{b}$ is the position of the Coulomb barrier) are adjusted to the
experimental data for the fusion or capture cross sections. To calculate the
nucleus-nucleus interaction potential $V(R)$, we use the procedure presented
in Refs. EPJSub ; EPJSub1 . For the nuclear part of the nucleus-nucleus
potential, the double-folding formalism with the Skyrme-type density-dependent
effective nucleon-nucleon interaction is used.
To analyze the experimental date on fusion cross section, it is useful to use
the so called universal fusion function (UFF) $F_{0}$ GomesUFF . The
advantages of UFF appear clearly when one wants to compare fusion cross
sections for systems with quite different Coulomb barrier heights and
positions. In the reactions where the capture and fusion cross sections
coincide, the comparison of experimental cross sections with the UFF allows us
to make conclusions about the role of deformation of colliding nuclei and the
nucleon transfer between interacting nuclei in the capture cross section
because the UFF (the consequence of the Wong’s formula) does not contain these
effects. In Ref. GomesUFF a reduction procedure was proposed to eliminate the
influence of the nucleus-nucleus potential on the fusion cross section. It
consists of the following transformations:
$E_{\rm c.m.}\rightarrow x=\dfrac{E_{\rm
c.m.}-V_{b}}{\hbar\omega},\qquad\sigma^{exp}\rightarrow F(x)=\dfrac{2E_{\rm
c.m.}}{\hbar\omega R_{b}^{2}}\sigma^{exp}.$
The frequency $\omega=\sqrt{V^{{}^{\prime\prime}}(R_{b})/\mu}$ is related with
the second derivative $V^{{}^{\prime\prime}}(R_{b})$ of the total nucleus-
nucleus potential $V(R)$ (the Coulomb + nuclear parts) at the barrier radius
$R_{b}$ and the reduced mass parameter $\mu$. With these replacements one can
compare the experimental data for different reactions. After these
transformations, the reduced calculated fusion cross section takes the simple
form
$F_{0}=\ln[1+\exp(2\pi x)].$
To take into consideration the deviation of the real potential from the
inverted oscillator, we modify the reduction procedure as follows:
$E_{\rm c.m.}\rightarrow x=S/(\hbar\pi),$ $\qquad\sigma^{exp}\rightarrow
F(x)=\dfrac{2SE_{\rm c.m.}}{\hbar\pi R_{b}^{2}(V_{b}-E_{\rm
c.m.})}\sigma^{exp}.$
In this case
$F_{0}=\ln[1+\exp(-2S/\hbar)],$
where $S(E_{\rm c.m.})$ is the classical action. At energies above the Coulomb
barrier, we have $S=\pi(V_{b}-E_{\rm c.m.})/\omega$.
## III Results of calculations
### III.1 Effect of quadrupole deformation
In Fig. 1 (upper part), one can see the comparisons of dependencies $F$ and
$F_{0}$ on $S/(\hbar\pi)$ for some reactions considered in present paper. As
expected, at sub-barrier energies the deviation from the UFF is larger in the
case of reactions with strongly deformed target-nuclei and large factor
$Z_{1}\times Z_{2}$ (16O,40Ar,48Ca+ 154Sm, 74Ge + 74Ge). For the reactions
16O,40Ar+144Sm with spherical targets the experimental cross sections are
rather close to the UFF.
Figure 1: Comparison of modified UFF $F_{0}$ with the experimental values of
$\dfrac{2E_{\rm c.m.}S(E_{\rm c.m.})}{\hbar\pi R_{b}^{2}(V_{b}-E_{\rm
c.m.})}\sigma^{exp}$ for the indicated reactions. The experimental data for
$\sigma^{exp}$ are from Refs. KnyazevaCa48Sm154 ; LeighO16Sm ; DiGregO16Sm ;
BeckermanGe74Ge74 ; Reisd40ArSmSn ; TimmersCa40Zr96 ; 4048Ca208Pb ; 48CaPb208
.
To separate the effects of deformation and neutron transfer, firstly we
consider the reactions with deformed nuclei in which $Q$-value for the neutron
transfer are small, i.e. the neutron transfers can be disregarded. In Figs. 2
and 3, the calculated capture cross sections for the reactions
16O,48Ca,40Ar+154Sm, and 74Ge+74Ge are in a good agreement with the available
experimental data KnyazevaCa48Sm154 ; LeighO16Sm ; BeckermanGe74Ge74 ;
Reisd40ArSmSn showing that the quadrupole deformations of the interacting
nuclei are the main reasons for the enhancement of the capture cross section
at sub-barrier energies. The quadrupole deformation parameters $\beta_{2}$ are
taken from Ref. Ram for the deformed even-even nuclei. In Ref. Ram the
quadropole deformation parameters $\beta_{2}$ for the first excited $2^{+}$
states of nuclei are given. For the nuclei deformed in the ground state, the
$\beta_{2}$ in $2^{+}$ state is similar to the $\beta_{2}$ in the ground state
and we use $\beta_{2}$ from Ref. Ram in the calculations. For double magic
nuclei, in the ground state we take $\beta_{2}=0$. In Ref. GomesRec the
experimentally observed enhancement of sub-barrier fusion for the reactions
16O,48Ca+154Sm, and 74Ge+74Ge was explained by the nucleon transfer and neck
formation effects. However, in the present article we demonstrate that a good
agreement with the experimental data at sub-barrier energies could be reached
taking only the quadrupole deformations of interacting nuclei into
consideration.
Figure 2: The calculated capture cross sections versus $E_{\rm c.m.}$ for the
indicated reactions 16O,48Ca + 154Sm (solid lines), and 16O + 144Sm (dashed
line). The experimental data (symbols) are from Refs. KnyazevaCa48Sm154 ;
LeighO16Sm ; DiGregO16Sm . The following quadrupole deformation parameters are
used: $\beta_{2}$(154Sm)=0.341 Ram , $\beta_{2}$(144Sm)=0.05, and
$\beta_{2}$(16O)=$\beta_{2}$(48Ca)=0. Figure 3: The same as Fig. 2, for the
indicated reactions 74Ge+74Ge, 40Ar + 154Sm (solid lines), and 40Ar + 144Sm
(dashed line). The experimental data (symbols) are from Ref. BeckermanGe74Ge74
; Reisd40ArSmSn . The following quadrupole deformation parameters are used:
$\beta_{2}$(40Ar)=0.25 Ram , $\beta_{2}$(74Ge)=0.2825 Ram ,
$\beta_{2}$(154Sm)=0.341 Ram , and $\beta_{2}$(144Sm)=0.05. Figure 4: The
same as Fig. 2, for the indicated reactions 28Si+94Zr,154Sm (solid lines), and
28Si + 90Zr,144Sm (dashed lines). The experimental data (symbols) are from
Refs. Kalkal28SiZr9490 ; GilSi28154sm ; NobreSi28144sm . The following
quadrupole deformation parameters are used: $\beta_{2}$(154Sm)=0.341 Ram ,
$\beta_{2}$(144Sm)=0.05, and $\beta_{2}$(28Si)=0.3. Figure 5: The same as
Fig. 2, for the indicated reactions 40Ar + 112,122Sn (solid lines), and 40Ar +
116Sn (dashed line). The experimental data (symbols) are from Ref.
Reisd40ArSmSn . The following quadrupole deformation parameters are used:
$\beta_{2}$(112Sn)=0.1227 Ram , $\beta_{2}$(116Sn)=0.1118 Ram ,
$\beta_{2}$(122Sn)=0.1036 Ram , and $\beta_{2}$(40Ar)=0.25 Ram . Figure 6:
The same as Fig. 2, for the indicated reactions 36,32S + 90Zr (solid lines),
and 36,32S + 96Zr (dashed lines). The experimental data (symbols) are from
Refs. ZhangS32Zn9096 ; StefaniniS36Zn9096 . The following quadrupole
deformation parameters are used: $\beta_{2}$(32S)=0.312 Ram ,
$\beta_{2}$(34S)=0.252 Ram , $\beta_{2}$(96Zr)=0.08, and
$\beta_{2}$(36S)=$\beta_{2}$(90Zr)=0. Figure 7: The same as Fig. 2, for the
indicated reactions 34S + 168Er and 64Ni + 132Sn. The experimental data
(symbols) are from Refs. Morton34S168Er ; LiangNi64Sn132 . The following
quadrupole deformation parameters are used: $\beta_{2}$(168Er)=0.3381 Ram ,
$\beta_{2}$(66Ni)=0.158 Ram , $\beta_{2}$(130Sn)=0, and
$\beta_{2}$(34S)=0.125. Figure 8: The same as Fig. 2, for the indicated
reactions 64Ni + 100Mo,150Nd (solid lines) and 60Ni + 100Mo,150Nd (dashed
lines). The experimental data (symbols) for the 64Ni + 100Mo reaction are from
Ref. Jiang64Ni100Mo . The following quadrupole deformation parameters are
used: $\beta_{2}$(62Ni)=0.1978 Ram , $\beta_{2}$(98Mo)=0.1684 Ram ,
$\beta_{2}$(100Mo)=0.2309 Ram , $\beta_{2}$(148Nd)=0.2036 Ram ,
$\beta_{2}$(150Nd)=0.2848 Ram , and $\beta_{2}$(64Ni)=0.087. Figure 9: The
same as Fig. 2, for the indicated reactions 40Ca + 96Zr,208Pb (dashed lines),
40Ca + 90Zr (solid line), and 48Ca + 208Pb (solid line and open squares and
triangles). For the reactions 40Ca + 96Zr,208Pb, the calculated capture cross
sections without taking into consideration the neutron transfer process are
shown by dotted lines. The experimental data (symbols) are from Refs.
TimmersCa40Zr96 ; 4048Ca208Pb ; 48CaPb208 . The following quadrupole
deformation parameters are used: $\beta_{2}$(42Ca)=0.247 Ram ,
$\beta_{2}$(94Zr)=0.09 Ram , $\beta_{2}$(96Zr)=0.08, and
$\beta_{2}$(40Ca)=$\beta_{2}$(48Ca)=$\beta_{2}$(90Zr)=$\beta_{2}$(206,208Pb)=0.
We should mention, that for the sub-barrier energies the results of
calculations are very sensitive to the quadrupole deformation parameters
$\beta_{2}$ of the interacting nuclei. Since there are uncertainties in the
definition of the values of $\beta_{2}$ in the light- and the medium-mass
nuclei, one can extract the quadrupole deformation parameters of these nuclei
from the comparison of the calculated capture cross sections with the
experimental data. The best case is when the projectile or target is the
spherical double magic nucleus and there are no neutron transfer channels with
large positive $Q$-values. In this way by describing the reactions 28Si +
90Zr,144Sm, 34S + 168Er, 36S + 90,96Zr, 40Ar + 112,116,122Sn,144Sm, 58Ni +
58Ni, 64Ni + 100Mo,74Ge (Figs. 5–10), we extract the following values of the
quadrupole deformation parameter $\beta_{2}$=0.30, 0.125, 0, 0.25, 0.05,
0.087, 0, 0.08, 0.12, 0.11, 0.1, and 0.05 for the nuclei 28Si, 34S, 36S, 40Ar,
58Ni, 64Ni, 90Zr, 96Zr, 112Sn, 116Sn, 122Sn, and 144Sm, respectively. Note
that almost the same values of quadrupole deformations parameters of nuclei in
the ground state were predicted within the mean-field and the macroscopic-
microscopic models Pet . For 40Ar, 96Zr, 112Sn, 116Sn, and 122Sn the extracted
$\beta_{2}$ for are equal to the experimental ones from Ref. Ram . These
extracted deformation parameters we use in calculations in next subsection.
Note that almost the same values of quadrupole deformations parameters of
nuclei in the ground state were predicted within the mean-field and the
macroscopic-microscopic models Pet . For 40Ar, 96Zr, 112Sn, 116Sn, and 122Sn
the extracted $\beta_{2}$ for are equal to the experimental ones from Ref. Ram
. These extracted deformation parameters we use in calculations in next
subsection.
### III.2 Effect of neutron transfer
Several experiments were performed to understand the effect of neutron
transfer in the fusion (capture) reactions.
Figure 10: The same as Fig. 2, for the indicated reactions 58Ni + 64Ni,74Ge
(dashed lines) and 58Ni + 58Ni, 64Ni + 74Ge (solid lines). The experimental
data (symbols) are from Ref. Beckerman58Ni5864Ni74Ge . The following
quadrupole deformation parameters are used: $\beta_{2}$(60Ni)=0.207 Ram ,
$\beta_{2}$(72Ge)=0.2424 Ram , $\beta_{2}$(74Ge)=0.2825 Ram ,
$\beta_{2}$(58Ni)=0.05, and $\beta_{2}$(62Ni)$\approx$
$\beta_{2}$(64Ni)=0.087. Figure 11: The same as Fig. 2, for the indicated
reactions 40Ca + 94Zr (solid line), 32S + 96Zr (dashed line and solid
squares), and 36S + 96Zr (solid line and open squares). For the 40Ca + 94Zr
reaction, the calculated capture cross sections without taking into
consideration the neutron transfer process are shown by dotted line. The
experimental data (symbols) are from Refs. ZhangS32Zn9096 ; StefaniniS36Zn9096
; StefaniniCa40Zn94 . The following quadrupole deformation parameters are
used: $\beta_{2}$(42Ca)=0.247 Ram , $\beta_{2}$(94Zr)=0.09 Ram ,
$\beta_{2}$(92Zr)=0.1028 Ram , $\beta_{2}$(96Zr)=0.08, and
$\beta_{2}$(36S)=$\beta_{2}$(40Ca)=0. Figure 12: The same as Fig. 2, for the
indicated reactions 40Ca + 192Os,194Pt (solid lines). The calculated capture
cross sections without taking into consideration the neutron transfer process
are shown by dotted lines. The experimental data (symbols) are from Ref.
Bierman40ca192Os194Pt . The following quadrupole deformation parameters are
used: $\beta_{2}$(42Ca)=0.247 Ram , $\beta_{2}$(192Os)=0.1667 Ram ,
$\beta_{2}$(190Os)=0.1775 Ram , $\beta_{2}$(194Pt)=0.1426 Ram ,
$\beta_{2}$(192Pt)=0.1532 Ram , and $\beta_{2}$(40Ca)=0. Figure 13: The same
as Fig. 2, for the indicated reactions 40Ca + 48Ca,116Sn (solid lines), and
40Ca + 124Sn (dashed line). The experimental data (symbols) are from Refs.
Aljuwair40Ca48Ca ; trotta40ca48ca ; Stefanini40ca116124sn . The following
quadrupole deformation parameters are used: $\beta_{2}$(42Ca)=0.247 Ram ,
$\beta_{2}$(116Sn)=0.1118 Ram , $\beta_{2}$(122Sn)=0.1036 Ram , and
$\beta_{2}$(46Ca)=$\beta_{2}$(40Ca)=0.
The choice of the projectile-target combination is crucial, and for the
systems studied one can make unambiguous statements regarding the neutron
transfer process with a positive $Q$-value when the interacting nuclei are
double magic or semi-magic spherical nuclei. In this case one can disregard
the strong nuclear deformation effects. The good examples are the reactions
with the spherical nuclei: 40Ca + 208Pb ($Q_{2n}$=5.7 MeV) and 40Ca + 96Zr
($Q_{2n}$=5.5 MeV). In Fig. 1 (lower part), one can see that the reduced
capture cross sections in these reactions strongly deviate from the UFF in
contrast to those in the reactions 48Ca + 208Pb and 48Ca + 96Zr, where the
neutron transfer channels are suppressed (the negative $Q$-values). Since the
transfer of protons is shielded by the Coulomb barrier, it occurs when two
nuclei almost touch each other obzor , i.e. after a capture. Thus, the proton
transfer can be disregarded in the calculations of capture cross sections.
Following the hypothesis of Ref. Broglia , we assume that the sub-barrier
capture mainly depends on the two-neutron transfer with the positive and
relatively large $Q$-value. Our assumption is that, before the projectile is
captured by target-nucleus (before the crossing of the Coulomb barrier) which
is the slow process, the two-neutron transfer occurs at larger separations
that can lead to the population of the first 2+ state in the recipient nucleus
SSzilner .
Figure 14: The same as Fig. 2, for the indicated reactions 32S + 110Pd (dashed
line and closed squares) and 36S + 110Pd (solid line and open squares). The
experimental data (symbols) are from Ref. Stefanini3236s110pd . The following
quadrupole deformation parameters are used: $\beta_{2}$(34S)=0.252 Ram ,
$\beta_{2}$(108Pd)=0.243 Ram , $\beta_{2}$(110Pd)=0.257 Ram , and
$\beta_{2}$(36S)=0. Figure 15: The same as Fig. 2, for the indicated
reactions 32S + 154Sm,208Pb. The experimental data (symbols) are from Refs.
Stefanini3236s110pd . The following quadrupole deformation parameters are
used: $\beta_{2}$(34S)=0.252 Ram , $\beta_{2}$(152Sm)=0.3064 Ram , and
$\beta_{2}$(206Pb)=0. Figure 16: The same as Fig. 2, for the indicated
reactions 28Si + 142Ce,208Pb (solid lines), and 28Si + 198Pt (dashed line).
The experimental data (symbols) are from Refs. GilSi28Ce142 ; NishioSi28Pt198
; HindeSi28Pb208 . The following quadrupole deformation parameters are used:
$\beta_{2}$(30Si)=0.315 Ram , $\beta_{2}$(140Ce)=0.1012 Ram ,
$\beta_{2}$(196Pt)=0.1296 Ram , and $\beta_{2}$(206Pb)=0.
Since after two-neutron transfer the mass numbers, the deformation parameters
of interacting nuclei, and, respectively, the height and shape of the Coulomb
barrier are changed, one can expect the enhancement or suppression of the
capture. For example, after the neutron transfer in the reaction
40Ca($\beta_{2}=0$) + 208Pb($\beta_{2}=0$)$\to^{42}$Ca($\beta_{2}=0.247$) +
206Pb($\beta_{2}=0$) (40Ca($\beta_{2}=0$) +
96Zr($\beta_{2}=0.08$)$\to^{42}$Ca($\beta_{2}=0.247$) +
94Zr($\beta_{2}=0.09$)) the deformation of the nuclei increases and the mass
asymmetry of the system decreases and thus the value of the Coulomb barrier
decreases and the capture cross section becomes larger (Fig. 10).
Figure 17: The same as Fig. 2, for the indicated reactions 58Ni + 207Pb
(dashed line), 64Ni + 64Ni (solid line), and 64Ni + 207Pb (solid line). For
the 58Ni + 207Pb reaction, the calculated capture cross sections without
taking into consideration the neutron transfer process are shown by dotted
line. The experimental data (symbols) are from Refs. Beckerman58Ni5864Ni74Ge ;
Jiang64Ni64Ni . The following quadrupole deformation parameters are used:
$\beta_{2}$(60Ni)=0.207 Ram , $\beta_{2}$(58Ni)=0.05, $\beta_{2}$(64Ni)=0.087,
and $\beta_{2}$(205,207Pb)=0.
We observe the same behavior in the reactions 64Ni + 132Sn (Fig. 7),
58Ni+64Ni,74Ge (Fig. 9), 32S+96Zr, 40Ca+94Zr (Fig. 11), 40Ca+192Os,198Pt (Fig.
12), and 40Ca + 48Ca,116,124Sn (Fig. 13). One can see a good agreement between
the calculated results and the experimental data. For some reactions at
energies above the Coulomb barrier, the small deviation between the calculated
results and experimental data probably arises from the fact that the fusion-
fission channel was not taken into consideration in the experimental capture
cross sections. So, our results show that the observed capture enhancement at
sub-barrier energies for the reactions mentioned above is related to the two-
neutron transfer channel. For these reactions there is a large deflection from
the UFF (see lower part of Fig. 1). Note that strong population of the yrast
states, and in particular of the first 2+ state of even Ar (Ca) isotopes via
the neutron pick-up channels in the 40Ar + 208Pb (40Ca + 96Zr) reaction is
experimentally found in Ref. SSzilner . In the calculations, for such excited
recipient nuclei we use the experimental deformation parameters $\beta_{2}$
related to the first 2+ states from the table of Ref. Ram . We assume that
after two neutron transfer the residues of donor nuclei remain in the ground
state with corresponding quadrupole deformation.
One can find the reactions with large positive two-neutron transfer $Q$-values
where the transfer weakly influences or even suppresses the capture process.
This happens if after transfer the deformations of nuclei almost do not change
or even decrease. For instance, in the reactions 32S($\beta_{2}=0.312$) +
96Zr($\beta_{2}=0.08$)$\to^{34}$S($\beta_{2}=0.252$) + 94Zr($\beta_{2}=0.09$),
60Ni($0.05<\beta_{2}\lesssim 0.1$) +
100Mo($\beta_{2}=0.231$)$\to^{62}$Ni($\beta_{2}=0.198$) +
98Mo($\beta_{2}=0.168$) and 60Ni($0.05<\beta_{2}\lesssim 0.1$) +
150Nd($\beta_{2}=0.285$)$\to^{62}$Ni($\beta_{2}=0.198$) +
148Nd($\beta_{2}=0.204$) one can expect weak dependence of the capture cross
section on the neutron transfer (Figs. 8 and 11). There is the experimental
indication of such effect for the 60Ni + 100Mo reaction Scarlassara . The weak
influence of neutron transfer on the capture process is also found in the
reactions 32S + 110Pd ,154Sm,208Pb (Figs. 14 and 15), 28Si +
94Zr,142Ce,154Sm,208Pb (Figs. 4 and 16). The same behaviour is expected in the
reactions 84Kr + 138Ce,140Nd. For these reactions, the effect of quadrupole
deformations of interacting nuclei is much stronger than the effect of neutron
transfer between the interacting nuclei.
Note that our model predicts almost the same capture cross sections for the
reactions with positive $Q$-values 6He,9Li,11Be + 206Pb, 18O + 58Ni and for
the reactions without neutron transfer 4He,7Li,9Be + 208Pb, 16O + 60Ni,
respectively.
In Fig. 17, the capture cross sections for the reactions 58,64Ni + 207Pb are
predicted. As seen, there is considerable difference between the capture cross
sections in these two reactions because of the existence of the two-neutron
transfer channel ($Q_{2n}$=5.6 MeV) in the reaction 58Ni + 207Pb$\to^{60}$Ni +
205Pb. Thus, the study of these reactions could be a good test for the
conclusion about the effect of neutron transfer. It will be interesting to
compare the role of the neutron transfer channel in the reactions with
spherical nuclei mentioned above (Fig. 10) and with deformed targets, 40Ca +
154Sm,238U (Fig. 18).
Due to a change of the regime of interaction (the turning-off of the nuclear
forces and friction) at sub-barrier energies EPJSub ; EPJSub1 ; Conf , the
curve related to the capture cross section as a function of bombarding energy
has smaller slope (see Figs. 2–8,10,11,13–16). This effect is more visible in
the capture of spherical nuclei without the neutron transfer. However, the
present experimental data at strongly sub-barrier energies are rather poor.
## IV Origin of fusion hindrance in reactions with medium-mass nuclei at deep
sub-barrier energies
Since the sum of the fusion cross section $\sigma_{fus}$ and the quasifission
cross section $\sigma_{qf}$ gives the capture cross section
$\sigma_{cap}=\sigma_{fus}+\sigma_{qf},$
one can estimate the relative contributions of $\sigma_{fus}$ and
$\sigma_{qf}$ to $\sigma_{cap}$. In Figs. 17, 13 and 19 the calculated capture
cross section are presented for the reactions 40Ca + 48Ca, 64Ni + 64Ni and 36S
+ 48Ca,64Ni.
Figure 18: The same as Fig. 2, for the indicated reactions 40Ca + 154Sm,238U
(dashed lines), and 48Ca + 154Sm,238U (solid lines). For the reactions 40Ca +
154Sm,238U, the calculated capture cross sections without taking into
consideration the neutron transfer process are shown by dotted line. The
experimental data (symbols) for the reactions 48Ca + 154Sm,238U are from Refs.
KnyazevaCa48Sm154 ; Shen . The following quadrupole deformation parameters are
used: $\beta_{2}$(42Ca)=0.247 Ram , $\beta_{2}$(152Sm)=0.3055 Ram ,
$\beta_{2}$(154Sm)=0.341 Ram , $\beta_{2}$(236U)=0.2821 Ram ,
$\beta_{2}$(238U)=0.2863 Ram , and $\beta_{2}$(48Ca)=0. Figure 19: The same
as Fig. 2, for the indicated reactions 36S + 48Ca,64Ni. The experimental data
(symbols) are from Refs. StefaniniS36Ca48 ; MontagnoliS36Ni64 . The following
quadrupole deformation parameters are used: $\beta_{2}$(64Ni)=0.087 and
$\beta_{2}$(36S)=$\beta_{2}$(48Ca)=0.
As seen, at energies above and just below the Coulomb barriers
$\sigma_{cap}=\sigma_{fus}$. The difference between the sub-barrier capture
and fusion cross sections becomes larger with decreasing bombarding energy
$E_{\rm c.m.}$. The same effect one can see for the 16O + 208Pb reaction
EPJSub . Assuming that the estimated capture and the measured fusion cross
sections are correct, the small fusion cross section at energies well below
the Coulomb barrier may indicate that other reaction channel is preferable and
the system goes to this channel after the capture. The observed hindrance
factor may be understood in term of quasifission whose cross section should be
added to the $\sigma_{fus}$ to obtain a meaningful comparison with the
calculated capture cross section. At deep sub-barrier energies, the
quasifission event corresponds to the formation of a nuclear-molecular state
or dinuclear system with small excitation energy that separates (in the
competition with the compound nucleus formation process) by the quantum
tunneling through the Coulomb barrier in a binary event with mass and charge
close to the entrance channel. In this sense the quasifission is the general
phenomenon which takes place in the reactions with the massive Volkov ; nasha
; Avaz ; GSI , medium-mass and, probably, light nuclei. For the medium-mass
and light nuclei, this reaction channel is expected to be at deep sub-barrier
energies and has to be studied in the future experiments: from the measurement
of the mass (charge) distribution in the collisions with total momentum
transfer one can show the distinct components due to the quasifission. Because
these energies the angular momentum $J<10$, the angular distribution would
have small anisotropy. The low-energy experimental data would probably provide
straight information since the high-energy data may be shaded by competing
nucleon transfer processes. Note that the binary decay events were already
observed experimentally in Ref. Wolfs for the 58Ni + 124Sn reaction at
energies below the Coulomb barrier but assumed to be related to deep-inelastic
scattering. At energies above the Coulomb barrier the hindrance to fusion was
revealed in Ref. Pollar for the reactions 58Ni + 124Sn and 16O + 208Pb.
## V Summary
The quantum diffusion approach was applied to study the capture process in the
reactions with deformed and spherical nuclei at sub-barrier energies. The
available experimental data at energies above and below the Coulomb barrier
are well described. As shown, the experimentally observed sub-barrier fusion
enhancement is mainly related to the quadrupole deformation of the colliding
nuclei and neutron transfer with large positive $Q$-value. The change of the
magnitude of the capture cross section after the neutron transfer occurs due
to the change of the deformations of nuclei. When after the neutron transfer
the deformations of nuclei do not change or slightly decrease, the neutron
transfer weakly influences or even suppresses the capture process. It would be
interesting to study such-type of reactions.
The importance of quasifission near the entrance channel was noticed for the
reactions with medium-mass nuclei at extreme sub-barrier energies. The
quasifission can explain the difference between the capture and fusion cross
sections. One can try to check experimentally these predictions.
## VI acknowledgements
We thank H. Jia, J.Q. Li, C.J. Lin, and S.-G. Zhou for fruitful discussions
and suggestions. This work was supported by DFG, NSFC, and RFBR. The
IN2P3(France)-JINR(Dubna), MTA(Hungary)-JINR(Dubna) and Polish - JINR(Dubna)
Cooperation Programmes are gratefully acknowledged.
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{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V. V. Sargsyan, G. G. Adamian, N. V. Antonenko, W. Scheid, and H. Q.\n Zhang",
"submitter": "Vazgen Sargsyan Dr.",
"url": "https://arxiv.org/abs/1110.3410"
}
|
1110.3456
|
# Dissipative dynamics of few-photons superposition states: A dynamical
invariant
Hong-Yan Wen Quantum Optoelectronics Laboratory, School of Physics and
Technology, Southwest Jiaotong University, Chengdu 610031, China Jing Cheng
Department of Physics, South China University of Technology, Guangzhou 510640,
China Y. Yang State Key Laboratory of Optoelectronic Materials and
Technologies, School of Physics and Engineering, Sun Yat-Sen University,
Guangzhou 510275, China L. F. Wei111weilianfu@gmail.com;
weilianf@mail.sysu.edu.cn Quantum Optoelectronics Laboratory, School of
Physics and Technology, Southwest Jiaotong University, Chengdu 610031, China
State Key Laboratory of Optoelectronic Materials and Technologies, School of
Physics and Engineering, Sun Yat-Sen University, Guangzhou 510275, China
###### Abstract
By numerically calculating the time-evolved Wigner functions, we investigate
the dynamics of a few-photon superposed (e.g., up to two ones) state in a
dissipating cavity. It is shown that, the negativity of the Wigner function of
the photonic state unquestionably vanishes with the cavity’s dissipation. As a
consequence, the nonclassical effects related to the negativity of the Wigner
function should be weakened gradually. However, it is found that the value of
the second-order correlation function $g^{(2)}(0)$ (which serves usually as
the standard criterion of a typical nonclassical effect, i.e., $g^{(2)}(0)<1$
implies that the photon is anti-bunching) is a dynamical invariant during the
dissipative process of the cavity. This feature is also proven analytically
and suggests that $g^{(2)}(0)$ might not be a good physical parameter to
describe the photonic decays. Alternatively, we find that the anti-normal-
order correlation function $g^{(2A)}(0)$ changes with the cavity’s dissipation
and thus is more suitable to describe the dissipative-dependent cavity.
Finally, we propose an experimental approach to test the above arguments with
a practically-existing cavity QED system.
PACS: 42.50.Ar 03.65.Yz, 42.50.Xa
## I Introduction
It is well-known that the Wigner function, introduced 70 years ago by Wigner
to describe the quasi-probability distribution of a quantum particle in its
phase space, is a very popular tool to study the statistical properties of
various quantum states [1]. Basically, once the Wigner function has been
determined, all the knowable information on the quantum state (such as its
nonclassical statistical properties) can be extracted [2-5]. Typically,
differing from the standard probability distribution, such a quasi-probability
distribution can be assigned by a negative value [6]. Therefore, a quantum
state with the negative Wigner function should be nonclassical and thus
certain nonclassical effects (such as the photon anti-buchings) [7-11]
demonstrate. This indicates that, determining the Wigner function of a
selected quantum state plays an important role both fundamentally and
practically in quantum-state engineerings.
Usually, any selected quantum system is always surrounded by the classical
environments. Thus, dissipation of the artificially-prepared quantum sate is
one of the central topics in quantum coherence science. Roughly, due to the
existence of various dissipations and fluctuations from the environments, any
excited quantum state will decay to the ground state and the relevant system
finally becomes classical. Under the standard logic, people pay the most
attention to calculate either decoherence or decay time of a superposition
quantum state, rather than cares on the process of the decoherence or decay
[12-15]. Alternatively, in the present work we investigate exactly the
dissipative dynamics for a prepared quantum state by calculating its
dissipative-dependent Wigner function. Our discussions are based on the
typical few-photon quantum state in a cavity, but can be directly generalized
to other quantum systems such as qubits and qutrits.
The paper is organized as: in Sec. 2, we describe how the Wigner function for
a few-photon superposed state changes with the cavity’s dissipation. Our
numerical results show naturally that the negativity of the Wigner function
weakens gradually with the dissipation and the final state of the cavity
should be “classical” with positive Wigner function. With the calculated
Wigner function we investigate how the nonclassical properties, such as the
anti-bunching effect of photons, changes with the cavity dissipation. It is
surprised that the value of the second-order correlation function $g^{(2)}(0)$
(which serves usually as the standard criterion of a nonclassical effect,
i.e., $g^{(2)}(0)<1$ implies that the photon is anti-bunching) is an invariant
during the dissipative process of the cavity. We prove such an argument
analytically by directly solving the relevant master equation and suggests
that $g^{(2)}(0)$ is not a good parameter to describe dissipative-dependent
non-classicality of the photonic decays. Alternatively, we find that the anti-
normal-order correlation function $g^{(2A)}(0)$ changes with the cavity’s
dissipation and thus could be more suitable to describe the dissipative-
dependent cavity. With an experimentally-demonstrated cavity QED system we
propose an approach to test our results, including how to prepare the
investigated few-photon superposed state of the cavity and measure its Wigner
function. Finally, our conclusions and discussions are given in Sec. 4.
## II Dissipative dynamics of Wigner functions for few-photons superposition
states
Generally, the quasi-probability distribution $W(\alpha,\alpha^{*})$ can be
defined by the Fourier transform of the symmetrical-ordered characteristic
function $C(\lambda,\lambda^{*})$ [16], i.e.,
$W(\alpha,\alpha^{*})=\frac{1}{\pi^{2}}\int
d^{2}\lambda\,\,C(\lambda,\lambda^{*})e^{\alpha\lambda^{*}-\alpha^{*}\lambda},$
(1)
with $\lambda$ and $\alpha$ being the complex parameters in phase space. The
expression of the symmetrical-ordered characteristic function is defined as
$C(\lambda,\lambda^{*})=Tr[\rho
e^{\lambda\hat{a}^{\dagger}-\hat{a}\lambda^{*}}],$ (2)
where $\rho$ is the density matrix of the cavity state $|\psi\rangle$, and
$\hat{a}$ and $\hat{a}^{\dagger}$ the usual annihilation and creation
operators of the photons, respectively.
For the simplicity and without loss of the generality, let us assume that the
cavity is initially prepared in the following few-photon superposition state
$|\psi(0)\rangle=C_{0}|0\rangle+C_{1}|1\rangle+C_{2}|2\rangle,$ (3)
with the complex amplitudes: $C_{0}=|C_{0}|e^{i\phi},C_{1}=|C_{1}|$, and
$C_{2}=|C_{2}|e^{i\varphi}$. Then, with the matrix elements of Wigner
operator: $\vartriangle(\alpha,\alpha^{*})=\int
d^{2}ze^{[z(\hat{a}^{\dagger}-\alpha^{*})-z^{*}(\hat{a}-\alpha)]}/2\pi^{2}$,
in the Fock representation [17]
$\langle
n|\vartriangle(\alpha,\alpha^{*})|m\rangle=\frac{(-1)^{m}}{\pi}\sqrt{\frac{m!}{n!}}(2\alpha)^{n-m}e^{(-2|\alpha|^{2})}L^{(n-m)}_{m}(4|\alpha|^{2}),\,n,m=0,1,2,...$
(4)
here $n>m,\alpha_{0}=|\alpha_{0}|e^{i\theta}$, one can easily obtain the
Wigner function of the initial state
$\displaystyle W(\alpha_{0},\alpha^{*}_{0},0)$ $\displaystyle=$
$\displaystyle\frac{2}{\pi}[|C_{0}|^{2}-|C_{1}|^{2}L^{0}_{1}(4|\alpha_{0}|^{2})+|C_{2}|^{2}L^{0}_{2}(4|\alpha_{0}|^{2})]e^{(-2|\alpha_{0}|^{2})}$
(5) $\displaystyle+$
$\displaystyle\frac{8\sqrt{2}}{\pi}e^{(-2|\alpha_{0}|^{2})}|C_{0}C_{2}||\alpha_{0}|^{2}\cos(2\theta-\varphi+\phi)$
$\displaystyle-$
$\displaystyle\frac{4\sqrt{2}}{\pi}e^{(-2|\alpha_{0}|^{2})}|C_{1}C_{2}||\alpha_{0}|\cos(\theta-\varphi)L^{1}_{1}(4|\alpha_{0}|^{2})$
$\displaystyle+$
$\displaystyle\frac{8}{\pi}e^{(-2|\alpha_{0}|^{2})}|C_{0}C_{1}||\alpha_{0}|\cos(\theta+\phi).$
for the above superposition initial state. Above, $L^{J}_{n}(x)$ is an
associated Laguerre polynomial defined by [18]
$L^{(J)}_{n}(x)=\sum\limits_{\kappa=0}^{n}(-1)^{\kappa}\frac{(n+J)!}{(n-\kappa)!(J+\kappa)!}\frac{x^{\kappa}}{\kappa!}.$
(6)
In what follows we discuss how such a state decay in a loss cavity by
investigating the time-evolutions of the above initial Wigner function.
### II.1 Dissipative dynamics for the Wigner function
We now consider how the above few-photons superposition state dissipates in a
loss cavity without any thermal photon (i.e., $\langle n\rangle_{\rm
th}=1/[\exp(\hbar\nu/k_{B}T)-1]\rightarrow 0$, for the present optical
frequency photons and at the room temperature: $\hbar\nu/k_{B}T\gg 1$), which
is described simply by the following master equation [19-20]
$\frac{d\rho}{dt}=-\kappa(\hat{a}^{\dagger}\hat{a}\rho+\rho\hat{a}^{\dagger}\hat{a}-2\hat{a}\rho\hat{a}^{\dagger}),$
(7)
with $k$ being the loss coefficient. Our discussion is based on the time-
evolutions of the Wigner function, i.e.,
$\frac{d}{dt}W(\alpha,\alpha^{*})=\frac{1}{\pi^{2}}\int
d^{2}\lambda\,\,\frac{d\,C(\lambda,\lambda^{*})}{dt}e^{\alpha\lambda^{*}-\alpha^{*}\lambda},$
(8)
with
$\frac{d\,C(\lambda,\lambda^{*})}{dt}=Tr[\frac{d\rho}{dt}e^{\lambda\hat{a}^{\dagger}-\hat{a}\lambda^{*}}]=\kappa
Tr[(2\hat{a}\rho\hat{a}^{\dagger}-\hat{a}^{\dagger}\hat{a}\rho-\rho\hat{a}^{\dagger}\hat{a})e^{\lambda\hat{a}^{\dagger}-\hat{a}\lambda^{*}}].$
(9)
Formally, Eq. (8) can be rewritten as
$\displaystyle\frac{d}{dt}W(\alpha,\alpha^{*})=2\kappa
W^{[\hat{a}\rho\hat{a}^{\dagger}]}(\alpha,\alpha^{*})-\kappa
W^{[\hat{a}^{\dagger}\hat{a}\rho]}(\alpha,\alpha^{*})-\kappa
W^{[\rho\hat{a}^{\dagger}\hat{a}]}(\alpha,\alpha^{*}),$ (10)
where the symbol $W^{[x]}(\alpha,\alpha^{*})$ is defined as
$\displaystyle W^{[x]}(\alpha,\alpha^{*})$ $\displaystyle=$
$\displaystyle\frac{1}{\pi^{2}}\int
d^{2}\lambda\,\,C^{[x]}(\lambda,\lambda^{*},t)e^{\alpha\lambda^{*}-\alpha^{*}\lambda},\,\,C^{[x]}(\lambda,\lambda^{*})=Tr[x\,e^{\lambda\hat{a}^{\dagger}-\hat{a}\lambda^{*}}],$
(11)
with $W^{[\rho]}(\alpha,\alpha^{*})=W(\alpha,\alpha^{*})$, and
$C^{[\rho]}(\lambda,\lambda^{*})=C(\lambda,\lambda^{*})$. Note that
$\displaystyle C^{[\rho\hat{a}^{\dagger}\hat{a}]}(\lambda,\lambda^{*})$
$\displaystyle=$
$\displaystyle[\frac{1}{2}+\frac{\partial}{\partial\lambda}(-\frac{\partial}{\partial\lambda^{*}})]C(\lambda,\lambda^{*})=(\frac{\partial}{\partial\lambda}+\frac{\lambda^{*}}{2})(\frac{\lambda}{2}-\frac{\partial}{\partial\lambda^{*}})C(\lambda,\lambda^{*}),$
(12)
and
$\displaystyle\frac{1}{\pi^{2}}\int
d^{2}\lambda\,\,e^{(\alpha\lambda^{*}-\alpha^{*}\lambda)}\frac{\partial}{\partial\lambda}C(\lambda,\lambda^{*})$
$\displaystyle=$ $\displaystyle\alpha^{*}W(\alpha,\alpha^{*}),$
$\displaystyle\frac{1}{\pi^{2}}\int
d^{2}\lambda\,\,e^{(\alpha\lambda^{*}-\alpha^{*}\lambda)}\frac{\partial}{\partial\lambda^{*}}C(\lambda,\lambda^{*})$
$\displaystyle=$ $\displaystyle-\alpha W(\alpha,\alpha^{*}),$
$\displaystyle\frac{1}{\pi^{2}}\int
d^{2}\lambda\,\,e^{(\alpha\lambda^{*}-\alpha^{*}\lambda)}\lambda^{*}C(\lambda,\lambda^{*})$
$\displaystyle=$
$\displaystyle\frac{\partial}{\partial\alpha}W(\alpha,\alpha^{*}),$
$\displaystyle\frac{1}{\pi^{2}}\int
d^{2}\lambda\,\,(-\lambda)e^{(\alpha\lambda^{*}-\alpha^{*}\lambda)}C(\lambda,\lambda^{*})$
$\displaystyle=$
$\displaystyle\frac{\partial}{\partial\alpha^{*}}W(\alpha,\alpha^{*}),$ (13)
we then have
$\displaystyle W^{[\hat{a}\rho\hat{a}^{\dagger}]}(\alpha,\alpha^{*})$
$\displaystyle=$ $\displaystyle\frac{1}{\pi^{2}}\int
d^{2}\lambda\,\,C^{[\hat{a}\rho\hat{a}^{\dagger}]}(\lambda,\lambda^{*})e^{\alpha\lambda^{*}-\alpha^{*}\lambda}$
(14) $\displaystyle=$ $\displaystyle\frac{1}{\pi^{2}}\int
d^{2}\lambda\,\,[\alpha\alpha^{*}+\frac{1}{2}-\alpha^{*}\frac{\partial}{\partial\alpha^{*}}-\frac{1}{4}\frac{\partial}{\partial\alpha}\frac{\partial}{\partial\alpha^{*}}+\frac{\alpha}{2}\frac{\partial}{\partial\alpha}]C(\lambda,\lambda^{*})e^{\alpha\lambda^{*}-\alpha^{*}\lambda}$
$\displaystyle=$ $\displaystyle\frac{1}{\pi^{2}}\int
d^{2}\lambda\,\,[\alpha^{*}+\frac{1}{2}\frac{\partial}{\partial\alpha}][\alpha-\frac{1}{2}\frac{\partial}{\partial\alpha^{*}}]C(\lambda,\lambda^{*})e^{\alpha\lambda^{*}-\alpha^{*}\lambda}$
$\displaystyle=$
$\displaystyle[\alpha^{*}+\frac{1}{2}\frac{\partial}{\partial\alpha}][\alpha-\frac{1}{2}\frac{\partial}{\partial\alpha^{*}}]W(\alpha,\alpha^{*}).$
Similarly,
$\displaystyle W^{[\hat{a}\rho\hat{a}^{\dagger}]}(\alpha,\alpha^{*})$
$\displaystyle=$
$\displaystyle[\alpha+\frac{1}{2}\frac{\partial}{\partial\alpha^{*}}][\alpha^{*}+\frac{1}{2}\frac{\partial}{\partial\alpha}]W^{[\rho]}(\alpha,\alpha^{*})$
$\displaystyle W^{[\hat{a}^{\dagger}\hat{a}\rho]}(\alpha,\alpha^{*})$
$\displaystyle=$
$\displaystyle[\alpha^{*}-\frac{1}{2}\frac{\partial}{\partial\alpha}][\alpha+\frac{1}{2}\frac{\partial}{\partial\alpha^{*}}]W^{[\rho]}(\alpha,\alpha^{*}).$
(15)
As a consequence, Eq. (10) reduces to
$\frac{dW(\alpha,\alpha^{*})}{dt}=k[\frac{\partial^{2}}{\partial\alpha\partial\alpha^{*}}+\frac{\partial}{\partial\alpha}\alpha+\frac{\partial}{\partial\alpha^{*}}\alpha^{*}]W(\alpha,\alpha^{*}),$
(16)
whose solution reads [12]
$W(\alpha,\alpha^{*},t)=\frac{2}{1-e^{-2\kappa
t}}\int\frac{d^{2}\alpha_{0}}{\pi}e^{[-\frac{2}{1-e^{-2\kappa
t}}|\alpha-\alpha_{0}e^{-\kappa t}|^{2}]}W(\alpha_{0},\alpha^{*}_{0},0),$ (17)
For the cavity initial state $|\psi(0)\rangle$ we substitute Eq. (5) into Eq.
(17) and get
$\displaystyle W(\alpha,\alpha^{*},t)$ $\displaystyle=$
$\displaystyle\frac{2}{\pi}e^{(-2|\alpha|^{2})}[|C_{0}|^{2}-|C_{1}|^{2}(2e^{-2\kappa
t}-1)]L_{1}^{0}[-\frac{|2\alpha e^{-2\kappa t}|^{2}}{1-2e^{-2\kappa t}}]$ (18)
$\displaystyle+$
$\displaystyle\frac{2}{\pi}e^{-2|\alpha|^{2}}|C_{2}|^{2}(2e^{-2\kappa
t}-1)^{2}L_{2}^{0}[-\frac{|2\alpha e^{-2\kappa t}|^{2}}{1-2e^{-2\kappa t}}]$
$\displaystyle+$
$\displaystyle\frac{8\sqrt{2}}{\pi}|C_{0}C_{2}|e^{(-2|\alpha|^{2}-2\kappa
t)}|\alpha|^{2}\cos(2\theta-\varphi+\phi)$ $\displaystyle+$
$\displaystyle\frac{8}{\pi}|C_{0}C_{1}|e^{(-2|\alpha|^{2}-\kappa
t)}|\alpha|\cos(\theta+\phi)$ $\displaystyle+$
$\displaystyle\frac{8\sqrt{2}}{\pi}|C_{1}C_{2}|e^{(-2|\alpha|^{2}-kt)}|\alpha|\cos(\theta-\varphi)[2(|\alpha|^{2}-1)e^{-2\kappa
t}+1].$
Above, an integral formula [21]
$\displaystyle\int$
$\displaystyle\frac{d^{2}z}{\pi}z^{n}z^{*m}e^{\\{x_{1}|z|^{2}+x_{2}z+x_{3}z^{*}\\}}$
(19)
$\displaystyle=e^{(-\frac{x_{2}x_{3}}{x_{1}})}\sum\limits_{\kappa=0}^{min(m,n)}\frac{n!m!}{\kappa!(n-\kappa)!(m-\kappa)!(-x_{1})^{m+n-\kappa+1}}x_{2}^{m-\kappa}x_{3}^{n-\kappa},\,\,Re(x_{1})<0,$
has been used and the unassociated Laguerre Polynomial $L_{m}(x,y)$:
$\displaystyle
L_{m}(x,y)=\frac{(-1)^{m}}{m!}H_{m,n}(x,y),\,\,H_{m,n}=\frac{\partial^{m+n}}{\partial
T^{m}\partial T^{\prime
n}}e^{[-TT^{\prime}+Tx+T^{\prime}y]}|_{T=T^{\prime}=0},$ (20)
was introduced [22-23] with $H_{m,n}(x,y)$ being the generating function of
two-variable Hermite polynomial.
### II.2 Time-dependent negativity of the Wigner function
With the above time-evolution Wigner function, we next check how its
negativity changes with the cavity loss. Fig. 1 numerically shows these
changes with the effective time $\kappa t$ for the parameters:
$|C_{1}|=1/3,|C_{2}|=\sqrt{2}/2,\theta=\varphi=\pi,\phi=0$. Here, for
convenience we define the Wigner function $W(x,p,t)$ in the $(x,p)$-space with
$x=(\alpha+\alpha^{*})/2$ and $p=(\alpha-\alpha^{*})/(2i)$. One can see:
Figure 1: Wigner functions versus phase space points, $(x,p)$ (upper line) and
$(x,p=0)$ (lower line), of the few-photon superposed state (3) for different
decay times, i.e., $\kappa
t=0(a,a^{\prime}),0.2(b,b^{\prime}),0.35(c,c^{\prime}),3(d,d^{\prime})$. Here,
the parameters in $|\psi(0)\rangle$ are taken as:
$|C_{1}|=1/3,|C_{2}|=\sqrt{2}/2,\theta=\varphi=\pi,\phi=0$.
(i) Initially, the Wigner function shows obviously a negativity, i.e., at
certain phase space points, $W(x,p)<0$. This means that certain nonclassical
effects (such as the anti-bunching of photons) can be revealed in this initial
cavity state.
(ii) With the cavity dissipation, the state of the cavity decays and the
negativity of its time-dependent Wigner function vanishes gradually. This
implies that the nonclassical properties possessed initially would be weakened
with the dissipation of the cavity.
(iii) After certain times, e.g., $\kappa t\geq 0.35$ in Fig. 1(c), the values
of the Wigner functions reveal the expected non-negativity, i.e., $W(x,p)\geq
0$. In this evolved state the decayed cavity should be classical and the
corresponding Wigner functions could be explained as the usual probabilistic
distributions.
(iv) After the sufficiently-long dissipative time, the cavity state will decay
to the expectable vacuum state or thermal state with the mean photon number
being zero (i.e., $\bar{n}=0$). The Wigner function for such a dissipated
final state should be a Gaussian distribution. Indeed, from Eq. (18), we have
$\displaystyle W(\alpha,\alpha^{*},\infty)$ $\displaystyle=$
$\displaystyle\frac{2}{\pi}e^{(-2|\alpha|^{2})}[|C_{0}|^{2}+|C_{1}|^{2}L_{1}^{0}(0)+|C_{2}|^{2}L_{2}^{0}(0)]$
(21) $\displaystyle=$
$\displaystyle\frac{2}{\pi}e^{(-2|\alpha|^{2})}[|C_{0}|^{2}+|C_{1}|^{2}+|C_{2}|^{2}]$
$\displaystyle=$ $\displaystyle\frac{2}{\pi}e^{(-2|\alpha|^{2})}.$
## III Dissipative-dependent quantum statistical properties of the few-
photons cavity initial state
Various nonclassical effects, e.g., squeezings on quantum fluctuations and
sub-Poisson photon statistics, in quantum optical states have attracted
considerable and continuing interests[24-26]. Many attentions have been paid
to find various non-classical optical states, while how these non-classical
effects change with the decays of the selected non-classical states is a
relatively-new topic. Recently, Biswas and Agarwal discussed how the Mandel
$Q$-factor decreases with the decays of the photon-subtracted squeezed
states[12]. Their numerical results showed that the $Q$-factor vanishes at the
long dissipative times (i.e., $\kappa t\rightarrow\infty$) and the initial
cavity state will decay to the vacuum. With the dissipative-dependent Wigner
functions obtained in the previous section, we can investigate how the
photonic anti-bunching effect changes with the decay of the few-photon
superposition state$|\psi(0)$ defined in Eq. (3).
It is well-known that, if the second-order correlation function
$g^{(2)}(0)=\frac{\langle\hat{a}^{\dagger
2}\hat{a}^{2}\rangle}{\langle\hat{a}^{\dagger}\hat{a}\rangle^{2}}$ (22)
is less than $1$, then the photonic distribution in the state $|\psi\rangle$
is anti-bunching; otherwise, it is bunching. The symbol
$\langle\hat{O}\rangle$ represents the expectation value of the operator
$\hat{O}$ in a quantum state $\rho$. For the present case we need to calculate
the time-dependent expectation values of the operators $\hat{a}^{\dagger
2}\hat{a}^{2}$ and $\hat{a}^{\dagger}\hat{a}$ for the decaying cavity state
with time-dependent Wigner function $W(\alpha,\alpha^{*},t)$.
Formally, for an operator function [16]
$\displaystyle O(\hat{a},\hat{a}^{\dagger})(t)$ $\displaystyle=$
$\displaystyle\sum\limits_{n,m}C_{n,m}\hat{a}^{\dagger n}(t)\hat{a}^{m}(t),$
(23)
we have
$\displaystyle\langle O(\hat{a},\hat{a}^{\dagger})\rangle(t)$ $\displaystyle=$
$\displaystyle Tr[O(\hat{a},\hat{a}^{\dagger})\rho(t)]=\int d^{2}\alpha
O_{S}(\alpha,\alpha^{*})W(\alpha,\alpha^{*},t).$ (24)
On the other hand, from
$\displaystyle\langle\hat{a}^{\dagger}\rangle(t)$ $\displaystyle=$
$\displaystyle[\frac{\partial}{\partial\lambda}+\frac{\lambda^{*}}{2}]C(\lambda,\lambda^{*},t)|_{\lambda=\lambda^{*}=0}$
$\displaystyle\langle\hat{a}\rangle(t)$ $\displaystyle=$
$\displaystyle[-\frac{\partial}{\partial\lambda^{*}}-\frac{\lambda}{2}]C(\lambda,\lambda^{*},t)|_{\lambda=\lambda^{*}=0},$
(25)
we can find that
$\displaystyle\langle O(\hat{a},\hat{a}^{\dagger})\rangle(t)$ $\displaystyle=$
$\displaystyle\sum\limits_{n,m}C_{n,m}[\frac{\partial}{\partial\lambda}+\frac{\lambda^{*}}{2}]^{n}[-\frac{\partial}{\partial\lambda^{*}}-\frac{\lambda}{2}]^{m}C(\lambda,\lambda^{*},t)|_{\lambda=\lambda^{*}=0}$
(26) $\displaystyle=$ $\displaystyle\int
d^{2}\alpha\sum\limits_{n,m}C_{n,m}[\frac{\partial}{\partial\lambda}+\frac{\lambda^{*}}{2}]^{n}[-\frac{\partial}{\partial\lambda^{*}}-\frac{\lambda}{2}]^{m}e^{(-\alpha\lambda^{*}+\alpha^{*}\lambda)}|_{\lambda=\lambda^{*}=0}W(\alpha,\alpha^{*},t)$
$\displaystyle=$ $\displaystyle\int d^{2}\alpha
O_{S}(\alpha,\alpha^{*})W(\alpha,\alpha^{*},t).$
Comparing (24) and (26), we obtain
$O_{S}(\alpha,\alpha^{*})=\sum\limits_{n,m}C_{n,m}[\frac{\partial}{\partial\lambda}+\frac{\lambda^{*}}{2}]^{n}[-\frac{\partial}{\partial\lambda^{*}}-\frac{\lambda}{2}]^{m}e^{(-\alpha\lambda^{*}+\alpha^{*}\lambda)}|_{\lambda=\lambda^{*}=0}.$
(27)
Specifically, if $\hat{O}=\hat{a}^{\dagger}\hat{a}$, then
$O_{S}(\alpha,\alpha^{*})|_{\hat{O}=\hat{a}^{\dagger}\hat{a}}=|\alpha|^{2}-\frac{1}{2},$
(28)
and thus
$\displaystyle\langle\hat{a}^{\dagger}\hat{a}\rangle(t)$ $\displaystyle=$
$\displaystyle\int d^{2}\alpha
W(\alpha,\alpha^{*},t)O_{S}(\alpha,\alpha^{*})|_{\hat{O}=\hat{a}^{\dagger
2}\hat{a}^{2}}$ (29) $\displaystyle=$ $\displaystyle 4|C_{2}|^{2}e^{(-4\kappa
t)}+2|C_{2}|^{2}e^{(-2\kappa t)}[1-2e^{(-2\kappa t)}]+|C_{1}|^{2}e^{(-2\kappa
t)},$
Also, if $\hat{O}=\hat{a}^{\dagger 2}\hat{a}^{2}$, then
$O_{S}(\alpha,\alpha^{*})|_{\hat{O}=\hat{a}^{\dagger
2}\hat{a}^{2}}=\frac{1}{2}-2|\alpha|^{2}+|\alpha|^{4},$ (30)
and thus
$\displaystyle\langle\hat{a}^{\dagger 2}\hat{a}^{2}\rangle(t)$
$\displaystyle=$ $\displaystyle\int d^{2}\alpha
W(\alpha,\alpha^{*},t)O_{S}(\alpha,\alpha^{*})|_{\hat{O}=\hat{a}^{\dagger}\hat{a}}$
(31) $\displaystyle=$ $\displaystyle 2|C_{2}|^{2}e^{(-4\kappa t)}.$
Above, the dissipative-dependent Wigner function shown in Eq. (18) was used.
Consequently, we have
$\displaystyle g^{(2)}(0;t)$ $\displaystyle=$
$\displaystyle\frac{\langle\hat{a}^{\dagger
2}\hat{a}^{2}\rangle(t)}{[\langle\hat{a}^{\dagger}\hat{a}\rangle(t)]^{2}}$
(32) $\displaystyle=$ $\displaystyle\frac{2|C_{2}|^{2}e^{(-4\kappa
t)}}{\\{4|C_{2}|^{2}e^{(-4\kappa t)}+2|C_{2}|^{2}e^{(-2\kappa
t)}[1-2e^{(-2\kappa t)}]+|C_{1}|^{2}e^{(-2\kappa t)}\\}^{2}}$ $\displaystyle=$
$\displaystyle\frac{2|C_{2}|^{2}}{[|C_{1}|^{2}+2|C_{2}|^{2}]^{2}}=g^{(2)}(0).$
This indicates that the normally-order correlation function $g^{(2)}(0;t)$ is
cavity-loss-invariant; its value depends only on the initial cavity state!.
This is a surprise argument; imagining that the photons in the initial cavity
state is anti-bunching (i.e., $g^{(2)}(0;t)<1$), then such a non-classical
feature is kept unchanged even the state approached finally to the vacuum with
non-negative Wigner function. This argument is verified numerically by Fig.
2(a), which really shows that the value of $g^{(2)}(0;t)$ is really unchanged
with the decay. It is noted that, at the exact vacuum $|0\rangle$ the expected
value of operator $\langle\hat{a}^{\dagger}\hat{a}\rangle$ is zero and thus
the definition of $g^{(2)(0)}$ for this state is bizarre and insignificant.
Therefore, our discussion always works for the dissipative process approaching
to (but not arriving at) the exact vacuum.
Figure 2: (a): Normal-ordered correlation function $g^{(2)}(t)$ is unchanged
with the decay of the few-photons cavity state; (b): Anti-normally-order
correlation function $g^{(2A)}(t)$ versus the effective decay time of the
cavity. Here, the relevant parameters are taken as:
$\theta=\varphi=\pi,\phi=0$, and $|C_{1}|=\sqrt{6}/6,|C_{2}|=\sqrt{6}/3$ (blue
line), $|C_{1}|=2/9,|C_{2}|=2/3$ (red line), $|C_{1}|=1/3,|C_{2}|=1/3$ (gray
line), and $|C_{1}|=1/5,|C_{2}|=1/3$ (green line), respectively.
The dissipative-independence of the normally-correlation function $g^{(2)}$
can also be proven analytically from the master equation (7). In fact, at any
time $t$ we have
$\displaystyle\langle\hat{a}^{\dagger}\hat{a}\rangle(t)$ $\displaystyle=$
$\displaystyle Tr[\hat{a}^{\dagger}\hat{a}\rho(t)]$ (33) $\displaystyle=$
$\displaystyle\langle 0|\hat{a}^{\dagger}\hat{a}\rho(t)|0\rangle+\langle
1|\hat{a}^{\dagger}\hat{a}\rho(t)|1\rangle+\langle
2|\hat{a}^{\dagger}\hat{a}\rho(t)|2\rangle+...+\langle
n|\hat{a}^{\dagger}\hat{a}\rho(t)|n\rangle+...$ $\displaystyle=$
$\displaystyle 0+\langle 1|\rho(t)|1\rangle+2\langle
2|\rho(t)|2\rangle+...+n\langle n|\rho(t)|n\rangle+...$ $\displaystyle=$
$\displaystyle\rho_{11}(t)+2\rho_{22}(t)+...+n\rho_{nn}(t)+...$
$\displaystyle=$ $\displaystyle\sum\limits_{n=0}^{\infty}n\rho_{n,n}(t),$
and
$\displaystyle\langle\hat{a}^{\dagger 2}\hat{a}^{2}\rangle(t)$
$\displaystyle=$ $\displaystyle
Tr[\hat{a}^{\dagger}\hat{a}^{\dagger}\hat{a}\hat{a}\rho(t)]=Tr[\hat{a}^{\dagger}(\hat{a}\hat{a}^{\dagger}-1)\hat{a}\rho(t)]$
(34) $\displaystyle=$ $\displaystyle[\langle
0|\hat{a}^{\dagger}\hat{a}\hat{a}^{\dagger}\hat{a}\rho(t)|0\rangle+\langle
1|\hat{a}^{\dagger}\hat{a}\hat{a}^{\dagger}\hat{a}\rho(t)|1\rangle+...+\langle
n|\hat{a}^{\dagger}\hat{a}\hat{a}^{\dagger}\hat{a}\rho(t)|n\rangle+...]-\langle\hat{a}^{\dagger}\hat{a}\rangle(t)$
$\displaystyle=$
$\displaystyle[\rho_{11}(t)+2^{2}\rho_{22}(t)+...+n^{2}\rho_{nn}(t)+...]-\langle\hat{a}^{\dagger}\hat{a}\rangle(t)$
$\displaystyle=$ $\displaystyle\sum\limits_{n=0}^{\infty}n(n-1)\rho_{n,n}(t),$
and thus
$\displaystyle g^{(2)}(0;t)$ $\displaystyle=$
$\displaystyle\frac{\langle\hat{a}^{\dagger
2}\hat{a}^{2}\rangle(t)}{[\langle\hat{a}^{\dagger}\hat{a}\rangle(t)]^{2}}$
(35) $\displaystyle=$ $\displaystyle\frac{\langle n^{2}\rangle(t)-\langle
n\rangle(t)}{[\langle n\rangle(t)]^{2}}$ $\displaystyle=$
$\displaystyle\frac{[\rho_{11}(t)+2^{2}\rho_{22}(t)+...+n^{2}\rho_{nn}(t)+...]-[\rho_{11}(t)+2\rho_{22}(t)+...+n\rho_{nn}(t)+...]}{[\rho_{1}(t)+2\rho_{22}(t)+...+n\rho_{nn}(t)+...]^{2}}$
$\displaystyle=$
$\displaystyle\frac{\sum\limits_{n=0}(n^{2}-n)\rho_{nn}(t)}{[\sum\limits_{n=0}n\rho_{nn}(t)]^{2}}.$
Above, $\rho_{n,n}(t)$ is the diagonal elements of the density matrix
$\rho(t)$ in the Fock space. For the loss cavity initially prepared in the
few-photon superposition state (3), one can easily see that $\rho_{n,n}=0$,
for $n>2$, and the other non-zero diagonal elements are determined by the
following equation (from Eq. (7)),
$\displaystyle\dot{\rho}_{00}(t)=2\kappa\rho_{11}(t),$
$\displaystyle\dot{\rho}_{11}(t)=-2\kappa\rho_{11}(t)+4\kappa\rho_{22}(t),$
$\displaystyle\dot{\rho}_{22}(t)=-4\kappa\rho_{22}(t).$ (36)
The solutions to these equations are
$\displaystyle\rho_{11}(t)=[\rho_{11}(0)+2\rho_{22}(0)]e^{-2\kappa
t}-2\rho_{22}(0)e^{-4\kappa t}$
$\displaystyle\rho_{22}(t)=\rho_{22}(0)e^{-4\kappa t}.$ (37)
Consequently,
$\displaystyle g^{(2)}(0;t)$ $\displaystyle=$
$\displaystyle\frac{2\rho_{22}(t)}{[\rho_{11}(t)+2\rho_{22}(t)]^{2}}$ (38)
$\displaystyle=$ $\displaystyle\frac{2\rho_{22}(0)e^{-4\kappa
t}}{[\rho_{11}(0)e^{-2\kappa t}+2\rho_{22}(0)e^{-2\kappa t}]^{2}}$
$\displaystyle=$
$\displaystyle\frac{2\rho_{22}(0)}{[\rho_{11(0)}+2\rho_{22}(0)]^{2}}=g^{(2)}(0).$
Suppose that any non-classical effect should vanish due to the dissipation,
the dissipative-independence of the normally-correlation function implies that
such a parameter should not be a good physical quantity to describe the cavity
loss. Alternatively, the anti-normal ordered correlation function, defined as
$g^{(2A)}(0)=\frac{\langle\hat{a}^{2}\hat{a}^{\dagger
2}\rangle}{\langle\hat{a}\hat{a}^{\dagger}\rangle^{2}}=\frac{\langle\hat{a}^{\dagger
2}\hat{a}^{2}\rangle+4\langle\hat{a}^{\dagger}\hat{a}\rangle+2}{[\langle\hat{a}^{\dagger}\hat{a}\rangle+1]^{2}},$
(39)
could be utilized to describe the dissipative process of the few-photon
cavity. Indeed, with Eqs. (29) and (31) we have
$\displaystyle g^{(2A)}(0;t)$ $\displaystyle=$
$\displaystyle\frac{\langle\hat{a}^{\dagger
2}\hat{a}^{2}\rangle(t)+4\langle\hat{a}^{\dagger}\hat{a}\rangle(t)+2}{[\langle\hat{a}^{\dagger}\hat{a}\rangle(t)+1]^{2}}$
(40) $\displaystyle=$ $\displaystyle\frac{2|C_{2}|^{2}e^{(-4\kappa
t)}+4\\{4|C_{2}|^{2}e^{(-4\kappa t)}+2|C_{2}|^{2}e^{(-2\kappa
t)}[1-2e^{(-2\kappa t)}]+|C_{1}|^{2}e^{(-2\kappa
t)}\\}+2}{\\{4|C_{2}|^{2}e^{(-4\kappa t)}+2|C_{2}|^{2}e^{(-2\kappa
t)}[1-2e^{(-2\kappa t)}]+|C_{1}|^{2}e^{(-2\kappa t)}+1\\}^{2}}$
$\displaystyle=$ $\displaystyle\frac{4|C_{1}|^{2}e^{(-2\kappa
t)}+8|C_{2}|^{2}e^{(-2\kappa t)}+2|C_{2}|^{2}e^{(-4\kappa
t)}+2}{[|C_{1}|^{2}e^{(-2\kappa t)}+2|C_{2}|^{2}e^{(-2\kappa t)}+1]^{2}},$
which is not an invariant during the cavity dissipation. One can see also from
Fig. 2(b) that, the value of the anti-normal correlation function changes with
the cavity loss. After a sufficiently-long decay time the value of
$g^{(2A)}(0;t)$ should limit to $2$, whatever its initial value is less than
$2$ or not. Certainly, such a dissipative-dependent behavior of the
$g^{(2A)}(0;t)$-parameter can also be exactly verified by using the analytic
solutions, i.e., Eq. (37). In fact, we can see that
$\displaystyle g^{(2A)}(0;t)$ $\displaystyle=$
$\displaystyle\frac{\langle\hat{a}^{\dagger
2}\hat{a}^{2}\rangle(t)+4\langle\hat{a}^{\dagger}\hat{a}\rangle(t)+2}{[\langle\hat{a}^{\dagger}\hat{a}\rangle(t)+1]^{2}}$
(41) $\displaystyle=$ $\displaystyle\frac{4\rho_{11}(0)e^{-2\kappa
t}+8\rho_{22}(0)e^{-2\kappa t}+2\rho_{22}(0)e^{-4\kappa
t}+2}{[\rho_{11}(0)e^{-2\kappa t}+2\rho_{22}(0)e^{-2\kappa t}+1]^{2}}.$
It is consistent with the Eq. (41), as if $t\rightarrow\infty$, Eq.(42) can be
shown
$g^{(2A)}(0;t\rightarrow\infty)=2.$ (42)
## IV Possible experimental verification: the preparation of few-photon
superposed states and measurement of its Wigner function
We now discuss how to test the above arguments with a typical cavity QED
system, i.e., highly excited Rydberg atoms in a high-Q microwave cavity [28].
An ideal setup is schematized in Fig. 4, wherein an atom is emitted from the
source O and then flies across sequentially a quantized cavity, a classical
microwave field, and finally is detected in the detector $I$. When the atom
passes through the quantized cavity, the usual Jaynes-Cummings model with the
Hamiltonian ($\hbar=1$)
$H=\omega_{a}S_{z}+\omega_{c}\hat{a}^{\dagger}\hat{a}+g(\hat{a}S_{+}+\hat{a}^{+}S_{-}),$
(43)
works. Here, $\omega_{a}$, $\omega_{c}$ are the atomic transition frequency
and the cavity field frequency, respectively. $S_{z}$, $S_{\pm}$ are the
atomic operators, such that $[S_{+},S_{-}]=2S_{z},[S_{z},S_{\pm}]=\pm
S_{\pm}$. $\hat{a}$ and $\hat{a}^{\dagger}$ are the annihilation and creation
operators of the cavity field, respectively. And, $g$ is the atom-field
coupling strength.
Figure 3: An experimental setup for preparing the superposition states of
$|0\rangle,|1\rangle$ and $|2\rangle$. Here, an atom is emitted from the
source O, then it flies sequentially across the J-C cavity, the classical
microwave field, and at last is detected in the detector $I$.
Initially, the atom is in the ground state $|e_{1}\rangle$ and the cavity mode
in the vacuum state, i.e., the wave function of the atom-cavity system is
$|\psi(0)\rangle=|0,e_{1}\rangle$. Next, the atom is injected into the cavity
and the state of the atom-cavity system evolves to
$|\psi(t)\rangle_{1}=\cos(gt_{1})|0,e_{1}\rangle-i\sin(gt_{1})|1,g_{1}\rangle,$
(44)
after the passage time $t_{1}$. Then, we let the atom continuously across a
classical microwave field for evolving the atomic states as:
$|e_{1}\rangle\longrightarrow\cos(\theta_{1}/2)|e_{1}\rangle\
+ie^{-i\varphi_{1}}\sin(\theta_{1}/2)|g_{1}\rangle$ and
$|g_{1}\rangle\longrightarrow\cos(\theta_{1}/2)|g_{1}\rangle\
+ie^{i\varphi_{1}}\sin(\theta_{1}/2)|e_{1}\rangle$. Here, the values of
$\theta_{1}$ and $\varphi_{1}$ are adjustable. Therefore, before arriving at
the atomic detector I, the state of the atom-cavity system reads
$\displaystyle|\psi(t)\rangle_{1}$ $\displaystyle=$
$\displaystyle[\cos(gt_{1})\cos(\frac{\theta_{1}}{2})|0\rangle+e^{i\varphi_{1}}\sin(gt_{1})\sin(\frac{\theta_{1}}{2})|1\rangle]|e_{1}\rangle$
(45) $\displaystyle+$
$\displaystyle[ie^{-i\varphi_{1}}\cos(gt_{1})\sin(\frac{\theta_{1}}{2})|0\rangle-i\sin(gt_{1})\cos(\frac{\theta_{1}}{2})|1\rangle]|g\rangle.$
In order to generate the desirable superposition of the states
$|0\rangle,|1\rangle$ and $|2\rangle$, we must let another atom (as the same
of the former one) pass sequentially across the cavity and the microwave
field. Finally, the state of the whole system including two atoms and a cavity
mode can be expressed as:
$\displaystyle|\psi(t)\rangle_{2}$ $\displaystyle=$
$\displaystyle|0\rangle\\{\cos(gt_{1})\cos(gt_{2})\cos\frac{\theta_{1}}{2}\cos\frac{\theta_{2}}{2}|e_{1}\rangle|e_{2}\rangle$
(46) $\displaystyle+$ $\displaystyle
ie^{-i\varphi_{2}}\cos(gt_{1})\cos(gt_{2})\cos\frac{\theta_{1}}{2}\sin\frac{\theta_{2}}{2}]|e_{1}\rangle|g_{2}\rangle\\}$
$\displaystyle+$
$\displaystyle|1\rangle\\{\sin(gt_{1})\cos(gt_{2})\sin\frac{\theta_{1}}{2}\cos\frac{\theta_{2}}{2}e^{i\varphi_{1}}|e_{1}\rangle|e_{2}\rangle$
$\displaystyle+$ $\displaystyle
ie^{-i\varphi_{1}}e^{i\varphi_{2}}\sin(gt_{1})\cos(gt_{2})\sin\frac{\theta_{1}}{2}\sin\frac{\theta_{2}}{2}|e_{1}\rangle|g_{2}\rangle$
$\displaystyle-$ $\displaystyle
i\cos(gt_{1})\sin(gt_{2})\cos\frac{\theta_{1}}{2}\cos\frac{\theta_{2}}{2}|e_{1}\rangle|g_{2}\rangle$
$\displaystyle+$ $\displaystyle
e^{i\varphi_{2}}\cos(gt_{1})\sin(gt_{2})\cos\frac{\theta_{1}}{2}\sin\frac{\theta_{2}}{2}|e_{1}\rangle|e_{2}\rangle\\}$
$\displaystyle+$
$\displaystyle|2\rangle\\{e^{i\varphi_{1}}e^{i\varphi_{2}}\sin(gt_{1})\sin(gt_{2})\sin\frac{\theta_{1}}{2}\sin\frac{\theta_{2}}{2}|e_{1}\rangle|e_{2}\rangle$
$\displaystyle-$ $\displaystyle
ie^{i\varphi_{1}}\sin(gt_{1})\sin(gt_{2})\sin\frac{\theta_{1}}{2}\cos\frac{\theta_{2}}{2}|e_{1}\rangle|g_{2}\rangle\\},$
As a consequence, the desirable few-photon superposed state can be generated
by the state-selective measurements on the atoms. For example, if the atoms
are detected at the state $|e_{1}\rangle|e_{2}\rangle$, then the cavity mode
collapses into
$\displaystyle|\psi(t)\rangle_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N}}\\{|0\rangle[\cos(gt_{1})\cos(gt_{2})\cos\frac{\theta_{1}}{2}\cos\frac{\theta_{2}}{2}]$
(47) $\displaystyle+$
$\displaystyle|1\rangle[e^{i\varphi_{1}}\sin(gt_{1})\cos(gt_{2})\sin\frac{\theta_{1}}{2}\cos\frac{\theta_{2}}{2}+e^{i\varphi_{2}}\cos(gt_{1})\sin(gt_{2})\cos\frac{\theta_{1}}{2}\sin\frac{\theta_{2}}{2}]$
$\displaystyle+$
$\displaystyle|2\rangle[e^{i\varphi_{1}}e^{i\varphi_{2}}\sin(gt_{1})\sin(gt_{2})\sin\frac{\theta_{1}}{2}\sin\frac{\theta_{2}}{2}]\\}|e_{1}\rangle|e_{2}\rangle,$
with
$\displaystyle N$ $\displaystyle=$ $\displaystyle[\cos gt_{1}\cos
gt_{2}\cos\frac{\theta_{1}}{2}\cos\frac{\theta_{2}}{2}]^{2}+[\sin gt_{1}\sin
gt_{2}\sin\frac{\theta_{1}}{2}\sin\frac{\theta_{2}}{2}]^{2}$ (48)
$\displaystyle+$ $\displaystyle[\sin gt_{1}\cos
gt_{2}\sin\frac{\theta_{1}}{2}\cos\frac{\theta_{2}}{2}]^{2}+[\cos gt_{1}\sin
gt_{2}\cos\frac{\theta_{1}}{2}\sin\frac{\theta_{2}}{2}]^{2}$ $\displaystyle+$
$\displaystyle\frac{1}{8}\cos(\varphi_{1}-\varphi_{2})\sin 2gt_{1}\sin
2gt_{2}\sin\theta_{1}\sin\theta_{2}$
being the normalized coefficient. If the relevant parameters are set properly
as:
$\varphi_{1}=\pi,\varphi_{2}=0,gt_{1}=gt_{2}=\theta_{2}/2=\pi/4,\theta_{1}/2=7\pi/4$,
then a typical few-photon state discussed above
$\displaystyle|\psi(t)\rangle_{2}=\frac{\sqrt{6}}{6}|0\rangle+\frac{\sqrt{6}}{3}|1\rangle+\frac{\sqrt{6}}{6}|2\rangle$
(49)
can be obtained.
The method to measure the Wigner function for a given cavity state is relative
standard. Here, we recommend the approach proposed by Lutterbach and
Davidovich [28] by directly detecting the the negativity of Wigner function
via the atomic Ramsey interferometries. In fact, at a phase space point
$\alpha$, Wigner function for the cavity state with the density matrix $\rho$
can be simply expressed by [29]
$W(\alpha)=2Tr[D(-\alpha)\rho D(\alpha)P]=2\langle P\rangle.$ (50)
Here, $P=\exp(i\pi\hat{a}^{+}\hat{a})$ and
$D(\alpha)=\exp(\alpha\hat{a}^{+}-\alpha^{*}\hat{a})$. Furthermore, the
quantity $\langle P\rangle$ can be determined by measuring the probability
$P_{e}$ (or $P_{g}$) of the atom is detected at its excited state $|e\rangle$
(or $|g\rangle$), i.e.,
$\displaystyle P_{e}(\phi,\alpha)=\frac{1}{2}[1+\langle P\rangle\cos\phi].$
(51)
Therefore, the Wigner function is determined by
$\displaystyle W(\alpha)$ $\displaystyle=$ $\displaystyle
2[P_{e}(0,\alpha)-P_{e}(\pi,\alpha)].$ (52)
Consequently, if we have
$P_{e}(0,\alpha)<P_{e}(\pi,\alpha),$ (53)
then the Wigner function attains a negative value. With these preparations and
measurements, the dissipative dynamics presented above could be tested
experimentally.
## V Discussions and Conclusions
With the few-photon superposed state, in this paper we have investigated the
dissipative dynamics of the quantized mode without any thermal photon. By
numerical method, we discuss how the Wiginer function of the cavity state
changes with the dissipation of the cavity. Our results show clearly that the
initial negativity of the Wigner function vanishes with the cavity
dissipation. With the dissipative-dependent Wigner function, we further
discuss how a typical quantum statistical property, the second-order
correlation function $g^{(2)}(0)$, changes with the dissipation of the cavity.
It is surprised that such a quantity is an invariant during the dissipation of
the cavity. This argument was also verified by analytical method directly
solving the master equation of the dissipative cavity. This implies that the
$g^{(2)}(0)$ should not be a good physical quantity to describe the
dissipative dynamics of the cavity, at least for the few-photon state.
The discussion in the present work is limited to the photons in optical
cavity, and thus the mean thermal photons at room temperature can be really
negligible. This implies that the final state of the dissipative optical
cavity is exactly vacuum, at which the standard definition of the second-order
correlation function is bizarre and insignificant. The generalization to the
dissipative cavity with non-zero thermal photons is in progress.
Given the few-photon superposed state of the cavity is not difficult to be
prepared and its relevant Wigner function can also be easily measured in the
usual cavity QED system, we believe that our arguments are testable with the
current experimental technique.
## Acknowledgments
One of us (Wen) thanks Dr. W. Z. Jia for useful discussions. This work was
supported in part by the National Science Foundation grant No. 10874142,
90921010, 11174373 and the National Fundamental Research Program of China
through Grant No. 2010CB923104. J. Cheng thanks the supports from the National
Basic Research Program of China under Grant No. 2012CB921904, the National
Natural Science Foundation of China (11174084, 10934011), the Fundamental
Research Funds for Central University (SCUT), and the State Key Laboratory of
Precision Spectroscopy.
## References
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* (3) D. Bouwmeester, A. Ekert, A. Zeilinger, The Physics of Quantum Information, (Springer-Verlag, Berlin, 2000); P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, Phys. Rev. Lett. 75, 4337 (1995).
* (4) V. Buzek and P. L. Knight, Progress in Optics, edited by E. Wolf (North Holland, Amsterdam, 1195), Vol. XXXIV, p. 1.
* (5) Y. Yang and F. L. Li, J. Opt. Soc. Am. B 26, 830 (2009).
* (6) M. Hillery, R. F. O Connell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 (1984).
* (7) L. F. Wei , S. J. Wang , Q. L. Jie , 42, 1686 (1997)
* (8) M. S. Kim, E. Park, P. L. Knight, and H. Jeong., Phys. Rev. A 71, 043805 (2005)
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* (14) L. Y. Hu, X. X. Xu, Z. S. Wang, and X. F. Xu, Phys. Rev. A 82, 043842 (2010)
* (15) X. M. Liu and C. Quesne, Phys. Lett. A 317, 210 (2003).
* (16) M. O. Scully, M. S. Zubairy, Quantum Optics, (Cambridge University Press, Cambridge, 1997)
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* (19) L. H. William, Quantum Statistical Properties of Radiation, (John Wiley, New York, 1973).
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* (22) A. Wünsche, J. Comput. and Appl. Math. 133, 665 (2001).
* (23) A. Wünsche, J . Phys. A: Math. and Gen., 33, 1603 (2000).
* (24) V. V. Dodonov, J. Opt. B: Quantum Semiclass. Opt 4, R1 (2002).
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|
arxiv-papers
| 2011-10-16T05:01:32 |
2024-09-04T02:49:23.169018
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hong-Yan Wen, Jing Cheng, Y. Yang, L. F. Wei",
"submitter": "Hao Yuan",
"url": "https://arxiv.org/abs/1110.3456"
}
|
1110.3548
|
# Full Spark Frames
Boris Alexeev Department of Mathematics, Princeton University, Princeton, New
Jersey 08544, USA; E-mail: balexeev@math.princeton.edu , Jameson Cahill
Department of Mathematics, University of Missouri, Columbia, Missouri 65211,
USA; E-mail: jameson.cahill@gmail.com and Dustin G. Mixon Program in
Applied and Computational Mathematics, Princeton University, Princeton, New
Jersey 08544, USA; E-mail: dmixon@princeton.edu
###### Abstract.
Finite frame theory has a number of real-world applications. In applications
like sparse signal processing, data transmission with robustness to erasures,
and reconstruction without phase, there is a pressing need for deterministic
constructions of frames with the following property: every size-$M$
subcollection of the $M$-dimensional frame elements is a spanning set. Such
frames are called full spark frames, and this paper provides new constructions
using the discrete Fourier transform. Later, we prove that full spark Parseval
frames are dense in the entire set of Parseval frames, meaning full spark
frames are abundant, even if one imposes an additional tightness constraint.
Finally, we prove that testing whether a given matrix is full spark is hard
for ${\mathsf{NP}}$ under randomized polynomial-time reductions, indicating
that deterministic full spark constructions are particularly significant
because they guarantee a property which is otherwise difficult to check.
###### Key words and phrases:
Frames, spark, sparsity, erasures
###### 2000 Mathematics Subject Classification:
42C15, 68Q17
The authors thank Profs. Peter G. Casazza and Matthew Fickus for discussions
on full spark frames, Prof. Dan Edidin and Will Sawin for discussions on
algebraic geometry, and the anonymous referees for very helpful comments and
suggestions. BA was supported by the NSF Graduate Research Fellowship under
Grant No. DGE-0646086, JC was supported by NSF Grant No. DMS-1008183, DTRA/NSF
Grant No. DMS-1042701 and AFOSR Grant No. FA9550-11-1-0245, 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
A _frame_ over a Hilbert space $\mathcal{H}$ is a collection of vectors
$\\{f_{i}\\}_{i\in\mathcal{I}}\subseteq\mathcal{H}$ with _frame bounds_
$0<A\leq B<\infty$ such that
$A\|x\|^{2}\leq\sum_{i\in\mathcal{I}}|\langle x,f_{i}\rangle|^{2}\leq
B\|x\|^{2}\qquad\forall x\in\mathcal{H}.$ (1)
For a finite frame $\\{f_{n}\\}_{n=1}^{N}$ in an $M$-dimensional Hilbert space
$\mathbb{H}_{M}$, the upper frame bound $B$ which satisfies (1) trivially
exists. In this finite-dimensional setting, the notion of being a frame solely
depends on the lower frame bound $A$ being strictly positive, which is
equivalent to having the frame elements $\\{f_{n}\\}_{n=1}^{N}$ span the
Hilbert space.
In practice, frames are chiefly used in two ways. The _synthesis operator_
$F\colon\mathbb{C}^{N}\rightarrow\mathbb{H}_{M}$ of a finite frame
$F=\\{f_{n}\\}_{n=1}^{N}$ is defined by $Fy:=\sum_{n=1}^{N}y[n]f_{n}$. As
such, the $M\times N$ matrix representation $F$ of the synthesis operator has
the frame elements $\\{f_{n}\\}_{n=1}^{N}$ as columns, with the natural
identification $\mathbb{H}_{M}\cong\mathbb{C}^{M}$. Note that here and
throughout, we make no notational distinction between a frame $F$ and its
synthesis operator. The adjoint of the synthesis operator is called the
_analysis operator_ $F^{*}\colon\mathbb{H}_{M}\rightarrow\mathbb{C}^{N}$,
defined by $(F^{*}x)[n]:=\langle x,f_{n}\rangle$.
Classically, frames are used to redundantly decompose signals, $y=F^{*}x$,
before synthesizing the corresponding frame coefficients, $z=Fy=FF^{*}x$, and
so the _frame operator_ $FF^{*}\colon\mathbb{H}_{M}\rightarrow\mathbb{H}_{M}$
is often analyzed to determine how well this process preserves information
about the original signal $x$. In particular, if the frame bounds are equal,
the frame operator has the form $FF^{*}=AI_{M}$, and so signal reconstruction
is rather painless: $x=\frac{1}{A}FF^{*}x$; in this case, the frame is called
_tight_. Oftentimes, it is additionally desirable for the frame elements to
have unit norm, in which case the frame is a _unit norm frame_. Moreover, the
_worst-case coherence_ between unit norm frame elements $\mu:=\max_{i\neq
j}|\langle f_{i},f_{j}\rangle|$ satisfies $\mu^{2}\geq\frac{N-M}{M(N-1)}$, and
equality is achieved precisely when the frame is tight with $|\langle
f_{i},f_{j}\rangle|=\mu$ for all distinct pairs $i,j\in\\{1,\ldots,N\\}$ [44];
in this case, the frame is called an _equiangular tight frame_ (ETF).
The utility of unit norm tight frames and ETFs is commonly expressed in terms
of a scenario in which frame coefficients $\\{(F^{*}x)[n]\\}_{n=1}^{N}$ are
transmitted over a noisy or lossy channel before reconstructing the signal:
$y=\mathcal{D}(F^{*}x),\qquad\tilde{x}=(FF^{*})^{-1}Fy,$ (2)
where $\mathcal{D}(\cdot)$ represents the channel’s random and not-
necessarily-linear deformation process. Using an additive white Gaussian noise
model, Goyal [23] established that, of all unit norm frames, unit norm tight
frames minimize mean squared error in reconstruction. For the case of a lossy
channel, Holmes and Paulsen [26] established that, of all tight frames, unit
norm tight frames minimize worst-case error in reconstruction after one
erasure, and that equiangular tight frames minimize this error after two
erasures. We note that the reconstruction process in (2), namely the
application of $(FF^{*})^{-1}F$, is inherently blind to the effect of the
deformation process of the channel. This contrasts with Püschel and
Kovačević’s more recent work [36], which describes an adaptive process for
reconstruction after multiple erasures; we will return to this idea later.
Another application of frames is sparse signal processing. This field concerns
signal classes which are sparse in some basis. As an example, natural images
tend to be nearly sparse in the wavelet basis [14]. Some applications have
signal sparsity in the identity basis [32]. Given a signal $x$, let $\Psi$
represent its sparsifying basis and consider
$y=F\Psi^{*}x+e,$ (3)
where $F$ is the $M\times N$ synthesis operator of a frame, $\Psi^{*}x$ has at
most $K$ nonzero entries, and $e$ is some sort of noise. When given
measurements $y$, one typically wishes to reconstruct the original vector $x$;
viewing $F$ as a sensing matrix, this process is commonly referred to as
_compressed sensing_ since we often take $M\ll N$. Note that $y$ can be viewed
as a noisy linear combination of a few unknown frame elements, and the goal of
compressed sensing is to determine which frame elements are active in the
combination, and further estimate the scalar multiples used in this
combination. This problem setup is very related to that of sparse
approximation, in which signals are known to be expressible as a sparse
combination of elements from an _overcomplete_ dictionary; in this case, the
dictionary elements form the columns of $F$, and we again wish to determine
the scalar multiples used in a given sparse combination.
In order to have any sort of inversion process for (3), even in the noiseless
case where $e=0$, $F$ must map $K$-sparse vectors injectively. That is, we
need $Fx_{1}\neq Fx_{2}$ for any distinct $K$-sparse vectors $x_{1}$ and
$x_{2}$. Subtraction then gives that $2K$-sparse vectors like $x_{1}-x_{2}$
cannot be in the nullspace of $F$. This identification led to the following
definition [16]:
###### Definition 1.
The _spark_ of a matrix $F$ is the size of the smallest linearly dependent
subset of columns, i.e.,
$\mathrm{Spark}(F)=\min\\{\|x\|_{0}:Fx=0,~{}x\neq 0\\}.$
Using the above analysis, Donoho and Elad [16] showed that a signal $x$ with
sparsity level $<\mathrm{Spark}(F)/2$ is necessarily the unique sparsest
solution to $y=Fx$, and furthermore, there exist signals $x$ of sparsity level
$\geq\mathrm{Spark}(F)/2$ which are not the unique sparsest solution to
$y=Fx$. This demonstrates that matrices with larger spark are naturally
equipped to support signals with larger sparsity levels. One is naturally led
to consider matrices that support the largest possible sparse signal class; we
say an $M\times N$ matrix $F$ is _full spark_ if its spark is as large as
possible, i.e., $\mathrm{Spark}(F)=M+1$. Equivalently, $M\times N$ full spark
matrices have the property that every $M\times M$ submatrix is invertible; as
such, a full spark matrix is necessarily full rank, and therefore a frame.
That being said, while the submatrices of a full spark matrix will be
invertible, they may not be well-conditioned. Moreover, the conditioning of
these submatrices is an important feature for compressed sensing; Candès [8]
gives an elegant proof that sensing matrices with well-conditioned
submatrices, specifically, satisfying the _restricted isometry property (RIP)_
, allow for stable and efficient recovery. Unfortunately, for deterministic
matrices, it is difficult to determine the conditioning of every submatrix; to
date, no deterministic RIP matrix is known to perform optimally [19], and in
one respect, this difficulty is precisely the barrier to significant progress
on the Kadison-Singer problem [13]. However, some work has been done to test
whether _most_ submatrices are well-conditioned [47], and better yet, other
work has managed to achieve RIP-like performance without requiring all
submatrices to be well-conditioned [1]. Regardless, good sensing matrices for
compressed sensing must necessarily have large spark to enable reconstruction
[16], and so building such matrices could serve as one step toward optimal
deterministic RIP matrices.
In sparse signal processing, the specific application of full spark frames has
been studied for some time. In 1997, Gorodnitsky and Rao [22] first considered
full spark frames, referring to them as matrices with the _unique
representation property_ (for reasons discussed above). Since [22], the unique
representation property has been explicitly used to find a variety of
performance guarantees for sparse signal processing [6, 33, 50]. Tang and
Nehorai [45] also obtain performance guarantees using full spark frames, but
they refer to them as _non-degenerate measurement matrices_.
Consider a matrix whose entries are independent continuous random variables.
Intuitively, the matrix is full spark with probability one, and this fact was
recently proved in [5]. However, this random process does not allow one to
control certain features of the matrix, such as unit-norm tightness or
equiangularity. Indeed, ETFs are notoriously difficult to construct, but they
appear to be particularly well-suited for sparse signal processing [1, 19,
32]. For example, Bajwa et al. [1] uses Steiner ETFs to recover the support of
$\Psi^{*}x$ in (3); given measurements $y$, the largest entries in the back-
projection $F^{*}y$ often coincide with the support of $\Psi^{*}x$. However,
Steiner ETFs have particularly small spark [20], and back-projection will
correctly identify the support of $x$ even when the corresponding columns in
$F$ are dependent; in this case, there is no way to estimate the nonzero
entries of $\Psi^{*}x$. This illustrates one reason to build deterministic
full spark frames: their submatrices are necessarily invertible, making it
possible to estimate these nonzero entries.
For another application of full spark frames, we return to the problem (2) of
reconstructing a signal from distorted frame coefficients. Specifically, we
consider Püschel and Kovačević’s work [36], which focuses on an Internet-like
channel that is prone to multitudes of erasures. In this context, they
reconstruct the signal after first identifying which frame coefficients were
not erased; with this information, the signal can be estimated provided the
corresponding frame elements span. In this sense, full spark frames are
_maximally robust to erasures_ , as coined in [36]. In particular, an $M\times
N$ full spark frame is robust to $N-M$ erasures since any $M$ of the frame
coefficients will uniquely determine the original signal.
Yet another application of full spark frames is phaseless reconstruction,
which can be viewed in terms of a channel, as in (2); in this case,
$\mathcal{D}(\cdot)$ is the entrywise absolute value function. Phaseless
reconstruction has a number of real-world applications including speech
processing [4], X-ray crystallography [10], and quantum state estimation [39].
As such, there has been a lot of work to reconstruct an $M$-dimensional vector
(up to an overall phase factor) from the magnitudes of its frame coefficients,
most of which involves frames in operator space, which inherently require
$N=\Omega(M^{2})$ measurements [3, 39]. However, Balan et al. [4] show that if
an $M\times N$ real frame $F$ is full spark with $N\geq 2M-1$, then
$\mathcal{D}\circ F^{*}$ is injective, meaning an inversion process is
possible with only $N=O(M)$ measurements. This result prompted an ongoing
search for efficient phaseless reconstruction processes [2, 10], but no
reconstruction process can succeed without a good family of frames, such as
full spark frames.
Despite the fact that full spark frames have a multitude of applications, to
date, there has not been much progress in constructing deterministic full
spark frames, let alone full spark frames with additional desirable
properties. A noteworthy exception is Püschel and Kovačević’s work [36], in
which real full spark tight frames are constructed using polynomial
transforms. In the present paper, we start by investigating Vandermonde
frames, harmonic frames, and modifications thereof. While the use of certain
Vandermonde and harmonic frames as full spark frames is not new [6, 9, 21],
the fruits of our investigation are new: For instance, we demonstrate that
certain classes of ETFs are full spark, and we characterize the $M\times N$
full spark harmonic frames for which $N$ is a prime power. The remainder of
the paper proves two results which might be considered folklore-type
observations—while their proofs are new, the results are not surprising.
First, in Section 3, we show that $M\times N$ full spark Parseval frames form
a dense subset of the entire collection of $M\times N$ Parseval frames. As
such, full spark frames are abundant, even after imposing the additional
condition of tightness. This result is balanced with Section 4, in which we
prove that verifying whether a matrix is full spark is hard for
${\mathsf{NP}}$ under randomized polynomial-time reductions. In other words,
assuming ${\mathsf{NP}}\not\subseteq{\mathsf{BPP}}$ (a computational
complexity assumption slightly stronger than ${\mathsf{P}}\neq{\mathsf{NP}}$
and nearly as widely believed), then there is no method by which one can
efficiently test whether matrices are full spark. As such, the deterministic
constructions in Section 2 are significant in that they guarantee a property
which is otherwise difficult to check.
## 2\. Deterministic constructions of full spark frames
A square matrix is invertible if and only if its determinant is nonzero, and
in our quest for deterministic constructions of full spark frames, this
characterization will reign supreme. One class of matrices has a particularly
simple determinant formula: Vandermonde matrices. Specifically, Vandermonde
matrices have the following form:
$V=\begin{bmatrix}1&1&\cdots&1\\\ \alpha_{1}&\alpha_{2}&\cdots&\alpha_{N}\\\
\vdots&\vdots&\cdots&\vdots\\\
\alpha_{1}^{M-1}&\alpha_{2}^{M-1}&\cdots&\alpha_{N}^{M-1}\end{bmatrix},$ (4)
and square Vandermonde matrices, i.e., with $N=M$, have the following
determinant:
$\mathrm{det}(V)=\prod_{1\leq i<j\leq M}(\alpha_{j}-\alpha_{i}).$ (5)
Consider (4) in the case where $N\geq M$. Since every $M\times M$ submatrix of
$V$ is also Vandermonde, we can modify the indices in (5) to calculate the
determinant of the submatrices. These determinants are nonzero precisely when
the bases $\\{\alpha_{n}\\}_{n=1}^{N}$ are distinct, yielding the following
result:
###### Lemma 2.
A Vandermonde matrix is full spark if and only if its bases are distinct.
To be clear, this result is not new. In fact, the full spark of Vandermonde
matrices was first exploited by Fuchs [21] for sparse signal processing.
Later, Bourguignon et al. [6] specifically used the full spark of Vandermonde
matrices whose bases are sampled from the complex unit circle. Interestingly,
when viewed in terms of frame theory, Vandermonde matrices naturally point to
the discrete Fourier transform:
###### Theorem 3.
The only $M\times N$ Vandermonde matrices that are equal norm and tight have
bases in the complex unit circle. Among these, the frames with the smallest
worst-case coherence have bases that are equally spaced in the complex unit
circle, provided $N\geq 2M$.
###### Proof.
Suppose a Vandermonde matrix is equal norm and tight. Note that a zero base
will produce the zeroth identity basis element $\delta_{0}$. Letting
$\mathcal{P}$ denote the indices of the nonzero bases, the fact that the
matrix is full rank implies $|\mathcal{P}|\geq M-1$. Also, equal norm gives
that the frame element length
$\|f_{n}\|^{2}=\sum_{m=0}^{M-1}|f_{n}[m]|^{2}=\sum_{m=0}^{M-1}|\alpha_{n}^{m}|^{2}=\sum_{m=0}^{M-1}|\alpha_{n}|^{2m}$
is constant over $n\in\mathcal{P}$. Since $\sum_{m=0}^{M-1}x^{2m}$ is strictly
increasing over $0<x<\infty$, there exists $c>0$ such that
$|\alpha_{n}|^{2}=c$ for all $n\in\mathcal{P}$. Next, tightness gives that the
rows have equal norm, implying that the first two rows have equal norm, i.e.,
$|\mathcal{P}|c=|\mathcal{P}|c^{2}$. Thus $c=1$, and so the nonzero bases are
in the complex unit circle. Furthermore, since the zeroth and first rows have
equal norm by tightness, we have $|\mathcal{P}|=N$, and so every base is in
the complex unit circle.
Now consider the inner product between Vandermonde frame elements whose bases
$\\{e^{2\pi ix_{n}}\\}_{n=1}^{N}$ come from the complex unit circle:
$\langle f_{n},f_{n^{\prime}}\rangle=\sum_{m=0}^{M-1}(e^{2\pi
ix_{n}})^{m}\overline{(e^{2\pi ix_{n^{\prime}}})^{m}}=\sum_{m=0}^{M-1}e^{2\pi
i(x_{n}-x_{n^{\prime}})m}.$
We will show that the worst-case coherence comes from the two closest bases.
Consider the following function:
$g(x):=\bigg{|}\sum_{m=0}^{M-1}e^{2\pi ixm}\bigg{|}^{2}.$ (6)
Figure 1 gives a plot of this function in the case where $M=5$. We will prove
two things about this function:
* (i)
$\tfrac{d}{dx}g(x)<0$ for every $x\in(0,\tfrac{1}{2M})$,
* (ii)
$g(x)\leq g(\tfrac{1}{2M})$ for every $x\in(\tfrac{1}{2M},1-\tfrac{1}{2M})$.
Figure 1. Plot of $g$ defined by (6) in the case where $M=5$. Observe (i) that
$g$ is strictly decreasing on the interval $(0,\frac{1}{10})$, and (ii) that
$g(x)\leq g(\frac{1}{10})$ for every $x\in(\frac{1}{10},\frac{9}{10})$. As
established in the proof of Theorem 3, $g$ behaves in this manner for general
values of $M$.
First, we claim that (i) and (ii) are sufficient to prove our result. To
establish this, we first show that the two closest bases $e^{2\pi
ix_{n^{\prime}}}$ and $e^{2\pi ix_{n^{\prime\prime}}}$ satisfy
$|x_{n^{\prime}}-x_{n^{\prime\prime}}|\leq\frac{1}{2M}$. Without loss of
generality, the $n$’s are ordered in such a way that
$\\{x_{n}\\}_{n=0}^{N-1}\subseteq[0,1)$ are nondecreasing. Define
$d(x_{n},x_{n+1}):=\left\\{\begin{array}[]{ll}x_{n+1}-x_{n},&n=0,\ldots,N-2\\\
x_{0}-(x_{N-1}-1),&n=N-1,\end{array}\right.$
and let $n^{\prime}$ be the $n$ which minimizes $d(x_{n},x_{n+1})$. Since the
minimum is less than the average, we have
$d(x_{n^{\prime}},x_{n^{\prime}+1})\leq\frac{1}{N}\bigg{(}(x_{0}-(x_{N-1}-1))+\sum_{n=0}^{N-1}(x_{n+1}-x_{n})\bigg{)}=\frac{1}{N}\leq\frac{1}{2M},$
(7)
provided $N\geq 2M$. Note that if we view $\\{x_{n}\\}_{n\in\mathbb{Z}_{N}}$
as members of $\mathbb{R}/\mathbb{Z}$, then $d(x_{n},x_{n+1})=x_{n+1}-x_{n}$.
Since $g(x)$ is even, then (i) implies that $|\langle
f_{n^{\prime}+1},f_{n^{\prime}}\rangle|^{2}=g(x_{n^{\prime}+1}-x_{n^{\prime}})$
is larger than any other $g(x_{p}-x_{p^{\prime}})=|\langle
f_{p},f_{p^{\prime}}\rangle|^{2}$ in which
$x_{p}-x_{p^{\prime}}\in[0,\tfrac{1}{2M}]\cup[1-\tfrac{1}{2M},1)$. Next, (7)
and (ii) together imply that $|\langle
f_{n^{\prime}+1},f_{n^{\prime}}\rangle|^{2}=g(x_{n^{\prime}+1}-x_{n^{\prime}})\geq
g(\tfrac{1}{2M})$ is larger than any other $g(x_{p}-x_{p^{\prime}})=|\langle
f_{p},f_{p^{\prime}}\rangle|^{2}$ in which
$x_{p}-x_{p^{\prime}}\in(\tfrac{1}{2M},1-\tfrac{1}{2M})$, provided $N\geq 2M$.
Combined, (i) and (ii) give that $|\langle
f_{n^{\prime}+1},f_{n^{\prime}}\rangle|$ achieves the worst-case coherence of
$\\{f_{n}\\}_{n\in\mathbb{Z}_{N}}$. Additionally, (i) gives that the worst-
case coherence $|\langle f_{n^{\prime}+1},f_{n^{\prime}}\rangle|$ is minimized
when $x_{n^{\prime}+1}-x_{n^{\prime}}$ is maximized, i.e., when the $x_{n}$’s
are equally spaced in the unit interval.
To prove (i), note that the geometric sum formula gives
$g(x)=\bigg{|}\sum_{m=0}^{M-1}e^{2\pi ixm}\bigg{|}^{2}=\bigg{|}\frac{e^{2M\pi
ix}-1}{e^{2\pi ix}-1}\bigg{|}^{2}=\frac{2-2\cos(2M\pi x)}{2-2\cos(2\pi
x)}=\bigg{(}\frac{\sin(M\pi x)}{\sin(\pi x)}\bigg{)}^{2},$ (8)
where the final expression uses the identity $1-\cos(2z)=2\sin^{2}z$. To show
that $g$ is decreasing over $(0,\frac{1}{2M})$, note that the base of (8) is
positive on this interval, and performing the quotient rule to calculate its
derivative will produce a fraction whose denominator is nonnegative and whose
numerator is given by
$M\pi\sin(\pi x)\cos(M\pi x)-\pi\sin(M\pi x)\cos(\pi x).$ (9)
This factor is zero at $x=0$ and has derivative:
$-(M^{2}-1)\pi^{2}\sin(\pi x)\sin(M\pi x),$
which is strictly negative for all $x\in(0,\tfrac{1}{2M})$. Hence, (9) is
strictly negative whenever $x\in(0,\tfrac{1}{2M})$, and so $g^{\prime}(x)<0$
for every $x\in(0,\tfrac{1}{2M})$.
For (ii), note that for every $x\in(\frac{1}{2M},1-\frac{1}{2M})$, we can
individually bound the numerator and denominator of what the geometric sum
formula gives:
$g(x)=\bigg{|}\sum_{m=0}^{M-1}e^{2\pi ixm}\bigg{|}^{2}=\frac{|e^{2M\pi
ix}-1|^{2}}{|e^{2\pi ix}-1|^{2}}\leq\frac{|e^{\pi i}-1|^{2}}{|e^{\pi
i/M}-1|^{2}}=\bigg{|}\sum_{m=0}^{M-1}e^{\pi
im/M}\bigg{|}^{2}=g(\tfrac{1}{2M}).\qed$
Consider the $N\times N$ discrete Fourier transform (DFT) matrix, scaled to
have entries of unit modulus:
$\begin{bmatrix}1&1&1&\cdots&1\\\ 1&\omega&\omega^{2}&\cdots&\omega^{N-1}\\\
1&\omega^{2}&\omega^{4}&\cdots&\omega^{2(N-1)}\\\
\vdots&\vdots&\vdots&\cdots&\vdots\\\
1&\omega^{N-1}&\omega^{2(N-1)}&\cdots&\omega^{(N-1)(N-1)}\end{bmatrix},$
where $\omega=e^{-2\pi i/N}$. The first $M$ rows of the DFT form a Vandermonde
matrix of distinct bases $\\{\omega^{n}\\}_{n=0}^{N-1}$; as such, this matrix
is full spark by Lemma 2. In fact, the previous result says that this is in
some sense an optimal Vandermonde frame, but this might not be the best way to
pick rows from a DFT. Indeed, several choices of DFT rows could produce full
spark frames, some with smaller coherence or other desirable properties, and
so the remainder of this section focuses on full spark DFT submatrices. First,
we note that not every DFT submatrix is full spark. For example, consider the
$4\times 4$ DFT:
$\begin{bmatrix}1&1&1&1\\\ 1&-i&-1&i\\\ 1&-1&1&-1\\\ 1&i&-1&-i\end{bmatrix}.$
Certainly, the zeroth and second rows of this matrix are not full spark, since
the zeroth and second columns of this submatrix form the all-ones matrix,
which is not invertible. So what can be said about the set of permissible row
choices? The following result gives some necessary conditions on this set:
###### Theorem 4.
Take an $N\times N$ discrete Fourier transform matrix, and select the rows
indexed by $\mathcal{M}\subseteq\mathbb{Z}_{N}$ to build the matrix $F$. If
$F$ is full spark, then so is the matrix built from rows indexed by
* (i)
any translation of $\mathcal{M}$,
* (ii)
any $A\mathcal{M}$ with $A$ relatively prime to $N$,
* (iii)
the complement of $\mathcal{M}$ in $\mathbb{Z}_{N}$.
###### Proof.
For (i), we first define $D$ to be the $N\times N$ diagonal matrix whose
diagonal entries are $\\{\omega^{n}\\}_{n=0}^{N-1}$. Note that, since
$\omega^{(m+1)n}=\omega^{n}\omega^{mn}$, translating the row indices
$\mathcal{M}$ by $1$ corresponds to multiplying $F$ on the right by $D$. For
some set $\mathcal{K}\subseteq\mathbb{Z}_{N}$ of size $M:=|\mathcal{M}|$, let
$F_{\mathcal{K}}$ denote the $M\times M$ submatrix of $F$ whose columns are
indexed by $\mathcal{K}$, and let $D_{\mathcal{K}}$ denote the $M\times M$
diagonal submatrix of $D$ whose diagonal entries are indexed by $\mathcal{K}$.
Then since $D_{\mathcal{K}}$ is unitary, we have
$|\mathrm{det}((FD)_{\mathcal{K}})|=|\mathrm{det}(F_{\mathcal{K}}D_{\mathcal{K}})|=|\mathrm{det}(F_{\mathcal{K}})||\mathrm{det}(D_{\mathcal{K}})|=|\mathrm{det}(F_{\mathcal{K}})|.$
Thus, if $F$ is full spark,
$|\mathrm{det}((FD)_{\mathcal{K}})|=|\mathrm{det}(F_{\mathcal{K}})|>0$, and so
$FD$ is also full spark. Using this fact inductively proves (i) for all
translations of $\mathcal{M}$.
For (ii), let $G$ denote the submatrix of rows indexed by $A\mathcal{M}$. Then
for any set $\mathcal{K}\subseteq\mathbb{Z}_{N}$ of size $M$,
$\mathrm{det}(G_{\mathcal{K}})=\mathrm{det}(\omega^{(Am)k})_{m\in\mathcal{M},k\in\mathcal{K}}=\mathrm{det}(\omega^{m(Ak)})_{m\in\mathcal{M},k\in\mathcal{K}}=\mathrm{det}(F_{A\mathcal{K}}).$
Since $A$ is relatively prime to $N$, multiplication by $A$ permutes the
elements of $\mathbb{Z}_{N}$, and so $A\mathcal{K}$ has exactly $M$ distinct
elements. Thus, if $F$ is full spark, then
$\mathrm{det}(G_{\mathcal{K}})=\mathrm{det}(F_{A\mathcal{K}})\neq 0$, and so
$G$ is also full spark.
For (iii), we let $G$ be the $(N-M)\times N$ submatrix of rows indexed by
$\mathcal{M}^{\mathrm{c}}$, so that
$NI_{N}=\begin{bmatrix}F^{*}&G^{*}\end{bmatrix}\begin{bmatrix}F\\\
G\end{bmatrix}=F^{*}F+G^{*}G.$ (10)
We will use contraposition to show that $F$ being full spark implies that $G$
is also full spark. To this end, suppose $G$ is not full spark. Then $G$ has a
collection of $N-M$ linearly dependent columns
$\\{g_{i}\\}_{i\in\mathcal{K}}$, and so there exists a nontrivial sequence
$\\{\alpha_{i}\\}_{i\in\mathcal{K}}$ such that
$\sum_{i\in\mathcal{K}}\alpha_{i}g_{i}=0.$
Considering $g_{i}=G\delta_{i}$, where $\delta_{i}$ is the $i$th identity
basis element, we can use (10) to express this linear dependence in terms of
$F$:
$0=G^{*}0=G^{*}\sum_{i\in\mathcal{K}}\alpha_{i}g_{i}=\sum_{i\in\mathcal{K}}\alpha_{i}G^{*}G\delta_{i}=\sum_{i\in\mathcal{K}}\alpha_{i}(NI_{N}-F^{*}F)\delta_{i}.$
Rearranging then gives
$x:=N\sum_{i\in\mathcal{K}}\alpha_{i}\delta_{i}=\sum_{i\in\mathcal{K}}\alpha_{i}F^{*}F\delta_{i}.$
(11)
Here, we note that $x$ is nonzero since $\\{\alpha_{i}\\}_{i\in\mathcal{K}}$
is nontrivial, and that $x\in\mathrm{Range}(F^{*}F)$. Furthermore, whenever
$j\not\in\mathcal{K}$, we have from (11) that
$\langle x,F^{*}F\delta_{j}\rangle=\langle
F^{*}Fx,\delta_{j}\rangle=N\bigg{\langle}F^{*}F\sum_{i\in\mathcal{K}}\alpha_{i}\delta_{i},\delta_{j}\bigg{\rangle}=N^{2}\bigg{\langle}\sum_{i\in\mathcal{K}}\alpha_{i}\delta_{i},\delta_{j}\bigg{\rangle}=0,$
and so
$x\perp\mathrm{Span}\\{F^{*}F\delta_{j}\\}_{j\in\mathcal{K}^{\mathrm{c}}}$.
Thus, the containment
$\mathrm{Span}\\{F^{*}F\delta_{j}\\}_{j\in\mathcal{K}^{\mathrm{c}}}\subseteq\mathrm{Range}(F^{*}F)$
is proper, and so
$M=\mathrm{Rank}(F)=\mathrm{Rank}(F^{*}F)>\mathrm{Rank}(F^{*}F_{\mathcal{K}^{\mathrm{c}}})=\mathrm{Rank}(F_{\mathcal{K}^{\mathrm{c}}}).$
Since the $M\times M$ submatrix $F_{\mathcal{K}^{\mathrm{c}}}$ is rank-
deficient, it is not invertible, and therefore $F$ is not full spark. ∎
We note that our proof of (iii) above uses techniques from Cahill et al. [7],
and can be easily generalized to prove that the Naimark complement of a full
spark tight frame is also full spark. Theorem 4 tells us quite a bit about the
set of permissible choices for DFT rows. For example, not only can we pick the
first $M$ rows of the DFT to produce a full spark Vandermonde frame, but we
can also pick any consecutive $M$ rows, by Theorem 4(i). We would like to
completely characterize the choices that produce full spark harmonic frames.
The following classical result does this in the case where $N$ is prime:
###### Theorem 5 (Chebotarëv, see [41]).
Let $N$ be prime. Then every square submatrix of the $N\times N$ discrete
Fourier transform matrix is invertible.
As an immediate consequence of Chebotarëv’s theorem, every choice of rows from
the DFT produces a full spark harmonic frame, provided $N$ is prime. This
application of Chebotarëv’s theorem was first used by Candès et al. [9] for
sparse signal processing. Note that each of these frames are equal-norm and
tight by construction. Harmonic frames can also be designed to have minimal
coherence; Xia et al. [49] produces harmonic equiangular tight frames by
selecting row indices which form a difference set in $\mathbb{Z}_{N}$.
Interestingly, most known families of difference sets in $\mathbb{Z}_{N}$
require $N$ to be prime [27], and so the corresponding harmonic equiangular
tight frames are guaranteed to be full spark by Chebotarëv’s theorem. In the
following, we use Chebotarëv’s theorem to demonstrate full spark for a class
of frames which contains harmonic frames, namely, frames which arise from
concatenating harmonic frames with any number of identity basis elements:
###### Theorem 6 (cf. [46, Theorem 1.1]).
Let $N$ be prime, and pick any $M\leq N$ rows of the $N\times N$ discrete
Fourier transform matrix to form the harmonic frame $H$. Next, pick any $K\leq
M$, and take $D$ to be the $M\times M$ diagonal matrix whose first $K$
diagonal entries are $\sqrt{\frac{N+K-M}{MN}}$, and whose remaining $M-K$
entries are $\sqrt{\frac{N+K}{MN}}$. Then concatenating $DH$ with the first
$K$ identity basis elements produces an $M\times(N+K)$ full spark unit norm
tight frame.
As an example, when $N=5$ and $K=1$, we can pick $M=3$ rows of the $5\times 5$
DFT which are indexed by $\\{0,1,4\\}$. In this case, $D$ makes the entries of
the first DFT row have size $\sqrt{\frac{1}{5}}$ and the entries of the
remaining rows have size $\sqrt{\frac{2}{5}}$. Concatenating with the first
identity basis element then produces an equiangular tight frame which is full
spark:
$F=\left[\begin{array}[]{llllll}\sqrt{\frac{1}{5}}&\sqrt{\frac{1}{5}}&\sqrt{\frac{1}{5}}&\sqrt{\frac{1}{5}}&\sqrt{\frac{1}{5}}&1\\\
\sqrt{\frac{2}{5}}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}/5}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}2/5}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}3/5}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}4/5}&0\\\
\sqrt{\frac{2}{5}}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}4/5}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}3/5}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}2/5}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}/5}&0\\\
\end{array}\right].$ (12)
###### Proof of Theorem 6.
Let $F$ denote the resulting $M\times(N+K)$ frame. We start by verifying that
$F$ is unit norm. Certainly, the identity basis elements have unit norm. For
the remaining frame elements, the modulus of each entry is determined by $D$,
and so the norm squared of each frame element is
$K(\tfrac{N+K-M}{MN})+(M-K)(\tfrac{N+K}{MN})=1.$
To demonstrate that $F$ is tight, it suffices to show that
$FF^{*}=\frac{N+K}{M}I_{M}$. The rows of $DH$ are orthogonal since they are
scaled rows of the DFT, while the rows of the identity portion are orthogonal
because they have disjoint support. Thus, $FF^{*}$ is diagonal. Moreover, the
norm squared of each of the first $K$ rows is
$N(\frac{N+K-M}{MN})+1=\frac{N+K}{M}$, while the norm squared of each of the
remaining rows is $N(\frac{N+K}{MN})=\frac{N+K}{M}$, and so
$FF^{*}=\frac{N+K}{M}I_{M}$.
To demonstrate that $F$ is full spark, first note that every $M\times M$
submatrix of $DH$ is invertible since
$|\mathrm{det}((DH)_{\mathcal{K}})|=|\mathrm{det}(DH_{\mathcal{K}})|=|\mathrm{det}(D)||\mathrm{det}(H_{\mathcal{K}})|>0,$
by Chebotarëv’s theorem. Also, in the case where $K=M$, we note that the
$M\times M$ submatrix of $F$ composed solely of identity basis elements is
trivially invertible. The only remaining case to check is when identity basis
elements and columns of $DH$ appear in the same $M\times M$ submatrix
$F_{\mathcal{K}}$. In this case, we may shuffle the rows of $F_{\mathcal{K}}$
to have the form
$\begin{bmatrix}A&0\\\ B&I_{K}\end{bmatrix}.$
Since shuffling rows has no impact on the size of the determinant, we may
further use a determinant identity on block matrices to get
$|\mathrm{det}(F_{\mathcal{K}})|=\bigg{|}\mathrm{det}\begin{bmatrix}A&0\\\
B&I_{K}\end{bmatrix}\bigg{|}=|\mathrm{det}(A)\mathrm{det}(I_{K})|=|\mathrm{det}(A)|.$
Since $A$ is a multiple of a square submatrix of the $N\times N$ DFT, we are
done by Chebotarëv’s theorem. ∎
As an example of Theorem 6, pick $N$ to be a prime congruent to $1\bmod 4$,
and select $\frac{N+1}{2}$ rows of the $N\times N$ DFT according to the index
set $\mathcal{M}:=\\{k^{2}:k\in\mathbb{Z}_{N}\\}$. If we take $K=1$, the
process in Theorem 6 produces an equiangular tight frame of redundancy $2$,
which can be verified using quadratic Gauss sums; in the case where $N=5$,
this construction produces (12). Note that this corresponds to a special case
of a construction in Zauner’s thesis [51], which was later studied by Renes
[38] and Strohmer [43]. Theorem 6 says that this construction is full spark.
Maximally sparse frames have recently become a subject of active research [12,
20]. We note that when $K=M$, Theorem 6 produces a maximally sparse
$M\times(N+K)$ full spark frame, having a total of $M(M-1)$ zero entries. To
see that this sparsity level is maximal, we note that if the frame had any
more zero entries, then at least one of the rows would have $M$ zero entries,
meaning the corresponding $M\times M$ submatrix would have a row of all zeros
and hence a zero determinant. Similar ideas were studied previously by
Nakamura and Masson [34].
Another interesting case is where $K=M=N$, i.e., when the frame constructed in
Theorem 6 is a union of the unitary DFT and identity bases. Unions of
orthonormal bases have received considerable attention in the context of
sparse approximation [16, 47]. In fact, when $N$ is a perfect square,
concatenating the DFT with an identity basis forms the canonical example $F$
of a dictionary with small spark [16]. To be clear, the Dirac comb of
$\sqrt{N}$ spikes is an eigenvector of the DFT, and so concatenating this comb
with the negative of its Fourier transform produces a $2\sqrt{N}$-sparse
vector in the nullspace of $F$. In stark contrast, when $N$ is prime, Theorem
6 shows that $F$ is full spark.
The vast implications of Chebotarëv’s theorem leads one to wonder whether the
result admits any interesting generalization. In this direction, Candès et al.
[9] note that any such generalization must somehow account for the nontrivial
subgroups of $\mathbb{Z}_{N}$ which are not present when $N$ is prime.
Certainly, if one could characterize the full spark submatrices of a general
DFT, this would provide ample freedom to optimize full spark frames for
additional considerations. While we do not have a characterization for the
general case, we do have one for the case where $N$ is a prime power. Before
stating the result, we require a definition:
###### Definition 7.
We say a subset $\mathcal{M}\subseteq\mathbb{Z}_{N}$ is _uniformly distributed
over the divisors of $N$_ if, for every divisor $d$ of $N$, the $d$ cosets of
$\langle d\rangle$ partition $\mathcal{M}$ into subsets, each of size
$\lfloor\frac{|\mathcal{M}|}{d}\rfloor$ or
$\lceil\frac{|\mathcal{M}|}{d}\rceil$.
At first glance, this definition may seem rather unnatural, but we will
discover some important properties of uniformly distributed rows from the DFT.
As an example, we take a short detour by considering the _restricted isometry
property (RIP)_ , which has received considerable attention recently for its
use in compressed sensing. We say a matrix $F$ is $(K,\delta)$-RIP if
$(1-\delta)\|x\|^{2}\leq\|Fx\|^{2}\leq(1+\delta)\|x\|^{2}\qquad\mbox{whenever
}\|x\|_{0}\leq K.$
Candès and Tao [11] demonstrated that the sparsest $\Psi^{*}x$ which satisfies
(3) can be found using $\ell_{1}$-minimization, provided $F$ is
$(2K,\sqrt{2}-1)$-RIP. Later, Rudelson and Vershynin [40] showed that a matrix
of random rows from a DFT and normalized columns is RIP with high probability.
We will show that harmonic frames satisfy RIP only if the selected row indices
are nearly uniformly distributed over sufficiently small divisors of $N$.
To this end, recall that for any divisor $d$ of $N$, the Fourier transform of
the $d$-sparse normalized Dirac comb
$\frac{1}{\sqrt{d}}\chi_{\langle\frac{N}{d}\rangle}$ is the
$\frac{N}{d}$-sparse normalized Dirac comb $\sqrt{\frac{d}{N}}\chi_{\langle
d\rangle}$. Let $F$ be the $N\times N$ unitary DFT, and let $H$ be the
harmonic frame which arises from selecting rows of $F$ indexed by
$\mathcal{M}$ and then normalizing the columns. In order for $H$ to be
$(K,\delta)$-RIP, $\mathcal{M}$ must contain at least one member of $\langle
d\rangle$ for every divisor $d$ of $N$ which is $\leq K$, since otherwise
$H\tfrac{1}{\sqrt{d}}\chi_{\langle\frac{N}{d}\rangle}=\sqrt{\tfrac{N}{|\mathcal{M}|}}(F\tfrac{1}{\sqrt{d}}\chi_{\langle\frac{N}{d}\rangle})_{\mathcal{M}}=\sqrt{\tfrac{N}{|\mathcal{M}|}}\Big{(}\sqrt{\tfrac{d}{N}}\chi_{\langle
d\rangle}\Big{)}_{\mathcal{M}}=\sqrt{\tfrac{d}{|\mathcal{M}|}}\chi_{\mathcal{M}\cap\langle
d\rangle}=0,$
which violates the lower RIP bound at
$x=\frac{1}{\sqrt{d}}\chi_{\langle\frac{N}{d}\rangle}$. In fact, the RIP
bounds indicate that
$\|Hx\|^{2}=\|H\tfrac{1}{\sqrt{d}}\chi_{\langle\frac{N}{d}\rangle}\|^{2}=\Big{\|}\sqrt{\tfrac{d}{|\mathcal{M}|}}\chi_{\mathcal{M}\cap\langle
d\rangle}\Big{\|}^{2}=\tfrac{d}{|\mathcal{M}|}|\mathcal{M}\cap\langle
d\rangle|$
cannot be more than $\delta$ away from $\|x\|^{2}=1$. Similarly, taking $x$ to
be $\frac{1}{\sqrt{d}}\chi_{\langle\frac{N}{d}\rangle}$ modulated by $a$,
i.e., $x[n]:=\frac{1}{\sqrt{d}}\chi_{\langle\frac{N}{d}\rangle}[n]e^{2\pi
ian/N}$ for every $n\in\mathbb{Z}_{N}$, gives that
$\|Hx\|^{2}=\frac{d}{|\mathcal{M}|}|\mathcal{M}\cap(a+\langle d\rangle)|$ is
also no more than $\delta$ away from $1$. This observation gives the following
result:
###### Theorem 8.
Select rows indexed by $\mathcal{M}\subseteq\mathbb{Z}_{N}$ from the $N\times
N$ discrete Fourier transform matrix and then normalize the columns to produce
the harmonic frame $H$. Then $H$ satisfies the $(K,\delta)$-restricted
isometry property only if
$\Big{|}\big{|}\mathcal{M}\cap(a+\langle
d\rangle)\big{|}-\tfrac{|\mathcal{M}|}{d}\Big{|}\leq\tfrac{|\mathcal{M}|}{d}\delta$
for every divisor $d$ of $N$ with $d\leq K$ and every $a=0,\ldots,d-1$.
Now that we have an intuition for uniform distribution in terms of modulated
Dirac combs and RIP, we take this condition to the extreme by considering
uniform distribution over all divisors. Doing so produces a complete
characterization of full spark harmonic frames when $N$ is a prime power:
###### Theorem 9.
Let $N$ be a prime power, and select rows indexed by
$\mathcal{M}\subseteq\mathbb{Z}_{N}$ from the $N\times N$ discrete Fourier
transform matrix to build the submatrix $F$. Then $F$ is full spark if and
only if $\mathcal{M}$ is uniformly distributed over the divisors of $N$.
Note that, perhaps surprisingly, an index set $\mathcal{M}$ can be uniformly
distributed over $p$ but not over $p^{2}$, and vice versa. For example,
$\mathcal{M}=\\{0,1,4\\}$ is uniformly distributed over $2$ but not $4$, while
$\mathcal{M}=\\{0,2\\}$ is uniformly distributed over $4$ but not $2$.
Since the first $M$ rows of a DFT form a full spark Vandermonde matrix, let’s
check that this index set is uniformly distributed over the divisors of $N$.
For each divisor $d$ of $N$, we partition the first $M$ indices into the $d$
cosets of $\langle d\rangle$. Write $M=qd+r$ with $0\leq r<d$. The first $qd$
of the $M$ indices are distributed equally amongst all $d$ cosets, and then
the remaining $r$ indices are distributed equally amongst the first $r$
cosets. Overall, the first $r$ cosets contain
$q+1=\lfloor\frac{M}{d}\rfloor+1$ indices, while the remaining $d-r$ cosets
have $q=\lfloor\frac{M}{d}\rfloor$ indices; thus, the first $M$ indices are
indeed uniformly distributed over the divisors of $N$. Also, when $N$ is
prime, _every_ subset of $\mathbb{Z}_{N}$ is uniformly distributed over the
divisors of $N$ in a trivial sense. In fact, Chebotarëv’s theorem follows
immediately from Theorem 9. In some ways, portions of our proof of Theorem 9
mirror recurring ideas in the existing proofs of Chebotarëv’s theorem [15, 18,
41, 46]. For the sake of completeness, we provide the full argument and save
the reader from having to parse portions of proofs from multiple references.
We start with the following lemmas, whose proofs are based on the proofs of
Lemmas 1.2 and 1.3 in [46].
###### Lemma 10.
Let $N$ be a power of some prime $p$, and let $P(z_{1},\ldots,z_{M})$ be a
polynomial with integer coefficients. Suppose there exists $N$th roots of
unity $\\{\omega_{m}\\}_{m=1}^{M}$ such that
$P(\omega_{1},\ldots,\omega_{M})=0$. Then $P(1,\ldots,1)$ is a multiple of
$p$.
###### Proof.
Denoting $\omega:=e^{-2\pi i/N}$, then for every $m=1,\ldots,M$, we have
$\omega_{m}=\omega^{k_{m}}$ for some $0\leq k_{m}<N$. Defining the polynomial
$Q(z):=P(z^{k_{1}},\ldots,z^{k_{M}})$, we have $Q(\omega)=0$ by assumption.
Also, $Q(z)$ is a polynomial with integer coefficients, and so it must be
divisible by the minimal polynomial of $\omega$, namely, the cyclotomic
polynomial $\Phi_{N}(z)$. Evaluating both polynomials at $z=1$ then gives that
$p=\Phi_{N}(1)$ divides $Q(1)=P(1,\ldots,1)$. ∎
###### Lemma 11.
Let $N$ be a power of some prime $p$, and pick
$\mathcal{M}=\\{m_{i}\\}_{i=1}^{M}\subseteq\mathbb{Z}_{N}$ such that
$\frac{\displaystyle{\prod_{1\leq i<j\leq
M}(m_{j}-m_{i})}}{\displaystyle{\prod_{m=0}^{M-1}m!}}$ (13)
is not a multiple of $p$. Then the rows indexed by $\mathcal{M}$ in the
$N\times N$ discrete Fourier transform form a full spark frame.
###### Proof.
We wish to show that $\mathrm{det}(\omega_{n}^{m})_{m\in\mathcal{M},1\leq
n\leq M}\neq 0$ for all $M$-tuples of distinct $N$th roots of unity
$\\{\omega_{n}\\}_{n=1}^{M}$. Define the polynomial
$D(z_{1},\ldots,z_{M}):=\mathrm{det}(z_{n}^{m})_{m\in\mathcal{M},1\leq n\leq
M}$. Since columns $i$ and $j$ of $(z_{n}^{m})_{m\in\mathcal{M},1\leq n\leq
M}$ are identical whenever $z_{i}=z_{j}$, we know that $D$ vanishes in each of
these instances, and so we can factor:
$D(z_{1},\ldots,z_{M})=P(z_{1},\ldots,z_{M})\prod_{1\leq i<j\leq
M}(z_{j}-z_{i})$
for some polynomial $P(z_{1},\ldots,z_{M})$ with integer coefficients. By
Lemma 10, it suffices to show that $P(1,\ldots,1)$ is not a multiple of $p$,
since this implies $D(\omega_{1},\ldots,\omega_{M})$ is nonzero for all
$M$-tuples of distinct $N$th roots of unity $\\{\omega_{n}\\}_{n=1}^{M}$.
To this end, we proceed by considering
$A:=\bigg{(}z_{1}\frac{\partial}{\partial
z_{1}}\bigg{)}^{0}\bigg{(}z_{2}\frac{\partial}{\partial
z_{2}}\bigg{)}^{1}\cdots\bigg{(}z_{M}\frac{\partial}{\partial
z_{M}}\bigg{)}^{M-1}D(z_{1},\ldots,z_{M})\bigg{|}_{z_{1}=\cdots=z_{M}=1}.$
(14)
To compute $A$, we note that each application of
$z_{j}\frac{\partial}{\partial z_{j}}$ produces terms according to the product
rule. For some terms, a linear factor of the form $z_{j}-z_{i}$ or
$z_{i}-z_{j}$ is replaced by $z_{j}$ or $-z_{j}$, respectively. For each the
other terms, these linear factors are untouched, while another factor, such as
$P(z_{1},\ldots,z_{M})$, is differentiated and multiplied by $z_{j}$. Note
that there are a total of $M(M-1)/2$ linear factors, and only $M(M-1)/2$
differentiation operators to apply. Thus, after expanding every product rule,
there will be two types of terms: terms in which every differentiation
operator was applied to a linear factor, and terms which have at least one
linear factor remaining untouched. When we evaluate at $z_{1}=\cdots=z_{M}=1$,
the terms with linear factors vanish, and so the only terms which remain came
from applying every differentiation operator to a linear factor. Furthermore,
each of these terms before the evaluation is of the form
$P(z_{1},\ldots,z_{M})\prod_{1\leq i<j\leq M}z_{j}$, and so evaluation at
$z_{1}=\cdots=z_{M}=1$ produces a sum of terms of the form $P(1,\ldots,1)$; to
determine the value of $A$, it remains to count these terms. The $M-1$ copies
of $z_{M}\frac{\partial}{\partial z_{M}}$ can only be applied to linear
factors of the form $z_{M}-z_{i}$, of which there are $M-1$, and so there are
a total of $(M-1)!$ ways to distribute these operators. Similarly, there are
$(M-2)!$ ways to distribute the $M-2$ copies of
$z_{M-1}\frac{\partial}{\partial z_{M-1}}$ amongst the $M-2$ linear factors of
the form $z_{M-1}-z_{i}$. Continuing in this manner produces an expression for
$A$:
$A=(M-1)!(M-2)!\cdots 1!0!~{}P(1,\ldots,1).$ (15)
For an alternate expression of $A$, we substitute the definition of
$D(z_{1},\ldots,z_{M})$ into $\eqref{eq.A defn}$. Here, we exploit the
multilinearity of the determinant and the fact that
$(z_{n}\frac{\partial}{\partial z_{n}})z_{n}^{m}=mz_{n}^{m}$ to get
$A=\mathrm{det}(m^{n-1})_{m\in\mathcal{M},1\leq n\leq M}=\prod_{1\leq i<j\leq
M}(m_{j}-m_{i}),$ (16)
where the final equality uses the fact that $(m^{n-1})_{m\in\mathcal{M},1\leq
n\leq M}$ is the transpose of a Vandermonde matrix. Equating (15) to (16)
reveals that (13) is an expression for $P(1,\ldots,1)$. Thus, by assumption,
$P(1,\ldots,1)$ is not a multiple of $p$, and so we are done. ∎
###### Proof of Theorem 9.
($\Leftarrow$) We will use Lemma 11 to demonstrate that $F$ is full spark. To
apply this lemma, we need to establish that (13) is not a multiple of $p$, and
to do this, we will show that there are as many $p$-divisors in the numerator
of (13) as there are in the denominator. We start by counting the $p$-divisors
of the denominator:
$\prod_{m=0}^{M-1}m!=\prod_{m=1}^{M-1}\prod_{\ell=1}^{m}\ell=\prod_{\ell=1}^{M-1}\prod_{m=1}^{M-l}\ell.$
(17)
For each pair of integers $k,a\geq 1$, there are $\max\\{M-ap^{k},~{}0\\}$
factors in (17) of the form $\ell=ap^{k}$. By adding these, we count each
factor $\ell$ as many times as it can be expressed as a multiple of a power of
$p$, which equals the number of $p$-divisors in $\ell$. Thus, the number of
$p$-divisors of (17) is
$\sum_{k=1}^{\lfloor\log_{p}M\rfloor}\sum_{a=1}^{\lfloor\frac{M}{p^{k}}\rfloor}(M-ap^{k}).$
(18)
Next, we count the $p$-divisors of the numerator of (13). To do this, we use
the fact that $\mathcal{M}$ is uniformly distributed over the divisors of $N$.
Since $N$ is a power of $p$, the only divisors of $N$ are smaller powers of
$p$. Also, the cosets of $\langle p^{k}\rangle$ partition $\mathcal{M}$ into
subsets $S_{k,b}:=\\{m_{i}\equiv b\mod p^{k}\\}$. We note that $m_{j}-m_{i}$
is a multiple of $p^{k}$ precisely when $m_{i}$ and $m_{j}$ belong to the same
subset $S_{k,b}$ for some $0\leq b<p^{k}$. To count $p$-divisors, we again
count each factor $m_{j}-m_{i}$ as many times as it can be expressed as a
multiple of a prime power:
$\sum_{k=1}^{\lfloor\log_{p}M\rfloor}\sum_{b=0}^{p^{k}-1}\binom{|S_{k,b}|}{2}.$
(19)
Write $M=qp^{k}+r$ with $0\leq r<p^{k}$. Then
$q=\lfloor\frac{M}{p^{k}}\rfloor$. Since $\mathcal{M}$ is uniformly
distributed over $p^{k}$, there are $r$ subsets $S_{k,b}$ with $q+1$ elements
and $p^{k}-r$ subsets with $q$ elements. We use this to get
$\sum_{b=0}^{p^{k}-1}\binom{|S_{k,b}|}{2}=\binom{q+1}{2}r+\binom{q}{2}(p^{k}-r)=\frac{q}{2}\Big{(}(q-1)p^{k}+2r+(qp^{k}-qp^{k})\Big{)}.$
Rearranging and substituting $M=qp^{k}+r$ then gives
$\sum_{b=0}^{p^{k}-1}\binom{|S_{k,b}|}{2}=\frac{q}{2}\Big{(}2M-(q+1)p^{k}\Big{)}=Mq-\binom{q+1}{2}p^{k}=\sum_{a=1}^{\lfloor\frac{M}{p^{k}}\rfloor}(M-ap^{k}).$
Thus, there are as many $p$-divisors in the numerator (19) as there are in the
denominator (18), and so (13) is not divisible by $p$. Lemma 11 therefore
gives that $F$ is full spark.
($\Rightarrow$) We will prove that this direction holds regardless of whether
$N$ is a prime power. Suppose $\mathcal{M}\subseteq\mathbb{Z}_{N}$ is not
uniformly distributed over the divisors of $N$. Then there exists a divisor
$d$ of $N$ such that one of the cosets of $\langle d\rangle$ intersects
$\mathcal{M}$ with $\leq\lfloor\frac{M}{d}\rfloor-1$ or
$\geq\lceil\frac{M}{d}\rceil+1$ indices. Notice that if a coset of $\langle
d\rangle$ intersects $\mathcal{M}$ with $\leq\lfloor\frac{M}{d}\rfloor-1$
indices, then the complement $\mathcal{M}^{\mathrm{c}}$ intersects the same
coset with
$\geq\lceil\frac{N-M}{d}\rceil+1=\lceil\frac{|\mathcal{M}^{\mathrm{c}}|}{d}\rceil+1$
indices. By Theorem 4(iii), $\mathcal{M}$ produces a full spark harmonic frame
precisely when $\mathcal{M}^{\mathrm{c}}$ produces a full spark harmonic
frame, and so we may assume without loss of generality that there exists a
coset of $\langle d\rangle$ which intersects $\mathcal{M}$ with
$\geq\lceil\frac{M}{d}\rceil+1$ indices.
To prove that the rows with indices in $\mathcal{M}$ are not full spark, we
find column entries which produce a singular submatrix. Writing $M=qd+r$ with
$0\leq r<d$, let $\mathcal{K}$ contain $q=\lfloor\frac{M}{d}\rfloor$ cosets of
$\langle\frac{N}{d}\rangle$ along with $r$ elements from an additional coset.
We claim that the DFT submatrix with row entries $\mathcal{M}$ and column
entries $\mathcal{K}$ is singular. To see this, shuffle the rows and columns
to form a matrix $A$ in which the row entries are grouped into common cosets
of $\langle d\rangle$ and the column entries are grouped into common cosets of
$\langle\frac{N}{d}\rangle$. This breaks $A$ into rank-1 submatrices: each
pair of cosets $a+\langle d\rangle$ and $b+\langle\frac{N}{d}\rangle$ produces
a submatrix
$(\omega^{(a+id)(b+j\frac{N}{d})})_{i\in\mathcal{I},j\in\mathcal{J}}=\omega^{ab}(\omega^{bdi}\omega^{a\frac{N}{d}j})_{i\in\mathcal{I},j\in\mathcal{J}}$
for some index sets $\mathcal{I}$ and $\mathcal{J}$; this is a rank-1 outer
product. Let $\mathcal{L}$ be the largest intersection between $\mathcal{M}$
and a coset of $\langle d\rangle$. Then
$|\mathcal{L}|\geq\lceil\frac{M}{d}\rceil+1$ is the number of rows in the
tallest of these rank-1 submatrices. Define $A_{\mathcal{L}}$ to be the
$M\times M$ matrix with entries $A_{\mathcal{L}}[i,j]=A[i,j]$ whenever
$i\in\mathcal{L}$ and zero otherwise. Then
$\mathrm{Rank}(A)=\mathrm{Rank}(A_{\mathcal{L}}+A-A_{\mathcal{L}})\leq\mathrm{Rank}(A_{\mathcal{L}})+\mathrm{Rank}(A-A_{\mathcal{L}}).$
(20)
Since $A-A_{\mathcal{L}}$ has $|\mathcal{L}|$ rows of zero entries, we also
have
$\mathrm{Rank}(A-A_{\mathcal{L}})\leq M-|\mathcal{L}|\leq
M-(\lceil\tfrac{M}{d}\rceil+1).$ (21)
Moreover, since we can decompose $A_{\mathcal{L}}$ into a sum of
$\lceil\frac{M}{d}\rceil$ zero-padded rank-1 submatrices, we have
$\mathrm{Rank}(A_{\mathcal{L}})\leq\lceil\frac{M}{d}\rceil$. Combining this
with (20) and (21) then gives that $\mathrm{Rank}(A)\leq M-1$, and so the DFT
submatrix is not invertible. ∎
Note that our proof of Theorem 9 establishes the necessity of having row
indices uniformly distributed over the divisors of $N$ in the general case.
This leaves some hope for completely characterizing full spark harmonic
frames. Naturally, one might suspect that the uniform distribution condition
is sufficient in general, but this suspicion fails when $N=10$. Indeed, the
following DFT submatrix is singular despite the row indices being uniformly
distributed over the divisors of $10$:
$(e^{-2\pi imn/10})_{m\in\\{0,1,3,4\\},n\in\\{0,1,2,6\\}}.$
Also, just as we used Chebotarëv’s theorem to analyze the harmonic equiangular
tight frames from Xia et al. [49], we can also use Theorem 9 to determine
whether harmonic equiangular tight frames with a prime power number of frame
elements are full spark. Unfortunately, none of the infinite families in [49]
have the number of frame elements in the form of a prime power (other than
primes). Luckily, there is at least one instance in which the number of frame
elements happens to be a prime power: the harmonic frames that arise from
Singer difference sets have $M=\frac{q^{d}-1}{q-1}$ and
$N=\frac{q^{d+1}-1}{q-1}$ for a prime power $q$ and an integer $d\geq 2$; when
$q=3$ and $d=4$, the number of frame elements $N=11^{2}$ is a prime power. In
this case, the row indices we select are
$\displaystyle\mathcal{M}=$
$\displaystyle\\{1,2,3,6,7,9,11,18,20,21,25,27,33,34,38,41,44,47,53,54,55,56,$
$\displaystyle~{}~{}~{}58,59,60,63,64,68,70,71,75,81,83,89,92,99,100,102,104,114\\},$
but these are not uniformly distributed over 11, and so the corresponding
harmonic frame is not full spark by Theorem 9.
## 3\. Full spark Parseval frames are dense
Recently, Lu and Do [29] showed that full spark frames are dense in the entire
set of matrices. This corresponds to our intuition that a matrix whose entries
are independent continuous random variables is full spark with probability
one, which was also recently proved by Blumensath and Davies [5]. By contrast,
as noted by Gorodnitsky and Rao [22], certain classes of frames which arise in
practice, such as in physical tomography, are never full spark. This issue
also occurs in frame theory: Steiner ETFs form one of the largest known
classes of ETFs, and yet none of them are full spark [20]. As such,
Bourguignon et al. [6] were prompted to prove that, among all Vandermonde
frames with bases in the complex unit circle, full spark frames are dense. In
this section, we consider a very important class of frames, namely, those
which exhibit Parseval tightness, where the frame bound is $1$. Specifically,
we show that full spark Parseval frames are dense in the entire set of
Parseval frames. Unlike the previous work in this vein, our techniques exploit
general concepts in algebraic geometry, lending themselves to future
application in proving further density results.
In order to make our arguments rigorous, we view each entry $F_{mn}$ of the
$M\times N$ matrix $F$ in terms of its real and imaginary parts:
$x_{mn}+iy_{mn}$; this decomposition will become helpful later when we
consider inner products between rows of $F$, which are not algebraic
operations on complex vectors. Recall that $F$ is full spark precisely when
each $M\times M$ submatrix has nonzero determinant. We note that for each
submatrix, the determinant is a polynomial in the $x_{mn}$’s and $y_{mn}$’s,
and having this polynomial be nonzero is equivalent to having either its real
or imaginary part be nonzero. This naturally leads us to the following
definition from algebraic geometry: A _real algebraic variety_ is the set of
common zeros of a finite set of polynomials, that is, given polynomials
$p_{1},\ldots,p_{r}\in\mathbb{R}[x_{1},\ldots,x_{k}]$, we define the
corresponding real algebraic variety by
$V(p_{1},\ldots,p_{r}):=\\{x\in\mathbb{R}^{k}:p_{1}(x)=\cdots=p_{r}(x)=0\\}.$
Each submatrix determinant corresponds to a real algebraic variety
$V\subseteq\mathbb{R}^{2MN}$ of two polynomials, and having this determinant
be nonzero is equivalent to restricting to the complement of $V$.
In general, a variety is equipped with a topology known as the _Zariski
topology_ , in which subvarieties are the closed sets. As such, the set of
matrices with a nonzero determinant forms a Zariski-open set, since it is the
complement of a variety. In order to exploit this Zariski-openness, we require
the additional concept of irreducibility: A variety is said to be
_irreducible_ if it cannot be written as a finite union of proper
subvarieties. As an example, the entire space $\mathbb{R}^{k}$ is the variety
which corresponds to the zero polynomial; in this case, every proper
subvariety is lower-dimensional, and so $\mathbb{R}^{k}$ is trivially
irreducible. On the other hand, the variety in $\mathbb{R}^{2}$ defined by
$xy=0$ is not irreducible because it can be expressed as the union of
varieties defined by $x=0$ and $y=0$. Irreducibility is important because it
says something about Zariski-open sets:
###### Theorem 12.
If $V$ is an irreducible algebraic variety, then every nonempty Zariski-open
subset of $V$ is dense in $V$ in the standard topology.
For example, the hyperplane defined by $x_{1}=0$ is a subvariety of
$\mathbb{R}^{k}$, and since $\mathbb{R}^{k}$ is irreducible, the complement of
the hyperplane is dense in the standard topology. Going back to the variety
defined by $xy=0$, we know it can be expressed as a union of proper
subvarieties, namely the $x$\- and $y$-axes, and complementing one gives a
subset of the other, neither of which is dense in the entire variety.
We are now ready to prove the following result:
###### Theorem 13.
Every matrix is arbitrarily close to a full spark frame.
###### Proof.
In an $M\times N$ matrix, there are $\binom{N}{M}$ submatrices of size
$M\times M$, and the determinant of each of these submatrices corresponds to a
variety of two real polynomials. Since the set of $M\times N$ full spark
frames is defined as the (finite) intersection of the complements of these
varieties, it is a Zariski-open subset of the irreducible variety
$\mathbb{R}^{2MN}$ of all $M\times N$ matrices. Moreover, this set is nonempty
since it contains the matrix formed by the first $M$ rows of the $N\times N$
DFT, and so we are done by Theorem 12. ∎
We now focus on the set of _Parseval frames_ , that is, tight frames with
frame bound $1$. Note that $M\times N$ Parseval frames are characterized by
the rows forming an orthonormal system of size $M$ in $N$-dimensional space.
The set of all such orthonormal systems is known as the _Stiefel manifold_ ,
denoted $\mathrm{St}(M,N)$. In general, a _manifold_ is a set of vectors with
a well-defined tangent space at every point in the set. Since this manifold
property is nice, we would like to think of varieties as manifolds, but there
exist varieties with points at which tangent spaces are not well-defined; such
points are called _singularities_. Note that our definition of singularity is
geometric, i.e., where the variety fails to be a real differentiable manifold,
as opposed to algebraic. For example, the variety defined by $xy=0$ is the
union of the $x$\- and $y$-axes, and as such, has a singularity at $x=y=0$.
Certainly with different types of varieties, there are other types of
singularities which may arise, but there are also many varieties which do not
have singularities at all, and we call these varieties _nonsingular_.
Nonsingularity is a useful property for the following reason:
###### Theorem 14.
An algebraic variety which is nonsingular and connected is necessarily
irreducible.
This result follows from Theorem I.5.1 and Remark III.7.9.1 of Hartshorne
[25], which assumes that the variety is over an algebraically closed field,
unlike $\mathbb{R}$; however, the proof is unaffected when removing the
algebraically-closed assumption.
Note that the orthonormality conditions which characterize Parseval frames $F$
can be expressed as polynomial equations in the real and imaginary parts of
the entries $F_{mn}$. As such, we may view the Stiefel manifold as a real
algebraic variety; this variety is nonsingular because it is a manifold.
Moreover, the variety is connected because the (connected) unitary group
$\mathrm{U}(N)$ acts transitively on $\mathrm{St}(M,N)$. Thus by Theorem 14,
the variety of Parseval frames is irreducible. Having established this, we can
now prove the following result:
###### Theorem 15.
Every Parseval frame is arbitrarily close to a full spark Parseval frame.
###### Proof.
Proceeding as in the proof of Theorem 13, we have that the set of $M\times N$
full spark Parseval frames is Zariski-open in the irreducible variety of all
$M\times N$ Parseval frames. Again considering the first $M$ rows of the
$N\times N$ DFT, we know this set is nonempty, and so we are done by Theorem
12. ∎
We note that Theorems 13 and 15 are also true when we further require the
frames to be real. In this case, we cannot use the first $M$ rows of the
$N\times N$ DFT to establish that the Zariski-open sets are nonempty. For the
real version of Theorem 13, we use an $M\times N$ Vandermonde matrix with
distinct real bases; see Lemma 2. However, this construction must be modified
to use it in the proof of the real version of Theorem 15, since such
Vandermonde matrices will not be tight; see Theorem 3. Given an $M\times N$
full spark frame $F$, the modification $G:=(FF^{*})^{-1/2}F$ is full spark and
Parseval; indeed, $GG^{*}=I_{M}$, and since $F$ is a frame, $(FF^{*})^{-1/2}$
is full rank, and so the columns of $G$ are linearly independent precisely
when the corresponding columns of $F$ are linearly independent.
Another way that the proof of Theorem 15 changes in the real case is in
verifying that the real Stiefel manifold is irreducible. Just as in the
complex case, this follows from Theorem 14 since $\mathrm{St}(M,N)$ is
connected [37], but the fact that $\mathrm{St}(M,N)$ is connected in the real
case is not immediate. By analogy, the orthogonal group $\mathrm{O}(N)$
certainly acts transitively on $\mathrm{St}(M,N)$, but unlike the unitary
group, the orthogonal group has two connected components. Intuitively, this is
resolved by the fact that $N>M$, granting additional freedom of movement
throughout $\mathrm{St}(M,N)$.
In addition to Theorems 13 and 15, we would like a similar result for unit
norm tight frames, i.e., that every unit norm tight frame is arbitrarily close
to a full spark unit norm tight frame. Certainly, the set of unit norm tight
frames is a real algebraic variety, but it is unclear whether this variety is
irreducible. Without knowing whether the variety is irreducible, we can follow
the proofs of Theorems 13 and 15 to conclude that full spark unit norm tight
frames are dense in the irreducible components in which they exist—a far cry
from the density result we seek. This illustrates a significant gap in our
current understanding of the variety of unit norm tight frames. It should be
mentioned that Strawn [42] showed that the variety of $M\times N$ unit norm
tight frames (over real or complex space) is nonsingular precisely when $M$
and $N$ are relatively prime. Additionally, Dykema and Strawn [17] proved that
the variety of $2\times N$ real unit norm tight frames is connected, and so by
Theorem 14, this variety is irreducible when $N$ is odd. It is unknown whether
the variety of $M\times N$ unit norm tight frames is connected in general.
Finally, we note that a Theorem 15 gives a weaker version of the result we
would like: every unit norm tight frame is arbitrarily close to a full spark
tight frame with frame element lengths arbitrarily close to $1$.
## 4\. The computational complexity of verifying full spark
In the previous section, we demonstrated the abundance of full spark frames,
even after imposing the additional condition of tightness. But how much
computation is required to check whether a particular frame is full spark? At
the heart of the matter is computational complexity theory, which provides a
rigorous playing field for expressing how hard certain problems are. In this
section, we consider the complexity of the following problem:
###### Problem 16 (Full Spark).
Given a matrix, is it full spark?
For the lay mathematician, Full Spark is “obviously” ${\mathsf{NP}}$-hard
because the easiest way he can think to solve it for a given $M\times N$
matrix is by determining whether each of the $M\times M$ submatrices is
invertible; computing $\binom{N}{M}$ determinants would do, but this would
take a lot of time, and so Full Spark must be ${\mathsf{NP}}$-hard. However,
computing $\binom{N}{M}$ determinants may not necessarily be the fastest way
to test whether a matrix is full spark. For example, perhaps there is an easy-
to-calculate expression for the product of the determinants; after all, this
product is nonzero precisely when the matrix is full spark. Recall that
Theorem 9 gives a very straightforward litmus test for Full Spark in the
special case where the matrix is formed by rows of a DFT of prime-power
order—who’s to say that a version of this test does not exist for the general
case? If such a test exists, then it would suffice to find it, but how might
one disprove the existence of any such test? Indeed, since we are concerned
with the necessary amount of computation, as opposed to a sufficient amount,
the lay mathematician’s intuition is a bit misguided.
To discern how much computation is necessary, the main feature of interest is
a problem’s _complexity_. We use complexity to compare problems and determine
whether one is harder than the other. As an example of complexity,
intuitively, doubling an integer is no harder than adding integers, since one
can use addition to multiply by $2$; put another way, the complexity of
doubling is somehow “encoded” in the complexity of adding, and so it must be
lesser (or equal). To make this more precise, complexity theorists use what is
called a _polynomial-time reduction_ , that is, a polynomial-time algorithm
that solves problem $A$ by exploiting an oracle which solves problem $B$; the
reduction indicates that solving problem $A$ is no harder than solving problem
$B$ (up to polynomial factors in time), and we say “$A$ reduces to $B$,” or
$A\leq B$. Since we can use the polynomial-time routine $x+x$ to produce $2x$,
we conclude that doubling an integer reduces to adding integers, as expected.
In complexity theory, problems are categorized into complexity classes
according to the amount of resources required to solve them. For example, the
complexity class ${\mathsf{P}}$ contains all problems which can be solved in
polynomial time, while problems in ${\mathsf{EXP}}$ may require as much as
exponential time. Problems in ${\mathsf{NP}}$ have the defining quality that
solutions can be verified in polynomial time given a certificate for the
answer. As an example, the graph isomorphism problem is in ${\mathsf{NP}}$
because, given an isomorphism between graphs (a certificate), one can verify
that the isomorphism is legit in polynomial time. Clearly,
${\mathsf{P}}\subseteq{\mathsf{NP}}$, since we can ignore the certificate and
still solve the problem in polynomial time. Finally, a problem $B$ is called
${\mathsf{NP}}$-_hard_ if every problem $A$ in ${\mathsf{NP}}$ reduces to $B$,
and a problem is called ${\mathsf{NP}}$-_complete_ if it is both
${\mathsf{NP}}$-hard and in ${\mathsf{NP}}$. In plain speak,
${\mathsf{NP}}$-hard problems are harder than every problem in
${\mathsf{NP}}$, while ${\mathsf{NP}}$-complete problems are the hardest of
problems in ${\mathsf{NP}}$.
At this point, it should be clear that ${\mathsf{NP}}$-hard problems are not
merely problems that seem to require a lot of computation to solve. Certainly,
${\mathsf{NP}}$-hard problems have this quality, as an ${\mathsf{NP}}$-hard
problem can be solved in polynomial time only if ${\mathsf{P}}={\mathsf{NP}}$;
this is an open problem, but it is widely believed that
${\mathsf{P}}\neq{\mathsf{NP}}$. However, there are other problems which seem
hard but are not known to be ${\mathsf{NP}}$-hard (e.g., the graph isomorphism
problem). Rather, to determine whether a problem is ${\mathsf{NP}}$-hard, one
must find a polynomial-time reduction that compares the problem to all
problems in ${\mathsf{NP}}$. To this end, notice that $A\leq B$ and $B\leq C$
together imply $A\leq C$, and so to demonstrate that a problem $C$ is
${\mathsf{NP}}$-hard, it suffices to show that $B\leq C$ for some
${\mathsf{NP}}$-hard problem $B$.
Unfortunately, it can sometimes be difficult to find a deterministic reduction
from one problem to another. One example is reducing the satisfiability
problem (SAT) to the unique satisfiability problem (Unique SAT). To be clear,
SAT is an ${\mathsf{NP}}$-hard problem [28] that asks whether there exists an
input for which a given Boolean function returns “true,” while Unique SAT asks
the same question with an additional promise: that the given Boolean function
is satisfiable only if there is a _unique_ input for which it returns “true.”
Intuitively, Unique SAT is easier than SAT because we might be able to exploit
the additional structure of uniquely satisfiable Boolean functions; thus, it
could be difficult to find a reduction from SAT to Unique SAT. Despite this
intuition, there is a _randomized_ polynomial-time reduction from SAT to
Unique SAT [48]. Defined over all Boolean functions of $n$ variables, the
reduction maps functions that are not satisfiable to other functions that are
not satisfiable, and with probability $\geq\frac{1}{8n}$, it maps satisfiable
functions to uniquely satisfiable functions. After applying this reduction to
a given Boolean function, if a Unique SAT oracle declares “uniquely
satisfiable,” then we know for certain that the original Boolean function was
satisfiable. But the reduction will only map a satisfiable problem to a
uniquely satisfiable problem with probability $\geq\frac{1}{8n}$, so what good
is this reduction? The answer lies in something called _amplification_ ; since
the success probability is, at worst, polynomially small in $n$ (i.e.,
$\geq\frac{1}{p(n)}$), we can repeat our oracle-based randomized algorithm a
polynomial number of times $np(n)$ and achieve an error probability
$\leq(1-\frac{1}{p(n)})^{np(n)}\sim e^{-n}$ which is exponentially small.
In this section, we give a randomized polynomial-time reduction from a problem
in matroid theory. Before stating the problem, we first briefly review some
definitions. To each bipartite graph with bipartition $(E,E^{\prime})$, we
associate a _transversal matroid_ $(E,\mathcal{I})$, where $\mathcal{I}$ is
the collection of subsets of $E$ whose vertices form the ends of a matching in
the bipartite graph; subsets in $\mathcal{I}$ are called independent. Hall’s
marriage theorem [24] gives a remarkable characterization of the independent
sets in a transversal matroid: $B\in\mathcal{I}$ if and only if every subset
$A\subseteq B$ has $\geq|A|$ neighbors in the bipartite graph. Next, just as
spark is the size of the smallest linearly dependent set, the _girth_ of a
matroid is the size of the smallest subset of $E$ that is not in
$\mathcal{I}$. In fact, this analogy goes deeper: A matroid is _representable
over a field_ $\mathbb{F}$ if, for some $M$, there exists a mapping
$\varphi\colon E\rightarrow\mathbb{F}^{M}$ such that $\varphi(A)$ is linearly
independent if and only if $A\in\mathcal{I}$; as such, the girth of
$(E,\mathcal{I})$ is the spark of $\varphi(E)$. In our reduction, we make use
of the fact that every transversal matroid is representable over $\mathbb{R}$
[35]. We are now ready to state the problem from which we will reduce Full
Spark:
###### Problem 17.
Given a bipartite graph, what is the girth of its transversal matroid?
Before giving the reduction, we will show that Problem 17 is
${\mathsf{NP}}$-hard. The result comes from McCormick’s thesis [31], which
credits the proof to Stockmeyer; since [31] is difficult to access and the
proof is instructive, we include it below:
###### Theorem 18.
Problem 17 is ${\mathsf{NP}}$-hard.
###### Proof.
We will reduce from the ${\mathsf{NP}}$-complete clique decision problem,
which asks “Given a graph, does it contain a clique of $K$ vertices?” [28].
First, we may assume $K\geq 4$ without loss of generality, since any such
clique can otherwise be found in cubic time by an exhaustive search. Take a
graph $G=(V,E)$, and consider the bipartite graph $G^{\prime}$ between
disjoint sets $E$ and $V\sqcup\\{1,\ldots,\binom{K}{2}-K-1\\}$, in which
$e\leftrightarrow v$ for every $e\in E$ and $v\in e$, and $e\leftrightarrow k$
for every $e\in E$ and $k\in\\{1,\ldots,\binom{K}{2}-K-1\\}$. We claim that
the girth of the transversal matroid of $G^{\prime}$ is $\binom{K}{2}$
precisely when there exists a $K$-clique in $G$.
We start by analyzing the girth of a transversal matroid. Consider any
dependent set $C\subseteq E$ with $\leq|C|-2$ neighbors in $G^{\prime}$. Then
removing any member $x$ of $C$ will produce a smaller set $C\setminus\\{x\\}$
with $\leq|C|-2=|C\setminus\\{x\\}|-1$ neighbors in $G^{\prime}$, which is
necessarily dependent by the pigeonhole principle. Now consider any dependent
set $C\subseteq E$ with $\geq|C|$ neighbors in $G^{\prime}$. By the Hall’s
marriage theorem, there exists a proper subset $C^{\prime}\subseteq C$ with
$<|C^{\prime}|$ neighbors in $G^{\prime}$, meaning $C^{\prime}$ is a smaller
dependent set. Thus, the girth $c$ is the size of the smallest subset
$C\subseteq E$ with $|C|-1$ total neighbors in $G^{\prime}$.
Suppose $c=\binom{K}{2}$. Then since $C$ is adjacent to every vertex in
$\\{1,\ldots,\binom{K}{2}-K-1\\}$, $C$ has $(c-1)-(\binom{K}{2}-K-1)=K$
neighbors in $V$. These are precisely the vertices $D$ in $G$ which are
induced by the edges in $C$, and so $C$ is contained in the set $C^{\prime}$
of edges induced by $D$, of which there are $\leq\binom{|D|}{2}=\binom{K}{2}$,
with equality only if $D$ induces a $K$-clique in $G$. Since
$\binom{K}{2}=|C|\leq|C^{\prime}|\leq\binom{|D|}{2}=\binom{K}{2}$ implies
equality, there exists a $K$-clique in $G$.
Now suppose there exists a $K$-clique with edges $C$. Then $C$ has
$\binom{K}{2}$ elements and $K+(\binom{K}{2}-K-1)=\binom{K}{2}-1$ neighbors in
$G^{\prime}$. To prove that $c=\binom{K}{2}$, it suffices to show that there
is no smaller subset $C^{\prime}\subseteq E$ with $|C^{\prime}|-1$ total
neighbors in $G^{\prime}$. Suppose, to the contrary, that
$c=\binom{K}{2}-\ell$ for some $\ell>0$. Then there exists
$C^{\prime}\subseteq E$ with $c$ elements and
$(c-1)-(\binom{K}{2}-K-1)=K-\ell$ neighbors in $V$. Note that each $e\in E$
contains two vertices in $G$, and so by the definition of $G^{\prime}$,
$C^{\prime}$ necessarily has $K-\ell\geq 2$ neighbors in $V$. Also, since the
$K-\ell$ neighbors of $C^{\prime}$ in $V$ arise from the subgraph of $G$
induced by $C^{\prime}$, and since those neighbors induce at most
$\binom{K-\ell}{2}$ edges including $C^{\prime}$, we have
$\binom{K}{2}-\ell=c\leq\binom{K-\ell}{2}$. This inequality simplifies to
$\ell\geq 2K-3$, which combines with $K-\ell\geq 2$ to contradict the fact
that $\ell>0$. ∎
Having established that Problem 17 is ${\mathsf{NP}}$-hard, we reduce from it
the main problem of this section. Our proof is specifically geared toward the
case where the matrix in question has integer entries; this is stronger than
manipulating real (complex) numbers exactly as well as with truncations and
tolerances.
###### Theorem 19.
Full Spark is hard for ${\mathsf{NP}}$ under randomized polynomial-time
reductions.
###### Proof.
We will give a randomized polynomial-time reduction from Problem 17 to Full
Spark. As such, suppose we are given a bipartite graph $G$, in which every
edge is between the disjoint sets $A$ and $B$. Take $M:=|B|$ and $N:=|A|$.
Using this graph, we randomly draw an $M\times N$ matrix $F$ using the
following process: for each $i\in B$ and $j\in A$, pick the entry $F_{ij}$
randomly from $\\{1,\ldots,N2^{N+1}\\}$ if $i\leftrightarrow j$ in $G$;
otherwise set $F_{ij}=0$. In Proposition 3.11 of [30], it is shown that the
columns of $F$ form a representation of the transversal matroid of $G$ with
probability $\geq\frac{1}{2}$. For the moment, we assume that $F$ succeeds in
representing the matroid.
Since the girth of the original matroid equals the spark of its
representation, for each $K=1,\ldots,M$, we test whether
$\mathrm{Spark}(F)>K$. To do this, take $H$ to be some $M\times P$ full spark
frame. We will determine an appropriate value for $P$ later, but for
simplicity, we can take $H$ to be the Vandermonde matrix formed from bases
$\\{1,\ldots,P\\}$; see Lemma 2. We claim we can randomly select $K$ indices
$\mathcal{K}\subseteq\\{1,\ldots,P\\}$ and test whether $H_{\mathcal{K}}^{*}F$
is full spark to determine whether $\mathrm{Spark}(F)>K$. Moreover, after
performing this test for each $K=1,\ldots,M$, the probability of incorrectly
determining $\mathrm{Spark}(F)$ is $\leq\frac{1}{2}$, provided $P$ is
sufficiently large.
We want to test whether $H_{\mathcal{K}}^{*}F$ is full spark and use the
result as a proxy for whether $\mathrm{Spark}(F)>K$. For this to work, we need
to have $\mathrm{Rank}(H_{\mathcal{K}}^{*}F_{\mathcal{K}^{\prime}})=K$
precisely when $\mathrm{Rank}(F_{\mathcal{K}^{\prime}})=K$ for every
$\mathcal{K}^{\prime}\subseteq\\{1,\ldots,N\\}$ of size $K$. To this end, it
suffices to have the nullspace $\mathcal{N}(H_{\mathcal{K}}^{*})$ of
$H_{\mathcal{K}}^{*}$ intersect trivially with the column space of
$F_{\mathcal{K}^{\prime}}$ for every $\mathcal{K}^{\prime}$. To be clear, it
is always the case that
$\mathrm{Rank}(H_{\mathcal{K}}^{*}F_{\mathcal{K}^{\prime}})\leq\mathrm{Rank}(F_{\mathcal{K}^{\prime}})$,
and so $\mathrm{Rank}(F_{\mathcal{K}^{\prime}})<K$ implies
$\mathrm{Rank}(H_{\mathcal{K}}^{*}F_{\mathcal{K}^{\prime}})<K$. If we further
assume that
$\mathcal{N}(H_{\mathcal{K}}^{*})\cap\mathrm{Span}(F_{\mathcal{K}^{\prime}})=\\{0\\}$,
then the converse also holds. To see this, suppose
$\mathrm{Rank}(H_{\mathcal{K}}^{*}F_{\mathcal{K}^{\prime}})<K$. Then by the
rank-nullity theorem, there is a nontrivial
$x\in\mathcal{N}(H_{\mathcal{K}}^{*}F_{\mathcal{K}^{\prime}})$. Since
$H_{\mathcal{K}}^{*}F_{\mathcal{K}^{\prime}}x=0$, we must have
$F_{\mathcal{K}^{\prime}}x\in\mathcal{N}(H_{\mathcal{K}}^{*})$, which in turn
implies $x\in\mathcal{N}(F_{\mathcal{K}^{\prime}})$ since
$\mathcal{N}(H_{\mathcal{K}}^{*})\cap\mathrm{Span}(F_{\mathcal{K}^{\prime}})=\\{0\\}$
by assumption. Thus, $\mathrm{Rank}(F_{\mathcal{K}^{\prime}})<K$ by the rank-
nullity theorem.
Now fix $\mathcal{K}^{\prime}\subseteq\\{1,\ldots,N\\}$ of size $K$ such that
$\mathrm{Rank}(F_{\mathcal{K}^{\prime}})=K$. We will show that the vast
majority of choices $\mathcal{K}\subseteq\\{1,\ldots,P\\}$ of size $K$ satisfy
$\mathcal{N}(H_{\mathcal{K}}^{*})\cap\mathrm{Span}(F_{\mathcal{K}^{\prime}})=\\{0\\}$.
To do this, we consider the columns $\\{h_{k}\\}_{k\in\mathcal{K}}$ of
$H_{\mathcal{K}}$ one at a time, and we make use of the fact that
$\mathcal{N}(H_{\mathcal{K}}^{*})=\bigcap_{k\in\mathcal{K}}\mathcal{N}(h_{k}^{*})$.
In particular, since $H$ is full spark, there are at most $M-K$ columns of $H$
in the orthogonal complement of $\mathrm{Span}(F_{\mathcal{K}^{\prime}})$, and
so there are at least $P-(M-K)$ choices of $h_{k_{1}}$ for which
$\mathcal{N}(h_{k_{1}}^{*})$ does not contain
$\mathrm{Span}(F_{\mathcal{K}^{\prime}})$, i.e.,
$\mathrm{dim}\Big{(}\mathcal{N}(h_{k_{1}}^{*})\cap\mathrm{Span}(F_{\mathcal{K}^{\prime}})\Big{)}=K-1.$
Similarly, after selecting the first $J$ $h_{k}$’s, we have
$\mathrm{dim}(S)=K-J$, where
$S:=\bigcap_{j=1}^{J}\mathcal{N}(h_{k_{j}}^{*})\cap\mathrm{Span}(F_{\mathcal{K}^{\prime}}).$
Again, since $H$ is full spark, there are at most $M-(K-J)$ columns of $H$ in
the orthogonal complement of $S$, and so the remaining $P-(M-(K-J))$ columns
are candidates for $h_{k_{J+1}}$ that give
$\mathrm{dim}\bigg{(}\bigcap_{j=1}^{J+1}\mathcal{N}(h_{k_{j}}^{*})\cap\mathrm{Span}(F_{\mathcal{K}^{\prime}})\bigg{)}=\mathrm{dim}\Big{(}\mathcal{N}(h_{k_{J+1}}^{*})\cap
S\Big{)}=K-(J+1).$
Overall, if we randomly pick $\mathcal{K}\subseteq\\{1,\ldots,P\\}$ of size
$K$, then
$\displaystyle\mathrm{Pr}\Big{(}\mathcal{N}(H_{\mathcal{K}}^{*})\cap\mathrm{Span}(F_{\mathcal{K}^{\prime}})=\\{0\\}\Big{)}$
$\displaystyle\geq(1-\tfrac{M-K}{P})(1-\tfrac{M-(K-1)}{P})\cdots(1-\tfrac{M-1}{P})$
$\displaystyle\geq(1-\tfrac{M}{P})^{K}$ $\displaystyle\geq 1-\tfrac{MK}{P},$
where the final step is by Bernoulli’s inequality. Taking a union bound over
all choices of $\mathcal{K}^{\prime}\subseteq\\{1,\ldots,N\\}$ and all values
of $K=1,\ldots,M$ then gives
$\displaystyle\mathrm{Pr}\bigg{(}\begin{array}[]{c}\mbox{fail to determine}\\\
\mbox{$\mathrm{Spark}(F)$}\end{array}\bigg{)}$
$\displaystyle\leq\sum_{K=1}^{M}\binom{N}{K}\mathrm{Pr}\Big{(}\mathcal{N}(H_{\mathcal{K}}^{*})\cap\mathrm{Span}(F_{\mathcal{K}^{\prime}})\neq\\{0\\}\Big{)}$
$\displaystyle\leq\sum_{K=1}^{M}\binom{N}{K}\frac{MK}{P}$
$\displaystyle\leq\frac{M^{3}2^{N}}{P}.$
Thus, to make the probability of failure $\leq\frac{1}{2}$, it suffices to
have $P=M^{3}2^{N+1}$.
In summary, we succeed in representing the original matroid with probability
$\geq\frac{1}{2}$, and then we succeed in determining the spark of its
representation with probability $\geq\frac{1}{2}$. The probability of overall
success is therefore $\geq\frac{1}{4}$. Since our success probability is, at
worst, polynomially small, we can apply amplification to achieve an
exponentially small error probability. ∎
Our use of random linear projections in the above reduction to Full Spark is
similar in spirit to Valiant and Vazirani’s use of random hash functions in
their reduction to Unique SAT [48]. Since their randomized reduction is the
canonical example thereof, we find our reduction to be particularly natural.
As a final note, we clarify that Theorem 19 is a statement about the amount of
computation necessary in the _worst case_. Indeed, the hardness of Full Spark
does not rule out the existence of smaller classes of matrices for which full
spark is easily determined. As an example, Theorem 9 determines Full Spark in
the special case where the matrix is formed by rows of a DFT of prime-power
order. This illustrates the utility of applying additional structure to
efficiently solve the Full Spark problem, and indeed, such classes of matrices
are rather special for this reason.
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|
arxiv-papers
| 2011-10-17T00:56:33 |
2024-09-04T02:49:23.180847
|
{
"license": "Public Domain",
"authors": "Boris Alexeev, Jameson Cahill, Dustin G. Mixon",
"submitter": "Dustin Mixon",
"url": "https://arxiv.org/abs/1110.3548"
}
|
1110.3571
|
# Models of $G$-spectra as presheaves of spectra
Bertrand Guillou bertguillou@uky.edu Department of Mathematics, University of
Kentucky, Lexington, KY 40506 USA and J.P. May may@math.uchicago.edu
Department of Mathematics, The University of Chicago, Chicago, IL 60637 USA
(Date: August 21, 2011)
###### Abstract.
Let $G$ be a finite group. We give Quillen equivalent models for the category
of $G$-spectra as categories of spectrally enriched functors from explicitly
described domain categories to nonequivariant spectra. Our preferred model is
based on equivariant infinite loop space theory applied to elementary
categorical data. It recasts equivariant stable homotopy theory in terms of
point-set level categories of $G$-spans and nonequivariant spectra. We also
give a more topologically grounded model based on equivariant Atiyah duality.
###### Contents
1. 1 The $\scr{S}$-category $G\scr{B}$ and the $\scr{S}_{G}$-category $\scr{B}_{G}$
1. 1.1 The bicategory $G\scr{E}$ of $G$-spans
2. 1.2 The precise statement of the main theorem
3. 1.3 The $G$-bicategory $\scr{E}_{G}$ of spans: intuitive definition
4. 1.4 The $G$-bicategory $\scr{E}_{G}$ of spans: working definition
5. 1.5 The categorical duality maps
2. 2 The proof of the main theorem
1. 2.1 The equivariant approach to 1.9
2. 2.2 Results from equivariant infinite loop space theory
3. 2.3 The self-duality of $\Sigma^{\infty}_{G}(A_{+})$
4. 2.4 The proof that $\scr{B}_{G}$ is equivalent to $\scr{D}_{G}$
5. 2.5 Identifications of suspension $G$-spectra and of tensors with spectra
3. 3 Atiyah duality for finite $G$-sets
1. 3.1 The categories $G\scr{Z}$, $G\scr{D}$, and $\scr{D}_{G}$
2. 3.2 Space level Atiyah duality for finite $G$-sets
3. 3.3 The weakly unital categories $G\scr{A}$ and $\scr{A}_{G}$
4. 3.4 The category of presheaves with domain $G\scr{A}$
## Introduction
The equivariant stable homotopy category is of fundamental importance in
algebraic topology. It is the natural home in which to study equivariant
stable homotopy theory, a subject that has powerful and unexpected
nonequivariant applications. For recent examples, it plays a central role in
the solution of the Kervaire invariant problem by Hill, Hopkins, and Ravenel,
it is central to calculations of topological cyclic homology and therefore to
calculations in algebraic K-theory made by Angeltveit, Gerhardt, Hesselholt,
Lindenstrauss, Madsen, and others, and it plays an interesting role by analogy
and comparision in the work of Voevodsky and others in motivic stable homotopy
theory. It is also of great intrinsic interest.
Setting up the equivariant stable homotopy category with its attendant model
structures takes a fair amount of work. The original version was due to Lewis
and May [11] and more modern versions that we shall start from are given in
[12]. A result of Schwede and Shipley [20], reproven in [5], asserts that any
stable model category $\scr{M}$ is equivalent to a category
$\mathbf{Pre}(\scr{D},\scr{S})$ of spectrally enriched presheaves with values
in a chosen category $\scr{S}$ of spectra. However, the domain
$\scr{S}$-category $\scr{D}$ is a full $\scr{S}$-subcategory of $\scr{M}$ and
typically is as inexplicit and mysterious as $\scr{M}$ itself. From the point
of view of applications and calculations, this is therefore only a starting
point. One wants a more concrete understanding of the category $\scr{D}$. We
shall give explicit equivalents to the domain category $\scr{D}$ in the case
when $\scr{M}=G\scr{S}$ is the category of $G$-spectra for a finite group $G$,
and we fix a finite group $G$ throughout.
We shall define an $\scr{S}$-category (or spectral category) $G\scr{B}$ by
applying a suitable infinite loop space machine to simply defined categories
of finite $G$-sets. The letter $\scr{B}$ stands for “Burnside”, and $G\scr{B}$
is a spectrally enriched version of the Burnside category of $G$. We shall
prove the following result.
###### Theorem 0.1 (Main theorem).
There is a zig-zag of Quillen equivalences
$G\scr{S}\simeq\mathbf{Pre}(G\scr{B},\scr{S})$
relating the category of $G$-spectra to the category of spectrally enriched
contravariant functors $G\scr{B}\longrightarrow\scr{S}$.
As usual, we call such functors presheaves. We reemphasize the simplicity of
our spectral category $G\scr{B}$: no prior knowledge of $G$-spectra is
required to define it.
We give a precise description of the relevant categorical input and restate
the main theorem more precisely in §1. The central point of the proof is to
use equivariant infinite loop space theory to construct the spectral category
$G\scr{B}$ from elementary categories of finite $G$-sets. We prove our main
theorem in §2, using the equivariant Barratt-Priddy-Quillen (BPQ) theorem to
compare $G\scr{B}$ to the spectral category $G\scr{D}$ given by the suspension
$G$-spectra $\Sigma^{\infty}_{G}(A_{+})$ of based finite $G$-sets $A_{+}$. It
is crucial to our work that these $G$-spectra are self-dual. Our original
proof (§3.2) took this as a special case of equivariant Atiyah duality,
thinking of $A$ as a trivial example of a smooth closed $G$-manifold. We later
found a direct categorical proof (§2.3) of this duality based on equivariant
infinite loop space theory and the equivariant BPQ theorem. This allows us to
give an illuminating new proof of the required self-duality as we go along. We
give an alternative model for the category of $G$-spectra in terms of
classical Atiyah duality in §3.
We take what we need from equivariant infinite loop space theory as a black
box in this paper, deferring the proofs of all but one detail to a sequel [7],
with that detail deferred to another sequel [18].
We thank a diligent referee for demanding a reorganization of our original
paper. We also thank Angelica Osorno and Inna Zakharevich for very helpful
comments.
## 1\. The $\scr{S}$-category $G\scr{B}$ and the $\scr{S}_{G}$-category
$\scr{B}_{G}$
We first define the $\scr{S}$-category $G\scr{B}$ and restate our main
theorem. We shall avoid categorical apparatus, but conceptually $G\scr{B}$ is
obtained by applying a nonequivariant infinite loop space machine $\mathbb{K}$
to a category $G\scr{E}$ “enriched in permutative categories”. The term in
quotes can be made categorically precise [4, 9, 19], but we shall use it just
as an informal slogan since no real categorical background is necessary to our
work: we shall give direct elementary definitions of the examples we use, and
they do satisfy the axioms specified in the cited sources. We then define a
$G$-category $\scr{E}_{G}$ “enriched in permutative $G$-categories”, from
which $G\scr{E}$ is obtain by passage to $G$-fixed subcategories. Finally, we
outline the proof of the main theorem, which is obtained by applying an
equivariant infinite loop space machine $\mathbb{K}_{G}$ to $\scr{E}_{G}$.
### 1.1. The bicategory $G\scr{E}$ of $G$-spans
In any category $\scr{C}$ with pullbacks, the bicategory of spans in $\scr{C}$
has $0$-cells the objects of $\scr{C}$. The $1$-cells and $2$-cells
$A\longrightarrow B$ are the diagrams
$\textstyle{B}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A}$
and
$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{B}$$\textstyle{A}$$\textstyle{E.\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
Composites of $1$-cells are given by (chosen) pullbacks
(1.1)
$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C}$$\textstyle{B}$$\textstyle{A.}$
The identity $1$-cells are the diagrams
$\textstyle{A}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\scriptstyle{=}$$\textstyle{A}$.
The associativity and unit constraints are determined by the universal
property of pullbacks. Observe that the $1$-cells $A\longrightarrow B$ can
just as well be viewed as objects over $B\times A$. Viewed this way, the
identity $1$-cells are given by the diagonal maps $A\longrightarrow A\times
A$.
Our starting point is the bicategory of spans of finite $G$-sets. Here the
disjoint union of $G$-sets over $B\times A$ gives us a symmetric monoidal
structure on the category of $1$-cells and $2$-cells $A\longrightarrow B$ for
each pair $(A,B)$. We can think of the bicategory of spans as a category
“enriched in the category of symmetric monoidal categories”. Again, the notion
in quotes does not make obvious mathematical sense since there is no obvious
monoidal structure on the category of symmetric monoidal categories, but
category theory due to the first author [4] (see also [9, 19]) explains what
these objects are and how to rigidify them to categories enriched in
permutative categories. We repeat that we have no need to go into such
categorical detail. Rather than apply such category theory, we give a direct
elementary construction of a strict structure that is equivalent to the
intuitive notion of the category “enriched in symmetric monoidal categories”
of spans of finite $G$-sets.
###### Definition 1.1.
We first define a bipermutative category $G\scr{E}(1)$ equivalent to the
symmmetric bimonoidal category of finite $G$-sets. Any finite $G$-set is
isomorphic to a finite $G$-set of the form $A=(\mathbf{n},\alpha)$, where
$\mathbf{n}=\\{1,\cdots,n\\}$, $\alpha$ is a homomorphism
$G\longrightarrow\Sigma_{n}$, and $G$ acts on $\mathbf{n}$ by $g\cdot
i=\alpha(g)(i)$ for $1\leq i\leq n$. We understand finite $G$-sets to be of
this specific restricted form from now on. A $G$-map
$f\colon(\mathbf{m},\alpha)\longrightarrow(\mathbf{n},\beta)$ is a function
$f\colon\mathbf{m}\longrightarrow\mathbf{n}$ such that
$f\circ\alpha(g)=\beta(g)\circ f$ for $g\in G$. The morphisms of $G\scr{E}(1)$
are the isomorphisms $(\mathbf{n},\alpha)\longrightarrow(\mathbf{n},\beta)$ of
$G$-sets. The disjoint union $D\amalg E$ of finite $G$-sets
$D=(\mathbf{s},\sigma)$ and $E=(\mathbf{t},\tau)$ is
$(\mathbf{s+t},\sigma+\tau)$, with $\sigma+\tau$ being the evident block sum
$G\longrightarrow\Sigma_{s+t}$. With the evident commutativity isomorphism,
this gives the permutative category $G\scr{E}(1)$ of finite $G$-sets; the
empty finite $G$-set is the unit for $\amalg$. Similarly, the cartesian
product $D\times E$ of $D$ and $E$ is $(\mathbf{st},\sigma\times\tau)$ where
the set $\mathbf{st}$ is identified with $\mathbf{{s}\times{t}}$, ordered
lexicographically, and $\sigma\times\tau$ is the evident block product. There
is again an evident commutativity isomorphism, and $\amalg$ and $\times$ give
$G\scr{E}(\ast)$ a structure of bipermutative category in the sense of [17];
the multiplicative unit is the trivial $G$-set $1=(\mathbf{1},\varepsilon)$,
where $\varepsilon(g)=1$ for $g\in G$.
We may view $G\scr{E}(1)$ as the category of finite $G$-sets over the one
point $G$-set $1$, and we generalize the definition as follows.
###### Definition 1.2.
For a finite $G$-set $A$, we define a permutative category $G\scr{E}(A)$ of
finite $G$-sets over $A$. The objects of $G\scr{E}(A)$ are the $G$-maps
$p\colon D\longrightarrow A$. The morphisms $p\longrightarrow q$, $q\colon
E\longrightarrow A$, are the $G$-isomorphisms $f\colon D\longrightarrow E$
such that $q\circ f=p$. Disjoint union of $G$-sets over $A$ gives
$G\scr{E}(A)$ a structure of permutative category; its unit is the empty set
over $A$. When $A=1$, $G\scr{E}(A)$ is the (“additive”) permutative category
of the previous definition.
###### Remark 1.3.
There is also a product $\times\colon G\scr{E}(A)\times
G\scr{E}(B)\longrightarrow G\scr{E}(A\times B)$. It takes $(D,E)$ to $D\times
E$, where $D$ and $E$ are finite $G$-sets over $A$ and $B$, respectively. This
product is also strictly associative and unital, with unit the unit of
$G\scr{E}(1)$, and it has an evident commutativity isomorphism. Restriction to
the object $1$ gives the “multiplicative” permutative category of 1.1. This
product distributes over $\amalg$ and makes the enriched category $G\scr{E}$
of the next definition into a “strict symmetric monoidal category enriched in
permutative categories” in a sense defined in [4].
###### Definition 1.4.
We define a category $G\scr{E}$ “enriched in permutative categories” as
follows. The $0$-cells of $G\scr{E}$ are the finite $G$-sets, which may be
thought of as the categories $G\scr{E}(A)$. The permutative category
$G\scr{E}(A,B)$ of $1$-cells and $2$-cells $A\longrightarrow B$ is
$G\scr{E}(B\times A)$, as defined in 1.2. The composition
$\circ\colon G\scr{E}(B,C)\times G\scr{E}(A,B)\longrightarrow G\scr{E}(A,C)$
is defined via pullbacks, as in the diagram (1.1). Precisely, the pullback $F$
is the sub $G$-set of $E\times D$ consisting of the elements $(e,d)$ such that
$d$ and $e$ map to the same element $b\in B$. This composition is strictly
associative and unital.
###### Remark 1.5.
We are suppressing some categorical details. The composition distributes over
coproducts, and it should be defined on a “tensor product” rather than a
cartesian product of permutative categories. Such a tensor product does in
fact exist [9], but we shall not use the relevant category theory. Rather we
will change notation to $\wedge$ since the composition is a pairing that gives
rise to a pairing defined on the smash product of the spectra constructed from
$G\scr{E}(B,C)$ and $G\scr{E}(A,B)$.
###### Remark 1.6.
It is helpful to observe that the composition just defined can be viewed as a
composite of maps of finite $G$-sets induced contravariantly and covariantly
by the maps of finite $G$-sets
$\textstyle{C\times B\times B\times
A}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces C\times
B\times
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\times\Delta\times\operatorname{id}}$$\scriptstyle{\pi}$$\textstyle{C\times
A,}$
where $\pi:C\times B\times A\longrightarrow C\times A$ is the projection.
Before beginning work, we recall an old result that motivated this paper. The
category $[G\scr{E}]$ of $G$-spans is obtained from the bicategory $G\scr{E}$
of $G$-spans by identifying spans from $A$ to $B$ if there is an isomorphism
between them. Composition is again by pullbacks. We add spans from $A$ to $B$
by taking disjoint unions, and that gives the morphism set $[G\scr{E}](A,B)$ a
structure of abelian monoid. We apply the Grothendieck construction to obtain
an abelian group of morphisms $A\longrightarrow B$. This gives an additive
category $\scr{A}\\!b[G\scr{E}]$. The following result is [11, V.9.6]. Let
$\text{Ho}G\scr{D}$ denote the full subcategory of the homotopy category
$\text{Ho}G\scr{S}$ of $G$-spectra whose objects are the $G$-spectra
$\Sigma^{\infty}_{G}(A_{+})$, where $A$ runs over the finite $G$-sets.
###### Theorem 1.7.
The categories $\text{Ho}G\scr{D}$ and $\scr{A}\\!b[G\scr{E}]$ are isomorphic.
### 1.2. The precise statement of the main theorem
Infinite loop space theory associates a spectrum $\mathbb{K}\scr{A}$ to a
permutative category $\scr{A}$. There are several equivalent machines
available. For definiteness, and because we have used it in working out the
details, we use a modernized version of [14, 16] that lands in the category
$\scr{S}$ of orthogonal spectra [13]. Precise details are given in [7]. With
this choice, the zeroth space of $\mathbb{K}\scr{A}$ is the classifying space
$B\scr{A}$. The objects $a\in\scr{A}$ are the vertices of the nerve of
$\scr{A}$ and thus are points of $B\scr{A}$. Therefore each $a$ determines a
map $S\longrightarrow\mathbb{K}\scr{A}$, where $S$ is the sphere spectrum. For
any $\scr{A}$, $\mathbb{K}\scr{A}$ is a positive $\Omega$-spectrum ([13, §14])
such that its structure map
$B\scr{A}\longrightarrow\Omega(\mathbb{K}\scr{A})_{1}$ is a group completion.
Since $\scr{S}$ is closed symmetric monoidal under the smash product, it makes
sense to enrich categories in $\scr{S}$. Our preferred version of spectral
categories is categories enriched in $\scr{S}$, abbreviated
$\scr{S}$-categories. Model theoretically, $\scr{S}$ is a particularly nice
enriching category since its unit $S$ is cofibrant in the stable model
structure and $\scr{S}$ satisfies the monoid axiom [13, 12.5].
When a spectral category $\scr{D}$ is used as the domain category of a
presheaf category, the objects and maps of the underlying category are
unimportant. The important data are the morphism spectra $\scr{D}(A,B)$, the
unit maps $S\longrightarrow\scr{D}(A,A)$, and the composition maps
$\scr{D}(B,C)\wedge\scr{D}(A,B)\longrightarrow\scr{D}(A,C).$
The presheaves $\scr{D}^{op}\longrightarrow\scr{S}$ can be thought of as
(right) $\scr{D}$-modules.
###### Definition 1.8.
We define a spectral category $G\scr{B}$. Its objects are the finite $G$-sets
$A$, which may be viewed as the spectra $\mathbb{K}G\scr{E}(A)$. Its morphism
spectra $G\scr{B}(A,B)$ are the spectra $\mathbb{K}G\scr{E}(B\times A)$. Its
unit maps $S\longrightarrow G\scr{B}(A,A)$ are induced by the points
$\operatorname{id}_{A}\in G\scr{E}(A,A)$ and its composition
$G\scr{B}(B,C)\wedge G\scr{B}(A,B)\longrightarrow G\scr{B}(A,C)$
is induced by composition in $G\scr{E}$.
As written, the definition makes little sense: to make the word “induced”
meaningful requires properties of the infinite loop space machine $\mathbb{K}$
that we will spell out in §2.2. Once this is done, we will have the presheaf
category $\mathbf{Pre}(G\scr{B},\scr{S})$ of $\scr{S}$-functors
$(G\scr{B})^{op}\longrightarrow\scr{S}$ and and $\scr{S}$-natural
transformations. As shown for example in [5], it is a cofibrantly generated
model category enriched in $\scr{S}$, or $\scr{S}$-model category for short.
As shown in [12], the category $G\scr{S}$ of (genuine) orthogonal $G$-spectra
is also an $\scr{S}$-model category. Our main theorem can be restated as
follows.
###### Theorem 1.9 (Main theorem).
There is a zigzag of enriched Quillen equivalences connecting the
$\scr{S}$-model categories $G\scr{S}$ and $\mathbf{Pre}(G\scr{B},\scr{S})$.
Therefore $G$-spectra can be thought of as constructed from the very
elementary category $G\scr{E}$ enriched in permutative categories, ordinary
nonequivariant spectra, and the black box of infinite loop space theory. The
following reassuring result falls out of the proof. Let $\scr{O}\\!rb$ denote
the orbit category of $G$. For a $G$-spectrum $X$, passage to $H$-fixed point
spectra for $H\subset G$ defines a functor
$X^{\bullet}\colon\scr{O}\\!rb^{op}\longrightarrow\scr{S}$. Analogously, a
presheaf $Y\in\mathbf{Pre}(G\scr{B},\scr{S})$ restricts to a functor
$\scr{O}\\!rb^{op}\longrightarrow\scr{S}$.
###### Corollary 1.10.
The zigzag of equivalences induces a natural zigzag of equivalences between
the fixed point orbit functor on $G$-spectra and the restriction to orbits of
presheaves; thus, if $X$ corresponds to $Y$, then $X^{H}$ is equivalent to
$Y(G/H)$.
###### Remark 1.11.
There is an important missing ingredient needed for a fully satisfactory
theory: we have not described the behavior of smash products under the
equivalences of 1.9. This problem deserves study both in our work and in
related work of others. The obvious guess is that
$\mathbf{Pre}(G\scr{B},\scr{S})$ is symmetric monoidal and the zigzag
connecting it to $G\scr{S}$ is a zigzag of symmetric monoidal Quillen
equivalences. We see how the problem can be attacked, but we also have reason
to believe that the obvious guess may be wrong. We intend to return to this
question elsewhere.
###### Remark 1.12.
Much of what we do applies to $G$-spectra indexed on an incomplete universe,
provided that we restrict attention to those finite $G$-sets $A$ that embed in
that universe, so that Atiyah duality applies to the orbit $G$-spectra
$\Sigma^{\infty}_{G}(A_{+})$. By [10], duality fails for orbits that do not
embed in the universe. Unfortunately, however, the cited restriction leads to
the wrong weak equivalences, since we are then only entitled to see the
homotopy groups of $H$-fixed point spectra for those $H$ that embed in the
given universe.
### 1.3. The $G$-bicategory $\scr{E}_{G}$ of spans: intuitive definition
Everything we do depends on first working equivariantly and then passing to
fixed points. Following [6, §1.2], we fix some generic notations. For a
category $\scr{C}$, let $G\scr{C}$ be the category of $G$-objects in $\scr{C}$
and $G$-maps between them. Let $\scr{C}_{G}$ be the $G$-category of
$G$-objects and nonequivariant maps, with $G$ acting by conjugation. The two
categories are related conceptually by $G\scr{C}=(\scr{C}_{G})^{G}$. The
objects, being $G$-objects, are already $G$-fixed; we apply the $G$-fixed
point functor to hom sets. More generally, we can start with a category
$\scr{C}$ with actions by $G$ on its objects and again define a category
$G\scr{C}$ of $G$-maps and a $G$-category $\scr{C}_{G}$ with $G$-fixed
category $G\scr{C}$.
We apply this framework to the category of finite $G$-sets. We have already
defined the $G$-fixed bicategory $G\scr{E}$, and we shall give two definitions
of $G$-bicategories $\scr{E}_{G}$ with fixed point bicategories equivalent to
$G\scr{E}$. The first, given in this section, is more intuitive, but the
second is more convenient for the proof of our main theorem.
Let $U$ be a countable $G$-set that contains all orbit types $G/H$ infinitely
many times. Again let $A$, $B$, and $C$ denote finite $G$-sets, but now let
the $D$, $E$ and $F$ of §1.1 be finite subsets of the $G$-set $U$; these
subsets need not be $G$-subsets. The action of $G$ on $U$ gives rise to an
action of $G$ on the finite subsets of $U$: for a finite subset $D$ of $U$ and
$g\in G$, $gD$ is another finite subset of $U$.
###### Definition 1.13.
We define a $G$-category $\scr{E}_{G}(A)$. The objects of $\scr{E}_{G}(A)$ are
the nonequivariant maps $p\colon D\longrightarrow A$, where $A$ is a finite
$G$-set and $D$ is a finite subset of $U$. The morphisms $f\colon
p\longrightarrow q$, $q\colon E\longrightarrow A$, are the bijections $f\colon
D\longrightarrow E$ such that $q\circ f=p$. The group $G$ acts on morphisms
via the maps $g\colon D\longrightarrow gD$ and the formula $(gf)(gd)=gf(d)$.
###### Definition 1.14.
We define a bicategory $\scr{E}_{G}$ with objects the finite $G$-sets and with
$G$-categories of morphisms between objects specified by
$\scr{E}_{G}(A,B)=\scr{E}_{G}(B\times A)$. Thinking of the objects of
$\scr{E}_{G}(A,B)$ as nonequivariant spans $B\longleftarrow D\longrightarrow
A$, composition and units are defined as in 1.4.
Observe that taking disjoint unions of finite sets over $A$ will not keep us
in $U$ and is thus not well-defined. Therefore the $\scr{E}_{G}(A)$ are not
symmetric monoidal (let alone permutative) $G$-categories in the naive sense
of symmetric monoidal categories with $G$ acting compatibly on all data. In
fact, the notion of a genuine permutative $G$-category, one that provides
input for an equivariant infinite loop space machine, is subtle. We shall give
two solutions to that categorical problem in [7]. In both, genuine permutative
$G$-categories are described in terms of actions by an $E_{\infty}$ operad of
$G$-categories, to which equivariant infinite loop space theory applies. One
solution gives each of the $\scr{E}_{G}(A)$ such a structure, but that is not
the solution we shall use.
### 1.4. The $G$-bicategory $\scr{E}_{G}$ of spans: working definition
The other solution starts from a less intuitive definition of $\scr{E}_{G}$
and gives an equivalent way of solving that categorical problem. It uses a
more convenient $E_{\infty}$ operad of $G$-categories, denoted $\scr{O}_{G}$.
We give details of this operad in [7], where we define a genuine permutative
$G$-category to be an algebra over $\scr{O}_{G}$. To give the idea, we apply
our general point of view on equivariant categories to the category
$\scr{C}\\!at$ of small categories. Thus, for $G$-categories $\scr{A}$ and
$\scr{B}$, let $\scr{C}\\!at_{G}(\scr{A},\scr{B})$ be the $G$-category of
functors $\scr{A}\longrightarrow\scr{B}$ and natural transformations, with $G$
acting by conjugation, and let $G\scr{C}\\!at(\scr{A},\scr{B})$ be the
category of $G$-functors and $G$-natural transformations.
###### Definition 1.15.
Let $\tilde{G}$ (sometimes denoted $EG$ in the literature111While $\tilde{G}$
is isomorphic as a $G$-category to the translation category of $G$, the action
of $G$ on that category is defined differently, as is explained in [8, Lemma
1.7].) be the groupoid with object set $G$ and a unique morphism, denoted
$(h,k)$, from $k$ to $h$ for each pair of objects. Let $G$ act from the right
on $\tilde{G}$ by $h\cdot g=hg$ on objects and $(h,k)\cdot g=(hg,kg)$ on
morphisms. The objects of $\scr{E}_{G}$ are the finite $G$-sets
$A=(\mathbf{n},\alpha)$, regarded as discrete (identity morphisms only)
$G$-categories. Define $\scr{O}(j)=\tilde{\Sigma}_{j}$; this is the $j$th
category of an $E_{\infty}$ operad of categories whose algebras are the
permutative categories [16]. Define $\scr{O}_{G}(j)$ to be the $G$-category
$\scr{C}\\!at_{G}(\tilde{G},\tilde{\Sigma}_{j})=\scr{C}\\!at_{G}(\tilde{G},\scr{O}(j)).$
Here $G$ acts trivially on $\tilde{\Sigma}_{j}$. The left action of $G$ on
$\scr{O}_{G}(j)$ is induced by the right action of $G$ on $\tilde{G}$, and the
right action of $\Sigma_{j}$ is induced by the right action of $\Sigma_{j}$ on
$\tilde{\Sigma}_{j}$. The functor $\scr{C}\\!at_{G}(\tilde{G},-)$ is product
preserving and the operad structure maps are induced from those of $\scr{O}$.
We interpret $\scr{O}(0)$ and $\scr{O}_{G}(0)$ to be trivial categories;
$\scr{O}_{G}(1)$ is also trivial, with unique object denoted
$\operatorname{id}$.
###### Definition 1.16.
Define the $G$-category $\scr{E}_{G}(A)$ by
(1.2) $\scr{E}_{G}(A)=\coprod_{n\geq
0}\scr{O}_{G}(n)\times_{\Sigma_{n}}A^{n}=(\coprod_{n\geq
1}\scr{O}_{G}(n)\times_{\Sigma_{n}}A^{n})_{+}.$
We interpret the term with $n=0$ to be a trivial base category $\ast$, which
explains the second equality, and we identify the term with $n=1$ with $A$. An
alternative formulation is $\scr{E}_{G}(A)=\mathbb{O}_{G}(A_{+})$, where
$\mathbb{O}_{G}$ denotes the monad in the category of based $G$-categories
whose algebras are the same as the $\scr{O}_{G}$-algebras. Thus
$\mathbb{O}_{G}(A_{+})$ is the free $\scr{O}_{G}$-algebra (= genuine
permutative $G$-category) generated by the based $G$-category $A_{+}$, with
unit given by a disjoint trivial base category added to $A$.
The following result is neither obvious nor difficult. It is proven in [7].
###### Theorem 1.17.
The $G$-fixed permutative category $\scr{E}_{G}(A)^{G}$ is naturally
isomorphic to the permutative category $G\scr{E}(A)$.
The starting point of the proof is the observation that a functor
$\tilde{G}\longrightarrow\tilde{\Sigma}_{n}$ is uniquely determined by its
object function $G\longrightarrow\Sigma_{n}$. In particular, for a finite
$G$-set $B=(\mathbf{n},\beta)$ we may view the $G$-map $\beta\colon
G\longrightarrow\Sigma_{n}$ as a $G$-fixed object of the category
$\scr{O}_{G}(n)$, and all $G$-fixed objects of $\scr{O}_{G}(n)$ are of this
form. With a little care, we see that a $G$-fixed object
$(\beta;a_{1},\cdots,a_{n})$ of $\scr{O}_{G}(n)\times_{\Sigma_{n}}A^{n}$ can
be interpreted as a $G$-map $B\longrightarrow A$ and that all finite $G$-sets
over $A$ are of this form.
The following is a sketch definition whose details will be fleshed out below.
###### Definition 1.18.
The $G$-category $\scr{E}_{G}$ “enriched in permutative $G$-categories” has
$0$-cells the finite $G$-sets $A$, which may be thought of as the
$G$-categories $\scr{E}_{G}(A)$. The permutative $G$-category
$\scr{E}_{G}(A,B)$ of $1$-cells and $2$-cells $A\longrightarrow B$ is
$\scr{E}_{G}(B\times A)$. The unit $\operatorname{id}_{A}$ of
$A=(\mathbf{n},\alpha)$ is the object $(\alpha;(1,1),\cdots,(n,n))$ of
$\scr{O}_{G}(n)\times_{\Sigma_{n}}(A\times A)^{n}$; it can be thought of as a
$G$-map $1\longrightarrow\scr{E}_{G}(A,A)$ of $G$-categories, where $1$ is the
trivial $G$-category. Composition is given by the following composite; its
first map is a specialization of a pairing of free $\scr{O}_{G}$-algebras, and
its second and third maps are specializations of contravariant functoriality
of the free $\scr{O}_{G}$-algebra functor on inclusions and covariant
functoriality on surjections that we shall shortly make precise.
$\textstyle{\scr{E}_{G}(C\times B)\wedge\scr{E}_{G}(B\times
A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega}$$\textstyle{\scr{E}_{G}(C\times
B\times B\times
A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\operatorname{id}\times\Delta\times\operatorname{id})^{*}}$$\textstyle{\scr{E}_{G}(C\times
B\times
A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{!}}$$\textstyle{\scr{E}_{G}(C\times
A).}$
We shall place the following ad hoc definition of the required pairing
$\omega$ in a suitable general context in [7], modernizing part of [15]. We
first comment on its domain; compare 1.5.
###### Remark 1.19.
We can define the smash product of based $G$-categories in the same way as the
smash product of based $G$-spaces. We are most interested in examples of the
form $\scr{A}_{+}$ and $\scr{B}_{+}$ for unbased $G$-categories $\scr{A}$ and
$\scr{B}$, and then $\scr{A}_{+}\wedge\scr{B}_{+}$ can be identified with
$(\scr{A}\times\scr{B})_{+}$. In particular,
$(\coprod_{m\geq
1}\scr{O}_{G}(m)\times_{\Sigma_{m}}A^{m})_{+}\wedge(\coprod_{n\geq
1}\scr{O}_{G}(n)\times_{\Sigma_{n}}B^{n})_{+}$
is isomorphic to
$(\coprod_{m\geq 1,n\geq
1}\scr{O}_{G}(m)\times\scr{O}_{G}(n)\times_{\Sigma_{m}\times\Sigma_{n}}A^{m}\times
B^{n})_{+}.$
We do not claim that this is an $\scr{O}_{G}$-category, but an equivariant
infinite loop space machine nevertheless constructs from it the smash product
of the spectra constructed from $\scr{E}_{G}(A)$ and $\scr{E}_{G}(B)$.
###### Definition 1.20.
Identify the ordered set $\mathbf{mn}$ with the set of pairs $(i,j)$, $1\leq
i\leq m$ and $1\leq j\leq n$, ordered lexicographically. This fixes a
homomorphism $\Sigma_{m}\times\Sigma_{n}\longrightarrow\Sigma_{mn}$ and
therefore a functor
$\tilde{\Sigma}_{m}\times\tilde{\Sigma}_{n}\longrightarrow\tilde{\Sigma}_{mn}$.
Applying the functor $\scr{C}\\!at_{G}(\tilde{G},-)$, we obtain pairings
$\omega_{m,n}\colon\scr{O}_{G}(m)\times\scr{O}_{G}(n)\longrightarrow\scr{O}_{G}(mn)$.
For finite $G$-sets $A$ and $B$, we have the injection $A^{m}\times
B^{n}\longrightarrow(A\times B)^{mn}$ that sends
$(a_{1},\cdots,a_{m})\times(b_{1},\cdots,b_{n})$ to the set of pairs
$(a_{i},b_{j})$, ordered lexicographically. Combining, there result functors
$\omega_{m,n}\colon(\scr{O}_{G}(m)\times_{\Sigma_{m}}A^{m})\times(\scr{O}_{G}(n)\times_{\Sigma_{n}}B^{n})\longrightarrow\scr{O}_{G}(mn)\times_{\Sigma_{mn}}(A\times
B)^{mn}.$
Distributing products over disjoint unions, these specify pairings of
$G$-categories
$\omega\colon\scr{E}_{G}(A)\wedge\scr{E}_{G}(B)\longrightarrow\scr{E}_{G}(A\times
B).$
The naturality maps in 1.18 are both applications of the free
$\scr{O}_{G}$-category functor to maps $f$ of based finite $G$-sets.
Conceptually, the definition (1.2) hides an extension of functors from
$\scr{E}_{G}(A)$, which a priori appears to be a functor on unbased finite
$G$-sets, to $\mathbb{O}_{G}(A_{+})$, which is a functor on based finite
$G$-sets.
###### Definition 1.21.
For a map $f\colon A_{+}\longrightarrow B_{+}$ of based finite $G$-sets, we
obtain a functor $f_{!}\colon\scr{E}_{G}(A)\longrightarrow\scr{E}_{G}(B)$ by
taking the disjoint union over $n$ of the functors
$\operatorname{id}\times_{\Sigma_{n}}f^{n}$. This is unproblematical if $f$ is
obtained from a map $A\longrightarrow B$ of unbased finite $G$-sets, so that
$f^{-1}(\ast)=\ast$.222With the intuitive version of $\scr{E}_{G}$,
$f_{!}\colon\scr{E}_{G}(A)\longrightarrow\scr{E}_{G}(B)$ is just the
pushforward functor obtained by composing $f$ with maps over $A$. In general,
however, the specification of $f_{!}$ depends on implicit basepoint
identifications that are invisible to (1.2) but become visible when evaluating
$\scr{E}_{G}f$. Because $\scr{O}_{G}(0)$ is the trivial category $\ast$, there
is a degeneracy $G$-functor
$\sigma_{i}\colon\scr{O}_{G}(n)\longrightarrow\scr{O}_{G}(n-1)$ associated to
the ordered inclusion $\mathbf{n-1}\colon\longrightarrow\mathbf{n}$ that
misses $i$. As in [14, 2.3], if $\gamma$ is the structural map of the operad
and $\nu\in\scr{O}_{G}(n)$,
$\sigma_{i}(\nu)=\gamma(\nu;\operatorname{id}^{i-1},\ast,\operatorname{id}^{n-i}).$
If $a_{i}=\ast$, then $(\nu,a_{1},\cdots,a_{n})$ must be identified with
$(\sigma_{i}(\nu),a_{1},\cdots,\hat{a}_{i},\cdots,a_{n})$, where $\hat{a}_{i}$
means delete $a_{i}$. In particular, if $i\colon A\longrightarrow B$ is an
inclusion of unbased finite $G$-sets, define an associated retraction $r\colon
B_{+}\longrightarrow A_{+}$ of based finite $G$-sets by setting $ri(a)=a$ and
$r(b)=\ast$ if $b\notin\operatorname{im}(A)$. Then define
$i^{*}=r_{!}\colon\scr{E}_{G}(B)\longrightarrow\scr{E}_{G}(A)$.333With the
intuitive version of $\scr{E}_{G}$,
$i^{*}\colon\scr{E}_{G}(B)\longrightarrow\scr{E}_{G}(A)$ is just the functor
obtained by pulling back maps over $B$ to maps over $A$. By 2.14 below, we may
think of $i^{*}$ as the dual of $i$.
The associativity of the composition defined in 1.18 is an easy diagram chase,
starting from the associativity of the pairing on $\scr{O}_{G}$. The
verification that composition with the prescribed unit objects
${\operatorname{id}_{A}}$ gives identity functors illustrates how 1.21 works.
Set $B=A$ and consider the composite
$(\mu;(c_{1},a_{1}),\cdots,(c_{m},a_{m}))\circ\operatorname{id}_{A}.$
We are focusing on objects, and $\mu\in\scr{O}_{G}(m)$, $c_{i}\in C$,
$a_{i}\in A$, and $A=(\mathbf{n},\alpha)$. Applying the pairing we get the
object
$(\omega_{m,n}(\mu,\alpha);(c_{i},a_{i},j,j))\in\scr{O}_{G}(mn)\times_{\Sigma_{mn}}(C\times
A\times A\times A)^{mn}.$
The four-tuple $(c_{i},a_{i},j,j)$ is in the image of
$\operatorname{id}\times\Delta\times\operatorname{id}$ if and only if
$a_{i}=j$. The $r$ corresponding to this inclusion maps all other
$(c_{i},a_{i},j,j)$ to the basepoint, and we have an accompanying iterated
degeneracy $\sigma\colon\scr{O}_{G}(mn)\longrightarrow\scr{O}_{G}(m)$ such
that $\sigma(\omega_{m,n}(\mu,\alpha))=\mu$. Therefore our composite is
$(\mu;(c_{1},a_{1}),\cdots,(c_{m},a_{m}))$, as required. The proof that
composition on the left with $\operatorname{id}_{A}$ is the identity functor
is similar.
1.17 has the following corollary by direct comparison of definitions.
###### Corollary 1.22.
The $G$-fixed category $(\scr{E}_{G})^{G}$ enriched in permutative categories
is isomorphic to the category $G\scr{E}$ enriched in permutative categories.
### 1.5. The categorical duality maps
Since various specializations are central to our work, we briefly recall how
duality works categorically, following [11, III§1] for example. We then define
maps of $\scr{O}_{G}$-algebras that will lead in §2.3 to the proof that the
objects of $G\scr{B}$ are self-dual.
Let $\scr{V}$ be a closed symmetric monoidal category with product $\wedge$,
unit $S$, and hom objects $F(X,Y)$; write $DX=F(X,S)$. A pair of objects
$(X,Y)$ in $\scr{V}$ is a dual pair if there are maps $\eta\colon
S\longrightarrow X\wedge Y$ and $\varepsilon\colon Y\wedge X\longrightarrow S$
such that the composites
$\textstyle{X\cong S\wedge
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta\wedge\operatorname{id}}$$\textstyle{X\wedge
Y\wedge
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\varepsilon}$$\textstyle{X\wedge
S\cong X}$ $\textstyle{Y\cong Y\wedge
S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\eta}$$\textstyle{Y\wedge
X\wedge
Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon\wedge\operatorname{id}}$$\textstyle{S\wedge
Y\cong Y}$
are identity maps. For any such pair, the adjoint $\tilde{\varepsilon}\colon
Y\longrightarrow DX$ of $\varepsilon$ is an isomorphism. We have a natural map
(1.3) $\zeta\colon Y\wedge DX=Y\wedge F(X,S)\longrightarrow F(X,Y)$
in $\scr{V}$, namely the adjoint of
$\operatorname{id}\wedge\varepsilon\colon Y\wedge DX\wedge X\longrightarrow
Y\wedge S\cong Y,$
where $\varepsilon$ is the evident evaluation map. The map $\zeta$ is an
isomorphism when either $X$ or $Y$ is dualizable [11, III.1.3]. When $X$ is
dualizable and $Y$ is arbitrary, we have the composite isomorphism
(1.4) $\delta=\zeta\circ(\operatorname{id}\wedge\tilde{\varepsilon})\colon
Y\wedge X\longrightarrow Y\wedge DX\longrightarrow F(X,Y).$
This map in various categories will play a central role in our work. When
$(X,Y)$ and $(X^{\prime},Y^{\prime})$ are dual pairs, the dual of a map
$f\colon X\longrightarrow X^{\prime}$ is the composite
(1.5) $\textstyle{Y^{\prime}\cong Y^{\prime}\wedge
S_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\eta}$$\textstyle{Y^{\prime}\wedge
X\wedge
Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge
f\wedge\operatorname{id}}$$\textstyle{Y^{\prime}\wedge X^{\prime}\wedge
Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon^{\prime}\wedge\operatorname{id}}$$\textstyle{S_{G}\wedge
Y\cong Y.}$
There are two maps of $\scr{O}_{G}$-algebras that are central to duality and
therefore to everything we do. Let $S^{0}=\\{\ast,1\\}$, where $\ast$ is the
basepoint and ${1}$ is not. We think of $S^{0}$ as $1_{+}$, where $1$ is the
one-point $G$-set. Remember that $\scr{E}_{G}(A)=\mathbb{O}_{G}(A_{+})$ is the
free $\scr{O}_{G}$-algebra generated by $A_{+}$, where we view finite $G$-sets
as categories with only identity morphisms. We have already seen the first map
implicitly.
###### Definition 1.23.
For a finite $G$-set $A$, define based $G$-maps
$\varepsilon\colon(A\times A)_{+}\longrightarrow S^{0},\ \ r\colon(A\times
A)_{+}\longrightarrow A_{+}\ \ \text{and}\ \ \pi\colon A_{+}\longrightarrow
S^{0}$
by $r(a,b)=\ast$ if $a\neq b$ and $r(a,a)=a$, $\pi(a)=1$, and
$\varepsilon=\pi\circ r$, so that $\varepsilon(a,b)=\ast$ if $a\neq b$ and
$\varepsilon(a,a)=1$. Note that $r\circ\Delta=\operatorname{id}$ and that
$\varepsilon$ is just an example of a Kronecker $\delta$-function. We agree to
again write $\varepsilon$ for the induced map of $\scr{O}_{G}$-algebras
$\varepsilon=\scr{E}_{G}\varepsilon\colon\scr{E}_{G}(A\times
A)\longrightarrow\scr{E}_{G}(1).$
###### Definition 1.24.
For a finite $G$-set $A$, regard the object
$\operatorname{id}_{A}\in\scr{E}_{G}(A)$ as the map of $G$-categories
$i_{A}\colon 1\longrightarrow\scr{E}_{G}(A)$ that sends the object $1$ to the
object $\operatorname{id}_{A}$. By freeness, there results a map of
$\scr{O}_{G}$-algebras
$\eta\colon\scr{E}_{G}(1)\longrightarrow\scr{E}_{G}(A\times A).$
If $A=(\mathbf{n},\alpha)$, then $\eta$ is the disjoint union of maps
$\scr{O}_{G}(m)/\Sigma_{m}\cong\scr{O}_{G}(m)\times_{\Sigma_{m}}1^{m}\longrightarrow\scr{O}_{G}(mn)\times_{\Sigma_{mn}}(A\times
A)^{mn}.$
These are obtained by composing $\scr{O}_{G}(m)\times i_{A}^{m}$ with the map
induced on passage to orbits from the maps
$\textstyle{\scr{O}_{G}(m)\times(\scr{O}_{G}(n)\times(A\times
A)^{n})^{m}\cong(\scr{O}_{G}(m)\times\scr{O}_{G}(n)^{m})\times((A\times
A)^{n})^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\scr{O}_{G}(mn)\times(A\times
A)^{mn}}$
given by shuffling and applying the structure map
$\gamma\colon\scr{O}_{G}(m)\times\scr{O}_{G}(n)^{m}\longrightarrow\scr{O}_{G}(mn)$.
The following categorical observation will lead to our proof in §2.3 that the
$G$-spectra $\Sigma^{\infty}_{G}(A_{+})$ are self-dual. Since care of
basepoints is crucial, we use the alternative notation
$\mathbb{O}_{G}(A_{+})$. Remember that $(A\times A)_{+}$ can be identified
with $A_{+}\wedge A_{+}$. We identify $1_{+}\wedge A_{+}$ and $A_{+}\wedge
1_{+}$ with $A_{+}$ at the bottom center of our diagrams.
###### Proposition 1.25.
The left and right squares commute in the following diagrams, and
(1.6)
$\mathbb{O}_{G}(\operatorname{id}\wedge\varepsilon)\circ\zeta_{\ell}=\operatorname{id}=\mathbb{O}_{G}(\varepsilon\wedge\operatorname{id})\circ\zeta_{r}.$
Therefore the diagrams obtained by removing the maps $\zeta_{\ell}$ and
$\zeta_{r}$ commute.
$\textstyle{\mathbb{O}_{G}(A_{+}\wedge
A_{+})\wedge\mathbb{O}_{G}(A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega}$$\textstyle{\mathbb{O}_{G}(A_{+}\wedge
A_{+}\wedge
A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{O}_{G}(\operatorname{id}\wedge\varepsilon)}$$\textstyle{\mathbb{O}_{G}(A_{+})\wedge\mathbb{O}_{G}(A_{+}\wedge
A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\varepsilon}$$\scriptstyle{\omega}$$\textstyle{\mathbb{O}_{G}(1_{+})\wedge\mathbb{O}_{G}(A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega}$$\scriptstyle{\eta\wedge\operatorname{id}}$$\textstyle{\mathbb{O}_{G}(A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta_{\ell}}$$\textstyle{\mathbb{O}_{G}(A_{+})\wedge\mathbb{O}_{G}(1_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega}$
$\textstyle{\mathbb{O}_{G}(A_{+})\wedge\mathbb{O}_{G}(A_{+}\wedge
A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega}$$\textstyle{\mathbb{O}_{G}(A_{+}\wedge
A_{+}\wedge
A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{O}_{G}(\varepsilon\wedge\operatorname{id})}$$\textstyle{\mathbb{O}_{G}(A_{+}\wedge
A_{+})\wedge\mathbb{O}_{G}(A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon\wedge\operatorname{id}}$$\scriptstyle{\omega}$$\textstyle{\mathbb{O}_{G}(A_{+})\wedge\mathbb{O}_{G}(1_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega}$$\scriptstyle{\operatorname{id}\wedge\eta}$$\textstyle{\mathbb{O}_{G}(A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta_{r}}$$\textstyle{\mathbb{O}_{G}(1_{+})\wedge\mathbb{O}_{G}(A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega}$
###### Proof.
In the right vertical arrows, $\varepsilon$ means
$\mathbb{O}_{G}(\varepsilon)$. Since the right squares are just naturality
diagrams, they clearly commute. For the rest, we must first define the maps
$\zeta_{\ell}$ and $\zeta_{r}$. Remember that the elements of $A$ are the
elements of $\mathbf{n}=\\{1,\cdots,n\\}$, permuted according to $\alpha\colon
G\longrightarrow\Sigma_{n}$. Define $j_{\ell}:A\longrightarrow(A\times A\times
A)^{n}$ by $j_{\ell}(a)=\big{(}(1,1,a),\cdots,(n,n,a)\big{)}$. Then define
$J_{\ell}\colon A_{+}\longrightarrow\scr{O}_{G}(A_{+}\wedge A_{+}\wedge
A_{+})$ by
$J_{\ell}(a)=(\alpha,j_{\ell}(a))\in\scr{O}_{G}(n)\times_{\Sigma_{n}}(A\times
A\times A)^{n}.$
Define
$\zeta_{\ell}\colon\mathbb{O}_{G}(A_{+})\longrightarrow\scr{O}_{G}(A_{+}\wedge
A_{+}\wedge A_{+})$
to be the map of $\scr{O}_{G}$-algebras induced by freeness. For
$\mu\in\scr{O}_{G}(m)$ and $\nu\in\scr{O}_{G}(q)$,
(1.7)
$\zeta_{\ell}(\omega(\mu,\nu);(a_{1},\cdots,a_{q})^{m})=(\gamma(\omega(\mu,\nu);\alpha^{mq});(j_{\ell}(a_{1}),\cdots,j_{\ell}(a_{q}))^{m})$
where $\gamma$ is the structural map of the operad $\scr{O}_{G}$. Define
$j_{r}$, $J_{r}$, and $\zeta_{r}$ by symmetry.
Clearly $\mathbb{O}_{G}(\operatorname{id}\wedge\varepsilon)$ sends
$J_{\ell}(a)$ to $a$. Indeed, $a$ is one of the elements $j\in\mathbf{n}$ and
$\operatorname{id}\wedge\varepsilon$ sends the coordinates $(i,i,a)$ with
$i\neq j$ to the basepoint and the coordinate $(j,j,a)$ to $a$. Since
$\mathbb{O}_{G}(\operatorname{id}\wedge\varepsilon)\circ\zeta_{\ell}$ is a map
of $\scr{O}_{G}$-algebras with domain the free $\scr{O}_{G}$-algebra
$\mathbb{O}_{G}(A_{+})$, this implies the first equality in (1.6); the
symmetric argument proves the second equality. It remains to prove that the
left squares of our diagrams commute; by symmetry it suffices to consider the
first diagram. Consider an element
$x=((\mu;1^{m}),(\nu;a_{1},\cdots,a_{q}))\in(\scr{O}_{G}(m)\times_{\Sigma_{m}}1^{m})\times(\scr{O}_{G}(q)\times_{\Sigma_{q}}A^{q}),$
where $m\geq 1$, $q\geq 1$, $\mu\in\scr{O}_{G}(m)$, $\nu\in\scr{O}_{G}(q)$,
and $a_{k}\in A$ for $1\leq k\leq q$. Write $[j,j,a_{k}]$ for the element of
$(A^{3})^{mnq}$ with $(i,j,k)th$ coordinate $(j,j,a_{k})$, $1\leq i\leq m$,
$1\leq j\leq n$, and $1\leq k\leq q$. Then
(1.8)
$\omega\circ(\eta\wedge\operatorname{id})(x)=(\omega(\gamma(\mu;\alpha^{m}),\nu);[j,j,a_{k}])\in\scr{O}_{G}(mnq)\times_{\Sigma_{mnq}}(A^{3})^{mnq}.$
On the other hand,
$\omega(x)=(\omega(\mu,\nu);(a_{1},\cdots,a_{q})^{m})\in\scr{O}_{G}(mq)\times_{\Sigma_{mq}}A^{mq}$
and therefore
(1.9)
$\zeta_{\ell}\omega(x)=(\gamma(\omega(\mu,\nu);\alpha^{mq});(j_{\ell}(a_{1}),\cdots,j_{\ell}(a_{q}))^{m})\in\scr{O}_{G}(mnq)\times_{\Sigma_{mnq}}(A^{3})^{mnq}.$
The coordinates in $A^{3}$ of the element on the right side of (1.9) differ
from those of the right side of (1.8) by a permutation
$\sigma\in\Sigma_{mnq}$, and it is a special case of the formula relating the
pairing $\omega$ to the structure map $\gamma$ of the operad $\scr{O}_{G}$
that
(1.10)
$\gamma(\omega(\mu,\nu);\alpha^{mq})\sigma=\omega(\gamma(\mu;\alpha^{m}),\nu).$
Therefore the right sides of (1.8) and (1.9) are equal and
$\omega\circ(\eta\wedge\operatorname{id})=\zeta_{\ell}\circ\omega$. ∎
###### Remark 1.26.
A more general form of (1.10) is the key defining property [15, 1.4(ii)] of a
pairing of operads, such as $\omega$. We have proven that the left and right
squares of our diagrams are examples of maps of pairings of algebras over a
permutative operad, as defined in [17, IX.1.3] and [15, 1.1]. Those sources
are nonequivariant and outdated, but a modern treatment of equivariant
pairings will be included in [18].
## 2\. The proof of the main theorem
### 2.1. The equivariant approach to 1.9
As we will explain in [7], equivariant infinite loop space theory associates
an orthogonal $G$-spectrum $\mathbb{K}_{G}\scr{A}_{G}$ to a (genuine)
permutative $G$-category $\scr{A}_{G}$. The $0$th space of
$\mathbb{K}_{G}\scr{A}_{G}$ is the classifying $G$-space $B\scr{A}_{G}$. The
$0$th structure map
$B\scr{A}_{G}\longrightarrow\Omega(\scr{B}_{G}\scr{A}_{G})_{1}$ is an
equivariant group completion.444The papers from around 1990, such as [2, 21]
are not adequate for our purposes, in part because the target category of
$G$-spectra was not yet well understood then. A full dress modern treatment of
equivariant infinite loop space theory, complementing [7], is in progress
[18]. The category $G\scr{S}$ of orthogonal $G$-spectra is the $G$-fixed
category of a $G$-category $\scr{S}_{G}$ of $G$-spectra and non-equivariant
maps with the same objects as $\scr{S}_{G}$ and with $G$ acting by
conjugation. Applying the functor $\mathbb{K}_{G}$ to $\scr{E}_{G}$, we obtain
the following equivariant analogue of 1.8.
###### Definition 2.1.
We define a $G$-spectral category, or $\scr{S}_{G}$-category555There is a
slight abuse of language here since the notion of a category enriched in
$\scr{S}_{G}$ (alias a $G$-spectral category) does not quite make sense in
classical enriched category theory because the smash product of $G$-spectra is
only functorial on $G$-maps, not on the more general maps in $\scr{S}_{G}$.
The terminology is explained and justified in [6, 1.9]. $\scr{B}_{G}$. Its
objects are the finite $G$-sets $A$, which may be viewed as the $G$-spectra
$\mathbb{K}_{G}\scr{E}_{G}(A)$. Its morphism $G$-spectra $\scr{B}_{G}(A,B)$
are the $G$-spectra $\mathbb{K}_{G}\scr{E}_{G}(B\times A)$. Its unit $G$-maps
$S_{G}\longrightarrow\scr{B}_{G}(A,A)$ are induced by the points
$\operatorname{id}_{A}\in G\scr{E}(A,A)$ and its composition $G$-maps
$\scr{B}_{G}(B,C)\wedge\scr{B}_{G}(A,B)\longrightarrow\scr{B}_{G}(A,C)$
are induced by composition in $\scr{E}_{G}$.
Again, as written, the definition makes little sense: to make the word
“induced” meaningful requires properties of the equivariant infinite loop
space machine $\mathbb{K}_{G}$ that we will spell out in §2.2. This depends on
having a functor that takes pairings of free $\scr{O}_{G}$-algebras to
pairings of $G$-spectra.
The equivariant and non-equivariant infinite loop space functors are related
by the following result.
###### Theorem 2.2 ([7]).
There is a natural equivalence of spectra
$\iota\colon\mathbb{K}(G\scr{A})\longrightarrow(\mathbb{K}_{G}\scr{A}_{G})^{G}$
for permutative $G$-categories $\scr{A}_{G}$ with $G$-fixed permutative
categories $G\scr{A}$.
In view of 1.22, there results an equivalence of $\scr{S}$-categories
$\textstyle{G\scr{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{(\scr{B}_{G})^{G}.}$
The proof of 1.9 goes as follows. We start with the following specialization
of a general result about stable model categories; it is discussed in [6,
§3.1].
###### Theorem 2.3.
Let $G\scr{D}$ be the full $\scr{S}$-subcategory of $G\scr{S}$ whose objects
are fibrant approximations of the suspension $G$-spectra
$\Sigma^{\infty}_{G}(A_{+})$, where $A$ runs through the finite $G$-sets. Then
there is an enriched Quillen adjunction
$\textstyle{\mathbf{Pre}(G\scr{D},\scr{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{T}}$$\textstyle{G\scr{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces,}$$\scriptstyle{\mathbb{U}}$
and it is a Quillen equivalence.
Here $G\scr{D}$ is isomorphic to $(\scr{D}_{G})^{G}$, where $\scr{D}_{G}$ is a
full $\scr{S}_{G}$-subcategory $\scr{D}_{G}$ of $\scr{S}_{G}$.
###### Theorem 2.4 (Equivariant version of the main theorem).
There is a zigzag of weak equivalences connecting the $\scr{S}_{G}$-categories
$\scr{B}_{G}$ and $\scr{D}_{G}$.
A weak equivalence between $\scr{S}_{G}$-categories with the same object sets
is just an $\scr{S}_{G}$-functor that induces weak equivalences on morphism
$G$-spectra.666A more general definition is given in [5, 2.3]. On passage to
$G$-fixed categories, this equivariant zigzag induces a zigzag of weak
$\scr{S}$-equivalences connecting the $\scr{S}$-categories $G\scr{B}$ and
$G\scr{D}$. In turn, by [5, 2.4], this zigzag induces a zigzag of Quillen
equivalences between $\mathbf{Pre}(G\scr{B},\scr{S})$ and
$\mathbf{Pre}(G\scr{D},\scr{S})$. Since $\mathbf{Pre}(G\scr{D},\scr{S})$ is
Quillen equivalent to $G\scr{S}$, it follows that 2.4 implies 1.9.
###### Remark 2.5.
The functor $\mathbb{U}$ sends $G/H$ to
$F_{G}(\Sigma^{\infty}_{G}G/H_{+},X)^{G}\cong X^{H}$. Keeping that fact in
mind shows why 1.10 follows from the proof of 1.9.
To understand $G\scr{S}$ as an $\scr{S}$-category, we must first understand
$\scr{S}_{G}$ as an $\scr{S}_{G}$-category. That is, to understand the
$G$-fixed spectra $F_{G}(X,Y)^{G}$, we must first understand the function
$G$-spectra $F_{G}(X,Y)$. Using infinite loop space theory to model function
spectra implicitly raises a conceptual issue: there is no known infinite loop
space machine that knows about function spectra. That is, given input data $X$
and $Y$ (permutative $G$-categories, $E_{\infty}$-$G$-spaces,
$\Gamma$-$G$-spaces, etc) for an infinite loop space machine $\mathbb{K}_{G}$,
we do not know what input data will have as output the function $G$-spectra
$F_{G}(\mathbb{K}_{G}X,\mathbb{K}_{G}Y)$. The problem does not even make sense
as just stated because the output $G$-spectra $\mathbb{K}_{G}X$ are always
connective, whereas $F_{G}(\mathbb{K}_{G}X,\mathbb{K}_{G}Y)$ is generally not.
The most that one could hope for in general is to detect the connective cover
of $F(\mathbb{K}_{G}X,\mathbb{K}_{G}Y)$. In our case, the relevant function
$G$-spectra are connective since the suspension $G$-spectra
$\Sigma^{\infty}_{G}(A_{+})$ are self-dual, as we shall reprove in §2.3.
### 2.2. Results from equivariant infinite loop space theory
The proof of 2.4 is the heart of this paper, and of course it depends on
equivariant infinite loop space theory and in particular on the relationship
between the $G$-spectra $\scr{B}_{G}(A)=\mathbb{K}_{G}\scr{E}_{G}(A)$ and the
suspension $G$-spectra $\Sigma^{\infty}_{G}(A_{+})$. We collect the results
that we need from [7] in this section, making Definitions 1.8 and 2.1 precise
and expanding on Theorems 1.17 and 2.2. We warn the skeptical reader that the
results of this paper depend on the two results just cited and on Theorems 2.6
and 2.7 below. The knowledgable expert will immediately accept the
plausibility of these results, especially since those of the results which
make sense when $G=e$ have been known for decades. However, their proofs
require work that is far afield from the applications in this paper.
In fact, 2.4 is an application of a categorical version of the equivariant
Barratt-Priddy-Quillen (BPQ) theorem for the identification of suspension
$G$-spectra.777For $A=\ast$, Carlsson [1, p.6] mentions a space level version
of the BPQ theorem. Shimakawa [21, p. 242] states and gives an incomplete
sketch proof of a $G$-spectrum level version. We state the theorem in full
generality before restricting attention to finite $G$-sets. We shall find use
for the full generality in §2.5.
Recall from 1.16 that $\scr{E}_{G}(A)$ is the category
$\mathbb{O}_{G}(A_{+})$, where $\mathbb{O}_{G}$ is the free
$\scr{O}_{G}$-category functor. We may view any based $G$-space $X$ as a
topological category888We understand a topological category to mean an
internal category in the category of spaces, not just a category enriched in
spaces. that is discrete in the categorical sense: its morphism and object
spaces are both $X$, and its source, target, identity, and composition maps
are all just the identity map of $X$. The functor $\mathbb{O}_{G}$ applies
equally well to based topological $G$-categories, hence we have the
topological $\scr{O}_{G}$-category $\mathbb{O}_{G}(X)$. The geometric
realization of its nerve is the free $E_{\infty}$ $G$-space generated by $X$.
Henceforward, we use the term stable equivalence, rather than weak
equivalence, for the weak equivalences in our model categories of spectra and
$G$-spectra.
###### Theorem 2.6 (Equivariant Barratt-Quillen Theorem, [7]).
For based $G$-spaces $X$, there is a natural stable equivalence
$\alpha\colon\Sigma^{\infty}_{G}X\longrightarrow\mathbb{K}_{G}\mathbb{O}_{G}(X).$
Of course, the naturality statement says that the following diagram commutes
for a map $f\colon X\longrightarrow Y$ of based $G$-spaces.
(2.1)
$\textstyle{\Sigma^{\infty}_{G}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Sigma^{\infty}_{G}f}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{K}_{G}\mathbb{O}_{G}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{K}_{G}\mathbb{O}_{G}(f)}$$\textstyle{\Sigma^{\infty}_{G}(B_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{K}_{G}\mathbb{O}_{G}(Y)}$
There is a companion theorem that relates $\alpha$ to smash products. The
pairing $\omega$ of 1.20 generalizes to give a natural pairing
$\omega\colon\mathbb{O}_{G}(X)\wedge\mathbb{O}_{G}(Y)\longrightarrow\mathbb{O}_{G}(X\wedge
Y)$
for based $G$-spaces $X$ and $Y$.
###### Theorem 2.7.
[7] The pairing $\omega$ induces a natural stable equivalence
$\wedge\colon\mathbb{K}_{G}\mathbb{O}_{G}(X)\wedge\mathbb{K}_{G}\mathbb{O}_{G}(Y)\longrightarrow\mathbb{K}_{G}\mathbb{O}_{G}(X\wedge
Y)$
such that the following diagram commutes.
(2.2)
$\textstyle{\Sigma^{\infty}_{G}X\wedge\Sigma^{\infty}_{G}Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wedge}$$\scriptstyle{\cong}$$\scriptstyle{\alpha\wedge\alpha}$$\textstyle{\mathbb{K}_{G}\mathbb{O}_{G}(X)\wedge\mathbb{K}_{G}\mathbb{O}_{G}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wedge}$$\textstyle{\Sigma^{\infty}_{G}(X\wedge
Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{K}_{G}\mathbb{O}_{G}(X\wedge
Y)}$
The left map $\wedge$ in (2.2) is a canonical natural isomorphism, and this
diagram says that the natural map $\alpha$ is lax monoidal. The result that we
need to prove 2.4 is an immediate specialization.
###### Theorem 2.8.
For finite $G$-sets $A$, there is a lax monoidal natural stable equivalence
$\alpha\colon\Sigma^{\infty}_{G}(A_{+})\longrightarrow\mathbb{K}_{G}\scr{E}_{G}(A).$
Identifying $A_{+}\wedge B_{+}$ with $(A\times B)_{+}$, (2.2) specializes to
the commutative diagram
(2.3)
$\textstyle{\Sigma^{\infty}_{G}(A_{+})\wedge\Sigma^{\infty}_{G}(B_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wedge}$$\scriptstyle{\cong}$$\scriptstyle{\alpha\wedge\alpha}$$\textstyle{\mathbb{K}_{G}\scr{E}_{G}(A)\wedge\mathbb{K}_{G}\scr{E}_{G}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wedge}$$\textstyle{\Sigma^{\infty}_{G}(A\times
B)_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{K}_{G}\scr{E}_{G}(A\times
B).}$
We restate the naturality of $\alpha$ with respect to $G$-maps $f\colon
A\longrightarrow B$ in the diagram
(2.4)
$\textstyle{\Sigma^{\infty}_{G}(A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Sigma^{\infty}_{G}f_{+}}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{K}_{G}\scr{E}_{G}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{K}_{G}f_{!}}$$\textstyle{\Sigma^{\infty}_{G}(B_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{K}_{G}\scr{E}_{G}(B).}$
If $i\colon A\longrightarrow B$ is an inclusion with retraction $r\colon
B_{+}\longrightarrow A_{+}$, we have the induced map of $G$-spectra
$\mathbb{K}_{G}i^{*}=\mathbb{K}_{G}r_{!}\colon\mathbb{K}_{G}\scr{E}_{G}(B)\longrightarrow\mathbb{K}_{G}\scr{E}_{G}(A),$
and (2.4) specializes to
(2.5)
$\textstyle{\Sigma^{\infty}_{G}(B_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Sigma^{\infty}_{G}r}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{K}_{G}\scr{E}_{G}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{K}_{G}i^{*}}$$\textstyle{\Sigma^{\infty}_{G}(A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{K}_{G}\scr{E}_{G}(A)}$
By 2.14 below, we may identify $\mathbb{K}_{G}i^{*}$ as the dual of
$\mathbb{K}_{G}i$ and thus $\Sigma^{\infty}_{G}r$ as the dual of
$\Sigma^{\infty}_{G}i_{+}$.
We combine these diagrams to construct those that we need to prove 2.4. Let
$A$, $B$, and $C$ be finite $G$-sets and recall 1.18.
###### Proposition 2.9.
The following diagram of $G$-spectra commutes.
(2.6) $\textstyle{\Sigma^{\infty}_{G}(C\times
B)_{+}\wedge\Sigma^{\infty}_{G}(B\times
A)_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wedge}$$\scriptstyle{\cong}$$\scriptstyle{\alpha\wedge\alpha}$$\textstyle{\mathbb{K}_{G}\scr{E}_{G}(C\times
B)\wedge\mathbb{K}_{G}\scr{E}_{G}(B\times
A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wedge}$$\textstyle{\Sigma^{\infty}(C\times
B\times B\times
A)_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\Sigma^{\infty}_{G}r}$$\textstyle{\mathbb{K}_{G}\scr{E}_{G}(C\times
B\times B\times
A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{K}_{G}(\operatorname{id}\times\Delta\times\operatorname{id})^{*}}$$\textstyle{\Sigma^{\infty}(C\times
B\times
A)_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\Sigma^{\infty}\pi}$$\textstyle{\mathbb{K}_{G}\scr{E}_{G}(C\times
B\times
A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{K}_{G}\pi_{!}}$$\textstyle{\Sigma^{\infty}_{G}(C\times
A)_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{K}_{G}\scr{E}_{G}(C\times
A)}$
Here $r$ is the retraction which sends the complement of the image of
$\operatorname{id}\times\Delta\times\operatorname{id}$ to the basepoint.
###### Definition 2.10.
To make 2.1 and therefore 1.8 precise, define the composition
$\scr{B}_{G}(B,C)\wedge\scr{B}_{G}(A,B)\longrightarrow\scr{B}_{G}(A,C)$
to be the right vertical composite in the diagram (2.6).
The diagram (2.6) relates the composition pairing of the
$\scr{S}_{G}$-category $\scr{B}_{G}$ to remarkably simple and explicit maps
between suspension $G$-spectra. In fact, recalling 1.23 and again writing
$\varepsilon=\Sigma^{\infty}_{G}\varepsilon$, we see that the left vertical
composite in (2.6) can be identified with
$\operatorname{id}\wedge\varepsilon\wedge\operatorname{id}$. We have proven
the following result.
###### Theorem 2.11.
The following diagram of $G$-spectra commutes.
$\textstyle{\Sigma^{\infty}_{G}(C\times
B)_{+}\wedge\Sigma^{\infty}_{G}(B\times
A)_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\alpha\wedge\alpha}$$\textstyle{\scr{B}_{G}(B,C)\wedge\scr{B}_{G}(A,B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\textstyle{\Sigma^{\infty}_{G}(C_{+})\wedge\Sigma^{\infty}_{G}(B\times
B)_{+}\wedge\Sigma^{\infty}_{G}(A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\varepsilon\wedge\operatorname{id}}$$\textstyle{\Sigma^{\infty}_{G}(C_{+})\wedge
S_{G}\wedge\Sigma^{\infty}_{G}(A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\Sigma^{\infty}_{G}(C\times
A)_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\scr{B}_{G}(A,C)}$
### 2.3. The self-duality of $\Sigma^{\infty}_{G}(A_{+})$
Let $A$ be a finite $G$-set and write $\mathbb{A}=\Sigma^{\infty}_{G}(A_{+})$
for brevity of notation. As recalled in §1.5, we must define maps $\eta\colon
S_{G}\longrightarrow\mathbb{A}\wedge\mathbb{A}$ and
$\varepsilon\colon\mathbb{A}\wedge\mathbb{A}\longrightarrow S_{G}$ in the
stable homotopy category $HoG\scr{S}$ such that the composites
(2.7)
$\textstyle{\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta\wedge\operatorname{id}}$$\textstyle{\mathbb{A}\wedge\mathbb{A}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\varepsilon}$$\textstyle{\mathbb{A}}$
and
$\textstyle{\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\eta}$$\textstyle{\mathbb{A}\wedge\mathbb{A}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon\wedge\operatorname{id}}$$\textstyle{\mathbb{A}}$
are the identity map in $HoG\scr{S}$. Using the stable equivalence $\alpha$
and the definitions of $\eta$ and $\varepsilon$ from Definitions 1.23 and
1.24, we let $\eta$ and $\varepsilon$ be the composites
$\textstyle{S_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{K}_{G}\scr{E}_{G}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{K}_{G}\eta}$$\textstyle{\mathbb{K}_{G}\scr{E}_{G}(A\times
A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{-1}}$$\textstyle{\Sigma^{\infty}_{G}(A\times
A)_{+}\cong\mathbb{A}\wedge\mathbb{A}}$
and
$\textstyle{\mathbb{A}\wedge\mathbb{A}\cong\Sigma^{\infty}_{G}(A\times
A)_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{K}_{G}\scr{E}_{G}(A\times
A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{K}_{G}\varepsilon}$$\textstyle{\mathbb{K}_{G}\scr{E}_{G}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{-1}}$$\textstyle{S_{G}.}$
The following commutative diagram proves that the first composite in (2.7) is
the identity map in $HoG\scr{S}$; the second is dealt with similarly. We
abbreviate notation by setting $\scr{B}_{G}A=\mathbb{K}_{G}\scr{E}_{G}(A)$.
Remember that $\scr{E}_{G}(A)=\mathbb{O}_{G}(A_{+})$. The center two squares
are derived by use of the diagrams from 1.25.
$\textstyle{\scr{B}_{G}(A^{2})\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\alpha}$$\textstyle{(\mathbf{A^{2}})\wedge\mathbb{A}\cong\mathbf{A^{3}}\cong\mathbb{A}\wedge(\mathbf{A^{2}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\wedge\operatorname{id}}$$\scriptstyle{\operatorname{id}\wedge\alpha}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{A}\wedge\scr{B}_{G}(A^{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\wedge\operatorname{id}}$$\scriptstyle{\operatorname{id}\wedge\varepsilon}$$\textstyle{\scr{B}_{G}(A^{2})\wedge\scr{B}_{G}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wedge}$$\textstyle{\scr{B}_{G}(A^{3})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\times\varepsilon}$$\textstyle{\scr{B}_{G}A\wedge\scr{B}_{G}(A^{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wedge}$$\scriptstyle{\operatorname{id}\wedge\varepsilon}$$\textstyle{\scr{B}_{G}1\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta\wedge\alpha}$$\scriptstyle{\eta\wedge\operatorname{id}}$$\textstyle{\mathbb{A}\wedge\scr{B}_{G}1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\wedge\operatorname{id}}$$\textstyle{\scr{B}_{G}1\wedge\scr{B}_{G}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta\wedge\operatorname{id}}$$\scriptstyle{\wedge}$$\textstyle{\scr{B}_{G}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta_{\ell}}$$\textstyle{\scr{B}_{G}A\wedge\scr{B}_{G}1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wedge}$$\textstyle{S_{G}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\wedge\alpha}$$\scriptstyle{\cong}$$\scriptstyle{\alpha\wedge\operatorname{id}}$$\textstyle{\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{A}\wedge
S_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\wedge\alpha}$$\scriptstyle{\cong}$$\scriptstyle{\operatorname{id}\wedge\alpha}$
Given 2.8, it is trivial that the outer parts of the diagram commute. We
comment on the passage from the diagrams of 1.25 to the central squares of the
diagram; compare 1.26.
###### Remark 2.12.
Nonequivariantly, the passage from pairings on the category level to pairings
on the spectrum level is worked out in [15], implicitly using orthogonal
spectra. The sequel [7] to this paper constructs the pairing $\wedge$ from the
pairing $\omega$ of free $\scr{O}_{G}$-categories used here, but it does not
treat its naturality with respect to maps of pairings that are not induced by
maps of finite $G$-sets. Modernized generalizations and details will be
supplied in [18].
Specializing general observations about duality recalled in §1.5, we have the
following corollary. This homotopical input is the crux of the proof of 2.4.
###### Corollary 2.13.
For finite $G$-sets $A$ and $B$, the canonical map
$\delta=\zeta\circ(\operatorname{id}\wedge\tilde{\varepsilon})\colon\mathbb{B}\wedge\mathbb{A}\longrightarrow\mathbb{B}\wedge
D\mathbb{A}\longrightarrow F_{G}(\mathbb{A},\mathbb{B})$
of (1.4) is a stable equivalence.
We insert a mild digression concerning the identification of some of our maps.
###### Remark 2.14.
For an inclusion $i\colon A\longrightarrow B$ of finite $G$-sets, (1.5) and
the precise constructions of $\eta$ and $\varepsilon$ starting from
Definitions 1.23 and 1.24 imply that the dual of $i$ is the map
$\mathbb{B}\longrightarrow\mathbb{A}$ induced by the evident retraction
$r\colon B_{+}\longrightarrow A_{+}$. A $G$-map $\pi\colon G/H\longrightarrow
G/K$ is a bundle, and the dual of $\Sigma^{\infty}\pi_{+}$ is the associated
transfer map (see e.g. [11, IV.pp 182 and 192]). It can be identified
explicitly by a similar (but not especially illuminating) inspection of
definitions.
### 2.4. The proof that $\scr{B}_{G}$ is equivalent to $\scr{D}_{G}$
We will have to chase large diagrams, and we again abbreviate notations by
writing
$\mathbb{A}=\Sigma^{\infty}_{G}(A_{+}),\ \ \
\mathbb{B}=\Sigma^{\infty}_{G}(B_{+}),\ \ \ \text{and}\ \ \
\mathbb{C}=\Sigma^{\infty}_{G}(C_{+})$
for finite $G$-sets $A$, $B$, and $C$. We also abbreviate notation by writing
$\scr{B}_{G}(A)=\scr{B}_{G}(\ast,A).$
It is the $G$-spectrum $\scr{B}_{G}(A)=\mathbb{K}_{G}\scr{E}_{G}(A)$, which is
equivalent to $\mathbb{A}$ by 2.8. Remember that we are free to choose any
bifibrant equivalents of the $G$-spectra $\mathbb{A}$ as the objects of
$\scr{D}_{G}$.
###### Proof of 2.4.
We use model categorical arguments, and we work with the stable model
structure on $G\scr{S}$. We use [5, §2.4] to obtain a model structure on the
category $G\scr{S}\mathbb{O}$-$\scr{C}\\!at$ of $G\scr{S}$-categories with the
same object set $\mathbb{O}$ as $G\scr{E}$. Maps are weak equivalences or
fibrations if they induce weak equivalences or fibrations on hom objects in
$G\scr{S}$. Here the nature of the objects is irrelevant; we are concerned
with $G\scr{S}$-categories with one object for each finite $G$-set $A$.
Let $\lambda\colon Q{\scr{B}_{G}}\longrightarrow\scr{B}_{G}$ be a cofibrant
approximation of $\scr{B}_{G}$. By [5, 2.16], since $S_{G}$ is cofibrant in
the stable model structure each morphism $G$-spectrum $Q\scr{B}_{G}(A,B)$ is
cofibrant in $G\scr{S}$. The maps $\lambda\colon
Q\scr{B}_{G}(A,B)\longrightarrow\scr{B}_{G}(A,B)$ are stable acyclic
fibrations. Digressively, since the $\scr{B}_{G}(A,B)$ are fibrant in the
positive stable model structure, that is also true of the $Q\scr{B}_{G}(A,B)$;
we will use this fact later, in §2.5.
Let $\rho\colon Q\scr{B}_{G}\longrightarrow RQ\scr{B}_{G}$ be a fibrant
approximation of $Q\scr{B}_{G}$. The morphism $G$-spectra $RQ\scr{B}_{G}(A,B)$
are then bifibrant in the stable model structure. Therefore $RQ\scr{B}_{G}(A)$
is bifibrant for each $A$, and it is stably equivalent to $\mathbb{A}$. We
take the $RQ\scr{B}_{G}(A)$ as the bifibrant approximations of the
$\mathbb{A}$ that we use to define the full $G\scr{S}$-subcategory
$\scr{D}_{G}$ of $G\scr{S}$.
We define $\scr{C}_{G}$ to be the full $G\scr{S}$-subcategory of $G\scr{S}$
with objects the $Q\scr{B}_{G}(A)$. To abbreviate notation, we agree to write
$Q\scr{B}_{G}(\ast,A)=Q\scr{B}_{G}A\ \ \ \text{and}\ \ \
RQ\scr{B}_{G}(\ast,A)=RQ\scr{B}_{G}A.$
With our notational conventions, it is consistent to write
$Q\scr{B}_{G}(B\times A)=Q\scr{B}_{G}(A,B)$.
For finite $G$-sets $A$ and $B$, let
$\beta\colon
Q\scr{B}_{G}(A,B)\longrightarrow\scr{C}_{G}(A,B)=F_{G}(Q\scr{B}_{G}A,Q\scr{B}_{G}B)$
be the adjoint of the composition map
$\circ\colon Q\scr{B}_{G}(A,B)\wedge Q\scr{B}_{G}A\longrightarrow
Q\scr{B}_{G}B$
and let
$\gamma\colon
RQ\scr{B}_{G}(A,B)\longrightarrow\scr{D}_{G}(A,B)=F_{G}(RQ\scr{B}_{G}A,RQ\scr{B}_{G}B)$
be the adjoint of the composition map
$\circ\colon RQ\scr{B}_{G}(A,B)\wedge RQ\scr{B}_{G}A\longrightarrow
RQ\scr{B}_{G}B.$
By [5, 5.6], these define $G\scr{S}$-functors
$\beta\colon Q\scr{B}_{G}\longrightarrow\scr{C}_{G}\ \ \ \text{\ \ and\ \ }\ \
\ \gamma\colon RQ\scr{B}_{G}\longrightarrow\scr{D}_{G}.$
It suffices to prove that each of the maps $\gamma$ is a stable equivalence.
For each finite $G$-set $A$, $\mathbb{A}$ is cofibrant and $\lambda\colon
Q\scr{B}_{G}A\longrightarrow\scr{B}_{G}A$ is an acyclic fibration in the
stable model structure. Therefore there is a map
$\mu\colon\mathbb{A}\longrightarrow Q\scr{B}_{G}A$ such that the diagram
$\textstyle{Q\scr{B}_{G}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda}$$\textstyle{\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{\alpha}$$\textstyle{\scr{B}_{G}A}$
commutes. Since $\alpha$ and $\lambda$ are stable equivalences, so is $\mu$.
We claim that the following diagram of $G$-spectra commutes in $HoG\scr{S}$.
Indeed, modulo inversion of maps which are stable equivalences, it commutes on
the nose. As before, we identify
$\mathbb{B}\wedge\mathbb{A}=\Sigma^{\infty}_{G}B_{+}\wedge\Sigma^{\infty}_{G}A_{+}$
with $\Sigma^{\infty}_{G}(B\times A)_{+}$.
$\textstyle{RQ\scr{B}_{G}(A,B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{F_{G}(RQ\scr{B}_{G}A,RQ\scr{B}_{G}B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{G}(\rho,\operatorname{id})}$$\scriptstyle{\simeq}$$\textstyle{F_{G}(Q\scr{B}_{G}A,RQ\scr{B}_{G}B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{G}(\mu,\operatorname{id})}$$\scriptstyle{\simeq}$$\textstyle{Q\scr{B}_{G}(A,B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{\simeq}$$\scriptstyle{\beta}$$\textstyle{F_{G}(Q\scr{B}_{G}A,Q\scr{B}_{G}B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{G}(\operatorname{id},\rho)}$$\scriptstyle{F_{G}(\mu,\operatorname{id})}$$\textstyle{F_{G}(\mathbb{A},RQ\scr{B}_{G}B)}$$\textstyle{\mathbb{B}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{\simeq}$$\scriptstyle{\delta}$$\scriptstyle{\simeq}$$\textstyle{F_{G}(\mathbb{A},\mathbb{B})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{G}(\operatorname{id},\mu)}$$\scriptstyle{\simeq}$$\textstyle{F_{G}(\mathbb{A},Q\scr{B}_{G}B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{G}(\operatorname{id},\rho)}$$\scriptstyle{\simeq}$
The map $\delta$ is the stable equivalence of 2.13. The maps $\mu$ and $\rho$
are also stable equivalences. The maps $F_{G}(\rho,\operatorname{id})$ and
$F_{G}(\mu,\operatorname{id})$ that are labeled $\simeq$ are stable
equivalences by [5, 1.22] since $\rho$ and $\mu$ are maps between cofibrant
objects and $RQ\scr{B}_{G}B$ is fibrant. The maps
$F_{G}(\operatorname{id},\mu)$ and $F_{G}(\operatorname{id},\rho)$ that are
labeled $\simeq$ are stable equivalences by [12, III.3.9], which shows that
the functor $F_{G}(\mathbb{A},-)$ preserves stable equivalences. Granting that
the diagram commutes, it follows that $\gamma$ is a stable equivalence since
all of the other outer arrows of the diagram are stable equivalences.
To prove that the diagram commutes in $HoG\scr{S}$, we consider its adjoint.
Remembering that $\lambda\circ\mu=\alpha$, we see that the adjoint can be
written in the following expanded form. Here we have inserted the map
$\circ\colon\scr{B}_{G}(A,B)\wedge\scr{B}_{G}A\longrightarrow\scr{B}_{G}B$ and
wrong way arrows into its source and target for purposes of proof.
---
$\textstyle{RQ\scr{B}_{G}(A,B)\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\mu}$$\textstyle{RQ\scr{B}_{G}(A,B)\wedge
Q\scr{B}_{G}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\rho}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
RQ\scr{B}_{G}(A,B)\wedge
RQ\scr{B}_{G}A}$$\scriptstyle{\circ}$$\textstyle{Q\scr{B}_{G}(A,B)\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\mu}$$\scriptstyle{\rho\wedge\operatorname{id}}$$\scriptstyle{\rho\wedge\mu}$$\textstyle{Q\scr{B}_{G}(A,B)\wedge
Q\scr{B}_{G}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\scriptstyle{\rho\wedge\operatorname{id}}$$\scriptstyle{\rho\wedge\rho}$$\scriptstyle{\lambda\wedge\lambda}$$\textstyle{Q\scr{B}_{G}B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{\lambda}$$\textstyle{RQ\scr{B}_{G}B}$$\textstyle{\scr{B}_{G}(A,B)\wedge\scr{B}_{G}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\textstyle{\scr{B}_{G}B}$$\textstyle{\mathbb{B}\wedge\mathbb{A}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\Sigma^{\infty}_{G}\varepsilon}$$\scriptstyle{\alpha\wedge\alpha}$$\scriptstyle{\mu\wedge\operatorname{id}}$$\scriptstyle{\mu\wedge\mu}$$\textstyle{\mathbb{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\mu}$$\scriptstyle{\rho\mu}$
Since $\lambda$ and $\rho$ are maps of $G\scr{S}$-categories, it is apparent
that all parts of the diagram commute except for the bottom trapezoid. Taking
$(A,B,C)=(\ast,A,B)$ in 2.11, we see that the trapezoid commutes. Since the
wrong way maps $\alpha$ and $\lambda$ are stable equivalences and can be
inverted upon passage to the homotopy category, this diagram and its adjoint
commute there. ∎
### 2.5. Identifications of suspension $G$-spectra and of tensors with
spectra
We expand the adjoint $\scr{S}$-equivalences in 1.9 more explicitly as
follows.
(2.8)
$\textstyle{G\scr{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{U}}$$\textstyle{\mathbf{Pre}(G\scr{D},\scr{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma^{*}}$$\scriptstyle{\mathbb{T}}$$\textstyle{\mathbf{Pre}((RQ\scr{B}_{G})^{G},\scr{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{*}}$$\scriptstyle{\gamma_{!}}$$\textstyle{\mathbf{Pre}(G\scr{B},\scr{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{!}}$$\textstyle{\mathbf{Pre}((\scr{B}_{G})^{G},\scr{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda^{*}}$$\scriptstyle{\iota^{*}}$$\textstyle{\mathbf{Pre}((Q\scr{B}_{G})^{G},\scr{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{!}}$$\scriptstyle{\rho_{!}}$
The map $\iota:G\scr{B}\longrightarrow(\scr{B}_{G})^{G}$ is the equivalence of
2.2. Before passage to $G$-fixed points, the proof in §2.4 gives stable
equivalences of $\scr{S}_{G}$-categories
$\rho\colon Q\scr{B}_{G}\longrightarrow RQ\scr{B}_{G},\ \
\gamma:RQ\scr{B}_{G}\longrightarrow\scr{D}_{G},\ \text{and}\ \lambda\colon
Q\scr{B}_{G}\longrightarrow\scr{B}_{G},$
and these maps give stable equivalences of $\scr{S}$-categories after passage
to fixed points.
For a finite $G$-set $B$, $\Sigma^{\infty}_{G}B_{+}$ corresponds under this
zigzag to the presheaf $\mathbf{B}$ that sends $A$ to $G\scr{B}(A,B)$. This is
almost a tautology since, for $E\in G\scr{S}$, $\mathbb{U}(E)$ is the presheaf
represented by $E$, while $G\scr{E}(-,B)$ is the functor represented by $B$.
In the proof of 2.4, we chose the bifibrant approximation of
$\Sigma^{\infty}_{G}B_{+}$ in $G\scr{D}_{G}$ to be $RQ\scr{B}_{G}(B)$. With
$B$ fixed, that proof shows that $\gamma$ gives an equivalence of presheaves
$RQ\scr{B}_{G}(-,B)\longrightarrow\gamma^{*}\mathbb{U}RQ\scr{B}_{G}(B)$
(before passage to $G$-fixed points). The functors $\rho^{*}$ and
$\lambda_{!}$ and the isomorphism $\iota^{*}$ preserve representable functors,
and therefore $\iota^{*}\lambda_{!}\rho^{*}RQ\scr{B}_{G}(-,B)\simeq
K_{G}\scr{E}_{G}(-,B)$.
This observation can be generalized from finite based $G$-sets $B_{+}$ to
arbitrary based $G$-spaces $X$. To see this, we mix general based $G$-spaces
$X$ with finite based $G$-sets $A$ to obtain a functorial construction of a
presheaf $\scr{P}_{G}(X)$.
###### Definition 2.15.
Define a presheaf
$\scr{P}_{G}(X)\colon(\scr{B}_{G})^{op}\longrightarrow\scr{S}_{G}$ by letting
$\scr{P}_{G}(X)(A)=\scr{K}_{G}\mathbb{O}_{G}(X\wedge A_{+}).$
The contravariant functoriality map
$\scr{P}_{G}(X)\colon\scr{B}_{G}(A,B)\longrightarrow
F_{G}(\scr{B}_{G}(X)(B),\scr{B}_{G}(X)(A))$
is the adjoint of the right vertical composite in the commutative diagram
(2.9) $\textstyle{\Sigma^{\infty}_{G}(X\wedge
B_{+})\wedge\Sigma^{\infty}_{G}(B_{+}\wedge
A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wedge}$$\scriptstyle{\cong}$$\scriptstyle{\alpha\wedge\alpha}$$\textstyle{\mathbb{K}_{G}\mathbb{O}_{G}(X\wedge
B_{+})\wedge\mathbb{K}_{G}\mathbb{O}_{G}(B_{+}\wedge
A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wedge}$$\textstyle{\Sigma^{\infty}(X\wedge
B_{+}\wedge B_{+}\wedge
A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\Sigma^{\infty}_{G}r}$$\textstyle{\mathbb{K}_{G}\mathbb{O}_{G}(X\wedge
B_{+}\wedge B_{+}\wedge
A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{K}_{G}\mathbb{O}_{G}(r)}$$\textstyle{\Sigma^{\infty}(X\wedge
B_{+}\wedge
A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\Sigma^{\infty}\pi}$$\textstyle{\mathbb{K}_{G}\mathbb{O}_{G}(X\wedge
B_{+}\wedge
A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{K}_{G}\mathbb{O}_{G}\pi}$$\textstyle{\Sigma^{\infty}_{G}(X\wedge
A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{K}_{G}\mathbb{O}_{G}(X\wedge
A_{+}).}$
Here $r$ is the evident left inverse of
$\operatorname{id}\wedge\Delta\wedge\operatorname{id}$ and $\pi$ is the
projection. The diagram commutes by the same concatenation of commutative
diagrams as in 2.9.
###### Theorem 2.16.
Let $X$ be a based $G$-space. Under our zigzag of equivalences,
$\Sigma^{\infty}_{G}X$ corresponds naturally to the presheaf
$(\scr{P}_{G}(X))^{G}$ that sends $A$ to
$\mathbb{K}\big{(}\mathbb{O}_{G}(X\wedge A_{+})^{G}\big{)}$.
###### Proof.
Note that $\mathbb{K}_{G}\mathbb{O}_{G}(X\wedge-_{+})$ is no longer a
representable presheaf. We again work with $G$-spectra and obtain the
conclusion after passage to $G$-fixed spectra. According to 2.6, we may
replace $\Sigma_{G}^{\infty}X$ by the positive fibrant $G$-spectrum
$\mathbb{K}_{G}\mathbb{O}_{G}(X)$, which we abbreviate to $\scr{B}_{G}(X)$ by
a slight abuse of notation. After this replacement, the presheaf
$\mathbb{U}(\Sigma_{G}^{\infty}X)$ may be computed as
$\mathbb{U}(\Sigma_{G}^{\infty}X)(A)=F_{G}(RQ\scr{B}_{G}(A),\scr{B}_{G}(X)).$
Therefore, following the chain of (2.8), we may compute
$\rho^{*}\gamma^{*}\mathbb{U}(\Sigma_{G}^{\infty}X)$ as
$\rho^{*}\gamma^{*}\mathbb{U}(\Sigma_{G}^{\infty}X)\simeq
F_{G}(Q\scr{B}_{G}(-),\scr{B}_{G}(X)).$
Thinking of $(B,A)$ above replaced by $(A,\ast)$, the adjoint to the composite
(2.10) $\scr{P}_{G}(X)(A)\wedge
Q\scr{B}_{G}(A)\xrightarrow{\operatorname{id}\wedge\lambda}\scr{P}_{G}(X)(A)\wedge\scr{B}_{G}(A)\xrightarrow{\circ}\scr{P}_{G}(X)(\ast)=\scr{B}_{G}(X)$
defines a map of presheaves
(2.11) $\lambda^{*}\scr{P}_{G}(X)\longrightarrow
F_{G}(Q\scr{B}_{G}(-),\scr{B}_{G}(X))$
with domain $Q\scr{B}_{G}$. It remains to show that this map is an
equivalence. To compute the adjoint (2.11), observe that the composite (2.10)
is the top horizontal composite in the commutative diagram
---
$\textstyle{\scr{P}_{G}(X)(A)\wedge
Q\scr{B}_{G}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\lambda}$$\textstyle{\scr{P}_{G}(X)(A)\wedge\scr{B}_{G}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\textstyle{\scr{B}_{G}(X)}$$\textstyle{\Sigma^{\infty}_{G}(X\wedge
A_{+})\wedge
Q\scr{B}_{G}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\wedge\operatorname{id}}$$\textstyle{\scr{B}_{G}(A,X)\wedge\Sigma^{\infty}_{G}A_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\alpha}$$\textstyle{\Sigma^{\infty}_{G}(X\wedge
A_{+})\wedge\Sigma^{\infty}_{G}A_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\mu}$$\scriptstyle{\alpha\wedge\operatorname{id}}$$\scriptstyle{\cong}$$\textstyle{\Sigma^{\infty}_{G}X\wedge\Sigma^{\infty}_{G}(A_{+}\wedge
A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\varepsilon}$$\textstyle{\Sigma^{\infty}_{G}X.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$
We have used that $\lambda\circ\mu=\alpha$. The pentagon on the right is a
special case of (2.9).
Therefore the map (2.11) is the top horizontal composite in the diagram
$\textstyle{\scr{P}_{G}(X)(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{G}(\scr{B}_{G}(A),\scr{B}_{G}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{G}(\lambda,\operatorname{id})}$$\textstyle{F_{G}(Q\scr{B}_{G}(A),\scr{B}_{G}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{G}(\mu,\operatorname{id})}$$\textstyle{\Sigma^{\infty}_{G}(X\wedge
A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\delta}$$\textstyle{F_{G}(\Sigma^{\infty}_{G}A_{+},\Sigma^{\infty}_{G}X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{G}(\operatorname{id},\alpha)}$$\textstyle{F_{G}(\Sigma^{\infty}_{G}A_{+},\scr{B}_{G}(X)).}$
The map $\alpha$ is a stable equivalence by 2.6. The map $\delta$ is the
stable equivalence of (1.4). The map $F_{G}(\operatorname{id},\alpha)$ is a
stable equivalence by [12, III.3.9]. Finally, the map
$F_{G}(\mu,\operatorname{id})$ is a stable equivalence by [5, 1.22]. ∎
There is another visible identification. The category $G\scr{S}$ and our
presheaf categories are $\scr{S}$-complete, so that they have tensors and
cotensors over $\scr{S}$ (see [5, §5.1]). It is formal that the left adjoint
of an $\scr{S}$-adjunction preserves tensors and the right adjoint preserves
cotensors. A quick chase of our zigzag of Quillen $\scr{S}$-equivalences gives
the following conclusion.
###### Theorem 2.17.
For $G$-spectra $Y$ and spectra $X$, if $Y$ corresponds to a presheaf
$\scr{P}Y$ under our zigzag of weak equivalences, then the tensor $Y\odot X$
corresponds to the tensor $\scr{P}Y\odot X$.
## 3\. Atiyah duality for finite $G$-sets
It is illuminating to see that we can come very close to constructing an
alternative model for the spectrally enriched category $G\scr{D}$ just by
applying the suspension $G$-spectrum functor $\Sigma^{\infty}_{G}$ to the
category of based $G$-spaces and $G$-maps and then passing to $G$-fixed
points. This is based on a close inspection of classical Atiyah duality
specialized to finite $G$-sets. However, it depends on working in the
alternative category $G\scr{Z}$ of $S_{G}$-modules [3, 12] rather than in the
category $G\scr{S}$ of orthogonal $G$-spectra. Because every object of
$G\scr{Z}$ is fibrant and its suspension $G$-spectra are easily understood, it
is more convenient than $G\scr{S}$ for comparison with space level
constructions. This leads us to a variant, 3.6, of 0.1 that does not invoke
infinite loop space theory. It is more topological and less categorical. It is
also more elementary.
### 3.1. The categories $G\scr{Z}$, $G\scr{D}$, and $\scr{D}_{G}$
Relevant background about $G\scr{Z}$ appears in [6, §3.4] and we just give a
minimum of notation here. In analogy with 2.3, we have the following
specialization of the same general result about stable model categories. It is
discussed in [6, §3.1].
###### Theorem 3.1.
Let $G\scr{D}$ be the full $\scr{Z}$-subcategory of $G\scr{Z}$ whose objects
are cofibrant approximations of the suspension $G$-spectra
$\Sigma^{\infty}_{G}(A_{+})$, where $A$ runs through the finite $G$-sets. Then
there is an enriched Quillen adjunction
$\textstyle{\mathbf{Pre}(G\scr{D},\scr{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{T}}$$\textstyle{G\scr{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces,}$$\scriptstyle{\mathbb{U}}$
and it is a Quillen equivalence.
Here $G\scr{D}$ is isomorphic to $(\scr{D}_{G})^{G}$, where $\scr{D}_{G}$ is a
full $\scr{Z}_{G}$-subcategory $\scr{D}_{G}$ of $\scr{S}_{G}$. All objects of
$G\scr{Z}$ are fibrant, and we need to choose cofibrant approximations of the
$\Sigma^{\infty}_{G}(A_{+})$. The construction of $G\scr{Z}$ starts from the
Lewis-May category $G\scr{S}\\!p$ of $G$-spectra, and $S_{G}$-modules are
$G$-spectra with additional structure. We have an elementary suspension
$G$-spectrum functor $\Sigma^{\infty}_{G}\colon G\scr{T}\longrightarrow
G\scr{S}\\!p.$ There is a left adjoint $\mathbb{F}\colon
G\scr{S}\\!p\longrightarrow G\scr{Z}$, which is a Quillen equivalence [3, 12].
Define $\mathbf{\Sigma}^{\infty}_{G}\colon G\scr{T}\longrightarrow G\scr{Z}$
to be the composite $\mathbb{F}\circ\Sigma^{\infty}_{G}$. Suspension
$G$-spectra have natural structures as $S_{G}$-modules, and there is a natural
stable equivalence of $S_{G}$-modules
$\gamma\colon\mathbf{\Sigma^{\infty}_{G}}X\longrightarrow\Sigma^{\infty}_{G}X.$
Viewing $\Sigma^{\infty}_{G}$ as a functor $G\scr{T}\longrightarrow G\scr{Z}$,
it is strong symmetric monoidal. However, the $\Sigma^{\infty}_{G}X$ are not
cofibrant. The functor $\mathbf{\Sigma^{\infty}_{G}}$ takes based $G$-CW
complexes $X$, such as $A_{+}$ for a finite $G$-set $A$, to cofibrant
$S_{G}$-modules. Therefore $\mathbf{\Sigma^{\infty}_{G}}$ may be viewed as a
cofibrant replacement functor for $\Sigma^{\infty}_{G}$. In particular, we
write $\mathbf{S_{G}}=\mathbf{\Sigma^{\infty}_{G}}S^{0}$ and have a cofibrant
approximation $\gamma\colon\mathbf{S_{G}}\longrightarrow S_{G}$ of the unit
object $S_{G}$. Moreover, the cofibrant approximation
$\mathbf{\Sigma^{\infty}_{G}}(A_{+})$ is isomorphic to
$\mathbf{S_{G}}\wedge\Sigma^{\infty}_{G}(A_{+})$ over
$\Sigma^{\infty}_{G}(A_{+})$.
As before, we consider finite $G$-sets $A$, $B$, and $C$, but we now agree to
write
$\mathbb{A}=\mathbf{\Sigma^{\infty}_{G}}A_{+},\ \ \
\mathbb{B}=\mathbf{\Sigma^{\infty}_{G}}B_{+},\ \ \ \text{and}\ \ \
\mathbb{C}=\mathbf{\Sigma^{\infty}_{G}}C_{+}.$
The $\mathbb{A}$ are bifibrant objects of $G\scr{Z}$ and we let $G\scr{D}$ and
$\scr{D}_{G}$ be the full subcategories of $G\scr{Z}$ and $\scr{Z}_{G}$ whose
objects are the $S_{G}$-modules $\mathbb{A}$, where $A$ runs over the finite
$G$-sets. Then $\scr{D}_{G}$ is enriched in $G\scr{Z}$ and
$G\scr{D}=(\scr{D}_{G})^{G}$ is enriched in the category $\scr{Z}$ of
$S$-modules. The functor $\mathbf{\Sigma^{\infty}_{G}}$ is almost strong
symmetric monoidal. Precisely, by [6, 3.9] there is a natural isomorphism
(3.1)
$\mathbb{A}\wedge\mathbb{B}\cong\mathbf{S_{G}}\wedge\mathbf{\Sigma^{\infty}_{G}}(A\times
B)_{+}$
with appropriate coherence properties with respect to associativity and
commutativity. Since $S_{G}$ is the unit for the smash product, we can compose
with
$\gamma\wedge\operatorname{id}\colon\mathbf{S_{G}}\wedge\mathbf{\Sigma^{\infty}_{G}}(A\times
B)_{+}\longrightarrow\mathbf{\Sigma^{\infty}_{G}}(A\wedge B)_{+}$
to give a pairing as if $\mathbf{\Sigma^{\infty}_{G}}$ were a lax symmetric
monoidal functor. However, the map $\gamma\colon\mathbf{S_{G}}\longrightarrow
S_{G}$ points the wrong way for the unit map of such a functor.
### 3.2. Space level Atiyah duality for finite $G$-sets
To lift the self-duality of $Ho\scr{D}_{G}$ to obtain a new model for
$\scr{D}_{G}$, we need representatives in $G\scr{Z}$ for the maps
$\eta\colon S_{G}\longrightarrow\mathbb{A}\wedge\mathbb{A}\ \ \ \text{and}\ \
\ \varepsilon\colon\mathbb{A}\wedge\mathbb{A}\longrightarrow S_{G}$
in $\text{Ho}G\scr{Z}$ that express the duality there. The map $\varepsilon$
is induced from the elementary map $\varepsilon$ of 1.23. The observation that
it plays a key role in Atiyah duality seems to be new. The definition of
$\eta$ requires desuspension by representation spheres.
Let $A$ be a finite $G$-set and let $V=\mathbb{R}[A]$ be the real
representation generated by $A$, with its standard inner product, so that
$|a|=1$ for $a\in A$. Since we are working on the space level, we may view
$A_{+}\wedge S^{V}$ as the wedge over $a\in A$ of the spaces (not $G$-spaces)
$\\{a\\}_{+}\wedge S^{V}$, with $G$ acting by $g(a,v)=(ga,gv)$. There is no
such wedge decomposition after passage to $G$-spectra.
###### Definition 3.2.
Recall that $\varepsilon\colon(A\times A)_{+}\longrightarrow S^{0}$ is the
$G$-map defined by $\varepsilon(a,b)=\ast$ if $a\neq b$ and
$\varepsilon(a,a)=1$. Recall too that $(A\times B)_{+}$ can be identified with
$A_{+}\wedge B_{+}$ and that the functor $\mathbf{\Sigma^{\infty}_{G}}$ is
almost strong symmetric monoidal. We shall also write $\varepsilon$ for the
composite map of $S_{G}$-modules
(3.2)
$\textstyle{\mathbb{A}\wedge\mathbb{A}\cong\mathbf{S_{G}}\wedge\mathbf{\Sigma^{\infty}_{G}}(A\times
A)_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\mathbf{\Sigma^{\infty}_{G}}\varepsilon}$$\textstyle{\mathbf{S_{G}}\wedge\mathbf{S_{G}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma\wedge\gamma}$$\textstyle{S_{G}\wedge
S_{G}\cong S_{G},}$
where the unlabeled isomorphisms are two instances of (3.1).
###### Definition 3.3.
Embed $A$ as the basis of the real representation $V=\mathbb{R}[A]$. The
normal bundle of the embedding is just $A\times V$, and its Thom complex is
$A_{+}\wedge S^{V}$. We obtain an explicit tubular embedding $\nu\colon
A\times V\longrightarrow V$ by setting
$\nu(a,v)=a+(\rho(|v|)/|v|)v,$
where $\rho\colon[0,\infty)\longrightarrow[0,d)$ is a homeomorphism for some
$d<1/2$; $\nu$ is a $G$-map since $|gv|=|v|$ for all $g$ and $v$. Applying the
Pontryagin-Thom construction, we obtain a $G$-map $t\colon
S^{V}\longrightarrow A_{+}\wedge S^{V}$, which is an equivariant pinch map
$S^{V}\longrightarrow\vee_{a\in A}S^{V}\cong A_{+}\wedge S^{V}.$
To be more precise, after collapsing the complement of the tubular embedding
to a point, we use $\nu^{-1}$ to expand each small homeomorphic copy of
$S^{V}$ to the canonical full-sized one; explicitly, if $|w|<d$, then
$\nu^{-1}(a+w)=(a,(\rho^{-1}(|w|)/|w|)w).$
The diagonal map on $A$ induces the Thom diagonal $\Delta\colon A_{+}\wedge
S^{V}\longrightarrow A_{+}\wedge A_{+}\wedge S^{V}$, and we let
(3.3) $\eta\colon S^{V}\longrightarrow A_{+}\wedge A_{+}\wedge S^{V}$
be the composite $\Delta\circ t$. Explicitly,
(3.4) $\eta(v)=\left\\{\begin{array}[]{ll}(a,a,(\rho^{-1}(|w|)/|w|)w)&\mbox{if
$v=a+w$ where $a\in A$ and $|w|<d$}\\\
\ast&\mbox{otherwise.}\end{array}\right.$
The negative sphere $G$-spectrum $S^{-V}$ in $G\scr{S}\\!p$ is obtained by
applying the left adjoint of the $V^{th}$-space functor to $S^{0}$, and
$S_{G}$ is isomorphic to $S^{V}\odot S^{-V}$ (see [11, I.4.2] and [12,
IV.2.2]). Taking the tensor of $\eta$ with $S^{-V}$ we obtain a map of
$G$-spectra
$S_{G}\cong S^{V}\odot S^{-V}\longrightarrow(A_{+}\wedge A_{+}\wedge
S^{V})\odot S^{-V}\cong(A_{+}\wedge A_{+})\odot
S_{G}\cong\Sigma^{\infty}_{G}(A_{+}\wedge A_{+}).$
Applying the functor $\mathbb{F}$ to this map and smashing with
$\mathbf{S_{G}}$ we obtain the second map in the diagram
(3.5) $\textstyle{S_{G}\cong S_{G}\wedge
S_{G}}$$\textstyle{\mathbf{S_{G}}\wedge\mathbf{S_{G}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma\wedge\gamma}$$\scriptstyle{\eta}$$\textstyle{\mathbf{S_{G}}\wedge\mathbf{\Sigma^{\infty}_{G}}{(A\times
A)_{+}}\cong\mathbb{A}\wedge\mathbb{A}.}$
The following result is a reminder about space level Atiyah duality. The
notion of a $V$-duality was defined and explained for smooth $G$-manifolds in
[11, §III.5].
###### Proposition 3.4.
The maps
$\eta\colon S^{V}\longrightarrow A_{+}\wedge A_{+}\wedge S^{V}\ \ \text{and}\
\ \varepsilon\wedge\operatorname{id}\colon A_{+}\wedge A_{+}\wedge
S^{V}\longrightarrow S^{V}$
specify a $V$-duality between $A_{+}$ and itself.
###### Proof.
This could be proven from scratch by proving the required triangle identities,
but in fact it is a special case of equivariant Atiyah duality for smooth
$G$-manifolds, $A$ being a $0$-dimensional example. Our specification of
$\eta$ is a specialization of the description of $\eta$ for a general smooth
$G$-manifold $M$ given in [11, p. 152]. We claim that our
$\varepsilon\wedge\operatorname{id}$ is a specialization of the definition of
$\varepsilon$ for a general smooth $G$-manifold given there. Indeed, letting
$s$ be the zero section of the normal bundle $\nu$ of the embedding
$A\subset\mathbb{R}[A]=V$, we have the composite embedding
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$$\textstyle{A\times
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s\times\mathrm{id}}$$\textstyle{(A\times
V)\times A\cong A\times A\times V.}$
The normal bundle of this embedding is $A\times V$, and we may view
$\Delta\times{\mathrm{id}}\colon A\times V\longrightarrow A\times A\times V$
as giving a big tubular neighborhood. The Pontryagin-Thom map here is obtained
by smashing the map $r\colon(A\times A)_{+}\longrightarrow A_{+}$ that sends
$(a,b)$ to $a$ if $a=b$ and to $\ast$ if $a\neq b$ with the identity map of
$S^{V}$. Composing with the map induced by the projection $\pi\colon
A_{+}\longrightarrow S^{0}$ that sends $a$ to $1$, this gives
$\varepsilon\wedge\operatorname{id}$. We observed this factorization of
$\varepsilon$ in 1.23 and we have used it before, in the proof of 2.11. ∎
Tensoring with $S^{-V}$, applying the functor
$\mathbf{S_{G}}\wedge\mathbb{F}$, and composing with $\gamma$, we obtain the
explicit duality maps in $G\scr{Z}$ displayed in (3.2) and (3.5).
### 3.3. The weakly unital categories $G\scr{A}$ and $\scr{A}_{G}$
Since the $G$-spectra $\mathbb{A}$ are self-dual,
$F_{G}(\mathbb{A},\mathbb{B})$ is naturally isomorphic to
$\mathbb{B}\wedge\mathbb{A}$ in $\text{Ho}G\scr{Z}$, and the composition and
unit
(3.6) $F_{G}(\mathbb{B},\mathbb{C})\wedge
F_{G}(\mathbb{A},\mathbb{B})\longrightarrow F_{G}(\mathbb{A},\mathbb{C})\ \ \
\text{and}\ \ \ S_{G}\longrightarrow F_{G}(\mathbb{B},\mathbb{B})$
can be expressed as maps
(3.7)
$\mathbb{C}\wedge\mathbb{B}\wedge\mathbb{B}\wedge\mathbb{A}\longrightarrow\mathbb{C}\wedge\mathbb{A}\
\ \ \text{and}\ \ \ S_{G}\longrightarrow\mathbb{A}\wedge\mathbb{A}$
in $\text{Ho}G\scr{Z}$. We want to understand these maps in terms of duality
in $G\scr{Z}$, without use of infinite loop space theory. However, since we
are working in $G\scr{Z}$, we must take the isomorphisms (3.1) and the
cofibrant approximation $\gamma\colon\mathbf{S}_{G}\longrightarrow S_{G}$ into
account, and we cannot expect to have strict units. The notion of a weakly
unital enriched category was introduced in [5, §3.5] to formalize what we see
here.
Thus we shall construct a weakly unital $G\scr{Z}$-category $\scr{A}_{G}$ and
compare it with $\scr{D}_{G}$. The $G$-fixed category $G\scr{A}$ will be a
weakly unital $\scr{Z}$-category.999Mnemonically, the $\scr{A}$ stands for
Atiyah.
The objects of $\scr{A}_{G}$ and $G\scr{A}$ are the $S_{G}$-modules
$\mathbb{A}$ for finite $G$-sets $A$. The morphism $S_{G}$-modules of
$\scr{A}_{G}$ are
$\scr{A}_{G}(\mathbb{A},\mathbb{B})=\mathbb{B}\wedge\mathbb{A}$. Composition
is given by the maps
(3.8)
$\operatorname{id}\wedge\varepsilon\wedge\operatorname{id}\colon\mathbb{C}\wedge\mathbb{B}\wedge\mathbb{B}\wedge\mathbb{A}\longrightarrow\mathbb{C}\wedge\mathbb{A},$
where $\varepsilon$ is the map of (3.2); compare 2.11.
As recalled in §1.5, the adjoint
$\tilde{\varepsilon}\colon\mathbb{A}\longrightarrow
D\mathbb{A}=F_{G}(\mathbb{A},S_{G})$ of $\varepsilon$ is a stable equivalence,
and we have the composite stable equivalence
(3.9)
$\delta=\zeta\circ(\operatorname{id}\wedge\tilde{\varepsilon})\colon\mathbb{B}\wedge\mathbb{A}\longrightarrow\mathbb{B}\wedge
D\mathbb{A}\longrightarrow F_{G}(\mathbb{A},\mathbb{B}).$
Formal properties of the adjunction ($\wedge$,$F_{G}$) give the following
commutative diagram in $G\scr{Z}$, which uses $\delta$ to compare composition
in $\scr{A}_{G}$ with composition in $\scr{D}_{G}$.
(3.10)
$\textstyle{\mathbb{C}\wedge\mathbb{B}\wedge\mathbb{B}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\varepsilon\wedge\operatorname{id}}$$\scriptstyle{\mathrm{id}\wedge\tilde{\varepsilon}\wedge\mathrm{id}\wedge\tilde{\varepsilon}}$$\textstyle{\mathbb{C}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}\wedge\tilde{\varepsilon}}$$\textstyle{\mathbb{C}\wedge
D\mathbb{B}\wedge\mathbb{B}\wedge
D\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}\wedge\varepsilon\wedge\mathrm{id}}$$\scriptstyle{\zeta\wedge\zeta}$$\textstyle{\mathbb{C}\wedge
D\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta}$$\textstyle{F_{G}(\mathbb{B},\mathbb{C})\wedge
F_{G}(\mathbb{A},\mathbb{B})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\textstyle{F_{G}(\mathbb{A},\mathbb{C})}$
At the bottom, we do not know that the function $S_{G}$-modules or their smash
product are cofibrant, but all objects at the top are cofibrant and thus
bifibrant. In general, to compute the smash product of $G$-spectra $X$ and $Y$
in the homotopy category, we should take the smash product of cofibrant
approximations $QX$ and $QY$ of $X$ and $Y$. Since all objects of $G\scr{Z}$
are fibrant, to compute a map $X\wedge Y\longrightarrow Z$ in the homotopy
category, we should represent it by a map $QX\wedge QY\longrightarrow QZ$ and
take its homotopy class. The diagram displays such a cofibrant approximation
of the composition in $\scr{D}_{G}$.
The unit $S_{G}\longrightarrow F_{G}(\mathbb{A},\mathbb{A})$ of $\scr{A}_{G}$
is represented by the (formal) composite
(3.11)
$\textstyle{S_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{\mathbb{A}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\tilde{\varepsilon}}$$\textstyle{\mathbb{A}\wedge
D\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta}$$\textstyle{F_{G}(\mathbb{A},\mathbb{A})}$
that is obtained by inverting the map $\gamma\wedge\gamma$ in (3.5) to obtain
the map denoted $\eta$. The weak unital property is a way of expressing the
unital property by maps in $\scr{Z}_{G}$, without use of inverses in
$Ho\scr{Z}_{G}$. This is a bit tedious. Here are the details.
###### Definition 3.5.
Let $V=\mathbb{R}[A]$. For $a\in A$, define $\xi_{a}\colon\\{a\\}_{+}\wedge
S^{V}\longrightarrow\\{a\\}_{+}\wedge S^{V}$ by
(3.12)
$\xi_{a}(a,v)=\left\\{\begin{array}[]{ll}(a,(\rho^{-1}(|w|)/|w|)w)&\mbox{if
$v=a+w$ and $|w|<d$}\\\ \ast&\mbox{otherwise,}\end{array}\right.$
where $\rho$ is as in 3.3. Then the wedge of the $\xi_{a}$ is a $G$-map
(3.13) $\xi\colon A_{+}\wedge S^{V}\longrightarrow A_{+}\wedge S^{V};$
$\xi$ is $G$-homotopic to the identity map of $A_{+}\wedge S^{V}$ via the
explicit $G$-homotopy
$h(a,v,t)=\left\\{\begin{array}[]{ll}(a,v)&\mbox{if $t=0$ or $v=a$}\\\
(a,(1-t)v+t(\rho^{-1}(t|w|)/|w|)w)&\mbox{if $v=a+w$ and $t|w|<d$}\\\
\ast&\mbox{otherwise.}\end{array}\right.$
With $\eta$ as specified in (3.3), easy and perhaps illuminating inspections
show that the following unit diagrams already commute in $G\scr{T}$, before
passage to homotopy. In both, $A$ and $B$ are finite $G$-sets. In the first,
$V=\mathbb{R}[A]$. In the second, $V=\mathbb{R}[B]$ and we move $S^{V}$ from
the right to the left for clarity.
$\textstyle{B_{+}\wedge A_{+}\wedge
S^{V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\eta_{A}}$$\scriptstyle{\operatorname{id}\wedge\xi_{A}}$$\textstyle{B_{+}\wedge
A^{3}_{+}\wedge
S^{V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\operatorname{id}\wedge\varepsilon\wedge\operatorname{id}}$$\textstyle{B_{+}\wedge
A_{+}\wedge S^{V}}$ and
---
$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces S^{V}\wedge
B_{+}\wedge
A_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{B}\wedge\operatorname{id}_{A}}$$\scriptstyle{\eta_{B}\wedge\operatorname{id}}$$\textstyle{S^{V}\wedge
B^{3}_{+}\wedge
A_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\operatorname{id}\wedge\varepsilon\wedge\operatorname{id}}$$\textstyle{S^{V}\wedge
B_{+}\wedge A_{+}}$
Tensoring with $S^{-V}$ and using the natural isomorphisms
$(X\wedge S^{V})\odot S^{-V}\cong X\odot S_{G}\cong\Sigma^{\infty}_{G}X$
for based $G$-spaces $X$, we see that the space level $G$-equivalence $\xi$
induces a spectrum level $G$-equivalence
$\xi\colon\mathbb{A}\longrightarrow\mathbb{A}$.
Tensoring with $S^{-V}$ and using (3.1) to pass to smash products of
$S_{G}$-modules, a little diagram chase shows that the previous pair of
diagrams in $G\scr{T}$ gives rise to the following pair of commutative
diagrams in $G\scr{Z}$. These express the unit laws for a weakly unital
$G\scr{Z}$-category $\scr{A}_{G}$ [5, §3.5] with objects the $\mathbb{A}$ and
composition as specified in (3.8). The cited unit laws allow us to start with
any chosen cofibrant approximation $\gamma\colon QS_{G}\longrightarrow S_{G}$
of the unit $S_{G}$, and we are led by (3.5) to choose our cofibrant
approximation to be
$\gamma\wedge\gamma\colon\mathbf{S_{G}}\wedge\mathbf{S_{G}}\longrightarrow
S_{G}\wedge S_{G}\cong S_{G}.$ Using the notation $\gamma\colon
QS_{G}\longrightarrow S_{G}$ for this map, we obtain the required diagrams
$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathbb{B}\wedge\mathbb{A}\wedge
QS_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\xi\wedge\gamma}$$\scriptstyle{\operatorname{id}\wedge\eta}$$\textstyle{\mathbb{B}\wedge\mathbb{A}\wedge\mathbb{A}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\textstyle{\mathbb{B}\wedge\mathbb{A}\wedge
S_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathbb{B}\wedge\mathbb{A}}$
and $\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
QS_{G}\wedge\mathbb{B}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma\wedge\xi\wedge\operatorname{id}}$$\scriptstyle{\eta\wedge\operatorname{id}}$$\textstyle{\mathbb{B}\wedge\mathbb{B}\wedge\mathbb{B}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\textstyle{S_{G}\wedge\mathbb{B}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathbb{B}\wedge\mathbb{A}.}$
Taking $A=S^{0}$ in our second space level diagram and changing $B$ to $A$, we
also obtain the following commutative diagrams in $G\scr{Z}$, where the second
diagram is adjoint to the first.
(3.14) $\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
QS_{G}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma\wedge\xi}$$\scriptstyle{\eta\wedge\operatorname{id}}$$\textstyle{\mathbb{A}\wedge\mathbb{A}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\varepsilon}$$\textstyle{S_{G}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathbb{A}}$
and
$\textstyle{QS_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{\eta}$$\textstyle{\mathbb{A}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\tilde{\varepsilon}}$$\textstyle{bA\wedge
D\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta}$$\textstyle{S_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{F_{G}(\mathbb{A},\mathbb{A})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{G}(\xi,\operatorname{id})}$$\textstyle{F_{G}(\mathbb{A},\mathbb{A})}$
Here $\eta$ at the bottom right is adjoint to the identity map of
$\mathbb{A}$. In effect, this uses
$\delta=\zeta\circ(\operatorname{id}\wedge\tilde{\varepsilon})$ to compare the
actual unit $\eta$ in $\scr{D}_{G}$ at the top with the weak unit in
$\scr{A}_{G}$, which is given by the interrelated maps $\eta$, $\gamma$, and
$\xi$.
### 3.4. The category of presheaves with domain $G\scr{A}$
The diagrams (3.10) and (3.14) show that the maps
$\delta\colon\mathbb{A}\wedge\mathbb{B}\longrightarrow
F_{G}(\mathbb{A},\mathbb{B})$ specify a map of weakly unital
$\scr{Z}_{G}$-categories from the weakly unital $\scr{Z}_{G}$-category
$\scr{A}_{G}$ to the (unital) $\scr{Z}_{G}$-category $\scr{D}_{G}$. Passing to
$G$-fixed points, we obtain a weakly unital $\scr{Z}$-category $G\scr{A}$ and
a map $\delta\colon G\scr{A}\longrightarrow G\scr{D}$ of weakly unital
$\scr{Z}$-categories. Weakly unital presheaves and presheaf categories are
defined in [5, 3.25]. By [5, 3.26], we obtain the same category of presheaves
$\scr{Z}^{G\scr{D}}$ using unital or weakly unital presheaves. Since $\delta$
is an equivalence, we can adapt the methodology of [5, §2] to prove the
following result. However, since we find the use of weakly unital categories
unpleasant and our main result 1.9 more satisfactory, we shall leave the
details to the interested reader. Nevertheless, it is this equivalence that
best captures the geometric intuition behind our results.
###### Theorem 3.6.
The categories $\mathbf{Pre}(G\scr{A},\scr{Z})$ and
$\mathbf{Pre}(G\scr{D},\scr{Z})$ are Quillen equivalent.
## References
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|
arxiv-papers
| 2011-10-17T03:41:19 |
2024-09-04T02:49:23.196009
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Bertrand Guillou and J.P. May",
"submitter": "Bertrand Guillou",
"url": "https://arxiv.org/abs/1110.3571"
}
|
1110.3676
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
LHCb-PAPER-2011-008 CERN-PH-EP-2011-150
First observation of the decay $\kern
3.73305pt\overline{\kern-3.73305ptB}{}^{0}_{s}\\!\rightarrow D^{0}K^{*0}$ and
a measurement of the ratio of branching fractions $\frac{{\cal B}\left(\kern
2.61313pt\overline{\kern-2.61313ptB}{}^{0}_{s}\\!\rightarrow
D^{0}K^{*0}\right)}{{\cal B}\left(\kern
2.61313pt\overline{\kern-2.61313ptB}{}^{0}\\!\rightarrow
D^{0}\rho^{0}\right)}$
Submitted to Phys. Lett. B
The LHCb Collaboration 111Authors are listed on the following pages.
The first observation of the decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow D^{0}K^{*0}$
using $pp$ data collected by the LHCb detector at a centre-of-mass energy of 7
TeV, corresponding to an integrated luminosity of 36 pb-1, is reported. A
signal of $34.4\pm 6.8$ events is obtained and the absence of signal is
rejected with a statistical significance of more than nine standard
deviations. The $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow D^{0}K^{*0}$
branching fraction is measured relative to that of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow D^{0}\rho^{0}$:
$\frac{{\cal B}\left(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\\!\rightarrow
D^{0}K^{*0}\right)}{{\cal B}\left(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}\\!\rightarrow
D^{0}\rho^{0}\right)}=1.48\pm 0.34\pm 0.15\pm 0.12$, where the first
uncertainty is statistical, the second systematic and the third is due to the
uncertainty on the ratio of the $B^{0}$ and $B^{0}_{s}$ hadronisation
fractions.
The LHCb Collaboration
R. Aaij23, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi42, C. Adrover6, A.
Affolder48, Z. Ajaltouni5, J. Albrecht37, F. Alessio37, M. Alexander47, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr22, S. Amato2, Y. Amhis38, J.
Anderson39, R.B. Appleby50, O. Aquines Gutierrez10, F. Archilli18,37, L.
Arrabito53, A. Artamonov 34, M. Artuso52,37, E. Aslanides6, G. Auriemma22,m,
S. Bachmann11, J.J. Back44, D.S. Bailey50, V. Balagura30,37, W. Baldini16,
R.J. Barlow50, C. Barschel37, S. Barsuk7, W. Barter43, A. Bates47, C. Bauer10,
Th. Bauer23, A. Bay38, I. Bediaga1, K. Belous34, I. Belyaev30,37, E. Ben-
Haim8, M. Benayoun8, G. Bencivenni18, S. Benson46, J. Benton42, R. Bernet39,
M.-O. Bettler17, M. van Beuzekom23, A. Bien11, S. Bifani12, A. Bizzeti17,h,
P.M. Bjørnstad50, T. Blake49, F. Blanc38, C. Blanks49, J. Blouw11, S. Blusk52,
A. Bobrov33, V. Bocci22, A. Bondar33, N. Bondar29, W. Bonivento15, S.
Borghi47, A. Borgia52, T.J.V. Bowcock48, C. Bozzi16, T. Brambach9, J. van den
Brand24, J. Bressieux38, D. Brett50, S. Brisbane51, M. Britsch10, T.
Britton52, N.H. Brook42, H. Brown48, A. Büchler-Germann39, I. Burducea28, A.
Bursche39, J. Buytaert37, S. Cadeddu15, J.M. Caicedo Carvajal37, O. Callot7,
M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P. Campana18,37, A. Carbone14,
G. Carboni21,k, R. Cardinale19,i,37, A. Cardini15, L. Carson36, K. Carvalho
Akiba23, G. Casse48, M. Cattaneo37, M. Charles51, Ph. Charpentier37, N.
Chiapolini39, K. Ciba37, X. Cid Vidal36, G. Ciezarek49, P.E.L. Clarke46,37, M.
Clemencic37, H.V. Cliff43, J. Closier37, C. Coca28, V. Coco23, J. Cogan6, P.
Collins37, A. Comerma-Montells35, F. Constantin28, G. Conti38, A. Contu51, A.
Cook42, M. Coombes42, G. Corti37, G.A. Cowan38, R. Currie46, B. D’Almagne7, C.
D’Ambrosio37, P. David8, I. De Bonis4, S. De Capua21,k, M. De Cian39, F. De
Lorenzi12, J.M. De Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M.
Deckenhoff9, H. Degaudenzi38,37, M. Deissenroth11, L. Del Buono8, C.
Deplano15, O. Deschamps5, F. Dettori15,d, J. Dickens43, H. Dijkstra37, P.
Diniz Batista1, F. Domingo Bonal35,n, S. Donleavy48, A. Dosil Suárez36, D.
Dossett44, A. Dovbnya40, F. Dupertuis38, R. Dzhelyadin34, S. Easo45, U.
Egede49, V. Egorychev30, S. Eidelman33, D. van Eijk23, F. Eisele11, S.
Eisenhardt46, R. Ekelhof9, L. Eklund47, Ch. Elsasser39, D.G. d’Enterria35,o,
D. Esperante Pereira36, L. Estève43, A. Falabella16,e, E. Fanchini20,j, C.
Färber11, G. Fardell46, C. Farinelli23, S. Farry12, V. Fave38, V. Fernandez
Albor36, M. Ferro-Luzzi37, S. Filippov32, C. Fitzpatrick46, M. Fontana10, F.
Fontanelli19,i, R. Forty37, M. Frank37, C. Frei37, M. Frosini17,f,37, S.
Furcas20, A. Gallas Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini51, Y.
Gao3, J-C. Garnier37, J. Garofoli52, J. Garra Tico43, L. Garrido35, D.
Gascon35, C. Gaspar37, N. Gauvin38, M. Gersabeck37, T. Gershon44,37, Ph.
Ghez4, V. Gibson43, V.V. Gligorov37, C. Göbel54, D. Golubkov30, A.
Golutvin49,30,37, A. Gomes2, H. Gordon51, M. Grabalosa Gándara35, R. Graciani
Diaz35, L.A. Granado Cardoso37, E. Graugés35, G. Graziani17, A. Grecu28, E.
Greening51, S. Gregson43, B. Gui52, E. Gushchin32, Yu. Guz34, T. Gys37, G.
Haefeli38, C. Haen37, S.C. Haines43, T. Hampson42, S. Hansmann-Menzemer11, R.
Harji49, N. Harnew51, J. Harrison50, P.F. Harrison44, J. He7, V. Heijne23, K.
Hennessy48, P. Henrard5, J.A. Hernando Morata36, E. van Herwijnen37, E.
Hicks48, W. Hofmann10, K. Holubyev11, P. Hopchev4, W. Hulsbergen23, P. Hunt51,
T. Huse48, R.S. Huston12, D. Hutchcroft48, D. Hynds47, V. Iakovenko41, P.
Ilten12, J. Imong42, R. Jacobsson37, A. Jaeger11, M. Jahjah Hussein5, E.
Jans23, F. Jansen23, P. Jaton38, B. Jean-Marie7, F. Jing3, M. John51, D.
Johnson51, C.R. Jones43, B. Jost37, S. Kandybei40, M. Karacson37, T.M.
Karbach9, J. Keaveney12, U. Kerzel37, T. Ketel24, A. Keune38, B. Khanji6, Y.M.
Kim46, M. Knecht38, S. Koblitz37, P. Koppenburg23, A. Kozlinskiy23, L.
Kravchuk32, K. Kreplin11, M. Kreps44, G. Krocker11, P. Krokovny11, F. Kruse9,
K. Kruzelecki37, M. Kucharczyk20,25,37, S. Kukulak25, R. Kumar14,37, T.
Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G. Lafferty50, A. Lai15,
D. Lambert46, R.W. Lambert37, E. Lanciotti37, G. Lanfranchi18, C.
Langenbruch11, T. Latham44, R. Le Gac6, J. van Leerdam23, J.-P. Lees4, R.
Lefèvre5, A. Leflat31,37, J. Lefran$c$cois7, O. Leroy6, T. Lesiak25, L. Li3,
L. Li Gioi5, M. Lieng9, M. Liles48, R. Lindner37, C. Linn11, B. Liu3, G.
Liu37, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-March38, J. Luisier38, F.
Machefert7, I.V. Machikhiliyan4,30, F. Maciuc10, O. Maev29,37, J. Magnin1, S.
Malde51, R.M.D. Mamunur37, G. Manca15,d, G. Mancinelli6, N. Mangiafave43, U.
Marconi14, R. Märki38, J. Marks11, G. Martellotti22, A. Martens7, L. Martin51,
A. Martín Sánchez7, D. Martinez Santos37, A. Massafferri1, Z. Mathe12, C.
Matteuzzi20, M. Matveev29, E. Maurice6, B. Maynard52, A. Mazurov16,32,37, G.
McGregor50, R. McNulty12, C. Mclean14, M. Meissner11, M. Merk23, J. Merkel9,
R. Messi21,k, S. Miglioranzi37, D.A. Milanes13,37, M.-N. Minard4, S. Monteil5,
D. Moran12, P. Morawski25, R. Mountain52, I. Mous23, F. Muheim46, K. Müller39,
R. Muresan28,38, B. Muryn26, M. Musy35, J. Mylroie-Smith48, P. Naik42, T.
Nakada38, R. Nandakumar45, J. Nardulli45, I. Nasteva1, M. Nedos9, M.
Needham46, N. Neufeld37, C. Nguyen-Mau38,p, M. Nicol7, S. Nies9, V. Niess5, N.
Nikitin31, A. Nomerotski51, A. Oblakowska-Mucha26, V. Obraztsov34, S.
Oggero23, S. Ogilvy47, O. Okhrimenko41, R. Oldeman15,d, M. Orlandea28, J.M.
Otalora Goicochea2, P. Owen49, K. Pal52, J. Palacios39, A. Palano13,b, M.
Palutan18, J. Panman37, A. Papanestis45, M. Pappagallo13,b, C. Parkes47,37,
C.J. Parkinson49, G. Passaleva17, G.D. Patel48, M. Patel49, S.K. Paterson49,
G.N. Patrick45, C. Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos Alvarez36,
A. Pellegrino23, G. Penso22,l, M. Pepe Altarelli37, S. Perazzini14,c, D.L.
Perego20,j, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35, P. Perret5, M.
Perrin-Terrin6, G. Pessina20, A. Petrella16,37, A. Petrolini19,i, E. Picatoste
Olloqui35, B. Pie Valls35, B. Pietrzyk4, T. Pilar44, D. Pinci22, R.
Plackett47, S. Playfer46, M. Plo Casasus36, G. Polok25, A. Poluektov44,33, E.
Polycarpo2, D. Popov10, B. Popovici28, C. Potterat35, A. Powell51, T. du
Pree23, J. Prisciandaro38, V. Pugatch41, A. Puig Navarro35, W. Qian52, J.H.
Rademacker42, B. Rakotomiaramanana38, M.S. Rangel2, I. Raniuk40, G. Raven24,
S. Redford51, M.M. Reid44, A.C. dos Reis1, S. Ricciardi45, K. Rinnert48, D.A.
Roa Romero5, P. Robbe7, E. Rodrigues47, F. Rodrigues2, P. Rodriguez Perez36,
G.J. Rogers43, S. Roiser37, V. Romanovsky34, M. Rosello35,n, J. Rouvinet38, T.
Ruf37, H. Ruiz35, G. Sabatino21,k, J.J. Saborido Silva36, N. Sagidova29, P.
Sail47, B. Saitta15,d, C. Salzmann39, M. Sannino19,i, R. Santacesaria22, R.
Santinelli37, E. Santovetti21,k, M. Sapunov6, A. Sarti18,l, C. Satriano22,m,
A. Satta21, M. Savrie16,e, D. Savrina30, P. Schaack49, M. Schiller11, S.
Schleich9, M. Schmelling10, B. Schmidt37, O. Schneider38, A. Schopper37, M.-H.
Schune7, R. Schwemmer37, B. Sciascia18, A. Sciubba18,l, M. Seco36, A.
Semennikov30, K. Senderowska26, I. Sepp49, N. Serra39, J. Serrano6, P.
Seyfert11, B. Shao3, M. Shapkin34, I. Shapoval40,37, P. Shatalov30, Y.
Shcheglov29, T. Shears48, L. Shekhtman33, O. Shevchenko40, V. Shevchenko30, A.
Shires49, R. Silva Coutinho54, H.P. Skottowe43, T. Skwarnicki52, A.C. Smith37,
N.A. Smith48, E. Smith51,45, K. Sobczak5, F.J.P. Soler47, A. Solomin42, F.
Soomro49, B. Souza De Paula2, B. Spaan9, A. Sparkes46, P. Spradlin47, F.
Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S. Stone52,37, B.
Storaci23, M. Straticiuc28, U. Straumann39, N. Styles46, V.K. Subbiah37, S.
Swientek9, M. Szczekowski27, P. Szczypka38, T. Szumlak26, S. T’Jampens4, E.
Teodorescu28, F. Teubert37, C. Thomas51,45, E. Thomas37, J. van Tilburg11, V.
Tisserand4, M. Tobin39, S. Topp-Joergensen51, N. Torr51, M.T. Tran38, A.
Tsaregorodtsev6, N. Tuning23, A. Ukleja27, P. Urquijo52, U. Uwer11, V.
Vagnoni14, G. Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36, S.
Vecchi16, J.J. Velthuis42, M. Veltri17,g, K. Vervink37, B. Viaud7, I. Videau7,
X. Vilasis-Cardona35,n, J. Visniakov36, A. Vollhardt39, D. Voong42, A.
Vorobyev29, H. Voss10, K. Wacker9, S. Wandernoth11, J. Wang52, D.R. Ward43,
A.D. Webber50, D. Websdale49, M. Whitehead44, D. Wiedner11, L. Wiggers23, G.
Wilkinson51, M.P. Williams44,45, M. Williams49, F.F. Wilson45, J. Wishahi9, M.
Witek25, W. Witzeling37, S.A. Wotton43, K. Wyllie37, Y. Xie46, F. Xing51, Z.
Xing52, Z. Yang3, R. Young46, O. Yushchenko34, M. Zavertyaev10,a, L. Zhang52,
W.C. Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, E. Zverev31, A. Zvyagin 37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
24Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, Netherlands
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Cracow, Poland
26Faculty of Physics & Applied Computer Science, Cracow, Poland
27Soltan Institute for Nuclear Studies, Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
43Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
44Department of Physics, University of Warwick, Coventry, United Kingdom
45STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
46School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
47School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
48Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
49Imperial College London, London, United Kingdom
50School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
51Department of Physics, University of Oxford, Oxford, United Kingdom
52Syracuse University, Syracuse, NY, United States
53CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oInstitució Catalana de Recerca i Estudis Avan$c$cats (ICREA), Barcelona,
Spain
pHanoi University of Science, Hanoi, Viet Nam
## 1 Introduction
A theoretically clean extraction of the Cabibbo-Kobayashi-Maskawa (CKM)
unitarity triangle angle $\gamma$ can be performed using time-integrated
$B\\!\rightarrow DX$ decays by exploiting the interference between Cabibbo-
suppressed $b\\!\rightarrow u$ and Cabibbo-allowed $b\\!\rightarrow c$
transitions [1, 2, 3, 4, 5, 6]. One of the most promising channels for this
purpose is $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow
D\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$, where $D$ represents a
$D^{0}$ or a $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ meson.222In
this Letter the mention of a decay will refer also to its charge-conjugate
state. Although this channel involves the decay of a neutral $B$ meson, the
final state is self-tagged by the flavour of the $K^{*0}$ so that a time-
dependent analysis is not required. In the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow D\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay, both the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow D^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ are colour suppressed. Therefore,
although the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow
D\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay has a lower
branching fraction compared to the $B^{+}\\!\rightarrow DK^{+}$ mode, it could
exhibits an enhanced interference.
The Cabibbo-allowed $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow D^{0}K^{*0}$ and
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow
D^{*0}K^{*0}$ decays potentially provide a significant background to the
Cabibbo-suppressed $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay. The expected size of this
background is unknown, since the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow D^{(*)0}K^{*0}$
decay has not yet been observed. In addition, a measurement of the branching
fraction of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow D^{0}K^{*0}$ is
of interest as a probe of $\mathrm{SU}(3)$ breaking in colour suppressed
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(d,s)}\\!\rightarrow
D^{0}V$ decays [7, 8], where $V$ denotes a neutral vector meson. Thus, the
detailed study of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow D^{0}K^{*0}$ is
an important goal with the first LHCb data.
The LHCb detector [9] is a forward spectrometer constructed to measure decays
of hadrons containing $b$ and $c$ quarks. The detector elements, placed along
the collision axis of the Large Hadron Collider (LHC), start with the Vertex
Locator, a silicon strip device that surrounds the $pp$ interaction region
with its innermost sensitive part positioned $8\text{\,}\mathrm{mm}$ from the
beam. It precisely determines the locations of the primary $pp$ interaction
vertices, the locations of the decay vertices of long-lived hadrons, and
contributes to the measurement of track momenta. Other tracking detectors
include a large-area silicon strip detector located upstream of the
$4\text{\,}\Tm$ dipole magnet and a combination of silicon strip detectors and
straw drift chambers placed downstream. Two Ring-Imaging Cherenkov (RICH)
detectors are used to identify charged hadrons. Further downstream an
electromagnetic calorimeter is used for photon detection and electron
identification, followed by a hadron calorimeter and a muon system consisting
of alternating layers of iron and gaseous chambers. LHCb operates a two stage
trigger system. In the first stage hardware trigger the rate is reduced from
the visible interaction rate to about $1\text{\,}\mathrm{MHz}$ using
information from the calorimeters and muon system. In the second stage
software trigger the rate is further reduced to $2\text{\,}\mathrm{kHz}$ by
performing a set of channel specific selections based upon a full event
reconstruction. During the 2010 data taking period, several trigger
configurations were used for both stages in order to cope with the varying
beam conditions.
The results reported here uses of $pp$ data collected at the LHC at a centre-
of-mass energy $\sqrt{s}=$7\text{\,}\mathrm{TeV}$$ in 2010. The strategy of
the analysis is to measure a ratio of branching fractions in which most of the
potentially large systematic uncertainties cancel. The decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow D^{0}\rho^{0}$ is
used as the normalisation channel. In both decay channels, the $D^{0}$ is
reconstructed in the Cabibbo-allowed decay mode $D^{0}\\!\rightarrow
K^{-}\pi^{+}$; the contribution from the doubly Cabibbo-suppressed
$D^{0}\\!\rightarrow K^{+}\pi^{-}$ decay is negligible. The $K^{*0}$ is
reconstructed in the $K^{*0}\\!\rightarrow K^{+}\pi^{-}$ decay mode and the
$\rho^{0}$ in the $\rho^{0}\\!\rightarrow\pi^{+}\pi^{-}$ decay mode. The main
systematic uncertainties arise from the different particle identification
requirements and the pollution of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow D^{0}\rho^{0}$ peak
by $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow
D^{0}\pi^{+}\pi^{-}$ decays where the $\pi^{+}\pi^{-}$ pairs do not originate
from a $\rho^{0}$ resonance. In addition, the normalisation of the $B^{0}_{s}$
decay to a $B^{0}$ decay suffers from a systematic uncertainty of
$8\text{\,}\mathrm{\char 37\relax}$ due to the current knowledge of the ratio
of the fragmentation fractions $f_{s}/f_{d}=0.267^{+0.021}_{-0.020}$ [10].
## 2 Events selection
Monte Carlo samples of signal and background events are used to optimize the
signal selection and to parametrize the probability density functions (PDFs)
used in the fit. Proton beam collisions are generated with PYTHIA [11] and
decays of hadronic particles are provided by EvtGen [12]. The generated
particles are traced through the detector with GEANT4 [13], taking into
account the details of the geometry and material composition of the detector.
$B^{0}$ and $B^{0}_{s}$ mesons are reconstructed from a selected $D^{0}$ meson
combined with a vector particle ($\rho^{0}$ or $K^{*0}$). The selection
requirements are kept as similar as possible for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow D^{0}K^{*0}$ and
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow
D^{0}\rho^{0}$. The four charged particles in the decay are each required to
have a transverse momentum $p_{T}>$300\text{\,}\MeVoverc$$ for the daughters
of the vector particle and $p_{T}>$250\text{\,}\MeVoverc$$
($400\text{\,}\MeVoverc$) for the pion (kaon) from the $D^{0}$ meson decay.
The $\chi^{2}$ of the track impact parameter with respect to any primary
vertex is required to be greater than 4. A cut on the absolute value of the
cosine of the helicity angle of the vector meson greater than 0.4 is applied.
The tracks of the $D^{0}$ meson daughters are combined to form a vertex with a
goodness of fit $\chi^{2}/\textrm{ndf}$ smaller than 5. The $B$ meson vertex
formed by the $D^{0}$ and the tracks of the $V$ meson daughters is required to
satisfy $\chi^{2}/\textrm{ndf}<4$. The smallest impact parameter of the $B$
meson with respect to all the primary vertices is required to be smaller than
9 and defines uniquely the primary vertex associated to the $B$ meson. Since
the $B^{0}$ or $B^{0}_{s}$ should point towards the primary vertex, the angle
between the $B$ momentum and the $B$ line of flight defined by the line
between the $B$ vertex and the primary vertex is required to be less than
$10\text{\,}\rm\,\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{d}$. Finally, since the
measured $z$ position (along the beam direction) of the $D$ vertex
($z_{\textrm{\scriptsize{$D$}}}$) is not expected to be situated significantly
upstream of the $z$ position of the vector particle vertex ($z_{V}$), a
requirement of
$(z_{\textrm{\scriptsize{$D$}}}-z_{V})/\sqrt{\sigma^{2}_{z\textrm{\scriptsize{,
$D$}}}+\sigma^{2}_{z\textrm{\scriptsize{, $V$}}}}>-2$ is applied, where
$\sigma_{z\textrm{\scriptsize{, $D$}}}$ and $\sigma_{z\textrm{\scriptsize{,
$V$}}}$ are the uncertainties on the $z$ positions of the $D$ and $V$ vertices
respectively.
The selection criteria for the $V$ candidates introduce some differences
between the signal and normalisation channel due to the particle
identification (PID) and mass window requirements. The $K^{*0}$ ($\rho^{0}$)
reconstructed mass is required to be within $50\text{\,}\MeVovercsq$
($150\text{\,}\MeVovercsq$) of its nominal value [14]. The selection criteria
for the $D^{0}$ and vector mesons include identifying kaon and pion candidates
using the RICH system. This analysis uses the comparison between the kaon and
pion hypotheses, $\mathrm{DLL_{K\pi}}$, which represents the difference in
logarithms of likelihoods for the $K$ with respect to the $\pi$ hypothesis.
The particle identification requirements for both kaon and pion hypotheses
have been optimized on data. The thresholds are set at $\mathrm{DLL_{K\pi}}>0$
and $\mathrm{DLL_{K\pi}}<4$, respectively, for the kaon and the pion from the
$D^{0}$. The misidentification rate is kept low by setting the thresholds for
the vector meson daughters to $\mathrm{DLL_{K\pi}}>3$ and
$\mathrm{DLL_{K\pi}}<3$ for the kaon and pion respectively. In order to remove
the potential backgrounds due to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow D^{+}_{s}\pi^{-}$
and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow
D^{+}\pi^{-}$ with $D^{-}_{s}\\!\rightarrow K^{*0}K^{-}$ and
$D^{-}\\!\rightarrow K^{*0}K^{-}$, vetoes around the nominal $D^{-}$ and
$D^{-}_{s}$ meson masses [14] of $\pm$15\text{\,}\MeVovercsq$$ are applied.
Monte Carlo studies suggest that these vetoes are more than 99.5% efficient on
the signal.
Finally, multiple candidates in an event (about 5%) are removed by choosing
the $B$ candidate with the largest $B$ flight distance significance and which
lies in the mass windows of the $D^{0}$ and the vector meson resonance.
## 3 Extraction of the ratio of branching fractions
The ratio of branching fractions is calculated from the number of signal
events in the two decay channels $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow D^{0}K^{*0}$ and
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow
D^{0}\rho^{0}$,
$\displaystyle\frac{{\cal B}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow
D^{0}K^{*0}\right)}{{\cal B}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow
D^{0}\rho^{0}\right)}=\frac{N^{\rm sig.}_{\textrm{\scriptsize{$\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\\!\rightarrow
D^{0}K^{*0}$}}}}{N^{\rm sig.}_{\textrm{\scriptsize{$\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}\\!\rightarrow
D^{0}\rho^{0}$}}}}\times\frac{{\cal
B}\left(\rho^{0}\\!\rightarrow\pi^{+}\pi^{-}\right)}{{\cal
B}\left(K^{*0}\\!\rightarrow
K^{+}\pi^{-}\right)}\times\frac{f_{d}}{f_{s}}\times\frac{\epsilon_{\textrm{\scriptsize{$\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}\\!\rightarrow
D^{0}\rho^{0}$}}}}{\epsilon_{\textrm{\scriptsize{$\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\\!\rightarrow D^{0}K^{*0}$}}}}$
(1)
where the $\epsilon$ parameters represent the total efficiencies, including
acceptance, trigger, reconstruction and selection, and $f_{s}/f_{d}$ is the
ratio of $B^{0}$ and $B^{0}_{s}$ hadronization fractions in $pp$ collisions at
$\sqrt{s}=7$ TeV. Since a given event can either be triggered by tracks from
the signal or by tracks from the other B hadron decay, absolute efficiencies
cannot be obtained with a great precision from the Monte Carlo simulation due
to improper modelling of the generic $B$ hadron decays. In order to reduce the
systematic uncertainty related to the Monte Carlo simulation of the trigger,
the data sample is divided into two categories: candidates that satisfy only
the hadronic hardware trigger333Events passing only the muon trigger on the
signal candidate tracks are rejected. (TOSOnly, since they are Triggered On
the Signal (TOS) exclusively and not on the rest of the event) and events
which are Triggered by the rest of the event Independent of the Signal
candidate $B$ decay (TIS). Approximately 6% of candidates do not enter either
of these two categories, and are vetoed in the analysis. The $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow D^{0}\rho^{0}$ signal
yield is extracted separately for the two trigger categories TOSOnly and TIS;
the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow
D^{0}K^{*0}$ signal yield is extracted from the sum of both data samples. The
ratio of efficiencies are sub-divided into the contributions arising from the
selection requirements (including acceptance effects, but excluding PID),
$r_{\textrm{\scriptsize{sel}}}$, the PID requirements,
$r_{\textrm{\scriptsize{PID}}}$, and the trigger requirements,
$r_{\textrm{\scriptsize{{TOSOnly}}}}$ and $r_{\textrm{\scriptsize{{TIS}}}}$.
The ratio of the branching fractions can therefore be expressed as
$\displaystyle\frac{{\cal B}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow
D^{0}K^{*0}\right)}{{\cal B}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow
D^{0}\rho^{0}\right)}=\frac{{\cal
B}\left(\rho^{0}\\!\rightarrow\pi^{+}\pi^{-}\right)}{{\cal
B}\left(K^{*0}\\!\rightarrow
K^{+}\pi^{-}\right)}\times\frac{f_{d}}{f_{s}}\times
r_{\textrm{\scriptsize{sel}}}\times
r_{\textrm{\scriptsize{PID}}}\times\frac{N^{\rm
sig.}_{\textrm{\scriptsize{$\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\\!\rightarrow
D^{0}K^{*0}$}}}}{\alpha\left(\frac{N^{\textrm{\scriptsize{{TOSOnly}}}}_{\textrm{\scriptsize{$\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}\\!\rightarrow
D^{0}\rho^{0}$}}}}{r_{\textrm{\scriptsize{{TOSOnly}}}}}+\frac{N^{\textrm{\scriptsize{{TIS}}}}_{\textrm{\scriptsize{$\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}\\!\rightarrow
D^{0}\rho^{0}$}}}}{r_{\textrm{\scriptsize{{TIS}}}}}\right)},$ (2)
where $\alpha$ represents a correction factor for the “non-$\rho^{0}$”
contribution in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow D^{0}\rho^{0}$
decays.
The values of the efficiency ratios are measured using simulated events,
except for $r_{\textrm{\scriptsize{PID}}}=1.09\pm 0.08$ which is obtained from
data using the $D^{*}\\!\rightarrow D^{0}\pi$ decay with $D^{0}\\!\rightarrow
K^{-}\pi^{+}$ where clean samples of kaons and pions can be obtained using a
purely kinematic selection. Since the event selection is identical for the
$D^{0}$ in the two channels of interest, many factors cancel out in
$r_{\textrm{\scriptsize{sel}}}=0.784\pm 0.024$ thereby reducing the systematic
uncertainties. The values of the trigger efficiency ratios,
$r_{\textrm{\scriptsize{{TOSOnly}}}}=1.20\pm 0.08$ and
$r_{\textrm{\scriptsize{{TIS}}}}=1.03\pm 0.03$, depend on the trigger
configurations and are therefore computed from a luminosity-weighted average.
The quoted uncertainties reflect the difference between data and Monte Carlo
simulation mainly caused by the energy calibration of the trigger.
The numbers of events in the two $D^{0}\rho^{0}$ trigger categories,
$N^{\textrm{\scriptsize{{TOSOnly}}}}_{\textrm{\scriptsize{$\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}\\!\rightarrow D^{0}\rho^{0}$}}}$
and $N^{\textrm{\scriptsize{{TIS}}}}_{\textrm{\scriptsize{$\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}\\!\rightarrow D^{0}\rho^{0}$}}}$,
and $N^{\rm sig.}_{\textrm{\scriptsize{$\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\\!\rightarrow D^{0}K^{*0}$}}}$
are extracted from a simultaneous unbinned maximum likelihood fit to the data.
In order to simplify the description of the partially reconstructed
background, the lower edge of the $B$ meson mass window is restricted to
$5.1\text{\,}\GeVovercsq$ for the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow D^{0}\rho^{0}$ decay
mode and to $5.19\text{\,}\GeVovercsq$ for the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow D^{0}K^{*0}$
decay mode. There are four types of events in each category: signal,
combinatorial background, partially reconstructed background and cross-
feed.444The cross-feed events are due to particle misidentification on one of
the vector daughters; some $D^{0}\rho^{0}$ events can be selected as
$D^{0}K^{*0}$ and vice versa. The signal $B$ meson mass PDFs for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow D^{0}\rho^{0}$ and
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow
D^{0}K^{*0}$ are parametrized for each channel using the sum of two Gaussians
sharing the same mean value. The mean and width of the core Gaussian
describing the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow
D^{0}\rho^{0}$ mass distribution are allowed to vary in the fit. The fraction
of events in the core Gaussian, $0.81\pm 0.02$, and the ratio of the tail and
core Gaussian widths, $2.04\pm 0.05$, are fixed to the values obtained from
Monte Carlo simulation. In order to take into account the difference in mass
resolution for the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow D^{0}\rho^{0}$ and
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow
D^{0}K^{*0}$ decay modes, the value of the ratio of core Gaussian widths
$\frac{\sigma_{\textrm{\scriptsize{$D^{0}K^{*0}$}}}}{\sigma_{\textrm{\scriptsize{$\kern
1.39998pt\overline{\kern-1.39998ptD}{}^{0}\rho^{0}$}}}}=0.89\pm 0.03$ is fixed
from the Monte Carlo simulation. The mass difference between the means of the
$B^{0}$ and $B^{0}_{s}$ signals is fixed to the nominal value [14].
The combinatorial background mass distribution is modelled by a flat PDF and
the partially reconstructed background is parametrized by an exponential
function; the exponential slope is different in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow D^{0}\rho^{0}$ and
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow
D^{0}K^{*0}$ categories. Since the number of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow D^{0}\rho^{0}$ decays
is larger than that of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow D^{0}K^{*0}$, the
contribution from misidentified pions as kaons from real $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow D^{0}\rho^{0}$ has to
be taken into account. The fractions of the cross-feed events,
$f_{D^{0}\rho^{0}\rightarrow D^{0}K^{*0}}=0.062\pm 0.031$ and
$f_{D^{0}K^{*0}\rightarrow D^{0}\rho^{0}}=0.095\pm 0.047$, are constrained
using the results from a Monte Carlo study corrected by the PID
misidentication rates measured in data. The PDF for the cross-feed is
empirically parametrised by a Crystal Ball function [15], whose width and
other parameters are taken from a fit to simulated events in which $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow D^{0}K^{*0}$
events are misidentified as $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow D^{0}\rho^{0}$ and
vice versa; the width is fixed to $1.75$ times the signal resolution. For the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow D^{0}\rho^{0}$
decay mode, the events are further split according to the TOSOnly and TIS
categories.
In summary, 13 parameters are free in the fit. Four shape parameters are used,
two for the signal and two for the partially reconstructed backgrounds. In
addition, nine event yields are extracted, three (signal, combinatorial and
partially reconstructed backgrounds) in each of the three categories: $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow D^{0}\rho^{0}$
(TOSOnly and TIS) and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow D^{0}K^{*0}$.
The results of the fit for $D^{0}\rho^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow D^{0}K^{*0}$ are
shown in Fig. 1 and Fig. 2. The overall signal yields are $154.1\pm 15.1$ and
$34.4\pm 6.8$ respectively. The yields for the different components are
summarised in Table 1.
Figure 1: The invariant mass distribution for the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow D^{0}\rho^{0}$ decay mode for the TOSOnly (left) and TIS (right) trigger categories with the result of the fit superimposed. The black points correspond to the data and the fit result is represented as a solid line. The signal is fitted with a double Gaussian (dashed line), the partially reconstructed background with an exponential function (light grey area) and the combinatorial background with a flat distribution (dark grey area) as explained in the text. The contributions from cross-feed are too small to be visible. Figure 2: The invariant mass distribution for the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow D^{0}K^{*0}$ decay mode with the result of the fit superimposed. The black points correspond to the data and the fit result is represented as a solid line. The signal is fitted with a double Gaussian (dashed line), the partially reconstructed background with an exponential function (light grey area), the combinatorial background with a flat distribution (dark grey area) and the cross-feed from $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow D^{0}\rho^{0}$ (intermediate grey area) as explained in the text. Table 1: Summary of the fitted yields for the different categories. The background yields are quoted for the full mass regions. Decay mode | Signal yield | Part. rec. bkgd yield | Comb. bkgd yield
---|---|---|---
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow D^{0}K^{*0}$ | $34.4\pm 6.8$ | $17.5\pm 11.4$ | $29.8\pm 8.4$
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow D^{0}\rho^{0}$ (TOSOnly) | $77.0\pm 10.1$ | $55.4\pm 10.1$ | $95.5\pm 13.1$
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow D^{0}\rho^{0}$ (TIS) | $77.1\pm 11.2$ | $85.6\pm 12.9$ | $176.0\pm 17.5$
In order to check the existence of other contributions under the vector mass
peaks, the sPlot technique [16] has been used to obtain background subtracted
invariant mass distributions. The sWeights are calculated from the
reconstructed $B$ invariant mass distribution using the same parametrization
as in the analysis, the selection being the same except for the $V$ invariant
mass ranges which are widened. It was checked that there is no correlation
between the $B$ and the $V$ invariant mass. The resulting plots are shown in
Fig. 3, where the resonant component is fitted with a Breit-Wigner convoluted
with a Gaussian and the non-resonant part with a second order polynomial.
While the $K^{*0}$ region shows no sign of an extra contribution, the
$\rho^{0}$ region shows a more complicated structure. An effective
“non-$\rho^{0}$” contribution is estimated using a second-order polynomial:
$30.1\pm 7.9$ events contribute in the $\rho^{0}$ mass window
($\pm$150\text{\,}\MeVovercsq$$). The measured $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow D^{0}\rho^{0}$ yields
are corrected by a factor $\alpha=0.805\pm 0.054$ (see Eq. 2), consistent with
expectations based on previous studies of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow D^{0}\pi^{+}\pi^{-}$
Dalitz plot [17, 18].
Figure 3: The $\rho^{0}$ (on the left) and $K^{*0}$ (on the right) invariant
mass distributions obtained from data using an sPlot technique. The level of
non $K^{*0}$ combinations in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow D^{0}K^{*0}$ peak
is negligible. Despite being mainly due to $D^{0}\rho^{0}$ combinations, the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow D^{0}\rho^{0}$
contains a significant contribution of “non-$\rho^{0}$” events. The black
points correspond to the data and the fit result is represented as a solid
line. The resonant component is fitted with a Breit-Wigner convoluted with a
Gaussian (dashed line) and the non-resonant part, if present, with a second-
order polynomial (grey area).
The ratio of branching fractions, $\frac{{\cal B}\left(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\\!\rightarrow
D^{0}K^{*0}\right)}{{\cal B}\left(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}\\!\rightarrow
D^{0}\rho^{0}\right)}$, is calculated using the measured yields of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow D^{0}\rho^{0}$ signal
in the two trigger categories, corrected for the “non-$\rho^{0}$” events and
assumed to contribute proportionally to the TOSOnly and TIS samples, the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow
D^{0}K^{*0}$ yield and the values of the $r$ ratios quoted above. The result
is $\frac{{\cal B}\left(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\\!\rightarrow
D^{0}K^{*0}\right)}{{\cal B}\left(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}\\!\rightarrow
D^{0}\rho^{0}\right)}=1.48\pm 0.34$, where the uncertainty is statistical
only. The small statistical correlation between the two yields due to the
cross-feed has been neglected.
## 4 Systematic uncertainties
A summary of the contributions to the systematic uncertainty is given in Table
2. The PID performances are determined with a $D^{*}\\!\rightarrow D^{0}\pi$
data calibration sample reweighted according to the kinematical properties of
our signals obtained from Monte Carlo simulation. The systematic uncertainty
has been assigned using the kinematical distributions directly obtained from
the data. However, due to the small signal yield in the $B^{0}_{s}$ case, this
systematic uncertainty suffers from large statistical fluctuations which
directly translate into a large systematic uncertainty on the kaon
identification. The statistical uncertainty obtained on the number of
“non-$\rho^{0}$” events present in the $\rho^{0}$ the mass window
($\pm$150\text{\,}\MeVovercsq$$) has been propagated in the systematic
uncertainty. The differences observed between Monte Carlo simulation and data
on the values of the $D^{0}$ and vector mesons reconstructed masses, as well
as on the transverse momentum spectra, have been propagated into the
uncertainty quoted on $r_{\textrm{\scriptsize{sel}}}$. The relative abundances
of TOSOnly and TIS triggered events determined from simulated signal are in
good agreement with those measured from data. This provides confidence in the
description of the trigger in the Monte Carlo simulation. Since these relative
abundances are directly measured in data, they do not enter the systematic
uncertainty evaluation. However, the difference in trigger efficiency between
the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow
D^{0}\rho^{0}$ and the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow D^{0}K^{*0}$
decay modes is taken from Monte Carlo simulation; this is considered reliable
since the difference arises due to the kinematical properties of the decays
which are well modelled in the simulation. The difference in the energy
measurement between the hardware trigger clustering and the offline
reconstruction clustering is conservatively taken as a systematic uncertainty
due to the hadronic trigger threshold. The systematic uncertainty due to the
TIS trigger performances on the two decay modes is obtained assuming that it
does not depend on the decay mode ($r_{\textrm{\scriptsize{{TIS}}}}=1$).
The systematic uncertainty due to the PDF parametrizations has been evaluated
using toy Monte Carlo simulations where the different types of background have
been generated using an alternative parametrization (wide Gaussians for the
partially reconstructed backgrounds, first order polynomial for the
combinatorial backgrounds) but fitted with the default PDFs.
The total systematic uncertainty is obtained by combining all sources in
quadrature. The dominant sources of systematic uncertainty are of statistical
nature and will be reduced with more data. The error on the ratio of the
fragmentation fractions [10] is quoted as a separate systematic uncertainty.
Table 2: Summary of the contributions to the systematic uncertainties. The uncertainty on the $r$ ratio gives the range used for the systematic uncertainty extraction on the ratios of the branching fractions. Source | Relative uncertainty
---|---
Difference between data and MC to compute $r_{\textrm{\scriptsize{PID}}}=1.09\pm 0.06$ | $5.8\text{\,}\mathrm{\char 37\relax}$
Uncertainty on the “non-$\rho^{0}$” component $\alpha=0.805\pm 0.054$ | $6.8\text{\,}\mathrm{\char 37\relax}$
MC selection efficiencies $r_{\textrm{\scriptsize{sel.}}}=0.784\pm 0.024$ | $3.1\text{\,}\mathrm{\char 37\relax}$
L0 Hadron threshold $r_{\textrm{\scriptsize{{TOSOnly}}}}=1.20\pm 0.08$ | $3.0\text{\,}\mathrm{\char 37\relax}$
TIS triggering efficiency $r_{\textrm{\scriptsize{{TIS}}}}=1.03\pm 0.03$ | $1.6\text{\,}\mathrm{\char 37\relax}$
PDF parametrisations | $1.0\text{\,}\mathrm{\char 37\relax}$
Overall relative systematic uncertainty | $10.2\text{\,}\mathrm{\char 37\relax}$
Fragmentation fractions | $7.9\text{\,}\mathrm{\char 37\relax}$
## 5 Summary
A signal of $34.4\pm 6.8$ $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow D^{0}K^{*0}$
events is observed for the first time. The significance of the background
fluctuating to form the $B^{0}_{s}$ signal corresponds to approximately nine
standard deviations, as determined from the change in twice the natural
logarithm of the likelihood of the fit without signal. Although this
significance includes the statistical uncertainty only, the result is
unchanged if the small sources of systematic error that affect the yields are
included. The branching fraction for this decay is measured relative to that
for $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow
D^{0}\rho^{0}$, after correcting for the “non-$\rho^{0}$” component, to be
$\frac{{\cal B}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow
D^{0}K^{*0}\right)}{{\cal B}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow
D^{0}\rho^{0}\right)}=1.48\pm 0.34\pm 0.15\pm 0.12,$ (3)
where the first uncertainty is statistical, the second systematic and the
third is due to the uncertainty in the hadronisation fraction ($f_{s}/f_{d}$).
The result is in agreement with other measurements of similar ratios and
supports the $\mathrm{SU}(3)$ breaking observation in colour suppressed $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(d,s)}\\!\rightarrow D^{0}V$
decays. Using ${\cal B}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow
D^{0}\rho^{0}\right)=(3.2\pm 0.5)\times 10^{-4}$ [14] for the branching
fraction of the normalising decay, a measurement of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow D^{0}K^{*0}$
branching fraction,
${\cal B}\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow
D^{0}K^{*0}\right)=(4.72\pm 1.07\pm 0.48\pm 0.37\pm 0.74)\times 10^{-4},$ (4)
is obtained, where the first uncertainty is statistical, the second
systematic, the third due to the uncertainty in the hadronisation fraction
($f_{s}/f_{d}$) and the last is due to the uncertainty of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow D^{0}\rho^{0}$
branching fraction. A future, larger data sample will allow the use of the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow D^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay as the normalising channel,
which will reduce the systematic uncertainty.
## Acknowledgments
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, 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
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* [2] M. Gronau and D. London, “How to determine all the angles of the unitarity triangle from $B^{0}_{d}\\!\rightarrow DK^{0}_{\rm\scriptscriptstyle S}$ and $B^{0}_{s}\\!\rightarrow D\Phi$, Phys. Lett. B 253 (1991) 483.
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|
arxiv-papers
| 2011-10-17T14:21:06 |
2024-09-04T02:49:23.213369
|
{
"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, 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, 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, S.\n Brisbane, M. Britsch, T. Britton, N.H. Brook, H. Brown, A. B\\\"uchler-Germann,\n I. Burducea, A. Bursche, J. Buytaert, S. Cadeddu, J.M. Caicedo Carvajal, 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, M. Charles, Ph. Charpentier, N. Chiapolini, K. Ciba, X. Cid Vidal,\n G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V.\n Coco, J. Cogan, P. Collins, A. Comerma-Montells, F. Constantin, G. Conti, A.\n Contu, A. Cook, M. Coombes, G. Corti, G.A. Cowan, R. Currie, B. D'Almagne, C.\n D'Ambrosio, P. David, I. De Bonis, S. De Capua, M. De Cian, F. De Lorenzi,\n J.M. De Miranda, L. De Paula, P. De Simone, D. Decamp, M. Deckenhoff, H.\n Degaudenzi, M. Deissenroth, L. Del Buono, C. Deplano, O. Deschamps, F.\n Dettori, J. Dickens, H. Dijkstra, P. Diniz Batista, F. Domingo Bonal, S.\n Donleavy, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R.\n Dzhelyadin, S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, F.\n Eisele, S. Eisenhardt, R. Ekelhof, L. Eklund, Ch. Elsasser, D.G. d'Enterria,\n D. Esperante Pereira, L. Est\\'eve, A. Falabella, E. Fanchini, 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, M. Frank,\n C. Frei, M. Frosini, S. Furcas, A. Gallas Torreira, D. Galli, M. Gandelman,\n P. Gandini, Y. Gao, J-C. Garnier, J. Garofoli, J. Garra Tico, L. Garrido, D.\n Gascon, C. Gaspar, 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, G.\n Graziani, A. Grecu, E. Greening, S. Gregson, B. Gui, E. Gushchin, Yu. Guz, T.\n Gys, G. Haefeli, C. Haen, S.C. Haines, T. Hampson, S. Hansmann-Menzemer, R.\n Harji, N. Harnew, J. Harrison, P.F. Harrison, J. He, V. Heijne, K. Hennessy,\n P. Henrard, J.A. Hernando Morata, E. van Herwijnen, E. Hicks, W. Hofmann, K.\n Holubyev, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, R.S. Huston, D.\n Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J. Imong, R. Jacobsson, A.\n Jaeger, M. Jahjah Hussein, E. Jans, F. Jansen, P. Jaton, B. Jean-Marie, F.\n Jing, M. John, D. Johnson, C.R. Jones, B. Jost, S. Kandybei, M. Karacson,\n T.M. Karbach, J. Keaveney, U. Kerzel, T. Ketel, A. Keune, B. Khanji, Y.M.\n Kim, M. Knecht, S. Koblitz, P. Koppenburg, A. Kozlinskiy, L. Kravchuk, K.\n Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, K. Kruzelecki, M.\n Kucharczyk, S. Kukulak, R. Kumar, 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, 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.H. Lopes, E.\n Lopez Asamar, N. Lopez-March, J. Luisier, F. Machefert, I.V. Machikhiliyan,\n F. 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, C. Mclean, M. Meissner, M. Merk, J. Merkel,\n R. Messi, S. Miglioranzi, D.A. Milanes, M.-N. Minard, S. Monteil, D. Moran,\n P. Morawski, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B.\n Muryn, M. Musy, J. Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, J.\n Nardulli, I. Nasteva, M. Nedos, M. Needham, N. Neufeld, C. Nguyen-Mau, M.\n Nicol, S. Nies, V. Niess, N. Nikitin, A. Nomerotski, 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. Petrella, A.\n Petrolini, E. Picatoste Olloqui, B. Pie Valls, B. Pietrzyk, T. Pilar, D.\n Pinci, R. Plackett, S. Playfer, M. Plo Casasus, G. Polok, A. Poluektov, E.\n Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, T. du Pree, 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, K. Rinnert, D.A. Roa Romero, P. Robbe, E.\n Rodrigues, F. Rodrigues, P. Rodriguez Perez, G.J. Rogers, S. Roiser, V.\n Romanovsky, M. Rosello, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino, J.J.\n Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, M. Sannino, R.\n Santacesaria, R. Santinelli, E. Santovetti, M. Sapunov, A. Sarti, C.\n Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M. Schiller, S.\n Schleich, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune,\n R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K.\n Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, B. Shao, M. Shapkin,\n I. Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O.\n Shevchenko, V. Shevchenko, A. Shires, R. Silva Coutinho, H.P. Skottowe, T.\n Skwarnicki, A.C. Smith, 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, N. Styles, V.K. Subbiah, S. Swientek, M.\n 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, M.T. Tran, A. Tsaregorodtsev, N. Tuning, A. Ukleja,\n P. Urquijo, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez\n Regueiro, S. Vecchi, J.J. Velthuis, M. Veltri, K. Vervink, B. Viaud, I.\n Videau, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt, D. Voong, A.\n Vorobyev, H. Voss, K. Wacker, S. Wandernoth, J. Wang, D.R. Ward, A.D. Webber,\n D. Websdale, M. Whitehead, D. Wiedner, L. Wiggers, G. Wilkinson, M.P.\n Williams, M. Williams, F.F. Wilson, J. Wishahi, M. Witek, W. Witzeling, S.A.\n Wotton, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R. Young, O.\n Yushchenko, M. Zavertyaev, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, L.\n Zhong, E. Zverev, A. Zvyagin",
"submitter": "Aur\\'elien Martens",
"url": "https://arxiv.org/abs/1110.3676"
}
|
1110.3850
|
# On the Power of Adaptivity in Sparse Recovery
Piotr Indyk Eric Price David P. Woodruff
The goal of (stable) sparse recovery is to recover a $k$-sparse approximation
$x^{*}$ of a vector $x$ from linear measurements of $x$. Specifically, the
goal is to recover $x^{*}$ such that
$\left\lVert x-x^{*}\right\rVert_{p}\leq C\min_{k\text{-sparse
}x^{\prime}}\left\lVert x-x^{\prime}\right\rVert_{q}$
for some constant $C$ and norm parameters $p$ and $q$. It is known that, for
$p=q=1$ or $p=q=2$, this task can be accomplished using $m=O(k\log(n/k))$ non-
adaptive measurements [CRT06] and that this bound is tight [DIPW10, FPRU10,
PW11].
In this paper we show that if one is allowed to perform measurements that are
adaptive , then the number of measurements can be considerably reduced.
Specifically, for $C=1+\epsilon$ and $p=q=2$ we show
* •
A scheme with $m=O(\frac{1}{\epsilon}k\log\log(n\epsilon/k))$ measurements
that uses $O(\log^{*}k\cdot\log\log(n\epsilon/k))$ rounds. This is a
significant improvement over the best possible non-adaptive bound.
* •
A scheme with $m=O(\frac{1}{\epsilon}k\log(k/\epsilon)+k\log(n/k))$
measurements that uses two rounds. This improves over the best possible non-
adaptive bound.
To the best of our knowledge, these are the first results of this type.
As an independent application, we show how to solve the problem of finding a
duplicate in a data stream of $n$ items drawn from $\\{1,2,\ldots,n-1\\}$
using $O(\log n)$ bits of space and $O(\log\log n)$ passes, improving over the
best possible space complexity achievable using a single pass.
## 1 Introduction
In recent years, a new “linear” approach for obtaining a succinct approximate
representation of $n$-dimensional vectors (or signals) has been discovered.
For any signal $x$, the representation is equal to $Ax$, where $A$ is an
$m\times n$ matrix, or possibly a random variable chosen from some
distribution over such matrices. The vector $Ax$ is often referred to as the
measurement vector or linear sketch of $x$. Although $m$ is typically much
smaller than $n$, the sketch $Ax$ often contains plenty of useful information
about the signal $x$.
A particularly useful and well-studied problem is that of stable sparse
recovery. We say that a vector $x^{\prime}$ is $k$-sparse if it has at most
$k$ non-zero coordinates. The sparse recovery problem is typically defined as
follows: for some norm parameters $p$ and $q$ and an approximation factor
$C>0$, given $Ax$, recover an “approximation” vector $x^{*}$ such that
$\left\lVert x-x^{*}\right\rVert_{p}\leq C\min_{k\text{-sparse
}x^{\prime}}\left\lVert x-x^{\prime}\right\rVert_{q}$ (1)
(this inequality is often referred to as $\ell_{p}/\ell_{q}$ guarantee). If
the matrix $A$ is random, then Equation (1) should hold for each $x$ with some
probability (say, 2/3). Sparse recovery has a tremendous number of
applications in areas such as compressive sensing of signals [CRT06, Don06],
genetic data acquisition and analysis [SAZ10, BGK+10] and data stream
algorithms111In streaming applications, a data stream is modeled as a sequence
of linear operations on an (implicit) vector x. Example operations include
increments or decrements of $x$’s coordinates. Since such operations can be
directly performed on the linear sketch $Ax$, one can maintain the sketch
using only $O(m)$ words. [Mut05, Ind07]; the latter includes applications to
network monitoring and data analysis.
It is known [CRT06] that there exist matrices $A$ and associated recovery
algorithms that produce approximations $x^{*}$ satisfying Equation (1) with
$p=q=1$, constant approximation factor $C$, and sketch length
$m=O(k\log(n/k))$ (2)
A similar bound, albeit using random matrices $A$, was later obtained for
$p=q=2$ [GLPS10] (building on [CCF02, CM04, CM06]). Specifically, for
$C=1+\epsilon$, they provide a distribution over matrices $A$ with
$m=O(\frac{1}{\epsilon}k\log(n/k))$ (3)
rows, together with an associated recovery algorithm.
It is also known that the bound in Equation (2) is asymptotically optimal for
some constant $C$ and $p=q=1$, see [DIPW10] and [FPRU10] (building on [GG84,
Glu84, Kas77]). The bound of [DIPW10] also extends to the randomized case and
$p=q=2$. For $C=1+\epsilon$, a lower bound of
$m=\Omega(\frac{1}{\epsilon}k\log(n/k))$ was recently shown [PW11] for the
randomized case and $p=q=2$, improving upon the earlier work of [DIPW10] and
showing the dependence on $\epsilon$ is optimal. The necessity of the “extra”
logarithmic factor multiplying $k$ is quite unfortunate: the sketch length
determines the “compression rate”, and for large $n$ any logarithmic factor
can worsen that rate tenfold.
In this paper we show that this extra factor can be greatly reduced if we
allow the measurement process to be adaptive. In the adaptive case, the
measurements are chosen in rounds, and the choice of the measurements in each
round depends on the outcome of the measurements in the previous rounds. The
adaptive measurement model has received a fair amount of attention in the
recent years [JXC08, CHNR08, HCN09, HBCN09, MSW08, AWZ08], see also [Def10].
In particular [HBCN09] showed that adaptivity helps reducing the approximation
error in the presence of random noise. However, no asymptotic improvement to
the number of measurements needed for sparse recovery (as in Equation (1)) was
previously known.
#### Results
In this paper we show that adaptivity can lead to very significant
improvements in the number of measurements over the bounds in Equations (2)
and (3). We consider randomized sparse recovery with $\ell_{2}/\ell_{2}$
guarantee, and show two results:
1. 1.
A scheme with $m=O(\frac{1}{\epsilon}k\log\log(n\epsilon/k))$ measurements and
an approximation factor $C=1+\epsilon$. For low values of $k$ this provides an
exponential improvement over the best possible non-adaptive bound. The scheme
uses $O(\log^{*}k\cdot\log\log(n\epsilon/k))$ rounds.
2. 2.
A scheme with $m=O(\frac{1}{\epsilon}k\log(k/\epsilon)+k\log(n/k))$ and an
approximation factor $C=1+\epsilon$. For low values of $k$ and $\epsilon$ this
offers a significant improvement over the best possible non-adaptive bound,
since the dependence on $n$ and $\epsilon$ is “split” between two terms. The
scheme uses only two rounds.
#### Implications
Our new bounds lead to potentially significant improvements to efficiency of
sparse recovery schemes in a number of application domains. Naturally, not all
applications support adaptive measurements. For example, network monitoring
requires the measurements to be performed simultaneously, since we cannot ask
the network to “re-run” the packets all over again. However, a surprising
number of applications are capable of supporting adaptivity. For example:
* •
Streaming algorithms for data analysis: since each measurement round can be
implemented by one pass over the data, adaptive schemes simply correspond to
multiple-pass streaming algorithms (see [McG09] for some examples of such
algorithms).
* •
Compressed sensing of signals: several architectures for compressive sensing,
e.g., the single-pixel camera of [DDT+08], already perform the measurements in
a sequential manner. In such cases the measurements can be made adaptive222We
note that, in any realistic sensing system, minimizing the number of
measurements is only one of several considerations. Other factors include:
minimizing the computation time, minimizing the amount of communication needed
to transfer the measurement matrices to the sensor, satisfying constraints on
the measurement matrix imposed by the hardware etc. A detailed cost analysis
covering all of these factors is architecture-specific, and beyond the scope
of this paper. . Other architectures supporting adaptivity are under
development [Def10].
* •
Genetic data analysis and acqusition: as above.
Therefore, it seems likely that the results in this paper will be applicable
in a wide variety of scenarios.
As an example application, we show how to solve the problem of finding a
duplicate in a data stream of $n$ arbitrarily chosen items from the set
$\\{1,2,\ldots,n-1\\}$ presented in an arbitrary order. Our algorithm uses
$O(\log n)$ bits of space and $O(\log\log n)$ passes. It is known that for a
single pass, $\Theta(\log^{2}n)$ bits of space is necessary and sufficient
[JST11], and so our algorithm improves upon the best possible space complexity
using a single pass.
#### Techniques
On a high-level, both of our schemes follow the same two-step process. First,
we reduce the problem of finding the best $k$-sparse approximation to the
problem of finding the best $1$-sparse approximation (using relatively
standard techniques). This is followed by solving the latter (simpler)
problem.
The first scheme starts by “isolating” most of of the large coefficients by
randomly sampling $\approx\epsilon/k$ fraction of the coordinates; this mostly
follows the approach of [GLPS10] (cf. [GGI+02]). The crux of the algorithm is
in the identification of the isolated coefficients. Note that in order to
accomplish this using $O(\log\log n)$ measurements (as opposed to $O(\log n)$
achieved by the “standard” binary search algorithm) we need to “extract”
significantly more than one bit of information per measurements. To achieve
this, we proceed as follows. First, observe that if the given vector (say,
$z$) is exactly $1$-sparse, then one can extract the position of the non-zero
entry (say $z_{j}$) from two measurements $a(z)=\sum_{i}z_{i}$, and
$b(z)=\sum_{i}iz_{i}$, since $b(z)/a(z)=j$. A similar algorithm works even if
$z$ contains some “very small” non-zero entries: we just round $b(z)/a(z)$ to
the nearest integer. This algorithm is a special case of a general algorithm
that achieves $O(\log n/\log SNR)$ measurements to identify a single
coordinate $x_{j}$ among $n$ coordinates, where
$SNR=x_{j}^{2}/\|x_{[n]\setminus j}\|^{2}$ (SNR stands for signal-to-noise
ratio). This is optimal as a function of $n$ and the SNR [DIPW10].
A natural approach would then be to partition $[n]$ into two sets
$\\{1,\ldots,n/2\\}$ and $\\{n/2+1,\ldots n\\}$, find the heavier of the two
sets, and recurse. This would take $O(\log n)$ rounds. The key observation is
that not only do we recurse on a smaller-sized set of coordinates, but the SNR
has also increased since $x_{j}^{2}$ has remained the same but the squared
norm of the tail has dropped by a constant factor. Therefore in the next round
we can afford to partition our set into more than two sets, since as long as
we keep the ratio of $\log(\\#\textrm{ of sets })$ and $\log SNR$ constant, we
only need $O(1)$ measurements per round. This ultimately leads to a scheme
that finishes after $O(\log\log n)$ rounds.
In the second scheme, we start by hashing the coordinates into a universe of
size polynomial in $k$ and $1/\epsilon$, in a way that approximately preserves
the top coefficients without introducing spurious ones, and in such a way that
the mass of the tail of the vector does not increase significantly by hashing.
This idea is inspired by techniques in the data stream literature for
estimating moments [KNPW10, TZ04] (cf. [CCF02, CM06, GI10]). Here, though, we
need stronger error bounds. This enables us to identify the positions of those
coefficients (in the hashed space) using only
$O(\frac{1}{\epsilon}k\log(k/\epsilon))$ measurements. Once this is done, for
each large coefficient $i$ in the hash space, we identify the actual large
coefficient in the preimage of $i$. This can be achieved using the number of
measurements that does not depend on $\epsilon$.
## 2 Preliminaries
We start from a few definitions. Let $x$ be an $n$-dimensional vector.
###### Definition 2.1.
Define
$H_{k}(x)=\operatorname*{arg\,max}_{\begin{subarray}{c}S\in[n]\\\
\left|S\right|=k\end{subarray}}\left\lVert x_{S}\right\rVert_{2}$
to be the largest $k$ coefficients in $x$.
###### Definition 2.2.
For any vector $x$, we define the “heavy hitters” to be those elements that
are both (i) in the top $k$ and (ii) large relative to the mass outside the
top $k$. We define
$H_{k,\epsilon}(x)=\\{j\in H_{k}(x)\mid x_{j}^{2}\geq\epsilon\left\lVert
x_{\overline{H_{k}(x)}}\right\rVert_{2}^{2}\\}$
###### Definition 2.3.
Define the error
$\operatorname{Err^{2}}(x,k)=\left\lVert
x_{\overline{H_{k}(x)}}\right\rVert_{2}^{2}$
For the sake of clarity, the analysis of the algorithm in section 4 assumes
that the entries of $x$ are sorted by the absolute value (i.e., we have
$|x_{1}|\geq|x_{2}|\geq\ldots\geq|x_{n}|$). In this case, the set $H_{k}(x)$
is equal to $[k]$; this allows us to simplify the notation and avoid double
subscripts. The algorithms themselves are invariant under the permutation of
the coordinates of $x$.
#### Running times of the recovery algorithms
In the non-adaptive model, the running time of the recovery algorithm is well-
defined: it is the number of operations performed by a procedure that takes
$Ax$ as its input and produces an approximation $x^{*}$ to $x$. The time
needed to generate the measurement vectors $A$, or to encode the vector $x$
using $A$, is not included. In the adaptive case, the distinction between the
matrix generation, encoding and recovery procedures does not exist, since new
measurements are generated based on the values of the prior ones. Moreover,
the running time of the measurement generation procedure heavily depends on
the representation of the matrix. If we suppose that we may output the matrix
in sparse form and receive encodings in time bounded by the number of non-zero
entries in the matrix, our algorithms run in $n\log^{O(1)}n$ time.
## 3 Full adaptivity
This section shows how to perform $k$-sparse recovery with $O(k\log\log(n/k))$
measurements. The core of our algorithm is a method for performing $1$-sparse
recovery with $O(\log\log n)$ measurements. We then extend this to $k$-sparse
recovery via repeated subsampling.
### 3.1 $1$-sparse recovery
This section discusses recovery of $1$-sparse vectors with $O(\log\log n)$
adaptive measurements. First, in Lemma 3.1 we show that if the heavy hitter
$x_{j}$ is $\Omega(n)$ times larger than the $\ell_{2}$ error ($x_{j}$ is
“$\Omega(n)$-heavy”), we can find it with two non-adaptive measurements. This
corresponds to non-adaptive $1$-sparse recovery with approximation factor
$C=\Theta(n)$; achieving this with $O(1)$ measurements is unsurprising,
because the lower bound [DIPW10] is $\Omega(\log_{1+C}n)$.
Lemma 3.1 is not directly very useful, since $x_{j}$ is unlikely to be that
large. However, if $x_{j}$ is $D$ times larger than everything else, we can
partition the coordinates of $x$ into $D$ random blocks of size $N/D$ and
perform dimensionality reduction on each block. The result will in expectation
be a vector of size $D$ where the block containing $j$ is $D$ times larger
than anything else. The first lemma applies, so we can recover the block
containing $j$, which has a $1/\sqrt{D}$ fraction of the $\ell_{2}$ noise.
Lemma 3.2 gives this result.
We then have that with two non-adaptive measurements of a $D$-heavy hitter we
can restrict to a subset where it is an $\Omega(D^{3/2})$-heavy hitter.
Iterating $\log\log n$ times gives the result, as shown in Lemma 3.3.
###### Lemma 3.1.
Suppose there exists a $j$ with $\left|x_{j}\right|\geq
C\frac{n}{\sqrt{\delta}}\left\lVert x_{[n]\setminus\\{j\\}}\right\rVert_{2}$
for some constant $C$. Then two non-adaptive measurements suffice to recover
$j$ with probability $1-\delta$.
###### Proof.
Let $s\colon[n]\to\\{\pm 1\\}$ be chosen from a $2$-wise independent hash
family. Perform the measurements $a(x)=\sum s(i)x_{i}$ and
$b(x)=\sum(n+i)s(i)x_{i}$. For recovery, output the closest integer to
$b/a-n$.
Let $z=x_{[n]\setminus\\{j\\}}$. Then
$\operatorname{\mathbb{E}}[a(z)^{2}]=\left\lVert z\right\rVert_{2}^{2}$ and
$\operatorname{\mathbb{E}}[b(z)^{2}]\leq 4n^{2}\left\lVert
z\right\rVert_{2}^{2}$. Hence with probability at least $1-2\delta$, we have
both
$\displaystyle\left|a(z)\right|\leq\sqrt{1/\delta}\left\lVert
z\right\rVert_{2}$ $\displaystyle\left|b(z)\right|\leq
2n\sqrt{1/\delta}\left\lVert z\right\rVert_{2}$
Thus
$\displaystyle\frac{b(x)}{a(x)}=$
$\displaystyle\frac{s(j)(n+j)x_{j}+b(z)}{s(j)x_{j}+a(z)}$
$\displaystyle\left|\frac{b(x)}{a(x)}-(n+j)\right|=$
$\displaystyle\left|\frac{b(z)-(n+j)a(z)}{s(j)x_{j}+a(z)}\right|$
$\displaystyle\leq$
$\displaystyle\frac{\left|b(z)\right|+(n+j)\left|a(z)\right|}{\left|\left|x_{j}\right|-\left|a(z)\right|\right|}$
$\displaystyle\leq$ $\displaystyle\frac{4n\sqrt{1/\delta}\left\lVert
z\right\rVert_{2}}{\left|\left|x_{j}\right|-\left|a(z)\right|\right|}$
Suppose $\left|x_{j}\right|>(8n+1)\sqrt{1/\delta}\left\lVert
z\right\rVert_{2}$. Then
$\displaystyle\left|\frac{b(x)}{a(x)}-(n+j)\right|<$
$\displaystyle\frac{4n\sqrt{1/\delta}\left\lVert
z\right\rVert_{2}}{8n\sqrt{1/\delta}\left\lVert z\right\rVert_{2}}$
$\displaystyle=$ $\displaystyle 1/2$
so $\hat{\imath}=j$. ∎
###### Lemma 3.2.
Suppose there exists a $j$ with $\left|x_{j}\right|\geq
C\frac{B^{2}}{\delta^{2}}\left\lVert x_{[n]\setminus\\{j\\}}\right\rVert_{2}$
for some constant $C$ and parameters $B$ and $\delta$. Then with two non-
adaptive measurements, with probability $1-\delta$ we can find a set
$S\subset[n]$ such that $j\in S$ and $\left\lVert
x_{S\setminus\\{j\\}}\right\rVert_{2}\leq\left\lVert
x_{[n]\setminus\\{j\\}}\right\rVert_{2}/B$ and $\left|S\right|\leq 1+n/B^{2}$.
###### Proof.
Let $D=B^{2}/\delta$, and let $h\colon[n]\to[D]$ and $s\colon[n]\to\\{\pm
1\\}$ be chosen from pairwise independent hash families. Then define
$S_{p}=\\{i\in[n]\mid h(i)=p\\}$. Define the matrix $A\in\mathbb{R}^{D\times
n}$ by $A_{h(i),i}=s(i)$ and $A_{p,i}=0$ elsewhere. Then
$(Az)_{p}=\sum_{i\in S_{p}}s(i)z_{i}.$
Let $p^{*}=h(j)$ and $y=x_{[n]\setminus\\{j\\}}$. We have that
$\displaystyle\operatorname{\mathbb{E}}[\left|S_{p^{*}}\right|]=$
$\displaystyle 1+(n-1)/D$
$\displaystyle\operatorname{\mathbb{E}}[(Ay)_{p^{*}}^{2}]=\operatorname{\mathbb{E}}[\left\lVert
y_{S_{p^{*}}}\right\rVert_{2}^{2}]=$ $\displaystyle\left\lVert
y\right\rVert_{2}^{2}/D$ $\displaystyle\operatorname{\mathbb{E}}[\left\lVert
Ay\right\rVert_{2}^{2}]=$ $\displaystyle\left\lVert y\right\rVert_{2}^{2}$
Hence by Chebyshev’s inequality, with probability at least $1-4\delta$ all of
the following hold:
$\displaystyle\left|S_{p^{*}}\right|\leq$ $\displaystyle 1+(n-1)/(D\delta)\leq
1+n/B^{2}$ (4) $\displaystyle\left\lVert y_{S_{p^{*}}}\right\rVert_{2}\leq$
$\displaystyle\left\lVert y\right\rVert_{2}/\sqrt{D\delta}$ (5)
$\displaystyle\left|(Ay)_{p^{*}}\right|\leq$ $\displaystyle\left\lVert
y\right\rVert_{2}/\sqrt{D\delta}$ (6) $\displaystyle\left\lVert
Ay\right\rVert_{2}\leq$ $\displaystyle\left\lVert
y\right\rVert_{2}/\sqrt{\delta}.$ (7)
The combination of (6) and (7) imply
$\displaystyle\left|(Ax)_{p^{*}}\right|\geq$
$\displaystyle\left|x_{j}\right|-\left|(Ay)_{p^{*}}\right|\geq(CD/\delta-1/\sqrt{D\delta})\left\lVert
y\right\rVert_{2}\geq(CD/\delta-1/\sqrt{D\delta})\sqrt{\delta}\left\lVert
Ay\right\rVert_{2}\geq\frac{CD}{2\sqrt{\delta}}\left\lVert Ay\right\rVert_{2}$
and hence
$\left|(Ax)_{p^{*}}\right|\geq\frac{CD}{2\sqrt{\delta}}\left\lVert(Ax)_{[D]\setminus
p^{*}}\right\rVert_{2}.$
As long as $C/2$ is larger than the constant in Lemma 3.1, this means two non-
adaptive measurements suffice to recover $p^{*}$ with probability $1-\delta$.
We then output the set $S_{p^{*}}$, which by (5) has
$\displaystyle\left\lVert x_{S_{p^{*}}\setminus\\{j\\}}\right\rVert_{2}=$
$\displaystyle\left\lVert y_{S_{p^{*}}}\right\rVert_{2}\leq\left\lVert
y\right\rVert_{2}/\sqrt{D\delta}=\left\lVert
x_{[n]\setminus\\{j\\}}\right\rVert_{2}/\sqrt{D\delta}=\left\lVert
x_{[n]\setminus\\{j\\}}\right\rVert_{2}/B$
as desired. The overall failure probability is $1-5\delta$; rescaling $\delta$
and $C$ gives the result. ∎
Algorithm 1 Adaptive $1$-sparse recovery
procedure NonAdaptiveShrink($x$, $D$) $\triangleright$ Find smaller set $S$
containing heavy coordinate $x_{j}$
For $i\in[n]$, $s_{1}(i)\leftarrow\\{\pm 1\\},h(i)\leftarrow[D]$
For $i\in[D]$, $s_{2}(i)\leftarrow\\{\pm 1\\}$
$a\leftarrow\sum s_{1}(i)s_{2}(h(i))x_{i}$$\triangleright$ Observation
$b\leftarrow\sum s_{1}(i)s_{2}(h(i))x_{i}(D+h(i))$$\triangleright$ Observation
$p^{*}\leftarrow\textsc{Round}(b/a-D)$.
return $\\{j^{*}\mid h(j^{*})=p^{*}\\}$.
end procedure
procedure AdaptiveOneSparseRec($x$)$\triangleright$ Recover heavy coordinate
$x_{j}$
$S\leftarrow[n]$
$B\leftarrow 2$, $\delta\leftarrow 1/4$
while $\left|S\right|>1$ do
$S\leftarrow\textsc{NonAdaptiveShrink}(x_{S},4B^{2}/\delta)$
$B\leftarrow B^{3/2}$, $\delta\leftarrow\delta/2$.
end while
return $S[0]$
end procedure
###### Lemma 3.3.
Suppose there exists a $j$ with $\left|x_{j}\right|\geq C\left\lVert
x_{[n]\setminus\\{j\\}}\right\rVert_{2}$ for some constant $C$. Then
$O(\log\log n)$ adaptive measurements suffice to recover $j$ with probability
$1/2$.
###### Proof.
Let $C^{\prime}$ be the constant from Lemma 3.2. Define $B_{0}=2$ and
$B_{i}=B_{i-1}^{3/2}$ for $i\geq 1$. Define $\delta_{i}=2^{-i}/4$ for $i\geq
0$. Suppose $C\geq 16C^{\prime}B_{0}^{2}/\delta_{0}^{2}$.
Define $r=O(\log\log n)$ so $B_{r}\geq n$. Starting with $S_{0}=[n]$, our
algorithm iteratively applies Lemma 3.2 with parameters $B=4B_{i}$ and
$\delta=\delta_{i}$ to $x_{S_{i}}$ to identify a set $S_{i+1}\subset S_{i}$
with $j\in S_{i+1}$, ending when $i=r$.
We prove by induction that Lemma 3.2 applies at the $i$th iteration. We chose
$C$ to match the base case. For the inductive step, suppose $\left\lVert
x_{S_{i}\setminus\\{j\\}}\right\rVert_{2}\leq\left|x_{j}\right|/(C^{\prime}16\frac{B_{i}^{2}}{\delta_{i}^{2}})$.
Then by Lemma 3.2,
$\left\lVert
x_{S_{i+1}\setminus\\{j\\}}\right\rVert_{2}\leq\left|x_{j}\right|/(C^{\prime}64\frac{B_{i}^{3}}{\delta_{i}^{2}})=\left|x_{j}\right|/(C^{\prime}16\frac{B_{i+1}^{2}}{\delta_{i+1}^{2}})$
so the lemma applies in the next iteration as well, as desired.
After $r$ iterations, we have $S_{r}\leq 1+n/B_{r}^{2}<2$, so we have uniquely
identified $j\in S_{r}$. The probability that any iteration fails is at most
$\sum\delta_{i}<2\delta_{0}=1/2$. ∎
### 3.2 $k$-sparse recovery
Given a $1$-sparse recovery algorithm using $m$ measurements, one can use
subsampling to build a $k$-sparse recovery algorithm using $O(km)$
measurements and achieving constant success probability. Our method for doing
so is quite similar to one used in [GLPS10]. The main difference is that, in
order to identify one large coefficient among a subset of coordinates, we use
the adaptive algorithm from the previous section as opposed to error-
correcting codes.
For intuition, straightforward subsampling at rate $1/k$ will, with constant
probability, recover (say) 90% of the heavy hitters using $O(km)$
measurements. This reduces the problem to $k/10$-sparse recovery: we can
subsample at rate $10/k$ and recover 90% of the remainder with $O(km/10)$
measurements, and repeat $\log k$ times. The number of measurements decreases
geometrically, for $O(km)$ total measurements. Naively doing this would
multiply the failure probability and the approximation error by $\log k$;
however, we can make the number of measurements decay less quickly than the
sparsity. This allows the failure probability and approximation ratios to also
decay exponentially so their total remains constant.
To determine the number of rounds, note that the initial set of $O(km)$
measurements can be done in parallel for each subsampling, so only $O(m)$
rounds are necessary to get the first 90% of heavy hitters. Repeating $\log k$
times would require $O(m\log k)$ rounds. However, we can actually make the
sparsity in subsequent iterations decay super-exponentially, in fact as a
power tower. This give $O(m\log^{*}k)$ rounds.
###### Theorem 3.4.
There exists an adaptive $(1+\epsilon)$-approximate $k$-sparse recovery scheme
with $O(\frac{1}{\epsilon}k\log\frac{1}{\delta}\log\log(n\epsilon/k))$
measurements and success probability $1-\delta$. It uses
$O(\log^{*}k\log\log(n\epsilon))$ rounds.
To prove this, we start from the following lemma:
###### Lemma 3.5.
We can perform $O(\log\log(n/k))$ adaptive measurements and recover an
$\hat{\imath}$ such that, for any $j\in H_{k,1/k}(x)$ we have
$\Pr[\hat{\imath}=j]=\Omega(1/k)$.
###### Proof.
Let $S=H_{k}(x)$. Let $T\subset[n]$ contain each element independently with
probability $p=1/(4C^{2}k)$, where $C$ is the constant in Lemma 3.3. Let $j\in
H_{k,1/k}(x)$. Then we have
$\operatorname{\mathbb{E}}[\left\lVert x_{T\setminus
S}\right\rVert_{2}^{2}]=p\left\lVert x_{\overline{S}}\right\rVert_{2}^{2}$
so
$\left\lVert x_{T\setminus S}\right\rVert_{2}\leq\sqrt{4p}\left\lVert
x_{\overline{S}}\right\rVert_{2}=\frac{1}{C\sqrt{k}}\left\lVert
x_{\overline{S}}\right\rVert_{2}\leq\left|x_{j}\right|/C$
with probability at least $3/4$. Furthermore we have
$\operatorname{\mathbb{E}}[\left|T\setminus S\right|]<pn$ so $\left|T\setminus
S\right|<n/k$ with probability at least $1-1/(4C^{2})>3/4$. By the union
bound, both these events occur with probability at least $1/2$.
Independently of this, we have
$\Pr[T\cap S=\\{j\\}]=p(1-p)^{k-1}>p/e$
so all these events hold with probability at least $p/(2e)$. Assuming this,
$\left\lVert x_{T\setminus\\{j\\}}\right\rVert_{2}\leq\left|x_{j}\right|/C$
and $\left|T\right|\leq 1+n/k$. But then Lemma 3.3 applies, and
$O(\log\log\left|T\right|)=O(\log\log(n/k))$ measurements can recover $j$ from
a sketch of $x_{T}$ with probability $1/2$. This is independent of the
previous probability, for a total success chance of $p/(4e)=\Omega(1/k)$. ∎
###### Lemma 3.6.
With $O(\frac{1}{\epsilon}k\log\frac{1}{f\delta}\log\log(n\epsilon/k))$
adaptive measurements, we can recover $T$ with $\left|T\right|\leq k$ and
$\operatorname{Err^{2}}(x_{\overline{T}},fk)\leq(1+\epsilon)\operatorname{Err^{2}}(x,k)$
with probability at least $1-\delta$. The number of rounds required is
$O(\log\log(n\epsilon/k))$.
###### Proof.
Repeat Lemma 3.5 $m=O(\frac{1}{\epsilon}k\log\frac{1}{f\delta})$ times in
parallel with parameters $n$ and $k/\epsilon$ to get coordinates
$T^{\prime}=\\{t_{1},t_{2},\dotsc,t_{m}\\}$. For each $j\in
H_{k,\epsilon/k}(x)\subseteq H_{k/\epsilon,\epsilon/k}(x)$ and $i\in[m]$, the
lemma implies $\Pr[j=t_{i}]\geq\epsilon/(Ck)$ for some constant $C$. Then
$\Pr[j\notin T^{\prime}]\leq(1-\epsilon/(Ck))^{m}\leq e^{-\epsilon m/(Ck)}\leq
f\delta$ for appropriate $m$. Thus
$\displaystyle\operatorname{\mathbb{E}}[\left|H_{k,\epsilon/k}(x)\setminus
T^{\prime}\right|]\leq f\delta\left|H_{k,\epsilon/k}(x)\right|\leq$
$\displaystyle f\delta k$
$\displaystyle\Pr\left[\left|H_{k,\epsilon/k}(x)\setminus
T^{\prime}\right|\geq fk\right]\leq$ $\displaystyle\delta.$
Now, observe $x_{T^{\prime}}$ directly and set $T\subseteq T^{\prime}$ to be
the locations of the largest $k$ values. Then, since
$H_{k,\epsilon/k}(x)\subseteq H_{k}(x)$, $\left|H_{k,\epsilon/k}(x)\setminus
T\right|=\left|H_{k,\epsilon/k}(x)\setminus T^{\prime}\right|\leq fk$ with
probability at least $1-\delta$.
Suppose this occurs, and let $y=x_{\overline{T}}$. Then
$\displaystyle\operatorname{Err^{2}}(y,fk)=$
$\displaystyle\min_{\left|S\right|\leq fk}\left\lVert
y_{\overline{S}}\right\rVert_{2}^{2}$ $\displaystyle\leq$
$\displaystyle\left\lVert y_{\overline{H_{k,\epsilon/k}(x)\setminus
T}}\right\rVert_{2}^{2}$ $\displaystyle=$ $\displaystyle\left\lVert
x_{\overline{H_{k,\epsilon/k}(x)}}\right\rVert_{2}^{2}$ $\displaystyle=$
$\displaystyle\left\lVert
x_{\overline{H_{k}(x)}}\right\rVert_{2}^{2}+\left\lVert x_{H_{k}(x)\setminus
H_{k,\epsilon/k}(x)}\right\rVert_{2}^{2}$ $\displaystyle\leq$
$\displaystyle\left\lVert
x_{\overline{H_{k}(x)}}\right\rVert_{2}^{2}+k\left\lVert x_{H_{k}(x)\setminus
H_{k,\epsilon/k}(x)}\right\rVert_{\infty}^{2}$ $\displaystyle\leq$
$\displaystyle(1+\epsilon)\left\lVert
x_{\overline{H_{k}(x)}}\right\rVert_{2}^{2}$ $\displaystyle=$
$\displaystyle(1+\epsilon)\operatorname{Err^{2}}(x,k)$
as desired. ∎
Algorithm 2 Adaptive $k$-sparse recovery
procedure AdaptiveKSparseRec($x$, $k$, $\epsilon$, $\delta$)$\triangleright$
Recover approximation $\hat{x}$ of $x$
$R_{0}\leftarrow[n]$
$\delta_{0}\leftarrow\delta/2$, $\epsilon_{0}\leftarrow\epsilon/e$,
$f_{0}\leftarrow 1/32$, $k_{0}\leftarrow k$.
$J\leftarrow\\{\\}$
for $i\leftarrow 0,\dotsc,O(\log^{*}k)$ do$\triangleright$ While $k_{i}\geq 1$
for $t\leftarrow
0,\dotsc,\Theta(\frac{1}{\epsilon_{i}}k_{i}\log\frac{1}{\delta_{i}})$ do
$S_{t}\leftarrow\textsc{Subsample}(R_{i},\Theta(\epsilon_{i}/k_{i}))$
$J.\text{add}(\textsc{AdaptiveOneSparseRec}(x_{S_{t}}))$
end for
$R_{i+1}\leftarrow[n]\setminus J$
$\delta_{i+1}\leftarrow\delta_{i}/8$
$\epsilon_{i+1}\leftarrow\epsilon_{i}/2$
$f_{i+1}\leftarrow 1/2^{1/(4^{i+1}f_{i})}$
$k_{i+1}\leftarrow k_{i}f_{i}$
end for
$\hat{x}\leftarrow x_{J}$ $\triangleright$ Direct observation
return $\hat{x}$
end procedure
###### Theorem 3.7.
We can perform
$O(\frac{1}{\epsilon}k\log\frac{1}{\delta}\log\log(n\epsilon/k))$ adaptive
measurements and recover a set $T$ of size at most $2k$ with
$\left\lVert x_{\overline{T}}\right\rVert_{2}\leq(1+\epsilon)\left\lVert
x_{\overline{H_{k}(x)}}\right\rVert_{2}.$
with probability $1-\delta$. The number of rounds required is
$O(\log^{*}k\log\log(n\epsilon))$.
###### Proof.
Define $\delta_{i}=\frac{\delta}{2\cdot 2^{i}}$ and
$\epsilon_{i}=\frac{\epsilon}{e\cdot 2^{i}}$. Let $f_{0}=1/32$ and
$f_{i}=2^{-1/(4^{i}f_{i-1})}$ for $i>0$, and define $k_{i}=k\prod_{j<i}f_{j}$.
Let $R_{0}=[n]$.
Let $r=O(\log^{*}k)$ such that $f_{r-1}<1/k$. This is possible since
$\alpha_{i}=1/(4^{i+1}f_{i})$ satisfies the recurrence $\alpha_{0}=8$ and
$\alpha_{i}=2^{\alpha_{i-1}-2i-2}>2^{\alpha_{i-1}/2}$. Thus $\alpha_{r-1}>k$
for $r=O(\log^{*}k)$ and then $f_{r-1}<1/\alpha_{r-1}<1/k$.
For each round $i=0,\dotsc,r-1$, the algorithm runs Lemma 3.6 on $x_{R_{i}}$
with parameters $\epsilon_{i}$, $k_{i}$, $f_{i}$, and $\delta_{i}$ to get
$T_{i}$. It sets $R_{i+1}=R_{i}\setminus T_{i}$ and repeats. At the end, it
outputs $T=\cup T_{i}$.
The total number of measurements is
$\displaystyle
O(\sum\frac{1}{\epsilon_{i}}k_{i}\log\frac{1}{f_{i}\delta_{i}}\log\log(n\epsilon_{i}/k_{i}))\leq$
$\displaystyle
O(\sum\frac{2^{i}(k_{i}/k)\log(1/f_{i})}{\epsilon}k(i+\log\frac{1}{\delta})\log(\log(k/k_{i})+\log(n\epsilon/k)))$
$\displaystyle\leq$ $\displaystyle
O(\frac{1}{\epsilon}k\log\frac{1}{\delta}\log\log(n\epsilon/k)\sum
2^{i}(k_{i}/k)\log(1/f_{i})(i+1)\log\log(k/k_{i}))$
using the very crude bounds $i+\log(1/\delta)\leq(i+1)\log(1/\delta)$ and
$\log(a+b)\leq 2\log a\log b$ for $a,b\geq e$. But then
$\displaystyle\sum 2^{i}(k_{i}/k)\log(1/f_{i})(i+1)\log\log(k/k_{i})\leq$
$\displaystyle\sum 2^{i}(i+1)f_{i}\log(1/f_{i})\log\log(1/f_{i})$
$\displaystyle\leq$ $\displaystyle\sum 2^{i}(i+1)O(\sqrt{f_{i}})$
$\displaystyle=$ $\displaystyle O(1)$
since $f_{i}<O(1/16^{i})$, giving
$O(\frac{1}{\epsilon}k\log\frac{1}{\delta}\log\log(n\epsilon/k)$ total
measurements. The probability that any of the iterations fail is at most
$\sum\delta_{i}<\delta$. The result has size $\left|T\right|\leq\sum k_{i}\leq
2k$. All that remains is the approximation ratio $\left\lVert
x_{\overline{T}}\right\rVert_{2}=\left\lVert x_{R_{r}}\right\rVert_{2}$.
For each $i$, we have
$\displaystyle\operatorname{Err^{2}}(x_{R_{i+1}},k_{i+1})=$
$\displaystyle\operatorname{Err^{2}}(x_{R_{i}\setminus
T_{i}},f_{i}k_{i})\leq(1+\epsilon_{i})\operatorname{Err^{2}}(x_{R_{i}},k_{i}).$
Furthermore, $k_{r}<kf_{r-1}<1$. Hence
$\displaystyle\left\lVert
x_{R_{r}}\right\rVert_{2}^{2}=\operatorname{Err^{2}}(x_{R_{r}},k_{r})\leq$
$\displaystyle\left(\prod_{i=0}^{r-1}(1+\epsilon_{i})\right)\operatorname{Err^{2}}(x_{R_{0}},k_{0})=\left(\prod_{i=0}^{r-1}(1+\epsilon_{i})\right)\operatorname{Err^{2}}(x,k)$
But $\prod_{i=0}^{r-1}(1+\epsilon_{i})<e^{\sum\epsilon_{i}}<e$, so
$\prod_{i=0}^{r-1}(1+\epsilon_{i})<1+\sum e\epsilon_{i}\leq 1+2\epsilon$
and hence
$\left\lVert x_{\overline{T}}\right\rVert_{2}=\left\lVert
x_{R_{r}}\right\rVert_{2}\leq(1+\epsilon)\left\lVert
x_{\overline{H_{k}(x)}}\right\rVert_{2}$
as desired. ∎
Once we find the support $T$, we can observe $x_{T}$ directly with $O(k)$
measurements to get a $(1+\epsilon)$-approximate $k$-sparse recovery scheme,
proving Theorem 3.4
## 4 Two-round adaptivity
The algorithms in this section are invariant under permutation. Therefore, for
simplicity of notation, the analysis assumes our vectors $x$ is sorted:
$\left|x_{1}\right|\geq\dotsc\geq\left|x_{n}\right|=0$.
We are given a $1$-round $k$-sparse recovery algorithm for $n$-dimensional
vectors $x$ using $m(k,\epsilon,n,\delta)$ measurements with the guarantee
that its output $\hat{x}$ satisfies
$\|\hat{x}-x\|_{p}\leq(1+\epsilon)\cdot\|x_{\overline{[k]}}\|_{p}$ for a
$p\in\\{1,2\\}$ with probability at least $1-\delta$. Moreover, suppose its
output $\hat{x}$ has support on a set of size $s(k,\epsilon,n,\delta)$. We
show the following black box two-round transformation.
###### Theorem 4.1.
Assume $s(k,\epsilon,n,\delta)=O(k)$. Then there is a $2$-round sparse
recovery algorithm for $n$-dimensional vectors $x$, which, in the first round
uses $m(k,\epsilon/5,{\mathrm{poly}}(k/\epsilon),1/100)$ measurements and in
the second uses $O(k\cdot m(1,1,n,\Theta(1/k)))$ measurements. It succeeds
with constant probability.
###### Corollary 4.2.
For $p=2$, there is a $2$-round sparse recovery algorithm for $n$-dimensional
vectors $x$ such that the total number of measurements is
$O(\frac{1}{\epsilon}k\log(k/\epsilon)+k\log(n/k))$.
###### Proof of Corollary 4.2..
In the first round it suffices to use CountSketch with
$s(k,\epsilon,n,1/100)=2k$, which holds for any $\epsilon>0$ [PW11]. We also
have that
$m(k,\epsilon/5,{\mathrm{poly}}(k/\epsilon),1/100)=O(\frac{1}{\epsilon}k\log(k/\epsilon))$.
Using [CCF02, CM06, GI10], in the second round we can set
$m(1,1,n,\Theta(1/k))=O(\log n)$. The bound follows by observing that
$\frac{1}{\epsilon}k\log(k/\epsilon)+k\log(n)=O(\frac{1}{\epsilon}k\log(k/\epsilon)+k\log(n/k))$.
∎
###### Proof of Theorem 4.1..
In the first round we perform a dimensionality reduction of the
$n$-dimensional input vector $x$ to a
${\mathrm{poly}}(k/\epsilon)$-dimensional input vector $y$. We then apply the
black box sparse recovery algorithm on the reduced vector $y$, obtaining a
list of $s(k,\epsilon/5,{\mathrm{poly}}(k/\epsilon),1/100)$ coordinates, and
show for each coordinate in the list, if we choose the largest preimage for it
in $x$, then this list of coordinates can be used to provide a $1+\epsilon$
approximation for $x$. In the second round we then identify which heavy
coordinates in $x$ map to those found in the first round, for which it
suffices to invoke the black box algorithm with only a constant approximation.
We place the estimated values of the heavy coordinates obtained in the first
pass in the locations of the heavy coordinates obtained in the second pass.
Let $N={\mathrm{poly}}(k/\epsilon)$ be determined below. Let
$h:[n]\rightarrow[N]$ and $\sigma:[n]\rightarrow\\{-1,1\\}$ be $\Theta(\log
N)$-wise independent random functions. Define the vector $y$ by
$y_{i}=\sum_{j\ \mid\ h(j)=i}\sigma(j)x_{j}$. Let $Y(i)$ be the vector $x$
restricted to coordinates $j\in[n]$ for which $h(j)=i$. Because the algorithm
is invariant under permutation of coordinates of $y$, we may assume for
simplicity of notation that $y$ is sorted:
$\left|y_{1}\right|\geq\dotsc\geq\left|y_{N}\right|=0$.
We note that such a dimensionality reduction is often used in the streaming
literature. For example, the sketch of [TZ04] for $\ell_{2}$-norm estimation
utilizes such a mapping. A “multishot” version (that uses several functions
$h$) has been used before in the context of sparse recovery [CCF02, CM06] (see
[GI10] for an overview). Here, however, we need to analyze a “single-shot”
version.
Let $p\in\\{1,2\\}$, and consider sparse recovery with the $\ell_{p}/\ell_{p}$
guarantee. We can assume that $\|x\|_{p}=1$. We need two facts concerning
concentration of measure.
###### Fact 4.3.
(see, e.g., Lemma 2 of [KNPW10]) Let $X_{1},\ldots,X_{n}$ be such that $X_{i}$
has expectation $\mu_{i}$ and variance $v_{i}^{2}$, and $X_{i}\leq K$ almost
surely. Then if the $X_{i}$ are $\ell$-wise independent for an even integer
$\ell\geq 2$,
$\Pr\left[\left|\sum_{i=1}^{n}X_{i}-\mu\right|\geq\lambda\right]\leq
2^{O(\ell)}\left(\left(v\sqrt{\ell}/\lambda\right)^{\ell}+\left(K\ell/\lambda\right)^{\ell}\right)$
where $\mu=\sum_{i}\mu_{i}$ and $v^{2}=\sum_{i}v_{i}^{2}$.
###### Fact 4.4.
(Khintchine inequality) ([Haa82]) For $t\geq 2$, a vector $z$ and a $t$-wise
independent random sign vector $\sigma$ of the same number of dimensions,
${\bf E}[|\langle z,\sigma\rangle|^{t}]\leq\|z\|_{2}^{t}(\sqrt{t})^{t}.$
We start with a probabilistic lemma. Let $Z(j)$ denote the vector $Y(j)$ with
the coordinate $m(j)$ of largest magnitude removed.
###### Lemma 4.5.
Let $r=O\left(\|x_{\overline{[k]}}\|_{p}\cdot\frac{\log N}{N^{1/6}}\right)$
and $N$ be sufficiently large. Then with probability $\geq 99/100$,
1. 1.
$\forall j\in[N]$, $\|Z(j)\|_{p}\leq r$.
2. 2.
$\forall i\in[N^{1/3}]$, $|\sigma(i)\cdot y_{h(i)}-x_{i}|\leq r$,
3. 3.
$\|y_{\overline{[k]}}\|_{p}\leq(1+O(1/\sqrt{N}))\cdot\|x_{\overline{[k]}}\|_{p}+O(kr)$,
4. 4.
$\forall j\in[N]$, if $h^{-1}(j)\cap[N^{1/3}]=\emptyset$, then $|y_{j}|\leq
r$,
5. 5.
$\forall j\in[N]$, $\|Y(j)\|_{0}=O(n/N+\log N)$.
###### Proof.
We start by defining events $\mathcal{E}$, $\mathcal{F}$ and $\mathcal{G}$
that will be helpful in the analysis, and showing that all of them are
satisfied simultaneously with constant probability.
Event $\mathcal{E}$: Let $\mathcal{E}$ be the event that
$h(1),h(2),\ldots,h(N^{1/3})$ are distinct. Then $\Pr_{h}[\mathcal{E}]\geq
1-1/N^{1/3}$.
Event $\mathcal{F}$: Fix $i\in[N]$. Let $Z^{\prime}$ denote the vector
$Y(h(i))$ with the coordinate $i$ removed. Applying Fact 4.4 with
$t=\Theta(\log N)$,
$\displaystyle\Pr_{\sigma}[|\sigma(i)y_{h(i)}-x_{i}|\geq
2\sqrt{t}\cdot\|Z(h(i))\|_{2}]$ $\displaystyle\leq$
$\displaystyle\Pr_{\sigma}[|\sigma(i)y_{h(i)}-x_{i}|\geq
2\sqrt{t}\cdot\left\lVert Z^{\prime}\right\rVert_{2}]$ $\displaystyle\leq$
$\displaystyle\Pr_{\sigma}[|\sigma(i)y_{h(i)}-x_{i}|^{t}\geq
2^{t}(\sqrt{t})^{t}\cdot\|Z^{\prime}\|_{2}^{t}]$ $\displaystyle\leq$
$\displaystyle\Pr_{\sigma}[|\sigma(i)y_{h(i)}-x_{i}|^{t}\geq
2^{t}\operatorname{\mathbb{E}}[\left|\langle\sigma,Z^{\prime}\rangle\right|^{t}]]$
$\displaystyle=$ $\displaystyle\Pr_{\sigma}[|\sigma(i)y_{h(i)}-x_{i}|^{t}\geq
2^{t}\operatorname{\mathbb{E}}[|\sigma(i)y_{h(i)}-x_{i}|^{t}]\leq 1/N^{4/3}.$
Let $\mathcal{F}$ be the event that for all $i\in[N]$,
$|\sigma(i)y_{h(i)}-x_{i}|\leq 2\sqrt{t}\cdot\|Z(h(i))\|_{2},$ so
$\Pr_{\sigma}[\mathcal{F}]\geq 1-1/N$.
Event $\mathcal{G}$: Fix $j\in[N]$ and for each
$i\in\\{N^{1/3}+1,\ldots,n\\}$, let $X_{i}=|x_{i}|^{p}{\bf 1}_{h(i)=j}$ (i.e.,
$X_{i}=|x_{i}|^{p}$ if $h(i)=j$). We apply Lemma 4.3 to the $X_{i}$. In the
notation of that lemma, $\mu_{i}=|x_{i}|^{p}/N$ and
$v_{i}^{2}\leq|x_{i}|^{2p}/N$, and so
$\mu=\|x_{\overline{[N^{1/3}]}}\|_{p}^{p}/N$ and
$v^{2}\leq\|x_{\overline{[N^{1/3}]}}\|_{2p}^{2p}/N$. Also,
$K=|x_{N^{1/3}+1}|^{p}$. Function $h$ is $\Theta(\log N)$-wise independent, so
by Fact 4.3,
$\displaystyle\Pr\left[\left|\sum_{i}X_{i}-\frac{\|x_{\overline{[N^{1/3}]}}\|_{p}^{p}}{N}\right|\geq\lambda\right]\leq$
$\displaystyle
2^{O(\ell)}\left(\left(\|x_{\overline{[N^{1/3}]}}\|_{2p}^{p}\sqrt{\ell}/(\lambda\sqrt{N})\right)^{\ell}+\left(|x_{N^{1/3}+1}|^{p}\ell/\lambda\right)^{\ell}\right)$
for any $\lambda>0$ and an $\ell=\Theta(\log N)$. For $\ell$ large enough,
there is a
$\lambda=\Theta(\|x_{\overline{[N^{1/3}]}}\|_{2p}^{p}\sqrt{(\log
N)/N}+|x_{N^{1/3}+1}|^{p}\cdot\log N)$
for which this probability is $\leq N^{-2}$. Let $\mathcal{G}$ be the event
that for all $j\in[N]$, $\|Z(j)\|_{p}^{p}\leq
C\left(\frac{\|x_{\overline{[N^{1/3}]}}\|_{p}^{p}}{N}+\lambda\right)$ for some
universal constant $C>0$. Then $\Pr[\mathcal{G}\mid\mathcal{E}]\geq 1-1/N$.
By a union bound, $\Pr[\mathcal{E}\wedge\mathcal{F}\wedge\mathcal{G}]\geq
999/1000$ for $N$ sufficiently large.
We know proceed to proving the five conditions in the lemma statement. In the
analysis we assume that the event
$\mathcal{E}\wedge\mathcal{F}\wedge\mathcal{G}$ holds (i.e., we condition on
that event).
First Condition: This condition follows from the occurrence of $\mathcal{G}$,
and using that
$\|x_{\overline{[N^{1/3}]}}\|_{2p}\leq\|x_{\overline{[N^{1/3}]}}\|_{p}$, and
$\|x_{\overline{[N^{1/3}]}}\|_{p}\leq\|x_{\overline{[k]}}\|_{p}$, as well as
$(N^{1/3}-k+1)|x_{N^{1/3}+1}|^{p}\leq\|x_{\overline{[k]}}\|_{p}^{p}$. One just
needs to make these substitutions into the variable $\lambda$ defining
$\mathcal{G}$ and show the value $r$ serves as an upper bound (in fact, there
is a lot of room to spare, e.g., $r/\log N$ is also an upper bound).
Second Condition: This condition follows from the joint occurrence of
$\mathcal{E}$, $\mathcal{F}$, and $\mathcal{G}$.
Third Condition: For the third condition, let $y^{\prime}$ denote the
restriction of $y$ to coordinates in the set
$[N]\setminus\\{h(1),h(2),...,h(k)\\}$. For $p=1$ and for any choice of $h$
and $\sigma$, $\|y^{\prime}\|_{1}\leq\|x_{\overline{[k]}}\|_{1}$. For $p=2$,
the vector $y$ is the sketch of [TZ04] for $\ell_{2}$-estimation. By their
analysis, with probability $\geq 999/1000$,
$\|y^{\prime}\|_{2}^{2}\leq(1+O(1/\sqrt{N}))\|x^{\prime}\|_{2}^{2}$, where
$x^{\prime}$ is the vector whose support is
$[n]\setminus\cup_{i=1}^{k}h^{-1}(i)\subseteq[n]\setminus[k]$. We assume this
occurs and add $1/1000$ to our error probability. Hence,
$\|y^{\prime}\|_{2}^{2}\leq(1+O(1/\sqrt{N}))\|x_{\overline{[k]}}\|_{2}^{2}$.
We relate $\|y^{\prime}\|_{p}^{p}$ to $\|y_{\overline{[k]}}\|_{p}^{p}$.
Consider any $j=h(i)$ for an $i\in[k]$ for which $j$ is not among the top $k$
coordinates of $y$. Call such a $j$ lost. By the first condition of the lemma,
$|\sigma(i)y_{j}-x_{i}|\leq r$. Since $j$ is not among the top $k$ coordinates
of $y$, there is a coordinate $j^{\prime}$ among the top $k$ coordinates of
$y$ for which $j^{\prime}\notin h([k])$ and
$|y_{j^{\prime}}|\geq|y_{j}|\geq|x_{i}|-r.$ We call such a $j^{\prime}$ a
substitute. We can bijectively map substitutes to lost coordinates. It follows
that
$\|y_{\overline{[k]}}\|_{p}^{p}\leq\|y^{\prime}\|_{p}^{p}+O(kr)\leq(1+O(1/\sqrt{N}))\|x_{\overline{[k]}}\|_{p}^{p}+O(kr).$
Fourth Condition: This follows from the joint occurrence of
$\mathcal{E},\mathcal{F}$, and $\mathcal{G}$, and using that
$|x_{m(j)}|^{p}\leq\|x_{\overline{[k]}}\|_{p}^{p}/(N^{1/3}-k+1)$ since
$m(j)\notin[N^{1/3}]$.
Fifth Condition: For the fifth condition, fix $j\in[N]$. We apply Fact 4.3
where the $X_{i}$ are indicator variables for the event $h(i)=j$. Then ${\bf
E}[X_{i}]=1/N$ and ${\bf Var}[X_{i}]<1/N$. In the notation of Fact 4.3,
$\mu=n/N$, $v^{2}<n/N$, and $K=1$. Setting $\ell=\Theta(\log N)$ and
$\lambda=\Theta(\log N+\sqrt{(n\log N)/N})$, we have by a union bound that for
all $j\in[N]$, $\|Y(j)\|_{0}\leq\frac{n}{N}+\Theta(\log N+\sqrt{(n\log
N)/N})=O(n/N+\log N)$, with probability at least $1-1/N$.
By a union bound, all events jointly occur with probability at least $99/100$,
which completes the proof. ∎
Event $\mathcal{H}$: Let $\mathcal{H}$ be the event that the algorithm returns
a vector $\hat{y}$ with
$\|\hat{y}-y\|_{p}\leq(1+\epsilon/5)\|y_{\overline{[k]}}\|_{p}.$ Then
$\Pr[\mathcal{H}]\geq 99/100$. Let $S$ be the support of $\hat{y}$, so
$|S|=s(k,\epsilon/5,N,1/100)$. We condition on $\mathcal{H}$.
In the second round we run the algorithm on $Y(j)$ for each $j\in S$, each
using $m(1,1,\|Y(j)\|_{0},\Theta(1/k)))$\- measurements. Using the fifth
condition of Lemma 4.5, we have that $\|Y(j)\|_{0}=O(\epsilon
n/k+\log(k)/\epsilon)$ for $N={\mathrm{poly}}(k/\epsilon)$ sufficiently large.
For each invocation on a vector $Y(j)$ corresponding to a $j\in S$, the
algorithm takes the largest (in magnitude) coordinate HH$(j)$ in the output
vector, breaking ties arbitrarily. We output the vector $\hat{x}$ with support
equal to $T=\\{\textrm{HH}(j)\mid j\in S\\}$. We assign the value
$\sigma(x_{j})\hat{y}_{j}$ to HH$(j)$. We have
$\displaystyle\|x-\hat{x}\|_{p}^{p}=$
$\displaystyle\|(x-\hat{x})_{T}\|_{p}^{p}+\|(x-\hat{x})_{[n]\setminus
T}\|_{p}^{p}=\|(x-\hat{x})_{T}\|_{p}^{p}+\|x_{[n]\setminus T}\|_{p}^{p}.$ (8)
The rest of the analysis is devoted to bounding the RHS of equation 8.
###### Lemma 4.6.
For $N={\mathrm{poly}}(k/\epsilon)$ sufficiently large, conditioned on the
events of Lemma 4.5 and $\mathcal{H}$,
$\|x_{[n]\setminus
T}\|_{p}^{p}\leq(1+\epsilon/3)\|x_{\overline{[k]}}\|_{p}^{p}.$
###### Proof.
If $[k]\setminus T=\emptyset$, the lemma follows by definition. Otherwise, if
$i\in([k]\setminus T)$, then $i\in[k]$, and so by the second condition of
Lemma 4.5, $|x_{i}|\leq|y_{h(i)}|+r.$ We also use the third condition of Lemma
4.5 to obtain
$\|y_{\overline{[k]}}\|_{p}\leq(1+O(1/\sqrt{N}))\cdot\|x_{\overline{[k]}}\|_{p}+O(kr).$
By the triangle inequality,
$\displaystyle\left(\sum_{i\in[k]\setminus T}|x_{i}|^{p}\right)^{1/p}$
$\displaystyle\leq k^{1/p}r+\left(\sum_{i\ \in\ [k]\setminus
T}|y_{h(i)}|^{p}\right)^{1/p}\leq k^{1/p}r+\left(\sum_{i\ \in\ [N]\setminus
S}|y_{i}|^{p}\right)^{1/p}$ $\displaystyle\leq$ $\displaystyle
k^{1/p}r+(1+\epsilon/5)\cdot\|y_{\overline{[k]}}\|_{p}.$
The lemma follows using that $r=O(\|x_{\overline{[k]}}\|_{2}\cdot(\log
N)/N^{1/6})$ and $N={\mathrm{poly}}(k/\epsilon)$ is sufficiently large. ∎
We bound $\|(x-\hat{x})_{T}\|_{p}^{p}$ using Lemma 4.5,
$|S|\leq{\mathrm{poly}}(k/\epsilon)$, and that $N={\mathrm{poly}}(k/\epsilon)$
is sufficiently large.
$\displaystyle\|(x-\hat{x})_{T}\|_{p}\leq$ $\displaystyle\left(\sum_{j\in
S}|x_{HH(j)}-\sigma(HH(j))\cdot\hat{y}_{j}|^{p}\right)^{1/p}\leq\left(\sum_{j\in
S}(|y_{j}-\hat{y}_{j}|+|\sigma(HH(j))\cdot x_{HH(j)}-y_{j}|)^{p}\right)^{1/p}$
$\displaystyle\leq$ $\displaystyle\left(\sum_{j\in
S}|y_{j}-\hat{y}_{j}|^{p}\right)^{1/p}+\left(\sum_{j\in S}|\sigma(HH(j))\cdot
x_{HH(j)}-y_{j}|^{p}\right)^{1/p}$ $\displaystyle\leq$
$\displaystyle(1+\epsilon/5)\|y_{\overline{[k]}}\|_{p}+\left(\sum_{j\in
S}|\sigma(HH(j))\cdot x_{HH(j)}-y_{j}|^{p}\right)^{1/p}$ $\displaystyle\leq$
$\displaystyle(1+\epsilon/5)(1+O(1/\sqrt{N}))\|x_{\overline{[k]}}\|_{p}+O(kr)+\left(\sum_{j\in
S}|\sigma(HH((j))\cdot x_{HH(j)}-y_{j}|^{p}\right)^{1/p}$ $\displaystyle\leq$
$\displaystyle(1+\epsilon/4)\|x_{\overline{[k]}}\|_{p}+\left(\sum_{j\in
S}|\sigma(HH(j))\cdot x_{HH(j)}-y_{j}|^{p}\right)^{1/p}$
Event $\mathcal{I}$: We condition on the event $\mathcal{I}$ that all second
round invocations succeed. Note that $\Pr[\mathcal{I}]\geq 99/100$.
We need the following lemma concerning $1$-sparse recovery algorithms.
###### Lemma 4.7.
Let $w$ be a vector of real numbers. Suppose
$|w_{1}|^{p}>\frac{9}{10}\cdot\|w\|^{p}_{p}$. Then for any vector $\hat{w}$
for which $\|w-\hat{w}\|^{p}_{p}\leq 2\cdot\|w_{\overline{[1]}}\|^{p}_{p}$, we
have $|\hat{w}_{1}|^{p}>\frac{3}{5}\cdot\|w\|_{p}^{p}$. Moreover, for all
$j>1$, $|\hat{w}_{j}|^{p}<\frac{3}{5}\cdot\|w\|_{p}^{p}$.
###### Proof.
$\|w-\hat{w}\|^{p}_{p}\geq|w_{1}-\hat{w}_{1}|^{p}$, so if
$|\hat{w}_{1}|^{p}<\frac{3}{5}\cdot\|w\|^{p}_{p}$, then
$\|w-\hat{w}\|^{p}_{p}>\left(\frac{9}{10}-\frac{3}{5}\right)\|w\|^{p}_{p}=\frac{3}{10}\cdot\|w\|^{p}_{p}$.
On the other hand,
$\|w_{\overline{[1]}}\|_{p}^{p}<\frac{1}{10}\cdot\|w\|_{p}^{p}$. This
contradicts that $\|w-\hat{w}\|^{p}_{p}\leq
2\cdot\|w_{\overline{[1]}}\|_{p}^{p}$. For the second part, for $j>1$ we have
$|w_{j}|^{p}<\frac{1}{10}\cdot\|w\|^{p}_{p}$. Now,
$\|w-\hat{w}\|^{p}_{p}\geq|w_{j}-\hat{w}_{j}|^{p}$, so if
$|\hat{w}_{j}|^{p}\geq\frac{3}{5}\cdot\|w\|_{p}^{p}$, then
$\|w-\hat{w}\|^{p}_{p}>\left(\frac{3}{5}-\frac{1}{10}\right)\|w\|_{p}^{p}=\frac{1}{2}\cdot\|w\|_{p}^{p}$.
But since $\|w_{\overline{[1]}}\|_{p}^{p}<\frac{1}{10}\cdot\|w\|_{p}^{p}$,
this contradicts that $\|w-\hat{w}\|^{p}_{p}\leq
2\cdot\|w_{\overline{[1]}}\|_{p}^{p}$. ∎
It remains to bound $\sum_{j\in S}|\sigma(HH(j))\cdot x_{HH(j)}-y_{j}|^{p}$.
We show for every $j\in S$, $|\sigma(HH(j))\cdot x_{HH(j)}-y_{j}|^{p}$ is
small.
Recall that $m(j)$ is the coordinate of $Y(j)$ with the largest magnitude.
There are two cases.
Case 1: $m(j)\notin[N^{1/3}]$. In this case observe that
$HH(j)\notin[N^{1/3}]$ either, and $h^{-1}(j)\cap[N^{1/3}]=\emptyset$. It
follows by the fourth condition of Lemma 4.5 that $|y_{j}|\leq r$. Notice that
$|x_{HH(j)}|^{p}\leq|x_{m(j)}|^{p}\leq\frac{\|x_{\overline{[k]}}\|_{p}^{p}}{N^{1/3}-k}.$
Bounding $|\sigma(HH(j))\cdot x_{HH(j)}-y_{j}|$ by $|x_{HH(j)}|+|y_{j}|$, it
follows for $N={\mathrm{poly}}(k/\epsilon)$ large enough that
$|\sigma(HH(j))\cdot
x_{HH(j)}-y_{j}|^{p}\leq\epsilon/4\cdot\|x_{\overline{[k]}}\|_{p}/|S|)$.
Case 2: $m(j)\in[N^{1/3}]$. If $HH(j)=m(j)$, then $|\sigma(HH(j))\cdot
x_{HH(j)}-y_{j}|\leq r$ by the second condition of Lemma 4.5, and therefore
$|\sigma(HH(j))\cdot x_{HH(j)}-y_{j}|^{p}\leq
r^{p}\leq\epsilon/4\cdot\|x_{\overline{[k]}}\|_{p}/|S|$
for $N={\mathrm{poly}}(k/\epsilon)$ large enough.
Otherwise, $HH(j)\neq m(j)$. From condition 2 of Lemma 4.5 and
$m(j)\in[N^{1/3}]$, it follows that
$\displaystyle|\sigma(HH(j)))x_{HH(j)}-y_{j}|\leq$
$\displaystyle|\sigma(HH(j))x_{HH(j)}-\sigma(m(j))x_{m(j)}|+|\sigma(m(j))x_{m(j)}-y_{j}|\leq|x_{HH(j)}|+|x_{m(j)}|+r$
Notice that $|x_{HH(j)}|+|x_{m(j)}|\leq 2|x_{m(j)}|$ since $m(j)$ is the
coordinate of largest magnitude. Now, conditioned on $\mathcal{I}$, Lemma 4.7
implies that $|x_{m(j)}|^{p}\leq\frac{9}{10}\cdot\|Y(j)\|_{p}^{p}$, or
equivalently, $|x_{m(j)}|\leq 10^{1/p}\cdot\|Z(j)\|_{p}.$ Finally, by the
first condition of Lemma 4.5, we have $\|Z(j)\|_{p}=O(r)$, and so
$|\sigma(HH(j))x_{HH(j)}-y_{j}|^{p}=O(r^{p})$, which as argued above, is small
enough for $N={\mathrm{poly}}(k/\epsilon)$ sufficiently large.
The proof of our theorem follows by a union bound over the events that we
defined. ∎
## 5 Adaptively Finding a Duplicate in a Data Stream
We consider the following FindDuplicate problem. We are given an adversarially
ordered stream $\mathcal{S}$ of $n$ elements in $\\{1,2,\ldots,n-1\\}$ and the
goal is to output an element that occurs at least twice, with probability at
least $1-\delta$. We seek to minimize the space complexity of such an
algorithm. We improve the space complexity of [JST11] for FindDuplicate from
$O(\log^{2}n)$ bits to $O(\log n)$ bits, though we use $O(\log\log n)$ passes
instead of a single pass. Notice that [JST11] also proves a lower bound of
$\Omega(\log^{2}n)$ bits for a single pass.
We use Lemma 3.3 of our multi-pass sparse recovery algorithm:
###### Fact 5.1.
Suppose there exists an $i$ with $|x_{i}|\geq
C\|x_{[n]\setminus\\{i\\}}\|_{2}$ for some constant $C$. Then $O(\log\log n)$
adaptive measurements suffice to recover a set $T$ of constant size so that
$i\in T$ with probability at least $1/2$. Further, all adaptive measurements
are linear combinations with integer coefficients of magnitude bounded by
${\mathrm{poly}}(n)$.
Our algorithm DuplicateFinder for this problem considers the equivalent
formulation of FindDuplicate in which we think of an underlying frequency
vector $x\in\\{-1,0,1,\ldots,n-1\\}^{n}$. We start by initializing $x_{i}=-1$
for all $i$. Each time item $i$ occurs in the stream, we increment its
frequency by $1$. The task is therefore to output an $i$ for which $x_{i}>0$.
###### Theorem 5.2.
There is an $O(\log\log n)$-pass, $O(\log n\log 1/\delta)$ bits of space per
pass algorithm for solving the FindDuplicate problem with probability at least
$1-\delta$.
###### Proof.
We describe an algorithm DuplicateFinder which succeeds with probability at
least $1/8$. Since it knows whether or not it succeeds, the probability can be
amplified to $1-\delta$ by $O(\log 1/\delta)$ independent parallel
repetitions. It is easy to see that the pass and space complexity are as
claimed, so we prove correctness.
DuplicateFinder($\mathcal{S}$) 1. Repeat the following procedure $C=O(1)$
times independently. (a) Select $O(1)$-wise independent uniform
$t_{i}\in[0,1]$ for $i\in[n]$. (b) Let $\epsilon>0$ be a sufficiently small
constant. Let $m=O(\log 1/\epsilon)$. (c) Let $z^{1},\ldots,z^{4m}$ be a
pairwise-independent partition of the coordinates of $z$, where
$z=x_{i}/t_{i}$ for all $i$. (d) Run algorithm $A$ independently on vectors
$z^{1},\ldots,z^{4m}$. (e) Let $T_{1},\ldots,T_{4m}$ be the outputs of
algorithm $A$ on $z^{1},\ldots,z^{4m}$, respectively, as per Fact 5.1. (f)
Compute each $x_{i}$ for $i\in\cup_{j=1}^{4m}T_{j}$ in an extra pass. If there
is an $i$ for which $x_{i}>0$, then output $i$. 2. If no coordinate $i$ has
been output, then output fail.
We use the following fact shown in the proof of Lemma 3 of [JST11].
###### Lemma 5.3.
(see first paragraph of Lemma 3 of [JST11]) For a single index $i\in[n]$ and
$t$ arbitrary, we have
$\Pr[\|z_{\overline{H_{m}(z)}}\|_{2}>\frac{1}{20}\sqrt{m}\|x\|_{1}\mid
t_{i}=t]=O(\epsilon),$
where $H_{m}(z)$ denotes the set of $m$ largest (in magnitude) coordinates of
$z$. Suppose $|z_{i}|>\|x\|_{1}$ for some value of $i$. This happens if
$t_{i}<\frac{|x_{i}|}{\|x\|_{1}}$ and occurs with probability equal to
$\frac{|x_{i}|}{\|x\|_{1}}$. Conditioned on this event, by Lemma 5.3 we have
that with probability $1-O(\epsilon)$,
$\|z_{\overline{H_{m}(z)}}\|_{2}\leq\frac{1}{20}\sqrt{m}\|x\|_{1}.$
Suppose $i$ occurs in $z^{j}$ for some value of $j\in[4m]$. Since the
partition is pairwise-independent, the expected number of $\ell\in
H_{m}(z)\setminus\\{i\\}$ which occur in $z^{j}$ is at most $\frac{m}{4m}$,
and so with probability at least $3/4-O(\epsilon)$, the norm of $z^{j}$ with
coordinate corresponding to coordinate $i$ in $z$ removed is at most
$\|z_{\overline{H_{m}(z)}}\|_{2}\leq\frac{1}{20}\sqrt{m}\|x\|_{1}\leq\frac{1}{20}\sqrt{m}|z_{i}|.$
Since $m$ is a constant, by Fact 5.1, with probability at least $1/2$, $A$
outputs a set $T$ which contains coordinate $i$. Hence, with probability at
least $3/8-O(\epsilon)$, if there is an $i$ for which $|z_{i}|>\|x\|_{1}$, it
is found by DuplicateFinder.
Let $p_{i}=\frac{|x_{i}|}{\|x\|_{1}}$. Then $|z_{i}|>\|x\|_{1}$ with
probability $p_{i}$. Since $\sum_{i}x_{i}>0$, we have $\sum_{i\ \mid\
x_{i}>0}p_{i}>\frac{1}{2}$. Consider one of the $C=O(1)$ independent
repetitions of step 1.
For coordinates $i$ for which $x_{i}>0$, let $W_{i}=1$ if $|z_{i}|>\|x\|_{1}$,
and let $W=\sum_{i}W_{i}$. Then ${\bf E}[W]>1/2$ and by pairwise-independence,
${\bf Var}[W]\leq{\bf E}[W]$.
Let $W^{\prime}$ be the average of the random variable $W$ over $C$
independent repetitions. Then ${\bf E}[W^{\prime}]={\bf E}[W]>\frac{1}{2}$ and
${\bf Var}[W^{\prime}]\leq\frac{{\bf E}[W]}{C}$, and so by Chebyshev’s
inequality for $C=O(1)$ sufficiently large we have that with probability at
least $\frac{1}{2}$, $W^{\prime}>0$, which means that in one of the $C$
repetitions there is a coordinate $i$ for which $x_{i}>0$ and
$|z_{i}|>\|x\|_{1}$.
Hence, the overall probability of success is at least
$1/2\cdot(3/8-O(\epsilon))>1/8$, for $\epsilon$ sufficiently small. This
completes the proof. ∎
## Acknowledgements
This material is based upon work supported by the Space and Naval Warfare
Systems Center Pacific under Contract No. N66001-11-C-4092, David and Lucille
Packard Fellowship, MADALGO (Center for Massive Data Algorithmics, funded by
the Danish National Research Association) and NSF grant CCF-1012042. E. Price
is supported in part by an NSF Graduate Research Fellowship.
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|
arxiv-papers
| 2011-10-17T23:35:11 |
2024-09-04T02:49:23.227563
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Piotr Indyk, Eric Price, and David P. Woodruff",
"submitter": "Eric Price",
"url": "https://arxiv.org/abs/1110.3850"
}
|
1110.3865
|
# A Closer Look at the LkCa 15 Protoplanetary Disk
Sean M. Andrews11affiliation: Harvard-Smithsonian Center for Astrophysics, 60
Garden Street, Cambridge, MA 02138 , Katherine A. Rosenfeld11affiliation:
Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA
02138 , David J. Wilner11affiliation: Harvard-Smithsonian Center for
Astrophysics, 60 Garden Street, Cambridge, MA 02138 , and Michael
Bremer22affiliation: IRAM, 300 Rue de la Piscine, F-38406 Saint Martin
d’Hères, France
###### Abstract
We present 870 $\mu$m observations of dust continuum emission from the LkCa 15
protoplanetary disk at high angular resolution (with a characteristic scale of
0$\farcs$25 = 35 AU), obtained with the IRAM Plateau de Bure interferometer
and supplemented by slightly lower resolution observations from the
Submillimeter Array. We fit these data with simple morphological models to
characterize the spectacular ring-like emission structure of this disk. Our
analysis indicates that a small amount of 870 $\mu$m dust emission ($\sim$5
mJy) originates inside a large (40-50 AU radius) low optical depth cavity.
This result can be interpreted either in the context of an abrupt decrease by
a factor of $\sim$5 in the radial distribution of millimeter-sized dust grains
or as indirect evidence for a gap in the disk, in agreement with previous
inferences from the unresolved infrared spectrum and scattered light images. A
preliminary model focused on the latter possibility suggests the presence of a
low-mass (planetary) companion, having properties commensurate with those
inferred from the recent discovery of LkCa 15b.
circumstellar matter — protoplanetary disks — planet-disk interactions —
submillimeter: planetary systems — stars: individual (LkCa 15)
## 1 Introduction
Hundreds of exoplanets have been discovered around main-sequence stars, and
substantial effort is being invested to explain their demographics with
formation models (e.g., Ida & Lin, 2004; Mordasini et al., 2009). But
associating exoplanet properties with their formation epoch is problematic:
dramatic evolutionary processes that occur at early times are closely tied to
the unknown physical conditions in the progenitor circumstellar disk. Ideally,
mature exoplanets could be compared with their younger counterparts, still
embedded in their natal disks. However, detecting planets around young stars
is difficult. Radial velocity and transit searches are hindered by stellar
variability (e.g., Huélamo et al., 2008), and direct imaging is limited by
contrast with the bright star and disk emission. However, the presence of a
young planet can be inferred indirectly through its dynamical imprint on the
structure of the disk material. A sufficiently massive planet ($\geq 1$ MJup)
opens a gap that impedes the inward flow of mass through the disk, decreasing
the densities at the disk center (e.g., Lin & Papaloizou, 1986; Bryden et al.,
1999; Quillen et al., 2004). The location of the gap marks the planet orbit,
and the amount of material that flows across it depends on the planet mass
(Lubow & D’Angelo, 2006; Varnière et al., 2006). In principle, the orbit and
mass of a $\sim$Myr-old giant planet can be estimated from observations of its
disk birthsite, through constraints on the gap location and the amount of
material interior to it, respectively.
The disk around the young star LkCa 15 is considered an excellent candidate
for planet-induced disk clearing, based on its distinctive infrared spectrum
(Espaillat et al., 2007) as well as the ring-like morphology of its mm-wave
dust emission (Piétu et al., 2006; Andrews et al., 2011) and scattered light
in the infrared (Thalmann et al., 2010). Those observations confirm that the
LkCa 15 disk has a large central “cavity”, with significantly diminished dust
optical depths on Solar System size-scales. However, the cavity is not empty.
A faint infrared signal is detected in excess of the stellar photosphere,
indicating that at least a small amount of warm dust resides near the star
(Espaillat et al., 2008). That excess verifies the presence of a tenuous inner
disk – and therefore a gap – although it provides only minimal bounds on its
size (and therefore the gap width) and mass. Based on an attempt to model a
high resolution Submillimeter Array (SMA) observation of the LkCa 15 disk,
Andrews et al. (2011) identified preliminary evidence for weak, optically thin
870 $\mu$m emission from dust inside the disk cavity. If confirmed, that
emission can be used to estimate the inner disk mass, a key diagnostic of the
flow rate across the gap.
In this Letter, we present new 870 $\mu$m continuum observations of the LkCa
15 protoplanetary disk, with a 50% improvement in angular resolution
facilitated by the recent commissioning of high-frequency receivers at the
Plateau de Bure interferometer (PdBI). In §2, we provide a brief overview of
the new data and describe how their combination with previous SMA observations
provide the sharpest view yet of the thermal emission from the LkCa 15 disk.
In §3 we use simple models to explore the properties of the disk cavity and
its contents. And in §4 we discuss those modeling results in the contexts of
planet formation around LkCa 15 and the potential future utility of similar
observations as an independent check on the properties of young exoplanets.
## 2 Observations and Data Reduction
LkCa 15 was observed for 5 hours with the most extended configuration (A:
baselines of 130-760 m) of the PdBI on 2011 January 27. The observations were
conducted in “shared-risk” mode since they used the new Band 4 receivers at an
effective continuum frequency of 345.8 GHz (868 $\mu$m) and the new WideX
correlator to sample the continuum emission with a total bandwidth of 3.6 GHz
(per polarization). The observations cycled between LkCa 15 and two nearby
quasars, J0530+1331 and J0336+3218, every 22 minutes. The data were calibrated
with the CLIC software in the GILDAS package. Short observations of the bright
quasars 3C 454.3 and 3C 273 were used to set the bandpass and absolute flux
scale, and the nearby quasars that were interleaved in the observing cycle
were utilized to calibrate the time-dependent complex gain response of the
system. At the time of the observations, the new Band 4 LO system perturbed
the first channel (of 3) in the PdBI water vapor radiometer (WVR) phase
correction system. We reduced the WVR system to a dual channel mode in the
post-processing, and smoothed the WVR data on 5 s intervals. The differential
phase correction determined on 45 s intervals was extended over each source
cycle by fitting and removing linear instrumental drifts. This process
requires a stable atmosphere, with water vapor fluctuations that average to
near zero over the source cycle. These conditions were generally met, due to
the low water vapor levels ($<$2 mm) present throughout the observations.
To improve the Fourier coverage on short spacings, we supplemented these PdBI
observations with the SMA data described by Andrews et al. (2011, baselines of
8-508 m). After adjusting the datasets to account for the small proper motion
of LkCa 15 (Ducourant et al., 2005), the disk centroid was estimated in each
dataset by minimizing the imaginary components of the visibilities (see
Andrews et al., 2011). The inferred reference centers for the two datasets
agree within $\sim$10 mas and are $<$70 mas from the expected stellar position
(within the absolute astrometric uncertainty in each dataset), at RA =
4h39m17$\fs$80 and DEC = $+$22°21′03$\farcs$20\. The SMA and PdBI calibrations
were compared over their redundant Fourier coverage, and were found to be in
excellent agreement on 150-500 k$\lambda$ baselines: deviations between the
visibility amplitudes in each dataset are random, with an RMS difference of
$<$5%. The combined SMA and PdBI visibilities were Fourier inverted assuming
natural weighting, deconvolved with the CLEAN algorithm, and restored with a
$0\farcs 33\times 0\farcs 22$ synthesized beam using the MIRIAD software
package. The resulting synthesized continuum map is shown in Figure 1, with an
effective wavelength of 870 $\mu$m, RMS noise of 0.7 mJy beam-1, peak flux
density of 27 mJy beam-1, and integrated flux density of 380 mJy.
## 3 Results
The 870 $\mu$m image in Figure 1 provides the sharpest view yet of cool dust
emission from the LkCa 15 disk. As noted previously at lower resolution (Piétu
et al., 2006; Andrews et al., 2011), this emission has an inclined ring
morphology with a large and prominent central depression in intensity. The
emission ring peaks at semimajor separations of $\sim$0$\farcs$4 (56 AU for an
assumed distance of 140 pc) and has an aspect ratio and orientation in good
agreement with the inclination ($i=51\arcdeg$) and major axis position angle
(PA = 61°) inferred from its molecular line emission (Piétu et al., 2007).
Figure 2 shows the azimuthally-averaged visibilities as a function of the
deprojected baseline length (accounting for the disk viewing geometry). The
real part of this visibility profile exhibits the classic oscillation pattern
expected from the Fourier transform of a ring in the sky-plane, with distinct
nulls (sign changes) at deprojected baselines near 150, 350, and 700
k$\lambda$. The imaginary terms are negligible on all baselines, consistent
with an axisymmetric emission distribution. Although subtle, two qualitative
features in the data can serve as useful benchmarks in a refined effort to
characterize the LkCa 15 disk structure. First, the continuum intensities
inside the ring are small, but not zero (see Figure 1). And second, the
oscillations in the continuum visibility profile are relatively muted, with a
maximum amplitude of only $\sim$5 mJy between the second and third nulls. This
latter property suggests that the emission peak near the inner ring edge is
not very sharp.
With those features in mind, we attempted to reproduce these LkCa 15 disk
observations with simple 870 $\mu$m emission models. We adopted a radial
surface brightness prescription that assumes optically thin thermal emission,
$I_{\nu}\propto B_{\nu}(T_{d})(1-e^{-\tau})\approx B_{\nu}(T_{d})\tau$, where
$B_{\nu}$ is the Planck function, $T_{d}$ the dust temperature, and $\tau$ the
optical depth. The temperature profile was fixed to $T_{d}(R)=100(R/{\rm
1\,AU})^{-0.5}$ K, based on a crude approximation of the midplane temperatures
derived in a more sophisticated treatment of radiative transfer (Andrews et
al., 2011). Assuming the dust emissivity is independent of radius, we utilized
a parametric form for the base optical depth profile motivated by the surface
densities in idealized viscous accretion disks:
$\tau_{b}(R)\propto(R/R_{c})^{-\gamma}\exp{[-(R/R_{c})^{2-\gamma}]}$ (e.g.,
Hartmann et al., 1998). Modifications to that base model were also considered,
including an optical depth cavity where $\tau(R\leq R_{\rm
cav})=\delta\tau_{b}$. Three model permutations were investigated: the base
model ($\delta=1$, $R_{\rm cav}$ undefined; Model A), the base model with an
empty cavity ($\delta=0$; Model B), and the base model with a partially filled
cavity ($0<\delta<1$; Model C). All models have three base parameters – a
gradient ($\gamma$), characteristic size ($R_{c}$), and normalization (defined
as the flux density, $F_{\rm tot}=\int I_{\nu}d\Omega$) – and can utilize up
to two additional parameters, {$R_{\rm cav}$, $\delta$}.
For a given model type and parameter set, synthetic visibilities were computed
for the appropriate viewing geometry at the spatial frequencies observed by
the SMA and PdBI. Those model visibilities were compared with the data and
assigned a fit quality statistic, the sum of the (real and imaginary)
$\chi^{2}$ values over all spatial frequencies. The best-fit parameter values
for a given model were determined by minimizing $\chi^{2}$ with the Metropolis
algorithm, utilizing multiple Monte Carlo Markov chains and an initial period
of simulated annealing (see Gregory, 2005). The results are compiled in Table
1. The estimated parameter uncertainties do not consider correlated errors
from the (fixed) temperature profile or viewing geometry, and therefore are
clearly under-estimated. The best-fit synthetic data products for each model
type are directly compared with the observations in Figures 2 and 3. The
corresponding radial brightness profiles are shown together in Figure 4.
For Model A, the observed emission morphology can only be reproduced with a
large and negative optical depth gradient parameter, $\gamma$ (e.g., see
Isella et al., 2009). The Model A fit does a relatively poor job accounting
for the breadth of the observed ring structure: there is a tendency to over-
predict the emission in the disk center and prematurely cut off at larger
radii. Significant improvement is made with Model B, when a cavity is added to
the base model. This is effectively the same structure assumed by Andrews et
al. (2011). That preliminary work used a fixed $\gamma=1$, which tends to
maximize the peak-to-cavity emission contrast in the fits, leading to higher
positive residuals at the disk center. Similar results were obtained when that
effort was repeated here, with strong centralized residuals ($\sim$11
$\sigma=8.2$ mJy). The best-fit Model B parameters are different than the
fixed-gradient case (see Table 1; $\chi^{2}_{\rm A}-\chi^{2}_{\rm
B}=135$)111The $\chi^{2}$ differences in our progression of models are large
enough (the best-fit likelihood ratios are significantly greater than unity)
to warrant the complexity of adding a parameter at each step from Models A
through C. – but those same residuals remain significant ($\sim$7 $\sigma$ =
4.8 mJy). Naturally, this motivated the addition of an emission component
inside the disk cavity, cast for simplicity as an adjustment to $\delta$
(Model C). The inclusion of that weak emission improved the fit quality
($\chi^{2}_{\rm B}-\chi^{2}_{\rm C}=111$), leaving no significant residuals
compared to the data. In this scenario, dust inside the disk cavity produces
$\sim$5 mJy of 870 $\mu$m emission, corresponding to 20% of the peak surface
brightness and only 1% of the integrated flux density.
A gap structure represents an alternative model that naturally produces dust
emission inside a disk cavity. To explore that possibility with a more
physically motivated prescription, we modeled the data with the treatment of
gap profiles advocated by Crida et al. (2006) and Crida & Morbidelli (2007).
In this scenario (Model D), we utilized a semi-analytic approximation for the
surface density perturbation produced by an embedded low-mass companion to
modify the base optical depth profile. The depth, width, and shape of the gap
profile perturbation were characterized by Crida et al. (2006, their Eq. 14)
in terms of the companion-to-star mass ratio ($q=M_{s}/M_{\ast}$), the
semimajor axis of the companion ($R_{s}$), the disk viscosity ($\nu$), and the
local disk aspect ratio ($H/R$, where $H$ is the vertical scale height of the
gas). Following Crida and his colleagues, we fixed $H/R=0.05$ and only
investigated models where $\nu=10^{-5}$ in the Crida et al. (2006) normalized
units (for our fixed $T_{d}$ profile, this corresponds to a typical viscosity
coefficient $\alpha\sim 0.001$ in the formulation of Shakura & Sunyaev, 1973).
Furthermore, we fixed $R_{s}=16$ AU, in line with the recent detection of a
faint companion (Kraus & Ireland, 2011, see §4). With these simplifying
assumptions, Model D has four parameters, {$\gamma$, $R_{c}$, $F_{\rm tot}$,
$q$}. The Model D structure also has improved fit quality relative to the
empty cavity model ($\chi^{2}_{\rm B}-\chi^{2}_{\rm D}=116$, comparable to
Model C). The estimate of $q$ implies a companion mass of $M_{s}=9\pm 1$ MJup,
given the LkCa 15 stellar mass of $M_{\ast}=1.01\pm 0.03$ M⊙ that was
determined dynamically by Piétu et al. (2007). We should again caution that
these represent formal parameter uncertainty estimates that are only
applicable under the restrained assumptions of this particular model: the true
uncertainties could be significantly larger. As for Model C, there is roughly
5 mJy of 870 $\mu$m emission interior to the gap of the favored Model D
structure.
## 4 Discussion
We have used high angular resolution 870 $\mu$m PdBI+SMA observations to
investigate the radial distribution of cool dust in the LkCa 15 protoplanetary
disk with simple emission models. Although grounded in more sophisticated
techniques, these models are inherently more morphological than physical.
Their advantage lies in computation speed, which facilitated a broader
exploration of dust structures that would have been prohibitive for a complex
radiative transfer analysis. Despite their limitations, these simple models
provide some fundamental qualitative insights on the LkCa 15 disk properties:
(1) there is a substantial decrease in the dust optical depths inside
$R\approx 40$-50 AU; (2) the emission just outside that cavity edge is not
sharply peaked, as attested by the smooth intensity profiles produced by the
favored negative optical depth gradients ($\gamma$); and (3) there is a small
amount of dust located inside the disk cavity. Given our limited resolution,
the spatial distribution of that weak emission in the cavity is unclear. It
may fill the cavity (Model C), or it may be more centrally concentrated in the
form of a gap structure (Model D) similar to what was inferred from models of
the unresolved infrared spectrum (Espaillat et al., 2008).
If the latter is true, the gap is most likely opened by the resonant torques
generated by interactions between the disk and a low-mass companion (Lin &
Papaloizou, 1986; Bryden et al., 1999). Alternative gap-opening mechanisms –
for example, photoevaporation – are unlikely given the properties of the LkCa
15 system (Alexander & Armitage, 2009; Owen et al., 2011). High-contrast
imaging has ruled out stellar and brown dwarf companions around LkCa 15,
hinting that the gap may be opened by a young giant planet (Thalmann et al.,
2010; Pott et al., 2010; Kraus et al., 2011). Recently, Kraus & Ireland (2011)
used a non-redundant masking technique to detect a faint, co-moving companion
$\sim$0$\farcs$07 from LkCa 15. If that object is co-planar with the disk and
on a circular orbit, it has a semimajor axis of 16 AU. Using a simple emission
model based on the prescription of Crida et al. (2006), we have shown that a
gap at this location can reproduce well the resolved 870 $\mu$m emission
morphology we observe if the companion mass is $\sim$9 MJup. At ages of 1-3
Myr, the Baraffe et al. (2003) evolution models suggest that this object
should have an infrared contrast of $\Delta K=6.4$-7.2, in reasonable
agreement with the $\Delta K=6.8$ measured by Kraus & Ireland (2011). However,
the Marley et al. (2007) models suggest it would be $\sim$150$\times$ fainter:
a substantial accretion luminosity would be required to account for the
observed infrared emission.
Ultimately, improved constraints on the companion mass could be based on the
disk contents interior to the gap. A crude estimate of the dust mass in that
region can be made from the luminosity of the optically thin 870 $\mu$m
emission that was inferred in Models C and D. Assuming a dust opacity of 3 cm2
g-1 and a fiducial $T_{d}=45$ K, the estimated flux density of 5 mJy
corresponds to 10-6 M⊙ (0.4 M⊕). If that dust traces the gas at a mass
fraction of $\sim$1%, then the accretion rate onto LkCa 15
($\dot{M}_{\ast}\approx 2\times 10^{-9}$ M⊙ yr-1; Ingleby et al., 2009)
implies that this inner disk material would rapidly drain onto the star (in
$<$0.05 Myr). Given the system age of 1-3 Myr, the inner disk must be
continually replenished from the massive reservoir outside the gap. There is
some notable tension with theoretical expectations here: it is not clear how a
$\sim$9 MJup companion can be reconciled with the inferred inner disk mass and
stellar accretion rate in numerical simulations of gap-crossing flows (Lubow
et al., 1999; Lubow & D’Angelo, 2006). If LkCa 15b has a much lower mass, it
likely cannot sculpt the deep, wide gap needed to explain the observations: an
additional companion with a longer orbital period must also be present. Zhu et
al. (2011) and Dodson-Robinson & Salyk (2011) have effectively argued for this
latter possibility. They suggested that multi-planet systems can alleviate the
apparent discrepancy between large transition disk cavities and accretion
rates, implying that LkCa 15b is but one component in a young planetary
system.
Robust, quantitative constraints on the properties of LkCa 15b based on the
structure of the LkCa 15 disk requires more work, including a stronger link
between numerical simulations, an improved modeling effort, and observations
that can probe the inner disk at even higher angular resolution. Nevertheless,
the PdBI+SMA data presented here offer a tantalizing foreshadowing of the new
roles mm-wave observations of disk structures can play in exoplanet science.
We are very grateful to Adam Kraus for his advice and for kindly sharing
results prior to publication. This article is based on observations carried
out with the IRAM Plateau de Bure Interferometer and the Submillimeter Array.
IRAM is supported by INSU/CNRS (France), MPG (Germany), and IGN (Spain). The
SMA 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.
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Table 1: Model Parameters
Model | A | B | C | D
---|---|---|---|---
$F_{\rm tot}$ (mJy) | $363\pm 2$ | $373\pm 2$ | $367\pm 3$ | $385\pm 2$
$\gamma$ | $-1.7\pm 0.1$ | $-1.0\pm 0.1$ | $-0.5\pm 0.1$ | $-0.3\pm 0.1$
$R_{c}$ (AU) | $107\pm 2$ | $113\pm 1$ | $114\pm 1$ | $113\pm 1$
$R_{\rm cav}$ (mJy) | $\cdots$ | $36\pm 1$ | $49\pm 1$ | $\cdots$
$\delta$ | 1 (fixed) | 0 (fixed) | $0.18\pm 0.02$ | $\cdots$
$R_{s}$ (AU) | $\cdots$ | $\cdots$ | $\cdots$ | 16 (fixed)
$q$ | $\cdots$ | $\cdots$ | $\cdots$ | $0.009\pm 0.001$
$\chi^{2}$ | 516,735 | 516,600 | 516,489 | 516,484
Note. — Parameter estimates, formal uncertainties, and $\chi^{2}$ values for
the models discussed in §3. There are 776,966 independent visibilities used in
the model fits.
Figure 1: Aperture synthesis image of the 870 $\mu$m continuum emission from
the LkCa 15 disk, made from the naturally-weighted combination of PdBI and SMA
datasets. The synthesized beam, with dimensions of $0\farcs 33\times 0\farcs
22$ ($46\times 31$ AU), is shown in the lower left. The wedge on the right
marks the conversion from color to surface brightness. Each side of the image
corresponds to 560 AU projected on the sky.
Figure 2: The real and imaginary 870 $\mu$m visibilities as a function of
baseline length, deprojected to account for the LkCa 15 disk viewing geometry
and azimuthally averaged. The inset in the top panel is a detailed view of the
gray-filled region. The best-fit models visibilities for different emission
prescriptions are overlaid in color (all models have zero imaginary fluxes, by
definition). Figure 3: Comparison of the data and models in the image plane.
The top left panel shows the same image as in Figure 1. To the right, the top
panels display the best-fit model images, and the bottom panels the imaged
residual visibilities. All panels show the same color scale and contour
levels, starting at 1.4 mJy beam-1 (2 $\sigma$) and increasing in 2.5 mJy
beam-1 (3.5 $\sigma$) increments. As noted in Figure 2, Models C and D – which
emulate a low-density (but not empty) cavity and a gap structure for the LkCa
15 disk, respectively – provide the best matches to the data. Figure 4:
Radial surface brightness profiles for the best-fit parameters of each model
type, cast for simplicity into a brightness temperature format. The combined
PdBI+SMA data provide a maximum projected radial resolution of $\sim$17 AU,
marked here by the shaded gray region.
|
arxiv-papers
| 2011-10-18T02:19:56 |
2024-09-04T02:49:23.239110
|
{
"license": "Public Domain",
"authors": "Sean M. Andrews, Katherine A. Rosenfeld, David J. Wilner, and Michael\n Bremer",
"submitter": "Sean Andrews",
"url": "https://arxiv.org/abs/1110.3865"
}
|
1110.3873
|
# Anharmonic Phonons and Magnons in BiFeO3
O. Delaire1 M.B. Stone1 J. Ma1 A. Huq1 D. Gout1 C. Brown2 K.F. Wang3 Z.F. Ren3
1\. Oak Ridge National Laboratory; Oak Ridge TN 37831 USA
2\. NIST Center for Neutron Research; Gaithersburg MD 20899 USA
3\. Department of Physics; Boston College; Boston MA 02467 USA
###### Abstract
The phonon density of states (DOS) and magnetic excitation spectrum of
polycrystalline BiFeO3 were measured for temperatures $200\leq T\leq 750\,$K ,
using inelastic neutron scattering (INS). Our results indicate that the
magnetic spectrum of BiFeO3 closely resembles that of similar Fe perovskites,
such as LaFeO3, despite the cycloid modulation in BiFeO3. We do not find any
evidence for a spin gap. A strong $T$-dependence of the phonon DOS was found,
with a marked broadening of the whole spectrum, providing evidence of strong
anharmonicity. This anharmonicity is corroborated by large-amplitude motions
of Bi and O ions observed with neutron diffraction. A clear anomaly is seen in
the $T$ dependence of Bi-dominated modes across the Néel transition. These
results highlight the importance of spin-phonon coupling in this material.
###### pacs:
63.20.kk, 75.30.Ds, 75.85.+t, 78.70.Nx
## I Introduction
Multiferroic materials exhibiting a strong magneto-electric coupling are of
great interest for potential applications in spintronic devices and actuator
systems Wang-Liu-Ren ; Catalan . BiFeO3 (BFO) is one of the few known systems
exhibiting simultaneous magnetic and ferroelectric ordering at $T>300\,$K, and
as such is a strong candidate for applications Wang-Liu-Ren ; Catalan ;
Lebeugle-PRB2007 . BFO crystallizes in a rhombohedrally-distorted perovskite
structure (space group $R3c$) Kubel ; Palewicz-synchrotron ; Palewicz-neutron
; Sosnowska-review . The Bi lone-pair is thought to be responsible for the
off-centering of Bi atoms, which induces the ferroelectricity, with a high
Curie temperature $T_{\rm C}\simeq 1100\,$K Catalan . The Fe ions, inside
oxygen octahedra, carry large magnetic moments $\simeq 4\mu_{\rm B}$ Catalan ,
and order antiferromagnetically (AF) below the Néel temperature, $T_{\rm
N}\simeq 640\,$K, with some canting of the spins, and a long-period cycloid
modulation Catalan ; Sosnowska ; Ramazanoglu . Unravelling the behavior of
phonons and magnons, and their interactions, is crucial to understanding and
controlling multiferroic properties Wang-Liu-Ren . Phonons couple to the
ferroelectric order, and magnons to the magnetic order, and it is expected
that phonons and magnons strongly interact in a system exhibiting simultaneous
ferroelectric and antiferromagnetic order, such as BFO Wang-Liu-Ren . This
interaction also gives rise to mixing of the excitations, resulting in
electromagnons, for example Wang-Liu-Ren ; Catalan . To our knowledge, there
are currently no reported experimental data of the full magnon and phonon
spectra in BFO, however. Multiple Raman measurements have been performed, but
these only probe modes at small wavevectors $q\rightarrow 0$ Haumont-Raman ;
Cazayous-Raman ; Rovillain-Raman ; Singh2008-Raman ; Singh2011-Raman ;
Shimizu-Raman ; Hlinka-Raman ; Porporati-Raman ; Fukumura-Raman-lowT ;
Fukumura-Raman-highT ; Palai-Raman . Here, we report the first INS measurement
of the phonon DOS in polycrystalline BiFeO3, as a function of $T$, as well as
more detailed measurements of the magnon spectrum than previously reported
Loewenhaupt . From these data, we extract the exchange coupling constant of
BFO, and we identify a strong anharmonicity of the phonons, providing evidence
for strong spin-phonon coupling.
## II Neutron Diffraction
A stoichiometric mixture of Bi2O3 (99.99%, Aldrich) and Fe2O3 (99.99%,
Aldrich) was ball-milled for 10 hours NIST-waiver . The resulting powder was
hot-pressed at $900^{\circ}$C for $5\,$min in a half-inch diameter graphite
die, with a $2\,$ton force applied, and a heating rate of
$300^{\circ}$C$/$min. The pressed pellets were annealed in vacuum at $750\,$K
for $24\,$h. The total sample weight was about $15\,$g.
Table 1: Results of Rietveld refinements of time-of-flight neutron diffraction data for BiFeO3 ($R3c$). $T$ | $\chi^{2}$ | $x_{\rm Fe_{2}O_{3}}$ | $a,b$ | $c$ | ${\rm rms}\,U_{{\rm Bi}\perp c}$ | ${\rm rms}\,U_{{\rm Bi}\parallel c}$ | ${\rm rms}\,U_{{\rm Fe}\perp c}$ | ${\rm rms}\,U_{{\rm Fe}\parallel c}$ | ${\rm rms}\,U_{{\rm O,long}}$ | ${\rm rms}\,U_{{\rm O,mid}}$ | ${\rm rms}\,U_{{\rm O,short}}$
---|---|---|---|---|---|---|---|---|---|---|---
(K) | | (%) | ($\AA$) | ($\AA$) | ($\AA$) | ($\AA$) | ($\AA$) | ($\AA$) | ($\AA$) | ($\AA$) | ($\AA$)
300 | 1.285 | 2.0 | 5.5752 | 13.8654 | 0.0977 | 0.0819 | 0.0694 | 0.0722 | 0.1038 | 0.0899 | 0.0698
473 | 2.306 | 2.0 | 5.5862 | 13.9013 | 0.1291 | 0.1035 | 0.0914 | 0.0919 | 0.1303 | 0.1148 | 0.0956
573 | 2.151 | 1.9 | 5.5930 | 13.9222 | 0.1455 | 0.1163 | 0.1008 | 0.1014 | 0.1458 | 0.1261 | 0.1058
623 | 2.099 | 2.0 | 5.5966 | 13.9332 | 0.1527 | 0.1189 | 0.1046 | 0.1088 | 0.1531 | 0.1327 | 0.1094
723 | 1.714 | 1.7 | 5.6033 | 13.9513 | 0.1645 | 0.1301 | 0.1153 | 0.1170 | 0.1658 | 0.1437 | 0.1189
773 | 1.749 | 1.7 | 5.6065 | 13.9594 | 0.1718 | 0.1353 | 0.1190 | 0.1225 | 0.1704 | 0.1493 | 0.1264
Figure 1: Neutron dffraction patterns (markers) and Rietveld refinements (red
line) for BiFeO3 at $300\,$K (a) and $773\,$K (b). Blues curves are the bottom
of each panel are difference curves. Upper and lower tick marks are the
reflection positions for the BiFeO3 and Fe2O3 phases, respectively. Peak
labels are for the BiFeO3 phase. The refinements were done with the $R3c$
space-group. The data for $T<640\,$K were refined with a G-type
antiferromagnetic structure (without cycloid modulation).
Neutron diffraction measurements were performed using the POWGEN time-of-
flight diffractometer at the Spallation Neutron Source (SNS), at Oak Ridge
National Laboratory. The sample was placed inside a thin-wall vanadium
container, and loaded in a radiative vacuum furnace. Data were collected at
$\rm T=300,473,573,623,723,773\,$K. The time-of-flight diffractometer uses a
broad incident spectrum of neutrons, and was configured with a center-
wavelength $\lambda=1.066\,\AA$. The diffraction data were refined with GSAS
Larson2004 , using the $R3c$ space-group. The fits indicated good
crystallinity and good sample purity with $\sim$98% BiFeO3 and $\sim$2% of a
secondary phase, indexed as Fe2O3, at all temperatures. For $T<640\,$K, the
data were refined with a G-type antiferromagnetic order, without cycloid
modulation. Since the inclusion of the G-type AF order did not change the
refinements significantly, it is likely that the further incorporation of the
cycloid structure would only have a minimal effect on our results. Results are
summarized in Table 1. The diffraction data and Rietveld fits for $T=300\,$K
and $773\,$K are shown in Fig. 1 (for one of three detector banks). We observe
a constant intensity ratio for (104) and (110) reflections at all temperatures
measured, and thus our data do not support the $R3c$–$R3m$ transition reported
in Ref. Jeong .
Figure 2: Results of Rietveld refinements (space-group $R3c$) for lattice
parameters and anisotropic atomic mean-square displacements, from POWGEN
neutron diffraction data. Straight lines in (b) are fits to the data, while
straight lines in (c) are guides to the eye.
The refined lattice parameters and atomic positions are in good agreement with
prior reports Palewicz-synchrotron ; Palewicz-neutron . The temperature
dependences of the lattice constants and mean square thermal displacements
(squares of quantities in Table 1) are shown in Fig. 2. The mean-square
displacements were refined with an anisotropic harmonic model for all atoms,
which assumes Gaussian probability distributions for atom positions. The
atomic mean-square displacements are largest for Bi, followed by O. Our
results for thermal displacements are also in good agreement with Ref.
Palewicz-neutron , although we find displacements that are larger for Bi than
for Fe, both along the (trigonal) $c$-axis and in the (basal) $a,b$ plane. For
Bi and Fe vibration modes, the behavior of $\langle u^{2}\rangle$ is linear in
$T$, as expected in the high-$T$ regime of an harmonic oscillator (from the
theoretical partial phonon DOS reported in Wang-DFT , the average phonon
energies for Bi and Fe vibrations are $12$ and $28\,$meV, respectively
equivalent to $140\,$K and $320\,$K). In this regime, the amplitude of
vibrations does not depend on the mass, but only on an effective force-
constant, $K$, $\langle u^{2}\rangle\propto T/K$ Willis-Pryor . Linear fits to
the Bi and Fe data indicate that the effective $K$ for Bi motions in the
($a,b$) plane is about half of the force-constant for Bi motions along $c$,
which itself is comparable to those for either types of Fe motions. We note
that our fits did not use anharmonic displacement models, which could slightly
affect the results. The results for $\langle u^{2}\rangle$ of oxygen atoms in
Fig. 2-c shows a departure from linearity around $600\,$K for the short and
long axes of the thermal ellipsoids. This effect may be related to the
magnetic transition at $T_{\rm N}=640\,$K, although there could also be
effects of phonon thermal occupations, since the average energy of O
vibrations is $45\,$meV, corresponding to $520\,$K Wang-DFT .
## III Inelastic Neutron Scattering
INS spectra were measured using the ARCS direct-geometry time-of-flight
chopper spectrometer at the SNS arcspaper . In the ARCS measurements, the
sample was encased in a $12\,$mm diameter, thin-walled Al can, and mounted
inside a low-background furnace for measurements at
$T=300,470,570,690,750\,$K. All measurements were performed under high vacuum.
An incident neutron energy $E_{i}=110\,$meV was used and the energy resolution
(full width at half max.) was $\sim$3 meV at $80\,$meV neutron energy loss,
increasing to $\sim$7 meV at the elastic line.
Additional INS measurements were performed with the Disk Chopper Spectrometer
(DCS) at the NIST Center for Neutron Research, with incident energy
$E_{i}=3.55\,$meV, in up-scattering mode (excitation annihilation) dcspaper .
In these conditions, the energy resolution was $\sim$0.12 meV at the elastic
line, increasing to $\sim$1.2 meV at 25 meV neutron energy gain. For DCS
measurements, the sample was encased in the same Al can as in the ARCS
measurements. The empty Al sample container was measured in identical
conditions to the sample at all temperatures, and subtracted from the data.
DCS measurements at $T=200,300,470,570\,$K were performed in a high-
temperature He refrigerator, and measurements at $T=570,670,720\,$K were
performed in a radiative furnace.
### III.1 Magnetic Spectrum
Figure 3: $S(Q,E)$ for BiFeO3 at different temperatures, measured using ARCS
(logarithmic intensity). Figure 4: $S(Q,E)$ for BiFeO3 at different
temperatures, measured using DCS (logarithmic intensity).
Figure 3 shows the orientation-averaged scattering function, $S(Q,E)$,
obtained with ARCS, as a function of temperature ($Q$ and $E$ are the momentum
and energy transfer to the sample, respectively). At $T=300\,$K, the data for
$Q<4\rm\AA^{-1}$ clearly show steep spin-waves, emanating from strong magnetic
Bragg peaks at $Q=1.37$ and $2.62\rm\AA^{-1}$, and extending to $E\sim
70\,$meV. This range of energies overlaps with much of the phonon spectrum
(see below). The intensity of the magnetic Bragg peaks decreases with
increasing $T$, vanishing for $T\geq 670\,$K, in good agreement with the
reported $T_{\rm N}=640\,$K. The magnetic scattering nearly vanishes for
$Q>6\rm\AA^{-1}$, owing to the magnetic form factor of Fe3+ ions. The
high-$E$, optical part of the spin-waves is strongly damped for
$T=470,570\,$K, well below $T_{\rm N}$. The low-$E$ part of the spin-waves is
also clearly seen in the DCS data in Fig. 4-a,b, corresponding to sharp
vertical streaks emanating from the magnetic Bragg peaks. The acoustic part of
the spin-waves shows some $Q$-broadening with increasing $T$ below $T_{\rm
N}$. Magnetic correlations remain for $T>T_{\rm N}$, but are much broader
(Figs. 3/4-c,d). We note that the orientation-averaged spin-waves show a
strong similarity between BiFeO3 and other AFeO3 perovskites, such as ErFeO3,
LaFeO3 and YFeO3 shapiro1974 ; McQueeney ; JieMa_phd .
An important question is whether low-energy spin-waves are present in BFO, and
whether an energy gap exists, potentially associated with an anisotropy in the
exchange coupling, or single-ion anisotropy. Figure 5-b shows the magnetic
scattering intensity, $S_{\rm mag}(E)$, measured with DCS, integrated over the
spin-wave dispersing from $Q=1.37\rm\AA^{-1}$, and compared to the phonon
background on either side. This figure clearly shows that the $S_{\rm mag}(E)$
intensity from the spin-wave persists down to $|E|\simeq 0.3\,$meV, where it
merges with the elastic scattering signal. Thus, we can estimate an upper-
bound for any magnetic anisotropy gap, $E_{g}<0.3\,$meV, if any such gap
exists. This is at odds with reports of a gap, $E_{g}=6\,$meV, derived from
modeling the low-$T$ specific heat $C_{P}$ Lu-SpecificHeat .
Figure 5: (a) Integrated INS intensity (DCS, $300\,$K) from spin-wave,
$1.3\leqslant Q\leqslant 1.45\,\rm\AA^{-1}$, compared with background phonon
signal ($1.15\leqslant Q\leqslant 1.3\,\rm\AA^{-1}$ and $1.45\leqslant
Q\leqslant 1.6\,\rm\AA^{-1}$). (b) Magnetic excitation spectra obtained from
INS data (ARCS), integrated over $2\leqslant Q\leqslant 6\,\rm\AA^{-1}$,
corrected for phonon scattering ($6\leqslant Q\leqslant 10\,\rm\AA^{-1}$,
scaled). The thick line is the magnon spectrum calculation (see text). Error
bars indicate one standard deviation.
Although the powder average of excitations does not allow us to determine if
multiple spin-wave branches exist within the magnon DOS, we can compare our
results with recent Raman scattering studies that reported multiple
electromagnon modes in BFO between 1.5 and 7.5 meV Rovillain-Raman ;
Singh2008-Raman . We do not see these excitations in the powder spectrum at
either $200$ or $300\,$K (see Fig. 5-a), although high-resolution INS
measurements on single-crystals may be needed to observe such effects. Our
results for the magnetic spectrum show a single maximum, around $65\,$meV at
$300\,$K, see Fig. 5-b. This is at odds with the magnetic spectrum reported in
Ref. Loewenhaupt , which showed additional maxima around $30$ and $55\,$meV.
However, the extra peaks in Loewenhaupt are likely due to peaks in the phonon
DOS at these energies (see below). We use the magnetic DOS to estimate the
exchange coupling, with a simple spin-wave model for a collinear Heisenberg
G-type antiferromagnet in an undistorted perovskite structure. A similar model
successfully described the spin waves in the orthoferrite compounds ErFeO3,
TmFeO3 shapiro1974 , as well as LaFeO3 and YFeO3 McQueeney ; JieMa_phd , which
have close magnetic structures. This does not capture possible effects from
the spiral spin structure in BFO, but these are expected to be limited, owing
to the long period of the modulation. Within this model, two exchange
constants, $J$ and $J^{{}^{\prime}}$, describe a gapless spin-wave dispersion,
with acoustic and optic modes that meet at the magnetic zone boundaries. This
behavior can be seen in the large patch of scattering intensity in the ARCS
$300\,$K data near 65 meV (Fig. 3-a). The $J$[$J^{{}^{\prime}}$] exchange
constant corresponds to coupled moments with spin-spin distances of
3.968[5.613$\pm 0.025$] Å. Assuming $S=5/2$ Fe3+ moments, and comparing the
maximum energy of the measured and model spin-wave spectra, we are able to
place limits on the values of $J$ and $J^{\prime}$. We find that
$J=1.6^{+0.4}_{-0.2}$ and $J^{\prime}=-0.25^{+0.07}_{-0.17}$ where there is a
linear dependence on these parameters within these bounds
$J^{\prime}=0.90(2)-J0.40(1)$. The calculated magnetic DOS for $J=1.6$ and
$J^{\prime}=-0.253$ meV agrees well with the phonon subtracted $T=300$ K data
shown in Fig. 5-b.
There is a clear softening and broadening of the spin-wave spectrum with
increasing temperature from $300\,$K to $570\,$K. Magnon-magnon and magnon-
phonon interactions are likely both responsible for this softening
Singh2008-Raman ; lovesey_vol2 . It is possible that the strong softening of
$S_{\rm mag}(E)$ in this range is related to the an anomalous magnetization
below $T_{\rm N}$ Lu-SpecificHeat . Above $T_{\rm N}$, we observe Lorentzian
scattering intensity centered at $E=0\,$meV, typical of paramagnetic behavior
(the dip below $20\,$meV is a result of imperfect phonon subtraction).
### III.2 Phonons
In both Figs. 3 and 4, the horizontal bands of intensity increasing as $Q^{2}$
are orientation-averaged phonon dispersions. In the DCS data, three main
horizontal bands are clearly observed at $|E|\simeq 6.5,8,11\,$meV,
corresponding to the top of acoustic phonon branches and low-$E$ optical
branches, which mainly involve Bi vibrations (see below). The acoustic
branches are seen dispersing out of a nuclear Bragg peak at $Q\simeq
2.25\rm\AA^{-1}$. The phonon cutoff corresponding to the top of oxygen-
dominated optic branches is seen at $E\simeq 85\,$meV in Fig. 3-a. A strong
broadening of the phonon modes with increasing temperature can be seen in both
Figs. 3 and 4. This broadening indicates a strong damping of phonons, over the
full spectrum. The $T$ range over which this occurs is in agreement with prior
Raman measurements Singh2011-Raman , and points to a spin-phonon coupling
effect.
Figure 6: (a) Generalized phonon DOS obtained from ARCS data. (b) Low-$E$ part
of generalized DOS from DCS data, showing strong broadening of Bi-dominated
modes. Curves for different temperatures are vertically offset for clarity.
The $S(Q,E)$ data were analyzed to extract the generalized phonon DOS, $g(E)$,
in the incoherent scattering approximation. The data from ARCS were integrated
over $6\leqslant Q\leqslant 10\rm\AA^{-1}$, which minimizes any contribution
from magnetic scattering, and the elastic peak was subtracted, and
extrapolated with a quadratic $E$ dependence for $E<5\,$meV. A correction for
multiphonon scattering was performed at all $T$ DANSE-ref ; Kresch-Ni . The
gDOS from the DCS data was obtained by integrating over the full range of $Q$
(no multiphonon correction) dave . The measured signal from the empty
container was much weaker than from the sample, and was easily subtracted. The
results are shown in Fig. 6(a) for the full gDOS (ARCS) and panel (b) for
$E<20\,$meV (DCS). Although the DCS data include a magnetic contribution, this
effect is limited, and the DOSs form ARCS and DCS are in excellent agreement,
considering the difference in instrument resolution (see Fig. 7). The coarser
energy resolution in ARCS data washes out the three Bi-dominated peaks at
$E\leqslant 15\,$meV, but the two DOS curves are otherwise very similar. In
BFO, the different elements have different ratios of cross-section over mass,
$\sigma/M$, resulting in a weighted phonon DOS (gDOS). The values of
$\sigma/M$ for (Bi, Fe, O) are (0.044, 0.208, 0.265), in units of barns/amu,
respectively. Thus, the modes involving primarily Bi motions are under-
emphasized in $S(Q,E)$ and $g(E)$ (but there is relatively little weighting of
Fe modes compared to O modes).
While the energy-range of the spectrum is comparable with other Fe perovskites
JieMa_phd , what is striking is the severe broadening of the spectrum with
increasing $T$. The gDOS measured at $300\,$K is in good agreement overall
with the first-principles calculation of Wang et al. within spin-polarized
DFT+U Wang-DFT , as can be seen in Fig. 7. This agreement allows for a clear
identification of the main features in the DOS. The three peaks at $E\simeq
6.5,8,11\,$meV (Fig. 6(b)) arise from the top of acoustic branches and the
lowest-$E$ optic modes, and they are dominated by Bi motions. The lower two Bi
peaks ($6.5\,$meV and $8\,$meV) are softer than predicted by DFT by about 10%
Wang-DFT . These Bi modes are responsible for the peak in $C_{P}/T^{3}$ around
$25\,$K reported in Ref. Lu-SpecificHeat . The phonon spectrum for $E>40\,$meV
is mainly comprised of oxygen vibrations, and is stiffer in the measured DOS
than in the DFT calculation Wang-DFT . While the agreement is generally good
between the DFT calculation of Wang et al. and the phonon DOS measured at low
temperature, the $T$ dependence of the DOS is strongly affected by
anharmonicity and spin-phonon coupling, as we discuss in the next section.
Figure 7: Generalized phonon DOS of BiFeO3 measured with INS (ARCS and DCS) at
300 K, compared with first-principles calculation of Wang et al. Wang-DFT .
## IV Discussion
Figure 8: (a) Centroids of the three low-energy peaks in DOS around $6.5$,
$8.5$, $11\,$meV (resp. $E_{1}$, $E_{2}$, $E_{3}$), as a function of
temperature. (b) Relative change in these energies with respect to their value
at $200\,$K. The vertical dashed line at $T=640\,$K denotes the Néel
temperature.
As already pointed out above, the phonon DOS exhibits a severe broadening with
increasing $T$, which affects the whole spectrum. In particular, the
broadening of Bi modes at low-$E$ is obvious in the DCS data, shown in Fig.
6-b. It is particularly strong between $470\,$K and $570\,$K, with little
additional broadening observed further above $T_{\rm N}$. This indicates a
coupling between anharmonicity and the loss of the AF order. These results are
consistent with the reported observations of strong broadening of Raman phonon
modes Singh2011-Raman . The oxygen modes, dominating the DOS between $40\,$meV
and $85\,$meV, are also strongly broadened in this $T$-range.
We now analyze in more detail the temperature dependence of the three peaks at
$6.5$, $8.5$, $11\,$meV (resp. $E_{1}$, $E_{2}$, $E_{3}$). The peaks were
fitted with Gaussians, and the resulting peak centers are plotted as a
function of temperature in Fig. 8. As may be seen on this figure, all three
peaks undergo a pronounced but gradual softening between $200\,$K and
$570\,$K, besides the strong broadening. However, this softening stops around
$T_{\rm N}$. This is compatible with the recently reported behavior of $A_{1}$
Raman modes Singh2011-Raman . $E_{3}$ actually appears to stiffen above
$T_{\rm N}$, but this may partly be due to the influence of broadening and
softening of modes at $E\geqslant 13\,$meV, causing a skewed contribution to
the $E_{3}$ peak.
The change in phonon softening behavior around $T_{\rm N}$ is further evidence
of the influence of spin-phonon coupling. The softening of phonon modes with
increasing $T$, as observed here for $T\leqslant 570\,$K, can generally be
related to the thermal expansion of the system through the quasiharmonic
approximation $d\ln E=-\gamma d\ln V$, with $\gamma$ the Grüneisen parameter
Grimvall-TPM99 . We determined the relative change in volume to be about 1.4%
between $200\,$K and $570\,$K from our diffraction measurements (this was
linearly extrapolated over the range $200-570\,$K, since our diffraction data
were limited to $T\geqslant 300\,$K). The average relative decrease of
$E_{1}$, $E_{2}$, and $E_{3}$ is $-4.2\pm 0.7$% over the same $T$ range,
yielding a Grüneisen parameter $\gamma=3.0\pm 0.5$. Such a large value of
$\gamma$ is a further corroboration of the anharmonicity of these Bi-dominated
modes. The lack of softening above $570\,$K indicates that an increase in
interatomic force-constants associated with the loss of magnetic order
compensates for the effect of thermal expansion.
We note that Bi and O atoms undergo large amplitude vibrations, according to
both our measurements and reports of others Palewicz-neutron ; Palewicz-
synchrotron . Since the Bi and O modes are sharp at $T\leqslant 300\,$K, the
large displacements observed in diffraction data are dynamic in nature, rather
than associated with static disorder. These large amplitudes of vibration for
Bi and O are related to the anharmonic scattering of phonons, which leads to
the broadening and softening of features in the DOS. We have also performed
measurements of the Fe-partial phonon DOS with nuclear-resonant inelastic
x-ray scattering (NRIXS) on 57Fe-enriched samples, and observed a more limited
broadening of Fe modes, in agreement with the smaller thermal displacements of
these atoms Delaire-BFO-NRIXS .
We suggest that the large thermal displacements and anharmonicity of Bi and O
modes lead to structural fluctuations, such as variations in Fe-O-Fe bond
lengths and bond angles (tuning the superexchange interaction) through tilts
and rotations of FeO6 octahedra, that could lead to fluctuations in magnetic
coupling. The magnitude of O thermal motions perpendicular to Fe-O bonds
actually leads to fluctuations in the Fe-O-Fe bond angle that are larger
($\simeq 6-10^{\circ}$) than the variation of average angle with $T$.
Reciprocally, the loss of magnetic order induces a change in interatomic
force-constants, stiffening the Bi vibration modes at low $E$. The motion of
Bi atoms is also directly related to the ferroelectricity. The large-amplitude
structural fluctuations could thus lead to steric effects between Bi motions
and the rotations of oxygen octahedra, yielding a complex coupling between
ferroelectric and AF magnetic orders.
## V Summary
We have systematically investigated the temperature dependence of the magnetic
excitation spectrum and phonon density of states of BiFeO3 over the range
$200\leqslant T\leqslant 750K$, using inelastic neutron scattering. In
addition, we performed neutron diffraction measurements and refined the
lattice parameters and thermal displacement parameters over $300\leqslant
T\leqslant 770K$. We separated the magnon and phonon contributions in the
$S(Q,E)$ data, and observed a strong resemblance of the magnon spectrum with
that of related compounds LaFeO3 and YFeO3. The magnon spectrum was fit
satisfactorily with a simple collinear Heisenberg model for a G-type
antiferromagnet, indicating the limited role of the cycloid on the spin
dynamics, as expected from the long period of the cycloid modulation. The
phonon DOS obtained from the high-$Q$ part of the $S(Q,E)$ data is in good
agreement at low temperatures with the first-principles calculation of Wang et
al. Wang-DFT . However, the phonon DOS shows a strong temperature dependence,
with in particular a pronounced broadening. Also, both the softening and
broadening of features in the DOS correlate with the loss of antiferromagnetic
order around $T_{N}=640\,$K, indicating the presence of significant spin-
phonon coupling, in agreement with recently reported Raman measurements
Singh2011-Raman . The potential influence of large atomic displacements on the
modulation of the superexchange interaction, and the concomitant effect of the
change in force-constants from the loss of magnetic order were pointed out.
## VI Acknowledgements
The Research at Oak Ridge National Laboratory’s Spallation Neutron Source was
sponsored by the Scientific User Facilities Division, Office of Basic Energy
Sciences, US DOE. This work utilized facilities supported in part by the
National Science Foundation under Agreement No. DMR-0944772. The work
performed at Boston College is funded by the US Department of Energy under
contract number DOE DE-FG02-00ER45805 (ZFR).
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|
arxiv-papers
| 2011-10-18T04:57:12 |
2024-09-04T02:49:23.247014
|
{
"license": "Public Domain",
"authors": "Olivier Delaire, Matthew B. Stone, Jie Ma, Ashfia Huq, Delphine Gout,\n Craig Brown, Kefeng Wang, and Zhifeng Ren",
"submitter": "Olivier Delaire",
"url": "https://arxiv.org/abs/1110.3873"
}
|
1110.3885
|
# Equivalence of three different kinds of optimal control problems for heat
equations and its applications
Gengsheng Wang Yashan Xu School of Mathematics and Statistics, Wuhan
University, Wuhan, 430072, China. (wanggs62@yeah.net) The author was partially
supported by National Basis Research Program of China (973 Program) under
grant 2011CB808002 and the National Natural Science Foundation of China under
grant 11161130003 and 11171264.School of Mathematical Sciences, Fudan
University, KLMNS, Shanghai 200433, China. (yashanxu@fudan.edu.cn) This work
was partially supported by NNSF Grant 10801041, 10831007.
###### Abstract
This paper presents an equivalence theorem for three different kinds of
optimal control problems, which are optimal target control problems, optimal
norm control problems and optimal time control problems. Controlled systems in
this study are internally controlled heat equations. With the aid of this
theorem, we establish an optimal norm feedback law and build up two algorithms
for optimal norms (together with optimal norm controls) and optimal time
(along with optimal time controls), respectively.
AMS Subject Classifications. 35K05, 49N90
Keywords. optimal controls, optimal norm, optimal time, feedback law, heat
equations
## 1 Introduction
We begin with introducing the controlled system. Let $T$ be a positive number
and $\Omega\subseteq{\mathbb{R}}^{d}$ be a bounded domain with a smooth
boundary $\partial\Omega$. Let $\omega$ be an open and non-empty subset of
$\Omega$. Write $\chi_{\omega}$ for the characteristic function of $\omega$.
Consider the following controlled heat equation:
$\left\\{\begin{array}[]{lll}\partial_{t}y-\triangle
y=\chi_{\omega}\chi_{(\tau,T)}u&\mbox{in}&\Omega\times(0,T),\\\
y=0&\mbox{on}&\partial\Omega\times(0,T).\\\
y(0)=y_{0}&\mbox{in}&\Omega,\end{array}\right.$ (1.1)
Here $y_{0}\in L^{2}(\Omega)$, $u\in L^{\infty}(0,T;L^{2}(\Omega))$,
$\tau\in[0,T)$ and $\chi_{(\tau,T)}$ stands for the characteristic function of
$(\tau,T)$. In this equation, controls are restricted over
$\omega\times(\tau,T)$. It is well known that for each $u\in
L^{\infty}(0,T;L^{2}(\Omega))$ and each $y_{0}\in L^{2}(\Omega)$, Equation
(1.1) has a unique solution in $C([0,T];L^{2}(\Omega))$. We denote, by
$y(\cdot;\chi_{(\tau,T)}u,y_{0})$, the solution of Equation (1.1)
corresponding to the control $u$ and the initial state $y_{0}$. Throughout
this paper, $\|\cdot\|$ and $<\cdot,\cdot>$ stand for the usual norm and inner
product of the space $L^{2}(\Omega)$, respectively.
Next, we will set up, for each $y_{0}\in L^{2}(\Omega)$, three kinds of
optimal control problems associated with Equation (1.1). For this purpose, we
take a target $z_{d}\in L^{2}(\Omega)$ such that
$\displaystyle z_{d}\notin\Bigr{\\{}y(T;\chi_{(0,T)}u,0):~{}u\in
L^{\infty}(0,T;L^{2}(\Omega))\Bigl{\\}}.$ (1.2)
The set on the right hand side of (1.2) is called the attainable set of
Equation (1.1). Then we introduce the following target sets:
$B(z_{d},r)=\\{\hat{y}\in L^{2}(\Omega):\|\hat{y}-z_{d}\|\leq r\\},\;\;r>0.$
For each $M\geq 0$, each $r>0$ and each $\tau\in[0,T)$, we define three sets
of controls as follows:
* •
$\mathcal{U}_{\tau,M}=\\{v\in L^{\infty}(0,T;L^{2}(\Omega)):\|v(t)\|\leq
M\;\;\mbox{for a.e.}\;\;t\in(\tau,T)\\}$;
* •
$\mathcal{U}_{M,r}=\\{v:\exists\;\tau\in[0,T)\;\mbox{s.t.}\;v\in\mathcal{U}_{\tau,M}\;\;\mbox{and}\;\;y(T;\chi_{(\tau,T)}v,y_{0})\in
B(z_{d},r)\\}$;
* •
$\mathcal{U}_{r,\tau}=\\{v\in
L^{\infty}(0,T;L^{2}(\Omega)):y(T;\chi_{(\tau,T)}v,y_{0})\in B(z_{d},r)\\}.$
For each $u\in\mathcal{U}_{M,r}$, we set
$\displaystyle\widetilde{\tau}_{M,r}(u)=\sup\\{\tau\in[0,T):u\in\mathcal{U}_{\tau,M}\;\mbox{and}\;y(T;\chi_{(\tau,T)}u,y_{0})\in
B(z_{d},r)\\}.$ (1.3)
Three kinds of optimal control problems studied in this paper are as follows:
* •
$(OP)^{\tau,M}$:
$\inf\\{\|y(T;\chi_{(\tau,T)}u,y_{0})-z_{d}\|^{2}:u\in\mathcal{U}_{\tau,M}\\}$;
* •
$(TP)^{M,r}$: $\sup\\{\widetilde{\tau}_{M,r}(u):u\in\mathcal{U}_{M,r}\\}$;
* •
$(NP)^{r,\tau}$:
$\inf\\{\|u\|_{L^{\infty}(\tau,T;L^{2}(\Omega))}:u\in\mathcal{U}_{r,\tau}\\}$.
We call $(OP)^{\tau,M}$ as an optimal target control problem, which is a kind
of optimal control problem with the observation of the final state (see [9],
page 177). The problem $(NP)^{r,\tau}$ is an optimal norm control problem,
which is related to the approximate controllability (see [4]). The problem
$(TP)^{M,r}$ is an optimal time control problem. The aim of controls in
$(TP)^{M,r}$ is to delay initiation of active control as late as possible,
such that the corresponding solution reaches the target $B(z_{d},r)$ at the
ending time $T$ (see [11]).
The above three problems provide the following three values, respectively:
* •
$r(\tau,M)\equiv\inf\\{\|y(T;\chi_{(\tau,T)}u,y_{0})-z_{d}\|:u\in\mathcal{U}_{\tau,M}\\};$
* •
$\tau(M,r)\equiv\sup\\{\widetilde{\tau}_{M,r}(u):u\in\mathcal{U}_{M,r}\\}$;
* •
$M(r,\tau)\equiv\inf\\{\|u\|_{L^{\infty}(\tau,T;L^{2}(\Omega))}:u\in\mathcal{U}_{r,\tau}\\}.$
The value $r(\tau,M)$ is called the optimal distance to the target for
$(OP)^{\tau,M}$; while values $\tau(M,r)$ and $M(r,\tau)$ are called the
optimal time for $(TP)^{M,r}$ and the optimal norm for $(NP)^{r,\tau}$,
respectively. The optimal controls to these problems are defined as follows:
* •
$u^{*}$ is called an optimal control to $(OP)^{\tau,M}$ if
$u^{*}=\chi_{(\tau,T)}v^{*}$ for some $v^{*}\in\mathcal{U}_{\tau,M}$ such that
$\|y(T;\chi_{(\tau,T)}v^{*},y_{0})-z_{d}\|=r(\tau,M)$;
* •
$u^{*}$ is called an optimal control to $(TP)^{M,r}$ if
$u^{*}=\chi_{(\tau(M,r),T)}v^{*}$ for some $v^{*}\in\mathcal{U}_{\tau(M,r),M}$
such that $y(T;\chi_{(\tau(M,r),T)}v^{*},y_{0})\in B(z_{d},r).$
* •
$u^{*}$ is called an optimal control to $(NP)^{r,\tau}$ if
$u^{*}=\chi_{(\tau,T)}v^{*}$ for some $v^{*}\in\mathcal{U}_{r,\tau}$ and
$\|u^{*}\|_{L^{\infty}(\tau,T;L^{2}(\Omega))}=M(r,\tau)$.
Throughout the paper, the following notation will be used frequently:
$\displaystyle r_{T}(y_{0})\equiv\|y(T;0,y_{0})-z_{d}\|.$ (1.4)
The main purpose of this study is to present an equivalence theorem for the
above-mentioned three kinds of optimal control problems and its applications.
This theorem can be stated, in plain language, as follows:
* •
$(OP)^{\tau,M}\Leftrightarrow(TP)^{M,r(\tau,M)}\Leftrightarrow(NP)^{r(\tau,M),\tau}$
when $M>0$ and $\tau\in[0,T)$;
* •
$(NP)^{r,\tau}\Leftrightarrow(OP)^{\tau,M(r,\tau)}\Leftrightarrow(TP)^{M(r,\tau),r}$
when $r\in(0,r_{T}(y_{0}))$ and $\tau\in[0,T)$;
* •
$(TP)^{M,r}\Leftrightarrow(NP)^{r,\tau(M,r)}\Leftrightarrow(OP)^{\tau(M,r),M}$
when $M>0$ and $r\in[r(0,M),r_{T}(y_{0}))$.
Here, by $(P_{1})\Leftrightarrow(P_{2})$, we mean that problems $(P_{1})$ and
$(P_{2})$ have the same optimal controls. Based on the equivalence theorem,
the study of one kind of optimal control problem can be carried out by
investigating one of the other two kinds of optimal control problems. In
particular, one can use some existing fine properties for optimal target
controls to derive properties of optimal norm controls and optimal time
controls.
An important application of the equivalence theorem is to build up a feedback
law for norm optimal control problems. We will roughly present this result in
what follows. Notice that Problem $(NP)^{r,\tau}$ depends on $\tau\in[0,T)$
and $y_{0}\in L^{2}(\Omega)$, when $r$ and $z_{d}$ are fixed. To stress this
dependence, we denote, by $(NP)^{r,\tau}_{y_{0}}$, the problem $(NP)^{r,\tau}$
with the initial state $y_{0}$. Throughout this paper, we let $A$ be the
operator on $L^{2}(\Omega)$ with domain $D(A)=H^{1}_{0}(\Omega)\bigcap
H^{2}(\Omega)$ and defined by $Ay=\triangle y$ for each $y\in D(A)$. Write
$\\{e^{t\triangle}:\;t\geq 0\\}$ for the semigroup generated by $A$. By the
equivalence theorem and some characteristics of the target optimal control
problems, we construct a map $F:[0,T)\times L^{2}(\Omega)\rightarrow
L^{2}(\Omega)$ holding properties:
$(i)$ For each $y_{0}\in L^{2}(\Omega)$ and each $\tau\in[0,T)$, the evolution
equation
$\displaystyle\left\\{\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\dot{y}(t)-Ay(t)=\chi_{\omega}\chi_{(\tau,T)}(t)F(t,y(t)),&t\in(0,T)\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr y(0)=y_{0},\end{array}\right.$
has a unique mild solution, which will be denoted by
$y_{F,\tau,y_{0}}(\cdot)$. Here $\chi_{\omega}$ is treated as an operator on
$L^{2}(\Omega)$ in the usual way.
${(ii)}$ For each $y_{0}\in L^{2}(\Omega)$ and each $\tau\in[0,T)$,
$\chi_{(\tau,T)}(\cdot)F(\cdot,y_{F,\tau,y_{0}}(\cdot))$ is the optimal
control to Problem $(NP)^{r,\tau}_{y_{0}}$.
Consequently, the map $F$ is an optimal feedback law for the family of optimal
norm control problems as follows:
$\big{\\{}(NP)^{r,\tau}_{y_{0}}\;:\;\tau\in[0,T),\,y_{0}\in
L^{2}(\Omega)\big{\\}}.$
With the aid of the equivalence theorem, we also build up two algorithms for
the optimal norm, along with the optimal control, to $(NP)^{r,\tau}$ and the
optimal time, together with the optimal control, to $(TP)^{M,r}$,
respectively. These algorithms show that the optimal norm and the optimal
control to $(NP)^{r,\tau}$ can be approximated through solving a series of
two-point boundary value problems, and the same can be said about the optimal
time and the optimal control to $(TP)^{M,r}$.
It deserves to mention that all results obtained in this paper still stand
when Equation (1.1) is replaced by
$\left\\{\begin{array}[]{lll}\partial_{t}y-\triangle
y+ay=\chi_{\omega}\chi_{(\tau,T)}u&\mbox{in}&\Omega\times(0,T),\\\
y=0&\mbox{on}&\partial\Omega\times(0,T),\\\
y(0)=y_{0}&\mbox{in}&\Omega,\end{array}\right.$ (1.6)
where $a\in L^{\infty}(\Omega\times(0,T))$ and $\Omega$ is convex (see Remark
2.13).
The equivalence between optimal time and norm control problems have been
studied in [13], [7] and [5] and the references therein. The optimal time
control problem studied in these papers is to initiate control from the
beginning such that the corresponding solution (to a controlled system)
reaches a target set in the shortest time. Though problems studied in the
current paper differ from those in [13], our study is partially inspired by
[13]. To the best of our knowledge, the equivalence theorem of the above-
mentioned three kinds of optimal control problems has not been touched upon.
Moreover, the feedback law and the algorithms established in this paper seem
to be new.
The rest of the paper is organized as follows: Section 2 presents the
equivalence theorem and its proof. Section 3 provides the above-mentioned two
algorithms. In section 4, we build up an optimal norm feedback law.
## 2 Equivalence of three optimal control problems
Throughout this section, the initial state $y_{0}$ is fixed in
$L^{2}(\Omega)$. For simplicity, we write $y(\cdot;\chi_{(\tau,T)}u)$ and
$r_{T}$ for $y(\cdot;\chi_{(\tau,T)}u,y_{0})$ and $r_{T}(y_{0})$ (which is
defined by (1.4)), respectively. The purpose of this section is to prove the
following equivalence theorem:
###### Theorem 2.1.
When $M>0$ and $\tau\in[0,T)$, the problems $(OP)^{\tau,M}$,
$(TP)^{M,r(\tau,M)}$ and $(NP)^{r(\tau,M),\tau}$ have the same optimal
control; When $r\in(0,r_{T})$ and $\tau\in[0,T)$, the problems
$(NP)^{r,\tau}$, $(OP)^{\tau,M(r,\tau)}$ and $(TP)^{M(r,\tau),r}$ have the
same optimal control; When $M>0$ and $r\in[r(0,M),r_{T})$, the problems
$(TP)^{M,r}$, $(NP)^{r,\tau(M,r)}$ and $(OP)^{\tau(M,r),M}$ have the same
optimal control.
### 2.1 Some properties on optimal target control problems
###### Lemma 2.2.
Let $M\geq 0$ and $\tau\in[0,T)$. Then, $(i)$ $(OP)^{\tau,M}$ has optimal
controls; $(ii)$ $r(\tau,M)>0$; $(iii)$ $u^{*}$ is an optimal control to
$(OP)^{\tau,M}$ if and only if $u^{*}\in L^{\infty}(0,T;L^{2}(\Omega))$, with
$u^{*}=0$ over $(\tau,T)$, satisfies
$\displaystyle\int_{0}^{T}<\chi_{(\tau,T)}(t)\chi_{\omega}p^{*}(t),u^{*}(t)>dt=\max_{v(\cdot)\in\mathcal{U}_{\tau,M}}\int_{0}^{T}<\chi_{(\tau,T)}(t)\chi_{\omega}p^{*}(t),v(t)>dt,$
(2.1)
where $p^{*}$ is the solution to the equation:
$\left\\{\begin{array}[]{lll}\partial_{t}p^{*}+\triangle
p^{*}=0&\mbox{in}&\Omega\times(0,T),\\\
p^{*}=0&\mbox{on}&\partial\Omega\times(0,T),\\\
p^{*}(T)=-(y^{*}(T)-z_{d})&\mbox{in}&\Omega\end{array}\right.$ (2.2)
with $y^{*}(\cdot)$ solving the equation:
$\left\\{\begin{array}[]{lll}\partial_{t}y^{*}-\triangle
y^{*}=\chi_{\omega}\chi_{(\tau,T)}u^{*}&\mbox{in}&\Omega\times(0,T),\\\
y^{*}=0&\mbox{on}&\partial\Omega\times(0,T),\\\
y^{*}(0)=y_{0}&\mbox{in}&\Omega.\end{array}\right.$ (2.3)
###### Proof.
$(i)$ and $(iii)$ have been proved in [9] (see the proof of Theorem 7.2,
Chapter III in [9]). The remainder is to show $(ii)$. For this purpose, we let
$u^{*}$ be an optimal control to $(OP)^{\tau,M}$ and write $y^{*}(\cdot)$ for
$y(\cdot;\chi_{(\tau,T)}u^{*},y_{0})$. Then it holds that
$r(\tau,M)=\|y^{*}(T)-z_{d}\|$ and
$\displaystyle y^{*}(T)\in\big{\\{}y(T;\chi_{(0,T)}u,y_{0})\;:\;u\in
L^{\infty}(0,T;L^{2}(\Omega))\big{\\}}.$ (2.4)
On the other hand, by the null controllability for the heat equation (see, for
instance, [3] or [6]), one can easily check that
$\big{\\{}y(T;\chi_{(0,T)}u,y_{0})\;:\;u\in
L^{\infty}(0,T;L^{2}(\Omega))\big{\\}}=\big{\\{}y(T;\chi_{(0,T)}u,0)\;:\;u\in
L^{\infty}(0,T;L^{2}(\Omega))\big{\\}}.$
This, along with (2.4) and the assumption (1.2), indicates that that
$y^{*}(T)\neq z_{d},$ which implies that $r(\tau,M)>0.$ This completes the
proof. ∎
###### Lemma 2.3.
Let $M\geq 0$ and $\tau\in[0,T)$. Then, $(i)$ $u^{*}$ is an optimal control to
$(OP)^{\tau,M}$ if and only if $u^{*}\in L^{\infty}(0,T;L^{2}(\Omega))$, with
$u^{*}=0$ over $(0,\tau)$, satisfies the following equality:
$\displaystyle
u^{*}(t)=M\frac{\chi_{\omega}p^{*}(t)}{\|\chi_{\omega}p^{*}(t)\|},\;\;\mbox{for
a.e.}\;\;t\in(\tau,T),$ (2.5)
where $p^{*}$ is the solution to (2.2), with $y^{*}(\cdot)$ solving the
equation (2.3); $(ii)$ $(OP)^{\tau,M}$ holds the bang-bang property: any
optimal control $u^{*}$ satisfies that $\|u^{*}(t)\|=M$ for a.e.
$t\in(\tau,T)$; $(iii)$ the optimal control of $(OP)^{\tau,M}$ is unique.
###### Proof.
First, the maximal condition (2.1) is equivalent to the following condition:
$\displaystyle<\chi_{\omega}p^{*}(t),u^{*}(t)>=\max_{v^{0}\in
B(0,M)}<\chi_{\omega}p^{*}(t),v^{0}>\;\;\mbox{for a.e.}\;\;t\in(\tau,T),$
(2.6)
where $B(0,M)$ is the closed ball (in $L^{2}(\Omega)$), centered at the origin
and of radius $M$. Since $p^{*}(T)=-(y^{*}(T)-z_{d})\neq 0$ (see $(ii)$ of
Lemma 2.2), it follows from the unique continuation property of the heat
equation (see [8]) that
$\displaystyle\|\chi_{\omega}p^{*}(t)\|\neq 0\;\;\mbox{for
each}\;\;t\in[0,T).$ (2.7)
Thus, the condition (2.6) is equivalent to the condition (2.5). This, along
with $(iii)$ of Lemma 2.2, yields $(i)$. Next, $(ii)$ follows at once from
(2.5). Finally, $(iii)$ follows from $(ii)$ (see [5] or [14]). This completes
the proof. ∎
###### Lemma 2.4.
Let $M\geq 0$ and $\tau\in[0,T)$. Then the two-point boundary value problem:
$\left\\{\begin{array}[]{ccll}\partial_{t}\varphi-\Delta\varphi=M\chi_{(\tau,T)}\displaystyle\frac{\chi_{\omega}\psi}{\|\chi_{\omega}\psi\|},&\partial_{t}\psi+\triangle\psi=0&\mbox{in}&\Omega\times(0,T),\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\varphi=0,&\psi=0&\mbox{on}&\partial\Omega\times(0,T),\\\ \vskip 3.0pt
plus 1.0pt minus
1.0pt\cr\varphi(0)=y_{0},&\psi(T)=-(\varphi(T)-z_{d})&\mbox{in}&\Omega\end{array}\right.$
(2.8)
admits a unique solution $(\varphi^{\tau,M},\psi^{\tau,M})$ in
$C([0,T];L^{2}(\Omega))\times C([0,T];L^{2}(\Omega))$. Furthermore, the
control, defined by
$\displaystyle
u^{\tau,M}(t)=M\chi_{(\tau,T)}(t)\frac{\chi_{\omega}\psi^{\tau,M}(t)}{\|\chi_{\omega}\psi^{\tau,M}(t)\|},\;\;t\in[0,T),$
(2.9)
is the optimal control to $(OP)^{\tau,M}$, while $\varphi^{\tau,M}$ is the
corresponding optimal state. Consequently, it holds that
$\|\varphi^{\tau,M}(T)-z_{d}\|=r(\tau,M).$ (2.10)
###### Proof.
By Lemma 2.2, $(OP)^{\tau,M}$ has an optimal control $u^{*}$. Let $y^{*}$ and
$p^{*}$ be the corresponding solutions to Equation (2.3) and Equation (2.2),
respectively. Clearly, they belong to $C([0,T];L^{2}(\Omega))$. It follows
from $(i)$ of Lemma 2.3 that $u^{*}$ satisfies (2.5). This, together with
(2.2) and (2.3), shows that $(y^{*},p^{*})$ solves Equation (2.8).
Next, we prove the uniqueness. Suppose that $(\varphi_{1},\psi_{1})$ and
$(\varphi_{2},\psi_{2})$ are two solutions of Equation (2.8). Define $u_{1}$
and $u_{2}$ by (2.5), where $p^{*}$ is replaced by $\psi_{1}$ and $\psi_{2}$,
respectively. It follows from $(i)$ of Lemma 2.3 that $u_{1}$ and $u_{2}$ are
the optimal control to $(OP)^{\tau,M}$ and
$\varphi_{i}(\cdot)=y(\cdot;\chi_{(\tau,T)}u_{i})$, $i=1,2$. Then by $(iii)$
of Lemma 2.3, $\varphi_{1}=\varphi_{2}$. Thus, it holds that
$\psi_{1}(T)=\psi_{2}(T)$, from which, it follows that $\psi_{1}=\psi_{2}$.
Finally, if $(\varphi^{\tau,M},\psi^{\tau,M})$ is the solution of Equation
(2.8), then it follows from $(i)$ of Lemma 2.3 that $u^{\tau,M}$ (defined by
(2.9)) and $\varphi^{\tau,M}$ are the optimal control and the optimal state to
$(OP)^{\tau,M}$. This completes the proof. ∎
###### Remark 2.5.
The unique continuation property (2.7) for the adjoint equation plays a very
important role in this paper. This property also holds for the adjoint
equation of Equation (1.6), where $\Omega$ is convex (see [12]). With the help
of this fact, one can easily check that all results in previous lemmas still
stand when the controlled system is Equation (1.6).
### 2.2 Equivalence of optimal target and norm control problems
###### Lemma 2.6.
Let $\tau\in[0,T)$. Then the map $M\rightarrow r(\tau,M)$ is strictly
monotonically decreasing and Lipschitz continuous from $[0,\infty)$ onto
$(0,r_{T}]$. Furthermore, it holds that
$\displaystyle r=r(\tau,M(r,\tau))\;\;\mbox{for each}\;\;r\in(0,r_{T}]$ (2.11)
and
$\displaystyle M=M(r(\tau,M),\tau)\;\;\mbox{for each}\;\;M\geq 0.$ (2.12)
Consequently, for each $\tau\in[0,T)$, the maps $M\rightarrow r(\tau,M)$ and
$r\rightarrow M(r,\tau)$ are the inverse of each other.
###### Proof.
The proof will be carried out by several steps as follows:
Step 1. It holds that $r(\tau,0)=r_{T}$ and
$\lim_{M\rightarrow\infty}r(\tau,M)=0$.
The first equality above follows directly from the definitions of $r_{T}$ and
$r(\tau,0)$. Now, we prove the second one. Let $\varepsilon>0$. By the
approximate controllability for the heat equation (see [4]), there is a
control $u_{\varepsilon}\in L^{\infty}(\tau,T;L^{2}(\Omega))$ such that
$y(T;\chi_{(\tau,T)}u_{\varepsilon})\in B(z_{d},\varepsilon)$. Clearly,
$u_{\varepsilon}\in\mathcal{U}_{\tau,M}$ for all
$M\geq\|u_{\varepsilon}\|_{L^{\infty}(\tau,T;L^{2}(\Omega))}$. Then, by the
optimality of $r(\tau,M)$, we deduce that
$r(\tau,M)\leq\|y(T;\chi_{(\tau,T)}u_{\varepsilon})-z_{d}\|\leq\varepsilon$
for each $M\geq\|u_{\varepsilon}\|_{L^{\infty}(\tau,T;L^{2}(\Omega))},$ from
which, it follows that $\lim_{M\rightarrow\infty}r(\tau,M)=0$.
Step 2. The map $M\rightarrow r(\tau,M)$ is strictly monotonically decreasing.
Let $0\leq M_{1}<M_{2}$. We claim that $r(\tau,M_{2})<r(\tau,M_{1})$. Seeking
for a contradiction, suppose that $r(\tau,M_{2})\geq r(\tau,M_{1})$. Then
optimal control $u_{1}$ to $(OP)^{\tau,M_{1}}$ would satisfy that
$\|y(T;\chi_{(\tau,T)}u_{1})-z_{d}\|=r(\tau,M_{1})\leq r(\tau,M_{2})$ and
$u_{1}\in\mathcal{U}_{\tau,M_{1}}\subset\mathcal{U}_{\tau,M_{2}}.$ These yield
that $u_{1}$ is the optimal control to $(OP)^{\tau,M_{2}}$. By the bang-bang
property of $(OP)^{\tau,M_{2}}$ (see $(ii)$ of Lemma 2.3), it holds that
$\|u_{1}(t)\|=M_{2}$ for a.e. $t\in(\tau,T)$. This contradicts to that
$u_{1}\in\mathcal{U}_{\tau,M_{1}}$, since $M_{1}<M_{2}$.
Step 3. The map $M\rightarrow r(\tau,M)$ is Lipschitz continuous.
Let $M_{1},M_{2}\in[0,\infty)$. Without loss of generality, we can assume that
$0\leq M_{1}<M_{2}$. Let $u^{*}$ be optimal control to $(OP)^{\tau,M_{2}}$.
Then by the monotonicity of the map $M\rightarrow r(\tau,M)$ and the
optimality of $u^{*}$ to $(OP)^{\tau,M_{2}}$, we see that
$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\cr&\displaystyle
r(\tau,M_{1})>r(\tau,M_{2})=\left\|e^{T\triangle}y_{0}+\int^{T}_{\tau}e^{(T-s)\triangle}u^{*}(s)ds-
z_{d}\right\|\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\geq&\displaystyle\left\|e^{T\triangle}y_{0}+\int^{T}_{\tau}e^{(T-s)\triangle}\frac{M_{1}}{M_{2}}u^{*}(s)ds-
z_{d}\right\|-\frac{(M_{2}-M_{1})}{M_{2}}\left\|\int^{T}_{\tau}e^{(T-s)\triangle}u^{*}(s)ds\right\|.\end{array}$
Since $\displaystyle\frac{M_{1}}{M_{2}}u^{*}\in\mathcal{U}_{\tau,M_{1}}$, it
follows from the definition of $r(\tau,M_{1})$ that
$\displaystyle\left\|e^{T\triangle}y_{0}+\int^{T}_{\tau}e^{(T-s)\triangle}\frac{M_{1}}{M_{2}}u^{*}(s)ds-
z_{d}\right\|\geq r(\tau,M_{1}).$
Because $\|u^{*}\|_{L^{\infty}(\tau,T;L^{2}(\Omega))}\leq M_{2}$, we find that
$\int_{\tau}^{T}\|e^{(T-s)\triangle}\|\|u^{*}(s)\|ds\leq M_{2}(T-\tau).$
Putting the above three estimates together leads to the estimate as follows:
$r(\tau,M_{1})>r(\tau,M_{2})\geq r(\tau,M_{1})-(M_{2}-M_{1}),$
from which, it follows that
$|r(\tau,M_{1})-r(\tau,M_{2})|\leq|M_{1}-M_{2}|(T-\tau)\;\;\mbox{for
all}\;\;M_{1},M_{2}\in[0,\infty).$
Step 4. The proof of (2.11)
First of all, by the definitions of $r_{T}$ , one can easily check that
$M(r_{T},\tau)=0$ and
$\displaystyle r_{T}=r(\tau,0)=r(\tau,M(r_{T},\tau)).$ (2.13)
Then, let $r\in(0,r_{T})$. By Step 2, $M(r,\tau)>0$ for this case. We are
going to prove the following two claims:
Claim one: $r\geq r(\tau,M(r,\tau))$ and Claim two: $r\leq r(\tau,M(r,\tau))$.
Clearly, these claims, together with (2.13), lead to (2.11). To prove the
first claim, we let $u$ be an optimal control to $(NP)^{r,\tau}$ (the
existence of such a control is provided in [4]). Then it holds that
$\|y(T;\chi_{(\tau,T)}u)-z_{d}\|\leq r$ and
$u\in\mathcal{U}_{\tau,M(r,\tau)}$. These, along with the definition of
$r(\tau,M)$, shows Claim one.
Now we show the second claim. Seeking a contradiction, suppose that
$r>r(\tau,M(r,\tau))$. Since the map $M\rightarrow r(\tau,M)$ is continuous
and strictly monotonically decreasing, there would be a
$M_{1}\in(0,M(r,\tau))$ such that $r(\tau,M_{1})=r$. Thus, the optimal control
$u_{1}$ to $(OP)^{\tau,M_{1}}$ satisfies that
$\displaystyle\|u_{1}\|_{L^{\infty}(\tau,T;L^{2}(\Omega))}=M_{1}<M(r,\tau)\;\;\mbox{and}\;\;\|y(T;\chi_{(\tau,T)}u_{1})-z_{d}\|=r(\tau,M_{1})=r.$
(2.14)
The second equality in (2.14) implies that $u_{1}\in\mathcal{U}_{r,\tau}$,
which, together with the optimality of $M(r,\tau)$, indicates that
$M(r,\tau)\leq\|u_{1}\|_{L^{\infty}(\tau,T;L^{2}(\Omega))}$. This contradicts
to the first inequality in (2.14).
Step 5. The proof of (2.12).
One can easily check that $r(\tau,M)\in(0,r_{T}]$ whenever $M\geq 0$ and
$\tau\in[0,T)$. Thus, we can make use of (2.11) to get that
$\displaystyle r(\tau,M)=r(\tau,M(r(\tau,M),\tau)),\;\;M\geq 0,\tau\in[0,T).$
(2.15)
Since the map $M\rightarrow r(\tau,M)$ is strictly monotone, (2.12) follows
from (2.15) at once.
In summary, we complete the proof. ∎
###### Proposition 2.7.
$(i)$ The optimal control to $(OP)^{\tau,M}$, where $M\geq 0$ and
$\tau\in[0,T)$, is an optimal control to $(NP)^{r(\tau,M),\tau}$. $(ii)$ Any
optimal control to $(NP)^{r,\tau}$, where $\tau\in[0,T)$ and $r\in(0,r_{T}]$,
is the optimal control to $(OP)^{\tau,M(r,\tau)}$. $(iii)$ For each
$\tau\in[0,T)$ and each $r\in(0,r_{T}]$, $(NP)^{r,\tau}$ holds the bang-bang
property (i.e., any optimal control $u^{*}$ satisfies that
$\|u^{*}(t)\|=M(r,\tau)$ for a.e. $t\in(\tau,T)$) and the optimal control to
$(NP)^{r,\tau}$ is unique.
###### Proof.
$(i)$ The optimal control $u$ to $(OP)^{\tau,M}$ satisfies that
$y(T;\chi_{(\tau,T)}u)\in B(z_{d},r(\tau,M))$,
$\|u\|_{L^{\infty}(\tau,T;L^{2}(\Omega))}=M$ and $u=0$ over $(0,\tau)$. These,
together with (2.12), indicate that $u$ is an optimal control to
$(NP)^{r(\tau,M),\tau}$. $(ii)$ An optimal control $v$ to $(NP)^{r,\tau}$,
where $\tau\in[0,T)$ and $r\in(0,r_{T}]$, satisfies that
$\|v\|_{L^{\infty}(\tau,T;L^{2}(\Omega))}=M(r,\tau)$,
$\|y(T;\chi_{(\tau,T)}v)-z_{d}\|\leq r$ and $v=0$ over $(0,\tau)$. These,
along with (2.11), yields that that $v$ is the optimal control to
$(OP)^{\tau,M(r,\tau)}$. $(iii)$ The bang-bang property and the uniqueness of
$(NP)^{r,\tau}$ follow from $(ii)$ and Lemma 2.3. This completes the proof.
∎
### 2.3 Equivalence of optimal norm and time control problems
###### Lemma 2.8.
Let $r\in(0,r_{T})$ and $M\geq M(r,0)$. Then, $(TP)^{M,r}$ has optimal
controls. Moreover, it holds that $\tau(M,r)<T$.
###### Proof.
We first claim that when $u\in\mathcal{U}_{M,r}$, the supremum in (1.3) can be
reached, i.e.
$\displaystyle y(T;\chi_{(\widetilde{\tau}(u),T)}u)\in
B(z_{d},r)\;\mbox{and}\;u\in\mathcal{U}_{\widetilde{\tau}(u),M}\;\;\mbox{when
}\;\;u\in\mathcal{U}_{M,r}.$ (2.16)
Here, we simply write $\widetilde{\tau}(u)$ for $\widetilde{\tau}_{M,r}(u)$,
which is defined by (1.3). To this end, we let $u\in\mathcal{U}_{M,r}$. Then
by the definition of $\widetilde{\tau}(u)$, there is a sequence
$\\{\tau_{n}\\}\subset[0,T)$ such that
$\tau_{n}\rightarrow\widetilde{\tau}(u)$, $y(T;\chi_{(\tau_{n},T)}u)\in
B(z_{d},r)$ and $u\in\mathcal{U}_{\tau_{n},M}$. From these, (2.16) follows at
once.
Next we notice that $(NP)^{r,0}$ has optimal controls (see [4]) and any
optimal control to $(NP)^{r,0}$ belongs to
$\mathcal{U}_{M(r,0),r}\subset\mathcal{U}_{M,r}$ (since $M\geq M(r,0)$). These
imply that $\mathcal{U}_{M,r}\neq\emptyset$. Thus, there is a sequence
$\\{u_{n}\\}\subset\mathcal{U}_{M,r}$ such that
$\widetilde{\tau}(u_{n})\rightarrow\tau(M,r)$. On the other hand, by (2.16),
$y(T;\chi_{(\widetilde{\tau}(u_{n}),T)}u_{n})\in B(z_{d},r)$ and
$u_{n}\in\mathcal{U}_{\widetilde{\tau}(u_{n}),M}$. Hence, there exist a
subsequence of $\\{u_{n}\\}$, still denoted in the same way, and a control
$v^{*}\in L^{\infty}(0,T;L^{2}(\Omega))$ such that
$\chi_{(\widetilde{\tau}(u_{n}),T)}u_{n}\rightarrow\chi_{(\tau(M,r),T)}v^{*}\;\;\mbox{weakly
star in}\;L^{\infty}(0,T;L^{2}(\Omega))$
and
$y(T;\chi_{(\widetilde{\tau}(u_{n}),T)}u_{n})\rightarrow
y(T;\chi_{(\tau(M,r),T)}v^{*}).$
From these, it follows that $y(T;\chi_{(\tau(M,r),T)}v^{*})\in B(z_{d},r)$ and
$v^{*}\in\mathcal{U}_{\tau(M,r),M}$. Hence, $\chi_{(\tau(M,r),T)}v^{*}$ is an
optimal control to $(TP)^{M,r}$.
Finally, since $r<r_{T}$ and $y(T;\chi_{(\tau(M,r),T)}v^{*})\in B(z_{d},r)$,
it follows that $\tau(M,r)<T$. This completes the proof. ∎
###### Lemma 2.9.
Let $r\in(0,r_{T})$. Then the map $\tau\rightarrow M(r,\tau)$ is strictly
monotonically increasing and continuous from $[0,T)$ onto
$[M(0,\tau),\infty)$. Furthermore, it holds that
$\displaystyle M=M(r,\tau(M,r))\;\;\mbox{for each}\;\;M\in[M(r,0),\infty)$
(2.17)
and
$\displaystyle\tau=\tau(M(r,\tau),r)\;\;\mbox{for each}\;\;\tau\in[0,T).$
(2.18)
Consequently, the maps $\tau\rightarrow M(r,\tau)$ and $M\rightarrow\tau(M,r)$
are the inverse of each other.
###### Proof.
We carry out the proof by several steps as follows:
Step 1. This map is strictly monotonically increasing over $[0,T)$.
Let $0\leq\tau_{1}<\tau_{2}<T$. We claim that $M(r,\tau_{1})<M(r,\tau_{2})$.
Seeking for a contradiction, suppose that $M(r,\tau_{2})\leq M(r,\tau_{1})$.
Then the optimal control $u_{2}$ to $(NP)^{r,\tau_{2}}$ would satisfy
$\displaystyle\|\chi_{(\tau_{2},T)}u_{2}\|_{L^{\infty}(0,T;L^{2}(\Omega))}=M(r,\tau_{2})\leq
M(r,\tau_{1})\;\;\mbox{and}\;\;y(T;\chi_{(\tau_{2},T)}u_{2})\in B(z_{d},r).$
These imply that $\chi_{(\tau_{2},T)}u_{2}$ is the optimal control to
$(NP)^{r,\tau_{1}}$. Then, it follows from the bang-bang property of
$(NP)^{r,\tau_{1}}$ (see Proposition 2.7) that
$\|\chi_{(\tau_{2},T)}u_{2}(t)\|=M(r,\tau_{1})$ over $(\tau_{1},\tau_{2})$.
This contradicts to the facts that $\tau_{1}<\tau_{2}$ and $M(r,\tau_{1})>0$
(which follows from $r<r_{T}$).
Step 2. $0\leq\tau_{1}<\tau_{2}<\cdots<\tau_{n}\rightarrow\tau<T\Rightarrow
M(r,\tau_{n})\rightarrow M(r,\tau)$.
If this did not hold, then by the monotonicity of the map $\tau\rightarrow
M(r,\tau)$, we would have
$\displaystyle\lim_{n\rightarrow\infty}M(r,\tau_{n})=M(r,\tau)-\delta\;\;\mbox{for
some}\;\;\delta>0.$ (2.19)
Let $u_{n}$ and $y_{n}$ be the optimal control and the optimal state to
$(OP)^{\tau_{n},M(r,\tau_{n})}$, respectively. Then, it follows from Lemma 2.2
that
$\displaystyle\int_{0}^{T}<\chi_{\omega}p_{n},\chi_{(\tau_{n},T)}u_{n}>dt\geq\int_{0}^{T}<\chi_{\omega}p_{n},\chi_{(\tau_{n},T)}v_{n}>dt\;\;\mbox{for
each}\;\;v_{n}\in\mathcal{U}_{\tau_{n},M(r,\tau_{n})},$ (2.20)
$\displaystyle\left\\{\begin{array}[]{ll}\partial_{t}y_{n}-\triangle
y_{n}=\chi_{\omega}\chi_{(\tau_{n},T)}u_{n}&\textrm{in }\Omega\times(0,T),\\\
y_{n}=0&\textrm{on }\partial\Omega\times(0,T),\\\ y_{n}(0)=y_{0}&\textrm{in
}\Omega,\end{array}\right.$
$\displaystyle\left\\{\begin{array}[]{ll}\partial_{t}p_{n}+\triangle
p_{n}=0&\textrm{in }\Omega\times(0,T),\\\ p_{n}=0&\textrm{on
}\partial\Omega\times(0,T),\\\ p_{n}(T)=-(y_{n}(T)-z_{d})&\textrm{in
}\Omega.\end{array}\right.$
Besides, by the optimality of $y_{n}$ and (2.11) (in Lemma 2.6), we see that
$\displaystyle\|y_{n}(T)-z_{d}\|=r(M(r,\tau_{n}),\tau_{n})=r\;\;\mbox{for
all}\;\;n\in\mathbb{N}.$ (2.23)
Since $\tau_{n}\rightarrow\tau$ and
$\|u_{n}\|_{L^{\infty}(\tau_{n},T;L^{2}(\Omega))}=M(r,\tau_{n})\leq
M(r,\tau)-\delta$, there exist a subsequence, still denoted in the same way,
and a control $\widetilde{u}\in L^{\infty}(0,T;L^{2}(\Omega))$ such that
$\displaystyle\chi_{(\tau_{n},T)}u_{n}\rightarrow\chi_{(\tau,T)}\widetilde{u}\;\;\mbox{weakly
star in}\;\;L^{\infty}(0,T;L^{2}(\Omega)).$ (2.24)
This, together with the equations satisfied by $y_{n}$ and $p_{n}$
respectively, indicates that
$\displaystyle
y_{n}\rightarrow\widetilde{y}\;\;\mbox{and}\;\;p_{n}\rightarrow\widetilde{p}\;\;\mbox{in}\;\;C([0,T];L^{2}(\Omega)),$
(2.25)
$\displaystyle\left\\{\begin{array}[]{ll}\partial_{t}\widetilde{y}-\triangle\widetilde{y}=\chi_{\omega}\chi_{(\tau,T)}\widetilde{u}&\textrm{in
}\Omega\times(0,T),\\\ \widetilde{y}=0&\textrm{on
}\partial\Omega\times(0,T),\\\ \widetilde{y}(0)=y_{0}&\textrm{in
}\Omega\end{array}\right.$ (2.29)
and
$\displaystyle\left\\{\begin{array}[]{ll}\partial_{t}\widetilde{p}+\triangle\widetilde{p}=0&\textrm{in
}\Omega\times(0,T),\\\ \widetilde{p}=0&\textrm{on
}\partial\Omega\times(0,T),\\\
\widetilde{p}(T)=-(\widetilde{y}(T)-z_{d})&\textrm{in
}\Omega.\end{array}\right.$ (2.33)
In addition, it follows from (2.23) and (2.25) that
$\|\widetilde{y}(T)-z_{d}\|=r$. By making use of (2.11) again, we deduce that
$\displaystyle\|\widetilde{y}(T)-z_{d}\|=r(\tau,M(r,\tau)).$ (2.34)
Now, we take a $v\in\mathcal{U}_{\tau,M(r,\tau)-\delta}$. Since
$M(r,\tau)-\delta>0$, it holds that
$\frac{M(r,\tau_{n})}{M(r,\tau)-\delta}\chi_{(\tau_{n},T)}v\in\mathcal{U}_{\tau_{n},M(r,\tau_{n})}.$
Then, it follows from (2.20) that
$\displaystyle\int_{0}^{T}<\chi_{\omega}p_{n},\chi_{(\tau_{n},T)}u_{n})>dt\geq\int_{0}^{T}<\chi_{\omega}p_{n},\frac{M(r,\tau_{n})}{M(r,\tau)-\delta}\chi_{(\tau_{n},T)}v>dt.$
By (2.19), (2.24) and (2.25), we can pass to the limit in the above to get
that
$\displaystyle\int_{0}^{T}<\chi_{\omega}\widetilde{p}\;,\chi_{(\tau,T)}\widetilde{u}>dt\geq\int_{0}^{T}<\chi_{\omega}\widetilde{p}\;,\chi_{(\tau,T)}v>dt\;\;\mbox{for
all}\;\;v\in\mathcal{U}_{\tau,M(r,\tau)-\delta,}.$
This, along with the fact that
$\widetilde{u}\in\mathcal{U}_{\tau,M(r,\tau)-\delta,}$ (which follows from
(2.24)), indicates that
$\displaystyle\int_{0}^{T}<\chi_{\omega}\widetilde{p}\;,\chi_{(\tau,T)}\widetilde{u}>dt=\displaystyle{\max_{v\in\mathcal{U}_{\tau,M(r,\tau)-\delta,}}}\int_{0}^{T}<\chi_{\omega}\widetilde{p}\;,\chi_{(\tau,T)}>dt.$
According to Lemma 2.2, the above equality, together with (2.29) and (2.33),
shows that $\chi_{(\tau,T)}\widetilde{u}$ and $\widetilde{y}$ are the optimal
control and the optimal state to $(OP)^{\tau,M(r,\tau)-\delta,}$. Therefore,
it stands that $\|\widetilde{y}(T)-z_{d}\|=r(\tau,M(r,\tau)-\delta),$ which,
combined with (2.34), indicates that
$r(\tau,M(r,\tau))=r(\tau,M(r,\tau)-\delta).$ This contradicts with the strict
monotonicity of the map $M\rightarrow r(\tau,M)$ (see Lemma 2.6).
Step 3. $T>\tau_{1}>\cdots>\tau_{n}\rightarrow\tau\geq 0\Rightarrow
M(r,\tau_{n})\rightarrow M(r,\tau)$.
If this did not hold, then by the monotonicity of the map $\tau\rightarrow
M(r,\tau)$, we would have that
$\lim_{n\rightarrow\infty}M(r,\tau_{n})=M(r,\tau)+\delta\;\;\mbox{for
some}\;\;\delta>0.$ Following the same argument as that in Step 2, we can
derive that $r(\tau,M(r,\tau))=r(\tau,M(r,\tau)+\delta).$ This contradicts to
the strict monotonicity of the map $M\rightarrow r(\tau,M)$.
Step 4. $\lim_{\tau\rightarrow T}M(r,\tau)=\infty.$
Seeking for a contradiction, we suppose that
$0<\tau_{1}<\cdots<\tau_{n}\rightarrow T$ and
$\lim_{n\rightarrow\infty}M(r,\tau_{n})=M<\infty$. Let $u_{n}$ and $y_{n}$ be
the optimal control and state for $(NP)^{r,\tau_{n}}.$ Then we would have that
$\chi_{(\tau_{n},T)}u_{n}\rightarrow 0$ weakly star in
$L^{\infty}(0,T;L^{2}(\Omega))$ and $y_{n}(\cdot)\rightarrow y(\cdot;0)$ in
$C([0,T];L^{2}(\Omega))$. Thus, it holds that
$r_{T}\equiv\|y(T;0)-z_{d}\|=\lim_{n\rightarrow\infty}\|y_{n}(T)-z_{d}\|\leq
r$, which contradicts to the assumption that $r<r_{T}$.
Step 5. The proof of (2.17)
By Lemma 2.8, the problem $(TP)^{M,r}$ has an optimal control $u$. It holds
that
$\displaystyle y(T;\chi_{(\tau(M,r),T)}u)\in B(z_{d},r)\;\;\mbox{
and}\;\;\|u\|_{L^{\infty}(\tau(M,r),T;L^{2}(\Omega))}\leq M.$ (2.35)
From the first fact in (2.35), we see that $u\in\mathcal{U}_{r,\tau(M,r)}$.
This, together with the optimality of $M(r,\tau)$ and the second fact in
(2.35), shows that
$\displaystyle M\geq M(r,\tau(M,r)).$ (2.36)
Seeking for a contradiction, suppose that $M>M(r,\tau(M,r))$. Since the map
$\tau\rightarrow M(r,\tau)$ is continuous and strictly monotonically
increasing, there would be a $\tau_{1}$, with $\tau_{1}\in(\tau(M,r),T)$, such
that $M(r,\tau_{1})=M$. Clearly, the optimal control $u_{1}$ to
$(NP)^{r,\tau_{1}}$ satisfies that
$\displaystyle\|u_{1}\|_{L^{\infty}(\tau_{1},T;L^{2}(\Omega))}=M(r,\tau_{1})=M\;\;\mbox{and}\;\;y(T;\chi_{(\tau_{1},T)}u_{1})\in
B(z_{d},r).$ (2.37)
From these, it follows that $u_{1}\in\mathcal{U}_{M,r}$. Then, by the
optimality of $\tau(M,r)$ , (1.3) and (2.37), we deduce that
$\tau(M,r)\geq\widetilde{\tau}(u_{1})\geq\tau_{1},$ which contradicts with
that $\tau_{1}\in(\tau(M,r),T)$.
Step 6. The proof of (2.18).
Let $\tau\in[0,T)$. By Step 1, it follows that $M(r,\tau)\geq M(r,0)$. Then we
can apply (2.17) to deduce that $M(r,\tau)=M(r,\tau(M(r,\tau),r))$. By making
use of Step 1 again, we obtain that $\tau=\tau(M(r,\tau),r)$.
In summary, we complete the proof. ∎
###### Proposition 2.10.
$(i)$ Any optimal control to $(TP)^{M,r}$, where $r\in(0,r_{T})$ and $M\geq
M(r,0)$, is the optimal control to $(NP)^{r,\tau(M,r)}$. $(ii)$ The optimal
optimal control to $(NP)^{r,\tau}$, with $\tau\in[0,T)$ and $r\in(0,r_{T})$,
is an optimal control to $(TP)^{M(r,\tau),r}$. $(iii)$ For each
$r\in(0,r_{T})$ and each $M\geq M(r,0)$, $(TP)^{M,r}$ holds the bang-bang
property (i.e., any optimal control $u^{*}$ satisfies that $\|u^{*}(t)\|=M$
for a.e. $t\in(\tau(M,r),T)$) and the optimal control to $(TP)^{M,r}$ is
unique.
###### Proof.
$(i)$ An optimal control $u$ to $(TP)^{M,r}$ satisfies that $u=0$ over
$(\tau(M,r),T)$,
$y(T;\chi_{(\tau(M,r),T)}u)\in B(z_{d},r)\;\mbox{ and
}\;\|u\|_{L^{\infty}(\tau(M,r),T;L^{2}(\Omega))}\leq M.$
These, together with (2.17), yields that $u$ is the optimal control to
$(NP)^{r,\tau(M,r)}$. $(ii)$ The optimal control $v$ to $(NP)^{r,\tau}$
satisfies that $v=0$ over $(\tau,T)$,
$\|u\|_{L^{\infty}(\tau,T;L^{2}(\Omega))}=M(r,\tau)$ and
$y(T;\chi_{(\tau,T)}u)\in B(z_{d},r)$. These, together with (2.18), yields
that $u$ is an optimal control to $(TP)^{M(r,\tau),r}$. $(iii)$ The bang-bang
property and the uniqueness of $(TP)^{M,r}$ follow from $(iii)$ of Proposition
2.7 and $(i)$ above. This completes the proof. ∎
### 2.4 Equivalence of optimal target and time control problems
Though the equivalence between optimal target and time control problems can be
derived from Proposition 2.7 and Proposition 2.10, the properties of maps
$\tau\rightarrow r(\tau,M)$ and $r\rightarrow\tau(M,r)$ are independently
interesting and will be used in the next section. This is why we introduce
what follows.
###### Lemma 2.11.
Let $M>0$. Then the map $\tau\rightarrow r(\tau,M)$ is strictly monotonically
increasing and continuous from $[0,T)$ onto $[r(0,M),r_{T})$. Furthermore, it
holds that
$\displaystyle r=r(\tau(M,r),M)\;\;\mbox{for each}\;\;r\in[r(0,M),r_{T}),$
(2.38) $\displaystyle\tau=\tau(M,r(\tau,M))\;\;\mbox{for
each}\;\;\tau\in[0,T).$ (2.39)
Consequently, the maps $\tau\rightarrow r(\tau,M)$ and $r\rightarrow\tau(M,r)$
are the inverse of each other.
###### Proof.
We carry out the proof by several steps as follows:
Step 1. The map $\tau\rightarrow r(\tau,M)$ is strictly monotonically
increasing.
Let $0\leq\tau_{1}<\tau_{2}<T$. It follows from (2.12) that
$\displaystyle M(r(\tau_{1},M),\tau_{1})=M(r(\tau_{2},M),\tau_{2}).$ (2.40)
We first claim that
$\displaystyle r(\tau_{2},M)\in(0,r_{T})\;\;\mbox{when}\;\;M>0.$ (2.41)
In fact, on one hand, it is clear that $r(\tau_{2},M)>0$ (see Lemma 2.2). On
the other hand, since the map $M\rightarrow r(\tau_{2},M)$ is strictly
monotonically decreasing (see Lemma 2.6), it holds that
$r(\tau_{2},M)<r(\tau_{2},0)=\|y(T;0)-z_{d}\|=r_{T}$. Then by (2.41), we can
apply Lemma 2.9 to get that
$M(r(\tau_{2},M),\tau_{2})>M(r(\tau_{2},M),\tau_{1})$. This, together with
(2.40), yields that
$\displaystyle M(r(\tau_{1},M),\tau_{1})>M(r(\tau_{2},M),\tau_{1}).$ (2.42)
Since the map $r\rightarrow M(r,\tau_{1})$ is strictly monotonically
decreasing (see Lemma 2.6), it follows from (2.42) that
$r(\tau_{1},M)<r(\tau_{2},M)$.
Step 2. The map $\tau\rightarrow r(\tau,M)$ is continuous.
Since for each $\tau\in[0,T)$, the map $r\rightarrow M(r,\tau)$ is continuous
and monotonic over $(0,r_{T})$ (see Lemma 2.6), and for each $r\in(0,r_{T})$,
the map $\tau\rightarrow M(r,\tau)$ is continuous (and monotonic) over $[0,T)$
(see Lemma 2.9), it follows that
$\displaystyle\mbox{the map}\;\;(r,\tau)\rightarrow M(r,\tau)\;\;\mbox{is
continuous over}\;\;(0,r_{T})\times[0,T).$ (2.43)
Now we prove that the map $\tau\rightarrow r(\tau,M)$ is continuous from left.
For this purpose, we let
$0\leq\tau_{1}<\tau_{2}<\cdots<\tau_{n}\rightarrow\tau<T$. Then by the
monotonicity of $\\{\tau_{n}\\}$, $\lim_{n\rightarrow\infty}r(\tau_{n},M)$
exists. Thus, it follows from (2.43) that
$\lim_{n\rightarrow\infty}M(r(\tau_{n},M),\tau_{n})=M(\lim_{n\rightarrow\infty}r(\tau_{n},M),\tau).$
On the other hand, by (2.12), it stands that
$M(r(\tau_{n},M),\tau_{n})=M=M(r(\tau,M),\tau)\;\;\mbox{for all}\;\;n.$
These yield that
$M(\lim_{n\rightarrow\infty}r(\tau_{n},M),\tau)=M(r(\tau,M),\tau)$. This,
together with the strict monotonicity of the map $r\rightarrow M(r,\tau)$ (see
Lemma 2.6), indicates that $\lim_{n\rightarrow\infty}r(\tau_{n},M)=r(\tau,M)$.
Thus, the map $\tau\rightarrow r(\tau,M)$ is continuous from left. Similarly,
we can prove that it is continuous from right.
Step 3. It holds that $\lim_{\tau\rightarrow T}r(\tau,M)=r_{T}$.
Clearly, the optimal control $u_{\tau}$ to $(OP)^{\tau,M}$ satisfies that
$\|y(T;\chi_{(\tau,T)}u_{\tau})-z_{d}\|=r(\tau,M)$ and
$\|u_{\tau}\|_{L^{\infty}(\tau,T;L^{2}(\Omega))}\leq M.$ One can easily see
that $\chi_{(\tau,T)}u_{\tau}\rightarrow
0\;\;\mbox{in}\;\;L^{\infty}(0,T;L^{2}(\Omega))$ as $\tau$ tends to $T$, from
which, it follows that $y(T;\chi_{(0,T)}u_{\tau})\rightarrow y(T;0)$ as $\tau$
tends to $T$. Therefore, it holds that
$r_{T}\equiv\|y(T;0)-z_{d}\|=\lim_{\tau\rightarrow
T}\|y(T;\chi_{(0,T)}u_{\tau})-z_{d}\|=\lim_{\tau\rightarrow T}r(\tau,M).$
Step 4. The proof of (2.38) and (2.39).
We start with proving the following:
$\displaystyle\mathcal{A}_{1}=\mathcal{A}_{2},$ (2.44)
where $\mathcal{A}_{1}=\\{(M,r):r\in(0,r_{T}),M\geq M(r,0)\\}$ and
$\mathcal{A}_{2}=\\{(M,r):M>0,r\in[r(0,M),r_{T})\\}.$ In fact, if
$(M,r)\in\mathcal{A}_{1}$, since $r>0$, it follows that $M>0$. On the other
hand, because $M\geq M(r,0)$, we can apply Lemma 2.6 to get that $r(0,M)\geq
r(0,M(r,0))=r.$ Thus, it stands that $(M,r)\in\mathcal{A}_{2}$. Similarly, we
can prove that $\mathcal{A}_{2}\subset\mathcal{A}_{1}$.
Next, it follows from (2.44) and (2.17) that $M=M(r,\tau(M,r))$ when $M>0$ and
$r\in[r(0,M),r_{T})$. This, together with (2.11), indicates that
$r(\tau(M,r),M)=r(\tau(M,r),M(r,\tau(M,r)))=r\;\;\mbox{for
each}\;r\in[r(0,M),r_{T}),$
which leads to (2.38).
Finally, because $r(\tau,M)\in(0,r_{T})$ (see (2.41)), we can make use of
(2.18) to get that $\tau(M(r(\tau,M),\tau),r(\tau,M))=\tau,$ which, along with
(2.12), gives (2.39).
In summary, we complete the proof. ∎
###### Proposition 2.12.
The optimal control to $(TP)^{M,r}$, where $M>0$ and $r\in[r(0,M),r_{T})$, is
the optimal control to $(OP)^{\tau(M,r),M}$. Conversely, the optimal control
to $(OP)^{\tau,M}$, where $M>0$ and $\tau\in[0,T)$, is the optimal control to
$(TP)^{M,r(\tau,M)}$.
This proposition can be directly derived from Lemma 2.11. Also it is a
consequence of Proposition 2.7, Proposition 2.10 and (2.44). We omit its
proof.
### 2.5 Proof of Theorem 2.1
Let $(P_{1})$ and $(P_{2})$ be two optimal control problems. By
$(P_{1})\Rightarrow(P_{2})$, we mean that the optimal control to $(P_{1})$ is
the optimal control to $(P_{2})$. The proof will be carried out by several
steps as follows:
Step 1.
$(OP)^{\tau,M}\Rightarrow(TP)^{M,r(\tau,M)}\Rightarrow(NP)^{r(\tau,M),\tau}\Rightarrow(OP)^{\tau,M}$,
$M>0$, $\tau\in[0,T)$.
$(OP)^{\tau,M}\Rightarrow(TP)^{M,r(\tau,M)}$: It follows from Proposition 2.7.
$(TP)^{M,r(\tau,M)}\Rightarrow(NP)^{r(\tau,M),\tau}$: We first claim that
$\displaystyle
r(\tau,M)\in(0,r_{T})\;\;\mbox{when}\;\;M>0\;\;\mbox{and}\;\;\tau\in[0,T).$
(2.45)
In fact, it follows from Lemma 2.2 that $r(\tau,M)>0$. On the other hand,
since $M>0$ and the map $M\rightarrow r(\tau,M)$ is strictly monotonically
decreasing (see Lemma 2.6), it holds that $r(\tau,M)<r(0,\tau)=r_{T}$. These
lead to (2.45).
We next claim that
$\displaystyle M\geq
M(r(\tau,M),0)\;\;\mbox{when}\;\;M>0\;\;\mbox{and}\;\;\tau\in[0,T).$ (2.46)
Indeed, since the map $\tau\rightarrow r(\tau,M)$ is monotonically increasing
(see Lemma 2.9), it holds that $r(0,M)\leq r(\tau,M)$. Because the map
$r\rightarrow M(r,0)$ is monotonically decreasing (see Lemma 2.6), it stands
that $M(r(0,M),0)\geq M(r(\tau,M),0)$. This, combined with (2.12), shows
(2.46). Now, by (2.45) and (2.46), we can apply Proposition 2.10, together
with (2.18), to get $(TP)^{M,r(\tau,M)}\Rightarrow(NP)^{r(\tau,M),\tau}$.
$(NP)^{r(\tau,M),\tau}\Rightarrow(OP)^{\tau,M}$: By (2.45), we can make use of
Proposition 2.7, together with (2.12), to get
$(NP)^{r(\tau,M),\tau}\Rightarrow(OP)^{\tau,M}$.
Step 2.
$(NP)^{r,\tau}\Rightarrow(OP)^{\tau,M(r,\tau)}\Rightarrow(TP)^{M(r,\tau),r}\Rightarrow(NP)^{r,\tau}$,
$r\in(0,r_{T})$, $\tau\in[0,T)$.
$(NP)^{r,\tau}\Rightarrow(OP)^{\tau,M(r,\tau)}$: It follows from Proposition
2.7.
$(OP)^{\tau,M(r,\tau)}\Rightarrow(TP)^{M(r,\tau),r}$: We first claim that
$\displaystyle
M(r,\tau)>0\;\;\mbox{when}\;\;r\in(0,r_{T})\;\;\mbox{and}\;\;\tau\in[0,T).$
(2.47)
In fact, since $r_{T}=\|y(T;0)-z_{d}\|$, it holds that $M(r_{T},\tau)=0$. On
the other hand, since $r<r_{T}$ and the map $r\rightarrow M(r,\tau)$ is
strictly monotonically decreasing (see Lemma 2.6), we see that
$M(r,\tau)>M(r_{T},\tau)$. Thus, (2.47) follows immediately. Now, by (2.47),
we can apply Proposition 2.12, along with (2.11), to derive
$(OP)^{\tau,M(r,\tau)}\Rightarrow(TP)^{M(r,\tau),r}$.
$(TP)^{M(r,\tau),r}\Rightarrow(NP)^{r,\tau}$: Since $r\in(0,r_{T})$, the map
$\tau\rightarrow M(r,\tau)$ is monotonically increasing (see Lemma 2.9). Thus,
it holds that $M(r,\tau)\geq M(r,0)$. Then we can make use of Proposition
2.10, together with (2.18), to yield
$(TP)^{M(r,\tau),r}\Rightarrow(NP)^{r,\tau}$.
Step 3.
$(TP)^{M,r}\Rightarrow(NP)^{r,\tau(M,r)}\Rightarrow(OP)^{\tau(M,r),M}\Rightarrow(TP)^{M,r}$,
$M>0$, $r\in[r(0,M),r_{T})$.
$(TP)^{M,r}\Rightarrow(NP)^{r,\tau(M,r)}$: It follows from (2.44) and
Proposition 2.10.
$(NP)^{r,\tau(M,r)}\Rightarrow(OP)^{\tau(M,r),M}$: Since $r>0$ in this case
(see (2.44)), we can apply Proposition 2.7, together with (2.12), to get
$(NP)^{r,\tau(M,r)}\Rightarrow(OP)^{\tau(M,r),M}$.
$(OP)^{\tau(M,r),M}\Rightarrow(TP)^{M,r}$: It follows from Proposition 2.12,
together with (2.38).
In summary, we complete the proof of Theorem 2.1.
###### Remark 2.13.
All results in this section hold for the case where the controlled system is
Equation (1.6). In fact, these results hold for the three kinds of optimal
control problems studied in this paper, when the adjoint equation of the
controlled heat equation has the unique continuation property (2.7).
## 3 Applications I: Algorithms for $M(r,\tau)$ and $\tau(M,r)$
Throughout this section, we fix an initial state $y_{0}\in L^{2}(\Omega)$ and
write $r_{T}$ for $r_{T}(y_{0})$. For each $M>0$ and $\tau\in[0,T)$,
$(\varphi^{\tau,M},\psi^{\tau,M})$ denotes the unique solution to the two-
point boundary value problem (2.8) and $\varphi^{\tau,M}$ (or
$\psi^{\tau,M})$) stands for the first (or second) component of this solution
when it appears alone.
###### Proposition 3.1.
Let $\tau\in[0,T)$ and $r\in(0,r_{T})$. Then $M^{*}$, $u^{*}$ and $y^{*}$ are
the optimal norm, the optimal control and the optimal state to $(NP)^{r,\tau}$
if and only if $M^{*}$, $u^{*}$ and $y^{*}$ satisfy that $M^{*}>0$,
$\displaystyle\|y^{*}(T)-z_{d}\|=r,$ (3.1) $\displaystyle
u^{*}(t)=M^{*}\chi_{(\tau,T)}(t)\displaystyle\frac{\chi_{\omega}\psi^{\tau,M^{*}}(t)}{\|\chi_{\omega}\psi^{\tau,M^{*}}(t)\|},\;\;t\in[\tau,T)$
(3.2)
and
$\displaystyle y^{*}(t)=\varphi^{\tau,M^{*}}(t),\;\;t\in[0,T].$ (3.3)
###### Proof.
Suppose that $M^{*}$, $u^{*}$ and $y^{*}$ are the optimal norm, the optimal
control and the optimal state to $(NP)^{r,\tau}$. Clearly, $M^{*}=M(r,\tau)$.
It follows from Lemma 2.6 that $M(r,\tau)>M(r_{T},\tau)$. Hence, $M^{*}>0$.
Then, by Theorem 2.1, $u^{*}$ and $y^{*}$ are the optimal control and the
optimal state to $(OP)^{\tau,M(r,\tau)}=(OP)^{\tau,M^{*}}$, respectively. On
the other hand, it follows from Lemma 2.4 that
$M^{*}\chi_{(\tau,T)}\displaystyle\frac{\chi_{\omega}\psi^{\tau,M^{*}}}{\|\chi_{\omega}\psi^{\tau,M^{*}}\|}$
and $y^{\tau,M^{*}}$ are also the optimal control and the optimal state to
$(OP)^{\tau,M^{*}}$. Then, by the uniqueness of the optimal control to this
problem, (3.2) and (3.3) follow at once. Besides, by the optimality of $y^{*}$
to $(OP)^{\tau,M(r,\tau)}$, we see that
$\|y^{*}(T)-z_{d}\|=r(\tau,M(r,\tau))$. This, together with (2.11), gives
(3.1).
Conversely, suppose that a triplet $(M^{*},u^{*},y^{*})$, with $M^{*}>0$,
enjoys (3.1), (3.2) and (3.3). According to Lemma 2.4, it follows from (3.2)
and (3.3) that $u^{*}$ and $y^{*}$ are the optimal control and the optimal
state to $(OP)^{\tau,M^{*}}$ and that $\|y^{*}(T)-z_{d}\|=r(M^{*},\tau)$,
which, together with (3.1), shows that $r=r(M^{*},\tau)$. Then, by Theorem
2.1, $u^{*}$ and $y^{*}$ are the optimal control and the optimal state to
$(NP)^{r(M^{*},\tau),\tau}=(NP)^{r,\tau}$. Hence,
$\|u^{*}\|_{L^{\infty}(\tau,T;L^{2}(\Omega))}=M(r,\tau)$, which, along with
(3.2), indicates that $M^{*}=M(r,\tau)$, i.e., $M^{*}$ is the optimal norm to
$(NP)^{r,\tau}$. This completes the proof. ∎
By Theorem 2.1, Lemma 2.4 and Lemma 2.11, following a very similar argument
used to prove Proposition 3.1, we can verify the following property for
$(TP)^{M,r}$.
###### Proposition 3.2.
Let $r\in(0,r_{T})$ and $M\geq M(r,0)$. Then $\tau^{*}$, $u^{*}$ and $y^{*}$
are the optimal time, the optimal control and the optimal state to
$(TP)^{M,r}$ if and only if $\tau^{*}$, $u^{*}$ and $y^{*}$ satisfy that
$\tau^{*}\in[0,T)$,
$\displaystyle\|y^{*}(T)-z_{d}\|=r,$ $\displaystyle
u^{*}(t)=M\chi_{(\tau^{*},T)}(t)\displaystyle\frac{\chi_{\omega}\psi^{\tau^{*},M}(t)}{\|\chi_{\omega}\psi^{\tau^{*},M}(t)\|},\;\;t\in(\tau^{*},T)$
and
$\displaystyle y^{*}(t)=\varphi^{\tau^{*},M}(t),\;\;t\in[0,T].$
The above two propositions not only are independently interesting, but also
hint us to find two algorithms for the optimal norm, together with the optimal
control, to $(NP)^{r,\tau}$ and the optimal time, along with the optimal
control, to $(TP)^{M,r}$, respectively. First of all, we build up,
corresponding to each $r\in(0,r_{T})$ and each $\tau\in(0,T)$, a sequence of
numbers as follows:
* •
Structure of $\\{M_{n}\\}_{n=0}^{\infty}$: Let $M_{0}>0$ be arbitrarily taken.
Let $K\in\mathbb{N}$ be such that
$K=\min\\{k:r(\tau,kM_{0})<r,k=1,2,\cdots\\}.$
( The existence of such a $K$ is guaranteed by Lemma 2.6.) Set $a_{0}=0$ and
$b_{0}=KM_{0}$. Write $M_{1}=\displaystyle\frac{a_{0}+b_{0}}{2}$. In general,
when $M_{n}=\displaystyle\frac{a_{n-1}+b_{n-1}}{2}$ with $a_{n-1}$ and
$b_{n-1}$ being given, it is defined that
$\displaystyle\\{a_{n},b_{n}\\}=\left\\{\begin{array}[]{ll}\\{M_{n},b_{n-1}\\}&\;\mbox{if}\;\;r(\tau,M_{n})>r,\\\
\\{a_{n-1},M_{n}\\}&\;\mbox{if}\;\;r(\tau,M_{n})\leq r\end{array}\right.$
and $M_{n+1}=\displaystyle\frac{a_{n}+b_{n}}{2}$.
###### Remark 3.3.
Let $\tau\in[0,T)$ and $r\in(0,r_{T})$ be given. For each $M\geq 0$, we can
determine the value $r(\tau,M)$ by solving the two-point boundary value
problem (2.8) corresponding to $(\tau,M)$, since
$r(\tau,M)=\|\varphi^{\tau,M}(T)-z_{d}\|$ (see Lemma 2.4). Clearly, $M_{1}$ is
determined by $K$. Since the map $M\rightarrow r(\tau,M)$ is strictly
monotonically decreasing and $r(\tau,M)$ tends to $0$ as $M$ goes to $\infty$
(see Lemma 2.6), $K$ can be determined by solving limited number of two-point
boundary value problems (2.8) corresponding to $(\tau,M)$ with $M=kM_{0}$,
$k=1,2,\cdots,K$. On the other hand, when $n\geq 1$ $M_{n+1}$ is determined by
$\varphi^{M_{n},\tau}$, which can be solved from (2.8) corresponding to
$(\tau,M_{n})$. In summary, we conclude that the sequence
$\\{M_{n}\\}_{n=0}^{\infty}$ can be solved from a series of two-point boundary
value problems (2.8) corresponding to $(\tau,M)$, with $M=kM_{0}$,
$k=1,2,\cdots,K$ and with $M=M_{n}$, $n=1,2,\cdots$.
###### Theorem 3.4.
Suppose that $r\in(0,r_{T})$ and $\tau\in[0,T)$. Let
$\\{M_{n}\\}_{n=0}^{\infty}$ be the sequence built up above. Let
$u_{n}=M_{n}\chi_{(\tau,T)}\displaystyle\frac{\chi_{\omega}\psi^{\tau,M_{n}}}{\|\chi_{\omega}\psi^{\tau,M_{n}}\|}$
and $u^{*}$ be the optimal control to $(NP)^{r,\tau}$. Then it holds that
$\displaystyle M_{n}\rightarrow M(r,\tau)$ (3.5)
and
$\displaystyle u_{n}\rightarrow
u^{*}\;\;\mbox{in}\;\;L^{2}(\tau,T;L^{2}(\Omega))\;\;\mbox{and
in}\;\;C([\tau,T-\delta];L^{2}(\Omega))\;\;\mbox{for
each}\;\;\delta\in(0,T-\tau).$ (3.6)
###### Proof.
For simplicity, we write $(\varphi_{n},\psi_{n})$ for the solution
$(\varphi^{\tau,M_{n}},\psi^{\tau,M_{n}})$ with $n=1,2,\cdots$. We start with
proving (3.5). From the structure of $\\{M_{n}\\}$, it follows that
$M_{n}\in[a_{n},b_{n}]\subset[a_{n-1},b_{n-1}]$ and
$b_{n}-a_{n}=\displaystyle\frac{b_{n-1}-a_{n-1}}{2}$. Thus, it holds that
$\lim_{n\rightarrow\infty}a_{n}=\lim_{n\rightarrow\infty}b_{n}=\lim_{n\rightarrow\infty}M_{n}$.
Since the map $M\rightarrow r(\tau,M)$ is continuous (see Lemma 2.6) and
$r(\tau,a_{n})>r\geq r(\tau,b_{n})$ (which follows also from the structure of
$\\{M_{n}\\}$), we find that $r(\tau,\lim_{n\rightarrow\infty}M_{n})=r$. This,
along with (2.11), indicates that
$\displaystyle r(\tau,\lim_{n\rightarrow\infty}M_{n})=r(\tau,M(r,\tau)).$
(3.7)
Then, (3.5) follows from (3.7) and the strict monotonicity of the map
$M\rightarrow r(\tau,M)$ (see Lemma 2.6).
Next, write $y^{*}(\cdot)$ and $y_{n}(\cdot)$ for the solutions
$y(\cdot;\chi_{(\tau,T)}u^{*})$ and $y(\cdot;\chi_{(\tau,T)}u_{n})$,
respectively. We claim that
$\displaystyle u_{n}\rightarrow u^{*}\;\;\mbox{weakly star
in}\;\;L^{\infty}(\tau,T;L^{2}(\Omega))\;\;\mbox{and}\;\;y_{n}\rightarrow
y^{*}\;\;\mbox{in}\;\;C([0,T];L^{2}(\Omega)).$ (3.8)
In fact, by the definitions of $u_{n}$ and $y_{n}$, it follows from Lemma 2.4
that they are the optimal control and the optimal state to
$(OP)^{\tau,M_{n}}$, respectively. We arbitrarily take subsequences of
$\\{u_{n}\\}$ and $\\{y_{n}\\}$, denoted by $\\{u_{n_{k}}^{\prime}\\}$ and
$\\{y_{n_{k}}^{\prime}\\}$, respectively. Clearly, there are subsequences
$\\{u_{n_{k}}\\}$ of $\\{u_{n_{k}}^{\prime}\\}$ and $\\{y_{n_{k}}\\}$ of
$\\{y_{n_{k}}^{\prime}\\}$ such that
$\displaystyle u_{n_{k}}\rightarrow\widetilde{u}\;\;\mbox{weakly star in
}\;L^{\infty}(\tau,T;L^{2}(\Omega))\;\;\mbox{and}\;\;y_{n_{k}}\rightarrow\widetilde{y}\;\;\mbox{in}\;\;C([0,T];L^{2}(\Omega)),$
(3.9)
where $\widetilde{y}(\cdot)=y(\cdot;\chi_{(\tau,T)}\widetilde{u})$. These,
along with (3.5) and (3.7), indicate that
$\|\widetilde{u}\|_{L^{\infty}(\tau,T;L^{2}(\Omega))}\leq\mathop{\underline{\rm
lim}}_{k\rightarrow\infty}\|u_{n_{k}}\|_{L^{\infty}(\tau,T;L^{2}(\Omega))}=\mathop{\underline{\rm
lim}}_{k\rightarrow\infty}M_{n_{k}}=M(r,\tau)$
and
$\|\widetilde{y}(T)-z_{d}\|=\lim_{k\rightarrow\infty}\|y_{n_{k}}(T)-z_{d}\|=\lim_{n\rightarrow\infty}r(\tau,M_{n_{k}})=r(\tau,\lim_{k\rightarrow\infty}M_{n_{k}})=r(\tau,M(r,\tau)).$
From these, we see that $\widetilde{u}$ and $\widetilde{y}$ are the optimal
control and the optimal state to $(OP)^{\tau,M(r,\tau)}$. Then, according to
Theorem 2.1, they are the optimal control and the optimal state to
$(NP)^{r,\tau}$. Since the optimal control to $(NP)^{r,\tau}$ is unique, (3.8)
follows from (3.9).
Now we verify the first convergence in (3.6). By the first convergence in
(3.8), we see that
$\displaystyle u_{n}\rightarrow u^{*}\;\;\mbox{weakly
in}\;\;L^{2}(\tau,T;L^{2}(\Omega)).$ (3.10)
On the other hand, according to Proposition 3.1, it stands that
$\displaystyle
u^{*}(t)=M(r,\tau)\chi_{(\tau,T)}(t)\frac{\chi_{\omega}\psi^{\tau,M(r,\tau)}(t)}{\|\chi_{\omega}\psi^{\tau,M(r,\tau)}(t)\|},\;\;t\in[0,T)$
(3.11) $\displaystyle
y^{*}=\varphi^{\tau,M(r,\tau)}\;\;\mbox{and}\;\;\|y^{*}(T)-z_{d}\|=r.$ (3.12)
By the definition of $u_{n}$, (3.11) and (3.5), we see that
$\|u_{n}\|_{L^{2}(\tau,T;L^{2}(\Omega))}\rightarrow\|u^{*}\|_{L^{2}(\tau,T;L^{2}(\Omega))}.$
This, along with (3.10), yields the first convergence in (3.6).
Finally, we show the second convergence in (3.6). By the first equality of
(3.12) and the second convergence in (3.9), we see that
$y_{n}(T)\rightarrow\varphi^{\tau,M(r,\tau)}(T)$ strongly in $L^{2}(\Omega)$.
This, together with the equations satisfied by $\psi_{n}$ and
$\psi^{\tau,M(r,\tau)}$, respectively, indicates that
$\displaystyle\psi_{n}\rightarrow\psi^{\tau,M(r,\tau)}\;\;\mbox{in}\;\;C([0,T];L^{2}(\Omega)).$
(3.13)
Then we arbitrarily fix a $\delta\in(0,T-\tau)$. By (3.11) and by the
definition of $u_{n}$, after some simple computation, we deduce that for each
$t\in[0,T-\delta]$,
$\displaystyle\begin{array}[]{ll}\|u_{n}(t)-u^{*}(t)\|&\leq|M_{n}-M(r,\tau)|+\displaystyle\frac{2M_{n}}{\|\chi_{\omega}\psi^{\tau,M(r,\tau)}(t)\|}\|\chi_{\omega}\psi_{n}(t)-\chi_{\omega}\psi^{\tau,M(r,\tau)}(t)\|.\end{array}$
(3.15)
On the other hand, by the second equality of (3.12) and the unique
continuation property (see [8]), it follows that
$\|\chi_{\omega}\psi^{\tau,M(r,\tau)}(t)\|\neq 0$ for all $t\in[0,T)$. This,
together with the continuity of $\psi^{\tau,M(r,\tau)}(\cdot)$ over
$[0,T-\delta]$, yields that
$\displaystyle\max_{t\in[0,T-\delta]}\displaystyle\frac{1}{\|\chi_{\omega}\psi^{\tau,M(r,\tau)}(t)\|}\leq
C_{\delta}\;\;\mbox{for some positive}\;\;C_{\delta}.$ (3.16)
Now, the second convergence in (3.6) follows immediately from (3.15), (3.5),
(3.13), and (3.16). This completes the proof.
∎
We end this section by introducing an algorithm for the optimal time and the
optimal control to $(TP)^{M,r}$. For each pair $(M,r)$ with $r\in(0,r_{T})$
and $M\geq M(r,0)$, we construct a sequence
$\\{\tau_{n}\\}_{n=0}^{\infty}\subset[0,T)$ as follows.
* •
Structure of $\\{\tau_{n}\\}_{n=1}^{\infty}$: Let $a_{0}=0$ and $b_{0}=T$. Set
$\tau_{1}=\displaystyle\frac{a_{0}+b_{0}}{2}$. In general, when
$\tau_{n}=\displaystyle\frac{a_{n-1}+b_{n-1}}{2}$ with $a_{n-1}$ and $b_{n-1}$
being given, it is defined that
$\displaystyle\\{a_{n},b_{n}\\}=\left\\{\begin{array}[]{ll}\\{a_{n-1},\tau_{n}\\}&\;\mbox{if}\;\;r(\tau_{n},M)>r,\\\
\\{\tau_{n},b_{n-1}\\}&\;\mbox{if}\;\;r(\tau_{n},M)\leq r\end{array}\right.$
and $\tau_{n+1}=\displaystyle\frac{a_{n}+b_{n}}{2}$.
###### Remark 3.5.
Since $r(\tau_{n},M)=\|\varphi^{\tau_{n},M}(T)-z_{d}\|$, $\tau_{n+1}$ is
determined by $\varphi^{\tau_{n},M}$, which can be solved from (2.8)
corresponding to $\tau=\tau_{n}$.
By Theorem 2.1, Lemma 2.4, Lemma 2.11 and Proposition 3.2, following a very
similar argument to prove Theorem 3.4, we can verify the next approximation
result.
###### Theorem 3.6.
Suppose that $r\in(0,r_{T})$ and $M\geq M(r,0)$. Let
$\\{\tau_{n}\\}_{n=1}^{\infty}$ be the sequence built up above. Let
$u_{n}=M\chi_{(\tau_{n},T)}\displaystyle\frac{\chi_{\omega}\psi^{\tau_{n},M}}{\|\chi_{\omega}\psi^{\tau_{n},M}\|}$
and $u^{*}$ be the optimal control to $(TP)^{M,r}$. Then it holds that
$\displaystyle\tau_{n}\rightarrow\tau(M,r)\;\;\mbox{as}\;\;n\rightarrow\infty$
and
$\displaystyle u_{n}\rightarrow
u^{*}\;\;\mbox{in}\;\;L^{2}(\tau(M,r),T;L^{2}(\Omega))\;\;\mbox{and
in}\;\;C([\tau(M,r),T-\delta];L^{2}(\Omega))$
for each $\delta\in(0,T-\tau(M,r))$.
###### Remark 3.7.
$(i)$ From the above-mentioned two algorithms, we observe that the optimal
norm and the optimal control to $(NP)^{r,\tau}$ and the optimal time and the
optimal control to $(TP)^{M,r}$ can be numerically solved, through numerically
solving the two-point boundary value problems (2.8) with parameters $M$ and
$\tau$ suitably chosen.
$(ii)$ All results obtained in this section hold for the case where the
controlled system is Equation (1.6) (see Remark 2.13).
## 4 Application II: Optimal Normal Feedback Law
Throughout this section, we arbitrarily fix a $r>0$. We aim to build up a
feedback law for norm optimal control problems.
### 4.1 Main results
We first introduce the following controlled equation:
$\left\\{\begin{array}[]{cl}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\partial_{t}y-\triangle y=\chi_{\omega}u{}{}&{\rm
in}~{}\Omega\times(t_{0},T),\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr
y=0{}&{\rm on}~{}\partial\Omega\times(t_{0},T),\\\ \vskip 3.0pt plus 1.0pt
minus 1.0pt\cr y(t_{0})=y_{0},{}&{\rm
in}~{}\Omega\times(t_{0},T).\end{array}\right.$ (4.1)
where $(t_{0},y_{0})\in[0,T)\times L^{2}(\Omega)$. Denote by
$y(\cdot;u,t_{0},y_{0})$ the solution to Equation (4.1) corresponding to the
control $u$ and the initial data $(t_{0},y_{0})$. Then, we define the
following optimal target control and optimal norm control problems.
* •
$(OP)^{M}_{t_{0},y_{0}}$: $\inf\\{\|y(T;u,t_{0},y_{0})-z_{d}\|^{2}:u\in
L^{\infty}(t_{0},T;B(0,M))\\}$;
* •
$(NP)_{t_{0},y_{0}}$: $\inf\\{\|u\|_{L^{\infty}(t_{0},T;L^{2}(\Omega))}:u\in
L^{\infty}(t_{0},T;L^{2}(\Omega)),y(T;u,t_{0},y_{0})\in B(z_{d},r)\\}$.
Throughout this section,
* •
$\bar{u}^{M}_{t_{0},y_{0}}$ stands for the optimal control to
$(OP)^{M}_{t_{0},y_{0}}$;
* •
$N(t_{0},y_{0})$ denotes the optimal norm to $(NP)_{t_{0},y_{0}}$.
Thus $N(\cdot,\cdot)$ defines an optimal norm functional over $[0,T)\times
L^{2}(\Omega)$.
The only difference between optimal target control problems $(OP)^{0,M}$
(which was introduced in Section 1) and $(OP)^{M}_{t_{0},y_{0}}$ is that the
initial data for the first one is $(0,y_{0})$ while the initial data for the
second one is $(t_{0},y_{0})$. The same can be said about the norm optimal
control problems. Therefore, corresponding to each result about $(OP)^{0,M}$
or $(NP)^{r,0}$, obtained in Section 2 or Section 3, there is an analogous
version for $(OP)^{M}_{t_{0},y_{0}}$ or $(NP)_{t_{0},y_{0}}$.
A feedback law for the norm optimal control problems will be established, with
the aid of the equivalence between norm and target optimal controls and some
properties of $(OP)^{M}_{t_{0},y_{0}}$. Those properties are related to the
following two-point boundary value problem associated with $M\geq 0$,
$t_{0}\in[0,T)$ and $y_{0}\in L^{2}(\Omega)$:
$\left\\{\begin{array}[]{ccll}\partial_{t}y-\Delta
y=M\displaystyle\frac{\chi_{\omega}\psi}{\|\chi_{\omega}\psi\|},&\partial_{t}\psi+\triangle\psi=0&\mbox{in}&\Omega\times(t_{0},T),\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
y=0,&\psi=0&\mbox{on}&\partial\Omega\times(t_{0},T),\\\ \vskip 3.0pt plus
1.0pt minus 1.0pt\cr
y(t_{0})=y_{0},&\psi(T)=-(y(T)-z_{d})&\mbox{in}&\Omega.\end{array}\right.$
(4.2)
Similar to Lemma 2.5, for each triplet
$(M,t_{0},y_{0})\in[0,\infty)\times[0,T)\times L^{2}(\Omega)$, Equation (4.2)
has a unique solution in $C([0,T];L^{2}(\Omega))$. Throughout this section,
* •
$(\bar{y}^{M}_{t_{0},y_{0}},\bar{\psi}^{M}_{t_{0},y_{0}})$ denotes the
solution of (4.2) corresponding to $M$, $t_{0}$ and $y_{0}$;
* •
$\bar{y}^{M}_{t_{0},y_{0}}$ and $\bar{\psi}^{M}_{t_{0},y_{0}}$ denote the
first and the second component of the above solution, respectively, when one
of them appears alone.
Because of the assumption (1.2),
$\bar{\psi}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(T)=-(\bar{y}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(T)-z_{d})\neq
0$ (see the proof of Lemma 2.2). Thus, it follows from the unique continuation
property of the heat equation (see [8]) that
$\displaystyle{\chi}_{\omega}\bar{\psi}^{M(t_{0},y_{0})}_{t_{0},y_{0}}(t_{0})\neq
0\;\;\mbox{for all}\;\;(t_{0},y_{0})\in[0,T)\times L^{2}(\Omega).$ (4.3)
Now we define a feedback law $F:[0,T)\times L^{2}({\Omega})\mapsto
L^{2}({\Omega})$ by setting
$\displaystyle
F(t_{0},y_{0})=N(t_{0},y_{0})\frac{{\chi}_{\omega}\bar{\psi}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(t_{0})}{\|{\chi}_{\omega}\bar{\psi}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(t_{0})\|},\;(t_{0},y_{0})\in[0,T)\times
L^{2}({\Omega}).$ (4.4)
Because of the existence and uniqueness of the solution to (4.2), as well as
(4.3), the map $F$ is well defined. For each $(t_{0},y_{0})\in[0,T)\times
L^{2}(\Omega)$, consider the evolution equation:
$\left\\{\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\dot{y}(t)-Ay(t)=\chi_{\omega}F(t,y(t)),&t\in(t_{0},T),\\\ \vskip
6.0pt plus 2.0pt minus 2.0pt\cr y(t_{0})=y_{0},\end{array}\right.$ (4.5)
where the operator $A$ was defined in Section 1. Two main results in this
section are as follows:
###### Theorem 4.1.
For each pair $(t_{0},y_{0})\in[0,T)\times L^{2}({\Omega})$, Equation (4.5)
has a unique (mild) solution. Furthermore, this solution is exactly
$\bar{y}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(\cdot)$.
###### Theorem 4.2.
For each pair $(t_{0},y_{0})\in[0,T)\times L^{2}(\Omega)$,
$F(\cdot,y_{F}(\cdot;t_{0},y_{0}))$ is the optimal control to
$(NP)_{t_{0},y_{0}}$, where $y_{F}(\cdot;t_{0},y_{0}))$ is the unique solution
to Equation (4.5) corresponding to the initial data $(t_{0},y_{0})$.
It follows directly from Theorem 4.1 that for each $y_{0}\in L^{2}(\Omega)$
and each $\tau\in[0,T)$, the evolution equation
$\left\\{\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\dot{y}(t)-Ay(t)=\chi_{\omega}\chi_{(\tau,T)}F(t,y(t)),&t\in(0,T),\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr y(0)=y_{0}\end{array}\right.$ (4.6)
admits a unique (mild) solution, denoted by $y_{F,\tau,y_{0}}(\cdot)$. Thus,
the following result is a direct consequence of Theorem 4.2:
###### Corollary 4.3.
For each $y_{0}\in L^{2}(\Omega)$ and each $\tau\in[0,T)$,
$\chi_{(\tau,T)}(\cdot)F(\cdot,y_{F,\tau,y_{0}}(\cdot))$ is the optimal
control to Problem $(NP)^{r,\tau}$ with the initial state $y_{0}$.
### 4.2 Proof of Theorem 4.1 (Part 1): The existence of solutions
By a very similar argument to prove Lemma 2.4, we can obtain that
$\displaystyle\bar{u}^{M}_{t_{0},y_{0}}(t)=M\frac{\chi_{\omega}\bar{\psi}^{M}_{t_{0},y_{0}}(t)}{||\chi_{\omega}\bar{\psi}^{M}_{t_{0},y_{0}}(t)||},\;t\in[t_{0},T).$
(4.7)
By the uniqueness and existence of the solution to (4.2), we can easily derive
the following consequence, which, in some sense, is a dynamic programming
principle.
###### Lemma 4.4.
Let $(t_{0},y_{0})\in[0,T)\times L^{2}({\Omega})$ and $M\geq 0$. Then, for
each $s\in(t_{0},T)$,
$\displaystyle(\bar{y}^{M}_{t_{0},y_{0}},\bar{\psi}^{M}_{t_{0},y_{0}}){\bigg{|}}_{[s,T]}=\bigr{(}\bar{y}^{M}_{s,\bar{y}^{M}_{t_{0},y_{0}}(s)},\bar{\psi}^{M}_{s,\bar{y}^{M}_{t_{0},y_{0}}(s)}\bigl{)}.$
(4.8)
###### Lemma 4.5.
Let $(t_{0},y_{0})\in[0,T)\times L^{2}({\Omega})$. Then
$M=N(t_{0},y_{0})\;\;\mbox{if and only
if}\;\;\displaystyle\left\|\bar{y}^{M}_{t_{0},y_{0}}(T)-z_{d}\right\|=r\wedge\left\|e^{(T-t_{0})\triangle}y_{0}-z_{d}\right\|,$
(4.9)
where ”$\wedge$” is the symbol taking the smaller. Moreover, the control,
defined by
$\displaystyle\bar{u}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(t)=N(t_{0},y_{0})\frac{\chi_{\omega}\psi^{N(t_{0},y_{0})}_{t_{0},y_{0}}(t)}{||\chi_{\omega}\psi^{N(t_{0},y_{0})}_{t_{0},y_{0}}(t)||},\;t\in(t_{0},T),$
(4.10)
is the unique optimal control of Problem $(NP)_{t_{0},y_{0}}$.
###### Proof.
First, we show (4.9) for the case where
$\|e^{(T-t_{0})\Delta}y_{0}-z_{d}\|>r$. In this case, we can apply the
analogous version of Proposition 3.1 for Problem $(NP)_{t_{0},y_{0}}$ to get
that $M=N(t_{0},y_{0})$ if and only if
$\|\bar{y}^{M}_{t_{0},y_{0}}(T)-z_{d}\|=r$. This leads to (4.9) for this case.
Next, we prove (4.9) for the case where
$\|e^{(T-t_{0})\Delta}y_{0}-z_{d}\|\leq r$. In this case, one can easily check
that $N(t_{0},y_{0})=0$, the null control is the optimal control to
$(OP)^{0}_{t_{0},y_{0}}$, and
$\bar{y}^{0}_{t_{0},y_{0}}(\cdot)=y(\cdot;0,t_{0},y_{0})=e^{(\cdot-
t_{0})\Delta}y_{0}$ over $[t_{0},T]$. Suppose that $M=N(t_{0},y_{0})$. Then it
holds that $M=0$ and
$\|\bar{y}^{0}_{t_{0},y_{0}}(T)-z_{d}\|=\|e^{(T-t_{0})\Delta}y_{0}-z_{d}\|$.
These lead to the statement on the right hand side of (4.9). Conversely,
suppose that there is an $M_{0}\geq 0$ such that
$\displaystyle\|\bar{y}^{M_{0}}_{t_{0},y_{0}}(T)-z_{d}\|=\|e^{(T-t_{0})\Delta}y_{0}-z_{d}\|.$
(4.11)
To show the statement on the left side of (4.9), it suffices to prove that
$M_{0}=0$. By the analogous version of Lemma 2.4 for
$(OP)^{M_{0}}_{t_{0},y_{0}}$ (see (2.10)), it holds that
$\displaystyle\|\bar{y}^{M_{0}}_{t_{0},y_{0}}(T)-z_{d}\|=r_{t_{0},y_{0}}(M_{0}),$
(4.12)
where $r_{t_{0},y_{0}}(\cdot)$ corresponds to the map $M\rightarrow r(0,M)$
given in Section 1, namely,
$r_{t_{0},y_{0}}(M)=\inf\\{\|y(T;u,t_{0},y_{0})-z_{d}\|:u\in
B(0,M))\\},\;M\geq 0.$
Since the null control is the optimal control to $(OP)^{0}_{t_{0},y_{0}}$, we
find that
$r_{t_{0},y_{0}}(0)=\|y(T;0,t_{0},y_{0})-z_{d}\|=\|e^{(T-t_{0})\Delta}y_{0}-z_{d}\|.$
Along with (4.11) and (4.12), this indicates that
$\displaystyle r_{t_{0},y_{0}}(0)=r_{t_{0},y_{0}}(M_{0}).$ (4.13)
By the analogous version of Lemma 2.6 for $(OP)^{M}_{t_{0},y_{0}}$, the map
$M\rightarrow r_{t_{0},y_{0}}(M)$ is strictly monotonically decreasing. This,
together with (4.13), yields that $M_{0}=0$.
In summary, we conclude that (4.9) stands.
Finally, we prove (4.10). In the case that $\|e^{(T-t_{0})}\Delta
y_{0}-z_{d}\|>r$, according to the analogous version of Proposition 3.1 for
Problem $(NP)_{t_{0},y_{0}}$, the control defined by (4.10) is the unique
optimal control of Problem $(NP)_{t_{0},y_{0}}$. In the case where
$\|e^{(T-t_{0})}\Delta y_{0}-z_{d}\|\leq r$, it is clear that
$N(t_{0},y_{0})=0$ and the null control is the optimal control to
$(NP)_{t_{0},y_{0}}$. Hence, (4.10) holds for this case. This completes the
proof.
∎
The following result shows that the functional $N(\cdot,\cdot)$ holds the
dynamic programming principle.
###### Lemma 4.6.
Let $(t_{0},y_{0})\in[0,T)\times L^{2}({\Omega})$. Then it stands that
$\displaystyle
N(t_{0},y_{0})=N\biggr{(}s,~{}\bar{y}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(s)\biggl{)}\;\mbox{for
each}\;s\in(t_{0},T).$ (4.14)
Proof. In the case where $e^{(T-t_{0})\triangle}y_{0}\in B(z_{d},r)$, it is
clear that
$N(t_{0},y_{0})=0\;\;\mbox{and}\;\;\bar{y}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(\cdot)=e^{(\cdot-
t_{0})\triangle}y_{0}.$
Because
$e^{(T-s)\triangle}\bar{y}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(s)=e^{(T-s)\triangle}e^{(s-t_{0})\triangle}y_{0}=e^{(T-t_{0})\triangle}y_{0}\in
B(z_{d},r)\;\;\mbox{for each}\;\;s\in(t_{0},T),$
we see that
$N\bigr{(}s,~{}\bar{y}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(s)\bigl{)}=0$. Therefore
the equality (4.14) holds for this case.
In the case where $e^{(T-t_{0})\triangle}y_{0}\notin B(z_{d},r)$, it is clear
that $r\wedge\left\|e^{(T-t_{0})\triangle}y_{0}-z_{d}\right\|=r$. By the
analogous version of Proposition 3.1 for Problem $(NP)_{t_{0},y_{0}}$, it
holds that $\|\bar{y}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(T)-z_{d}\|=r$. Thus, it
follows from (4.9) that
$\displaystyle\bar{y}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(T)\in\partial
B(z_{d},r).$ (4.15)
We claim that
$e^{(T-s)\triangle}\bar{y}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(s)\notin
B(z_{d},r)\;\;\mbox{for all}\;\;s\in(t_{0},T).$ (4.16)
If (4.16) did not hold, then there would exist a $\hat{s}\in(t_{0},T)$ such
that
$\displaystyle
e^{(T-\hat{s})\triangle}\bar{y}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(\hat{s})\notin
B(z_{d},r).$ (4.17)
We construct a control $\hat{u}$ by setting
$\displaystyle\hat{u}(s)=\displaystyle\chi_{(t_{0},\hat{s})}(s)N(t_{0},y_{0})\frac{\chi_{\omega}\bar{\psi}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(s)}{\|\chi_{\omega}\bar{\psi}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(s)\|},\;\;s\in[t_{0},T).$
(4.18)
Clearly, the solution $y(\cdot;\hat{u},t_{0},y_{0})$ to (4.1), where
$u=\hat{u}$, coincides with $\bar{y}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(\cdot)$
over $[t_{0},\hat{s}]$. This, along with (4.18) and (4.17), indicates that
$\displaystyle
y(T;\hat{u},t_{0},y_{0})=e^{(T-\hat{s})\triangle}y(\hat{s};\hat{u},t_{0},y_{0})\in
B(z_{d},r).$ (4.19)
On the other hand, it follows from (4.18) that
$\displaystyle\|\hat{u}||_{L^{\infty}(t_{0},T;L^{2}(\Omega))}=N(t_{0},y_{0}).$
This, together with (4.19), yields that $\hat{u}$ is the optimal control to
$(NP)_{t_{0},y_{0}}$. However, the problem $(NP)_{t_{0},y_{0}}$ holds the
bang-bang property (it follows from the analogous version of Proposition 2.7
for $(NP)_{t_{0},y_{0}}$). This implies that $\|\hat{u}(s)\|=N(t_{0},y_{0})$
for a.e. $s\in(t_{0},T)$, which contradicts to the structure of $\hat{u}$.
Hence, (4.16) stands.
Next, by (4.8) and (4.15), we see that
$\displaystyle\bar{y}^{N(t_{0},y_{0})}_{s,\bar{y}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(s)}(T)=\bar{y}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(T)\in\partial
B(z_{d},r)\;\mbox{for each}\;s\in(t_{0},T).$
This, together with (4.16), implies that
$\displaystyle\|\displaystyle\bar{y}^{N(t_{0},y_{0})}_{s,\bar{y}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(s)}(T)-z_{d}\|=r\wedge\|e^{(T-s)\triangle}\bar{y}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(s)-z_{d}\|\;\mbox{for
each}\;s\in(t_{0},T).$ (4.20)
Now, we arbitrarily fix a $s\in(t_{0},T)$. By (4.20), we can apply (4.9), with
$t_{0}=s$ and $y_{0}=\bar{y}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(s)$, to get that
$N(t_{0},y_{0})=N(s,\bar{y}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(s)),$
which gives the equality (4.14) for the second case. In summary, we finish the
proof.
Proof of Theorem 4.1 (Part 1): The existence. It follows from (4.4) (the
definition of $F$) that
$\displaystyle\displaystyle
F(t,~{}\bar{y}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(t))=N(t,~{}\bar{y}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(t))\frac{\phi(t)}{\|\phi(t\|},\;t\in(t_{0},T),$
where
$\phi(t)={\chi}_{\omega}\bar{\psi}^{N(t,~{}\bar{y}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(t))}_{t,~{}\bar{y}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(t)}(t)$.
This, together with (4.14) and (4.8), yields that
$\displaystyle
F(t,~{}\bar{y}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(t))=N(t_{0},y_{0})\frac{{\chi}_{\omega}\bar{\psi}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(t)}{\|{\chi}_{\omega}\bar{\psi}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(t)\|}=\frac{d}{dt}\bar{y}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(t)+A\bar{y}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(t),\;t\in(t_{0},T).$
Here, we used that $\chi_{\omega}\circ\chi_{\omega}=\chi_{\omega}$. From the
above equality and the fact that
$\bar{y}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(t_{0})=y_{0}$ , it follows that
$\bar{y}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(\cdot)$ is a solution to (4.5). This
completes the proof.
### 4.3 Proof of Theorem 4.1 (Part 2): The uniqueness
The key to prove the uniqueness is showing the following properties of the
feedback law $F(\cdot,\cdot)$.
###### Proposition 4.7.
$(i)$ For each pair $(\bar{t}_{0},\bar{y}_{0})\in[0,T)\times L^{2}(\Omega)$,
there is a $\bar{\rho}>0$ such that $F(t_{0},\cdot)$ is Lipschitz continuous
in $B(\bar{y}_{0},\bar{\rho})$ uniformly with respect to
$t_{0}\in[(\bar{t}_{0}-\bar{\rho})^{+},\bar{t}_{0}+\bar{\rho}]$. $(ii)$ For
each $\bar{y}_{0}\in L^{2}(\Omega)$, $F(\cdot\,,\bar{y}_{0})$ is continuous
over $[0,T)$.
When it is proved, the uniqueness of the solution to Equation (4.5) follows
immediately from the generalized Picard-Lindelof Theorem (see [15]) and
Proposition 4.7. Consequently, the proof of Theorem 4.1 is completed.
The remainder is showing Proposition 4.7. To serve such purpose, we first
study some continuity properties of $N(\cdot,\cdot)$. These properties will be
concluded in Lemma 4.10. Two lemmas before it will play important roles in its
proof.
###### Lemma 4.8.
For each $t_{0}\in[0,T)$, the functional $N(t_{0},\cdot)$ is convex over
$L^{2}(\Omega)$.
Proof. Let $y_{0}^{1}$ and $y_{0}^{2}$ belong to $L^{2}(\Omega)$. The optimal
controls $\bar{u}^{i}$ to $(NP)_{t_{0},y_{0}^{i}}$, $i=1,2$, satisfy that
$N(t_{0},y_{0}^{i})=\|\bar{u}^{i}\|_{L^{\infty}(t_{0},T;L^{2}(\Omega))}$ and
$y(T;\bar{u}^{i},t_{0},y_{0}^{i})\in B(z_{d},r)$, $i=1,2.$ Since for each
$\lambda\in(0,1)$,
$y(T;\lambda\bar{u}^{1}+(1-\lambda)\bar{u}^{2},t_{0},\lambda
y_{0}^{1}+(1-\lambda)y_{0}^{2})=\lambda
y(T;\bar{u}^{1},t_{0},y_{0}^{1})+(1-\lambda)y(T;\bar{u}^{2},t_{0},y_{0}^{2})\in
B(z_{d},r),$
we obtain that
$\begin{array}[]{l }\vskip 3.0pt plus 1.0pt minus 1.0pt\cr N(t_{0},\lambda
y_{0}^{1}+(1-\lambda)y_{0}^{2})\leq\|\lambda\bar{u}^{1}+(1-\lambda)\bar{u}^{2}\|_{L^{\infty}(t_{0},T;L^{2}(\Omega))}\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\leq\lambda\|\bar{u}^{1}\|_{L^{\infty}(t_{0},T;L^{2}(\Omega))}+(1-\lambda)\|\bar{u}^{2}\|_{L^{\infty}(t_{0},T;L^{2}(\Omega))}\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr=\lambda
N(t_{0},y_{0}^{1})+(1-\lambda)N(t_{0},y_{0}^{2}).\end{array}.$
This completes the proof.
###### Lemma 4.9.
For each $\bar{t}_{0}\in[0,T)$ and each bounded subset $E$ of $L^{2}(\Omega)$,
the functional $N(\cdot,\cdot)$ is bounded on
$\Bigr{[}(\bar{t}_{0}-\bar{\delta})^{+},\bar{t}_{0}+\bar{\delta}\Bigl{]}\times
E$, where $\bar{\delta}=(T-\bar{t}_{0})/2$.
Proof. Write $C_{E}=\sup\\{\|y_{0}\|:y_{0}\in E\\}$. Let
$(t_{0},y_{0})\in\Bigr{[}(\bar{t}_{0}-\bar{\delta})^{+},\bar{t}_{0}+\bar{\delta}\Bigl{]}\times
E$. By the null controllability of the heat equation over
$(t_{0},\bar{t}_{0}+3\bar{\delta}/2)$ (see, for instance, [6]), there is a
control $u_{1}$ with
$\|u_{1}\|_{L^{\infty}(t_{0},\bar{t}_{0}+3\bar{\delta}/2;L^{2}(\Omega))}\leq
C_{1}\|y_{0}\|\leq C_{1}C_{E},$ (4.21)
where $C_{1}>0$ is independent of $t_{0}$ and $y_{0}$, such that
$\bar{y}\equiv y(\bar{t}_{0}+3\bar{\delta}/2;u_{1},t_{0},y_{0})=0.$
Here we used that $t_{0}\leq\bar{t}_{0}+\bar{\delta}$. Then, by the
approximate controllability of the heat equation over
$(\bar{t}_{0}+3\bar{\delta}/2,T)$ (see, for instance, [4]), there is another
control $u_{2}$ with
$\|u_{2}\|_{L^{\infty}(\bar{t}_{0}+3\bar{\delta}/2,T;L^{2}(\Omega))}\leq
C_{2},$
where $C_{2}>0$ is independent of $t_{0}$ and $y_{0}$, such that
$y(T;u_{2},\bar{t}_{0}+3\bar{\delta}/2,\bar{y})\in B(z_{d},r).$
Clearly, the control
$v\equiv\chi_{(t_{0},\bar{t}_{0}+3\bar{\delta}/2)}u_{1}+\chi_{(\bar{t}_{0}+3\bar{\delta}/2,T)}u_{2}$
satisfies that $y(T;v,t_{0},y_{0})\in B(z_{d},r)$. Therefore, it holds that
$N(t_{0},y_{0})\leq\|v\|_{L^{\infty}(t_{0},T;L^{2}(\Omega))}\leq\max\\{C_{1}C_{E},C_{2}\\}.$
This completes the proof.
###### Lemma 4.10.
$(i)$ For each $(\bar{t}_{0},\bar{y}_{0})\in[0,T)\times L^{2}(\Omega)$ and
each $\rho\in(0,1/2)$, $N(t_{0},\cdot)$ is Lipschitz continuous over
$B(\bar{y}_{0},\rho)$ uniformly w.r.t.
$t_{0}\in\Bigr{[}(\bar{t}_{0}-\bar{\delta})^{+},\bar{t}_{0}+\bar{\delta}\Bigl{]}$,
where $\bar{\delta}=(T-\bar{t}_{0})/2$; $(ii)$ For each $\bar{y}_{0}\in
L^{2}(\Omega)$, $N(\cdot,\bar{y}_{0})$ is continuous over $[0,T)$.
Proof. $(i)$ Let $(\bar{t}_{0},y_{0})\in[0,T)\times L^{2}(\Omega)$ and let
$\rho\in(0,1/2)$. We arbitrarily take two different points $y_{0}^{1}$ and
$y_{0}^{2}$ from $B(\bar{y}_{0},\rho)\subset B(\bar{y}_{0},1)$. Denote by
$\mathcal{L}$ the straight line passing through $y_{0}^{1}$ and $y_{0}^{2}$,
namely,
$\mathcal{L}\equiv\biggr{\\{}y_{0}^{\lambda}\mathop{\buildrel\Delta\over{=}}(1-\lambda)y_{0}^{1}+\lambda
y_{0}^{2}\bigm{|}\lambda\in(-\infty,+\infty)\biggl{\\}}$. Clearly,
$\mathcal{L}$ intersects with $B(\bar{y}_{0},1)$ at two different points,
denoted by $y_{0}^{\lambda_{1}}$ and $y_{0}^{\lambda_{2}}$, with
$\lambda_{1}<\lambda_{2}$. Since the segment
$\\{\,y_{0}^{\lambda}|\lambda\in[0,1]\,\\}\subseteq B(\bar{y}_{0},\rho)$ and
$B(\bar{y},\rho)\bigcap\partial B(\bar{y},1)=\emptyset$, it holds that
$\lambda_{1}<0<1<\lambda_{2}.$ Moreover, one can easily check that
$\displaystyle\frac{\|y_{0}^{1}-y_{0}^{\lambda_{1}}\|}{0-\lambda_{1}}=\frac{\|y_{0}^{2}-y_{0}^{1}\|}{1-0}=\frac{\|y_{0}^{\lambda_{2}}-y_{0}^{2}\|}{\lambda_{2}-1}.$
(4.22)
For each
$t_{0}\in\Bigr{[}(\bar{t}_{0}-\bar{\delta})^{+},\bar{t}_{0}+\bar{\delta}\Bigl{]}$,
we define a function $g_{t_{0}}(\cdot;y_{0}^{1},y_{0}^{2})$ by setting
$g_{t_{0}}(\lambda;y_{0}^{1},y_{0}^{2})=N(t_{0},(1-\lambda)y_{0}^{1}+\lambda
y_{0}^{2}),\qquad\lambda\in(-\infty,+\infty).$
Obviously, the convexity of $N(t_{0},\cdot)$ (see Lemma 4.8) implies the
convexity of $g_{t_{0}}(\cdot;y_{0}^{1},y_{0}^{2})$. By the property of convex
functions, one has that
$\displaystyle\frac{g_{t_{0}}(0;y_{0}^{1},y_{0}^{2})-g_{t_{0}}(\lambda_{1};y_{0}^{1},y_{0}^{2})}{0-\lambda_{1}}\leq\frac{g_{t_{0}}(1;y_{0}^{1},y_{0}^{2})-g_{t_{0}}(0;y_{0}^{1},y_{0}^{2})}{1-0}\leq\frac{g_{t_{0}}(\lambda_{2};y_{0}^{1},y_{0}^{2})-g_{t_{0}}(1;y_{0}^{1},y_{0}^{2})}{\lambda_{2}-1}.$
This, along with (4.22) and the nonnegativity of
$g_{t_{0}}(1;y_{0}^{1},y_{0}^{2})$, indicates that
$\displaystyle\begin{array}[]{l}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle~{}~{}~{}\frac{N(t_{0},y_{0}^{2})-N(t_{0},y_{0}^{1})}{\|y_{0}^{2}-y_{0}^{1}\|}=\frac{g_{t_{0}}(1;y_{0}^{1},y_{0}^{2})-g_{t_{0}}(0;y_{0}^{1},y_{0}^{2})}{\|y_{0}^{2}-y_{0}^{1}\|}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\leq\frac{g_{t_{0}}(\lambda_{2};y_{0}^{1},y_{0}^{2})-g_{t_{0}}(1;y_{0}^{1},y_{0}^{2})}{(\lambda_{2}-1)\|y_{0}^{2}-y_{0}^{1}\|}\leq\frac{g_{t_{0}}(\lambda_{2};y_{0}^{1},y_{0}^{2})}{(\lambda_{2}-1)\|y_{0}^{2}-y_{0}^{1}\|}\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle=\frac{g_{t_{0}}(\lambda_{2};y_{0}^{1},y_{0}^{2})}{\|y_{0}^{\lambda_{2}}-y_{0}^{2}\|}.\end{array}$
(4.26)
Two observations are as follows. The triangle inequality implies that
$\|y_{0}^{\lambda_{2}}-y_{0}^{2}\|\geq\|y_{0}^{\lambda_{2}}-\bar{y}_{0}\|-\|y_{0}^{2}-\bar{y}_{0}\|\geq
1-\rho;$
The boundedness of $N(\cdot,\cdot)$ (see Lemma 4.9) gives that for each
$t_{0}\in\Bigr{[}(\bar{t}_{0}-\bar{\delta})^{+},\bar{t}_{0}+\bar{\delta}\Bigl{]}$,
$y_{0}^{1},y_{0}^{2}\in B(\bar{y}_{0},\rho)$,
$g_{t_{0}}(\lambda_{2};y_{0}^{1},y_{0}^{2})=N(t_{0},y_{0}^{\lambda_{2}})\leq\sup\biggr{\\{}N(s_{0},z_{0})\Bigm{|}(s_{0},z_{0})\in\Bigr{[}(\bar{t}_{0}-\bar{\delta})^{+},\bar{t}_{0}+\bar{\delta}\Bigl{]}\times
B(\bar{y}_{0},1)\biggl{\\}}\equiv C.$
Along with these two observations, (4.26) yields that
${N(t_{0},y_{0}^{1})-N(t_{0},y_{0}^{2})}\leq\frac{C}{1-\rho}{\|y_{0}^{2}-y_{0}^{1}\|}\equiv
C(\rho){\|y_{0}^{2}-y_{0}^{1}\|}.$
Similarly, we can obtain ${N(t_{0},y_{0}^{2})-N(t_{0},y_{0}^{1})}\leq
C(\rho){\|y_{0}^{2}-y_{0}^{1}\|}.$ These lead to the desired Lipschitz
continuity.
$(ii)$ Let $\bar{y}_{0}\in L^{2}(\Omega)$. Arbitrarily take $\bar{t}_{0}$ from
$[0,T)$ and write $\bar{\delta}=(T-\bar{t}_{0})/2$. It suffices to show that
$N(\cdot,\bar{y}_{0})$ is continuous over
$\Bigr{[}(\bar{t}_{0}-\bar{\delta})^{+},\bar{t}_{0}+\bar{\delta}\Bigl{]}$. For
this purpose, we arbitrarily take two different $t_{0}^{1}$ and $t_{0}^{2}$
from this interval. Without lose of generality, we can assume that
$t_{0}^{1}<t_{0}^{2}$. Then by (4.14) (see Lemma 4.6), the part $(i)$ of the
current lemma and Lemma 4.9, we can easily deduce that
$\begin{array}[]{l}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr~{}~{}~{}|N(t_{0}^{1},\bar{y}_{0})-N(t_{0}^{2},\bar{y}_{0})|\\\ \vskip
3.0pt plus 1.0pt minus
1.0pt\cr=\left|N\Bigr{(}t_{0}^{2},\,\bar{y}^{N(t_{0}^{1},\bar{y}_{0})}_{t_{0}^{1},\bar{y}_{0}}(t_{0}^{2})\Bigr{)}-N(t_{0}^{2},\bar{y}_{0})\right|\leq
C\left\|\bar{y}^{N(t_{0}^{1},\bar{y}_{0})}_{t_{0}^{1},\bar{y}_{0}}(t_{0}^{2})-\bar{y}_{0})\right\|\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle=C\left\|\Bigr{[}e^{(t_{0}^{2}-t_{0}^{1})\triangle}-I\Bigl{]}\bar{y}_{0}+\int^{t_{0}^{2}}_{t_{0}^{1}}e^{(t_{0}^{2}-s)\triangle}\bar{u}^{N(t_{0}^{1},\bar{y}_{0})}_{t_{0}^{1},\bar{y}_{0}}(s)ds\right\|\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle\leq
C\left\|e^{(t_{0}^{2}-t_{0}^{1})\triangle}-I\right\|\|\bar{y}_{0}\|+C|t_{0}^{2}-t_{0}^{1}|\end{array}$
where $C$ stands for a positive constant independent of $t_{0}^{1}$ and
$t_{0}^{2}$. It varies in different contexts. Clearly, the continuity of
$N(\cdot,\bar{y}_{0})$ over
$\Bigr{[}(\bar{t}_{0}-\bar{\delta})^{+},\bar{t}_{0}+\bar{\delta}\Bigl{]}$
follows from the above inequality at once.
In summary, we finish the proof.
Next, we study some properties for the map ${\cal N}:[0,T)\times
L^{2}(\Omega)\times[0,+\infty)\mapsto L^{2}(\Omega)$ defined by
$\displaystyle{\cal
N}(t_{0},y_{0},M)=\bar{u}^{M}_{t_{0},y_{0}}(t_{0})=M\frac{\chi_{\omega}\bar{\psi}^{M}_{t_{0},y_{0}}(t_{0})}{\|\chi_{\omega}\bar{\psi}^{M}_{t_{0},y_{0}}(t_{0})\|}.$
(4.27)
###### Lemma 4.11.
$(i)$ For each $(\bar{t}_{0},\bar{y}_{0},\bar{M})\in[0,T)\times
L^{2}(\Omega)\times[0,\infty)$, there is a $\bar{\rho}>0$ such that ${\cal
N}(t_{0},\cdot,\cdot)$ is Lipschitz continuous over
$B(\bar{y}_{0},\bar{\rho})\times[(\bar{M}-\bar{\rho})^{+},\bar{M}+\bar{\rho}]$
uniformly with respect to $t_{0}\in B(\bar{t}_{0},\bar{\rho})\bigcap[0,T)$.
$(ii)$ ${\cal N}(\cdot,\bar{y}_{0},\cdot)$ is continuous over
$[0,T)\times[0,\infty)$.
Proof. $(i)$ Let $(\bar{t}_{0},\bar{y}_{0},\bar{M})\in[0,T)\times
L^{2}(\Omega)\times[0,\infty)$. The proof of the first continuity will be
carried by several steps as follows:
Step 1. For all $t_{0}\in[0,T)$, $0\leq M_{1}\leq M_{2}$ and
$y_{0}^{1},~{}y_{0}^{2}\in L^{2}(\Omega)$, it holds that
$\begin{array}[]{rl}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr&\displaystyle\|{\cal N}(t_{0},y_{0}^{1},M_{1})-{\cal
N}(t_{0},y_{0}^{2},M_{2})\|\leq|M_{1}-M_{2}|+\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr&\displaystyle\frac{4M_{1}}{\|\chi_{\omega}\bar{\psi}^{M_{1}}_{t_{0},y^{1}_{0}}\|}\left[M_{1}\|y_{0}^{1}-y_{0}^{2}\|+(\|y_{0}^{2}\|+\|z_{d}\|){|M_{1}-M_{2}|}\right]\;\;\mbox{when}\;M_{1}>0;\end{array}$
(4.28) $\|{\cal N}(t_{0},y_{0}^{1},M_{1})-{\cal
N}(t_{0},y_{0}^{2},M_{2})\|=|M_{1}-M_{2}|\;\;\mbox{when}\;M_{1}=0,$ (4.29)
The equality (4.29) follows directly from the definition of ${\cal N}$. Now we
prove (4.28). For simplification of notation, we write
$\bar{y}^{i}\mathop{\buildrel\Delta\over{=}}\bar{y}^{M_{i}}_{t_{0},y^{i}_{0}},\quad\bar{\psi}^{i}\mathop{\buildrel\Delta\over{=}}\bar{\psi}^{M_{i}}_{t_{0},y^{i}_{0}},\quad\bar{u}^{i}\mathop{\buildrel\Delta\over{=}}\bar{u}^{M_{i}}_{t_{0},y^{i}_{0}},\qquad
i=1,2.$
It is clear that that
$(1-\varepsilon)\bar{u}^{1}+\varepsilon\frac{M_{1}}{M_{2}}\bar{u}^{2}\in
L^{\infty}(t_{0},T;B(0,M_{1}))$ for any $\varepsilon\in[0,1],$ which, together
with the optimality of $\bar{u}^{1}$ to $(OP)^{M_{1}}_{t_{0},y_{0}^{1}}$,
shows that
$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
0&\displaystyle\leq\mathop{\underline{\rm lim}}\limits_{\varepsilon\rightarrow
0+}\frac{1}{2\varepsilon}\left\\{\biggr{\|}e^{(T-t_{0})\triangle}y_{0}^{1}+\int^{T}_{t_{0}}e^{(T-s)\triangle}[(1-\varepsilon)\bar{u}^{1}+\varepsilon\frac{M_{1}}{M_{2}}\bar{u}^{2}]ds-
z_{d}\biggl{\|}^{2}\right.\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.\displaystyle-\biggr{\|}e^{(T-t_{0})\triangle}y_{0}^{1}+\int^{T}_{t_{0}}e^{(T-s)\triangle}\bar{u}^{1}ds-
z_{d}\biggl{\|}^{2}\right\\}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr&\displaystyle=\left\langle
e^{(T-t_{0})\triangle}y_{0}^{1}+\int^{T}_{t_{0}}e^{(T-s)\triangle}\bar{u}^{1}ds-
z_{d}\,,~{}\int^{T}_{t_{0}}e^{(T-s)\triangle}\left[\frac{M_{1}}{M_{2}}\bar{u}^{2}-\bar{u}^{1}\right]ds\right\rangle.\end{array}$
Similarly, we can prove that
$\displaystyle 0\leq\left\langle
e^{(T-t_{0})\triangle}y_{0}^{2}+\int^{T}_{t_{0}}e^{(T-s)\triangle}\bar{u}^{2}ds-
z_{d}\,,~{}\int^{T}_{t_{0}}e^{(T-s)\triangle}\left[\frac{M_{2}}{M_{1}}\bar{u}^{1}-\bar{u}^{2}\right]ds\right\rangle.$
Dividing the first inequality above by $M_{1}^{2}$ and the second one by
$M_{2}^{2}$, then adding them together, we obtain that
$\begin{array}[]{l}\displaystyle~{}~{}~{}\left\langle\frac{e^{(T-t_{0})\triangle}y_{0}^{1}-z_{d}}{M_{1}}-\frac{e^{(T-t_{0})\triangle}y_{0}^{2}-z_{d}}{M_{2}}\,,~{}\int^{T}_{t_{0}}e^{(T-s)\triangle}\left[\frac{\bar{u}^{2}}{M_{2}}-\frac{\bar{u}^{1}}{M_{1}}\right]ds\right\rangle\\\
\geq\displaystyle\biggr{\|}\int^{T}_{t_{0}}e^{(T-s)\triangle}\left[\frac{\bar{u}^{2}}{M_{2}}-\frac{\bar{u}^{1}}{M_{1}}\right]ds\biggl{\|}^{2},\end{array}$
which implies that
$\biggr{\|}\frac{e^{(T-t_{0})\triangle}y_{0}^{1}-z_{d}}{M_{1}}-\frac{e^{(T-t_{0})\triangle}y_{0}^{2}-z_{d}}{M_{2}}\biggl{\|}\geq\biggr{\|}\int^{T}_{t_{0}}e^{(T-s)\triangle}\left[\frac{\bar{u}^{2}}{M_{2}}-\frac{\bar{u}^{1}}{M_{1}}\right]ds\biggl{\|}.$
(4.30)
Since $(\bar{y}^{i},\bar{\psi}^{i})$, $i=1,2$, solve (4.2) and the semigroup
$\\{e^{t\triangle}:t\geq 0\\}$ is contractive, we can use (4.30) to derive
that
$\begin{array}[]{rl}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr&\displaystyle\biggr{\|}\frac{\bar{\psi}^{1}(t_{0})}{M_{1}}-\frac{\bar{\psi}^{2}(t_{0})}{M_{2}}\biggl{\|}=\biggr{\|}e^{(T-t_{0})\triangle}\left(\frac{\bar{\psi}^{1}(T)}{M_{1}}-\frac{\bar{\psi}^{2}(T)}{M_{2}}\right)\biggl{\|}\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\leq&\displaystyle\biggr{\|}\frac{\bar{\psi}^{1}(T)}{M_{1}}-\frac{\bar{\psi}^{2}(T)}{M_{2}}\biggl{\|}=\biggr{\|}\frac{\bar{y}^{1}(T)-z_{d}}{M_{1}}-\frac{\bar{y}^{2}(T)-z_{d}}{M_{2}}\biggl{\|}\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\leq&\displaystyle\left\|\frac{e^{(T-t_{0})\triangle}y^{1}_{0}-z_{d}}{M_{1}}-\frac{e^{(T-t_{0})\triangle}y^{2}_{0}-z_{d}}{M_{2}}\right\|+\left\|\int^{T}_{t_{0}}e^{(T-s)\triangle}\left[\frac{\bar{u}^{1}}{M_{1}}-\frac{\bar{u}^{2}}{M_{2}}\right]ds\right\|\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\leq&\displaystyle\frac{2}{M_{1}}\|y_{0}^{1}-y_{0}^{2}\|+2(\|y_{0}^{2}\|+\|z_{d}\|)\frac{|M_{1}-M_{2}|}{M_{1}M_{2}}.\end{array}$
(4.31)
By direct computation, we obtain that
$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr&\displaystyle\|{\cal N}(t_{0},y_{0}^{1},M_{1})-{\cal
N}(t_{0},y_{0}^{2},M_{2})\|=\biggr{\|}M_{1}\frac{\chi_{\omega}\bar{\psi}^{1}(t_{0})}{\|\chi_{\omega}\bar{\psi}^{1}(t_{0})\|}-M_{2}\frac{\chi_{\omega}\bar{\psi}^{2}(t_{0})}{\|\chi_{\omega}\bar{\psi}^{2}(t_{0})\|}\biggl{\|}\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\leq&\displaystyle~{}M_{1}\biggr{\|}\frac{\chi_{\omega}\bar{\psi}^{1}(t_{0})}{\|\chi_{\omega}\bar{\psi}^{1}(t_{0})\|}-\frac{\chi_{\omega}\bar{\psi}^{2}(t_{0})}{\|\chi_{\omega}\bar{\psi}^{2}(t_{0})\|}\biggl{\|}+|M_{1}-M_{2}|\biggr{\|}\frac{\chi_{\omega}\bar{\psi}^{2}(t_{0})}{\|\chi_{\omega}\bar{\psi}^{2}(t_{0})\|}\biggl{\|}\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr=&\displaystyle\frac{M_{1}}{\|{\frac{\chi_{\omega}\bar{\psi}^{1}(t_{0})}{M_{1}}}\|\|{\frac{\chi_{\omega}\bar{\psi}^{2}(t_{0})}{M_{2}}}\|}{\biggr{\|}}\left[\|\frac{\chi_{\omega}\bar{\psi}^{2}(t_{0})}{M_{2}}\|\frac{\chi_{\omega}\bar{\psi}^{1}(t_{0})}{M_{1}}-\|\frac{\chi_{\omega}\bar{\psi}^{1}(t_{0})}{M_{1}}\|\frac{\chi_{\omega}\bar{\psi}^{2}(t_{0})}{M_{2}}\right]\biggl{\|}+|M_{1}-M_{2}|\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\leq&\displaystyle\frac{2M_{1}}{\|{\frac{\chi_{\omega}\bar{\psi}^{1}(t_{0})}{M_{1}}}\|\|{\frac{\chi_{\omega}\bar{\psi}^{2}(t_{0})}{M_{2}}}\|}\left\|\frac{\chi_{\omega}\bar{\psi}^{2}(t_{0})}{M_{2}}\right\|\left\|\frac{\chi_{\omega}\bar{\psi}^{1}(t_{0})}{M_{1}}-\frac{\chi_{\omega}\bar{\psi}^{2}(t_{0})}{M_{2}}\right\|+|M_{1}-M_{2}|\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\leq&\displaystyle\frac{2M_{1}^{2}}{\|\chi_{\omega}\bar{\psi}^{1}(t_{0})\|}\left\|\frac{\bar{\psi}^{1}(t_{0})}{M_{1}}-\frac{\bar{\psi}^{2}(t_{0})}{M_{2}}\right\|+|M_{1}-M_{2}|.\end{array}$
This, together with (4.31), shows (4.28).
Step 2. When $\bar{M}>0$, there is $\bar{\rho}>0$ such that for each
$(t_{0},y_{0},M)\in[(\bar{t}_{0}-\bar{\rho})^{+},\bar{t}_{0}+\bar{\rho}]\times
B(\bar{y}_{0},\bar{\rho})\times[\bar{M}-\bar{\rho},\bar{M},\bar{\rho}]$,
$\displaystyle\left\|\chi_{\omega}\bar{\psi}^{M}_{t_{0},y_{0}}(t_{0})\right\|\geq\frac{1}{2}\left\|\chi_{\omega}\bar{\psi}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(\bar{t}_{0})\right\|>0.$
(4.32)
The second inequality in (4.32) follows from (4.3). The first one will be
proved by the following two cases:
Case 1: $t_{0}\leq\bar{t}_{0}$. In this case, the following three estimates
hold for all $y_{0}\in L^{2}(\Omega)$ and $M\in[\bar{M}/2,(3\bar{M})/2]$:
$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr&\displaystyle\left\|\bar{\psi}^{M}_{t_{0},y_{0}}(t_{0})-\bar{\psi}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(\bar{t}_{0})\right\|=\displaystyle\left\|e^{(\bar{t}_{0}-t_{0})\triangle}\bar{\psi}^{M}_{t_{0},y_{0}}(\bar{t}_{0})-\bar{\psi}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(\bar{t}_{0})\right\|\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\leq&\left\|e^{(\bar{t}_{0}-t_{0})\triangle}\right\|\cdot\left\|\bar{\psi}^{M}_{t_{0},y_{0}}(\bar{t}_{0})-\bar{\psi}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(\bar{t}_{0})\right\|+\left\|e^{(\bar{t}_{0}-t_{0})\triangle}-I\right\|\cdot\|\bar{\psi}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(\bar{t}_{0})\|\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\leq&\left\|\bar{\psi}^{M}_{t_{0},y_{0}}(\bar{t}_{0})-\bar{\psi}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(\bar{t}_{0})\right\|+\left\|e^{(\bar{t}_{0}-t_{0})\triangle}-I\right\|\cdot\|\bar{\psi}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(\bar{t}_{0})\|;\end{array}$
(Here $I$ denotes the identity operator on $L^{2}(\Omega)$.)
$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr&\left\|\bar{\psi}^{M}_{t_{0},y_{0}}(\bar{t}_{0})-\bar{\psi}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(\bar{t}_{0})\right\|=\left\|\bar{\psi}^{M}_{\bar{t}_{0},\bar{y}^{M}_{t_{0},y_{0}}(\bar{t}_{0})}(\bar{t}_{0})-\bar{\psi}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(\bar{t}_{0})\right\|\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\leq&\displaystyle
M\left\|\frac{\bar{\psi}^{M}_{\bar{t}_{0},\bar{y}^{M}_{t_{0},y_{0}}(\bar{t}_{0})}(\bar{t}_{0})}{M}-\frac{\bar{\psi}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(\bar{t}_{0})}{\bar{M}}\right\|+\frac{|M-\bar{M}|}{\bar{M}}\left\|\bar{\psi}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(\bar{t}_{0})\right\|\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\leq&\displaystyle
2\left\|\bar{y}^{M}_{t_{0},y_{0}}(\bar{t}_{0})-\bar{y}_{0}\right\|+\left(2\|\bar{y}_{0}\|+2\|z_{d}\|+\left\|\bar{\psi}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(\bar{t}_{0})\right\|\right)\frac{|M-\bar{M}|}{\bar{M}};\end{array}$
(Here, (4.8) and (4.31) have been used.) and
$\begin{array}[]{ll}{}{}{}{}&\displaystyle\left\|\bar{y}^{M}_{t_{0},y_{0}}(\bar{t}_{0})-\bar{y}_{0}\right\|=\left\|e^{(\bar{t}_{0}-t_{0})\triangle}y_{0}-\bar{y}_{0}+\int^{\bar{t}_{0}}_{t_{0}}e^{(\bar{t}_{0}-s)\triangle}\bar{u}^{M}_{t_{0},y_{0}}(s)ds\right\|\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\leq&\displaystyle\left\|e^{(\bar{t}_{0}-t_{0})\triangle}\right\|\cdot\|y_{0}-\bar{y}_{0}\|+\left\|e^{(\bar{t}_{0}-t_{0})\triangle}-I\right\|\cdot\|\bar{y}_{0}\|+\int^{\bar{t}_{0}}_{t_{0}}\left\|e^{(\bar{t}_{0}-s)\triangle}\right\|\cdot\left\|\bar{u}^{M}_{t_{0},y_{0}}(s)\right\|ds\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\leq&\displaystyle\|y_{0}-\bar{y}_{0}\|+\left\|e^{(\bar{t}_{0}-t_{0})\triangle}-I\right\|\cdot\|\bar{y}_{0}\|+M(\bar{t}_{0}-{t_{0}}).\end{array}$
Combining the above-mentioned three inequalities together leads to
$\begin{array}[]{l}~{}~{}\displaystyle\left\|\bar{\psi}^{M}_{t_{0},y_{0}}(t_{0})-\bar{\psi}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(\bar{t}_{0})\right\|\leq
2\|y_{0}-\bar{y}_{0}\|+2M(\bar{t}_{0}-{t_{0}})\\\ \vskip 3.0pt plus 1.0pt
minus
1.0pt\cr\displaystyle\par+\left\|e^{(\bar{t}_{0}-t_{0})\triangle}-I\right\|\cdot\left(2\|\bar{y}_{0}\|+\|\bar{\psi}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(\bar{t}_{0})\|\right)\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle+\left(2\|\bar{y}_{0}\|+2\|z_{d}\|+\left\|\bar{\psi}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(\bar{t}_{0})\right\|\right)\frac{|M-\bar{M}|}{\bar{M}}.\end{array}$
(4.33)
Clearly, the right hand side of (4.33) is continuous with respect to
$(t_{0},y_{0},M)$. This, along with the second inequality in (4.32), indicates
that there exists a $\rho_{1}$ with
$\displaystyle 0<\rho_{1}<\frac{T-\bar{t}_{0}}{2}\wedge\frac{\bar{M}}{2},$
such that for each
$(t_{0},y_{0},M)\in[(\bar{t}_{0}-\rho_{1})^{+},\bar{t}_{0}]\times
B(\bar{y}_{0},\rho_{1})\times[\bar{M}-\rho_{1},\bar{M}+\rho_{1}]$,
$\left\|\bar{\psi}^{M}_{t_{0},y_{0}}(t_{0})-\bar{\psi}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(\bar{t}_{0})\right\|\leq\displaystyle\frac{1}{2}\left\|\bar{\psi}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(\bar{t}_{0})\right\|,$
(4.34)
Case 2: $t_{0}\geq\bar{t}_{0}$. In this case, the following two estimates hold
for all $y_{0}\in L^{2}(\Omega)$ and $M\in[\bar{M}/2,(3\bar{M})/2]$:
$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr&\displaystyle\left\|\bar{\psi}^{M}_{t_{0},y_{0}}(t_{0})-\bar{\psi}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(t_{0})\right\|\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\leq&\displaystyle
M\left\|\frac{\bar{\psi}^{M}_{t_{0},y_{0}}(t_{0})}{M}-\frac{\bar{\psi}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(t_{0})}{\bar{M}}\right\|+\frac{|M-\bar{M}|}{\bar{M}}\left\|\bar{\psi}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(t_{0})\right\|\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr=&\displaystyle
M\left\|\frac{\bar{\psi}^{M}_{t_{0},y_{0}}(t_{0})}{M}-\frac{\bar{\psi}^{\bar{M}}_{t_{0},\bar{y}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(t_{0})}(t_{0})}{\bar{M}}\right\|+\frac{|M-\bar{M}|}{\bar{M}}\left\|\bar{\psi}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(t_{0})\right\|\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\leq&\displaystyle
2\left\|y_{0}-\bar{y}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(t_{0})\right\|+\left(2\left\|\bar{y}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(t_{0})\right\|+2\|z_{d}\|+\left\|\bar{\psi}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(t_{0})\right\|\right)\frac{|M-\bar{M}|}{\bar{M}}\end{array}$
(Here, (4.8) and (4.31) have been used.) and
$\begin{array}[]{ll}{}{}{}{}&\displaystyle\left\|\bar{y}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(t_{0})-y_{0}\right\|=\left\|e^{(t_{0}-\bar{t}_{0})\triangle}\bar{y}_{0}-y_{0}+\int^{t_{0}}_{\bar{t}_{0}}e^{(t_{0}-s)\triangle}\bar{u}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(s)ds\right\|\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr=&\displaystyle\left\|e^{(t_{0}-\bar{t}_{0})\triangle}\bar{y}_{0}-y_{0}+\int^{t_{0}}_{\bar{t}_{0}}e^{(t_{0}-s)\triangle}\bar{u}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(s)ds\right\|\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\leq&\displaystyle\left\|e^{(\bar{t}_{0}-t_{0})\triangle}-I\right\|\cdot\|\bar{y}_{0}\|+\|y_{0}-\bar{y}_{0}\|+\int^{t_{0}}_{\bar{t}_{0}}\left\|e^{(t_{0}-s)\triangle}\right\|\cdot\left\|\bar{u}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(s)\right\|ds\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\leq&\displaystyle\|y_{0}-\bar{y}_{0}\|+\left\|e^{(\bar{t}_{0}-t_{0})\triangle}-I\right\|\cdot\|\bar{y}_{0}\|+\bar{M}(t_{0}-\bar{t_{0}}).\end{array}$
From the above-mentioned two estimates, we derive that
$\begin{array}[]{c}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr~{}~{}~{}~{}\left\|\bar{\psi}^{M}_{t_{0},y_{0}}(t_{0})-\bar{\psi}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(\bar{t}_{0})\right\|\leq
2\|y_{0}-\bar{y}_{0}\|+2\bar{M}(\bar{t}_{0}-{t_{0}})+2\left\|e^{(t_{0}-\bar{t}_{0})\triangle}-I\right\|\cdot\|\bar{y}_{0}\|\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\par+\left(2\left\|\bar{y}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(t_{0})\right\|+\left\|\bar{\psi}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(\bar{t}_{0})\right\|+2\|z_{d}\|\right)\frac{|M-\bar{M}|}{\overline{M}}+\left\|\bar{\psi}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(t_{0})-\bar{\psi}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(\bar{t}_{0})\right\|\end{array}$
(4.35)
By the same argument used to get (4.34) (notice the continuity of
$\bar{\psi}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(\cdot)$), we can find a
$\rho_{2}$ with $\displaystyle
0<\rho_{2}<\frac{T-\bar{t}_{0}}{2}\wedge\frac{\bar{M}}{2}$, such that for each
triplet $(t_{0},y_{0},M)\in[\bar{t}_{0},\bar{t}_{0}+\rho_{2}]\times
B(\bar{y}_{0}),\rho_{2})\times[\bar{M}-\rho_{2},\bar{M}+\rho_{2}]$,
$\left\|\bar{\psi}^{M}_{t_{0},y_{0}}(t_{0})-\bar{\psi}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(\bar{t}_{0})\right\|\leq\displaystyle\frac{1}{2}\left\|\bar{\psi}^{\bar{M}}_{\bar{t}_{0},\bar{y}_{0}}(\bar{t}_{0})\right\|.$
(4.36)
Now we set $\bar{\rho}=\rho_{1}\wedge\rho_{2}$. Then the first inequality of
(4.32) follows from (4.34) and (4.36).
Step 3. When $\bar{M}=0$, there is $\bar{\rho}>0$ such that for each
$t_{0}\in[(\bar{t}_{0}-\bar{\rho})^{+},\bar{t}_{0}+\bar{\rho}]$, each
$y_{0}\in B(\bar{y}_{0},\bar{\rho})$ and each $M\in[0,\bar{\rho}]$,
$\displaystyle\left\|\chi_{\omega}\bar{\psi}^{M}_{t_{0},y_{0}}(t_{0})\right\|\geq\frac{1}{2}\left\|\chi_{\omega}\bar{\psi}^{0}_{\bar{t}_{0},\bar{y}_{0}}(\bar{t}_{0})\right\|>0.$
(4.37)
The second inequality in (4.37) follows from (4.3). The remainder is to show
the first one. The following two inequalities can be checked by direct
computation:
$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr&\displaystyle\left\|\bar{\psi}^{M}_{t_{0},y_{0}}(t_{0})-\bar{\psi}^{0}_{t_{0},\bar{y}_{0}}(t_{0})\right\|=\left\|e^{(T-\bar{t}_{0})\triangle}\left[\bar{\psi}^{M}_{t_{0},y_{0}}(T)-\bar{\psi}^{0}_{t_{0},\bar{y}_{0}}(T)\right]\right\|\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr=&\displaystyle\left\|e^{(T-\bar{t}_{0})\triangle}\left[\bar{y}^{M}_{t_{0},y_{0}}(T)-\bar{y}^{0}_{t_{0},\bar{y}_{0}}(T)\right]\right\|\leq\left\|\bar{y}^{M}_{t_{0},y_{0}}(T)-\bar{y}^{0}_{t_{0},\bar{y}_{0}}(T)\right\|\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr=&\displaystyle\left\|e^{(T-t_{0})\triangle}(y_{0}-\bar{y}_{0})+\int^{T}_{t_{0}}e^{(T-s)\triangle}\bar{u}^{M}_{t_{0},y_{0}}(s)ds\right\|\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\leq&\left\|y_{0}-\bar{y}_{0}\right\|+M(T-t_{0})\end{array}$
and
$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr&\left\|\bar{\psi}^{0}_{t_{0},\bar{y}_{0}}(t_{0})-\bar{\psi}^{0}_{\bar{t}_{0},\bar{y}_{0}}(\bar{t}_{0})\right\|\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr=&\left\|\left[e^{2(T-t_{0})\triangle}-e^{2(T-\bar{t}_{0})\triangle}\right]\bar{y}_{0}+\left[e^{(T-t_{0})\triangle}-e^{(T-\bar{t}_{0})\triangle}\right]z_{d}\right\|\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\leq&\left\|e^{2(|t_{0}-\bar{t}_{0}|)\triangle}-I\right\|\|\bar{y}_{0}\|+\left[e^{(|t_{0}-\bar{t}_{0}|)\triangle}-I\right]\|z_{d}\|.\end{array}$
From these, we deduce that
$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr&\left\|\bar{\psi}^{M}_{t_{0},y_{0}}(t_{0})-\bar{\psi}^{0}_{\bar{t}_{0},\bar{y}_{0}}(\bar{t}_{0})\right\|\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\leq&\displaystyle\left\|\bar{\psi}^{M}_{t_{0},y_{0}}(t_{0})-\bar{\psi}^{0}_{t_{0},\bar{y}_{0}}(t_{0})\right\|+\left\|\bar{\psi}^{0}_{t_{0},\bar{y}_{0}}(t_{0})-\bar{\psi}^{0}_{\bar{t}_{0},\bar{y}_{0}}(\bar{t}_{0})\right\|\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\leq&\displaystyle\|y_{0}-\bar{y}_{0}\|+M(T-t_{0})+\left\|e^{2(|t_{0}-\bar{t}_{0}|)\triangle}-I\right\|\|\bar{y}_{0}\|+\left[e^{(|t_{0}-\bar{t}_{0}|)\triangle}-I\right]\|z_{d}\|.\end{array}$
(4.38)
Clearly, the right hand side of (4.38) is continuous with respect to
$t_{0},y_{0}$ and $M$. This, together with the second inequality of (4.37),
yields that there exists
$\bar{\rho}\in(0,\displaystyle\frac{T-\bar{t}_{0}}{2})$ such that for each
$(t_{0},y_{0},M)\in[(\bar{t}_{0}-\bar{\rho})^{+},\bar{t}_{0}+\bar{\rho}]\times
B(\bar{y}_{0},\bar{\rho})\times[0,\bar{\rho}]$,
$\displaystyle\left\|\bar{\psi}^{M}_{t_{0},y_{0}}(t_{0})-\bar{\psi}^{0}_{\bar{t}_{0},\bar{y}_{0}}(\bar{t}_{0})\right\|\leq\displaystyle\frac{1}{2}\left\|\bar{\psi}^{0}_{\bar{t}_{0},\bar{y}_{0}}(\bar{t}_{0})\right\|,$
from which, the first inequality in (4.37) follows at once.
Step 4. The required Lipschitz continuity of the map ${\cal N}$
Clearly, we can take the same constant $\bar{\rho}$ in Step 2 and Step 3 such
that (4.32) and (4.37) stand. When $\bar{M}=0$, it follows from (4.28), (4.29)
and (4.37) that the map ${\cal N}(t_{0},\cdot,\cdot)$ is Lipschitz continuous
over $B(y_{0},\bar{\rho})\times[(\bar{M}-\bar{\rho})^{+},\bar{M}+\bar{\rho}]$
uniformly with respect to
$t_{0}\in[(\bar{t}_{0}-\bar{\rho})^{+},\bar{t}_{0}+\bar{\rho}]$; When
$\bar{M}>0$, the same conclusion follows from (4.28), (4.29) and (4.32).
$(ii)$ Fix a $\bar{y}_{0}\in L^{2}(\Omega)$. Let
$(\bar{M},\bar{t}_{0})\in[0,\infty)\times[0,T)$. Since
$\|\chi_{\omega}\bar{\psi}^{M}_{\bar{t}_{0},\bar{y}_{0}}(\bar{t}_{0})\|\neq 0$
(see (4.3)), the continuity of the map $(t_{0},M)\rightarrow{\cal
N}(t_{0},\bar{y}_{0},M)$ at $(\bar{t}_{0},\bar{M})$ follows from the
continuity of the map
$(t_{0},M)\rightarrow\bar{\psi}^{M}_{t_{0},\bar{y}_{0}}(t_{0})$ at
$(\bar{t}_{0},\bar{M})$. When $\bar{M}>0$, the continuity of the map
$(t_{0},M)\rightarrow\bar{\psi}^{M}_{t_{0},\bar{y}_{0}}(t_{0})$ at
$(\bar{t}_{0},\bar{M})$ follows from (4.33) and (4.35). When $\bar{M}=0$, the
continuity of this map at $(\bar{t}_{0},\bar{M})$ follows from (4.38). Thus,
${\cal N}(\cdot,\bar{y}_{0},\cdot)$ is continuous over
$[0,T)\times[0,\infty)$.
In summary, we complete the proof.
Proof of Proposition 4.7. $(i)$ Let $(\bar{t}_{0},\bar{y}_{0})\in[0,T)\times
L^{2}(\Omega)$. By the definition of maps ${\cal N}$ and $F$ (see (4.4) and
(4.27)), we see that
$\displaystyle F(t_{0},y_{0})={\cal N}(t_{0},y_{0},N(t_{0},y_{0}))\;\;{\rm
for~{}all}~{}(t_{0},y_{0})\in[0,T)\times L^{2}(\Omega).$ (4.39)
According to Lemma 4.11, there are $\bar{\rho}_{1}>0$ and $C_{1}>0$ such that
when $(t_{0},y_{0}^{1},M_{1})$ and $(t_{0},y_{0}^{2},M_{2})$ belong to
$[(\bar{t}_{0}-\bar{\rho}_{1})^{+},\bar{t}_{0}+\bar{\rho}_{1}]\times
B(\bar{y}_{0},\bar{\rho}_{1})\times[(\bar{M}-\bar{\rho}_{1})^{+},\bar{M}+\bar{\rho}_{1}]$,
$\|{\cal N}(t_{0},y_{0}^{1},M_{1})-{\cal N}(t_{0},y_{0}^{2},M_{2})\|\leq
C_{1}\Bigr{(}\|y_{0}^{1}-y_{0}^{2}\|+|M_{1}-M_{2}|\Big{)}.$ (4.40)
According to Lemma 4.10, there are $\bar{\rho}_{2}>0$ and $C_{2}>0$ such that
$|N(t_{0},y_{0}^{1})-N(t_{0},y_{0}^{2})|\leq C_{2}\|y_{0}^{1}-y_{0}^{2}\|,$
for all $(t_{0},y_{0}^{1})$ and $(t_{0},y_{0}^{2})$ belong to
$[(\bar{t}_{0}-\bar{\delta})^{+},\bar{t}_{0}+\bar{\delta}]\times
B(\bar{y}_{0},\bar{\rho}_{2})$, where $\bar{\delta}=(T-\bar{t}_{0})/2$.
Let
$\bar{\rho}=\min\\{\bar{\rho}_{1},\bar{\rho}_{2},\displaystyle\frac{\bar{\rho}_{1}}{2C_{2}},\bar{\delta}\\}$.
Then it follows from the above inequality that
$|N(t_{0},y_{0}^{1})-N(t_{0},\bar{y}_{0}^{2})|\leq
2C_{2}\bar{\rho}\leq\bar{\rho}_{1}\;\;\mbox{for all}\;y_{0}^{1},y_{0}^{2}\in
B(\bar{y}_{0},\bar{\rho})\;\mbox{and}\;t_{0}\in[(\bar{t}_{0}-\bar{\rho})^{+},\bar{t}_{0}+\bar{\rho}].$
This, along with (4.40), indicates that
$\begin{array}[]{l}~{}~{}~{}\|F(t_{0},y_{0}^{1})-F(t_{0},y_{0}^{2})\|\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr=\|{\cal
N}(t_{0},y_{0}^{1},N(t_{0},y_{0}^{1}))-{\cal
N}(t_{0},y_{0}^{2},N(t_{0},y_{0}^{2}))\|\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\leq
C_{1}\Bigr{(}\|y_{0}^{1}-y_{0}^{2}\|+|N(t_{0},y_{0}^{1})-N(t_{0},y_{0}^{2})|\Bigr{)}\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\leq
C_{1}\Bigr{(}\|y_{0}^{1}-y_{0}^{2}\|+C_{2}\|y_{0}^{1}-y_{0}^{2}\|\Bigr{)}=C_{1}(1+C_{2})\|y_{0}^{1}-y_{0}^{2}\|\end{array}.$
for all $y_{0}^{1},y_{0}^{2}\in B(\bar{y}_{0},\bar{\rho})$ and
$t_{0}\in[(\bar{t}_{0}-\bar{\rho})^{+},\bar{t}_{0}+\bar{\rho}]$. The desired
Lipschitz continuity follows from the above inequality at once.
$(ii)$ Let $\bar{y}_{0}\in L^{2}(\Omega)$. Since $F(t_{0},\bar{y}_{0})={\cal
N}(t_{0},\bar{y}_{0},N(t_{0},\bar{y}_{0}))$ for all $t_{0}\in[0,T)$ (see
(4.39)), the desired continuity of $F(\cdot,\bar{y}_{0})$ follows directly
from the continuity of $N(\cdot,\bar{y}_{0})$ and ${\cal
N}(\cdot,\bar{y}_{0},\cdot)$.
In summary, we complete the proof.
### 4.4 Proof of Theorem 4.2
Proof of Theorem 4.2. Let $(t_{0},y_{0})\in[0,T)\times L^{2}({\Omega})$. By
theorem 4.1, $\bar{y}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(\cdot)$ is the unique
solution to Equation (4.5), i.e.,
$y_{F}(\cdot;t_{0},y_{0})=\bar{y}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(\cdot)$ over
$[t_{0},T)$. Then, by (4.4), (4.14), (4.8) and (4.10), we see that
$F\Bigr{(}t,y_{F}(t;t_{0},y_{0})\Bigl{)}=F\left(t,\bar{y}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(t)\right)=N(t_{0},y_{0})\frac{{\chi}_{\omega}\bar{\psi}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(t)}{\|{\chi}_{\omega}\bar{\psi}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(t)\|}=\bar{u}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(t),\quad\forall~{}t\in(t_{0},T).$
Since $\bar{u}^{N(t_{0},y_{0})}_{t_{0},y_{0}}(\cdot)$ is optimal norm control
for Problem $(NP)_{t_{0},y_{0}}$ (see Lemma4.5), the above equality implies
that $F(\cdot;y_{F}(\cdot;t_{0},y_{0})$ is the optimal control to
$(NP)_{t_{0},y_{0}}$. This completes the proof.
## References
* [1] M. Bardi, Boundary value problem for the minimum time function, SIAM J. Control Optim., Vol 27 (1989) 776-785.
* [2] O. Carja, The minimal time function in infinite dimensions, SIAM J. Control Optim. Vol. 31, No. 5 (1993), 1103-1114.
* [3] T. Duyckaerts, X. Zhang, E. Zuazua, On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), 1-41.
* [4] C. Fabre, J.-P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation, Proc.Royal Soc.Edinburgh, 125 A (1995), 31-61.
* [5] H. O. Fattorini, Infinite Dimensional Linear Control Systems, The Time Optimal and Norm Optimal Problems, North-Holland Mathematics Studies 201, ELSEVIER, 2005.
* [6] E. Fernandez-Cara and E. Zuazua, Null and approximate controllability for weakly blowing-up semilinear heat equations, Annales Inst. Henri Poincare, Analyse non-lineaire, 17 (5) (2000), 5 83-616.
* [7] F. Gozzi and P. Loreti, Regularity of the minimum time function and minimum energy problems: The linear case, SIAM J. Control Optim. Vol. 37, No. 4 (1999), 1195-1221.
* [8] F. H. Lin, A uniqueness theorem for the parabolic equation, Comm. Pure Appl. Math., 43 (1990), 127-136.
* [9] J. L. Lions, Optimal control of systems governed by partial differential equations. Translated from the French by S. K. Mitter. Die Grundlehren der mathematischen Wissenschaften, Band 170 Springer-Verlag, New York-Berlin, 1971.
* [10] X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhauser Boston, Boston, MA, 1995.
* [11] Mizel, V., Seidman, T.: An abstract ’bang-bang principle’ and time optimal boundary control of the heat equation, SIAM J. Control Optim. 35 (1997), 1204–1216
* [12] K. D. Phung and G. Wang, Quantitative unique continuation for the semilinear heat equations in a convex domain, Journal of Functional Analysis, 259(5) (2010), 1230-1247.
* [13] G. Wang and E. Zuazua, On the equivalence between time and norm optimal controls for heat equations, preprint.
* [14] G. S. Wang, $L^{\infty}$-null controllability for the heat equation and its consequences for the time optimal control problem, SIAM, J. Control Optim. Vol. 47, No. 4 (2008) 1701-1720.
* [15] E. Zeidler, Nonlinear functional Analysis and its application. I. Fixed-point theorem. Translated from German by Peter R. Wadsack. Springer-Verlag, New York, 1986.
|
arxiv-papers
| 2011-10-18T06:37:06 |
2024-09-04T02:49:23.256872
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Gengsheng Wang, Yashan Xu",
"submitter": "Yashan Xu",
"url": "https://arxiv.org/abs/1110.3885"
}
|
1110.3893
|
# Generalization of Rindler Potential at Cluster Scales in Randers-Finslerian
Spacetime: a Possible Explanation of the Bullet Cluster 1E0657-558 ?
Zhe Chang Institute of High Energy Physics, Chinese Academy of Sciences,
100049 Beijing, China, and
Theoretical Physics Center for Science Facilities, Chinese Academy of
Sciences, 100049 Beijing, China
changz@ihep.ac.cn Ming-Hua Li Institute of High Energy Physics, Chinese
Academy of Sciences, 100049 Beijing, China
limh@ihep.ac.cn Hai-Nan Lin Institute of High Energy Physics, Chinese
Academy of Sciences, 100049 Beijing, China
limh@ihep.ac.cn Xin Li Institute of High Energy Physics, Chinese Academy of
Sciences, 100049 Beijing, China, and
Theoretical Physics Center for Science Facilities, Chinese Academy of
Sciences, 100049 Beijing, China
lixin@ihep.ac.cn
(Day Month Year; Day Month Year)
###### Abstract
The data of the Bullet Cluster 1E0657-558 released on November 15, 2006 reveal
that the strong and weak gravitational lensing convergence $\kappa$-map has an
$8\sigma$ offset from the $\Sigma$-map. The observed $\Sigma$-map is a direct
measurement of the surface mass density of the Intracluster medium(ICM) gas.
It accounts for $83\%$ of the averaged mass-fraction of the system. This
suggests a modified gravity theory at large distances different from Newton’s
inverse-square gravitational law. In this paper, as a cluster scale
generalization of Grumiller’s modified gravity model (D. Grumiller, Phys. Rev.
Lett. 105, 211303 (2010)), we present a gravity model with a generalized
linear Rindler potential in Randers-Finslerian spacetime without invoking any
dark matter. The galactic limit of the model is qualitatively consistent with
the MOND and Grumiller’s. It yields approximately the flatness of the
rotational velocity profile at the radial distance of several kpcs and gives
the velocity scales for spiral galaxies at which the curves become flattened.
Plots of convergence $\kappa$ for a galaxy cluster show that the peak of the
gravitational potential has chances to lie on the outskirts of the baryonic
mass center. Assuming an isotropic and isothermal ICM gas profile with
temperature $T=14.8$ keV (which is the center value given by observations), we
obtain a good match between the dynamical mass $M_{\textmd{T}}$ of the main
cluster given by collisionless Boltzmann equation and that given by the King
$\beta$-model. We also consider a Randers$+$dark matter scenario and a
$\Lambda$-CDM model with the NFW dark matter distribution profile. We find
that a mass ratio $\eta$ between dark matter and baryonic matter about 6 fails
to reproduce the observed convergence $\kappa$-map for the isothermal
temperature $T$ taking the observational center value.
###### keywords:
Modified Gravity; Rindler potential; Finsler geometry; Randers spacetime;
Bullet Cluster.
Managing Editor
## 1 Introduction
It has long been known that the gravitational potentials of some galaxy
clusters are too deep to be generated by the observed baryonic matter
according to Newton’s inverse-square law of gravitation [1]. This violation of
Newton’s law is further confirmed by a great variety of observations. To name
a few: the Oort discrepancy in the disk of the Milky Way [2], the velocity
dispersions of dwarf Spheroidal galaxies [3], and the flat rotation curves of
spiral galaxies [4]. The most widely adopted way to solve these mysteries is
to assume that all our galaxies and clusters are surrounded by massive non-
luminous dark matter [5]. Despite its phenomenological success in explaining
the flat rotation curves of spiral galaxies, the hypothesis has its own
deficiencies. No theory predicts these matters, and they behave in such ad hoc
way like existing as a halo without undergoing gravitational collapse. There
are a lot of possible candidates for dark matter (such as axions, neutrinos et
al.), but none of them are sufficiently satisfactory. Up to now, all of them
are either undetected or excluded by experiments and observations.
Because of all these troubles, some models have been built as alternatives of
the dark matter hypothesis. Their main ideas are to suggest that Newton’s
dynamics is invalid in the galactic scale. A famous example is the MOND [6].
It supposes that in the galactic scale, the Newton’s dynamics appears as
$\begin{array}[]{l}m\mu\left(\displaystyle\frac{a}{a_{0}}\right)\mathbf{a}=\mathbf{F},\\\\[11.38092pt]
\displaystyle\lim_{x\gg 1}\mu(x)=1,~{}~{}~{}\lim_{x\ll 1}\mu(x)=x,\end{array}$
(1)
where $a_{0}$ is a constant and the value of which is of order $10^{-8}$
cm/s2. Dwarf and low surface brightness galaxies provide a good test for the
MOND [7]. With a simple formula and the one-and-only-one constant parameter
$a_{0}$, the MOND yields the observed luminosity-rotation velocity relation,
the Tully-Fisher relation [8]. By introducing several scalar, vector and
tensor fields, Bekenstein developed a relativistic version of the MOND [9].
The covariant MOND satisfies all four classical tests on Einstein’s general
relativity in Solar system.
Although the MOND successfully reduces the discrepancy between the visible and
the Newtonian dynamical mass (which is also quantified in terms of mass-to-
light ratio) to a factor of $2/3$, there still remains a missing mass problem,
particularly in the cores of clusters of galaxies [10]. The data release of
the Bullet Cluster 1E0657-558 in November of 2006 posed a serious challenge
for modified gravity theories such as the MOND.
The Bullet Cluster 1E0657-558 was first spotted by the Chandra X-ray
Observatory in 2002 [11]. Located at a redshift $z=0.296$ (Gpc scale), it has
exceptionally high X-ray luminosity and is one of the largest and hottest
luminous galaxy clusters in the sky. A high-resolution map of the ICM gas,
i.e. the surface mass density $\Sigma(x,y)$, was reconstructed by Clowe et al.
[12, 13] in 2006. It exhibits a supersonic shock front in the plane of the
merger, which is just aligned with our sky. The high-resolution and absolutely
calibrated convergence $\kappa$-map of the sky region that surrounds the
“bullet” was also reconstructed by Bradač and Clowe et al. in their
gravitational lensing surveys [14, 15, 16]. The $\kappa$-map is evidently
offset from the $\Sigma$-map. The peak of the $\kappa$-map lies on the region
of galaxies instead of tracing the ICM gas of the main cluster, which makes up
about $83\%$ of the total baryonic mass of the merging system.
Clowe et al. [12, 15, 16] took it as a direct empirical evidence of the
existence of dark matter, while whether the MOND could fit the X-ray
temperature profiles without dark matter component is still in issue [10, 17,
18, 19, 20]. Using their modified gravity (MOG), Brownstein and Moffat partly
explained the steepened peaks of the $\kappa$-map, while attributing the rest
differences to the MOG’s effect of the galaxies [21].
On the other hand, Grumiller [22] presented an effective model for gravity of
a central object at large scales recently. To leading order in the large
radius expansion, the action of his model leads to an additional “Rindler
term” in the gravitational potential. This extra term gives rise to a constant
acceleration towards or away from the source. The scale where the velocity
profile flattens is $v\sim 300$ km/s, in reasonable agreement with the
observational data.
In this paper, inspired by these prominent work, we try to construct a
modified gravity model at large distances with a generalized Rindler potential
without invoking any dark matter. This is carried out in a Randers-Finslerian
spacetime in Zermelo’s navigation scenario [23, 24, 26]. Finslerian geometry
is a generalization of Riemannian geometry without quadratic restrictions on
the line element [27]. It is intriguing to investigate the possible physical
implication in such a general geometrical background. In fact, precedent work
have yielded some interesting results [25, 28, 29, 30, 31, 32, 33, 34, 35, 36,
37, 38, 39, 40]. The work in this paper is a cluster-scale generalization of
Grumiller’s model and it is ensured that in the galactic limit, it agrees with
both the Grumiller’s model and the observational data. An approximately
flattened velocity profile predicted by our model makes it qualitatively
consistent with the MOND at the distance scale of several kpcs. The Newtonian
limit and the gravitational deflection of light are particularly investigated
and the deflection angle is given explicitly.
We use the isothermal King $\beta$-model to describe the observed $\Sigma$-map
of a galaxy main cluster. The convergence $\kappa$ is obtained. It is found
that the gravitational potential peak does not always lie on the center of the
baryonic material center. Chances are that it will has a bigger value in the
outskirts rather than the center. This is one of the distinguishing features
of the reconstructed $\kappa$-map of the Bullet Cluster system. Besides, the
gravity provided by the baryonic material is somehow “enlarged”. It is
reasonable to suggest that these results may ameliorate the conundrum between
the gravity theory and the observations of the Bullet Cluster 1E0657-558.
The rest of the paper is organized as follows. Section 2 is divided into four
parts: in Section 1, we introduce the basic concepts of Finsler geometry; in
Section 2.1, we use the the second Bianchi identities to get the gravitational
field equation in Berwald-Finslerian space; in Section 2.3, we consider a
Randers-type spacetime in a navigation scenario with a vector field in the
radial direction; in Section 2.4, we integrate the geodesic equation to get
the deflection angle in Randers-Finslerian spacetime with a generalized
Rindler potential at cluster scales. Section 3 is divided into five parts: in
Section 3.1, we give the Poisson’s equation by which the effective lens
potential obeys; in Section 3.2, by making use of the effective lens
potential, we obtain the convergence $\bar{\kappa}$ of the Bullet Cluster
1E0657-558. The cross section of the calculated $\bar{\kappa}$-map is
presented; in Section 3.3, the isothermal temperature of the main cluster is
calculated; in Section 3.4, we consider a Randers$+$dark matter model for
comparison; in Section 3.5, we investigate the performance of our model at
galactic scales. Conclusions and discussions are presented in Section 4.
Appendix is in the last section.
## 2 Finslerian Geometry
### 2.1 Basic Concepts
Finslerian geometry is a natural generalization of Riemannian geometry without
quadratic restrictions on the metric [27]. It is based on a real function $F$
called Finsler structure (or Finslerian norm in some literature) with the
property $F(x,\lambda y)=\lambda F(x,y)$ for all $\lambda>0$, where
$y^{\mu}\equiv dx^{\mu}/{d\tau}$ ($\mu=0,1,2,~{}...~{},n$). In physics,
$x^{\mu}$ stands for position and $y^{\mu}$ stands for velocity. The metric of
Finslerian space is given by [41]
$\displaystyle g_{\mu\nu}\equiv\frac{\partial}{\partial
y^{\mu}}\frac{\partial}{\partial y^{\nu}}\left(\frac{1}{2}F^{2}\right)\,.$ (2)
Finslerian geometry has its genesis in the integral of the form
$\displaystyle\int^{r}_{s}F(x^{1},\cdots,x^{n};y^{1},\cdots,y^{n})d\tau~{}\,.$
(3)
It represents the arc length of a curve in a Finslerian manifold. The first
variation of (3) gives the geodesic equation in a Finslerian space [41]
$\displaystyle\frac{d^{2}x^{\mu}}{d\tau^{2}}+G^{\mu}=0\ ,$ (4)
where
$\displaystyle
G^{\mu}\equiv\frac{1}{2}g^{\mu\nu}\left(\frac{\partial^{2}F^{2}}{\partial
x^{\lambda}\partial y^{\nu}}y^{\lambda}-\frac{\partial F^{2}}{\partial
x^{\nu}}\right)$ (5)
is called the geodesic spray coefficient. Obviously, if $F$ is Riemannian
metric, then
$G^{\mu}=\tilde{\gamma}^{\mu}_{~{}\nu\lambda}y^{\nu}y^{\lambda},$ (6)
where $\tilde{\gamma}^{\mu}_{~{}\nu\lambda}$ is the Riemannian Christoffel
symbol.
In a Finslerian manifold, there exists a unique linear connection - the Chern
connection [42]. It is torsion freeness and almost metric-compatibility,
$\Gamma^{\alpha}_{~{}\mu\nu}=\gamma^{\alpha}_{~{}\mu\nu}-g^{\alpha\lambda}\left(A_{\lambda\mu\beta}\frac{N^{\beta}_{~{}\nu}}{F}-A_{\mu\nu\beta}\frac{N^{\beta}_{~{}\lambda}}{F}+A_{\nu\lambda\beta}\frac{N^{\beta}_{~{}\mu}}{F}\right),$
(7)
where $N^{\mu}_{~{}\nu}$ is defined as
$N^{\mu}_{~{}\nu}\equiv\gamma^{\mu}_{~{}\nu\alpha}y^{\alpha}-A^{\mu}_{~{}\nu\lambda}\gamma^{\lambda}_{~{}\alpha\beta}y^{\alpha}y^{\beta}$
and $A_{\lambda\mu\nu}\equiv\frac{F}{4}\frac{\partial}{\partial
y^{\lambda}}\frac{\partial}{\partial y^{\mu}}\frac{\partial}{\partial
y^{\nu}}(F^{2})$ is the Cartan tensor (regarded as a measurement of deviation
from the Riemannian Manifold). In terms of Chern connection, the curvature of
Finsler space is given as
$R^{~{}\lambda}_{\kappa~{}\mu\nu}=\frac{\delta\Gamma^{\lambda}_{~{}\kappa\nu}}{\delta
x^{\mu}}-\frac{\delta\Gamma^{\lambda}_{~{}\kappa\mu}}{\delta
x^{\nu}}+\Gamma^{\lambda}_{~{}\alpha\mu}\Gamma^{\alpha}_{~{}\kappa\nu}-\Gamma^{\lambda}_{~{}\alpha\nu}\Gamma^{\alpha}_{~{}\kappa\mu},$
(8)
where $\frac{\delta}{\delta x^{\mu}}=\frac{\partial}{\partial
x^{\mu}}-N^{\nu}_{~{}\mu}\frac{\partial}{\partial y^{\nu}}$.
### 2.2 Field Equations
Constructing a physical Finslerian theory of gravity in an arbitrary
Finslerian spacetime is a difficult task. However, it has been pointed out
that constructing a Finslerian theory of gravity in a Finlserian spacetime of
Berwald type is viable [37]. A Finslerian spacetime is said to be of Berwald
type if the Chern connection (7) have no $y$ dependence[41]. In the light of
the research of Tavakol et al. [37], the gravitational field equation in
Berwald-Finslerian space has been studied in [28, 32]. In Berwald-Finslerian
space, the Ricci tensor reduces to
$Ric_{\mu\nu}=\frac{1}{2}(R^{~{}\alpha}_{\mu~{}\alpha\nu}+R^{~{}\alpha}_{\nu~{}\alpha\mu})\,.$
(9)
It is manifestly symmetric and covariant. Apparently it will reduce to the
Riemann-Ricci tensor if the metric tensor $g_{\mu\nu}$ does not depend on $y$.
We starts from the second Bianchi identities in Berwald-Finslerian space [41]
$R^{~{}\alpha}_{\mu~{}\lambda\nu|\beta}+R^{~{}\alpha}_{\mu~{}\nu\beta|\lambda}+R^{~{}\alpha}_{\mu~{}\beta\lambda|\nu}=0\
,$ (10)
where the “$|$” means the covariant derivative. The metric-compatibility
$g_{\mu\nu|\alpha}=0$ and $g^{\mu\nu}_{~{}~{}~{}|\alpha}=0$ and contraction of
(10) with $g^{\mu\beta}$ gives that
$R^{\mu\alpha}_{~{}~{}~{}\lambda\nu|\mu}+R^{\mu\alpha}_{~{}~{}~{}\nu\mu|\lambda}+R^{\mu\alpha}_{~{}~{}~{}\mu\lambda|\nu}=0\,.$
(11)
Lowering the index $\alpha$ and contracting with $g^{\alpha\lambda}$, we
obtain
$\left[Ric_{\mu\nu}-\frac{1}{2}g_{\mu\nu}S\right]_{|\mu}+\left\\{\frac{1}{2}B^{~{}\alpha}_{\alpha~{}\mu\nu}+B^{~{}\alpha}_{\mu~{}\nu\alpha}\right\\}_{|\mu}=0,$
(12)
where
$\displaystyle B_{\mu\nu\alpha\beta}$ $\displaystyle=$ $\displaystyle-
A_{\mu\nu\lambda}R^{~{}\lambda}_{\theta~{}\alpha\beta}y^{\theta}/F\ ,$ (13)
$\displaystyle R$ $\displaystyle=$
$\displaystyle\frac{y^{\mu}}{F}R^{~{}\kappa}_{\mu~{}\kappa\nu}\frac{y^{\nu}}{F}\
,$ (14) $\displaystyle S$ $\displaystyle=$ $\displaystyle
g^{\mu\nu}Ric_{\mu\nu}\,.$ (15)
Thus, we get the counterpart of the Einstein’s field equation in Berwald -
Finslerian space
$\left[Ric_{\mu\nu}-\frac{1}{2}g_{\mu\nu}S\right]+\left\\{\frac{1}{2}B^{~{}\alpha}_{\alpha~{}\mu\nu}+B^{~{}\alpha}_{\mu~{}\nu\alpha}\right\\}=8\pi
GT_{\mu\nu}\,.$ (16)
In Eq. (16), the term in “[ ]” is symmetrical tensor, and the term in “{}” is
asymmetrical tensor. By making use of Eq. (16), the vacuum field equation in
Finslerian spacetime of Berwald type implies
$Ric_{\mu\nu}=\frac{1}{2}(R^{~{}\alpha}_{\mu~{}\alpha\nu}+R^{~{}\alpha}_{\nu~{}\alpha\mu})=0\,.$
(17)
### 2.3 Randers type space with a “Wind”
Randers space is a special kind of Finslerian geometry with the Finsler
structure $F$ defined on the slit tangent bundle $TM\backslash 0$ of a
manifold $M$ as [41, 43],
$\displaystyle F(x,y)=\alpha(x,y)+\beta(x,y)\ ,$ (18)
where
$\displaystyle\alpha(x,y)$ $\displaystyle\equiv$
$\displaystyle\sqrt{\tilde{a}_{\mu\nu}(x)y^{\mu}y^{\nu}}\ ,$ (19)
$\displaystyle\beta(x,y)$ $\displaystyle\equiv$
$\displaystyle\tilde{b}_{\mu}(x)y^{\mu}\ .$ (20)
Here, $\tilde{a}_{\mu\nu}$ is a Riemannian metric and $\tilde{b}_{\mu}$ is an
1-form. Here and after, if not specified, lower case Greek indices (i.e.
$\mu,\nu,\alpha,...$) run from $0$ to $3$ and the Latin ones (i.e.
$i,j,k,...$) run from $1$ to $3$. Positivity of $F$ holds if and only if [41]
$\displaystyle|\tilde{b}|\equiv\sqrt{\tilde{b}_{\mu}\tilde{b}^{\mu}}~{}~{}<1\
,$ (21)
where
$\displaystyle\tilde{b}^{\mu}\equiv\tilde{a}^{\mu\nu}\tilde{b}_{\nu}\,.$ (22)
Stavrinos et al. [25] constructed a generalized FRW model based on a
Lagrangian identified to be the Randers-type metric function. New Friedman
equations and a physical generalization of the Hubble and other cosmological
parameters were obtained. Zermelo [26] aimed to find minimum-time trajectories
in a Riemannian manifold $(M,h)$ under the influence of a “wind” represented
by a vector field $W$ . Shen [Shen2003] proved that the minimum time
trajectories are exactly the geodesics of Randers space, if the wind is time
independent.
In this paper, we consider a Randers-Finslerian structure $F(x,y)$ under the
influence of a “wind” in the radial direction $W\equiv
W_{\mu}dx^{\mu}=W_{r}dr$, to wit
$\displaystyle\tilde{a}_{\mu\nu}=\frac{\lambda
h_{\mu\nu}+W_{\mu}W_{\nu}}{\lambda^{2}},~{}~{}\tilde{b}_{\mu}=-\frac{W_{\mu}}{\lambda},~{}~{}\lambda=1-h_{\mu\nu}W^{\mu}W^{\nu}\
,$ (23)
where $W^{\mu}=h^{\mu\nu}W_{\nu}$ and
$\tilde{a}^{\mu\nu}=\lambda(h^{\mu\nu}-W^{\mu}W^{\nu})$. Here $h_{\mu\nu}$ is
the Schwarzschild metric
$\displaystyle
h_{ij}dx^{i}dx^{j}=\left(1-\frac{2GM}{r}\right)^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}sin^{2}\theta
d\varphi^{2}\,.$ (24)
From (23), we have
$\displaystyle\tilde{b}_{r}=-\frac{1}{\lambda}\sqrt{\frac{1-\lambda}{\left(1-\frac{2GM}{r}\right)^{-1}}}\
,$ (25)
where $\lambda$ is a function of $r$, i.e. $\lambda=\lambda(r)$. Zermelo [26]
said little about the $\lambda$ except for the condition that the size of the
component $\tilde{b}_{r}$ must be suitably controlled, i.e.
$|\tilde{b}_{r}|<1$, for $F$ to be positive on $TM\backslash 0$. But for a
physical model, the specific form of $\lambda(r)$ is determined not only by
the local symmetry of the spacetime but also constrained by the experiments
and observations.
The explicit form of $F(x,y)$ reads
$\displaystyle Fd\tau$ $\displaystyle=$
$\displaystyle\sqrt{\lambda^{-1}\left(\left(1-\frac{2GM}{r}\right)^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}sin^{2}\theta
d\varphi^{2}\right)+\lambda^{-2}W_{r}^{2}dr^{2}}-\lambda^{-1}W_{r}dr$ (26)
$\displaystyle=$
$\displaystyle\sqrt{\lambda^{-2}\left(1-\frac{2GM}{r}\right)^{-1}dr^{2}+\lambda^{-1}\left(r^{2}d\theta^{2}+r^{2}sin^{2}\theta
d\varphi^{2}\right)}-\lambda^{-1}W_{r}dr\ ,$
where the second equation exploits the expression of $\lambda$ in (23)
assuming $|\frac{GM}{r}|\ll 1$. The relativistic form of (26) is given as
111In Chapter 8 of [Weinberg1972], the standard form of the proper time
interval of a static isotropic or approximately static isotropic gravitational
field is given as $\displaystyle
d\tau^{2}=B(r)dt^{2}-A(r)dr^{2}-r^{2}\left(d\theta^{2}+sin^{2}\theta
d\varphi^{2}\right)\,.$ The field equations for empty space $R_{\mu\nu}=0$
requires that $A(r)B(r)=\textmd{constant}$. And the metric tensor must
approach the Minkowski tensor in spherical coordinates, that is, for
$r\rightarrow\infty$, $A(r)=B(r)=1$. Thus we have $\displaystyle
A(r)B(r)=1\,.$ For the Randers-Finslerian metric (18) and (26), that is
$\displaystyle\tilde{a}_{00}=-\lambda^{2}\left(1-\frac{2GM}{r}\right)\,.$
$\displaystyle
Fd\tau=\sqrt{-\lambda^{2}\left(1-\frac{2GM}{r}\right)dt^{2}+\lambda^{-2}\left(1-\frac{2GM}{r}\right)^{-1}dr^{2}+\lambda^{-1}\left(r^{2}d\theta^{2}+r^{2}sin^{2}\theta
d\varphi^{2}\right)}-\lambda^{-1}W_{r}dr\ .$ (27)
Discussions in the next subsection are based on the geodesic equation which
stems from a Lagrangian identified to be the Randers-type metric function (27)
in four-dimensional spacetime.
### 2.4 Equations of Montion and Deflection Angle
In a Randers space, the geodesic equation (4) takes the form of 222We just
consider the case that the $\beta$ in (18) is a closed 1-form, i.e.
$d\beta=0$.
$\displaystyle\frac{d^{2}x^{\mu}}{d\tau^{2}}+\left(\tilde{\gamma}^{\mu}_{~{}\nu\alpha}+\ell^{\mu}\tilde{b}_{\nu|\alpha}\right)y^{\nu}y^{\alpha}=0\
,$ (28)
where
$\displaystyle\ell^{\mu}\equiv\frac{y^{\mu}}{F},~{}~{}~{}~{}\tilde{b}_{\nu|\alpha}\equiv\frac{\partial\tilde{b}_{\nu}}{\partial
x^{\alpha}}-\tilde{\gamma}^{\mu}_{~{}\nu\alpha}\tilde{b}_{\mu}\ ,$ (29)
and $\tilde{\gamma}^{\mu}_{~{}\nu\alpha}$ is the Christoffel symbols of the
Riemannian metric $\tilde{a}_{\mu\nu}$. Given the Finslarian structure in
(27), the non-vanishing components of the geodesic equations (28) (i.e. the
equation of motion) give rise to the relation between the radial distance $r$
and the angle $\varphi$ of the orbits of free particles, to wit [44]
$\displaystyle\left(\frac{1}{r^{2}}\frac{dr}{d\varphi}\right)^{2}=\left(\frac{E}{J\lambda(r)}\right)^{2}-\frac{\lambda(r)}{r^{2}}\left(1-\frac{2GM}{r}\right)\
,$ (30)
where $E$ and $J$ are the integral constants of motion. Introducing a new
quantity
$\displaystyle u\equiv\frac{GM}{r}\ ,$ (31)
Eq. (30) can be rewritten in terms of $u$ as
$\displaystyle\left(\frac{du}{d\varphi}\right)^{2}=\left(\frac{EGM}{J\lambda(r)}\right)^{2}-\lambda
u^{2}(1-2u)\,.$ (32)
It should be noticed that the only difference between Eq. (32) and its
Riemmanian counterpart is the $\lambda(r)$. For $\lambda\rightarrow 1$, Eq.
(32) returns to that in the general relativity.
To describe a real physical system, one has to give a specific form of
$\lambda(r)$. In this paper we consider
$\displaystyle\lambda(r)=1-\frac{GM}{r_{s}}\left(1+\frac{r}{r_{e}}\right)e^{-\frac{r}{r_{e}}}\,.$
(33)
$r_{s}$ and $r_{e}$ parameterize the physical scales of the system. As we
stated before, one of the restrictions for $\lambda(r)$ is to ensure that
$|\tilde{b}_{r}|<1$. It is shown in Section 4 that (33) satisfies this
condition. Given (33), one can solve 333See the Appendix for details. the
equation of motion (32), which is derived from (27). The result is
$\phi_{M}=-\frac{GM}{r}-\frac{GM}{r_{s}}\left(1+\frac{r}{r_{e}}\right)e^{-\frac{r}{r_{e}}}\,.$
(34)
The first term in (34) is the usual Newtonian potential and the last linear
term with an exponential cutoff is novel.
The particular function form (33) of the parameter $\lambda$ is inspired by
Grumiller’s work [22]. The effective potential in his paper was given as
$\phi_{M}=-\frac{GM}{r}+Dr\ ,$ (35)
where $D$ is constant and the linear term $Dr$ is called the Rindler
acceleration term. A more general form of (35) can be written as
$\phi_{M}=-\frac{GM}{r}+\tilde{f}(r)\ ,$ (36)
where $\tilde{f}(r)$ is a function of the distance scale $r$. For Grumiller’s
model, $\tilde{f}(r)=Dr$. And for the specific form of $\lambda(r)$ in (34),
$\tilde{f}(r)$ takes a form as
$\tilde{f}(r)=-\frac{GM}{r_{s}}\left(1+\frac{r}{r_{e}}\right)e^{-\frac{r}{r_{e}}}\,.$
(37)
It is a rescaled linear potential of $r$ like the Grumiller’s, but an
exponential cutoff is imposed to avoid possible divergence at large distances.
Grumiller’s potential does not confront with such a difficulty because he only
discussed the galactic physics. While we try to extrapolate the potential (35)
to the cluster scale, we do need to consider this problem. The effective
acceleration $a_{M}$ has two terms, also
$\displaystyle
a_{M}=-\frac{GM}{r^{2}}-\frac{GM}{r_{e}^{2}}\cdot\frac{r}{r_{s}}e^{-\frac{r}{r_{e}}}\,.$
(38)
At sufficiently large distances, the second term may become dominant and
provides a linear acceleration towards the source.
As in the general relativity, one integrates Eq. (32) and obtains the
deflection angle of light $\alpha_{\textmd{\small R}}$ in a modified Rindler
potential in Randers-Finslerian spacetime, to wit
$\displaystyle\alpha_{\textmd{\small R}}(r)=\frac{4GM}{r}f(r;r_{s},r_{e})\ ,$
(39)
where
$\displaystyle f(r;r_{s},r_{e})\equiv
1-\frac{1}{2r_{s}}\int_{r}^{\infty}\frac{\frac{r^{2}}{r^{\prime
2}}}{\sqrt{1-\frac{r^{2}}{r^{\prime 2}}}}\frac{\left(2+\frac{r^{2}}{r^{\prime
2}}\right)\left(1+\frac{r^{\prime}}{r_{e}}\right)e^{-\frac{r^{\prime}}{r_{e}}}-3\left(1+\frac{r}{r_{e}}\right)e^{-\frac{r}{r_{e}}}}{2\left(1-\frac{r^{2}}{r^{\prime
2}}\right)}dr^{\prime}\,.$ (40)
This integration can be computed numerically. The model parameters $r_{s}$ and
the cutoff scale $r_{e}$ depend on the specific gravitational system and are
to be determined by observations. For $r\gg r_{e}$,
$\phi_{M}\rightarrow-\frac{GM}{r}$ and $\alpha_{\textmd{\small
R}}\rightarrow\frac{4GM}{r}$. This is what we expect in general relativity and
the Newtonian limit.
## 3 Comparing with the Observations
In this section, we use the modified gravity model to calculate the
convergence $\kappa$ of the Bullet Cluster 1E0657-558. Before this, we first
get the effective lens potential in the Randers-Finslerian spacetime (27). We
then use the potential to calculate the convergence.
### 3.1 Effective Lens Potential
We take a “leap” here. We do not deduce but give the the effective lens
potential $\bar{\psi}$ in the Randers-Finslerian spacetime that will generate
the deflection angle (39). Then we use $\bar{\psi}$ to calculate the
corresponding convergence $\kappa$. Hereafter, we use natural units in
calculations, i.e. setting the speed of light $c=1$.
Einstein’s general relativity predicts that a light ray passing by a spherical
body of mass $M$ at a minimum distance $\xi$ is deflected by the angle
$\displaystyle\alpha=\frac{4GM}{\xi}\
,~{}~{}~{}~{}~{}\xi\equiv\sqrt{x^{2}+y^{2}}\,.$ (41)
The mass of the lens $M$ can be given as
$\displaystyle
M(\xi)=2\pi\int_{0}^{\xi}\Sigma(\xi^{\prime})\xi^{\prime}d\xi^{\prime}\ ,$
(42)
where $\Sigma(\xi^{\prime})$ is the surface mass density distribution. It
results from projecting the volume mass distribution of the “lens” $\rho(r)$
onto the lens plane (i.e. the $(x,y)$-plane) which is orthogonal to the line-
of-sight direction (i.e. the $z$-direction) of the observer, to wit
$\displaystyle\Sigma(\xi)=\int_{-z_{\textmd{\small out}}}^{z_{\textmd{\small
out}}}\rho(r)dz\ ,$ (43)
where $z\equiv\sqrt{r^{2}-x^{2}-y^{2}}=\sqrt{r^{2}-\xi^{2}}$ and
$z_{\textmd{\small out}}\equiv\sqrt{r_{\textmd{\small out}}^{2}-\xi^{2}}$.
$r_{\textmd{\small out}}$ denotes the outer radial extent of the galaxy
cluster, which is defined as when $\rho$ drops to $\rho(r_{\textmd{\small
out}})\simeq 10^{-28}~{}\textmd{g}/\textmd{cm}^{3}$.
The “Einstein angle” (41) can be rewritten in a vector form as [45]
$\displaystyle\hat{\alpha}=4G\int_{\textmd{\small
R}^{2}}d^{2}\vec{\xi}^{\prime}\Sigma(\vec{\xi}^{\prime})\frac{\vec{\xi}-\vec{\xi}^{\prime}}{~{}|\vec{\xi}-\vec{\xi}^{\prime}|^{2}}\
,$ (44)
where
$\displaystyle
d^{2}\vec{\xi}^{\prime}=\int_{0}^{2\pi}d\varphi\int_{0}^{\xi}d\xi^{\prime}\vec{\xi}^{\prime}$
(45)
is the surface element of the lens plane.
With $\vec{\theta}=\frac{\vec{\xi}}{D_{\textmd{\small L}}}$, one can easily
check that (44) satisfies 444 In the two-dimensional polar coordinates,
$\nabla_{\vec{\xi}}\equiv\frac{\partial~{}}{\partial\vec{\xi}}=\hat{\mathbf{e}}_{\xi}\frac{\partial~{}}{\partial\xi}$
. (see Section 4.1 in [46])
$\displaystyle\hat{\alpha}=\frac{D_{\textmd{\small
S}}}{D_{\textmd{\small{LS}}}}\nabla_{\theta}\psi(\vec{\theta})=\frac{D_{\textmd{\small
S}}D_{\textmd{\small
L}}}{D_{\textmd{\small{LS}}}}\nabla_{\vec{\xi}}~{}\psi(\vec{\xi})\ ,$ (46)
where
$\displaystyle\psi(\vec{\xi})=\frac{1}{\pi\Sigma_{\textmd{c}}}\int_{\textmd{\small
R}^{2}}\Sigma(\vec{\xi}^{\prime})~{}\textmd{ln}|\vec{\xi}-\vec{\xi}^{\prime}|~{}d^{2}\vec{\xi}^{\prime}\
,~{}~{}~{}~{}~{}\Sigma_{\textmd{c}}\equiv\frac{D_{\textmd{\small S}}}{4\pi
GD_{\textmd{\small L}}D_{\textmd{\small{LS}}}}\,.$ (47)
$\Sigma_{\textmd{c}}$ is the critical surface density of the lens.
$D_{\textmd{\small S}}$ is the angular distance between the observer and the
source galaxy, i.e. the background. $D_{\textmd{\small L}}$ is the angular
distance between the observer and the lens, i.e. the Bullet Cluster
1E0657-558, and $D_{\textmd{\small LS}}$ denotes the angular distance between
the lens and the source galaxy. The lens potential $\psi(\vec{\xi})$ obeys the
two-dimensional Poisson’s equation 555In general, the Laplacian $\Delta$ in
polar coordinates is given as
$\displaystyle\Delta\equiv\frac{1}{\xi}\frac{\partial~{}}{\partial\xi}\left(\xi\frac{\partial~{}}{\partial\xi}\right)+\frac{1}{\xi^{2}}\frac{\partial^{2}~{}}{\partial\varphi^{2}}\
.$ For a $\varphi$-independent $\psi(\vec{\xi})$, one has
$\frac{\partial\psi(\vec{\xi})}{\partial\varphi}=0$, and
$\displaystyle\Delta\psi\equiv\frac{1}{\xi}\frac{\partial~{}}{\partial\xi}\left(\xi\frac{\partial\psi}{\partial\xi}\right)\
.$
$\displaystyle\Delta\psi\equiv\nabla^{2}\psi=2\frac{\Sigma}{\Sigma_{\textmd{c}}}\
,~{}~{}~{}~{}\Delta\equiv\frac{1}{\xi}\frac{\partial~{}}{\partial\xi}\left(\xi\frac{\partial~{}}{\partial\xi}\right)\
,$ (48)
In astronomy and astrophysics, the quantity
$\frac{\Sigma}{\Sigma_{\textmd{c}}}$ in Eq. (48) is defined as the convergence
$\kappa$, which is also called the scaled surface mass density, i.e.
$\displaystyle\kappa\equiv\frac{\Sigma}{\Sigma_{\textmd{c}}}\,.$ (49)
Consider a lens potential
$\displaystyle\bar{\psi}(\vec{\xi})\equiv\psi(\vec{\xi})f(\vec{\xi};r_{s},r_{e})=\frac{1}{\pi\Sigma_{\textmd{c}}}\int_{\textmd{\small
R}^{2}}\Sigma(\vec{\xi}^{\prime})f(\vec{\xi};r_{s},r_{e})~{}\textmd{ln}|\vec{\xi}-\vec{\xi}^{\prime}|~{}d^{2}\vec{\xi}^{\prime}\
,$ (50)
where
$\displaystyle f(\vec{\xi};r_{s},r_{e})\equiv\int_{-z_{\textmd{\small
out}}}^{z_{\textmd{\small out}}}f(r;r_{s},r_{e})dz$ (51)
and $f(r;r_{s},r_{e})$ is given by (40). For the inner of the lens system, we
have $\xi=\xi^{\prime}$. Thus, the potential (50) can be rewritten as
$\displaystyle\bar{\psi}(\vec{\xi})$ $\displaystyle=$
$\displaystyle\frac{1}{\pi\Sigma_{\textmd{c}}}\int_{\textmd{\small
R}^{2}}\Sigma(\vec{\xi}^{\prime})f(\vec{\xi}^{\prime};r_{s},r_{e})~{}\textmd{ln}|\vec{\xi}-\vec{\xi}^{\prime}|~{}d^{2}\vec{\xi}^{\prime}$
(52) $\displaystyle\equiv$
$\displaystyle\frac{1}{\pi\Sigma_{\textmd{c}}}\int_{\textmd{\small
R}^{2}}\bar{\Sigma}(\vec{\xi}^{\prime})~{}\textmd{ln}|\vec{\xi}-\vec{\xi}^{\prime}|~{}d^{2}\vec{\xi}^{\prime}\,.$
(53)
Given the potential (53) and using Eq. (46), one can reproduce the deflection
angle $\alpha_{\textmd{\small R}}$ in the model, i.e.
$\displaystyle\alpha_{\textmd{\small
R}}(\xi)=\frac{4GM}{\xi}f(\xi;r_{s},r_{0})\,.$ (54)
### 3.2 The $\Sigma$\- and $\kappa$-Map of Bullet Cluster 1E0657-558
(a) $\Sigma$-Map
(b) Section of $\Sigma$-Map
Figure 1: The $\Sigma$-map from X-ray imaging observations of the Bullet
Cluster 1E0657-558, November 15, 2006 data release. (a) The entire
$\Sigma$-map is presented in the equatorial coordinate system J2000. DEC in
the $y$-axis is short for “Declination” and the RA in the $x$-axis is short
for “Right Ascension”. The bright shockwave region at the right half of the
map is the ICM gas of the subcluster. The main cluster gas locates at the
brightly glowing region to the left of the subcluster gas. The released
$\Sigma$-map has $185\times 185$ pixels and a resolution of $8.5$ kpc/pixel.
(b) A subset of the $\Sigma$-map on a straight-line connecting the peak of the
main cluster to that of the subcluster. The peak of the main cluster is taken
to be the referential center of the system, i.e. $\xi=0$ . The peak of the
subcluster is located at $\xi\simeq 398$ kpc.
To calculate the convergence $\kappa$, one needs the surface mass density
distribution $\Sigma(\xi)$ of the specific system. The $\Sigma$-map
reconstructed from X-ray imaging observations of the Bullet Cluster 1E0657-558
is shown in Figure 1a. There are two distinct glowing peaks in Figure 1a – the
left one of the main cluster and the right one of the subcluster. A subset of
the $\Sigma$-map on a straight-line connecting the peak of the main cluster to
that of the subcluster is shown in Figure 1b.
For the Bullet Cluster system, the volume mass distribution of the ICM gas of
the main cluster $\rho(r)$ is phenomenologically described by the King
$\beta$-model [47, 48, 49]
$\displaystyle\rho(r)=\rho_{0}\left[1+\left(\frac{r}{r_{c}}\right)^{2}\right]^{-3\beta/2}\
,~{}~{}~{}~{}r=\sqrt{x^{2}+y^{2}+z^{2}}\equiv\sqrt{\xi^{2}+z^{2}}\ ,$ (55)
where the parameters $\rho_{0}$, $r_{c}$ and $\beta$ are determined to be [21]
$\displaystyle\rho_{0}$ $\displaystyle=$ $\displaystyle 3.34\times
10^{5}~{}~{}M_{\odot}/\textmd{kpc}^{3}\ ,$ (56) $\displaystyle\beta$
$\displaystyle=$ $\displaystyle 0.803\pm 0.013\ ,$ (57) $\displaystyle r_{c}$
$\displaystyle=$ $\displaystyle 278.0\pm 6.8~{}~{}\textmd{kpc}\,.$ (58)
$M_{\odot}$ denotes the mass of the sun.
The outer radial extent of the Bullet Cluster system is given as
$\displaystyle r_{\textmd{\small
out}}=r_{c}\left[\left(\frac{\rho_{0}}{10^{-28}~{}\textmd{g}/\textmd{cm}^{3}}\right)^{-2/3\beta}-1\right]^{1/2}\simeq
2620~{}~{}\textmd{kpc}\,.$ (59)
The radius of the main cluster is $\sim 1000$ kpc, thus we have
$\xi=\xi^{\prime}$. The potential (53) now becomes
$\displaystyle\bar{\psi}(\vec{\xi})=\frac{1}{\pi\Sigma_{\textmd{c}}}\int_{\textmd{\small
R}^{2}}\bar{\Sigma}(\vec{\xi}^{\prime})~{}\textmd{ln}|\vec{\xi}-\vec{\xi}^{\prime}|~{}d^{2}\vec{\xi}^{\prime}\
,$ (60)
where the effective surface mass density $\bar{\Sigma}(\xi)$ is defined as
$\displaystyle\bar{\Sigma}(\xi)$ $\displaystyle\equiv$
$\displaystyle\int_{-z_{\textmd{\small out}}}^{z_{\textmd{\small
out}}}\rho(r)f(r;r_{s},r_{e})dz\ ,~{}~{}~{}~{}~{}z_{\textmd{\small
out}}=\sqrt{r_{\textmd{\small
out}}^{2}-\xi^{2}}=\sqrt{2620^{2}-\xi^{2}}~{}~{}~{}\textmd{kpc}\,.$ (61)
Making use of (40), (49), (55) and (61), one finally obtains the convergence
$\kappa$-map of the Bullet Cluster system
$\displaystyle\bar{\kappa}(\xi)\equiv\frac{\bar{\Sigma}(\xi)}{\Sigma_{\textmd{c}}}$
$\displaystyle=$
$\displaystyle\frac{\rho_{0}}{\Sigma_{\textmd{c}}}\int_{-z_{\textmd{\small
out}}}^{z_{\textmd{\small
out}}}\left[1+\left(\frac{r}{r_{c}}\right)^{2}\right]^{-3\beta/2}f(r;r_{s},r_{e})dz\
,$ (62)
where
$\displaystyle f(r;r_{s},r_{e})\equiv
1-\frac{1}{2r_{s}}\int_{r}^{\infty}\frac{\frac{r^{2}}{r^{\prime
2}}}{\sqrt{1-\frac{r^{2}}{r^{\prime 2}}}}\frac{\left(2+\frac{r^{2}}{r^{\prime
2}}\right)\left(1+\frac{r^{\prime}}{r_{e}}\right)e^{-\frac{r^{\prime}}{r_{e}}}-3\left(1+\frac{r}{r_{e}}\right)e^{-\frac{r}{r_{e}}}}{2\left(1-\frac{r^{2}}{r^{\prime
2}}\right)}dr^{\prime}$ (63)
and
the parameters $\rho_{0}$, $r_{c}$ and $\beta$ are given in (56) to (58),
$z=\sqrt{r^{2}-x^{2}-y^{2}}\equiv\sqrt{r^{2}-\xi^{2}}$ and $z_{\textmd{\small
out}}$ is given by (61),
for the Bullet Cluster 1E0657-558, one has $\frac{D_{\textmd{\small
L}}D_{\textmd{\small LS}}}{D_{\textmd{\small S}}}\simeq 540$ kpc. So
$\Sigma_{\textmd{c}}$ in (62) takes a value of
$\displaystyle\Sigma_{\textmd{c}}\equiv\frac{D_{\textmd{\small S}}}{4\pi
GD_{\textmd{\small L}}D_{\textmd{\small LS}}}\simeq 3.1\times
10^{9}~{}~{}M_{\odot}/\textmd{kpc}^{2}\ ,$ (64)
$r_{s}$ and $r_{e}$ are model parameters to be determined by fitting (62) to
the $\kappa$-map reconstructed from the gravitational lensing survey.
(a) $\kappa$-Map
(b) Section of $\kappa$-Map
Figure 2: The $\kappa$-map reconstructed from the strong and weak
gravitational lensing survey of the Bullet Cluster 1E0657-558, November 15,
2006 data release. (a) The entire $\kappa$-map is presented in the equatorial
coordinate system J2000. DEC in the $y$-axis is short for “Declination” and
the RA in the $x$-axis is short for “Right Ascension”. The bright blurred
region at the left half of the map illuminates the convergence of the main
cluster, while the smaller glowing one to the left corresponds to that of the
subcluster. The released $\kappa$-map has $110\times 110$ pixels and a
resolution of $15.4$ kpc/pixel. (b) A section of the $\kappa$-map on a
straight-line connecting the peak of the main cluster to that of the
subcluster. The peak of the main cluster is located at $\xi\simeq-180$ kpc and
that of the subcluster is located at $\xi\simeq 522$ kpc. The $\xi=0$ point is
chosen to be the same with that of the $\Sigma$-map in Figure 2b.
The $\kappa$-map obtained from the strong and weak gravitational lensing
survey of the Bullet Cluster 1E0657-558 is presented in Figure 2a. One can see
that the two distinct glowing regions in Figure 2a – the left one of the main
cluster and the right one of the subcluster – somewhat depart from those shown
in Figure 1a. A subset of the $\kappa$-map on a straight-line connecting the
peak of the main cluster to that of the subcluster is also shown in Figure 2b.
A section of the $\bar{\kappa}$-map (62), which crossing the two peaks is
plotted in Figure 3. For a qualitative illustration, different values of
parameter set $(r_{s},r_{e})$ are plotted for comparison instead of carrying a
best-fit. The “best-fit” values of parameters $r_{s}$ and $r_{e}$ with a $5\%$
error666Variation of $r_{s}$ and $r_{e}$ from their “best-fit” values leads to
a deviation of $\Delta M/M_{K}$ and $\kappa$ from their extremal points (see
Table 3.3). We consider both of these deviations of $\Delta M/M_{K}$ and
$\kappa$ within a $5\%$ level to obtain the corresponding confidence regions
of $r_{s}$ and $r_{e}$. Gaussian prior distributions of the parameters are
assumed. are presented in Table 3.3. Plot for $f(r;r_{s},r_{e})$ is presented
in Figure 6(a). Our approach follows a sequence of approximations:
* •
Take the main cluster thermal profile to be isothermal.
* •
Neglect the subcluster for zeroth order approximation.
* •
Perform the fit using a section of the $\kappa$-map on a straight-line
connecting the peak of the main cluster to that of the subcluster and then
extrapolating it to the entire map.
* •
Take the $\Sigma$-peak of the main cluster as the center of the gravitational
system, and project the section of the $\kappa$-map onto that of the
$\Sigma$-map to make the two overlay for comparison.
Figure 3: Cross sections of the model-predicted $\bar{\kappa}$-map and the
$\Sigma$-, $\kappa$-map reconstructed from the November 15, 2006 data release.
The solid and dashed lines denote the sections of the $\bar{\kappa}$-map (62)
predicted by the Randers-Finslerian model with a modified Rindler potential
(34) for parameters ($r_{s}$, $r_{e}$) listed in Table 3.3. The sections of
the $\Sigma$\- and $\kappa$-map obtained by observations are respectively
represented by small black dots and circles as in Figure 1b and 2b.
### 3.3 The Isothermal Temperature Profile
Besides the convergence $\kappa$, the surface temperature $T$ of the cluster
obtained from the X-ray spectrum analysis is also another observed quantity
which should be used to constrain a model. Assuming an isotropic and
isothermal gas profile with temperature $T$, one can calculate dynamical mass
$M_{\textmd{T}}$ of the main cluster as a function of the radial position $r$
and the temperature $T$. By comparing it with the result given by integrating
the King $\beta$-model (55), we obtain a more rigorous constraint on the model
parameters $r_{s}$ and $r_{e}$.
The collisionless Boltzmann equation of a spherical system in hydrostatic
equilibrium reads
$\displaystyle\frac{d}{dr}(\rho(r)\sigma_{r}^{2})+\frac{2\rho(r)}{r}\left(\sigma_{r}^{2}-\sigma_{\theta,\phi}^{2}\right)=-\rho(r)\frac{d\Phi(r)}{dr}\
,$ (65)
where $\Phi(r)$ is the gravitational potential of the system and $\sigma_{r}$
and $\sigma_{\theta,\phi}$ are respectively the mass-weighted velocity
dispersions in the radial and ($\theta,\phi$) directions. Given an isotropic
gas sphere distribution $\rho(r)$ with a temperature profile $T(r)$, one has
$\displaystyle\sigma_{r}^{2}=\sigma_{\theta,\phi}^{2}=\frac{k_{B}T(r)}{\mu_{A}m_{p}}\
,$ (66)
where $k_{B}$ is Boltzmann’s constant, $\mu_{A}\simeq 0.609$ is the mean
atomic weight and $m_{p}$ is the proton mass. Eq. (65) becomes
$\displaystyle\frac{d}{dr}\left(\frac{k_{B}T(r)}{\mu_{A}m_{p}}\rho(r)\right)=-\rho(r)\frac{d\Phi(r)}{dr}\,.$
(67)
For the main cluster of the Bullet Cluster system, the ICM gas distribution
$\rho(r)$ is fit by an isotropic and isothermal King $\beta$-model (55) with
the temperature $T(r)=T$. Solving Eq. (67) for the gravitational acceleration,
one obtains
$\displaystyle a(r)\equiv-\frac{d\Phi(r)}{dr}$ $\displaystyle=$
$\displaystyle\frac{k_{B}T}{\mu_{A}m_{p}r}\left[\frac{d\ln(\rho(r))}{d\ln(r)}\right]$
(68) $\displaystyle=$ $\displaystyle-\frac{3\beta
k_{B}T}{\mu_{A}m_{p}}\left(\frac{r}{r^{2}+r_{c}^{2}}\right)\,.$
Replacing $a(r)$ in (68) with the effective acceleration $a_{M}$ in (38), to
wit
$\displaystyle
a(r)=a_{M}(r)=-\frac{GM_{T}}{r^{2}}\left(1+\frac{r^{3}}{r_{e}^{2}r_{s}}e^{-\frac{r}{r_{e}}}\right)\
,$ (69)
we obtain the relation between the dynamical mass $M_{\textmd{T}}$ as a
function of the radial position $r$ and the temperature $T$, to wit
$\displaystyle M_{\textmd{T}}(r)=\frac{3\beta
k_{B}T}{\mu_{A}m_{p}G}\left(\frac{r^{3}}{r^{2}+r_{c}^{2}}\right)\cdot\left(1+\frac{r^{3}}{r_{e}^{2}r_{s}}e^{-\frac{r}{r_{e}}}\right)^{-1}\,.$
(70)
On the other hand, the mass profile of the main cluster is given by the King
$\beta$-model as
$\displaystyle M_{\textmd{K}}(r)$ $\displaystyle=$ $\displaystyle
4\pi\int_{0}^{r}\rho(r^{\prime})r^{\prime 2}dr^{\prime}$ (71) $\displaystyle=$
$\displaystyle
4\pi\rho_{0}\int_{0}^{r}\left[1+\left(\frac{r^{\prime}}{r_{c}}\right)^{2}\right]^{-3\beta/2}r^{\prime
2}dr^{\prime}\,.$
The detection in X-ray by the Einstein IPC, ROSAT and ASCA observations
constrained the temperature of the main cluster to be $T=17.4\pm 2.5$ keV
(with $12.3\%$ error) [50] and $T=14.5_{-2.0}^{+1.7}$ keV (with $6.5\%$ error)
[51]. It was later reported by Markevitch [11] that $T=14.8_{-1.2}^{+1.7}$ keV
(with $4.5\%$ error). Fixing the temperature $T$ in (70) to be the observed
center value $T=14.8$ keV and by comparing the two $M(r)$ in (71) and (70) at
the radial distance $r=1000$ kpc (which is also the boundary of the
reconstructed $\kappa$\- and $\Sigma$-map), one can put a constraint on the
model parameters $r_{s}$ and $r_{e}$. The results are presented in Table 3.3
and Figure 3.
Mass discrepancies of different parameter set $(r_{s},r_{e})$. The isothermal
temperature of main cluster is fixed to be $T=14.8$ keV as reported by
Markevitch. ‘$\Delta M$’ represents the mass difference between (70) and (71),
i.e. $\Delta m\equiv|M_{T}-M_{K}|$. The last column presents the peak values
of the $\kappa$-map given by (62) at $r\sim 180$ kpc. The “best-fit” result of
parameters $(r_{s},r_{e})$ are highlighted in boldface in the second row. The
first and third row show that a variation of $r_{s}$ near
$(r_{s},r_{e})=(25,207)$ will leads to a bad $\Delta M$ and $kappa$. The
fourth and fifth row show the same result for a variation of $r_{e}$ near
$(r_{s},r_{e})=(25,207)$. The errors of the “best-fit” result are given by
considering a $5\%$ deviation of both $\Delta M/M_{K}$ and $\kappa$ from their
extremal values. $T$ $r_{s}$ $r_{e}$ $\Delta M/M_{K}$ $\kappa$ (keV) (kpc)
(kpc) (%) (peak values) 14.8 20 207 21.37 0.48 14.8 25$\pm$2.40 207$\pm$11.15
0.01 0.38 14.8 30 207 15.16 0.34 14.8 25 180 32.05 0.37 14.8 25 235 32.74 0.43
### 3.4 A Randers Plus Dark Matter Model
Stavrinos et al.’s work [25] showed that the Randers-type spacetime does not
forbid the existence of dark matter in cosmology. Thus it would be interesting
to consider dark matter in the Randers-Finslerian spacetime. We consider the
most popular Navarro-Frenk-White(NFW) profile of the dark matter [52, 53]. The
mass density in (61) is now given as
$\displaystyle\rho(r)=\rho_{\textmd{K}}(r)+\rho_{\textmd{DM}}(r)\ ,$ (72)
where
$\displaystyle\rho_{\textmd{DM}}(r)=\frac{\rho_{d}r_{d}^{3}}{r^{3}+r_{d}^{3}}\,.$
(73)
$\rho_{d}$ is the central dark matter density and $r_{d}$ is the core radius.
Now the convergence $\bar{\kappa}$ is given as
$\displaystyle\bar{\kappa}(\xi)\equiv\frac{\bar{\Sigma}(\xi)}{\Sigma_{\textmd{c}}}$
$\displaystyle=$
$\displaystyle\frac{1}{\Sigma_{\textmd{c}}}\int_{-z_{\textmd{\small
out}}}^{z_{\textmd{\small
out}}}\left[\rho_{\textmd{K}}(r)+\rho_{\textmd{DM}}(r)\right]f(r;r_{s},r_{e})dz\
,$ (74)
where $\rho_{\textmd{K}}(r)$ is given by (55) and $\rho_{\textmd{DM}}(r)$ is
given by (73). From WMAP’s seven-year result [54], we know that the total
amount of matter (or energy) in the universe in the form of dark energy about
$73\%$ and dark matter about $23\%$ . This leaves the ratio of baryonic matter
at only $\sim 4\%$. The ratio can be parameterized as $\eta\equiv
M_{\textmd{DM}}/M_{\textmd{b}}$, where $M_{\textmd{DM}}$ denotes the total
volume mass of dark matter in a region and $M_{\textmd{b}}$ refers to that for
ordinary baryonic matter. In this paper, we fix this ratio to be $\eta=6$.
Given (73) and (55), one can integrate to get $\rho_{d}$ as a function of
$\rho_{0}$ and $\eta$, i.e. $\rho_{d}=\rho_{d}(\rho_{0},\eta;r_{d})$, leaving
$r_{d}$ the only free parameter in the NFW profile in our model. The
convergence $\bar{\kappa}$ and the mass profile of the main cluster are now
given as
$\displaystyle\bar{\kappa}(\xi)\equiv\frac{\bar{\Sigma}(\xi)}{\Sigma_{\textmd{c}}}$
$\displaystyle=$
$\displaystyle\frac{1}{\Sigma_{\textmd{c}}}\int_{-z_{\textmd{\small
out}}}^{z_{\textmd{\small
out}}}\left[\rho_{\textmd{K}}(r)+\rho_{\textmd{DM}}(r;r_{d},\eta)\right]f(r;r_{s},r_{e})dz\
,$ (75)
and
$\displaystyle M_{\textmd{K}}(r)$ $\displaystyle=$ $\displaystyle
4\pi\int_{0}^{r}\left(\rho_{\textmd{K}}(r^{\prime})+\rho_{\textmd{DM}}(r^{\prime};r_{d},\eta)\right)r^{\prime
2}dr^{\prime}\,.$ (76)
Thus for the Randers$+$dark matter model, we have three free parameters
$r_{s}$, $r_{e}$ and $r_{d}$. The numerical results are given in Figure 4 and
Table 3.4.
In Table 3.4, the first three rows show that we take a declining journey of
$r_{d}$ to get a less mass discrepancy $\Delta M$ at the cost of a rapidly
rising $\kappa$. (A small $r_{d}$ means a more condense dark matter core and a
more sparse outskirt for the NFW profile.) Such a result means that we have
added too much dark matter into the core of the main cluster thus the $\kappa$
flies. Then in the fourth row, we strip out the Finslerian effect, leaving
only the dark matter and the baryonic matter, by setting
$(r_{d},r_{s},r_{e})=(440,1000,2)$ (large $r_{s}$ and small $r_{e}$ will
radically suppress the Finslerian effect at large distances, for in (33)
$\lambda\rightarrow 1$.) It still yields too large $\kappa$ ($\simeq 0.48$)
compared to the observed value $\kappa\simeq 0.38$. This result implies that
an averaged distribution density of cold dark matter in cosmological senses
fails to reproduce the observed convergence $\kappa$ of the Bullet Cluster.
Instead we take another approach to fill up the mass discrepancy shown in the
first row: we tune up $r_{e}$ to “turn on” the Finslerian effect to fill up
the “mass gap” $\Delta M$ at the cluster center. But the results in the last
two rows demonstrate that this way does not work too. For one to obtain a
ideal $\Delta M$, the convergence $\kappa$ have greatly exceeded the observed
value. Thus for a Randers$+$dark matter model, the mass discrepancy $\Delta M$
and the convergence $\kappa$ is like the two ends of a see saw. It can not
both be lowered at the same time. One possible reason for this may be that a
dark matter-to-baryonic matter ratio $\eta\simeq 6$ is too large. Another sign
of this is that at the center of the main cluster, the compound model fails to
reproduce the gravitational potential offset from the mass center. The
Finslerian effect seems to be overwhelmed by the dark matter background. Since
the mass ratio of dark matter and its type are not the subjects of this paper,
we will not discuss it here. $f(r;r_{s},r_{e})$ for different parameters are
plotted in Figure 6.
To compare with the Randers and Randers$+$dark matter models, we also plot the
results for the concordance $\Lambda$-CDM cosmological model [54]. This can be
implemented by setting $r_{s}\rightarrow\infty$ or/and $r_{e}\rightarrow 0$ in
the equation (75). It will result in $f(r;r_{s},r_{e})=1$ and leave us the
convergence $\kappa$ in a $\Lambda$-CDM model:
$\displaystyle\bar{\kappa}(\xi)\equiv\frac{\bar{\Sigma}(\xi)}{\Sigma_{\textmd{c}}}$
$\displaystyle=$
$\displaystyle\frac{1}{\Sigma_{\textmd{c}}}\int_{-z_{\textmd{\small
out}}}^{z_{\textmd{\small
out}}}\left[\rho_{\textmd{K}}(r)+\rho_{\textmd{DM}}(r;r_{d},\eta)\right]dz\ ,$
(77)
Together with (76), we give our numerical results in Table 3.4. Two comments
should be given about the results: First, it fails to give a reasonable ($\leq
5\%$) mass discrepancy $\Delta M/M_{\textmd{K}}$ together with an
observations-compatible convergence $\kappa$ (highlighted in boldface
respectively in Table 3.4). Second, the $\Delta M/M_{\textmd{K}}$ and $\kappa$
we get for $(r_{s},r_{e})=(\infty,0)$ are not so much different from those in
the first and fourth row in Table 3.4. One reason for this is that setting
$(r_{s},r_{e})=(1000,2)$ is already enough for one to strip out the Finslerian
impacts on the dynamical mass and the convergence $\kappa$. The other one is
that at the center of the main cluster, the Finsler effects are “drowned” by
the dark matter background with a dark matter-to-baryons mass ratio $\eta\sim
6$, just like the case in the Randers+dark matter model. We plot the results
of the $\Lambda$-CDM model in Figure 4 for comparison.
Figure 4: Cross sections of the model-predicted $\bar{\kappa}$-map and the
$\Sigma$-, $\kappa$-map reconstructed from the November 15, 2006 data release.
The solid and dashed lines except the bottom one denote the sections of the
$\bar{\kappa}$-map (75) predicted by the Randers$+$dark matter model with
parameters ($r_{d}$, $r_{s}$, $r_{e}$) listed in Table 3.4. The red dashed
line represents the “best-fit” result for the $\Lambda$-CDM model (see Table
3.4 and the discussions in the last paragraph of subsection 3.4). The sections
of the $\Sigma$\- and $\kappa$-map obtained by observations are respectively
represented by small black dots and circles as in Figure 1b and 2b.
Mass discrepancies of different parameter set $(r_{d},r_{s},r_{e})$. The
isothermal temperature of main cluster is fixed to be $T=14.8$ keV as reported
by Markevitch. ‘$\eta$’ is mass ratio between the baryonic matter and the non-
baryonic dark matter ‘$\Delta M$’ represents the mass difference between (70)
and (76), i.e. $\Delta M=|M_{\textmd{T}}-M_{\textmd{K}}|$. The last column
presents the peak values of the $\kappa$-map given by (75) at $r\sim 180$ kpc.
$T$ $\eta$ $r_{d}$ $r_{s}$ $r_{e}$ $\Delta M/M_{\textmd{K}}$ $\kappa$ (keV)
(kpc) (kpc) (kpc) (%) (peak values) 14.8 6 530 490 25 9.13 0.39 14.8 6 470 490
25 3.08 0.42 14.8 6 440 490 25 0.05 0.49 14.8 6 440 1000 2 0.04 0.48 14.8 6
530 490 100 8.29 0.43 14.8 6 530 490 148 0.07 0.46
(a) Randers model only
(b) Randers model $+$ dark matter
Figure 5: Plot for the dimensionless Finslerian factor $f(r)$ in Eq. (40) vs.
the radial distance $r$ in unit of kpc. (a) is for Randers model without any
dark matter. (b) is for the Randers$+$dark matter model. The parameter values
are the “best-fit” value which are presented in boldface in Table 3.3 and
Table 3.4.
Mass discrepancies for the $\Lambda$-CDM model. The isothermal temperature of
main cluster is fixed to be $T=14.8$ keV as reported by Markevitch. ‘$\eta$’
is mass ratio between the baryonic matter and the non-baryonic dark matter
‘$\Delta M$’ represents the mass difference between (70) and (76), i.e.
$\Delta M=|M_{\textmd{T}}-M_{\textmd{K}}|$. The last column presents the peak
values of the $\kappa$-map given by (77) at $r\sim 180$ kpc. Reasonable
results are highlighted in boldface. The result in the first row is plotted in
Figure 4 for comparison. $T$ $\eta$ $r_{d}$ $r_{s}$ $r_{e}$ $\Delta
M/M_{\textmd{K}}$ $\kappa$ (keV) (kpc) (kpc) (kpc) (%) (peak values) 14.8 6
530 $\infty$ 0 9.25 0.38 14.8 6 440 $\infty$ 0 0.04 0.48
### 3.5 The Galactic Regime
The specific form of $\lambda(r)$ in (33) is postulated at cluster scales. It
would be interesting to see its galactic-scale behaviors. The potential (34)
is given by solving the equation of motion (32) which is derived from (27)
(see the Appendix). It recovers some features of the galactic rotation curves
predicted by Grumiller’s model, which was considered to be a good
phenomenological fit to the observational data [22]. From $v\sim\sqrt{ar}$ and
(38), we obtain a new formula for the velocity profile of a galaxy:
$v(r)\simeq\sqrt{\frac{GM}{r}+\frac{GM}{r_{s}}\left(\frac{r}{r_{e}}\right)^{2}e^{-\frac{r}{r_{e}}}}\,.$
(78)
To describe galaxies, we assume that the total mass $M\simeq 10^{11}M_{\odot}$
(instead of $M\simeq 10^{14}M_{\odot}$ for the Bullet Cluster system). For a
qualitative illustration, the plot of profile (78) for $r_{s}\simeq 1$ kpc and
the cutoff scale $r_{e}\simeq 80$ kpc is shown in Figure 6. From the figure,
we can see that our model in galactic limit yields an approximately flattened
rotation curve of spiral galaxy. It is qualitatively consistent with the MOND
and Grumiller’s model. The velocity scale where the rotation curve flattens is
$\sim 240$ km/s, which is in reasonable agreement with Grumiller’s prediction
and the observational data. A possible divergence of the velocity (78) at
large radial distances is reconciled by the exponential factor to yield a
physical result.
Figure 6: Rotation curves of a spiral galaxy $v(r)$ vs. $r$ in unit m/s vs. m
($1$ kpc $\simeq 3\times 10^{19}$ m). The dashed line denotes the velocity
profile predicted by Grumiller (which is qualitatively compatible with the
MOND at the distance scale of several kpcs and considered a good
phenomenological fit to the observational data). The solid line denotes the
results of our model for the same total galactic mass. The dotted line which
sinks into the bottom is given by Newton’s theory, which fails to account for
the observations.
## 4 Conclusions and Discussions
As a cluster-scale generalization of Grumiller’s gravity model, we presented a
gravity model in a navigation scenario in Finslerian geometry [23, 26]. The
galactic limit of the model shared some qualitative features of Gumiller’s
result and the MOND. It yielded approximately the flatness of the rotational
velocity profile at the radial distance of several kpcs. It also gave
observations-compatible velocity scales for spiral galaxies at which the
curves become flattened.
We also studied the gravitational deflection of light in such a framework and
the deflection angle was obtained. The modified convergence $\kappa$ formula
of a galaxy cluster showed that the peak of the gravitational potential has
chances to lie on the outskirts of the baryonic mass center. For the Bullet
Cluster 1E0657-558 system, the later refers to the center of the ICM gas
profile of the main cluster. Taking the mass ratio between dark matter and
baryonic matter $\eta$ to be a factor of 6 and assuming an isotropic and
isothermal ICM gas profile with temperature $T=14.8$ keV (which is the center
value given by Markevitch et al.’s observations [11]), we used the
collisionless Boltzmann equation to calculate the dynamical mass
$M_{\textmd{T}}$ of the main cluster. We obtained a good match between
$M_{\textmd{T}}$ and that given by King $\beta$-model and simultaneously
ameliorated the shape of the convergence $\kappa$ curve. For comparison, we
also consider a Randers$+$dark matter model. Numerical results showed that it
fails to fill up the mass difference between $M_{\textmd{T}}$ and that given
by King $\beta$-model. A smaller $\eta$ seems to be able to reconcile this
dilemma. Similar results were also obtained for the concordance $\Lambda$-CDM
model. More careful investigations are needed for drawing a confirmative
conclusion.
A few comments should be given on the $\lambda$ in the action (27). First, for
a time-independent radial “wind” in the manifold, $\lambda$ is a function of
$r$. Any $\lambda(r)$ that giving a small-enough $|\tilde{b}^{r}|$ would be
considered valid in Finslerian geometry. But not all these mathematically
valid $\lambda(r)$ would be acceptable for constructing a physical model. A
both mathematically and physically valid $\lambda(r)$ should at least satisfy
the following conditions: 1)
$|\tilde{b}^{r}|=\sqrt{(1-\lambda(r))/h_{rr}}/\lambda(r)<1$, such that the
positivity of $F$ holds; 2) experiments- and observations-compatibility. The
$\lambda$ in (33) satisfies both of these conditions. For
$(r_{s},r_{e})=(25,207)$ and $(1,80)$ , $|\tilde{b}^{r}|\sim 10^{-13}\ll 1$.
Second, besides the mathematical validity of $\lambda(r)$ we chose, from
Appendix one can see that if we redefine $\lambda$ as
$\lambda=1-\frac{GM}{r_{s}}(1+\frac{r}{r_{e}})e^{-\frac{r}{r_{e}}}\equiv
1+\phi_{\lambda}$, the non-vanishing component of the geodesic equation will
give that the gravitational potential in Finslerian spacetime is
$\phi_{M}\equiv\left(\phi_{N}+\phi_{\lambda}\right)$ and
$\phi_{N}\equiv-\frac{GM}{r}$ is the Newtonian potential. It means that the
results have a close relationship with $\lambda$. On the other hand, as a
physical model, the specific form of $\lambda$ should be determined by the
local spacetime symmetry, which cannot be deduced from the gravity theory. It
is not the fruit but a prior stipulation of the theory. There is no physical
principle or equation to constrain its form. Professor Shen’s description of
Finsler geometry (private conversation) may help us in understanding this —
“Riemann geometry is ‘a white egg’, for the tangent manifold at each point on
the Riemannian manifold is isometric to a Minkowski spacetime. However,
Finsler geometry is ‘a colorful egg’, for the tangent manifolds at different
points of the Finsler manifold are not isometric to each other in general.” In
physics, it implies that our nature does not always prefer an isotropic
gravitational force. It is also “colorful”, as we have seen in case of Bullet
Cluster 1E0657-558.
Last but not the least, as a physical model at cluster scales, the
$\lambda(r)$ should be subject to more observational tests, just like the NFW
profile of the dark matter [52, 53]. A new challenge is posed by the Abell 520
cluster [55]. A combined constraint on the model should be carried out.
Relevant research are currently undertaken. We hope that it would help to
constrain the form of $\lambda$, which embodies the symmetry of Finsler
spacetime.
## Appendix
By combining the non-vanishing components of the geodesic equations (28), one
obtains the relation between the radial distant $r$ and the time $t$ [44],
$\displaystyle\frac{AE^{2}}{B^{2}}\left(\frac{dr}{dt}\right)^{2}+\frac{J^{2}\lambda}{r^{2}}-\frac{E^{2}}{B}=-C\
,$
where $A(r)\equiv\lambda^{-2}\left(1-\frac{2GM}{r}\right)^{-1}$ and
$B(r)\equiv\lambda^{2}\left(1-\frac{2GM}{r}\right)$. $E$ is an integration
constant (see [44] for details). For photons, the constant $C=0$. The above
equation can be rewritten as
$\displaystyle
A^{3}\left(\frac{dr}{dt}\right)^{2}+\frac{J^{2}\lambda}{r^{2}E^{2}}-\frac{1}{B}=0\,.$
(79)
In the Newtonian limit and the weak-field approximation, the quantities
$\frac{J^{2}}{r^{2}},\left(\frac{dr}{dt}\right)^{2},E^{2}-1,\frac{GM}{r}$ are
small. To first order of these quantities (remembering that the leading order
terms of $A$ and $B$ are 1), Eq. (79) becomes
$\displaystyle\left(\frac{dr}{dt}\right)^{2}+\frac{J^{2}}{r^{2}}-\frac{1}{B}=0\,.$
(80)
Redefining $\lambda$ in (33) as
$\lambda=1-\frac{GM}{r_{s}}(1+\frac{r}{r_{e}})e^{-\frac{r}{r_{e}}}\equiv
1+\phi_{\lambda}$, one has
$\displaystyle-\frac{1}{B}$ $\displaystyle\equiv$
$\displaystyle-\lambda^{-2}\frac{1}{1-\frac{2GM}{r}}$ (81) $\displaystyle=$
$\displaystyle-\frac{1}{\left(1+\phi_{\lambda}\right)^{2}}\frac{1}{1-\frac{2GM}{r}}$
$\displaystyle\simeq$
$\displaystyle-\left(1-2\phi_{\lambda}\right)\left(1+\frac{2GM}{r}\right)$
$\displaystyle\simeq$
$\displaystyle-\left(1+2\frac{GM}{r}-2\phi_{\lambda}\right)$ $\displaystyle=$
$\displaystyle-1+2\phi_{M}\ ,$
where $\phi_{M}\equiv\left(\phi_{N}+\phi_{\lambda}\right)$ and
$\phi_{N}\equiv-\frac{GM}{r}$ is the Newtonian potential. Substituting (81)
back into (80), one obtains
$\displaystyle\frac{1}{2}\left(\frac{dr}{dt}\right)^{2}+\frac{J^{2}}{2r^{2}}+\phi_{M}=\frac{1}{2}\
,$ (82)
where the effective Newtonian potential $\phi_{M}$ is given as
$\displaystyle\phi_{M}=-\frac{GM}{r}-\frac{GM}{r_{s}}\left(1+\frac{r}{r_{e}}\right)e^{-\frac{r_{e}}{r}}\,.$
(83)
We are grateful to Y.-G. Jiang and S. Wang for useful discussions. This work
was supported by the National Natural Science Fund of China under Grant No.
10875129 and No. 11075166.
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|
arxiv-papers
| 2011-10-18T07:26:07 |
2024-09-04T02:49:23.272213
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhe Chang, Ming-Hua Li, Hai-Nan Lin, and Xin Li",
"submitter": "Ming-Hua Li",
"url": "https://arxiv.org/abs/1110.3893"
}
|
1110.3970
|
# Search for $C\\!P$ violation in $D^{+}\rightarrow K^{-}K^{+}\pi^{+}$ decays
R. Aaij23, B. Adeva36, M. Adinolfi42, C. Adrover6, A. Affolder48, Z.
Ajaltouni5, J. Albrecht37, F. Alessio37, M. Alexander47, G. Alkhazov29, P.
Alvarez Cartelle36, A.A. Alves Jr22, S. Amato2, Y. Amhis38, J. Anderson39,
R.B. Appleby50, O. Aquines Gutierrez10, F. Archilli18,37, L. Arrabito53, A.
Artamonov 34, M. Artuso52,37, E. Aslanides6, G. Auriemma22,m, S. Bachmann11,
J.J. Back44, D.S. Bailey50, V. Balagura30,37, W. Baldini16, R.J. Barlow50, C.
Barschel37, S. Barsuk7, W. Barter43, A. Bates47, C. Bauer10, Th. Bauer23, A.
Bay38, I. Bediaga1, K. Belous34, I. Belyaev30,37, E. Ben-Haim8, M. Benayoun8,
G. Bencivenni18, S. Benson46, J. Benton42, R. Bernet39, M.-O. Bettler17, M.
van Beuzekom23, A. Bien11, S. Bifani12, A. Bizzeti17,h, P.M. Bjørnstad50, T.
Blake49, F. Blanc38, C. Blanks49, J. Blouw11, S. Blusk52, A. Bobrov33, V.
Bocci22, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi47, A. Borgia52,
T.J.V. Bowcock48, C. Bozzi16, T. Brambach9, J. van den Brand24, J.
Bressieux38, D. Brett50, S. Brisbane51, M. Britsch10, T. Britton52, N.H.
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Buytaert37, S. Cadeddu15, J.M. Caicedo Carvajal37, O. Callot7, M. Calvi20,j,
M. Calvo Gomez35,n, A. Camboni35, P. Campana18,37, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,37, A. Cardini15, L. Carson36, K. Carvalho
Akiba23, G. Casse48, M. Cattaneo37, M. Charles51, Ph. Charpentier37, N.
Chiapolini39, K. Ciba37, X. Cid Vidal36, G. Ciezarek49, P.E.L. Clarke46,37, M.
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Collins37, F. Constantin28, G. Conti38, A. Contu51, A. Cook42, M. Coombes42,
G. Corti37, G.A. Cowan38, R. Currie46, B. D’Almagne7, C. D’Ambrosio37, P.
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Voss10, K. Wacker9, S. Wandernoth11, J. Wang52, D.R. Ward43, A.D. Webber50, D.
Websdale49, M. Whitehead44, D. Wiedner11, L. Wiggers23, G. Wilkinson51, M.P.
Williams44,45, M. Williams49, F.F. Wilson45, J. Wishahi9, M. Witek25,37, W.
Witzeling37, S.A. Wotton43, K. Wyllie37, Y. Xie46, F. Xing51, Z. Yang3, R.
Young46, O. Yushchenko34, M. Zavertyaev10,a, F. Zhang3, L. Zhang52, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, E. Zverev31, A. Zvyagin 37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
24Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, Netherlands
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Cracow, Poland
26Faculty of Physics & Applied Computer Science, Cracow, Poland
27Soltan Institute for Nuclear Studies, Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
43Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
44Department of Physics, University of Warwick, Coventry, United Kingdom
45STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
46School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
47School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
48Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
49Imperial College London, London, United Kingdom
50School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
51Department of Physics, University of Oxford, Oxford, United Kingdom
52Syracuse University, Syracuse, NY, United States
53CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oInstitució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain
pHanoi University of Science, Hanoi, Viet Nam
###### Abstract
A model-independent search for direct $C\\!P$ violation in the Cabibbo
suppressed decay $D^{+}\rightarrow K^{-}K^{+}\pi^{+}$ in a sample of
approximately 370,000 decays is carried out. The data were collected by the
LHCb experiment in 2010 and correspond to an integrated luminosity of 35 pb-1.
The normalized Dalitz plot distributions for $D^{+}$ and $D^{-}$ are compared
using four different binning schemes that are sensitive to different
manifestations of $C\\!P$ violation. No evidence for $C\\!P$ asymmetry is
found.
###### pacs:
13.25.Ft, 11.30.Er, 14.40.Lb
††preprint: APS/123-QED
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
| |
---|---|---
| | LHCb-PAPER-2011-017
| | CERN-PH-EP-2011-163
## I Introduction
To date $C\\!P$ violation (CPV) has been observed only in decays of neutral
$K$ and $B$ mesons. All observations are consistent with CPV being generated
by the phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix Cabibbo:1963yz ;
Kobayashi:1973fv of the Standard Model (SM). In the charm sector, CKM
dynamics can produce direct $C\\!P$ asymmetries in Cabibbo suppressed
$D^{\pm}$ decays of the order of 10-3 or less Bianco:2003vb . Asymmetries of
up to around 1% can be generated by new physics (NP) Artuso:2008vf ;
Grossman:2006jg . In most extensions of the SM, asymmetries arise in processes
with loop diagrams. However, in some cases CPV could occur even at tree level,
for example in models with charged Higgs exchange.
In decays of hadrons, CPV can be observed when two different amplitudes with
non-zero relative weak and strong phases contribute coherently to a final
state. Three-body decays are dominated by intermediate resonant states, and
the requirement of a non-zero relative strong phase is fulfilled by the phases
of the resonances. In two-body decays, CPV leads to an asymmetry in the
partial widths. In three-body decays, the interference between resonances in
the two-dimensional phase space can lead to observable asymmetries which vary
across the Dalitz plot.
$C\\!P$-violating phase differences of $10^{\circ}$ or less do not, in
general, lead to large asymmetries in integrated decay rates, but they could
have clear signatures in the Dalitz plot, as we will show in Sect. III. This
means that a two-dimensional search should have higher sensitivity than an
integrated measurement. In addition, the distribution of an asymmetry across
phase space could hint at the underlying dynamics.
At present, no theoretical tools for computing decay fractions and relative
phases of resonant modes in $D$ decays have been applied to multibody $D^{+}$
decay modes, and no predictions have been made for how asymmetries might vary
across their Dalitz plots. A full Dalitz plot analysis of large data samples
could, in principle, measure small phase differences. However, rigorous
control of the much larger strong phases would be required. For this to be
achieved, better understanding of the amplitudes, especially in the scalar
sector, will be needed, and effects like three-body final state interactions
should be taken into account.
This paper describes a model-independent search for direct CPV in the Cabibbo
suppressed decay $D^{+}\rightarrow K^{-}K^{+}\pi^{+}$ in a binned Dalitz
plot.111Throughout this paper charge conjugation is implied, unless otherwise
stated. A direct comparison between the $D^{+}$ and the $D^{-}$ Dalitz plots
is made on a bin-by-bin basis. The data sample used is approximately 35 pb-1
collected in 2010 by the LHCb experiment at a centre of mass energy of
$\sqrt{s}=7$ TeV. This data set corresponds to nearly 10 and 20 times more
signal events than used in previous studies of this channel performed by the
BABAR Aubert:2005gj and CLEO-c :2008zi collaborations, respectively. It is
comparable to the dataset used in a more recent search for CPV in
$D^{+}\rightarrow\phi\pi^{+}$ decays at BELLE :2011en .
The strategy is as follows. For each bin in the Dalitz plot, a local $C\\!P$
asymmetry variable is defined Bediaga:2009tr ; Aubert:2008yd ,
$\mathcal{S}_{CP}^{i}=\frac{N^{i}(D^{+})-\alpha
N^{i}(D^{-})}{\sqrt{N^{i}(D^{+})+\alpha^{2}N^{i}(D^{-})}}\ ,\hskip
14.22636pt\alpha=\frac{N_{\mathrm{tot}}(D^{+})}{N_{\mathrm{tot}}(D^{-})},$ (1)
where $N^{i}(D^{+})$ and $N^{i}(D^{-})$ are the numbers of $D^{\pm}$
candidates in the $i^{\mathrm{th}}$ bin and $\alpha$ is the ratio between the
total $D^{+}$ and $D^{-}$ yields. The parameter $\alpha$ accounts for global
asymmetries, i.e. those that are constant across the Dalitz plot.
In the absence of Dalitz plot dependent asymmetries, the
$\mathcal{S}_{CP}^{i}$ values are distributed according to a Gaussian
distribution with zero mean and unit width. CPV signals are, therefore,
deviations from this behaviour. The numerical comparison between the $D^{+}$
and the $D^{-}$ Dalitz plots is made with a $\chi^{2}$ test
($\chi^{2}=\sum(\mathcal{S}_{CP}^{i})^{2}$). The number of degrees of freedom
is the number of bins minus one (due to the constraint of the overall
$D^{+}/D^{-}$ normalization). The $p$-value that results from this test is
defined as the probability of obtaining, for a given number of degrees of
freedom and under the assumption of no CPV, a $\chi^{2}$ that is at least as
high as the value observed lyons1989statistics . It measures the degree to
which we are confident that the differences between the $D^{+}$ and $D^{-}$
Dalitz plots are driven only by statistical fluctuations.
If CPV is observed, the $p$-value from this test could be converted into a
significance for a signal using Gaussian statistics. However, in the event
that no CPV is found, there is no model-independent mechanism for setting
limits on CPV within this procedure. In this case, the results can be compared
to simulation studies in which an artificial $C\\!P$ asymmetry is introduced
into an assumed amplitude model for the decay. Since such simulations are
clearly model-dependent, they are only used as a guide to the sensitivity of
the method, and not in the determination of the $p$-values that constitute the
results of the analysis.
The technique relies on careful accounting for local asymmetries that could be
induced by sources such as, the difference in the $K$–nucleon inelastic cross-
section, differences in the reconstruction or trigger efficiencies, left-right
detector asymmetries, etc. These effects are investigated in the two control
channels $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ and $D^{+}_{s}\rightarrow
K^{-}K^{+}\pi^{+}$.
The optimum sensitivity is obtained with bins of nearly the same size as the
area over which the asymmetry extends in the Dalitz plot. Since this is a
search for new and therefore unknown phenomena, it is necessary to be
sensitive to effects restricted to small areas as well as those that extend
over a large region of the Dalitz plot. Therefore two types of binning scheme
are employed. The first type is simply a uniform grid of equally sized bins.
The second type takes into account the fact that the $D^{+}\rightarrow
K^{-}K^{+}\pi^{+}$ Dalitz plot is dominated by the $\phi\pi^{+}$ and $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*}(892)^{0}K^{+}$ resonances, so the
event distribution is highly non-uniform. This “adaptive binning” scheme uses
smaller bins where the density of events is high, aiming for a uniform bin
population. In each scheme, different numbers of bins are used in our search
for localized asymmetries.
The paper is organized as follows. In Sect. II, a description of the LHCb
experiment and of the data selection is presented. In Sect. III, the methods
and the binnings are discussed in detail. The study of the control channels
and of possible asymmetries generated by detector effects or backgrounds is
presented in Sect. IV. The results of our search are given in Sect. V, and the
conclusions in Sect. VI.
## II Detector, dataset and selection
The LHCb detector Alves:2008zz is a single-arm forward spectrometer with the
main purpose of measuring CPV and rare decays of hadrons containing $b$ and
$c$ quarks. A vertex locator (VELO) determines with high precision the
positions of the vertices of primary $pp$ collisions (PVs) and the decay
vertices of long-lived particles. The tracking system also includes a large
area silicon strip detector located in front of a dipole magnet with an
integrated field of around 4 Tm, and a combination of silicon strip detectors
and straw drift chambers placed behind the magnet. Charged hadron
identification is achieved with two ring-imaging Cherenkov (RICH) detectors.
The calorimeter system consists of a preshower, a scintillator pad detector,
an electromagnetic calorimeter and a hadronic calorimeter. It identifies high
transverse energy ($E_{\rm T}$) 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 leptonic and hadronic final states with high tranverse momentum, using
the subset of the detectors that are able to reduce the rate at which the
whole detector is read out to a maximum of 1 MHz. The second level, the High
Level Trigger (HLT), is subdivided into two software stages that can use the
information from all parts of the detector. The first stage, HLT1, performs a
partial reconstruction of the event, reducing the rate further and allowing
the next stage, HLT2, to fully reconstruct the individual channels. At each
stage, several selections designed for specific types of decay exist. As
luminosity increased throughout 2010 several changes in the trigger were
required. To match these, the datasets for signal and control modes are
divided into three parts according to the trigger, samples 1, 2 and 3, which
correspond to integrated luminosities of approximately 3, 5 and 28 pb-1
respectively. The magnet polarity was changed several times during data
taking.
The majority of the signal decays come via the hadronic hardware trigger,
which has an $E_{\rm T}$ threshold that varied between 2.6 and 3.6 GeV in
2010. In the HLT1, most candidates also come from the hadronic selections
which retain events with at least one high transverse momentum ($p_{\rm T}$)
track that is displaced from the PV. In the HLT2, dedicated charm triggers
select most of the signal. However, the signal yield for these channels can be
increased by using other trigger selections, such as those for decays of the
form $B\rightarrow DX$. To maintain the necessary control of Dalitz plot-
dependent asymmetries, only events from selections which have been measured
not to introduce charge asymmetries into the Dalitz plot of the
$D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ control mode are accepted.
The signal ( $D^{+}\rightarrow K^{-}K^{+}\pi^{+}$) and control
($D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ and $D^{+}_{s}\rightarrow
K^{-}K^{+}\pi^{+}$) mode candidates are selected using the same criteria,
which are chosen to maximize the statistical significance of the signal.
Moreover, care is taken to use selection cuts that do not have a low
efficiency in any part of the Dalitz plot, as this would reduce the
sensitivity in these areas. The selection criteria are the same regardless of
the trigger conditions.
The event selection starts by requiring at least one PV with a minimum of five
charged tracks to exist. To control CPU consumption each event must also have
fewer than 350 reconstructed tracks. The particle identification system
constructs a relative log-likelihood for pion and kaon hypotheses,
$\mathrm{DLL}_{K\pi}$, and we require $\mathrm{DLL}_{K\pi}$ $>$ 7 for kaons
and $<$ 3 for pions. Three particles with appropriate charges are combined to
form the $D^{+}_{(s)}$ candidates. The corresponding tracks are required to
have a good fit quality ($\chi^{2}/{\rm ndf}<5$), $\mbox{$p_{\rm T}$}>$ 250
MeV$/c$, momentum $p>$ 2000 MeV$/c$ and the scalar sum of their $p_{\rm T}$
above 2800 MeV$/c$. Because a typical $D^{+}$ travels for around 8 mm before
decaying, the final state tracks should not point to the PV. The smallest
displacement from each track to the PV is computed, and a $\chi^{2}$
($\chi^{2}_{\mathrm{IP}}$), formed by using the hypothesis that this distance
is equal to zero, is required to be greater than 4 for each track. The three
daughters should be produced at a common origin, the charm decay vertex, with
vertex fit $\chi^{2}/{\rm ndf}<$ 10.
This ‘secondary’ vertex must be well separated from any PV, thus a flight
distance variable ($\chi^{2}_{\mathrm{FD}}$) is constructed. The secondary
vertex is required to have $\chi^{2}_{\mathrm{FD}}>100$, and to be downstream
of the PV. The $p_{\rm T}$ of the $D^{+}_{(s)}$ candidate must be greater than
1000 MeV$/c$, and its reconstructed trajectory is required to originate from
the PV ($\chi^{2}_{\mathrm{IP}}<12$).
Figure 1: Fitted mass spectra of (a) $K^{-}\pi^{+}\pi^{+}$ and (b)
$K^{-}K^{+}\pi^{+}$ candidates from samples 1 and 3, $D^{+}$ and $D^{-}$
combined. The signal mass windows and sidebands defined in the text are
labelled. Figure 2: Dalitz plot of the $D^{+}\rightarrow K^{-}K^{+}\pi^{+}$
decay for selected candidates in the signal window. The vertical $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*}(892)^{0}$ and horizontal
$\phi(1020)$ contributions are clearly visible in the data.
In order to quantify the signal yields ($S$), a simultaneous fit to the
invariant mass distribution of the $D^{+}$ and $D^{-}$ samples is performed. A
double Gaussian is used for the $K^{-}K^{+}\pi^{+}$ signal, whilst the
background ($B$) is described by a quadratic component and a single Gaussian
for the small contamination from $D^{*+}\rightarrow D^{0}(K^{-}K^{+})\pi^{+}$
above the $D^{+}_{s}$ peak. The fitted mass spectrum for samples 1 and 3
combined is shown in Fig. 1, giving the yields shown in Table 1. A weighted
mean of the widths of the two Gaussian contributions to the mass peaks is used
to determine the overall widths, $\sigma$, as 6.35 MeV/$c^{2}$ for
$D^{+}\rightarrow K^{-}K^{+}\pi^{+}$, 7.05 MeV/$c^{2}$ for
$D^{+}_{s}\rightarrow K^{-}K^{+}\pi^{+}$, and 8.0 MeV/$c^{2}$ for
$D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$. These values are used to define signal
mass windows of approximately 2$\sigma$ in which the Dalitz plots are
constructed. The purities, defined as $S/(B+S)$ within these mass regions, are
also shown in Table 1 for samples 1 and 3 in the different decay modes.
For sample 2, the yield cannot be taken directly from the fit, because there
is a mass cut in the HLT2 line that accepts the majority of the signal,
selecting events in a $\pm 25$ MeV$/c^{2}$ window around the nominal value.
However, another HLT2 line with a looser mass cut that is otherwise identical
to the main HLT2 line exists, although only one event in 100 is retained. In
this line the purity is found to be the same in sample 2 as in sample 3. The
yield in sample 2 is then inferred as the total $(S+B)$ in all allowed
triggers in the mass window times the purity in sample 3. Thus the overall
yield of signal $D^{+}\rightarrow K^{-}K^{+}\pi^{+}$ candidates in the three
samples within the mass window is approximately 370,000. The total number of
candidates ($S+B$) in each decay mode used in the analysis are given in Table
2. The Dalitz plot of data in the $D^{+}$ window is shown in Fig. 2.
Table 1: Yield ($S$) and purity for samples 1 and 3 after the final selection. The purity is estimated in the 2$\sigma$ mass window. Decay | Yield | Purity
---|---|---
| Sample 1+3 | Sample 1 | Sample 3
$D^{+}\rightarrow K^{-}K^{+}\pi^{+}$ | $(3.284\pm 0.006)\times 10^{5}$ | 88% | 92%
$D^{+}_{s}\rightarrow K^{-}K^{+}\pi^{+}$ | $(4.615\pm 0.012)\times 10^{5}$ | 89% | 92%
$D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ | $(3.3777\pm 0.0037)\times 10^{6}$ | 98% | 98%
Table 2: Number of candidates $(S+B)$ in the signal windows shown in Fig. 1 after the final selection, for use in the subsequent analysis. | sample 1 | sample 2 | sample 3 | Total
---|---|---|---|---
$D^{+}\rightarrow K^{-}K^{+}\pi^{+}$ | 84,667 | 65,781 | 253,446 | 403,894
$D^{+}_{s}\rightarrow K^{-}K^{+}\pi^{+}$ | 126,206 | 91,664 | 346,068 | 563,938
$D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ | 858,356 | 687,197 | 2,294,315 | 3,839,868
Table 3: The CLEO-c amplitude model “B” :2008zi used in the simulation studies. The uncertainties are statistical, experimental systematic and model systematic respectively. Resonance | Amplitude | Relative phase | Fit fraction
---|---|---|---
$\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*}(892)^{0}$ | 1 (fixed) | 0 (fixed) | $25.7\pm 0.5^{+0.4+0.1}_{-0.3-1.2}$
$\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*}_{0}(1430)^{0}$ | $\phantom{0}4.56\pm 0.13^{+0.10+0.42}_{-0.01-0.39}$ | $\phantom{-}\phantom{0}70\pm 6^{+1+16}_{-6-23}$ | $18.8\pm 1.2^{+0.6+3.2}_{-0.1-3.4}$
$\kappa(800)$ | $\phantom{0}2.30\pm 0.13^{+0.01+0.52}_{-0.11-0.29}$ | $\phantom{}-87\pm 6^{+2+15}_{-3-10}$ | $\phantom{0}7.0\pm 0.8^{+0.0+3.5}_{-0.6-1.9}$
$\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*}_{2}(1430)^{0}$ | $\phantom{00}7.6\pm 0.8^{+0.5+2.4}_{-0.6-4.8}$ | $\phantom{-}171\pm 4^{+0+24}_{-2-11}$ | $\phantom{0}1.7\pm 0.4^{+0.3+1.2}_{-0.2-0.7}$
$\phi(1020)$ | $1.166\pm 0.015^{+0.001+0.025}_{-0.009-0.009}$ | $-163\pm 3^{+1+14}_{-1-5}$ | $27.8\pm 0.4^{+0.1+0.2}_{-0.3-0.4}$
$a_{0}(1450)^{0}$ | $\phantom{0}1.50\pm 0.10^{+0.09+0.92}_{-0.06-0.33}$ | $\phantom{-}116\pm 2^{+1+7}_{-1-14}$ | $\phantom{0}4.6\pm 0.6^{+0.5+7.2}_{-0.3-1.8}$
$\phi(1680)$ | $\phantom{0}1.86\pm 0.20^{+0.02+0.62}_{-0.08-0.77}$ | $-112\pm 6^{+3+19}_{-4-12}$ | $0.51\pm 0.11^{+0.01+0.37}_{-0.04-0.15}$
Within the $2\sigma$ $D^{+}\rightarrow K^{-}K^{+}\pi^{+}$ mass window, about
8.6% of events are background. Apart from random three-body track
combinations, charm backgrounds and two-body resonances plus one track are
expected. Charm reflections appear when a particle is wrongly identified in a
true charm three-body decay and/or a track in a four-body charm decay is lost.
The main three-body reflection in the $K^{-}K^{+}\pi^{+}$ spectrum is the
Cabibbo-favoured $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$, where the incorrect
assignment of the kaon mass to the pion leads to a distribution that partially
overlaps with the $D^{+}_{s}\rightarrow K^{-}K^{+}\pi^{+}$ signal region, but
not with $D^{+}\rightarrow K^{-}K^{+}\pi^{+}$. The four body, Cabibbo-favoured
mode $D^{0}\rightarrow K^{-}\pi^{+}\pi^{-}\pi^{+}$ where a $\pi^{+}$ is lost
and the $\pi^{-}$ is misidentified as a $K^{-}$ will appear broadly
distributed in $K^{-}K^{+}\pi^{+}$ mass, but its resonances could create
structures in the Dalitz plot. Similarly, $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*}(892)^{0}$ and $\phi$ resonances
from the PV misreconstructed with a random track forming a three-body vertex
will also appear.
## III Methods and binnings
Figure 3: $\mathcal{S}_{CP}$ across the Dalitz plot in a Monte Carlo pseudo-
experiment with a large number of events with (a) no CPV and (b) a 4∘ CPV in
the $\phi\pi$ phase. Note the difference in colour scale between (a) and (b).
Figure 4: Layout of the (a) “Adaptive I” and (b) “Adaptive II” binnings on
the Dalitz plot of data.
Monte Carlo pseudo-experiments are used to verify that we can detect CPV with
the strategy outlined in Sect. I without producing fake signals, and to devise
and test suitable binning schemes for the Dalitz plot. They are also used to
quantify our sensitivity to possible manifestations of CPV, where we define
the sensitivity to a given level of CPV as the probability of observing it
with $3\sigma$ significance.
For the $D^{+}\rightarrow K^{-}K^{+}\pi^{+}$ Dalitz plot model, the result of
the CLEO-c analysis (fit B) :2008zi is used. The amplitudes and phases of the
resonances used in this model are reproduced in Table 3. For simplicity, only
resonant modes with fit fractions greater than $2\%$ are included in the
pseudo-experiments. The fit fraction for a resonance is defined as the
integral of its squared amplitude over the Dalitz plot divided by the integral
of the square of the overall complex amplitude. An efficiency function is
determined from a two-dimensional second order polynomial fit to the Dalitz
plot distribution of triggered events that survive the selection cuts in the
GEANT-based Agostinelli:2002hh LHCb Monte Carlo simulation for nonresonant
$D^{+}\rightarrow K^{-}K^{+}\pi^{+}$. A simple model for the background is
inferred from the Dalitz plots of the sidebands of the $D^{+}\rightarrow
K^{-}K^{+}\pi^{+}$ signal. It is composed of random combinations of $K^{-}$,
$K^{+}$, and $\pi^{+}$ tracks, $\phi$ resonances with $\pi^{+}$ tracks, and
$\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*}(892)^{0}$ resonances with
$K^{+}$ tracks. The CLEO-c Dalitz plot analysis has large uncertainties, as do
the background and efficiency simulations (due to limited numbers of MC
events), so the method is tested on a range of different Dalitz plot models.
Pseudo-experiments with large numbers of events are used to investigate how
CPV would be observed in the Dalitz plot. These experiments are simple “toy”
simulations that produce points in the Dalitz plot according to the
probability density function determined from the CLEO-c amplitude model with
no representation of the proton-proton collision, detector, or trigger. Figure
3(a) illustrates the values of $\mathcal{S}_{CP}^{i}$ observed with $8\times
10^{7}$ events and no CPV. This dataset is approximately 50 times larger than
the data sample under study. The resulting $\chi^{2}/{\rm ndf}$ is
$253.4/218$, giving a $p$-value for consistency with no CPV of 5.0%. This test
shows that the method by itself is very unlikely to yield false positive
results. Figure 3(b) shows an example test using $5\times 10^{7}$ events with
a $C\\!P$ violating phase difference of $4^{\circ}$ between the amplitudes for
the $\phi(1020)\pi^{+}$ component in $D^{+}$ and $D^{-}$ decays. The $p$-value
in this case is less than $10^{-100}$. The CPV effect is clearly visible, and
is spread over a broad area of the plot, changing sign from left to right.
This sign change means the CPV causes only a 0.1% difference in the total
decay rate between $D^{+}$ and $D^{-}$. This illustrates the strength of our
method, as the asymmetry would be much more difficult to detect in a
measurement that was integrated over the Dalitz plot. Even with no systematic
uncertainties, to see a 0.1% asymmetry at the $3\sigma$ level would require
$2.25\times 10^{6}$ events. With the method and much smaller dataset used here
we would observe this signal at the $3\sigma$ level with 76% probability, as
shown in Table 4 below.
The sensitivity to a particular manifestation of CPV depends on the choice of
binning. The fact that the $C\\!P$-violating region in most of the pseudo-
experiments covers a broad area of the Dalitz plot suggests that the optimal
number of bins for this type of asymmetry is low. Each bin adds a degree of
freedom without changing the $\chi^{2}$ value for consistency with no CPV.
However, if $C\\!P$ asymmetries change sign within a bin, they will not be
seen. Similarly, the sensitivity is reduced if only a small part of a large
bin has any CPV in it. To avoid effects due to excessive fluctuations, bins
that contain fewer than 50 candidates are not used anywhere in the analysis.
Such bins are very rare.
The binnings are chosen to reflect the highly non-uniform structure of the
Dalitz plot. A simple adaptive binning algorithm was devised to define
binnings of approximately equal population without separating $D^{+}$ and
$D^{-}$. Two binnings that are found to have good sensitivity to the simulated
asymmetries contain 25 bins (“Adaptive I”) arranged as shown in Fig. 4(a), and
106 bins (“Adaptive II”) arranged as shown in Fig. 4(b). For Adaptive I, a
simulation of the relative value of the strong phase across the Dalitz plot in
the CLEO-c amplitude model is used to refine the results of the algorithm: if
the strong phase varies significantly across a bin, $C\\!P$ asymmetries are
more likely to change sign. Therefore the bin boundaries are adjusted to
minimise changes in the strong phase within bins. The model-dependence of this
simulation could, in principle, influence the binning and therefore the
sensitivity to CPV, but it cannot introduce model-dependence into the final
results as no artificial signal could result purely from the choice of
binning. Two further binning schemes, “Uniform I” and “Uniform II”, are
defined. These use regular arrays of rectangular bins of equal size.
Table 4: Results from sets of 100 pseudo-experiments with different $C\\!P$ asymmetries and Adaptive I and II binnings. $p(3\sigma)$ is the probability of a 3$\sigma$ observation of CPV. $\langle S\rangle$ is the mean significance with which CPV is observed. CPV | Adaptive I | Adaptive II
---|---|---
| $p(3\sigma)$ | $\langle S\rangle$ | $p(3\sigma)$ | $\langle S\rangle$
no CPV | 0 | 0.84$\sigma$ | 1% | 0.84$\sigma$
$6^{\circ}$ in $\phi(1020)$ phase | 99% | 7.0$\sigma$ | 98% | 5.2$\sigma$
$5^{\circ}$ in $\phi(1020)$ phase | 97% | 5.5$\sigma$ | 79% | 3.8$\sigma$
$4^{\circ}$ in $\phi(1020)$ phase | 76% | 3.8$\sigma$ | 41% | 2.7$\sigma$
$3^{\circ}$ in $\phi(1020)$ phase | 38% | 2.8$\sigma$ | 12% | 1.9$\sigma$
$2^{\circ}$ in $\phi(1020)$ phase | 5% | 1.6$\sigma$ | 2% | 1.2$\sigma$
$6.3\%$ in $\kappa(800)$ magnitude | 16% | 1.9$\sigma$ | 24% | 2.2$\sigma$
$11\%$ in $\kappa(800)$ magnitude | 83% | 4.2$\sigma$ | 95% | 5.6$\sigma$
Table 5: Results from sets of 100 pseudo-experiments with $4^{\circ}$ CPV in the $\phi(1020)$ phase and different Dalitz plot models. $p(3\sigma)$ is the probability of a 3$\sigma$ observation of CPV. $\langle S\rangle$ is the mean significance with which CPV is observed. The sample size is comparable to that seen in data. Model | Adaptive I | Adaptive II
---|---|---
| $p(3\sigma)$ | $\langle S\rangle$ | $p(3\sigma)$ | $\langle S\rangle$
B (baseline) | 76% | 3.8$\sigma$ | 41% | 2.7$\sigma$
A | 84% | 4.3$\sigma$ | 47% | 2.9$\sigma$
B2 (add $f_{0}(980)$) | 53% | 3.2$\sigma$ | 24% | 2.2$\sigma$
B3 (vary $\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*}_{0}(1430)^{0}$ magn.) | 82% | 4.0$\sigma$ | 41% | 2.8$\sigma$
B4 (vary $\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*}_{0}(1430)^{0}$ phase) | 73% | 3.7$\sigma$ | 38% | 2.7$\sigma$
The adaptive binnings are used to determine the sensitivity to several
manifestations of CPV. With 200 test experiments of approximately the same
size as the signal sample in data, including no asymmetries, no
$C\\!P$-violating signals are observed at the 3$\sigma$ level with Adaptive I
or Adaptive II. The expectation is 0.3.
With the chosen binnings, a number of sets of 100 pseudo-experiments with
different $C\\!P$-violating asymmetries are produced. The probability of
observing a given signal in either the $\phi(1020)$ or $\kappa(800)$
resonances with 3$\sigma$ significance is calculated in samples of the same
size as the dataset. The results are given in Table 4. The CPV shows up both
in the $\chi^{2}/{\rm ndf}$ and in the width of the fitted $\mathcal{S}_{CP}$
distribution.
For comparison, the asymmetries in the $\phi$ phase and $\kappa$ magnitude
measured by the CLEO collaboration using the same amplitude model were $(6\pm
6^{+0+6}_{-2-2})^{\circ}$ and $(-12\pm 12^{+6+2}_{-1-10})\%$,222The
conventions used in the CLEO paper to define asymmetry are different, so the
asymmetries in Table II of :2008zi have been multiplied by two in order to be
comparable with those given above. where the uncertainties are statistical,
systematic and model-dependent, respectively. Table 4 suggests that, assuming
their model, we would be at least 95% confident of detecting the central
values of these asymmetries.
The sensitivity of the results to variations in the Dalitz plot model and the
background is investigated, and example results for the $C\\!P$ asymmetry in
the $\phi(1020)$ phase are shown in Table 5. In this table, models A and B are
taken from the CLEO paper, model B2 includes an $f_{0}(980)$ contribution that
accounts for approximately 8% of events, and models B3 and B4 are variations
of the $\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*}_{0}(1430)^{0}$
amplitude and phase within their uncertainties. As expected, the sensitivity
to CPV in the resonances of an amplitude model depends quite strongly on the
details of the model. This provides further justification for our model-
independent approach. However, a reasonable level of sensitivity is retained
in all the cases we tested. Thus, when taken together, the studies show that
the method works well. It does not yield fake signals, and should be sensitive
to any large CPV that varies significantly across the Dalitz plot even if it
does not occur precisely in the way investigated here.
## IV Control modes
It is possible that asymmetries exist in the data that do not result from CPV,
for example due to production, backgrounds, instrumental effects such as left-
right differences in detection efficiency, or momentum-dependent differences
in the interaction cross-sections of the daughter particles with detector
material. Our sensitivity to such asymmetries is investigated in the two
Cabibbo favoured control channels, where there is no large CPV predicted. The
$D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ control mode has an order of magnitude
more candidates than the Cabibbo-suppressed signal mode, and is more sensitive
to detector effects since there is no cancellation between $K^{+}$ and $K^{-}$
reconstruction efficiencies. Conversely, the $D^{+}_{s}\rightarrow
K^{-}K^{+}\pi^{+}$ control mode is very similar to our signal mode in terms of
resonant structure, number of candidates, kinematics, detector effects, and
backgrounds.
Figure 5: (a) Distribution of $\mathcal{S}_{CP}$ values from $D^{+}\rightarrow
K^{-}\pi^{+}\pi^{+}$ from a test with 900 uniform bins. The mean of the fitted
Gaussian distribution is $0.015\pm 0.034$ and the width is $0.996\pm 0.023$.
(b) Distribution of $\mathcal{S}_{CP}$ values from $D^{+}_{s}\rightarrow
K^{-}K^{+}\pi^{+}$ with 129 bins. The fitted mean is $-0.011\pm 0.084$ and the
width is $0.958\pm 0.060$.
Figure 6: Dalitz plots of (a) $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$, showing
the 25-bin adaptive scheme with the $\mathcal{S}_{CP}$ values, and (b)
$D^{+}_{s}\rightarrow K^{-}K^{+}\pi^{+}$, showing the three regions referred
to in the text. The higher and lower $K^{-}\pi^{+}$ invariant mass
combinations are plotted in (a) as there are identical pions in the final
state.
The control modes and their mass sidebands defined in Fig. 1 are tested for
asymmetries using the method described in the previous section. Adaptive and
uniform binning schemes are defined for $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$
and $D^{+}_{s}\rightarrow K^{-}K^{+}\pi^{+}$. They are applied to samples 1–3
and each magnet polarity separately. In the final results, the asymmetries
measured in data taken with positive and negative magnet polarity are combined
in order to cancel left-right detector asymmetries. The precise number of bins
chosen is arbitrary, but care is taken to use a wide range of tests with
binnings that reflect the size of the dataset for the decay mode under study.
For $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$, five different sets of bins in
each scheme are used. A very low $p$-value would indicate a local asymmetry.
One test with 25 adaptive bins in one of the subsamples (with negative magnet
polarity) has a $p$-value of 0.1%, but when combined with the positive
polarity sample the $p$-value increases to 1.7%. All other tests yield
$p$-values ranging from 1–98%. Some example results are given in Table 6. A
typical distribution of the $\mathcal{S}_{CP}$ values with a Gaussian fit is
shown in Fig. 5(a) for a test with 900 uniform bins. The fitted values of the
mean and width are consistent with one and zero respectively, suggesting that
the differences between the $D^{+}$ and the $D^{-}$ Dalitz plots are driven
only by statistical fluctuations.
Table 6: Results ($p$-values, in %) from tests with the $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ control channel using the uniform and adaptive binning schemes. The values correspond to tests performed on the whole dataset in the mass windows defined in Sect. II. | 1300 bins | 900 bins | 400 bins | 100 bins | 25 bins
---|---|---|---|---|---
Uniform | 73.8 | 17.7 | 72.6 | 54.6 | 1.7
Adaptive | 81.7 | 57.4 | 65.8 | 30.0 | 11.8
For the $D^{+}_{s}\rightarrow K^{-}K^{+}\pi^{+}$ mode a different procedure is
followed due to the smaller sample size and to the high density of events
along the $\phi$ and the $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*}(892)^{0}$ bands. The Dalitz plot is
divided into three zones, as shown in Fig. 6. Each zone is further divided
into 300, 100 and 30 bins of same size. The results are given in Table 7. In
addition, a test is performed on the whole Dalitz plot using 129 bins chosen
by the adaptive algorithm, and a version of the 25-bin scheme outlined in
Sect. III scaled by the ratio of the available phase space in the two modes.
These tests yield $p$-values of 71.5% and 34.3% respectively.
Table 7: Results ($p$-values, in %) from tests with the $D^{+}_{s}\rightarrow K^{-}K^{+}\pi^{+}$ control channel using the uniform binning scheme. The values correspond to tests performed separately on Zones A-C, with samples 1-3 and both magnet polarities combined. bins | Zone A | Zone B | Zone C
---|---|---|---
300 | 20.1 | 25.3 | 14.5
100 | 41.7 | 84.6 | 89.5
30 | 66.0 | 62.5 | 24.6
Other possible sources of local charge asymmetry in the signal region are the
charm contamination of the background, and asymmetries from CPV in
misreconstructed $B$ decays. In order to investigate the first possibility,
similar tests are carried out in the mass sidebands of the
$D^{+}_{(s)}\rightarrow K^{-}K^{+}\pi^{+}$ signal (illustrated in Fig. 1).
There is no evidence for asymmetries in the background.
From a simulation of the decay $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ the
level of secondary charm ($B\rightarrow DX$) in our selected sample is found
to be 4.5%. The main discriminating variable to distinguish between prompt and
secondary charm is the impact parameter (IP) of the $D$ with respect to the
primary vertex. Given the long $B$ lifetime, the IP distribution of secondary
charm candidates is shifted towards larger values compared to that of prompt
$D^{+}$ mesons.
The effect of secondary charm is investigated by dividing the data set
according to the value of the candidate IP significance ($\chi^{2}_{IP}$). The
subsample with events having larger $\chi^{2}_{IP}$ are likely to be richer in
secondary charm. The results are shown in Table 8. No anomalous effects are
seen in the high $\chi^{2}_{IP}$ sample, so contamination from secondary charm
with CPV does not affect our results for studies with our current level of
sensitivity.
Table 8: Results ($p$-values, in %) from tests with the $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ and $D^{+}_{s}\rightarrow K^{-}K^{+}\pi^{+}$ samples divided according to the impact parameter with respect to the primary vertex. The tests are performed using the adaptive binning scheme with 25 bins. | $\chi^{2}_{IP}<6$ | $\chi^{2}_{IP}>6$
---|---|---
$D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ | 8.5 | 88.9
$D^{+}_{s}\rightarrow K^{-}K^{+}\pi^{+}$ | 52.0 | 30.6
The analysis on the two control modes and on the sidebands in the final states
$K^{-}K^{+}\pi^{+}$ and $K^{-}\pi^{+}\pi^{+}$ gives results from all tests
that are fully consistent with no asymmetry. Therefore, any asymmetry observed
in $D^{+}\rightarrow K^{-}K^{+}\pi^{+}$ is likely to be a real physics effect.
Table 9: Fitted means and widths, $\chi^{2}/{\rm ndf}$ and $p$-values for consistency with no CPV for the $D^{+}\rightarrow K^{-}K^{+}\pi^{+}$ decay mode with four different binnings. Binning | Fitted mean | Fitted width | $\chi^{2}/{\rm ndf}$ | $p$-value (%)
---|---|---|---|---
Adaptive I | $\phantom{-}0.01\pm 0.23$ | $1.13\pm 0.16$ | 32.0/24 | 12.7
Adaptive II | $-0.024\pm 0.010$ | $1.078\pm 0.074$ | 123.4/105 | 10.6
Uniform I | $-0.043\pm 0.073$ | $0.929\pm 0.051$ | 191.3/198 | 82.1
Uniform II | $-0.039\pm 0.045$ | $1.011\pm 0.034$ | 519.5/529 | 60.5
Figure 7: Distribution of ${\mathcal{S}}^{i}_{\it CP}$ in the Dalitz plot for
(a) “Adaptive I”, (b) “Adaptive II”, (c) “Uniform I” and (d) “Uniform II”. In
(c) and (d) bins at the edges are not shown if the number of entries is not
above a threshold of 50 (see Sect. III).
Figure 8: Distribution of ${\mathcal{S}}^{i}_{\it CP}$ fitted to Gaussian
functions, for (a) “Adaptive I”, (b) “Adaptive II”, (c) “Uniform I” and (d)
“Uniform II”. The fit results are given in Table 9.
## V Results
The signal sample with which we search for $C\\!P$ violation consists of
403,894 candidates selected within the $K^{-}K^{+}\pi^{+}$ mass window from
1856.7 to 1882.1 MeV$/c^{2}$, as described in Sect. II. There are 200,336 and
203,558 $D^{+}$ and $D^{-}$ candidates respectively. This implies a
normalization factor $\alpha=N_{\rm tot}(D^{+})/N_{\rm tot}(D^{-})=0.984\pm
0.003$, to be used in Eq. 1.
The strategy for looking for signs of localized CPV is discussed in the
previous sections. In the absence of local asymmetries in the control channels
$D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ and $D^{+}_{s}\rightarrow
K^{-}K^{+}\pi^{+}$ and in the sidebands of the $K^{-}K^{+}\pi^{+}$ mass
spectrum, we investigate the signal sample under different binning choices.
First, the adaptive binning is used with 25 and 106 bins in the Dalitz plot as
illustrated in Fig. 4. Then CPV is investigated with uniform binnings, using
200 and 530 bins of equal size. For each of these binning choices, the
significance ${\cal S}^{i}_{\it CP}$ of the difference in $D^{+}$ and $D^{-}$
population is computed for each bin $i$, as defined in Eq. 1. The
$\chi^{2}/{\rm ndf}=\sum_{i}({\cal S}^{i}_{\it CP})^{2}/{\rm ndf}$ is
calculated and the $p$-value is obtained. The distributions of
${\mathcal{S}}^{i}_{\it CP}$ are fitted to Gaussian functions.
The $p$-values are shown in Table 9. The Dalitz plot distributions of
${\mathcal{S}}^{i}_{\it CP}$ are shown in Fig. 7. In Fig. 8 the distributions
of ${\mathcal{S}}^{i}_{\it CP}$ and the corresponding Gaussian fits for the
different binnings are shown. The $p$-values obtained indicate no evidence for
CPV. This is corroborated by the good fits of the ${\mathcal{S}}^{i}_{\it CP}$
distributions to Gaussians, with means and widths consistent with 0 and 1,
respectively.
As further checks, many other binnings are tested. The number of bins in the
adaptive and uniform binning schemes is varied from 28 to 106 and from 21 to
530 respectively. The samples are separated according to the magnet polarity
and the same studies are repeated. In all cases the $p$-values are consistent
with no CPV, with values ranging from 4% to 99%. We conclude that there is no
evidence for CPV in our data sample of $D^{+}\rightarrow K^{-}K^{+}\pi^{+}$.
## VI Conclusion
Due to the rich structure of their Dalitz plots, three body charm decays are
sensitive to $C\\!P$ violating phases within and beyond the Standard Model.
Here, a model-independent search for direct $C\\!P$ violation is performed in
the Cabibbo suppressed decay $D^{+}\rightarrow K^{-}K^{+}\pi^{+}$ with 35 pb-1
of data collected by the LHCb experiment, and no evidence for CPV is found.
Several binnings are used to compare normalised $D^{+}$ and $D^{-}$ Dalitz
plot distributions. This technique is validated with large numbers of
simulated pseudo-experiments and with Cabibbo favoured control channels from
the data: no false positive signals are seen. To our knowledge this is the
first time a search for CPV is performed using adaptive bins which reflect the
structure of the Dalitz plot.
Monte Carlo simulations illustrate that large localised asymmetries can occur
without causing detectable differences in integrated decay rates. The
technique used here is shown to be sensitive to such asymmetries. Assuming the
decay model, efficiency parameterisation and background model described in
Sect. III we would be 90% confident of seeing a $C\\!P$ violating difference
of either $5^{\circ}$ in the phase of the $\phi\pi^{+}$ or 11% in the
magnitude of the $\kappa(800)K^{+}$ with $3\sigma$ significance. Since we find
no evidence of CPV, effects of this size are unlikely to exist.
## VII Acknowledgments
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, 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
| 2011-10-18T13:21:27 |
2024-09-04T02:49:23.289100
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb Collaboration: R. Aaij, B. Adeva, M. Adinolfi, C. Adrover, A.\n Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, G. Alkhazov,\n P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, Y. Amhis, J. Anderson, R.B.\n Appleby, O. Aquines Gutierrez, F. Archilli, L. Arrabito, A. Artamonov, M.\n Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J.J. Back, D.S. Bailey, V.\n Balagura, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, A.\n Bates, C. Bauer, Th. Bauer, A. Bay, I. Bediaga, K. Belous, I. Belyaev, E.\n Ben-Haim, M. Benayoun, G. Bencivenni, S. Benson, J. Benton, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, A. Bizzeti, P.M. Bj{\\o}rnstad,\n T. Blake, F. Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A.\n Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, C.\n Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, S. Brisbane, M.\n Britsch, T. Britton, N.H. Brook, H. Brown, A. B\\\"uchler-Germann, I. Burducea,\n A. Bursche, J. Buytaert, S. Cadeddu, J.M. Caicedo Carvajal, O. Callot, M.\n Calvi, M. Calvo Gomez, A. Camboni, P. Campana, A. Carbone, G. Carboni, R.\n Cardinale, A. Cardini, L. Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo,\n M. Charles, Ph. Charpentier, N. Chiapolini, K. Ciba, X. Cid Vidal, G.\n Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V.\n Coco, J. Cogan, P. Collins, F. Constantin, G. Conti, A. Contu, A. Cook, M.\n Coombes, G. Corti, G.A. Cowan, R. Currie, B. D'Almagne, C. D'Ambrosio, P.\n David, I. De Bonis, S. De Capua, M. De Cian, F. De Lorenzi, J.M. De Miranda,\n L. De Paula, P. De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi, M.\n Deissenroth, L. Del Buono, C. Deplano, O. Deschamps, F. Dettori, J. Dickens,\n H. Dijkstra, P. Diniz Batista, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D.\n Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, C. Eames, S. Easo, U.\n Egede, V. Egorychev, S. Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R.\n Ekelhof, L. Eklund, Ch. Elsasser, D.G. d'Enterria, D. Esperante Pereira, L.\n Est\\`eve, A. Falabella, E. Fanchini, C. F\\\"arber, G. Fardell, C. Farinelli,\n S. Farry, V. Fave, V. Fernandez Albor, M. Ferro-Luzzi, S. Filippov, C.\n Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, M. Frank, C. Frei, M.\n Frosini, S. Furcas, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini,\n Y. Gao, J-C. Garnier, J. Garofoli, J. Garra Tico, L. Garrido, C. Gaspar, 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, 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, J. He, V. Heijne, K. Hennessy, P. Henrard, J.A.\n Hernando Morata, E. van Herwijnen, E. Hicks, W. Hofmann, 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, S. Kandybei, M. Karacson, T.M. Karbach, J.\n Keaveney, U. Kerzel, T. Ketel, A. Keune, B. Khanji, Y.M. Kim, M. Knecht, S.\n Koblitz, P. Koppenburg, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G.\n Krocker, P. Krokovny, F. Kruse, K. Kruzelecki, M. Kucharczyk, S. Kukulak, R.\n Kumar, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty, A. Lai, D.\n Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch, T.\n Latham, 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.H. Lopes, E. Lopez Asamar, N.\n Lopez-March, J. Luisier, 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, D. Martins Tostes, A.\n Massafferri, Z. Mathe, C. Matteuzzi, M. Matveev, E. Maurice, B. Maynard, A.\n Mazurov, G. McGregor, R. McNulty, C. Mclean, M. Meissner, M. Merk, J. Merkel,\n R. Messi, S. Miglioranzi, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez, S.\n Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller,\n R. Muresan, B. Muryn, M. Musy, J. Mylroie-Smith, P. Naik, T. Nakada, R.\n Nandakumar, J. Nardulli, I. Nasteva, M. Nedos, M. Needham, N. Neufeld, C.\n Nguyen-Mau, M. Nicol, S. Nies, V. Niess, N. Nikitin, A. Oblakowska-Mucha, V.\n Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M.\n Otalora Goicochea, P. Owen, B. Pal, J. Palacios, 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, B. Pie Valls, B. Pietrzyk, T. Pilar, D. Pinci, R. Plackett, S.\n Playfer, M. Plo Casasus, G. Polok, A. Poluektov, E. Polycarpo, D. Popov, B.\n Popovici, C. Potterat, A. Powell, T. du Pree, J. Prisciandaro, V. Pugatch, A.\n Puig 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, J. Rouvinet, T. Ruf, H. Ruiz,\n 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. Schmelling, B. Schmidt, O.\n Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, A. Sciubba, M. Seco, A.\n Semennikov, K. Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, B.\n Shao, M. Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L.\n Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva Coutinho, H.P.\n Skottowe, T. Skwarnicki, A.C. Smith, N.A. 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, N. Styles, V.K. Subbiah, S. Swientek, M.\n 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, M.T. Tran, A. Tsaregorodtsev, N. Tuning, M. Ubeda Garcia, A.\n Ukleja, P. Urquijo, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P.\n Vazquez Regueiro, S. Vecchi, J.J. Velthuis, M. Veltri, K. Vervink, B. Viaud,\n I. Videau, D. Vieira, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt, D.\n Voong, A. Vorobyev, H. Voss, K. Wacker, S. Wandernoth, J. Wang, D.R. Ward,\n A.D. Webber, D. Websdale, M. Whitehead, D. Wiedner, L. Wiggers, G. Wilkinson,\n M.P. Williams, M. Williams, F.F. Wilson, J. Wishahi, M. Witek, W. Witzeling,\n S.A. Wotton, K. Wyllie, Y. Xie, F. Xing, Z. Yang, R. Young, O. Yushchenko, M.\n Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong,\n E. Zverev, A. Zvyagin",
"submitter": "Hamish Gordon",
"url": "https://arxiv.org/abs/1110.3970"
}
|
1110.3980
|
# High speed shadowgrpah of a L/D cavity at Mach 0.7 and 1.5
Ryan Schmit, Frank Semmelmayer, Mitch Haverkamp and James Grove
Air Force Research Laboratory,
Wright-Patterson Air Force Base, OH 45433, USA
###### Abstract
This is article highlights the fluid dynamics video of a rectangular cavity
with an L/D of 5.67 at Mach 0.7 and 1.5.
A rectangular cavity with an L/D of 5.67 at Mach 0.7 and 1.5, Reynolds number
$2x10^{6}$ and $2.3x10^{6}$ respectively, was examined using high speed
shadowgraph imaging. The cavity motion is shown in the video. The three movies
clips presented were sampled at at 75kHz and played back at 20 Hz. The camera
shutter-speed was 0.37$\mu$sec. The first clip shows the side and top view of
the cavity at Mach 0.7. Note that these two views are not in sync. The second
clip shows the side and top view of the cavity at Mach 1.5. Again the views
are not in sync. The third clip shows a zoomed out side view of the cavity at
Mach 1.5.
For more information please refer to Schmit, Semmelmayer, Haverkamp and Grove,
”Fourier Analysis of High Speed Shadowgraph Images around a Mach 1.5 Cavity
Flow Field”, 29th AIAA Applied Aerodyanmic Conference, Honolulu, Hi, pp 1-24,
2011, AIAA 2011-3961
|
arxiv-papers
| 2011-10-17T14:32:19 |
2024-09-04T02:49:23.299960
|
{
"license": "Public Domain",
"authors": "Ryan Schmit, Frank Semmelmayer, Mitch Haverkamp and James Grove",
"submitter": "Ryan Schmit",
"url": "https://arxiv.org/abs/1110.3980"
}
|
1110.4031
|
# Scaling of Seismic Memory with Earthquake Size
Zeyu Zheng Department of Environmental Sciences, Tokyo University of
Information Sciences, Chiba 265-8501, Japan Kazuko Yamasaki Department of
Environmental Sciences, Tokyo University of Information Sciences, Chiba
265-8501, Japan Center for Polymer Studies and Department of Physics, Boston
University, Boston, MA 02215, USA Joel Tenenbaum Center for Polymer Studies
and Department of Physics, Boston University, Boston, MA 02215, USA Boris
Podobnik Center for Polymer Studies and Department of Physics, Boston
University, Boston, MA 02215, USA Faculty of Civil Engineering, University of
Rijeka, Rijeka, Croatia H. Eugene Stanley Center for Polymer Studies and
Department of Physics, Boston University, Boston, MA 02215, USA
###### Abstract
It has been observed that the earthquake events possess short-term memory,
i.e. that events occurring in a particular location are dependent on the short
history of that location. We conduct an analysis to see whether real-time
earthquake data also possess long-term memory and, if so, whether such
autocorrelations depend on the size of earthquakes within close spatiotemporal
proximity. We analyze the seismic waveform database recorded by 64 stations in
Japan, including the 2011 “Great East Japan Earthquake”, one of the five most
powerful earthquakes ever recorded which resulted in a tsunami and devastating
nuclear accidents. We explore the question of seismic memory through use of
mean conditional intervals and detrended fluctuation analysis (DFA). We find
that the waveform sign series show long-range power-law anticorrelations while
the interval series show long-range power-law correlations. We find size-
dependence in earthquake auto-correlations—as earthquake size increases, both
of these correlation behaviors strengthen. We also find that the DFA scaling
exponent $\alpha$ has no dependence on earthquake hypocenter depth or
epicentral distance.
###### pacs:
PACS numbers:89.65.Gh, 89.20.-a, 02.50.Ey
## I Introduction
Many complex physical systems exhibit complex dynamics in which subunits of
the system interact at widely varying scales of time and space MFS ; Bunde .
These complex interactions often generate very noisy output signals which
still exhibit scale-invariant structure. Such complex systems span areas
studied in physiology Ashk20 , finance Engle82 , and seismology Bak ; Corral04
; Corral05 ; Lippiello07 ; Lippiello08 ; Bottiglieri ; Kagan ; Lennartz10 ;
Livina05 ; Lennartz08 .
In seismology the study of seismic waves is both scientifically interesting
and of practical concern, particularly in such applied areas as engineering. A
better understanding of seismic waves is immediately applicable in the design
of structures for earthquake-prone areas Mayeda ; Hisada ; Hisada2 . It also
allows scientists to better understand the underlying mechanisms that drive
earthquakes Bensen ; Xu2 ; Okada ; Ide ; Shapiro ; Campillo . In seismology,
temporal and spatial clustering are considered important properties of seismic
occurrences and, together with the Omori law (dictating aftershock timing) and
the Gutenberg-Richter law (specifying the distribution of earthquake size),
comprise the main starting requirements to be fulfilled in any reasonable
seismic probabilistic model. Analyzing the timing of individual earthquakes,
Ref. Bak introduces the scaling concept to statistical seismology. The
recurrence times are defined as the time intervals between consecutive events,
$\tau_{i}=t_{i}-t_{i-1}$. In the case of stationary seismicity, the
probability density $P(\tau)$ of the occurrence times was found to follow a
universal scaling law
$P(\tau)=Rf(R\tau)$ (1)
where $f$ is a scaling function and $R$ is the rate of seismic occurrence,
defined as the mean number of events with $M\geq M_{c}$ Corral04 . Reference
Corral05 ; Lippiello07 has demonstrated how the structure of seismic
occurrence in time and magnitude can be treated within the framework of
critical phenomena.
Recently, a few papers have analyzed the existence of correlations between
magnitudes of subsequent earthquakes Corral05 ; Lippiello07 . Analyzing
earthquakes with $\tau$ greater than 30 minutes, Ref. Corral05 reported
possible magnitude correlations in the Southern California catalog. Magnitude
correlations have often been interpreted as a spurious effect due to so called
short-term aftershock incompleteness (STAI) Kagan . This hypothesis assumes
that some aftershocks, especially small events, are not reported in the
experimental catalogs, which is in agreement with the standard approach that
assumes interdependence of earthquake magnitudes implying no memory in
earthquakes.
However, recent work has also challenged this interpretation. Reference
Lippiello08 reports the existence of magnitude clustering in which
earthquakes of a given magnitude are more likely to occur close in time and
space to other events of similar magnitude. They find that a subsequent
earthquake tends to have a magnitude similar to but smaller than the previous
earthquake. Reference Lippiello07 also reports the existence of magnitude
correlations and additionally demonstrates the structure of these correlations
and their relationship to $\Delta t$ and $\Delta r$, where the latter
represents the distance between subsequent epicenters. Reference Lennartz10
creates a model to explain these magnitude correlations. They note that the
Omori law and “background tectonic cycles” are responsible for clustering in
interoccurrence times. Additionally, Refs. Livina05 and Lennartz08 find that
the distribution of recurrence times strongly depends on the previous
recurrence time such that small and large recurrence times tend to cluster in
time. This dependence on the past is reflected in both the conditional mean
recurrence time and the conditional mean residual time until the next
earthquake.
Since it is our hypothesis that long-range autocorrelations exist in seismic
waves, we first note that long-range power-law autocorrelations are quite
common in a large number of natural phenomena ranging from weather Yamasaki3 ;
Gozolchiani ; BP05 , and physiological systems Lennartz ; Ashk20 ;
Kantelhardt2 ; Stanley3 ; Karasik , to financial markets Mantegna ; Yamasaki ;
Wang2 ; Wang3 ; Stanley ; BPPnas09 ; BPPnas10 .
In addition to analyzing the raw waveform, it is also common to analyze
related time series, such the time series generated by taking the sign or
magnitude of the waveform Ashk20 . Reference Ashk20 reports an empirical
approximate relation at small time scales for the scaling exponents calculated
for sign, magnitude, and the original time series, $\alpha_{\rm
sign}=1/2(\alpha_{\rm magnitude}+\alpha_{\rm original})$, in physiology. The
study of magnitude and sign time series is important in physiology because the
magnitude time series exhibits weaker autocorrelations and a scaling exponent
closer to the exponent of an uncorrelated series found when a subject is
unhealthy Ashk20 . Diagnostic power in physiology has been confirmed for sign
time series as well—the sign time series of heart failure subjects exhibit
scaling behavior similar to that observed in the original time series, but
significantly different that of healthy subjects Ashk20 . Understanding the
correlation properties of these three time series allows us to also understand
the underlying processes generating them.
Our investigation and discussion is organized as follows. First, we study the
autocorrelations of interval series by using the mean conditional technique.
Second, we employ detrended fluctuation analysis (DFA) CKP ; Hu ; Chen and
find long-range power-law autocorrelations in the sign and interval time
series. For the interval time series we find a positive regression between the
DFA scaling exponent $\alpha$ and earthquake size (measured by the Richter
magnitude scale $M$ or seismic moment $M_{0}$), while for the sign time series
we find an inverted regression between $\alpha$ and earthquake magnitude. Thus
we report that the observed autocorrelation depends on earthquake size, both
in the sign and interval time series. We also find that the scaling exponent
$\alpha$ has no dependence on hypocenter depth or epicentral distance.
## II Data
Seismic waves are unique in that they have non-stationarities of a much larger
order than those of any other known natural signal. Large earthquakes are
characterized by a maximum amplitude that is often $>100$ times larger than
the mean amplitude [see Fig. 1(a)]. This is a limitation that makes seismic
waves difficult to analyze using traditional analysis. Although we might want
to use detrended fluctuation analysis (DFA) CKP ; Hu ; Chen ; Chen2 ,
originally proposed to study the correlations in a time series in the presence
of non-stationarities commonly observed in natural phenomena, the level of
non-stationarity in earthquakes is so large that DFA is inappropriate
regardless of the order of the polynomial fit applied Chen . Thus, due to lack
of methods for highly non-stationary signals, we do not analyze correlations
in the series of magnitudes, but instead analyze the correlations in the sign
series [Fig. 1(c)] and interval series [Fig. 1(d)]. For our data, we use the
seismic waveform database from the National Research Institute for Earth
Science and Disaster Prevention (NIED) F-net (Full Range Seismograph Network
of Japan), which records continuous seismic waveform data $w_{t}$ by using
broadband sensors in 64 stations in Japan [see Fig. 1(a)]. In our study we
select 46 stations (ADM, AOG, ASI, HID, HJO, HRO, IGK, IMG, INN, IYG, IZH,
KGM, KMU, KNM, KNP, KNY, KSK, KSN, KSR, KYK, MMA, NKG, NOK, NOP, NRW, NSK,
OSW, SAG, SHR, SIB, TAS, TGA, TGW, TKO, TMC, TSA, TYM, TYS, UMJ, WTR, YAS,
YNG, YSI , YTY, YZK, ZMM), based on locations and integrity of data series.
Seismic signals are recorded in three directions: (1) U (up-down with up
positive), N (north-south with north positive), and E (east-west with east
positive) Okada . In this paper, we report results from the vertical dimension
only (U data), since the results for the horizontal data (N and E) data are
very similar. Sampling intervals have five recording frequencies: 80Hz, 20Hz,
1Hz, 0.1Hz, and 0.01Hz. We study earthquake coda wave data with 1Hz sampling
interval for the year 2003, together with selected earthquake coda wave data
from 11 March 2011. We note that, because of the interaction between
earthquakes, not all earthquakes can be employed in our analysis (see Appendix
A). The data from 11 March 2011 is selected because it contains the notable
2011 Tohoku earthquake (“Great East Japan Earthquake”) which resulted in the
tsunami that caused a number of nuclear accidents. We also add two large
earthquakes ($M=7.3$ and $M=7.6$) to our study, which also occurred the same
day as aftershocks.
We employ the following procedure to create our time series:
* (i)
For each selected earthquake (see Appendix A) we create a new time series, the
normalized waveform denoted by $w_{t}$ out of the raw seismic acceleration
waveform data
$w_{norm}\equiv(w_{t}-\overline{w})/\sqrt{\overline{w_{t}^{2}}-\overline{w}^{2}}.$
(2)
* (ii)
From the time series $w_{norm}$ we define a new sub-series $w_{t}^{\prime}$,
starting at time coordinate where maximum $w_{t}$ occurs and terminating at
the end of the normalized waveform $w_{t}^{\prime}$ (see inset in Fig1(a)).
* (iii)
Let the time series $t_{i}$ denote the points in time when $w_{t}^{\prime}$
changes sign, with $t_{i}<t_{i+1}$. We define (see Fig 1(c)) the interval
series by
$\tau_{i}\equiv t_{i}-t_{i-1}.$ (3)
* (iv)
The sign series (see Fig. 1(d)) is defined by
$s_{t}\equiv sgn(w_{t}^{\prime})$ (4)
Note that our definition of interval is different than that recently defined
in several papers, where the return intervals $\tau$ have studied between
consecutive fluctuations above a volatility threshold $q$ in different complex
systems. The probability density function (pdf) of return intervals
$P_{q}(\tau)$ scales with the mean return interval as
$P_{q}(\tau)=\overline{\tau}^{-1}f(\tau/\overline{\tau})$ (5)
where $f()$ is a stretched exponential Yamasaki ; Wang2 ; Wang3 . Since, on
average, there is one volatility above the threshold $q$ for every
$\overline{\tau}_{q}$ volatilities, then it holds that BPPnas09
$1/\overline{\tau}_{q}\approx\int_{q}^{\infty}P(|R|)d|R|=P(|R|>q)\sim
q^{-\alpha}.$ (6)
For the time intervals $\tau_{q}$ between events given by fluctuations $R$
where $R>q$ Ref. BPPnas09 derived that $\overline{\tau_{q}}$, the average of
$\tau_{q}$, obeys a scaling law,
$\overline{\tau_{q}}=q^{\alpha}$ (7)
where by $\alpha$ denotes our estimate of the tail exponent probability
density function, $P(|R|^{1+\alpha})$. Similarly, if $P(|R|)$ follows an
exponential function $P(|R|)\propto\exp(-\beta|R|)$, then employing Eq. (6) we
easily derive
$\overline{\tau}_{q}\propto\exp{(\beta q)}.$ (8)
Eq. (8) can be used as a new method for estimation of the exponential
parameter $\beta$.
## III Memory of interval time series
Returning to waveform data, we begin analyzing the series by studying the
conditional mean
$\langle\tau|\tau_{0}\rangle/\overline{\tau}$ (9)
which gives the mean value of $\tau$ (see Eq. (3)) immediately following a
given term $\tau_{0}$, normalized in units of $\overline{\tau}$. The
conditional mean gives evidence of whether seismic memory exists in the
intervals in the form of correlations or anticorrelations. For example, should
correlations exist, one would expect the mean interval to be shorter in the
window immediately following a small interval.
Indeed, Fig. 2 shows that the large intervals $\tau$ tend to follow large
initial $\tau_{0}$ and small $\tau$ follow small $\tau_{0}$ indicating the
existence of (positive) correlations in the interval time series. We also note
that the autocorrelations tend to be stronger for the subset associated with
larger earthquakes than for those associated with smaller earthquakes.
To expand on this we also extend our investigation to longer range effects. We
investigate the mean interval after a cluster of $n$ consecutive intervals
that are either entirely above the series mean or entirely below it. We denote
clusters that are entirely above the series mean with a “$+$” and clusters
below the series mean with a “$-$”. Fig. 3 shows the mean interval $\tau$ that
follows a $\tau_{0}(n)$ defined as a cluster size of $n$. We find that for
“$+$” clusters—shown by open symbols—the mean interval increases with the size
of the cluster $n$. This is the opposite of what we find for “$-$”
clusters—shown as closed symbols. The results indicate the existence of at
least short-term memory in the interval time series. Furthermore, we find that
the mean interval increases with the seismic magnitude. However, this
relationship breaks at the high end of the Richter magnitude scale $M>6.5$.
## IV Detrended fluctuation analysis
Many physical, physiological, biological, and social systems are characterized
by complex interactions between a large number of individual components, which
manifest in scale-invariant correlations MFS ; Bunde ; Tak ; Kob82 . Since the
resulting observable at each moment is the product of a magnitude and a sign,
many recent investigations have focused on the study of correlations in
magnitude and sign time series Ashk20 ; Kantelhardt2 ; Kant2002 ; Hu ;
Plamen04itt ; Livina ; Engle82 . For example, the time series of changes
$\delta\tau_{i}$ of heartbeat intervals Ashk20 ; Kant2002 ; Kantelhardt2 ,
physical activity levels Hu , intratrading times in the stock market
Plamen04itt , and river flux values Livina all exhibit power-law
anticorrelations, while their magnitudes $|\delta\tau_{i}|$ are positively
correlated. A common means of finding autocorrelations hidden within a noisy
non-stationary time series is detrended fluctuation analysis (DFA)CKP ; Hu ;
Chen . In the DFA method, the time series is partitioned into pieces of equal
size $n$. For each piece, the local trend is subtracted and the resulting
standard deviation over the entire series is obtained. In general, the
standard deviation $F(n)$ of the detrended fluctuations depends on $n$, with
smaller $n$ resulting in trends that more closely match the data. The
dependence of $F$ on $n$ can generally be represented as a power law such that
$F(n)\propto n^{\alpha},$ (10)
where $\alpha$ is the scaling exponent—sometimes referred to as the Hurst
exponent—to be obtained empirically. DFA therefore can conceptually be
understood as characterizing the motion of a random walker whose steps are
given by the time series. $F(n)$ gives the walker’s deviation from the local
trend as a function of the trend window. Because the root mean square
displacement of a walker with no correlations between his steps scales like
$\sqrt{(}n)$, we can expect a time series with no autocorrelations to yield an
$\alpha$ of 0.5. Similarly, long-range power-law correlations in the signal
(i.e. large terms follow large terms and small terms follow small terms)
manifest as $\alpha>0.5$. Power-law anticorrelations within a signal will
result in $\alpha<0.5$. Additionally, DFA can be related to the
autocorrelation as follows: if the autocorrelation function $C(L)$ can be
approximated by a power law with exponent $\gamma$ such that
$C(L)\propto L^{-\gamma},$ (11)
then $\gamma$ is related to $\alpha$ by CKP
$\alpha\approx 1-\gamma/2.$ (12)
Another reason we employ the DFA method is that it is appropriate for sign
time series Kantelhardt2 . Other techniques for the detection of correlations
in non-stationary time series are not appropriate for sign time series. Also,
because the sign and interval time series have affine relations, the analysis
of sign will be helpful in understanding the intervals. However, the DFA gives
biased estimates for the power-law exponent in analysis of anticorrelated
series Hu , and so in order to improve the accuracy of analysis, we integrate
the time series before we employ the standard DFA procedure.
For the 2011 Tohoku earthquake, also known as the “Great East Japan
Earthquake”, we present the fluctuation function $F(n)$ of the coda wave,
measured at KSN station, as typical examples of sign and intervals time series
(Fig. 4). By using DFA, we find, for most coda waves after earthquakes, that
the time series of the intervals are consistent with a power-law correlated
behavior $\alpha=0.69$, while the sign time series of Eq. (4) are consistent
with a power-law anti-correlated behavior ($\alpha=0.32$). The results
therefore indicate that for the interval series large increments are more
likely to be followed by large increments and small increments by small
increments. These results are in agreement with the results of the correlation
analysis reported in Section 3. In contrast, anticorrelations in the sign time
series indicate that positive increments are more likely to be followed by
negative increments and vice versa.
For the entire set of sign time series comprising our sample we calculate the
average DFA scaling exponent $\overline{\alpha}=0.34\pm 0.09$ indicating
anticorrelations, and for the interval time series we calculate the average
DFA scaling exponent $\overline{\alpha}=0.58\pm 0.08$ indicating correlations.
For the different stations measuring the 2011 Tohoku earthquake we find that
for the sign time series, $\overline{\alpha}=0.29\pm 0.05$ and for the
interval time series, $\overline{\alpha}=0.66\pm 0.07$.
## V Relation between earthquake moments and scaling exponents of sign and
interval series
Because large earthquake events release such extraordinary amounts of energy,
it is reasonable to ask whether their occurrence influences local wave
dynamics. To this end, we study interval time series of coda waves from
earthquakes occurring in 2003, also including the particularly large events of
11 March 2011, when three events $M>7$ occurred in the same day. Fig. 5(a)
shows the DFA scaling exponent of the sign series versus seismic moment, where
seismic moment is a quantity used to measure the size of an earthquake. We
find a decreasing functional dependence between the DFA exponent of the sign
series and the seismic moment of the proximal earthquake with slope
$\gamma=-0.028\pm 0.002$, indicating that the DFA exponent decreases
approximately with seismic moment. Note that because most of the exponents are
$<0.5$, this indicates the presence of ever stronger anticorrelations in the
time series as earthquake magnitude increases. Note, however, that the data
break with this trend for very large earthquakes (Richter magnitude scale
$>6.6$ or seismic moment $>10^{19}$).
We also find similar results in the interval series, the difference being that
the anticorrelations become correlations. Fig. 5(b) shows that the DFA
interval exponent and seismic moment exhibit a positive functional dependence
with slope $\gamma=0.025\pm 0.002$ so that the DFA exponent increases with
increasing seismic moment. Because most of the exponents for the interval
series are $>0.5$, this indicates that the series show stronger correlations
for increasing seismic moment. Again, as with the sign series, we find a
deviation from this trend for very large earthquakes.
Having observed the influence of seismic moment on autocorrelations, we now
investigate whether other readily observable factors such hypocenter depth and
epicentral distance (the distance from the event to the recording station)
also contribute. Specifically, we would like to explore whether there is
evidence that such long-term memory is affected by the spreading process as
seismic waves disseminate outward from their epicenter to a recording station
or whether the memory observed is strictly due to the seismic activity. Fig. 6
shows that the DFA exponent for both interval and sign series are independent
of both hypocenter depth and epicentral distance. From these results we
speculate that the DFA exponent is mainly a result of the characteristics of
the hypocenter rather than the process by which the seismic waves are spread.
For moderately large earthquakes ($M_{0}=10^{14}\sim 10^{19}$), we approximate
the relation between the DFA scaling exponent and seismic moment through the
empirical formula
$\alpha\approx a~{}log_{10}(M_{0})+c$ (13)
where $a=-0.028$, $c=0.797$ for the sign time series and where $a=0.025$,
$c=0.174$ for the interval time series. Since
$M=(log(M_{0})-9.1)/1.5,$ (14)
we can also write
$\alpha\approx a(1.5M+9.1)+c=a^{\prime}~{}M+c^{\prime},$ (15)
where $a^{\prime}=-0.042$, $c^{\prime}=0.542$ for the sign series, and
$a^{\prime}=0.037$, $c^{\prime}=0.398$ for the interval series.
We note that similar size dependence in Hurst exponent was found in Ref.
Eisler where Hurst exponents of financial time series increase
logarithmically with company size.
## VI Summary
We analyze seismic coda waves during earthquakes, finding long-range power-law
autocorrelations in both the interval and sign time series. The sign series
generally display power-law anticorrelated behavior, with anticorrelations
becoming stronger with larger earthquake events, while the interval series
generally display power-law correlated behavior, with correlations also
becoming stronger with larger earthquake events. We also show that while the
DFA autocorrelation exponent is influenced by the size of the earthquake
seismic moment, it is unaffected by earthquake depth or epicentral distance.
Our findings are in contrast with a standard approach which assumes
independence in earthquake signals and thus have strong implications on the
ongoing debate about earthquake predictability SornettePNAS .
## VII Acknowledgements
We thank S. Havlin for his constructive suggestions, and thank JSPS for grant
of ”Research project for a sustainable development of economic and social
structure dependent on the environment of the eastern coast of Asia” that made
it possible to complete this study. We also thank the National Science
Foundation and the Ministry of Science of Croatia for financial support.
## VIII Appendix: The Selection of Earthquakes
In some regions it is common for multiple earthquakes to occur in short
succession. In many cases, because the interoccurrence times are so short, the
coda waves can be derived from more than one earthquake. This is especially
true for large earthquakes with many aftershocks Utu . In order to make sure
that the coda waves we study are the effects of only one earthquake, we need a
way of determining which earthquakes are independent. We use the following two
functions to determine the sphere of influence and duration of each earthquake
by using the Richter magnitude scale M Utu . We select only those earthquakes
that have no larger earthquake in their spatiotemporal sphere of influence,
$t\approx 10^{(M-4.71)/1.67}$ (16)
and
$R\approx 2\times 10^{(M+1)/2.7},$ (17)
where $t$ is the duration and $R$ is the sphere radius of influence. The two
functions are empirical formulas based on an analysis of earthquakes in Japan
Utu . The $10^{M+1.0}/2.7$ is an empirical formula that indicates the maximum
radius that a human can feel an earthquake, especially for the earthquakes in
Japan.
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Figure 1: (a) Location map for the 46 broadband stations of Full Range
Seismograph Network of Japan (F-net) (red snow marks). Inset: An example of a
record of a seismic wave (Up-Down component). (b) A part of the coda wave
series indicated in inset of (a), as an example. (c) An example sign time
series where the positive sign (+1) represents a positive waveform, and the
negative sign (-1) represents a negative waveform in coda wave series of
seismic wave. (d) Interval time series ($\tau$) of the coda wave series for a
subset of the record shown in (b).
Figure 2: Scaled mean conditional interval
$\langle\tau|\tau_{0}\rangle/\overline{\tau}$ vs $\tau_{0}/\overline{\tau}$ .
Five groups, one with no proximal earthquake and earthquakes with Richter
magnitude scale $M<4.5$, $M=4.5\sim 5.5$, $M=6.5\sim 6.5$, $M>6.5$. An
increasing trend implies a short-range correlation in the interval series.
Figure 3: Long-range memory in interval clusters. $\tau_{0}$ signifies a
cluster of intervals, consisting of $n$ consecutive values that all are above
(denote as ”$+$”) or below (denote as ”$-$”) the median of the entire interval
records. Plots display the scaled mean interval conditioned on a cluster,
$\langle\tau|\tau_{0}\rangle/\overline{\tau}$ vs the size $n$ of the cluster
for five group intervals. The upper part (overplotted) of curves is for ”$+$”
clusters while the lower part is for ”$-$” clusters. The plots show that ”$+$”
clusters are likely to be followed by large intervals and ”$-$” clusters by
small intervals, consistent with long-term correlations in interval records.
Similar to Fig.2, the long-term correlation increases with earthquake size,
with exceptions for very large earthquakes.
Figure 4: DFA fluctuation function $F(n)$ of 2011 Tohoku earthquake as a
function of time scale $n$ ($F(n)\propto n^{\alpha}$) for (a) sign time series
($\alpha+1=1.32$, ($\alpha<0.5$), indicates anticorrelations) and (b) interval
time series ($\alpha+1=1.69$, ($\alpha>0.5$), indicates correlations).
Figure 5: Scaling exponent $\alpha$ vs seismic moment (Richter magnitude
scale) for (a) sign time series (correlation coefficient $Cor=-0.3604$), and
(b) interval time series (correlation coefficient $Cor=0.3602$). The values of
$\gamma$ show negative slope in the regression $\alpha$ vs seismic moment of
the sign series, and positive slope in the regression of the interval series.
Triangular symbols show the mean of exponent within each bin ( bins:
$<1e+15,1e+15\sim 1e+16,1e+16\sim 1e+17,1e+17\sim 1e+18,1e+18\sim
1e+19,1e+19\sim 1e+20,>1e+21$), the error bar shows the $\pm$ standard
deviation. The plots show a linear relationship between logarithmic earthquake
moment and scaling exponent $\alpha$ in the sign and interval series, with
exceptions for very large earthquakes.
Figure 6: Scaling exponent $\alpha$ vs hypocenter depth for events where
Richter magnitude scale $M<5$ for (a) sign time series (b) interval time
series. Inset: scaling exponent $\alpha$ vs hypocenter depth requiring that
Richter magnitude scale $M>5$. (c) and (d) show Scaling exponent $\alpha$ vs
epicentral distance for events where Richter magnitude scale $M<5$. Inset:
scaling exponent $\alpha$ vs epicentral distance requiring that Richter
magnitude scale $M>5$. (c) sign time series, (d) interval series. All absolute
values of correlation coefficient are smaller than $0.1$, showing that
$\alpha$ is uncorrelated with both hypocenter depth and epicentral distance.
|
arxiv-papers
| 2011-10-18T15:49:38 |
2024-09-04T02:49:23.309616
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zeyu Zheng, Kazuko Yamasaki, Joel Tenenbaum, Boris Podobnik, and H.\n Eugene Stanley",
"submitter": "Zeyu Zheng",
"url": "https://arxiv.org/abs/1110.4031"
}
|
1110.4058
|
# ¿Como se afecta la descripción termodinámica de los sistemas físicos cuando
se incluye la gravedad?
W. A. Rojas C warojasc@unal.edu.co Universidad Nacional de Colombia R.
ARENAS S Universidad Nacional de Colombia
###### Abstract
This reseach aims at revising the thermal concepts and their relationship with
the General Theory of Relativity (GRT) and how physical systems are affected
when gravity is included in their description thermodynamics. The study found
that in the case of light, the entropy in an extreme scenario is proportional
to the area, not to the volume. This is due to a reduction of freedom degrees
of the system, since the gravity imposes a constraint limiting the number of
microstates which are accessible to the system and this is consistent with the
holographic principle.
###### pacs:
04.,04.20-q,04.70Bw,05.20.Gg
## I Introducción
La termodinámica estándar es el campo de la física que se encarga de estudiar
los procesos de intercambio de energía entre los sistemas y el medio que los
rodea. Y en esta termodinámica las variables que sirven para caracterizar el
estado de cierto sistema físico no está incluida la gravedad. ¿Pero cómo se
afecta la descripción termodinámica de los sistemas físicos, por ejemplo la
luz cuando se incluye la gravedad?
## II LA CONEXIÓN ENTRE LA GRAVITACION Y LA TERMODINÁMICA
Newton dio cuenta de ley fundamental que predecía el movimiento de los cuerpos
celestes. Una ley que es directamente proporcional al producto de las masas de
cuerpos que interactúan e inversamente proporcional al cuadrado de la
distancia de separación que hay entre ellos
$\vec{F}=G\frac{m_{1}m_{2}}{r^{2}}\hat{r},$ (1)
con $G$ siendo la constante de gravitación universal. Esta descripción es
buena y sirve de común para estudiar casi cualquier fenómeno mecánico de la
vida diaria. Solo fue hasta los primeros años del siglo XX, cuando Einstein
nos explico con su Teoría General de la Relatividad (TGR), el porqué de la
gravedad
$G_{ab}=\frac{8\pi G}{c^{4}}T_{ab}$ (2)
donde $G_{ab}$ es el tensor de Einstein, $c$ la velocidad de la luz y $T_{ab}$
el tensor momentum energía. La ecuación (2) es conocida como la ecuación de
campo de gravitatorio de Einstein; que conectan por un lado la curvatura del
espacio-tiempo $G_{ab}$ y la distribución de materia-energía $T_{ab}$.
En la descripción termodinámica de los sistemas físicos se notan las
siguientes limitaciones
* •
Todos los sistemas analizados son considerados en reposo respecto a un
observador. No se estudian sistemas que estén acelerados.
* •
Tales descripciones térmicas no toman en cuenta los efectos gravitatorios.
Tales restricciones en la descripción térmica de los sistemas se deben remover
al considerar los efectos de la curvatura espacio-tiempo para escenarios donde
esta no sea despreciable Tolman . Una forma que históricamente ha servido para
vincular la TGR y la termodinámica clásica ha sido considerar la primera ley
$\Delta E=\Delta Q-\Delta W,$ (3)
que hace referencia a la conservación de la energía de cualquier sistema
físico. Esta ley nos muestra el mecanismo de transferencia de energía entre el
medio y el sistema, ya sea por calor $Q$ o por trabajo $W$. Si consideramos el
equivalente relativista de la primera ley de termodinámica se puede obtener
vía el tensor momentum-energía $T_{ab}$, dado que tal tensor incluye todas las
formas de energía y materia presentes. Así, tendremos que sí la derivada
covariante del tensor momentum-energía es igual a cero ello implica la
conservación de la energía en TGRTolman .
$\nabla_{a}T^{ab}=0$ (4)
La ley cero establece que existe un parámetro de equilibrio que llamamos
temperatura, $T$. Sí un sistema físico por ejemplo un gas ideal está en
equilibrio térmico, tal se caracteriza por que todas sus partes exhiben la
misma temperatura. En el contexto gravitacional, debemos hacer la distinción
entre dos tipos de temperatura una local y otra medida en el infinito.
Consideremos un espacio-tiempo del tipo Schwarzschild; un agujero negro de
masa $M$ y de simetría esférica, no rotante y sin carga. Se halla que la
temperatura local es una función que solo depende de la distancia radial, es
decir varía con el potencial gravitacional. La cual no significa que un
sistema dado inmerso en un campo gravitacional intenso no se halle en
equilibrio, si no que la temperatura local , $T(r)$ se ve afectada por la
curvatura espacio-temporal (Este fenómeno, es más conocido como la Ley de
Tolman. Dado que $T(r)=T_{\infty}f(r)^{-1/2}$ con $f(r)$ siendo una función
dependiente de la coordenada radial $r$; que en el caso de Schwarzschild es
$f(r)=1-\frac{2Gm}{c^{2}r}$). Para un observador que se halle muy alejado del
horizonte del agujero negro medirá una temperatura $T_{\infty}$. Si mide la
temperatura de un objeto cae en dirección radial hacia el agujero negro vera
que esta aumenta; el objeto se ha termalizado. De lo que llega a concluir que
el agujero negro actúa como una fuente de calor Tolman ; SusskindL .
Cualquier objeto con una temperatura diferente del cero absoluto posee un
cierto grado de desorden al cual llamamos entropía. La entropía es una función
de estado que sirve para caracterizar un sistema físico, por ejemplo un gas
ideal contenido en un recipiente de paredes adiabáticas. Su entropía es
proporcional al volumen del contenedor (En pleno acuerdo al principio de
Boltzmann que relaciona la entropía $S$ con el logarítmo de la probabilidad de
hallar el sistema en un cierto microestado $\Omega$. Así el principio de
Boltzmann queda determinado por $S=k_{B}ln\left|\Omega\right|$ con $k_{B}$
igual a la constante de Boltzmann).
$S\propto ln\left|\frac{V}{V_{0}}\right|^{N},$ (5)
donde $V_{0}$ y $V$ son los volúmenes iniciales y finales en que es posible
hallar el gas en momento determinado dentro del recipiente y $N$ el número de
partículas del gas. Tal entropía en últimas depende del número de grados de
libertad que hay por partícula, que para un gas ideal monoatómico es igual a
$\frac{3}{2}k_{B}T$, con$k_{B}$ siendo la constante de Boltzmann.
Esto permite caracterizar el número de microestados compatibles con un
macroestado; que finalmente corresponde a la información que se puede conocer
del sistema pues a mayor entropía, menor es la información disponible sobre el
estado del sistema. Es decir la entropía estadística debe dar cuenta de la
información que tenemos sobre el estado del sistema y esta nunca puede
decrecer en el tiempo, a lo sumo para un sistema aislado debe permanecer
constante SusskindL ; Benkenstein .
En la década de los 70’s, con los trabajos de Hawking Hawking y Bekenstein
Benkenstein lograron establecer algunas propiedades termodinámicas de los
agujeros, como relacionar la gravedad superficial con la temperatura $T$ y el
área del horizonte con la entropía. Así, la entropía termodinámica de un
agujero negro es proporcional al área del horizonte Corichi . Dar cuenta de
este tipo de entropía es uno de los paradigmas de la Física Teórica en la
actualidad, y su explicación yace en teorías tan sofisticadas como la Teoría
de Cuerdas o la Gravedad Cuántica de Bucles.
Nuestro siguiente paso en la profundización de una termodinámica que incluya
la gravedad en la descripción de los sistemas físicos, es tomar un ejemplo
ampliamente aceptado como lo es el de la luz (radiación electromagnética) y
estudiarla en un escenario gravitacional extremo.
## III El método de Einstein
Maxwell demostró que la luz era de naturaleza ondulatoria y casi medio siglo
después Planck y Einstein mostraron que la radiación posee estructura
granular, cuantos de radiación indivisible son emitidos y absorbidos
continuamente cuando la luz interactúa con la materia Einstein .
Una de las bondades del método de Einstein, consiste en demostrar que partir
de la función de distribución de cuerpo negro de Wien, del principio de
Boltzmann y de la termodinámica conocida para aquella época, establecer la
estructura granular de la luz en el espacio de Minkowski. Sin necesidad de
establecer hipótesis adicionales como si lo hace Planck. Seguiremos este
método para establecer si la luz, en un espacio-tiempo con alta curvatura;
presenta estructura corpuscular.
De acuerdo al segundo principio de termodinámica, suponemos a la luz como un
sistema físico que está en un definido estado con una densidad de entropía
$S=V\phi$ donde $V$ es el volumen del sistema físico y $\phi$ la densidad de
entropía. Tal entropía consiste en la suma de las entropías monocromáticas (es
decir para una frecuencia especifica) que están separadas las unas de las
otras y que se puede obtener por adición
$S=\int^{\infty}_{0}V\phi d\nu$
Esto es válido en el espacio plano. Con un espacio-tiempo curvo debemos
considerar como la gravedad afecta el volumen del sistema físico. Sea
$dV=\frac{4\pi r^{2}}{\sqrt{f(r)}}dr$, el elemento diferencial de volumen
corregido gravitacionalmente con $\rho(\nu)$ siendo la distribución de cuerpo
negro para un espacio-tiempo curvo. Por lo que la entropía total de la
radiación electromagnética en tales condiciones es
$S=\int^{R}_{0}\int^{\infty}_{0}\phi\left(\rho(\nu),\nu\right)d\nu\frac{4\pi
r^{2}}{\sqrt{f(r)}}dr.$ (6)
Para el modelo tipo cuerpo negro, $\delta S=0$, se obtiene la ley
$\frac{\partial\phi}{\partial\rho}=\frac{1}{T_{\infty}}.$ (7)
Lo cual significa que todas las radiaciones con distintas frecuencias están
caracterizadas por tener la misma temperatura y que tal temperatura está
afectada por el campo gravitacional
$\frac{\partial\phi}{\partial\rho}=\frac{1}{T(r)}=\frac{1}{T_{\infty}}f(r)^{1/2}.$
(8)
Sabemos que la frecuencia de la luz también se halla afectada por la presencia
del campo gravitacional. Por lo que podemos escribir la función de
distribución de Wien para la radiación electromagnética
$\rho(\nu,r)=\frac{8\pi
h(\nu_{\infty}f(r)^{-1/2})^{3}}{c^{3}}e^{-\frac{h\nu_{\infty}}{k_{B}T_{\infty}}}.$
(9)
Despejando de (9) el término $\frac{1}{T_{\infty}}$ e incertandolo en (8)
$\frac{d\phi}{d\rho}=-\frac{k_{B}}{h\nu_{\infty}}ln\left|\frac{\rho
c^{3}}{8\pi h\nu_{\infty}^{3}f(r)^{-3/2}}\right|f(r)^{1/2}.$ (10)
Integrando
$\phi=-\frac{k_{B}f(r)^{1/2}\rho}{h\nu_{\infty}}\left[ln\left|\frac{\rho
c^{3}f(r)^{3/2}}{8\pi h\nu_{\infty}^{3}}\right|-1\right].$ (11)
Tenemos que la entropía en un intervalo de frecuencia $\nu$ y $\nu+d\nu$ está
dada por
$S=V\phi\Delta\nu,$ (12)
y la energía por unidad de volumen y frecuencia en la forma
$E=V\rho\Delta\nu.$ (13)
Por lo anterior,tenemos que (11) se convierte en
$S=-\frac{k_{B}f(r)^{1/2}E}{h\nu_{\infty}}\left[ln\left|\frac{c^{3}f(r)^{3/2}E}{8\pi
h\nu_{\infty}^{3}V\Delta\nu}\right|-1\right].$ (14)
Sea $S_{0}$, la entropía de la radiación electromagnética confinada a un
volumen $V_{0}$
$S_{0}=-\frac{k_{B}f(r)^{1/2}E}{h\nu_{\infty}}\left[ln\left|\frac{c^{3}f(r)^{3/2}E}{8\pi
h\nu_{\infty}^{3}V_{0}\Delta\nu}\right|-1\right].$ (15)
Entonces la variación en la entropía $\Delta S$ de un volumen $V_{0}$ a un
volumen $V$ para el sistema en consideración
$\Delta
S=-\frac{k_{B}f(r)^{1/2}E}{h\nu_{\infty}}ln\left|\frac{V}{V_{0}}\right|,$ (16)
Si el principio de Boltzmann se considera siempre válido incluso en el
gravitatorio. En donde la entropía es proporcional al logartimo de la
probabilidad de hallar el sistema en un microestado dado $\Delta
S=k_{B}ln\left|\Omega\right|$. Y que tal probabilidad para un gas ideal es
$\Omega=\left[\frac{V}{V_{0}}\right]^{N}$, donde $N$ es el número de moléculas
del gas. Luego (16) se puede escribir como
$\Delta
S=k_{B}ln\left|\frac{V}{V_{0}}\right|^{\frac{Ef(r)^{1/2}}{h\nu_{\infty}}}.$
(17)
Einstein en su trabajo original encontró
$\Delta S=k_{B}ln\left|\frac{V}{V_{0}}\right|^{\frac{E}{h\nu}},$ (18)
se obtiene que
$\frac{E}{N}=h\nu_{\infty}f(r)^{-1/2}=h\nu(r).$ (19)
La entropía para un gas ideal a temperatura constante es de la forma
$pdV=TdS=nRT\frac{dV}{V}.$ (20)
En el límite cuando $\Delta S\rightarrow 0$, (18) de transforma se reduce a
$dS=\frac{k_{B}Ef(r)^{1/2}}{h\nu_{\infty}}\frac{dV}{V},$ (21)
Luego
$T_{\infty}dS=\frac{k_{B}Ef(r)^{1/2}}{h\nu_{\infty}}T_{\infty}\frac{dV}{V},$
(22)
la comparación entre las ecuaciones (20) y (22) nos permite obtener más
evidencias a cerca de la estructura granular de la radiación electromagnética
cerca de la superficie de Schwarzschild. En la aproximación de Wien, que
funciona bien el rango de altas energías, la luz en un campo gravitacional
intenso se comporta como un gas ideal con cuantos de energía $hv(r)$. El
método de Einstein ha mostrado ser eficaz incluso en un escenario
gravitacional intenso pues la luz exhibe una estructura granular.
## IV UNA APROXIMACIÓN A LA DESCRIPCIÓN TERMODINÁMICA DE LA LUZ EN UN
ESPACIO-TIEMPO CURVO
Es obvio que para considerar una curvatura espacio-tiempo alta, que no sea
despreciable, el escenario más indicado son objetos celestes que sean más
masivos que la tierra tales como el sol, enanas blancas, estrellas de
neutrones o agujeros negros Rojas ; Mukohyama .
Consideremos una masa con simetría esférica de magnitud estelar. Sean dos
casquetes esféricos de superficies reflectoras concéntricas que rodean esta
masa. Tal que los radios $R$, $L$ de cada uno de los casquetes sean mayores
que el radio de Schwarzschild $R_{0}$, de tal manera que $R\geq R_{0}$ y
$R=R_{0}+\epsilon$ con $\epsilon\ll R_{0}$ para la primera superficie
reflectora y $L\gg R_{0}$. Tal como se puede ver en la Figura (1).
Figure 1: Cuerpo gravitacional rodeado por dos superficies reflectoras
En el espacio comprendido entre las dos superficies reflectoras se coloca un
gas de fotones, que alcanza una temperatura $T_{\infty}$ cuando es medida
sobre el casquete exterior. Con la aproximación de altas energías para la
radiación, de tal forma que la longitud de onda es pequeña en comparación con
los radios de las superficies reflectoras o a la curvatura espacio-tiempo, se
tendrá una aproximación a la física estadística clásica. El espacio
comprendido entre las dos superficies reflectoras esta descrito por una
métrica de la forma
$ds^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}sin^{2}\theta
d\phi^{2},$ (23)
así, se tiene que la ecuación (20) corresponde al elemento de línea que
describe el exterior de la masa. Por lo que la forma que adopta la energía
libre de Helmholtz cerca de la superficie interna es Fursaev
$F=-\frac{\pi^{2}k^{4}_{B}}{90\hbar^{3}c^{3}}\int T^{4}\sqrt{-g}d^{3}x.$ (24)
Donde $\sqrt{-g}$ es el determinante del tensor métrico $g_{ab}$ asociado al
elemento de línea dado por (21) y $d^{3}x$ el elemento diferencial de volumen.
Cerca del horizonte se puede reemplazar la coordenada $r$ por la coordenada
$\zeta$ Susskind , que mide la distancia propia desde el radio de
Schwarzschild, $R_{0}=\frac{2Gm}{c^{2}}$. Con lo que la energía libre de
Helmholtz se puede escribir como
$F=-\frac{\pi^{2}k^{4}_{B}c^{3}}{90\hbar^{3}}T^{4}_{\infty}\kappa^{-3}\int
d^{2}\sigma\int\zeta^{-3}d\zeta,$ (25)
con $\kappa$ siendo la gravedad superficial y $d^{2}\sigma=dx^{2}+dy^{2}$.
Integrando (22) con $d^{2}\sigma=A$
$F=-\frac{\pi^{2}k^{4}_{B}c^{3}}{90\hbar^{3}}T^{4}_{\infty}\kappa^{-3}A\int^{\zeta=\delta}_{\zeta=\epsilon}\zeta^{-3}d\zeta,$
(26)
con la aproximación de $\delta\gg\epsilon$, se halla
$F=-\frac{\pi^{2}k^{4}_{B}c^{3}}{180\hbar^{3}\epsilon^{2}}T^{4}_{\infty}\kappa^{-3}A.$
(27)
Recordemos que el parámetro $\epsilon$, corresponde a la distancia de
separación entre el radio de Schwarzschild $(R_{0})$ y la primera superficie
reflectora de radio $R$. El siguiente paso es calcular las demás propiedades
térmicas de la luz con la misma receta del espacio plano
$S=-\left(\frac{\partial F}{\partial
T_{\infty}}\right)_{V}=\frac{\pi^{2}k^{4}_{B}c^{3}}{45\hbar^{3}\epsilon^{2}}T^{3}_{\infty}\kappa^{-3}A$
(28)
la entropía de la luz cerca del radio de Schwarzschild queda descrita por
(25), nótese que sigue siendo una función extensiva, pero ya no es
proporcional al volumen sino que es proporcional al área $A$.Y no significa
que el sistema se haya reducido a un área. La energía interna de la luz es
$E=\frac{\pi^{2}k^{4}_{B}c^{3}}{60\hbar^{3}\epsilon^{2}}T^{4}_{\infty}\kappa^{-3}A.$
(29)
La energía interna de la luz es proporcional a $T^{4}_{\infty}\kappa^{-3}A$.
Lo que confirma que este contexto la ley de Stephan- Boltzmann también es
válida. La capacidad calorífica es
$C_{v}=-\left(\frac{\partial E}{\partial
T_{\infty}}\right)_{V}=\frac{\pi^{2}k^{4}_{B}c^{3}}{15\hbar^{3}\epsilon^{2}}T^{3}_{\infty}\kappa^{-3}A.$
(30)
Si consideramos un agujero negro de una masa solar su temperatura Hawking
$T_{H}\propto 10^{-8}K$, que corresponde a un temperatura muy cerca del cero
absoluto. Zemanski y Dittman comentan al respecto: si $T\rightarrow 0$,
tendremos que $C_{P}\rightarrow C_{V}$ Zemansky . La presión que ejerce la
radiación electromagnética es
$P=-\frac{1}{\epsilon}\left(\frac{\partial F}{\partial
A}\right)_{T_{\infty}}=\frac{\pi^{2}k^{4}_{B}c^{3}}{180\hbar^{3}\epsilon^{3}}T^{4}_{\infty}\kappa^{-3}A.$
(31)
La presión que ejerce la radiación electromagnética es proporcional a
$T_{\infty}^{4}\kappa^{-3}A$ y está fuertemente ligada al tipo de $\epsilon$
que se escoja
## V Conclusiones
El resultado de (18) indica que la noción de fotón introducida por Einstein,
considerando la aproximación de Wien corregida gravitacionalmente (9), sigue
siendo válida en presencia de la gravedad. Así, se tiene que los fotones
poseen una energía $h\nu(r)$ que incluye la corrección gravitacional. De igual
forma la comparación entre las expresiones (18) y (19) nos permite evidenciar
la estructura granular de la radiación electromagnética cerca del radio
gravitacional. Todo lo anterior se ha logrado considerando siempre valido el
principio de Boltzmann.
Cerca de la superficie de Schwarzschild, el campo gravitacional es muy fuerte
si $\epsilon$ es pequeño comparado con las dimensiones del sistema por efecto
de la relación (24). Se halló que la energía libre de Helmholtz, es
proporcional a $T_{\infty}^{4}\kappa^{-3}A$ con $A$ siendo el área del
horizonte y no al volumen del sistema como ocurre en el espacio-tiempo de
Minkowski (Una completa descripción de la termodinámica de la radiación
electromagnética se halla en L. Landau and E. LifshitzLandau ). Ello es
importante dada la relación existente entre la energía de Helmholtz (F) y la
función de partición Z de la termodinámica estadística pues
$F\propto lnZ.$
La expresión (25), corresponde a la entropía de la radiación electromagnética
cerca del radio gravitacional es proporcional al área y no al volumen. Ello se
justifica en el hecho que siempre hemos considerado valido el principio de
Boltzmann, que nos indica que la entropía es proporcional al logaritmo de la
probabilidad y tal está ligada al conjunto de microestados que son accesibles
al sistema. Tal número de configuraciones es asociado necesariamente al número
de grados de libertad del sistema. Considerando un escenario gravitacional
intenso, la entropía de la radiación electromagnética exhibe un comportamiento
proporcional al área y no al volumen. Lo cual significa que el número de
microestados que son accesibles al sistema ha disminuido cuando se ha
incorporado la gravedad en la descripción termodinámica de la luz. ¿Qué ha
pasado con esos microestados que ya no son accesibles al sistema? En
condiciones de equilibrio térmico todos los microestados son equiprobables
para que se cumpla la condición de máxima entropía. Cuando es considerada la
gravedad en la descripción estadística de la luz ciertos microestados dejan de
ser equiprobables y por lo tanto ya no son accesibles al sistema. Ello ocurre
pues el número de grados de libertad de la radiación ha disminuido. La
gravedad lo que hace es imponer una ligadura sobre el sistema. Limitando sus
grados de libertad y sus microestados. La idea que la materia ordinaria
también pueda exhibir una entropía proporcional al area cuando en la
descripción termodinámica se incorpora la gravedad es consistente con el
principio holográfico. Tesis que fue por primera vez expresada por ‘t Hooft y
Susskind en 1993. Y expresa que ”la máxima entropía posible depende del área
de la superficie que delimita el volumen y no de este…Si un sistema
tridimensional completo puede ser descrito plenamente por una teoría física
definida solo en su contorno bidimensional se espera que el contenido de
información del sistema no exceda del contenido de la descripción limitado al
contorno” Bekensteinj .
## References
## References
* (1) R. C. Tolman. Relativity Thermodynamics and Cosmology. Dover Publications Inc., New York (1987).
* (2) L. Susskind. Temas: Investigación y ciencia (Barcelona). 36 (2004):36-41.
* (3) S. W. Hawking. Comm. Math. Phys. 43 (1975):199.
* (4) J. D. Benkenstein. Phys. Rev. D 7 (1973):2333.
* (5) A. Corichi and D. Sudarky. Mod. Phys. Lett A17 (2002):1431.
* (6) A. Einstein. Ann. Phys. 17(1905):132.
* (7) L. Landau and E. Lifshitz. Curso de Física Teórica. Física Estadística, volumen 5. Editorial Reverte S.A., Barcelona, 1973.
* (8) R. P. Feynman. Statistical Mechanics: A set of lectures. The Benjamin/Cummings Publishing Company, Inc, Massachusetts, 1961.
* (9) M. W. Zemansky and R. H Dittman.Heat and Thermodynamics. The McGraw-Hill Companies, Inc, New York, 1997.
* (10) D. V. Fursaev. Phys. Part. Nucl. 36 (2005):81.
* (11) L. Susskind and J. Lindesay. An Introduction to Black Holes, Information, and String Theory Revolution. World Scientific Publising Co. Pte. Ltd., London, 2005.
* (12) K. S. Thorne, C. W. Misner and Wheeler. Gravitation. W.H. Freedman and Company. San Francisco. 1973.
* (13) D. McHamon. Relativity Demystified. Mc Graw-Hill. New York. 2006.
* (14) J. D. Bekenstein. Temas: Investigación y ciencia (Barcelona). 36 (2004):16-23.
* (15) S. M. Carroll, e-Print: arXiv:9712019v1 [gr-qc].
* (16) W. A. Rojas C. Tesis de Maestría Termodinámica de un gas de fotones en la vecindad de una superficie de Schwarzschild. Observatorio Astronómico Nacional. Universidad Nacional de Colombia. Director: J. R. Arenas S. Disponible en:
www.observatorio.unal.edu.co
/archivos/tesisOAN/2010/wRojas.pdf
* (17) S. Mukohyama and W. Israel. Phys. Rev D58 (1998):104005.
|
arxiv-papers
| 2011-10-18T17:27:22 |
2024-09-04T02:49:23.318697
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "W. A. Rojas C. and R. Arenas S",
"submitter": "Alexis Larranaga PhD",
"url": "https://arxiv.org/abs/1110.4058"
}
|
1110.4129
|
# Four IRAC Sources with an Extremely Red H$-$[3.6] Color: Passive or Dusty
Galaxies at z$>$4.5?
J.-S. Huang11affiliation: Harvard-Smithsonian Center for Astrophysics, 60
Garden Str., Cambridge, MA02138, USA , X. Z. Zheng22affiliation: Purple
Mountain Observatory, 2 West Beijing Rd., Nanjing, Jiangsu Province, PRC , D.
Rigopoulou33affiliation: Department of Physics, Denys Wilkinson Building,
Keble Road, Oxford, OX1 3RH, UK , and G. Magdis33affiliation: Department of
Physics, Denys Wilkinson Building, Keble Road, Oxford, OX1 3RH, UK ,G. G.
Fazio11affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden
Str., Cambridge, MA02138, USA , T. Wang11affiliation: Harvard-Smithsonian
Center for Astrophysics, 60 Garden Str., Cambridge, MA02138, USA
,44affiliation: Department of Astronomy, Nanjing University, Nanjing, Jiangsu
Province, PRC
###### Abstract
We report detection of four IRAC sources in the GOODS-South field with an
extremely red color of H$-$[3.6]$>$4.5. The four sources are not detected in
the deep HST WFC3 H-band image with Hlimit=28.3 mag. We find that only 3 types
of SED templates can produce such a red H$-$[3.6] color: a very dusty SED with
the Calzetti extinction of AV=16 mag at z=0.8; a very dusty SED with the SMC
extinction of AV=8 mag at z=2.0$\sim$2.2; and an 1Gyr SSP with
A${}_{V}\sim$0.8 at z=5.7. We argue that these sources are unlikely dusty
galaxies at z$\leq$2.2 based on absent strong MIPS 24$\mu$m emission. The old
stellar population model at z$>$4.5 remains a possible solution for the 4
sources. At z$>$4.5, these sources have stellar masses of
Log(M∗/M⊙)=10.6$\sim$11.2. One source, ERS-1, is also a type-II X-ray QSO with
L2-8keV=1.6$\times$1044 erg s-1. One of the four sources is an X-ray QSO and
another one is a HyperLIRG, suggesting a galaxy-merging scenario for the
formation of these massive galaxies at high redshifts.
cosmology: observations — galaxies: evolution — galaxies:formation — infrared:
galaxies
## 1 Introduction
Extremely Red Objects(ERO) are of great interests to modern astrophysics. A
simple red color criterion generally selects two types of galaxies: those at
high redshifts and those with heavy dust extinction. With rapid progress in
telescope apertures and detectors, this red color selection always leads to
new types of galaxies or galaxies at record-high redshifts. After large format
near-infrared array cameras became available for astronomical surveys, people
started to detect galaxies with very red R$-$K colors (Elston et al., 1988,
EROs, R$-$K$>$5 or I$-$K$>$4). EROs were so rare in the early days that they
were thought to be abnormal objects at very high redshifts. There have been
more and more EROs detected by larger aperture telescopes with more advanced
IR array cameras. Most EROs with R$-$K$>$5 are now identified as elliptical
and dusty galaxies at $0.6<z<1.5$ (Thompson et al., 1999; McCarthy et al.,
2001; Cimatti et al., 2002). One extreme case, an ERO with I$-$K=6.5, was
spectroscopically identified as a dusty Ultra-Luminous InfraRed Galaxy (ULIRG)
at z=1.44 (Elbaz et al., 2002). This source is analog to a local ULIRG,
Arp220. Smail et al. (2002) suggested that most dusty EROs at high redshifts
are LIRG/ULIRGs. The Spitzer IRAC permits very fast imaging of sky in mid-
infrared bands with great depth. Wilson et al. (2004) found that 17% of their
IRAC 3.6$\mu$m selected sample are EROs at z$\geq$1.
Red color criteria are practically diversified, and applied to almost all
kinds of photometry in optical and IR bands. But the physics for this type of
criteria are limited to following: (1) the Lyman break at 912Å; (2) the Balmer
and the accumulated absorption line breaks at 3648Å and 4000Å; or (3) dust
extinction. Red color caused by the Lyman/Balmer break can be used to estimate
redshifts. In most deep broad band imaging surveys, one could not tell if a
red color is due to Lyman/Balmer Break or dust extinction (Steidel et al.,
2003). An additional color in longer wavelength bands is usually applied
together with the red color criterion to select galaxies at high redshifts.
For example, U$-$g and g$-$R colors were used to select galaxies at z=3 where
the Lyman break shifts between U and g bands, commonly known as U drop-out for
red U$-$g color(Steidel et al., 2003). The drop-out technique was applied in
much longer wavelength bands to select galaxies at z=6$\sim$9\. Franx et al.
(2003) used the NIR color J$-$K$>$2.3 to select Distant Red Galaxies (DRGs)
with the strong Balmer/4000Å break shifting in between the J and K at
z$\sim$2\. The NIR spectroscopy for DRGs by Kriek et al. (2007) shows that
about half of their sample are passive evolved galaxies at z$\sim$2, and the
rest are dusty galaxies in a much wider redshift range. Several groups
idenitfied 24$\mu$m luminous and optically faint or invisible sources with
R$-$[24]$>$14.2 ($f_{24}/f_{R}>1000$) as very dusty ULIRGs at z$\sim$2\. These
sources are confirmed spectroscopically by Spitzer IRS and ground-based
optical spectroscopy (Houck et al., 2005; Yan et al., 2007; Dey et al., 2008;
Huang et al., 2009).
In this paper we report detection of four galaxies with extremely red colors
of H$-$[3.6]$>$4.5 in the GOODS-South field. Only one similar source, a
submillimeter galaxy (SMG) called GOODS 850-5 (aka GN10) in the GOODS-North
field, was ever found to have H$-$[3.6]$>$4.5. This SMG was also detected by
the Submillimeter Array (Wang et al., 2007) with a high angular resolution of
$\sim$2”, permitting identification of its counterparts in shorter wavelength
bands. Wang et al. (2009) performed ultra-deep J and H band imaging for this
source with NIC3 camera on HST. A total of 16 orbits of HST observation,
reaching a nanoJansky depth in the F160W band, yields no detection for this
source. Based on this red H$-$[3.6] color, they argue that its redshift is at
z=4$\sim$6.5. Later, detection of CO(4-3) from this source confirms its
redshift at z=4.05 (Daddi et al., 2009). This study provides the first look at
properties of this new type of object. More sources of this kind will be
detected in the Cosmic Asembly Near-infrared Deep Extragalactic Legacy Survey
(Grogin et al., 2011, CANDELS).
## 2 Deep IR Imaging of GOODS-South
The Great Observatories Origins Deep Survey(GOODS) is the deepest multi-
wavelength survey with space telescopes including HST, Spitzer and
Chandra(Dickinson, 2004). The depth of GOODS IRAC 3.6$\mu$m imaging reaches
sub-microJansky level. The deep NIR imaging of the GOODS-South field was
selected for the Early Released Science (O’Connell, 2010, ERS) for the Wide
Field Camera 3 (WFC3), a fourth-generation UVIS/IR imager aboard HST. We
construct an H-selected multi-wavelength catalog including YJH+IRAC photometry
in the ERS covered region. The IR images have very different angular
resolutions: 0.03” for the HST WFC3 YJH band images and $\sim$2” for the
Spitzer IRAC 3.6-8.0 $\mu$m images. A photometry program called TFIT is
specifically designed to perform photometry on a lower resolution image with
input information of object positions and light distributions measured in a
high resolution image (Laidler et al., 2007). The TFIT program convolves a PSF
kernel to the high angular resolution stamp image for each object to construct
lower angular resolution image templates and fit them to the lower angular
resolution image. In our case, we ran the TFIT to perform photometry on the
IRAC 3.6$\mu$m image for the H-band selected galaxies detected in the ERS
F160W image. The TFIT also produces a residual image after subtracting all
H-band detected galaxies in the 3.6$\mu$m image. We visually inspected the
residual image and found four IRAC sources detected at 3.6$\mu$m with no
H-band counterparts shown in Figure 1. The limiting magnitude for the input
H-selected sample is H$=$28.3 mag at 3$\sigma$ level, therefore these sources
are fainter than H$=$28.3 mag and have colors redder than H$-$[3.6]=4.5.
We searched for counterparts of these sources in all available wavelength
bands in the GOODS-South field. All four sources are detected in the remaining
3 IRAC bands. None of these sources is detected in the GOOD-South ACS BVIZ
images with the 5$\sigma$ limiting magnitudes of 28.65,28.76,28.17.and 27.93
respectively. The K-band is the only band available in between H and
3.6$\mu$m, permitting further constraint on its SED and photometric redshift.
The 5$\sigma$ limiting magnitude for the K-band image of the GOOD-South field
(Retzlaff et al., 2010) is 24.4 mag and none of our sources is detected in K
band. FIR observation is also critical in determining properties of these red
sources (Wang et al., 2009). ERS-3 is clearly detected at 24$\mu$m and ERS-2
is marginally detected at $\sim$3$\sigma$ level. We inspected the Herschel
SPIRE deep imaging of the CDFS, and found only ERS-3 is marginally detected at
250$\mu$m and 350$\mu$m. The PSF for the SPIRE 500 $\mu$m band image is too
broad ( 36.6”) to permit accurate extraction of flux density for ERS-3 (Huang
et al., 2011). ERS-3 is also detected at 1.4gHz with
f1.4gHz=29.2$\pm$8$\mu$Jy. The remaining three sources are not detected in
radio with f${}_{1.4gHz}<$24$\mu$Jy. No submillimeter/millimemter source is
detected in the locations of these four sources (Scott et al., 2009;
Wei$\beta$ et al., 2009). Another source, ERS-1, is an X-ray source in the
Chandra 2Ms catalog(Alexander et al., 2003). ERS-2 and ERS-4 are detected only
in 4 IRAC bands (Table.1).
## 3 SEDs, Photometric Redshifts, and Properties of the Extremely Red Objects
For three out of the four sources in this study, only NIR+IRAC flux densities
are available for their photometric redshift estimation. The most predominant
feature in their SEDs is the extremely red color of H$-$[3.6]$>$4.5. We first
rule out that those sources are brown dwarves. A brown dwarf with T=600K has
H$-$[4.5]$>$4.0 (Legget et al., 2010), such a brown dwarf should have
[3.6]$-$[4.5]$>$1 due to the methane absorption at 3.6$\mu$m in its
photosphere. All our sources have [3.6]$-$[4.5]$<$0.5. It is unlikely that
this red color is due to the Lyman break at z$>$15\. Galaxies with a strong
Balmer/4000Å break at 3$<$z$<$8 can have very red H$-$[3.6] colors. Recently
Richard et al. (2011) detected a lensed source at z=6.02 behind cluster A383
with H$-$[3.6]=1.5, arguing that this is a possible passive galaxy. A few more
lensed galaxies with extreme red optical-MIR color are identified to be dusty
galaxies at either z$\sim$2 or z$>$6 (Boone et al., 2011; laporte2011). Their
H$-$[3.6] colors are only in range of 0.5$<$H$-$[3.6]$<$2.
### 3.1 SED Fitting
We model the H$-$[3.6] color using stellar population models of both BC03
(Bruzual & Charlot, 2003, BC03) and the upgraded model CB07 (Charlot &
Bruzual, 2007) emphasizing on AGB star contribution in the rest-frame NIR
bands. The templates are constructed with various stellar populations of the
solar metallicity and a very wide range of dust extinction of
0$<$A${}_{V}<$25\. The star formation history used in constricting the
template set includes single burst, exponential decreasing with various e-fold
times, and constant rates. Two types of extinction curves are used in the SED
templates: the widely used Calzetti extinction curve for galaxies (Calzetti et
al., 2000) and the SMC extinction curves (Gordon et al., 2003). Both
extinction curves are only up to 2.2$\mu$m, while our detected photometry
points for the four sources are in 3.6$<\lambda<$8.0$\mu$m. We extend both
curve to the IRAC bands using the MIR dust extinction curve in
3.6$<\lambda<$24$\mu$m (Chapman et al., 2009).
We first fitted our model templates to GOODS 850-5 to investigate what kind of
stellar population and how much amount of dust extinction can make such a red
H$-$[3.6] color. GOODS 850-5 is already known at z=4.05, thus provides better
constraint on stellar population and dust extinction. Both BC03 and CB07
models with either Calzetti or SMC extinction yields a similar result for
GOODS 850-5 : 1Gyr old single stellar population with modest extinction of
AV=2.4$\sim$3.6. The best fit template is an 1Gyr single stellar population
model with the Calzetti extinction of AV=3.6. Wang et al. (2009) obtained a
similar model template for their best fitting but yielding much higher
redshifts at z=6.9.
We argue that the four objects in this study are at the same redshifts: they
have very similar SEDs, and their positions are very close to each other, with
a mean distance of $\sim$1.5’ to their closest neighbors. We fitted the SED
templates to the six IR photometry points (H, K, and 4 IRAC bands) for each
object in the sample. Our fitting yields two extreme solutions with the
Calzetti exinction: a very dusty template with AV=16$\sim$18 at z$\sim$0.8 and
an old stellar population template with z$\sim$5.7 and A${}_{V}\sim$0.8
(Figure 2). The templates with the SMC extinction yields a similar dusty
solution with AV=7$\sim$8 at z$\sim$2.2. By applying heavy dusty extinction
with A${}_{V}>$7 to templates, its shape and the resulting photometric
redshift are only determined by extinction curves. For example, the SMC
extinction curve yields a photometric redshift of zp=2.2 for our objects,
which is caused by a dip at 1.25$\mu$m in the SMC extinction
curve(A(1.65$\mu$m)/AV=0.169, A(1.25$\mu$m)/AV=0.131, and
A(0.81$\mu$m)/AV=0.567, Gordon et al. (2003)). With a very high AV value, this
feature is amplified. At z=2.2, this extinction dip shifts to the IRAC
3.6$\mu$m band to make H$-$[3.6] redder and [3.6]$-$[8.0] bluer. The
photometric redshifts obtained with the SMC extinction curve are mainly driven
by this feature.
There are generally three final solutions (Figure 2) in our SED fitting for
the 4 objects: a dusty template at z=0.8 with the Calzetti extinction of
AV=16; a dusty template at z=2.2 with the SMC extinction of AV=8; and an old
stellar population template with age of 1Gyr and A${}_{V}<$1\. Though each
solution has a slightly different minimum $\chi$ of 3$<\chi_{min}<$6, we
consider each solution equally possible. All three SED models are able to
produce H$-$[3.6]$>$4.5, and require extreme conditions in the galaxies:
either extremely dusty of A${}_{V}>$7 or even AV=16 or very massive galaxies
at z$>$4.5, both of which are very rare in current extragalactic surveys.
### 3.2 Extremely Dusty Galaxies at z$<$3?
In the first solution, a galaxy with the Calzetti extinction of
A${}_{V}\sim$16 at z=0.8 can have H$-$[3.6]$>$4.5. At z=0.8, the IRAC
3.6$\mu$m band probes the rest-frame K-band. Our sources have 3.6$\mu$m flux
density of f3.6=0.6$\sim$1.5$\mu$Jy, implying that their stellar masses are
less than 5$\times$109 M⊙. Most galaxies with such a small stellar mass at
z=0.8 are blue galaxies with no dust extinction. M82 is a dusty galaxy with
lower stellar mass of 4$\times$109 M⊙, with heavy dust obscuration
(5$<$A${}_{V}<$51) only occurring in its center (Beirão et al., 2008). The
whole M82 appears much bluer than our objects. The H$-$[3.6] color at z=0.8 is
equivalent to the rest-frame I$-$K color. M82 has I$-$K=0.82 (Dale et al.,
2007), because most stars in the disk of M82 are in the outside of its dusty
region. Thus an object like M82 at z=0.8 would be detected in the ERS H-band
imaging. This solution, however, requires that the whole galaxy should be in
heavy obscuration. Only ULIRGs have such a obscured morphology. Using M82
central region SEDs, we predict that the MIPS 24$\mu$m flux densities for the
first 3 sources would have f24=50$\sim$70$\mu$Jy, well above the FIDEL MIPS
24$\mu$m limiting flux density. Only ERS3 is marginally detected at 24$\mu$m
with f24=39$\mu$Jy, the rest are not detected at 24$\mu$m. We argue that this
scenario is least possible.
The second solution is the template with the SMC extinction of AV=7$\sim$8 at
z=2$\sim$2.2. There are many Dust Obscured Galaxies(DOG) identified at
z$\sim$2 (Houck et al., 2005; Dey et al., 2008; Bussmann et al., 2009).
Several groups (Houck et al., 2005; Yan et al., 2007; Huang et al., 2009)
performed mid-infrared spectroscopy for DOGs and detected a very strong
silicate absorption feature at 9.8$\mu$m in their spectra, indicating a very
heavy dust extinction. Bussmann et al. (2009) took high resolution H-band
imaging of these sources in the Bootes field with NICMOS on HST to study their
rest-frame optical morphologies. We identified these DOGs in the Bootes IRAC
photometry catalog (Ashby et al., 2009), and found that DOGs were generally
red with 1.5$<$H$-$[3.6]$<$3.3, and luminous at 3.6$\mu$m with
f3.6=5$\sim$60$\mu$Jy. The four sources in this study are much fainter at
3.6$\mu$m, and have a much redder color of H$-$[3.6]$>$4.5. On the other hand,
the DOGs in Bussmann et al. (2009) have a 24-to-8$\mu$m flux ratio of
f24/f8=40$\sim$350\. If the 4 sources were indeed fainter DOGs at z$\sim$2,
they should have a 24$\mu$m flux density of f${}_{24}>$120$\mu$Jy, much higher
than the 24$\mu$m limiting flux density in GOODS-South. Based on much redder
H$-$[3.6] and fainter 24$\mu$m flux, we argue that these objects are at higher
redshifts than the DOGs at z$\sim$2.
### 3.3 Massive Galaxies at z$>$4.5?
The 1Gyr SSP template with the Calzetti extinction of AV=0.8 at z$\sim$5.7 can
also fit the SEDs of the four objects, very similar to the best-fit template
for GOODS 850-5. In this scenario, the red H$-$[3.6] is mainly due to the
Balmer/4000Å jump shifting in between H and 3.6$\mu$m bands at z$>$4\. Figure
3 shows that the old stellar template fits to SEDs of the 4 objects. The
resulting photometric redshifts have a large error of 0.4$<\sigma$(z)$<$1.1.
We argue that the 4 objects are at the same redshifts. Thus by adopting the
best $\sigma(z)$ in the 4 objects, these objects should be at z$>$4.5 at
3$\sigma$ level.
ERS-3 is detected at 250 and 350$\mu$m, thus has a very strong FIR emission,
very similar to GOODS 850-5 (Huang et al., 2011). The best fit SED models for
both GOODS 850-5 and ERS-3 are old stellar population models. We propose a
two-component SED model to reconcile the old stellar population and FIR
emission in these objects: an old stellar population and a very dusty star-
forming component. The star-forming component is so dusty, simliar to those
dusty galaxies detected at z$\sim$2 (Houck et al., 2005; Yan et al., 2007; Dey
et al., 2008; Huang et al., 2009), that its optical/NIR SED is dominated by
the old stellar population component. For example, a dusty component with the
same stellar mass and AV=6 at z$>$5 would only contribute 10% increase at 8
micron, and much lower percentage in the shorter IRAC bands. Assuming a
typical dust temperature of Tdust=40K and redshift of z=5.7, we calculate the
FIR luminosity for ERS-3 as Log(LFIR/L${}_{\odot})$=13.1. The FIR-to-Radio
flux ratio is about q=2.26, consistent with q values for submillimeter
galaxies at $z>4$ (Huang et al., 2011). The remaining 3 objects are not be
detected by Herschel SPIRE, but may still be IR luminous galaxies with just
Log(LFIR/L${}_{\odot})<$13.1.
The SSP model fitting also yields stellar mass of Log(M∗/M⊙)=10.6$\sim$11.2
for our sources (Figure 3). Spectroscopic confirmed galaxies at z$\sim$5.7
including both Lyman-break (LBGs) and Ly-$\alpha$ (LAE) galaxies have a
typical stellar mass of Log(M∗/M⊙)$\leq$10 (Yan et al., 2006; Lai et al.,
2007; Younger et al., 2007; Richard et al., 2011). Recently Marchesini et al.
(2010) argued that very massive galaxies were already formed at 3$<$z$<$4\.
Theoretically, Li et al. (2007) argued that QSOs at z$>$6 resided in a massive
halo of M$\sim$8$\times$1012M⊙, and stellar mass for their host galaxies can
be as high as 1012M⊙. We have another piece of evidence consistent with these
systems being massive. ERS-1 is also an X-ray source detected in the Chandra
2Ms survey (Alexander et al., 2003). It has x-ray flux densities of
$f_{0.5-2keV}=7.55\pm 2.14\times 10^{-17}erg~{}s^{-1}~{}cm^{-2}$ and
$f_{2-8keV}=6.50\pm 1.50\times 10^{-16}erg~{}s^{-1}~{}cm^{-2}$. The hard X-ray
Luminosity for this source is L2-8keV=1.6$\times$10${}^{44}erg~{}s^{-1}$ at
$z_{p}=5.7$ assuming no absorption correction. Its X-ray-to-optical-flux ratio
($f_{x}/f_{R}$) is higher than 60 and the hardness ratio is $\sim$1, thus
ERS-1 is an obscured type-II QSO. A typical black hole mass for such a QSO at
z=3$\sim$6 is 109 $\sim$ 1010 M⊙ (Netzer, 2003; Shemmer et al., 2004; Fan et
al., 2006). Assuming a typical H${\alpha}$ FWHM of 2000 km/s for a QSO, we
convert the L2-8keV to black hole mass for ERS-1 as MBH=5$\times$108 M⊙ using
the relation proposed by Sarria et al. (2010). Trakhtenbrot & Netzer (2010)
argued that a host galaxy with 109 M⊙ black hole has a typical stellar mass of
M${}_{*}\sim$1011 M⊙, consistent with the stellar mass we derived for ERS-1.
## 4 Summary and Discussion
We identified four IRAC sources in the GOODS-South field with extremely red
color of H$-$[3.6]$>$4.5. The only known source with a similar H$-$[3.6] color
is GOODS 850-5, a SMG in the GOODS-North field. We argue that the four sources
must be at the same redshift based on the following facts: they have similar
rest-frame optical/NIR SEDs; and they are spatially very close to each other
with a mean angular distance $\sim 1.5^{\prime}$. Only 3 types of templates
can produce H$-$[3.6]$>$4.5: a very dusty template with the Calzetti
extinction of AV=16 mag at z=0.8; a very dusty templates with the SMC
extinction of AV=8 mag at z=2.0; and an 1Gyr SSP model with A${}_{V}\sim$0.8
at z=5.7. By comparing the 4 objects with local dusty galaxies and DOGs at
z$\sim$2, we argue that they are unlikely dusty galaxies at z=0.8 or z=2.2
based on absent strong 24$\mu$m emission. The old stellar population model at
z$>$4.5, with the best fit at z=5.7, remain a possible solution for the 4
sources. One of our sources, ERS-3, is also detected by Herschel at 250$\mu$m
and 350$\mu$m, yielding Log(LFIR/L⊙)=13.2. We propose a two-component SED
model for these sources: an old SSP component dominating their optical-to-MIR
SEDs and a very dusty star-forming component mainly contributing to their FIR
SEDs. The SED fitting yields stellar masses of Log(M∗/M⊙)=10.6$\sim$11.2 for
the four sources. One source, ERS-1, is also a type-II x-ray QSO with
L2-8keV=1.6$\times$10${}^{44}erg~{}s^{-1}$. Based on the MBH-Mbulge relation
for high-z QSOs, ERS-1 should have a massive bulge of Log(M∗/M⊙)=11. One of
the four sources is an X-ray QSO and another one is a HyperLIRG, suggesting a
galaxy-merging scenario for the formation of these massive galaxies at high
redshifts.
This work is based on observations made with the Spitzer Space Telescope,
which is operated by the Jet Propulsion Laboratory, California Institute of
Technology under NASA contract 1407, and with the NASA/ESA HST obtained at the
Space Telescope Science Institute, which is operated by the association of
Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555.
Facilities: Spitzer(IRAC), HST(STIS), CXO(ASIS).
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Figure 1: Stamp images for the 4 H-band drop-out objects in the GOODS-South
field in the F160W and 3.6$\mu$m bands. ERS-3 is marginally detected at 24,
250, 350$\mu$m, and 20cm. ERS-1 is an X-ray source detected in the Chandra 2Ms
imaging (Alexander et al., 2003).
Figure 2: The likelihood contours as a function of redshift and dust
extinction AV. for the best-fit SED of ERS-1. SED fitting for the remaining
objects yields the same solutions. The left panel is the contour with the 1Gyr
stellar population model and the Calzetti extinction and the right panel with
the SMC extinction. The 1Gyr stellar population model with the Calzetti
extinction of AV=3.6 is also the best fit for GOODS 850-5 at z=4.05. Figure
3: Optical-to-MIR SEDs for the H-band drop-out sources in ERS. The best-fit
templates to the SEDs are dusty SSP models with E(B-V)=0.2$\sim$0.35 and
$z_{p}=5.7$. The fitting also yields stellar mass of Log(M∗/M⊙)=10.6$\sim$11.2
for the 4 sources. We also plot two dusty model templates against observed
SEDs.
Table 1: Infrared flux densities for the H-band Dropout sources
Name | RA | DEC | F160W | K | 3.6$\mu$m | 4.5$\mu$m | 5.8$\mu$m | 8.0$\mu$m | 24$\mu$m
---|---|---|---|---|---|---|---|---|---
ERS-1 | 53.084726 | -27.707964 | 0.0066$\pm$0.002 | 0.2904$\pm$0.1398 | 1.23$\pm$0.03 | 1.97$\pm$0.04 | 3.04$\pm$0.24 | 3.25$\pm$0.26 | -11.7$\pm$3.8
ERS-2 | 53.132749 | -27.720144 | 0.0036$\pm$0.002 | -0.0431$\pm$0.1681 | 1.54$\pm$0.20 | 1.56$\pm$0.10 | 2.03$\pm$0.26 | 3.21$\pm$0.28 | 14.7$\pm$4.0
ERS-3 | 53.060827 | -27.718263 | 0.0019$\pm$0.002 | 0.0727$\pm$0.1398 | 1.05$\pm$0.03 | 1.41$\pm$0.05 | 1.75$\pm$0.23 | 2.24$\pm$0.25 | 39.5$\pm$4.3
ERS-4 | 53.167161 | -27.715316 | 0.0054$\pm$0.002 | 0.2628$\pm$0.1382 | 0.57$\pm$0.06 | 0.66$\pm$0.06 | 1.25$\pm$0.25 | 0.87$\pm$0.28 | 5.9$\pm$3.6
Note. — All flux densities in this table are in unit of $\mu$Jy.
|
arxiv-papers
| 2011-10-18T21:26:28 |
2024-09-04T02:49:23.330774
|
{
"license": "Public Domain",
"authors": "J.-S. Huang, X. Z. Zheng, D. Rigopoulou, G. Magdis, G. G. Fazio, T.\n Wang",
"submitter": "Jiasheng Huang",
"url": "https://arxiv.org/abs/1110.4129"
}
|
1110.4176
|
# Elliptically distributed lozenge tilings of a hexagon
Dan Betea
###### Abstract
We present a detailed study of a 4 parameter family of elliptic weights on
tilings of a hexagon introduced by Borodin, Gorin and Rains, and generalize
some of their results. In the process, we connect the combinatorics of the
model with the theory of elliptic special functions.
We first analyze some properties of the measure and introduce canonical
coordinates that are useful for combinatorially interpreting results. We then
show how the computed $n$-point function (called the elliptic Selberg density)
and transitional probabilities connect to the theory of $BC_{n}$-symmetric
multivariate elliptic special functions and difference operators discovered by
Rains. In particular, the difference operators intrinsically capture the
combinatorial model under study, while the elliptic Selberg density is a
generalization (deformation) of probability distributions pervasive in the
theory of random matrices and interacting particle systems.
Based on quasi-commutation relations between elliptic difference operators, we
construct certain natural measure-preserving Markov chains on such tilings. We
then immediately obtain and describe an exact sampling algorithm from such
distributions. We present sample random tilings from these measures showing an
arctic boundary phenomenon. Interesting examples include a 1 parameter family
of tilings where the arctic curve acquires 3 nodes.
Finally, we show that the particle process associated to such tilings is
determinantal with correlation kernel given in terms of the univariate
elliptic biorthogonal functions of Spiridonov and Zhedanov.
###### Contents
1. 1 Introduction
2. 2 The model
1. 2.1 Interpretations
2. 2.2 Probabilistic model
3. 2.3 Positivity of the weight
4. 2.4 Degenerations of the weight
5. 2.5 Canonical coordinates
3. 3 Distributions and transition probabilities
4. 4 Elliptic difference operators
1. 4.1 Definitions and some properties
2. 4.2 Interpretation of difference operators and their properties
5. 5 Perfect Markov chain sampling algorithm
1. 5.1 The $S\mapsto S+1$ step
2. 5.2 Algorithmic description of the $S\mapsto S+1$ step
3. 5.3 Algorithmic description of the $S\mapsto S-1$ step
6. 6 Correlation kernel and determinantal representations
1. 6.1 A brief overview of elliptic biorthogonal functions
2. 6.2 Determinantal representations
7. 7 Computer simulations
8. 8 Appendix
## 1 Introduction
This paper examines work began by Borodin, Gorin and Rains in [BGR10]. In op.
cit., the authors examined $q$-distributed boxed plane partitions from several
perspectives, but the $q$-distributions were obtained as limits of the
elliptic distribution briefly appearing in the Appendix. The present paper
takes the Appendix of [BGR10] and expands upon it, following the steps in
[BGR10] and [BG09]. However, since we are working at the elliptic level
(rather than a degeneration as in [BGR10]), new tools are needed to generalize
the results of [BGR10]. These tools belong to the area of elliptic special
functions, an active area of research in algebra and analysis generalizing,
among other things, the Askey and $q$-Askey schemes of orthogonal polynomials
(as described in [KS] for example). Thus, in some complementary sense, while
being a generalization of [BGR10], the paper is an application of multivariate
tools introduced by Rains in [Rai10] and [Rai06] (the first is more analytic,
the second being more algebraic) which build upon univariate elliptic
biorthogonal functions found by Spiridonov and Zhedanov a few years earlier in
[SZ00]. Work in the area of elliptic special functions started with Frenkel
and Turaev’s discovery of elliptic (theta) hypergeometric series ([FT97]) -
the authors of op. cit. cite Baxter’s work (see his book [Bax82]) as the
genesis of the theory.
The history of the problem starts with random uniformly distributed boxed
plane partitions. Much is known about these: asymptotics and frozen boundary
behavior ([CKP01], [CLP98], [KO07]); correlation kernel via Hahn orthogonal
polynomials (see [Joh05], [BG09], [Gor08]); exact sampling algorithms
([BG09]). Somewhat central to the subject is the topic of discrete Hahn
orthogonal polynomials (which themselves are terminating generalized
hypergeometric series). One level up and we arrive at $q$-distributions on
boxed plane partitions ([BGR10], and [KO07] for the variational problem used
to derive the limit shape for the $q^{\pm Volume}$ distributions). Almost as
much as above is known about these, and central to the subject are certain
discrete $q$-orthogonal polynomials ($q$-Racah, $q$-Hahn) from the $q$-Askey
scheme, which themselves are terminating $q$-hypergeometric series (see [GR04]
for a full description, [KS] for a distillation of the formulas).
The present work analyzes the elliptic level (the distribution was introduced
in the Appendix of [BGR10], but also independently from a slightly different
perspective in [Sch07]). We look at two aspects: exact sampling algorithm and
correlation kernel. The third aspect in [BG09] and [BGR10] is obtaining
asymptotics of the correlation kernel and through this obtaining the frozen
boundary behavior in the large scale limit. While we indeed see a frozen
boundary behavior in our case and can characterize it via variational
techniques (and we present computer simulations of the results), we cannot yet
analyze the asymptotics of elliptic biorthogonal functions (techniques used in
previous works - e.g., in [BGR10] \- fail if we replace orthogonal polynomials
by elliptic biorthogonal functions). More direct techniques like solving the
variational problem described in [KO07] for the $q$-Hahn case and in Section
2.4 of [BGR10] seem computationally intractable so far. The reason is the
associated complex Burgers equation one has to solve becomes considerably more
complicated. Nevertheless, it is a (new) feature of the elliptic model that
the frozen boundary can have 3 nodal points (as seen in the computer
simulations).
From a different perspective, we try to create a bridge between elliptic
special functions discussed in the references given above and combinatorics of
tilings of hexagons (equivalently, dimer coverings of the appropriate graph
etc. See Section 2 for interpretations). To wit, we give a combinatorial
interpretation to several objects appearing in the theory of elliptic special
functions: the ($t=q$ case) multivariate elliptic difference operators
discovered by Rains ([Rai10]), the $\Delta$-symbols of [Rai06] and the
(univariate) elliptic biorthogonal functions of Spiridonov and Zhedanov
([SZ00]).
This paper tries to emulate the organization of [BG09] and [BGR10], but with
notation heavily influenced by [Rai06]. It is organized as follows: in the
remainder of the introduction, we set up most the important notation and
terminology.
We set up the combinatorial and probabilistic aspects in Section 2 (in the
Appendix we consider a different way of assigning weights to rhombi that is
manifestly more symmetric. We prefer to use the formulas from Section 2
though, at the cost of symmetry breaking since they lead to shorter
computations and arguments). Also in this section we study positivity of our a
priori complex measure and introduce various coordinate systems used
throughout the paper, including the important canonical coordinates (which
embed our model in a certain square of an elliptic curve).
In Section 3 we compute relevant distributions and transition probabilities.
Sections 2 and 3 are an expansion and in depth analysis of the Appendix in
[BGR10].
Section 4 recalls some definitions and properties of elliptic tools introduced
by Rains ([Rai10], [Rai06] \- we refer the reader to these works for the
proofs we omit) and then connects these with the probability and combinatorics
being studied. We show that the constraints of the model are intrinsically
captured by the elliptic difference operators under discussion.
Section 5 describes a perfect sampling algorithm for such elliptic distributed
boxed plane partitions. It is based on the idea of forming a new measure-
preserving Markov chain out of two old quasi-commuting ones (as in [BF10]; see
also [DF90]). The algorithm starts from a deterministic parallelogram shape
and samples relatively easy distributions to successively transform the
parallelogram into a hexagon accordingly distributed (by increasing one side
by 1, and decreasing another side by 1; a parallelogram can be seen as a
hexagon with two sides of length 0). We use the quasi-commutation relations
for the elliptic difference operators of Section 4 to construct this
algorithm.
Section 6 deals with the correlation kernel. We start by recalling facts about
univariate elliptic biorthogonal functions and show that the time increasing
(decreasing) Markov process is indeed determinantal, with correlation kernel
given as a determinant of (a matrix composed of) elliptic biorthogonal
functions. These replace the orthogonal polynomials discussed above.
In Section 7 we present some computer simulations obtained from the algorithm
described in Section 5. Choice of parameters for obtaining the trinodal cases
(surfaces where the arctic circle has 3 nodes at 3 vertices of the hexagon)
are also explained.
We end with the Appendix, which provides a highly symmetric view of the entire
picture
For the remainder of the section, we will set the notation that will appear in
the rest of the paper. We define the theta and elliptic Gamma functions
([Rui97]) as follows:
$\displaystyle\theta_{p}(x):=\prod_{k\geq 0}(1-p^{k}x)(1-\frac{p^{k+1}}{x})$
$\displaystyle\Gamma_{p,q}(x)=\prod_{k,l\geq
0}\frac{1-p^{k+1}q^{l+1}/x}{p^{k}q^{l}x}$
Note the elliptic gamma function is symmetric in $p$ and $q$. The theta-
Pochhammer symbol (a generalization of the $q$-Pochhammer symbol) is defined,
for $m\geq 0$, as
$\theta_{p}(x;q)_{m}=\prod_{0\leq i<m}\theta_{p}(q^{i}x).$
As is usual in this area, presence of multiple arguments before the semicolon
(inside theta or elliptic Gamma functions) will mean multiplication. To wit:
$\theta_{p}(uz^{\pm 1};q)_{m}=\theta_{p}(uz;q)_{m}\theta_{p}(u/z;q)_{m};\
\Gamma_{p,q}(a,b)=\Gamma_{p,q}(a)\Gamma_{p,q}(b).$
We have the following important identities ($n\geq 0$ an integer):
$\begin{split}&\theta_{p}(x)=\theta_{p}(p/x)\\\
&\theta_{p}(px)=\theta_{p}(1/x)=-(1/x)\theta_{p}(x)\\\
&\Gamma_{p,q}(q^{n}x)=\theta_{p}(x;q)_{n}\Gamma_{p,q}(x)\end{split}$ (1)
The last identity in (1) can be extended for $n<0$ or even for non integer $n$
to provide a generalization of the theta-Pochhammer symbol for negative or
even non-integer lengths. These identities also extend to the following among
theta-Pochhammer symbols:
$\begin{split}&\theta_{p}(a;q)_{n+k}=\theta_{p}(a;q)_{n}\theta_{p}(aq^{n};q)_{k}\\\
&\theta_{p}(a;q)_{n}=\theta_{p}(q^{1-n}/a;q)_{n}(-a)^{n}q^{\binom{n}{2}}\\\
&\theta_{p}(a;q)_{n-k}=\frac{\theta_{p}(a;q)_{n}}{\theta_{p}(q^{1-n}/a;q)_{k}}(-\frac{q}{a})^{k}q^{\binom{k}{2}-nk}\\\
&\theta_{p}(aq^{-n};q)_{k}=\frac{\theta_{p}(a;q)_{k}\theta_{p}(q/a;q)_{n}}{\theta_{p}(q^{1-k}/a;q)_{n}}q^{-nk}\\\
&\theta_{p}(a;q)_{-n}=\frac{1}{\theta_{p}(aq^{-n};q)_{n}}=\frac{1}{\theta_{p}(q/a;q)_{n}}(-\frac{q}{a})^{n}q^{\binom{n}{2}}\\\
&\theta_{p}(aq^{n};q)_{k}=\frac{\theta_{p}(a;q)_{k}\theta_{p}(aq^{k};q)_{n}}{\theta_{p}(a;q)_{n}}=\frac{\theta_{p}(a;q)_{n+k}}{\theta_{p}(a;q)_{n}}\end{split}$
(2)
We will use the above identities throughout for simplifying computations
without explicitly referring to them.
If $f(x_{1},...,x_{n})$ is a function of $n$ variables defined on
$(\mathbb{C}^{*})^{n}$, we call it $BC_{n}$-symmetric if it is symmetric (does
not change under permutation of the variables) and invariant under $x_{k}\to
1/x_{k}$ for all $k$. We will call it a $BC_{n}$-symmetric theta function of
degree $m$ if in addition, it satisfies the following:
$f(px_{1},...,x_{n})=(\frac{1}{px_{1}^{2}})^{m}f(x_{1},...,x_{n}).$
The prototypical example of a $BC_{n}$-symmetric theta function of degree 1
is:
$\prod_{1\leq k\leq n}\theta_{p}(ux_{k}^{\pm 1}).$
We now define the following function (which will play an important subsequent
role):
$\displaystyle\varphi(z,w)=z^{-1}\theta_{p}(zw,z/w)$ (3)
Note $\varphi$ is $BC_{2}$-antisymmetric ($\varphi(z,w)=-\varphi(w,x)$) of
degree 1. It is a consequence of the addition formula for Riemann theta
functions that
$\displaystyle\varphi(x,y)=\left(\frac{\varphi(z,x)}{\varphi(w,x)}-\frac{\varphi(z,y)}{\varphi(w,y)}\right)\frac{\varphi(w,x)\varphi(w,y)}{\varphi(z,w)}$
for arbitrary $z,w$. We observe that the expression in parentheses appearing
above is a Vandermonde-like factor in transcendental coordinates
$X=\frac{\varphi(z,x)}{\varphi(w,x)},Y=\frac{\varphi(z,y)}{\varphi(w,y)}$, so
$\varphi(z_{i},z_{j})$ is an “elliptic analogue” of the (Vandermonde)
difference $z_{i}-z_{j}$. This is indeed the case if one takes the right limit
and then a product over $i<j$. To wit:
$\displaystyle\lim_{q\to 1}\frac{\lim_{p\to
0}\varphi(q^{x_{i}},q^{x_{j}})}{q-q^{-1}}=x_{i}-x_{j}.$
Notationally, for a function $f$ of $n$ variables, we will use the
abbreviation $f(...x_{k}...)$ to stand for $f(x_{1},...,x_{n})$.
We will make reference to the delta symbols defined in [Rai10] (see also
[Rai06] \- we are in the case $t=q$ in the notation from both references),
which we define here. We fix $\lambda\in m^{n}$ a partition (that is, a
partition with at most $n$ parts all bounded by $m$). Define the partition
$2\lambda^{2}$ by $(2\lambda^{2})_{i}=2(\lambda_{\lceil i/2\rceil})$.
$\displaystyle\mathcal{C}^{0}_{\lambda}(x;q)=\prod_{1\leq
i}\theta_{p}(q^{1-i}x;q)_{\lambda_{i}}$
$\displaystyle\mathcal{C}^{0}_{2\lambda^{2}}(x;q)=\prod_{1\leq
i}\theta_{p}(q^{1-2i}x,q^{2-2i}x;q)_{2\lambda_{i}}$
$\displaystyle\mathcal{C}^{+}_{\lambda}(x;q)=\prod_{1\leq i\leq
j}\frac{\theta_{p}(q^{2-i-j}x;q)_{\lambda_{i}+\lambda_{j}}}{\theta_{p}(q^{2-i-j}x;q)_{\lambda_{i}+\lambda_{j+1}}}=\prod_{i<j}\frac{\theta_{p}(q^{2-i-j}x)}{\theta_{p}(q^{2-i-j+\lambda_{i}+\lambda_{j}}x)}\prod_{1\leq
i}\frac{\theta_{p}(q^{2-2i}x;q)_{2\lambda_{i}}}{\theta_{p}(q^{2-i-n}x;q)_{\lambda_{i}}}$
$\displaystyle\mathcal{C}^{-}_{\lambda}(x;q)=\prod_{1\leq i\leq
j}\frac{\theta_{p}(q^{j-i}x;q)_{\lambda_{i}-\lambda_{j+1}}}{\theta_{p}(q^{j-i}x;q)_{\lambda_{i}-\lambda_{j}}}=\prod_{i<j}\frac{\theta_{p}(q^{j-i-1}x)}{\theta_{p}(q^{j-i+\lambda_{i}-\lambda_{j}-1}x)}\prod_{1\leq
i}\theta_{p}(q^{n-i}x;q)_{\lambda_{i}}$
$\displaystyle\Delta_{\lambda}(a|...b_{i}...;q)=\frac{\mathcal{C}^{0}(...b_{i}...;q)}{\mathcal{C}^{0}(...\frac{pqa}{b_{i}}...;q)}\cdot\frac{\mathcal{C}^{0}_{2\lambda^{2}}(pqa;q)}{\mathcal{C}^{-}_{\lambda}(pq,q;q)\mathcal{C}^{+}_{\lambda}(pa,a;q)}$
Of interest will be the $\Delta$-symbol with six parameters
$t_{0},t_{1},t_{2},t_{3},u_{0},u_{1}$ satisfying the balancing condition
$q^{2n-2}t_{0}t_{1}t_{2}t_{3}u_{0}u_{1}=q$. Because the usual balancing
condition has $pq$ on the right hand side (the reader should consult the
Appendix of [Rai10] for more on why this is necessary), we multiply $u_{1}$ by
$p$ (this choice is arbitrary, so a priori some symmetry is broken, but this
will not affect our results). We define the discrete elliptic Selberg density
as:
$\displaystyle\Delta_{\lambda}(q^{2n-2}t_{0}^{2}|q^{n},q^{n-1}t_{0}t_{1},q^{n-1}t_{0}t_{2},q^{n-1}t_{0}t_{3},q^{n-1}t_{0}u_{0},q^{n-1}t_{0}(pu_{1});q)=const\cdot\prod_{i<j}(\varphi(z_{i},z_{j}))^{2}$
(4) $\displaystyle\prod_{1\leq
i}q^{l_{i}(2n-1)}\theta_{p}(z_{i}^{2})\frac{\theta_{p}(t_{0}^{2},t_{0}t_{1},t_{0}t_{2},t_{0}t_{3},t_{0}u_{0},t_{0}u_{1};q)_{l_{i}}}{\theta_{p}(q,q\frac{t_{0}}{t_{1}},q\frac{t_{0}}{t_{2}},q\frac{t_{0}}{t_{3}},q\frac{t_{0}}{u_{0}},q\frac{t_{0}}{u_{1}};q)_{l_{i}}}$
(5)
where $l_{i}=n-i+\lambda_{i}$, $z_{i}=q^{l_{i}}t_{0}$ and the constant is
independent of $\lambda$ and present in the formula to make the
$\Delta$-symbol elliptic in all of its arguments (the value of the constant
can be computed, but such constants will be immaterial for the rest of the
paper). This discrete elliptic Selberg density is the weight function for the
discrete elliptic multivariate biorthogonal functions defined in [Rai06].
Notice it can be written more symmetrically in terms of the $z_{i}$’s and the
elliptic Gamma functions as
$const\cdot\prod_{i<j}(\varphi(z_{i},z_{j}))^{2}\cdot\prod_{i}z_{i}^{2n-1}\theta_{p}(z_{i}^{2})\frac{\Gamma_{p,q}(t_{0}z_{i},t_{1}z_{i},t_{2}z_{i},t_{3}z_{i},u_{0}z_{i},u_{1}z_{i})}{\Gamma_{p,q}(\frac{q}{t_{0}}z_{i},\frac{q}{t_{1}}z_{i},\frac{q}{t_{2}}z_{i},\frac{q}{t_{3}}z_{i},\frac{q}{u_{0}}z_{i},\frac{q}{u_{1}}z_{i})}.$
We will denote by $\mathbb{E}$ the elliptic (Tate) curve
$\mathbb{C}^{*}/\langle p\rangle$ for some $|p|<1$. An elliptic function $f$
(of 1 variable) will just be a function defined on $\mathbb{E}$ (that is,
$f(px)=f(x)$).
Throughout the remainder, constants (by which we mean factors independent of
the variables usually denoted by $x_{k},y_{k},z_{k}$) will largely be ignored
(and we will write $const$ wherever this appears), but they are there to make
measures into probability measures (i.e., normalizing factors) or to make
certain functions elliptic (i.e., invariant under $p$-shifts). Their values
can often be recovered, and we comment on how to recover them whenever
possible.
Finally, throughout this paper we will freely use two different systems of
coordinates for our model (related by a simple affine transformation - see the
next section). While this may seem redundant, coordinatizing in two different
ways will more aptly reveal different features of the elliptic special
functions and difference operators under study.
Acknowledgements
The author would like to thank Alexei Borodin, Fokko van de Bult, Vadim Gorin,
and Eric Rains for their help through numerous conversations.
## 2 The model
### 2.1 Interpretations
We consider random tilings of an $a\times b\times c$ regular hexagon embedded
in the triangular lattice (with Cartesian coordinates $(i,j)$) by tiles of
three types, as can be seen in the Figure 1. The probabilistic details are set
out in Section 2.2. We will find it more convenient to encode the hexagon via
the following three numbers:
$N=a,T=b+c,S=c.$
Figure 1: A tiling of a $3\times 2\times 3$ hexagon and the associated stepped
surface
Equivalently, these tilings can be thought of as dimer matchings on the dual
honeycomb lattice (every rhombus in a tiling is a line matching two vertices
in the dual lattice), stepped surfaces, boxed plane partitions ($b\times c$
rectangles with positive integers $\leq a$ filled in that decrease weakly
along rows and columns starting from the top left corner box) or 3D Young
diagrams (any section parallel to one of the three bounding walls is a Young
diagram).
A yet different way of viewing such tilings, important hereinafter, is as
collections of non-intersecting paths in the square lattice. The paths start
at $N$ consecutive points on the vertical axis (counting from the origin
upwards) and end at $N$ consecutive points on the vertical line with
coordinate $T$. Each path is composed of horizontal segments or diagonal
(Southwest to Northeast, slope 1) segments, and the paths are required not to
intersect. Figure 2 explains this, and also introduces the coordinate frame
$(t,x)$ that will be used for computational convenience in various sections to
follow:
$(i,j)=(t,x-t/2).$
Figure 2: Duality between tilings and non-intersecting paths
Following the notation in [BG09], let $\Omega(N,S,T)$ denote the set of $N$
non-intersecting paths in the lattice $\mathbb{N}^{2}$ starting from positions
$(0,0),...,(0,N-1)$ and ending at positions $(T,S),...,(T,S+N-1)$. Each path
has segments of slope 0 or 1 (paths go either horizontally or diagonally
upwards from left to right). Set
$\displaystyle\mathfrak{X}^{S,t}_{N,T}=\\{x\in\mathbb{Z}:\max(0,t+S-T)\leq
x\leq\min(t+N-1,S+N-1)\\}$
$\displaystyle\mathpzc{X}^{S,t}_{N,T}=\\{X=(x_{1},...,x_{N})\in(\mathfrak{X}^{S,t}_{N,T})^{N}:x_{1}<x_{2}<...<x_{N}\\}.$
$\mathfrak{X}^{S,t}_{N,T}$ is the set of all possible particle positions in a
section vertical section of our hexagon with horizontal coordinate $t$ (in
$(t,x)$ coordinates). $\mathpzc{X}^{S,t}_{N,T}$ is the set of all possible
$N$-tuples of particles in the same vertical section.
For $X\in\Omega(N,S,T)$, we have $X=(X(t))_{0\leq t\leq T}$ and each
$X(t)\in\mathpzc{X}^{S,t}_{N,T}$. $X$ is a discrete time Markov chain as it
will be shown.
### 2.2 Probabilistic model
We will now define the probability measure on $\Omega(N,S,T)$ that will be the
object of study. For a tiling $\mathcal{T}$ corresponding to an
$X\in\Omega(N,S,T)$ we define its weight to be:
$\displaystyle w(\mathcal{T})=\prod_{l\in\\{\text{horizontal \
lozenges}\\}}w(l)$
where by a horizontal lozenge we mean a lozenge whose diagonals are parallel
to the $i$ and $j$ axes. The probability of such a tiling would then simply
be:
$\displaystyle
Prob(\mathcal{T})=\frac{w(\mathcal{T})}{\sum_{\mathcal{S}\in\Omega(N,S,T)}w(\mathcal{S})}.$
The weight function $w$ on horizontal lozenges $l$ is defined by
$\begin{split}w(l)&=\frac{(u_{1}u_{2})^{1/2}q^{j-1/2}\theta_{p}(q^{2j-1}u_{1}u_{2})}{\theta_{p}(q^{j-3i/2-1}u_{1},q^{j-3i/2}u_{1},q^{j+3i/2-1}u_{2},q^{j+3i/2}u_{2})}\\\
&=\frac{(v_{1}v_{2})^{1/2}q^{j-S/2-1/2}\theta_{p}(q^{2j-S-1}v_{1}v_{2})}{\theta_{p}(q^{j-3i/2-S-1}v_{1},q^{j-3i/2-S}v_{1},q^{j+3i/2-1}v_{2},q^{j+3i/2}v_{2})}\end{split}$
(6)
where $(i,j)$ is the coordinate of the top vertex of the horizontal lozenge
$l$, $u_{1},u_{2},q,p$ are complex parameters ($|p|<1$) and
$u_{1}=q^{-S}v_{1},u_{2}=v_{2}$ (the reason for this break in symmetry is that
it will make other formulas throughout the paper more symmetric).
###### Remark 2.1.
Only considering weights of horizontal lozenges for a tiling of a hexagon is
equivalent to considering all types of lozenges but assigning the other two
types weight 1 (i.e., each lozenge that is not horizontal has weight 1). This
is a break in symmetry that can easily be fixed. However, for the remainder of
the paper we prefer this non-symmetric weight assignment system as it makes
computations easier. Nevertheless, we show in Appendix 8 that we can assign
weights to the 3 types of lozenges in a $S_{3}$-invariant way (i.e., invariant
under permuting the 3 types of lozenges or equivalently the 3 spatial
directions).
This weight on dimer coverings of a hexagon was derived in [BGR10] (see also
[Sch07] for elliptic enumeration of lattice paths).
The connection with elliptic functions will now be explained. Fix a horizontal
coordinate $i$, denote by $w(i,j)$ the weight of the horizontal lozenge with
top vertex coordinates $(i,j)$, and observe that for two consecutive vertical
positions we have ($u_{1}u_{2}u_{3}=1$):
$\begin{split}r(i,j)&=\frac{w(i,j)}{w(i,j-1)}=\frac{q^{3}\theta_{p}(q^{j-3i/2-1}u_{1},q^{j+3i/2-1}u_{2},q^{-2j-1}u_{3})}{\theta_{p}(q^{j-3i/2+1}u_{1},q^{j+3i/2+1}u_{2},q^{-2j+1}u_{3})}\\\
&=\frac{q^{3}\theta_{p}(q^{j-3i/2-S-1}v_{1},q^{j+3i/2-1}v_{2},q^{-2j+S-1}/v_{1}v_{2})}{\theta_{p}(q^{j-3i/2-S+1}v_{1},q^{j+3i/2+1}v_{2},q^{-2j+S+1}/v_{1}v_{2})}\end{split}$
(7)
Figure 3: Going from 3 dimensions to 2 dimensions
Figure 4: A full $1\times 1\times 1$ box (left) and an empty one (right)
In 3-dimensional coordinates $(x,y,z)$ pictured in Figure 3 (note we only
consider surfaces in 3 dimensions that are stepped, meaning there is a 1-1
correspondence between the 2D tiling picture and the 3D surface picture) with
$i=x-y,j=z-(x+y)/2$, the weight ratio looks like
$\displaystyle r(x,y,z)=\frac{w(\text{full \ box})}{w(\text{empty \
box})}=\frac{q^{3}\theta_{p}(\tilde{u}_{1}/q,\tilde{u}_{2}/q,\tilde{u}_{3}/q)}{\theta_{p}(\tilde{u}_{1}q,\tilde{u}_{2}q,\tilde{u}_{3}q)}$
(8)
where
$\displaystyle\tilde{u}_{1}=q^{y+z-2x}u_{1},\
\tilde{u}_{2}=q^{x+z-2y}u_{2},\tilde{u}_{3}=q^{x+y-2z}u_{3},\
u_{1}u_{2}u_{3}=1$
and $(x,y,z)$ is the 3-dimensional centroid of the $1\times 1\times 1$ full
cube (on the left in Figure 4) with top lid the horizontal lozenge with top
vertex coordinate $(i,j)$.
The word elliptic now becomes clear as $r$ in (8) is an elliptic function of
$q$ (that is, defined on $\mathbb{E}$ \- see the Introduction for details).
Moreover, $r$ is the unique elliptic function of $q$ with zeros at
$\tilde{u}_{1},\tilde{u}_{2},\tilde{u}_{3}$ and poles at
$1/\tilde{u}_{1},1/\tilde{u}_{2},1/\tilde{u}_{3}$ normalized such that
$r(1)=1$. Of interest is also that $r$ is elliptic in $\tilde{u}_{k}$ for
$k=1,2,3$ subject to the condition that $\prod_{k=1}^{3}\tilde{u}_{k}=1$.
###### Remark 2.2.
$r$ is invariant under the natural action of $S_{3}$ permuting the
$\tilde{u}_{k}$’s (and of course the 3 axes: $x,y,z$).
We can view our tilings as stepped surfaces composed of $1\times 1\times 1$
cubes bounded by the 6 planes $x=0,y=0,z=0,x=b,y=c,z=a$. Then the 2
dimensional picture in Figure 1 can be viewed as a projection of the 3
dimensional stepped surface onto the plane $x+y+z=0$.
For $\mathcal{T}$ a tiling, we have
$\displaystyle
wt(\mathcal{T})=\prod_{\includegraphics[scale={0.10}]{hor_lozenge}\ \in\
\mathcal{T}}w(i,j)$
where $(i,j)$ are the coordinates of the top vertex of a horizontal lozenge.
Grouping all $1\times 1\times 1$ cubes into columns in the $z$ direction with
fixed $(x,y)$ coordinates (see Figure 3), we obtain:
$\displaystyle
wt(\mathcal{T})=const\cdot\prod_{\includegraphics[scale={0.02}]{full_box}}\frac{w(i,j)}{w(i,j-1)}$
where the product is taken over all cubes (visible and hidden) of the boxed
plane partition and $(i,j)$ is the top coordinate of the bounding hexagon of a
$1\times 1\times 1$ cube. Note to get to this equality we have merely observed
that $wt(\text{empty \ box})$ is a constant independent of $i$ and $j$. We can
further refine this (rearranging the terms in the product and gauging away
more constants - see Section 2.3 of [BGR10] for more details) as:
$\displaystyle
wt(\mathcal{T})=const\cdot\prod_{v}\left(\frac{w(i,j)}{w(i,j-1)}\right)^{h(v)}=const\cdot\prod_{v}r(i,j)^{h(v)}$
where $v=(x_{0},y_{0},z_{0})$ ranges over all vertices on the border (but not
on the bounding hexagon) of the stepped surface with $x_{0},y_{0},z_{0}$
integers (equivalently, $v$ ranges over all vertices of the triangular lattice
inside the hexagon, but we view $v$ in 3 dimensions). $h(v)$ is the distance
from $v$ to the plane $x+y+z=0$ divided by $\sqrt{3}$ :
$h(v)=(x_{0}+y_{0}+z_{0})/3$.
### 2.3 Positivity of the weight
The content of the previous subsection shows that in order to make the whole
model well defined as a probabilistic model, it suffices to establish
positivity of the elliptic weight ratio $r(i,j)=w(i,j)/w(i,j-1)$ defined in
(7) (where $(i,j)$ is the location of a given horizontal tiling and ranges
over all possible horizontal tilings inside the hexagon). Recall that
$r(i,j)=\frac{q^{3}\theta_{p}(\tilde{u}_{1}/q,\tilde{u}_{2}/q,\tilde{u}_{3}/q)}{\theta_{p}(q\tilde{u}_{1},q\tilde{u}_{2},q\tilde{u}_{3})}$
where
$\tilde{u}_{1}=q^{j-3i/2}u_{1},\tilde{u}_{2}=q^{j+3i/2}u_{2},\tilde{u}_{3}=q^{-2j}u_{3}$
and $u_{1}u_{2}u_{3}=1$. We recall that $r$ is elliptic in $\tilde{u}_{k}$ for
$k=1,2,3$ as well as in $q$. In order to make $r$ positive, we will first
restrict ourselves to the case where $r$ is real valued. This means $r$ is
defined over a real elliptic curve, and we have $-1<p\neq 0<1$ (a priori, $p$
is complex of modulus less than 1; $p\in(-1,1)-\\{0\\}$ is equivalent to
$\mathbb{E}$ being defined over $\mathbb{R}$ \- for more on real elliptic
curves, we refer the reader to Chapter 5 of [Sil94]). We then ensure
positivity of $r$ by an explicit computation. We will of course have two
cases: $p<0$ and $p>0$. We deal with the case $p>0$ throughout (and make
remarks when necessary for $p<0$).
Now that we have restricted ourselves to real elliptic curves $\mathbb{E}$, we
first note that $q\in\mathbb{E}$ (i.e., $r$ is elliptic as a function of $q$).
For a chosen $0<p<1$ there are two non-isomorphic elliptic curves defined over
$\mathbb{R}$ (since $Gal(\mathbb{C}/\mathbb{R})=\mathbb{Z}/2\mathbb{Z}$), both
homeomorphic to a disjoint union of two circles (every real elliptic curve is
topologically homeomorphic to a circle if $p<0$ or with a disjoint union of
two circles if $p>0$ \- one can just see this by plotting the Weierstrass
equation in $\mathbb{R}^{2}$ and compactifying):
$\displaystyle\mathbb{E}\cong_{\mathbb{R}}\mathbb{R}^{*}/p^{\mathbb{Z}}\
\mathrm{and}$
$\displaystyle\mathbb{E}\cong_{\mathbb{R}}\\{u\in\mathbb{C}^{*}/p^{\mathbb{Z}}:|u|^{2}\in\\{1,p\\}\\}$
We will call the first case real and the second trigonometric (abusing
terminology, since both are real elliptic curves). We will analyze the
trigonometric case, but the real case is similar. In the trigonometric case,
the curve has two connected components (circles): the identity component (it
contains the points 1 and -1) and another component that contains the other
2-torsion points: $\pm\sqrt{p}$. There will be 3 cases to be analyzed which we
list now and motivate after (if $p<0$ there is only one component so the 3
cases coalesce to only one - Case 2.):
* •
Case 1. $q$ lies on the non identity component ($|q|=\sqrt{p}$).
* •
Case 2. $q$ and all the $u_{k}$’s (and so the $\tilde{u}_{k}$’s) lie on the
identity component ($|q|=|u_{1}|=|u_{2}|=|u_{3}|=1$)
* •
Case 3. $q$ and one of the $u_{k}$’s lies on the identity component, the other
two $u_{k}$’s lie on the non identity component
To analyze positivity at a fixed site $(i,j)$ inside the hexagon, we note that
$r(q)$ has zeros at $\tilde{u}_{k}$ and poles at $1/\tilde{u}_{k}$
($k=1,2,3$). We note $r=\pm 1$ at $q=\pm 1$ so at least one $u_{k}$ (along
with its reciprocal/complex conjugate $1/u_{k}$) needs to be on the identity
component (so that $r$ can change signs on the identity component). Since
$r=-1$ at $q=\pm\sqrt{p}$ and $u_{1}u_{2}u_{3}=1$, either exactly one or all
three of the $u$’s need to be on the identity component. This motivates the
three choices above.
Case 1. will never lead to positivity for all four admissible sites $(i,j)$
inside a $1\times 2\times 2$ hexagon (see Figure 5), so we can eliminate it
(if a $1\times 2\times 2$ hexagon is never positive, much larger ones which
are of interest to us will also never be as they contain the $1\times 2\times
2$ case). For a proof, we suppose that $u_{1}$ is on the identity component,
and $u_{2},u_{3}$ are (along with $q$) on the non identity component (the case
where all three $u$’s are on the identity component is handled similarly). The
$\tilde{u}$’s differ from the $u$’s by integer powers of $q$ given in the last
three columns of the following table (for the four admissible $(i,j)$ pairs in
the $1\times 2\times 2$ hexagon):
$j$ | $i$ | $j-3i/2$ | $j+3i/2$ | $-2j$
---|---|---|---|---
1/2 | 1 | -1 | 2 | -1
1 | 2 | -2 | 4 | -2
0 | 2 | -3 | 3 | 0
1/2 | 3 | -4 | 5 | -1
Notice mod 2 (and we only care about mod 2 as $q^{2}$ is on the identity
component), the four vectors (from the last 3 columns of the table) above are
$(1,0,1),(0,0,0),(1,1,0),(0,1,1)$. The corresponding $\tilde{u}_{k}$’s we get
by multiplying each $u_{k}$ by $q$ to the power coming from the vector
$(0,1,1)$ \-
$(\tilde{u}_{1},\tilde{u}_{2},\tilde{u}_{3})=(q^{-4}v_{1},q^{5}v_{2},q^{-1}v_{3})$
\- will all be on the identity component, which means the elliptic weight
ratio will be negative at the site $(i,j)=(1/2,3)$ as $q$ is on the non
identity component. This is a contradiction. The other cases are handled
similarly, leading to contradictions. This proves $q$ must be on the identity
component, so only cases 2. and 3. above can lead to positive hexagons.
Figure 5: The admissible sites $(i,j)$ inside a $1\times 2\times 2$ hexagon
Figure 6: The identity component of
$\mathbb{E}\cong_{\mathbb{R}}\\{u\in\mathbb{C}^{*}/p^{\mathbb{Z}}:|u|^{2}\in\\{1,p\\}\\}$.
For positivity of $r$ throughout the hexagon (i.e., for all admissible
$\tilde{u}_{k}$’s), $q$ must always be closer to 1 than any
$\tilde{u}_{k}^{\pm 1}$ as depicted.
I will next discuss the case where $q$ and all $u_{k}$ are on the identity
component (case 2. above; for case 3. the reasoning is similar). For a fixed
site $(i,j)$ inside the hexagon, the 3 $\tilde{u}_{k}$’s and their reciprocals
(complex conjugates) break down the unit circle into 6 arcs (see Figure 6) and
$q$ must be on one of the three arcs where $r$ is positive (as depicted in the
figure). If we want to ensure positivity of the ratio for all 4 admissible
sites $(i,j)$ within a given $1\times 2\times 2$ hexagon (Figure 5), we first
observe that for $|x|=1$:
$\theta_{p}(x)=(1-x)\prod_{i\geq 1}|1-p^{i}x|^{2}.$
So we reduce to positivity of the corresponding four functions
$\prod_{i=1,2,3}\frac{1-\tilde{u}_{i}/q}{1-\tilde{u}_{i}q}$. Through standard
trigonometric manipulations we thus want positivity of each of the following
functions:
$\displaystyle\frac{\sin\pi(\alpha_{1}-\alpha)}{\sin\pi(\alpha_{1}+\alpha)}\cdot\frac{\sin\pi(\alpha_{2}-\alpha)}{\sin\pi(\alpha_{2}+\alpha)}\cdot\frac{\sin\pi(\alpha_{3}-\alpha)}{\sin\pi(\alpha_{3}+\alpha)}$
$\displaystyle\frac{\sin\pi(\alpha_{1})}{\sin\pi(\alpha_{1}+2\alpha)}\cdot\frac{\sin\pi(\alpha_{2}-3\alpha)}{\sin\pi(\alpha_{2}-\alpha)}\cdot\frac{\sin\pi(\alpha_{3})}{\sin\pi(\alpha_{3}+2\alpha)}$
$\displaystyle\frac{\sin\pi(\alpha_{1}-3\alpha)}{\sin\pi(\alpha_{1}-\alpha)}\cdot\frac{\sin\pi(\alpha_{2})}{\sin\pi(\alpha_{2}+2\alpha)}\cdot\frac{\sin\pi(\alpha_{3}+\alpha)}{\sin\pi(\alpha_{3}+3\alpha)}$
$\displaystyle\frac{\sin\pi(\alpha_{1}+\alpha)}{\sin\pi(\alpha_{1}+3\alpha)}\cdot\frac{\sin\pi(\alpha_{2}-2\alpha)}{\sin\pi(\alpha_{2})}\cdot\frac{\sin\pi(\alpha_{3}-2\alpha)}{\sin\pi(\alpha_{3})}$
where $2\pi\alpha_{i}=\arg
u_{i},\alpha_{1}+\alpha_{2}+\alpha_{3}\in\\{0,1,2\\},2\pi\alpha=\arg q$ and
$(\alpha,\alpha_{1},\alpha_{2})$ are defined on the 3-dimensional unit torus
$\mathbb{R}^{3}/\mathbb{Z}^{3}$. If we restrict to the fundamental domain
$[0,1]^{3}$ and look at all the regions (polytopes) cut out by the planes
(linear functions) in the arguments of the sines above (divided by $\pi$), we
find (by solving the appropriate linear programs via Mathematica) that there
exists only one region of positivity for all 4 functions. We can characterize
the region best in terms of Figure 6. That is, as $(i,j)$ range over all 4
sites inside a $1\times 2\times 2$ hexagon, there should not be any
$\tilde{u}_{k}$ ($k=1,2,3$) or any $\tilde{u}_{k}^{-1}$ on the arc subtended
by 1 and $q$ (and that does not contain -1).
###### Remark 2.3.
In view of the above, for any choice of a reasonably large hexagon (say one
that contains a $1\times 2\times 2$ hexagon) and parameters
$u_{1},u_{2},u_{3}$ (satisfying the balancing condition), the set of $q$
giving rise to nonnegative weights is a symmetric closed arc containing 1.
### 2.4 Degenerations of the weight
Certain degenerations of the weight have been studied before (among the
relevant sources for our purposes are [BG09], [BGR10], [Joh05], [KO07],
[Gor08]) from many angles. For example, when $q=1$ the weight in (6) becomes a
constant independent of the position of the horizontal lozenges, and so we are
looking at uniformly distributed tilings of the appropriate hexagon. An exact
sampling algorithm to sample such a tiling was constructed in [BG09] and the
theory behind this (as well as behind other results for such tilings) is
closely connected to the theory of discrete Hahn orthogonal polynomials (see
[Joh05], [BG09], [Gor08]). The frozen boundary phenomenon (the shape of a
“typical boxed plane partition”) was first proven in [CLP98] and then via
alternate techniques (and generalized) in [CKP01] and [KO07].
A more general limit than the above is the following: in (6) we let
$v_{1}=v_{2}=\kappa\sqrt{p}$ and then let $p\to 0$. This is the $q$-Racah
limit (named after the discrete orthogonal polynomials that appear in the
analysis). This limit is the most general limit that can be analyzed by
orthogonal polynomials (as $q$-Racah polynomials sit at the top of the
$q$-Askey scheme - see [KS]). Up to gauge equivalence, we obtain the weight of
a horizontal lozenge with top corner $(i,j)$ as:
$\displaystyle w(i,j)=\kappa q^{j}-\frac{1}{\kappa q^{j}}.$ (9)
This weight was studied in [BGR10]. If we take $\kappa$ to 0 or $\infty$, we
see the $q$-Racah weight is an interpolation between two types of weights:
$w(i,j)=q^{j}\ \text{and}\ w(i,j)=q^{-j}.$
A direct alternative limit from the elliptic level is given by
$v_{1}=v_{2}=p^{1/3},p\to 0$ (and then replace $q^{2}$ by $q$ or $1/q$). These
two weights give rise to tilings weighted proportional to $q^{\text{Volume}}$
or $q^{\text{-Volume}}$ (where Volume = number of $1\times 1\times 1$ cubes in
the stepped surface representing a tiling). This is the $q$-Hahn weight (as
the analysis leads to $q$-Hahn orthogonal polynomials). The frozen boundary
phenomenon for this type of weight was first studied in [KO07], and then via
alternative methods in [BGR10].
Finally, the Racah weight is the limit $q\to 1$ in (9) (we denote
$k=\log_{q}(\kappa)$ and need $\kappa\to 1$ as $q\to 1$). The weight function
becomes
$w(i,j)=k+j.$
Notice in all these limits the weight of a horizontal lozenge is independent
of the horizontal coordinate of its top vertex.
Taking these limits corresponds to the hypergeometric hierarchy of special
functions involved in the analysis via the orthogonal polynomial (OP) or
biorthogonal elliptic functions (down arrows denote limits):
Elliptic hypergeometric (elliptic weights; elliptic biorthogonal ensembles)
$\Downarrow$
$q$-hypergeometric ($q$-weights; $q$-OP ensembles)
$\Downarrow$
Hypergeometric (uniform/Racah weight; Hahn/Racah OP ensemble)
As a side final note, the most general degeneration of the weight is the top
level trigonometric limit $p\to 0$, which gives rise to a 3 parameter family
of weights (the use of the word trigonometric here should not be confused with
its usage in Section 2.3). Being more general (more parameters) than the
$q$-Racah limit, its analysis requires $q$ rational biorthogonal functions
rather than orthogonal polynomials. We will not use this limit hereinafter, as
we can approximate the trigonometric level by choosing $p$ really small at the
elliptic level.
### 2.5 Canonical coordinates
It will be convenient for various computations to express the geometry of an
elliptic lozenge tiling in terms of coordinates on a certain product of
elliptic curves. First we will introduce 6 parameters $A,B,C,D,E,F$ depending
on $q,t,S,T,N,v_{1},v_{2}$ (note we have listed, other than $q$, 6 parameters,
of which 4 are discrete and dictate the geometry: $t,S,T,N$). $t$ here is a
(discrete) time parameter and ranges from $0$ to $T$. It will be explained
better in Section 3. It corresponds to the fact that we will be interested in
distributions of particles on a certain vertical line: that is, tilings of
hexagons that have prescribed positions of particles (or holes) on the
vertical line with horizontal coordinate $t$. The set of parameters is:
$\begin{split}&A=q^{t/2+S/2-T+1/2}\sqrt{v_{1}v_{2}}\\\
&B=q^{t/2+S/2+T+1/2}\sqrt{\frac{v_{2}}{v_{1}}}\\\
&C=q^{t/2-S/2-N+1/2}\frac{1}{\sqrt{v_{1}v_{2}}}\\\
&D=q^{-t/2+S/2-N+1/2}\frac{1}{\sqrt{v_{1}v_{2}}}\\\
&E=q^{-t/2-S/2+1/2}\sqrt{\frac{v_{1}}{v_{2}}}\\\
&F=q^{-t/2-S/2+1/2}\sqrt{v_{1}v_{2}}\\\ \end{split}$ (10)
Observe that
$\displaystyle q^{2N-2}ABCDEF=q.$ (11)
Recall that the weight function (to be more precise, the ratio of weights of a
full to an empty $1\times 1\times 1$ box - see (8)) depends on the geometry of
the hexagon via the three parameters
$\tilde{u}_{1},\tilde{u}_{2},\tilde{u}_{3}$ ($\prod\tilde{u}_{k}=1$) which in
the $(i,j)$ coordinates are:
$\displaystyle\tilde{u}_{1}=q^{j-3i/2-S}v_{1}$
$\displaystyle\tilde{u}_{2}=q^{j+3i/2}v_{2}$
$\displaystyle\tilde{u}_{3}=q^{-2j+S}/v_{1}v_{2}$
What we want is to change coordinates from $(i,j)$ (2 dimensional) or
$(x,y,z)$ (3 dimensional) to $(\tilde{u}_{1},\tilde{u}_{2},\tilde{u}_{3})$ via
the above formula. We call these new coordinates canonical. In practice each
line of interest in the geometry has an equation in the $(i,j)$ plane which
can then be translated in terms of the $\tilde{u}_{k}$’s by solving in (10)
for $t,S,T,N,v_{1},v_{2}$ in terms of $A,B,C,D,E,F$. We thus find the
following equations for the relevant edges of our hexagon:
$\begin{split}&\mathrm{Left\ vertical\ edge\ (corresp.\ equation:\
}i=0):\frac{\tilde{u}_{1}}{\tilde{u}_{2}}=q^{-S}v_{1}/v_{2}=\left(\frac{ABC}{DEF}\right)^{1/2}E^{3}q^{-3/2}\\\
&\mathrm{Right\ vertical\ edge\ (corresp.\ equation:\
}i=T):\frac{\tilde{u}_{1}}{\tilde{u}_{2}}=q^{-3T-S}v_{1}/v_{2}=\left(\frac{ABC}{DEF}\right)^{1/2}B^{-3}q^{3/2}\\\
&\mathrm{NW\ edge\ (corresp.\ equation:\
}j=i/2+N):\frac{\tilde{u}_{3}}{\tilde{u}_{1}}=q^{2S-3N}1/v_{1}^{2}v_{2}=\left(\frac{ABC}{DEF}\right)^{1/2}D^{3}q^{-3/2}\\\
&\mathrm{SE\ edge\ (corresp.\ equation:\
}j=i/2-(T-S)):\frac{\tilde{u}_{3}}{\tilde{u}_{1}}=q^{3T-S}1/v_{1}^{2}v_{2}=\left(\frac{ABC}{DEF}\right)^{1/2}A^{-3}q^{3/2}\\\
&\mathrm{NE\ edge\ (corresp.\ equation:\
}j=-i/2+S+N):\frac{\tilde{u}_{2}}{\tilde{u}_{3}}=q^{2S+3N}v_{1}v_{2}^{2}=\left(\frac{ABC}{DEF}\right)^{1/2}C^{-3}q^{3/2}\\\
&\mathrm{SW\ edge\ (corresp.\ equation:\
}j=-i/2):\frac{\tilde{u}_{2}}{\tilde{u}_{3}}=q^{-S}v_{1}v_{2}^{2}=\left(\frac{ABC}{DEF}\right)^{1/2}F^{3}q^{-3/2}\\\
&\mathrm{Vertical\ particle\ line\ (corresp.\ equation:\
}i=t):\frac{\tilde{u}_{1}}{\tilde{u}_{2}}=q^{-3t-S}v_{1}/v_{2}=\frac{DEF}{ABC}\end{split}$
(12)
###### Remark 2.4.
We can see from above that there exists a bijection between the six bounding
edges of our hexagon and the 6 parameters $A,B,C,D,E,F$. That is, to an edge
we assign the parameter that appears to the power $\pm 3$ above. The 6
parameters are not independent (they satisfy one balancing condition
$ABCDEF=q^{3-2N}$), but then neither are the 6 edges (they must satisfy the
condition that the hexagon they form is tillable by the three types of rhombi,
which in this case tautologically means the edges form the 6 visible frame-
edges of a rectangular parallelepiped). See Figure 7.
Figure 7: Correspondence between edges and the 6 parameters.
With (12) in mind we have a (local) map $\mathbb{R}^{2}\to\mathbb{E}^{2}$
(where $\mathbb{E}^{2}$ is isomorphic to the subvariety of $\mathbb{E}^{3}$
with coordinates $(\tilde{u}_{1},\tilde{u}_{2},\tilde{u}_{3})$ and relation
$\prod\tilde{u}_{i}=1$) which embeds our hexagon in $\mathbb{E}^{2}$:
$(i,j)\mapsto(\tilde{u}_{1},\tilde{u}_{2},\tilde{u}_{3}).$
Note that $\mathbb{E}^{2}$ is the square of a real elliptic curve if
parameters are chosen so that the weight ratio (of full to empty $1\times
1\times 1$ box) is real positive. Hence as $\mathbb{E}$ is homeomorphic to a
circle or a disjoint union of two circles, the above embeds our hexagon in a
2-dimensional real torus (base field = $\mathbb{R}$).
## 3 Distributions and transition probabilities
In this section we compute the $N$-point correlation function and transitional
probabilities for the model under study. We refer the reader to the Appendix
of [BGR10] for the relevant application of Kasteleyn’s theorem which makes
these computations easy and to Kasteleyn’s original paper for the theory
itself ([Kas67]).
Take a collection of $N$ non-intersecting lattice paths in $\Omega(N,S,T)$.
Fix a vertical line inside the hexagon with horizontal integer coordinate $t$
($0\leq t\leq T$). This vertical line will contain $N$ particles
$X=(x_{1}<...<x_{N})\in\mathpzc{X}^{S,t}_{N,T}$. Depending on the geometry of
our hexagon, there are four ways in which we can fix a vertical line with
horizontal coordinate $t$ inside a collection of $N$ non-intersecting paths in
$\Omega(N,S,T)$. They are described below (see also Figure 8 in which the four
cases are depicted - we only depict the outside bounding hexagon and the
middle vertical line which is the desired particle line):
$\begin{split}&\mathrm{Case\ 1.}\ t<S,\ t<T-S,\ 0\leq x_{k}\leq t+N-1\\\
&\mathrm{Case\ 2.}\ S\leq t\leq T-S,\ 0\leq x_{k}\leq S+N-1\\\ &\mathrm{Case\
3.}\ T-S\leq t<S,\ t+S-T\leq x_{k}\leq t+N-1\\\ &\mathrm{Case\ 4.}\ t\geq
T-S,\ t\geq S,\ t+S-T\leq x_{k}\leq S+N-1\end{split}$ (13)
Figure 8: The four ways of choosing a vertical particle line (dotted) inside a
hexagon. In all cases $N=5$ particles, $T=8,S\in\\{3,5\\}$. The middle
vertical line in any hexagon is the particle line.
We make the following notations:
$L_{t}(X)=$ sum of products of weights corresponding to holes (horizontal
lozenges) to the left of the vertical line with coordinate $t$. The sum is
taken over all possible ways of tiling the region to the left of this line.
Equivalently, it is taken over all families of paths starting at
$((0,0),...,(0,N-1))$ and ending at $((t,x_{1}),...,(t,x_{N}))$.
$R_{t}(X)=$ sum of products of weights corresponding to holes to the right of
the vertical line with coordinate $t$. The sum is taken over all possible ways
of tiling the region to the right of this line. Equivalently, it is taken over
all families of paths starting at $((t,x_{1}),...,(t,x_{N}))$ and ending at
$((T,S),...,(T,S+N-1))$.
$C_{t}(X)=$ product of weights corresponding to the holes on this vertical
line.
Let
$\displaystyle\varphi_{t,S}(x_{k},x_{l})=q^{-x_{k}}\theta_{p}(q^{x_{k}-x_{l}},q^{x_{k}+x_{l}+1-t-S}v_{1}v_{2}).$
(14)
###### Remark 3.1.
As observed in the introduction, $\varphi_{t,S}(x,y)=-\varphi_{t,S}(y,x)$ so
the product $\prod_{k<l}\varphi_{t,S}(x_{k},x_{l})$ is the “elliptic” analogue
of the Vandermonde product $\prod_{k<l}(x_{k}-x_{l})$ (to which it tends in
the limit $p\to 0,q\to 1$ as explained in the Introduction).
###### Proposition 3.2.
We have
$\displaystyle L_{t}(X=(x_{1},...,x_{N}))$
$\displaystyle=const\cdot\prod_{k<l}\varphi_{t,S}(x_{k},x_{l})\times$
$\displaystyle\prod_{1\leq k\leq
N}q^{Nx_{l}}\theta_{p}(q^{2x_{l}+1-t-S}v_{1}v_{2})\frac{\theta_{p}(q^{1-N-t},q^{1-t-S}v_{1},q^{t}v_{2},q^{1-t-S}v_{1}v_{2};q)_{x_{l}}}{\theta_{p}(q,q^{2-2t-S}v_{1},qv_{2},q^{1+N-S}v_{1}v_{2};q)_{x_{l}}}.$
###### Proof.
This follows from an elaborate calculation and Lemma 10.2 in Appendix A of
[BGR10] (which is in essence an application of Kasteleyn’s theorem and the
computation of the appropriate inverse Kasteleyn matrix).
First, as is in the case of the aforementioned lemma, we restrict ourselves to
the case $S<t<T-S$ (Case 2. in (13); Computations are similar for the other 3
cases). Note in such a case we have $N$ particles and $S$ holes on the line
with abscissa $t$. We then apply the particle-hole involution (as the weight
in Lemma 10.2 in Appendix A of [BGR10] is given in terms of the positions of
the holes = horizontal lozenges on the $t$-line). There are two types of
products appearing in the total weight in question: a univariate one over the
holes and a bivariate Vandermonde-like (again over the holes). For the first
product, we just reciprocate to turn it into a product over particles (as the
total product over holes and particles of the functions involved is a constant
dependent only on $t,S,T,N,q,p,v_{1},v_{2}$). For the Vandermonde-like
product, we note for a function $f$ satisfying
$f(y_{i},y_{j})=-f(y_{j},y_{i})$ we have (up to a possible sign not depending
on holes or particles):
$\displaystyle\prod_{1\leq i<j\leq S}f(y_{i},y_{j})=\prod_{1\leq i<j\leq
N}f(x_{i},x_{j})\times\prod_{0\leq u<v\leq S+N-1}f(u,v)\times$
$\displaystyle\prod_{1\leq i\leq N}\frac{1}{\prod_{0\leq
u<x_{i}}f(x_{i},u)\prod_{x_{i}<u\leq S+N-1}f(u,x_{i})}$ (15)
where $y$’s represent locations of holes (top vertices of horizontal lozenges)
and $x$’s locations of particles. We take $f=\varphi_{t,S}$ as defined in
(14). Finally, in Appendix A of [BGR10], the convention is that particles and
holes are counted from the top going down. This is opposite to the convention
in this paper, so we substitute $x_{l}\mapsto S+N-1-x_{l}$. After standard
manipulations with theta-Pochhammer symbols we arrive at the desired result. ∎
###### Proposition 3.3.
We have
$\displaystyle R_{t}(X=(x_{1},...,x_{N}))$
$\displaystyle=const\cdot\prod_{k<l}\varphi_{t,S}(x_{k},x_{l})\times$
$\displaystyle\prod_{1\leq k\leq
N}q^{Nx_{l}}\theta_{p}(q^{2x_{l}+1-t-S}v_{1}v_{2})\frac{\theta_{p}(q^{1-N-S},q^{-2t-S}v_{1},q^{1+T}v_{2},q^{1-T}v_{1}v_{2};q)_{x_{l}}}{\theta_{p}(q^{1-S-t+T},q^{1-t-S-T}v_{1},q^{2+t}v_{2},q^{1+N-t}v_{1}v_{2};q)_{x_{l}}}.$
###### Proof.
Similar to the previous proof except we use Lemma 10.3 in Appendix A of
[BGR10]. ∎
###### Proposition 3.4.
We have
$\displaystyle C_{t}(X=(x_{1},...,x_{N}))=const\cdot\prod_{1\leq k\leq
N}\frac{\theta_{p}(q^{x_{l}-2t-S}v_{1},q^{x_{l}-2t-S+1}v_{1},q^{x_{l}+t}v_{2},q^{x_{l}+t+1}v_{2})}{q^{x_{l}}\theta_{p}(q^{2x_{l}+1-t-S}v_{1}v_{2})}.$
###### Proof.
This weight is (up to a constant not depending on holes or particles) the
reciprocal of the total weight of the $S$ holes (horizontal lozenges) on the
$t$ line and the latter is readily computed from the definition (6). ∎
###### Theorem 3.5.
$\begin{split}Prob(X(t)=(x_{1},...,x_{N}))=const\cdot\prod_{k<l}(\varphi_{t,S}(x_{k},x_{l}))^{2}\times\prod_{1\leq
k\leq N}q^{(2N-1)x_{k}}\theta_{p}(q^{2x_{k}+1-t-S}v_{1}v_{2})\times\\\
\prod_{1\leq k\leq
N}\frac{\theta_{p}(q^{1-N-t},q^{1-N-S},q^{1-t-S}v_{1},q^{1+T}v_{2},q^{1-T}v_{1}v_{2},q^{1-t-S}v_{1}v_{2};q)_{x_{k}}}{\theta_{p}(q,q^{1-S-t+T},q^{1-t-T-S}v_{1},qv_{2},q^{1+N-S}v_{1}v_{2},q^{1+N-t}v_{1}v_{2};q)_{x_{k}}}\\\
=const\cdot\prod_{k<l}(\varphi_{t,S}(x_{k},x_{l}))^{2}\times\prod_{1\leq k\leq
N}q^{(2N-1)x_{k}}\theta_{p}(q^{2x_{k}}F^{2})\frac{\theta_{p}(AF,BF,CF,DF,EF,F^{2};q)_{x_{k}}}{\theta_{p}(q,q\frac{A}{F},q\frac{B}{F},q\frac{C}{F},q\frac{D}{F},q\frac{E}{F};q)_{x_{k}}}.\end{split}$
(16)
###### Proof.
$Prob(X(t)=(x_{1},...,x_{N}))\propto L_{t}(X)C_{t}(X)R_{t}(X).$
∎
###### Remark 3.6.
The above distribution is what was called in the Introduction the discrete
elliptic Selberg density. That is to say,
$Prob(X(t)=(x_{1},...,x_{N}))=\Delta_{\lambda}(q^{2N-2}F^{2}|q^{N},q^{N-1}AF,q^{N-1}(pB)F,q^{N-1}CF,q^{N-1}DF,q^{N-1}EF)$
(17)
where $\lambda\in m^{n}$ ($m=S+N-1,n=N$) and $\lambda_{i}+N-i=x_{N+1-i}$ (to
account for the fact that $x_{1}<x_{2}<...<x_{N}$ whereas partitions are
always listed in non-increasing order). The particle-hole involution invoked
in Proposition 3.2 then takes the following form. If $\lambda_{p}$ is the
partition associated to the particle positions (at time $t$) via the above
equation and $\lambda_{h}$ is the partition associated to the whole positions
at the same time (in the case above, there are $S$ holes), then
$\lambda_{h}=(\lambda_{p}^{c})^{\prime}$
where $\lambda^{c}$ denotes the complemented partition corresponding to
$\lambda\in m^{n}$ ($\lambda^{c}_{i}=m-\lambda_{n+1-i}$) and
$\lambda^{\prime}$ denotes the dual (transposed) partition
($\lambda_{i}^{\prime}=$ number of parts of $\lambda$ that are $\geq i$). The
fact that both probabilities (in terms of holes and in terms of particles) are
$\Delta$-symbols can be observed directly as shown in Proposition 3.2 or using
the following relations appearing in [Rai06]:
$\displaystyle\Delta_{\lambda^{\prime}}(a|...b_{i}...;1/q)=\Delta_{\lambda}(a/q^{2}|...b_{i}...;q)$
$\displaystyle\frac{\Delta_{\lambda^{c}}(a|...b_{i}...;q)}{\Delta_{m^{n}}(a|...b_{i}...;q)}=\Delta_{\lambda}(\frac{q^{2m-2}}{q^{2n}a}|...\frac{q^{n-1}b_{i}}{q^{m}a}...,q^{n},pq^{n},q^{-m},pq^{-m};q).$
We will for brevity denote the measure described in Theorem (3.5) by
$\rho_{S,t}$ (note it also depends on $N,T,v_{1},v_{2},p,q$, but it is the
dependence on $S$ and $t$ that will be of most interest to us). Observe we can
transform the factor
$\displaystyle
q^{x}q^{(2N-2)x}\frac{\theta_{p}(q^{1-t-S}v_{1},q^{1+T}v_{2})}{\theta_{p}(q^{1-t-S-T}v_{1},qv_{2})}$
appearing in the univariate product of the above probability into something
proportional to
$\displaystyle
q^{x}\frac{\theta_{p}(q^{N-t-S}v_{1},q^{N+T}v_{2})}{\theta_{p}(q^{2-N-t-
S-T}v_{1},q^{2-N}v_{2})}\cdot\frac{1}{\theta_{p}(q^{x+1-t-S}v_{1},q^{-x+t+S+T}/v_{1},q^{x+1+T}v_{2},q^{-x}/v_{2})_{N-1}}$
by using
$\theta_{p}(Aq^{N-1};q)_{x}=\frac{\theta_{p}(A;q)_{x}\theta_{p}(Aq^{x};q)_{N-1}}{\theta_{p}(A;q)_{N-1}}\text{\
and \
}\theta_{p}(Aq^{1-N};q)_{x}=\frac{q^{(1-N)x}\theta_{p}(A;q)_{x}\theta_{p}(q/A;q)_{N-1}}{\theta_{p}(q^{1-x}/A;q)_{N-1}}$
and absorbing into the initial constant anything independent of $x$ (of the
particle positions $x_{k}$). After using (10) our probability distribution
becomes
$\begin{split}&Prob(X(t)=(x_{1},...,x_{N}))=\\\
&const\cdot\prod_{k<l}(\varphi_{t,S}(x_{k},x_{l}))^{2}\times\prod_{1\leq k\leq
N}\frac{1}{\theta_{p}(B(Fq^{x_{k}})^{\pm 1},E(Fq^{x_{k}})^{\pm
1};q)_{N-1}}\times\prod_{1\leq k\leq N}w(x_{k}).\end{split}$ (18)
where
$\displaystyle w(x)$
$\displaystyle=\frac{q^{x}\theta_{p}(q^{2x+1-t-S}v_{1}v_{2})\theta_{p}(q^{1-N-t},q^{1-N-S},q^{N-t-S}v_{1},q^{N+T}v_{2},q^{1-T}v_{1}v_{2},q^{1-t-S}v_{1}v_{2};q)_{x}}{\theta_{p}(q^{1-t-S}v_{1}v_{2})\theta_{p}(q,q^{1-S-t+T},q^{2-N-t-
T-S}v_{1},q^{2-N}v_{2},q^{1+N-S}v_{1}v_{2},q^{1+N-t}v_{1}v_{2};q)_{x}}$
$\displaystyle=\frac{q^{x}\theta_{p}(F^{2}q^{2x})\theta_{p}(AF,BF\left(\frac{q}{ABCDEF}\right)^{\frac{1}{2}},CF,DF,EF\left(\frac{q}{ABCDEF}\right)^{\frac{1}{2}},F^{2};q)_{x}}{\theta_{p}(F^{2})\theta_{p}(\frac{F}{A}q,\frac{F}{B}q\left(\frac{ABCDEF}{q}\right)^{\frac{1}{2}},\frac{F}{C}q,\frac{F}{D}q,\frac{F}{E}q\left(\frac{ABCDEF}{q}\right)^{\frac{1}{2}},q;q)_{x}}$
$w$ is the weight function for the discrete elliptic univariate biorthogonal
functions discovered by Spiridonov and Zhedanov (see [SZ00], [SZ01]). It is of
course also the discrete elliptic Selberg density for $N=1$ (hence a
$\Delta$-symbol in $n=1$ variable as seen in (4)). Notice in (18) above $B$
and $E$ play a special role, as does $F$. This will become more transparent in
Section 6. $w$ is elliptic in $q,v_{1},v_{2}$ and $q^{\\{t,S,T,N\\}}$ (or,
analogously, in $A,B,C,D,E,F,q$).
###### Remark 3.7.
Note that in the definition of $w$ above, the first line is given in terms of
the geometry of the hexagon and the choice of the particular particle line
(Case 2. in (13) as previously discussed), while the second line is intrinsic
and the geometry of the hexagon only comes in after using (10). We can also
define the equivalent of (10) in the other 3 cases described in (13) (and the
three other choices of 6 parameters differ from (10) by (a): interchanging $S$
ant $t$, (b): shifting the 6 parameters in (10) by $q^{\pm(t+S-T)})$, or (c):
a combination of both (a) and (b)). We will not use this any further, as all
calculations will be done in Case 2. from (13).
###### Remark 3.8.
The limit $v_{1}=v_{2}=\kappa\sqrt{p},\ p\to 0$ gives the distributions
present in [BGR10] at the $q$-Racah level. Also, as will be seen in Section 6,
such probabilities are structurally a product of a “Vandermonde-like”
determinant squared (the first two products in (18)) and a product over the
particles of univariate weights of elliptic biorthogonal functions. Indeed,
under the appropriate limits, one can arrive from (18) to a much simpler
(prototypical) such $N$-point function: the joint density of the $N$
eigenvalues of a GUE $N\times N$ random matrix.
The transition and co-transition probabilities for the Markov chain $X(t)$ are
given by the next two statements.
###### Theorem 3.9.
If $Y=(y_{1},...,y_{N})$ and $X=(x_{1},...,x_{N})$ such that
$y_{k}-x_{k}\in\\{0,1\\}\ \forall k$, then
$\displaystyle
Prob(X(t+1)=Y|X(t)=X)=const\cdot\prod_{k<l}\frac{\varphi_{t+1,S}(y_{k},y_{l})}{\varphi_{t,S}(x_{k},x_{l})}\times\prod_{k:y_{k}=x_{k}+1}w_{1}(x_{k})\prod_{k:y_{k}=x_{k}}w_{0}(x_{k})$
where
$\displaystyle w_{0}(x)=\frac{q^{-x-N+1}\theta_{p}(q^{x+T-t-S},q^{x-T-
t-S}v_{1},q^{x+t+1}v_{2},q^{x+N-t}v_{1}v_{2})}{\theta_{p}(q^{2x+1-t-S}v_{1}v_{2})}$
$\displaystyle
w_{1}(x)=-\frac{q^{-x}\theta_{p}(q^{x+1-N-S},q^{x-2t-S}v_{1},q^{x+T+1}v_{2},q^{x-T+1}v_{1}v_{2})}{\theta_{p}(q^{2x+1-t-S}v_{1}v_{2})}.$
###### Proof.
The formula
$\displaystyle Prob(X(t+1)=Y|X(t)=X)$
$\displaystyle=\frac{L_{t}(X)C_{t}(X)C_{t+1}(Y)R_{t+1}(Y)}{L_{t}(X)C_{t}(X)R_{t}(X)}=$
$\displaystyle=\frac{C_{t+1}(Y)R_{t+1}(Y)}{R_{t}(X)}$
along with the formulas for $L,R$ and $C$ yield the result. ∎
###### Theorem 3.10.
If $Y=(y_{1},...,y_{N})$ and $X=(x_{1},...,x_{N})$ such that
$y_{k}-x_{k}\in\\{0,-1\\}\ \forall k$, then
$\displaystyle
Prob(X(t-1)=Y|X(t)=X)=const\cdot\prod_{k<l}\frac{\varphi_{t-1,S}(y_{k},y_{l})}{\varphi_{t,S}(x_{k},x_{l})}\times\prod_{k:y_{k}=x_{k}-1}w^{\prime}_{1}(x_{k})\prod_{k:y_{k}=x_{k}}w^{\prime}_{0}(x_{k})$
where
$\displaystyle
w^{\prime}_{0}(x)=-\frac{q^{-x}\theta_{p}(q^{x-N-t+1},q^{x-t-S+1}v_{1},q^{x+t}v_{2},q^{x-t-S+1}v_{1}v_{2})}{\theta_{p}(q^{2x+1-t-S}v_{1}v_{2})}$
$\displaystyle
w^{\prime}_{1}(x)=\frac{q^{-x-N+1}\theta_{p}(q^{x},q^{x-2t-S+1}v_{1},q^{x}v_{2},q^{x+N-S}v_{1}v_{2})}{\theta_{p}(q^{2x+1-t-S}v_{1}v_{2})}.$
###### Proof.
$\displaystyle Prob(X(t-1)=Y|X(t)=X)$
$\displaystyle=\frac{L_{t-1}(X)C_{t-1}(X)C_{t}(Y)R_{t}(Y)}{L_{t}(X)C_{t}(X)R_{t}(X)}=$
$\displaystyle=\frac{L_{t-1}(Y)C_{t-1}(Y)}{L_{t}(X)}.$
∎
We are now in a position to define six stochastic matrices (Markov chains)
needed in what will follow. Their stochasticity along with other properties
will be proven in Section 4, although we know the first two are stochastic as
they represent the transition probabilities obtained in this section. To
condense notation, we denote
$z_{k}=Fq^{x_{k}}.$
Let:
$\displaystyle
P^{S,t}_{t+}:\mathpzc{X}^{S,t}\times\mathpzc{X}^{S,t+1}\to[0,1]$
$\displaystyle
P^{S,t}_{t-}:\mathpzc{X}^{S,t}\times\mathpzc{X}^{S,t-1}\to[0,1]$
${}_{t+}P^{S,t}_{S+}:\mathpzc{X}^{S,t}\times\mathpzc{X}^{S+1,t}\to[0,1]$
${}_{t+}P^{S,t}_{S-}:\mathpzc{X}^{S,t}\times\mathpzc{X}^{S-1,t}\to[0,1]$
${}_{t-}P^{S,t}_{S+}:\mathpzc{X}^{S,t}\times\mathpzc{X}^{S+1,t}\to[0,1]$
${}_{t-}P^{S,t}_{S-}:\mathpzc{X}^{S,t}\times\mathpzc{X}^{S-1,t}\to[0,1]$
be defined by:
$P^{S,t}_{t+}(X,Y)=\left\\{\begin{array}[]{lll}const\cdot\prod_{k<l}\frac{\varphi_{t+1,S}(y_{k},y_{l})}{\varphi_{t,S}(x_{k},x_{l})}\cdot\prod_{k:y_{k}=x_{k}+1}-\frac{q^{-x_{k}}\theta_{p}(Az_{k},Bz_{k},Cz_{k},q^{1-N}z_{k}/ABC)}{\theta_{p}(z_{k}^{2})}\times\\\
\prod_{k:y_{k}=x_{k}}\frac{q^{-x_{k}-N+1}\theta_{p}(z_{k}/A,z_{k}/B,z_{k}/C,q^{N-1}z_{k}ABC)}{\theta_{p}(z_{k}^{2})}\text{\
if \ }y_{k}-x_{k}\in\\{0,1\\}\ \forall k\\\ 0,\text{ \
otherwise}\end{array}\right.$ (19)
$P^{S,t}_{t-}(X,Y)=\left\\{\begin{array}[]{lll}const\cdot\prod_{k<l}\frac{\varphi_{t-1,S}(y_{k},y_{l})}{\varphi_{t,S}(x_{k},x_{l})}\cdot\prod_{k:y_{k}=x_{k}-1}\frac{q^{-x_{k}-N+1}\theta_{p}(z_{k}/D,z_{k}/E,z_{k}/F,q^{N-1}z_{k}DEF)}{\theta_{p}(z_{k}^{2})}\times\\\
\prod_{k:y_{k}=x_{k}}-\frac{q^{-x_{k}}\theta_{p}(Dz_{k},Ez_{k},Fz_{k},q^{1-N}z_{k}/DEF)}{\theta_{p}(z_{k}^{2})}\text{\
if \ }y_{k}-x_{k}\in\\{0,-1\\}\ \forall k\\\ 0,\text{ \
otherwise}\end{array}\right.$ (20)
$_{t+}P^{S,t}_{S+}(X,Y)=\left\\{\begin{array}[]{lll}const\cdot\prod_{k<l}\frac{\varphi_{t,S+1}(y_{k},y_{l})}{\varphi_{t,S}(x_{k},x_{l})}\cdot\prod_{k:y_{k}=x_{k}+1}-\frac{q^{-x_{k}}\theta_{p}(Az_{k},Bz_{k},Dz_{k},q^{1-N}z_{k}/ABD)}{\theta_{p}(z_{k}^{2})}\times\\\
\prod_{k:y_{k}=x_{k}}\frac{q^{-x_{k}-N+1}\theta_{p}(z_{k}/A,z_{k}/B,z_{k}/D,q^{N-1}z_{k}ABD)}{\theta_{p}(z_{k}^{2})}\text{\
if \ }y_{k}-x_{k}\in\\{0,1\\}\ \forall k\\\ 0,\text{ \
otherwise}\end{array}\right.$ (21)
$_{t+}P^{S,t}_{S-}(X,Y)=\left\\{\begin{array}[]{lll}const\cdot\prod_{k<l}\frac{\varphi_{t,S-1}(y_{k},y_{l})}{\varphi_{t,S}(x_{k},x_{l})}\cdot\prod_{k:y_{k}=x_{k}+1}-\frac{q^{-x_{k}}\theta_{p}(Bz_{k},Cz_{k},Fz_{k},q^{1-N}z_{k}/BCF)}{\theta_{p}(z_{k}^{2})}\times\\\
\prod_{k:y_{k}=x_{k}}\frac{q^{-x_{k}-N+1}\theta_{p}(z_{k}/B,z_{k}/C,z_{k}/F,q^{N-1}z_{k}BCF)}{\theta_{p}(z_{k}^{2})}\text{\
if \ }y_{k}-x_{k}\in\\{0,-1\\}\ \forall k\\\ 0,\text{ \
otherwise}\end{array}\right.$ (22)
$_{t-}P^{S,t}_{S+}(X,Y)=\left\\{\begin{array}[]{lll}const\cdot\prod_{k<l}\frac{\varphi_{t,S+1}(y_{k},y_{l})}{\varphi_{t,S}(x_{k},x_{l})}\cdot\prod_{k:y_{k}=x_{k}-1}\frac{q^{-x_{k}-N+1}\theta_{p}(z_{k}/D,z_{k}/E,z_{k}/A,q^{N-1}z_{k}DEA)}{\theta_{p}(z_{k}^{2})}\times\\\
\prod_{k:y_{k}=x_{k}}-\frac{q^{-x_{k}}\theta_{p}(Dz_{k},Ez_{k},Az_{k},q^{1-N}z_{k}/DEA)}{\theta_{p}(z_{k}^{2})}\text{\
if \ }y_{k}-x_{k}\in\\{0,1\\}\ \forall k\\\ 0,\text{ \
otherwise}\end{array}\right.$ (23)
$_{t-}P^{S,t}_{S-}(X,Y)=\left\\{\begin{array}[]{lll}const\cdot\prod_{k<l}\frac{\varphi_{t,S-1}(y_{k},y_{l})}{\varphi_{t,S}(x_{k},x_{l})}\cdot\prod_{k:y_{k}=x_{k}-1}\frac{q^{-x_{k}-N+1}\theta_{p}(z_{k}/E,z_{k}/F,z_{k}/C,q^{N-1}z_{k}EFC)}{\theta_{p}(z_{k}^{2})}\times\\\
\prod_{k:y_{k}=x_{k}}-\frac{q^{-x_{k}}\theta_{p}(Ez_{k},Fz_{k},Cz_{k},q^{1-N}z_{k}/EFC)}{\theta_{p}(z_{k}^{2})}\text{\
if \ }y_{k}-x_{k}\in\\{0,-1\\}\ \forall k\\\ 0,\text{ \
otherwise}\end{array}\right.$ (24)
The normalizing constants are independent of the $x_{k}$’s and the $y_{k}$’s.
They will become explicit in Section 4.
Note that ${}_{t-}P^{S,t}_{S-}$, under interchanging $t$ and $S$, becomes
$P^{S,t}_{t-}$. Under the same procedure ${}_{t+}P^{S,t}_{S+}$ becomes
$P^{S,t}_{t+}$. We can think of $P^{S,t}_{t+}$ ($P^{S,t}_{t-}$) as a Markov
chain that increases (decreases) $t$, while ${}_{t\pm}P^{S,t}_{S+}$
(${}_{t\pm}P^{S,t}_{S-}$) increases (decreases) $S$.
###### Remark 3.11.
In the $q$-Racah limit $v_{1}=v_{2}=\kappa\sqrt{p},\ p\to 0$, the chains
${}_{t\pm}P^{S,t}_{S+}$ coalesce into one ($P^{S,t}_{S+}$ in [BGR10]).
Likewise for ${}_{t\pm}P^{S,t}_{S-}$.
## 4 Elliptic difference operators
In this section we explain how recent results on elliptic special functions
and elliptic difference operators intrinsically capture the model we described
thus far. The main two references are [Rai10] and [Rai06] and we will state
results from these without going into the proofs (with a few exceptions where
the proofs are short and revealing of common techniques employed in the area).
The focus will be on certain elliptic difference operators satisfying
normalization, quasi-commutation and quasi-adjointness relations. We define
them abstractly in the first subsection. We then turn to motivating the
definitions and interpreting the operators probabilistically.
### 4.1 Definitions and some properties
In [Rai10] (see also [Rai06] for an algebraic description) Rains has
introduced a family of difference operators acting nicely on various classes
of $BC_{n}$-symmetric functions. To define it, we let
$r_{0},r_{1},r_{2},r_{3}\in\mathbb{C}^{*}$ satisfy
$r_{0}r_{1}r_{2}r_{3}=pq^{1-n}$. Then define
$\mathpzc{D}(r_{0},r_{1},r_{2},r_{3})$ (also depending on $q,p,n$) by:
$\displaystyle(\mathpzc{D}(r_{0},r_{1},r_{2},r_{3})f)(...z_{k}...)=\sum_{\sigma\in\\{\pm
1\\}^{n}}\prod_{1\leq k\leq n}\frac{\prod_{0\leq s\leq
3}\theta_{p}(r_{s}z_{k}^{\sigma_{k}})}{\theta_{p}(z_{k}^{2\sigma_{k}})}\prod_{1\leq
k<l\leq
n}\frac{\theta_{p}(qz_{k}^{\sigma_{k}}z_{l}^{\sigma_{l}})}{\theta_{p}(z_{k}^{\sigma_{k}}z_{l}^{\sigma_{l}})}f(...q^{\sigma_{k}/2}z_{k}...).$
(25)
###### Remark 4.1.
The difference operator above described is the special case $t=q$ of the more
general elliptic $(q,t)$ difference operator mentioned in the references.
In view of $r_{0}r_{1}r_{2}r_{3}=pq^{1-n}$ we will break symmetry and denote
the difference operator by $\mathpzc{D}(r_{0},r_{1},r_{2})$, the fourth
parameter being implicit from the balancing condition.
###### Remark 4.2.
$\mathpzc{D}$ takes $BC_{n}$-symmetric functions to $BC_{n}$-symmetric
functions.
By letting $\mathpzc{D}$ act on the function $f\equiv 1$, we obtain the
following important lemma, whose proof we sketch following [Rai10]:
###### Lemma 4.3.
For $r_{0}r_{1}r_{2}r_{3}=pq^{1-n}$ we have
$\displaystyle\sum_{\sigma\in\\{\pm 1\\}^{n}}\prod_{1\leq k\leq
n}\frac{\prod_{0\leq s\leq
3}\theta_{p}(r_{s}z_{k}^{\sigma_{k}})}{\theta_{p}(z_{k}^{2\sigma_{k}})}\prod_{1\leq
k<l\leq
n}\frac{\theta_{p}(qz_{k}^{\sigma_{k}}z_{l}^{\sigma_{l}})}{\theta_{p}(z_{k}^{\sigma_{k}}z_{l}^{\sigma_{l}})}=\prod_{0\leq
k<n}\theta_{p}(q^{k}r_{0}r_{1},q^{k}r_{0}r_{2},q^{k}r_{1}r_{2})$
###### Proof.
By direct computation the LHS above is invariant under $z_{k}\to pz_{k}$ for
all $k$ (this is insured by the fact $r_{0}r_{1}r_{2}r_{3}=pq^{1-n}$). It is
also $BC_{n}$-symmetric (invariant under permutations of $z_{k}$’s and under
$z_{k}\to 1/z_{k}$). Finally, by multiplying LHS by
$R=\prod_{k}z_{k}^{-1}\theta_{p}(z_{k}^{2})\prod_{k<l}\varphi(z_{k},z_{l})$ we
will have cleared potential poles of the LHS. Because $R$ is
$BC_{n}$-antisymmetric the result will end up being a multiple of $R$:
$R\cdot\mathrm{LHS}=const\cdot R$ showing LHS has no singularities in the
variables and is thus independent of the $z_{i}$’s. Evaluating then at
$z_{i}=r_{0}q^{n-i}$ yields the result. Observe the main point here was to
prove the LHS is elliptic and has no poles in the variables, and indeed any
analysis that shows this will prove the result. ∎
Hereinafter we will use $\mathpzc{D}$ for the “normalized” difference operator
(so that $\mathpzc{D}(r_{0},r_{1},r_{2})1=1$) following Lemma 4.3.
The difference operators described above satisfy a number of identities,
including a series of commutation relations. For an elegant proof which relies
on the action of these operators on a suitably large space of functions (more
precisely, on the action of the difference operators on $BC_{n}$-symmetric
interpolation abelian functions), see [Rai10] or [Rai06].
###### Lemma 4.4.
If $U,V,W,Z$ are 4 parameters, then
$\displaystyle\mathpzc{D}(U,V,W)\mathpzc{D}(q^{1/2}U,q^{1/2}V,q^{-1/2}Z)=\mathpzc{D}(U,V,Z)\mathpzc{D}(q^{1/2}U,q^{1/2}V,q^{-1/2}W)$
Next we look at the action of the difference operators on special classes of
functions. For $\lambda\in m^{n}$ a partition, let
$\displaystyle\mathpzc{d}_{\lambda}(...x_{k}...)=\prod_{1\leq k\leq
n}\frac{\prod_{1\leq l\leq m+n}\theta_{p}(uq^{l-1}x_{k}^{\pm 1})}{\prod_{1\leq
l\leq n}\theta_{p}(uq^{\lambda_{l}+n-l}x_{k}^{\pm 1})}$
By direct computation, we see that
$\mathpzc{d}_{\lambda}(...uq^{\mu_{k}+n-k}...)=\delta_{\lambda,\mu}c_{\lambda}$.
###### Remark 4.5.
$\mathpzc{d}_{\lambda}$ is a special version of the interpolation theta
functions $P_{\lambda}^{*(m,n)}(...x_{k}...;a,b;q;p)$ defined in [Rai06]
(matching the notation in the reference with ours, $a=u,b=q^{-m-n+1}/a)$).
They are defined (up to normalization) by two properties: being
$BC_{n}$-symmetric of degree $m$ (which happens for $\mathpzc{d}_{\lambda}$’s)
and vanishing at $\mu\neq\lambda$ (which trivially happens in our case).
If we now define
$\mathfrak{d}_{\lambda}=\frac{\mathpzc{d}_{\lambda}}{c_{\lambda}}$ we see that
$\displaystyle\mathfrak{d}_{\lambda}(...uq^{\mu_{k}+n-k}...)=\delta_{\lambda,\mu}$
(26)
so in a precise way, $\mathfrak{d}_{\lambda}$ is an interpolation Kronecker-
delta theta-function. We then immediately have the following proposition:
###### Proposition 4.6.
Fix $\tau\in\\{\pm 1\\}^{n}$. Let $z_{k}=uq^{\lambda_{k}+n-k}$. Then
$\displaystyle(\mathpzc{D}(r_{0},r_{1},r_{2})\mathfrak{d}_{\lambda})(...q^{-\tau_{k}/2}z_{k}...)=\prod_{k}\frac{\theta_{p}(r_{0}z_{k}^{\tau_{k}},r_{1}z_{k}^{\tau_{k}},r_{2}z_{k}^{\tau_{k}},(pq^{1-n}/r_{0}r_{1}r_{2})z_{k}^{\tau_{k}})}{\theta_{p}(z_{k}^{2\tau_{k}})}\prod_{k<l}\frac{\theta_{p}(qz_{k}^{\tau_{k}}z_{l}^{\tau_{l}})}{\theta_{p}(z_{k}^{\tau_{k}}z_{l}^{\tau_{l}})}.$
###### Proof.
Immediate by substituting into the definition of the difference operator (25).
For any $\sigma\neq\tau$, $q^{\sigma_{k}/2-\tau_{k}/2}z_{k}$ will be of the
form $uq^{\mu_{k}+n-k}$ with $\mu\neq\lambda$ and the corresponding summand
will be 0. ∎
A useful final property of the difference operators is their quasi-
adjointness. It was shown in [Rai10] that the $\mathpzc{D}$’s satisfy a
certain “adjointness” relation that we will need in the next section. We start
with 6 parameters $t_{0},t_{1},t_{2},t_{3},u_{0},u_{1}$ satisfying the
balancing condition
$q^{2n-2}t_{0}t_{1}t_{2}t_{3}u_{0}u_{1}=pq.$
We fix the number of variables at $n$ and $\lambda$ will be a partition in
$m^{n}$. As in the introduction, we denote $l_{i}=\lambda_{i}+n-i$. We define
the discrete Selberg inner product $\langle,\rangle$ (depending on $p,q$ and
the 6 parameters) by
$\displaystyle\langle f,g\rangle=\frac{1}{Z}\sum_{\lambda\subseteq
m^{n}}f(...t_{0}q^{l_{i}}...)g(...t_{0}q^{l_{i}}...)\Delta_{\lambda}(q^{2n-2}t_{0}^{2}|q^{n},q^{n-1}t_{0}t_{1},q^{n-1}t_{0}t_{2},q^{n-1}t_{0}t_{3},q^{n-1}t_{0}u_{0},q^{n-1}t_{0}u_{1};q)$
(27)
where $f,g$ belong to some sufficiently nice set of functions (we will assume
they are $BC_{n}$-symmetric) and $Z$ is an explicit constant that makes
$\langle 1,1\rangle=1$. This is a discrete analogue of the continuous inner
product introduced in [Rai10] and can be obtained from that by residue
calculus.
If the above conditions are satisfied, then ([Rai10]):
$\displaystyle\langle\mathpzc{D}(u_{0},t_{0},t_{1})f,g\rangle=\langle
f,\mathpzc{D}(u_{1}^{\prime},t_{2}^{\prime},t_{3}^{\prime})g\rangle^{\prime}$
(28)
where
$(t_{0}^{\prime},t_{1}^{\prime},t_{2}^{\prime},t_{3}^{\prime},u_{0}^{\prime},u_{1}^{\prime})=(q^{1/2}t_{0},q^{1/2}t_{1},q^{-1/2}t_{2},q^{-1/2}t_{3},q^{1/2}u_{0},q^{-1/2}u_{1})$
and $\langle,\rangle^{\prime}$ is the inner product defined in (27) with
primed parameters inserted throughout.
### 4.2 Interpretation of difference operators and their properties
We now show how the difference operators and their properties discussed in the
previous section can be given probabilistic interpretations.
###### Remark 4.7.
Observe from (10) that $q^{2n-3}ABCDEF=1$.
In what follows $h_{k}$ ($h^{\prime}_{k}$) is the location of the $k$-th
particle on the vertical line $i=t$ ($i=t+1$) in the $(i,j)$ frame (note
according to the $t\to t+1$ dynamics the particles move either up or down by
$1/2$). We can prove the following proposition:
###### Proposition 4.8.
For $A,B,C,D,E,F$ and $z_{k}=Fq^{h_{k}}$ given by (10), the summands in
$(\mathpzc{D}(A,B,C)1)(...z_{k}...)$
(see (25)), appropriately normalized using (4.3), are equal to the transition
probabilities (entries in the stochastic matrix) $P^{S,t}_{t+}(H,H^{\prime})$
defined in (19) (after switching coordinates from $(x,y)$ back to $(i,j)$).
This statement also holds for:
$\displaystyle\mathpzc{D}(D,E,F)\text{\ and \ }P^{S,t}_{t-}$
$\displaystyle\mathpzc{D}(A,B,D)\text{\ and \ }_{t+}P^{S,t}_{S+}$
$\displaystyle\mathpzc{D}(B,C,F)\text{\ and \ }_{t+}P^{S,t}_{S-}$
$\displaystyle\mathpzc{D}(D,E,A)\text{\ and \ }_{t-}P^{S,t}_{S+}$
$\displaystyle\mathpzc{D}(E,F,C)\text{\ and \ }_{t-}P^{S,t}_{S-}$
###### Proof.
I will only prove the statement for $\mathpzc{D}(A,B,C)$ and $t+$ (the
equivalent statement for $\mathpzc{D}(D,E,F)$ and $t-$ is proved much the same
way). The proof is immediate in view of (10), the change of variables
$(X,Y)\mapsto(H,H^{\prime})$ in (19) (to the $(i,j)$ coordinates) and the
following observations.
First, a choice of $\sigma_{k}\in\\{\pm 1\\}$ for all $k$ in the definition of
$\mathpzc{D}(A,B,C)$ is equivalent to a choice of which particles move up/down
from the position vector $H$ (at vertical line $t$) to the position vector
$H^{\prime}$ (at vertical line $t+1$). If $\sigma_{k}=1$, the corresponding
$k$-th particle at vertical position $h_{k}$ moves up to
$h^{\prime}_{k}=h_{k}+1/2$ (and if $\sigma_{k}=-1$, the $k$-th particle moves
down). Next observe that in the univariate product appearing in any term of
$(\mathpzc{D}(A,B,C)1)(...z_{k}...)$, we can change $\theta_{p}(uz_{i}^{-b})$
($b=1,2$) to $\theta_{p}(z_{i}^{b}/u)$ by the reflection formula for theta
functions and it will now match with the univariate product appearing in
$P_{t+}^{S,t}$. The product
$\prod_{k:y_{k}=x_{k}+1}(...)\prod_{y_{k}=x_{k}}(...)$ now indeed is identical
(modulo constants independent of the particle positions) to
$\prod_{k:h^{\prime}_{k}=h_{k}+1/2}(...)\prod_{k:h^{\prime}_{k}=h_{k}-1/2}(...)$
which is nothing more than
$\prod_{k:\sigma_{k}=1}(...)\prod_{k:\sigma_{k}=-1}(...)$ in (25).
The elliptic Vandermonde product $\prod_{k<l}$ appearing in (19) is the same
product (modulo constants independent of the particles) as the Vandermonde-
like product in any term of $(\mathpzc{D}(A,B,C)1)(...z_{k}...)$ once we’ve
transformed (in the latter product) $\theta_{p}(z_{l}/z_{k})$ into
$\theta_{p}(z_{k}/z_{l})$ and $\theta_{p}(1/z_{k}z_{l})$ into
$\theta_{p}(z_{k}z_{l})$ (picking up appropriate multipliers in front that
will be powers of $q$ appearing the Vandermonde-like product in (19)). The
extra powers of $q$ appearing in (19) will also surface in the difference
operator once we’ve performed the aforementioned transformations. Finally
observe that the ratio
$\frac{\varphi_{t+1,S}(h_{k}^{\prime},h_{l}^{\prime})}{\varphi_{t,S}(h_{k},h_{l})}$
reduces (modulo the power of $q$ up front already accounted for) to a ratio of
only 2 theta functions (of the 4 initially present) because either
$h_{k}^{\prime}-h_{l}^{\prime}=h_{k}-h_{l}$ or
$h_{k}^{\prime}+h_{l}^{\prime}=h_{k}+h_{l}$ (depending whether particles $k$
and $l$ moved both in the same or in different directions). ∎
###### Remark 4.9.
We describe how the difference operators capture the particle interpretation
of the model intrinsically. In their definition specialized appropriately as
in the statement of the above proposition, if two consecutive particles
$k,k+1$ are 1 unit apart ($h_{k+1}-h_{k}=1$), the bottom one cannot move up
and the top one down to collide because the summand in the difference operator
is 0 (indeed $\theta_{p}(qz_{k}z_{k+1}^{-1})=\theta_{p}(1)=0$ in the cross
terms). Thus, the non-intersecting condition on the paths is intrinsically
built into the difference operator. A similar reasoning shows that top-most
and bottom-most particles are not allowed to leave the bounding hexagon
either. To exemplify, for the difference operator $\mathpzc{D}(A,B,C)$
corresponding to the $t\to t+1$ transition (particles moving from left most
vertical line to the right), we observe that the restriction on top (bottom)
particle is not to cross the NE (SE) edge labeled $C$ ($A$) in Figure 7 (or
indeed not to “walk too far” to the right by crossing the $B$ edge). However
$A$ and $C$ are two of the parameters of the difference operator, and the
corresponding terms in the univariate product in the appropriate summand in
(25) become 0 once the top (bottom) particle tries to leave the hexagon. Same
reasoning applies to the particles not being able to “walk too far right”.
Hence the difference operators intrinsically capture the boundary constraints
of our model.
###### Remark 4.10.
Proposition 4.8 is even more general, as we obtain $\binom{6}{3}=20$ different
stochastic matrices (Markov chains) from the 20 different difference operators
(6 of them already described).
We are now in a position to prove that the 6 matrices defined in section 3 are
indeed stochastic and measure preserving.
###### Theorem 4.11.
$\displaystyle\sum_{Y}P_{t\pm}^{S,t}(X,Y)=1$
$\displaystyle\sum_{Y}{}_{t\pm}P_{S\pm}^{S,t}(X,Y)=1$
$\displaystyle\rho_{S,t\pm
1}(Y)=\sum_{X}P_{t\pm}^{S,t}(X,Y)\cdot\rho_{S,t}(X)$ $\displaystyle\rho_{S\pm
1,t}(Y)=\sum_{X}{}_{t\pm}P_{S\pm}^{S,t}(X,Y)\cdot\rho_{S,t}(X)$
###### Proof.
There is one way to prove these statements which works for 4 of the 6
matrices. Observe that the results for $t\pm$ follow from Theorems 3.9 and
3.10, and then to observe that under $t\leftrightarrow S$, we have
$\mathpzc{X}^{S,t}=\mathpzc{X}^{t,S},\ \text{and}\ \rho_{S,t}=\rho_{t,S}$
and then under interchanging $S$ and $t$, $P^{S,t}_{t+}$ becomes
${}_{t+}P^{S,t}_{S+}$ (and $P^{S,t}_{t-}$ becomes ${}_{t-}P^{S,t}_{S-}$,
respectively). This idea worked both at the $q$-Racah level and Hahn level
(see [BGR10] and [BG09]).
Alternatively we can observe that the first two equalities are, by using (10)
and Proposition 4.8, restatements of Lemma 4.3 for difference operators
corresponding to parameters $(A,B,C)$ (for $P^{S,t}_{t+}$), $(D,E,F)$ (for
$P^{S,t}_{t-}$), $(A,B,D)$ (for ${}_{t+}P^{S,t}_{S+}$), $(B,C,F)$ (for
${}_{t+}P^{S,t}_{S-}$), $(D,E,A)$ (for ${}_{t-}P^{S,t}_{S+}$), $(E,F,C)$ (for
${}_{t-}P^{S,t}_{S-}$). Moreover, the normalizing constants that we omitted in
defining the transition matrices can be recovered easily from Proposition 4.8.
The last two statements are a special case of the adjointness relation. We
will prove the third statement for the $t+$ operator. Similar results exist
for the other 5 operators. We recall that $\rho_{S,t}(X)$ is nothing more than
the discrete elliptic Selberg density
$\Delta_{\lambda_{X}}(q^{2N-2}F^{2}|q^{N},q^{N-1}AF,q^{N-1}(pB)F,q^{N-1}CF,q^{N-1}DF,q^{N-1}EF)$
defined in the introduction, with $\lambda_{X,k}+n-k=x_{n+1-k}$. We also
define the partition $\lambda_{Y}$ to be the one corresponding to vertical
line $t+1$ and particle positions given by $Y$: $\lambda_{Y,k}+n-k=y_{n+1-k}$.
Then one sees $\rho_{S,t+1}(Y)=\sum_{X}P_{t+}^{S,t}(X,Y)\cdot\rho_{S,t}(X)$ is
equivalent to:
$\displaystyle\langle\mathpzc{D}(A,B,C)\mathfrak{d}_{\lambda_{Y}},1\rangle=\langle\mathfrak{d}_{\lambda_{Y}},\mathpzc{D}(D^{\prime},E^{\prime},F^{\prime})1\rangle^{\prime}$
(29)
where the prime parameters and $\langle,\rangle^{\prime}$ are defined in the
previous section. The above equality (29) is only “morally correct” as we
encounter the following issue: the (summands in the) difference operators
$\mathpzc{D}$ correspond to transitional probabilities in the $(i,j)$
coordinates where particles move up or down by 1/2 from the $t$ vertical line
to the $t+1$ vertical line (from Proposition 4.8). $\mathfrak{d}_{\lambda}$,
$\Delta_{\lambda}$ as well as the definition of the inner product (27)
correspond to coordinates $(x,t)$ where particles either move horizontally 1
step to the right or diagonally up by 1 from vertical line $t$ to vertical
line $t+1$ (see the previous subsection and recall
$\lambda_{k}+n-k=x_{n+1-k}$). But this can be easily fixed since
$(i,j)=(t,x-t/2)$. However, writing an “approximate” version conveys the
meaning of the quasi-adjointness of the difference operators in a notationally
uncluttered way.
With the previous comment in mind, the right hand side in (29) equals
$\sum_{\mu}\mathfrak{d}_{\lambda_{Y}}(...Fq^{\mu_{k}+n-k}...)\Delta_{\mu}^{\prime}=\Delta_{\lambda_{Y}}^{\prime}=\rho_{S,t+1}(Y)$
(observe $\Delta^{\prime}$ = $\Delta$ with prime parameters corresponds to the
distribution of particles at the line $t+1$) while the left hand side equals
$\sum_{\lambda_{X}}Prob(\lambda_{Y}|\lambda_{X})\cdot\Delta_{\lambda_{X}}=\sum_{X}P_{t+}^{S,t}(Y|X)\cdot\rho_{S,t}(X)$.
The result follows. ∎
We finish this section with a graphical description of the 6 Markov processes
described thus far. The key is to look at the domain and codomain of the
difference operators in canonical coordinates. We will exemplify with the
difference operator $\mathpzc{D}(A,B,D)$, corresponding to Markov chain
${}_{t+}P^{S,t}_{S+}$. Recall this Markov chain quasi-commutes with the $t\to
t+1$ chain. The key is the following relation (a restatement of Theorem 4.11):
Figure 9: Action of the difference operator $\mathpzc{D}(A,B,D)$ on a tiling
of a $N=2,S=4,T=7$ hexagon drawn in canonical coordinates. The source is
marked 1 and the destination 2. Only edges relevant to the model are
considered: the 6 bordering edges and the particle line at horizontal
displacement $t$ from the leftmost vertical edge. Note the slight shifting,
the increase in $S$ by 1, and the fact that the particle line’s displacement
from the left vertical edge ($=t$) is kept constant (though particle positions
are shifted by a third step).
$\displaystyle\sum_{X}Prob(Y|X;A,B,D)Prob(X;A,B,C,D,E,F)=Prob(Y;A^{\prime},B^{\prime},C^{\prime},D^{\prime},E^{\prime},F^{\prime})\
\mathrm{where}$
$\displaystyle(A^{\prime},B^{\prime},C^{\prime},D^{\prime},E^{\prime},F^{\prime})=(q^{\frac{1}{2}}A,q^{\frac{1}{2}}B,q^{-\frac{1}{2}}C,q^{\frac{1}{2}}D,q^{-\frac{1}{2}}E,q^{-\frac{1}{2}}F)$
We note ${}_{t+}P^{S,t}_{S+}$ corresponding to difference operator
$\mathpzc{D}(A,B,D)$ maps marked random tilings of hexagons determined by
parameters $(A,B,C,D,E,F)$ to random tilings of hexagons determined by
parameters
$(A^{\prime},B^{\prime},C^{\prime},D^{\prime},E^{\prime},F^{\prime})$ (marked
here refers to the particle line corresponding to parameter $t$). We figure
what happens to the edges of such hexagons when parameters get shifted by
$q^{\pm 1/2}$ by using (12) (canonical coordinates). Figure 9 is a graphical
description. In particular, we observe ${}_{t+}P^{S,t}_{S+}$ increases $S$ by
1. Similarly for the other difference operators: they increase (decrease) $S$
or $t$ by 1 while leaving the other constant.
## 5 Perfect Markov chain sampling algorithm
### 5.1 The $S\mapsto S+1$ step
In this section, which follows closely the notation and proofs of [BG09] (see
also [BGR10]), we will define a stochastic matrix
$\displaystyle P^{S}_{S\mapsto
S+1}:\Omega(N,S,T)\times\Omega(N,S+1,T)\to[0,1]$
that is measure preserving: it preserves the elliptic measure $\mu(N,S,T)$ \-
the total mass of a hexagon tiling (collection of $N$ non-intersecting lattice
paths) in $\Omega(N,S,T)$. Recall $\mu$ is defined as a product of the weights
of the individual horizontal lozenges inside the hexagon. Viewed as a Markov
chain, the input for $P^{S}_{S\mapsto S+1}$ is a hexagon of size $a\times
b\times c$ and the output a hexagon of size $a\times(b-1)\times(c+1)$. Both
the input and the output will turn out to be distributed according to
$\mu(N,S,T)$ and $\mu(N,S+1,T)$ respectively.
Given a collection of non-intersecting paths
$X=(X(0),...,X(T))\in\Omega(N,S,T)$, we will construct a (random) new
collection $Y=(Y(0),...,Y(T))\in\Omega(N,S+1,T)$ by defining a stochastic
transition matrix $P^{S}_{S\mapsto S+1}(X,Y)$. Observe that
$Y(0)\in\mathpzc{X}^{S+1,0}=(0,...,N-1)$ is unambiguously defined. Next we
perform the sequential (inductive) update. That is, the procedure which
produces a random $Y(t+1)$ given knowledge of $Y(0),...,Y(t)$ and $X$ which we
assume to have already been obtained. $Y(t+1)$ will be defined according to
the distribution
$\begin{split}Prob(Y(t+1)=Z)&=\frac{P_{t+}^{S+1,t}(Y(t),Z)\cdot{}_{t+}P_{S-}^{S+1,t+1}(Z,X(t+1))}{(P_{t+}^{S+1,t}\cdot{}_{t+}P_{S-}^{S+1,t+1})(Y(t),X(t+1))}\\\
&=\frac{{}_{t-}P_{S+}^{S,t+1}(X(t+1),Z)\cdot
P_{t-}^{S+1,t+1}(Z,Y(t))}{({}_{t-}P_{S+}^{S,t+1}\cdot
P_{t-}^{S+1,t+1})(X(t+1),Y(t))}\end{split}$ (30)
where the last equality follows from the fact that
$\rho_{S+1,t+1}(A)P^{S+1,t+1}_{t-}(A,B)=\rho_{S+1,t}(B)P^{S+1,t}_{t+}(B,A)$
(this is nothing more than the equality $Prob(A\cap
B)=Prob(A)Prob(B|A)=Prob(B)Prob(A|B)$).
We define the matrix $P_{S\mapsto
S+1}:\Omega(N,S,T)\times\Omega(N,S+1,T)\to[0,1]$ by
$P_{S\mapsto
S+1}=\left\\{\begin{array}[]{lll}&\prod_{t=0}^{T-1}\frac{P_{t+}^{S+1,t}(Y(t),Y(t+1))\cdot{}_{t+}P_{S-}^{S+1,t+1}(Y(t+1),X(t+1))}{(P_{t+}^{S+1,t}\cdot{}_{t+}P_{S-}^{S+1,t+1})(Y(t),X(t+1))}\\\
&\text{if \
}\prod_{t=0}^{T-1}(P_{t+}^{S+1,t}\cdot{}_{t+}P_{S-}^{S+1,t+1})(Y(t),X(t+1))>0\\\
&0,\ \text{otherwise}\end{array}\right.$ (31)
###### Theorem 5.1.
The matrix $P_{S\mapsto S+1}$ is stochastic and $\mu$-measure preserving, in
the sense that
$\displaystyle\mu(N,S+1,T)(Y)=\sum_{X\in\Omega(N,S,T)}P_{S\mapsto
S+1}(X,Y)\mu(N,S,T)(X).$ (32)
###### Proof.
(following [BG09]) We want to show that
$\displaystyle\sum_{Y}P_{S\mapsto
S+1}(X,Y)=\sum_{Y}\prod_{t=0}^{T-1}\frac{P_{t+}^{S+1,t}(Y(t),Y(t+1))\cdot{}_{t+}P_{S-}^{S+1,t+1}(Y(t+1),X(t+1))}{(P_{t+}^{S+1,t}\cdot{}_{t+}P_{S-}^{S+1,t+1})(Y(t),X(t+1))}=1$
where the sum is taken over all $Y=(Y(0),...,Y(T))\in\Omega(N,S+1,T)$ such
that
$\displaystyle\prod_{t=0}^{T-1}(P_{t+}^{S+1,t}\cdot{}_{t+}P_{S-}^{S+1,t+1})(Y(t),X(t+1))>0$
(33)
We first sum over $Y(T)$ and because $Y(T)$ is distributed according to a
singleton measure, the respective sum is 1. Next we deal with the sum
$\displaystyle\sum_{Y(T-1)}\frac{P_{t+}^{S+1,T-2}(Y(T-2),Y(T-1))\cdot{}_{t+}P_{S-}^{S+1,T-1}(Y(T-1),X(T-1))}{(P_{t+}^{S+1,T-2}\cdot{}_{t+}P_{S-}^{S+1,T-1})(Y(T-2),X(T-1))}$
over $Y(T-1)$ satisfying
$(P_{t+}^{S+1,T-1}\cdot{}_{t+}P_{S-}^{S+1,T})(Y(T-1),X(T))>0$ (because of
(33)). Because of the quasi-commutation relations from Theorem 4.4, we have
$\displaystyle(P_{t+}^{S+1,T-1}\cdot{}_{t+}P_{S-}^{S+1,T})(Y(T-1),X(T))=(P_{S-}^{S+1,T-1}\cdot{}_{t+}P_{S-}^{S,T-1})(Y(T-1),X(T))$
$\displaystyle\geq
P_{S-}^{S+1,T-1}(Y(T-1),X(T-1))P_{t+}^{S,T-1}(X(T-1),X(T)).$
We are summing over $Y(T-1)$ such that the LHS above is non-vanishing, but if
it vanishes, then by the above inequality so does
${}_{t+}P_{S-}^{S+1,T-1}(Y(T-1),X(T))$ (one of the numerator terms in the sum
over $Y(T-1)$ considered). This means we can drop the condition that
$(P_{t+}^{S+1,T-1}\cdot{}_{t+}P_{S-}^{S+1,T})(Y(T-1),X(T))>0$ and sum over all
$Y(T-1)$. We obtain 1 for this sum (the denominator is independent of the
summation variable, and summing the numerator over $Y(T-1)$ we obtain the
denominator). We next sum inductively over $Y(T-2)$ and so on until we are
left over with a sum over $Y(0)$. This sum only has 1 term, so we obtain the
desired result.
To show $P_{S\mapsto S+1}$ preserves the measure $\mu$, observe first that
$\mu(N,S,T)(X)=m_{0}(X(0))\cdot
P_{t+}^{S,0}(X(0),X(1))...P^{S,T-1}_{t+}(X(T-1),X(T))$
where $m_{0}$ is the unique probability measure on any singleton set (in this
case $\mathpzc{X}^{S,0}$). Then the RHS of (32) becomes
$\displaystyle\sum_{X}m_{0}(X(0))\prod_{t=0}^{T-1}P_{t+}^{S,t}(X(t),X(t+1))\times\prod_{t=0}^{T-1}\frac{P_{t+}^{S+1,t}(Y(t),Y(t+1))\cdot{}_{t+}P_{S-}^{S+1,t+1}(Y(t+1),X(t+1))}{(P_{t+}^{S+1,t}\cdot{}_{t+}P_{S-}^{S+1,t+1})(Y(t),X(t+1))}.$
(34)
Pulling out factors independent of the summation variables, replacing
$1=m_{0}(X(0))$ with $1=m_{0}(Y(0))$, using
${}_{t+}P_{S-}^{S+1,T}(Y(T),X(T))={}_{t+}P_{S-}^{S+1,0}(Y(0),X(0))=1$ and
$P_{t+}^{S,t}\cdot{}_{t+}P_{S-}^{S,t+1}={}_{t+}P_{S-}^{S,t}\cdot
P_{t+}^{S-1,t}$, we transform (34) into
$\displaystyle
m_{0}(Y(0))\prod_{t=0}^{T-1}P_{t+}^{S+1,t}(Y(t),Y(t+1))\times\sum_{X}\prod_{t=0}^{T-1}\frac{{}_{t+}P_{S-}^{S+1,t}(Y(t),X(t))\cdot
P_{t+}^{S,t}(X(t),X(t+1))}{({}_{t+}P_{S-}^{S,t}\cdot
P_{t+}^{S+1,t})(Y(t),X(t+1))}.$
Now we sum first over $X(T)$, then over $X(T-1)$ and so on like in the
previous argument to finally obtain on the LHS the desired result:
$\displaystyle
m_{0}(Y(0))\prod_{t=0}^{T-1}P_{t+}^{S+1,t}(Y(t),Y(t+1))=\mu(N,S+1,T)(Y).$
∎
### 5.2 Algorithmic description of the $S\mapsto S+1$ step
As before, whenever possible, we try to keep the notation similar to [BG09].
For $x\in\mathbb{N}$ we define
$\displaystyle\mathpzc{p}(x)=\frac{q\theta_{p}(q^{x-t-S+T-1},q^{x-t-T-1}v_{1},q^{x+t+1}v_{2},q^{x-t-S-1}v_{1}v_{2})}{\theta_{p}(q^{x+1},q^{x-2t-S-1}v_{1},q^{x-S+T+1}v_{2},q^{x-T+1}v_{1}v_{2})}\times\frac{\theta_{p}(q^{2x-t-S+1}v_{1}v_{2})}{\theta_{p}(q^{2x-t-S-1}v_{1}v_{2})}.$
Note $\mathpzc{p}$ also depends on $S,T,v_{1},v_{2},q,p$, but we will omit
these for simplicity of notation. Also note $p$ is an elliptic function of
$q,q^{S},q^{T},q^{t},v_{1},v_{2},q^{x}$. Consider (again omitting most
parameter dependence)
$\displaystyle P(x;s)=\prod_{i=1}^{s}\mathpzc{p}(x+i-1).$
$P$ is just a ratio of 5 length $s$ theta-Pochhammer symbols over 5 others
(multiplied by $q^{s-1}$ to make everything elliptic). We define the following
probability distribution on the set $\\{0,1,...,n\\}$.
$\displaystyle Prob(s)=D(x;n)(s)=\frac{P(x;s)}{\sum_{j=0}^{n}P(x;j)}.$ (35)
For the exact sampling algorithm, given $X=(X(0),...,X(T))\in\Omega(N,S,T)$,
we will construct $Y=(Y(0),...,Y(T))\in\Omega(N,S+1,T)$ by first observing
that $Y(0)=(0,...,N-1)$ is uniquely defined. We then perform $T$ sequential
updates. At step $t+1$ we obtain $Y(t+1)$ based on $Y(t)$ and $X(t+1)$.
Suppose $X(t+1)=(x_{1},...,x_{N})\in\mathpzc{X}^{S,t+1}$ and
$Y(t)=(y_{1},...,y_{N})\in\mathpzc{X}^{S+1,t}$. We want to define/sample
$Y(t+1)=(z_{1},...,z_{N})\in\mathpzc{X}^{S+1,t+1}$. $Y(t)$ and $X(t+1)$
satisfy $x_{i}-y_{i}\in\\{0,-1,1\\}$ (follows by construction from
$(P_{t+}^{S+1,t}\cdot{}_{t+}P_{S-}^{S+1,t+1})(Y(t),X(t+1))>0$). We thus have
three cases, and we describe how to choose $z_{i}$ in each:
* •
Case 1. Consider all $i$ such that $x_{i}-y_{i}=1$. Then $z_{i}=x_{i}$ is
forced.
* •
Case 2. Consider all $i$ such that $x_{i}-y_{i}=-1$. Then $z_{i}=y_{i}$ is
forced.
* •
Case 3. For the remaining indices, group them in blocks and consider one such
called a $(k,l)$ block (where $k$ is the smallest particle location in the
block, and $l$ is the number of particles in the block). That is, we have
$y_{i-1}<k-1$, $y_{i+l}>k+l$ and the block consists of
$x_{i}=y_{i}=k,\ x_{i+1}=y_{i+1}=k+1,...,x_{i+l-1}=y_{i+l-1}=k+l-1.$
For each such block independently, we sample a random variable $\xi$ according
to the distribution $D(k;l)$. We set $z_{i}=x_{i}$ for the first $\xi$
consecutive positions in the block, and we set $z_{i}=x_{i}+1$ for the
remainder of the $l-\xi$ positions. We provide an example in Figure 10 below:
Figure 10: Sample block split. Same picture appears in [BG09] for uniformly
distributed tilings.
###### Theorem 5.2.
By constructing $Y$ this way, we have simulated a $S\mapsto S+1$ step of the
Markov chain $P_{S\mapsto S+1}$.
###### Proof.
We perform the following computation (and are interested in Case 3. described
above, that is on how to split a $(k,l)$ block; note $x_{i}=y_{i}$ in the case
of interest):
$\displaystyle
Prob(Y(t+1)=Z)=\frac{P_{t+}^{S+1,t}(Y(t),Z)\cdot{}_{t+}P_{S-}^{S+1,t+1}(Z,X(t+1))}{(P_{t+}^{S+1,t}\cdot{}_{t+}P_{S-}^{S+1,t+1})(Y(t),X(t+1))}=(\mathrm{factors\
independent\ of}Z)$ (36)
$\displaystyle\times\prod_{i:z_{i}=y_{i}}q^{-y_{i}-N+1}\frac{\theta_{p}(q^{y_{i}+T-S-t-1},q^{y_{i}-T-S-t-1}v_{1},q^{y_{i}+t+1}v_{2},q^{y_{i}+N-t}v_{1}v_{2})}{\theta_{p}(q^{2y_{i}-t-S}v_{1}v_{2})}$
(37)
$\displaystyle\times\prod_{i:z_{i}=y_{i}+1}q^{-y_{i}}\frac{\theta_{p}(q^{y_{i}-S-N},q^{y_{i}-2t-S-1}v_{1},q^{y_{i}+T+1}v_{2},q^{y_{i}-T+1}v_{1}v_{2})}{\theta_{p}(q^{2y_{i}-t-S}v_{1}v_{2})}$
(38)
$\displaystyle\times\prod_{i:z_{i}=x_{i}}q^{-x_{i}}\frac{\theta_{p}(q^{x_{i}-S-N},q^{x_{i}-t-T-1}v_{1},q^{x_{i}+T+1}v_{2},q^{x_{i}-t-S-1}v_{1}v_{2})}{\theta_{p}(q^{2x_{i}-t-S-1}v_{1}v_{2})}$
(39)
$\displaystyle\times\prod_{i:z_{i}=x_{i}+1}q^{-x_{i}-N}\frac{\theta_{p}(q^{x_{i}+1},q^{x_{i}-T-S-t-1}v_{1},q^{x_{i}-S+T+1}v_{2},q^{x_{i}+N-t}v_{1}v_{2})}{\theta_{p}(q^{2x_{i}-t-S+1}v_{1}v_{2})}.$
(40)
We thus see the blocks split independently. The probability that the first $j$
particles in a $(k,l)$ block stay put from $Y(t)$ to $Y(t+1)$ (and the rest of
$l-j$ jump by 1) is, by using the above formula:
$\displaystyle\prod_{i=0}^{j-1}\frac{q\theta_{p}(q^{k+i-t-S+T-1},q^{k+i-t-T-1}v_{1},q^{k+i+t+1}v_{2},q^{k+i-t-S-1}v_{1}v_{2})}{\theta_{p}(q^{2k+2i-t-S-1}v_{1}v_{2})}$
$\displaystyle\times\prod_{i=j}^{l-1}\frac{\theta_{p}(q^{k+i+1},q^{k+i-2t-S-1}v_{1},q^{k+i-S+T+1}v_{2},q^{k+i-T+1}v_{1}v_{2})}{\theta_{p}(q^{2k+2i-t-S+1}v_{1}v_{2})}\times(\mathrm{factors\
independent\ of\ }j)$
where in (36) we have gauged away everything independent of the split position
$j$. This probability is nothing more than the distribution $D$ we defined in
(35). This finishes the proof. ∎
### 5.3 Algorithmic description of the $S\mapsto S-1$ step
Similar to the $P_{S\mapsto S+1}$ matrix described in the previous two
sections, we can construct a $P_{S\mapsto S-1}$ measure preserving Markov
chain that takes random tilings in $\Omega(N,S,T)$ and maps them to random
tilings in $\Omega(N,S-1,T)$. We proceed exactly as in Section 5.1 and will
omit most details and theorems as they transfer verbatim from Section 5.1 and
the previous section (we refer the reader to [BG09] for more details on this
algorithm). Given $X\in\Omega(N,S,T)$ and $Y(0),Y(1),...,Y(t)$ already defined
inductively, we choose $Y(t+1)$ from the distribution:
$\displaystyle
Prob(Y(t+1)=Z)=\frac{P_{t+}^{S-1,t}(Y(t),Z)\cdot{}_{t+}P_{S+}^{S-1,t+1}(Z,X(t+1))}{(P_{t+}^{S-1,t}\cdot{}_{t+}P_{S+}^{S-1,t+1})(Y(t),X(t+1))}.$
(41)
We define
$P_{S\mapsto
S-1}=\left\\{\begin{array}[]{lll}&\prod_{t=0}^{T-1}\frac{P_{t+}^{S-1,t}(Y(t),Y(t+1))\cdot{}_{t+}P_{S+}^{S-1,t+1}(Y(t+1),X(t+1))}{(P_{t+}^{S-1,t}\cdot{}_{t+}P_{S+}^{S-1,t+1})(Y(t),X(t+1))}\\\
&\text{if \
}\prod_{t=0}^{T-1}(P_{t+}^{S-1,t}\cdot{}_{t+}P_{S+}^{S-1,t+1})(Y(t),X(t+1))>0\\\
&0,\ \text{otherwise}\end{array}\right.$ (42)
We will also sketch the algorithm for sampling using $P_{S\mapsto S-1}$. We
need to define the equivalent for $\mathpzc{p}$ from the previous section. For
$x\in\mathbb{N}$ we define
$\displaystyle\mathpzc{p}^{\prime}(x)=\frac{q\theta_{p}(q^{x-t-N-1},q^{x-t-2S}v_{1},q^{x+t}v_{2},q^{x-t+N-1}v_{1}v_{2})}{\theta_{p}(q^{x-S-N+1},q^{x-2t-S}v_{1},q^{x+S}v_{2},q^{x-S+N+1}v_{1}v_{2})}\times\frac{\theta_{p}(q^{2x-t-S+1}v_{1}v_{2})}{\theta_{p}(q^{2x-t-S-1}v_{1}v_{2})}.$
As before, $\mathpzc{p}^{\prime}$ is an elliptic in
$q,q^{S},q^{N},q^{t},v_{1},v_{2},q^{x}$. We also have
$P^{\prime}(x;s)=\prod_{i=1}^{s}\mathpzc{p}^{\prime}(x+i-1)$ and the following
distribution on $\\{0,1,...,n\\}$:
$\displaystyle
Prob(s)=D^{\prime}(x;n)(s)=\frac{P^{\prime}(x;s)}{\sum_{j=0}^{n}P^{\prime}(x;j)}.$
(43)
Assuming we have $X\in\Omega(N,S,T)$ with $X(t+1)=(x_{1}<...<x_{N})$ and
inductively $Y(0),...,Y(t)=(y_{1}<...<y_{N})$, we sample
$Y(t+1)=(z_{1}<...<z_{N})$ by first observing that $x_{i}-y_{i}\in\\{0,1,2\\}$
(because $(P_{t+}^{S-1,t}\cdot{}_{t+}P_{S+}^{S-1,t+1})(Y(t),X(t+1))>0$) and
then performing appropriate updates for the following three simple cases:
* •
Case 1. For all $i$ with $x_{i}-y_{i}=0$ we set $z_{i}=x_{i}$.
* •
Case 2. For all $i$ with $x_{i}-y_{i}=2$ we set $z_{i}=y_{i}+1$.
* •
Case 3. For the remaining indices (for which $x_{i}-y_{i}=1$), group them in
blocks and consider one such called a $(k,l)$ block (where $k$ is the smallest
particle location in the block, and $l$ is the number of particles in the
block). That is, we have $y_{i-1}<k-1$, $y_{i+l}>k+l$ and the block consists
of
$x_{i}=y_{i}+1=k,\ x_{i+1}=y_{i+1}+1=k+1,...,x_{i+l-1}=y_{i+l-1}+1=k+l-1.$
For each such block independently, we sample a random variable $\xi$ according
to the distribution $D^{\prime}(k;l)$. We set $z_{i}=y_{i}$ for the first
$\xi$ consecutive positions in the block, and we set $z_{i}=y_{i}+1$ for the
remainder of the $l-\xi$ positions. See Figure 10.
An analogous of Theorem 5.2 exists and is proved in a similar way to show the
above 3 steps are all that is necessary to simulate the Markov chain
$P_{S\mapsto S-1}$.
## 6 Correlation kernel and determinantal representations
In this section we will show the process $X(t)$ corresponding to a tiling of
the hexagon is determinantal with correlation kernel given in terms of
elliptic biorthogonal functions due to Spiridonov and Zhedanov
([SZ00],[Rai06]). We start by a brief overview of the necessary facts about
biorthogonal functions, and continue with the heart of the proof: an
application of the Eynard-Mehta theorem.
### 6.1 A brief overview of elliptic biorthogonal functions
We will first gather together a few results about univariate discrete elliptic
biorthogonal functions. The notation and exposition will mostly be following
[Rai06]. We will need to make brief use of univariate interpolation abelian
functions. They were introduced in [Rai10] (see also [Rai06] for a description
closer to our purposes). They are, for a fixed integer $l$, $BC_{1}$-symmetric
(i.e., symmetric under $x\mapsto 1/x$) ratios of $BC_{1}$-symmetric theta
functions of degree $l$ with prescribed poles and zeros. To wit:
$\displaystyle R^{*}_{l}(x;a,b)=\frac{\theta_{p}(ax^{\pm
1};q)_{l}}{\theta_{p}(bq^{-l}x^{\pm 1};q)_{l}}.$
Observe $R^{*}_{l}$ has zeros at finitely many $q$-shifts of $a$ and poles at
finitely many $q$-shifts of $b$ (up to taking reciprocals and shifting by
$p$). The biorthogonal functions
$R_{l}(x;t_{0}:t_{1},t_{2},t_{3};u_{0},u_{1})$ discovered by Spiridonov and
Zhedanov ([SZ00] in the univariate case; see [Rai10] for continuous and
[Rai06] for discrete multivariate analogs) can be defined in terms of the
interpolation functions as follows ([Rai06]). Fix $|p|<1,q$ as well as six
parameters $t_{0},t_{1},t_{2},t_{3},u_{0},u_{1}$ such that
$t_{0}t_{1}t_{2}t_{3}u_{0}u_{1}=pq$. Then (dependence on $p,q$ implied but not
written)
$\displaystyle R_{l}(x;t_{0}:t_{1},t_{2},t_{3};u_{0},u_{1})=\sum_{0\leq k\leq
l}d_{k}R^{*}_{k}(x;t_{0},u_{0})=d_{l}R^{*}_{l}+\text{lower \ order \ terms}$
where the formula for the $d_{k}$’s is explicitly given in [Rai06] and is
independent of $x$ (but of course depends on
$t_{0},t_{1},t_{2},t_{3};u_{0},u_{1},q,p$ and $k$). These functions have poles
at shifts of $u_{0}^{\pm 1}$ (we will say $u_{0}$ controls the poles of
$R_{l}(x;t_{0}:t_{1},t_{2},t_{3};u_{0},u_{1})$). They are elliptic in the 6
parameters (provided the balancing condition is satisfied) as well as in the
variable $x$. Furthermore, if in addition to the balancing condition, one also
has
$\displaystyle t_{0}t_{1}=q^{-m}$ (44)
(for some $m>0$ an integer), the functions with poles controlled by $u_{0}$
and those with poles controlled by $u_{1}$ satisfy the following discrete
biorthogonality relation on $\\{0,...,m\\}$:
$\displaystyle\sum_{0\leq s\leq
m}R_{l}(t_{0}q^{s};t_{0}:t_{1},t_{2},t_{3};u_{0},u_{1})R_{k}(t_{0}q^{s};t_{0}:t_{1},t_{2},t_{3};u_{1},u_{0})\Delta_{s}(t_{0}^{2}|q,t_{0}t_{1},t_{0}t_{2},t_{0}t_{3},t_{0}u_{0},t_{0}u_{1})=\delta_{l,k}c_{l}$
where $\Delta_{s}$ is the univariate weight discussed in the Introduction
(also appearing in section 3) and
$c_{l}=const\cdot\Delta_{l}(1/u_{0}u_{1}|q,t_{0}t_{1},t_{0}t_{2},t_{0}t_{3},1/t_{0}u_{0},1/t_{0}u_{1})^{-1}=const\cdot\Delta(\hat{t}_{0}^{2}|q,\hat{t}_{0}\hat{t}_{1},\hat{t}_{0}\hat{t}_{2},\hat{t}_{0}\hat{t}_{3},\hat{t}_{0}\hat{u}_{0},\hat{t}_{0}\hat{u}_{1})^{-1}.$
(45)
The “hat” parameters are defined by the relations
$\displaystyle\hat{t}_{0}=\sqrt{\frac{t_{0}t_{1}t_{2}t_{3}}{pq}},\
\hat{t}_{0}\hat{t_{i}}=t_{0}t_{i},\
\frac{\hat{u}_{j}}{\hat{t}_{0}}=\frac{u_{j}}{t_{0}}$ (46)
for $i=1,2,3$ and $j=0,1$. The “hat” is an involution. Also observe the hat
parameters satisfy the same balancing conditions the original parameters
satisfy. They are important because by hatting we can exchange the variable
and the index of the biorthogonal functions as follows:
$\displaystyle
R_{l}(t_{0}q^{s};t_{0}:t_{1},t_{2},t_{3};u_{0},u_{1})=R_{s}(\hat{t}_{0}q^{l};\hat{t}_{0}:\hat{t}_{1},\hat{t}_{2},\hat{t}_{3};\hat{u}_{0},\hat{u}_{1}).$
(47)
The biorthogonal functions described above have $t_{0}$ as a special
normalization parameter (distinguished among the $t_{i}$’s). That is,
$R_{l}(t_{0};t_{0}:t_{1},t_{2},t_{3};u_{0},u_{1})=1$. The normalized
difference operators of section 4 act on the biorthogonal functions as follows
(note $u_{0}$ is special - it controls the poles, and $t_{0}$ is also special
as the choice of normalization):
$\displaystyle\mathpzc{D}(u_{0},t_{0},t_{1})R_{l}((q^{1/2}t_{0})q^{s};q^{1/2}t_{0}:q^{1/2}t_{1},q^{-1/2}t_{2},q^{-1/2}t_{3};q^{1/2}u_{0},q^{-1/2}u_{1})=R_{l}(t_{0}q^{s};t_{0}:t_{1},t_{2},t_{3};u_{0},u_{1}).$
(48)
Finally, we can exchange $t_{0}$ with another $t_{j}$ at the choice of picking
up a factor (this is in essence a renormalization so that $R$ takes value 1 at
$t_{j}$ rather than $t_{0}$):
$\displaystyle
R_{l}(x;t_{1}:t_{0},t_{2},t_{3};u_{0},u_{1})=\frac{R_{l}(x;t_{0}:t_{1},t_{2},t_{3};u_{0},u_{1})}{R_{l}(t_{1};t_{0}:t_{1},t_{2},t_{3};u_{0},u_{1})}$
(49)
### 6.2 Determinantal representations
In this section we will show the processes $t\mapsto t\pm 1$ are determinantal
point processes. For a review of such processes we direct the reader to
[Bor11]. We will do the calculation for the $t\mapsto t-1$ Markov process as
it leads to less complicated formulas, but analogous results hold for
$t\mapsto t+1$.
For the remainder, it is now convenient to relabel and rescale the parameter
set $\\{A,B,C,D,E,F\\}$ as $\\{t_{0},t_{1},t_{2},t_{3},u_{0},u_{1}\\}$ in
order for certain symmetries to become more prominent (and in doing so, we
will use the notation set forth in the previous section). To wit:
$\displaystyle A=t_{2},\ q^{N-1}B=u_{1},\ C=t_{3},\ D=t_{1},\ q^{N-1}E=u_{0},\
F=t_{0}.$ (50)
Note these parameters depend on $t$ (the time parameter), and such dependence
will be made more explicit when it becomes important. Notation is as in the
previous section. Note $u_{0}u_{1}t_{0}t_{1}t_{2}t_{3}=q$. Since the balancing
condition for the biorthogonal functions requires a $pq$ on the right hand
side, we will again multiply $u_{1}$ by $p$. These are the parameters of the
univariate biorthogonal functions discussed in the previous section. $u_{0}$
and $u_{1}$ control the poles of the pair of biorthogonal functions.
At the core of the computations will be the Eynard-Mehta theorem, which we now
state in a “decreasing-time” form convenient for our computations (see [EM98],
[Bor11] for a review and [BR05] for an elementary proof):
###### Theorem 6.1.
Assume we are given the following:
* •
a discrete biorthonormal system $(f_{l}^{t},g_{l}^{t})_{l\geq 0}$ on
$l_{2}(\\{0,1,...,L\\})$ for each time $t=0,...,T$
* •
a matrix
$v_{t\to t-1}(x,y)=\sum_{l\geq
0}f_{l}^{t-1}(t_{0}^{t-1}q^{x})g_{l}^{t}(t_{0}^{t}q^{y}),$
for $n\geq 0$, $t=1,...,T$ and a parameter $t_{0}$ changing with time
* •
a discrete time Markov chain $X(t)$ (with time decreasing from $T$ to 0)
taking values in state spaces $\mathpzc{X}^{t}$ (set of possible particle
positions at time $t$) with one dimensional distributions proportional to
$\displaystyle\det_{1\leq k,l\leq
N}(f^{t}_{k-1}(t_{0}^{t}q^{x_{l}}))\det_{1\leq k,l\leq
N}(g^{t}_{k-1}(t_{0}^{t}q^{x_{l}}))$
and transition probabilities proportional to
$\displaystyle\frac{\det_{1\leq k,l\leq N}(v_{t\to
t-1}(x_{k},y_{l}))\det_{1\leq k,l\leq
N}(f^{t-1}_{k-1}(t_{0}^{t-1}q^{y_{l}}))}{\det_{1\leq k,l\leq
N}(f^{t}_{k-1}(t_{0}^{t}q^{x_{l}}))}$
Then
$\displaystyle Prob(x_{1}\in X(\tau_{1}),...,x_{s}\in X(\tau_{s}))=\det_{1\leq
k,l\leq s}(K(\tau_{k},x_{k};\tau_{l},x_{l}))$
where
$\displaystyle
K(\tau_{1},x_{1};\tau_{2},x_{2})=\left\\{\begin{array}[]{lll}\sum_{s\geq
0}f_{s}^{\tau_{1}}(t_{0}^{\tau_{1}}q^{x_{1}})g_{s}^{\tau_{2}}(t_{0}^{\tau_{2}}q^{x_{2}}),\mathrm{\
if\ }\tau_{1}\geq\tau_{2}\\\ \\\ -\sum_{s\geq
N}f_{s}^{\tau_{1}}(t_{0}^{\tau_{1}}q^{x_{1}})g_{s}^{\tau_{2}}(t_{0}^{\tau_{2}}q^{x_{2}}),\mathrm{\
if\ }\tau_{1}<\tau_{2}\end{array}\right.$
The first step in showing the required determinantal formulas needed to apply
the Eynard-Mehta theorem is the following determinantal formula, a version of
which was discovered by Warnaar (see [War02] Lemma 5.3 and Corollary 5.4 for
comparison):
###### Lemma 6.2.
$\displaystyle\det_{1\leq k,l\leq
n}R_{l-1}(z_{k};t_{0}:t_{1},t_{2},t_{3};u_{0},pu_{1})=const\cdot\prod_{k<l}\varphi(z_{k},z_{l})\prod_{k}\frac{1}{\theta_{p}(q^{1-n}u_{0}z_{k}^{\pm
1};q)_{n-1}}$
where $z_{k}=t_{0}q^{x_{k}}$, the constant is independent of the $z_{k}$’s and
nonzero.
###### Proof.
This proof is essentially the same as that of Lemma 5.3 in [War02], but is
reproduced here for clarity. A first observation is that the constant in front
of the right hand side will not matter much, and because it is ignored, the
proof is somewhat simpler (of course, something has to be said about it not
being 0). If we denote the left hand side by $L$ and the right hand side by
$R$, we notice both $L$ and $R$ are elliptic in the $z_{k}$’s (for $R$ this is
a direct calculation, and for $L$ the biorthogonal functions inside the
determinant are elliptic as mentioned in the previous section though one can
just see this from the definition in terms of abelian interpolation
functions). Fixing a variable $z_{k}$, we see poles for $L/R$ come from the
zeros of $R$ or the poles of $L$. For the latter, the poles are controlled by
$u_{0}$ but are exactly canceled by the zeros of $1/R$ appearing in the
univariate product (one can see this from the definition of biorthogonal
functions in terms of abelian interpolation functions). For the former the
zeros of $R$ possibly leading to poles are $z_{k}=z_{l},z_{k}=1/z_{l}$ for
$l\neq k$ (and $p$ shifts thereof). Plugging in $z_{k}=z_{l}$ into $L$ makes
two columns the same, so $L$ vanishes. Since univariate biorthogonal functions
are $BC_{1}$-symmetric in the variable (a fact made explicit in the previous
section in the definition in terms of abelian interpolation functions; see
also [Rai06]), $L$ also vanishes if $z_{k}z_{l}=1$ for some $l\neq k$. Hence
all the poles of $L/R$ are removable, and since $L/R$ is elliptic, it must be
constant. To show the constant is nonzero, we notice that the functions inside
the determinant are linearly independent, so the columns of the determinant
are linearly independent. This concludes the proof. ∎
###### Remark 6.3.
A more convoluted way to arrive at such determinantal representations (but the
way that nevertheless suggested the formula above) would be to take the right
hand side of the above formula and observe it appears in Corollary 5.4 of
[War02]. What appear in the determinant on the left are the abelian
interpolation functions $R^{*}_{l}$ discussed in the previous section:
$\displaystyle\det_{1\leq k,l\leq n}(\frac{\theta_{p}(az_{k}^{\pm
1};q)_{n-l}}{\theta_{p}(bz_{k}^{\pm
1};q)_{n-l}})=a^{\binom{n}{2}}q^{\binom{n}{3}}\prod_{k<l}\varphi(z_{k},z_{l})\prod_{k}\frac{\theta_{p}(b/a,abq^{2n-2k};q)_{k-1}}{\theta_{p}(bz_{k}^{\pm
1};q)_{n-1}}$
The above formula in fact allows us to compute the constant explicitly by
expanding the biorthogonal functions in terms of abelian interpolation
functions (only leading coefficient is of interest for the determinant, and it
is explicitly given in [Rai06]).
To simplify notation hereinafter we let
$\displaystyle\Phi_{l}^{t}(t_{0}q^{s}):=R_{l}(t_{0}q^{s};t_{0}:t_{1},t_{2},t_{3};u_{0},pu_{1})$
$\displaystyle\Psi_{l}^{t}(t_{0}q^{s}):=R_{l}(t_{0}q^{s};t_{0}:t_{1},t_{2},t_{3};pu_{1},u_{0}).$
The $t$ superscript for these functions stands for the fact their arguments,
as it will become apparent in the next proposition, are essentially locations
of the particles at time $t$. Likewise the parameters depend on $t$ ($t_{i}$
and $u_{j}$ are implicit for $t_{i}^{t}$, $u_{j}^{t}$ respectively; see (50)
and (10)). We’ll also denote
$\displaystyle\tilde{\Psi}_{l}(t_{0}q^{s})=\Psi_{l}(t_{0}q^{s})\Delta_{s}(t_{0}^{2}|q,t_{0}t_{1},t_{0}t_{2},t_{0}t_{3},t_{0}u_{0},pt_{0}u_{1})/c_{l}\
\mathrm{so\ that}$ $\displaystyle\sum_{s\geq
0}\Phi_{k}(t_{0}q^{s})\tilde{\Psi}_{l}(t_{0}q^{s})=\delta_{k,l}$ (51)
Thus Lemma 6.2 along with (18) and (50) yields:
###### Proposition 6.4.
$\displaystyle Prob(X(t)=(x_{1},...,x_{N}))$
$\displaystyle=const\cdot\det_{1\leq k,l\leq
n}\Phi_{l-1}^{t}(t_{0}q^{x_{k}})\cdot\det_{1\leq k,l\leq
n}\Psi_{l-1}^{t}(t_{0}q^{x_{k}})\cdot\prod_{k}\Delta_{x_{k}}$
$\displaystyle=const\cdot\det_{1\leq k,l\leq
n}\Phi_{l-1}^{t}(t_{0}q^{x_{k}})\cdot\det_{1\leq k,l\leq
n}\tilde{\Psi}_{l-1}^{t}(t_{0}q^{x_{k}}).$
###### Proposition 6.5.
We have
$\displaystyle v_{t\to t-1}(k,l):=\sum_{s\geq
0}\Phi_{s}^{t-1}(t_{0}^{t-1}q^{k})\tilde{\Psi}_{s}^{t}(t_{0}^{t}q^{l})=\frac{1}{Z}(w_{0}^{\prime}\delta_{k,l}+w_{1}^{\prime}\delta_{k+1,l})$
(52)
with $w_{0}^{\prime}$ and $w_{1}^{\prime}$ as in Theorem 3.10 and
$Z=\frac{1}{\theta_{p}(u_{0}^{t-1}t_{0}^{t-1},u_{0}^{t-1}t_{1}^{t-1},t_{0}^{t-1}t_{1}^{t-1})}$.
###### Proof.
We observe that
$\displaystyle\sum_{s\geq
0}\Phi_{s}^{t}(t_{0}^{t}q^{k})\tilde{\Psi}_{s}^{t}(t_{0}^{t}q^{l})=\delta_{k,l}$
which expresses the relation $BA=1$ where
$A(k,l)=\Phi_{k}^{t}(t_{0}q^{l}),B(k,l)=\tilde{\Psi}_{l}^{t}(t_{0}q^{k})$ and
we know $AB=1$ by definition (see (6.2)). We now apply the difference operator
$\mathpzc{D}(u_{0}^{t-1},t_{0}^{t-1},t_{1}^{t-1})$ (corresponding to the
Markov transition $t\mapsto t-1$) to both sides and observe the parameters at
time $t$ are the required $q$ shifts of the parameters at time $t-1$ (see
(48)). Finally on the right hand side we have a delta function which is acted
upon by the difference operator to produce the desired result (see Proposition
4.6). ∎
###### Remark 6.6.
In [BGR10] and [BG09] formulas as in the above proposition involved discrete
orthogonal polynomials ($q$-Racah and Hahn respectively) and were proven via
the three term recurrence relation satisfied by these polynomials (which is an
identity between hypergeometric or $q$-hypergeometric series). Such a relation
exists for biorthogonal functions as well (we refer the reader to [SZ00] for
an explicit form, though with different notation) and can be used to prove the
above proposition, but the computations are more involved.
###### Remark 6.7.
A similar result holds if we apply the transition $t\mapsto t+1$ which
corresponds to the operator $\mathpzc{D}(u_{1},t_{2},t_{3})$. For that though,
we have to renormalize the biorthogonal functions at either $t_{2}$ or $t_{3}$
(see (48) and (49)), so the bidiagonal matrix that will appear on the RHS will
be of the above form conjugated by two diagonal matrices (coming from the
renormalization coefficients). This is an artifact of our choice of
coordinates (we are counting particles going up from the bottom left edge of
the hexagon).
Finally, in applying Theorem 6.1 to the $t\to t-1$ Markov chain $X(t)$ we need
to check that the transition probabilities have the required determinantal
form. This is a consequence of Theorem 3.10, Lemma 6.2 and the following
computation (the proof of which is immediate from Theorem 3.10 and Proposition
6.5; we use the notation from 3.10 for $w_{0}^{\prime},w_{1}^{\prime},X,Y$):
$\displaystyle\det_{1\leq k,l\leq N}(v_{t\to
t-1}(x_{k},y_{l}))=const\cdot\prod_{k:y_{k}=x_{k}-1}w^{\prime}_{1}(x_{k})\prod_{k:y_{k}=x_{k}}w^{\prime}_{0}(x_{k})$
(53)
We thus obtain:
###### Proposition 6.8.
$\displaystyle Prob(X(t-1)=Y|X(t)=X)=const\cdot\frac{\det_{1\leq k,l\leq
N}(v_{t\to t-1}(x_{k},y_{l}))\det_{1\leq k,l\leq
N}(\Phi_{k-1}^{t-1}(t_{0}^{t-1}q^{y_{l}}))}{\det_{1\leq k,l\leq
N}(\Phi_{k-1}^{t}(t_{0}^{t}q^{x_{l}}))}$ (54)
###### Theorem 6.9.
The Markov processes $t\mapsto t\pm 1$ discussed in Section 3 meet the
assumptions of Theorem 6.1 and are therefore determinantal.
###### Proof.
This follows from all the results gathered in this Section for the $t-$ Markov
chain with $f=\Phi$ and $g=\tilde{\Psi}$ in the notation of Theorem 6.1. For
$t+$ see Remark 6.7. ∎
###### Remark 6.10.
For obtaining quantitative results about the artic boundary, one can try to
look at the asymptotics of the diagonal of the correlation kernel of the
process (which is of course the probability that a particle is present at that
site):
$\displaystyle K(x,x)=$
$\displaystyle\sum_{i=0}^{S+N-1}R_{i}^{t}(t_{0}q^{x}|t_{0}:t_{1},t_{2},t_{3};u_{0},pu_{1})R_{i}^{t}(t_{0}q^{x}|t_{0}:t_{1},t_{2},t_{3};pu_{1},u_{0})\times$
$\displaystyle\Delta_{x}(t_{0}^{2}|q,t_{0}t_{1},t_{0}t_{2},t_{0}t_{3},t_{0}u_{0},pt_{0}u_{1})\Delta_{i}(1/(pu_{0}u_{1})|q,t_{0}t_{1},t_{0}t_{2},t_{0}t_{3},1/(t_{0}u_{0}),1/(pt_{0}u_{1})).$
## 7 Computer simulations
In this section we present computer simulations of the exact sampling
algorithm from Section 5. We are (with one exception) looking at $200\times
200\times 200$ hexagons, and parameters are chosen so the elliptic measure
sampled is positive throughout the range of the algorithm (recall that the
algorithm starts with a $200\times 400\times 0$ box and increases $c$ while
decreasing $b$ by 1, until it reaches the desired size - after 200 iterations
in our case). Under each figure we list the values of the four parameters
$p,q,v_{1},v_{2}$. Computations and simulations are done using double
precision, the $S\mapsto S+1$ algorithm polynomial algorithm described above,
and a custom program written in Java and that can handle large hexagons (in
excess of $N=1000$ particles) fast enough on modern CPUs.
In Figure 11 we observe that the sample looks like one from the uniform
measure with the arctic ellipse theoretically predicted in [CLP98] clearly
visible.
Figure 11: $p=10^{-7},q=0.999999995,v_{1}=0.0000214,v_{2}=1.00675$. $400\times
400\times 400$. Because $q$ is very close to 1, the limit shape looks uniform
(recall that $q=1$ gives rise to the uniform measure).
Figures 12 and 13 exhibit a new behavior for the arctic circle: the curve
seems to acquire 3 nodes at the 3 vertices of the hexagon seen in the
pictures. To obtain these shapes, the parameters have been tweaked so that the
elliptic weight ratio vanishes (or $=\infty$) at the respective corners (in
other words, the weight ratio (7) is “barely positive” as described in Section
2.3). To be more precise, we have:
$\displaystyle q=e^{\frac{2\pi i}{T-1}}$ $\displaystyle v_{1}=q^{2T-1}$
$\displaystyle v_{2}=1/q.$
This fixes 3 of the 4 parameters of the measure and we have the extra degree
of freedom $p$ and so we obtain a 1-parameter family of trinodal arctic
boundaries. All simulations are taken from the trigonometric positivity case
($q,v_{1},v_{2}$ are of unit modulus - see Section 2.3). While the first
arctic boundary looks like an equilateral “flat” triangle, the second looks
like an equilateral “thin/hyperbolic” triangle. The change from Figure 12 to
13 is an increase in $p$ (and indeed if we increase $p$ further the triangle
will get thinner and thinner, until it will degenerate into a union of the 3
coordinate axes as $p\to 1$). The limit $p\to 0$ yields the same “thinning
behavior” in the real positivity case.
Figure 12: An instance of a trinodal arctic boundary. $p=0.00186743,\arg
q=0.000835422,\arg v_{1}=0.667502,\arg v_{2}=-0.000835422$. Figure 13: Another
instance of a trinodal arctic boundary. $p=0.2,\arg q=0.000835422,\arg
v_{1}=0.667502,\arg v_{2}=-0.000835422$. Note $p$ is larger in this case than
in the previous.
Finally in Figure 14 we exhibit a trinodal case in the top level trigonometric
case $p=0$ when $q,v_{1},v_{2}$ are of unit modulus (in the case $q$ and
$v_{i}$ are real, arctic boundary is the union of the coordinate axes as
stated above).
Figure 14: Top level trigonometric $p=0$ case. As above, $\arg
q=0.000835422,\arg v_{1}=0.667502,\arg v_{2}=-0.000835422$.
## 8 Appendix
In this Appendix we show how one can assign $S_{3}$-invariant weights to the
three types of rhombi (lozenges) that make up a tiling of a hexagon in the
triangular lattice. We start with the $2\times 2\times 2$ triangle (inside the
triangular lattice) depicted in Figure 15 that contains an overlapping of the
3 types of rhombi considered for our tilings.
Figure 15: A $2\times 2\times 2$ triangle composed of 3 overlapping lozenges
of each type.
To each such type of rhombus we assign a label from the set
$\\{\tilde{u}_{1},\tilde{u}_{2},\tilde{u}_{3}\\}$ (see Figure 16) such that if
the rhombi are as described overlapping inside a $2\times 2\times 2$ triangle
we have
$\tilde{u}_{1}\tilde{u}_{2}\tilde{u}_{3}=1.$
Figure 16: The 3 types of rhombi (lozenges) and their labels.
Each $\tilde{u}_{i}$ will eventually be a power of $q$ times $u_{i}$ (see
Section 2.2). First, we can obviously shift any of such rhombi along the
directions given by their edges, either upwards or downwards. If we shift the
horizontal lozenge labeled $\tilde{u}_{3}$ upwards-right or upwards-left, the
label of the new lozenges will be multiplied by $q^{-1}$. If we shift it
downwards-right/left, the label will get multiplied by $q$. Naturally, if we
shift directly upwards, the label will be multiplied by $q^{-2}$ as a
composite of an upwards-right and and upwards-left shift. A similar rule is
used for lozenges with labels $\tilde{u}_{2}$ and $\tilde{u}_{3}$. The process
is depicted in Figure 17, with the caveat that for labels $\tilde{u}_{1}$ and
$\tilde{u}_{2}$ we only show the directions in which the label gets multiplied
by $q$ (it gets multiplied by $q^{-1}$ in the opposite two directions than the
ones depicted). Clearly translating any lozenge along its long diagonal does
not change its label.
Figure 17: Shifting lozenges in the triangular lattice, we shift the labels by
$q$ or $q^{-1}$ as depicted.
To a lozenge with label $\tilde{u}_{i}$ ($i=1,2,3$) we assign the following
weight:
$\displaystyle wt(\mathrm{lozenge\ with\ label\
}\tilde{u}_{i})=\tilde{u}_{i}^{-1/2}\theta_{p}(\tilde{u}_{i}),\ i=1,2,3.$
where
$\tilde{u}_{1}=q^{y+z-2x}u_{1},\tilde{u}_{2}=q^{x+z-2y}u_{2},\tilde{u}_{3}=q^{x+y-2z}u_{3},u_{1}u_{2}u_{3}=1,$
$u_{1},u_{2},u_{3}$ are three complex numbers that multiply to 1 and $(x,y,z)$
is the 3-dimensional coordinate of the center (intersection of the diagonals)
of a lozenge. At this point we need to fix a choice of square roots:
$\sqrt{q},\sqrt{u_{1}},\sqrt{u_{2}},\sqrt{u_{3}}$ such that
$\sqrt{u_{1}}\sqrt{u_{2}}\sqrt{u_{3}}=1$. Note the 3-dimensional coordinates
are only defined up to the diagonal action of $\mathbb{Z}$. Figure 18 depicts
the 3 lozenges with labels $u_{i}$ ($x=y=z=0$) in the chosen coordinate
system.
This way of assigning weights is manifestly $S_{3}$-invariant. To recover the
same probability distribution as in Section 2.2 (i.e., a gauge-equivalent
weight for tilings) we again require that the weight of a tiling of a hexagon
is the product of weights of lozenges inside it. To check this, one can simply
check the weight ratio of a full $1\times 1\times 1$ box to an empty $1\times
1\times 1$ box (this is a gauge-invariant quantity) under the present
assumptions and observe the result is the same as in (7).
Figure 18: Choice of a coordinate system and the 3 special parameters
(lozenges centered at the origin) needed.
The $S_{3}$ invariance can be viewed at the level of the partition function
(the sum of weights of all tilings in a hexagon written in this gauge) as
follows. We start with an $\alpha\times\beta\times\gamma$ hexagon. The origin
is at the hidden corner of the 3D box. In the canonical coordinates
$(\tilde{u}_{1}=q^{y+z-2x}u_{1},\tilde{u}_{2}=q^{x+z-2y}u_{2},\tilde{u}_{3}=q^{x+y-2z}u_{3})$
the 6 bounding edges have the following equations (see Figure 19 for
correspondence between edges and $L_{i}$’s):
$\begin{split}&\tilde{u}_{1}/\tilde{u}_{2}:=L_{0}:=q^{3\beta}u_{1}/u_{2}\\\
&\tilde{u}_{3}/\tilde{u}_{1}:=L_{1}:=q^{-3\gamma}u_{3}/u_{1}\\\
&\tilde{u}_{2}/\tilde{u}_{3}:=L_{2}:=q^{3\gamma}u_{2}/u_{3}\\\
&\tilde{u}_{1}/\tilde{u}_{2}:=L_{3}:=q^{-3\alpha}u_{1}/u_{2}\\\
&\tilde{u}_{3}/\tilde{u}_{1}:=L_{4}:=q^{3\alpha}u_{3}/u_{1}\\\
&\tilde{u}_{2}/\tilde{u}_{3}:=L_{5}:=q^{-3\beta}u_{2}/u_{3}\end{split}$ (55)
Figure 19: An $\alpha\times\beta\times\gamma$ hexagon with canonical
coordinates of the edges on the outside and edge lengths on the inside.
We then have the following proposition. Throughout, the $S_{3}$-invariant
weight is assumed.
###### Proposition 8.1.
The partition function for an $\alpha\times\beta\times\gamma$ hexagon is equal
to:
$\displaystyle P\times\lim_{\rho\to
1}\frac{\Gamma_{p,q,q}(q^{1+\alpha+\beta+\gamma}\rho,q^{1+\alpha}\rho,q^{1+\beta}\rho,q^{1+\gamma}\rho)}{\Gamma_{p,q,q}(q\rho,q^{1+\alpha+\beta}\rho,q^{1+\alpha+\gamma}\rho,q^{1+\beta+\gamma}\rho)}\times$
$\displaystyle\frac{\Gamma_{p,q,q}(q^{1-\alpha+\beta+\gamma}u_{1},q^{1-\alpha}u_{1},q^{1-\beta+\alpha+\gamma}u_{2},q^{1-\beta}u_{2},q^{1-\gamma+\alpha+\beta}u_{3},q^{1-\gamma}u_{3})}{\Gamma_{p,q,q}(q^{1-\alpha+\beta}u_{1},q^{1-\alpha+\gamma}u_{1}),q^{1-\beta+\alpha}u_{2},q^{1-\beta+\gamma}u_{2},q^{1-\gamma+\alpha}u_{3},q^{1-\gamma+\beta}u_{3})}=$
$\displaystyle P\times\lim_{\rho\to
1}\frac{\Gamma_{p,q,q}(q(L_{0}L_{2}L_{4})^{1/3}\rho,q(L_{0}L_{4}L_{5})^{1/3}\rho,q(L_{0}L_{1}L_{2})^{1/3}\rho,q(L_{2}L_{3}L_{4})^{1/3}\rho)}{\Gamma_{p,q,q}(q\rho,q(L_{0}/L_{3})^{1/3}\rho,q(L_{4}/L_{1})^{1/3}\rho,q(L_{2}/L_{5})^{1/3}\rho)}\times$
$\displaystyle\frac{\Gamma_{p,q,q}(q(L_{0}L_{2}L_{3})^{1/3},q(L_{0}L_{3}L_{5})^{1/3},q(L_{2}L_{4}L_{5})^{1/3},q(L_{1}L_{2}L_{5})^{1/3},q(L_{0}L_{1}L_{4})^{1/3},q(L_{1}L_{3}L_{4})^{1/3})}{\Gamma_{p,q,q}(q(L_{0}/L_{4})^{1/3},q(L_{3}/L_{1})^{1/3},q(L_{5}/L_{3})^{1/3},q(L_{2}/L_{0})^{1/3},q(L_{4}/L_{2})^{1/3},q(L_{1}/L_{5})^{1/3})}$
where
$\displaystyle
P=q^{\alpha\beta\gamma-\frac{\alpha\beta^{2}+\beta\alpha^{2}+\alpha\gamma^{2}+\gamma\alpha^{2}+\beta\gamma^{2}+\gamma\beta^{2}}{4}}u_{1}^{-\frac{\beta\gamma}{2}}u_{2}^{-\frac{\alpha\gamma}{2}}u_{3}^{-\frac{\alpha\beta}{2}},$
$\displaystyle\Gamma_{p,q,t}=\prod_{i,j,k\geq
0}(1-p^{i+1}q^{j+1}t^{k+1}/x)(1-p^{i}q^{j}t^{k}x).$
It is left invariant by $S_{3}$ permuting the coordinates $\tilde{u}_{i}$;
equivalently, the tuple $((x,\alpha,u_{1}),(y,\beta,u_{2}),(z,\gamma,u_{3})).$
Furthermore, this invariance can be expanded to the group
$W(G_{2})=S_{3}\times\mathbb{Z}_{2}=Dih_{6}$ (the symmetry group of a regular
hexagon) with the missing involution being the transformation:
$\displaystyle(u_{1},u_{2},u_{3})\to(\frac{1}{q^{A}u_{1}},\frac{1}{q^{B}u_{2}},\frac{1}{q^{C}u_{3}})$
where $A=-2\alpha+\beta+\gamma,B=\alpha-2\beta+\gamma,C=\alpha+\beta-2\gamma$.
###### Proof.
We start with the elliptic MacMahon identity derived in the Appendix of
[BGR10]:
$\displaystyle\frac{\sum_{\mathrm{tilings\
}T}wt(T,G)}{wt(0,G)}=q^{\alpha\beta\gamma}\prod_{1\leq x\leq\alpha,1\leq
y\leq\beta,1\leq
z\leq\gamma}\frac{\theta_{p}(q^{x+y+z-1},q^{y+z-x-1}u_{1},q^{x+z-y-1}u_{2},q^{x+y-z-1}u_{3})}{\theta_{p}(q^{x+y+z-2},q^{y+z-x}u_{1},q^{x+z-y}u_{2},q^{x+y-z}u_{3})}$
where $0$ denotes the empty tiling (box) and $G$ is any gauge equivalent to
the ones used in this paper (that is to say, both sides are gauge-
independent). For $G$ the $S_{3}$ invariant gauge herein discussed, the
formula for the empty tiling multiplied by the right hand side above
simplifies the partition function via straightforward computations. We arrive
at the desired result using the following transformations for $\Gamma$
functions:
$\displaystyle\Gamma_{p,q}(qx)=\theta_{p}(x)\Gamma_{p,q}(x),$
$\displaystyle\Gamma_{p,q,t}(tx)=\Gamma_{p,q}(x)\Gamma_{p,q,t}(x)$
The limit $\rho\to 1$ is needed for technical reasons to avoid zeros of triple
$\Gamma$ functions.
For the $S_{3}$-invariance, it suffices to show how the edges transform under
the 3-cycle
$(\tilde{u}_{1},\tilde{u}_{2},\tilde{u}_{3})\to(\tilde{u}_{2},\tilde{u}_{3},\tilde{u}_{1})$
(a $120^{\circ}$ clockwise rotation) and the transposition
$\tilde{u}_{1}\leftrightarrow\tilde{u}_{2}$ (a reflection in the $z$ axis).
For the 3-cycle, the new edges (denoted with primes) have equations:
$L_{i}^{\prime}=L_{i+2}$
where $+2$ is taken modulo 6, while for the transposition we have:
$L_{0}^{\prime}=1/L_{3},L_{1}^{\prime}=1/L_{2},L_{2}^{\prime}=1/L_{1},L_{3}^{\prime}=1/L_{0},L_{4}^{\prime}=1/L_{5},L_{5}^{\prime}=1/L_{4}.$
Both these transformations leave the partition function invariant. The extra
involution giving the group $W(G_{2})$ is a reflection through the centroid of
the hexagon having coordinates:
$\displaystyle(q^{A/2}u_{1},q^{B/2}u_{2},q^{C/2}u_{3})$
The edges transform as:
$L_{i}^{\prime}=1/L_{i+3}$
where addition is mod 6. We look at the first form of the partition function
written in the statement. We use the following two difference equations to
simplify the calculations and arrive at the original form:
$\displaystyle\Gamma_{p,q,q}(q/x)=\Gamma_{p,q,q}(pqx)=\Gamma_{q,q}(qx)\Gamma_{p,q,q}(qx)$
$\displaystyle\frac{\Gamma_{q,q}(q^{l}q^{m}x,x)}{\Gamma_{q,q}(q^{l}x,q^{m}x)}=(-x)^{ml}q^{-(l\binom{m}{2}+m\binom{l}{2})}$
∎
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|
arxiv-papers
| 2011-10-19T04:16:23 |
2024-09-04T02:49:23.340132
|
{
"license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/",
"authors": "Dan Betea",
"submitter": "Dan Betea",
"url": "https://arxiv.org/abs/1110.4176"
}
|
1110.4191
|
# Time Dependence of Advection Dominated Accretion Flow with a Toroidal
Magnetic Field
Alireza Khesali and Kazem Faghei
Department of Physics, Mazandaran University, Babolsar, Iran E-mail:
khesali@umz.ac.irE-mail: faghei@umz.ac.ir
###### Abstract
The present study examines self-similarity evolution of advection dominated
accretion flow (ADAF) in the presence of a toroidal magnetic field. In this
research, it was assumed that the angular momentum transport is due to viscous
turbulence and $\alpha$-prescription was used for kinematics coefficient of
viscosity. The flow does not have a good cooling efficiency and so, a fraction
of energy accretes with matter on central object. The effect of a toroidal
magnetic field on such systems in a dynamical behavior was investigated. In
order to solve the integrated equations which govern the dynamical behavior of
the accretion flow, self-similar solution was used. The solution provides some
insights into the dynamics of quasi-spherical accretion flow and avoids many
of the strictures of the steady self-similar solutions. The solutions show
that the behavior of physical quantities in a dynamical ADAF are different
from steady accretion flow and a disk with polytropic approach. The effect of
the toroidal magnetic field is considered with additional variable
$\beta[=p_{mag}/p_{gas}]$, where $p_{mag}$ and $p_{gas}$ are the magnetic and
gas pressure, respectively. Also to consider the effect of advection in these
systems, the advection parameter $f$ was introduced that stands for a fraction
of energy that accretes by matter to the central object. The solution
indicates a transonic point in the accretion flow for all selected amounts of
$f$ and $\beta$. Also, by adding strength of the magnetic field and the degree
of advection, the radial-thickness of the disk decreased and the disk
compressed. The model implies that the flow has differential rotation and is
sub-Keplerian at small radii and is super-Keplerian in large radii and that
different result was obtained using a polytropic accretion flow. The obtained
$\beta$ parameter was used a function of position that increases by increasing
radii. Also, The behavior of ADAF in a large toroidal magnetic field implies
that different result was obtained using steady self-similar models in large
magnetic field.
###### keywords:
accretion, accretion disks, magnetohydrodynamics: MHD
## 1 Introduction
During recent years one type of accretion disks has been studied, in which it
is assumed that the energy released through viscous processes in the disk may
be trapped within the accreting gas. This kind of flow is known as advection-
dominated accretion flow (ADAF). The basic ideas of such ADAF models have been
developed by a number of researchers (e.g., Ichimaru 1977; Rees et al. 1982;
Narayan & Yi 1994, 1995; Abramowicz et al. 1995; Ogilvie 1998; Akizuki & Fukue
2006; hereafter AF06).
It is thought that accretion disks, whether in star-forming regions, in X-ray
binaries, in cataclysmic variables, or in the centers of active galactic
nuclei, are likely to be threaded by magnetic fields. Consequently, the role
of magnetic fields on ADAF has been analyzed in detail by a number of
investigators (Bisnovatyi-kogan & Lovelace 2001; Shadmehri 2004; AF06;
Ghanbari et al. 2007, Abbassi et al. 2008). The existence of the toroidal
magnetic fields have been proven in the outer regions of YSO discs (Greaves et
al. 1997; Aitken et al. 1993; Wright et al. 1993) and in the Galactic center
(Novak et al. 2003; Chuss et al. 2003). Thus, considering the accretion disks
with a toroidal magnetic field have been studied by several authors (AF06;
Begelman & Pringle 2007; abbassi et al. 2008; Khesali & Faghei 2008 and
references within; hereafter KF08). KF08 considered dynamic behavior of a
polytropic accretion flow in presence of a toroidal magnetic field. In a
dynamic approach they showed the radial behavior of the physical quantities
were different with results achieved by those who considered the accretion
flow in a steady self-similar methods (Shadmehri 2004; AF06; Ghanbari et al
2007; Abbassi et al. 2008). For example, KF08 presented that ratio of the
magnetic pressure to the gas pressure is not constant and varies by radius.
The results of KF08 were assembled on polytropic equation that implies the
accreting gas has a good cooling efficiency, while results of some authors
have shown that the behavior of physical quantities are very sensible to
fraction of the energy that traps within the accreting gas (AF06). So,in the
present study it is intended to investigate dynamic behavior of an ADAF in
presence of a toroidal magnetic field. The paper is organized as follows. In
section 2, the general problem of constructing a model for quasi-spherical
magnetized advection dominated accretion flow will be defined. In section 3,
self-similar method for solving the integrated equations which govern the
dynamic behavior of the accreting gas was utilized. The summary of the model
will appear in section 4.
## 2 Basic equations
In this section, we derive the basic equations which describe the physics of
accretion flow with a toroidal magnetic field. We use the spherical
coordinates $(r,\theta,\phi)$ centred on the accreting object and make the
following standard assumptions:
* (i)
The accreting gas is a highly ionized gas with infinitive conductivity;
* (ii)
The magnetic field has only an azimuthal component;
* (iii)
The gravitational force on a fluid element is characterized by the Newtonian
potential of a point mass, $\Psi=-GM_{*}/r$, with $G$ representing the
gravitational constant and $M_{*}$ standing for the mass of the central star;
* (iv)
The equations written in spherical coordinates are considered in the
equatorial plane $\theta=\pi/2$ and terms with any $\theta$ and $\varphi$
dependence are neglected, hence all quantities will be expressed in terms of
spherical radius $r$ and time $t$;
* (v)
For simplicity, the self-gravity and general relativistic effects have been
neglected.
Under the assumptions and the approximation of quasi-spherical symmetry and
the ideal magnetohydrodynamics treatment, the dynamics of a magnetized
accretion flow is described by the following equations:
the continuity equation
$\frac{\partial\rho}{\partial t}+\frac{1}{r^{2}}\frac{\partial}{\partial
r}(r^{2}\rho v_{r})=0,$ (1)
the radial force equation
$\frac{\partial v_{r}}{\partial t}+v_{r}\frac{\partial v_{r}}{\partial
r}+\frac{1}{\rho}\frac{\partial p}{\partial
r}+\frac{GM_{*}}{r^{2}}=r\Omega^{2}-\frac{B_{\varphi}}{4\pi
r\rho}\frac{\partial}{\partial r}(rB_{\varphi}),$ (2)
the azimuthal force equation
$\rho\left[\frac{\partial}{\partial
t}(r^{2}\Omega)+v_{r}\frac{\partial}{\partial
r}(r^{2}\Omega)\right]=\frac{1}{r^{2}}\frac{\partial}{\partial r}\left[\nu\rho
r^{4}\frac{\partial\Omega}{\partial r}\right],$ (3)
the energy equation
$\displaystyle\frac{1}{\gamma-1}\left[\frac{\partial p}{\partial
t}+v_{r}\frac{\partial p}{\partial
r}\right]+\frac{\gamma}{\gamma-1}\frac{p}{r^{2}}\frac{\partial}{\partial
r}\left(r^{2}v_{r}\right)=$ $\displaystyle f\nu\rho
r^{2}\left(\frac{\partial\Omega}{\partial r}\right)^{2}$ (4)
and the field freezing equation
$\displaystyle\frac{\partial B_{\varphi}}{\partial
t}+\frac{1}{r}\frac{\partial}{\partial r}(rv_{r}B_{\varphi})=0,$ (5)
Here $\rho$ is the density, $v_{r}$ the radial velocity, $\Omega$ the angular
velocity, $M_{*}$ the mass of the central object, $p$ the gas pressure,
$B_{\varphi}$ the toroidal component of magnetic field, $\nu$ the kinematic
viscosity coefficient and it is given, as in Narayan & Yi (1995), by an
$\alpha$-model
$\nu=\alpha\frac{p_{gas}}{\rho\Omega_{K}}$ (6)
where $\Omega_{K}=({GM_{*}}/{r^{3}})^{1/2}$ is the Keplerian angular velocity.
The parameters $\gamma$ and $\alpha$ are assumed to be constant and $f$
measures the degree to which the flow is advection-dominated (Narayan & Yi
1994), and is assumed to be constant.
## 3 Self-similar solutions
### 3.1 Analysis
Self-similar models have proved very useful in astrophysics because the
similarity assumption reduces the complexity of the partial differential
equations. Even greater simplification is achieved in the case of spherical
symmetry since the governing equations then reduce to comparatively simple
ordinary differential equations. We introduce a similarity variable $\eta$ and
assume that each physical quantity is given by the following form:
$r=r_{0}(t)\eta$ (7) $\rho(r,t)=\rho_{0}(t)R(\eta)$ (8)
$p(r,t)=p_{0}(t)P(\eta)$ (9) $v_{r}(r,t)=v_{0}(t)V(\eta)$ (10)
$\Omega(r,t)=\Omega_{0}(t)\omega(\eta)$ (11)
$B_{\varphi}(r,t)=b_{0}(t)B(\eta).$ (12)
By assuming power-law time dependent of $r_{0}(t)=at^{n}$, where $n=2/3$, we
find the following relations:
$r_{0}(t)=at^{2/3}$ (13) $p_{0}(t)/\rho_{0}(t)=\frac{GM_{*}}{a}t^{-2/3}$ (14)
$v_{0}(t)=\sqrt{\frac{GM_{*}}{a}}t^{-1/3}$ (15)
$\Omega_{0}(t)=\sqrt{\frac{GM_{*}}{a^{3}}}t^{-1}$ (16)
$b_{0}^{2}(t)/8\pi\rho_{0}(t)=\frac{GM_{*}}{a}t^{-2/3}.$ (17)
The above results imply that $p_{0}(t)$ and $b_{0}(t)$ are dependent on timely
behavior of $\rho_{0}(t)$. So, for specifying time dependent of $\rho_{0}(t)$,
and then $p_{0}(t)$ and $b_{0}(t)$, we introduce the mass accretion rate
$\dot{M}$
$\dot{M}=-4\pi r^{2}\rho v_{r}.$ (18)
Similar to equations (7)-(12) for the mass accretion rate we can write
$\dot{M}(r,t)=\dot{M}_{0}(t)\dot{m}(\eta).$ (19)
Under transformations of equations (7), (8) and (10), equation (19) becomes
$\dot{M}(r,t)=\left[r_{0}^{2}(t)\rho_{0}(t)v_{0}(t)\right]\times\left[-4\pi\eta^{2}R(\eta)V(\eta)\right]$
(20)
in which implies
$\dot{M}_{0}(t)=r_{0}^{2}(t)\rho_{0}(t)v_{0}(t)$ (21)
$\dot{m}(\eta)=-4\pi\eta^{2}R(\eta)V(\eta).$ (22)
Now, we consider a set of solutions that $\dot{M}_{0}(t)$ is a constant
(KF08), thus we can write
$\rho_{0}(t)=(\dot{M}_{0}/\sqrt{GM_{*}a^{3}})t^{-1}$ (23)
that implies
$p_{0}(t)=(\dot{M}_{0}\sqrt{GM_{*}/a^{5}})t^{-5/3}$ (24)
and
$b^{2}_{0}(t)/8\pi=(\dot{M}_{0}\sqrt{GM_{*}/a^{5}})t^{-5/3}.$ (25)
Substituting equations (6)-(12) and (13)-(17) into the basic equations
(1)-(6), the similarity equations are obtained as
$-R+\left(V-\frac{2\eta}{3}\right)\frac{dR}{d\eta}+\frac{R}{\eta^{2}}\frac{d}{d\eta}\left(\eta^{2}V\right)=0,$
(26)
$\displaystyle-\frac{V}{3}+\left(V-\frac{2\eta}{3}\right)\frac{dV}{d\eta}+\frac{1}{R}\frac{dP}{d\eta}+\frac{1}{\eta^{2}}=~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\eta\omega^{2}-\frac{2B}{\eta R}\frac{d\left(\eta
B\right)}{d\eta},$ (27) $\displaystyle
R\left[\frac{1}{3}\left(\eta^{2}\omega\right)+\left(V-\frac{2\eta}{3}\right)\frac{d}{d\eta}\left(\eta^{2}\omega\right)\right]=~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\frac{\alpha}{\eta^{2}}\frac{d}{d\eta}\left[P\eta^{11/2}\frac{d\omega}{d\eta}\right],$
(28)
$\displaystyle\frac{1}{\gamma-1}\left[-\frac{5}{4}P+\left(V-\frac{2\eta}{3}\right)\frac{dP}{d\eta}\right]+\frac{\gamma}{\gamma-1}\frac{P}{\eta^{2}}\frac{d}{d\eta}\left(\eta^{2}V\right)$
$\displaystyle=\alpha fP\eta^{7/2}\left(\frac{d\omega}{d\eta}\right)^{2},$
(29)
$-\frac{5}{4}B+\left(V-\frac{2\eta}{3}\right)\frac{dB}{d\eta}+\frac{B}{\eta}\frac{d}{d\eta}\left(\eta
V\right)=0.$ (30)
To investigate existence of transonic point, the square of the sound velocity
is introduced that subsequently can be expressed as
$v_{s}^{2}\equiv\frac{p}{\rho}=\frac{GM_{*}}{a}\frac{P}{R}~{}t^{-2/3}$ (31)
Here, $S=\left(P/R\right)^{1/2}$ the _sound velocity_ in self-similar flow,
which is rescaled in the course of time. The _Mach number_ referred to the
reference frame is defined as (Fukue 1984; Gaffet & Fukue 1983)
$\mu\equiv\frac{v_{r}-v_{F}}{v_{s}}=\frac{V-n\eta}{S}$ (32)
where
$v_{F}=\frac{dr}{dt}=n\frac{r}{t}$ (33)
is the velocity of the reference frame which is moving outward as time goes
by. The Mach number introduced so far, represents the _instantaneous_ and
_local_ Mach number of the unsteady self-similar flow. We will consider
transonic points of accretion flow in next subsection.
In order to consider the strength of the magnetic field in the plasma, the
$\beta$ parameter is introduced that is ratio of the magnetic to the gas
pressures
$\beta(r,t)=\frac{B^{2}_{\varphi}(r,t)/8\pi}{p(r,t)}=\frac{B^{2}(\eta)}{P(\eta)}.$
(34)
In completing this section, we also summarize the main results here. Solving
equations (1), (10), (11), and (19) under transformations (12)-(15) in non-
magnetically state, makes it clear that time behavior of physical quantities
in the non- magnetically and the magnetically disk are the same. This result
is one of the strictures of time-dependent self-similar solution. on the other
hand, the fact that timely- dependent behavior of the magnetic and gas
pressures becomes same is one of limits the self-similarity solution. On the
other hand, the physical quantities with a same physical dimension have
similar behaviors in self similar solution.
### 3.2 Asymptotic behavior
In this subsection, the asymptotic behavior of the equations (22), (26)-(30),
and (34) at $\eta\rightarrow 0$ and $\gamma<5/3$ is investigated. the
asymptotic solutions are given by
$R(\eta)\sim R_{0}\eta^{-3/2}$ (35) $P(\eta)\sim P_{0}\eta^{-5/2}$ (36)
$V(\eta)\sim V_{0}\eta^{-1/2}$ (37) $\omega(\eta)\sim\omega_{0}\eta^{-3/2}$
(38) $B(\eta)\sim B_{0}\eta^{-1/2}$ (39) $\dot{m}(\eta)\sim-4\pi R_{0}V_{0}$
(40) $\beta(\eta)\sim({B^{2}_{0}}/{P_{0}})\eta^{3/2}$ (41)
in which
$R_{0}=-\frac{3}{8\pi}\alpha
f\dot{m}_{in}\left(\frac{\gamma-1}{\gamma-5/3}\right)\left(\frac{g_{1}}{g_{3}}\right)$
(42) $P_{0}=\frac{\dot{m}_{in}}{6\pi\alpha}$ (43) $V_{0}=\frac{2}{3\alpha
f}\left(\frac{\gamma-5/3}{\gamma-1}\right)\left(\frac{g_{3}}{g_{1}}\right)$
(44) $\omega_{0}=-\frac{2}{3\alpha
f}\left(\frac{\gamma-5/3}{\gamma-1}\right)\left(\frac{g_{3}}{g_{1}}\right)^{1/2}$
(45) $B^{2}_{0}=\beta_{0}\frac{\dot{m}_{in}}{6\pi\alpha}$ (46)
where
$\frac{1}{g_{1}}=1-\frac{5f}{2}\left(\frac{\gamma-1}{\gamma-5/3}\right)$ (47)
$g_{2}=\frac{3}{2}\alpha f\left(\frac{\gamma-1}{\gamma-5/3}\right)$ (48)
$g_{3}=-1+\sqrt{1+2g^{2}_{1}g^{2}_{2}}$ (49)
$\beta_{0}=\beta_{in}/\eta^{3/2}_{in}.$ (50)
The achieved results for asymptotic behavior of physical quantities show that
the physical quantities of accretion flow are very sensible to parameters of
$\alpha$, $\gamma$, $f$, $\beta_{in}$, and $\dot{m}_{in}$. The $\beta_{in}$
and $\dot{m}_{in}$ are amounts of $\beta$ and $\dot{m}$ at $\eta_{in}$ that
$\eta_{in}$ is a point near of the center. The affects of the viscous
parameter $\alpha$ and the advection parameter $f$ on accretion flow are
plotted in figure 1. The angular velocity profiles indicate that by increasing
the viscous parameter $\alpha$, the angular velocity of accretion flow
decreases, because we increase the viscous torque by increasing parameter
$\alpha$. Also increasing the advection parameter $f$ decreases the angular
velocity that is qualitatively consistent with AF06. Figure 1 shows the radial
infall velocity increases by adding $\alpha$ and $f$ that are similar to the
results of AF06 and KF08. Also the density profiles represent density
decreases by adding $f$ and $\alpha$.
### 3.3 Numerical solutions
If the value of $\eta_{in}$ is guessed, that is a point very near to the
center, the equations can be integrated from this point to the outward through
the use of the above expansion. Examples of such solutions are presented in
figures 2, 3, and 4. The profiles in figure 2 are plotted for different
$\beta_{in}$, the profiles in figure 3 are plotted for different $f$ and in
figure 4 transonic behavior of the accreting gas for different amount of $f$
and $\beta_{in}$ is considered. The delineated quantities ($Log(\eta^{3/2}R)$,
$Log(-\eta^{1/2}V)$, …) in figures 2, 3, and 4 are constant in steady self-
similar solutions (Narayan & Yi 1994; Narayan & Yi 1995; Shadmehri 2004; AF06;
Ghanbari et. al. 2007; Abbassi et al. 2008), while here, they vary by
position.
Figure 2 informs us that density and the radial thickness of disk decreases by
adding strength of the toroidal magnetic field, these results are well
consistent with KF08. Also, by decreasing amount of magnetic field, the
behavior of density becomes similar to non-magnetic case (Ogilvie 1999). The
behavior of the gas pressure in KF08 had polytropic behavior and this
selection caused the gas pressure follow the density behavior, while here we
see behavior of the gas pressure does not follow the density behavior. Also,
by adding the $\beta$ parameter, the radial infall velocity increases; such
property is qualitatively consistent with AF07 and KF08. This is due to the
magnetic tension terms, which dominate the magnetic pressure term in the
radial momentum equation that assist the radial infall motion. The profiles of
the angular velocity imply that the disk is sub-Keplerican in inner part of
the disk and is super-Keplerian in outer part of it, while in polytropic
accreting flow (KF08) and non-magnetic accretion flow (Ogilvie 1999) the
angular velocity is sub-Keplerian in all radii (KF08). Similar to the results
KF08 the $\beta$ parameter, the ratio of the magnetic pressure to the gas
pressure, is a function of position and arises from inner to outer that the
result is well consistent with observational evidence obtained by some authors
(Aitken et al. 1993; Wright et al. 1993; Greaves et al. 1997). While the
$\beta$ parameter in steady self-similar solution becomes constant at all
radii (AF06) that is one of restriction of steady self-similar solution.
Figure 3 is plotted for different amounts of the advection parameter $f$. The
advection parameter $f$ has slight effect on the toroidal magnetic field, the
parameter of $\beta$, and the Mach number, however has outstanding effect on
the density, the gas pressure, the radial infall velocity, and the angular
velocity. The density and the radial thickness of disk decrease by more
advecting of accreting gas that is same at all part of the disk, the result
can be achieved by assuming of $f$ as a constant amount. Also we see by
increasing the amount of the advection parameter $f$, the gas pressure
decreases. By increasing $f$, the radial infall velocity increases and the
angular velocity decreases. The results are qualitatively consistent with the
results of AF06.
The Mach number profiles in figure 4 imply that the flow of outer part for all
selected amounts of the magnetic field become super sonic. We can see this
result in polytropic accretion flow by KF08. The advection parameter decreases
the amount of the Mach number slightly.
The profiles of physical quantities in figure 2 imply that they have the power
of law dependency to $\eta$ in magnetical domination ($\beta_{in}>1$). So, by
fitting a power function on data in magnetical domination ($\beta_{in}=10$),
we can write
$R(\eta)\propto\eta^{-1.66}$ (51) $P(\eta)\propto\eta^{-2.58}$ (52)
$V(\eta)\propto\eta^{-0.01}$ (53) $\omega(\eta)\propto\eta^{-1.25}$ (54)
$B(\eta)\propto\eta^{-0.83}$ (55) $\beta(\eta)\propto\eta^{-0.92}$ (56)
$\mu(\eta)\propto\eta^{0.93}$ (57) $\dot{m}(\eta)\propto\eta^{-0.33}.$ (58)
The achieved results are different with steady magnetical dominated accretion
flow (Meier 2005, Shadmehri & Khajenabi 2005).
## 4 Summary and Discussion
In the paper, the equations of time-dependent of advection dominated accretion
flow with a toroidal magnetic field have been solved by semi-analytical
similarity methods. The flow is not able to radiate efficiency, so we
substituted the energy equation instead of polytropic equation that KF08 had
used. A solution was found for the case $\gamma<5/3$ that has differential
rotation and viscous dissipation. The flow avoids many of the strictures of
steady self-similar solutions (Narayan & Yi 1994; AF06; Ghanbari et al. 2007;
Abbassi et al. 2008). Thus, the radial-dependence of calculated physical
quantities in this approach are different from steady self-similar solution.
Increase of the advection parameter $f$ and the parameter $\beta_{in}$ will
separately increase the infall radial velocity and decrease the angular
velocity. The flow has differential rotation and is sub-Keplerian in inner
part and is super-Keplerian in large radii in which the behavior is seen in
some astrophysical objects such as M81, M87 and Milky Way (Sofue 1998; Ford &
Tsvetanov 1999). The solution showed that the flow for all selected amounts of
$f$ and $\beta_{in}$ becomes super sonic in large radii and sub-sonic in small
radii that are qualitatively consistent with the results of KF08. The
parameter of $\beta$ is a function of position that raises from inner to outer
and states the magnetic field is more important in large radii. It is also
consistent with observational evidences in the outer regions of YSO discs
(Greaves et al. 1997; Aitken et al. 1993; Wright et al. 1993) and in the
Galactic center (Novak et al. 2003; Chuss et al. 2003).
Here, latitudinal dependence of physical quantities is ignored, while some
authors showed that latitudinal dependence is important in the structure of a
disk (Narayan & Yi 1995; Ghanbari et. al. 2007). Latitudinal behavior of such
disks can be investigated in other studies. Also we did not consider
relativity effect, If the central object is relativistic, the gravitational
field should be changed. Furthermore, in a realistic model the advection
parameter $f$ is a function of position and time, other researchers can
consider such disks.
## Acknowledgments
We wish to thank the anonymous referee for his/her very constructive comments
which helped us to improve the initial version of the paper; we would also
like to thank Wilhelm Kley and Serena Arena for their helpful discussion.
## References
* [1]
* [2] [] Abbassi, S., Ghanbari, J., Najjar, S., 2008, MNRAS, 388, 663
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* [4] [] Abramowicz, M., Chen, X., Kato, S., Lasota, J. P., Regev, O., 1995, ApJ, 438, L37
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* [16] [] Ford, H., Tsvetanov, Z., 1999, in The Radio Galaxy Messier 87, ed. H.-J. Röser & K. Meisenheimer (Berlin: Springer), 278
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* [22] [] Ghanbari, J., Salehi, F., Abbassi, S., 2007, MNRAS, 381, 159
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* [24] [] Greaves, J. S., Holland, W. S., Ward-Thompson D., 1997, ApJ, 480, 255
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* [26] [] Ichimaru, S., 1977, ApJ, 214, 840
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* [32] [] Narayan, R., Yi, I., 1994, ApJ, 428, L13
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* [34] [] Narayan, R., Yi, I., 1995, ApJ, 452, 710
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* [44] [] Shadmehri, M., Khajenabi, F. 2005, MNRAS, 361, 719
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* [46] [] Sofue, Y., 1998, PASJ, 50, 227
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* [49]
Figure 1: Numerical coefficient $\omega_{0}$ (dotted lines), $R_{0}$ (solid
lines) and $V_{0}$ (dashed lines) as functions of advection parameter $f$ or
the the viscous parameter $\alpha$. The ratio of specific heats is set to be
$\gamma=1.5$ and the inner mass accretion rate is $\dot{m}_{in}=0.001$.
Figure 2: Time-dependent self-similar solution for $\gamma=1.5$, $\alpha=0.5$,
$f=1.0$, and $\dot{m}_{in}=0.001$. The lines represent
$\beta_{in}=0.1,0.5,1.0,10$ that $\beta_{in}$ is value of $\beta$ in
$\eta_{in}$.
Figure 3: Time-dependent self-similar solution for $\gamma=1.5$, $\alpha=0.5$,
$\beta_{in}=1.0$, and $\dot{m}_{in}=0.001$. lines represent $f=0.1,0.5,1.0$.
Figure 4: Left panel: Mach number profiles for $\gamma=1.5$, $\alpha=0.5$,
$f=1.0$, and $\dot{m}_{in}=0.001$. Right panel: Mach number profiles for
$\gamma=1.5$, $\alpha=0.5$, $\beta_{in}=1.0$, and $\dot{m}_{in}=0.001$.
|
arxiv-papers
| 2011-10-19T06:14:59 |
2024-09-04T02:49:23.357165
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alireza Khesali, Kazem Faghei",
"submitter": "Kazem Faghei",
"url": "https://arxiv.org/abs/1110.4191"
}
|
1110.4240
|
# Strange and identified hadron production at the LHC with ALICE
L. S. Barnby for the ALICE Collaboration
###### Abstract
The ALICE detector was designed to identify hadrons over a wide range of
transverse momentum at mid-rapidity. Here measurements of light charged
($\pi$, $\mathrm{K}$, p) and neutral ($\mathrm{\Lambda}$,
$\mathrm{K^{0}_{S}}$) hadrons in Pb–Pb collisions at $\sqrt{s_{\mathrm{NN}}}$
= 2.76 ${\rm TeV}$ are presented with additional data from a pp reference at
$\sqrt{s}=7$ ${\rm TeV}$. Such measurements are crucial for understanding the
properties of the fireball produced in heavy-ion collisions at the LHC. The
particle-type dependence of the spectra and the yields of particles extracted
give information on the expansion dynamics and chemical composition
respectively. In addition studying the ratio of baryons to mesons may help in
understanding the mechanisms by which hadronisation takes place. We find that,
when comparing to data from $\sqrt{s_{\mathrm{NN}}}$ = 200 ${\rm GeV}$ Au+Au
collisions at RHIC, a more strongly expanding system is created with a similar
relative population of hadron species. We also see that collective effects or
complex mechanisms responsible for a relative enhancement of baryons have an
influence at a much higher $p_{\mathrm{T}}$ than was previously seen.
###### Keywords:
QGP, hadron production
###### :
25.75-q,25.75.Dw,13.85.Hd
## 1 Introduction
The aim of relativistic heavy-ion collision experiments is to detect and
understand the properties of the bulk QCD matter created in these collisions.
Past experiments have shown that the ability to perform measurements
differentially with respect to the identity of the final state hadrons is
crucial to a full understanding of the evolution and dynamics of the produced
fireball Braun-Munzinger et al. (1995, 1996); Kolb and Rapp (2003). Such
measurements have also revealed anomalies challenging the detailed modelling
of the collision Adler et al. (2003); Adams et al. (2006). The ALICE
experiment was designed with the goal of maximising the particle
identification capability using transition and Cherenkov radiation detectors,
calorimetry and, in particular for the analysis presented here, identifying
the most abundant species of charged hadrons over a wide range of
$p_{\mathrm{T}}$ at mid-rapidity using $\mathrm{d}E/\mathrm{d}x$ and time-of-
flight techniques Aamodt et al. (2008). The excellent tracking down to low
$p_{\mathrm{T}}$ also allows the reconstruction of weakly decaying neutral
strange particles via their charged decay modes.
## 2 Experiment
The ALICE central barrel performs tracking of charged particles in a 0.5 T
magnetic field using a Time Projection Chamber (TPC) and Inner Tracking System
(ITS). Particles with large enough $p_{\mathrm{T}}$ pass through the outer
wall of the TPC and can go on to hit a surrounding Time-of-Flight detector
(TOF). Pb–Pb events were collected using a minimum bias trigger and several
million events are used in this analysis. The Pb–Pb data sample can be
separated into centrality bins using the event-wise multiplicity in the VZERO
forward scintillator detectors in combination with a Monte Carlo Glauber study
Aamodt et al. (2011a). Charged particle identification is achieved using two
techniques. The specific energy loss, $\mathrm{d}E/\mathrm{d}x$, can be
calculated for each track from the ionisation in the TPC gas (or ITS silicon)
and compared to theoretical values from the Bethe-Bloch formula which predicts
the regions in momentum where $\pi$, $\mathrm{K}$, and p signals can be
separated. This separation between species can be used at low $p_{\mathrm{T}}$
but near to the minimum of $\mathrm{d}E/\mathrm{d}x$ all three species are
merged. In this range however the TOF can separate these species so a combined
$p_{\mathrm{T}}$ spectrum can be extracted Aamodt et al. (2010a). In this
analysis the primary yield of charged particles is reported; that is those
emerging directly from the collision or the decay of short-lived resonances
and not the charged particles from the weak decay of strange hadrons nor
secondaries from the material. These are both excluded using the distribution
of the distance of closest approach to the primary interaction vertex, which
can be fitted to a template obtained from Monte Carlo events, where the origin
of the particle is known. The decay of neutral strange particles decaying into
charged daughters; $\Lambda\rightarrow p\mbox{$\mathrm{\pi^{-}}$}$ and
$\mbox{$\mathrm{K^{0}_{S}}$}\rightarrow\mbox{$\mathrm{\pi^{+}}$}\mbox{$\mathrm{\pi^{-}}$}$,
can be reconstructed and the invariant mass distributions used for
identification. The analysis follows the method used for $\mathrm{pp}$
collisions but tighter cuts are made to further reduce the combinatorial
background in Pb–Pb events Aamodt et al. (2011b). For both neutral and charged
particle analyses the spectra are corrected using efficiencies from Monte
Carlo events having equivalent mean multiplicities.
## 3 Results
### 3.1 Charged Particles
The combined $p_{\mathrm{T}}$ spectra are obtained for each of eight
centrality bins for $\mathrm{\pi^{\pm}}$, $\mathrm{K^{\pm}}$, p and
$\mathrm{\overline{p}}$ and are shown, for positive particles only, in figure
1. The most noticeable features are: the dramatic change in the shape of the
spectra going from $\pi$ through $\mathrm{K}$ to p; and the shifting of the
most probable values to higher $p_{\mathrm{T}}$, particularly for p but also
for $\mathrm{K}$. A direct comparison of the most central spectra to Au–Au
data at $\sqrt{s_{\mathrm{NN}}}$ = 200 ${\rm GeV}$ is made in figure 2. This
shows how the spectra at LHC energy are much less steeply falling. A first
attempt at quantifying the changes using a parameterised blast wave function
Schnedermann et al. (1993) was made. The resulting fit parameters for the
freeze-out temperature, $T_{\mathrm{fo}}$, and mean transverse velocity,
$\mathrm{\beta}$, are shown in figure 3 as 1-$\sigma$ contours for each
centrality class. Fits ranges 0.3–1.0 GeV$\kern-1.49994pt/\kern-1.19995ptc$,
0.2–1.5 GeV$\kern-1.49994pt/\kern-1.19995ptc$ and 0.3–3.0
GeV$\kern-1.49994pt/\kern-1.19995ptc$ were used for $\pi$, $\mathrm{K}$ p
respectively in order to avoid the region where a hard component of the
spectum might be expected and, at low $p_{\mathrm{T}}$, to avoid a strong
contribution of resonances to $\pi$. There appears to be a larger
$\mathrm{\beta}$, corresponding to stronger flow, than observed by STAR at
lower energy Adams et al. (2005). $T_{\mathrm{fo}}$ is very sensitive to the
fit range so any change with respect to RHIC needs further study. A blast wave
was also fitted to each individual spectrum in order to obtain
$p_{\mathrm{T}}$-integrated yields, including the unmeasured part. These can
be used to form the ratios p/$\pi$ and $\mbox{$\mathrm{K}$}/\pi$ for each
centrality bin. The ratio p/$\pi$ is almost constant with centrality and is
consistent with similar measurements in Au–Au collisions at RHIC Adler et al.
(2004). The ratio $\mbox{$\mathrm{K}$}/\pi$ shows a small rise from
$\mathrm{pp}$ and peripheral collisions to the most central collisions and is
also consistent with previous lower energy data Abelev et al. (2009).
Figure 1: The centrality-selected $p_{\mathrm{T}}$ spectra for identified
$\mathrm{\pi^{+}}$(top) $\mathrm{K^{+}}$(middle) and p (bottom). Fits are to a
parameterised blast wave.
Figure 2: The $p_{\mathrm{T}}$ spectra for $\mathrm{\pi^{-}}$,
$\mathrm{K^{0}_{S}}$, $\mathrm{K^{-}}$, and $\mathrm{\overline{p}}$ for the
most central Pb–Pb (0-5%) collisions (solid markers) plotted with those
measured in $\sqrt{s_{\mathrm{NN}}}$ = 200 ${\rm GeV}$Au–Au collisions (open
symbols.) Figure 3: 1-$\sigma$ contours in the T-$\mathrm{\beta}$ plane from a
simultaneous fit of a parameterised blast wave function to the
$\mathrm{\pi^{\pm}}$, $\mathrm{K}$, and p $p_{\mathrm{T}}$ spectra for various
centrality classes. Pb–Pb collisions from the ALICE experiment in red, Au–Au
collisions from the STAR experiment in blue. Most central data lie to the
right.
### 3.2 Neutral Particles
The $p_{\mathrm{T}}$ spectra of $\mathrm{\Lambda}$ and $\mathrm{K^{0}_{S}}$
have also been extracted for each centrality bin. As the systematic
uncertainties on the efficiency correction are still under study the
preliminary spectra themselves are not yet ready. However the study reveals
that the ratio of the efficiencies for each particle, as a function of
$p_{\mathrm{T}}$ , is rather stable with respect to changing the centrality of
the collision. In particular in the $p_{\mathrm{T}}$ range 2.5-5.5
GeV$\kern-1.49994pt/\kern-1.19995ptc$ the variation of the efficiency ratio
between the most central and the most peripheral centrality selections is
below 2%. This allows the $\mathrm{\Lambda}$/ $\mathrm{K^{0}_{S}}$ ratio to be
calculated with an estimated systematic uncertainty of 10% and the resulting
curves for each centrality are shown in figure 4 (upper). Also shown are the
ratios in $\mathrm{pp}$ collisions at $\sqrt{s}=0.9$ and 7 ${\rm TeV}$ Aamodt
et al. (2011b). The $\mathrm{pp}$ data demonstrate that in the ${\rm TeV}$
range the maximum value of the ratio is almost constant and it is reasonable
to assume that $\mathrm{pp}$ collisions at $\sqrt{s}=2.76$ ${\rm TeV}$ would
show the same maximum. Taking this $\mathrm{pp}$ baseline the ratio is
observed to have a maximum which rises strongly going to peripheral and then
to central events, with a total increase up to a factor of three. The value of
$p_{\mathrm{T}}$ at which the maximum is reached is also increasing by several
hundred MeV$\kern-1.49994pt/\kern-1.19995ptc$. The data are compared to a
similar measurement previously made by STAR in figure 4 (lower) Lamont and the
STAR Collaboration (2006). To facilitate the comparison the lower energy data
were scaled by the $\mathrm{\overline{\Lambda}}$/$\mathrm{\Lambda}$ ratio
measured for each centrality Adams et al. (2007), assuming it is constant in
$p_{\mathrm{T}}$, because it has previously been noted that there is a
$\sqrt{s}$-dependence of the ratio Aggarwal et al. (2011), presumably due to
the change in the baryo-chemical potential. The ALICE data were not scaled in
this way because the anti-baryon/baryon ratio in LHC collisions is very close
to one Aamodt et al. (2010b). The ratio in peripheral 60-80% collisions is
very similar in shape for the two collision systems with only a small change
in the magnitude. In the most central 0-10% however the shape is quite
different with the enhancement of the $\mathrm{\Lambda}$ extending to a much
larger $p_{\mathrm{T}}$ in the higher energy data. This is qualitatively in
agreement with some predictions Fries and Müller (2004).
Figure 4: (Upper panel.) The ratio of $\mathrm{\Lambda}$ to
$\mathrm{K^{0}_{S}}$ as a function of $p_{\mathrm{T}}$ for five centrality
classes in Pb–Pb collisions. Also shown the same ratio in $\mathrm{pp}$
collisions at two energies. (Lower panel.) A comparison between the ratio
measured by ALICE (solid markers) and STAR (open symbols) for selected
centralities.
## 4 Conclusions
Pb–Pb collisions at $\sqrt{s_{\mathrm{NN}}}$ = 2.76 ${\rm TeV}$ reveal a
number of similarities to Au–Au collisions at RHIC; the ratios of the yields
are the same within the experimental uncertainties, the spectra are compatible
with a strong collective motion which increases going to more central
collisions and there is a growth of the $\mathrm{\Lambda}$/
$\mathrm{K^{0}_{S}}$ ratio in the $p_{\mathrm{T}}$ region 2-4
GeV$\kern-1.49994pt/\kern-1.19995ptc$, also with centrality. There are however
some notable differences; the $p_{\mathrm{T}}$ spectra are much flatter giving
a transverse flow velocity in a blast wave parameterisation 10% larger than
that in $\sqrt{s_{\mathrm{NN}}}$ = 200 ${\rm GeV}$Au–Au collisions and the
enhanced baryon-to-meson ratio extends to a $p_{\mathrm{T}}$ of around 6
GeV$\kern-1.49994pt/\kern-1.19995ptc$. This may imply that the influence of
particles participating in the collective dynamics of the system extends to a
higher $p_{\mathrm{T}}$ than has previously been observed.
## References
* Braun-Munzinger et al. (1995) P. Braun-Munzinger, J. Stachel, J. Wessels, and N. Xu, _Physics Letters B_ 344, 43 – 48 (1995), URL http://www.sciencedirect.com/science/article/pii/037026939401534J.
* Braun-Munzinger et al. (1996) P. Braun-Munzinger, J. Stachel, J. Wessels, and N. Xu, _Physics Letters B_ 365, 1 – 6 (1996), URL http://www.sciencedirect.com/science/article/pii/0370269395012583.
* Kolb and Rapp (2003) P. F. Kolb, and R. Rapp, _Phys. Rev. C_ 67, 044903 (2003), URL http://link.aps.org/doi/10.1103/PhysRevC.67.044903.
* Adler et al. (2003) S. S. Adler, et al., _Phys. Rev. Lett._ 91, 172301 (2003), URL http://link.aps.org/doi/10.1103/PhysRevLett.91.172301.
* Adams et al. (2006) J. Adams, et al. (2006), nucl-ex/0601042.
* Aamodt et al. (2008) K. Aamodt, et al., _Journal of Instrumentation_ 3, S08002 (2008), URL http://stacks.iop.org/1748-0221/3/i=08/a=S08002.
* Aamodt et al. (2011a) K. Aamodt, et al., _Phys. Rev. Lett._ 106, 032301 (2011a), URL http://link.aps.org/doi/10.1103/PhysRevLett.106.032301.
* Aamodt et al. (2010a) K. Aamodt, et al., _Physics Letters B_ 693, 53–68 (2010a).
* Aamodt et al. (2011b) K. Aamodt, et al., _The European Physical Journal C_ 71, 1–24 (2011b), URL http://dx.doi.org/10.1140/epjc/s10052-011-1594-5.
* Schnedermann et al. (1993) E. Schnedermann, J. Sollfrank, and U. Heinz, _Phys. Rev. C_ 48, 2462–2475 (1993), URL http://link.aps.org/doi/10.1103/PhysRevC.48.2462.
* Adams et al. (2005) J. Adams, et al., _Nuclear Physics A_ 757, 102 – 183 (2005), URL http://www.sciencedirect.com/science/article/pii/S0375947405005294.
* Adler et al. (2004) S. S. Adler, et al., _Phys. Rev. C_ 69, 034909 (2004), URL http://link.aps.org/doi/10.1103/PhysRevC.69.034909.
* Abelev et al. (2009) B. I. Abelev, et al., _Phys. Rev. C_ 79, 034909 (2009), URL http://link.aps.org/doi/10.1103/PhysRevC.79.034909.
* Lamont and the STAR Collaboration (2006) M. A. C. Lamont, and the STAR Collaboration, _Journal of Physics G: Nuclear and Particle Physics_ 32, S105 (2006), URL http://stacks.iop.org/0954-3899/32/i=12/a=S13.
* Adams et al. (2007) J. Adams, et al., _Phys. Rev. Lett._ 98, 062301 (2007), URL http://link.aps.org/doi/10.1103/PhysRevLett.98.062301.
* Aggarwal et al. (2011) M. M. Aggarwal, et al., _Phys. Rev. C_ 83, 024901 (2011).
* Aamodt et al. (2010b) K. Aamodt, et al., _Phys. Rev. Lett._ 105, 072002 (2010b).
* Fries and Müller (2004) R. Fries, and B. Müller, _The European Physical Journal C - Particles and Fields_ 34, s279–s285 (2004), URL http://dx.doi.org/10.1140/epjcd/s2004-04-026-6.
|
arxiv-papers
| 2011-10-19T11:19:18 |
2024-09-04T02:49:23.365567
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "L. S. Barnby (for the ALICE Collaboration)",
"submitter": "Lee Barnby",
"url": "https://arxiv.org/abs/1110.4240"
}
|
1110.4357
|
# Non-WKB Models of the FIP Effect: The Role of Slow Mode Waves
J. Martin Laming Space Science Division, Naval Research Laboratory Code
7674L, Washington, D.C. 20375
###### Abstract
A model for element abundance fractionation between the solar chromosphere and
corona is further developed. The ponderomotive force due to Alfvén waves
propagating through, or reflecting from the chromosphere in solar conditions
generally accelerates chromospheric ions, but not neutrals, into the corona.
This gives rise to what has become known as the First Ionization Potential
(FIP) Effect. We incorporate new physical processes into the model. The
chromospheric ionization balance is improved, and the effect of different
approximations is discussed. We also treat the parametric generation of slow
mode waves by the parallel propagating Alfvén waves. This is also an effect of
the ponderomotive force, arising from the periodic variation of the magnetic
pressure driving an acoustic mode, which adds to the background longitudinal
pressure. This can have subtle effects on the fractionation, rendering it
quasi-mass independent in the lower regions of the chromosphere. We also
briefly discuss the change in the fractionation with Alfvén wave frequency,
relative to the frequency of the overlying coronal loop resonance.
Sun:abundances – Sun:chromosphere – turbulence – waves
## 1 Introduction
The First Ionization Potential (FIP) effect is the by now well known
enhancement in abundance by a factor of 3-4 over photospheric values of
elements in the solar corona with FIP less than about 10 eV. These low FIP
elements include Fe, Si, Mg, etc. Elements with FIP greater than 10 eV (high-
FIP) mostly retain their photospheric composition. This was actually first
observed in the 1960’s (Pottasch 1963), making it nearly as old as the problem
of coronal heating. It has been taken seriously as a phenomenon since the mid
1980’s (Meyer 1985ab). There are a number of studies of the FIP effect in
different regions of the solar corona and wind, reviewed in Feldman & Laming
(2000) and Laming (2004), and references therein. With the launch of the
Extreme Ultraviolet Explorer (EUVE) in the 1990’s, it became clear that
element abundances in late-type stellar coronae also do not always resemble
the stellar photospheric composition. The FIP effect is also observed in many
solar-like late type stars. At higher activity levels and/or later spectral
types, a so called “inverse FIP” effect is observed, where the low FIP
elements are depleted in the corona (e.g. Wood & Linsky, 2010).
A variety of models have been proposed to explain these phenomena. Laming
(2004, 2009) review many of these different scenarios, and argue that the
ponderomotive force is the most likely agent of FIP fractionation. This force
arises as Alfvén waves propagate through the chromosphere, and acts on
chromospheric ions, but not neutrals. Physically, it corresponds to the
interaction of waves and plasma through the refractive index of the medium.
Waves carrying significant energy and momentum refracting or reflecting in a
plasma must exert a force, in this case on the charged particles that
contribute to the dielectric tensor, but not on the neutrals. The
ponderomotive force in the chromosphere may in principle be directed upwards
or downwards, giving rise to FIP or so-called “inverse FIP” effects
respectively (i.e. a coronal enhancement or depletion of low FIP ions).
The chromosphere-corona interface is generally a barrier to Alfvén wave
propagation; upcoming waves from the chromosphere are usually reflected back
down again, and downward directed waves from the corona reflect back upwards,
as illustrated in the right hand footpoint (chromosphere B) in Figure 1.
Alfvén waves with predominantly coronal origin generally give rise to the
positive (i.e. solar-like) FIP effect, while waves generated by upward
propagating acoustic waves associated with stellar convection may produce
inverse FIP effect. The association of coronal abundance anomalies with Alfvén
waves gives us a unique and unexpected diagnostic with which to explore the
behavior of MHD turbulence in solar and stellar upper atmospheres. While we
will argue that the coronal Alfvén waves themselves are actually byproducts of
processes that heat solar and stellar coronae, (most likely accomplished by
various forms of “nanoflares”), an understanding of coronal abundance
anomalies still becomes far more central to exploring coronal heating than
would be the case in prior models for the fractionation invoking thermal
processes such as diffusion.
In this paper we seek to develop the ponderomotive force model incorporating
the parametric generation of parallel propagating slow mode waves by the
Alfvén waves themselves, together with revisions to some of the atomic data.
First, in section 2, we review the important features of the Laming (2004,
2009) model. Section 3 outlines the various theoretical refinements made in
this paper, and section 4 describes new results for fractionations in open and
closed magnetic field configurations. Section 5 discusses in more detail the
effect of the parametric generation of slow mode waves and its implications
for fractionation in closed coronal loops and in the slow speed solar wind. We
also consider limits on the upward flow speed through the chromosphere, and
make final conclusions.
## 2 The Ponderomotive Model Revisited
The ponderomotive force stems from second order terms in $\delta{\bf
J}\times\delta{\bf B}/c$ and $\rho\delta{\bf v}\cdot\nabla\delta{\bf v}$ in
the MHD momentum equation. It can be manipulated (e.g Litwin & Rosner 1998,
Laming 2009) for waves of frequency $\omega_{A}<<\Omega$, the ion
gyrofrequency, into the time averaged form
$F={\partial\over\partial z}\left(q^{2}\delta E_{\perp}^{2}\over
4m\Omega^{2}\right)={mc^{2}\over 4}{\partial\over\partial z}\left(\delta
E_{\perp}^{2}\over B^{2}\right),$ (1)
where $\delta E_{\perp}$ is the perpendicular wave electric field and $q$ is
the ion charge. The ponderomotive acceleration, $F/m$, is independent of the
ion mass. The Laming (2004, 2009) model comes about as a natural extension of
existing work on Alfvén wave propagation in the solar atmosphere with
essentially no extra physics required. It is also the model most worked out in
detail to interpret observations, giving it unique potential for diagnosing
wave processes in the corona and chromosphere.
The basic model builds on Hollweg (1984), where upward propagating Alfvén
waves were introduced at one loop footpoint. Here they could reflect back down
into the chromosphere, or be transmitted into the loop, where they propagated
back and forth with a small probability of leaking back into the chromosphere
at each end. With reference to Figure 1, we initiate our simulations with one
downward propagating wave at the $\beta\sim 1.2$ layer in chromosphere A, and
integrate the Alfvén wave transport equations (see below) to chromosphere B to
evaluate the standing wave pattern there. The chromosphere at each footpoint
can be based on any of the Vernazza, Avrett, & Loeser (1981) models or
similar. Here we use the Avrett & Loeser (2008) update of VALC. The ionization
balance of the minor ions is computed at each height in the chromosphere using
the model temperature and electron density, and a coronal UV-X-ray spectrum
appropriately absorbed in the intervening chromospheric layers. We have tried
a number of different spectra based on Vernazza & Reeves (1978), or model
flare spectra computed using CHIANTI (see e.g. Huba et al., 2005, for the 2000
Bastille Day flare). The atomic data are as in Laming (2004) and Laming
(2009), with estimates for the charge exchange ionization for Si, Fe, and
other low FIP elements added (see subsection 3.3). The chromospheric magnetic
field is taken to be a 2D force free field from Athay (1981) and designed to
match chromospheric magnetic fields in Gary (2001) and Campos & Mendes (1995),
which represents the expansion of the field from the high $\beta$ photosphere
where the field is concentrated into small network segments, into the low
$\beta$ chromosphere where the field expands to fill much more of the volume.
We model the Alfvén waves in a non-WKB approximation. The procedure follows
that described in detail by Cranmer & van Ballegooijen (2005), but applied to
closed rather than open magnetic field structures. The curvature of the loop
is neglected. For Alfvén waves, where the energy flux is necessarily directed
along the field direction, this is unlikely to have a significant effect. Some
extra damping may result as the wave develops a component of its wavevector
perpendicular to the field, but we neglect this and all other damping
mechanisms in this work. The transport equations are
${\partial z_{\pm}\over\partial t}+\left(u\pm V_{A}\right){\partial
z_{\pm}\over\partial r}=\left(u\pm V_{A}\right)\left({z_{\pm}\over
4H_{D}}+{z_{\mp}\over 2H_{A}}\right),$ (2)
where $z_{\pm}=\delta v_{\perp}\pm\delta B_{\perp}/\sqrt{4\pi\rho}$ are the
Elsässer variables for Alfvén waves propagating against or along the magnetic
field respectively, and are valid for torsional or planar Alfvén waves. The
Alfvén speed is $V_{A}$, the upward flow speed is $u$ and the density is
$\rho$. The signed scale heights are $H_{D}=\rho/\left(\partial\rho/\partial
r\right)$ and $H_{A}=V_{A}/\left(\partial V_{A}/\partial r\right)$. In the
solar chromosphere and corona $u<<V_{A}$, and we put $u=0$. The calculation of
$V_{A}$ uses the total (ionized and neutral) gas density, since the wave
frequencies of interest here are well below the charge exchange rate, and
neutrals are well coupled to the wave. Charge changing collisions involving
electrons (impact ionization, and radiative and dielectronic recombination)
are generally slower than charge exchange in chromospheric conditions. Hence
in considering the wave propagation, ion-neutral friction is neglected, though
it is included in the evaluation of the fractionation.
The fractionations are calculated by postprocessing the non-WKB wave. This is
valid because the fractionation has a negligible effect on the wave
propagation. The degree of fractionation is given by the formula
${\rm
fractionation}=\exp\left(2\int_{z_{l}}^{z_{u}}{\xi_{s}a\nu_{eff}/\nu_{s,i}/v_{s}^{2}}dz\right)$
(3)
(see equation 9, Laming 2009, equation 12, Laming 2004, which follow Schwadron
et al. 1999), where $\xi_{s}$ is the ionization fraction of element $s$, $a$
is the ponderomotive acceleration,
$\nu_{eff}=\nu_{s,i}\nu_{s,n}/\left[\xi_{s}\nu_{s,n}+\left(1-\xi_{s}\right)\nu_{s,i}\right]$
where $\nu_{s,i}$ and $\nu_{s,n}$ are the collision rates of ions and
neutrals, respectively, of element $s$ with the ambient gas. Also
$v_{s}^{2}=kT/m_{s}+v_{\mu turb}^{2}+v_{turb}^{2}$, where $v_{\mu turb}$ is
the amplitude of microturbulence coming from the Avrett & Loeser (2008)
chromospheric model, and $v_{turb}$, discussed further in section 3.2, is the
amplitude of longitudinal waves induced by the Alfvén waves themselves. The
limits of integration, $z_{l}$ and $z_{u}$ are the lower and upper limits over
which the ponderomotive force acts. We take $z_{l}$ to be where $\beta\simeq
1$, and $z_{u}$ is in the transition region at an altitude where all elements
are ionized.
## 3 New Model Features
### 3.1 Slow Mode Waves
We have introduced an extra longitudinal pressure associated with the Alfvén
waves proportional to $v_{turb}^{2}$, which has the effect of causing some
saturation of the FIP fractionation. Here we give the physical motivation for
this extra term, which arises from the generation of slow mode waves.
Physically, the periodic variation in magnetic pressure of the Alfvén wave
drives longitudinal compressional waves. These generated acoustic waves can
act back on the Alfvén driver, as the compressional wave introduces a periodic
variation in the Alfvén speed, which generates new Alfvén waves. We illustrate
the generation of slow mode or acoustic waves by the ponderomotive force
associated with plane Alfvén waves with a simple 1D calculation. The
linearized momentum equation keeping terms to all orders in perturbed
quantities is (all symbols have their usual meanings),
$\left(\rho+\delta\rho\right){\partial\delta v_{z}\over\partial
t}+\left(\rho+\delta\rho\right)\delta v_{z}{\partial\delta v_{z}\over\partial
z}=\left(\rho+\delta\rho\right){\partial\over\partial z}{\delta B^{2}\over
8\pi\left(\rho+\delta\rho\right)}-{\partial\delta P\over\partial
z}-g\delta\rho,$ (4)
where
$\displaystyle\delta\rho$
$\displaystyle=-\rho\nabla\cdot{\bf\xi}-\xi_{z}{\partial\rho\over\partial
z}=-\rho\xi\left(ik_{s}+{1\over L_{\rho}}\right)$ (5) $\displaystyle\delta P$
$\displaystyle=-\gamma P\nabla\cdot{\bf\xi}-\xi_{z}{\partial P\over\partial
z}=-P\xi\left(ik_{s}\gamma+{1\over L_{P}}\right)$ (6)
for Eulerian displacement ${\bf\xi}\propto\exp
i\left(\omega_{s}t+k_{s}z\right)$, where $L_{p}=P/\left(\partial P/\partial
z\right)$ and $L_{\rho}=\rho/\left(\partial\rho/\partial z\right)$ (signed)
pressure and density scale heights respectively. The first term on the right
hand side of equation 4 represents the instantaneous ponderomotive force. In
cases where the WKB approximation applies, $\delta B\propto\rho^{1/4}$, and
this expression is equivalent to the more usual form $-\partial\left(\delta
B^{2}/8\pi\right)/\partial z$. Substituting for $\delta\rho$ and $\delta P$
from equations 5, keeping terms as high as second order in small quantities,
equation 3 becomes
$-i{\omega_{s}\over L_{\rho}}\delta
v_{z}^{2}+\left(-\omega_{s}^{2}+k_{s}^{2}c_{s}^{2}-i{k_{s}c_{s}^{2}\over
L_{P}}-{c_{s}^{2}\over\gamma L_{P}^{2}}-i{k_{s}c_{s}^{2}\over\gamma
L_{P}}-ik_{s}g-{g\over L_{\rho}}\right)\delta
v_{z}-i\omega_{s}{\partial\over\partial z}{\delta B^{2}\over 8\pi\rho}=0.$ (7)
This is considerably simplified in isothermal conditions, $\gamma=1$,
$L_{P}=L_{\rho}=-c_{s}^{2}/g$, so that
$-i{\omega_{s}\over L_{\rho}}\delta
v_{z}^{2}+\left(-\omega_{s}^{2}+k_{s}^{2}c_{s}^{2}+ik_{s}g\right)\delta
v_{z}-i\omega_{s}{\partial\over\partial z}{\delta B^{2}\over 8\pi\rho}=0.$ (8)
We put $\Im k_{s}=-g/2c_{s}^{2}$, and $\sqrt{\left(\Re
k_{s}\right)^{2}+g^{2}/4c_{s}^{4}}=2\Re k_{A}/n$, $\omega=2\omega_{A}/n$,
where $k_{A}$ and $\omega_{A}$ are the wavevector and angular frequency of the
Alfvén wave with $n=1,2,3,..$ (anticipating the result below). We find
$\delta v_{z}^{2}-\delta v_{z}iL_{\rho}\omega_{s}\left(1-{c_{s}^{2}\over
V_{A}^{2}}\right)+L_{\rho}{\partial\over\partial z}{\delta B^{2}\over
8\pi\rho}=0$ (9)
with solution
$\delta v_{z}={-i\over 2}\left[\sqrt{\delta
v_{A}^{2}+L_{\rho}^{2}\omega_{s}^{2}\left(1-{c_{s}^{2}\over
V_{A}^{2}}\right)^{2}}-L_{\rho}\omega_{s}\left(1-{c_{s}^{2}\over
V_{A}^{2}}\right)\right]$ (10)
where we have put ${\partial\over\partial z}\left(\delta
B^{2}/8\pi\rho\right)=\left(\delta B^{2}/4\pi\rho\right)/4L_{\rho}=\delta
v_{A}^{2}/4L_{\rho}$ using the WKB result for Alfvén waves in a density
gradient, and assuming $1/L_{\rho}>>\Re k_{A}$. When $c_{s}^{2}\sim V_{A}^{2}$
or $L_{\rho}\rightarrow 0$, $\delta v_{z}\simeq-i\delta v_{A}/2$. In the
opposite limit $\delta v_{z}\simeq-i\delta
v_{A}^{2}/4L_{\rho}\omega_{s}\left(1-c_{s}^{2}/V_{A}^{2}\right)$. In these two
cases $\omega_{s}=\omega_{A}$ or $\omega_{s}=2\omega_{A}$ respectively. In
fact acoustic waves can be generated with $\omega_{s}=2\omega_{A}/n$, with
higher $n$ becoming more intense as the nonlinearity increases (Landau &
Lifshitz, 1976). Vasheghani Farahani et al. (2011) treat the case of slow mode
wave generation by a torsional Alfvén wave. This is subtly different to the
case of a plane Alfvén wave considered here, and the FIP fractionation
resulting from such a model will be investigated in a future paper.
Anticipating applications to possibly nonlinear Alfvén and slow mode wave
amplitudes, we revisit the analysis above retaining more higher order terms.
From $\delta\rho=-\rho\xi\left(ik_{s}+1/L\right)$ we derive
${\partial\delta\rho\over\partial z}=\delta\rho\left(ik_{s}+{1\over
L}\right).$ (11)
which when substituted into the linearized continuity equation,
${\partial\delta\rho\over\partial t}+{\partial\over\partial z}\left(\rho\delta
v_{z}+\delta\rho\delta v_{z}\right)=0,$ (12)
with the time derivatives $\partial\delta v_{z}/\partial t=i\omega_{s}\delta
v_{z}$ and $\partial\delta\rho/\partial t=i\omega_{s}\delta\rho$, gives
$\delta\rho=-\left(ik_{s}+1/L_{\rho}\right){\rho\delta v_{z}\over
i\omega_{s}}\left(1+{2k_{s}\delta v_{z}\over\omega}+{\delta v_{z}\over i\omega
L_{\rho}}\right)^{-1}.$ (13)
Writing $\delta P=\gamma P\left(\delta\rho/\rho+\delta v_{z}/i\omega
L_{\rho}\right)-P\delta v_{z}/i\omega L_{P}$ we similarly derive
${\partial\delta P\over\partial z}=\left({1\over
L_{P}}+ik_{s}\right)\left(c_{s}^{2}\delta\rho+P{\delta v_{z}\over
i\omega}\left({\gamma\over L_{\rho}}-{1\over L_{P}}\right)\right).$ (14)
We now eliminate $\delta\rho$ and $\delta P$ in favor of $\delta v_{z}$ in
equation 3 to derive a quartic equation in $\delta v_{z}$ for the driven slow
mode wave with angular frequency $\omega_{s}$ and wavevector $k_{s}$;
$\displaystyle\delta v_{z}^{4}\left[-{k_{s}^{3}\over\omega_{s}}\right]+\delta
v_{z}^{3}\left[-3k_{s}^{2}+\left({c_{s}^{2}\over
L_{\rho}}-{c_{s}^{2}\over\gamma L_{p}}\right)\left({1\over
L_{p}}+ik_{s}\right)\left({2k_{s}^{2}\over\omega_{s}^{2}}+{k_{s}\over
i\omega_{s}^{2}L_{\rho}}\right)\right]$ (15) $\displaystyle+$
$\displaystyle\delta v_{z}^{2}\left[-3k_{s}\omega_{s}+\left({c_{s}^{2}\over
L_{\rho}}-{c_{s}^{2}\over\gamma L_{p}}\right)\left({1\over
L_{p}}+ik_{s}\right)\left({2k_{s}\over\omega_{s}}+{1\over
i\omega_{s}L_{\rho}}\right)+{k_{s}^{3}c_{s}^{2}\over\omega_{s}}-i{k_{s}^{2}c_{s}^{2}\over\omega_{s}L_{p}}-{k_{s}c_{s}^{2}\over\gamma\omega
L_{p}^{2}}\right]$ (16) $\displaystyle+$ $\displaystyle\delta
v_{z}^{2}\left[-i{k_{s}^{2}c_{s}^{2}\over\gamma\omega_{s}L_{p}}-i{k_{s}^{2}g\over\omega_{s}}-{gk_{s}\over\omega_{s}L_{\rho}}+\left(2ik_{s}+{1\over
L_{\rho}}\right)\left(ik_{A}\delta v_{A}^{2}{k_{s}\over\omega_{s}}+{\delta
v_{A}^{2}k_{s}\over 2\omega L_{\rho}}+{\delta v_{A}^{2}ik_{s}^{2}\over
2\omega_{s}}\right)\right]$ (17) $\displaystyle+$ $\displaystyle\delta
v_{z}\left[-\omega_{s}^{2}+k_{s}^{2}c_{s}^{2}-i{k_{s}c_{s}^{2}\over
L_{p}}-{c_{s}^{2}\over\gamma L_{p}^{2}}-ik_{s}{c_{s}^{2}\over\gamma
L_{p}}-ik_{s}g-{g\over L_{\rho}}+\left(3ik_{s}+{1\over
L_{\rho}}\right)ik_{A}\delta v_{A}^{2}+{i\delta v_{A}^{2}k_{s}\over
2L_{\rho}}-{\delta v_{A}^{2}k_{s}^{2}\over 2}\right]$ (18)
$\displaystyle-\omega_{s}k_{A}\delta v_{A}^{2}=0.$ (19)
To lowest order, the terms in $\delta v_{z}$ and the constant are the same as
in equation 6. The quadratic and higher terms are changed, because of the
difference between equations 12 and 13, and those in equation 5. Inserting
even accurate spatial derivatives in place of those in equations 12 and 13
would generate yet more higher order terms, extending $n$ in principle without
limit. Landau & Lifshitz (1976) give a similar conclusion in their treatment
of parametric resonance.
We solve equation 14 numerically, with the same $k_{s}$ (real and imaginary
parts) as above. We select the solution with the lowest absolute magnitude as
the physically correct solution for our problem, this being the solution that
goes to zero as $\delta v_{A}\rightarrow 0$. This is always close to the
solution obtained discarding all terms of order higher than $\delta v_{z}^{2}$
in equation 14, and usually close to the case when the term in $\delta
v_{z}^{2}$ is also neglected. This describes turbulence for parallel
propagating waves. Zaqarashvili & Roberts (2006) give a detailed treatment of
the interaction between weak Alfvén and sound waves in a homogeneous medium,
where acoustic and Alfvén speeds are equal. The stronger generation of
acoustic waves by the ponderomotive force in a density gradient is
demonstrated by the simulations of Del Zanna et al. (2005), where slow mode
waves are seen propagating up from loop footpoints with properties consistent
with the solutions of equations 9 or 14.
Closer to the layer where $c_{s}^{2}=V_{A}^{2}$, the magnetic field becomes
more curved, giving rise to higher perpendicular components of Alfvén wave
wavevectors, and potentially stronger turbulent cascade and/or mode
conversion. In this case we expect stronger slow mode waves. In Laming (2009)
we assumed $\delta v_{s}=\delta v_{A}$. However considerations of mode
conversion for initially upward propagating acoustic waves suggest that higher
slow mode intensities than this should be present. Khomenko & Cally (2011)
report that at conditions for maximum conversion of a high $\beta$ fast mode
wave to a low $\beta$ Alfvén wave, the low $\beta$ slow mode wave has 2-3
times more flux. In this paper, we make the approximation
$\delta v_{s}^{2}=\delta v_{z}^{2}+6\delta v_{A}^{2}c_{s}^{2}/V_{A}^{2},$ (20)
where $\delta v_{z}$ represents the solution of equation 14, and the factor 6
in the last term is motivated by calculations illustrated in Cranmer et al.
(2007) and Khomenko (2010). Slow mode waves governed by equation 15 have the
effect of suppressing fractionation in the low chromosphere close to where the
plasma $\beta\simeq 1.2$. Studies of mode conversion between acoustic and
Alfvén waves generally show the upward moving acoustic wave beginning to
convert to Alfvén waves at the $\beta\simeq 1.2$ layer, and mode conversion
continuing over a range of altitudes of order 100’s of km (e.g Cally &
Goossens, 2008). The explicit incorporation of such effects is well beyond the
scope of the work here, and the prescription in equation 15 should be
sufficient to avoid the occurrence of unphysical fractionations. Even so, it
remains a feature of this work requiring further investigation in future
papers.
### 3.2 Normalization Relative to Oxygen
In previous papers Laming (2004, 2009) we have discussed FIP fractionations
relative to H. However the derivation of 3 has assumed fractionated elements
are minor species, with the fractionation having no back reaction on the flow
due to the neglect of inertial terms. Thus it is more appropriate to present
and describe element fractionations with respect to another minor element. We
choose O, which is a common choice for observers also.
### 3.3 Charge Exchange Ionization
Charge exchange rates have been previously taken from the compilation of
Kingdon & Ferland (1996). These have been supplemented more recently with
charge exchange ionization rates for Si and Fe, colliding with protons. We
implement an estimate of the charge exchange ionization rate for all ions with
lower FIP than H as follows. We estimate the radius $R$ at which the sum of
the binding energy of the electron in the target neutral and the polarization
potential energy of the target neutral in the electric field of the incoming
proton and equal to the equivalent sum for the resulting neutral H atom in the
electric field of the newly formed ion, (known as the radius of the potential
crossing) from
$V=-{\left(\alpha_{s}-\alpha_{H}\right)\over R^{4}}=-I_{H}+I_{s}$ (21)
where $V$ represents the difference in potential energy of the proton in terms
of the polarizability $\alpha_{s}$ of the target atom and resulting ion in
terms of $\alpha_{H}$, the polarizability of the resulting hydrogen atom.
$I_{H}$ and $I_{s}$ are the ionization potentials of hydrogen and the target
atom, $s$, respectively. Polarizabilities and ionization potential here are in
atomic units. The cross section is then $\sigma_{cxi}=\pi
R^{2}/2=\pi\sqrt{\left(\alpha_{s}-\alpha_{H}\right)/\left(I_{H}-I_{s}\right)}/2$,
assuming the maximum probability for a reaction is 1/2. This estimate is a
slightly different form of the Langevin formula given by Ferland et al.
(1997). This approximation gives good agreement with the calculations of Allan
et al. (1988) for charge exchange ionization of Mg. These authors comments
that similar rates should exist for all other elements with ionization
potentials below that of H, and we apply it to all of these elements. The
charge exchange implemented is just the thermal process. No account is taken
of any possible effects of the waves on the chromospheric ionization balance.
### 3.4 Comparison of Different Approximations
Carlsson & Stein (2002) have argued that the concept of an “average”
chromosphere pursued by Avrett & Loeser (2008) and its antecedents is invalid,
due to the extreme dynamics associated with chromospheric shock waves, arguing
that “the mean value of a dynamic property is not the same as that property
evaluated for the mean atmosphere”. Avrett & Loeser (2008) derive mean values
for plasma properties based on observations, not on calculated mean
atmospheres. More importantly, Carlsson & Stein (2002) show that the
ionization fraction for H varies very little about its mean, due to the length
of ionization and recombination times compared to the frequencies of shocks,
and that their average electron density agrees very well with that in the
Vernazza, Avrett, & Loeser (1981) model C. It is easy to see that other high
FIP elements should behave similarly, and that the ionization balance we
calculate (the overwhelmingly most important chromospheric property to us)
should not be greatly in error, if at all. The inclusion of charge exchange
ionization (section 3.3) increases the ionization level for all other elements
as the ionization of hydrogen is increased above its thermal equilibrium
level. The Ca+ to Ca2+ ionization balance is considered by Wedemeyer-Böhm &
Carlsson (2011). This is more variable than that for H to H+ in Carlsson &
Stein (2002), but is less of a concern to us, since the ponderomotive force
experienced by an ion is independent of ion charge, so long as
$\omega_{A}<<\Omega$.
We give sample calculations that go some way to quantifying the effect of
these issues. Figure 2 shows the coronal section of the 100,000 km long loop
with magnetic field $B=20$G. The density at the loop apex is $5\times 10^{8}$
cm-3. This gives a resonant angular frequency of 0.07 rad s-1. A wave of this
frequency propagates on the loop, which is thus half a wavelength long. The
top panel gives the Elsässer variables (real and imaginary parts), the middle
panel gives the wave energy fluxes and their difference, and the third panel
gives the ponderomotive acceleration. Figure 3 shows the chromospheric section
of the loop where the fractionations are evaluated. This has the same three
panels as for Figure 2, where the third panel also gives the slow mode wave
amplitude, with a fourth panel (bottom right) that gives FIP fractionation and
the ionization fraction of elements Fe, Mg, S, and He relative to O. In Table
1 fractionations for He, C, N, Ne, Mg, Si, S, Ar, and Fe relative to O are
displayed, calculated according to our basic model described above, labeled
“baseline”. In the succeeding columns, we give FIP fractionations calculated
with different modifications to the model. In the first case the density given
by Avrett & Loeser (2008) is modified so that the model density is consistent
with the H ionization fraction and the assumption of photoionization-
recombination equilibrium. The second variation gives fractionations
calculated assuming the ionization fractions to be given by the Saha equation
at the temperature and density in the chromospheric model. The first variation
reduces the degree of ionization in the chromosphere, while the second one
increases it. This has the opposite effect on the fractionations, since these
are given relative to O, and the increase or decrease of the ionization of O
has a bigger effect on its fractionation than is the case with the other
elements. As can be seen, the basic phenomenon of the FIP effect remains
unaffected by the choice of approximation, but some of the details, e.g. the
Mg/Fe ratio are subtly different. We return to discuss this in more detail
below, in subsection 4.2.
The final column in Table 1 gives the FIP fractionation calculated with the
assumption of slow mode wave amplitude $\delta v_{z}=\delta v_{A}$, the
amplitude of the Alfvén wave, as taken in Laming (2009), instead of
implementing the solution of equation 14. The overall fractionation is reduced
for the same Alfvén wave amplitude, due to the increase in longitudinal
pressure with the higher slow mode wave amplitude. In Laming (2009),we found
that the FIP Effect saturated in this case at values around 3-4 for
arbitrarily high Alfvén wave amplitude. In section 4, we will consider the
behavior of the FIP Effect with resonant and nonresonant waves.
## 4 Results
### 4.1 Coronal Hole
A coronal hole is modeled int he same fashion. The chromosphere is identical
to the case above, and the density evolves smoothly off-limb, declining to
about $2\times 10^{6}$ cm-3 at an altitude of $5\times 10^{5}$ km (as in e.g.
Figure 5 of Cranmer, 2009). The magnetic field follows the model of
Banaskiewicz et al. (1998). We take the coronal hole Alfvén wave spectrum
calculated by Cranmer et al. (2007) as our starting point. It is illustrated
in their Figure 3 for a position in the solar transition region. It is
represented by the five wave frequencies and amplitudes at the starting
position of the non-WKB integration of the wave transport equations, in this
case at an altitude of 500,000 km. Parameters are chosen to match Figure 9 in
Cranmer et al. (2007), and are given in Table 2 as the spectrum labeled “v0”.
Figure 4 shows the wave amplitudes in the coronal hole section of the
calculation, with the same three panels as for Figure 2, but illustrated up to
an altitude of 500,000 km. Figure 5 shows the chromospheric response with the
same four panels as in Figure 3. The ponderomotive acceleration (Figure 5, top
right) has a similar “spike” at the top of the chromosphere as in Figure 3 for
the closed loop, but is about an order of magnitude weaker. Lower down, the
Alfvén wave amplitudes, and the corresponding slow mode wave amplitudes are
larger than before. The net effect of smaller ponderomotive acceleration and
higher slow mode wave amplitude is to reduce the FIP effect from the case in
Figure 3. The resulting fractionation here is very similar to that found by
Cranmer et al. (2007). In models v1-2 in Table 2, we increase the amplitude of
the highest and lowest frequency wave in the spectrum respectively. Increasing
the highest frequency wave amplitude has the effect of increasing the
fractionation predicted, while increasing the low frequency wave has very
little effect. The results from all three models are summarized in Table 3,
and compared with observations for various open field regions. Reasonable
agreement for the fractionation of low FIP elements is seen. A small depletion
of He is seen, but not as large as observed. Our purpose here has not been to
provide a definitive calculation of the FIP effect in a coronal hole, but
merely to illustrate that the observed difference between the FIP effect in
closed and open field lines arises naturally in this model.
### 4.2 Closed Loop
We now explore fractionation in a closed coronal loop, illustrating the
difference that a coronal resonant frequency can make to the fractionation. In
Table 4 we give the fractionations computed for a closed loop with length
100,000 km, and a magnetic field of 20 G. A coronal wave at the loop resonant
angular frequency of 0.07 rad s-1 is modeled. The wave transport equations
(equation 2) are integrated from the $\beta=1$ layer in one chromosphere,
where and initially downgoing wave amplitude is specified, through the loop to
the opposite chromospheric footpoint, where the FIP fractionations are
evaluated. The initial wave amplitudes are 0.4, 0.5, and 0.6 km s-1, giving
coronal wave peak amplitudes of approximately 55, 70, and 82 km s-1
respectively.
These cases illustrate the variation of the FIP effect with wave amplitude.
The six columns on the right hand side give various observations of FIP
fractionation. Zurbuchen et al. (2002) and von Steiger et al. (2000) both give
fractionations measured in situ in the solar wind over relatively long periods
of time. Giammanco et al. (2007, 2008) give fractionations also measured in
the solar wind, but over time periods selected such that the wind speed was
close to 380, 390 or 400 km s-1. Phillips et al. (2003); Sylwester et al.
(2010a, b); McKenzie & Feldman (1994) give abundance ratios measured
spectroscopically in solar flares, and Bryans et al. (2009) observe quiet
solar corona above the western limb, also spectroscopically. While the
agreement between the FIP fractionations calculated for the initial Alfvén
wave amplitude corresponding to 0.5 km s-1 is generally good, there are some
important discrepancies. The ratio C/O is typically observed in the solar wind
to be higher than calculated, as is S/O for some of the observations. While
Fe/O and Mg/O are reasonably well reproduced, the direct ratio between Fe and
Mg is not, except in the case of Bryans et al. (2009). The last case, with
initial wave amplitude 0.6 km s-1, is designed to match these observations,
and does quite well. Only K is seriously discrepant, with S also somewhat
underestimated. In the solar wind and flare observations, Fe and Mg are often
fractionated by the same amount.
Table 5 gives FIP fractionations for varying Alfvén wave frequency, with the
amplitudes chosen to keep the Fe/O abundance ratio close to 4 as is often
observed. Here, we have also included a wave field of chromospheric origin
designed to have the same properties as that in the v0 model for the open
field case. Downgoing wave amplitudes are specified at the $\beta=1$ layer in
one chromosphere, and then the wave transport equations are integrated back to
the opposite footpoint where FIP fractionations are evaluated. The column
corresponding to the resonant frequency of 0.07 rad s-1 is the same as in
Table 4, but for the new model. The chromospheric waves can be seen to have
rather little effect, with the biggest changes being seen in the increased
fractionations of C/O and S/O. Moving away from this resonance, either to
lower or higher frequency, the C/O and S/O ratios increase to better agree
with observations. At higher frequency so too does the Mg/O ratio, so that Mg
and Fe fractionate more closely to the same degree; quasi-mass independent
fractionation is achieved. Going to yet higher frequency, all high FIP
elements remain unchanged, while all low FIP elements are enhanced by a factor
3-4. The depletions of He and Ne are lost. The quiet sun observations of
Bryans et al. (2009) are best matched at the frequency of 0.075 rad s-1 and
with the exception of K, the consistency between their measurements and the
model is excellent at an initial wave amplitude of 0.7 km s-1.
Figures 6 and 7 illustrate the ponderomotive acceleration and FIP
fractionation within the chromosphere for the cases of frequencies 0.06, 0.07,
0.085 and 0.105 rad s-1. At 0.06 rad s-1, the ponderomotive acceleration is
positive from about 800 km up, and has a “spike” at about 2150 km, with
maximum close to $10^{6}$ cm s-2. Fractionation of Fe, Mg, and S is similar
low down, but the fractionation hierarchy Fe $>$ Mg $>$ S is established in
the range 1500-2000 km. For 0.07 rad s-1, the ponderomotive acceleration has a
similar, but slightly larger maximum at about 2150 km. In response to this Fe,
Mg and S undergo a similar and small inverse FIP fractionation up to 1500 km,
giving way to positive FIP higher up. The fractionation pattern Fe $>$ Mg $>$
S is even stronger here than for 0.06 rad s-1, and is mainly occurring at the
“spike” in the ponderomotive acceleration. In the last two cases, the
ponderomotive force is stronger lower down in the chromospheric, and the
“spike” at 2150 km becomes less pronounced. The slow mode wave amplitude is
also stronger lower down. At 0.105 rad s-1, all fractionation occurs by 1600
km, and the local maximum in the ponderomotive acceleration at 2150 km has no
effect.
These fractionation patterns have simple qualitative explanations. First we
display in Figure 8, the 0.07 rad s-1 case again to illustrate the relation of
the fractionation to important features of the chromosphere. The left panels
give the ponderomotive acceleration (bottom) and the density and temperature
structure of the chromosphere (top). The “spike” in the ponderomotive
acceleration can be seen to stem from the steep density gradient beginning at
an altitude of about 2100 km. The solid and dashed lines in the bottom plot
show the ponderomotive acceleration with and without the energy loss to slow
mode waves. In the regions where significant fractionation occurs, the slow
mode wave do not affect the ponderomotive acceleration very much, and their
main effect on the fractionation is through the additional longitudinal
pressure they provide. The panels on the right hand side show the same
fractionations as before (bottom), and on the top an expanded view of the
ionization fraction of the elements C, S, Mg, and Fe.
With reference to equation 3, in regions where H is predominantly neutral,
(below about 1500 km altitude) $\nu_{s,i}\sim\nu_{s,n}$ and similar
fractionation results for elements where $\xi_{s}$ is reasonably close to
unity. Where H is predominantly ionized, $\nu_{s,i}>>\nu_{s,n}$, and small
departures in $\xi_{s}$ from unity can make a big difference to the
fractionation. This is the reason why S fractionates similarly to Fe and Mg in
the low chromosphere, but markedly less so in higher regions. This is also the
reason why Mg fractionates less than Fe higher up. At an altitude of 2000 km,
the charge state fractions of Fe, Mg, and S are 0.9995, 0.9981, and 0.9942
respectively (see Figure 8). Even though these are close to unity, the
differences from unity result in different fractionations where the H
ionization fraction (which closely follows that of O) is about 0.6. Lower
down, where H is mostly neutral, the different ionization fractions matter
much less in the fractionation. Recalling the results calculated using the
Saha approximation for the ionization fractions, we can now see why Mg and Fe
fractionate much more similarly in this case. The ionization fractions at 2000
km altitude are now 0.999974, 0.999995, and 0.9983 respectively, for Fe, Mg,
and S. These are much closer to unity than before, so Fe and Mg now
fractionate to a more similar degree. This is to be expected, since the
assumption of LTE in the Saha equation suppresses radiative recombination
rates, since the photons so produced cannot escape, and so the Saha ionization
fractions will be higher than a more realistic calculation would predict.
However in Table 1 using Saha equilibrium, Fe and Mg do not behave exactly
identically. Recalling equation 3,
${\rm
fractionation}=\exp\left(2\int_{z_{l}}^{z_{u}}{\xi_{s}a\nu_{s,n}\over\left[\xi_{s}\nu_{s,n}+\left(1-\xi_{s}\right)\nu_{s,i}\right]}{1\over\left[kT/m_{s}+v_{\mu
turb}^{2}+v_{turb}^{2}\right]}dz\right)$ (22)
and remembering that $v_{turb}$ is the amplitude of slow mode waves generated
by the Alfvén waves themselves, we can see that when $v_{turb}$ and $v_{\mu
turb}$ dominate over the ion thermal speed (usually $v_{turb}>v_{\mu turb}$),
the mass dependence disappears from this part of the equation, and will only
reside, if at all, in the collision frequencies. In fractionation occurring
high in the chromosphere associated with the “spike”, where the plasma
temperature is increasing rapidly up to coronal values, this condition may not
be met and mass dependent fractionation can occur. In fractionation occurring
lower down near the chromospheric temperature minimum, for example in the
0.085 rad s-1 case, this condition is met, and quasi-mass independent
fractionation results.
## 5 Discussion
### 5.1 The Effect of Slow Mode Waves
The parametric generation of slow mode wave is a crucial part of the
fractionation process by ponderomotive forces. One important effect has been
to render the fractionation quasi mass independent as is often observed. This
is demonstrated most clearly in Figure 7a, corresponding to a 0.085 rad s-1
Alfvén wave, the case which also has the highest slow mode wave amplitude,
staying close to 10 km s-1 for large regions of the chromosphere where thermal
speeds are $\sim 1$ km s-1. Fractionation occurring at the top of the
chromosphere in the location of the “spike” in the ponderomotive acceleration
often retains some mass dependence, because the plasma temperature is
increasing rapidly here while the slow mode wave amplitude is usually small.
In the case that $\delta v_{s}\sim\delta v_{A}$ as in Laming (2009), the
increased slow mode wave amplitude relative to this work suppresses all FIP
fractionation except when the wave frequency coincides precisely with the loop
resonance, and then all fractionation occurs at the loop top and hence is mass
dependent. For reasons we discuss below, this coincidence is probably not
realized ubiquitously in the solar corona. Moreover the assumption of
isotropic turbulence probably requires a well developed turbulent cascade,
which is unlikely to develop with purely parallel propagating waves (e.g. Luo
& Melrose, 2006). Lower down in the chromosphere as the plasma beta approaches
unity, the magnetic field becomes more concentrated in network segments. The
increased curvature of field lines will lead initially parallel propagating
waves to develop perpendicular components to their wavevectors, and hence
turbulent cascade or mode conversion become more likely. Our equation 15 above
is an attempt to capture this behavior, and obviously needs to be revisited
with greater rigor.
### 5.2 The Alfvén Wave Frequency
Table 5 displays FIP fractionations for a range of frequencies close to the
fundamental of a loop with length 100,000 km and magnetic field 20 G, with
wave amplitudes chosen such that the fractionation of Fe/O is close to the
usually observed value of 4. We commented above how the fractionation details
of other elements vary slightly as the coronal wave moves from being in
coincidence with the loop resonance, to a position well off resonance. This
arises because resonant waves reflect from the top of the chromosphere, and
this is then the sole location of FIP fractionation, but nonresonant waves
penetrate further down, allowing FIP fractionation to occur over a greater
range of altitudes in the chromosphere.
When FIP fractionation is concentrated at the top of the chromosphere, the
different ionization structures of the various high FIP elements becomes
important, and fractionation occurs among them. Most significantly, He becomes
depleted relative to O, with this depletion being strongest for a frequency
0.075 rad s-1, just higher than the resonance, at a value of 0.60. This gives
an abundance close to the He abundance frequently observed in the slow speed
solar wind (Aellig et al., 2001; Kasper et al., 2007). In this frequency
region too, C and possibly S also have minima in their fractionations. These
elements have ionization potentials of 11.26 and 10.36 eV respectively (on the
boundary between low FIP and high FIP elements). Although they are highly
ionized throughout the chromosphere, as described above, they have sufficient
neutral component that they fractionate well when H is predominantly neutral,
but not when H is ionized. When fractionation is restricted to the top of the
chromosphere where H is ionized, they behave more like high FIP elements. This
is commonly seen in spectroscopic observations of S (e.g. Laming et al., 1995;
Feldman et al., 2009; Widing & Feldman, 2008; Brooks & Warren, 2011).
Off resonance, when FIP fractionation can occur over a more extended range of
heights, including those where H is mainly neutral, C and S might be expected
to behave more like low FIP elements. Such behavior is more apparent in the
solar wind observations of Zurbuchen et al. (2002) and von Steiger et al.
(2000). Here, the FIP bias is variable, so that the time average over an
extended period gives Fe/O $\sim 2$ instead of $\sim 4$ as modeled. Even so,
S/O has a similar value to Fe/O. These observations are best matched in Table
5 for a frequency 0.085 rad s-1.
We have previously suggested that Alfvén waves of coronal origin probably
derive from coronal heating mechanisms such as nanoflares or Alfvén resonance.
The coronal Alfvén amplitudes required above ($\sim 50-100$ km s-1) are larger
than nonthermal mass motions observed through spectral linewidths by factors
2-3. This suggests that the Alfvén wave must be confined to a small fraction
of the loop cross-sectional area, which would also be a natural consequence of
nanoflare or Alfvén resonance heating.
In as far as the heating can be considered a small perturbation to the coronal
loop, the waves so generated should be eigenfunctions of the loop, with
frequencies constrained to coincide with the loop resonance(s). The fact that
many observations are better matched by Alfvén wave frequencies slightly
higher than the resonance possibly suggests a dynamic system. The loop
releases waves at its resonance as part of the heating process. The heat
conducts down to the chromosphere, and heated plasma there evaporates back
upwards into the loop (e.g. Patsourakos & Klimchuk, 2006), thus increasing its
density and reducing the coronal Alfvén speed, and hence also reducing the
loop resonance frequency. The waves produced at the original resonance
continue to propagate until damped. So we might naturally expect a mismatch
between Alfvén wave frequency and the loop resonance, and also expect this
mismatch to become larger in more strongly heated region of the corona, e.g.
active region and flares, as opposed to quiet sun. Of course this discussion
presupposes that the heating is a weak perturbation to the coronal loop, and
so its eigenfunctions are well defined, and can be excited. In flares and
CMEs, this might no longer be true, and the heating mechanism itself will
determine which waves are produced, irrespective of the loop boundary
conditions.
Such ideas will be investigated in greater detail in a separate paper. For the
time being, we restrict ourselves to some simple predictions. The coronal
helium abundance should increase with increasing solar activity, as it appears
to do both in solar wind observations (Aellig et al., 2001; Kasper et al.,
2007) and in spectroscopic measurements of the quiet sun (Laming & Feldman,
2001, 2003) compared to flares (Feldman et al., 2005). The S and C abundance
should also vary. In the solar wind (e.g. von Steiger et al., 2000) it appears
to vary as a low FIP ion, whereas in spectroscopic observations, (e.g. Laming
et al., 1995; Feldman et al., 2009, of quiet and active regions) sulfur is
observed to behave as a high FIP element.
### 5.3 The Upward Flow Speed
We have suggested conduction driven chromospheric evaporation as the source of
the plasma upflow that populates the corona with fractionated gas. Here we
estimate the flow speed in the chromosphere, and show that it is consistent
with limits set by the operation of the ponderomotive force producing the FIP
fractionation. With $d\rho/dt=0$ we write
${\partial\over\partial z}\left(\rho v\right)=-{\partial\rho\over\partial
t}=-{\mu gz\over k_{\rm B}T^{2}}{\partial T\over\partial t}\rho$ (23)
where the density $\rho\propto\exp\left(-\mu gz/k_{\rm B}T\right)$ is a
gravitationally stratified solution. The mean molecular mass is $\mu$, and
$k_{\rm B}$ is Boltzmann’s constant. Integrating between $z_{l}$ and $z_{u}$
$\rho\left(z_{u}\right)v\left(z_{u}\right)-\rho\left(z_{l}\right)v\left(z_{l}\right)=-\int_{z_{l}}^{z_{u}}{\mu
gz\over k_{\rm B}T^{2}}{\partial T\over\partial t}\rho dz.$ (24)
We choose the upper limit $z_{u}$ to be where $v\left(z_{u}\right)=0$ in the
corona, and so
$v\left(z_{l}\right)=\int_{z_{l}}^{z_{u}}{z\over
L_{\rho}}{\rho\left(z\right)\over\rho\left(z_{l}\right)}{\partial\ln
T\over\partial t}dz=\int_{z_{l}}^{z_{u}}{z\over L_{\rho}}\exp\left(-{z\over
L_{\rho}}\right){\partial\ln T\over\partial t}dz$ (25)
where $L_{\rho}=k_{\rm B}T/\mu g$ is the density scale height. This integral
will be dominated by the integrand near $z=z_{l}$, so
$v\left(z_{l}\right)\simeq L_{\rho}{\partial\ln T\over\partial
t}\simeq{2L_{\rho}\over 5nk_{\rm B}T}{\partial\over\partial
z}\left(10^{-6}T^{5/2}{\partial T\over\partial z}\right)\sim
10^{17}{T^{1/2}\over n}\left(\Delta T\over L_{T}\right)^{2}$ (26)
where we have put $2.5nk_{\rm B}\partial T/\partial t=-\nabla\cdot{\bf Q}$
with the heat flux ${\bf Q}=-10^{-6}T^{5/2}\nabla T$. The temperature gradient
has been replaced by $\Delta T/L_{T}$, where $\Delta T$ may be the coronal
peak temperature, and $L_{T}$ the loop half-length. Taking the chromospheric
temperature $T\sim 10^{4}$,
$v\left(z_{l}\right)\simeq{10^{13}\over n}\left(T_{c}/5\times 10^{6}~{}{\rm
K}\over L_{T}/5\times 10^{9}~{}{\rm cm}\right)^{2}{\rm cms}^{-1},$ (27)
which suggests a velocity of $10^{3}$ cm s-1 at a density of $10^{10}$ cm-3.
This may well be an underestimate due to our approximation for the temperature
gradient, but as discussed in Laming (2004), is sufficiently high that
gravitational settling should not occur. If $v\left(z_{l}\right)$ approaches
$\sim 1$ km s-1, as might happen in flares, some further discussion is
required.
The derivation of equation 17 neglected inertial terms in the momentum
equations for ions and neutrals. Reinstating these, in the limit that
$u_{si}-u_{sn}<<u_{s}\sim u$ the fractionation becomes
${\rm
fractionation}=\exp\left(2\int_{z_{l}}^{z_{u}}{\xi_{s}a\nu_{s,n}\over\left[\xi_{s}\nu_{s,n}+\left(1-\xi_{s}\right)\nu_{s,i}\right]}{1\over\left[kT/m_{s}+v_{\mu
turb}^{2}+v_{turb}^{2}+u_{flow}^{2}\right]}dz\right).$ (28)
When the magnitude of $u_{flow}^{2}$ approaches those of the other terms in
the second square bracket in the denominator of the integrand, some reduction
in the FIP effect will result. This is most likely to have an impact on
fractionation occurring at the top of the chromosphere, taking $\rho
u_{flow}\sim$ constant through the chromosphere, and will possibly reduce the
amount of mass dependent fractionation occurring there. This will happen for
relatively large flow speeds $u_{flow}\sim 1$ km s-1 or larger at the top of
the chromosphere, still significantly lower than the Alfvén speed in this
region.
### 5.4 Conclusions
We have further developed the model of element fractionation to give rise to
the FIP effect by the ponderomotive force, paying careful attention to the
generation of slow mode waves by the primary Alfvén oscillations. When
considering a realistic wave spectrum with both chromospheric and and coronal
contributions, the extra longitudinal pressure due to the slow mode waves is
crucial in producing the correct fractionation. With this extra ingredient,
together with the improvements to the ionization balance and the normalization
of the fractionation, a rather comprehensive description of the coronal
fractionation has emerged. In seeking to understand the FIP effect as usually
described, we have also found an explanation for the depletion of He in the
solar wind, and also possibly its variation. The Ne abundance also appears to
vary in a similar manner, but to a lesser degree. It is also more sensitive to
assumptions about the ionization balance, but further investigation of this is
expected to resolve current controversy surrounding the solar photospheric
abundance of this element.
The theory now appears to be developed to the point where variations in the
FIP fractionation from place to place in the solar corona or wind may now be
interpreted in terms of their physical origins. The element abundances in the
solar corona may therefore be considered as diagnostics of the behavior of MHD
turbulence, and also thereby of the mechanisms that heat the solar corona.
These ideas will be further developed in subsequent papers.
This work was supported by NASA Contracts NNH10A055I, NNH11AQ23I, and by basic
research funds of the Office of Naval Research. I am also grateful to Cara
Rakowski for a critical reading of an earlier draft of this paper.
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Table 1: FIP Fractionations in Different Approximations ratio | baselinea | density mod.b | Saha ionizationc | $\delta v_{z}=\delta V_{A}$
---|---|---|---|---
He/O | 0.67 | 0.83 | 0.73 | 0.85
C/O | 0.99 | 1.26 | 1.17 | 1.03
N/O | 0.82 | 1.02 | 0.95 | 0.93
Ne/O | 0.74 | 0.93 | 0.90 | 0.89
Mg/O | 1.98 | 2.52 | 2.33 | 1.43
Si/O | 1.89 | 2.37 | 2.41 | 1.41
S/O | 1.40 | 1.75 | 1.47 | 1.23
Ar/O | 0.92 | 1.16 | 1.03 | 0.97
Fe/O | 3.29 | 4.17 | 2.79 | 1.87
Table 2: Open Magnetic Field Wave Spectra at 500,000 km Altitude ang. freq. | v0 | v1 | v2
---|---|---|---
0.010 | 12.5 | 12.5 | 125
0.031 | 150 | 150 | 150
0.062 | 75 | 75 | 75
0.093 | 50 | 50 | 50
0.124 | 12.5 | 125 | 12.5
Table 3: FIP Fractionations in Open Magnetic Field | models | observations
---|---|---
ratio | v0 | v1 | v2 | a | b | c
He/O | 0.90 | 0.85 | 0.85 | 0.60-0.58 | 0.37-0.47 | 0.45-0.55
C/O | 1.13 | 1.18 | 1.10 | 1.50-1.41 | 1.17-1.35 | 0.9 - 1.1
N/O | 0.96 | 0.94 | 0.94 | 1.19-0.9 | 0.64-0.99 |
Ne/O | 0.95 | 0.92 | 0.92 | 0.48-0.47 | 0.40-0.56 | 0.3 - 0.4
Na/O | 1.97 | 2.99 | 2.04 | | |
Mg/O | 1.65 | 2.21 | 1.67 | 1.73-1.92 | 0.98-1.60 | 0.95 - 2.45
Al/O | 1.72 | 2.37 | 1.75 | | |
Si/O | 1.51 | 1.89 | 1.53 | 2.07-1.92 | 1.20-2.09 | 0.9 - 1.8
S/O | 1.26 | 1.39 | 1.26 | 1.53-1.56 | 1.38-2.57 |
Ar/O | 1.00 | 0.99 | 0.99 | | |
K/O | 2.03 | 3.14 | 2.12 | | |
Ca/O | 1.88 | 2.74 | 1.93 | | |
Fe/O | 1.97 | 2.97 | 2.04 | 1.42-1.73 | 1.04-1.69 | 0.65 - 1.35
Ni/O | 1.85 | 2.67 | 1.91 | | |
Kr/O | 1.01 | 1.01 | 1.01 | | |
Rb/O | 1.97 | 2.94 | 2.04 | | |
W/O | 1.99 | 2.99 | 2.07 | | |
Table 4: FIP Fractionations in Closed Magnetic Field I | models | observations
---|---|---
| 0.4 | 0.5 | 0.6 | a | b | c | d | e | f
ratio | (km s-1) | | | | | |
He/O | 0.72 | 0.61 | 0.47 | 0.68-0.60 | 0.29-0.75 | | | |
C/O | 1.01 | 1.03 | 0.97 | 1.36-1.41 | 1.06-1.37 | | | |
N/O | 0.86 | 0.81 | 0.69 | 0.72-1.32 | 0.22-0.89 | | | |
Ne/O | 0.86 | 0.71 | 0.57 | 0.58 | 0.38-0.75 | | | |
Na/O | 2.42 | 3.96 | 6.74 | | | 7.8${+13\atop-5}$ | 1.8${+2\atop-1}$ | |
Mg/O | 1.76 | 2.35 | 3.25 | 2.58-2.61 | 1.08-2.36 | 2.8${+2.3\atop-1.3}$ | 2.7$\pm 0.3$ | |
Al/O | 1.95 | 2.82 | 4.12 | | | 3.6${+1.7\atop-1.2}$ | 5.6${+3.3\atop-2.1}$ | |
Si/O | 1.70 | 2.27 | 3.02 | 2.49-3.11 | 1.36-3.24 | 4.9${+2.9\atop-1.8}$ | | |
S/O | 1.34 | 1.57 | 1.78 | 1.62-1.92 | 1.23-2.68 | 2.2$\pm 0.2$ | 2.1$\pm 0.2$ | $1.7\pm 0.3$ |
Ar/O | 0.96 | 0.94 | 0.86 | | | | | $1.1\pm 0.1$ | 1.12$\pm 0.15$
K/O | 2.72 | 4.70 | 8.51 | | | 1.8${+0.4\atop-0.6}$ | 4.7${+7.0\atop-2.8}$ | $3.5\pm 0.9$ | 6
Ca/O | 2.38 | 3.80 | 6.27 | | | 3.5${+4.3\atop-1.9}$ | 2.7$\pm 0.25$ | | 3.0-9.7
Fe/O | 2.65 | 4.44 | 7.85 | 2.28-2.90 | 0.96-2.46 | 7.0${+1.4\atop-1.2}$ | | |
Ni/O | 2.59 | 4.10 | 6.92 | | | | | |
Kr/O | 0.99 | 1.00 | 0.92 | | | | | |
Rb/O | 2.84 | 4.96 | 9.01 | | | | | |
W/O | 3.01 | 5.35 | 9.85 | | | | | |
Table 5: FIP Fractionations in Closed Magnetic Field II ang. freq. (rad s-1) | 0.055 | 0.06 | 0.065 | 0.07 | 0.075 | 0.08 | 0.085 | 0.09 | 0.105
---|---|---|---|---|---|---|---|---|---
$v_{init}$ (km s-1) | 0.255 | 0.25 | 0.35 | 0.45 | 0.55 | 0.5 | 0.45 | 0.36 | 0.193
$v_{cor}$ (km s-1) | 40 | 40 | 45 | 60 | 70 | 60 | 50 | 38 | 20
ratio | | | | | | | | |
He/O | 0.84 | 0.81 | 0.74 | 0.63 | 0.60 | 0.74 | 0.86 | 0.94 | 0.98
C/O | 1.43 | 1.38 | 1.32 | 1.18 | 1.23 | 1.40 | 1.53 | 1.59 | 1.56
N/O | 0.93 | 0.92 | 0.89 | 0.83 | 0.80 | 0.88 | 0.94 | 0.97 | 0.99
Ne/O | 0.90 | 0.89 | 0.84 | 0.75 | 0.72 | 0.83 | 0.91 | 0.96 | 0.98
Na/O | 4.74 | 4.47 | 4.61 | 4.31 | 4.47 | 4.36 | 4.69 | 4.73 | 4.60
Mg/O | 3.29 | 3.13 | 3.11 | 2.74 | 2.89 | 3.13 | 3.49 | 3.59 | 3.51
Al/O | 3.59 | 3.43 | 3.49 | 3.15 | 3.30 | 3.42 | 3.73 | 3.81 | 3.75
Si/O | 2.70 | 2.63 | 2.69 | 2.51 | 2.69 | 2.74 | 2.88 | 2.89 | 2.84
S/O | 1.79 | 1.77 | 1.79 | 1.72 | 1.83 | 1.86 | 1.92 | 1.90 | 1.87
Ar/O | 0.98 | 0.98 | 0.97 | 0.96 | 0.94 | 0.97 | 0.99 | 0.99 | 1.00
K/O | 5.08 | 4.85 | 5.13 | 4.84 | 4.92 | 4.68 | 4.95 | 4.98 | 4.94
Ca/O | 4.31 | 4.13 | 4.32 | 4.00 | 4.12 | 4.06 | 4.34 | 4.39 | 4.36
Fe/O | 4.78 | 4.60 | 4.87 | 4.52 | 4.57 | 4.44 | 4.73 | 4.79 | 4.79
Ni/O | 4.20 | 4.09 | 4.35 | 4.12 | 4.23 | 4.06 | 4.24 | 4.25 | 4.27
Kr/O | 1.00 | 1.00 | 1.00 | 1.00 | 0.99 | 1.00 | 1.00 | 1.00 | 1.00
Rb/O | 4.73 | 4.61 | 4.98 | 4.79 | 4.80 | 4.50 | 4.67 | 4.67 | 4.72
W/O | 4.84 | 4.73 | 5.15 | 4.83 | 4.83 | 4.56 | 4.76 | 4.79 | 4.89
Figure 1: Cartoon illustrating the model. Alfvén waves generated in the corona
either reflect from each footpoint or precipitate down, depending on their
frequency with respect to the loop resonance. Resonant waves reflect,
nonresonant waves precipitate down. Upcoming waves in the chromosphere,
deriving from the mode or parametric conversion of p-modes at the $\beta=1.2$
layer, are generally reflected back downwards, as illustrated at footpoint B.
In our models, we specify wave amplitudes at footpoint A, and integrate the
non-WKB transport equation back to footpoint B, where FIP fractionations are
evaluated. Figure 2: Coronal section of loop, length 100,000 km, magnetic
field 20 G, (half wavelength long for a 0.07 rad s-1 angular frequency Alfvén
wave) showing from top: Elsässer variables in km s-1 ($\delta
B/\sqrt{4\pi\rho}$ solid lines, $\delta v$ dashed lines), with black lines for
real parts and gray lines for imaginary parts. Middle; wave energy fluxes in
ergs cm-2 s-1, the thin solid line shows the difference in energy fluxes
divided by the magnetic field strength and should be a horizontal line if
energy is properly conserved. Bottom, the ponderomotive acceleration in cm
s-2. Positive acceleration means positive along the $z$ axis, which is upwards
pointing near $z=0$ and downwards near $z=100,000$. Figure 3: Same as figure 2
giving the first three panels for the left hand chromosphere “B”, where waves
leak down from the corona. The extra bottom right panel shows the FIP
fractionations (in black) for the ratios Fe/O, Mg/O, S/Oand He/O.
Chromospheric ionization fractions are also shown in the fourth panel (in
gray, to be read on the left hand axis) in the same linestyles as for the
fractionation. The extra dash-triple dot gives the O ionization fraction, the
long dash curve gives the H ionization fraction. Fe and Mg are essentially
fully ionized throughout the fractionation region. Figure 4: Same as figure 2
for an open field region. Five waves corresponding to the baseline model in
Table 2 are considered, initiated at 500,000 km altitude and integrated back
to the chromosphere. Figure 5: Chromospheric section of open field calculation
of Figure 4. Fractionations are shown for Fe/O, Ar/O and He/O. Figure 6:
Ponderomotive force (top panels) and FIP fractionations (bottom panels) for
the chromosphere including five chromospheric waves and a coronal wave with
angular frequency 0.06 (left) and 0.07 rad s-1 (right) respectively.
Fractionations for Fe/O, Mg/O, S/O, and He/O are shown. Figure 7:
Ponderomotive force (top panels) and FIP fractionations (bottom panels) for
the chromosphere including five chromospheric waves and a coronal wave with
angular frequency 0.085 (left) and 0.0105 rad s-1 (right) respectively.
Fractionations for Fe/O, Mg/O, S/O, and He/O are shown. The 0.085 coronal wave
gives a more similar FIP effect for Fe/O and Mg/O, as frequently observed. The
He/O depletion reduces as the coronal wave moves off resonance, as the “spike”
in the ponderomotive acceleration decreases in prominence. Figure 8:
Illustration of FIP fractionation for the 0.07 rad s-1 wave, showing the
correspondence with important features of the chromosphere. Top left gives the
chromospheric density and temperature with height. Bottom left gives the
ponderomotive acceleration as before, with solid and dashed curves showing the
acceleration with and without energy loss to slow mode waves. The dotted curve
shows the slow mode wave amplitude. The left panels show the “spike” in the
ponderomotive acceleration at the altitude where the chromospheric density
gradient is strongest. Bottom right is the same as before, with FIP
fractionations for Fe/O, Mg/O, He/O, and S/O, together with ionization
fractions. Top right shows the ionization fractions for C, S, Mg, and Fe in an
expanded view. Thick curves correspond to the “baseline” charge state
fractions used for FIP fractionation throughout this paper, thin curves give
the results of the Saha approximation. This overestimates ionization fraction
close to the top of the chromosphere, because photons resulting from radiative
recombinations are not allowed to escape, but remain trapped to cause further
photoionization.
|
arxiv-papers
| 2011-10-19T19:29:31 |
2024-09-04T02:49:23.374818
|
{
"license": "Public Domain",
"authors": "J. Martin Laming",
"submitter": "Martin Laming",
"url": "https://arxiv.org/abs/1110.4357"
}
|
1110.4414
|
# $(1+\epsilon)$-approximate Sparse Recovery
Eric Price
MIT David P. Woodruff
IBM Almaden
(2011-08-12)
###### Abstract
The problem central to sparse recovery and compressive sensing is that of
_stable sparse recovery_ : we want a distribution $\mathcal{A}$ of matrices
$A\in\mathbb{R}^{m\times n}$ such that, for any $x\in\mathbb{R}^{n}$ and with
probability $1-\delta>2/3$ over $A\in\mathcal{A}$, there is an algorithm to
recover $\hat{x}$ from $Ax$ with
$\displaystyle\left\lVert\hat{x}-x\right\rVert_{p}\leq C\min_{k\text{-sparse
}x^{\prime}}\left\lVert x-x^{\prime}\right\rVert_{p}$ (1)
for some constant $C>1$ and norm $p$.
The measurement complexity of this problem is well understood for constant
$C>1$. However, in a variety of applications it is important to obtain
$C=1+\epsilon$ for a small $\epsilon>0$, and this complexity is not well
understood. We resolve the dependence on $\epsilon$ in the number of
measurements required of a $k$-sparse recovery algorithm, up to
polylogarithmic factors for the central cases of $p=1$ and $p=2$. Namely, we
give new algorithms and lower bounds that show the number of measurements
required is $k/\epsilon^{p/2}\textrm{polylog}(n)$. For $p=2$, our bound of
$\frac{1}{\epsilon}k\log(n/k)$ is tight up to _constant_ factors. We also give
matching bounds when the output is required to be $k$-sparse, in which case we
achieve $k/\epsilon^{p}\textrm{polylog}(n)$. This shows the distinction
between the complexity of sparse and non-sparse outputs is fundamental.
## 1 Introduction
Over the last several years, substantial interest has been generated in the
problem of solving underdetermined linear systems subject to a sparsity
constraint. The field, known as _compressed sensing_ or _sparse recovery_ ,
has applications to a wide variety of fields that includes data stream
algorithms [Mut05], medical or geological imaging [CRT06, Don06], and genetics
testing [SAZ10]. The approach uses the power of a _sparsity_ constraint: a
vector $x^{\prime}$ is _$k$ -sparse_ if at most $k$ coefficients are non-zero.
A standard formulation for the problem is that of _stable sparse recovery_ :
we want a distribution $\mathcal{A}$ of matrices $A\in\mathbb{R}^{m\times n}$
such that, for any $x\in\mathbb{R}^{n}$ and with probability $1-\delta>2/3$
over $A\in\mathcal{A}$, there is an algorithm to recover $\hat{x}$ from $Ax$
with
$\displaystyle\left\lVert\hat{x}-x\right\rVert_{p}\leq C\min_{k\text{-sparse
}x^{\prime}}\left\lVert x-x^{\prime}\right\rVert_{p}$ (2)
for some constant $C>1$ and norm $p$111Some formulations allow the two norms
to be different, in which case $C$ is not constant. We only consider equal
norms in this paper.. We call this a _$C$ -approximate $\ell_{p}/\ell_{p}$
recovery scheme_ with _failure probability $\delta$_. We refer to the elements
of $Ax$ as _measurements_.
It is known [CRT06, GLPS10] that such recovery schemes exist for
$p\in\\{1,2\\}$ with $C=O(1)$ and $m=O(k\log\frac{n}{k})$. Furthermore, it is
known [DIPW10, FPRU10] that any such recovery scheme requires
$\Omega(k\log_{1+C}\frac{n}{k})$ measurements. This means the measurement
complexity is well understood for $C=1+\Omega(1)$, but not for $C=1+o(1)$.
A number of applications would like to have $C=1+\epsilon$ for small
$\epsilon$. For example, a radio wave signal can be modeled as $x=x^{*}+w$
where $x^{*}$ is $k$-sparse (corresponding to a signal over a narrow band) and
the noise $w$ is i.i.d. Gaussian with $\left\lVert w\right\rVert_{p}\approx
D\left\lVert x^{*}\right\rVert_{p}$ [TDB09]. Then sparse recovery with
$C=1+\alpha/D$ allows the recovery of a $(1-\alpha)$ fraction of the true
signal $x^{*}$. Since $x^{*}$ is concentrated in a small band while $w$ is
located over a large region, it is often the case that $\alpha/D\ll 1$.
The difficulty of $(1+\epsilon)$-approximate recovery has seemed to depend on
whether the output $x^{\prime}$ is required to be $k$-sparse or can have more
than $k$ elements in its support. Having $k$-sparse output is important for
some applications (e.g. the aforementioned radio waves) but not for others
(e.g. imaging). Algorithms that output a $k$-sparse $x^{\prime}$ have used
$\Theta(\frac{1}{\epsilon^{p}}k\log n)$ measurements [CCF02, CM04, CM06,
Wai09]. In contrast, [GLPS10] uses only $\Theta(\frac{1}{\epsilon}k\log(n/k))$
measurements for $p=2$ and outputs a non-$k$-sparse $x^{\prime}$.
| | Lower bound | Upper bound
---|---|---|---
$k$-sparse output | $\ell_{1}$ | $\Omega(\frac{1}{\epsilon}(k\log\frac{1}{\epsilon}+\log\frac{1}{\delta}))$ | $O(\frac{1}{\epsilon}k\log n)$[CM04]
| $\ell_{2}$ | $\Omega(\frac{1}{\epsilon^{2}}(k+\log\frac{1}{\delta}))$ | $O(\frac{1}{\epsilon^{2}}k\log n)$[CCF02, CM06, Wai09]
Non-$k$-sparse output | $\ell_{1}$ | $\Omega(\frac{1}{\sqrt{\epsilon}\log^{2}(k/\epsilon)}k)$ | $O(\frac{\log^{3}(1/\epsilon)}{\sqrt{\epsilon}}k\log n)$
| $\ell_{2}$ | $\Omega(\frac{1}{\epsilon}k\log(n/k))$ | $O(\frac{1}{\epsilon}k\log(n/k))$[GLPS10]
Figure 1: Our results, along with existing upper bounds. Fairly minor
restrictions on the relative magnitude of parameters apply; see the theorem
statements for details.
#### Our results
We show that the apparent distinction between complexity of sparse and non-
sparse outputs is fundamental, for both $p=1$ and $p=2$. We show that for
sparse output, $\Omega(k/\epsilon^{p})$ measurements are necessary, matching
the upper bounds up to a $\log n$ factor. For general output and $p=2$, we
show $\Omega(\frac{1}{\epsilon}k\log(n/k))$ measurements are necessary,
matching the upper bound up to a constant factor. In the remaining case of
general output and $p=1$, we show $\widetilde{\Omega}(k/\sqrt{\epsilon})$
measurements are necessary. We then give a novel algorithm that uses
$O(\frac{\log^{3}(1/\epsilon)}{\sqrt{\epsilon}}k\log n)$ measurements, beating
the $1/\epsilon$ dependence given by all previous algorithms. As a result, all
our bounds are tight up to factors logarithmic in $n$. The full results are
shown in Figure 1.
In addition, for $p=2$ and general output, we show that thresholding the top
$2k$ elements of a Count-Sketch [CCF02] estimate gives
$(1+\epsilon)$-approximate recovery with $\Theta(\frac{1}{\epsilon}k\log n)$
measurements. This is interesting because it highlights the distinction
between sparse output and non-sparse output: [CM06] showed that thresholding
the top $k$ elements of a Count-Sketch estimate requires
$m=\Theta(\frac{1}{\epsilon^{2}}k\log n)$. While [GLPS10] achieves
$m=\Theta(\frac{1}{\epsilon}k\log(n/k))$ for the same regime, it only succeeds
with constant probability while ours succeeds with probability
$1-n^{-\Omega(1)}$; hence ours is the most efficient known algorithm when
$\delta=o(1),\epsilon=o(1),$ and $k<n^{0.9}$.
#### Related work
Much of the work on sparse recovery has relied on the Restricted Isometry
Property [CRT06]. None of this work has been able to get better than
$2$-approximate recovery, so there are relatively few papers achieving
$(1+\epsilon)$-approximate recovery. The existing ones with $O(k\log n)$
measurements are surveyed above (except for [IR08], which has worse dependence
on $\epsilon$ than [CM04] for the same regime).
A couple of previous works have studied the $\ell_{\infty}/\ell_{p}$ problem,
where every coordinate must be estimated with small error. This problem is
harder than $\ell_{p}/\ell_{p}$ sparse recovery with sparse output. For $p=2$,
[Wai09] showed that schemes using Gaussian matrices $A$ require
$m=\Omega(\frac{1}{\epsilon^{2}}k\log(n/k))$. For $p=1$, [CM05] showed that
any sketch requires $\Omega(k/\epsilon)$ bits (rather than measurements).
Independently of this work and of each other, multiple authors [CD11, IT10,
ASZ10] have matched our $\Omega(\frac{1}{\epsilon}k\log(n/k))$ bound for
$\ell_{2}/\ell_{2}$ in related settings. The details vary, but all proofs are
broadly similar in structure to ours: they consider observing a large set of
“well-separated” vectors under Gaussian noise. Fano’s inequality gives a lower
bound on the mutual information between the observation and the signal; then,
an upper bound on the mutual information is given by either the Shannon-
Hartley theorem or a KL-divergence argument. This technique does not seem
useful for the other problems we consider in this paper, such as lower bounds
for $\ell_{1}/\ell_{1}$ or the sparse output setting.
#### Our techniques
For the upper bounds for non-sparse output, we observe that the hard case for
sparse output is when the noise is fairly concentrated, in which the
estimation of the top $k$ elements can have $\sqrt{\epsilon}$ error. Our goal
is to recover enough mass from outside the top $k$ elements to cancel this
error. The upper bound for $p=2$ is a fairly straightforward analysis of the
top $2k$ elements of a Count-Sketch data structure.
The upper bound for $p=1$ proceeds by subsampling the vector at rate $2^{-i}$
and performing a Count-Sketch with size proportional to
$\frac{1}{\sqrt{\epsilon}}$, for $i\in\\{0,1,\dotsc,O(\log(1/\epsilon))\\}$.
The intuition is that if the noise is well spread over many (more than
$k/\epsilon^{3/2}$) coordinates, then the $\ell_{2}$ bound from the first
Count-Sketch gives a very good $\ell_{1}$ bound, so the approximation is
$(1+\epsilon)$-approximate. However, if the noise is concentrated over a small
number $k/\epsilon^{c}$ of coordinates, then the error from the first Count-
Sketch is proportional to $1+\epsilon^{c/2+1/4}$. But in this case, one of the
subsamples will only have $O(k/\epsilon^{c/2-1/4})<k/\sqrt{\epsilon}$ of the
coordinates with large noise. We can then recover those coordinates with the
Count-Sketch for that subsample. Those coordinates contain an
$\epsilon^{c/2+1/4}$ fraction of the total noise, so recovering them decreases
the approximation error by exactly the error induced from the first Count-
Sketch.
The lower bounds use substantially different techniques for sparse output and
for non-sparse output. For sparse output, we use reductions from communication
complexity to show a lower bound in terms of bits. Then, as in [DIPW10], we
embed $\Theta(\log n)$ copies of this communication problem into a single
vector. This multiplies the bit complexity by $\log n$; we also show we can
round $Ax$ to $\log n$ bits per measurement without affecting recovery, giving
a lower bound in terms of measurements.
We illustrate the lower bound on bit complexity for sparse output using $k=1$.
Consider a vector $x$ containing $1/\epsilon^{p}$ ones and zeros elsewhere,
such that $x_{2i}+x_{2i+1}=1$ for all $i$. For any $i$, set
$z_{2i}=z_{2i+1}=1$ and $z_{j}=0$ elsewhere. Then successful
$(1+\epsilon/3)$-approximate sparse recovery from $A(x+z)$ returns $\hat{z}$
with $\operatorname{supp}(\hat{z})=\operatorname{supp}(x)\cap\\{2i,2i+1\\}$.
Hence we can recover each bit of $x$ with probability $1-\delta$, requiring
$\Omega(1/\epsilon^{p})$ bits222For $p=1$, we can actually set
$\left|\operatorname{supp}(z)\right|=1/\epsilon$ and search among a set of
$1/\epsilon$ candidates. This gives
$\Omega(\frac{1}{\epsilon}\log(1/\epsilon))$ bits.. We can generalize this to
$k$-sparse output for $\Omega(k/\epsilon^{p})$ bits, and to $\delta$ failure
probability with $\Omega(\frac{1}{\epsilon^{p}}\log\frac{1}{\delta})$.
However, the two generalizations do not seem to combine.
For non-sparse output, we split between $\ell_{2}$ and $\ell_{1}$. In
$\ell_{2}$, we consider $A(x+w)$ where $x$ is sparse and $w$ has uniform
Gaussian noise with $\left\lVert w\right\rVert_{2}^{2}\approx\left\lVert
x\right\rVert_{2}^{2}/\epsilon$. Then each coordinate of $y=A(x+w)=Ax+Aw$ is a
Gaussian channel with signal to noise ratio $\epsilon$. This channel has
channel capacity $\epsilon$, showing $I(y;x)\leq\epsilon m$. Correct sparse
recovery must either get most of $x$ or an $\epsilon$ fraction of $w$; the
latter requires $m=\Omega(\epsilon n)$ and the former requires
$I(y;x)=\Omega(k\log(n/k))$. This gives a tight
$\Theta(\frac{1}{\epsilon}k\log(n/k))$ result. Unfortunately, this does not
easily extend to $\ell_{1}$, because it relies on the Gaussian distribution
being both stable and maximum entropy under $\ell_{2}$; the corresponding
distributions in $\ell_{1}$ are not the same.
Therefore for $\ell_{1}$ non-sparse output, we have yet another argument. The
hard instances for $k=1$ must have one large value (or else $0$ is a valid
output) but small other values (or else the $2$-sparse approximation is
significantly better than the $1$-sparse approximation). Suppose $x$ has one
value of size $\epsilon$ and $d$ values of size $1/d$ spread through a vector
of size $d^{2}$. Then a $(1+\epsilon/2)$-approximate recovery scheme must
either locate the large element or guess the locations of the $d$ values with
$\Omega(\epsilon d)$ more correct than incorrect. The former requires
$1/(d\epsilon^{2})$ bits by the difficulty of a novel version of the
Gap-$\ell_{\infty}$ problem. The latter requires $\epsilon d$ bits because it
allows recovering an error correcting code. Setting $d=\epsilon^{-3/2}$
balances the terms at $\epsilon^{-1/2}$ bits. Because some of these reductions
are very intricate, this extended abstract does not manage to embed $\log n$
copies of the problem into a single vector. As a result, we lose a $\log n$
factor in a universe of size $n=\text{poly}(k/\epsilon)$ when converting to
measurement complexity from bit complexity.
## 2 Preliminaries
#### Notation
We use $[n]$ to denote the set $\\{1\ldots n\\}$. For any set $S\subset[n]$,
we use $\overline{S}$ to denote the complement of $S$, i.e., the set
$[n]\setminus S$. For any $x\in\mathbb{R}^{n}$, $x_{i}$ denotes the $i$th
coordinate of $x$, and $x_{S}$ denotes the vector
$x^{\prime}\in\mathbb{R}^{n}$ given by $x^{\prime}_{i}=x_{i}$ if $i\in S$, and
$x^{\prime}_{i}=0$ otherwise. We use $\operatorname{supp}(x)$ to denote the
support of $x$.
## 3 Upper bounds
The algorithms in this section are indifferent to permutation of the
coordinates. Therefore, for simplicity of notation in the analysis, we assume
the coefficients of $x$ are sorted such that
$\left|x_{1}\right|\geq\left|x_{2}\right|\geq\dotsc\geq\left|x_{n}\right|\geq
0$.
#### Count-Sketch
Both our upper bounds use the Count-Sketch [CCF02] data structure. The
structure consists of $c\log n$ hash tables of size $O(q)$, for $O(cq\log n)$
total space; it can be represented as $Ax$ for a matrix $A$ with $O(cq\log n)$
rows. Given $Ax$, one can construct $x^{*}$ with
$\displaystyle\left\lVert
x^{*}-x\right\rVert_{\infty}^{2}\leq\frac{1}{q}\left\lVert
x_{\overline{[q]}}\right\rVert_{2}^{2}$ (3)
with failure probability $n^{1-c}$.
### 3.1 Non-sparse $\ell_{2}$
It was shown in [CM06] that, if $x^{*}$ is the result of a Count-Sketch with
hash table size $O(k/\epsilon^{2})$, then outputting the top $k$ elements of
$x^{*}$ gives a $(1+\epsilon)$-approximate $\ell_{2}/\ell_{2}$ recovery
scheme. Here we show that a seemingly minor change—selecting $2k$ elements
rather than $k$ elements—turns this into a $(1+\epsilon^{2})$-approximate
$\ell_{2}/\ell_{2}$ recovery scheme.
###### Theorem 3.1.
Let $\hat{x}$ be the top $2k$ estimates from a Count-Sketch structure with
hash table size $O(k/\epsilon)$. Then with failure probability
$n^{-\Omega(1)}$,
$\left\lVert\hat{x}-x\right\rVert_{2}\leq(1+\epsilon)\left\lVert
x_{\overline{[k]}}\right\rVert_{2}.$
Therefore, there is a $1+\epsilon$-approximate $\ell_{2}/\ell_{2}$ recovery
scheme with $O(\frac{1}{\epsilon}k\log n)$ rows.
###### Proof.
Let the hash table size be $O(ck/\epsilon)$ for constant $c$, and let $x^{*}$
be the vector of estimates for each coordinate. Define $S$ to be the indices
of the largest $2k$ values in $x^{*}$, and $E=\left\lVert
x_{\overline{[k]}}\right\rVert_{2}$.
By (3), the standard analysis of Count-Sketch:
$\left\lVert x^{*}-x\right\rVert_{\infty}^{2}\leq\frac{\epsilon}{ck}E^{2}.$
so
$\displaystyle\left\lVert x^{*}_{S}-x\right\rVert_{2}^{2}-E^{2}=\left\lVert
x^{*}_{S}-x\right\rVert_{2}^{2}-\left\lVert
x_{\overline{[k]}}\right\rVert_{2}^{2}\leq$
$\displaystyle\left\lVert(x^{*}-x)_{S}\right\rVert_{2}^{2}+\left\lVert
x_{[n]\setminus S}\right\rVert_{2}^{2}-\left\lVert
x_{\overline{[k]}}\right\rVert_{2}^{2}$ $\displaystyle\leq$
$\displaystyle\left|S\right|\left\lVert
x^{*}-x\right\rVert_{\infty}^{2}+\left\lVert x_{[k]\setminus
S}\right\rVert_{2}^{2}-\left\lVert x_{S\setminus[k]}\right\rVert_{2}^{2}$
$\displaystyle\leq$ $\displaystyle\frac{2\epsilon}{c}E^{2}+\left\lVert
x_{[k]\setminus S}\right\rVert_{2}^{2}-\left\lVert
x_{S\setminus[k]}\right\rVert_{2}^{2}$ (4)
Let $a=\max_{i\in[k]\setminus S}x_{i}$ and $b=\min_{i\in S\setminus[k]}x_{i}$,
and let $d=\left|[k]\setminus S\right|$. The algorithm passes over an element
of value $a$ to choose one of value $b$, so
$a\leq b+2\left\lVert x^{*}-x\right\rVert_{\infty}\leq
b+2\sqrt{\frac{\epsilon}{ck}}E.$
Then
$\displaystyle\left\lVert x_{[k]\setminus S}\right\rVert_{2}^{2}-\left\lVert
x_{S\setminus[k]}\right\rVert_{2}^{2}\leq$ $\displaystyle da^{2}-(k+d)b^{2}$
$\displaystyle\leq$ $\displaystyle
d(b+2\sqrt{\frac{\epsilon}{ck}}E)^{2}-(k+d)b^{2}$ $\displaystyle\leq$
$\displaystyle-
kb^{2}+4\sqrt{\frac{\epsilon}{ck}}dbE+\frac{4\epsilon}{ck}dE^{2}$
$\displaystyle\leq$
$\displaystyle-k(b-2\sqrt{\frac{\epsilon}{ck^{3}}}dE)^{2}+\frac{4\epsilon}{ck^{2}}dE^{2}(k-d)$
$\displaystyle\leq$
$\displaystyle\frac{4d(k-d)\epsilon}{ck^{2}}E^{2}\leq\frac{\epsilon}{c}E^{2}$
and combining this with (4) gives
$\left\lVert
x^{*}_{S}-x\right\rVert_{2}^{2}-E^{2}\leq\frac{3\epsilon}{c}E^{2}$
or
$\left\lVert x^{*}_{S}-x\right\rVert_{2}\leq(1+\frac{3\epsilon}{2c})E$
which proves the theorem for $c\geq 3/2$. ∎
### 3.2 Non-sparse $\ell_{1}$
###### Theorem 3.2.
There exists a $(1+\epsilon)$-approximate $\ell_{1}/\ell_{1}$ recovery scheme
with $O(\frac{\log^{3}1/\epsilon}{\sqrt{\epsilon}}k\log n)$ measurements and
failure probability $e^{-\Omega(k/\sqrt{\epsilon})}+n^{-\Omega(1)}$.
Set $f=\sqrt{\epsilon}$, so our goal is to get $(1+f^{2})$-approximate
$\ell_{1}/\ell_{1}$ recovery with $O(\frac{\log^{3}1/f}{f}k\log n)$
measurements.
For intuition, consider 1-sparse recovery of the following vector $x$: let
$c\in[0,2]$ and set $x_{1}=1/f^{9}$ and $x_{2},\dotsc,x_{1+1/f^{1+c}}\in\\{\pm
1\\}$. Then we have
$\displaystyle\left\lVert x_{\overline{[1]}}\right\rVert_{1}$
$\displaystyle=1/f^{1+c}$
and by (3), a Count-Sketch with $O(1/f)$-sized hash tables returns $x^{*}$
with
$\displaystyle\left\lVert x^{*}-x\right\rVert_{\infty}\leq\sqrt{f}\left\lVert
x_{\overline{[1/f]}}\right\rVert_{2}\approx 1/f^{c/2}=f^{1+c/2}\left\lVert
x_{\overline{[1]}}\right\rVert_{1}.$
The reconstruction algorithm therefore cannot reliably find any of the $x_{i}$
for $i>1$, and its error on $x_{1}$ is at least $f^{1+c/2}\left\lVert
x_{\overline{[1]}}\right\rVert_{1}$. Hence the algorithm will not do better
than a $f^{1+c/2}$-approximation.
However, consider what happens if we subsample an $f^{c}$ fraction of the
vector. The result probably has about $1/f$ non-zero values, so a
$O(1/f)$-width Count-Sketch can reconstruct it exactly. Putting this in our
output improves the overall $\ell_{1}$ error by about $1/f=f^{c}\left\lVert
x_{\overline{[1]}}\right\rVert_{1}$. Since $c<2$, this more than cancels the
$f^{1+c/2}\left\lVert x_{\overline{[1]}}\right\rVert_{1}$ error the initial
Count-Sketch makes on $x_{1}$, giving an approximation factor better than $1$.
This tells us that subsampling can help. We don’t need to subsample at a scale
below $k/f$ (where we can reconstruct well already) or above $k/f^{3}$ (where
the $\ell_{2}$ bound is small enough already), but in the intermediate range
we need to subsample. Our algorithm subsamples at all $\log 1/f^{2}$ rates in
between these two endpoints, and combines the heavy hitters from each.
First we analyze how subsampled Count-Sketch works.
###### Lemma 3.3.
Suppose we subsample with probability $p$ and then apply Count-Sketch with
$\Theta(\log n)$ rows and $\Theta(q)$-sized hash tables. Let $y$ be the
subsample of $x$. Then with failure probability
$e^{-\Omega(q)}+n^{-\Omega(1)}$ we recover a $y^{*}$ with
$\left\lVert y^{*}-y\right\rVert_{\infty}\leq\sqrt{p/q}\left\lVert
x_{\overline{[q/p]}}\right\rVert_{2}.$
###### Proof.
Recall the following form of the Chernoff bound: if $X_{1},\dotsc,X_{m}$ are
independent with $0\leq X_{i}\leq M$, and $\mu\geq\operatorname{E}[\sum
X_{i}]$, then
$\Pr[\sum X_{i}\geq\frac{4}{3}\mu]\leq e^{-\Omega(\mu/M)}.$
Let $T$ be the set of coordinates in the sample. Then
$\operatorname{E}[\left|T\cap[\frac{3q}{2p}]\right|]=3q/2$, so
$\Pr\left[\left|T\cap[\frac{3q}{2p}]\right|\geq 2q\right]\leq e^{-\Omega(q)}.$
Suppose this event does not happen, so $\left|T\cap[\frac{3q}{2p}]\right|<2q$.
We also have
$\left\lVert
x_{\overline{[q/p]}}\right\rVert_{2}\geq\sqrt{\frac{q}{2p}}\left|x_{\frac{3q}{2p}}\right|.$
Let $Y_{i}=0$ if $i\notin T$ and $Y_{i}=x_{i}^{2}$ if $i\in T$. Then
$\operatorname{E}[\sum_{i>\frac{3q}{2p}}Y_{i}]=p\left\lVert
x_{\overline{[\frac{3q}{2p}]}}\right\rVert_{2}^{2}\leq p\left\lVert
x_{\overline{[q/p]}}\right\rVert_{2}^{2}$
For $i>\frac{3q}{2p}$ we have
$Y_{i}\leq\left|x_{\frac{3q}{2p}}\right|^{2}\leq\frac{2p}{q}\left\lVert
x_{\overline{[q/p]}}\right\rVert_{2}^{2}$
giving by Chernoff that
$\displaystyle\Pr[\sum Y_{i}\geq\frac{4}{3}p\left\lVert
x_{\overline{[q/p]}}\right\rVert_{2}^{2}]\leq e^{-\Omega(q/2)}$
But if this event does not happen, then
$\displaystyle\left\lVert
y_{\overline{[2q]}}\right\rVert_{2}^{2}\leq\sum_{i\in
T,i>\frac{3q}{2p}}x_{i}^{2}=\sum_{i>\frac{3q}{2p}}Y_{i}\leq\frac{4}{3}p\left\lVert
x_{\overline{[q/p]}}\right\rVert_{2}^{2}$
By (3), using $O(2q)$-size hash tables gives a $y^{*}$ with
$\left\lVert y^{*}-y\right\rVert_{\infty}\leq\frac{1}{\sqrt{2q}}\left\lVert
y_{\overline{[2q]}}\right\rVert_{2}\leq\sqrt{p/q}\left\lVert
x_{\overline{[q/p]}}\right\rVert_{2}$
with failure probability $n^{-\Omega(1)}$, as desired. ∎
Let $r=2\log 1/f$. Our algorithm is as follows: for $j\in\\{0,\dotsc,r\\}$, we
find and estimate the $2^{j/2}k$ largest elements not found in previous $j$ in
a subsampled Count-Sketch with probability $p=2^{-j}$ and hash size $q=ck/f$
for some parameter $c=\Theta(r^{2})$. We output $\hat{x}$, the union of all
these estimates. Our goal is to show
$\displaystyle\left\lVert\hat{x}-x\right\rVert_{1}-\left\lVert
x_{\overline{[k]}}\right\rVert_{1}\leq O(f^{2})\left\lVert
x_{\overline{[k]}}\right\rVert_{1}.$
For each level $j$, let $S_{j}$ be the $2^{j/2}k$ largest coordinates in our
estimate not found in $S_{1}\cup\dotsb\cup S_{j-1}$. Let $S=\cup S_{j}$. By
Lemma 3.3, for each $j$ we have (with failure probability
$e^{-\Omega(k/f)}+n^{-\Omega(1)}$) that
$\displaystyle\left\lVert(\hat{x}-x)_{S_{j}}\right\rVert_{1}$
$\displaystyle\leq\left|S_{j}\right|\sqrt{\frac{2^{-j}f}{ck}}\left\lVert
x_{\overline{[2^{j}ck/f]}}\right\rVert_{2}$ $\displaystyle\leq
2^{-j/2}\sqrt{\frac{fk}{c}}\left\lVert x_{\overline{[2k/f]}}\right\rVert_{2}$
and so
$\displaystyle\left\lVert(\hat{x}-x)_{S}\right\rVert_{1}$
$\displaystyle=\sum_{j=0}^{r}\left\lVert(\hat{x}-x)_{S_{j}}\right\rVert_{1}\leq\frac{1}{(1-1/\sqrt{2})\sqrt{c}}\sqrt{fk}\left\lVert
x_{\overline{[2k/f]}}\right\rVert_{2}$ (5)
By standard arguments, the $\ell_{\infty}$ bound for $S_{0}$ gives
$\displaystyle\left\lVert x_{[k]}\right\rVert_{1}\leq\left\lVert
x_{S_{0}}\right\rVert_{1}+k\left\lVert\hat{x}_{S_{0}}-x_{S_{0}}\right\rVert_{\infty}\leq\sqrt{fk/c}\left\lVert
x_{\overline{[2k/f]}}\right\rVert_{2}$ (6)
Combining Equations (5) and (6) gives
$\displaystyle\left\lVert\hat{x}-x\right\rVert_{1}-\left\lVert
x_{\overline{[k]}}\right\rVert_{1}=$
$\displaystyle\left\lVert(\hat{x}-x)_{S}\right\rVert_{1}+\left\lVert
x_{\overline{S}}\right\rVert_{1}-\left\lVert
x_{\overline{[k]}}\right\rVert_{1}$ $\displaystyle=$
$\displaystyle\left\lVert(\hat{x}-x)_{S}\right\rVert_{1}+\left\lVert
x_{[k]}\right\rVert_{1}-\left\lVert x_{S}\right\rVert_{1}$ $\displaystyle=$
$\displaystyle\left\lVert(\hat{x}-x)_{S}\right\rVert_{1}+(\left\lVert
x_{[k]}\right\rVert_{1}-\left\lVert
x_{S_{0}}\right\rVert_{1})-\sum_{j=1}^{r}\left\lVert
x_{S_{j}}\right\rVert_{1}$ $\displaystyle\leq$
$\displaystyle\left(\frac{1}{(1-1/\sqrt{2})\sqrt{c}}+\frac{1}{\sqrt{c}}\right)\sqrt{fk}\left\lVert
x_{\overline{[2k/f]}}\right\rVert_{2}-\sum_{j=1}^{r}\left\lVert
x_{S_{j}}\right\rVert_{1}$ $\displaystyle=$ $\displaystyle
O(\frac{1}{\sqrt{c}})\sqrt{fk}\left\lVert
x_{\overline{[2k/f]}}\right\rVert_{2}-\sum_{j=1}^{r}\left\lVert
x_{S_{j}}\right\rVert_{1}$ (7)
We would like to convert the first term to depend on the $\ell_{1}$ norm. For
any $u$ and $s$ we have, by splitting into chunks of size $s$, that
$\displaystyle\left\lVert
u_{\overline{[2s]}}\right\rVert_{2}\leq\sqrt{\frac{1}{s}}\left\lVert
u_{\overline{[s]}}\right\rVert_{1}$ $\displaystyle\left\lVert
u_{\overline{[s]}\cap[2s]}\right\rVert_{2}\leq\sqrt{s}\left|u_{s}\right|.$
Along with the triangle inequality, this gives us that
$\displaystyle\sqrt{kf}\left\lVert x_{\overline{[2k/f]}}\right\rVert_{2}$
$\displaystyle\leq\sqrt{kf}\left\lVert
x_{\overline{[2k/f^{3}]}}\right\rVert_{2}+\sqrt{kf}\sum_{j=1}^{r}\left\lVert
x_{\overline{[2^{j}k/f]}\cap[2^{j+1}k/f]}\right\rVert_{2}$ $\displaystyle\leq
f^{2}\left\lVert
x_{\overline{[k/f^{3}]}}\right\rVert_{1}+\sum_{j=1}^{r}k2^{j/2}\left|x_{2^{j}k/f}\right|$
so
$\displaystyle\left\lVert\hat{x}-x\right\rVert_{1}-\left\lVert
x_{\overline{[k]}}\right\rVert_{1}\leq$ $\displaystyle
O(\frac{1}{\sqrt{c}})f^{2}\left\lVert
x_{\overline{[k/f^{3}]}}\right\rVert_{1}+\sum_{j=1}^{r}O(\frac{1}{\sqrt{c}})k2^{j/2}\left|x_{2^{j}k/f}\right|-\sum_{j=1}^{r}\left\lVert
x_{S_{j}}\right\rVert_{1}$ (8)
Define $a_{j}=k2^{j/2}\left|x_{2^{j}k/f}\right|$. The first term grows as
$f^{2}$ so it is fine, but $a_{j}$ can grow as $f2^{j/2}>f^{2}$. We need to
show that they are canceled by the corresponding $\left\lVert
x_{S_{j}}\right\rVert_{1}$. In particular, we will show that $\left\lVert
x_{S_{j}}\right\rVert_{1}\geq\Omega(a_{j})-O(2^{-j/2}f^{2}\left\lVert
x_{\overline{[k/f^{3}]}}\right\rVert_{1})$ with high probability—at least
wherever $a_{j}\geq\left\lVert a\right\rVert_{1}/(2r)$.
Let $U\in[r]$ be the set of $j$ with $a_{j}\geq\left\lVert
a\right\rVert_{1}/(2r)$, so that $\left\lVert
a_{U}\right\rVert_{1}\geq\left\lVert a\right\rVert_{1}/2$. We have
$\displaystyle\left\lVert x_{\overline{[2^{j}k/f]}}\right\rVert_{2}^{2}$
$\displaystyle=\left\lVert
x_{\overline{[2k/f^{3}]}}\right\rVert_{2}^{2}+\sum_{i=j}^{r}\left\lVert
x_{\overline{[2^{j}k/f]}\cap[2^{j+1}k/f]}\right\rVert_{2}^{2}$
$\displaystyle\leq\left\lVert
x_{\overline{[2k/f^{3}]}}\right\rVert_{2}^{2}+\frac{1}{kf}\sum_{i=j}^{r}a_{j}^{2}$
(9)
For $j\in U$, we have
$\displaystyle\sum_{i=j}^{r}a_{i}^{2}\leq a_{j}\left\lVert
a\right\rVert_{1}\leq 2ra_{j}^{2}$
so, along with $(y^{2}+z^{2})^{1/2}\leq y+z$, we turn Equation (9) into
$\displaystyle\left\lVert x_{\overline{[2^{j}k/f]}}\right\rVert_{2}$
$\displaystyle\leq\left\lVert
x_{\overline{[2k/f^{3}]}}\right\rVert_{2}+\sqrt{\frac{1}{kf}\sum_{i=j}^{r}a_{j}^{2}}$
$\displaystyle\leq\sqrt{\frac{f^{3}}{k}}\left\lVert
x_{\overline{[k/f^{3}]}}\right\rVert_{1}+\sqrt{\frac{2r}{kf}}a_{j}$
When choosing $S_{j}$, let $T\in[n]$ be the set of indices chosen in the
sample. Applying Lemma 3.3 the estimate $x^{*}$ of $x_{T}$ has
$\displaystyle\left\lVert x^{*}-x_{T}\right\rVert_{\infty}$
$\displaystyle\leq\sqrt{\frac{f}{2^{j}ck}}\left\lVert
x_{\overline{[2^{j}k/f]}}\right\rVert_{2}$
$\displaystyle\leq\sqrt{\frac{1}{2^{j}c}}\frac{f^{2}}{k}\left\lVert
x_{\overline{[k/f^{3}]}}\right\rVert_{1}+\sqrt{\frac{2r}{2^{j}c}}\frac{a_{j}}{k}$
$\displaystyle=\sqrt{\frac{1}{2^{j}c}}\frac{f^{2}}{k}\left\lVert
x_{\overline{[k/f^{3}]}}\right\rVert_{1}+\sqrt{\frac{2r}{c}}\left|x_{2^{j}k/f}\right|$
for $j\in U$.
Let $Q=[2^{j}k/f]\setminus(S_{0}\cup\dotsb\cup S_{j-1})$. We have
$\left|Q\right|\geq 2^{j-1}k/f$ so $\operatorname{E}[\left|Q\cap T\right|]\geq
k/2f$ and $\left|Q\cap T\right|\geq k/4f$ with failure probability
$e^{-\Omega(k/f)}$. Conditioned on $\left|Q\cap T\right|\geq k/4f$, since
$x_{T}$ has at least $\left|Q\cap T\right|\geq k/(4f)=2^{r/2}k/4\geq
2^{j/2}k/4$ possible choices of value at least $\left|x_{2^{j}k/f}\right|$,
$x_{S_{j}}$ must have at least $k2^{j/2}/4$ elements at least
$\left|x_{2^{j}k/f}\right|-\left\lVert x^{*}-x_{T}\right\rVert_{\infty}$.
Therefore, for $j\in U$,
$\displaystyle\left\lVert x_{S_{j}}\right\rVert_{1}$
$\displaystyle\geq-\frac{1}{4\sqrt{c}}f^{2}\left\lVert
x_{\overline{[k/f^{3}]}}\right\rVert_{1}+\frac{k2^{j/2}}{4}(1-\sqrt{\frac{2r}{c}})\left|x_{2^{j}k/f}\right|$
and therefore
$\displaystyle\sum_{j=1}^{r}\left\lVert
x_{S_{j}}\right\rVert_{1}\geq\sum_{j\in U}\left\lVert
x_{S_{j}}\right\rVert_{1}\geq$ $\displaystyle\sum_{j\in
U}-\frac{1}{4\sqrt{c}}f^{2}\left\lVert
x_{\overline{[k/f^{3}]}}\right\rVert_{1}+\frac{k2^{j/2}}{4}(1-\sqrt{\frac{2r}{c}})\left|x_{2^{j}k/f}\right|$
$\displaystyle\geq$ $\displaystyle-\frac{r}{4\sqrt{c}}f^{2}\left\lVert
x_{\overline{[k/f^{3}]}}\right\rVert_{1}+\frac{1}{4}(1-\sqrt{\frac{2r}{c}})\left\lVert
a_{U}\right\rVert_{1}$ $\displaystyle\geq$
$\displaystyle-\frac{r}{4\sqrt{c}}f^{2}\left\lVert
x_{\overline{[k/f^{3}]}}\right\rVert_{1}+\frac{1}{8}(1-\sqrt{\frac{2r}{c}})\sum_{j=1}^{r}k2^{j/2}\left|x_{2^{j}k/f}\right|$
(10)
Using (8) and (3.2) we get
$\displaystyle\left\lVert\hat{x}-x\right\rVert_{1}-\left\lVert
x_{\overline{[k]}}\right\rVert_{1}\leq$
$\displaystyle\left(\frac{r}{4\sqrt{c}}+O(\frac{1}{\sqrt{c}})\right)f^{2}\left\lVert
x_{\overline{[k/f^{3}]}}\right\rVert_{1}+\sum_{j=1}^{r}\left(O(\frac{1}{\sqrt{c}})+\frac{1}{8}\sqrt{\frac{2r}{c}}-\frac{1}{8}\right)k2^{j/2}\left|x_{2^{j}k/f}\right|$
$\displaystyle\leq$ $\displaystyle f^{2}\left\lVert
x_{\overline{[k/f^{3}]}}\right\rVert_{1}\leq f^{2}\left\lVert
x_{\overline{[k]}}\right\rVert_{1}$
for some $c=O(r^{2})$. Hence we use a total of $\frac{rc}{f}k\log
n=\frac{\log^{3}1/f}{f}k\log n$ measurements for $1+f^{2}$-approximate
$\ell_{1}/\ell_{1}$ recovery.
For each $j\in\\{0,\dotsc,r\\}$ we had failure probability
$e^{-\Omega(k/f)}+n^{-\Omega(1)}$ (from Lemma 3.3 and $\left|Q\cap
T\right|\geq k/2f$). By the union bound, our overall failure probability is at
most
$(\log\frac{1}{f})(e^{-\Omega(k/f)}+n^{-\Omega(1)})\leq
e^{-\Omega(k/f)}+n^{-\Omega(1)},$
proving Theorem 3.2.
## 4 Lower bounds for non-sparse output and $p=2$
In this case, the lower bound follows fairly straightforwardly from the
Shannon-Hartley information capacity of a Gaussian channel.
We will set up a communication game. Let
$\mathcal{F}\subset\\{S\subset[n]\mid\left|S\right|=k\\}$ be a family of
$k$-sparse supports such that:
* •
$\left|S\Delta S^{\prime}\right|\geq k$ for $S\neq S^{\prime}\in\mathcal{F}$,
* •
$\Pr_{S\in\mathcal{F}}[i\in S]=k/n$ for all $i\in[n]$, and
* •
$\log\left|\mathcal{F}\right|=\Omega(k\log(n/k))$.
This is possible; for example, a random linear code on $[n/k]^{k}$ with
relative distance $1/2$ has these properties [Gur10].333This assumes $n/k$ is
a prime power larger than 2. If $n/k$ is not prime, we can choose
$n^{\prime}\in[n/2,n]$ to be a prime multiple of $k$, and restrict to the
first $n^{\prime}$ coordinates. This works unless $n/k<3$, in which case a
bound of $\Theta(\min(n,\frac{1}{\epsilon}k\log(n/k)))=\Theta(k)$ is trivial.
Let $X=\\{x\in\\{0,\pm 1\\}^{n}\mid\operatorname{supp}(x)\in\mathcal{F}\\}$.
Let $w\sim N(0,\alpha\frac{k}{n}I_{n})$ be i.i.d. normal with variance $\alpha
k/n$ in each coordinate. Consider the following process:
#### Procedure
First, Alice chooses $S\in\mathcal{F}$ uniformly at random, then $x\in X$
uniformly at random subject to $\operatorname{supp}(x)=S$, then $w\sim
N(0,\alpha\frac{k}{n}I_{n})$. She sets $y=A(x+w)$ and sends $y$ to Bob. Bob
performs sparse recovery on $y$ to recover $x^{\prime}\approx x$, rounds to
$X$ by $\hat{x}=\operatorname*{arg\,min}_{\hat{x}\in
X}\left\lVert\hat{x}-x^{\prime}\right\rVert_{2}$, and sets
$S^{\prime}=\operatorname{supp}(\hat{x})$. This gives a Markov chain $S\to
x\to y\to x^{\prime}\to S^{\prime}$.
If sparse recovery works for any $x+w$ with probability $1-\delta$ as a
distribution over $A$, then there is some specific $A$ and random seed such
that sparse recovery works with probability $1-\delta$ over $x+w$; let us
choose this $A$ and the random seed, so that Alice and Bob run deterministic
algorithms on their inputs.
###### Lemma 4.1.
$I(S;S^{\prime})=O(m\log(1+\frac{1}{\alpha}))$.
###### Proof.
Let the columns of $A^{T}$ be $v^{1},\dotsc,v^{m}$. We may assume that the
$v^{i}$ are orthonormal, because this can be accomplished via a unitary
transformation on $Ax$. Then we have that $y_{i}=\langle
v^{i},x+w\rangle=\langle v^{i},x\rangle+w^{\prime}_{i}$, where
$w^{\prime}_{i}\sim N(0,\alpha k\left\lVert
v^{i}\right\rVert_{2}^{2}/n)=N(0,\alpha k/n)$ and
$\operatorname{E}_{x}[\langle
v^{i},x\rangle^{2}]=\operatorname{E}_{S}[\sum_{j\in
S}(v^{i}_{j})^{2}]=\frac{k}{n}$
Hence $y_{i}=z_{i}+w^{\prime}_{i}$ is a Gaussian channel with power constraint
$\operatorname{E}[z_{i}^{2}]\leq\frac{k}{n}\left\lVert
v^{i}\right\rVert_{2}^{2}$ and noise variance
$\operatorname{E}[(w^{\prime}_{i})^{2}]=\alpha\frac{k}{n}\left\lVert
v^{i}\right\rVert_{2}^{2}$. Hence by the Shannon-Hartley theorem this channel
has information capacity
$\max_{v_{i}}I(z_{i};y_{i})=C\leq\frac{1}{2}\log(1+\frac{1}{\alpha}).$
By the data processing inequality for Markov chains and the chain rule for
entropy, this means
$\displaystyle I(S;S^{\prime})$ $\displaystyle\leq I(z;y)=H(y)-H(y\mid
z)=H(y)-H(y-z\mid z)$ $\displaystyle=H(y)-\sum H(w^{\prime}_{i}\mid
z,w^{\prime}_{1},\dotsc,w^{\prime}_{i-1})$ $\displaystyle=H(y)-\sum
H(w^{\prime}_{i})\leq\sum H(y_{i})-H(w^{\prime}_{i})$ $\displaystyle=\sum
H(y_{i})-H(y_{i}\mid z_{i})=\sum I(y_{i};z_{i})$
$\displaystyle\leq\frac{m}{2}\log(1+\frac{1}{\alpha}).$ (11)
∎
We will show that successful recovery either recovers most of $x$, in which
case $I(S;S^{\prime})=\Omega(k\log(n/k))$, or recovers an $\epsilon$ fraction
of $w$. First we show that recovering $w$ requires $m=\Omega(\epsilon n)$.
###### Lemma 4.2.
Suppose $w\in\mathbb{R}^{n}$ with $w_{i}\sim N(0,\sigma^{2})$ for all $i$ and
$n=\Omega(\frac{1}{\epsilon^{2}}\log(1/\delta))$, and $A\in\mathbb{R}^{m\times
n}$ for $m<\delta\epsilon n$. Then any algorithm that finds $w^{\prime}$ from
$Aw$ must have $\left\lVert
w^{\prime}-w\right\rVert_{2}^{2}>(1-\epsilon)\left\lVert
w\right\rVert_{2}^{2}$ with probability at least $1-O(\delta)$.
###### Proof.
Note that $Aw$ merely gives the projection of $w$ onto $m$ dimensions, giving
no information about the other $n-m$ dimensions. Since $w$ and the $\ell_{2}$
norm are rotation invariant, we may assume WLOG that $A$ gives the projection
of $w$ onto the first $m$ dimensions, namely $T=[m]$. By the norm
concentration of Gaussians, with probability $1-\delta$ we have $\left\lVert
w\right\rVert_{2}^{2}<(1+\epsilon)n\sigma^{2}$, and by Markov with probability
$1-\delta$ we have $\left\lVert w_{T}\right\rVert_{2}^{2}<\epsilon
n\sigma^{2}$.
For any fixed value $d$, since $w$ is uniform Gaussian and
$w^{\prime}_{\overline{T}}$ is independent of $w_{\overline{T}}$,
$\displaystyle\Pr[\left\lVert w^{\prime}-w\right\rVert_{2}^{2}<d]$
$\displaystyle\leq\Pr[\left\lVert(w^{\prime}-w)_{\overline{T}}\right\rVert_{2}^{2}<d]\leq\Pr[\left\lVert
w_{\overline{T}}\right\rVert_{2}^{2}<d].$
Therefore
$\displaystyle\Pr[\left\lVert
w^{\prime}-w\right\rVert_{2}^{2}<(1-3\epsilon)\left\lVert
w\right\rVert_{2}^{2}]\leq$ $\displaystyle\Pr[\left\lVert
w^{\prime}-w\right\rVert_{2}^{2}<(1-2\epsilon)n\sigma^{2}]$
$\displaystyle\leq$ $\displaystyle\Pr[\left\lVert
w_{\overline{T}}\right\rVert_{2}^{2}<(1-2\epsilon)n\sigma^{2}]$
$\displaystyle\leq$ $\displaystyle\Pr[\left\lVert
w_{\overline{T}}\right\rVert_{2}^{2}<(1-\epsilon)(n-m)\sigma^{2}]\leq\delta$
as desired. Rescaling $\epsilon$ gives the result. ∎
###### Lemma 4.3.
Suppose $n=\Omega(1/\epsilon^{2}+(k/\epsilon)\log(k/\epsilon))$ and
$m=O(\epsilon n)$. Then $I(S;S^{\prime})=\Omega(k\log(n/k))$ for some
$\alpha=\Omega(1/\epsilon)$.
###### Proof.
Consider the $x^{\prime}$ recovered from $A(x+w)$, and let $T=S\cup
S^{\prime}$. Suppose that $\left\lVert w\right\rVert_{\infty}^{2}\leq
O(\frac{\alpha k}{n}\log n)$ and $\left\lVert w\right\rVert_{2}^{2}/(\alpha
k)\in[1\pm\epsilon]$, as happens with probability at least (say) $3/4$. Then
we claim that if recovery is successful, one of the following must be true:
$\displaystyle\left\lVert x^{\prime}_{T}-x\right\rVert_{2}^{2}$
$\displaystyle\leq 9\epsilon\left\lVert w\right\rVert_{2}^{2}$ (12)
$\displaystyle\left\lVert x^{\prime}_{\overline{T}}-w\right\rVert_{2}^{2}$
$\displaystyle\leq(1-2\epsilon)\left\lVert w\right\rVert_{2}^{2}$ (13)
To show this, suppose $\left\lVert
x^{\prime}_{T}-x\right\rVert_{2}^{2}>9\epsilon\left\lVert
w\right\rVert_{2}^{2}\geq 9\left\lVert w_{T}\right\rVert_{2}^{2}$ (the last by
$\left|T\right|=2k=O(\epsilon n/\log n)$). Then
$\displaystyle\left\lVert(x^{\prime}-(x+w))_{T}\right\rVert_{2}^{2}$
$\displaystyle>(\left\lVert x^{\prime}-x\right\rVert_{2}-\left\lVert
w_{T}\right\rVert_{2})^{2}$ $\displaystyle\geq(2\left\lVert
x^{\prime}-x\right\rVert_{2}/3)^{2}\geq 4\epsilon\left\lVert
w\right\rVert_{2}^{2}.$
Because recovery is successful,
$\left\lVert x^{\prime}-(x+w)\right\rVert_{2}^{2}\leq(1+\epsilon)\left\lVert
w\right\rVert_{2}^{2}.$
Therefore
$\displaystyle\left\lVert
x^{\prime}_{\overline{T}}-w_{\overline{T}}\right\rVert_{2}^{2}+\left\lVert
x^{\prime}_{T}-(x+w)_{T}\right\rVert_{2}^{2}$ $\displaystyle=\left\lVert
x^{\prime}-(x+w)\right\rVert_{2}^{2}$ $\displaystyle\left\lVert
x^{\prime}_{\overline{T}}-w_{\overline{T}}\right\rVert_{2}^{2}+4\epsilon\left\lVert
w\right\rVert_{2}^{2}$ $\displaystyle<(1+\epsilon)\left\lVert
w\right\rVert_{2}^{2}$ $\displaystyle\left\lVert
x^{\prime}_{\overline{T}}-w\right\rVert_{2}^{2}-\left\lVert
w_{T}\right\rVert_{2}^{2}$ $\displaystyle<(1-3\epsilon)\left\lVert
w\right\rVert_{2}^{2}\leq(1-2\epsilon)\left\lVert w\right\rVert_{2}^{2}$
as desired. Thus with $3/4$ probability, at least one of (12) and (13) is
true.
Suppose Equation (13) holds with at least $1/4$ probability. There must be
some $x$ and $S$ such that the same equation holds with $1/4$ probability. For
this $S$, given $x^{\prime}$ we can find $T$ and thus
$x^{\prime}_{\overline{T}}$. Hence for a uniform Gaussian $w_{\overline{T}}$,
given $Aw_{\overline{T}}$ we can compute $A(x+w_{\overline{T}})$ and recover
$x^{\prime}_{\overline{T}}$ with $\left\lVert
x^{\prime}_{\overline{T}}-w_{\overline{T}}\right\rVert_{2}^{2}\leq(1-\epsilon)\left\lVert
w_{\overline{T}}\right\rVert_{2}^{2}$. By Lemma 4.2 this is impossible, since
$n-\left|T\right|=\Omega(\frac{1}{\epsilon^{2}})$ and $m=\Omega(\epsilon n)$
by assumption.
Therefore Equation (12) holds with at least $1/2$ probability, namely
$\left\lVert x^{\prime}_{T}-x\right\rVert_{2}^{2}\leq 9\epsilon\left\lVert
w\right\rVert_{2}^{2}\leq 9\epsilon(1-\epsilon)\alpha k<k/2$ for appropriate
$\alpha$. But if the nearest $\hat{x}\in X$ to $x$ is not equal to $x$,
$\displaystyle\left\lVert x^{\prime}-\hat{x}\right\rVert_{2}^{2}=$
$\displaystyle\left\lVert
x^{\prime}_{\overline{T}}\right\rVert_{2}^{2}+\left\lVert
x^{\prime}_{\overline{T}}-\hat{x}\right\rVert_{2}^{2}\geq\left\lVert
x^{\prime}_{\overline{T}}\right\rVert_{2}^{2}+(\left\lVert
x-\hat{x}\right\rVert_{2}-\left\lVert
x^{\prime}_{\overline{T}}-x\right\rVert_{2})^{2}$ $\displaystyle>$
$\displaystyle\left\lVert
x^{\prime}_{\overline{T}}\right\rVert_{2}^{2}+(k-k/2)^{2}>\left\lVert
x^{\prime}_{\overline{T}}\right\rVert_{2}^{2}+\left\lVert
x^{\prime}_{\overline{T}}-x\right\rVert_{2}^{2}=\left\lVert
x^{\prime}-x\right\rVert_{2}^{2},$
a contradiction. Hence $S^{\prime}=S$. But Fano’s inequality states
$H(S|S^{\prime})\leq 1+\Pr[S^{\prime}\neq S]\log\left|\mathcal{F}\right|$ and
hence
$I(S;S^{\prime})=H(S)-H(S|S^{\prime})\geq-1+\frac{1}{4}\log\left|\mathcal{F}\right|=\Omega(k\log(n/k))$
as desired. ∎
###### Theorem 4.4.
Any $(1+\epsilon)$-approximate $\ell_{2}/\ell_{2}$ recovery scheme with
$\epsilon>\sqrt{\frac{k\log n}{n}}$ and failure probability $\delta<1/2$
requires $m=\Omega(\frac{1}{\epsilon}k\log(n/k))$.
###### Proof.
Combine Lemmas 4.3 and 4.1 with $\alpha=1/\epsilon$ to get
$m=\Omega(\frac{k\log(n/k)}{\log(1+\epsilon)})=\Omega(\frac{1}{\epsilon}k\log(n/k))$,
$m=\Omega(\epsilon n)$, or $n=O(\frac{1}{\epsilon}k\log(k/\epsilon))$. For
$\epsilon$ as in the theorem statement, the first bound is controlling. ∎
## 5 Bit complexity to measurement complexity
The remaining lower bounds proceed by reductions from communication
complexity. The following lemma (implicit in [DIPW10]) shows that lower
bounding the number of bits for approximate recovery is sufficient to lower
bound the number of measurements. Let $B_{p}^{n}(R)\subset\mathbb{R}^{n}$
denote the $\ell_{p}$ ball of radius $R$.
###### Definition 5.1.
Let $X\subset\mathbb{R}^{n}$ be a distribution with
$x_{i}\in\\{-n^{d},\dotsc,n^{d}\\}$ for all $i\in[n]$ and $x\in X$. We define
a $1+\epsilon$-approximate $\ell_{p}/\ell_{p}$ sparse recovery _bit scheme_ on
$X$ with $b$ bits, precision $n^{-c}$, and failure probability $\delta$ to be
a deterministic pair of functions $f\colon X\to\\{0,1\\}^{b}$ and
$g\colon\\{0,1\\}^{b}\to\mathbb{R}^{n}$ where $f$ is linear so that $f(a+b)$
can be computed from $f(a)$ and $f(b)$. We require that, for $u\in
B_{p}^{n}(n^{-c})$ uniformly and $x$ drawn from $X$, $g(f(x))$ is a valid
result of $1+\epsilon$-approximate recovery on $x+u$ with probability
$1-\delta$.
###### Lemma 5.2.
A lower bound of $\Omega(b)$ bits for such a sparse recovery bit scheme with
$p\leq 2$ implies a lower bound of $\Omega(b/((1+c+d)\log n))$ bits for
regular $(1+\epsilon)$-approximate sparse recovery with failure probability
$\delta-1/n$.
###### Proof.
Suppose we have a standard $(1+\epsilon)$-approximate sparse recovery
algorithm $\mathcal{A}$ with failure probability $\delta$ using $m$
measurements $Ax$. We will use this to construct a (randomized) sparse
recovery bit scheme using $O(m(1+c+d)\log n)$ bits and failure probability
$\delta+1/n$. Then by averaging some deterministic sparse recovery bit scheme
performs better than average over the input distribution.
We may assume that $A\in\mathbb{R}^{m\times n}$ has orthonormal rows
(otherwise, if $A=U\Sigma V^{T}$ is its singular value decomposition,
$\Sigma^{+}U^{T}A$ has this property and can be inverted before applying the
algorithm). When applied to the distribution $X+u$ for $u$ uniform over
$B_{p}^{n}(n^{-c})$, we may assume that $\mathcal{A}$ and $A$ are
deterministic and fail with probability $\delta$ over their input.
Let $A^{\prime}$ be $A$ rounded to $t\log n$ bits per entry for some parameter
$t$. Let $x$ be chosen from $X$. By Lemma 5.1 of [DIPW10], for any $x$ we have
$A^{\prime}x=A(x-s)$ for some $s$ with $\left\lVert s\right\rVert_{1}\leq
n^{2}2^{-t\log n}\left\lVert x\right\rVert_{1}$, so $\left\lVert
s\right\rVert_{p}\leq n^{2.5-t}\left\lVert x\right\rVert_{p}\leq n^{3.5+d-t}$.
Let $u\in B_{p}^{n}(n^{5.5+d-t})$ uniformly at random. With probability at
least $1-1/n$, $u\in B_{p}^{n}((1-1/n^{2})n^{5.5+d-t})$ because the balls are
similar so the ratio of volumes is $(1-1/n^{2})^{n}>1-1/n$. In this case
$u+s\in B_{p}^{n}(n^{5.5+d-t})$; hence the random variable $u$ and $u+s$
overlap in at least a $1-1/n$ fraction of their volumes, so $x+s+u$ and $x+u$
have statistical distance at most $1/n$. Therefore
$\mathcal{A}(A(x+u))=\mathcal{A}(A^{\prime}x+Au)$ with probability at least
$1-1/n$.
Now, $A^{\prime}x$ uses only $(t+d+1)\log n$ bits per entry, so we can set
$f(x)=A^{\prime}x$ for $b=m(t+d+1)\log n$. Then we set
$g(y)=\mathcal{A}(y+Au)$ for uniformly random $u\in B_{p}^{n}(n^{5.5+d-t})$.
Setting $t=5.5+d+c$, this gives a sparse recovery bit scheme using
$b=m(6.5+2d+c)\log n$. ∎
## 6 Non-sparse output Lower Bound for $p=1$
First, we show that recovering the locations of an $\epsilon$ fraction of $d$
ones in a vector of size $n>d/\epsilon$ requires $\widetilde{\Omega}(\epsilon
d)$ bits. Then, we show high bit complexity of a distributional product
version of the Gap-$\ell_{\infty}$ problem. Finally, we create a distribution
for which successful sparse recovery must solve one of the previous problems,
giving a lower bound in bit complexity. Lemma 5.2 converts the bit complexity
to measurement complexity.
### 6.1 $\ell_{1}$ Lower bound for recovering noise bits
###### Definition 6.1.
We say a set $C\subset[q]^{d}$ is a $(d,q,\epsilon)$ code if any two distinct
$c,c^{\prime}\in C$ agree in at most $\epsilon d$ positions. We say a set
$X\subset\\{0,1\\}^{dq}$ represents $C$ if $X$ is $C$ concatenated with the
trivial code $[q]\to\\{0,1\\}^{q}$ given by $i\to e_{i}$.
###### Claim 6.2.
For $\epsilon\geq 2/q$, there exist $(d,q,\epsilon)$ codes $C$ of size
$q^{\Omega(\epsilon d)}$ by the Gilbert-Varshamov bound (details in [DIPW10]).
###### Lemma 6.3.
Let $X\subset\\{0,1\\}^{dq}$ represent a $(d,q,\epsilon)$ code. Suppose
$y\in\mathbb{R}^{dq}$ satisfies $\left\lVert
y-x\right\rVert_{1}\leq(1-\epsilon)\left\lVert x\right\rVert_{1}$. Then we can
recover $x$ uniquely from $y$.
###### Proof.
We assume $y_{i}\in[0,1]$ for all $i$; thresholding otherwise decreases
$\left\lVert y-x\right\rVert_{1}$. We will show that there exists no other
$x^{\prime}\in X$ with $\left\lVert
y-x\right\rVert_{1}\leq(1-\epsilon)\left\lVert x\right\rVert_{1}$; thus
choosing the nearest element of $X$ is a unique decoder. Suppose otherwise,
and let $S=\operatorname{supp}(x),T=\operatorname{supp}(x^{\prime})$. Then
$\displaystyle(1-\epsilon)\left\lVert x\right\rVert_{1}$
$\displaystyle\geq\left\lVert x-y\right\rVert_{1}$ $\displaystyle=\left\lVert
x\right\rVert_{1}-\left\lVert y_{S}\right\rVert_{1}+\left\lVert
y_{\overline{S}}\right\rVert_{1}$ $\displaystyle\left\lVert
y_{S}\right\rVert_{1}$ $\displaystyle\geq\left\lVert
y_{\overline{S}}\right\rVert_{1}+\epsilon d$
Since the same is true relative to $x^{\prime}$ and $T$, we have
$\displaystyle\left\lVert y_{S}\right\rVert_{1}+\left\lVert
y_{T}\right\rVert_{1}$ $\displaystyle\geq\left\lVert
y_{\overline{S}}\right\rVert_{1}+\left\lVert
y_{\overline{T}}\right\rVert_{1}+2\epsilon d$ $\displaystyle 2\left\lVert
y_{S\cap T}\right\rVert_{1}$ $\displaystyle\geq 2\left\lVert
y_{\overline{S\cup T}}\right\rVert_{1}+2\epsilon d$ $\displaystyle\left\lVert
y_{S\cap T}\right\rVert_{1}$ $\displaystyle\geq\epsilon d$
$\displaystyle\left|S\cap T\right|$ $\displaystyle\geq\epsilon d$
This violates the distance of the code represented by $X$. ∎
###### Lemma 6.4.
Let $R=[s,cs]$ for some constant $c$ and parameter $s$. Let $X$ be a
permutation independent distribution over $\\{0,1\\}^{n}$ with $\left\lVert
x\right\rVert_{1}\in R$ with probability $p$. If $y$ satisfies $\left\lVert
x-y\right\rVert_{1}\leq(1-\epsilon)\left\lVert x\right\rVert_{1}$ with
probability $p^{\prime}$ with $p^{\prime}-(1-p)=\Omega(1)$, then
$I(x;y)=\Omega(\epsilon s\log(n/s))$.
###### Proof.
For each integer $i\in R$, let $X_{i}\subset\\{0,1\\}^{n}$ represent an
$(i,n/i,\epsilon)$ code. Let $p_{i}=\Pr_{x\in X}[\left\lVert
x\right\rVert_{1}=i]$. Let $S_{n}$ be the set of permutations of $[n]$. Then
the distribution $X^{\prime}$ given by (a) choosing $i\in R$ proportional to
$p_{i}$, (b) choosing $\sigma\in S_{n}$ uniformly, (c) choosing $x_{i}\in
X_{i}$ uniformly, and (d) outputting $x^{\prime}=\sigma(x_{i})$ is equal to
the distribution $(x\in X\mid\left\lVert x\right\rVert_{1}\in R)$.
Now, because $p^{\prime}\geq\Pr[\left\lVert x\right\rVert_{1}\notin
R]+\Omega(1)$, $x^{\prime}$ chosen from $X^{\prime}$ satisfies $\left\lVert
x^{\prime}-y\right\rVert_{1}\leq(1-\epsilon)\left\lVert
x^{\prime}\right\rVert_{1}$ with $\delta\geq p^{\prime}-(1-p)$ probability.
Therefore, with at least $\delta/2$ probability, $i$ and $\sigma$ are such
that
$\left\lVert\sigma(x_{i})-y\right\rVert_{1}\leq(1-\epsilon)\left\lVert\sigma(x_{i})\right\rVert_{1}$
with $\delta/2$ probability over uniform $x_{i}\in X_{i}$. But given $y$ with
$\left\lVert y-\sigma(x_{i})\right\rVert_{1}$ small, we can compute
$y^{\prime}=\sigma^{-1}(y)$ with $\left\lVert
y^{\prime}-x_{i}\right\rVert_{1}$ equally small. Then by Lemma 6.3 we can
recover $x_{i}$ from $y$ with probability $\delta/2$ over $x_{i}\in X_{i}$.
Thus for this $i$ and $\sigma$, $I(x;y\mid
i,\sigma)\geq\Omega(\log\left|X_{i}\right|)=\Omega(\delta\epsilon s\log(n/s))$
by Fano’s inequality. But then $I(x;y)=\operatorname{E}_{i,\sigma}[I(x;y\mid
i,\sigma)]=\Omega(\delta^{2}\epsilon s\log(n/s))=\Omega(\epsilon s\log(n/s))$.
∎
### 6.2 Distributional Indexed Gap $\ell_{\infty}$
Consider the following communication game, which we refer to as
$\mathsf{Gap}\ell_{\infty}^{B}$, studied in [BYJKS04]. The legal instances are
pairs $(x,y)$ of $m$-dimensional vectors, with
$x_{i},y_{i}\in\\{0,1,2,\ldots,B\\}$ for all $i$ such that
* •
NO instance: for all $i$, $y_{i}-x_{i}\in\\{0,1\\}$, or
* •
YES instance: there is a _unique_ $i$ for which $y_{i}-x_{i}=B$, and for all
$j\neq i$, $y_{i}-x_{i}\in\\{0,1\\}$.
The distributional communication complexity $D_{\sigma,\delta}(f)$ of a
function $f$ is the minimum over all deterministic protocols computing $f$
with error probability at most $\delta$, where the probability is over inputs
drawn from $\sigma$.
Consider the distribution $\sigma$ which chooses a random $i\in[m]$. Then for
each $j\neq i$, it chooses a random $d\in\\{0,\ldots,B\\}$ and $(x_{i},y_{i})$
is uniform in $\\{(d,d),(d,d+1)\\}$. For coordinate $i$, $(x_{i},y_{i})$ is
uniform in $\\{(0,0),(0,B)\\}$. Using similar arguments to those in [BYJKS04],
Jayram [Jay02] showed
$D_{\sigma,\delta}(\mathsf{Gap}\ell_{\infty}^{B})=\Omega(m/B^{2})$ (this is
reference [70] on p.182 of [BY02]) for $\delta$ less than a small constant.
We define the one-way distributional communication complexity
$D^{1-way}_{\sigma,\delta}(f)$ of a function $f$ to be the smallest
distributional complexity of a protocol for $f$ in which only a single message
is sent from Alice to Bob.
###### Definition 6.5 (Indexed $\mathsf{Ind}\ell_{\infty}^{r,B}$ Problem).
There are $r$ pairs of inputs
$(x^{1},y^{1}),(x^{2},y^{2}),\ldots,(x^{r},y^{r})$ such that every pair
$(x^{i},y^{i})$ is a legal instance of the $\mathsf{Gap}\ell_{\infty}^{B}$
problem. Alice is given $x^{1},\ldots,x^{r}$. Bob is given an index $I\in[r]$
and $y^{1},\ldots,y^{r}$. The goal is to decide whether $(x^{I},y^{I})$ is a
NO or a YES instance of $\mathsf{Gap}\ell_{\infty}^{B}$.
Let $\eta$ be the distribution $\sigma^{r}\times U_{r}$, where $U_{r}$ is the
uniform distribution on $[r]$. We bound
$D^{1-way}_{\eta,\delta}(\mathsf{Ind}\ell_{\infty})^{r,B}$ as follows. For a
function $f$, let $f^{r}$ denote the problem of computing $r$ instances of
$f$. For a distribution $\zeta$ on instances of $f$, let
$D_{\zeta^{r},\delta}^{1-way,*}(f^{r})$ denote the minimum communication cost
of a deterministic protocol computing a function $f$ with error probability at
most $\delta$ in each of the $r$ copies of $f$, where the inputs come from
$\zeta^{r}$.
###### Theorem 6.6.
(special case of Corollary 2.5 of [BR11]) Assume $D_{\sigma,\delta}(f)$ is
larger than a large enough constant. Then
$D^{1-way,*}_{\sigma^{r},\delta/2}(f^{r})=\Omega(rD_{\sigma,\delta}(f))$.
###### Theorem 6.7.
For $\delta$ less than a sufficiently small constant,
$D^{1-way}_{\eta,\delta}(\mathsf{Ind}\ell_{\infty}^{r,B})=\Omega(\delta^{2}rm/(B^{2}\log
r))$.
###### Proof.
Consider a deterministic $1$-way protocol $\Pi$ for
$\mathsf{Ind}\ell_{\infty}^{r,B}$ with error probability $\delta$ on inputs
drawn from $\eta$. Then for at least $r/2$ values $i\in[r]$,
$\Pr[\Pi(x^{1},\ldots,x^{r},y^{1},\ldots,y^{r},I)=\mathsf{Gap}\ell_{\infty}^{B}(x^{I},y^{I})\mid
I=i]\geq 1-2\delta.$ Fix a set $S=\\{i_{1},\ldots,i_{r/2}\\}$ of indices with
this property. We build a deterministic $1$-way protocol $\Pi^{\prime}$ for
$f^{r/2}$ with input distribution $\sigma^{r/2}$ and error probability at most
$6\delta$ in each of the $r/2$ copies of $f$.
For each $\ell\in[r]\setminus S$, independently choose
$(x^{\ell},y^{\ell})\sim\sigma$. For each $j\in[r/2]$, let $Z_{j}^{1}$ be the
probability that
$\Pi(x^{1},\ldots,x^{r},y^{1},\ldots,y^{r},I)=\mathsf{Gap}\ell_{\infty}^{B}(x^{i_{j}},y^{i_{j}})$
given $I=i_{j}$ and the choice of $(x^{\ell},y^{\ell})$ for all
$\ell\in[r]\setminus S$.
If we repeat this experiment independently $s=O(\delta^{-2}\log r)$ times,
obtaining independent $Z_{j}^{1},\ldots,Z_{j}^{s}$ and let
$Z_{j}=\sum_{t}Z_{j}^{t}$, then $\Pr[Z_{j}\geq s-s\cdot 3\delta]\geq
1-\frac{1}{r}.$ So there exists a set of $s=O(\delta^{-1}\log r)$ repetitions
for which for each $j\in[r/2]$, $Z_{j}\geq s-s\cdot 3\delta$. We hardwire
these into $\Pi^{\prime}$ to make the protocol deterministic.
Given inputs $((X^{1},\ldots,X^{r/2}),(Y^{1},\ldots,Y^{r/2}))\sim\sigma^{r/2}$
to $\Pi^{\prime}$, Alice and Bob run $s$ executions of $\Pi$, each with
$x^{i_{j}}=X^{j}$ and $y^{i_{j}}=Y^{j}$ for all $j\in[r/2]$, filling in the
remaining values using the hardwired inputs. Bob runs the algorithm specified
by $\Pi$ for each $i_{j}\in S$ and each execution. His output for
$(X^{j},Y^{j})$ is the majority of the outputs of the $s$ executions with
index $i_{j}$.
Fix an index $i_{j}$. Let $W$ be the number of repetitions for which
$\mathsf{Gap}\ell_{\infty}^{B}(X^{j},Y^{j})$ does not equal the output of
$\Pi$ on input $i_{j}$, for a random $(X^{j},Y^{j})\sim\sigma$. Then, ${\bf
E}[W]\leq 3\delta$. By a Markov bound, $\Pr[W\geq s/2]\leq 6\delta$, and so
the coordinate is correct with probability at least $1-6\delta$.
The communication of $\Pi^{\prime}$ is a factor $s=\Theta(\delta^{-2}\log r)$
more than that of $\Pi$. The theorem now follows by Theorem 6.6, using that
$D_{\sigma,12\delta}(\mathsf{Gap}\ell_{\infty}^{B})=\Omega(m/B^{2})$. ∎
### 6.3 Lower bound for sparse recovery
Fix the parameters $B=\Theta(1/\epsilon^{1/2}),r=k$, $m=1/\epsilon^{3/2}$, and
$n=k/\epsilon^{3}$. Given an instance $(x^{1},y^{1}),\ldots,(x^{r},y^{r}),I$
of $\mathsf{Ind}\ell_{\infty}^{r,B}$, we define the input signal $z$ to a
sparse recovery problem. We allocate a set $S^{i}$ of $m$ disjoint coordinates
in a universe of size $n$ for each pair $(x^{i},y^{i})$, and on these
coordinates place the vector $y^{i}-x^{i}$. The locations are important for
arguing the sparse recovery algorithm cannot learn much information about the
noise, and will be placed uniformly at random.
Let $\rho$ denote the induced distribution on $z$. Fix a
$(1+\epsilon)$-approximate $k$-sparse recovery bit scheme $Alg$ that takes $b$
bits as input and succeeds with probability at least $1-\delta/2$ over
$z\sim\rho$ for some small constant $\delta$. Let $S$ be the set of top $k$
coordinates in $z$. $Alg$ has the guarantee that if it succeeds for
$z\sim\rho$, then there exists a small $u$ with $\left\lVert
u\right\rVert_{1}<n^{-2}$ so that $v=Alg(z)$ satisfies
$\displaystyle\left\lVert v-z-u\right\rVert_{1}$
$\displaystyle\leq(1+\epsilon)\left\lVert(z+u)_{[n]\setminus
S}\right\rVert_{1}$ $\displaystyle\left\lVert v-z\right\rVert_{1}$
$\displaystyle\leq(1+\epsilon)\left\lVert z_{[n]\setminus
S}\right\rVert_{1}+(2+\epsilon)/n^{2}$
$\displaystyle\leq(1+2\epsilon)\left\lVert z_{[n]\setminus S}\right\rVert_{1}$
and thus
$\displaystyle\left\lVert(v-z)_{S}\right\rVert_{1}+\left\lVert(v-z)_{[n]\setminus
S}\right\rVert_{1}\leq(1+2\epsilon)\|z_{[n]\setminus S}\|_{1}.$ (14)
###### Lemma 6.8.
For $B=\Theta(1/\epsilon^{1/2})$ sufficiently large, suppose that
$\Pr_{z\sim\rho}[\|(v-z)_{S}\|_{1}\leq 10\epsilon\cdot\|z_{[n]\setminus
S}\|_{1}]\geq 1-\delta$. Then $Alg$ requires $b=\Omega(k/(\epsilon^{1/2}\log
k))$.
###### Proof.
We show how to use $Alg$ to solve instances of
$\mathsf{Ind}\ell_{\infty}^{r,B}$ with probability at least $1-C$ for some
small $C$, where the probability is over input instances to
$\mathsf{Ind}\ell_{\infty}^{r,B}$ distributed according to $\eta$, inducing
the distribution $\rho$. The lower bound will follow by Theorem 6.7. Since
$Alg$ is a deterministic sparse recovery bit scheme, it receives a sketch
$f(z)$ of the input signal $z$ and runs an arbitrary recovery algorithm $g$ on
$f(z)$ to determine its output $v=Alg(z)$.
Given $x^{1},\ldots,x^{r}$, for each $i=1,2,\ldots,r$, Alice places $-x^{i}$
on the appropriate coordinates in the block $S^{i}$ used in defining $z$,
obtaining a vector $z_{Alice}$, and transmits $f(z_{Alice})$ to Bob. Bob uses
his inputs $y^{1},\ldots,y^{r}$ to place $y^{i}$ on the appropriate coordinate
in $S^{i}$. He thus creates a vector $z_{Bob}$ for which
$z_{Alice}+z_{Bob}=z$. Given $f(z_{Alice})$, Bob computes $f(z)$ from
$f(z_{Alice})$ and $f(z_{Bob})$, then $v=Alg(z)$. We assume all coordinates of
$v$ are rounded to the real interval $[0,B]$, as this can only decrease the
error.
We say that $S^{i}$ is bad if either
* •
there is no coordinate $j$ in $S^{i}$ for which $|v_{j}|\geq\frac{B}{2}$ yet
$(x^{i},y^{i})$ is a YES instance of $\mathsf{Gap}\ell_{\infty}^{r,B}$, or
* •
there is a coordinate $j$ in $S^{i}$ for which $|v_{j}|\geq\frac{B}{2}$ yet
either $(x^{i},y^{i})$ is a NO instance of $\mathsf{Gap}\ell_{\infty}^{r,B}$
or $j$ is not the unique $j^{*}$ for which $y^{i}_{j^{*}}-x^{i}_{j^{*}}=B$
The $\ell_{1}$-error incurred by a bad block is at least $B/2-1$. Hence, if
there are $t$ bad blocks, the total error is at least $t(B/2-1)$, which must
be smaller than $10\epsilon\cdot\|z_{[n]\setminus S}\|_{1}$ with probability
$1-\delta$. Suppose this happens.
We bound $t$. All coordinates in $z_{[n]\setminus S}$ have value in the set
$\\{0,1\\}$. Hence, $\|z_{[n]\setminus S}\|_{1}<rm$. So $t\leq 20\epsilon
rm/(B-2)$. For $B\geq 6$, $t\leq 30\epsilon rm/B$. Plugging in $r$, $m$ and
$B$, $t\leq Ck$, where $C>0$ is a constant that can be made arbitrarily small
by increasing $B=\Theta(1/\epsilon^{1/2})$.
If a block $S^{i}$ is not bad, then it can be used to solve
$\mathsf{Gap}\ell_{\infty}^{r,B}$ on $(x^{i},y^{i})$ with probability $1$. Bob
declares that $(x^{i},y^{i})$ is a YES instance if and only if there is a
coordinate $j$ in $S^{i}$ for which $|v_{j}|\geq B/2$.
Since Bob’s index $I$ is uniform on the $m$ coordinates in
$\mathsf{Ind}\ell_{\infty}^{r,B}$, with probability at least $1-C$ the players
solve $\mathsf{Ind}\ell_{\infty}^{r,B}$ given that the $\ell_{1}$ error is
small. Therefore they solve $\mathsf{Ind}\ell_{\infty}^{r,B}$ with probability
$1-\delta-C$ overall. By Theorem 6.7, for $C$ and $\delta$ sufficiently small
$Alg$ requires $\Omega(mr/(B^{2}\log r))=\Omega(k/(\epsilon^{1/2}\log k))$
bits. ∎
###### Lemma 6.9.
Suppose $\Pr_{z\sim\rho}[\|(v-z)_{[n]\setminus
S}\|_{1}]\leq(1-8\epsilon)\cdot\|z_{[n]\setminus S}\|_{1}]\geq\delta/2$. Then
$Alg$ requires $b=\Omega(\frac{1}{\sqrt{\epsilon}}k\log(1/\epsilon))$.
###### Proof.
The distribution $\rho$ consists of $B(mr,1/2)$ ones placed uniformly
throughout the $n$ coordinates, where $B(mr,1/2)$ denotes the binomial
distribution with $mr$ events of $1/2$ probability each. Therefore with
probability at least $1-\delta/4$, the number of ones lies in $[\delta
mr/8,(1-\delta/8)mr]$. Thus by Lemma 6.4, $I(v;z)\geq\Omega(\epsilon
mr\log(n/(mr)))$. Since the mutual information only passes through a $b$-bit
string, $b=\Omega(\epsilon mr\log(n/(mr)))$ as well. ∎
###### Theorem 6.10.
Any $(1+\epsilon)$-approximate $\ell_{1}/\ell_{1}$ recovery scheme with
sufficiently small constant failure probability $\delta$ must make
$\Omega(\frac{1}{\sqrt{\epsilon}}k/\log^{2}(k/\epsilon))$ measurements.
###### Proof.
We will lower bound any $\ell_{1}/\ell_{1}$ sparse recovery bit scheme $Alg$.
If $Alg$ succeeds, then in order to satisfy inequality (14), we must either
have $\|(v-z)_{S}\|_{1}\leq 10\epsilon\cdot\|z_{[n]\setminus S}\|_{1}$ or we
must have $\|(v-z)_{[n]\setminus
S}\|_{1}\leq(1-8\epsilon)\cdot\|z_{[n]\setminus S}\|_{1}$. Since $Alg$
succeeds with probability at least $1-\delta$, it must either satisfy the
hypothesis of Lemma 6.8 or the hypothesis of Lemma 6.9. But by these two
lemmas, it follows that $b=\Omega(\frac{1}{\sqrt{\epsilon}}k/\log k)$.
Therefore by Lemma 5.2, any $(1+\epsilon)$-approximate $\ell_{1}/\ell_{1}$
sparse recovery algorithm requires
$\Omega(\frac{1}{\sqrt{\epsilon}}k/\log^{2}(k/\epsilon))$ measurements. ∎
## 7 Lower bounds for $k$-sparse output
###### Theorem 7.1.
Any $1+\epsilon$-approximate $\ell_{1}/\ell_{1}$ recovery scheme with
$k$-sparse output and failure probability $\delta$ requires
$m=\Omega(\frac{1}{\epsilon}(k\log\frac{1}{\epsilon}+\log\frac{1}{\delta}))$,
for $32\leq\frac{1}{\delta}\leq n\epsilon^{2}/k$.
###### Theorem 7.2.
Any $1+\epsilon$-approximate $\ell_{2}/\ell_{2}$ recovery scheme with
$k$-sparse output and failure probability $\delta$ requires
$m=\Omega(\frac{1}{\epsilon^{2}}(k+\log\frac{\epsilon^{2}}{\delta}))$, for
$32\leq\frac{1}{\delta}\leq n\epsilon^{2}/k$.
These two theorems correspond to four statements: one for large $k$ and one
for small $\delta$ for both $\ell_{1}$ and $\ell_{2}$.
All the lower bounds proceed by reductions from communication complexity. The
following lemma (implicit in [DIPW10]) shows that lower bounding the number of
bits for approximate recovery is sufficient to lower bound the number of
measurements.
###### Lemma 7.3.
Let $p\in\\{1,2\\}$ and $\alpha=\Omega(1)<1$. Suppose $X\subset\mathbb{R}^{n}$
has $\left\lVert x\right\rVert_{p}\leq D$ and $\left\lVert
x\right\rVert_{\infty}\leq D^{\prime}$ for all $x\in X$, and all coefficients
of elements of $X$ are expressible in $O(\log n)$ bits. Further suppose that
we have a recovery algorithm that, for any $\nu$ with
$\left\lVert\nu\right\rVert_{p}<\alpha D$ and
$\left\lVert\nu\right\rVert_{\infty}<\alpha D^{\prime}$, recovers $x\in X$
from $A(x+\nu)$ with constant probability. Then $A$ must have
$\Omega(\log\left|X\right|)$ measurements.
###### Proof.
[††margin: xxx Use lemma 5.2] First, we may assume that
$A\in\mathbb{R}^{m\times n}$ has orthonormal rows (otherwise, if $A=U\Sigma
V^{T}$ is its singular value decomposition, $\Sigma^{+}U^{T}A$ has this
property and can be inverted before applying the algorithm). Let $A^{\prime}$
be $A$ rounded to $c\log n$ bits per entry. By Lemma 5.1 of [DIPW10], for any
$v$ we have $A^{\prime}v=A(v-s)$ for some $s$ with $\left\lVert
s\right\rVert_{1}\leq n^{2}2^{-c\log n}\left\lVert v\right\rVert_{1}$, so
$\left\lVert s\right\rVert_{p}\leq n^{2.5-c}\left\lVert v\right\rVert_{p}$.
Suppose Alice has a bit string of length $r\log\left|X\right|$ for
$r=\Theta(\log n)$. By splitting into $r$ blocks, this corresponds to
$x_{1},\dotsc,x_{r}\in X$. Let $\beta$ be a power of $2$ between $\alpha/2$
and $\alpha/4$, and define
$z_{j}=\sum_{i=j}^{r}\beta^{i}x_{i}.$
Alice sends $A^{\prime}z_{1}$ to Bob; this is $O(m\log n)$ bits. Bob will
solve the _augmented indexing problem_[††margin: xxx citation?]—given
$A^{\prime}z_{1}$, arbitrary $j\in[r]$, and $x_{1},\dotsc,x_{j-1}$, he must
find $x_{j}$ with constant probability. This requires $A^{\prime}z_{1}$ to
have $\Omega(r\log\left|X\right|)$ bits, giving the result.
Bob receives $A^{\prime}z_{1}=A(z_{1}+s)$ for $\left\lVert
s\right\rVert_{1}\leq n^{2.5-c}\left\lVert z_{1}\right\rVert_{p}\leq
n^{2.5-c}D$. Bob then chooses $u\in B_{p}^{n}(n^{4.5-c}D)$ uniformly at
random. With probability at least $1-1/n$, $u\in
B_{p}^{n}((1-1/n^{2})n^{4.5-c}D)$ by a volume argument. In this case $u+s\in
B_{p}^{n}(n^{4.5-c}D)$; hence the random variables $u$ and $u+s$ overlap in at
least a $1-1/n$ fraction of their volumes, so $z_{j}+s+u$ and $z_{j}+u$ have
statistical distance at most $1/n$. The distribution of $z_{j}+u$ is
independent of $A$ (unlike $z_{j}+s$) so running the recovery algorithm on
$A(z_{j}+s+u)$ succeeds with constant probability as well.
We also have $\left\lVert
z_{j}\right\rVert_{p}\leq\frac{\beta^{j}-\beta^{r+1}}{1-\beta}D<2(\beta^{j}-\beta^{r+1})D$.
Since $r=O(\log n)$ and $\beta$ is a constant, there exists a $c=O(1)$ with
$\left\lVert
z_{j}+s+u\right\rVert_{p}<(2\beta^{j}+n^{4.5-c}+n^{2.5-c}-2\beta^{r})D\leq\beta^{j-1}\alpha
D$
for all $j$.
Therefore, given $x_{1},\dotsc,x_{j-1}$, Bob can compute
$\frac{1}{\beta^{j}}(A^{\prime}z_{1}+Au-A^{\prime}\sum_{i<j}\beta^{i}x_{i})=A(x_{j}+\frac{1}{\beta^{j}}(z_{j+1}+s+u))=A(x_{j}+y)$
for some $y$ with $\left\lVert y\right\rVert_{p}\leq\alpha D$. Hence Bob can
use the recovery algorithm to recover $x_{j}$ with constant probability.
Therefore Bob can solve augmented indexing, so the message $A^{\prime}z_{1}$
must have $\Omega(\log n\log\left|X\right|)$ bits, so
$m=\Omega(\log\left|X\right|)$. ∎
We will now prove another lemma that is useful for all four theorem
statements.
Let $x\in\\{0,1\\}^{n}$ be $k$-sparse with $\operatorname{supp}(x)\subseteq S$
for some known $S$. Let $\nu\in\mathbb{R}^{n}$ be a noise vector that roughly
corresponds to having $O(k/\epsilon^{p})$ ones for $p\in\\{1,2\\}$, all
located outside of $S$. We consider under what circumstances we can use a
$(1+\epsilon)$-approximate $\ell_{p}/\ell_{p}$ recovery scheme to recover
$\operatorname{supp}(x)$ from $A(x+\nu)$ with (say) $90\%$ accuracy.
Lemma 7.4 shows that this is possible for $p=1$ when $\left|S\right|\leq
O(k/\epsilon)$ and for $p=2$ when $\left|S\right|\leq 2k$. The algorithm in
both instances is to choose a parameter $\mu$ and perform sparse recovery on
$A(x+\nu+z)$, where $z_{i}=\mu$ for $i\in S$ and $z_{i}=0$ otherwise. The
support of the result will be very close to $\operatorname{supp}(x)$.
###### Lemma 7.4.
Let $S\subset[n]$ have $\left|S\right|\leq s$, and suppose $x\in\\{0,1\\}^{n}$
satisfies $\operatorname{supp}(x)\subseteq S$ and $\left\lVert
x_{S}\right\rVert_{1}=k$. Let $p\in\\{1,2\\}$, and $\nu\in\mathbb{R}^{n}$
satisfy $\left\lVert\nu_{S}\right\rVert_{\infty}\leq\alpha$,
$\left\lVert\nu\right\rVert_{p}^{p}\leq r$, and
$\left\lVert\nu\right\rVert_{\infty}\leq D$ for some constants $\alpha\leq
1/4$ and $D=O(1)$. Suppose $A\in\mathbb{R}^{m\times n}$ is part of a
$(1+\epsilon)$-approximate $k$-sparse $\ell_{p}/\ell_{p}$ recovery scheme with
failure probability $\delta$.
Then, given $A(x_{S}+\nu)$, Bob can with failure probability $\delta$ recover
$\hat{x_{S}}$ that differs from $x_{S}$ in at most $k/c$ locations, as long as
either
$\displaystyle
p=1,s=\Theta(\frac{k}{c\epsilon}),r=\Theta(\frac{k}{c\epsilon})$ (15)
or
$\displaystyle p=2,s=2k,r=\Theta(\frac{k}{c^{2}\epsilon^{2}})$ (16)
###### Proof.
For some parameter $\mu\geq D$, let $z_{i}=\mu$ for $i\in S$ and $z_{i}=0$
elsewhere. Consider $y=x_{S}+\nu+z$. Let $U=\operatorname{supp}(x_{S})$ have
size $k$. Let $V\subset[n]$ be the support of the result of running the
recovery scheme on $Ay=A(x_{S}+\nu)+Az$. Then we have that $x_{S}+z$ is
$\mu+1$ over $U$, $\mu$ over $S\setminus U$, and zero elsewhere. Since
$\left\lVert u+v\right\rVert_{p}^{p}\leq p(\left\lVert
u\right\rVert_{p}^{p}+\left\lVert v\right\rVert_{p}^{p})$ for any $u$ and $v$,
we have
$\displaystyle\left\lVert y_{\overline{U}}\right\rVert_{p}^{p}$
$\displaystyle\leq
p(\left\lVert(x_{S}+z)_{\overline{U}}\right\rVert_{p}^{p}+\left\lVert\nu\right\rVert_{p}^{p})$
$\displaystyle\leq p((s-k)\mu^{p}+r)$ $\displaystyle<p(r+s\mu^{p}).$
Since $\left\lVert\nu_{S}\right\rVert_{\infty}\leq\alpha$ and
$\left\lVert\nu_{\overline{S}}\right\rVert_{\infty}<\mu$, we have
$\displaystyle\left\lVert y_{U}\right\rVert_{\infty}$
$\displaystyle\geq\mu+1-\alpha$ $\displaystyle\left\lVert
y_{\overline{U}}\right\rVert_{\infty}$ $\displaystyle\leq\mu+\alpha$
We then get
$\displaystyle\left\lVert y_{\overline{V}}\right\rVert_{p}^{p}$
$\displaystyle=\left\lVert y_{\overline{U}}\right\rVert_{p}^{p}+\left\lVert
y_{U\setminus V}\right\rVert_{p}^{p}-\left\lVert y_{V\setminus
U}\right\rVert_{p}^{p}$ $\displaystyle\geq\left\lVert
y_{\overline{U}}\right\rVert_{p}^{p}+\left|V\setminus
U\right|((\mu+1-\alpha)^{p}-(\mu+\alpha)^{p})$ $\displaystyle=\left\lVert
y_{\overline{U}}\right\rVert_{p}^{p}+\left|V\setminus
U\right|(1+(2p-2)\mu)(1-2\alpha)$
where the last step can be checked for $p\in\\{1,2\\}$. So
$\displaystyle\left\lVert y_{\overline{V}}\right\rVert_{p}^{p}$
$\displaystyle\geq\left\lVert
y_{\overline{U}}\right\rVert_{p}^{p}(1+\left|V\setminus
U\right|\frac{(1+(2p-2)\mu)(1-2\alpha)}{p(r+s\mu^{p})})$
However, $V$ is the result of $1+\epsilon$-approximate recovery, so
$\displaystyle\left\lVert y_{\overline{V}}\right\rVert_{p}$
$\displaystyle\leq\left\lVert
y-\hat{y}\right\rVert_{p}\leq(1+\epsilon)\left\lVert
y_{\overline{U}}\right\rVert_{p}$ $\displaystyle\left\lVert
y_{\overline{V}}\right\rVert_{p}^{p}$
$\displaystyle\leq(1+(2p-1)\epsilon)\left\lVert
y_{\overline{U}}\right\rVert_{p}^{p}$
for $p\in\\{1,2\\}$. Hence
$\displaystyle\left|V\setminus
U\right|\frac{(1+(2p-2)\mu)(1-2\alpha)}{p(r+s\mu^{p})}$
$\displaystyle\leq(2p-1)\epsilon$
for $\alpha\leq 1/4$, this means
$\displaystyle\left|V\setminus U\right|$
$\displaystyle\leq\frac{2\epsilon(2p-1)p(r+s\mu^{p})}{1+(2p-2)\mu}.$
Plugging in the parameters $p=1,s=r=\frac{k}{d\epsilon},\mu=D$ gives
$\left|V\setminus U\right|\leq\frac{2\epsilon((1+D^{2})r)}{1}=O(\frac{k}{d}).$
Plugging in the parameters
$p=2,q=2,r=\frac{k}{d^{2}\epsilon^{2}},\mu=\frac{1}{d\epsilon}$ gives
$\left|V\setminus U\right|\leq\frac{12\epsilon(3r)}{2\mu}=\frac{18k}{d}.$
Hence, for $d=O(c)$, we get the parameters desired in the lemma statement, and
$\left|V\setminus U\right|\leq\frac{k}{2c}.$
Bob can recover $V$ with probability $1-\delta$. Therefore he can output
$\hat{x}$ given by $\hat{x}_{i}=1$ if $i\in V$ and $\hat{x}_{i}=0$ otherwise.
This will differ from $x_{S}$ only within $(V\setminus U\cup U\setminus V)$,
which is at most $k/c$ locations. ∎
### 7.1 $k>1$
Suppose $p,s,3r$ satisfy Lemma 7.4 for some parameter $c$, and let $q=s/k$.
The Gilbert-Varshamov bound implies that there exists a code $V\subset[q]^{r}$
with $\log\left|V\right|=\Omega(r\log q)$ and minimum Hamming distance $r/4$.
Let $X\subset\\{0,1\\}^{qr}$ be in one-to-one correspondence with $V$: $x\in
X$ corresponds to $v\in V$ when $x_{(a-1)q+b}=1$ if and only if $v_{a}=b$.
Let $x$ and $v$ correspond. Let $S\subset[r]$ with $\left|S\right|=k$, so $S$
corresponds to a set $T\subset[n]$ with $\left|T\right|=kq=s$. Consider
arbitrary $\nu$ that satisfies
$\left\lVert\nu\right\rVert_{p}<\alpha\left\lVert x\right\rVert_{p}$ and
$\left\lVert\nu\right\rVert_{\infty}\leq\alpha$ for some small constant
$\alpha\leq 1/4$. We would like to apply Lemma 7.3, so we just need to show we
can recover $x$ from $A(x+\nu)$ with constant probability. Let
$\nu^{\prime}=x_{\overline{T}}+\nu$, so
$\displaystyle\left\lVert\nu^{\prime}\right\rVert_{p}^{p}$ $\displaystyle\leq
p(\left\lVert
x_{\overline{T}}\right\rVert_{p}^{p}+\left\lVert\nu\right\rVert_{p}^{p})\leq
p(r-k+\alpha^{p}r)\leq 3r$
$\displaystyle\left\lVert\nu^{\prime}_{\overline{T}}\right\rVert_{\infty}$
$\displaystyle\leq 1+\alpha$
$\displaystyle\left\lVert\nu^{\prime}_{T}\right\rVert_{\infty}$
$\displaystyle\leq\alpha$
Therefore Lemma 7.4 implies that with probability $1-\delta$, if Bob is given
$A(x_{T}+\nu^{\prime})=A(x+\nu)$ he can recover $\hat{x}$ that agrees with
$x_{T}$ in all but $k/c$ locations. Hence in all but $k/c$ of the $i\in S$,
$x_{\\{(i-1)q+1,\dotsc,iq\\}}=\hat{x}_{\\{(i-1)q+1,\dotsc,iq\\}}$, so he can
identify $v_{i}$. Hence Bob can recover an estimate of $v_{S}$ that is
accurate in $(1-1/c)k$ characters with probability $1-\delta$, so it agrees
with $v_{S}$ in $(1-1/c)(1-\delta)k$ characters in expectation. If we apply
this in parallel to the sets $S_{i}=\\{k(i-1)+1,\dotsc,ki\\}$ for $i\in[r/k]$,
we recover $(1-1/c)(1-\delta)r$ characters in expectation. Hence with
probability at least $1/2$, we recover more than $(1-2(1/c+\delta))r$
characters of $v$. If we set $\delta$ and $1/c$ to less than $1/32$, this
gives that we recover all but $r/8$ characters of $v$. Since $V$ has minimum
distance $r/4$, this allows us to recover $v$ (and hence $x$) exactly. By
Lemma 7.3 this gives a lower bound of
$m=\Omega(\log\left|V\right|)=\Omega(r\log q)$. Hence
$m=\Omega(\frac{1}{\epsilon}k\log\frac{1}{\epsilon})$ for $\ell_{1}/\ell_{1}$
recovery and $m=\Omega(\frac{1}{\epsilon^{2}}k)$ for $\ell_{2}/\ell_{2}$
recovery.
### 7.2 $k=1,\delta=o(1)$
To achieve the other half of our lower bounds for sparse outputs, we restrict
to the $k=1$ case. A $k$-sparse algorithm implies a $1$-sparse algorithm by
inserting $k-1$ dummy coordinates of value $\infty$, so this is valid.
Let $p,s,51r$ satisfy Lemma 7.4 for some $\alpha$ and $D$ to be determined,
and let our recovery algorithm have failure probability $\delta$. Let
$C=1/(2r\delta)$ and $n=Cr$. Let $V=[(s-1)C]^{r}$ and let
$X^{\prime}\in\\{0,1\\}^{(s-1)Cr}$ be the corresponding binary vector. Let
$X=\\{0\\}\times X^{\prime}$ be defined by adding $x_{0}=0$ to each vector.
Now, consider arbitrary $x\in X$ and noise $\nu\in\mathbb{R}^{1+(s-1)Cr}$ with
$\left\lVert\nu\right\rVert_{p}<\alpha\left\lVert x\right\rVert_{p}$ and
$\left\lVert\nu\right\rVert_{\infty}\leq\alpha$ for some small constant
$\alpha\leq 1/20$. Let $e^{0}/5$ be the vector that is $1/5$ at $0$ and $0$
elsewhere. Consider the sets
$S_{i}=\\{0,(s-1)(i-1)+1,(s-1)(i-1)+2,\dotsc,(s-1)i\\}$. We would like to
apply Lemma 7.4 to recover $(x+\nu+e^{0}/5)_{S_{i}}$ for each $i$.
To see what it implies, there are two cases: $\left\lVert
x_{sSi}\right\rVert_{1}=1$ and $\left\lVert x_{S_{i}}\right\rVert_{1}=0$
(since $S_{i}$ lies entirely in one character, $\left\lVert
x_{S_{i}}\right\rVert_{1}\in\\{0,1\\}$). In the former case, we have
$\nu^{\prime}=x_{\overline{S_{i}}}+\nu+e^{0}/5$ with
$\displaystyle\left\lVert\nu^{\prime}\right\rVert_{p}^{p}$
$\displaystyle\leq(2p-1)(\left\lVert
x_{\overline{S_{i}}}\right\rVert_{p}^{p}+\left\lVert\nu\right\rVert_{p}^{p}+\left\lVert
e^{0}/5\right\rVert_{p}^{p})\leq 3(r+\alpha^{p}r+1/5^{p})<4r$
$\displaystyle\left\lVert\nu^{\prime}_{\overline{S_{i}}}\right\rVert_{\infty}$
$\displaystyle\leq 1+\alpha$
$\displaystyle\left\lVert\nu^{\prime}_{S_{i}}\right\rVert_{\infty}$
$\displaystyle\leq 1/5+\alpha\leq 1/4$
Hence Lemma 7.4 will, with failure probability $\delta$, recover
$\hat{x}_{S_{i}}$ that differs from $x_{S_{i}}$ in at most $1/c<1$ positions,
so $x_{S_{i}}$ is correctly recovered.
Now, suppose $\left\lVert x_{S_{i}}\right\rVert_{1}=0$. Then we observe that
Lemma 7.4 would apply to recovery from $5A(x+\nu+e^{0}/5)$, with
$\nu^{\prime}=5x+5\nu$ and $x^{\prime}=e^{0}$, so
$\displaystyle\left\lVert\nu^{\prime}\right\rVert_{p}^{p}$ $\displaystyle\leq
5^{p}p(\left\lVert
x\right\rVert_{p}^{p}+\left\lVert\nu\right\rVert_{p}^{p})\leq
5^{p}p(r+\alpha^{p}r)<51r$
$\displaystyle\left\lVert\nu^{\prime}_{\overline{S_{i}}}\right\rVert_{\infty}$
$\displaystyle\leq 5+5\alpha$
$\displaystyle\left\lVert\nu^{\prime}_{S_{i}}\right\rVert_{\infty}$
$\displaystyle\leq 5\alpha.$
Hence Lemma 7.4 would recover, with failure probability $\delta$, an
$\hat{x}_{S_{i}}$ with support equal to $\\{0\\}$.
Now, we observe that the algorithm in Lemma 7.4 is robust to scaling the input
$A(x^{\prime}+\nu^{\prime})$ by $5$; the only difference is that the effective
$\mu$ changes by the same factor, which increases the number of errors $k/c$
by a factor of at most $5$. Hence if $c>5$, we can apply the algorithm once
and have it work regardless of whether $\left\lVert x_{S_{i}}\right\rVert_{1}$
is $0$ or $1$: if $\left\lVert x_{S_{i}}\right\rVert_{1}=1$ the result has
support $\operatorname{supp}(x_{i})$, and if $\left\lVert
x_{S_{i}}\right\rVert_{1}=0$ the result has support $\\{0\\}$. Thus we can
recover $x_{S_{i}}$ exactly with failure probability $\delta$.
If we try this to the $Cr=1/(2\delta)$ sets $S_{i}$, we recover all of $x$
correctly with failure probability at most $1/2$. Hence Lemma 7.3 implies that
$m=\Omega(\log\left|X\right|)=\Omega(r\log\frac{s}{r\delta})$. For
$\ell_{1}/\ell_{1}$, this means
$m=\Omega(\frac{1}{\epsilon}\log\frac{1}{\delta})$; for $\ell_{2}/\ell_{2}$,
this means $m=\Omega(\frac{1}{\epsilon^{2}}\log\frac{\epsilon^{2}}{\delta})$.
Acknowledgment: We thank T.S. Jayram for helpful discussions.
## References
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* [BR11] Mark Braverman and Anup Rao. Information equals amortized communication. In STOC, 2011.
* [BY02] Ziv Bar-Yossef. The Complexity of Massive Data Set Computations. PhD thesis, UC Berkeley, 2002.
* [BYJKS04] Ziv Bar-Yossef, T. S. Jayram, Ravi Kumar, and D. Sivakumar. An information statistics approach to data stream and communication complexity. J. Comput. Syst. Sci., 68(4):702–732, 2004.
* [CCF02] M. Charikar, K. Chen, and M. Farach-Colton. Finding frequent items in data streams. ICALP, 2002.
* [CD11] E.J. Candès and M.A. Davenport. How well can we estimate a sparse vector? Arxiv preprint arXiv:1104.5246, 2011.
* [CM04] G. Cormode and S. Muthukrishnan. Improved data stream summaries: The count-min sketch and its applications. LATIN, 2004.
* [CM05] Graham Cormode and S. Muthukrishnan. Summarizing and mining skewed data streams. In SDM, 2005.
* [CM06] G. Cormode and S. Muthukrishnan. Combinatorial algorithms for compressed sensing. Sirocco, 2006.
* [CRT06] E. J. Candès, J. Romberg, and T. Tao. Stable signal recovery from incomplete and inaccurate measurements. Comm. Pure Appl. Math., 59(8):1208–1223, 2006.
* [DIPW10] K. Do Ba, P. Indyk, E. Price, and D. Woodruff. Lower bounds for sparse recovery. SODA, 2010.
* [Don06] D. L. Donoho. Compressed Sensing. IEEE Trans. Info. Theory, 52(4):1289–1306, Apr. 2006.
* [FPRU10] S. Foucart, A. Pajor, H. Rauhut, and T. Ullrich. The gelfand widths of lp-balls for $0<p\leq 1$. 2010\.
* [GLPS10] Anna C. Gilbert, Yi Li, Ely Porat, and Martin J. Strauss. Approximate sparse recovery: optimizing time and measurements. In STOC, pages 475–484, 2010.
* [Gur10] V. Guruswami. Introduction to coding theory. Graduate course notes, available at http://www.cs.cmu.edu/~venkatg/teaching/codingtheory/, 2010.
* [IR08] Piotr Indyk and Milan Ruzic. Near-optimal sparse recovery in the l1 norm. In FOCS, pages 199–207, 2008.
* [IT10] MA Iwen and AH Tewfik. Adaptive group testing strategies for target detection and localization in noisy environments. IMA Preprint Series, (2311), 2010.
* [Jay02] T.S. Jayram. Unpublished manuscript, 2002.
* [Mut05] S. Muthukrishnan. Data streams: Algorithms and applications). FTTCS, 2005.
* [SAZ10] N. Shental, A. Amir, and Or Zuk. Identification of rare alleles and their carriers using compressed se(que)nsing. Nucleic Acids Research, 38(19):1–22, 2010.
* [TDB09] J. Treichler, M. Davenport, and R. Baraniuk. Application of compressive sensing to the design of wideband signal acquisition receivers. In Proc. U.S./Australia Joint Work. Defense Apps. of Signal Processing (DASP), 2009.
* [Wai09] Martin J. Wainwright. Information-theoretic limits on sparsity recovery in the high-dimensional and noisy setting. IEEE Transactions on Information Theory, 55(12):5728–5741, 2009\.
|
arxiv-papers
| 2011-10-19T22:44:28 |
2024-09-04T02:49:23.388681
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Eric Price and David P. Woodruff",
"submitter": "Eric Price",
"url": "https://arxiv.org/abs/1110.4414"
}
|
1110.4420
|
# Long-lived dipolar molecules and Feshbach molecules in a 3D optical lattice
Amodsen Chotia,∗ Brian Neyenhuis,∗ Steven A. Moses, Bo Yan, Jacob P. Covey,
Michael Foss-Feig, Ana Maria Rey, Deborah S. Jin,† and Jun Ye† JILA, National
Institute of Standards and Technology and University of Colorado, Department
of Physics, University of Colorado, Boulder, CO 80309-0440, USA
###### Abstract
We have realized long-lived ground-state polar molecules in a 3D optical
lattice, with a lifetime of up to 25 s, which is limited only by off-resonant
scattering of the trapping light. Starting from a 2D optical lattice, we
observe that the lifetime increases dramatically as a small lattice potential
is added along the tube-shaped lattice traps. The 3D optical lattice also
dramatically increases the lifetime for weakly bound Feshbach molecules. For a
pure gas of Feshbach molecules, we observe a lifetime of $>$20 s in a 3D
optical lattice; this represents a 100-fold improvement over previous results.
This lifetime is also limited by off-resonant scattering, the rate of which is
related to the size of the Feshbach molecule. Individually trapped Feshbach
molecules in the 3D lattice can be converted to pairs of K and Rb atoms and
back with nearly 100$\%$ efficiency.
###### pacs:
03.75.-b, 37.10.Pq, 67.85.-d, 33.20.-t
Controllable long-range and anisotropic dipole-dipole interactions can enable
novel applications of quantum gases in investigating strongly correlated many-
body systems Baranov (2008); Pupillo et al. (2009); Lahaye et al. (2009);
Gorshkov et al. (2011); Carr et al. (2009); Bloch et al. (2008). Recent
experiments have realized an ultracold gas of polar molecules in the ro-
vibrational ground state Ni et al. (2008) with high-resolution, single-state
control at the level of hyperfine structure Ospelkaus et al. (2010a). However,
an obstacle to creating long-lived quantum gases of polar molecules was
encountered with the observation of bimolecular chemical reactions in the
quantum regime Ospelkaus et al. (2010b). Even with the demonstrated strong
suppression of the reaction rate for spin-polarized fermionic KRb molecules,
the lifetime of a 300 nK sample with a peak density of 1012/cm3 was limited to
$\sim$1 s. Furthermore, when an external electric field is applied to polarize
the molecules in the lab frame, the attractive part of the dipole-dipole
interaction dramatically increases the ultracold chemical reaction rate,
reducing the lifetime of the dipolar gas to a few ms when the lab-frame
molecular dipole moment reaches 0.2 Debye Ni et al. (2010). A promising recent
development was the demonstration that confining fermionic polar molecules in
a 1D optical lattice suppresses the rate of chemical reactions even in the
presence of dipolar interactions. Here, the spatial anisotropy of the dipolar
interaction was exploited by confining a gas of oriented KRb molecules in a
two-dimensional geometry to suppress the attractive part of the dipolar
interaction and thus achieve control of the stereodynamics of the bimolecular
reactions de Miranda et al. (2011). In this regime, the lifetime of a trapped
gas of polar molecules with a lab-frame dipole moment of $\sim$0.2 Debye, a
temperature of 800 nK, and a number density of 107 cm-2, was $\sim$1 s.
In this letter, we study KRb molecules confined in 2D and 3D optical lattice
traps, where we explore the effects of the lattice confinement on the lifetime
of the ultracold gas. We note that a lifetime of 8 s has been achieved for
homonuclear Cs2 molecules in a 3D lattice Danzl et al. (2010). In our work, we
find that long lifetimes are achieved for the molecules in a strong 3D lattice
trap, even when there is a significant dipole moment in the lab frame. In
addition, we observe that adding a weak axial corrugation to a 2D lattice can
result in long lifetimes for the trapped molecules.
The experiments start with an ultracold mixture of $2.9\times 10^{5}$ 40K
atoms and $2.3\times 10^{5}$ 87Rb atoms in a crossed optical dipole trap (ODT)
at 1064 nm, at a temperature that is twice the Rb condensation temperature
$T_{c}$. The trap frequencies for Rb are $21$ Hz in the horizontal ($x,y$)
plane and $165$ Hz in the vertical ($z$) direction; the trap frequencies for K
are 1.37 times larger. The atoms are transferred into a 3D optical lattice in
three steps. We first turn on a retro-reflected vertical beam in $150$ ms to
create a weak 1D lattice. In the second step, the ODT is ramped off in $100$
ms so that the two beams used for the ODT (which propagate along $x$ and $y$)
can be converted to lattice beams by allowing them to be retro-reflected. The
intensities along all three directions are then ramped to their final values
in $100$ ms. The three lattice beams are derived from a common laser but
individually frequency shifted. The $x$ and $y$ beams are elliptical with a
200 $\times$ 40 $\mu$m waist and are linearly polarized orthogonal in the
$x$-$y$ plane; the $z$-beam has a circular waist of 250 $\mu$m and is linearly
polarized along $x$. We calibrate the lattice strength using Rb atoms with two
different methods (Kapitza-Dirac scattering pulse in a BEC Denschlag et al.
(2002) and parametric modulation of the lattice) and then account for the
differences in mass and ac polarizability to determine the lattice strength
for KRb molecules. The values reported herein for the lattice depth in each
direction are expressed in units of the molecule recoil energy, $E_{R}$, and
have an estimated $10\%$ uncertainty.
Once the atoms are loaded in the 3D lattice, we ramp an external magnetic
field across an $s$-wave Feshbach resonance at $546.78$ G to form loosely
bound 40K87Rb molecules with an efficiency of about 10$\%$. With the Feshbach
molecules at $B=545.8$ G, where their binding energy is $h\times$400 kHz, we
use two-photon stimulated Raman adiabatic passage (STIRAP) to coherently
transfer the Feshbach molecules to the ro-vibrational ground state Ni et al.
(2008), with a typical one-way transfer efficiency of 80%. All the molecules
are in a single nuclear spin state in the rotational ground state,
$|N=0,m_{N}=0,m_{I}^{K}=-4,m_{I}^{Rb}=1/2\rangle$, following the notation
defined in Ospelkaus et al. (2010a). During this procedure, unpaired K and Rb
atoms are removed using resonant light pulses. To measure the number of
ground-state molecules in the lattice, we reverse the STIRAP process and then
image the resultant Feshbach molecules using absorption of a probe beam that
is tuned to the imaging transition for K atoms.
Figure 1: Loss of ground-state KRb molecules as a function of time in a 3D
lattice with depths of 56, 56, and 70 $E_{R}$ in $x$, $y$, and $z$,
respectively, where $E_{R}=\hbar^{2}k^{2}/2m$ is the KRb recoil energy, $k$ is
the magnitude of the lattice beam wave vector, and $m$ the molecular mass.
Neglecting the very short time points (red solid circles), the number of
molecules for times larger than 1 s (black solid circles) are fit to a single
exponential decay, yielding a 1/$e$ lifetime of 16.3$\pm$1.5 s. Inset:
Lifetime in an isotropic lattice with a depth of 50 $E_{R}$, with (blue open
squares, $0.17$ Debye) and without (black squares, 0 Debye) an applied
electric field. The lifetimes at $0.17$ Debye (15$\pm$4 s) and 0 Debye agree
within uncertainty.
Figure 1 shows a time-dependent evolution of the ground-state molecule
population in the 3D lattice. In the first few 100’s of ms, the measured
number of molecules exhibits relatively large variations in repeated
iterations of the experiment and is consistent with some fast initial decay.
In all our measurements of ground-state molecules in deep 3D lattices (for
example, in the data for Fig. 2), we observe a similar feature. One possible
explanation for this fast decay is collisions of the ground-state molecules
with impurities, such as molecules in excited internal states that might be
produced in the STIRAP process. Fitting the data for times greater than 1 s to
an exponential decay, which is consistent with a single-body loss mechanism,
gives a $1/e$ lifetime of 16.3$\pm$1.5 s. This is much longer than previously
measured lifetimes of trapped ultracold polar molecules of about 1 s in an ODT
Ospelkaus et al. (2010b) or in a 1D lattice de Miranda et al. (2011).
The long lifetime for ground-state molecules in a reasonably deep 3D lattice
can be understood simply from the fact that the optical lattice localizes the
molecules and therefore prevents bimolecular reactions. It was previously seen
that an applied electric field strongly increased the chemical reaction rate
Ni et al. (2010). However, for molecules individually isolated in a 3D
lattice, we expect no dependence of the lifetime on the strength of an applied
electric field. In the inset to Fig. 1, we show that indeed we do not observe
any decrease of the lifetime for polarized molecules with an induced dipole
moment of $0.17$ Debye.
To understand what limits the lifetime of the molecules in the 3D lattice, we
investigate its dependence on the lattice strength as summarized in Fig. 2.
First, we explore the transition from a 2D lattice (an array of one-
dimensional tubes) to a 3D lattice. For a molecular gas confined in the tubes
with no lattice in $z$, we find a lifetime of $\sim$1 s. However, as soon as a
small lattice potential is added along $z$, the lifetime is dramatically
increased, reaching 5 s at 12 $E_{R}$ and 20 s at $17$$E_{R}$ (point a in Fig.
2). To verify that bimolecular reactions are the dominant loss mechanism, we
have checked that the lifetime in uncorrugated tubes decreases significantly
(to 0.1 s) when we apply an electric field (oriented along the tubes) that
gives an induced dipole moment of 0.17 Debye. In addition, the fact that we
can place an upper limit of 10% of the initial number remaining at long times
puts a limit on the contribution to our signal from tubes that are occupied
with only one molecule.
Figure 2: Lifetime of KRb ground-state molecules in an optical lattice. Black
circles: $x$ and $y$ lattice beams are fixed at 56 $E_{R}$ per beam, while $z$
is varied from 0 to 136 $E_{R}$ (1 $E_{R}$ corresponds to a lattice intensity
$I$ = $0.025$ kW/cm2). The lifetime reaches a maximum of 25$\pm$5 s when the
$z$ lattice depth is 34 $E_{R}$ (point b). For higher lattice intensities, the
lifetime decreases, which is consistent with loss due to off-resonant light
scattering (dashed line). The open circles correspond to a 3D lattice where
the radial confinement was also varied. The red squares correspond to
lifetimes measured with an additional traveling-wave beam at 1064 nm
illuminating the molecules in the 3D lattice. Point c (d) corresponds to the
3D lattice of point a with an intensity of 3.2 kW/cm2 (b with 3.7 kW/cm2) plus
the additional beam intensity of 2.3 kW/cm2 (3.5 kW/cm2). Solid lines: see
text.
We consider a number of factors (Supplementary Information) to understand the
rapid suppression of loss as a function of the lattice strength in $z$. In
general, Pauli blocking for identical fermions and dissipation blockade
effects (suppression of loss when the loss rate for particles on the same site
is much larger than the tunneling rate Syassen et al. (2008)) can play a role
in the lifetime of KRb molecules in an optical lattice. However, in the
measurements reported here, the optical lattice is sparsely filled. For our
identical fermionic molecules, we expect Pauli blocking would give an even
steeper function than we observe. Moreover, for a 5 $E_{R}$ lattice in $z$,
the lifetime of KRb molecules in the tube does not change significantly in the
presence of an applied electric field. This observation suggests that an
incoherent process, such as heating of the trapped gas, limits the lifetime in
this regime of weak lattice confinement. Instabilities in the optical phase of
the lattice beams directly give rise to translational noise of the lattice,
which can promote molecules to higher bands, where they have increased
mobility and could then collide with other molecules. A simple theoretical
model taking into account a constant heating rate, with collisions in higher
bands happening on a timescale much shorter than the heating time, is
consistent with the experimental observation (red and blue solid lines in Fig.
2, with heating rates of 1 $E_{R}$/s (66 nK/s) and 2 $E_{R}$/s, respectively).
In Fig. 2, the lifetime reaches a maximum of 25$\pm$5 s indicated by point b.
As the intensity is increased further, the lifetime starts to decrease,
consistent with off-resonant photon scattering becoming the dominant loss
mechanism. The rich internal state structure of molecules ensures that each
off-resonant photon scattering event has a high probability of causing the
loss of a molecule from the ground state. To explore this effect, we added an
additional traveling-wave beam with a wavelength of 1064 nm; this increases
the photon scattering rate without increasing the trap depth and we observe a
significant reduction of the lifetime due to the additional light. We can
extract the imaginary part of the polarizability of KRb molecules by fitting
the lifetime as a function of the light intensity to $1/(\alpha I)$. Here,
$\alpha$ is the imaginary part of the polarizability at 1064 nm, which we
determine to be $(2.052\pm 0.009)\times 10^{-12}$ MHz/(W/cm2); this is
consistent with a theory estimate for KRb Kotochigova (2011); Kotochigova et
al. (2009).
We have also explored the lifetime of KRb Feshbach molecules in the 3D optical
lattice. It has been shown that these weakly bound molecules can be rapidly
lost from an ODT due to collisions with atoms Zirbel et al. (2008). Even with
removal of the Rb atoms and the RF transfer of the K atoms to a different
hyperfine state, all previously measured lifetimes for KRb Feshbach molecules
were less than 10 ms Ospelkaus et al. (2008). However, with the ability to
create ground-state molecules, which do not scatter light that is resonant
with the single-atom transitions, we can more efficiently use light pulses to
remove any residual atoms. A few ms after the atom removal, we reverse the
STIRAP process and recreate a Feshbach molecule gas. For the case of molecules
in the ODT (no lattice), we find that this extends the lifetime of the
Feshbach molecules to 150 ms. Therefore, we can conclude that previous
lifetime measurements were likely limited by collisions with residual atoms.
When we perform the procedure described above for KRb molecules in an optical
lattice, we find that the purified gas of Feshbach molecules can have a
lifetime as long as 10 s. In Fig. 3 we explore the lifetime of the Feshbach
molecules in the 3D lattice as a function of the magnetic-field detuning from
the resonance ($B_{0}$ = 546.78 G), where varying the magnetic field, $B$,
changes the binding energy and the size of the Feshbach molecules. We start by
forming a purified sample of the Feshbach molecules in a strong 3D lattice
with an intensity of 50 $E_{R}$ per beam at $B$ = 545.8 G. The magnetic field
is then ramped to its final value in 1 ms. At the end of the hold time in the
3D lattice, $B$ is ramped back to 545.8 G where we image the molecules.
Figure 3: Lifetime of Feshbach molecules and confinement-induced molecules
measured as a function of $B$. A purified sample of Feshbach molecules is held
in an isotropic 3D optical lattice (50 $E_{R}$ per beam, 20 kHz trap
frequency). Near the Feshbach resonance, the loss rate due to photon
scattering can be modeled (solid line) as a weighted sum of the free atom loss
rate $\Gamma_{atom}$ and a higher loss rate for tightly bound molecules
$\Gamma_{molecule}$ . The grey shaded area indicates the single atom lifetime,
and its uncertainty, measured for the same experimental conditions. Inset:
Lifetime of Feshbach molecules in a 3D lattice as a function of the trap
intensity, at 545.8 G (blue stars) and 543.18 G (green diamonds). The dashed
and solid line fits are used to extract the scattering rates.
Above the Feshbach resonance, the lattice potential allows for the existence
of confinement-induced molecules that do not exist in free space Stöferle et
al. (2006). We find that confinement-induced molecules have a lifetime (25 s)
that is comparable to that for K or Rb atoms in the same trap. Below the
Feshbach resonance, the molecule lifetime decreases quickly when the magnetic
field is ramped to lower values. Several Gauss below the resonance, the
Feshbach molecule lifetime is reduced to $\sim$1 s, which is still
significantly longer than in the ODT. Overall, these results represent a two-
orders-of-magnitude improvement in the lifetime of Feshbach molecules compared
to a previous measurement of KRb molecules in an optical lattice Ospelkaus et
al. (2006).
To understand the dependence of the lifetime on $B$, we can assume that the
lifetime is limited by off-resonant photon scattering from the lattice light
and consider two limiting cases. For $B\gg B_{0}$, the photon scattering limit
is simply that for free atoms $\Gamma_{atom}$; for $B\ll B_{0}$, we have a
higher photon scattering rate $\Gamma_{molecule}$ due to a larger wavefunction
overlap with electronically excited molecules Danzl et al. (2009). In a two-
channel model of the Feshbach resonance Chin et al. (2010), the Feshbach
molecule wavefunction can be written as an amplitude $Z^{1/2}$ times the bare
“closed-channel” molecule wavefunction plus an amplitude $(1-Z)^{1/2}$ times
the “open channel” wavefunction that describes the scattering state of two
free atoms. We then take the total photon scattering rate to be given by
$Z\Gamma_{molecule}+(1-Z)\Gamma_{atom}$. With pairs of atoms confined in an
optical trap with a known depth, $Z$ can be calculated straightforwardly with
a coupled-channel theory Chin et al. (2010); Julienne (2009). Using the
measured loss rates for the limiting cases, $\Gamma_{atom}$ and
$\Gamma_{molecule}$, this simple theory (solid line in Fig. 3) without any
additional adjustable parameters describes very well the experimental results
(filled circles). We note that the rate of atom-molecule collisions has been
analyzed in a similar way Deuretzbacher et al. (2008); Ospelkaus et al.
(2006).
The assumption that the lifetime of the purified gas of Feshbach molecules in
a 3D lattice is limited by only the photon scattering can be checked by
varying the lattice beam intensity. This is shown in the inset of Fig. 3 for
two values of $B$. Similar to the case of ground-state molecules, we observe a
rapid initial increase in the lifetime going from no lattice (only the ODT) to
a weak lattice. Following this initial rise, we observe a decrease in the
Feshbach molecule lifetime as the lattice intensity is increased, consistent
with loss due to off-resonant scattering of the lattice light. From the fits
of Fig. 3 we extract an imaginary part of the Feshbach molecule’s ac
polarizability at 1064 nm of 15.9$\pm$1.6 MHz/(W/cm2) for $B$ = 545.8 G and
30$\pm$3 MHz/(W/cm2) for $B$ = 543.18 G.
In a simple model for the conversion of atoms to Feshbach molecules in a 3D
lattice, one could assume 100$\%$ conversion efficiency for individual lattice
sites that are occupied by exactly one K atom and one Rb atom. Starting with a
purified sample of Feshbach molecules prepared with round-trip STIRAP and atom
removal as discussed above, we dissociate the molecules in the lattice by
ramping $B$ above $B_{0}$ to 548.97 G. This should ideally produce only pre-
formed pairs of atoms. We then ramp $B$ back down to 545.8 G and measure the
molecular conversion efficiency. The result is $(87\pm 13)\%$, where the
uncertainty is dominated by fluctuations in STIRAP efficiency in successive
runs of the experiment. This high efficiency is far above the maximum of
$25\%$ observed in an ODT Zirbel et al. (2008); this indicates that optimizing
the loading procedure in order to have a larger number of sites with exactly
one Rb and one K atom would increase the overall conversion efficiency. For
heteronuclear Bose-Fermi mixtures, optimizing the number of pre-formed atom
pairs in a lattice remains a challenge. Recent progress in this direction
includes the characterization of a dual Bose-Fermi Mott insulator with two
isotopes of Yb Sugawa et al. (2011), as well as proposals to use interaction
effects to optimize the lattice loading Jaksch et al. (2002); Damski et al.
(2003); Freericks et al. (2010).
The capability demonstrated here for using a 3D lattice to freeze out chemical
reactions, and thus prepare an ensemble of long-lived dipolar molecules, opens
the door for studying many-body interactions in a gas of polar molecules in a
lattice. For example, spectroscopy of rotational states of polar molecules in
a lattice is a possible approach to access correlation functions Hazzard et
al. (2011). While individual molecules may experience long rotational
coherence times, dipolar interactions between neighboring lattice sites, which
could have an interaction energy on the order of a few hundred Hz, will
certainly require a systematic understanding of the many-body system to
understand the resulting complex response. Future challenges in studying these
systems include achieving higher lattice filling factors, or correspondingly,
lower entropy for a dipolar gas in a lattice.
We thank P. Julienne for the calculation based on the two-channel model of the
Feshbach resonance and S. Kotochigova for the calculation of the complex
polarizability of KRb ground-state molecules. We thank G. Quéméner and J. L.
Bohn for stimulating discussions and also their estimates of the on-site
bimolecular loss rate. We gratefully acknowledge financial support for this
work from NIST, NSF, AFOSR-MURI, DOE, and DARPA. S.A.M. acknowledges funding
from the NDSEG Graduate Fellowship.
∗These authors contributed equally to this work.
†To whom correspondence should be addressed;
E-mail: Jin@jilau1.colorado.edu; Ye@jila.colorado.edu
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|
arxiv-papers
| 2011-10-20T01:29:31 |
2024-09-04T02:49:23.401365
|
{
"license": "Public Domain",
"authors": "Amodsen Chotia, Brian Neyenhuis, Steven A. Moses, Bo Yan, Jacob P.\n Covey, Michael Foss-Feig, Ana Maria Rey, Deborah S. Jin, and Jun Ye",
"submitter": "Amodsen Chotia",
"url": "https://arxiv.org/abs/1110.4420"
}
|
1110.4445
|
# On the Applications of Cyclotomic Fields in Introductory Number Theory
Kabalan Gaspard
(Date: June 22, 2011, re-edited February 11, 2012)
###### Abstract.
In this essay, we see how prime cyclotomic fields (cyclotomic fields obtained
by adjoining a primitive $p$-th root of unity to $\mathbb{Q}$, where $p$ is an
odd prime) can lead to elegant proofs of number theoretical concepts. We
namely develop the notion of primary units in a cyclotomic field, demonstrate
their equivalence to real units in this case, and show how this leads to a
proof of a special case of Fermat’s Last Theorem. We finally modernize
Dirichlet’s solution to Pell’s Equation.
Throughout this paper, unless specified otherwise, $\zeta\equiv\zeta_{p}\equiv
e^{\frac{2\pi\sqrt{-1}}{p}}$ where $p$ is an odd prime.
$K\equiv\mathbb{Q}(\zeta)$ and $\mathcal{O}_{K}$ is the ring of integers of
$K$. We assume knowledge of the basic properties of prime cyclotomic fields
that can be found in any introductory algebraic number theory textbook, namely
that:
* •
$Gal(K:\mathbb{Q})\simeq U(\mathbb{Z}/p\mathbb{Z})$ (the group of units of
$\mathbb{Z}/p\mathbb{Z}$), which is cyclic and of order $p-1$.
* •
$\mathcal{O}_{K}=\mathbb{Z}[\zeta_{p}]=\left\langle
1,\zeta_{p},...,\zeta_{p}^{p-2}\right\rangle_{\mathbb{Z}}$, where
$\left\\{1,\zeta_{p},...,\zeta_{p}^{p-2}\right\\}$ is a $\mathbb{Z}$-basis for
$\mathcal{O}_{K}$.
* •
The only roots of unity in $\mathcal{O}_{K}$ (i.e. solutions in $\mathbb{C}$
to $x^{n}=1$ for some $n\in\mathbb{N}$) are of the form $\pm\zeta_{p}^{i}$,
$i\in\mathbb{Z}$.
We also assume elementary knowledge of quadratic characters, quadratic
reciprocity, and the Legendre symbol $\left(\dfrac{k}{p}\right)$.
## 1\. Primary elements in $\mathcal{O}_{K}$
###### Definition 1.
Let $\alpha\in\mathcal{O}_{K}$ with $\alpha$ prime to $p$. Then $\alpha$ is
_primary_ iff $\alpha$ is congruent to a rational integer modulo
$(1-\zeta_{p})^{2}$.
The definition of primary elements has historically been ambiguous in Number
Theory. In [2], Dalawat shows that definitions of primary elements in
$\mathcal{O}_{K}$ even differ by country (”$p$-primary”, ”primaire” and
”primär”) and, even though these definitions do form a chain of implications,
they are not equivalent.
We also note that it is not true that if $p$ an arbitrary odd prime and $\mu$
prime in $\mathcal{O}_{K}$, only one associate of $\mu$ is primary (for
example, according to the above definition, both $\pm(4+3\omega)$ are primary
in the ring of integers of $\mathbb{Q}(\omega)$ where
$\omega=e^{\frac{2\pi\sqrt{-1}}{3}}$).
###### Proposition 1.
Let $\alpha$ $\in\mathcal{O}_{K}$ (not necessarily prime) and suppose $\alpha$
prime to $p$ in $\mathcal{O}_{K}$. Then there exists a $k\in\mathbb{Z}$,
unique (modulo $p$), such that $\zeta_{p}^{k}\alpha$ is primary.
###### Proof.
Consider the ideal $P=(1-\zeta_{p})$ in $\mathcal{O}_{K}$. Then the norm of
the ideal $N(P)=\prod\limits_{i=1}^{p-1}(1-\zeta_{p}^{i})=p$ by the fact that
$Gal(K:\mathbb{Q})\simeq U(\mathbb{Z}/p\mathbb{Z})$. So $P$ is a prime ideal
and is thus of degree $1$. So by Dedekind’s Theorem in Algebraic Number
Theory, any element of $\mathcal{O}_{K}$ is the root of a monic polynomial of
degree $1$ in $\mathcal{O}_{K}/P$. So in the particular case of $\alpha$,
$\alpha-a_{0}=\overline{0}$ in $\mathcal{O}_{K}/P$ for some
$a_{0}\in\mathbb{Z}$. In other words, $\alpha\equiv a_{0}$ $(1-\zeta_{p})$. So
$\frac{\alpha-a_{0}}{(1-\zeta_{p})}\in\mathcal{O}_{K}$ and so, by the same
argument, $\frac{\alpha-a_{0}}{(1-\zeta_{p})}\equiv a_{1}$ $(1-\zeta_{p})$ for
some $a_{1}\in\mathbb{Z}$. We stop repeating this here because multiplying the
congruence by $(1-\zeta_{p})$, we now have a congruence modulo
$(1-\zeta_{p})^{2}$, which is what we want to consider. More precisely, we now
have $\alpha-a_{0}\equiv a_{1}(1-\zeta_{p})$ $\ (1-\zeta_{p})^{2}$, so
$\alpha\equiv a_{0}+a_{1}(1-\zeta_{p})$ $\ (1-\zeta_{p})^{2}$.
We want to eliminate the $(1-\zeta_{p})$ term by multiplying both sides by
$\zeta_{p}^{n}$ for some $n\in\mathbb{Z}$. Notice that
$\zeta_{p}=(1-(1-\zeta_{p}))$. So modulo $(1-\zeta_{p})^{2}$,
$\displaystyle\zeta_{p}^{n}\alpha$ $\displaystyle\equiv$
$\displaystyle\zeta_{p}^{n}a_{0}+a_{1}\zeta_{p}^{n}(1-\zeta_{p})$
$\displaystyle\equiv$ $\displaystyle
a_{0}(1-(1-\zeta_{p}))^{n}+a_{1}(1-\zeta_{p})(1-(1-\zeta_{p}))^{n}$
$\displaystyle\equiv$ $\displaystyle
a_{0}(1-n(1-\zeta_{p}))+a_{1}(1-\zeta_{p})(1-n(1-\zeta_{p}))\text{ }$
since considering $(1-(1-\zeta_{p}))^{n}$ as a polynomial in $(1-\zeta_{p})$,
$(1-\zeta_{p})^{2}$ divides $(1-\zeta_{p})^{i}$ for $i\geq 2$. So
$\zeta_{p}^{n}\alpha\equiv a_{0}+(a_{1}-na_{0})(1-\zeta_{p})\text{ \ \
}(1-\zeta_{p})^{2}$
Now $\alpha$ prime to $p$, so if $a_{0}\equiv 0$ $(p)$, then $a_{0}\equiv 0$
$(1-\zeta_{p})$, and so $\alpha\equiv 0$ $(1-\zeta_{p})$, which is a
contradiction. So $a_{0}\not\equiv 0$ $(p)$, and so $a_{1}-na_{0}\equiv 0$ has
a unique solution $k$ modulo $p$. Now $(1-\zeta_{p})\mid(1-\zeta_{p}^{2})$,
and
$N(\frac{1-\zeta_{p}^{2}}{1-\zeta_{p}})=\frac{N(1-\zeta_{p}^{2})}{N(1-\zeta_{p})}=1$,
so $(1-\zeta_{p}^{2})$ is associate to $(1-\zeta_{p})$. It follows that
$(1-\zeta_{p})^{2}\mid p$, and so $k$ is (still, since
$a_{1}-na_{0}\in\mathbb{Z}$) the unique integral solution modulo $p$ to
$a_{1}-na_{0}\equiv 0$ $(1-\zeta_{p})^{2}$. Then $\zeta_{p}^{k}\alpha\equiv
a_{0}$ $(1-\zeta_{p})^{2}$, and therefore $\zeta_{p}^{k}\alpha$ is primary.
###### Lemma 1.
Let $u$ be a unit in $\mathcal{O}_{K}$. Then
$\frac{u}{\overline{u}}=\zeta^{t}$ for some $t\in\mathbb{Z}$
###### Proof.
Write $\upsilon=\frac{u}{\overline{u}}$. Conjugation is a Galois automorphism
on $\mathcal{O}_{K}$ since $\overline{\zeta}=\zeta^{-1}=\zeta^{p-1}$. So
$\overline{u}$ is also a unit, and so $\upsilon\in\mathcal{O}_{K}$. Now let
$\sigma_{k}$ be the $(p-1)$ Galois automorphisms on $\mathcal{O}_{K}$ such
that $\sigma_{k}(\zeta)=\zeta^{k}$, $k\in\mathbb{Z}$. Then for all $1\leq
k\leq(p-1)$, $\sigma_{k}\upsilon=\frac{\sigma_{k}u}{\sigma_{k}\overline{u}}=$
$\frac{\sigma_{k}u}{\overline{\sigma_{k}u}}$ by the above remark. So
$\left|\sigma_{k}\upsilon\right|=\sigma_{k}\upsilon\overline{\sigma_{k}\upsilon}=1$.
So $\left|\sigma_{k}\upsilon\right|^{n}=1$ for any $n\in\mathbb{N}$.
Now consider the polynomial
$f(x)=\prod\limits_{k=1}^{p-1}(x-\sigma_{k}\upsilon)$. The coefficients of
this polynomial are elementary symmetric polynomials in
$\\{\sigma_{k}\upsilon:1\leq k\leq p-1\\}$, and so are invariant by action by
$Gal(K:Q)=\\{\sigma_{k}\upsilon:1\leq k\leq p-1\\}$. So
$f(x)\in\mathbb{Z}[x]$. But then the coefficient of $x^{k}$ is $s_{(p-1)-k}$
where $s_{j}$ is the $j^{th}$ elementary symmetric polynomial. But by the
previous paragraph,
$\left|s_{(p-1)-k}\right|\leq\sum\limits_{j=1}^{p-1-k}\left|\sigma_{k}\upsilon\right|^{k}\leq
p-1-k$. So there are finitely many possible such $f(x)\in\mathbb{Z}[x]$ since
the coefficients are bounded. So there are finitely many possible roots since
a polynomial of finite degree has a finite number of roots. But
$\left|\sigma_{k}\upsilon^{n}\right|=1$ for any $n\in\mathbb{N}$, so
$\\{\upsilon^{n}:n\in\mathbb{N}\\}$ satisfy the same argument. So we must have
$\upsilon^{n}=\upsilon^{n^{\prime}}$ for some $n,n^{\prime}\in\mathbb{Z}$. So
$\upsilon^{n-n^{\prime}}=1$, and it follows that $\upsilon$ is a root of unity
in $\mathcal{O}_{K}$.
So by the basic properties of prime cyclotomic fields, we must have
$\upsilon=\pm\zeta^{t}$ for some $t\in\mathbb{Z}$. Now consider congruence
modulo $\lambda=1-\zeta$. Then since
$\dfrac{1-\zeta^{k}}{1-\zeta}=\sum\limits_{i=1}^{k-1}\zeta^{i}\in\left\langle
1,\zeta_{p},...,\zeta_{p}^{p-2}\right\rangle_{\mathbb{Z}}=\mathcal{O}_{K}$,
$\zeta^{k}\equiv 1$ $(\lambda)$ for all $k\in\mathbb{Z}$. So since
$\overline{\zeta^{k}}=\zeta^{-k}\equiv 1\equiv\zeta^{k}$ $(\lambda)$,
$\alpha\equiv\overline{\alpha}$ $(\lambda)$ for all
$\alpha\in\mathcal{O}_{K}$. Namely,
$u\equiv\overline{u}=\pm\zeta^{-t}u\equiv\pm u$ $(\lambda)$. So if
$\upsilon=-\zeta^{t}$, $u\equiv-u$ $(\lambda)\Rightarrow 2u\equiv 0$
$(\lambda)$ which is impossible since $N(\lambda)=p\nmid N(2u)=2^{p-1}$ since
$p$ is odd. So $\upsilon=+\zeta^{t}$.
###### Theorem 1.
Let $u$ be a unit in $\mathcal{O}_{K}$. Then $u$ is real $\Leftrightarrow$ $u$
is primary in $\mathcal{O}_{K}$.
###### Proof.
Since $\mathcal{O}_{K}=\mathbb{Z}[\zeta_{p}]=\left\langle
1,\zeta_{p},...,\zeta_{p}^{p-2}\right\rangle_{\mathbb{Z}}$, we can write $u$
as $\sum\limits_{k=0}^{p-2}a_{k}\zeta^{k}$ for unique
$a_{0},...,a_{p-2}\in\mathbb{Z}$. And so, noting that
$\zeta^{p-1}=-\sum\limits_{i=0}^{p-2}\zeta^{i}$,
$\zeta^{-t}u=\sum\limits_{k=0}^{p-2}a_{k}\zeta^{k-t}=\sum\limits_{k=0}^{p-2}(a_{k+t}-a_{(p-1)+t})\zeta^{k}$
where $a_{k}$ is defined to be $a_{(k\text{ }\mathop{\mathrm{m}od}p)}$ for all
$k\notin\\{0,...,p-1\\}$ ($a_{p-1}=0$, trivially). And so
$\sum\limits_{k=0}^{p-2}(a_{p-k}-a_{1})\zeta^{k}=\overline{u}=$
$\zeta^{-t}u=\sum\limits_{k=0}^{p-2}(a_{k+t}-a_{(p-1)+t})\zeta^{k}$ by 1 and
therefore, since this representation is unique, we get
(1.1) $a_{k+t}-a_{(p-1)+t}=a_{p-k}-a_{1}\text{ for all }0\leq k\leq p-1$
Letting $k_{0}$ be the $\mathop{\mathrm{m}od}p$ solution to $k+t\equiv p-k$
$(p)$, we get $a_{k_{0}+t}=a_{p-k_{0}}$ and so (1.1) yields
$a_{(p-1)+t}=a_{1}$. (1.1) then becomes
(1.2) $a_{k+t}=a_{p-k}=a_{-k}\text{ for all }0\leq k\leq p-1$
Since replacing $k$ by $-(k+t)$ in (1.2) leaves the equation invariant, we get
$\frac{p-1}{2}$ pairs of equal terms with distinct indices amongst
$a_{0},...,a_{p-1}$ (the ’remaining’ term being $a_{k_{0}+t}$). Let
$b_{1},...,b_{\frac{p-1}{2}}$ be representatives of these distinct pairs, and
let $b_{k_{0}+t}=a_{k_{0}+t}$ (we have simply selected and reordered the
$a_{i}$’s).
Now by the proof of 1, there is a unique $c$ modulo $p$ such that $\zeta^{c}u$
is primary, and this $c$ is the solution to $ax\equiv b$ $(p)$ where $u\equiv
a+b\lambda$ $(\lambda^{2})$ where $\lambda=(1-\zeta)$. Now $u=$
$\sum\limits_{k=0}^{p-2}a_{k}\zeta^{k}$. Writing, as a polynomial,
$f(x)=\sum\limits_{k=0}^{p-2}a_{k}x^{k}$, we can find $a$ and $b$ by finding
the coefficients of $1$ and $x$ respectively of $f(1-x)$ since
$\zeta=1-\lambda$. Making elementary use of the Binomial Theorem, we see that
$f(1-x)=\sum\limits_{k=0}^{p-2}a_{k}(1-x)^{k}=\sum\limits_{k=0}^{p-2}a_{k}-\sum\limits_{k=0}^{p-2}ka_{k}x+...$
(we only need the first two terms). So $c$ is the solution to
(1.3)
$\left(\sum\limits_{k=0}^{p-2}a_{k}\right)x\equiv-\sum\limits_{k=0}^{p-2}ka_{k}\text{
}(p)$
Which, since $a_{p-1}=0$, is equivalent to
(1.4)
$\left(\sum\limits_{k=0}^{p-1}a_{k}\right)x\equiv-\sum\limits_{k=0}^{p-1}ka_{k}\text{
}(p)$
Now $k_{0}+t\equiv p-k_{0}$ $(p)\Rightarrow k_{0}+t\equiv-(k_{0}+t)+t$
$(p)\Rightarrow(k_{0}+t)\equiv 2^{-1}t\Rightarrow
b_{k_{0}+t}=a_{k_{0}+t}=a_{2^{-1}t}$. Finally, note that for
$a_{i}=a_{t-i}=b_{l}$ for $1\leq l\leq\frac{p-1}{2}$ by (1.2),
$ia_{i}+(t-i)a_{t-i}=tb_{l}$.
(1.4) then becomes
$\left(b_{k_{0}+t}+2\sum\limits_{k=1}^{\frac{p-1}{2}}b_{k}\right)x\equiv-\left((2^{-1}t\mathop{\mathrm{m}od}p)b_{k_{0}}+\sum\limits_{k=1}^{\frac{p-2}{2}}tb_{k}\right)$
$(p)$. It is clear that $c\equiv-2^{-1}t$ $(p)$ is the solution to this
congruence. By its uniqueness, we see that $u$ is primary $\Leftrightarrow
t\equiv 0$ $(p)\Leftrightarrow u=\zeta^{t}\overline{u}$ is real.
## 2\. Application to a Special Case of Fermat’s Last Theorem
Fermat’s well-known final theorem, proved by Andrew Wiles and Richard Taylor
in 1994, states that
$x^{n}+y^{n}=z^{n}$
where $x,y,z,n\in\mathbb{Z}$ has no non-trivial solutions $(x,y,z)$ for $n\geq
3$.
In fact, to prove this theorem, it suffices to prove that $x^{p}+y^{p}=z^{p}$
has no integral solutions for any positive odd prime $p$, since
$x_{0}^{n}+y_{0}^{n}=z_{0}^{n}\Rightarrow x_{1}^{p}+y_{1}^{p}=z_{1}^{p}$ where
$p$ is an odd prime dividing $n$ (exists since $n\geq 3$) and
$(x_{1},y_{1},z_{1})=(x_{0}^{n/p},y_{0}^{n/p},z_{0}^{n/p})$. In other words,
we can restrict our study to the case where $n$ is an odd prime.
There is a very elegant proof of a special case of this theorem using
cyclotomy. The main use of the concept here is that it allows us to transform
a ”sum of $n$-th powers” problem into a ”divisibility” problem since we can
now factor $x^{p}+y^{p}$ as $\prod\limits_{i=0}^{p-1}(x+\zeta_{p}^{i}y)$.
In this section, we shall lay out said proof. Let $K=\mathbb{Q}(\zeta)$ where
$\zeta=\zeta_{p}$. We will suppose that for some $(x_{0,}y_{0},z_{0})$ is a
solution to $x^{p}+y^{p}=z^{p}$ for some odd prime $p$. Then
(2.1) $x_{0}^{p}+y_{0}^{p}=z_{0}^{p}$
WLOG, we can take $x_{0}$, $y_{0}$ and $z_{0}$ to be pairwise relatively
prime, for if some $d\in\mathbb{Z}$ divides two of them, it must divide the
3${}^{\text{rd}}$, and then $x_{0}^{p}+y_{0}^{p}=z_{0}^{p}\Leftrightarrow
x_{1}^{p}+y_{1}^{p}=z_{1}^{p}$ where $x_{0},y_{0},z_{0}=dx_{1},dy_{1}$
$,dz_{1}$ respectively, with $x_{1},y_{1},z_{1}\in\mathbb{Z}$.
We shall now reduce the problem to a special case and suppose that $p$_does
not divide the class number_ $h$_of_ $O_{K}$, and that $p\nmid
x_{0}y_{0}z_{0}$. From (2.1), we shall reach a contradiction.
This case has been treated in Number Theory textbooks such as [1]. However,
using the equivalence of primary and real units in $\mathcal{O}_{K}$ when $K$
is a prime cyclotomic field, we can prove the result more rapidly.
###### Lemma 2.
Let $i\not\equiv j$ $(p)$. Then the ideals $I=(x_{0}+\zeta^{i}y_{0})$ and
$J=(x_{0}+\zeta^{j}y_{0})$ are relatively prime.
###### Proof.
Consider the ideal $I+J$. $J$ contains the element $-(x_{0}+\zeta^{j}y_{0})$,
so $x_{0}+\zeta^{i}y_{0}-(x_{0}+\zeta^{j}y_{0})=(\zeta^{i}-\zeta^{j})y_{0}\in
I+J$. Likewise, since $\mathcal{O}_{K}=\mathbb{Z}[\zeta]$,
$-\zeta^{j}(x_{0}+\zeta^{i}y_{0})=\zeta^{j}x_{0}+\zeta^{i+j}y_{0}\in I$ and
$\zeta^{i}(x_{0}+\zeta^{j}y_{0})=\zeta^{i}x_{0}+\zeta^{i+j}y_{0}\in J$. So
$\zeta^{i}x_{0}+\zeta^{i+j}y_{0}-\zeta^{j}(x_{0}+\zeta^{i}y_{0})=(\zeta^{i}-\zeta^{j})x_{0}\in
I+J$. Now $(x_{0},y_{0})=1\Rightarrow$ there exist $a,b\in\mathbb{Z}$ such
that $ax_{0}+by_{0}=1$. So
$a(\zeta^{i}-\zeta^{j})x_{0}+b(\zeta^{i}-\zeta^{j})y_{0}=(\zeta^{i}-\zeta^{j})\in
I+J$.
Now $N(\zeta^{i}-\zeta^{j})=p$ since $(N(\zeta^{i}-\zeta^{j}))^{2}=$
$\prod\limits_{k=1}^{p-1}(\zeta^{ik}-\zeta^{jk})^{2}=\prod\limits_{k=1}^{p-1}(-\zeta^{-k(j-i)})(1-\zeta^{k(j-i)})^{2}=\prod\limits_{k=1}^{p-1}(-\zeta^{-k})(1-\zeta^{k})^{2}=+\zeta^{-p\frac{p-1}{2}}\prod\limits_{k=1}^{p-1}(1-\zeta^{k})^{2}=1\cdot\left(\sum\limits_{k=1}^{p-1}1\right)^{2}=p^{2}$.
So $N(I+J)\mid p$. If $N(I+J)=p$, then since $I\subseteq I+J$, $p=N(I+J)\mid
N(I)=\prod\limits_{i=0}^{p-1}(x_{0}+\zeta^{i}y_{0})=x_{0}^{p}+y_{0}^{p}=z_{0}^{p}$.
So since $p$ is prime, $p\mid z_{0}\Rightarrow$ contradiction. So $N(I+J)=1$,
and therefore $I+J=\mathcal{O}_{K}$. So $I$ and $J$ are coprime since $P\mid
I$ and $P\mid J\Rightarrow P\mid I+J\Rightarrow P=\mathcal{O}_{K}$.
Now
$x_{0}^{p}+y_{0}^{p}=z_{0}^{p}\Rightarrow\prod\limits_{i=0}^{p-1}(x_{0}+\zeta^{i}y_{0})=(z_{0})^{p}$
as ideals. But $\\{(x_{0}+\zeta^{i}y_{0}):0\leq i\leq p-1\\}$ are pairwise
coprime. So by unique factorization of ideals, each of these ideals must be a
$p$-th power. So in particular, taking $i=1$, $(x_{0}+\zeta
y_{0})=\mathfrak{I}^{p}$ for some ideal $\mathfrak{I}$. So since $(x_{0}+\zeta
y_{0})$ is principal, $[\mathfrak{I]}$ has order dividing $p$ in the ideal
class group, but since $p\nmid h$, we must have that the order of
$[\mathfrak{I]}$ is $1$. So $\mathfrak{I}$ is principal. Let
$\mathfrak{I}=(\alpha)$. Then $(x_{0}+\zeta y_{0})=(\alpha^{p})$, and so
$x_{0}+\zeta y_{0}$ is associate to $\alpha^{p}$. We write $x_{0}+\zeta
y_{0}=u\alpha^{p}$ where $u$ is a unit in $\mathcal{O}_{K}$.
Then by 1 there exists a unique $c$ modulo $p$ such that $\zeta^{-c}u$ is
primary. Let $\zeta^{-c}u=u_{0}$ so that $u=\zeta^{c}u_{0}$ where $u_{0}$ is
primary. But $u_{0}$ is trivially a unit, and is therefore real by 1.
So $x_{0}+\zeta y_{0}=\zeta^{c}u_{0}\alpha^{p}$ where $u_{0}$ is real. Note
that modulo $p$,
$\alpha^{p}\equiv\left(\sum\limits_{i=0}^{p-2}a_{i}\zeta^{i}\right)^{p}\equiv\sum\limits_{i=0}^{p-2}a_{i}^{p}\zeta^{ip}\equiv\sum\limits_{i=0}^{p-2}a_{i}^{p}\in\mathbb{Z}$
$(p)$. So $\alpha^{p}\equiv\overline{\alpha^{p}}$ $(p)$. It follows that
$x_{0}+\zeta y_{0}=\zeta^{c}u_{0}\alpha^{p}\Rightarrow x_{0}+\zeta
y_{0}\equiv\zeta^{c}u_{0}\alpha^{p}$ $(p)\Rightarrow\overline{x_{0}+\zeta
y_{0}}\equiv\overline{\zeta^{c}u_{0}\alpha^{p}}$ $(p)\Rightarrow
x_{0}+\zeta^{-1}y_{0}\equiv\zeta^{-c}u_{0}\alpha^{p}$ $(p)$. So we now have
$x_{0}+\zeta y_{0}\equiv\zeta^{c}u_{0}\alpha^{p}$
$(p)\Rightarrow\zeta^{-c}x_{0}+\zeta^{1-c}y_{0}\equiv u_{0}\alpha^{p}$ $(p)$
and $x_{0}+\zeta^{-1}y_{0}\equiv\zeta^{-c}u_{0}\alpha^{p}$
$(p)\Rightarrow\zeta^{c}x_{0}+\zeta^{c-1}y_{0}\equiv u_{0}\alpha^{p}$ $(p)$.
Subtracting the latter congruence from the former yields
(2.2) $\zeta^{-c}x_{0}+\zeta^{1-c}y_{0}-\zeta^{c}x_{0}-\zeta^{c-1}y_{0}\equiv
0\text{ }(p)$
Now an element of $\mathcal{O}_{K}=\mathbb{Z}[\zeta]$ is divisible by $p$ if
and only if all of the coefficients as a polynomial in $\zeta$ are divisible
by $p$. $p\nmid x_{0},y_{0}$ since $p\nmid x_{0}y_{0}z_{0}$, so we must check
the cases where one of $\\{c,-c,1-c,c-1\\}$ is congruent to $-1$ modulo $p$ or
where two of $\\{c,-c,1-c,c-1\\}$ are equal modulo $p$. These cases can be
split as follows:
* •
$c\equiv 0$ $(p)$ (so that $c\equiv-c$ $(p)$). Then $p\mid
y_{0}(\zeta-\zeta^{-1})=y_{0}(\sum\limits_{i=2}^{p-2}\zeta^{i}+1)\Rightarrow
p\mid y_{0}$ (even if $p=3$) $\Rightarrow$ contradiction.
* •
$c\equiv 1$ $(p)$ (so that $1-c\equiv c-1$ $(p)$). Then $p\mid
x_{0}(\zeta^{-1}-\zeta)\Rightarrow p\mid x_{0}$ as in the previous case
$\Rightarrow$ contradiction.
* •
$c\equiv 2^{-1}$ $(p)$ (so that $c\equiv 1-c$ $(p)$). Then
$p\mid(y_{0}-x_{0})\zeta^{c}+\zeta^{-c}(x_{0}-y_{0})$. So
$p\mid(x_{0}-y_{0})$. We then rewrite 2.1 as
$x_{0}^{p}+(-z_{0})^{p}=(-y_{0})^{p}$ (since $p$ is odd). Then with the same
argument we will get $p\mid(x_{0}+z_{0})$. But 2.1 yields
$x_{0}^{p}+y_{0}^{p}-z_{0}^{p}\equiv 0$ $(p)$ and so $x_{0}+y_{0}-z_{0}\equiv
0$ $(p)$. This yields $3x_{0}\equiv 0$ $(p)$. We suppose for now that $p>3$.
Then this yields $p\mid x_{0}\Rightarrow$ contradiction.
* •
Letting one of $\\{c,-c,1-c,c-1\\}$ be congruent to $-1$ modulo $p$ will yield
one of the coefficients of the terms of (2.2) as $\pm(x_{0}-y_{0})$, giving
the same contradiction as in the previous case.
We therefore obtain a contradiction in all cases. We have, however, supposed
that $p>3\,$. A general study of the case where $p=3$ is done elegantly in
[4].
## 3\. An Approach to Pell’s Equation using cyclotomy
Pell’s Equation is
$x^{2}-dy^{2}=1\text{, \ \ }x,y\in\mathbb{Z}$
in $x$ and $y$, where $d\in\mathbb{Z}^{+}$. $d\leq 0$ trivially yields the
single solution $(1,0)$, and we can consider $d$ to be square-free, since any
square factor of $d$ can be incorporated into $y$.
The equation can be solved using cyclotomy and quadratic residues. A partial
solution was found by Dirichlet using this method, building upon the work of
Gauss [3]. In this section, we build upon Dirichlet’s work, explicitly writing
the solution and using the modern machinery of Galois Theory to streamline the
approach. Again, we let $p$ be an odd prime, and define
$p^{\ast}=(-1)^{\frac{p-1}{2}}p$, $i=\sqrt{-1}$, and start by introducing an
important lemma.
###### Lemma 3.
$\left\\{\begin{array}[]{l}q_{1}(x)=2\prod\limits_{\begin{subarray}{c}1\leq
k<p\\\
\QOVERD(){k}{p}=1\end{subarray}}(x-\zeta^{k})=f(x)+\sqrt{p^{\ast}}g(x)\\\
q_{-1}(x)=2\prod\limits_{\begin{subarray}{c}1\leq k<p\\\
\QOVERD(){k}{p}=-1\end{subarray}}(x-\zeta^{k})=f(x)-\sqrt{p^{\ast}}g(x)\end{array}\right.$
where $f(x),g(x)$ are polynomials in $\mathbb{Z}[x]$.
###### Proof.
Note that the product of the 2 above polynomials (on the left-hand side) is
$4\prod\limits_{1\leq k<p}(x-\zeta^{k})=4m_{p}(x)\in\mathbb{Z}[x]$. It is
therefore fixed by any Galois automorphism in $Gal(K:\mathbb{Q})$. Now taking
$\theta=\zeta^{\frac{p^{2}-1}{8}}\prod\limits_{k=1}^{\frac{p-1}{2}}(1-\zeta^{k})^{2}$,
we see that $\theta^{2}=p^{\ast}$ since
$(-1)^{\frac{p^{2}-1}{8}}\equiv\left(\frac{2}{p}\right)$ $($mod $2)$, and
trivially $\theta\in\mathcal{O}_{K}$. So $\sqrt{p^{\ast}}\in\mathcal{O}_{K}$,
Now an automorphism $\sigma$ in the Galois group fixes $p^{\ast}$ if and only
if $\sigma$ is a square. But this is if and only if $\sigma$ fixes all (and
only) the $\zeta^{k}$ such that $k$ is a quadratic residue modulo $p$. So
$\prod\limits_{\begin{subarray}{c}1\leq k<p\\\
\QOVERD(){k}{p}=1\end{subarray}}(x-\zeta^{k})\in L[x]$ where
$L=\mathbb{Q}(\sqrt{p^{\ast}})$. All the coefficients in $L[x]$ are of the
form $a+b\sqrt{p^{\ast}}$ where $a$ and $b$ are both rational, and
$\frac{1}{2}\cdot$ an algebraic integer (allowing for the fact that
$p^{\ast}\equiv 1$ $(4)$). The coefficients of
$2\prod\limits_{\begin{subarray}{c}1\leq k<p\\\
\QOVERD(){k}{p}=1\end{subarray}}(x-\zeta^{k})$ are therefore rational
algebraic integers and thus in $\mathbb{Z}$. We can now expand $q_{1}(x)$ and
rewrite it as $q_{1}(x)=f(x)+\sqrt{p^{\ast}}g(x)$ where $f(x),g(x)$ are
polynomials in $\mathbb{Z}[x]$.
A similar argument shows that $q_{-1}(x)\in L[x]$. Now let $\tau$ be the
Galois automorphism in $Gal(K:Q)$ defined by
$\tau(\sqrt{p^{\ast}})=-\sqrt{p^{\ast}}$ (noting that $K:L:\mathbb{Q}$ is a
tower of fields). Then by the above, and since $\tau^{2}$ must fix $q_{1}(x)$,
we must have that $\tau(\zeta^{k})=\zeta^{l}$ where
$\QOVERD(){k}{p}\QOVERD(){l}{p}=-1$. So since $\tau$ is a Galois automorphism
over $K$, we must have $\tau(q_{1}(x))=q_{-1}(x)$. This yields that
$q_{-1}(x)=f(x)-\sqrt{p^{\ast}}g(x)$.
We will primarily consider the case where $d$ is an odd prime. Pell’s Equation
then becomes
(3.1) $x^{2}-py^{2}=1$
By Lemma 3,
$4m_{p}(x)=q_{1}(x)q_{-1}(x)=f(x)^{2}-(p^{\ast})g(x)^{2}$
And so, replacing $x$ by $1$, we get
(3.2) $4p=x_{1}^{2}-p^{\ast}y_{1}^{2}\text{ where }x_{1}=f(1)\text{,
}y_{1}=g(1)$
Since $f(x),g(x)\in\mathbb{Z}[x]$, $x_{1},y_{1}\in\mathbb{Z}$, and we can see
that Lemma 3 relates to Pell’s Equation insofar as it gives us a pair
$(x_{1},y_{1})$ that verifies an equation very similar to (3.1).
$4p=x_{1}^{2}-p^{\ast}y_{1}^{2}\Rightarrow
x_{1}^{2}=4p+p^{\ast}y_{1}^{2}\Rightarrow p\mid x_{1}^{2}\Rightarrow p\mid
x_{1}$ since $p$ is prime. So letting $p\xi_{1}=x_{1}$, we can rewrite
equation (3.2) as $4p=p^{2}\xi_{1}^{2}-p^{\ast}y_{1}^{2}$, and so, dividing by
$p$,
(3.3) $p\xi_{1}^{2}-(-1)^{\frac{p-1}{2}}y_{1}^{2}=4$
We now analyze $q_{1}(x)$ and $q_{-1}(x)$ to obtain some insight as to the
values $x_{1}$ and $y_{1}$. $x^{2}\equiv(p-x)^{2}$ $(p)$, so all quadratic
residues are in $\\{x^{2}$ $(p):1\leq x\leq\frac{p-1}{2}\\}$. We can therefore
reorder the terms in $q_{1}(x)$ and write it as
$2\prod\limits_{k=1}^{\frac{p-1}{2}}(x-\zeta^{k^{2}})$, and so
$q_{1}(1)=2\prod\limits_{k=1}^{\frac{p-1}{2}}(1-\zeta^{k^{2}})$.
The value of $p^{\ast}$ depends on the value of $p$ modulo $4$ so we will
consider the two cases separately for simplicity.
Case 1:__ $p\equiv 1$ $(4)$.
Then (3.3) becomes $p\xi_{1}^{2}-y_{1}^{2}=4$ (or, to emphasize the similarity
to Pell’s Equation, $y_{1}^{2}-p\xi_{1}^{2}=-4$).
We then have two subcases.
If $p\equiv 1$ $(8)$, then $y_{1}^{2}-\xi_{1}^{2}\equiv 4$ $(8)$. Trivially
$y_{1}$ and $\xi_{1}$ must either be both odd or both even. But $1^{2}\equiv
3^{2}\equiv 5^{2}\equiv 7^{2}\equiv 1$ $(8)$, so if $y_{1}$ and $\xi_{1}$ were
both odd we would have $y_{1}^{2}-\xi_{1}^{2}\equiv 0$ $(8)\Rightarrow$
contradiction. It follows that $y_{1}$ and $\xi_{1}$ are both even, and we can
thus write $y_{2}=\frac{y_{1}}{2},\xi_{2}=\frac{\xi_{1}}{2}\in\mathbb{Z}$.
Then $y_{2}-p\xi_{2}^{2}=-1$. We can use the fact that
$(\sqrt{p})^{2}\in\mathbb{Z}$ to get rid of the minus sign in front of $1$.
$y_{2}^{2}-p\xi_{2}^{2}=-1$ yields
$(y_{2}-\sqrt{p}\xi_{2})(y_{2}+\sqrt{p}\xi_{2})=-1$, and so
$(y_{2}-\sqrt{p}\xi_{2})^{2}(y_{2}+\sqrt{p}\xi_{2})^{2}=1$. But
$(y_{2}\pm\sqrt{p}\xi_{2})^{2}=a\pm b\sqrt{p}$, $a,b\in\mathbb{Z}$. Taking
$(x,y)=(a,b)$, we have solved (3.1). Summarizing, we get a solution from
$\displaystyle(a,b)$ $\displaystyle=$
$\displaystyle\left(\frac{1}{4}(g(1)^{2}+\frac{f(1)^{2}}{p})\text{ },\text{
}\frac{f(1)g(1)}{2p}\right)\text{ }$ $\displaystyle\text{where we can directly
compute }f(1)\text{ and }g(1)$
If $p\equiv 5$ $(8)$, $y_{1}^{2}+3\xi_{1}^{2}\equiv 4$ $(8)$. Given that the
only quadratic residues modulo $8$ are $0,1,4$, we must have
$(y_{1}^{2},\xi_{1}^{2})\equiv(1,1),(0,4)$ or $(4,0)$ $\ (8)$.
We now use the fact that $8^{2}=2^{2\cdot 3}=4^{3}$ and consider
$(y_{1}+\sqrt{p}\xi_{1})^{3}=(y_{1}^{3}+3p\xi_{1}^{2}y_{1})+\sqrt{p}(p\xi_{1}^{3}+3y_{1}^{2}\xi_{1})=y_{2}+\sqrt{p}\xi_{2}$
and see that $y_{2}^{2}-p\xi_{2}^{2}=(y_{1}^{2}-p\xi_{1}^{2})^{3}=-4^{3}$.
But $y_{2}=y_{1}(y_{1}^{2}+3p\xi_{1}^{2})\equiv y_{1}(y_{1}^{2}-\xi_{1}^{2})$
$(8)$. $(y_{1}^{2},\xi_{1}^{2})\equiv(1,1)$ $(8)\Rightarrow$ $y_{2}\equiv 0$
$(8)$. $(y_{1}^{2},\xi_{1}^{2})\equiv(0,4)$ or $(4,0)$ $(8)\Rightarrow
y_{2}\equiv 4\cdot 4,$ $0\cdot 4$ or $\pm 2\cdot 4\equiv 0$ $(8)$. So in any
case $y_{2}\equiv 0$ $(8)$.
Similarly
$\xi_{2}=\xi_{1}(p\xi_{1}^{2}+3y_{1}^{2})\equiv\xi_{1}(5\xi_{1}^{2}+3y_{1}^{2})$
$(8)$. $(y_{1}^{2},\xi_{1}^{2})\equiv(1,1)$ $(8)\Rightarrow$
$\xi_{2}\equiv\xi_{2}(5+3)\equiv 0$ $(8)$.
$(y_{1}^{2},\xi_{1}^{2})\equiv(0,4)$ or $(4,0)$
$(8)\Rightarrow\xi_{2}\equiv\pm 2\cdot 4,$ $0\cdot 4$ or $4\cdot 0\equiv 0$
$(8)$. So in any case $\xi_{2}\equiv 0$ $(8)$.
So $8\mid y_{2},\xi_{2}$ and thus, writing
$y_{3}=\frac{y_{2}}{8},\xi_{3}=\frac{\xi_{2}}{8}\in\mathbb{Z}$, we get
$(y_{3}^{2}-p\xi_{3}^{2})=\frac{-4^{3}}{8^{2}}=-1$. As in the case where
$p\equiv 1$ $(8)$, writing $(y_{3}\pm\sqrt{p}\xi_{3})^{2}=a\pm b\sqrt{p}$,
$a,b\in\mathbb{Z}$, $(x,y)=(a,b)$ is a solution of (3.1). Summarizing, we get
a solution from
$(a,b)=\left(\begin{array}[]{c}\frac{1}{64}((g(1)^{3}+\frac{3f(1)^{2}g(1)}{p})^{2}+p(\frac{f(1)^{3}}{p^{2}}+3\frac{g(1)^{2}f(1)}{p})^{2})\text{
},\\\ \text{
}\frac{1}{32}(g(1)^{3}+3\frac{f(1)^{2}g(1)}{p})(\frac{f(1)^{3}}{p^{2}}+3\frac{g(1)^{2}f(1)}{p})\end{array}\right)$
Case 2: $p\equiv 3$ $(4)$.
Let $l=\frac{p-1}{2}$. $p\equiv 3$ $(4)\Rightarrow l$ is odd. We see that
$f(x)=\frac{1}{2}(q_{1}(x)+q_{-1}(x))=\prod\limits_{\begin{subarray}{c}1\leq
k<p\\\
\QOVERD(){k}{p}=1\end{subarray}}(x-\zeta^{k})+\prod\limits_{\begin{subarray}{c}1\leq
k<p\\\ \QOVERD(){k}{p}=-1\end{subarray}}(x-\zeta^{k})$. $f$ is of degree $l$.
We shall find a relation amongst the coefficients of $f$ by comparing
$f(\zeta)$ and $f(\overline{\zeta})=\overline{f(\zeta)}$ (since
$f(x)\in\mathbb{Z}[x]$). Trivially $\QOVERD(){1}{p}=1$, so
$\prod\limits_{\begin{subarray}{c}1\leq k<p\\\
\QOVERD(){k}{p}=1\end{subarray}}(\zeta-\zeta^{k})=0$ and so
$f(\zeta)=\prod\limits_{\begin{subarray}{c}1\leq k<p\\\
\QOVERD(){k}{p}=-1\end{subarray}}(\zeta-\zeta^{k})$. Also note that
$\QOVERD(){-1}{p}=(-1)^{\frac{p-1}{2}}=-1$, and so
$\QOVERD(){k}{p}=-\QOVERD(){-k}{p}$ for all $1\leq k\leq p-1$. So
$f(\zeta)=\prod\limits_{\begin{subarray}{c}1\leq k<p\\\
\QOVERD(){k}{p}=1\end{subarray}}(\zeta-\zeta^{-k})$. By the same line of
reasoning, $f(\overline{\zeta})=f(\zeta^{-1})=$
$\prod\limits_{\begin{subarray}{c}1\leq k<p\\\
\QOVERD(){k}{p}=1\end{subarray}}(\zeta^{-1}-\zeta^{k})$. So
$\displaystyle\frac{f(\zeta)}{f(\zeta^{-1})}$ $\displaystyle=$
$\displaystyle\prod\limits_{\begin{subarray}{c}1\leq k<p\\\
\QOVERD(){k}{p}=1\end{subarray}}\frac{(\zeta-\zeta^{-k})}{(\zeta^{-1}-\zeta^{k})}=(-1)^{l}\prod\limits_{\begin{subarray}{c}1\leq
k<p\\\ \QOVERD(){k}{p}=1\end{subarray}}\zeta^{1-k}$ $\displaystyle\text{since
there are precisely }l\text{ quadratic residues modulo }p$ $\displaystyle=$
$\displaystyle-\zeta^{l}\prod\limits_{\begin{subarray}{c}1\leq k<p\\\
\QOVERD(){k}{p}=1\end{subarray}}\zeta^{-k}$ $\displaystyle=$
$\displaystyle-\zeta^{l}$ $\displaystyle\text{since
}\sum\limits_{\begin{subarray}{c}1\leq k<p\\\
\QOVERD(){k}{p}=1\end{subarray}}k=p\frac{p-1}{2}+0\text{ since the Legendre
symbol is a }$ $\displaystyle\text{quadratic character modulo }p\text{ and
since }\left(\frac{0}{p}\right)=0\text{.}$
So $f(\zeta)=-\zeta^{l}f(\zeta^{-1})$. So writing
$f(x)=a_{l}x^{l}+a_{l-1}x^{l-1}+...+a_{1}x+a_{0}$, this yields
$a_{l}\zeta^{l}+a_{l-1}\zeta^{l-1}+...+a_{1}\zeta+a_{0}=-a_{0}\zeta^{l}-a_{1}\zeta^{l-1}-...-a_{l-1}\zeta-
a_{l}$, i.e.
(3.4)
$\sum\limits_{k=0}^{l}a_{k}\zeta^{k}=\sum\limits_{k=0}^{l}(-a_{k})\zeta^{l-k}$
Now it is trivial to see that $a_{l}=2$ by the above formula for $f(x)$. Also,
$q_{1}(x)=2\prod\limits_{\begin{subarray}{c}1\leq k<p\\\
\QOVERD(){k}{p}=1\end{subarray}}(x-\zeta^{k})$. The constant term of $q_{1}$
is $2(-1)^{l}\prod\limits_{\begin{subarray}{c}1\leq k<p\\\
\QOVERD(){k}{p}=1\end{subarray}}\zeta^{k}=-2\prod\limits_{1\leq k\leq
l}\zeta^{k^{2}}=-2\zeta^{\frac{l(l+1)(2l+1)}{6}}=-2\zeta^{p\frac{p^{2}-1}{24}}$.
Now $3\mid p^{2}-1$ since $p\neq 3$ ($p\equiv 3$ $(4)$), and $p^{2}\equiv 1$
$(8)$ since $p$ is odd. So $3\cdot 8=24\mid p^{2}-1$. So The constant term of
$q_{1}$ is $-2\cdot 1=-2$. But $q_{1}(x)=f(x)+\sqrt{p^{\ast}}g(x)$ where
$f(x),g(x)\in\mathbb{Z}[x]$. So we must have $a_{0}=-2$. Therefore
$a_{l}=-a_{0}$. So (3.4) now yields
$\sum\limits_{k=1}^{l-1}a_{k}\zeta^{k}=\sum\limits_{k=1}^{l-1}(-a_{k})\zeta^{l-k}=\sum\limits_{k=1}^{l-1}(-a_{l-k})\zeta^{k}$
(after replacing $k$ by $l-k$), and $\\{\zeta,...,\zeta^{l-1}\\}$ is a
$\mathbb{Z}$-linearly independent subset. So $a_{l-k}=-a_{l}$ for $1\leq k\leq
l-1$, and so by the above, $a_{l-k}=-a_{l}$ for all $0\leq k\leq l$. We can
therefore rewrite $f(x)$ as
$2(x^{l}-1)+b_{1}x(x^{l-2}-1)+b_{2}x^{2}(x^{l-4}-1)+...+b_{\frac{l-1}{2}}x^{\frac{l-1}{2}}(x-1)=\sum\limits_{k=0}^{\frac{l-1}{2}}b_{k}x^{k}(x^{l-2k}-1)$,
$b_{k}\in\mathbb{Z}$ for all $0\leq k\leq\frac{l-1}{2}$ (with $b_{0}=2$).
Replacing $x$ by $i=\sqrt{-1}$, we see that $x^{k}(x^{l-2k}-1)$ depends on
whether $p\equiv 3$ or $7$ $(8)$.
Let $p\equiv 3$ $(8)$. Then $l\equiv 1$ $(4)$ and simple calculation yields
$i^{k}(i^{l-2k}-1)=\left\\{\begin{array}[]{ll}1-i&\text{if }k\equiv 1,2\text{
}(4)\\\ -(1-i)&\text{if }k\equiv 0,3\text{ }(4)\end{array}\right.$
$p\equiv 7$ $(8)\Rightarrow l\equiv 3$ $(4)$, and the same type of calculation
yields
$i^{k}(i^{l-2k}-1)=\left\\{\begin{array}[]{ll}1+i&\text{if }k\equiv 3,2\text{
}(4)\\\ -(1+i)&\text{if }k\equiv 0,1\text{ }(4)\end{array}\right.$
Writing $i^{\ast}=\left\\{\begin{array}[]{ll}-i&\text{if }p\equiv 3\text{
}(8)\\\ +i&\text{if }p\equiv 7\text{ }(8)\end{array}\right.$, we see that
$f(i)=\sum\limits_{k=0}^{\frac{l-1}{2}}\pm
b_{k}(1+i^{\ast})=y_{2}(1+i^{\ast})$ where $y_{2}\in\mathbb{Z}$.
Now,
$\displaystyle g(x)$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{p^{\ast}}}(q_{1}(x)-q_{-1}(x))$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{p^{\ast}}}\left(\prod\limits_{\begin{subarray}{c}1\leq
k<p\\\
\QOVERD(){k}{p}=1\end{subarray}}(x-\zeta^{k})-\prod\limits_{\begin{subarray}{c}1\leq
k<p\\\ \QOVERD(){k}{p}=-1\end{subarray}}(x-\zeta^{k})\right)$
And so
$\displaystyle g(\zeta)$ $\displaystyle=$
$\displaystyle\boldsymbol{-}\frac{1}{\sqrt{p^{\ast}}}\left(\prod\limits_{\begin{subarray}{c}1\leq
k<p\\\ \QOVERD(){k}{p}=1\end{subarray}}(\zeta-\zeta^{-k})\right)$
$\displaystyle\text{and }g(\zeta^{-1})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{p^{\ast}}}\left(\prod\limits_{\begin{subarray}{c}1\leq
k<p\\\ \QOVERD(){k}{p}=1\end{subarray}}(\zeta^{-1}-\zeta^{k})\right)$
A similar line of reasoning as for $f(x)$ gives us that
$g(\zeta)=+\zeta^{l}g(\zeta^{-1})$. Following the same steps as for $f(x)$, we
find that, writing $g(x)$ as
$\frac{1}{\sqrt{p^{\ast}}}\sum\limits_{k=0}^{l}a_{k}x^{k}$, we get
$a_{l-k}=+a_{l}$ for all $0\leq k\leq l$ (with $a_{l}=a_{0}=0$ this time). We
can therefore similarly rewrite $g(x)$ as
$\sum\limits_{k=0}^{\frac{l-1}{2}}b_{k}x^{k}(x^{l-2k}+1)$,
$b_{k}\in\mathbb{Z}$ (remembering that $g(x)\in\mathbb{Z}[x]$ by 3). A similar
argument shows that $g(i)=\sum\limits_{k=0}^{\frac{l-1}{2}}\pm
b_{k}(1-i^{\ast})=\xi_{2}(1-i^{\ast})$ where $\xi_{2}\in\mathbb{Z}$.
Now $l\equiv 3$ $(4)$, so
$q_{1}(i)q_{-1}(i)=4m_{p}(i)=4(1+i+...+i^{l})=4\cdot((1+i-1-i)+(1+i-1-i)+...+(1+i-1))=4i$
So $f(i)^{2}-p^{\ast}g(i)^{2}=f(i)^{2}+pg(i)^{2}=4i$, and so
$y_{2}^{2}(1+i^{\ast})^{2}+p\xi_{2}(1-i^{\ast})^{2}=2y_{2}^{2}i^{\ast}-2p\xi_{2}^{2}i^{\ast}=4i$
or, dividing by $2i^{\ast}=\pm 2i$,
$\displaystyle y_{2}^{2}-p\xi_{2}^{2}=\pm 2$ $\displaystyle\Rightarrow$
$\displaystyle(y_{2}+\sqrt{p}\xi_{2})^{2}(y_{2}-\sqrt{p}\xi_{2})^{2}=4$
Now $y_{2},\xi_{2}$ are odd, else $y_{2}^{2}-p\xi_{2}^{2}\equiv
y_{2}^{2}+\xi_{2}^{2}\equiv 0\not\equiv\pm 2$ $(4)$. So the coefficients of
$(y_{2}+\sqrt{p}\xi_{2})^{2}=(y_{2}^{2}+p\xi_{2}^{2})+2y_{2}\xi_{2}\sqrt{p}$
are even. We can thus write
$a=\frac{(y_{2}^{2}+p\xi_{2}^{2})}{2},b=y_{2}\xi_{2}\in\mathbb{Z}$ and get
$a^{2}-pb^{2}=\frac{(y_{2}+\sqrt{p}\xi_{2})^{2}(y_{2}-\sqrt{p}\xi_{2})^{2}}{2\cdot
2}=\frac{4}{4}=1$
This solves the equation, where
$\displaystyle(a,b)$ $\displaystyle=$
$\displaystyle\left(\frac{i^{\ast}}{4}(pg(i)^{2}-f(i)^{2})\text{ },\text{
}\frac{1}{2}g(i)f(i)\right)\text{ }$ $\displaystyle\text{where we can directly
compute }f(i)\text{ and }g(i)$
To apply this method to the general case of Pell’s Equation (where $d$ is
square-free but not necessarily prime), since $d$ is square-free, it can be
written as $d=\prod\limits_{k=1}^{r}p_{k}$ where the $p_{k}$’s are rational
primes. So it suffices to study the case where $d=pq$ for primes $p$ and $q$
and deduce the general case by induction. We will not describe said case in
depth here since this paper mainly focuses on prime cyclotomic fields, but we
remark that taking $\mathbb{Q}(\zeta_{pq})$,
$m_{pq}(x)=m_{p}(x)m_{q}(x)\frac{(x^{pq}-1)/(x-1)}{((x^{p}-1)/(x-1))\cdot((x^{q}-1)/(x-q))}=\frac{(x^{pq}-1)(x-1)}{(x^{p}-1)(x^{q}-1)}$
which can be shown to be irreducible by a similar method as the simple proof
for showing that $\sum\limits_{k=0}^{p-1}x^{k}$ is the minimal polynomial of
$\zeta_{p}$ in $\mathbb{Z}[x]$. Following the same reasoning as in the case
where $d=p$, we can write $4m_{pq}(x)=f(x)^{2}\pm pqg(x)^{2}$ where
$f(x),g(x)\in\mathbb{Z}$. The rest of the problem is solved in a similar
fashion as well.
Using some interesting approximation methods and quadratic number fields,
Ireland & Rosen [5] show that $x^{2}-dy^{2}=1$ has _infinitely many solutions_
for any square-free integer $d$ (including $d=2$), and that every solution has
the form $\pm(x_{n},y_{n})$ where
$x_{n}+\sqrt{d}y_{n}=(x_{1}+\sqrt{d}y_{1})^{n}$ for some solution
$(x_{1},y_{1})$ and $n\in\mathbb{Z}$.
Acknowledgment
_Many thanks to Professor Dan Segal, All-Souls College, Oxford, for his
advice._
## References
* [1] Borevich, Z. I., and Shafarevich I. R., Number Theory, Academic Press, New York, 1973.
* [2] C. S. Dalawat, Primary units in cyclotomic fields, Annales des sciences mathématiques du Québec to appear, 2011.
* [3] G. L. Dirichlet, Sur la manière de résoudre l’équation $t^{2}-pu^{2}=1$ au moyen des fonctions circulaires, Journal für die reine und angewandte Mathematik 17, pp. 286-290, 1837.
* [4] V. Flynn, Algebraic Number Theory Lecture Notes. University of Oxford. Oxford Mathematical Institute, Oxford, UK. 2011. Lecture Notes.
* [5] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, New York, 1982.
* [6] S. Lang, Algebraic Number Theory, Springer-Verlag, New York, 1986.
* [7] L. C. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, New York, 1982.
|
arxiv-papers
| 2011-10-20T05:09:30 |
2024-09-04T02:49:23.410071
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kabalan Gaspard",
"submitter": "Kabalan Gaspard",
"url": "https://arxiv.org/abs/1110.4445"
}
|
1110.4569
|
# Efficiency measurement of b-tagging algorithms developed by the CMS
experiment
Saptaparna Bhattacharya for the CMS collaboration Department of Physics and
Astronomy, Brown University, Providence, RI, USA
###### Abstract
Identification of jets originating from b quarks (b-tagging) is a key element
of many physics analyses at the LHC. Various algorithms for b-tagging have
been developed by the CMS experiment to identify b-tagged jets with a typical
efficiency between 40$\%$ and 70$\%$ while keeping the rate of misidentified
light quark jets between 0.1$\%$ and 10$\%$. An important step, in order to be
able to use these tools in physics analysis, is the determination of the
efficiency for tagging b-jets. Several methods to measure the efficiencies of
the life-time based b-tagging algorithms are presented. Events that have jets
with muons are used to enrich a jet sample in heavy flavor content. The
efficiency measurement relies on the transverse momentum of the muon relative
to the jet axis or on solving a system of equations which incorporate two
uncorrelated taggers. Another approach uses the number of b-tagged jets in top
pair events to estimate the efficiency. The results obtained in 2010 data and
the uncertainties obtained with the different techniques are reported. The
rate of misidentified light quarks have been measured using the “negative”
tagging technique.
## I Introduction
B tagging or the identification of b-jets is of crucial importance in event
topologies involving b-quarks. Many standard model processes entail b-quark
production in the intermediate state, for example, in top physics b-tagging is
imperative to distinguish between signal and background processes. Higgs
physics is heavily b-tagging dependent when the Higgs primarily decays to
$b{\bar{b}}$ pairs at a mass of 120 GeV. Hence, for such processes, the
efficiency of tagging b-jets is an important variable in the analysis. The CMS
detector has performed remarkably well. There is good agreement between data
and simulations. However, b-tagging is a complex tool that relies on many
aspects of detector performance and hence it is essential to measure the
b-tagging efficiency on data and not rely exclusively on input from
simulations.
The algorithms for b-jet identification utilize several salient features of B
hadron decays. B hadrons have a relatively high lifetime of $\sim$1.5 ps
(c$\tau$ = 450 $\mu$m). They have a mass of $\sim$ 5.2 GeV, which is higher
than the mass of the light quarks. They typically tend to decay into a large
number of charged particles, the average decay multiplicity being $\sim 5$.
Due to the high mass, the fragmentation is hard, hence the $p_{T}$ of decay
products is high. The semi-leptonic branching ratio of B hadrons is $\sim$
11$\%$ for each lepton flavor. This branching ratio is as high as $\sim$
20$\%$ when $b\rightarrow c$ cascade decays are taken into account. These
properties allow b-jets to be distinguished from light jets (u, d, s) or gluon
jets and to a lesser extent c-jets.
## II B tagging Algorithms
The inputs to b-tagging are particle flow jets Particle_Flow_paper , charged
particle tracks and vertices, both primary and secondary. The jets are
reconstructed by the anti-$k_{T}$ clustering method, with a cone radius
parameter of $\Delta R$=0.5, where $\Delta R$ is defined in terms of the
azimuthal angle $\phi$ and pseudorapidity $\eta$ as $\Delta
R=\sqrt{{\Delta\eta}^{2}+{\Delta\phi}^{2}}$. The tracks are reconstructed with
a Kalman Filter based method Kalman . The vertices are reconstructed from
tracks compatible with the beam spot using the Adaptive Vertex Fitter
algorithm AVF . The output of the b-tagging algorithms is a discriminator.
This is a variable which is sensitive to the flavor content of the jet and is
computed from tracks associated with the jets. The next step is to choose a
working point. A loose operating point implies a 10$\%$ light quark fraction,
while medium and tight correspond to 1$\%$ and 0.1$\%$ light quark fractions
respectively.
The algorithms for b-jet identification utilize the unique features of B
hadron decays. The impact parameter (IP) is defined as the two dimensional or
three dimensional distance between the track and the vertex at the point of
closest approach as shown in Fig. 1. Since the uncertainty, $\sigma_{IP}$,
varies with the number of tracks, the preferred b-tagging variable is
$IP/\sigma_{IP}$ . The lifetime based taggers rely on tracks with large impact
parameters or on the presence of a reconstructed secondary vertex within a
jet. Track Counting (TC) and Jet Probability (JP) are impact parameter based
taggers. The TC discriminator is based on finding $N$ tracks with
$IP/\sigma_{IP}>S$, where $S$ is a threshold. In the high efficiency (HE)
version of this tagger, the value of $N$ is set at two, while the high purity
(HP) tagger utilizes the first three tracks. The HP version of the tagger,
hence, has a lower b-tagging efficiency due to the application of a stringent
cut. Consequently, the mis-tag rate is also low. The JP tagger combines
information from all tracks and computes the probability of these tracks to
come from the primary vertex. An alternate version of the JP tagger used in
analyses is based on enhancing the b flavor content by associating a higher
weight to the four most displaced tracks. This form of the JP tagger is
analogous to a HP version of the tagger. The next set of b tagging algorithms
involve a secondary vertex in ${\bf B}$ hadron decays. The simple secondary
vertex (SSV) tagger is based on the reconstruction of at least one secondary
vertex. The discriminating variable for this tagger is obtained from the
significance of the 3D flight distance. SSVHE is obtained by associating two
tracks with the vertex, while SSVHP relies on three tracks associated with the
vertex.
These taggers are simple taggers that do not require calibration, therefore,
ideal for early data taking. In addition to these taggers, the complex
secondary vertex tagger (CSV) is used. This tagger uses various track and
vertex information combined through a multi-variate technique.
Figure 1: Definition of positive and negative impact parameters
### II.1 Efficiency measurement from muon-jet events : $p_{Trel}$ method
The $p_{Trel}$ method utilizes semi-leptonic B hadron decays giving rise to
b-jets that contain a muon (“muon jet”). $p_{Trel}$ is defined as the
transverse momentum of the muon with respect to the jet direction as
pictorially described in Fig. 2. Due to the high b quark mass, $p_{Trel}$ is
larger for muons from B hadron decays. A sample, with an enhanced b-jet
purity, is constructed by asking for two reconstructed jets : the muon-jet and
another fulfilling the b-tagging criterion. The $p_{Trel}$ spectra for muon
jets originating from $b$, $c$ and light flavor partons are obtained from
simulations. $f_{b}^{tag}$ ($f_{b}^{untag}$) are defined as fractions of jets
that pass (fail) the b-tagging requirement. From the $p_{Trel}$ spectra of $b$
and non-$b$ ($c$ \+ light flavor jets), these fractions are extracted with a
maximum likelihood fit. The fractions and the total number of tagged and
untagged muon jets ($N_{data}^{tag}$, $N_{data}^{untag}$) are used to
calculate the efficiency:
$\varepsilon_{b}^{tag}=\frac{f_{b}^{tag}.N_{data}^{tag}}{f_{b}^{tag}.N_{data}^{tag}+f_{b}^{untag}.N_{data}^{untag}}$.
The plots of the fits to the $p_{Trel}$ distributions are in Fig. 3.
Figure 2: $p_{Trel}$ is defined as the transverse momentum of the muon with
respect to the jet direction.
Figure 3: Fits of the muon $p_{Trel}$ distributions to b and light flavor
templates for jets containing muons that (left) pass or (right) fail the
b-tagging algorithm: SSVHPT (Simple Secondary Vertex High Purity Tight
Operating Point). The fractions and the total yields ($N_{data}^{tag}$,
$N_{data}^{untag}$) are used to calculate the efficiency.
### II.2 “System 8”
“System 8” is a data driven method with minimal dependence on simulations.
System8, like the $p_{Trel}$ method, takes advantage of semi-leptonic B hadron
decays. It is applied to a sample of muon jet events. A system of 8 non-linear
equations are set up and solved using numerical methods. Two data samples are
used:
* •
The muon jet+ away-jet sample : Contains two reconstructed jets and a muon
within $\Delta R<0.4$ of one of the jets. The highest $p_{T}$ muon is taken
when there exist more muons in the jet. If there exist two jets with muons in
them in an event, both are counted as muon jets.
* •
The muon jet+tagged-away-jet sample : This sample is created by tagging a b
quark in the away jet. Since b quarks are produced in pairs a b quark can be
tagged in the same event in another jet.
The first two equations, hence are:
$\displaystyle n=n_{b}+n_{cl}$ (1) $\displaystyle p=p_{b}+p_{cl}$ (2)
Here, $(n,p)$ are the muon-in-jets in each sample.
Two different taggers are used: A test tagger (“tag”) which in this case is
chosen to be a lifetime based tagger and a cut on $p_{Trel}$. This choice is
dictated by the requirement that these taggers be minimally correlated.
Hence the next set of equations are:
$\displaystyle
n^{tag}=\varepsilon_{b}^{tag}n_{b}+\varepsilon_{cl}^{tag}n_{cl}$ (3)
$\displaystyle~{}~{}~{}p^{tag}=\beta_{12}\varepsilon_{b}^{tag}p_{b}+\alpha_{12}\varepsilon_{cl}^{tag}p_{cl}$
(4)
Here, $(n^{tag},p^{tag})$ are lifetime tagged.
$\displaystyle
n^{p_{Trel}}=\varepsilon_{b}^{p_{Trel}}n_{b}+\varepsilon_{cl}^{p_{Trel}}n_{cl}$
(5)
$\displaystyle~{}~{}~{}p^{p_{Trel}}=\beta_{23}\varepsilon_{b}^{p_{Trel}}p_{b}+\alpha_{23}\varepsilon_{cl}^{p_{Trel}}p_{cl}$
(6)
Here, $(n^{p_{{Trel}}},p^{p_{Trel}})$ are obtained by applying a cut on the
$p_{Trel}$ distribution.
$\displaystyle
n^{tag,p_{Trel}}=\beta_{13}\varepsilon_{b}^{tag}\varepsilon_{b}^{p_{Trel}}n_{b}+\alpha_{13}\varepsilon_{cl}^{tag}\varepsilon_{cl}^{p_{Trel}}n_{cl}$
(7)
$\displaystyle~{}~{}~{}p^{tag,p_{Trel}}=\beta_{123}\varepsilon_{b}^{tag}\varepsilon_{b}^{p_{Trel}}p_{b}+\alpha_{123}\varepsilon_{cl}^{tag}\varepsilon_{cl}^{p_{Trel}}p_{cl}$
(8)
The last set of equations are a result of the application of both tags.
The correlation factors are
$(\alpha_{12},\beta_{12},\alpha_{23},\beta_{23},\alpha_{13},\beta_{13},\alpha_{123},\beta_{123})$
obtained from simulations. They are defined as:
$\beta_{12}=\frac{{\varepsilon}_{b}^{tag}\text{from muon jet+tagged-away-jet
sample}}{{\varepsilon}_{b}^{tag}\text{from muon-jet+away-jet sample}}$ (9)
$\alpha_{12}=\frac{{\varepsilon}_{cl}^{tag}\text{from muon jet+tagged-away-jet
sample}}{{\varepsilon}_{cl}^{tag}\text{from muon-jet+away-jet sample}}$ (10)
$\beta_{23}=\frac{{\varepsilon}_{b}^{p_{Trel}}\text{from muon jet+tagged-away-
jet sample}}{{\varepsilon}_{b}^{p_{Trel}}\text{from muon-jet+away-jet
sample}}$ (11) $\alpha_{23}=\frac{{\varepsilon}_{cl}^{p_{Trel}}\text{from muon
jet+tagged-away-jet sample}}{{\varepsilon}_{cl}^{p_{Trel}}\text{from muon-
jet+away-jet sample}}$ (12)
$\beta_{13}=\frac{{\varepsilon}_{b}^{tag,p_{Trel}}}{{\varepsilon}_{b}^{tag}{\varepsilon}_{b}^{p_{Trel}}}\hskip
7.22743pt\text{and}\hskip
7.22743pt\alpha_{13}=\frac{{\varepsilon}_{cl}^{tag,p_{Trel}}}{{\varepsilon}_{cl}^{tag}{\varepsilon}_{cl}^{p_{Trel}}}$
(13)
for the muon jet and away-jet sample and,
$\beta_{123}=\frac{{\varepsilon}_{b}^{tag,p_{Trel}}}{{\varepsilon}_{b}^{tag}{\varepsilon}_{b}^{p_{Trel}}}\hskip
7.22743pt\text{and}\hskip
7.22743pt\alpha_{123}=\frac{{\varepsilon}_{cl}^{tag,p_{Trel}}}{{\varepsilon}_{cl}^{tag}{\varepsilon}_{cl}^{p_{Trel}}}$
(14)
for the muon jet and tagged-away-jet sample.
These definitions are obtained by writing the left hand side of the equations
in terms of a composite efficiency term (${\varepsilon}_{b}^{tag,p_{Trel}}$)
and equating the $b$ and $c$ and light jet terms on each side of the equation.
These correlation factors are the only variables that are obtained from
simulations, hence, justifying the claim that this method is data-driven.
### II.3 Measured $b$-tagging efficiencies
This section contains the measured b-tagging efficiencies, with the use of the
$p_{Trel}$ and the System8 method, parametrized in jet $p_{T}$. Table 1
contains the efficiency values along with the statistical and systematic
uncertainty. The sources of systematic uncertainties are described in the next
section. The left panel of Fig. 4 shows that there is good agreement between
the two methods and also with Monte Carlo (MC) generator level information.
However, the plot on the right panel shows considerable disagreement in the
high $p_{T}$ region. This can be attributed to low statistics in high $p_{T}$
bins when a high purity tight operating point is used.
In all cases, the ratio of data to MC generator level information (scale
factor, SF) is calculated. The scale factor is a measure of the departure from
ideality, hence they are expected to be close to $\sim$ 1\. The scale factors
along with the efficiencies are used for various physics analysis involving
b-jets. In Table 2 the scale factors are parametrized as a function of the
pseudorapidity, $\eta$. No major variation with respect to $\eta$ is observed.
Figure 4: b-tagging for the TCHEL (left panel) and SSVHPT (right panel) taggers as a function of muon-jet $p_{T}$. Both lower panels show data/MC scale factors. Table 1: Measured b-tagging efficiencies and data/MC scale factors for several b-tagging algorithms. Uncertainties are statistical for $\epsilon_{b}^{tag}$ and statistical+systematic for $SF_{b}$. b-tagger | $\epsilon_{b}^{tag}$ | $SF_{b}^{tag}$ | $\epsilon_{b}^{tag}$ | $SF_{b}^{tag}$
---|---|---|---|---
50-80 GeV | PtRel | Ptrel | System8 | System8
JPL | 0.82 $\pm$ 0.01 | 0.97 $\pm$ 0.01 $\pm$ 0.05 | 0.85 $\pm$ 0.02 | 1.00 $\pm$ 0.02 $\pm$ 0.07
TCHEL | 0.76 $\pm$ 0.01 | 0.95 $\pm$ 0.01 $\pm$ 0.05 | 0.77 $\pm$ 0.01 | 0.96 $\pm$ 0.02 $\pm$ 0.05
TCHEM | 0.63 $\pm$ 0.01 | 0.93 $\pm$ 0.02 $\pm$ 0.06 | 0.63 $\pm$ 0.02 | 0.93 $\pm$ 0.02 $\pm$ 0.07
TCHPM | 0.48 $\pm$ 0.01 | 0.92 $\pm$ 0.02 $\pm$ 0.05 | 0.49 $\pm$ 0.01 | 0.93 $\pm$ 0.03 $\pm$ 0.09
SSVHEM | 0.62 $\pm$ 0.01 | 0.95 $\pm$ 0.02 $\pm$ 0.07 | 0.60 $\pm$ 0.01 | 0.94 $\pm$ 0.02 $\pm$ 0.06
SSVHPT | 0.38 $\pm$ 0.01 | 0.89 $\pm$ 0.02 $\pm$ 0.06 | 0.37 $\pm$ 0.01 | 0.90 $\pm$ 0.03 $\pm$ 0.05
TCHPT | 0.36 $\pm$ 0.01 | 0.88 $\pm$ 0.02 $\pm$ 0.05 | 0.37 $\pm$ 0.01 | 0.88 $\pm$ 0.03 $\pm$ 0.07
Table 2: Measured data/MC scale factors for several b-tagging algorithms in the overall jet $p_{T}$ range from 20 to 240 GeV for pseudorapidity $|\eta|<$ 2.4, $|\eta|<$ 1.2, 1.2 $<|\eta|<$ 2.4. Uncertainties are statistical for $\epsilon_{b}^{tag}$ and statistical+systematic for $SF_{b}$. Both $p_{Tel}$ and System8 provide values compatible with each other. b-tagger | $SF^{tag}_{b}$ | $SF^{tag}_{b}$ | $SF^{tag}_{b}$
---|---|---|---
20-240 GeV | $|\eta|<$ 2.4 | $|\eta|<$ 1.2 | 1.2 $<|\eta|<$ 2.4
JPL | 0.99 $\pm$ 0.01$\pm$ 0.10 | 0.99 $\pm$ 0.01 $\pm$ 0.10 | 0.98 $\pm$ 0.01$\pm$ 0.10
TCHEL | 0.95 $\pm$ 0.01$\pm$ 0.10 | 0.95 $\pm$ 0.01 $\pm$ 0.10 | 0.95 $\pm$ 0.02$\pm$ 0.10
TCHEM | 0.94 $\pm$ 0.01$\pm$ 0.09 | 0.94 $\pm$ 0.01 $\pm$ 0.09 | 0.93 $\pm$ 0.02$\pm$ 0.09
TCHPM | 0.91 $\pm$ 0.01$\pm$ 0.09 | 0.91 $\pm$ 0.02 $\pm$ 0.09 | 0.90 $\pm$ 0.03$\pm$ 0.09
SSVHEM | 0.95 $\pm$ 0.01$\pm$ 0.10 | 0.95 $\pm$ 0.01 $\pm$ 0.10 | 0.93 $\pm$ 0.02$\pm$ 0.09
SSVHPT | 0.90 $\pm$ 0.02$\pm$ 0.09 | 0.89 $\pm$ 0.02 $\pm$ 0.09 | 0.90 $\pm$ 0.03$\pm$ 0.09
TCHPT | 0.88 $\pm$ 0.02$\pm$ 0.09 | 0.88 $\pm$ 0.02 $\pm$ 0.09 | 0.87 $\pm$ 0.03$\pm$ 0.09
### II.4 Systematic Uncertainties
Several sources of systematic uncertainties were identified. Some of these
were method dependent, while most of the systematic uncertainties are common
to both methods. A $p_{Trel}$-method specific systematic uncertainty was from
the mismodeling of the light jet $p_{Trel}$ spectra. This was determined by
constructing a collision data sample with the application of basic kinematic
cuts and quoting the disagreement between data and simulations as the
uncertainty. For the System8 method, the dependence on various event
topologies, was a source of uncertainty. Essentially, this allowed one to vary
the MC parameters in the equations and obtain the uncertainty due to their
variation. Also, the $p_{Trel}$ cut was changed from 0.5 to 1.2 GeV to
estimate the uncertainty due to this requirement. The rest of the sources of
systematic uncertainty discussed below are applicable to both methods. The
average systematic uncertainty varied between 6$\%$-7$\%$. The contributions
from each source of systematic uncertainty is listed in Table 3 for the
$p_{Trel}$ method and in Table 4 for the System8 method.
* •
Pile-up: The distribution of primary vertices from simulations were reweighted
to match data. Systematic uncertainties were estimated by constructing two
samples with high and low pileup regions.
* •
Away jet tagger: Dependency of the away-jet tagger on btagging efficiency was
obtained by changing the taggers and the operating points.
* •
Muon $p_{T}$: Muon $p_{T}$ cut was varied from its central value at 5 GeV to 7
and 10 GeV.
* •
Gluon splitting: To account for the error in mismodeling gluon to $b\bar{b}$
pairs. The number of events with gluon splitting was artificially changed by a
factor of two to calculate this effect.
* •
Closure test: The methods were checked for self-consistency. The difference
between the efficiency measurement from data and simulation was quoted as the
uncertainty.
Table 3: Sources of systematic uncertainties for the Ptrel method. b-tagger | pile-up | away jet | muon pT | light | $g\rightarrow b{\bar{b}}$
---|---|---|---|---|---
JPL | 0.2$\%$ | 3.0$\%$ | 2.3$\%$ | 2.8$\%$ | 0.3$\%$
TCHEM | 2.4$\%$ | 3.6$\%$ | 1.5$\%$ | 3.3$\%$ | 0.2$\%$
TCHEM | 0.9$\%$ | 5.1$\%$ | 1.5$\%$ | 3.7$\%$ | 0.1$\%$
TCHPM | 1.8$\%$ | 3.3$\%$ | 2.6$\%$ | 3.4$\%$ | 0.4$\%$
SSVHEM | 1.4$\%$ | 5.8$\%$ | 1.9$\%$ | 3.4$\%$ | 0.6$\%$
SSVHPT | 1.1$\%$ | 4.8$\%$ | 2.8$\%$ | 3.4$\%$ | 0.6$\%$
TCHPT | 0.6$\%$ | 4.3$\%$ | 2.3$\%$ | 3.7$\%$ | 0.3$\%$
Table 4: Sources of systematic uncertainties for the System8 method b-tagger | pile-up | away jet | muon pT | pTrel | $g\rightarrow b{\bar{b}}$ | sample
---|---|---|---|---|---|---
JPL | 5.1$\%$ | 1.3$\%$ | 0.8$\%$ | 2.2$\%$ | 0.1$\%$ | 3.8$\%$
TCHEM | 3.3$\%$ | 2.4$\%$ | 2.8$\%$ | 0.9$\%$ | 0.6$\%$ | 1.9$\%$
TCHEM | 5.8$\%$ | 2.6$\%$ | 0.9$\%$ | 2.0$\%$ | 0.7$\%$ | 2.4$\%$
TCHPM | 4.8$\%$ | 3.9$\%$ | 4.9$\%$ | 1.7$\%$ | 2.1$\%$ | 4.0$\%$
SSVHEM | 3.5$\%$ | 4.6$\%$ | 0.4$\%$ | 1.8$\%$ | 0.2$\%$ | 3.0$\%$
SSVHPT | 1.2$\%$ | 2.9$\%$ | 2.8$\%$ | 2.4$\%$ | 0.2$\%$ | 3.0$\%$
TCHPT | 3.5$\%$ | 3.1$\%$ | 4.0$\%$ | 2.8$\%$ | 2.5$\%$ | 2.5$\%$
## III Cross-checks with $t{\bar{t}}$ events
In the standard model, $t$ decays to $Wb$ at least 99.8$\%$ of the time. The
measurement of heavy flavor content, can lead to a measurement of
$R_{b}=\big{(}\frac{B(t\rightarrow Wb)}{B(t\rightarrow Wq)}\big{)}$, where $q$
is any down type quark. $R_{b}$, if assumed to be 1, can be used to extract
the b tagging efficiency. Several methods were used for the determination of
b-tagging efficiencies:
* •
The Profile Likelihood Ratio method : This method uses dilepton $t{\bar{t}}$
events. The distribution of jet multiplicity versus b-tagged jet multiplicity
in dilepton $t{\bar{t}}$ events is used to construct a likelihood function.
* •
The $R_{b}$ method : The methods also replies dilepton $t{\bar{t}}$ events.
The observed b-tagged jets is proportional to the fraction of b-jets present,
the proportionality factor being $\epsilon_{b}^{tag}$. The number of b-tagged
jets is modeled probabilistically using $\epsilon_{b}^{tag}$ and
$\epsilon^{mistag}$ for dilepton $t\bar{t}$ events.
* •
The Flavor Tag Consistency Method : lepton+jets $t{\bar{t}}$ events from top
decays are used as input to this method. The procedure requires consistency
between observed and expected number of identified jets in an event in
$t\bar{t}$ lepton+jets decays . A dedicated likelihood function is built based
on $\epsilon_{b}^{tag}$, $\epsilon_{c}^{tag}$ and $\epsilon_{mistag}$,
$t{\bar{t}}$ cross section and acceptance obtained from simulations.
* •
The Simultaneous Heavy Flavor and Top method : This method also uses
lepton+jets $t{\bar{t}}$ events. $\epsilon_{b}^{tag}$ is obtained from two-
dimensional fit with the number of jets and the invariant mass of the tracks
forming the secondary vertex.
All of these methods give efficiency values compatible with Ptrel and System8
methods and are also consistent with each other.
## IV Estimation of mis-tag rate with Negative Taggers
The mis-tag rate is obtained from tracks with negative impact parameters or
secondary vertices with negative decay lengths. The TC discriminators are
plotted in Fig. 5. The negative IPs are ordered from the most negative
upwards. The ordering on the positive side remains unchanged. The negative
taggers are used in the same way as the current b-tagging algorithms. The mis-
tag rate is evaluated as:
${\varepsilon}_{data}^{mistag}={\varepsilon}^{-}_{data}.R^{light}$, where
$\varepsilon^{-}_{data}$ is the negative tag rate in data and
$R_{light}=\varepsilon_{MC}^{mistag}/\varepsilon_{MC}^{-}$ is the ratio
between the light flavor mis-tag rate and negative tag rate of all jets in the
simulation. The measured mis-tag rates are in Table 5. The light jet scale
factors are also included.
Figure 5: Signed $b$-tag discriminators in data (dots) and simulation of light
flavor jets (blue), c-jets(green) and b-jets (red area) with a $p_{T}$
threshold of 30 GeV.
### IV.1 Systematic Uncertainties
The following sources of systematic errors were taken into consideration:
* •
b and c fractions: The b+c flavor fraction is varied in the QCD simulations
and a systematic uncertainty is obtained on $R_{light}$ (1.9$\%$).
* •
Gluon fraction: Uncertainty is extracted from comparison of simulation with
data (0.2$\%$).
* •
Long lived $K_{s}^{0}$ and $\Lambda$ decays (displaced vertices) and photon
conversion and nuclear interactions (2.0$\%$). QCD simulation events are re-
weighted to take into account the observed yields of $K_{s}^{0}$ and $\Lambda$
in data since these processes involve displaced vertices.
* •
Mismeasured tracks: Spurious tracks increase the number of positive over
negative tags (0.3$\%$).
* •
Sign flip: The ratio of the number of negative and positive tagged jets is
computed in a muon-jet sample with a larger than 80$\%$ b purity (4.3$\%$).
* •
Event sample (dominant systematic): Using jets originating from different
event topologies. Dominant systematic (10$\%$).
* •
Pile up: Uncertainty estimated in the same way as described above (0.7$\%$).
Table 5: Mis-tag rate and data/MC scale factor for different b-taggers with $p_{T}$ between 50 and 80 GeV. The statistical+systematic uncertainties are quoted. b-tagger | mis-tag rate (${\varepsilon}_{data}^{mistag}$) | Scale Factor for light jets (${\varepsilon}_{data}^{mistag}/{\varepsilon}_{MC}^{mistag}$)
---|---|---
JPL | 0.077 $\pm$ 0.001$\pm$ 0.016 | 0.98 $\pm$ 0.01 $\pm$ 0.11
TCHEL | 0.128 $\pm$ 0.001$\pm$ 0.026 | 1.11 $\pm$ 0.01 $\pm$ 0.12
TCHEM | 0.0175 $\pm$ 0.0003$\pm$ 0.0038 | 1.21 $\pm$ 0.02 $\pm$ 0.17
SSVHEM | 0.0144 $\pm$ 0.0003$\pm$ 0.0029 | 0.91 $\pm$ 0.02 $\pm$ 0.15
SSVHPT | 0.0012 $\pm$ 0.0001$\pm$ 0.0002 | 0.93 $\pm$ 0.09 $\pm$ 0.12
TCHPT | 0.0017 $\pm$ 0.0001$\pm$ 0.0004 | 1.21 $\pm$ 0.10 $\pm$ 0.18
## V Conclusion
Several methods have been used to obtain the tagging efficiency of b jets
using an integrated luminosity of 0.50 to 0.89 fb-1 collected by the CMS
experiment in 2011. The data/MC scale factor is measured with an uncertainty
of 10$\%$ for b jets with $p_{T}$ up to 240 GeV. For light flavor jets with
$p_{T}$ up to 500 GeV the mis-tag rate is measured with an uncertainty of
10-20$\%$. B-tagging efficiencies are cross checked with independent analyses
using $t{\bar{t}}$ events. B tagging is of crucial importance in events with
topologies involving b quarks single_top .
## References
* (1) CMS Collaboration, “Performance of b-jet identi cation in CMS”, CMS PAS BTV_11_001 (2011).
* (2) CMS “Status of b-tagging tools for 2011 data analysis”, CMS PAS BTV_11_002 (2011).
* (3) CMS Collaboration, Commissioning of the Particle-Flow reconstruction in Minimum-Bias and Jet Events from pp Collisions at 7 TeV , CMS PAS PFT_10_002 (2010).
* (4) R. Fruhwirth et al.: Adaptive vertex fitting, Report CMS-NOTE-2007-008, CERN, Geneva 2007 (submitted to J.Phys.G).
* (5) R. Fruhwirth, Application Of Kalman Filtering To Track And Vertex Fitting, Nucl. Instrum. Meth. A 262 (1987) 444.
* (6) CMS Collaboration, “Algorithms for b jet Identification in CMS”, CMS PAS BTV_09_001 (2009).
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|
arxiv-papers
| 2011-10-20T16:20:53 |
2024-09-04T02:49:23.430099
|
{
"license": "Public Domain",
"authors": "Saptaparna Bhattacharya (for the CMS collaboration)",
"submitter": "Saptaparna Bhattacharya",
"url": "https://arxiv.org/abs/1110.4569"
}
|
1110.4724
|
# Flexible and robust patterning by centralized gene networks.
S. Vakulenko1 and O. Radulescu2
3 Saint Petersburg State University of Technology and Design, St.Petersburg,
Russia,
4 DIMNP UMR CNRS 5235, University of Montpellier 2, Montpellier, France.
Abstract
We investigate the possibility of programming arbitrarily complex space-time
patterns, and transitions between such patterns, by gene networks. We consider
networks with two types of nodes. The $v$-nodes, called centers, are
hyperconnected and interact one to another via $u$-nodes, called satellites.
This centralized architecture realizes a bow-tie scheme and possesses
interesting properties. Namely, this organization creates feedback loops that
are capable to generate any prescribed patterning dynamics, chaotic or
periodic, or stabilize a number of prescribed equilibrium states. We show that
activation or silencing of a node can sharply switch the network dynamics,
even if the activated or silenced node is weakly connected. Centralized
networks can keep their flexibility, and still be protected against
environmental noises. Finding an optimized network that is both robust and
flexible is a computationally hard problem in general, but it becomes feasible
when the number of satellites is large. In theoretical biology, this class of
models can be used to implement the Driesch-Wolpert program, allowing to go
from morphogen gradients to multicellular organisms.
## 1 Introduction
The richness of Alan Turing’s ideas hides somehow their unity. Is there a
relation between the “chemical theory of morphogenesis” (Turing 1952) and the
“universal machine”, or other, less known works, such as “Intelligent
machinery” (Turing 1969, Teuscher and Sanchez 2001) in which he anticipates
random binary networks? As emphasized by M.H.A. Newman (Newman 1955), a common
denominator of Turing’s scientific work is the quest for a mechanical
explanation of nature. However, an even deeper unifying idea concerns the
computability of nature and reciprocally, how nature computes. Turing looked
for a mechanical support for natural pattern computation and found an analog
machine, working by the chemical morphogens. Had he had known about gene
networks, he would have probably analyzed the computational capacity of these
networks to make patterns and multicellular organisms. In this paper we
discuss a particular class of gene networks, the centralized or bow tie
networks, and show that they can “compute” multicellular organisms and comply
with important desiderata of life such as flexibility and robustness.
Flexibility and robustness are important properties of living systems in
general, most particularly observed during development from egg or embryo to a
large, fully organized organism. Flexibility means the capacity to change,
when environmental conditions vary. Opposite to this, robustness is the
capacity to support homeostasis in spite of external changes. Intriguingly,
biological systems are in the same time robust and flexible. Organization of
the body plan in development, should be robust under unavoidable fluctuations
of maternal gradients, embryo size, and environment conditions (for instance,
temperature). This is a viability condition. On the other hand, developmental
systems should be flexible in order to produce a number of different patterns
and complicated dynamics.
Pattern formation processes are important for the body plan establishment via
cell fate decisions in multicellular organisms. Mathematical modeling of these
phenomena (Turing 1952, Meinhardt 1982, Murray 1993, Wolpert et al. 2002,
Mjolness et al. 1991, Reinitz and Sharp 1995, Page et al. 2005) is based on
systems of reaction-diffusion equations, sometimes with spatial inhomogeneous
reaction terms. Two different approaches, both considering that laws of
physics and chemistry are sufficient to account for making of a multicellular
organism, are fundamental in modeling of body plan organization. The first
approach, pioneered by the seminal work of Turing (1952), uses diffusion-
driven instability as a patterning mechanism. For Turing’s mechanism, a
spatial dependence of the reaction term is not necessary, and the translation
symmetry breaking needed for body plan organization results from the Turing
instability. The second approach is based on Driesch-Wolpert positional
information (Wolpert et al. 2002, Wolpert 1970). The corresponding models can
be also based on reaction-diffusion equations, but in this case the models
have space dependent reaction terms and no translation symmetry, therefore
Turing instability is not needed. The main example of this type of models is
the gene circuit model (Mjolness et al 1991, Reinitz and Sharp 1995). In this
case, spatial organization is triggered by pre-patterns of signaling
molecules, generically called morphogen gradients. It is remarkable that germs
of this second approach can be found in the conclusion of Turing’s 1952 paper,
where it is suggested that “most of a organism, most of the time, is
developing from one pattern into another, rather than from homogeneity into a
pattern”.
The body plan, considered as well defined, stable sequence of transitions from
one pattern to another, can be encoded in a system of gene-gene interactions
or gene network. For instance, in the segmentation along the anterior-
posterior axis of Drosophila (fruit-fly) embryos, the chemical support of the
pre-pattern is the maternal bicoid gradient developed in eggs soon after
fecundation. This gradient induces spatially localized expression of
segmentation genes (hunchback, krüppel, giant, knirps, tailless, fushi tarazu,
even skipped, runt, hairy, odd skipped, paired, sloppy paired, etc.) forming a
gene network. This gene network employs several types of interactions to
stabilize the segmentation pattern. Some of these interactions originate in
trans, i.e., far, on the DNA sequence, from the gene, and are due to
transcription factors (TF) and microRNAs (miRNA). Other interactions originate
in cis, i.e., close, on the DNA sequence, to the gene. Indeed, the zygotic
genome contains cis-regulatory elements (CREM) controlling expression of the
segmentation genes. The gene circuit model (Mjolness et al 1991, Reinitz and
Sharp 1995) accounts for part of the trans interactions, considering that
segmentation genes are mutually regulated transcription factors. The miRNA and
the CREMs interactions are not represented in this model. Although the role in
stabilizing the development has been experimentally proven for miRNAs(Li et
al. 2009) and for CREMs (Ludwig et al. 2011), the mechanistic details of these
interactions are still unknown. MiRNAs and CREMs can be abstractly considered
as intermediate nodes in a gene network, mediating interactions between
transcription factors. In such networks, TFs can be target hubs, being
controlled by many CREMs, and also source hubs, because they can bind to many
CREMs. Similarly, a few bioinformatics studies (Shalgi et al. 2007) suggest
the existence of many genes submitted to extensive miRNA regulation with many
TF among these target hubs. Direct testing of these interactions have recently
shown that important transcription factors can be regulated by multiple miRNA
(Tu and Bassler 2006, Martinez et al 2008, Wu et al 2010, Peter 2010). Without
excluding other applications of our mathematical framework, we consider the
TF-miRNAs networks as well as the TF-CREMs networks as possible examples of
centralized networks, or bow-tie networks.
The main goal of this paper is to show, by rigorous mathematical methods, the
following new results:
i centralized networks can create “a multicellular organism” consisting of
many specialized cells where the network dynamics within each cell can have a
different attractor,
ii this pattern is robust under variations of morphogenetic fields; our system
performs trade-offs between flexibility and robustness,
iii bifurcations between attractors can be obtained by gene silencing or
reactivation.
These results, however, would be useless, without algorithms that can resolve,
in polynomial time $Poly(N)$, (where $N$ is the gene number), the problem of a
prescribed complicated and robust pattern construction (“computation of a
robust organism”, CRO problem). It is one of the key questions in development,
why evolution had a sufficient time to construct complicated organs and
organisms (Darwin, Origins of Species, Chapter 6). This problem is, in fact, a
hard combinatorial one. Using new ideas in such problems, we show, under some
assumptions, that
iv for centralized networks with large hub connectivity, the CRO problem is
feasible in polynomial $Poly(N)$ time.
Similar ideas, that bow-tie connectivity can play a role in flexibility and
robustness, have been proposed by (Csete and Doyle 2004, Ma et al. 2007) in
the context of metabolism, but lacked mathematical proofs. In theoretical
computer science, it was shown that artificial neural networks can simulate
any Turing machine (Siegelmann and Sontag 1991, 1995). Also, it was shown that
networks can simulate any time trajectories (Funahashi and Nakamura 1993) and
any attractors (Vakulenko 1994, 2000, Vakulenko and Gordon 1998).
We extend these results to simulation of any spatio-temporal structure, with
any attractors. Since the pioneering ideas of Delbrück (1949), it became well
accepted that differentiation and specialization of initially undifferentiated
clone cells can be understood via multiple dynamical structures and attractors
(Thomas 1998). In particular, differences between gene expression programs can
be understood as differences between attractors of dynamical gene networks. At
least mathematically, the possibility to control any spatio-temporal pattern
is equivalent to the possibility to organize any multicellular organism. Thus,
we show that centralized networks can be used to implement Driesch-Wolpert
positional information paradigm in order to organize a multicellular organism.
This organism consists of a number of specialized cells, each cell type being
dynamically characterized by distinct attractors. The complexity of the
attractors, that can be arbitrarily large, can be programmed by gradients of
morphogens. Transitions between attractors can be performed by acting on key
nodes of the network. Contrary to previous theories of random networks
(Kauffman 1969, Aldana 2003, Aldana and Cluzel 2003), these key nodes do not
have to be hubs. Furthermore, we show that patterning in such networks is
maximally flexible in the sense that it can produce any structurally stable
attractor. We also prove that (and show how) optimally flexible and robust
structures can be computed in polynomial time and can thus be easily attained
by evolution.
The paper is organized as follows. Centralized networks are introduced in
Section 2. A first theorem (Proposition 2.3) concerns with the flexibility of
general centralized networks. We show that these networks are capable to
generate practically all structurally stable prescribed dynamics, chaotic or
periodic, and can have any number of equilibrium states. Another key result
(Theorem 2.5) can be interpreted, in biological terms, as follows. The
centralized networks are capable to create a “multicellular organism”, where
each cell have a prescribed time dynamics. This assertion can be considered as
a mathematical realization of the Wolpert approach since this intrinsic
dynamics in a cell is predetermined only by the morphogen concentration in
this cell. In Section 3 we show that gene activation or silencing can produce
a sharp change of dynamics even if this gene is weakly connected in the
network (it is well known that a mutation in a hub can sharply change the
dynamics, see Aldana 2003). We show that in such a way one can obtain
arbitrary bifurcations. In Section 4 we consider the robustness of centralized
networks and show how these can acquire protection against environmental
perturbations. We show that the design of a network that is both flexible and
robust can be stated as an optimization problem for a discrete spin
hamiltonian. When the number of satellites $N$ is large, the optimization
problem can be solved in polynomial time, $Poly(N)$.
## 2 Centralized networks
By centralized networks we mean networks that contain a few strongly connected
nodes (hubs) and a number of less connected, satellite nodes. A typical
example is given by scale-free networks (Albert and Barabasi 2002, Lesne
2006), that 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 hubs and a large 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 does not act directly on another satellite. We also
assume that satellites respond more rapidly to perturbations and are more
diffusive/mobile than the centers.
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 gene circuit model reads:
$\frac{\partial u_{i}}{\partial t}=\tilde{d}_{i}\Delta
u_{i}+\tilde{r}_{i}\sigma\left({\bf
A}_{i}v+\tilde{b}_{i}m(x)-\tilde{h}_{i}\right)-\tilde{\lambda}_{i}u_{i},$ (1)
$\frac{\partial v_{j}}{\partial t}=d_{j}\Delta v_{j}+r_{j}\sigma\left({\bf
B}_{j}v+{\bf C}_{j}u+b_{j}m(x)-h_{j}\right)-\lambda_{j}v_{j},$ (2)
where $m(x)$ represents the maternal morphogen gradient, $i=1,...,N,\
j=1,...,M$. We assume that the diffusion coefficient $d_{i},\tilde{d}_{i}$ and
maximal production rates $r_{i},\tilde{r}_{i}$ are non-negative:
$d_{i},\tilde{d}_{i},r_{i},\tilde{r}_{i}\geq 0$. Here the morphogenetic field
$m(x)$ and unknown gene concentrations $u_{i}(x,t),v_{j}(x,t)$ are defined in
a compact domain $x\in\Omega$ ($dim(\Omega)\leq 3$) having smooth boundary
$\partial\Omega$, $x\in\Omega$ 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
$\sigma(h)=\frac{1}{1+\exp(-h)},\quad\sigma(h)=\frac{1}{2}\left(\frac{h}{\sqrt{1+h^{2}}}+1\right).$
(4)
The function $\sigma(\beta x)$ becomes a step-like function as its sharpness
$\beta$ tends to $\infty$.
We also set the Neumann boundary conditions
$\nabla u_{i}(x,t)\cdot{\bf n}(x)=0,\quad\nabla v_{j}(x,t)\cdot{\bf
n}(x)=0,\quad(x\in\partial\Omega).$ (5)
They mean that the flux of each reagent through the boundary is zero (here
$\bf n$ denotes the unit normal vector towards the boundary $\partial\Omega$
at the point $x$). Moreover, we set the initial conditions
$u_{i}(x,0)=\tilde{\phi}_{i}(x)\geq 0,\quad v_{j}(x,0)=\phi_{j}(x)\geq
0\quad(x\in\Omega).$ (6)
It is natural to assume that all concentrations are non-negative at the
initial point, and it is easy to show that they stay non-negative for all
times (see below).
Neglecting diffusion effects we obtain from (1),(2) the following shorted
system:
$\frac{\partial u_{i}}{\partial t}=\tilde{r}_{i}\sigma\left({\bf
A}_{i}v+\tilde{b}_{i}m(x)-\tilde{h}_{i}\right)-\tilde{\lambda}_{i}u_{i},$ (7)
$\frac{\partial v_{j}}{\partial t}=r_{j}\sigma\left({\bf B}_{j}v+{\bf
C}_{j}u+b_{j}m(x)-h_{j}\right)-\lambda_{j}v_{j}.$ (8)
This is a Hopfield-like network model (Hopfield 1982) with thresholds
depending on $x$ (contrary to the Hopfield model, the interaction matrices are
not necessarily symmetric). In this case we remove all boundary conditions
(5). If only $d_{i}=0$ we remove the corresponding boundary conditions for
$v_{i}$.
### 2.1 Existence of solutions
Let us introduce some special functional spaces (Henry, 1981). Let us denote
$H=L_{2}(\Omega)^{n}$ the Hilbert space of the vector value functions $w$.
This space is enabled by the standard $L_{2}$\- norm defined by
$||w||^{2}=\int_{\Omega}|w(x)|^{2}dx$, where $|w|^{2}=\sum w_{i}^{2}$. For
$\alpha>0$ we denote by $H_{\alpha}$ the space consisting of all functions
$w\in H$ such that the norm $||w||_{\alpha}$ is bounded, here
$||w||^{2}_{\alpha}=||(-\Delta+I)^{\alpha}w||^{2}.$ These spaces have been
well studied (see Henry 1981 and references therein). The phase space of our
system is ${\cal H}=\\{w=(u,v):u\in H,\ v\in H\\}$, the corresponding natural
fractional spaces are denoted by $H_{\alpha}$ and ${\cal H}_{\alpha}$, here
$H_{0}=H$ and ${\cal H}_{0}={\cal H}$. Denote by $B_{\alpha}(R)$ the
$n$-dimensional ball in $H_{\alpha}$ centered at the origin with the radius
$R$: $B_{\alpha}(R)=\\{w:w\in H_{\alpha},\ ||w||_{\alpha}<R\\}$.
In our case all $f_{i}(w,x)$ are smooth in $w,x$, therefore, the standard
technique (Henry 1981) shows that solutions of (1), (2) exist locally in time
and are unique. In fact, our system can be rewritten as an evolution equation
of the form
$w_{t}=Aw+f(w),$ (9)
where $f$ is a uniformly bounded $C^{1}$ map from ${\cal H}_{\alpha}$ to
${\cal H}$ (since $\sup_{x\in\Omega}|w|\leq c||w||_{\alpha},\,\alpha>3/4$, and
the derivative $\sigma^{\prime}(z)$ is uniformly bounded in $z$) and a linear
self-adjoint negatively defined operator $A$ generates a semigroup satisfying
the estimate $||\exp(At)w||\leq\exp(-\beta t)||w||$ with a $\beta>0$.
Let us prove that the gene network dynamics is correctly defined for all $t$
and solutions are non-negative and bounded. 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 super and subsolutions, that
$\begin{split}0\leq
u_{i}(x,t)\leq\tilde{\phi}_{i}(x)\exp(-\tilde{\lambda}_{i}t)+\tilde{r}_{i}\tilde{\lambda}_{i}^{-1}(1-\exp(-\tilde{\lambda}_{i}t)),\\\
0\leq
v_{i}(x,t)\leq\phi_{i}(x)\exp(-\lambda_{i}t)+r_{i}\lambda_{i}^{-1}(1-\exp(-\lambda_{i}t)).\end{split}$
(10)
Therefore, solutions of (1), (2) not only exist for all times $t$ but also
they enter the set ${\cal B}$ at a time moment $t_{0}$ and then they stay in
this set for all $t>t_{0}$. So, our system defines a global dissipative
semiflow (Henry, 1981).
### 2.2 Reduced dynamics
The key idea is to find a simpler asymptotic description of system dynamics.
It is possible under some assumptions, we suppose here that the $u$-variables
are fast and the $v$-ones are slow. We show then that the fast $u$ variables
are captured, for large times, by the slow $v$ modes. More precisely, 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:
$\ |A_{jl}|,|B_{il}|,|C_{ij}|,|\tilde{h}_{i}|,|h_{j}|<C_{0},$ (11)
where $i=1,2,...,N,\ \ i,l=1,...,M,\ j=1,...,N$,
$0<C_{1}<\tilde{\lambda}_{j},\quad\tilde{d}_{j}<C_{2},$ (12)
$|b_{j}|,|\tilde{b}_{i}|<C_{3},\quad\sup|m(x)|<C_{4},$ (13)
and
$r_{i}=\kappa R_{i},\quad\tilde{r}_{i}=\kappa\tilde{R}_{i},$ (14)
where
$|R_{i}|,|\tilde{R}_{i}|<C_{5},\quad\lambda_{i}=\kappa\bar{\lambda}_{i},\
|\bar{\lambda}|<C_{6},$ (15) $d_{j}=\kappa\bar{d}_{j},\quad
0<\bar{d}_{j}<C_{7},$ (16)
where $\kappa$ is a small parameter, and where all positive constants $C_{k}$
are independent of $\kappa$.
Proposition 2.1. Assume the space dimension $Dim\Omega\leq 3$. Under
assumptions (11), (12), (13), (14) for sufficiently small $\kappa<\kappa_{0}$
solutions $(u,v)$ of (1), (2), (5), and (6) satisfy
$u=U(x,v(x,t))+\tilde{u}(x,t),$ (17)
where the $j$-th component $U_{j}$ of $U$ is defined as a unique solution of
the equation
$\tilde{d}_{j}\Delta U_{j}-\tilde{\lambda}_{j}U_{j}=\kappa G_{j}(v),$ (18)
under the boundary conditions (5), where
$G_{j}=\tilde{R}_{j}\sigma\left({\bf
A}_{j}v(x,t)+\tilde{b}_{j}m(x)-\tilde{h}_{j}\right)$
The function $\tilde{u}$ satisfies the estimates
$||\tilde{u}||+||\nabla\tilde{u}||<c_{1}\kappa^{2}+R\exp(-\beta
t),\quad\beta>0.$ (19)
The $v$ dynamics for large times $t>C_{1}|\log\kappa|$ takes the form
$\frac{\partial v_{i}}{\partial t}=\kappa F_{i}(u,v)+w_{i},$ (20)
where $w_{i}$ satisfy
$||w_{i}||<c_{0}\kappa^{2}$
and
$F_{i}(u,v)=\bar{d}_{i}\Delta v_{i}+R_{i}\sigma\left({\bf B}_{i}v+{\bf
C}_{i}U(x,v)+b_{i}m-h_{i}\right)-\bar{\lambda}_{i}v_{i}.$
Constants $c_{0},c_{1}$ do not depend on $\kappa$ as $\kappa\to 0$ but they
may depend on $R_{i},\tilde{R}_{i},C_{i}$.
A tedious proof of this assertion is basically straightforward; it is based on
well known results (Henry 1981) and is relegated to the Appendix.
An analogous assertion holds for shorted system (7),(8). In this case the
functions $U_{i}$ can be found by an explicit formula. Namely, one has
$U_{i}(x,v(x,t))=\kappa V_{i},\quad
V_{i}=R_{i}\tilde{\lambda}_{i}^{-1}\sigma\left({\bf
A}_{j}v(x,t)+\tilde{b}_{j}m(x)-\tilde{h}_{j}\right).$ (21)
For large times the reduced $v$ dynamics has the same form (20) with
$d_{i}=0$.
### 2.3 Realization of prescribed dynamics by networks
Our next goal is to show that dynamics (20) can realize, in a sense, arbitrary
structurally stable 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 and
Poláčik 1999, Rybakowski 1994, Vakulenko 2000). One can show that some systems
possess the following properties:
A These systems generate global semiflows $S_{\cal P}^{t}$ in an ambient
Hilbert or Banach phase space $H$. These semiflows depend on some parameters
$\cal P$ (which could be elements of another Banach space $\cal B$). They have
global attractors and finite dimensional local attracting invariant $C^{1}$ \-
manifolds $\cal M$ , at least for some $\cal P$.
(Remark: in some cases, these manifolds can be even globally attracting, i.e.,
inertial. Theory of invariant and inertial manifold is well developed, see
(Marion 1989, Mane 1977, Constantin et al 1989, Chow and Lu 1988, Babin and
Vishik 1988).
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
$\frac{dp}{dt}=F(p),\quad F\in C^{1}(B^{n})$ (22)
define a dynamical system in the unit ball ${B}^{n}\subset{\bf R}^{n}$.
For any prescribed dynamics (22) 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 ${B}^{n}$;
B2 The reduced dynamics $S_{\cal P}^{t}|_{{\cal M}_{\cal P}}$ is defined by
equations
$\frac{dp}{dt}=\tilde{F}(p,{\cal P}),\quad\tilde{F}\in C^{1}(B^{n})$ (23)
where the estimate
$|F-\tilde{F}|_{C^{1}({B}^{n})}<\delta$ (24)
holds. In other words, one can say that, by $\cal P$, the inertial dynamics
can be specified to within an arbitrarily small error.
Thus, all robust dynamics (stable under small perturbations) can occur as
inertial forms of these systems. Such systems can be named maximally
dynamically flexible, or, for brevity, MDF systems.
Such structurally stable dynamics can be “chaotic”. There is a rather wide
variation in different definitions of “chaos”. In principle, one can use here
any concept of chaos, provided that this is stable under small $C^{1}$
-perturbations. To fix ideas, we shall use here, following classical tradition
(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 it generates a hyperbolic invariant set $\Gamma$, which is not a periodic
cycle or a rest point. For a definition of hyperbolic sets see, for example,
(Ruelle 1989); a famous example is given by the Smale horseshoe. If, moreover,
this set $\Gamma$ is attracting we say that $\Gamma$ is a chaotic (strange)
attractor. 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 (22) generates
the hyperbolic set $\Gamma$ and $\delta$ is sufficiently small, then dynamics
(23) also generates another hyperbolic set $\tilde{\Gamma}$. Dynamics (22) and
(23) restricted to $\Gamma$ and $\tilde{\Gamma}$ respectively, are
topologically orbitally equivalent (on definition of this equivalence, see
Ruelle 1989, Anosov 1995).
Thus, any kind of the chaotic hyperbolic sets can occur in the dynamics of 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). Examples of systems satisfying these properties can be given by
some reaction diffusion systems (Dancer and Poláčik 1999, Rybakowski 1994,
Vakulenko 2000). Although not yet observed in gene networks, structurally
stable chaotic itineracy is thought to play a functional role in neuroscience
(Rabinovitch 1998).
Let us apply this approach to network dynamics using the results of the
previous section. To this end, assume that (14), 15) and (16) hold. Moreover,
let us assume
$b_{i}=\kappa\bar{b}_{i},\quad h_{i}=\kappa\bar{h}_{i}$ (25)
$\lambda_{i}=\kappa^{2}\bar{\lambda}_{i},\quad d_{i}=\kappa^{2}\bar{d}_{i}$
(26)
where all coefficients $\bar{b}_{i}$ and $\bar{h}_{i}$ are uniform in $\kappa$
as $\kappa\to 0$. These assumptions are useful for technical reasons. We also
assume that all direct interactions between centers are absent, ${\bf B}={\bf
0}$. This constraint is not essential but facilitates notation and
calculations.
Since $U_{j}=O(\kappa)$ for small $\kappa$, we can use the Taylor expansion
for $\sigma$ in (20). Then these equations reduce to
$\frac{\partial v_{i}(x,\tau)}{\partial\tau}=\bar{d}_{i}\Delta
v_{i}+\rho_{i}({\bf
C}_{i}V(x,v)+\bar{b}_{i}m(x)-\bar{h}_{i})-\bar{\lambda}_{i}v_{i}+\tilde{w}_{i}(x,t),$
(27)
where $\rho_{i}(x)=\bar{r}_{i}\sigma^{\prime}(0)$, $i=1,2,...,M$ and $\tau$ is
a slow rescaling time: $\tau=\kappa^{2}t$. Due to conditions (25) and (26)
corrections $\tilde{w}_{i}$ satisfy
$||\tilde{w}_{i}||<c\kappa.$
Let us focus now our attention to non-perturbed equation (27) 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 lemma follows from known approximation theorems of
multilayered network theory, see, for example, (Barron 1993, Funahashi and
Nakamura 1993).
Lemma 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 $N$, a $N\times M$ matrix ${\bf A}$, a
$M\times N$ matrix ${\bf C}$ and coefficients $h_{i}$, where $i=1,2,...,N$,
such that
$|F_{j}(\cdot)-{\bf C}_{j}W(\cdot)|_{C^{1}(B^{M})}<\delta,$ (28)
where
$W_{i}(v)=\sigma\left({\bf A}_{i}v-h_{i}\right),$ (29)
where $v=(v_{1},...,v_{M})\in{\bf R}^{M}$.
This lemma gives us a tool to control network dynamics and patterns. First we
consider the case when the morphogens are absent. Formally, we can set
$\tilde{b}_{i}=\bar{b}_{j}=\bar{d}_{i}=0$. Assume $\bar{h}_{i}=0$. Then
equations (27) with $\tilde{w}_{i}=0$ reduce to the Hopfield-like equations
for variables $v_{i}\equiv v_{i}(\tau)$ that depend only on $\tau$:
$\frac{dv_{l}}{d\tau}={\bf K}_{l}W(v)-\bar{\lambda}_{l}v_{l},$ (30)
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 (30) are $\bf K$, $M$, $h_{j}$ and $\bar{\lambda}_{j}$.
In this case one can formulate the following result.
Proposition 2.3. Let us consider a $C^{1}$-smooth vector field $Q(p)$ defined
on a ball $B^{M}\subset{\bf R}^{M}$ and directed strictly inside this ball at
the boundary $\partial B^{M}$:
$F(p)\cdot p<0,\quad p\in\partial B^{M}.$ (31)
Then, for each $\delta>0$, there is a choice of parameters $\cal P$ such that
(30) $\delta$ -realizes the system (22). This means that (30) is a MDF system.
This proposition follows from the Prop. 2.1 and Lemma 2.2.
Prop. 2.3 implies the following important corollary: all structurally stable
dynamics, including periodic and chaotic dynamics can be realized by
centralized networks. The proof of this fact uses the classical results on the
persistence of hyperbolic sets, and on the existence of invariant manifolds
(Ruelle 1989), see (Vakulenko 2000).
### 2.4 Pattern and attractor control by Wolpert positional information
Above we have considered a spatially homogeneous case. Proposition 2.3 shows
that a centralized network can approximate an arbitrary prescribed dynamics.
Thus, it is shown that cells can be programmed to have arbitrarily complex
dynamics. By network rewiring or by interaction tuning, one can switch between
various types of dynamics. During development these switches are position
dependent, and induce cell differentiation into specific spatial arrangements.
Let us show that the centralized networks, coupled to morphogen gradients, can
generate any spatio-temporal pattern as support for multicellular
organization. We consider shorted dynamics (7), (8) that is reasonable for
cellularized developmental stages, where cell walls prevent a free diffusion
of regulatory molecules. Although other phenomena such as cell signalling can
also lead to cell coupling, we do not discuss these effects here.
Assume cell positions are centered at the points $x\in{\cal
X}=\\{x_{1},x_{2},...,x_{k}\\}$, $dim\Omega=1$, ${\cal X}$ is a discrete
subset of $[0,L]$. Let us show that eqs. (7)-(8) can realize different
dynamics at different points $x_{l}$ of the domain $\Omega=[0,L]$.
We can formulate now the following,
Theorem 2.5. (On translation of positional information into complex and
variegated cell dynamics, or programming of multicellular organism). Suppose
$x\in[0,L]\subset{\bf R}$ and $m(x)$ is a strictly monotone smooth function.
Assume that $0<x_{1}<x_{2}<...<x_{k}<L$ and that $F^{(l)}(p),\ l=1,2,...,k$ is
a family of $C^{1}$-smooth vector fields defined on a unit ball
$B^{M}\subset{\bf R}^{M}$. We assume that each field defines a dynamical
system, i.e., $F^{(l)}$ are directed inwards on the boundary $\partial B^{M}$.
Then, for each $\delta>0$ there is a parameter $\cal P$ choice such that for
shorted dynamics (7)-(8) one has
$u=U(x,v)+\tilde{u},$
where
$|\tilde{u}|<C\exp(-\beta\tau)+c\kappa^{2}.$
For $x=x_{l}$ and for sufficiently large times the dynamics for $v(x_{l},t)$
can be reduced to the form
$\frac{dp_{i}}{d\tau}=\bar{F}_{i}(x_{l},p),$ (32)
where
$\sup_{p\in B^{M}}|\bar{F}(x_{l},p)-F^{(l)}(p)|<\delta.$ (33)
Here $p_{i}(\tau)$ can be expressed in a linear way via $v_{i}(x_{l},\tau)$ by
$v_{i}(x_{l},\tau)-\bar{b}m(x_{l})=\rho_{0}p_{i}(\tau).$
This theorem can be considered as a mathematical formalization of positional
information ideas. It extends Driesch-Wolpert theory by incorporating gene
networks and coping with their information processing role. Flexible gene
networks have different dynamics and attractors, for different local
concentrations of morphogens. The attractor selection ensures the cell fate
decision. Concerning the relation between attractors and cell fate
determination, see (Delbrück 1949, Thomas 1998).
To prove this assertion, let us turn to eqs. (27), where, taking into account
biological arguments given above, we set $d_{i}=0,\tilde{b}_{i}=0$. Let us
set, to simplify formulas, $\rho_{j}=1,\bar{h}_{j}=0$ and
$\bar{\lambda}_{j}=1$. Then
$V_{j}(v)=R_{j}\tilde{\lambda}_{j}^{-1}\sigma({\bf A}_{j}v-\tilde{h}_{j}).$
Denote by $Q_{i}$ the sums $Q_{i}(v)=\sum_{j=1}^{M}C_{ij}V_{j}(v)={\bf
C}_{i}V$. Removing the terms $\tilde{w}_{i}$ in (27), one obtains that eqs.
(27) reduce to
$\frac{\partial
v_{i}(x,\tau)}{\partial\tau}=Q_{i}(v(x,\tau))+\bar{b}_{i}m(x)-v_{i}(x,\tau).$
(34)
Let us fix a $x=x_{l}\in{\cal X}$. Let us make the substitution
$v_{i}(x_{l},\tau)=z_{i}(\tau)+\bar{b}m(x_{l})$ in (34) that gives
$\frac{dz_{i}(\tau)}{d\tau}=Q_{i}(z+\bar{b}m(x_{l}))-z_{i},$ (35)
where $\bar{b}=(\bar{b}_{1},...,\bar{b}_{M})$, $i=1,...,M$.
Now we again use approximation Lemma 2.2. Let us consider a family of vector
fields $C^{1}$-smooth vector field $F$ defined on a unit ball
$B^{M}=\\{z\in{\bf R}^{M},\ |z|\leq 1\\}$ and directed strictly inside this
ball at the boundary $\partial B^{M}$:
$F^{(l)}(z)\cdot z<0,\quad z\in\partial B^{M}.$ (36)
Assume $m(x)$ is a strictly monotone function in $x$. The main idea is as
follows: since all $m(x_{l})=\mu_{l}$ and $m(x_{j})=\mu_{j}$ are different for
$j\neq l$, the vector fields $Q^{(l)}(z)=Q(z+\bar{b}\mu_{l})$ can approximate
different vector fields $F^{(l)}(z)$ for $l=1,...,k$ and for $z$ such that
$|z|<\rho_{0}$, where $\rho_{0}=\frac{1}{2}\min_{i,j,l,j\neq
l}|\bar{b}_{i}||\mu(x_{j})-\mu(x_{l})|$.
For each $\epsilon>0$ we can find an approximation $Q$ satisfying
$\sup_{|z|<\rho_{0}}|Q(z+\bar{b}m(x_{l}))-(\rho_{0}F^{(l)}\rho_{0}^{-1}z)+z)|<\rho_{0}\epsilon,$
(37)
and
$\sup_{|z|<\rho_{0}}|\nabla(Q(z+\bar{b}m(x_{l})))-\nabla\rho_{0}F^{(l)}(\rho_{0}^{-1}z)+z)|<\rho_{0}\epsilon.$
(38)
Then equation (35) reduces to
$\frac{dz}{d\tau}=\rho_{0}F^{(l)}(\rho_{0}^{-1}z)+\rho_{0}\epsilon\tilde{F}^{(l)}(z),$
where
$\sup_{z\in B^{M}}|\tilde{F}^{(l)}(z)|<1,\quad\sup_{z\in
B^{M}}|\nabla\tilde{F}^{(l)}(z)|<1.$
We set $z_{i}=\rho_{0}p_{i}$. This gives
$\frac{dp}{d\tau}=F^{(l)}(p)+\epsilon\tilde{F}^{(l)}(p).$ (39)
Let us notice that, if $\epsilon$ is small enough, then for each index $l$,
due to assumption (36), the trajectory $p(t,p(0))$ stays in $B^{M}$ when the
starting point lies in $B^{M}$: $p(0)\in B^{M}$. Consequently, our
approximations (37) give vector fields that, by (39), realize different
dynamics for each $x_{l}$.
## 3 Sharp genetic switch by satellite silencing/reactivation
In the context of scale-free random networks, it was proposed (Aldana 2003)
that removing of a strong connected center can sharply change the network
attractor. Here we will show that one can obtain transitions between all
possible structurally stable attractors by a single event acting on a
specially chosen weakly connected satellite. Such a satellite interacts only
to one or two centers. Such event may be, either deletion, silencing, or
reactivation. Therefore, such a node can serve as a switch between two kinds
of network behavior. Each of the type of behavior can be defined, for example,
by an attractor or several coexisting attractors that can be fixed points,
periodic or chaotic attractors.
To formalize these ideas mathematically, let us consider a system of ordinary
differential equations
$\frac{dp}{dt}=F(p,s),\quad p\in B^{n}\subset{\bf R}^{n}$ (40)
depending on a real parameter $s$. Here $p=(p_{1},p_{2},...,p_{n})$, $B^{n}$
is the unit ball centered at $p=0$, and $F$ is $C^{1}$-smooth vector field
directed inside the ball at the ball boundary for each $s$ (see (36)). Let us
consider $s_{0},s_{1}$ such that $s_{0}\neq s_{1}$ and suppose that (40) has
different attractors ${\cal A}_{0}$ and ${\cal A}_{1}$ for $s=s_{0},s=s_{1}$
respectively.
Consider, for simplicity, the gene circuit model (1), (2) without diffusion
and space variables:
$\frac{du_{i}}{dt}=\tilde{r}_{i}\sigma\left({\bf
A}_{i}v-\tilde{h}_{i}\right)-\tilde{\lambda}_{i}u_{i},$ (41)
$\frac{dv_{j}}{dt}=r_{j}\sigma\left({\bf
C}_{j}u-h_{j}\right)-\lambda_{j}v_{j},$ (42)
where $i=1,...,M+1$, $j=1,...,N$. The parameters $\cal P$ of this system are
$M,N,h_{i},\tilde{h}_{i}$,
$\lambda_{i},r_{i},\tilde{r}_{j},\tilde{\lambda}_{j}$ and the matrices ${\bf
A,C}$. We can assume, without loss of generality, that we eliminate (by
silencing) the $M+1$-th satellite node, $i=M+1$. As a result of this
elimination, we obtain a similar system with $i=1,...,M$ and shorted matrices
${\bf A},{\bf C}$. Of course, we can also consider the opposite event, which
is to reactivate the $M+1$-th node and recover the initial system this way.
Theorem 3.1 For each $\epsilon>0$ there is a choice of the parameters $\cal P$
such that system (41), (42) with $M$ satellite nodes $\epsilon$ -realizes (40)
with $s=s_{0}$ and system (41), (42) with $M+1$ satellite nodes $\epsilon$
-realizes (40) with $s=s_{1}$.
To prove it, we use the following extended system
$\frac{dp}{dt}=\rho F(p,s),\quad p\in B^{n}$ (43)
$\frac{ds}{dt}=f(s,\beta)-\nu s,\quad s\in{\bf R}$ (44)
where $\nu>1$ and $f(s)$ is a smooth function, $\beta,\rho>0$ are parameters.
Then equilibrium points $s_{eq}$ of (44) are solutions of
$f(s,\beta)=\nu s$ (45)
The point $s_{eq}$ is a local attractor if $f^{\prime}_{s}(s_{eq})<\nu$. Let
us denote $s_{eq}(\beta_{k})=s_{k}$, where $k=0,1$, and let these roots of
(45) be stable, i.e., $f^{\prime}(s_{k})<\nu$.
Then, if $\rho>0$ is small enough, and $s_{k}$ is a single stable rest point,
the fast variable $s$ approaches at $s_{eq}(\beta)$ and for large times $t$
the dynamics of our system (43), (44) is defined by the reduced equations
$\frac{dp}{dt}=\rho F(p,s_{eq}(\beta)).$ (46)
Now let us set
$f(s,\beta)=2\beta^{2}\sigma(b(s-h_{0})),\quad h_{0}<0$ (47)
where $b$ is a large parameter. Then $f$ is close to a step function with the
step $2\beta^{2}$. Therefore for $s_{eq}(\beta)$ one has the asymptotics
$s_{eq}=2\beta^{2}\nu^{-1}+O(\exp(-b))$ as $b\to\infty$. Thus, we can adjust
parameters $\beta,b>0$ in such a way that (45) has a single stable root
$s_{0}$ and the equation
$f(s,\beta/\sqrt{2})=\nu s$ (48)
also has a single root $s_{1}=\beta^{2}/\nu\neq s_{0}$.
Dynamics (43), (44) with $f$ from (47) can be realized by a network (41), (42)
in such a way. We decompose all satellites $u_{i}$ into two subsets. The first
set contains satellites $u_{1},u_{2},...,u_{M-1}$, the second one consists of
the satellites $u_{M},u_{M+1}$ ( to single out this variables, let us denote
$u_{M}=y_{1},u_{M+1}=y_{2}$). The main idea of this decomposition is as
follows. We can linearize equations for the centers $v_{j}$ assuming that the
matrix ${\bf C}$ is small and ${\bf B}=0$ (as above in Section 2).
The $y$ satellites realizes the dynamics (44) by a center $s$:
$\frac{ds}{dt}=-\nu s+\beta(y_{1}+y_{2}),$ (49)
$\frac{dy_{k}}{dt}=-y_{k}+\beta\sigma(b(s-h_{0})),\quad k=1,2.$ (50)
Here we assume that $\nu,\beta$ is small enough, therefore, for large times
this system reduces to (44) with $f$ defined by (47). We see that this
dynamics bifurcates into (44) with $f=\beta^{2}\sigma(b(s-h_{0}))$ if we
remove $y_{2}$ in the right hand side of (49).
The rest of the equations, after a notation modification and linearization,
take the following form
$\frac{du_{i}}{dt}=\tilde{r}_{i}\sigma\left({\bf
A}_{i}v+D_{i}s-\tilde{h}_{i}\right)-\tilde{\lambda}_{i}u_{i},\quad
i=1,...,M-1$ (51) $\frac{dv_{j}}{dt}=-\lambda_{j}v_{j}+r_{j}{\bf
C}_{j}u-h_{j},$ (52)
where $i=1,...,M+1$, $j=1,...,N$.
Equations (51), (52) can $\epsilon$-realize arbitrary systems (40) with the
parameter $s$ which can be shown as above (see Section 2), and this completes
the proof.
## 4 Robust dynamics
Our definition of robust dynamics is inspired from similar ideas in viability
theory (Aubin et al. 2005). Let us suppose that the dynamics (the global
semiflow $S^{t}$), generates a number of attractors. Each attractor ${\cal A}$
has an attraction basin $B({\cal A})$, that is an open set in the phase space.
Assume that our initial data $\phi$ lie in an attractor, $\phi\in{\cal A}$,
and let us add some noise $\xi$ to the dynamics, representing the effect of
the environment. Trajectories become random, and then it is possible that,
under this noise, the trajectory leaves $B({\cal A})$.
We can now define the following characteristic of stability under the noise.
Let us denote $P(T,B({\cal A}),\phi)$ the probability that the trajectory
$u(t,\phi)$ such that $u(0)=\phi\in{\cal A}$ stays in $B({\cal A})$ within the
time interval $[0,T]$.
Definition. Let us consider a network dynamics depending on some parameters
$\cal P$ and on the noise $\xi$. We say that the dynamics is robust under the
noise $\xi$, if for each $T$ and $\delta>0$ there is a choice of the
parameters such that
$P(T,B({\cal A}),\phi)<\delta$
for each attractor $\cal A$ and $\phi\in{\cal A}$.
### 4.1 Centralized motif with noise
For the rest of this section we consider a simplified network, with a single
central node interacting with many satellites. This motif can appear as a
subnetwork in a larger centralized network. In order to study robustness, we
consider the case when the satellites and the center are under the influence
of noise. More general situations, including perturbations of several centers
and satellites, will be studied elsewhere.
The network dynamics is described by the following equations:
$\frac{\partial u_{i}}{\partial t}=d_{i}\Delta
u_{i}-\lambda_{i}u_{i}+\sigma(b_{i}v-h_{i}+\xi_{i}(x,t)),\quad i=1,...,N,$
(53) $\frac{\partial v}{\partial t}=d_{0}\Delta
v-\lambda_{0}v+\sigma(\sum_{i=1}^{N}a_{i}u_{i}-h_{0}+\xi_{0}(x,t)),$ (54)
The random fields $\xi_{i}(x,t)$ summarize the effect of various extrinsic
noise sources. These can be random variations of the morphogen, or environment
noise, or genetic variability affecting network interactions.
Intrinsic noise, resulting from stochastic gene expression, could be
represented as supplementary terms outside the sigmoid function. In order to
avoid further some tedious technical difficulties, we postpone the discussion
of intrinsic noise to future work. Some aspects of the robustness of patterns
with respect to intrinsic noise was studied numerically by Scott et al.
(2010).
The flexibility of such simple networks results from the preceding section.
We can formulate the following problem: how to choose a network motif, robust
under a given environmental noise, and simultaneously flexible? The choice can
result either from genetic changes (for instance mutations, deletions or
duplications of DNA regions) or from network plasticity (epigenetic changes,
such as methylation and chromatin remodeling).
Assume that the considered process is a choice of satellites $u_{i}$ from a
large pool of possible regulators. We can present this process as a choice of
$n$ indices ${j_{i}},i=1,...,n$ from a larger set $I_{N}=\\{1,2,...,N\\}$ of
indices, where $N\geq n$. This choice can be done by boolean variables $s_{j}$
that multiply the coefficients $a_{j}$: the $j$-th reagent participates in the
network if $s_{j}=1$ and does not participate if $s_{i}=0$. Let us make an
important assumption allowing us to obtain a thermodynamical limit as
$N\to\infty$. We assume that
$|a_{i}|<CN^{-1},\quad N\to\infty.$ (55)
Now we transform eqs. (53),(54), using the results of the previous subsection.
Let $v=q(x)$ and $u_{i}=U_{i}(x)$ be equilibrium solutions of (53),(54) where
the noises $\xi_{i}$ are removed. We suppose that the assumptions of the
previous subsection hold. Let us set
$v=q+\tilde{v},\quad u_{i}=U_{i}+\tilde{u}_{i},$
and $U,u$ denote vectors $(U_{1},...,U_{N})$, $(u_{1},...,u_{N})$. Let us set
temporarily $\xi_{0}=0$ ( below we shall show how one can stabilize the system
state, when $\xi_{0}\neq 0$). This gives
$\frac{\partial\tilde{v}}{\partial
t}=d_{0}\Delta\tilde{v}-\lambda_{0}\tilde{v}+\sigma(\rho(U+\tilde{u})-\bar{h})-\sigma(\rho(U)-\bar{h}),$
(56)
where we use, for brevity, the notation
$\rho(u)=\sum_{i=1}^{N}s_{i}a_{i}u_{i}.$
The second part of equations takes then the form
$\frac{\partial\tilde{u}_{i}}{\partial
t}=-d_{i}\Delta\tilde{u}_{i}-\lambda_{i}\tilde{u}_{i}+\sigma(b_{i}(q+\tilde{v})+\xi_{i}-h_{i})-\sigma(b_{i}q-h_{i}).$
(57)
To investigate equations (56), (57), we use a special method justified in a
rigorous way in Appendix. This holds under the following assumption:
Assumption 4.4. The “morphogenetic” noises $\xi_{i}(x,t)$ are independent on
$t$:
$\xi_{i}(x,t)=\xi_{i}(x),\quad i=0,1....,N.$
The functions $\xi_{i}$ are continuous in $x$ and a priori bounded
$\sup_{x\in\Omega}|\xi_{i}(x)|<C_{*},\quad i=0,1,...,N.$ (58)
where a positive constant $C_{*}$ may be large but it is independent of $N$
for large $N$.
Notice that Assumption 4.4 guarantees global existence of solutions
$\tilde{u}_{i}(x,t),\tilde{v}(x,t)$ of eqs. (56), (57) for all $t>0$.
Intuitively, one can expect that the term $\tilde{v}$ in (57) in $\sigma$ is
small and can be, thus, removed. Following this idea, let us introduce
$\eta_{i}$ as solutions of
$\frac{\partial\eta_{i}}{\partial
t}=-d_{i}\Delta\eta_{i}-\lambda_{i}\eta_{i}+\sigma(b_{i}q+\xi_{i}-h_{i})-\sigma(b_{i}q-h_{i}).$
(59)
If $\xi_{i}$ are independent of $t$, and sufficiently regular in $x$ then
arguments of the previous section show that for large $t$
$\eta_{i}(x,t)\to\bar{\eta}_{i}(x),$ (60)
where $\bar{\eta}_{i}$ are solutions of elliptic equations
$d_{i}\Delta\bar{\eta}_{i}+\lambda_{i}\bar{\eta}_{i}=G_{i}(\xi_{i}(x)),\quad
G_{i}(\xi_{i}(x))=\sigma(b_{i}q+\xi_{i}-h_{i})-\sigma(b_{i}q-h_{i})$ (61)
under zero Neumann boundary conditions.
Let us consider equation (56) for $v$. Assume $\rho$ is small. Then we can
linearize the nonlinear contributions in the right hand side of this equation:
$\sigma(\rho(U+\tilde{u})-\bar{h})-\sigma(\rho(U)-\bar{h})=\sigma^{\prime}(U-\bar{h})\rho(\tilde{u})+O(\rho(\tilde{u})^{2}).$
We assume that $\tilde{u}_{i}$ are close to $\bar{\eta}_{i}$. Thus,
$\rho(\tilde{u})\approx\rho(\bar{\eta})$. Calculations presented in the
Appendix show that the fluctuation influence can be estimated through the
quantity
$\delta(s,T)=\sup_{t\in[0,T]}{\bf H}(s,t),\quad{\bf
H}(s,t)=||\rho(\bar{\eta})||^{2}.$ (62)
If $\xi_{i}$ are independent of $t$, for large $t$ one has
${\bf H}(s,t)\to\bar{\bf H}(s)=||\rho(\bar{\eta})||^{2}.$ (63)
Notice that $\bar{\bf H}$ can be rewritten in the form
$\bar{\bf
H}(s)=\sum_{m=1}^{N}\sum_{m^{\prime}=1}^{N}W_{mm^{\prime}}(\bar{\eta}(\cdot))s_{m}s_{m^{\prime}},$
(64)
where $W_{mm}$ are random and
$W_{mm^{\prime}}(\bar{\eta}(\cdot))=a_{m}a_{m^{\prime}}\langle\bar{\eta}_{m},\
\bar{\eta}_{m^{\prime}}\rangle,$ (65)
here $\langle f,g\rangle$ denotes the inner scalar product in $H$: $\langle
f,g\rangle=\int_{\Omega}fgdx$, where $dx$ is the standard Lebesgue measure.
### 4.2 Hard combinatorial problems in network evolution
We assume that Assumption 4.4 holds. The minimization of $\bar{\bf H}(s)$ with
respect to $s$ should be done under the condition that at least one satellite
is involved, i.e.,
$R_{0}(s)=N^{-1}\sum_{i=1}^{N}s_{i}>0.$ (66)
The analysis of the minimization problem for this random Hamiltonian is a
computationally hard problem advanced firstly by methods from statistical
physics of spin glasses (see, for example, (Mezard Zecchina 2002) for
applications to hard combinatorial problems, and (Talagrand 2003) for rigorous
justification). To make the analogy with spin glasses more transparent, we can
make change $s_{i}=2S_{i}+1$, where spin variables $S_{i}$ take values $1$ or
$-1$. However, our problem is even more complicated because, besides (66),
some other restrictions should be taken into account.
In addition to (66), we must take into account restrictions connected with
generation of several steady states $q_{1}$, $q_{2}$, …, $q_{M}$, to provide
flexibility. Let us take a small $\epsilon>0$. By adjusting $s_{i}$ we would
like to obtain a set of equilibria close to $q_{l}$. This gives the following
restrictions
$\sup_{x\in\Omega}\sigma(\sum_{i=1}^{N}s_{i}a_{i}\sigma(b_{i}q_{l}-\bar{h})-\lambda_{0}q_{l}-h_{0})<\epsilon,\quad
l=1,2,...,M$ (67)
or, in a simpler form,
$\sup_{x\in\Omega}|R_{l}(s,x)-B_{l}(x)|<c\epsilon,\quad l=1,2,...,M$ (68)
where
$R_{l}=\sum_{i=1}^{N}M_{li}s_{i},$
$M_{li}=a_{i}\sigma(b_{i}q_{l}-\bar{h}),\quad
B_{l}=\sigma^{-1}(\lambda_{0}q_{l}-h_{0}).$
Although $R_{l}$ are linear in $s_{i}$ functions, the left hand side of (68)
is, in general, a nonlinear function of a complicated form. To overcome this
difficulty, we replace the $\sup$ in (68) by the $L_{2}$\- norm that gives
quadratic in $s$ functionals:
${\bf R}_{l}(s)=||R_{l}(s,x)-B_{l}(x)||^{2}<c\epsilon^{2}.\quad l=1,2,...,M$
(69)
We use Lagrange multipliers $\beta_{l}$ to take into account conditions (69).
This leads to the following Lagrange function:
${\bf F}(s)={\bf H}(s)+\sum_{l=1}^{L}\beta_{l}{\bf R}_{l}(s)^{2}.$ (70)
Let us remind that the matrix $W_{mm^{\prime}}$, that determines our
hamiltonian ${\bf H}$, is a random matrix depending on random fields
$\xi_{i}(x)$ through $\rho(\bar{\eta})$. Let us consider these fields as
elements of the Banach space $C^{0}(\Omega)$ of all bounded continuous in $x$
vector valued functions. Let $\mu_{\xi}$ be a probability measure defined on
the subset of all such functions satisfying (58).
We also propose that variables $s$ are chosen by a stochastic algorithm. The
stochastic algorithm depends on some set of parameters ${P}$ that can be
adjusted. Let $\mu_{P}$ be a probability measure associated with this
algorithm (this measure is defined below).
We would like to have a small value of ${\bf F}$ for a “most part” of field
$\xi$ and $s$ values, with respect to the product measure
$\mu=\mu_{\xi}\times\mu_{P}$.
Finally, the combinatorial problem can be formulated as follows: for a small
number $\delta$, find parameters $P$ such that the probability (computed by
the measure $\mu$),
$Prob\\{{\bf F}(\xi(\cdot),s)>\delta\\}$ (71)
is small enough.
### 4.3 Mean field solution can be obtained by quadratic optimization
We show here that the optimization problem is feasible when $N$ is large. To
this end, we define the mean field Lagrange function $\bar{\bf F}$ that is
obtained from ${\bf F}(\xi(\cdot),s)$ by averaging with respect to $\mu$. In
order to estimate the deviations of ${\bf F}$ from $\bar{\bf F}$ we use the
Chebyshev inequality:
$P(F,s)=Prob\\{|{\bf F}(\xi(\cdot),s)-\bar{\bf F}(s)|>a\\}\leq a^{-2}Var{\bf
F},$ (72)
where the probability, the average and the variance should be computed by
$\mu$.
The stochastic algorithm for choosing the satellites can be a simple Bernoulli
scheme. Namely, let us consider $s_{i}$ as mutually independent random
variables such that
$Prob\\{s_{i}=1\\}=p_{i}.$
Thus, the mean field Lagrange function reads
$\bar{\bf
F}(p)=\sum_{i=1}^{N}\sum_{j=1}^{N}\bar{W}_{ij}p_{i}p_{j}+\sum_{l=1}^{L}\beta_{l}{\bf
R}_{l}(p),$ (73)
where $\bar{W}_{ij}$ is obtained from $W_{ij}$ by averaging with respect to
$\mu_{\xi}$. Our main idea is as follows.
Step 1: Quadratic programming for the mean field Lagrange function
First, we minimize $\bar{\bf F}$ with respect to $p_{i}$. This is a quadratic
programming problem that can be solved in polynomial time.
QP to find a minimum $\bar{\bf F}(p)$ under conditions
$0\leq p_{i}\leq 1,$ (74) $R_{0}(p)=\sum_{i=1}^{N}p_{i}>0.$ (75)
The last condition is trivial and can be omitted. Therefore, we look for a
minimum of a positively defined quadratic form on the multidimensional box.
The well-known L.Khachiyan ellipsoid algorithm for this problem runs in
$Poly(N)$ time. This proves such a lemma:
Lemma 4.5. If a solution of the problem QP exists, then it can be found in
$Poly(N)$ time.
Step 2: Obtain a small variance of the Lagrange function in the limit N large
Let us suppose that $\bar{\bf F}<\delta/2$. Then, using (72) we get
$Prob\\{{\bf F}(\xi(\cdot),s)>\delta\\}<Prob\\{|{\bf F}(\xi(\cdot),s)-\bar{\bf
F}(s)|>\delta/2\\}\leq 4\delta^{-2}Var{\bf F}.$ (76)
Now let us estimate $Var{\bf F}$. We consider $Var{\bf H}$, the rest terms
${\bf R}_{l}$ can be considered in a similar way. First we estimate variation
with respect to $s$ by the measure $\mu_{P}$. One notices that
$Var{\bf
H}=E\sum_{iji^{\prime}j^{\prime}}s_{i}s_{j}s_{i^{\prime}}s_{j^{\prime}}W_{ij}W_{i^{\prime}j^{\prime}}-E\sum_{ij}s_{i}s_{j}W_{ij}E\sum_{i^{\prime}j^{\prime}}s_{i^{\prime}}s_{j^{\prime}}W_{i^{\prime}j^{\prime}}.$
Notice that if $i\neq i^{\prime}$ and $j\neq j^{\prime}$ then
$Es_{i}s_{j}s_{i^{\prime}}s_{j^{\prime}}=Es_{i}s_{j}Es_{i^{\prime}}s_{j^{\prime}}.$
Moreover, $|W_{ij}|=O(N^{-2})$ due to our assumption $(\ref{aih})$ on $a_{i}$
and Assumption 4.4. Thus we have maximum $N^{3}$ of non-zero terms in $DH$,
which have the order $O(N^{-4})$. Thus, the complete variation satisfies
$Var{\bf H}<C_{0}N^{-1},$ (77)
where $C_{0}$ is uniform in $N$ as $N\to\infty$.
Thus, for large $N$ one has $Var{\bf F}\to 0$, thus the probability (71) is
arbitrarily small. This shows that the problem of minimization (72) is
feasible in polynomial time $Poly(N)$, when $N$ is large enough. More
precisely, we have the following
Proposition 4.6. If a solution of the problem QP exists and $N$ is large
enough, then a solution $s$ satisfying all restrictions and minimizing $F$ at
level $\delta$ with a probability, arbitrarily close to $1$, can be found in
$Poly(N)$ time.
Remark. Above we have studied the case $\xi_{0}=0$. For smooth $\xi_{0}(x)$ we
can obtain a robustness with respect to $\xi_{0}$ variations in a simple way.
Namely, for large $N$ one can choose the constant $C$ in (55)) large enough,
then $\sum_{i=1}^{N}{a_{i}}u_{i}-h_{0}>>|\xi_{0}(x)|$.
There arises, however, a natural question: how genetic networks can realize
these sophisticated algorithms which are capable to optimize the network
robustness? A possible answer to this question is that $s_{i}$ could be
themselves involved in a gene network of the form (1), (2). We showed that
gene networks are capable to simulate all structurally stable dynamics. The
fact that this is equivalent to simulating arbitrary Turing machines and thus
arbitrary algorithms follows from results of Koiran and Moore (1999).
## 5 Conclusion
We are concerned with dynamical properties of 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 rigorous
mathematical methods, that this centralized architecture, widespread in gene
networks, allow to realize two fundamental biological strategies: flexible and
robust bow-tie control and Wolpert positional information concepts.
We show how a combination of these strategies leads to the remarkable
possibility to create a “multicellular organism”, where each “cell” can
exhibit a complicated time behaviour, different for different cells.
Centralized network architectures provide the flexibility important in
developmental processes and for adaptive functions.
Contrary to previous works on centralized boolean networks (Aldana 2003), we
show that arbitrary bifurcations between attractors can be controlled by
action on satellites, instead of actions on centers.
To check the robustness of such architectures we have considered a simplified
example of a centralized network with a single center. Such system produces
many equilibria, and this dynamical structure can be protected against large
space dependent, random perturbations. We show that in general, designing an
optimal network that is protected against such perturbations boils down to
finding the minimum energy of a spin glass hamiltonian, which is a
computationally hard problem. However, for a large number of satellites, the
randomness is filtered and reliable protection against perturbations results
as a solution to a quadratic programming problem, that can be solved in
polynomial time. We expect that similar results hold more generally, for
networks with any number of centers. This suggests an evolutionary bias
towards centralized networks where hubs are subjected to control from many
satellites.
These findings can be interpreted in terms of gene networks. The flexibility
control by satellites, and not by transcription factors (centers) can be a
major property of such networks. It may be easier to act on a satellite (by
silencing or reactivating it), then to perform similar actions on a center
(deletion of a hub proves most of the time to be lethal). We have proposed
miRNAs and CREMs as possible candidates for satellite nodes in gene networks
controlling pattering in development. A few examples of such centralized
motifs are known, such for instance the enhancer system of the even-skipped
gene of Drosophila (Ludwig et al 2011). The process of reconstruction of such
networks is only at the beginning (see for instance (Berezikov 2011)). One
could expect that many more examples of centralized motifs and networks will
be found during this process.
Acknowledgements. The authors are grateful to John Reinitz, Maria Samsonova
and Vitaly Gursky for useful discussions. We are thankful to M. S. Gelfand and
his colleagues for stimulating discussions in Moscow.
SV 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) and by
a visiting professorship grant from the University of Montpellier 2.
Appendix: Proofs and estimates
I. The proof of Proposition 2.1
To outline the proof, let us notice that our system has a typical form, where
slow ($v$) and fast ($u$) components are separated:
$v_{t}=\kappa F(v,u),\quad u_{t}=Au+\kappa G(v).$ (78)
Let us present $u$ as $u=U+\tilde{u}$, where $U=-\kappa A^{-1}G(v)$ and
$\tilde{u}$ is a new unknown. Let us notice that $U_{i}$ are solutions of (18)
under boundary conditions (5) and that $|U_{i}|<c\kappa$.
By substituting $u=U+\tilde{u}$ into (78), we obtain
$v_{t}=\kappa
F(v,U+\tilde{u}),\quad\tilde{u}_{t}=A\tilde{u}+\kappa^{2}G_{1}(v,U+\tilde{u}),$
(79)
where $G_{1}=\kappa^{-1}A^{-1}G^{\prime}(v)v_{t}=A^{-1}F(v,U+\tilde{u})$, the
operator $A=diag\\{\tilde{d}_{i}\Delta-\tilde{\lambda}_{i}\\}$. Let us show
that $G_{1}(\tilde{u},v)$ is a uniformly bounded map in $\cal H$ for all $u,v$
satisfying a priori estimates (10). For sufficiently smooth initial data
$\phi,\tilde{\phi}\in C^{2}$ these estimates and evolution equation (9) imply
$||v(t)||_{\alpha}\leq C_{1},\quad t\geq 0,\ \alpha\in(0,1).$ (80)
The Sobolev embedding gives then
$||\nabla v(t)||_{L_{4}(\Omega)}\leq c||v(t)||_{\alpha}\leq C_{2},\quad t\geq
0,\ \alpha\in(1/2,1).$ (81)
To estimate now $G_{1}=(w_{1},...,w_{N})^{tr}$, let us notice that $w_{i}$
satisfy the following equations:
$(\tilde{d}_{i}\Delta-\tilde{\lambda}_{i})w_{i}=g_{i}(x,v)(d_{i}\Delta-\lambda_{i})v_{i},$
(82)
where $g_{i}$ are smooth functions with uniformly bounded derivatives. Our
goal is, thus, to estimate $||\nabla w||$ through $||\nabla v||_{L_{4}}$ and
$||v||_{\alpha}$. Let us multiply (82) through $w_{i}$ and then integrate the
left hand and the right hand sides of the obtained equations by parts. We find
$||\nabla w_{i}||^{2}\leq c_{1}||\nabla w_{i}||||\nabla
v_{i}||+c_{2}\langle(\nabla v)^{2},|w|\rangle.$ (83)
To estimate $\langle(\nabla v)^{2},|w|\rangle$, we use the Cauchy-Schwartz
inequality
$|\langle(\nabla v)^{2},|w|\rangle|\leq c||\nabla v||_{L_{4}(\Omega)}||w||,$
Now we can apply (81) and the Cauchy inequality with a parameter $a>0$ that
gives
$||\nabla w||\ ||\nabla v||\leq c_{1}a||\nabla w||^{2}+Ca^{-1}||\nabla
v||^{2},$ (84)
and if $a>0$ is small enough ($c_{1}a<1$), we obtain, by (83) and (84), the
need estimate:
$||\nabla w||<C_{3}.$
The second equation in (79) then entails
$||\tilde{u}||_{t}\leq-\beta||\tilde{u}||+\kappa^{2}\sup||G_{1}||,$
where $\beta=\min\\{\tilde{\lambda}_{i}\\}>0$ is independent of $\kappa$. This
gives
$||\tilde{u}(t)||\leq||\tilde{u}(0)||\exp(-\beta t)+C_{4}\kappa^{2}.$
In a similar way one can obtain the same estimate for $||\nabla\tilde{u}||$.
This completes the proof.
II. Estimates for network viability via spin hamiltonian
Assume that for some $\xi(x)=(\xi_{1}(x),...,\xi_{N}(x))$ there holds
${\bf H}(x,\xi(\cdot))<\delta.$ (85)
Let us obtain estimates of deviations $\tilde{v}=v-q$ and
$\tilde{u}_{i}=u_{i}-U_{i}(q)$, where $v=q(x),\ u=U_{i}$ define an equilibrium
stationary solution for $\xi_{i}(x)\equiv 0$. These estimates hold only due to
the special structure of our network: we admit that $\xi_{i}$ are not small,
nonetheless, the summarized effect of these perturbations is small. We assume
$\tilde{u}_{i}(x,0)=0,\quad\tilde{v}(x,0)=0.$ (86)
Let us present the functions $\tilde{u}_{i}$ as sums
$\tilde{u}_{i}=\bar{\eta}_{i}+w_{i}$, where $\bar{\eta}_{i}$ are defined by
(61). For $w_{i},\tilde{v}$ we then obtain
$\frac{\partial\tilde{v}}{\partial
t}=d_{0}\Delta\tilde{v}-\lambda_{0}v+\sigma(\rho(U+\tilde{u}(\tau))-\bar{h})-\sigma(\rho(U)-\bar{h}),$
(87) $\frac{\partial w_{i}(t)}{\partial
t}=d_{i}\Delta\tilde{v}-\lambda_{i}v+F_{i}(\tilde{v}(\tau),\xi)d\tau,$ (88)
where
$F_{i}(\tilde{v},\xi)=\sigma(b_{i}(q+\tilde{v})-h_{i}+\xi_{i})-\sigma(b_{i}q-h_{i}+\xi_{i}).$
Let us observe that
$||\tilde{F}_{i}||<c||\tilde{v}||.$ (89)
Condition (85) implies that
$||\rho(\eta(\cdot))||<\delta.$ (90)
Let us introduce $||w||$, by $||w||^{2}=\sum_{i=1}^{N}||w_{i}||^{2}$ and $|a|$
by $|a|^{2}=\sum_{i=1}^{N}|a_{i}|^{2}.$ Then
$||\rho(w)||\leq|a|||w||.$
By (87), (88), (89) and (90) now one obtains inequalities for
$||\tilde{v}||,||w_{i}||$:
$\frac{d||\tilde{v}||^{2}}{2dt}\leq-\lambda_{0}||\tilde{v}||^{2}+c_{3}(||\rho(\bar{\eta})||+|a|||w||),$
(91) $\frac{d||w||^{2}}{2dt}\leq-\bar{\lambda}||w||^{2}+c_{4}||\tilde{v}||.$
(92)
where $\min_{i}\lambda_{i}=\bar{\lambda}>0$. Assume that
$\min_{i}\lambda_{i}>0$ are large enough. Moreover, for large $N$ the
coefficient $c_{3}|a|<1$. Combining (91), (92) one obtains the inequality for
$||Y||^{2}=||w||^{2}+||\tilde{v}||^{2}$:
$\frac{d||Y||^{2}}{2dt}\leq-\lambda_{0}||Y||^{2}+(\lambda_{0}-\bar{\lambda})||w||^{2}+c_{3}\delta||\tilde{v}||+c_{5}||w||\
||\tilde{v}||.$ (93)
We apply now the Cauchy inequality $xy<ax^{2}+a^{-1}y^{2}$ to the two terms in
the right hand side of this last inequality. This gives
$\frac{d||Y||^{2}}{2dt}\leq-\lambda_{0}||Y||^{2}+(\lambda_{0}-\bar{\lambda})||w||^{2}+a^{-1}||w||^{2}+c_{6}a||\tilde{v}||^{2}+c_{7}a^{-1}\delta^{2}||\tilde{v}||.$
(94)
We adjust an $a$ such that $c_{0}a<\lambda_{0}/2$. If $\bar{\lambda}$ is large
enough, (94) gives then
$||Y(t)||\leq c\delta+||Y(0)||\exp(-\lambda_{0}t/2).$ (95)
Then (95) implies that for large $t$ there holds
$\sup_{t>0}||\tilde{v}(t)||_{\alpha}\leq c_{9}\delta$
with a constant $c_{9}>0$. This gives us the need estimate of $\tilde{v}$ via
the spin hamiltonian.
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|
arxiv-papers
| 2011-10-21T08:00:27 |
2024-09-04T02:49:23.441452
|
{
"license": "Public Domain",
"authors": "Sergei Vakulenko, Ovidiu Radulescu",
"submitter": "Ovidiu Radulescu",
"url": "https://arxiv.org/abs/1110.4724"
}
|
1110.4732
|
# Maxwell’s Demon and Data Compression
Akio Hosoya ahosoya@th.phys.titech.ac.jp Department of Physics, Tokyo
Institute of Technology, Tokyo, Japan Koji Maruyama maruyama@sci.osaka-
cu.ac.jp Department of Chemistry and Materials Science, Osaka City University,
Osaka, Japan Yutaka Shikano shikano@th.phys.titech.ac.jp Department of
Physics, Tokyo Institute of Technology, Tokyo, Japan
###### Abstract
In an asymmetric Szilard engine model of Maxwell’s demon, we show the
equivalence between information theoretical and thermodynamic entropies when
the demon erases information optimally. The work gain by the engine can be
exactly canceled out by the work necessary to reset demon’s memory after
optimal data compression a la Shannon before the erasure.
###### pacs:
89.70.Cf, 05.70.-a
## I Introduction
Entropy is one of the most cardinal concepts in the modern science. The idea
of entropy plays a crucial role in not only thermodynamics, but also the
physics of black holes and information science, including quantum information,
to name a few. The interplay of entropy in classical physics and information
science has been studied intensively since it was first pointed out by
Brillouin in the general context brillouin ; Brillouin . Then, this idea was
clarified by Landauer in the form of information erasure principle landauer .
Landauer’s work opened up a way to relate the information theoretic and
thermodynamic entropies. In order to obtain further insights into the
relation, we need a simple and specific model. In this sense, the most well-
studied is Szilard’s engine and Maxwell’s demon.
Since Maxwell mentioned an apparent violation of the second law of
thermodynamics by a fictitious intelligent being in his textbook in 1871
maxwell , this paradoxical problem has been debated intensively under the name
of Maxwell’s demon demon2 ; maruyama09 . Towards its solution, Szilard devised
in 1929 a one-molecule engine model, which ingeniously distilled the essence
of the problem and made him realize the significance of information in the
thermodynamic process szilard . Although it still took some time after
Szilard, a satisfactory solution that lets the demon down was eventually
reached, based on the idea of Landauer landauer and Bennett bennett82 . The
overall consensus we share today is that erasing information in demon’s memory
causes an entropy increase, which, with demon’s best effort, precisely cancels
out the work gain when closing the thermodynamic cycle.
The physical process of information erasure has been investigated from various
aspects: noteworthy examples are two derivations of the entropy increase by
Shizume shizume95 and Piechocinska piechocinska00 . They both showed that the
lower bound of the entropy increase for erasing one bit of information should
be $k_{B}\ln 2$, where $k_{B}$ is the Boltzmann constant. This entropy
increase is exactly the minimum amount to circumvent the contradiction with
the second law in the demonic paradox.
This specific example suggests a possible way to link a certain entropy like
quantity with the information entropy. This could be achieved by considering
an operational model to carry out information erasure with a dynamical process
intrinsic to the system of interest.
In the present paper, on the basis of the Shannon compression of demon’s
memory before erasure in the asymmetric Szilard engine model, we prove that
the optimal cost of information erasure is
$k_{B}\ln 2\cdot H(p),$ (1)
where $H(p)$ is the Shannon information entropy
$H(p)=-p\log_{2}p-(1-p)\log_{2}(1-p)$ (2)
with $p$ being the ratio of the proportional division of the cylinder by the
partition. Therefore, the entropy decrease of the Szilard engine exactly
cancels out the entropy increase by the optimal information erasure of the
demon.
This paper is organized as follows. In Sec. II, we recapitulate the resolution
of the Maxwell’s demon paradox by Landauer and Bennett in the standard
symmetric Szilard engine model. In Sec. III, we propose the protocol of the
demon in the case of an asymmetric Szilard engine. We show that the erasure
work can be minimized by Shannon’s data compression. In Sec. IV, we consider a
different scenario, where heat baths of different temperatures are used for
the engine-demon system. Section V is devoted to summary and discussions.
## II Symmetric Szilard engine and erasure of memory
In this section we briefly review the Landauer principle of information
erasure in the standard symmetric Szilard engine model landauer . The Szilard
engine consists of a one-dimensional cylinder, whose volume is $V_{0}$,
containing a single-molecule gas and a partition that works as a movable
piston.
The operator, i.e., a demon, of the engine inserts the partition into the
cylinder, measures the position of the molecule, and connects to the partition
a string with a weight at its end. These actions by the demon are optimally
performed without energy consumption bennett82 . Throughout this paper, the
demon’s memory is also modeled as a single-molecule gas in a box with a
partition in the middle. Binary information, $0$ and $1$, is represented by
the position of the molecule in the box, the left and the right, respectively.
This model of symmetric memory has an advantage that reading, encoding, and
computing over bits require no energy, making it consistent with the scenario
of reversible computation.
The following is the protocol to extract work from the engine by information
processing of the demon (see Fig. 1), where we denote “SzE” for the Szilard
engine and “DM” for the demon’s memory at each step of the protocol.
Initially, the molecule in the cylinder moves freely over the volume $V_{0}$.
Step 1 (SzE)
The partition is inserted at the center of the cylinder.
Step 2 (SzE, DM)
The demon measures the location of the molecule, either the left (“L”) or the
right (“R”) side of the partition. The demon records the measurement outcome
in his memory. When it is L (R), his memory is recorded as “$0$” (“$1$”).
Step 3 (SzE)
Depending on the measurement outcome, the demon arranges the device
differently. That is, when the molecule was found on the left (right) hand
side, i.e., the record is $0$ ($1$), he attaches the string to the partition
from the left (right). In either case, by putting the cylinder in contact with
the heat bath of temperature $T$, the molecule pushes the partition, thus
exerting work on the weight, until the partition reaches the end of the
cylinder. The amount of work extracted by the engine is
$W=k_{B}T\ln 2,$ (3)
as can be seen by applying the combined gas law in one dimension.
Step 4 (SzE)
The demon removes the partition of the engine, letting the molecule return to
its initial state.
Step 5 (DM)
The demon removes the partition of his memory to erase information.
Step 6 (DM)
In order to reset the memory to its initial state, the demon compresses the
volume of the gas by half.
Figure 1: (Color online). A protocol of symmetric Szilard engine (black/left
side) and demon’s memory (red/right side). This figure shows an example in
which the molecule was found in the right hand side of the cylinder. In
demon’s memory, the state after removing the partition is denoted by “$\ast$”.
In order to complete the cycle for both the Szilard engine and the memory, the
demon has to reset the memory, which follows the erasure of one-bit
information. Following is a more precise explanation about the physical
process of information erasure and memory resetting described in Steps 5 and
6. The box is in contact with the thermal bath at the same temperature $T$ as
that of the engine. The record in the memory can be erased simply by removing
the partition, since the location of the molecule becomes completely
uncertain. To bring the memory back to its initial state, e.g., $0$, one has
to compress the gas by half by sliding a piston from the right end to the
middle. The necessary work for this compression is $k_{B}T\ln 2$, which
exactly cancels out the work gain by the engine (3). Here, we have taken the
result by Piechocinska for granted that the erasure of a single bit of
information requires a work of at least $k_{B}T\ln 2$ piechocinska00 .
Let us look at the same process in terms of thermodynamic entropy. By Steps 1
and 2, the volume of the gas in engine is halved, regardless of the
measurement outcome. As the entropy change of an ideal gas under the
isothermal process is given by $\Delta
S:=S(V^{\prime})-S(V)=k_{B}\ln(V^{\prime}/V)$, the entropy of the engine is
lowered by $k_{B}\ln 2$. The isothermal expansion in Step 3 increases the
entropy of the gas by $k_{B}\ln 2$, while that of the heat bath is decreased
by the same amount. As far as the Szilard engine and its heat bath are
concerned, the net result is an entropy decrease of $k_{B}\ln 2$.
Nevertheless, this is exactly canceled out by the entropy increase due to
information erasure and reset performed in Steps 5 and 6.
These last two steps are of crucial importance when closing a cycle of the
memory. Information erasure in Step 5 is an irreversible process and increases
thermodynamic entropy by $k_{B}\ln 2$. The isothermal compression to reset the
memory in Step 6 requires work and dissipates entropy of $k_{B}\ln 2$ to its
heat bath. This is the essence of Landauer-Bennett mechanism that resolves the
Maxwell’s demon paradox.
Now let us slightly generalize the Szilard engine model to an asymmetric one
in such a way that the partition is inserted to divide the whole volume
$V_{0}$ into $pV_{0}$ and $(1-p)V_{0}$ with $0<p<1$ (See Fig. 2). A
straightforward calculation shows that the work extracted by the asymmetric
Szilard engine is
$k_{B}TS(p),$ (4)
where $S(p)=-p\ln p-(1-p)\ln(1-p)$ Feynman . If the memory is reset after
every cycle of the engine, the amount of work consumption is
$\Delta W=k_{B}T\ln 2-k_{B}TS(p)\geq 0.$ (5)
However, one may wonder if the gap, $\Delta W$, could be smaller by employing
a better strategy. In the following section, we show an information
theoretical protocol that fills the gap optimally.
Figure 2: (Color online). The model of an asymmetric Szilard engine. The
position of the partition in demon’s memory is the same as that in the case of
the symmetric version.
## III Asymmetric Szilard engine and erasure of compressed memory
We are going to show a protocol in which the demon is clever enough to reduce
the work for the erasure by using the Shannon data compression shannon in the
asymmetric Szilard engine introduced in the previous section. First, the demon
accumulates the data of $N$ cycles, which we assume is very large. The data
contains uneven number of $0$’s and $1$’s corresponding to the measured
position of the molecule in the engine. The relative frequency of $0$’s is
obviously $p$, while that of $1$’s is $1-p$. According to Shannon’s noiseless
coding theorem, the demon can compress the data to a shorter one, whose length
will be $N_{s}:=NH(p)\leq N$ at shortest. Coding does not cost any work if we
employ reversible computation bennett_rev , provided that the memory is
symmetric as remarked before. If asymmetric memory were used, even the NOT
gate, and therefore generic computation, cost energy, which makes our task
less transparent. See, e.g., Refs. barkeshli ; sagawa . Then, he erases the
shortened data string with the work $k_{B}T\ln 2\cdot N_{s}=k_{B}TNS(p)$.
Therefore, the difference between the work to reset the memory and the work
extracted by the engine approaches zero,
$\Delta W(optimal)=k_{B}TN_{s}-k_{B}TS(p)=0,$ (6)
for a very large $N$.
To be more precise, we write down the optimal protocol below.
Step 1 (SzE)
The partition is inserted to divide the volume into two parts, $pV_{0}$ and
$(1-p)V_{0}$, in the initial configuration of the cylinder and a single
molecule is either on the left or the right of the partition.
Step 2 (SzE, DM)
The demon measures the location of the molecule and records either $0$ for the
left (L) or $1$ for the right (R) and keep the result in his memory.
Step 3 (SzE)
Depending on the recorded information, the demon arranges the device
differently. That is, when the molecule was found on the left (right) hand
side, i.e., the record is $0$ ($1$), he attaches the end of the string to the
partition from the left (right). In either case, the molecule pushes the
partition which is now movable to the very end of the cylinder.
Step 4 (SzE)
In order to go back to the initial configuration, the demon disconnects the
cylinder from the attached device.
Step 5 (DM)
The demon repeats Steps from 1 to 4 for $N$ times, keeping the $N$-bit string
in his memory. Then, he compresses the $N$-bit string to the minimum length
$NH(p)$, according to Shannon’s noiseless coding theorem. We break up the
$N=mn$ bit string into $m$ blocks of $n$ bits.
A brief description of the data compression is the following.
First, punctuate the $N$-bit string by $n$ bits so that we have $2^{n}$
sequences with relative frequencies $p^{n},(1-p)p^{n-1},\dots,(1-p)^{n}$ so
that we can encode the sequences into strings of bit length of
$-\log_{2}p^{n},-\log_{2}(1-p)p^{n-1},\dots,-\log_{2}(1-p)^{n}$, if they were
integers. Roughly speaking, the average bit length would be
$\displaystyle\sim\sum^{n}_{k=0}{n\choose
k}\left\\{-(1-p)^{k}p^{n-k}\log_{2}(1-p)^{k}p^{n-k}\right\\}$
$\displaystyle=nH(p).$ (7)
The right hand side of Eq. (7) coincides with the shortest average bit length
shown by Shannon shannon .
Step 6 (DM)
The demon repeats the process of $n$ turns $m$ times so that the total amount
of bits to be erased is
$\tilde{S}\approx mnH(p)=NH(p).$ (8)
Figure 3: The protocol of data compression for demon’s memory. The dashed
blocks express the trivial initial memory state “0” after data compression.
The change in thermodynamic entropy is calculated in the same manner as in the
above case of symmetric engine. The volume of the gas after Step 2 becomes
$pV_{0}$ with probability $p$ or $(1-p)V_{0}$ with probability $1-p$. The
entropy of the gas after Step 2 is thus decreased by $-p\ln p-(1-p)\ln(1-p)$,
which is equal to $S(p)$ in Eq. (4) and is to be canceled out by the later
steps of information erasure and memory resetting.
Let us treat Steps 5 and 6 more precisely. According to the Shannon source
coding theorem for the symbol codes shannon , for $n$-bit string, there always
exists an optimal code such that the averaged code length $\bar{S}$ satisfies
$nH(p)\leq\bar{S}<nH(p)+1$. The well-known example of optimal codes is the
Huffman code for the encoding procedure, see Ref. huffman . It is reminded
that the demon breaks up the $N$-bit string into $n$-bit block, i.e.,
$N=nm+\delta$, where $0\leq\delta<n$. The extra $\delta$-bit string cannot be
encoded. When the demon optimally encodes an $N$-bit string, its length is
longer than $NH(p)$ bits by $m+\delta$ bits at worst. As the discrepancy is
bounded as
$m+\delta=N-(n-1)m\geq N-\frac{(n-1)^{2}+m^{2}}{2},$ (9)
it can be minimized when $n-1=m$. Thus, we obtain $n=m={\cal O}(\sqrt{N})$.
The extra bits are at worst $\delta={\cal O}(\sqrt{N})$. Therefore, we can
more precisely express the average length of the optimally compressed data
string as
$\tilde{S}=mnH(p)+\delta=NH(p)+{\cal O}(\sqrt{N}).$ (10)
The averaged work necessary to erase information in this string over $N$ bits
is
$\displaystyle W(erasure)$ $\displaystyle=\frac{k_{B}T\ln 2\cdot\tilde{S}}{N}$
$\displaystyle=k_{B}T\ln 2\cdot H(p)+{\cal O}\left(\frac{1}{\sqrt{N}}\right).$
(11)
On the other hand, in Step 3, the amount of average work extracted by the
engine over $N$ cycles is given by
$W(engine)=k_{B}TS(p)+{\cal O}\left(\frac{1}{\sqrt{N}}\right),$ (12)
where $S(p)=\ln 2\cdot H(p)$ as can be seen by applying the combined gas law.
It is reminded that $S(p)$ is the thermodynamic entropy as discussed before.
It is now clear that the work for information erasure of demons’s memory (11)
and the work from the asymmetric Szilard engine (12) agree for sufficiently
large $N$. The above argument leads us to conclude that the information
theoretical entropy is equivalent to the thermodynamic entropy when optimal
information processing is physically executed.
Note that, for the symmetric Szilard engine, we do not need to compress data
because the number of $0$’s and $1$’s are equal. Also, the erasure model for
non-equiprobability distribution of the memory was considered in a simple
thermodynamic process maruyama09 ; maruyama_phd . The amount of work for this
process coincides with the optimal one (11).
## IV Another heat bath at a lower temperature
The reader might question why the demon resets his memory at the same
temperature (say, $T_{H}$) as the heat bath for the Szilard engine and wonder
what if the erasure is executed at a lower temperature, $T_{L}(<T_{H})$,
because the compression of the memory space would then require less work. With
two heat baths of different temperatures, some nonzero work $W$ can indeed be
extracted, however, the amount of entropy increase is always larger than or
equal to $k_{B}\ln 2$. That is, in terms of entropy balance there is no
difference from the case with a single heat bath. Hence, the demon’s attempt
to outdo the second law ends up in vain, as we naturally expect. An example of
erasing process with two heat baths that achieves the bound is depicted in
Fig. 4 and explained in its caption.
When the optimality is achieved, the entire compound system, the engine and
the memory, simply works as a single engine; it converts a part of heat
absorbed at $T_{H}$ into the work $W$ and throws the residual energy away to
the heat bath at $T_{L}$. The overall thermal efficiency is equal to
$\eta=W/Q_{H}=1-T_{L}/T_{H}$, where $Q_{H}$ is the amount of heat flowed from
the heat bath at $T_{H}$ to the Szilard engine, thus effectively the same as
the Carnot engine.
Figure 4: (Color online). The $p$-$V$ diagram of an alternative erasing
process with a heat bath of lower temperature. The erasure is realized with
any path from the “$\ast$” state to the “$0$” state, which are denoted by a
diamond $\diamond$ and a circle $\circ$, respectively. The solid black line
represents the standard erasure by an isothermal compression (at temperature
$T_{H}$). Although the demon may want to use a colder heat bath of temperature
$T_{L}$, the entropy increase due to the erasure cannot be smaller than
$k_{B}\ln 2$. The path, consisting of two adiabatic processes (red dashed
lines) and one isothermal compression (blue dot-dashed line), in the figure is
the optimal one in terms of entropy increase and attains the Landauer limit of
$k_{B}\ln 2$. Naturally, the entropy cost is the same even if the temperature
of the memory is always $T_{L}$, while the Szilard engine is operated at
$T_{H}$.
Let us make a remark to avoid a potential confusion. Despite being equivalent
to the Carnot engine, the engine-memory system does not work reversibly in the
context of information erasure. While the system can be run in the reverse
direction, the erased information can never be restored reliably.
## V Summary and Discussions
We have shown in the asymmetric Szilard engine that the work extracted by
Maxwell’s demon is asymptotically canceled out after a large number of cycles
by the work to reset the memory after optimal data compression. We have
described an explicit protocol and shown its optimality by making use of
Shannon’s noiseless coding theorem. The key point is data compression before
information erasure of the memory and this argument makes the seminal work by
Landauer and Bennett more general and precise. The coincidence between
information and thermodynamic entropies is now very clear, thanks to the
demon’s cleverest strategy. As a slight generalization we have also considered
the case of information erasure at lower temperature to see that the
efficiency of the whole system can be only as efficient as the Carnot cycle
and that there is no net gain for the demon.
We would like to stress that the thermo-informational cycle has to be
completed to correctly address the apparent violation of the second law in the
context of Maxwell’s demon. This means that no residual information should be
left outside the engine-demon system after the cycle bennett03 . What makes
the argument of demon important and interesting is this physical loss of
information, otherwise it is merely a sequence of normal measurements.
As briefly remarked in the introduction, the present work in the specific
model suggests a general method to relate information and physical entropies
by considering an operational process to erase information in the physical
system. One such example is the original derivation of the black hole entropy
by Beckenstein beckenstein . He considered a gedanken experiment of dropping a
“particle of one bit” for information erasure which increases the area of the
event horizon as a back action. He identified the amount of information loss
with the change of the intrinsic entropy of black hole. Also, there is a well-
known derivation of the Boltzmann distribution on the basis of the principle
of the maximum Shannon entropy under the energy constraint Jaynes . However,
the physical meaning of the Shannon entropy there is not clear, though the
optimal value coincides with the thermodynamic entropy. It would be nice if we
could clarify the meaning of the maximization of the Shannon entropy in terms
of the optimal memory reset.
The optimal information erasure would help us fully understand physical
entropy in terms of information entropy, as Brillouin envisioned Brillouin .
## Acknowledgments
The authors would like to thank Haruka Kibe for her contribution in the early
stage of the present investigation and Charles Bennett for useful discussion.
The authors (A.H. and Y.S.) are supported by the Global Center of Excellence
Program “Nanoscience and Quantum Physics” at Tokyo Institute of Technology.
K.M. is supported by Grant-in-Aid for Scientific Research (C) (No. 22540405).
Y.S. is also supported by JSPS (Grant No. 21008624).
## References
* (1) L. Brillouin, J. Appl. Phys. 22, 334 (1951).
* (2) L. Brillouin, Science and Information Theory (Dover, Mineola, N.Y., [1956, 1962] 2004).
* (3) R. Landauer, IBM J. Res. Dev. 5, 183 (1961).
* (4) J. C. Maxwell, Theory of Heat (Longmans, Green, London) pp. 308–309 (1871).
* (5) H. S. Leff and A. F. Rex, Maxwell’s Demon 2 (IOP, Bristol, 2003).
* (6) K. Maruyama, F. Nori, and V. Vedral, Rev. Mod. Phys. 81, 1 (2009).
* (7) L. Szilard, Z. Phys. 53, 840 (1929).
* (8) C. H. Bennett, Int. J. Theor. Phys. 21, 905 (1982).
* (9) K. Shizume, Phys. Rev. E 52, 3495 (1995).
* (10) B. Piechocinska, Phys. Rev. A 61, 062314 (2000).
* (11) Feynman defined the amount of information by Eq. (4) instead of the equivalence between Eqs. (1) and (4) in Ch. 5, R. P. Feynman, Feynman Lectures on Computation, edited by A. J. G. Hey and R. W. Allen (Perseus, Cambridge, MA, 1999).
* (12) C. E. Shannon, Bell System Technical Journal 27, 379 (1948); 623 (1948).
* (13) C. H. Bennett, IBM J. Res. Dev. 17, 525 (1973).
* (14) M. M. Barkeshli, arXiv:cond-mat/0504323v3.
* (15) T. Sagawa and M. Ueda, Phys. Rev. Lett. 102, 250602 (2009).
* (16) D. Huffman, Proc. of IRE 40, 1098 (1952); see more details in T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithm (MIT Press, Cambridge, USA, 2001).
* (17) K. Maruyama, Ph.D. thesis, Imperial College London (2004).
* (18) C. H. Bennett, Stud. Hist. Philos. Mod. Phys. 34, 501 (2003).
* (19) J. D. Beckenstein, Phys. Rev. D 7, 2333 (1973).
* (20) E. T. Jaynes, Phys. Rev. 106, 620 (1957).
|
arxiv-papers
| 2011-10-21T09:03:30 |
2024-09-04T02:49:23.454085
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Akio Hosoya, Koji Maruyama, Yutaka Shikano",
"submitter": "Yutaka Shikano",
"url": "https://arxiv.org/abs/1110.4732"
}
|
1110.4779
|
# Conditions of coincidence of central extensions of von Neumann algebras and
algebras of measurable operators
S. Albeverio1, K. K. Kudaybergenov2 and R. T. Djumamuratov3 1 Institut für
Angewandte Mathematik and HCM, Rheinische Friedrich-Wilhelms-Universität Bonn,
Endenicher Allee 60, D-53115 Bonn, Germany albeverio@uni-bonn.de 2 Department
of Mathematics, Karakalpak state university
Ch. Abdirov 1, 230113, Nukus, Uzbekistan
karim2006@mail.ru 3 Department of Mathematics, Karakalpak state university
Ch. Abdirov 1, 230113, Nukus, Uzbekistan
rauazh@mail.ru
###### Abstract.
Given a von Neumann algebra $M$ we consider the central extension $E(M)$ of
$M.$ We describe class of von Neumann algebras $M$ for which the algebra
$E(M)$ coincides with the algebra $S(M)$ – the algebra of all measurable
operators with respect to $M,$ and with $S(M,\tau)$ – the algebra of all
$\tau$-measurable operators with respect to $M.$
###### Key words and phrases:
von Neumann algebras, measurable operators, central extensions
###### 2000 Mathematics Subject Classification:
46L51, 46L10
## 1\. Introduction
In the series of paper [1, 3, 2, 4] we have considered derivations on the
algebra $LS(M)$ of locally measurable operators affiliated with a von Neumann
algebra $M,$ and on various subalgebras of $LS(M).$ A complete description of
derivations has been obtained in the case of von Neumann algebras of type I
and III. A comprehensive survey of recent results concerning derivations on
various algebras of unbounded operators affiliated with von Neumann algebras
is presented in [4]. The general form of automorphisms on the algebra $LS(M)$
in the case of von Neumann algebras of type I has been obtained in [2]. In the
proof of the main results of the above papers the crucial role is played by
the central extensions of von Neumann algebras and also by various topologies
considered in [3].
Let $M$ be an arbitrary von Neumann algebra with the center $Z(M)$ and let
$LS(M)$ denote the algebra of all locally measurable operators with respect
$M.$ We consider the set $E(M)$ of all elements $x$ from $LS(M)$ for which
there exists a sequence of mutually orthogonal central projections
$\\{z_{i}\\}_{i\in I}$ in $M$ with $\bigvee\limits_{i\in I}z_{i}=\textbf{1},$
such that $z_{i}x\in M$ for all $i\in I.$ It is known [3] that $E(M)$ is a
*-subalgebra in $LS(M)$ with the center $S(Z(M)),$ where $S(Z(M))$ is the
algebra of all measurable operators with respect to $Z(M),$ moreover,
$LS(M)=E(M)$ if and only if $M$ does not have direct summands of type II. A
similar notion (i.e. the algebra $E(\mathcal{A})$) for arbitrary *-subalgebras
$\mathcal{A}\subset LS(M)$ was independently introduced by M.A. Muratov and
V.I. Chilin [8]. The algebra $E(M)$ is called the central extension of $M.$
In section 2 we recall the notions of the algebras $S(M)$ of measurable
operators and $LS(M)$ of locally measurable operators affiliated with a von
Neumann algebra $M.$ We also consider the central extension $E(M)$ of the von
Neumann algebra $M.$
In section 3 we describe the classes of von Neumann algebras $M$ for which the
algebra $E(M)$ coincides with the algebras $S(M),S(M,\tau)$ and $M.$
## 2\. Central extensions of von Neumann algebras
In this section we recall the notions of the algebras $S(M)$ of measurable
operators and respectively $LS(M)$ of locally measurable operators affiliated
with a von Neumann algebra $M.$ We also consider the central extension $E(M)$
of the von Neumann algebra $M.$
Let $H$ be a complex Hilbert space and let $B(H)$ be the algebra of all
bounded linear operators on $H.$ Consider a von Neumann algebra $M$ in $B(H)$
with the operator norm $\|\cdot\|_{M}.$ Denote by $P(M)$ the lattice of
projections in $M.$
A linear subspace $\mathcal{D}$ in $H$ is said to be _affiliated_ with $M$
(denoted as $\mathcal{D}\eta M$), if $u(\mathcal{D})\subset\mathcal{D}$ for
every unitary $u$ from the commutant
$M^{\prime}=\\{y\in B(H):xy=yx,\,\forall x\in M\\}$
of the von Neumann algebra $M.$
A linear operator $x$ on $H$ with the domain $\mathcal{D}(x)$ is said to be
_affiliated_ with $M$ (denoted as $x\eta M$) if $\mathcal{D}(x)\eta M$ and
$u(x(\xi))=x(u(\xi))$ for all $\xi\in\mathcal{D}(x)$ and for every unitary
$u\in M^{\prime}.$
A linear subspace $\mathcal{D}$ in $H$ is said to be _strongly dense_ in $H$
with respect to the von Neumann algebra $M,$ if
1) $\mathcal{D}\eta M;$
2) there exists a sequence of projections $\\{p_{n}\\}_{n=1}^{\infty}$ in
$P(M)$ such that $p_{n}\uparrow\textbf{1},$ $p_{n}(H)\subset\mathcal{D}$ and
$p^{\perp}_{n}=\textbf{1}-p_{n}$ is finite in $M$ for all $n\in\mathbb{N},$
where 1 is the identity in $M.$
A closed linear operator $x$ acting in the Hilbert space $H$ is said to be
_measurable_ with respect to the von Neumann algebra $M,$ if $x\eta M$ and
$\mathcal{D}(x)$ is strongly dense in $H.$
Denote by $S(M)$ the set of all linear operators on $H,$ which are measurable
with respect to the von Neumann algebra $M.$ If $x\in S(M),$
$\lambda\in\mathbb{C},$ where $\mathbb{C}$ is the field of complex numbers,
then $\lambda x\in S(M)$ and the operator $x^{\ast},$ adjoint to $x,$ is also
measurable with respect to $M$ (see [10]). Moreover, if $x,y\in S(M),$ then
the operators $x+y$ and $xy$ are defined on dense subspaces and admit closures
that are called, correspondingly, the strong sum and the strong product of the
operators $x$ and $y,$ and are denoted by $x\stackrel{{\scriptstyle.}}{{+}}y$
and $x\ast y,$ respectively. It was shown in [10] that
$x\stackrel{{\scriptstyle.}}{{+}}y$ and $x\ast y$ belong to $S(M)$ and these
algebraic operations make $S(M)$ a $\ast$-algebra with the identity 1 over the
field $\mathbb{C}.$ Here, $M$ is a $\ast$-subalgebra of $S(M).$ In what
follows, the strong sum and the strong product of operators $x$ and $y$ will
be denoted in the same way as the usual operations, by $x+y$ and $xy,$
respectively.
A closed linear operator $x$ in $H$ is said to be _locally measurable_ with
respect to the von Neumann algebra $M,$ if $x\eta M$ and there exists a
sequence $\\{z_{n}\\}_{n=1}^{\infty}$ of central projections in $M$ such that
$z_{n}\uparrow\textbf{1}$ and $z_{n}x\in S(M)$ for all $n\in\mathbb{N}$ (see
[11]).
Denote by $LS(M)$ the set of all linear operators that are locally measurable
with respect to $M.$ It was proved in [11] that $LS(M)$ is a $\ast$-algebra
over the field $\mathbb{C}$ with identity $\textbf{1},$ the operations of
strong addition, strong multiplication, and passing to the adjoint. In such a
case, $S(M)$ is a $\ast$-subalgebra in $LS(M).$ In the case where $M$ is a
finite von Neumann algebra or a factor, the algebras $S(M)$ and $LS(M)$
coincide. This is not true in the general case. In [7] the class of von
Neumann algebras $M$ has been described for which the algebras $LS(M)$ and
$S(M)$ coincide.
Let $\tau$ be a faithful normal semi-finite trace on $M.$ We recall that a
closed linear operator $x$ is said to be $\tau$-measurable with respect to the
von Neumann algebra $M,$ if $x\eta M$ and $\mathcal{D}(x)$ is $\tau$-dense in
$H,$ i.e. $\mathcal{D}(x)\eta M$ and given $\varepsilon>0$ there exists a
projection $p\in M$ such that $p(H)\subset\mathcal{D}(x)$ and
$\tau(p^{\perp})<\varepsilon.$ Denote by $S(M,\tau)$ the set of all
$\tau$-measurable operators with respect to $M$ (see [9]).
It is well-known that $S(M,\tau)$ is a $\ast$-subalgebras in $LS(M)$ (see
[9]).
Consider the topology $t_{\tau}$ of convergence in measure or measure topology
on $S(M,\tau),$ which is defined by the following neighborhoods of zero:
$V(\varepsilon,\delta)=\\{x\in S(M,\tau):\exists\,e\in
P(M),\tau(e^{\perp})\leq\delta,xe\in M,\|xe\|_{M}\leq\varepsilon\\},$
where $\varepsilon,\delta$ are positive numbers, and $\|\cdot\|_{M}$ denotes
the operator norm on $M$.
It is well-known [9]) that $S(M,\tau)$ equipped with the measure topology is a
complete metrizable topological $\ast$-algebra.
Let $(\Omega,\Sigma,\mu)$ be a measure space and suppose that the measure
$\mu$ has the direct sum property, i.e. there is a family
$\\{\Omega_{i}\\}_{i\in J}\subset\Sigma,$ $0<\mu(\Omega_{i})<\infty,\,i\in J,$
such that for any $A\in\Sigma,$ $\mu(A)<\infty,$ there exist a countable
subset $J_{0}\subset J$ and a set $B$ with zero measure such that
$A=\bigcup\limits_{i\in J_{0}}(A\cap\Omega_{i})\cup B.$
It is well-known (see e.g. [10]) that every commutative von Neumann algebra
$M$ is $\ast$-isomorphic to the algebra $L^{\infty}(\Omega,\Sigma,\mu)$ of all
(equivalence classes of) complex essentially bounded measurable functions on
$(\Omega,\Sigma,\mu)$ and in this case $LS(M)=S(M)\cong
L^{0}(\Omega,\Sigma,\mu),$ where $L^{0}(\Omega,\Sigma,\mu)$ the algebra of all
(equivalence classes of) complex measurable functions on
$(\Omega,\Sigma,\mu).$
Further we consider the algebra $S(Z(M))$ of operators which are measurable
with respect to the center $Z(M)$ of the von Neumann algebra $M.$ Since $Z(M)$
is an abelian von Neumann algebra it is $\ast$-isomorphic to
$L^{\infty}(\Omega,\Sigma,\mu)$ for an appropriate measure space
$(\Omega,\Sigma,\mu)$. Therefore the algebra $S(Z(M))$ coincides with
$Z(LS(M))$ and can be identified with the algebra $L^{0}(\Omega,\Sigma,\mu).$
The basis of neighborhoods of zero in the topology of convergence locally in
measure on $L^{0}(\Omega,\Sigma,\mu)$ consists of the sets
$W(A,\varepsilon,\delta)=\\{f\in L^{0}(\Omega,\Sigma,\mu):\exists
B\in\Sigma,\,B\subseteq A,\,\mu(A\setminus B)\leq\delta,$ $f\cdot\chi_{B}\in
L^{\infty}(\Omega,\Sigma,\mu),\,||f\cdot\chi_{B}||_{L^{\infty}(\Omega,\Sigma,\mu)}\leq\varepsilon\\},$
where $\varepsilon,\delta>0,\,A\in\Sigma,\,\mu(A)<+\infty,$ and $\chi_{B}$ is
the characteric function of the set $B\in\Sigma.$
Let us recall the definition of the dimension functions $d$ on the lattice
$P(M)$ of projection from $M$ (see [6], [10]).
Let $L_{+}$ denote the set of all measurable functions
$f:(\Omega,\Sigma,\mu)\rightarrow[0,{\infty}]$ (modulo functions equal to zero
$\mu$-almost everywhere).
Let $M$ be an arbitrary von Neumann algebra with the center $Z(M)\equiv
L^{\infty}(\Omega,\Sigma,\mu).$ Then there exists a map $d:P(M)\rightarrow
L_{+}$ with the following properties:
(i) $d(e)$ is a finite function if only if the projection $e$ is finite;
(ii) $d(e+q)=d(e)+d(q)$ for $p,q\in P(M),$ $eq=0;$
(iii) $d(uu^{*})=d(u^{*}u)$ for every partial isometry $u\in M;$
(iv) $d(ze)=zd(e)$ for all $z\in P(Z(M)),\,\,e\in P(M);$
(v) if $\\{e_{\alpha}\\}_{\alpha\in J},\,\,\,e\in P(M)$ and
$e_{\alpha}\uparrow e,$ then $d(e)=\sup\limits_{\alpha\in J}d(e_{\alpha}).$
This map $d:P(M)\rightarrow L_{+},$ is a called the _dimension functions_ on
$P(M).$
The basis of neighborhoods of zero in the topology $t(M)$ of _convergence
locally in measure_ on $LS(M)$ consists (in the above notations) of the
following sets
$\displaystyle V(A,\varepsilon,\delta)=\\{x\in LS(M):\exists p\in
P(M),\,\exists z\in P(Z(M)),\,xp\in M,$
$\displaystyle||xp||_{M}\leq\varepsilon,\,\,z^{\bot}\in
W(A,\varepsilon,\delta),\,\,d(zp^{\bot})\leq\varepsilon z\\},$
where $\varepsilon,\delta>0,\,A\in\Sigma,\,\mu(A)<+\infty$ (see [11]).
The topology $t(M)$ is metrizable if and only if the center $Z(M)$ is
$\sigma$-finite (see [6]).
Given an arbitrary family $\\{z_{i}\\}_{i\in I}$ of mutually orthogonal
central projections in $M$ with $\bigvee\limits_{i\in I}z_{i}=\textbf{1}$ and
a family of elements $\\{x_{i}\\}_{i\in I}$ in $LS(M)$ there exists a unique
element $x\in LS(M)$ such that $z_{i}x=z_{i}x_{i}$ for all $i\in I.$ This
element is denoted by $x=\sum\limits_{i\in I}z_{i}x_{i}.$
We denote by $E(M)$ the set of all elements $x$ from $LS(M)$ for which there
exists a sequence of mutually orthogonal central projections
$\\{z_{i}\\}_{i\in I}$ in $M$ with $\bigvee\limits_{i\in I}z_{i}=\textbf{1},$
such that $z_{i}x\in M$ for all $i\in I,$ i.e.
$E(M)=\\{x\in LS(M):\exists z_{i}\in P(Z(M)),z_{i}z_{j}=0,i\neq
j,\bigvee\limits_{i\in I}z_{i}=\textbf{1},z_{i}x\in M,i\in I\\},$
where $Z(M)$ is the center of $M.$
It is known [3] that $E(M)$ is *-subalgebras in $LS(M)$ with the center
$S(Z(M)),$ where $S(Z(M))$ is the algebra of all measurable operators with
respect to $Z(M),$ moreover, $LS(M)=E(M)$ if and only if $M$ does not have
direct summands of type II.
A similar notion (i.e. the algebra $E(\mathcal{A})$) for arbitrary
*-subalgebras $\mathcal{A}\subset LS(M)$ was independently introduced recently
by M.A. Muratov and V.I. Chilin [8]. The algebra $E(M)$ is called the central
extension of $M.$
It is known [3], [8] that an element $x\in LS(M)$ belongs to $E(M)$ if and
only if there exists $f\in S(Z(M))$ such that $|x|\leq f.$ Therefore for each
$x\in E(M)$ one can define the following vector-valued norm
$||x||=\inf\\{f\in S(Z(M)):|x|\leq f\\}$
and this norm satisfies the following conditions:
$1)\|x\|\geq 0;\|x\|=0\Longleftrightarrow x=0;$
$2)\|fx\|=|f|\|x\|;$
$3)\|x+y\|\leq\|x\|+\|y\|;$
$4)||xy||\leq||x||||y||;$
$5)||xx^{\ast}||=||x||^{2}$
for all $x,y\in E(M),f\in S(Z(M)).$
Let $M$ be an arbitrary von Neumann algebra with the center $Z(M)\equiv
L^{\infty}(\Omega,\Sigma,\mu).$ On the space $E(M)$ we consider the following
sets:
$O(A,\varepsilon,\delta)=\left\\{x\in E(M):||x||\in
W(A,\varepsilon,\delta)\right\\},$
where $\varepsilon,\delta>0,\,\,\,A\in\sum,\,\,\,\mu\left(A\right)<+\infty$.
The system of sets
$\\{x+O(A,\varepsilon,\delta)\\},$ (2.1)
where $x\in E(M),\varepsilon>0,\delta>0,A\in\Sigma,\mu(A)<\infty$, defines on
$E(M)$ a Hausdorff vector topology $t_{c}(M),$ for which the sets (2.1) form
the base of neighborhoods of the element $x\in E(M).$ Moreover in this
topology the involution is continuous and the multiplication is jointly
continuous, i.e. $(E(M),t_{c}(M))$ is a topological $\ast$-algebra. It is
known [5, Proposition 3.2] that $(E(M),t_{c}(M))$ is a complete topological
$\ast$-algebra and $M$ is a $t_{c}(M)$-dense in $E(M).$
## 3\. Conditions of coincidence of central extensions of von Neumann
algebras and algebras of measurable operators
In this section we describe class of von Neumann algebras $M$ for which the
algebra $E(M)$ coincide with algebra $S(M),S(M,\tau)$ and $M.$
It should be noted that [3, Proposition 1.1] and [5, Theorem 3.1] imply the
following result.
###### Theorem 3.1.
The following conditions on a given von Neumann algebra $M$ are equivalent:
(1) $E(M)=LS(M);$
(2) $M$ does not have direct summands of type II.
In this case the topologies $t_{c}(M)$ and $t(M)$ coincide.
Now we describe a class of von Neumann algebras $M$ for which the algebras
$E(M)$ and $S(M)$ coincide.
###### Theorem 3.2.
The following conditions on a given von Neumann algebra $M$ are equivalent:
(1) $E(M)=S(M);$
(2) $M=\bigoplus\limits_{k=0}^{n}M_{k},$ where $M_{0}$ be a von Neumann
algebra of type $I_{fin},$ $M_{k}$ be a factors of type $I_{\infty}$ or III,
$k=\overline{1,n}.$
In this case the topologies $t_{c}(M)$ and $t(M)$ coincide.
For the proof of Theorem 3.2 we need following result.
###### Lemma 3.3.
If $M$ be a von Neumann algebra of type II with a faithful normal semifinite
trace $\tau,$ then $E(M)\neq S(M,\tau)$ and $E(M)\neq S(M).$
###### Proof.
Suppose that $M$ is a type II von Neumann algebra. First assume that $M$ is of
type II1 and admits a faithful normal tracial state $\tau$ on $M.$ Without
loss generality we assume that $\tau(\textbf{1})=1.$ Let $\Phi$ be the
canonical center-valued trace on $M.$ Since $M$ is of type II, there exists a
projection $p_{1}\in M$ such that
$p_{1}\sim\textbf{1}-p_{1}.$
Then $\Phi(p_{1})=\Phi(p_{1}^{\perp}).$ From
$\Phi(p_{1})+\Phi(p_{1}^{\perp})=\Phi(\textbf{1})=\textbf{1}$ it follows that
$\Phi(p_{1})=\Phi(p_{1}^{\perp})=\frac{\textstyle 1}{\textstyle 2}\textbf{1}.$
Suppose that there exist mutually orthogonal projections
$p_{1},\,p_{2},\cdots,p_{n}$ in $M$ such that
$\Phi(p_{k})=\frac{\textstyle 1}{\textstyle
2^{k}}\textbf{1},\,k=\overline{1,n}.$
Set $e_{n}=\sum\limits_{k=1}^{n}p_{k}.$ Then
$\Phi(e_{n}^{\perp})=\frac{\textstyle 1}{\textstyle 2^{n}}\textbf{1}.$ Take a
projection $p_{n+1}<e_{n}^{\perp}$ such that
$p_{n+1}\sim e_{n}^{\perp}-p_{n+1}.$
Then
$\Phi(p_{n+1})=\frac{\textstyle 1}{\textstyle 2^{n+1}}.$
Hence there exists a sequence a mutually orthogonal projections
$\\{p_{n}\\}_{n\in\mathbb{N}}$ in $M$ such that
$\Phi(p_{n})=\frac{\textstyle 1}{\textstyle
2^{n}}\textbf{1},\,n\in\mathbb{N}.$
Note that $\tau(p_{n})=\frac{\textstyle 1}{\textstyle 2^{n}}.$ Indeed
$\tau(p_{n})=\tau(\Phi(p_{n}))=\tau\left(\frac{\textstyle 1}{\textstyle
2^{n}}\textbf{1}\right)=\frac{\textstyle 1}{\textstyle 2^{n}}.$
Since
$\sum\limits_{n=1}^{\infty}n\tau(p_{n})=\sum\limits_{n=1}^{\infty}\frac{n}{2^{n}}<+\infty$
it follows that the series
$\sum\limits_{n=1}^{\infty}np_{n}$
converges in measure in $S(M,\tau).$ Therefore there exists
$x=\sum\limits_{n=1}^{\infty}np_{n}\in S(M,\tau).$
Let us show that $x\in S(M,\tau)\setminus E(M).$ Suppose that $zx\in M,$ where
$z$ is a nonzero central projection. Since any $p_{n}$ is a faithful
projection we have that $zp_{n}\neq 0$ for all $n.$ Thus
$||zx||_{M}=1||zx||_{M}1=||p_{n}||_{M}\cdot||zx||_{M}\cdot||p_{n}||_{M}\geq||zp_{n}xp_{n}||_{M}=||zp_{n}n||_{M}=n,$
i.e.
$||zx||_{M}\geq n$
for all $n\in\mathbb{N}.$ From this contradiction it follows that $x\in
S(M,\tau)\setminus E(M).$
For a general type II von Neumann algebra $M$ take a non zero finite
projection $e\in M$ and consider the finite type II von Neumann algebra $eMe$
which admits a separating family of normal tracial states. Now if $f\in eMe$
is the support projection of some tracial state $\tau$ on $eMe$ then $fMf$ is
a type II1 von Neumann algebra with a faithful normal tracial state. Hence as
above one can construct an element $x\in S(M,\tau)\setminus E(M),$ moreover
$x\in S(M)\setminus E(M).$ The proof is complete. ∎
The proof of the theorem 3.2. (1) $\Rightarrow$ (2). If the algebra $M$ has a
direct summand of type II, then by Lemma 3.3 we have that $E(M)\neq S(M).$
Hence if $E(M)=S(M),$ then there exist mutually orthogonal central projections
$z_{0},z_{1},z_{2}$ with $z_{0}+z_{1}+z_{2}=\textbf{1}$ such that $z_{0}M$ is
a type I${}_{fin},$ $z_{1}M$ is a type I∞ and $z_{2}M$ is a type III.
Suppose that $zZ(M)$ is infinite dimensional, where $Z(M)$ is the center $M$
and $z=z_{1}+z_{2}.$ Then there exists a sequence of nonzero mutually
orthogonal projections $\\{p_{n}\\}_{n=1}^{\infty}$ in $zZ(M).$ Put
$x=\sum\limits_{n=1}^{\infty}np_{n}.$ (3.1)
Then $0\leq x\in E(M)$ and $e_{n}(x)=\sum\limits_{k=1}^{n}p_{k},$ where
$e_{n}(x)$ is a spectral projection $x$ corresponding to the interval $[0,n].$
Since $zM$ is a properly infinite algebra, then
$e_{n}(x)^{\perp}=\sum\limits_{k=n+1}^{\infty}p_{k}$ is an infinite projection
for all $n\in\mathbb{N}.$ This means that $x\notin S(M),$ i.e. $E(M)\neq
S(M).$ This is a contradiction with $E(M)=S(M).$ Therefore $zZ(M)$ is a finite
dimensional algebra. Thus
$zM=\bigoplus\limits_{k=1}^{n}M_{k},$
where $M_{k}$ is a factor of type I∞ or III, $k=\overline{1,n}.$ Put
$M_{0}=z_{0}M.$ Then
$M=z_{0}M\oplus zM=\bigoplus\limits_{k=0}^{n}M_{k},$
where $M_{0}$ is a type I${}_{fin},$ $M_{k}$ is a factor of type I∞ or III,
$k=\overline{1,n}.$
(2) $\Rightarrow$ (1) Let
$M=\bigoplus\limits_{k=0}^{n}M_{k},$
where $M_{0}$ be a type I${}_{fin},$ $M_{k}$ be a factor of type I∞ or III,
$k=\overline{1,n}.$ Since $M_{0}$ is a type I${}_{fin},$ then from [3,
Proposition 1.1] it follows that
$E(M_{0})=LS(M_{0})=S(M_{0}).$
Since $M_{k}$ are factors of type I∞ or III, $k=\overline{1,n},$ then by [7,
Theorem 1] we obtain that
$S(M_{k})=M_{k}=E(M_{k}),$
for all $k=\overline{1,n}.$ Hence
$E(M)=\bigoplus\limits_{k=0}^{n}E(M_{k})=\bigoplus\limits_{k=0}^{n}S(M_{k})=S(M),$
i.e. $E(M)=S(M).$
Now let $M=\bigoplus\limits_{k=0}^{n}M_{k},$ where $M_{0}$ is a von Neumann
algebra of type I${}_{fin},$ $M_{k}$ are factors of type I∞ or III,
$k=\overline{1,n}.$ Then $LS(M)=S(M)=E(M)$ and $M$ does not have direct
summands of type II. Therefore from Theorem 3.1 we obtain that the topologies
$t(M)$ and $t_{c}(M)$ coincide. The proof is complete. $\Box$
We now describe a class of von Neumann algebras $M$ for which the algebras
$E(M)$ and $S(M,\tau)$ coincide.
###### Theorem 3.4.
Let $M$ be a von Neumann algebra with a faithful normal semifinite trace
$\tau.$ The following conditions are equivalent:
(1) $E(M)=S(M,\tau);$
(2) $M=\bigoplus\limits_{k=0}^{n}M_{k},$ where $M_{0}$ be a type $I_{fin}$
algebra, $M_{k}$ be a factors of type $I_{\infty},$ $k=\overline{1,n},$ and
restriction of trace $\tau$ on $M_{0}$ is finite.
In this case the topologies $t_{c}(M)$ and $t_{\tau}$ coincide.
###### Proof.
(1) $\Rightarrow$ (2). If $M$ has a direct summand of type II, then by Lemma
3.3 we obtain that $E(M)\neq S(M,\tau).$ Hence if $E(M)=S(M,\tau),$ then there
exist orthogonal central projections $z_{0},z_{1}$ with
$z_{0}+z_{1}=\textbf{1}$ such that $z_{0}M$ is a type I${}_{fin},$ $z_{1}M$ is
a type I${}_{\infty}.$
If we assume that $z_{1}Z(M)$ is infinite dimensional then the element $x\in
E(M)$ defined similarly as in (3.1) we have that $x\notin S(M,\tau),$ i.e.
$E(M)\neq S(M,\tau).$ Hence, $z_{1}Z(M)$ is finite dimensional. Thus
$zM=\bigoplus\limits_{k=1}^{n}M_{k},$
where $M_{k}$ is of type I${}_{\infty},$ $k=\overline{1,n}.$ Put
$M_{0}=z_{0}M.$ Then
$M=z_{0}M\oplus z_{1}M=\bigoplus\limits_{k=0}^{n}M_{k},$
where $M_{0}$ is of type I${}_{fin},$ $M_{k}$ are factors of type
I${}_{\infty},$ $k=\overline{1,n}.$ Now by Theorem 3.2 we have that
$E(M_{0})=S(M_{0}).$ At the same time by conditions of theorem it follows that
$E(M_{0})=S(M_{0},\tau_{0}),$ where $\tau_{0}$ is the restriction of $\tau$ on
$M_{0}.$ Thus $S(M_{0})=S(M_{0},\tau_{0}).$ Therefore by [7, Proposition 9]
the restriction of trace $\tau$ on $M_{0}$ is finite.
(2) $\Rightarrow$ (1). Let
$M=\bigoplus\limits_{k=0}^{n}M_{k},$
where $M_{0}$ is an algebra of type I${}_{fin},$ $M_{k}$ are factors of type
I${}_{\infty},$ $k=\overline{1,n}$ and the restriction $\tau$ on $M_{0}$ be a
finite. Let $\tau_{0}$ be the restriction of $\tau$ on $M_{0}.$ Since $M_{0}$
is of type Ifin then by [3, Proposition 1.1] it follows that
$E(M_{0})=LS(M_{0})=S(M_{0}).$
Now since the trace $\tau_{0}$ is a finite then $S(M_{0})=S(M,\tau_{0}).$ Thus
$E(M_{0})=S(M_{0},\tau_{0}).$ Since $M_{k}$ is a factor of type
I${}_{\infty},$ $k=\overline{1,n},$ then from [6, Theorem 2.2.9] we obtain
that $S(M_{k},\tau_{k})=M_{k}=E(M_{k}),$ where $\tau_{k}$ is the restriction
of $\tau$ on $M_{k},$ $k=\overline{1,n}.$ Therefore
$E(M)=\bigoplus\limits_{k=0}^{n}E(M_{k})=\bigoplus\limits_{k=0}^{n}S(M_{k},\tau_{k})=S(M,\tau),$
i.e. $E(M)=S(M,\tau).$
Now let $M=\bigoplus\limits_{k=0}^{n}M_{k},$ where $M_{0}$ is an algebra of
type I${}_{fin},$ $M_{k}$ are factors of type I${}_{\infty},$
$k=\overline{1,n},$ and the restriction $\tau_{0}$ of the trace $\tau$ on
$M_{0}$ is finite. Since the restriction of the trace $\tau$ on $M_{0}$ is
finite then $E(M_{0})=S(M_{0})=S(M_{0},\tau_{0})$ and the restrictions of the
topologies $t_{c}(M)$ and $t_{\tau}$ on $E(M_{0})$ coincide. Further since
$M_{k}$ are factors of type I${}_{\infty},$ $k=\overline{1,n},$ then
$E\left(\bigoplus\limits_{k=1}^{n}M_{k}\right)=S\left(\bigoplus\limits_{k=1}^{n}M_{k},\tau|_{\bigoplus\limits_{k=1}^{n}M_{k}}\right)=\bigoplus\limits_{k=1}^{n}M_{k}$
and the restrictions of the topologies $t_{c}(M)$ and $t_{\tau}$ on
$\bigoplus\limits_{k=1}^{n}M_{k}$ coincide with the uniform topology on
$\bigoplus\limits_{k=1}^{n}M_{k}.$ Therefore the topologies $t_{c}(M)$ and
$t_{\tau}$ coincide. The proof is complete. ∎
Finally we describe a class of von Neumann algebras $M$ for which the algebras
$E(M)$ and $M$ coincide.
###### Theorem 3.5.
Let $M$ be a von Neumann algebra. The following conditions are equivalent:
(1) $E(M)=M;$
(2) $M=\bigoplus\limits_{k=1}^{n}M_{k},$ where $M_{k}$ are von Neumann factors
for all $k=\overline{1,n}.$
In this case the topology $t_{c}(M)$ coincides with uniform topology.
###### Proof.
Let $E(M)=M.$ Suppose that $Z(M)$ is infinite dimensional. Then there exists a
sequence of nonzero central orthogonal projections
$\\{p_{n}\\}_{n=1}^{\infty}$ in $M.$ Put
$x=\sum\limits_{n=1}^{\infty}np_{n}.$
Then $x\in E(M)\setminus M,$ this is contradiction. This means that $Z(M)$ is
finite dimensional. Thus
$M=\bigoplus\limits_{k=1}^{n}M_{k},$
where $M_{k}$ are von Neumann factors for all $k=\overline{1,n}.$
(2) $\Rightarrow$ (1). Let
$M=\bigoplus\limits_{k=1}^{n}M_{k},$
where $M_{k}$ are von Neumann factors for all $k=\overline{1,n}.$ Then by the
definition of central extensions it follows that $E(M_{k})=M_{k}.$ Therefore
$E(M)=\bigoplus\limits_{k=1}^{n}E(M_{k})=\bigoplus\limits_{k=1}^{n}M_{k}=M,$
i.e. $E(M)=M.$
Now let $M=\bigoplus\limits_{k=1}^{n}M_{k},$ where $M_{k}$ are von Neumann
factors for all $k=\overline{1,n}.$ Since $M_{k}$ are von Neumann factors for
all $k=\overline{1,n}$ then the restriction of the topology $t_{c}(M)$ on
$M_{k}$ coincides with the uniform topology. Therefore the topology $t_{c}(M)$
coincides with the uniform topology. The proof is complete. ∎
## Acknowledgments
The second named author would like to acknowledge the hospitality of the
”Institut fur Angewandte Mathematik”, Universitat Bonn (Germany). This work is
supported in part by the DFG AL 214/36-1 project (Germany). This work is
supported in part by the German Academic Exchange Service – DAAD.
## References
* [1] Albeverio S., Ayupov Sh. A., Kudaybergenov K. K., Structure of derivations on various algebras of measurable operators for type I von Neumann algebras, J. Func. Anal., 256 (2009) 2917–2943.
* [2] Albeverio S., Ayupov Sh. A., Kudaybergenov K. K., Djumamuratov R. T., Automorphisms of central extensions of type I von Neumann algebras, http://arxiv.org/abs/1104.4698.
* [3] Ayupov Sh. A., Kudaybergenov K. K., Additive derivations on algebras of measurable operators, ICTP, Preprint, No IC/2009/059, – Trieste, 2009. – 16 p. (accepted in Journal of operator theory).
* [4] Ayupov Sh. A., Kudaybergenov K. K., Derivations on algebras of measurable operators, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 13 (2010) 305–337.
* [5] Ayupov Sh. A., Kudaybergenov K. K., Djumamuratov R. T., Topologies on central extensions of von Neumann algebras, http://arxiv.org/abs/1107.5153.
* [6] Muratov M. A., Chilin V. I., Algebras of measurable and locally measurable operators, Institute of Mathematics Ukrainian Academy of Sciences, Kiev, 2007.
* [7] Muratov M. A., Chilin V. I., $\ast$-Algebras of unbounded operators affiliated with a von Neumann algebra, J. Math. Sciences, 140 (2007), 445–451.
* [8] Muratov M. A., Chilin V. I., Central extensions of *-algebras of measurable operators, Doklady AN Ukraine, 2009, 24–28.
* [9] Nelson E., Notes on non-commutative integration, J. Funct. Anal. 15 (1974) 103–116.
* [10] Segal I., A non-commutative extension of abstract integration, Ann. Math., 57 (1953) 401–457.
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|
arxiv-papers
| 2011-10-21T12:50:24 |
2024-09-04T02:49:23.462482
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. Albeverio, K. K. Kudaybergenov, R. T. Djumamuratov",
"submitter": "Karimbergen Kudaybergenov",
"url": "https://arxiv.org/abs/1110.4779"
}
|
1110.4790
|
# Effect of packing fraction on the jamming of granular flow through small
apertures
Rodolfo Omar Uñac1, Ana María Vidales1 and Luis A. Pugnaloni2 1 Departamento
de Física, Instituto de Física Aplicada (UNSL-CONICET), Universidad Nacional
de San Luis, Ejército de los Andes 950, D5700HHW San Luis, Argentina.
2 Instituto de Física de Líquidos y Sistemas Biológicos (CONICET La Plata,
UNLP), Calle 59 Nro 789, 1900 La Plata, Argentina. runiac@unsl.edu.ar (R O
Uñac), avidales@unsl.edu.ar (A M Vidales), luis@iflysib.unlp.edu.ar (L A
Pugnaloni)
###### Abstract
We investigate the flow and jamming through small apertures of a column of
granular disks via a pseudo-dynamic model. We focus on the effect that the
preparation of the granular assembly has on the size of the avalanches
obtained. Ensembles of packings with different mean packing fractions are
created by tapping the system at different intensities. Surprisingly, packing
fraction is not a good indicator of the ability of the deposit to jam a given
orifice. Different mean avalanche sizes are obtained for deposits with the
same mean packing fraction that were prepared with very different tap
intensities. It has been speculated that the number and size of arches in the
bulk of the granular column should be correlated with the ability of the
system to jam a small opening. We show that this correlation, if exists, is
rather poor. A comparison between bulk arches and jamming arches (i.e., arches
that block the opening) reveals that the aperture imposes a lower cut-off on
the horizontal span of the arches which is greater than the actual size of the
opening. This is related to the fact that blocking arches have to have the
appropriate orientation to fit the gap between two piles of grains resting at
each side of the aperture.
## 1 Introduction
The flow of granular materials through apertures is commonplace in a variety
of industrial applications. Studies in this respect can be separated into two
major areas: (1) continuous flow, and (2) jamming. Continuous granular flow is
observed for dry non-cohesive materials if the size of the aperture is large
(typically above five grain diameters for spherical particles). Jamming is
observed whenever the opening is smaller, which requires the input of external
perturbations in order to restart the flow. The cause of jamming is the
formation of a blocking arch. We distinguish two type of arches: blocking
arches (or jamming arches) and bulk arches. Blocking arches form at the
orifice during discharge and prevent the flow. Bulk arches form at any place
inside the packing during the dynamical process that leads the grains to reach
mechanical equilibrium. Arches are set of particles that are mutually stable.
The removal of any particle in the arch leads to the destabilization of the
others.
A number of studies have considered the jamming of an aperture during the
discharge of grains. These include experimental studies on two-dimensional
(2D) hoppers using circular grains [1, 2] and three-dimensional (3D) silos
using spherical and non-spherical particles [3, 4], numerical simulations
using discrete element methods in 2D [7, 8], experiments with quasi-2D silos
and spherical grains [5], 3D vibrated silos [6] and experiments with tilted
[9] and wedge-shaped hoppers [10]. Also, the properties of blocking arches and
bulk arches have been considered in the past [1, 14, 11, 12, 13, 15]. However,
none of these studies have considered the effect of the preparation of the
granular column prior to the discharge. It is known that the number and size
of arches inside a granular assembly are dependent on the packing fraction.
Therefore, it is expected that the jamming of columns prepared at different
packing fractions may occur with different probability. A related issue is the
question of to what extent the arches formed in the bulk of the system are
comparable with the arches that effectively block the aperture during
drainage.
In this paper we use a 2D pseudo-dynamic simulation scheme previously
developed by Manna and Khakhar [16, 17] to study the effect of the initial
packing fraction on the jamming probability and the correlation between bulk
arches and blocking arches. The jamming probability is directly connected with
the mean size of the avalanches [5]. An avalanche is the flow of grains that
occurs between the initiation of the discharge and the arrest of the flow due
to the formation of a blocking arch [3]. We will show that there is a strong
dependence of the size of the avalanches with the packing fraction. However,
there is not a monotonic relation between these two quantities. We also find
that there is a poor correlation between the size of arches in the bulk and
the size of the avalanches. A comparison between bulk arches and jamming
arches reveals that the aperture not only imposes a cut-off on arches of
horizontal span below the opening size, but also prevents the formation of
some blocking arches that, in principle, are wide enough to induce jamming.
## 2 The pseudo-dynamic algorithm
Our simulations are based on an algorithm for inelastic massless hard disks
designed by Manna and Khakhar [16, 17]. This is a pseudo-dynamics that
consists in small falls and rolls of the grains until they come to rest by
contacting other particles or the system boundaries. We use a container formed
by a flat base and two flat vertical walls. No periodic boundary conditions
are applied.
The deposition algorithm consists in choosing a disk in the system and
allowing a free fall of length $\delta$ if the disk has no supporting
contacts, or a roll of arc-length $\delta$ over its supporting disk if the
disk has one single supporting contact [16, 17, 18]. Disks with two supporting
contacts are considered stable and left in their positions. If in the course
of a fall of length $\delta$ a disk collides with another disk (or the base),
the falling disk is put just in contact and this contact is defined as its
first supporting contact. Analogously, if in the course of a roll of length
$\delta$ a disk collides with another disk (or a wall), the rolling disk is
put just in contact. If the first supporting contact and the second contact
are such that the disk is in a stable position, the second contact is defined
as the second supporting contact; otherwise, the lowest of the two contacting
particle is taken as the first supporting contact of the rolling disk and the
second supporting contact is left undefined. If, during a roll, a particle
reaches a lower position than the supporting particle over which it is
rolling, its first supporting contact is left undefined (in this way the
particle will fall vertically in the next step instead of rolling underneath
the first contact). A moving disk can change the stability state of other
disks supported by it, therefore, this information is updated after each move.
The deposition is over once each particle in the system has both supporting
contacts defined or is in contact with the base (particles at the base are
supported by a single contact). Then, the coordinates of the centers of the
disks and the corresponding labels of the two supporting particles, wall, or
base, are saved for analysis.
Figure 1: (Color online). Sample configurations of the granular column for the
steady state corresponding to $\Gamma=0.39$ before (a) and after (b)
discharging through and opening of width $D=2.75$. The red segments indicate
the arches in the system.
An important point in these simulations is the effect that the parameter
$\delta$ has in the results since particles do not move simultaneously but one
at a time. One might expect that in the limit $\delta\rightarrow 0$ we should
recover a fairly ”realistic” dynamics for fully inelastic non-slipping disk
dragged downwards at constant velocity. This should represent particles
deposited in a viscous medium or carried by a conveyor belt. Although this
dynamics contrasts with the dynamics of dry granulates, experiments on the
jamming of fully submerged grains in gels [19] have shown remarkable
similarities with the more widely available data on dry systems. We chose
$\delta=0.0062d$ (with $d$ the particle diameter) since we have observed that
for smaller values of $\delta$ results are indistinguishable from those
obtained here [18].
The pseudo-dynamics approach has been chosen in view of the low CPU time
demanded by this scheme. In this study, we need to prepare a large number of
samples (through tapping) with a given packing fraction and then trigger
discharges for different aperture sizes. More realistic simulations such as
granular dynamics (or discrete element method) would require a much higher
computational effort. In spite of the simplifications of the pseudo-dynamics,
it has been shown that results on tapping agree qualitatively with granular
dynamics simulations [21].
## 3 Initial packings
In order to study the effect of packing fraction on the jamming of the flow
through an aperture, we first need to prepare packings at reproducible packing
fractions. To achieve this, several techniques can be applied. For example,
the sequential deposition of grains submerged in a viscous liquid yield
reproducible packing fractions that can be tuned by changing the friction
coefficient of the particles or the density mismatch [22]. We have chosen
another well known technique to generate reproducible ensembles of packings:
tapping. Nowak et al. have shown that an appropriate tapping protocol can lead
to reproducible states in the sense that an ensemble of configurations with
well defined mean packing fraction is recovered if the same protocol is
followed irrespective of the initial state [23]. This has been more carefully
discussed by Ribière et al. [20]. Dijksman et al. showed how different states
can be obtained not only by changing the tap intensity but also by changing
the tap duration [24]. A similar effect was investigated by Pica Ciamarra et
al. in submerged samples where a fluid pulse is used to excite the granular
column [25]. Hence, we use a simulated tapping protocol (see below) to
generate sets of initial configurations that have well defined mean packing
fractions.
Figure 2: Mean packing fraction $\phi$ in the steady state of the tapping
protocol as a function of the tap intensity $\Gamma$.
The simulations are carried out in a rectangular box of width $24.78d$
containing $1500$ equal-sized disks of diameter $d$. Initially, disks are
placed at random in the simulation box (with no overlaps) and deposited using
the pseudo-dynamic algorithm. Once all the grains come to rest, the system is
expanded in the vertical direction and randomly shaken to simulate a vertical
tap. Then, a new deposition cycle begins. After many taps of given amplitude,
the system achieves a steady state where all characterizing parameters
fluctuate around equilibrium values independently of the previous history of
the granular bed. The existence of such “equilibrium” states has been
previously reported in experiments [20].
The tapping of the system is simulated by multiplying the vertical coordinate
of each particle by a factor $A$ (with $A>1$). Then, the particles are
subjected to several (about $20$) Monte Carlo loops where positions are
changed by displacing particles a random length $\Delta r$ uniformly
distributed in the range $0<\Delta r<A-1$. New configurations that correspond
to overlaps are rejected. This disordering phase is crucial to avoid particles
falling back again into the same positions. Moreover, the upper limit for
$\Delta r$ (i.e. $A-1$) is deliberately chosen so that a larger tap promotes
larger random changes in the particle positions. The expansion amplitude $A$
ranges from $1.03$ up to $3.0$. Following Refs. [26, 21] we quantify the tap
intensity by the parameter $\Gamma=\sqrt{A-1}$. For each value of $\Gamma$
studied, $10^{3}$ taps are carried out for equilibration followed by $5\times
10^{3}$ taps for production. $500$ deposited configurations are stored which
are obtained by saving every $10$ taps during the production run after
equilibration. These deposits will be used later as initial conditions for the
discharge and flow through an opening.
The deposited configurations are analyzed in search of bulk arches. We first
identify all mutually stable particles —which we define as directly connected—
and then we find the arches as chains of connected particles. Two disks A and
B are mutually stable if A is the left supporting particle of B and B is the
right supporting particle of A, or viceversa. We measure the total number of
arches, arch size distribution $n(k)$, and the horizontal span distribution of
the arches $n_{k}(x)$. The latter is the probability density of finding an
arch consisting of $k$ disks with horizontal span between $x$ and $x+dx$. The
horizontal span (or lateral extension) is defined as the projection onto the
horizontal axis of the segment that joins the centers of the right-end disk
and the left-end disk in the arch. In Fig. 1(a), we show an example of a
deposited configuration with arches indicated by segments (for a description
of Fig. 1(b) see next section). Notice that the pseudo-dynamics mimics the
behavior of disks that roll without slipping. This corresponds to a system
with infinite static friction which is expected to yield a large number and
variety of arches. The arch structure of frictionless systems may differ
significantly from the one seen in Fig. 1(a). However, simulations with
realistic discrete element methods with finite friction yield similar
structures [13].
In Fig. 2, we present the steady state mean packing fraction, $\phi$, of our
granular deposits as a function of $\Gamma$. There exists a rather sharp
decrease of $\phi$ as the tapping intensity is increased followed by a minimum
and a very smooth increase. The sharp drop of $\phi$ is associated to a
discontinuous order-disorder transition previously reported for this model
[18] and also observed in granular dynamics of polygonal grains [27]. The
appearance of the minimum packing fraction as a function of tap intensity has
been reported for several models (including a frustrated lattice gas model
[28], a Monte Carlo type deposition [21] and a realistic discrete element
method simulation [21]) and in experiments of tapping with a quasi-2D system
[30, 29]. For the model we use in this paper, the minimum $\phi$ has been
shown to exist even if bidisperse systems are considered [31].
Figure 3: (Color online). Number of arches per particle (red circles) and mean
size of the arches in terms of the number of grains involved in an arch (black
squares) as a function of $\Gamma$.
The minimum in $\phi$ is caused to a large extent by the formation of arches
[21]. Figure 3 shows the number and mean size of arches found in the system as
a function of $\Gamma$. When $\Gamma$ is increased considerably (above $0.7$),
every tap expands the assembly in such a way that particles get well apart
from each other. During deposition, particles will reach the free surface of
the bed almost sequentially (one at a time), reducing the chances of mutual
stabilization. Therefore, arches are less probable to form as $\Gamma$
increases and so $\phi$ must grow since fewer voids get trapped. Indeed, we
see in Fig. 3 that the number and size of arches decrease at large $\Gamma$
($\gtrsim 0.7$) for increasing tap intensities. Eventually, for very large
$\Gamma$, no arches are formed after each tap and $\phi$ will reach a limiting
value. At lower $\Gamma$ ($0.4\lesssim\Gamma\lesssim 0.7$), the free volume
injection due to a tap creates very narrow gaps between particles. For a given
arch to grow by the insertion of a new particle, it is necessary to create a
gap between two particles in the existing arch where the new particle can fit
in. This explains why increasing $\Gamma$ will promote the formation of larger
arches and reduce $\phi$ in this regime. For very low tapping intensities
($\Gamma\lesssim 0.3$), we find a rather constant $\phi$. However, the number
and size of the arches decrease with $\Gamma$ (see Fig. 3). This would imply
that a maximum in the packing fraction should be observed at such light
tapping. This maximum has been recently reported in other models [27, 32], but
is not present in our pseudo-dynamics. Notice that the number and size of the
arches give only a rough indication for the free volume in the sample since
the actual shape of the arches will also be important.
## 4 Flow and jamming
For each deposit generated as described in the previous section, we trigger a
discharge by opening an aperture of width $D$ relative to the diameter $d$ of
the disk in the center of the base of the containing box. Grains will flow out
of the box following the pseudo-dynamics until a blocking arch forms or until
the entire system is discharged (with the exception of two piles resting on
each side of the aperture). During the dynamics, disks that reach the bottom
and whose centers lie on the interval that defines the opening will fall
vertically (even if the surface of the disk touches the edge of the aperture).
This prevents the formation of arches with end disks sustained by the vertical
edge of the orifice. Although such arrangements happen in real experiments,
they are uncommon [14]. After each discharge, we record the size of the
avalanche (i.e. the number of grains flowed out) and the final arrangement of
the grains left in the box. Averages are taken over 500 discharges for each
value of $\Gamma$ used to prepare the initial packings.
One single discharge attempt is carried out for each initial deposit. This
allows us to assure that the initial preparation of the pack belongs to the
ensemble of deposits corresponding to the steady state of the particular tap
intensity chosen. In many experiments and most industrial applications,
discharges are triggered one after another from the same deposit without
preparing the system in the initial condition again [3, 4, 5, 6]. However,
some experiments do fill the container anew before each discharge [1, 2, 33].
We can see in Fig. 1(b) an image of the system after a discharge that resulted
in a jam. It is clear that the structure of the packing is greatly affected by
the partial discharge in our simulations. Therefore, these final structures
are not used for new discharges.
In Fig. 4 we plot the avalanche size distribution $p(s)$ for a few values of
$\Gamma$ and $D=2.25$. We obtain this by counting the number of grains $s$
that flow in each of the $500$ discharges corresponding to each initial
deposit generated for each $\Gamma$. An exponential tail in $p(s)$ has been
observed in several previous studies, both two-dimensional [5, 2, 8, 7] and
three-dimensional [3, 4]. Manna and Herrmann [34], using the same model, found
avalanches with a power law distribution. Notice however, that in Ref. [34]
the authors trigger one avalanche after the other by simply removing a grain
of the blocking arch. This minute perturbation to trigger avalanches may
induce strong correlations between successive discharges in contrast with the
strong rearrangements induced in most experiments. Based in our limited number
of discharges, we are unable to assert if an exponential or a power-law decay
is at play in our simulations (see log-lin and log-log plots in Fig. 4).
Although we report $p(s)$ up to $s=100$, larger avalanches of up to $1000$
disks are observed, whose statistics can be largely affected by the finite
size of the system ($N=1500$).
Figure 4: (Color online). Avalanche size distribution $p(s)$ for several
values of $\Gamma$ for $D=2.25$. (a) Semilog plot, (b) log-log plot.
The mean avalanche size, $\langle s\rangle$, as a function of $\Gamma$ is
shown in Fig. 5 for various aperture sizes. As it can be expected, $\langle
s\rangle$ increases if $D$ increases. As we can see, for small apertures,
$\langle s\rangle$ grows monotonically as $\Gamma$ increases. However, for
$D>2.0$, the mean avalanche size presents a local maximum and a local minimum
as a function of $\Gamma$.
It has been speculated [35, 11] that the size of the avalanches can be
connected with the arches inside the granular deposit. Although arches in the
initial configuration are not dragged to the aperture during flow since arches
actually break and form all the time in the process, it is believed that the
ability of the system to form arches in the initial deposit is connected with
the ability to form blocking arches during flow. Indeed, the features observed
in Fig. 5 are somewhat correlated with the number and size of the bulk arches.
As $\Gamma$ is increased from the lower values, the size of the arches remains
initially rather constant (whereas the number of arches decreases, Fig. 3).
This reduction in the number of arches leads to a smaller jamming probability
and a rapid increase in the size of the avalanches (see Fig. 5). This regime
ends when the sharp drop in $\phi$ ends ($\Gamma\approx 0.4$). For larger
$\Gamma$ and up to the packings with minimum packing fractions (i.e.
$0.4\lesssim\Gamma\lesssim 0.7$), the number and size of the arches increase.
As a consequence, $\langle s\rangle$ decreases due to the increased likelihood
of jamming. Finally, for $\Gamma\gtrsim 0.7$, the number and size of the
arches fall and a corresponding increase of the avalanche sizes is observed.
From these observations we can assert that the number and size of arches in a
given packing give an overall indication of the chances that the system will
jam if it is left to flow through a small aperture.
Figure 5: (Color online). Mean avalanche size $\langle s\rangle$ as a function
of $\Gamma$ for several sizes of the aperture $D$. Notice that $\langle
s\rangle$ is affected to a large extent by the rare large avalanches not
reported in Fig. 4.
Notwithstanding the previous analysis, the size of the avalanches is not only
dependent on the preparation protocol —defined in our case by $\Gamma$— but
also on the actual size of the outlet imposed. For example, as we mentioned,
for $D=2.0$ the mean avalanche size does not present the maximum and minimum
suggested by the number and size of arches. In order to take into account the
effect of the size of the aperture we measure the probability
$P_{\mathrm{arch}}(D)$ of finding an arch in the bulk of the deposits wide
enough to block a given aperture $D$. We measure the horizontal span of each
arch as the projection on the horizontal axis of the segment that joins the
centers of the end particles of the arch. An arch of span $x$ can jam an
opening of width $x+d$ (with $d$ the diameter of a grain). In this analysis we
include the grains that do not form arches, which can jam any orifice with
$D\leq 1$. In Fig. 6 we plot $P_{\mathrm{arch}}(D)$ as a function of $\Gamma$
and compare with the corresponding $\langle s\rangle$. Overall, the
probability of finding an arch wide enough to block an aperture of size $D$
decreases with $D$ in correspondence with the overall increase of $\langle
s\rangle$. However, for a given $D$, the dependence with $\Gamma$ does not
show a clear anti-correlation between the probability and the mean avalanche
size. This implies that the bulk arches can give only a rough indication of
the eventual size of the avalanches that would discharge if an aperture is
opened.
Figure 6: (Color online). Mean avalanche size $\langle s\rangle$ as a function
of $\Gamma$ and probability $P_{\mathrm{arch}}(D)$ of finding an arch of
horizontal span $D-1$ for several apertures: (a) $D=2.0$, (b) $D=2.25$, (c)
$D=2.5$, (d) $D=2.75$.
It is important to mention that we have always opened the aperture at the
center of the bottom of the container. The ordering observed for $\Gamma<0.5$
(see Fig. 1) suggests that an effect related to the relative position of the
aperture and the first layer of grains may be expected. The main effect would
be related to the fact that particles just at the edges of the orifice do not
flow in the pseudo-dynamics and these will produce a reduced effective
aperture. If all discharges start from an initial packing so ordered that
grains at the first layer always sit on the same horizontal positions, the
mean avalanche size would depend on the horizontal position of the aperture.
When this happens, one observes oscillations in the avalanche size as a
function of the aperture size [33]. However, this does not happen in our
simulations as we can see in Fig. 5. This effect is observed only for highly
ordered structures obtained with frictionless particles [33]. Our particles
model non-slipping grains and the small deviations in position of the disks of
the first layer with respect to a truly crystalline structure are sufficiently
large to mask any systematic effect due to ordering that may require a
detailed study on the position chosen for the aperture.
## 5 Effect of packing fraction
It is generally believed that packing fraction is a good parameter to
characterize many properties of a granular bed [36, 37]. Results obtained for
deposits prepared at a given $\phi$ are not necessarily general and must be
repeated for different packing fractions. However, this does not mean that
packing fraction is the only or the main factor that can affect the results.
In Table 1 we present part of the data of Fig. 5 but ordered by mean packing
fractions $\phi$ corresponding to the steady state obtained for each given
$\Gamma$ (see Fig. 2). As we can see, the mean avalanche size depends on
$\phi$ in a non trivial way. At the highest $\phi$, obtained by light tapping,
the mean avalanche size can range from a few grains to a hundred grains
depending on the value of $\Gamma$ used to create the packings. The larger the
aperture $D$, the wider the range in $\langle s\rangle$ within this regime
where the system is rather ordered but the number and size of the arches fall
with increasing $\Gamma$ (see Fig. 3). For low $\phi$, deposits with similar
packing fractions but created with low and high tap intensities display
different values of $\langle s\rangle$ for any given $D$.
As we can see, deposits with the same $\phi$ can present different values of
$\langle s\rangle$. Therefore, the steady state ensembles of packings with
equal $\phi$ obtained by tapping may behave differently. This has been pointed
out in Refs. [29, 30] where steady states with the same $\phi$ but bearing
different stresses were obtained through tapping. In our simulations, forces
are not calculated and therefore the stress tensor cannot be obtained.
However, a clear difference in the response of the granular columns with same
$\phi$ is observed in the sense that avalanches are, in average, of different
size.
| | | | $\langle s\rangle$ |
---|---|---|---|---|---
$\phi$ | $\Gamma$ | $D=2.00$ | $D=2.25$ | $D=2.50$ | $D=2.75$
0.8425 | 0.173 | 7.2 | 7 | 7.1 | 8
0.8425 | 0.224 | 7 | 9 | 8 | 9
0.8424 | 0.274 | 9 | 11 | 12 | 33
0.8424 | 0.316 | 7 | 29 | 74 | 107
0.7393 | 0.548 | 26 | 27 | 41 | 85
0.7354 | 1.000 | 47 | 59 | 78 | 104
0.7540 | 0.447 | 10 | 36 | 80 | 130
0.7603 | 1.414 | 50 | 60 | 71 | 110
Table 1: Mean avalanche size $\langle s\rangle$ for different tap intensities
$\Gamma$ that yield similar packing fractions $\phi$.
## 6 Connection between bulk arches and jamming arches
Although we have shown in the previous section that arches found in the bulk
of the granular packing give a rough indication as to whether the system would
be more or less likely to jam, arches actually formed at the aperture during
discharge are different. Detailed studies of such blocking arches have been
reported for two-dimensional experimental setups [14, 38]. Since we have
access to both bulk and blocking arches in our simulations, we compare a few
properties and discuss on the implications for the jamming probability.
In Fig. 7 we compare the arch size distribution $n(k)$ for the bulk arches and
for the jamming arches for different values of $D$. $n(k)$ is the probability
of finding an arch of $k$ grains ($k\geq 2$). For bulk arches, $n(k)$ is
calculated as the number of arches of $k$ disks found in all initial packings
divided by the total number of arches (i.e., summing for all $k\geq 2$). For
jamming arches, $n(k)$ is calculated as the number of discharges that led to a
blocking arch of $k$ disks divided by the total number of discharges
(discharges that ended without producing a jam were not considered). The plot
is presented as a function of $k-D$ since this produces a collapse of the
curves by subtracting a quantity ($D$) proportional to the lower cut-off
imposed by the orifice on the arch sizes (see Ref. [14]). The fact that all
the normalized histograms fall on the same curve indicates that the nature of
the arches that jam the aperture is the same for small and big orifices. In
general, an exponential tail is observed and a cut off for small arches is
imposed by the orifice. The large arches generally form upstream supported by
two stationary piles resting on both sides of the opening. It is clear from
Fig. 7 that blocking arches tend to be larger than bulk arches (even after the
correction due to the cut-off imposed by $D$). We believe this is due to the
fact that many small arches that are stable in the bulk thanks to the many
neighbors cannot accommodate in the conical shaped funnel created by the two
stationary piles. This is best demonstrated in the next paragraph.
Figure 7: (Color online). Distribution $n(k)$ of arch sizes for
$\Gamma=0.387$. The black symbols correspond to arches found in the bulk prior
to the discharge, the color data correspond to the blocking arches for
different values of $D$ as indicated in the legend. Notice that the horizontal
axis is shifted by the size of the aperture $D$ for each set of data.
The horizontal span of the arches for a given number of grains $k$ is shown in
Fig. 8. We include data for the bulk arches found in the initial deposits and
for the jamming arches found for an aperture of width $D=2.5$. As we can see,
for a given $k$, arches are more likely to be wider in the case of jamming
arches as compared to bulk arches. This is to be expected for small $k$ since
small arches with small span might not be able to jam the aperture. However,
even arches with bigger $k$ are biased in the distribution of blocking arches.
This is due to the two piles formed at each side of the orifice. Blocking
arches that are wider than the aperture $D$ must span this funnel. Arches of
$k$ disks with horizontal span sufficient to jam the orifice might still be
unable to span the funnel (see the inset in Fig. 8(c)). This results in
blocking arches generally wider, for a given $k$, than the corresponding bulk
arches. Notice that arches of horizontal span $x<D-1$ may also jam the orifice
(see Fig. 8(a)) due to the grains sitting at the edges of the aperture that
reduce the effective size of the opening. This effect has been studied in more
detail by Pournin et al. [33].
Figure 8: (Color online). Horizontal span distribution $n_{k}(x)$ for arches
formed by different number of grains $k$ at $\Gamma=0.707$. (a) $k=2$, (b)
$k=3$, (c) $k=4$, (d) $k=5$. The red dashed lines correspond to blocking
arches for an aperture $D=2.5$, whereas the black solid lines correspond to
arches found in the bulk of the initial deposits. The inset of panel (c) shows
two schematic arches of $k=4$ which are wide enough to jam the orifice but
only one of them fits in the gap left by the two piles at rest on each side of
the aperture.
## 7 Conclusions
We have considered granular avalanches discharged though small apertures at
the bottom of a container by using a 2D pseudo-dynamic model. We have focused
on the effect of the packing fraction of the granular deposit prior to the
avalanche discharge. The results indicate that the initial packing fraction
has an important effect on the mean avalanche size for a given opening size.
However, similar $\langle s\rangle$ can be obtained for packings with very
different $\phi$. Most importantly, very different values of $\langle
s\rangle$ can correspond to initial packings with the same packing fractions
that were prepared by using different tap intensities. It is important to note
that these results are obtained not for single packings but for ensembles of
deposits representative of steady states corresponding to a particular tap
intensity.
Our main conclusion is that packing fraction is not a good macroscopic
parameter to predict the size of the avalanches that would flow through a
given aperture. It seems that further information is necessary. Although this
information is expected to reside in the size and number of arches, we have
seen that the correlation of these with $\langle s\rangle$ is not consistent
for all openings $D$. It seems that is not possible to predict the jamming
probability of a granular column as it flows through a small aperture based on
a few global properties of the initial deposits.
A side result from our study is that blocking arches are generally wider than
the arches found in bulk. This is not only due to the fact that the aperture
imposes a lower cut-off for the possible jamming arches, but also to the fact
that even arches formed in the bulk which are wide enough to block the outlet
have shapes not compatible with the conical boundary effectively created by
the two piles of stationary grains at the sides of the aperture.
LAP acknowledges valuable discussion with Angel Garcimartín. This work has
been partially supported by CONICET (Argentina).
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|
arxiv-papers
| 2011-10-21T13:37:13 |
2024-09-04T02:49:23.470747
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rodolfo Omar U\\~nac, Ana Mar\\'ia Vidales and Luis A. Pugnaloni",
"submitter": "Luis Ariel Pugnaloni",
"url": "https://arxiv.org/abs/1110.4790"
}
|
1110.4793
|
11institutetext: Instituto de Física de Líquidos y Sistemas Biológicos
(CONICET La Plata, UNLP), Calle 59 Nro 789, 1900 La Plata, Argentina
22institutetext: Universidad Tecnológica Nacional - FRBA, UDB Física, Mozart
2300, C1407IVT Buenos Aires, Argentina.
# Arches and contact forces in a granular pile
C. Manuel Carlevaro 1122 Luis A. Pugnaloni 11
(Received: date / Revised version: date)
###### Abstract
Assemblies of granular particles mechanically stable under their own weight
contain arches. These are structural units identified as sets of mutually
stable grains. It is generally assumed that these arches shield the weight
above them and should bear most of the stress in the system. We test such
hypothesis by studying the stress born by in-arch and out-of-arch grains. We
show that, indeed, particles in arches withstand larger stresses. In
particular, the isotropic stress tends to be larger for in-arch-grains whereas
the anisotropic component is marginally distinguishable between the two types
of particles. The contact force distributions demonstrate that an exponential
tail (compatible with the maximization of entropy under no extra constraints)
is followed only by the out-of-arch contacts. In-arch contacts seem to be
compatible with a Gaussian distribution consistent with a recently introduced
approach that takes into account constraints imposed by the local force
balance on grains.
## 1 Introduction
The study of mechanically stable granular beds has become a major point of
interest in granular Physics. Studies range from analysis of force network
properties ostojic ; behringer ; mueth ; peters to structural
characterization latzel ; aste ; torquato to thermodynamic and statistical
descriptions edwards1 ; henkes ; tighe ; snoeijer1 ; pugnaloni1 . A recurrent
question in the subject is related to the existence or not of a relation
between force chains and arches mehta1 ; mehta3 .
Arches (or bridges) are multiparticle structures where all grains are mutually
stable mehta1 ; pugnaloni2 ; pugnaloni3 ; jenkins , i.e., fixing the positions
of all other particles in the assembly, the removal of any particle in the
arch leads to the destabilization of the other particles in it. For an arch to
be formed, it is necessary (although not sufficient) that two or more falling
particles be in contact at the time they reach mechanical equilibrium in order
to create mutually stabilizing structures arevalo . Like arches in
architecture, granular arches are assumed to sustain the weight of the
material above.
Highly stressed grains in static deposits are generally found to form linear
structures: the so-called force chains. Notice that, the term “arch” is
sometimes used lovoll ; dorbolo ; nicodemi to refer to these force chains and
not to the mutually stabilizing structures defined above. Likewise, the term
“dynamic arch” has been used to refer to ephemeral structures that choke the
flow of grains luding1 . We have to distinguish between these usages and the
classical meaning we adhere to in this work: an arch is a mechanically stable
structure of mutually stabilizing bodies.
Force chains are a clear spatial heterogeneity of the contact force network.
The distribution of contact forces (both normal and tangential) shows no
bimodal character. However, the spatial distribution of large and small forces
is heterogeneous with large forces developing a somewhat open stringy network
that encloses regions of grains that sustain little weight radjai .
To what extent the bimodal spatial distribution of forces is related to the
mutually stable structures (arches) has not been assessed so far. A
correlation as been pointed out mehta1 ; pugnaloni2 ; mehta2 between the
distribution of horizontal span of arches in a granular pile and the
distribution of normal forces at the grain contacts. Therefore, it is assumed
that a strong correlation has to be present between arches and highly stressed
grains in a granular deposit. In this paper, we assess this general belief in
the frame of a simulation of granular packings prepared by tapping. The
results provide additional information on the validity of basic assumptions
made in the statistical description of granular packings.
## 2 Simulation model
To simulate packings of gains we have followed the standard techniques on
discrete element methods (see for example Refs. cundall ; poschel ; schafer ).
We used a velocity Verlet algorithm to integrate the Newton equations for $N$
monosized disks (diameter $d$) in a rectangular box of width $L$. We studied
two system sizes: (i) $N=512$ with $L=12.39d$ and (ii) $N=2048$ with
$L=24.78d$. The non-commensurate box is chosen to prevent crystallization to
some extent. The larger system is roughly twice as tall as the smaller system.
However, this depends on the actual packing fraction obtained for a given
preparation protocol.
The disk–disk and disk–wall contact interaction comprises a linear
spring–dashpot in the normal direction
$F_{\rm{n}}=k_{\rm{n}}\xi-\gamma_{\rm{n}}v_{i,j}^{\rm{n}}$ (1)
and a tangential friction force
$F_{\rm{t}}=-\min\left(\mu|F_{\rm{n}}|,|F_{\rm{s}}|\right)\cdot\rm{sign}\left(\zeta\right)$
(2)
that implements the Coulomb criterion to switch between dynamic and static
friction arevalo ; pugnaloni4 .
In Eqs. (1)–(2), $\xi=d-\left|\mathbf{r}_{ij}\right|$ is the particle–particle
overlap, $\mathbf{r}_{ij}$ represents the center-to-center vector between
particles $i$ and $j$, $v_{i,j}^{\rm{n}}$ is the relative normal velocity,
$F_{\rm{s}}=-k_{\rm{s}}\zeta-\gamma_{\rm{s}}v_{i,j}^{\rm{t}}$ is the static
friction force,
$\zeta\left(t\right)=\int_{t_{0}}^{t}v_{i,j}^{\rm{t}}\left(t^{\prime}\right)dt^{\prime}$
is the relative shear displacement,
$v_{i,j}^{\rm{t}}=\dot{\mathbf{r}}_{ij}\cdot\mathbf{s}+\frac{1}{2}d\left(\omega_{i}+\omega_{j}\right)$
is the relative tangential velocity, and $\mathbf{s}$ is a unit vector normal
to $\mathbf{r}_{ij}$. The shear displacement $\zeta$ is calculated by
integrating $v_{i,j}^{\rm{t}}$ from the beginning of the contact (i.e.,
$t=t_{0}$). The disk–wall interaction is calculated considering the wall as an
infinite radius, infinite mass disk. Other than these, the interaction
parameters are the same as for the disk-disk interaction.
In these simulation we used the following set of parameters: dynamic friction
coefficient $\mu=0.5$, normal spring stiffness $k_{n}=10^{5}(mg/d)$, normal
viscous damping $\gamma_{n}=300(m\sqrt{g/d})$, tangential spring stiffness
$k_{s}=\frac{2}{7}k_{n}$, and tangential viscous damping
$\gamma_{s}=200(m\sqrt{g/d})$. The integration time step is
$\delta=10^{-4}\sqrt{d/g}$. Units are reduced with the diameter of the disks,
$d$, the disk mass, $m$, and the acceleration of gravity, $g$.
In order to investigate reproducible states, we implement a tapping protocol.
The system is initially deposited from a dilute configuration in which
particles have no contacts nor overlaps. After the grains reach mechanical
equilibrium, the system is tapped with a given amplitude and left to come back
to mechanical equilibrium. After many taps of given amplitude, the system
reaches a steady state where the properties of the static configurations
generated have well defined mean values and fluctuations. The steady state
properties are independent of the initial conditions and are reproducible
pugnaloni1 ; ribiere ; pugnaloni5 . We generate a number of static packings
after the steady state is reached to average quantities. Different steady
states are generated by changing the tap amplitude.
Tapping is simulated by moving the confining box in the vertical direction
following a half sine wave trajectory of frequency $\nu=\pi/2(g/d)^{1/2}$. The
intensity of the excitation is controlled through the amplitude, $A$, of the
sinusoidal trajectory; and it is characterized by the parameter
$\Gamma=A(2\pi\nu)^{2}/g$. A new tap is applied only after the system has come
to mechanical equilibrium, which is defined via the stability of each
particle-particle contact arevalo . Averages were taken over 500 taps
(configurations) in the steady state corresponding to each value of $\Gamma$
after the $500$ initial taps were discarded to avoid any transient.
## 3 Identification of arches
Figure 1: (Color online). Sample images of the simulated granular columns
($N=512$) for different $\Gamma$: (a) $\Gamma=2.19$, (b) $\Gamma=4.93$ and (c)
$\Gamma=15.39$. The color code indicates the trace, Tr$(\sigma)$, of the
stress tensor for each particle in units of $mg/d$. The joining segments
indicate the arches detected in the system. (d) A closeup on a 5-particle
arch. See main text for a description on the supporting contacts of particle
4.
Details on the algorithms used to identify arches can be found in previous
works pugnaloni2 ; pugnaloni3 ; arevalo . Briefly, we need first to identify
the supporting grains of each particle in the packing. In 2D, there are two
disks that support any given grain. Two grains in contact with a given
particle are able to provide support if the segment defined by the contact
points lies below the center of mass of the particle. Some of these supporting
contacts may be provided by the walls of the container. Then, we find all
mutually stable particles. Two grains A and B are mutually stable if A
supports B and B supports A. Arches are defined as sets of particles connected
through these mutually stabilizing contacts (MSC).
The fact that the supporting particles of each grain have to be known implies
that contacts, and the chronological order in which they form, have to be
clearly defined in the model. Figure 1 shows some examples of the packings
generated where arches are indicated by joining segments. In Fig. 1(d), a
close view of an arch detected in a given packing (formed by particles 1 to 5)
is displayed. Particle 4 is an example of a grain whose pair of stabilizing
particles is ambiguous form the limited information provided by the snapshot;
discrimination requires chronological information. The pairs 3-5, 5-6 and 6-7
comply with the condition that the center of mass of grain 4 is above the
segment that joins the corresponding contact points. However, the contacts
with grains 3 and 5 where formed first and for that reason these particles are
considered to be the supporting pair of grain 4. To identify the contacts that
support each particle we use an algorithm that has been previously designed to
work with molecular dynamic type simulations arevalo .
## 4 Single particle stress tensor
We measure the stress tensor $\sigma_{i}$ for grain $i$ as latzel :
$\sigma_{i}^{\alpha\beta}=\frac{1}{\pi(d/2)^{2}}\sum_{j=1}^{N_{c}}{f_{ij}^{\alpha}b_{ij}^{\beta}},$
(3)
where, $f_{ij}$ is the force exerted by grain $j$ on grain $i$ and $b_{ij}$ is
the branch vector that goes from the center of grain $i$ to the contact point
with grain $j$. The sum runs over the $N_{c}$ particles in contact with
particle $i$.
The pressure (or isotropic stress) is given by the trace, $\rm{Tr}(\sigma)$,
of $\sigma$ whereas the anisotropic component is characterized by the
deviatoric stress $s$
$s_{i}^{\alpha\beta}=\sigma_{i}^{\alpha\beta}-\frac{\delta_{\alpha\beta}}{3}\sum_{\gamma}{\sigma_{i}^{\gamma\gamma}}$
(4)
We use $\rm{Dev}(\sigma)=\sigma^{zz}-\sigma^{xx}$ to characterize the
anisotropic component. In average, our packings under gravity present
$\sigma^{xx}<\sigma^{zz}$ (i.e., the vertical component is higher than the
horizontal component).
The principal directions of the stress vary form configuration to
configuration during tapping. However, these fluctuations are very small since
the shear component $\sigma^{xz}$ is less than 1% of $\rm{Tr}(\sigma)$ in all
our packings.
Figure 2: (a) The mean packing fraction, $\phi$, as a function of tap
intensity $\Gamma$. (b) Fraction of in-arch grains as a function of $\Gamma$.
(c) The arch size distribution $n(s)$ for $\Gamma=2.19$. Results obtained for
the two system sizes: $N=512$ (solid triangles) and $N=2048$ (open circles).
## 5 General properties of the deposits
In Fig. 2(a), we report the mean packing fraction, $\phi$, as a function of
the tap intensity, $\Gamma$. The packing fraction was measured in a slab of
the bed that covers 50% of the height of the column and is vertically centered
with the center of mass of the system. The values of $\phi$ for the smaller
system is affected by the presence of the lateral walls, which tend to reduce
the packing fraction. $\phi$ presents a minimum at intermediate $\Gamma$ as
previously observed in various models pugnaloni4 ; gago and experiments
pugnaloni1 ; pugnaloni5 . This minimum in $\phi$ is related with the existence
of a maximum in the number of grains involved in arches [see Fig. 2(b)]. A
description of the mechanisms that lead to the existence of a maximum in the
number of grains involved in arches can be found in Ref. pugnaloni4 .
In spite of the system being monodisperse, the packings obtained present only
partial crystallization. This is due to the non-commensurate simulation box.
Even if very ordered packings were obtained, the contact forces (the main
focus of this paper) have been found to display similar statistics to the one
shown by disordered packings blair . We have also assessed the structural
anisotropy through the fabric tensor. We found that all our packings present a
deviatoric fabric of less than 5% of the fabric trace. Therefore, the
structural anisotropy is rather small.
The distribution, $n(s)$, of the sizes of the arches found in the packings is
shown in Fig. 2(c). Here, $n(s)$ is the fraction of arches consisting in $s$
grains, with $n(s=1)$ the fraction of grains that do not belong to any arch.
As we can see, $n(s)$ is not affected by the system boundaries and arches of
more than 10 disks have not been detected even in the 24-disk-wide system
(i.e., $N=2048$).
Figure 1 shows some examples of the distribution of pressures and arches
inside a granular pile. As it is to be expected, particles are subjected to
higher pressures, in average, in the lower part of the pile as compared with
the upper layers. The system does not display force chains that span the
system from top to bottom as is commonly seen in many experiments and
simulations. This is due to the fact that the system is in mechanical
equilibrium under its own weight; no external compression is applied to the
sample in any direction.
Figure 3: (Color online). (a) Mean normal contact force $\langle F_{n}\rangle$
as a function of the depth into the granular column. (b) PDF of the normal
contact force. We consider two system sizes: $N=512$ (solid symbols) and
$N=2048$ (open symbols). Results for three different tap intensities are
reported (see legend). The intermediate value corresponds to the value of
$\Gamma$ that yields the minimum $\phi$ for the given system size (i.e.,
$\Gamma=4.93$ for $N=512$ and $\Gamma=6.59$ for $N=2048$). The inset in part
(b) is a close up for forces below the mean.
In Fig. 3, we show the normal component of the contact forces for three
different tap intensities (the lower and the higher $\Gamma$ studied, and the
value $\Gamma_{\rm{min}}$ that leads to the minimum $\phi$ for the given
system size). The mean normal contact force $\langle F_{\rm{n}}\rangle$
increases rather linearly with the depth into the packing and is little
dependent on the system size for any given depth [see Fig. 3(a)]. Only the
system with 2048 grains display a hint of Janssen saturation in the deeper
layers. Small differences in $\langle F_{\rm{n}}\rangle$ can be observed
between packings obtained with different $\Gamma$. In particular, for the
lowest tap intensity considered, $\langle F_{\rm{n}}\rangle$ is smaller at all
depths.
Figure 3(b) presents the normal contact force distribution for a depth of
$35d$ (this corresponds to the lower part of the smaller system and to the
middle section of the larger system). All grain-grain contacts that lay within
a slab $10d$-wide centered at a depth $35d$ are considered. Taking narrower
slabs leads to similar results. As we can see, the PDF of $F_{\rm{n}}$
coincides for both system sizes. We have seen that the tangential contact
forces also show consistent results when systems of different sizes are
compared by looking into slabs at the same depth.
There exist a current debate on whether the tail of these distributions are or
not exponentials tighe ; eerd . Exponential tails for forces above the mean
contact force have been reported by a number of authors considering granular
packs subjected to external compression behringer ; mueth ; snoeijer2 or
stable under their own weight lovoll . As we can see in Fig. 3(b), for low
$\Gamma$, we observe a clear exponential tail. However, a faster than
exponential decay seem to be followed by the rest of the packings. Tighe et
al. tighe have argued that some reported exponential tails are perhaps
Gaussians. The behavior for very small forces [see inset to Fig. 3(b)]
resembles the weak divergence found for packings without external compression
when bulk contacts (as opposed to contacts made between the grains and the
container) are considered snoeijer2 .
In order to compare results from different system sizes, the remaining of the
paper, unless otherwise specified, will refer to measurements made in a slab
$10d$-wide centered at a depth $35d$.
## 6 Contact forces and arches
Figure 4: (Color online). The mean value of the contact force for mutually
stabilizing contacts (blue) and non-mutually stabilizing contacts (red). (a)
Normal contact forces, and (b) tangential contact forces. Results obtained for
the two system sizes: $N=512$ (solid triangles) and $N=2048$ (open circles).
.
It can be observed from Fig. 1 that, at any depth into the pile, grains can
present high and low stress irrespective of whether they belong to an arch or
not. Also, force chains do not coincide with arches although arches form part
of portions of these chains. For a more quantitative analysis we plot in Fig.
4 the mean value of the contact forces (normal and tangential to the contact
in a slab at $35d$ of depth) as a function of $\Gamma$. MSC (mutually
stabilizing contacts) and non-MSC have been separated in the analysis.
Although some small differences are observed between the results for the two
system sizes studied, the general trends are quite similar. It is clear that
MSC (i.e., contacts within arches) have, in average, larger (roughly 50%)
normal and tangential forces. This supports the idea that arches bear most of
the stress in the system and that force chains and arches must be correlated.
Figure 5: (Color online). The PDF of contact forces for MSC (blue) and non-MSC
(red) for $\Gamma=\Gamma_{min}$. (a) Normal contact forces, and (b) tangential
contact forces. Results obtained for the two system sizes: $N=512$ (solid
triangles) and $N=2048$ (open circles).
Figure 5(a) shows that the distribution of contact forces for MSC and non-MSC
are clearly distinguishable for the normal component. Although, we present the
distribution obtained for $\Gamma_{\rm{min}}$, most packings display the same
general features (some exceptions regarding packings prepared at low $\Gamma$
are discussed below). The mild divergence for very small forces is still
present for both distributions. Despite the difference, there is not a clear
separation of the two populations of contacts. The bimodal character observed
in the spatial distribution of contacts seems to be poorly correlated with
MSC.
As we can see from Fig. 5(a), the PDF for non-MSC present a clear exponential
tail, whereas MSC present a faster decay. The non-MSC PDF corresponds to an
exponential decay even for forces below the mean. The exponential PDF has a
well established statistical explanation. If the mean force is set and all
contact force states are equally probable, an exponential distribution of
contact forces maximizes the entropy (defined as the logarithm of the number
of contact force states) edwards2 ; evesque .
MSC seem to have a distribution compatible with a Gaussian tail, or at least a
faster than exponential tail. It seems that the deviation form an exponential
in the full PDFs reported in the literature [and in Fig. 3(b)] seem to be due
to the presence of MSC (and therefore the presence of arches). The immediate
conclusion is that the presence of arches prevents us from making some of the
basic assumptions on the contact forces to render a simplified statistical
analysis. In particular, arches introduce force balance constraints that need
to be accounted for. Tighe et al. tighe have shown that force balance
constraints (the conservation of the total area of the Maxwell reciprocal
tiling) can be introduced in a force ensemble. These have led to Gaussian
contact force distributions. Notice however that this theoretical approach
yields the same Gaussian distribution irrespective of the existence of arches
in the packing.
Figure 5(b) shows that the tangential components of the contact forces have a
much subtle difference between the distribution for MSC and non-MSC. Again,
MSC present a somewhat faster-than-exponential tail in contrast with the non-
MSC.
We now focus in the results for the smallest tap intensity reported. As we
mentioned, Fig. 3(b) shows that for $\Gamma=2.19$ the PDF for normal contacts
presents a clear exponential tail, in contrast with the packings generated
with stronger taps. Separating MSC and non-MSC in the analysis leads to two
exponential tails (presenting slightly different slopes). We speculated that
there could be fewer MSC in these packings than in packings obtained with
stronger taps. However, these packings present similar number of MSC as
compared with packings that show a faster-than-exponential tail. The main
difference we have been able to find is that these packings have, in
comparison, fewer arches composed of three or more grains. It seems that
arches composed of three or more particles are the responsible for introducing
strong force balance constraints that render the PDF non-exponential. Some
reports of pure exponential decays can be found in previous studies. Blair et
al. mentioned that a pure exponential was found in some cases depending on the
history of the packing blair . Makse et al. found pure exponentials too in
simulations of isotropically compressed grains, which may develop structures
without arches makse . We believe the preparation history of these packings
may have lead to a small presence of arches composed of three or more grains.
## 7 Stress tensor and arches
Figure 6: (Color online). Mean stress tensor for in-arch (blue) and out-of-
arch (red) grains. (a) The trace $\rm{Tr}(\sigma)$ of the stress, and (b) the
deviator $\rm{Dev}(\sigma)$. Results obtained for the two system sizes:
$N=512$ (solid triangles) and $N=2048$ (open circles). Figure 7: (Color
online). The PDF of the stress tensor for in-arch (blue) and out-of-arch (red)
for $\Gamma=\Gamma_{min}$. (a) The trace, $\rm{Tr}(\sigma)$, of the stress
tensor, and (b) the deviator $\rm{Dev}(\sigma)$. Results obtained for the two
system sizes: $N=512$ (solid triangles) and $N=2048$ (open circles).
In Fig. 6, we show the results of an analysis similar to the previous section
but now the stress tensor on each particle, as defined in Eq. (3), is
considered. The stress tensor accounts for both MSC and non-MSC on each grain.
We separate in-arch grains from out-of-arch grains in the analysis. In-arch
grains support, in average, isotropic pressures [see $\rm{Tr}(\sigma)$ in Fig.
6(a)] about 20% higher than out-of-arch grains. In contrast, the anisotropic
component of the stress, $\rm{Dev}(\sigma)$, seems to be rather similar for
both types of grains. This implies that the actual difference between the
stress tensor of in-arch and out-of-arch grains corresponds to the addition of
a constant to the diagonal components (in contrast to an increse given by a
multiplicative constant). We have seen that the shear stress $\sigma_{xz}$ is
the same for both types of grains.
Figure 7(a) shows that the distribution of the isotropic stress is markedly
different for in-arch and out-of-arch grains. In-arch grains present a clear
maximum in the PDF of $\rm{Tr}(\sigma)$ at around the mean. Although there is
not a strong separation, the maximum in the PDFs suggest that the well known
bimodal character of the force network is driven by the presence of arches to
some extent.
The distribution of the stress deviator is presented in Fig. 7(b). As we can
see, the PDFs for in-arch and out-of-arch grains are almost identical. The
negative values are due to the fact that some grains have
$\sigma_{xx}>\sigma_{yy}$. However, in average $\sigma_{xx}<\sigma_{yy}$ and
the mean deviator as defined above is always positive in our packings.
## 8 Conclusions
We have shown that MSC, which define arches, present higher normal and
tangential components of the contact forces as compared with non-MSC. Grains
belonging to arches are generally subjected to larger isotropic stresses but
similar anisotropic stress. Therefore, particles in arches are, to some
extent, different from particles that do not form arches when their contact
forces are considered. This is in line with the common assumption that arches
carry most of the weight in a granular deposit.
The PDF of normal contact forces show that non-MSC follow an exponential decay
whereas the MSC present a faster-than-exponential fall. This has strong
implications for the basic statistical models of force distribution. In
particular, it seems that MSC are the main cause for the constraints in force
balance not considered in simplistic approaches. These constraints lead to the
deviation of the overall-contacts PDF from the expected exponential. Indeed,
packings containing a low number of large arches (arches of three or more
grains) seem to fit better the exponential law.
The bimodal spatial distribution of stresses seems to be related to some
extent with the presence of arches. Particles in arches present a clear
maximum around the mean stress in the PDF of isotropic stress.
It is worth mentioning that despite the correlations found between arches and
force chains, there is not a one-to-one correspondence. Arches that sustain
little weight can always exist in the structure since they are shielded by
other arches above. This leads to the preponderance of very small forces in
the distributions for MSC. Also, force chains can develop without the need of
arches. A deposit carefully built by sequential deposition of grains contains
no arches in the structure, yet it will present force chains.
## Acknowledgements
LAP thanks fruitful discussions with Gary C. Barker and Anita Mehta. This work
was supported by CONICET (Argentina). We thank an anonymous reviewer for a
insightful suggestion on the first version of the manuscript.
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|
arxiv-papers
| 2011-10-21T13:53:42 |
2024-09-04T02:49:23.480366
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "C. Manuel Carlevaro and Luis A. Pugnaloni",
"submitter": "Luis Ariel Pugnaloni",
"url": "https://arxiv.org/abs/1110.4793"
}
|
1110.4834
|
# Nonlinear Synchronization on Connected Undirected Networks
S. Orange∗ and N. Verdière111LMAH (Laboratoire de Mathématiques Appliquées du
Havre), Université du Havre, 25 rue Philippe Lebon, BP 540, 76058 Le Havre,
France. Sebastien.Orange@univ-lehavre.fr, Nathalie.Verdiere@univ-lehavre.fr
###### Abstract
This paper gives sufficient conditions for having complete synchronization of
oscillators in connected undirected networks. The considered oscillators are
not necessarily identical and the synchronization terms can be nonlinear. An
important problem about oscillators networks is to determine conditions for
having complete synchronization that is the stability of the synchronous
state. The synchronization study requires to take into account the graph
topology. In this paper, we extend some results to non linear cases and we
give an existence condition of trajectories. Sufficient conditions given in
this paper are based on the study of a Lyapunov function and the use of a
pseudometric which enables us to link network dynamics and graph theory.
Applications of these results are presented.
_AMS Subject Classification 2010: 93D20, 93D30, 68R10_.
_Keywords: Nonlinear systems, Synchronization, Networks, Graph topology,
Dynamical Systems_
## 1 Introduction
The study of the dynamics of coupled nonlinear dynamical systems are the
subject of a growing interest in various communities like in theoretical
physic, in information technology or in neuronal biology. The literature on
this topic shows different kinds of synchronization (see [10]). Classically,
two coupled limit-cycle are said synchronized when their time evolution is
periodic with the same period and perhaps the same phase. From the discover of
synchronization of chaotic systems (see [1, 5, 8]), the word synchronization
recovered different meanings such as having identical or functional related
solutions, eventually with a delay. The definition has also been modulated by
considering strong forms like complete, cluster form or weaker forms like
phase and lag synchronization (see [11]).
An important question about synchronization of a network of oscillators is to
determine the stability of the synchronisation state. This question leads to
consider some properties of networks and state vectors of oscillators (see,
for example, [4, 13, 14, 15, 17]). For this purpose, two methods are proposed
in the literature. The first one called master stability function is based on
the computation of a Lyapunov exponent and the eigenvalues of the connectivity
matrix [9]. However, this method is adapted when the coupling terms are linear
and the computation of eigenvalues can become a difficult task. A second
proposed method is the connection graph stability method (see [4]). It links
the study of a Lyapunov function and the graph topology. This productive
method has been extended to unbalance and undirected graph (see [2, 3]).
The results presented in this paper generalize some results of [4] to the non
linear synchronization case. For this, we introduce a notion of pseudometric
in the graph. The determination of the sign of the Lyapunov function
derivative requires two steps. The first one is to use assumptions allowing
comparisons between oscillators and synchronization terms. The second step
consists in using pseudometrics which enable us to use some graph properties.
For the complete synchronization, we present two results. The first one gives
a condition on synchronization strength for having a global synchronization of
oscillators. The second result is a local versus of the first one, that is
when the oscillators are closed to the synchronization variety. In these two
cases, we give sufficient conditions that insure existence of trajectories.
This paper is organized as follows. The problem statements are presented in
Section 2. First, we precise the kind of systems and the kind of
synchronizations considered. Then, we recall the definition and some
properties of pseudometrics defined on a graph. In Section 3, after precising
the assumptions on the synchronization term, main results, that is conditions
for having complete synchronization of the system of oscillators, are
presented. These results are applied in Section 4.
## 2 Problem statements
Thereafter, $Y^{T}$ is the transpose of the vector
$Y=(Y^{1},\ldots,Y^{m})\in\mathbb{R}^{m}$.
### 2.1 Systems and synchronizations considered
Let $G$ be a connected undirected graph and $n$ its number of vertex. The
graph $G$ describes the set of interactions between the oscillators. We denote
by $\mathcal{E}$ the set of its edges. If $G$ contains an undirected edge from
a vertex $i$ to a vertex $j$, we denote it by $(i,j)$.
The considered dynamical systems are defined by the following system of
equations:
$\left\\{\begin{array}[]{l}\displaystyle\dot{X}_{1}=F_{1}(X_{1},t)-\epsilon\sum_{(1,j)\in\mathcal{E}}h(X_{1},X_{j}),\\\
\phantom{\dot{X}_{1}\leavevmode\nobreak\ \,}\vdots\\\
\displaystyle\dot{X}_{n}=F_{n}(X_{n},t)-\epsilon\sum_{(n,j)\in\mathcal{E}}h(X_{n},X_{j}),\\\
\end{array}\right.$ (1)
where
* •
$X_{i}=(X_{i}^{1},\ldots,X_{i}^{d})^{T}$ is the vector composed of the $d$
coordinates of the $i$-th oscillator,
* •
$F_{i}=(F_{i}^{1},\ldots,F_{i}^{d})^{T}$ is the vectorial function defining
one oscillator,
* •
$h=(h^{1},\ldots,h^{d})^{T}$ is the synchronization function which defines the
vector coupling between oscillators,
* •
the real parameter $\epsilon$ corresponds to the synchronization strength
Recall that, for a given initial state of the set of oscillators
$(X_{1}(0),\,X_{2}(0),\cdots X_{n}(0))^{T}\,,$ system (1) synchronizes
completely if, for all
$(i,j)\in\text{\textlbrackdbl}1,n\text{\textrbrackdbl}$,
$\|X_{i}(t)-X_{j}(t)\|\xrightarrow[t\rightarrow+\infty]{}0\,.$
This means that the vector $(X_{1},\ldots,X_{n})$ approaches the
synchronization manifold defined by $X_{1}(t)=X_{2}(t)=\cdots=X_{n}(t)$. In
particular, this implies that the oscillators have the same asymptotic
behavior (such as chaotic trajectories, stable and periodic solutions). The
complete synchronization of all oscillators can occur whatever their initial
states are, in this case, the synchronization is said global; otherwise it is
said local.
In this paper, we focus naturally on the differences
$\Delta_{i,j}=X_{i}^{T}-X_{j}^{T}$ and therefore on the vector
$\Delta=(\Delta_{1,2},\,\cdots,\,\Delta_{1,n},\,\Delta_{2,3},\,\cdots,\,\Delta_{2,n},\,\cdots,\,\Delta_{n-1,n})^{T}\;.$
Thus, proving the complete synchronization of system (1) is equivalent to
prove that $\|\Delta(t)\|\xrightarrow[t\rightarrow+\infty]{}0\,.$
### 2.2 Quasimetrics defined on a graph
In the following, we consider pseudometric verifying the $\rho$-relaxed
triangle inequality for a positive real $\rho$, that is an application
$\varphi:D\times D\rightarrow\mathbb{R}^{+}$, where $D$ is an non empty set,
satisfying the following three axioms:
* •
$\varphi(z_{1},z_{1})=0$;
* •
$\varphi(z_{1},z_{2})=\varphi(z_{2},z_{1})$ (symmetry property);
* •
$\varphi(z_{1},z_{3})\leq\rho\,(\varphi(z_{1},z_{2})+\varphi(z_{2},z_{3}))$
($\rho$-relaxed triangle inequality).
Remark that any classical metric is such a pseudometric with $\rho=1$.
Let $\varphi$ be a pseudometric on a set $D$. Let’s set, for all
$m\in\mathbb{N}^{*}$, $\rho(m)$ the smallest real such that
$\varphi(z_{1},z_{m+1})\leq\rho(m)\,\left[\varphi(z_{1},z_{2})+\cdots+\varphi(z_{m},z_{m+1})\right]\,.$
(2)
Note that $\rho(1)=1$.
In the following examples, expressions of $\rho(m)$ appearing in inequalities
(2) are direct consequences of the convexity of functions
$x\rightarrow(x^{2})^{\alpha}$ and $x\rightarrow x^{2}\,e^{1-|x|}$.
###### Example 2.1.
1. 1.
The application
$\varphi_{\alpha}:\mathbb{R}^{2}\times\mathbb{R}^{2}\rightarrow\mathbb{R}^{+}$
defined by
$\varphi_{\alpha}\left(\left(\begin{array}[]{l}x_{1}\\\
y_{1}\end{array}\right),\left(\begin{array}[]{l}x_{2}\\\
y_{2}\end{array}\right)\right)=\left((x_{1}-x_{2})^{2}\right)^{\alpha}$
with $\alpha\geq 1/2$ is a pseudometric for which $\rho(m)=m^{2\alpha-1}$.
2. 2.
Let $D$ be the closed ball of center $0$ and radius $2-\sqrt{2}$. The
application $\varphi:D\times D\rightarrow\mathbb{R}^{+}$ defined by
$\varphi\left(\left(\begin{array}[]{l}x_{1}\\\ y_{1}\\\
z_{1}\end{array}\right),\left(\begin{array}[]{l}x_{2}\\\ y_{2}\\\
z_{2}\end{array}\right)\right)=(x_{1}-x_{2})^{2}e^{1-|x_{1}-x_{2}|}$
is a pseudometric for which $\rho(m)=m$.
We have the following properties.
###### Proposition 2.1.
1. 1.
The sequence of reals $(\rho(m))_{m\geq 1}$ is increasing.
2. 2.
For all $m\in\mathbb{N}^{*}$, we have $\rho(m)\leq\rho^{m-1}$ (see [16]).
3. 3.
Let $\varphi_{1}$ and $\varphi_{2}$ be two pseudometrics on $D$ and
$\rho_{1}(m)$ and $\rho_{2}(m)$ be the smallest respective reals verifying
(2). For all $\alpha>0$ and $\beta>0$, the application
$\alpha\,\varphi_{1}+\beta\,\varphi_{2}$ is a pseudometric on $D$ satisfying
$\rho(m)=Max\\{\rho_{1}(m),\rho_{2}(m)\\}$.
We now apply pseudometrics to networks of oscillators. Recall that a state
vector $z_{i}$ of an oscillator is associated to $i$-th vertex of $G$. Let’s
consider a pseudometric $\varphi$ defined on the set of state vectors of
oscillators. This pseudometric enables one to define the pseudolength
$\varphi(z_{i},z_{j})$ between vertices $i$ and $j$ and also the pseudolength
$\varphi(z_{i_{1}},z_{i_{2}})+\cdots+\varphi(z_{i_{m-1}},z_{i_{m}})$ of any
path $P_{i,j}=(i=i_{1},i_{2},\cdots,i_{m}=j)$ from vertex $i$ to vertex $j$.
In the following proposition, we bound, up to a multiplicative constant
$C(G)$, the sum of pseudolengths between any two oscillators by the sum of
pseudolengths of paths joining any two oscillators. This constant plays an
important role in Theorems 3.1 and 3.2 since the synchronization strenght
$\epsilon$ appearing in these theorems is proportionnal to this constant.
###### Proposition 2.2.
Let $G$ be a connected graph, $\mathcal{E}$ be the set of its edges and
$\varphi$ be a pseudometric on a set $D$. For any vertex $i$, let $z_{i}\in D$
be a vector associated to vertex $i$. There exists a constant $C$ depending
only on $G$ so that we have
$\sum_{i,j}\varphi(z_{i},z_{j})\leq
C\sum_{(i,j)\in\mathcal{E}}\varphi(z_{i},z_{j})\,.$ (3)
Moreover, the smallest real $C$ satisfying (3), $C(G)$, is bounded by
$\dfrac{n(n-1)}{2}\delta(G)\,\rho(\delta(G))\,,$ (4)
where $\delta(G)$ is the diameter of $G$.
###### Proof.
Let $i$ and $j$ be two vertices of $G$ and let’s denote
$P_{i,j}=(i=i_{1},i_{2},\cdots,i_{s+1}=j)$
a path of $G$ from the vertex $i$ to vertex $j$ (recall that $G$ is
connected). Since $\varphi$ is a pseudometric on $D$, we have
$\varphi(z_{i},z_{j})\leq\rho(s)\sum_{\ell=1}^{s}\varphi(z_{i_{\ell}},z_{i_{\ell+1}})\,.$
The path $P_{i,j}$ can be chosen so that $s\leq\delta(G)$. Suppose that this
choice is done for any vertices $i$ and $j$; since the sequence
$(\rho(n))_{n\in\mathbb{N}^{*}}$ is increasing, we have
$\rho(s)\leq\rho(\delta(G))$. Consequently, for any vertices $i$ and $j$, we
have
$\varphi(z_{i},z_{j})\leq\rho(\delta(G))\;\delta(G)\;Max\left(\\{\varphi(z_{i},z_{j})\mid(i,j)\in\mathcal{E}\\}\right)$
which implies the result. ∎
In Theorem 3.1, we need to determine the lowest bound $C(G)$ of the set of
reals $C$ satisfying inequality (3). The bound (4) of $C(G)$ may not lead to a
good estimation of $C(G)$ for a particular graph; nevertheless, this bound is
valid for any graph with $n$ vertices.
In the case of a pseudometric satisfying the classical triangle inequality,
i.e. when $\rho(n)=n$ for all $n\in\mathbb{N}^{*}$, a method taking $G$ as
input and returning a bound of $C(G)$ is proposed in [3]. Its two main steps
are:
1. 1.
for all $(i,j)$ with $i>j$, choose a path $P_{i,j}$; this path is usually
chosen with minimal length (number of edges in the path);
2. 2.
for each edge $e$ of the connection graph, determine the sum $B(e)$ of the
lengths of all chosen paths $P_{i,j}$ containing $e$. A bound for $C(G)$ is
then $Max\\{B(e):e\in\mathcal{E}\\}$.
For each choice of paths, these two steps return a bound for $C(G)$. Clearly,
the number of possible paths is huge but computations of bounds for $C(G)$ are
possible since most of these choices are suboptimal. Up to a slight
modification of the first step, this method can be applied here: its consists
in considering, for all path $P_{i,j}$, the pseudolength $\rho(|P_{i,j}|)$
instead of its length $|P_{i,j}|$.
###### Remark 2.1.
In the case of pseudometrics $\varphi$ satisfying $\rho(m)=m$, explicit bounds
of $C(G)$ for specific graphs and the method proposed in [4, 3] for computing
$C(G)$ from $G$ can be directly used. This is the case of the second function
in Example 2.1.
## 3 Complete synchronizations
### 3.1 Hypothesis
Afterwards, two cases are considered. The first one is the global complete
synchronization for which oscillators $X_{1},\ldots,X_{n}$ lies in
$D=\mathbb{R}^{d}$. The second one is the complete synchronization for which
oscillators are in a neighborhood $D$ of the variety
$X_{1}=X_{2}=\cdots=X_{n}$.
Thereafter, we will suppose the following assumptions on system (1).
* •
For all $(i,j)\in\mathcal{E}$, there exist some non negative reals
$a_{1},\,\ldots,\,a_{d}$ such that
$\forall(X_{i},X_{j})\in
D,\;\varphi(X_{i},X_{j})=\sum_{k=1}^{d}a_{k}(X_{i}^{k}-X_{j}^{k})h^{k}(X_{i},X_{j})$
(5)
are pseudometrics where $h=(h^{1},\ldots,h^{d})^{T}$ is the synchronization
function.
* •
For all $(i,j)\in\text{\textlbrackdbl}1,n\text{\textrbrackdbl}^{2}$ and, for
all $t\geq t_{0}$ where $t_{0}\in\mathbb{R}$,
$\forall(X_{i},X_{j})\in
D,\;{\sum_{k=1}^{d}a_{k}(X_{i}^{k}-X_{j}^{k})\left(F_{i}^{k}(X_{i},t)-F_{j}^{k}(X_{j},t)\right)}\leq{\varphi(X_{i},X_{j})}\,.$
(6)
* •
For all $(i,j)\in\text{\textlbrackdbl}1,n\text{\textrbrackdbl}^{2}$,
$\forall(X_{i},X_{j})\in D,\;$
$\begin{array}[]{c}\varphi(X_{i},X_{j})=0\text{ and/or
}\sum_{k=1}^{d}a_{k}(X_{i}^{k}-X_{j}^{k})\left(F_{i}^{k}(X_{i},t)-F_{j}^{k}(X_{j},t)\right)=0\nobreak\leavevmode\hfill\\\
\leavevmode\nobreak\ \hfill\Rightarrow(X_{i}=X_{j})\,.\end{array}$ (7)
###### Remark 3.1.
1. 1.
Notice that hypothesis (5) implies that,
$\forall(i,j)\in\mathcal{E},\;\forall(X_{i},X_{j})\in
D,\;h(X_{i},X_{j})=-h(X_{j},X_{i})\,\text{(antisymmetry)}.$ (8)
2. 2.
The assumption (7) is necessary for proving the complete synchronisation of
system (1) in Theorems 3.1 and 3.2. The condition $\varphi(X_{i},X_{j})=0$ in
this assumption is not always sufficient when it does not imply equalities of
all the components of oscillators. In this case, the second condition is
necessary for proving the complete synchronization.
For practical cases, a first problem is to prove the existence of trajectories
of system (1) for a sufficient large $t$. For this goal, the following
proposition enables us to link existence of trajectories between synchronized
and non synchronized systems.
###### Proposition 3.1.
For all $(i,j)\in\text{\textlbrackdbl}1,n\text{\textrbrackdbl}^{2}$, suppose
that assumptions (5), (6) and (7) are satisfied and that, for all $t\geq
t_{0}$,
$X_{i}^{T}F_{i}(X_{i},t)\leq\Psi(\mid\mid X_{i}\mid\mid)$
where $\Psi$ satifies the conditions
$\displaystyle\int_{s=s_{0}}^{+\infty}\dfrac{ds}{\Psi(t)}=+\infty$ and
$\Psi(s)>0$ for all $s\geq s_{0}\geq 0$.
Then, the Cauchy’s problem defined by system (1) and an initial condition
$\left(\begin{array}[]{c}X_{1}(t_{0})\\\ \vdots\\\
X_{n}(t_{0})\end{array}\right)\in\mathbb{R}^{nd}$ has a solution on the
complete semi-axis $[t_{0};+\infty)$ .
###### Proof.
Let’s set $X=\left(\begin{array}[]{c}X_{1}\\\ \vdots\\\
X_{n}\end{array}\right)\in\mathbb{R}^{nd}$ and
$F(X,t)=\left(\begin{array}[]{c}F_{1}({X}_{1},t)\\\ \vdots\\\
F_{n}({X}_{n},t)\end{array}\right)\in\mathbb{R}^{nd}$. In a first step, we
prove that there exists a real $\beta$ such that the following inequality
between the scalar products holds:
$X^{T}\dot{X}\leq\beta X^{T}F(X,t).$ (9)
For this, we consider the $dn\times dn$ diagonal matrix $M=Diag(a_{1},\ldots
a_{d},\ldots,a_{1},\ldots a_{d}).$ We have:
$\begin{array}[]{rcl}X^{T}M\dot{X}&=&\displaystyle\sum_{i=1}^{n}\sum_{k=1}^{d}a_{k}X_{i}^{k}F_{i}^{k}(X_{i},t)-\epsilon\sum_{i=1}^{n}\sum_{k=1}^{d}a_{k}\sum_{\\{j|(i,j)\in\mathcal{E}\\}}X_{i}^{k}h^{k}(X_{i},X_{j})\\\
&=&\displaystyle
X^{T}MF(X,t)-\epsilon\sum_{k=1}^{d}\sum_{(i,j)\in\mathcal{E}}a_{k}X_{i}^{k}h^{k}(X_{i},X_{j})\\\
\end{array}$
and, since to any edge $(i,j)\in\mathcal{E}$ corresponds the edge
$(j,i)\in\mathcal{E}$, we obtain
$\begin{array}[]{rcl}X^{T}M\dot{X}&=&\displaystyle
X^{T}MF(X,t)-\frac{\epsilon}{2}\sum_{k=1}^{d}a_{k}\sum_{(i,j)\in\mathcal{E}}X_{i}^{k}h^{k}(X_{i},X_{j})+X_{j}^{k}h^{k}(X_{j},X_{i})\\\
&=&\displaystyle
X^{T}MF(X,t)-\frac{\epsilon}{2}\sum_{k=1}^{d}a_{k}\sum_{(i,j)\in\mathcal{E}}(X_{i}^{k}-X_{j}^{k})h^{k}(X_{i},X_{j})\text{
(see\leavevmode\nobreak\ equality\leavevmode\nobreak\ (\ref{hyp1}))}\\\
&=&\displaystyle
X^{T}MF(X,t)-\frac{\epsilon}{2}\sum_{(i,j)\in\mathcal{E}}\varphi(X_{i},X_{j})\\\
&\leq&\displaystyle X^{T}MF(X,t).\text{ (see\leavevmode\nobreak\
assumption\leavevmode\nobreak\ (\ref{not_varphi}))}\\\ \end{array}$
Inequality (9) is then a direct consequence of the fact that the reals $a_{i}$
are non negative.
If the conditions of the proposition are verified, inequality (9) shows that
we have, for all $t\geq t_{0}$,
$X^{T}\dot{X}\leq\widetilde{\Psi}(\mid\mid X\mid\mid)$
where $\widetilde{\Psi}$ is a application satifying the conditions
$\displaystyle\int_{s=s_{0}}^{+\infty}\dfrac{ds}{\widetilde{\Psi}(t)}=+\infty$
and $\widetilde{\Psi}(s)>0$ for all $s\geq s_{0}\geq 0$.
Thus, system (1) satisfies the conditions of Wintner’s theorem ([12]) and,
consequently, solutions of system (1) are defined for any $t\geq t_{0}$.
∎
### 3.2 Global synchronization
###### Theorem 3.1.
Suppose that the assumptions done in Section 3.1 are satisfied for
$D=(\mathbb{R}^{d})^{2}$. If $\epsilon>\dfrac{C_{G}}{2n}$, where $C_{G}$ is
the optimal bound such that inequality (3) holds, then system (1) synchronizes
completely.
###### Proof.
In order to show this result, we will apply the second method of Lyapunov.
Let’s consider the Lyapunov candidate function:
$V=\dfrac{1}{2}\sum_{k=1}^{d}\sum_{i\leq j}a_{k}(X^{k}_{i}-X^{k}_{j})^{2}\,.$
Clearly, this function is non negative if $\Delta\neq\overrightarrow{0}$ and
equal to $0$ iff $\Delta=\overrightarrow{0}$ that is when the system (1) is
synchronized.
The derivative of $V$ gives:
$\begin{array}[]{rcl}\displaystyle\dot{V}&=&\displaystyle\sum_{k=1}^{d}a_{k}\dfrac{1}{2}\sum_{i=1}^{n}\dfrac{\partial
V}{\partial X^{k}_{i}}\;\dot{X}^{k}_{i}\\\
&=&\displaystyle\sum_{k=1}^{d}a_{k}\sum_{i=1}^{n}(nX_{i}^{k}-\sum_{j=1}^{n}X^{k}_{j})\dot{X}^{k}_{i}\\\
&=&\displaystyle\sum_{k=1}^{d}a_{k}\left(n\sum_{i=1}^{n}X^{k}_{i}\dot{X}^{k}_{i}-\sum_{j=1}^{n}X^{k}_{j}\sum_{i=1}^{n}\dot{X}^{k}_{i}\right)\\\
&=&\displaystyle\sum_{k=1}^{d}a_{k}\left[n\left(\sum_{i=1}^{n}X_{i}^{k}F_{i}^{k}(X_{i},t)-\epsilon\sum_{i=1}^{n}\sum_{\\{j|(i,j)\in\mathcal{E}\\}}X^{k}_{i}\,h^{k}(X_{i},X_{j})\right)\right.\\\
&&\displaystyle\left.\qquad-\sum_{j=1}^{n}X^{k}_{j}\left(\sum_{i=1}^{n}F_{i}^{k}(X_{i},t)-\epsilon\sum_{i=1}^{n}\sum_{\\{j|(i,j)\in\mathcal{E}\\}}h^{k}(X_{i},X_{j})\right)\right]\\\
&=&\displaystyle\sum_{k=1}^{d}a_{k}\left[\sum_{i=1}^{n}\left(nX^{k}_{i}-\sum_{j=1}^{n}X^{k}_{j}\right)F_{i}^{k}(X_{i},t)\right.\\\
&&\displaystyle\left.\qquad-n\epsilon\sum_{(i,j)\in\mathcal{E}}X^{k}_{i}h^{k}(X_{i},X_{j})+\epsilon\left(\sum_{j=1}^{n}X^{k}_{j}\right)\sum_{(i,j)\in\mathcal{E}}h^{k}(X_{i},X_{j})\right]\\\
&=&\displaystyle\sum_{k=1}^{d}a_{k}\left[\sum_{(i,j)\in\text{\textlbrackdbl}1,n\text{\textrbrackdbl}}^{n}\left(X^{k}_{i}-X^{k}_{j}\right)F_{i}^{k}(X_{i},t)\right.\\\
&&\displaystyle\left.\qquad-n\epsilon\sum_{(i,j)\in\mathcal{E}}X^{k}_{i}h^{k}(X_{i},X_{j})+\epsilon\left(\sum_{j=1}^{n}X^{k}_{j}\right)\sum_{(i,j)\in\mathcal{E}}h^{k}(X_{i},X_{j})\right]\,.\\\
\end{array}$
Since each edge $(i,j)\in\mathcal{E}$ corresponds to an edge $(j,i)$ and using
equality (8), we have, for all
$k\in\text{\textlbrackdbl}1,n\text{\textrbrackdbl}$,
$\begin{array}[]{rcl}\displaystyle
2\sum_{(i,j)\in\mathcal{E}}h^{k}(X_{i},X_{j})&=&\displaystyle\sum_{(i,j)\in\mathcal{E}}h^{k}(X_{i},X_{j})+\sum_{(i,j)\in\mathcal{E}}h^{k}(X_{j},X_{i})\\\
&=&\displaystyle\sum_{(i,j)\in\mathcal{E}}h^{k}(X_{i},X_{j})+\sum_{(i,j)\in\mathcal{E}}-h^{k}(X_{i},X_{j})\\\
&=&0\end{array}$
and
$\begin{array}[]{rcl}\displaystyle
2\sum_{k=1}^{d}a_{k}\sum_{(i,j)\in\mathcal{E}}X^{k}_{i}h^{k}(X_{i},X_{j})&=&\displaystyle\sum_{k=1}^{d}a_{k}\left[\sum_{(i,j)\in\mathcal{E}}X^{k}_{i}h^{k}(X_{i},X_{j})+\sum_{(i,j)\in\mathcal{E}}X^{k}_{j}h^{k}(X_{j},X_{i})\right]\\\
&=&\displaystyle\sum_{k=1}^{d}a_{k}\left[\sum_{(i,j)\in\mathcal{E}}X^{k}_{i}h^{k}(X_{i},X_{j})+\sum_{(i,j)\in\mathcal{E}}-X^{k}_{j}h^{k}(X_{i},X_{j})\right]\\\
&=&\displaystyle\sum_{(i,j)\in\mathcal{E}}\varphi(X_{i},X_{j})\,\text{(see\leavevmode\nobreak\
\ref{not_varphi})}.\end{array}$
Moreover, we have
$\begin{array}[]{rcl}\displaystyle
2\sum_{i,j}(X^{k}_{i}-X^{k}_{j})F_{i}^{k}(X_{i},t)&=&\displaystyle\sum_{i,j}(X^{k}_{i}-X^{k}_{j})F_{i}^{k}(X_{i},t)+\sum_{i,j}(X^{k}_{j}-X^{k}_{i})F_{j}^{k}(X_{j},t)\\\
&=&\displaystyle\sum_{i,j}(X^{k}_{i}-X^{k}_{j})(F_{i}^{k}(X_{i},t)-F_{j}^{k}(X_{j},t))\,.\end{array}$
These three equalities gives
$\displaystyle\dot{V}=\displaystyle\displaystyle\sum_{i,j}\sum_{k=1}^{d}\frac{a_{k}}{2}(X^{k}_{i}-X^{k}_{j})\left({F_{i}^{k}(X_{i},t)-F_{j}^{k}(X_{j},t)}\right)-n\epsilon\sum_{(i,j)\in\mathcal{E}}\varphi(X_{i},X_{j})$
(10)
With assumption (6) and inequality (3), we obtain
$\begin{array}[]{rcl}\displaystyle\dot{V}&\leq&\displaystyle\frac{1}{2}\sum_{i,j}\varphi(X_{i},X_{j})-n\epsilon\sum_{(i,j)\in\mathcal{E}}\varphi(X_{i},X_{j})\\\
&\leq&\displaystyle\left(\frac{C_{G}}{2}-n\epsilon\right)\sum_{(i,j)\in\mathcal{E}}\varphi(X_{i},X_{j})\\\
\end{array}$
Since $\varphi$ is a pseudometric the right factor of this last expression is
non negative. Therefore, if $\epsilon>\dfrac{C_{G}}{2n}$ then $\dot{V}\leq 0$.
To prove that $\dot{V}$ is negative definite, it remains to show that if
$\dot{V}=0$ then $X_{1}=X_{2}=\cdots=X_{n}$. Suppose that $\dot{V}=0$. Since
$\left(\frac{C_{G}}{2}-n\epsilon\right)<0$, the last inequality implies that
we have $\varphi(X_{i},X_{j})=0$ for all $(i,j)\in\mathcal{E}$. From equality
(10), we obtain
$\sum_{i,j}\sum_{k=1}^{d}{a_{k}}(X^{k}_{i}-X^{k}_{j})\left({F_{i}^{k}(X_{i},t)-F_{j}^{k}(X_{j},t)}\right)=0\,.$
Consequently, assumption (7) is satisfied and system (1) synchronizes. ∎
### 3.3 Local synchronization
Let $H$ be the diagonal matrix $Diag(a_{1},\ldots,a_{d})$ and
$\mathcal{H}=\left(\begin{array}[]{cccc}H&0&\cdots&0\\\ 0&H&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&H\end{array}\right)$ the matrix
composed with $\frac{n(n-1)}{2}$ matrices $H$. The application
$\begin{array}[]{lccc}\|.\|_{V}:&\mathbb{R}^{\frac{n(n-1)}{2}d}&\rightarrow&\mathbb{R}^{+}\\\
&X&\rightarrow&\sqrt{\frac{1}{2}X^{T}\mathcal{H}X}\end{array}$ (11)
is a norm since $a_{1},\ldots,a_{d}$ are non negative. Let’s set
$V(t)=\|\Delta(t)\|_{V}^{2}={\frac{1}{2}\sum_{k=1}^{d}\sum_{i<j\leq
n}a_{k}(X_{i}^{k}(t)-X_{j}^{k}(t))^{2}}\,.$
###### Theorem 3.2.
Let $\mathcal{B}$ the closed ball
$\\{X\in\mathbb{R}^{\frac{n(n-1)}{2}d}\mid\|X\|_{V}\leq{r}\\}$ where $r$ is a
non negative real. Suppose that assumptions of Section 3.1 are satisfied when
$\Delta$ belongs to the inner $\stackrel{{\scriptstyle\circ}}{{\mathcal{B}}}$
of $\mathcal{B}$ and suppose that, for an instant $t_{0}$,
$\Delta(t_{0})\in\;\stackrel{{\scriptstyle\circ}}{{\mathcal{B}}}$.
If $\epsilon>\dfrac{C_{G}}{2n}$, where $C_{G}$ is the optimal bound such that
inequality (3) holds, then system (1) synchronizes.
###### Proof.
Let’s show that if
$\Delta(t_{0})\in\stackrel{{\scriptstyle\circ}}{{\mathcal{B}}}$ then $\forall
t>t_{0}$, $\Delta(t)\in\mathcal{B}$. If
$\Delta(t_{0})\in\stackrel{{\scriptstyle\circ}}{{\mathcal{B}}}$, by definition
of $\mathcal{B}$, we have $V(t_{0})<r^{2}$. Suppose that there exists
$t_{1}>t_{0}$ such that $\Delta(t_{1})\notin\mathcal{B}$; by definition of
$\mathcal{B}$, we have $V(t_{1})>r^{2}$. Since $t\rightarrow V(t)$ is
continuous, there exists a real $t_{2}=Inf\\{t\in[t_{0},t_{1}]|V(t)=r^{2}\\}$.
The mean value theorem shows that there exists $t_{3}\in(t_{0},t_{2})$ such
that $V^{\prime}(t_{3})=\frac{V(t_{0})-V(t_{2})}{t_{0}-t_{2}}>0.$
On the other side, since $t_{3}<t_{2}=Inf\\{t\in[t_{0},t_{1}]|V(t)=r^{2}\\}$,
we have $V(t_{3})<r^{2}$ and
$\Delta(t_{3})\in\;\stackrel{{\scriptstyle\circ}}{{\mathcal{B}}}$.
Consequently, the hypothesis of Section 3.1 are satisfied by $\Delta(t_{3})$
and we can proceed like in the proof of Theorem 3.1 to show that
$V^{\prime}(t_{3})\leq 0$. This brings to a contradiction.
Finally, we have $\forall t\geq t_{0}$, $\Delta(t)\in\mathcal{B}$ and the
assumptions of Section 3.1 are satisfied for any $t\geq t_{0}$. Now, we can
proceed like in the proof of Theorem 3.1 to conclude. ∎
## 4 Applications
In this section, we focus on applications of Theorems 3.1 and 3.2 in order to
have a sufficient condition for global synchronization of two systems. The
fact that solutions of these two systems are defined on $\mathbb{R}$ is a
direct consequence of Proposition 3.1.
### 4.1 Global synchronization of a network of neurons
In this section, we apply Theorem 3.1 to a network of neurons satisfying the
FitzHugh-Nagumo model (See [6]). Recall that the dynamic of a single neuron is
modelised by the equation $\dot{X}=F(X)$ where
* •
$X=\left(\begin{array}[]{c}x\\\ y\end{array}\right)$;
* •
$F(X)=\left(\begin{array}[]{c}-x^{3}+x-y+a\\\ bx-cy-d\end{array}\right)$ for
some real parameters $a$, $b$, $c$ and $d$.
In the following, we suppose that $b$ is positive. Let’s set $G$ the connected
graph describing the interaction between the oscillators, $n$ its number of
vertices and $\mathcal{E}$ the set of its edges. For the synchronization
terms, we consider the function $h$ defined by
$\forall(i,j)\in\text{\textlbrackdbl}1,n\text{\textrbrackdbl}^{2},\;h(X_{i},X_{j})=\left(\begin{array}[]{c}\alpha(x_{i}-x_{j})+\beta\sqrt[3]{(x_{i}-x_{j})^{5}}\\\
\gamma(y_{i}-y_{j})\end{array}\right)$
with $\alpha\geq 1$, $\beta\geq 0$ and $\gamma\geq Max\\{0,-c\\}$. The system
of equations for the network of oscillators is then
$\left\\{\begin{array}[]{l}\displaystyle\dot{X}_{1}=F_{1}(X_{1})-\epsilon\sum_{(1,j)\in\mathcal{E}}h(X_{1},X_{j}),\\\
\phantom{\dot{x}_{1}\leavevmode\nobreak\ \,}\vdots\\\
\displaystyle\dot{X}_{n}=F_{n}(X_{n})-\epsilon\sum_{(n,j)\in\mathcal{E}}h(X_{n},X_{j}).\\\
\end{array}\right.$ (12)
The three hypothesis of Section 3.1 are satisfied with $a_{1}=1$ and
$a_{2}=1/b$. Indeed,
1. 1.
assumption (7) is obvious;
2. 2.
the fact that the application $\varphi$ corresponding to $h$, explicitly
defined by
$\varphi(X_{i},X_{j})=\alpha(x_{i}-x_{j})^{2}+\beta\sqrt[3]{(x_{i}-x_{j})^{8}}+\gamma/b(y_{i}-y_{j})^{2},$
is a pseudometric satisfying $\rho(m)=m^{5/3}$ is a consequence of Example 2.1
and Proposition 2.1. Therefore, assumption (5) is satisfied;
3. 3.
the following inequalities shows assumption (6), for all $(X_{i},X_{j})\in D$,
$\begin{array}[]{l}\sum_{k=1}^{2}a_{k}(X_{i}^{k}-X_{j}^{k})\left(F_{i}^{k}(X_{i})-F_{j}^{k}(X_{j})\right)\nobreak\leavevmode\hfill\nobreak\leavevmode\hfill\\\
\leavevmode\nobreak\ \hskip 91.0631pt=\left(\begin{array}[]{c}x_{i}-x_{j}\\\
y_{i}-y_{j}\end{array}\right).\left(\begin{array}[]{c}-(x_{i}^{3}-x_{j}^{3})+(x_{i}-x_{j})-(y_{i}-y_{j})\\\
(x_{i}-x_{j})-c/b(y_{i}-y_{j})\end{array}\right)\\\ \leavevmode\nobreak\
\hskip
91.0631pt=-(x_{i}-x_{j})(x_{i}^{3}-x_{j}^{3})+(x_{i}-x_{j})^{2}-c/b(y_{i}-y_{j})^{2}\\\
\leavevmode\nobreak\ \hskip 91.0631pt\leq\varphi(X_{i},X_{j})\,.\end{array}$
For any connected graph $G$ with $n$ vertex, inequality (3) is verified for
the bound of $C(G)$ given by $C=\dfrac{n(n-1)}{2}\delta(G)\,\rho(\delta(G))$.
Theorem 3.1 shows then that, for any connected graph $G$ with $n$ vertex, if
$\epsilon>\dfrac{(n-1)\,\delta(G)^{8/3}}{4}$ then system (12) synchronizes.
### 4.2 Local synchronization of a network of oscillators
In this section, we apply Theorem 3.2 to a network of Chua oscillators. We
consider the simplified version suggested by Chua for these oscillators (see
[7]): if we set $X=(x,y,z)^{T}$, the state equation for a single oscillator is
given by $\dot{X}=F(X)$ where
$F(x,y,z)=\left(\begin{array}[]{c}a[y-x-f(x)]\\\ x-y+z\\\ -by-
cz\end{array}\right),\,$
$a>0$, $b>0$, $c>0$ and $f$ is a piece-wise function
$f(x)=dx+1/2(d-e)(|x+1|-|x-1|)$ with $2d<e$.
Since $f$ is a piece-wise function, a real $\delta\geq 0$ bounds the set of
slopes $\left\\{\frac{f(x)-f(y)}{x-y}\,\mid\,0<|x-y|\leq 1\right\\}$. In the
following, we suppose that:
1. 1.
the set of vertex of $G$ is $\mathcal{E}=\\{(1;2),(1;3),\,\ldots,\,(1;n)\\}$.
In other words, we consider a star configuration of oscillators;
2. 2.
the synchronization function $h$ is given by
$h((x_{i},y_{i},z_{i}),(x_{j},y_{j},z_{j}))=\left(\begin{array}[]{c}a\delta(x_{i}-x_{j})e^{1-|x_{i}-x_{j}|}\\\
0\\\ 0\end{array}\right)\;.$
The equation for the $i$-th oscillator of the network is then
$\left(\begin{array}[]{l}\dot{x_{i}}\\\ \dot{y_{i}}\\\
\dot{z_{i}}\end{array}\right)=\left(\begin{array}[]{c}a[y_{i}-x_{i}-f(x_{i})]\\\
x_{i}-y_{i}+z_{i}\\\
-by_{i}-cz_{i}\end{array}\right)+\epsilon\sum_{j\,\mid\,(i,j)\in\mathcal{E}}\left(\begin{array}[]{c}a\delta(x_{i}-x_{j})e^{1-|x_{i}-x_{j}|}\\\
0\\\ 0\end{array}\right)\,.$
Assumptions of Section 3.1 have to be verified in order to apply Theorem 3.2.
The first one is obvious. For the second and the third one, let’s set
$a_{1}=1/a$, $a_{2}=1$ and $a_{3}=1/b$.
Let’s consider a closed ball
$\mathcal{B}=\left\\{X\in\mathbb{R}^{\frac{n(n-1)}{2}d}\mid\|X\|_{V}\leq{(\sqrt{2}-1)}{\sqrt{a}}\right\\}$
where $\|.\|_{V}$ is defined by (11) and the norm $\|.\|_{\tilde{V}}$ given by
$\begin{array}[]{lccc}\|.\|_{\tilde{V}}:&\mathbb{R}^{d}&\rightarrow&\mathbb{R}^{+}\\\
&Y&\rightarrow&\sqrt{\frac{1}{2}Y^{T}{H}Y}\end{array}$
where $H$ is the diagonal matrix $Diag(a_{1},\ldots,a_{d})$. If we have
$\Delta\in\mathcal{B}$ then
$\|\Delta_{i,j}\|_{\tilde{V}}<{(\sqrt{2}-1)}{\sqrt{a}}$. This implies that
$\mid x_{i}-x_{j}\mid<2-\sqrt{2}$ and, according to Example 2.1, the
application $\varphi$ corresponding to $h$ satisfies assumption (5).
Let’s verify assumption (6). We have
$\begin{array}[]{l}\sum_{k=1}^{3}a_{k}(X_{i}^{k}-X_{j}^{k})\left(F_{i}^{k}(X_{i})-F_{j}^{k}(X_{j})\right)\nobreak\leavevmode\hfill\\\
\leavevmode\nobreak\ \hskip
39.0242pt=\left(\begin{array}[]{c}\dfrac{x_{i}-x_{j}}{a}\\\ y_{i}-y_{j}\\\
\dfrac{z_{i}-z_{j}}{b}\end{array}\right).\left(\begin{array}[]{c}a[(y_{i}-y_{j})-(x_{i}-x_{j})-(f(x_{i})-f(x_{j}))]\\\
(x_{i}-x_{j})-(y_{i}-y_{j})+(z_{i}-z_{j})\\\
-b(y_{i}-y_{j})-c(z_{i}-z_{j})\end{array}\right)\\\ \leavevmode\nobreak\
\hskip
39.0242pt=(x_{i}-x_{j})(f(x_{i})-f(x_{j}))-(x_{i}-x_{j})^{2}-(y_{i}-y_{j})^{2}-c/b(z_{i}-z_{j})^{2}\,.\\\
\end{array}$
By definition of $\delta$, we have
$\begin{array}[]{l}(x_{i}-x_{j})(f(x_{i})-f(x_{j}))\leq\delta(x_{i}-x_{j})^{2}e^{1-|x_{i}-x_{j}|}\,.\end{array}$
This shows inequality (6).
Moreover, if $\varphi(x_{i},x_{j})=0$ and
$\sum_{k=1}^{3}a_{k}(X_{i}^{k}-X_{j}^{k})\left(F_{i}^{k}(X_{i})-F_{j}^{k}(X_{j})\right)=0$
then we have $X_{i}=X_{j}$. Consequently, assumption (7) holds.
Since the induced pseudometric $\varphi$ satisfies $\forall
m\in\mathbb{N}^{*},\;\rho(m)=m$ (see Example 2.1), the bound $C_{G}$ is given
explicitly by $2n-3$ (See Remark 2.1 and [4]).
Theorem 3.2 can now be applied : if
$\Delta(t_{0})\in\;\stackrel{{\scriptstyle\circ}}{{\mathcal{B}}}$ for an
instant $t_{0}$ and if $\epsilon>\dfrac{2n-3}{2n}$ then system (1)
synchronizes.
## 5 Conclusion
In this paper, sufficient conditions for proving complete synchronization of
oscillators in a connected undirected network are presented. The contribution
of this paper lies in the extension of results established in the case of
linear synchronization to the non linear case. For this, we have introduced
pseudometrics which enable us to link graph topology and minimal
synchronization strength between oscillators. Under our assumptions, a
criterion proving the existence of trajectories is given. Two results for
proving the complete synchronization are then proposed: the first one gives a
global criterion and the second one deals with local synchronization, that is
when the trajectories lie in a neighborhood of the synchronization variety. To
illustrate these results, two applications are treated.
## References
* [1] VS Afraimovich, NN Verichev, and MI Rabinovich. Stochastically synchronized oscillators in dissipative systems. Radiophys. Quant. Electron, 29:795–803, 1986.
* [2] I. Belykh, V. Belykh, and M. Hasler. Synchronization in asymmetrically coupled networks with node balance. Chaos: An Interdisciplinary Journal of Nonlinear Science, 16:015102, 2006.
* [3] I. Belykh, M. Hasler, M. Lauret, and H. Nijmeijer. Synchronization and graph topology. Int. J. Bifurcation and Chaos, 15(11):3423–3433, 2005.
* [4] V.N. Belykh, I.V. Belykh, and M. Hasler. Connection graph stability method for synchronized coupled chaotic systems. Physica D: nonlinear phenomena, 195(1-2):159–187, 2004.
* [5] H. Fujisaka and T. Yamada. Stability theory of synchronized motion in coupled dynamical systems. Prog. Theor. Phys, 69(1):32–47, 1983.
* [6] JL Hindmarsh and RM Rose. A model of the nerve impulse using two first-order differential equations. 1982\.
* [7] T. Matsumoto. A chaotic attractor from chua’s circuit. Circuits and Systems, IEEE Transactions on, 31(12):1055–1058, 1984\.
* [8] L.M. Pecora and T.L. Carroll. Synchronization in chaotic systems. Physical review letters, 64(8):821–824, 1990.
* [9] L.M. Pecora and T.L. Carroll. Master stability functions for synchronized coupled systems. Physical Review Letters, 80(10):2109–2112, 1998.
* [10] A. Pikovsky, M. Rosenblum, and J. Kurths. Synchronization: A universal concept in nonlinear sciences, volume 12. Cambridge Univ Pr, 2003.
* [11] M.G. Rosenblum, A.S. Pikovsky, and J. Kurths. From phase to lag synchronization in coupled chaotic oscillators. Physical Review Letters, 78(22):4193–4196, 1997.
* [12] A. Wintner. The non-local existence problem of ordinary differential equations. American Journal of Mathematics, 67(2):277–284, 1945.
* [13] C.W. Wu. Synchronization in coupled chaotic circuits and systems, volume 41. World Scientific Pub Co Inc, 2002.
* [14] C.W. Wu. Synchronization in networks of nonlinear dynamical systems coupled via a directed graph. Nonlinearity, 18:1057, 2005.
* [15] C.W. Wu and L.O. Chua. Synchronization in an array of linearly coupled dynamical systems. Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on, 42(8):430–447, 1995.
* [16] Q. Xia. The geodesic problem in quasimetric spaces. J. Geom. Anal., 19(2):452–479, 2009.
* [17] J. Zhou, J. Lu, and J. Lü. Pinning adaptive synchronization of a general complex dynamical network. Automatica, 44(4):996–1003, 2008.
|
arxiv-papers
| 2011-10-21T16:33:11 |
2024-09-04T02:49:23.489692
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S\\'ebastien Orange and Nathalie Verdi\\`ere",
"submitter": "S\\'ebastien Orange Mr",
"url": "https://arxiv.org/abs/1110.4834"
}
|
1110.4999
|
# Capacity of the Gaussian Relay Channel with Correlated Noises to Within a
Constant Gap
Lei Zhou, Student Member, IEEE and Wei Yu, Senior Member, IEEE
###### Abstract
This paper studies the relaying strategies and the approximate capacity of the
classic three-node Gaussian relay channel, but where the noises at the relay
and at the destination are correlated. It is shown that the capacity of such a
relay channel can be achieved to within a constant gap of
$\frac{1}{2}\log_{2}3=0.7925$ bits using a modified version of the noisy
network coding strategy, where the quantization level at the relay is set in a
correlation dependent way. As a corollary, this result establishes that the
conventional compress-and-forward scheme also achieves to within a constant
gap to the capacity. In contrast, the decode-and-forward and the single-tap
amplify-and-forward relaying strategies can have an infinite gap to capacity
in the regime where the noises at the relay and at the destination are highly
correlated, and the gain of the relay-to-destination link goes to infinity.
###### Index Terms:
Relay channel, approximate capacity, noise correlation, noisy network coding.
## I Introduction
The relay channel models a communication scenario where an intermediate relay
is deployed to assist the direct communication between a source and the
destination. Although the capacity of the relay channel is still not known
exactly even for the Gaussian case, much progress has been made recently in
the characterization of its approximate capacity [1, 2, 3].
In the classic Gaussian relay channel, the noises at the relay and at the
destination are independent. In many practical systems, however, the noises at
the relay and at the destination may be correlated. This may arise, for
example, due to the presence of a common interference, which in a practical
system is often treated as a part of the background noise, but nevertheless
contributes to the correlation between the noises.
The Gaussian relay channel with correlated noises has been studied in [4],
where relaying strategies such as the decode-and-forward and the compress-and-
forward schemes are studied in full-duplex or half-duplex modes. Likewise, the
effect of noise correlation for the single-tap amplify-and-forward scheme has
been studied for the diamond network and the two-hop parallel relay network in
[5]. In both papers, noise correlation has been found to be beneficial.
Neither [4] nor [5], however, addresses the question of whether the classic
relaying strategies are able to achieve to within constant bits of the
capacity for the relay channel with correlated noises.
Inspired by the recent work [1] and [3], where the quantize-map-and-forward
and the noisy network coding strategies with fixed quantization level at the
relays are shown to achieve the capacity of arbitrary Gaussian relay networks
with uncorrelated noises to within a constant gap, this paper shows that such
strategies are also capable of approximating the capacity of the three-node
Gaussian relay channel with correlated noises. However, unlike the existing
schemes of [1] and [3], this paper shows that the relay quantization level
needs to be modified to be noise-correlation dependent in the correlated-noise
case. As a corollary, this paper also establishes that the conventional
compress-and-forward scheme [6] achieves to within constant bits of the
capacity for the Gaussian relay channel in the correlated-noise case as well.
Finally, in contrast to the case with uncorrelated noises, the decode-and-
forward and the single-tap amplify-and-forward strategies can have an infinite
gap to capacity, when the noise correlation goes to $\pm 1$ and the gain of
the relay-to-destination link goes to infinity.
## II Channel Model
Figure 1: Three-node Gaussian relay channel with correlated noises
This paper considers a real-valued discrete-time three-node Gaussian relay
channel as depicted in Fig. 1, which consists of a source $X$, a destination
$Y$, and a relay. The relay observes a noise-corrupted version of the source
signal, denoted by $Y_{R}$, and transmits $X_{R}$ to the destination. The
source-to-destination channel is denoted $h_{SD}$, the relay-to-destination
channel $h_{RD}$, and the source-to-relay channel $h_{SR}$. The additive
Gaussian noises at the relay and at the destination are denoted as $Z_{R}$ and
$Z$ respectively. Mathematically, the channel model is:
$\displaystyle Y_{R}$ $\displaystyle=$ $\displaystyle h_{SR}X+Z_{R},$ (1)
$\displaystyle Y$ $\displaystyle=$ $\displaystyle h_{SD}X+h_{RD}X_{R}+Z.$ (2)
Without loss of generality, the power constraints at the source and at the
relay can both be normalized to one, i.e., $\mathbb{E}[X^{2}]\leq 1$ and
$\mathbb{E}[X_{R}^{2}]\leq 1$, and so can the noise variances, i.e.,
$Z_{R}\sim\mathcal{N}(0,1)$ and $Z\sim\mathcal{N}(0,1)$. Different from most
of the literature that assumes independence between $Z_{R}$ and $Z$, this
paper introduces a correlation between the two noises
$\rho_{z}\triangleq\frac{\mathbb{E}\left[Z_{R}Z\right]}{\sqrt{\mathbb{E}[|Z_{R}|^{2}]\mathbb{E}[|Z|^{2}]}}.$
(3)
Note that $Z$ and $Z_{R}$ are both i.i.d. in time. Further, the relay
operation is causal.
## III Within Constant Bits of the Capacity
To approach capacity, the relaying strategy must take advantage of the noise
correlation. Consider the limiting scenario of $\rho_{z}\rightarrow\pm 1$. The
relay’s observation becomes more and more useful to the destination in this
case, thus an increasingly fine quantization resolution at the relay is
required — the fixed quantization strategy of [1] and [3] would result in
significant inefficiency. The main contribution of this paper is to introduce
a correlation-aware quantization strategy at the relay, which better exploits
the noise correlation and achieves to within $\frac{1}{2}\log_{2}3$ bits of
the capacity of the Gaussian relay channel with correlated noises.
###### Theorem 1.
The capacity of the three-node Gaussian relay channel with correlated noises,
as shown in Fig. 1, can be achieved to within $\frac{1}{2}\log_{2}3$ bits to
capacity using a noisy network coding strategy with independent Gaussian
inputs $X\sim\mathcal{N}(0,1)$, $X_{R}\sim\mathcal{N}(0,1)$ and Gaussian
quantization at the relay with quantization variance
$\mathsf{q}^{*}=2(1-\rho_{z}^{2})$.
###### Proof:
First, the capacity of the relay channel is upper bounded by the cut-set
bound, i.e.,
$\displaystyle\overline{C}$ $\displaystyle=$
$\displaystyle\max_{p(x,x_{R})}\min\\{I(X,X_{R};Y),I(X;Y,Y_{R}|X_{R})\\}$ (4)
$\displaystyle=$
$\displaystyle\max_{\rho_{x}}\min\left\\{\frac{1}{2}\log(1+h_{SD}^{2}+h_{RD}^{2}+2\rho_{x}h_{SD}h_{RD}),\right.$
$\displaystyle\left.\frac{1}{2}\log\left(1+\frac{(1-\rho_{x}^{2})(h_{SD}^{2}+h_{SR}^{2}-2\rho_{z}h_{SD}h_{SR})}{1-\rho_{z}^{2}}\right)\right\\}$
$\displaystyle\leq$
$\displaystyle\min\left\\{\frac{1}{2}\log(1+h_{SD}^{2}+h_{RD}^{2}+2h_{SD}h_{RD}),\right.$
$\displaystyle\quad\quad\;\;\left.\frac{1}{2}\log\left(1+\frac{h_{SD}^{2}+h_{SR}^{2}-2\rho_{z}h_{SD}h_{SR}}{1-\rho_{z}^{2}}\right)\right\\}$
$\displaystyle=$ $\displaystyle\min\\{R_{UB1},R_{UB2}\\},$
where $\rho_{x}$ is the correlation between $X$ and $X_{R}$.
The achievable rate by noisy network coding or compress-and-forward with joint
decoding can be readily obtained from [7, Proposition 2] and [3, Theorem 1]:
$\displaystyle R$ $\displaystyle=$
$\displaystyle\min\\{I(X,X_{R};Y)-I(Y_{R};\hat{Y}_{R}|X,X_{R},Y),$ (5)
$\displaystyle\quad\quad\;I(X;Y,\hat{Y}_{R}|X_{R})\\}$ $\displaystyle=$
$\displaystyle\min\\{R_{1},R_{2}\\}$
for any distribution
$p(x,x_{R},y_{R},\hat{y}_{R})=p(x)p(x_{R})p(y_{R}|x,x_{R})p(\hat{y}_{R}|x_{R},y_{R}).$
Substitute independent Gaussian distributions $X\sim\mathcal{N}(0,1)$ and
$X_{R}\sim\mathcal{N}(0,1)$ into (5), and set $\hat{Y}_{R}=Y_{R}+e$, where the
quantization noise $e\sim\mathcal{N}(0,\mathsf{q})$ is independent with
everything else, we have
$\displaystyle R_{1}=I(X,X_{R};Y)-I(Y_{R};\hat{Y}_{R}|X,X_{R},Y)$ (6)
$\displaystyle=$
$\displaystyle\frac{1}{2}\log(1+h_{SD}^{2}+h_{RD}^{2})-\frac{1}{2}\log\left(1+\frac{1-\rho_{z}^{2}}{\mathsf{q}}\right),$
and
$\displaystyle R_{2}$ $\displaystyle=$ $\displaystyle
I(X;Y,\hat{Y}_{R}|X_{R})$ (7) $\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}$
$\displaystyle\frac{1}{2}\log(1+h_{SD}^{2})$
$\displaystyle+\frac{1}{2}\log\left(\frac{\mathsf{q}+\sigma^{2}_{h_{SR}X+Z_{R}|h_{SD}X+Z}}{\mathsf{q}+1-\rho_{z}^{2}}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{q}+(\mathsf{q}+1)h_{SD}^{2}+h_{SR}^{2}-2\rho_{z}h_{SD}h_{SR}}{1-\rho_{z}^{2}}\right)$
$\displaystyle-\frac{1}{2}\log\left(1+\frac{\mathsf{q}}{1-\rho_{z}^{2}}\right),$
where in $(a)$ the conditional variance of $h_{SR}X+Z_{R}$ given $h_{SD}X+Z$
is calculated as
$\displaystyle\sigma^{2}_{h_{SR}X+Z_{R}|h_{SD}X+Z}$ (8) $\displaystyle=$
$\displaystyle\mathbb{E}[|h_{SR}X+Z_{R}|^{2}]-\frac{|\mathbb{E}[(h_{SR}X+Z_{R})(h_{SD}X+Z)]|^{2}}{\mathbb{E}[|h_{SD}X+Z|^{2}]}$
$\displaystyle=$
$\displaystyle\frac{1-\rho_{z}^{2}+h_{SR}^{2}+h_{SD}^{2}-2\rho_{z}h_{SR}h_{SD}}{1+h_{SD}^{2}}.$
Comparing $R_{1}$ and the upper bound $R_{UB1}$, we have
$\displaystyle R_{UB1}-R_{1}$ (9) $\displaystyle=$
$\displaystyle\frac{1}{2}\log(1+h_{SD}^{2}+h_{RD}^{2}+2h_{SD}h_{RD})$
$\displaystyle-\frac{1}{2}\log(1+h_{SD}^{2}+h_{RD}^{2})+\frac{1}{2}\log\left(1+\frac{1-\rho_{z}^{2}}{\mathsf{q}}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(\frac{1+h_{SD}^{2}+h_{RD}^{2}+2h_{SD}h_{RD}}{2+2h_{SD}^{2}+2h_{RD}^{2}}\right)$
$\displaystyle+\frac{1}{2}\log\left(1+\frac{1-\rho_{z}^{2}}{\mathsf{q}}\right)+\frac{1}{2}$
$\displaystyle<$
$\displaystyle\frac{1}{2}\log\left(1+\frac{1-\rho_{z}^{2}}{\mathsf{q}}\right)+\frac{1}{2}.$
Comparing $R_{2}$ and the upper bound $R_{UB2}$, we have
$\displaystyle R_{UB2}-R_{2}$ (10) $\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(1+\frac{h_{SD}^{2}+h_{SR}^{2}-2\rho_{z}h_{SD}h_{SR}}{1-\rho_{z}^{2}}\right)$
$\displaystyle-\frac{1}{2}\log\left(1+\frac{\mathsf{q}+(\mathsf{q}+1)h_{SD}^{2}+h_{SR}^{2}-2\rho_{z}h_{SD}h_{SR}}{1-\rho_{z}^{2}}\right)$
$\displaystyle+\frac{1}{2}\log\left(1+\frac{\mathsf{q}}{1-\rho_{z}^{2}}\right)$
$\displaystyle<$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{q}}{1-\rho_{z}^{2}}\right).$
The gap between the cut-set bound $\overline{C}$ and the achievable rate $R$
is then upper bounded by the maximum of (9) and (10), i.e.
$\displaystyle\overline{C}-R$ $\displaystyle\leq$
$\displaystyle\min\\{R_{UB1},R_{UB2}\\}-\min\\{R_{1},R_{2}\\}$ (11)
$\displaystyle\leq$ $\displaystyle\max\\{R_{UB1}-R_{1},R_{UB2}-R_{2}\\}$
$\displaystyle<$
$\displaystyle\max\left\\{\frac{1}{2}\log\left(1+\frac{1-\rho_{z}^{2}}{\mathsf{q}}\right)+\frac{1}{2},\right.$
$\displaystyle\quad\quad\;\;\left.\frac{1}{2}\log\left(1+\frac{\mathsf{q}}{1-\rho_{z}^{2}}\right)\right\\}.$
The first term above monotonically decreases with $\mathsf{q}$, while the
second term monotonically increases with $\mathsf{q}$. To minimize the maximum
of the two terms, we set
$\displaystyle\frac{1}{2}\log\left(1+\frac{1-\rho_{z}^{2}}{\mathsf{q}^{*}}\right)+\frac{1}{2}=\frac{1}{2}\log\left(1+\frac{\mathsf{q}^{*}}{1-\rho_{z}^{2}}\right),$
(12)
which results in $\mathsf{q}^{*}=2(1-\rho_{z}^{2})$. Substituting
$\mathsf{q}^{*}$ into (11), we have
$\overline{C}-R<\frac{1}{2}\log_{2}3=0.7925$. ∎
In addition, it can be shown that the conventional compress-and-forward rate
is also within the same constant gap to capacity. To prove this directly would
have been quite involved (see [2] for the computation of the gap for the case
of $\rho_{z}=0$). Instead, we obtain the result as a direct consequence of
Theorem 1.
###### Corollary 1.
The following rate, which is achieved by the classic compress-and-forward
strategy on the three-node Gaussian relay channel with correlated noises shown
in Fig. 1:
$R_{CF}=\frac{1}{2}\log\left(1+h_{SD}^{2}+\frac{(h_{SR}-\rho_{z}h_{SD})^{2}}{1-\rho_{z}^{2}+\mathsf{q}_{c}}\right),$
(13)
where
$\mathsf{q}_{c}=\frac{(1-\rho_{z}^{2})(1+h_{SD}^{2})+(h_{SR}-\rho_{z}h_{SD})^{2}}{h_{RD}^{2}}$
(14)
is within $\frac{1}{2}\log_{2}3$ bits to the capacity.
###### Proof:
The rate expression $R_{CF}$ for the correlated-noise Gaussian relay channel
has been obtained in [4, Proposition 5]. The derivation is based on the
classic compress-and-forward rate for the relay channel by Cover and El Gamal
[6, Theorem 6], which is $R_{CF}=I(X;\hat{Y}_{R},Y|X_{R})$ subject to
$I(X_{R};Y)\geq I(Y_{R};\hat{Y}_{R}|X_{R},Y)$ for some joint distribution
$p(x)p(x_{R})p(y_{R}|x,x_{R})p(\hat{y}_{R}|x_{R},y_{R})$. Using the same
signaling scheme as in Theorem 1, i.e., $X\sim\mathcal{N}(0,1)$ and
$X_{R}\sim\mathcal{N}(0,1)$ are independent, and $\hat{Y}_{R}=Y_{R}+e$, where
$e\sim\mathcal{N}(0,\mathsf{q}_{c})$ is chosen to satisfy the relay-
destination rate constraint, we obtain (13).
In the following, we prove the constant gap result for the compress-and-
forward rate by showing that $R_{CF}$ in (13) is greater than the noisy
network coding rate, i.e., $R_{CF}\geq\min(R_{1},R_{2})$, where $R_{1}$ and
$R_{2}$ are as in (6) and (7) respectively. Substituting $\mathsf{q}_{c}$ in
(14) as $\mathsf{q}$ in $R_{1}$ and $R_{2}$, it is easy to verify that
$R_{1}(\mathsf{q}_{c})=R_{2}(\mathsf{q}_{c})=R_{CF}$. Since $R_{1}$ increases
with $\mathsf{q}$ and $R_{2}$ decreases with $\mathsf{q}$, we have
$R_{CF}=\min\\{R_{1}(\mathsf{q}_{c}),R_{2}(\mathsf{q}_{c})\\}=\max_{\mathsf{q}}\min\\{R_{1}(\mathsf{q}),R_{2}(\mathsf{q})\\}\geq\min\\{R_{1}(\mathsf{q}^{*}),R_{2}(\mathsf{q}^{*})\\}$
for any $\mathsf{q}*$ and in particular for
$\mathsf{q}^{*}=2(1-\rho_{z}^{2})$. Since it has been show in Theorem 1 that
$\min\\{R_{1}(\mathsf{q}^{*}),R_{2}(\mathsf{q}^{*})\\}$ is within
$\frac{1}{2}\log 3$ bits of the cut-set upper bound, so is $R_{CF}$. ∎
## IV Suboptimality of Decode-and-Forward and Single-Tap Amplify-and-Forward
The decode-and-forward and the single-tap amplify-and-forward strategies have
been shown to achieve to within a constant gap to the capacity of the Gaussian
relay channel with uncorrelated noises [1, 2]. In this section, we show that
this is no longer the case when noises are correlated.
### IV-A Decode-and-Forward
Consider a decode-and-forward strategy as described in [1, Appendix A], in
which when the source-to-relay link is weaker than the source-to-destination
link, i.e., $h_{SR}\leq h_{SD}$, the relay is simply ignored, otherwise the
relay decodes and forwards a bin index to the destination as in the original
scheme of [6]. The following rate is achievable:
$R_{DF}=\max\left\\{\frac{1}{2}\log(1+h_{SD}^{2}),\right.\\\
\left.\min\left\\{\frac{1}{2}\log(1+h_{SR}^{2}),\frac{1}{2}\log(1+h_{SD}^{2}+h_{RD}^{2})\right\\}\right\\}$
(15)
In the extreme scenario where $\rho_{z}=1$ and
$h_{RD}^{2}\gg h_{SR}^{2}\gg h_{SD}^{2}\gg 1,$ (16)
the above decode-and-forward rate (15) becomes
$\displaystyle R_{DF}=\frac{1}{2}\log(1+h_{SR}^{2}).$ (17)
Meanwhile, when $\rho_{z}=1$, the cut-set bound (4) becomes
$\overline{C}=\frac{1}{2}\log(1+h_{SD}^{2}+h_{RD}^{2}+2h_{SD}h_{RD}).$ (18)
Comparing (17) with (18), we observe that
$\displaystyle\overline{C}-R_{DF}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(\frac{1+h_{SD}^{2}+h_{RD}^{2}+2h_{SD}h_{RD}}{1+h_{SR}^{2}}\right)$
(19) $\displaystyle\rightarrow$
$\displaystyle\frac{1}{2}\log\left(\frac{h_{RD}^{2}}{h_{SR}^{2}}\right),$
which is unbounded in the asymptotic regime (16). This is not unexpected,
because the decoding at the relay eliminates the noise. Therefore, noise
correlation is not exploited.
### IV-B Single-Tap Amplify-and-Forward
In the single-tap amplify-and-forward, the relay scales the current
observation and forwards to the destination in the next time instance, i.e.,
$X_{R}[i]=\alpha(h_{SR}X[i-1]+Z_{R}[i-1]),$ (20)
where $\alpha\leq\frac{1}{\sqrt{1+h_{SR}^{2}}}$ is chosen to satisfy the power
constraint at the relay. Since $Y[i]=h_{SD}X[i]+h_{RD}X_{R}[i]+Z[i]$, the
relay channel is now converted into a single-tap inter-symbol-interference
(ISI) channel:
$Y[i]=h_{SD}X[i]+\alpha h_{RD}h_{SR}X[i-1]+Z[i]+\alpha h_{RD}Z_{R}[i-1].$ (21)
The capacity of the Gaussian ISI channel is given by
$R_{AF}=\max_{S(\omega)}\frac{1}{2\pi}\int_{0}^{2\pi}\frac{1}{2}\log\left(1+S(\omega)\frac{|H(\omega)|^{2}}{N(\omega)}\right)d\omega,$
(22)
subject to
$\frac{1}{2\pi}\int_{0}^{2\pi}S(\omega)d\omega\leq
1,\;\;\textrm{and}\;\;S(\omega)\geq 0,\quad 0\leq\omega\leq 2\pi,$ (23)
where $N(\omega)=1+\alpha^{2}h_{RD}^{2}+2\rho_{z}\alpha h_{RD}\cos(\omega)$ is
the power spectrum density of the noise, and $H(\omega)=h_{SD}+\alpha
h_{RD}h_{SR}e^{j\omega}$ is the Fourier transform of the channel coefficients,
and $S(\omega)=\left(\lambda-\frac{N(\omega)}{|H(\omega)|^{2}}\right)^{+}$ is
the water-filling power allocation over the frequencies.
Consider again the case of $\rho_{z}=1$ and the asymptotic regime of (16),
i.e. $h_{RD}^{2}\gg h_{SR}^{2}\gg h_{SD}^{2}\gg 1$. In this high signal-to-
noise ratio regime, it is easy to verify that the water-filling power spectrum
converges to an equal power allocation, i.e., $S(\omega)=1$, $0\leq\omega\leq
2\pi$. Substituting $N(\omega)$, $H(\omega)$ and $S(\omega)=1$ into (22) and
using table of integrals, after some algebra, it is possible to show that
$\displaystyle R_{AF}\leq\frac{1}{2}\log(2+h_{SR}^{2}+h_{SD}^{2}).$
Comparing the above with the cut-set bound, we see that
$\displaystyle\overline{C}-R_{AF}$ $\displaystyle\geq$
$\displaystyle\frac{1}{2}\log\left(\frac{1+h_{SD}^{2}+h_{RD}^{2}+2h_{SD}h_{RD}}{2+h_{SR}^{2}+h_{SD}^{2}}\right)$
(24) $\displaystyle\rightarrow$
$\displaystyle\frac{1}{2}\log\left(\frac{h_{RD}^{2}}{h_{SR}^{2}}\right)$
in the asymptotic regime of (16), which is unbounded.
## V Numerical Simulation
This section numerically compares the cut-set upper bound and the achievable
rates of different relaying schemes. Here, the noisy network coding rate is
computed with $\mathsf{q}^{*}=2(1-\rho_{z}^{2})$. We consider two examples:
Fig. 2 shows the case for $h_{SD}^{2}=20$dB, $h_{SR}^{2}=40$dB and
$h_{RD}^{2}=60$dB, corresponding to an extreme scenario of $h_{RD}^{2}\gg
h_{SR}^{2}\gg h_{SD}^{2}\gg 1$. Fig. 3 shows the case for $h_{SD}^{2}=5$dB,
$h_{SR}^{2}=10$dB, and $h_{RD}^{2}=10$dB. It is clear that in both cases,
compress-and-forward is always better than the noisy network coding scheme
with the specific $\mathsf{q}^{*}$, and both are within a constant gap to the
cut-set upper bound for all values of $\rho_{z}$.
The decode-and-forward rate is always independent of $\rho_{z}$. In the
asymptotic regime as shown in Fig. 2, the single-tap amplify-and-forward rate
is almost independent of $\rho_{z}$ as well, and it coincides with the decode-
and-forward rate. Both can have an unbounded gap to the cut-set bound as
$h^{2}_{RD}\rightarrow\infty$ and $\rho_{z}\rightarrow\pm 1$. Compress-and-
forward, on the other hand, closely tracks the cut-set bound. (Note that the
above observations are not true in the non-asymptotic SNR regime as shown in
Fig. 3.) The noisy-network-coding scheme, although not as good as compress-
and-forward, nevertheless is always within a constant gap to the cut-set
bound.
Figure 2: Comparison of different relaying schemes for $h_{SD}^{2}=20$dB,
$h_{SR}^{2}=40$dB and $h_{RD}^{2}=60$dB Figure 3: Comparison of different
relaying schemes for $h_{SD}^{2}=5$dB, $h_{SR}^{2}=10$dB, and
$h_{RD}^{2}=10$dB
It is interesting to see that the noisy-network-coding rate resembles the
shape of the cut-set upper bound as shown in both Fig. 2 and Fig. 3. It is
also interesting to note that the decode-and-forward curve touches the cut-set
bound at a particular value of $\rho_{z}$. This is because at this value of
$\rho_{z}$, the relay channel becomes degraded [4, Theorem 1].
## VI Conclusion
This paper investigates different relaying strategies for the three-node
Gaussian relay channel with correlated noises. It is shown that both the
proposed correlation-aware noisy network coding scheme and the conventional
compress-and-forward relaying scheme can achieve to within a constant gap to
the capacity, while the decode-and-forward scheme and the single-tap amplify-
and-forward scheme cannot.
## References
* [1] S. Avestimehr, S. Diggavi, and D. Tse, “Wireless network information flow: a deterministic approach,” _IEEE Trans. Inf. Theory_ , vol. 57, no. 4, pp. 1872–1905, Apr. 2011.
* [2] W. Chang, S.-Y. Chung, and Y. H. Lee, “Gaussian relay channel capacity to within a fixed number of bits,” 2010. [Online]. Available: http://arxiv.org/abs/1011.5065
* [3] S. H. Lim, Y.-H. Kim, A. El Gamal, and S.-Y. Chung, “Noisy network coding,” _IEEE Trans. Inf. Theory_ , vol. 57, no. 5, pp. 3132–3152, May 2011.
* [4] L. Zhang, J. Jiang, A. J. Goldsmith, and S. Cui, “Study of gaussian relay channels with correlated noises,” _IEEE Trans. Commun._ , vol. 59, no. 3, pp. 863–876, Mar. 2011.
* [5] K. S. Gomadam and S. A. Jafar, “The effect of noise correlation in amplify-and-forward relay networks,” _IEEE Trans. Inf. Theory_ , vol. 55, no. 2, pp. 731 –745, Feb. 2009.
* [6] T. M. Cover and A. El Gamal, “Capacity theorems for the relay channel,” _IEEE Trans. Inf. Theory_ , vol. 25, no. 5, pp. 572–584, Sep. 1979.
* [7] R. Dabora and S. Servetto, “On the role of estimate-and-forward with time sharing in cooperative communication,” _IEEE Trans. Inf. Theory_ , vol. 54, no. 10, pp. 4409 –4431, Oct. 2008.
* [8] A. El Gamal, M. Mohseni, and S. Zahedi, “Bounds on capacity and minimum energy-per-bit for AWGN relay channels,” _IEEE Trans. Inf. Theory_ , vol. 52, no. 4, pp. 1545–1561, Apr. 2006.
|
arxiv-papers
| 2011-10-22T20:45:05 |
2024-09-04T02:49:23.501517
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lei Zhou and Wei Yu",
"submitter": "Lei Zhou",
"url": "https://arxiv.org/abs/1110.4999"
}
|
1110.5000
|
# On Noisy Network Coding for a Gaussian Relay Chain Network with Correlated
Noises
Lei Zhou and Wei Yu
Department of Electrical and Computer Engineering,
University of Toronto, Toronto, Ontario M5S 3G4, Canada
emails: {zhoulei, weiyu}@comm.utoronto.ca
###### Abstract
Noisy network coding, which elegantly combines the conventional compress-and-
forward relaying strategy and ideas from network coding, has recently drawn
much attention for its simplicity and optimality in achieving to within
constant gap of the capacity of the multisource multicast Gaussian network.
The constant-gap result, however, applies only to Gaussian relay networks with
independent noises. This paper investigates the application of noisy network
coding to networks with correlated noises. By focusing on a four-node Gaussian
relay chain network with a particular noise correlation structure, it is shown
that noisy network coding can no longer achieve to within constant gap to
capacity with the choice of Gaussian inputs and Gaussian quantization. The
cut-set bound of the relay chain network in this particular case, however, can
be achieved to within half a bit by a simple concatenation of a correlation-
aware noisy network coding strategy and a decode-and-forward scheme.
## I Introduction
The capacity region of the Gaussian relay network has been open for decades.
Recently, the capacities of several relay networks with simple structures have
been approximated to within constant number of bits. For example, for the
three-node Gaussian relay channel, Avestimehr and Tse [1] showed that the
decode-and-forward strategy achieves to within half a bit of the capacity;
Chang, Chung, and Lee [2] proved that the compress-and-forward rate is within
half a bit of the capacity, and the amplify-and-forward rate is within one
bit.
In their breakthrough work, Avestimehr and Tse [1] further showed that, the
capacity of the single-source single-destination Gaussian relay network in
general can be achieved to within constant bits via a universal relaying
scheme called quantize-map-and-forward (QMF). They also showed that, the gap
to capacity is only related to the number of nodes in the network.
Parallel to Avestimehr and Tse’s work, Lim, Kim, El Gamal, and Chung [3]
proposed a noisy network coding strategy that naturally extends the
conventional compress-and-forward scheme of Cover and El Gamal [4] and the
classic network coding by Ahlswede, Cai, Li, and Yeung [5] to noisy networks.
The main idea of noisy network coding is to derive an explicit expression of
the achievable rate for each cut-set of the network. Then, by comparing with
the cut-set upper bound, noisy network coding can be shown to achieve to
within constant gap to the capacity of general multisource multicast Gaussian
networks.
A key assumption made in both [1] and [3] is that the noises in the Gaussian
relay network are independent with each other. This assumption may not hold in
practical systems, where common interferences from other sources play a role.
In this paper, we are interested in the following question. In the context of
Gaussian relay networks with correlated noises, can noisy network coding
achieve within constant bits to the capacity as well? This paper gives a
negative answer by studying a four-node Gaussian chain network with correlated
noises. It is shown that, in a certain scenario, the noisy-network-coding rate
(with Gaussian input and Gaussian quantization) has an unbounded gap to the
cut-set bound, whereas a concatenation of a modified correlation-aware noisy
network coding strategy and a conventional decode-and-forward scheme achieves
to within half a bit of the cut-set bound in this specific case.
## II Channel Model
Figure 1: A four-node Gaussian relay chain network
The four-node Gaussian relay chain, as depicted in Fig. 1, consists of a
source node, a destination node, and two relay nodes. The source communicates
with the destination with the help from the two relays in between. Information
passes from the source to the neighboring relay, and to the next, then finally
to the destination. The input-output relationship can be described as follows:
$\displaystyle Y_{1}$ $\displaystyle=$ $\displaystyle h_{1}X+Z_{1},$
$\displaystyle Y_{2}$ $\displaystyle=$ $\displaystyle h_{2}X_{1}+Z_{2},$
$\displaystyle Y_{3}$ $\displaystyle=$ $\displaystyle h_{3}X_{2}+Z_{3}.$
Without loss of generality, assume that the transmit power of all nodes are
normalized to one, and the variances of the receiver noises are also
normalized to one, i.e., $Z_{i}\sim\mathcal{N}(0,1)$. The receiver noises are
i.i.d. in time, but the noises $[Z_{1},Z_{2},Z_{3}]$ are correlated with the
following correlation matrix:
$K_{Z}=\left[\begin{array}[]{ccc}1&\rho_{12}&\rho_{13}\\\
\rho_{12}&1&\rho_{23}\\\ \rho_{13}&\rho_{23}&1\end{array}\right],$
where $K_{z}$ is positive semidefinite and $\rho_{ij}$ is the correlation
coefficient between $Z_{i}$ and $Z_{j}$. Note that the relay operation must be
causal in time.
## III Suboptimality of the Noisy Network Coding
We begin by showing that using noisy network coding with the choices of
Gaussian inputs and Gaussian quantization noises, the gap between the
achievable rate and the cut-set bound can be unbounded for a Gaussian relay
chain network with a certain noise correlation structure. First, an upper
bound to the cut-set bound of the four-node relay chain can be computed as
follows:
$\displaystyle\max_{p(x,x_{1},x_{2})}\min\\{I(X;Y_{1}Y_{2}Y_{3}|X_{1}X_{2}),I(XX_{1};Y_{2}Y_{3}|X_{2}),$
(1) $\displaystyle\quad\quad\quad
I(XX_{1}X_{2};Y_{3}),I(XX_{2};Y_{1}Y_{3}|X_{1})\\}$ $\displaystyle\leq$
$\displaystyle\min\\{\max I(X;Y_{1}Y_{2}Y_{3}|X_{1}X_{2}),\max
I(XX_{1};Y_{2}Y_{3}|X_{2}),$ $\displaystyle\quad\quad\;\max
I(XX_{1}X_{2};Y_{3}),\max I(XX_{2};Y_{1}Y_{3}|X_{1})\\}$ $\displaystyle=$
$\displaystyle\min\\{\overline{C}(\mathcal{S}_{1}),\overline{C}(\mathcal{S}_{2}),\overline{C}(\mathcal{S}_{3}),\overline{C}(\mathcal{S}_{4})\\}$
$\displaystyle\triangleq$ $\displaystyle\overline{C}$
where the cut-sets are defined as $\mathcal{S}_{1}=\\{X\\}$,
$\mathcal{S}_{2}=\\{X,X_{1}\\}$, $\mathcal{S}_{3}=\\{X,X_{1},X_{2}\\}$, and
$\mathcal{S}_{4}=\\{X,X_{2}\\}$, and $\overline{C}(\mathcal{S}_{i}),i=1,2,3,4$
are the four cut-set upper bounds, which can be calculated as
$\displaystyle\overline{C}(\mathcal{S}_{1})$ $\displaystyle=$
$\displaystyle\max I(X;Y_{1}Y_{2}Y_{3}|X_{1}X_{2})$ (2) $\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(1+\frac{(1-\rho_{23}^{2})h_{1}^{2}}{|K_{Z}|}\right),$
and
$\displaystyle\overline{C}(\mathcal{S}_{2})$ $\displaystyle=$
$\displaystyle\max I(XX_{1};Y_{2}Y_{3}|X_{2})$ (3) $\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(1+\frac{h_{2}^{2}}{1-\rho_{23}^{2}}\right),$
and
$\displaystyle\overline{C}(\mathcal{S}_{3})$ $\displaystyle=$
$\displaystyle\max I(XX_{1}X_{2};Y_{3})$ (4) $\displaystyle=$
$\displaystyle\frac{1}{2}\log(1+h_{3}^{2}),$
and
$\displaystyle\overline{C}(S_{4})$ $\displaystyle=$ $\displaystyle\max
I(XX_{2};Y_{1}Y_{3}|X_{1})$ (5) $\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(1+\frac{h_{1}^{2}+h_{3}^{2}+h_{1}^{2}h_{3}^{2}}{1-\rho_{13}^{2}}\right)$
$\displaystyle\geq$ $\displaystyle\overline{C}(\mathcal{S}_{3}),$
which is redundant.
The main point of noisy network coding is that an achievable rate can be
derived for each of the cut-sets $\mathcal{S}_{1}$, $\mathcal{S}_{2}$, and
$\mathcal{S}_{3}$ using a generalization of the compress-and-forward scheme.
For convenience, we state the achievable rates as follows.
###### Theorem 1 (Noisy Network Coding Theorem [3]).
Let $\mathcal{D}=\mathcal{D}_{1}=\mathcal{D}_{2}=\cdots=\mathcal{D}_{N}$. A
rate tuple $(R_{1},\cdots,R_{N})$ is achievable for the DMN $p(y^{N}|x^{N})$
if there exists some joint pmf
$p(q)\prod_{k=1}^{N}p(x_{k}|q)p(\hat{y}_{k}|y_{k},x_{k},q)$ such that
$\displaystyle R(\mathcal{S})$ $\displaystyle<$
$\displaystyle\min_{d\in\mathcal{\mathcal{S}}^{c}\cap\mathcal{D}}I(X(\mathcal{S});\hat{Y}(\mathcal{S}^{c}),Y_{d}|X(\mathcal{S}^{c}),Q)$
(6)
$\displaystyle\quad-I(Y(\mathcal{S});\hat{Y}(\mathcal{S})|X^{N},\hat{Y}(\mathcal{S}^{c}),Y_{d},Q)$
for all cutsets $\mathcal{S}\subseteq[1:N]$ with
$S^{c}\cap\mathcal{D}\neq\emptyset$, where
$R(\mathcal{S})=\sum_{k\in\mathcal{\mathcal{S}}}R_{k}$.
Although the quantization in the above noisy network coding theorem can in
theory have arbitrary distributions, Gaussian inputs and Gaussian quantization
noises are usually adopted for Gaussian networks [Wang_ReceiverCooperation]
[6], and are shown to achieve constant gap to capacity for networks with
uncorrelated noises [3]. Thus, this paper follows the same choice, i.e.,
$\displaystyle\hat{Y}_{i}=Y_{i}+\hat{Z}_{i}$ (7)
where the quantization noise $\hat{Z}_{i}\sim\mathcal{N}(0,\mathsf{q}_{i})$ is
independent with everything else. Now applying the noisy network coding
theorem with Gaussian inputs and Gaussian quantization noises, the following
achievable rates for cut-sets $\mathcal{S}_{1}$, $\mathcal{S}_{2}$, and
$\mathcal{S}_{3}$ can be derived:
$\displaystyle R(\mathcal{S}_{1})$ $\displaystyle=$ $\displaystyle
I(X;\hat{Y}_{1}\hat{Y}_{2}Y_{3}|X_{1}X_{2})$ (14) $\displaystyle=$
$\displaystyle h\left(\begin{array}[]{r}h_{1}X+Z_{1}+\hat{Z}_{1}\\\
Z_{2}+\hat{Z}_{2}\\\
Z_{3}\end{array}\right)-h\left(\begin{array}[]{r}Z_{1}+\hat{Z}_{1}\\\
Z_{2}+\hat{Z}_{2}\\\ Z_{3}\end{array}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(1+\frac{(1+\mathsf{q}_{2}-\rho_{23}^{2})h_{1}^{2}}{|K_{\beta}|}\right),$
(15)
where
$|K_{\beta}|=\left|\begin{array}[]{ccc}1+\mathsf{q}_{1}&\rho_{12}&\rho_{13}\\\
\rho_{12}&1+\mathsf{q}_{2}&\rho_{23}\\\
\rho_{13}&\rho_{23}&1\end{array}\right|,$ (16)
and
$\displaystyle R(\mathcal{S}_{2})$ $\displaystyle=$ $\displaystyle
I(XX_{1};\hat{Y_{2}}Y_{3}|X_{2})-I(Y_{1};\hat{Y}_{1}|XX_{1}X_{2}\hat{Y}_{2}Y_{3})$
(17) $\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(1+\frac{h_{2}^{2}}{1+\mathsf{q}_{2}-\rho_{23}^{2}}\right)$
$\displaystyle\qquad-\frac{1}{2}\log\left(1+\frac{1-\rho_{13}^{2}}{\mathsf{q}_{1}}\right).$
and
$\displaystyle R(\mathcal{S}_{3})$ $\displaystyle=$ $\displaystyle
I(XX_{1}X_{2};Y_{3})-I(Y_{1}Y_{2};\hat{Y}_{1}\hat{Y}_{2}|XX_{1}X_{2}Y_{3})$
(20) $\displaystyle=$ $\displaystyle\frac{1}{2}\log(1+h_{3}^{2})$
$\displaystyle-\frac{1}{2}\log\frac{\left|\begin{array}[]{cc}1-\rho_{13}^{2}+\mathsf{q}_{1}&\rho_{12}-\rho_{13}\rho_{23}\\\
\rho_{12}-\rho_{13}\rho_{23}&1-\rho_{23}^{2}+\mathsf{q}_{2}\end{array}\right|}{\mathsf{q}_{1}\mathsf{q}_{2}}.$
The achievable rate is then upper bounded by the minimum of the three:
$R\leq\min\\{R(\mathcal{S}_{1}),R(\mathcal{S}_{2}),R(\mathcal{S}_{3})\\}.$
(21)
Now, consider a special scenario when the noise $Z_{3}$ is independent with
both $Z_{1}$ and $Z_{2}$, and channel strengths $h_{2}^{2}$ and $h_{3}^{2}$
scale with $h_{1}^{2}$, i.e.,
$\rho_{13}=\rho_{23}=0,$ (22)
and
$h_{2}^{2}=h_{3}^{2}=\frac{h_{1}^{2}}{1-\rho_{12}^{2}}.$ (23)
This special setting gives us the following cut-set bounds:
$\overline{C}(\mathcal{S}_{1})=\overline{C}(\mathcal{S}_{2})=\overline{C}(\mathcal{S}_{3})=\frac{1}{2}\log\left(1+\frac{h_{1}^{2}}{1-\rho_{12}^{2}}\right),$
(24)
and the following achievable rates for cut-sets $\mathcal{S}_{1}$ to
$\mathcal{S}_{3}$:
$\displaystyle R(\mathcal{S}_{1})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(1+\frac{h_{1}^{2}}{1+\mathsf{q}_{1}-\frac{\rho_{12}^{2}}{1+\mathsf{q}_{2}}}\right),$
$\displaystyle R(\mathcal{S}_{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(1+\frac{h_{1}^{2}}{(1-\rho_{12}^{2})(1+\mathsf{q}_{2})}\right)$
$\displaystyle-\frac{1}{2}\log\left(1+\frac{1}{\mathsf{q}_{1}}\right),$
$\displaystyle R(\mathcal{S}_{3})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(1+\frac{h_{1}^{2}}{1-\rho_{12}^{2}}\right)$
$\displaystyle-\frac{1}{2}\log\left(1+\frac{\mathsf{q}_{1}+\mathsf{q}_{2}+1-\rho_{12}^{2}}{\mathsf{q}_{1}\mathsf{q}_{2}}\right).$
Next, we show that, the gaps
$\overline{C}(\mathcal{S}_{1})-R(\mathcal{S}_{1})$,
$\overline{C}(\mathcal{S}_{2})-R(\mathcal{S}_{2})$,
$\overline{C}(\mathcal{S}_{3})-R(\mathcal{S}_{3})$ cannot be made all finite
when $\rho_{12}^{2}\rightarrow 1$. First, the gap on the cut-set
$\mathcal{S}_{1}$ is given by
$\displaystyle\Delta(\mathcal{S}_{1})$ $\displaystyle=$
$\displaystyle\overline{C}(\mathcal{S}_{1})-R(\mathcal{S}_{1})$ (25)
$\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(1+\frac{h_{1}^{2}}{1-\rho_{12}^{2}}\right)$
$\displaystyle-\frac{1}{2}\log\left(1+\frac{h_{1}^{2}}{1+\mathsf{q}_{1}-\frac{\rho_{12}^{2}}{1+\mathsf{q}_{2}}}\right),$
and the gap on the cut-set $\mathcal{S}_{2}$ is lower bounded by
$\displaystyle\Delta(\mathcal{S}_{2})$ $\displaystyle=$
$\displaystyle\overline{C}(\mathcal{S}_{2})-R(\mathcal{S}_{2})$ (26)
$\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(1+\frac{h_{1}^{2}}{1-\rho_{12}^{2}}\right)+\frac{1}{2}\log\left(1+\frac{1}{\mathsf{q}_{1}}\right)$
$\displaystyle-\frac{1}{2}\log\left(1+\frac{h_{1}^{2}}{(1-\rho_{12}^{2})(1+\mathsf{q}_{2})}\right)$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{1}{\mathsf{q}_{1}}\right),$
and the gap on the cut-set $\mathcal{S}_{3}$ is lower bounded by the same
number as well, i.e.,
$\displaystyle\Delta(\mathcal{S}_{3})$ $\displaystyle=$
$\displaystyle\overline{C}(\mathcal{S}_{3})-R(\mathcal{S}_{3})$ (27)
$\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{q}_{1}+\mathsf{q}_{2}+1-\rho_{12}^{2}}{\mathsf{q}_{1}\mathsf{q}_{2}}\right)$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\log\left(1+\frac{1}{\mathsf{q}_{1}}\right).$
Now, since
$\overline{C}-R\geq\max\\{\Delta(\mathcal{S}_{1}),\Delta(\mathcal{S}_{2}),\Delta(\mathcal{S}_{3})\\},$
(28)
in order to make $\overline{C}-R$ finite, all three gaps have to be upper
bounded by a finite number. Inspecting the gap of $\Delta(\mathcal{S}_{1})$ in
(25), in order to make it finite when $\rho_{12}\rightarrow\infty$, both
$\mathsf{q}_{1}$ and $\mathsf{q}_{2}$ have to go to zero. However, in this
case, $\Delta(\mathcal{S}_{2})$ and $\Delta(\mathcal{S}_{3})$ are apparently
unbounded. Therefore, in the scenario of (22) and (23), as $\rho_{12}^{2}$
goes to $1$, it is impossible to keep all three gaps
$\Delta(\mathcal{S}_{1})$, $\Delta(\mathcal{S}_{2})$, and
$\Delta(\mathcal{S}_{3})$ finite simultaneously. As a consequence, for the
four-node Gaussian chain network with correlated noises, the noisy network
coding achievable rate with the choice of Gaussian inputs and Gaussian
quantization noises has an unbounded gap to the cut-set upper bound.
## IV An Optimal Concatenated Scheme
It is known that the cut-set upper bound is not always tight for the relay
channel [7, 8], but for the four-node Gaussian chain network, does the cut-set
bound have an infinite gap to capacity? Or, is it the noisy networking coding
achievable rate that has an infinite gap to capacity? To answer this question,
we show in the following that the cut-set bound (24) can actually be achieved
to within half a bit for this four-node relay network with the particular
noise correlation structure (22) by a simple concatenation of a correlation-
aware noisy network coding strategy and a conventional decode-and-forward
scheme. This justifies the suboptimality of the noisy network coding for
Gaussian relay networks with correlated noises.
Inspecting the structure of the four-node relay network in Fig. 1 and the
special correlation structure (22), it is easy to see that the last node is
essentially independent of the first three nodes in this example. Now the
first three nodes (from the source node $X$ to the second relay node $Y_{2}$)
forms a three-node Gaussian relay channel, so we can apply the noisy network
coding theorem just to the first three nodes. With the source message decoded
at $Y_{2}$, the second relay node can then re-encode the source information
and forward to the destination $Y_{3}$. With $Y_{1}$ serving as a noisy-
network-coding type of relay and $Y_{2}$ serving as a decode-and-forward type
of relay, this is essentially a concatenation of the noisy network coding and
the decode-and-forward scheme. The achievable rate of this concatenated scheme
can be derived as follows.
In the first relaying stage where $Y_{1}$ serves as a noisy-network-coding
type of relay, according to the noisy network coding theorem [3, Theorem 1],
$Y_{2}$ can decode the source message if the following rate is satisfied:
$\displaystyle R$ $\displaystyle\leq$
$\displaystyle\min\\{I(X,X_{1};Y_{2})-I(Y_{1};\hat{Y}_{1}|X,X_{1},Y_{2}),$
(29) $\displaystyle\quad\quad\;I(X;Y_{2},\hat{Y}_{1}|X_{1})\\},$
for some distribution
$p(x,x_{1},y_{1},\hat{y}_{1})=p(x)p(x_{1})p(y_{1}|x,x_{1})p(\hat{y}_{1}|x_{1},y_{1}).$
Substituting Gaussian inputs $X\sim\mathcal{N}(0,1)$,
$X_{1}\sim\mathcal{N}(0,1)$, and Gaussian quantization signal
$\hat{Y}_{1}=Y_{1}+\hat{Z}_{1}$, where
$\hat{Z}_{1}\sim\mathcal{N}(0,\mathsf{q}_{1})$ is independent with everything
else, we have the following achievable rate in the first stage:
$\displaystyle R$ $\displaystyle\leq$
$\displaystyle\min\left\\{\frac{1}{2}\log\left(1+\frac{h_{1}^{2}}{1-\rho_{12}^{2}}\right)-\frac{1}{2}\log\left(1+\frac{\mathsf{q}_{1}}{1-\rho_{12}^{2}}\right),\right.$
(30)
$\displaystyle\left.\quad\quad\;\;\frac{1}{2}\log(1+h_{2}^{2})-\frac{1}{2}\log\left(1+\frac{1-\rho_{12}^{2}}{\mathsf{q}_{1}}\right)\right\\}.$
Next, with the source message decoded at $Y_{2}$, the second relay node acts
as a decode-and-forward type of relay, which re-encodes and forwards the
source message to the destination $Y_{3}$ through the Gaussian channel of
channel gain $h_{3}$. The destination can successfully decode the source
message if
$\displaystyle R\leq\frac{1}{2}\log(1+h_{3}^{2}).$ (31)
Combining the above rate constraints (30) and (31) gives us the following
achievable rate by the concatenated scheme:
$\displaystyle R$ $\displaystyle\leq$
$\displaystyle\min\left\\{\frac{1}{2}\log\left(1+\frac{h_{1}^{2}}{1-\rho_{12}^{2}}\right)-\frac{1}{2}\log\left(1+\frac{\mathsf{q}_{1}}{1-\rho_{12}^{2}}\right),\right.$
(32)
$\displaystyle\quad\quad\;\;\frac{1}{2}\log(1+h_{2}^{2})-\frac{1}{2}\log\left(1+\frac{1-\rho_{12}^{2}}{\mathsf{q}_{1}}\right),$
$\displaystyle\quad\quad\;\;\left.\frac{1}{2}\log(1+h_{3}^{2})\right\\}.$
Comparing the above achievable rate with the cut-set upper bound with
$\rho_{12}$ and $\rho_{23}$ set to zero:
$\displaystyle\overline{C}$ $\displaystyle=$
$\displaystyle\min\left\\{\frac{1}{2}\log\left(1+\frac{h_{1}^{2}}{1-\rho_{12}^{2}}\right),\frac{1}{2}\log(1+h_{2}^{2}),\right.$
(33) $\displaystyle\quad\quad\;\;\left.\frac{1}{2}\log(1+h_{3}^{2})\right\\},$
we have the difference upper bounded by
$\overline{C}-R\leq\max\left\\{\frac{1}{2}\log\left(1+\frac{\mathsf{q}_{1}}{1-\rho_{12}^{2}}\right),\frac{1}{2}\log\left(1+\frac{1-\rho_{12}^{2}}{\mathsf{q}_{1}}\right)\right\\}$
(34)
It is easy to see that the first term monotonically increases with
$\mathsf{q}_{1}$ while the second term monotonically decreases. As a result,
to minimize the maximum of the two terms, we need
$\displaystyle\frac{1}{2}\log\left(1+\frac{1-\rho_{12}^{2}}{\mathsf{q}_{1}^{*}}\right)=\frac{1}{2}\log\left(1+\frac{\mathsf{q}_{1}^{*}}{1-\rho_{12}^{2}}\right),$
(35)
which results in the optimal correlation-aware quantization level
$\mathsf{q}_{1}^{*}=1-\rho_{12}^{2}$. Substituting $\mathsf{q}_{1}^{*}$ into
(34) gives us $\overline{C}-R<\frac{1}{2}$. Therefore, for the four-node
Gaussian chain network as shown in Fig. 1, in the scenario where
$\rho_{13}=\rho_{23}=0$, the cut-set upper bound can be achieved to within
constant gap.
Figure 2: Noisy network coding vs. concatenated scheme
Fig. 2 shows a numerical example for comparing noisy network coding and the
concatenated scheme. In the simulation, we let $h_{1}^{2}=20$dB and all other
channel parameters are set to satisfy (22) and (23). For the quantization
parameters, we choose the optimal quantization level
$\mathsf{q}_{1}=1-\rho_{12}^{2}$ and let $\mathsf{q}_{2}=1$. As can be seen
from the figure, when $\rho_{12}$ approaches to $+1$ or $-1$, both the cut-set
upper bound and the achievable rate by the concatenated scheme go to infinity.
However, the achievable rate by the noisy network coding scheme remains
finite, making the gap to the cut-set bound unbounded.
## V Conclusion
This paper studies the optimality of the noisy network coding for a four-node
Gaussian chain network with correlated noises. It is shown that, under a
certain noise correlation structure, noisy network coding with Gaussian inputs
and Gaussian quantization noises has an infinite gap to the cut-set upper
bound. But, the upper bound can be achieved to within half a bit in this
specific case by a simple concatenation of a correlation-aware noisy network
coding strategy and a decode-and-forward scheme.
## References
* [1] S. Avestimehr, S. Diggavi, and D. Tse, “Wireless network information flow: a deterministic approach,” _Submitted to IEEE Trans. Inf. Theory_ , 2009.
* [2] W. Chang, S.-Y. Chung, and Y. H. Lee, “Gaussian relay channel capacity to within a fixed number of bits,” 2010. [Online]. Available: http://arxiv.org/abs/1011.5065
* [3] S.-Y. Lim, Y.-H. Kim, A. El Gamal, and S.-Y. Chung, “Noisy network coding,” _Submitted to IEEE Trans. Inf. Theory_ , 2010.
* [4] T. M. Cover and A. El Gamal, “Capacity theorems for the relay channel,” _IEEE Trans. Inf. Theory_ , vol. 25, no. 5, pp. 572–584, Sep. 1979.
* [5] R. Ahlswede, N. Cai, R. Li, and R. W. Yeung, “Network information flow,” _IEEE Trans. Inf. Theory_ , vol. 46, pp. 1004 –1016, Jul 2000.
* [6] L. Zhou and W. Yu, “Gaussian z-interference channel with a relay link: achievability region and asymptotic sum capacity,” 2010. [Online]. Available: http://arxiv.org/abs/1006.5087
* [7] M. Aleksic, P. Razaghi, and W. Yu, “Capacity of a class of modulo-sum relay channels,” _IEEE Trans. Inf. Theory_ , vol. 55, no. 3, pp. 921–930, Mar. 2009.
* [8] Z. Zhang, “Partial converse for a relay channel,” _IEEE Trans. Inf. Theory_ , vol. 34, no. 5, pp. 1106–1110, Sep. 1988.
|
arxiv-papers
| 2011-10-22T20:52:17 |
2024-09-04T02:49:23.508460
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lei Zhou and Wei Yu",
"submitter": "Lei Zhou",
"url": "https://arxiv.org/abs/1110.5000"
}
|
1110.5002
|
Testing the approximations described in
“Asymptotic formulae for likelihood-based tests of new physics”
Eric Burns, Wade Fisher
Department of Physics and Astronomy, Michigan State University, East Lansing,
MI 48825
###### Abstract
“Asymptotic formulae for likelihood-based tests of new physics” presents a
mathematical formalism for a new approximation for hypothesis testing in high
energy physics. The approximations are designed to greatly reduce the
computational burden for such problems. We seek to test the conditions under
which the approximations described remain valid. To do so, we perform parallel
calculations for a range of scenarios and compare the full calculation to the
approximations to determine the limits and robustness of the approximation. We
compare this approximation against values calculated with the Collie
framework, which for our analysis we assume produces true values.
Keywords: systematic uncertainties, profile likelihood, hypothesis test,
confidence interval, frequentist methods, asymptotic methods, asimov data set,
Collie, AWW approximation
###### Contents
1. 1 Introduction
2. 2 Mathematical Formalism
1. 2.1 Basic Statistics
2. 2.2 The Likelihood Function and Maximization
3. 2.3 Likelihood Approximation for Binned Data
4. 2.4 Particle Physics Statistics
3. 3 The Asimov Data Set Approximation
1. 3.1 The Wald Equation
2. 3.2 The Tevatron Test Statistic
4. 4 Pseudo-data Tests
1. 4.1 Background-only Rate Systematic Uncertainty
2. 4.2 Signal and Background Rate Systematic Uncertainties
3. 4.3 Asymmetric Gaussian “Flat” Systematic Uncertainties
4. 4.4 Uncertainty on Background Shape
5. 4.5 Varying the Number of Histogram Bins
6. 4.6 Variation in the Number of Events
5. 5 Conclusion
## 1 Introduction
One of the primary goals in experimental particle physics is the search for
new particles. In order to determine whether or not a particle has been
discovered statistical hypothesis tests are used. The probability of finding
an outcome as extreme as the one observed can be compared to a predetermined
threshold to ascertain whether or not discovery has occured.
Unfortunately, due to the sheer magnitude of the amount of data involved in
the search for the new particles, determining probabilities is often
computationally intensive. In this paper we examine the approximation
presented in “Asymptotic formulae for likelihood-based tests of new physics,”
to find the limits of its applicability. This approximation is evaluated to
determine when it successfully reproduces the results from a full semi-
frequentist computation with no approximations (Section 4). Conclusions based
on these findings are presented in Section 5.
Presented below is the necessary prerequisite knowledge; this includes general
statistics (Section 2.1), such as hypothesis testing and the likelihood ratio
(Section 2.2), as well as an explanation of how these techniques are used in
particle physics (Section 2.4). We then explain the mathematical basis for the
Asimov data set based upon results from Wilks and Wald (Section 3), as given
by the authors of [1]. The Asimov data set is a representative set of values
that theoretically represents the true parameters of the full ensemble. This
set contains represents an ensemble of simulated data; later it is described
in greater depth (Section 3). Henceforth the three approximations together
will be abbreivated as the AWW approximation, an acronym of their names. This
allows us to examine the possibility that the approximation generates valid
parameters. The approximation, the full mathematical formalism and subsequent
evidence are presented in (arXiv:1007.1727v2), upon which our explanation and
formalism are based [1].
## 2 Mathematical Formalism
Presented here are some basic statistical principles, such as hypothesis
testing and test statistics, as well as more complex ideas like the likelihood
function and it’s application to binned data. This section ends with a brief
overview of statistical methods used in particle physics.
### 2.1 Basic Statistics
A hypothesis is a suggested solution to explain a given phenomenom. One often
compares the validity of two hypotheses through statistical testing, where one
decides whether a given null hypothesis, $H_{0}$, should be rejected in favor
of the alternate hypotheses, $H_{1}$. In particle physics the null hypothesis
typically contains all known processes and the alternate hypothesis may also
contain a new process or particle. Meaning, the null hypothesis would be
background-only and the alternate hypothesis would then be signal-plus-
background.
A test statistic is a function of the sample and assumed to be a numerical
summary of the data that can be used to reject, or fail to reject, a
hypothesis. This can be done by calculating the probability of obtaining a
test statistic as extreme as the one observed, which is called a $p$-value.
This represents the level of agreement between the data and a single
hypothesis. The $p$-value can be measured against a significance level
$\alpha$, defined as the critical $p$-value; i.e. $p$ must be less than or
equal to $\alpha$ to reject a given hypothesis.
The $p$-value can also be converted to a standardized value, such as a
$Z$-score, the number of standard deviations a datum is from the mean; $Z$ is
given as a function of $p$ by
$Z(p)=\Phi^{-1}(1-p)\,,$ (1)
where $\Phi^{-1}(p)=\sqrt{2}Erf^{-1}(2p-1)$, the quantile of the standard
Gaussian111Erf is the error function.
$Erf^{-1}(z)=\displaystyle\sum_{k=0}^{\infty}\frac{c_{k}}{2k+1}(\frac{\sqrt{\pi}}{2}z)^{2k+1}$
where
$c_{k}=\displaystyle\sum_{m=0}^{k-1}\frac{c_{m}c_{k-1-m}}{(m+1)(2m+1)}=\\{1,1,\frac{7}{6},\frac{127}{90}\\},...$.
At $\alpha=0.05$, a commonly used signficance level, the $Z$-score is equal to
1.64 for a one-sided test; a one-sided test is used when the critical outcomes
capable of rejecting a hypothesis occur on only one side of the distribution.
Because we can distinguish between positive and negative fluctuations in our
tests we use a one-sided test, with a 95% confidence level (CL) exclusion.
### 2.2 The Likelihood Function and Maximization
The likelihood of a given observation given a set of parameters is equal to
the probability of a set of parameter values given an observation.
Consider a set of N observables, contained in
$\mathchoice{\mbox{\boldmath$\displaystyle\bf
x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf
x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}}=(x_{1},...,x_{N})$, described
by probability distribution function (p.d.f.)
$f(\mathchoice{\mbox{\boldmath$\displaystyle\bf
x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf
x$}}{\mbox{\boldmath$\scriptscriptstyle\bf
x$}};\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})$,
where
$\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}=(\theta_{1},...,\theta_{n})$
are the unknown parameters, which also known as the nuisance parameters.
Assuming statistical independence between the measurements $x_{i}$, then the
likelihood function L($\bf\theta$) is
$L(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})=\prod_{i=1}^{N}{f(x_{i};\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})}.$
(2)
The $\textstyle\bf\theta$ values that maximize this function are denoted
$\hat{\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}}$.
In order to find the maximum likelihood (ML) estimators one can solve the
formula [6]
$\frac{\partial\ln L}{\partial\theta_{i}}=0,\indent i=1,...,n.$ (3)
The covariance matrix of the ML estimators,
$V_{ij}=\mbox{cov}[\hat{\theta_{i}},\hat{\theta_{j}}]$ can be used to estimate
the standard deviation, $\sigma$. We can find this by first finding the
inverse covariance matrix, which can be approximated as
$(\hat{V^{-1}})_{ij}=-\frac{\partial^{2}\ln
L}{\partial\theta_{i}\partial\theta_{j}}\bigg{|}_{\mathchoice{\mbox{\boldmath$\displaystyle\bf\hat{\theta}$}}{\mbox{\boldmath$\textstyle\bf\hat{\theta}$}}{\mbox{\boldmath$\scriptstyle\bf\hat{\theta}$}}{\mbox{\boldmath$\scriptscriptstyle\bf\hat{\theta}$}}},$
(4)
and then invert the resulting matrix to find the standard deviation. This is
also known as the curvature matrix, and can only be used when the positive and
negative deviations are equal.
### 2.3 Likelihood Approximation for Binned Data
If a sample size is large it is often easier bin the data into a histogram.
This results in a vector $\mathchoice{\mbox{\boldmath$\displaystyle\bf
n$}}{\mbox{\boldmath$\textstyle\bf n$}}{\mbox{\boldmath$\scriptstyle\bf
n$}}{\mbox{\boldmath$\scriptscriptstyle\bf n$}}=(n_{1},...,n_{N})$ with
expectation value
$\mathchoice{\mbox{\boldmath$\displaystyle\bf\nu$}}{\mbox{\boldmath$\textstyle\bf\nu$}}{\mbox{\boldmath$\scriptstyle\bf\nu$}}{\mbox{\boldmath$\scriptscriptstyle\bf\nu$}}=E[\mathchoice{\mbox{\boldmath$\displaystyle\bf
n$}}{\mbox{\boldmath$\textstyle\bf n$}}{\mbox{\boldmath$\scriptstyle\bf
n$}}{\mbox{\boldmath$\scriptscriptstyle\bf n$}}]$ and p.d.f.
$f(\mathchoice{\mbox{\boldmath$\displaystyle\bf
n$}}{\mbox{\boldmath$\textstyle\bf n$}}{\mbox{\boldmath$\scriptstyle\bf
n$}}{\mbox{\boldmath$\scriptscriptstyle\bf
n$}},\mathchoice{\mbox{\boldmath$\displaystyle\bf\nu$}}{\mbox{\boldmath$\textstyle\bf\nu$}}{\mbox{\boldmath$\scriptstyle\bf\nu$}}{\mbox{\boldmath$\scriptscriptstyle\bf\nu$}})$.
Maximizing the likelihood ratio is equivalent to minimizing the quantity
$-2\ln\lambda(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})$.
For independent, Poisson distributed $n_{i}$ this quantity is [5]
$-2\ln\lambda(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})=2\sum_{i=1}^{N}\left[\nu_{i}(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})-n_{i}+n_{i}\ln\frac{n_{i}}{\nu_{i}(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})}\right],$
(5)
where the last term is zero when $n_{i}=0$. According to Wilks’ theorem, for
sufficiently large samples that meet certain regularity conditions, the
minimum of Eq. (5) follows a $\chi^{2}$ distribution, allowing the usage of
goodness-of-fit tests [2].
### 2.4 Particle Physics Statistics
This subsection describes how the forementioned statistical principles are
often applied in particle physics. In particle physics a $Z$-score greater
than or equal to 5, or $p=2.87\times 10^{-7}$ for a one-sided tail, is usually
required for discovery, which results from the rejection of the background-
only hypothesis.
For binned data with a histogram of variable $x$ and information
$\mathchoice{\mbox{\boldmath$\displaystyle\bf
n$}}{\mbox{\boldmath$\textstyle\bf n$}}{\mbox{\boldmath$\scriptstyle\bf
n$}}{\mbox{\boldmath$\scriptscriptstyle\bf n$}}=(n_{1},....,n_{N})$, the
expectation value
$E[n_{i}]=\mu s_{i}+b_{i},$ (6)
where $\mu$ is the signal strength, and $s_{i}$ and $b_{i}$ are the mean
number of entries in the $i$th bin, meaning [1]
$\displaystyle s_{i}=s_{\rm tot}\int_{{\rm
bin}\,i}f_{s}(x;\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}_{s})\,dx\,,$
(7) $\displaystyle b_{i}=b_{\rm tot}\int_{{\rm
bin}\,i}f_{b}(x;\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}_{b})\,dx\,.$
(8)
Here
$f_{s}(x;\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}_{s}$)
and
$f_{b}(x;\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}_{b}$)
are the p.d.f.s of the variable $x$ for signal and background events
respectively. The signal strength is equal to zero for the background-only
hypothesis and one for the nominal signal hypothesis. Henceforth,
$\textstyle\bf\theta$ contains all nuisance parameters, i.e.
$\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}=(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta_{s}$}}{\mbox{\boldmath$\textstyle\bf\theta_{s}$}}{\mbox{\boldmath$\scriptstyle\bf\theta_{s}$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta_{s}$}},\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta_{b}$}}{\mbox{\boldmath$\textstyle\bf\theta_{b}$}}{\mbox{\boldmath$\scriptstyle\bf\theta_{b}$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta_{b}$}},b_{tot})$;
$s_{tot}$ is not contained in $\textstyle\bf\theta$ because it’s value is
fixed by the prediction from the nominal signal hypothesis.
One can create a control sample that measures only background events, with
information contained in histogram
$\mathchoice{\mbox{\boldmath$\displaystyle\bf
m$}}{\mbox{\boldmath$\textstyle\bf m$}}{\mbox{\boldmath$\scriptstyle\bf
m$}}{\mbox{\boldmath$\scriptscriptstyle\bf m$}}=(m_{1},...,m_{M})$ the
expectation value of $m_{i}$ is
$E[m_{i}]=u_{i}(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}),$
(9)
where $u_{i}$ is dependent on the nuisance parameters. The purpose of the
control sample is to add useful constraints to the nuisance parameters.
Using the signal-plus-background and background-only information, the
likelihood function can be written as a product of two Poisson probabilities
$L(\mu,\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})=\prod_{j=1}^{N}\frac{(\mu
s_{j}+b_{j})^{n_{j}}}{n_{j}!}e^{-(\mu
s_{j}+b_{j})}\;\;\prod_{k=1}^{M}\frac{u_{k}^{m_{k}}}{m_{k}!}\,e^{-u_{k}}\;.$
(10)
The test statistic we are interested in is $\gamma=-2\ln\lambda(\mu)$, where
$\lambda(\mu)=\frac{L(\mu,\hat{\hat{\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}}})}{L(\hat{\mu},\hat{\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}})}$
(11)
is the profile likelihood ratio. Here
$\hat{\hat{\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}}}$
denotes the conditional maximum-likelihood estimator for the specified $\mu$;
$\hat{\mu}$ and
$\hat{\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}}$
are the unconditional maximum-likelihood estimators.
Assigning our value as $\gamma$’, we can calculate the $p$-value from
$p(\mu)=\int_{\gamma}^{\infty}f(\gamma|\mu)dt,$ (12)
where $f(\gamma|\mu)$ is the p.d.f. of $\gamma$ for the given signal strength
$\mu$ [1].
## 3 The Asimov Data Set Approximation
The conditional definition of the Asimov data set is that when one uses it to
evaluate the estimators for all parameters one obtains the true parameter
values, i.e. it represents the maximum likelihood for the parent p.d.f. In
order to test if the Asimov condition holds one can use the generic likelihood
function Eq. (2). Using the simplified notation
$\nu_{i}=\mu^{\prime}s_{i}+b_{i}$, and setting $\theta_{0}=\mu$, then Eq. (3)
becomes
$\frac{\partial\ln
L}{\partial\theta_{j}}=\displaystyle\sum_{i=1}^{N}\left(\frac{n_{i}}{\nu_{i}}-1\right)\frac{\partial\nu_{i}}{\partial\theta_{j}}+\displaystyle\sum_{i=1}^{M}\left(\frac{m_{i}}{u_{i}}-1\right)\frac{\partial
u_{i}}{\partial\theta_{j}}=0.$ (13)
If $n_{i,A}=E(n_{i})$ and $m_{i,A}=E[m_{i}]$, where the subscript A denotes
Asimov values, then the Asimov condition is met. We cannot calculate the
Asimov likelihood $L_{A}$ because it contains factorial dependence on Asimov
values that can be non-integer. However, these factorials are canceled in the
Asimov profile likelihood ratio
$\lambda_{\rm A}(\mu)=\frac{L_{\rm
A}(\mu,\hat{\hat{\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}}})}{L_{\rm
A}(\hat{\mu},\hat{\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}})}=\frac{L_{\rm
A}(\mu,\hat{\hat{\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}}})}{L_{A}(\mu^{\prime},\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})}\;,$
(14)
where the substitution in the denominator of the final equality is allowed by
the definition of the Asimov data set [1].
### 3.1 The Wald Equation
Suppose we have a test with strength parameter $\mu$ and the data is
distributed by strength parameter $\mu^{{}^{\prime}}$, then according to Wald
[3]
$-2\ln\lambda(\mu)=\frac{(\mu-\hat{\mu})^{2}}{\sigma^{2}}+{\cal
O}(1/\sqrt{N})\;,$ (15)
where N is the sample size and $\hat{\mu}$ is a Gaussian distribution with
mean $\mu^{\prime}$. Here $\sigma$ is found using the covariance matrix.
Substituting the Asimov data set with strength parameter $\mu^{\prime}$ into
the Wald approximation equation, it follows from Eq. (15) that
$-2\ln\lambda_{A}(\mu)\approx\frac{(\mu-\mu^{{}^{\prime}})^{2}}{\sigma^{2}}$
(16)
for large samples. We provide an alternate way to find the standard deviation
via the Asimov data set, defining $q_{\mu,A}=-2\ln\lambda_{A}(\mu)$,
$\sigma_{A}^{2}=\frac{(\mu-\mu^{\prime})^{2}}{q_{\mu,A}}.$ (17)
To find the median exclusion significance assuming there is no signal
$\mu^{\prime}=0$, Eq. (17) reduces to
$\sigma_{A}^{2}=\frac{\mu^{2}}{q_{\mu,A}}.$ (18)
Similarly for the case of discovery where $\mu=0$, Eq. (17) is
$\sigma_{A}^{2}=\frac{\mu^{\prime 2}}{q_{\mu,A}}.$ (19)
### 3.2 The Tevatron Test Statistic
The test statistic
$q=-2\ln\frac{L_{s+b}}{L_{b}},$ (20)
is often used in analyses at the Fermilab Tevatron Collider. Here $L_{s+b}$ is
the nominal signal model with strength parameter $\mu=1$, and $L_{b}$ is the
background-only hypothesis with $\mu=0$. Rewriting Eq. (20),
$q=-2\ln\frac{L(\mu=1,\hat{\hat{\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}}}(1))}{L(\mu=0,\hat{\hat{\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}}}(0))}=-2\ln\lambda(1)+2\ln\lambda(0).$
(21)
If the Wald appromixation holds, then
$q=\frac{(\hat{\mu}-1)^{2}}{\sigma^{2}}-\frac{\hat{\mu}^{2}}{\sigma^{2}}=\frac{1-2\hat{\mu}}{\sigma^{2}}.$
(22)
Since $\hat{\mu}$ is Gaussian and $q$ is dependent on $\hat{\mu}$ then $q$ is
also Gaussian. Therefore, the expectation value and standard deviation of $q$
are [1]
$\displaystyle E[q]=\frac{1-2\mu}{\sigma^{2}},$ (23)
$\displaystyle\sigma[q]=\frac{2}{\sigma(\mu)}.$ (24)
ASince $q$ is Gaussian we can use the cumulative distribution function222For a
normal variable with mean $\mu$, variance $\sigma^{2}$ and observation x the
cumulative distribution function is
$\Phi(\frac{x-\mu}{\sigma})=\frac{1}{2}[1+erf(\frac{x-\mu}{\sigma/\sqrt{2}})]$
to determine the $p$-value. Plugging in what we know of the signal strengths
of the two hypotheses, as well as the mean and standard deviation of q,333The
original paper contains confusing notation and a substitution error in their
derivation; the formulas presented here are correct.
$\displaystyle
p_{s+b}=\int_{q_{obs}}^{\infty}f(q|s+b)dq=1-\Phi\left(\frac{q_{obs}+1/\sigma_{s+b}^{2}}{2/\sigma_{s+b}}\right),$
(25) $\displaystyle
p_{b}=\int_{-\infty}^{q_{obs}}f(q|b)dq=\Phi\left(\frac{q_{obs}-1/\sigma_{b}^{2}}{2/\sigma_{b}^{2}}\right).$
(26)
## 4 Pseudo-data Tests
In order to test if the AWW approximation reproduces the real distributions of
$\gamma$ we created a set of test data, applied various systematic
uncertainties and compared with the values produced by the Collie framework.
We calculate the signal strength required to achieve a given significance
level in both models and compare.
The pseudo-data generated has least likelihood ratios similar to a set of
Tevatron data by construction, and is displayed in Fig. 1. We define the data
as equal to the background before systematic uncertainties.
Figure 1: On the left is a plot of the test input data generated in root with
1,000,000 events placed into 1500 bins. The background was filled with an
exponential decay function, $e^{\tau x}$, with $\tau=0.203$ and weight
135000/event count, and the signal was filled by a mirrored exponential decay
function, $1-e^{\tau x}$, with $\tau=0.215$ and weight 214/event count. For
ease of intepretation the signal is displayed with a scale factor of 500 and
the y-axis is linear. This histogram has 1500 bins and 1,000,000 events, which
are used in this paper unless stated otherwise. On the right we display the
ratio of signal over background.
The Collie software suite generates semi-frequentist confidence intervals with
an output designed for Root [4]. Here we will consider the Collie confidence
level value to be true for the sake of evaluating the Asimov conditions.
Collie also outputs the observed, signal plus background, and background-only
least likelihood ratios, which are used to calculate the AWW approximation.
From Eq. (18), with $\mu=1$ from the nominal signal hypotheses we have
$\sigma_{s+b,A}^{2}=\frac{1}{q_{s+b}}.$ (27)
Substituting this value into Eq. (25)
$P_{s+b}=1-\Phi\left(\frac{q_{obs}+q_{s+b}}{2\sqrt{q_{s+b}}}\right),$ (28)
which provides a simple calculation of the AWW approximation using the Collie
output. We report results in terms of a ratio; this ratio is always the
approximation value divided by the Collie value. We keep this standard because
the Wald approximation should result in underestimation, thus the ratio should
stay below one.
### 4.1 Background-only Rate Systematic Uncertainty
The first systematic uncertainty applied was a rate systematic uncertainty on
only the background. Our results are plotted in Fig. (2). As expected we see
no discrepancy when there is no uncertainty, i.e. when the background rate
systematic uncertainty is set at 0%, meaning the data and background are
equal.
Figure 2: This experiment measured the two methods against each other while
they accounted for a background-only rate systematic uncertainty that varied
from zero to fifty percent in five percent increments. (a) Shows the signal
scale necessary to achieve the CL for both methods, (b) shows the ratio of the
approximation value over the Collie value. (c) shows the fractional
uncertainty of the background at 25% uncertainty in the rate systematic
uncertainty.
As we apply the rate systematic uncertainty we get up to around 5% deviation
from the “true” value, as well as no obvious trend as a function of systematic
uncertainty percent. Therefore the AWW approximation is valid.
### 4.2 Signal and Background Rate Systematic Uncertainties
Perhaps the most striking results were the three dimensional plots where the
axes in the horizontal plane represent the percent rate systematic
uncertainties of the signal and background histograms. We created plots of
both uncorrelated and correlated systematic uncertainties. No systematic
uncertainty plots are shown as they are equivalent to the systematic
uncertainty plot in the background rate systematic uncertainty, only now
applied to signal as well as background.
Figure 3: The horizontal plane contains the axes for the background and signal
rate systematic uncertainties percent, varying from 5% to 45% in 10%
increments. The variable we are interested in is the ratio of the AWW
approximation over the Collie value, which is presented in the z-axis. The
color is a gradient and the scale is held in this manner for comparison to the
correlated plot. Figure 4: The axes, scale and confidence level are equivalent
to that of Fig. (3). These plots appear fairly similar
Fig. (3) displays the uncorrelated data set and Fig. (4) the correlated. With
relatively flat signal scale ratios at the C.L. we conclude that the AWW
approximation is valid for these systematic uncertainties.
### 4.3 Asymmetric Gaussian “Flat” Systematic Uncertainties
For the next experiment we ran two tests with a flat systematic uncertainties
with a discontinuity at the center. Fig. (5) shows the way in which Collie
approximates a solution for an asymmetric Gaussian as well as the systematic
uncertainty itself. The first test had the positive systematic uncertainty
constant and the negative varied, while the second reversed the roles.
Figure 5: The plot in (a) shows the collie approxmation to an asymmetric
Gaussian. (b) shows the flat systematic uncertainty at negative fluctuations
of 5% and positive fluctuations of 10%. Figure 6: Here the negative side of
the flat systematic uncertainty was held at 5% while the positive varied from
0% to 35%. (a) The signal scale necessary required is 90% confidence, (b)
shows the ratio Figure 7: This displays the same information as Fig. (6)
except for this experiment the positive component is held fixed while the
negative is varied.
One notable difference here from the other tests is that we had to use the
observed Collie confidence level instead of the calculated median, which
results in slightly greater random variability. This is due to the systematic
uncertainty being non-Gaussian.
Both sets were run from 0% to 50% on the uncertainty that varies, but are only
plotted up to 35%. This is because the data at and above 35% return unusable
values due to a failure in the AWW approximation. This occurs because at this
level and type of systematic uncertainty the histograms are no longer
Gaussian. When there is 5% negative and no positive uncertainty the AWW
approximation overestimates the value. Other than this, at low uncertainty
differences the AWW approximation is still valid, however above 35% on the
varying systematic uncertainty it is invalid as the model breaks down.
### 4.4 Uncertainty on Background Shape
Next, we tested the resilience of the AWW approximation against deviations in
the tau of the exponential decay function of the background. The initial
$\tau$ value we used, 0.203, was chosen in order to simulate the least
likelihood ratio values found in a set of real Tevatron data (this is also
true for the case of the signal tau formula, where $\tau=0.215$). Fig. (8)
displays these findings.
Figure 8: A background-only shape systematic uncertainty with the $\tau$ value
is deviated from 0% to 10%, for both positive and negative directions. (a)
Again 95% is used for the confidence level requirement and (b) displays the
ratio. (c) shows the fractional uncertainty when the $\tau$ value deviates by
5%.
The AWW approximation stays consistently below the Collie value by around 1.5%
and follows the same trend. This test was run with a 5% rate systematic
uncertainty on the background, which holds the ratio maximum at around 0.95.
The ratio varies within a percent of 95%, therefore the approximation is
valid.
### 4.5 Varying the Number of Histogram Bins
An inherent loss of information occurs when data is binned. Due to this, we
want to test the ability of the AWW approximation to reproduce the level of
information loss of the full calculation by varying the number of bins. Our
results are presented in Fig. (9).
Figure 9: The number of bins was varied from 100 to 1500 in 100 step
increments. (a) Displays the signal scale required to achieve 95% confidence
and (b) the ratios for both methods.
This test was run with a 5% background-only rate systematic uncertainty. As is
consistent with this additional uncertainty the ratio stays around 95%; the
AWW approximation is valid in reproducing equivalent information loss.
### 4.6 Variation in the Number of Events
The last test of the system we built was by varying the number of data used.
We wanted to find how many data points were necessary in order to achieve a
usable approximation. These results are plotted in Fig. (10).
Figure 10: The Collie and the approximation values for the signal strength
required for 95% confidence are plotted as a function if the number of data
points. The X-axis is labeled as events/50,000. 50,000 is the lowest number of
data points that returned usable values for both calculations; the number of
iterations is equal to the number of data points. (b) shows the ratio of the
two. These plots were generated with a 5% rate systematic uncertainty on the
background.
When the number is too small the conditions for Wilks’ Theorem are not met,
which invalidates the AWW approximation under these conditions. This is
evident on the ratio plot, where there is an asymptotic behavior as the number
of events increases. This was applied with a 5% background-only rate
systematic uncertainty, so the limit approaches about 0.95.
## 5 Conclusion
In summary, we tested the AWW approximation against the full semi-frequentist
calculation, with no approximations, as calculated in Collie. We ran
background-only rate systematic uncertainties, background-only and signal
shape systematic uncertainties, asymmetric Gaussian flat systematic
uncertainties, varied the background shape itself, varied the number of bins,
and varied the number of events. The AWW approximation behaved as expected
based on the results from [1].
The tests where the model correctly reproduces the parameter values of the
full calculation include the rate systematic uncertainties, the background and
signal shape uncertainties, the number of histograms bins, and the uncertainty
in the background shape. The shape systematic uncertainties on only
background, and the combined shape and background systematic uncertainties run
at about 95% of the true value, i.e. the AWW approximation would exclude with
95% the signal strength required of the full calculation. When there are no
systematic uncertainties the two methods returned nearly equal values. None of
the figures for these tests show any absolute trend.
The tests where the AWW model breaks down occur where expected. The first of
these are the asymmetric Gaussian tests. In the case where the asymmetry is
small, roughly at or below 25% difference (A=2/3), the AWW approximation and
Collie agree. But when the difference is greater the AWW approximation fails.
The second test where the model fails to reproduce the full calculation value
is where the number of data points is varied. At low numbers the model fails
to reproduce the full calculation, but as the number increases it approches an
asymptotic value close to that of the full calculation.
These results are as expected given the two approximations, Wilks and Wald,
combined to form the new approximation, the Asimov data set, and is consistent
with the report this paper examines. One of the conditions for Wilks’ theorem
is using a sufficiently large sample and one of the conditions for Wald’s
theorem is that the data uncertainties follow a Gaussian distribution (There
are more conditions necessary to use either theorem, but these are the two
that explain the behavior found in tests where the AWW approximation fails).
In the case where an asymmetric Gaussian becomes non-Gaussian the model fails,
as expected according to Wald’s theorem and as the number of data points
falls, the mentioned condition for Wilks’s theorem fails (as well as
increasing the neglected term in the Wald formula).
Therefore, we conclude that when the conditional definitions of Wilks and Wald
are met, then the approximation presented in Asymptotic formulae for
likelihood-based tests of new physics does reproduce the full calculation
reliably within 5-10%. Our results suggest that the approximations, published
by Cowan, Cranmer, Gross, and Vitells, has the correct asymptotic behavior as
designed. Though this approximation has limitations when any of the component
approximations are explicitly invalidated, also as expected.
## References
* [1] Glen Cowan, Kyle Cranmer, Eilam Gross, Ofer Vitells, Asymptotic formulae for likelihood-based tests of new physics, Eur.Phys.J.C71:1554,2001 (3 Oct 2010).
* [2] S.S. Wilks, The large-sample distribution of the likelihood ratio for testing composite hypotheses, Ann. Math. Statist. 9 (1938) 60-2.
* [3] A. Wald, Tests of Statistical Hypotheses Concerning Several Parameters When the Number of Observations is Large, Transactions of the American Mathematical Society, Vol. 54, No. 3 (Nov., 1943), pp. 426-482.
* [4] Wade Fisher Collie: A Confidence Level Estimator, Fermilab, (Feb. 2010)
* [5] S. Baker and R. Cousins, Nucl. Instrum. Methods 221, 437 (1984)
* [6] K. Nakamura et al., JPG 37, 075021 (2010) (http://pdg.lbl.gov)
|
arxiv-papers
| 2011-10-22T20:59:40 |
2024-09-04T02:49:23.515774
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Eric Burns, Wade Fisher",
"submitter": "Eric Burns",
"url": "https://arxiv.org/abs/1110.5002"
}
|
1110.5006
|
CytoSaddleSum: a functional enrichment analysis plugin for Cytoscape based on
sum-of-weights scores
Aleksandar Stojmirović , Alexander Bliskovsky and Yi-Kuo Yu***to whom
correspondence should be addressed
National Center for Biotechnology Information
National Library of Medicine
National Institutes of Health
Bethesda, MD 20894
United States
### Summary:
CytoSaddleSum provides Cytoscape users with access to the functionality of
SaddleSum, a functional enrichment tool based on sum-of-weight scores. It
operates by querying SaddleSum locally (using the standalone version) or
remotely (through an HTTP request to a web server). The functional enrichment
results are shown as a term relationship network, where nodes represent terms
and edges show term relationships. Furthermore, query results are written as
Cytoscape attributes allowing easy saving, retrieval and integration into
network-based data analysis workflows.
### Availability:
www.ncbi.nlm.nih.gov/CBBresearch/Yu/downloads
The source code is placed in Public Domain.
### Contact:
yyu@ncbi.nlm.nih.gov
## 1 Introduction
CytoSaddleSum is a Cytoscape (Smoot et al., 2011) plugin to access the
functionality of SaddleSum, an enrichment analysis tool based on sum-of-
weights-score (Stojmirović and Yu, 2010). Unlike most other enrichment tools,
SaddleSum does not require users to directly select significant genes or
perform extensive simulations to compute statistics. Instead, it uses weights
derived from measurements, such as log-expression ratios, to produce a score
for each database term. It then estimates, depending on the number of genes
involved, the P-value for that score by using the saddlepoint approximation
(Lugannani and Rice, 1980) to the empirical distribution function derived from
all weights. This approach was shown (Stojmirović and Yu, 2010) to yield
accurate P-values and internally consistent retrievals.
As a popular and flexible platform for visualization, integration and analysis
of network data, Cytoscape allows gene expression data import and hosts
numerous plugins for functional enrichment analysis. However, none of these
plugins are based on the ‘gene set analysis approach’ that takes into account
gene weights. Therefore, to fill this gap, we have developed CytoSaddleSum, a
Cytoscape interface to SaddleSum. To enable several desirable features of
CytoSaddleSum, however, we had to significantly extend the original SaddleSum
code (see descriptions below).
## 2 Implementation
While CytoSaddleSum is implemented in Java using Cytoscape API, it functions
by running either locally or remotely a separate instance of SaddleSum,
written in C. In either mode, CytoSaddleSum takes the user input through a
graphical user interface, validates it, and passes a query to SaddleSum. Upon
receiving the entire query results, CytoSaddleSum stores them as the node and
network attributes of the newly-created term relationship graph. Consequently,
the query output can be edited or manipulated within Cytoscape. Furthermore,
saving term graph through Cytoscape also preserves the results for later use.
The most important extension to SaddleSum involved construction of extended
term databases (ETDs). Each ETD contains the mappings of genes to Gene
Ontology (Gene Ontology Consortium, 2010) terms and KEGG (Kanehisa et al.,
2008) pathways, as well as an abbreviated version of the NCBI Gene (Maglott et
al., 2011) database for all genes mapped to terms. Thanks to the latter, when
using an ETD, SaddleSum is able to interpret the provided gene labels as NCBI
Gene IDs, as gene symbols and as gene aliases. Each ETD also contains
relations among terms that are used by SaddleSum for term graph construction.
Figure 1: CytoSaddleSum user interface consists of the query form (left), the
results panel (right) and the term relationship network (center), which here
partially covers the original network. The results stored as attributes of the
term network can be edited through Cytoscape Data Panel.
## 3 Usage
CytoSaddleSum operates on the currently selected Cytoscape network whose nodes
represent genes or gene products. The queries are submitted through the query
form embedded as a tab into the Cytoscape Control Panel, on the left of the
screen. The selected network must contain at least one node mapped to a
floating-point Cytoscape attribute, which would provide node weights.
CytoSaddleSum considers only the selected nodes within the network. The user
can select the weight attribute through a dropdown box on the query form. Any
selected node without specified weight is assumed to have weight 0. The user-
settable cannonicalName attribute, automatically created by Cytoscape for each
network node, serves as the gene label.
After selecting the network and the nodes within it, the user needs to select
a term database and set the statistical and weight processing parameters. The
latter enable users to transform the supplied weights within SaddleSum. This
includes changing the sign of the weights, as well as applying a cutoff, by
weight or by rank. All weights below the cutoff are set to 0. The statistical
parameters are E-value cutoff, minimum term size, effective database size and
statistical method. We define the effective database size as the number of
terms in the term database that map to at least $k$ genes among the selected
nodes, where $k$ is the minimum term size. Apart from the default ‘Lugannani-
Rice’ statistics, it is also possible to select ‘One-sided Fisher’s Exact
test’ statistics, which are based on the hypergeometric distribution. In that
case, the user must select a cutoff under the weight processing parameters.
To run local queries, a user needs the command-line version of SaddleSum and
the term databases, both available for download from our website, and install
them on the same machine that runs Cytoscape. The advantages of running local
queries include speed, independence of Internet connection, and support of
queries to custom databases in the GMT file format used by the GSEA tool
(Subramanian et al., 2005). Furthermore, the standalone program can be used
outside of Cytoscape for large sets of queries. On the other hand, running
remote queries require no installation of additional software, since queries
are passed to the SaddleSum server over an HTTP connection. The disadvantage
of running remote queries is that it can take much longer to run and that the
choice of term databases is restricted to ETDs available only for some model
organisms.
CytoSaddleSum also displays warning or error messages reported by SaddleSum.
For example, when a provided gene label is ambiguous, depending on whether the
ambiguity could be resolved, CytoSaddleSum will relay a warning or an error
message reported by SaddleSum. CytoSaddleSum presents query results as a term
relationship network (Fig. 1), consisting of significant terms or their
ancestors linked by hierarchical relations available in the term database. The
statistical significance of each term is indicated by the color of its
corresponding node. To facilitate browsing of the results, CytoSaddleSum
generates a set of summary tables, which contain the lists of significant
terms and various details about the query. These summary tables are embedded
into Cytoscape Results Panel, on the right of the screen. Clicking on a
significant term in a summary table will select that term in the term
relationship network and select all nodes mapping to it in the original
network. The results can be exported as text or tab-delimited files and can be
restored from tab-delimited files through the Export and Import menus of
Cytoscape.Detailed instructions and explanations can be found in SaddleSum
manual available from our website.
## Acknowledgments
This work was supported by the Intramural Research Program of the National
Library of Medicine at National Institutes of Health.
## References
* Gene Ontology Consortium (2010) Gene Ontology Consortium (2010). The gene ontology in 2010: extensions and refinements. Nucleic Acids Res, 38(Database issue), D331–5.
* Kanehisa et al. (2008) Kanehisa, M. et al. (2008). KEGG for linking genomes to life and the environment. Nucleic Acids Res, 36(Database issue), D480–4.
* Lugannani and Rice (1980) Lugannani, R. and Rice, S. (1980). Saddle point approximation for the distribution of the sum of independent random variables. Adv. in Appl. Probab., 12(2), 475–490.
* Maglott et al. (2011) Maglott, D. et al. (2011). Entrez Gene: gene-centered information at NCBI. Nucleic Acids Res, 39(Database issue), D52–7.
* Smoot et al. (2011) Smoot, M. E. et al. (2011). Cytoscape 2.8: new features for data integration and network visualization. Bioinformatics, 27(3), 431–2.
* Stojmirović and Yu (2010) Stojmirović, A. and Yu, Y.-K. (2010). Robust and accurate data enrichment statistics via distribution function of sum of weights. Bioinformatics, 26(21), 2752–9.
* Subramanian et al. (2005) Subramanian, A. et al. (2005). Gene set enrichment analysis: a knowledge-based approach for interpreting genome-wide expression profiles. Proc Natl Acad Sci USA, 102(43), 15545–15550.
|
arxiv-papers
| 2011-10-22T22:46:52 |
2024-09-04T02:49:23.524347
|
{
"license": "Public Domain",
"authors": "Aleksandar Stojmirovic, Alexander Bliskovsky and Yi-Kuo Yu",
"submitter": "Aleksandar Stojmirovi\\'c",
"url": "https://arxiv.org/abs/1110.5006"
}
|
1110.5051
|
# Wikipedia Edit Number Prediction based on Temporal Dynamics Only
Dell Zhang DCSIS
Birkbeck, University of London
Malet Street
London WC1E 7HX, UK
Email: dell.z@ieee.org
###### Abstract
In this paper, we describe our approach to the Wikipedia Participation
Challenge which aims to predict the number of edits a Wikipedia editor will
make in the next 5 months. The best submission from our team, “zeditor”,
achieved 41.7% improvement over WMF’s baseline predictive model and the final
rank of 3rd place among 96 teams. An interesting characteristic of our
approach is that only temporal dynamics features (i.e., how the number of
edits changes in recent periods, etc.) are used in a self-supervised learning
framework, which makes it easy to be generalised to other application domains.
###### Index Terms:
social media; user modelling, data mining; machine learning.
## I Introduction
Wikipedia is “a free, web-based, collaborative, multilingual encyclopaedia
project” supported by the non-profit Wikimedia Foundation (WMF). Started in
2001, Wikipedia has become the largest and most popular general reference
knowledge source on the Internet. Almost all of its 19.7 million articles can
be edited by anyone with access to the site, and it has about 90,000 regularly
active volunteer editors around the world. However, it has recently been
observed that Wikipedia growth has slowed down significantly [1]. In
particular, WMF has reported
that111http://strategy.wikimedia.org/wiki/March_2011_Update:
> Between 2005 and 2007, newbies started having real trouble successfully
> joining the Wikimedia community. Before 2005 in the English Wikipedia,
> nearly 40% of new editors would still be active a year after their first
> edit. After 2007, only about 12-15% of new editors were still active a year
> after their first edit. Post-2007, lots of people were still trying to
> become Wikipedia editors. What had changed, though, is that they were
> increasingly failing to integrate into the Wikipedia community, and failing
> increasingly quickly. The Wikimedia community had become too hard to
> penetrate.
It is therefore of utter importance to understand quantitatively what factors
determine editors’ future editing behaviour (why they continue editing, change
the pace of editing, or stop editing), in order to ensure that the Wikipedia
community can continue to grow in terms of size and diversity.
The Wikipedia Participation Challenge222http://www.kaggle.com/c/wikichallenge,
sponsored by WMF and hosted by Kaggle, request contestants to build a
predictive model that could accurately predict the number of edits a Wikipedia
editor would make in the next 5 months based on his edit history so far. Such
a predictive model may be able to help WMF in figuring out how people can be
encouraged to become, and remain, active contributors to Wikipedia.
The ‘training’ dataset consists of randomly sampled active editors with their
full history of editing activities on the English Wikipedia (the first 6
namespaces only) in the period from 2001-01-01 to 2010-09-01. An editor is
considered “active” if he or she made at least one edit in the last one year
period, i.e., from 2009-09-01 to 2010-09-01. For each edit, the available
information includes its user_id, article_id, revision_id, namespace,
timestamp, etc.
The predictive model to be constructed should predict, for each editor from
the ‘training’ dataset, how many edits would be made in the 5 months after the
end date of the ‘training’ dataset, i.e., from 2010-09-01 to 2011-02-01. The
predictive model’s accuracy is going to be measured by the Root Mean Squared
Logarithmic Error (RMSLE):
$\epsilon=\sqrt{\frac{1}{n}\sum_{i=1}^{n}{(\log(1+p_{i})-\log(1+a_{i}))^{2}}}\
,$ (1)
where $n$ is total number of editors in the dataset, $\log(\cdot)$ is the
natural logarithm function, $p_{i}$ and $a_{i}$ are the predicted and actual
edit numbers respectively for editor $i$ in the next 5 month period.
The best submission from our team, “zeditor”, achieved 41.7% improvement over
WMF’s baseline predictive model and the final rank of 3rd place among 96
teams. An interesting characteristic of our approach is that only temporal
dynamics features (i.e., how the number of edits changes in recent periods,
etc.) are used in a self-supervised learning framework, which makes it easy to
be generalised to other application domains.
The rest of this paper is organised as follows. In Section II, we present our
approach in details. In Section III, we show the experimental results. In
Section IV, we review the related work. In Section V, we make conclusions.
## II Approach
Our basic idea is to build a predictive model $f$ (that estimates an active
editor’s future number of edits based on his recent edit history) through
self-supervised learning, as illustrated schematically in Figure 1. The
approach is called “self-supervised” to emphasise the fact that it does not
require any manual labelling of data (as in standard _supervised learning_
[2]) but extracts the needed labels from data automatically.
Figure 1: Our self-supervised learning framework.
To facilitate the description of our approach, we shall from now on talk about
any time-length in the unit of months and refer to any time-point as the real
number of months passed since the beginning date of the dataset. So for the
official dataset ‘training’, the timestamp “2001-06-16 00:00:00” would be 5.5
because it is five and a half months since 2001-01-01.
Let $t_{\text{test}}$ denote the time-point when we would like to predict each
active editor’s number of edits in the next 5 months. To train the predictive
model, we would move 5 months backwards and assume that we were at the time-
point $t_{\text{train}}=t_{\text{test}}-5$. Thus we could know the actual
number of edits made by each active editor in those 5 months after
$t_{\text{train}}$, i.e., the label for our machine learning (regression)
methods. Specifically, the target value for regression would be set as
$y_{i}=\log(1+a_{i})$ where $a_{i}$ is the actual number of edits in the next
5 months. In this way, the _squared error_ loss function
$L(f(\mathbf{x}),y)=(f(\mathbf{x})-y)^{2}$ used by most machine learning
methods (including those in our experiments and final submission) would
connect the _empirical risk_ [2] directly to the evaluation metric RMSLE:
$R_{emp}(f)=\frac{1}{n}\sum_{i=1}^{n}L(f(\mathbf{x}_{i}),y_{i})=\epsilon^{2}\
.$ (2)
Given a time-point (either $t_{\text{train}}$ or $t_{\text{test}}$), each
active editor $i$ would be represented as a vector $\mathbf{x}_{i}$ that
consists of the following temporal dynamics features:
* •
the number of edits in recent periods of time;
* •
the number of edited articles in recent periods of time;
* •
the length of time between the first edit and the last edit, scaled
logarithmically.
The periods used in our final submission for the above temporal dynamics
features are
$\frac{1}{16},\frac{1}{8},\frac{1}{4},\frac{1}{2},1,2,4,12,36,108$
where the length of period first doubles at each step from $\frac{1}{16}$ to 4
and then triples at each step from 4 to 108. The usage of such temporal
dynamics features was inspired by the decent performance of the “most-
recent-5-months-benchmark” — if using the exact number of edits in just one
period (the last 5 months) for prediction could work reasonably well, we
should be able to achieve a better performance by using many more recent
periods. The periods were chosen to be at exponentially increasing temporal
scales, because we conjecture that the influence of an editing activity to the
editor’s future editing behaviour would be exponentially decaying along with
the time distance away from now. The process of _exponential decay_
333http://en.wikipedia.org/wiki/Exponential_decay occurs in numerous natural
phenomena, and it has been widely used in temporal applications where it is
desirable to gradually discount the history of past events [3]. One reason for
changing from doubling to tripling midway through is to include the special
period of 12-months (i.e., one year) that has been used to define the “active”
editors. The periods will be capped by the time scope of the given dataset
(e.g., 106 for the additional dataset ‘moredata’) in case they are out of
range.
We have also introduced a constant _drift_ term (i.e., how much the average
number of edits would change after 5 months) into the formula of making final
predictions, which is a crude way to cover the global shift of target values
along with time. Again, its value is estimated from the situation 5 months
ago.
The concise pseudo-code of our algorithms for learning and predicting is shown
in Figure 2. The complete source code will be made available at the author’s
homepage444http://www.dcs.bbk.ac.uk/~dell/.
Learning
* •
$t=t_{\text{train}}$ (i.e., $t_{\text{test}}-5$)
* •
for each active editor $i$ who made at least one edit in $[t-12,t)$:
* –
represent the editor as a vector $\mathbf{x}_{i}$ consisting of temporal
dynamics features (please refer to the above description)
* –
label the editor by $y_{i}=\log(1+a_{i})$ where $a_{i}$ is the actual number
of edits in $[t,t+5)$
* •
learn a predictive model/function $f:x\rightarrow y$ from
$(\mathbf{x}_{i},y_{i})$ pairs using a regression technique such as GBT
* •
estimate the drift $d$ by comparing the average number of edits in $[t-5,t)$
and that in $[t,t+5)$
Predicting
* •
$t=t_{\text{test}}$ (e.g., $116$ for the dataset ‘training’)
* •
for each active editor $i$ who made at least one edit in $[t-12,t)$:
* –
represent the editor as a vector $\mathbf{x}_{i}$ consisting of temporal
dynamics features (please refer to the above description)
* –
compute $\hat{y}_{i}$ = $f(\mathbf{x}_{i})$ using the learnt $f$
* –
output $p_{i}=\exp(\max(\hat{y}_{i}+d,0))-1$ as the predicted number of edits
in $[t,t+5)$
Figure 2: Our algorithms for learning and predicting.
## III Experiments
### III-A Datasets
There are three datasets available to all contestants:
* •
‘training’ is the official dataset for training and testing;
* •
‘validation’ is the official dataset for validation;
* •
‘moredata’ is the additional dataset generously provided by Twan van
Laarhoven555http://www.kaggle.com/c/wikichallenge/forums/t/719/more-training-
data.
The characteristics of each dataset are shown in Table I.
TABLE I: The characteristics of each dataset. dataset | $t_{\text{train}}$ | $t_{\text{test}}$ | #editors | #edits
---|---|---|---|---
‘validation’ | 79 | 84 | 4856 | 274820
‘moredata’ | 106 | 111 | 23584 | 5717049
‘training’ | 111 | 116 | 44514 | 22326031
Since we did not have local access to the true labels (target values) of the
dataset ‘training’, we only used it to make the final submission, but
conducted our experiments (for parameter tuning etc.) on the other two
datasets ‘validation’ and ‘moredata’. It is noteworthy that these two datasets
‘validation’ and ‘moredata’ had been filtered to contain only active editors
(who made at least one edit in the last one year period) in order to make them
exhibit the same _survivorship bias_
666http://en.wikipedia.org/wiki/Survivorship_bias as the dataset ‘training’.
This might (partially) ensure that the experimental findings on the former two
datasets could be transferred to the latter one.
### III-B Tools
We have only used Python777http://www.python.org/ (equipped with
Numpy888http://numpy.scipy.org/) to write small programs for analysing data
and making predictions. The machine learning methods that we have tried for
our regression task all come from two _open-source_ Python modules: one is
scikit-learn999http://scikit-learn.sourceforge.net/, and the other is
OpenCV101010http://opencv.willowgarage.com/wiki/.
### III-C Results
First, we compare different machine learning methods (with their default
parameter values) in terms of their prediction performances (RMSLE). The
methods being compared include:
* •
Ordinary Least Squares (OLS)111111http://scikit-
learn.sourceforge.net/modules/linear_model.html#ordinary-least-squares-ols,
* •
Support Vector Machine (SVM)121212http://scikit-
learn.sourceforge.net/modules/svm.html,
* •
K Nearest Neighbours
(KNN)131313http://opencv.itseez.com/modules/ml/doc/k_nearest_neighbors.html,
* •
Artificial Neural Network
(ANN)141414http://opencv.itseez.com/modules/ml/doc/neural_networks.html,
* •
Gradient Boosted Trees
(GBT)151515http://opencv.itseez.com/modules/ml/doc/gradient_boosted_trees.html.
The experimental results are shown in Table II and Figure 3. Gradient Boosted
Trees (GBT)161616http://en.wikipedia.org/wiki/Gradient_boosting [4, 5] clearly
outperformed all the other machine learning methods on both datasets. GBT (aka
GBM, MART and TreeNet) represents a general and powerful machine learning
method that builds an ensemble of _weak_ tree learners in a greedy fashion. It
evolved from the application of boosting to regression trees [2]. The general
idea is to compute a sequence of very simple trees, where each successive tree
is built for the prediction residuals of all preceding trees on a randomly
selected subsample of the full training dataset. Eventually a “weighted
additive expansion” of those trees can produce an excellent fit of the
predicted values to the observed values. It allows optimisation of any
differentiable loss function. Here we just use the _squared error_ for the
reasons given in Section II. The success of GBT in our task is probably
attributable to (i) its ability to capture the complex nonlinear relationship
between the target variable and the features, (ii) its insensitivity to
different feature value ranges as well as outliers, and (iii) its resistance
to overfitting via regularisation mechanisms such as shrinkage and subsampling
[4, 5].
TABLE II: The prediction performances of different machine learning methods (with their default parameter values). learning method | ‘validation’ | ‘moredata’
---|---|---
OLS | 0.832351 | 0.869779
SVM | 0.901698 | 0.732814
KNN | 0.833288 | 0.690832
ANN | 0.987345 | 1.040396
GBT | 0.820805 | 0.635807
(a) validation
(b) moredata
Figure 3: The prediction performances of different machine learning methods
(with their default parameter values).
Second, we investigate how GBT’s most important parameter weak_count — the
number of weak tree learners — affects its prediction performance for our
task. Tuning weak_count is our major means of controlling the model complexity
to avoid underfitting or overfitting. The experimental results are shown in
Table III and Figure 4. It seems that on big datasets like ‘moredata’, a
higher value of weak_count (i.e., more weak tree learners) would be
beneficial, but on small datasets like ‘validation’, it might increase the
risk of overfitting.
TABLE III: The prediction performances of GBT with different number of weak tree learners (weak_count). GBT weak_count | ‘validation’ | ‘moredata’
---|---|---
200 | 0.820805 | 0.635807
400 | 0.817789 | 0.616876
600 | 0.817483 | 0.614507
800 | 0.817614 | 0.613757
1000 | 0.818726 | 0.613530
1200 | 0.819804 | 0.613465
1400 | 0.819998 | 0.613671
(a) validation
(b) moredata
Figure 4: The prediction performances of GBT with different number of weak
tree learners (weak_count).
Third, we demonstrate how the prediction performance changes when we use more
and more periods to generate temporal dynamics features: we start from just
the shortest period ($\frac{1}{16}$) and then each time we add the next longer
period to the series (see Section II). The experimental results are shown in
Table IV and Figure 5. It seems that making use of more periods for temporal
dynamics features usually helps, but the pay-off gradually diminishes.
TABLE IV: The prediction performances of GBT using different number of periods for temporal dynamics features. GBT #periods | ‘validation’ | ‘moredata’
---|---|---
1 | 0.861111 | 0.788450
2 | 0.857575 | 0.760365
3 | 0.849440 | 0.728888
4 | 0.841127 | 0.696196
5 | 0.836116 | 0.669754
6 | 0.830619 | 0.647883
7 | 0.829393 | 0.629062
8 | 0.816459 | 0.614429
9 | 0.818515 | 0.613749
10 | 0.818726 | 0.613530
(a) validation
(b) moredata
Figure 5: The prediction performances of GBT using different number of periods
for temporal dynamics features.
### III-D Submissions
Since ‘moredata’ is more similar than ‘validation’ to the official dataset
‘training’ in terms of the time scope and the number of editors, we applied
the best working algorithm, GBT, with the optimal parameter setting on
‘moredata’ (weak_count = 1000), to make the final submission based on
‘training’. It got an RMSLE score of 0.862582 on the private leaderboard,
which is roughly 41.7% better than WMF’s baseline predictive model. The final
rank of our team, “zeditor”, is the 3rd place among 96 teams.
## IV Related Work
The global slowdown of Wikipedia’s growth rate (both in the number of editors
and the number of edits per month) has been studied [1]. It is found that
medium-frequency editors now cover a lower percentage of the total population
while high frequency editors continue to increase the number of their edits.
Moreover, there are increased patterns of conflict and dominance (e.g.,
greater resistance to new edits in particular those from occasional editors),
which may be the consequence of the increasingly limited opportunities in
making novel contributions. These findings could guide us to generate other
kinds of useful features to tackle the problem of edit number prediction.
Furthermore, researchers have also investigated other activities of
Wikipedia’s editors, such as voting on the promotion of Wikipedia admins [6].
In addition to Wikipedia, the temporal dynamics of online users’ behaviour has
been explored and exploited in web search [7, 8, 9, 10], social tagging [11,
12], blogging [13], twittering [14], and collaborative filtering [15]. The
_power law_ [16] and the _exponential decay_ [3] seem to be recurrent themes
across application domains.
## V Conclusions
Our most important insight is that a Wikipedia editor’s future behaviour can
be largely determined by the temporal dynamics of his recent behaviour. We are
a bit surprised that just temporal dynamics features can go such a long way
when we choose proper temporal scales and employ a powerful machine learning
method. Human beings seem to be working and living in a more mechanical way
than one might have thought. Since such temporal dynamics features are
actually independent of any semantics or knowledge about this specific
problem, our approach could be easily generalised to other application
domains, such as predicting the future supermarket spendings of shoppers
(e.g., the dunnhumby’s Shopper
Challenge171717http://www.kaggle.com/c/dunnhumbychallenge), predicting the
future hospital admissions of patients (e.g., the Heritage Health Prize
Competition181818http://www.heritagehealthprize.com/c/hhp), and so on, based
on historical behavioural data.
Have we answered the question that we asked at the beginning of this paper?
Yes and No. On one hand, we have built a predictive model which can be used to
identify those editors who are likely to become inactive, or in other words,
who need special care and attention to be kept — if an editor is going to
leave the Wikipedia community, there would probably be early signals in the
temporal dynamics of his recent behaviour. On the other hand, that predictive
model is pretty much a black box — it does not reveal the underlying reasons
why editors become inactive, and therefore it cannot tell us how to encourage
editors to remain active. For the ultimate purpose of Wikipedia’s sustainable
growth, we will need to investigate which attributes of an editor (his
articles’ category distribution, his relationship with other editors, etc.)
and also which recent events happened to him (his articles being deleted, his
revisions being reverted, unfair comments about his edits being received,
etc.) could affect his behaviour. Due to the time constraints and the dataset
limitations (for example, the lack of information about articles and comments
in the datasets ‘validation’ and ‘moredata’), we have to leave it to future
work.
Long live Wikipedia!
## Acknowledgment
We would like to thank WMF and Kaggle for their wonderful job in organising
this interesting contest. We are grateful to Twan van Laarhoven for creating
and sharing the additional dataset ‘moredata’. We also appreciate the
reviewers’ helpful comments.
## References
* [1] B. Suh, G. Convertino, E. H. Chi, and P. Pirolli, “The singularity is not near: Slowing growth of Wikipedia,” in _Proceedings of the 2009 International Symposium on Wikis (WikiSym)_ , Orlando, FL, USA, 2009.
* [2] T. Hastie, R. Tibshirani, and J. Friedman, _The Elements of Statistical Learning: Data Mining, Inference, and Prediction_ , 2nd ed. Springer, 2009.
* [3] C. C. Aggarwal, J. Han, J. Wang, and P. S. Yu, “A framework for projected clustering of high dimensional data streams,” in _Proceedings of the 13th International Conference on Very Large Data Bases (VLDB)_ , Toronto, Canada, 2004, pp. 852–863.
* [4] J. Friedman, “Greedy function approximation: A gradient boosting machine,” IMS 1999 Reitz Lecture, Tech. Rep., Feb 1999.
* [5] ——, “Stochastic gradient boosting,” Stanford University, Tech. Rep., Mar 1999\.
* [6] J. Leskovec, D. P. Huttenlocher, and J. M. Kleinberg, “Governance in social media: A case study of the wikipedia promotion process,” in _Proceedings of the 4th International Conference on Weblogs and Social Media (ICWSM)_ , Washington, DC, USA, 2010.
* [7] D. Zhang, J. Lu, R. Mao, and J.-Y. Nie, “Time-sensitive language modelling for online term recurrence prediction,” in _In Proceedings of the 2nd International Conference on the Theory of Information Retrieval (ICTIR)_ , Cambridge, UK, 2009, pp. 128–138.
* [8] D. Zhang and J. Lu, “What queries are likely to recur in web search?” in _Proceedings of the 32nd Annual International ACM SIGIR Conference on Research and Development in Information Retrieval (SIGIR)_ , Boston, MA, USA, 2009\.
* [9] J. L. Elsas and S. T. Dumais, “Leveraging temporal dynamics of document content in relevance ranking,” in _Proceedings of the 3rd International Conference on Web Search and Web Data Mining (WSDM)_ , New York, NY, USA, 2010, pp. 1–10.
* [10] A. Kulkarni, J. Teevan, K. M. Svore, and S. T. Dumais, “Understanding temporal query dynamics,” in _Proceedings of the 4th International Conference on Web Search and Web Data Mining (WSDM)_ , Hong Kong, China, 2011, pp. 167–176.
* [11] D. Zhang, R. Mao, and W. Li, “The recurrence dynamics of social tagging,” in _Proceedings of the 18th International Conference on World Wide Web (WWW)_ , Madrid, Spain, 2009, pp. 1205–1206.
* [12] H. Halpin, V. Robu, and H. Shepherd, “The complex dynamics of collaborative tagging,” in _Proceedings of the 16th International Conference on World Wide Web (WWW)_ , Banff, Alberta, Canada, 2007, pp. 211–220.
* [13] Y.-R. Lin, H. Sundaram, Y. Chi, J. Tatemura, and B. L. Tseng, “Detecting splogs via temporal dynamics using self-similarity analysis,” _ACM Transactions on the Web (TWEB)_ , vol. 2, no. 1, pp. 1–35, 2008.
* [14] F. Abel, Q. Gao, G.-J. Houben, and K. Tao, “Analyzing temporal dynamics in twitter profiles for personalized recommendations in the social web,” in _Proceedings of the 3rd International Conference on Web Science (WebSci)_ , Koblenz, Germany, 2011.
* [15] Y. Koren, “Collaborative filtering with temporal dynamics,” in _Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD)_ , Paris, France, 2009, pp. 447–456.
* [16] A. Clauset, C. R. Shalizi, and M. E. J. Newman, “Power-law distributions in empirical data,” _SIAM Review_ , vol. 51, no. 4, pp. 661–703, 2009.
|
arxiv-papers
| 2011-10-23T14:41:21 |
2024-09-04T02:49:23.534292
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dell Zhang",
"submitter": "Dell Zhang",
"url": "https://arxiv.org/abs/1110.5051"
}
|
1110.5055
|
# Theory of “Weak Value” and Quantum Mechanical Measurements
Yutaka Shikano
Department of Physics, Tokyo Institute of Technology, Tokyo, Japan
Center of Quantum Studies, Schmid College of Science and Technology,
Chapman University, CA, USA email: yshikano@ims.ac.jpMy current affiliation is
Institute for Molecular Science located at Okazaki, Aichi, Japan.
###### Contents
1. 1 Introduction
2. 2 Review of Quantum Operation
1. 2.1 Historical Remarks
2. 2.2 Operator-Sum Representation
3. 2.3 Indirect Quantum Measurement
3. 3 Review of Weak Value
4. 4 Historical Background – Two-State Vector Formalism
1. 4.1 Time Symmetric Quantum Measurement
2. 4.2 Protective Measurement
3. 4.3 Weak Measurement
5. 5 Weak-Value Measurement for a Qubit System
6. 6 Weak Values for Arbitrary Coupling Quantum Measurement
7. 7 Weak Value with Decoherence
8. 8 Weak Measurement with Environment
9. 9 Summary
## 1 Introduction
Quantum mechanics provides us many perspectives and insights on Nature and our
daily life. However, its mathematical axiom initiated by von Neumann [121] is
not satisfied to describe nature phenomena. For example, it is impossible not
to explain a non self-adjoint operator, i.e., the momentum operator on a half
line (See, e.g., Ref. [154].), as the physical observable. On considering
foundations of quantum mechanics, the simple and specific expression is
needed. One of the candidates is the weak value initiated by Aharonov and his
colleagues [4]. It is remarked that the idea of their seminal work is written
in Ref. [3]. Furthermore, this quantity has a potentiality to explain the
counter-factual phenomena, in which there is the contradiction under the
classical logic, e.g., the Hardy paradox [64]. If so, it may be possible to
quantitatively explain quantum mechanics in the particle picture. In this
review based on the author thesis [152], we consider the theory of the weak
value and construct a measurement model to extract the weak value. See the
other reviews in Refs. [14, 15, 20, 12].
Let the weak value for an observable $A$ be defined as
$\,{}_{f}\langle{A}\rangle_{i}^{w}:=\frac{\langle{f}|A|{i}\rangle}{\langle{f}|{i}\rangle},$
(1)
where $|{i}\rangle$ and $|{f}\rangle$ are called a pre- and post-selected
state, respectively. As the naming of the “weak value”, this quantity is
experimentally accessible by the weak measurement as explained below. As seen
in Fig. 1, the weak value can be measured as the shift of a meter of the probe
after the weak interaction between the target and the probe with the specific
post-selection of the target. Due to the weak interaction, the quantum state
of the target is only slightly changed but the information of the desired
observable $A$ is encoded in the probe by the post-selection. While the
previous studies of the weak value since the seminal paper [4], which will be
reviewed in Sec. 3, are based on the measurement scheme, there are few works
that the weak value is focused on and is independent of the measurement
scheme. Furthermore, in these 20 years, we have not yet understood the
mathematical properties of the weak value. In this chapter, we review the
historical backgrounds of the weak value and the weak measurement and recent
development on the measurement model to extract the weak value.
Figure 1: Schematic figure of the weak measurement.
## 2 Review of Quantum Operation
The time evolution for the quantum state and the operation for the measurement
are called a quantum operation. In this section, we review a general
description of the quantum operation. Therefore, the quantum operation can
describe the time evolution for the quantum state, the control of the quantum
state, the quantum measurement, and the noisy quantum system in the same
formulation.
### 2.1 Historical Remarks
Within the mathematical postulates of quantum mechanics [121], the state
change is subject to the Schrödinger equation. However, the state change on
the measurement is not subject to this but is subject to another axiom,
conventionally, von Neumann-Lüders projection postulate [105]. See more
details on quantum measurement theory in the books [31, 40, 194].
Let us consider a state change from the initial state $|{\psi}\rangle$ on the
projective measurement 111This measurement is often called the von Neumann
measurement or the strong measurement. for the operator
$A=\sum_{j}a_{j}|{a_{j}}\rangle\\!\langle{a_{j}}|$. From the Born rule, the
probability to obtain the measurement outcome, that is, the eigenvalue of the
observable $A$, is given by
$\Pr[A=a_{m}]=|\langle{a_{m}}|{\psi}\rangle|^{2}=\operatorname{Tr}\left[|{\psi}\rangle\\!\langle{\psi}|\cdot|{a_{m}}\rangle\\!\langle{a_{m}}|\right]=\operatorname{Tr}\rho
P_{a_{m}},$ (2)
where $\rho:=|{\psi}\rangle\\!\langle{\psi}|$ and
$P_{a_{m}}=|{a_{m}}\rangle\\!\langle{a_{m}}|$. After the measurement with the
measurement outcome $a_{m}$, the quantum state change is given by
$|{\psi}\rangle\to|{a_{m}}\rangle,$ (3)
which is often called the “collapse of wavefunction” or “state reduction”.
This implies that it is necessary to consider the non-unitary process even in
the isolated system. To understand the measuring process as quantum dynamics,
we need consider the general theory of quantum operations.
### 2.2 Operator-Sum Representation
Let us recapitulate the general theory of quantum operations of a finite
dimensional quantum system [122]. All physically realizable quantum operations
can be generally described by a completely positive (CP) map [127, 128], since
the isolated system of a target system and an auxiliary system always
undergoes the unitary evolution according to the axiom of quantum mechanics
[121]. Physically speaking, the operation of the target system should be
described as a positive map, that is, the map from the positive operator to
the positive operator, since the density operator is positive. Furthermore, if
any auxiliary system is coupled to the target one, the quantum dynamics in the
compound system should be also described as the positive map since the
compound system should be subject to quantum mechanics. Given the positive
map, the positive map is called a CP map if and only if the positive map is
also in the compound system coupled to any auxiliary system. One of the
important aspects of the CP map is that all physically realizable quantum
operations can be described only by operators defined in the target system.
Furthermore, the auxiliary system can be environmental system, the probe
system, and the controlled system. Regardless to the role of the auxiliary
system, the CP map gives the same description for the target system. On the
other hand, both quantum measurement and decoherence give the same role for
the target system.
Let ${\cal E}$ be a positive map from ${\cal L}({\cal H}_{s})$, a set of
linear operations on the Hilbert space ${\cal H}_{s}$, to ${\cal L}({\cal
H}_{s})$. If ${\cal E}$ is completely positive, its trivial extension ${\cal
K}$ from ${\cal L}({\cal H}_{s})$ to ${\cal L}({\cal H}_{s}\otimes{\cal
H}_{e})$ is also positive such that
${\cal K}(|{\alpha}\rangle):=({\cal E}\otimes{\bf
1})(|{\alpha}\rangle\\!\langle{\alpha}|)>0,$ (4)
for an arbitrary state $|{\alpha}\rangle\in{\cal H}_{s}\otimes{\cal H}_{p}$,
where ${\bf 1}$ is the identity operator. We assume without loss of generality
${\rm dim}{\cal H}_{s}={\rm dim}{\cal H}_{e}<\infty$. Throughout this chapter,
we concentrate on the case that the target state is pure though the
generalization to mixed states is straightforward. From the complete
positivity, we obtain the following theorem for quantum state changes.
###### Theorem 2.1.
Let ${\cal E}$ be a CP map from ${\cal H}_{s}$ to ${\cal H}_{s}$. For any
quantum state $|{\psi}\rangle_{s}\in{\cal H}_{s}$, there exist a map $\sigma$
and a pure state $|{\alpha}\rangle\in{\cal H}_{s}\otimes{\cal H}_{e}$ such
that
${\cal E}(|{\psi}\rangle_{s}\langle{\psi}|)=\,_{e}\langle{\tilde{\psi}}|{\cal
K}(|{\alpha}\rangle)|{\tilde{\psi}}\rangle_{e},$ (5)
where
$|{\psi}\rangle_{s}=\sum_{k}\psi_{k}|{k}\rangle_{s},\ \ \
|{\tilde{\psi}}\rangle_{e}=\sum_{k}\psi^{\ast}_{k}|{k}\rangle_{e},$ (6)
which represents the state change for the density operator.
###### Proof.
We can write in the Schmidt form as
$|{\alpha}\rangle=\sum_{m}|{m}\rangle_{s}|{m}\rangle_{e}.$ (7)
We rewrite the right hand sides of Eq. (5) as
$\displaystyle{\cal K}(|{\alpha}\rangle)$ $\displaystyle=({\cal E}\otimes{\bf
1})\left(\sum_{m,n}|{m}\rangle_{s}|{m}\rangle_{e}\,{}_{s}\langle{n}|_{e}\langle{n}|\right)$
$\displaystyle=\sum_{m,n}|{m}\rangle_{e}\langle{n}|{\cal
E}(|{m}\rangle_{s}\langle{n}|),$ (8)
to obtain
$\,{}_{e}\langle{m}|{\cal K}(|{\alpha}\rangle)|{n}\rangle_{e}={\cal
E}(|{m}\rangle_{s}\langle{n}|).$ (9)
By linearity, the desired equation (5) can be derived. ∎
From the complete positivity, ${\cal K}(|{\alpha}\rangle)>0$ for all
$|{\alpha}\rangle\in{\cal H}_{s}\otimes{\cal H}_{e}$, we can express
$\sigma(|{\alpha}\rangle)$ as
${\cal
K}(|{\alpha}\rangle)=\sum_{m}s_{m}|{\hat{s}_{m}}\rangle\\!\langle{\hat{s}_{m}}|=\sum_{m}|{s_{m}}\rangle\\!\langle{s_{m}}|,$
(10)
where $s_{m}$’s are positive and $\\{|{\hat{s}_{m}}\rangle\\}$ is a complete
orthonormal set with $|{s_{m}}\rangle:=\sqrt{s_{m}}|{\hat{s}_{m}}\rangle$. We
define the Kraus operator $E_{m}$ [95] as
$E_{m}|{\psi}\rangle_{s}:=\,_{e}\langle{\tilde{\psi}}|{s_{m}}\rangle.$ (11)
Then, the quantum state change becomes the operator-sum representation,
$\sum_{m}E_{m}|{\psi}\rangle_{s}\langle{\psi}|E^{\dagger}_{m}=\sum_{m}\,{}_{e}\langle{\tilde{\psi}}|{s_{m}}\rangle\langle{s_{m}}|{\tilde{\psi}}\rangle_{e}=\,_{e}\langle{\tilde{\psi}}|{\cal
K}(|{\alpha}\rangle)|{\tilde{\psi}}\rangle_{e}\\\ ={\cal
E}(|{\psi}\rangle_{s}\langle{\psi}|).$
It is emphasized that the quantum state change is described solely in terms of
the quantities of the target system.
### 2.3 Indirect Quantum Measurement
In the following, the operator-sum representation of the quantum state change
is related to the indirect measurement model. Consider the observable $A_{s}$
and $B_{p}$ for the target and probe systems given by
$A_{s}=\sum_{j}a_{j}|{a_{j}}\rangle_{s}\langle{a_{j}}|,\ \ \
B_{p}=\sum_{j}b_{j}|{b_{j}}\rangle_{p}\langle{b_{j}}|,$ (12)
respectively. We assume that the interaction Hamiltonian is given by
$H_{int}(t)=g(A_{s}\otimes B_{p})\ \delta(t-t_{0}),$ (13)
where $t_{0}$ is measurement time. Here, without loss of generality, the
interaction is impulsive and the coupling constant $g$ is scalar. The quantum
dynamics for the compound system is given by
$|{s_{m}}\rangle\\!\langle{s_{m}}|=U(|{\psi}\rangle_{s}\langle{\psi}|\otimes|{\phi}\rangle_{p}\langle{\phi}|)U^{\dagger},$
(14)
where $|{\psi}\rangle_{s}$ and $|{\phi}\rangle_{p}$ are the initial quantum
state on the target and probe systems, respectively. For the probe system, we
perform the projective measurement for the observable $B_{p}$. The probability
to obtain the measurement outcome $b_{m}$ is given by
$\displaystyle\Pr[B_{p}=b_{m}]$
$\displaystyle=\operatorname{Tr}_{s}\langle{b_{m}}|U(|{\psi}\rangle_{s}\langle{\psi}|\otimes|{\phi}\rangle_{p}\langle{\phi}|)U^{\dagger}|{b_{m}}\rangle,$
$\displaystyle=\operatorname{Tr}_{s}E_{m}|{\psi}\rangle_{s}\langle{\psi}|E^{\dagger}_{m}=\operatorname{Tr}_{s}|{\psi}\rangle_{s}\langle{\psi}|M_{m},$
(15)
where the Kraus operator $E_{m}$ is defined as
$E_{m}:=\,_{p}\langle{b_{m}}|U|{\phi}\rangle_{p},$ (16)
and $M_{m}:=E^{\dagger}_{m}E_{m}$ is called a positive operator valued measure
(POVM) [45]. The POVM has the same role of the spectrum of the operator
$A_{s}$ in the case of the projective measurement. To derive the projective
measurement from the indirect measurement, we set the spectrum of the operator
$A_{s}$ as the POVM, that is, $M_{m}=|{a_{m}}\rangle_{s}\langle{a_{m}}|$.
Since the sum of the probability distribution over the measurement outcome
equals to one, we obtain
$\displaystyle\sum_{m}\Pr[B_{p}=b_{m}]=1$
$\displaystyle\Longleftrightarrow\sum_{m}\operatorname{Tr}|{\psi}\rangle_{s}\langle{\psi}|M_{m}=\operatorname{Tr}|{\psi}\rangle_{s}\langle{\psi}|\sum_{m}M_{m}=1$
$\displaystyle\to\sum_{m}M_{m}={\bf 1}.$ (17)
Here, the last line uses the property of the density operator,
$\operatorname{Tr}|{\psi}\rangle_{s}\langle{\psi}|=1$ for any
$|{\psi}\rangle$.
## 3 Review of Weak Value
In Secs. 2.1 and 2.3, the direct and indirect quantum measurement schemes, we
only get the probability distribution. However, the probability distribution
is not the only thing that is experimentally accessible in quantum mechanics.
In quantum mechanics, the phase is also an essential ingredient and in
particular the geometric phase is a notable example of an experimentally
accessible quantity [150]. The general experimentally accessible quantity
which contains complete information of the probability and the phase seems to
be the weak value advocated by Aharonov and his collaborators [4, 14]. They
proposed a model of weakly coupled system and probe, see Sec. 4.3, to obtain
information to a physical quantity as a “weak value” only slightly disturbing
the state. Here, we briefly review the formal aspects of the weak value.
For an observable $A$, the weak value $\langle{A}\rangle_{w}$ is defined as
$\langle{A}\rangle_{w}:=\frac{{\langle{f}|}U(t_{f},t)AU(t,t_{i})|{i}\rangle}{\langle{f}|U(t_{f},t_{i})|{i}\rangle}\in{\mathbb{C}},$
(18)
where $|{i}\rangle$ and $\langle{f}|$ are normalized pre-selected ket and
post-selected bra state vectors, respectively [4]. Here, $U(t_{2},t_{1})$ is
an evolution operator from the time $t_{1}$ to $t_{2}$. The weak value
$\langle{A}\rangle_{w}$ actually depends on the pre- and post-selected states
$|{i}\rangle$ and $\langle{f}|$ but we omit them for notational simplicity in
the case that we fix them. Otherwise, we write them explicitly as
${}_{f}\langle{A}\rangle_{i}^{w}$ instead for $\langle{A}\rangle_{w}$. The
denominator is assumed to be non-vanishing. This quantity is, in general, in
the complex number ${\mathbb{C}}$. Historically, the terminology “weak value”
comes from the weak measurement, where the coupling between the target system
and the probe is weak, explained in the following section. Apart from their
original concept of the weak value and the weak measurement, we emphasize that
the concept of the weak value is independent of the weak measurement 222This
concept is shared in Refs. [81, 82, 117, 1, 78, 130, 49, 51].. To take the
weak value as a priori given quantity in quantum mechanics, we will construct
the observable-independent probability space. In the conventional quantum
measurement theory, the probability space, more precisely speaking, the
probability measure, depends on the observable [151, Sec. 4.1] 333Due to this,
the probability in quantum mechanics cannot be applied to the standard
probability theory. As another approach to resolve this, there is the quantum
probability theory [138]..
Let us calculate the expectation value in quantum mechanics for the quantum
state $|{\psi}\rangle$ as
$\displaystyle\operatorname{Ex}[A]=\langle{\psi}|A|{\psi}\rangle$
$\displaystyle=\int
d\phi\,\langle{\psi}|{\phi}\rangle\langle{\phi}|A|{\psi}\rangle=\int
d\phi\,\langle{\psi}|{\phi}\rangle\cdot\langle{\phi}|{\psi}\rangle\frac{\langle{\phi}|A|{\psi}\rangle}{\langle{\phi}|{\psi}\rangle},$
$\displaystyle=\int
d\phi\,|\langle{\psi}|{\phi}\rangle|^{2}\,_{\phi}\langle{A}\rangle_{\psi}^{w},$
(19)
where $h_{A}[|{\phi}\rangle]=\,_{\phi}\langle{A}\rangle_{\psi}^{w}$ is complex
random variable and $dP:=|\langle{\phi}|{\psi}\rangle|^{2}d\phi$ is the
probability measure and is independent of the observable $A$. Therefore, the
event space $\Omega=\\{|{\phi}\rangle\\}$ is taken as the set of the post-
selected state. This formula means that the extended probability theory
corresponds to the Born rule. From the conventional definition of the variance
in quantum mechanics, we obtain the variance as
$\displaystyle\operatorname{Var}[A]$
$\displaystyle=\int|h_{A}[|{\phi}\rangle]|^{2}dP-\left(\int
h_{A}[|{\phi}\rangle]dP\right)^{2}$
$\displaystyle=\int\left|\frac{\langle{\phi}|A|{\psi}\rangle}{\langle{\phi}|{\psi}\rangle}\right|^{2}|\langle{\phi}|{\psi}\rangle|^{2}d\phi-\left(\int\frac{\langle{\phi}|A|{\psi}\rangle}{\langle{\phi}|{\psi}\rangle}|\langle{\phi}|{\psi}\rangle|^{2}d\phi\right)^{2}$
$\displaystyle=\int\left|\langle{\phi}|A|{\psi}\rangle\right|^{2}d\phi-\left(\int\langle{\psi}|{\phi}\rangle\langle{\phi}|A|{\psi}\rangle
d\phi\right)^{2}$
$\displaystyle=\int\langle{\psi}|A|{\phi}\rangle\\!\langle{\phi}|A|{\psi}\rangle
d\phi-(\langle{\psi}|A|{\psi}\rangle)^{2}$
$\displaystyle=\langle{\psi}|A^{2}|{\psi}\rangle-(\langle{\psi}|A|{\psi}\rangle)^{2}.$
(20)
This means that the observable-independent probability space can be
characterized by the weak value [155]. From another viewpoint of the weak
value, the statistical average of the weak value coincides with the
expectation value in quantum mechanics [7]. This can be interpreted as the
probability while this allows the “negative probability” 444The concept of
negative probability is not new, e.g., see Refs. [47, 57, 65, 71, 66]. The
weak value defined by Eq. (18) is normally called the transition amplitude
from the state $|{\psi}\rangle$ to $\langle{\phi}|$ via the intermediate state
$|{a}\rangle$ for $A=|{a}\rangle\\!\langle{a}|$, the absolute value squared of
which is the probability for the process. But the three references quoted
above seem to suggest that they might be interpreted as probabilities in the
case that the process is counter-factual, i.e., the case that the intermediate
state $|{a}\rangle$ is not projectively measured. The description of
intermediate state $|{a}\rangle$ in the present work is counter-factual or
virtual in the sense that the intermediate state would not be observed by
projective measurements. Feynman’s example is the counter-factual
“probability” for an electron to have its spin up in the $x$-direction and
also spin down in the $z$-direction [57].. On this idea, the uncertainty
relationship was analyzed on the Robertson inequality [58, 163] and on the
Ozawa inequality [106], which the uncertainty relationships are reviewed in
Ref. [151, Appendix A]. Also, the joint probability for the compound system
was analyzed in Refs. [27, 30]. Furthermore, if the operator $A$ is a
projection operator $A=|{a}\rangle\\!\langle{a}|$, the above identity becomes
an analog of the Bayesian formula,
$|\langle{a}|{\psi}\rangle|^{2}=\int\,_{\phi}\langle{|{a}\rangle\\!\langle{a}|}\rangle_{\psi}^{w}|\langle{\phi}|{\psi}\rangle|^{2}d\phi.$
(21)
The left hand side is the probability to obtain the state $|{a}\rangle$ given
the initial state $|{\psi}\rangle$. From this, one may get some intuition by
interpreting the weak value
$\,{}_{\phi}\langle{|{a}\rangle\\!\langle{a}|}\rangle_{\psi}^{w}$ as the
complex conditional probability of obtaining the result $|{a}\rangle$ under an
initial condition $|{i}\rangle$ and a final condition $|{f}\rangle$ in the
process $|{i}\rangle\rightarrow|{a}\rangle\rightarrow|{f}\rangle$ [171, 170]
555The interpretation of the weak value as a complex probability is suggested
in the literature [118].. Of course, we should not take the strange weak
values too literally but the remarkable consistency of the framework of the
weak values due to Eq. (21) and a consequence of the completeness relation,
$\sum_{a}\langle{|{a}\rangle\langle{a}|}\rangle_{w}=1,$ (22)
may give a useful concept to further push theoretical consideration by
intuition.
This interpretation of the weak values gives many possible examples of strange
phenomena like a negative kinetic energy [11], a spin $100\hbar$ for an
electron [4, 52, 60, 23] and a superluminal propagation of light [142, 162]
and neutrino [176, 28] motivated by the OPERA experiment [125]. The framework
of weak values has been theoretically applied to foundations of quantum
physics, e.g., the derivation of the Born rule from the alternative assumption
for a priori measured value [74], the relationship to the uncertainty
relationship [72], the quantum stochastic process [190], the tunneling
traverse time [171, 170, 135], arrival time and time operator [146, 21, 147,
39], the decay law [46, 187], the non-locality [181, 180, 32], especially,
quantum non-locality, which is characterized by the modular variable,
consistent history [188, 87], Bohmian quantum mechanics [98], semi-classical
weak values on the tunneling [175], the quantum trajectory [192], and
classical stochastic theory [177]. Also, in quantum information science, the
weak value was analyzed on quantum computation [126, 35], quantum
communications [36, 29], quantum estimation, e.g., state tomography [67, 68,
69, 158, 111] and the parameter estimation [70, 73, 157], the entanglement
concentration [113], the quasi-probability distribution [24, 148, 183, 61] and
the cloning of the unknown quantum state with hint [161]. Furthermore, this
was applied to the cosmological situations in quantum-mechanical region, e.g.,
the causality [22], the inflation theory [42], backaction of the Hawking
radiation from the black hole [54, 55, 34], and the new interpretation of the
universe [9, 62, 53]. However, the most important fact is that the weak value
is experimentally accessible so that the intuitive argument based on the weak
values can be either verified or falsified by experiments. There are many
experimental proposals to obtain the weak value in the optical [101, 88, 159,
2, 44, 112, 197] and the solid-state [144, 143, 94, 115, 84, 191, 83, 200]
systems. Recently, the unified viewpoint was found in the weak measurement
[92].
On the realized experiments on the weak value, we can classify the three
concepts: (i) testing the quantum theory, (ii) the amplification of the tiny
effect in quantum mechanics, and (iii) the quantum phase.
* (i)
Testing the quantum theory. The weak value can solve many quantum paradoxes
seen in the book [14]. The Hardy paradox [64], which there occurs in two Mach-
Zehnder interferometers of the electron and the position, was resolved by the
weak value [8] and was analyzed deeper [75]. This paradoxical situation was
experimentally demonstrated in the optical setup [107, 198]. By the
interference by the polarization [131] and shifting the optical axis [141],
the spin beyond the eigenvalue is verified. By the latter technique, the
three-box paradox [16, 188] was realized [139]. Thereafter, the theoretical
progresses are the contextuality on quantum mechanics [178], the generalized
N-box paradox [99], and the relationship to the Kirkpatrick game [137]. The
weak value is used to show the violation of the Leggett-Garg inequality [191,
110]. This experimental realizations were demonstrated in the system of the
superconducting qubit [97], the optical systems [134, 50]. Furthermore, since
the weak value for the position observable $|{x}\rangle\\!\langle{x}|$ with
the pre-selected state $|{\psi}\rangle$ and the post-selection $|{p}\rangle$
is given by
$\langle{|{x}\rangle\\!\langle{x}|}\rangle_{w}=\frac{\langle{p}|{x}\rangle\langle{x}|{\psi}\rangle}{\langle{p}|{\psi}\rangle}=\frac{e^{ixp}\psi(x)}{\phi(p)},$
(23)
we obtain the wavefunction $\psi(x):=\langle{x}|{\psi}\rangle$ as the weak
value with the multiplication factor $1/\phi(0)$ with
$\phi(p):=\langle{p}|{\psi}\rangle$ in the case of $p=0$. Using the photon
transverse wavefunction, there are experimentally demonstrated by replacing
the weak measurement for the position as the polarization measurement [109].
This paper was theoretically criticized to compare the standard quantum state
tomography for the phase space in Ref. [63] and was generalized to a
conventionally unobservable [108]. As other examples, there are the detection
of the superluminal signal [37], the quantum non-locality [165], and the
Bohmian trajectory [149, 91] on the base of the theoretical analysis [193].
* (ii)
Amplification of the tiny effect in quantum mechanics. Since the weak value
has the denominator, the weak value is very large when the pre- and post-
selected states are almost orthogonal666Unfortunately, the signal to noise
ratio is not drastically changed under the assumption that the probe
wavefunction is Gaussian on a one-dimensional parameter space.. This is
practical advantage to use the weak value. While the spin Hall effect of light
[124] is too tiny effect to observe its shift in the conventional scheme, by
almost orthogonal polarizations for the input and output, this effect was
experimentally verified [76] to be theoretically analyzed from the viewpoint
of the spin moments [96]. Also, some interferometers were applied. The beam
deflection on the Sagnac interferometer [48] was shown to be supported by the
classical and quantum theoretical analyses [77] 777Unfortunately, the
experimental data are mismatched to the theoretical prediction. While the
authors claimed that this differences results from the stray of light, the
full-order calculation even is not mismatched [93]. However, this difference
remains the open problem.. Thereafter, optimizing the signal-to-noise ratio
[184, 166], the phase amplification [168, 169], and the precise frequency
measurement [167] were demonstrated. As another example, there is shaping the
laser pulse beyond the diffraction limit [136]. According to Steinberg [172],
in his group, the amplification on the single-photon nonlinearity has been
progressed to be based on the theoretical proposal [56]. While the charge
sensing amplification was proposed in the solid-state system [200], there is
no experimental demonstration on the amplification for the solid-state system.
Furthermore, the upper bound of the amplification has not yet solved.
Practically, this open problem is so important to understand the relationship
to the weak measurement regime.
* (iii)
Quantum phase. The argument of the weak value for the projection operator is
the geometric phase as
$\displaystyle\gamma$
$\displaystyle:=\arg\langle{\psi_{1}}|{\psi_{2}}\rangle\langle{\psi_{2}}|{\psi_{3}}\rangle\langle{\psi_{3}}|{\psi_{1}}\rangle$
$\displaystyle=\arg\frac{\langle{\psi_{1}}|{\psi_{2}}\rangle\langle{\psi_{2}}|{\psi_{3}}\rangle\langle{\psi_{3}}|{\psi_{1}}\rangle}{|\langle{\psi_{3}}|{\psi_{1}}\rangle|^{2}}=\arg\frac{\langle{\psi_{1}}|{\psi_{2}}\rangle\langle{\psi_{2}}|{\psi_{3}}\rangle}{\langle{\psi_{1}}|{\psi_{3}}\rangle}$
$\displaystyle=\arg\,_{\psi_{1}}\langle{|{\psi_{2}}\rangle\\!\langle{\psi_{2}}|}\rangle_{\psi_{3}}^{w}.$
(24)
where the quantum states, $|{\psi_{1}}\rangle,|{\psi_{2}}\rangle$, and
$|{\psi_{3}}\rangle$, are the pure states [160]. Here, the quantum states,
$|{\psi_{1}}\rangle$ and $|{\psi_{3}}\rangle$, are the post- and pre-selected
states, respectively. Therefore, we can evaluate the weak value from the phase
shift [174]. Of course, vice versa [38]. Tamate et al. proposal was
demonstrated on the relationship to quantum eraser [90] and by the a three-
pinhole interferometer [89]. The phase shift from the zero mode to $\pi$ mode
was observed by using the interferometer with a Cs vapor [41] and the phase
shift in the which-way path experiment was demonstrated [116]. Furthermore, by
the photonic crystal, phase singularity was demonstrated [164].
* (iv)
Miscellaneous. The backaction of the weak measurement is experimentally
realized in the optical system [79]. Also, the parameter estimation using the
weak value is demonstrated [73].
## 4 Historical Background – Two-State Vector Formalism
In this section, we review the original concept of the two-state vector
formalism. This theory is seen in the reviewed papers [20, 15].
### 4.1 Time Symmetric Quantum Measurement
While the fundamental equations of the microscopic physics are time symmetric,
for example, the Newton equation, the Maxwell equation, and the Schrödinger
equation 888It is, of course, noted that thermodynamics does not have the time
symmetric properties from the second law of thermodynamics., the quantum
measurement is not time symmetric. This is because the quantum state after
quantum measurement depends on the measurement outcome seen in Sec. 2. The
fundamental equations of the microscopic physics can be solved to give the
initial boundary condition. To construct the time symmetric quantum
measurement, the two boundary conditions, which is called pre- and post-
selected states, are needed. The concept of the pre- and post-selected states
is called the two-state vector formalism [6]. In the following, we review the
original motivation to construct the time symmetric quantum measurement.
Let us consider the projective measurement for the observable
$A=\sum_{i}a_{i}|{a_{i}}\rangle\\!\langle{a_{i}}|$ with the initial boundary
condition denoted as $|{i}\rangle$ at time $t_{i}$. To take quantum
measurement at time $t_{0}$, the probability to obtain the measurement outcome
$a_{j}$ is given by
$\Pr[A=a_{j}]=\parallel\langle{a_{j}}|U|{i}\rangle\parallel^{2},$ (25)
with the time evolution $U:=U(t_{0},t_{i})$. After the projective measurement,
the quantum state becomes $|{a_{j}}\rangle$. Thereafter, the quantum state at
$t_{f}$ is given by $|{\varphi_{j}}\rangle:=V|{a_{j}}\rangle$ with
$V=U(t_{f},t_{0})$. the probability to obtain the measurement outcome $a_{j}$
can be rewritten as
$\Pr[A=a_{j}]=\frac{\parallel\langle{\varphi_{j}}|V|{a_{j}}\rangle\parallel^{2}\parallel\langle{a_{j}}|U|{i}\rangle\parallel^{2}}{\sum_{j}\parallel\langle{\varphi_{j}}|V|{a_{j}}\rangle\parallel^{2}\parallel\langle{a_{j}}|U|{i}\rangle\parallel^{2}}.$
(26)
It is noted that
$\parallel\langle{\varphi_{j}}|V|{a_{j}}\rangle\parallel^{2}=1$. Here, we
consider the backward time evolution from the quantum state
$|{\varphi_{j}}\rangle$ at time $t_{f}$. We always obtain the quantum state
$|{a_{j}}\rangle$ after the projective measurement at time $t_{0}$. Therefore,
the quantum state at time $t_{i}$ is given by
$|{{\tilde{i}}}\rangle:=U^{\dagger}|{a_{j}}\rangle\\!\langle{a_{j}}|V^{\dagger}|{\varphi_{j}}\rangle=U^{\dagger}|{a_{j}}\rangle.$
(27)
In general, $|{{\tilde{i}}}\rangle$ is different from $|{i}\rangle$.
Therefore, projective measurement is time asymmetric.
To construct the time-symmetric quantum measurement, we add the boundary
condition at time $t_{f}$. Substituting the quantum state
$|{\varphi_{j}}\rangle$ to the specific one denoted as $|{f}\rangle$, which is
called the post-selected state, the probability to obtain the measurement
outcome $a_{j}$, Eq. (26), becomes
$\Pr[A=a_{j}]=\frac{\parallel\langle{f}|V|{a_{j}}\rangle\parallel^{2}\parallel\langle{a_{j}}|U|{i}\rangle\parallel^{2}}{\sum_{j}\parallel\langle{f}|V|{a_{j}}\rangle\parallel^{2}\parallel\langle{a_{j}}|U|{i}\rangle\parallel^{2}}.$
(28)
This is called the Aharonov-Bergmann-Lebowitz (ABL) formula [6]. From the
analogous discussion to the above, this measurement is time symmetric.
Therefore, describing quantum mechanics by the pre- and post-selected states,
$|{i}\rangle$ and $\langle{f}|$, is called the “two-state vector formalism”.
### 4.2 Protective Measurement
In this subsection, we will see the noninvasive quantum measurement for the
specific quantum state on the target system. Consider a system of consisting
of a target and a probe defined in the Hilbert space ${\cal H}_{s}\otimes{\cal
H}_{p}$. The interaction between the target and the probe is given by
$H_{int}(t)=g(t)(A\otimes\hat{P}),$ (29)
where
$\int^{T}_{0}g(t)dt=:g_{0}.$ (30)
The total Hamiltonian is given by
$H_{tot}(t)=H_{s}(t)+H_{p}(t)+H_{int}(t).$ (31)
Here, we suppose that $H_{s}(t)$ has discrete and non-degenerate eigenvalues
denoted as $E_{i}(t)$. Its corresponding eigenstate is denoted as
$|{E_{i}(t)}\rangle$ for any time $t$. Furthermore, we consider the
discretized time from the time interval $[0,T]$;
$t_{n}=\frac{n}{N}T\ (n=0,1,2,\dots,N),$ (32)
where $N$ is a sufficiently large number. We assume that the initial target
state is the energy eigenvalue $|{E_{i}(t)}\rangle$ 999Due to this assumption,
it is impossible to apply this to the arbitrary quantum state. Furthermore,
while we seemingly need the projective measurement, that is, destructive
measurement, for the target system to confirm whether the initial quantum
state is in the eigenstates [186, 145], they did not apply this to the
arbitrary state. For example, if the system is cooled down, we can pickup the
ground state of the target Hamiltonian $H_{s}(0)$. the initial probe state is
denoted as $|{\xi(0)}\rangle$. Under the adiabatic condition, the compound
state for the target and probe systems at time $T$ is given by
$\displaystyle|{\Phi(T)}\rangle$
$\displaystyle:=|{E_{i}(t_{N})}\rangle\\!\langle{E_{i}(t_{N})}|e^{-i\frac{T}{N}H_{tot}(t_{N})}|{E_{i}(t_{N-1})}\rangle\\!\langle{E_{i}(t_{N-1})}|e^{-i\frac{T}{N}H_{tot}(t_{N-1})}\cdots$
$\displaystyle\ \ \ \ \
\times|{E_{i}(t_{2})}\rangle\\!\langle{E_{i}(t_{2})}|e^{-i\frac{T}{N}H_{tot}(t_{2})}|{E_{i}(t_{1})}\rangle\\!\langle{E_{i}(t_{1})}|e^{-i\frac{T}{N}H_{tot}(t_{1})}|{E_{i}(0)}\rangle\otimes|{\xi(0)}\rangle.$
(33)
Applying the Trotter-Suzuki theorem [182, 173], one has
$\displaystyle|{\Phi(T)}\rangle$
$\displaystyle:=|{E_{i}(t_{N})}\rangle\\!\langle{E_{i}(t_{N})}|e^{-i\frac{T}{N}H_{int}(t_{N})}|{E_{i}(t_{N})}\rangle\\!\langle{E_{i}(t_{N-1})}|e^{-i\frac{T}{N}H_{int}(t_{N-1})}\cdots$
$\displaystyle\ \ \ \ \
\times|{E_{i}(t_{3})}\rangle\\!\langle{E_{i}(t_{2})}|e^{-i\frac{T}{N}H_{int}(t_{2})}|{E_{i}(t_{2})}\rangle\\!\langle{E_{i}(t_{1})}|e^{-i\frac{T}{N}H_{int}(t_{1})}|{E_{i}(1)}\rangle\otimes|{\xi(T)}\rangle.$
(34)
By the Taylor expansion with the respect to $N$, the expectation value is
$\displaystyle\langle{E_{i}(t_{n})}|e^{-i\frac{T}{N}g(t_{n})A\otimes\hat{P}}|{E_{i}(t_{n})}\rangle$
$\displaystyle=1-i\frac{T}{N}g(t_{n})\operatorname{Ex}[A(t_{n})]\hat{P}-\frac{1}{2}\frac{T^{2}}{N^{2}}g^{2}(t_{n})(\operatorname{Ex}[A(t_{n})])^{2}\hat{P}^{2}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-\frac{1}{2}\frac{T^{2}}{N^{2}}g^{2}(t_{n})\operatorname{Var}[A(t_{n})]\hat{P}^{2}+O\left(\frac{1}{N^{3}}\right)$
$\displaystyle\sim
e^{-i\frac{T}{N}g(t_{n})\operatorname{Ex}[A(t_{n})]\hat{P}}\left(1-\frac{1}{2}\frac{T^{2}}{N^{2}}g^{2}(t_{n})\operatorname{Var}[A(t_{n})]\hat{P}^{2}\right).$
(35)
In the limit of $N\to\infty$, by quadrature by parts, we obtain
$\displaystyle|{\Phi(T)}\rangle$
$\displaystyle\sim|{E_{i}(T)}\rangle\exp\left[-i\left(\int^{T}_{0}g(t)\operatorname{Ex}[A(t)]dt\right)\hat{P}\right]$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\times\left[1-\frac{T}{N}\left(\int^{T}_{0}g^{2}(t)\operatorname{Var}[A(t)]dt\right)\hat{P}^{2}\right]|{\xi(T)}\rangle+O\left(\frac{1}{N}\right)$
$\displaystyle=|{E_{i}(T)}\rangle\exp\left[-i\left(\int^{T}_{0}g(t)\operatorname{Ex}[A(t)]dt\right)\hat{P}\right]|{\xi(T)}\rangle.$
(36)
Therefore, the shift of the expectation value for the position operator on the
probe system is given by
$\Delta[Q]=\int^{T}_{0}g(t)\operatorname{Ex}[A(t)]dt.$ (37)
It is emphasized that the quantum state on the target system remains to be the
energy eigenstate of $H_{s}$. Therefore, this is called the protective
measurement [18, 5]. It is remarked that the generalized version of the
protective measurement in Ref. [19] by the pre- and post-selected states and
in Ref. [10] by the meta-stable state.
### 4.3 Weak Measurement
From the above discussions, is it possible to combine the above two concepts,
i.e., the time-symmetric quantum measurement without destroying the quantum
state [189]? This answer is the weak measurement [4]. Consider a target system
and a probe defined in the Hilbert space ${\cal H}_{s}\otimes{\cal H}_{p}$.
The interaction of the target system and the probe is assumed to be weak and
instantaneous,
$H_{int}(t)=g(A\otimes\hat{P})\delta(t-t_{0}),$ (38)
where an observable $A$ is defined in ${\cal H}_{s}$, while $\hat{P}$ is the
momentum operator of the probe. The time evolution operator becomes
$e^{-ig(A\otimes\hat{P})}$. Suppose the probe initial state is
$|{\xi}\rangle$. For the transition from the pre-selected state $|{i}\rangle$
to the post-selected state $|{f}\rangle$, the probe wave function becomes
$|{\xi^{\prime}}\rangle=\langle{f}|Ve^{-ig(A\otimes\hat{P})}U|{i}\rangle|{\xi}\rangle$,
which is in the weak coupling case,
$\displaystyle|{\xi^{\prime}}\rangle$
$\displaystyle=\langle{f}|Ve^{-ig(A\otimes\hat{P})}U|{i}\rangle|{\xi}\rangle$
$\displaystyle=\langle{f}|V[{\bf
1}-ig(A\otimes\hat{P})]U|{i}\rangle|{\xi}\rangle+O(g^{2})$
$\displaystyle=\langle{f}|VU|{i}\rangle-
ig\langle{f}|VAU|{i}\rangle\otimes\hat{P}|{\xi}\rangle+O(g^{2})$
$\displaystyle=\langle{f}|VU|{i}\rangle\left(1-ig\langle{A}\rangle_{w}\hat{P}\right)|{\xi}\rangle+O(g^{2})$
(39)
where
$\langle{f}|VAU|{i}\rangle/\langle{f}|VU|{i}\rangle=\langle{A}\rangle_{w}$.
Here, the last equation uses the approximation that $g\langle{A}\rangle_{w}\ll
1$ 101010It is remarked that Wu and Li showed the second-order correction of
the weak measurement [196]. A further analysis was shown in Refs. [129, 132]..
We obtain the shifts of the expectation values for the position and momentum
operators on the probe as the following theorem:
###### Theorem 4.1 (Jozsa [85]).
We obtain the shifts of the expectation values for the position and momentum
operators on the probe after the weak measurement with the post-selection as
$\displaystyle\Delta[\hat{Q}]$ $\displaystyle=g{\rm
Re}\langle{A}\rangle_{w}+mg{\rm
Im}\langle{A}\rangle_{w}\left.\frac{d\operatorname{Var}[\hat{Q}]}{dt}\right|_{t=t_{0}},$
(40) $\displaystyle\Delta[\hat{P}]$ $\displaystyle=2g{\rm
Im}\langle{A}\rangle_{w}\operatorname{Var}[\hat{P}],$ (41)
where
$\displaystyle\Delta[\hat{Q}]$
$\displaystyle:=\frac{\langle{\xi^{\prime}}|\hat{Q}|{\xi^{\prime}}\rangle}{\langle{\xi^{\prime}}|{\xi^{\prime}}\rangle}-\langle{\xi}|\hat{Q}|{\xi}\rangle,$
(42) $\displaystyle\Delta[\hat{P}]$
$\displaystyle:=\frac{\langle{\xi^{\prime}}|\hat{P}|{\xi^{\prime}}\rangle}{\langle{\xi^{\prime}}|{\xi^{\prime}}\rangle}-\langle{\xi}|\hat{P}|{\xi}\rangle,$
(43) $\displaystyle\operatorname{Var}[\hat{Q}]$
$\displaystyle:=\langle{\xi}|\hat{Q^{2}}|{\xi}\rangle-(\langle{\xi}|\hat{Q}|{\xi}\rangle)^{2},$
(44) $\displaystyle\operatorname{Var}[\hat{P}]$
$\displaystyle:=\langle{\xi}|\hat{P^{2}}|{\xi}\rangle-(\langle{\xi}|\hat{P}|{\xi}\rangle)^{2}.$
(45)
Here, the probe Hamiltonian is assumed as
$\hat{H}=\frac{\hat{P}^{2}}{2m}+V(Q),$ (46)
where $V(Q)$ is the potential on the coordinate space.
###### Proof.
For the probe observable $\hat{M}$, we obtain
$\displaystyle\frac{\langle{\xi^{\prime}}|\hat{M}|{\xi^{\prime}}\rangle}{\langle{\xi^{\prime}}|{\xi^{\prime}}\rangle}$
$\displaystyle=\frac{\langle{\xi}|\hat{M}|{\xi}\rangle-
ig\langle{A}\rangle_{w}\langle{\xi}|\hat{M}\hat{P}|{\xi}\rangle+ig\overline{\langle{A}\rangle_{w}}\langle{\xi}|\hat{P}\hat{M}|{\xi}\rangle}{\langle{\xi}|{\xi}\rangle-
ig\langle{A}\rangle_{w}\langle{\xi}|\hat{P}|{\xi}\rangle+ig\overline{\langle{A}\rangle_{w}}\langle{\xi}|\hat{P}|{\xi}\rangle}$
$\displaystyle=\frac{\langle{\xi}|\hat{M}|{\xi}\rangle+ig{\rm
Re}\langle{A}\rangle_{w}\langle{\xi}|[\hat{P},\hat{M}]|{\xi}\rangle+g{\rm
Im}\langle{A}\rangle_{w}\langle{\xi}|\\{\hat{P},\hat{M}\\}|{\xi}\rangle}{\langle{\xi}|{\xi}\rangle+2g{\rm
Im}\langle{A}\rangle_{w}\langle{\xi}|\hat{P}|{\xi}\rangle}$
$\displaystyle=\left(\langle{\xi}|\hat{M}|{\xi}\rangle+ig{\rm
Re}\langle{A}\rangle_{w}\langle{\xi}|[\hat{P},\hat{M}]|{\xi}\rangle+g{\rm
Im}\langle{A}\rangle_{w}\langle{\xi}|\\{\hat{P},\hat{M}\\}|{\xi}\rangle\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \times\left(1-2g{\rm
Im}\langle{A}\rangle_{w}\langle{\xi}|\hat{P}|{\xi}\rangle\right)+O(g^{2})$
$\displaystyle=\langle{\xi}|\hat{M}|{\xi}\rangle+ig{\rm
Re}\langle{A}\rangle_{w}\langle{\xi}|[\hat{P},\hat{M}]|{\xi}\rangle$
$\displaystyle\ \ \ \ \ \ \ \ +g{\rm
Im}\langle{A}\rangle_{w}\left(\langle{\xi}|\\{\hat{P},\hat{M}\\}|{\xi}\rangle-2\langle{\xi}|\hat{M}|{\xi}\rangle\langle{\xi}|\hat{P}|{\xi}\rangle\right)+O(g^{2}).$
(47)
If we set $\hat{M}=\hat{P}$, one has
$\Delta[\hat{P}]=2g{\rm Im}\langle{A}\rangle_{w}\operatorname{Var}[\hat{P}].$
(48)
If instead we set $\hat{M}=\hat{Q}$, one has
$\Delta[\hat{Q}]=g{\rm Re}\langle{A}\rangle_{w}+g{\rm
Im}\langle{A}\rangle_{w}\left(\langle{\xi}|\\{\hat{P},\hat{Q}\\}|{\xi}\rangle-2g\langle{\xi}|\hat{Q}|{\xi}\rangle\langle{\xi}|\hat{P}|{\xi}\rangle\right)$
(49)
since $[\hat{P},\hat{Q}]=-i$. From the Heisenberg equation with the probe
Hamiltonian (46), we obtain the Ehrenfest theorem;
$\displaystyle i\frac{d}{dt}\langle{\xi}|\hat{Q}|{\xi}\rangle$
$\displaystyle=\langle{\xi}|[\hat{Q},\hat{H}]|{\xi}\rangle=i\frac{\langle{\xi}|\hat{P}|{\xi}\rangle}{m}$
(50) $\displaystyle i\frac{d}{dt}\langle{\xi}|\hat{Q}^{2}|{\xi}\rangle$
$\displaystyle=\langle{\xi}|[\hat{Q}^{2},\hat{H}]|{\xi}\rangle=i\frac{\langle{\xi}|\\{\hat{P},\hat{Q}\\}|{\xi}\rangle}{m}.$
(51)
Substituting them into Eq. (49), we derive
$\Delta[\hat{Q}]=g{\rm Re}\langle{A}\rangle_{w}+mg{\rm
Im}\langle{A}\rangle_{w}\left.\frac{d\operatorname{Var}[\hat{Q}]}{dt}\right|_{t=t_{0}}$
(52)
since the interaction to the target system is taken at time $t=t_{0}$. ∎
Putting together, we can measure the weak value $\langle{A}\rangle_{w}$ by
observing the shift of the expectation value of the probe both in the
coordinate and momentum representations. The shift of the probe position
contains the future information up to the post-selected state.
###### Corollary 4.2.
When the probe wavefunction is real-valued in the coordinate representation,
Eq. (40) can be reduced to
$\Delta[\hat{Q}]=g{\rm Re}\langle{A}\rangle_{w}.$ (53)
###### Proof.
From the Schrödinger equation in the coordinate representation;
$i\frac{\partial}{\partial t}\xi(Q)=\frac{1}{2m}\frac{\partial^{2}}{\partial
Q^{2}}\xi(Q)+V(Q)\xi(Q),$ (54)
where $\xi(Q)\equiv\langle{Q}|{\xi}\rangle$, putting $\xi(Q)=R(Q)e^{iS(Q)}$,
we obtain the equation for the real part as
$\frac{\partial}{\partial t}R(Q)+\frac{\partial}{\partial
Q}\left(\frac{R(Q)\frac{\partial}{\partial Q}S(Q)}{m}\right)=0.$ (55)
Therefore, if the probe wavefunction is real-valued in the coordinate
representation, one has $\frac{\partial}{\partial Q}S(Q)=0$ to obtain
$\frac{\partial}{\partial t}R=0$. Therefore, we obtain
$\frac{d\operatorname{Var}[\hat{Q}]}{dt}=0$ (56)
for any time $t$. Vice versa. From this statement, we obtain the desired
result from Eq. (40). ∎
It is noted that there are many analyses on the weak measurement, e.g., on the
phase space [102], on the finite sample [179], on the counting statics [26,
104], on the non-local observable [32, 33], and on the complementary
observable [197].
Summing up this section, the two-state vector formalism is called if the pre-
and post-selected states are prepared and the weak or strong measurement is
taken in the von-Neumann type Hamiltonian, $H=gA\hat{P}\delta(t-t_{0})$
between the pre- and post-selected states. In the case of the strong
measurement, we obtain the expectation value $\operatorname{Ex}(A)$ in the
probe. On the other hand, in the case of the weak measurement, we obtain the
weak value $\langle{A}\rangle_{w}$ in the probe.
## 5 Weak-Value Measurement for a Qubit System
In this subsection, we consider the weak measurement in the case that the
probe system is a qubit system [195]. In general, the interaction Hamiltonian
is given by
$H_{int}=g[A\otimes(\vec{v}\cdot\vec{\sigma})]\delta(t-t_{0}),$ (57)
where $\vec{v}$ is a unit vector. Expanding the interaction Hamiltonian for
the pre- and post-selected states, $|{\psi}\rangle$ and $|{\phi}\rangle$,
respectively up to the first order for $g$, we obtain the shift of the
expectation value for $\vec{q}\cdot\vec{\sigma}$ as
$\displaystyle\Delta[\vec{q}\cdot\vec{\sigma}]$
$\displaystyle=\frac{\langle{\xi^{\prime}}|[\vec{q}\cdot\vec{\sigma}]|{\xi^{\prime}}\rangle}{\langle{\xi^{\prime}}|{\xi^{\prime}}\rangle}-\langle{\xi}|[\vec{q}\cdot\vec{\sigma}]|{\xi}\rangle$
$\displaystyle=g\langle{\xi}|i[\vec{v}\cdot\vec{\sigma},\vec{q}\cdot\vec{\sigma}]|{\xi}\rangle{\rm
Re}\langle{A}\rangle_{w}$ $\displaystyle\ \ \
+g\left(\langle{\xi}|\left\\{\vec{v}\cdot\vec{\sigma},\vec{q}\cdot\vec{\sigma}\right\\}|{\xi}\rangle-2\langle{\xi}|\vec{v}\cdot\vec{\sigma}|{\xi}\rangle\\!\langle{\xi}|\vec{q}\cdot\vec{\sigma}|{\xi}\rangle\right){\rm
Im}\langle{A}\rangle_{w}+O(g^{2})$
$\displaystyle=2g\\{(\vec{q}\times\vec{v})\cdot\vec{m}\\}{\rm
Re}\langle{A}\rangle_{w}+2g\\{\vec{v}\cdot\vec{q}-(\vec{v}\cdot\vec{m})(\vec{q}\cdot\vec{m})\\}{\rm
Im}\langle{A}\rangle_{w}+O(g^{2}),$ (58)
where
$\displaystyle|{\xi^{\prime}}\rangle$
$\displaystyle=\langle{\phi}|e^{-ig[A\otimes(\vec{v}\cdot\vec{\sigma})]}|{\psi}\rangle|{\xi}\rangle,$
(59) $\displaystyle|{\xi}\rangle\\!\langle{\xi}|$
$\displaystyle=:\frac{1}{2}({\bf 1}+\vec{m}\cdot\vec{\sigma}).$ (60)
Furthermore, the pre- and post-selected states are assumed to be
$|{\psi}\rangle\\!\langle{\psi}|=:\frac{1}{2}({\bf
1}+\vec{r}_{i}\cdot\vec{\sigma}),\ \ \
|{\phi}\rangle\\!\langle{\phi}|=:\frac{1}{2}({\bf
1}+\vec{r}_{f}\cdot\vec{\sigma}).$ (61)
Since the weak value of the observable $\vec{n}\cdot\vec{\sigma}$ is
$\langle{\vec{n}\cdot\vec{\sigma}}\rangle_{w}=\frac{\langle{\phi}|\vec{n}\cdot\vec{\sigma}|{\psi}\rangle\langle{\psi}|{\phi}\rangle}{|\langle{\phi}|{\psi}\rangle|^{2}}=\vec{n}\cdot\frac{\vec{r}_{i}+\vec{r}_{f}+i(\vec{r}_{i}\times\vec{r}_{f})}{1+\vec{r}_{i}\cdot\vec{r}_{f}},$
(62)
we obtain
$\Delta[\vec{q}\cdot\vec{\sigma}]=2g\\{(\vec{q}\times\vec{v})\cdot\vec{m}\\}\frac{\vec{n}\cdot(\vec{r}_{i}+\vec{r}_{f})}{1+\vec{r}_{i}\cdot\vec{r}_{f}}+2g\\{\vec{v}\cdot\vec{q}-(\vec{v}\cdot\vec{m})(\vec{q}\cdot\vec{m})\\}\frac{\vec{n}\cdot(\vec{r}_{i}\times\vec{r}_{f})}{1+\vec{r}_{i}\cdot\vec{r}_{f}}+O(g^{2}).$
(63)
From Eq. (63), we can evaluate the real and imaginary parts of the weak value
changing the parameter of the measurement direction $\vec{q}$. This
calculation is used in the context of the Hamiltonian estimation [157].
Next, as mentioned before, we emphasize that the weak measurement is only one
of the methods to obtain the weak value. There are many other approaches to
obtain the weak value, e.g., on changing the probe state [59, 103, 80, 119],
and on the entangled probe state [114]. Here, we show another method to obtain
the weak value in the case that the target and the probe systems are both
qubit systems [133].
Let $|{\psi}\rangle_{s}:=\alpha|{0}\rangle_{s}+\beta|{1}\rangle_{s}$ be the
pre-selected state for the target system. The initial probe state can
described as $|{\xi}\rangle_{p}:=\gamma|{0}\rangle_{p}+\eta|{1}\rangle_{p}$.
It is emphasized that the initial probe state is controllable. Here, the
initial states are normalized, that is, $|\alpha|^{2}+|\beta|^{2}=1$ and
$|\gamma|^{2}+|\eta|^{2}=1$. Applying the Controlled-NOT (C-NOT) gate, we make
a transform of the quantum state for the compound system to
$|{\psi}\rangle_{s}\otimes|{\xi}\rangle_{p}\xlongrightarrow[]{{\rm
C-NOT}}|{\Psi_{c}}\rangle:=(\alpha\gamma|{0}\rangle_{s}+\beta\eta|{1}\rangle_{s})|{0}\rangle_{p}+(\alpha\eta|{0}\rangle_{s}+\beta\gamma|{1}\rangle_{s})|{1}\rangle_{p}.$
(64)
In the case of $\gamma\sim 1$, we obtain the compound state as
$\alpha|{0}\rangle_{s}|{0}\rangle_{p}+\beta|{1}\rangle_{s}|{1}\rangle_{p},$
(65)
and similarly, in the case of $\eta\sim 1$, one has
$\alpha|{0}\rangle_{s}|{1}\rangle_{p}+\beta|{1}\rangle_{s}|{0}\rangle_{p}.$
(66)
Those cases can be taken as the standard von Neumann projective measurement.
For the post-selected state $|{\phi}\rangle$, the probability to obtain the
measurement outcome $k$ on the probe is
$\displaystyle\Pr[k]$
$\displaystyle:=\frac{\parallel\left(\,{}_{s}\langle{\phi}|\otimes\,_{p}\langle{k}|\right)|{\Psi_{c}}\rangle\parallel^{2}}{\sum_{m\in\\{0,1\\}}\parallel\left(\,{}_{s}\langle{\phi}|\otimes\,_{p}\langle{m}|\right)|{\Psi_{c}}\rangle\parallel^{2}}$
$\displaystyle=\frac{\left|\left(\,{}_{s}\langle{\phi}|{0}\rangle_{s}\langle{0}|{\psi}\rangle_{s}\gamma+\,_{s}\langle{\phi}|{1}\rangle_{s}\langle{1}|{\psi}\rangle_{s}\eta\right)\delta_{k,0}+\left(\,{}_{s}\langle{\phi}|{0}\rangle_{s}\langle{0}|{\psi}\rangle_{s}\eta+\,_{s}\langle{\phi}|{1}\rangle_{s}\langle{1}|{\psi}\rangle_{s}\gamma\right)\delta_{k,1}\right|^{2}}{\sum_{m\in\\{0,1\\}}\parallel\left(\,{}_{s}\langle{\phi}|\otimes\,_{p}\langle{m}|\right)|{\Psi_{c}}\rangle\parallel^{2}}$
$\displaystyle=\frac{|(\gamma-\eta)\,_{s}\langle{\phi}|{k}\rangle_{s}\langle{k}|{\psi}\rangle_{s}+\eta\,_{s}\langle{\phi}|{\psi}\rangle_{s}|^{2}}{|(\gamma-\eta)\,_{s}\langle{\phi}|{0}\rangle_{s}\langle{0}|{\psi}\rangle_{s}+\eta\,_{s}\langle{\phi}|{\psi}\rangle_{s}|^{2}+|(\gamma-\eta)\,_{s}\langle{\phi}|{1}\rangle_{s}\langle{1}|{\psi}\rangle_{s}+\eta\,_{s}\langle{\phi}|{\psi}\rangle_{s}|^{2}}$
$\displaystyle=\frac{|(\gamma-\eta)\,_{\phi}\langle{|{k}\rangle_{s}\langle{k}|}\rangle_{\psi}^{w}+\eta|^{2}}{1-(\gamma-\eta)^{2}(1-\sum_{m\in\\{0,1\\}}|\,_{\phi}\langle{|{m}\rangle_{s}\langle{m}|}\rangle_{\psi}^{w}|^{2})}.$
(67)
Here, in the last line, the parameters $\gamma$ and $\eta$ are assumed to be
real. Without the post-selection, the POVM to obtain the measurement outcome
$k$ is
$E_{k}=(\gamma^{2}-\eta^{2})|{k}\rangle_{s}\langle{k}|+\eta^{2}.$ (68)
Here, the coefficient of the first term means that the strength of measurement
and the second term is always added. Therefore, we define the quantity to
distinguish the probability for the measurement outcome $k$ as
$R[k]:=\frac{\Pr[k]-\eta^{2}}{(\gamma^{2}-\eta^{2})}.$ (69)
Putting together Eqs. (67) and (69), we obtain
$R[k]=\frac{2\eta(\gamma-\eta){\rm
Re}\,_{\phi}\langle{|{k}\rangle_{s}\langle{k}|}\rangle_{\psi}^{w}+(\gamma-\eta)^{2}[|\,_{\phi}\langle{|{k}\rangle_{s}\langle{k}|}\rangle_{\psi}^{w}|^{2}+\eta^{2}(1-|\,_{\phi}\langle{|{k}\rangle_{s}\langle{k}|}\rangle_{\psi}^{w}|^{2})]}{(\gamma^{2}-\eta^{2})[1-(\gamma-\eta)^{2}(1-\sum_{m\in\\{0,1\\}}|\,_{\phi}\langle{|{m}\rangle_{s}\langle{m}|}\rangle_{\psi}^{w}|^{2})]}.$
(70)
Setting the parameters;
$\gamma=\sqrt{\frac{1}{2}+\epsilon},\ \ \ \eta=\sqrt{\frac{1}{2}-\epsilon},$
(71)
one has
$\displaystyle R[k]$ $\displaystyle=\frac{(1-\epsilon){\rm
Re}\,_{\phi}\langle{|{k}\rangle_{s}\langle{k}|}\rangle_{\psi}^{w}+\epsilon\left[|\,_{\phi}\langle{|{k}\rangle_{s}\langle{k}|}\rangle_{\psi}^{w}|^{2}+\left(\frac{1}{2}-\epsilon\right)(1-|\,_{\phi}\langle{|{k}\rangle_{s}\langle{k}|}\rangle_{\psi}^{w}|^{2})\right]}{2\left[1-\epsilon^{2}\left(1-\sum_{m\in\\{0,1\\}}|\,_{\phi}\langle{|{m}\rangle_{s}\langle{m}|}\rangle_{\psi}^{w}|^{2}\right)\right]}+O(\epsilon^{2}),$
$\displaystyle=\frac{1}{2}{\rm
Re}\,_{\phi}\langle{|{k}\rangle_{s}\langle{k}|}\rangle_{\psi}^{w}-\frac{\epsilon}{2}\left({\rm
Re}\,_{\phi}\langle{|{k}\rangle_{s}\langle{k}|}\rangle_{\psi}^{w}-\frac{1}{2}|\,_{\phi}\langle{|{k}\rangle_{s}\langle{k}|}\rangle_{\psi}^{w}|^{2}\right)+O(\epsilon^{2}).$
(72)
From Eq. (72), it is possible to obtain the real part of the weak value from
the first term and its imaginary part from the second term. Since the first
order of the parameter $\epsilon$ is the gradient on changing the initial
probe state from
$|{\xi}\rangle_{p}=\frac{1}{\sqrt{2}}(|{0}\rangle_{p}+|{1}\rangle_{p})$,
realistically, we can evaluate the imaginary part of the weak value from the
gradient of the readout. This method is also used in Ref. [198] on the joint
weak value. It is emphasized that the weak value can be experimentally
accessible by changing the initial probe state while the interaction is not
weak 111111This point seems to be misunderstood. According to Ref. [134], the
violation of the Leggett-Garg inequality [100] was shown, but the macroscopic
realism cannot be denied since the noninvasive measurability is not realized..
## 6 Weak Values for Arbitrary Coupling Quantum Measurement
We just calculate an arbitrary coupling between the target and the probe
systems [199, 120, 93]. Throughout this section, we assume that the desired
observable is the projection operator to be denoted as $A^{2}=A$ [153]. In the
case of the von-Neumann interaction motivated by the original work [4], when
the pre- and post-selected states are $|{i}\rangle$ and $|{f}\rangle$,
respectively, and the probe state is $|{\xi}\rangle$, the probe state
$|{\xi^{\prime}}\rangle$ after the interaction given by $H_{int}=gA\hat{P}$
becomes
$\displaystyle|{\xi^{\prime}}\rangle$
$\displaystyle=\langle{f}|e^{-igA\hat{P}}|{i}\rangle|{\xi}\rangle=\langle{f}|\left(1+\sum_{k=1}^{\infty}\frac{1}{k!}(-igA\hat{P})^{k}\right)|{i}\rangle|{\xi}\rangle=\langle{f}|\left(1+A\sum_{k=1}^{\infty}\frac{1}{k!}(-ig\hat{P})^{k}\right)|{i}\rangle|{\xi}\rangle$
$\displaystyle=\langle{f}|\left(1-A+A\sum_{k=0}^{\infty}\frac{1}{k!}(-ig\hat{P})^{k}\right)|{i}\rangle|{\xi}\rangle=\langle{f}|\left(1-A+Ae^{-ig\hat{P}}\right)|{i}\rangle|{\xi}\rangle$
$\displaystyle=\langle{f}|{i}\rangle\left(1-\langle{A}\rangle_{w}+\langle{A}\rangle_{w}e^{-ig\hat{P}}\right)|{\xi}\rangle.$
(73)
It is remarked that the desired observable $B$, which satisfies $B^{2}=1$
[120, 93], corresponds to $B=2A-1$. Analogous to Theorem 4.1, we can derive
the expectation values of the position and the momentum after the weak
measurement. These quantities depends on the weak value
$\langle{A}\rangle_{w}$ and the generating function for the position and the
momentum of the initial probe state $|{\xi}\rangle$.
## 7 Weak Value with Decoherence
The decoherence results from the coupled system to the environment and leads
to the transition from the quantum to classical systems. The general framework
of the decoherence was discussed in Sec. 2. In this section, we discuss the
analytical expressions for the weak value.
While we directly discuss the weak value with decoherence, the weak value is
defined as a complex number. To analogously discuss the density operator
formalism, we need the operator associated with the weak value. Therefore, we
define a W operator $W(t)$ as
$W(t):=U(t,t_{i})|{i}\rangle{\langle{f}|}U(t_{f},t).$ (74)
To facilitate the formal development of the weak value, we introduce the ket
state $|{\psi(t)}\rangle$ and the bra state $\langle{\phi(t)}|$ as
$|{\psi(t)}\rangle=U(t,t_{i})|{i}\rangle,\
\langle{\phi(t)}|=\langle{f}|U(t_{f},t),$ (75)
so that the expression for the W operator simplifies to
$W(t)=|{\psi(t)}\rangle\\!\langle{\phi(t)}|.$ (76)
By construction, the two states $|{\psi(t)}\rangle$ and $\langle{\phi(t)}|$
satisfy the Schrödinger equations with the same Hamiltonian with the initial
and final conditions $|{\psi(t_{i})}\rangle=|{i}\rangle$ and
$\langle{\phi(t_{f})}|=\langle{f}|$. In a sense, $|{\psi(t)}\rangle$ evolves
forward in time while $\langle{\phi(t)}|$ evolves backward in time. The time
reverse of the W operator (76) is
$W^{\dagger}=|{\phi(t)}\rangle\\!\langle{\psi(t)}|$. Thus, we can say the W
operator is based on the two-state vector formalism formally described in
Refs. [16, 17]. Even an apparently similar quantity to the W operator (76) was
introduced by Reznik and Aharonov [140] in the name of “two-state” with the
conceptually different meaning. This is because the W operator acts on a
Hilbert space ${\cal H}$ but the two-state vector acts on the Hilbert space
$\overrightarrow{{\cal H}_{1}}\otimes\overleftarrow{{\cal H}_{2}}$.
Furthermore, while the generalized two-state, which is called a multiple-time
state, was introduced [13], this is essentially reduced to the two-state
vector formalism. The W operator gives the weak value of the observable $A$
121212While the original notation of the weak values is
$\langle{A}\rangle_{w}$ indicating the “w”eak value of an observable $A$, our
notation is motivated by one of which the pre- and post-selected states are
explicitly shown as $\,{}_{f}\langle{A}\rangle_{i}^{w}$. as
$\langle{A}\rangle_{W}=\frac{\operatorname{Tr}(WA)}{\operatorname{Tr}W},$ (77)
in parallel with the expectation value of the observable $A$ by
$\operatorname{Ex}[A]=\frac{\operatorname{Tr}(\rho A)}{\operatorname{Tr}\rho}$
(78)
from Born’s rule. Furthermore, the W operator (74) can be regarded as a
special case of a standard purification of the density operator [185]. In our
opinion, the W operator should be considered on the same footing of the
density operator. For a closed system, both satisfy the Schrödinger equation.
In a sense, the W operator $W$ is the square root of the density operator
since
$W(t)W^{\dagger}(t)=|{\psi(t)}\rangle\langle{\psi(t)}|=U(t,t_{i})|{i}\rangle\\!\langle{i}|U^{\dagger}(t,t_{i}),$
(79)
which describes a state evolving forward in time for a given initial state
$|{\psi(t_{i})}\rangle\langle{\psi(t_{i})}|=|{i}\rangle\langle{i}|$, while
$\displaystyle
W^{\dagger}(t)W(t)=|{\phi(t)}\rangle\langle{\phi(t)}|=U(t_{f},t)|{f}\rangle\\!\langle{f}|U^{\dagger}(t_{f},t),$
(80)
which describes a state evolving backward in time for a given final state
$|{\phi(t_{f})}\rangle\langle{\phi(t_{f})}|=|{f}\rangle\langle{f}|$. The W
operator describes the entire history of the state from the past ($t_{i}$) to
the future ($t_{f}$) and measurement performed at the time $t_{0}$ as we shall
see in Appendix 4.3. This description is conceptually different from the
conventional one by the time evolution of the density operator. From the
viewpoint of geometry, the W operator can be taken as the Hilbert-Schmidt
bundle. The bundle projection is given by
$\Pi:W(t)\to\rho_{i}(t):=W(t)W^{\dagger}(t).$ (81)
When the dimension of the Hilbert space is $N$: ${\rm dim}{\cal H}=N$, the
structure group of this bundle is $U(N)$ [25, Sec. 9.3]. Therefore, the W
operator has richer information than the density operator formalism as we
shall see a typical example of a geometric phase [155]. Furthermore, we can
express the probability to get the measurement outcome $a_{n}\in A$ due to the
ABL formula (28) using the W operator $W$ as
$\Pr[A=a_{n}]=\frac{|\operatorname{Tr}WP_{a_{n}}|^{2}}{\sum_{n}|\operatorname{Tr}WP_{a_{n}}|^{2}},$
(82)
where
$A=\sum_{n}a_{n}|{a_{n}}\rangle\\!\langle{a_{n}}|=:\sum_{n}a_{n}P_{a_{n}}$.
This shows the usefulness of the W operator.
Let us discuss a state change in terms of the W operator and define a map
${\cal X}$ as
${\cal X}(|{\alpha}\rangle,|{\beta}\rangle):=({\cal E}\otimes{\bf
1})\left(|{\alpha}\rangle\\!\langle{\beta}|\right),\\\ $ (83)
for an arbitrary $|{\alpha}\rangle,|{\beta}\rangle\in{\cal H}_{s}\otimes{\cal
H}_{e}$. Then, we obtain the following theorem on the change of the W operator
such as Theorem 2.1.
###### Theorem 7.1.
For any W operator $W=|{\psi(t)}\rangle_{s}\langle{\phi(t)}|$, we expand
$|{\psi(t)}\rangle_{s}=\sum_{m}\psi_{m}|{\alpha_{m}}\rangle_{s},\
|{\phi(t)}\rangle_{s}=\sum_{m}\phi_{m}|{\beta_{m}}\rangle_{s},$ (84)
with fixed complete orthonormal sets $\\{|{\alpha_{m}}\rangle_{s}\\}$ and
$\\{|{\beta_{m}}\rangle_{s}\\}$. Then, a change of the W operator can be
written as
${\cal
E}\left(|{\psi(t)}\rangle_{s}\langle{\phi(t)}|\right)=\,_{e}\langle{\tilde{\psi}(t)}|{\cal
X}(|{\alpha}\rangle,|{\beta}\rangle)|{\tilde{\phi}(t)}\rangle_{e},$ (85)
where
$|{\tilde{\psi}(t)}\rangle_{e}=\sum_{k}\psi^{\ast}_{k}|{\alpha_{k}}\rangle_{e},\
|{\tilde{\phi}(t)}\rangle_{e}=\sum_{k}\phi^{\ast}_{k}|{\beta_{k}}\rangle_{e},$
(86)
and $|{\alpha}\rangle$ and $|{\beta}\rangle$ are maximally entangled states
defined by
$|{\alpha}\rangle:=\sum_{m}|{\alpha_{m}}\rangle_{s}|{\alpha_{m}}\rangle_{e},\
|{\beta}\rangle:=\sum_{m}|{\beta_{m}}\rangle_{s}|{\beta_{m}}\rangle_{e}.$ (87)
Here, $\\{|{\alpha_{m}}\rangle_{e}\\}$ and $\\{|{\beta_{m}}\rangle_{e}\\}$ are
complete orthonormal sets corresponding to $\\{|{\alpha_{m}}\rangle_{s}\\}$
and $\\{|{\beta_{m}}\rangle_{s}\\}$, respectively.
The proof is completely parallel to that of Theorem 2.1.
###### Theorem 7.2.
For any W operator $W=|{\psi(t)}\rangle_{s}\langle{\phi(t)}|$, given the CP
map ${\cal E}$, the operator-sum representation is written as
${\cal E}(W)=\sum_{m}E_{m}WF^{\dagger}_{m},$ (88)
where $E_{m}$ and $F_{m}$ are the Kraus operators.
It is noted that, in general, ${\cal E}(W){\cal E}(W^{\dagger})\neq{\cal
E}(\rho)$ although $\rho=WW^{\dagger}$.
###### Proof.
We take the polar decomposition of the map $X$ to obtain
${\cal X}={\cal K}u,$ (89)
noting that
${\cal X}{\cal X}^{\dagger}={\cal K}uu^{\dagger}{\cal K}={\cal K}^{2}.$ (90)
The unitary operator $u$ is well-defined on ${\cal H}_{s}\otimes{\cal H}_{e}$
because ${\cal K}$ defined in Eq. (4) is positive. This is a crucial point to
obtain this result (88), which is the operator-sum representation for the
quantum operation of the W operator. From Eq. (10), we can rewrite ${\cal X}$
as
${\cal
X}=\sum_{m}|{s_{m}}\rangle\\!\langle{s_{m}}|u=\sum_{m}|{s_{m}}\rangle\\!\langle{t_{m}}|,$
(91)
where
$\langle{t_{m}}|=\langle{s_{m}}|u.$ (92)
Similarly to the Kraus operator (16), we define the two operators, $E_{m}$ and
$F^{\dagger}_{m}$, as
$E_{m}|{\psi(t)}\rangle_{s}:=\,_{e}\langle{\tilde{\psi}(t)}|{s_{m}}\rangle,\ \
\
\,_{s}\langle{\phi(t)}|F^{\dagger}_{m}:=\langle{t_{m}}|{\tilde{\phi}(t)}\rangle_{e},$
(93)
where $|{\tilde{\psi}(t)}\rangle_{e}$ and $|{\tilde{\phi}(t)}\rangle_{e}$ are
defined in Eq. (86). Therefore, we obtain the change of the W operator as
$\displaystyle\sum_{m}E_{m}|{\psi(t)}\rangle_{s}\langle{\phi(t)}|F^{\dagger}_{m}$
$\displaystyle=\sum_{m}\,{}_{e}\langle{\tilde{\psi}(t)}|{s_{m}}\rangle\langle{t_{m}}|{\tilde{\phi}(t)}\rangle_{e}=\,_{e}\langle{\tilde{\psi}(t)}|{\cal
X}|{\tilde{\phi}(t)}\rangle_{e}$ $\displaystyle={\cal
E}\left(|{\psi(t)}\rangle_{s}\langle{\phi(t)}|\right),$ (94)
using Theorem 7.1 in the last line. By linearity, we obtain the desired
result. ∎
Summing up, we have introduced the W operator (74) and obtained the general
form of the quantum operation of the W operator (88) in an analogous way to
the quantum operation of the density operator assuming the complete positivity
of the physical operation. This can be also described from information-
theoretical approach [43] to solve the open problem listed in Ref. [13, Sec.
XII]. However, this geometrical meaning has still been an open problem.
It is well established that the trace preservation, $\operatorname{Tr}({\cal
E}(\rho))=\operatorname{Tr}\rho=1$ for all $\rho$, implies that
$\sum_{m}E^{\dagger}_{m}E_{m}=1$. As discussed in Eq. (17), the proof goes
through as
$1=\operatorname{Tr}({\cal E}(\rho))=\operatorname{Tr}\left(\sum_{m}E_{m}\rho
E^{\dagger}_{m}\right)=\operatorname{Tr}\left(\sum_{m}E^{\dagger}_{m}E_{m}\rho\right)\;(\forall\rho).$
(95)
This argument for the density operator $\rho=WW^{\dagger}$ applies also for
$W^{\dagger}W$ to obtain $\sum_{m}F^{\dagger}_{m}F_{m}=1$ because this is the
density operator in the time reversed world in the two-state vector
formulation as reviewed in Sec. 4. Therefore, we can express the Kraus
operators,
$E_{m}=\,_{e}\langle{e_{m}}|U|{e_{i}}\rangle_{e},\
F_{m}^{\dagger}=\,_{e}\langle{e_{f}}|V|{e_{m}}\rangle_{e},$ (96)
where
$U=U(t,t_{i}),\ V=U(t_{f},t),$ (97)
are the evolution operators, which act on ${\cal H}_{s}\otimes{\cal H}_{e}$.
$|{e_{i}}\rangle$ and $|{e_{f}}\rangle$ are some basis vectors and
$|{e_{m}}\rangle$ is a complete set of basis vectors with
$\sum_{m}|{e_{m}}\rangle\\!\langle{e_{m}}|=1$. We can compute
$\sum_{m}F^{\dagger}_{m}E_{m}=\sum_{m}\,{}_{e}\langle{e_{f}}|V|{e_{m}}\rangle_{e}\langle{e_{m}}|U|{e_{i}}\rangle_{e}=\,_{e}\langle{e_{f}}|VU|{e_{i}}\rangle_{e}.$
(98)
The above equality (98) may be interpreted as a decomposition of the history
in analogy to the decomposition of unity because
$\,{}_{e}\langle{e_{f}}|VU|{e_{i}}\rangle_{e}=\,_{e}\langle{e_{f}}|S|{e_{i}}\rangle_{e}=S_{fi}$
(99)
is the S-matrix element. On this idea, Ojima and Englert have developed the
formulation on the S-matrix in the context of the algebraic quantum field
theory [123] and the backaction of the Hawking radiation [55], respectively.
## 8 Weak Measurement with Environment
Let us consider a target system coupled with an environment and a general weak
measurement for the compound of the target system and the environment. We
assume that there is no interaction between the probe and the environment and
the same interaction between the target and probe systems (38). The
Hamiltonian for the target system and the environment is given by
$H=H_{0}\otimes{\bf 1}_{e}+H_{1},$ (100)
where $H_{0}$ acts on the target system ${\cal H}_{s}$ and the identity
operator ${\bf 1}_{e}$ is for the environment ${\cal H}_{e}$, while $H_{1}$
acts on ${\cal H}_{s}\otimes{\cal H}_{e}$. The evolution operators
$U:=U(t,t_{i})$ and $V:=U(t_{f},t)$ as defined in Eq. (97) can be expressed by
$U=U_{0}K(t_{0},t_{i}),\ V=K(t_{f},t_{0})V_{0},$ (101)
where $U_{0}$ and $V_{0}$ are the evolution operators forward in time and
backward in time, respectively, by the target Hamiltonian $H_{0}$. $K$’s are
the evolution operators in the interaction picture,
$K(t_{0},t_{i})={\cal
T}e^{-i\int^{t_{0}}_{t_{i}}dtU_{0}^{\dagger}H_{1}U_{0}},\
K(t_{f},t_{0})=\overline{{\cal
T}}e^{-i\int^{t_{f}}_{t_{0}}dtV_{0}H_{1}V_{0}^{\dagger}},$ (102)
where ${\cal T}$ and $\overline{{\cal T}}$ stand for the time-ordering and
anti time-ordering products.
Let the initial and final environmental states be $|{e_{i}}\rangle$ and
$|{e_{f}}\rangle$, respectively. The probe state now becomes
$|{\xi^{\prime}}\rangle=\langle{f}|\langle{e_{f}}|VU|{e_{i}}\rangle|{i}\rangle\left({\bf
1}-g\frac{\langle{f}|\langle{e_{f}}|VAU|{e_{i}}\rangle|{i}\rangle}{\langle{f}|\langle{e_{f}}|VU|{e_{i}}\rangle|{i}\rangle}\hat{P}+O(g^{2})\right)|{\xi}\rangle.$
(103)
Plugging the expressions for $U$ and $V$ into the above, we obtain the probe
state as
$|{\xi^{\prime}}\rangle=N\xi\left({\bf
1}-g\frac{\langle{f}|\langle{e_{f}}|K(t_{f},t_{0})V_{0}AU_{0}K(t_{0},t_{i})|{e_{i}}\rangle|{i}\rangle}{N}\hat{P}\right)|{\xi}\rangle+O(g^{2}),$
(104)
where
$N=\langle{f}|\langle{e_{f}}|K(t_{f},t_{0})V_{0}U_{0}K(t_{0},t_{i})|{e_{i}}\rangle|{i}\rangle$
is the normalization factor. We define the dual quantum operation as
${\cal
E}^{\ast}(A):=\langle{e_{f}}|K(t_{f},t_{0})V_{0}AU_{0}K(t_{0},t_{i})|{e_{i}}\rangle=\sum_{m}V_{0}F^{\dagger}_{m}AE_{m}U_{0},$
(105)
where
$\displaystyle F^{\dagger}_{m}$
$\displaystyle:=V^{\dagger}_{0}\langle{e_{f}}|K(t_{f},t_{0})|{e_{m}}\rangle
V_{0},$ (106) $\displaystyle E_{m}$
$\displaystyle:=U_{0}\langle{e_{m}}|K(t_{0},t_{i})|{e_{i}}\rangle
U^{\dagger}_{0}$ (107)
are the Kraus operators. Here, we have inserted the completeness relation
$\sum_{m}|{e_{m}}\rangle\langle{e_{m}}|=1$ with $|{e_{m}}\rangle$ being not
necessarily orthogonal. The basis $|{e_{i}}\rangle$ and $|{e_{f}}\rangle$ are
the initial and final environmental states, respectively. Thus, we obtain the
wave function of the probe as
$\displaystyle|{\xi^{\prime}}\rangle$ $\displaystyle=N\left({\bf
1}-g\frac{\langle{f}|{\cal
E}^{*}(A)|{i}\rangle}{N}\hat{P}\right)|{\xi}\rangle+O(g^{2})$
$\displaystyle=N\left({\bf
1}-g\frac{\sum_{m}\langle{f}|V_{0}F^{\dagger}_{m}AE_{m}U_{0}|{i}\rangle}{\sum_{m}\langle{f}|V_{0}F^{\dagger}_{m}E_{m}U_{0}|{i}\rangle}\hat{P}\right)|{\xi}\rangle+O(g^{2})$
$\displaystyle=N\left({\bf
1}-g\frac{\operatorname{Tr}\left[A\sum_{m}E_{m}U_{0}|{i}\rangle\\!\langle{f}|V_{0}F^{\dagger}_{m}\right]}{\operatorname{Tr}\left[\sum_{m}E_{m}U_{0}|{i}\rangle\\!\langle{f}|V_{0}F^{\dagger}_{m}\right]}\hat{P}\right)|{\xi}\rangle+O(g^{2})$
$\displaystyle=N\left({\bf 1}-g\frac{\operatorname{Tr}[{\cal
E}(W)A]}{\operatorname{Tr}[{\cal
E}(W)]}\hat{P}\right)|{\xi}\rangle+O(g^{2})=N({\bf
1}-g\langle{A}\rangle_{{\cal E}(W)}\hat{P})|{\xi}\rangle+O(g^{2}),$ (108)
Analogous to Theorem 4.1, the shift of the expectation value of the position
operator on the probe is
$\Delta[Q]=g\cdot{\rm Re}[\langle{A}\rangle_{{\cal E}(W)}]+mg\cdot{\rm
Im}[\langle{A}\rangle_{{\cal
E}(W)}]\left.\frac{d\operatorname{Var}[Q]}{dt}\right|_{t=t_{0}}.$ (109)
From an analogous discussion, we obtain the shift of the expectation value of
the momentum operator on the probe as
$\Delta[P]=2g\cdot\operatorname{Var}[P]\cdot{\rm Im}[\langle{A}\rangle_{{\cal
E}(W)}].$ (110)
Thus, we have shown that the probe shift in the weak measurement is exactly
given by the weak value defined by the quantum operation of the W operator due
to the environment.
## 9 Summary
We have reviewed that the weak value is defined independent of the weak
measurement in the original idea [4] and have explained its properties.
Furthermore, to extract the weak value, we have constructed some measurement
model to extract the weak value. I hope that the weak value becomes the
fundamental quantity to describe quantum mechanics and quantum field theory
and has practical advantage in the quantum-mechanical world.
## Acknowledgment
The author acknowledges useful collaborations and discussion with Akio Hosoya,
Yuki Susa, and Shu Tanaka. The author thanks Yakir Aharonov, Richard Jozsa,
Sandu Popescu, Aephraim Steinberg, and Jeff Tollaksen for useful discussion.
The author would like to thank the use of the utilities of Tokyo Institute of
Technology and Massachusetts Institute of Technology and many technical and
secretary supports. The author is grateful to the financial supports from JSPS
Research Fellowships for Young Scientists (No. 21008624), JSPS Excellent Young
Researcher Overseas Visit Program, Global Center of Excellence Program
“Nanoscience and Quantum Physics” at Tokyo Institute of Technology during his
Ph.D study.
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|
arxiv-papers
| 2011-10-23T15:43:01 |
2024-09-04T02:49:23.542770
|
{
"license": "Public Domain",
"authors": "Yutaka Shikano",
"submitter": "Yutaka Shikano",
"url": "https://arxiv.org/abs/1110.5055"
}
|
1110.5207
|
# Gauss-Newton Filtering incorporating Levenberg-Marquardt Methods for Radar
Tracking
Roaldje Nadjiasngar Roaldje.Nadjiasngar@uct.ac.za Michael Inggs
Michael.Inggs@uct.ac.za
###### Abstract
This paper shows that the Levenberg-Marquardt Algorithms (LMA) algorithms can
be merged into the Gauss Newton Filters (GNF) to track difficult, non-linear
trajectories, without divergence. The GNF discusssed in this paper is an
iterative filter with memory that was introduced by Norman Morrison [1]. The
filter uses back propagation of the predicted state to compute the Jacobian
matrix over the filter memory length. The LMA are optimisation techniques
widely used for data fitting [2]. These optimisation techniques are iterative
and guarantee local convergence. We also show through simulation studies that
this filter performance is not affected by the process noise whose knowledge
is central to the family of Kalman filters.
###### keywords:
Gauss-Newton, filter, tracking, Levenberg Marquardt
## 1 Introduction
This paper shows that the Levenberg-Marquardt Algorithms (LMA) can be merged
into the Gauss Newton Filters (GNF) to track difficult, non-linear
trajectories, without divergence . [3, 4, 5, 6]. In the past, the LMA has been
used for initialising tracking filters [7, 8, 9]. In this paper we show that
the LMA can be merged into the flexible GNF filters to produce a hybrid
formulation with very powerful convergence properties even in highly non-
linear input data situations. The hybrid filter is also self initialising.
The LMA are optimisation techniques widely used for data fitting [2]. These
optimisation techniques are iterative and guarantee convergence in a specified
region i.e. they do necessarily produce global minima[10]. They are also used
in most neural networks algorithm [11, 12, 13].
The Gauss Newton filter (GNF) discussed in this paper was introduced by
Morrison[1] to tracking and smoothing at about the same period as the Kalman
filter, but it received little attention due to its computational
requirements, problematic for the limited computers of the time. The GNF is
iterative and non-recursive, with memory that can be adaptively controlled.
The GNF differs from the Gauss-Newton optimisation methods discussed in the
literature as it provides a different method for computing the Hessian matrix
[14]. This flexibility makes the GNF filter highly suitable for tracking in
strongly non-linear situations.
In this paper we adapt the GNF to the LMA method (which we call the Morrison
LMA Filter) and we state that this filter can be used for radar target
tracking without the risk of divergence. We also show through simulation
studies that this filter performance is not affected by the process noise
whose knowledge is central to the family of Kalman filters. The literature on
the use of LMA as a tracking algorithm are rare, possibly due to lack of
exposure to Morrison approach in the GNF.
The LMA is well known as an aid for track initiation [7, 8, 9]. We make it
clear here that the LMA is not applied as an initiation tool in our hybrid
filter, but rather as an integral part of the filter. The paper starts in
Section 2 to define a state space model based on nonlinear differential
equations.
Section 3 is important as it describes the incorporation of the LMA methods
into the GNF to produce the Morrison LMA, which converges very robustly . The
performance of the new filters is demonstrated in a series of simulations
described in Section 4. The paper concludes with a summary and indication of
future work.
## 2 State space model based on nonlinear differential equations
Consider the following autonomous, nonlinear differential equation (DE)
governing the process state:
$DX(t)=F(X(t))$ (1)
in which $F$ is a non linear vector function of the state vector $X$
describing a process, such as the position of a target in space. We assume the
observation scheme of the process is a nonlinear function of the process state
with expression :
$Y(t)=G(X(t))+v(t)$ (2)
where $G$ is a nonlinear function of $X$ and $v(t)$ is a random Gaussian
vector. The goal is to estimate the process state from the given state
nonlinear models. For linear DEs, the state transition matrix could be easily
obtained. This, however, is not the case with a nonlinear DEs. Nevertheless,
there is a procedure, based on local linearisation, that enables us to get
around this obstacle, which we will now present.
### 2.1 The method of local linearisation
The solution of the DE gives rise to infinitely many trajectories that are
dependent on the initial condition. However there will be one trajectory whose
state vector the filter will attempt to identify from the observations. We
assume that there is a known nominal trajectory with state vector $\bar{X}(t)$
that has the following properties:
* 1.
$\bar{X}(t)$ satisfies the same DE as $X(t)$
* 2.
$\bar{X}(t)$ is close to $X(t)$
The above-mentioned properties result in the following expression:[2].
$X(t)=\bar{X}(t)+\delta X(t)$ (3)
where $\delta X(t)$ is a vector of time-dependent functions that are small in
relation to the corresponding elements of either $\bar{X}(t)$ or $X(t)$, The
vector $\delta X(t)$ is called the perturbation vector and is governed by the
following DE (The derivation is shown in Appendix A):
$D(\delta X(t))=A(\bar{X}(t))\delta X(t)$ (4)
where $A(\bar{X}(t))$ is called a sensitivity matrix defined as follows:
$A(\bar{X}(t))=\left.\frac{\partial
F(X(t))}{\partial(X(t))}\right|_{\bar{X}(t)}.$ (5)
.
This equation is therefore a linear DE, with a time varying coefficient and
has a the following transition equation:
$\delta X(t+\zeta)=\Phi(t_{n}+\zeta,t_{n},\bar{X})\delta X(t)subsec:local$ (6)
in which $\Phi(t_{n}+\zeta,t_{n},\bar{X})$ is the transition matrix from time
$t_{n}$ to $t_{n}+\zeta$ (increment $\zeta$). The transition matrix is [2].
governed by the following DE:
$\frac{\partial}{\partial\zeta}\Phi(t_{n+\zeta},t_{n},\bar{X})=A(\bar{X}(t_{n}+\zeta))\Phi(t_{n+\zeta},t_{n},\bar{X})$
(7)
$\Phi(t_{n},t_{n},\bar{X})=I$ (8)
The transition matrix is a function of $\bar{X}(t)$ and can be evaluated by
numerical integration and in order to fill the values of
$A(\bar{X}(t_{n}+\zeta))$, $\bar{X}(t)$ has to be integrated numerically.
### 2.2 The observation perturbation vector
In this section we will adopt the notation $X_{n}$ and $Y_{n}$ for $X(t_{n})$
and $Y(t_{n})$ respectively. We define a simulated noise free observation
vector $\bar{Y}_{n}$ as follows:
$\bar{Y}_{n}=G(\bar{X}_{n})$ (9)
Subtracting $\bar{Y}_{n}$ from the actual observation $Y_{n}$ gives the
observation perturbation vector:
$\delta Y_{n}=Y_{n}-\bar{Y}_{n}$ (10)
In Appendix A we show that the observation perturbation vector is related to
the state perturbation vector as follows:
$\delta Y_{n}=M(\bar{X}_{n})\delta X_{n}+v_{n}$ (11)
where $M(\bar{X}_{n})$ is the Jacobean matrix of G, evaluated at
$\bar{X}_{n}$. The matrix is also called the observation sensitivity matrix
and is defined as follows:
$M(\bar{X}_{n})=\left.\frac{\partial
F(X_{n})}{\partial(X_{n})}\right|_{\bar{X}_{n}}$ (12)
We now examine the sequence of observations.
### 2.3 Sequence of observation
We assume that $L+1$ observation are obtained with time stamps
$t_{n},t_{n-1},...,t_{n-L}$,Theses observations are assembled as follows :
$\left[\begin{array}[]{c}\delta Y_{n}\\\ \delta Y_{n-1}\\\ .\\\ .\\\ .\\\
\delta Y_{n-L}\end{array}\right]=\left[\begin{array}[]{c}M(\bar{X}_{n})\delta
X_{n}\\\ M(\bar{X}_{n-1})\delta X_{n-1}\\\ .\\\ .\\\ .\\\
M(\bar{X}_{n-L})\delta
X_{n-L}\end{array}\right]+\left[\begin{array}[]{c}v_{n}\\\ v_{n-1}\\\ ..\\\
.\\\ .\\\ v_{n-L}\end{array}\right]$ (13)
Using the relationship:
$\delta X_{m}=\Phi(t_{m},t_{n},\bar{X})\delta X_{n}$ (14)
then, substituting Equation 13 the observation sensitivity equation can be
written as :
$\mathbf{\delta Y}_{n}=\mathbf{T}_{n}\delta X_{n}+\mathbf{V}_{n}$ (15)
in which $\mathbf{T}_{n}$, the total observation matrix is defined as follows:
$\mathbf{T}_{n}=\left[\begin{array}[]{c}M(\bar{X}_{n})\\\
M(\bar{X}_{n-1})\Phi(t_{n-1},t_{n};\bar{X})\\\ .\\\ .\\\ .\\\
M(\bar{X}_{n-L})\Phi(t_{n-L},t_{n};\bar{X})\end{array}\right]$ (16)
The vectors $\mathbf{\delta Y}_{n}$ and $\mathbf{V}_{n}$ are large. The cost
function we would want to minimize is :
$efE(\delta X_{n})=(\mathbf{\delta
Y}_{n}-\mathbf{V}_{n})^{T}\mathbf{R}_{n}^{-1}(\mathbf{\delta
Y}_{n}-\mathbf{V}_{n})=\delta
X_{n}^{T}(\mathbf{T}_{n}^{T}\mathbf{R}_{n}^{-1}\mathbf{T}_{n})\delta X_{n}$
(17)
The solution that minimises the cost function can be obtained from the minimum
variance estimation as follows:
$\delta\hat{X}_{n}=(\mathbf{T}_{n}^{T}\mathbf{R}_{n}^{-1}\mathbf{T}_{n})^{-1}\mathbf{T}_{n}^{T}\mathbf{R}_{n}^{-1}\mathbf{\delta
Y}_{n}$ (18)
The estimate $\delta\hat{X}_{n}$ has a covariance matrix:
$S_{n}=(\mathbf{T}_{n}^{T}\mathbf{R}_{n}^{-1}\mathbf{T}_{n})^{-1}$ (19)
where $\mathbf{R}_{n}^{-1}$ is a block diagonal weight matrix, also called the
least squares weight matrix, but, in fact, if we define $R_{n}$ as the
covariance matrix of the the error vector $v_{n}$, then $\mathbf{R}_{n}^{-1}$
is expressed as:
$\mathbf{R}_{n}^{-1}=\left[\begin{array}[]{cccccc}R_{n}^{-1}&0&.&.&.&0\\\
0&R_{n-1}^{-1}&&&&.\\\ .&&.&&&.\\\ .&&&.&&.\\\ .&&&&.\\\
0&.&.&.&0&R_{n-L}^{-1}\end{array}\right]$ (20)
In this section we arrived at a form of filter that uses the minimum variance
estimation method, initiated by Gauss in ”Theoria Combinationis Observationum
Erroribus Minimis Obnoxiae,” and the local linearisation technique championed
by Newton to estimate the state of the process from the non linear observation
scheme. This filter is called Gauss-Newton filter (GNF) and is described in
detail in Morrison’s work [1].
## 3 Adaptation to Levenberg Marquard
This section represents the key step in the development of the Morrison LMA
Filter. For simplicity in adaptation of the GNF to the Levenberg Marquard
method we assume the dynamic of of the process we want to track is governed by
linear differential equations and the observation scheme is non linear. The
process transition equation will be:
$X_{n+\varsigma}=\Phi(\varsigma)X_{n}$ (21)
.
The GNF will fail to converge if the matrix
$(\mathbf{T}_{n}^{T}\mathbf{R}_{n}^{-1}\mathbf{T}_{n})^{-1}$ is singular. By
definition this matrix is positive definite However, it can loose this
property due to numerical inaccuracy or high non-linearity. To avoid the
singularity, a damping factor is introduced in equation as follows:
$\delta\hat{X}_{n}=(\mathbf{T}_{n}^{T}\mathbf{R}_{n}^{-1}\mathbf{T}_{n}+\mu
I)^{-1}\mathbf{T}_{n}^{T}\mathbf{R}_{n}^{-1}\mathbf{\delta Y}_{n}$ (22)
which is the form suggested by Levenberg and Marquardt [2]
The effect of the damping factor is as follows:
* 1.
For all positive $\mu$ the matrix
$(\mathbf{T}_{n}^{T}\mathbf{R}_{n}^{-1}\mathbf{T}_{n}+\mu I)$ is positive
definite, ensuring that $\delta X$ is in the descent direction;
* 2.
When $\mu$ is large we have:
$\delta\hat{X}=\frac{1}{\mu}\mathbf{T}_{n}^{T}\mathbf{R}_{n}^{-1}\delta\mathbf{Y}_{n}$
(23)
The algorithm behaves as a steepest descent which is ideal when the current
solution is far from the local minimum. The convergence will be slow but
however guaranteed. When $\mu$ is small, the algorithm has faster convergence
and behaves like the Gauss-Newton.
The damping factor can be updated by the gain ratio:
$\varrho=\frac{\mathbf{\delta Y}_{n}^{T}\mathbf{R}_{n}^{-1}\mathbf{\delta
Y}_{n}-(\mathbf{Y}_{n}-\bar{\mathbf{Y}})^{T}\mathbf{R}_{n}^{-1}(\mathbf{Y}_{n}-\bar{\mathbf{Y}})}{\mathbf{E}(\delta
X_{n})}$ (24)
where $\mathbf{\delta Y}_{n}$ is the long vector of L sequences of
obserservation including the current observation.
$\bar{\mathbf{Y}}$ is the long error free observation computed by back
propagation of the current iterate $X_{new}$. If we sample at constant rate
$\varsigma$ then:
$\bar{\mathbf{Y}}=\left[\begin{array}[]{c}G(X_{new})\\\
G(\Phi(-\varsigma)X_{new})\\\ \vdots\\\
G(\Phi(-(L-1)\varsigma)X_{new})\end{array}\right]$ (25)
The numerator is the actual computed gain and the denominator is the predicted
gain. Recalling equation 17 and replacing $\delta X_{n}$ by the expression in
equation 18 we have:
$E(\delta X_{n})=\delta
X_{n}^{T}(\mathbf{T}_{n}^{T}\mathbf{R}_{n}^{-1}\mathbf{T}_{n})(\mathbf{T}_{n}^{T}\mathbf{R}_{n}^{-1}\mathbf{T}_{n}+\mu
I)^{-1}\mathbf{T}_{n}^{T}\mathbf{R}_{n}^{-1}\delta\mathbf{Y}_{n}$ (26)
which reduces to:
$E(\delta X_{n})=\delta
X_{n}^{T}(\mathbf{T}_{n}^{T}\mathbf{R}_{n}^{-1}\delta\mathbf{Y}_{n}+\mu\delta
X_{n})$ (27)
A large value of $\varrho$ indicates that $E(\delta X_{n})$ is a good
approximation of $\bar{\mathbf{Y}}$, and $\mu$ can be decreased so that the
next Levenberg-Marquardt step is closer to the Gauss-Newton step. If $\varrho$
is small or negative then $E(\delta X_{n})$ is a poor approximation, then
$\mu$ should be increased to move closer to the steepest descent direction.
The algorithm adapted from [15] is presented as follows:
$k:=0$,$\nu:=2$,$X:=X_{n-1/n}$
$A:=\mathbf{T}_{n}^{T}\mathbf{R}_{n}^{-1}\mathbf{T}_{n}$;$\delta\mathbf{Y}_{n}:=\mathbf{Y}_{n}-\mathbf{\bar{Y}}_{n}$;
$g:=\mathbf{T}_{n}^{T}\mathbf{R}_{n}^{-1}\delta\mathbf{Y}_{n}$;
$\mathbf{\bar{Y}}_{n}$ is computed using $X$
$stop:=false$;$\mu=\tau*max(diag(A))$;
While (not stop) and ($k\leq k_{max}$)
$k:=k+1$
repeat
solve $(A+\mu I)\delta\hat{X}_{n}=g$
if ($||\delta\hat{X}_{n}||\leq\varepsilon||X||$)
stop:=true;
else
$X_{new}:=X+\delta\hat{X}_{n}$;
$\varrho=[\delta\mathbf{Y}_{n}^{T}\mathbf{R}_{n}^{-1}\delta\mathbf{Y}_{n}-(\mathbf{Y}_{n}-\mathbf{\bar{Y}})^{T}\mathbf{R}_{n}^{-1}(\mathbf{Y}_{n}-\mathbf{\bar{Y}})]/[\delta\hat{X}_{n}^{T}(g+\mu\delta\hat{X}_{n})]$;
$\mathbf{\bar{Y}}$ evaluated at $X_{new}$
if $\varrho>0$
$X=X_{new}$;
$A:=\mathbf{T}_{n}^{T}\mathbf{R}_{n}^{-1}\mathbf{T}_{n}$;$\delta\mathbf{Y}_{n}:=\mathbf{Y}_{n}-\mathbf{\bar{Y}}_{n}$;
$g:=\mathbf{T}_{n}^{T}\mathbf{R}_{n}^{-1}\delta\mathbf{Y}_{n}$;
$\mathbf{\bar{Y}}_{n}$ is computed using X
$\mu=\mu*max(1/3,1-(2\varrho+1)^{3})$;$\nu:=2$;
else
$\mu:=\nu*\mu$;
$\nu:=2*\nu$;
endif
endif
until($\varrho>0$)or(stop);
endwhile
$X_{n/n}=X$;
$X_{n/n+1}=\Phi(s)X_{n/n}$
Algorithm 1 L-M algorithm for tracking system
subsec:local
## 4 Simulations
In the simulation studies we adopt multiple target dynamics:
* 1.
Case 1 : The target is moving at constant velocity under the process noise of
constant standard deviation.
* 2.
Case 2 : The target moving at constant velocity with process noise standard
deviation varying
In all the cases the observation scheme is non linear. The observables are
range $\rho$,bearing $\phi$,elevation $\theta$ and Doppler $f_{d}$. The
observation equation is therefore defined as follows:
$Y=\left[\begin{array}[]{c}\sqrt{x^{2}+y^{2}+z^{2}}\\\ tan^{-1}(y/x)\\\
tan^{-1}(z/\sqrt{x^{2}+y^{2}})\\\
K_{d}\frac{x\dot{x}+y\dot{y}+z\dot{z}}{\sqrt{x^{2}+y^{2}+z^{2}}}\end{array}\right]+v(t)$
(28)
where $v(t)$ is vector of random variables with covariance
$R=\left[\begin{array}[]{cccc}60^{2}&0&0&0\\\ 0&0.001^{2}&0&0\\\
0&0&0.001^{2}&0\\\ 0&0&0&2^{2}\end{array}\right]$
throughout the simulations. $K_{d}=-2\pi/\lambda=-200$. The constants
$\tau=10^{-1}$, $\varepsilon=10^{-20}$, $k_{max}=200$, $\zeta=1s$ are used in
all the cases.
### 4.1 Case 1
In this example, we seek to demonstrate that the filter does not diverge in
the presence of constant variance process which is unknown to its model. : The
target state vector $X=[x,\dot{x},y,\dot{y},z,\dot{z}]^{T}$ is defined by the
following transition equation:
$X_{n+1}=\left[\begin{array}[]{cccccc}1&\varsigma&0&0&0&0\\\ 0&1&0&0&0&0\\\
0&0&1&\varsigma&0&0\\\ 0&0&0&1&0&0\\\ 0&0&0&0&1&\varsigma\\\
0&0&0&0&0&1\end{array}\right]X_{n}+\left[\begin{array}[]{c}\frac{1}{2}a_{1}\varsigma^{2}\\\
a_{1}\varsigma\\\ \frac{1}{2}a_{2}\varsigma^{2}\\\ a_{2}\varsigma\\\
\frac{1}{2}a_{3}\varsigma^{2}\\\ a_{3}\varsigma\end{array}\right]$ (29)
where $a_{1}$, $a_{1}$, $a_{1}$ are independent, Gaussian random variables,
with standard deviation $\sigma=0.001$. The state vector is used in to
generate measurements for the simulation. The filter, however, does not depend
on the process noise, it assumes the target is moving at constant speed
without process noise.
The initial value of the state vector is $X=[800,25,1000,-25,400,14]$. Two
thousand samples are generated and the process is repeated 50 times. The
position root mean squared error (RMSE) after the 50 Monte Carlo runs is
presented in Figure [1]. The position RMSE is computed as follows :
$RMSE=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\left((x_{n}^{i}-\hat{x_{n}})^{2}+(y_{n}^{i}-\hat{y_{n}})^{2}+(z_{n}^{i}-\hat{z_{n}})^{2}\right)}$
(30)
where $(x_{n}^{i},y_{n}^{i},z_{n}^{i})$ and
$(\hat{x_{n}},\hat{y_{n}},\hat{z_{n}})$ true and estimated position
coordinates respectively.
We see from Figure [1] that there is no divergence in position despite the
presence of the process noise, which is unknown to the filter. The filter with
the smallest memory exhibits the largest RMSE. The average number of
iterations is presented in Fgure [2]. All the filters have about the same
value of $k$, which is around 34, meaning the computation time of the
algorithm is primarily dependent on the computation of the $\mathbf{T}_{n}$
matrix.Therefore if we want to reduce the computation time of the algorithm,
we would choose a small memory length (the $\mathbf{T}_{n}$ matrix will be
small and hence less computation), but this would result in less accuracy in
the estimates.
### 4.2 Case 2
Here we show the effect of higher variation in the process noise on the filter
performance. In this case the target dynamic model is the same as in Case 1.
The standard deviation($\sigma$) of the process noise is varied. From sample 0
to 200 $\sigma=0.001$, between samples 201 and 260 $\sigma=0.05$ and finally
from sample 261 to 400 $\sigma=0.001$. The position RMSE after 200 Montecarlo
runs is shown in Figure [3]. All the filters reset to the original RMSE when
the process noise standard deviation returned to the former value. The RMSE of
filter with the smallest memory length is less affected by these changes.
However the number of iterations during high disturbance is higher for the
smaller memory length filter (Figure [4]). Theses results highlight the
adaptiveness of the algorithm to disturbance.They also give a hint about the
ability of the filter to track manoeuvre. This will be the subject of our next
pubiblication.
## 5 Conclusion
This paper introduced the standard Gauss Newton filter that uses the back
propagation of the predicted state vector over a finite memory length to
compute the Jacobian matrix. It then computes the current estimate of the
state vector through the minimum variance estimation. The Gauss Newton was
then adapted to the Levenberg and Marquard method to guarantee its convergence
all the time.
The adapted algorithm was used in simulations to track targets in Cartesian
coordinates when the observations consist of range, bearing , elevation and
Doppler. The results highlight the robustness of the new, Morrison LMA filter
which can withstand strong random disturbance and nonlinear trajectories. We
observed from simulations studies that by adaptively changing the memory
length, the filter will be able to track maneuvres. Such memory control
algorithm will be a subject of our next publication.
## Appendix A
### A.1 The differential equation governing $\delta X(t)$
Starting from :
$\delta X(t)=X(t)-\bar{X}(t)$ (31)
The differentiation rule is applied:
$D\delta X(t)=F(\bar{X}(t)+\delta X(t))-F(\bar{X}(t))$ (32)
Let $F$ be defined as follows :
$F=\left[\begin{array}[]{c}f_{1}\\\ .\\\ .\\\ .\\\ f_{n}\end{array}\right]$
(33)
Equation becomes :
$D\delta X(t)=\left[\begin{array}[]{c}f_{1}(\bar{X}(t)+\delta X(t))\\\ .\\\
.\\\ .\\\ f_{n}(\bar{X}(t)+\delta
X(t))\end{array}\right]-\left[\begin{array}[]{c}f_{1}(\bar{X}(t))\\\ .\\\ .\\\
.\\\ f_{n}(\bar{X}(t))\end{array}\right]$ (34)
The Taylor first order approximation is applied:
$\displaystyle D\delta X(t)$ $\displaystyle{}=$
$\displaystyle{}\left[\begin{array}[]{c}f_{1}(\bar{X}(t))\\\ .\\\ .\\\ .\\\
f_{n}(\bar{X}(t))\end{array}\right]+\left[\begin{array}[]{c}\nabla
f_{1}(\bar{X}(t))^{T}\\\ .\\\ .\\\ .\\\ \nabla
f_{n}(\bar{X}(t))^{T}\end{array}\right]\delta X(t)$ (51)
$\displaystyle{-}\>\left[\begin{array}[]{c}f_{1}(\bar{X}(t))\\\ .\\\ .\\\ .\\\
f_{n}(\bar{X}(t))\end{array}\right]$
The following relation is obtained :
$D\delta X(t)=A(\bar{X}(t))\delta X(t)$ (52)
Where:
$A(\bar{X}(t))=\left[\begin{array}[]{c}\nabla f_{1}(\bar{X}(t))^{T}\\\ .\\\
.\\\ .\\\ \nabla f_{n}(\bar{X}(t))^{T}\end{array}\right]=\left.\frac{\partial
F(X(t))}{\partial(X(t))}\right|_{\bar{X}(t)}$ (53)
### A.2 The relation between $\delta X_{n}$ and $\delta Y_{n}$
$\delta Y_{n}=G(\bar{X}_{n}+\delta X_{n})-G(\bar{X}_{n})$ (54)
As direct consequence of A.1 the following relationship is obtained:
$\delta Y_{n}=M(\bar{X}_{n})\delta X_{n}+v_{n}$ (55)
## Acknowledgment
The authors would like to thank our colleague Dr Norman Morrison for his
contribution in introducing us to the GNF and his tireless enthusiasm for
teaching and providing insights into the fundamentals of Filter Engineering.
We wish him well for his soon to be published book, which provides new
material on these remarkable filters.
## References
* [1] N. Morrison, Introduction to sequential smoothing and prediction, McGraw-Hill Book Company, 1969.
* [2] D. W. Marquardt, An algorithm for least-squares estimation of nonlinear parameters, Journal of the Society for Industrial and Applied Mathematics 11 (2) (1963) pp. 431–441.
* [3] L. Perea, J. How, L. Breger, P. Elosegui, Nonlinearity in sensor fusion: Divergence issues in ekf, modified truncated sof, and ukf, in: AIAA Guidance, Navigation and Control Conference and Exhibit, 2007, p. 6514.
* [4] S. Blackman, R. Popoli, Design and Analysis of Modern Tracking Systems, Artech House, Boston.London, 1999.
* [5] R. Niu, P. Varshney, M. Alford, A. Bubalo, E. Jones, M. Scalzo, Curvature nonlinearity measure and filter divergence detector for nonlinear tracking problems, in: Information Fusion, 2008 11th International Conference on, 2008, pp. 1 –8.
* [6] X. Wang, Y. Huang, Convergence study in extended kalman filter-based training of recurrent neural networks, Neural Networks, IEEE Transactions on 22 (4) (2011) 588 –600.
* [7] L. Zong-xiang, X. Wei-xin, A new method for track initiation in a distributed passive sensor network, in: Signal Processing, 2008. ICSP 2008. 9th International Conference on, 2008, pp. 2616 –2619.
* [8] M. Yeddanapudi, Y. Bar-Shalom, K. Pattipati, S. Deb, Ballistic missile track initiation from satellite observations, Aerospace and Electronic Systems, IEEE Transactions on 31 (3) (1995) 1054 –1071. doi:10.1109/7.395236.
* [9] P. Howland, Target tracking using television-based bistatic radar, Radar, Sonar and Navigation, IEE Proceedings - 146 (3) (1999) 166 –174. doi:10.1049/ip-rsn:19990322.
* [10] C. Ma, L. Jiang, Some research on levenberg-marquardt method for the nonlinear equations, Applied Mathematics and Computation 184 (2) (2007) 1032 – 1040. doi:10.1016/j.amc.2006.07.004.
URL http://www.sciencedirect.com/science/article/pii/S00963%00306007910
* [11] B. G. Kermani, S. S. Schiffman, H. T. Nagle, Performance of the levenberg-marquardt neural network training method in electronic nose applications, Sensors and Actuators B: Chemical 110 (1) (2005) 13 – 22. doi:10.1016/j.snb.2005.01.008.
URL http://www.sciencedirect.com/science/article/pii/S09254%00505000961
* [12] E. Derya, Übeyli, Analysis of eeg signals by implementing eigenvector methods/recurrent neural networks, Digital Signal Processing 19 (1) (2009) 134 – 143. doi:10.1016/j.dsp.2008.07.007.
URL http://www.sciencedirect.com/science/article/pii/S10512%00408001243
* [13] V. Singh, I. Gupta, H. Gupta, Ann-based estimator for distillation using levenberg–marquardt approach, Engineering Applications of Artificial Intelligence 20 (2) (2007) 249 – 259. doi:10.1016/j.engappai.2006.06.017.
URL http://www.sciencedirect.com/science/article/pii/S09521%9760600114X
* [14] T. Dahlin, M. Loke, Resolution of 2d wenner resistivity imaging as assessed by numerical modelling, Journal of Applied Geophysics 38 (4) (1998) 237 – 249. doi:10.1016/S0926-9851(97)00030-X.
URL http://www.sciencedirect.com/science/article/pii/S09269%8519700030X
* [15] O. T. K. Madsen, H.B. Nielsen, Method of non-linear least squares problems, 2nd Edition, Informatics and Mathematical Modelling, Technical University of Denmark, 2004.
|
arxiv-papers
| 2011-10-24T12:11:43 |
2024-09-04T02:49:23.562591
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Roaldje Nadjiasngar, Michael Inggs",
"submitter": "Roaldje Nadjiasngar",
"url": "https://arxiv.org/abs/1110.5207"
}
|
1110.5212
|
# The Recursive Gauss-Newton Filter
Roaldje Nadjiasngar Roaldje.Nadjiasngar@uct.ac.za Michael Inggs
Michael.Inggs@uct.ac.za
###### Abstract
This paper presents a compact, recursive, non-linear, filter, derived from the
Gauss-Newton (GNF), which is an algorithm that is based on weighted least
squares and the Newton method of local linearisation. The recursive form
(RGNF), which is then adapted to the Levenberg-Maquardt method is applicable
to linear / nonlinear of process state models, coupled with the linear /
nonlinear observation schemes. Simulation studies have demonstrated the
robustness of the RGNF, and a large reduction in the amount of computational
memory required, identified in the past as a major limitation on the use of
the GNF.
###### keywords:
Gauss-Newton, filter, tracking, recursive
## 1 Introduction
The minimum variance algorithm has been used to estimate parameters from
batches of observations, accumulated over a defined period of time. The most
popular version of the minimum variance methods is the weighted least squares,
which are at the heart of adaptive filtering [1] [2]. The recursive least
squares (RLS) methods are efficient versions of the least squares approach,
and are applicable to estimation of future states from scalar input data
streams. However, recent studies [3] have seen the development of state space
recursive least squares (SSRLS) methods that show robustness in the estimation
of linear state space models. For the estimation of non-linear state space
models, a non-recursive filter called the Gauss-Newton filter (GNF) was
developed and has been successfully used in many applications [4] [5] . The
GNF algorithm is a combination of the Newton method of local linearization and
the least squares-like version of the minimum variance method[4]. It is used
to estimate process states that are governed by non-linear, autonomous,
differential equations, coupled with linear or non-linear observation schemes.
The GNF algorithm, although robust, requires significant processing power,
i.e. the amount of memory required. To improve the computational efficiency of
the GNF, studies of the use of Field Programmable Gate Arrays (FPGA) and other
co-processor technology have been made [6, 7]. Memory requirements were
identified in these studies as being the major stumbling block in
implementations on both on FPGA (low power and parallelism) and coprocessor
(ease of use) technology. This paper obtains a recursive form of the GNF with
zero memory. We then adapt the recursive filter to the Levenberg-Maquardt
method, renown for its robustness [8, 9, 10, 11, 12], widely used in non
linear curve fitting problems and neural networks algorithms. The contribution
of this paper is the derivation of a compact recursive form of the GNF that is
applicable to four major scenarios:
Case 1 : linear process dynamic and linear observation scheme.
Case 2 : linear process dynamic and non-linear observation scheme.
Case 3 : non-linear process dynamic and linear observation scheme.
Case 4 : non-linear process dynamic and non linear observation scheme.
The paper begins with an exposition of a state space model based on non-
linear, differential equations. This is followed, in Section 3, by the
derivation of a recursive GNF. In Section 4 we describe the adaptation of the
recursive equations of the filter to the Levenberg-Maquardt method. A complete
filter algorithm is presented. In Section 5, the state space situations to
which we can apply this new recursive form are demonstrated, with a look at
stability. We then demonstrate the power of the new recursive GNF in an
application to range and bearing only tracking of a manoeuvring target
(Section 6), before concluding with a summary of results achieved.
## 2 State space model based on non-linear differential equations
Consider the following autonomous, non-linear differential equation (DE)
governing the process state:
$DX(t)=F(X(t))$ (1)
in which $F$ is a non linear vector function of the state vector $X$
describing a process, such as the position of a target in space. We assume the
observation scheme of the process is a non-linear function of the process
state with expression :
$Y(t)=G(X(t))+v(t)$ (2)
where $G$ is a non-linear function of $X$ and $v(t)$ is a random Gaussian
vector. The goal is to estimate the process state from the given non-linear
state models. For linear differential equations (DEs), the state transition
matrix could be easily obtained. This, however, is not the case with non-
linear DEs. Nevertheless, there is a procedure, based on local linearization,
that enables us to get around this obstacle, which we will now present.
### 2.1 The method of local linearisation
The solution of the DE gives rise to infinitely many trajectories that are
dependent on the initial condition. However there will be one trajectory whose
state vector the filter will attempt to identify from the observations. We
assume that there is a known nominal trajectory with state vector $\bar{X}(t)$
that has the following properties:
* 1.
$\bar{X}(t)$ satisfies the same DE as $X(t)$
* 2.
$\bar{X}(t)$ is close to $X(t)$
The above-mentioned properties result in the following expression:
$X(t)=\bar{X}(t)+\delta X(t)$ (3)
where $\delta X(t)$ is a vector of time-dependent functions that are small in
relation to the corresponding elements of either $\bar{X}(t)$ or $X(t)$ . The
vector $\delta X(t)$ is called the perturbation vector and is governed by the
following DE (the derivation is shown in Appendix A):
$D(\delta X(t))=A(\bar{X}(t))\delta X(t)$ (4)
where $A(\bar{X}(t))$ is called a sensitivity matrix defined as follows:
$A(\bar{X}(t))=\left.\frac{\partial
F(X(t))}{\partial(X(t))}\right|_{\bar{X}(t)}$ (5)
.
Equation is thus a linear DE, with a time varying coefficient and has a the
following transition equation:
$\delta X(t+\zeta)=\Phi(t_{n}+\zeta,t_{n},\bar{X})\delta X(t)$ (6)
in which $\Phi(t_{n}+\zeta,t_{n},\bar{X})$ is the transition matrix from time
$t_{n}$ to $t_{n}+\zeta$ (increment $\zeta$). The transition matrix is
governed by the following DE:
$\frac{\partial}{\partial\zeta}\Phi(t_{n+\zeta},t_{n},\bar{X})=A(\bar{X}(t_{n}+\zeta))\Phi(t_{n+\zeta},t_{n},\bar{X})$
(7)
$\Phi(t_{n},t_{n},\bar{X})=I$ (8)
The transition matrix is a function of $\bar{X}(t)$ and can be evaluated by
numerical integration and in order to fill the values of
$A(\bar{X}(t_{n}+\zeta))$, $\bar{X}(t)$ has to be integrated numerically. We
will soon present a recursive algorithm that will avoid the computation of the
transition matrix. We have shown in this section that we can estimate the true
state of process by estimating the perturbation vector, which is governed by a
linear differential equation. The next task is to obtain a linear perturbation
observation from the non-linear observation scheme.
### 2.2 The observation perturbation vector
In this section we will adopt the notation $X_{n}$ and $Y_{n}$ for $X(t_{n})$
and $Y(t_{n})$ respectively. We define a simulated noise free observation
vector $\bar{Y}_{n}$ as follows:
$\bar{Y}_{n}=G(\bar{X}_{n})$ (9)
Subtracting $\bar{Y}_{n}$ from the actual observation $Y_{n}$ gives the
observation perturbation vector:
$\delta Y_{n}=Y_{n}-\bar{Y}_{n}$ (10)
In appendix A we show that the observation perturbation vector is related to
the state perturbation vector as follows:
$\delta Y_{n}=M(\bar{X}_{n})\delta X_{n}+v_{n}$ (11)
where $M(\bar{X}_{n})$ is the Jacobean matrix of G, evaluated at
$\bar{X}_{n}$. The matrix is also called the observation sensitivity matrix
and is defined as follows:
$M(\bar{X}_{n})=\left.\frac{\partial
F(X_{n})}{\partial(X_{n})}\right|_{\bar{X}_{n}}$ (12)
We now examine the sequence of observations.
### 2.3 Sequence of observation
We assume that $L+1$ observation are obtained with time stamps
$t_{n},t_{n-1},...,t_{n-L}$. Theses observations are assembled as follows :
$\left[\begin{array}[]{c}\delta Y_{n}\\\ \delta Y_{n-1}\\\ .\\\ .\\\ .\\\
\delta Y_{n-L}\end{array}\right]=\left[\begin{array}[]{c}M(\bar{X}_{n})\delta
X_{n}\\\ M(\bar{X}_{n-1})\delta X_{n-1}\\\ .\\\ .\\\ .\\\
M(\bar{X}_{n-L})\delta
X_{n-L}\end{array}\right]+\left[\begin{array}[]{c}v_{n}\\\ v_{n-1}\\\ .\\\
.\\\ .\\\ v_{n-L}\end{array}\right]$ (13)
Using the relationship:
$\delta X_{m}=\Phi(t_{m},t_{n},\bar{X})\delta X_{n}$ (14)
then, substituting Equation 13 the observation sensitity equation can be
written as:
$\mathbf{\delta Y}_{n}=\mathbf{T}_{n}\delta X_{n}+\mathbf{V}_{n}$ (15)
in which $\mathbf{T}_{n}$, the total observation matrix is defined as follows:
$\mathbf{T}_{n}=\left[\begin{array}[]{c}M(\bar{X}_{n})\\\
M(\bar{X}_{n-1})\Phi(t_{n-1},t_{n};\bar{X})\\\ .\\\ .\\\ .\\\
M(\bar{X}_{n-L})\Phi(t_{n-L},t_{n};\bar{X})\end{array}\right]$ (16)
The vectors $\mathbf{\delta Y}_{n}$ and $\mathbf{V}_{n}$ are large. The
solution of the equation can be obtained from the minimum variance estimation
as follows:
$\delta\hat{X}_{n}=(\mathbf{T}_{n}^{T}\mathbf{R}_{n}^{-1}\mathbf{T}_{n})^{-1}\mathbf{T}_{n}^{T}\mathbf{R}_{n}^{-1}\mathbf{\delta
Y}_{n}$ (17)
The estimate $\delta\hat{X}_{n}$ has a covariance matrix:
$S_{n}=(\mathbf{T}_{n}^{T}\mathbf{R}_{n}^{-1}\mathbf{T}_{n})^{-1}$ (18)
where $\mathbf{R}_{n}^{-1}$ is a block diagonal weight matrix, also called the
least squares weight matrix, but, in fact, if we define $R_{n}$ as the
covariance matrix of the the error vector $v_{n}$. Then $\mathbf{R}_{n}^{-1}$
is expressed as:
$\mathbf{R}_{n}^{-1}=\left[\begin{array}[]{cccccc}R_{n}^{-1}&0&.&.&.&0\\\
0&R_{n-1}^{-1}&&&&.\\\ .&&.&&&.\\\ .&&&.&&.\\\ .&&&&.\\\
0&.&.&.&0&R_{n-L}^{-1}\end{array}\right]$ (19)
In this section we arrived at a form of filter that uses the minimum variance
estimation initiated by Gauss and the local linearisation technique championed
by Newton, to estimate the estate of the process from the non linear
observation scheme. This filter is called Gauss-Newton filter (GNF) and is
described in detail in Morrison’s work [4, 13]. The GNF has been successfully
implemented in some practical applications:
[5] showing strong stability. The memory nature of the filter has made it
unattractive to researchers in the past, and even now, challenging [7] .
However recent developments have presented recursive form of the linear least-
squares for state space model [3] . We derive a recursive form of GNF using a
similar approach to M. B. Malik [3]. However, before we derive a recursive
form of the GNF filter, we rewrite the expression of $\mathbf{T}_{n}$ using
the backward differentiation:
$\Phi(t_{n-L},t_{n},\bar{X})=A(\bar{X}_{n-L})^{-1}\Phi(t_{n-L+1},t_{n},\bar{X})$
(20)
The expression is thus:
$\mathbf{\delta Y}_{n}\mathbf{T}_{n}=\left[\begin{array}[]{c}M_{0}\\\
M_{1}A_{1}\\\ M_{2}A_{2}\\\ .\\\ .\\\ .\\\ M_{L}A_{L}\end{array}\right]$ (21)
where
$A_{L}=\prod_{i=1}^{L}A(\bar{X}_{n-i})^{-1}$ (22)
and
$M_{L}=M(\bar{X}_{n-L})$ (23)
with
$A_{0}=I$ (24)
We now move to derive the Recursive Gauss Newton Filter in the next section.
## 3 The Recursive Gauss-Newton filter
To obtain the recursive form, we use an approach similar to M. B. Malik in
[3]. Suppose that the observations start arriving at $n=0$ and that all
initial values of the filter are available. In in order to maintain the filter
adaptiveness, a weight matrix function using a fading parameter $\lambda<1$ is
adopted, and is defined as follows:
$\mathbf{R}_{n}^{-1}=\left[\begin{array}[]{cccccc}R^{-1}&0&.&.&.&0\\\
0&\lambda R^{-1}&&&&.\\\ .&&.&&&.\\\ .&&&.&&.\\\ .&&&&.\\\
0&.&.&.&0&\lambda^{n}R^{-1}\end{array}\right]$ (25)
The following, further definitions are adopted:
$\mathbf{W}_{n}=\mathbf{T}_{n}^{T}\mathbf{R}_{n}^{-1}\mathbf{T}_{n}$ (26)
$\mathbf{\mathbf{\xi}}_{n}=\mathbf{T}_{n}^{T}\mathbf{R}_{n}^{-1}\delta\mathbf{Y}$
(27)
Resulting in:
$\delta\hat{X}_{n}=\mathbf{W}_{n}^{-1}\xi_{n}$ (28)
.
In the next section, the recursive update of the perturbation vector is
demonstrated.
### 3.1 The recursive update of $\mathbf{W}_{n}$
Using equation (21) and the definitions in equations (26) and (25) we have:
$\displaystyle\mathbf{W}_{n}$ $\displaystyle{}=$
$\displaystyle{}\sum_{j=1}^{L}\lambda^{j}R^{-1}\prod_{i=1}^{j}A(\bar{X}_{n-i})^{-T}M(\bar{X}_{n-j})^{T}$
(29)
$\displaystyle\times{}M(\bar{X}_{n-j})\prod_{i=0}^{j}A(\bar{X}_{n-i})^{-1}$
$\displaystyle{+}\>M(\bar{X}_{n})^{T}R^{-1}M(\bar{X}_{n})$
and
$\displaystyle\mathbf{W}_{n-1}$ $\displaystyle{}=$
$\displaystyle{}\sum_{j=1}^{L-1}\lambda^{j}R^{-1}\prod_{i=1}^{j}A(\bar{X}_{n-1-i})^{-T}M(\bar{X}_{n-1-j})^{T}$
(30)
$\displaystyle\times{}^{-1}M(\bar{X}_{n-1-j})\prod_{i=0}^{j}A(\bar{X}_{n-1-i})^{-1}$
$\displaystyle{+}\>M(\bar{X}_{n-1})^{T}R^{-1}M(\bar{X}_{n-1})$
Comparing equations (29) and (30) the following recursive equation is
obtained:
$\mathbf{W}_{n}=\lambda
A(\bar{X}_{n-1})^{-T}\mathbf{W}_{n-1}A(\bar{X}_{n-1})^{-1}+M(\bar{X}_{n})^{T}R^{-1}M(\bar{X}_{n})$
(31)
which is the discrete, quadratic, Lyapunov, difference equation.
### 3.2 The recursive form of $\mathbf{\mathbf{\xi}}_{n}$
Using equations (21) (25) (27) $\mathbf{\mathbf{\xi}}_{n}$ can be expressed
as:
$\displaystyle\xi_{n}$ $\displaystyle{}=$
$\displaystyle{}\sum_{j=0}^{L}\lambda^{j}R^{-1}\prod_{i=1}^{j}A(\bar{X}_{n-i})^{-T}M(\bar{X}_{n-j})^{T}\delta
Y_{n-j}$ (32) $\displaystyle{+}\>M(\bar{X}_{n})^{T}R^{-1}\delta Y_{n}$
and
$\displaystyle\xi_{n-1}$ $\displaystyle{}=$
$\displaystyle{}\sum_{j=1}^{L-1}\lambda^{j}R^{-1}\prod_{i=1}^{j}A(\bar{X}_{n-1-i})^{-T}M(\bar{X}_{n-1-j})^{T}$
(33) $\displaystyle{\times}\>\delta Y_{n-1-j}+M(\bar{X}_{n-1})^{T}R^{-1}\delta
Y_{n-1}$
Comparing equations (32) and (33) the following recursive equation is
obtained:
$\xi_{n}=\lambda A(\bar{X}_{n-1})^{-T}\xi_{n-1}+M(\bar{X}_{n})^{T}R^{-1}\delta
Y_{n}$ (34)
## 4 Adaptation to Levenberg and Maquardt
In order to guarantee local convergence of the recursive filter and also to
avoid the singularity of $\mathbf{W}_{n}$. we replace it by
$\mathbf{W}_{n}+\mu I$ as suggested by Levenberg and Maquardt. The presence of
the damping factor $\mu$ will have two effects:
* 1.
for large value of $\mu$ the algorithm behaves as a steepest descent which is
ideal when the current solution is far from the local minimum. The convergence
will be slow but however guaranteed. We therefore have
$\delta\hat{X}_{n}=\frac{1}{\mu}\xi_{n}$ (35)
.
* 2.
for $\mu$ very small the algorithm will behave as gauss newton with faster
convergence. The current step will be
$\delta\hat{X}_{n}=\mathbf{W}_{n}^{-1}\xi_{n}$ (36)
.
### 4.1 The Gain Ratio
The $\mu$ can be updated by the so called gain ratio. We consider the
following cost function which is
$E(\delta X_{n})=(\mathbf{\delta Y}_{n}-\mathbf{T}_{n}\delta
X_{n})^{T}{R}^{-1}(\mathbf{\delta Y}_{n}-\mathbf{T}_{n}\delta X_{n})$ (37)
The denominator of gain ratio is :
$E(0)-E(\delta X_{n})=\delta X_{n}^{T}(\xi_{n}+\mu\delta X_{n})$ (38)
We define :
$F(\delta X_{n})=(Y_{n}-G(\bar{X}_{n}+\delta
X_{n}))^{T}R^{-1}(Y_{n}-G(\bar{X}_{n}+\delta X_{n}))$ (39)
The gain ratio is therefore:
$\varrho=\frac{F(0)-F(\delta X_{n})}{E(0)-E(\delta X_{n})}$ (40)
A large value of $\varrho$ indicates that $E(\delta X_{n})$ is a good
approximation of $\bar{Y}$, and $\mu$ can be decreased so that the next
Levenberg-Marquardt step is closer to the Gauss-Newton step. If $\varrho$ is
small or negative then $E(\delta X_{n})$ is a poor approximation, then $\mu$
should be increased to move closer to the steepest descent direction. The
complete filter algorithm adapted from [9] is presented in Algorithm 1
$k:=0$;$\nu:=2$;$\bar{X}_{n}:=X_{n/n-1}$;
$\delta{Y}_{n}:={Y}_{n}-G(\bar{X}_{n})$;
${W}_{temp}=M(\bar{X}_{n})^{T}R^{-1}M(\bar{X}_{n})$;
$\mathbf{W}_{n}=\mathbf{W}_{n-1/n}+{W}_{temp}$;
$\xi_{temp}=M(\bar{X}_{n})^{T}R^{-1}\delta Y_{n}$;
$\xi_{n}=\xi_{n/n-1}+\xi_{temp}$;
$stop:=false$;$\mu=\tau*max(diag(\mathbf{W}_{n/n-1}))$;
While (not stop) and ($k\leq k_{max}$)
$k:=k+1$;
repeat;
solve $(\mathbf{W}_{n}+\mu I)\delta\hat{X}_{n}=\xi_{n}$;
if ($||\delta\hat{X}_{n}||\leq\varepsilon||\bar{X}_{n}||$)
stop:=true;
else
$X_{new}:=\bar{X}_{n}+\delta\hat{X}_{n}$;
$F(\delta X)=Y_{n}-G(X_{new})$;$F(0)=\delta Y_{n}^{T}R^{-1}\delta Y_{n}$;
$E(0)-E(\delta X_{n})=\delta X_{n}^{T}(\xi_{n}+\mu\delta X_{n})$;
$\varrho=\frac{F(0)-F(\delta X_{n})}{E(0)-E(\delta X)}$;
if $\varrho>0$
$\bar{X}_{n}=X_{new}$;
$\delta{Y}_{n}:={Y}_{n}-G(\bar{X}_{n})$;
${W}_{temp}=M(\bar{X}_{n})^{T}R^{-1}M(\bar{X}_{n})$;
$\mathbf{W}_{n}=\mathbf{W}_{n/n-1}+{W}_{temp}$;
$\xi_{temp}=M(\bar{X}_{n})^{T}R^{-1}\delta Y_{n}$;
$\xi_{n}=\xi_{n/n-1}+\xi_{temp}$;
$\mu=\mu*max(1/3,1-(2\varrho+1)^{3})$;$\nu:=2$;
else
$\mu:=\nu*\mu$;
$\nu:=2*\nu$;ssm
endif
endif
until($\varrho>0$)or(stop);
endwhile
$X_{n/n}=X_{new}$;
$X_{n/n+1}=\Phi(s)X_{n/n}$;
$\mathbf{W}_{n/n+1}=\lambda A({X}_{n/n})^{-T}\mathbf{W}_{n}A({X}_{n/n})^{-1}$;
$\xi_{n/n+1}=\lambda A({X}_{n/n})^{-T}\xi_{n}$;
Algorithm 1 L-M algorithm for tracking system
## 5 State Space Models
We will present four possible models to which the recursive GNF can be
applied:
* 1.
Model 1, with linear process dynamic and linear observation scheme. In this
model the recursive formulation is similar to the derived forms except the
estimation is made directly for $X_{n}$ and that the observed perturbation
vector $\delta Y_{n}$ is replaced by the actual observation vector $Y_{n}$.The
sensitivity matrices in this case become the measurement and transition
matrices of the process. In this case the LM algorithm is not required.
* 2.
Model 2, with linear process dynamic and non-linear observation scheme. The
recursive model of the filter remains the same except the state sensitivity
matrix becomes a the transition matrix of the process. The state perturbation
is estimated to obtain the estimate of the process state.
* 3.
Model 3, with non-linear process dynamic and linear observation scheme. The
measurement sensitivity matrix has become the measurement matrix.
* 4.
Model 4, with a non-linear process dynamic and non linear observation scheme.
The derived recursive form without any further modification is applicable to
this case.
### 5.1 Stability of the Recursive GNF
The matrix $\mathbf{W}_{n}$ is the inverse of of the covariance matrix of the
filter and is therefore positive definite. As a consequence the solution of
the derived discrete Ly5apunov equation in (31) is unique with the sensitivity
matrix being stable. The eigenvalues of the inverse of the sensitivity matrix
are within an open unit circle and therefore the stability of athe system is
ensured by having $\lambda<1$.
## 6 Simulation: Range and Bearing tracking
In these simulation studies, we consider an example of a vehicle executing
various manoeuvres. During turn manoeuvres of unknown constant turn rate, the
aircraft dynamic model is :
$X_{n}=\left[\begin{array}[]{ccccc}1&\frac{sin(\Omega
T)}{\Omega}&0&-(\frac{1-cos(\Omega T)}{\Omega})&0\\\ 0&cos(\Omega
T)&0&-sin(\Omega T)&0\\\ 0&\frac{1-cos(\Omega T)}{\Omega}&1&\frac{sin(\Omega
T)}{\Omega}&0\\\ 0&sin(\Omega T)&0&cos(\Omega T)&0\\\
0&0&0&0&1\end{array}\right]X_{n-1}+v_{n}$ (41)
where the state of the vehicle is $X_{n}=[x,\dot{x},y,\dot{y},\Omega]$, with
$x$,$y$ the position coordinates and $\dot{x}$,$\dot{y}$ their corresponding
velocity components.The process noise $v_{k}\sim\mathcal{N}(0,Q)$ with
covariance matrix
$Q=diag\left[\begin{array}[]{ccc}q1BB^{T}&q1BB^{T}&q2T\end{array}\right]$
where,
$BB^{T}=\left[\begin{array}[]{cc}\frac{T^{4}}{4}&\frac{T^{3}}{2}\\\
\frac{T^{3}}{2}&T^{2}\end{array}\right]$ (42)
When the vehicle moves at nearly constant velocity its dynamic model is:
$X_{n}=\left[\begin{array}[]{ccccc}1&T&0&0&0\\\ 0&1&0&0&0\\\ 0&0&1&T&0\\\
0&0&0&1&0\\\ 0&0&0&0&1\end{array}\right]X_{n-1}+v_{n}$ (43)
The vehicle is observed by a radar located at the origin of the plane, capable
of measuring the range $r$ and and the bearing angle $\theta$. The measurement
equation is therefore:
$\left[\begin{array}[]{c}r_{n}\\\
\theta_{n}\end{array}\right]=\left[\begin{array}[]{c}\sqrt{x^{2}+y^{2}}\\\
tan^{-1}(\frac{y}{x})\end{array}\right]+w_{n}$ (44)
where the measurement noise is $w_{k}\sim\mathcal{N}(0,R)$ with covariance
$R=diag\left[\begin{array}[]{cc}\sigma_{r}^{2}&\sigma_{\theta}^{2}\end{array}\right]$
The following constants were used for data generation: $T=1s$;
$\Omega=-3^{0}s^{-1}$; $q1=0.01$m${}^{2}\rm{s}^{-4}$; $q2=1.75\times
10^{-4}\rm{s}^{-4}$; $\sigma_{r}$=10m; $\sigma_{\theta}=\sqrt{0.1}$mrad.
The vehicle starts at true initial state $X_{n}$=[10m, 25ms-1,400m,
0ms-1,-3ms-1] and moves at nearly constant velocity for $100$s, Then it
executes a turn manoeuvre from time index $n=101$ to $n=150$. After the
manoeuvre, the vehicle’s velocity remains nearly constant from $n=151$ to
$n=250$. At $n=251$ it starts a new turn manoeuvre at rate $\Omega$=3ms-1
until $n=400$. Finally from $n=400$ to $n=500$ it moves at nearly constant
velocity. Figure [1] describes the complete trajectory of the vehicle.
The filter uses a single model of a constant velocity to track the entire
manoeuvre:
$A(X_{n})=\left[\begin{array}[]{ccccc}1&T&0&0&0\\\ 0&1&0&0&0\\\ 0&0&1&T&0\\\
0&0&0&1&0\\\ 0&0&0&0&1\end{array}\right]$ (45)
The initial value $\mathbf{W}_{-1/0}=10^{-2}I$, where $I$ is an identity
matrix. The filter parameters are the following $k_{max}=200$,
$\varepsilon=1\times 10^{-24}$, $\tau=1\times 10^{-3}$, $\lambda=0.4$ The
filter initial state is generated randomly and then ensuring that it has the
same sign as the true state. This procedure guarantees the local convergence
of the first estimate. The experiment was repeated for 250 Monte Carlo runs
and the root means squared error (RMSE) is used as a performance metric. The
position RMSE is computed using the following expression:
$RMSE=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\left((x_{n}^{i}-\hat{x_{n}})^{2}+(y_{n}^{i}-\hat{y_{n}})^{2}\right)}$
(46)
where $(x_{n}^{i},y_{n}^{i})$ and $(\hat{x_{n}},\hat{y_{n}})$ true and
estimated position coordinates respectively. The velocity root mean square
error (RMSE) is computed similarly. Figures [2] and [3] show the RMSE of the
position and velocity respectively. The position RMSE is not affected by
different manoeuvres while the velocity RMSE shows variation from different
manoeuvre states. The average values of the damping factor after complete
cycles of iteration is presented in Figure [3]. The damping factor increases
rapidly at the transition between manoeuvres. The average number of iterations
$k$ at convergence from Figure [4] shows similar variations.
## 7 Conclusions
The GNF with memory combines the minimum variance estimation and the Newton
method of local linearisation to estimate the process true state. The
recursive form for the Gauss-Newton filter has been derived in one compact
form that can be applied to all the four state and observation linearity and
nonlinearity scenarios:
Case 1 : linear process dynamic and linear observation scheme.
Case 2 : linear process dynamic and non-linear observation scheme.
Case 3 : with non-linear process dynamic and linear observation scheme.
Case 4 : non-linear process dynamic and non linear observation scheme.
The Hessian matrix of the filter which is computed recursively is augmented by
a damping factor as suggested earlier by Levenberg-Maquardt for non linear
curve fitting problems. The new filter is therefore a combination of Newtons
steepest descent and the Gauss-newton, ensuring its robustness. The presence
of a forgetting factor in the filter equations renders it capable of tracking
manoeuvring targets with a single filter dynamic model.
## Appendix A
### A.1 The differential equation governing $\delta X(t)$
Starting from:
$\delta X(t)=X(t)-\bar{X}(t)$ (47)
The differentiation rule is applied:
$D\delta X(t)=F(\bar{X}(t)+\delta X(t))-F(\bar{X}(t))$ (48)
Let $F$ be defined as follows :
$F=\left[\begin{array}[]{c}f_{1}\\\ .\\\ .\\\ .\\\ f_{n}\end{array}\right]$
(49)
Equation becomes:
$D\delta X(t)=\left[\begin{array}[]{c}f_{1}(\bar{X}(t)+\delta X(t))\\\ .\\\
.\\\ .\\\ f_{n}(\bar{X}(t)+\delta
X(t))\end{array}\right]-\left[\begin{array}[]{c}f_{1}(\bar{X}(t))\\\ .\\\ .\\\
.\\\ f_{n}(\bar{X}(t))\end{array}\right]$ (50)
The Taylor first order approximation is applied:
$\displaystyle D\delta X(t)$ $\displaystyle{}=$
$\displaystyle{}\left[\begin{array}[]{c}f_{1}(\bar{X}(t))\\\ .\\\ .\\\ .\\\
f_{n}(\bar{X}(t))\end{array}\right]+\left[\begin{array}[]{c}\nabla
f_{1}(\bar{X}(t))^{T}\\\ .\\\ .\\\ .\\\ \nabla
f_{n}(\bar{X}(t))^{T}\end{array}\right]\delta X(t)$ (67)
$\displaystyle{-}\>\left[\begin{array}[]{c}f_{1}(\bar{X}(t))\\\ .\\\ .\\\ .\\\
f_{n}(\bar{X}(t))\end{array}\right]$
The following relation is obtained :
$D\delta X(t)=A(\bar{X}(t))\delta X(t)$ (68)
Where:
$A(\bar{X}(t))=\left[\begin{array}[]{c}\nabla f_{1}(\bar{X}(t))^{T}\\\ .\\\
.\\\ .\\\ \nabla f_{n}(\bar{X}(t))^{T}\end{array}\right]=\left.\frac{\partial
F(X(t))}{\partial(X(t))}\right|_{\bar{X}(t)}$ (69)
### A.2 The relation between $\delta X_{n}$ and $\delta Y_{n}$
$\delta Y_{n}=G(\bar{X}_{n}+\delta X_{n})-G(\bar{X}_{n})$ (70)
As direct consequence of A.1 the following relationship is obtained:
$\delta Y_{n}=M(\bar{X}_{n})\delta X_{n}+v_{n}$ (71)
## Appendix B Figure captions list
Figure 1: Target complete trajectory with manoeuvres
Figure 2: The Position RMSE is unaffected by the manoeuvres.
Figure 3: The velocity RMSE varies with manoeuvres.
Figure 4: The damping factor shows sharp peaks at start of manoeuvres.
Figure 5: The number of iterations increases during manoeuvres.
The figure numbering appears in the same order as the figures in the pdf
document
## Acknowledgment
The authors would like to thank Dr Norman Morrison for his contribution during
the research that leads to obtaining a recursive form of GNF. Dr Morrison has
been working on the GNF throughout his career and even in his retirement is
enthusiastic in providing teaching and insights into the fundamentals of
filter Engineering.
## References
* [1] Recursive Least Squares With Linear Constraints.
* [2] T. Kailath, A. H. Sayed, B. Hassibi, Linear Estimation, 2nd Edition, Prentice Hall, New Jersey, USA, 2000.
* [3] M. B. Malik, State-space recursive least-squares: Part i, Signal Processing 84 (2004) 1709–1718.
* [4] N. Morrison, Filter engineering -a practical approach: The Gauss-Newton and polynomial filters, to be published.
* [5] N. Morrison, R. T. Lord, M. R. Inggs, The Gauss-Newton algorithm applied to track-while-scan radar, in: Proceedings of the IET International Conference on Radar Systems (RADAR 2007), Institution for Engineering and Technology, 2007\.
* [6] J.-P. da Conceicao, Accelerating Gauss-Newton filters on FPGAs, Master’s thesis, University of Cape Town (Dec. 2011).
* [7] J. Milburn, Co-processor offloading applied to passive coherent location with doppler and bearing data, Master’s thesis, University of Cape Town - RRSG (February 2010).
* [8] D. W. Marquardt, An algorithm for least-squares estimation of nonlinear parameters, Journal of the Society for Industrial and Applied Mathematics 11 (2) (1963) pp. 431–441.
* [9] O. T. K. Madsen, H.B. Nielsen, Method of non-linear least squares problems, 2nd Edition, Informatics and Mathematical Modelling, Technical University of Denmark, 2004.
* [10] B. G. Kermani, S. S. Schiffman, H. T. Nagle, Performance of the Levenberg-Marquardt neural network training method in electronic nose applications, Sensors and Actuators B: Chemical 110 (1) (2005) 13 – 22. doi:10.1016/j.snb.2005.01.008.
URL http://www.sciencedirect.com/science/article/pii/S09254%00505000961
* [11] E. Derya, beyli, Analysis of EEG signals by implementing eigenvector methods/recurrent neural networks, Digital Signal Processing 19 (1) (2009) 134 – 143. doi:10.1016/j.dsp.2008.07.007.
URL http://www.sciencedirect.com/science/article/pii/S10512%00408001243
* [12] V. Singh, I. Gupta, H. Gupta, ANN-based estimator for distillation using Levenberg-Marquardt approach, Engineering Applications of Artificial Intelligence 20 (2) (2007) 249 – 259. doi:10.1016/j.engappai.2006.06.017.
URL http://www.sciencedirect.com/science/article/pii/S09521%9760600114X
* [13] N. Morrison, Introduction to Sequential Smoothing and Prediction, McGraw-Hill Book Company, 1969.
|
arxiv-papers
| 2011-10-24T12:20:28 |
2024-09-04T02:49:23.569971
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Roaldje Nadjiasngar, Michael Inggs",
"submitter": "Roaldje Nadjiasngar",
"url": "https://arxiv.org/abs/1110.5212"
}
|
1110.5379
|
# The atmospheric dispersion corrector for the Large Sky Area Multi–object
Fibre Spectroscopic Telescope (LAMOST)
Ding-qiang Su1,2,3, Peng Jia1,2,3 and Genrong Liu3
1Department of Astronomy, Nanjing University, 22 Hankou Road, Nanjing 210093,
China
2Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University),
Ministry of Education, Nanjing 210093, China
3National Astronomical Observatories / Nanjing Institute of Astronomical
Optics & Technology (NIAOT), Chinese Academy of Science,
188 Bancang Street, Nanjing 210042, China E-mail: dqsu@nju.edu.cn
###### Abstract
The Large Sky Area Multi–object Fibre Spectroscopic Telescope (LAMOST) is the
largest (aperture 4 $\mathrm{m}$) wide field of view (FOV) telescope and is
equipped with the largest amount (4000) of optical fibres in the world. For
the LAMOST North and the LAMOST South the FOV are 5 ∘ and 3.5 ∘, the linear
diameters are 1.75 $\mathrm{m}$ and 1.22 $\mathrm{m}$, respectively. A new
kind of atmospheric dispersion corrector (ADC) is put forward and designed for
LAMOST. It is a segmented lens which consists of many lens–prism strips.
Although it is very big, its thickness is only 12 $\mathrm{mm}$. Thus the
difficulty of obtaining big optical glass is avoided, and the aberration
caused by the ADC is small. Moving this segmented lens along the optical axis,
the different dispersions can be obtained. The effects of ADC’s slits on the
diffraction energy distribution and on the obstruction of light are discussed.
The aberration caused by ADC is calculated and discussed. All these results
are acceptable. Such an ADC could also be used for other optical fibre
spectroscopic telescopes, especially those which a have very large FOV.
###### keywords:
Telescopes – Instrumentation: spectrographs – Atmospheric effects
††pagerange: The atmospheric dispersion corrector for the Large Sky Area
Multi–object Fibre Spectroscopic Telescope (LAMOST)–14††pubyear: 2011
## 1 Introduction
Large Sky Area Multi–object Fibre Spectroscopic Telescope (LAMOST) is a new
type telescope (Wang et al., 1996; Cui et al., 2000; Su & Cui, 2004). The main
parameters of LAMOST are the following: clear aperture 4 $\mathrm{m}$
(average), f–ratio 5, and field of view (FOV) 5 ∘. The linear diameter of FOV
is 1.75 $\mathrm{m}$. 4000 optical fibres (Xing et al., 1998), which introduce
the light of different celestial objects to 16 spectrographs (Zhu et al.,
2006), are put on such a big focal surface. The dedication ceremony of LAMOST
was held on 2008 October 16. We call it the LAMOST North. At present, another
telescope of this kind, the LAMOST South designed to survey the southern sky,
is under consideration by China and other countries, with a view to
international cooperation (Cui et al., 2010). The clear aperture and f–ratio
of the LAMOST South are still 4 $\mathrm{m}$ and $5$. But its FOV will be
reduced to 3.5 ∘. The linear diameter of FOV of this telescope is 1.22
$\mathrm{m}$. Many years ago some atmospheric dispersion correctors for small
FOV had been designed and used. Since 1980’s some atmospheric dispersion
correctors for larger FOV have been designed (Epps et al., 1984; Su, 1986; Su
& Liang, 1986; Wynne, 1986; Willstrop, 1987; Liang & Su, 1988; Su et al.,
1988; Bingham, 1988; Wynne, 1988; Wang & Su, 1990). In these correctors there
is a pair of prisms or lens–prisms (lensms), each of which is a cemented lens
with a tilted cemented surface and consists of two different glasses. Rotating
these two lens–prisms, we could obtain different dispersions. Using this
method in LAMOST would require very big lens–prisms, and it is difficult to
obtain optical glass of such large size and to support the big lens–prisms.
Liu & Yuan have designed several kinds of small corrector each for an optical
fibre (Liu & Yuan, 2005). But it is difficult to install and move them for
4000 optical fibres. In this paper, as an example a detailed design and
discussion of this ADC are given for the LAMOST South. It is a big but thin
segmented lens which consists of many lens–prism strips. Thus the difficulty
of obtaining such big optical glass is avoided, and since it is thin, the
introduced aberration is small. Moving this segmented lens along the optical
axis, we could obtain different dispersions. The entire design of this ADC
could be used for the LAMOST North too—we only need to extend its diameter to
about 1.78 $\mathrm{m}$. In this ADC the clear aperture of a portion of
celestial objects is divided into two parts, i.e., a slit is added in it. Here
the word “silt” means silt plus chamfer at the edge of each lens–prism strip
and they are covered with black paint, i.e., in this paper “slit” means a
light-obstructing black belt with a width of 1 $\mathrm{mm}$. In this case the
diffraction spot is enlarged. But in LAMOST, as the following discussion will
show, it is not serious and it is acceptable. Apart from this, there is some
loss from light obstruction by the silt. For different celestial objects, the
loss is different in the whole FOV, and also different when ADC is at
different positions. For these two reasons mentioned above, such a segmented
ADC is not suitable for the diffraction–limited high resolution telescope and
photometry. Such a segmented ADC is mainly used for the optical fibre
spectroscopic telescope, especially if this telescope has a very large FOV.
## 2 A brief introduction of LAMOST
LAMOST is shown in Figure 1. It lies on the ground along the South–North
direction. Mb is a spherical mirror. Ma is an aspherical reflecting plate,
which corrects the spherical aberration of Mb and reflects the light of
celestial objects to the Mb. Ma and Mb consists of 24 and 37 hexagonal mirrors
respectively and each mirror has diagonal 1.1 $\mathrm{m}$. In a segmented
mirror telescope PSF is enlarged, especially when a segmented ADC is added. At
a good astronomical site, if we do not use adaptive optics, atmospheric seeing
only allows a telescope with an aperture under 30 $\mathrm{cm}$ to obtain the
diffraction–limited image. Because adaptive optics cannot be used for a large
FOV telescope, therefore, in this case for a segmented mirror telescope a sub-
area of diameter 50–60 $\mathrm{cm}$ is enough. The other reason that LAMOST
North uses two segmented mirrors is to save the cost. By the way, it is
possible that LAMOST South will use one segmented mirror consisting of 1.1
$\mathrm{m}$ hexagonal sub–mirrors. On the other hand, in many optical fibre
spectroscopic telescopes the diameter of optical fibre is more than 1.5
$\mathrm{arcsec}$. Compared with the seeing and diameter of the optical fibre,
the effect of enlarged PSF in LAMOST including segmented ADC is minor and
acceptable. During the observation, for a particular observation direction (a
particular sky area), LAMOST is a reflecting Schmidt telescope. But when
observing different directions (i.e. different celestial objects or same
celestial objects in different times), LAMOST should be different reflecting
Schmidt telescopes. This means the shape of Ma should be different for
correcting the spherical aberration. Traditionally, such an optical system can
not be realized. Active optics is a key technology for correcting the
telescope errors–gravitation deformation and thermal deformation. This
technology was developed mainly by Wilson and Lemaitre et.al. (Wilson, 1999;
Lemaitre, 2009) for the thin mirror, and by Nelson and Mast et.al. (Nelson,
1980; Mast & Nelson, 1980, 1982) for the segmented mirror. Chinese experts
have creatively applied active optics technology to Ma to make such a
telescope possible and developed the thin–mirror and segmented–mirror combined
active optics (Su et al., 1986, 1998; Cui et al., 2000; Su & Cui, 2004; Cui,
2008). LAMOST is a wide FOV telescope with the largest aperture and has the
strongest fibre spectroscopic obtaining capability in the world. The observing
sky area of LAMOST is $-10^{\circ}\leqslant\delta\leqslant+90^{\circ}$. The
celestial objects are observed for an average of 1.5 hours before and after
they pass through the meridian. During observation only the mounting of Ma
does the tracking and the focal surface does the rotation. LAMOST North was
set up in Xing Long Station (latitude 40.4 ∘ N, height above sea level 900
$\mathrm{m}$), National Astronomical Observatories, Chinese Academy of
Sciences. The biggest zenith distance (with the celestial object in the
meridian, it is the same in the following text) is 50.4 ∘. For waveband $380$
– $1000\mathrm{nm}$ the atmospheric dispersion is 2.2 $\mathrm{arcsec}$. At
Xing Long Station FWHM of seeing is about 2 $\mathrm{arcsec}$. The biggest
spread of aberration is 1.84 $\mathrm{arcsec}$. The diameter of optical fibre
adopted is 3.3 $\mathrm{arcsec}$. In this situation, although it is better to
correct the atmospheric dispersion, leaving it uncorrected will not present
serious problem. Through the development of the LAMOST North, Chinese experts
found that the size of each optical fibre positioner could be significantly
reduced, so in the LAMOST South the FOV will be reduced to 3.5 ∘, the linear
diameter of it is 1.22 $\mathrm{m}$, and 6000 optical fibres could be put on
this focal surface. The observing sky area of the LAMOST South is
$0^{\circ}\geqslant\delta\geqslant-90^{\circ}$. The LAMOST South may be
installed on the the NOAO Las Campanas Observatory or the ESO Paranal
Observatory. NOAO Las Campanas Observatory is situated at latitude 29.9 ∘ S,
altitude 2400 $\mathrm{m}$ above the sea level and the biggest zenith distance
observed is 60.1 ∘. ESO Paranal Observatory is situated at latitude 24.6 ∘ S,
altitude 2635 $\mathrm{m}$ above the sea level and the biggest zenith distance
observed is 65.4 ∘. For waveband $380$ – 1000 $\mathrm{nm}$ the corresponding
atmospheric dispersion at these two observatories are 2.75 $\mathrm{arcsec}$
and 3.35 $\mathrm{arcsec}$ respectively. Since image quality (due to the
reduction of FOV to 3.5 ∘) and two observatories’seeing are better than the
LAMOST North, the diameter of optical fibre used will be 1.6
$\mathrm{arcsec}$. In the LAMOST South the atmospheric dispersion should be
corrected. Since at ESO Paranal Observatory the biggest atmospheric dispersion
is bigger, it is chosen as an example in this paper.
## 3 The structure of this ADC
The layout and specification of this ADC are shown in Figure 2,3 and Table 1.
Each lens–prism strip consists of Schott glass PSK3 and LLF1. The refractive
index of PSK3 and LLF1 are given in Table 2. In $\lambda$ $=$ 441.8
$\mathrm{nm}$ the refractive indices of the two kinds of glass are the same.
In LAMOST the Mb and the focal surface are concentric. The radius of focal
surface is 20 $\mathrm{m}$. When ADC is at the farthest position, we take its
two outside surfaces and the focal surface to be concentric, i.e. the radius
of the outside surface equals 20 $\mathrm{m}$ \+ 250 $\mathrm{mm}$. We require
that in this position for waveband $380$ – 1000 $\mathrm{nm}$ this ADC can
produce dispersion of 3.35 $\mathrm{arcsec}$. From it the tilt angle of
lens–prism strip can be obtained which is 6.89 ∘. We set the width of each
lens–prism strip to be 50 $\mathrm{mm}$ which equals to the light beam
diameter of each celestial object on this ADC. Thus in this position only one
slit is added for each object. Since the thickness difference of one lens of
lens–prism strip is 6.04 $\mathrm{mm}$, we take the total thickness of ADC is
12 $\mathrm{mm}$. Since the f–ratio of LAMOST is 5, the diameter of ADC should
equal the linear FOV diameter plus 50 $\mathrm{mm}$ $(250/5=50)$, i.e., 1.27
$\mathrm{m}$ for the LAMOST South. So the maximum length of the strip is also
1.27 $\mathrm{m}$ for the LAMOST South. A similar result for the LAMOST North
is 1.78 $\mathrm{m}$. If one feels that some lens–prism strips are too long in
the LAMOST North, those strips longer than 1 $\mathrm{m}$ could be divided
into two parts. In this case, only about $1/20$ celestial objects will meet
two slits when ADC is at the farthest position. Although such an ADC is very
big, it is very thin and only includes one segmented lens. As it is moved
along the optical axis its dispersion is changed. Thus the atmospheric
dispersion for $z<65.4$ ∘ celestial objects can be compensated. The dispersion
produced by this ADC is direct proportion to the distance from it to focal
surface. And in these situations only one slit is met for a part of celestial
objects. When the atmospheric dispersion is small enough which needs not to be
compensated, this ADC can be moved out easily. With regard to the
manufacturing of the ADC, we plan to glue these lens–prism stripes together
(with removable glue) to form a disk, then grind and polish it. We have
already had some experience with such a method. Since the maximum light beam
of each celestial object on ADC is only 50 $\mathrm{mm}$, the figure tolerance
of lens–prism strip is loose. As a whole this ADC’s tolerance of position, tip
and tilts are loose. All lens–prism strips are fixed on the edge of the frame.
So long as each lens-prism strip is well fixed, its tip and tilts will be
small. Liquid glue may be used for the cemented surface to reduce the
reflecting loss. Nevertheless, moderate difficulties do exist for the
manufacturing and mounting of the ADC, thus it is still necessary to conduct
more research and testing in this respect.
## 4 Some discussions on special topics
### 4.1 The effect of ADC’s slit on diffraction energy distribution
In LAMOST both Ma and Mb consist of hexagonal mirrors each with a diagonal of
1.1 $\mathrm{m}$. The surface area of such a hexagonal mirror equals to a
circular area with a diameter of 1 $\mathrm{m}$. For different celestial
objects these sub–mirrors of Ma and Mb are covered by each other in the clear
aperture with different states. Since in LAMOST all sub–mirrors only co–focus,
the light from different sub–areas is non–coherent and these shapes of
sub–areas divided by edges of hexagonal mirrors are complex. First we discuss
the case when both Ma and Mb are approximately perpendicular to the optical
axis, i.e., the angle between the light of the celestial object and the
optical axis pointing to Mb is small. It can be found that each hexagonal
sub–mirror is mainly divided into 3 to 4 sub–areas. If we assume 4 sub–areas,
an important conclusion can be obtained: the average surface area of sub–area
equals a circular area with a diameter of 0.5 $\mathrm{m}$. As a rough
estimate we think that the diffraction energy distribution of LAMOST in this
case is like a circular hole with a diameter of 0.5 $\mathrm{m}$, i.e., 84
$\mathrm{percent}$ of the light energy spreads in about 0.5 $\mathrm{arcsec}$
area (Airy disk). Xu made a detailed calculation for four special situations,
and a similar conclusion was obtained (Xu, 1997). Since LAMOST’s clear
aperture is 4 $\mathrm{m}$, its surface area is 64 times of a 0.5 $\mathrm{m}$
circular area. We could think that about 64 sub–areas are included in LAMOST
clear aperture. For a particular celestial object in the worst situation its
clear aperture is divided by an ADC’s slit along the diameter direction, thus
about eight sub–areas are divided. As an average result the diffraction energy
of the eight sub–areas will distribute in two times its original length in
perpendicular direction to the slit, i.e., the 84 $\mathrm{percent}$
diffraction energy will spread in an area 0.5 $\mathrm{arcsec}$ wide and 1
$\mathrm{arcsec}$ long, i.e., half of the energy will disperse beyond the 0.5
$\mathrm{arcsec}$ area. Thus in the 0.5 $\mathrm{arcsec}$ area the light
energy will reduce by $4/64$, i.e., about 6 $\mathrm{percent}$. The energy
loss is small and it still distributes in 1 $\mathrm{arcsec}$ area. Given that
the diameter of optical fibre is 1.6 $\mathrm{arcsec}$, the energy loss is
acceptable. Secondly, we discuss the situation where the celestial objects
observed are near the celestial pole. In this case, in a plane perpendicular
to the optical axis the projective width of sub–mirrors of Ma will reduce to
about 1/2, but the projective width of sub–mirrors of Mb is unchangeable.
Considering these two factors and using a method of analysis similar to the
above, we obtain the following conclusion: in this case in LAMOST 84
$\mathrm{percent}$ of the light energy spreads in an area about 0.5
$\mathrm{arcsec}$ wide and 0.75 $\mathrm{arcsec}$ long and in this area the
light energy will reduce about 4.5 $\mathrm{percent}$ due to an ADC’s slit.
The energy loss is small and it distributes in a 1.5 $\mathrm{arcsec}$ area.
Given that the diameter of optical fibre is 1.6 $\mathrm{arcsec}$, the energy
loss is acceptable. In LAMOST South either Ma or Mb may adopt a non–segmented
(monolithic) mirror. It is easy to find that if either Ma or Mb is a
non–segmented mirror or consists of larger sub–mirrors, with such a segmented
ADC the total diffraction energy distribution will be more concentrated than
in the above situation.
### 4.2 The probability of objects meet slit
In the largest zenith distance $\mathrm{z}$ = 65.4 ∘, each object will meet a
slit of strips. When $\mathrm{z}<$65.4 ∘, i.e., the distance between ADC and
focal surface is less than 250 $\mathrm{mm}$, only a part of objects meet a
slit of lens–prism strips. For example, if the distance between ADC and focal
surface is 100 $\mathrm{mm}$ (according to $\mathrm{z}$ = 39.6 ∘, see section
5 and Table 3) only $2/5$ objects meet a slit and if the distance between ADC
and focal surface is 50 $\mathrm{mm}$ (according to $\mathrm{z}=$20.6 ∘), only
$1/5$ objects meet a slit.
### 4.3 The light obstructed by slits of ADC
According to a technical requirement, the edge of strips of ADC should be
chamfered to a projective width of 0.5 $\mathrm{mm}$. In order to reduce
scattered light, all chamfers and slits of ADC should be covered with black
paint. Thus all slits will become black belts with a width of 1 $\mathrm{mm}$
between two strips. These slits will obstruct light. The width of each strip
is 50 $\mathrm{mm}$. The approximate average obstructed ratio equals 1/50 = 2
$\mathrm{percent}$. The thickness of ADC is 12 $\mathrm{mm}$. The f–ratio of
LAMOST is 5 and the maximum inclination angle of ray is 1/10 to ADC’s surface.
Considering these situations, we obtain that about an average of 2.5
$\mathrm{percent}$ of light will be obstructed. This is the average light
loss. For a specific celestial object, this loss may be zero when the object
does not meet a slit, and it may be several times the average loss when ADC is
near the focal surface and the object meets a slit. Since the loss of light
obstruction by a slit is uneven, such a segmented ADC is not suitable for
photometry.
### 4.4 The ADC’s orientation
Both the atmospheric dispersion and ADC’s dispersion are vectors. The
compensating error equals ADC dispersion vector plus the atmospheric
dispersion vector. For compensation not only the amounts of the two vectors
should be equal but also their directions should be opposite to each other. In
a telescope the ADC should be rotated to make its dispersion direction
opposite to the atmospheric dispersion direction. From spherical astronomy the
ADC’s orientation formula can be derived easily. The tolerance of ADC’s
orientation angular is loose.
### 4.5 The aberrations
First, we ignore the tilt of cemented surface of ADC. Since the radii of ADC
is very large, it can be considered as a parallel glass plate. From the
third–order aberration formula the least circle of spherical aberration is
only 0.01 $\mathrm{arcsec}$ and it could be corrected by active optics. From
the chromatic aberration formula the spread circle of it is 0.17
$\mathrm{arcsec}$ at the extreme wavelengths 380 $\mathrm{nm}$ and 1000
$\mathrm{nm}$. For a glass parallel plate the spherical aberration and
chromatic aberration are unchanged when it moved along optical axis. When the
tilt of cemented surface of ADC is considered the aberration mainly coma will
be added. It increases with the increasing difference of the refractive index
of two glasses, i.e., it is maximum in two extreme wavelengths. And this
aberration is direct proportion to the distance from ADC to the focal surface.
Here we do not use formula to calculate it. In next section by using Zemax
software all aberrations, indicated by spot diagrams, will be given including
this aberration.
### 4.6 The compensation error of atmospheric dispersion
Due to the difference between the atmosphere and the glass dispersion, the
compensation error is existent. It is just like as secondary spectrum in an
achromatic optical system. Apparently it is direct proportion to the amount of
the compensated atmospheric dispersion.
## 5 The calculation and following discussion
The Zemax is used for the following calculations.
In this paper only the aberrations caused by the ADC are analyzed i.e. the
aberrations of original optical system are ignored.
We take the two outside surfaces of the ADC to be concentric with the focal
surface when this ADC is at the farthest position. In this situation if we
ignore the structure of strip in the whole FOV the images are the same. We
only need to calculate and discuss the case where the object is at the centre
of FOV but with different relations to the strips. For centre objects we take
two states: (1) the light beam area of object is at the middle of a lens–prism
strip; (2) a slit of two lens–prism strips is at the centre of the light beam
area of this object, i.e., the light of this object is half in one strip and
half in another strip. In this section all spot diagrams are calculated for
these two states. For state (1) we adjust the tilt angle of the cemented
surface of lens–prism strip to make the image centroids of $\lambda$ $=$ 380
$\mathrm{nm}$ and $\lambda$ $=$ 1000 $\mathrm{nm}$ coincide at $\mathrm{z}$
$=$ 65.4 ∘, i.e., to remove the atmospheric dispersion at these two extreme
wavelengths. Thus the tilt angle 6.89 ∘ mentioned above is obtained. In this
situation we found at wavelength about 500 $\mathrm{nm}$, the image centroid
is farthest from the co–centre of $\lambda$ $=$ 380 $\mathrm{nm}$ and
$\lambda$ = $1000$ $\mathrm{nm}$, the angular distance of them is the maximum
compensation error. Since all images should be in the one same focal surface
in calculation of state (2) the focal surface position of state (1) is used.
Since the radii of ADC are fixed when the distance between ADC and the focal
surface is less than 250 $\mathrm{mm}$, the two outside surfaces of ADC are
not concentric with the focal surface. In this situation image quality in
centre and off–axis of FOV are different. But the difference is small. We
still only calculate and discuss where the object is at the FOV centre and in
the above two states, i.e., (1) the light beam area of object is at the middle
of a lens–prism strip. For each decided position of ADC the zenith distance is
chosen to make the image centroids of $\lambda$ = 380 $\mathrm{nm}$ and
$\lambda$ $=$ 1000 $\mathrm{nm}$ coinciding, i.e., eliminate the atmospheric
dispersion at these two extreme wavelengths. (2) a slit of two lens–prism
strips is at the centre of the light beam area of this object. In this state
the focal surface position of state (1) is used.
These main calculation results are shown in Figure 4, 5, 6, 7, 8 and Table
3,4,5. It is clear that the predictions in 4.5 and 4.6 are proved and some
specific values are obtained: the largest monochromatic image at extreme
wavelength is about 0.18 $\mathrm{arcsec}$, mainly is chromatic aberration.
This figure shows no apparent change as the ADC moves along the optical axis.
From Figure 4, 5, 6, 7 and 8 we can find some coma induced by tilt cemented
surface and it reduces with the ADC’s moves towards the focal surface. The
maximum compensation error of atmospheric dispersion is 0.29 $\mathrm{arcsec}$
for the atmospheric dispersion 3.35 $\mathrm{arcsec}$, i.e., the compensation
error is about $1/12$ of the atmospheric dispersion. The total spread in the
waveband between $\lambda$ $=$ 380 $\mathrm{nm}$ and $\lambda$ = 1000
$\mathrm{nm}$ is about 0.4 $\mathrm{arcsec}$. In Figure 4, 5, 6, 7 and 8, the
same wavelength images are separated about 0.04 $\mathrm{arcsec}$ because the
light of this object is half in one strip and half in the other strip. These
two tilt cemented surfaces have different distances to focal surface. They
produce different dispersions, one bigger and the other smaller than the
average dispersion. By the way, the most part of one semi–circle of lens–prism
strip is LLF1, the other most part is PSK3, they produce the difference
chromatic aberration, so the sizes of chromatic spread are different. Even
though a 0.4 $\mathrm{arcsec}$ geometrical aberration is brought, it is worthy
to use such a simple ADC to correct 3.35 $\mathrm{arcsec}$ atmospheric
dispersion. Based on these analysis and calculation above, some extending
results could be obtained. For example if the thickness of ADC increases to 18
$\mathrm{mm}$, the chromatic aberration increase about 0.085 $\mathrm{arcsec}$
and the total spread from waveband $\lambda$ $=$ 380 $\mathrm{nm}$ to
$\lambda$ $=$ 1000 $\mathrm{nm}$ increases to about 0.45 $\mathrm{arcsec}$. In
this situation the wide of each strip can increase to 75 $\mathrm{mm}$ or even
100 $\mathrm{mm}$, thus ADC at the farthest position only $2/3$ or even $1/2$
objects will meet a slit.
## 6 Conclusion
In this paper a new kind of atmospheric dispersion corrector (ADC) is put
forward. It is a segmented lens which consists of many lens–prism strips. As
an example, such an ADC is discussed and designed for the LAMOST South. Its
linear diameter of FOV is 1.22 $\mathrm{m}$ and its f–ratio is 5. Although
this ADC’s diameter is 1.27 $\mathrm{m}$, its thickness is only 12
$\mathrm{mm}$. Thus the difficulty of obtaining big optical glass is avoided,
and the aberration caused by the ADC is small. When we move this segmented
lens along the optical axis, the different dispersions can be obtained. These
slits of ADC will produce about 2.5 $\mathrm{percent}$ average obstruction
loss of light. In the largest zenith distance each celestial object’s light
only meets one slit of this ADC, and at other zenith distance only a part
object light meet a slit. The effects of ADC slits on the diffraction energy
distribution are discussed. Since LAMOST is used for optical fibre
spectroscopic observation and the diameter of optical fibre is 1.6
$\mathrm{arcsec}$ (LAMOST South) and 3.3 $\mathrm{arcsec}$ (LAMOST North),
these effects are acceptable. From waveband $\lambda$ $=$ 380 $\mathrm{nm}$ to
$\lambda$ $=$ 1000 $\mathrm{nm}$ a total spread of aberration 0.4
$\mathrm{arcsec}$, which mainly is the compensation error of dispersions and
achromatic aberration, will be brought when we use such an ADC for
compensating 3.35 $\mathrm{arcsec}$ atmospheric dispersion. In this segmented
ADC the diffraction spot is enlarged and the loss of light obstruction by slit
is uneven. For these two reasons, such a segmented ADC is not suitable for the
diffraction-limited high resolution telescope and photometry. It is mainly
used for the optical fibre spectroscopic telescope. Since this ADC is thin,
the aberration caused is small, there is no difficulty for obtaining its
glasses, its thickness does not increase with the enlarging of FOV, and it has
almost the same image quality for the center and off–axis of FOV. Such an ADC
is especially suitable for very large FOV optical fibre spectroscopic
telescopes.
## Acknowledgments
Thanks to Professor Xiangqun Cui for her enthusiastic support and helpful
discussion, and to Professor Xiangyan Yuan for her helpful discussion and
assistance during calculation.
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Figure 1: The LAMOST and the position of the ADC in LAMOST
Figure 2: A sectional drawing of the ADC and focal surface. In this figure the
radii of ADC and focal surface have been reduced (i.e., more bended), and the
thickness of the ADC has been enlarged.
Figure 3: The figure (a) shows the light beam area of the object is at the
middle of a lens–prism strip and the figure (b) shows a slit of two lens–prism
strips is at the centre of the light beam area of the object.
Table 1: The structure parameters of the ADC Surface | Radius ($\mathrm{mm}$) | Separation ($\mathrm{mm}$) | Glass | Tilt Angle (∘)
---|---|---|---|---
1 | 20250 | | |
| | 6 | LLF1 |
2 | 20244 | | | 6.89
| | 6 | PSK3 |
3 | 20238 | | |
| | 238 | |
Focal Surface | 20000 | | |
Table 2: The refractive index of the Schott glass PSK3 and LLF1 $\lambda$ ($\mathrm{nm}$) | 380 | 441.8 | 500 | 587.6 | 656.3 | 1000
---|---|---|---|---|---|---
LLF1 | 1.574977 | 1.562383 | 1.555066 | 1.548138 | 1.544564 | 1.535632
PSK3 | 1.570682 | 1.562383 | 1.557313 | 1.552320 | 1.549650 | 1.542388
Figure 4: The spot diagram when the distance from the first surface of the ADC
to the focal surface is 250 $\mathrm{mm}$. The upper figure shows the spot
diagram when the light beam area of the object is at the middle of a
lens–prism strip and the lower figure shows when a slit of two lens–prism
strips is at the centre of the light beam area of the object. The length of
the scale line (the line beside the spot diagram) represents 40
$\mu\mathrm{m}$ (0.413 $\mathrm{arcsec}$). The different colour of the
diagrams in the picture represent different wavelength: blue, $\lambda$ 380
$\mathrm{nm}$, the spot diagram in the top of the upper figure; green,
$\lambda$ 500 $\mathrm{nm}$, the spot diagram in the lowermost of the upper
figure; red, $\lambda$ 1000 $\mathrm{nm}$, the spot diagram in the middle of
the upper figure.
Figure 5: The spot diagram when the distance from the first surface of ADC to
the focal surface is 200 $\mathrm{mm}$. The rest of explanation is the same as
in Figure 4.
Figure 6: The spot diagram when the distance from the first surface of ADC to
the focal surface is 150 $\mathrm{mm}$. The rest of explanation is the same as
in Figure 4.
Figure 7: The spot diagram when the distance from the first surface of ADC to
the focal surface is 100 $\mathrm{mm}$. The rest of explanation is the same as
in Figure 4.
Figure 8: The spot diagram when the distance from the first surface of ADC to
the focal surface is 50 $\mathrm{mm}$. The rest of explanation is the same as
in Figure 4.
Table 3: The distance from the first surface of ADC to the focal surface, corresponding to the object’s zenith distance at Paranal Observatory and the diameter of the light beam area of the object on the ADC. Distance from the first surface | | | | |
---|---|---|---|---|---
of ADC to the focal surface ($\mathrm{mm}$) | 250 | 200 | 150 | 100 | 50
The object’s zenith distance | | | | |
at Paranal Observatory (∘) | 65.4 | 59.9 | 51.9 | 39.6 | 20.6
The diameter of the light beam area of | | | | |
the object on the ADC ($\mathrm{mm}$) | 50 | 40 | 30 | 20 | 10
Table 4: The correction results of the ADC when the light beam area of the object is at the middle of a lens–prism strip. Largest spread of spot diagram and compensation error are in $\mathrm{arcsec}$. Distance from the first surface | | | | |
---|---|---|---|---|---
of ADC to focal surface ($\mathrm{mm}$) | 250 | 200 | 150 | 100 | 50
Largest spread of spot diagram $\lambda$ 380 $\mathrm{nm}$ | 0.180 | 0.180 | 0.182 | 0.183 | 0.185
Largest spread of spot diagram $\lambda$ 500 $\mathrm{nm}$ | 0.033 | 0.029 | 0.027 | 0.024 | 0.021
Largest spread of spot diagram $\lambda$ 1000 $\mathrm{nm}$ | 0.172 | 0.168 | 0.165 | 0.162 | 0.160
Largest spread of spot diagram | | | | |
of the whole waveband $\lambda$ 380 $\mathrm{nm}$ – 1000 $\mathrm{nm}$ | 0.401 | 0.343 | 0.279 | 0.215 | 0.185
Compensation error | 0.289 | 0.227 | 0.175 | 0.113 | 0.041
Table 5: The correction results of the ADC when a slit of two lens–prism strips is at the centre of the light beam area of the object. Largest spread of spot diagram and compensation error are in $\mathrm{arcsec}$. Distance from the first surface | | | | |
---|---|---|---|---|---
of ADC to focal surface ($\mathrm{mm}$) | 250 | 200 | 150 | 100 | 50
Largest spread of spot diagram $\lambda$ 380 $\mathrm{nm}$ | 0.192 | 0.193 | 0.194 | 0.196 | 0.200
Largest spread of spot diagram $\lambda$ 500 $\mathrm{nm}$ | 0.028 | 0.025 | 0.023 | 0.026 | 0.027
Largest spread of spot diagram $\lambda$ 1000 $\mathrm{nm}$ | 0.188 | 0.187 | 0.185 | 0.183 | 0.184
Largest spread of spot diagram | | | | |
of the whole waveband $\lambda$ 380 $\mathrm{nm}$ – 1000 $\mathrm{nm}$ | 0.397 | 0.339 | 0.276 | 0.212 | 0.200
Compensation error | 0.289 | 0.227 | 0.175 | 0.108 | 0.046
|
arxiv-papers
| 2011-10-24T23:36:20 |
2024-09-04T02:49:23.581458
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ding-qiang Su, Peng Jia, Genrong Liu",
"submitter": "Jia Peng",
"url": "https://arxiv.org/abs/1110.5379"
}
|
1110.5436
|
# On the origin of galactic cosmic rays
Ya. N. Istomin istomin@lpi.ru P. N. Lebedev Physical Institute, Leninsky
Prospect 53, Moscow, 119991 Russia
###### Abstract
It is shown that the relativistic jet, emitted from the center of the Galaxy
during its activity, possessed power and energy spectrum of accelerated
protons sufficient to explain the current cosmic rays distribution in the
Galaxy. Proton acceleration takes place on the light cylinder surface formed
by the rotation of a massive black hole carring into rotation the radial
magnetic field and the magnetosphere. Observed in gamma, x-ray and radio bands
bubbles above and below the galactic plane can be remnants of this bipolar
get. The size of the bubble defines the time of the jet’s start, $\simeq
2.4\cdot 10^{7}$ years ago. The jet worked more than $10^{7}$ years, but less
than $2.4\cdot 10^{7}$ years.
###### keywords:
cosmic rays , galactic center , relativistic jets
###### PACS:
98.70.Sa , 98.62.Nx
††journal: Astroparticle Physics
,
## 1 Introduction
The traditional point of view on the origin of cosmic rays in the Galaxy is
the concept of acceleration of charged particles at fronts of shocks from
supernova explosions. Arguments in favour of this mechanism are sufficient
mechanical energy that is released when the supernova explodes, as well as
universal index of power law spectrum of particles, accelerated by strong
shocks. Total power of cosmic ray sources in order to maintain their observed
density of energy is $5\cdot 10^{40}erg/s$, which equals approximately 15% of
the kinetic energy of supernova explosions. When the gas compression in a
shock is equal to 4, the index of the power law energy spectrum of accelerated
particles is equal to -2, $N(E)\propto E^{-2}$. It is in a good agreement with
observed cosmic ray spectrum at energies $E<3\cdot 10^{15}eV$. Beginning from
the first works by Krymskii, 1977; Bell, 1978; Blandford & Ostriker, 1978, who
proposed the mechanism of acceleration of charged particles on fronts of
shocks propagating in the turbulent environment, much progress has been made
to explain the observed characteristics of galactic cosmic rays in the belief
that they are accelerated at the front of shocks.
On the other hand, there is no objections to generate galactic cosmic rays in
one source in the Galaxy (Ptuskin & Khazan, 1981). This potential source can
be the center of the Galaxy, which is the massive black hole of $M\simeq
4\cdot 10^{6}M_{\odot}$ mass. And while the luminosity of Sgr A* is small now,
it is only $10^{36}$ erg/s, in the past the center could be much brighter
because its Eddington luminosity equals $L_{Edd}=5.2\cdot 10^{44}$ erg/s. On
the past activity of the center of the Galaxy shows newly discovered by Fermi
Gamma-ray Space Telescope above and below the galactic plane big bubbles
emitting gamma radiation in the range of $0.1-1000$ GeV (Su et al., 2010).
Such formations was previously observed in the x-ray range $(1.5-2)$ KeV by
ROSAT All-Sky Survey (Snowden et al., 1997) and in the microwave range
$(20-40)$ GHz by WMAP (Finkbeiner, 2004). Estimated energy stored in bubbles
is of $10^{54}-10^{55}$ erg (Sofue, 2000). As we will see below, bubbles of a
relativistic gas could be formed by the jet, emitted from surroundings of the
massive black hole. Here we provide an alternative mechanism of origin
galactic cosmic rays, in which the nucleus of the Galaxy in the active phase
injected the relativistic jet, which was the source of cosmic rays.
In the following sections we will calculate the power of the jet and the
energy spectrum of protons in the relativistic jet, as well as describe the
remnants of the relativistic jet injected from the center of the Galaxy,
having the form of bubbles above and below the galactic plane. In the final
section we will discuss correspondence of the scenario of the galactic cosmic
rays origin provided with cosmic rays characteristics observed.
## 2 Relativistic jet
Sources of energy of active galactic nuclei are the accretion on a massive
black hole, in which the gravitational energy of a falling gas transforms into
radiation and heat, as well as the rotation of a black hole. Mechanism of
extraction of energy and angular momentum from the black hole is called as the
mechanism of Blandford-Znajek (1977). The energy of a rotating black hole is a
large value, $E_{rot}=Mr_{H}^{2}\Omega_{H}^{2}/2=a^{2}Mc^{2}/8=2.25\cdot
10^{53}a^{2}(M/M_{\odot})$ erg. For the Galaxy $E_{rot}\simeq 9\cdot
10^{59}a^{2}$ erg. Here we have introduced the dimensionless parameter of $a$,
describing rotation of the black hole, $a=Jc/M^{2}G,\,a<1$. $J$ is the black
hole angular momentum, $G$ is the gravitational constant. Angular velocity of
rotation of a black hole is proportional to the value of $a$,
$\Omega_{H}=ac/2r_{H}$, $r_{H}$ is the gravitational radius of not rotating
black hole $(a=0)$, $r_{H}=2MG/c^{2}$. Energy extraction is possible when
there is a poloidal magnetic field $B$ near the black hole horizon. In this
case, rotating black hole acts as a Dynamo machine, creating a voltage
$U=f_{H}\Omega_{H}/2\pi c$ (Landau & Lifshits, 1984). The value of $f_{H}$ is
the flux of the poloidal magnetic field reaching the horizon of a black hole,
$f_{H}\simeq\pi Br_{H}^{2}$. Voltage $U$ generates the electric current
$I=U/(R+R_{H})$, which on the one hand is closed on the horizon of a black
hole that has the resistance $R_{H}=4\pi/c\simeq 377$ ohm. Resistance of the
outer part of the current loop is $R$. Thus, the power, extracting from a
rotating black hole, is
$L=RI^{2}=U^{2}R/(R+R_{H})^{2}=a^{2}B^{2}r_{H}^{2}R/16(R+R_{H})^{2}$, and
reaches the maximum $L_{m}$ at $R=R_{H}$, $L_{m}=a^{2}B^{2}r_{H}^{2}c/256\pi$.
The value of $L$ is proportional to the energy of the poloidal magnetic field
near a black hole and can reach the Eddington luminosity at sufficiently large
magnetic fields $B\simeq 10^{6}a^{-1}$ Gauss in the center of the Galaxy. This
field is accumulated near the horizon of a black hole in the process of
accretion of a disk matter in which the magnetic field is frozen. Thus, for
the effective work of the mechanism of Blandford-Znajek an accretion disk
around a massive black hole is required, not as a source of the energy, but as
the agent bearing the magnetic field to a black hole. In addition, the
electric current $I$ flows in the disk, this is the part of the current loop:
in the disk, in the black hole horizon, then in the jet, closing at large
distances in the interstellar matter (see Figure 1).
Figure 1: Configuration of the magnetic field and electric currents in the jet
and in the disk.
In the disk, in addition to the radial electric current $I_{\rho}=\int
j_{\rho}ds=-I$, stronger toroidal current $j_{\phi}$, $j_{\phi}\simeq
10^{2}j_{\rho}$, flows also (Istomin & Sol, 2011), it generates the radial
magnetic field $B$.
The rotating black hole brings into rotation the radial magnetic field in the
magnetosphere of a black hole above and below the disk. Angular velocity of
rotation of the magnetic field lines, the same as rotation of the
magnetospheric plasma, $\Omega_{F}$, is proportional to the angular velocity
of rotation of the black hole $\Omega_{H}$, $\Omega_{F}=\Omega_{H}R/(R+R_{H})$
(Thorne et al., 1986). Plasma rotation is the drift motion in crossed radial
magnetic field and electric field of plasma polarization. Thus, there appears
so-called the light cylinder surface in the black hole magnetosphere, where
the magnitude of the electric field is compared with that of the magnetic
field and the rotation velocity approaches the speed of light $c$. The radius
of the light surface is $r_{L}=c/\Omega_{F}=2a^{-1}r_{H}(R+R_{H})/R>r_{H}$. On
the light surface charged particles get considerable energy and angular
momentum of rotation. Energy density of particles on the light surface is
compared with the energy density of the electromagnetic field
$(E_{L}^{2}+B_{L}^{2})/8\pi=B_{L}^{2}/4\pi$ (Istomin, 2010). All energy passes
to protons, $\gamma={\cal E}_{p}/m_{p}c^{2}>>1$. Electrons are practically not
accelerated due to large synchrotron losses in a strong magnetic field
(Istomin & Sol, 2009). Energetic protons, accelerated near the light surface,
and whose energy is mainly in the azimuthal motion, create the jet. Jet’s
power is $L_{J}=B^{2}r_{H}^{2}c(\omega_{cH}r_{H}/c)^{-1/4}/2$ (Istomin & Sol,
2011). Here $\omega_{cH}$ is the non relativistic cyclotron frequency of
protons in the magnetic field near the black hole, $\omega_{cH}=eB/m_{p}c$.
Jet arises when the power extracted from the rotating black hole $L$ becomes
greater than the jet power $L_{J}$, $L>L_{J}$. This imposes a limitation on
the value of the magnetic field
$\frac{\omega_{cH}r_{H}}{c}\geq(128\pi)^{4}a^{-8}\left[\frac{(R+R_{H})^{2}}{4RR_{H}}\right]^{4}.$
(1)
For $R=R_{H}$ it gives
$B\geq 2.7\cdot 10^{11}a^{-8}\frac{M_{\odot}}{M}{\rm Gauss}.$ (2)
For the center of the Galaxy, the magnetic field must satisfy the condition
$B\geq 6.75\cdot 10^{4}a^{-8}$ Gauss. We see that to generate a jet, less
massive black holes should have a stronger poloidal magnetic field near the
horizon, $B\propto M^{-1}$. In addition, rotation must be fast, close to the
critical value of $a\simeq 1$, because of strong dependence of the expression
(2) on $a$. It should also be noted that the resistance of the external
current loop $R$, on which the jet power $L_{J}$ depends, is not the ohmic one
$R_{c}$, which turns out to be small, $R_{c}<<R_{H}$ (Istomin & Sol, 2011),
but is the effective resistance $R_{J}$, which can be attributed to the jet,
receiving energy from the rotating black hole. If $L=L_{J}$ the resistance of
the jet is $R_{J}=R_{H}$. Under the equality in the expression (1) when a
rotating black hole can begin to generate a jet, the expression for the jet’s
power becomes universal
$L_{J}=2^{48}\pi^{7}m_{p}c^{2}\left(\frac{m_{p}c^{3}}{e^{2}}\right)a^{-14}=2.5\cdot
10^{41}a^{-14}\,erg/s.$ (3)
All jet energy are in the energy of protons, and, as we can see, is sufficient
for the production of galactic cosmic rays.
## 3 Energy spectrum of fast particles in jet
Istomin and Sol (2009) had shown that on the light surface, produced by the
rotating radial magnetic field, which is carried into rotation by a black
hole, protons gain considerable energy. The Lorentz factor $\gamma$ becomes
equal to $\gamma=(\gamma_{0}\gamma_{i})^{1/2}$. The value of $\gamma_{i}$ is
the Lorentz factor of particles in the magnetosphere of a black hole before
crossing the light surface. And the value of $\gamma_{0}$ is the maximum of
the Lorentz factor, which could be achieved by a particle in this acceleration
mechanism, $\gamma_{0}=\omega_{cL}/\Omega_{F}$. Here $\omega_{cL}$ is the
cyclotron frequency of rotation of protons in the poloidal magnetic field near
the light surface. When $\gamma=\gamma_{0}$ the cyclotron radius of a proton
is compared with the radius of the light surface. For non relativistic
particles of the black hole magnetosphere, $\gamma_{i}\simeq 1$, the Lorentz
factor of accelerated particles is equal to
$\gamma=\gamma_{0}^{1/2}=(\omega_{cL}/\Omega_{F})^{1/2}$. Crossing the light
surface at different distances $z$ from the accretion disk plane, particles
gain different energies, since the magnetic field decreases with distance from
the black hole. For a radial magnetic field $B\propto(z^{2}+r_{L}^{2})^{-1}$.
Thus, $\gamma=\gamma_{m}(1+z^{2}/r_{L}^{2})^{-1/2}$, where $\gamma_{m}$ is the
maximal Lorentz factor of accelerated particles near the accretion disk.
Accelerated protons of the jet are collected from various parts of the light
cylinder surface of $r_{L}$ radius, but located at different distances $z$.
Therefore, the number of particles is $dN\propto ndz$, where $n$ is the
density of protons in the magnetosphere near the light surface. Connecting
values $z$ and $\gamma$, we get
$\frac{dz}{r_{L}}=-\frac{\gamma_{m}d\gamma}{\gamma^{2}(1-\gamma^{2}/\gamma_{m}^{2})^{1/2}}.$
Considering that the vertical size of the magnetosphere is larger than the
light surface radius, the density $n$ can be taken as constant. As a result we
get the distribution function of relativistic protons in the jet,
$F(\gamma)=dN/d\gamma$,
$F(\gamma)=const\cdot\gamma^{-2}(1-\gamma^{2}/\gamma_{m}^{2})^{-1/2},\,\gamma<\gamma_{m}.$
(4)
We see that in the range $\gamma<<\gamma_{m}$ the spectrum of relativistic
protons is the power law spectrum with the index -2. This spectrum is observed
in gamma radiation from bubbles above and below the Galactic plane by Fermi
Gamma-ray Space Telescope (Su et al., 2010). Considering that the gamma
radiation occurs due to collisions of relativistic protons with interstellar
gas through meson production (Crocker & Aharonian, 2011), and the distribution
of photons is similar to the distribution of protons, one can conclude that
the jet from the center of the Galaxy actually existed, and bubbles are filled
with relativistic protons of the jet. The value of $\gamma_{m}$ is (Istomin &
Sol, 2011)
$\gamma_{m}=\left(\frac{\omega_{cH}r_{H}}{c}\right)^{1/2},$ (5)
and for $R=R_{H}$ equals (see the expression (1))
$\gamma_{m}=(128\pi)^{2}a^{-4}=1.6\cdot 10^{5}a^{-4}.$
The spectrum of protons (4) breaks at $\gamma=\gamma_{m}$ and has there the
root singularity (integrable) that is smoothed considering the thermal
dispersion of particles in the magnetosphere of the black hole,
$\Delta\gamma_{i}={\cal E}_{p}/m_{p}c^{2}$. The distribution (4) is shown on
Figure 2. The value of $\gamma_{m}$ is chosen to be equal to the Lorentz
factor of the break in the observed spectrum of cosmic rays at the energy
$E=3\cdot 10^{15}$ GeV, $\gamma_{m}=3.2\cdot 10^{6}$. This corresponds to the
rotation parameter $a=0.47$. The power of the jet (3) is $L_{J}\simeq 8.9\cdot
10^{45}$ erg/s. That is in agreement with estimated from observations powers
of jets ejected from active galactic nuclei–$10^{45}-10^{46}$ erg/s (Mao-Li et
al., 2008).
Figure 2: Distribution function of relativistic protons in the jet,
$\gamma<\gamma_{m}$. The value of $\gamma_{m}$ corresponds to the break in the
spectrum of the cosmic ray. The slope is equal to -2.
Relativistic protons with the spectrum (4) are formed from thermal particles
of the black hole magnetosphere, $\gamma_{i}\simeq 1$. But in addition to
thermal particles in the magnetosphere there can exist accelerated protons.
The turbulent motion of the accreting disk matter in the presence of the
frozen magnetic field leads to acceleration of particles, which have the power
law energy spectrum,
$f(\gamma)=const\cdot\gamma^{-\beta},\,\gamma<\gamma_{1},\,\beta\simeq 1$
(Istomin & Sol, 2009). The disk must be turbulent to provide for the abnormal
gas transport. Getting onto the light surface, accelerated protons are
converted to more energetic, $\gamma\rightarrow(\gamma_{0}\gamma)^{1/2}$.
Their distribution function becomes equal (Istomin & Sol 2009)
$f^{\prime}(\gamma)=2const\cdot\gamma_{0}^{\beta-1}\gamma^{-2\beta+1},\,\gamma_{0}^{1/2}<\gamma<(\gamma_{0}\gamma_{1})^{1/2}$.
Thus, there is another component of jet relativistic protons, their number is
equal to
$N\propto\int_{0}^{\infty}dz\int_{\gamma_{0}^{1/2}}^{(\gamma_{0}\gamma_{1})^{1/2}}\gamma_{0}^{\beta-1}\gamma^{-2\beta+1}d\gamma,\,\gamma_{0}=\gamma_{m}^{2}(1+z^{2}/r_{L}^{2})^{-1}.$
(6)
Transforming the integration area in (6), we get
$\displaystyle
N\propto\int_{1}^{\gamma_{m}}\gamma^{-2\beta+1}d\gamma\int_{(\gamma_{m}^{2}/\gamma^{2}-1)^{1/2}}^{(\gamma_{m}^{2}\gamma_{1}/\gamma^{2}-1)^{1/2}}\left(1+\frac{z^{2}}{r_{L}^{2}}\right)^{1-\beta}\frac{dz}{r_{L}}+$
$\displaystyle\int_{\gamma_{m}}^{\gamma_{m}\gamma_{1}^{1/2}}\gamma^{-2\beta+1}d\gamma\int_{0}^{(\gamma_{m}^{2}\gamma_{1}/\gamma^{2}-1)^{1/2}}\left(1+\frac{z^{2}}{r_{L}^{2}}\right)^{1-\beta}\frac{dz}{r_{L}}.$
(7)
The first term in Eq. (7) corresponds to relativistic protons with energies
$\gamma<\gamma_{m}$ similar to protons (4), accelerated from the thermal gas.
Their distribution function equals
$F(\gamma)=const\cdot\gamma^{-2\beta+1}\int_{(\gamma_{m}^{2}/\gamma^{2}-1)^{1/2}}^{(\gamma_{m}^{2}\gamma_{1}/\gamma^{2}-1)^{1/2}}\left(1+\frac{z^{2}}{r_{L}^{2}}\right)^{1-\beta}\frac{dz}{r_{L}}.$
In the energy range $\gamma<<\gamma_{m}$ this distribution has the same power
law spectrum (4), $F(\gamma)\propto\gamma^{-2}$. But since the number of
accelerated particles in the magnetosphere of the black hole is much less than
that of thermal particles, the contribution of these particles into the total
distribution at $\gamma<\gamma_{m}$ can be neglected. The second term in Eq.
(7) describes the distribution of relativistic protons at
$\gamma<\gamma_{m}<\gamma_{m}\gamma_{1}^{1/2}$
$\displaystyle
F(\gamma)=const\cdot\gamma^{-2\beta+1}\int_{0}^{(\gamma_{m}^{2}\gamma_{1}/\gamma^{2}-1)^{1/2}}\left(1+\frac{z^{2}}{r_{L}^{2}}\right)^{1-\beta}\frac{dz}{r_{L}}=$
$\displaystyle\frac{1}{2}const\cdot\gamma^{-2\beta+1}\left[B(1,\beta-3/2,1/2)-B(\gamma^{2}/(\gamma_{m}^{2}\gamma_{1}),\beta-3/2,1/2)\right].$
(8)
Here $B(x,a,b)=\int_{0}^{x}t^{a-1}(1-t)^{b-1}dt$ is the incomplete Beta
function, $B(1,\beta-3/2,1/2)=\pi^{1/2}\Gamma(\beta-3/2)/\Gamma(\beta-1)$,
$\Gamma(x)$ is the Gamma function. The distribution (8) with
$\beta=1.7,\,\gamma_{m}=3.2\cdot 10^{6}$ and $\gamma_{1}=10^{5}$ is shown on
Figure 3. For energies $\gamma<\gamma_{m}\gamma_{1}^{1/2}$ the distribution of
relativistic protons is the power law with index $-(2\beta-1)$. At
$\gamma\simeq\gamma_{m}\gamma_{1}^{1/2}$ the distribution falls down, for
$\gamma_{m}=3.2\cdot 10^{6}$ and $\gamma_{1}=10^{5}$ the maximum energy is
$\simeq 10^{18}$ eV.
Figure 3: The distribution function of relativistic protons in the jet,
$\gamma>\gamma_{m}$. The value of $\gamma_{m}$ corresponds to the break in the
cosmic ray spectrum. The spectral index equals -2.4. The maximum energy is
$10^{18}$ eV.
We have chosen the value of $\beta=1.7$ from the fact that indices of spectrum
of cosmic rays before and after the break at energy $3\cdot 10^{15}$ eV differ
on the value of 0.4 – the spectrum becomes more soft with the index $-3.1$.
The same difference in indices must be in the source of cosmic rays, which in
our case is the relativistic jet. At $\beta=1.7$ the index of the spectrum (8)
equals -2.4, while the index of the spectrum (4) is equal to -2. It should be
noted that the distribution function of relativistic protons (4) at energies
$\gamma<\gamma_{m}$, and (8) at energies $\gamma>\gamma_{m}$, is continuous,
i.e. $F(\gamma=\gamma_{m}-0)=F(\gamma=\gamma_{m}+0)$. This is because the
acceleration on the light surface undergo protons of the black hole
magnetosphere with a unique spectrum – thermal at low energies, turning into
the tail of fast particles up to the energy $\gamma=\gamma_{1}$. If their
energy distribution function is $F_{i}({\cal E}_{i})$, then the distribution
of particles, accelerated on the light surface, have also the continuous
distribution $F({\cal E})=F_{i}[{\cal E}_{i}({\cal E})]d{\cal E}_{i}/d{\cal
E,}\,{\cal E}_{i}={\cal E}^{2}/\gamma_{0}m_{p}c^{2}$.
Here and hereafter, we talk about protons, bearing in mind that they are
’heavy’ particles, unlike electrons. Nuclei of $m$ mass and $Ze$ charge will
be accelerated effectively also on the light surface. As we saw above, the
efficiency of acceleration depends on the value of
$\gamma_{0}^{1/2}=(\omega_{c}/\Omega_{F})^{1/2}$. Thus, nuclei will receive
energy per nucleon $(Zm_{p}/m)^{1/2}\simeq 2^{-1/2}$ times less than protons.
## 4 Jet’s remnants
The center of the Galaxy, being active, created a relativistic jet, particles
of which spread in the Galaxy, forming isotropic background of cosmic rays. If
one consider that the angular momentum of the massive black hole in the center
of the Galaxy coincides with that of the Galaxy, than the direction of the jet
propagation is perpendicular to the plane of the Galaxy. The jet length can be
quite large, so jet in M87 extends $\simeq 2$ kpc. Therefore, above and below
the Galactic plane (if we have two almost symmetrical jets) one can see traces
of the jet.
Let us consider a simple diffusive model of propagation of relativistic
particles, whose source is located in the center of the Galaxy $({\bf r}=0)$,
in an interstellar medium
$\frac{\partial N}{\partial t}+{\bf u}\nabla N-\nabla\hat{D}\nabla
N=Q(t)\delta({\bf r}).$ (9)
Here ${\bf u}$ is the velocity of the interstellar matter. We consider the
region outside the stellar galactic disk, then the velocity ${\bf u}$ is the
speed of the galactic wind, which is along the coordinates $z$ which is
perpendicular to the plane of the Galaxy. The value of $\hat{D}$ is the
diffusion coefficient, which can be anisotropic. Diffusion depends on the
intensity of the magnetic field $B$, falling exponentially with the distance
$z$ from the galactic plane, $B=B_{0}\exp(-z/z_{1}),\,z_{1}\simeq 2$ kpc.
Diffusion of charged particles is determined by their motion in the magnetic
field, and the diffusion coefficient is inversely proportional to the magnetic
field strength, $D\propto B^{-\alpha}$. So, for the most strong Bohm diffusion
$D=cr_{c}/3$, $r_{c}$ is the proton cyclotron radius, $\alpha=1$. Thus, the
particle diffusion increases exponentially with the coordinate $z$,
$D=D_{0}\exp(z/z_{0}),\,z_{0}=z_{1}/\alpha$. Such a strong dependence of the
diffusion on the coordinate $z$ leads to effective non diffusion expansion of
particles along this coordinate and decreasing its density. To take into
account this effect we insert the function $\varphi=N\exp(z/z_{0})$. Eq. (9)
becomes
$\displaystyle\exp\left(-\frac{z}{z_{0}}\right)\frac{\partial\varphi}{\partial
t}+u\exp\left(-\frac{z}{z_{0}}\right)\left(\frac{\partial\varphi}{\partial
z}-\frac{\varphi}{z_{0}}\right)+\frac{D_{0}}{z_{0}}\frac{\partial\varphi}{\partial
z}-$ $\displaystyle
D_{0}\left[\frac{\kappa}{\rho}\frac{\partial}{\partial\rho}\left(\rho\frac{\partial\varphi}{\partial\rho}\right)+\frac{\partial^{2}\varphi}{\partial
z^{2}}\right]=Q(t)\frac{\delta(\rho)\delta(z)}{2\pi\rho}.$ (10)
We consider the distribution of particles as azimuthal symmetric depending on
the distance $z$ and the cylindrical radius $\rho$. The value of $\kappa$ is
the ratio of the transverse diffusion coefficient $D_{\perp}$, perpendicular
to $z$, to the longitudinal one $D_{\parallel}$, along $z$,
$\kappa=D_{\perp}/D_{\parallel}$. We see that in the equation of particle
motion there appears the effective velocity along $z$, $u_{0}=D_{0}/z_{0}$. It
arises as a result of the exponential growth of the particle diffusion over
$z$. For characteristic values of $D_{0}=5\cdot 10^{28}$ $cm^{2}$/s and
$z_{0}=2$ kpc the velocity $u_{0}$ is of $u_{0}\simeq 10^{2}$ km/s. This speed
is much larger than the speed of the galactic wind $u$, which at distances of
several kpc from the galactic plane is less than $30$ km/s (Ptuskin, 2007).
Although the wind velocity increases with distance $z$, the exponential factor
$\exp(-z/z_{0})$ in Eq. (10), allows us to ignore the velocity of the galactic
wind in comparison with the velocity $u_{0}$. Values of $D_{0}$ and $z_{0}$
specify scales of length and time, so it is convenient to go to the
dimensionless variables in Eq. (10)
$z^{\prime}=z/z_{0},\,\rho^{\prime}=\rho/z_{0},\,t^{\prime}=t(D_{0}/z_{0}^{2}),\,\varphi^{\prime}=\varphi
z_{0}^{3}$. Eq. (10) becomes (primes are omitted)
$\exp(-z)\frac{\partial\varphi}{\partial t}+\frac{\partial\varphi}{\partial
z}-\frac{\partial^{2}\varphi}{\partial
z^{2}}-\frac{\kappa}{\rho}\frac{\partial}{\partial\rho}\left(\rho\frac{\partial\varphi}{\partial\rho}\right)=Q(t)\frac{\delta(\rho)\delta(z)}{2\pi\rho}.$
(11)
At $z>1$ the propagation (first derivative over $z$) predominates over the
diffusion (second derivative), and Eq. (11) makes easy
$\varphi=\frac{Q[t-(1-e^{-z})]}{4\pi\kappa
z}\exp\left(-\frac{\rho^{2}}{4\kappa z}\right).$
From this solution one can see that to the point $z$ come particles, which
were emitted by the source at the retarded time $t^{\prime}=t-(1-e^{z})$. If
the time of the jet’s start is $t=0$, particles will lift to the maximum
height of $z_{m}=-\ln(1-t),\,t<1$. Formally at $t=1$ particles come to
infinity during the finite time, that is impossible. The velocity limitation
implies the condition $z_{m}<(cz_{0}/D_{0})t$, $c$ is the speed of the light,
which is not difficult to hold because $cz_{0}/D_{0}\simeq 3.6\cdot 10^{3}$.
Knowing the value of $z_{m}$ from observation one can estimate the time
$t_{1}$ when the power source of cosmic rays in the center of the Galaxy
begins to work, i.e. when the jet starts, $t_{1}=1-\exp(z_{m})$. In
dimensional units $t_{1}=t_{0}[1-\exp(z_{m}/z_{0})],\,t_{0}=z_{0}^{2}/D_{0}$.
When $z_{0}=2$ kpc and $D_{0}=5\cdot 10^{28}\,cm^{2}/s$ the time $t_{0}$ is
$t_{0}=7.6\cdot 10^{14}\,s=2.4\cdot 10^{7}$ yr. The gamma radiation observed
above and below the galactic plane extends to the height of about 8 kpc (Su et
al., 2010), i.e. $z_{m}\simeq 4$. This means that in fact $t_{1}=t_{0}$. In
addition, one can find the time $t_{2}$ when the source turns off. If the jet
worked a short time, all particles would have lifted in height at $z\geq 1$,
and we would see their absence near the galactic plane. Since this is not
observed in bubbles, then $t_{2}<t_{0}(1-1/e)\simeq 0.6t_{0}$.
Knowing the solution for $\varphi$ at $z>1$ we can find the density of
relativistic particles $N(t,z,\rho)$ in the same region
$N=\frac{Q[t-(1-e^{-z})]}{4\pi\kappa
z_{0}^{3}}\exp\left[-\left(\frac{\rho^{2}}{4\kappa z}+z+\ln(z)\right)\right].$
(12)
The distribution $N(z,\rho)$ (12) for the permanent source, $Q=const(t)$, is
shown on Figure 4.
Figure 4: The density distribution of relativistic particles (12) above the
galactic plane in dimensionless coordinates: $z$, the distance from the plane
of the Galaxy, and $\rho$, the cylindrical radius.
We also draw levels of the constant density $N,\,\rho^{2}/4\kappa
z+z+ln(z)=const$, Figure 5.
Figure 5: Levels of the constant density of relativistic particles specified
by the distribution (12).
The value of $\kappa$, the anisotropy of diffusion, is chosen to be equal to
$\kappa=0.14$ from those considerations that the observed ratio of bubble’s
scales $z/\rho\simeq 8kpc/3kpc=8/3$ would be consistent with the forms of the
constant density profiles painted on Figure 5. It seems that such value of
$\kappa$ indicates that the magnetic field in the halo of the Galaxy near the
center is mostly vertical. And this is natural because for the cylindrical
symmetry radial and azimuthal magnetic field components should approach zero
on the axis $\rho=0$.
## 5 Discussion
We have shown that whereas in the past the center of the Galaxy was active and
radiated the jet, its energy and composition are sufficient to explain the
origin of cosmic rays in the Galaxy. Bubbles of a relativistic gas observed in
gamma, x-ray and radio bands above and below the galactic plane, apparently,
are remnants of the bipolar jet emitted from the vicinity of the massive black
hole in the center of the Galaxy. The vertical size of the bubble $z\simeq 8$
kpc permits us to estimate the time of switching on of the jet, $t_{1}\simeq
t_{0}=2.4\cdot 10^{7}$ years ago. We can also estimate the lower limit of the
jet’s work, $\Delta t=t_{1}-t_{2}>0.4t_{0}\simeq 10^{7}$ yr. During the time
$\Delta t$ the jet got the energy $L_{J}\Delta t\simeq 8.9\cdot 10^{45}$ erg/s
$\times$ $3\cdot 10^{14}$ s $\simeq 2.7\cdot 10^{60}$ erg, that is slightly
larger than the energy stored in the black hole rotation $\simeq 10^{60}$ erg.
However, if we estimate the mass of the accreted matter, absorbed by the black
hole at the same time, it can reach a large part of the mass of the black
hole, $\Delta M\simeq{\dot{M}}_{Edd}\Delta t=9.2\cdot
10^{-2}M_{\odot}\,yr^{-1}\times 10^{7}\,yr\simeq 10^{6}M_{\odot}$, $\Delta
M\simeq M/4$. Transmitted to the black hole by the accreted matter, the
angular momentum $\Delta J$ can even exceed its initial value (Istomin, 2004).
The jet’s energy $\simeq 10^{60}$ erg is enough to fill by cosmic rays as the
disk ($10^{55}$ erg), as the halo ($10^{57}-10^{58}$ erg) of the Galaxy.
Filling of the disk of the Galaxy by relativistic particles is described by
the same Eq. (11), but there should be $|z|<1$. Therefore, we can ignore the
dependence of the diffusion coefficient on the distances $z$, and solve the
pure diffusion equation
$\frac{\partial N}{\partial t}-D_{\parallel}\frac{\partial^{2}N}{\partial
z^{2}}-D_{\perp}\frac{1}{\rho}\frac{\partial}{\partial\rho}\left(\rho\frac{\partial
N}{\partial\rho}\right)=Q(t)\frac{\delta(\rho)\delta(z)}{2\pi\rho}.$
Here we are interested in distribution of particles in the disk on the
transverse distances $\rho$, so we average this equation over $z$ and get
$\frac{\partial{\bar{N}}}{\partial
t}-\frac{D_{g}}{\rho}\frac{\partial}{\partial\rho}\left(\rho\frac{\partial{\bar{N}}}{\partial\rho}\right)=Q(t)\frac{\delta(\rho)}{2\pi\rho},$
(13)
where the value of $D_{\perp}=D_{g}$ is the diffusion coefficient of cosmic
rays in the galactic disk. The solution is
${\bar{N}}=\frac{1}{4\pi
D_{g}}\int_{0}^{t_{1}}\frac{1}{\tau}\exp\left(-\frac{\rho^{2}}{4D_{g}\tau}\right)Q(t_{1}-\tau)d\tau.$
The time $t_{1}$ is the start time of the jet. If the jet had worked with
constant power $Q$ and switched off at the time $t_{2}$, then the density
distribution of cosmic rays in the disk is
${\bar{N}}=\frac{Q}{4\pi
D_{g}}\int_{\rho^{2}/4D_{g}t_{1}}^{\rho^{2}/4D_{g}(t_{1}-t_{2})}x^{-1}e^{-x}dx.$
(14)
The solution (14) shows that if $\rho^{2}/4D_{g}t_{1}<1$ then the distribution
of the density over the radius $\rho$ is almost uniform. For
$\rho^{2}/4D_{g}(t_{1}-t_{2})>1$ it is logarithmic,
${\bar{N}}\propto-\ln(\rho^{2}/4D_{g}t_{1})$, and for
$\rho^{2}/4D_{g}(t_{1}-t_{2})<1$ it is constant,
$N\propto\ln(t_{1}/(t_{1}-t_{2}))$. The diffusion coefficient of cosmic rays
in the disk equals $D_{g}=2.2\cdot 10^{28}\gamma^{0.6}\,cm^{2}/s$ (Ptuskin,
2007). The condition $R^{2}/4D_{g}<t_{0},\,R\simeq 15$ kpc is the disk radius,
imposes a lower limit on the energy of protons, homogeneously filling the
galactic disc, $\gamma>400$. Apart the diffusion particles can move freely
along the regular magnetic field of spiral arms of the galactic disk. The
necessary for filling velocity $R/t_{0}\simeq 6\cdot 10^{7}$ cm/s $=2\cdot
10^{-3}$ c does not contradict the observed anisotropy of cosmic rays
$\delta$, $\delta\simeq 10^{-3}$. Since the dependence of the distribution
(14) over the energy is determined not only by the energy spectrum of the
source $Q(\gamma)$ but also the dependence of the diffusion coefficient
$D_{g}\propto\gamma^{0.6}$ over the energy, the spectrum of particles in the
disk will be softer than that in the source. Thus, the discussed mechanism of
origin of galactic cosmic rays by the jet, emitted from the center of the
Galaxy, satisfactorily explains the observed spectrum, the index -2.7 (-2.6
for the jet) before the break, and the index -3.1 (-3.0 for the jet) after the
break.
Cosmic rays, filling simultaneously the galactic disk and the halo, flow out
from the Galaxy. Their lifetime $\tau$ is determined by as the energy loss
time $\tau_{E}$, as the time of diffusion leakage from the Galaxy after the
source of relativistic particles stopped, $\tau_{D}$. The time $\tau_{E}$ is
estimated as $\tau_{E}\simeq 3\cdot 10^{7}$ yr (Strong & Moskalenko, 1998). It
is larger than the time of the jet’s start $t_{1}$, i.e. beginning of the
filling of the Galaxy by cosmic rays, $\tau_{E}>t_{1}$. The diffusion time
$\tau_{D}=r^{2}/4D$ is determined by the distance $r$ that particles travel
during the period from the switching on of the source $r=(4Dt_{1})^{1/2}$.
Thus, $\tau_{D}\simeq t_{1}$ does not depend on the energy of particles. This
time is also larger than the time of jet’s switching off $t_{2}$,
$t_{2}<0.6t_{1}$. We see that to now the distribution of relativistic
particles, generated by the jet, does not change noticeable.
## Aknowlegements
This work was done under support of the Russian Foundation for Fundamental
Research (grant number 11-02-01021).
## References
* [1] Bell, A.R., 1978. MNRAS, 182, 147.
* [2] Blandford, R.D., Znajek, R.L., 1977. MNRAS, 179, 423.
* [3] Blandford, R.D., Ostriker, J.R., 1978. Astrophys. J. (Lett.), 221, L29.
* [4] Crocker, R.M., Aharonian, F., 2011. Phys., Rev. Lett., 106, 101102.
* [5] Finkbeiner, D.P., 2004. Astrophys. J., 614, 186.
* [6] Istomin, Ya.N., 2004. New Astronomy, 10, 157.
* [7] Istomin, Ya.N., Sol, H., 2009. Astrophys. Space Science, 321, 57.
* [8] Istomin, Ya.N., 2010. MNRAS, 408, 1307
* [9] Istomin, Ya.N., Sol, H., 2011. Astron.& Astrophys. 527, A22.
* [10] Krymskii, G.F., 1977. Soviet Physics-Doklady, 22, 327.
* [11] Landau, L.D., Lifshits, E.M., 1984. in Course of Theoretical Physics, Electrodynamics of Continuous Media, Oxford, 308.
* [12] Mao-Li, M., Xin-Wu, C., Dong-Rong, J, Min-Feng, G., 2008. Chin. J. Astron. Astrophys., 8, 39.49.
* [13] Ptuskin, V.S., Khazan, Y.M., 1982. Soviet Astronomy, 25, 547.
* [14] Ptuskin, V.S., 2007. Physics Uspehi, 50, 534.
* [15] Snowden, S.L., Egger, R., Freyberg, M.J., McCammon, D., Plucinsky, P.P., Sanders, W.T., Schmitt, J.H.M.M., Truemper, J., Voges, W., 1997\. Astrophys. J., 485, 125.
* [16] Sofue, Y., 2000. Astrophys. J., 540, 224.
* [17] Strong, A.W., Moskalenko, I.V., 1998. Astrophys. J., 509, 212.
* [18] Su, M., Slatyer, T.R., Finkbeiner, D.P., 2010. Astrophys. J., 724, 1044.
* [19] Thorne, K.S., Price, R.H., MacDonald, D.A., 1986. Black Holes: the Membrane Paradigm, Yale University Press.
|
arxiv-papers
| 2011-10-25T08:14:49 |
2024-09-04T02:49:23.590938
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ya. N. Istomin",
"submitter": "Ya. N. Istomin",
"url": "https://arxiv.org/abs/1110.5436"
}
|
1110.5466
|
# $\eta^{\prime}$ photoproduction on the nucleons in the quark model
Xian-Hui Zhong1 111E-mail: zhongxh@ihep.ac.cn and Qiang Zhao2,3 222E-mail:
zhaoq@ihep.ac.cn 1) Department of Physics, Hunan Normal University, and Key
Laboratory of Low-Dimensional Quantum Structures and Quantum Control of
Ministry of Education, Changsha 410081, China 2) Institute of High Energy
Physics, Chinese Academy of Sciences, Beijing 100049, China 3) Theoretical
Physics Center for Science Facilities, Chinese Academy of Sciences, Beijing
100049, China
###### Abstract
A chiral quark-model approach is adopted to study the $\gamma
p\rightarrow\eta^{\prime}p$ and $\gamma n\rightarrow\eta^{\prime}n$. Good
descriptions of the recent observations from CLAS and CBELSA/TAPS are
obtained. Both of the processes are governed by $S_{11}(1535)$ and $u$ channel
background. Strong evidence of an $n=3$ shell resonance $D_{15}(2080)$ is
found in the reactions, which accounts for the bump-like structure around
$W=2.1$ GeV observed in the total cross section and excitation functions at
very forward angles. The $S_{11}(1920)$ seems to be needed in the reactions,
with which the total cross section near threshold for the $\gamma
p\rightarrow\eta^{\prime}p$ is improved slightly. The polarized beam
asymmetries show some sensitivities to $D_{13}(1520)$, although its effects on
the differential cross sections and total cross sections are negligible. There
is no obvious evidence of the $P$-, $D_{13}$-, $F$\- and $G$-wave resonances
with a mass around 2.0 GeV in the reactions.
###### pacs:
13.60.Le, 14.20.Gk, 12.39.Jh, 12.39.Fe
## I Introduction
The threshold energy of the $\gamma p\rightarrow\eta^{\prime}p$ and $\gamma
n\rightarrow\eta^{\prime}n$ reactions is above the second resonance region,
which might be a good place to extract information of the less-explored higher
nucleon resonances around $2.0$ GeV. Thus, the study of $\eta^{\prime}$
photoproduction becomes an interest topic in both experiment and theory.
However, due to the small production rate for the $\eta^{\prime}$ via an
electromagnetic probe, it had been a challenge for experiment to measure the
$\eta^{\prime}$ production cross section in the photoproduction reaction
:1968ke ; Struczinski:1975ik ; Plotzke:1998ua .
Theoretical analyses can be found in the literature which were performed to
interpret the old data of $\gamma p\rightarrow\eta^{\prime}p$ :1968ke ;
Struczinski:1975ik ; Plotzke:1998ua . Zhang _et al._ Zhang:1995uha first
analyzed the old data with an effective Lagrangian approach, in which the off-
shell contributions from the low-lying resonances in $(1.5\sim 1.7)$ GeV were
excluded. They considered that the main contribution to the photoproduction
amplitude came from $D_{13}(2080)$. Li Li:1996wj and Zhao Zhao:2001kk also
studied the reaction within a constituent quark model approach. They found the
dominance of $S$ wave in the $\eta^{\prime}$ production, and the off-shell
$S_{11}(1535)$ excitation played an important role near the $\eta^{\prime}$
threshold. They also predicted that effects of higher resonances in the $n=3$
shell might be observable in experiment. The dominant role of $S_{11}(1535)$
was also suggested by Borasoy with the $U(3)$ baryon chiral perturbation
theory Borasoy:2001pj , and Sibirtsev _et al._ with a hadronic model
Sibirtsev:2003ng . Considering the interferences between $S_{11}(1535)$ and
the background ($t$ channel vector meson exchanges), they gave a reasonable
description of the old data. In 2003 Chiang and Yang developed a Reggeized
model for $\eta$ and $\eta^{\prime}$ photoproduction on protons Chiang:2002vq
. In this model, the differential cross section data from Plotzke:1998ua can
be well described by the interference of an $S_{11}$ resonance with a mass in
the range of $(1.932\sim 1.959)$ GeV and the $t$ channel Regge trajectory
exchanges. In 2004 Nakayama and Haberzett Nakayama:2004ek analyzed the
differential cross section data from Plotzke:1998ua within a relativistic
meson exchange model of hadronic interactions. They predicted that the
observed angular distribution is due to the interference between the
$t$-channel and the nucleon resonances $S_{11}(1650)$ and $P_{11}(1880)$.
Although there are some hints of higher nucleon resonances in the
$\eta^{\prime}$ photoproduction process, it is not straightforward to extract
them based on the old data with large uncertainties.
With the rapid development in experiment, recently, high-statistics and large-
angle-coverage data for the $\gamma p\rightarrow\eta^{\prime}p$ reaction have
been reported by the CLAS Collaboration Dugger:2005my ; Williams:2009yj and
CBELSA/TAPS Collaboration Crede:2009zzb , respectively. More recently, the
measurements of the quasi-free photoproduction of $\eta^{\prime}$ mesons off
nucleons bound in the deuteron were also carried out by the CBELSA/TAPS
Collaboration Jaegle:2010jg . The recent new data not only provide us a good
opportunity to better understand the reaction mechanism but also allows us to
carry out a detailed investigation of the less-explored higher nucleon
resonances. Motivated by the new high-precision cross-section data obtained by
the CLAS Collaboration Dugger:2005my , Nakayama and Haberzett Nakayama:2005ts
updated their fits and found that higher resonances with $J=3/2$ might play
important roles in reproducing the details of the measured angular
distribution. A bump structure in the total cross around $W=2.09$ GeV is
predicted and might be caused by $D_{13}(2080)$ and/or $P_{13}(2100)$. In the
quark model Li Li:1996wj and Zhao Zhao:2001kk also found a bump structure
around $W=2.1$ GeV ($E_{\gamma}\simeq 2.0$ GeV) in the cross section by
analyzing the old data. This structure comes from the $n=3$ terms in the
harmonic oscillator basis. The later higher-precision free proton data from
the CLAS Collaboration Dugger:2005my ; Williams:2009yj indeed show a broad
bump structure in the cross section around $W=2.1$ GeV. This structure seems
to also appear in the very recent quasi-free proton data and the data for
inclusive quasi-free $\gamma d\rightarrow(np)\eta^{\prime}$ process
Jaegle:2010jg .
To clarify the structures from the above analyses and observations, we present
a systemic analysis of the recent experimental data for $\gamma p\rightarrow
p\eta^{\prime}$ and $\gamma n\rightarrow\eta^{\prime}n$ in the framework of a
chiral quark model as an improvement of the previous studies Li:1996wj ;
Zhao:2001kk . The chiral quark model has been well developed and widely
applied to meson photoproduction reactions qk2 ; qkk ; Li:1997gda ; qkk2 ; qk3
; qk4 ; qk5 ; Li:1995vi ; Li:1998ni ; Saghai:2001yd ; He:2008ty ; He:2009zzi .
The details about the model can be found in Li:1997gda ; qk3 . Recently, we
applied this model to study $\eta$ photoproduction on the free and quasifree
nucleons Zhong:2011ti . Good descriptions of the observations were obtained.
In this work, we extend this approach to $\eta^{\prime}$ photoproduction.
Given that the $\eta^{\prime}$ and $\eta$ are mixing states of flavor singlet
and octet in the SU(3) flavor symmetry, we expect that some flavor symmetry
relation can be applied to these two channels as a constraint on the model
parameters. Moreover, since $\eta^{\prime}$ production has a higher threshold,
the determinations of the low-lying resonances in $(1.5\sim 1.7)$ GeV in the
$\eta$ photoproduction would be useful for estimating their off-shell
contributions in the $\eta^{\prime}$ photoproduction.
Similar to the $\eta$ production, an interesting difference between $\gamma
p\rightarrow\eta^{\prime}p$ and $\gamma n\rightarrow\eta^{\prime}n$ is that in
the $\gamma p$ reactions, contributions from states of representation
$[70,^{4}8]$ will be forbidden by the Moorhouse selection rule
Moorhouse:1966jn in the SU(6)$\otimes$O(3) symmetry. As a consequence, only
states of $[56,^{2}8]$ and $[70,^{2}8]$ would contribute to $\gamma
p\rightarrow\eta^{\prime}p$. In contrast, all the octet states can contribute
to the $\gamma n$ reactions. In another word, more states will be present in
the $\gamma n$ reactions. Therefore, a combined study of the $\eta^{\prime}$
meson photoproduction on the proton and neutron should provide some
opportunities for disentangling the role played by intermediate baryon
resonances.
The paper is organized as follows. In Sec. II, a brief introduction of the
chiral quark model approach is given. The numerical results are presented and
discussed in Sec. III. Finally, a summary is given in Sec. IV.
## II framework
In the chiral quark model, the $s$\- and $u$-channel transition amplitudes for
pseudoscalar-meson photoproduction on the nucleons have been worked out in the
harmonic oscillator basis in Ref. Li:1997gda . The $t$-channel contributions
from vector meson exchange are not considered in this work. If a complete set
of resonances are included in the $s$ and $u$ channels, the introduction of
$t$-channel contributions might result in double counting Dolen:1967jr ;
Williams:1991tw .
It should be remarked that the amplitudes in terms of the harmonic oscillator
principle quantum number $n$ are the sum of a set of SU(6) multiplets with the
same $n$. To see the contributions of individual resonances, we need to
separate out the single-resonance-excitation amplitudes within each principle
number $n$ in the $s$-channel. Taking into account the width effects of the
resonances, the resonance transition amplitudes of the $s$-channel can be
generally expressed as Li:1997gda
$\displaystyle\mathcal{M}^{s}_{R}=\frac{2M_{R}}{s-M^{2}_{R}+iM_{R}\Gamma_{R}}\mathcal{O}_{R}e^{-(\textbf{k}^{2}+\textbf{q}^{2})/6\alpha^{2}},$
(1)
where $\sqrt{s}=E_{i}+\omega_{\gamma}$ is the total energy of the system,
$\alpha$ is the harmonic oscillator strength, $M_{R}$ is the mass of the
$s$-channel resonance with a width $\Gamma_{R}(\mathbf{q})$, and
$\mathcal{O}_{R}$ is the separated operators for individual resonances in the
$s$-channel. In the Chew-Goldberger-Low-Nambu (CGLN) parameterization
Chew:1957tf , the transition amplitude can be written with a standard form:
$\displaystyle\mathcal{O}_{R}$ $\displaystyle=$ $\displaystyle
if^{R}_{1}\mbox{\boldmath$\sigma$\unboldmath}\cdot\mbox{\boldmath$\epsilon$\unboldmath}+f^{R}_{2}\frac{(\mbox{\boldmath$\sigma$\unboldmath}\cdot\mathbf{q})\mbox{\boldmath$\sigma$\unboldmath}\cdot(\mathbf{k}\times\mbox{\boldmath$\epsilon$\unboldmath})}{|\mathbf{q}||\mathbf{k}|}$
(2)
$\displaystyle+if^{R}_{3}\frac{(\mbox{\boldmath$\sigma$\unboldmath}\cdot\mathbf{k})(\mathbf{q}\cdot\mbox{\boldmath$\epsilon$\unboldmath})}{|\mathbf{q}||\mathbf{k}|}+if^{R}_{4}\frac{(\mbox{\boldmath$\sigma$\unboldmath}\cdot\mathbf{q})(\mathbf{q}\cdot\mbox{\boldmath$\epsilon$\unboldmath})}{|\mathbf{q}|^{2}},$
where $\sigma$ is the spin operator of the nucleon, $\epsilon$ is the
polarization vector of the photon, and $\mathbf{k}$ and $\mathbf{q}$ are
incoming photon and outgoing meson momenta, respectively.
The $\mathcal{O}_{R}$ for the $n\leq 2$ shell resonances have been extracted
in Li:1997gda . For the $n=3$ shell resonances are just around the
$\eta^{\prime}$ production threshold, which might play important roles in the
reaction. Thus, in this work we can not treat them as degenerate any more.
Their transition amplitudes, $\mathcal{O}_{R}$, for $S_{11}$, $D_{13}$,
$D_{15}$, $G_{17}$ and $G_{19}$ waves are derived in the SU(6)$\otimes$O(3)
symmetric quark model limit, which have been given in Tab. 1. The $g$-factors
that appear in Tab. 1 can be extracted from the quark model in the
SU(6)$\otimes$O(3) symmetry limit, and are defined by
$\displaystyle g_{3}^{v}$ $\displaystyle\equiv$ $\displaystyle\langle
N_{f}|\sum_{j}e_{j}I_{j}\sigma_{jz}|N_{i}\rangle,$ (3) $\displaystyle
g_{3}^{s}$ $\displaystyle\equiv$ $\displaystyle\langle
N_{f}|\sum_{j}e_{j}I_{j}|N_{i}\rangle,$ (4) $\displaystyle g_{2}^{s}$
$\displaystyle\equiv$ $\displaystyle\langle N_{f}|\sum_{i\neq
j}e_{j}I_{i}\mbox{\boldmath$\sigma$\unboldmath}_{i}\cdot\mbox{\boldmath$\sigma$\unboldmath}_{j}|N_{i}\rangle/3,$
(5) $\displaystyle g_{2}^{v}$ $\displaystyle\equiv$ $\displaystyle\langle
N_{f}|\sum_{i\neq
j}e_{j}I_{i}(\mbox{\boldmath$\sigma$\unboldmath}_{i}\times\mbox{\boldmath$\sigma$\unboldmath}_{j})_{z}|N_{i}\rangle/2,$
(6) $\displaystyle g_{2}^{v^{\prime}}$ $\displaystyle\equiv$
$\displaystyle\langle N_{f}|\sum_{i\neq j}e_{j}I_{i}\sigma_{iz}|N_{i}\rangle,$
(7)
where $|N_{i}\rangle$ and $|N_{f}\rangle$ stand for the initial and final
states, respectively, and $I_{j}$ is the isospin operator, which has been
defined in Li:1997gda . For the $\eta$ and $\eta^{\prime}$ production, the
isospin operator is $I_{j}=1$.
From Tab. 1 we can see that the $n=3$ resonance amplitudes
$f^{R}_{i}(i=1,2,3,4)$ for $S$ and $D$ waves contain two terms, which are in
proportion to $x^{2}$ and $x^{3}$, respectively. The term $\mathcal{O}(x^{3})$
is a higher order term in the amplitudes for
$x\equiv|\mathbf{k}||\mathbf{q}|/(3\alpha^{2})\ll 1$. For the $G_{17}$ and
$G_{19}$ waves, their amplitudes only contain the high order term
$\mathcal{O}(x^{3})$, thus their contributions to the reactions should be
small in the $n=3$ shell resonances. Comparing the resonance amplitudes
$f^{R}_{i}(i=1,2,3,4)$ for $D_{13}$ with those for $D_{15}$, we find that
$\displaystyle\left|f^{R}_{1}[D_{15}(n=3)]\right|$ $\displaystyle>$
$\displaystyle\left|f^{R}_{1}[D_{13}(n=3)]\right|P_{3}^{\prime}(\cos\theta),$
(8) $\displaystyle\left|f^{R}_{i}[D_{15}(n=3)]\right|$ $\displaystyle>$
$\displaystyle\left|f^{R}_{i}[D_{13}(n=3)]\right|\ \ \ \ (i=2,3,4),$ (9)
for the $\eta^{\prime}$ and $\eta$ photoproduction processes. The amplitude
$f^{R}_{1}$ for $D_{13}$ is reaction angle independent, while the $f^{R}_{1}$
for $D_{15}$ depends on the reaction angle $\theta$ (i.e. $\propto
P_{3}^{\prime}(\cos\theta)$). According to Eq. 8, at very forward and backward
angles [i.e. $\cos\theta\simeq\pm 1$] we obtain
$\displaystyle\left|f^{R}_{1}[D_{15}(n=3)]\right|_{\cos\theta\simeq\pm 1}$
$\displaystyle>$ $\displaystyle 6\left|f^{R}_{1}[D_{13}(n=3)]\right|.$ (10)
It shows that the magnitude of $f^{R}_{1}$ at very forward and backward angles
for $D_{15}$ is about an order larger than that of $D_{13}$. Thus, the
$D_{15}$ partial wave is the main contributor to the $\eta^{\prime}$ and
$\eta$ photoproduction processes in the $n=3$ shell resonances. At very
forward and backward angle regions, the angle distributions might be sensitive
to the $D_{15}$ partial wave. We note that due to lack of experimental
information and high density of states above 2 GeV, different representations
that contribute to the same partial wave quantum number in the $n=3$ shell are
treated degenerately as one state as listed in Tab. 1.
Table 1: CGLN amplitudes for $s$-channel resonances of the $n=3$ shell in the SU(6)$\otimes$O(3) symmetry limit. We have defined $A\equiv(\frac{\omega_{m}}{E_{f}+M_{N}}+1)|\mathbf{q}|$, $x\equiv\frac{|\mathbf{k}||\mathbf{q}|}{3\alpha^{2}}$, $P_{l}^{\prime}(z)\equiv\frac{\partial P_{l}(z)}{\partial z}$, $P_{l}^{\prime\prime}(z)\equiv\frac{\partial^{2}P_{l}(z)}{\partial z^{2}}$, $g_{1}\equiv g_{3}^{v}-\frac{1}{8}g_{2}^{v}$, $g_{2}\equiv g_{3}^{v}-\frac{1}{8}g_{2}^{v^{\prime}}$ and $g_{3}\equiv g_{3}^{s}-\frac{1}{8}g_{2}^{s}$. $\omega_{\gamma}$, $\omega_{m}$ and $E_{f}$ stand for the energies of the incoming photon, outgoing meson and final nucleon, respectively, $m_{q}$ is the constitute $u$ or $d$ quark mass, $1/\mu_{q}$ is a factor defined by $1/\mu_{q}=2/m_{q}$, and $P_{l}(z)$ is the Legendre function with $z=\cos\theta$. | $f^{R}_{1}$ | $f^{R}_{2}$ | $f^{R}_{3}$ | $f^{R}_{4}$
---|---|---|---|---
$S_{11}$ | $-\frac{i}{36}\frac{\omega_{m}\omega_{\gamma}}{\mu_{q}}(g_{2}+\frac{k}{2m_{q}}g_{1})x^{2}$ | | |
| +$\frac{i}{60}(g_{1}\frac{k}{m_{q}}+2g_{2})Ax^{3}$ | 0 | 0 | 0
$D_{13}$ | $\frac{i}{90}\frac{\omega_{m}\omega_{\gamma}}{\mu_{q}}(g_{2}+\frac{k}{2m_{q}}g_{1})x^{2}$ | $\frac{i}{180}\frac{\omega_{m}\omega_{\gamma}^{2}}{\mu_{q}m_{q}}g_{1}x^{2}P_{2}^{\prime}(z)-\frac{i}{105}$ | | $-\frac{i}{90}\frac{\omega_{m}\omega_{\gamma}}{\mu_{q}m_{q}}g_{2}x^{2}P_{2}^{\prime\prime}(z)+\frac{i}{420}Ax^{3}$
| $-\frac{i}{60}(g_{1}\frac{k}{m_{q}}+2g_{2})Ax^{3}$ | $\frac{k}{m_{q}}(g_{1}+g_{3}/2)Ax^{3}P_{2}^{\prime}(z)$ | 0 | $[14g_{2}-(g_{1}-g_{3})\frac{k}{m_{q}}]P_{2}^{\prime\prime}(z)$
$D_{15}$ | $\\{-\frac{i}{90}\frac{\omega_{m}\omega_{\gamma}}{\mu_{q}}(g_{2}+\frac{k}{2m_{q}}g_{1})x^{2}+\frac{i}{105}$ | $-\frac{i}{180}\frac{\omega_{m}\omega_{\gamma}^{2}}{\mu_{q}m_{q}}g_{1}x^{2}P_{2}^{\prime}(z)+\frac{i}{420}$ | $-\frac{i}{90}\frac{\omega_{m}\omega_{\gamma}}{\mu_{q}}g_{2}x^{2}P_{3}^{\prime\prime}(z)+\frac{i}{420}$ | $\frac{i}{90}\frac{\omega_{m}\omega_{\gamma}}{\mu_{q}}g_{2}x^{2}P_{2}^{\prime\prime}(z)-\frac{i}{420}$
| $[(g_{1}-\frac{1}{2}g_{3})\frac{k}{m_{q}}+g_{2}]Ax^{3}\\}P_{3}^{\prime}(z)$ | $\frac{k}{m_{q}}(5g_{1}-3g_{3})Ax^{3}P_{2}^{\prime}(z)$ | $[4g_{2}-(g_{1}-g_{3})\frac{k}{m_{q}}]Ax^{3}P_{3}^{\prime\prime}(z)$ | $[4g_{2}-(g_{1}-g_{3})\frac{k}{m_{q}}]Ax^{3}P_{2}^{\prime\prime}(z)$
$G_{17}$ | $\frac{-i}{1890}[(4g_{1}+5g_{3})\frac{k}{m_{q}}+18g_{2}]Ax^{3}P_{3}^{\prime}(z)$ | $\frac{-i}{210}(8g_{2}-g_{1}\frac{k}{m_{q}})Ax^{3}P_{4}^{\prime}(z)$ | $\frac{i}{1890}[(g_{1}-g_{3})\frac{k}{m_{q}}-18g_{2}]Ax^{3}P_{3}^{\prime\prime}(z)$ | $\frac{-i}{1890}[(g_{1}-g_{3})\frac{k}{m_{q}}-18g_{2}]Ax^{3}P_{4}^{\prime\prime}(z)$
$G_{19}$ | $i\frac{2k}{945m_{q}}(g_{1}-g_{3})Ax^{3}P_{5}^{\prime}(z)$ | $i\frac{k}{378m_{q}}(g_{1}-g_{3})Ax^{3}P_{4}^{\prime}(z)$ | $-i\frac{k}{1890m_{q}}(g_{1}-g_{3})Ax^{3}P_{5}^{\prime\prime}(z)$ | $i\frac{k}{1890m_{q}}(g_{1}-g_{3})Ax^{3}P_{4}^{\prime\prime}(z)$
Table 2: The $g$-factor in the amplitudes. reaction | $g_{3}^{v}$ | | $g_{3}^{s}$ | | $g_{2}^{s}$ | | $g_{2}^{v}$ | | $g_{2}^{v^{\prime}}$ | | $g_{1}$ | | $g_{2}$ | | $g_{3}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\gamma p\rightarrow\eta^{\prime}(\eta)p$ | $1$ | | $1$ | | $0$ | | $0$ | | $0$ | | 1 | | 1 | | 1
$\gamma n\rightarrow\eta^{\prime}(\eta)n$ | $-\frac{2}{3}$ | | $0$ | | $-\frac{2}{3}$ | | $0$ | | $-\frac{2}{3}$ | | $-\frac{2}{3}$ | | $-\frac{3}{4}$ | | $\frac{1}{12}$
Finally, the physical observables, differential cross section and photon beam
asymmetry, are given by the following standard expressions Walker:1968xu :
$\displaystyle\frac{d\sigma}{d\Omega}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{e}\alpha_{\eta^{\prime}}(E_{i}+M_{N})(E_{f}+M_{N})}{16sM_{N}^{2}}\frac{1}{2}\frac{|\mathbf{q}|}{|\mathbf{k}|}\sum^{4}_{i=1}|H_{i}|^{2},$
(11) $\displaystyle\Sigma$ $\displaystyle=$ $\displaystyle
2\mathrm{Re}(H_{4}^{*}H_{1}-H_{3}^{*}H_{2})/\sum^{4}_{i=1}|H_{i}|^{2},$ (12)
where the helicity amplitudes $H_{i}$ can be expressed by the CGLN amplitudes
$f_{i}$ Walker:1968xu ; Fasano:1992es .
## III CALCULATIONS AND ANALYSIS
### III.1 Parameters
In our previous work, we have studied $\eta$ photoproduction off the quasi-
free neutron and proton from a deuteron target, where the masses, widths and
coupling strength parameters $C_{R}$ of the $n\leq 2$ shell resonances had
been determined Zhong:2011ti . In this work, the same parameter set is
adopted. For the $n=3$ shell resonances, $S_{11}$, $D_{13}$, $D_{15}$,
$G_{17}$ and $G_{19}$ waves, their transition amplitudes, $\mathcal{O}_{R}$,
have been derived in the SU(6)$\otimes$O(3) symmetric quark model limit, which
are given in Tab. 1. The various $g$-factors in these amplitudes for
$\eta^{\prime}$ photoproduction on the nucleons have been derived in the
SU(6)$\otimes$O(3) symmetry limit, which are listed in Tab. 2. Their resonance
parameters are determined by the experimental data. The determined mass and
width for $D_{15}$ are $M\simeq 2080$ MeV and $\Gamma\simeq 80$ MeV,
respectively, while the determined mass and width of $S_{11}$ are $M\simeq
1920$ MeV and $\Gamma\simeq 90$ MeV. It should be pointed out that the
reactions are insensitive to the masses and widths of $G$\- and $D_{13}$\-
wave states in the $n=3$ shell. Thus, in the calculation we roughly take their
mass and width with $M=2100$ MeV and $\Gamma=150$ GeV, respectively.
There are two overall parameters, the constituent quark mass $m_{q}$ and the
harmonic oscillator strength $\alpha$, from the transition amplitudes. In the
calculations we adopt the standard values in the the quark model, $m_{q}=330$
MeV and $\alpha^{2}=0.16$ GeV2.
To take into account the relativistic effects, the commonly applied Lorentz
boost factor is introduced in the resonance amplitude for the spatial
integrals qkk , which is
$\displaystyle\mathcal{O}_{R}(\textbf{k},\textbf{q})\rightarrow\gamma_{k}\gamma_{q}\mathcal{O}_{R}(\gamma_{k}\textbf{k},\gamma_{q}\textbf{q}),$
(13)
where $\gamma_{k}=M_{N}/E_{i}$ and $\gamma_{q}=M_{N}/E_{f}$.
The $\eta^{\prime}NN$ coupling is a free parameter in the present calculations
and to be determined by the experimental data. In the present work the overall
parameter $\eta^{\prime}NN$ coupling $\alpha_{\eta^{\prime}}$ is set to be the
same for both $\gamma n\rightarrow\eta^{\prime}n$ and $\gamma
p\rightarrow\eta^{\prime}p$. The fitted value $g_{\eta^{\prime}NN}\simeq 1.86$
(i.e. $\alpha_{\eta^{\prime}}\equiv g^{2}_{\eta^{\prime}NN}/4\pi=0.275$) is in
agreement with that in Ref. Nakayama:2005ts , where the upper limit of
$g_{\eta^{\prime}NN}$ was suggested to be $g_{\eta^{\prime}NN}\lesssim 2$. In
our previous work we determined the $\eta NN$ coupling, i.e. $g_{\eta
NN}\simeq 2.13$ Zhong:2011ti . This allows us to examine the
$\eta-\eta^{\prime}$ mixing relation for their non-strange components
production,
$\displaystyle\tan\phi_{P}=\frac{g_{\eta^{\prime}NN}}{g_{\eta NN}}\ ,$ (14)
which gives $\phi_{P}\simeq 41.2^{\circ}$. This value is within the range of
$\phi_{P}=\theta_{P}+\arctan\sqrt{2}\simeq 34^{\circ}\sim 44^{\circ}$, where
$\theta_{P}\simeq-20^{\circ}\sim-10^{\circ}$ is the flavor singlet and octet
mixing angle. The favored value for $\phi_{P}$ implies a flavor symmetry
between the $\eta$ and $\eta^{\prime}$ production.
Since the single resonance excitation amplitudes can be separated out for
$n\leq 2$ shells, the $\eta^{\prime}N^{*}N$ coupling form factor in principle
can be extracted by taking off the EM helicity amplitudes. The expressions are
similar to those extracted in $\eta$ meson photoproduction Zhong:2011ti apart
from the overall $g_{\eta^{\prime}NN}$ coupling constant. For higher excited
states in $n=3$, due to the lack of information about their EM excitation
amplitudes and high density of states above the 2 GeV mass region, we treat
all SU(6) multiplets that contribute to the same quantum number in $n=3$ to be
degenerate. In this sense, the partial waves in Tab. 1 are collective
amplitudes from both 56 and 70 representations.
Figure 1: (Color online)Differential cross sections for the $\eta^{\prime}$
photoproduction off the free proton at various beam energies. The data are
taken from Crede:2009zzb (solid circles), Williams:2009yj (open circles),
Dugger:2005my (diamonds). The quasi-free data from Jaegle:2010jg (squares)
are also included. The bold solid curves stand for the full model
calculations. The thin solid and dotted curves stand for the results without
$S_{11}(1535)$ and background $u$ channel contributions, respectively.
### III.2 $\gamma p\rightarrow\eta^{\prime}p$
Figure 2: (Color online) Same as Fig. 1. The dashed curves stand for the
results without $D_{15}(2080)$. Figure 3: (Color online) Fixed-angle
excitation functions for $\gamma p\rightarrow\eta^{\prime}p$ as a function of
center mass energy $W$ for eight $\cos\theta$, which have been labeled on the
plot. The stars stand for the data from Williams:2009yj for $\cos\theta=0.7$.
The chiral quark model studies of $\gamma p\to\eta^{\prime}p$ have been
carried out in Refs. Li:1996wj ; Zhao:2001kk , where a bump structure around
$E_{\gamma}=2$ GeV is found arising from the $n=3$ terms in the harmonic
oscillator basis. However, which partial wave contributes to this structure
can not be studied in detail since only a few datum points were available at
that time. The improvement of the experimental situations not only gives us a
good opportunity to better understand the $\gamma p\to\eta^{\prime}p$ process,
but also allows us to carry out a detailed investigation of the resonances in
the higher mass region.
In Fig. 1, we have plotted the differential cross sections. It shows that our
calculations are in good agreement with the data from threshold up to
$E_{\gamma}\simeq 2.4$ GeV. $S_{11}(1535)$ plays a dominant role in the
reaction, switching off its contributions the differential cross sections are
underestimated drastically. The important role of $S_{11}(1535)$ in the
$\gamma p\to\eta^{\prime}p$ is also predicted in the previous chiral quark
model study Li:1996wj ; Zhao:2001kk and the hadronic model study with the
exchange of vector mesons Sibirtsev:2003ng ; Nakayama:2005ts . It should be
mentioned that the $S_{11}(1535)$ is treated as a mixed state by the mixing of
$[70,^{2}8]$ and $[70,^{4}8]$ Zhong:2011ti , where the mixing angle is in
agreement with the recent study An:2011sb .
Furthermore, the $u$ channel plays an important role in the reactions as well.
The dotted curves in Fig. 1 show that without the contributions of the $u$
channel, the cross sections will be underestimated significantly. It should be
pointed out that the forward peaks in the differential cross sections are
mainly caused by the $u$ channel backgrounds. The crucial role of non-resonant
background contributions in the $\gamma p\to\eta^{\prime}p$ is also predicted
in Refs. Sibirtsev:2003ng ; Nakayama:2005ts , where the $t$ channel vector
meson exchanges are the main non-resonant contributions. In this work, the $t$
channel contributions are not considered. Since a complete set of resonances
in the $s$ and $u$ channels is included and the $\eta^{\prime}$ threshold is
rather high, we do not include the $t$ channel exchanges to avoid the double
counting problem Dolen:1967jr ; Williams:1991tw ; Li:1995vi .
It is interesting to see that $D_{15}(2080)$ in the $n=3$ shell plays a
crucial role in the reaction. It causes a shape change in the differential
cross section around the $D_{15}(2080)$ mass region (i.e. $E_{\gamma}\simeq
1.8$ GeV). In Fig. 2 we demonstrate the interfering effects of $D_{15}(2080)$
by switching off it in the differential cross section below and above the mass
of $D_{15}(2080)$. It could be obvious evidence of $D_{15}(2080)$ in the
$\gamma p\to\eta^{\prime}p$ process. We have noted that another $D$-wave
state, $D_{13}(2080)$, was predicted to have significant effects on the
reaction in Zhang:1995uha ; Nakayama:2005ts . However, in our approach the
contributions of the $D$-wave states with $J^{P}=3/2^{-}$ in the $n=3$ shell
are negligible. The dominant features of $D_{15}$ in the $D$ wave states can
be well understood from their amplitudes, which has been discussed in Sec. II.
The amplitude $f^{R}_{1}$ for $D_{15}$ is in proportion to
$P_{3}^{\prime}(\cos\theta)=(15\cos^{2}\theta-3)/2$, which can naturally
explain the strong effects of $D_{15}(2080)$ on the deferential cross sections
at forward and backward angles (i.e. $\cos\theta\simeq\pm 1$).
The effects of $D_{15}(2080)$ can be expected in $\gamma p\rightarrow\eta p$
taking into account the mixing between $\eta^{\prime}$ and $\eta$. A recent
quark model study of $\eta$ photoproduction in the high energy region has
reported effects from $D_{15}(2080)$ He:2008ty ; He:2009zzi . Evidence of
$D_{15}(2080)$ was also found by a partial wave analysis of the $\eta$
photoproduction data from CB-ELSA Crede:2003ax in the Bonn-Gatchina (BnGa)
model Anisovich:2005tf . Its contribution to $\gamma p\rightarrow
K^{+}\Lambda$ was also reported Anisovich:2011ye . Our analysis of the partial
wave amplitudes in Sec. II also suggests that the $D_{15}$ amplitude plays a
dominant role in the $n=3$ shell $D$ wave states in $K$ photoproduction.
We also mention that $P_{13}(1900)$ can slightly enhances the differential
cross sections around the $\eta^{\prime}$ production threshold as found in the
previous studies as well Zhao:2001kk ; Chiang:2002vq . It has a similar
behavior to the $u$ channel, although its contribution is much less than that
of the $u$ channel. It could be difficult to identify $P_{13}(1900)$ in the
$\gamma p\to\eta^{\prime}p$ process in the cross section measurement. Similar
conclusion is found in Ref. Chiang:2002vq . In our study, contributions from
other individual resonances are rather small, and we do not find obvious
signals for states, such as higher $S_{11}$ states.
Figure 4: (Color online) The cross sections for the $\eta^{\prime}$
photoproduction off the free proton. The data are taken from Crede:2009zzb
(solid circles), Williams:2009yj (stars). The quasi-free data from
Jaegle:2010jg (squares) are also included. In the upper panel the bold solid
curve corresponds to the full model result, while the thin solid, dotted,
dash-dotted, dash-dot-dotted and dashed curves are for the results by
switching off the contributions from $S_{11}(1535)$, $S_{11}(1650)$,
$S_{11}(1920)$, $D_{15}(2080)$ and $u$ channel, respectively. In the lower
panel the partial cross sections for the main contributors are indicated
explicitly by different legends.
In Fig. 3 we have plotted the fixed-angle excitation functions for $\gamma
p\rightarrow\eta^{\prime}p$. Our calculations show that at very forward (e.g.
$\cos\theta=0.7$) and backward scattering angles (e.g. $\cos\theta=-0.7$),
there is a bump around $W=2.1$ GeV. At forward angles, a similar structure
appears clearly in the recent data from the CLAS Collaboration Williams:2009yj
(see the stars in Fig. 3). In our approach the bump structure is caused by
$D_{15}(2080)$. At backward angles, due to the small $\eta^{\prime}$
production cross section, it might be difficult to observe such an enhancement
in the excitation functions around $W=2.1$ GeV.
Finally, the total cross section and exclusive cross sections for several
single resonances are illustrated in Fig. 4. The data can be reasonably well
described. The recent data show a small bump-like structure around $W=2.1$ GeV
(see the stars) Williams:2009yj , which in our approach is due to the
interferences of $D_{15}(2080)$ with other partial waves. Switching off the
contribution of $D_{15}(2080)$, we find that the bump-like structure
disappears (see the dash-dot-dotted curve in the upper panel of Fig. 4). It
should be mentioned that the bump-like structure around $W=2.1$ GeV was
explained by the effects of $D_{13}(2080)$ and/or $P_{11}2100$ in
Nakayama:2005ts .
In Fig. 4, the dominant role of $S_{11}(1535)$ and $u$ channel background can
be obviously seen from their exclusive cross sections, which are much larger
than that of other resonances. The large cross section around the
$\eta^{\prime}$ production threshold mainly comes from the interferences of
$S_{11}(1535)$ and $u$ channel. Switching off either of them, we find that the
cross section will be underestimated drastically. The calculation shows that
both $S_{11}(1650)$ and $S_{11}(1920)$ have rather small effects on the cross
section around the $\eta^{\prime}$ production threshold (see the dotted and
dash-dotted curves in the upper panel of Fig. 4). It should be noted that,
although $S_{11}(1920)$ has a small contribution to the cross section, its
mass favors to be less than $1950$ MeV. Otherwise, we can not reproduce the
present cross sections in the region of $W<2.0$ GeV. The mass of
$S_{11}(1920)$ extracted here is close to that obtained in Ref. Chiang:2002vq
. $S_{11}(1920)$ might correspond to the $S_{11}(2090)$ listed by the Particle
Data Group as a one-star resonance with a mass varying from 1880 to 2180 MeV
Nakamura:2010zzi .
In brief, the $\gamma p\rightarrow\eta^{\prime}p$ reaction is dominated by
$S_{11}(1535)$ and $u$ channel contributions. The constructive interference
between them accounts for the large cross section near threshold.
$D_{15}(2080)$ plays an important role in the reaction. It has obvious effects
on the angle distributions, and is responsible for the bump-like structure
around $W=2.1$ GeV observed in the cross section. Weak signal of
$S_{11}(1920)$ might be extracted from the cross section near threshold. The
reaction is less sensitive to the other intermediate states.
Figure 5: (Color online) The differential cross sections for the $\gamma
n\rightarrow\eta^{\prime}n$ at various beam energies. The data are taken from
Jaegle:2010jg (squares). The bold solid curves stand for the full model
calculations. The thin solid and dotted curves stand for the results without
$S_{11}(1535)$ and background $u$ channel contributions, respectively. Figure
6: (Color online) The cross sections for the $\gamma
n\rightarrow\eta^{\prime}n$ process. The data are taken from Jaegle:2010jg .
In the upper panel the bold solid curve corresponds to the full model result,
while the dotted, thin solid, dash-dot-dotted, dash-dotted, and dashed curves
are for the results by switching off the contributions from $S_{11}(1535)$,
$S_{11}(1650)$, $S_{11}(1920)$, $D_{15}(2080)$ and $u$ channel, respectively.
In the lower panel the partial cross sections for the main contributors are
indicated explicitly by different legends. Figure 7: (Color online) The data
for inclusive quasi-free $\gamma d\rightarrow np\eta^{\prime}$ cross section (
$\sigma_{np}$) and the sum of quasi-free proton and quasi-free neutron cross
section ($\sigma_{p}$+$\sigma_{n}$). The solid curve corresponds to our
results of the sum of free proton and free neutron cross section.
### III.3 $\gamma n\to\eta^{\prime}n$
Recently, the CBELSA/TAPS collaboration had observed the $\gamma
n\to\eta^{\prime}n$ process for the first time Jaegle:2010jg . The data had
been compared to fits with the NH Nakayama:2005ts and MAID model
Chiang:2002vq . There exists large disagreement between model fits and the
experimental observations. As mentioned earlier, in $\gamma
n\to\eta^{\prime}n$ states of $[70,^{4}8]$ representation can contribute here
while they are forbidden in $\gamma p\to\eta^{\prime}p$ by the Moorhouse
selection rule Moorhouse:1966jn . Therefore, we expect that more information
about the $s$-channel resonances can be gained in the study of $\gamma
n\to\eta^{\prime}n$. For instance, as the only $D_{15}$ state in the first
orbital excitations and belonging to $[70,^{4}8]$, $D_{15}(1675)$ can only be
excited by $\gamma n$ instead of $\gamma p$. We also note that in this work
the nuclear Fermi motion effects have been neglected since they are negligible
according to the recent analysis Jaegle:2010jg .
In Fig. 5, the differential cross sections at various beam energies have been
plotted. It shows that our quark model fits are in good agreement with the
recent CBELSA/TAPS measurements in the beam energy region $E_{\gamma}>1.9$ GeV
Jaegle:2010jg . However, in the region $E_{\gamma}<1.9$ GeV, we can not
reproduce the data well, especially at the forward angles. In this region, our
results are close to the fits of NH model Nakayama:2005ts .
Similar to $\gamma p\to\eta^{\prime}p$, the differential cross sections for
$\gamma n\to\eta^{\prime}n$ are governed by the $S_{11}(1535)$ and $u$ channel
contributions. Switching off either of them (see thin solid and dashed
curves), we find that the cross sections would be underestimated
significantly. It shows that $S_{11}(1535)$ dominates near threshold
($E_{\gamma}<1.9$ GeV), and strongly enhances the cross section. At higher
energies ($E_{\gamma}>2.0$ GeV), the $u$ channel becomes the main contributor
in the differential cross sections. The role of $D_{15}(2080)$ in the
$\eta^{\prime}n$ channel is similar to that in the $\eta^{\prime}p$ channel.
It slightly enhances the cross sections at forward angles in the higher energy
region ($E_{\gamma}>1.9$ GeV). However, the present data for $\gamma
n\to\eta^{\prime}n$ seems not precise enough to confirm $D_{15}(2080)$ in the
reaction. Again, we find that the contribution from $P_{13}(1900)$ is
negligibly small and might be difficult to identify in the cross section
measurement.
Figure 8: (Color online) The fixed-angle excitation functions for $\gamma
n\rightarrow\eta^{\prime}n$ as a function of center mass energy $W$ for eight
values of $\cos\theta$, which have been labeled on the plot.
In Fig. 6, the total cross section and the exclusive cross sections of several
single resonances are shown. Again, we see the dominance of $S_{11}(1535)$ and
$u$ channel in the cross sections. Some interfering effects between
$S_{11}(1650)$/$S_{11}(1920)$ and $S_{11}(1535)$ can be seen near threshold.
There also exist some discrepancies in the low energy region, i.e.
$E_{\gamma}\simeq(1.6\sim 2.0)$ GeV, between our model results and
experimental data. Our model suggests two bump structures in the total cross
section. The first one around $W=1.95$ GeV is mainly caused by $S_{11}(1535)$,
while the second around $W=2.1$ GeV is caused by $D_{15}(2080)$. The data
Jaegle:2010jg seem to show a bump structure around $W=1.95$ GeV, while the
second bump structure around $W=2.1$ GeV can not be identified due to the
large experimental uncertainties.
In Ref. Jaegle:2010jg , the data for the inclusive quasi-free $\gamma
d\rightarrow np\eta^{\prime}$ cross section, $\sigma_{np}$, are also
presented. It shows that the $\sigma_{np}$ is nearly equal to the sum of the
free proton ($\sigma_{p}$) and free neutron cross sections ($\sigma_{n}$).
Interestingly, the data indicate two broad bump structures in the cross
section around $W=1.95$ and $W=2.1$ GeV, respectively. To compare with the
data we plot our calculations of $(\sigma_{p}+\sigma_{n})$ in Fig. 7, which
appears to be compatible with the data, although the cross section around
$W=2.05$ GeV is slightly overestimated. In our approach the second bump
structure in the inclusive quasi-free $\gamma d\rightarrow np\eta^{\prime}$
cross section is caused by $D_{15}(2080)$. This contribution seems to be
highlighted in $\gamma d\rightarrow np\eta^{\prime}$ as a summed-up effects
from both proton and neutron reactions. Further improved measurement should be
able to clarify the under-lying mechanisms that produces the bump structures.
In Fig. 8 the excitation functions for $\gamma n\rightarrow\eta^{\prime}n$ as
a function of the center-of-mass energy $W$ at various angles are plotted. It
is sensitive to the presence of $D_{15}(2080)$ as shown by the drastic
enhancement at very forward angles around $W=2.1$ GeV. This feature is similar
to that in $\gamma p\rightarrow\eta^{\prime}p$ (see Figs. 3 and 8).
Polarization observables should be more sensitive to the underlying
mechanisms. In Fig. 9, we plot the polarized beam asymmetries for $\gamma
p\rightarrow\eta^{\prime}p$ (left) and $\gamma n\rightarrow\eta^{\prime}n$
(right), respectively. The beam asymmetries for both of the precesses are
sensitive to $S_{11}(1535)$, $D_{13}(1520)$, $D_{15}(2080)$ and $u$ channel
contributions (see the bottom of Fig. 9). A sudden change of the beam
asymmetries around $E_{\gamma}\simeq 1.8$ GeV (i.e. the threshold of
$D_{15}(2080)$) can be seen, which is mainly caused by the $D_{15}(2080)$.
Furthermore, it shows that the beam asymmetry for $\gamma
n\rightarrow\eta^{\prime}n$ ($\Sigma_{n}$) is quite similar to that of $\gamma
p\rightarrow\eta^{\prime}p$ ($\Sigma_{p}$) up to $E_{\gamma}\simeq 1.8$ GeV.
In this energy region the beam asymmetry is nearly symmetric in the forward
and backward directions. Above $E_{\gamma}\simeq 1.9$ GeV, obvious differences
show up between $\Sigma_{n}$ and $\Sigma_{p}$. It should be noted that the
contribution of $D_{13}(1520)$ does not appear to be significant in the
hadronic model studies. Therefore, experimental measurement of the polarized
beam asymmetries should provide a test for various models.
In brief, $\gamma n\rightarrow\eta^{\prime}n$ has features similar to those of
$\gamma p\rightarrow\eta^{\prime}p$. Both reactions are dominated by
$S_{11}(1535)$ and $u$ channel contributions. We predict that $D_{15}(2080)$
should have significant contributions to $\gamma n\rightarrow\eta^{\prime}n$,
and the polarized beam asymmetries might be sensitive to its presence in the
transition amplitude. Finally, we should point out that although
$D_{15}(1675)$ has a significant contribution to $\gamma n\rightarrow\eta n$
process, its contributions to $\gamma n\rightarrow\eta^{\prime}n$ is
negligible.
Figure 9: (Color online) Predicted beam asymmetries at nine beam energies
($E_{\gamma}=1.575\sim 2.375$ GeV) for $\gamma p\rightarrow\eta^{\prime}p$ and
$\gamma n\rightarrow\eta^{\prime}n$.
## IV Summary
In this work, we have studied the $\eta^{\prime}$ photo-production off the
proton and neutron within a chiral quark model. A good description of the
recent experimental data for both processes is achieved. Due to the similar
reaction mechanism for both processes it is understandable that some similar
features exist in both reactions as manifested in the cross sections,
excitation functions and polarized beam asymmetries.
The large peak of the cross section around threshold for both processes mainly
accounts for the constructive interferences between $S_{11}(1535)$ and the
$u$-channel background. Strong evidence of $D_{15}(2080)$ has been found in
the reactions, with which we can naturedly explain the following recent high-
statistics observations for the $\gamma p\rightarrow\eta^{\prime}p$ reaction
from the CLAS Collaboration: (i) the sudden change of the shape of the
differential cross section around $E_{\gamma}=1.8$ GeV, (ii) the bump-like
structure in the total cross section around $W=2.1$ GeV ($E_{\gamma}\simeq
1.9$ GeV), and (iii) the peak around $W=2.1$ GeV in the excitation functions
at very forward angles. Furthermore, $D_{15}(2080)$ also accounts for the
bump-like structure at $W\simeq 2.1$ GeV in the inclusive quasi-free $\gamma
d\rightarrow np\eta^{\prime}$ cross section measured by CBELSA/TAPS.
$S_{11}(1920)$ seems to be needed in the reaction, with which the total cross
section near threshold for $\gamma p\rightarrow\eta^{\prime}p$ is improved
slightly. However, the differential cross sections, excitation functions, and
beam asymmetries are not sensitive to $S_{11}(1920)$. To confirm
$S_{11}(1920)$, more accurate observations are needed.
Furthermore, it should be mentioned that the polarized beam asymmetries are
found to be sensitive to $D_{13}(1520)$, although its effects on the
differential cross sections and total cross sections are negligible. There is
no obvious evidence of the $P$-, $D_{13}$-, $F$-, and $G$-wave resonances with
a mass around 2.0 GeV in the reactions.
To better understand the physics in the $\gamma p\rightarrow\eta^{\prime}p$
and $\gamma n\rightarrow\eta^{\prime}n$ reactions, we expect more accurate
measurements of the following observables for both of the processes: (i) the
total cross section in the energy region $E_{\gamma}\simeq(1.55\sim 2.1)$ GeV,
(ii) the fixed-angle excitation functions at very forward angles from
threshold up to $W\simeq 2.3$ GeV, (iii) the differential cross sections in
the energy region $E_{\gamma}\simeq(1.6\sim 1.9)$ GeV, and (iv) the beam
asymmetries in the energy region $E_{\gamma}\simeq(1.6\sim 2.0)$ GeV.
## Acknowledgements
The authors thank B. Krusche for providing us the data of $\eta^{\prime}$
photoproduction off quasi-free nucleons. This work is supported, in part, by
the National Natural Science Foundation of China (Grants 10775145, 11075051
and 11035006), Chinese Academy of Sciences (KJCX2-EW-N01), Ministry of Science
and Technology of China (2009CB825200), the Program for Changjiang Scholars
and Innovative Research Team in University (PCSIRT, No. IRT0964), the Program
Excellent Talent Hunan Normal University, and the Hunan Provincial Natural
Science Foundation (11JJ7001).
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|
arxiv-papers
| 2011-10-25T11:18:45 |
2024-09-04T02:49:23.600225
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xian-Hui Zhong and Qiang Zhao",
"submitter": "Xianhui Zhong",
"url": "https://arxiv.org/abs/1110.5466"
}
|
1110.6008
|
QUANTUM TUNNELING
IN
BLACK HOLES
Thesis submitted for the degree of
Doctor of Philosophy (Science)
of
University of Calcutta, India
December, 2010
Bibhas Ranjan Majhi
Department of Theoretical Sciences
$\mathcal{TO}$
$\mathcal{MY\ NEPHEW}$
## Acknowledgements
This thesis is the culminative outcome of four years work, which has been made
possible by the blessings and support of many individuals. I take this
opportunity to express my sincere gratitude to all of them.
First and foremost, I would like to thank Prof. Rabin Banerjee, my thesis
supervisor. His uncanny ability to select a particular problem, a keen and
strategic analysis of it and deep involvement among the students makes him
something special. Thank you Sir for giving me all that could last my entire
life.
In addition to this, I like to express my sincere thanks to Prof. Claus
Kiefer, Universität zu Köln, Germany and Dr. Elias C. Vagenas, Academy of
Athens, Greece for their constant academic help and stimulating collaboration
throughout my research activity. I thank Prof. Subir Ghosh, ISI Kolkata. It
was a quite nice experience to work with them.
I would also like to express my gratitude Prof. Sandip K. Chakrabarti, Dr.
Debashis Gangopadhyay, Dr. Archan S. Majumdar, Dr. Biswajit Chakraborty and
Prof. Subhrangshu Sekhar Manna for helping me in my academics at Satyendra
Nath National Center for Basic Sciences (SNBNCBS).
Prof. Jayanta Kumar Bhattacharjee and Prof. Binayak Dutta Roy have always been
available to clarify very elementary but at same time subtle concepts in
physics. I sincerely thank them for giving me their valuable attention.
I thank Prof. Arup K. Raychaudhuri, Director of SNBNCBS, for providing an
excellent academic atmosphere.
I am also thankful to the Library staff of SNBNCBS for their support.
I am indebted to Prof. Manoranjan Saha, Prof. Amitava Raychaudhuri, Prof.
Anirban Kundu, Dr. Debnarayana Jana, University of Calcutta, for giving me
crucial suggestions and guidance, whenever I required. I would also like to
thank Prof. Tapan Kumar Das , University of Calcutta and my college teachers
Dr. Debabrata Das, Prof. Maynak Gupta.
My life during this voyage has been made cherishable and interesting by some
colourful and memorable personalities to whom I would like to express my
heartiest feelings. I am indebted to my seniors - Chandrashekhar da and Mitali
di. Things, which I gained from them, turn out to be most important in my
thesis work. I would like to thank all of them for their brotherly support.
Saurav Samanta and Shailesh Kulkarni, seniors and friends, deserve special
mention for those strategic discussions, on both the academic as well as non-
academic fronts. Then come my brilliant and helpful group mates, ranging from
enthusiastic Sujoy, Debraj, Sumit, Dibakar to Biswajit. I am grateful to
Himadri for his sometimes meaningless but still delightful gossips and also to
his group for sharing precious moments. I would also like to thank all the
cricket and football players of SNBNCBS for making my stay enjoyable.
I owe my deepest gratitude to each and every member of my of family.
Especially, I would like to express my heartiest love and respect to my
parents for their constant care, encouragement and blessings. Another two
pairs of my family, who stands in the same footing as my parents, are my uncle
(Jethu) and aunt (Jethima) and my would be father-in-law and mother-in-law. I
give paranams to all. I express my love to my elder brother Pinku and elder
sisters Tinku, Rinku and my brother-in-law, Subikash and sister-in-law, Dona
for understanding and helping me, on many occasions.
I am indebted to my better halves: my grandmothers (Didima and Thakurma) and
my beloved nephew, Oishik (Sona) for making my life fill of joy and greetings.
Finally, I express my heartiest love to Priyanka (Mousam, my would be wife)
with whom I shared all my moments (sweet or bitter) for making everything
possible from my post-graduate life.
## List of publications
1. 1.
Gauge Theories on A(dS) space and Killing Vectors.
Rabin Banerjee and Bibhas Ranjan Majhi
Annals Phys. 323, 705 (2008) [arXiv:hep-th/0703207].
2. 2.
Crypto-Harmonic Oscillator in Higher Dimensions: Classical and Quantum
Aspects.
Subir Ghosh and Bibhas Ranjan Majhi
J. Phys. A41, 065306 (2008) [arXiv:0709.4325].
3. 3.
Quantum Tunneling and Back Reaction.
Rabin Banerjee and Bibhas Ranjan Majhi
Phys. Lett. B662, 62 (2008) [arXiv:0801.0200].
4. 4.
Noncommutative Black Hole Thermodynamics.
Rabin Banerjee, Bibhas Ranjan Majhi and Saurav Samanta
Phys. Rev. D77, 124035 (2008) [arXiv:0801.3583].
5. 5.
Noncommutative Schwarzschild Black Hole and Area Law.
Rabin Banerjee, Bibhas Ranjan Majhi and Sujoy Kumar Modak
Class. Quant. Grav. 26, 085010 (2009) [arXiv:0802.2176].
6. 6.
Quantum Tunneling Beyond Semiclassical Approximation.
Rabin Banerjee and Bibhas Ranjan Majhi
JHEP 0806, 095 (2008) [arXiv:0805.2220].
7. 7.
Quantum Tunneling and Trace Anomaly.
Rabin Banerjee and Bibhas Ranjan Majhi
Phys. Lett. B674, 218 (2009) [arXiv:0808.3688].
8. 8.
Fermion Tunneling Beyond Semiclassical Approximation.
Bibhas Ranjan Majhi
Phys. Rev. D79, 044005 (2009) [arXiv:0809.1508].
9. 9.
Connecting anomaly and tunneling methods for Hawking effect through chirality.
Rabin Banerjee and Bibhas Ranjan Majhi
Phys. Rev. D79, 064024 (2009) [arXiv:0812.0497].
10. 10.
Hawking Radiation due to Photon and Gravitino Tunneling.
Bibhas Ranjan Majhi and Saurav Samanta
Annals Phys. 325, 2410 (2010) [arXiv:0901.2258].
11. 11.
Hawking black body spectrum from tunneling mechanism.
Rabin Banerjee and Bibhas Ranjan Majhi
Phys. Lett. B675, 243 (2009) [arXiv:0903.0250].
12. 12.
Quantum tunneling and black hole spectroscopy.
Rabin Banerjee, Bibhas Ranjan Majhi and Elias C. Vagenas
Phys. Lett. B686, 279 (2010) [arXiv:0907.4271].
13. 13.
Hawking radiation and black hole spectroscopy in Horava-Lifshitz gravity.
Bibhas Ranjan Majhi
Phys. Lett. B686, 49 (2010) [arXiv:0911.3239] .
14. 14.
New Global Embedding Approach to Study Hawking and Unruh Effects.
Rabin Banerjee and Bibhas Ranjan Majhi
Phys. Lett. B690, 83 (2010) [arXiv:1002.0985].
15. 15.
Statistical Origin of Gravity.
Rabin Banerjee and Bibhas Ranjan Majhi
Phys. Rev. D81, 124006 (2010) [arXiv:1003.2312].
16. 16.
A Note on the Lower Bound of Black Hole Area Change in Tunneling Formalism.
Rabin Banerjee, Bibhas Ranjan Majhi and Elias C. Vagenas
Europhys. Lett. 92, 20001 (2010) [arXiv:1005.1499].
17. 17.
Quantum gravitational correction to the Hawking temperature from the Lemaitre-
Tolman-Bondi model.
Rabin Banerjee, Claus Kiefer and Bibhas Ranjan Majhi
Phys. Rev. D82, 044013 (2010) [arXiv:1005.2264].
18. 18.
Killing Symmetries and Smarr Formula for Black Holes in Arbitrary Dimensions.
Rabin Banerjee, Bibhas Ranjan Majhi, Sujoy Kumar Modak and Saurav Samanta
Phys. Rev. D82, 124002 (2010) [arXiv:1007.5204].
This thesis is based on the papers numbered by [3,4,6,9,11,12,14,15] whose
reprints are attached at the end of the thesis.
##
QUANTUM TUNNELING
IN
BLACK HOLES
###### Contents
1. 1 Introduction
1. 1.1 Overview
2. 1.2 Outline of the thesis
2. 2 The tunneling mechanism
1. 2.1 Hamilton-Jacobi method
1. 2.1.1 Schwarzschild like coordinate system
2. 2.1.2 Painleve coordinate system
2. 2.2 Radial null geodesic method
3. 2.3 Calculation of Hawking temperature
1. 2.3.1 Schwarzschild black hole
2. 2.3.2 Kerr black hole
4. 2.4 Discussions
3. 2.A Ingoing and outgoing modes
4. 3 Null geodesic approach
1. 3.1 Back reaction effect
2. 3.2 Inclusion of noncommutativity
1. 3.2.1 Schwarzschild black hole in noncommutative space
2. 3.2.2 Noncommutative Hawking temperature, tunneling rate and entropy in the presence of back reaction
3. 3.3 Discussions
5. 3.A Incomplete gamma function
6. 3.B Some useful formulas
7. 4 Tunneling mechanism and anomaly
1. 4.1 Metric and null coordinates
2. 4.2 Chirality conditions
3. 4.3 Chirality, gravitational anomaly and Hawking flux
4. 4.4 Chirality, quantum tunneling and Hawking temperature
5. 4.5 Discussions
8. 5 Black body spectrum from tunneling mechanism
1. 5.1 Black body spectrum and Hawking flux
2. 5.2 Discussions
9. 6 Global embedding and Hawking-Unruh effect
1. 6.1 Reduced global embedding
1. 6.1.1 Schwarzschild metric
2. 6.1.2 Reissner-Nordstr$\ddot{\textrm{o}}$m metric
3. 6.1.3 Schwarzschild-AdS metric
2. 6.2 Kerr-Newman metric
3. 6.3 Conclusion
10. 6.A Dimensional reduction technique
11. 7 Quantum tunneling and black hole spectroscopy
1. 7.1 Near horizon modes
2. 7.2 Entropy and area spectrum
3. 7.3 Discussions
12. 8 Statistical origin of gravity
1. 8.1 Partition function and the relation $S_{bh}=\frac{E}{2T_{H}}$
2. 8.2 Identification of $E$ in Einstein’s gravity
3. 8.3 Discussions
13. 9 Conclusions
## Chapter 1 Introduction
### 1.1 Overview
This is a short overview of the vast subject of black holes. It specifically
highlights those issues which are relevant for the present thesis. The search
for a theory of quantum gravity drives a great deal of research in theoretical
physics today, and much has been learned along the way, but convincing success
remains elusive. There are two parts of general relativity: the framework of
space-time curvature and its influence on matter, and the dynamics of the
metric in response to energy-momentum (as described by Einstein’s equation).
Lacking the true theory of quantum gravity, we may still take the first part
of GR - the idea that matter fields propagate on a curved space-time
background - and consider the case where those matter fields are quantum
mechanical. In other words, we take the metric to be fixed, rather than
obeying some dynamical equations, and study quantum field theory in the curved
space-time.
Classical solutions of Einstein’s equation gives several metrics of space-time
in absence (Schwarzschild metric) or in presence (e.g. Reissner-Nordstrom
metric) of matter fields. Both of these solutions show there exists a region
of space-time in which information can enter, but nothing can come out from
it. The partition that separates this region (known as black hole) is usually
called the event horizon. The black holes are usually formed from the collapse
of star etc. According to the No Hair theorem, collapse leads to a black hole
endowed with small number of macroscopic parameters (mass, charge, angular
momentum) with no other free parameters. All these are classical pictures.
Hawking showed that the area of a black hole never decreases \- known as area
theorem [1]. This fact attracted Bekenstein a lot. A simple thought experiment
led him to associate entropy with the black hole. Then he [2] proposed that a
black hole has an entropy $S_{bh}$ which is some finite multiple $\eta$ of its
area of the event horizon $A$. He was not able to determine the exact value of
$\eta$, but gave heuristic arguments for conjecturing that it was
$\frac{ln2}{8\pi}$. Also, several investigations reveled that classical black
hole mechanics can be summarized by the following three basic laws [3],
1. 1.
Zeroth law : The surface gravity $\kappa$ of a black hole is constant on the
horizon.
2. 2.
First law : The variations in the black hole parameters, i.e mass $M$, area
$A$, angular momentum $L$, and charge $Q$, obey
$\delta M=\frac{\kappa}{8\pi}\delta A+\Omega\delta L-V\delta Q$ (1.1)
where $\Omega$ and $V$ are the angular velocity and the electrostatic
potential, respectively.
3. 3.
Second law : The area of a black hole horizon $A$ is nondecreasing in time
[1],
$\delta A\geq 0.$ (1.2)
These laws have a close resemblance to the corresponding laws of
thermodynamics. The zeroth law of thermodynamics says that the temperature is
constant throughout a system in thermal equilibrium. The first law states that
in small variations between equilibrium configurations of a system, the
changes in the energy and entropy of the system obey equation (1.1), if the
surface gravity $\kappa$ is replaced by a term proportional to temperature of
the system (other terms on the right hand side are interpreted as work terms).
The second law of thermodynamics states that, for a closed system, entropy
always increases in any (irreversible or reversible) process. Therefore from
Bekenstein’s argument and the first law of black hole mechanics one might say
$T_{H}=\epsilon\kappa$ and $S_{bh}=\eta A$ with $8\pi\eta\epsilon=1$.
Bekenstein proposed that $\eta$ is finite and it is equal to
$\frac{ln2}{8\pi}$. Then one would get $\epsilon=\frac{1}{ln2}$ and so
$T_{H}=\frac{\kappa}{ln2}$.
Later on, the study of QFT in curved space-time by Hawking in 1974-75 [4, 5]
showed that black holes are not really black, instead emit thermal radiation
at temperature ($T_{H}$) proportional to surface gravity ($\kappa$) of black
hole - popularly known as Hawking effect. The exact expression was found to be
[5]:
$\displaystyle T_{H}=\frac{\hbar c\kappa}{2\pi k_{B}},$ (1.3)
where $c$, $\hbar$ and $k_{B}$ are respectively the velocity of light, plank
constant and Boltzmann constant. This is known as Hawking temperature
111Although in (1.3) we keep all the fundamental constants explicitly, for
later analysis, whenever any particular unit will be chosen, that will be
mentioned there.. For the Schwarzschild black hole $\kappa=\frac{c^{2}}{4GM}$
where $M$ is the mass of the black hole and $G$ is the gravitational constant.
All these reflects the fact that Hawking effect incorporates quantum
mechanics, gravity as well as thermodynamics. The key idea behind quantum
particle production in curved space-time is that the definition of a particle
is vacuum dependent. It depends on the choice of reference frame. Since the
theory is generally covariant, any time coordinate, possibly defined only
locally within a patch, is a legitimate choice with which to define positive
and negative frequency modes. Hawking considered a massless quantum scalar
field moving in the background of a collapsing star. If the quantum field was
initially in the vacuum state (no particle state) defined in the asymptotic
past, then at late times it will appear as if particles are present in that
state. Hawking showed [5], by explicit computation of the Bogoliubov
coefficients (see also [6, 7] for detailed calculation of Bogoliubov
coefficients) between the two sets of vacuum states defined at asymptotic past
and future respectively, that the spectrum of the emitted particles is
identical to that of black body with the temperature (1.3). This remarkable
discovery indeed helps us to get various physical information about the
classically forbidden region inside the horizon. Since then people thought
that the black holes may play a major role in the attempts to shed some light
on the nature of quantum theory of gravity as the role played by atoms in
early development of quantum mechanics. Hence QFT on curved space-time and
Hawking effect attracted the physicists for their beauty and usefulness in
various aspects.
Hawking then realised that Bekenstein’s idea was consistent. In fact, since
the black hole temperature is given by (1.3), $\epsilon=\frac{1}{2\pi}$ and
hence $\eta=\frac{1}{4}$. This leads to the famous Bekenstein-Hawking area law
for entropy of black hole
$\displaystyle S_{bh}=\frac{A}{4},$ (1.4)
where $A$ is the area of the event horizon 222Here all the fundamental
constants are chosen to be unity.. This astonishing result is obtained using
the approximation that the matter field behaves quantum mechanically but the
gravitational field (metric) satisfy the classical Einstein equation. This
semi-classical approximation holds good for energies below the Planck scale
[5]. Although it is a semi-classical result, Hawking’s computation is
considered an important clue in the search for a theory of quantum gravity.
Any theory of quantum gravity that is proposed must predict black hole
evaporation.
Apart from Hawking’s original calculation there are other semi-classical
approaches. We summarise these briefly. S. Hawking and G. Gibbons, in 1977 [8]
developed an approach based on the Euclidean quantum gravity. In this approach
they computed an action for gravitational field, including the boundary term,
on the complexified space-time. The purely imaginary values of this action
gives a contribution of the metrics to the partition function for a grand
canonical ensemble at Hawking temperature (1.3). Using this, they were able to
show that the entropy associated with these metrics is always equal to (1.4).
Almost at the same time, Christensen and Fulling [9], by exploiting the
structure of trace anomaly, were able to obtain the expectation value for each
component of the stress tensor $\langle T_{\mu\nu}\rangle$, which eventually
lead to the Hawking flux. This approach is exact in $(1+1)$ dimensions,
however in $3+1$ dimensions, the requirements of spherical symmetry, time
independence and covariant conservation are not sufficient to fix completely
the flux of Hawking radiation in terms of the trace anomaly [6, 9]. There is
an additional arbitrariness in the expectation values of the angular
components of the stress tensor.
Later on, S. Robinson and F. Wilczek [10, 11, 12] gave a new approach to
compute the Hawking flux from a black hole. This approach is based on gauge
and gravitational or diffeomorphism anomalies. Basic and essential fact used
in their analysis is that the theory of matter fields (scalar or fermionic) in
the $3+1$ dimensional static black hole background can effectively be
represented, in the vicinity of event horizon, by an infinite collection of
free massless $1+1$ dimensional fields, each propagating in the background of
an effective metric given by the $r-t$ sector of full $3+1$ dimensional metric
333Such a dimensional reduction of matter fields has been already used in the
analysis of [13, 14] to compute the entropy of $2+1$ dimensional $BTZ$ black
hole.. By definition the horizon is a null surface and hence the region inside
it is causally disconnected from the exterior. Thus, in the region near to the
horizon the modes which are going into the black hole do not affect the
physics outside the horizon. In other words, the theory near the event horizon
acquires a definite chirality. Any two dimensional chiral theory in general
curved background possesses both gauge and gravitational anomaly [15]. This
anomaly is manifested in the nonconservation of the current or the stress
tensor. The theory far away from the event horizon is $3+1$ dimensional and
anomaly free and the stress tensor in this region satisfies the usual
conservation law. Consequently, the total energy-momentum tensor, which is a
sum of two contribution from the two different regions, is also anomalous.
However, it becomes anomaly free once we take into account the contribution
from classically irrelevant ingoing modes. This imposes restrictions on the
structure of the energy-momentum tensor and is ultimately responsible for the
Hawking radiation [10]. The expression for energy-momentum flux obtained by
this anomaly cancellation approach is in exact agreement with the flux from
the perfectly black body kept at Hawking temperature [10]. In this approach
they used consistent expression for anomaly (satisfying Wess-Zumino
consistency condition) but used a covariant boundary condition.
Recently, a technically simple (only one Ward identity) and conceptually
cleaner (covariant expression for anomaly with covariant BC) derivation of
Hawking flux was introduced by Banerjee and Kulkarni [16, 17]. In addition to
this, a new method [18], to obtain the Hawking flux using chiral effective
action, was put forwarded by them. In all these approaches, the covariant
boundary condition is applied by hand. Later on, it was shown again by them
that such a boundary condition is compatible with the choice of Unruh vacuum
[19]. The connection of the diffeomorphism anomaly approach with the earlier
trace anomaly approach [9] was also elaborated [20, 21].
Interestingly, none of the existing approaches to study Hawking effect,
however, corresponds directly to one of the heuristic pictures that visualises
the source of radiation as tunneling, first stated in [5]. Later on, this
picture was mathematically introduced to discuss the Hawking effect [22, 23].
This picture is similar to an electron-positron pair creation in a constant
electric field. The idea is that pair production occurs inside the event
horizon of a black hole. One member of the pair corresponds to the ingoing
mode and other member corresponds to the outgoing mode. The outgoing mode is
allowed to follow classically forbidden trajectories, by starting just behind
the horizon onward to infinity. So this mode travels back in time, since the
horizon is locally to the future of the external region. The actual physical
picture is that the tunneling occures by the shrinking of the horizon so that
the particle effectively moves out. Thus the classical one particle action
becomes complex and so the tunneling amplitude is governed by the imaginary
part of this action for the outgoing mode. However, the action for the ingoing
mode must be real, since classically a particle can fall behind the horizon.
This is an important point of this mechanism as will be seen later. Also,
since it is a near horizon theory and the tunneling occures radially, the
phenomenon is effectively dominated by the two dimensional ($t-r$) metric.
This follows form the fact that near the horizon all the angular part can be
neglected and the solution of the field equation corresponds to angular
quantum number $l=0$ which is known as $s$-wave [22]. Hence, the essence of
tunneling based calculations is, thus, the computation of the imaginary part
of the action for the process of $s$-wave emission across the horizon, which
in turn is related to the Boltzmann factor for the emission at the Hawking
temperature. It also reveals that the presence of the event horizon is
necessary and the Hawking effect is a completely quantum mechanical
phenomenon. There are two different methods in literature to calculate the
imaginary part of the action: one is by Srinivasan et al [22] \- the Hamilton-
Jacobi (HJ) method 444For more elaborative discussions and further development
on HJ method see [24, 25, 26]. and another is radial null geodesic method
which was first given by Parikh - Wilczek [23] 555To find the basis of this
method see [27, 28, 29].. Both these approaces will be discussed in this
thesis.
Historically, another phenomenon was discovered by Unruh [30] \- Known as
Unruh effect \- in an attempt to understand the physics underlying the Hawking
effect [5]. The basic idea of the Unruh effect is based on the equivalence
principle \- locally gravitational effect can be ignored by choosing a
uniformly accelerated frame and the observers with different notions of
positive and negative frequency modes will disagree on the particle content of
a given state. A uniformly accelerated observer on the Minkowski space-time
percives a horizon. The space-time seen by the observer is known as Rindler
space-time and so the observer is usually called as the Rindler observer.
Although, an inertial observer would describe the Minkowski vacuum as being
completely empty, the Rindler observer will detect particles in that vacuum. A
detailed calculation tells that the emission spectrum exactly matches with
that of the black body with the temperature given by [30],
$\displaystyle T=\frac{{\hbar{\tilde{a}}}}{2\pi}$ (1.5)
where $a$ is the accleration of the Rindler observer. The similarity with
Hawking temperature is obvious with $a\rightarrow\kappa$. It is now well
understood that Hawking effect is related to the event horizon of a black hole
intrinsic to the space-time geometry while Unruh effect connects the horizon
associated with a uniformly accelerated observer on the Minkowski space-time.
A unified description of them was first put forward by Deser and Levin [31,
32] followed from an earlier attempt [33]. This is called the global embedding
Minkowskian space (GEMS) approach. In this approach, the relevant detector in
curved space-time (namely Hawking detector) and its event horizon map to the
Rindler detector in the corresponding higher dimensional flat embedding space
[34, 35] and its event horizon. Then identifying the acceleration of the Unruh
detector and using (1.5), the Unruh temperature (or local Hawking temperature)
was calculated. Finally, use of the Tolman relation [36] yields the Hawking
temperature. Subsequently, this unified approach to determine the Hawking
temperature using the Unruh effect was applied for several black hole space-
times [37, 38, 39]. However the results were confined to four dimensions and
the calculations were done case by case, taking specific black hole metrics.
It was not clear whether the technique was applicable to complicated examples
like the Kerr-Newman metric which lacks spherical symmetry.
In the mean time, after the discovery of Hawking effect, it was believed that
the black holes may give some hints to find the quantum theory of gravity. It
is then natural to consider quantization of a black hole. This was first
pioneered by Bekenstein [40, 41]. The idea was based on the remarkable
observation that the horizon area of a non-extremal black hole behaves as a
classical adiabatic invariant quantity. In the spirit of the Ehrenfest
principle, any classical adiabatic invariant corresponds to a quantum entity
with discrete spectrum, Bekenstein conjectured that the horizon area of a non-
extremal black hole should have a discrete eigenvalue spectrum. To elucidate
the spacing of the area levels he used Christodoulou’s reversible process [42]
\- the assimilation of a neutral point particle by a non-extremal black hole.
Bekenstein pointed out that the limit of a point particle is not a legal one
in quantum theory. Because, according to the Heisenberg’s uncertainty
principle, the particle cannot be both at the horizon and at a turning point
of its motion. Considering a finite size of the particle - not smaller than
the Compton wavelength - he found a lower bound on the increase in the black
hole surface area [2, 43]:
$\displaystyle(\Delta A)_{min}=8\pi l_{p}^{2}$ (1.6)
where $l_{p}=(\frac{G\hbar}{c^{3}})^{1/2}$ is the Planck length (we use
gravitational units in which $G=c=1$). The independence of the black hole
parameters in the lower bound shows its universality and hence it is a strong
evidence in favor of a uniformly spaced area spectrum for a quantum black
holes.
These ideas led to a new research direction; namely the derivation of the area
and thus the entropy spectrum of black holes utilizing the quasinormal modes
(QNM) of black holes [44]. According to this method, since QNM frequencies are
the characteristic of the black hole itself, the latter must have an adiabatic
invariant quantity. Its form is given by energy of the black hole divided by
this frequency, as happens in classical mechanics. Hod showed for
Schwarzschild black hole that if one considers the real part of the QNM
frequency only, then this adiabatic invariant quantity is actually related to
area of the black hole horizon. Now use of Bohr-Sommerfield quantization rule
gives the spectrum for the area which is equispaced. Then by the well known
Bekenstein-Hawking area law, the entropy spectrum is obtained. In this case
the spacing of this entropy spectrum is given by $\Delta S_{bh}=\ln 3$.
Another significant attempt was to fix the Immirzi parameter in the framework
of Loop Quantum Gravity [45] but it was unsuccessful [46]. Later on Kunstatter
[47] gave an explicit form of the adiabatic invariant quantity for the black
hole:
$\displaystyle I_{adiab}=\int\frac{dW}{\Delta f(W)},\,\,\,\ \Delta
f=f_{n+1}-f_{n}$ (1.7)
where ‘$W$’ and ‘$f$’ are the energy and the frequency of the QNM
respectively. The Borh-Sommerfield quantization rule is given by,
$\displaystyle I_{adiab}=n\hbar$ (1.8)
which is valid for semi-classical (large $n$) limit. For the real part of the
frequency of QNM, (1.7) can be shown to be related to black hole entropy
which, ultimately by (1.8), yields the entropy spectrum. For Schwarzschild
black hole it yields the same spacing as obtained by Hod [44]. This, however,
disagrees with Bekenstein’s result, $\Delta S_{bh}=2\pi$ [2]. In a recent work
[48], Maggiore told that a black hole behaves like a damped harmonic
oscillator whose frequency is given by
$f=(f_{R}^{2}+f_{I}^{2})^{\frac{1}{2}}$, where $f_{R}$ and $f_{I}$ are the
real and imaginary parts of the frequency of the QNM. In the large $n$ limit
$f_{I}>>f_{R}$. Consequently one has to use $f_{I}$ rather than $f_{R}$ in the
adiabatic quantity (1.7). It then leads to Bekenstein’s result. With this new
interpretation, entropy spectrum for the most general black hole has been
calculated in [49], which leads to an identical conclusion. In addition, it
has been tested that the entropy spectrum is equidistance even for more
general gravity theory (e.g. Einstein-Gauss-Bonnet theory), but that of area
is not alaways equispaced, particularly, if the entropy is not proportional to
area [50]. In this sense quantization of entropy is more fundamental than that
of area.
A universal feature for black hole solutions, in a wide class of theories, is
that the notions of entropy and temperature can be attributed to them [2, 4,
3, 51]. Also, of all forces of nature gravity is clearly the most universal.
Gravity influences and is influenced by everything that carries an energy, and
is intimately connected with the structure of space-time. The universal nature
of gravity is also demonstrated by the fact that its basic equations closely
resemble the laws of thermodynamics [3, 51, 52, 53]. So far, there has not
been a clear explanation for this resemblance. Gravity is also considerably
harder to combine with quantum mechanics than all the other forces. The quest
for unification of gravity with these other forces of nature, at a microscopic
level, may therefore not be the right approach. It is known to lead to many
problems, paradoxes and puzzles. Many physicists believe that gravity and
space-time geometry are emergent. Also string theory and its related
developments have given several indications in this direction. Particularly
important clues come from the AdS/CFT correspondence. This correspondence
leads to a duality between theories that contain gravity and those that don’t.
It therfore provides evidence for the fact that gravity can emerge from a
microscopic description that doesn’t know about its existence 666Such a
prediction was first given long ago by Sakharov [54].. The universality of
gravity suggests that its emergence should be understood from general
principles that are independent of the specific details of the underlying
microscopic theory.
### 1.2 Outline of the thesis
This thesis, based on the work [55, 56, 57, 58, 59, 60, 61, 62], is focussed
towards the applications of field theory, classical as well as quantum, to
study black holes – mainly the Hawking effect. This is discussed by the
quantum tunneling mechanism. Here we give a general frame work of the existing
tunneling mechanism, both the radial null geodesic and Hamilton – Jacobi
methods. On the radial null geodesic method side, we study the modifications
to the tunneling rate, Hawking temperature and the Bekenstein-Hawking area law
by including the back reaction as well as non-commutative effects in the
space-time.
A major part of the thesis is devoted to the different aspects of the
Hamilton-Jacobi (HJ) method. A reformulation of this method is first
introduced. Based on this, a close connection between the quantum tunneling
and the gravitational anomaly mechanisms to discuss Hawking effect, is put
forwarded. An interesting advantage of this reformulated HJ method is that one
can get directly the emission spectrum from the event horizon of the black
hole, which was missing in the earlier literature. Also, the quantization of
the entropy and area of a black hole is discussed in this method.
Another part of the thesis is the introduction of a new type of global
embedding of curved space-time to higher dimensional Minkowskian space-time
(GEMS). Using this a unified description of the Hawking and Unruh effects is
given. Advantage of this approach is, it simplifies as well as generalises the
conventional embedding. In addition to the spherically symmetric space-times,
the Kerr-Newman black hole is exemplified.
Finally, following the above ideas and the definition of partition function
for gravity, it is shown that extremization of entropy leads to the Einstein’s
equations of motion. In this frame work, a relation between the entropy,
energy and the temperature of a black hole is given where energy is shown to
be the Komar expression. Interestingly, this relation is the generalized Smarr
formula. In this analysis, the GEMS method provides the law of equipartition
of energy as an intermediate step.
The whole thesis is consists of $9$ \- chapters, including this introductory
part. Chapter wise summary is given below.
Chapter -2: The tunneling mechanism: In this chapter, we present a general
framework of tunneling mechanism within the semi-classical approximation. The
black hole is considered to be a general static, spherically symmetric one.
First, the HJ method is discussed both in Schwarzschild like coordinates and
Painleve coordinates. Then a general methodology of the radial null geodesic
method is presented. Here the tunneling rate, which is related to the
imaginary part of the action, is shown to be equal to the exponential of the
entropy change of the black hole. In both the methods, a general expression
for Hawking temperature is obtained, which ultimately reduces to the Hawking
expression (1.3). Finally, using this general expression, calculation of
Hawking temperature for some particular black hole metrics, is explicitly
done. Chapter -3: Null geodesic approach: In this chapter, we provide an
application of the general frame work, discussed in the previous chapter, for
the radial null geodesic method, to incorporate back reaction as well as
noncommutative effects in the space-time. Here the main motivation is to find
their effects on the thermodynamic quantities. First, starting from a modified
surface gravity of a black hole due to one loop back reaction effect, the
tunneling rate is obtained. From this, the temperature and the area law are
derived. The semi-classical Hawking temperature is altered. Interestingly, the
leading order correction to the area law is logarithmic of the horizon area of
the black hole while the non-leading corrections are the inverse powers of the
area. The coefficient of the logarithmic term is related to the trace anomaly.
Similar type of corrections were also obtained earlier [63, 64, 65, 66, 67,
68, 69, 70, 71] by different methods.
Next, we shall apply our general formulation to discuss various thermodynamic
properties of a black hole defined in a noncommutative Schwarzschild space
time where back reaction is also taken into account. In particular, we are
interested in the black hole temperature when the radius is very small. Such a
study is relevant because noncommutativity is supposed to remove the so called
“information paradox” where for a standard black hole, temperature diverges as
the radius shrinks to zero. The Hawking temperature is obtained in a closed
form that includes corrections due to noncommutativity and back reaction.
These corrections are such that, in some examples, the “information paradox”
is avoided. Expressions for the entropy and tunneling rate are also found for
the leading order in the noncommutative parameter. Furthermore, in the absence
of back reaction, we show that the entropy and area are algebraically related
in the same manner as occurs in the standard Bekenstein-Hawking area law.
Chapter -4: Tunneling mechanism and anomaly: Several existing methods to study
Hawking effect yield similar results. The universality of this phenomenon
naturally tempts us to find the underlying mechanism which unifies the
different approaches. Recently, two widely used approaches – gravitational
anomaly method and quantum tunneling method – can be described in a unified
picture, since these two have several similarities in their techniques. One of
the most important and crucial step in the tunneling approach (in both the
methods) is that the tunneling of the particle occurs radially and its a near
horizon phenomenon. This enforces that only the near horizon ($t-r$) sector of
the original metric is relevant. Also, the ingoing mode is completely trapped
inside the horizon. Similar step is also invoked in the gravitational (chiral)
anomaly approach [10, 11, 12, 16]. Here, since near the event horizon the
theory is dominated by the two dimensional, ($t-r$) sector of the metric, and
the ingoing mode is trapped inside the horizon, the theory is chiral. Hence
one should has the gravitational anomaly in the quantum level. Therefore, one
might thought that these two approaches - quantum tunneling and anomaly
methods - can be discussed in an unified picture.
We begin this exercise by introducing the chirality conditions on the modes
and the energy-momentum tensor in chapter-4. The Klein-Gordon equation under
the effective ($t-r$) sector of the original metric shows that the there exits
a general solution which is a linear combination of two solutions. One is left
moving and function of only one null tortoise coordinate ($v$) while other is
right moving which is function of the other null tortoise coordinate ($u$).
From this information it is easy to find the chirality conditions. Then use of
these conditions on the usual expressions for the anomaly in the non-chiral
theory in two dimensions leads to the chiral anomaly expression. Finally,
following the approach by Banerjee et al [16], it is easy to find the
expression for the Hawking flux. Another portion of this chapter is dedicated
to show that the same chirality conditions are enough to find the Hawking
temperature in quantum tunneling method. First, the Hamilton-Jacobi equations
are obtained from these conditions, which are derived in the usual analysis
from the field equations. Then a reformulation of tunneling method is given in
which the trapping of the left mode is automatically satisfied. The right mode
tunnels through the horizon with a finite probability which is exactly the
Boltzmann factor. This immediately leads to the Hawking temperature. Thus,
this analysis reflects the crucial role of the chirality to give a unified
description of both the approaches to discuss Hawking effect.
Chapter -5: Black body spectrum from tunneling mechanism: So far, in the
tunneling mechanism only the Hawking temperature was obtained by comparing the
tunneling rate with the Boltzmann factor. The discussion of the emission
spectrum is absent and hence it is not clear whether this temperature really
corresponds to the emission spectrum from the black hole event horizon. This
shortcoming is addressed in chapter -5. Following the modified tunneling
approach, introduced in the previous chapter, the reduced density matrix for
the outgoing particles, as seen from the asymptotic observer, is constructed.
Then determination of the average number of outgoing particles yields the Bose
or Fermi distribution depending on the nature of the particles produced inside
the horizon. The distributions come out to be exactly similar to those in the
case of black body radiation. It is now easy to identify the temperature
corresponding to the emission spectrum. The temperature here we obtain is just
the Hawking expression. Thereby we provide a complete description of the
Hawking effect in the tunneling mechanism.
Chapter -6: Global embedding and Hawking - Unruh Effect: After Hawking’s
discovery, Unruh showed that an uniformly accelerated observer on the
Minkowski space-time sees a thermal radiation from the Minkowski vacuum. Later
on, Levin and Deser gave a unified picture of these two effects by using the
globally embedding of the curved space-time in the higher dimensional
Minkowski space-time. Such an interesting analysis was done using the
embedding of the full curved metric and was confined within the spherically
symmetric black hole space-time. The main difficulty to discuss for more
general space-times is the finding of the embeddings.
This issue is addressed in chapter - 6. Since, the thermodynamic quantities of
a black hole are determined by the horizon properties and near the horizon the
effective theory is dominated by the two dimensional ($t-r$) metric, it is
sufficient to consider the embedding of this two dimensional metric.
Considering this fact, a new type of global embedding of curved space-times in
higher dimensional flat ones is introduced to present a unified description of
Hawking and Unruh effects. Our analysis simplifies as well as generalises the
conventional embedding approach.
Chapter -7: Quantum tunneling and black hole spectroscopy: The entropy-area
spectrum of a black hole has been a long-standing and challenging problem. In
chapter - 7, based on the modified tunneling mechanism, introduced in the
previous chapters, we obtain the entropy spectrum of a black hole. In
Einstein’s gravity, we show that both entropy and area spectrum are evenly
spaced. But in more general theories (like Einstein-Gauss-Bonnet gravity),
although the entropy spectrum is equispaced, the corresponding area spectrum
is not. In this sense, quantization of entropy is more fundamental than that
of area.
Chapter -8: Statistical origin of gravity: Based on the above conceptions and
findings, we explore in chapter - 8 an intriguing possibility that gravity can
be thought as an emergent phenomenon. Starting from the definition of entropy,
used in statistical mechanics, we show that it is proportional to the gravity
action. For a stationary black hole this entropy is expressed as
$S_{bh}=E/2T_{H}$, where $T_{H}$ is the Hawking temperature and $E$ is shown
to be the Komar energy. This relation is also compatible with the generalised
Smarr formula for mass.
Chapter -9: Conclusions: Finally, in chapter-9 we present our conclusion and
outlook.
## Chapter 2 The tunneling mechanism
Classical general relativity gives the concept of black hole from which
nothing can escape. This picture was changed dramatically when Hawking [4, 5]
incorporated the quantum nature into this classical problem. In fact he showed
that black hole radiates a spectrum of particles which is quite analogous with
a thermal black body radiation, popularly known as Hawking effect. Thus
Hawking radiation emerges as a nontrivial consequence of combining gravity and
quantum mechanics. People then started thinking that this may give some
insight towards quantum nature of gravity. Since the original derivation,
based on the calculation of Bogoliubov coefficients in the asymptotic states,
was technically very involved, several derivations of Hawking radiation were
subsequently presented in the literature to give fresh insights. For example,
Path integral derivation [8], Trace anomaly approach [9] and chiral
(gravitational) anomaly approach [10, 11, 12, 16, 17, 18, 19], each having its
merits and demerits.
Interestingly, none of the existing approaches to study Hawking effect,
however, corresponds directly to one of the heuristic pictures that visualises
the source of radiation as tunneling. This picture is similar to an electron-
positron pair creation in a constant electric field. The idea is that pair
production occurs inside the event horizon of a black hole. One member of the
pair corresponds to the ingoing mode and other member corresponds to the
outgoing mode. The outgoing mode is allowed to follow classically forbidden
trajectories, by starting just behind the horizon onward to infinity. So this
mode travels back in time, since the horizon is locally to the future of the
external region. Unitarity is not violated since physically it is possible to
envisage the tunneling as the shrinking of the horizon forwarded in time
rather than the particle travelling backward in time [23]. The classical one
particle action becomes complex and so the tunneling amplitude is governed by
the imaginary part of this action for the outgoing mode. However, the action
for the ingoing mode must be real, since classically a particle can fall
behind the horizon. Another essential fact is that tunneling occurs radially
and it is a near horizon phenomenon where the theory is driven by only the
effective ($t-r$) metric [22]. Under this circumstance the solution of a field
equation corresponds to $l=0$ mode which is actually the $s$ \- wave. These
are all important points of this mechanism as will be seen later. The essence
of tunneling based calculations is, thus, the computation of the imaginary
part of the action for the process of $s$-wave emission across the horizon,
which in turn is related to the Boltzmann factor for the emission at the
Hawking temperature. Also, it reveals that the presence of the event horizon
is necessary and the Hawking effect is a completely quantum mechanical
phenomenon, determined by properties of the event horizon.
There are two different methods in the literature to calculate the imaginary
part of the action: one is by Parikh-Wilczek [23] \- radial null geodesic
method and another is the Hamilton-Jacobi (HJ) method which was first used by
Srinivasan et. al. [22]. Later, many people [72, 73] used the radial null
geodesic method as well as HJ method to find out the Hawking temperature for
different space-time metrics. Also, several issues and aspects of these
methods have been discussed extensively [74, 75, 76, 77, 78, 79, 80, 81, 82,
83, 84, 85, 86, 87, 88].
In this chapter, we will give a short review of both the HJ and radial null
geodesic methods. While most of the material is available in the rather
extensive literature [22, 23, 24, 25, 26, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 88, 89, 90, 91] on tunneling, there are some new insights
and clarifications. The organization of the chapter is the following. First we
will discuss the HJ method within a semi-classical approximation to find the
Hawking temperature both in Schwarzschild like coordinate system and Painleve
coordinate system. A general static, spherically symmetric black hole metric
will be considered. In the next section, the radial null geodesic method will
be introduced. A general derivation of the Hawking temperature of this black
hole will be presented. Both these expressions will be shown identical. Then
using this obtained expression, the Hawking temperature will be explicitly
calculated for some known black hole metrics. Final section will be devoted
for the concluding remarks.
### 2.1 Hamilton-Jacobi method
Usually, calculations of the Hawking temperature, based on the tunneling
formalism, for different black holes conform to the general formula
$T_{H}=\frac{\hbar\kappa}{2\pi}$. This relation is normally understood as a
consequence of the mapping of the second law of black hole thermodynamics
$dM=\frac{\kappa}{8\pi}dA$ with $dE=T_{H}dS_{bh}$, coupled with the
Bekenstein-Hawking area law $S_{bh}=\frac{A}{4\hbar}$.
Using the tunneling approach, we now present a derivation of
$T_{H}=\frac{\hbar\kappa}{2\pi}$ where neither the second law of black hole
thermodynamics nor the area law are required. In this sense our analysis is
general.
In this section we will briefly discuss about the HJ method [22] to find the
temperature of a black hole using the picture of Hawking radiation as quantum
tunneling. The analysis will be restricted to the semi-classical limit.
Equivalent results are obtained by using either the standard Schwarzschild
like coordinates or other types, as for instance, the Painleve ones. We
discuss both cases in this section.
#### 2.1.1 Schwarzschild like coordinate system
First, we consider a general class of static (i.e. invariant under time
reversal as well as stationary), spherically symmetric space-time of the form
$\displaystyle ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{g(r)}+r^{2}d\Omega^{2}$ (2.1)
where the horizon $r=r_{H}$ is given by $f(r_{H})=g(r_{H})=0$.
Let us consider a massless particle in the space-time (2.1) described by the
massless Klein-Gordon equation
$\displaystyle-\frac{\hbar^{2}}{\sqrt{-g}}\partial_{\mu}[g^{\mu\nu}\sqrt{-g}\partial_{\nu}]\phi=0~{}.$
(2.2)
Since tunneling across the event horizon occurs radially, only the radial
trajectories will be considered here. Also, it is an near horizon phenomenon
and so the theory is effectively dominated by the two dimensional ($r-t$)
sector of the full metric. Here the modes corresponds to angular quantum
number $l=0$, which is actually the $s$ \- wave [22, 10, 11]. In this regard,
only the $(r-t)$ sector of the metric (2.1) is important. Therefore under this
metric the Klein-Gordon equation reduces to
$\displaystyle-\frac{1}{\sqrt{f(r)g(r)}}\partial^{2}_{t}\phi+\frac{1}{2}\Big{(}f^{\prime}(r)\sqrt{\frac{g(r)}{f(r)}}+g^{\prime}(r)\sqrt{\frac{f(r)}{g(r)}}\Big{)}\partial_{r}\phi+\sqrt{f(r)g(r)}\partial_{r}^{2}\phi=0~{}.$
(2.3)
The semi-classical wave function satisfying the above equation is obtained by
making the standard ansatz for $\phi$ which is
$\displaystyle\phi(r,t)={\textrm{exp}}\Big{[}-\frac{i}{\hbar}S(r,t)\Big{]},$
(2.4)
where $S(r,t)$ is a function which will be expanded in powers of $\hbar$.
Substituting into the wave equation (2.3), we obtain
$\displaystyle\frac{i}{\sqrt{f(r)g(r)}}\Big{(}\frac{\partial S}{\partial
t}\Big{)}^{2}-i\sqrt{f(r)g(r)}\Big{(}\frac{\partial S}{\partial
r}\Big{)}^{2}-\frac{\hbar}{\sqrt{f(r)g(r)}}\frac{\partial^{2}S}{\partial
t^{2}}+\hbar\sqrt{f(r)g(r)}\frac{\partial^{2}S}{\partial r^{2}}$
$\displaystyle+\frac{\hbar}{2}\Big{(}\frac{\partial f(r)}{\partial
r}\sqrt{\frac{g(r)}{f(r)}}+\frac{\partial g(r)}{\partial
r}\sqrt{\frac{f(r)}{g(r)}}\Big{)}\frac{\partial S}{\partial r}=0~{}.$ (2.5)
Expanding $S(r,t)$ in a powers of $\hbar$, we find,
$\displaystyle S(r,t)$ $\displaystyle=$ $\displaystyle S_{0}(r,t)+\hbar
S_{1}(r,t)+\hbar^{2}S_{2}(r,t)+...........$ (2.6) $\displaystyle=$
$\displaystyle S_{0}(r,t)+\sum_{i}\hbar^{i}S_{i}(r,t).$
where $i=1,2,3,......$. In this expansion the terms from ${\cal{O}}(\hbar)$
onwards are treated as quantum corrections over the semi-classical value
$S_{0}$. Here, as mentioned earlier, we will restrict only upto the semi-
classical limit, i.e. $\hbar\rightarrow 0$. The effects due to inclusion of
higher order terms are discussed in [55, 82, 83, 84, 85, 86, 87, 88] 111For
extensive literature on the discussion of the higher order terms see [93, 94].
Substituting (2.6) in (2.5) and taking the semi-classical limit
$\hbar\rightarrow 0$, we obtain the following equation:
$\displaystyle\frac{\partial S_{0}}{\partial
t}=\pm\sqrt{f(r)g(r)}\frac{\partial S_{0}}{\partial r}~{}.$ (2.7)
This is the usual semi-classical Hamilton-Jacobi equation [22]. Now, to obtain
a solution for $S_{0}(r,t)$, we will proceed in the following manner. Since
the metric (2.1) is static it has a time-like Killing vector. Thus we will
look for a solution of (2.7) which behaves as
$\displaystyle S_{0}=\omega t+\tilde{S}_{0}(r),$ (2.8)
where $\omega$ is the conserved quantity corresponding to the time-like
Killing vector. This ultimately is identified as the energy of the particle as
seen by an observer at infinity. Substituting this in (2.7) and then
integrating we obtain,
$\displaystyle\tilde{S_{0}}(r)=\pm\omega\int\frac{dr}{\sqrt{f(r)g(r)}}$ (2.9)
where the limits of the integration are chosen such that the particle just
goes through the horizon $r=r_{H}$. So the one can take the range of
integration from $r=r_{H}-\epsilon$ to $r=r_{H}+\epsilon$, where $\epsilon$ is
a very small constant. The $+(-)$ sign in front of the integral indicates that
the particle is ingoing ($L$) (outgoing ($R$)) (For elaborate discussion to
determine the nature of the modes, see Appendix 2.A). Using (2.9) in (2.8) we
obtain
$\displaystyle S_{0}(r,t)=\omega t\pm\omega\int\frac{dr}{\sqrt{f(r)g(r)}}~{}.$
(2.10)
Therefore the ingoing and outgoing solutions of the Klein-Gordon equation
(2.2) under the back ground metric (2.1) is given by exploiting (2.4) and
(2.10),
$\displaystyle\phi^{(L)}={\textrm{exp}}\Big{[}-\frac{i}{\hbar}\Big{(}\omega
t+\omega\int\frac{dr}{\sqrt{f(r)g(r)}}\Big{)}\Big{]}$ (2.11)
and
$\displaystyle\phi^{(R)}={\textrm{exp}}\Big{[}-\frac{i}{\hbar}\Big{(}\omega
t-\omega\int\frac{dr}{\sqrt{f(r)g(r)}}\Big{)}\Big{]}.$ (2.12)
In the rest of the analysis we will call $\phi^{(L)}$ as the left mode and
$\phi^{(R)}$ as the right mode.
A point we want to mention here that if one expresses the above modes in terms
of null coordinates ($u,v$), then $\phi^{(L)}$ becomes function of “$v$” only
while $\phi^{(R)}$ becomes that of “$u$”. These are call holomorphic modes.
Such modes satisfies chirality condition. This will be elaborated and used in
the later discussions.
Now for the tunneling of a particle across the horizon the nature of the
coordinates change. The time-like coordinate $t$ outside the horizon changes
to space-like coordinate inside the horizon and likewise for the outside
space-like coordinate $r$. This indicates that ‘$t$’ coordinate may have an
imaginary part on crossing the horizon of the black hole and correspondingly
there will be a temporal contribution to the probabilities for the ingoing and
outgoing particles along with the spacial part. This has similarity with [78]
where they show for the Schwarzschild metric that two patches across the
horizon are connected by a discrete imaginary amount of time.
The ingoing and outgoing probabilities of the particle are, therefore, given
by,
$\displaystyle
P^{(L)}=|\phi^{(L)}|^{2}={\textrm{exp}}\Big{[}\frac{2}{\hbar}(\omega{\textrm{Im}}~{}t+\omega{\textrm{Im}}\int\frac{dr}{\sqrt{f(r)g(r)}}\Big{)}\Big{]}$
(2.13)
and
$\displaystyle
P^{(R)}=|\phi^{(R)}|^{2}={\textrm{exp}}\Big{[}\frac{2}{\hbar}\Big{(}\omega{\textrm{Im}}~{}t-\omega{\textrm{Im}}\int\frac{dr}{\sqrt{f(r)g(r)}}\Big{)}\Big{]}$
(2.14)
Now the ingoing probability $P^{(L)}$ has to be unity in the classical limit
(i.e. $\hbar\rightarrow 0$) - when there is no reflection and everything is
absorbed - instead of zero or infinity [89].Thus, in the classical limit,
(2.13) leads to,
$\displaystyle{\textrm{Im}}~{}t=-{\textrm{Im}}\int\frac{dr}{\sqrt{f(r)g(r)}}~{}.$
(2.15)
It must be noted that the above relation satisfies the classical condition
$\frac{\partial S_{0}}{\partial\omega}=$ constant. This is understood by the
following argument. Calculating the left side of this condition from (2.10) we
obtain,
$\displaystyle t={\textrm{constant}}\mp\int\frac{dr}{\sqrt{f(r)g(r)}}$ (2.16)
where $-(+)$ sign indicates that the particle is ingoing ($L$) (outgoing
($R$)). So for an ingoing particle this condition immediately yields (2.15)
considering that “constant” is always real. On the other hand a naive
substitution of ‘Im$~{}t$’ in (2.14) from (2.16) for the outgoing particle
gives $P^{(R)}=1$. But it must be noted that according to classical general
theory of relativity, a particle can be absorbed in the black hole, while the
reverse process is forbidden. In this regard, ingoing classical trajectory
exists while the outgoing classical trajectory is forbidden. Hence use of the
classical condition for outgoing particle is meaningless.
Now to find out ‘Im$~{}t$’ for the outgoing particle we will take the help of
the Kruskal coordinates which are well behaved throughout the space-time. The
Kruskal time ($T$) and space ($X$) coordinates inside and outside the horizon
are defined in terms of Schwarzschild coordinates as [95]
$\displaystyle T_{in}=e^{\kappa r^{*}_{in}}\cosh\\!\left(\kappa
t_{in}\right)~{}~{};\hskip 17.22217ptX_{in}=e^{\kappa
r^{*}_{in}}\sinh\\!\left(\kappa t_{in}\right)$ (2.17) $\displaystyle
T_{out}=e^{\kappa r^{*}_{out}}\sinh\\!\left(\kappa t_{out}\right)~{}~{};\hskip
17.22217ptX_{out}=e^{\kappa r^{*}_{out}}\cosh\\!\left(\kappa t_{out}\right)$
(2.18)
where $\kappa$ is the surface gravity defined by
$\displaystyle\kappa=\frac{1}{2}\sqrt{f^{\prime}(r_{H})g^{\prime}(r_{H})}~{}.$
(2.19)
Here ‘$in(out)$’ stands for inside (outside) the event horizon while $r^{*}$
is the tortoise coordinate, defined by
$\displaystyle r^{*}=\int\frac{dr}{\sqrt{f(r)g(r)}}~{}.$ (2.20)
These two sets of coordinates are connected through the following relations
$\displaystyle t_{in}=t_{out}-i\frac{\pi}{2\kappa}$ (2.21) $\displaystyle
r^{*}_{in}=r^{*}_{out}+i\frac{\pi}{2\kappa}$ (2.22)
so that the Kruskal coordinates get identified as $T_{in}=T_{out}$ and
$X_{in}=X_{out}$. This indicates that when a particle travels from inside to
outside the horizon, ‘$t$’ coordinate picks up an imaginary term
$-\frac{\pi}{2{\kappa}}$. This fact will be used elaborately in later
chapters. Below we shall show that this is precisely given by (2.15). Near the
horizon one can expand $f(r)$ and $g(r)$ about the horizon $r_{H}$:
$\displaystyle f(r)=f^{\prime}(r_{H})(r-r_{H})+{\cal{O}}((r-r_{H})^{2})$
$\displaystyle g(r)=g^{\prime}(r_{H})(r-r_{H})+{\cal{O}}((r-r_{H})^{2})~{}.$
(2.23)
Substituting these in (2.15) and using (2.19) we obtain,
$\displaystyle{\textrm{Im}}~{}t=-\frac{1}{2\kappa}{\textrm{Im}}~{}\int_{r_{H}-\epsilon}^{r_{H}+\epsilon}\frac{dr}{r-r_{H}}~{}.$
(2.24)
Here we explicitly mentioned the integration limits. Now to evaluate the above
integration we make a substitution $r-r_{H}=\epsilon e^{i\theta}$ where
$\theta$ runs from $\pi$ to $2\pi$. Hence,
$\displaystyle{\textrm{Im}}~{}t=-\frac{1}{2\kappa}{\textrm{Im}}~{}\int_{\pi}^{2\pi}id\theta=-\frac{\pi}{2\kappa}~{}.$
(2.25)
For the Schwarzschild space-time, since $\kappa=\frac{1}{4M}$, one can easily
show that ${\textrm{Im}}~{}t=-2\pi M$ which is precisely the value given in
[78].
Therefore, substituting (2.15) in (2.14), the probability of the outgoing
particle is
$\displaystyle
P^{(R)}={\textrm{exp}}\Big{[}-\frac{4}{\hbar}\omega{\textrm{Im}}\int\frac{dr}{\sqrt{f(r)g(r)}}\Big{]}.$
(2.26)
Now using the principle of “detailed balance” [22]
$\displaystyle
P^{(R)}={\textrm{exp}}\Big{(}-\frac{\omega}{T_{H}}\Big{)}P^{(L)}={\textrm{exp}}\Big{(}-\frac{\omega}{T_{H}}\Big{)},$
(2.27)
we obtain the temperature of the black hole as
$\displaystyle
T_{H}=\frac{\hbar}{4}\Big{(}{\textrm{Im}}\int\frac{dr}{\sqrt{f(r)g(r)}}\Big{)}^{-1}~{}.$
(2.28)
This is the standard semi-classical Hawking temperature of the black hole.
Using this expression and knowing the metric coefficients $f(r)$ and $g(r)$
one can easily find out the temperature of the corresponding black hole.
Some comments are now in order. The first point is that (2.28) yields a novel
form of the semi-classical Hawking temperature. Also, we will later show that
(2.28) can be applied to non-spherically symmetric metrics. This will be
exemplified in the case of the Kerr metric. For a spherically symmetric metric
it is possible to show that (2.28) reproduces the familiar form
$\displaystyle T_{H}=\frac{\hbar\kappa}{2\pi}~{}.$ (2.29)
This can be done in the following way. The near horizon expansions for $f(r)$
and $g(r)$ are given by (2.23). Inserting these in (2.28) and performing the
contour integration, as done earlier, (2.29) is obtained. Note that this form
is the standard Hawking temperature found [90, 91] by the Hamilton-Jacobi
method. There is no ambiguity regarding a factor of two in the Hawking
temperature as reported in the literature [90, 91, 92]. This issue is
completely avoided in the present analysis where the standard expression for
the Hawking temperature is reproduced.
The other point is that the form of the solution (2.8) of (2.7) is not unique,
since any constant multiple of ‘$S_{0}$’ can be a solution as well. For that
case one can easily see that the final expression (2.28) for the temperature
still remains unchanged. It is only a matter of rescaling the particle energy
‘$\omega$’. This shows the uniqueness of the expression (2.28) for the Hawking
temperature.
#### 2.1.2 Painleve coordinate system
Here we will discuss the Hamilton-Jacobi method in Painleve coordinates and
explicitly show how one can obtain the standard Hawking temperature. Consider
a metric of the form (2.1), which describes a general class of static,
spherically symmetric space time. There is a coordinate singularity in this
metric at the horizon $r=r_{H}$ where $f(r_{H})=g(r_{H})=0$. This singularity
is avoided by the use of Painleve coordinate transformation [96],
$\displaystyle dt\to dt-\sqrt{\frac{1-g(r)}{f(r)g(r)}}dr~{}.$ (2.30)
Under this transformation, the metric (2.1) takes the following form,
$\displaystyle
ds^{2}=-f(r)dt^{2}+2f(r)\sqrt{\frac{1-g(r)}{f(r)g(r)}}dtdr+dr^{2}+r^{2}d\Omega^{2}.$
(2.31)
Note that the metric (2.1) looks both stationary and static, whereas the
transformed metric (2.31) is stationary but not static which reflects the
correct nature of the space time.
As before, consider a massless scalar particle in the spacetime metric (2.31)
described by the Painleve coordinates. Since the Klein-Gordon equation (2.2)
is in covariant form, the scalar particle in the background metric (2.31) also
satisfies (2.2). Therefore under this metric the Klein-Gordon equation reduces
to
$\displaystyle-$
$\displaystyle(\frac{g}{f})^{\frac{3}{2}}\partial^{2}_{t}\phi+\frac{2g\sqrt{1-g}}{f}\partial_{t}\partial_{r}\phi-\frac{gg^{\prime}}{2f\sqrt{1-g}}\partial_{t}\phi+g\sqrt{\frac{g}{f}}\partial^{2}_{r}\phi$
(2.32) $\displaystyle+$
$\displaystyle\frac{1}{2}\sqrt{\frac{g}{f}}(3g^{\prime}-\frac{f^{\prime}g}{f})\partial_{r}\phi=0.$
As before, substituting the standard ansatz (2.4) for $\phi$ in the above
equation, we obtain,
$\displaystyle-(\frac{g}{f})^{\frac{3}{2}}\Big{[}-\frac{i}{\hbar}\Big{(}\frac{\partial
S}{\partial t}\Big{)}^{2}+\frac{\partial^{2}S}{\partial
t^{2}}\Big{]}+\frac{2g\sqrt{1-g}}{f}\Big{[}-\frac{i}{\hbar}\frac{\partial
S}{\partial t}\frac{\partial S}{\partial r}+\frac{\partial^{2}S}{\partial
r\partial t}\Big{]}-\frac{gg^{\prime}}{2f\sqrt{1-g}}\frac{\partial S}{\partial
t}$
$\displaystyle+g\sqrt{\frac{g}{f}}\Big{[}-\frac{i}{\hbar}\Big{(}\frac{\partial
S}{\partial r}\Big{)}^{2}+\frac{\partial^{2}S}{\partial
r^{2}}\Big{]}+\frac{1}{2}(3g^{\prime}-\frac{f^{\prime}g}{f})\frac{\partial
S}{\partial r}=0.$ (2.33)
Substituting (2.6) in the above and then neglecting the terms of order $\hbar$
and greater we find to the lowest order,
$\displaystyle(\frac{g}{f})^{\frac{3}{2}}\Big{(}\frac{\partial S_{0}}{\partial
t}\Big{)}^{2}-\frac{2g\sqrt{1-g}}{f}\frac{\partial S_{0}}{\partial
t}\frac{\partial S_{0}}{\partial r}-g\sqrt{\frac{g}{f}}\Big{(}\frac{\partial
S_{0}}{\partial r}\Big{)}^{2}=0.$ (2.34)
It has been stated earlier that the metric (2.31) is stationary. Therefore
following the same argument as before it has a solution of the form (2.8).
Inserting this in (2.34) yields,
$\displaystyle\frac{d\tilde{S}_{0}(r)}{dr}=\omega\sqrt{\frac{1-g(r)}{f(r)g(r)}}\Big{(}-1\pm\frac{1}{\sqrt{1-g(r)}}\Big{)}$
(2.35)
Integrating,
$\displaystyle\tilde{S}_{0}(r)=\omega\int\sqrt{\frac{1-g(r)}{f(r)g(r)}}\Big{(}-1\pm\frac{1}{\sqrt{1-g(r)}}\Big{)}dr.$
(2.36)
The $+(-)$ sign in front of the integral indicates that the particle is
ingoing (outgoing). Therefore the actions for ingoing and outgoing particles
are
$\displaystyle S_{0}^{(L)}(r,t)=\omega
t+\omega\int\frac{1-\sqrt{1-g}}{\sqrt{fg}}dr$ (2.37)
and
$\displaystyle S_{0}^{(R)}(r,t)=\omega
t-\omega\int\frac{1+\sqrt{1-g}}{\sqrt{fg}}dr$ (2.38)
Since in the classical limit (i.e. $\hbar\rightarrow 0$) the probability for
the ingoing particle ($P^{(L)}$) has to be unity, $S_{0}^{(L)}$ must be real.
Following identical steps employed in deriving (2.15) we obtain, starting from
(2.37), the analogous condition,
$\displaystyle{\textrm{Im}}~{}t=-{\textrm{Im}}\int\frac{1-\sqrt{1-g}}{\sqrt{fg}}dr$
(2.39)
Substituting this in (2.38) we obtain the action for the outgoing particle:
$\displaystyle
S_{0}^{(R)}(r,t)=\omega{\textrm{Re}}~{}t-\omega{\textrm{Re}}\int\frac{1+\sqrt{1-g}}{\sqrt{fg}}dr-2i\omega{\textrm{Im}}\int\frac{dr}{\sqrt{fg}}$
(2.40)
Therefore the probability for the outgoing particle is
$\displaystyle
P^{(R)}=|e^{-\frac{i}{\hbar}S_{0}^{(R)}}|^{2}=e^{-\frac{4}{\hbar}\omega{\textrm{Im}}\int\frac{dr}{\sqrt{f(r)g(r)}}}$
(2.41)
Now using the principle of “detailed balance” (2.27) we obtain the same
expression (2.28) for the standard Hawking temperature which was calculated in
Schwarzschild like coordinates by the Hamilton-Jacobi method.
### 2.2 Radial null geodesic method
So far, we gave a general discussion on the HJ method both in Scwarzschild
like coordinates as well as Painleve coordinates and obtained the expression
of the temperature for a static, spherically symmetric black hole. Also, this
has been reduced to the famous Hawking expression - temperature is
proportional to the surface gravity.
In this section, we will give a general discussion on the radial null geodesic
method. A derivation of the Hawking temperature by this method will be
explicitly performed for the metric (2.1).
In this method, the first step is to find the radial null geodesic. To do that
it is necessary to remove the apparent singularity at the event horizon. This
is done by going to the Painleve coordinates. In these coordinates, the metric
(2.1) takes the form (2.31). Then the radial null geodesics are obtained by
setting $ds^{2}=d\Omega^{2}=0$ in (2.31),
$\displaystyle\dot{r}\equiv\frac{dr}{dt}=\sqrt{\frac{f(r)}{g(r)}}\Big{(}\pm
1-\sqrt{1-g(r)}\Big{)}$ (2.42)
where the positive (negative) sign gives outgoing (incoming) radial geodesics.
At the neighbourhood of the black hole horizon, the trajectory (2.42) of an
outgoing shell is written as,
$\displaystyle\dot{r}=\frac{1}{2}\sqrt{f^{\prime}(r_{H})g^{\prime}(r_{H})}(r-r_{H})+{\mathcal{O}}((r-r_{H})^{2})$
(2.43)
where we have used the expansions (2.23) of the functions $f(r)$ and $g(r)$.
Now we want to write (2.43) in terms of the surface gravity of the black hole.
The reason is that in some cases, for example in the presence of back
reaction, one may not know the exact form of the metric but what one usually
knows is the surface gravity of the problem. Also, the Hawking temperature is
eventually expressed in terms of the surface gravity. The form of surface
gravity for the transformed metric (2.31) at the horizon is given by,
$\displaystyle\kappa=\Gamma{{}^{0}}{{}_{00}}|_{r=r_{H}}=\frac{1}{2}\Big{[}\sqrt{\frac{1-g(r)}{f(r)g(r)}}g(r)\frac{df(r)}{dr}\Big{]}|_{r=r_{H}}.$
(2.44)
Using the Taylor series (2.23), the above equation reduces to the familiar
form of surface gravity (2.19). This expression of surface gravity is used to
write (2.43) in the form,
$\displaystyle\dot{r}=\kappa(r-r_{H})+{\mathcal{O}}((r-r_{H})^{2}).$ (2.45)
We consider a positive energy shell which crosses the horizon in the outward
direction from $r_{{\textrm{in}}}$ to $r_{{\textrm{out}}}$. The imaginary part
of the action for that shell is given by [23],
$\displaystyle\textrm{Im}~{}{\cal{S}}=\textrm{Im}\int_{r_{{\textrm{in}}}}^{r_{{\textrm{out}}}}p_{r}dr=\textrm{Im}\int_{r_{in}}^{r_{out}}\int_{0}^{p_{r}}dp_{r}^{\prime}dr.$
(2.46)
Using the Hamilton’s equation of motion $\dot{r}=\frac{dH}{dp_{r}}|_{r}$ the
last equality of the above equation is written as,
$\displaystyle\textrm{Im}~{}{\cal{S}}$ $\displaystyle=$
$\displaystyle\textrm{Im}\int_{r_{{\textrm{in}}}}^{r_{{\textrm{out}}}}\int_{0}^{H}\frac{dH^{\prime}}{\dot{r}}dr$
(2.47)
where, instead of momentum, energy is used as the variable of integration.
Now we consider the self gravitation effect [27] of the particle itself, for
which (2.45) and (2.47) will be modified. Following [23], under the $s$\- wave
approximation, we make the replacement $M\rightarrow M-\omega$ in (2.45) to
get the following expression
$\dot{r}=(r-r_{H})\kappa[M-\omega]$ (2.48)
where $\omega$ is the energy of a shell moving along the geodesic of space-
time 222Here $\kappa[M-\omega]$ represents that $\kappa$ is a function of
($M-\omega$). This symbol will be used in the later part of the chapter for a
similar purpose..
Now we use the fact [23], for a black hole of mass $M$, the Hamiltonian
$H=M-\omega$. Inserting in (2.47) the modified expression due to the self
gravitation effect is obtained as,
$\displaystyle\textrm{Im}~{}{\cal{S}}=\textrm{Im}\int_{r_{{\textrm{in}}}}^{r_{{\textrm{out}}}}\int_{M}^{M-\omega}\frac{d(M-\omega^{\prime})}{\dot{r}}dr=-\textrm{Im}\int_{r_{{\textrm{in}}}}^{r_{{\textrm{out}}}}\int_{0}^{\omega}\frac{d\omega^{\prime}}{\dot{r}}dr$
(2.49)
where in the final step we have changed the integration variable from
$H^{\prime}$ to $\omega^{\prime}$. Substituting the expression of $\dot{r}$
from (2.48) into (2.49) we find,
$\displaystyle\textrm{Im}~{}{\cal{S}}=-\textrm{Im}\int_{0}^{\omega}\frac{d\omega^{\prime}}{{\kappa}[M-\omega^{\prime}]}\int_{r_{{\textrm{in}}}}^{r_{{\textrm{out}}}}\frac{dr}{r-r_{H}}~{}.$
(2.50)
The $r$-integration is done by deforming the contour. Ensuring that the
positive energy solutions decay in time (i.e. into the lower half of
$\omega^{\prime}$ plane and $r_{{\textrm{in}}}>r_{{\textrm{out}}}$) we have
after $r$ integration333One can also take the contour in the upper half plane
with the replacement $M\rightarrow M+\omega$ [27].,
$\displaystyle\textrm{Im}~{}{\cal{S}}=\pi\int_{0}^{\omega}\frac{d\omega^{\prime}}{{\kappa}[M-\omega^{\prime}]}~{}.$
(2.51)
To understand the ordering $r_{{\textrm{in}}}>r_{{\textrm{out}}}$ \- which
supplies the correct sign, let us do the following analysis. For simplicity,
we consider the Schwarzschild black hole whose surface gravity is given by
$\kappa[M]=\frac{1}{4M}$. Substituting this in (2.51) and performing the
$\omega^{\prime}$ integration we obtain
$\displaystyle{\textrm{Im}}~{}{\cal{S}}=4\pi\omega(M-\frac{\omega}{2}).$
(2.52)
Now let us first perform the $\omega^{\prime}$ integration before $r$
integration in (2.50). For Schwarzschild black hole this will give
$\displaystyle{\textrm{Im}}~{}{\cal{S}}=4~{}{\textrm{Im}}~{}\int_{r_{\textrm{in}}}^{r_{\textrm{out}}}dr\int_{M}^{M-\omega}\frac{M^{\prime}}{r-2M^{\prime}}dM^{\prime}$
(2.53)
where substitution of $M^{\prime}=M-\omega^{\prime}$ has been used. Evaluation
of $M^{\prime}$ integration and then $r$ integration in the above yields,
$\displaystyle{\textrm{Im}}~{}{\cal{S}}=\frac{\pi}{2}(r_{\textrm{in}}^{2}-r_{\textrm{out}}^{2})$
(2.54)
Hence (2.52) and (2.54) to be equal we must have $r_{\textrm{in}}=2M$ and
$r_{\textrm{out}}=2(M-\omega)$, which clearly shows that
$r_{{\textrm{in}}}>r_{{\textrm{out}}}$.
The tunneling amplitude following from the WKB approximation is given by,
$\displaystyle\Gamma\sim
e^{-\frac{2}{\hbar}{\textrm{Im}}~{}{\cal{S}}}=e^{\Delta S_{bh}}$ (2.55)
where the result is expressed more naturally in terms of the black hole
entropy change [23]. To understand the last identification ($\Gamma=e^{\Delta
S_{bh}}$), consider a process where a black hole emits a shell of energy. We
denote the initial state and final state by the levels $i$ and $f$. In thermal
equilibrium,
$\displaystyle\frac{dP_{i}}{dt}=P_{i}P_{i\rightarrow f}-P_{f}P_{f\rightarrow
i}=0$ (2.56)
where $P_{a}$ denotes the probability of getting the system in the macrostate
$a(a=i,f)$ and $P_{a\rightarrow b}$ denotes the transition probability from
the state $a$ to $b$ ($a,b=i,f$). According to statistical mechanics, the
entropy of a given state (specified by its macrostates) is a logarithmic
function of the total number of microstates ($S_{bh}={\textrm{log}}\Omega$).
So the number of microstates $\Omega$ for a given black hole is
$e^{S_{{bh}}}$. Since the probability of getting a system in a particular
macrostate is proportional to the number of microstates available for that
configuration, we get from (2.56),
$\displaystyle
e^{S_{i}}P_{{\textrm{emission}}}=e^{S_{f}}P_{{\textrm{absorption}}}$ (2.57)
where $P_{{\textrm{emission}}}$ is the emission probability $P_{i\rightarrow
f}$ and $P_{{\textrm{absorption}}}$ is the absorption probability
$P_{f\rightarrow i}$. So the tunneling amplitude is given by,
$\displaystyle\Gamma=\frac{P_{{\textrm{emission}}}}{P_{{\textrm{absorption}}}}=e^{S_{f}-S_{i}}=e^{\Delta
S_{{bh}}}$ (2.58)
thereby leading to the correspondence,
$\displaystyle\Delta S_{bh}=-\frac{2}{\hbar}{\textrm{Im}}~{}{\cal{S}}$ (2.59)
that follows from (2.55). We mention that the above relation (2.59) has also
been shown using semi-classical arguments based on the second law of
thermodynamics [97] or on the assumption of entropy being proportional to area
[29, 98]. But such arguments are not used in our derivation. Rather our
analysis has some points of similarity with the physical picture suggested in
[23] leading to a general validity of (2.58). This implies that when quantum
effects are taken into consideration, both sides of (2.59) are modified
keeping the functional relationship identical. In our analysis we will show
that self consistency is preserved by (2.59).
In order to write the black hole entropy in terms of its mass alone we have to
substitute the value of $\omega$ in terms of $M$ for which the black hole is
stable i. e.
$\displaystyle\frac{d(\Delta S_{bh})}{d\omega}=0~{}.$ (2.60)
Using (2.51) and (2.59) in the above equation we get,
$\displaystyle\frac{1}{{\kappa}[M-\omega]}=0~{}.$ (2.61)
The roots of this equation are written in the form
$\displaystyle\omega=\psi[M]$ (2.62)
which means
$\displaystyle\frac{1}{{\kappa}[M-\psi[M]]}=0.$ (2.63)
This value of $\omega$ from eq. (2.62) is substituted back in the expression
of $\Delta S_{bh}$ to yield,
$\displaystyle\Delta
S_{bh}=-\frac{2\pi}{\hbar}\int_{0}^{\psi[M]}\frac{d\omega^{\prime}}{{\kappa}[M-\omega^{\prime}]}~{}.$
(2.64)
Having obtained the form of entropy change, we are now able to give an
expression of entropy for a particular state. We recall the simple definition
of entropy change
$\displaystyle\Delta S_{bh}=S_{{\textrm{final}}}-S_{{\textrm{initial}}}~{}.$
(2.65)
Now setting the black hole entropy at the final state to be zero we get the
expression of entropy as
$\displaystyle S_{bh}=S_{{\textrm{initial}}}=-\Delta
S_{bh}=\frac{2\pi}{\hbar}\int_{0}^{\psi[M]}\frac{d\omega^{\prime}}{{\kappa}[M-\omega^{\prime}]}~{}.$
(2.66)
From the law of thermodynamics, we write the inverse black hole temperature
as,
$\displaystyle\frac{1}{T_{H}}$ $\displaystyle=$
$\displaystyle\frac{dS_{bh}}{dM}$ (2.67) $\displaystyle=$
$\displaystyle\frac{2\pi}{\hbar}\frac{d}{dM}\int_{0}^{\psi[M]}\frac{d\omega^{\prime}}{{\kappa}[M-\omega^{\prime}]}~{}.$
(2.68)
Using the identity,
$\displaystyle\frac{dF[x]}{dx}=f[x,b[x]]b^{\prime}[x]-f[x,a[x]]a^{\prime}[x]+\int_{a[x]}^{b[x]}\frac{\partial}{\partial
x}f[x,t]dt$ (2.69)
for,
$\displaystyle F[x]=\int_{a[x]}^{b[x]}f[x,t]dt$ (2.70)
we find,
$\displaystyle\frac{1}{T_{H}}=\frac{2\pi}{\hbar}\big{[}\frac{1}{{\kappa}[M-\psi[M]]}\psi^{\prime}[M]-\int_{0}^{\psi[M]}\frac{1}{[{\kappa}[M-\omega^{\prime}]]^{2}}\frac{\partial{\kappa}[M-\omega^{\prime}]}{\partial(M-\omega^{\prime})}d\omega^{\prime}\big{]}~{}.$
(2.71)
Making the change of variable $x=M-\omega^{\prime}$ in the second integral we
obtain,
$\displaystyle\frac{1}{T_{H}}=\frac{2\pi}{\hbar}\big{[}\frac{\psi^{\prime}[M]-1}{{\kappa}[M-\psi[M]]}+\frac{1}{{\kappa}[M]}\big{]}~{}.$
(2.72)
Finally, making use of (2.63), the cherished expression (2.29) for the Hawking
temperature follows.
For a consistency check, consider the second law of thermodynamics which is
now written as,
$\displaystyle
dM=d\omega^{\prime}=T_{h}dS_{bh}=\frac{\hbar{\kappa}[M]}{2\pi}dS_{bh}~{}.$
(2.73)
Inserting in (2.51), yields,
$\displaystyle{\textrm{Im}}~{}{\cal
S}=\frac{\hbar}{2}\int_{S_{bh}[M]}^{S_{bh}[M-\omega]}dS_{bh}=-\frac{\hbar}{2}\Delta
S_{bh}$ (2.74)
thereby reproducing (2.59). This shows the internal consistency of the
tunneling approach.
### 2.3 Calculation of Hawking temperature
In this section we will consider some standard metrics to show how the semi-
classical Hawking temperature can be calculated from (2.28). For instance we
consider a spherically symmetric space-time, the Schwarzschild metric and a
non-spherically symmetric space-time, the Kerr metric.
#### 2.3.1 Schwarzschild black hole
The spacetime metric is given by
$\displaystyle
ds^{2}=-(1-\frac{2M}{r})dt^{2}+(1-\frac{2M}{r})^{-1}dr^{2}+r^{2}d\Omega^{2}.$
(2.75)
So the metric coefficients are
$\displaystyle f(r)=g(r)=(1-\frac{r_{H}}{r});\,\,\,r_{H}=2M.$ (2.76)
Since this metric is spherically symmetric we use the formula (2.28) to
compute the semi-classical Hawking temperature. This is found to be,
$\displaystyle T_{H}=\frac{\hbar}{4\pi r_{H}}=\frac{\hbar}{8\pi M}.$ (2.77)
which is the standard expression (2.29) where the surface gravity, calculated
by (2.19), is $\kappa=1/4M$.
#### 2.3.2 Kerr black hole
This example provides a nontrivial application of our formula (2.28) for
computing the semi-classical Hawking temperature. Here the metric is not
spherically symmetric, invalidating the use of (2.29).
In Boyer-Linquist coordinates the form of the Kerr metric is given by
$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\Big{(}1-\frac{2Mr}{\rho^{2}}\Big{)}dt^{2}-\frac{2Mar~{}{\textrm{sin}}^{2}\theta}{\rho^{2}}(dtd\phi+d\phi
dt)$ (2.78) $\displaystyle+$
$\displaystyle\frac{\rho^{2}}{\Delta}dr^{2}+\rho^{2}d\theta^{2}+\frac{{\textrm{sin}}^{2}\theta}{\rho^{2}}~{}\Big{[}(r^{2}+a^{2})^{2}-a^{2}\Delta~{}{\textrm{sin}}^{2}\theta\Big{]}d\phi^{2}$
where
$\displaystyle\Delta(r)$ $\displaystyle=$ $\displaystyle
r^{2}-2Mr+a^{2};\,\,\,\rho^{2}(r,\theta)=r^{2}+a^{2}~{}{\textrm{cos}}^{2}\theta$
$\displaystyle a$ $\displaystyle=$ $\displaystyle\frac{J}{M}$ (2.79)
and $J$ is the Komar angular momentum. We have chosen the coordinates for Kerr
metric such that the event horizons occur at those fixed values of $r$ for
which $g^{rr}=\frac{\Delta}{\rho^{2}}=0$. Therefore the event horizons are
$\displaystyle r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}.$ (2.80)
This metric is not spherically symmetric and static but stationary. So it must
have time-like Killing vectors. Subtleties in employing the tunneling
mechanism for such (rotating) black holes were first discussed in [90, 91]. In
the present discussion we will show that, although the general formulation was
based only on the static, spherically symmetric metrics, it is still possible
to apply this methodology for such a metric. The point is that for radial
trajectories, the Kerr metric simplifies to the following form
$\displaystyle
ds^{2}=-\Big{(}\frac{r^{2}+a^{2}-2Mr}{r^{2}+a^{2}}\Big{)}dt^{2}+\Big{(}\frac{r^{2}+a^{2}}{r^{2}+a^{2}-2Mr}\Big{)}dr^{2}$
(2.81)
where, for simplicity, we have taken $\theta=0$ (i.e. particle is going along
$z$-axis). This is exactly the form of the $(r-t)$ sector of the metric (2.1).
Since in our formalism only the $(r-t)$ sector is important, our results are
applicable here. In particular if the metric has no terms like $(drdt)$ then
we can apply (2.28) to find the semi-classical Hawking temperature. Here,
$\displaystyle f(r)=g(r)=\Big{(}\frac{r^{2}+a^{2}-2Mr}{r^{2}+a^{2}}\Big{)}$
(2.82)
Substituting these in (2.28) we obtain,
$\displaystyle
T_{H}=\frac{\hbar}{4}\Big{(}{\textrm{Im}}\int\frac{r^{2}+a^{2}}{(r-r_{+})(r-r_{-})}\Big{)}^{-1}.$
(2.83)
The integrand has simple poles at $r=r_{+}$ and $r=r_{-}$. Since we are
interested only with the event horizon at $r=r_{+}$, we choose the contour as
a small half-loop going around this pole from left to right. Integrating, we
obtain the value of the semi-classical Hawking temperature as
$\displaystyle T_{H}=\frac{\hbar}{4\pi}\frac{r_{+}-r_{-}}{r_{+}^{2}+a^{2}}.$
(2.84)
which is the result quoted in the literature [99]. This can also be expressed
in standard expression (2.29) where
$\kappa=\frac{r_{+}-r_{-}}{2(r_{+}^{2}+a^{2})}$.
### 2.4 Discussions
In this chapter, we introduced the tunneling method to study the Hawking
effect within the semi-classical limit (i.e. $\hbar\rightarrow 0$),
particularly to find the familiar form of the semi-classical Hawking
temperature. There exist two approaches: Hamilton-Jacobi method [22] (HJ) and
radial null geodesic method [23]. For simplicity, a general form of the
static, spherically symmetric black hole metric was considered.
First, discussions on HJ method in both the Schwarzschild like coordinates and
Painleve coordinates have been done. In both coordinate systems, we obtained
identical results. A general expression (2.28) for the semi-classical Hawking
temperature was obtained. For the particular case of a spherically symmetric
metric, our expression reduces to the standard form (2.29). The factor of two
problem in the Hawking temperature has been taken care of by considering the
contribution from the imaginary part of the temporal coordinate since it
changes its nature across the horizon. Also, this method is free of the rather
ad hoc way of introducing an integration constant, as reported in [89]. Our
approach, on the other hand, is similar in spirit to [78] where it has been
shown that ‘$t$’ changes by an imaginary discrete amount across the horizon.
Indeed, the explicit expression for this change, in the case of Schwarzschild
metric, calculated from our general formula (2.15), agrees with the findings
of [78]. Then, a general discussion on the other method, the radial null
geodesic method, was given. In this method, again the standard form of the
Hawking temperature was obtained. Finally, as an application, we calculated
the semi-classical temperature of the Schwarzschild black hole from the
general expression (2.28). Also, use of this expression to find the
temperature of a non-spherically symmetric metric, for instance Kerr metric,
has been shown.
As a final remark, we want to mention that our derivation of Hawking
temperature in terms of the surface gravity by considering the action of an
outgoing particle crossing the black hole horizon due to quantum mechanical
tunneling is completely general. The expression of temperature was known long
before [100, 3, 2, 43] from a comparison between two classical laws. One is
the law of black hole thermodynamics which states that the mass change is
proportional to the change of horizon area multiplied by surface gravity at
the horizon. The other is the area law according to which the black hole
entropy is proportional to the surface area of the horizon. The important
point of our derivation is that it is not based on either of these two
classical laws. The only assumption is that the metric is static and
spherically symmetric. Hence it is useful to apply this method to study the
Hawking effect for the black holes which incorporates both the back reaction
and noncommutative effects but still are in static, spherically symmetric
form. This will be done in the next chapter.
## Appendix
## Appendix 2.A Ingoing and outgoing modes
Our convention is such that, a mode will be called ingoing (outgoing) if its
radial momentum eigenvalue is negative (positive). For a wave function $\phi$,
the momentum eigenvalue equation is
$\displaystyle{\hat{p}_{r}}\phi=p_{r}\phi,$ (2a.1)
where ${\hat{p}_{r}}=-i\hbar\frac{\partial}{\partial r}$. So according to our
convention, if $p_{r}<0$ for a mode, then it is ingoing and vice versa.
Now the mode solutions are given by (2.11) and (2.12). So according to (2a.1),
the momentum eigenvalue for $\phi^{(L)}$ is
$p_{r}^{(L)}=-\frac{\omega}{\sqrt{fg}}$ which is negative. So this mode is
ingoing. Similarly, the momentum eigenvalue for $\phi^{(R)}$ mode comes out to
be positive and hence it is a outgoing mode.
## Chapter 3 Null geodesic approach
In the previous chapter, a systematic analysis on tunneling mechanism, both by
HJ and radial null geodesic methods, to find the Hawking temperature has been
presented. The temperature was found to be proportional to the surface gravity
of a black hole represented by a general static, spherically symmetric metric.
This indicates that such an analysis can be extended to the cases in which the
space-time metric is modified by effects like back reaction and
noncommutivity, provided these are still in the static, spherically symmetric
form.
To investigate the last stage evolution of black hole evaporation back
reaction in space-time has a significant influence. An approach to this
problem is to solve the semiclassical Einstein equations in which the matter
fields including the graviton, are quantized at the one-loop level and coupled
to (c -number) gravity through Einstein’s equation. The space-time geometry
$g_{\mu\nu}$, generates a non-zero vacuum expectation value of the energy-
momentum tensor ($<T_{\mu\nu}>$) which in turn acts as a source of curvature
(this is the so-called ”back-reaction problem”). With this energy momentum
tensor and an ansatz for the metric, the solutions of Einstein’s equation
yields the metric solution, which is static and spherically symmetric [63].
Using the conformal anomaly method the modifications to the space-time metric
by the one loop back reaction was computed [101, 63]. Later it was shown [64,
65] that the Bekenstein-Hawking area law was modified, in the leading order,
by logarithmic corrections. Similar conclusions were also obtained by using
quantum gravity techniques [66, 71, 102]. Likewise, corrections to the semi-
classical Hawking temperature were derived [67, 68, 69, 70].
It is known that for the usual cases, the Hawking temperature diverges as the
radius of the event horizon decreases. This uncomfortable situation leads to
the “information paradox”. To avoid this one of the attempts is inclusion of
the noncommutative effect in the space-time. There exits two methods: (i)
directly take the space-time as noncommutative,
$[x_{\mu},x_{\nu}]=i\theta_{\mu\nu}$ and use Seibarg-Witten map to recast the
gravitational theory (in noncommutative space) in terms of the corresponding
theory in usual (commutative space) variables, and (ii) incorporate the effect
of noncommutativity in the mass term of the gravitating object.
In this chapter, we shall include the back reaction as well as noncommutative
effects in the space-time metric. Following the radial null geodesic method
presented in the previous chapter, the thermodynamic entities will be
calculated. Although there have been sporadic attempts in this direction [74,
75] a systematic, thorough and complete analysis was lacking.
The organization of the chapter is as follows. In the first section, we
compute the corrections to the semi-classical tunneling rate by including the
effects of self gravitation and back reaction. The usual expression found in
[23], given in the Maxwell-Boltzmann form $e^{-\frac{\omega}{T_{H}}}$, is
modified by a prefactor. This prefactor leads to a modified Bekenstein-Hawking
entropy. The semi-classical Bekenstein-Hawking area law connecting the entropy
to the horizon area is altered. As obtained in other approaches [64, 65, 66,
67, 68, 69, 70, 71], the leading correction is found to be logarithmic while
the nonleading one is a series in inverse powers of the horizon area (or
Bekenstein-Hawking entropy). We also compute the appropriate modification to
the Hawking temperature. Explicit results are given for the Schwarzschild
black hole.
Next, we shall apply our general formulation to discuss various thermodynamic
properties of a black hole defined in a noncommutative Schwarzschild space
time where back reaction is also taken into account. A short introduction of
the noncommutative Schwarzschild black hole is presented at the beginning of
the section (3.2). In particular we are interested in the black hole
temperature when the radius is of the order $\sqrt{\theta}$, where $\theta$ is
the noncommutative parameter. Such a study is relevant because
noncommutativity is supposed to remove the so called information paradox where
for a standard black hole, temperature diverges as the radius shrinks to zero.
The Hawking temperature is obtained in a closed form that includes corrections
due to noncommutativity and back reaction. These corrections are such that, in
some examples, the information paradox is avoided. Expressions for the entropy
and tunneling rate are also found for the leading order in the noncommutative
parameter. Furthermore, in the absence of back reaction, we show that the
entropy and area are algebraically related in the same manner as occurs in the
standard Bekenstein-Hawking area law.
### 3.1 Back reaction effect
In this section we shall derive the modifications in the Hawking temperature
and Bekenstein-Hawking area law due to the one loop back reaction effect in
the space-time. Back reaction is essentially the effect of the Hawking
radiation on the horizon. For simplicity, only the Schwarzschild black hole
will be considered. One way to include the back reaction effect into the
problem is to solve Einstein’s equation with an appropriate source. In this
case one considers the renormalized energy-momentum tensor due to one loop
back reaction effect on the right hand side of the Einstein’s equation. Then
solution of this equation gives the black hole metric given by the form (2.1)
[63]. Therefore it is feasible to apply the tunneling method developed in the
previous chapter for this case to find the modifications to the usual
thermodynamical entities. Here our discussions will be based on the radial
null geodesic method.
Here our starting point is the expression for the imaginary part of the action
(2.51), since in the present problem the form of the modified surface gravity
of the black hole is known. The modified surface gravity due to one loop back
reaction effects is given by [63],
$\displaystyle\kappa[M]=\kappa_{0}[M]\Big{(}1+\frac{\alpha}{M^{2}}\Big{)}$
(3.1)
where $\kappa_{0}$ is the classical surface gravity at the horizon of the
black hole. Such a form is physically dictated by simple scaling arguments. As
is well known, a loop expansion is equivalent to an expansion in powers of the
Planck constant $\hbar$. Therefore, the one loop back reaction effect in the
surface gravity is written as,
$\displaystyle\kappa=\kappa_{0}+\xi\kappa_{0}$ (3.2)
where $\xi$ is a dimensionless constant having magnitude of the order $\hbar$.
Now, in natural units $G=c=k_{B}=1$, Planck lenght $l_{p}=$ Planck mass
$M_{p}=\sqrt{\hbar}$ 111Planck length $l_{p}=\sqrt{\frac{\hbar G}{c^{3}}}$,
Planck mass $M_{p}=\sqrt{\frac{\hbar c}{G}}$. On the other hand, for
Schwarzschild black hole, mass $M$ is the only macroscopic parameter.
Therefore, $\xi$ must be function of $\frac{M_{p}}{M}$ which vanishes in the
limit $M_{p}<<M$. Since, as stated earlier, $\xi$ is a dimensionless constant
with magnitude of order $\hbar$, the leading term has the following quadratic
form,
$\displaystyle\xi=\beta\frac{M_{p}^{2}}{M^{2}}~{}.$ (3.3)
In the above, $\beta$ is a pure numerical factor. Taking $\alpha=\beta
M_{p}^{2}$ and then substituting (3.3) in (3.2) we obtain (3.1). The constant
$\beta$ is related to the trace anomaly coefficient taking into account the
degrees of freedom of the fields [103, 63, 64]. Its explicit form is given by
[103, 64],
$\displaystyle\beta=\frac{1}{360\pi}\Big{(}-N_{0}-\frac{7}{4}N_{\frac{1}{2}}+13N_{1}+\frac{233}{4}N_{\frac{3}{2}}-212N_{2})$
(3.4)
where $N_{s}$ denotes the number of fields with spin ‘$s$’.
For the classical Schwarzschild space-time the metric coefficients are given
by (2.76) and so by equation (2.19) the value of $\kappa_{0}[M]$ is
$\displaystyle\kappa_{0}[M]=\frac{f^{\prime}(r_{H}=2M)}{2}=\frac{1}{4M}~{}.$
(3.5)
Substituting (3.1) with $\kappa_{0}$ is given by (3.5) in (2.51) and then
integrating over $\omega^{\prime}$ we have
$\displaystyle
Im~{}{\cal{S}}=4\pi\omega(M-\frac{\omega}{2})+2\pi\alpha\ln{\Big{[}\frac{(M-\omega)^{2}+\alpha}{M^{2}+\alpha}\Big{]}}~{}.$
(3.6)
Now according to the WKB-approximation method the tunneling probability is
given by (2.55). So the modified tunneling probability due to back reaction
effects is,
$\displaystyle\Gamma\sim\Big{[}1-\frac{2\omega(M-\frac{\omega}{2})}{M^{2}+\alpha}\Big{]}^{-\frac{4\pi\alpha}{\hbar}}e^{-\frac{8\pi\omega}{\hbar}(M-\frac{\omega}{2})}$
(3.7)
The exponential factor of the tunneling probability was previously obtained by
Parikh and Wilczek [23]. The factor before the exponential is new. It is
actually due the effect of back reaction. It will eventually give the
correction to the Bekenstein-Hawking entropy, area law and the Hawking
temperature as will be shown below.
It was shown in the previous chapter and also in the literature [23, 80, 97]
that a change in the Bekenstein-Hawking entropy due to the tunneling through
the horizon is related to $Im~{}{\cal{S}}$ by the relation (2.59). Therefore
the corrected change in Bekenstein-Hawking entropy is
$\displaystyle\Delta
S_{bh}=-\frac{8\pi\omega}{\hbar}(M-\frac{\omega}{2})-\frac{4\pi\alpha}{\hbar}\ln\Big{[}(M-\omega)^{2}+\alpha\Big{]}+\frac{4\pi\alpha}{\hbar}\ln(M^{2}+\alpha)$
(3.8)
Next using the stability criterion $\frac{d(\Delta S_{bh})}{d\omega}=0$ for
the black hole, one obtains the following condition
$\displaystyle(\omega-M)^{3}=0$ (3.9)
which gives the only solution as $\omega=M$. Substituting this value of
$\omega$ in (3.8) we will have the change in entropy of the black hole from
its initial state to final state:
$\displaystyle S_{final}-S_{initial}=-\frac{4\pi
M^{2}}{\hbar}+\frac{4\pi\alpha}{\hbar}\ln{(\frac{M^{2}}{\alpha}+1)}~{}.$
(3.10)
Setting $S_{final}=0$, the Bekenstein-Hawking entropy of the black hole with
mass $M$ is
$\displaystyle S_{bh}=S_{initial}$ $\displaystyle=$ $\displaystyle\frac{4\pi
M^{2}}{\hbar}-4\pi\beta\ln{(\frac{M^{2}}{\beta\hbar}+1)}$ (3.11)
where we have substituted $\alpha=\beta M_{p}^{2}=\beta\hbar$.
Now the area of the black hole horizon given by
$\displaystyle A=4\pi r^{2}_{H}=16\pi M^{2}~{}.$ (3.12)
Putting (3.12) in (3.11) and expanding the logarithm, we obtain the final
form,
$\displaystyle S_{bh}$ $\displaystyle=$
$\displaystyle\frac{A}{4\hbar}-4\pi\beta\ln\frac{A}{4\hbar}-64\pi^{2}\hbar\beta^{2}\Big{[}\frac{1}{A}-\frac{16\pi\hbar\beta}{2A^{2}}+\frac{(16\pi\hbar\beta)^{2}}{3A^{3}}-.....\Big{]}$
(3.13) $\displaystyle+$ $\displaystyle\textrm{const.(independent~{} of~{}
$A$)}~{}.$
The first term is the usual semi-classical area law [2, 5] and other terms are
the corrections due to the one loop back reaction effect. The leading
correction is the well known logarithmic correction [64, 65, 66, 67, 68, 69,
70, 71]. Quantum gravity calculations lead to a prefactor $-\frac{1}{2}$ for
the $\ln\frac{A}{4\hbar}$ term which would correspond to choosing
$\beta=\frac{1}{8\pi}$. But here on the contrary $\beta$ is given by (3.4).
Also, the nonleading corrections are found to be expressed as a series in
inverse powers of $A$, exactly as happens in quantum gravity inspired analysis
[66, 71]. Now using the first law of black hole mechanics, $T_{H}dS_{bh}=dM$,
or the relation (2.29) between the Hawking temperature and surface gravity, we
can find the corrected form of the Hawking temperature $T_{H}$ due to back
reaction. This is obtained from (3.1) as,
$\displaystyle T_{H}=T_{0}\Big{(}1+\frac{\beta\hbar}{M^{2}}\Big{)}$ (3.14)
where $T_{0}=\frac{\hbar\kappa_{0}}{2\pi}=\frac{\hbar}{8\pi M}$ is the semi-
classical Hawking temperature and the other term is the correction due to the
back reaction. A similar expression was obtained previously in [64] by the
conformal anomaly method.
It is also possible to obtain the corrected Hawking temperature (3.14) in the
standard tunneling method to leading order [23] where this temperature is read
off from the coefficient of ‘$\omega$’ in the exponential of the probability
amplitude (3.7). Recasting this amplitude as,
$\displaystyle\Gamma\sim
e^{-\frac{8\pi\omega}{\hbar}(M-\frac{\omega}{2})-\frac{4\pi\alpha}{\hbar}\ln(1-\frac{2\omega(M-\frac{\omega}{2})}{M^{2}+\alpha})}$
(3.15)
and retaining terms upto leading order in $\omega$, we obtain,
$\displaystyle\Gamma$ $\displaystyle\sim$ $\displaystyle e^{-\frac{8\pi
M\omega}{\hbar}+4\pi\beta(\frac{2M\omega}{M^{2}+\beta\hbar})}$ (3.16)
$\displaystyle=$ $\displaystyle e^{-(\frac{8\pi
M^{3}}{\hbar(M^{2}+\beta\hbar)})\omega}=e^{-\frac{\omega}{T_{H}}}.$
The inverse Hawking temperature, indentified with the coefficient of
‘$\omega$’, reproduces (3.14).
The above analysis showed how the effects of back reaction in the space-time
can be discussed in a general frame work of tunneling mechanism. The only
assumption was that the modified metric must be static, spherically symmetric.
In particular, the modifications to the temperature and entropy for the
Schwarzschild case were explicitly evaluated. The results agree with earlier
findings by different methods. Next, we shall discuss the noncommutative
effect in addition to the back reaction effect in the space-time using our
general frame work.
### 3.2 Inclusion of noncommutativity
Here we shall apply our previous formulations to find the modifications to the
Hawking temperature and Bekenstein-Hawking area law due to noncommutative as
well as back reaction effects. In the vanishing limit of noncommutative
parameter, the results reduce to those obtained in the previous section. First
a short introduction on the noncommutative Schwarzschild black hole will be
given. Then the modifications to the thermodynamic entities will be
calculated.
#### 3.2.1 Schwarzschild black hole in noncommutative space
The fact is that gravitation is a manifestation of the structure of spacetime
as dictated by the presence of gravitating objects. Therefore, inclusion of
noncommutative effects in gravity can be done in two ways. Directly take the
spacetime as noncommutative, $[x_{\mu},x_{\nu}]=i\theta_{\mu\nu}$ and use the
Seibarg-Witten map to recast the gravitational theory (in noncommutative
space) in terms of the corresponding theory in usual (commutative space)
variables. This leads to correction terms (involving powers of
$\theta{\mu\nu}$) in the various expressions like the metric, Riemann tensor
etc. This approach has been adopted in [104, 105, 106, 107, 108] 222For a
detailed discussions of this approach and a list of references see [109]..
Alternatively, incorporate the effect of noncommutativity in the mass term of
the gravitating object. Here the mass density, instead of being represented by
a Dirac delta function, is replaced by a Gaussian distribution. This approach
has been adopted in [110, 111, 112, 113, 114, 115] 333For a review and list of
references, see [116].. The two ways of incorporating noncommutative effects
in gravity are, in general, not equivalent. Here we follow the second
approach, for our investigation on the computation of thermodynamic entities
and area law for the noncommutative Schwarzschild black hole.
The usual definition of mass density in terms of the Dirac delta function in
commutative space does not hold good in noncommutative space because of the
position-position uncertainty relation. In noncommutative space mass density
is defined by replacing the Dirac delta function by a Gaussian distribution of
minimal width $\sqrt{\theta}$ in the following way [110]
$\displaystyle\rho_{\theta}(r)=\dfrac{M}{{(4\pi\theta)}^{3/2}}e^{-{\frac{r^{2}}{4\theta}}};\,\,\,\
{\displaystyle{Lim}}_{\theta\rightarrow
0}\rho_{\theta}(r)=\frac{M\delta(r)}{4\pi r^{2}}$ (3.17)
where the noncommutative parameter $\theta$ is a small ($\sim$ Plank length2)
positive number. This mass distribution is inspired from the coherent state
approach, where one has to consider the Voros star product instead of the
Moyal star product [88]. Using this expression one can write the mass of the
black hole of radius $r$ in the following way
$\displaystyle m_{\theta}(r)=\int_{0}^{r}{4\pi r^{\prime
2}\rho_{\theta}(r^{\prime})dr^{\prime}}=\frac{2M}{\sqrt{\pi}}\gamma(3/2,r^{2}/4\theta)$
(3.18)
where $\gamma(3/2,r^{2}/4\theta)$ is the lower incomplete gamma function,
which is discussed in the appendix. In the limit $\theta\rightarrow 0$ it
becomes the usual gamma function $(\Gamma_{{\textrm{total}}})$. Therefore
$m_{\theta}(r)\rightarrow M$ is the commutative limit of the noncommutative
mass $m_{\theta}(r)$.
To find a solution of Einstein equation with the noncommutative mass density
of the type (3.17), the temporal component of the energy momentum tensor
${(T_{\theta})}_{\mu}^{\nu}$ is identified as,
${(T_{\theta})}_{t}^{t}=-\rho_{\theta}$. Now demanding the condition on the
metric coefficients ${(g_{\theta})}_{tt}=-{(g_{\theta})}^{rr}$ for the
noncommutative Schwarzschild metric and using the covariant conservation of
energy momentum tensor ${(T_{\theta})}_{\mu}^{\nu}~{}_{;\nu}=0$, the energy
momentum tensor can be fixed to the form,
$\displaystyle{(T_{\theta})}_{\mu}^{\nu}={\textrm{diag}}{[-\rho_{\theta},p_{r},p^{\prime},p^{\prime}]},$
(3.19)
where, $p_{r}=-\rho_{\theta}$ and
$p^{\prime}=p_{r}-\frac{r}{2}\partial_{r}\rho_{\theta}$. This form of energy
momentum tensor is different from the perfect fluid because here $p_{r}$ and
$p^{\prime}$ are not same,
$\displaystyle
p^{\prime}=\Big{[}\frac{r^{2}}{4\theta}-1\Big{]}\frac{M}{(4\pi\theta)^{\frac{3}{2}}}e^{-\frac{r^{2}}{4\theta}}$
(3.20)
i.e. the pressure is anisotopic.
The solution of Einstein equation (in $c=G=1$ unit)
${(G_{\theta})}^{\mu\nu}=8\pi{{(T_{\theta})}^{\mu\nu}}$, using (3.19) as the
matter source, is given by the line element [110],
$\displaystyle
ds^{2}=-f_{\theta}(r)dt^{2}+\frac{dr^{2}}{f_{\theta}(r)}+r^{2}d\Omega^{2};\,\,\,f_{\theta}(r)=-{(g_{\theta})}_{tt}=\left(1-\frac{4M}{r\sqrt{\pi}}\gamma(\frac{3}{2},\frac{r^{2}}{4\theta})\right)~{}.$
(3.21)
Incidentally, this is same if one just replaces the mass term in the usual
commutative Schwarzschild space-time by the noncommutative mass
$m_{\theta}(r)$ from (3.18). Also observe that for $r>>\sqrt{\theta}$ the
above noncommutative metric reduces to the standard Schwarzschild form.
It is interesting to note that the noncommutative metric (3.21) is still
stationary, static and spherically symmetric as in the commutative case. One
or more of these properties is usually violated for other approaches [105,
106, 107] of introducing noncommutativity, particularly those based on
Seiberg-Witten maps that relate commutative spaces with noncommutative ones.
The event horizon can be found where $g^{rr}(r_{H})=0$, that is
$\displaystyle
r_{H}=\frac{4M}{\sqrt{\pi}}\gamma\Big{(}\frac{3}{2},\frac{r_{H}^{2}}{4\theta}\Big{)}.$
(3.22)
This equation cannot be solved for $r_{H}$ in a closed form. In the large
radius regime ($\frac{r_{H}^{2}}{4\theta}>>1$) we use the expanded form of the
incomplete $\gamma$ function given in the Appendix (eq. (3A.4)) to solve eq.
(3.22) by iteration. Keeping upto the order
$\frac{1}{\sqrt{\theta}}e^{-\frac{M^{2}}{\theta}}$, we find
$\displaystyle r_{H}\simeq
2M\Big{(}1-\frac{2M}{\sqrt{\pi\theta}}e^{-\frac{M^{2}}{\theta}}\Big{)}.$
(3.23)
#### 3.2.2 Noncommutative Hawking temperature, tunneling rate and entropy in
the presence of back reaction
Here the one loop back reaction effect on the space-time will be considered.
As explained earlier the modified surface gravity will be of the form given by
(3.2). But in this case since the only macroscopic parameter is $m_{\theta}$,
$\xi$ will has the following structure:
$\displaystyle\xi=\beta\frac{M_{\textrm{p}}^{2}}{m^{2}_{\theta}}$ (3.24)
where, as earlier, $\beta$ is a pure numerical factor. In the commutative
picture $\beta$ is known to be related to the trace anomaly coefficient [63,
64]. Putting this form of $\xi$ in (3.2) we get,
$\displaystyle\kappa=\kappa_{0}[r_{H}]\Big{(}1+\beta\frac{M_{\textrm{p}}^{2}}{m^{2}_{\theta}}\Big{)}.$
(3.25)
Equation (3.25) is recast as,
$\displaystyle\kappa=\kappa_{0}[r_{H}]\Big{(}1+\frac{\alpha}{m^{2}_{\theta}(r_{h})}\Big{)}$
(3.26)
where $\alpha=\beta M_{\textrm{p}}^{2}$. Since as mentioned already, the
noncommutative parameter $\theta$ is of the order of $l_{\textrm{p}}^{2}$,
$\alpha$ and $\theta$ are of the same order. This fact will be used later when
doing the graphical analysis.
In order to calculate the right hand side of (3.26), we need to obtain an
expression for noncommutative classical surface gravity at the horizon of the
black hole $(\kappa_{0}[r_{H}])$. This is done by using (2.19). For the
classical noncommutative Schwarzschild spacetime the metric coefficients are
given by (3.21). The value of $\kappa_{0}[r_{H}]$ is thus found to be,
$\displaystyle\kappa_{0}[r_{H}]=\frac{f^{\prime}(r_{H})}{2}=\frac{1}{2}\Big{[}\frac{1}{r_{H}}-\frac{r^{2}_{H}}{4\theta^{\frac{3}{2}}}\frac{e^{-\frac{r^{2}_{H}}{4\theta}}}{\gamma\Big{(}\frac{3}{2},\frac{r^{2}_{H}}{4\theta}\Big{)}}\Big{]}.$
(3.27)
Inserting (3.27) in (3.26) we get,
$\displaystyle\kappa=\frac{1}{2}\Big{[}\frac{1}{r_{H}}-\frac{r^{2}_{H}}{4\theta^{\frac{3}{2}}}\frac{e^{-\frac{r^{2}_{H}}{4\theta}}}{\gamma\Big{(}\frac{3}{2},\frac{r^{2}_{H}}{4\theta}\Big{)}}\Big{]}\Big{(}1+\frac{\alpha}{m^{2}_{\theta}(r_{H})}\Big{)}.$
(3.28)
In order to write the above equation completely in terms of $r_{H}$ we have to
express the mass $m_{\theta}$ in terms of $r_{H}$. For that we compare
equations (3.18) and (3.22) to get,
$\displaystyle m_{\theta}(r_{H})=\frac{r_{H}}{2}.$ (3.29)
This relation is the noncommutative deformation of the standard radius-mass
relation for the usual (commutative space) Schwarzschild black hole.
Expectedly in the limit $\theta\rightarrow 0$ eq. (3.29) reduces to its
commutative version $r_{H}=2M$.
Substituting (3.29) in (3.28) we get the value of modified noncommutative
surface gravity
$\displaystyle\kappa=\frac{1}{2}\Big{[}\frac{1}{r_{H}}-\frac{r^{2}_{H}}{4\theta^{\frac{3}{2}}}\frac{e^{-\frac{r^{2}_{H}}{4\theta}}}{\gamma\Big{(}\frac{3}{2},\frac{r^{2}_{H}}{4\theta}\Big{)}}\Big{]}\Big{(}1+\frac{4\alpha}{r^{2}_{H}}\Big{)}.$
(3.30)
So from (2.29), the modified noncommutative Hawking temperature including the
effect of back reaction is given by,
$\displaystyle
T_{H}=\frac{\hbar\kappa}{2\pi}=\frac{\hbar}{4\pi}\Big{[}\frac{1}{r_{H}}-\frac{r^{2}_{H}}{4\theta^{\frac{3}{2}}}\frac{e^{-\frac{r^{2}_{H}}{4\theta}}}{\gamma\Big{(}\frac{3}{2},\frac{r^{2}_{H}}{4\theta}\Big{)}}\Big{]}\Big{(}1+\frac{4\alpha}{r^{2}_{H}}\Big{)}.$
(3.31)
If the back reaction is ignored (i. e. $\alpha=0$), the expression for the
Hawking temperature agrees with that given in [110]. Also for the
$\theta\rightarrow 0$ limit, one can recover the standard result (3.14) [63,
64].
In the standard (commutative) case $T_{H}$ diverges as $M\rightarrow 0$ and
this puts a limit on the validity of the conventional description of Hawking
radiation. Against this scenario, temperature (3.31) includes noncommutative
and back reaction effects which are relevant at distances comparable to
$\sqrt{\theta}$. The behaviour of the temperature $T_{H}$ as a function of
horizon radius $r_{H}$ is plotted in fig.(3.1) (with positive $\alpha$) and in
fig.(3.2) (with negative $\alpha$).
Figure 3.1: $T_{H}$ Vs. $r_{H}$ plot (Here $\alpha=\theta$, $\alpha$ and
$\theta$ are positive).
$r_{H}$ is plotted in units of $\sqrt{\theta}$ and $T_{H}$ is plotted in units
of $\frac{1}{\sqrt{\theta}}$.
Red curve: $\alpha\neq 0,\theta=0$.
Blue curve: $\alpha=0,\theta=0$.
Black curve: $\alpha\neq 0,\theta\neq 0$.
Yellow curve: $\alpha=0,\theta\neq 0$.
Fig.(3.1) shows that in the region $r_{H}\simeq\sqrt{\theta}$, the effect of
noncommutativity significantly changes the nature of commutative space curves.
Interestingly two noncommutative curves, whether including back reaction or
not are qualitatively same. Both of them attain a maximum value at
$r_{H}={\tilde{r}}_{0}\simeq 4.7\sqrt{\theta}$ and then sharply drop to zero
forming an extremal black hole. In the region $r_{H}<r_{0}\simeq
3.0\sqrt{\theta}$ there is no black hole, because physically $T_{H}$ cannot be
negative. The only difference between them is that the back reaction effect
increases the maximum temperature by $20\%$. Infact, in the commutative space
also, back reaction effect increases the value of Hawking temperature. But
quite contrary to the noncommutative curves, both of them diverge as
$r_{H}\rightarrow 0$. As easily observed, the Hawking paradox is circumvented
by noncommutativity, with or without back reaction. This was also noted in
[110] where, however, the quantitative effects of back reaction were not
considered.
On the other hand fig.(3.2) shows that if any of the two effects (i.e. either
noncommutativity or back reaction) is present $T_{H}$ drops to zero. For
$\alpha=0,\theta\neq 0$ (yellow curve) $T_{H}$ becomes zero at
$r_{H}=r_{0}\simeq 3.0\sqrt{\theta}$ and for $\alpha\neq 0,\theta=0$ (red
curve) it becomes zero at $r_{H}=r_{0}\simeq 2.0\sqrt{\theta}$ . These cases
therefore bypass the Hawking paradox. But for noncommutative black hole with
back reaction ($\alpha\neq 0,\theta\neq 0$), $T_{H}$ is zero for two values of
$r_{H}$: $r_{H}\simeq 3.0\sqrt{\theta}$ and $r_{H}=2.0\sqrt{\theta}$ and then
it diverges towards positive infinity. This is not physically possible since
after entering the forbidden zone it resurfaces on the allowed sector. So for
both noncommutativity and back reaction effect, $\alpha$ can never be
negative.
Figure 3.2: $T_{H}$ Vs. $r_{H}$ plot (Here $|\alpha|=\theta$, $\alpha$ is
negative but $\theta$ is positive).
$r_{H}$ is plotted in units of $\sqrt{\theta}$ and $T_{H}$ is plotted in units
of $\frac{1}{\sqrt{\theta}}$.
Red curve: $\alpha\neq 0,\theta=0$.
Blue curve: $\alpha=0,\theta=0$.
Black curve: $\alpha\neq 0,\theta\neq 0$.
Yellow curve: $\alpha=0,\theta\neq 0$.
Having obtained the Hawking temperature of the black hole we calculate the
Bekenstein-Hawking entropy. The expression of entropy can be obtained from the
second law of thermodynamics. But instead of using it we employ the formula
(2.59) to calculate the entropy. Using (3.23) the modified surface gravity
(3.30) can be approximately expressed in terms of $M$. To the leading order,
we obtain,
$\displaystyle\kappa(M)$ $\displaystyle=$
$\displaystyle\frac{M^{2}+\alpha}{4M^{3}}\Big{[}1-\frac{4M^{5}}{(M^{2}+\alpha)\theta\sqrt{\pi\theta}}e^{-\frac{M^{2}}{\theta}}\Big{]}+{\cal{O}}(\frac{1}{\sqrt{\theta}}e^{-\frac{M^{2}}{\theta}}).$
(3.32)
Substituting this in (2.51) and then integrating over $\omega^{\prime}$ we
have,
$\displaystyle{\textrm{Im}}~{}{\cal{S}}$ $\displaystyle=$ $\displaystyle
4\pi\omega(M-\frac{\omega}{2})+2\pi\alpha\ln{\Big{[}\frac{(M-\omega)^{2}+\alpha}{M^{2}+\alpha}\Big{]}}-8\sqrt{\frac{\pi}{\theta}}M^{3}e^{-\frac{M^{2}}{\theta}}$
(3.33) $\displaystyle+$ $\displaystyle
8\sqrt{\frac{\pi}{\theta}}(M-\omega)^{3}e^{-\frac{(M-\omega)^{2}}{\theta}}$
$\displaystyle+$ $\displaystyle\textrm{const.(independent of $M$)}+{\cal
O}(\sqrt{\theta}e^{-\frac{M^{2}}{\theta}}).$
So by the relation (2.55) the modified tunneling probability due to
noncommutativity and back reaction effects is,
$\displaystyle\Gamma$ $\displaystyle\sim$
$\displaystyle\Big{[}1-\frac{2\omega(M-\frac{\omega}{2})}{M^{2}+\alpha}\Big{]}^{-\frac{4\pi\alpha}{\hbar}}\textrm{exp}\Big{[}\frac{16}{\hbar}\sqrt{\frac{\pi}{\theta}}M^{3}e^{-\frac{M^{2}}{\theta}}-\frac{16}{\hbar}\sqrt{\frac{\pi}{\theta}}(M-\omega)^{3}e^{-\frac{(M-\omega)^{2}}{\theta}}$
(3.34) $\displaystyle+$ $\displaystyle\textrm{const.(independent of
$M$)}\Big{]}\textrm{exp}\Big{[}-\frac{8\pi\omega}{\hbar}(M-\frac{\omega}{2})\Big{]}.$
The last exponential factor of the tunneling probability was previously
obtained by Parikh and Wilczek [23] where neither noncommutativity nor back
reaction effects were considered. The factors before this exponential are
actually due the effect of back reaction and noncommutativity. It will
eventually give the correction to the Bekenstein-Hawking entropy and the
Hawking temperature as will be shown below. Taking $\theta\rightarrow 0$ limit
we can immediately reproduce the commutative tunneling rate for Schwarzschild
black hole with back reaction effect [57].
We are now in a position to obtain the noncommutative deformation of the
Bekenstein-Hawking area law. The first step is to compute the entropy change
$\Delta S_{bh}$. Using (2.55) and (3.34) we obtain, to the leading order,
$\displaystyle\Delta S_{bh}=S_{final}-S_{initial}$ $\displaystyle\simeq$
$\displaystyle-\frac{8\pi\omega}{\hbar}(M-\frac{\omega}{2})-\frac{4\pi\alpha}{\hbar}\ln{\Big{[}\frac{(M-\omega)^{2}+\alpha}{M^{2}+\alpha}\Big{]}}+\frac{16}{\hbar}\sqrt{\frac{\pi}{\theta}}M^{3}e^{-\frac{M^{2}}{\theta}}$
(3.35) $\displaystyle-$
$\displaystyle\frac{16}{\hbar}\sqrt{\frac{\pi}{\theta}}(M-\omega)^{3}e^{-\frac{(M-\omega)^{2}}{\theta}}+\textrm{const.(independent
of $M$)}.$
Next using the stability criterion $\frac{d(\Delta S_{bh})}{d\omega}=0$ for
the black hole, one obtains the only physically possible solution for $\omega$
as $\omega=M$. Substituting this value of $\omega$ in (3.35) and setting
$S_{{\textrm{final}}}=0$ we have the Bekenstein-Hawking entropy
$\displaystyle S_{bh}=S_{{\textrm{initial}}}$ $\displaystyle\simeq$
$\displaystyle\frac{4\pi
M^{2}}{\hbar}-\frac{4\pi\alpha}{\hbar}\ln{(\frac{M^{2}}{\alpha}+1)}$ (3.36)
$\displaystyle-$
$\displaystyle\frac{16}{\hbar}\sqrt{\frac{\pi}{\theta}}M^{3}e^{-\frac{M^{2}}{\theta}}+\textrm{const.(independent
of $M$)}.$
Neglecting the back reaction effect ($\alpha=0$) the above expression of black
hole entropy is written as
$\displaystyle S_{bh}\simeq\frac{4\pi
M^{2}}{\hbar}-\frac{16}{\hbar}\sqrt{\frac{\pi}{\theta}}M^{3}e^{-\frac{M^{2}}{\theta}}.$
(3.37)
Now in order to write the above equation in terms of the noncommutative
horizon area ($A_{\theta}$), we use (3.23) to obtain,
$\displaystyle A_{\theta}=4\pi r_{H}^{2}=16\pi
M^{2}-64\sqrt{\frac{\pi}{\theta}}M^{3}e^{-\frac{M^{2}}{\theta}}+{\cal{O}}(\sqrt{\theta}e^{-\frac{M^{2}}{\theta}}).$
(3.38)
Comparing equations (3.37) and (3.38) we see that at the leading order the
noncommutative black hole entropy satisfies the area law
$\displaystyle S_{bh}=\frac{A_{\theta}}{4\hbar}.$ (3.39)
This is functionally identical to the Bekenstein-Hawking area law in the
commutative space.
Considering $\theta\rightarrow 0$ limit in (3.36) we have the corrected form
of Bekenstein-Hawking entropy for commutative Schwarzschild black hole with
back reaction effect [64, 57]. The well known logarithmic correction [71] is
reproduced (see equation (3.13)).
Now using the second law of thermodynamics (2.67) we can find the corrected
form of the Hawking temperature $T_{H}$ due to back reaction. This is obtained
from (3.36) as,
$\displaystyle\frac{1}{T_{H}}=\frac{dS_{bh}}{dM}=\frac{8\pi
M^{3}}{\hbar(M^{2}+\alpha)}+\frac{32}{\hbar}\frac{\sqrt{\pi}}{\theta^{\frac{3}{2}}}M^{4}e^{-\frac{M^{2}}{\theta}}+{\cal{O}}(\frac{1}{\sqrt{\theta}}e^{-\frac{M^{2}}{\theta}}).$
(3.40)
Therefore the back reaction corrected noncommutative Hawking temperature is
given by
$\displaystyle T_{H}$ $\displaystyle=$
$\displaystyle\frac{\hbar(M^{2}+\alpha)}{8\pi M^{3}}-\frac{\hbar
M^{2}}{2(\pi\theta)^{\frac{3}{2}}}e^{-\frac{M^{2}}{\theta}}+{\cal{O}}(\frac{1}{\sqrt{\theta}}e^{-\frac{M^{2}}{\theta}}).$
(3.41)
We now provide a simple consistency check on the relation (3.31). The Hawking
temperature is recalculated using this relation and showing that it reproduces
(3.41). For the large radius limit, (3.31) takes the value,
$\displaystyle
T_{H}\simeq\frac{\hbar}{4\pi}\big{[}\frac{1}{r_{H}}-\frac{r_{H}^{2}}{2\sqrt{\pi}\theta^{3/2}}e^{-\frac{r_{H}^{2}}{4\theta}}\big{]}\big{(}1+\frac{4\alpha}{r_{H}^{2}}\big{)}.$
(3.42)
Now the approximated form of $r_{H}$ in terms of $M$ (3.23) is substituted in
(3.42) to get the relation (3.41) upto the leading order in the noncommutative
parameter. This shows the self consistency of our calculation.
For $\alpha=\theta=0$, the expression (3.41) reduces to the usual Hawking
temperature $T_{H}=\frac{\hbar}{8\pi M}$ for a Schwarzschild black hole. Also,
keeping the back reaction ($\alpha$) but taking $\theta\rightarrow 0$ limit,
we reproduce the commutative Hawking temperature (3.14) [63, 64, 57].
### 3.3 Discussions
We have considered self-gravitation and (one loop) back reaction effects in
tunneling formalism for Hawking radiation. The modified tunneling rate was
computed. From this modification, corrections to the semiclassical expressions
for entropy and Hawking temperature were obtained. Also, the logarithmic
correction to the semiclassical Bekenstein-Hawking area law was reproduced.
The other significant part of this chapter was the application of our
formulation to a noncommutative Schwarzschild metric, keeping in mind the
consequence of back reaction. Several thermodynamic entities like the
temperature and entropy were computed. The tunneling rate was also derived.
The temperature, in particular, was obtained in a closed form. This result was
analyzed in detail using two graphical representations. We gave particular
attention to the small scale behaviour of black hole temperature where the
effects of both noncommutativity and back reaction are highly nontrivial. The
graphs presented here are naturally more general than [110, 63], because in
[110] the effect of back reaction was not included and in [63] space time was
taken to be commutative in nature. Expectedly in suitable limits, the results
of our paper reduced to that of [110, 63], but the combination of
noncommutativity and back reaction, as shown here, gave new results at small
scale. In particular, it was shown that in the presence of both
noncommutativity and back reaction, the back reaction parameter $\alpha$
cannot be negative. Interestingly, even for the commutative case, arguments
based on quantum geometry [57, 71, 66] fix a positive value for $\alpha$.
In the noncommutative analysis, with positive $\alpha$, (Fig 3.1), the maximum
Hawking temperature got enhanced in the presence of back reaction. However,
the Hawking paradox was avoided whether or not the back reaction is included.
Apart from the temperature, other variables like the tunneling rate and
entropy were given upto the leading order in the noncommutative parameter. The
entropy was expressed in terms of the area. The result was a noncommutative
deformation of the Bekenstein-Hawking area law, preserving the usual
functional form. Since both $T_{H}=\frac{\hbar\kappa}{2\pi}$ and the area law
retained their standard forms it suggests that the laws of noncommutative
black hole thermodynamics are a simple noncommutative deformation of the usual
laws. However, it must be remembered this result was obtained only in the
leading order approximation. For $r\sim\sqrt{\theta}$ this approximation is
expected to be significant.
As a final remark we mention that although our results are presented for the
Schwarzschild metric, the formulation is resilient enough to discuss both back
reaction and noncommutativity in other types of black holes.
## Appendix
## Appendix 3.A Incomplete gamma function
The lower incomplete gamma function is given by
$\displaystyle\gamma(a,x)=\int_{0}^{x}t^{a-1}e^{-t}dt$ (3A.1)
whereas the upper incomplete gamma function is
$\displaystyle\Gamma(a,x)=\int_{x}^{\infty}t^{a-1}e^{-t}dt$ (3A.2)
and they are related to the total gamma function through the following
relation
$\displaystyle\Gamma_{{\textrm{total}}}(a)=\gamma(a,x)+\Gamma(a,x)=\int_{0}^{\infty}t^{a-1}e^{-t}dt.$
(3A.3)
Furthermore, for large $x$, i.e. $x>>1$, the asymptotic expansion of the lower
incomplete gamma function is given by
$\displaystyle\gamma(\frac{3}{2},x)$ $\displaystyle=$
$\displaystyle\Gamma_{{\textrm{total}}}(\frac{3}{2})-\Gamma(\frac{3}{2},x)$
(3A.4) $\displaystyle\simeq$
$\displaystyle\frac{\sqrt{\pi}}{2}\Big{[}1-e^{-x}\sum_{p=0}^{\infty}\frac{x^{\frac{1-2p}{2}}}{\Gamma_{{\textrm{total}}}(\frac{3}{2}-p)}\Big{]}.$
Using the definition (3A.1) and then integrating by parts we have
$\displaystyle\gamma(a+1,x)=\int_{0}^{x}t^{a}e^{-t}dt$ $\displaystyle=$
$\displaystyle-t^{a}e^{-t}|_{0}^{x}+a\int_{o}^{x}t^{a-1}e^{-t}dt$ (3A.5)
$\displaystyle=$ $\displaystyle-x^{a}e^{-x}+a\gamma(a,x).$
Similarly by the definition (3A.2) one can show
$\displaystyle\Gamma(a+1,x)=x^{a}e^{-x}+a\Gamma(a,x).$ (3A.6)
## Appendix 3.B Some useful formulas
$\displaystyle I_{1}=\int_{a}^{b}e^{-\alpha
x^{2}}dx=\frac{1}{2{\alpha}^{\frac{1}{2}}}\Big{[}\sqrt{\pi}-\gamma(\frac{1}{2},\alpha
a^{2})-\Gamma(\frac{1}{2},\alpha b^{2})\Big{]}$ (3B.1) $\displaystyle
I_{2}=\int_{a}^{b}x^{2}e^{-\alpha
x^{2}}dx=\frac{1}{2{\alpha}^{\frac{3}{2}}}\Big{[}\frac{\sqrt{\pi}}{2}-\gamma(\frac{3}{2},\alpha
a^{2})-\Gamma(\frac{3}{2},\alpha b^{2})\Big{]}$ (3B.2) $\displaystyle
I_{3}=\int_{a}^{b}x^{4}e^{-\alpha
x^{2}}dx=\frac{1}{2{\alpha}^{\frac{5}{2}}}\Big{[}\frac{3\sqrt{\pi}}{4}-\gamma(\frac{5}{2},\alpha
a^{2})-\Gamma(\frac{5}{2},\alpha b^{2})\Big{]}$ (3B.3)
## Chapter 4 Tunneling mechanism and anomaly
Ever since Hawking’s original observation [4, 5] that black holes radiate,
there have been several derivations [8, 9, 22, 23, 10, 11, 16, 17, 18] of this
effect. A common feature in these derivations is the universality of the
phenomenon; the Hawking radiation is determined by the horizon properties of
the black hole leading to the same answer. This, in the absence of direct
experimental evidence, definitely reinforces Hawking’s original conclusion.
Moreover, it strongly suggests that there is some fundamental mechanism which
could, in some sense, unify the various approaches.
In this chapter we show that chirality is the common property which connects
the tunneling formalism [22, 23] and the anomaly method [9, 10, 11, 16, 17,
18, 19, 20, 21, 117, 118] in studying Hawking effect. Apart from being among
the most widely used approaches, interest in both the anomaly and tunneling
methods has been revived recently leading to different variations and
refinements in them [16, 18, 20, 21, 117, 118, 72, 73, 74, 78, 55, 119]. The
calculation will be performed using a family of metrics that includes a subset
of the stationary, spherically symmetric space-times which are asymptotically
flat. Also, the results are derived using mostly physical reasoning and do not
require any specific technical skill.
Before commencing on our analysis we briefly recapitulate the basic tenets of
the tunneling and anomaly methods. The idea of a tunneling description, quite
akin to what we know in usual quantum mechanics where classically forbidden
processes might be allowed through quantum tunneling, dates back to 1976
[120]. Present day computations generally follow either the null geodesic
method [23] or the Hamilton-Jacobi method [22], both of which rely on the
semi-classical WKB approximation yielding equivalent results. The essential
idea, as explained earlier, is that a particle-antiparticle pair forms close
to the event horizon. The ingoing negative energy mode is trapped inside the
horizon while the outgoing positive energy mode is observed at infinity as the
Hawking flux.
Although the notion of an anomaly, which represents the breakdown of some
classical symmetry upon quantisation, is quite old, its implications for
Hawking effect were first studied in [9]. It was based on the conformal
(trace) anomaly but the findings were confined only to two dimensions. However
it is possible to apply this method to general dimensions. Recently a new
method was put forward in [10, 11] where a general (any dimensions) derivation
was given. It was based on the well known fact that the effective theory near
the event horizon is a two dimensional conformal theory. The ingoing modes are
trapped within the horizon and cannot contribute to the effective theory near
the horizon. Thus the near horizon theory becomes a two dimensional chiral
theory. Such a chiral theory suffers from a general coordinate
(diffeomorphism) anomaly manifested by a nonconservation of the stress tensor.
Using this gravitational anomaly and a suitable boundary condition the Hawking
flux was obtained. A covariant version of this method, that was also
technically simpler, was given in [16]. This was followed by another, new,
effective action based approach in [18, 17].
The first step in our procedure is to derive the two dimensional gravitational
anomaly using the notion of chirality. This is a new method of obtaining the
gravitational anomaly. Once this anomaly is obtained, the flux is easily
deduced. Exploiting the same notion of chirality the probability of the
outgoing mode in the tunneling approach will be computed. The Hawking
temperature then follows from this probability. At an intermediate stage of
this computation we further show that the chiral modes obtained in the
tunneling formalism reproduce the gravitational anomaly thereby completing the
circle of arguments regarding the connection of the two approaches.
### 4.1 Metric and null coordinates
Consider a black hole characterised by a spherically symmetric, static space-
time and asymptotically flat metric of the form (2.1). For simplicity we
consider here $f(r)=g(r)=F(r)$ and hence the event horizon $r=r_{H}$ is
defined by $F(r_{H})=0$. Now it is well known [10, 11, 14, 121] that near the
event horizon the effective theory reduces to a two dimensional conformal
theory whose metric is given by the ($r-t$) sector of the original metric
(2.1).
It is convenient to express (2.1) in the null tortoise coordinates which are
defined as,
$\displaystyle u=t-r^{*},\,\,\,v=t+r^{*};$ (4.1)
where $r^{*}$ is defined by the relation (2.20). Under these set of
coordinates the relevant ($r-t$)-sector of the metric (2.1) takes the form,
$\displaystyle ds^{2}=-\frac{F(r)}{2}(du~{}dv+dv~{}du)$ (4.2)
Chiral conditions, to be discussed in the next section, are most appropriately
described in these coordinates.
### 4.2 Chirality conditions
Consider the Klein-Gordon (KG) equation (2.2) for a massless scalar particle
governed by the metric (4.2). Then the KG equation reduces to the following
form:
$\displaystyle 2\partial_{u}\partial_{v}\phi(u,v)=0.$ (4.3)
The general solution of this can be taken as
$\phi(u,v)=\phi^{(R)}(u)+\phi^{(L)}(v)$ where $\phi^{(R)}(u)$ and
$\phi^{(L)}(v)$ are the right (outgoing) and left (ingoing) modes (see
Appendix 2.A) satisfying
$\displaystyle\nabla_{v}\phi^{(R)}=0,\,\,\nabla_{u}\phi^{(R)}\neq 0;\,\,\,\
\nabla_{u}\phi^{(L)}=0,\,\,\,\nabla_{v}\phi^{(L)}\neq 0.$ (4.4)
These equations are expressed simultaneously as,
$\displaystyle\nabla_{\mu}\phi=\pm\bar{\epsilon}_{\mu\nu}\nabla^{\nu}\phi=\pm\sqrt{-g}\epsilon_{\mu\nu}\nabla^{\nu}\phi$
(4.5)
where $+(-)$ stand for left (right) mode and $\epsilon_{\mu\nu}$ is the
numerical antisymmetric tensor with $\epsilon_{uv}=\epsilon_{tr}=-1$. This is
the chirality condition 111In analogy with studies in 2d CFT this condition is
usually referred as holomorphy condition and the chiral modes $\phi^{(L,R)}$
are called the holomorphic modes.. In fact the condition (4.5) holds for any
chiral vector $J_{\mu}$ in which case
$J_{\mu}=\pm{\bar{\epsilon}}_{\mu\nu}J^{\nu}$. Likewise, the chirality
condition for the energy-momentum tensor is [19],
$\displaystyle
T_{\mu\nu}=\pm\frac{1}{2}(\bar{\epsilon}_{\mu\sigma}T^{\sigma}_{\nu}+\bar{\epsilon}_{\nu\sigma}T^{\sigma}_{\mu})+\frac{1}{2}g_{\mu\nu}T^{\alpha}_{\alpha}$
(4.6)
The $+$ ($-$) sign corresponds to the left (right) mode satisfying,
$\displaystyle T^{(R)}_{vv}=0,\,\,\,\,T^{(R)}_{uu}\neq 0$ (4.7) $\displaystyle
T^{(L)}_{uu}=0,\,\,\,\,T^{(L)}_{vv}\neq 0~{}.$ (4.8)
These are the analogous of (4.4). They manifest the symmetry under the
interchange $u\leftrightarrow v$ and $L\leftrightarrow R$. In the next
section, using these chirality conditions we will derive the explicit form for
the gravitational anomaly that reproduces the Hawking flux.
### 4.3 Chirality, gravitational anomaly and Hawking flux
It is well known that for a non-chiral (vector like) theory it is not possible
to simultaneously preserve, at the quantum level, general coordinate
invariance as well as conformal invariance. Since the former invariance is
fundamental in general relativity, conformal invariance is sacrificed leading
to a nonvanishing trace of the stress tensor, called the trace anomaly. Using
this trace anomaly and the chirality condition we will derive an expression
for the chiral gravitational (diffeomophism) anomaly from which the Hawking
flux is computed.
The energy-momentum tensor near an evaporating black hole is split into a
traceful and traceless part by [122],
$\displaystyle T_{\mu\nu}=\frac{R}{48\pi}g_{\mu\nu}+\theta_{\mu\nu}$ (4.9)
where $\theta_{\mu\nu}$ is symmetric (i.e. $\theta_{\mu\nu}=\theta_{\nu\mu}$),
so that it preserves the symmetricity of $T_{\mu\nu}$, and traceless (i.e.
$\theta_{\mu}^{\mu}=0$ so that in $u,v$ coordinates $\theta_{uv}=0$). The
traceful part is contained in the first piece leading to the trace anomaly,
$T^{\mu}_{\mu}=\frac{R}{24\pi}$. Also, since general coordinate invariance is
preserved, $\nabla^{\mu}T_{\mu\nu}=0$, from which it follows that the
solutions of $\theta_{\mu\nu}$ satisfy,
$\displaystyle\nabla^{\mu}\theta_{\mu\nu}=-\frac{1}{48\pi}\nabla_{\nu}R$
(4.10)
Now the energy-momentum tensor (4.9) can be regarded as the sum of the
contributions from the right and left moving modes. Symmetry principle tells
that the contribution from one mode is exactly equal to that from the other
mode, only that $u,v$ have to be interchanged. Since $T_{\mu\nu}$ is symmetric
we have $T_{\mu\nu}=T_{\mu\nu}^{(R)}+T_{\mu\nu}^{(L)}$ with
$\displaystyle
T_{\mu\nu}^{(R/L)}=\frac{R}{96\pi}g_{\mu\nu}+\theta_{\mu\nu}^{(R/L)}$ (4.11)
where $\theta_{\mu\nu}=\theta_{\mu\nu}^{(R)}+\theta_{\mu\nu}^{(L)}$ (in
analogy with $T_{\mu\nu}$). Therefore the chirality condition (4.7) and the
traceless condition of $\theta_{\mu\nu}$ immediately show
$\displaystyle\theta_{uv}^{(R)}=0,\,\,\,\theta_{vv}^{(R)}=0,\,\,\,\,\theta_{uu}^{(R)}\neq
0;\,\,\,\theta_{uv}^{(L)}=0,\,\,\,\theta_{uu}^{(L)}=0,\,\,\,\,\theta_{vv}^{(L)}\neq
0~{}.$ (4.12)
The trace anomaly for the chiral components follows from (4.11) and (4.12),
$\displaystyle
T{{}^{\mu}_{\mu}}{{}^{(R)}}=T{{}^{\mu}_{\mu}}{{}^{(L)}}=\frac{1}{2}T^{\mu}_{\mu}=\frac{R}{48\pi}~{}.$
(4.13)
To find out the diffeomorphism anomaly for the chiral components we will use
(4.11). Considering only the right mode, for example, we have
$\displaystyle\nabla^{\mu}T_{\mu\nu}^{(R)}=\frac{1}{96\pi}\nabla_{\nu}R+\nabla^{\mu}\theta_{\mu\nu}^{(R)}~{}.$
(4.14)
Next, using (4.10) and (4.12) for the right mode we obtain,
$\displaystyle\nabla^{\mu}\theta_{\mu
u}^{(R)}=-\frac{1}{48\pi}\nabla_{u}R;\,\,\,\nabla^{\mu}\theta_{\mu v}^{(R)}=0$
(4.15)
Substituting these in (4.14) we get, once for $\nu=u$ and then $\nu=v$,
$\displaystyle\nabla^{\mu}T_{\mu
u}^{(R)}=-\frac{1}{96\pi}\nabla_{u}R;\,\,\,\,\,\nabla^{\mu}T_{\mu
v}^{(R)}=\frac{1}{96\pi}\nabla_{v}R.$ (4.16)
Therefore, combining both the above results yields
$\displaystyle\nabla^{\mu}T_{\mu\nu}^{(R)}=\frac{1}{96\pi}\bar{\epsilon}_{\nu\lambda}\nabla^{\lambda}R$
(4.17)
which is the chiral (gravitational) anomaly for the right mode. Similarly the
chiral anomaly for left mode can also be obtained which has a similar form
except for a minus sign on the right side of (4.17). This anomaly is in
covariant form and so it is also called the covariant gravitational anomaly.
The structure, including the normalization, agrees with that found by using
explicit regularization of the chiral stress tensor [123, 124].
From (4.17) and (4.13) a simple relation follows between the gravitational
anomaly (${\cal{A}}_{\nu}$) and the trace anomaly ($T$),
$\displaystyle{\cal{A}}_{\nu}=\frac{1}{2}\bar{\epsilon}_{\nu\lambda}\nabla^{\lambda}T.$
(4.18)
Such a relation is not totally unexpected since covariant expressions must
involve the Ricci scalar. However (4.18) should not be interpreted as a Wess-
Zumino consistency condition which involves only ‘consistent’ expressions
[124]. Here, on the contrary, we are dealing with covariant expressions.
The covariant anomaly (4.17) is now used to obtain the Hawking flux. As was
mentioned earlier the effective two dimensional theory near the horizon
becomes chiral. The chiral theory has the anomaly (4.17). Taking its $\nu=u$
component we obtain,
$\displaystyle\partial_{r}T_{uu}^{(R)}=\frac{F}{96\pi}\partial_{r}R=\frac{F}{96\pi}\partial_{r}(F^{\prime\prime})=\frac{1}{96\pi}\partial_{r}(FF^{\prime\prime}-\frac{F^{\prime
2}}{2})$ (4.19)
which yields,
$\displaystyle
T_{uu}^{(R)}=\frac{1}{96\pi}\Big{(}FF^{{}^{\prime\prime}}-\frac{F^{\prime
2}}{2}\Big{)}+C_{uu}$ (4.20)
where $C_{uu}$ is an integration constant.
Now, in the coordinates $U=-\kappa e^{-\kappa u}$ and $V=\kappa e^{\kappa v}$,
we have the following relations for components of the energy-momentum tensor:
$\displaystyle T_{UU}^{(R)}=\frac{T_{uu}^{(R)}}{(\kappa U)^{2}}$ (4.21)
$\displaystyle T_{VV}^{(R)}=\frac{T_{vv}^{(R)}}{(\kappa V)^{2}}~{}.$ (4.22)
According to the definition of Unruh vacuum (proper vacuum for studying
Hawking effect) for outgoing mode $T_{UU}$ must be finite at future horizon
($U\rightarrow 0$), implying that a freely falling observer sees a finite
amount of flux at the outer horizon. This requires $T_{uu}^{(R)}(r\rightarrow
r_{H})=0$, leads to $C_{uu}=\frac{F^{\prime 2}(r_{H})}{192\pi}$. The
corresponding condition on the ingoing mode for the Unruh vacuum - $T_{VV}$ is
finite at infinity - is satisfied by default since, due to chirality, these
are absent ($T_{vv}^{(R)}=0$). This choice of the Unruh vacuum is similar to
imposing the covariant boundary condition [19]. Note, however, that the Unruh
condition on the ingoing modes $T_{vv}^{(R)}(r\rightarrow\infty)=0$ is applied
at asymptotic infinity where the theory is non-chiral. This does not affect
our interpretation since, asymptotically, the anomaly (4.17) vanishes. Hence
the results from the chiral expressions will agree with the non-chiral ones at
asymptotic infinity. Indeed, the Hawking flux, obtained by taking the
asymptotic infinity limit ($r\rightarrow\infty$) of (4.20),
$\displaystyle T_{uu}^{(R)}(r\rightarrow\infty)=C_{uu}=\frac{F^{\prime
2}(r_{H})}{192\pi}=\frac{\kappa^{2}}{48\pi}$ (4.23)
where $\kappa$ is the surface gravity of the black hole given by (2.19),
reproduces the known result corresponding to the Hawking temperature (2.29) in
$\hbar=1$ unit [10, 11, 12, 16, 17, 18, 19, 20, 21, 118, 119]. The other terms
in (4.20) drop out due to asymptotic flatness.
### 4.4 Chirality, quantum tunneling and Hawking temperature
Here, using the chirality condition (4.5), we will derive the tunneling
probability, which will eventually yield the Hawking temperature. Under the
($t-r$) sector of the metric (2.1), this condition corresponds to,
$\displaystyle\partial_{t}\phi(r,t)=\pm F(r)\partial_{r}\phi(r,t)$ (4.24)
As before $+(-)$ stand for left (right) mode. Putting the standard WKB ansatz
(2.4) and the expansion for $S(r,t)$ (2.6) in (4.24), we get in the
$\hbar\rightarrow 0$ limit the familiar semiclassical Hamilton-Jacobi equation
(2.7), which is the basic equation in the tunneling mechanism for studying
Hawking radiation. This has been derived earlier from the Klein-Gordon
equation with the background metric (2.1) and the ansatz (2.4) [22, 55].
Now proceeding in the similar way as earlier, we obtain the solution for
$S_{0}(r,t)$ as,
$\displaystyle S_{0}(r,t)=\omega t\pm\omega\int\frac{dr}{F(r)}$ (4.25)
which is nothing but (2.10) for $f(r)=g(r)=F(r)$. Expressing (4.25) in the
null tortoise coordinates (see (4.1)), defined inside and outside of the event
horizon, we obtain
$\displaystyle\Big{(}S_{0}^{(R)}(r,t)\Big{)}_{in}=\omega(t_{in}-r^{*}_{in})=\omega
u_{in};$ (4.26)
$\displaystyle\Big{(}S_{0}^{(L)}(r,t)\Big{)}_{in}=\omega(t_{in}+r^{*}_{in})=\omega
v_{in}$ (4.27)
$\displaystyle\Big{(}S_{0}^{(R)}(r,t)\Big{)}_{out}=\omega(t_{out}-r^{*}_{out})=\omega
u_{out};$ (4.28)
$\displaystyle\Big{(}S_{0}^{(L)}(r,t)\Big{)}_{out}=\omega(t_{out}+r^{*}_{out})=\omega
v_{out}~{}.$ (4.29)
Substituting these in (2.4) one can obtain the right and left modes for both
sectors:
$\displaystyle\Big{(}\phi^{(R)}\Big{)}_{in}=e^{-\frac{i}{\hbar}\omega
u_{in}};\,\,\,\Big{(}\phi^{(L)}\Big{)}_{in}=e^{-\frac{i}{\hbar}\omega v_{in}}$
(4.30) $\displaystyle\Big{(}\phi^{(R)}\Big{)}_{out}=e^{-\frac{i}{\hbar}\omega
u_{out}};\,\,\,\Big{(}\phi^{(L)}\Big{)}_{out}=e^{-\frac{i}{\hbar}\omega
v_{out}}$ (4.31)
which satisfy the condition (4.4). Precisely these modes were used previously
to find the trace anomaly [122] as well as the chiral (gravitational) anomaly
[123] by the point splitting regularization technique. In our formulation
these modes (4.31) are a natural consequence of chirality.
Now in the tunneling formalism, as stated earlier, a virtual pair of particles
is produced in the black hole. One of this pair can quantum mechanically
tunnel through the horizon. This particle is observed at infinity while the
other goes towards the center of the black hole. While crossing the horizon
the nature of the coordinates changes. This can be explained in the following
way. The Kruskal time ($T$) and space ($X$) coordinates inside and outside the
horizon are defined by (2.17) and (2.18) respectively. In section 2.1.1 of
chapter 2 it has been shown that these two sets of coordinates are connected
by the relations (2.21) and (2.22), so that the Kruskal coordinates get
identified as $T_{in}=T_{out}$ and $X_{in}=X_{out}$. In particular, for the
Schwarzschild metric, the surface gravity is $\kappa=\frac{1}{4M}$ and thus
the extra term connecting $t_{in}$ and $t_{out}$ is given by ($-2\pi iM$).
Such a result (for the Schwarzschild case) was earlier discussed in [78]. It
should be mentioned that instead of Kruskal coordinates one can do the
analysis employing the Painleve coordinates [96] since in these coordinates
the apparent singularity at the horizon is also removed. Nevertheless it is
noteworthy that the coordinate transformation from the Schwarzschild-like to
the Painleve coordinates contains a singularity at the horizon while
transformations (2.17) and (2.18) do not have such singularity. Therefore,
Painleve coordinates are not suitable for the present analysis. In addition,
there is an arbitrariness in the mapping $T_{in}=T_{out}$ and $X_{in}=X_{out}$
because they can also be obtained if, instead of (2.21) and (2.22), we use the
following relations
$\displaystyle
t_{in}=t_{out}+i\frac{\pi}{2\kappa};\,\,\,\,r^{*}_{in}=r^{*}_{out}-i\frac{\pi}{2\kappa}~{}.$
(4.32)
However, this set of coordinates gives unphysical results. This issue will be
clarified in the subsequent analysis. Therefore, we can exclude the set of
coordinates given by equation (4.32).
Employing equations (2.21) and (2.22) in equation (4.1), we can obtain the
relations that connect the null coordinates defined inside and outside the
black hole event horizon
$\displaystyle u_{in}=t_{in}-r^{*}_{in}=u_{out}-i\frac{\pi}{\kappa}$ (4.33)
$\displaystyle v_{in}=t_{in}+r^{*}_{in}=v_{out}~{}.$ (4.34)
Under these transformations the modes in equations (4.30) and (4.31) which are
travelling in the “$in$” and “$out$” sectors of the black hole horizon are
connected through the expressions
$\displaystyle\phi^{(R)}_{in}=e^{-\frac{\pi\omega}{\hbar\kappa}}\phi^{(R)}_{out}$
(4.35) $\displaystyle\phi^{(L)}_{in}=\phi^{(L)}_{out}~{}.$ (4.36)
Since the left moving mode travels towards the center of the black hole, its
probability to go inside, as measured by an external observer, is expected to
be unity. This is easily verified by computing
$\displaystyle P^{(L)}=|\phi^{(L)}_{in}|^{2}=|\phi^{(L)}_{out}|^{2}=1$ (4.37)
where we have used (4.36) to recast $\phi^{(L)}_{in}$ in terms of
$\phi^{(L)}_{out}$ since measurements are done by an outside observer. This
shows that the left moving (ingoing) mode is trapped inside the black hole, as
expected.
On the other hand the right moving mode, i.e. $\phi^{(R)}_{in}$, tunnels
through the event horizon. So to calculate the tunneling probability as seen
by an external observer one has to use the transformation (4.35) to recast
$\phi^{(R)}_{in}$ in terms of $\phi^{(R)}_{out}$. Then we find
$\displaystyle
P^{(R)}=|\phi^{(R)}_{in}|^{2}=|e^{-\frac{\pi\omega}{\hbar\kappa}}\phi^{(R)}_{out}|^{2}=e^{-\frac{2\pi\omega}{\hbar\kappa}}~{}.$
(4.38)
Finally, using the principle of “detailed balance” [22], i.e.
$P^{(R)}=e^{-\frac{\omega}{T_{H}}}P^{(L)}=e^{-\frac{\omega}{T_{H}}}$, and
making comparison with equation (4.38), one immediately reproduces the Hawking
temperature (2.29). This is the standard expression corresponding to the flux
(4.23) in units of $\hbar=1$.
It should be pointed out that the tunneling probability given by equation
(4.38) goes to zero in the classical limit ($\hbar\rightarrow 0$), which is
expected since classically a black hole cannot radiate. On the other hand, if
the above analysis is repeated by utilizing the set of coordinates given in
equation (4.32), then $P^{(R)}=e^{\frac{2\pi\omega}{\hbar\kappa}}$. This
probability diverges in the classical limit which is unphysical. Therefore,
the set of coordinates presented in equation (4.32) are not appropriate for
our study.
As we observe the ingoing modes are trapped and do not play any role in the
computation of the Hawking temperature. A similar feature occurs in the
anomaly approach where the ingoing modes are neglected leading to a chiral
theory that eventually yields the flux. These observations provide a physical
picture of chirality connecting the tunneling and anomaly methods.
### 4.5 Discussions
We have shown that the notion of chirality pervades the anomaly and tunneling
formalisms thereby providing a close connection between them. This is true
both from a physical as well as algebraic perspective. The chiral restrictions
play a pivotal role in the abstraction of the anomaly from which the flux is
computed. The same restrictions, in the tunneling formalism, lead to the
Hawking temperature corresponding to that flux.
A dimensional reduction is known to reduce the theory effectively to a two
dimensional conformal theory near the event horizon. The ingoing (left moving)
modes are lost inside the horizon. They cannot contribute to the near horizon
theory thereby rendering it chiral and, hence, anomalous. Using the
restrictions imposed by chirality we obtained a form for this (gravitational)
anomaly, manifested by a nonconservation of the stress tensor, by starting
from the familiar form of the trace anomaly. From a knowledge of the
gravitational anomaly we were able to obtain the flux.
The chirality constraints were then exploited to obtain the equations for the
ingoing and outgoing modes in the tunneling formalism, following the standard
geometrical (WKB) approximation. We reformulated the tunneling mechanism to
highlight the role of coordinate systems in the chiral framework. A specific
feature of this reformulation is that explicit treatment of the singularity in
(4.25) is not required since we do not carry out the integration. Only the
modes inside ($\phi_{in}$) and outside ($\phi_{out}$) the horizon, both of
which are well defined, are required. The singularity now manifests in the
complex transformations (2.21) and (2.22) that connect these modes across the
horizon. In this way our formalism, contrary to the traditional approaches
[22, 23] avoids explicit complex path analysis. It is implicit only in the
expression for $S_{0}(r,t)$ (4.25). The probability for finding the ingoing
modes was shown to be unity. These modes do not play any role in the tunneling
approach which is the exact analogue of omitting them when considering the
effective near horizon theory in the anomaly method.
It is useful to observe that the crucial role of chirality in both approaches
is manifested in the near horizon regime. This reaffirms the universality of
the Hawking effect being governed by the properties of the event horizon.
## Chapter 5 Black body spectrum from tunneling mechanism
So far we have discussed the Hawking effect by the tunneling mechanism.
However, the analysis was confined to obtaining the Hawking temperature only
by comparing the tunneling probability of an outgoing particle with the
Boltzmann factor. There was no discussion of the spectrum. Hence it is not
clear whether this temperature really corresponds to the temperature of a
black body spectrum associated with black holes. One has to take recourse to
other results to really justify the fact that the temperature found in the
tunneling approach is indeed the Hawking black body temperature. Indeed, as
far as we are aware, there is no discussion of the spectrum in the different
variants of the tunneling formalism. In this sense the tunneling method,
presented so far, is incomplete.
In this chapter we rectify this shortcoming. Using density matrix techniques
we will directly find the spectrum from a reformulation of the tunneling
mechanism discussed in the previous chapter. For both bosons and fermions we
obtain a black body spectrum with a temperature that corresponds to the
familiar semi-classical Hawking expression. Our results are valid for black
holes with spherically symmetric geometry.
### 5.1 Black body spectrum and Hawking flux
Here the emission spectrum of the black hole will be calculated by the density
matrix technique. It has been shown in chapter 4 that a pair created inside
the black hole is represented by the modes (4.30). Since the Hawking effect is
observed from outside the black hole, one must recast these modes in terms of
the outside coordinates. This will yield the relations between the “$in$” and
“$out$” modes. These are given by (4.35) and (4.36). These transformations are
the essential ingredients of constructing all the physical observables
regarding the Hawking effect, because the observer is situated outside the
event horizon of the black hole.
Now to find the black body spectrum and Hawking flux, we first consider $n$
number of non-interacting virtual pairs that are created inside the black
hole. Each of these pairs is represented by the modes defined by (4.30). Then
the physical state of the system, observed from outside, is given by,
$\displaystyle|\Psi>=N\sum_{n}|n^{(L)}_{in}>\otimes|n^{(R)}_{in}>=N\sum_{n}e^{-\frac{\pi
n\omega}{\hbar\kappa}}|n^{(L)}_{out}>\otimes|n^{(R)}_{out}>$ (5.1)
where use has been made of the transformations (4.35) and (4.36). Here
$|n^{(L)}_{out}>$ corresponds to $n$ number of left going modes and so on
while $N$ is a normalization constant which can be determined by using the
normalization condition $<\Psi|\Psi>=1$. This immediately yields,
$\displaystyle N=\frac{1}{\Big{(}\displaystyle\sum_{n}e^{-\frac{2\pi
n\omega}{\hbar\kappa}}\Big{)}^{\frac{1}{2}}}~{}.$ (5.2)
The above sum will be calculated for both bosons and fermions. For bosons
$n=0,1,2,3,....$ whereas for fermions $n=0,1$. With these values of $n$ we
obtain the normalization constant (5.2) as
$\displaystyle
N_{(\textrm{boson})}=\Big{(}1-e^{-\frac{2\pi\omega}{\hbar\kappa}}\Big{)}^{\frac{1}{2}}$
(5.3) $\displaystyle
N_{(\textrm{fermion})}=\Big{(}1+e^{-\frac{2\pi\omega}{\hbar\kappa}}\Big{)}^{-\frac{1}{2}}~{}.$
(5.4)
Therefore the normalized physical states of the system for bosons and fermions
are, respectively,
$\displaystyle|\Psi>_{(\textrm{boson})}=\Big{(}1-e^{-\frac{2\pi\omega}{\hbar\kappa}}\Big{)}^{\frac{1}{2}}\sum_{n}e^{-\frac{\pi
n\omega}{\hbar\kappa}}|n^{(L)}_{out}>\otimes|n^{(R)}_{out}>,$ (5.5)
$\displaystyle|\Psi>_{(\textrm{fermion})}=\Big{(}1+e^{-\frac{2\pi\omega}{\hbar\kappa}}\Big{)}^{-\frac{1}{2}}\sum_{n}e^{-\frac{\pi
n\omega}{\hbar\kappa}}|n^{(L)}_{out}>\otimes|n^{(R)}_{out}>~{}.$ (5.6)
From here on our analysis will be only for bosons since for fermions the
analysis is identical. For bosons the density matrix operator of the system is
given by,
$\displaystyle{\hat{\rho}}_{(\textrm{boson})}$ $\displaystyle=$
$\displaystyle|\Psi>_{(\textrm{boson})}<\Psi|_{(\textrm{boson})}$ (5.7)
$\displaystyle=$
$\displaystyle\Big{(}1-e^{-\frac{2\pi\omega}{\hbar\kappa}}\Big{)}\sum_{n,m}e^{-\frac{\pi
n\omega}{\hbar\kappa}}e^{-\frac{\pi
m\omega}{\hbar\kappa}}|n^{(L)}_{out}>\otimes|n^{(R)}_{out}><m^{(R)}_{out}|\otimes<m^{(L)}_{out}|~{}.$
Now since, as explained in the previous chapter, the ingoing ($L$) modes are
completely trapped, they do not contribute to the emission spectrum from the
black hole event horizon. Hence tracing out the ingoing (left) modes we obtain
the density matrix for the outgoing modes,
$\displaystyle{\hat{\rho}}^{(R)}_{(\textrm{boson})}=\Big{(}1-e^{-\frac{2\pi\omega}{\hbar\kappa}}\Big{)}\sum_{n}e^{-\frac{2\pi
n\omega}{\hbar\kappa}}|n^{(R)}_{out}><n^{(R)}_{out}|~{}.$ (5.8)
Therefore the average number of particles detected at asymptotic infinity is
given by,
$\displaystyle<n>_{(\textrm{boson})}={\textrm{trace}}({\hat{n}}{\hat{\rho}}^{(R)}_{(\textrm{boson})})$
$\displaystyle=$
$\displaystyle\Big{(}1-e^{-\frac{2\pi\omega}{\hbar\kappa}}\Big{)}\sum_{n}ne^{-\frac{2\pi
n\omega}{\hbar\kappa}}$ (5.9) $\displaystyle=$
$\displaystyle\Big{(}1-e^{-\frac{2\pi\omega}{\hbar\kappa}}\Big{)}(-\frac{\hbar\kappa}{2\pi})\frac{\partial}{\partial\omega}\Big{(}\sum_{n}e^{-\frac{2\pi
n\omega}{\hbar\kappa}}\Big{)}$ $\displaystyle=$
$\displaystyle\Big{(}1-e^{-\frac{2\pi\omega}{\hbar\kappa}}\Big{)}(-\frac{\hbar\kappa}{2\pi})\frac{\partial}{\partial\omega}\Big{(}\frac{1}{1-e^{-\frac{2\pi\omega}{\hbar\kappa}}}\Big{)}$
$\displaystyle=$ $\displaystyle\frac{1}{e^{\frac{2\pi\omega}{\hbar\kappa}}-1}$
where the trace is taken over all $|n^{(R)}_{out}>$ eigenstates. This is the
Bose distribution. Similar analysis for fermions leads to the Fermi
distribution:
$\displaystyle<n>_{(\textrm{fermion})}=\frac{1}{e^{\frac{2\pi\omega}{\hbar\kappa}}+1}~{}.$
(5.10)
Note that both these distributions correspond to a black body spectrum with a
temperature given by the Hawking expression (2.29). Correspondingly, the
Hawking flux can be obtained by integrating the above distribution functions
over all $\omega$’s. For fermions it is given by,
$\displaystyle{\textrm{Flux}}=\frac{1}{\pi}\int_{0}^{\infty}\frac{\omega~{}d\omega}{e^{\frac{2\pi\omega}{\hbar
K}}+1}=\frac{\hbar^{2}\kappa^{2}}{48\pi}$ (5.11)
Similarly, the Hawking flux for bosons can be calculated, leading to the same
answer.
### 5.2 Discussions
We have adopted a novel formulation of the tunneling mechanism which was
elaborated in the previous chapter to find the emission spectrum from the
black hole event horizon. Here the computations were done in terms of the
basic modes obtained earlier in Chapter 4. From the density matrix constructed
from these modes we were able to directly reproduce the black body spectrum,
for either bosons or fermions, from a black hole with a temperature
corresponding to the standard Hawking expression. We feel that the lack of
such an analysis was a gap in the existing tunneling formulations [22, 23, 72,
73, 74, 75, 77] which yield only the temperature rather that the actual black
body spectrum. Finally, although our analysis was done for a static
spherically symmetric space-time in Einstein gravity, this can be applied as
well for a stationary black hole, for example Kerr-Newman metric [125] and
also for black holes in other gravity theory like Hořava-Lifshit theory [126].
## Chapter 6 Global embedding and Hawking-Unruh effect
After Hawking’s famous work [4] \- radiation of black holes - known as Hawking
effect, it is now well understood that this is related to the event horizon of
a black hole. A closely related effect is the Unruh effect [30], where a
similar type of horizon is experienced by a uniformly accelerated observer on
the Minkowski space-time. A unified description of them was first put
forwarded by Deser and Levin [31, 32] which was a sequel to an earlier attempt
[33]. This is called the global embedding Minkowskian space (GEMS) approach.
In this approach, the relevant detector in curved space-time (namely Hawking
detector) and its event horizon map to the Rindler detector in the
corresponding flat higher dimensional embedding space [34, 35] and its event
horizon. Then identifying the acceleration of the Unruh detector, the Unruh
temperature can be calculated. Finally, use of the Tolman relation [36] yields
the Hawking temperature. In this picture the Unruh temperature is interpreted
as a local Hawking temperature. Subsequently, this unified approach to
determine the Hawking temperature using the Unruh effect was applied for
several black hole space-times [37, 38, 39, 127]. However the results were
confined to four dimensions and the calculations were done case by case,
taking specific black hole metrics. It was not clear whether the technique was
applicable to complicated examples like the Kerr-Newman metric which lacks
spherical symmetry.
The motivation of this chapter is to give a modified presentation of the GEMS
approach that naturally admits generalization. Higher dimensional black holes
with different metrics, including Kerr-Newman, are considered. Using this new
embedding, the local Hawking temperature (Unruh temperature) will be derived.
Then the Tolman formula leads to the Hawking temperature.
We shall first introduce a new global embedding which embeds only the
($t-r$)-sector of the curved metric into a flat space. It will be shown that
this embedding is enough to derive the Hawking result using the Deser-Levin
approach [31, 32], instead of the full embedding of the curved space-time.
Hence we might as well call this the reduced global embedding. This is
actually motivated from the fact that an $N$-dimensional black hole metric
effectively reduces to a $2$ -dimensional metric (only the ($t-r$)-sector)
near the event horizon by the dimensional reduction technique [10, 14, 121,
12, 125] (for examples see Appendix 6.A). Furthermore, this $2$-dimensional
metric is enough to find the Hawking quantities if the back scattering effect
is ignored. Several spherically symmetric static metrics will be exemplified.
Also, to show the utility of this reduced global embedding, we shall discuss
the most general solution of the Einstein gravity - Kerr-Newman space-time,
whose full global embedding is difficult to find. Since the reduced embedding
involves just the two dimensional ($t-r$)-sector, black holes in arbitrary
dimensions can be treated. In this sense our approach is valid for any higher
dimensional black hole.
The organization of the chapter is as follows. In section 6.1 we shall find
the reduced global embedding of several black hole space-times which are
spherically symmetric. In the next section the power of this approach will be
exploited to find the Unruh/Hawking temperature for the Kerr-Newman black
hole. Finally, we shall give our concluding remarks. One appendix, briefly
reviewing dimensional reduction, is also included.
### 6.1 Reduced global embedding
A unified picture of Hawking effect [4] and Unruh effect [30] was established
by the global embedding of a curved space-time into a higher dimensional flat
space [32]. Subsequently, this unified approach to determine the Hawking
temperature using the Unruh effect was applied for several black hole space-
times [37, 38], but usually these are spherically symmetric. For instance, no
discussion on the Kerr-Newman black hole has been given, because it is
difficult to find the full global embedding.
Since the Hawking effect is governed solely by properties of the event
horizon, it is enough to consider the near horizon theory. As already stated,
this is a two dimensional theory obtained by dimensional reduction of the full
theory. Its metric is just the ($t-r$)-sector of the original metric.
In the following sub-sections we shall find the global embedding of the near
horizon effective $2$-dimensional theory. Then the usual local Hawking
temperature will be calculated. Technicalities are considerably simplified and
our method is general enough to include different black hole metrics.
#### 6.1.1 Schwarzschild metric
Near the event horizon the physics is given by just the two dimensional
($t-r$) -sector of the full Schwarzschild metric [10] (see also Appendix 6.A):
$\displaystyle
ds^{2}=g_{tt}dt^{2}+g_{rr}dr^{2}=\Big{(}1-\frac{2M}{r}\Big{)}dt^{2}-\frac{dr^{2}}{1-\frac{2M}{r}}.$
(6.1)
It is interesting to see that this can be globally embedded in a flat $D=3$
space as,
$\displaystyle ds^{2}=(dz^{0})^{2}-(dz^{1})^{2}-(dz^{2})^{2}$ (6.2)
by the following relations among the flat and curved coordinates:
$\displaystyle
z^{0}_{out}=\kappa^{-1}\Big{(}1-\frac{2M}{r}\Big{)}^{1/2}\textrm{sinh}(\kappa
t),\,\,\,\
z^{1}_{out}=\kappa^{-1}\Big{(}1-\frac{2M}{r}\Big{)}^{1/2}\textrm{cosh}(\kappa
t),$ $\displaystyle
z^{0}_{in}=\kappa^{-1}\Big{(}\frac{2M}{r}-1\Big{)}^{1/2}\textrm{cosh}(\kappa
t),\,\,\,\
z^{1}_{in}=\kappa^{-1}\Big{(}\frac{2M}{r}-1\Big{)}^{1/2}\textrm{sinh}(\kappa
t),$ $\displaystyle z^{2}=\int
dr\Big{(}1+\frac{r_{H}r^{2}+r_{H}^{2}r+r_{H}^{3}}{r^{3}}\Big{)}^{1/2},$ (6.3)
where the surface gravity $\kappa=\frac{1}{4M}$ and the event horizon is
located at $r_{H}=2M$. The suffix “$in$” (“$out$”) refer to the inside
(outside) of the event horizon while variables without any suffix (like
$z^{2}$) imply that these are valid on both sides of the horizon. We shall
follow these notations throughout the chapter.
Now if a detector moves according to constant $r$ (Hawking detector) outside
the horizon in the curved space, then the detector corresponding to the $z$
coordinates, moves on the constant $z^{2}$ plane and it will follow the
hyperbolic trajectory
$\displaystyle\Big{(}z^{1}_{out}\Big{)}^{2}-\Big{(}z^{0}_{out}\Big{)}^{2}=16M^{2}\Big{(}1-\frac{2M}{r}\Big{)}$
(6.4)
Such a detector is usually called as the Unruh detector, since the metric
corresponding to $z^{2}$ constant plane:
$\displaystyle ds^{2}_{(z^{0},z^{1})}$ $\displaystyle=$
$\displaystyle(dz^{0}_{out})^{2}-(dz^{1}_{out})^{2}$ (6.5) $\displaystyle=$
$\displaystyle\Big{(}1-\frac{2M}{r}\Big{)}dt^{2}-\frac{16M^{4}}{r^{4}}\Big{(}1-\frac{2M}{r}\Big{)}^{-1}dr^{2}$
is in generalized Rindler form,
$ds^{2}_{Rind}=\alpha^{2}H(r)^{2}dt^{2}-H^{\prime}(r)^{2}dr^{2}$ (6.6)
with
$\displaystyle H(r)=\kappa^{-1}\Big{(}1-\frac{2M}{r}\Big{)}^{1/2};\,\,\,\
\alpha=\kappa~{}.$ (6.7)
For the generalized Rindler metric (6.6) the acceleration of the Unruh
detector is given by [99],
$\displaystyle{\tilde{a}}=\frac{1}{H(r)}$ (6.8)
and according to Unruh [30], the accelerated detector will see a thermal
spectrum in the Minkowski vacuum with the local Hawking (Unruh) temperature
given by (1.5). This shows that the Unruh detector is moving in the
($z^{0}_{out},z^{1}_{out}$) flat plane with a uniform acceleration
${\tilde{a}}=\frac{1}{4M}\Big{(}1-\frac{2M}{r}\Big{)}^{-1/2}$ and it will see
a thermal spectrum in the Minkowski vacuum with local Hawking temperature
given by,
$\displaystyle T=\frac{\hbar{\tilde{a}}}{2\pi}=\frac{\hbar}{8\pi
M}\Big{(}1-\frac{2M}{r}\Big{)}^{-1/2}.$ (6.9)
So we see that with the help of the reduced global embedding the local Hawking
temperature near the horizon can easily be obtained. The same analysis can
also be done in the upcoming discussions, although we shall not mention
explicitly. We shall only read off the acceleration of the Unruh detector by
finding the appropriate hyperbolic trajectory and thereby the local Hawking
(Unruh) temperature will be derived.
Now the temperature measured by any observer away from the horizon can be
obtained by using the Tolman formula [36] which ensures constancy between the
product of temperatures and corresponding Tolman factors measured at two
different points in space-time. This formula is given by [36]:
$\displaystyle\sqrt{g_{tt}}~{}T=\sqrt{g_{0_{tt}}}~{}T_{0}$ (6.10)
where, in this case, the quantities on the left hand side are measured near
the horizon whereas those on the right hand side are measured away from the
horizon (say at $r_{0}$). Since away from the horizon the space-time is given
by the full metric, $g_{0_{tt}}$ must correspond to the $dt^{2}$ coefficient
of the full (four dimensional) metric.
For the case of Schwarzschild metric $g_{tt}=1-2M/r$, $g_{0_{tt}}=1-2M/r_{0}$.
Now the Hawking effect is observed at infinity ($r_{0}=\infty$), where
$g_{0_{tt}}=1$. Hence, use of the Tolman formula (6.10) immediately yields the
Hawking temperature:
$\displaystyle T_{H}\equiv T_{0}={\sqrt{g_{tt}}}~{}T=\frac{\hbar}{8\pi M}.$
(6.11)
Thus, use of the reduced embedding instead of the embedding of the full metric
is sufficient to get the answer.
#### 6.1.2 Reissner-Nordstr$\ddot{\textrm{o}}$m metric
In this case, the effective metric near the event horizon is given by [10]
(see also appendix 6.A),
$\displaystyle
ds^{2}=\Big{(}1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\Big{)}dt^{2}-\frac{dr^{2}}{1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}}.$
(6.12)
This metric can be globally embedded into the $D=4$ dimensional flat metric
as,
$\displaystyle ds^{2}=(dz^{0})^{2}-(dz^{1})^{2}-(dz^{2})^{2}+(dz^{3})^{2}$
(6.13)
where the coordinate transformations are:
$\displaystyle
z^{0}_{out}=\kappa^{-1}\Big{(}1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\Big{)}^{1/2}\textrm{sinh}(\kappa
t),\,\,\,\
z^{1}_{out}=\kappa^{-1}\Big{(}1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\Big{)}^{1/2}\textrm{cosh}(\kappa
t),$ $\displaystyle
z^{0}_{in}=\kappa^{-1}\Big{(}\frac{2M}{r}-\frac{Q^{2}}{r^{2}}-1\Big{)}^{1/2}\textrm{cosh}(\kappa
t),\,\,\,\
z^{1}_{in}=\kappa^{-1}\Big{(}\frac{2M}{r}-\frac{Q^{2}}{r^{2}}-1\Big{)}^{1/2}\textrm{sinh}(\kappa
t),$ $\displaystyle z^{2}=\int
dr\Big{[}1+\frac{r^{2}(r_{+}+r_{-})+r_{+}^{2}(r+r_{+})}{r^{2}(r-r_{-})}\Big{]}^{1/2},$
$\displaystyle z^{3}=\int
dr\Big{[}\frac{4r_{+}^{5}r_{-}}{r^{4}(r_{+}-r_{-})^{2}}\Big{]}^{1/2}.$ (6.14)
Here in this case the surface gravity $\kappa=\frac{r_{+}-r_{-}}{2r_{+}^{2}}$
and $r_{\pm}=M\pm\sqrt{M^{2}-Q^{2}}$. The black hole event horizon is given by
$r_{H}=r_{+}$. Note that for $Q=0$, the above transformations reduce to the
Schwarzschild case (6.3). The Hawking detector moving in the curved space
outside the horizon, following a constant $r$ trajectory, maps to the Unruh
detector on the constant ($z^{2},z^{3}$) surface. The trajectory of the Unruh
detector is given by
$\displaystyle\Big{(}z^{1}_{out}\Big{)}^{2}-\Big{(}z^{0}_{out}\Big{)}^{2}=\Big{(}\frac{r_{+}-r_{-}}{2r_{+}^{2}}\Big{)}^{-2}\Big{(}1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\Big{)}=\frac{1}{{\tilde{a}}^{2}}.$
(6.15)
This, according to Unruh [30], immediately leads to the local Hawking
temperature
$\displaystyle T=\frac{\hbar{\tilde{a}}}{2\pi}=\frac{\hbar(r_{+}-r_{-})}{4\pi
r_{+}^{2}\sqrt{1-2M/r+Q^{2}/r^{2}}}$ (6.16)
which was also obtained from the full global embedding [32]. Again, since in
this case $g_{0_{tt}}=1-2M/r_{0}+Q^{2}/r_{0}^{2}$ which reduces to unity at
$r_{0}=\infty$ and $g_{tt}=1-2M/r+Q^{2}/r^{2}$, use of Tolman formula (6.10)
leads to the standard Hawking temperature
$\displaystyle T_{H}\equiv
T_{0}=\sqrt{g_{tt}}~{}T=\frac{\hbar(r_{+}-r_{-})}{4\pi r_{+}^{2}}~{}.$ (6.17)
#### 6.1.3 Schwarzschild-AdS metric
Near the event horizon the relevant effective metric is [10] (see also
Appendix 6.A),
$\displaystyle
ds^{2}=\Big{(}1-\frac{2M}{r}+\frac{r^{2}}{R^{2}}\Big{)}dt^{2}-\frac{dr^{2}}{\Big{(}1-\frac{2M}{r}+\frac{r^{2}}{R^{2}}\Big{)}},$
(6.18)
where $R$ is related to the cosmological constant $\Lambda=-1/R^{2}$. This
metric can be globally embedded in the flat space (6.13) with the following
coordinate transformations:
$\displaystyle
z^{0}_{out}=\kappa^{-1}\Big{(}1-\frac{2M}{r}+\frac{r^{2}}{R^{2}}\Big{)}^{1/2}\textrm{sinh}(\kappa
t),\,\,\
z^{1}_{out}=\kappa^{-1}\Big{(}1-\frac{2M}{r}+\frac{r^{2}}{R^{2}}\Big{)}^{1/2}\textrm{cosh}(\kappa
t),$ $\displaystyle
z^{0}_{in}=\kappa^{-1}\Big{(}\frac{2M}{r}-\frac{r^{2}}{R^{2}}-1\Big{)}^{1/2}\textrm{cosh}(\kappa
t),\,\,\,\
z^{1}_{in}=\kappa^{-1}\Big{(}\frac{2M}{r}-\frac{r^{2}}{R^{2}}-1\Big{)}^{1/2}\textrm{sinh}(\kappa
t),$ $\displaystyle z^{2}=\int
dr\Big{[}1+\Big{(}\frac{R^{3}+Rr_{H}^{2}}{R^{2}+3r_{H}^{2}}\Big{)}^{2}\frac{r^{2}r_{H}+rr_{H}^{2}+r_{H}^{3}}{r^{3}(r^{2}+rr_{H}+r_{H}^{2}+R^{2})}\Big{]}^{1/2},$
$\displaystyle z^{3}=\int
dr\Big{[}\frac{(R^{4}+10R^{2}r_{H}^{2}+9r_{H}^{4})(r^{2}+rr_{H}+r_{H}^{2})}{(r^{2}+rr_{H}+r_{H}^{2}+R^{2})(R^{2}+3r_{H}^{2})^{2}}\Big{]}^{1/2}$
(6.19)
where the surface gravity $\kappa=\frac{R^{2}+3r_{H}^{2}}{2r_{H}R^{2}}$ and
the event horizon $r_{H}$ is given by the largest root of the equation
$1-\frac{2M}{r_{H}}+\frac{r^{2}_{H}}{R^{2}}=0$. Note that in the
$R\rightarrow\infty$ limit these transformations reduce to those for the
Schwarzschild case (6.3). We observe that the Unruh detector on the
($z^{2},z^{3}$) surface (i.e. the Hawking detector moving outside the event
horizon on a constant $r$ surface) follows the hyperbolic trajectory:
$\displaystyle\Big{(}z^{1}_{out}\Big{)}^{2}-\Big{(}z^{0}_{out}\Big{)}^{2}=\Big{(}\frac{R^{2}+3r_{H}^{2}}{2r_{H}R^{2}}\Big{)}^{-2}\Big{(}1-\frac{2M}{r}+\frac{r^{2}}{R^{2}}\Big{)}=\frac{1}{{\tilde{a}}^{2}}$
(6.20)
leading to the local Hawking temperature
$\displaystyle
T=\frac{\hbar{\tilde{a}}}{2\pi}=\frac{\hbar\kappa}{2\pi\Big{(}1-\frac{2M}{r}+\frac{r^{2}}{R^{2}}\Big{)}^{1/2}}.$
(6.21)
This result was obtained earlier [32], but with more technical complexities,
from the embedding of the full metric.
It may be pointed out that for the present case, the observer must be at a
finite distance away from the event horizon, since the space-time is
asymptotically AdS. Therefore, if the observer is far away from the horizon
($r_{0}>>r$) where $g_{0_{tt}}=1-2M/r_{0}+r_{0}^{2}/R^{2}$, then use of (6.10)
immediately leads to the temperature measured at $r_{0}$:
$\displaystyle
T_{0}=\frac{\hbar\kappa}{2\pi\sqrt{1-2M/r_{0}+r_{0}^{2}/R^{2}}}.$ (6.22)
Now, this shows that $T_{0}\rightarrow 0$ as $r_{0}\rightarrow\infty$; i.e. no
Hawking particles are present far from horizon.
### 6.2 Kerr-Newman metric
So far we have discussed a unified picture of Unruh and Hawking effects using
our reduced global embedding approach for spherically symmetric metrics,
reproducing standard results. However, our approach was technically simpler
since it involved the embedding of just the two dimensional near horizon
metric. Now we shall explore the real power of this new embedding.
The utility of the reduced embedding approach comes to the fore for the Kerr-
Newman black hole which is not spherically symmetric. The embedding for the
full metric, as far as we are aware, is not done in the literature.
The effective $2$-dimensional metric near the event horizon is given by
(6A.14) [12, 125] (see Appendix 6.A), This metric can be embedded in the
following $D=5$-dimensional flat space:
$\displaystyle
ds^{2}=\Big{(}dz^{0}\Big{)}^{2}-\Big{(}dz^{1}\Big{)}^{2}-\Big{(}dz^{2}\Big{)}^{2}+\Big{(}dz^{3}\Big{)}^{2}+\Big{(}dz^{4}\Big{)}^{2},$
(6.23)
where the coordinate transformations are
$\displaystyle
z^{0}_{out}=\kappa^{-1}\Big{(}1-\frac{2Mr}{r^{2}+a^{2}}+\frac{Q^{2}}{r^{2}+a^{2}}\Big{)}^{1/2}\textrm{sinh}(\kappa
t),$ $\displaystyle
z^{1}_{out}=\kappa^{-1}\Big{(}1-\frac{2Mr}{r^{2}+a^{2}}+\frac{Q^{2}}{r^{2}+a^{2}}\Big{)}^{1/2}\textrm{cosh}(\kappa
t),$ $\displaystyle
z^{0}_{in}=\kappa^{-1}\Big{(}\frac{2Mr}{r^{2}+a^{2}}-\frac{Q^{2}}{r^{2}+a^{2}}-1\Big{)}^{1/2}\textrm{cosh}(\kappa
t),$ $\displaystyle
z^{1}_{in}=\kappa^{-1}\Big{(}\frac{2Mr}{r^{2}+a^{2}}-\frac{Q^{2}}{r^{2}+a^{2}}-1\Big{)}^{1/2}\textrm{sinh}(\kappa
t),$ $\displaystyle z^{2}=\int
dr\Big{[}1+\frac{(r^{2}+a^{2})(r_{+}+r_{-})+r_{+}^{2}(r+r_{+})}{(r^{2}+a^{2})(r-r_{-})}\Big{]}^{1/2},$
$\displaystyle z^{3}=\int
dr\Big{[}\frac{4r_{+}^{5}r_{-}}{(r^{2}+a^{2})^{2}(r_{+}-r_{-})^{2}}\Big{]}^{1/2},$
$\displaystyle z^{4}=\int
dra\Big{[}\frac{r_{+}+r_{-}}{(a^{2}+r_{-}^{2})(r_{-}-r)}+\frac{4(a^{2}+r_{+}^{2})(a^{2}-r_{+}r_{-}+(r_{+}+r_{-})r)}{(r_{+}-r_{-})^{2}(a^{2}+r^{2})^{3}}$
$\displaystyle+\frac{4r_{+}r_{-}(a^{2}+2r_{+}^{2})}{(r_{+}-r_{-})^{2}(a^{2}+r^{2})^{2}}+\frac{rr_{-}-a^{2}+r_{+}(r+r_{-})}{(a^{2}+r_{-}^{2})(a^{2}+r^{2})}\Big{]}^{1/2}.$
(6.24)
Here the surface gravity $\kappa=\frac{r_{+}-r_{-}}{2(r_{+}^{2}+a^{2})}$. For
$Q=0,a=0$, as expected, the above transformations reduce to the Schwarzschild
case (6.3) while only for $a=0$ these reduce to the Reissner-
Nordstr$\ddot{\textrm{o}}$m case (6.14).
As before, the trajectory adopted by the Unruh detector on the constant
($z^{2},z^{3},z^{4}$) surface corresponding to the Hawking detector on the
constant $r$ surface is given by the hyperbolic form,
$\displaystyle\Big{(}z^{1}_{out}\Big{)}^{2}-\Big{(}z^{0}_{out}\Big{)}^{2}=\kappa^{-2}\Big{(}1-\frac{2Mr}{r^{2}+a^{2}}+\frac{Q^{2}}{r^{2}+a^{2}}\Big{)}=\frac{1}{{\tilde{a}}^{2}}.$
(6.25)
Hence the Unruh or local Hawking temperature is
$\displaystyle
T=\frac{\hbar{\tilde{a}}}{2\pi}=\frac{\hbar\kappa}{2\pi\sqrt{\Big{(}1-\frac{2Mr}{r^{2}+a^{2}}+\frac{Q^{2}}{r^{2}+a^{2}}\Big{)}}}.$
(6.26)
Finally, since $g_{tt}=1-\frac{2Mr}{r^{2}+a^{2}}+\frac{Q^{2}}{r^{2}+a^{2}}$
(corresponding to the near horizon reduced two dimensional metric) and
$g_{0_{tt}}=\frac{r_{0}^{2}-2Mr_{0}+a^{2}+Q^{2}-a^{2}{\textrm{sin}}^{2}\theta}{r_{0}^{2}+a^{2}{\textrm{cos}}^{2}\theta}$
(corresponding to the full four dimensional metric), use of the Tolman
relation (6.10) leads to the Hawking temperature
$\displaystyle T_{H}\equiv
T_{0}=\frac{\sqrt{g_{tt}}}{\sqrt{(g_{0}{{}_{tt}})_{r_{0}\rightarrow\infty}}}~{}T=\frac{\hbar\kappa}{2\pi}=\frac{\hbar(r_{+}-r_{-})}{4\pi(r_{+}^{2}+a^{2})},$
(6.27)
which is the well known result [12].
### 6.3 Conclusion
We provided a new approach to the study of Hawking/Unruh effects including
their unification, initiated in [31, 32, 33], popularly known as global
embedding Minkowskian space-time (GEMS). Contrary to the usual formulation
[31, 32, 33, 37, 38, 39], the full embedding was avoided. Rather, we required
the embedding of just the two dimensional ($t-r$)-sector of the theory. This
was a consequence of the fact that the effective near horizon theory is
basically two dimensional. Only near horizon theory is significant since
Hawking/Unruh effects are governed solely by properties of the event horizon.
This two dimensional embedding ensued remarkable technical simplifications
whereby the treatment of more general black holes (e.g. those lacking
spherical symmetry like the Kerr-Newman) was feasible. Also, black holes in
any dimensions were automatically considered since the embedding just required
the ($t-r$)-sector.
## Appendix
## Appendix 6.A Dimensional reduction technique
Dimensional reduction has been discussed in various contexts in the literature
[10, 121, 12, 125]. Here we briefly summarise the technique and findings
relevant for our study. Two specific examples are considered.
Spherically symmetric static metric:
Let us consider a spherically symmetric static metric
$\displaystyle
ds^{2}=f(r)dt^{2}-\frac{dr^{2}}{f(r)}-r^{2}(d\theta^{2}+\textrm{sin}^{2}\theta
d\phi^{2})$ (6A.1)
whose event horizon is given by $f(r=r_{H})=0$. Now in terms of the tortoise
coordinate (2.20) the above metric takes the following form
$\displaystyle
ds^{2}=f(r(r^{*}))\Big{(}dt^{2}-dr^{*^{2}}\Big{)}-r^{2}(r^{*})(d\theta^{2}+\textrm{sin}^{2}\theta
d\phi^{2})$ (6A.2)
Then the free action for massless scalar field under this background is given
by
$\displaystyle A$ $\displaystyle=$ $\displaystyle-\int
d^{4}x~{}\sqrt{-g}~{}\Phi\nabla_{\mu}\nabla^{\mu}\Phi$ (6A.3) $\displaystyle=$
$\displaystyle-\int dtdr^{*}d\theta
d\phi~{}\textrm{sin}\theta~{}\Phi\Big{[}r^{2}(r^{*})(\partial^{2}_{t}-\partial^{2}_{r^{*}})-2r(r^{*})f(r(r^{*}))\partial_{r^{*}}\Big{]}\Phi$
$\displaystyle-$ $\displaystyle\int dtdr^{*}d\theta
d\phi~{}f(r(r^{*}))\textrm{sin}\theta~{}\Phi L^{2}\Phi,$
where
$\displaystyle
L^{2}=-\frac{1}{\textrm{sin}^{2}\theta}\partial^{2}_{\phi}-\textrm{cot}\theta\partial_{\theta}-\partial^{2}_{\theta}.$
(6A.4)
Substituting the partial wave decomposition for $\Phi$
$\displaystyle\Phi(t,r^{*},\theta,\phi)=\displaystyle\sum_{l,n}\phi_{ln}(t,r^{*})Y_{ln}(\theta,\phi)$
(6A.5)
in (6A.3) and using the eigenvalue equation
$L^{2}Y_{ln}(\theta,\phi)=l(l+1)Y_{ln}(\theta,\phi)$ followed by the
orthonormality condition, $\int d\theta
d\phi~{}{\textrm{sin}}\theta~{}Y_{l^{\prime}n^{\prime}}Y_{ln}=\delta_{l^{\prime}l}\delta_{n^{\prime}n}$,
we obtain,
$\displaystyle A$ $\displaystyle=$ $\displaystyle-\displaystyle\sum_{l,n}\int
dtdr^{*}r^{2}(r^{*})\phi_{ln}\Big{[}\partial^{2}_{t}-\partial^{2}_{r^{*}}\Big{]}\phi_{ln}$
(6A.6) $\displaystyle+$ $\displaystyle\displaystyle\sum_{l,n}\int
dtdr^{*}r^{2}(r^{*})\phi_{ln}f(r(r^{*}))\Big{[}\frac{l(l+1)}{r^{2}(r^{*})}+\frac{1}{r(r^{*})}\partial_{r}f(r)\Big{]}\phi_{ln}.$
Now near the horizon ($r\rightarrow r_{H}$), $f(r)\rightarrow 0$, and hence
the above action reduces to the following form:
$\displaystyle A\simeq-\displaystyle\sum_{l,n}\int
dtdr^{*}r_{H}^{2}(r^{*})\phi_{ln}\Big{[}\partial^{2}_{t}-\partial^{2}_{r^{*}}\Big{]}\phi_{ln}.$
(6A.7)
Transforming back to the original coordinates ($t,r$), yields
$\displaystyle A\simeq-\displaystyle\sum_{l,n}\int
dtdrr_{H}^{2}\phi_{ln}\Big{[}\frac{1}{f(r)}\partial^{2}_{t}-\partial_{r}(f\partial_{r})\Big{]}\phi_{ln}.$
(6A.8)
It must be noted that the above action is the original action for the infinite
collection of free scalar fields under the metric [12],
$\displaystyle ds^{2}=f(r)dt^{2}-\frac{dr^{2}}{f(r)},$ (6A.9)
which is just the ($t-r$)-sector of the ($3+1$)-dimensional metric (6A.1). It
is simple to extend this analysis for arbitrary dimensions [128]. The
effective theory is again given by the metric (6A.9).
Kerr-Newman metric:
The most general black hole in four dimensional Einstein theory is given by
the Kerr-Newman metric,
$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{\Delta-a^{2}\sin^{2}\theta}{\Sigma}dt^{2}+\frac{2a\sin^{2}\theta}{\Sigma}(r^{2}+a^{2}-\Delta)dtd\varphi$
(6A.10) $\displaystyle-$
$\displaystyle\frac{a^{2}\Delta\sin^{2}\theta-(r^{2}+a^{2})^{2}}{\Sigma}\sin^{2}\theta
d\varphi^{2}-\frac{\Sigma}{\Delta}dr^{2}-\Sigma d\theta^{2}$
where
$\displaystyle a\equiv\frac{J}{M};\,\,\ \Sigma\equiv
r^{2}+a^{2}\cos^{2}\theta;\,\,\ \Delta\equiv
r^{2}-2Mr+a^{2}+Q^{2}=(r-r_{+})(r-r_{-}),$ $\displaystyle
r_{\pm}=M\pm\sqrt{M^{2}-a^{2}-Q^{2}},$ (6A.11)
while $M,J,Q$ and $r_{+(-)}$ are the mass, angular momentum, electrical charge
and the outer (inner) horizon of the Kerr-Newman black hole, respectively. The
event horizon is located at $r=r_{+}$.
Proceeding in a similar way as above, the action for a massless complex scalar
field, in the near horizon limit, reduces to the following form [12, 125]:
$\displaystyle A$ $\displaystyle=$ $\displaystyle-\int
d^{4}x\sqrt{-g}\Phi^{*}(\nabla_{\mu}+iA_{\mu})(\nabla^{\mu}-iA^{\mu})\Phi$
(6A.12) $\displaystyle=$ $\displaystyle-\displaystyle{\sum_{l,n}}\int
dtdr(r^{2}+a^{2})\phi^{*}_{ln}\Big{[}\frac{r^{2}+a^{2}}{\Delta}\Big{(}\partial_{t}-iA_{t}\Big{)}^{2}$
$\displaystyle-$
$\displaystyle\partial_{r}\frac{\Delta}{r^{2}+a^{2}}\partial_{r}\Big{]}\phi_{ln},$
where
$\displaystyle
A_{t}=-eV(r)-n\Omega(r);\,\,\,V(r)=\frac{Qr}{r^{2}+a^{2}},\Omega(r)=\frac{a}{r^{2}+a^{2}}.$
(6A.13)
Here $e$ is the charge of the scalar field. This shows that each partial wave
mode of the fields can be described near the horizon as a ($1+1$) dimensional
complex scalar field with two $U(1)$ gauge potentials $V(r)$, $\Omega(r)$ and
the dilaton field $\psi=r^{2}+a^{2}$. It should be noted that the above action
for each $l,n$ can also be obtained from the complex scalar field action in
the background of the metric
$\displaystyle ds^{2}=F(r)dt^{2}-\frac{dr^{2}}{F(r)};\,\,\
F(r)=\frac{\Delta}{r^{2}+a^{2}}$ (6A.14)
with the dilaton field $\psi=r^{2}+a^{2}$. Thus, the effective near horizon
theory is two dimensional with a metric given by (6A.14). Although here we
have presented the dimensional reduction technique for the $4$ dimensional
case, it can also be generalized to higher dimensional black holes. In that
case one again gets a two dimensional ($t-r$) metric near the event horizon.
For example see [129].
## Chapter 7 Quantum tunneling and black hole spectroscopy
Since the birth of Einstein’s theory of gravitation, black holes have been one
of the main topics that attracted the attention and consumed a big part of the
working time of the scientific community. In particular, the computation of
black hole entropy in the semi-classical and furthermore in the quantum regime
has been a very difficult and (in its full extent) unsolved problem that has
created a lot of controversy. A closely related issue is the spectrum of this
entropy as well as that of the horizon area. This will be our main concern.
Bekenstein was the first to show that there is a lower bound (quantum) in the
increase of the area of the black hole horizon when a neutral (test) particle
is absorbed [2]
$\displaystyle(\Delta{A})_{min}=8\pi\l_{pl}^{2}$ (7.1)
where we use gravitational units, i.e. $G=c=1$, and
$\l_{pl}=(G\hbar/c^{3})^{1/2}$ is the Planck length. Later on, Hod considered
the case of a charged particle assimilated by a Reissner-Nordström black hole
and derived a smaller bound for the increase of the black hole area [130]
$\displaystyle(\Delta{A})_{min}=4\l_{pl}^{2}~{}.$ (7.2)
At the same time, a new research direction was pursued; namely the derivation
of the area as well as the entropy spectrum of black holes utilizing the
quasinormal modes of black holes [44] 111For some works on this direction see,
for instance, [131] and references therein.. In this framework, the result
obtained is of the form
$\displaystyle(\Delta{A})_{min}=4\l_{pl}^{2}\ln k$ (7.3)
where $k=3$. A similar expression was first put forward by Bekenstein and
Mukhanov [132] who employed the “bit counting” process. However in that case
$k$ is equal to $2$. Such a spectrum can also be derived in the context of
quantum geometrodynamics [133]. Furthermore, using this result one can find
the corrections to entropy consistent with Gibbs’ paradox [134].
Another significant attempt was to fix the Immirzi parameter in the framework
of Loop Quantum Gravity [45] but it was unsuccessful [46]. Furthermore,
contrary to Hod’s statement for a uniformly spaced area spectrum of generic
Kerr-Newman black holes, it was proven that the area spacing of Kerr black
hole is not equidistant [135]. However, a new interpretation for the black
hole quasinormal modes was proposed [48] which rejuvenated the interest in
this direction. In this framework the area spectrum is evenly spaced and the
area quantum for the Schwarschild as well as for the Kerr black hole is given
by (7.1) [49]. While this is in agreement with the old result of Bekenstein,
it disagrees with (7.2).
In this chapter, we will use a modified version of the tunneling mechanism,
discussed in chapter 4, to derive the entropy-area spectrum of a black hole.
In this formalism, as explained earlier, a virtual pair of particles is
produced just inside the black hole. One member of this pair is trapped inside
the black hole while the other member can quantum mechanically tunnel through
the horizon. This is ultimately observed at infinity, giving rise to the
Hawking flux. Now the uncertainty in the energy of the emitted particle is
calculated from a simple quantum mechanical point of view. Then exploiting
information theory (entropy as lack of information) and the first law of
thermodynamics, we infer that the entropy spectrum is evenly spaced for both
Einstein’s gravity as well as Einstein-Gauss-Bonnet gravity. Now, since in
Einstein gravity, entropy is proportional to horizon area of black hole, the
area spectrum is also evenly spaced and the spacing is shown to be exactly
identical with one computed by Hod [130] who studied the assimilation of
charged particle by a Reissner-Nordström black hole. On the contrary, in more
general theories like Einstein-Gauss-Bonnet gravity, the entropy is not
proportional to the area and therefore area spacing is not equidistant. This
also agrees with recent conclusions [50, 136].
The organization of the chapter goes as follows. In section 7.1, we briefly
review the results of dimensional reduction presented earlier in Appendix 6.A
which will be used in this chapter. In section 7.2, we compute the entropy and
area spectrum of black hole solutions of both Einstein gravity and Einstein-
Gauss-Bonnet gravity. Finally, section 7.3 is devoted to a brief summary of
our results and concluding remarks.
### 7.1 Near horizon modes
According to the no hair theorem, collapse leads to a black hole endowed with
mass, charge, angular momentum and no other free parameters. The most general
black hole in four dimensional Einstein theory is given by the Kerr-Newman
metric (6A.10).
Now considering complex scalar fields in the Kerr-Newmann black hole
background and then substituting the partial wave decomposition of the scalar
field in terms of spherical harmonics it has been shown in Appendix 6.A that
near the horizon the action reduces to an effective 2-dimensional action
(6A.12) for free complex scalar field. From (6A.12) one can easily derive the
equation of motion of the field $\phi_{lm}$ for the $l=0$ mode. We will denote
this mode as $\phi$. This equation is given by the Klein-Gordon equation:
$\displaystyle\Big{[}\frac{1}{F(r)}(\partial_{t}-iA_{t})^{2}-F(r)\partial^{2}_{r}-F^{\prime}(r)\partial_{r}\Big{]}\phi=0~{}.$
(7.4)
Now proceeding in a similar way as presented in chapter 4, we obtain the
relations between the modes defined inside and outside the black hole event
horizon, which are given by (4.35) and (4.36). In this case, the surface
gravity $\kappa$ is defined by,
$\displaystyle\kappa=\frac{1}{2}\frac{dF(r)}{dr}\Big{|}_{r=r_{+}}$ (7.5)
and the energy of the particle ($\omega$) as seen from an asymptotic observer
is identified as,
$\displaystyle\omega=E-eV(r_{+})-m\Omega(r_{+}).$ (7.6)
Here $E$ is the conserved quantity corresponding to a timelike Killing vector
($1,0,0,0$). The other variables $V(r_{+})$ and $\Omega(r_{+})$ are the
electric potential and the angular velocity calculated on the horizon.
The same analysis also goes through for a D-dimensional spherically symmetric
static black hole which is a solution for Einstein-Gauss-Bonnet theory [137]:
$\displaystyle
ds^{2}=F(r)dt^{2}-\frac{dr^{2}}{F(r)}-r^{2}d\Omega^{2}_{(D-2)}.$ (7.7)
Here $F(r)$ is given by
$\displaystyle
F(r)=1+\frac{r^{2}}{2\alpha}\Big{[}1-\Big{(}1+\frac{4\alpha\bar{\omega}}{r^{D-1}}\Big{)}^{\frac{1}{2}}\Big{]}$
(7.8)
with
$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle(D-3)(D-4)\alpha_{GB}$
(7.9) $\displaystyle\bar{\omega}$ $\displaystyle=$
$\displaystyle\frac{16\pi}{(D-2)\Sigma_{D-2}}M$ (7.10)
where $\alpha_{GB}$, $\Sigma_{D-2}$ and $M$ are the coupling constant for the
Gauss-Bonnet term in the action, the volume of unit ($D-2$) sphere and the ADM
mass, respectively. Approaching in a similar manner for the dimensional
reduction near the horizon, as discussed in Appendix 6.A (also see [128]) for
arbitrary dimensional case), one can show that the physics can be effectively
described by the 2-dimensional form (6A.14). Therefore, in the Einstein-Gauss-
Bonnet theory one will obtain the same transformations, namely equations
(4.35) and (4.36), between the inside and outside modes.
In the analysis to follow, using the aforementioned transformations, i.e.
equations (4.35) and (4.36), we will discuss about the spectroscopy of the
entropy and area of black holes.
### 7.2 Entropy and area spectrum
In this section we will derive the spectrum for the entropy as well as the
area of the black hole defined both in Einstein and Einstein-Gauss-Bonnet
gravity. It has already been mentioned that the pair production occurs inside
the horizon. The relevant modes are $\phi_{in}^{(L)}$ and $\phi_{in}^{(R)}$.
It has also been shown in chapter 4 that the left mode is trapped inside the
black hole while the right mode can tunnel through the horizon which is
observed at asymptotic infinity. Therefore, the average value of $\omega$ will
be computed as
$\displaystyle<\omega>=\frac{\displaystyle{\int_{0}^{\infty}\left(\phi^{(R)}_{in}\right)^{*}\omega\phi^{(R)}_{in}d\omega}}{\displaystyle{\int_{0}^{\infty}\left(\phi^{(R)}_{in}\right)^{*}\phi^{(R)}_{in}d\omega}}~{}.$
(7.11)
It should be stressed that the above definition is unique since the pair
production occurs inside the black hole and it is the right moving mode that
eventually escapes (tunnels) through the horizon.
To compute this expression it is important to recall that the observer is
located outside the event horizon. Therefore it is essential to recast the
“$in$” expressions into their corresponding “$out$” expressions using the map
(4.35) and then perform the integrations. Consequently, using (4.35) in the
above we will obtain the average energy of the particle, as seen by the
external observer. This is given by,
$\displaystyle<\omega>$ $\displaystyle=$
$\displaystyle\frac{\displaystyle{\int_{0}^{\infty}e^{-\frac{\pi\omega}{\hbar\kappa}}\left(\phi^{(R)}_{out}\right)^{*}\omega
e^{-\frac{\pi\omega}{\hbar\kappa}}\phi^{(R)}_{out}d\omega}}{\displaystyle{\int_{0}^{\infty}e^{-\frac{\pi\omega}{\hbar\kappa}}\left(\phi^{(R)}_{out}\right)^{*}e^{-\frac{\pi\omega}{\hbar\kappa}}\phi^{(R)}_{out}d\omega}}$
(7.12) $\displaystyle=$
$\displaystyle\frac{\displaystyle{\int_{0}^{\infty}\omega
e^{-\beta\omega}d\omega}}{\displaystyle{\int_{0}^{\infty}e^{-\beta\omega}d\omega}}$
$\displaystyle=$
$\displaystyle\frac{\displaystyle{-\frac{\partial}{\partial\beta}\left(\int_{0}^{\infty}e^{-\beta\omega}d\omega\right)}}{\displaystyle{\int_{0}^{\infty}e^{-\beta\omega}d\omega}}=\beta^{-1}$
where $\beta$ is the inverse Hawking temperature
$\displaystyle\beta=\frac{2\pi}{\hbar\kappa}=\frac{1}{T_{H}}.$ (7.13)
In a similar way one can compute the average squared energy of the particle
detected by the asymptotic observer
$\displaystyle<\omega^{2}>$ $\displaystyle=$
$\displaystyle\frac{\displaystyle{\int_{0}^{\infty}e^{-\frac{\pi\omega}{\hbar\kappa}}\left(\phi^{(R)}_{out}\right)^{*}\omega^{2}e^{-\frac{\pi\omega}{\hbar\kappa}}\phi^{(R)}_{out}d\omega}}{\displaystyle{\int_{0}^{\infty}e^{-\frac{\pi\omega}{\hbar\kappa}}\left(\phi^{(R)}_{out}\right)^{*}e^{-\frac{\pi\omega}{\hbar\kappa}}\phi^{(R)}_{out}d\omega}}=\frac{2}{\beta^{2}}~{}.$
(7.14)
Now it is straightforward to evaluate the uncertainty, employing equations
(7.12) and (7.14), in the detected energy $\omega$
$\displaystyle\left(\Delta\omega\right)=\sqrt{<\\!\\!\omega^{2}\\!\\!>-<\\!\\!\omega\\!\\!>^{2}}\,=\,\beta^{-1}=T_{H}$
(7.15)
which is nothing but the Hawking temperature $T_{H}$. Hence the characteristic
frequency of the outgoing mode is given by,
$\displaystyle\Delta f=\frac{\Delta\omega}{\hbar}=\frac{T_{H}}{\hbar}.$ (7.16)
Now the uncertainty (7.15) in $\omega$ can be seen as the lack of information
in energy of the black hole due to the particle emission. This is because
$\omega$ is the effective energy defined in (7.6). Also, since in information
theory the entropy is lack of information, then the first law of black hole
mechanics can be exploited to connect these quantities,
$\displaystyle S_{bh}=\int\frac{\Delta\omega}{T_{H}}.$ (7.17)
Substituting the value of $T_{H}$ from (7.16) in the above we obtain
$\displaystyle S_{bh}=\frac{1}{\hbar}\int\frac{\Delta\omega}{\Delta f}.$
(7.18)
Now according to the Bohr-Sommerfeld quantization rule
$\displaystyle\int\frac{\Delta\omega}{\Delta f}=n\hbar$ (7.19)
where $n=1,2,3....$. Hence, combining (7.18) and (7.19), we can immediately
infer that the entropy is quantized and the spectrum is given by
$\displaystyle S_{bh}=n.$ (7.20)
This shows that the entropy of the black hole is quantized in units of the
identity, $\Delta S_{bh}=(n+1)-n=1$. Thus the corresponding spectrum is
equidistant for both Einstein as well as Einstein-Gauss-Bonnet theory.
Moreover, since the entropy of a black hole in Einstein theory is given by the
Bekenstein-Hawking formula,
$\displaystyle S_{bh}=\frac{A}{4\l_{pl}^{2}}.$ (7.21)
the area spectrum is evenly spaced and given by,
$\displaystyle A_{n}=4\l_{pl}^{2}\,n\,$ (7.22)
with $n=1,2,3,\ldots$ . Consequently, the area of the black hole horizon is
also quantized with the area quantum given by,
$\displaystyle\Delta A=4\l_{pl}^{2}~{}.$ (7.23)
A couple of comments are in order here. First, in Einstein gravity, the area
quantum is universal in the sense that it is independent of the black hole
parameters. This universality was also derived in the context of a new
interpretation of quasinormal moles of black holes [48, 49]. Second, the same
value was also obtained earlier by Hod by considering the Heisenberg
uncertainty principle and Schwinger-type charge emission process [130].
On the contrary, in Einstein-Gauss-Bonnet theory, the black hole entropy is
given by
$\displaystyle
S_{bh}=\frac{A}{4}\Big{[}1+2\alpha\Big{(}\frac{D-2}{D-4}\Big{)}\Big{(}\frac{A}{\Sigma_{D-2}}\Big{)}^{-\frac{2}{D-2}}\Big{]}$
(7.24)
which shows that entropy is not proportional to area. Therefore in this case
the area spacing is not equidistant. The explicit form of the area spectrum is
not be given here since (7.24) does not have any analytic solution for $A$ in
terms of $S_{bh}$. This is compatible with recent findings [50, 136].
### 7.3 Discussions
We have calculated the entropy and area spectra of a black hole which is a
solution of either Einstein or Einstein-Gauss-Bonnet (EGB) theory. The
computations were pursued in the framework of the tunneling method as
reformulated in chapter 4. In both cases entropy spectrum is equispaced and
the quantum of spacing is identical. Since in Einstein gravity, the entropy is
proportional to the horizon area, the spectrum for the corresponding area is
also equally spaced. The area quantum obtained here is equal to
$4\l^{2}_{pl}$. This exactly reproduces the result of Hod who studied the
assimilation of a charged particle by a Reissner-Nordström black hole [130].
In addition, the area quantum $4\l^{2}_{pl}$ is smaller than that given by
Bekenstein for neutral particles [2] as well as the one computed in the
context of black hole quasinormal modes [48, 49].
Furthermore, for the computation of the area quantum obtained here, concepts
from statistical physics, quantum mechanics and black hole physics were
combined in the following sense. First the uncertainty in energy of a emitted
particle from the black hole horizon was calculated from the simple quantuam
mechanical averaging process. Then exploiting the statistical information
theory (entropy is lack of information) in the first law of black hole
mechanics combined with Bohr-Sommerfeld quantization rule, the entropy/area
quantization has been discussed. Since this is done on the basis of the
fundamental concepts of physics, it seems that the result reached in our
analysis is a better approximation (since a quantum theory of gravity which
will give a definite answer to the quantization of black hole entropy/area is
still lacking). Finally, the equality between our result and that of Hod for
the area quantum may be due to the similarity between the tunneling mechanism
and the Schwinger mechanism (for a further discussion on this similarity see
[22, 138]). On the other hand in Einstein - Gauss - Bonnet gravity, since
entropy is not proportional to area, the spectrum of area is not evenly
spaced. This method is general enough to discuss entropy and area spectra for
the black holes in other type of gravity theories like Hořava-Lifshitz gravity
[126]. Here also the entropy spectrum comes out to be evenly spaced while that
of area is not. Hence, it may be legitimate to say that for gravity theories,
in general, the notion of the quantum of entropy is more natural than the
quantum of area. However, one should mention that since our calculations are
based on a semi-classical approximation, the spacing obtained here is valid
for large values of $n$ and for $s$-wave ($l=0$ mode).
## Chapter 8 Statistical origin of gravity
There are numerous evidences [2, 4, 3] which show that gravity and
thermodynamics are closely connected to each other. Recently, there has been a
growing consensus [52, 53, 139] that gravity need not be interpreted as a
fundamental force, rather it is an emergent phenomenon just like
thermodynamics and hydrodynamics. The fundamental role of gravity is replaced
by thermodynamical interpretations leading to similar or equivalent results.
Nevertheless, understanding the entropic or thermodynamic origin of gravity is
far from complete since the arguments are more heuristic than concrete and
depend upon specific ansatz or assumptions.
In this chapter, using certain basic results derived in the earlier chapters
(also see [58, 61]) in the context of tunneling mechanism, we are able to
provide a statistical interpretation of gravity. The starting point is the
standard definition of entropy given in statistical mechanics. We show that
this entropy gets identified with the action for gravity. Consequently the
Einstein equations obtained by a variational principle involving the action
can be equivalently obtained by an extremisation of the entropy.
Furthermore, for a black hole with stationary metric we derive the relation
$S_{bh}=E/2T_{H}$, connecting the entropy ($S_{bh}$) with the Hawking
temperature ($T_{H}$) and energy ($E$). We prove that this energy corresponds
to Komar’s expression [140, 141]. Using this fact we show that the relation
$S_{bh}=E/2T_{H}$ is also compatible with the generalised Smarr formula [142,
3, 8]. We mention that this relation was also obtained and discussed in [143,
144].
### 8.1 Partition function and the relation $S_{bh}=\frac{E}{2T_{H}}$
We start with the partition function for the space-time with matter field [8],
$\displaystyle{\cal{Z}}=\int~{}D[g,\Phi]~{}e^{iI[g,\Phi]}$ (8.1)
where $I[g,\Phi]$ is the action representing the whole system and $D[g,\Phi]$
is the measure of all field configurations ($g,\Phi$). Now consider small
fluctuations in the metric ($g$) and the matter field ($\Phi$) in the
following form:
$\displaystyle g=g_{0}+{\tilde{g}};\,\,\,\,\ \Phi=\Phi_{0}+{\tilde{\Phi}}$
(8.2)
where $g_{0}$ and $\Phi_{0}$ are the stable background fields satisfying the
periodicity conditions and which extremise the action. So they satisfy the
classical field equations. Whereas ${\tilde{g}}$ and $\tilde{\Phi}$, the
fluctuations around these classical values, are very very small. Expanding
$I[g,\Phi]$ around ($g_{0},\Phi_{0}$) we obtain
$\displaystyle
I[g,\Phi]=I[g_{0},\Phi_{0}]+I_{2}[\tilde{g}]+I_{2}[\tilde{\Phi}]+{\textrm{higher
order terms}}.$ (8.3)
The dominant contribution to the path integral (8.1) comes from fields that
are near the background fields ($g_{0},\Phi_{0}$). Hence one can neglect all
the higher order terms. The first term $I[g_{0},\Phi_{0}]$ leads to the usual
Einstein equations and gives rise to the standard area law [8]. On the other
hand the second and third terms give the contributions of thermal gravitation
and matter quanta respectively on the background contribution
$I[g_{0},\Phi_{0}]$. They lead to the (logarithmic) corrections to the usual
area law [145]. Here, since we want to confine ourself within the usual semi-
classical regime, we shall neglect these quadratic terms for the subsequent
analysis. Therefore, keeping only the term $I[g_{0},\Phi_{0}]$ in (8.3) we
obtain the partition function (8.1) as [8],
$\displaystyle{\cal{Z}}\simeq e^{iI[g_{0},\Phi_{0}]}.$ (8.4)
Therefore, adopting the standard definition of entropy in statistical
mechanics,
$\displaystyle S_{bh}=\ln{\cal{Z}}+\frac{E}{T_{H}}$ (8.5)
and using (8.4), the entropy of the gravitating system is given by 111In this
chapter we have chosen units such that $k_{B}=G=\hbar=c=1$.,
$\displaystyle S_{bh}=iI[g_{0},\Phi_{0}]+\frac{E}{T_{H}}$ (8.6)
where $E$ and $T_{H}$ are respectively the energy and temperature of the
system.
It may be pointed out that it is possible to interpret (8.4) as defining the
partition function of an emergent theory without specifying the detailed
configuration of the gravitating system. The validity of such an
interpretation is borne out by the subsequent analysis.
In order to get an explicit expression for $E$, let us consider a specific
system - a black hole. Now thermodynamics of a black hole is universally
governed by its properties near the event horizon. It is also well understood
that near the event horizon the effective theory becomes two dimensional whose
metric is given by the two dimensional ($t-r$)- sector of the original metric
[121, 10]. Correspondingly, the left ($L$) and right ($R$) moving
(holomorphic) modes are obtained by solving the appropriate field equation
using the geometrical (WKB) approximation. Furthermore, the modes inside and
outside the horizon are related by the transformations (4.35) and (4.36) [58,
61]. Concentrating on the modes inside the horizon, the $L$ mode gets trapped
while the $R$ mode tunnels through the horizon and is eventually observed at
asymptotic infinity as Hawking radiation [58, 61] 222For a unified treatment
of these issues, see [146]. The average value of the energy, measured from
outside, is given by (7.12). Therefore if we consider that the energy $E$ of
the system is encoded near the horizon and the total number of pairs created
is $n$ among which this energy is distributed, then we must have,
$\displaystyle E=nT_{H}$ (8.7)
where only the $R$ mode of the pair is significant.
Now to proceed further, it must be realised that the effective two dimensional
curved metric can always be embedded in a flat space which has exactly two
space-like coordinates. This is a consequence of a modification in the
original GEMS (globally embedding in Minkowskian space) approach of [32] and
has been elaborated by us in Chapter 6. Hence we may associate each $R$ mode
with two degrees of freedom. Then the total number of degrees of freedom for
$n$ number of $R$ modes is $N=2n$. Hence, from (8.7), we obtain the energy of
the system as
$\displaystyle E=\frac{1}{2}NT_{H}.$ (8.8)
As a side remark, it may be noted that (8.8) can be interpreted as a
consequence of the usual law of equipartition of energy. For instance, if we
consider that the energy $E$ is distributed equally over each degree of
freedom, then (8.8) implies that each degree of freedom should contain an
energy equal to $T_{H}/2$, which is nothing but the equipartition law of
energy. The fact that the energy is equally distributed among the degrees of
freedom may be understood from the symmetry of two space-like coordinates
($z^{1}\longleftrightarrow z^{2}$) such that the metric is unchanged [60] (see
chapter 6). In our subsequent analysis, however, we only require (8.8) rather
than its interpretation as the law of equipartition of energy.
Now since there are $N$ number of degrees of freedom in which all the
information is encoded, the entropy ($S_{bh}$) of the system must be
proportional to $N$. Hence using (8.6) we obtain
$\displaystyle N=N_{0}S_{bh}=N_{0}(iI[g_{0},\Phi_{0}]+\frac{E}{T_{H}}),$ (8.9)
where $N_{0}$ is a proportionality constant, which will be determined later.
Substituting the value of $N$ from (8.8) in (8.9) we obtain the expression for
the energy of the system as
$\displaystyle E=\frac{N_{0}}{2-N_{0}}iT_{H}I[g_{0},\Phi_{0}].$ (8.10)
This shows that in the absence of any fluctuations, the energy of a system is
actually given by the classical action representing the system. In the
following we shall use this expression to find the energy of a stationary
black hole. Before that let us substitute the value of $I[g_{0},\Phi_{0}]$
from (8.10) in (8.6). This immediately leads to a simple relation between the
entropy, temperature and energy of the black hole:
$\displaystyle S_{bh}=\frac{2E}{N_{0}T_{H}}.$ (8.11)
Now in order to fix the value of “$N_{0}$” we consider the simplest example,
the Schwarzschild black hole for which the entropy, energy and temperature are
given by,
$\displaystyle S_{bh}=\frac{A}{4}=4\pi M^{2},\,\,\ E=M,\,\,\
T_{H}=\frac{1}{8\pi M},$ (8.12)
where “$M$” is the mass of the black hole. Substitution of these in (8.11)
leads to $N_{0}=4$.
At this point we want to make a comment on the value of $N_{0}$. According to
standard statistical mechanics one would have thought that $1/N_{0}=\ln c$,
where $c$ is an integer. Whereas to keep our analysis consistent with semi-
classical area law, we obtained $c=e^{1/4}$, which is clearly not an integer.
Indeed, any departure from this value of $N_{0}$ would invalidate the semi-
classical area law and hence our analysis. Such a disparity is not peculiar to
our approach and has also occurred elsewhere [48, 147]. This may be due to the
fact that our analysis is confined within the semi-classical regime, which is
valid for large degrees of freedom. In this regime, it is not obvious that a
semi-classical computation can reproduce $c$ to be an integer. Furthermore,
the above value of $N_{0}$ is still valid even for very small number of
degrees of freedom ($N$), where this semi-classical calculation is
unjustified. This also happens in the semi-classical computation of the
entropy spectrum of a black hole [48]. The entropy spectrum is found there to
be $S_{bh}=2\pi N$ rather than $S_{bh}=N\ln c$ and this discrepancy is
identified with the semi-classical approximation. A possible way to resolve
such disagreement from standard statistical mechanics may be the full quantum
theoretical computation of the number of microstates which is beyond the scope
of the present chapter.
Finally, putting back $N_{0}=4$ in (8.11) we obtain,
$\displaystyle S_{bh}=\frac{E}{2T_{H}}.$ (8.13)
Such a relation was later obtained by us for higher dimensional Einstein
gravity where $E$ is the Komar conserved quantity [148]. Before discussing the
significance and implications of this relation, we observe that substituting
the value of $E$ from (8.13) in (8.10) with $N_{0}=4$, we obtain
$\displaystyle S_{bh}=-iI[g_{0},\Phi_{0}].$ (8.14)
Consequently, extremization of entropy leads to Einstein’s equations.
### 8.2 Identification of $E$ in Einstein’s gravity
The relation (8.13) is significant for various reasons which will become
progressively clear. It is valid for all black hole solutions in Einstein
gravity with appropriate identifications consistent with the area law. Here
$S_{bh}$ and $T_{H}$ are easy to identify. These are, respectively, the
entropy and Hawking temperature of the black hole. Since energy is one of the
most diversely defined entities in general theory of relativity, special care
is needed to identify $E$ in (8.13). We now show that this $E$ corresponds to
Komar’s definition [140, 141]. Simplifying (8.10) using $N_{0}=4$ and
$T_{H}=\kappa/2\pi$, we obtain,
$\displaystyle E=-\frac{i\kappa I[g_{0},\Phi_{0}]}{\pi}.$ (8.15)
The classical action $I[g_{0},\Phi_{0}]$ has already been calculated in [8].
The result is,
$\displaystyle I[g_{0},\Phi_{0}]$ $\displaystyle=$ $\displaystyle
2i\pi\kappa^{-1}\Big{[}\frac{1}{16\pi}\int_{\Sigma}R\xi^{a}d\Sigma_{a}+\int_{\Sigma}(T_{ab}-\frac{1}{2}Tg_{ab})\xi^{b}d\Sigma^{a}$
(8.16) $\displaystyle-$
$\displaystyle\frac{1}{16\pi}\int_{\cal{H}}\epsilon_{abcd}\nabla^{c}\xi^{d}\Big{]},$
where $\xi^{a}\partial/\partial x^{a}=\partial/\partial t$ is the time
translation Killing vector and $\Sigma$ is the space-like hypersurface whose
boundary is given by ${\cal{H}}$. Here $T_{ab}$ is the energy-momentum tensor
of the matter field whose trace is given by $T$. Now for a stationary
geometry, $\xi^{a}\nabla_{a}R=0$ [99]. Hence for a volume ${\cal{A}}$, we have
$\displaystyle
0=\int_{\cal{A}}\xi^{a}\nabla_{a}Rd{\cal{A}}=\int_{\cal{A}}\Big{[}\nabla_{a}(\xi^{a}R)-(\nabla_{a}\xi^{a})R\Big{]}d{\cal{A}}=\int_{\cal{A}}\nabla_{a}(\xi^{a}R)d{\cal{A}}$
(8.17)
where in the last step the Killing equation
$\nabla_{a}\xi_{b}+\nabla_{b}\xi_{a}=0$ has been used. Finally, the Gauss
theorem yields,
$\displaystyle\int_{\Sigma}\xi^{a}Rd{\Sigma_{a}}=0.$ (8.18)
Using this in (8.16) we obtain,
$\displaystyle
I[g_{0},\Phi_{0}]=2i\pi\kappa^{-1}\Big{[}\int_{\Sigma}(T_{ab}-\frac{1}{2}Tg_{ab})\xi^{b}d\Sigma^{a}-\frac{1}{16\pi}\int_{\cal{H}}\epsilon_{abcd}\nabla^{c}\xi^{d}\Big{]}.$
(8.19)
Substituting this in (8.15) we obtain the expression for the energy of the
gravitating system as
$\displaystyle
E=2\int_{\Sigma}(T_{ab}-\frac{1}{2}Tg_{ab})\xi^{b}d\Sigma^{a}-\frac{1}{8\pi}\int_{\cal{H}}\epsilon_{abcd}\nabla^{c}\xi^{d}$
(8.20)
which is the Komar expression for energy [140, 141] corresponding to the time
translation Killing vector. Similarly, if there is a rotational Killing
vector, then there must be a Komar expression for rotational energy [99, 149]
and the total energy will be their sum.
Incidentally, (8.13) was obtained earlier in [143] for static space-time and
its implications were discussed in [144]. However a specific ‘ansatz’ for
entropy compatible with the area law was taken and, more importantly, the
Komar energy expression was explicitly used as an input in the derivation.
Hence our analysis is different, since we do not invoke any ansatz for the
entropy; neither is the Komar expression required at any stage. Rather we
prove its occurence in the relation (8.13).
As an explicit check of (8.13) for different black hole solutions, we consider
a couple of examples. Take the Reissner-Nordstr$\ddot{\textrm{o}}$m (RN) black
hole. In this case the entropy and temperature are given by,
$\displaystyle S_{bh}=\pi r_{+}^{2},\,\,\,\ T_{H}=\frac{r_{+}-r_{-}}{4\pi
r_{+}^{2}};\,\,\ r_{\pm}=M\pm\sqrt{M^{2}-Q^{2}}$ (8.21)
where “$Q$” is the charge of the black hole. Substitution of these in (8.13)
yields,
$\displaystyle E=M-\frac{Q^{2}}{r_{+}},$ (8.22)
which is the Komar energy of RN black hole [86].
Next we consider the Kerr black hole for which the entropy and temperature are
respectively,
$\displaystyle S_{bh}$ $\displaystyle=$
$\displaystyle\pi(r_{+}^{2}+a^{2}),\,\,\,\
T_{H}=\frac{r_{+}-r_{-}}{4\pi(r_{+}^{2}+a^{2})};$ $\displaystyle r_{\pm}$
$\displaystyle=$ $\displaystyle M\pm\sqrt{M^{2}-a^{2}},\,\,\,\ a=\frac{J}{M}.$
(8.23)
Here “$J$” is the angular momentum of the black hole. Substituting (8.23) in
(8.13) we obtain,
$\displaystyle E=M-2J\Omega$ (8.24)
which is the total Komar energy for Kerr black hole [150, 86]. Here
$\Omega=\frac{a}{r_{+}^{2}+a^{2}}$ is the angular velocity at the event
horizon $r=r_{+}$.
We thus find that, in all cases where $S_{bh}$, $E$, $T$ are known, they
satisfy (8.13) apart from the area law. In fact, it is possible to take (8.13)
as the defining relation for the Komar energy. Such an instance is provided by
the Kerr-Newman black hole. The entropy and temperature of Kerr-Newman black
hole are given by,
$\displaystyle S_{bh}=\pi(r_{+}^{2}+a^{2});\,\,\,\
T_{H}=\frac{r_{+}-r_{-}}{4\pi(r_{+}^{2}+a^{2})}$ (8.25)
where
$\displaystyle r_{\pm}=M\pm\sqrt{M^{2}-Q^{2}-a^{2}};\,\,\,\ a=\frac{J}{M}.$
(8.26)
Now substituting (8.25) in (8.13) and then using (8.26) we obtain the total
Komar energy of Kerr-Newman black hole:
$\displaystyle
E=\sqrt{M^{2}-Q^{2}-a^{2}}=M-\frac{Q^{2}}{r_{+}}-2J\Omega\Big{(}1-\frac{Q^{2}}{2Mr_{+}}\Big{)}=M-QV-2J\Omega,$
(8.27)
where $\Omega=\frac{a}{r_{+}^{2}+a^{2}}$ is the angular velocity at the event
horizon and $V=\frac{Q}{r_{+}}-\frac{QJ\Omega}{Mr_{+}}$. This value exactly
matches with the direct evaluations of Komar expressions for energies [149,
150, 86]. It is also reassuring to note that the definition of $M$ following
from (8.13) and (8.27) reproduces the generalised Smarr formula [142, 3, 8],
$\displaystyle\frac{M}{2}=\frac{\kappa A}{8\pi}+\frac{VQ}{2}+\Omega J.$ (8.28)
### 8.3 Discussions
In this chapter we have further clarified the possibility of considering
gravity as an emergent phenomenon. Taking the standard definition of entropy
from statistical mechanics we were able to show the equivalence of entropy
with the action. Consequently, extremisation of the action leading to
Einstein’s equations is equivalent to the extremisation of the entropy. We
derived the relation $S_{bh}=E/2T_{H}$ for stationary black holes with
$S_{bh}$ and $T_{H}$ being the entropy and Hawking temperature. The nature of
energy $E$ appearing in this relation was clarified. It was proved to be
Komar’s expression valid for stationary asymptotically flat space-time. An
explicit check of $S_{bh}=E/2T_{H}$ was done for all black hole solutions of
Einstein gravity. This relation was also seen to reproduce the generalised
mass formula of Smarr [142, 3, 8]. In this sense the Smarr formula can be
interpreted as a thermodynamic relation further illuminating the emergent
nature of gravity. As a final remark we feel that although our results were
derived for Einstein gravity, the methods are general enough to include other
possibilities like higher order theories.
## Chapter 9 Conclusions
The motivation of this thesis was to study certain field theory aspects of
black holes, with particular emphasis on the Hawking effect, using various
semi-classical techniques. We now summarize the results obtained in last seven
chapters and briefly comment on future prospects.
In the second chapter, we gave a general framework of tunneling mechanism for
a static, spherically symmetric black hole metric. Both Hamilton - Jacobi and
radial null geodesic approaches were elaborated. The tunneling rate was found
to be the Boltzmann factor. Then Hawking’s expression for the temperature of a
black hole - proportional to surface gravity - was derived.
In the third chapter, we provided an application of this general framework for
null geodesic method. Back reaction as well as noncommutative effects in the
space-time were incorporated. Here the main motivation was to find the
modifications to the thermodynamic entities, such as temperature, entropy etc.
First the back reaction, which is just the effect of space-time fluctuations,
was considered. It was shown in [63] that even in the presence of this effect
the metric remains in the static, spherically symmetric form, but with a
modified surface gravity. So it was possible to use the method elaborated in
the previous chapter. In this case, we showed the following results:
* •
The temperature was modified and also the entropy received corrections. The
leading order correction was found to be the logarithmic of area while the
non-leading corrections are just the inverse powers of area.
* •
The coefficient of the logarithmic term was related to the trace anomaly of
energy-momentum tensor.
Both these results agreed with the earlier findings [63, 64] by other methods.
We also discussed the effect of noncommutativity in addition to the back
reaction effect in the black hole space-time. Here again the corrections to
the thermodynamic quantities were given. For consistency, we showed that in
the proper limit the usual (commutative space-time) results were recovered.
In the fourth chapter we discussed another method, the chiral anomaly method,
to derive the fluxes of Hawking radiation. Here the chiral anomaly expressions
were obtained from the non-chiral theory by using the trace anomaly and the
chirality conditions. Then the Hawking flux was derived following the path
prescribed in [16, 17]. Another portion of this chapter was dedicated to show
that the same chirality conditions were enough to find the Hawking temperature
in the quantum tunneling method. Here the explicit form of modes created
inside the black hole were obtained by solving the chirality condition. Then
using the Kruskal coordinates relations between the “inside” modes and
“outside” modes were established. Finally, calculation of the respective
probabilities yielded that the left moving mode was actually trapped inside
the horizon while right moving mode can come out from the horizon with a
finite probability. Thus this analysis manifested the crucial role of the
chirality to give a unified description of both tunneling and anomaly
approaches.
In the fifth chapter, the Hawking emission spectrum from the event horizon was
derived based on our reformulated tunneling mechanism introduced in the
previous chapter. Using the density matrix technique the average number of
emitted particle from the horizon was computed. The spectrum was exactly that
of the black body with the Hawking temperature. Thereby we provided a complete
description of the Hawking effect in the tunneling mechanism. The absence of
any derivation of the spectrum was a glaring omission within the tunneling
paradigm.
In the next chapter, a unified description of Unruh and Hawking effects was
discussed by introducing a new type of global embedding. Since the
thermodynamic quantities of a black hole are determined by the horizon
properties and near the horizon the effective theory is dominated by the two
dimensional ($t-r$) metric, it is sufficient to consider the embedding of this
two dimensional metric. Considering this fact, a new reduced global embedding
of two dimensional curved space-times in higher dimensional flat ones was
introduced to present a unified description of Hawking and Unruh effects. Our
analysis simplified as well as generalised the conventional embedding
approach.
In chapter - 7, based on the modified tunneling mechanism, introduced in the
previous chapters, we obtained the entropy spectrum of a black hole. Our
conclusions were following:
* •
In Einstein’s gravity, both entropy and area spectrum are evenly spaced.
* •
On the other hand in more general theories (like Einstein-Gauss-Bonnet
gravity), although the entropy spectrum is equispaced, the corresponding area
spectrum is not.
In this sense, it was legitimate to say that quantization of entropy is more
fundamental than that of area.
Finally, based on the above conceptions and findings, we explored an
intriguing possibility that gravity can be thought as an emergent phenomenon.
Starting from the definition of entropy, used in statistical mechanics, we
showed that it was proportional to the gravity action. For a stationary black
hole this entropy was expressed as $S_{bh}=E/2T_{H}$, where $T_{H}$ and $E$
were the Hawking temperature and the Komar energy respectively. This relation
was also compatible with the generalised Smarr formula for mass.
There are certain issues which are worthwhile for future study.
* •
The inclusion of grey body effect within the tunneling approach would be an
interesting exercise. The analysis given here did not include the grey body
effect. Consequently, the flux obtained was compared with that associated with
the perfect black body.
* •
Another important issue is the computation of black hole entropy by using the
anomaly approach. There are strong reasons to believe that the black hole
entropy, like Hawking flux can be related to the diffeomorphism anomaly [14,
151, 152, 153, 154, 155, 156]. For example, in the analysis of [151, 152] the
counting of microstates was done by imposing the “horizon constraints”. The
algebra among these “horizon constraints” commutes only after modifying the
generators for diffeomorphism symmetry. This modification in the generators
give rise to the desired central charge, which ultimately leads to the
Bekenstein-Hawking entropy. This is roughly similar to the diffeomorphism
anomaly mechanism.
* •
So far, not much progress has been achieved in the understanding of the Unruh
effect by the gravitational anomaly method. The main difficulty lies in the
fact that the Unruh effect is basically related to flat space-time and the
observer must be uniformly accelerated. So a naive use of the anomaly
expressions is unjustified. In this thesis it was shown that the flat space
embedding of the near horizon effective two dimensional ($t-r$) metric was
enough for giving a unified description of Hawking and Unruh effects and it
simplified as well as generalized earlier facts. The local Hawking temperature
was exactly equivalent to the one detected by the Unruh observer. Again, in
the gravitational (chiral) anomaly expressions the metric that contributed was
the aforesaid effective metric. It may be possible to translate these
expressions for the anomaly in the embedded space and establish a connection
with the Unruh effect.
* •
The last point I want to mention is that in chapter-8, an emergent nature of
gravity was illustrated from a statistical point of view. These discussions
were confined to the four dimensional Einstein gravity without cosmological
constant. It would be fascinating to extend our discussion to higher
dimensional Einstein gravity (with or without cosmological constant) and more
general gravity theories (e.g. Lovelock gravity). If this attempt is
successful, then one will be able to give a unified form of the Smarr formula
for all such theories.
It is thus clear that the quantum tunneling mechanism, provided in this
thesis, could illuminate the subject of thermodynamics of gravity, more
precisely, the black hole.
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REPRINTS
|
arxiv-papers
| 2011-10-27T08:30:41 |
2024-09-04T02:49:23.624006
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bibhas Ranjan Majhi",
"submitter": "Bibhas Majhi Ranjan",
"url": "https://arxiv.org/abs/1110.6008"
}
|
1110.6190
|
# The Okubo-Weiss Criteria in Two-Dimensional Hydrodynamic and
Magnetohydrodynamic Flows
B. K. Shivamoggi111Permanent Address: University of Central Florida, Orlando,
FL 32816-1364, USA and G. J. F. van Heijst
J. M. Burgers Centre and Fluid Dynamics Laboratory
Department of Physics
Eindhoven University of Technology
NL-5600MB Eindhoven, The Netherlands
Abstract
The “slow-variation” restriction on the straining flow-velocity gradient field
used in the Okubo [1]-Weiss [2] criterion is quantified via the Beltrami
condition with the divorticity framework in 2D hydrodynamic flows. This turns
out to provide interesting interpretations of the Okubo-Weiss criterion in
terms of the topological properties of the underlying vorticity manifold.
These developments are then extended to 2D quasi-geostrophic flows (via the
potential divorticity framework) and magnetohydrodynamic flows and the Okubo-
Weiss criteria for these cases are considered.
1\. Introduction
A central question in the problem of transport in two-dimensional (2D)
turbulent flows is how to divide a vorticity field into hyperbolic (cascading
turbulence) and elliptic (coherent vortex) regions because the topology of 2D
turbulence is parameterized in terms of the relative dominance of flow
deformation or flow rotation. Okubo [1] and Weiss [2] gave a kinematic
criterion to serve as a diagnostic tool towards this goal which has been
widely used in numerical simulations (Brachet et al. [3], Ohkitani [4],
Babiano and Provenzale [5]) and laboratory experiments (Ouelette and Gollub
[6]) of 2D hydrodynamic flows.222The Okubo-Weiss parameter describing the
local strain-vorticity balance in the horizontal flow field of a shallow fluid
layer turns out also to quantify the deviations from two-dimensionality of
this flow (Cieslik et al. [7]). More specifically, the Okubo-Weiss parameter
turns out to be the source turn in the Poisson equation for the pressure (Kamp
[8]). A key assumption underlying the Okubo-Weiss criterion is that the
vorticity gradient field evolves quasi-adiabatically with respect to the
underlying straining flow-velocity gradient field. This issue was explored by
Basdevant and Philipovitch [9] who tried to improve on it by invoking the
topological properties of the pressure field, while Hua and Klein [10] tried
to include the strain-rate time evolution explicitly. The purpose of this
paper is to seek to quantify the “slow-variation” restriction on the straining
flow-velocity gradient field used in the Okubo-Weiss criterion via the
Beltrami condition with the divorticity framework in 2D hydrodynamic flows
(Shivamoggi et al. [11]). This also turns out to provide interesting
interpretations of the Okubo-Weiss criterion in terms of the topological
properties of the underlying vorticity manifold. These developments are then
extended to 2D quasi-geostrophic flows (via the potential divorticity
framework) and magnetohydrodynamic flows and the Okubo-Weiss criteria for
these cases are considered.
2\. Beltrami Condition Interpretation of the Okubo-Weiss Criterion
The vorticity dynamics in 2D hydrodynamic flows is governed by the following
equation (Kida [12], Kuznetsov et al. [13])
$\frac{\partial\boldsymbol{\mathscr{B}}}{\partial t}=\nabla\times\left({\bf
v}\times\boldsymbol{\mathscr{B}}\right)$ (1a)
or
$\frac{D\boldsymbol{\mathscr{B}}}{Dt}\equiv\frac{\partial\boldsymbol{\mathscr{B}}}{\partial
t}+\left({\bf
v}\cdot\nabla\right)\boldsymbol{\mathscr{B}}=\left(\boldsymbol{\mathscr{B}}\cdot\nabla\right){\bf
v}$ (1b)
where ${\bf v}=\left<u,v\right>$ is the flow velocity, $\boldsymbol{\omega}$
is the vorticity,
$\boldsymbol{\omega}\equiv\nabla\times{\bf v}$ (2a)
and $\boldsymbol{\mathscr{B}}$ is the divorticity,
$\boldsymbol{\mathscr{B}}\equiv\nabla\times\boldsymbol{\omega}.$ (2b)
Equation (1b) may be rewritten as
$\frac{D\boldsymbol{\mathscr{B}}}{Dt}=\boldsymbol{\mathscr{A}}\cdot\boldsymbol{\mathscr{B}}$
(1c)
where $\boldsymbol{\mathscr{A}}$ is the velocity gradient matrix,
$\boldsymbol{\mathscr{A}}\equiv\left[\begin{matrix}\partial u/\partial
x&\partial u/\partial y\\\ \partial v/\partial x&\partial v/\partial
y\end{matrix}\right]=\frac{1}{2}\left[\begin{matrix}s_{1}&s_{2}-\omega\\\
s_{2}+\omega&-s_{1}\end{matrix}\right]$ $s_{1}\equiv-2\frac{\partial
v}{\partial y},~{}s_{2}\equiv\frac{\partial v}{\partial x}+\frac{\partial
u}{\partial y},~{}\omega\equiv\frac{\partial v}{\partial x}-\frac{\partial
u}{\partial y}.$ (3)
If the straining flow velocity gradient tensor $\nabla{\bf v}$ is assumed,
following Okubo [1] and Weiss [2], to temporally evolve slowly so the
divorticity field evolves quasi-adiabatically with respect to the straining
flow-velocity gradient field, equation (1c) may be locally approximated by an
eigenvalue problem with eigenvalues given by
$\lambda^{2}={s_{1}}^{2}+{s_{2}}^{2}-\omega^{2}\equiv Q.$ (4)
The Okubo-Weiss parameter Q is a measure of the relative importance of flow
strain
(Q $>$ 0, hyperbolic) and vorticity (Q $<$ 0, elliptic). Numerical simulations
(Brachet et al. [3], Ohkitani [4], Babiano and Provenzale [5]) and laboratory
experiments (Ouellette and Gollub [6]) of 2D hydrodynamic flows confirmed that
coherent vortices are indeed located in elliptic regions while divorticity
sheets are located333It may be noted that divorticity sheets are also more
likely to occur near vorticity nulls due to selective rapid viscous decay of
vorticity in these layers (Shivamoggi et al. [11]), just as vortex sheets are
more likely to form near velocity nulls in 3D hydrodynamic flows. in
hyperbolic regions.
The “slow-variation” restriction on the straining flow-velocity gradient field
used above may be quantified via the Beltrami condition444A similar approach
was taken previously (Shivamoggi and van Heijst [14]) in the quantification of
the “slow variation” restriction used in Flierl-Stern-Whitehead [15] zero
angular momentum theorem for localized nonlinear structures in 2D quasi-
geostrophic flows on the $\beta$-plane. with the divorticity framework in 2D
hydrodynamic flows (Shivamoggi et al. [11]). Equations (1a-c) yield for the
Beltrami state (Shivamoggi et al. [11]),
$\boldsymbol{\mathscr{B}}=a{\bf v}$ (5)
$a$ being an arbitrary constant. Using (5), the Okubo-Weiss parameter Q
becomes
$Q=\frac{4}{a^{2}}\left[\left(\frac{\partial^{2}\omega}{\partial x\partial
y}\right)^{2}-\frac{\partial^{2}\omega}{\partial
x^{2}}\frac{\partial^{2}\omega}{\partial y^{2}}\right].$ (6)
(6) implies that the Okubo-Weiss parameter also characterizes the topological
properties of the vorticity manifold - it is in fact the negative of the
Gaussian curvature of the vorticity surface. Thus, the character of the
ensuing time-dependent 2D flow behavior appears to be rooted in the local
topological properties of the underlying equilibrium vorticity manifold. It
may be mentioned that the above reduction was pointed out by Larcheveque [16]
on the premise of replacing streamlines by isovorticity lines which lacked, as
Larcheveque [16] admitted, any dynamical meaning - streamlines are actually
isomorphic to divorticity lines (as implied by the Beltrami condition (5)).
3\. The Okubo-Weiss Criterion for Quasi-geostrophic Flows
Consider a 2D quasi-geostrophic flow in which the baroclinic effects are
produced by the deformed free surface of the ocean. The governing equation (in
appropriate units) is (Charney [17])
$\frac{\partial{\bf q}}{\partial t}+\left({\bf v}\cdot\nabla\right){\bf q}=0$
(7)
where q is the potential vorticity vector,
${\bf q}\equiv\boldsymbol{\omega}-k^{2}\boldsymbol{\psi}+{\bf f}$ (8)
f is the Coriolis parameter, $k$ is the inverse Rossby radius of deformation,
$k\equiv\sqrt{{f_{0}}^{2}/gH}$, ($f_{0}$ being the local value of $|{\bf f}|$
and H the depth of the ocean, which is taken to be uniform), and
${\bf v}\equiv-\nabla\times\boldsymbol{\psi}$ (9)
Upon taking the curl of equation (7), we obtain
$\frac{\partial\boldsymbol{\mathscr{D}}}{\partial t}=\nabla\times\left({\bf
v}\times\boldsymbol{\mathscr{D}}\right)$ (10a)
where $\boldsymbol{\mathscr{D}}$ is the potential divorticity vector (in
analogy to the potential vorticity vector q),
$\boldsymbol{\mathscr{D}}\equiv\nabla\times{\bf
q}=\boldsymbol{\mathscr{B}}+k^{2}{\bf v}+{\bf h}$ (11)
and
${\bf h}=\nabla\times{\bf f}=\left<\beta,0,0\right>$
$\beta$ being the planetary vorticity gradient.
Equation (10a) may be rewritten as
$\frac{D\boldsymbol{\mathscr{D}}}{Dt}=\boldsymbol{\mathscr{A}}\cdot\boldsymbol{\mathscr{D}}.$
(10b)
Following Okubo [1] and Weiss [2], and assuming that the potential divorticity
field evolves quasi-adiabatically with respect to the straining flow velocity
gradient field, equation (10b) may again be locally approximated by an
eigenvalue problem with eigenvalues given by,
$\lambda^{2}=\frac{1}{4}\left(u_{y}v_{x}+{v_{y}}^{2}\right)\equiv Q.$ (12)
Equation (10a) yields for the Beltrami state,
$\boldsymbol{\mathscr{D}}=b{\bf v}$ (13)
$b$ being an arbitrary constant. Using (13), the Okubo-Weiss parameter Q
becomes
$Q\equiv\frac{1}{4b^{2}}\left[\left(\frac{\partial^{2}\omega}{\partial
x\partial y}\right)^{2}-\frac{\partial^{2}\omega}{\partial
x^{2}}\frac{\partial^{2}\omega}{\partial y^{2}}\right]$ (14)
which is the same as (6) for 2D hydrodynamic case. This shows that the Okubo-
Weiss parameter is robust and remains intact under extension to 2D quasi-
geostrophic flows (in the $\beta$-plane approximation to the Coriolis
parameter). The inclusion of the nonlinear variation in the Coriolis parameter
(the so-called $\gamma$-effect) will, however, lead to changes in the Okubo-
Weiss parameter,
$Q=\frac{1}{4b^{2}}\left[\left(\frac{\partial^{2}\omega}{\partial x\partial
y}\right)^{2}-\frac{\partial^{2}\omega}{\partial
x^{2}}\left(\frac{\partial^{2}\omega}{\partial y^{2}}+\gamma\right)\right].$
(15)
4\. The Okubo-Weiss Criterion for Magnetohydrodynamic Flows
Consider a 2D incompressible magnetohydrodynamic (MHD) flow. The equation
governing the transport of the magnetic field ${\bf
B}=\left<B_{1},B_{2}\right>$ is (Goedbloed and Poedts [18])
$\frac{D{\bf B}}{Dt}=\left({\bf B}\cdot\nabla\right){\bf v}$ (16a)
which may be rewritten as
$\frac{D{\bf B}}{Dt}=\boldsymbol{\mathscr{A}}\cdot{\bf B}.$ (16b)
If we now assume that the magnetic field evolves quasi-adiabatically with
respect to the straining flow velocity gradient field, equation (16b) may
again be locally approximated by an eigenvalue problem with eigenvalues given
by,
$\lambda^{2}=\frac{1}{4}\left(u_{y}v_{x}+{v_{y}}^{2}\right)\equiv Q.$ (17)
Noting that the MHD Beltrami state (Shivamoggi [19]) corresponds to the so-
called Alfvénic state (Hasegawa [20])
${\bf v}=c{\bf B}$ (18)
$c$ being an arbitrary constant, (17) becomes
$Q=\frac{c^{2}}{4}\left({B_{1}}_{y}{B_{2}}_{x}+{B_{2y}}^{2}\right).$ (19)
In terms of the magnetic vector potential A given by
${\bf B}\equiv\nabla\times{\bf A},~{}{\bf A}=A{\bf\hat{i}}_{z}$ (20)
(19) becomes
$Q=\frac{c^{2}}{4}\left[\left(\frac{\partial^{2}A}{\partial x\partial
y}\right)^{2}-\frac{\partial^{2}A}{\partial
x^{2}}\frac{\partial^{2}A}{\partial y^{2}}\right].$ (21)
(21) implies that the Okubo-Weiss parameter Q for the MHD case characterizes
the topological properties of the magnetic flux surface - it is the negative
of the Gaussian curvature of the magnetic flux surface. As with the case of 2D
hydrodynamic flows, (21) can serve as a useful diagnostic tool to parameterize
the magnetic field topology in 2D MHD flows.
5\. Discussion
The “slow variation” restriction on the straining flow-velocity gradient field
used in the Okubo-Weiss criterion may be quantified via the Beltrami condition
with the divorticity framework in 2D hydrodynamic flows (Shivamoggi et al.
[11]). This also turns out to provide interesting interpretations of the
Okubo-Weiss criterion in terms of the topological properties of the underlying
vorticity manifold. Extension of these considerations to 2D quasi-geostrophic
flows (via the potential divorticity framework) shows the robustness of the
Okubo-Weiss parameter under varying 2D hydrodynamic flow situations. Extension
to 2D MHD flows, on the other hand, provides one again with a useful
diagnostic tool to parameterize the magnetic field topology in 2D MHD flows.
6\. Acknowledgments
The authors are thankful to Dr. Leon Kamp for helpful discussions. BKS would
like to thank The Netherlands Organization for Scientific Research (NWO) for
the financial support.
## References
* [1] A. Okubo: Deep Sea Res. 17, 445, (1970).
* [2] J. Weiss: Physica D 48, 273, (1991).
* [3] M. E. Brachet, M. Meneguzzi, H. Politano and P. L. Sulem: J. Fluid Mech. 194, 333, (1988).
* [4] K. Ohkitani: Phys. Fluids A 3, 1598, (1991).
* [5] A. Babiano and A. Provenzale: J. Fluid Mech. 574, 429, (2007).
* [6] N. T. Ouelette and J. P. Gollub: Phys. Rev. Lett. 99, 194502, (2007).
* [7] A. R. Cieslik, L. P. J. Kamp, H. J. H. Clercx and G. J. F. van Heijst: Europhys. Lett. 85, 54001, (2009).
* [8] L. P. J. Kamp: Phys. Fluids, Submitted, (2011).
* [9] C. Basdevant and T. Philipovitch: Physica D 73, 17, (1994).
* [10] B. L. Hua and P. Klein: Physica D 113, 98, (1998).
* [11] B. K. Shivamoggi, G. J. F. van Heijst and J. Juul Rasmussen: Phys. Lett. A 374, 2309, (2010).
* [12] S. Kida: J. Phys. Soc. Japan 54, 2840, (1985).
* [13] E. A. Kuznetsov, V. Naulin, A. H. Nielsen and J. Juul Rasmussen: Phys. Fluids 19, 105110, (2007).
* [14] B. K. Shivamoggi and G. J. F. van Heijst: Geophys. Astrophys. Fluid Dyn. 103, 293, (2009).
* [15] G. R. Flierl, M. E. Stern and J. A. Whitehead: Dyn. Atmos. Oceans 7, 233, (1983).
* [16] M. Larcheveque: Theor. Comp. Fluid Dyn. 5, 215, (1993).
* [17] J. G. Charney: Geophys. Publ. Kosjones. Nor. Vidensk. Akad. Oslo 17, 1, (1947).
* [18] H. Goedbloed and S. Poedts: Principles of Magnetohydrodynamics, Ch. 4, Cambridge Univ. Press, (2004).
* [19] B. K. Shivamoggi: Euro. Phys. J. D 64, 393, (2011).
* [20] A. Hasegawa: Adv. Phys. 34, 1, (1985).
|
arxiv-papers
| 2011-10-27T20:16:50 |
2024-09-04T02:49:23.658769
|
{
"license": "Creative Commons Zero - Public Domain - https://creativecommons.org/publicdomain/zero/1.0/",
"authors": "B. K. Shivamoggi, G. J. F. van Heijst and L.P.J. Kamp",
"submitter": "Bhimsen Shivamoggi",
"url": "https://arxiv.org/abs/1110.6190"
}
|
1110.6420
|
# Imaging with HST the time evolution of Eta Carinae’s colliding winds
11affiliation: Support for program 12013 was provided by NASA through a grant
from the Space Telescope Science Institute, which is operated by the
Association of Universities for Research in Astronomy, Inc., under NASA
contract NAS 5-26555.
Theodore R. Gull Code 667, Astrophysics Science Division, Goddard Space
Flight Center, Greenbelt, MD 20771, USA; Theodore.R.Gull@nasa.gov Thomas I.
Madura and Jose H. Groh Max-Planck-Institut fur Radioastronomie, Auf dem
Hugel 69, D-53121 Bonn, Germany Michael F. Corcoran22affiliation:
Universities Space Research Association, 10211 Wincopin Circle, Ste 500,
Columbia, MD 21044 CRESST and X-ray Astrophysics Laboratory, Goddard Space
Flight Center, Greenbelt, MD 20771, USA
###### Abstract
We report new HST/STIS observations that map the high-ionization forbidden
line emission in the inner arcsecond of Eta Car, the first that fully image
the extended wind-wind interaction region of the massive colliding wind
binary. These observations were obtained after the 2009.0 periastron at
orbital phases 0.084, 0.163, and 0.323 of the 5.54-year spectroscopic cycle.
We analyze the variations in brightness and morphology of the emission, and
find that blue-shifted emission ($-$400 to $-$200 ${\rm km\ s^{-1}}$) is
symmetric and elongated along the northeast-southwest axis, while the red-
shifted emission ($+$100 to $+$200 ${\rm km\ s^{-1}}$) is asymmetric and
extends to the north-northwest. Comparison to synthetic images generated from
a 3-D dynamical model strengthens the 3-D orbital orientation found by Madura
et al. (2011), with an inclination $i\approx\ $ 138°, argument of periapsis
$\omega\approx\ $ 270°, and an orbital axis that is aligned at the same PA on
the sky as the symmetry axis of the Homunculus, 312°. We discuss the potential
that these and future mappings have for constraining the stellar parameters of
the companion star and the long-term variability of the system.
stars: atmospheres — stars: mass-loss — stars: winds, outflows — stars:
variables: general — supergiants — stars: individual (Eta Carinae)
††slugcomment: Draft version
## 1 Introduction
Eta Carinae, one of the most luminous, variable objects in our Milky Way, is
sufficiently close ($D=2.3\pm 0.1$ kpc, Smith 2006) that we can study many of
its properties throughout the electromagnetic spectrum. As noticed by Damineli
(1996), the object exhibits a 5.54-year orbital period characterized by a
lengthy high ionization111Low and high ionization are used here to describe
atomic species with ionization potentials (IPs) below and above 13.6 eV, the
IP of hydrogen. state with multiple high ionization forbidden lines that
disappear during months-long low ionization state (Damineli et al. 2008b). Eta
Car is considered to be a massive, highly eccentric ($e\sim 0.9$, Corcoran
2005; Nielsen et al. 2005) binary consisting of $\eta_{\mathrm{A}}$, a
luminous blue variable (LBV), and $\eta_{\mathrm{B}}$, a hot, less massive
companion not directly seen, but whose properties have been inferred from its
effects on the wind of $\eta_{\mathrm{A}}$ and the photoionization of nearby
ejecta (Verner et al. 2005; Teodoro et al. 2008; Mehner et al. 2010, hereafter
Me10; Groh et al. 2010a, b)
The total luminosity, dominated by $\eta_{\mathrm{A}}$, is $\geq$ 5$\times$106
L⊙ (Davidson & Humphreys 1997), with the total mass of the binary exceeding
120 M⊙ (Hillier et al. 2001, hereafter H01). Radiative transfer modeling of
HST/STIS spatially-resolved spectroscopic observations suggests that
$\eta_{\mathrm{A}}$ has a mass $\gtrsim 90\ M_{\odot}$, and a stellar wind
with a mass-loss rate of $\sim 10^{-3}M_{\odot}\ \mathrm{yr}^{-1}$ and
terminal speed of $\sim 500-600\ \mathrm{km\ s}^{-1}$ (Hillier et al. 2001;
Hillier et al. 2006, hereafter H06). Models of the observed X-ray spectrum
require the wind terminal velocity of $\eta_{\mathrm{B}}$ to be $\sim 3000$
${\rm km\ s^{-1}}$ with a mass-loss rate of $\sim$ 10${}^{-5}M_{\odot}$ yr-1
(Pittard & Corcoran 2002). The spectral type of $\eta_{\mathrm{B}}$ has been
loosely constrained via modeling of the inner ejecta to be a mid-O supergiant
(Verner et al. 2005; Teodoro et al. 2008; Me10).
3-D numerical simulations suggest that the wind of $\eta_{\mathrm{B}}$
strongly influences the very dense wind of $\eta_{\mathrm{A}}$, creating a
low-density cavity and inner wind-wind collision zone (WWCZ) (Pittard &
Corcoran 2002; Okazaki et al. 2008; Parkin et al. 2009). The geometry and
physical conditions of this inner region have been constrained from spatially
unresolved X-ray (Henley et al. 2008), optical (Nielsen et al. 2007; Damineli
et al. 2008a), and near-infrared (Groh et al. 2010a, b) observations.
In addition to the interaction between the two winds in the inner region (at
spatial scales comparable to the semi-major axis length, $a\approx 15.4$ AU =
0$\farcs$0067 at 2.3 kpc), the 3-D hydrodynamical simulations predict an
outer, extended, ballistic WWCZ that stretches to distances several orders of
magnitude larger than the size of the orbit (Okazaki et al. 2008; Madura 2010,
hereafter M10). Observational evidence for an extended WWCZ comes from the
analysis of previous HST/STIS longslit observations (G09; M10; Madura et al.
2011, hereafter M11) which revealed spatially-extended forbidden line emission
from low- and high-ionization species at $\sim$ 0$\farcs$1 to 0$\farcs$7 (230
to 1600 AU) from the central core. During the high state, [Fe II] line
emission extends up to $\pm$500 ${\rm km\ s^{-1}}$ along the STIS slit, while
[Fe III] line emission extends to $-$400 ${\rm km\ s^{-1}}$ for STIS slit
position angles close to 45°. Radiative transfer modeling of the extended [Fe
III] emission (; ) tightly constrains the orbital inclination, $i\approx
138\arcdeg$, close to the axis of inclination of the Homunculus, and the
argument of periapsis 240° $\lesssim\omega\lesssim 270$° in agreement with
most researchers (Damineli et al. 2008b; Groh et al. 2010a; Parkin et al. 2009
and references therein). This constraint invalidates the claim by several
groups (Falceta-Gonçalves & Abraham 2009; Kashi & Soker 2009 and references
therein) that periastron occurs on the near side of $\eta_{\mathrm{A}}$
($\omega=90\arcdeg$).
Here we report new HST/STIS observations, the first that fully map the inner
arcsecond high-ionization, forbidden line emission of Eta Car. Maps of [Fe
III] $\lambda\lambda$4659.35, 4702.85222All wavelengths are measured in
vacuum. and [N II] $\lambda$5756.19 recorded in early phases following the
2009.0 periastron event show changes in the wind structures excited by FUV
radiation from $\eta_{\mathrm{B}}$. These results demonstrate that structural
changes can be followed using specific forbidden lines, leading to increased
knowledge about interacting wind properties, the parameters of the binary
orbit and, most importantly, the stellar properties of $\eta_{\mathrm{B}}$.
## 2 Observations
The HST/STIS mapping observations were obtained after the successful repair of
STIS during Service Mission 4. The first visit occurred in June 2009
($\phi=12.084$333All observations are referenced by cycle number relative to
cycle 1 beginning 1948 February, following the convention introduced by Groh &
Damineli (2004). The phase $\phi$ is zeroed to JD2482819.8 $\pm$ 0.5 with
period $P=2022.7\pm 1.3$ days (Damineli et al. 2008b).) as an early release
observation demonstrating the repaired-STIS capabilities (Program 11506
PI=Noll). The second and third visits were scheduled in December 2009
($\phi=12.163$) and October 2010 ($\phi=12.323$) under a CHANDRA/HST grant
(Program 12013, PI Corcoran).
All observations were performed with the $52\arcsec\times 0\farcs 1$ longslit.
The strongest, most isolated, high-ionization forbidden emission lines from
the inner and outer WWCZs are [Fe III] $\lambda\lambda$ 4659, 4702 and [N II]
$\lambda$5756 (G09). The STIS gratings, G430M, centered at $\lambda$4706, and
G750M, centered at $\lambda$5734, provide a spectral resolving power of about
8000.
Spatial mapping was accomplished with the standard STIS-PERP-TO-SLIT mosaic
routine using the 52″$\times$0$\farcs$1 aperture with multiple 0$\farcs$1
offset position spacings centered on Eta Carinae. The size of the map, given
limited foreknowledge of the extended forbidden emission structure, was
adjusted with each visit based upon the anticipated HST/STIS longslit position
angle (PA), pre-determined by the HST solar panel orientation. As buffer dumps
impact the total integration time, only the central CCD rows, typically 64
(3$\farcs$2) or 128 (6$\farcs$4), were read out. The PAs for each visit were
PA = 79°($\phi=12.084$), -121°($\phi=12.163$), and -167°($\phi=12.323$). Since
a full spatial map was obtained during each visit, the PA has little effect on
the results presented here (see Figures 1 and 2).
Figure 1: Comparison of red and blue images for isolated high-ionization
forbidden lines from the $\phi=$12.084 observations (June 2009). (a) HST/ACS
image shows the 2$\farcs$2$\times$2$\farcs$2 box centered on Eta Carinae
located within the 18″ Homunculus as indicated in the small inset (HST
archives). Strong continuum (b) has been subtracted from each forbidden
emission map. [Fe III] $\lambda$4659 emission (c), integrated from $-$400 to
$+$200 ${\rm km\ s^{-1}}$, has a very different spatial distribution from the
continuum. Blue images, extracted from $-$400 to $-$200 ${\rm km\ s^{-1}}$ for
[Fe III] $\lambda$4659 (d), $\lambda$4702 (f) and [N II] $\lambda$5756 (h) are
similar for each ion, as are red images extracted from $+$100 to $+$200 ${\rm
km\ s^{-1}}$ for [Fe III] $\lambda$4659 (e), $\lambda$4702 (g) and [N II]
$\lambda$5756 (i). Images are displayed as sqrt(ergs cm${}^{-2}\ $s-1). North
is up, and east is left.
The data were reduced with STIS GTO CALSTIS software. While data quality is
similar to previous HST/STIS observations of Eta Car obtained from 1998 to
2004 (Davidson et al. 2005; G09), the CCD detector has increased number of hot
pixels, some bad columns, and increased charge transfer inefficiencies. Bright
local continuum (Figure 1b) was subtracted from each pixel, isolating the
faint forbidden line emission (Figures 1c-i, 2). Velocity channels were co-
added to produce blue ($-400$ to $-200$ ${\rm km\ s^{-1}}$), low-velocity
($-90$ to $+30$ ${\rm km\ s^{-1}}$), and red ($+$100 to $+$200 ${\rm km\
s^{-1}}$) images for each of the high-ionization forbidden lines (Figure 2).
Only the high velocity blue and red maps are sensitive to the wind wind
interaction that we model in this present work. The low velocity maps are
dominated by slow-moving, extended ejecta produced in the 19th century
eruptions, and so are not discussed in detail here. A refinement to the
current model will include a screen of condensations to account for the low-
velocity emission.
## 3 Results
### 3.1 Morphology and time evolution of the extended wind-wind collision
For each phase, we compared velocity-separated images of [Fe III]
$\lambda\lambda$4659, 4702 and [N II] $\lambda$5756, and found remarkable
similarities in the blue and red images between the three emission lines (see
Figure 1 for June 2009, $\phi=$12.084). Hereafter we focus on the [Fe III]
$\lambda$4659 emission, which cannot be formed by the primary star alone.
Emission of [Fe III] requires 16.2 eV photons from $\eta_{\mathrm{B}}$, plus
thermal collisions at electron densities approaching Ne = 107 cm-3 (; ; ). By
comparison, [N II] emission is produced by 14.6 eV photons at electron
densities approaching Ne = 3$\times$107 cm-3. As the primary star,
$\eta_{\mathrm{A}}$, produces significant numbers of 14.6 eV photons (H01), [N
II] emission does not fully disappear during periastron (Damineli et al.
2008a; G09). However, the red emission from [Fe III] $\lambda$4659.35 can be
contaminated by blue emission from [Fe II] $\lambda$4665.75. Likewise, the red
emission image of [Fe III] $\lambda$4702.85 may be depressed by He I
$\lambda$4714.47 absorption. Hence, we examined the [N II] maps to ensure
little or no red high-ionization emission is present.
Figure 2 shows the time evolution of the blue, low-velocity, and red
components of [Fe III] $\lambda$4659 at orbital phases $\phi=12.084$, 12.163,
and 12.323. The morphology and geometry of the extended [Fe III] $\lambda$4659
emission resolved by HST/STIS changes conspicuously as a function of velocity
and time. The blue emission extends along the NE–SW direction, along
$\mathrm{PA}\simeq 45\arcdeg$, which is similar to what has been suggested
from previous sparse HST/STIS long-slit observations obtained at different
orbital phases across cycle 11 (G09, Me10, M10, M11). At $\phi=$12.084, the
linear structure is nearly symmetrical about the central region, but at later
phases becomes more diffuse, shifting to the S and SE. The red emission is
fuzzier, asymmetric and extends primarily to the NNW at each phase. In
contrast, the low-velocity structure is larger and extends diffusely
northward. The low-velocity emission is heavily dominated by emission from the
Weigelt blobs (Weigelt & Ebersberger 1986) and a screen of fainter
condensations (Me10), located within the $\eta_{\mathrm{B}}$ wind-blown cavity
and thusly obscuring the much fainter WWCZ contributions. While we describe
the qualitative changes of the low-velocity component, we defer the detailed
modeling of this equatorial emission to a future paper.
For discussion purposes, we now isolate the central core (inner
0$\farcs$3$\times$0$\farcs$3) as representative of the inner WWCZ, and a time-
variant extended ($>$0$\farcs$3$\times$0$\farcs$3) structure as representative
of the outer WWCZ. These two regions have very different physical drivers. The
central core exhibits X-ray (Pittard & Corcoran 2002) and He I emission, along
with strong forbidden line emission. The outer WWCZ, expanding ballistically,
is best traced by strong forbidden line emission. The spatially-extended blue
and red emission components are thought to arise in the outer WWCZ of Eta Car
(G09), composed of material which was earlier part of the inner WWCZ, but over
the past 5.5-year period streamed outward (; ). While the primary wind is
estimated to have a terminal velocity of $500-600$ ${\rm km\ s^{-1}}$, the
peak radial velocity component of the forbidden emission lines appears to be
$\sim 400$ ${\rm km\ s^{-1}}$. At terminal velocity, the outer WWCZ expands at
0$\farcs$25 per 5.5-year cycle, hence the current WWCZ, even at $\phi=$0.323,
is within the 0$\farcs$3$\times$0$\farcs$3 core.
Figure 2: The changing shape of high-ionization [Fe III] $\lambda$4659 early
in Eta Carinae’s binary period. Top row: $\phi=$12.084. Middle row:
$\phi=$12.163. Bottom row: $\phi=$12.323. Left column: blue emission ($-$400
to $-$200 ${\rm km\ s^{-1}}$). Middle column: low-velocity emission ($-$90 to
$+$30 ${\rm km\ s^{-1}}$). Right column: red emission ($+$100 to $+$200 ${\rm
km\ s^{-1}}$). Gaps between the velocity intervals are purposefully excluded
to show very separate velocity fields. The color bars show flux scaled by
sqrt(ergs cm-2s-1.)
Both the central and extended structures brighten with phase, but they change
differently. At $\phi=$12.084, the central core accounts for 1/3 of the flux,
but brightens only thirty percent by $\phi=$12.323. The extended emission more
than doubles in brightness by $\phi=$12.323. Brightening of the velocity
components within the core and extended structures are likewise different. The
brightness of the red component is nearly constant for both the core and the
extended structure. The core blue component increases by seventy percent while
the extended blue component doubles in brightness. The core low-velocity
component increases only by fifty percent, but the low-velocity extended
component triples in brightness and appears to shift further outward from the
core. We note that between $\phi=$ 12.163 and 12.323 the brightest low-
velocity component shifts from the vicinity of Weigelt C, noted by Me10, to
Weigelt B and D.
These brightness changes in the core and extended structures support a
scenario in which the current WWCZ, namely the direct collision between the
winds of $\eta_{\mathrm{A}}$ and $\eta_{\mathrm{B}}$, is contained within the
0$\farcs$3 diameter core. After each periastron passage, a new secondary-wind-
blown cavity must form and expand outward. The cavity rapidly approaches a
balance between the FUV flux of $\eta_{\mathrm{B}}$ and the cavity wall
structure at critical density. However, the outer cavity wall is very thin,
ionizes rapidly and drops in density allowing FUV radiation to pass outward
into the much larger, ballistically expanding outer cavity formed in the
previous cycle. Within this cavity, the FUV photons encounter dense walls of
primary wind. The growth in brightness in the blue images, with little change
in the red images, indicates expansion in the general direction of the
observer. The larger increase in brightness of the low-velocity images shows
where the FUV radiation escapes through the multiple cavities built up by the
wind of $\eta_{\mathrm{B}}$ over many cycles.
### 3.2 Comparison with a 3-D Dynamical Model
Proper interpretation of the mapping observations requires a full 3-D
dynamical model that accounts for the effects of orbital motion on the WWCZ.
Here we use full 3-D Smoothed Particle Hydrodynamics (SPH) simulations of Eta
Car’s colliding winds and radiative transfer codes to compute the intensity in
the [Fe III] $\lambda$4659 line projected on the sky for a specified orbital
orientation (; ). The numerical simulations were performed using the same 3-D
SPH code as that in Okazaki et al. (2008) with identical parameters except for
the mass loss rate of $\eta_{\mathrm{A}}$, which we changed to 10-3 M⊙ yr-1 (;
). The two stellar winds in our simulation are also taken to be adiabatic. In
order to allow for a more direct comparison to the HST observations, the
computational domain is a factor of ten larger than that of Okazaki et al.
(2008) (i.e. $\pm$ 1600 $\mathrm{AU}\approx\pm$ 0$\farcs$7). Details on the
radiative transfer calculations can be found in M10, M11.
Figures 3 and 4 compare the observed blue and red images at $\phi=$ 12.163 and
12.323 with those predicted by the model for the same velocity intervals. For
simplicity, the zero reference phase of the spectroscopic cycle (Damineli et
al. 2008a), is assumed to coincide with the zero reference phase of the
orbital cycle (i.e. periastron passage) in the 3-D SPH simulation. In a
highly-eccentric binary system like Eta Car, the two values should be within a
few weeks, which will not affect the overall conclusions (Groh et al. 2010b).
The binary orbit is assumed to be oriented with an inclination $i=$ 138°,
argument of periapsis $\omega=$ 270°, and an orbital axis that is aligned at
the same PA on the sky as the symmetry axis of the Homunculus, 312° (Davidson
et al. 2001)444Davidson et al. (2001) determined the Homunculus axis of
symmetry to be tilted 42° into the sky plane. We refer the reader to M11 for
detailed discussion of the binary orbital inclination at 138°=180°-42°..
The relatively compact central core produces little [Fe III] emission as
densities in the WWCZ walls greatly exceed the critical density for efficient
emission. The low-velocity maps, displayed on a flux scale similar to the
scales for the blue and red images, would be blank while the observed low
velocity emission, heavily dominated by flux from the Weigelt blobs and
fainter slow-moving clumps, extends to the northwest. As mentioned in section
3.1, we are refining the model to include such a screen, which will be a topic
in a much more encompassing paper. Hence only the red and blue components,
successfully replicated in this study, are presented in Figures 3 and 4.
The spatial extent of the emission compares quite favorably between the
observations and the models (Figures 3 and 4), with the blue structures
extending projected distances of $\sim$ 1″ (2300 AU) along $\mathrm{PA}\sim$
45° , and the red structures displaced to the NE of the core by $\sim$
0$\farcs$1 to 0$\farcs$4 (230 to 1000 AU). We display unreddened fluxes for
the model structures due to known uncertainties of reddening. Model fluxes,
reddened by $\approx$5–20 using typical *interstellar* reddening values for
stars in the vicinity of Eta Car (; ) agree with the observations within a
factor of a few. This discrepancy could arise due to uncertainties in the
assumed stellar parameters of both stars, the reddening law and atomic
physics, or systematics in the radiative transfer and hydrodynamical modeling.
However, reddening is highly variable across the Carinae complex. Moreover,
reddening by dust in the Homunculus and within the extended core of Eta Car
may change on very small scales. Hence we chose to display unreddened model
fluxes in Figures 3 and 4.
Figure 3: Comparison of $\phi=$12.163 blue and red components to 3-D
dynamical model. Top row: Observed blue and red images. Bottom Row: 3-D
SPH/radiative transfer images. Left column: $-$400 to $-$200 ${\rm km\
s^{-1}}$. Right column: $+$100 to $+$200 ${\rm km\ s^{-1}}$. Color display in
all images is on a square root scale of ergs cm-2 s-1. North is up.
Figure 4: Comparison of $\phi=$ 12.323 blue and red components to 3-D
dynamical model as in Figure 3. Changes are subtle as $\eta_{\mathrm{B}}$
physically is close to the position of apastron; the ionization structure is
primarily expanding.
## 4 Discussion
This work represents the first time the extended WWCZ of a massive colliding
wind binary system has been imaged using high-ionization forbidden emission
lines. Spatial- and velocity-extended emission, recorded by individual
HST/STIS longslit observations at various phases and PAs, provided impetus to
expand 3-D models to simulate the wind dynamics leading to this emission.
Indeed, the initial 3-D dynamical model above produces red and blue images
that are similar to those observed. From multiple longslit observations, G09,
M10 and M11 demonstrated that the binary orbit could be fully constrained in
3-D. The noticeable symmetry in velocity for observations taken at PA=38°
(G09) is now reinforced by the spatial symmetry about the central core in the
blue maps. Our modeling, of the observed maps suggests that the argument of
periapsis must be closer to $\omega=$270° than 240°, thus further reinforcing
the result that $\eta_{\mathrm{B}}$ is on the near side of $\eta_{\mathrm{A}}$
at apastron, with periastron passage on the far side (Damineli et al. 1997;
Pittard & Corcoran 2002; Okazaki et al. 2008; Parkin et al. 2009; G09, M10;
M11).
These and future spatial maps of Eta Car’s high-ionization forbidden emission
have the potential to determine the nature of the unseen companion star
$\eta_{\mathrm{B}}$. The mass-loss rate of $\eta_{\mathrm{A}}$ and ionizing
flux of photons from $\eta_{\mathrm{B}}$ determine which regions of Eta Car’s
WWCZ are photoionized and capable of producing high-ionization forbidden line
emission like the forbidden emission from Fe++, due to 16.2 eV radiation.
Comparing this mass model loss rates and UV fluxes to those of stellar models
for a range of O (Martins et al. 2005; Me10) and WR (Crowther 2007) stars
would allow one to obtain a luminosity and temperature for
$\eta_{\mathrm{B}}$. Both the current model (; ) and previous individual
HST/STIS longslit observations (G09) show major changes with orbital phase,
especially near periastron. Mappings at multiple phases around periastron are
therefore essential in order to determine when the FUV radiation from
$\eta_{\mathrm{B}}$ becomes trapped in the dense wind of $\eta_{\mathrm{A}}$
and the extended high-ionization emission vanishes, and likewise when
$\eta_{\mathrm{B}}$ emerges from $\eta_{\mathrm{A}}$’s wind and the extended
emission returns.
This approach has a number of advantages over previous 1-D modeling efforts to
constrain $\eta_{\mathrm{B}}$’s properties (Verner et al. 2005; ), which probe
the ionization structure of the Weigelt blobs. Such 1-D models make
considerable assumptions about the physical conditions within the blobs and
intervening material, leading to poor constraints on the luminosity of
$\eta_{\mathrm{B}}$.
Eta Car is variable, not only on a 5.5-year period, but has a centuries-long
history of variation, including two major eruptions (Davidson & Humphreys
1997; Humphreys et al. 2008; Smith & Frew 2010). These high-ionization
forbidden emission lines are powerful tools for monitoring changes in the
WWCZ, providing quantitative information on the properties of the individual
binary components and changes thereof, including a historical record of the
recent decade-long mass loss from the primary. Following this system will
provide unique information on how a massive star, during the LBV stage, loses
much of its mass on its way to becoming a supernova.
We sincerely thank G. Weigelt, S. Owocki, A. Damineli and A. Okazaki for many
fruitful discussions and encouragements. TG acknowledges the hospitality of
MPIR during his multiple visits. We thank the referee for insightful comments
leading to an improved presentation.
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|
arxiv-papers
| 2011-10-28T18:42:20 |
2024-09-04T02:49:23.673122
|
{
"license": "Public Domain",
"authors": "Theodore R. Gull, Thomas I. Madura, Jose H. Groh and Michael F.\n Corcoran",
"submitter": "Theodore Gull",
"url": "https://arxiv.org/abs/1110.6420"
}
|
1110.6526
|
# Hyperbolic spaces in Teichmüller spaces††thanks: This work is in the public
domain. The first author was supported by NSF grants DMS 0905748 and DMS
1207183. The second author was supported by EPSRC grant EP/I028870/1.
Christopher J. Leininger and Saul Schleimer
###### Abstract
We prove, for any $n$, that there is a closed connected orientable surface $S$
so that the hyperbolic space $\mathbb{H}^{n}$ almost-isometrically embeds into
the Teichmüller space of $S$, with quasi-convex image lying in the thick part.
As a consequence, $\mathbb{H}^{n}$ quasi-isometrically embeds in the curve
complex of $S$.
## 1 Introduction
We denote the Teichmüller space of a surface $S$ by $\mathcal{T}(S)$, and the
$\epsilon$–thick part by $\mathcal{T}_{\epsilon}(S)$; see Section 4. An
almost-isometric embedding of one metric space into another is a
$(1,C)$–quasi-isometric embedding, for some $C\geq 0$; see Section 2. Let
$\mathbb{H}^{n}$ denote hyperbolic $n$–space. The main result of this paper is
the following.
###### Theorem 1.1.
For any $n\geq 2$, there exists a surface of finite type $S$ and an almost-
isometric embedding
$\mathbb{H}^{n}\to\mathcal{T}(S).$
Moreover, the image is quasi-convex and lies in $\mathcal{T}_{\epsilon}(S)$
for some $\epsilon>0$.
According to Proposition 4.4 below, Theorem 1.1 remains true if we replace
“surface of finite type” with “closed surface”. Our work is motivated, in
part, by the following open question (see [7] for the case $n=2$).
###### Question 1.2.
Does there exist a closed surface $S$ of genus at least $2$, a closed
hyperbolic $n$–manifold $B$ with $n\geq 2$, and an $S$–bundle $E$ over $B$ for
which $\pi_{1}(E)$ is Gromov hyperbolic?
To explain the relationship with our theorem, suppose that
$S\to E\to B$
is an $S$–bundle over $B=\mathbb{H}^{n}/\Gamma$, for some closed surface $S$
and some torsion free cocompact lattice
$\Gamma<\mathrm{Isom}(\mathbb{H}^{n})$. The monodromy is a homomorphism to the
mapping class group of $S$, $\rho\colon\pi_{1}(B)=\Gamma\to\mathrm{Mod}(S)$.
The mapping class group $\mathrm{Mod}(S)$ acts on $\mathcal{T}(S)$ by
isometries with respect to the Teichmüller metric, and according to work of
Farb-Mosher [7] and Hamenstädt [12], $\pi_{1}(E)$ is $\delta$-hyperbolic if
and only if we can construct a $\Gamma$–equivariant quasi-isometric embedding
$f\colon\mathbb{H}^{n}\to\mathcal{T}(S)$
with quasi-convex image lying in $\mathcal{T}_{\epsilon}(S)$ for some
$\epsilon>0$; see also [25]. (In fact the $\Gamma$–equivariance and quasi-
isometric embedding assumptions imply that the image lies in
$\mathcal{T}_{\epsilon}(S)$.)
Our main theorem states that if we drop the assumption of equivariance, then
quasi-isometric embeddings with all the remaining properties exist. On the
other hand, as was shown in [6], one can find cocompact lattices
$\Gamma<\mathrm{Isom}(\mathbb{H}^{2})$ and $\Gamma$–equivariant quasi-
isometries into $\mathcal{T}(S)$ with image in $\mathcal{T}_{\epsilon}(S)$—for
these examples the image is not quasi-convex.
The main theorem for $n=2$ also contrasts with the situation of isometrically
embedding hyperbolic planes in $\mathcal{T}(S)$. More precisely, every
geodesic in $\mathcal{T}(S)$ is contained in an isometrically embedded
hyperbolic plane (with the Poincaré metric) called a Teichmüller disk.
However, it is well-known that no Teichmüller disk lies in any thick part—this
follows from [21] which guarantees that along a dense set of geodesic rays in
the Teichmüller disk the hyperbolic length of some curve on $S$ tends to zero.
The curve complex of $S$ is a metric simplicial complex $\mathcal{C}(S)$ whose
vertices are isotopy classes of essential simple closed curves, and for which
$k+1$ distinct isotopy classes of curves span a $k$–simplex if they can be
realized disjointly. In [23], Masur and Minsky proved that $\mathcal{C}(S)$ is
$\delta$–hyperbolic. One of the key ingredients in their proof is the
construction of a coarsely Lipschitz map $\mathcal{T}(S)\to\mathcal{C}(S)$.
The restriction of this map to any quasi-convex subset of
$\mathcal{T}_{\epsilon}(S)$ is a quasi-isometry (see for example [27, Lemma
4.4] or [15, Theorem 7.6]). Composing the almost-isometry of Theorem 1.1 with
the map $\mathcal{T}(S)\to\mathcal{C}(S)$ we have the following corollary.
###### Corollary 1.3.
For every $n\geq 2$, there exists a surface of finite type $S$ and a quasi-
isometric embedding
$\mathbb{H}^{n}\to\mathcal{C}(S).$
The case of $n=2$ here can be compared to the result of Bonk and Kleiner [5]
in which it is shown that every $\delta$–hyperbolic group which is not
virtually free contains a quasi-isometrically embedding hyperbolic plane. The
assumption that the group is not virtually free implies the existence of an
arc in the boundary. According to [9] (see also [19, 18]) with the exception
of a few small surfaces, there are indeed arcs in the boundary of
$\mathcal{C}(S)$. In [5] however, essential use is made of the fact that there
is an action of the group, and so even in the case $n=2$, Corollary 1.3 does
not follow from [5].
We now explain the idea for the construction in the case $n=2$. Given a closed
Riemann surface $Z$ and a point $z\in Z$, the Teichmüller space
$\mathcal{T}(Z,z)$ is naturally a $\mathbb{H}^{2}$–bundle over
$\mathcal{T}(Z)$; see Section 4.3. Given a biinfinite geodesic $\tau$ in
$\mathcal{T}(Z)$, the preimage of $\tau$ in $\mathcal{T}(Z,z)$ is a
$3$–manifold. The parameterization $t\mapsto\tau(t)$ lifts to a flow on the
preimage of $\tau$ for which the flow lines are geodesics in
$\mathcal{T}(Z,z)$. The fiber over $\tau(0)$ admits a pair of transverse
$1$–dimensional singular foliations—these are naturally associated to the
vertical and horizontal foliations of the quadratic differential defining
$\tau$. Any two flow lines meeting the same nonsingular leaf of the vertical
foliation are forward asymptotic. Therefore, we have a $1$–parameter family of
forward asymptotic geodesics in $\mathcal{T}(Z,z)$. We use this to define a
map from $\mathbb{H}^{2}$ to $\mathcal{T}(Z,z)$: we pick a horocycle
$C\subset\mathbb{H}^{2}$ and send the pencil of geodesics perpendicular to $C$
to our set of forward asymptotic geodesics in $\mathcal{T}(Z,z)$.
At the beginning of Section 5.2 we give a brief explanation of how this can be
modified to give the construction for $n=3$. The idea for $n\geq 4$ is then a
straightforward inductive construction.
Acknowledgements. We thank Richard Kent for useful conversations as well as
having originally asked about the existence of quasi-isometric embeddings of
hyperbolic planes into $\mathcal{C}(S)$. We thank the referee for their
comments.
## 2 Hyperbolic geometry
Suppose that $(X,d_{X})$ and $(Y,d_{Y})$ are metric spaces.
###### Definition 2.1.
A map $F\colon X\to Y$ is a $K$–almost-isometric embedding if for all
$x,x^{\prime}\in X$ we have
$|d_{X}(x,x^{\prime})-d_{Y}(F(x),F(x^{\prime}))|\leq K.$
We use the exponential model for hyperbolic space:
$\mathbb{H}^{n}=\mathbb{R}^{n-1}\times\mathbb{R}$ with length element
$ds^{2}=e^{-2t}\mathopen{}\mathclose{{}\left(dx_{1}^{2}+\ldots+dx_{n-1}^{2}}\right)+dt^{2}.$
For two points $p,q\in\mathbb{H}^{n}$ we use $d_{\mathbb{H}}(p,q)$ to denote
the distance between them. The exponential model of hyperbolic space is
related to the upper-half space model $U=\mathbb{R}^{n-1}\times(0,\infty)$ by
the map $\mathbb{H}^{n}\to U$ given by $(x,t)\mapsto(x,e^{t})$. In the
exponential model, for every $x\in\mathbb{R}^{n-1}$ the path
$\eta_{x}(t)=(x,t)$ is a vertical geodesic and is parameterized by arc-length.
###### Lemma 2.2.
Suppose $(X,d_{X})$ is a geodesic metric space and $\delta,\epsilon,R>0$ are
constants. Suppose $F\colon\mathbb{H}^{n}\to X$ is a function with the
following properties.
1. 1.
$F\circ\eta_{x}$ is a geodesic for all $x\in\mathbb{R}^{n-1}$.
2. 2.
For distinct $x,x^{\prime}\in\mathbb{R}^{n-1}$ the geodesics $F\circ\eta_{x}$
and $F\circ\eta_{x^{\prime}}$ are two sides of an ideal $\delta$–slim triangle
in $(X,d_{X})$.
3. 3.
For any $x,x^{\prime}\in\mathbb{R}^{n-1}$ if $e^{-t}|x-x^{\prime}|<\epsilon$
then $d_{X}(F(x,t),F(x^{\prime},t))\leq R$.
4. 4.
If $(x_{k},t_{k}),(x_{k}^{\prime},t_{k})\in\mathbb{H}^{n}$ satisfy
$\displaystyle{\lim_{k\to\infty}e^{-t_{k}}|x_{k}-x_{k}^{\prime}|=\infty}$,
then
$\displaystyle{\lim_{k\to\infty}d_{X}\mathopen{}\mathclose{{}\left(F(x_{k},t_{k}),F(x_{k}^{\prime},t_{k})}\right)=\infty}$.
Then there exists a constant $K$ so that $F$ is a $K$–almost isometric
embedding.
A useful consequence of Property 3 is that for any
$x,x^{\prime},t\in\mathbb{R}$ we have
$d\mathopen{}\mathclose{{}\left(F(x,t),F(x^{\prime},t)}\right)\leq\frac{R}{\epsilon}e^{-t}|x-x^{\prime}|+R.$
(1)
The remainder of this section gives the proof of Lemma 2.2. We begin by
controlling how $F$ moves the centers of ideal triangles. To be precise:
Suppose that
$T=\mathcal{P}\cup\mathcal{Q}\cup\mathcal{R}\subset\mathbb{H}^{n}$ is an ideal
triangle where $\mathcal{P}$ and $\mathcal{Q}$ are distinct vertical
geodesics. Let $r$ denote the point of $\mathcal{R}$ with maximal
$t$–coordinate. We call $r$ the midpoint of $\mathcal{R}$. Thus $r$ serves as
a center for $T$. Define $x=x(\mathcal{P}),x^{\prime}=x(\mathcal{Q})$.
Observe, say from the upper-half space model, that for all $t\geq t(r)$ we
have
$d_{\mathbb{H}}\mathopen{}\mathclose{{}\left((x,t),(x^{\prime},t)}\right)\leq
e^{-t}|x-x^{\prime}|\leq e^{-t(r)}|x-x^{\prime}|=2.$ (2)
Thus, by Inequality (1) we have $d_{X}(F(x,t),F(x^{\prime},t))\leq
2R/\epsilon+R$. Define $\Delta=\max\\{3\delta,2R/\epsilon+R\\}$ and define the
displaced height of $T$ to be
$h_{T}=h(T)=\min\Big{\\{}t\in\mathbb{R}\,\,\Big{|}\,\,d_{X}\mathopen{}\mathclose{{}\left(F(x,t),F(\mathcal{Q})}\right)\leq\Delta\,\,\mbox{or}\,\,d_{X}\mathopen{}\mathclose{{}\left(F(\mathcal{P}),F(x^{\prime},t)}\right)\leq\Delta\Big{\\}}.$
It follows that $h(T)\leq t(r)$. Note that for any vertical triangle $T$,
Property 2 implies that $h(T)>-\infty$.
###### Claim 2.3.
For any vertical triangle
$T=\mathcal{P}\cup\mathcal{Q}\cup\mathcal{R}\subset\mathbb{H}^{n}$,
$d_{X}\mathopen{}\mathclose{{}\left(F(x,h_{T}),F(x^{\prime},h_{T})}\right)\leq
3\Delta,$
where $x=x(\mathcal{P})$, $x^{\prime}=x(\mathcal{Q})$.
###### Proof.
Breaking symmetry, in this setting, allows us to assume that there is some
$s\in\mathbb{R}$ so that $d_{X}(F(x^{\prime},s),F(x,h_{T}))\leq\Delta$. Let
$t^{\prime}=\max\\{s,t(r)\\}$. Using the triangle inequality, Inequality 1 and
Property 1 we have
$\displaystyle t^{\prime}-h_{T}$
$\displaystyle=d_{X}\mathopen{}\mathclose{{}\left(F(x,t^{\prime}),(x,h_{T})}\right)$
$\displaystyle\leq
d_{X}\mathopen{}\mathclose{{}\left(F(x,t^{\prime}),F(x^{\prime},t^{\prime})}\right)+d_{X}\mathopen{}\mathclose{{}\left(F(x^{\prime},t^{\prime}),F(x^{\prime},s)}\right)+d_{X}\mathopen{}\mathclose{{}\left(F(x^{\prime},s),F(x,h_{T})}\right)$
$\displaystyle\leq(2R/\epsilon+R)+(t^{\prime}-s)+\Delta$ and similarly
$\displaystyle t^{\prime}-s$ $\displaystyle\leq
2R/\epsilon+R+t^{\prime}-h_{T}+\Delta.$
Thus $|h_{T}-s|\leq 2R/\epsilon+R+\Delta$. Another application of the triangle
inequality and Property 1 implies that
$d_{X}\mathopen{}\mathclose{{}\left(F(x,h_{T}),F(x^{\prime},h_{T})}\right)\leq
2R/\epsilon+R+2\Delta\leq 3\Delta$, as desired. ∎
As mentioned above, for every vertical triangle $T$ we have $h(T)>-\infty$ and
hence $t(r)-h(T)<\infty$. We now obtain a uniform bound on this quantity.
###### Claim 2.4.
There is a constant $C_{0}=C_{0}(F)$ so that $t(r)-h(T)\leq C_{0}$ for all
vertical triangles $T\subset\mathbb{H}^{n}$.
###### Proof.
Suppose not. Then we are given a sequence of vertical triangles
$T_{k}=\mathcal{P}_{k}\cup\mathcal{Q}_{k}\cup\mathcal{R}_{k}$ where
$t(r_{k})-h(T_{k})$ tends to infinity with $k$. Here $r_{k}$ is the midpoint
of $\mathcal{R}_{k}$, the non-vertical side. Define $t_{k}=t(r_{k})$,
$h_{k}=h(T_{k})$. Define $x_{k}=x(\mathcal{P}_{k})$,
$x_{k}^{\prime}=x(\mathcal{Q}_{k})$ to be the horizontal coordinates of the
vertical sides of $T_{k}$.
Note that by Equation (2)
$\displaystyle e^{-t_{k}}|x_{k}-x_{k}^{\prime}|$ $\displaystyle=2$ and so
$\displaystyle e^{-h_{k}}|x_{k}-x_{k}^{\prime}|$
$\displaystyle=e^{-h_{k}}\cdot 2e^{t_{k}}=2e^{t_{k}-h_{k}}.$
Thus $e^{-h_{k}}|x_{k}-x_{k}^{\prime}|$ tends to infinity with $k$. From
Property 4 we deduce that the quantity
$d_{X}\mathopen{}\mathclose{{}\left(F(x_{k},h_{k}),F(x_{k}^{\prime},h_{k})}\right)$
also tends to infinity with $k$. This last, however, contradicts Claim 2.3. ∎
We give the proof of Lemma 2.2. Fix any $p,q\in\mathbb{H}^{n}$. If $x(p)=x(q)$
then we are done by Property 1. Suppose instead that $x(p)\neq x(q)$. Let
$\mathcal{P}\cup\mathcal{Q}\cup\mathcal{R}$ denote the vertical triangle
having vertical sides $\mathcal{P}$ and $\mathcal{Q}$ so that
$x(\mathcal{P})=x(p)$, $x(\mathcal{Q})=x(q)$; let $r\in\mathcal{R}$ be the
midpoint of the non-vertical side. Define $C_{1}=2C_{0}+5\Delta+1$. There are
now two cases to consider.
###### Case.
Suppose that $t(p)\geq h(T)-C_{1}$.
Let $p^{\prime}\in\mathcal{P}$ and $q^{\prime}\in\mathcal{Q}$ be the points
with $t(p^{\prime})=t(q^{\prime})=\max\\{t(p),t(r)\\}$. Then by the triangle
inequality and Equation (2) we have
$\displaystyle d_{\mathbb{H}}(p,q^{\prime})$ $\displaystyle\leq
d_{\mathbb{H}}(p,p^{\prime})+d_{\mathbb{H}}(p^{\prime},q^{\prime})$
$\displaystyle\leq t(p^{\prime})-t(p)+2$ $\displaystyle\leq t(r)-h(T)+C_{1}+2$
$\displaystyle\leq C_{0}+C_{1}+2.$
It follows that $d_{\mathbb{H}}(p,q)$ is estimated by
$d_{\mathbb{H}}(q^{\prime},q)=|t(q^{\prime})-t(q)|$ up to an additive error at
most $C_{0}+C_{1}+2$. Appealing to Property 1, Inequality (1), and the
triangle inequality we similarly have
$\displaystyle d_{X}\mathopen{}\mathclose{{}\left(F(p),F(q^{\prime})}\right)$
$\displaystyle\leq
d_{X}\mathopen{}\mathclose{{}\left(F(p),F(p^{\prime})}\right)+d_{X}\mathopen{}\mathclose{{}\left(F(p^{\prime}),F(q^{\prime})}\right)$
$\displaystyle\leq t(p^{\prime})-t(p)+2R/\epsilon+R$ $\displaystyle\leq
C_{0}+C_{1}+2R/\epsilon+R.$
Thus $d_{X}\mathopen{}\mathclose{{}\left(F(p),F(q)}\right)$ is estimated by
$d_{X}\mathopen{}\mathclose{{}\left(F(q^{\prime}),F(q)}\right)=d_{\mathbb{H}}(q^{\prime},q)$
with an additive error at most $C_{0}+C_{1}+2R/\epsilon+R$. This completes the
proof in this case.
###### Case.
Suppose that $t(p),t(q)\leq h(T)-C_{1}$.
In this case, since the triangle $T=\mathcal{P}\cup\mathcal{Q}\cup\mathcal{R}$
is slim in $\mathbb{H}^{n}$, we find that that $d_{\mathbb{H}}(p,q)$ is
estimated by $t(r)-t(p)+t(r)-t(q)$ up to an additive error of at most $2$. We
now show that $d_{X}(F(p),F(q))$ is also estimated by the latter quantity,
with a uniformly bounded error. Using Property 1 and Inequality (1) deduce
$d_{X}\mathopen{}\mathclose{{}\left(F(p),F(q)}\right)\leq
t(r)-t(p)+2R/\epsilon+R+t(r)-t(q).$
We now give a lower bound for
$d_{X}\mathopen{}\mathclose{{}\left(F(p),F(q)}\right)$. Recall that
$F(\mathcal{P})$ and $F(\mathcal{Q})$ are two sides of a $\delta$–slim
triangle in $X$. Let $\mathcal{R}_{X}$ be the third side of this triangle.
Since
$d_{X}\mathopen{}\mathclose{{}\left(F(p),F(\mathcal{Q})}\right),d_{X}\mathopen{}\mathclose{{}\left(F(\mathcal{P}),F(q)}\right)>\Delta\geq\delta$
it follows that there are points $p_{X},q_{X}\in\mathcal{R}_{X}$ so that
$d_{X}\mathopen{}\mathclose{{}\left(F(p),p_{X}}\right),d_{X}\mathopen{}\mathclose{{}\left(q_{X},F(q)}\right)\leq\delta$.
Thus the distance $d_{X}(p_{X},q_{X})$ is an estimate for
$d_{X}\mathopen{}\mathclose{{}\left(F(p),F(q)}\right)$ with an additive error
at most $2\delta$.
Define $a=(x,h_{T}),b=(x^{\prime},h_{T})$. Again, as in the previous
paragraph, there are points $a_{X},b_{X}\in\mathcal{R}_{X}$ within distance
$\delta$ of $F(a),F(b)$. Since $d_{\mathbb{H}}(a,b)\leq 2(t(r)-h(T))+2$ we
find
$\displaystyle d_{X}(a_{X},b_{X})$ $\displaystyle\leq
2\delta+2(t(r)-h(T))+2R/\epsilon+R$ $\displaystyle\leq
2\delta+2C_{0}+2R/\epsilon+R.$
Note that the geodesic segments
$[p_{X},a_{X}],[b_{X},q_{X}]\subset\mathcal{R}_{X}$ have length at least
$h(T)-t(p)-2\delta$ and $h(T)-t(q)-2\delta$ respectively. Each of these is
greater than $C_{1}-2\delta$.
If $p_{X}\in[a_{X},b_{X}]$ then $C_{1}-2\delta\leq
2\delta+2C_{0}+2R/\epsilon+R$ and this is a contradiction. Similarly, deduce
$q_{X}\not\in[a_{X},b_{X}]$. If $p_{X}=q_{X}$ then $d_{X}(F(p),F(q))\leq
2\delta<\Delta$, contradicting our assumption that $t(p)<h(T)$. Finally, if
$p_{X}\in(b_{X},q_{X})$ then an intermediate value argument using the fact
that $\mathcal{R}_{X}$ is a geodesic implies $d_{X}(F(p),F(\mathcal{Q}))\leq
3\delta$, again a contradiction. Similarly $q_{X}$ is not in $(p_{X},a_{X})$.
Thus, $[p_{X},a_{X}]\cap[b_{X},q_{X}]$ is either empty or is equal to
$[a_{X},b_{X}]$. We deduce that
$\displaystyle d_{X}(p_{X},q_{X})$ $\displaystyle\geq
2h(T)-t(p)-t(q)-4\delta-2\delta-2C_{0}-2R/\epsilon-R$ $\displaystyle\geq
2t(r)-t(p)-t(q)-7\Delta-4C_{0}.$
The proof of Lemma 2.2 is complete. ∎
## 3 Foliations and projections
Let $Z$ be a closed surface of genus at least $2$ and ${\bf z}$ a set of
marked points. A measured singular foliation $\mathcal{F}$ on $(Z,{\bf z})$ is
a singular topological foliation so that
* •
$\mathcal{F}$ has only prong-type singularties,
* •
all one-prong singularties of $\mathcal{F}$ appear at points of ${\bf z}$, and
* •
$\mathcal{F}$ is equipped with a transverse measure of full support.
We refer the reader to [8, 20] for a detailed discussion of measured
foliations. Two measured (respectively, topological) foliations are measure
equivalent (respectively, topologically equivalent) if they differ by isotopy
and Whitehead moves. We will only be concerned with those foliations which
appear as the vertical foliation for some meromorphic quadratic differential
on $Z$ (see Section 4.1). Every measured singular foliation is measure
equivalent to such a foliation for a fixed complex structure on $Z$; see [13].
The space of measure classes of measured foliation on $(Z,{\bf z})$ is denoted
by $\mathcal{MF}(Z,{\bf z})$ and its projectivization by
$\mathbb{P}\mathcal{MF}(Z,{\bf z})$. A measured foliation
$\mathcal{F}\in\mathcal{MF}(Z,{\bf z})$ is arational if it has no closed leaf
cycles. We say that $\mathcal{F}$ is uniquely ergodic if whenever
$\mathcal{F}^{\prime}\in\mathcal{MF}(Z,{\bf z})$ is topologically equivalent
to $\mathcal{F}$, then $\mathcal{F}$ and $\mathcal{F}^{\prime}$ project to the
same point in $\mathbb{P}\mathcal{MF}(Z,{\bf z})$. Both of these notions
depend only on the topological classes of the foliations, and not the
transverse measures.
If $\mathcal{F}$ is a measured foliation representing an element of
$\mathcal{MF}(Z)$, and ${\bf z}\subset Z$ is a set of marked points, then
$\mathcal{F}$ also determines an element of $\mathcal{MF}(Z,{\bf z})$. We note
that it is important in this case that $\mathcal{F}$ be a foliation, and not
an equivalence class of foliations. If $\mathcal{F}$ is arational as an
element of $\mathcal{MF}(Z)$, and if ${\bf z}=\\{z\\}$ is a single point, then
$\mathcal{F}$ is also arational as an element of $\mathcal{MF}(Z,z)$; see
[19].
By a strict subsurface $Y\subset Z-{\bf z}$ we mean a properly embedded
surface with nonempty boundary and a set of punctures, possibly empty, such
that every component of $\partial Y$ is an essential curve in $Z-{\bf z}$;
that is, homotopically nontrivial and nonperipheral. We also assume that $Y$
is not a sphere with $k$ punctures and $j$ boundary components where $k+j=3$.
We will only refer to subsurfaces in one context, and that is as follows.
Given a pair of arational measured foliation
$\mathcal{F},\mathcal{G}\in\mathcal{MF}(Z,{\bf z})$ and a proper subsurface
$Y\subset Z-{\bf z}$, we have the projection distance
$d_{Y}(\mathcal{F},\mathcal{G})\in\mathbb{Z}_{\geq 0}$
between $\mathcal{F}$ and $\mathcal{G}$ in $Y$. This is the distance in the
arc-and-curve complex of $Y$ between the the subsurface projections of
$\mathcal{F}$ and $\mathcal{G}$ to $Y$. For a detailed discussion, see [23,
24]. All we use is that $d_{Y}$ satisfies a triangle inequality
$d_{Y}(\mathcal{F}_{1},\mathcal{F}_{2})\leq
d_{Y}(\mathcal{F}_{1},\mathcal{G})+d_{Y}(\mathcal{G},\mathcal{F}_{2})$
for all arational measured foliations
$\mathcal{F}_{1},\mathcal{F}_{2},\mathcal{G}\in\mathcal{MF}(Z,{\bf z})$. This
relates to Teichmüller geometry by Theorem 4.2 below.
## 4 Teichmüller spaces
Here we set notation and recall some basic properties of Teichmüller space.
For background on Teichmüller space, we refer the reader to any of [2, 10, 1,
14].
### 4.1 Teichmüller space, quadratic differentials and geodesics
Given a closed Riemann surface $Z$ with a finite (possibly empty) set of
marked points ${\bf z}\subset Z$, let $\mathcal{T}(Z,{\bf z})$ denote the
Teichmüller space of equivalence classes of marked Riemann surfaces
$\mathcal{T}(Z,{\bf z})=\mathopen{}\mathclose{{}\left\\{[f\colon(Z,{\bf
z})\to(X,{\bf x})]\,\mathopen{}\mathclose{{}\left|\,\begin{array}[]{l}f\mbox{
is an orientation preserving homeo-}\\\ \mbox{morphism to the Riemann surface
}X\end{array}}\right.}\right\\}.$
The equivalence relation is defined by
$\big{(}f\colon(Z,{\bf z})\to(X,{\bf x})\big{)}\sim\big{(}g\colon(Z,{\bf
z})\to(Y,{\bf y})\big{)}$
if $f\circ g^{-1}\colon(Y,{\bf y})\to(X,{\bf x})$ is isotopic (rel marked
points) to a conformal map. If ${\bf z}=\emptyset$, then we write
$\mathcal{T}(Z)=\\{[f\colon Z\to X]\\}$.
The Teichmüller distance on $\mathcal{T}(Z,{\bf z})$ is defined by
$d_{\mathcal{T}}\big{(}[f\colon(Z,{\bf z})\to(X,{\bf x})],[g\colon(Z,{\bf
z})\to(Y,{\bf
y})]\big{)}=\inf\mathopen{}\mathclose{{}\left\\{\mathopen{}\mathclose{{}\left.\frac{1}{2}\log\mathopen{}\mathclose{{}\left(K_{h}}\right)\,}\right|\,h\simeq
f\circ g^{-1}}\right\\}$
where $K_{h}$ is the dilatation of $h$ and where $h\colon(Y,{\bf y})\to(X,{\bf
x})$ ranges over all quasi-conformal maps isotopic (rel marked points) to
$f\circ g^{-1}$.
Given $\epsilon>0$, the $\epsilon$–thick part of Teichmüller space
$\mathcal{T}_{\epsilon}(Z,{\bf z})\subset\mathcal{T}(Z,{\bf z})$ is the set of
points $[f\colon(Z,{\bf z})\to(X,{\bf x})]\in\mathcal{T}(Z,{\bf z})$ where the
unique complete hyperbolic surface in the conformal class of $X-{\bf x}$ has
its shortest geodesic of length at least $\epsilon$. When $\epsilon$ is
understood from context we will simply refer to $\mathcal{T}_{\epsilon}(Z,{\bf
z})$ as the the thick part of Teichmüller space.
Let $\mathcal{T}(Z,{\bf z})\to\mathcal{M}(Z,{\bf z})$ denote the projection to
moduli space obtained by forgetting the marking
$[f\colon(Z,{\bf z})\to(X,{\bf x})]\mapsto[(X,{\bf x})]$
or, equivalently, by taking the quotient by the mapping class group. Mumford’s
compactness criterion [3] now implies: For any $\epsilon>0$, the thick part
$\mathcal{T}_{\epsilon}(Z,{\bf z})$ projects to a compact subset of
$\mathcal{M}(Z,{\bf z})$. Conversely, the preimage of any compact subset of
$\mathcal{M}(Z,{\bf z})$ is contained in $\mathcal{T}_{\epsilon}(Z,{\bf z})$
for some $\epsilon>0$.
Suppose $(X,{\bf x})$ is a closed Riemann surface with marked points and
$q\in\mathcal{Q}(X,{\bf x})$ is a unit norm, meromorphic quadratic
differential with all poles simple and contained in ${\bf x}$. We also use $q$
to denote the associated Euclidean cone metric on $X$. We note that
$\mathcal{Q}(X)\subset\mathcal{Q}(X,{\bf x})$, for any set of marked point
${\bf x}\subset X$. Given $q\in\mathcal{Q}(X)$ we view it as an element of
$\mathcal{Q}(X,{\bf x})$ whenever it is convenient.
Given $q\in\mathcal{Q}(X,{\bf x})$ and $t\in\mathbb{R}$, let
$g_{t}\colon(X,{\bf x})\to(X_{t},g_{t}({\bf x}))$ denote the $e^{2t}$–quasi-
conformal Teichmüller mapping defined by $(q,t)$. Let
$q_{t}\in\mathcal{Q}(X_{t},g_{t}({\bf x}))$ denote the terminal quadratic
differential. For any point $p\in X$ which is not a zero or pole of $q$ we
have a preferred coordinate $z_{0}$ for $(X,q)$ near $p$ and preferred
coordinate $z_{t}$ for $(X_{t},q_{t})$ near $g_{t}(p)$. In these coordinates
$q=dz_{0}^{2}$ and $q_{t}=dz_{t}^{2}$, and $g_{t}$ is given by
$(u,v)\mapsto(e^{t}u,e^{-t}v)$. If we mark $(X,{\bf x})$ by $f\colon(Z,{\bf
z})\to(X,{\bf x})$, then setting $f_{t}=g_{t}\circ f$ we have
$\tau_{q}(t)=[f_{t}\colon(Z,{\bf z})\to(X_{t},g_{t}({\bf x}))]$
being a Teichmüller geodesic through $[f\colon(Z,{\bf z})\to(X,{\bf x})]$;
note that every Teichmüller geodesic can be described in this way. The
Teichmüller geodesic $\tau$ is $\epsilon$–thick if the image of $\tau$ lies in
$\mathcal{T}_{\epsilon}(Z,{\bf z})$. We also simply say a geodesic $\tau$ is
thick if it is $\epsilon$–thick for some $\epsilon>0$. A collection of
geodesics $\\{\tau_{\alpha}\\}$ is uniformly thick if there is an $\epsilon>0$
so that each $\tau_{\alpha}$ is $\epsilon$–thick.
Given $q\in\mathcal{Q}(X,{\bf x})$ we will let $\mathcal{F}(q),\mathcal{G}(q)$
denote the vertical and horizontal foliations respectively; that is, the
preimage in preferred coordinates of the foliations of $\mathbb{C}$ by
vertical and horizontal lines. For $q\in\mathcal{Q}(X,{\bf x})$ and
$t\in\mathbb{R}$ consider the associated Teichmüller mapping
$g_{t}\colon(X,{\bf x})\to(X_{t},g_{t}({\bf x}))$ as above. If
$c\colon\mathbb{R}\to X$ is a nonsingular leaf of $\mathcal{F}(q)$
parameterized by arc-length with respect to the $q$–metric, then composing
with $g_{t}$ we obtain a nonsingular leaf of the vertical foliation for the
terminal quadratic differential $\mathcal{F}(q_{t})$,
$g_{t}\circ c\colon\mathbb{R}\to X_{t}.$
From the description of $g_{t}$ in local coordinates we see that this is
parameterized proportional to arc-length and, in fact, the $q_{t}$–length is
given by
$\ell_{q_{t}}\mathopen{}\mathclose{{}\left(g_{t}\circ
c|_{[x,x^{\prime}]}}\right)=e^{-t}|x^{\prime}-x|.$ (3)
### 4.2 Properties of Teichmüller geodesics
Suppose $\tau=\tau_{q}$ is the Teichmüller geodesic determined by
$[f\colon(Z,{\bf z})\to(X,{\bf x})]\in\mathcal{T}(Z,{\bf z})$ and
$q\in\mathcal{Q}(X,{\bf x})$. The forward asymptotic behavior of $\tau$ is
reflected in the structure of the vertical foliation $\mathcal{F}(q)$. For us,
the most important instance of this is a result of Masur [22].
###### Theorem 4.1 (Masur).
If there exists $\epsilon>0$ and $\\{t_{k}\\}_{k=1}^{\infty}$ such that
* •
$t_{k}\to\infty$ as $k\to\infty$ and
* •
$\tau_{q}(t_{k})\in\mathcal{T}_{\epsilon}(Z,{\bf z})$ for all $k$
then $\mathcal{F}(q)$ is arational and uniquely ergodic.
In particular, if $\tau_{q}$ is thick then both $\mathcal{F}(q)$ and
$\mathcal{G}(q)$ are uniquely ergodic. We say a pair of arational foliations
$\mathcal{F}$ and $\mathcal{G}$ are $K$–cobounded if for all strict
subsurfaces $Y\subset X-{\bf x}$ we have $d_{Y}(\mathcal{F},\mathcal{G})\leq
K$. A result of Rafi [26, Theorem 1.5] relates the thickness of a geodsic
$\tau_{q}\subset\mathcal{T}$ to the coboundedness of the associated vertical
and horizontal foliations.
###### Theorem 4.2 (Rafi).
For all $\epsilon>0$ there exists $K>0$ so that if $q\in\mathcal{Q}(X,{\bf
x})$ has $\tau_{q}$ being $\epsilon$–thick then $\mathcal{F}(q)$ and
$\mathcal{G}(q)$ are $K$–cobounded.
Conversely, for all $K>0$ there exists $\epsilon>0$ so that if
$q\in\mathcal{Q}(X,{\bf x})$ has $\mathcal{F}(q)$ and $\mathcal{G}(q)$ being
$K$–cobounded then $\tau_{q}$ is $\epsilon$–thick.
### 4.3 Forgetting the marked point: the Bers fibration
Suppose now that $Z$ is a closed surface and $z\in Z$ is a single marked
point; we use $(Z,z)$ to denote $(Z,\\{z\\})$. Let $p\colon\widetilde{Z}\to Z$
denote the universal covering. Given $[f\colon(Z,z)\to(X,f(z))]$ we can forget
the marked point to obtain an element $[f\colon Z\to X]\in\mathcal{T}(Z)$.
This defines a holomorphic map
$\Pi\colon\mathcal{T}(Z,z)\to\mathcal{T}(Z)$
called the Bers fibration [4]. The fiber of this map over $[f\colon Z\to X]$
is holomorphically identified with $\widetilde{X}$, the universal covering of
$X$. Moreover, this identification is canonical, up to the action of the
covering group on $\widetilde{X}$.
The projection of Teichmüller spaces
$\Pi\colon\mathcal{T}(Z,z)\to\mathcal{T}(Z)$ descends to a projection of
moduli spaces $\hat{\Pi}\colon\mathcal{M}(Z,z)\to\mathcal{M}(Z)$. The fiber of
$\hat{\Pi}$ over $X\in\mathcal{M}(Z)$ is just $X/\mathop{\rm Aut}(X)$ and this
is compact.
Recall that puncturing a closed surface once increases the hyperbolic systole.
(Lift to universal covers and apply the Schwarz-Pick lemma.) It follows that
the preimage of $\mathcal{T}_{\epsilon}(Z)$ by $\Pi^{-1}$ is contained in
$\mathcal{T}_{\epsilon}(Z,z)$.
By a theorem of Royden [28] the Teichmüller metric agrees with the Kobayashi
metric on Teichmüller space. Recall that the inclusion of the universal
covering $\widetilde{X}\to\mathcal{T}(Z,z)$ is a holomorphic embedding [4].
Thus, if we give $\widetilde{X}$ the Poincaré metric $\rho_{0}$ — one-half the
hyperbolic metric — then
$(\widetilde{X},\rho_{0})\to(\mathcal{T}(Z,z),d_{\mathcal{T}})$ is a
contraction [16]. Kra [17] further proved the following.
###### Theorem 4.3 (Kra).
There exists a homeomorphism $h\colon[0,\infty)\to[0,\infty)$ so that for any
$[f\colon Z\to X]\in\mathcal{T}(Z)$, and any
$\tilde{x}_{1},\tilde{x}_{2}\in\widetilde{X}\subset\mathcal{T}(Z,z)$, we have
$h(\rho_{0}(\tilde{x}_{1},\tilde{x}_{1}))\leq
d_{\mathcal{T}}(\tilde{x}_{1},\tilde{x}_{2})\leq\rho_{0}(\tilde{x}_{1},\tilde{x}_{2}).$
The function $h$ can be described concretely in terms of the solution to a
certain extremal mapping problem for the hyperbolic plane which was solved by
Teichmüller [29] and Gehring [11]. We will extend $h$ to a nondecreasing
function, $h\colon\mathbb{R}\to[0,\infty)$ by declaring $h(t)=0$ for all
$t\leq 0$.
### 4.4 Branched covers
Here we use branched covers to induce maps on Teichmüller space.
Suppose $P\colon\Sigma\to Z$ is a branched cover, branched over some finite
set of points ${\bf z}\subset Z$. Then any complex structure on $Z$ pulls back
to a complex structure on $\Sigma$, and thus induces a map
$P^{*}\colon\mathcal{T}(Z,{\bf z})\to\mathcal{T}(\Sigma)$. Regarding
Teichmüller space as the space of marked Riemann surfaces, $\mathcal{T}(Z,{\bf
z})=\\{[f\colon(Z,{\bf z})\to(X,{\bf x})]\\}$, the embedding is described as
follows. The branched covering $P\colon\Sigma\to(Z,{\bf z})$ induces a
branched covering $U\colon\Omega\to(X,{\bf x})$, for some Riemann surface
$\Omega$, namely the branched cover induced by the subgroup $(f\circ
P)_{*}(\pi_{1}(\Sigma-P^{-1}({\bf z})))<\pi_{1}(X-{\bf x})$. By construction,
there is a lift of the marking homeomorphism $\phi\colon\Sigma\to\Omega$. This
is described by the following commutative diagram.
$\textstyle{\Sigma\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P}$$\scriptstyle{\phi}$$\textstyle{\Omega\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U}$$\textstyle{(Z,z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{(X,x).}$
Then, we have
$P^{*}([f\colon(Z,{\bf z})\to(X,{\bf x})])=[\phi\colon\Sigma\to\Omega].$
We now give a well-known consequence of these definitions.
###### Proposition 4.4.
If $P\colon\Sigma\to Z$ is nontrivially branched at every point of
$P^{-1}({\bf z})$, then $P^{*}\colon\mathcal{T}(Z,{\bf
z})\to\mathcal{T}(\Sigma)$ is an isometric embedding. Moreover, for all
$\epsilon>0$ there exists $\epsilon^{\prime}>0$ so that
$P^{*}(\mathcal{T}_{\epsilon}(Z,{\bf
z}))\subset\mathcal{T}_{\epsilon^{\prime}}(\Sigma)$.
###### Proof.
When $P$ is a covering then $P^{*}$ is an isometric embedding; see [27,
Section 7]. The proof is identical in the presence of nontrivial branching, as
a one-prong singularity at a point of ${\bf z}$ lifts to a regular point or to
a three-prong or higher singularity.
Let $\widetilde{\mathcal{M}}(Z,{\bf z})$ be the quotient of
$\mathcal{T}(Z,{\bf z})$ by the group of mapping classes of $(Z,{\bf z})$ that
lift to $\Sigma$. Note that $\widetilde{\mathcal{M}}(Z,{\bf
z})\to\mathcal{M}(Z,{\bf z})$ is a finite sheeted (orbifold) covering. The
embedding $P^{*}\colon\mathcal{T}(Z,{\bf z})\to\mathcal{T}(\Sigma)$ descends
to a map $\widetilde{\mathcal{M}}(Z,{\bf z})\to\mathcal{M}(\Sigma)$, giving a
commutative square.
$\textstyle{\mathcal{T}(Z,{\bf
z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P^{*}}$$\textstyle{\mathcal{T}(\Sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\mathcal{M}}(Z,{\bf
z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{M}(\Sigma)}$
By Mumford’s compactness criteria [3], the image of
$\mathcal{T}_{\epsilon}(Z,{\bf z})$ in $\widetilde{\mathcal{M}}(Z,{\bf z})$ is
compact, and hence so is the image in $\mathcal{M}(\Sigma)$. Appealing to
Mumford’s criteria again (for $\mathcal{M}(\Sigma)$), it follows that for some
$\epsilon^{\prime}>0$ we have $P^{*}(\mathcal{T}_{\epsilon}(Z,{\bf
z}))\subset\mathcal{T}_{\epsilon^{\prime}}(\Sigma)$. ∎
In general, for any branched cover $P\colon\Sigma\to Z$, branched over ${\bf
z}\subset Z$, consider ${\bf\sigma}=P^{-1}({\bf z})$ as a set of marked points
on $\Sigma$. Then again there is an isometric embedding
$P^{*}\colon\mathcal{T}(Z,{\bf z})\to\mathcal{T}(\Sigma,{\bf\sigma}).$
If ${\bf\omega}\subset{\bf\sigma}$ then define
$\Pi_{\omega}\colon\mathcal{T}(\Sigma,{\bf\sigma})\to\mathcal{T}(\Sigma,{\bf\omega})$
by forgetting the points of ${\bf\sigma}$ not in ${\bf\omega}$. When
${\bf\omega}$ is empty we may omit the subscript. In this notation, the
composition $\Pi\circ P^{*}$ gives the map of Proposition 4.4. So, if $P$ is
non-trivially branched at all points of ${\bf\sigma}$ then $\Pi\circ P^{*}$ is
an isometric embedding. If $P$ is not branched at all points of ${\bf\sigma}$
then $\Pi\circ P^{*}$ fails to be an isometric embedding; however it remains
$1$–Lipschitz.
###### Proposition 4.5.
If $P\colon\Sigma\to Z$ is branched over ${\bf z}$ and if
${\bf\omega}\subset{\bf\sigma}=P^{-1}({\bf z})$ is any subset then
$\Pi_{\omega}\circ P^{*}\colon\mathcal{T}(Z,{\bf
z})\to\mathcal{T}(\Sigma,{\bf\omega})$
is $1$–Lipschitz.
###### Proof.
The Bers fibration is a holomorphic map [4] and, by forgetting the points of
${\bf\sigma}-{\bf\omega}$ one at a time, we see that
$\Pi_{\bf\omega}\colon\mathcal{T}(\Sigma,{\bf\sigma})\to\mathcal{T}(\Sigma,{\bf\omega})$
is a composition of holomorphic maps, hence holomorphic. In particular,
because the Teichmüller metric agrees with the Kobayashi metric [28], it
follows that $\Pi_{\bf\omega}$ is $1$–Lipschitz [16]. Since $P^{*}$ is an
isometric embedding, the composition is $1$–Lipschitz. ∎
## 5 An inductive construction
The proof of Theorem 1.1 is constructive, but also appeals to an inductive
procedure. We begin by constructing the required embedding of $\mathbb{H}^{2}$
into some Teichmüller space as the base case of the induction, then produce an
embedding of $\mathbb{H}^{3}$ into some other Teichmüller space, then an
embedding of $\mathbb{H}^{4}$, and so on. All the main ideas and technical
difficulties are present in the construction of the embedding of
$\mathbb{H}^{2}$ and then the embedding of $\mathbb{H}^{3}$ from that of
$\mathbb{H}^{2}$. The only further complications which arise to describe the
embedding of $\mathbb{H}^{n}$ from $\mathbb{H}^{n-1}$ for $n\geq 4$ are in the
notation, which becomes increasingly messy as $n$ increases. This is due to
the fact that the proof for $n$ really depends on the proof for all $2\leq
k<n$ (rather than just $n-1$). For this reason, we carefully describe the
cases $n=2$ and $n=3$, and sketch the general inductive step indicating only
those things that require modification.
### 5.1 The hyperbolic plane case
Let $Z$ be a closed hyperbolic surface. Let $q\in\mathcal{Q}(Z)$ be a nonzero
holomorphic quadratic differential on $Z$ so that the associated Teichmüller
geodesic $[g_{t}\colon Z\to Z_{t}]$ is thick. Write
$\mathcal{F}=\mathcal{F}(q)$ and $\mathcal{G}=\mathcal{G}(q)$ for the vertical
and horizontal foliations of $q$, respectively. Next, let
$c\colon\mathbb{R}\to Z$ be a nonsingular leaf of $\mathcal{F}$ parameterized
by arc-length with respect to $q$ and let $z=c(0)$ be a marked point on $Z$;
see Section 4.
Our goal is to construct an almost-isometric embedding
${\bf Z}\colon\mathbb{H}^{2}\to\mathcal{T}(Z,z).$
We consider an isotopy $Z\times\mathbb{R}\to Z$, written $(w,x)\mapsto
f^{x}(w)$, where $f^{x}\colon Z\to Z$ is a homeomorphism for all
$x\in\mathbb{R}$, $f^{0}$ is the identity and $f^{x}(z)=c(x)$ for all
$x\in\mathbb{R}$. We further assume that $f^{x}$ preserves $\mathcal{F}$ for
all $x\in\mathbb{R}$.
We can construct such an isotopy by piecing together isotopies defined on
small balls. More precisely, we start with some $\epsilon$–ball around $z$,
and construct a vector field tangent to $\mathcal{F}$ supported in the ball
with with norm identically equal to $1$ on the $\epsilon/2$ ball. The flow for
time $t\in(-\epsilon/2,\epsilon/2)$ is an isotopy of the correct form. Now we
repeat this for a ball around $c(\epsilon/2)$. Since the arc of $c$ from $z$
to any point $c(x)$ is compact, we can cover it with finitely many such balls
to produce the required isotopy.
We think of the isotopy as “pushing $z$ along $c$”. This determines the
horocyclic coordinate
$\widetilde{c}\colon\mathbb{R}\to\mathcal{T}(Z,z)$
given by
$\widetilde{c}(x)=[f^{x}\colon(Z,z)\to(Z,c(x))].$
The image of $\widetilde{c}$ lies in the Bers fiber over the basepoint
$[\mathop{\rm Id}\colon Z\to Z]\in\mathcal{T}(Z)$; the fiber is identified
with the universal cover $\widetilde{Z}$ of $Z$. As such, we can identify
$\widetilde{c}$ with a lift of $c$ to $\widetilde{Z}$ and write
$\widetilde{c}\colon\mathbb{R}\to\widetilde{Z}\subset\mathcal{T}(Z,z).$
Applying the Teichmüller mapping $g_{t}\colon Z\to Z_{t}$ determined by $q$
and $t\in\mathbb{R}$ gives the height coordinate. These coordinates together
define ${\bf Z}\colon\mathbb{H}^{2}\to\mathcal{T}(Z,z)$ where
${\bf Z}(x,t)=[g_{t}\circ f^{x}\colon(Z,z)\to(Z_{t},g_{t}(c(x)))].$
Here we are using the coordinates $(x,t)$ on $\mathbb{H}^{2}$ described in
Section 2.
Since the marking homeomorphisms are determined by $x$ and $t$, we simplify
notation and denote the values in Teichmüller space by
${\bf Z}(x,t)=\widetilde{c}_{t}(x)=(Z_{t},g_{t}(c(x))).$ (4)
We also write
${\bf Z}(x,0)=\widetilde{c}(x)=(Z,c(x)).$
As the notation suggests,
$\widetilde{c}_{t}\colon\mathbb{R}\to\widetilde{Z}_{t}\subset\mathcal{T}(Z,z)$
is a lift of $g_{t}\circ c\colon\mathbb{R}\to Z_{t}$ to the universal cover
$\widetilde{Z}_{t}$, thought of as the fiber over $[g_{t}\colon Z\to Z_{t}]$.
###### Theorem 5.1.
The map ${\bf Z}\colon\mathbb{H}^{2}\to\mathcal{T}(Z,z)$ is an almost-
isometric embedding. Moreover, the image lies in the thick part and is quasi-
convex.
###### Proof.
We verify the hypothesis of Lemma 2.2 to prove that ${\bf Z}$ is an almost-
isometric embedding and, along the way, prove that the image is quasi-convex
and lies in the thick part.
First, fix any $x\in\mathbb{R}$ so that $\eta_{x}(t)=(x,t)$ is a vertical
geodesic in $\mathbb{H}^{2}$. Then $t\mapsto{\bf Z}\circ\eta_{x}(t)={\bf
Z}(x,t)=(Z_{t},g_{t}(c(x)))$ is a Teichmüller geodesic, and hence Property 1
of Lemma 2.2 holds. Furthermore, since $t\mapsto Z_{t}$ is a thick geodesic,
we see that $\\{{\bf Z}\circ\eta_{x}(t)\\}_{x\in\mathbb{R}}$ are uniformly
thick geodesics. That is, that the union of these geodesics, over all
$x\in\mathbb{R}$, project into a compact subset of $\mathcal{M}(Z,z)$; namely,
the preimage of the compact subset of $\mathcal{M}(Z)$ containing the image of
$t\mapsto Z_{t}$ (see Section 4.3). In particular, the image of ${\bf Z}$ lies
in the thick part of $\mathcal{T}(Z,z)$.
For each $x\in\mathbb{R}$, the geodesic ${\bf Z}\circ\eta_{x}$ is defined by
the quadratic differential $q\in\mathcal{Q}(Z)$ viewed as a quadratic
differential in $\mathcal{Q}(Z,c(x))$. We denote the vertical and horizontal
foliations of $q\in\mathcal{Q}(Z,c(x))$ by $\mathcal{F}^{x}$ and
$\mathcal{G}^{x}$, respectively, and consider them as measured foliations in
$\mathcal{MF}(Z,z)$ by pulling them back via $f^{x}$. Since $f^{x}$ preserves
$\mathcal{F}$, it follows that
$\mathcal{F}^{x}=\mathcal{F}^{0}\in\mathcal{MF}(Z,z)$ for all
$x\in\mathbb{R}$.
Now, since $t\mapsto Z_{t}$ is a thick geodesic, by Theorem 4.1 the foliations
$\mathcal{F}$ and $\mathcal{G}$ are arational. Puncturing an arational
foliation once gives an arational foliation in the punctured surface. Hence
$\mathcal{F}^{x}$ and $\mathcal{G}^{x}$ are also arational for all $x$. Since
$\mathcal{F}^{x}=\mathcal{F}^{0}$ for all $x\in\mathbb{R}$ and since the
geodesics $\\{{\bf Z}\circ\eta_{x}\\}_{x\in\mathbb{R}}$ are uniformly thick,
Theorem 4.2 implies that there exists $K>0$ so that the pairs
$(\mathcal{F}^{0},\mathcal{G}^{x})=(\mathcal{F}^{x},\mathcal{G}^{x})$ are
$K$–cobounded for all $x$. By the triangle inequality (applied to each
subsurface $Y$) we see that for all $x,x^{\prime}\in\mathbb{R}$ the pair
$(\mathcal{G}^{x},\mathcal{G}^{x^{\prime}})$ is $2K$–cobounded (to see that
$\mathcal{G}^{x}$ and $\mathcal{G}^{x^{\prime}}$ are different foliations,
note that $(\mathcal{F}^{0},\mathcal{G}^{x})$ and
$(\mathcal{F}^{0},\mathcal{G}^{x^{\prime}})$ define different geodesics ${\bf
Z}\circ\eta_{x}$ and ${\bf Z}\circ\eta_{x^{\prime}}$, respectively).
Appealing to the other direction in Theorem 4.2 the geodesic
$\Gamma^{x,x^{\prime}}$, determined by $\mathcal{G}^{x}$ and
$\mathcal{G}^{x^{\prime}}$ for distinct $x,x^{\prime}\in\mathbb{R}$, is
uniformly thick, independent of $x$ and $x^{\prime}$. From this and [15,
Theorem 4.4] it follows that there is a $\delta>0$ so that ${\bf
Z}\circ\eta_{x}$, ${\bf Z}\circ\eta_{x^{\prime}}$ and $\Gamma^{x,x^{\prime}}$
are the sides of a $\delta$–slim triangle for every pair of distinct points
$x,x^{\prime}\in\mathbb{R}$, and hence Property 2 of Lemma 2.2 holds. From
this, it follows that ${\bf Z}(\mathbb{H}^{2})$ (is contained in and) has
Hausdorff distance at most $\delta$ from the union of the geodesics
${\bf Z}(\mathbb{H}^{2})\cup\mathopen{}\mathclose{{}\left(\bigcup_{x\neq
x^{\prime}\in\mathbb{R}}\Gamma^{x,x^{\prime}}}\right)=\mathopen{}\mathclose{{}\left(\bigcup_{x\in\mathbb{R}}{\bf
Z}\circ\eta_{x}}\right)\cup\mathopen{}\mathclose{{}\left(\bigcup_{x\neq
x^{\prime}\in\mathbb{R}}\Gamma^{x,x^{\prime}}}\right).$
This is precisely the weak hull of
$\\{\mathcal{G}^{x}\\}_{x\in\mathbb{R}}\cup\\{\mathcal{F}^{0}\\}\subset\mathbb{P}\mathcal{MF}(Z,z)$,
and so according to [15, Theorem 4.5], this set, hence also ${\bf
Z}(\mathbb{H}^{2})$, is quasi-convex (the assumption in [15] that the subset
of $\mathbb{P}\mathcal{MF}(Z)$ be closed was not used in the proof).
Finally, we must prove that Properties 3 and 4 of Lemma 2.2 hold. For this we
can appeal directly to Theorem 4.3. More precisely, observe that because
$\\{Z_{t}\\}_{t\in\mathbb{R}}$ lies in the thick part, the pull-back of the
flat metric on $\widetilde{Z}_{t}$ (which we also denote $q_{t}$) is uniformly
quasi-isometric to the Poincaré metric $\rho_{0}$ on $\widetilde{Z}_{t}$. That
is, there exist constants $A,B\geq 0$ so that
$\frac{1}{A}\mathopen{}\mathclose{{}\left(d_{q_{t}}(\widetilde{z},\widetilde{z}^{\prime})-B}\right)\leq\rho_{0}(\widetilde{z},\widetilde{z}^{\prime})\leq
A\,d_{q_{t}}(\widetilde{z},\widetilde{z}^{\prime})+B$ (5)
for all $t\in\mathbb{R}$ and $\widetilde{z},\widetilde{z}^{\prime}\in Z_{t}$
(see for example [7, Lemma 2.2]).
Applying (4), the upper bound of Theorem 4.3, (5) and (3), in that order, we
find
$\displaystyle d_{\mathcal{T}}\mathopen{}\mathclose{{}\left({\bf Z}(x,t),{\bf
Z}(x^{\prime},t)}\right)$
$\displaystyle=d_{\mathcal{T}}(\widetilde{c}_{t}(x),\widetilde{c}_{t}(x^{\prime}))$
$\displaystyle\leq\rho_{0}(\widetilde{c}_{t}(x),\widetilde{c}_{t}(x^{\prime}))$
$\displaystyle\leq
A\,d_{q_{t}}(\widetilde{c}_{t}(x),\widetilde{c}_{t}(x^{\prime}))+B$
$\displaystyle=Ae^{-t}|x^{\prime}-x|+B.$
So, setting $\epsilon=1$ and $R=A+B$, Property 3 of Lemma 2.2 holds.
On the other hand, (4), the lower bound of Theorem 4.3, monotonicity of $h$,
and (3) gives
$\displaystyle d_{\mathcal{T}}\mathopen{}\mathclose{{}\left({\bf Z}(x,t),{\bf
Z}(x^{\prime},t)}\right)$
$\displaystyle=d_{\mathcal{T}}(\widetilde{c}_{t}(x),\widetilde{c}_{t}(x^{\prime}))$
$\displaystyle\geq
h(\rho_{0}(\widetilde{c}_{t}(x),\widetilde{c}_{t}(x^{\prime})))$
$\displaystyle\geq
h\mathopen{}\mathclose{{}\left(\frac{1}{A}\mathopen{}\mathclose{{}\left(d_{q_{t}}(\widetilde{c}_{t}(x),\widetilde{c}_{t}(x^{\prime}))-B}\right)}\right)$
$\displaystyle=h\mathopen{}\mathclose{{}\left(\frac{1}{A}(e^{-t}|x^{\prime}-x|-B)}\right).$
From this, and because $h$ is a homeomorphism on $[0,\infty)$ and hence
proper, Property 4 of Lemma 2.2 also holds. This completes the proof of
Theorem 5.1. ∎
### 5.2 Hyperbolic $3$–space
Before diving into the construction, we explain the basic idea. Our embedding
of the hyperbolic plane in Section 5.1 sends $(x,t)$ to ${\bf
Z}(x,t)\in\mathcal{T}(Z,z)$ by pushing the marked point $z$ distance $x$ along
a leaf of the vertical foliation of a quadratic differential then travelling
distance $t$ along the Teichmüller flow. There is a simple extension of this
construction which produces a map of hyperbolic $3$–space into Teichmüller
space $\mathcal{T}(Z,\\{z,w\\})$. Take $z$ and $w$ to lie on distinct leaves
and send $(x,y,t)$ to the point of $\mathcal{T}(Z,\\{z,w\\})$ obtained by
pushing $z$ a distance $x$ along its leaf, pushing $w$ a distance $y$ along
its leaf, and applying the Teichmüller flow for time $t$.
The problem is that whenever $z$ and $w$ move close to each other on $Z$, the
corresponding point in $\mathcal{T}(Z,\\{z,w\\})$ is in the thin part of
Teichmüller space; if $z$ and $w$ are very close to each other then there is a
simple closed curve surrounding $z$ and $w$ having an annular neighborhood of
large modulus. This also shows that this map
$(x,y,t)\mapsto\mathcal{T}(Z,\\{z,w\\})$ is not a quasi-isometric embedding.
In fact the map is not even coarsely Lipschitz.
A more subtle construction is required. We first choose a branched cover
$P\colon\Sigma\to Z$, nontrivially branched at each point of $P^{-1}(z)$.
According to Proposition 4.4, this induces an isometric embedding of
$\mathcal{T}(Z,z)$ into $\mathcal{T}(\Sigma)$. Fix a suitably generic point
$w\in(Z,z)$ and pick a point $\sigma\in P^{-1}(w)$. Roughly, we map our three
parameters $(x,y,t)$ into $\mathcal{T}(\Sigma,\sigma)$ as follows. The
coordinates $(x,t)$ determine ${\bf Z}(x,t)\in\mathcal{T}(Z,z)$ as in Section
5.1. The map $P^{*}$ applied to ${\bf Z}(x,t)$ gives a point in
$\mathcal{T}(\Sigma)$ as in Section 4.4. Finally use $y$ to determine a point
${\bf\Sigma}(x,y,t)\in\mathcal{T}(\Sigma,\sigma)$, lying in the Bers fiber
above $P^{*}\circ{\bf Z}(x,t)$. On its face, this new construction avoids the
problem we had before. In $(Z,z)$ we have only one marked point; after taking
the branched covering over $z$ we forget all of the branch points over $z$.
The single image of $\sigma$ can now move freely enough so that we stay in the
thick part of $\mathcal{T}(\Sigma,\sigma)$. We now explain this construction
in more detail and prove that the resulting map has all the required
properties.
#### 5.2.1 The construction
The notation from Section 5.1 carries over to this section without change. Let
$P\colon\Sigma\to Z$ be a branched cover, branched over the marked point $z\in
Z$ so that $P$ is nontrivially branched at every point of $P^{-1}(z)$. This
determines an isometric embedding of Teichmüller spaces
$P^{*}\colon\mathcal{T}(Z,z)\to\mathcal{T}(\Sigma)$
by Proposition 4.4. We write
$P^{*}([g_{t}\circ
f^{x}\colon(Z,z)\to(Z_{t},g_{t}(c(x)))])=[\phi_{t}^{x}\colon\Sigma\to\Sigma_{t}^{x}]$
so that $\phi_{t}^{x}$ is a lift of the marking $g_{t}\circ f^{x}$, and
$P_{t}^{x}$ is the induced branched cover making the following commute:
$\textstyle{\Sigma\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P}$$\scriptstyle{\phi_{t}^{x}}$$\textstyle{\Sigma_{t}^{x}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P_{t}^{x}}$$\textstyle{(Z,z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{t}\circ
f^{x}}$$\textstyle{(Z_{t},g_{t}(c(x))).}$
The quadratic differentials $q_{t}$ pull back to quadratic differentials
$\lambda_{t}^{x}$ on $\Sigma_{t}^{x}$, and $g_{t}$ lifts to Teichmüller
mappings of the covers
$\psi_{t}^{x}\colon\Sigma_{0}^{x}\to\Sigma_{t}^{x}$
so that $t\mapsto\Sigma_{t}^{x}$ is a Teichmüller geodesic for all $x$. The
lifts satisfy $\phi_{t}^{x}=\psi_{t}^{x}\circ\phi_{0}^{x}$. We have another
commutative diagram which may be helpful in organizing all the maps:
$\textstyle{\Sigma\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P}$$\scriptstyle{\phi_{0}^{x}}$$\textstyle{\Sigma_{0}^{x}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{t}^{x}}$$\scriptstyle{P_{0}^{x}}$$\textstyle{\Sigma_{t}^{x}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P_{t}^{x}}$$\textstyle{(Z,z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{x}}$$\textstyle{(Z,c(x))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{t}}$$\textstyle{(Z_{t},g_{t}(c(x))).}$
Denote the vertical foliation for $\lambda_{t}^{x}$ by $\Phi_{t}^{x}$. Each
nonsingular leaf of $\Phi_{t}^{x}$ maps isometrically to a nonsingular leaf of
the vertical foliation $\mathcal{F}_{t}$ for $q_{t}$ via the branched covering
$\Sigma_{t}^{x}\to Z_{t}$ since $\lambda_{t}^{x}$ is the pull back of $q_{t}$.
Choose any nonsingular leaf
$\gamma_{0}^{0}\colon\mathbb{R}\to\Sigma_{0}^{0}=\Sigma$, parameterized by
arc-length. Observe that $\gamma^{0}_{0}$ maps isometrically by $P$ to a leaf
$\gamma\colon\mathbb{R}\to Z$ for $\mathcal{F}$. Note that $c$ and $\gamma$
are distinct leaves; the preimage of $c$ in $\Sigma$ consists entirely of
singular leaves, namely the leaves that meet the branch points of $P$.
As we vary $x$, we can continuously choose lifts of $\gamma$ to leaves
$\gamma_{0}^{x}\colon\mathbb{R}\to\Sigma_{0}^{x}$ which agrees with our
initial leaf $\gamma_{0}^{0}$ when $x=0$. Specifically, we define the lift to
be
$\gamma_{0}^{x}=\phi_{0}^{x}\circ\mathopen{}\mathclose{{}\left(P|_{\gamma_{0}^{0}(\mathbb{R})}}\right)^{-1}\circ(f^{x})^{-1}\circ\gamma.$
Composing with the lifts $\psi_{t}^{x}$, we obtain leaves
$\gamma_{t}^{x}=\psi_{t}^{x}\circ\gamma_{0}^{x}$. Observe that via the
branched covering $P_{t}^{x}\colon\Sigma_{t}^{x}\to Z_{t}$, $\gamma_{t}^{x}$
projects to the leaf $g_{t}\circ\gamma$, independent of $x$. Furthermore, this
shows that the $\lambda_{t}^{x}$–length of the arc
$\gamma_{t}^{x}([y,y^{\prime}])$ is the $q_{t}$–length of $g_{t}\circ\gamma$
which is $e^{-t}|y-y^{\prime}|$.
We pick a basepoint $\sigma=\gamma_{0}^{0}(0)\in\Sigma$, and consider the
surface $(\Sigma,\sigma)$, marked by the identity $\mathop{\rm
Id}=\phi_{0}^{0}$ as a point in $\mathcal{T}(\Sigma,\sigma)$. Just as we
constructed $f^{x}$ by pushing along $c$ to $c(x)$, we push $\sigma$ along
$\gamma_{t}^{x}$ to $\gamma_{t}^{x}(y)$ to obtain maps
$\xi_{t}^{x,y}\colon(\Sigma,\sigma)\to(\Sigma_{t}^{x},\gamma_{t}^{x}(y)).$
Specifically, we take $\xi_{0}^{x,y}$ to be the composition of $\phi_{0}^{x}$
and a map isotopic to the identity on $\Sigma_{0}^{x}$ which preserves the
foliation $\Phi_{0}^{x}$ and pushes $\phi_{0}^{x}(\sigma)$ along
$\gamma_{0}^{x}$ to $\gamma_{0}^{x}(y)$. Then
$\xi_{t}^{x,y}=\psi_{t}^{x}\circ\xi_{0}^{x,y}$ maps the foliation
$\Phi_{0}^{0}$ to $\Phi_{t}^{x}$.
We denote the associated point in Teichmüller space
$[\xi_{t}^{x,y}\colon(\Sigma,\sigma)\to(\Sigma_{t}^{x},\gamma_{t}^{x}(y))]\in\mathcal{T}(\Sigma,\sigma)$
simply by $(\Sigma_{t}^{x},\gamma_{t}^{x}(y))$ as this point is uniquely
determined in this construction by $(x,y,t)$.
We define
${\bf\Sigma}\colon\mathbb{H}^{3}\to\mathcal{T}(\Sigma,\sigma)$
in the coordinates $(x,y,t)$ for $\mathbb{H}^{3}$ from Section 2 by
${\bf\Sigma}(x,y,t)=(\Sigma_{t}^{x},\gamma_{t}^{x}(y)).$
#### 5.2.2 Fibration over $\mathbb{H}^{2}$ case
We also require a slightly different description of the map ${\bf\Sigma}$ to
take advantage of the construction in the $\mathbb{H}^{2}$ case. Observe that
$P^{*}\circ{\bf
Z}\colon\mathbb{H}^{2}\to\mathcal{T}(Z,z)\to\mathcal{T}(\Sigma)$ is an almost-
isometric embedding, and is given by
$P^{*}\mathopen{}\mathclose{{}\left({\bf Z}(x,t)}\right)=\Sigma_{t}^{x},$
where $\Sigma_{t}^{x}$ denotes the point
$[\phi_{t}^{x}\colon\Sigma\to\Sigma_{t}^{x}]$. Recall that
$\Pi\colon\mathcal{T}(\Sigma,\sigma)\to\mathcal{T}(\Sigma).$
is the Bers fibration. If we fix $(x,t)\in\mathbb{H}^{2}$, then for every $y$
we see that $(\Sigma_{t}^{x},\gamma_{t}^{x}(y))$ is contained the fiber
$\Pi^{-1}(\Sigma_{t}^{x})$. Since $\Pi^{-1}(\Sigma_{t}^{x})$ is identified
with the universal covering $\widetilde{\Sigma}_{t}^{x}$ of $\Sigma_{t}^{x}$,
just as in the case of $\mathbb{H}^{2}$ we see that
$t\mapsto(\Sigma_{t}^{x},\gamma_{t}^{x}(y))$ is a lift of $\gamma_{t}^{x}$ to
$\widetilde{\Sigma}_{t}^{x}\subset\mathcal{T}(\Sigma,\sigma)$. As such, we use
the alternative notation
$\widetilde{\gamma}_{t}^{x}\colon\mathbb{R}\to\widetilde{\Sigma}_{t}^{x}\subset\mathcal{T}(\Sigma,\sigma)$
with
$\widetilde{\gamma}_{t}^{x}(y)=(\Sigma_{t}^{x},\gamma_{t}^{x}(y))$
when it is convenient to do so.
Finally we record the equation
$\Pi\circ{\bf\Sigma}(x,y,t)=P^{*}\circ{\bf Z}(x,t)$ (6)
which holds for all $(x,y,t)\in\mathbb{H}^{3}$. The fact that $\Pi$ is
$1$–Lipschitz and $P^{*}\circ{\bf Z}$ is an almost-isometric embedding
provides us with useful metric information about ${\bf\Sigma}$.
###### Theorem 5.2.
The map ${\bf\Sigma}\colon\mathbb{H}^{3}\to\mathcal{T}(\Sigma,\sigma)$ is an
almost-isometric embedding. Moreover, the image lies in the thick part and is
quasi-convex.
###### Proof.
As before, we will verify the hypothesis of Lemma 2.2 to prove that
${\bf\Sigma}$ is an almost-isometry and, along the way, prove that the image
is quasi-convex and lies in the thick part.
For all $(x,y)\in\mathbb{R}^{2}$, the geodesic $\eta_{(x,y)}(t)$ in
$\mathbb{H}^{3}$ is sent to
${\bf\Sigma}\circ\eta_{(x,y)}(t)=\mathopen{}\mathclose{{}\left(\Sigma_{t}^{x},\gamma_{t}^{x}(y)}\right)=\mathopen{}\mathclose{{}\left(\psi_{t}^{x}(\Sigma_{0}^{x}),\psi_{t}^{x}(\gamma_{0}^{x}(y))}\right).$
This is a geodesic in $\mathcal{T}(\Sigma,\sigma)$ because
$\psi_{t}^{x}\colon\Sigma_{0}^{x}\to\Sigma_{t}^{x}$ is a Teichmüller mapping;
thus Property 1 follows. Furthermore, note that
${\bf\Sigma}\circ\eta_{(x,y)}(t)$ lies over $P^{*}\circ{\bf
Z}\circ\eta_{x}(t)$ for all $(x,y,t)$. Since $P$ is nontrivially branched over
every point, the uniform thickness of the set of geodesics $\\{{\bf
Z}\circ\eta_{x}(t)\\}_{x\in\mathbb{R}}$ implies the same for
$\\{P^{*}\circ{\bf Z}\circ\eta_{x}(t)\\}_{x\in\mathbb{R}}$ by Proposition 4.4,
and hence also for
$\\{{\bf\Sigma}\circ\eta_{(x,y)}(t)\mathbin{\mid}(x,y)\in\mathbb{R}^{2}\\}$ by
(6) as discussed in Section 4.3. That is, ${\bf\Sigma}(\mathbb{H}^{3})$ lies
in the thick part.
By our choice of maps $\xi_{0}^{x,y}$, if we pull back the vertical foliation
$\Phi_{0}^{x}$ of $\lambda_{0}^{x}$ to a foliation
$\Phi_{0}^{x,y}\in\mathcal{MF}(\Sigma,\sigma)$ the result is independent of
$x$ and $y$. Furthermore, Theorem 4.1 implies that these foliations, as well
as the pull backs of the horizontal foliations, are arational. Thus all strict
subsurface projection distances are defined. Theorem 4.2 and the results of
[15] can be applied as in the $\mathbb{H}^{2}$ case to prove that Property 2
of Lemma 2.2 is satisfied for some $\delta>0$. Furthermore,
${\bf\Sigma}(\mathbb{H}^{3})$ is quasi-convex.
We now come to the subtle point of the proof, which is verifying Properties 3
and 4 of Lemma 2.2. We start with Property 3.
###### Claim.
There exists $\epsilon>0$ and $R>0$ so that if
$e^{-t}\mathopen{}\mathclose{{}\left|(x,y)-(x^{\prime},y^{\prime})}\right|<\epsilon$
then
$d_{\mathcal{T}}\mathopen{}\mathclose{{}\left({\bf\Sigma}(x,y,t),{\bf\Sigma}(x^{\prime},y^{\prime},t)}\right)<R.$
Before we give the proof, we briefly explain the core technical difficulty.
Fix $t$ and define $C_{x}=g_{t}(c(x))$ and $\Gamma_{y}=g_{t}(\gamma(y))$.
Observe that, as before, when we vary $y$ we are simply point pushing; thus
the change in Teichmüller distance is controlled by Theorem 4.3. On the other
hand, varying $x$ means that we are varying the conformal stucture on the
closed surface $\Sigma_{t}^{x}$. This is obtained by varying $x$ in
$(Z_{t},C_{x})$ (which is also point pushing) then taking a branched cover.
However, while we vary $C_{x}$ in $Z_{t}$ we must also keep track of our $y$
coordinate, which means we should also project $\gamma_{t}^{x}(y)$ down to
$Z_{t}$—this is precisely the point $\Gamma_{y}$. Now if $C_{x}$ and
$\Gamma_{y}$ are close together and we vary $x$ so as to push these points
apart, then this can result in a large distance in the “auxiliary” Teichmüller
space $\mathcal{T}(Z,\\{z,w\\})$, even for small variation of $x$. The idea is
therefore to first vary $y$, if necessary, to move $\gamma_{t}^{x}(y)$ in
$\Sigma_{t}^{x}$ and so guaranteeing that $\Gamma_{y}$ is not too close to
$C_{x}$. We can then vary $x$ as required, then vary $y$ back to its original
value. Since the variation of $y$ can be carried out independent of $x$, this
will result in a uniformly bounded change in Teichmüller distance.
###### Proof of Claim..
Since the surfaces $\\{\Sigma_{t}^{x}\\}_{t,x\in\mathbb{R}}$ lie in the thick
part, the (pulled back) metrics $\lambda_{t}^{x}$ and the Poincaré metric(s)
$\rho_{0}$ on the universal cover $\widetilde{\Sigma}_{t}^{x}$ are uniformly
comparable. That is, there exist constants $A$ and $B$ so that for all
$\widetilde{\sigma},\widetilde{\sigma}^{\prime}\in\widetilde{\Sigma}_{t}^{x}$
$\frac{1}{A}\mathopen{}\mathclose{{}\left(d_{\lambda_{t}^{x}}(\widetilde{\sigma},\widetilde{\sigma}^{\prime})-B}\right)\leq\rho_{0}\mathopen{}\mathclose{{}\left(\widetilde{\sigma},\widetilde{\sigma}^{\prime}}\right)\leq
A\,d_{\lambda_{t}^{x}}\mathopen{}\mathclose{{}\left(\widetilde{\sigma},\widetilde{\sigma}^{\prime}}\right)+B.$
(7)
Applying Theorem 4.3, Equations (7) and (3) we have
$\displaystyle
d_{\mathcal{T}}\mathopen{}\mathclose{{}\left({\bf\Sigma}(x,y,t),{\bf\Sigma}(x,y^{\prime},t)}\right)$
$\displaystyle=d_{\mathcal{T}}\mathopen{}\mathclose{{}\left(\widetilde{\gamma}_{t}^{x}(y),\widetilde{\gamma}_{t}^{x}(y^{\prime})}\right)$
(8)
$\displaystyle\leq\rho_{0}\mathopen{}\mathclose{{}\left(\widetilde{\gamma}_{t}^{x}(y),\widetilde{\gamma}_{t}^{x}(y^{\prime})}\right)$
$\displaystyle\leq
Ad_{\lambda_{t}^{x}}\mathopen{}\mathclose{{}\left(\widetilde{\gamma}_{t}^{x}(y),\widetilde{\gamma}_{t}^{x}(y^{\prime})}\right)+B$
$\displaystyle=A\mathopen{}\mathclose{{}\left(e^{-t}\mathopen{}\mathclose{{}\left|y-y^{\prime}}\right|}\right)+B.$
We now fix $t$ and the notation $C_{x}=g_{t}(c(x))$,
$\Gamma_{y}=g_{t}(\gamma(y))$. To understand the effect of varying $x$ we must
consider the branched covering
$P_{t}^{x}\colon\Sigma_{t}^{x}\to(Z_{t},C_{x})$, but also keep track of the
image of our marked point $\gamma_{t}^{x}(y)=\psi_{t}^{x}(\gamma_{0}^{x}(y))$
down in $(Z_{t},C_{x})$; that is, the point $\Gamma_{y}$. This results in the
surface $Z_{t}$ with two marked points:
$(Z_{t},\\{C_{x},\Gamma_{y}\\}).$
Appealing to Proposition 4.5 we have
$d_{\mathcal{T}}\mathopen{}\mathclose{{}\left({\bf\Sigma}(x,y,t),{\bf\Sigma}(x^{\prime},y^{\prime},t)}\right)\leq
d_{\mathcal{T}}\mathopen{}\mathclose{{}\left((Z_{t},\\{C_{x},\Gamma_{y}\\}),(Z_{t},\\{C_{x^{\prime}},\Gamma_{y^{\prime}}\\})}\right).$
(9)
This is because we are taking a branched covering, $\Sigma_{t}^{x}\to Z_{t}$,
and then forgetting all but one of the marked points in $\Sigma_{t}^{x}$.
Since $Z_{t}$ lies in some fixed thick part of $\mathcal{T}(Z)$ for all
$t\in\mathbb{R}$, there exists $\epsilon>0$ so that the $2\epsilon$–ball about
$C_{x}$ in the $q_{t}$ metric, $B_{q_{t}}(C_{x},2\epsilon)$ is a disk for all
$t,x\in\mathbb{R}$ (that is, we have a lower bound on the $q_{t}$–injectivity
radius of $Z_{t}$, independent of $t$). Now suppose $\Gamma_{y}$ lies outside
this ball
$\Gamma_{y}\not\in B_{q_{t}}(C_{x},2\epsilon).$
Using again the fact that $Z_{t}$ lies in some thick part of $\mathcal{T}(Z)$
for all $t\in\mathbb{R}$, it follows that there is some $R^{\prime}>0$ with
the property that for any point $z^{\prime}\in B_{q_{t}}(C_{x},\epsilon)$ we
have
$d_{\mathcal{T}}\mathopen{}\mathclose{{}\left((Z_{t},\\{C_{x},\Gamma_{y}\\}),(Z_{t},\\{z^{\prime},\Gamma_{y}\\})}\right)<R^{\prime}.$
Here the marking homeomorphism for $(Z_{t},\\{z^{\prime},\Gamma_{y}\\})$ is
assumed to differ from that of $(Z_{t},\\{C_{x},\Gamma_{y}\\})$ by composition
with a homeomorphism of $Z_{t}$ that is the identity outside
$B_{q_{t}}(C_{x},2\epsilon)$. In particular, if
$e^{-t}|x-x^{\prime}|<\epsilon$ and, crucially, $\Gamma_{y}\not\in
B_{q_{t}}(C_{x},2\epsilon)$ then deduce that $C_{x^{\prime}}\in
B_{q_{t}}(C_{x},\epsilon)$ and, from Equation (9), that
$\displaystyle
d_{\mathcal{T}}\mathopen{}\mathclose{{}\left({\bf\Sigma}(x,y,t),{\bf\Sigma}(x^{\prime},y,t)}\right)$
$\displaystyle\leq
d_{\mathcal{T}}\mathopen{}\mathclose{{}\left((Z_{t},\\{C_{x},\Gamma_{y}\\}),(Z_{t},\\{C_{x^{\prime}},\Gamma_{y}\\})}\right)<R^{\prime}.$
(10)
On the other hand, because the leaves of $\mathcal{F}$ are geodesics for
$q_{t}$ and because $B_{q_{t}}(C_{x},2\epsilon)$ is a disk, if $\Gamma_{y}\in
B_{q_{t}}(C_{x},2\epsilon)$ then there exists $y^{\prime}\in\mathbb{R}$ so
that $e^{-t}|y^{\prime}-y|\leq 2\epsilon$ and
$\Gamma_{y^{\prime}}\not\in B_{q_{t}}(C_{x},2\epsilon).$
Then, from (10) we have
$\displaystyle
d_{\mathcal{T}}\mathopen{}\mathclose{{}\left({\bf\Sigma}(x,y^{\prime},t),{\bf\Sigma}(x^{\prime},y^{\prime},t)}\right)$
$\displaystyle\leq
d_{\mathcal{T}}\mathopen{}\mathclose{{}\left((Z_{t},\\{C_{x},\Gamma_{y^{\prime}}\\}),(Z_{t},\\{C_{x^{\prime}},\Gamma_{y^{\prime}}\\})}\right)<R^{\prime}.$
Combining this, inequalities (8) and (10), and the triangle inequality, it
follows that for any $x,y,x^{\prime},t$ with
$e^{-t}|x-x^{\prime}|\leq\epsilon$ there is some $y^{\prime}\in\mathbb{R}$
with $e^{-t}|y^{\prime}-y|\leq 2\epsilon$ such that
$\displaystyle
d_{\mathcal{T}}({\bf\Sigma}(x,y,t),{\bf\Sigma}(x^{\prime},y,t))$
$\displaystyle\leq
d_{\mathcal{T}}({\bf\Sigma}(x,y,t),{\bf\Sigma}(x,y^{\prime},t))+d_{\mathcal{T}}({\bf\Sigma}(x,y^{\prime},t),{\bf\Sigma}(x^{\prime},y^{\prime},t))$
(11)
$\displaystyle\phantom{\leq\,}+d_{\mathcal{T}}({\bf\Sigma}(x^{\prime},y^{\prime},t),{\bf\Sigma}(x^{\prime},y,t))$
$\displaystyle\leq 2(A(e^{-t}|y-y^{\prime}|)+B)+R^{\prime}$
$\displaystyle<2(A2\epsilon+B)+R^{\prime}$ $\displaystyle\leq
4(A\epsilon+B)+R^{\prime}.$
Now, let $\epsilon>0$ be as above and set $R=5(A\epsilon+B)+R^{\prime}$. Given
$(x,y,t),(x^{\prime},y^{\prime},t)\in\mathbb{H}^{3}$ with
$e^{-t}|(x,y)-(x^{\prime},y^{\prime})|<\epsilon$, then we have
$e^{-t}|x-x^{\prime}|,e^{-t}|y-y^{\prime}|\leq
e^{-t}|(x,y)-(x^{\prime},y^{\prime})|<\epsilon$. Applying Equations (8) and
(11) and the triangle inequality we obtain
$\displaystyle
d_{\mathcal{T}}\mathopen{}\mathclose{{}\left({\bf\Sigma}(x,y,t),{\bf\Sigma}(x^{\prime},y^{\prime},t)}\right)$
$\displaystyle\leq
d_{\mathcal{T}}\mathopen{}\mathclose{{}\left({\bf\Sigma}(x,y,t),{\bf\Sigma}(x^{\prime},y,t)}\right)+d_{\mathcal{T}}\mathopen{}\mathclose{{}\left({\bf\Sigma}(x^{\prime},y,t),{\bf\Sigma}(x^{\prime}y^{\prime},t)}\right)$
$\displaystyle\leq 4(A\epsilon+B)+R^{\prime}+A\epsilon+B$
$\displaystyle<5(A\epsilon+B)+R^{\prime}=R.$
This completes the proof of the claim, and so verifies Property 3 of Lemma
2.2. ∎
All that remains to show is Property 4 of Lemma 2.2. Suppose we have a
sequence of pairs
$\\{(x_{n},y_{n},t_{n}),(x_{n}^{\prime},y_{n}^{\prime},t_{n})\\}_{n=1}^{\infty}$
with $e^{t_{n}}|(x_{n},y_{n})-(x_{n}^{\prime},y_{n}^{\prime})|\to\infty$ as
$n\to\infty$. Then, up to subsequence, we must be in one of two cases.
###### Case.
$e^{t_{n}}|x_{n}-x_{n}^{\prime}|\to\infty$ as $n\to\infty$.
Forgetting the marked point is $1$–Lipschitz, and so we have
$\displaystyle
d_{\mathcal{T}}\mathopen{}\mathclose{{}\left({\bf\Sigma}(x_{n},y_{n},t_{n}),{\bf\Sigma}(x_{n}^{\prime},y_{n}^{\prime},t_{n})}\right)$
$\displaystyle\geq
d_{\mathcal{T}}\mathopen{}\mathclose{{}\left(\Sigma_{t_{n}}^{x_{n}},\Sigma_{t_{n}}^{x_{n}^{\prime}}}\right)$
$\displaystyle=d_{\mathcal{T}}\mathopen{}\mathclose{{}\left((Z_{t_{n}},g_{t_{n}}(x_{n})),(Z_{t_{n}},g_{t_{n}}(x_{n}^{\prime}))}\right)$
$\displaystyle=d_{\mathcal{T}}\mathopen{}\mathclose{{}\left({\bf
Z}(x_{n},t_{n}),{\bf Z}(x_{n}^{\prime},t_{n})}\right).$
However, we have already verified that ${\bf
Z}\colon\mathbb{H}^{2}\to\mathcal{T}(Z,z)$ satisfies Lemma 2.2. Therefore the
last expression tends to infinity, and hence
$\lim_{n\to\infty}d_{\mathcal{T}}({\bf\Sigma}(x_{n},y_{n},t_{n}),{\bf\Sigma}(x_{n}^{\prime},y_{n}^{\prime},t_{n}))=\infty$
as required.
###### Case.
$e^{t_{n}}|y_{n}-y_{n}^{\prime}|\to\infty$ as $n\to\infty$.
If we also have $e^{t_{n}}|x_{n}-x_{n}^{\prime}|\to\infty$, then we can appeal
to the previous case and we are done. So we assume, as we may, that
$e^{t_{n}}|x_{n}-x_{n}^{\prime}|<M$, for some constant $M>0$. Since we have
already shown that there are $\epsilon,R>0$ so that part 3 from Lemma 2.2
holds, it follows from (1)that
$d_{\mathcal{T}}\mathopen{}\mathclose{{}\left({\bf\Sigma}(x_{n},y_{n}^{\prime},t_{n}),{\bf\Sigma}(x_{n}^{\prime},y_{n}^{\prime},t_{n})}\right)\leq\frac{R}{\epsilon}\mathopen{}\mathclose{{}\left(e^{-t_{n}}|x_{n}-x_{n}^{\prime}|}\right)+R\leq\frac{R}{\epsilon}M+R.$
Now, by the triangle inequality we have
$\displaystyle
d_{\mathcal{T}}\mathopen{}\mathclose{{}\left({\bf\Sigma}(x_{n},y_{n},t_{n}),{\bf\Sigma}(x_{n}^{\prime},y_{n}^{\prime},t_{n})}\right)$
$\displaystyle\geq
d_{\mathcal{T}}\mathopen{}\mathclose{{}\left({\bf\Sigma}(x_{n},y_{n},t_{n}),{\bf\Sigma}(x_{n},y_{n}^{\prime},t_{n})}\right)$
(12)
$\displaystyle\phantom{\geq\,}-d_{\mathcal{T}}\mathopen{}\mathclose{{}\left({\bf\Sigma}(x_{n},y_{n}^{\prime},t_{n}),{\bf\Sigma}(x_{n}^{\prime},y_{n}^{\prime},t_{n})}\right)$
$\displaystyle\geq
d_{\mathcal{T}}\mathopen{}\mathclose{{}\left(\widetilde{\gamma}_{t_{n}}^{x_{n}}(y_{n}),\widetilde{\gamma}_{t_{n}}^{x_{n}}(y_{n}^{\prime})}\right)-\frac{R}{\epsilon}M-R$
We can now appeal to Theorem 4.3 as in our proof for ${\bf
Z}\colon\mathbb{H}^{2}\to\mathcal{T}(Z,z)$ to find $A,B$ so that
$d_{\mathcal{T}}(\widetilde{\gamma}_{t_{n}}^{x_{n}}(y_{n}),\widetilde{\gamma}_{t_{n}}^{x_{n}}(y_{n}^{\prime}))\geq
h\mathopen{}\mathclose{{}\left(\frac{1}{A}e^{-{t_{n}}}|y_{n}^{\prime}-y_{n}|-B}\right)$
The right-hand side tends to infinity by the properness of $h$, so we can
combine this with (12) to obtain
$\lim_{n\to\infty}d_{\mathcal{T}}({\bf\Sigma}(x_{n},y_{n},t_{n}),{\bf\Sigma}(x_{n}^{\prime},y_{n}^{\prime},t_{n}))=\infty$
as required. Therefore, Property 4 from Lemma 2.2 holds, and the proof of
Theorem 5.2 is complete. ∎
### 5.3 The general case
The previous arguments set up an inductive scheme for producing almost-
isometric embeddings of $\mathbb{H}^{n}$ into Teichmüller spaces. The idea is
as follows.
For $n-1\geq 3$, induction gives us an almost-isometric embedding ${\bf
W}\colon\mathbb{H}^{n-1}\to\mathcal{T}(W,w)$ satisfying all the hypotheses of
Lemma 2.2 for some closed surface $W$ with a marked point $w$. We again take a
branched cover
$P\colon\Omega\to W$
nontrivially branched over each point in $P^{-1}(w)\subset\Omega$. This
determines a map
$P^{*}\circ{\bf W}\colon\mathbb{H}^{n-1}\to\mathcal{T}(\Omega).$
Using the coordinates
$(x,t)=(x_{1},x_{2},\ldots,x_{n-2},t)\in\mathbb{H}^{n-1}$ we write
$P^{*}\circ{\bf W}(x,t)=\Omega_{t}^{x}.$
Inductively, we assume that the foliation of $\mathbb{H}^{n-1}$ by asymptotic
geodesics $\\{\eta_{x}(t)\\}_{x\in\mathbb{R}^{n-2}}$ are all mapped by ${\bf
W}$ to uniformly thick geodesics in $\mathcal{T}(W,w)$, so the same is true
for $P^{*}\circ W$. These geodesics are obtained by applying the Teichmüller
mapping $\psi_{t}^{x}\colon\Omega_{0}^{x}\to\Omega_{t}^{x}$ giving
$P^{*}\circ{\bf W}(x,t)=\psi_{t}^{x}\circ P^{*}\circ{\bf W}(x,0)$
for all $x\in\mathbb{R}^{n-2}$ and $t\in\mathbb{R}$. Furthermore, the defining
quadratic differentials all have the same vertical foliation.
We pick a leaf of this foliation $\gamma\colon\mathbb{R}\to\Omega$, and
arguing as before, this determines a leaf in each surface
$\gamma_{t}^{x}\colon\mathbb{R}\to\Omega_{t}^{x}$ with
$\gamma_{0}^{0}=\gamma\colon\mathbb{R}\to\Omega_{0}^{0}=\Omega$. Now, pick
$\omega=\gamma_{0}^{0}(0)$ to be our marked point, add a factor of
$\mathbb{R}$ to $\mathbb{H}^{n-1}$ with coordinate $y=x_{n-1}$ to obtain
$\mathbb{H}^{n}$ with coordinates $(x,y,t)=(x_{1},\ldots,x_{n-2},x_{n-1},t)$,
and define
${\bf\Omega}\colon\mathbb{H}^{n}\to\mathcal{T}(\Omega,\omega)$
by
${\bf\Omega}(x,y,t)=(\Omega_{t}^{x},\gamma_{t}^{x}(y)).$
So we are again pushing a point along a leaf of the vertical foliation.
###### Theorem 5.3.
The map ${\bf\Omega}\colon\mathbb{H}^{n}\to\mathcal{T}(\Omega,\omega)$ is an
almost-isometric embedding. Moreover, the image lies in the thick part and is
quasi-convex.
###### Sketch of proof.
Again, we must verify the hypotheses of Lemma 2.2 and prove that the image of
${\bf\Omega}$ is quasi-convex in the thick part, assuming that this is true in
all previous steps of the construction.
We can argue exactly as in the case of $\mathbb{H}^{3}$ to prove Properties 1
and 2 of Lemma 2.2 as well as the fact that the image of ${\bf\Omega}$ is
quasi-convex in the thick part. Property 3 requires more care. However, once
established, Property 4 follows formally, just as in the case of
$\mathbb{H}^{3}$.
We elaborate on the proof that Property 3 holds for some $\epsilon$ and $R$.
For this, we must give a more precise description of the construction. Write
$\Omega_{n-1}=\Omega$, $\Omega_{n-2}=W$ and
$P_{n-2}=P\colon\Omega_{n-1}\to\Omega_{n-2}$
for the branched cover used in the construction. Inductively, we have a tower
of branched covers
$\textstyle{\Omega_{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P_{n-2}}$$\textstyle{\Omega_{n-2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P_{n-3}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P_{1}}$$\textstyle{\Omega_{1}}$
In this tower, $P_{j}$ is nontrivially branched at every point
$P_{j}^{-1}(\omega_{j})$ where $\omega_{j}\in\Omega_{j}$ is the marked point.
To clarify, we note that $\Omega_{1}=Z$, $\omega_{1}=z$, $\Omega_{2}=\Sigma$
and $\omega_{2}=\sigma$ from the preceding discussion.
We also have a quadratic differential $\nu_{1}$ on $\Omega_{1}$ (this is
$\nu_{1}=q$ from before), which pulls back via all the branched covers to
quadratic differentials
$\nu_{i}=P_{i-1}^{*}(\nu_{i-1})\in\mathcal{Q}(\Omega_{i})$. On $\Omega_{1}$,
we have chosen $n-1$ distinct nonsingular leaves from the vertical foliation
of $\nu_{1}$ which we denote
$\\{\zeta_{i}\colon\mathbb{R}\to\Omega_{1}\\}_{i=1}^{n-1}$. These leaves are
parametrized by arc-length so that $\zeta_{j}(0)=P_{1}\circ
P_{2}\circ\cdots\circ P_{j-1}(\omega_{j})$.
Recall that $y=x_{n-1}$. We can now describe
$\Omega(x,y,t)=\Omega(x_{1},\ldots,x_{n-2},x_{n-1},t)$ for any
$(x,y,t)\in\mathbb{H}^{n}$. At the bottom of the tower we push $\omega_{1}$
along $\zeta_{1}$ to $\zeta_{1}(x_{1})$, then take the branched cover
$\Omega_{2}^{x_{1}}\to(\Omega_{1},\zeta_{1}(x_{1}))$ induced by $P_{1}$ (it is
the induced branched cover since it branches over $\zeta_{1}(x_{1})$ rather
than over $\zeta_{1}(0)=\omega_{1}$; see Section 4.4). Next, the lifted
marking identifies $\omega_{2}$ with a point in the preimage of
$\zeta_{2}(0)$, and we push this along an appropriate lift $\zeta_{2}^{x_{1}}$
of $\zeta_{2}$ to a point $\zeta_{2}^{x_{1}}(x_{2})$ in the preimage of
$\zeta_{2}(x_{2})$. At the next level, there is an branched cover
$\Omega_{3}^{x_{1},x_{2}}\to(\Omega_{2}^{x_{1}},\zeta_{2}^{x_{1}}(x_{2}))$
induced by $P_{2}$. The lifted marking identifies $\omega_{3}$ with a point in
the preimage of $\zeta_{3}(0)$ in the composition of branched covers
$\Omega_{3}^{x_{1},x_{2}}\to\Omega_{2}^{x_{1}}\to\Omega_{1}$ and we push this
along an appropriate lift $\zeta_{3}^{x_{1},x_{2}}$ of $\zeta_{3}$ to a point
$\zeta_{3}^{x_{1},x_{2}}(x_{3})$ in the preimage of $\zeta_{3}(x_{3})$. We
continue in this way to produce a tower of branched covers induced by
$P_{1},P_{2},\ldots,P_{n-3},P_{n-2}$:
$\textstyle{\Omega_{n-1}^{x_{1},\ldots,x_{n-2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega_{n-2}^{x_{1},\ldots,x_{n-3}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega_{3}^{x_{1},x_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega_{2}^{x_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega_{1}.}$
The point $\omega_{n-1}$ is identified with a marked point in
$\Omega_{n-1}^{x_{1},\ldots,x_{n-2}}$ in the preimage of $\zeta_{n-1}(0)$, and
then we push this point along an appropriate lift
$\zeta_{n-1}^{x_{1},\ldots,x_{n-2}}$ of $\zeta_{n-1}$ to the point
$\zeta_{n-1}^{x_{1},\ldots,x_{n-2}}(y)=\zeta_{n-1}^{x_{1},\ldots,x_{n-2}}(x_{n-1})$.
With this notation
${\bf\Omega}(x,y,0)={\bf\Omega}(x_{1},\ldots,x_{n-2},x_{n-1},0)=(\Omega_{n-1}^{x_{1},\ldots,x_{n-2}},\zeta_{n-1}^{x_{1},\ldots,x_{n-2}}(x_{n-1})).$
To find ${\bf\Omega}(x,y,t)$ for any $t$, we apply the appropriate Teichmüller
deformation to ${\bf\Omega}(x,y,0)$. This is the Teichmüller deformation
determined by $t$ and the pull back of $\nu_{1}$ (via the composition of
branched covers). We can pull back $\nu_{1}$ by any of the branched covers,
and since the resulting quadratic differential depends only on the surface in
this construction, we will simply write $\Phi_{t}$ for the associated
Teichmüller deformation on any of the surfaces
$\Omega_{j}^{x_{1},\ldots,x_{j-1}}$. In particular, we have
${\bf\Omega}(x,y,t)=\Phi_{t}({\bf\Omega}(x,y,0)).$
Set $x^{\prime}=(x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{n-2}^{\prime})$. We
now must find an $\epsilon$ and $R$ so that if
$e^{-t}|(x,y)-(x^{\prime},y^{\prime})|\leq\epsilon$
then
$d_{\mathcal{T}}\mathopen{}\mathclose{{}\left({\bf\Omega}(x,y,t),{\bf\Omega}(x^{\prime},y^{\prime},t)}\right)\leq
R.$
As in the case of $\mathbb{H}^{3}$, appealing to the triangle inequality it
suffices to find an $\epsilon$ and $R^{\prime}$ so that if
$(x_{1},\ldots,x_{n-2},y)$ and
$(x_{1}^{\prime},\ldots,x_{n-2}^{\prime},y^{\prime})$ agree in all but one
coordinate, and in that coordinate differ by at most $\epsilon$, then
$d_{\mathcal{T}}\mathopen{}\mathclose{{}\left({\bf\Omega}(x,y,t),{\bf\Omega}(x^{\prime},y^{\prime},t)}\right)\leq
R^{\prime}.$
If $(x,y)$ and $(x^{\prime},y^{\prime})$ differ only in the last coordinate,
then we can apply Theorem 4.3 just as before to produce $\epsilon=1$ and
$R^{\prime}=A+B$. Suppose instead that $y=y^{\prime}$ and $x$ differs from
$x^{\prime}$ in the $n-2$–coordinate only. We start at the highest coordinate,
$y=x_{n-1}$ and work two steps down to $x_{n-2}$. The idea is similar to what
was done in varying $x$ in $(x,y,t)\in\mathbb{H}^{3}$. We look on
$\Phi_{t}(\Omega_{n-2}^{x_{1},\ldots,x_{n-3}})$ as an “auxiliary” surface when
it is equipped with the two marked points
$\Phi_{t}(\zeta_{n-2}^{x_{1},\ldots,x_{n-3}}(x_{n-2}))$ and the image of
$\Phi_{t}(\zeta_{n-1}^{x_{1},\ldots,x_{n-2}}(x_{n-1}))$ via the branched
covering
$\Phi_{t}(\Omega_{n-1}^{x_{1},\ldots,x_{n-2}})\to\Phi_{t}(\Omega_{n-2}^{x_{1},\ldots,x_{n-3}}).$
If these two points are not too close, then we can move from
$\Phi_{t}(\zeta_{n-2}^{x_{1},\ldots,x_{n-3}}(x_{n-2}))$ to
$\Phi_{t}(\zeta_{n-2}^{x_{1},\ldots,x_{n-3}}(x_{n-2}^{\prime}))$ keeping the
other marked point fixed, and the distance between these two points in the
Teichmüller space of the auxiliary surface with two marked points is uniformly
bounded. Since the branched cover induces a $1$–Lipschitz map (compare (9)),
this means that
$d_{\mathcal{T}}\mathopen{}\mathclose{{}\left({\bf\Omega}(x_{1},\ldots,x_{n-3},x_{n-2},x_{n-1},t),{\bf\Omega}(x_{1},\ldots,x_{n-3},x_{n-2}^{\prime},x_{n-1},t)}\right)$
is uniformly bounded.
On the other hand, if the two marked points in
$\Phi_{t}(\Omega_{n-2}^{x_{1},\ldots,x_{n-3}})$ are close, we
$\mbox{move}\quad\Phi_{t}(\zeta_{n-1}^{x_{1},\ldots,x_{n-2}}(x_{n-1}))\quad\mbox{to}\quad\Phi_{t}(\zeta_{n-1}^{x_{1},\ldots,x_{n-2}}(x_{n-1}^{\prime})),$
$\mbox{move}\quad\Phi_{t}(\zeta_{n-2}^{x_{1},\ldots,x_{n-3}}(x_{n-2}))\quad\mbox{to}\quad\Phi_{t}(\zeta_{n-2}^{x_{1},\ldots,x_{n-3}}(x_{n-2}^{\prime})),$
and then
$\mbox{move}\quad\Phi_{t}(\zeta_{n-1}^{x_{1},\ldots,x_{n-2}^{\prime}}(x_{n-1}^{\prime}))\quad\mbox{back
to}\quad\Phi_{t}(\zeta_{n-1}^{x_{1},\ldots,x_{n-2}^{\prime}}(x_{n-1})).$
By the triangle inequality, we obtain the desired uniform bound on
$d_{\mathcal{T}}\mathopen{}\mathclose{{}\left({\bf\Omega}(x_{1},\ldots,x_{n-3},x_{n-2},x_{n-1},t),{\bf\Omega}(x_{1},\ldots,x_{n-3},x_{n-2}^{\prime},x_{n-1},t)}\right).$
Note that this required three point pushes in two different auxiliary
surfaces. We varied the $(n-1)^{\rm st}$ coordinate twice, in the highest
surface, and varied the $(n-2)^{\rm nd}$ coordinate once.
Now suppose that $x$ differs from $x^{\prime}$ in the $(n-3)^{\rm rd}$
coordinate only. We view $\Phi_{t}(\Omega_{n-3}^{x_{1},\ldots,x_{n-4}})$ as an
auxiliary surface with three marked points: the images of the points
$\Phi_{t}(\zeta_{n-1}^{x_{1},\ldots,x_{n-2}}(x_{n-1}))$ and
$\Phi_{t}(\zeta_{n-2}^{x_{1},\ldots,x_{n-3}}(x_{n-2}))$ under the respective
branched covers and the point
$\Phi_{t}(\zeta_{n-3}^{x_{1},\ldots,x_{n-4}}(x_{n-3}))$. We can move this last
point a small amount, changing the Teichmüller distance a bounded amount,
provided the other two points, higher in the tower, are not too close to it.
If they are too close, we first move them out of the way (as in the first two
pushes above), move the third point, then move the two higher points back. The
triangle inequality together with the $1$–Lipschitz property of the branched
cover map applied as before, implies a uniform bound on the change in
Teichmüller distance
$d_{\mathcal{T}}\mathopen{}\mathclose{{}\left({\bf\Omega}(x_{1},\ldots,x_{n-3},x_{n-2},x_{n-1},t),{\bf\Omega}(x_{1},\ldots,x_{n-3}^{\prime},x_{n-2},x_{n-1},t)}\right).$
It follows that varying $x_{n-3}$ requires at most five point pushes in the
three highest auxiliary surfaces.
In general, varying $x_{n-k}$ in this way requires $2k-1$ point pushes in the
$k$ highest auxiliary surfaces. Thus we can change any coordinate by a small
amount $\epsilon$ and change the Teichmüller distance by a bounded amount
$R^{\prime}$, as required. This completes the sketch of the proof of Theorem
5.3. ∎
## References
* [1] William Abikoff. The real analytic theory of Teichmüller space, volume 820 of Lecture Notes in Mathematics. Springer, Berlin, 1980.
* [2] Lars V. Ahlfors. Lectures on quasiconformal mappings. Manuscript prepared with the assistance of Clifford J. Earle, Jr. Van Nostrand Mathematical Studies, No. 10. D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966.
* [3] Lipman Bers. A remark on Mumford’s compactness theorem. Israel J. Math., 12:400–407, 1972.
* [4] Lipman Bers. Fiber spaces over Teichmüller spaces. Acta. Math., 130:89–126, 1973.
* [5] Mario Bonk and Bruce Kleiner. Quasi-hyperbolic planes in hyperbolic groups. Proc. Amer. Math. Soc., 133(9):2491–2494 (electronic), 2005.
* [6] Matt Clay, Christopher J Leininger, and Johanna Mangahas. The geometry of right-angled Artin subgroups of mapping class groups. To appear in Groups Geom. Dyn, arXiv:1007.1129.
* [7] Benson Farb and Lee Mosher. Convex cocompact subgroups of mapping class groups. Geom. Topol., 6:91–152 (electronic), 2002.
* [8] A. Fathi, F. Laudenbach, and V. Poénaru. Travaux de Thurston sur les surfaces. Société Mathématique de France, Paris, 1991. Séminaire Orsay, Reprint of Travaux de Thurston sur les surfaces, Soc. Math. France, Paris, 1979 Astérisque No. 66-67 (1991).
* [9] David Gabai. Almost filling laminations and the connectivity of ending lamination space. Geom. Topol., 13(2):1017–1041, 2009.
* [10] Frederick P. Gardiner. Teichmüller theory and quadratic differentials. Pure and Applied Mathematics (New York). John Wiley & Sons Inc., New York, 1987. A Wiley-Interscience Publication.
* [11] F. W. Gehring. Quasiconformal mappings which hold the real axis pointwise fixed. In Mathematical Essays Dedicated to A. J. Macintyre, pages 145–148. Ohio Univ. Press, Athens, Ohio, 1970.
* [12] Ursula Hamenstädt. Word hyperbolic extensions of surface groups. Preprint, arXiv:math.GT/0505244.
* [13] John Hubbard and Howard Masur. Quadratic differentials and foliations. Acta Math., 142(3-4):221–274, 1979.
* [14] Y. Imayoshi and M. Taniguchi. An introduction to Teichmüller spaces. Springer-Verlag, Tokyo, 1992. Translated and revised from the Japanese by the authors.
* [15] Richard P. Kent, IV and Christopher J. Leininger. Shadows of mapping class groups: capturing convex cocompactness. Geom. Funct. Anal., 18(4):1270–1325, 2008.
* [16] Shoshichi Kobayashi. Invariant distances on complex manifolds and holomorphic mappings. J. Math. Soc. Japan, 19:460–480, 1967.
* [17] Irwin Kra. On the Nielsen-Thurston-Bers type of some self-maps of Riemann surfaces. Acta Math., 146(3-4):231–270, 1981.
* [18] Christopher J. Leininger, Mahan Mj, and Saul Schleimer. The universal Cannon–Thurston map of the curve complex. To appear in Comm. Math. Helv., arXiv:0808.3521.
* [19] Christopher J. Leininger and Saul Schleimer. Connectivity of the space of ending laminations. Duke Math. J., 150(3):533–575, 2009.
* [20] Howard Masur. Interval exchange transformations and measured foliations. Ann. of Math. (2), 115(1):169–200, 1982.
* [21] Howard Masur. Closed trajectories for quadratic differentials with an application to billiards. Duke Math. J., 53(2):307–314, 1986.
* [22] Howard Masur. Hausdorff dimension of the set of nonergodic foliations of a quadratic differential. Duke Math. J., 66(3):387–442, 1992.
* [23] Howard A. Masur and Yair N. Minsky. Geometry of the complex of curves. I. Hyperbolicity. Invent. Math., 138(1):103–149, 1999.
* [24] Howard A. Masur and Yair N. Minsky. Geometry of the complex of curves. II. Hierarchical structure. Geom. Funct. Anal., 10(4):902–974, 2000.
* [25] Mahan Mj and Pranab Sardar. A combination theorem for metric bundles. Preprint, arXiv:0912.2715.
* [26] Kasra Rafi. A characterization of short curves of a Teichmüller geodesic. Geom. Topol., 9:179–202, 2005.
* [27] Kasra Rafi and Saul Schleimer. Covers and the curve complex. Geom. Topol., 13(4):2141–2162, 2009.
* [28] H. L. Royden. Automorphisms and isometries of Teichmüller space. In Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969), pages 369–383. Ann. of Math. Studies, No. 66. Princeton Univ. Press, Princeton, N.J., 1971.
* [29] Oswald Teichmüller. Ein Verschiebungssatz der quasikonformen Abbildung. Deutsche Math., 7:336–343, 1944.
|
arxiv-papers
| 2011-10-29T11:47:24 |
2024-09-04T02:49:23.684258
|
{
"license": "Public Domain",
"authors": "Christopher J. Leininger and Saul Schleimer",
"submitter": "Saul Schleimer",
"url": "https://arxiv.org/abs/1110.6526"
}
|
1110.6576
|
# Dynamical Casmir effect and Conductivity
Xiao-Min Bei Zhong-Zhu Liu xiaominbei@gmail.com Department of Physics,
Huazhong University of Science and Technology, Wuhan, 430074, China
###### Abstract
In this paper we find that the second law of thermodynamics requires an upper
limit of the conductivity. To begin with we present an ideal model, the cavity
with a mobile plate, for studying the thermodynamic properties of radiation
field. It is shown that the pressure fluctuation of thermal radiation field in
the cavity leads to the random motion of the plate and photons would be
generated by dynamical Casimir effect. Meanwhile, such photons obey a non-
thermal distribution. Then, to ensure the second law of thermodynamics, there
must be a upper limit of the conductivity.
?, ?
###### pacs:
?
## I Introduction
When the space symmetry of a system is broken the statistical fluctuation may
lead to macroscopic motion, which is called molecular motor 1 ; 2 ; 3 ; 4 ; 5
. At the same time, the system must obey the second law of thermodynamics.
This will give the system of molecular motor some rigorous limit. Contrasting
to the space symmetry, it has recently paid more attention to the time
asymmetry of the system. People want to understand what macroscopical
phenomena will result from thermal fluctuations and what constraints on the
system will be prescribed by the second law of thermodynamics.
In quantum field theory, time-dependent boundary conditions or uncharged
mirrors in accelerated motion may induce particle creation, even when the
initial state of a quantum field is the vacuum 6 ; 7 ; 8 . This purely quantum
effect has been known as the dynamical Casimir effect (DCE) 9 . In this case,
along with moving boundaries, the vacuum state of the electromagnetic field is
changed, and the changing vacuum results in the generation of photons.
However, the reverse process does not occur. This is the time symmetry
breaking of the system, which will lead to a phenomenon similar to the
molecular motor.
Generated photons in the DCE obey a non-thermal distribution. In some peculiar
case, the non-thermal distribution will lead to a system’s macroscopically
breaking of the thermal equilibrium. To keep the thermal equilibrium of the
system, the second law of thermodynamics requires a compensatory effect. In
this work, we shall study how need material to absorb photons generated by the
DCE for ensuring the second law of thermodynamics. Firstly we present an ideal
model, a conductor cavity with a mobile plate inside it. Assuming the system
to be initially at thermal equilibrium, the pressure fluctuation will result
in a pressure difference on both sides of plate. It leads to a random motion
of the conducting plate and photons would be generated in the cavity by the
DCE. Note that these photons created do not satisfy the thermal distribution
but the super-Poissonian distribution 10 , which may break the balance of
thermal radiation field. To ensure the second law of thermodynamics, created
photons should be absorbed by the cavity wall in the relaxaion time; that is
to say, the photon absorption rate has to be greater than that of the
generation rate. Meanwhile, both the photon absorption rate and the generation
rate are related with the matieral’s conductivity. Thus an upper limit of the
conductivity is prescribed.
It is confirmed experientially already that there is an upper limit of the
conductivity. However, this existence of the upper limit lacks a theoretical
proof. In this paper, we shall prove this upper limit according to the second
law of thermodynamics.
The article is organized as follows. In Sec. II, we start by analyzing the
pressure fluctuation of the thermal radiation in the cavity. Then we establish
a Langevin equation describing the plate motion and derive the time
correlation function of the acceleration. In Sec. III, according to DCE we
obtain the numbers of photons created by random motion of the conducting
plate, and present an expression of the relative photon generation rate per
volume. Next, the second law of thermodynamics is considered in Sec. IV, and
we show that requiring the photon absorption rate to be greater than the
generation rate, it could lead to an upper limit of the conductivity. Finally
we conclude our work in the last section.
## II The Model
In this section we will present a thermodynamic analysis of thermal radiation
field in the cavity with a mobile plate inside it.
We consider a three-dimensional model of a thermal radiation field within a
rectangular cavity with conducting walls. The cavity has the dimensions
$L_{x}$, $L_{y}$ and $L_{z}$. At the midpoint of the cavity
$\left(x=L_{x}/2\right)$ a thin mobile plate is located, which is made of the
same material as the cavity walls. Thus the cavity is divided into two
regions: region I $\left(0\leq x\leq{L_{x}/2}\right)$ and region II
$\left({L_{x}/2}\leq x\leq L_{x}\right)$. For the sake of simplicity we assume
the system to be initially at thermal equilibrium corresponding to some
nonvanishing temperature $T$ and the temperature and pressure within two
regions are equal, i.e., $T_{\texttt{I}}=T_{\texttt{II}}$, and
$p_{\texttt{I}}=p_{\texttt{II}}$. Owing to the pressure fluctuation in respect
of the thermal equilibrium, a pressure difference on both sides of plate
emerges 10 . The pressure difference is variable stochastically and leads to
the random motion of the conducting plate. Under the motion of the plate,
photons would be generated in the cavity by the DCE.
Initially, the pressure of back-body radiation in the cavity can be simply
expressed by 11
$p_{a}=\frac{4}{3}\sigma T^{4},$ (1)
where the coefficient $\sigma$ is the Stefan-Boltzmann constant and the
subscript $a=\texttt{I},\texttt{II}$ differentiate the radiation pressures in
the left or right cavity. Note that this pressure is proportional to the
fourth power of the temperature. At the same time, owing to the pressure
fluctuation, we can take the following expression for the mean square
fluctuation of the pressure 11
$\overline{\left(\Delta p_{a}\right)^{2}}=-k_{B}T\frac{\partial
p_{a}}{\partial V}\bigg{|}_{S},$ (2)
where $k_{B}$ is Boltzmann’s constant, and $V$ is the cavity volume.
Now we put Eq. (1) into the above formula (2) and take into account of Maxwell
relations 11 . Then the mean square fluctuation of this pressure can be
rewritten as
$\overline{\left(\Delta p_{a}\right)^{2}}=\frac{\alpha}{3V}k_{B}T^{5},$ (3)
with $\alpha=\frac{4\sigma}{c}$. It is shown that the mean square fluctuation
of the pressure is proportional to the fifth power of the temperature and
inversely proportional to the volume.
There is a correlation between $\Delta p_{a}\left(t\right)$ at different
instants. This means that the value of $\Delta p_{a}$ at a given instant $t$
affects the probabilities of its various values at a later instant
$t^{\prime}$. We can characterize the time correlation by the mean value of
the product 11
$\overline{\Delta p_{a}(t)\Delta p_{a}(t^{\prime})}=\overline{[\Delta
p_{a}]^{2}}e^{-\lambda(t^{\prime}-t)}.$ (4)
Here the constant $1/\lambda$ determines the order of magnitude of the
relaxation time for the establishment of complete equilibrium.
Since pressure fluctuations on both sides of the plate are obviously
irrelevant of each other, the fluctuations of the pressure $\Delta
p_{\texttt{I}}(t)$ and $\Delta p_{\texttt{II}}(t)$ are statistically
independent at each instant. Then the pressure difference on both sides of
plate can be simply written as $\Delta P(t)=\Delta p_{\texttt{I}}(t)-\Delta
p_{\texttt{II}}(t)$ which is also a random variable. Finally, using Eq. (4),
we obtain a formula for the time correlation of pressure difference:
$\overline{\Delta P(t)\Delta
P(t^{\prime})}=\frac{2\alpha}{3V}k_{B}T^{5}e^{-\lambda(t^{\prime}-t)},$ (5)
At the same time, in the formal limit $t^{\prime}\rightarrow t$, the mean
square fluctuation of the pressure difference is given simply as
$\overline{[\Delta P]^{2}}=\frac{2\alpha}{3V}k_{B}T^{5}.$ (6)
To illustrate how we may incorporate dynamics into the discussion of a
stochastic process, let $\dot{x}(t)$ be the velocity of the plate. The
Newtonian equation of the plate’s motion is given by
$\displaystyle M\ddot{x}+\beta\dot{x}=\Delta
P(t)S+\frac{1}{2}\left(\frac{\partial A\left(x,t\right)}{\partial
x}\right)\left(\big{|}_{x=L_{0}/2}-\big{|}_{x=L_{0}/2+\Delta x}\right),$ (7)
where $M$ and $S$ are the mass and area of the plate respectively, $\beta$ is
the friction coefficient, and $\Delta x$ is the thickness of the plate. This
is called the Langevin equation 12 ; 13 . The first term $\Delta P(t)S$
denotes a randomly fluctuating external force induced by the pressure
difference while the second term, which is enough small to can be neglected 14
, represents the recoil force due to photon radiation. For simplicity, the
above formulas can be equivalently expressed as
$\ddot{x}+\gamma\dot{x}=s\Delta P(t),$ (8)
where $s=\frac{S}{M}$ and $\gamma=\frac{\beta}{M}$ is a damping coefficient.
The above equation (8) is a random differential equation describing how the
plate takes the Brownian movement. Let us assume that at time $t=0$, the
velocity and position of the plate are $\dot{x}(0)$ and $x(0)$, respectively.
Then the solution of the Eq. (8) is
$\dot{x}(t)=\dot{x}(0)e^{-\gamma t}+e^{-\gamma t}\int_{0}^{t}s\Delta
P\left(\xi\right)e^{\gamma\xi}d\xi,$ (9)
This equation gives $\dot{x}(t)$ for a single realization of $\Delta
P\left(t\right)$. Since $\Delta P\left(t\right)$ is a stochastic variable,
$\dot{x}(t)$ and $\ddot{x}(t)$ are also stochastic whose properties are
determined by $\Delta P\left(t\right)$. The average velocity is
$\overline{\dot{x}(t)}=\dot{x}(0)e^{-\gamma t}$. Using Eq.(5) and Eq. (9) ,
the time correlation function of the velocity can be obtained as follows
$\displaystyle\overline{\dot{x}(t)\dot{x}(t^{\prime})}=$
$\displaystyle\left(\dot{x}(0)\right)^{2}e^{-\gamma(t+t^{\prime})}$ (10)
$\displaystyle+s^{2}\overline{[\Delta P]^{2}}\left(\frac{e^{\lambda
t}-e^{-\gamma t}}{\gamma+\lambda}\right)\left(\frac{e^{-\lambda
t^{\prime}}-e^{-\gamma t^{\prime}}}{\gamma-\lambda}\right).$
Then the time correlation function of the acceleration can be written in terms
of Eq.(8)
$\overline{\ddot{x}(t)\ddot{x}(t^{\prime})}=\gamma^{2}\overline{\dot{x}(t)\dot{x}(t^{\prime})}+s^{2}\overline{\Delta
P(t)\Delta P(t^{\prime})},$ (11)
where the formula $\overline{\Delta P}=0$ is used. Now we put Eq. (5) and Eq.
(10) into the above equation (11), and then derive the time correlation
function of the acceleration as
$\displaystyle\overline{\ddot{x}(t)\ddot{x}(t^{\prime})}=$
$\displaystyle\gamma^{2}\left(\dot{x}(0)\right)^{2}e^{-\gamma(t+t^{\prime})}$
(12) $\displaystyle+\gamma^{2}s^{2}\overline{[\Delta
P]^{2}}\left(\frac{e^{\lambda t}-e^{-\gamma
t}}{\gamma+\lambda}\right)\left(\frac{e^{-\lambda t^{\prime}}-e^{-\gamma
t^{\prime}}}{\gamma-\lambda}\right)$ $\displaystyle+s^{2}\overline{[\Delta
P]^{2}}e^{-\lambda(t^{\prime}-t)}.$
For time long enough, namely $t,t^{\prime}\gg 1/\gamma$, the initial velocity
of the plate can be neglected. Then Eq. (12) becomes simply,
$\displaystyle\overline{\ddot{x}(t)\ddot{x}(t^{\prime})}$
$\displaystyle=s^{2}\overline{[\Delta
P]^{2}}\left(\gamma^{2}\frac{e^{-\lambda(t^{\prime}-t)}-e^{\lambda t-\gamma
t^{\prime}}-e^{-\gamma t-\lambda
t^{\prime}}+e^{-\gamma(t+t^{\prime})}}{\gamma^{2}-\lambda^{2}}+e^{-\lambda(t^{\prime}-t)}\right).$
In the following we can take $\ddot{x}(t)$ to be any random function of $t$,
and shall write $t$ instead of $t^{\prime}$ for convenience. Then one has
$\displaystyle\overline{[\ddot{x}(t)]^{2}}=s^{2}\overline{[\Delta
P]^{2}}\left(\gamma^{2}\frac{1-e^{(\lambda-\gamma)t}-e^{-(\lambda+\gamma)t}+e^{-2\gamma
t}}{\gamma^{2}-\lambda^{2}}+1\right).$ (14)
Thus it is equal to the mean square value of the fluctuating function
$\ddot{x}(t)$.
Hence, the radiation pressure fluctuations in two different regions of the
cavity would lead to the random motion of the conducting plate. In contrast to
10 , we are interested here in whether it is likely to create photons. This
will be discussed in detail in the next section.
## III Random Motion And Photon Creation
In above section, we get the time correlations of the acceleration and the
velocity of the plate. The movement of the plate brings on the photon
generation according to the theory of the DCE 6 ; 7 . And the relative photon
generation rate per volume will be derived. First of all we will calculate the
numbers of photons created by the DCE in the right or left cavity, which is
due to random motion of the conducting plate. Comparing with Ref. 10 , we have
assumed that the type of motion of the plate may be described by
$\displaystyle x(t)=\left\\{\begin{array}[]{ll}x_{1}(t)=L&\hbox{$t\leq 0$}\\\
x_{2}(t)&\hbox{$0\leq t\leq t_{1}$}\\\
x_{3}(t)=x_{2}(t)=x_{0}&\hbox{$t>t_{1}$}\end{array}\right.,$ (18)
with $L=\frac{L_{x}}{2}$, and $x_{0}$ is a constant satisfied $0<x_{0}<L_{x}$.
Note that the cavity is at rest for times $t\leq 0$ and $t>t_{1}$, and we
require $x_{2}(0)=L$, $\dot{x}_{2}(0)=0$, $\dot{x}_{3}(t_{1})=0$, and
$x(t)\geq 0$ for all $t$Since the system is initially in thermal equilibrium
at temperature $T$, we may take a number state $|n_{k}\rangle$ to be the
initial quantum state of photons
$|n_{k}\rangle=\frac{\left(a_{k}^{(1){\dagger}}\right)^{n_{k}}}{\sqrt{n_{k}!}}|0\rangle,$
(19)
where it denotes a number state with $n_{k}$ quanta in the $k$th mode and
$a_{k}^{(1){\dagger}}$ are the creation operators at $t\leq 0$. Then the mean
number of photons created from vacuum in this state during the time interval
$[0,t_{1}]$ is 10
$\displaystyle\langle
n_{k}|a_{n}^{(3){\dagger}}a_{n}^{(3)}|n_{k}\rangle\approx
n_{k}\left\\{\delta_{nk}\right.$ $\displaystyle\
\left.+\frac{36n^{2}}{c^{4}\pi^{4}}\frac{1-\delta_{nk}}{(n-k)^{6}}\left[\sqrt{\frac{n}{k}}\frac{1}{6}x_{0}\ddot{x}(t_{1})-\sqrt{\frac{k}{n}}\frac{1}{6}x_{0}\ddot{x}(0)\right]^{2}\right\\}.$
(20)
where $a_{n}^{(3)}$ and $a_{n}^{(3){\dagger}}$ are the annihilation and
creation operators at $t>t_{1}$. Substituting Eqs. (II) and (14) into the
expansion (III) we can obtain the ensemble average of the expectation value of
the number operator
$\overline{\langle n_{k}|a_{n}^{(3){\dagger}}a_{n}^{(3)}|n_{k}\rangle}\approx
n_{k}\left\\{\delta_{nk}+\frac{1-\delta_{nk}}{(n-k)^{6}}A(n,k)\right\\},$ (21)
where
$\displaystyle
A(n,k)=\frac{n^{2}}{c^{4}\pi^{4}}x_{0}^{2}s^{2}\frac{2k_{B}\alpha
T^{5}}{3V}\left\\{\left(\frac{n}{k}+\frac{k}{n}-2e^{-\lambda
t_{1}}\right)\right.$ $\displaystyle\ \
\left.+\frac{\gamma^{2}}{\gamma^{2}-\lambda^{2}}\left[\frac{n}{k}\left(1-e^{(\lambda-\gamma)t_{1}}+e^{-2\gamma
t_{1}}-e^{-(\gamma+\lambda)t_{1}}\right)\right]\right\\}.$ (22)
Note that from Eq. (21) the statistics of photons created do not satisfy a
thermal distribution but a super-Poissonian distribution 10 . Therefore, it
turns out that photons generated in the cavity are different from thermal
photons. In this case the relative photon creation rate per volume in the
$k$th mode can be written in the form
$\displaystyle\tilde{P}_{k}(t_{1})$
$\displaystyle=\sum_{n}\frac{1}{t_{1}}\frac{1}{n_{k}}\left[\overline{\langle
n_{k}|a_{n}^{(3){\dagger}}a_{n}^{(3)}|n_{k}\rangle}-n_{k}\right]$ (23)
$\displaystyle\approx\sum_{n}\frac{1-\delta_{nk}}{t_{1}(n-k)^{6}}A(n,k).$
With the proviso that $t_{1}\gg 1/\gamma$ and $t_{1}\gg 1/\lambda$ the total
photon creation rate per volume in the lowest mode with $k=1$ and $n=2$ are
$\displaystyle\tilde{P}(t_{1})$ $\displaystyle=\sum_{k}\tilde{P}_{k}(t_{1})$
(24) $\displaystyle\approx\frac{2x_{0}^{2}s^{2}\overline{[\Delta
P]^{2}}}{t_{1}c^{4}\pi^{4}}\left[\frac{\gamma^{2}}{\lambda^{2}-\gamma^{2}}\left(e^{(\lambda-\gamma)t_{1}}-1\right)+\frac{5}{4}\right].$
We note that this equation is greater than zero in any case. Therefore, over
time the total number of photons created would continue to be generated and
increased by the DCE. To keep the thermal equilibrium of the system, it is
necessary to find a compensatory effect. This we shall discuss more fully in
Sec. .
## IV Upper Limit Of The Conductivity
From the previous discussion, we learn that thermal photons in the cavity
satisfy the Planck distribution, and photons created by the DCE satisfy the
super-Poissonian distribution. Thus, these two types of photons have different
distributions and are statistically independent each other. From Eq. (24) the
total number of photons generated would continue to be increased as time goes
on. If there are no other physical effects of compensation, the system will
transition to a non-equilibrium state. This will violate the second law of
thermodynamics.
Photons created by the DCE can not be directly converted into thermal photons.
Therefore, to meet the second law of thermodynamics, photons created must be
absorbed by conducting walls or plate. That is to say, the reflection
coefficient of the conducting walls and plate need be less than 1.
First the reflection coefficient of the conducting walls and plate is
expressed as $R$. Then the absorption rate per unit time can be written as
$1-R^{f}$ 15 , where $f=c/l$ is folding times of a light beam in the cavity.
At the same time, in order to satisfy the second law of thermodynamics, the
photon absorption rate per unit time must be greater than that of the
generation rate
$\left(1-R^{f}\right)\geq\tilde{P}(t).$ (25)
And according to Ref. 16 , the reflection coefficient can be written as
$R\approx 1-2\sqrt{\frac{2\omega\varepsilon_{0}}{\sigma}}$. Inserting Eq.(24)
into Eq.(25), we can get the expression of the conductivity as
$\displaystyle\sigma_{c}\leq\left(\frac{\sqrt{8\omega\varepsilon_{0}}ft_{1}c^{4}\pi^{4}}{2x_{0}^{2}s^{2}\overline{[\Delta
P]^{2}}\left[\frac{\gamma^{2}}{\lambda^{2}-\gamma^{2}}\left(e^{(\lambda-\gamma)t_{1}}-1\right)+\frac{5}{4}\right]}\right)^{2}.$
(26)
Note that the above expression (26) represents an upper limit for the
conductivity. For simplicity, different cases will be discussed. We shall
complete our study by considering various limiting cases. If
$\lambda\gg\gamma$, the relax time of radiation fluctuation $\lambda^{-1}$ is
much less than the characteristic time $\gamma^{-1}$ of the plate motion. Then
we can get
$\displaystyle\sigma_{c}\leq\left(\frac{\sqrt{8\omega\varepsilon_{0}}ft_{1}c^{4}\pi^{4}}{2x_{0}^{2}s^{2}\overline{[\Delta
P]^{2}}\left(\frac{\gamma^{2}}{\lambda^{2}}e^{\lambda
t_{1}}+\frac{5}{4}\right)}\right)^{2}.$ (27)
In the opposite limiting case $\lambda\ll\gamma$, the results are
$\displaystyle\sigma_{c}\leq\left(\frac{\sqrt{8\omega\varepsilon_{0}}ft_{1}c^{4}\pi^{4}}{2x_{0}^{2}s^{2}\overline{[\Delta
P]^{2}}}\frac{4}{9}\right)^{2}.$ (28)
By this means we can get an upper limit of the conductivity which is result
from the DCE. This result has no concern with the structure of conductor and
the property of its material. In nature, this is an inevitable consequence
that is caused by the compatibility between the second law of thermodynamics
and the dynamical Casimir Effect. Then one can simply estimate the value of
this upper limit.
Now we insert some explicit numbers to get the upper limits of the
conductivity. The parameters are given as follow: the mass of the plate $M\sim
0.01kg$, $\gamma=10^{-3}s^{-1}$ and the frequency of the lowest mode is
$\omega_{1}\sim 1GHz$ for $L\sim 0.1m$. Then in the former case approximately
$\sigma_{c}\leq 10^{22}S\cdot m^{-1}$ would be obtained at room temperature
$T\sim 290K$ during 1 hour, where we take $\lambda=\frac{3(1-R)c}{2L}$ (as in
the appendix). And in the latter case $\sigma_{c}\leq 10^{136}S\cdot m^{-1}$
would be gotten.
## V Conclusion
This paper discuss the possibility of upper limits of the conductivity by
virtue of the compatibility between the second law of thermodynamics and the
DCE. We first consider a three-dimensional model of a thermal radiation field
within a rectangular cavity with a thin conducting plate. The system to be
initially at thermal equilibrium, the pressure fluctuation in thermal
equilibrium will result in a pressure difference on both sides of plate. Then
we establish a Langevin equation describing the plate movement and derive the
time correlation function of the acceleration. It is important to notice that
the random motion of the conducting plate is in general nonzero, which may
lead to photon generation even from vacuum.
The relative photon generation rate per volume is derived in Sec.III. First of
all we calculate the numbers of photons created by the DCE in the right or
left cavity, which is due to random motion of the conducting plate. And it is
found that the statistics of photons created by DCE do not satisfy a thermal
distribution but a super-Poissonian distribution. Thus, it turns out that
Photons generated in this cavity obey a non-thermal distribution and present
an expression of the relative photon generation rate per volume. Finally, to
ensure the second law of thermodynamics, the photon absorption rate has to be
greater than that of the generation rate. Thus we obtain this upper limit of
the conductivity.
## Appendix
In this Appendix, we derive the expresssion of the coefficient $\lambda$ in
the rectangular cavity with a thin conducting plate.
The energy density in the thermal equilibrium can be expressed as 11
$\displaystyle u=\frac{E}{V}=\alpha T^{4}.$ (29)
And the intensity of emission is obtained as
$\displaystyle J=\frac{1}{4}c\alpha T^{4}.$ (30)
We study in this paper how the thermal radiation field with a pressure
fluctuation tends to equilibrium. The field only can exchange energy with the
cavity wall in relaxation time. The energy flux of the exchange energy is
expressed by the temperature difference between the cavity wall and the
thermal radiation field. So when the system comes back to equilibrium, the
change in the energy density is given by
$\displaystyle\Delta u=\alpha T^{4}-\alpha T_{0}^{4},$ (31)
where $T$ and $T_{0}$ are the temperature of radiation field and cavity wall
respectively. In this case the change in the intensity of absorption can be
read from Eq. (30) and Eq. (31)
$\displaystyle A\Delta J=\frac{1}{4}Ac\alpha T^{4}-\frac{1}{4}Ac\alpha
T_{0}^{4}=\frac{1}{4}Ac\Delta u,$ (32)
where $A$ is the absorptivity of the cavity wall. Then, for the rectangular
cavity, the variation of internal energy per unit time can be represented as
$\displaystyle\frac{\Delta E}{\Delta t}=\tilde{S}A\Delta J,$ (33)
where $\tilde{S}$ is the total surface area of the cavity. Thus we get the
relaxation time for the establishment of complete equilibrium
$\displaystyle\Delta t=\frac{\Delta E}{\tilde{S}A\Delta J}.$ (34)
Considering the coefficient $\lambda^{-1}$ being the order of magnitude of the
relaxation time 11 , we put Eq. (32) into the above formula (34) and then
obtain
$\displaystyle\lambda=\frac{3Ac}{2L_{x}}=\frac{3(1-R)c}{2L_{x}}.$ (35)
Here $R$ is the reflection coefficient and the equation $A=1-R$ has been used.
###### Acknowledgements.
This work is supported in part by the National Natural Science Foundation of
China (Grant No. 2010CB832800).
## References
* (1) F. Jülicher, A. Ajdari, and J. Prost, Rev. Mod. Phys. 69, 1269 (1997).
* (2) S. Muhuri and I. Pagonabarraga, Phys. Rev. E 82, 021925 (2010).
* (3) P. Reimann, Phys. Rep. 361, 57 (2002).
* (4) K. L. Sebastian, Phys. Rev. E 61, 937 (2000).
* (5) P. Hänggi and F. Marchesoni, Rev. Mod. Phys. 81, 387 (2009).
* (6) G. T. Moore, J. Math. Phys. 11, 2679 (1970); S. A. Fulling and P. C. W. Davies, Proc. R. Soc. London Ser. A 348, 393 (1976); P. C. W. Davies and S. A. Fulling, Proc. R. Soc. London Ser. A 356, 237 (1977).
* (7) V. V. Dodonov and A. B. Klimov, Phys. Rev. A 53, 2664 (1996).
* (8) A. Agnesi, C. Braggio, G. Bressi, G. Carugno, G. Galeazzi, F. Pirzio, G. Reali, G. Ruoso, and D. Zanello, J. Phys. A 41, 164024 (2008); A. Agnesi, C. Braggio, G. Bressi, G. Carugno, F. Della Valle, G. Galeazzi, G. Messineo, F. Pirzio, G. Reali,G. Ruoso, D. Scarpa, and D. Zanello, J. Phys.: Conf. Ser. 161, 012028 (2009).
* (9) C. K. Law, Phys. Rev. A 49, 433 (1994).
* (10) S. Sarkar, Quantum Opt. 4, 345 (1992).
* (11) L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1 (Pergamon, New York, 1968).
* (12) J. H. Weiner, Statistical Mechanics of Elasticity (Dover, New York, 2002).
* (13) F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965).
* (14) C. K. Law, Phys. Rev. A 51, 2537 (1995).
* (15) S. C. Wu, Z. Z. Wan, H. Li, and Z.-Z. Liu, Chin. Phys. Lett. 23, 3173 (2006)
* (16) J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
|
arxiv-papers
| 2011-10-30T03:20:57 |
2024-09-04T02:49:23.698401
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiao-Min Bei and Zhong-Zhu Liu",
"submitter": "Xiaomin Bei",
"url": "https://arxiv.org/abs/1110.6576"
}
|
1110.6587
|
# Nonclassicality and decoherence of photon-added squeezed thermal state in
thermal environment
Li-Yun Hu1,2 and Zhi-Ming Zhang2 E-mail: hlyun2008@126.com.E-mail:
zmzhang@scnu.edu.cn 1College of Physics & Communication Electronics, Jiangxi
Normal University, Nanchang 330022, China
2Key Laboratory of Photonic Information Technology of Guangdong Higher
Education Institutes,
SIPSE & LQIT, South China Normal University, Guangzhou 510006, China
###### Abstract
Theoretical analysis is given of nonclassicality and decoherence of the field
states generated by adding any number of photons to the squeezed thermal state
(STS). Based on the fact that the squeezed number state can be considered as a
single-variable Hermite polynomial excited state, the compact expression of
the normalization factor is derived, a Legendre polynomial. The
nonclassicality is investigated by exploring the sub-Poissonian and negative
Wigner function (WF). The results show that the WF of single photon-added STS
(PASTS) always has negative values at the phase space center. The decoherence
effect on 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 STS.
PACS number(s): 42.50.Dv, 03.65.Wj, 03.67.Mn
## I Introduction
Generation and manipulation of non-classical light field has been a topic of
great interest in quantum optics and quantum information science 1 . Many
experimental schemes have been proposed to generate nonclassical states of
optical field. Among them, subtracting photons from and/or adding photons to
quantum states have been paid much attention because these fields exhibit an
abundant of nonclassical properties and may give access to a complete
engineering of quantum states and to fundamental quantum phenomena 2 ; 3 ; 4 ;
5 ; 6 ; 7 ; 8 ; 9 ; 10 . For example, quantum-to-classical transition has been
realized experimentally through single-photon-added coherent states of light.
These states allow one to witness the gradual change from the spontaneous to
the stimulated regimes of light emission 4 . For $m$-photon-added coherent
state in the dissipative channel, the nonclassical properties are studied
theoretically 11 by deriving the analytical expression of the Wigner function
(WF), which turns out to be a Laguerre-Gaussian function. As another example,
photon addition and subtraction experimentally have been employed to probe
quantum commutation rules by Parigi et al. In fact, they have implemented
simple alternated sequences of photon creation (addition) and annihilation
(subtraction) on a thermal field and observed the noncommutativity of the
creation and annihilation operators 6 . In addition, photon
subtraction/addition can be applied to improve entanglement between Gaussian
states 12 ; 13 , loophole-free tests of Bell’s inequality 14 ; 15 , and
quantum computing 16 .
On the other hand, it is interesting to notice that subtracting or adding one
photon from/to pure squeezed vacuum can generate the same output state, i.e.,
squeezed single-photon state 17 . Actually, the photon addition is able to
generate a nonclassical state (e.g coherent and thermal states), which is
quite different from photon subtraction only from a nonclassical state 18 ; 19
; 20 . In addition, the resulting states obtained by successive photon
subtractions or additions are different from each other. For instance,
successive two-photon additions [$a^{{\dagger}2}$] and successive two-photon
subtractions [$a^{2}$] will result in the same state produced by using
subtraction-addition ($a^{{\dagger}}a$) and addition-subtraction
($aa^{{\dagger}}$), respectively. In Ref.21 , two photon-subtracted squeezed
vacuum is used to generate the squeezed superposition of coherent states with
high fidelities and large amplitudes.
In general, different non-Gaussian operators (e.g subtracting and adding
photon) will suffer different effects from the surroundings, thus it is
important to know which operator is more robust compared to the other under an
identical initial quantum state when the environment is taken into account.
Very recently, the robustness of several superposition states is studied by
using the linear entropy under a thermal environment 22 . In this paper, we
shall introduce a kind of nonclassical state—photon-addition squeezed thermal
state (PASTS), generated by adding photon to squeezed thermal state (STS)
which can be considered as a generalized Gaussian state. Then we shall
investigate the nonclassical properties and decoherence of single-mode any
number PASTS under the influence of thermal environment.
This paper is organized as follows. In Sec. II we introduce the single-mode
PASTS. By converting the PASTS to an Hermite polynomial excitation squeezed
vacuum state, we derive a compact expression for the normalization factor of
PASTS, which is an $m$-order Legendre polynomial of squeezing parameter
$\lambda$ and mean number $n_{c}$ of thermal state, where $m$ is the number of
added photons. In Sec III, we discuss the nonclassical properties of the PASTS
in terms of sub-Poissonian statistics and the negativity of its WF. We find
the negative region of WF in phase space and there is an upper bound value of
$\lambda$ for this state to exhibit sub-Poissonian statistics which increases
as $m$ increases. Then, in Sec. IV we derive the explicitly analytical
expression of time evolution of WF of the arbitrary PASTS in the thermal
channel and discuss the loss of nonclassicality in reference of the negativity
of WF. The threshold value of decay time corresponding to the transition of
the WF from partial negative to completely positive definite is obtained at
the center of the phase space, which is independent of parameters $\lambda$
and $n_{c}$. It shown that the WF for single PASTS (SPASTS) has always
negative value for all parameters $\lambda$ and $n_{c}$ if the decay time
$\kappa t<\frac{1}{2}\ln[(2\mathcal{N}+2)/(2\mathcal{N}+1)]\ $(see Eq.(46)
below), where $\mathcal{N}$ denotes the average thermal photon number in the
environment with dissipative coefficient $\kappa$. Comparing to the case of
single photon-subtraction STS (SPSSTS), the decoherence time of SPASTS is
longer. In this sense, the photon-addition non-Gaussian states present more
robust contrast to decoherence than photon-subtraction ones. The reason may be
that the amount of non-Gaussianity for SPASTS is larger than that for SPSSTS
as presented in Sec. V. Conclusions are involved in the last section.
## II Photon-addition squeezed thermal state (PASTS)
The $m$-photon-added scheme, denoted by the mapping $\rho\rightarrow
a^{{\dagger}m}\rho a^{m},$ was first proposed by Agarwal and Tara 18 . Here,
we introduce the PASTS. Theoretically, the PASTS can be obtained by repeatedly
operating the photon creation operator $a^{\dagger}$ on a STS, so its density
operator is given by
$\rho_{ad}=C_{a,m}^{-1}a^{{\dagger}m}S_{1}^{\dagger}\rho_{th}S_{1}a^{m},$ (1)
where $m$ is the added photon number (a non-negative integer), $C_{a,m}^{-1}$
is the normalization constant to be determined, and
$S_{1}=\exp[\lambda(a^{2}-a^{\dagger 2})/2]$ is the single-mode squeezing
operator with $\lambda$ being squeezing parameter 23 ; 24 . $\rho_{th}$ is a
single field mode with frequency $\omega$ in a thermal equilibrium state
corresponding to absolute temperature $T$, whose the density operator is 25
$\rho_{th}=\sum_{n=0}^{\infty}\frac{n_{c}^{n}}{\left(n_{c}+1\right)^{n+1}}\left|n\right\rangle\left\langle
n\right|=\frac{1}{n_{c}}\vdots e^{-\frac{1}{n_{c}}a^{{\dagger}}a}\vdots,$ (2)
($\vdots$ $\vdots$ denoting antinormally ordering) which implies that the
density operator $\rho_{th}$ can be expanded as
$\rho_{th}=\frac{1}{n_{c}}\int\frac{d^{2}\alpha}{\pi}e^{-\frac{1}{n_{c}}\left|\alpha\right|^{2}}\left|\alpha\right\rangle\left\langle\alpha\right|,$
(3)
where $n_{c}=[\exp(\omega/(kT))-1]^{-1}$ being the average photon number of
the thermal state $\rho_{th}$ and $k_{B}$ being Boltzmann’s constant. Eq.(3)
is useful for later calculation.
### II.1 Squeezed number state as a Hermite polynomial excited state
Recalling that the single-mode squeezed operator $S_{1}$ has its natural
expression in the coordinate representation 26 ,
$S_{1}=\frac{1}{\sqrt{\mu}}\int_{-\infty}^{\infty}dq\left|\frac{q}{\mu}\right\rangle\left\langle
q\right|,\mu=e^{\lambda},$ (4)
where $\left|q\right\rangle$ is the eigenstate of
$Q=(a+a^{{\dagger}})/\sqrt{2}$, $Q\left|q\right\rangle=q\left|q\right\rangle,$
and
$\left|q\right\rangle=\pi^{-1/4}\exp\left\\{-\frac{q^{2}}{2}+\sqrt{2}qa^{\dagger}-\frac{a^{\dagger
2}}{2}\right\\}\left|0\right\rangle.$ (5)
Thus, using Eq.(5) and the overlap relation
$\left\langle
q\right|\left.n\right\rangle=\frac{1}{\sqrt{2^{n}n!\sqrt{\pi}}}e^{-q^{2}/2}H_{n}\left(q\right),$
(6)
where $H_{n}\left(q\right)$ is the single-variable Hermite polynomial then
$S_{1}\left|n\right\rangle$ can be expressed as
$\displaystyle S_{1}\left|n\right\rangle$
$\displaystyle=\int_{-\infty}^{\infty}\frac{dq}{\sqrt{2^{n}n!\mu\sqrt{\pi}}}e^{-q^{2}/2}H_{n}\left(q\right)\left|\frac{q}{\mu}\right\rangle$
$\displaystyle=\frac{\text{sech}^{1/2}\lambda}{\sqrt{2^{n}n!}}\frac{\partial^{n}}{\partial\tau^{n}}\left.e^{\sqrt{2}a^{\dagger}\tau\text{sech}\lambda+(\tau^{2}-\frac{1}{2}a^{\dagger
2})\tanh\lambda}\left|0\right\rangle\right|_{\tau=0}$
$\displaystyle=\frac{\left(i\sqrt{\tanh\lambda}\right)^{n}}{\sqrt{2^{n}n!}}H_{n}\left(\frac{a^{\dagger}\text{sech}\lambda}{i\sqrt{2\tanh\lambda}}\right)S_{1}\left|0\right\rangle,$
(7)
where we have set $\text{sech}\lambda=2\mu/(\mu^{2}+1),$
$\tanh\lambda=(\mu^{2}-1)/(\mu^{2}+1),$ and we have used
$S_{1}\left|0\right\rangle=\text{sech}^{1/2}\lambda\exp[-a^{\dagger
2}/2\tanh\lambda]\left|0\right\rangle$ as well as the generating function of
$H_{n}\left(q\right)$ 27 :
$H_{n}\left(q\right)=\left.\frac{\partial^{n}}{\partial\tau^{n}}\exp\left(2q\tau-\tau^{2}\right)\right|_{\tau=0}.$
(8)
Eq.(7) indicates that the single-mode squeezed number state
$S_{1}\left|n\right\rangle$ is actually a Hermite polynomial excited squeezed
vacuum state 28 . Obviously, when $n=0,H_{0}\left(q\right)=1,$ Eq.(7) just
reduces to single-mode squeezed vacuum. While for $n=1,2,$noting
$H_{1}\left(q\right)=2q$ and $H_{2}\left(q\right)=4q^{2}-2,$ Eq.(7) become
$\displaystyle S_{1}\left|1\right\rangle$
$\displaystyle=a^{\dagger}\text{sech}\lambda\text{
}S_{1}\left|0\right\rangle,$ $\displaystyle S_{1}\left|2\right\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(a^{\dagger
2}\text{sech}^{2}\lambda+\tanh\lambda\right)S_{1}\left|0\right\rangle,$ (9)
respectively. It is interesting to notice that the single photon-added
squeezed vacuum (PASV) is equal to the squeezed number state
$S_{1}\left|1\right\rangle$, and the two PASV can be considered as a
superposition of the squeezed number state $S_{1}\left|2\right\rangle$ and the
squeezed vacuum.
### II.2 Normalization of PASTS
To fully describe a quantum state, its normalization is usually necessary.
Next, we shall employ the fact (7) to realize our aim. First, let us derive
the normally ordering form of STS $\rho_{s}\equiv
S_{1}^{\dagger}\rho_{th}S_{1}$, which is convenient for further calculation of
normalization.
Using Eqs.(2) and (7), we can rewrite the STS $\rho_{s}$ as
$\displaystyle\rho_{s}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{n_{c}^{n}}{\left(n_{c}+1\right)^{n+1}}S_{1}\left(-\lambda\right)\left|n\right\rangle\left\langle
n\right|S_{1}^{{\dagger}}\left(-\lambda\right)$
$\displaystyle=\frac{\text{sech}\lambda}{n_{c}+1}\sum_{n=0}^{\infty}\frac{\left(n_{c}\tanh\lambda\right)^{n}}{2^{n}n!\left(n_{c}+1\right)^{n}}\colon
H_{n}\left(\frac{-a^{\dagger}\text{sech}\lambda}{\sqrt{2\tanh\lambda}}\right)$
$\displaystyle\times\exp\left[\frac{1}{2}\left(a^{2}+a^{\dagger
2}\right)\tanh\lambda-a^{{\dagger}}a\right]H_{n}\left(\frac{-a\text{sech}\lambda}{\sqrt{2\tanh\lambda}}\right)\colon,$
(10)
where $S_{1}^{{\dagger}}\left(-\lambda\right)=S_{1}\left(\lambda\right)$ and
the vacuum projector $\left|0\right\rangle\left\langle
0\right|=\colon\exp\left[-a^{{\dagger}}a\right]\colon$ is used. Further using
the two-linear generating function of Hermite polynomial 29 ,
$\displaystyle\sum_{n=0}^{\infty}\frac{t^{n}}{2^{n}n!}H_{n}\left(x\right)H_{n}\left(y\right)$
$\displaystyle=\frac{1}{\sqrt{1-t^{2}}}\exp\left[\frac{2txy-t^{2}\left(x^{2}+y^{2}\right)}{1-t^{2}}\right],$
(11)
we can directly obtain the normally ordering form of STS,
$\rho_{s}=\frac{1}{\sqrt{A}}\colon\exp\left[\frac{C}{2}\left(a^{\dagger
2}+a^{2}\right)+\left(B-1\right)a^{\dagger}a\right]\colon,$ (12)
where we have set
$\displaystyle A$
$\displaystyle=n_{c}^{2}+\left(2n_{c}+1\right)\cosh^{2}\lambda,$
$\displaystyle B$ $\displaystyle=\frac{n_{c}}{A}\left(n_{c}+1\right),$
$\displaystyle C$ $\displaystyle=\frac{\allowbreak 2n_{c}+1}{2A}\sinh
2\lambda.$ (13)
By introducing $a=(Q+iP)/\sqrt{2}$ and $a^{\dagger}=(Q-iP)/\sqrt{2}$, Eq.(12)
can be put into another form
$\rho_{s}=\frac{1}{\tau_{1}\tau_{2}}\colon\exp\left[-\frac{Q^{2}}{2\tau_{1}^{2}}-\frac{P^{2}}{2\tau_{2}^{2}}\right]\colon,$
(14)
where $\tau_{1}\tau_{2}=\sqrt{A},$ and
$\displaystyle 2\tau_{1}^{2}$ $\displaystyle=\left(\allowbreak
2n_{c}+1\right)e^{2\lambda}+1,$ $\displaystyle 2\tau_{2}^{2}$
$\displaystyle=\left(\allowbreak 2n_{c}+1\right)e^{-2\lambda}+1.$ (15)
Eq.(12) or (14) is a compact expression of the STS, which is just a Gaussian
distribution within normal order for operators $Q$ and $P$ 30 .
Next, we shall derive the normalization factor for PASTS. Employing Eq.(12),
the PASTS reads as
$\rho_{ad}=\frac{C_{a,m}^{-1}}{\tau_{1}\tau_{2}}\colon
a^{{\dagger}m}\exp\left[\frac{C}{2}\left(a^{\dagger
2}+a^{2}\right)+\left(B-1\right)a^{\dagger}a\right]a^{m}\colon.$ (16)
Thus the normalization factor $C_{a,m}$ is
$\left(1=\mathtt{tr}\rho_{ad}\right)$
$\displaystyle C_{a,m}$
$\displaystyle=\frac{1}{\tau_{1}\tau_{2}}\int\frac{d^{2}\alpha}{\pi}\left|\alpha\right|^{2m}e^{-\left(1-B\right)\left|\alpha\right|^{2}+\frac{C}{2}\left(\alpha^{\ast
2}+\alpha^{2}\right)}$ $\displaystyle=\frac{\partial^{2m}}{\partial
s^{m}\partial
t^{m}}\int\frac{d^{2}\alpha}{\pi\tau_{1}\tau_{2}}\left.e^{-\left(1-B\right)\left|\alpha\right|^{2}+s\alpha^{\ast}+t\alpha+\frac{C}{2}\left(\alpha^{\ast
2}+\alpha^{2}\right)}\right|_{s=t=0}$
$\displaystyle=\frac{\partial^{2m}}{\partial s^{m}\partial
t^{m}}\exp\left[A\left(1-B\right)st+\frac{AC}{2}\left(s^{2}+t^{2}\right)\right]_{s=t=0},$
(17)
where we have used the completeness relation of coherent state, and
$[\left(1-B\right)^{2}-C^{2}]^{-1}=\tau_{1}^{2}\tau_{2}^{2}=A$, as well as the
integration formula 31
$\displaystyle\int\frac{d^{2}z}{\pi}\exp\left(\zeta\left|z\right|^{2}+\xi
z+\eta z^{\ast}+fz^{2}+gz^{\ast 2}\right)$
$\displaystyle=\frac{1}{\sqrt{\zeta^{2}-4fg}}\exp\left[\frac{-\zeta\xi\eta+\xi^{2}g+\eta^{2}f}{\zeta^{2}-4fg}\right],$
(18)
whose convergent condition is Re$\left(\zeta\pm f\pm g\right)<0,\
$Re($\zeta^{2}-4fg)/(\zeta\pm f\pm g)<0.$
Recalling the newly found formula of Legendre polynomial 32 ; 33 , i.e.,
$\displaystyle\frac{\partial^{2m}}{\partial
t^{m}\partial\tau^{m}}\left.\exp\left(-t^{2}-\tau^{2}+\frac{2x\tau
t}{\sqrt{x^{2}-1}}\right)\right|_{t,\tau=0}$
$\displaystyle=\frac{2^{m}m!}{\left(x^{2}-1\right)^{m/2}}P_{m}\left(x\right),$
(19)
and noticing $x^{2}-1=AC^{2},$ together with
$x=\sqrt{A}\left(1-B\right)=\left[A-n_{c}\left(n_{c}+1\right)\right]/\sqrt{A}$,
we have
$\displaystyle C_{a,m}$
$\displaystyle=\frac{\left(AC\right)^{m}}{2^{m}}\frac{\partial^{2m}}{\partial
s^{m}\partial
t^{m}}\exp\left[\frac{2}{C}\left(1-B\right)st-s^{2}-t^{2}\right]_{s=t=0}$
$\displaystyle=m!A^{m/2}P_{m}\left(\bar{B}/\sqrt{A}\right),$ (20)
which indicates that $C_{a,m}$ is also just related to Legendre polynomial,
and
$\bar{B}=n_{c}\cosh 2\lambda+\cosh^{2}\lambda.$ (21)
It is noted that, for the case of no-photon-addition with $m=0$, $C_{a,0}=1$
as expected. Under the case of $m$-photon-addition thermal state (no
squeezing) with $\bar{B}=\allowbreak n_{c}+1$,
$A=\allowbreak\left(n_{c}+1\right)^{2},$ and $P_{m}\left(1\right)=1$, then
$C_{a,m}=m!\left(n_{c}+1\right)^{m}.$ The same result as Eq.(32) can be found
in Ref.34 .
## III Nonclassical properties of PASTS
In this section, we shall discuss the nonclassical properties of PASTS in
terms of sub-Poissonian statistics and the negativity of its WF.
### III.1 Sub-Poissonian nature of PASTS
The nonclassicality of the PASTS can be analyzed by studying its sub-
Poissonian distribution. Using Eq.(20) we can directly calculate:
$\displaystyle\left\langle a^{{\dagger}}a\right\rangle$
$\displaystyle=\frac{C_{a,m+1}}{C_{a,m}}-1,$ (22) $\displaystyle\left\langle
a^{{\dagger}2}a^{2}\right\rangle$
$\displaystyle=\frac{C_{a,m+2}}{C_{a,m}}-4\frac{C_{a,m+1}}{C_{a,m}}+2.$ (23)
Thus the Mandel’s $\mathcal{Q}$-parameter 35 can be obtained by substituting
Eqs.(22) into $\mathcal{Q}\equiv\left\langle a^{\dagger
2}a^{2}\right\rangle/\left\langle a^{{\dagger}}a\right\rangle-\left\langle
a^{{\dagger}}a\right\rangle,$
$\mathcal{Q=}\frac{C_{a,m+2}-4C_{a,m+1}+2C_{a,m}}{C_{a,m+1}-C_{a,m}}-\frac{C_{a,m+1}-C_{a,m}}{C_{a,m}}.$
(24)
The negativity of the Mandel’s $\mathcal{Q}$-parameter refers to sub-
Poissonian statistics of the state. In order to see clearly the variation of
$\mathcal{Q}$-parameter with $\lambda$ and $n_{c}$, we show the plots of
$\mathcal{Q}$-parameter in Fig.1, from which one can clearly see that, for a
given small $n_{c}$ value, $\mathcal{Q}$-parameter becomes negative ($m\neq
0)$ when $\lambda$ is less than a certain threshold value which increases as
$m$ increases; while for $m=0\ $or a large $n_{c}$, $\mathcal{Q}$ is always
positive. This implies that the nonclassicality is enhanced by adding photon
to squeezed state. Here, we should emphasize that the WF has negative region
for all $\lambda$ and $n_{c},$ and thus the PASTS is nonclassical.
Figure 1: (Color online) The $\mathcal{Q}$-parameter as the function of
squeezing parameter$r$ for different $m=0,1,2,3,4,19,20$ with a small $n_{c}$
value.
### III.2 Photon-number distribution (PND) of the PASTS
The photon-number distribution (PND) is a key characteristic of every optical
field. For this purpose, we first calculate the PND of STS, then the PND of
PASTS can be directly obtain. The PND, i.e., the probability of finding $n$
photons in a quantum state described by the density operator $\rho$, is
$\mathcal{P}(n)=\left\langle n\right|\rho\left|n\right\rangle.$ So the PND of
the STS is
$\mathcal{P}(n)=\left\langle
n\right|S_{1}^{\dagger}\rho_{th}S_{1}\left|n\right\rangle.$ (25)
Using the fact in (7) and the P-representation of $\rho_{th}$ (3), Eq.(25) can
be directly written as
$\displaystyle\mathcal{P}(n)$
$\displaystyle=\frac{\text{sech}\lambda}{2^{n}n!n_{c}}\frac{\partial^{2n}}{\partial
t^{n}\partial\tau^{n}}\exp\left[\left(t^{2}+\tau^{2}\right)\tanh\lambda\right]$
$\displaystyle\times\int\frac{d^{2}\alpha}{\pi}\exp\left[\sqrt{2}\left(\alpha
t+\alpha^{\ast}\tau\right)\text{sech}\lambda-\frac{n_{c}+1}{n_{c}}\left|\alpha\right|^{2}\right]$
$\displaystyle\times\exp\left[-\frac{\tanh\lambda}{2}\left(\alpha^{2}+\alpha^{\ast
2}\right)\right]_{\tau=t=0}$
$\displaystyle=\frac{\text{sech}\lambda}{2^{n}n!\sqrt{A}}\frac{\partial^{2n}}{\partial
t^{n}\partial\tau^{n}}\exp\left[2Bt\tau+C\left(t^{2}+\tau^{2}\right)\right]_{\tau=t=0}.$
(26)
In a similar way to deriving Eq.(20), using Eq.(19) we have
$\mathcal{P}(n)=\frac{D^{n/2}}{\sqrt{A}}P_{n}\left(B/\sqrt{D}\right),$ (27)
where
$D=\frac{n_{c}^{2}-\left(2n_{c}+1\right)\sinh^{2}\lambda}{n_{c}^{2}+\left(2n_{c}+1\right)\cosh^{2}\lambda}.$
(28)
Eq.(27) shows that the PND of STS is the Legendre polynomial of $B/\sqrt{D}.$
In particular, when $\lambda=0,$ $A=(n_{c}+1)^{2}$ and
$B/\sqrt{D}=1,D=n_{c}^{2}/(n_{c}+1)^{2},$ then Eq.(27) becomes
$\mathcal{P}(n)=n_{c}^{n}/(n_{c}+1)^{n},$ corresponding to the PND of thermal
state 34 . In fact, we can also check Eq.(27) using the normalization
condition. Note that the Legendre polynomial can also be defined as the
coefficients in a Taylor series expansion 36
$\frac{1}{\sqrt{1-2xt+t^{2}}}=\sum_{n=0}^{\infty}P_{n}\left(x\right)t^{n},$
(29)
thus $\sum_{n=0}^{\infty}\mathcal{P}(n)=1/\sqrt{A(1-2B+D)}=1$ as expected.
Next, we turn to present the PND of PASTS. From Eq.(27) and noting
$a^{{\dagger}m}\left|n\right\rangle=\sqrt{(m+n)!/n!}\left|m+n\right\rangle$
and $a^{m}\left|n\right\rangle=\sqrt{n!/(n-m)!}\left|n-m\right\rangle$, it
then directly follows
$\displaystyle\mathcal{P}_{2}(n)$ $\displaystyle=C_{a,m}^{-1}\left\langle
n\right|a^{{\dagger}m}\rho_{s}a^{m}\left|n\right\rangle$
$\displaystyle=\frac{n!C_{a,m}^{-1}D^{(n-m)/2}}{(n-m)!\sqrt{A}}P_{n-m}\left(B/\sqrt{D}\right).$
(30)
Eq.(30) is the PND of PASTS, a Legendre polynomial with a condition
$n\geqslant m$ which implies that the photon-number ($n$) involved in PASTS is
always no-less than the photon-number ($m$) operated on the STS, and there is
no photon distribution when $n<m$). For some other non-Gaussian states, such
as
$a^{{\dagger}n}a^{m}\rho_{s}a^{{\dagger}m}a^{n},a^{m}a^{{\dagger}n}\rho_{s}a^{n}a^{{\dagger}m},$
and $a^{m}\rho_{s}a^{{\dagger}m},$ their PNDs can also be directly obtained by
using Eq.(27). In Fig. 2, the PND is shown for different values
$\left(\lambda,n_{c}\right)$ and $m.$ By adding photons, we have been able to
move the peak from zero photons to nonzero photons (see blue and red bar in
Fig.2). The position of peak depends on how many photons are created and how
much the state is squeezed initially. The probability of PND becomes smaller
with the increasement of squeezing parameter (see red and green bar in Fig.2).
Figure 2: (Color online) Photon-number distributions of PASTS with n̄=1 for
${\small\lambda}$=0.3, m=0 (blue bar); ${\small\lambda}$=0.3, m=1 (red bar),
${\small\lambda}$=0.3, m=5 (yellow bar), and ${\small\lambda}$=0.8, m=1 (green
bar).
## IV Wigner function of PASTS
Next, the normally ordering form Eq.(12) is applied to deduce the WF of PASTS.
The partial negativity of WF is indeed a good indication of the highly
nonclassical character of the state. Therefore it is worth of obtaining the WF
for any states. The WF $W\left(\alpha,\alpha^{\ast}\right)$ associated with a
quantum state $\rho$ can be derived as follows 23 :
$W\left(\alpha,\alpha^{\ast}\right)=e^{2\left|\alpha\right|^{2}}\int\frac{\mathtt{d}^{2}\beta}{\pi^{2}}\left\langle-\beta\right|\rho\left|\beta\right\rangle
e^{2\left(\alpha\beta^{\ast}-\alpha^{\ast}\beta\right)},$ (31)
where $\left|\beta\right\rangle=\exp(-\left|\beta\right|^{2}/2+\beta
a^{{\dagger}})\left|0\right\rangle$ is the coherent state.
Substituting Eq.(16) into Eq.(31), we can finally obtain the WF of PASTS (see
Appendix A),
$W\left(\alpha,\alpha^{\ast}\right)=F_{m}\left(\alpha,\alpha^{\ast}\right)W_{0}\left(\alpha,\alpha^{\ast}\right),$
(32)
where $W_{0}\left(\alpha,\alpha^{\ast}\right)$ is the WF of STS,
$\displaystyle W_{0}\left(\alpha,\alpha^{\ast}\right)$
$\displaystyle=\frac{1}{\pi\allowbreak\left(\allowbreak
2n_{c}+1\right)\allowbreak}\exp\left[-\frac{2\cosh
2r}{2n_{c}+1}\left|\alpha\right|^{2}\right.$ $\displaystyle+\left.\frac{\sinh
2r}{\allowbreak 2n_{c}+1}\left(\alpha^{2}+\alpha^{\ast}{}^{2}\right)\right],$
(33)
and
$\displaystyle F_{m}\left(\alpha,\alpha^{\ast}\right)$
$\displaystyle=\frac{\left(m!\right)^{2}C_{am}^{-1}\sinh^{m}2\lambda}{2^{2m}\left(2n_{c}+1\right)^{m}}$
$\displaystyle\times\sum_{l=0}^{m}\frac{\left(-1\right)^{l}2^{2l}\left(n_{c}+\cosh^{2}\lambda\right)^{l}}{l!\left[\left(m-l\right)!\right]^{2}\sinh^{l}2\lambda}\left|H_{m-l}(\bar{\gamma})\right|^{2},$
(34)
where $\bar{\gamma}=[\alpha^{\ast}\sinh
2\lambda-2\alpha(\cosh^{2}\lambda+n_{c})]/\\{i[\left(2n_{c}+1\right)\sinh
2\lambda]^{1/2}\\}.$ Eq.(32) is the analytical expression of WF for PASTS,
related to single-variable Hermite polynomials. In particular, when $m=0,$
$F_{0}\left(\alpha,\alpha^{\ast}\right)=1,$ Eq.(32) becomes
$W\left(\alpha,\alpha^{\ast}\right)=W_{0}\left(\alpha,\alpha^{\ast}\right)$;
while for $\lambda=0$, note $C_{am}=m!\left(n_{c}+1\right)^{m}$,
$W_{0}\left(\alpha,\alpha^{\ast}\right)=e^{-2\left|\alpha\right|^{2}/\allowbreak\left(\allowbreak
2n_{c}+1\right)}/\allowbreak[\pi\left(\allowbreak 2n_{c}+1\right)]$ and
$F_{m}\left(\alpha,\alpha^{\ast}\right)=\left(-1\right)^{m}/\left(2n_{c}+1\right)^{m}L_{m}[4\left(n_{c}+1\right)\left|\alpha\right|^{2}/\left(2n_{c}+1\right)]$,
Eq.(32) reduces to
$W\left(\alpha,\alpha^{\ast}\right)=\frac{\left(-1\right)^{m}e^{-\frac{2\left|\alpha\right|^{2}}{2\bar{n}+1}}}{\pi\allowbreak\allowbreak\left(2n_{c}+1\right)^{m+1}}L_{m}\left(\frac{4\left(n_{c}+1\right)}{2n_{c}+1}\left|\alpha\right|^{2}\right),$
(35)
which corresponds to the WF of $m$-photon added thermal state 34 , and can be
checked directly from Eq.(A3). In addition, for $m=1,$[single-photon-added
squeezed thermal state (SPASTS)], $C_{a1}=\bar{B}$ (20), the special WF of
SPASTS is
$W_{1}\left(\alpha,\alpha^{\ast}\right)=F_{1}\left(\alpha,\alpha^{\ast}\right)W_{0}\left(\alpha,\alpha^{\ast}\right),$
(36)
where
$F_{1}\left(\alpha,\alpha^{\ast}\right)=\frac{\sinh
2\lambda}{\left(2n_{c}+1\right)\bar{B}}\left[\left|\bar{\gamma}\right|^{2}-\frac{n_{c}+\cosh^{2}\lambda}{\sinh
2\lambda}\right].$ (37)
Noting $\bar{B}>0$, thus from Eq.(37) one can see that when the factor
$F_{1}\left(\alpha,\alpha^{\ast}\right)<0,$ the WF of SPASTS has its negative
distribution in phase space. This indicates that the WF of SPASTS always has
the negative values at the phase space center $\alpha=0$ ($\bar{\gamma}=0$),
which is different from the case of single-photon-subtracted STS with a
condition $n_{c}<\sinh^{2}\lambda$ 32 , but similar to the case of single-
photon-added/subtracted squeezed vacuum 28 ; 37 .
Figure 3: (Color online) Wigner function distributions ${\small
W}\left(\alpha,\alpha^{\ast}\right)$ of PASTS with $\lambda=0.3$ for different
$n_{c}$ and $m$ values (a) $n_{c}=0.1,m=1;$(b) $n_{c}=0.5,m=1;$ (c)
$n_{c}=0.1,m=2;$ (d) $n_{c}=0.1,m=3.$
Using Equations (32)-(34) we show the plots of WF in the phase space in Fig. 3
for the squeezing parameter ($\lambda=0.3$) with different photon-added
numbers $m$ and average number $n_{c}$ of the thermal state. One can see
clearly that there is some negative region of the WF in the phase space which
implies the nonclassicality of this state. In addition, the squeezing effect
in one of the quadratures is clear in the plots (see Figure 3(a)), which is
another evidence of the nonclassicality of this state. The WF has its minimum
value for $m=1,3$ at the center of phase space ($\alpha=0$) (see Fig. 2(a) and
(d)). The case is not true for $m=2$ (see Fig. 2(c)). For $m=2$, there are two
negative regions of the WF, which differs from the case of single PASTS. In
addition, the negative region of WF gradually decreases with the increasement
of $n_{c}$, but not disappear for $m=1$.
## V Decoherence of PASTS in thermal environment
In this section, we consider how this single-mode PASTS evolves at the
presence of thermal environment. In thermal channel, the evolution of the
density matrix for the $m$-PASV can be described by 38
$\displaystyle\frac{d\rho}{dt}$
$\displaystyle=\kappa\left(\mathcal{N}+1\right)\left(2a\rho
a^{\dagger}-a^{\dagger}a\rho-\rho a^{\dagger}a\right)$
$\displaystyle+\kappa\mathcal{N}\left(2a^{\dagger}\rho a-aa^{\dagger}\rho-\rho
aa^{\dagger}\right),$ (38)
where $\kappa$ represents the dissipative coefficient and $\mathcal{N}$
denotes the average thermal photon number of the environment. When
$\mathcal{N}=0,$ Eq.(38) reduces to the master equation describing the photon-
loss channel. The evolution formula of WF of the PASV can be derived as
follows 39
$W\left(\eta,\eta^{\ast},t\right)=\frac{2}{\left(2\mathcal{N}+1\right)\mathcal{T}}\int\frac{d^{2}\alpha}{\pi}W\left(\alpha,\alpha^{\ast},0\right)e^{-2\frac{\allowbreak\left|\eta-\alpha
e^{-\kappa t}\right|^{2}}{\left(2\mathcal{N}+1\right)\mathcal{T}}},$ (39)
where $W\left(\alpha,\alpha^{\ast},0\right)$ is the WF of the initial state,
and $\mathcal{T}=1-e^{-2\kappa t}$. Thus, in thermal channel, the WF at any
time can be obtained by performing the integration when the initial WF is
known.
In a similar way to deriving Eq.(32), substituting Eqs.(32)-(34) into Eq.(39)
and using the generating function of single-variable Hermite polynomials (8),
we finally obtain (see Appendix B)
$W\left(\eta,\eta^{\ast},t\right)=F_{m}\left(\eta,\eta^{\ast},t\right)W_{0}\left(\eta,\eta^{\ast},t\right),$
(40)
where
$\displaystyle W_{0}\left(\eta,\eta^{\ast},t\right)$
$\displaystyle=\frac{2/\left(2n_{c}+1\right)}{\pi\left(2\mathcal{N}+1\right)\mathcal{T}\sqrt{G}}$
$\displaystyle\times\exp\left[-\Delta_{1}\left|\eta\right|^{2}+\frac{g_{2}g_{3}^{2}}{G}\left(\eta^{\ast
2}+\eta^{2}\right)\right],$ (41)
$F_{m}\left(\eta,\eta^{\ast},t\right)=C_{am}^{-1}\sum_{l=0}^{m}\frac{\left(m!\right)^{2}\chi^{l}\Delta_{2}^{m-l}}{l!\left[\left(m-l\right)!\right]^{2}}\left|H_{m-l}\left(\frac{-i\omega/2}{\sqrt{\Delta_{2}}}\right)\right|^{2},$
(42)
and
$\displaystyle g_{0}$ $\displaystyle=\frac{\cosh 2\lambda}{2n_{c}+1},\text{
}g_{1}=\frac{n_{c}\mathcal{+}\cosh^{2}\lambda}{2n_{c}+1},$ $\displaystyle
g_{2}$ $\displaystyle=\frac{\sinh 2\lambda}{2n_{c}+1},\text{
}g_{3}=\frac{2e^{-\kappa t}}{\left(2\mathcal{N}+1\right)\mathcal{T}},$ (43)
as well as
$\displaystyle G$ $\displaystyle=\left(2g_{0}+g_{3}\allowbreak e^{-\kappa
t}\right)^{2}-4g_{2}^{2},$ $\displaystyle\Delta_{1}$
$\displaystyle=g_{3}e^{\kappa
t}\allowbreak-\frac{g_{3}^{2}}{G}\left(2g_{0}+g_{3}\allowbreak e^{-\kappa
t}\right),$ $\displaystyle\Delta_{2}$
$\displaystyle=\frac{g_{2}}{G}\left(g_{3}e^{-\kappa t}/2-1\right)^{2},$ (44)
$\displaystyle\omega$ $\displaystyle=\frac{2g_{3}}{g_{3}e^{-\kappa
t}-2}\left(2\Delta_{2}\eta^{\ast}+\chi\eta\right),$ $\displaystyle\chi$
$\displaystyle=\frac{2-g_{3}e^{-\kappa
t}}{G}\allowbreak\left(g_{0}+g_{1}g_{3}e^{-\kappa
t}+\frac{1}{\left(2n_{c}+1\right)^{2}}\right).$
Equation (40) is just the analytical expression of WF for PASTS in the thermal
channel. It is obvious that the WF loses its Gaussian property due to the
presence of single-variable Hermite polynomials. It is interesting to notice
that $W_{0}\left(\eta,\eta^{\ast},t\right)$ is actually the WF of squeezed
thermal state in thermal channel corresponding to the case without photon
addition ($m=0$), $F_{0}\left(\eta,\eta^{\ast},t\right)=1$; while
$F_{m}\left(\eta,\eta^{\ast},t\right)$ is just the non-Gaussian contribution
from photon-addition. The partial negativity of WF is fully determined by that
of $F_{m}\left(\eta,\eta^{\ast},t\right)$.
In particular, at the initial time $\left(t=0\right)$, noting
$\left(2\mathcal{N}+1\right)\mathcal{T}\sqrt{G}\rightarrow 2$,
$g_{3}^{2}/G\rightarrow 1,$ and $\Delta_{1}\rightarrow 2g_{0}$,
$\Delta_{2}\rightarrow\sinh 2\lambda/[4(2n_{c}+1)],$
$\chi\rightarrow-(\cosh^{2}\lambda+n_{c})/(2n_{c}+1),$ as well as
$\omega/(2i\sqrt{\Delta_{2}})\rightarrow\bar{\gamma}=[\eta^{\ast}\sinh
2\lambda-2\eta(\cosh^{2}\lambda+n_{c})]/\\{i[\left(2n_{c}+1\right)\sinh
2\lambda]^{1/2}\\},$ Eqs.(41) and (42) just do reduce to Eqs.(33) and (34),
respectively, i.e., the WF of the PASTS. On the other hand, when $\kappa
t\rightarrow\infty,$ noticing that $\mathcal{T}\rightarrow 1,G\rightarrow
4/\allowbreak\left(2n_{c}+1\right)^{2},\Delta_{1}\rightarrow
2/\left(2\mathcal{N}+1\right),\omega/(2i\sqrt{\Delta_{2}})\rightarrow
0,\Delta_{2}\rightarrow\frac{1}{4}\left(2n_{c}+1\right)\sinh 2\lambda,$ and
$\chi\rightarrow n_{c}\cosh 2\lambda+\cosh^{2}\lambda,$ as well as
$H_{m}\left(0\right)=\left(-1\right)^{j}\frac{m!}{j!}\delta_{m,2j},$ then
Eq.(40) becomes $\allowbreak
W\left(\eta,\eta^{\ast},\infty\right)=1/[\pi\left(2\mathcal{N}+1\right)]\exp[-2\left|\eta\right|^{2}/(2\allowbreak\mathcal{N}+1)],$
a Gaussian distribution, which is independent of photon-addition number $m$
and corresponds to the WF of thermal state with mean thermal photon number
$\mathcal{N}$. This indicates that the system state reduces to a thermal state
with mean photon number $\mathcal{N}$ after an enough long time interaction
with the environment.
In addition, for the case of $m=1$, corresponding to the case of SPASTS, Eq.
(40) just becomes
$W_{1}\left(\eta,\eta^{\ast},t\right)=C_{a1}^{-1}W_{0}\left(\eta,\eta^{\ast},t\right)\left(\left|\omega\right|^{2}+\chi\right).$
(45)
It is obvious that when $F_{1}\left(\eta,\eta^{\ast},t\right)<0,$ the WF of
SPASTS in thermal channel has its negative distribution in phase space. At the
center of phase space $\eta=\eta^{\ast}=0,$ the WF of SPASTS always has the
negative values when $\chi<0$, i.e., ($2-g_{3}e^{-\kappa
t})/(2g_{0}+g_{3}\allowbreak e^{-\kappa t}-2g_{2})<0$ (note
$2g_{0}+g_{3}\allowbreak e^{-\kappa t}-2g_{2}>0$) leading to the following
condition:
$\kappa t<\kappa t_{c}=\frac{1}{2}\ln\frac{2\mathcal{N}+2}{2\mathcal{N}+1},$
(46)
which is independent of the squeezing parameter $\lambda$ and the average
photon number $n_{c}$ of thermal state, there always exist negative region for
WF in phase space and the WF of PASTS is always positive in the whole phase
space when $\kappa t\ $exceeds the threshold value $\kappa t_{c}$. Due to this
and from Eq. (46), we can see how the thermal noise shortens the threshold
value of the decay time. Comparing to the time threshold value of SPSSTS 32
with the identical squeezed thermal state to that of SPASTS,
$\kappa
t_{cs}=\frac{1}{2}\ln\left[1-\frac{2n_{c}+1}{2\mathcal{N}+1}\frac{n_{c}-\sinh^{2}\lambda}{n_{c}\cosh
2\lambda+\sinh^{2}\lambda}\right],$ (47)
one can find a difference of $e^{2\kappa t_{c}}-e^{2\kappa t_{cs}}:$
$e^{2\kappa t_{c}}-e^{2\kappa
t_{cs}}=\allowbreak\frac{2n_{c}\left(n_{c}+1\right)}{\left(2N+1\right)\left(n_{c}\cosh
2\lambda+\sinh^{2}\lambda\right)},$ (48)
which implies that the decoherence time of SPASTS is longer than that of
SPSSTS. In this sense, the photon-addition Gaussian states present more robust
contrast to decoherence than photon-subtraction ones.
Figure 4: (Color online) Wigner function distributions ${\small
W}\left(\alpha,\alpha^{\ast}\right)$ of PASTS with $m=1,$ $n_{c}=0.3$ for
different $\mathcal{N},$ $\lambda$ and $\kappa t$ values (a)
$\mathcal{N}=0.2,\lambda=0.3,\kappa t=0.05;$(b)
$\mathcal{N}=0.2,\lambda=0.3,\kappa t=0.2;$ (c)
$\mathcal{N}=0.2,,\lambda=0.8,\kappa t=0.05;$ (d)
$\mathcal{N}=2,\lambda=0.3,\kappa t=0.05.$
In Fig.3, the WFs of PASTS with $m=1$ and $n_{c}=0.3$ are depicted in phase
space for several different $\mathcal{N},$ $\lambda$ and $\kappa t$ values. It
is easy to see that the negative region of WF gradually diminishes as the time
$\kappa t$ increases (see Fig.3 (a) and (b)). In addition, the partial
negativity of WF decreases gradually as $\mathcal{N}$ (or $\lambda$) increases
for a given time (see Fig.3 (c) and (d)). The squeezing effect in one of the
quadrature is shown in Fig.4(c). For the case of large squeezing value
$\lambda$ and small $n_{c}$ and $\mathcal{N}$ values, the single-PASTS becomes
similar to a Schodinger cat state. The WF becomes Gaussian with the time
evolution.
## VI Non-Gaussianity measure for PASTS
As well known, non-Gaussian operators (such as photon-adding/subtracting) can
improve the nonclassicality and entanglement between Gaussian states 12 ; 13 .
One reason of such an enhancement is their amount of non-Gaussianity 40 ; 41 .
Recently, an experimentally accessible criterion has been proposed to measure
the degree based on the conditional entropy of the state with a Gaussian
reference 42 . Therefore, it is of interest to evaluate the degree of the
resulting non-Gaussianity and assess this operation as a resource to obtain
non-Gaussian states starting from Gaussian ones. Noting that the STS can be
considered as a generalized Gaussian state, thus the fidelity between PASTS
and STS may be seen as a non-Gaussianity measure. For this purpose, we define
the fidelity by 32
$\mathcal{F}=\mathtt{tr}\left(\rho_{s}\rho\right)/\mathtt{tr}\left(\rho_{s}^{2}\right),$
(49)
where $\rho_{s}$ and $\rho$ are the STS (a generalized Gaussian state) and the
PASTS, respectively.
Noticing $\mathtt{tr}\left(\rho_{s}^{2}\right)=1/(2\bar{n}_{c}+1),$ and using
the formula (C1), we finally obtain (see Appendix C)
$\mathcal{F}=\frac{m!}{C_{a,m}}K_{2}^{m/2}P_{m}\left(\frac{K_{1}}{\sqrt{K_{2}}}\right)=\left(\frac{K_{2}}{A}\right)^{m/2}\frac{P_{m}\left(K_{1}/\sqrt{K_{2}}\right)}{P_{m}\left(\bar{B}/\sqrt{A}\right)},$
(50)
where
$K_{1}=\frac{n_{c}\left(n_{c}+1\right)}{2n_{c}+1}\cosh
2\lambda,K_{2}=\frac{n_{c}^{2}\left(n_{c}+1\right)^{2}}{\left(2n_{c}+1\right)^{2}}\allowbreak-\frac{\sinh^{2}2\lambda}{4}.$
(51)
Eq.(50) is just the analytical expression for the fidelity between PASTS and
STS. It is obvious that when $m=0$ (without photon-addition), $\mathcal{F}=1$.
Comparing to the fidelity $\mathcal{F}_{s}$ between PSSTS and STS (59) in
Ref.32 , one can clearly see that
$\frac{\mathcal{F}}{\mathcal{F}_{s}}=\left(\frac{Z}{A}\right)^{m/2}\frac{P_{m}\left(H/\sqrt{Z}\right)}{P_{m}\left(\bar{B}/\sqrt{A}\right)}=\frac{C_{s,m}}{C_{a,m}},$
(52)
where $Z=n_{c}^{2}-\left(2n_{c}+1\right)\sinh^{2}\lambda,$ $H=n_{c}\cosh
2\lambda+\sinh^{2}\lambda.$ Eq.(52) implies that the ratio to fidelities is
just that to the normalization factors. In particular, for $m=1$ (the case of
SPASTS), Eq.(50) reduces to
$\frac{\mathcal{F}}{\mathcal{F}_{s}}\mathfrak{=}\frac{n_{c}\cosh
2\lambda+\sinh^{2}\lambda}{n_{c}\cosh 2\lambda+\cosh^{2}\lambda}<1,$ (53)
from which one can see that $\mathcal{F}\mathfrak{<}\mathcal{F}_{s},$ i.e.,
the amount of non-Gaussianity for SPASTS is larger than that for SPSSTS.
This point is made clear in Fig.5, in which the fidelity $\mathcal{F}$ between
PASTS and STS as the function of squeezing parameter $\lambda$ for different
photon-addition number $m.$ As a comparision, the fidelity $\mathcal{F}_{s}$
between PSSTS and STS is also shown in Fig.5, from which one can see that the
fidelity decreases as the increment of photon-addition/subtraction number $m,$
as expected. The fidelity $\mathcal{F}$ increases monotonously with the
augment of the squeezing parameter $\lambda$. However, the case is not true
for the fidelity $\mathcal{F}_{s}.$ For a given $m$ value, the fidelity
$\mathcal{F}$ is always smaller than the fidelity $\mathcal{F}_{s}$ within the
region shown in Fig.5. In this sense, the amount of non-Gaussianity for PASTS
is larger than that for PSSTS.
Figure 5: (Color online) The fidelity $\mathcal{F}$ between PASTS (PSSTS) and
STS as the function of squeezing parameter $\lambda$ for different photon-
addition number $m=0,1,2,3(n_{c}=0.2).$
## VII Conclusions
In this paper, we investigate the nonclassical properties and decoherence of
single-mode PASTS when evolving under a thermal environment. Based on the fact
that squeezed number can be considered as an Hermite polynomial excitation
squeezed vacuum, the normally ordering form of PASTS is directly obtained,
from which one can expediently calculate some quasi-distributions, such as Q-,
P- and Wigner function; And the normalization factor of PASTS is analytically
derived, which is just proved to be an $m$-order Legendre polynomial of the
squeezing parameter $r$ and average photon number $n_{c}$ of the thermal
state, a remarkable result. Furthermore, for any photon-added number
$m$-PASTS, the explicit expression of WF is derived, which considered as a
product of the WF of STS in thermal channel and a non-Gaussian distribution
resulting from photon-addition. It is shown that the WF of SPASTS always has
the negative values at the phase space center, which is different from the
case of SPSSTS with a condition $n_{c}<\sinh^{2}\lambda$. Then the effects of
decoherence to the nonclassicality of PASTS in the thermal channel is also
demonstrated according to the compact expression for the WF. The threshold
value of the decay time corresponding to the transition of the WF from partial
negative to completely positive definite is obtained for SPASTS at the center
of phase space. It is found that the WF has always negative value for all
parameters $r,n_{c}$ if the decay time $\kappa t<\kappa
t_{c}=\frac{1}{2}\ln\frac{2\mathcal{N}+2}{2\mathcal{N}+1}$, a larger value
than that of SPSSTS.
A comparison between the nonclassicality and decoherence of PASTS and PSSTS
shows that the photon-addition non-Gaussian states present more robust
contrast to decoherence than photon-subtraction ones, which may be due to the
amount of non-Gaussianity for SPASTS is larger that that for SPSSTS. On the
other hand, in the limit of vanishing squeezing and $n_{c}=0$, the PASTS
reduces to a single-mode Fock state, remaining non-Gaussian, while the PSSTS
becomes Gaussian, as it reduces to the single mode vacuum. Entanglement
evaluation investigation for photon-subtracted/added two-mode squeezed thermal
state is a future problem.
Acknowledgments: This work was supported by the National Natural Science
Foundation of China (Grant Nos. 11047133, 60978009 ), the Major Research Plan
of the National Natural Science Foundation of China (Grant No. 91121023 ), and
the “973” Project (Grant No. 2011CBA00200), as well as the Natural Science
Foundation of Jiangxi Province of China (No. 2010GQW0027).
Appendix A: Derivation of WF (32) for PASTS
Substituting Eq.(16) into Eq.(31) and using the integration formula (18), we
have
$\displaystyle W\left(\alpha,\alpha^{\ast}\right)$
$\displaystyle=\frac{\left(-1\right)^{m}C_{am}^{-1}}{\tau_{1}\tau_{2}}e^{2\left|\alpha\right|^{2}}\int\frac{\mathtt{d}^{2}\beta}{\pi^{2}}\left|\beta\right|^{2m}\exp\left[-\left(1+B\right)\left|\beta\right|^{2}\right.$
$\displaystyle+\left.2\left(\alpha\beta^{\ast}-\alpha^{\ast}\beta\right)+\frac{C}{2}\left(\beta^{\ast
2}+\beta^{2}\right)\right]$
$\displaystyle=\frac{C_{am}^{-1}}{\tau_{1}\tau_{2}}e^{2\left|\alpha\right|^{2}}\frac{\partial^{2m}}{\partial
s^{m}\partial
t^{m}}\int\frac{\mathtt{d}^{2}\beta}{\pi^{2}}\exp\left[-\left(1+B\right)\left|\beta\right|^{2}\right.$
$\displaystyle+\left.\left(2\alpha+s\right)\beta^{\ast}-\left(2\alpha^{\ast}+t\right)\beta+\frac{C}{2}\left(\beta^{\ast
2}+\beta^{2}\right)\right]_{s=t=0}$
$\displaystyle=W_{0}\left(\alpha,\alpha^{\ast}\right)F_{m}\left(\alpha,\alpha^{\ast}\right),$
(A1)
where we have set
$\displaystyle W_{0}\left(\alpha,\alpha^{\ast}\right)$
$\displaystyle=\frac{\sqrt{A_{1}}}{\pi\tau_{1}\tau_{2}}\exp\left[A_{2}\left(\alpha^{2}+\alpha^{\ast
2}\right)-2A_{3}\left|\alpha\right|^{2}\right],$ (A2) $\displaystyle
F_{m}\left(\alpha,\alpha^{\ast}\right)$
$\displaystyle=C_{am}^{-1}\frac{\partial^{2m}}{\partial s^{m}\partial
t^{m}}\exp\left[\frac{A_{2}}{4}\left(s^{2}+t^{2}\right)-\frac{A_{4}}{2}st\right.$
$\displaystyle+\left.\allowbreak\left(A_{2}\alpha^{\ast}-A_{4}\alpha\right)t+\left(\allowbreak
A_{2}\alpha-A_{4}\alpha^{\ast}\right)s\right]_{s=t=0},$ (A3)
and
$\displaystyle A_{1}$
$\displaystyle=\frac{1}{\left(1+B\right)^{2}-C^{2}}=\frac{A}{\left(2n_{c}+1\right)^{2}},$
$\displaystyle A_{2}$
$\displaystyle=\frac{2C}{\left(1+B\right)^{2}-C^{2}}=\frac{\sinh
2\lambda}{2n_{c}+1},$ $\displaystyle A_{3}$
$\displaystyle=\frac{2\left(B+1\right)}{\left(1+B\right)^{2}-C^{2}}-1=\frac{\cosh
2\lambda}{2n_{c}+1},$ $\displaystyle A_{4}$
$\displaystyle=\frac{2\left(B+1\right)}{\left(1+B\right)^{2}-C^{2}}=A_{3}+1=2\frac{n_{c}\mathcal{+}\cosh^{2}\lambda}{2n_{c}+1}.$
(A4)
Substituting Eq.(A3) into Eq.(A2) yields Eq.(33), i.e., the WF of squeezed
thermal state.
Further expanding the exponential term $st$ included in (A3) into sum series,
and using the generating function of single-variable Hermite polynomials 27 ,
$H_{n}(x)=\left.\frac{\partial^{n}}{\partial
t^{n}}\exp\left(2xt-t^{2}\right)\right|_{t=0},$ (A5)
which leads to
$\displaystyle\left.\frac{\partial^{n}}{\partial
t^{n}}\exp\left(At+Bt^{2}\right)\right|_{t=0}$
$\displaystyle=\left(i\sqrt{B}\right)^{n}H_{n}\left[A/(2i\sqrt{B})\right]$
$\displaystyle=\left(-i\sqrt{B}\right)^{n}H_{n}\left[A/(-2i\sqrt{B})\right],$
(A6)
thus we can see
$\displaystyle F_{m}\left(\alpha,\alpha^{\ast}\right)$
$\displaystyle=C_{am}^{-1}\sum_{l=0}^{\infty}\frac{\left(-A_{4}\right)^{l}}{2^{l}l!}\frac{\partial^{2m}}{\partial
s^{m}\partial t^{m}}s^{l}t^{l}$
$\displaystyle\times\exp\left[\frac{A_{2}}{4}\left(s^{2}+t^{2}\right)+\gamma
t+\gamma^{\ast}s\right]_{s=t=0}$
$\displaystyle=C_{am}^{-1}\sum_{l=0}^{\infty}\frac{\left(-A_{4}\right)^{l}}{2^{l}l!}\frac{\partial^{2l}}{\partial\gamma^{l}\partial\gamma^{\ast
l}}\frac{\partial^{2m}}{\partial s^{m}\partial t^{m}}$
$\displaystyle\times\exp\left[\frac{A_{2}}{4}\left(s^{2}+t^{2}\right)+\gamma
t+\gamma^{\ast}s\right]_{s=t=0}$
$\displaystyle=\frac{A_{2}^{m}}{2^{2m}}C_{am}^{-1}\sum_{l=0}^{\infty}\frac{\left(-A_{4}\right)^{l}}{2^{l}l!}\frac{\partial^{2l}}{\partial\gamma^{l}\partial\gamma^{\ast
l}}\left|H_{m}\left(\bar{\gamma}\right)\right|^{2},$ (A7)
where $\gamma=A_{2}\alpha^{\ast}-A_{4}\alpha,$ and
$\bar{\gamma}=\gamma/(i\sqrt{A_{2}}),$ i.e.,
$\bar{\gamma}=\frac{\alpha^{\ast}\sinh
2\lambda-2\alpha\left(\cosh^{2}\lambda+n_{c}\right)}{i\sqrt{\left(2n_{c}+1\right)\sinh
2\lambda}},$ (A8)
Then using the recurrence relation of $H_{n}(x),$
$\frac{\mathtt{d}}{\mathtt{d}x^{l}}H_{n}(x)=\frac{2^{l}n!}{\left(n-l\right)!}H_{n-l}(x),$
(A9)
Eq.(A7) becomes
$\displaystyle F_{m}\left(\alpha,\alpha^{\ast}\right)$
$\displaystyle=\frac{A_{2}^{m}}{2^{2m}}C_{am}^{-1}\sum_{l=0}^{\infty}\frac{\left(-A_{4}/A_{2}\right)^{l}}{2^{l}l!}$
$\displaystyle\times\frac{\partial^{2l}}{\partial\bar{\gamma}^{l}}H_{m}\left(\bar{\gamma}\right)\frac{\partial^{2l}}{\partial\bar{\gamma}^{\ast
l}}H_{m}\left(\bar{\gamma}^{\ast}\right)$
$\displaystyle=\frac{A_{2}^{m}}{2^{2m}}C_{am}^{-1}\sum_{l=0}^{m}\frac{\left(m!\right)^{2}\left(-2A_{4}/A_{2}\right)^{l}}{l!\left[\left(m-l\right)!\right]^{2}}\left|H_{m-l}(\bar{\gamma})\right|^{2}.$
(A10)
Substituting Eq.(A4) into Eq.(A10) yields Eq.(34). Thus we complete the
derivation of WF Eq.(32) by combing Eqs. (A2) and (A10).
Appendix B: Derivation of WF (40) for PASTS in thermal channel
Substituting Eqs.(32)-(34) into Eq.(39), we have
$\displaystyle W\left(\eta,\eta^{\ast},t\right)$
$\displaystyle=\frac{C_{am}^{-1}g_{3}e^{\kappa
t}}{\pi\left(2n_{c}+1\right)}e^{-g_{3}e^{\kappa
t}\allowbreak\left|\eta\right|^{2}}\frac{\partial^{2m}}{\partial
s^{m}\partial\tau^{m}}$
$\displaystyle\times\exp\left[\frac{g_{2}}{4}\left(s^{2}+\tau^{2}\right)-g_{1}s\tau\right]$
$\displaystyle\times\int\frac{d^{2}\alpha}{\pi}\exp\left[-\left(2g_{0}+g_{3}\allowbreak
e^{-\kappa t}\right)\left|\alpha\right|^{2}\right.$
$\displaystyle+\left(g_{3}\eta^{\ast}+g_{2}s-2g_{1}\tau\right)\alpha$
$\displaystyle+\left.\allowbreak\left(g_{3}\eta+g_{2}\tau-2g_{1}s\right)\alpha^{\ast}+g_{2}\left(\alpha^{2}+\alpha^{\ast
2}\right)\right]_{s=\tau=0},$ (B1)
where we have set
$\displaystyle g_{0}$ $\displaystyle=A_{3}=\frac{\cosh
2\lambda}{2n_{c}+1},\text{
}g_{1}=\frac{A_{4}}{2}=\frac{n_{c}\mathcal{+}\cosh^{2}\lambda}{2n_{c}+1},$
$\displaystyle g_{2}$ $\displaystyle=A_{2}=\frac{\sinh
2\lambda}{2n_{c}+1},\text{ }g_{3}=\frac{2e^{-\kappa
t}}{\left(2\mathcal{N}+1\right)\mathcal{T}}.$ (B2)
Further using the integration (18), Eq.(B1) can be put into the form
$W\left(\eta,\eta^{\ast},t\right)=F_{m}\left(\eta,\eta^{\ast},t\right)W_{0}\left(\eta,\eta^{\ast},t\right),$
(B3)
where $W_{0}\left(\eta,\eta^{\ast},t\right)$ is defined in Eq.(41), and
$\displaystyle F_{m}\left(\eta,\eta^{\ast},t\right)$
$\displaystyle=C_{am}^{-1}\frac{\partial^{2m}}{\partial
s^{m}\partial\tau^{m}}\exp\left[\Delta_{2}\left(s^{2}+\tau^{2}\right)\right.$
$\displaystyle+\left.\omega\tau+\omega^{\ast}s+\chi s\tau\right]_{s=\tau=0},$
(B4)
here $\left(\Delta_{2},\omega,\chi\right)$ are defined in Eq. (44). In a
similar way to deriving Eq. (32), we can further insert Eq. (B4) into Eq.
(42).
Appendix C: Derivation of fidelity (50) between PASTS and STS
The fidelity ($\mathtt{tr}\left(\rho_{s}\rho\right)$) can be calculated as the
overlap between the two WFs:
$\mathtt{tr}\left(\rho_{s}\rho\right)=4\pi\int d^{2}\alpha
W_{0}\left(\alpha,\alpha^{\ast}\right)W_{\rho}\left(\alpha,\alpha^{\ast}\right),$
(C1)
where $W_{0}\left(\alpha,\alpha^{\ast}\right)$ is the WF of squeezed thermal
state $\rho_{s}$. Using Eq.(32) we may express Eq.(C1) as
$\mathtt{tr}\left(\rho_{s}\rho\right)=4\pi\int
F_{m}\left(\alpha,\alpha^{\ast}\right)W_{0}^{2}\left(\alpha,\alpha^{\ast}\right)d^{2}\alpha.$
(C2)
Then employing Eqs.(32) and (A2),(A3) as well as the integration formula (18),
we can put Eq.(C2) into the following form:
$\displaystyle\mathtt{tr}\left(\rho_{s}\rho\right)$
$\displaystyle=\frac{4C_{am}^{-1}}{\left(2n_{c}+1\right)^{2}}\frac{\partial^{2m}}{\partial
s^{m}\partial\tau^{m}}\exp\left[\frac{g_{2}}{4}\left(s^{2}+\tau^{2}\right)-g_{1}s\tau\right]$
$\displaystyle\int\frac{d^{2}\alpha}{\pi}\exp\left[-4g_{0}\left|\alpha\right|^{2}+2g_{2}\left(\alpha^{2}+\alpha^{\ast
2}\right)\right]$ $\displaystyle+\left.\left(\allowbreak
g_{2}s-2g_{1}\tau\right)\alpha+\allowbreak\left(g_{2}\tau-2g_{1}s\right)\alpha^{\ast}\right]_{s=\tau=0}$
$\displaystyle=\frac{C_{am}^{-1}}{2n_{c}+1}\frac{\partial^{2m}}{\partial
s^{m}\partial\tau^{m}}\exp\left[K_{1}s\tau+K_{0}\left(s^{2}+\tau^{2}\right)\right]_{s=\tau=0},$
(C3)
where $K_{1}$ is defined in Eq.(51), and
$K_{0}=\frac{2n_{c}^{2}+2n_{c}+1}{4\left(2n_{c}+1\right)}\sinh 2\lambda.$ (C4)
Similarly to deriving Eq.(20), we have
$\displaystyle\left.\frac{\partial^{2m}}{\partial
s^{m}\partial\tau^{m}}\exp\left[K_{0}\left(k^{2}+t^{2}\right)+\allowbreak
K_{1}kt\right]\right|_{k=t=0}$
$\displaystyle=m!K_{2}^{m/2}P_{m}\left(K_{1}/\sqrt{K_{2}}\right),$ (C5)
and $K_{2}\equiv K_{1}^{2}-4K_{0}^{2}$ given in Eq.(51), which leads to
Eq.(50).
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|
arxiv-papers
| 2011-10-30T08:39:09 |
2024-09-04T02:49:23.709900
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Li-Yun Hu and Zhi-Ming Zhang",
"submitter": "Liyun Hu",
"url": "https://arxiv.org/abs/1110.6587"
}
|
1110.6626
|
# CR electrons and positrons: what we have learned in the latest three years
and future perspectives
Daniele Gaggero Dario Grasso Department of Physics, Pisa University,
Largo B. Pontecorvo 3, 56127 Pisa Italy
∗E-mail: daniele.gaggero@pi.infn.it
###### Abstract
After the PAMELA finding of an increasing positron fraction above 10 GeV, the
experimental evidence of the presence of a new electron and positron spectral
component in the cosmic ray zoo has been recently confirmed by Fermi-LAT. We
show as a simple phenomenological model which assumes the presence of an
electron and positron extra component peaked at $\sim 1~{}{\rm TeV}$ allows a
consistent description of all available data sets. We then describe the most
relevant astrophysical uncertainties which still prevent to determine
$e^{\pm}$ source properties from those data and the perspectives of
forthcoming experiments.
###### keywords:
Proceedings; World Scientific Publishing.
## 1 Introdution
Recent experimental results raised a wide interest about the origin and the
propagation of the leptonic component of the cosmic radiation.
Among the most striking of those results, there is the observation performed
by the PAMELA satellite experiment that the positron to electron fraction
$e^{+}/(e^{-}+e^{+})$ rises with energy from 10 up to 100 GeV at least
(Adriani et al. 2008 [1]). This appeared in contrast with the predictions of
the standard cosmic ray scenario and could therefore be interpreted as the
smoking gun of new physics, unless a very soft electron spectrum was assumed.
The significance of this anomaly increased when the Fermi-LAT space
observatory measured the $e^{-}+e^{+}$ spectrum in the 7 GeV - 1 TeV energy
range with unprecedented accuracy and found it to be compatible with a power-
law with index $\gamma(e^{\pm})=-3.08\pm 0.05$ (Abdo et al. 2009 [2],
Ackermann et al. 2010 [3]); this slope is significantly harder than what
estimated on the basis of previous measurements: the hypothesis of a steep
spectrum was therefore excluded.
More recently, the same collaboration provided a further, and stronger,
evidence of the positron anomaly by providing direct measurement of the
absolute $e^{+}$ and $e^{-}$ spectra, and of their fraction, between 20 and
200 GeV using the Earth magnetic field. A steady rising of the positron
fraction was observed by this experiment up to that energy in agreement with
that found by PAMELA. In the same energy range, the $e^{-}$ spectrum was
fitted with a power-law with index $\gamma(e^{-})=-3.19\pm 0.07$ which is in
agreement with what recently measured by PAMELA between 1 and 625 GeV (Adriani
et al. 2011 [4]). Most importantly, Fermi-LAT measured, for the first time,
the $e^{+}$ spectrum in the 20 - 200 GeV energy interval and showed it is
fitted by a power-law with index $\gamma(e^{+})=-2.77\pm 0.14$.
We will show in the following paragraph how all those measurements rule out
the standard scenario in which the bulk of electrons reaching the Earth in the
GeV - TeV energy range are originated by Supernova Remnants (SNRs) and only a
small fraction of secondary positrons and electrons comes from the interaction
of CR nuclei with the interstellar medium (ISM). Then we will see how the
alternative scenario in which the presence of electron + positron component
peaked at $\sim 1$ TeV is invoked allows a consistent description of all the
available data sets. Finally we will discuss to which extent astrophysical and
particle physics uncertainties still affect our modeling of cosmic ray leptons
origin and propagation and how forthcoming measurements are expected to reduce
those uncertainties.
## 2 The necessity of a primary extra-component
After the release of Fermi-LAT $e^{-}+e^{+}$ spectrum, it was clearly pointed
out in several papers (see e.g. Grasso et al. 2009 [5] and Di Bernardo et al.
2011 [6]) that both Fermi-LAT and PAMELA measurements described in the
Introduction are in contrast with a standard single-component scenario in
which positrons are the secondary products of the nuclear component of cosmic
rays (CRs) interacting with the interstellar medium (ISM).
Figure 1: Fermi-LAT and PAMELA data on electrons + positrons and electrons are
compared to a double component phenomenological model. The absolute positron
spectrum is compared to a single and double component phenomenological model.
Red dotted line: $e^{+}$ in single-component scenario. Red dot-dashed line:
$e^{+}$ in double-component scenario. Blue triple dotted-dashed line, black
solid line: $e^{-}$ and $e^{-}+e^{+}$ in double-component scenario. Blue
dashed line: $e^{-}$ diffuse background in double-component scenario. The
Kolmogorov diffusion setup is adopted.
The main problems encountered by this kind of models can be summarized as
follows.
* •
As explained many times (see e.g. Serpico 2011 [7] for a recent review), they
cannot reproduce the rising positron-to-electron ratio measured by PAMELA and
recently confirmed by Fermi-LAT;
* •
They are unable to reproduce all the features revealed by Fermi-LAT in the CRE
spectrum, in particular the flattening observed at around 20 GeV and the
softening at $\sim 500$ GeV. In fact, if such models are normalized against
data in the 20 - 100 GeV energy range, where systematical and theoretical
uncertainties are the smallest, they clearly fail to match CRE Fermi-LAT and
PAMELA $e^{-}$ data outside that range. A different normalization results in
even worse fits.
With the release of the $e^{-}$ and $e^{+}$ separate spectra by the Fermi-LAT
collaboration the problems with the single component scenario became even
worse. In fact, the $e^{+}$ spectrum (Fig. 1) is clearly inconsistent with the
predictions of a single component scenario computed with DRAGON numerical
diffusion package (and similar results are obtained with GALPROP). Even
without considering numerical models, the simple consideration that the
reported positron spectral slope is $-2.77\pm 0.14$ reveals how these data are
incompatible with a purely secondary origin from proton spallation on
interstellar gas: the source slope should be the same as the proton spectrum,
i.e. $\simeq-2.75$ (Adriani et al. 2011 [8]) and no room is then left for the
unavoidable steepening due to energy-dependent diffusion and energy losses.
Figure 2: Fermi-LAT and PAMELA data on the positron ratio are compared to a
single and double component phenomenological model. Dot-dashed line: positron
ratio in single-component scenario. Dotted line: positron ratio in double-
component scenario due to conventional secondary positron production. Solid
line: positron ratio in double-component scenario including extra-component.
The progagation setup and modulation potential are the same of Fig. 1. The
solar modulation potential is taken $\Phi=550$ MV in all figures of this
paper.
A double component scenario is the most straightforward solution to these
problems.
The idea dates back to the pioneering work by F. Aharonian and A. Atoyan 1995
[9] and was extensively studied after the release of ATIC and PAMELA data in
2008 (see e.g. Hooper et al. 2009 [10] and Profumo 2008 [11]).
More recently, we contributed to several papers in which it was shown that a
consistent interpretation of the $e^{+}+e^{-}$ spectrum measured by Fermi-LAT
and the PAMELA positron fraction can be naturally obtained in that framework
(Grasso et al. 2009 [5], Ackermann et al. 2010 [3], Di Bernardo et al. 2011
[6]).
For example, in Fig. 1 and Fig. 2 we show that the double component model
proposed in Ackermann et al. 2010[3] reproduces the data mentioned above and
also the $e^{+}$ and $e^{-}$ separate spectra, and their ratio, recently
released by the Fermi-LAT collaboration and not yet available at the time. The
model represented in those figures assumes a propagation setup characterized
by a cylindrical diffusive halo with half-thikness of 4 kpc; a diffusion
coefficient scaling with rigidity like $\rho^{1/3}$ (corresponding to a
Kolmogorov-like diffusion within the quasi-linear approximation) and a
relatively strong reacceleration (the Alfvén velocity is $v_{A}=30~{}{\rm
kms^{-1}}$). Solar modulation is treated here as charge independent in the
force field approximation by fixing the modulation potential $\Phi$ against
proton data taken in the same solar phase. In that model, the standard $e^{-}$
primary component is tuned to fit Fermi-LAT data at low energy in the presence
of the extra-component becoming dominant at higher energies; the injection
slope for the primary electron component is set to $-2.70$ above 2 GeV, while
under that energy a slope of $-1.6$ is adopted, in accord with recent
constraints from the synchrotron spectra (see Jaffe et al. 2011 [12]). The
extra component, instead, originates from a primary source of
electron+positron pairs; it has an injection spectrum modelled in a simple way
as a power-law with index $-1.5$ plus an exponential cutoff at $1.2$ TeV; the
spatial distribution of this source is the same as the standard one and the
propagation parameters are also the same; the normalization is tuned so that
Fermi-LAT and PAMELA data at high energy are matched by the sum of standard +
extra component. Both components are computed with DRAGON (even if it was
checked that the same result can be obtained with GALPROP).
An issue remains open about the origin of the discrepancy between the
prediction of this, or similar, models and the positron fraction measured by
PAMELA below 10 GeV. In the next section we will show as that discrepancy may
be interpreted as the consequence of an incorrect choice of the propagation
setup and discuss other uncertainties which can affect the electron and
positron spectra in that low energy range.
## 3 LOW ENERGY. Impact of astrophysical uncertainties
Figure 3: Effect of changing the diffusion halo height. Solid line: h = 1 kpc;
dashed: h = 10 kpc. Figure 4: Effect of the diffusion setup. Solid line: KRA;
dashed line: KOL.
Cosmic ray electrons and positrons, either belonging to the standard or the
extra component, propagate in the Galaxy undergoing several physical
processes: diffusion, reacceleration, energy losses. Such complex motion is
effectively described by a well-known diffusion-loss equation (Berezinskii et
al. 1990 [13]). In this equation several free parameters are involved: the
height of the halo in which the propagation takes place, the normalization and
energy dependence of the diffusion coefficient (the latter parametrized by the
parameter $\delta$), the Alfvén velocity that influences the effectiveness of
reacceleration; moreover, several astrophysical inputs need to be considered:
the injection spectrum, the spatial distribution of the source term, the
interstellar radiation field, the gas distribution.
The free parameters that appear in the diffusion-loss equation are constrained
by some CR observables such as Boron-to-Carbon (B/C) or antiproton-to-proton
ratio; different diffusion setups exist in the literature, obtained through
comparison of experimental data with the prediction of semi-analytical codes
(Maurin et al. 2001 [14], Donato et al. 2004 [15]) or numerical packages such
as DRAGON or GALPROP (see e.g. Di Bernardo et al. 2010 [16] for DRAGON-related
models and Trotta et al. 2011 [17] for a GALPROP-based analysis).
The uncertainties related to the diffusion model and to the astrophysical
inputs were discussed in the latest years in several papers making use of
semi-analytic codes (e.g. Delahaye et al. 2010 [18]). In the following we will
briefly analyse the impact of these uncertainties adopting the DRAGON code.
One of the most relevant parameter is the halo height. According to the
analytical computations by Bulanov and Dogel [19], while at low energy the
electrons (or positrons) are distributed throughout all the diffusion halo, as
the energy increases the electrons occupy a smaller and smaller fraction of
the halo due to energy losses. This is relevant especially for the secondary
positron spectrum. In fact, since their injection power is determined by the
CR nuclei density in the Galactic disk, a thicker halo results in a larger
dilution of their density in the halo hence a in smaller flux on the Earth.
Numerical computations confirm the expectation of this heuristic argument as
shown in Fig. 3. From the plot it is also evident that large halo heights are
disfavoured by the data.
Even fixing the height of the diffusion halo, the choice of the diffusion
setup can also affect the low energy spectra of CR leptons. This is evident
from Fig. 4 where we compare the predictions of two different models which
both reproduce nuclear CR data:
* •
a Kraichnan-like diffusion setup with $\delta=0.5$ and moderate reacceleration
(that was pointed out as the preferred one in a DRAGON-based maximum
likelihood analysis with focus on both B/C and antiproton high energy
data[16])
* •
a Kolmogorov-like diffusion setup with $\delta=0.33$ and high reacceleration
(that was pointed out as the preferred one in a GALPROP-based maximum
likelihood analysis with focus on B/C data[17])
It is clear from that plot that the Kraichnan-like setups allows a better fit
of low-energy positron ratio measured by PAMELA; this consideration, together
with several other facts (high reacceleration models do not permit a good fit
of antiproton data and cannot reproduce the spectrum of the synchrotron
emission of the Galaxy), led us to conclude that models with strong
reacceleration are disfavoured.
## 4 High energy uncertainties and the nature of the extra-component
In the double component scenario discussed in Sec. 2, the positron spectrum
above $\sim 10$ GeV is dominated by the primary extra component. The nature of
its source is one of the hottest matter of debate in the CR physics.
Galactic pulsars were suggested as natural source candidates of a primary CR
positron component well before PAMELA results (Aharonian and Atoyan, 1995
[9].) More recently, it was noticed that a single, nearby, pulsar (such as
Monogem or Geminga) could explain the positrons fraction excess found by
PAMELA (Hooper et al. 2009 [10]).
In the Fermi-LAT era, we showed (Grasso et al. 2009 [5] and Di Bernardo et al.
2010 [6]) that also the $e^{+}+e^{-}$ measured by that experiment can
consistently be explained in the same terms: if one considers the observed
nearby pulsars within 2 kpc and assumes that a relevant fraction of their
rotational energy is transferred into $e^{+}+e^{-}$ pairs ($\simeq 30$%),
under reasonable assumpions on the injection spectrum and cutoff it is
possible to reproduce all existing data. In the cosmic ray channel, this
scenario has two possible testable consequences:
* •
the detection of a CR electron anisotropy towards the most relevant sources
(in our analysis, Monogem and Geminga [10]);
* •
the presence of some bumpiness in the $e^{-}$ and $e^{+}$ spectra in the TeV
region due to the contribution of several pulsars.
Those two signatures are somehow complementary: if a single pulsar give the
dominant contribution to the extra component a large anisotropy and a small
bumpiness should be expected; if several pulsars contribute the opposite
scenario is expected.
So far no positive detection of CRE anisotropy was reported by the Fermi-LAT
collaboration, but some stringent upper limits were published. In Di Bernardo
et al. 2010 [6] we showed that the pulsar scenario is still compatible with
these upper limits. Also, no evidence of spectral bumpiness has been found so
far in the $e^{+}+e^{-}$ spectrum.
It should be noted that several astrophysical uncertainties prevent accurate
predictions of the CRE anisotropy and of the spectral bumpiness. For example,
unknown irregularities in the local structure of the Galactic magnetic field
may distort the angular distribution of the CRE flux due to a nearby pulsar.
Furthermore, due to the stochastic nature of the $e^{-}$ emission of nearby
SNRs, the CRE standard component is expected to be subject to fluctuations
which may produce anisotropies and spectral bumpiness which may hide those due
to pulsars.
The other possible scenario to explain the origin of the extra component is
more exotic but very appealing as it invokes DM annihilihation/decay as the
origin of the $e^{\pm}$ extra component. Plenty of papers were published on
that subject after the release of PAMELA and Fermi-LAT results (see e.g. He
2009 [20] for a review). That scenario, however, present some problems. The
most important are the following ones.
* •
It requires a heavy DM particle mass – O(TeV) – and an annihilation cross
section much higher than that predicted by standard cosmology if one assumes
that DM is a thermal relic.
* •
Since no excess was detected for antiprotons, the annihilation/decay channels
must include only leptons (lepto-philic DM).
Although several DM models which may fulfil those conditions were developed,
another issue arises when electroweak corrections are taken into account.
Those corrections, in fact, give rise – even in a lepto-philic scenario – to
soft electroweak gauge bosons, and hence to antiprotons, at the end of their
decay chains (Ciafaloni et al. 2011 [21]). Since those exotic ${\bar{p}}$ are
produced mainly in the Galactic Center region, the flux reaching the Earth
strongly depends on the properties of CR propagation in the Galaxy. As we
discussed in Sec. 3, these properties are still subject to strong
uncertainties. It was shown in Evoli et al. 2011 [22] that, accounting for
those uncertainties, a scenario in which a heavy DM particle annihilates into
muons is still compatible with the antiproton constraints. In the same paper
it was also shown that AMS-02 is expected to constrain even more these models
since its sensitivity to antiprotons will be much higher.
## 5 Conclusions and future perspectives
In this contribution we argued as recent experimental data rule out the
standard scenario in which CR positrons are produced only by CR spallation
onto the ISM and showed as an empirical model which invokes an extra $e^{\pm}$
component fulfils all data sets. We also discussed several uncertainties which
still prevent to infer some of the properties of CR electron and positron
sources. We argued that at low energy those uncertainties are dominated by our
poor knowledge of CR propagation (which prevent an accurate determination of
the injection spectrum of the $e^{-}$ standard component) while at high energy
the effect of the stochastic nature of astrophysical sources prevails (which
makes more difficult to decide between the astrophysical and DM origin of the
extra component).
Forthcoming experiments like AMS-02 and CALET are expected to reduce
drastically the uncertainties on the propagation parameters by providing more
accurate measurements of the spectra of the nuclear components of CR. Fermi-
LAT and those experiments are also expected to provide more accurate
measurements of the CRE spectrum and anisotropy looking for features which may
give a clue of the nature of the extra component.
## References
* [1] O. A. et al. [PAMELA collaboration], Nature 458, 607(April 2009).
* [2] A. A. A. et al. [Fermi Collaboration], Physical Review Letters 102, p. 181101(May 2009).
* [3] M. A. et al. [Fermi Collaboration], Physical Review D 82, p. 092004(November 2010).
* [4] O. A. et al. [PAMELA collaboration], Physical Review Letters 106, p. 201101(May 2011).
* [5] D. Grasso, S. Profumo, A. W. Strong, L. Baldini, R. Bellazzini, E. D. Bloom, J. Bregeon, G. di Bernardo, D. Gaggero, N. Giglietto, T. Kamae, L. Latronico, F. Longo, M. N. Mazziotta, A. A. Moiseev, A. Morselli, J. F. Ormes, M. Pesce-Rollins, M. Pohl, M. Razzano, C. Sgro, G. Spandre and T. E. Stephens, Astroparticle Physics 32, 140(September 2009).
* [6] G. di Bernardo, C. Evoli, D. Gaggero, D. Grasso, L. Maccione and M. N. Mazziotta, Astroparticle Physics 34, 528(February 2011).
* [7] P. D. Serpico, ArXiv e-prints (August 2011).
* [8] O. A. et al. [PAMELA collaboration], Science 332, p. 69(April 2011).
* [9] A. M. Atoyan, F. A. Aharonian and H. J. Völk, Physical Review D 52, 3265(September 1995).
* [10] D. Hooper, P. Blasi and P. Dario Serpico, Journal of Cosmology and Astroparticle Physics 1, p. 25(January 2009).
* [11] S. Profumo, ArXiv e-prints (December 2008).
* [12] T. R. Jaffe, A. J. Banday, J. P. Leahy, S. Leach and A. W. Strong, Monthly Notices of the Royal Astronomical Society 416, 1152(September 2011).
* [13] V. S. Berezinskii, S. V. Bulanov, V. A. Dogiel and V. S. Ptuskin, Astrophysics of cosmic rays 1990\.
* [14] D. Maurin, F. Donato, R. Taillet and P. Salati, Astrophysical Journal 555, 585(July 2001).
* [15] F. Donato, N. Fornengo, D. Maurin, P. Salati and R. Taillet, Physical Review D 69, p. 063501(March 2004).
* [16] G. di Bernardo, C. Evoli, D. Gaggero, D. Grasso and L. Maccione, Astroparticle Physics 34, 274(December 2010).
* [17] R. Trotta, G. Jóhannesson, I. V. Moskalenko, T. A. Porter, R. Ruiz de Austri and A. W. Strong, Astrophysical Journal 729, p. 106(March 2011).
* [18] T. Delahaye, J. Lavalle, R. Lineros, F. Donato and N. Fornengo, Astronomy and Astrophysics 524, p. A51(December 2010).
* [19] S. V. Bulanov and V. A. Dogel, Astrophysics and Space Science 29, 305(August 1974).
* [20] X.-G. He, Modern Physics Letters A 24, 2139 (2009).
* [21] P. Ciafaloni, D. Comelli, A. Riotto, F. Sala, A. Strumia and A. Urbano, Journal of Cosmology and Astroparticle Physics 3, p. 19(March 2011).
* [22] C. Evoli, I. Cholis, D. Grasso, L. Maccione and P. Ullio, ArXiv e-prints (August 2011).
|
arxiv-papers
| 2011-10-30T18:15:52 |
2024-09-04T02:49:23.720751
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Daniele Gaggero, Dario Grasso",
"submitter": "Daniele Gaggero",
"url": "https://arxiv.org/abs/1110.6626"
}
|
1110.6653
|
# Graded Betti numbers of path ideals of cycles and lines
Ali Alilooee Department of Mathematics and Statistics, Dalhousie University,
Halifax, Canada, alilooee@mathstat.dal.ca. Sara Faridi Department of
Mathematics and Statistics, Dalhousie University, Halifax, Canada,
faridi@mathstat.dal.ca.
###### Abstract
We use purely combinatorial arguments to give a formula to compute all graded
Betti numbers of path ideals of line graphs and cycles. As a consequence we
can give new and short proofs for the known formulas of regularity and
projective dimensions of path ideals of line graphs.
## 1 Introduction
Path complexes are simplicial complexes whose facets encode paths of a fixed
length in a graph. These simplicial complexes in turn correspond to monomial
ideals called “path ideals”. Path ideals of graphs were first introduced by
Conca and De Negri [3] in a different algebraic context, but the study of
algebraic invariants corresponding to their minimal free resolutions has
become popular, with works of Bouchat, Hà and O’Keefe [2] and He and Van Tuyl
[6], and the authors [1].
The papers cited above gives partial information on Betti numbers of path
ideals. In this paper we use purely combinatorial arguments based on our
results in [1] to give an explicit formula for all the graded Betti numbers of
path ideals of line graphs and cycles. As a consequence we can give new and
short proofs for the known formulas of regularity and projective dimensions of
path ideals of line graphs.
## 2 Preliminaries
A simplicial complex on vertex set ${\mathcal{X}}=\\{x_{1},\dots,x_{n}\\}$ is
a collection $\Delta$ of subsets of ${\mathcal{X}}$ such that
$\\{x_{i}\\}\in\Delta$ for all i, and if $F\in\Delta$ and $G\subset F$, then
$G\in\Delta$. The elements of $\Delta$ are called faces of $\Delta$ and the
maximal faces under inclusion are called facets of $\Delta$. We denote the
simplicial complex $\Delta$ with facets $F_{1},\dots,F_{s}$ by $\langle
F_{1},\dots,F_{s}\rangle$. We call $\\{F_{1},\dots,F_{s}\\}$ the facet set of
$\Delta$ and is denoted by $F(\Delta)$. The vertex set of $\Delta$ is denoted
by $\mbox{Vert}(\Delta)$. A subcollection of $\Delta$ is a simplicial complex
whose facet set is a subset of the facet set of $\Delta$. For
${\mathcal{Y}}\subseteq{\mathcal{X}}$, an induced subcollection of $\Delta$ on
${\mathcal{Y}}$, denoted by $\Delta_{{\mathcal{Y}}}$, is the simplicial
complex whose vertex set is a subset of ${\mathcal{Y}}$ and facet set is
$\\{F\in F(\Delta)\ |\ F\subseteq{{\mathcal{Y}}}\\}.$
If $F$ is a face of $\Delta=\langle F_{1},\dots,F_{s}\rangle$, we define the
complement of $F$ in $\Delta$ to be
$\displaystyle F_{\mathcal{X}}^{c}={\mathcal{X}}\setminus F$ and
$\displaystyle\Delta_{\mathcal{X}}^{c}=\langle(F_{1})^{c}_{\mathcal{X}},\dots,(F_{s})^{c}_{\mathcal{X}}\rangle.$
Note that if ${\mathcal{X}}\subsetneqq\mbox{Vert}(\Delta)$, then
$\Delta^{c}_{\mathcal{X}}=(\Delta_{\mathcal{X}})^{c}_{\mathcal{X}}$.
From now on we assume that $R=K\left[x_{1},\dots,x_{n}\right]$ is a polynomial
ring over a field $K$. Suppose $I$ an ideal in $R$ minimally generated by
square-free monomials $M_{1},\ldots,M_{s}$. The facet complex $\Delta(I)$
associated to $I$ has vertex set $\\{x_{1},\dots,x_{n}\\}$ and is defined as
$\Delta(I)=\langle F_{1},\ldots,F_{s}\rangle\mbox{ where }F_{i}=\\{x_{j}\ |\
x_{j}|M_{i},\ 1\leq j\leq n\\},\ 1\leq i\leq s.$ Conversely if $\Delta$ is a
simplicial complex with vertices labeled $x_{1},\ldots,x_{n}$, the facet ideal
of $\Delta$ is defined as $I(\Delta)=(\prod_{x\in F}x\ |\ \ F\mbox{ is a facet
of}\Delta).$
Given a homogeneous ideal $I$ of the polynomial ring $R$ there exists a graded
minimal finite free resolution
$0\rightarrow{\displaystyle\bigoplus_{d}}R(-d)^{\beta_{p,d}}\rightarrow\cdots{\displaystyle\rightarrow\bigoplus_{d}}R(-d)^{\beta_{1,d}}\rightarrow
R\rightarrow R/I\rightarrow 0$
of $R/I$ in which $R(-d)$ denotes the graded free module obtained by shifting
the degrees of elements in $R$ by $d$. The numbers $\beta_{i,d}$ are the
$i$-th $\mathbb{N}$-graded Betti numbers of degree $d$ of $R/I$, and are
independent of the choice of graded minimal finite free resolution.
The first step to our computations of Betti numbers is a form of Hochster’s
formula for Betti numbers that was proved in [1].
###### Theorem 2.1 ([1] Theorem 2.8).
Let $R=K[x_{1},\dots,x_{n}]$ be a polynomial ring over a field $K$, and $I$ be
a pure square-free monomial ideal in $R$. Then the $\mathbb{N}$-graded Betti
numbers of $R/I$ are given by
$\beta_{i,d}(R/I)={\displaystyle\sum_{\Gamma\subset\Delta(I),|\mbox{Vert}(\Gamma)|=d}}\hskip
7.22743pt\dim_{K}\widetilde{H}_{i-2}(\Gamma^{c}_{\mbox{Vert}(\Gamma)})$
where the sum is taken over the induced subcollections $\Gamma$ of $\Delta(I)$
which have $d$ vertices.
Because of Theorem 2.1, to compute Betti numbers we only need to consider
induced subcollections $\Gamma=\Delta_{\mathcal{Y}}$ of a simplicial complex
$\Delta$ with ${\mathcal{Y}}=\mbox{Vert}(\Gamma)$.
## 3 Path complexes and runs
###### Definition 3.1.
Let $G=({\mathcal{X}},E)$ be a finite simple graph and $t$ be an integer such
that $t\geq 2$. If $x$ and $y$ are two vertices of $G$, a path of length
$(t-1)$ from $x$ to $y$ is a sequence of vertices
$x=x_{i_{1}},\dots,x_{i_{t}}=y$ of $G$ such that
$\\{x_{i_{j}},x_{i_{j+1}}\\}\in E$ for all $j=1,2,\dots,t-1$. We define the
path ideal of $G$, denoted by $I_{t}(G)$ to be the ideal of
$K[x_{1},\dots,x_{n}]$ generated by the monomials of the form
$x_{i_{1}}x_{i_{2}}\dots x_{i_{t}}$ where
$x_{i_{1}},x_{i_{2}},\dots,x_{i_{t}}$ is a path in $G$. The facet complex of
$I_{t}(G)$, denoted by $\Delta_{t}(G)$, is called the path complex of the
graph $G$.
Two special cases that we will be considering in this paper are when $G$ is a
cycle $C_{n}$, or a line graph $L_{n}$ on vertices $\\{x_{1},\dots,x_{n}\\}$.
$C_{n}=\langle x_{1}x_{2},\ldots,x_{n-1}x_{n},x_{n}x_{1}\rangle\mbox{\ and \
}L_{n}=\langle x_{1}x_{2},\ldots,x_{n-1}x_{n}\rangle.$
###### Example 3.2.
Consider the cycle $C_{5}$ with vertex set
${\mathcal{X}}=\\{x_{1},\dots,x_{5}\\}$ Then
$I_{4}(C_{5})=(x_{1}x_{2}x_{3}x_{4},\\\
x_{2}x_{3}x_{4}x_{5},x_{3}x_{4}x_{5}x_{1},x_{4}x_{5}x_{1}x_{2},x_{5}x_{1}x_{2}x_{3}).$
###### Notation 3.3.
Let $i$ and $n$ be two positive integers. For (a set of) labeled objects we
use the notation $\mod n$ to denote
$x_{i}\mod n\ =\\{x_{j}\ |\ 1\leq j\leq n,i\equiv j\mod n\\}$
and
$\\{x_{u_{1}},x_{u_{2}},\dots,x_{u_{t}}\\}\mod n\ =\\{x_{u_{j}}\mod n\ |\
j=1,2,\dots,n\\}.$
Let $C_{n}$ be a cycle on vertex set ${\mathcal{X}}=\\{x_{1},\dots,x_{n}\\}$
and $t<n$. The standard labeling of the facets of $\Delta_{t}(C_{n})$ is as
follows. We let $\Delta_{t}(C_{n})=\langle F_{1},\dots,F_{n}\rangle$ where
$F_{i}=\\{x_{i},x_{i+1},\dots,x_{i+t-1}\\}\mod n$ for all $1\leq i\leq n$.
Since for each $1\leq i\leq n$ we have
$\begin{array}[]{llll}F_{i+1}\setminus
F_{i}=\\{x_{t+i}\\}&\mbox{and}&F_{i}\setminus F_{i+1}=\\{x_{i}\\}&\mod
n,\end{array}$
it follows that $\begin{array}[]{lllll}\left|F_{i}\setminus
F_{i+1}\right|=1&\mbox{and}&\left|F_{i+1}\setminus F_{i}\right|=1&\mod
n&\mbox{for all $1\leq i\leq n-1$}.\end{array}$
###### Definition 3.4.
Given an integer $t$, we define a run to be the path complex of a line graph.
A run which has $p$ facets is called a run of length $p$ and corresponds to
$\Delta_{t}(L_{p+t-1})$. Therefore a run of length $p$ has $p+t-1$ vertices.
###### Example 3.5.
Consider the cycle $C_{7}$ on vertex set ${\mathcal{X}}=\\{x_{1},\dots
x_{7}\\}$ and the simplicial complex $\Delta_{4}(C_{7})$. The following
induced subcollections are two runs in $\Delta_{4}(C_{7})$
$\begin{array}[]{lll}\Delta_{1}&=&\langle\\{x_{1},x_{2},x_{3},x_{4}\\},\\{x_{2},x_{3},x_{4},x_{5}\\}\rangle\\\
\Delta_{2}&=&\langle\\{x_{1},x_{2},x_{6},x_{7}\\},\\{x_{1},x_{2},x_{3},x_{7}\\},\\{x_{1},x_{2},x_{3},x_{4}\\}\rangle.\end{array}$
In [1] we show that every induced subcollection of the path complex of a cycle
is a disjoint union of runs ([1] Proposition 3.6), and that two induced
subcollections of the path complex of a cycle composed of the same number of
runs of the same lengths are homeomorphic ([1] Lemma 3.8). Therefore all the
information we need to compute the homologies of induced subcollections of
$\Delta_{t}(C_{n})$ depends on the number and the lengths of the runs.
## 4 Graded Betti numbers of path ideals
We focus on Betti numbers of degree less than $n$, as those of degree $n$ were
computed in [1]. By Theorem 2.1 we need to count induced subcollections.
###### Definition 4.1.
Let $i$ and $j$ be positive integers. We call an induced subcollection
$\Gamma$ of $\Delta_{t}(C_{n})$ an $(i,j)$-eligible subcollection of
$\Delta_{t}(C_{n})$ if $\Gamma$ is composed of disjoint runs of lengths
$\displaystyle(t+1)p_{1}+1,\dots,(t+1)p_{\alpha}+1,(t+1)q_{1}+2,\ldots,(t+1)q_{\beta}+2$
(4.1)
for nonnegative integers
$\alpha,\beta,p_{1},p_{2},\dots,p_{\alpha},q_{1},q_{2},\dots,q_{\beta}$, which
satisfy the following conditions
$\begin{array}[]{lll}j&=&(t+1)(P+Q)+t(\alpha+\beta)+\beta\\\
i&=&2(P+Q)+2\beta+\alpha,\end{array}$
where $P=\sum_{i=1}^{\alpha}p_{i}$ and $Q=\sum_{i=1}^{\beta}q_{i}$.
Eligible subcollections count the graded Betti numbers.
###### Theorem 4.2 ( [1] Theorem 5.3).
Let $I=I(\Lambda)$ be the facet ideal of an induced subcollection $\Lambda$ of
$\Delta_{t}(C_{n})$. Suppose $i$ and $j$ are integers with $i\leq j<n$. Then
the ${\mathbb{N}}$-graded Betti number $\beta_{i,j}(R/I)$ is the number of
$(i,j)$-eligible subcollections of $\Lambda$.
The following corollary is a special case of Theorem 4.2.
###### Corollary 4.3.
Let $I=I(\Lambda)$ be the facet ideal of an induced subcollection $\Lambda$ of
$\Delta_{t}(C_{n})$. Then for every $i$, $\beta_{i,ti}(R/I)$, is the number of
induced subcollections of $\Lambda$ which are composed of $i$ runs of length
1.
###### Proof.
From Theorem 4.2 we have $\beta_{i,ti}(R/I)$ is the number of
$(i,ti)$-eligible subcollections of $\Lambda$. With notation as in Definition
4.1 we have
$\left\\{\begin{array}[]{ll}ti=(t+1)(P+Q)+t(\alpha+\beta)+\beta&\\\
i=2(P+Q)+(\alpha+\beta)+\beta&\Rightarrow ti=2t(P+Q)+t(\alpha+\beta)+t\beta\\\
\end{array}\right.$
Putting the two equations for $ti$ together, we conclude that
$(t-1)(P+Q+\beta)=0$. But $\beta$, $P$, $Q\geq 0$ and $t\geq 2$, so we must
have
$\beta=P=Q=0\Rightarrow p_{1}=p_{2}=\dots=p_{\alpha}=0.$
So $\alpha=i$ and $\Gamma$ is composed of $i$ runs of length one. ∎
Theorem 4.2 holds in particular for $\Lambda=\Delta_{t}(L_{m})$ and
$\Lambda=\Delta_{t}(C_{n})$ for any integers $m,n$. Our next statement is in a
sense a converse to Theorem 4.2.
###### Proposition 4.4.
Let $t$ and $n$ be integers such that $2\leq t\leq n$ and $I=I(\Lambda)$ be
the facet ideal of $\Lambda$ where $\Lambda$ is an induced subcollection of
$\Delta_{t}(C_{n})$. Then for each $i,j\in\mathbb{N}$ with $i\leq d<n$, if
$\beta_{i,j}(R/I)\neq 0$, there exist nonnegative integers $\ell,d$ such that
$\left\\{\begin{array}[]{lll}i&=&\ell+d\\\ j&=&t\ell+d\end{array}\right.$
###### Proof.
From Theorem 4.2 we know $\beta_{i,j}$ is equal to the number of
$(i,j)$-eligible subcollections of $\Lambda$, where with notation as in
Definition 4.1 we have
$\left\\{\begin{array}[]{lcr}j=(t+1)(P+Q)+t(\alpha+\beta)+\beta\\\
i=2(P+Q)+(\alpha+\beta)+\beta.\end{array}\right.$
It follows that
$\displaystyle j-i=(t-1)(P+Q+\alpha+\beta)$ and $\displaystyle
ti-j=(t-1)(P+Q+\beta).$ (4.2)
We now show that there exist positive integers $\ell,d$ such that $i=\ell+d$
and $j=t\ell+d$.
$\begin{array}[]{lll}\left\\{\begin{array}[]{lcr}i=\ell+d\\\
j=t\ell+d\end{array}\right.\Rightarrow\begin{array}[]{lll}\ell=\displaystyle\frac{j-i}{t-1}&\mbox{and}&d=\displaystyle\frac{ti-j}{t-1}\end{array}.\end{array}$
From (4.2) we can see that $i$ and $j$ as described above are nonnegative
integers. ∎
Theorem 4.2 tells us that to compute Betti numbers of induced subcollections
of $\Delta_{t}(C_{n})$ we need to count the number of its induced
subcollections which consist of disjoint runs of lengths one and two. The next
few pages are dedicated to counting such subcollections. We use some
combinatorial methods to generalize a helpful formula which can be found in
Stanley’s book [8] on page 73.
###### Lemma 4.5.
Consider a collection of $n$ points arranged on a line. The number of ways of
coloring $k$ points, when there are at least $t$ uncolored points on the line
between each colored point is
${{n-(k-1)t}\choose{k}}.$
###### Proof.
First label the points from $1,2,\dots,n$ from left to right, and let
$a_{1}<a_{2}<\dots<a_{k}$ be the colored points. For $1\leq i\leq k-1$, we
define $x_{i}$ to be the number of points, including $a_{i}$, which are
between $a_{i}$ and $a_{i+1}$, and $x_{0}$ to be the number of points which
exist before $a_{1}$, and $x_{k}$ the number of points, including $a_{k}$,
which are after $a_{k}$.
$\begin{array}[]{llllll}\overbrace{\cdots}^{x_{0}}&\overbrace{\bullet\
\cdots}^{x_{1}}&\overbrace{\bullet\ \cdots}^{x_{2}}&{\bullet}\
\cdots&\overbrace{\bullet\ \cdots}^{x_{k-1}}&\overbrace{\bullet\
\cdots}^{x_{k}}\\\ 1&a_{1}&a_{2}&a_{3}&a_{k-1}&a_{k}\ \ n\\\ \end{array}$
If we consider the sequence $x_{0},x_{1},\dots,x_{k}$ it is not difficult to
see that there is a one to one correspondence between the positive integer
solutions of the following equation and the ways of coloring $k$ points of $n$
points on a line with at least $t$ uncolored points between each two colored
points.
$\displaystyle x_{0}+x_{1}+\dots+x_{k}=n$ $\displaystyle\mbox{$x_{0}\geq 0$,
$x_{i}>t$, for $1\leq i\leq k-1$, and $x_{k}\geq 1$}.$
So we only need to find the number of positive integer solutions of this
equation. Consider the following equation
$(x_{0}+1)+(x_{1}-t)+\dots+(x_{k-1}-t)+x_{k}=n-(k-1)t+1$
where $x_{0}+1\geq 1$, $x_{i}-t\geq 1$, for $i=0\dots,k-1$ and $x_{k}\geq 1$.
The number of positive integer solution of this equation is (see for example
[5] page 29)
${{n-(k-1)t}\choose{k}}.$
∎
###### Corollary 4.6.
Let $C_{n}$ be a graph cycle and with the standard labeling let $\Gamma$ be a
proper subcollection of $\Delta_{t}(C_{n})$ with $k$ facets
$F_{a},\ldots,F_{a+k-1}\mod n$. The number of induced subcollections of
$\Gamma$ which are composed of $m$ runs of length one is
${k-(m-1)t\choose m}.$
###### Proof.
To compute the number of induced subcollections of $\Gamma$ which are composed
of $m$ runs of length one, it is enough to consider the facets
$F_{a},\ldots,F_{a+k-1}$ as points arranged on a line and compute the number
of ways which we can color $m$ points of these $k$ arranged points with at
least $t$ uncolored points between each two consecutive colored points.
Therefore, by Lemma 4.5 we have the number of induced subcollections of
$\Gamma$ which are composed of $m$ runs of length one is ${k-(m-1)t\choose
m}.$ ∎
###### Proposition 4.7.
Let $C_{n}$ be a graph cycle with vertex set
${\mathcal{X}}=\\{x_{1},\dots,x_{n}\\}$. The number of induced subcollections
of $\Delta_{t}(C_{n})$ which are composed of $m$ runs of length one is
$\frac{n}{n-mt}{n-mt\choose m}.$
###### Proof.
Recall that $\Delta_{t}(C_{n})=\langle F_{1},\dots,F_{n}\rangle$ with standard
labeling. First we compute the number of induced subcollections of
$\Delta_{t}(C_{n})$ which consist of $m$ runs of length one and do not contain
the vertex $x_{n}$. There are $t$ facets of $\Delta_{t}(C_{n})$ which contain
$x_{n}$, the remaining facets are $F_{1},\dots,F_{n-t}$, and so by Corollary
4.6 the number we are looking for is
$\displaystyle{n-t-(m-1)t\choose m}={n-mt\choose m}.$ (4.3)
Now we are going to compute the number of induced subcollections $\Gamma$
which consist of $m$ runs of length one and include $x_{n}$. We have $t$
facets which contain $x_{n}$, they are $F_{n-t+1}\dots,F_{n}$. Each such
$\Gamma$ will contain one $F_{i}\in\\{F_{n-t+1}\dots,F_{n}\\}$ as the run
containing $x_{n}$, and $m-1$ other runs of length one which have to be chosen
so that they are disjoint from $F_{i}$. So we are looking for $m-1$ runs of
length one in the subcollection $\Gamma^{\prime}=\langle
F_{i+t},\ldots,F_{i-t}\rangle\mod n$. The subcollection $\Gamma^{\prime}$ has
$n-2t-1$ facets, so by Corollary 4.6 it has
${n-2t-1-(m-2)t\choose m-1}={n-mt-1\choose m-1}$
induced subcollections that consist of runs of length one. Putting this
together with the number of ways to choose $F_{i}$ and with (4.3) we conclude
that the number of induced subcollections of $\Delta_{t}(C_{n})$ which are
composed of $m$ runs of length one is
$t{n-mt-1\choose m-1}+{n-mt\choose m}=\frac{n}{n-mt}{n-mt\choose m}.$
∎
We apply these counting facts to find Betti numbers in specific degrees; the
formula in (iii) below (that of a line graph) was also computed by Bouchat, Ha
and O’Keefe [2] using Eliahou-Kervaire techniques.
###### Corollary 4.8.
Let $n\geq 2$ and $t$ be an integer such that $2\leq t\leq n$. Then we have
1. i.
For the cycle $C_{n}$ we have
$\beta_{i,it}(R/I_{t}(C_{n}))=\frac{n}{n-it}{n-it\choose i}.$
2. ii.
For any proper induced subcollection $\Lambda$ of $\Delta_{t}(C_{n})$ with $k$
facets we have
$\beta_{i,ti}(R/I(\Lambda)={k-(i-1)t\choose i}.$
3. iii.
For the line graph $L_{n}$, we have
$\beta_{i,ti}(R/I_{t}(L_{n})={n-it+1\choose i}.$
###### Proof.
From Corollary 4.3 we have $\beta_{i,it}(R/I)$ in each of the three cases (i),
(ii) and (iii) is the number of induced subcollections of $\Delta_{t}(C_{n})$,
$\Lambda$ and $\Delta_{t}(L_{n})$, respectively, which are composed of $i$
runs of length 1. Case (i) now follows from Proposition 4.7, while (ii) and
(iii) follow directly from Corollary 4.6. ∎
The following Lemma is the core of our counting later on in this section.
###### Lemma 4.9.
Let $\Delta_{t}(C_{n})=\langle F_{1},F_{2},\dots,F_{n}\rangle$, $2\leq t\leq
n$, be the standard labeling of the path complex of a cycle $C_{n}$ on vertex
set ${\mathcal{X}}=\\{x_{1},\ldots,x_{n}\\}$. Let $i$ be a positive integer
and $\Gamma=\langle F_{c_{1}},F_{c_{2}},\dots,F_{c_{i}}\rangle$ be an induced
subcollection of $\Delta_{t}(C_{n})$ consisting of $i$ runs of length 1, with
$1\leq c_{1}<c_{2}<\dots<c_{i}\leq n$. Suppose $\Sigma$ is the induced
subcollection on $\mbox{Vert}(\Gamma)\cup\\{x_{c_{u}+t}\\}$ for some $1\leq
u\leq i$. Then
$|\Sigma|=\left\\{\begin{array}[]{lll}|\Gamma|+t&u<i\ \mbox{
and}&c_{u+1}=c_{u}+t+1\\\ |\Gamma|+1&u=i\ \mbox{
or}&c_{u+1}>c_{u}+t+1\end{array}\right.$
###### Proof.
Since $\Gamma$ consists of runs of length one and each
$F_{c_{u}}=\\{x_{c_{u}},x_{c_{u}+1},\dots,x_{c_{u}+t-1}\\}$ we must have
$c_{u+1}>c_{u}+t\mod n$ for $u\in\\{1,2,\dots,i-1\\}$. There are two ways that
$x_{c_{u}+t}$ could add facets to $\Gamma$ to obtain $\Sigma$.
1. 1.
If $c_{u+1}=c_{u}+t+1$ then
$F_{c_{u}},F_{c_{u}+1},\dots,F_{c_{u}+t+1}=F_{c_{u+1}}\in\Sigma$ or in other
words, we have added $t$ new facets to $\Gamma$.
2. 2.
If $c_{u+1}>c_{u}+t+1$ or $u=i$ then $F_{c_{u}+1}\in\Sigma$, and therefore one
new facet is added to $\Gamma$.
∎
The following propositions, which generalize Lemma 7.4.22 in [7], will help us
compute the remaining Betti numbers.
###### Proposition 4.10.
Let $\Delta_{t}(C_{n})=\langle F_{1},F_{2},\dots,F_{n}\rangle$, $2\leq t\leq
n$, be the standard labeling of the path complex of a cycle $C_{n}$ on vertex
set ${\mathcal{X}}=\\{x_{1},\ldots,x_{n}\\}$. Also let $i$, $j$ be positive
integers such that $j\leq i$ and $\Gamma=\langle
F_{c_{1}},F_{c_{2}},\dots,F_{c_{i}}\rangle$ be an induced subcollection of
$\Delta_{t}(C_{n})$ consisting of $i$ runs of length 1, with $1\leq
c_{1}<c_{2}<\dots<c_{i}\leq n$. Suppose $W=\mbox{Vert}(\Gamma)\cup
A\subsetneq{\mathcal{X}}$ for some subset $A$ of
$\\{x_{c_{1}+t},\dots,x_{c_{i}+t}\\}\mod n$ with $|A|=j$. Then the induced
subcollection $\Sigma$ of $\Delta_{t}(C_{n})$ on $W$ is an
$(i+j,ti+j)$-eligible subcollection.
###### Proof.
Since $\Gamma$ consists of runs of length one and each
$F_{c_{u}}=\\{x_{c_{u}},x_{c_{u}+1},\dots,x_{c_{u}+t-1}\\}$ we must have
$c_{u+1}>c_{u}+t\mod n$ for $u\in\\{1,2,\dots,i-1\\}$. The runs (or connected
components) of $\Sigma$ are of the form $\Sigma^{\prime}=\Sigma_{U}$ where
$U\subseteq W$, and can have one of the following possible forms.
1. a.
For some $a\leq i$:
$U=F_{c_{a}},$
and therefore $\Sigma^{\prime}=\langle F_{c_{a}}\rangle$ is a run of length 1.
2. b.
For some $a\leq i$:
$U=F_{c_{a}}\cup\\{x_{{c_{a}}+t}\\},$
and therefore $c_{a+1}>c_{a}+t+1$, so from Lemma 4.9 we have
$\Sigma^{\prime}=\langle F_{c_{a}},F_{c_{a}+1}\rangle$ is a run of length 2.
3. c.
For some $a\leq i$:
$U=F_{c_{a}}\cup F_{c_{a+1}}\cup\dots\cup
F_{c_{a+r}}\cup\\{x_{c_{a}+t},x_{c_{a+1}+t},\dots,x_{c_{a+r-1}+t}\\}\hskip
7.22743pt\mod n$
and $F_{c_{a+j}}=F_{{c_{a}}+j(t+1)}$ for $j=0,1,\dots,r$ and $r\geq 1$. Then
from Lemma 4.9 above we know $\Sigma^{\prime}$ is a run of length
$r+1+tr=(t+1)r+1$.
4. d.
For some $a\leq i$:
$U=F_{c_{a}}\cup F_{c_{a+1}}\cup\dots\cup
F_{c_{a+r}}\cup\\{x_{c_{a}+t},x_{c_{a+1}+t},\dots,x_{c_{a+r}+t}\\}\hskip
7.22743pt\mod n$
and $F_{c_{a+j}}=F_{{c_{a}}+j(t+1)}$ for $j=0,1,\dots,r$ and $r\geq 1$, and
$c_{a+r+1}>c_{a+r}+t+1$ or ${a+r}=i$. Then from Lemma 4.9 we have
$\Sigma^{\prime}$ is a run of length $r+1+tr+1=(t+1)r+2$.
So we have shown that $\Sigma$ consists of runs of length $1$ and $2$ $\mod
t+1$.
Suppose the runs in $\Sigma$ are of the form described in (4.1). By Definition
3.4 we have
$\begin{array}[]{ll}|\mbox{Vert}(\Sigma)|&=(t+1)p_{1}+t+\dots+(t+1)p_{\alpha}+t+(t+1)q_{1}+t+1+\dots+(t+1)q_{\beta}+t+1\vspace{.1in}{}\\\
&=(t+1)P+t\alpha+(t+1)Q+t\beta+\beta\vspace{.1 in}{}\\\
&=(t+1)(P+Q)+t(\alpha+\beta)+\beta.\end{array}$
On the other hand by the definition of $\Sigma$ we know that, $\Sigma$ has
$ti+j$ vertices and therefore
$ti+j=(t+1)(P+Q)+t(\alpha+\beta)+\beta.$
It remains to show that $i+j=2(P+Q)+(\alpha+\beta)+\beta$. Note that if $j=0$
then $\beta=P=Q=0$ and hence
$\displaystyle j=0$ $\displaystyle\Longrightarrow$ $\displaystyle
P+Q+\beta=0.$ (4.4)
Moreover each vertex $x_{{c_{v}}+t}\in A$ either increases the length of a run
in $\Gamma$ by one and hence increases $\beta$ (the number of runs of length 2
in $\Gamma$) by one, or increases the length of a run by $t+1$, in which case
$P+Q$ increases by 1. We can conclude that if we add $j$ vertices to $\Gamma$,
$P+Q+\beta$ increases by $j$. From this and (4.4) we have $j=P+Q+\beta$. Now
we solve the following system
$\left\\{\begin{array}[]{rllll}ti+j&=&(t+1)(P+Q)+t(\alpha+\beta)+\beta&\Longrightarrow&ti=t(P+Q)+t(\alpha+\beta)\\\
j&=&P+Q+\beta&\Longrightarrow&i=P+Q+\alpha+\beta\end{array}\right.$
$\Longrightarrow\left\\{\begin{array}[]{lll}i&=&P+Q+\alpha+\beta\\\
j&=&P+Q+\beta\end{array}\right.\Longrightarrow
i+j=2(P+Q)+(\alpha+\beta)+\beta.$
∎
###### Proposition 4.11.
Let $C_{n}$ be a cycle, $2\leq t\leq n$, and $i$ and $j$ be positive integers.
Suppose $\Sigma$ is an $(i+j,ti+j)$-eligible subcollection of
$\Delta_{t}(C_{n})$, $2\leq t\leq n$. Then with notation as in Definition 4.1,
there exists a unique induced subcollection $\Gamma$ of $\Delta_{t}(C_{n})$ of
the form $\langle F_{c_{1}},F_{c_{2}},\dots,F_{c_{i}}\rangle$ with $1\leq
c_{1}<c_{2}<\dots<c_{i}\leq n$ consisting of $i$ runs of length $1$, and a
subset $A$ of $\\{x_{c_{1}+t},\dots,x_{c_{i}+t}\\}$ $\mod\ n$, with $|A|=j$
such that $\Sigma=\Delta_{t}(C_{n})_{W}$ where $W=\mbox{Vert}(\Gamma)\cup A.$
Moreover if ${\mathcal{R}}=\langle F_{h},F_{h+1},\dots,F_{h+m}\rangle\mod n$
is a run in $\Sigma$ with $|{\mathcal{R}}|=2\mod(t+1)$, then
$F_{h+m}\notin\Gamma\mod n$.
###### Proof.
Suppose $\Sigma$ consists of runs
$R_{1}^{\prime},R_{2}^{\prime},\ldots,R_{\alpha+\beta}^{\prime}$ where for
$k=1,2,\ldots,\alpha+\beta$
$\begin{array}[]{ll}R_{k}^{\prime}=\langle
F_{h_{k}},F_{h_{k}+1},\dots,F_{h_{k}+m_{k}-1}\rangle&\mod n\vspace{.1 in}\\\
\mbox{Vert}(R_{k}^{\prime})=\\{x_{h_{k}},x_{h_{k}+1},\dots,x_{h_{k}+m_{k}+t-2}\\}&\mod
n\vspace{.1 in}\\\ h_{k+1}\geq t+h_{k}+m_{k}&\mod n\end{array}$
and
$\displaystyle
m_{k}=\left\\{\begin{array}[]{lll}(t+1)p_{k}+1&\mbox{for}&k=1,2,\dots,\alpha\\\
(t+1)q_{k-\alpha}+2&\mbox{for}&k=\alpha+1,\alpha+2,\dots,\alpha+\beta.\end{array}\right.$
(4.7)
For each $k$, we remove the following vertices from
$\mbox{Vert}(R_{k}^{\prime})$
$\displaystyle\begin{array}[]{lll}x_{h_{k}+t},x_{h_{k}+2t+1},\dots,x_{h_{k}+p_{k}t+(p_{k}-1)}&\mod
n&\mbox{ if }1\leq k\leq\alpha\mbox{ and }p_{k}\neq 0\\\
x_{h_{k}+t},x_{h_{k}+2t+1},\dots,x_{h_{k}+(q_{k-\alpha}+1)t+q_{k-\alpha}}&\mod
n&\mbox{ if }\alpha+1\leq k\leq\alpha+\beta\end{array}$ (4.10)
Let $\Gamma=\langle R_{1},R_{2},\dots R_{\alpha+\beta}\rangle$ be the induced
subcollection on the remaining vertices of $\Sigma$, where
$\displaystyle R_{k}=\left\\{\begin{array}[]{lll}\langle
F_{h_{k}},F_{h_{k}+t+1},\dots,F_{h_{k}+(t+1)p_{k}}\rangle&\mod n&\mbox{for
}1\leq k\leq\alpha\\\ \langle
F_{h_{k}},F_{h_{k}+t+1},\dots,F_{h_{k}+(t+1)q_{k-\alpha}}\rangle&\mod
n&\mbox{for }\alpha+1\leq k\leq\alpha+\beta.\end{array}\right.$ (4.13)
In other words,$\mod n$, $\Gamma$ has facets
$F_{h_{1}},F_{h_{1}+t+1},\dots,F_{h_{1}+(t+1)p_{1}},F_{h_{2}},F_{h_{2}+t+1},\dots,F_{h_{2}+(t+1)p_{2}},\dots,F_{h_{\alpha+\beta}},\dots,F_{h_{\alpha+\beta}+(t+1)q_{\beta}}.$
It is clear that each $R_{k}$ consists of runs of length one. Since $\Gamma$
is a subcollection of $\Sigma$, no runs of $R_{k}$ and $R_{k^{\prime}}$ are
connected to one another if $k\neq k^{\prime}$, and hence we can conclude
$\Gamma$ is an induced subcollection of $\Delta_{t}(C_{n})$ which is composed
of runs of length one. From (4.13) we have the number of runs of length 1 in
$\Gamma$ (or the number of facets of $\Gamma$) is equal to
$(p_{1}+1)+(p_{2}+1)+\dots+(p_{\alpha}+1)+(q_{1}+1)+\dots+(q_{\beta}+1)=P+Q+\alpha+\beta=i.$
Therefore, $\Gamma$ is an induced subcollection of $\Delta_{t}(C_{n})$ which
is composed of $i$ runs of length 1. We relabel the facets of $\Gamma$ as
$\Gamma=\langle F_{c_{1}},\dots,F_{c_{i}}\rangle$. Now consider the following
subset of $\\{x_{c_{1}+t},\dots,x_{c_{i}+t}\\}$ as $A$
$\bigcup_{k=1,p_{k}\neq
0}^{\alpha}\\{x_{h_{k}+t},x_{h_{k}+2t+1},\dots,x_{h_{k}+p_{k}t+(p_{k}-1)}\\}\cup\bigcup_{k={\alpha+1}}^{\alpha+\beta}\\{x_{h_{k}+t},x_{h_{k}+2t+1},\dots,x_{h_{k}+(q_{k-\alpha}+1)t+q_{k-\alpha}}\\}$
by (4.10) we have:
$|A|=(p_{1}+p_{2}+\dots+p_{\alpha})+(q_{1}+1\dots+q_{\beta}+1)=P+Q+\beta=j.$
Then if we set
$W=(\bigcup_{h=1}^{i}F_{c_{h}})\cup A$
we clearly have $\Sigma=(\Delta_{t}(C_{n}))_{W}$. This proves the existence of
$\Gamma$, we now prove its uniqueness. Let $\Lambda=\langle
F_{s_{1}},F_{s_{2}},\dots,F_{s_{i}}\rangle$ be an induced subcollection of
$\Delta_{t}(C_{n})$ which is composed of $i$ runs of length 1 such that $1\leq
s_{1}<s_{2}<\dots<s_{i}\leq n$. Also let $B$ be a $j$\- subset of the set
$\begin{array}[]{lll}\\{x_{s_{1}+t},x_{s_{2}+t},\dots,x_{s_{i}+t}\\}&\mod
n\end{array}$ such that
$\Sigma=({\Delta_{t}(C_{n})})_{\mbox{Vert}(\Lambda)\cup B}.$ (4.14)
Suppose $\Lambda=\langle S_{1},S_{2},\dots,S_{\alpha+\beta}\rangle$, such that
for $k=1,2,\dots,\alpha+\beta$, $S_{k}$ is an induced subcollection of
$R_{k}^{\prime}$ which consists of $y_{k}$ runs of length one. By (4.14) we
have $y_{k}\neq 0$ for all $k$. Now we prove the following claims for each
$k\in\\{1,2,\dots,\alpha+\beta\\}$.
1. a.
_$F_{h_{k}}\in\Lambda$_. Suppose $1\leq k\leq\alpha+\beta$. If $p_{k}=0$ we
are clearly done, so consider the case $p_{k}\neq 0$.
Assume $F_{h_{k}}\notin\Lambda$. Since $F_{h_{k}}$ is the only facet of
$\Sigma$ which contains $x_{h_{k}}$ we can conclude
$x_{h_{k}}\notin\mbox{Vert}(\Lambda)$. From (4.14), it follows that
$x_{h_{k}}\in\\{x_{s_{1}+t},x_{s_{2}+t},\dots,x_{s_{i}+t}\\}$, so
$\displaystyle x_{h_{k}}=x_{s_{a}+t}\mod n\mbox{ for some $a$}.$ (4.15)
On the other hand we know
$\displaystyle F_{s_{a}}=\\{x_{s_{a}},x_{s_{a}+1},\dots,x_{s_{a}+t-1}\\}$
$\displaystyle\mod\ n$ $\displaystyle
F_{s_{a}+1}=\\{x_{s_{a}+1},x_{s_{a}+2},\dots,x_{s_{a}+t}\\}$
$\displaystyle\mod\ n.$
Since $R_{k}^{\prime}$ is an induced connected component of $\Sigma$, by
(4.15) we can conclude $x_{h_{k}}\in F_{s_{a}+1}$ and
$F_{s_{a}},F_{s_{a}+1}\in R_{k}^{\prime}$. However, we know $F_{h_{k}}$ is the
only facet of $R_{k}^{\prime}$ which contains $x_{h_{k}}$ and so
$F_{s_{a}+1}=F_{h_{k}}$ and then $\begin{array}[]{ll}s_{a}+1=h_{k}&\mod
n\end{array}$. This and (4.15) imply that $t=1\mod n$, which contradicts our
assumption $2\leq t\leq n$.
2. b.
_If $F_{u}\in S_{k}$ for some $u$ and $F_{u+t+1}\in R_{k}^{\prime}$, then
$F_{u+t+1}\in S_{k}$._ Assume $F_{u+t+1}\notin S_{k}$ and $F_{u+t+1}\in
R_{k}^{\prime}$. Let
$r_{0}=\min\\{r:r>u,F_{r}\in S_{k}\mod n\\}.$
Since $S_{k}$ consists of runs of length one we can conclude $r_{0}\geq
u+t+1$. Since $r_{0}\neq u+t+1$ we have $r_{0}\geq u+t+2$. But then
$x_{u+t+1}\notin\mbox{Vert}(\Lambda)\cup\\{x_{s_{1}+t},x_{s_{2}+t},\dots,x_{s_{i}+t}\\}$
and therefore $x_{u+t+1}\notin\mbox{Vert}(\Sigma)$ which is a contradiction.
Now for each $k$, by (a) we have $F_{h_{k}}\in\Lambda$ and from repeated
applications of (b) we find that
$F_{h_{k}+f(t+1)}\in S_{k}\hskip 7.22743pt\mbox{ for
}f=\left\\{\begin{array}[]{ll}1,2,\dots,p_{k}&1\leq k\leq\alpha\\\
1,2,\dots,q_{k-\alpha}&\alpha+1\leq k\leq\alpha+\beta.\end{array}\right.$
So $R_{k}\subseteq S_{k}$. On the other hand $S_{k}$ consists of runs of
length one, so no other facet of $R^{\prime}_{k}$ can be added to it, and
therefore $S_{k}=R_{k}$ for all $k$. We conclude that $\Lambda=\Gamma$ and we
are therefore done. The last claim of the proposition is also apparent from
this proof. ∎
We are now ready to compute the remaining Betti numbers.
###### Theorem 4.12.
Let $n$, $i$, $j$ and $t$ be integers such that $n\geq 2$, $2\leq t\leq n$,
and $ti+j<n$. Then
1. i.
For the cycle $C_{n}$
$\beta_{i+j,ti+j}(R/I_{t}(C_{n}))=\frac{n}{n-it}{i\choose j}{n-it\choose i}$
2. ii.
For the line graph $L_{n}$
$\beta_{i+j,ti+j}(R/I_{t}(L_{n}))={i\choose j}{n-it\choose i}+{i-1\choose
j}{n-it\choose i-1}$
###### Proof.
If $I=I_{t}(C_{n})$ (or $I=I_{t}(L_{n})$), from Theorem 4.2,
$\beta_{i+j,ti+j}(R/I)$ is the number of $(i+j,ti+j)$-eligible subcollections
of $\Delta_{t}(C_{n})$ (or $\Delta_{t}(L_{n})$). We consider two separate
cases for $C_{n}$ and for $L_{n}$.
1. i.
For the cycle $C_{n}$, suppose ${\mathcal{R}}_{(i)}$ denotes the set of all
induced subcollections of $\Delta_{t}(C_{n})$ which are composed of $i$ runs
of length one. By propositions 4.10 and 4.11 there exists a one to one
correspondence between the set of all $(i+j,ti+j)$-eligible subcollections of
$\Delta_{t}(C_{n})$ and the set
${\mathcal{R}}_{(i)}\times{\left[i\right]\choose j}$
where ${\left[i\right]\choose j}$ is the set of all $j$-subsets of a set with
$i$ elements. By Corollary 4.3 we have $|{\mathcal{R}}_{(i)}|=\beta_{i,ti}$
and since $|{\left[i\right]\choose j}|={i\choose j}$ and so we apply Corollary
4.8 to observe that
$\beta_{i+j,ti+j}(R/I_{t}(C_{n}))={i\choose
j}\beta_{i,ti}(R/I_{t}(C_{n}))=\frac{n}{n-it}{i\choose j}{n-it\choose i}.$
2. ii.
For the line graph $L_{n}$, recall that
$\Delta_{t}(L_{n})=\langle F_{1},\ldots,F_{n-t+1}\rangle.$
Let $\Lambda=\langle F_{c_{1}},F_{c_{2}},\dots,F_{c_{i}}\rangle$ be the
induced subcollection of $\Delta_{t}(L_{n})$ which is composed of $i$ runs of
length 1 and $\mbox{Vert}(\Lambda)\subset{\mathcal{X}}\setminus\\{x_{n}\\}$,
so that it is also an induced subcollection of $\Delta_{t}(L_{n-1})$. Also let
$A$ be a $j$ \- subset of $\\{x_{c_{1}+t},x_{c_{2}+t},\dots,x_{c_{i}+t}\\}\mod
n$. So by Propositions 4.10 and 4.11 the induced subcollections on
$\mbox{Vert}(\Lambda)\cup A$ are $(i+j,ti+j)$-eligible and if one denotes
these induced subcollections by ${\mathcal{B}}$ we have the following
bijection
$\displaystyle{\mathcal{B}}\rightleftharpoons{[i]\choose
j}\times\\{\Gamma\subset\Delta_{t}(L_{n-1}):\Gamma$ $\displaystyle\mbox{is
composed of $i$ runs of length 1}\\}.$ (4.16)
We make the following claim:
Claim: _Let $\Gamma$ be an $(i+j,ti+j)$-eligible subcollection of
$\Delta_{t}(L_{n})$ which contains a run ${\mathcal{R}}$ with
$F_{n-t+1}\in{\mathcal{R}}$. Then $\Gamma\in{\mathcal{B}}$ if and only if
$|{\mathcal{R}}|=2\mod t+1$._
###### Proof of Claim.
Let $\Gamma\in{\mathcal{B}}$ and assume that $\Lambda=\langle
F_{c_{1}},F_{c_{2}},\dots,F_{c_{i}}\rangle$ is the subcollection of
$\Delta_{t}(L_{n-1})$ used to build $\Gamma$ as described above. Then we must
have $c_{i}=n-t$. Now, the run ${\mathcal{R}}$ contains $F_{n-t+1}$ and
$F_{n-t}$.
If $|{\mathcal{R}}|>2$, then $c_{i-1}=n-2t-1$ and $x_{c_{i-1}+t}=x_{n-t-1}\in
A$ and from Lemma 4.9 we can see that another $t+1$ facets
$F_{n-2t-1},\ldots,F_{n-t-1}$ are in ${\mathcal{R}}$. If we have all elements
of ${\mathcal{R}}$, we stop, and otherwise, we continue the same way. At each
stage $t+1$ new facets are added to ${\mathcal{R}}$ and therefore in the end
$|{\mathcal{R}}|=2\mod t+1$.
Conversely, if $|{\mathcal{R}}|=(t+1)q+2$ then let $\Lambda=\langle
F_{c_{1}},F_{c_{2}},\dots,F_{c_{i}}\rangle$ be the unique subcollection of
$\Delta_{t}(L_{n})$ consisting of $i$ runs of length one from which we can
build $\Gamma$. Since $F_{n-t-1}\in{\mathcal{R}}$, we must have $c_{i}=n-t$ or
$c_{i}=n-t+1$.
If $c_{i}=n-t$, then we are done, since $\Lambda$ will be subcollection of
$\Delta_{t}(L_{n-1})$ and so $\Gamma\in{\mathcal{B}}$. If $c_{i}=n-t+1$, then
$R$ has one facet $F_{n-t+1}$ and if $x_{c_{i-1}+t}\in A$, then by Lemma 4.9
${\mathcal{R}}$ gets an additional $t+1$ facets. And so on: for each $c_{u}$
either 0 or $t+1$ facets are contributed to ${\mathcal{R}}$. Therefore, for
some $p$, $|{\mathcal{R}}|=(t+1)p+1$ which is a contradiction. This settles
our claim. ∎
We now denote the set of remaining $(i+j,ti+j)$-eligible induced
subcollections of $\Delta_{t}(L_{n})$ by ${\mathcal{C}}$. First we note that
${\mathcal{C}}$ consists of those induced subcollections which contain
$F_{n-t+1}$ and are not in ${\mathcal{B}}$. Also, if $j=i$, then a
$(2i,(t+1)j)$-eligible subcollection $\Gamma$ of $\Delta_{t}(L_{n})$ would
have no runs of length $1$, as the equations in Definition 4.1 would give
$\alpha=0$. So $\Gamma\in{\mathcal{C}}$ and we can assume from now on that
$j<i$.
We consider $\Lambda=\langle
F_{c_{1}},F_{c_{2}},\dots,F_{c_{i-1}}\rangle\subset\Delta_{t}(L_{n})$ which is
composed of $i-1$ runs of length 1 with
$\mbox{Vert}(\Lambda)\subset{\mathcal{X}}\setminus
F_{n-t+1}\cup\\{x_{n-t}\\}$. If $A$ is a $j$-subset of the set
$\\{x_{c_{1}+t},x_{c_{2}+t},\dots,x_{c_{i-1}+t}\\}$, we claim that the induced
subcollection $\Gamma$ on $\mbox{Vert}(\Lambda)\cup A\cup F_{n-t+1}$ belongs
to ${\mathcal{C}}$.
Suppose ${\mathcal{R}}$ is the run in $\Gamma$ which includes $F_{n-t+1}$. If
$|{\mathcal{R}}|\neq 1$ then $c_{i-1}+t=n-t$ which implies that
$c_{i-1}=n-2t$. By Lemma 4.9 we see that $t+1$ facets
$F_{n-2t},F_{n-2t+1},\dots,F_{n-t}$ are added to ${\mathcal{R}}$. If these
facets are not all the facets of ${\mathcal{R}}$ then with the same method we
can see that in each step $t+1$ new facets will be added to ${\mathcal{R}}$
and since $F_{n-t+1}\in{\mathcal{R}}$ we can conclude $|{\mathcal{R}}|=1\mod
t+1$. Therefore $\Gamma\notin{\mathcal{B}}$.
Now we only need to show that $\Gamma$ is an $(i+j,ti+j)$-eligible induced
subcollection. By Proposition 4.10 the induced subcollection $\Gamma^{\prime}$
on $\mbox{Vert}(\Lambda)\cup A$ is an $(i-1+j,t(i-1)+j)$-eligible induced
subcollection. Suppose $\Gamma^{\prime}$ is composed of runs
${\mathcal{R}}_{1},{\mathcal{R}}_{2},\dots,{\mathcal{R}}_{\alpha^{\prime}+\beta^{\prime}}$
and then we have
$\displaystyle\left\\{\begin{array}[]{ll}t(i-1)+j&=(t+1)(P^{\prime}+Q^{\prime})+t(\alpha^{\prime}+\beta^{\prime})+\beta^{\prime}\\\
i-1+j&=2(P^{\prime}+Q^{\prime})+2\beta^{\prime}+\alpha^{\prime}\end{array}\right.\Longrightarrow\left\\{\begin{array}[]{ll}i-1&=P^{\prime}+Q^{\prime}+\alpha^{\prime}+\beta^{\prime}\\\
j&=P^{\prime}+Q^{\prime}+\beta^{\prime}\end{array}\right.$ (4.21)
So $\Gamma$ consists of all or all but one of the runs
${\mathcal{R}}_{1},{\mathcal{R}}_{2}\dots,{\mathcal{R}}_{\alpha^{\prime}+\beta^{\prime}}$
as well as ${\mathcal{R}}$ where ${\mathcal{R}}$ is the run which includes
$F_{n-t+1}$.
As we have seen $|{{\mathcal{R}}}|=1\mod t+1$. If we suppose
$|{\mathcal{R}}|=1$ then we can claim that $\Gamma$ is composed of
$\alpha=\alpha^{\prime}+1$ runs of length 1 and $\beta=\beta^{\prime}$ runs of
length 2 $\mod t+1$, and with $P=P^{\prime}$ and $Q=Q^{\prime}$, by (4.21) we
have $\Gamma$ is an $(i+j,ti+j)$-eligible induced subcollection. Now assume
$|{\mathcal{R}}|=(t+1)p+1$, so clearly we have $F_{n-2t}\in\Lambda$ and
$x_{n-t}\in A$. Let ${{\mathcal{R}}}^{\prime}$ be the induced subcollection on
$\mbox{Vert}{({\mathcal{R}})}\setminus F_{n-t+1}$. Then clearly we have
${\mathcal{R}}^{\prime}$ is a run in $\Gamma^{\prime}$ and since the only
facets which belong to ${{\mathcal{R}}}$ but not to ${{\mathcal{R}}}^{\prime}$
are the $t$ facets $F_{n-2t+2},\dots,F_{n-t+1}$ we have
$|{{\mathcal{R}}}^{\prime}|=(t+1)p+1-t=(t+1)(p-1)+2$ (4.22)
Therefore we have shown the run in $\Gamma$ which includes $F_{n-t+1}$ has
been generated by a run of length $2\mod(t+1)$ in $\Gamma^{\prime}$. Using
(4.21) we can conclude $\Gamma$ consists of $\alpha=\alpha^{\prime}+1$ runs of
length $1$ and $\beta=\beta^{\prime}-1$ runs of length $2$ $\mod t+1$. We set
$P=P^{\prime}+p$ and $Q=Q^{\prime}-(p-1)$, and use (4.22) to conclude that
$\displaystyle\left\\{\begin{array}[]{lll}P+Q+\alpha+\beta=(P^{\prime}+p)+(Q^{\prime}-p+1)+(\alpha^{\prime}+1)+(\beta^{\prime}-1)=i\\\
P+Q+\beta=(P^{\prime}+p)+(Q^{\prime}-p+1)+(\beta^{\prime}-1)=j.\end{array}\right.$
Therefore $\Gamma\in{\mathcal{C}}$ as we had claimed.
Conversely, let $\Gamma\in{\mathcal{C}}$ then one can consider the induced
subcollection $\Gamma^{\prime}$ on $\mbox{Vert}(\Gamma)\backslash F_{n-t+1}$.
Assume $\Gamma$ is composed of runs
${\mathcal{R}}_{1},{\mathcal{R}}_{2}\dots,{\mathcal{R}}_{\alpha+\beta}$, so
that $\mod t+1$, ${\mathcal{R}}_{h}$ is a run of length $1$ if $h\leq\alpha$
and length $2$ otherwise.
Suppose ${\mathcal{R}}_{h}$ is the run which includes $F_{n-t+1}$. By our
assumption we have $|{\mathcal{R}}_{h}|=1\mod t+1$, so $h\leq\alpha$. If
$|{\mathcal{R}}_{h}|=1$ then ${\mathcal{R}}_{h}\notin\Gamma^{\prime}$ and
therefore we delete one run of length one from $\Gamma$ to obtain
$\Gamma^{\prime}$, in which case $\Gamma^{\prime}$ is
$(i-1+j,t(i-1)+j)$-eligible.
If $|{\mathcal{R}}_{h}|=(t+1)p_{h}+1>1$ then the $t$ facets
$F_{n-2t+2},\dots,F_{n-t+1}\in{\mathcal{R}}_{h}$ do not belong to
$\Gamma^{\prime}$. So $\Gamma^{\prime}$ consists of $\alpha+\beta$ runs
${\mathcal{R}}_{1},\dots,\widehat{{\mathcal{R}}_{h}},\dots,{\mathcal{R}}_{\alpha+\beta},{\mathcal{R}}_{h}^{\prime}$
where
$|{\mathcal{R}}_{h}^{\prime}|=(t+1)p_{h}+1-t=(t+1)(p_{h}-1)+2.$
Setting $\alpha^{\prime}=\alpha-1$, $\beta^{\prime}=\beta+1$,
$P^{\prime}=P-p_{h}$ and $Q^{\prime}=Q+p_{h}-1$ it follows that
$\Gamma^{\prime}$ is $(i-1+j,t(i-1)+j)$-eligible. By Proposition 4.11 there
exists a unique induced subcollection $\Lambda=\left\langle
F_{c_{1}},F_{c_{2}},\dots,F_{c_{i-1}}\right\rangle$ of $\Delta_{t}(L_{n-t-1})$
which is composed of $i-1$ runs of length one and a $j$ subset $A$ of
$\\{x_{c_{1}+t},\dots,x_{c_{i-1}+t}\\}$ such that $\Gamma^{\prime}$ equals to
induced subcollection on $\mbox{Vert}(\Lambda)\cup A$. So $\Gamma$ is the
induced subcollection on $\mbox{Vert}(\Lambda)\cup A\cup F_{n-t+1}$. Therefore
there is a one to one correspondence between elements of ${\mathcal{C}}$ and
$\displaystyle{[i-1]\choose
j}\times\\{\Gamma\subset\Delta_{t}(L_{n-t-1}):\Gamma$ $\displaystyle\mbox{is
composed of $i-1$ runs of length 1}\\}$ (4.23)
By (4.16), (4.23) and Corollary 4.8 (iii) we have
$\begin{array}[]{lll}\beta_{i+j,ti+j}(R/I)&=&|{\mathcal{B}}|+|{\mathcal{C}}|\vspace{0.1
in}{}\\\ &=&{i\choose j}\beta_{i,ti}(R/I_{t}(L_{n-1}))+{i-1\choose
j}\beta_{i-1,t(i-1)}(R/I_{t}(L_{n-t-1})\vspace{0.1 in}{}\\\ &=&{i\choose
j}{n-it\choose i}+{i-1\choose j}{n-it\choose i-1}.\end{array}$
∎
Finally, we put together Theorem 4.2, Proposition 4.4, Theorem 5.1 of [1] and
Theorem 2.1. Note that the case $t=2$ is the case of graphs which appears in
Jacques [7]. Also note that $\beta_{i,j}(R/I_{t}(C_{n}))=0$ for all $i\geq 1$
and $j>ti$, see for example see for example [7] 3.3.4.
###### Theorem 4.13 (Betti numbers of path ideals of lines and cycles).
Let $n$, $t$, $p$ and $d$ be integers such that $n\geq 2$, $2\leq t\leq n$,
$n=(t+1)p+d$, where $p\geq 0$, $0\leq d\leq t$. Then
1. i.
The $\mathbb{N}$-graded Betti numbers of the path ideal of the graph cycle
$C_{n}$ are given by
$\beta_{i,j}(R/I_{t}(C_{n}))=\left\\{\begin{array}[]{ll}t&j=n,\ d=0,\
\displaystyle i=2\left(\frac{n}{t+1}\right)\\\ &\\\ 1&j=n,\ d\neq 0,\
\displaystyle i=2\left(\frac{n-d}{t+1}\right)+1\\\ &\\\
\displaystyle\frac{n}{{n-t\left(\frac{j-i}{t-1}\right)}}{\frac{j-i}{t-1}\choose\frac{ti-j}{t-1}}{n-t\left(\frac{j-i}{t-1}\right)\choose\frac{j-i}{t-1}}&\left\\{\begin{array}[]{l}j<n,\
i\leq j\leq ti,\mbox{ and }\\\ \\\ \displaystyle 2p\geq\frac{2(j-i)}{t-1}\geq
i\end{array}\right.\\\ &\\\ 0&\mbox{otherwise.}\end{array}\right.$
2. ii.
The $\mathbb{N}$-graded Betti numbers of the path ideal of the line graph
$L_{n}$ are nonzero and equal to
$\beta_{i,j}(R/I_{t}(L_{n}))=\displaystyle{\frac{j-i}{t-1}\choose\frac{ti-j}{t-1}}{n-t\left(\frac{j-i}{t-1}\right)\choose\frac{j-i}{t-1}}+{\frac{j-i}{t-1}-1\choose\frac{ti-j}{t-1}}{n-t\left(\frac{j-i}{t-1}\right)\choose\frac{j-i}{t-1}-1}$
if and only if
1. (a)
$j\leq n$ and $i\leq j\leq ti$;
2. (b)
If $d<t$ then $\displaystyle p\geq\frac{j-i}{t-1}\geq i/2$;
3. (c)
If $d=t$ then $\displaystyle(p+1)\geq\frac{j-i}{t-1}\geq(i+1)/2$.
###### Proof.
We only need to make sure we have the correct conditions for the Betti numbers
to be nonzero.
1. i.
When $j<n$, $\beta_{i,j}(R/I_{t}(C_{n}))\neq 0\Longleftrightarrow$
$\displaystyle\begin{array}[]{lll}&\Longleftrightarrow\left\\{\begin{array}[]{l}\displaystyle\frac{j-i}{t-1}\geq\frac{ti-j}{t-1}\vspace{.1in}{}\\\
\displaystyle n-\frac{t(j-i)}{t-1}\geq\frac{j-i}{t-1}\end{array}\right.&\\\
&&\\\ &\Longleftrightarrow\left\\{\begin{array}[]{l}\displaystyle
2j\geq(t+1)i\vspace{.1 in}{}\\\ \displaystyle
n\geq\left(\frac{t+1}{t-1}\right)(j-i)\end{array}\right.&\\\ &&\\\
&\Longleftrightarrow\left\\{\begin{array}[]{l}\displaystyle
2j\geq(t+1)i\vspace{.1 in}{}\\\
\displaystyle(t+1)p+d\geq\left(\frac{t+1}{t-1}\right)(j-i)\end{array}\right.&\\\
&&\\\ &\Longleftrightarrow\left\\{\begin{array}[]{l}\displaystyle
2j\geq(t+1)i\vspace{.1 in}{}\\\ \displaystyle
p+\frac{d}{t+1}\geq\frac{j-i}{t-1}\end{array}\right.&\\\ &&\\\
&\Longleftrightarrow\displaystyle 2p\geq\frac{2(j-i)}{t-1}\geq i&\mbox{ as
}d<t+1\\\ \end{array}$ (4.41)
2. ii.
$\beta_{i,j}(R/I_{t}(L_{n}))\neq 0\Longleftrightarrow$
$\displaystyle\begin{array}[]{ll}\Longleftrightarrow\left\\{\begin{array}[]{l}\displaystyle\frac{j-i}{t-1}\geq\frac{ti-j}{t-1}\vspace{.1in}{}\\\
\displaystyle n-\frac{t(j-i)}{t-1}\geq\frac{j-i}{t-1}\end{array}\right.&\mbox{
or }\hskip
28.90755pt\left\\{\begin{array}[]{l}\displaystyle\frac{j-i}{t-1}\geq\frac{ti-j}{t-1}+1\vspace{.1in}{}\\\
\displaystyle n-\frac{t(j-i)}{t-1}\geq\frac{j-i}{t-1}-1\end{array}\right.\\\
&\\\ \Longleftrightarrow\left\\{\begin{array}[]{l}\displaystyle
2j\geq(t+1)i\vspace{.1 in}{}\\\ \displaystyle
n\geq\left(\frac{t+1}{t-1}\right)(j-i)\end{array}\right.&\mbox{or}\hskip
28.90755pt\left\\{\begin{array}[]{l}\displaystyle 2j\geq(t+1)(i+1)\vspace{.1
in}{}\\\ \displaystyle
n+1\geq\left(\frac{t+1}{t-1}\right)(j-i)\end{array}\right.\\\ &\\\
\Longleftrightarrow\left\\{\begin{array}[]{l}\displaystyle
2j\geq(t+1)i\vspace{.1 in}{}\\\
\displaystyle(t+1)p+d\geq\left(\frac{t+1}{t-1}\right)(j-i)\end{array}\right.&\mbox{or}\hskip
28.90755pt\left\\{\begin{array}[]{l}\displaystyle 2j\geq(t+1)(i+1)\vspace{.1
in}{}\\\
\displaystyle(t+1)p+d+1\geq\left(\frac{t+1}{t-1}\right)(j-i)\end{array}\right.\\\
&\\\ \Longleftrightarrow\left\\{\begin{array}[]{l}\displaystyle
2j\geq(t+1)i\vspace{.1 in}{}\\\ \displaystyle
p+\frac{d}{t+1}\geq\frac{j-i}{t-1}\end{array}\right.&\mbox{or}\hskip
28.90755pt\left\\{\begin{array}[]{l}\displaystyle 2j\geq(t+1)(i+1)\vspace{.1
in}{}\\\ \displaystyle
p+\frac{d+1}{t+1}\geq\frac{j-i}{t-1}\end{array}\right.\\\ &\\\
\Longleftrightarrow\begin{array}[]{l}\displaystyle
p+\frac{d}{t+1}\geq\frac{(j-i)}{t-1}\geq i/2\end{array}&\mbox{ or }\hskip
28.90755pt\displaystyle
p+\frac{d+1}{t+1}\geq\frac{(j-i)}{t-1}\geq\frac{i+1}{2}\par\end{array}$ (4.68)
Then since $d<t+1$ and $(j-i)/(t-1)$ is an integer we can conclude that $i\leq
2p$ when $d\neq t$ and $i\leq 2p+1$ for $d=t$. Also we have $j-i\leq(t-1)p$
for $d\neq t$ and $j-i\leq(t-1)(p+1)$ for $d=t$. ∎
We can now easily derive the projective dimension and regularity of path
ideals of lines, which were known before. The projective dimension of lines
(Part i below) was computed by He and Van Tuyl in [6] using different methods.
The case $t=2$ is the case of graphs which appears in Jacques [7]. Part ii of
the following Corollary reproves Theorem 5.3 in [2] which computes the
Castelnuovo-Mumford regularity of path ideal of a line. The case of cycles was
done in [1].
###### Corollary 4.14 (Projective dimension and regularity of path ideals of
lines).
Let $n$, $t$, $p$ and $d$ be integers such that $n\geq 2$, $2\leq t\leq n$,
$n=(t+1)p+d$, where $p\geq 0$, $0\leq d\leq t$. Then
1. i.
The projective dimension of the path ideal of a line $L_{n}$ is given by
$pd(R/I_{t}(L_{n}))=\left\\{\begin{array}[]{ll}2p&d\neq t\vspace{.1 in}{}\\\
2p+1&d=t\\\ \end{array}\right.$
2. ii.
The regularity of the path ideal of a line $L_{n}$ is given by
$reg(R/I_{t}(L_{n}))=\left\\{\begin{array}[]{ll}p(t-1)&d<t\vspace{.1 in}\\\
(p+1)(t-1)&d=t\\\ \end{array}\right.$
###### Proof.
1. i.
By using Theorem 4.13 we know that if $\beta_{i,j}(R/I_{t}(L_{n})\neq 0$ then
$i\leq 2p+1$ when $d=t$ and therefore $pd(R/I_{t}(L_{n}))\leq 2p+1$. On the
other hand by applying Theorem 4.13 we have
$\beta_{2p+1,n}(R/I_{t}(L_{n}))=\displaystyle{p+1\choose p}{p\choose
p+1}+{p\choose p}{p\choose p}=1\neq 0.$
Then we can conclude that $pd(R/I_{t}(L_{n}))=2p+1$.
Now we suppose $d\neq t$. From (4.68) we can see that if
$\beta_{i,j}(R/I_{t}(L_{n}))\neq 0$ then $2p\geq i$ and therefore
$pd(R/I_{t}(L_{n}))\leq 2p$. On the other hand, by applying Theorem 4.13
again, we can see that
$\beta_{2p,p(t+1)}(R/I_{t}(L_{n}))=\displaystyle{p\choose p}{p+d\choose
p}+{p-1\choose p}{p\choose p}={p+d\choose p}\neq 0.$
Therefore $pd(R/I_{t}(L_{n}))\geq 2p$ and we have $pd(R/I_{t}(L_{n}))=2p$.
2. ii.
By definition, the regularity of a module $M$ is $\max\\{j-i\ |\
\beta_{i,j}(M)\neq 0\\}$. By Theorem 4.13, we know exactly when the graded
Betti numbers of $R/I_{t}(L_{n})$ are nonzero, and the formula follows
directly from (4.68).
∎
## Acknowledgement
We gratefully acknowledge the helpful computer algebra systems CoCoA [9] and
Macaulay2 [4], without which our work would have been difficult or impossible.
## References
* [1] A. Alilooee and S. Faridi. On the resolution of path ideals of cycles. to appear, 2011.
* [2] R. Bouchat, H. T. Ha, and A. O’ Keefe. Path ideals of rooted trees and their graded betti numbers. J. Combinatorial Theory, Ser. A, 118:2411–2425, 2010.
* [3] A. Conca and E. De Negri. M-sequences, graph ideals and ladder ideals of linear types. J. Algebra, 211:599–624, 1999.
* [4] D. R. Grayson and M. E. Stillman. Macaulay 2, a software system for research in algebraic geometry. Available http://www.math.uiuc.edu/Macaulay2/.
* [5] Ralph Grimaldi. Discrete and combinatorial mathematics - an applied introduction (3. ed.). Addison-Wesley, 1993.
* [6] J. He and A. Van Tuyl. Algebraic properties of path ideal of a tree. Comm. Algebra, 38:1725–1742, 2010.
* [7] S. Jacques. Betti numbers of graph ideals. PhD thesis, The University of Sheffield, arXiv.math.AC/0410107, 2004.
* [8] R. P. Stanley. Enumerative combinatorics. Volume 1. Cambridge Studies in Advanced Mathematics.
* [9] CoCoA Team. Cocoa: a system for doing computations in commutative algebra. Available at http://cocoa.dima.unige.it.
|
arxiv-papers
| 2011-10-30T20:24:23 |
2024-09-04T02:49:23.729332
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Ali Alilooee and Sara Faridi",
"submitter": "Ali Alilooee",
"url": "https://arxiv.org/abs/1110.6653"
}
|
1110.6720
|
# Calogero-Sutherland model in interacting fermion picture and explicit
construction of Jack states
Jian-Feng Wu wujf@itp.ac.cn Ming Yu yum@itp.ac.cn Institute of Theoretical
Physics, Chinese Academy of Sciences, Beijing, China, 100190
###### Abstract
The 40-year-old Calogero-Sutherland (CS) model remains a source of
inspirations for understanding 1d interacting fermions. At $\beta=1,\text{or
}0$, the CS model describes a free non-relativistic fermion, or boson theory,
while for generic $\beta$, the system can be interpreted either as interacting
fermions or bosons, or free anyons depending on the context. However, we shall
show in this letter that the fermionic picture is advantageous in
diagonalizing the CS Hamiltonian. Comparing to the previously known multi-
integral representation or the Dunkl operator formalism for the CS wave
functions, our method depends on the (upper or lower) triangular nature of the
fermion interaction, which is resolved in perturbation theory of the second
quantized form. The eigenstate is constructed from a multiplet of unperturbed
states and the perturbation is of finite order. The full construction is a
similarity transformation from the free fermion theory, in the same spirit as
the Landau Fermi liquid theory and the 1d Luttinger liquid theory. That means
quasi-particles or anyons can be represented in terms of free fermion modes
(or bosonic modes via bosonization). The method is applicable to other (higher
than one space dimension) systems for which the adiabatic theorem applies.
###### pacs:
71.10.Pm, 11.25.Hf, 02.30.Ik
In this letter we shall propose an explicit formula for solving a class of
Hamiltonian eigenequation and work out the explicit construction of the Jack
states for the CS modelCalogero:1969 ; Sutherland:1971 as a specific example.
Comparing to the previously known multi-integral representationAwata:1995ky ;
Wu:2011cya or the Dunkl operator formalismLapointe(1995) for the CS wave
functions, our method depends on the (upper or lower) triangular nature of the
fermion interaction, which is resolved in perturbation theory of the second
quantized form. The similar method has also been used in a different context
on the explicit construction of the AFLT statesShou:2011nu . The general
statement on the Hamiltonian system to which our construction applies is as
following: An interacting Hamiltonian system (with Hamiltonian $H$) or its
equivalent class by a similarity transformation, is exactly solvable through
finite order perturbation if its matrix form is (upper or lower) triangular
which will be abbreviated just as triangular hereafter. In practice, we choose
as the basis vectors the already solved eigenstates for an “unperturbed”
Hamiltonian $H^{(0)}$,
$H^{(0)}|E^{(0)}\rangle_{0}=E^{(0)}|E^{(0)}\rangle_{0}$. Although not
necessarily, one can choose the free theory as the unperturbed system, in
which the interaction is turned off. $H$ will contain perturbations away from
$H^{(0)}$, $H=H^{(0)}+H^{(I)}$. The crucial point is that we shall assume,
although not always guaranteed to be so, that $H^{(I)}$ can be decomposed into
$H^{(I)}=H^{\parallel}+H^{\perp}$. Such decomposition makes our method
differing substantially from the others’ triangulating the
HamiltonianLapointe(1995) ; Sutherland:Lecture . Here, $H^{\parallel}$ is
diagonal with diagonal entry $E^{(I)}$ and $H^{\perp}$ is strictly triangular
in the basis of the $H^{(0)}$ eigenstates. By “strictly triangular” we mean
the triangular matrix with zero diagonal entries. The hermiticity of $H$, if
lost, will be restored by the inverse-similarity transformation. We may write
$H^{d}=H^{(0)}+H^{\parallel},\,E=E^{(0)}+E^{(I)},\,|E\rangle=R(E)|E^{(0)}\rangle_{0}$.
Then the energy eigenequation $H|E\rangle=E|E\rangle$ is solved with the
following solution,
$R(E)=\bigl{(}1-(E-H^{d})^{-1}H^{\perp}\bigr{)}^{-1}=\sum_{n=0}^{\infty}\bigl{(}(E-H^{d})^{-1}H^{\perp}\bigr{)}^{n}$.
This can be checked by rewriting
$H=E+(H^{d}-E)\bigl{(}1-(E-H^{d})^{-1}H^{\perp}\bigr{)}$. A few assumptions
are in need: i) $|E\rangle$ ends up with a finite order perturbation in
$H^{\perp}$ powers if $H^{\perp}$ is nilpotent on the subspace in which an $H$
eigenstate is built. A matrix $A$ is said to be nilpotent if $A^{n}=0$ for
some positive integer $n>1$. ii) suppose i) is satisfied, then $|E\rangle$ is
constructed from a multiplet of member states ranked by the number of powers
of $H^{\perp}$ action ascending from a father state. We shall assume within
each multiplet the $H^{d}$ spectrum is not degenerate for generic perturbation
parameters. For simplicity we shall assume that $H^{\perp}$, when acts to the
right, actually maps a member state to its brother states with smaller $H^{d}$
eigenvalues. iii) With the above assumptions in mind, one can show that the
exact eigenstate is in fact obtained by a similarity transformation $S$ from
the corresponding father state. Thus the integrability of the $H$ system
inherits from that of the unperturbed $H^{(0)}$ system. In other words, any
diagonal action in the unperturbed system conjugated by $S$, will remain
mutually commutable when the prescribed perturbation is turned on. The
similarity transformation $S$ is defined by the following time ordered multi-
integration,
$\displaystyle S$ $\displaystyle=$ $\displaystyle
T\exp\bigl{(}\int_{-\infty}^{0}H^{\perp}(t)dt\bigr{)},$ (1) $\displaystyle
H^{\perp}(t)$ $\displaystyle=$
$\displaystyle\exp(-tH^{d})H^{\perp}\exp(tH^{d}).$
Here $T$ means time ordering with larger $t$ to the left, and
$H^{\perp}(-\infty)=0$ because of our convention that $H^{\perp}$ lowers the
maximal energy of the Hilbert space when acts to the right. It can be verified
that
$S|E^{(0)}\rangle_{0}=\bigl{(}1-(E-H^{d})^{-1}H^{\perp}\bigr{)}^{-1}|E^{(0)}\rangle_{0}=|E\rangle$.
One can think of $S$111The construction of $\log(S)$ resembles that of the
screening charges in 2d CFT, although in later cases the analog of
$H^{\perp}$, which is $V_{\alpha_{\pm}}(1)$, does not seem to be triangular.
as an action to the right by adiabatically turning on the perturbation
$H^{\perp}$ from time $-\infty$ to time $0$. Consequently,
$S^{-1}=T\exp\bigl{(}-\int_{0}^{\infty}H^{\perp}(-t)dt\bigr{)}.$ (2)
The orthogonality maintains if the conjugate state is defined by
${}_{0}\langle E^{(0)}|S^{-1}=_{0}\langle
E^{(0)}|\bigl{(}1-H^{\perp}(E-H^{d})^{-1}\bigr{)}^{-1}=\langle E|$. One can
further show that $H=H^{d}+H^{\perp}=SH^{d}S^{-1}$. Thus the conditions that
restrict our construction is about the same as for which the adiabatic theorem
in quantum mechanics could apply. We may identify the $S$ transformation as an
adiabatic transformation. We believe the procedure should work for a class of
integrable models. So in this letter we shall concentrate ourselves on the CS
model ($\beta>0$ will be assumed in this letter in accordance with our
convention of time ordering). The merits lying behind this construction is
that the interacting fermion system can be regarded as an adiabatic mapping by
a similarity transformation from the free fermion system. This is in the same
spirits as the Landau Fermi liquid theory and the 1d Luttinger liquid theory.
Being an integrable system, the CS model is exactly solvable. Though explicit
constructions of the eigenstates in the second quantized form has not appeared
prior to our present work. The integrability originates from the essentially
free quasiparticle spectrum which accounts for the fractional statistics. This
has been considered from various point of views elsewhereHaldane:1991 ;
Azuma:1993ra ; Pasquier:1994cs ; Wu:1994 ; Lapointe(1995) ;
Polychronakos:1992zk . Although we generally work on the CS model with
positive $\beta$, negative rational values of $\beta$ for the Jack polynomials
have been proposed in Bernervig:2008 to unify the FQHE wave functions of the
Laughlin, More-Read and Read-Rezayi type in one picture. Estinne:2010 also
relates the non-abelian statistics to the CS model through differential
equations for degenerate conformal blocks. Estienne:2011qk has gone even
further by unveiling a deep connection between 2d $WA_{k-1}$ minimal models
and the integrability of the generalized CS models.
Stanley contains a comprehensive review on the Jack symmetric function which
are the spectrum generation function for the CS model. Our recent
workWu:2011cya , in which more relevant references can be found, also
constitute a concise introduction on the subject. The CS model is introduced
for studying N interacting particles distributed on a circle of circumference
$L$ with the Hamiltonian,
$\displaystyle H_{CS}$ $\displaystyle=$
$\displaystyle-\sum_{i=1}^{N}\frac{1}{2}\partial_{x_{i}}^{2}+\sum_{i<j}\frac{\beta(\beta-1)}{\sin^{2}(x_{ij})}.$
(3)
Here for convenience, we have set $\hbar^{2}/m=1$, $L=\pi$. For simplicity, we
shall restrict ourselves to the following simple solutions of the
eigenfunctions (for more general boundary conditions, see Doyon:2006ph ;
Estienne:2011qk ),
$\Psi_{\lambda}(\\{x_{i}\\})=\Psi_{0}(\\{x_{i}\\})J_{\lambda}^{1/\beta}(\\{z_{i}\\})$.
Here, the ground state $\Psi_{0}(\\{x_{i}\\})$ is the Jastrow-like wave
function, $\Psi_{0}(\\{x_{i}\\})=\prod_{i<j}\sin^{\beta}(x_{ij})$,
$J_{\lambda}(\\{z_{i}\\})$ is the Jack symmetric polynomial with
$z_{j}=\exp(2ix_{j})$. For each Young tableau
$\lambda=\\{\lambda_{1},\lambda_{2},\cdots,\lambda_{N}\\}$, with
$\lambda_{i}\geq\lambda_{i+1}\geq 0$, we normalize the energy eigenvalue as
$2E_{\lambda}$, $E_{\lambda}=\sum
P_{i}^{2},\,\,P_{i}=\lambda_{i}+\beta\bigl{(}(N+1)/2-i\bigr{)}$. It is known
that the Jack polynomial is triangular in the sense that it is a linear
superposition of the squeezed states starting from a dominant symmetric
monomial. However, in this symmetric monomial basis, it is difficult to
separate the interacting Hamiltonian to $H^{\parallel}$ and $H^{\perp}$ parts.
See however, Kadell for diagonalization on this basis. So we have to find
other basis in which our method could apply. For this reason we prefer to work
on the second quantized form of the Hamiltonian for the collective motion of
the CS model,
$\displaystyle H$ $\displaystyle=$ $\displaystyle
k\sum_{n,m>0}(a_{-n}a_{-m}a_{n+m}+a_{-n-m}a_{n}a_{m})$ $\displaystyle+$
$\displaystyle\sum_{n>0}\bigl{(}N\beta+(1-\beta)n\bigr{)}a_{-n}a_{n}.$
Here $\beta=k^{2}$, $k$ is the charge unit of the $N$ identical particles,
$[a_{n},a_{m}]=n\delta_{n+m,0},\,[a_{0},q]=1$. The ground state energy $E_{0}$
is no longer included in $H$. This is the bosonic picture of the CS system
which describes the density fluctuation of the electrons. To transform this
Hamiltonian to the original CS Hamiltonian, we shall use the vertex operator
formalism defined by
$V_{k}(z_{i})=\exp\bigl{(}k\sum_{n>0}a_{-n}z_{i}^{n}/n\bigr{)}\exp\bigl{(}-k\sum_{n>0}a_{n}z_{i}^{-n}/n\bigr{)}e^{kq}z_{i}^{ka_{0}},$
and $\Psi_{\lambda}(\\{x_{i}\\})=\langle
k_{f}|J_{\lambda}\prod_{i=1}^{N}V_{k}(z_{i})|k_{in}\rangle$. Here
$a_{0}|k_{in}\rangle=k_{in}|k_{in}\rangle,\,k_{in}=-(N-1)k/2,\,k_{f}=(N+1)k/2$.
$J_{\lambda}$ solves the equation $\langle 0|J_{\lambda}H=\langle
0|J_{\lambda}(E_{\lambda}-E_{0})$. Then we have
$H_{CS}\Psi_{\lambda}(\\{x_{i}\\})=\langle
k_{f}|J_{\lambda}(H+E_{0})2\prod_{i=1}^{N}V_{k}(z_{i})|k_{in}\rangle=2E_{\lambda}\Psi_{\lambda}(\\{x_{i}\\})$.
$J_{\lambda}$’s are the Jack symmetric functions in the power sum basis,
$J_{\lambda}\equiv J_{\lambda}^{1/\beta}(\\{a_{n}/k\\})$, not in an apparently
squeezed form. So in the bosonic picture, $H$ does not warrant an explicit
decomposition into $H^{\parallel}$ and $H^{\perp}$ parts. However, we know
that for $\beta=1$, the Jack states reduce to the Schur states, which
corresponds to a free non-relativistic spinless “chiral” fermion theory. This
suggests that we may rewrite $H$ as an interacting fermion theory with
perturbation parameter $\beta-1$. The construction of the Schur states in the
fermionic picture is made possible by the standard bosonization (for
convenience we assume Neveu-Schwarz (NS) boundary condition for the moment),
$a_{n}=\sum_{r\in\mathbb{Z}+1/2}:b_{n-r}c_{r}:$, with
$\\{b_{r},c_{s}\\}=\delta_{r+s,0},\,r,s\in\mathbb{Z}+1/2$ and
$b_{r}|{0}\rangle=c_{r}|{0}\rangle=0,\,r>0$. Hereafter for simplicity we shall
drop the term $\beta N\sum_{n>0}a_{-n}a_{n}$ in the original $H$, for it just
adds a value $\beta N|\lambda|$. The Schur functions are the eigenstates of
$H$ at $\beta=1$, $H^{(0)}\equiv
H_{\beta=1}=\sum_{r>0}(r^{2}+\frac{3}{4})(b_{-r}c_{r}-c_{-r}b_{r}),\,E^{(0)}_{\lambda}=\sum_{i=1}^{d(\lambda)}(r_{i}^{2}-s_{i}^{2})=\sum_{i=1}^{\lambda^{t}_{1}}\lambda_{i}^{2}-\sum_{i=1}^{\lambda_{1}}(\lambda_{i}^{t})^{2}$.
Each Schur state is created by a monomial of equal number $d(\lambda)$ of
$b_{-r}$’s and $c_{-s}$’s acting on the vacuum state with $d(\lambda)$ the
number of squares along the diagonal line of $\lambda$,
$|{\lambda}\rangle\equiv
s_{\lambda}|{0}\rangle=(-1)^{\sum_{i=1}^{d(\lambda)}(1/2-s_{i})}\prod_{i=1}^{d(\lambda)}b_{-r_{i}}c_{-s_{i}}|{0}\rangle$.
The Schur function in the fermionic picture is labeled by the Maya diagram
which is translated into the Young tableauJimbo this way:
$r_{i}=\lambda_{i}-i+1/2,\,s_{i}=\lambda^{t}_{i}-i+1/2$. Here,
$\lambda=\\{\lambda_{1},\lambda_{2},\dots\\}$ denotes the Young tableau and
$\lambda^{t}=\\{\lambda_{1}^{t},\lambda^{t}_{2},\dots\\}$ its transposed Young
tableau. For $\beta\neq 1$ CS model, the two body interaction appears and the
interaction strength is proportional to $\beta-1$. Therefore we need to
eliminate any odd powers of $k$ in $H$, which make branch cuts in the coupling
space after fermionization. This can be done by the following redefinition,
$\tilde{a}_{-n}=a_{-n}/k,\,\tilde{a}_{n}=ka_{n},\,n>0$ and
$\tilde{a}_{0}=ka_{0},\,\tilde{q}=q/k$. We call the above non-unitary
similarity transformation the $D$ transformation, which keeps the bosonic
commutators invariant, and as we shall see, also makes the Hamiltonian
triangular in the fermionic picture. Making the standard bosonization to
$\tilde{a}_{n}$’s, one found that the Hamiltonian $H$ can be written as
$\displaystyle H$ $\displaystyle=$ $\displaystyle
H^{(0)}+H^{\parallel}+H^{\perp},$ (5) $\displaystyle H^{\parallel}$
$\displaystyle=$
$\displaystyle\sum_{r>0}(1-\beta)(r-\frac{1}{2})\bigl{(}\frac{1}{3}b_{-r}c_{r}+(r+\frac{1}{6})c_{-r}b_{r}\bigr{)}$
$\displaystyle+$
$\displaystyle\sum_{r+s>0}\frac{2}{3}(2r+s)(1-\beta):b_{-s}c_{-r}b_{r}c_{s}:,$
$\displaystyle H^{\perp}$ $\displaystyle=$
$\displaystyle(1-\beta)\sum_{\begin{subarray}{c}r+s>0,r+l<0\\\
k+l+r+s=0\end{subarray}}\bigl{(}2r+\frac{2}{3}(s+l)\bigr{)}:b_{k}c_{l}b_{r}c_{s}:.$
$H^{\perp}$ is strictly triangular. That is to say, it always squeezes the
original Young tableau $\lambda$ for a given Schur state to the “thinner” ones
$\lambda^{\prime}$’s for states after its action,
$\lambda^{\prime}<\lambda\Rightarrow\sum_{i=1}^{j}\lambda^{\prime}_{i}<\sum_{i=1}^{j}\lambda_{i},\,\text{for
}\,j=1,2,\cdots\,.$ (6)
To see this triangular nature in a more transparent form, $H^{\perp}$ is
simplified and decomposed into 5 subprocesses,
$\displaystyle\frac{1}{2(1-\beta)}H^{\perp}$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}r+s>0,r+l<0\\\
r>k,k+l+r+s=0\end{subarray}}(r-k):b_{k}c_{l}b_{r}c_{s}:$ $\displaystyle=$
$\displaystyle\sum_{n=1,r>s>0}^{n=s-1/2}(s-r)c_{-r-n}c_{-s+n}b_{s}b_{r}$
$\displaystyle+$
$\displaystyle\sum_{n=1,r>s>0}^{n=[(r-s-1)/2]}(s-r+2n)b_{-r+n}b_{-s-n}c_{s}c_{r}$
$\displaystyle+$
$\displaystyle\sum_{n=1,r,s>0}^{n=s-1/2}(r+s-n)b_{-s+n}c_{-r-n}b_{r}c_{s}$
$\displaystyle+$ $\displaystyle\sum_{r>l>0,s>0}(l-r)c_{-l-s-r}b_{l}b_{r}c_{s}$
$\displaystyle+$
$\displaystyle\sum_{l>r>0,s>0}(l-r)b_{-l}c_{-s}b_{-r}c_{r+s+l}\,,$
here $[x]$ stands for the integer part of the number $x$. Each line in the
above expression stands for a process of “squeezing” (moving downwards
plaquettes in) the Young tableau representing the fermion monomial in
agreement with (6). While process 1)-3) does not change $d(\lambda)$, process
4) or 5) make it changed by $\mp 1$. $H^{\parallel}$ shifts the energy-
eigenvalue of the Schur state from $E^{(0)}_{\lambda}$ to
$E_{\lambda}^{1/\beta}=\sum_{i=1}^{\lambda^{t}_{1}}\bigl{(}\lambda_{i}^{2}-\beta(2i-1)\lambda_{i}\bigr{)}$,
which is the eigenenergy for the Jack state. $H^{\perp}$, however, only
changes the fermion monomial (Schur state) to a fermion polynomial (Jack
state) and does not change the eigenvalue. Let’s first concentrate on the ket
state $|{P^{1/\beta}_{\lambda}}\rangle\equiv
S(k)|{\lambda}\rangle=DR_{\lambda}|{\lambda}\rangle$. Here,
$D=\exp\bigl{(}-\log(k)(qa_{0}+\sum_{n>0}a_{-n}a_{n}/n)\bigr{)}$ and
$R_{\lambda}=\bigl{(}1-(E_{\lambda}^{1/\beta}-H^{d})^{-1}H^{\perp}\bigr{)}^{-1},\,S(1)\equiv
S$ and $S(k)=DS$ has also scaled back the $1/k$ factor for the $a_{-n}$’s
($n>0$ and $a_{n}$’s will gain a factor $k$) are related to the fermionic
oscillators through standard bosonization. This way $H\equiv
S(k)H^{d}S^{-1}(k)$ will resume hermiticity (no longer triangular). Similarly,
$\langle{P_{\lambda^{t}}^{\beta}}|\equiv\langle{\lambda}|S^{-1}(k)$. Here we
have used the duality relation $P_{\lambda^{t}}^{\beta}({-ka_{n}})\propto
P_{\lambda}^{1/\beta}({a_{n}/k})$. The orthogonality is obvious:
$\langle{P_{\chi^{t}}^{\beta}}|P_{\lambda}^{1/\beta}\rangle=\langle\chi|{\lambda}\rangle=\delta_{\chi,\lambda}$.
For a standard-normalized Jack symmetric function,
$J_{\lambda}^{1/\beta}=(a_{-1}/k)^{|\lambda|}+\cdots$, we have
$\langle{J_{\chi}^{1/\beta}}|J_{\lambda}^{1/\beta}\rangle=\delta_{\chi,\lambda}j_{\lambda}$.
Here $j_{\lambda}=A_{\lambda}^{1/{\beta}}B_{\lambda}^{1/\beta}$,
$\displaystyle A_{\lambda}^{1/\beta}$ $\displaystyle=$
$\displaystyle\prod_{s\in\lambda}\left(a_{\lambda}(s)\beta^{-1}+l_{\lambda}(s)+1\right),$
(8) $\displaystyle B_{\lambda}^{1/\beta}$ $\displaystyle=$
$\displaystyle\prod_{s\in\lambda}\left((a_{\lambda}(s)+1)\beta^{-1}+l_{\lambda}(s)\right).$
$a_{\lambda}(s)$ and $l_{\lambda}(s)$ are called arm-length and leg-length of
the box $s$ in the Young tableau $\lambda$, $a_{\lambda}(s)=\lambda_{i}-j$,
$l_{\lambda}(s)=\lambda^{t}_{j}-i$. With this normalization, we have
$|J_{\lambda}^{1/\beta}\rangle=|P_{\lambda}^{1/\beta}\rangle
A_{\lambda}^{1/\beta}$ and
$\langle{J_{\lambda}^{1/\beta}}|=B_{\lambda}^{1/\beta}\langle{P_{\lambda^{t}}^{\beta}}|$.
We have checked this fermionic formalism for Jack states up to level 4, all of
them match with those obtained from the known bosonic examples (solved by
brute force) as desired. We now write down the level 3 results for readers’
reference,
$\displaystyle\left|\right.J^{1/\beta}_{\tiny\yng(1,1,1)}\left.\right\rangle$
$\displaystyle=$ $\displaystyle
6\left|\right.{\tiny\yng(1,1,1)}\left.\right\rangle,$
$\displaystyle\left|\right.J^{1/\beta}_{\tiny\yng(2,1)}\left.\right\rangle$
$\displaystyle=$
$\displaystyle\dfrac{2\beta+1}{\beta}\left|\right.{\tiny\yng(2,1)}\left.\right\rangle+\dfrac{2(\beta-1)}{\beta}\left|\right.{\tiny\yng(1,1,1)}\left.\right\rangle,$
$\displaystyle\left|\right.J^{1/\beta}_{\tiny\yng(3)}\left.\right\rangle$
$\displaystyle=$
$\displaystyle\dfrac{(\beta+2)(\beta+1)}{\beta^{2}}\left|\right.{\tiny\yng(3)}\left.\right\rangle$
$\displaystyle+$
$\displaystyle\dfrac{2(\beta-1)(\beta+1)}{\beta^{2}}\left|\right.{\tiny\yng(2,1)}\left.\right\rangle+\dfrac{(\beta-1)(\beta-2)}{\beta^{2}}\left|\right.{\tiny\yng(1,1,1)}\left.\right\rangle\,.$
The integrability of the CS model is also nicely incorporated in our
formalism. The usual Dunkl exchange operator or Sekiguchi differential
operator does not apply here since there is no simple way translating the
coordinate formalism to the collective mode formalism for higher order
invariants. To get the CS spectrum, put $N$ vertex operators $V_{k}(z_{i})$’s
acting successively on the $|k_{in}-k/2\rangle$ vacuum starting from
$V_{k}(z_{N})$. If only the creation modes are taken into account, the
resulting state is a linear superposition of the following modes labeled by
Young tableau (with maximal $N$ rows), $V_{(1-N)\beta/2-\lambda_{1}}\cdots
V_{(N-1)\beta/2-\lambda_{N}}|k_{in}-k/2\rangle$. Here the i-th mode carries
the momentum $\bigl{(}(N+1)/2-i\bigr{)}\beta+\lambda_{i}$ and the CS energy is
just summing over each momentum square. Notice that for $\beta=1$, we come
back to the free fermion theory (NS sector $\Rightarrow N\in even$). In this
case we can define a momentum operator for the specific fermionic mode
$b_{-r}$,
$P^{(0)}_{r}={\tiny{\times}\atop\tiny{\times}}rb_{-r}c_{r}{\tiny{\times}\atop\tiny{\times}}$.
Here the normal ordering
${\tiny{\times}\atop\tiny{\times}}\cdots{\tiny{\times}\atop\tiny{\times}}$ is
defined with respect to the “empty” vacuum $|k_{in}-1/2\rangle$ at $\beta=1$,
$b_{r}|k_{in}-1/2\rangle=0,\,r>N/2$.
${}^{\tiny{\times}}_{\tiny{\times}}{b_{r}c_{s}}^{\tiny{\times}}_{\tiny{\times}}=\begin{cases}b_{r}c_{s},&\text{if
}r<N/2;\\\ -c_{s}b_{r},&\text{if }r>N/2.\end{cases}$
For $\beta\neq 1$ CS model, we choose to stay in the fermionic picture, so
that the canonical commutation as well as the normal ordering just defined
remains valid. To produce the exact CS spectrum, we just need to define the
shifted momentum operator for each fermionic mode, in a way similar to the
minimal coupling of a self-generated fictitious gauge potential. This pseudo-
momentum operator for a specific mode is in fact for a collective motion,
since additional information on each electron’s relative position among the
total of $N$ electrons is needed, $P^{d}_{r}\equiv
P^{(0)}_{r}+P^{\parallel}_{r}={\tiny{\times}\atop\tiny{\times}}b_{-r}c_{r}{\tiny{\times}\atop\tiny{\times}}\bigl{(}r+(\beta-1)((N+1)/2-\sum_{s\geq
r}{\tiny{\times}\atop\tiny{\times}}b_{-s}c_{s}{\tiny{\times}\atop\tiny{\times}})\bigr{)}$.
For $\beta=1$, $P^{\parallel}_{r}$ vanishes and the “gauge” potential drops
out, and we get exactly the momentum operator $P^{(0)}_{r}$ for the mode
$b_{-r}$. For $\beta\neq 1$, a self-generated “gauge” potential has to be
coupled. The ground state is specified by the null Young tableau, and the
Fermi sea is filled up to momentum $(N-1)/2$. We call this filled Fermi sea
the perturbative vacuum state $|f\rangle$, $b_{r}|f\rangle=0$ for $r>-N/2$.
Since there are two vacuum states considered, there exists two kinds of normal
ordering each associated with different vacuum state, which one to choose
depends on the context. For example, in constructing the Schur or Jack state,
we are doing perturbation around the filled Fermi sea $|f\rangle$, so it is
better to work with the following normal ordering
$:b_{r}c_{s}:=\begin{cases}b_{r}c_{s},&\text{if }r<-N/2;\\\
-c_{s}b_{r},&\text{if }r>-N/2.\end{cases}$
Now define $H^{d}=\sum_{r}(P^{d}_{r})^{2}$. If the i-th electron’s momentum is
moved up exactly by $\lambda_{i}$ amount, then $H^{d}$ acts on this system
will produce the exact CS spectrum. $S(k)$ act on this fermion monomial state
will produce the exact Jack state. Since $[P^{d}_{r},P^{d}_{s}]=0$, the
conserved charges can now be constructed,
$W^{n}=S(k)\sum_{r}(P^{d}_{r})^{n}S^{-1}(k)\Rightarrow[W^{n},W^{m}]=0,\,n,m>0$.
We have shown in this letter that Jack symmetric function is triangular in the
basis of Schur functions. On the other hand, expanding in symmetric monomial
basis $m_{\mu}$,
$P_{\lambda}^{1/\beta}(\\{z_{i}^{n}\\})=(v^{1/\beta})_{\lambda}^{\mu}m_{\mu}=R_{\lambda}^{\nu}s_{\nu}(\\{z_{i}^{n}\\})\Rightarrow
R_{\lambda}^{\nu}=(v^{1/\beta})_{\lambda}^{\mu}((v^{1})^{-1})_{\mu}^{\nu}$.
Here, $\\{(v^{1/\beta})_{\lambda}^{\mu}\\}$ as well as
$\\{((v^{1})^{-1})_{\mu}^{\nu}\\}$ is a triangular matrix with unit diagonal
entry. This shows again that the matrix $\\{R_{\lambda}^{\nu}\\}$, as well as
$H$, with their explicit form given in this letter, is triangular. To get the
bosonic formalism we can use the well-known Frobenius formula to expand the
Schur polynomial in the basis of power-sum polynomials and the transition
coefficient is proportional to the character for the related representation
evaluated in the conjugacy class of symmetric groupLassalle .
This work is part of the CAS program “Frontier Topics in Mathematical Physics”
(KJCX3-SYW-S03) and is supported in part by a national grant NSFC(11035008).
## References
* (1) F. Calogero, J. Math. Phys. 10, 2191, 2197 (1969)
* (2) B. Sutherland, J. Math. Phys. 12, 246 (1971); Phys. Rev. A 4, 2019 (1971); Phys. Rev. A 5, 1372 (1972)
* (3) B. Sutherland, Lecture Notes in Physics, 242, 1-95 (1985)
* (4) B. Doyon and J. Cardy, J. Phys. A 40, 2509 (2007)
* (5) B. Estienne, V. Pasquier, R. Santachiara and D. Serban, arXiv:1110.1101.
* (6) B. Shou, J. F. Wu and M. Yu, arXiv:1107.4784.
* (7) J. F. Wu, Y. Y. Xu and M. Yu, arXiv:1107.4234.
* (8) B. Estienne, B. A. Bernevig and R. Santachiara, Phys. Rev. B 82, 205307 (2010)
* (9) B. A. Bernevig and F. D. M. Haldane, Phys. Rev. Lett. 100, 246802 (2008)
* (10) F. D. M. Haldane, Phys. Rev. Lett. 67, 937 (1991)
* (11) H. Azuma and S. Iso, Phys. Lett. B331, 107 (1994)
* (12) Y. S. Wu, Phys. Rev. Lett. 73, 922 (1994)
* (13) V. Pasquier, Lecture Notes in Physics, 436, 36 (1994)
* (14) L. Lapointe and L. Vinet, Commun. Math. Phys. 178, 425 (1996)
* (15) H. Awata, Y. Matsuo, S. Odake and J. Shiraishi, Nucl. Phys. B 449, 347 (1995)
* (16) A. P. Polychronakos, Phys. Rev. Lett. 69, 703 (1992)
* (17) K. W. J. Kadell, Adv. Math. 130, 33 (1997)
* (18) R. P. Stanley, Adv. Math. 77, 76 (1989)
* (19) M. Lassalle, Math. Ann. 340, 383 (2008)
* (20) T. Miwa, M. Jimbo and E. Date, “Solitons: Differential equations, symmetries and infinite dimensional algebras”, Cambridge University Press (2000)
|
arxiv-papers
| 2011-10-31T08:46:39 |
2024-09-04T02:49:23.739933
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jian-feng Wu, Ming Yu",
"submitter": "Ming Yu",
"url": "https://arxiv.org/abs/1110.6720"
}
|
1110.6749
|
# Nyström Methods in the RKQ Algorithm for Initial-value Problems
J.S.C. Prentice
Department of Applied Mathematics
University of Johannesburg
South Africa
###### Abstract
We incorporate explicit Nyström methods into the RKQ algorithm for stepwise
global error control in numerical solutions of initial-value problems. The
initial-value problem is transformed into an explicitly second-order problem,
so as to be suitable for Nyström integration. The Nyström methods used are
fourth-order, fifth-order and 10th-order. Two examples demonstrate the
effectiveness of the algorithm.
## 1 Introduction
In two previous papers we have considered the RK$rv$Q$z$ algorithm for
stepwise control of the global error in the numerical solution of an initial-
value problem (IVP), using Runge-Kutta methods [1][2]. In the current paper,
the third in the series, we focus our attention on the use of Nyström methods
in this error control algorithm for $n$-dimensional problems of the form
$\displaystyle\mathbf{y}^{\prime\prime}\left(x\right)$ $\displaystyle=$
$\displaystyle\mathbf{f}\left(x,\mathbf{y}\right)$ (1)
$\displaystyle\mathbf{y}\left(x_{0}\right)$ $\displaystyle=$
$\displaystyle\mathbf{y}_{0}$
$\displaystyle\mathbf{y}^{\prime}\left(x_{0}\right)$ $\displaystyle=$
$\displaystyle\mathbf{y}_{0}^{\prime}.$
Note that $\mathbf{f}$ is not dependent on $\mathbf{y}^{\prime}.$ We designate
this Nyström-based algorithm RKN$rv$Q$z,$ and we will show in a later section
how any first-order IVP can be written in the form (1), so that RKN$rv$Q$z$
is, in fact, generally applicable. The motivation for considering this
modification to RK$rv$Q$z$ is twofold: most physical systems are described by
second-order differential equations, and Nyström methods applied to (1) tend
to be more efficient than their Runge-Kutta counterparts.
## 2 Relevant Concepts, Terminology and Notation
Here we describe concepts, terminology and notation relevant to our work. Note
that boldface quantities are $n\times 1$ vectors, except for
$\mathbf{\alpha}_{i}^{r},\mathbf{I}_{n},\mathbf{F}_{y}^{r},\mathbf{F}_{y^{\prime}}^{r}$
and $\mathbf{g}_{y},$ which are $n\times n$ matrices.
### 2.1 Nyström Methods
The most general definition of a Nyström method (sometimes known as Runge-
Kutta-Nyström (RKN)) for solving (1) is
$\begin{array}[]{l}\mathbf{k}_{p}=\mathbf{f}\left(x_{i}+c_{p}h_{i},\mathbf{w}_{i}+c_{p}h_{i}\mathbf{w}_{i}^{\prime}+h_{i}^{2}\sum\limits_{q=1}^{m}a_{pq}\mathbf{k}_{q}\right)\text{
\ \ \ \ \ \ \ }p=1,2,...,m\\\
\mathbf{w}_{i+1}=\mathbf{w}_{i}+h_{i}\mathbf{w}_{i}^{\prime}+h_{i}^{2}\sum\limits_{p=1}^{m}b_{p}\mathbf{k}_{p}\equiv\mathbf{w}_{i}+h_{i}\mathbf{F}\left(x_{i},\mathbf{w}_{i}\right)\\\
\mathbf{w}_{i+1}^{\prime}=\mathbf{w}_{i}^{\prime}+h_{i}\sum\limits_{p=1}^{m}\widehat{b}_{p}\mathbf{k}_{p}.\end{array}$
(2)
The coefficients $c_{p},a_{pq},b_{p}$ and $\widehat{b}_{p}$ are unique to the
given method. If $a_{pq}=0$ for all $p\leqslant q,$ then the method is said to
be explicit; otherwise, it is known as an implicit RKN method. We will focus
our attention on explicit methods. In the second line of (2), we have
implicitly defined the function $\mathbf{F}$. We treat
$\mathbf{w}_{i}^{\prime}$ as an ‘internal parameter’; for our purposes here,
we do not identify $\mathbf{w}^{\prime}$ with $\mathbf{y}^{\prime},$ because
$\mathbf{f}$ is not dependent on $\mathbf{y}^{\prime}$. The symbol
$\mathbf{w}$ is used here and throughout to indicate the approximate numerical
solution, whereas the symbol $\mathbf{y}$ will be used to denote the exact
solution. We will denote an RKN method of order $r$ as RKN$r$ and, for such a
method, we write
$\mathbf{w}_{i+1}^{r}=\mathbf{w}_{i}^{r}+h_{i}\mathbf{F}^{r}\left(x_{i},\mathbf{w}_{i}^{r},\mathbf{w}_{i}^{r\prime}\right).$
(3)
The stepsize $h_{i}$ is given by
$h_{i}\equiv x_{i+1}-x_{i}$
and carries the subscript because it may vary from step to step. It is known
that RKN$r$ has a local error of order $r+1$ and a global error of order $r,$
just like its Runge-Kutta counterpart RK$r$.
### 2.2 IVPs in the form $y^{\prime\prime}=f\left(x,\mathbf{y}\right)$
Consider the $n$-dimensional IVP
$\displaystyle\mathbf{y}^{\prime}\left(x\right)$ $\displaystyle=$
$\displaystyle\mathbf{g}\left(x,\mathbf{y}\right)$ (4)
$\displaystyle\mathbf{y}\left(x_{0}\right)$ $\displaystyle=$
$\displaystyle\mathbf{y}_{0}.$
This gives
$y_{j}^{\prime\prime}=\sum\limits_{i=1}^{n}\frac{\partial
g_{j}\left(x,\mathbf{y}\right)}{\partial
y_{i}}\frac{dy_{i}}{dx}=\sum\limits_{i=1}^{n}\frac{\partial
g_{j}\left(x,\mathbf{y}\right)}{\partial y_{i}}g_{i}\left(x,\mathbf{y}\right)$
where $y_{i}$ is the $i$th component of $\mathbf{y,}$ and $g_{i}$ is the $i$th
component of $\mathbf{g.}$ Clearly, we have
$y_{j}^{\prime\prime}=\sum\limits_{i=1}^{n}\frac{\partial
g_{j}\left(x,\mathbf{y}\right)}{\partial
y_{i}}g_{i}\left(x,\mathbf{y}\right)\equiv f_{j}\left(x,\mathbf{y}\right)$
for all $j=1,2,\ldots,n,$ and so we can write
$\mathbf{y}^{\prime\prime}\left(x\right)=\mathbf{f}\left(x,\mathbf{y}\right).$
The initial values for this second-order problem are then given by
$\displaystyle\mathbf{y}\left(x_{0}\right)$ $\displaystyle=$
$\displaystyle\mathbf{y}_{0}$
$\displaystyle\mathbf{y}^{\prime}\left(x_{0}\right)$ $\displaystyle=$
$\displaystyle\mathbf{g}\left(x_{0},\mathbf{y}_{0}\right)\equiv\mathbf{y}_{0}^{\prime}.$
Hence, any first-order IVP can be transformed into an IVP of the form (1).
This is ideally suited to the Nyström methods, which are specifically designed
for this type of IVP. They are also more efficient than their Runge-Kutta
counterparts; for example, the methods to be used later, RKN4 and RKN5,
require three and four stage evaluations, respectively, as opposed to RK4 and
RK5, which require at least four and six stage evaluations, respectively.
### 2.3 Error Propagation in RKN
It can be shown [3] that, for RK$r$,
$\displaystyle\mathbf{\Delta}_{i+1}^{r}$ $\displaystyle\equiv$
$\displaystyle\mathbf{w}_{i+1}^{r}-\mathbf{y}_{i+1}=\mathbf{\varepsilon}_{i+1}^{r}+\mathbf{\alpha}_{i}^{r}\mathbf{\Delta}_{i}^{r}$
(5) $\displaystyle\mathbf{\alpha}_{i}^{r}$ $\displaystyle\equiv$
$\displaystyle\mathbf{I}_{n}+h_{i}\mathbf{F}_{y}^{r}\left(x_{i},\mathbf{\xi}_{i}\right),$
(6)
where $\mathbf{\varepsilon}_{i+1}^{r}=O\left(h_{i}^{r+1}\right)$ is the local
error, $\mathbf{\Delta}_{i+1}^{r}$ is the global error and
$\mathbf{F}_{y}^{r}$ is the Jacobian (with respect to $\mathbf{y})$ of the
function $\mathbf{F}^{r}\left(x_{i},\mathbf{w}_{i}^{r}\right)$ associated with
RK$r$. The term $h_{i}\mathbf{F}_{y}^{r}\left(x_{i},\mathbf{\xi}_{i}\right)$
in the matrix $\mathbf{\alpha}_{i}^{r}$ arises from a first-order Taylor
expansion of $\mathbf{F}^{r}\left(x_{i},\mathbf{w}_{i}\right)=$
$\mathbf{F}^{r}\left(x_{i},\mathbf{y}_{i}+\mathbf{\Delta}_{i}^{r}\right)$ with
respect to $\mathbf{y}_{i}$.
For a Nyström method RKN$r,$ we have
$\mathbf{F}^{r}=\mathbf{F}^{r}\left(x_{i},\mathbf{w}_{i}^{r}\right)$ and so,
as above,
$\mathbf{\alpha}_{i}^{r}\equiv\mathbf{I}_{n}+h_{i}\mathbf{F}_{y}^{r}\left(x_{i},\mathbf{\zeta}_{i}\right),$
where $\mathbf{\zeta}_{i}$ is an appropriate constant. Hence, the global error
in RKN$r$ is also given by (5).
### 2.4 RK$rv$Q$z$
We will not discuss RK$rv$Q$z$ in detail here; the reader is referred to our
previous work where the algorithm has been discussed extensively. It suffices
to say that RK$rv$Q$z$ uses RK$r$ and RK$v$ to control local error via local
extrapolation, while simultaneously using RK$z$ to keep track of the global
error in the RK$r$ solution. Such global error arises due to the propagation
of the RK$v$ global error. RK$rv$Q$z$ is designed to estimate the various
components of the global error in RK$r$ and RK$v$ at each node and, when the
global error is deemed too large, a quenching procedure is carried out. This
simply involves replacing the RK$r$ and RK$v$ solutions with the much more
accurate RK$z$ solution, whenever necessary, so that the RK$r$ and RK$v$
global errors do not accumulate beyond a desired tolerance.
### 2.5 RKN$rv$Q$z$
The algorithm RKN$rv$Q$z$ is nothing more than RK$rv$Q$z$ with RK$r$, RK$v$
and RK$z$ replaced with RKN$r$, RKN$v$ and RKN$z$. Of course, RKN$rv$Q$z$ is
applied to problems of the form (1), whereas RK$rv$Q$z$ is applied to problems
of the form (4).
We also report on a refinement to the algorithm: in RK$rv$Q$z,$ if the global
error at $x_{i}$ is too large, we replace $\mathbf{w}_{i}^{r}$ with
$\mathbf{w}_{i}^{z}$ and then recompute $\mathbf{w}_{i+1}^{r}$ and
$\mathbf{w}_{i+1}^{v},$ using $\mathbf{w}_{i}^{z}$ as input for both RK$r$ and
RK$v.$ This is the essence of the quenching procedure. However, in retrospect
it seems quite acceptable to simply replace $\mathbf{w}_{i+1}^{r}$ and
$\mathbf{w}_{i+1}^{v}$ with $\mathbf{w}_{i+1}^{z};$ this avoids the need for
recomputing $\mathbf{w}_{i+1}^{r}$ and $\mathbf{w}_{i+1}^{v},$ which improves
efficiency and, after all, it is the global error in $\mathbf{w}_{i+1}^{r}$
and $\mathbf{w}_{i+1}^{v},$ not $\mathbf{w}_{i}^{r}$ and $\mathbf{w}_{i}^{v},$
that is too large. Both approaches are effective, although one is more
efficient than the other. It is the more efficient approach that we have
employed in RKN$rv$Q$z.$
## 3 Numerical Examples
It is not our intention to compare methods or algorithms but, for the sake of
consistency, we will apply RKN$rv$Q$z$ to the same examples that we considered
in our previous work on RK$rv$Q$z.$ In our calculations, we use RKN4, RKN5 and
RKN10 which gives the algorithm RKN45Q10. RKN4 and RKN5 are taken from Hairer
et al [5], and RKN10 is from Dormand et al [4].
The first of these is the scalar problem
$\displaystyle y^{\prime}$ $\displaystyle=$ $\displaystyle\left(\frac{\ln
1000}{100}\right)y$ $\displaystyle y\left(0\right)$ $\displaystyle=$
$\displaystyle 1$
which transforms to
$\displaystyle y^{\prime\prime}$ $\displaystyle=$
$\displaystyle\left(\frac{\ln 1000}{100}\right)^{2}y$ $\displaystyle
y\left(0\right)$ $\displaystyle=$ $\displaystyle 1$ $\displaystyle
y^{\prime}\left(0\right)$ $\displaystyle=$ $\displaystyle\frac{\ln
1000}{100}.$
Solving this problem with RKN45 and RKN45Q10 with a tolerance of $10^{-10}$ on
the absolute local and global errors gives the error curves shown in Figure 1.
The global error obtained with RKN45 is clearly larger than the desired
tolerance on most of the interval, despite local error control via local
extrapolation. However, RKN45Q10 yields a solution with a global error always
less than the tolerance - the maximum global error in this case is $9.1\times
10^{-11}$. The points on the $x$-axis where this global error decreases
sharply correspond to the quenches carried out using RKN10.
The second example is the simple harmonic oscillator
$\begin{array}[]{c}y_{1}^{\prime}=y_{2}\\\ y_{2}^{\prime}=-y_{1}\\\
\mathbf{y}\left(0\right)=\left[\begin{array}[]{c}0\\\
1000\end{array}\right]\end{array}$
which has solution
$\displaystyle y_{1}\left(x\right)$ $\displaystyle=$ $\displaystyle 1000\sin
x$ $\displaystyle y_{2}\left(x\right)$ $\displaystyle=$ $\displaystyle
1000\cos x$
and becomes, in explicit second-order form,
$\begin{array}[]{c}\mathbf{y}^{\prime\prime}=\left[\begin{array}[]{c}y_{1}^{\prime\prime}\\\
y_{2}^{\prime\prime}\end{array}\right]=\left[\begin{array}[]{c}-y_{1}\\\
-y_{2}\end{array}\right]\equiv\mathbf{f}\left(x,\mathbf{y}\right)\\\
\mathbf{y}\left(0\right)=\left[\begin{array}[]{c}0\\\
1000\end{array}\right],\mathbf{y}^{\prime}\left(0\right)=\left[\begin{array}[]{c}1000\\\
0\end{array}\right].\end{array}$
Since the solution oscillates between $-1000$ and $1000$, there are regions
where the solution has magnitude less than unity - here, we implement absolute
error control - and regions where the solution has magnitude greater than
unity, where we implement relative error control. With an imposed tolerance of
$10^{-8}$ on the local and global errors (relative and absolute) we found a
maximum global error of $\sim 4\times 10^{-8}$ in each component when using
RKN45, and a global error no greater than $0.99\times 10^{-8}$ with RKN45Q10,
on $x\in\left[0,200\right].$ A total of 20 quenches were needed.
## 4 Conclusion
We have considered the use of Nyström methods in RK$rv$Q$z,$ wherein a
combination of local extrapolation and quenching result in stepwise global
error control in numerical solutions of IVPs. Two examples have demonstrated
the success of RKN45Q10.
## References
* [1] Prentice, J.S.C. (2011). Stepwise Global Error Control in an Explicit Runge-Kutta Method using Local Extrapolation with High-Order Selective Quenching, Journal of Mathematics Research, 3, 2, 126-136. [http://ccsenet.org/journal/index.php/jmr/article/view/8700/7481]
* [2] Prentice, J.S.C. (2011). Relative Global Error Control in the RKQ Method for Systems of Ordinary Differential Equations, Journal of Mathematics Research, 3, 4, 59-66. [http://ccsenet.org/journal/index.php/jmr/article/view/10491/8952]
* [3] Prentice, J.S.C. (2009). General error propagation in the RK$r$GL$m$ method, Journal of Computational and Applied Mathematics, 228, 344-354.
* [4] Dormand, J.R., El-Mikkawy, M.E.A., and Prince, J. (1987). High-Order Embedded Runge-Kutta-Nyström Formulae, IMA Journal of Numerical Analysis, 7, 423-430.
* [5] Hairer, E., Norsett, S.P., and Wanner, G. (2000). Solving Ordinary Differential Equations I: Nonstiff Problems, Berlin: Springer
|
arxiv-papers
| 2011-10-31T11:07:57 |
2024-09-04T02:49:23.748202
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "J. S. C. Prentice",
"submitter": "Justin Prentice",
"url": "https://arxiv.org/abs/1110.6749"
}
|
1110.6760
|
# Synchronous imaging for rapid visualization of complex vibration profiles in
electromechanical microresonators
Y. Linzon yoli@braude.ac.il; yoav.linzon@cornell.edu. Department of Physics
and Optical Engineering, Ort Braude College, PO Box 78, Karmiel 21982, Israel
D. J. Joe School of Applied and Engineering Physics, Cornell University,
Ithaca, New York 14853, USA S. Krylov School of Mechanical Engineering,
Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel B. Ilic
School of Applied and Engineering Physics, Cornell University, Ithaca, New
York 14853, USA J. Topolancik School of Applied and Engineering Physics,
Cornell University, Ithaca, New York 14853, USA J. M. Parpia School of
Applied and Engineering Physics, Cornell University, Ithaca, New York 14853,
USA H. G. Craighead School of Applied and Engineering Physics, Cornell
University, Ithaca, New York 14853, USA
###### Abstract
Synchronous imaging is used in dynamic space-domain vibration profile studies
of capacitively driven, thin n+ doped poly-silicon microbridges oscillating at
rf frequencies. Fast and high-resolution actuation profile measurements of
micromachined resonators are useful when significant device nonlinearities are
present. For example, bridges under compressive stress near the critical Euler
value often reveal complex dynamics stemming from a state close to the onset
of buckling. This leads to enhanced sensitivity of the vibration modes to
external conditions, such as pressure, temperatures, and chemical composition,
the global behavior of which is conveniently evaluated using synchronous
imaging combined with spectral measurements. We performed an experimental
study of the effects of high drive amplitude and ambient pressure on the
resonant vibration profiles in electrically-driven microbridges near critical
buckling. Numerical analysis of electrostatically driven post-buckled
microbridges supports the richness of complex vibration dynamics that are
possible in such micro-electromechanical devices.
###### pacs:
87.64.-t, 07.10.Cm, 85.85.+j, 05.45.-a
## I Introduction
Suspended resonant nano- and micro-electro-mechanical systems (NEMS and MEMS)
find use in versatile applications, such as ultra-sensitive mass detectors, rf
filters, and switching devices Craig (2000); Roukes1 (2005). As device
miniaturization advances, optimization of the overall characteristics in high-
frequency MEMS/NEMS resonators becomes increasingly complex and linked with
various mechanical, electrical, thermal and optical parameters of the system
and its environment. This compounds their seemingly superior sensitivity to
environmental conditions, such as the pressure, temperature and chemical
composition of the surrounding gas.
In the characterization of NEMS and MEMS under periodic electrical actuation,
vibration profile (VP) measurements are important in conjunction with
frequency-domain spectral studies Craig (2000); Roukes1 (2005); Pressure_dep
(2011); Carr (1999); Max (2000). While the latter yield important mechanical
properties, the former can be useful in many applications, including
optimization of the excitation parameters, aiding the identification of sites
most effective for localized functionalization to enable sensing, and in
studies of dissipation effects such as intrinsic and pressure-dependent
damping Pressure_dep (2011). Space-domain profiling is crucial in the presence
of significant nonlinearities where boundary conditions become critical RonLif
(2008). For instance, in typical capacitive electrical drive configuration,
the force between the grounded substrate and a device fabricated by patterned
suspended poly-crystalline silicon (polySi) film (serving as an electrode) is
inherently nonlinear with the drive amplitude and film stress Craig (2000);
Roukes1 (2005); Pressure_dep (2011); Carr (1999); Max (2000); RonLif (2008).
With interferometric reflection-mode optical transduction Carr (1999),
thermoacoustic effects can significantly modify the effective device stiffness
or induce autoparametric optical drive Max (2000). We observe all these
effects to be significant in NEMS/MEMS devices defined on films with low
compressive residual stress under applied loads near to the Euler critical
value Buckled_exp (1999). At the critical point, MEMS devices are most
sensitive to changes induced by stress variations in chemically-reactive
coatings SensorReview (2011); Darren (2010). Of additional practical interest
are possible non-uniformities of mechanical and electrical film properties
across the wafer, originating from growth processes and application of
anisotropic etch, which directly affect each circumferentially-clamped
microresonator ($\mu$R) Buckled_exp (1999). Fast space-domain visualization of
resonant VPs serves as a direct means to study the physics of all these
effects on the single device micro-scale during its actuation.
VPs in MEMS are traditionally imaged optically with vibrometric Pressure_dep
(2011), interferometric intMZ (2001); intHeterodyne (2008), or stroboscopic
Strobo (2002); YL (2010) microscopy. Recently, spatiotemporal evaluation of
resonant VPs in high-frequency MEMS $\mu$Rs was demonstrated using resonant
realtime synchronous imaging (RSI) with a pulsed low duty-cycle nanosecond
laser YL (2010). The main feature of stroboscopic imaging is a rapid
production of time-resolved interference pattern movies and static profiles,
as well as the fast evaluation of VPs, thus supplanting scanned motorized
probes that are expensive and inherently slow. This technique is applicable
with mechanical resonant frequencies up to $f_{0}\simeq 1$GHz. In this paper,
we use RSI YL (2010) in averaging mode to rapidly characterize the VPs in
bridge $\mu$Rs close to critical stress as a function of the drive amplitude
and ambient pressure. The effects of high drive nonlinearity and air damping
on the resonator VPs are directly monitored.
## II Experimental method
$\mu$Rs are fabricated by standard top-down micromachining methods, where
bridges are defined photolithographically on compressively-stressed n+ doped
polySi films, deposited by low pressure chemical vapor deposition over a
sacrificial oxide layer and wet-etch released. Upon release of doubly clamped
bridges, residual stress is relieved through buckling Buckled_exp (1999);
Darren (2010). The devices are driven capacitively with the moving $\mu$R
serving as an electrode and the silicon substrate serving as a bottom ground
electrode. The inset in Fig. 1(a) shows the schematics of a buckled $\mu$R
cross section, as well as definitions we use, and Fig. 1(b) shows SEM images
of our released bridges.
Figure 1: (a) Schematic cross section of the devices studied and definitions
of optical quantities used in the analysis. (b) SEM images of bridges of
dimensions: 25$\times$6$\times$0.12 $\mu$m3 (left, slightly post-buckled), and
20$\times$1$\times$0.14 $\mu$m3 (right, flat), both with $\sim$220 nm static
elevations. Figure 2: (Color online) (a) Schematics of the experimental setup.
(b) Calibration curves for synchronous imaging assuming a device with film
thickness $t$=138 nm and static midpoint elevation $d_{0}$=220 nm. Left:
absolute reflection coefficient $R$. Right: differential reflection $\Delta
R/R_{0}$. With negative values of $\Delta R$, the intensity contrast in the
image is negative.
In Fig. 2(a) a schematic of our RSI configuration is illustrated. A dual
channel pulse source feeds the $\mu$R and optical imaging pulse source
($\lambda_{0}$=661.5 nm) in synchrony. The collimated illumination at a
glancing angle $\theta\simeq 40^{\circ}$ is reflected off the $\mu$R and
collected by an objective lens followed by a $4f$ lens pair, the latter of
which is used for spatial waveform filtering at the Fourier plane with a phase
mask Fourier (1978). The outgoing light is finally imaged on a standard CCD
camera. Changes in the reflection with respect to the static image of the
$\mu$R, due to resonant motion, are monitored as a function of the rf source
frequency $f_{0}$, voltage and phase. The pressure within the chamber is set
with a vacuum pump and venting tubes, and monitored via a Pirani gauge. In
order to calibrate the physical VPs from measured reflection images, an
interferometric analysis is carried out in the out-of-plane direction (shown
in Fig. 1(a)), as detailed below. Application of a 50% duty-cycle to the
imaging pulses (full synchronization with the capacitive drive), high in-phase
sensitivity to _average_ differential actuation amplitudes is attained at the
expense of lost temporal resolution.
For calibration of the physical VPs from measured reflectivity images, a
Fabry-Pèrot interferometer multilayer analysis, as a function of the total
elevation Interference_book (1995), is performed using knowledge of the static
film elevation profile $d_{0}$, thickness $t$, and the refractive indices of
the film (n-doped polySi, $n$=3.916) and substrate $n_{S}$ (single crystal Si,
$n$=3.834). The reflectance coefficients are calculated from the effective
reflectivity matrix, assuming nearly normal incidence:
$\displaystyle M_{total}=M_{2}\cdot
M_{1}=\left(\begin{array}[]{cc}\cos\delta_{2}&\frac{i}{n_{2}}\sin\delta_{2}\\\
in_{2}\sin\delta_{2}&\cos\delta_{2}\\\ \end{array}\right)\cdot$ (3)
$\displaystyle\left(\begin{array}[]{cc}\cos\delta_{1}&\frac{i}{n_{1}}\sin\delta_{1}\\\
in_{1}\sin\delta_{1}&\cos\delta_{1}\\\ \end{array}\right)$ (6)
Figure 3: (Color online) Drive amplitude dependence of VPs in the fundamental
mode of a critically upward buckled resonator. (a) Reference image of the
static bridge. (b) Frequency domain spectra with low ac actuation voltage and
a constant 5 V dc bias. (c) Static height profile of the bridge along the $Y$
direction taken from AFM measurements (0 is defined as the height of the
trench and known film thickness of 140 nm is subtracted on the bridge).
(d),(e) Measured synchronous images at $f_{0}$=4.1 MHz with different ac
amplitudes; (f)-(i) corresponding VPs integrated along $Y$ [in (f),(g),
$X$-profiles], and along $X$ [in (h),(i), $Y$-profiles]. Vertical arrows in
(f),(g) indicate diminished actuation signals with high excitation.
where $\delta_{j}=k_{j}d_{j}$ is the effective phase of layer $j$ and $k_{j}$
is the wave number. Denoting $R_{0}(x,y)$ the reference image of the static
reflection, the contrast signal measured during actuation corresponds to:
$R_{meas}(x,y)=\frac{R(x,y)-R_{0}(x,y)}{R_{0}(x,y)}\equiv\frac{\Delta
R(x,y)}{R_{0}}$ (7)
Under full synchronization of the sampling beam with the drive frequency and
phase, the observed average amplitude $<A>$ at transverse position $(x,y)$ is
then:
$\langle A\rangle(x,y)=d_{1}(x,y)-d_{0}(x,y)$ (8)
With full sampling synchronization and a periodic unipolar square wave
excitation, the observed average amplitude $\langle A\rangle$ at transverse
position $(x,y)$ is $\langle A\rangle=A_{max}/2$. The glancing angle of
illumination (see Fig. 1(a)) introduces an additional scaling factor
$\cos\theta$. The peak VP average amplitude, as measured away from the static
beam position, is then:
$A_{max}(x,y)=\frac{2}{\cos\theta}[d_{1}(x,y)-d_{0}(x,y)]$ (9)
giving rise to a normalization factor of 2.83 in our implementation, with
$\theta=45^{\circ}$. A glancing angle also introduces shadow effects at the
resonator edges, which we can easily eliminate with appropriate phase masks at
the Fourier plane (see Fig. 1(a)). We consider only reflection variations at
the positions of the $\mu$R itself to constitute its real VPs. An example of
the total and differential reflectance curves as function of the total
elevations is shown in Fig. 2(b). Using Eqs. (1)-(4), together with the
calibration curve, the average maximum amplitude profiles are estimated. Here
we will concentrate on characterizations of the fundamental (lowest) resonant
mode.
Figure 4: (Color online) Pressure-dependent studies of VPs in the fundamental
mode of a slightly post-buckled resonator. (a) Reference image of the static
bridge. (b) Frequency domain spectra (inset) and inverse quality factors
(dissipation) as a function of chamber pressure, under continuous 315 mV ac
drive and a 5 V dc bias. (c) Measured interferometric images at $f_{0}$=2.4
MHz as a function of the pressure, and (d) overlaid $Y$-integrated
$X$-profiles of vibration.
## III Results and Discussion
In Fig. 3 we study a $\mu$R of dimensions (25$\times$6$\times$0.14 $\mu$m3)
and a midpoint elevation of 220 nm under low pressure settings (P$<$1 Torr).
The undriven $\mu$R is almost flat (see Fig. 3(c)) whereas other slightly
longer devices exhibit noticeable static upward buckling, suggesting the
existence of a compressive force whose magnitude is close to critical load.
Highly buckled resonators have been found as hard to drive electrostatically.
Figure 3(a) shows a static optical image of the unactuated device in its
initial reference configuration. In Fig. 3(b), the frequency response under
low-voltage actuation is shown. Even with drive amplitudes as low as 45 mV and
a dc bias of 5 V, we observe the formation of Duffing nonlinearity RonLif
(2008) and significant spectral broadening, with a sudden frequency detuning
between 35 and 45 mV drive voltage. An AFM measurement of the static bridge
height profile, in the transverse ($Y$) direction, is shown in Fig. 3(c), and
the profile is uniform in the axial ($X$) direction, indicating a shell-like
bridge profile. RSI images with intermediate and high ac drive voltages, at a
frequency corresponding to the maximum resonant amplitude, optimal phase and a
dc bias of 5 V, are shown in Figs. 3(d) and 3(e), respectively. Following the
image analysis for amplitude calibration, as detailed in the experimental
section, and integration along the beam width ($Y$), the peak VP amplitude
$X$-profiles are shown in Figs. 3(f) and 3(g), respectively. The peak VP
amplitude $Y$-profiles, integrated and averaged along $X$, are also shown in
Figs. 3(h) and 3(i). With intermediate drive amplitudes the VP shapes are as
shown in Fig. 3(f) and 3(h). With high drive amplitudes, central regions on
the beam appear to undergo diminished displacement at the original frequency
(Fig. 2(g)). However, detuning of the imaging frequency in these cases to
values near multiples of the fundamental mechanical frequency and the same
phase settings show some tiny components of vibration at these locations. We
interpret this observation as resulting from either nonlinear
electromechanical processes inducing transfer of energy to higher harmonics at
locations of high vibration amplitudes on the $\mu$R, or from optical
nonlinearity due to the measured response crossing extreme reflection points.
In any case, positions with diminished signal, such as the one indicated by
the vertical arrows in Figs. 3(f),(g) would clearly not be beneficial to
employ in phase-locked-loop (PLL) sensing applications, using this class of
$\mu$Rs, at this wavelength. Along the $Y$-profiles, slight localization of
the motion at the central region of the bridge is also observed with high
drives.
Figure 4 shows studies using a narrow microbridge of dimensions
(25$\times$1$\times$0.12 $\mu$m3) and a midpoint elevation of 660 nm ($t$=120
nm and $d_{0}$=660 nm) under varying ambient hydrostatic pressure and constant
driving conditions of 1.2 V ac voltage and 5 V dc bias. This bridge is
slightly buckled in the upward direction, as observed in the static optical
reference image of Fig. 4(a). Figure 4(b) shows the dissipation (inverse
quality factor, $Q^{-1}$) of the fundamental resonant mode as a function of
pressure, and corresponding spectra (inset). Different pressure ranges
correspond to well-known dominant dissipation mechanismsPressure_dep (2011);
Darren2 (2009). The total quality factor $Q$ is known to approximately scale
according to Darren2 (2009):
$1/Q=1/Q_{int}+\alpha P$ (10)
Where $Q_{int}$ is the intrinsic (material) quality factor, $\alpha$ is the
coefficient of viscous damping and $P$ is the pressure. In the data
corresponding to Fig. 4(b), a linear fit yields $Q_{int}=154$ and
$\alpha=1.83\times 10^{-4}$ $[\textmd{Torr}^{-1}]$ in this $\mu$R.
In the current experiment we have succeeded in recording RSI images of
sufficient contrast only at pressures below the viscous (gas-dominated)
regime, namely, corresponding to the intrinsic and molecular regimes in Fig.
4(b). It is estimated that the most significant limiting factors are the low
spectral signal-to-noise bandwidth (S/N) at low quality factors (below
$Q\approx$ 20) combined with diminished amplitudes of motion under external
air damping. Figure 4(c) shows RSI images of the $\mu$R as a function of
increasing pressure, with a transition from intrinsic to molecular damping.
Calibrated maximum amplitude $X$-profiles, integrated across the beam width
($Y$), are shown in Fig. 4(d). Increased errors in the VP estimations result
from diminished available S/N, giving rise to less accurate numerical fits. We
consistently find that with increasing pressure, the VPs in this $\mu$R become
suppressed around the regions close to the bridge overhang (see Fig. 4(d)).
This edge suppression effect is not observed in repeated experiments under low
pressure and drive conditions (0.3 V ac voltage and 5 V dc bias), that yield
available signal-to-noise close to the detection limit, with extracted
vibration amplitudes comparable to the highest pressure case shown here and
with more pronounced motion near the overhang.
## IV Numerical Model
The dynamics of a compressively stressed beam are described by the equation
Nayfeh (2004); Slava (2011):
$\displaystyle EI(\frac{\partial^{4}w}{\partial
x^{4}}-\frac{\partial^{4}w_{0}}{\partial
x^{4}})-[P-\frac{EA}{2L}\int^{L}_{0}((\frac{\partial
w}{dx})^{2}-(\frac{\partial w_{0}}{\partial x})^{2})dx]\times$
$\displaystyle\frac{\partial^{2}w}{\partial x^{2}}+\rho
d\frac{\partial^{2}w}{\partial
t^{2}}=-\frac{\epsilon_{0}bV^{2}}{2(g_{0}+w)^{2}}$
Figure 5: Schematics of the numerical model.
where now $E$ is the Young’s modulus of the beam material, $I=b\times
d^{3}/12$ is the moment of inertia of the beam cross section, $A=b\times d$ is
the sectional area, $L$ is the beam length, $\rho$ is the density, and $b$ and
$d$ are the thickness and width of the beam, respectively. In addition,
$g_{0}$ is distance between the ends of the flat side of the beam and the
electrode (electrostatic gap), $\epsilon_{0}$ is the vacuum permittivity and
$V(t)$ is the time-dependent actuation voltage. In accordance with the
definitions in Fig. 5, the elevation of the beam $w(x)$, as well as the
electrostatic force $f^{e}(x,t)=-\epsilon_{0}bV^{2}/2(g_{0}+w)^{2}$, which is
calculated using the simplest parallel capacitor approximation formula, are
considered positive upwards.
Equation (5) has been reduced to the system of coupled nonlinear ordinary
differential equations by means of the Galerkin decomposition with linear
undamped eigenmodes of a straight beam used as base functions. The equaations
were solved numerically using the ODE45-solver implemented in Matlab. The
details of the formulation and numerical approach used for the analysis are
found in Slava (2011) (see also Nayfeh (2004)).
## V Numerical Results
Figures 6-8 show numerical solutions of Eq. (5). In all cases, the actuation
voltage contained both ac and dc bias components, such that
$V(t)=V_{dc}+V_{ac}\cos(\omega t)$. Zero initial conditions, corresponding to
the post-buckled configuration of the beam in rest, were used. In all cases,
Young’s modulus $E$=150 GPa and density $\rho$=2300 kg/m3 corresponding to
polySi were used.
Figure 6: (a) Numerical frequency domain resonant response. Midpoint
deflection of the beam is shown. (b) Deflection profiles (difference between
the actual and initial elevations of the beam) averaged over a single period.
In (b) the operation frequency, $\omega$=2.45 MHz, is close to the fundamental
mode resonant frequency. In both simulations, the dimensions of the beam are
25$\times$1$\times$0.12 $\mu$m3; the electrostatic gap is $g_{0}$=660 nm; the
initial elevation of the midpoint above the beam’s ends, due to buckling, is
155 nm; input voltages are $V_{dc}=5$ V and $V_{ac}=1.2$ V, and (a) $Q$=7; (b)
$Q$=700.
Figure 6(a) shows the resonant response of the beam with dimensions
(25$\times$1$\times$0.12 $\mu$m3) and electrostatic gap $g_{0}$=660 nm. The
axial force was chosen such that the midpoint elevation of the beam above its
ends was 155 nm. The driving voltages were $V_{dc}=5$ V and $V_{ac}=1.2$ V,
and the quality factor was $Q$=7. It is observed that the fundamental resonant
frequency is 2.45 MHz, which is close to the experimentally observed value
(Fig. 4(b)). The corresponding resonant displacement profile, averaged over a
single period, is shown in Fig. 6(b). Small initial imperfection of 0.05 in
the initial buckled height, corresponding to an excitation of the second anti-
symmetric mode, was introduced in order to allow non-symmetric mode shapes of
the beam. Calculations show that while the actual beam profiles are dominated
by the fundamental mode, the resonant deflection profiles (i.e., the
differences between the initial buckled shapes and the actual, time dependent
shapes of the vibrating beam) could be more complex.
Figure 7: Dynamic snapshots of the deflection profiles (differences between
actual and initial elevations of the beam) corresponding to different time
sections within a single resonant cycle. The beam dimensions are
25$\times$1$\times$0.12 $\mu$m3; the electrostatic gap is $g_{0}$=220 nm; the
initial elevation of the midpoint above the beam’s ends, due to buckling, is
98 nm; input voltages are $V_{dc}=1$ V and $V_{ac}=350$ mV, and $Q$=1000. The
operation frequency is $\omega$=1.495 MHz. Nine (symmetric and skew-symmetric)
base functions are preserved in the reduced-order model.
A decrease in the initial separation between the beam and the electrode
results in an increased contribution of higher modes in the resonant VPs.
Figure 7 shows time-resolved snapshots of the VPs (relative displacements from
equilibrium) with the same dimensions as in Fig. 6, but with $g_{0}$=220 nm;
Figure 8 shows the same vibration profile as averaged over a single period.
Small initial imperfection of 0.05 in the initial buckled height,
corresponding to a contribution of the second buckled mode, was again used as
an initial condition. Complex displacement profiles are clearly observed in
this case as well.
Figure 8: Deflection profiles (difference between the actual and initial beam
elevations) averaged over a single period, corresponding to the results in
Fig. 7 (same parameters).
## VI Conclusion
Synchronous imaging has been demonstrated as a robust method for direct and
rapid observations of gradual changes in resonant vibration profiles of
electromechanical microresonators under varying conditions of drive and
ambient pressure. Synchronous imaging can serve as a useful tool for studying
fundamental processes in resonant MEMS/NEMS, as well as for identification of
favorable device positions most suitable for sensitive phase-locked
applications, such as sensors, filters and switches. Numerical analysis of
electrostatically driven post-buckled microbridges supports the richness of
the complex resonant vibrations that are possible in these micro-
electromechanical systems.
## VII Acknowledgments
This research was funded by the National Science Foundation (grants
DMR-0908634 and DMR-0520404) and Analog Devices. Fabrication was performed at
the Cornell Nanoscale science and technology Facility.
## References
* Craig (2000) H. G. Craighead, Science 290, 5496 (2000).
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* Pressure_dep (2011) R. C. Tung RC, J. W. Lee, H. Sumali, and A. Raman, J. Micromech. Microeng. 21, 025003 (2011); R. A. Bidkar, R. C. Tung, A. A. Alexeenko, H. Sumali, and A. Raman, Appl. Phys. Lett. 94, 163117 (2009); H. Sumali, J. Micromech. Microeng. 17, 2231 (2007).
* Carr (1999) D. W. Carr, S. Evoy, L. Sekaric, H. G. Craighead, and J. M. Parpia, Appl. Phys. Lett. 75, 920 (1999).
* Max (2000) M. Zalalutdinov, A. Zehnder, A. Olkhovets, S. Turner, L. Sekaric, B. Ilic, D. Czaplewski, J. M. Parpia, and H. G. Craighead, Appl. Phys. Lett. 79, 695 (2001); B. Ilic, S. Krylov, K. Aubin, R. Reichenbach, and H. G. Craighead, Appl. Phys. Lett. 86, 193114 (2005).
* RonLif (2008) R. Lifshitz and M. C. Cross, _Review of Nonlinear Dynamics and Complexity_ (Wiley, Meinheim, 2008), Vol. I, pp. 1-52.
* Buckled_exp (1999) W. Fang, C.-H. Lee, and H.-H. Hu, J. Micromech. Microeng. 9, 236 (1999).
* SensorReview (2011) A. Boisen, S. Dohn, S. S. Keller, S. Schmid, and M. Tenje, Rep. Prog. Phys. 74, 036101 (2011).
* Darren (2010) D. R. Southworth, L. M. Bellan, Y. Linzon, H. G. Craighead, and J. M. Parpia, Appl. Phys. Lett. 96, 163503 (2010).
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* Strobo (2002) C. Rembe and R. S. Muller, J. Microelectromech S. 11, 479 (2002).
* YL (2010) Y. Linzon, S. Krylov, B. Ilic, D. R. Southworth, R. A. Barton, B. R. Cipriany, J. D. Cross, J. M. Parpia, and H. G. Craighead, Opt. Lett. 15, 2654 (2010).
* Darren2 (2009) D. R. Southworth, H. G. Craighead, and J. M. Parpia, Appl. Phys. Lett. 94, 213506 (2009).
* Fourier (1978) J. D. Gaskill, _Linear Systems, Fourier Transforms, and Optics_ (Wiley, New York, 1978).
* Interference_book (1995) M. Bass, _Handbook of Optics_ , 2nd ed. (McGraw-Hill, San Francisco, 1995), Vol. I, pp. 42.10-42.14.
* Slava (2011) S. Krylov, B. R. Ilic, and S. Lulinsky, Nonlinear Dyn. 66, 403 (2011).
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|
arxiv-papers
| 2011-10-31T12:00:24 |
2024-09-04T02:49:23.755643
|
{
"license": "Public Domain",
"authors": "Yoav Linzon, Daniel J. Joe, Slava Krylov, Bojan Ilic, Juraj\n Topolancik, Jeevak M. Parpia, Halrod G. Craighead",
"submitter": "Yoav Linzon Dr.",
"url": "https://arxiv.org/abs/1110.6760"
}
|
1110.6815
|
11institutetext: Dipartimento di Fisica dell’Università degli Studi di Milano,
I-20133 Milano, Italia, EU 22institutetext: CNISM - Udr Milano, I-20133
Milano, Italia, EU. 33institutetext: 33email: matteo.paris@fisica.unimi.it
# The modern tools of quantum mechanics
A tutorial on quantum states, measurements, and operations
Matteo G A Paris 112233
###### Abstract
We address the basic postulates of quantum mechanics and point out that they
are formulated for a closed isolated system. Since we are mostly dealing with
systems that interact or have interacted with the rest of the universe one may
wonder whether a suitable modification is needed, or in order. This is indeed
the case and this tutorial is devoted to review the modern tools of quantum
mechanics, which are suitable to describe states, measurements, and operations
of realistic, not isolated, systems. We underline the central role of the Born
rule and and illustrate how the notion of density operator naturally emerges,
together with the concept of purification of a mixed state. In reexamining the
postulates of standard quantum measurement theory, we investigate how they may
be formally generalized, going beyond the description in terms of selfadjoint
operators and projective measurements, and how this leads to the introduction
of generalized measurements, probability operator-valued measures (POVMs) and
detection operators. We then state and prove the Naimark theorem, which
elucidates the connections between generalized and standard measurements and
illustrates how a generalized measurement may be physically implemented. The
”impossibility” of a joint measurement of two non commuting observables is
revisited and its canonical implementation as a generalized measurement is
described in some details. The notion of generalized measurement is also used
to point out the heuristic nature of the so-called Heisenberg principle.
Finally, we address the basic properties, usually captured by the request of
unitarity, that a map transforming quantum states into quantum states should
satisfy to be physically admissible, and introduce the notion of complete
positivity (CP). We then state and prove the Stinespring/Kraus-Choi-Sudarshan
dilation theorem and elucidate the connections between the CP-maps description
of quantum operations, together with their operator-sum representation, and
the customary unitary description of quantum evolution. We also address
transposition as an example of positive map which is not completely positive,
and provide some examples of generalized measurements and quantum operations.
###### Contents
1. 1 Introduction
2. 2 Quantum states
1. 2.1 Density operator and partial trace
1. 2.1.1 Conditional states
2. 2.2 Purity and purification of a mixed state
3. 3 Quantum measurements
1. 3.1 Probability operator-valued measure and detection operators
2. 3.2 The Naimark theorem
1. 3.2.1 Conditional states in generalized measurements
3. 3.3 Joint measurement of non commuting observables
4. 3.4 About the so-called Heisenberg principle
5. 3.5 The quantum roulette
4. 4 Quantum operations
1. 4.1 The operator-sum representation
1. 4.1.1 The dual map and the unitary equivalence
2. 4.2 The random unitary map and the depolarizing channel
3. 4.3 Transposition and partial transposition
5. 5 Conclusions
6. A Trace and partial trace
7. B Uncertainty relations
## 1 Introduction
Quantum information science is a novel discipline which addresses how quantum
systems may be exploited to improve the processing, transmission, and storage
of information. This field has fostered new experiments and novel views on the
conceptual foundations of quantum mechanics, and also inspired much current
research on coherent quantum phenomena, with quantum optical systems playing a
prominent role. Yet, the development of quantum information had so far little
impact on the way that quantum mechanics is taught, both at graduate and
undergraduate levels. This tutorial is devoted to review the mathematical
tools of quantum mechanics and to present a modern reformulation of the basic
postulates which is suitable to describe quantum systems in interaction with
their environment, and with any kind of measuring and processing devices.
We use Dirac braket notation throughout the tutorial and by system we refer to
a single given degree of freedom (spin, position, angular momentum,…) of a
physical entity. Strictly speaking we are going to deal with systems described
by finite-dimensional Hilbert spaces and with observable quantities having a
discrete spectrum. Some of the results may be generalized to the infinite-
dimensional case and to the continuous spectrum.
The postulates of quantum mechanics are a list of prescriptions to summarize
* 1.
how we describe the states of a physical system;
* 2.
how we describe the measurements performed on a physical system;
* 3.
how we describe the evolution of a physical system, either because of the
dynamics or due to a measurement.
In this section we present a picoreview of the basic postulates of quantum
mechanics in order to introduce notation and point out both i) the implicit
assumptions contained in the standard formulation, and ii) the need of a
reformulation in terms of more general mathematical objects. For our purposes
the postulates of quantum mechanics may be grouped and summarized as follows
###### Postulate 1 (States of a quantum system)
The possible states of a physical system correspond to normalized vectors
$|\psi\rangle$, $\langle\psi|\psi\rangle=1$, of a Hilbert space $H$. Composite
systems, either made by more than one physical object or by the different
degrees of freedom of the same entity, are described by tensor product
$H_{1}\otimes H_{2}\otimes...$ of the corresponding Hilbert spaces, and the
overall state of the system is a vector in the global space. As far as the
Hilbert space description of physical systems is adopted, then we have the
superposition principle, which says that if $|\psi_{1}\rangle$ and
$|\psi_{2}\rangle$ are possible states of a system, then also any (normalized)
linear combination $\alpha|\psi_{1}\rangle+\beta|\psi_{2}\rangle$,
$\alpha,\beta\in{\mathbbm{C}}$, $|\alpha|^{2}+|\beta|^{2}=1$ of the two states
is an admissible state of the system.
###### Postulate 2 (Quantum measurements)
Observable quantities are described by Hermitian operators $X$. Any hermitian
operator $X=X^{\dagger}$, admits a spectral decomposition $X=\sum_{x}xP_{x}$,
in terms of its real eigenvalues $x$, which are the possible value of the
observable, and of the projectors $P_{x}=|x\rangle\langle x|$,
$P_{x},P_{x^{\prime}}=\delta_{xx^{\prime}}P_{x}$ on its eigenvectors
$X|x\rangle=x|x\rangle$, which form a basis for the Hilbert space, i.e. a
complete set of orthonormal states with the properties $\langle
x|x^{\prime}\rangle=\delta_{xx^{\prime}}$ (orthonormality), and
$\sum_{x}|x\rangle\langle x|=\mathbbm{I}$ (completeness, we omitted to
indicate the dimension of the Hilbert space). The probability of obtaining the
outcome $x$ from the measurement of the observable $X$ is given by
$p_{x}=\left|\langle\psi|x\rangle\right|^{2}$, i.e
$\displaystyle
p_{x}=\langle\psi|P_{x}|\psi\rangle=\sum_{n}\langle\psi|\varphi_{n}\rangle\langle\varphi_{n}|P_{x}|\psi\rangle=\sum_{n}\langle\varphi_{n}|P_{x}|\psi\rangle\langle\psi|\varphi_{n}\rangle=\hbox{Tr}\left[|\psi\rangle\langle\psi|\,P_{x}\right]\,,$
(1)
and the overall expectation value by
$\langle
X\rangle=\langle\psi|X|\psi\rangle=\hbox{Tr}\left[|\psi\rangle\langle\psi|\,X\right]\,.$
This is the Born rule, which represents the fundamental recipe to connect the
mathematical description of a quantum state to the prediction of quantum
theory about the results of an actual experiment. The state of the system
after the measurement is the (normalized) projection of the state before the
measurement on the eigenspace of the observed eigenvalue, i.e.
$|\psi_{x}\rangle=\frac{1}{\sqrt{p_{x}}}\,P_{x}|\psi\rangle\,.$
###### Postulate 3 (Dynamics of a quantum system)
The dynamical evolution of a physical system is described by unitary
operators: if $|\psi_{0}\rangle$ is the state of the system at time $t_{0}$
then the state of the system at time $t$ is given by
$|\psi_{t}\rangle=U(t,t_{0})|\psi_{0}\rangle$, with
$U(t,t_{0})U^{\dagger}(t,t_{0})=U^{\dagger}(t,t_{0})U(t,t_{0})=\mathbbm{I}$.
We will denote by $L(H)$ the linear space of (linear) operators from $H$ to
$H$, which itself is a Hilbert space with scalar product provided by the trace
operation, i.e. upon denoting by $|A\rangle\rangle$ operators seen as elements
of $L(H)$, we have $\langle\langle A|B\rangle\rangle=\hbox{Tr}[A^{\dagger}B]$
(see Appendix A for details on the trace operation).
As it is apparent from their formulation, the postulates of quantum mechanics,
as reported above, are about a closed isolated system. On the other hand, we
are mostly dealing with system that interacts or have interacted with the rest
of the universe, either during their dynamical evolution, or when subjected to
a measurement. As a consequence, one may wonder whether a suitable
modification is needed, or in order. This is indeed the case and the rest of
his tutorial is devoted to review the tools of quantum mechanics and to
present a modern reformulation of the basic postulates which is suitable to
describe, design and control quantum systems in interaction with their
environment, and with any kind of measuring and processing devices.
## 2 Quantum states
### 2.1 Density operator and partial trace
Suppose to have a quantum system whose preparation is not completely under
control. What we know is that the system is prepared in the state
$|\psi_{k}\rangle$ with probability $p_{k}$, i.e. that the system is described
by the statistical ensemble $\\{p_{k},|\psi_{k}\rangle\\}$, $\sum_{k}p_{k}=1$,
where the states $\\{|\psi_{k}\rangle\\}$ are not, in general, orthogonal. The
expected value of an observable $X$ may be evaluated as follows
$\displaystyle\langle X\rangle$ $\displaystyle=\sum_{k}p_{k}\langle
X\rangle_{k}=\sum_{k}p_{k}\langle\psi_{k}|X|\psi_{k}\rangle=\sum_{n\,p\,k}p_{k}\langle\psi_{k}|\varphi_{n}\rangle\langle\varphi_{n}|X|\varphi_{p}\rangle\langle\varphi_{p}|\psi_{k}\rangle$
$\displaystyle=\sum_{n\,p\,k}p_{k}\langle\varphi_{p}|\psi_{k}\rangle\langle\psi_{k}|\varphi_{n}\rangle\langle\varphi_{n}|X|\varphi_{p}\rangle=\sum_{n\,p}\langle\varphi_{p}|\varrho|\varphi_{n}\rangle\langle\varphi_{n}|X|\varphi_{p}\rangle$
$\displaystyle=\sum_{p}\langle\varphi_{p}|\varrho\,X|\varphi_{p}\rangle=\hbox{Tr}\left[\varrho\,X\right]\,$
where
$\varrho=\sum_{k}p_{k}\,|\psi_{k}\rangle\langle\psi_{k}|$
is the statistical (density) operator describing the system under
investigation. The $|\varphi_{n}\rangle$’s in the above formula are a basis
for the Hilbert space, and we used the trick of suitably inserting two
resolutions of the identity
$\mathbbm{I}=\sum_{n}|\varphi_{n}\rangle\langle\varphi_{n}|$. The formula is
of course trivial if the $|\psi_{k}\rangle$’s are themselves a basis or a
subset of a basis.
###### Theorem 2.1 (Density operator)
An operator $\varrho$ is the density operator associated to an ensemble
$\\{p_{k},|\psi_{k}\rangle\\}$ is and only if it is a positive $\varrho\geq 0$
(hence selfadjoint) operator with unit trace $\hbox{\rm
Tr}\left[\varrho\right]=1$.
###### Proof
: If $\varrho=\sum_{k}p_{k}|\psi_{k}\rangle\langle\psi_{k}|$ is a density
operator then $\hbox{Tr}[\varrho]=\sum_{k}p_{k}=1$ and for any vector
$|\varphi\rangle\in H$,
$\langle\varphi|\varrho|\varphi\rangle=\sum_{k}p_{k}|\langle\varphi|\psi_{k}\rangle|^{2}\geq
0$. Viceversa, if $\varrho$ is a positive operator with unit trace than it can
be diagonalized and the sum of eigenvalues is equal to one. Thus it can be
naturally associated to an ensemble. ∎
As it is true for any operator, the density operator may be expressed in terms
of its matrix elements in a given basis, i.e.
$\varrho=\sum_{np}\varrho_{np}|\varphi_{n}\rangle\langle\varphi_{p}|$ where
$\varrho_{np}=\langle\varphi_{n}|\varrho|\varphi_{p}\rangle$ is usually
referred to as the density matrix of the system. Of course, the density matrix
of a state is diagonal if we use a basis which coincides or includes the set
of eigenvectors of the density operator, otherwise it contains off-diagonal
elements.
Different ensembles may lead to the same density operator. In this case they
have the same expectation values for any operator and thus are physically
indistinguishable. In other words, different ensembles leading to the same
density operator are actually the same state, i.e. the density operator
provides the natural and most fundamental quantum description of physical
systems. How this reconciles with Postulate 1 dictating that physical systems
are described by vectors in a Hilbert space?
In order to see how it works let us first notice that, according to the
postulates reported above, the action of ”measuring nothing” should be
described by the identity operator $\mathbbm{I}$. Indeed the identity it is
Hermitian and has the single eigenvalues $1$, corresponding to the persistent
result of measuring nothing. Besides, the eigenprojector corresponding to the
eigenvalue $1$ is the projector over the whole Hilbert space and thus we have
the consistent prediction that the state after the ”measurement” is left
unchanged. Let us now consider a situation in which a bipartite system
prepared in the state $|\psi_{\scriptscriptstyle\\!AB}\rangle\rangle\in
H_{{\scriptscriptstyle A}}\otimes H_{{\scriptscriptstyle B}}$ is subjected to
the measurement of an observable $X=\sum_{x}P_{x}\in L(H_{\scriptscriptstyle
A})$, $P_{x}=|x\rangle\langle x|$ i.e. a measurement involving only the degree
of freedom described by the Hilbert space $H_{\scriptscriptstyle A}$. The
overall observable measured on the global system is thus
$\boldsymbol{X}=X\otimes\mathbbm{I}_{\scriptscriptstyle B}$, with spectral
decomposition $\boldsymbol{X}=\sum_{x}x\,\boldsymbol{Q}_{x}$,
$\boldsymbol{Q}_{x}=P_{x}\otimes\mathbbm{I}_{\scriptscriptstyle B}$. The
probability distribution of the outcomes is then obtained using the Born rule,
i.e.
$\displaystyle
p_{x}=\hbox{Tr}_{\scriptscriptstyle\\!AB}\Big{[}|\psi_{\scriptscriptstyle\\!AB}\rangle\rangle\langle\langle\psi_{\scriptscriptstyle\\!AB}|\,P_{x}\otimes\mathbbm{I}_{\scriptscriptstyle
B}\Big{]}\,.$ (2)
On the other hand, since the measurement has been performed on the sole system
$A$, one expects the Born rule to be valid also at the level of the single
system $A$, and a question arises on the form of the object
$\varrho_{\scriptscriptstyle A}$ which allows one to write
$p_{x}=\hbox{Tr}_{\scriptscriptstyle A}\left[\varrho_{\scriptscriptstyle
A}\,P_{x}\right]$ i.e. the Born rule as a trace only over the Hilbert space
$H_{\scriptscriptstyle A}$. Upon inspecting Eq. (2) one sees that a suitable
mapping
$|\psi_{\scriptscriptstyle\\!AB}\rangle\rangle\langle\langle\psi_{\scriptscriptstyle\\!AB}|\rightarrow\varrho_{\scriptscriptstyle
A}$ is provided by the partial trace $\varrho_{\scriptscriptstyle
A}=\hbox{Tr}_{\scriptscriptstyle
B}\big{[}|\psi_{\scriptscriptstyle\\!AB}\rangle\rangle\langle\langle\psi_{\scriptscriptstyle\\!AB}|\big{]}$.
Indeed, for the operator $\varrho_{\scriptscriptstyle A}$ defined as the
partial trace, we have $\hbox{Tr}_{\scriptscriptstyle
A}[\varrho_{\scriptscriptstyle
A}]=\hbox{Tr}_{\scriptscriptstyle\\!AB}\left[|\psi_{\scriptscriptstyle\\!AB}\rangle\rangle\langle\langle\psi_{\scriptscriptstyle\\!AB}|\right]=1$
and, for any vector $|\varphi\rangle\in H_{\scriptscriptstyle A}$ ,
$\langle\varphi_{\scriptscriptstyle A}|\varrho_{\scriptscriptstyle
A}|\varphi_{\scriptscriptstyle
A}\rangle=\hbox{Tr}_{\scriptscriptstyle\\!AB}\left[|\psi_{\scriptscriptstyle\\!AB}\rangle\rangle\langle\langle\psi_{\scriptscriptstyle\\!AB}|\,|\varphi_{\scriptscriptstyle
A}\rangle\langle\varphi_{\scriptscriptstyle
A}|\otimes\mathbbm{I}_{\scriptscriptstyle B}\right]\geq 0$. Being a positive,
unit trace, operator $\varrho_{\scriptscriptstyle A}$ is itself a density
operator according to Theorem 1. As a matter of fact, the partial trace is the
unique operation which allows to maintain the Born rule at both levels, i.e.
the unique operation leading to the correct description of observable
quantities for subsystems of a composite system. Let us state this as a little
theorem nie00
###### Theorem 2.2 (Partial trace)
The unique mapping
$|\psi_{\scriptscriptstyle\\!AB}\rangle\rangle\langle\langle\psi_{\scriptscriptstyle\\!AB}|\rightarrow\varrho_{\scriptscriptstyle
A}=f(\psi_{\scriptscriptstyle\\!AB})$ from $H_{\scriptscriptstyle A}\otimes
H_{\scriptscriptstyle B}$ to $H_{\scriptscriptstyle A}$ for which $\hbox{\rm
Tr}_{\scriptscriptstyle\\!AB}\left[|\psi_{\scriptscriptstyle\\!AB}\rangle\rangle\langle\langle\psi_{\scriptscriptstyle\\!AB}|\,P_{x}\otimes\mathbbm{I}_{\scriptscriptstyle
B}\right]=\hbox{\rm Tr}_{\scriptscriptstyle
A}\left[f(\psi_{\scriptscriptstyle\\!AB})\,P_{x}\right]$ is the partial trace
$f(\psi_{\scriptscriptstyle\\!AB})\equiv\varrho_{\scriptscriptstyle
A}=\hbox{\rm Tr}_{\scriptscriptstyle
B}\left[|\psi_{\scriptscriptstyle\\!AB}\rangle\rangle\langle\langle\psi_{\scriptscriptstyle\\!AB}|\right]$.
###### Proof
Basically the proof reduces to the fact that the set of operators on
$H_{\scriptscriptstyle A}$ is itself a Hilbert space $L(H_{\scriptscriptstyle
A})$ with scalar product given by $\langle\langle
A|B\rangle\rangle=\hbox{Tr}[A^{\dagger}B]$. If we consider a basis of
operators $\\{M_{k}\\}$ for $L(H_{\scriptscriptstyle A})$ and expand
$f(\psi_{\scriptscriptstyle\\!AB})=\sum_{k}M_{k}\hbox{Tr}_{\scriptscriptstyle
A}[M_{k}^{\dagger}f(\psi_{\scriptscriptstyle\\!AB})]$, then since the map $f$
has to preserve the Born rule, we have
$f(\psi_{\scriptscriptstyle\\!AB})=\sum_{k}M_{k}\hbox{Tr}_{\scriptscriptstyle
A}[M_{k}^{\dagger}\,f(\psi_{\scriptscriptstyle\\!AB})]=\sum_{k}M_{k}\hbox{Tr}_{\scriptscriptstyle\\!AB}\left[M_{k}^{\dagger}\otimes\mathbbm{I}_{\scriptscriptstyle
B}\,|\psi_{\scriptscriptstyle\\!AB}\rangle\rangle\langle\langle\psi_{\scriptscriptstyle\\!AB}|\right]\,$
and the thesis follows from the fact that in a Hilbert space the decomposition
on a basis is unique. ∎
The above result can be easily generalized to the case of a system which is
initially described by a density operator $\varrho_{\scriptscriptstyle\\!AB}$,
and thus we conclude that when we focus attention to a subsystem of a
composite larger system the unique mathematical description of the act of
ignoring part of the degrees of freedom is provided by the partial trace. It
remains to be proved that the partial trace of a density operator is a density
operator too. This is a very consequence of the definition that we put in the
form of another little theorem.
###### Theorem 2.3
The partial traces $\varrho_{\scriptscriptstyle A}=\hbox{\rm
Tr}_{\scriptscriptstyle B}[\varrho_{\scriptscriptstyle\\!AB}]$,
$\varrho_{\scriptscriptstyle B}=\hbox{\rm Tr}_{\scriptscriptstyle
A}[\varrho_{\scriptscriptstyle\\!AB}]$ of a density operator
$\varrho_{\scriptscriptstyle\\!AB}$ of a bipartite system, are themselves
density operators for the reduced systems.
###### Proof
We have $\hbox{Tr}_{\scriptscriptstyle A}[\varrho_{\scriptscriptstyle
A}]=\hbox{Tr}_{\scriptscriptstyle B}[\varrho_{\scriptscriptstyle
B}]=\hbox{Tr}_{\scriptscriptstyle\\!AB}[\varrho_{\scriptscriptstyle\\!AB}]=1$
and, for any state $|\varphi_{\scriptscriptstyle A}\rangle\in
H_{\scriptscriptstyle A}$, $|\varphi_{\scriptscriptstyle B}\rangle\in
H_{\scriptscriptstyle B}$,
$\displaystyle\langle\varphi_{\scriptscriptstyle
A}|\varrho_{\scriptscriptstyle A}|\varphi_{\scriptscriptstyle A}\rangle$
$\displaystyle=\hbox{Tr}_{\scriptscriptstyle\\!AB}\left[\varrho_{\scriptscriptstyle\\!AB}\,|\varphi_{\scriptscriptstyle
A}\rangle\langle\varphi_{\scriptscriptstyle
A}|\otimes\mathbbm{I}_{\scriptscriptstyle B}\right]\geq 0$
$\displaystyle\langle\varphi_{\scriptscriptstyle
B}|\varrho_{\scriptscriptstyle B}|\varphi_{\scriptscriptstyle B}\rangle$
$\displaystyle=\hbox{Tr}_{\scriptscriptstyle\\!AB}\left[\varrho_{\scriptscriptstyle\\!AB}\,\mathbbm{I}_{\scriptscriptstyle
A}\otimes|\varphi_{\scriptscriptstyle
B}\rangle\langle\varphi_{\scriptscriptstyle B}|\right]\geq 0\,.\quad\qed$
#### 2.1.1 Conditional states
From the above results it also follows that when we perform a measurement on
one of the two subsystems, the state of the ”unmeasured” subsystem after the
observation of a specific outcome may be obtained as the partial trace of the
overall post measurement state, i.e. the projection of the state before the
measurement on the eigenspace of the observed eigenvalue, in formula
$\displaystyle\varrho_{{\scriptscriptstyle
B}x}=\frac{1}{p_{x}}\hbox{Tr}_{\scriptscriptstyle
A}\left[P_{x}\otimes\mathbbm{I}_{\scriptscriptstyle
B}\,\varrho_{\scriptscriptstyle\\!AB}\,P_{x}\otimes\mathbbm{I}_{\scriptscriptstyle
B}\right]=\frac{1}{p_{x}}\hbox{Tr}_{\scriptscriptstyle
A}\left[\varrho_{\scriptscriptstyle\\!AB}\,P_{x}\otimes\mathbbm{I}_{\scriptscriptstyle
B}\right]\,$ (3)
where, in order to write the second equality, we made use of the circularity
of the trace (see Appendix A) and of the fact that we are dealing with a
factorized projector. The state $\varrho_{{\scriptscriptstyle B}x}$ will be
also referred to as the ”conditional state” of system $B$ after the
observation of the outcome $x$ from a measurement of the observable $X$
performed on the system $A$.
###### Exercise 1
Consider a bidimensional system (say the spin state of a spin $\frac{1}{2}$
particle) and find two ensembles corresponding to the same density operator.
###### Exercise 2
Consider a spin $\frac{1}{2}$ system and the ensemble $\\{p_{k},|\psi_{k}\\}$,
$k=0,1$, $p_{0}=p_{1}=\frac{1}{2}$, $|\psi_{0}\rangle=|0\rangle$,
$|\psi_{1}\rangle=|1\rangle$, where $|k\rangle$ are the eigenstates of
$\sigma_{3}$. Write the density matrix in the basis made of the eigenstates of
$\sigma_{3}$ and then in the basis of $\sigma_{1}$. Then, do the same but for
the ensemble obtained from the previous one by changing the probabilities to
$p_{0}=\frac{1}{4}$, $p_{1}=\frac{3}{4}$.
###### Exercise 3
Write down the partial traces of the state
$|\psi\rangle\rangle=\cos\phi\,|00\rangle\rangle+\sin\phi\,|11\rangle\rangle$,
where we used the notation $|jk\rangle\rangle=|j\rangle\otimes|k\rangle$.
### 2.2 Purity and purification of a mixed state
As we have seen in the previous section when we observe a portion, say $A$, of
a composite system described by the vector
$|\psi_{\scriptscriptstyle\\!AB}\rangle\rangle\in H_{\scriptscriptstyle
A}\otimes H_{\scriptscriptstyle B}$, the mathematical object to be inserted in
the Born rule in order to have the correct description of observable
quantities is the partial trace, which individuates a density operator on
$H_{\scriptscriptstyle A}$. Actually, also the converse is true, i.e. any
density operator on a given Hilbert space may be viewed as the partial trace
of a state vector on a larger Hilbert space. Let us prove this constructively:
if $\varrho$ is a density operator on $H$, then it can be diagonalized by its
eigenvectors and it can be written as
$\varrho=\sum_{k}\lambda_{k}|\psi_{k}\rangle\langle\psi_{k}|$; then we
introduce another Hilbert space $K$, with dimension at least equal to the
number of nonzero eqigenvalues of $\varrho$ and a basis
$\\{|\theta_{k}\rangle\\}$ in $K$, and consider the vector
$|\varphi\rangle\rangle\in H\otimes K$ given by
$|\varphi\rangle\rangle=\sum_{k}\sqrt{\lambda_{k}}\,|\psi_{k}\rangle\otimes|\theta_{k}\rangle$.
Upon tracing over the Hilbert space $K$, we have
$\hbox{Tr}_{\scriptscriptstyle
K}\left[|\varphi\rangle\rangle\langle\langle\varphi|\right]=\sum_{kk^{\prime}}\sqrt{\lambda_{k}\lambda_{k^{\prime}}}\,|\psi_{k}\rangle\langle\psi_{k^{\prime}}|\,\langle\theta_{k^{\prime}}|\theta_{k}\rangle=\sum_{k}\lambda_{k}\,|\psi_{k}\rangle\langle\psi_{k}|=\varrho\>.$
Any vector on a larger Hilbert space which satisfies the above condition is
referred to as a purification of the given density operator. Notice that, as
it is apparent from the proof, there exist infinite purifications of a density
operator. Overall, putting together this fact with the conclusions from the
previous section, we are led to reformulate the first postulate to say that
quantum states of a physical system are described by density operators, i.e.
positive operators with unit trace on the Hilbert space of the system.
A suitable measure to quantify how far a density operator is from a projector
is the so-called purity, which is defined as the trace of the square density
operator $\mu[\varrho]=\hbox{Tr}[\varrho^{2}]=\sum_{k}\lambda_{k}^{2}$, where
the $\lambda_{k}$’s are the eigenvalues of $\varrho$. Density operators made
by a projector $\varrho=|\psi\rangle\langle\psi|$ have $\mu=1$ and are
referred to as pure states, whereas for any $\mu<1$ we have a mixed state.
Purity of a state ranges in the interval $1/d\leq\mu\leq 1$ where $d$ is the
dimension of the Hilbert space. The lower bound is found looking for the
minimum of $\mu=\sum_{k}\lambda_{k}^{2}$ with the constraint
$\sum_{k}\lambda_{k}=1$, and amounts to minimize the function
$F=\mu+\gamma\sum_{k}\lambda_{k}$, $\gamma$ being a Lagrange multipliers. The
solution is $\lambda_{k}=1/d$, $\forall k$, i.e. the maximally mixed state
$\varrho=\mathbbm{I}/d$, and the corresponding purity is $\mu=1/d$.
When a system is prepared in a pure state we have the maximum possible
information on the system according to quantum mechanics. On the other hand,
for mixed states the degree of purity is connected with the amount of
information we are missing by looking at the system only, while ignoring the
environment, i.e. the rest of the universe. In fact, by looking at a portion
of a composite system we are ignoring the information encoded in the
correlations between the portion under investigation and the rest of system:
This results in a smaller amount of information about the state of the
subsystem itself. In order to emphasize this aspect, i.e. the existence of
residual ignorance about the system, the degree of mixedness may be quantified
also by the Von Neumann (VN) entropy
$S[\varrho]=-\hbox{Tr}\left[\varrho\,\log\varrho\right]=-\sum_{n}\lambda_{n}\log\lambda_{n}$,
where $\\{\lambda_{n}\\}$ are the eigenvalues of $\varrho$. We have $0\leq
S[\varrho]\leq\log d$: for a pure state $S[|\psi\rangle\langle\psi|]=0$
whereas $S[\mathbbm{I}/d]=\log d$ for a maximally mixed state. VN entropy is a
monotone function of the purity, and viceversa.
###### Exercise 4
Evaluate purity and VN entropy of the partial traces of the state
$|\psi\rangle\rangle=\cos\phi\,|01\rangle\rangle+\sin\phi\,|10\rangle\rangle$.
###### Exercise 5
Prove that for any pure bipartite state the entropies of the partial traces
are equal, though the two density operators need not to be equal.
###### Exercise 6
Take a single-qubit state with density operator expressed in terms of the
Pauli matrices
$\varrho=\frac{1}{2}(\mathbbm{I}+r_{1}\sigma_{1}+r_{2}\sigma_{2}+r_{3}\sigma_{3})$
(Bloch sphere representation), $r_{k}=\hbox{\rm Tr}[\varrho\,\sigma_{k}]$, and
prove that the Bloch vector $(r_{1},r_{2},r_{3})$ should satisfies
$r_{1}^{2}+r_{2}^{2}+r_{3}^{3}\leq 1$ for $\varrho$ to be a density operator.
## 3 Quantum measurements
In this section we put the postulates of standard quantum measurement theory
under closer scrutiny. We start with some formal considerations and end up
with a reformulation suitable for the description of any measurement performed
on a quantum system, including those involving external systems or a noisy
environment Per93 ; Bergou .
Let us start by reviewing the postulate of standard quantum measurement theory
in a pedantic way, i.e. by expanding Postulate 2; $\varrho$ denotes the state
of the system before the measurement.
* [2.1]
Any observable quantity is associated to a Hermitian operator $X$ with
spectral decomposition $X=\sum_{x}\,x\,|x\rangle\langle x|$. The eigenvalues
are real and we assume for simplicity that they are nondegenerate. A
measurement of $X$ yields one of the eigenvalues $x$ as possible outcomes.
* [2.2]
The eigenvectors of $X$ form a basis for the Hilbert space. The projectors
$P_{x}=|x\rangle\langle x|$ span the entire Hilbert space,
$\sum_{x}P_{x}=\mathbbm{I}$.
* [2.3]
The projectors $P_{x}$ are orthogonal
$P_{x}P_{x^{\prime}}=\delta_{xx^{\prime}}P_{x}$. It follows that
$P_{x}^{2}=P_{x}$ and thus that the eigenvalues of any projector are $0$ and
$1$.
* [2.4]
(Born rule) The probability that a particular outcome is found as the
measurement result is
$p_{x}=\hbox{Tr}\left[P_{x}\varrho P_{x}\right]=\hbox{Tr}\left[\varrho
P_{x}^{2}\right]\stackrel{{\scriptstyle\bigstar}}{{=}}\hbox{Tr}\left[\varrho
P_{x}\right]\,.$
* [2.5]
(Reduction rule) The state after the measurement (reduction rule or projection
postulate) is
$\varrho_{x}=\frac{1}{p_{x}}\,P_{x}\varrho P_{x}\,,$
if the outcome is $x$.
* [2.6]
If we perform a measurement but we do not record the results, the post-
measurement state is given by
$\widetilde{\varrho}=\sum_{x}p_{x}\,\varrho_{x}=\sum_{x}P_{x}\varrho P_{x}$.
The formulations [2.4] and ${\bf[2.5]}$ follow from the formulations for pure
states, upon invoking the existence of a purification:
$\displaystyle p_{x}$
$\displaystyle=\hbox{Tr}_{\scriptscriptstyle\\!AB}\left[P_{x}\otimes\mathbbm{I}_{\scriptscriptstyle
B}\,|\psi_{\scriptscriptstyle\\!AB}\rangle\rangle\langle\langle\psi_{\scriptscriptstyle\\!AB}|\,P_{x}\otimes\mathbbm{I}_{\scriptscriptstyle
B}\right]=\hbox{Tr}_{\scriptscriptstyle\\!AB}\left[|\psi_{\scriptscriptstyle\\!AB}\rangle\rangle\langle\langle\psi_{\scriptscriptstyle\\!AB}|\,P_{x}^{2}\otimes\mathbbm{I}_{\scriptscriptstyle
B}\right]$ $\displaystyle=\hbox{Tr}_{\scriptscriptstyle
A}\left[\varrho_{\scriptscriptstyle A}P_{x}^{2}\right]\,$ (4)
$\displaystyle\varrho_{{\scriptscriptstyle A}x}$
$\displaystyle=\frac{1}{p_{x}}\hbox{Tr}_{\scriptscriptstyle
B}\left[P_{x}\otimes\mathbbm{I}_{\scriptscriptstyle
B}\,|\psi_{\scriptscriptstyle\\!AB}\rangle\rangle\langle\langle\psi_{\scriptscriptstyle\\!AB}|\,P_{x}\otimes\mathbbm{I}_{\scriptscriptstyle
B}\right]=\frac{1}{p_{x}}P_{x}\,\hbox{Tr}_{\scriptscriptstyle
B}\left[|\psi_{\scriptscriptstyle\\!AB}\rangle\rangle\langle\langle\psi_{\scriptscriptstyle\\!AB}|\right]P_{x}$
$\displaystyle=\frac{1}{p_{x}}P_{x}\,\varrho_{\scriptscriptstyle A}\,P_{x}\,.$
(5)
The message conveyed by these postulates is that we can only predict the
spectrum of the possible outcomes and the probability that a given outcome is
obtained. On the other hand, the measurement process is random, and we cannot
predict the actual outcome of each run. Independently on its purity, a density
operator $\varrho$ does not describe the state of a single system, but rather
an ensemble of identically prepared systems. If we perform the same
measurement on each member of the ensemble we can predict the possible results
and the probability with which they occur but we cannot predict the result of
individual measurement (except when the probability of a certain outcome is
either $0$ or $1$).
### 3.1 Probability operator-valued measure and detection operators
The set of postulates [2.*] may be seen as a set of recipes to generate
probabilities and post-measurement states. We also notice that the number of
possible outcomes is limited by the number of terms in the orthogonal
resolution of identity, which itself cannot be larger than the dimensionality
of the Hilbert space. It would however be often desirable to have more
outcomes than the dimension of the Hilbert space while keeping positivity and
normalization of probability distributions. In this section will show that
this is formally possible, upon relaxing the assumptions on the mathematical
objects describing the measurement, and replacing them with more flexible
ones, still obtaining a meaningful prescription to generate probabilities.
Then, in the next sections we will show that there are physical processes that
fit with this generalized description, and that actually no revision of the
postulates is needed, provided that the degrees of freedom of the measurement
apparatus are taken into account.
The Born rule is a prescription to generate probabilities: its textbook form
is the right term of the starred equality in ${\bf[2.4]}$. However, the form
on the left term has the merit to underline that in order to generate a
probability it sufficient if the $P_{x}^{2}$ is a positive operator. In fact,
we do not need to require that the set of the $P_{x}$’s are projectors, nor we
need the positivity of the underlying $P_{x}$ operators. So, let us consider
the following generalization: we introduce a set of positive operators
$\Pi_{x}\geq 0$, which are the generalization of the $P_{x}$ and use the
prescription $p_{x}=\hbox{Tr}[\varrho\,\Pi_{x}]$ to generate probabilities. Of
course, we want to ensure that this is a true probability distribution, i.e.
normalized, and therefore require that $\sum_{x}\Pi_{x}=\mathbbm{I}$, that is
the positive operators still represent a resolution of the identity, as the
set of projectors over the eigenstates of a selfadjoint operator. We will call
a decomposition of the identity in terms of positive operators
$\sum_{x}\Pi_{x}=\mathbbm{I}$ a probability operator-valued measure (POVM) and
$\Pi_{x}\geq 0$ the elements of the POVM.
Let us denote the operators giving the post-measurement states (as in
${\bf[2.5]}$) by $M_{x}$. We refer to them as to the detection operators. As
noted above, they are no longer constrained to be projectors. Actually, they
may be any operator with the constraint, imposed by ${\bf[2.4]}$ i.e.
$p_{x}=\hbox{Tr}[M_{x}\varrho\,M_{x}^{\dagger}]=\hbox{Tr}[\varrho\,\Pi_{x}]$.
This tells us that the POVM elements have the form
$\Pi_{x}=M_{x}^{\dagger}M_{x}$ which, by construction, individuate a set of a
positive operators. There is a residual freedom in designing the post-
measurement state. In fact, since $\Pi_{x}$ is a positive operator
$M_{x}=\sqrt{\Pi_{x}}$ exists and satisfies the constraint, as well as any
operator of the form $M_{x}=U_{x}\,\sqrt{\Pi_{x}}$ with $U_{x}$ unitary. This
is the most general form of the detection operators satisfying the constraint
$\Pi_{x}=M_{x}^{\dagger}M_{x}$ and corresponds to their polar decomposition.
The POVM elements determine the absolute values leaving the freedom of
choosing the unitary part.
Overall, the detection operators $M_{x}$ represent a generalization of the
projectors $P_{x}$, while the POVM elements $\Pi_{x}$ generalize $P_{x}^{2}$.
The postulates for quantum measurements may be reformulated as follows
* [II.1]
Observable quantities are associated to POVMs, i.e. decompositions of identity
$\sum_{x}\Pi_{x}=\mathbbm{I}$ in terms of positive $\Pi_{x}\geq 0$ operators.
The possible outcomes $x$ label the elements of the POVM and the construction
may be generalized to the continuous spectrum.
* [II.2]
The elements of a POVM are positive operators expressible as
$\Pi_{x}=M^{\dagger}_{x}\,M_{x}$ where the detection operators $M_{x}$ are
generic operators with the only constraint
$\sum_{x}M^{\dagger}_{x}\,M_{x}=\mathbbm{I}$.
* [II.3]
(Born rule) The probability that a particular outcome is found as the
measurement result is $p_{x}=\hbox{Tr}\left[M_{x}\varrho
M_{x}^{\dagger}\right]=\hbox{Tr}\left[\varrho
M_{x}^{\dagger}M_{x}\right]=\hbox{Tr}\left[\varrho\Pi_{x}\right]$.
* [II.4]
(Reduction rule) The state after the measurement is
$\varrho_{x}=\frac{1}{p_{x}}\,M_{x}\varrho M_{x}^{\dagger}$ if the outcome is
$x$.
* [II.5]
If we perform a measurement but we do not record the results, the post-
measurement state is given by
$\widetilde{\varrho}=\sum_{x}p_{x}\,\varrho_{x}=\sum_{x}M_{x}\varrho
M_{x}^{\dagger}$.
Since orthogonality is no longer a requirement, the number of elements of a
POVM has no restrictions and so the number of possible outcomes from the
measurement. The above formulation generalizes both the Born rule and the
reduction rule, and says that any set of detection operators satisfying
${\bf[II.2]}$ corresponds to a legitimate operations leading to a proper
probability distribution and to a set of post-measurement states. This scheme
is referred to as a generalized measurement. Notice that in ${\bf[II.4]}$ we
assume a reduction mechanism sending pure states into pure states. This may be
further generalized to reduction mechanism where pure states are transformed
to mixtures, but we are not going to deal with this point.
Of course, up to this point, this is just a formal mathematical generalization
of the standard description of measurements given in textbook quantum
mechanics, and few questions naturally arise: Do generalized measurements
describe physically realizable measurements? How they can be implemented? And
if this is the case, does it means that standard formulation is too
restrictive or wrong? To all these questions an answer will be provided by the
following sections where we state and prove the Naimark Theorem, and discuss
few examples of measurements described by POVMs.
### 3.2 The Naimark theorem
The Naimark theorem basically says that any generalized measurement satisfying
[II.*] may be viewed as a standard measurement defined by [2.*] in a larger
Hilbert space, and conversely, any standard measurement involving more than
one physical system may be described as a generalized measurement on one of
the subsystems. In other words, if we focus attention on a portion of a
composite system where a standard measurement takes place, than the statistics
of the outcomes and the post-measurement states of the subsystem may be
obtained with the tools of generalized measurements. Overall, we have
###### Theorem 3.1 (Naimark)
For any given POVM $\sum_{x}\Pi_{x}=\mathbbm{I}$, $\Pi_{x}\geq 0$ on a Hilbert
space $H_{\scriptscriptstyle A}$ there exists a Hilbert space
$H_{\scriptscriptstyle B}$, a state $\varrho_{\scriptscriptstyle
B}=|\omega_{\scriptscriptstyle B}\rangle\langle\omega_{\scriptscriptstyle
B}|\in L(H_{\scriptscriptstyle B})$, a unitary operation $U\in
L(H_{\scriptscriptstyle A}\otimes H_{\scriptscriptstyle B})$,
$UU^{\dagger}=U^{\dagger}U=\mathbbm{I}$, and a projective measurement $P_{x}$,
$P_{x}P_{x}^{\prime}=\delta_{xx^{\prime}}P_{x}$ on $H_{\scriptscriptstyle B}$
such that $\Pi_{x}=\hbox{\rm Tr}_{\scriptscriptstyle
B}[\mathbbm{I}\otimes\varrho_{\scriptscriptstyle
B}\,U^{\dagger}\mathbbm{I}\otimes P_{x}\,U]$. The setup is referred to as a
Naimark extension of the POVM. Conversely, any measurement scheme where the
system is coupled to another system, from now on referred to as the ancilla,
and after evolution, a projective measurement is performed on the ancilla may
be seen as the Naimark extension of a POVM, i.e. one may write the Born rule
$p_{x}=\hbox{\rm Tr}[\varrho_{\scriptscriptstyle A}\,\Pi_{x}]$ and the
reduction rule $\varrho_{\scriptscriptstyle
A}\rightarrow\varrho_{{\scriptscriptstyle
A}x}=\frac{1}{p_{x}}M_{x}\varrho_{\scriptscriptstyle A}M_{x}^{\dagger}$ at the
level of the system only, in terms of the POVM elements $\Pi_{x}=\hbox{\rm
Tr}_{\scriptscriptstyle B}[\mathbbm{I}\otimes\varrho_{\scriptscriptstyle
B}\,U^{\dagger}\mathbbm{I}\otimes P_{x}\,U]$ and the detection operators
$M_{x}|\varphi_{\scriptscriptstyle A}\rangle=\langle
x|U|\varphi_{\scriptscriptstyle A},\omega_{\scriptscriptstyle
B}\rangle\rangle$.
Let us start with the second part of the theorem, and look at what happens
when we couple the system under investigation to an additional system, usually
referred to as ancilla (or apparatus), let them evolve, and then perform a
projective measurement on the ancilla. This kind of setup is schematically
depicted in Figure 1.
Figure 1: Schematic diagram of a generalized measurement. The system of
interest is coupled to an ancilla prepared in a known state
$|\omega_{\scriptscriptstyle B}\rangle$ by the unitary evolution $U$, and then
a projective measurement is performed on the ancilla.
The Hilbert space of the overall system is $H_{\scriptscriptstyle A}\otimes
H_{\scriptscriptstyle B}$, and we assume that the system and the ancilla are
initially independent on each other, i.e. the global initial preparation is
$R=\varrho_{\scriptscriptstyle A}\otimes\varrho_{\scriptscriptstyle B}$. We
also assume that the ancilla is prepared in the pure state
$\varrho_{\scriptscriptstyle B}=|\omega_{\scriptscriptstyle
B}\rangle\langle\omega_{\scriptscriptstyle B}|$ since this is always possible,
upon a suitable purification of the ancilla degrees of freedom, i.e. by
suitably enlarging the ancilla Hilbert space. Our aim it to obtain information
about the system by measuring an observable $X$ on the ancilla. This is done
after the system-ancilla interaction described by the unitary operation $U$.
According to the Born rule the probability of the outcomes is given by
$p_{x}=\hbox{Tr}_{\scriptscriptstyle\\!AB}\left[U\varrho_{\scriptscriptstyle
A}\otimes\varrho_{\scriptscriptstyle
B}U^{\dagger}\,\mathbbm{I}\otimes|x\rangle\langle
x|\right]=\hbox{Tr}_{\scriptscriptstyle A}\left[\varrho_{\scriptscriptstyle
A}\,\underbrace{\hbox{Tr}_{\scriptscriptstyle
B}\left[\mathbbm{I}\otimes\varrho_{\scriptscriptstyle
B}\,U^{\dagger}\,\mathbbm{I}\otimes|x\rangle\langle
x|U\right]}\right]\vspace{-2mm}$
${\;\Pi_{x}}$
where the set of operators $\Pi_{x}=\hbox{Tr}_{\scriptscriptstyle
B}\left[\mathbbm{I}\otimes\varrho_{\scriptscriptstyle
B}\,U^{\dagger}\,\mathbbm{I}\otimes|x\rangle\langle
x|U\right]=\langle\omega_{\scriptscriptstyle B}|U^{\dagger}\mathbbm{I}\otimes
P_{x}U|\omega_{\scriptscriptstyle B}\rangle$ is the object that would permit
to write the Born rule at the level of the subsystem $A$, i.e. it is our
candidate POVM.
In order to prove this, let us define the operators $M_{x}\in
L(H_{\scriptscriptstyle A})$ by their action on the generic vector in
$H_{\scriptscriptstyle A}$
$M_{x}|\varphi_{\scriptscriptstyle A}\rangle=\langle
x|U|\varphi_{\scriptscriptstyle A},\omega_{\scriptscriptstyle
B}\rangle\rangle$
where $|\varphi_{\scriptscriptstyle A},\omega_{\scriptscriptstyle
B}\rangle\rangle=|\varphi_{\scriptscriptstyle
A}\rangle\otimes|\omega_{\scriptscriptstyle B}\rangle$ and the $|x\rangle$’s
are the orthogonal eigenvectors of $X$. Using the decomposition of
$\varrho_{\scriptscriptstyle
A}=\sum_{k}\lambda_{k}|\psi_{k}\rangle\langle\psi_{k}|$ onto its eigenvectors
the probability of the outcomes can be rewritten as
$\displaystyle p_{x}$
$\displaystyle=\hbox{Tr}_{\scriptscriptstyle\\!AB}\left[U\varrho_{\scriptscriptstyle
A}\otimes\varrho_{\scriptscriptstyle
B}U^{\dagger}\,\mathbbm{I}\otimes|x\rangle\langle
x|\right]=\sum_{k}\lambda_{k}\hbox{Tr}_{\scriptscriptstyle\\!AB}\left[U|\psi_{k},\omega_{\scriptscriptstyle
B}\rangle\rangle\langle\langle\omega_{\scriptscriptstyle
B},\psi_{k}|U^{\dagger}\,\mathbbm{I}\otimes|x\rangle\langle x|\right]$
$\displaystyle=\sum_{k}\lambda_{k}\hbox{Tr}_{\scriptscriptstyle
A}\left[\langle x|U|\psi_{k},\omega_{\scriptscriptstyle
B}\rangle\rangle\langle\langle\omega_{\scriptscriptstyle
B},\psi_{k}|U^{\dagger}|x\rangle\right]=\sum_{k}\lambda_{k}\hbox{Tr}_{\scriptscriptstyle
A}\left[M_{x}|\psi_{k}\rangle\langle\psi_{k}|M_{x}^{\dagger}\right]$
$\displaystyle=\hbox{Tr}_{\scriptscriptstyle
A}\left[M_{x}\varrho_{\scriptscriptstyle
A}M_{x}^{\dagger}\right]=\hbox{Tr}_{\scriptscriptstyle
A}\left[\varrho_{\scriptscriptstyle A}\,M_{x}^{\dagger}M_{x}\right]\,,$ (6)
which shows that $\Pi_{x}=M_{x}^{\dagger}M_{x}$ is indeed a positive operator
$\forall x$. Besides, for any vector $|\varphi_{\scriptscriptstyle A}\rangle$
in $H_{\scriptscriptstyle A}$ we have
$\displaystyle\langle\varphi_{\scriptscriptstyle
A}|\sum_{x}M^{\dagger}_{x}M_{x}|\varphi_{\scriptscriptstyle A}\rangle$
$\displaystyle=\sum_{x}\langle\langle\omega_{\scriptscriptstyle
B},\varphi_{\scriptscriptstyle A}|U^{\dagger}|x\rangle\langle
x|U|\varphi_{\scriptscriptstyle A},\omega_{\scriptscriptstyle
B}\rangle\rangle$ $\displaystyle=\langle\langle\omega_{\scriptscriptstyle
B},\varphi_{\scriptscriptstyle A}|U^{\dagger}U|\varphi_{\scriptscriptstyle
A},\omega_{\scriptscriptstyle B}\rangle\rangle=1\,,$ (7)
and since this is true for any $|\varphi_{\scriptscriptstyle A}\rangle$ we
have $\sum_{x}M_{x}^{\dagger}M_{x}=\mathbbm{I}$. Putting together Eqs. (6) and
(7) we have that the set of operators $\Pi_{x}=M^{\dagger}_{x}M_{x}$ is a
POVM, with detection operators $M_{x}$. In turn, the conditional state of the
system $A$, after having observed the outcome $x$, is given by
$\displaystyle\varrho_{{\scriptscriptstyle A}x}$
$\displaystyle=\frac{1}{p_{x}}\hbox{Tr}_{\scriptscriptstyle
B}\left[U\varrho_{\scriptscriptstyle A}\otimes|\omega_{\scriptscriptstyle
B}\rangle\langle\omega_{\scriptscriptstyle B}|U^{\dagger}\,\mathbbm{I}\otimes
P_{x}\right]=\frac{1}{p_{x}}\sum_{k}\lambda_{k}\langle
x|U|\psi_{k},\omega_{\scriptscriptstyle
B}\rangle\rangle\langle\langle\omega_{\scriptscriptstyle
B},\psi_{k}|U^{\dagger}|x\rangle$
$\displaystyle=\frac{1}{p_{x}}M_{x}\varrho_{\scriptscriptstyle
A}M_{x}^{\dagger}$ (8)
This is the half of the Naimark theorem: if we couple our system to an
ancilla, let them evolve and perform the measurement of an observable on the
ancilla, which projects the ancilla on a basis in $H_{\scriptscriptstyle B}$,
then this procedure also modify the system. The transformation needs not to be
a projection. Rather, it is adequately described by a set of detection
operators which realizes a POVM on the system Hilbert space. Overall, the
meaning of the above proof is twofold: on the one hand we have shown that
there exists realistic measurement schemes which are described by POVMs when
we look at the system only. At the same time, we have shown that the partial
trace of a spectral measure is a POVM, which itself depends on the projective
measurement performed on the ancilla, and on its initial preparation. Finally,
we notice that the scheme of Figure 1 provides a general model for any kind of
detector with internal degrees of freedom.
Let us now address the converse problem: given a set of detection operators
$M_{x}$ which realizes a POVM $\sum_{x}M^{\dagger}_{x}M_{x}=\mathbbm{I}$, is
this the system-only description of an indirect measurement performed a larger
Hilbert space? In other words, there exists a Hilbert space
$H_{\scriptscriptstyle B}$, a state $\varrho_{\scriptscriptstyle
B}=|\omega_{\scriptscriptstyle B}\rangle\langle\omega_{\scriptscriptstyle
B}|\in L(H_{\scriptscriptstyle B})$, a unitary $U\in L(H_{\scriptscriptstyle
A}\otimes H_{\scriptscriptstyle B})$, and a projective measurement
$P_{x}=|x\rangle\langle x|$ in $H_{\scriptscriptstyle B}$ such that
$M_{x}|\varphi_{\scriptscriptstyle A}\rangle=\langle
x|U|\varphi_{\scriptscriptstyle A},\omega_{\scriptscriptstyle
B}\rangle\rangle$ holds for any $|\varphi_{\scriptscriptstyle A}\rangle\in
H_{\scriptscriptstyle A}$ and $\Pi_{x}=\langle\omega_{\scriptscriptstyle
B}|U^{\dagger}\mathbbm{I}\otimes P_{x}U|\omega_{\scriptscriptstyle B}\rangle$?
The answer is positive and we will provide a constructive proof. Let us take
$H_{\scriptscriptstyle B}$ with dimension equal to the number of detection
operators and of POVM elements, and choose a basis $|x\rangle$ for
$H_{\scriptscriptstyle B}$, which in turn individuates a projective
measurement. Then we choose an arbitrary state $|\omega_{\scriptscriptstyle
B}\rangle\in H_{\scriptscriptstyle B}$ and define the action of an operator U
as
$U\,|\varphi_{\scriptscriptstyle A}\rangle\otimes|\omega_{\scriptscriptstyle
B}\rangle=\sum_{x}M_{x}\,|\varphi_{\scriptscriptstyle
A}\rangle\otimes|x\rangle$
where $|\varphi_{\scriptscriptstyle A}\rangle\in H_{\scriptscriptstyle A}$ is
arbitrary. The operator $U$ preserves the scalar product
$\displaystyle\langle\langle\omega_{\scriptscriptstyle
B},\varphi_{\scriptscriptstyle
A}^{\prime}|U^{\dagger}U|\varphi_{\scriptscriptstyle
A},\omega_{\scriptscriptstyle
B}\rangle\rangle=\sum_{xx^{\prime}}\langle\varphi_{\scriptscriptstyle
A}^{\prime}|M_{x^{\prime}}^{\dagger}M_{x}|\varphi_{\scriptscriptstyle
A}\rangle\langle
x^{\prime}|x\rangle=\sum_{x}\langle\varphi_{\scriptscriptstyle
A}^{\prime}|M_{x^{\prime}}^{\dagger}M_{x}|\varphi_{\scriptscriptstyle
A}\rangle=\langle\varphi_{\scriptscriptstyle
A}^{\prime}|\varphi_{\scriptscriptstyle A}\rangle$
and so it is unitary in the one-dimensional subspace spanned by
$|\omega_{\scriptscriptstyle B}\rangle$. Besides, it may be extended to a full
unitary operator in the global Hilbert space $H_{\scriptscriptstyle A}\otimes
H_{\scriptscriptstyle B}$, eg it can be the identity operator in the subspace
orthogonal to $|\omega_{\scriptscriptstyle B}\rangle$. Finally, for any
$|\varphi_{\scriptscriptstyle A}\rangle\in H_{\scriptscriptstyle A}$, we have
$\langle x|U|\varphi_{\scriptscriptstyle A},\omega_{\scriptscriptstyle
B}\rangle\rangle=\sum_{x^{\prime}}M_{x^{\prime}}|\varphi_{\scriptscriptstyle
A}\rangle\langle x|x^{\prime}\rangle=M_{x}|\varphi_{\scriptscriptstyle
A}\rangle\,,$
and
$\langle\varphi_{\scriptscriptstyle A}|\Pi_{x}|\varphi_{\scriptscriptstyle
A}\rangle=\langle\varphi_{\scriptscriptstyle
A}|M_{x}^{\dagger}M_{x}|\varphi_{\scriptscriptstyle
A}\rangle=\langle\langle\omega_{\scriptscriptstyle
B},\varphi_{\scriptscriptstyle A}|U^{\dagger}\mathbbm{I}\otimes
P_{x}U|\varphi_{\scriptscriptstyle A},\omega_{\scriptscriptstyle
B}\rangle\rangle\,,$
that is, $\Pi_{x}=\langle\omega_{\scriptscriptstyle
B}|U^{\dagger}\mathbbm{I}\otimes P_{x}U|\omega_{\scriptscriptstyle B}\rangle$.
∎
This completes the proof of the Naimark theorem, which asserts that there is a
one-to-one correspondence between POVM and indirect measurements of the type
describe above. In other words, an indirect measurement may be seen as the
physical implementation of a POVM and any POVM may be realized by an indirect
measurement.
The emerging picture is thus the following: In measuring a quantity of
interest on a physical system one generally deals with a larger system that
involves additional degrees of freedom, besides those of the system itself.
These additional physical entities are globally referred to as the apparatus
or the ancilla. As a matter of fact, the measured quantity may be always
described by a standard observable, however on a larger Hilbert space
describing both the system and the apparatus. When we trace out the degrees of
freedom of the apparatus we are generally left with a POVM rather than a PVM.
Conversely, any conceivable POVM, i.e. a set of positive operators providing a
resolution of identity, describe a generalized measurement, which may be
always implemented as a standard measurement in a larger Hilbert space.
Before ending this Section, few remarks are in order:
* R1
The possible Naimark extensions are actually infinite, corresponding to the
intuitive idea that there are infinite ways, with an arbitrary number of
ancillary systems, of measuring a given quantity. The construction reported
above is sometimes referred to as the canonical extension of a POVM. The
Naimark theorem just says that an implementation in terms of an ancilla-based
indirect measurement is always possible, but of course the actual
implementation may be different from the canonical one.
* R2
The projection postulate described at the beginning of this section, the
scheme of indirect measurement, and the canonical extension of a POVM have in
common the assumption that a nondemolitive detection scheme takes place, in
which the system after the measurement has been modified, but still exists.
This is sometimes referred to as a measurement of the first kind in textbook
quantum mechanics. Conversely, in a demolitive measurement or measurement of
the second kind, the system is destroyed during the measurement and it makes
no sense of speaking of the state of the system after the measurement. Notice,
however, that for demolitive measurements on a field the formalism of
generalized measurements provides the framework for the correct description of
the state evolution. As for example, let us consider the detection of photons
on a single-mode of the radiation field. A demolitive photodetector (as those
based on the absorption of light) realizes, in ideal condition, the
measurement of the number operator $a^{\dagger}a$ without leaving any photon
in the mode . If $\varrho=\sum_{np}\varrho_{np}|n\rangle\langle p|$ is the
state of the single-mode radiation field a photodetector of this kind gives a
natural number $n$ as output, with probability $p_{n}=\varrho_{nn}$, whereas
the post-measurement state is the vacuum $|0\rangle\langle 0|$ independently
on the outcome of the measurement. This kind of measurement is described by
the orthogonal POVM $\Pi_{n}=|n\rangle\langle n|$, made by the eigenvectors of
the number operator, and by the detection operator $M_{n}=|0\rangle\langle
n|$. The proof is left as an exercise.
* R3
We have formulated and proved the Naimark theorem in a restricted form,
suitable for our purposes. It should be noticed that it holds in more general
terms, as for example with extension of the Hilbert space given by direct sum
rather than tensor product, and also relaxing the hypothesis Pau .
#### 3.2.1 Conditional states in generalized measurements
If we have a composite system and we perform a projective measurement on, say,
subsystem $A$, the conditional state of the unmeasured subsystem $B$ after the
observation of the outcome $x$ is given by Eq. (3), i.e. it is the partial
trace of the projection of the state before the measurement on the eigenspace
of the observed eigenvalue. One may wonder whether a similar results holds
also when the measurement performed on the subsystem a $A$ is described by a
POVM. The answer is positive and the proof may be given in two ways. The first
is based on the observation that, thanks to the existence of a canonical
Naimark extension, we may write the state of the global system after the
measurement as
$\varrho_{{\scriptscriptstyle\\!AB}x}=\frac{1}{p_{x}}M_{x}\otimes\mathbbm{I}_{\scriptscriptstyle
B}\,\varrho_{\scriptscriptstyle\\!AB}\,M_{x}^{\dagger}\otimes\mathbbm{I}_{\scriptscriptstyle
B}\,,$
and thus the conditional state of subsystem $B$ is the partial trace
$\varrho_{{\scriptscriptstyle B}x}=\hbox{Tr}_{\scriptscriptstyle
A}[\varrho_{{\scriptscriptstyle\\!AB}x}]$ i.e.
$\varrho_{{\scriptscriptstyle
B}x}=\frac{1}{p_{x}}\hbox{Tr}_{\scriptscriptstyle
A}[M_{x}\otimes\mathbbm{I}_{\scriptscriptstyle
B}\,\varrho_{\scriptscriptstyle\\!AB}\,M_{x}^{\dagger}\otimes\mathbbm{I}_{\scriptscriptstyle
B}]=\frac{1}{p_{x}}\hbox{Tr}_{\scriptscriptstyle
A}[\varrho_{\scriptscriptstyle\\!AB}\,M_{x}^{\dagger}M_{x}\otimes\mathbbm{I}_{\scriptscriptstyle
B}]=\frac{1}{p_{x}}\hbox{Tr}_{\scriptscriptstyle
A}[\varrho_{\scriptscriptstyle\\!AB}\,\Pi_{x}\otimes\mathbbm{I}_{\scriptscriptstyle
B}]\,,$
where again we used the circularity of partial trace in the presence of
factorized operators. A second proof may be offered invoking the Naimark
theorem only to ensure the existence of an extension, i.e. a projective
measurement on a larger Hilbert space $H_{\scriptscriptstyle C}\otimes\
H_{\scriptscriptstyle A}$, which reduces to the POVM after tracing over
$H_{\scriptscriptstyle C}$. In formula, assuming that $P_{x}\in
L(H_{\scriptscriptstyle C}\otimes\ H_{\scriptscriptstyle A})$ is a projector
and $\sigma\in L(H_{\scriptscriptstyle C})$ a density operator
$\displaystyle\varrho_{{\scriptscriptstyle B}x}$
$\displaystyle=\frac{1}{p_{x}}\hbox{Tr}_{{\scriptscriptstyle
C}{\scriptscriptstyle A}}\left[P_{x}\otimes\mathbbm{I}_{\scriptscriptstyle
B}\,\varrho_{\scriptscriptstyle\\!AB}\otimes\sigma\,P_{x}\otimes\mathbbm{I}_{\scriptscriptstyle
B}\right]=\frac{1}{p_{x}}\hbox{Tr}_{{\scriptscriptstyle C}{\scriptscriptstyle
A}}\left[\varrho_{\scriptscriptstyle\\!AB}\otimes\sigma\,P_{x}\otimes\mathbbm{I}_{\scriptscriptstyle
B}\right]$ $\displaystyle=\frac{1}{p_{x}}\hbox{Tr}_{{\scriptscriptstyle
A}}\left[\varrho_{\scriptscriptstyle\\!AB}\Pi_{x}\otimes\mathbbm{I}_{\scriptscriptstyle
B}\right]\,.$
### 3.3 Joint measurement of non commuting observables
A common statement about quantum measurements says that it is not possible to
perform a joint measurement of two observables $Q_{\scriptscriptstyle A}$ and
$P_{\scriptscriptstyle A}$ of a given system $A$ if they do not commute, i.e.
$[Q_{\scriptscriptstyle A},P_{\scriptscriptstyle A}]\neq 0$. This is related
to the impossibility of finding any common set of projectors on the Hilbert
space $H_{\scriptscriptstyle A}$ of the system and to define a joint
observable. On the other hand, a question arises on whether common projectors
may be found in a larger Hilbert space, i.e. whether one may implement a joint
measurement in the form of a generalized measurement. The answer is indeed
positive art1 ; yue82 : This Section is devoted to describe the canonical
implementation of joint measurements for pair of observables having a
(nonzero) commutator $[Q_{\scriptscriptstyle A},P_{\scriptscriptstyle
A}]=c\,\mathbbm{I}\neq 0$ proportional to the identity operator.
The basic idea is to look for a pair of commuting observables
$[X_{\scriptscriptstyle\\!AB},Y_{\scriptscriptstyle\\!AB}]=0$ in a larger
Hilbert space $H_{\scriptscriptstyle A}\otimes H_{\scriptscriptstyle B}$ which
trace the observables $P_{\scriptscriptstyle A}$ and $Q_{\scriptscriptstyle
A}$, i.e. which have the same expectation values
$\displaystyle\langle
X_{\scriptscriptstyle\\!AB}\rangle\equiv\hbox{Tr}_{\scriptscriptstyle\\!AB}[X_{\scriptscriptstyle\\!AB}\,\varrho_{\scriptscriptstyle
A}\otimes\varrho_{\scriptscriptstyle B}]$
$\displaystyle=\hbox{Tr}_{\scriptscriptstyle A}[Q_{\scriptscriptstyle
A}\,\varrho_{\scriptscriptstyle A}]\equiv\langle Q_{\scriptscriptstyle
A}\rangle$ $\displaystyle\langle
Y_{\scriptscriptstyle\\!AB}\rangle\equiv\hbox{Tr}_{\scriptscriptstyle\\!AB}[Y_{\scriptscriptstyle\\!AB}\,\varrho_{\scriptscriptstyle
A}\otimes\varrho_{\scriptscriptstyle B}]$
$\displaystyle=\hbox{Tr}_{\scriptscriptstyle A}[P_{\scriptscriptstyle
A}\,\varrho_{\scriptscriptstyle A}]\equiv\langle P_{\scriptscriptstyle
A}\rangle$ (9)
for any state $\varrho_{\scriptscriptstyle A}\in H_{\scriptscriptstyle A}$ of
the system under investigation, and a fixed suitable preparation
$\varrho_{\scriptscriptstyle B}\in H_{\scriptscriptstyle B}$ of the system
$B$. A pair of such observables may be found upon choosing a replica system
$B$, identical to $A$, and considering the operators
$\displaystyle X_{\scriptscriptstyle\\!AB}$
$\displaystyle=Q_{\scriptscriptstyle A}\otimes\mathbbm{I}_{\scriptscriptstyle
B}+\mathbbm{I}_{\scriptscriptstyle A}\otimes Q_{\scriptscriptstyle B}$
$\displaystyle Y_{\scriptscriptstyle\\!AB}$
$\displaystyle=P_{\scriptscriptstyle A}\otimes\mathbbm{I}_{\scriptscriptstyle
B}-\mathbbm{I}_{\scriptscriptstyle A}\otimes P_{\scriptscriptstyle B}$ (10)
where $Q_{\scriptscriptstyle B}$ and $P_{\scriptscriptstyle B}$ are the
analogue of $Q_{\scriptscriptstyle A}$ and $P_{\scriptscriptstyle A}$ for
system $B$, see BV10 for more details involving the requirement of
covariance. The operators in Eq. (10), taken together a state
$\varrho_{\scriptscriptstyle B}\in H_{\scriptscriptstyle B}$ satisfying
$\displaystyle\hbox{Tr}_{\scriptscriptstyle B}[Q_{\scriptscriptstyle
B}\,\varrho_{\scriptscriptstyle B}]=\hbox{Tr}_{\scriptscriptstyle
B}[P_{\scriptscriptstyle B}\,\varrho_{\scriptscriptstyle B}]=0\,,$ (11)
fulfill the conditions in Eq. (9), i.e. realize a joint generalized
measurement of the noncommuting observables $Q_{\scriptscriptstyle A}$ and
$P_{\scriptscriptstyle A}$. The operators $X_{\scriptscriptstyle\\!AB}$ and
$Y_{\scriptscriptstyle\\!AB}$ are Hermitian by construction. Their commutator
is given by
$\displaystyle[X_{\scriptscriptstyle\\!AB},Y_{\scriptscriptstyle\\!AB}]=[Q_{\scriptscriptstyle
A},P_{\scriptscriptstyle A}]\otimes\mathbbm{I}_{\scriptscriptstyle
B}-\mathbbm{I}_{\scriptscriptstyle A}\otimes[Q_{\scriptscriptstyle
B},P_{\scriptscriptstyle B}]=0\,.$ (12)
Notice that the last equality, i.e. the fact that the two operators commute,
is valid only if the commutator $[Q_{\scriptscriptstyle
A},P_{\scriptscriptstyle A}]=c\,\mathbbm{I}$ is proportional to the identity.
More general constructions are needed if this condition does not hold jsp1 .
Since the $[X_{\scriptscriptstyle\\!AB},Y_{\scriptscriptstyle\\!AB}]=0$ the
complex operator
$Z_{\scriptscriptstyle\\!AB}=X_{\scriptscriptstyle\\!AB}+i\,Y_{\scriptscriptstyle\\!AB}$
is normal i.e.
$[Z_{\scriptscriptstyle\\!AB},Z_{\scriptscriptstyle\\!AB}^{\dagger}]=0$. For
normal operators the spectral theorem holds, and we may write
$\displaystyle Z_{\scriptscriptstyle\\!AB}=\sum_{z}z\,P_{z}\qquad
P_{z}=|z\rangle\\!\rangle\langle\\!\langle z|\qquad
Z_{\scriptscriptstyle\\!AB}|z\rangle\\!\rangle=z|z\rangle\\!\rangle$ (13)
where $z\in{\mathbbm{C}}$, and $P_{z}$ are orthogonal projectors on the
eigenstates
$|z\rangle\\!\rangle\equiv|z\rangle\\!\rangle_{\scriptscriptstyle\\!AB}$ of
$Z_{\scriptscriptstyle\\!AB}$. The set $\\{P_{z}\\}$ represents the common
projectors individuating the joint observable $Z_{\scriptscriptstyle\\!AB}$.
Each run of the measurement returns a complex number, whose real and imaginary
parts correspond to a sample of the $X_{\scriptscriptstyle\\!AB}$ and
$Y_{\scriptscriptstyle\\!AB}$ values, aiming at sampling
$Q_{\scriptscriptstyle A}$ and $P_{\scriptscriptstyle A}$. The statistics of
the measurement is given by
$\displaystyle p_{\scriptscriptstyle
Z}(z)=\hbox{Tr}_{\scriptscriptstyle\\!AB}[\varrho_{\scriptscriptstyle
A}\otimes\varrho_{\scriptscriptstyle B}\,P_{z}]=\hbox{Tr}_{\scriptscriptstyle
A}[\varrho_{\scriptscriptstyle A}\,\Pi_{z}]$ (14)
where the POVM $\Pi_{z}$ is given by
$\displaystyle\Pi_{z}=\hbox{Tr}_{\scriptscriptstyle
B}[\mathbbm{I}_{\scriptscriptstyle A}\otimes\varrho_{\scriptscriptstyle
B}\,P_{z}]\,.$ (15)
The mean values $\langle X_{\scriptscriptstyle\\!AB}\rangle=\langle
Q_{\scriptscriptstyle A}\rangle$ and $\langle
Y_{\scriptscriptstyle\\!AB}\rangle=\langle P_{\scriptscriptstyle A}\rangle$
are the correct ones by construction, where by saying ”correct” we intend the
mean values that one would have recorded by measuring the two observables
$Q_{\scriptscriptstyle A}$ and $P_{\scriptscriptstyle A}$ separately in a
standard (single) projective measurement on $\varrho_{\scriptscriptstyle A}$.
On the other hand, the two marginal distributions
$p_{\scriptscriptstyle X}(x)=\int\\!dy\,p_{\scriptscriptstyle Z}(x+iy)\qquad
p_{\scriptscriptstyle Y}(y)=\int\\!dx\,p_{\scriptscriptstyle Z}(x+iy)\,,$
need not to reproduce the distributions obtained in single measurements. In
particular, for the measured variances $\langle\Delta
X_{\scriptscriptstyle\\!AB}^{2}\rangle=\langle
X_{\scriptscriptstyle\\!AB}^{2}\rangle-\langle
X_{\scriptscriptstyle\\!AB}\rangle^{2}$ and $\langle\Delta
Y_{\scriptscriptstyle\\!AB}\rangle$ one obtains
$\displaystyle\langle\Delta X_{\scriptscriptstyle\\!AB}^{2}\rangle$
$\displaystyle=\hbox{\rm Tr}\left[(Q_{\scriptscriptstyle
A}^{2}\otimes\mathbbm{I}_{\scriptscriptstyle
B}+\mathbbm{I}_{\scriptscriptstyle A}\otimes Q_{\scriptscriptstyle
B}^{2}+2\,Q_{\scriptscriptstyle A}\otimes Q_{\scriptscriptstyle
B})\,\varrho_{\scriptscriptstyle A}\otimes\varrho_{\scriptscriptstyle
B}\right]-\langle Q_{\scriptscriptstyle A}\rangle^{2}$
$\displaystyle=\langle\Delta Q_{\scriptscriptstyle A}^{2}\rangle+\langle
Q_{\scriptscriptstyle B}^{2}\rangle$ $\displaystyle\langle\Delta
Y_{\scriptscriptstyle\\!AB}^{2}\rangle$ $\displaystyle=\langle\Delta
P_{\scriptscriptstyle A}^{2}\rangle+\langle P_{\scriptscriptstyle
B}^{2}\rangle\,$ (16)
where we have already taken into account that $\langle Q_{\scriptscriptstyle
B}\rangle=\langle P_{\scriptscriptstyle B}\rangle=0$. As it is apparent from
Eqs. (16) the variances of $X_{\scriptscriptstyle\\!AB}$ and
$Y_{\scriptscriptstyle\\!AB}$ are larger than those of the original, non
commuting, observables $Q_{\scriptscriptstyle A}$ and $P_{\scriptscriptstyle
A}$.
Overall, we may summarize the emerging picture as follows: a joint measurement
of a pair of non commuting observables corresponds to a generalized
measurement and may be implemented as the measurement of a pair of commuting
observables on an enlarged Hilbert space. Mean values are preserved whereas
the non commuting nature of the original observables manifests itself in the
broadening of the marginal distributions, i.e. as an additional noise term
appears to both the variances. The uncertainty product may be written as
$\displaystyle\langle\Delta
X_{\scriptscriptstyle\\!AB}^{2}\rangle\langle\Delta
Y_{\scriptscriptstyle\\!AB}^{2}\rangle$ $\displaystyle=\langle\Delta
Q_{\scriptscriptstyle A}^{2}\rangle\langle\Delta P_{\scriptscriptstyle
A}^{2}\rangle+\langle\Delta Q_{\scriptscriptstyle A}^{2}\rangle\langle
P_{\scriptscriptstyle B}^{2}\rangle+\langle Q_{\scriptscriptstyle
B}^{2}\rangle\langle\Delta P_{\scriptscriptstyle A}^{2}\rangle+\langle
Q_{\scriptscriptstyle B}^{2}\rangle\langle P_{\scriptscriptstyle
B}^{2}\rangle\,,$ $\displaystyle\geq\frac{1}{4}\big{|}[Q_{\scriptscriptstyle
A},P_{\scriptscriptstyle A}]\big{|}^{2}+\langle\Delta Q_{\scriptscriptstyle
A}^{2}\rangle\langle P_{\scriptscriptstyle B}^{2}\rangle+\langle
Q_{\scriptscriptstyle B}^{2}\rangle\langle\Delta P_{\scriptscriptstyle
A}^{2}\rangle+\langle Q_{\scriptscriptstyle B}^{2}\rangle\langle
P_{\scriptscriptstyle B}^{2}\rangle\,,$ (17)
where the last three terms are usually referred to as the added noise due to
the joint measurement. If we perform a joint measurement on a minimum
uncertainty state (MUS, see Appendix B) for a given pair of observables (e.g.
a coherent state in the joint measurement of a pair of conjugated quadratures
of the radiation field) and use a MUS also for the preparation of the replica
system (e.g. the vacuum), then Eq. (17) rewrites as
$\displaystyle\langle\Delta
X_{\scriptscriptstyle\\!AB}^{2}\rangle\langle\Delta
Y_{\scriptscriptstyle\\!AB}^{2}\rangle=\big{|}[Q_{\scriptscriptstyle
A},P_{\scriptscriptstyle A}]\big{|}^{2}\,.$ (18)
This is four times the minimum attainable uncertainty product in the case of a
measurement of a single observable (see Appendix B). In terms of rms’ $\Delta
X=\sqrt{\langle\Delta X^{2}\rangle}$ we have a factor $2$, which is usually
referred to as the $3$ dB of added noise in joint measurements. The
experimental realization of joint measurements of non commuting observables
has been carried out for conjugated quadratures of the radiation field in a
wide range of frequencies ranging from radiowaves to the optical domain, see
e.g. wal .
### 3.4 About the so-called Heisenberg principle
Let us start by quoting Wikipedia about the Heisenberg principle wikiHP
> Published by Werner Heisenberg in 1927, the principle implies that it is
> impossible to simultaneously both measure the present position while
> ”determining” the future momentum of an electron or any other particle with
> an arbitrary degree of accuracy and certainty. This is not a statement about
> researchers’ ability to measure one quantity while determining the other
> quantity. Rather, it is a statement about the laws of physics. That is, a
> system cannot be defined to simultaneously measure one value while
> determining the future value of these pairs of quantities. The principle
> states that a minimum exists for the product of the uncertainties in these
> properties that is equal to or greater than one half of the reduced Planck
> constant.
As is it apparent from the above formulation, the principle is about the
precision achievable in the measurement of an observable and the disturbance
introduced by the same measurement on the state under investigation, which, in
turn, would limit the precision of a subsequent measurement of the conjugated
observable. The principle, which has been quite useful in the historical
development of quantum mechanics, has been inferred from the analysis of the
celebrated Heisenberg’ gedanken experiments, and thus is heuristic in nature.
However, since its mathematical formulation is related to that of the
uncertainty relations (see Appendix B), it is often though as a theorem
following from the axiomatic structure of quantum mechanics. This is not the
case: here we exploit the formalism of generalized measurements to provide an
explicit example of a measurement scheme providing the maximum information
about a given observable, i.e. the statistics of the corresponding PVM, while
leaving the state under investigation in an eigenstate of the conjugated
observable.
Let us consider the two noncommuting observables $[A,B]=c\,\mathbbm{I}$ and
the set of detection operators $M_{a}=|b\rangle\langle a|$ where $|a\rangle$
and $|b\rangle$ are eigenstates of $A$ and $B$ respectively, i.e.
$A|a\rangle=a|a\rangle$, $B|b\rangle=b|b\rangle$. According to the Naimark
theorem the set of operators $\\{M_{a}\\}$ describe a generalized measurement
(e.g. an indirect measurement as the one depicted in Fig. 1) with statistics
$p_{a}=\hbox{\rm Tr}[\varrho\,\Pi_{a}]$ described by the POVM
$\Pi_{a}=M^{\dagger}_{a}M_{a}=|a\rangle\langle a|$ and where the conditional
states after the measurement are given by
$\varrho_{a}=\frac{1}{p_{a}}M_{a}\varrho M_{a}^{\dagger}=|b\rangle\langle b|$.
In other words, the generalized measurement described by the set $\\{M_{a}\\}$
has the same statistics of a Von-Neumann projective measurement of the
observable $A$, and leave the system under investigating in an eigenstate of
the observable $B$, thus determining its future value with an arbitrary degree
of accuracy and certainty and contrasting the formulation of the so-called
Heisenberg principle reported above. An explicit unitary realization of this
kind of measurement for the case of position, as well as a detailed discussion
on the exact meaning of the Heisenberg principle, and the tradeoff between
precision and disturbance in a quantum measurement, may be found in Ozawa02 .
### 3.5 The quantum roulette
Let us consider $K$ projective measurements corresponding to $K$ nondegenerate
isospectral observables $X_{k}$, $k=1,...,K$ in a Hilbert space $H$, and
consider the following experiment. The system is sent to a detector which at
random, with probability $z_{k}$, $\sum_{k}z_{k}=1$, perform the measurement
of the observable $X_{k}$. This is known as the quantum roulette since the
observable to be measured is chosen at random, eg according to the outcome of
a random generator like a roulette. The probability of getting the outcome $x$
from the measurement of the observable $X_{k}$ on a state $\varrho\in L(H)$ is
given by $p_{x}^{(k)}=\hbox{Tr}[\varrho\,P^{(k)}_{x}]$,
$P^{(k)}_{x}=|x\rangle_{k}{}_{k}\langle x|$, and the overall probability of
getting the outcome $x$ from our experiment is given by
$p_{x}=\sum_{k}z_{k}p_{x}^{(k)}=\sum_{k}z_{k}\hbox{Tr}[\varrho\,P^{(k)}_{x}]=\hbox{Tr}[\varrho\,\sum_{k}z_{k}P^{(k)}_{x}]=\hbox{Tr}[\varrho\,\Pi_{x}]\,,$
where the POVM describing the measurement is given by
$\Pi_{x}=\sum_{k}z_{k}P^{(k)}_{x}$. This is indeed a POVM and not a projective
measurement since
$\Pi_{x}\Pi_{x^{\prime}}=\sum_{kk^{\prime}}z_{k}z_{k^{\prime}}P^{(k)}_{x}P^{(k^{\prime})}_{x^{\prime}}\neq\delta_{xx^{\prime}}\Pi_{x}\,.$
Again, we have a practical situation where POVMs naturally arise in order to
describe the statistics of the measurement in terms of the Born rule and the
system density operator. A Naimark extension for the quantum roulette may be
obtained as follows. Let us consider an additional probe system described by
the Hilbert space $H_{\scriptscriptstyle P}$ of dimension $K$ equal to the
number of measured observables in the roulette, and the set of projectors
$Q_{x}=\sum_{k}P^{(k)}_{x}\otimes|\theta_{k}\rangle\langle\theta_{k}|$ where
$\\{|\theta_{k}\rangle\\}$ is a basis for $H_{\scriptscriptstyle P}$. Then,
upon preparing the probe system in the superposition
$|\omega_{P}\rangle=\sum_{k}\sqrt{z_{k}}|\theta_{k}\rangle$ we have that
$p_{x}=\hbox{Tr}_{{\scriptscriptstyle S}{\scriptscriptstyle
P}}[\varrho\otimes|\omega_{\scriptscriptstyle
P}\rangle\langle\omega_{\scriptscriptstyle P}|\,Q_{x}]$ and, in turn,
$\Pi_{x}=\hbox{Tr}_{\scriptscriptstyle P}[\mathbbm{I}_{\scriptscriptstyle
S}\otimes|\omega_{\scriptscriptstyle
P}\rangle\langle\omega_{\scriptscriptstyle
P}|\,Q_{x}]=\sum_{k}z_{k}P^{(k)}_{x}$. The state of the system after the
measurement may be obtained as the partial trace
$\displaystyle\varrho_{x}$
$\displaystyle=\frac{1}{p_{x}}\hbox{Tr}_{\scriptscriptstyle
P}\left[Q_{x}\,\varrho\otimes|\omega_{\scriptscriptstyle
P}\rangle\langle\omega_{\scriptscriptstyle P}|\,Q_{x}\right]$
$\displaystyle=\frac{1}{p_{x}}\sum_{k}\sum_{k^{\prime}}\hbox{Tr}_{\scriptscriptstyle
P}\left[P_{x}^{(k)}\otimes|\theta_{k}\rangle\langle\theta_{k}|\,\varrho\otimes|\omega_{\scriptscriptstyle
P}\rangle\langle\omega_{\scriptscriptstyle
P}|\,P_{x}^{(k^{\prime})}\otimes|\theta_{k^{\prime}}\rangle\langle\theta_{k^{\prime}}|\right]$
$\displaystyle=\frac{1}{p_{x}}\sum_{k}z_{k}P_{x}^{(k)}\varrho\,P_{x}^{(k)}\>.$
Notice that the presented Naimark extension is not the canonical one.
###### Exercise 7
Prove that the operators $Q_{x}$ introduced for the Naimark extension of the
quantum roulette, are indeed projectors.
###### Exercise 8
Take a system made by a single qubit system and construct the canonical
Naimark extension for the quantum roulette obtained by measuring the
observables $\sigma_{\alpha}=\cos\alpha\,\sigma_{1}+\sin\alpha\,\sigma_{2}$,
where $\sigma_{1}$ and $\sigma_{2}$ are Pauli matrices and $\alpha\in[0,\pi]$
is chosen at random with probability density $p(\alpha)=\pi^{-1}$.
## 4 Quantum operations
In this Section we address the dynamical evolution of quantum systems to see
whether the standard formulation in terms of unitary evolutions needs a
suitable generalization. This is indeed the case: we will introduce a
generalized description and see how this reconciles with what we call
Postulate 3 in the Introduction. We will proceed in close analogy with what we
have done for states and measurements. We start by closely inspecting the
physical motivations behind any mathematical description of quantum evolution,
and look for physically motivated conditions that a map, intended to transform
a quantum state into a quantum state, from now on a quantum operation, should
satisfy to be admissible. This will lead us to the concept of complete
positivity, which suitably generalizes the motivations behind unitarity. We
then prove that any quantum operation may be seen as the partial trace of a
unitary evolution in a larger Hilbert space, and illustrate a convenient form,
the so-called Kraus or operator-sum representation, to express the action of a
quantum operation on quantum states.
By quantum operation we mean a map $\varrho\rightarrow{\cal E}(\varrho)$
transforming a quantum state $\varrho$ into another quantum state ${\cal
E}(\varrho)$. The basic requirements on ${\cal E}$ to describe a physically
admissible operations are those captured by the request of unitarity in the
standard formulation, i.e.
* ${\boldsymbol{Q1}}$
The map is positive and trace-preserving, i.e. ${\cal E}(\varrho)\geq 0$
(hence selfadjoint) and $\hbox{Tr}[{\cal E}(\varrho)]=\hbox{Tr}[\varrho]=1$.
The last assumption may be relaxed to that of being trace non-increasing
$0\leq\hbox{Tr}[{\cal E}(\varrho)]\leq 1$ in order to include evolution
induced by measurements (see below).
* ${\boldsymbol{Q2}}$
The map is linear ${\cal E}(\sum_{k}p_{k}\varrho_{k})=\sum_{k}p_{k}{\cal
E}(\varrho_{k})$, i.e. the state obtained by applying the map to the ensemble
$\\{p_{k},\varrho_{k}\\}$ is the ensemble $\\{p_{k},{\cal E}(\varrho_{k})\\}$.
* ${\boldsymbol{Q3}}$
The map is completely positive (CP), i.e. besides being positive it is such
that if we introduce an additional system, any map of the form ${\cal
E}\otimes\mathbbm{I}$ acting on the extended Hilbert space is also positive.
In other words, we ask that the map is physically meaningful also when acting
on a portion of a larger, composite, system. As we will see, this request is
not trivial at all, i.e. there exist maps that are positive but not completely
positive.
### 4.1 The operator-sum representation
This section is devoted to state and prove a theorem showing that a map is a
quantum operation if and only if it is the partial trace of a unitary
evolution in a larger Hilbert space, and provides a convenient form, the so-
called Kraus decomposition or operator-sum representation Pre ; nota , to
express its action on quantum states.
###### Theorem 4.1 (Kraus)
A map ${\cal E}$ is a quantum operation i.e. it satisfies the requirements
$\boldsymbol{Q1}$-$\boldsymbol{Q3}$ if and only if is the partial trace of a
unitary evolution on a larger Hilbert space with factorized initial condition
or, equivalently, it possesses a Kraus decomposition i. e. its action may be
represented as ${\cal E}(\varrho)=\sum_{k}M_{k}\varrho M^{\dagger}_{k}$ where
$\\{M_{k}\\}$ is a set of operators satisfying
$\sum_{k}M_{k}^{\dagger}M_{k}=\mathbbm{I}$.
###### Proof
The first part of the theorem consists in assuming that ${\cal E}(\varrho)$ is
the partial trace of a unitary operation in a larger Hilbert space and prove
that it has a Kraus decomposition and, in turn, it satisfies the requirements
$\boldsymbol{Q1}$-$\boldsymbol{Q3}$. Let us consider a physical system $A$
prepared in the quantum state $\varrho_{\scriptscriptstyle A}$ and another
system $B$ prepared in the state $\varrho_{\scriptscriptstyle B}$. $A$ and $B$
interact through the unitary operation $U$ and we are interested in describing
the effect of this interaction on the system $A$ only, i.e. we are looking for
the expression of the mapping $\varrho_{\scriptscriptstyle
A}\rightarrow\varrho^{\prime}_{\scriptscriptstyle A}={\cal
E}(\varrho_{\scriptscriptstyle A})$ induced by the interaction. This may be
obtained by performing the partial trace over the system $B$ of the global
$AB$ system after the interaction, in formula
$\displaystyle{\cal E}(\varrho_{\scriptscriptstyle A})$
$\displaystyle=\hbox{Tr}_{\scriptscriptstyle
B}\left[U\,\varrho_{\scriptscriptstyle A}\otimes\varrho_{\scriptscriptstyle
B}U^{\dagger}\right]=\sum_{s}p_{s}\hbox{Tr}_{\scriptscriptstyle
B}\left[U\,\varrho_{\scriptscriptstyle
A}\otimes|\theta_{s}\rangle\langle\theta_{s}|U^{\dagger}\right]$
$\displaystyle=\sum_{st}p_{s}\langle\varphi_{t}|U|\theta_{s}\rangle\,\varrho_{\scriptscriptstyle
A}\langle\theta_{s}|U^{\dagger}|\varphi_{t}\rangle=\sum_{k}M_{k}\,\varrho_{\scriptscriptstyle
A}M^{\dagger}_{k}$ (19)
where we have introduced the operator
$M_{k}=\sqrt{p_{s}}\langle\varphi_{t}|U|\theta_{s}\rangle$, with the polyindex
$k\equiv st$ obtained by a suitable ordering, and used the spectral
decomposition of the density operator $\varrho_{\scriptscriptstyle
B}=\sum_{s}p_{s}|\theta_{s}\rangle\langle\theta_{s}|$. Actually, we could have
also assumed the additional system in a pure state
$|\omega_{\scriptscriptstyle B}\rangle$, since this is always possible upon
invoking a purification, i.e. by suitably enlarging the Hilbert space. In this
case the elements in the Kraus decomposition of our map would have be written
as $\langle\varphi_{t}|U|\omega_{\scriptscriptstyle B}\rangle$. The set of
operators $\\{M_{k}\\}$ satisfies the relation
$\sum_{k}M^{\dagger}M_{k}=\sum_{st}p_{s}\theta_{s}|U^{\dagger}|\varphi_{t}\rangle\langle\varphi_{t}|U|\theta_{s}\rangle=\sum_{s}p_{s}\langle\theta_{s}|U^{\dagger}U|\theta_{s}\rangle=\mathbbm{I}\,.$
Notice that the assumption of a factorized initial state is crucial to prove
the existence of a Kraus decomposition and, in turn, the complete positivity.
In fact, the dynamical map ${\cal E}(\varrho_{\scriptscriptstyle
A})=\hbox{Tr}_{\scriptscriptstyle
B}\left[U\,\varrho_{\scriptscriptstyle\\!AB}\,U^{\dagger}\right]$ resulting
from the partial trace of an initially correlated preparation
$\varrho_{\scriptscriptstyle\\!AB}$ needs not to be so. In this case, the
dynamics can properly be defined only on a subset of initial states of the
system. Of course, the map can be extended to all possible initial states by
linearity, but the extension may not be physically realizable, i.e. may be not
completely positive or even positive PP94 .
We now proceed to show that for map of the form (19) (Kraus decomposition) the
properties $\boldsymbol{Q1}$-$\boldsymbol{Q3}$ hold. Preservation of trace and
of the Hermitian character, as well as linearity, are guaranteed by the very
form of the map. Positivity is also ensured, since for any positive operator
$O_{\scriptscriptstyle A}\in L(H_{\scriptscriptstyle A})$ and any vector
$|\varphi_{\scriptscriptstyle A}\rangle\in H_{\scriptscriptstyle A}$ we have
$\displaystyle\langle\varphi_{\scriptscriptstyle A}|{\cal
E}(O_{\scriptscriptstyle A})|\varphi_{\scriptscriptstyle A}\rangle$
$\displaystyle=\langle\varphi_{\scriptscriptstyle
A}|\sum_{k}M_{k}\,O_{\scriptscriptstyle
A}M_{k}^{\dagger}|\varphi_{\scriptscriptstyle
A}\rangle=\langle\varphi_{\scriptscriptstyle A}|\hbox{Tr}_{\scriptscriptstyle
B}[U\,O_{\scriptscriptstyle A}\otimes\varrho_{\scriptscriptstyle
B}\,U^{\dagger}]|\varphi_{\scriptscriptstyle A}\rangle$
$\displaystyle=\hbox{Tr}_{{\scriptscriptstyle A}{\scriptscriptstyle
B}}[U^{\dagger}|\varphi_{\scriptscriptstyle
A}\rangle\langle\varphi_{\scriptscriptstyle
A}|\otimes\mathbbm{I}\,U\,O_{\scriptscriptstyle
A}\otimes\varrho_{\scriptscriptstyle B}\,]\geq
0\quad\forall\,O_{\scriptscriptstyle A},\forall\,\varrho_{\scriptscriptstyle
B},\forall\,|\varphi_{\scriptscriptstyle A}\rangle\,.$
Therefore it remains to be proved that the map is completely positive. To this
aim let us consider a positive operator $O_{\scriptscriptstyle\\!AC}\in
L(H_{\scriptscriptstyle A}\otimes H_{\scriptscriptstyle C})$ and a generic
state $|\psi_{{\scriptscriptstyle\\!AC}}\rangle\rangle$ on the same enlarged
space, and define
$|\omega_{k}\rangle\rangle=\frac{1}{\sqrt{N_{k}}}M_{k}\otimes\mathbbm{I}_{\scriptscriptstyle
C}|\psi_{{\scriptscriptstyle A}{\scriptscriptstyle C}}\rangle\rangle\qquad
N_{k}=\langle\langle\psi_{{\scriptscriptstyle A}{\scriptscriptstyle
C}}|M_{k}^{\dagger}M_{k}\otimes\mathbbm{I}_{\scriptscriptstyle
C}|\psi_{{\scriptscriptstyle A}{\scriptscriptstyle C}}\rangle\rangle\geq 0\,.$
Since $O_{\scriptscriptstyle\\!AC}$ is positive we have
$\langle\langle\psi_{{\scriptscriptstyle\\!AC}}|(M_{k}^{\dagger}\otimes\mathbbm{I}_{\scriptscriptstyle
C})\,O_{\scriptscriptstyle\\!AC}(M_{k}\otimes\mathbbm{I}_{\scriptscriptstyle
C})|\psi_{{\scriptscriptstyle\\!AC}}\rangle\rangle=N_{k}\langle\langle\omega_{k}|O_{\scriptscriptstyle\\!AC}|\omega_{k}\rangle\rangle\geq
0$
and therefore $\langle\langle\psi_{{\scriptscriptstyle\\!AC}}|{\cal
E}\otimes\mathbbm{I}_{\scriptscriptstyle
C}(O_{\scriptscriptstyle\\!AC})|\psi_{{\scriptscriptstyle\\!AC}}\rangle\rangle=\sum_{k}N_{k}\langle\langle\omega_{k}|O_{\scriptscriptstyle\\!AC}|\omega_{k}\rangle\rangle\geq
0$, which proves that for any positive $O_{\scriptscriptstyle\\!AC}$ also
${\cal E}\otimes\mathbbm{I}_{\scriptscriptstyle
C}(O_{\scriptscriptstyle\\!AC})$ is positive for any choice of
$H_{\scriptscriptstyle C}$, i.e. ${\cal E}$ is a CP-map.
Let us now prove the second part of the theorem, i.e. we consider a map ${\cal
E}:L(H_{\scriptscriptstyle A})\rightarrow L(H_{\scriptscriptstyle A})$
satisfying the requirements $\boldsymbol{Q1}$-$\boldsymbol{Q3}$ and show that
it may be written in the Kraus form and, in turn, that its action may be
obtained as the partial trace of a unitary evolution in a larger Hilbert. We
start by considering the state
$|\varphi\rangle\rangle=\frac{1}{\sqrt{d}}\sum_{k}|\theta_{k}\rangle\otimes|\theta_{k}\rangle\in
H_{\scriptscriptstyle A}\otimes H_{\scriptscriptstyle A}$ and define the
operator $\varrho_{{\scriptscriptstyle A}{\scriptscriptstyle A}}={\cal
E}\otimes\mathbbm{I}(|\varphi\rangle\rangle\langle\langle\varphi|)$. From the
complete positivity and trace preserving properties of ${\cal E}$ we have that
$\hbox{Tr}[\varrho_{{\scriptscriptstyle A}{\scriptscriptstyle A}}]=1$, and
$\varrho_{{\scriptscriptstyle A}{\scriptscriptstyle A}}\geq 0$, i.e.
$\varrho_{{\scriptscriptstyle A}{\scriptscriptstyle A}}$ is a density
operator. Besides, this establishes a one-to-one correspondence between maps
$L(H_{\scriptscriptstyle A})\rightarrow L(H_{\scriptscriptstyle A})$ and
density operators in $L(H_{\scriptscriptstyle A})\otimes
L(H_{\scriptscriptstyle A})$ which may be proved as follows: for any
$|\psi\rangle=\sum_{k}\psi_{k}|\theta_{k}\rangle\in H_{\scriptscriptstyle A}$
define $|\tilde{\psi}\rangle=\sum_{k}\psi_{k}^{*}|\theta_{k}\rangle$ and
notice that
$\langle\tilde{\psi}|\varrho_{{\scriptscriptstyle A}{\scriptscriptstyle
A}}|\tilde{\psi}\rangle=\frac{1}{d}\langle\tilde{\psi}|\sum_{kl}{\cal
E}(|\theta_{k}\rangle\langle\theta_{l}|)\otimes|\theta_{k}\rangle\langle\theta_{l}|\,|\tilde{\psi}\rangle=\frac{1}{d}\sum_{kl}\psi_{l}^{*}\psi_{k}\,{\cal
E}(|\theta_{k}\rangle\langle\theta_{l}|)=\frac{1}{d}\,{\cal
E}(|\psi\rangle\langle\psi|)\,,$
where we used linearity to obtain the last equality. Then define the operators
$M_{k}|\psi\rangle=\sqrt{dp_{k}}\langle\tilde{\psi}|\omega_{k}\rangle\rangle$,
where $|\omega_{k}\rangle\rangle$ are the eigenvectors of
$\varrho_{{\scriptscriptstyle A}{\scriptscriptstyle
A}}=\sum_{k}p_{k}|\omega_{k}\rangle\rangle\langle\langle\omega_{k}|$: this is
a linear operator on $H_{\scriptscriptstyle A}$ and we have
$\sum_{k}M_{k}|\psi\rangle\langle\psi|M_{k}^{\dagger}=d\sum_{k}p_{k}\langle\tilde{\psi}|\omega_{k}\rangle\rangle\langle\langle\omega_{k}|\tilde{\psi}\rangle=d\langle\tilde{\psi}|\varrho_{{\scriptscriptstyle
A}{\scriptscriptstyle A}}|\tilde{\psi}\rangle={\cal
E}(|\psi\rangle\langle\psi|)$
for all pure states. Using again linearity we have that ${\cal
E}(\varrho)=\sum_{k}M_{k}\varrho M^{\dagger}_{k}$ also for any mixed state. It
remains to be proved that a unitary extension exists, i.e. to prove that for
any map on $L(H_{\scriptscriptstyle A})$ which satisfies
$\boldsymbol{Q1}$-$\boldsymbol{Q3}$, and thus possesses a Kraus decomposition,
there exist: i) a Hilbert space $H_{\scriptscriptstyle B}$, ii) a state
$|\omega_{\scriptscriptstyle B}\rangle\in H_{\scriptscriptstyle B}$, iii) a
unitary $U\in L(H_{\scriptscriptstyle A}\otimes H_{\scriptscriptstyle B})$
such that ${\cal E}(\varrho_{\scriptscriptstyle
A})=\hbox{Tr}_{\scriptscriptstyle B}[U\,\varrho_{\scriptscriptstyle
A}\otimes|\omega_{\scriptscriptstyle
B}\rangle\langle\omega_{\scriptscriptstyle B}|U^{\dagger}]$ for any
$\varrho_{\scriptscriptstyle A}\in L(H_{\scriptscriptstyle A})$. To this aim
we proceed as we did for the proof of the Naimark theorem, i.e. we take an
arbitrary state $|\omega_{\scriptscriptstyle B}\rangle\in
H_{\scriptscriptstyle B}$, and define an operator $U$ trough its action on the
generic $\varphi_{\scriptscriptstyle
A}\rangle\otimes|\omega_{\scriptscriptstyle B}\rangle\in H_{\scriptscriptstyle
A}\otimes H_{\scriptscriptstyle B}$, $U\,|\varphi_{\scriptscriptstyle
A}\rangle\otimes|\omega_{\scriptscriptstyle
B}\rangle=\sum_{k}M_{k}\,|\varphi_{\scriptscriptstyle
A}\rangle\otimes|\theta_{k}\rangle$, where the $|\theta_{k}\rangle$’s are a
basis for $H_{\scriptscriptstyle B}$. The operator $U$ preserves the scalar
product
$\displaystyle\langle\langle\omega_{\scriptscriptstyle
B},\varphi_{\scriptscriptstyle
A}^{\prime}|U^{\dagger}U|\varphi_{\scriptscriptstyle
A},\omega_{\scriptscriptstyle
B}\rangle\rangle=\sum_{kk^{\prime}}\langle\varphi_{\scriptscriptstyle
A}^{\prime}|M_{k^{\prime}}^{\dagger}M_{k}|\varphi_{\scriptscriptstyle
A}\rangle\langle\theta_{k^{\prime}}|\theta_{k}\rangle=\sum_{k}\langle\varphi_{\scriptscriptstyle
A}^{\prime}|M_{k}^{\dagger}M_{k}|\varphi_{\scriptscriptstyle
A}\rangle=\langle\varphi_{\scriptscriptstyle
A}^{\prime}|\varphi_{\scriptscriptstyle A}\rangle$
and so it is unitary in the one-dimensional subspace spanned by
$|\omega_{\scriptscriptstyle B}\rangle$. Besides, it may be extended to a full
unitary operator in the global Hilbert space $H_{\scriptscriptstyle A}\otimes
H_{\scriptscriptstyle B}$, eg it can be the identity operator in the subspace
orthogonal to $|\omega_{\scriptscriptstyle B}\rangle$. Then, for any
$\varrho_{\scriptscriptstyle A}$ in $H_{\scriptscriptstyle A}$ we have
$\displaystyle\hbox{Tr}_{\scriptscriptstyle
B}\left[U\varrho_{\scriptscriptstyle A}\otimes|\omega_{\scriptscriptstyle
B}\rangle\langle\omega_{\scriptscriptstyle B}|\,U^{\dagger}\right]$
$\displaystyle=\sum_{s}p_{s}\,\hbox{Tr}_{\scriptscriptstyle
B}\left[U|\psi_{s}\rangle\langle\psi_{s}|\otimes|\omega_{\scriptscriptstyle
B}\rangle\langle\omega_{\scriptscriptstyle B}|\,U^{\dagger}\right]$
$\displaystyle=\sum_{skk^{\prime}}p_{s}\,\hbox{Tr}_{\scriptscriptstyle
B}\left[M_{k}|\psi_{s}\rangle\langle\psi_{s}|\,M_{k^{\prime}}^{\dagger}\otimes|\theta_{k}\rangle\langle\theta_{k^{\prime}}|\right]$
$\displaystyle=\sum_{sk}p_{s}\,M_{k}|\psi_{s}\rangle\langle\psi_{s}|\,M_{k}^{\dagger}=\sum_{k}M_{k}\varrho_{\scriptscriptstyle
A}M_{k}^{\dagger}\qquad\qed$
The Kraus decomposition of a quantum operation generalizes the unitary
description of quantum evolution. Unitary maps are, of course, included and
correspond to maps whose Kraus decomposition contains a single elements. The
set of quantum operations constitutes a semigroup, i.e. the composition of two
quantum operations is still a quantum operation:
${\cal E}_{2}({\cal E}_{1}(\varrho))=\sum_{k_{1}}M^{(1)}_{k_{1}}{\cal
E}_{2}(\varrho)M^{(1){\dagger}}_{k_{1}}=\sum_{k_{1}k_{2}}M^{(1)}_{k_{1}}M^{(2)}_{k_{2}}\varrho
M^{(2){\dagger}}_{k_{2}}M^{(1){\dagger}}_{k_{1}}=\sum_{\boldsymbol{k}}\boldsymbol{M}_{\boldsymbol{k}}\varrho\boldsymbol{M}_{\boldsymbol{k}}^{\dagger}\,,$
where we have introduced the polyindex $\boldsymbol{k}$. Normalization is
easily proved, since
$\sum_{\boldsymbol{k}}\boldsymbol{M}_{\boldsymbol{k}}^{\dagger}\boldsymbol{M}_{\boldsymbol{k}}=\sum_{k_{1}k_{2}}M^{(2){\dagger}}_{k_{2}}M^{(1){\dagger}}_{k_{1}}M^{(1)}_{k_{1}}M^{(2)}_{k_{2}}=\mathbbm{I}$.
On the other hand, the existence of inverse is not guaranteed: actually only
unitary operations are invertible (with a CP inverse).
The Kraus theorem also allows us to have a unified picture of quantum
evolution, either due to an interaction or to a measurement. In fact, the
modification of the state in the both processes is described by a set of
operators $M_{k}$ satisfying $\sum_{k}M^{\dagger}_{k}M_{k}=\mathbbm{I}$. In
this framework, the Kraus operators of a measurement are what we have referred
to as the detection operators of a POVM.
#### 4.1.1 The dual map and the unitary equivalence
Upon writing the generic expectation value for the evolved state ${\cal
E}(\varrho)$ and exploiting both linearity and circularity of trace we have
$\langle X\rangle=\hbox{Tr}[{\cal
E}(\varrho)\,X]=\sum_{k}\hbox{Tr}[M_{k}\varrho
M_{k}^{\dagger}\,X]=\sum_{k}\hbox{Tr}[\varrho\,M_{k}^{\dagger}XM_{k}]=\hbox{Tr}[\varrho{\cal
E}^{\vee}(X)]\,,$
where we have defined the dual map ${\cal
E}^{\vee}(X)=\sum_{k}M_{k}^{\dagger}XM_{k}$ which represents the ”Heisenberg
picture” for quantum operations. Notice also that the elements of the Kraus
decomposition $M_{k}=\langle\varphi_{k}|U|\omega_{\scriptscriptstyle
B}\rangle$ depend on the choice of the basis used to perform the partial
trace. Change of basis cannot have a physical effect and this means that the
set of operators
$N_{k}=\langle\theta_{k}|U|\omega_{\scriptscriptstyle
B}\rangle=\sum_{s}\langle\theta_{k}|\varphi_{s}\rangle\langle\varphi_{s}|U|\omega_{\scriptscriptstyle
B}\rangle=\sum_{s}V_{ks}M_{s}\,,$
where the unitary $V\in L(H_{\scriptscriptstyle B})$ describes the change of
basis, and the original set $M_{k}$ actually describe the same quantum
operations, i.e. $\sum_{k}N_{k}\varrho N_{k}^{\dagger}=\sum_{k}M_{k}\varrho
M_{k}^{\dagger}$, $\forall\varrho$. The same can be easily proved for the
system $B$ prepared in mixed state. The origin of this degree of freedom stays
in the fact that if the unitary $U$ on $H_{\scriptscriptstyle A}\otimes
H_{\scriptscriptstyle B}$ and the state $|\omega_{\scriptscriptstyle
B}\rangle\in H_{\scriptscriptstyle B}$ realize an extension for the map ${\cal
E}:L(H_{\scriptscriptstyle A})\rightarrow L(H_{\scriptscriptstyle A})$ then
any unitary of the form $(\mathbbm{I}\otimes V)U$ is a unitary extension too,
with the same ancilla state. A quantum operation is thus identified by an
equivalence class of Kraus decompositions. An interesting corollary is that
any quantum operation on a given Hilbert space of dimension $d$ may be
generated by a Kraus decomposition containing at most $d^{2}$ elements, i.e.
given a Kraus decomposition ${\cal E}(\varrho)=\sum_{k}M_{k}\varrho
M_{k}^{\dagger}$ with an arbitrary number of elements, one may exploit the
unitary equivalence and find another representation ${\cal
E}(\varrho)=\sum_{k}N_{k}\varrho N_{k}^{\dagger}$ with at most $d^{2}$
elements.
### 4.2 The random unitary map and the depolarizing channel
A simple example of quantum operation is the random unitary map, defined by
the Kraus decomposition ${\cal E}(\varrho)=\sum_{k}p_{k}U_{k}\varrho
U^{\dagger}_{k}$, i.e. $M_{k}=\sqrt{p_{k}}\,U_{k}$ and
$U_{k}^{\dagger}U_{k}=\mathbbm{I}$. This map may be seen as the evolution
resulting from the interaction of our system with another system of dimension
equal to the number of elements in the Kraus decomposition of the map via the
unitary $V$ defined by $V|\psi_{\scriptscriptstyle
A}\rangle\otimes|\omega_{\scriptscriptstyle
B}\rangle=\sum_{k}\sqrt{p_{k}}\,U_{k}|\psi_{\scriptscriptstyle
A}\rangle\otimes|\theta_{k}\rangle$, $|\theta_{k}\rangle$ being a basis for
$H_{\scriptscriptstyle B}$ which includes $|\omega_{\scriptscriptstyle
B}\rangle$. If ”we do not look” at the system $B$ and trace out its degree of
freedom the evolution of system $A$ is governed by the random unitary map
introduced above.
###### Exercise 9
Prove explicitly the unitarity of V.
The operator-sum representation of quantum evolutions have been introduced,
and finds its natural application, for the description of propagation in noisy
channels, i.e. the evolution resulting from the interaction of the system of
interest with an external environment, which generally introduces noise in the
system degrading its coherence. As for example, let us consider a qubit system
(say, the polarization of a photon), on which we have encoded binary
information according to a suitable coding procedure, traveling from a sender
to a receiver. The propagation needs a physical support (say, an optical
fiber) and this unavoidably leads to consider possible perturbations to our
qubit, due to the interaction with the environment. The resulting open system
dynamics is usually governed by a Master equation, i.e. the equation obtained
by partially tracing the Schroedinger (Von Neumann) equation governing the
dynamics of the global system, and the solution is expressed in form of a CP-
map. For a qubit $Q$ in a noisy environment a quite general description of the
detrimental effects of the environment is the so-called depolarizing channel
nie00 , which is described by the Kraus operator
$M_{0}=\sqrt{1-\gamma}\,\sigma_{0}$, $M_{k}=\sqrt{\gamma/3}\,\sigma_{k}$,
$k=1,2,3$, i.e.
${\cal
E}(\varrho)=(1-\gamma)\varrho+\frac{\gamma}{3}\sum_{k}\sigma_{k}\,\varrho\,\sigma_{k}\qquad
0\leq\gamma\leq 1\,.$
The depolarizing channel may be seen as the evolution of the qubit due to the
interaction with a four-dimensional system through the unitary
$V|\psi_{\scriptscriptstyle Q}\rangle\otimes|\omega_{\scriptscriptstyle
E}\rangle=\sqrt{1-\gamma}|\psi_{\scriptscriptstyle
Q}\rangle\otimes|\omega_{\scriptscriptstyle
E}\rangle+\sqrt{\frac{\gamma}{3}}\sum_{k=1}^{3}\sigma_{k}|\psi_{\scriptscriptstyle
Q}\rangle\otimes|\theta_{k}\rangle\,,$
$|\theta_{k}\rangle$ being a basis which includes $|\omega_{\scriptscriptstyle
E}\rangle$. From the practical point view, the map describes a situation in
which, independently on the underlying physical mechanism, we have a
probability $\gamma/3$ that a perturbation described by a Pauli matrix is
applied to the qubit. If we apply $\sigma_{1}$ we have the so-called spin-flip
i.e. the exchange $|0\rangle\leftrightarrow|1\rangle$, whereas if we apply
$\sigma_{3}$ we have the phase-flip, and for $\sigma_{2}$ we have a specific
combination of the two effects. Since for any state of a qubit
$\varrho+\sum_{k}\sigma_{k}\varrho\sigma_{k}=2\mathbbm{I}$ the action of the
depolarizing channel may be written as
${\cal
E}(\varrho)=(1-\gamma)\varrho+\frac{\gamma}{3}(2\mathbbm{I}-\varrho)=\frac{2}{3}\gamma\mathbbm{I}+(1-\frac{4}{3}\gamma)\varrho=p\varrho+(1-p)\frac{\mathbbm{I}}{2}\,,$
where $p=1-\frac{4}{3}\gamma$, i.e. $-\frac{1}{3}\leq p\leq 1$. In other
words, we have that the original state $\varrho$ is sent to a linear
combination of itself and the maximally mixed state $\frac{\mathbbm{I}}{2}$,
also referred to as the depolarized state.
###### Exercise 10
Express the generic qubit state in Bloch representation and explicitly write
the effect of the depolarizing channel on the Bloch vector.
###### Exercise 11
Show that the purity of a qubit cannot increase under the action of the
depolarizing channel.
### 4.3 Transposition and partial transposition
The transpose $T(X)=X^{\scriptscriptstyle T}$ of an operator $X$ is the
conjugate of its adjoint $X^{\scriptscriptstyle
T}=(X^{\dagger})^{*}=(X^{*})^{\dagger}$. Upon the choice of a basis we have
$X=\sum_{nk}X_{nk}|\theta_{n}\rangle\langle\theta_{k}|$ and thus
$X^{\scriptscriptstyle
T}=\sum_{nk}X_{nk}|\theta_{k}\rangle\langle\theta_{n}|=\sum_{nk}X_{kn}|\theta_{n}\rangle\langle\theta_{k}|$.
Transposition does not change the trace of an operator, neither its
eigenvalues. Thus it transforms density operators into density operators:
$\hbox{Tr}[\varrho]=\hbox{Tr}[\varrho^{\scriptscriptstyle T}]=1$
$\varrho^{\scriptscriptstyle T}\geq 0$ if $\varrho\geq 0$. As a positive,
trace preserving, map it is a candidate to be a quantum operation. On the
other hand, we will show by a counterexample that it fails to be completely
positive and thus it does not correspond to physically admissible quantum
operation. Let us consider a bipartite system formed by two qubits prepared in
the state
$|\varphi\rangle\rangle=\frac{1}{\sqrt{2}}\,|00\rangle\rangle+|11\rangle\rangle$.
We denote by $\varrho^{\tau}=\mathbbm{I}\otimes T(\varrho)$ the partial
transpose of $\varrho$ i.e. the operator obtained by the application of the
transposition map to one of the two qubits. We have
$\displaystyle\big{(}|\varphi\rangle\rangle\langle\langle\varphi|\big{)}^{\tau}$
$\displaystyle=\frac{1}{2}\left(\begin{array}[]{cccc}1&0&0&1\\\ 0&0&0&0\\\
0&0&0&0\\\ 1&0&0&1\end{array}\right)^{\tau}$ (24)
$\displaystyle=\frac{1}{2}\Big{(}|0\rangle\langle 0|\otimes|0\rangle\langle
0|+|1\rangle\langle 1|\otimes|1\rangle\langle 1|+|0\rangle\langle
1|\otimes|0\rangle\langle 1|+|1\rangle\langle 0|\otimes|1\rangle\langle
0|\Big{)}^{\tau}$ $\displaystyle=\frac{1}{2}\Big{(}|0\rangle\langle
0|\otimes|0\rangle\langle 0|+|1\rangle\langle 1|\otimes|1\rangle\langle
1|+|0\rangle\langle 1|\otimes|1\rangle\langle 0|+|1\rangle\langle
0|\otimes|0\rangle\langle 1|\Big{)}$
$\displaystyle=\frac{1}{2}\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&0&1&0\\\
0&1&0&0\\\ 0&0&0&1\end{array}\right)$ (29)
Using the last expression it is straightforward to evaluate the eigenvalues of
$\varrho^{\tau}$, which are $+\frac{1}{2}$ (multiplicity three) and
$-\frac{1}{2}$. In other words $\mathbbm{I}\otimes T$ is not a positive map
and the transposition is not completely positive. Notice that for a factorized
state of the form
$\varrho_{\scriptscriptstyle\\!AB}=\varrho_{\scriptscriptstyle
A}\otimes\varrho_{\scriptscriptstyle B}$ we have $\mathbbm{I}\otimes
T(\varrho_{\scriptscriptstyle\\!AB})=\varrho_{\scriptscriptstyle
A}\otimes\varrho_{\scriptscriptstyle B}^{\scriptscriptstyle T}\geq 0$ i.e.
partial transposition preserves positivity in this case .
###### Exercise 12
Prove that transposition is not a CP-map by its action on any state of the
form
$|\varphi\rangle\rangle=\frac{1}{\sqrt{d}}\sum_{k}|\varphi_{k}\rangle\otimes|\theta_{k}\rangle$.
Hint: the operator $\mathbbm{I}\otimes
T(|\varphi\rangle\rangle\langle\langle\varphi|)\equiv E$ is the so-called swap
operator since it ”exchanges” states as $E(|\psi\rangle_{\scriptscriptstyle
A}\otimes|\varphi\rangle_{\scriptscriptstyle
B})=|\varphi\rangle_{\scriptscriptstyle
A}\otimes|\psi\rangle_{\scriptscriptstyle B}$.
## 5 Conclusions
In this tutorial, we have addressed the postulates of quantum mechanics about
states, measurements and operations. We have reviewed their modern formulation
and introduced the basic mathematical tools: density operators, POVMs,
detection operators and CP-maps. We have shown how they provide a suitable
framework to describe quantum systems in interaction with their environment,
and with any kind of measuring and processing devices. The connection with the
standard formulation have been investigated in details building upon the
concept of purification and the Theorems of Naimark and Stinespring/Kraus-
Choi-Sudarshan.
The framework and the tools illustrated in this tutorial are suitable for the
purposes of quantum information science and technology, a field which has
fostered new experiments and novel views on the conceptual foundation of
quantum mechanics, but has so far little impact on the way that it is taught.
We hope to contribute in disseminating these notions to a larger audience, in
the belief that they are useful for several other fields, from condensed
matter physics to quantum biology.
###### Acknowledgements.
I’m grateful to Konrad Banaszek, Alberto Barchielli, Maria Bondani, Mauro
D’Ariano, Ivo P. Degiovanni, Marco Genoni, Marco Genovese, Paolo Giorda,
Chiara Macchiavello, Sabrina Maniscalco, Alex Monras, Stefano Olivares, Jyrki
Piilo, Alberto Porzio, Massimiliano Sacchi, Ole Steuernagel, and Bassano
Vacchini for the interesting and fruitful discussions about foundations of
quantum mechanics and quantum optics over the years. I would also like to
thank Gerardo Adesso, Alessandra Andreoni, Rodolfo Bonifacio, Ilario Boscolo,
Vlado Buzek, Berge Englert, Zdenek Hradil, Fabrizio Illuminati, Ludovico Lanz,
Luigi Lugiato, Paolo Mataloni, Mauro Paternostro, Mladen Pavičić, Francesco
Ragusa, Mario Rasetti, Mike Raymer, Jarda Řeháček, Salvatore Solimeno, and
Paolo Tombesi.
## References
* (1) M. Nielsen, E. Chuang, Quantum Computation and Quantum Information, (Cambridge University Press, 2000).
* (2) A. Peres, Quantum Theory: concepts and methods, (Kluwer Academic, Dordrecht, 1993).
* (3) J. Bergou, J. Mod. Opt. 57, 160 (2010).
* (4) V. Paulsen, Completely Bounded Maps and Operator Algebras (Cambridge University Press, 2003).
* (5) E. Arthurs, J. L. Kelly, Bell. Syst. Tech. J. 44, 725 (1965); J. P. Gordon, W. H. Louisell in Physics of Quantum Electronics (Mc-Graw-Hill, NY, 1966); E. Arthurs, M. S. Goodman, Phys. Rev. Lett. 60, 2447 (1988).
* (6) H. P. Yuen, Phys. Lett. A 91, 101 (1982).
* (7) B. Vacchini in Theoretical foundations of quantum information processing and communication, E. Bruening et al (Eds.), Lect. Not. Phys. 787, 39 (2010).
* (8) E. Prugovečki, J. Phys. A 10, 543 (1977).
* (9) N. G. Walker, J. E. Carrol, Opt. Quantum Electr. 18, 355 (1986); N. G. Walker, J. Mod. Opt. 34, 16 (1987).
* (10)
http://en.wikipedia.org/wiki/Uncertainty_principle
* (11) M. Ozawa, Phys. Lett. A 299, 17 (2002); Phys. Rev. A 67, 042105 (2003); J. Opt. B 7, S672 (2005).
* (12) J. Preskill, Lectures notes for Physics 229: Quantum information and computation available at www.theory.caltech.edu/$\,\,\widetilde{}$preskill/ph229/
* (13) Depending on the source, and on the context, the theorem is known as the Stinespring dilation theorem, or the Kraus-Choi-Sudarshan theorem.
* (14) P. Pechukas, Phys. Rev. Lett. 73, 1060 (1994).
* (15) R. Puri, Mathematical methods of quantum optics (Springer, Berlin, 2001).
* (16) K. E. Cahill, R. J. Glauber, Phys. Rev. 177, 1857 (1969); 177, 1882 (1969).
## Further readings
1. 1.
I. Bengtsson, K. Zyczkowski, Geometry of Quantum States, (Cambridge University
Press, 2006).
2. 2.
Lectures and reports by C. M. Caves, available at
http://info.phys.unm.edu/$\,\,\widetilde{}$caves/
3. 3.
P. Busch, M. Grabowski, P. J. Lahti,Operational Quantum Mechanics, Lect.
Notes. Phys. 31, (Springer, Berlin,1995).
4. 4.
T. Heinosaari, M. Ziman, Acta Phys. Slovaca 58, 487 (2008).
5. 5.
C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New
York, 1976)
6. 6.
A.S. Holevo, Statistical Structure of Quantum Theory, Lect. Not. Phys 61,
(Springer, Berlin, 2001).
7. 7.
M. Ozawa, J. Math. Phys. 25, 79 (1984).
8. 8.
M. G. A. Paris, J. Rehacek (Eds.), Quantum State Estimation Lect. Notes Phys.
649, (Springer, Berlin, 2004).
9. 9.
V. Gorini, A. Frigerio, M. Verri, A. Kossakowski, E. C. G. Sudarshan, Rep.
Math. Phys. 13, 149 (1978).
10. 10.
F. Buscemi, G. M. D’Ariano, and M. F. Sacchi, Phys. Rev. A 68. 042113 (2003).
11. 11.
K. Banaszek, Phys. Rev. Lett. 86, 1366 (2001).
## Appendix A Trace and partial trace
The trace of an operator $O$ is a scalar quantity equal to sum of diagonal
elements in a given basis
$\hbox{Tr}[O]=\sum_{n}\langle\varphi_{n}|O|\varphi_{n}\rangle$. The trace is
invariant under any change of basis, as it is proved by the following chain of
equalities
$\displaystyle\sum_{n}\langle\theta_{n}|O|\theta_{n}\rangle$
$\displaystyle=\sum_{njk}\langle\theta_{n}|\varphi_{k}\rangle\langle\varphi_{k}|O|\varphi_{j}\rangle\langle\varphi_{j}|\theta_{n}\rangle=\sum_{njk}\langle\varphi_{j}|\theta_{n}\rangle\langle\theta_{n}|\varphi_{k}\rangle\langle\varphi_{k}|O|\varphi_{j}\rangle$
$\displaystyle=\sum_{jk}\langle\varphi_{j}|\varphi_{k}\rangle\langle\varphi_{k}|O|\varphi_{j}\rangle=\sum_{k}\langle\varphi_{k}|O|\varphi_{k}\rangle\,,$
where we have suitably inserted and removed resolutions of the identity in
terms of both basis $\\{|\theta_{n}\rangle\\}$ and
$\\{|\varphi_{n}\rangle\\}$. As a consequence, using the basis of eigenvectors
of $O$, $\hbox{Tr}[O]=\sum_{n}o_{n}$, $o_{n}$ being the eigenvalues of $O$.
Trace is a linear operation, i.e.
$\hbox{Tr}[O_{1}+O_{2}]=\hbox{Tr}[O_{1}]+\hbox{Tr}[O_{2}]$ and
$\hbox{Tr}[\lambda\,O]=\lambda\hbox{Tr}[O]$ and thus
$\partial\hbox{Tr}[O]=\hbox{Tr}[\partial O]$ for any derivation. The trace of
any ”ket-bra” $\hbox{Tr}[|\psi_{1}\rangle\langle\psi_{2}|]$ is obtained by
”closing the sandwich”
$\hbox{Tr}[|\psi_{1}\rangle\langle\psi_{2}|]=\langle\psi_{2}|\psi_{1}\rangle$;
in fact upon expanding the two vectors in the same basis and taking the trace
in that basis
$\hbox{Tr}[|\psi_{1}\rangle\langle\psi_{2}|]=\sum_{nkl}\psi_{1k}\psi_{2l}^{*}\langle\theta_{n}|\theta_{k}\rangle\langle\theta_{l}|\theta_{n}\rangle=\sum_{n}\psi_{1n}\psi_{2n}^{*}=\langle\psi_{2}|\psi_{1}\rangle$.
Other properties are summarized by the following theorem.
###### Theorem A.1
For the trace operation the following properties hold
* i)
Given any pair of operators $\hbox{\rm Tr}[A_{1}A_{2}]=\hbox{\rm
Tr}[A_{2}A_{1}]$
* ii)
Given any set of operators $A_{1},...,A_{\scriptscriptstyle N}$ we $\hbox{\rm
Tr}[A_{1}A_{2}A_{3}...A_{\scriptscriptstyle N}]=\hbox{\rm
Tr}[A_{2}A_{3}...A_{\scriptscriptstyle N}A_{1}]=\hbox{\rm
Tr}[A_{3}A_{4}...A_{1}A_{2}]=...$ (circularity).
###### Proof
: left as an exercise.∎
Notice that the ”circularity” condition is essential to have property ii) i.e.
$\hbox{Tr}[A_{1}A_{2}A_{3}]=\hbox{Tr}[A_{2}A_{3}A_{1}]$, but
$\hbox{Tr}[A_{1}A_{2}A_{3}]\neq\hbox{Tr}[A_{2}A_{1}A_{3}]$
Partial traces $R_{\scriptscriptstyle B}\in L(H_{\scriptscriptstyle B})$
$R_{\scriptscriptstyle A}\in L(H_{\scriptscriptstyle A})$ of an operator $R$
in $L(H_{1}\otimes H_{2})$ are defined accordingly as
$R_{\scriptscriptstyle B}=\hbox{Tr}_{\scriptscriptstyle
A}\left[R\,\right]=\sum_{n}{}_{\scriptscriptstyle
A}\langle\varphi_{n}|R\,|\varphi_{n}\rangle_{\scriptscriptstyle A}\qquad
R_{\scriptscriptstyle A}=\hbox{Tr}_{\scriptscriptstyle
B}\left[R\,\right]=\sum_{n}{}_{\scriptscriptstyle
B}\langle\varphi_{n}|R\,|\varphi_{n}\rangle_{\scriptscriptstyle B}\,$
and circularity holds only for single-system operators, e.g., if
$R_{1},R_{2}\in L(H_{\scriptscriptstyle A}\otimes H_{\scriptscriptstyle B})$,
$A\in L(H_{\scriptscriptstyle A})$, $B\in L(H_{\scriptscriptstyle B})$
$\displaystyle\hbox{Tr}_{\scriptscriptstyle
A}\left[A\otimes\mathbbm{I}\,R_{1}R_{2}\right]$
$\displaystyle=\sum_{n}a_{n}\langle
a_{n}|R_{1}R_{2}|a_{n}\rangle=\hbox{Tr}_{\scriptscriptstyle
A}\left[R_{1}R_{2}\,A\otimes\mathbbm{I}\right]$
$\displaystyle\hbox{Tr}_{\scriptscriptstyle A}\left[A\otimes
B\,R_{1}R_{2}\right]$ $\displaystyle=\sum_{n}a_{n}\langle
a_{n}|\mathbbm{I}\otimes
B\,R_{1}R_{2}|a_{n}\rangle=\hbox{Tr}_{\scriptscriptstyle
A}\left[\mathbbm{I}\otimes B\,R_{1}R_{2}\,A\otimes\mathbbm{I}\right]$
$\displaystyle\neq\sum_{n}a_{n}\langle a_{n}|R_{1}R_{2}\,\mathbbm{I}\otimes
B|a_{n}\rangle=\hbox{Tr}_{\scriptscriptstyle A}\left[R_{1}R_{2}\,A\otimes
B\right]$
###### Exercise 13
Consider a generic mixed state $\varrho\in L(H\otimes H)$ and write the matrix
elements of the two partial traces in terms of the matrix elements of
$\varrho$.
###### Exercise 14
Prove that also partial trace is invariant under change of basis.
## Appendix B Uncertainty relations
Two non commuting observables $[X,Y]\neq 0$ do not admit a complete set of
common eigenvectors, and thus it not possible to find common eigenprojectors
and to define a joint observable. Two non commuting observables are said to be
incompatible or complementary, since they cannot assume definite values
simultaneously. A striking consequence of this fact is that when we measure an
observable $X$ the precision of the measurement, as quantified by the variance
$\langle\Delta X^{2}\rangle=\langle X^{2}\rangle-\langle X\rangle^{2}$, is
influenced by the variance of any observable which is non commuting with $X$
and cannot be made arbitrarily small. In order to determine the relationship
between the variances of two noncommuting observables, one of which is
measured on a given state $|\psi\rangle$, let us consider the two vectors
$|\psi_{1}\rangle=(X-\langle
X\rangle)|\psi\rangle\qquad|\psi_{2}\rangle=(Y-\langle
Y\rangle)|\psi\rangle\,,$
and write explicitly the Schwartz inequality
$\langle\psi_{1}|\psi_{1}\rangle\langle\psi_{2}|\psi_{2}\rangle\geq\left|\langle\psi_{1}|\psi_{2}\rangle\right|^{2}$,
i.e. Puri
$\displaystyle\langle\Delta X^{2}\rangle\langle\Delta
Y^{2}\rangle\geq\frac{1}{4}\left[\left|\langle
F\rangle\right|^{2}+\left|\langle
C\rangle\right|^{2}\right]\geq\frac{1}{4}\left|\langle C\rangle\right|^{2}\,,$
(30)
where $[X,Y]=iC$ and $F=XY-YX-2\langle X\rangle\langle Y\rangle$. Ineq. (30)
represents the uncertainty relation for the non commuting observables $X$ and
$Y$ and it is usually presented in the form involving the second inequality.
Uncertainty relations set a lower bound to the measured variance in the
measurement of a single observable, say $X$, on a state with a fixed,
intrinsic, variance of the complementary observable $Y$ (see Section 3.3 for
the relationship between the variance of two non commuting observables in a
joint measurement). The uncertainty product is minimum when the two vectors
$|\psi_{1}\rangle$ and $|\psi_{2}\rangle$ are parallel in the Hilbert space,
i.e. $|\psi_{1}\rangle=-i\lambda|\psi_{2}\rangle$ where $\lambda$ is a complex
number. Minimum uncertainty states (MUS) for the pair of observables $X,Y$ are
thus the states satisfying
$\left(X+i\lambda Y\right)|\psi\rangle=\left(\langle X\rangle+i\lambda\langle
Y\rangle\right)|\psi\rangle\,.$
If $\lambda$ is real then $\langle F\rangle=0$, i.e. the quantities $X$ and
$Y$ are uncorrelated when the physical system is prepared in the state
$|\psi\rangle$. If $|\lambda|=1$ then $\langle\Delta
X^{2}\rangle=\langle\Delta Y^{2}\rangle$ and the corresponding states are
referred to as equal variance MUS. Coherent states of a single-mode radiation
field cah69 are equal variance MUS, e. g. for the pair of quadrature
operators defined by $Q=\frac{1}{\sqrt{2}}(a^{\dagger}+a)$ and
$P=\frac{i}{\sqrt{2}}(a^{\dagger}-a)$.
|
arxiv-papers
| 2011-10-31T14:58:32 |
2024-09-04T02:49:23.765592
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Matteo G. A. Paris",
"submitter": "Matteo G. A. Paris",
"url": "https://arxiv.org/abs/1110.6815"
}
|
1110.6915
|
# Minimal period problems for brake orbits of nonlinear autonomous reversible
semipositive Hamiltonian systems
Duanzhi Zhang
School of Mathematical Sciences and LPMC, Nankai University
Tianjin 300071, People’s Republic of China Partially supported by National
Science Foundation of China (10801078, 11171314), LPMC of Nankai University.
E-mail: zhangdz@nankai.edu.cn
###### Abstract
In this paper, for any positive integer $n$, we study the Maslov-type index
theory of $i_{L_{0}}$, $i_{L_{1}}$ and $i_{\sqrt{-1}}^{L_{0}}$ with
$L_{0}=\\{0\\}\times{\bf R}^{n}\subset{\bf R}^{2n}$ and $L_{1}={\bf
R}^{n}\times\\{0\\}\subset{\bf R}^{2n}$. As applications we study the minimal
period problems for brake orbits of nonlinear autonomous reversible
Hamiltonian systems. For first order nonlinear autonomous reversible
Hamiltonian systems in ${\bf R}^{2n}$, which are semipositive, and
superquadratic at zero and infinity we prove that for any $T>0$, the
considered Hamiltonian systems possesses a nonconstant $T$ periodic brake
orbit $X_{T}$ with minimal period no less than $\frac{T}{2n+2}$. Furthermore
if $\int_{0}^{T}H^{\prime\prime}_{22}(x_{T}(t))dt$ is positive definite, then
the minimal period of $x_{T}$ belongs to $\\{T,\;\frac{T}{2}\\}$. Moreover, if
the Hamiltonian system is even, we prove that for any $T>0$, the considered
even semipositive Hamiltonian systems possesses a nonconstant symmetric brake
orbit with minimal period belonging to $\\{T,\;\frac{T}{3}\\}$.
MSC(2000): 58E05; 70H05; 34C25
Key words: symmetric, brake orbit, semipositive and reversible, Maslov-type
index, minimal period, Hamiltonian systems.
## 1 Introduction and main results
In this paper, let $J=\left(\begin{array}[]{cc}0&-I_{n}\\\
I_{n}&0\end{array}\right)$ and $N=\left(\begin{array}[]{cc}-I_{n}&0\\\
0&I_{n}\end{array}\right)$, where $I_{n}$ is the identity in ${\bf R}^{n}$ and
$n\in{\bf N}$. We suppose the following condition
(H1) $H\in C^{2}({\bf R}^{2n},{\bf R})$ and satisfies the following reversible
condition
$\displaystyle H(Nx)=H(x),\qquad\forall x\in{\bf R}^{2n}.$
We consider the following problem:
$\displaystyle\dot{x}=JH^{\prime}(x),\qquad x\in{\bf R}^{2n},$ (1.1)
$\displaystyle x(-t)=Nx(t),\;\;x(T+t)=x(t),\qquad\forall t\in{\bf R}.$ (1.2)
A solution $(T,x)$ of (1.1)-(1.2) is a special periodic solution of the
Hamiltonian system (1.1). We call it a brake orbit and $T$ the period of $x$.
Moreover, if $x({\bf R})=-x({\bf R})$, we call it a symmetric brake orbit. It
is easy to check that if $\tau$ is the minimal period of $x$, there must holds
$x(t+\frac{\tau}{2})=-x(t)$ for all $t\in{\bf R}$.
Since 1948, when H. Seifert in [47] proposed his famous conjecture of the
existence of $n$ geometrically different brake orbits in the potential well in
${\bf R}^{n}$ under certain conditions, many people began to study this
conjecture and related problems. Let ${}^{\\#}\tilde{\mathcal{O}}({\Omega})$
and ${}^{\\#}\tilde{\mathcal{J}}_{b}({\Sigma})$ the number of geometrically
distinct brake obits in ${\Omega}$ for the second order case and on ${\Sigma}$
for the first order case respectively. S. Bolotin proved first in [7](also see
[8]) of 1978 the existence of brake orbits in general setting. K. Hayashi in
[27], H. Gluck and W. Ziller in [25], and V. Benci in [5] in 1983-1984 proved
${}^{\\#}\tilde{\mathcal{O}}({\Omega})\geq 1$ if $V$ is $C^{1}$,
$\bar{{\Omega}}=\\{V\leq h\\}$ is compact, and $V^{\prime}(q)\neq 0$ for all
$q\in\partial{{\Omega}}$. In 1987, P. Rabinowitz in [45] proved that if $H$ is
$C^{1}$ and satisfies the reversible conditon, ${\Sigma}\equiv H^{-1}(h)$ is
star-shaped, and $x\cdot H^{\prime}(x)\neq 0$ for all $x\in{\Sigma}$, then
${}^{\\#}\tilde{\mathcal{J}}_{b}({\Sigma})\geq 1$. In 1987, V. Benci and F.
Giannoni gave a different proof of the existence of one brake orbit in [6].
In 1989, A. Szulkin in [49] proved that ${}^{\\#}\tilde{{\cal
J}_{b}}(H^{-1}(h))\geq n$, if $H$ satisfies conditions in [43] of Rabinowitz
and the energy hypersurface $H^{-1}(h)$ is $\sqrt{2}$-pinched. E. van Groesen
in [26] of 1985 and A. Ambrosetti, V. Benci, Y. Long in [1] of 1993 also
proved ${}^{\\#}\tilde{\mathcal{O}}({\Omega})\geq n$ under different pinching
conditions.
In [42] of 2006, Long , Zhu and the author of this paper proved that there
exist at least $2$ geometrically distinct brake orbits on any central
symmetric strictly convex hypersuface ${\Sigma}$ in ${\bf R}^{2n}$ for $n\geq
2$. Recently, in [35], Liu and the author of this paper proved that there
exist at least $[n/2]+1$ geometrically distinct brake orbits on any central
symmetric strictly convex hypersuface ${\Sigma}$ in ${\bf R}^{2n}$ for $n\geq
2$, if all brake orbits on ${\Sigma}$ are nondegenerate then there are at
least $n$ geometrically distinct brake orbits on ${\Sigma}$. For more details
one can refer to [42], [35] and the reference there in.
In his pioneering paper [43] of 1978, P. Rabinowitz proved the following
famous result via the variational method. Suppose $H$ satisfies the following
conditions:
($\rm{H}1^{\prime}$) $H\in C^{1}({\bf R}^{2n},{\bf R})$.
(H2) There exist constants $\mu>2$ and $r_{0}>0$ such that
$0<\mu H(x)\leq H^{\prime}(x)\cdot x,\quad\forall|x|\leq r_{0}.$
(H3) $H(x)=o(|x|^{2})$ at $x=0$.
(H4) $H(x)\geq 0$ for all $x\in{\bf R}^{2n}$.
Then for any $T>0$, the system (1.1) possesses a non-constant $T$-periodic
solution. Because a $T/k$ periodic function is also a $T$-periodic function,
in [43] Rabinowitz proposed a conjecture that under conditions (H1′) and
(H2)-(H4), there is a non-constant solution possessing any prescribed minimal
period. Since 1978, this conjecture has been deeply studied by many
mathematicians. A significant progress was made by Ekeland and Hofer in their
celebrated paper [16] of 1985, where they proved Rabinowitz’s conjecture for
the strictly convex Hamiltonian system. For Hamiltonian systems with convex or
weak convex assumptions, we refer to [2]-[3], [12]-[13], [15]-[17], [41],
[20]-[23], and references therein for more details. For the case without
convex condition we refer to [37]-[39] and Chapter 13 of [41] and references
therein. A interesting result is for the semipositive first order Hamiltonian
system, in [18] G. Fei, S.-T. Kim, and T. Wang proved the existence of a T
periodic solution of system (1.1) with minimal period no less than $T/2n$ for
any given $T>0$.
Note that in the second order Hamiltonian systems there are many results on
the minimal problem of brake orbits such us [37]-[39] and [50]. For the even
first order Hamiltonian system, in [51], the author of this paper studied the
minimal period problem of semipositive even Hamiltonian system and gave a
positive answer to Rabinowitz’s conjecture in that case. In [19], G. Fei,
S.-T. Kim, and T. Wang proved the same result for second order Hamiltonian
systems.
So it is natural to consider the minimal period problem of brake orbits in
reversible first order nonlinear Hamiltonian systems. In [32], Liu have
considered the strictly convex reversible Hamiltonian systems case and proved
the existence of nonconstant brake orbit of (1.1) with minimal period
belonging to $\\{T,T/2\\}$ for any given $T>0$.
Since [51], we also hope to obtain some interesting results in the even
Hamiltonian system for the minimal period problem of brake orbits.
It can be found in many papers mentioned above that the Maslov-type index
theory and its iteration theory play a important role in the study of minimal
period problems in Hamiltonian systems. In this paper we study some
monotonicity properties of Maslov-type index and apply it to prove our main
results.
In this paper we denote by $\mathcal{L}({\bf R}^{2n})$ and
$\mathcal{L}_{s}({\bf R}^{2n})$ the set of all real $2n\times 2n$ matrices and
symmetric matrices respectively. And we denote by $y_{1}\cdot y_{2}$ the usual
inner product for all $y_{1},\;y_{2}\in{\bf R}^{k}$ with $k$ being any
positive integer. Also we denote by ${\bf N}$ and ${\bf Z}$ the set of
positive integers and integers respectively.
Let ${\rm Sp}(2n)=\\{M\in\mathcal{L}({\bf R}^{2n})|M^{T}JM=J\\}$ be the
$2n\times 2n$ real symplectic group. For any $\tau>0$, Set
$\mathcal{P}_{\tau}=\\{{\gamma}\in C([0,\tau],{\rm
Sp}(2n))|{\gamma}(0)=I_{2n}\\}$ and $S_{\tau}={\bf R}/(\tau{\bf Z})$.
For any ${\gamma}\in\mathcal{P}_{\tau}$ and ${\omega}\in{\bf U}$, where ${\bf
U}$ is the unit circle of the complex plane ${\bf C}$, the Maslov-type index
$(i_{\omega}({\gamma}),\nu_{\omega}({\gamma}))\in{\bf Z}\times\\{0,1,...2n\\}$
was defined by Long in [40]. We have a brief review in Appendix of Section 6.
For convenience to introduce our results, we define the following (B1)
condition, since the Hamiltonian systems considered here are reversible, this
condition is natural.
(B1) Condition. For any $\tau>0$ and $B\in C([0,\tau],\mathcal{L}_{s}({\bf
R}^{2n})$ with the $n\times n$ matrix square block form
$B(t)=\left(\begin{array}[]{cc}B_{11}(t)&B_{12}(t)\\\
B_{21}(t)&B_{22}(t)\end{array}\right)$ satisfying
$B_{12}(0)=B_{21}(0)=0=B_{12}(\tau)=B_{21}(\tau)$, We will call $B$ satisfies
the condition (B1).
Throughout this paper, we denote by
$\displaystyle L_{0}=\\{0\\}\times{\bf R}^{n}\subset{\bf R}^{2n},\quad
L_{1}={\bf R}^{n}\times\\{0\\}\subset{\bf R}^{2n}.$ (1.3)
The definitions of Maslov-type indices
$(i_{\sqrt{-1}}^{L_{0}}({\gamma}),\nu_{\sqrt{-1}}^{L_{0}}({\gamma}))$ and
$(i_{L_{j}}({\gamma}),\nu_{L_{j}}({\gamma}))\in{\bf Z}\times\\{0,1,...,n\\}$
for $j=0,1$ and ${\gamma}\in\mathcal{P}_{\tau}(2n)$ with $\tau>0$ can be found
in [42] and Section 2 below. Also for $B\in C([0,\tau],\mathcal{L}_{s}({\bf
R}^{2n})$ satisfies condition (B1), the definitions of
$(i_{\sqrt{-1}}^{L_{0}}(B),\nu_{\sqrt{-1}}^{L_{0}}(B))$ and
$(i_{L_{j}}(B),\nu_{L_{j}}(B))\in{\bf Z}\times\\{0,1,...,n\\}$ for $j=0,1$ and
${\gamma}\in\mathcal{P}_{\tau}(2n)$ can be found in Section 2 and references
therein.
For any $B\in C([0,\tau],\mathcal{L}_{s}({\bf R}^{2n}))$, denote by
${\gamma}_{B}$ the fundamental solution of the following problem:
$\displaystyle\dot{{\gamma}}_{B}(t)$ $\displaystyle=$ $\displaystyle
JB(t){\gamma}_{B}(t),$ (1.4) $\displaystyle{\gamma}_{B}(0)$ $\displaystyle=$
$\displaystyle I_{2n}.$ (1.5)
Then ${\gamma}_{B}\in\mathcal{P}_{\tau}$. We call ${\gamma}_{B}$ the
symplectic path associated to $B$.
Definition 1.1. If $H\in C^{2}({\bf R}^{2n},{\bf R})$ is a reversible
function, for any $x_{\tau}$ be a $\tau$-periodic brake orbit solution of
(1.1), let $B(t)=H^{\prime\prime}(x(t))$, we define
${\gamma}_{x_{\tau}}={\gamma}_{B}|_{[0,\frac{\tau}{2}]}$ and call it the
symplectic path associated to $x_{\tau}$. We define
$i_{L_{0}}(x_{\tau})=i_{L_{0}}({\gamma}_{x_{\tau}}),\qquad\nu_{L_{0}}(x_{\tau})=i_{L_{0}}({\gamma}_{x_{\tau}}).$
(1.6)
Moreover, if $H$ is even and $x_{\tau}$ is a $\tau$-periodic symmetric brake
orbit solution of (1.1), let $B(t)=H^{\prime\prime}(x(t))$, we define
${\gamma}_{x_{\tau}}={\gamma}_{B}|_{[0,\frac{\tau}{4}]}$ and call it the
symplectic path associated to $x_{\tau}$. We define
$i_{\sqrt{-1}}^{L_{0}}(x_{\tau})=i_{\sqrt{-1}}^{L_{0}}({\gamma}_{x_{\tau}}),\qquad\nu_{\sqrt{-1}}^{L_{0}}(x_{\tau})=i_{\sqrt{-1}}^{L_{0}}({\gamma}_{x_{\tau}}).$
(1.7)
Definition 1.2. For any $\tau$-period and $k\in{\bf N}\equiv\\{1,2,...\\}$, we
define the $k$ times iteration $x^{k}$ of $x$ by
$x^{k}(t)=x(t-j\tau),\quad j\tau\leq t\leq(j+1)\tau,\quad 0\leq j\leq k.$
(1.8)
As in [35], for any ${\gamma}\in\mathcal{P}_{\tau}$ and $k\in{\bf
N}\equiv\\{1,2,...\\}$, in this paper the $k$-time iteration ${\gamma}^{k}$ of
${\gamma}\in\mathcal{P}_{\tau}(2n)$ in brake orbit boundary sense is defined
by $\tilde{{\gamma}}|_{[0,k\tau]}$ with
$\displaystyle\tilde{{\gamma}}(t)=\left\\{\begin{array}[]{l}{\gamma}(t-2j\tau)(N{\gamma}(\tau)^{-1}N{\gamma}(\tau))^{j},\;t\in[2j\tau,(2j+1)\tau],j=0,1,2,...\\\
N{\gamma}(2j\tau+2\tau-t)N(N{\gamma}(\tau)^{-1}N{\gamma}(\tau))^{j+1}\;t\in[(2j+1)\tau,(2j+2)\tau],j=0,1,2,...\end{array}\right.$
(1.11)
The followings are our main results of this paper.
Theorem 1.1. Suppose that $H$ satisfies conditions (H1)-(H4) and
(H5) $H^{\prime\prime}(x)$ is semipositive definite for all $x\in{\bf
R}^{2n}$.
Then for any $T>0$, the system (1.1)-(1.2) possesses a nonconstant $T$
periodic brake orbit solution $x_{T}$ with minimal period no less that
$\frac{T}{2n+2}$. Moreover, for $x=(x_{1},x_{2})$ with $x_{1},x_{2}\in{\bf
R}^{n}$, denote by $H^{\prime\prime}_{22}(x)$ the second order differential of
$H$ with respect to $x_{2}$, if
$\int_{0}^{\frac{T}{2}}H^{\prime\prime}_{22}(x_{T}(t))\,dt>0,$ (1.12)
then the minimal period of $x_{T}$ belongs to $\\{T,\frac{T}{2}\\}$.
Remark 1.1. (Theorem 1.1 of [32]) Suppose that $H$ satisfies conditions
(H1)-(H4) and if $x_{T}$ satisfies
(H5′) $\int_{0}^{\frac{T}{2}}H^{\prime\prime}(X_{T}(t))\,dt>0$.
Then the minimal period of $x_{T}$ belongs to $\\{T,\frac{T}{2}\\}$.
In the case $n=1$, the result can be better, i.e., the following
Theorem 1.2. For $n=1$, suppose that $H$ satisfies conditions (H1)-(H4).
Then for any $T>0$, the system (1.1)-(1.2) possesses a nonconstant $T$
periodic brake orbit solution with minimal period belong to
$\\{T,\frac{T}{2}\\}$.
Consider the minimal period problem for $H(x)=\frac{1}{2}B_{0}x\cdot
x+\hat{H}(x)$, where $B_{0}\in\mathcal{L}_{s}({\bf R}^{2n})$. This is
motivated by [18], [22], and [43], where in [18] $B_{0}$ was considered to be
semipositive, in [22] and [43] $B_{0}$ was considered to be positive.
We have the following general result.
Theorem 1.3. Let $2n\times 2n$ be real semipositive matrix $B_{0}={\rm
diag}(B_{11},B_{22})$ with $B_{11}$ and $B_{22}$ being $n\times n$ matrix.
Assume $H(x)=\frac{1}{2}B_{0}x\cdot x+\hat{H}(x)$ for all $x\in{\bf R}^{2n}$,
and $\hat{H}$ satisfies conditions (H1)-(H5).
Then for any $T>0$, (1.1) possesses a nonconstant $T$-periodic brake orbit
$x_{T}$ with minimal period no less than
$\frac{T}{2i_{L_{0}}(B_{0})+2\nu_{L_{0}}(B_{0})+2n+2}$, where we see $B_{0}$
as an element in $C([0,T/2],\mathcal{L}_{s}({\bf R}^{2n}))$ satisfies
condition (B1).
Remark 1.2. In section 3, we will show
$i_{L_{0}}(B_{0})+\nu_{L_{0}}(B_{0})\geq 0$.
As a direct consequence of Theorem 1.3, we have the following Corollary 1.1.
Corollary 1.1. For $T>0$ such that $i_{L_{0}}(B_{0})+\nu_{L_{0}}(B_{0})=0$,
where we see $B_{0}$ as an element in $C([0,T/2],\mathcal{L}_{s}({\bf
R}^{2n})$ satisfies condition (B1), under the same assumptions of Theorem 1.2,
the system (1.1) possesses a nonconstant $T$-periodic brake orbit with minimal
period no less that $\frac{T}{2n+2}$.
We can also prove the following Corollary 1.2 of Theorem 1.3.
Corollary 1.2. If $B_{0}\neq 0$, then for $0<T<\frac{\pi}{||B_{0}||}$ with
$||B_{0}||$ being the operator norm of $B_{0}$, under the same condition of
Theorem 1.2, possesses a nonconstant $T$-periodic brake orbit $x_{T}$ with
minimal period no less than $\frac{T}{2n+2}$. Moreover , if
$\displaystyle\int_{0}^{\frac{T}{2}}H^{\prime\prime}_{22}(x_{T}(t))\,dt>0,$
then the minimal period of $x_{T}$ belongs to $\\{T,\frac{T}{2}\\}$.
Theorem 1.4. Suppose that $H$ satisfies conditions (H1)-(H5) and
(H6) $H(-x)=H(x)$ for all $x\in{\bf R}^{2n}$.
Then for any $T>0$, the system (1.1)-(1.2) possesses a nonconstant symmetric
brake orbit with minimal period belonging to $\\{T,T/3\\}$.
Theorem 1.5. Let $2n\times 2n$ be real semipositive matrix $B_{0}={\rm
diag}(B_{11},B_{22})$ with $B_{11}$ and $B_{22}$ being $n\times n$ matrix,
assume $H(x)=\frac{1}{2}B_{0}x\cdot x+\hat{H}(x)$ for all $x\in{\bf R}^{2n}$,
and $\hat{H}$ satisfies conditions (H1)-(H6). Then for any $T>0$, the system
(1.1)-(1.2) possesses a nonconstant symmetric brake orbit $x_{T}$ with minimal
period no less than
$\frac{T}{4(i_{\sqrt{-1}}^{L_{0}}(B_{0})+\nu_{\sqrt{-1}}^{L_{0}}(B_{0}))+7}$.
Moreover, if $i_{\sqrt{-1}}^{L_{0}}(B_{0})+\nu_{\sqrt{-1}}^{L_{0}}(B_{0})$ is
even, then the minimal period of $x_{T}$ is no less than
$\frac{T}{4(i_{\sqrt{-1}}^{L_{0}}(B_{0})+\nu_{\sqrt{-1}}^{L_{0}}(B_{0}))+3}$
where we see $B_{0}$ as an element in $C([0,T/4],\mathcal{L}_{s}({\bf
R}^{2n})$ satisfies condition (B1).
Remark 1.3. In section 3, we will show that $i_{\sqrt{-1}}^{L_{0}}(B_{0})\geq
0$, hence $i_{\sqrt{-1}}^{L_{0}}(B_{0})+\nu_{\sqrt{-1}}^{L_{0}}(B_{0})\geq 0$.
As a direct consequence of Theorem 1.5, we have the following Corollary 1.3.
Corollary 1.3. For $T>0$ such that
$i_{\sqrt{-1}}^{L_{0}}(B_{0})+\nu_{\sqrt{-1}}^{L_{0}}(B_{0})=0$, under the
same assumptions of Theorem 1.4, the system (1.1) possesses a nonconstant
symmetric brake orbit with minimal period belonging to $\\{T,T/3\\}$.
We can also prove the following Corollary 1.4 of Theorem 1.5.
Corollary 1.4. If $B_{0}\neq 0$, then for $0<T<\frac{\pi}{||B_{0}||}$ with
$||B_{0}||$ being the operator norm of $B_{0}$, under the same condition of
Theorem 1.5, the system (1.1) possesses a nonconstant symmetric brake orbit
with minimal period belonging to $\\{T,T/3\\}$.
This paper is organized as follows. In section 2, we study the Maslov-type
index theory of $i_{L_{0}}$, $i_{L_{1}}$ and $i_{\sqrt{-1}}^{L_{0}}$. We
compute the difference between $i_{L_{0}}({\gamma})$ and
$i_{L_{1}}({\gamma})$. In Section 3, we study the relation between the Maslov-
type index $(i_{\sqrt{-1}}^{L_{0}}(B),\nu_{\sqrt{-1}}^{L_{0}}(B))$ for $B\in
C([0,\tau],\mathcal{L}_{s}({\bf R}^{2n})$ satisfies condition (B1) and the
Morse indices of the corresponding Galerkin approximation. As applications we
get some monotonicity properties of $i_{L_{0}}(B)$, $i_{L_{1}}(B)$ and
$i_{\sqrt{-1}}^{L_{0}}(B)$ and we prove Theorem 3.2 which is very important in
the proof of Theorems 1.4-1.5. In Section 4, based on the preparations in
Sections 2 and 3 we prove Theorems 1.1-1.3 and Corollary 1.2. In Section 5, we
prove Theorems 1.4-1.5 and corollary 1.4. In Section 6, we give a briefly
review of $(i_{\omega},\nu_{\omega})$ index theory with ${\omega}\in{\bf U}$
for symplectic paths starting with identity as appendix.
## 2 Maslov-type index theory associated with Lagrangian subspaces
### 2.1 A brief review of index function $(i_{L_{j}},\nu_{L_{j}})$ with
$j=0,1$ and $(i_{\sqrt{-1}}^{L_{0}},\nu_{\sqrt{-1}}^{L_{0}})$
Let
$F={\bf R}^{2n}\oplus{\bf R}^{2n}$ (2.1)
possess the standard inner product. We define the symplectic structure of $F$
by
$\\{v,w\\}=(\mathcal{J}v,w),\;\forall v,w\in F,\;{\rm
where}\;\mathcal{J}=(-J)\oplus J=\left(\begin{array}[]{cc}-J&0\\\
0&J\end{array}\right).\;$ (2.2)
We denote by ${\rm Lag}(F)$ the set of Lagrangian subspaces of $F$, and equip
it with the topology as a subspace of the Grassmannian of all $2n$-dimensional
subspaces of $F$.
It is easy to check that, for any $M\in{\rm Sp}(2n)$ its graph
${\rm Gr}(M)\equiv\left\\{\left(\begin{array}[]{c}x\\\
Mx\end{array}\right)|x\in{\bf R}^{2n}\right\\}$
is a Lagrangian subspace of $F$.
Let
$\displaystyle V_{1}=\\{0\\}\times{\bf R}^{n}\times\\{0\\}\times{\bf
R}^{n}\subset{\bf R}^{4n},\quad V_{2}={\bf R}^{n}\times\\{0\\}\times{\bf
R}^{n}\times\\{0\\}\subset{\bf R}^{4n}.$ (2.3)
By Proposition 6.1 of [35] and Lemma 2.8 and Definition 2.5 of [42], we give
the following definition.
Definition 2.1. For any continuous path ${\gamma}\in\mathcal{P}_{\tau}(2n)$,
we define the following Maslov-type indices:
$\displaystyle i_{L_{0}}({\gamma})=\mu^{CLM}_{F}(V_{1},{\rm
Gr}({\gamma}),[0,\tau])-n,$ (2.4) $\displaystyle
i_{L_{1}}({\gamma})=\mu^{CLM}_{F}(V_{2},{\rm Gr}({\gamma}),[0,\tau])-n,$ (2.5)
$\displaystyle\nu_{L_{j}}({\gamma})=\dim({\gamma}(\tau)L_{j}\cap L_{j}),\qquad
j=0,1,$ (2.6)
where we denote by $i^{CLM}_{F}(V,W,[a,b])$ the Maslov index for Lagrangian
subspace path pair $(V,W)$ in $F$ on $[a,b]$ defined by Cappell, Lee, and
Miller in [11].
For $\omega=e^{\sqrt{-1}\theta}$ with $\theta\in{\bf R}$, we define a Hilbert
space $E^{\omega}=E^{\omega}_{L_{0}}$ consisting of those $x(t)$ in
$L^{2}([0,\tau],{\bf C}^{2n})$ such that $e^{-\theta tJ}x(t)$ has Fourier
expending
$e^{-\frac{\theta t}{\tau}J}x(t)=\sum_{j\in{\bf Z}}e^{\frac{j\pi
t}{\tau}J}\left(\begin{array}[]{cc}0\\\ a_{j}\end{array}\right),\;a_{j}\in{\bf
C}^{n}$
with
$\|x\|^{2}:=\sum_{j\in{\bf Z}}\tau(1+|j|)|a_{j}|^{2}<\infty.$
For $\omega=e^{\sqrt{-1}\theta}$, $\theta\in(0,\pi)$, we define two self-
adjoint operators $A^{\omega},B^{\omega}\in\mathcal{L}(E^{\omega})$ by
$\displaystyle(A^{\omega}x,y)=\int^{1}_{0}\langle-J\dot{x}(t),y(t)\rangle
dt,\;\;(B^{\omega}x,y)=\int^{1}_{0}\langle B(t)x(t),y(t)\rangle dt$
on $E^{\omega}$. Then $B^{\omega}$ is also compact.
Definition 2.2. We define the index function
$\displaystyle
i_{\omega}^{L_{0}}(B)=I(A^{\omega},\;\;A^{\omega}-B^{\omega})\equiv-{\rm
sf}\\{A^{\omega}-sB^{\omega},0\leq s\leq 1\\},$
$\displaystyle\nu_{\omega}^{L_{0}}(B)=m^{0}(A^{\omega}-B^{\omega}),\;\forall\,\omega=e^{\sqrt{-1}\theta},\;\;\theta\in(0,\pi),$
where the definition of ${\rm sf}$ of spectral flow for the path of bounded
self-adjoint linear operators one can refer to [53] and references their in.
By (3.21) of [35], we have
$i_{L_{0}}(B)\leq i^{L_{0}}_{{\omega}}(B)\leq i_{L_{0}}(B)+n.$ (2.7)
Lemma 2.1. For ${\omega}=e^{\sqrt{-1}\theta}$ with $\theta\in(0,\pi)$, let
$V_{\omega}=L_{0}\times(e^{\theta J}L_{0})\subset{\bf R}^{4n}\equiv F$. There
holds
$i_{\omega}^{L_{0}}(B)=\mu_{F}^{CLM}(V_{\omega},{\rm
Gr}({\gamma}_{B}),[0,\tau]).$ (2.8)
Proof. Without loss of generality we can suppose the $C^{1}$ path ${\rm
Gr}({\gamma}_{B})$ of Lagrangian subspaces intersects $V_{\omega}$ regularly
(otherwise we can perturb it slightly with fixed endow points such that they
intersects regularly and the index dose not change by the homotopy invariant
property $\mu_{F}^{CLM}$ ), where the definition of intersection form can be
found in [46]. We denote by $\mu^{BF}$ the maslov index defined by Booss and
Furutani in [9].
By the spectral flow formula of Theorem 5.1 in [9] or Theorem 1.5 of [10] (cf.
also proof of Proposition 2.3 of [52]), we have
$\displaystyle{\rm sf}\\{A^{\omega}sB^{\omega},0\leq s\leq 1\\}$ (2.9)
$\displaystyle=$ $\displaystyle\mu^{BF}({\rm
Gr}({\gamma}_{B}),V_{\omega},[0,\tau])$ $\displaystyle=$
$\displaystyle\mu^{BF}((I\oplus e^{-\sqrt{-1}\theta J}){\rm
Gr}({\gamma}_{B}),(I\oplus e^{-\sqrt{-1}\theta J})V_{\omega},[0,\tau])$
$\displaystyle=$ $\displaystyle\mu^{BF}((I\oplus e^{-\sqrt{-1}\theta J}){\rm
Gr}({\gamma}_{B}),V_{1},[0,\tau])$ $\displaystyle=$
$\displaystyle-m^{-}(-{\Gamma}((I\oplus e^{-\sqrt{-1}\theta J}){\rm
Gr}({\gamma}_{B}),V_{1},0))+\sum_{0<t<\tau}{\rm sign}(-{\Gamma}((I\oplus
e^{-\sqrt{-1}\theta J}){\rm Gr}({\gamma}_{B}),V_{1},t))$
$\displaystyle+m^{+}(-{\Gamma}((I\oplus e^{-\sqrt{-1}\theta J}){\rm
Gr}({\gamma}_{B}),V_{1},\tau))$ $\displaystyle=$
$\displaystyle-m^{+}({\Gamma}((I\oplus e^{-\sqrt{-1}\theta J}){\rm
Gr}({\gamma}_{B}),V_{1},0))-\sum_{0<t<\tau}{\rm sign}({\Gamma}((I\oplus
e^{-\sqrt{-1}\theta J}){\rm Gr}({\gamma}_{B}),V_{1},t))$
$\displaystyle+m^{-}({\Gamma}((I\oplus e^{-\sqrt{-1}\theta J}){\rm
Gr}({\gamma}_{B}),V_{1},\tau))$ $\displaystyle=$
$\displaystyle-\mu_{F}^{CLM}(V_{1},(I\oplus e^{-\sqrt{-1}\theta J}){\rm
Gr}({\gamma}_{B}),[0,\tau])$ $\displaystyle=$
$\displaystyle-\mu_{F}^{CLM}((I\oplus e^{\sqrt{-1}\theta J})V_{1},{\rm
Gr}({\gamma}_{B}),[0,\tau])$ $\displaystyle=$
$\displaystyle-\mu_{F}^{CLM}(V_{\omega},{\rm Gr}({\gamma}_{B}),[0,\tau]),$
where in the fourth equality we have used Theorem 2.1 in [9] and the property
of index $\mu^{RS}$ for symplectic paths defined in [46](cf also (2.6)-(2.8)
of [52]), in the sixth equality we have used Lemma 2.6 of [42], in the second
and seventh equalities we used the symplectic invariance property of index
$\mu^{BF}$ and $\mu_{F}^{CLM}$ respectively.
Definition 2.3. Let $B\in C([0,\tau],\mathcal{L}_{s}({\bf R}^{2n})$ and
${\gamma}_{B}$ be the symplectic path associated to $B$. We define
$\displaystyle i_{{\omega}}^{L_{0}}({\gamma}_{B})=i_{{\omega}}^{L_{0}}(B),$
(2.10)
$\displaystyle\nu_{{\omega}}^{L_{0}}({\gamma}_{B})=\nu_{{\omega}}^{L_{0}}(B).$
(2.11)
By Lemma 2.1, in general we give the following definition.
Definition 2.4. For any ${\gamma}\in\mathcal{P}_{\tau}(2n)$ and
${\omega}=e^{\sqrt{-1}\theta}$ with $\theta\in(0,\pi)$, we define
$\displaystyle i_{\omega}^{L_{0}}({\gamma})=\mu_{F}^{CLM}(V_{\omega},{\rm
Gr}({\gamma}_{B}),[0,\tau]),$
$\displaystyle\nu_{\omega}^{L_{0}}({\gamma})=\dim\left({\gamma}(\tau)L_{0}\cap
e^{\sqrt{-1}\theta J}L_{0}\right).$ (2.12)
For any ${\gamma}\in\mathcal{P}_{\tau}(2n)$, we define a new symplectic path
$\tilde{{\gamma}}\in\mathcal{P}_{\tau}(2n)$ by
$\tilde{{\gamma}}(t)=\left\\{\begin{array}[]{l}I_{2n},\quad
t\in[0,\frac{\tau}{3}],\\\ {\gamma}(3t-\tau),\quad
t\in[\frac{\tau}{3},\frac{2\tau}{3}],\\\ {\gamma}(\tau),\quad
t\in[\frac{2\tau}{3},\tau].\end{array}\right.$ (2.13)
So we can perturb $\tilde{{\gamma}}$ slightly to a $C^{1}$ path
$\hat{{\gamma}}$ such that $\hat{{\gamma}}$ is homotopic to $\tilde{{\gamma}}$
with fixed end points and $\hat{{\gamma}}(t)=I_{2n}$ for
$t\in[0,\frac{\tau}{6}]$ and $\hat{{\gamma}}(t)={\gamma}(\tau)$ for
$t\in[\frac{5\tau}{6},\tau]$. Set
$\hat{B}(t)=-J\dot{\hat{{\gamma}}}(t)(\hat{{\gamma}}(t))^{-1}$. So we have
$\hat{B}(0)=\hat{B}(\tau)=0.$ (2.14)
Then this $\hat{B}\in C([0,\tau],\mathcal{L}_{s}({\bf R}^{2n})$ and satisfies
condition (B1). Also we have $\hat{{\gamma}}$ is is homotopic to ${\gamma}$
with fixed end points. So we have
$\displaystyle
i_{1}(\hat{{\gamma}}^{k})=i_{1}({\gamma}^{k})=i_{1}({\gamma}_{\hat{B}}^{k}),\qquad\forall
k\in{\bf N},$ (2.15)
$\displaystyle\nu_{1}(\hat{{\gamma}}^{k})=\nu_{1}({\gamma}^{k})=\nu_{1}({\gamma}_{\hat{B}}^{k}),\qquad\forall
k\in{\bf N}$ (2.16)
and
$\displaystyle
i_{L_{0}}(\hat{{\gamma}}^{k})=i_{L_{0}}({\gamma}^{k})=i_{L_{0}}({\gamma}_{\hat{B}}^{k}),\qquad\forall
k\in{\bf N},$ (2.17)
$\displaystyle\nu_{L_{0}}(\hat{{\gamma}}^{k})=\nu_{L_{0}}({\gamma}^{k})=\nu_{L_{0}}({\gamma}_{\hat{B}}^{k}),\qquad\forall
k\in{\bf N}.$ (2.18)
Also by the property of index $\mu_{F}^{CLM}$ and Definition 2.4 have
$\displaystyle
i_{\sqrt{-1}}^{L_{0}}({\gamma}^{k})=i_{\sqrt{-1}}^{L_{0}}(\hat{{\gamma}}^{k})=i_{\sqrt{-1}}^{L_{0}}({\gamma}_{\hat{B}}^{k}),\qquad\forall
k\in{\bf N},$
$\displaystyle\nu_{\sqrt{-1}}^{L_{0}}({\gamma}^{k})=\nu_{\sqrt{-1}}^{L_{0}}(\hat{{\gamma}}^{k})=\nu_{\sqrt{-1}}^{L_{0}}({\gamma}_{\hat{B}}^{k}),\qquad\forall
k\in{\bf N}.$
Hence, in [35] the authors essentially proved the following Bott-type
iteration formula.
Theorem 2.1. (Theorem 4.1 of [35]) Let ${\gamma}\in\mathcal{P}_{\tau}(2n)$ and
$\omega_{k}=e^{\pi\sqrt{-1}/k}$. For odd $k$ we have
$\displaystyle
i_{L_{0}}(\gamma^{k})=i_{L_{0}}(\gamma^{1})+\sum_{i=1}^{(k-1)/2}i_{\omega_{k}^{2i}}(\gamma^{2}),$
$\displaystyle\nu_{L_{0}}(\gamma^{k})=\nu_{L_{0}}(\gamma^{1})+\sum_{i=1}^{(k-1)/2}\nu_{\omega_{k}^{2i}}(\gamma^{2}),$
and for even $k$, we have
$\displaystyle
i_{L_{0}}(\gamma^{k})=i_{L_{0}}(\gamma^{1})+i^{L_{0}}_{\sqrt{-1}}(\gamma^{1})+\sum_{i=1}^{k/2-1}i_{\omega_{k}^{2i}}(\gamma^{2}),\;$
$\displaystyle\nu_{L_{0}}(\gamma^{k})=\nu_{L_{0}}(\gamma^{1})+\nu^{L_{0}}_{\sqrt{-1}}(\gamma^{1})+\sum_{i=1}^{k/2-1}\nu_{\omega_{k}^{2i}}(\gamma^{2}).$
Obviously we also have
$i_{L_{0}}({\gamma})\leq i^{L_{0}}_{\sqrt{-1}}({\gamma})\leq
i_{L_{0}}({\gamma})+n.$ (2.19)
### 2.2 The Bott-type iteration formula for
$(i_{\sqrt{-1}}^{L_{0}},\nu_{\sqrt{-1}}^{L_{0}})$
In order to study the minimal period problem for Even reversible Hamiltonian
systems, we need the iteration formula of the Maslov-type index of
$(i_{\sqrt{-1}}^{L_{0}},\nu_{\sqrt{-1}}^{L_{0}})$ for symplectic paths
starting with identity. We use Theorem 2.1 to obtain it.
Precisely we have the following Theorem.
Theorem 2.2. Let ${\gamma}\in\mathcal{P}_{\tau}(2n)$ and
$\omega_{k}=e^{\pi\sqrt{-1}/k}$. For odd $k$ we have
$\displaystyle
i_{\sqrt{-1}}^{L_{0}}(\gamma^{k})=i_{\sqrt{-1}}^{L_{0}}(\gamma^{1})+\sum_{i=1}^{(k-1)/2}i_{\omega_{k}^{2i-1}}(\gamma^{2}),$
(2.20)
$\displaystyle\nu_{\sqrt{-1}}^{L_{0}}(\gamma^{k})=\nu_{\sqrt{-1}}^{L_{0}}(\gamma^{1})+\sum_{i=1}^{(k-1)/2}\nu_{\omega_{k}^{2i-1}}(\gamma^{2}),$
(2.21)
and for even $k$, we have
$\displaystyle
i_{\sqrt{-1}}^{L_{0}}(\gamma^{k})=\sum_{i=1}^{k/2}i_{\omega_{k}^{2i-1}}(\gamma^{2}),\;$
(2.22)
$\displaystyle\nu_{\sqrt{-1}}^{L_{0}}(\gamma^{k})=\sum_{i=1}^{k/2}\nu_{\omega_{k}^{2i-1}}(\gamma^{2}).$
(2.23)
Proof. For odd $k$, since ${\gamma}^{2k}=({\gamma}^{k})^{2}$, by Theorem 2.1
we have
$\displaystyle
i_{L_{0}}({\gamma}^{2k})=i_{L_{0}}({\gamma}^{k})+i_{\sqrt{-1}}^{L_{0}}({\gamma}^{k}),$
(2.24)
$\displaystyle\nu_{L_{0}}({\gamma}^{2k})=\nu_{L_{0}}({\gamma}^{k})+\nu_{\sqrt{-1}}^{L_{0}}({\gamma}^{k}).$
(2.25)
Also by Theorem 2.1 we have
$\displaystyle
i_{L_{0}}(\gamma^{k})=i_{L_{0}}(\gamma^{1})+\sum_{i=1}^{(k-1)/2}i_{\omega_{k}^{2i}}(\gamma^{2}),$
(2.26)
$\displaystyle\nu_{L_{0}}(\gamma^{k})=\nu_{L_{0}}(\gamma^{1})+\sum_{i=1}^{(k-1)/2}\nu_{\omega_{k}^{2i}}(\gamma^{2}),$
(2.27) $\displaystyle
i_{L_{0}}(\gamma^{2k})=i_{L_{0}}(\gamma^{1})+i_{\sqrt{-1}}^{L_{0}}({\gamma})+\sum_{i=1}^{k-1}i_{\omega_{2k}^{2i}}(\gamma^{2}),$
(2.28)
$\displaystyle\nu_{L_{0}}(\gamma^{2k})=\nu_{L_{0}}(\gamma^{1})+\nu_{\sqrt{-1}}^{L_{0}}({\gamma})+\sum_{i=1}^{k-1}\nu_{\omega_{2k}^{2i}}(\gamma^{2}).$
(2.29)
Since ${\omega}_{k}={\omega}_{2k}^{2}$, by (2.24), (2.28) minus (2.26) yields
(2.20). By (2.25), (2.29) minus (2.27) yields (2.21).
For even k, by similar argument we obtain (2.22) and (2.23). The proof of
Theorem 2.2 is complete.
### 2.3 The difference of $i_{L_{0}}({\gamma})$ and $i_{L_{1}}({\gamma})$.
The precise difference of $i_{L_{0}}({\gamma})$ and $i_{L_{1}}({\gamma})$ for
${\gamma}\in\mathcal{P}_{\tau}$ with $\tau>0$ is very important in the proof
of the main results of this paper. In this subsection we use the H$\ddot{{\rm
o}}$rmander index (cf. [14]) to compute it. Note that in [42], in fact we have
already proved that $|i_{L_{0}}({\gamma})-i_{L_{1}}({\gamma})|\leq n$.
For any $P\in{\rm Sp}(2n)$ and $\varepsilon\in{\bf R}$, we set
$\displaystyle
M_{\varepsilon}(P)=P^{T}\left(\begin{array}[]{cc}\sin{2{\varepsilon}}I_{n}&-\cos{2{\varepsilon}I_{n}}\\\
-\cos{2{\varepsilon}}I_{n}&-\sin
2{\varepsilon}I_{n}\end{array}\right)P+\left(\begin{array}[]{cc}\sin{2{\varepsilon}}I_{n}&\cos{2{\varepsilon}}I_{n}\\\
\cos{2{\varepsilon}}I_{n}&-\sin 2{\varepsilon}I_{n}\end{array}\right).$ (2.34)
Then we have the following theorem.
Theorem 2.3. For ${\gamma}\in\mathcal{P}_{\tau}$ with $\tau>0$, we have
$i_{L_{0}}({\gamma})-i_{L_{1}}({\gamma})=\frac{1}{2}{\rm
sgn}M_{\varepsilon}({\gamma}(\tau)),$ (2.35)
where ${\rm sgn}M_{\varepsilon}({\gamma}(\tau))$ is the signature of the
symmetric matrix $M_{\varepsilon}({\gamma}(\tau))$ and ${\varepsilon}>0$ is
sufficiently small.
we also have,
$(i_{L_{0}}({\gamma})+\nu_{L_{0}}({\gamma}))-(i_{L_{1}}({\gamma})+\nu_{L_{1}}({\gamma}))=\frac{1}{2}{\rm
sign}M_{\varepsilon}({\gamma}(\tau)),$ (2.36)
where ${\varepsilon}<0$ and $|{\varepsilon}|$ is sufficiently small.
Proof. By the first geometrical definition of the Maslov-type index in Section
4 of [11], there exists an ${\varepsilon}>0$ small enough such that
$V_{1}\cap e^{-{\varepsilon}\mathcal{J}}{\rm Gr}({\gamma}(0))=\\{0\\},\qquad
V_{2}\cap e^{-{\varepsilon}\mathcal{J}}{\rm Gr}({\gamma}(\tau))=\\{0\\}.$
(2.37)
By definition 2.1, we have
$\displaystyle
i_{L_{0}}({\gamma})=\mu^{CLM}_{F}(V_{1},e^{-{\varepsilon}\mathcal{J}}{\rm
Gr}({\gamma}),[0,\tau])-n,$ (2.38) $\displaystyle
i_{L_{1}}({\gamma})=\mu^{CLM}_{F}(V_{2},e^{-{\varepsilon}\mathcal{J}}{\rm
Gr}({\gamma}),[0,\tau])-n.$ (2.39)
Define ${\gamma}_{1}(t)=e^{-{\varepsilon}\mathcal{J}}{\rm Gr}({\gamma}(t))$
and ${\gamma}_{2}(t)=e^{-{\varepsilon}\mathcal{J}}{\rm Gr}({\gamma}(\tau-t))$
for $t\in[0,\tau]$. Then ${\gamma}_{1}$ and ${\gamma}_{2}$ are two paths of
Lagrangian subspaces of the symplectic space $(F,\mathcal{J})$ defined in
(2.1) and (2.2). ${\gamma}_{1}$ connects $e^{-{\varepsilon}\mathcal{J}}{\rm
Gr}({\gamma}(0))$ and $e^{-{\varepsilon}\mathcal{J}}{\rm Gr}({\gamma}(\tau))$
and is transversal to $V_{1}$ and $V_{2}$. ${\gamma}_{2}$ connects
$e^{-{\varepsilon}\mathcal{J}}{\rm Gr}({\gamma}(\tau))$ and
$e^{-{\varepsilon}\mathcal{J}}{\rm Gr}({\gamma}(0))$ and is transversal to
$V_{1}$ and $V_{2}$. Denote by ${\gamma}$ the catenation of the paths
${\gamma}_{1}$ and ${\gamma}_{2}$. By Definition 3.4.2 of the
$H\ddot{o}rmande\;index$ $s(M_{1},M_{2};L_{1},L_{2})$ on p. 66 of [14] and
(2.38)-(2.39), we have
$\displaystyle s(V_{1},V_{2};e^{-{\varepsilon}\mathcal{J}}{\rm
Gr}({\gamma}(0)),e^{-{\varepsilon}\mathcal{J}}{\rm Gr}({\gamma}(\tau)))$
(2.40) $\displaystyle=$ $\displaystyle\langle{\gamma},{\alpha}\rangle$
$\displaystyle=$
$\displaystyle\mu^{CLM}_{F}(V_{1},{\gamma}_{1})+\mu^{CLM}_{F}(V_{2},{\gamma}_{2})$
$\displaystyle=$
$\displaystyle\mu^{CLM}_{F}(V_{1},e^{-{\varepsilon}\mathcal{J}}{\rm
Gr}({\gamma}))-\mu^{CLM}_{F}(V_{2},e^{-{\varepsilon}\mathcal{J}}{\rm
Gr}({\gamma}))$ (2.41) $\displaystyle=$ $\displaystyle
i_{L_{0}}({\gamma})-i_{L_{1}}({\gamma}),$ (2.42)
where ${\alpha}$ is the Maslov-Arnold index defined in Theorem 3.4.9 on p. 64
of [14]. Since ${\gamma}_{1}$ and ${\gamma}_{2}$ are transversal to $V_{1}$
and $V_{2}$ (2.40) holds, (2.41) holds from the definition of ${\gamma}_{1}$
and ${\gamma}_{2}$.
In the proof of Theorem 3.3 of [42], we have proved that for ${\varepsilon}>0$
small enough, there holds
${\rm sgn}(V_{1},e^{-{\varepsilon}\mathcal{J}}{\rm Gr}(I_{2n});V_{2})=0,$
(2.43)
where ${\rm sgn}(W_{1},W_{3};W_{2})$ for 3 Lagrangian spaces with $W_{3}$
transverses to $W_{1}$ and $W_{2}$ is introduced in Definition 3.2.3 on p. 67
of [14]. Note that by Claim 1 below, we can prove (2.43) at once.
Claim 1. For ${\varepsilon}>0$, small enough, there holds
${\rm sign}(V_{1},e^{-{\varepsilon}\mathcal{J}}{\rm
Gr}({\gamma}(\tau));V_{2})={\rm sgn}(M_{\varepsilon}({\gamma}(\tau))).$ (2.44)
Proof of Claim 1. In fact,
$e^{-\mathcal{J}}{\rm
Gr}({\gamma}(\tau))=\left\\{\left(\begin{array}[]{cc}e^{{\varepsilon}J}&0\\\
0&e^{-{\varepsilon}J}{\gamma}(\tau)\end{array}\right)\left(\begin{array}[]{c}p\\\
q\\\ p\\\ q\end{array}\right)=\left(\begin{array}[]{c}cp-sq\\\ sp+cq\\\
(c,s){\gamma}(\tau)(p,q)^{T}\\\
(-s,c){\gamma}(\tau)(p,q)^{T}\end{array}\right);\quad p,q\in{\bf
R}^{n}\right\\},$ (2.45)
where we denote by $c=\cos{\varepsilon}I_{n}$ and $s=\sin{\varepsilon}I_{n}$.
Hence the transformation $A:V_{1}\mapsto e^{-\mathcal{J}}{\rm Gr}(I_{2n})$
satisfies
$\displaystyle A(0,-sp-cq,0,-(-s,c){\gamma}(\tau)(p,q)^{T})$
$\displaystyle=(cp-
sq,sp+cq,(c,s){\gamma}(\tau)(p,q)^{T},(-s,c){\gamma}(\tau)(p,q)^{T}),\quad\forall
p,q\in{\bf R}^{n},$ (2.46)
where $A$ is introduced in Definition 3.4.3 of ${\rm sign}(M_{1},M_{2};L)$ on
p. 67 of [14]. For the convenience of our computation, we rewrite (2.46) as
follows.
$\displaystyle A\left(-\left(\begin{array}[]{cc}0&0\\\
s&c\end{array}\right)\left(\begin{array}[]{c}p\\\
q\end{array}\right),-\left(\begin{array}[]{cc}0&0\\\
-s&c\end{array}\right){\gamma}(\tau)\left(\begin{array}[]{c}p\\\
q\end{array}\right)\right)$ (2.55)
$\displaystyle=\left(\left(\begin{array}[]{cc}c&-s\\\
s&c\end{array}\right)\left(\begin{array}[]{c}p\\\
q\end{array}\right),\left(\begin{array}[]{cc}c&s\\\
-s&c\end{array}\right){\gamma}(\tau)\left(\begin{array}[]{c}p\\\
q\end{array}\right)\right).$ (2.64)
Then for $p_{1},p_{2},q_{1},q_{2}\in{\bf R}^{n}$, the symmetric bilinear form
$Q(V_{2}):(x,y)\mapsto\mathcal{J}(Ax,y)$ on $V_{1}$ defined in Definition
3.4.3 on p. 67 of [14] satisfies:
$\displaystyle Q(V_{2})\left(\left(-\left(\begin{array}[]{cc}0&0\\\
s&c\end{array}\right)\left(\begin{array}[]{c}p\\\
q\end{array}\right),-\left(\begin{array}[]{cc}0&0\\\
-s&c\end{array}\right){\gamma}(\tau)\left(\begin{array}[]{c}p\\\
q\end{array}\right)\right)\right)$ (2.73) $\displaystyle=$
$\displaystyle\left\langle((-J)\oplus J)\left[\left(\begin{array}[]{cc}c&-s\\\
s&c\end{array}\right)\left(\begin{array}[]{c}p\\\
q\end{array}\right),\left(\begin{array}[]{cc}c&s\\\
-s&c\end{array}\right){\gamma}(\tau)\left(\begin{array}[]{c}p\\\
q\end{array}\right)\right],\right.\;$ (2.91)
$\displaystyle\left[-\left(\begin{array}[]{cc}0&0\\\
s&c\end{array}\right)\left.\left(\begin{array}[]{c}p\\\
q\end{array}\right),-\left(\begin{array}[]{cc}0&0\\\
-s&c\end{array}\right){\gamma}(\tau)\left(\begin{array}[]{c}p\\\
q\end{array}\right)\right]\right\rangle.$ $\displaystyle=$
$\displaystyle\left\langle\left[\left(\begin{array}[]{cc}0&s\\\
0&c\end{array}\right)J\left(\begin{array}[]{cc}c&-s\\\
s&c\end{array}\right)-{\gamma}(\tau)^{T}\left(\begin{array}[]{cc}0&-s\\\
0&c\end{array}\right)J\left(\begin{array}[]{cc}c&s\\\
-s&c\end{array}\right){\gamma}(\tau)\right]\left(\begin{array}[]{c}p\\\
q\end{array}\right),\;\left(\begin{array}[]{c}p\\\
q\end{array}\right)\right\rangle.$ (2.104) $\displaystyle=$
$\displaystyle\left\langle\left[\left(\begin{array}[]{cc}sc&-s^{2}\\\
c^{2}&-sc\end{array}\right)+{\gamma}(\tau)^{T}\left(\begin{array}[]{cc}sc&s^{2}\\\
-c^{2}&-sc\end{array}\right){\gamma}(\tau)\right]\left(\begin{array}[]{c}p\\\
q\end{array}\right),\;\left(\begin{array}[]{c}p\\\
q\end{array}\right)\right\rangle.$ (2.113)
Let
$\tilde{M}_{\varepsilon}({\gamma}(\tau))=\left(\begin{array}[]{cc}sc&-s^{2}\\\
c^{2}&-sc\end{array}\right)+{\gamma}(\tau)^{T}\left(\begin{array}[]{cc}sc&s^{2}\\\
-c^{2}&-sc\end{array}\right){\gamma}(\tau)$. Then by definition of the
symmetric bilinear form $Q(V_{2})$, $\tilde{M}_{\varepsilon}({\gamma}(\tau)$
is an invertible symmetric $2n\times 2n$ matrix. We define
$M_{\varepsilon}({\gamma}(\tau))=2\tilde{M}_{\varepsilon}({\gamma}(\tau))=\tilde{M}_{\varepsilon}({\gamma}(\tau))+\tilde{M}_{\varepsilon}^{T}({\gamma}(\tau)).$
(2.114)
Then we have
$\displaystyle
M_{\varepsilon}({\gamma}(\tau))={\gamma}(\tau)^{T}\left(\begin{array}[]{cc}\sin{2{\varepsilon}}I_{n}&-\cos{2{\varepsilon}I_{n}}\\\
-\cos{2{\varepsilon}}I_{n}&-\sin
2{\varepsilon}I_{n}\end{array}\right){\gamma}(\tau)+\left(\begin{array}[]{cc}\sin{2{\varepsilon}}I_{n}&\cos{2{\varepsilon}}I_{n}\\\
\cos{2{\varepsilon}}I_{n}&-\sin 2{\varepsilon}I_{n}\end{array}\right).$
(2.119)
It is clear that
${\rm sgn}Q(V_{2})={\rm sgn}\tilde{M}_{\varepsilon}({\gamma}(\tau))={\rm
sgn}M_{\varepsilon}({\gamma}(\tau)).$ (2.120)
By the definition of ${\rm sgn}(V_{1},e^{-{\varepsilon}\mathcal{J}}{\rm
Gr}({\gamma}(\tau));V_{2})$, we have
${\rm sgn}(V_{1},e^{-{\varepsilon}\mathcal{J}}{\rm
Gr}({\gamma}(\tau));V_{2})={\rm sgn}Q(V_{2}).$ (2.121)
Then (2.44) holds from (2.120) and (2.121), and the proof of Claim 1 is
complete.
Thus by (2.42), (2.43) and Claim 1, we have
$\displaystyle i_{L_{0}}({\gamma})-i_{L_{1}}({\gamma})$ $\displaystyle=$
$\displaystyle s(V_{1},V_{2};e^{-{\varepsilon}\mathcal{J}}{\rm
Gr}({\gamma}(0)),e^{-{\varepsilon}\mathcal{J}}{\rm Gr}({\gamma}(\tau)))$
$\displaystyle=$ $\displaystyle\frac{1}{2}{\rm
sgn}(V_{1},e^{-{\varepsilon}\mathcal{J}}{\rm
Gr}({\gamma}(\tau));V_{2})-\frac{1}{2}{\rm
sgn}(V_{1},e^{-{\varepsilon}\mathcal{J}}{\rm Gr}({\gamma}(0));V_{2})$
$\displaystyle=$ $\displaystyle\frac{1}{2}{\rm
sgn}(V_{1},e^{-{\varepsilon}\mathcal{J}}{\rm
Gr}({\gamma}(\tau));V_{2})-\frac{1}{2}{\rm
sgn}(V_{1},e^{-{\varepsilon}\mathcal{J}}{\rm Gr}(I_{2n});V_{2})$
$\displaystyle=$ $\displaystyle\frac{1}{2}{\rm
sgn}(V_{1},e^{-{\varepsilon}\mathcal{J}}{\rm Gr}({\gamma}(\tau));V_{2})$
$\displaystyle=$ $\displaystyle\frac{1}{2}{\rm
sgn}M_{\varepsilon}({\gamma}(\tau)).$
Here in the second equality, we have used Theorem 3.4.12 of on p. 68 of [14].
Thus (2.35) holds.
Choose ${\varepsilon}<0$ such that $|{\varepsilon}|$ is sufficiently small, by
the discussion of $\mu^{CLM}_{F}$ index we have
$\displaystyle
i_{L_{0}}({\gamma})=\mu^{CLM}_{F}(V_{1},e^{-{\varepsilon}\mathcal{J}}{\rm
Gr}({\gamma}),[0,\tau])-\nu_{L_{0}}({\gamma}),$ (2.122) $\displaystyle
i_{L_{1}}({\gamma})=\mu^{CLM}_{F}(V_{2},e^{-{\varepsilon}\mathcal{J}}{\rm
Gr}({\gamma}),[0,\tau])-\nu_{L_{1}}({\gamma}).$ (2.123)
Then by the same proof as above, we have
$i_{L_{0}}({\gamma})+\nu_{L_{0}}({\gamma})-i_{L_{1}}({\gamma})-\nu_{L_{1}}({\gamma})=\frac{1}{2}{\rm
sgn}M_{\varepsilon}({\gamma}(\tau)),$ (2.124)
where ${\varepsilon}<0$ is small enough. Hence (2.36) holds. The proof of
Theorem 2.3 is complete.
We have the following consequence.
Corollary 2.1. (Theorem 2.3 of [35]) For ${\gamma}\in\mathcal{P}_{\tau}(2n)$
with $\tau>0$, there hold
$\displaystyle|i_{L_{0}}({\gamma}))-i_{L_{1}}({\gamma}))|\leq
n,\quad|i_{L_{0}}({\gamma})+\nu_{L_{0}}({\gamma})-i_{L_{1}}({\gamma})-\nu_{L_{1}}({\gamma})|\leq
n.$ (2.125)
Moreover if ${\gamma}(1)$ is a orthogonal matrix then there holds
$i_{L_{0}}({\gamma})=i_{L_{1}}({\gamma}).$ (2.126)
Proof. (2.125) holds directly from Theorem 2.3, so we only need to prove
(2.126). Since ${\gamma}(\tau)$ is an orthogonal and symplectic matrix, we
have
${\gamma}^{T}(\tau)J{\gamma}(\tau)=J,\quad{\gamma}^{T}(\tau){\gamma}(\tau)=I_{2n}.$
(2.127)
So we have
${\gamma}(\tau)J=J{\gamma}(\tau),\quad{\gamma}(\tau)^{T}J=J{\gamma}(\tau)^{T}.$
(2.128)
It is easy to check that for any ${\varepsilon}\in{\bf R}$, there holds
$J\left(\begin{array}[]{cc}\sin{2{\varepsilon}}I_{n}&\pm\cos{2{\varepsilon}I_{n}}\\\
\pm\cos{2{\varepsilon}}I_{n}&-\sin
2{\varepsilon}I_{n}\end{array}\right)J=\left(\begin{array}[]{cc}\sin{2{\varepsilon}}I_{n}&\pm\cos{2{\varepsilon}I_{n}}\\\
\pm\cos{2{\varepsilon}}I_{n}&-\sin 2{\varepsilon}I_{n}\end{array}\right).$
(2.129)
Hence by (2.128) and (2.129), we have
$\displaystyle JM_{\varepsilon}({\gamma}(\tau))J$ $\displaystyle=$
$\displaystyle
J\left[{\gamma}(\tau)^{T}\left(\begin{array}[]{cc}\sin{2{\varepsilon}}I_{n}&-\cos{2{\varepsilon}I_{n}}\\\
-\cos{2{\varepsilon}}I_{n}&-\sin
2{\varepsilon}I_{n}\end{array}\right){\gamma}(\tau)+\left(\begin{array}[]{cc}\sin{2{\varepsilon}}I_{n}&\cos{2{\varepsilon}}I_{n}\\\
\cos{2{\varepsilon}}I_{n}&-\sin 2{\varepsilon}I_{n}\end{array}\right)\right]J$
(2.134) $\displaystyle=$ $\displaystyle
J{\gamma}(\tau)^{T}\left(\begin{array}[]{cc}\sin{2{\varepsilon}}I_{n}&-\cos{2{\varepsilon}I_{n}}\\\
-\cos{2{\varepsilon}}I_{n}&-\sin
2{\varepsilon}I_{n}\end{array}\right){\gamma}(\tau)J+J\left(\begin{array}[]{cc}\sin{2{\varepsilon}}I_{n}&\cos{2{\varepsilon}}I_{n}\\\
\cos{2{\varepsilon}}I_{n}&-\sin 2{\varepsilon}I_{n}\end{array}\right)J$
(2.139) $\displaystyle=$
$\displaystyle{\gamma}(\tau)^{T}J\left(\begin{array}[]{cc}\sin{2{\varepsilon}}I_{n}&-\cos{2{\varepsilon}I_{n}}\\\
-\cos{2{\varepsilon}}I_{n}&-\sin
2{\varepsilon}I_{n}\end{array}\right)J{\gamma}(\tau)+J\left(\begin{array}[]{cc}\sin{2{\varepsilon}}I_{n}&\cos{2{\varepsilon}}I_{n}\\\
\cos{2{\varepsilon}}I_{n}&-\sin 2{\varepsilon}I_{n}\end{array}\right)J$
(2.144) $\displaystyle=$
$\displaystyle{\gamma}(\tau)^{T}\left(\begin{array}[]{cc}\sin{2{\varepsilon}}I_{n}&-\cos{2{\varepsilon}I_{n}}\\\
-\cos{2{\varepsilon}}I_{n}&-\sin
2{\varepsilon}I_{n}\end{array}\right){\gamma}(\tau)+\left(\begin{array}[]{cc}\sin{2{\varepsilon}}I_{n}&\cos{2{\varepsilon}}I_{n}\\\
\cos{2{\varepsilon}}I_{n}&-\sin 2{\varepsilon}I_{n}\end{array}\right)$ (2.149)
$\displaystyle=$ $\displaystyle M_{\varepsilon}({\gamma}(\tau)).$ (2.150)
So we have
$M_{\varepsilon}({\gamma}(\tau))J=-JM_{\varepsilon}({\gamma}(\tau)).$ (2.151)
Thus for any $x\in{\bf R}^{2n}$ and ${\lambda}\in{\bf R}$ satisfying
$M_{\varepsilon}({\gamma}(\tau))x={\lambda}x.$ (2.152)
By (2.151) we have
$M_{\varepsilon}({\gamma}(\tau))(Jx)=-JM_{\varepsilon}({\gamma}(\tau))x=-{\lambda}(Jx).$
(2.153)
Since for ${\varepsilon}>0$ small enough $M_{\varepsilon}({\gamma}(\tau))$ is
an invertible symmetric matrix, by (2.153) we have
$m^{+}(M_{\varepsilon}({\gamma}(\tau)))=m^{-}(M_{\varepsilon}({\gamma}(\tau)))=n$
(2.154)
which yields
${\rm
sgn}M_{\varepsilon}({\gamma}(\tau))=m^{+}(M_{\varepsilon}({\gamma}(\tau)))-m^{-}(M_{\varepsilon}({\gamma}(\tau)))=0.$
(2.155)
Then (2.126) holds from Theorem 2.3.
Lemma 2.2. For a symplectic path $P:[0,\tau]\to{\rm Sp}(2n)$ with $\tau>0$, if
for $j=0,1$ there holds $\nu_{L_{j}}(P(t))=constant$ for all $t\in[0,\tau]$,
then for ${\varepsilon}>0$ small enough we have
${\rm sgn}M_{\varepsilon}(P(0))={\rm sgn}M_{\varepsilon}(P(\tau)).$ (2.156)
Proof. Since ${\rm Sp}(2n)$ is path connected, we can choose a path
${\gamma}\in\mathcal{P}_{\tau}$ with ${\gamma}(\tau)=P(0)$. By Proposition
2.11 of [42] and the definition of $\mu_{j}$ for $j=1,2$ in [42], we have
$\mu_{F}^{CLM}(V_{j},{\rm Gr}(P),[0,\tau])=0,\qquad j=0,1.$ (2.157)
So by the Path Additivity and Reparametrization Invariance properties of
$\mu_{F}^{CLM}$ in [11], we have
$\displaystyle i_{L_{j}}(P*{\gamma})$ $\displaystyle=$
$\displaystyle\mu_{F}^{CLM}(V_{j},{\rm Gr}(P*{\gamma}),[0,\tau])-n$ (2.158)
$\displaystyle=$ $\displaystyle\mu_{F}^{CLM}(V_{j},{\rm
Gr}({\gamma}),[0,\tau])+\mu_{F}^{CLM}(V_{j},{\rm Gr}(P),[0,\tau])-n$
$\displaystyle=$ $\displaystyle\mu_{F}^{CLM}(V_{j},{\rm
Gr}({\gamma}),[0,\tau])-n$ $\displaystyle=$ $\displaystyle
i_{L_{j}}({\gamma}),$
where the definition of joint path $\eta*\xi$ is given by (6.1) in Section 6
below. Then by Theorem 2.3 we have
$\displaystyle i_{L_{0}}({\gamma})-i_{L_{1}}({\gamma})=\frac{1}{2}{\rm
sgn}(M_{\varepsilon}(P(0))),$ (2.159) $\displaystyle
i_{L_{0}}(P*{\gamma})-i_{L_{1}}(P*{\gamma})=\frac{1}{2}{\rm
sgn}(M_{\varepsilon}(P(\tau))).$ (2.160)
Then (2.156) holds from (2.158)-(2.160). The proof of Lemma 2.2 is complete.
Remark 2.1. It is easy to check that for $n_{j}\times n_{j}$ symplectic matrix
$P_{j}$ with $j=1,2$ and $n_{j}\in{\bf N}$, we have
$\displaystyle M_{\varepsilon}(P_{1}\diamond
P_{2})=M_{\varepsilon}(P_{1})\diamond M_{\varepsilon}(P_{2}),$
$\displaystyle{\rm sgn}M_{\varepsilon}(P_{1}\diamond P_{2})={\rm
sgn}M_{\varepsilon}(P_{1})+{\rm sgn}M_{\varepsilon}(P_{2}).$
By direct computation according to Theorem 2.3 and Corollary 2.1, for
${\gamma}\in\mathcal{P}_{\tau}(2)$, $b>0$, and ${\varepsilon}>0$ small enough
we have
$\displaystyle{\rm sgn}M_{\varepsilon}(R(\theta))=0,\quad{\rm
for}\;\theta\in{\bf R},$ (2.161) $\displaystyle{\rm
sgn}M_{\varepsilon}(P)=0,\quad{\rm if}\;P=\pm\left(\begin{array}[]{cc}1&b\\\
0&1\end{array}\right)\;{\rm or}\;\pm\left(\begin{array}[]{cc}1&0\\\
-b&1\end{array}\right),$ (2.166) $\displaystyle{\rm
sgn}M_{\varepsilon}(P)=2,\quad{\rm if}\;P=\pm\left(\begin{array}[]{cc}1&-b\\\
0&1\end{array}\right),$ (2.169) $\displaystyle{\rm
sgn}M_{\varepsilon}(P)=-2,\quad{\rm if}\;P=\pm\left(\begin{array}[]{cc}1&0\\\
b&1\end{array}\right).$ (2.172)
Also we give a example as follows to finish this section
${\rm sgn}M_{\varepsilon}(P)=2,\quad{\rm
if}\;P=\pm\left(\begin{array}[]{cc}2&-1\\\ -1&1\end{array}\right).$ (2.173)
## 3 Relation between $i_{L_{0}}$, $i_{L_{1}}$, $i_{\sqrt{-1}}^{L_{0}}$ and
the corresponding Morse indices, and their monotonicity properties.
In [31], Liu studied the relation between the $L$-index of solutions of
Hamiltonian systems with $L$-boundary conditions and the Morse index of the
corresponding functional defined via the Galerkin approximation method on the
finite dimensional truncated space at its corresponding critical points. In
order to prove the main results of this paper, in this section we use the
results of [31] to study some monotonicity properties of $i_{L_{0}}$ and
$i_{L_{1}}$. We also study the index $i_{\sqrt{-1}}^{L_{0}}(B)$ with $B$ being
a continuous symmetric matrices path satisfying condition (B1) defined in
Section 1 and the Morse index of the corresponding functional defined via the
Galerkin approximation method. Then as applications we study some monotonicity
properties of $i_{\sqrt{-1}}^{L_{0}}(B)$ which will be important in the proof
of Theorems 1.4-1.5 in Section 5 below.
For any $\tau>0$ and $B\in C([0,\tau/4],\mathcal{L}_{s}({\bf R}^{2n}))$ (in
order to apply the results in this section conveniently Section 5, we always
assume $B\in C([0,\tau/4],\mathcal{L}_{s}({\bf R}^{2n})$) satisfying condition
(B1). We extend $B$ to $[0,\frac{\tau}{2}]$ by
$B(\frac{\tau}{4}+t)=NB(\frac{\tau}{4}-t)N,\;\forall t\in[0,\frac{\tau}{4}].$
(3.1)
Then since $B(\frac{\tau}{2})=B(0)$, we can extend it
$\frac{\tau}{2}$-periodically to ${\bf R}$, so we can see $B$ as an element in
$C(S_{\tau/2},\mathcal{L}_{s}({\bf R}^{2n}))$.
Let $E_{\tau}=\\{x\in W^{1/2,2}(S_{\tau},{\bf
R}^{2n})|\,x(-t)=Nx(t)\;a.e.\;t\in{\bf R}\\}$ with the usual norm and inner
product denoted by $||\cdot||$ and $\langle\cdot\rangle$ respectively.
By the Sobolev embedding theorem, for any $s\in[1,+\infty)$, there is a
constant $C_{s}>0$ such that
$||z||_{L^{s}}\leq C_{s}||z||,\quad\forall z\in E_{2\tau}.$ (3.2)
Note that $B$ can also be seen as an element in
$C(S_{\tau},\mathcal{L}_{s}({\bf R}^{2n}))$. We define two selfadjoint
operators $A_{\tau}$ and $B_{\tau}$ on $E_{\tau}$ by the following bilinear
forms
$\displaystyle\langle A_{\tau}x,y\rangle=\int_{0}^{\tau}-J\dot{x}\cdot
y\,dt,\qquad\langle B_{\tau}x,y\rangle=\int_{0}^{\tau}B(t)x\cdot y\,dt.$ (3.3)
Then $A_{\tau}$ is a bounded operator on $E_{\tau}$ and dim $\ker A_{\tau}=n$,
the Fredholm index of $A_{\tau}$ is zero, and $B_{\tau}$ is a compact operator
on $E_{\tau}$.
Set
$E_{\tau}(j)=\left\\{z\in E_{\tau}\left|z(t)={\rm exp}(\frac{2j\pi
t}{\tau}J)a+{\rm exp}(-\frac{2j\pi t}{\tau}J)b,\;\forall t\in{\bf R};\;\forall
a,\,b\in L_{0}\right.\right\\}.$
and
$E_{\tau,m}=E_{\tau}(0)+E_{\tau}(1)+\cdots+E_{\tau}(m).$
Let ${\Gamma}_{\tau}=\\{P_{\tau,m}:m=0,1,2,...\\}$ be the usual Galerkin
approximation scheme w.r.t. $A_{\tau}$, just as in [31], i.e.,
${\Gamma}_{\tau}$ is a sequence of orthogonal projections satisfies:
(1) $E_{\tau,0}=P_{\tau,0}E_{\tau}=\ker
A_{\tau},\;E_{\tau,m}=P_{\tau,m}E_{\tau}$ is finite dimension for $m\geq 0$;
(2) $P_{\tau,m}\to x$ as $m\to\infty$ for any $x\in E_{\tau}$;
(3) $P_{\tau,m}A_{\tau}=A_{\tau}P_{\tau,m}$, $\forall m\geq 0$.
For $d>0$, we denote by $M^{+}_{d}(\cdot)$, $M^{-}_{d}(\cdot)$ and
$M^{0}_{d}(\cdot)$ the eigenspace corresponding to the eigenvalue ${\lambda}$
belong to $[d,+\infty)$, $(-\infty,-d]$ and $(-d,d)$ respectively, and
$M^{+}(\cdot)$, $M^{-}(\cdot)$ and $M^{0}(\cdot)$ the positive, negative and
null subspace of of the selfadjoint operator defining it respectively. For any
bounded selfadjoint linear operator on $E$, We denote
$L^{\\#}=(L|_{ImL})^{-1}$, and we also denote by
$P_{\tau,m}LP_{\tau,m}=(P_{\tau,m}LP_{\tau,m})|_{E_{\tau,m}}:E_{\tau,m}\to
E_{\tau,m}$.
Similarly we define two subspaces of $E_{\tau}$ by $\hat{E}=\\{x\in
E|x(t+\frac{\tau}{2})=-x(t),a.e.\,t\in{\bf R}\\}$ and $\tilde{E}=\\{x\in
E|x(t+\frac{\tau}{2})=x(t),a.e.\,t\in{\bf R}\\}$ be the symmetric ones and
$\frac{\tau}{2}$-periodic ones of $E_{\tau}$ respectively.
We define two selfadjoint operators $\hat{A}$ and $\hat{B}$ on $\hat{E}$ by
the following bilinear forms
$\displaystyle\langle\hat{A}x,y\rangle=\int_{0}^{\tau}-J\dot{x}\cdot
y\,dt,\qquad\langle\hat{B}x,y\rangle=\int_{0}^{\tau}B(t)x(t)\cdot y(t)\,dt.$
(3.4)
Then $\hat{A}$ is a bounded Fredholm operator on $\hat{E}$ and dim
$\ker\hat{A}=0$, the Fredholm index of $\hat{A}$ is zero. $\hat{B}$ is a
compact operator on $\hat{E}$.
For any positive integer $m$, we define
$\hat{E}_{m}={\Sigma}_{j=1}^{m}E_{\tau}(2j-1).$
For $m\geq 1$, let $\hat{P}_{m}$ be the orthogonal projection from $\hat{E}$
to $\hat{E}_{m}$. Then $\\{\hat{P}_{m}\\}$ is a Galerkin approximation scheme
w.r.t. $\hat{A}$.
Theorem 3.1. For any $B(t)\in C([0,\frac{\tau}{4}],\mathcal{L}_{s}({\bf
R}^{2n}))$ satisfying condition (B1) and
$0<d\leq\frac{1}{4}||(A_{\tau}-B_{\tau})^{\\#}||^{-1}$, there exists $m^{*}>0$
such that for $m\geq m^{*}$ there hold
$\displaystyle\dim M_{d}^{+}(\hat{P}_{m}(\hat{A}-\hat{B})\hat{P}_{m})$
$\displaystyle=$ $\displaystyle mn-
i_{\sqrt{-1}}^{L_{0}}(B)-\nu_{\sqrt{-1}}^{L_{0}}(B),$ (3.5) $\displaystyle\dim
M_{d}^{-}(\hat{P}_{m}(\hat{A}-\hat{B})\hat{P}_{m})$ $\displaystyle=$
$\displaystyle mn+i_{\sqrt{-1}}^{L_{0}}(B),$ (3.6) $\displaystyle\dim
M_{d}^{0}(\hat{P}_{m}(\hat{A}-\hat{B})\hat{P}_{m})$ $\displaystyle=$
$\displaystyle\nu_{\sqrt{-1}}^{L_{0}}(B).$ (3.7)
Proof. The method of the proof here is similar as that of Theorem 2.1 in [51].
For any positive integer $m$, we define
$\tilde{E}_{m}=\sum_{j=0}^{m}E_{\tau}(2j).$
For $m\geq 1$, let $\tilde{P}_{m}$ be the orthogonal projection from
$\tilde{E}$ to $\tilde{E}_{m}$. Then $\\{\tilde{P}_{m}\\}$ is a Galerkin
approximation scheme w.r.t. $\tilde{A}$.
For any $y\in\hat{E}_{m}$ and $z\in\tilde{E}_{m}$, it is easy to check that
$\displaystyle\langle(P_{\tau,m}(A_{\tau}-B_{\tau})P_{\tau,m}y,z)\rangle=0.$
(3.8)
So we have the following $P_{\tau,m}(A_{\tau}-B_{\tau})P_{\tau,m}$ orthogonal
decomposition
$E_{\tau,2m}=\hat{E}_{m}\oplus\tilde{E}_{m}.$ (3.9)
Similarly, we have the following $A_{\tau}-B_{\tau}$ orthogonal decomposition
$E_{\tau}=\hat{E}\oplus\tilde{E}.$ (3.10)
Hence, under above decomposition we have
$(A_{\tau}-B_{\tau})=(\hat{A}-\hat{B})\oplus(\tilde{A}-\tilde{B}).$ (3.11)
Thus
$\displaystyle||(A_{\tau}-B_{\tau})^{\\#}||^{-1}\leq||(\hat{A}-\hat{B})^{\\#}||^{-1}$
(3.12)
$\displaystyle||(A_{\tau}-B_{\tau})^{\\#}||^{-1}\leq||(\tilde{A}-\tilde{B})^{\\#}||^{-1}$
(3.13)
By the definitions of $M_{d}^{*}(\cdot)$ for
$P_{\tau,2m}(A_{\tau}-B_{\tau})P_{\tau,2m}$,
$\hat{P}_{m}(\hat{A}-\hat{B})\hat{P}_{m}$, and
$\tilde{P}_{m}(\tilde{A}-\tilde{B})\tilde{P}_{m}$ with $*=+,-,0$. So for
$*\in\\{+,-,0\\}$ we have
$\dim M_{d}^{*}(P_{\tau,2m}(A_{\tau}-B_{\tau})P_{\tau,2m})=\dim
M_{d}^{*}(\hat{P}_{m}(\hat{A}-\hat{B})\hat{P}_{m})+\dim
M_{d}^{*}(\tilde{P}_{m}(\tilde{A}-\tilde{B})\tilde{P}_{m}).$ (3.14)
Note that, the space $E_{\tau}$ and the operators $A_{\tau}$, $B_{\tau}$ and
$P_{\tau,m}$ are also defined in the same way. So by the definition we see
that $\tilde{E}$ is the $\tau$-periodic extending of $E_{\tau}$ from
$S_{\tau}$ to $S_{2\tau}$, and $\tilde{E}_{m}$ is the $\tau$-periodic
extending of $E_{\tau,2m}$ from $S_{\tau}$ to $S_{2\tau}$ too.
Thus we have
$||(A_{\tau}-B_{\tau})^{\\#}||^{-1}=||(\tilde{A}-\tilde{B})^{\\#}||^{-1}.$
(3.15)
By (3.13) and (3.15) we have
$||(A_{2\tau}-B_{2\tau})^{\\#}||^{-1}\leq||(A_{\tau}-B_{\tau})^{\\#}||^{-1}.$
(3.16)
For $*\in\\{+,-,0\\}$ we have
$\dim
M_{d}^{*}(P_{\tau,m}(A_{\tau}-B_{\tau})P_{\tau,m})=M_{d}^{*}(\tilde{P}_{m}(\tilde{A}-\tilde{B})\tilde{P}_{m}).$
(3.17)
Then for $0<d\leq\frac{1}{4}||(A_{\tau}-B_{\tau})^{\\#}||^{-1}$, by Theorem
2.1 in [31] there exists $m_{1}>0$ such that for $m\geq m_{1}$ we have
$\displaystyle\dim M_{d}^{+}(P_{\tau,2m}(A_{\tau}-B_{\tau})P_{\tau,2m})=2mn-
i_{L_{0}}({\gamma}_{B}^{2})-\nu_{L_{0}}({\gamma}_{B}^{2}),$ (3.18)
$\displaystyle\dim
M_{d}^{-}(P_{\tau,2m}(A_{\tau}-B_{\tau})P_{\tau,2m})=2mn+n+i_{L_{0}}({\gamma}_{B}^{2}),$
(3.19) $\displaystyle\dim
M_{d}^{0}(P_{\tau,2m}(A_{\tau}-B_{\tau})P_{\tau,2m})=\nu_{L_{0}}({\gamma}_{B}^{2}).$
(3.20)
By (3.16), we have $0<d\leq\frac{1}{4}||(A_{\tau}-B_{\tau})^{\\#}||^{-1}$. By
Theorem 2.1 in [31] again there exists $m_{2}>0$, such that for $m\geq m_{2}$
we have
$\displaystyle\dim M_{d}^{+}(P_{\tau,m}(A_{\tau}-B_{\tau})P_{\tau,m})=mn-
i_{L_{0}}({\gamma}_{B})-\nu_{L_{0}}({\gamma}_{B})),$ (3.21) $\displaystyle\dim
M_{d}^{-}(P_{\tau,m}(A_{\tau}-B_{\tau})P_{\tau,m})=mn+n+i_{L_{0}}({\gamma}_{B})),$
(3.22) $\displaystyle\dim
M_{d}^{0}(P_{\tau,m}(A_{\tau}-B_{\tau})P_{\tau,m})=\nu_{L_{0}}({\gamma}_{B})).$
(3.23)
Let $m^{*}=\max\\{m_{1},m_{2}\\}$. Then for $m\geq m^{*}$, all of
(3.18)-(3.23) hold.
So by (3.14), (3.17), and (3.18)-(3.23) we have
$\displaystyle\dim M_{d}^{+}(\hat{P}_{m}(\hat{A}-\hat{B})\hat{P}_{m})$
$\displaystyle=$ $\displaystyle
mn-(i_{L_{0}}({\gamma}_{B}^{2})-i_{L_{0}}({\gamma}_{B}))-(\nu_{L_{0}}({\gamma}_{B}^{2})-\nu_{L_{0}}({\gamma}_{B})),$
(3.24) $\displaystyle\dim M_{d}^{-}(\hat{P}_{m}(\hat{A}-\hat{B})\hat{P}_{m})$
$\displaystyle=$ $\displaystyle
mn+i_{L_{0}}({\gamma}_{B}^{2})-i_{L_{0}}({\gamma}_{B}),$ (3.25)
$\displaystyle\dim M_{d}^{0}(\hat{P}_{m}(\hat{A}-\hat{B})\hat{P}_{m})$
$\displaystyle=$
$\displaystyle\nu_{L_{0}}({\gamma}_{B}^{2})-\nu_{L_{0}}({\gamma}_{B}).$ (3.26)
Thus (3.5)-(3.7) hold from (3.24)-(3.26), Definition 2.3, and Theorem 2.2. The
proof of Theorem 3.1 is complete.
Remark 3.1. Let any $B\in C([0,\frac{\tau}{4}],\mathcal{L}_{s}({\bf R}^{2n}))$
be a constant matrix path satisfying condition (B1). By Theorem 5.1 of [42],
for $d=0$ the same conclusions of Theorem 2.1 of [31] still holds . Hence for
$d=0$ the same conclusions of Theorem 3.1 still hold, i.e., there exists
$m^{*}>0$ such that for $m\geq m^{*}$ there hold
$\displaystyle\dim M^{+}(\hat{P}_{m}(\hat{A}-\hat{B})\hat{P}_{m})$
$\displaystyle=$ $\displaystyle mn-
i_{\sqrt{-1}}^{L_{0}}(B)-\nu_{\sqrt{-1}}^{L_{0}}(B),$ $\displaystyle\dim
M^{-}(\hat{P}_{m}(\hat{A}-\hat{B})\hat{P}_{m})$ $\displaystyle=$
$\displaystyle mn+i_{\sqrt{-1}}^{L_{0}}(B),$ $\displaystyle\dim
M^{0}(\hat{P}_{m}(\hat{A}-\hat{B})\hat{P}_{m})$ $\displaystyle=$
$\displaystyle\nu_{\sqrt{-1}}^{L_{0}}(B).$
In the following, we study some monotonicity of the the Maslov-type
$i_{\sqrt{-1}}^{L_{0}}$ index. In this paper, for any two symmetric matrices
$B_{1}$ and $B_{2}$, we say $B_{1}>B_{2}$ if $B_{1}-B_{2}$ is positive
definite and we say $B_{1}\geq B_{2}$ if $B_{1}-B_{2}$ is semipositive.
Similarly for two symmetric matrix paths $B_{1}$, $B_{2}\in
C([0,\tau],\mathcal{L}_{s}(R^{2n}))$, we say $B_{1}>B_{2}$ if
$B_{1}(t)-B_{2}(t)$ is positive definite for all $t\in[0,\tau]$ and we say
$B_{1}\geq B_{2}$ if $B_{1}(t)-B_{2}(t)$ is semipositive definite for all
$t\in[0,\tau]$.
Lemma 3.1. For any $\tau>0$ and $B_{1},\;B_{2}\in
C([0,\frac{\tau}{4}],\mathcal{L}_{s}({\bf R}^{2n}))$ satisfying condition
(B1). If $B_{1}\geq B_{2}$, then there hold
$i_{\sqrt{-1}}^{L_{0}}(B_{1})\geq i_{\sqrt{-1}}^{L_{0}}(B_{2})$ (3.27)
and
$\displaystyle i_{\sqrt{-1}}^{L_{0}}(B_{1})+\nu_{\sqrt{-1}}^{L_{0}}(B_{1})\geq
i_{\sqrt{-1}}^{L_{0}}(B_{2})+\nu_{\sqrt{-1}}^{L_{0}}(B_{2}).$ (3.28)
Moreover, if
$\int_{0}^{\frac{\tau}{4}}(B_{1}(t)-B_{2}(t))dt>0,$ (3.29)
then there holds
$\displaystyle i_{\sqrt{-1}}^{L_{0}}(B_{1})\geq
i_{\sqrt{-1}}^{L_{0}}(B_{2})+\nu_{\sqrt{-1}}^{L_{0}}(B_{2}).$ (3.30)
Proof. Let the space $\hat{E}$ and the orthogonal projection operator
$\hat{P}_{m}$ be the ones defined in Section 2. Correspondingly we define the
compact operators $\hat{B}_{1}$ and $\hat{B}_{2}$. By Theorem 3.1, for $d>0$
small enough, there exists $m^{*}>0$ such that
$\displaystyle\dim M_{d}^{+}(\hat{P}_{m}(\hat{A}-\hat{B}_{1})\hat{P}_{m})$
$\displaystyle=$ $\displaystyle mn-
i_{\sqrt{-1}}^{L_{0}}(B_{1})-\nu_{\sqrt{-1}}^{L_{0}}(B_{1}),$ (3.31)
$\displaystyle\dim M_{d}^{-}(\hat{P}_{m}(\hat{A}-\hat{B}_{1})\hat{P}_{m})$
$\displaystyle=$ $\displaystyle mn+i_{\sqrt{-1}}^{L_{0}}(B_{1}),$ (3.32)
$\displaystyle\dim M_{d}^{0}(\hat{P}_{m}(\hat{A}-\hat{B}_{1})\hat{P}_{m})$
$\displaystyle=$ $\displaystyle\nu_{\sqrt{-1}}^{L_{0}}(B_{1}).$ (3.33)
and
$\displaystyle\dim M_{d}^{+}(\hat{P}_{m}(\hat{A}-\hat{B_{2}})\hat{P}_{m})$
$\displaystyle=$ $\displaystyle mn-
i_{\sqrt{-1}}^{L_{0}}(B_{2})-\nu_{\sqrt{-1}}^{L_{0}}(B_{2}),$ (3.34)
$\displaystyle\dim M_{d}^{-}(\hat{P}_{m}(\hat{A}-\hat{B}_{2})\hat{P}_{m})$
$\displaystyle=$ $\displaystyle mn+i_{\sqrt{-1}}^{L_{0}}(B_{2}),$ (3.35)
$\displaystyle\dim M_{d}^{0}(\hat{P}_{m}(\hat{A}-\hat{B}_{2})\hat{P}_{m})$
$\displaystyle=$ $\displaystyle\nu_{\sqrt{-1}}^{L_{0}}(B_{2}).$ (3.36)
If $B_{1}\geq B_{2}$, we have
$\hat{P}_{m}(\hat{A}-\hat{B}_{1})\hat{P}_{m}\leq\hat{P}_{m}(\hat{A}-\hat{B}_{2})\hat{P}_{m}$,
So
$\dim M_{d}^{-}(\hat{P}_{m}(\hat{A}-\hat{B}_{1})\hat{P}_{m})\geq\dim
M_{d}^{-}(\hat{P}_{m}(\hat{A}-\hat{B}_{2})\hat{P}_{m}).$ (3.37)
Then by (3.32) and (3.35), (3.27) holds. Also we have
$\dim M_{d}^{+}(\hat{P}_{m}(\hat{A}-\hat{B}_{1})\hat{P}_{m})\leq\dim
M_{d}^{+}(\hat{P}_{m}(\hat{A}-\hat{B}_{2})\hat{P}_{m}).$ (3.38)
Then by (3.31) and (3.34), (3.28) holds.
If $\int_{0}^{\frac{\tau}{4}}(B_{1}(t)-B_{2}(t))dt>0$, then
$\hat{P}_{m}(\hat{A}-\hat{B}_{1})\hat{P}_{m}<\hat{P}_{m}(\hat{A}-\hat{B}_{2})\hat{P}_{m}.$
(3.39)
So we have
$\dim M_{d}^{-}(\hat{P}_{m}(\hat{A}-\hat{B}_{1})\hat{P}_{m})\geq\dim
M_{d}^{-}(\hat{P}_{m}(\hat{A}-\hat{B}_{2})\hat{P}_{m})+M_{d}^{0}(\hat{P}_{m}(\hat{A}-\hat{B}_{2})\hat{P}_{m}).$
(3.40)
Then by (3.32), (3.35) and (3.36), (3.30) holds and the proof of Lemma 3.1 is
complete.
Corollary 3.1. For any $\tau>0$ and $B\in
C([0,\frac{\tau}{4}],\mathcal{L}_{s}({\bf R}^{2n}))$ satisfying condition (B1)
and $B\geq 0$, there holds
$i_{\sqrt{-1}}^{L_{0}}(B)\geq 0.$ (3.41)
proof. By Lemma 3.1, we have
$i_{\sqrt{-1}}^{L_{0}}(B)\geq i_{\sqrt{-1}}^{L_{0}}(0).$ (3.42)
Then the conclusion holds from the fact that
$i_{\sqrt{-1}}^{L_{0}}(0)=i_{\sqrt{-1}}^{L_{0}}({\gamma}_{0})=0,$ (3.43)
Where ${\gamma}_{0}$ is the identity symplectic path.
By Theorem 2.1 of [31] and the Remark below Theorem 2.1 in [31] and the
similar proof of Lemma 3.1 we have the following lemma.
Lemma 3.2. If $\tau>0$ and $B_{1},\;B_{2}\in
C([0,\frac{\tau}{4}],\mathcal{L}_{s}({\bf R}^{2n}))$ satisfying condition (B1)
and $B_{1}\geq B_{2}$, then for $j=0,1$ there hold
$i_{L_{j}}(B_{1})\geq i_{L_{j}}(B_{2})$ (3.44)
and
$\displaystyle i_{L_{j}}(B_{1})+\nu_{L_{j}}(B_{1})\geq
i_{L_{j}}(B_{2})+\nu_{L_{j}}(B_{2}).$ (3.45)
Moreover, if $\int_{0}^{\frac{\tau}{4}}(B_{1}(t)-B_{2}(t))dt>0$, then there
holds
$\displaystyle i_{L_{j}}(B_{1})\geq i_{L_{j}}(B_{2})+\nu_{L_{j}}(B_{2}).$
(3.46)
Since $i_{L_{j}}(0)=-n$ and $\nu_{L_{j}}(0)=n$ for $j=0,1$, a direct
consequence of Lemma 3.2 is the following
Corollary 3.2. If $\tau>0$ and $B\in C([0,\frac{\tau}{2}],\mathcal{L}_{s}({\bf
R}^{2n}))$ satisfying condition (B1) and $B\geq 0$, then for $j=0,1$ there
hold
$i_{L_{j}}(B)+\nu_{L_{j}}(B)\geq 0,\qquad i_{L_{j}}(B)\geq-n.$ (3.47)
Moreover if $\int_{0}^{\frac{\tau}{2}}B(t)dt>0$, there holds
$i_{L_{j}}(B)\geq 0.$ (3.48)
Moreover we can give a stronger version of Corollary 3.2, i.e., the following
Lemma 3.3.
Lemma 3.3. Let $\tau>0$ and $B\in C([0,\frac{\tau}{2}],\mathcal{L}_{s}({\bf
R}^{2n}))$ with the $n\times n$ matrix square block form
$B(t)=\left(\begin{array}[]{cc}B_{11}(t)&B_{12}(t)\\\
B_{21}(t)&B_{22}(t)\end{array}\right)$ satisfying condition (B1) and $B\geq
0$.
If $\int_{0}^{\frac{\tau}{2}}B_{22}(t)dt>0$, there holds
$i_{L_{0}}(B)\geq 0.$ (3.49)
If $\int_{0}^{\frac{\tau}{2}}B_{11}(t)dt>0$, there holds
$i_{L_{1}}(B)\geq 0.$ (3.50)
Proof. Without loss of generality, assume ${\lambda}>0$ such that
$\int_{0}^{\frac{\tau}{2}}B_{22}(t)\geq{\lambda}I_{n}.$ (3.51)
Also we can extend $B$ to $[0,\tau]$ by
$B(\frac{\tau}{2}+t)=NB(\frac{\tau}{2}-t)N,\;\forall t\in[0,\frac{\tau}{2}].$
(3.52)
Then since $B(\tau)=B(0)$, we can extend it $\tau$-periodically to ${\bf R}$,
so we can see $B$ as an element in $C(S_{\tau},\mathcal{L}_{s}({\bf
R}^{2n}))$. Then we have
$\int_{0}^{\tau}B_{22}(t)\geq 2{\lambda}I_{n}.$ (3.53)
For any $m\in{\bf N}$, we define two subspaces of $E$ as follows
$E^{-}_{\tau,m}=\left\\{z\in E_{\tau}\left|z(t)=\sum_{j=1}^{m}{\rm
exp}(-\frac{2j\pi t}{\tau}J)b_{j},\;\forall t\in{\bf R};\;\forall b_{j}\in
L_{0}\right.\right\\},$ $E_{\tau}(0)=\left\\{z\in E_{\tau}\left|z(t)\equiv
b,\;b\in L_{0}\right.\right\\}.$
Then for any $z=\alpha x+\beta y\in E_{\tau}(0)\oplus E^{-}_{\tau,m}$ with
$\alpha^{2}+\beta^{2}=1$ and $||x||=||y||=1$, we have
$\displaystyle\langle(A_{\tau}-B_{\tau})z,z\rangle$ $\displaystyle=$
$\displaystyle\langle(A_{\tau}-B_{\tau})(\alpha x+\beta y),\alpha x+\beta
y\rangle$ (3.54) $\displaystyle=$ $\displaystyle-\beta^{2}\langle
A_{\tau}y,y\rangle-\langle B_{\tau}(\alpha x+\beta y),\alpha x+\beta y\rangle$
$\displaystyle\leq$ $\displaystyle-||A_{\tau}^{\\#}||^{-1}\beta^{2}-\langle
B_{\tau}(\alpha x+\beta y),\alpha x+\beta y\rangle.$
Since $B\geq 0$, note that $x(t)\equiv b=(0,b_{1})\in L_{0}$ for all $t\in
S_{\tau}$ with $\tau|b_{1}|^{2}=1$, we have
$\displaystyle\langle B_{\tau}(\alpha x+\beta y),\alpha x+\beta y\rangle$
(3.55) $\displaystyle=$ $\displaystyle\int_{0}^{\tau}(\alpha^{2}Bx\cdot
x+\beta^{2}By\cdot y+2\alpha\beta Bx\cdot y)\,dt$ $\displaystyle\geq$
$\displaystyle\alpha^{2}\int_{0}^{\tau}Bx\cdot
x\,dt+\beta^{2}\int_{0}^{\tau}By\cdot
y\,dt-2|\alpha||\beta|(\int_{0}^{\tau}Bx\cdot
x\,dt)^{1/2}(\int_{0}^{\tau}By\cdot y\,dt)^{1/2}$ $\displaystyle\geq$
$\displaystyle\alpha^{2}\int_{0}^{\tau}Bx\cdot
x\,dt+\beta^{2}\int_{0}^{\tau}By\cdot
y\,dt-\frac{1}{1+{\varepsilon}}\alpha^{2}\int_{0}^{\tau}Bx\cdot
x\,dt-(1+{\varepsilon})\beta^{2}\int_{0}^{\tau}By\cdot y\,dt$ $\displaystyle=$
$\displaystyle\frac{{\varepsilon}\alpha^{2}}{1+{\varepsilon}}\int_{0}^{\tau}Bx\cdot
x\,dt-{\varepsilon}\beta^{2}\int_{0}^{\tau}By\cdot y\,dt$ $\displaystyle=$
$\displaystyle\frac{{\varepsilon}\alpha^{2}}{1+{\varepsilon}}\left(\int_{0}^{\tau}B(t)dt\right)b\cdot
b-{\varepsilon}\beta^{2}\int_{0}^{\tau}By\cdot y\,dt$ $\displaystyle=$
$\displaystyle\frac{{\varepsilon}\alpha^{2}}{1+{\varepsilon}}\left(\int_{0}^{\tau}B_{22}(t)dt\right)b_{1}\cdot
b_{1}-{\varepsilon}\beta^{2}\int_{0}^{\tau}By\cdot y\,dt$ $\displaystyle\geq$
$\displaystyle\frac{{\varepsilon}\alpha^{2}}{1+{\varepsilon}}2{\lambda}|b_{1}|^{2}-{\varepsilon}\beta^{2}||B_{\tau}||\,||y||^{2}$
$\displaystyle=$
$\displaystyle\frac{2{\varepsilon}{\lambda}\alpha^{2}}{(1+{\varepsilon})\tau}-{\varepsilon}\beta^{2}||B_{\tau}||$
for any ${\varepsilon}>0$.
Let
${\varepsilon}=\min\\{1,\frac{||A_{\tau}^{\\#}||^{-1}||B_{\tau}||^{-1}}{2}\\}$.
By (3.54) and (3.55), we have
$\displaystyle\langle(A_{\tau}-B_{\tau})z,z\rangle$ $\displaystyle\leq$
$\displaystyle-||A_{\tau}^{\\#}||^{-1}\beta^{2}-\frac{2{\varepsilon}{\lambda}\alpha^{2}}{(1+{\varepsilon})\tau}+{\varepsilon}\beta^{2}||B_{\tau}||$
(3.56) $\displaystyle\leq$
$\displaystyle-\frac{||A_{\tau}^{\\#}||^{-1}\beta^{2}}{2}-\frac{{\varepsilon}{\lambda}\alpha^{2}}{\tau}$
$\displaystyle\leq$ $\displaystyle-d_{0}(\alpha^{2}+\beta^{2})$
$\displaystyle=$ $\displaystyle-d_{0},$
where
$d_{0}=\min\\{\frac{||A_{\tau}^{\\#}||^{-1}}{2},\,\frac{{\varepsilon}{\lambda}}{\tau}\\}=\min\\{\frac{||A_{\tau}^{\\#}||^{-1}}{2},\,\frac{{\lambda}}{\tau},\,\frac{{\lambda}||A_{\tau}^{\\#}||^{-1}||B_{\tau}||^{-1}}{2\tau}\\}$.
Note that $d_{0}$ is independent of $m$, so for
$0<d\leq\min\\{d_{0},\frac{||(A_{\tau}-B_{\tau})^{\\#}||^{-1}}{4}\\}$, by
Theorem 2.1 of [31] there exists $m^{*}>0$ such that, for $m\geq m^{*}$, we
have
$\dim M^{-}_{d}(P_{\tau,m}(A_{\tau}-B_{\tau})P_{\tau,m})=mn+n+i_{L_{0}}(B).$
(3.57)
By (3.56) we have
$\displaystyle\dim
M^{-}_{d}(P_{\tau,m}(A_{\tau}-B_{\tau})P_{\tau,m})\geq\dim(E_{\tau}(0)\oplus
E^{-}_{\tau,m})=mn+n.$ (3.58)
Then by (3.57) and (3.58) we have $i_{L_{0}}(B)\geq 0$.
For $\int_{0}^{\frac{\tau}{2}}B_{11}(t)dt>0$, by similar proof we have
$i_{L_{1}}(B)\geq 0$. The proof of Lemma 3.3 is complete.
Now we give the following Theorem 3.2 which will play a important role in the
proof of our main results in Section 5. This results implies that the
corresponding Maslov-type index of a periodic symmetric solution of a first
order even semipositive Hamilton increases with the increasing of the
iteration time of the solution.
Theorem 3.2. If $\tau>0$ and $B\in C([0,\frac{\tau}{4}],\mathcal{L}_{s}({\bf
R}^{2n}))$ satisfying condition (B1) and $B\geq 0$, then for any two positive
integers $p>q$ there holds
$i_{\sqrt{-1}}^{L_{0}}({\gamma}_{B}^{p})\geq
i_{\sqrt{-1}}^{L_{0}}({\gamma}_{B}^{q}).$ (3.59)
Proof. Extend ${\gamma}_{B}(t)$ to $[0,\frac{p\tau}{4}]$ as
${\gamma}_{B}^{p}$, we still denote it by ${\gamma}_{B}$. By definition of
$i_{\sqrt{-1}}^{L_{o}}$ and the Path additivity and Symplectic invariance
property of $\mu_{F}^{CLM}$ in [11], we have
$\displaystyle
i_{\sqrt{-1}}^{L_{0}}({\gamma}_{B}^{p})-i_{\sqrt{-1}}^{L_{0}}({\gamma}_{B}^{q})$
(3.60) $\displaystyle=$ $\displaystyle\mu_{F}^{CLM}(L_{0}\times JL_{0},{\rm
Gr}({\gamma}_{B}),[0,\frac{p\tau}{4}])-\mu_{F}^{CLM}(L_{0}\times JL_{0},{\rm
Gr}({\gamma}_{B}),[0,\frac{q\tau}{4}])$ $\displaystyle=$
$\displaystyle\mu_{F}^{CLM}(L_{0}\times JL_{0},{\rm
Gr}({\gamma}_{B}),[\frac{q\tau}{4},\frac{p\tau}{4}])$ $\displaystyle=$
$\displaystyle\mu_{F}^{CLM}(L_{0}\times L_{0},{\rm
Gr}(-J{\gamma}_{B}),[\frac{q\tau}{4},\frac{p\tau}{4}]).$
By the first geometrical definition of the index $\mu_{F}^{CLM}$ in section 4
of [11], there is a ${\varepsilon}>0$ small enough such that
$\displaystyle(e^{-{\varepsilon}\mathcal{J}}{\rm
Gr}(-J{\gamma}_{B}(\frac{p\tau}{4}))\cap(L_{0}\times
L_{0})=\\{0\\}=(e^{-{\varepsilon}J}{\rm
Gr}({\gamma}_{B}(\frac{q\tau}{4}))\cap(L_{0}\times L_{0})$ (3.61)
and
$\displaystyle\mu_{F}^{CLM}(L_{0}\times L_{0},{\rm
Gr}(-J{\gamma}_{B}),[\frac{q\tau}{4},\frac{p\tau}{4}])$ (3.62)
$\displaystyle=$ $\displaystyle\mu_{F}^{CLM}(L_{0}\times
L_{0},e^{-{\varepsilon}\mathcal{J}}{\rm
Gr}(-J{\gamma}_{B}),[\frac{q\tau}{4},\frac{p\tau}{4}])$ $\displaystyle=$
$\displaystyle\mu_{F}^{CLM}(L_{0}\times L_{0},{\rm
Gr}(-e^{-{\varepsilon}J}J{\gamma}_{B}e^{-{\varepsilon}J}),[\frac{q\tau}{4},\frac{p\tau}{4}]),$
where in the second equality we have used Symplectic invariance property of
$\mu_{F}^{CLM}$ index in [11]. Choose a $C^{1}$ path
${\gamma}\in\mathcal{P}_{\frac{p\tau}{4}}$ such that
${\gamma}(t)=-e^{-{\varepsilon}J}J{\gamma}_{B}e^{-{\varepsilon}J}$ for all
$t\in[\frac{q\tau}{,}\frac{p\tau}{4}]$. Denote by
$D(t)=-J\dot{{\gamma}}(t){\gamma}(t)^{-1}$ for $t\in[0,\frac{p\tau}{4}]$. For
$t\in[\frac{q\tau}{,}\frac{p\tau}{4}]$, by direct computation we have
$\displaystyle
D(t)=-J\frac{d}{dt}(-e^{-{\varepsilon}J}J{\gamma}e^{-{\varepsilon}J})(-e^{-{\varepsilon}J}J{\gamma}e^{-{\varepsilon}J})^{-1}=-Je^{-{\varepsilon}J}B(t)e^{{\varepsilon}J}J.$
(3.63)
Since $B\geq 0$ we have $D(t)\geq 0$ for $t\in[q\tau,p\tau]$ and $D\in
C([0,\frac{p\tau}{4}],\mathcal{L}_{s}({\bf R}^{2n}))$. For $s\geq 0$, we
define $D_{s}(t)=D(t)+sI_{2n}$ and symplectic path ${\gamma}_{s}(t)$ by
$\displaystyle\frac{d}{dt}{\gamma}_{s}(t)=JD_{s}(t){\gamma}_{s}(t),\quad
t\in[0,\frac{p\tau}{4}]$ $\displaystyle{\gamma}_{s}(0)=I_{2n}.$
It is clear that
${\gamma}_{0}={\gamma}.$ (3.64)
By the same argument of step2 of the proof of Theorem 5.1 in [42], we have
$\displaystyle-J\frac{d}{ds}{\gamma}_{s}(t)({\gamma}_{s}(t))^{-1}>0,\quad{\rm
for}\,t=\frac{p\tau}{4},\frac{q\tau}{4}.$ (3.65)
By (3.61) and definition of ${\gamma}_{s}$ we have
$\nu_{L_{0}}({\gamma}_{0}(\frac{p\tau}{4}))=0=\nu_{L_{0}}({\gamma}_{0}(\frac{q\tau}{4})).$
(3.66)
So by (3.65), there is a ${\sigma}>0$ small enough such that
$\nu_{L_{0}}({\gamma}_{s}(\frac{p\tau}{4}))=0=\nu_{L_{0}}({\gamma}_{s}(\frac{q\tau}{4})),\quad\forall
s\in[0,{\sigma}].$ (3.67)
So we have
$\displaystyle\mu_{F}^{CLM}(L_{0}\times L_{0},{\rm
Gr}({\gamma}_{s}(\frac{p\tau}{4})),s\in[0,{\sigma}])=0,$
$\displaystyle\mu_{F}^{CLM}(L_{0}\times L_{0},{\rm
Gr}({\gamma}_{s}(\frac{q\tau}{4})),s\in[0,{\sigma}])=0.$ (3.68)
By the Homotopy invariance with respect to end points and Path additivity
properties of $\mu_{F}^{CLM}$ index in [11], we have
$\displaystyle\mu_{F}^{CLM}(L_{0}\times L_{0},{\rm
Gr}({\gamma}_{s}(\frac{p\tau}{4})),s\in[0,{\sigma}])+\mu_{F}^{CLM}(L_{0}\times
L_{0}a,{\rm Gr}({\gamma}_{\sigma}(t)),t\in[\frac{q\tau}{4},\frac{p\tau}{4}])$
(3.69) $\displaystyle=$ $\displaystyle\mu_{F}^{CLM}(L_{0}\times L_{0},{\rm
Gr}({\gamma}_{0}(t)),t\in[\frac{q\tau}{4},\frac{p\tau}{4}])+\mu_{F}^{CLM}(L_{0}\times
L_{0},{\rm Gr}({\gamma}_{s}(\frac{p\tau}{4})),s\in[0,{\sigma}]).$
So by (3.60), (3.62), (3.64),(3.68) and (3.69), we have
$i_{\sqrt{-1}}^{L_{0}}({\gamma}_{B}^{p})-i_{\sqrt{-1}}^{L_{0}}({\gamma}_{B}^{q})=\mu_{F}^{CLM}(L_{0}\times
L_{0},{\rm Gr}({\gamma}_{\sigma}(t)),t\in[\frac{q\tau}{4},\frac{p\tau}{4}]).$
(3.70)
Since $D(t)\geq 0$ for $t\in[\frac{q\tau}{4},\frac{p\tau}{4}]$, we have
$D_{\sigma}(t)>0,\quad\forall t\in[\frac{q\tau}{4},\frac{p\tau}{4}].$ (3.71)
So by the proof of Lemma 3.1 of [42] and Lemma 2.6 of [42], we have
$\mu_{F}^{CLM}(L_{0}\times L_{0},{\rm
Gr}({\gamma}_{\sigma}(t)),t\in[\frac{q\tau}{4},\frac{p\tau}{4}])=\sum_{t\in[\frac{q\tau}{4},\frac{p\tau}{4})}\nu_{L_{0}}({\gamma}_{\sigma}(t))\geq
0.$ (3.72)
Thus by (3.70) and (3.72), (3.59) holds. The proof of Theorem 3.1 is complete.
By similar proof of Theorem 3.2 we have the following Theorem 3.3.
Theorem 3.3. If $\tau>0$ and $B\in C([0,\frac{\tau}{4}],\mathcal{L}_{s}({\bf
R}^{2n}))$ satisfying condition (B1) and $B\geq 0$, then for $j=0,1$ and any
two positive integers $p\geq q$ there holds
$i_{L_{j}}({\gamma}_{B}^{p})\geq i_{L_{j}}({\gamma}_{B}^{q}).$ (3.73)
## 4 Proof of Theorems 1.1-1.3 and Corollary 1.2
In this section we study the minimal period problem for brake orbits of the
reversible Hamiltonian system (1.1) and complete the proof of Theorems 1.1-1.3
and Corollary 1.2.
For $T>0$, we set $E=W^{1/2,2}(S_{T},{\bf R}^{2n})$ with the usual norm and
inner product denoted by $||\cdot||$ and $\langle\cdot\rangle$ respectively,
and two subspaces of $E$ by $E_{T}=\\{x\in W^{1/2,2}(S_{\tau},{\bf
R}^{2n})|\,x(-t)=Nx(t)\;a.e.\;t\in{\bf R}\\}$ and $\check{E}_{T}=\\{x\in
W^{1/2,2}(S_{\tau},{\bf R}^{2n})|\,x(-t)=-Nx(t)\;a.e.\;t\in{\bf R}\\}$. Then
we have
$E=E_{T}\oplus\check{E}_{T}.$ (4.1)
As in Section 3, we define two selfadjoint operators $A_{T}$ on $E_{T}$ by the
same way as (3.3). We also define two selfadjoint operators $\check{A}_{T}$ on
$\check{E}_{T}$ by the following bilinear form:
$\displaystyle\langle\check{A}_{{\bf T}}x,y\rangle=\int_{0}^{T}-J\dot{x}\cdot
y\,dt.$ (4.2)
Then $A_{T}$ is a bounded operator on $E_{T}$ and dim $\ker A_{T}=n$, the
Fredholm index of $A_{T}$ is zero, and $\check{A}_{T}$ is a bounded operator
on $\check{E}_{T}$ and dim $\ker\check{A}_{T}=n$, the Fredholm index of
$\check{A}_{T}$ is zero.
Set
$E_{T}(j)=\left\\{z\in E_{T}\left|z(t)={\rm exp}(\frac{2j\pi t}{T}J)a+{\rm
exp}(-\frac{2j\pi t}{T}J)b,\;\forall t\in{\bf R};\;\forall a,\,b\in
L_{0}\right.\right\\},$ $E_{T,m}=E_{T}(0)+E_{T}(1)+\cdots+E_{T}(m)$
and
$\check{E}_{T}(j)=\left\\{z\in E_{T}\left|z(t)={\rm exp}(\frac{2j\pi
t}{T}J)a+{\rm exp}(-\frac{2j\pi t}{T}J)b,\;\forall t\in{\bf R};\;\forall
a,\,b\in L_{1}\right.\right\\},$
$\check{E}_{T,m}=\check{E}_{T}(0)+\check{E}_{T}(1)+\cdots+\check{E}_{T}(m).$
Let $P_{T,m}$ be the orthogonal projection from $E_{T}$ to $E_{T,m}$ and
$\check{P}_{T,m}$ be the orthogonal projection from $\check{E}_{T}$ to
$\check{E}_{T,m}$ for $m=0,1,2,...$, then
${\Gamma}_{T}=\\{P_{T,m}:m=0,1,2,...\\}$ and
$\check{{\Gamma}}_{T}=\\{\check{P}_{T,m}:m=0,1,2,...\\}$ are the usual
Galerkin approximation schemes w.r.t. $A_{T}$ and $\check{A}_{T}$
respectively.
For $z\in{E_{T}}$, we define
$f(z)=\frac{1}{2}\langle A_{T}z,z\rangle-\int_{0}^{T}H(z)dt.$ (4.3)
It is well known that $f\in C^{2}(E_{T},{\bf R})$ whenever,
$H\in C^{2}({\bf R}^{2n})\qquad{\rm and\qquad}|H^{\prime\prime}(x)|\leq
a_{1}|x|^{s}+a_{2}$ (4.4)
for some $s\in(1,+\infty)$ and all $x\in{\bf R}^{2n}$.
By similar argument of Lemma 4.1 of [51], looking for $T$-periodic brake orbit
solutions of (1.1) is equivalent to look for critical points of $f$.
In order to get the information about the Maslov-type indices, we need the
following theorem which was proved in [24, 28, 48].
Theorem 4.1. Let $W$ be a real Hilbert space with orthogonal decomposition
$E=X\oplus Y$, where $\dim X<+\infty$. Suppose $f\in C^{2}(W,{\bf R})$
satisfies (PS) condition and the following conditions:
(i) There exist $\rho,\;\delta>0$ such that $f(w)\geq\delta$ for any $w\in W$;
(ii) There exist $e\in\partial B_{1}(0)\cap Y$ and $r_{0}>\rho>0$ such that
for any $w\in\partial Q$, $f(w)<\delta$ where $Q=(B_{r_{0}}(0)\cap
X)\oplus\\{re:0\leq r\leq r_{0}\\}$, $B_{r}(0)=\\{w\in W:||w||\leq r\\}$.
Then (1) $f$ possesses a critical value $c\geq\delta$, which is given by
$c=\inf_{h\in{\Gamma}}\max_{w\in Q}f(h(w)),$
where ${\Gamma}=\\{h\in C(Q,E):h=id\;{\rm on}\;\partial Q\\}$;
(2) There exists $w_{0}\in\mathcal{K}_{c}\equiv\\{w\in
E:\,f^{\prime}(w)=0,\,f(w)=c\\}$ such that the Morse index $m^{-}(w_{0})$ of
$f$ at $w_{0}$ satisfies
$m^{-}(w_{0})\leq\dim X+1.$
Proof of Theorem 1.3. For any given $T>0$, we prove the existence of
$T$-periodic brake solution of (1.1) whose minimal period satisfies the
inequalities in the conclusion of Theorem 1.2. We divide the proof into five
steps.
Step 1. We truncate the function $\hat{H}$ suitably and evenly such that it
satisfies the growth condition (4.4). Hence corresponding new reversible
function $H$ satisfies condition (4.4).
We follow the method in Rabinowitz’s pioneering work [43] (cf. also [18], [44]
and [51]). Let $K>0$ and $\chi\in C^{\infty}({\bf R},{\bf R})$ such that
$\chi\equiv 1$ if $y\leq K$, $\chi\equiv 0$ if $y\geq K$ and
$\chi^{\prime}(y)<0$ if $y\in(K,K+1)$, Where $K$ will be determined later. Set
$\hat{H}_{K}(z)=\chi(|z|)\hat{H}(z)+(1-\chi(|z|))R_{K}|z|^{4}$ (4.5)
and
$H_{K}(z)=\frac{1}{2}B_{0}x\cdot x+\hat{H}_{K}(z),$ (4.6)
where the constant $R_{K}$ satisfies
$R_{K}\geq\max_{K\leq|z|\leq K+1}\frac{H(z)}{|z|^{4}}.$ (4.7)
Then $H_{K}\in C^{2}({\bf R}^{2n},{\bf R})$. Since $\hat{H}$ satisfies (H3),
$\forall\varepsilon>0$, there is a $\delta_{1}>0$ such that
$\hat{H}_{K}(z)\leq\varepsilon|z|^{2}$ for $|z|\leq\delta_{1}$. It is easy to
see that $H_{K}(z)|z|^{4}$ is uniformly bounded as $|z|\to+\infty$, there is
an $M_{1}=M_{1}(\varepsilon,K)$ such that $\hat{H}_{K}(z)\leq M_{1}|z|^{4}$
for $|z|\geq\delta_{1}$. So
$\hat{H}_{K}(z)\leq\varepsilon|z|^{2}+M_{1}|z|^{4},\quad\forall z\in{\bf
R}^{2n}.$ (4.8)
Set
$\displaystyle
f_{K}(z)=\frac{1}{2}\langle{A_{T}}z,z\rangle-\int_{0}^{T}H_{K}(z)dt,\qquad\forall
z\in\hat{E}.$
Then $f_{K}\in C^{2}(E_{T},{\bf R})$ and
$\displaystyle
f_{K}(z)=\frac{1}{2}\langle({A_{T}}-{B_{0}}_{T})z,z\rangle-\int_{0}^{T}\hat{H}_{K}(z)dt,\qquad\forall
z\in\hat{E},$
where ${B_{0}}_{T}$ is the selfadjoint linear compact operator on ${E_{T}}$
defined by
$\displaystyle\langle{B_{0}}_{T}z,z\rangle=\int_{0}^{T}B_{0}z(t)\cdot
z(t)\,dt.$
Step 2. For $m>0$, let $f_{Km}=f|E_{T,m}$. We show $f_{Km}$ satisfies the
hypotheses of Theorem 4.1.
We set
$\displaystyle X_{m}=M^{-}(P_{T,m}({A_{T}}-{B_{0}}_{T})P_{T,m})\oplus
M^{0}(P_{T,m}({A_{T}}-{B_{0}}_{T})P_{T,m}),$ $\displaystyle
Y_{m}=M^{+}(P_{T,m}({A_{T}}-{B_{0}}_{T})P_{T,m}).$
For $z\in Y_{m}$, by (4.8), (3.2), and the fact that
$P_{T,j}{B_{0}}_{T}=P_{T,j}{B_{0}}_{T}$ for $j>0$, we have
$\displaystyle f_{Km}(z)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\langle({A_{T}}-{B_{0}}_{T})z,z\rangle-\int_{0}^{T}\hat{H}_{K}(z)dt$
(4.9) $\displaystyle\geq$
$\displaystyle\frac{1}{2}||({A_{T}}-{B_{0}}_{T})^{\\#}||^{-1}||z||^{2}-(\varepsilon||z||_{L^{2}}^{2}+M_{1}||z||_{L^{4}}^{4})$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}||({A_{T}}-{B_{0}}_{T})^{\\#}||^{-1}||z||^{2}-(\varepsilon
C_{2}^{2}+M_{1}C_{4}^{4}||z||^{2})||z||^{2},$
where $C_{2}$ and $C_{4}$ are constants for $s=2,\,4$ for the Sobolev
embedding of inequality (3.2), and they are independent of $m$ and $K$.
So if choose $\varepsilon>0$ small enough such that $\varepsilon
C_{2}^{2}<\frac{1}{4}||(A_{T}-{B_{0}}_{T})^{\\#}||^{-1}$, then there exists
$\rho=\rho(K)>0$ small enough and $\delta=\delta(K)>0$, which are independent
of $m$, such that
$f_{m}(z)\geq\delta,\qquad\forall z\in\partial B_{\rho}(0)\cap Y_{m}.$ (4.10)
Let $e\in B_{1}(0)\cap Y_{m}$ and set
$\displaystyle Q_{m}=\\{re:0\leq r\leq r_{1}\\}\oplus(B_{r_{1}}(0)\cap
X_{m}),$
where $r_{1}$ will be determined later. Let $z=z_{-}+z_{0}\in B_{r_{1}}(0)\cap
X_{m}$, we have
$\displaystyle f_{Km}(z+re)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\langle({A_{T}}-{B_{0}}_{T})z_{,}z_{\rangle}+\frac{1}{2}r^{2}\langle({A_{T}}-{B_{0}}_{T})e,e\rangle-\int_{0}^{T}\hat{H}_{K}(z+re)dt$
(4.11) $\displaystyle\leq$
$\displaystyle\frac{1}{2}||{A_{T}}-{B_{0}}_{T}||r^{2}-\frac{1}{2}||(A_{T}-{B_{0}}_{T})^{\\#}||^{-1}||z_{-}||^{2}-\int_{0}^{T}\hat{H}_{K}(z+re)dt.$
Since $\hat{H}$ satisfies (H2) we have
$\displaystyle\hat{H}_{K}(x)\geq a_{1}|x|^{\alpha}-a_{2},\qquad\forall
x\in{\bf R}^{2n},$
where $\alpha=\min\\{\mu,4\\}$, $a_{1}>0$, $a_{2}$ are two constants
independent of $K$ and $m$. Then there holds
$\int_{0}^{T}\hat{H}_{K}(z+re)dt\geq
a_{1}\int_{0}^{T}|z+re|^{\alpha}-Ta_{2}\geq
a_{3}(||z_{0}||_{L^{\alpha}}^{\alpha}+r^{\alpha})-a_{4},$ (4.12)
where $a_{3}$ and $a_{4}$ are constants independent of $K$ and $m$. By (4.11)
and (4.12) we have
$\displaystyle
f_{Km}(z+re)\leq\frac{1}{2}||{A_{T}}-\hat{B_{0}}||r^{2}-\frac{1}{2}||(A-B_{0})^{\\#}||^{-1}||z_{-}||^{2}-a_{3}(||z_{0}||_{L^{\alpha}}^{\alpha}+r^{\alpha})+a_{4}.$
Since $\alpha>2$ there exists a constant $r_{1}>\rho>0$, which are independent
of $K$ and $m$, such that
$f_{Km}\leq 0,\qquad\forall z\in\partial Q_{m}.$ (4.13)
Then by Theorem 4.1, $f_{Km}$ has a critical value $c_{Km}$, which is given by
$c_{Km}=\inf_{g\in{\Gamma}_{m}}\max_{z\in Q_{m}}f_{Km}(g(z)),$ (4.14)
where ${\Gamma}_{m}=\\{g\in C(Q_{m},\hat{E}_{m}|g=id;{\rm on}\;\partial
Q_{m}\\}$. Moreover there is a critical point $x_{Km}$ of $f_{Km}$ which
satisfies
$\displaystyle m^{-}(x_{Km})\leq\dim X_{m}+1.$ (4.15)
Step 3. We prove that there exists a $T$-periodic brake orbit solution $x_{T}$
of (1.1) which satisfies $i_{L_{0}}(x_{T})\leq
i_{L_{0}}(B_{0})+\nu_{L_{0}}(B_{0})+1$.
Note that $id\in{\Gamma}_{m}$, by (4.11) and condition (H4), we have
$\displaystyle c_{Km}\leq\sup_{z\in
Q_{m}}f_{Km}(z)\leq\frac{1}{2}||{A_{T}}-{B_{0}}_{T}||r_{1}^{2}.$
Then $\\{c_{Km}\\}$ possesses a convergent subsequence, we still denote it by
$\\{c_{Km}\\}$ for convenience. So there is a $c_{K}\in[\delta,]$ such that
$c_{Km}\to c_{K}$.
By the same arguments as in section 6 of [44] we have $f_{K}$ satisfies
$(PS)_{c}^{*}$ condition for $c\in{\bf R}$, i.e., any sequence ${z_{m}}$ such
that $z_{m}\in E_{T,m}$, $f_{Km}^{\prime}(z_{m})\to 0$ and $f_{Km}(z_{m})\to
c$ possesses a convergent subsequence in $E_{T}$. Hence in the sense of
subsequence we have
$\displaystyle x_{Km}\to x_{K},\qquad f_{K}(x_{K})=c_{K},\qquad
f^{\prime}_{K}(x_{K})=0.$ (4.16)
By similar argument in [44], $x_{K}$ is a classical nonconstant symmetric
$T$-periodic solution of
$\displaystyle\dot{x}=JH_{K}^{\prime}(x),\quad x\in{\bf R}^{2n}.$
Set $B_{K}(t)=H^{\prime\prime}_{K}(x_{K}(t))$, Then $B_{K}\in
C([0,T/2],\mathcal{L}_{s}({\bf R}^{2n}))$ and satisfies condition (B1). Let
${B_{K}}_{T}$ be the operator defined by the same way of the definition of
${B_{0}}_{T}$. It is easy to show that
$\displaystyle||f^{\prime\prime}(z)-({A_{T}}-{B_{K}}_{T})||\to 0\qquad{\rm
as}\;\;||z-x_{K}||\to 0.$
So for $0<d\leq\frac{1}{4}||(A_{T}-B_{K_{T}})^{\\#}||^{-1}$, there exists
$r_{2}>0$ such that
$\displaystyle||f_{Km}^{\prime\prime}(z)-P_{T,m}({A_{T}}-{B_{K}}_{T})P_{T,m}||\leq||f^{\prime\prime}(z)-({A_{T}}-{B_{K}}_{T})||\leq\frac{1}{2}d,\;\forall
z\in\\{z\in E_{T}:||z-x_{K}||\leq r_{2}\\}.$
Then for $z\in\\{z\in E_{T}:||z-x_{K}||\leq r_{2}\\}\cap E_{T,m}$, $\forall
u\in M^{-}_{d}(P_{T,m}({A_{T}}-{B_{K}}_{T})P_{T,m})\setminus\\{0\\}$, we have
$\displaystyle\langle f_{Km}^{\prime\prime}(z)u,u\rangle$ $\displaystyle\leq$
$\displaystyle\langle
P_{T,m}({A_{T}}-{B_{K}}_{T})P_{T,m}u,u\rangle+\|f_{Km}^{\prime\prime}(z)-P_{T,m}({A_{T}}-{B_{K}}_{T})P_{T,m}\|\|u\|^{2}$
$\displaystyle\leq$ $\displaystyle-\frac{1}{2}d\|u\|^{2}.$
So we have
$m^{-}(f_{Km}^{\prime\prime}(z))\geq\dim
M^{-}_{d}(P_{T,m}({A_{T}}-{B_{K}}_{T})P_{T,m}).$ (4.17)
By Theorem 2.1 of [31] and Remark 3.1, there is $m^{*}>0$ such that for $m\geq
m^{*}$ we have
$\displaystyle\dim X_{m}=mn+n+i_{L_{0}}(B_{0})+\nu_{L_{0}}(B_{0}),$ (4.18)
$\displaystyle\dim
M^{-}_{d}(P_{T,m}({A_{T}}-{B_{K}}_{T})P_{T,m})=mn+n+i_{L_{0}}(B_{K}).$ (4.19)
Then by (4.15), (4.16), and (4.17)-(4.19), we have
$\displaystyle i_{L_{0}}(B_{K})\leq i_{L_{0}}(B_{0})+\nu_{L_{0}}(B_{0})+1.$
By the similar argument as in the section 6 of [44], there is a constant
$M_{2}$ independent of $K$ such that $||x_{K}||_{\infty}\leq M_{2}$. Choose
$K>M_{2}$. Then $x_{K}$ is a non-constant symmetric $T$-periodic solution of
the problem (1.1). From now on in the proof of Theorem 1.3, we write $B=B_{K}$
and $x_{T}=x_{K}$. Then $x_{T}$ is a non-constant symmetric $T$-periodic
solution of the problem (1.1), and $B$ satisfies
$\displaystyle i_{L_{0}}(x_{T})=i_{L_{0}}(B)\leq
i_{L_{0}}(B_{0})+\nu_{L_{0}}(B_{0})+1.$ (4.20)
Since $x_{T}$ obtained in Step 3 is a nonconstant and symmetric $T$-period
solution, its minimal period $\tau=\frac{T}{k}$ for some $k\in{\bf N}$.
We denote by $x_{\tau}=x_{T}|_{[0,\tau]}$, then it is a brake orbit solution
of (1.1) with the minimal $\tau$ and $X_{T}=x_{\tau}^{k}$ being the $k$ times
iteration of $x_{\tau}$. As in Section 1, let ${\gamma}_{x_{T}}$ and
${\gamma}_{x_{\tau}}$ be the symplectic path associated to $(\tau,x)$ and
$(T,x_{T})$ respectively. Then ${\gamma}_{x_{\tau}}\in
C([0,\frac{\tau}{2}],{\rm Sp}(2n))$ and ${\gamma}_{x_{T}}\in
C([0,\frac{T}{2}],{\rm Sp}(2n))$. Also we have
${\gamma}_{x_{T}}={\gamma}_{x_{\tau}}^{k}$.
Step 4. We prove that
$\displaystyle
i_{L_{1}}({\gamma}_{x_{\tau}})+\nu_{L_{1}}({\gamma}_{x_{\tau}})\geq 1.$
We follow the way of the proof of Theorem 1.2 of [18]. By the same way as
$\check{E}_{T}$ and $\check{A}_{T}$ we can define the space $\check{E}_{\tau}$
and the operator $\check{A}_{\tau}$ on it. Also we can define the orthogonal
projection $\check{P}_{\tau},m$ and the subspaces $\check{E}_{\tau,m}$ for
$m=0,1,2,...$. Let $\check{B}_{\tau}$ be the selfadjoint linear compact
operator on ${\check{E}_{T}}$ defined by:
$\displaystyle\langle\check{B}_{\tau}z,z\rangle=\int_{0}^{\tau}B(t)z(t)\cdot
z(t)\,dt,\quad\forall z\in\check{E}_{\tau}.$
For $z\in\check{E}_{\tau}$, set
$\displaystyle
f_{\tau}(z)=\frac{1}{2}\langle(\check{A}_{\tau}-\check{B}_{\tau})z,z\rangle=\frac{1}{2}\langle\check{A}_{\tau}z,z\rangle-\frac{1}{2}\int_{0}^{\tau}H^{\prime\prime}(x_{\tau}(t))z\cdot
z\,dt$
and
$\displaystyle f_{\tau m}(w)=f_{\tau}(w),\qquad\forall
w\in\check{E}_{\tau,m}.$
Let
$X=\\{z\in L_{1}|\,B_{0}z=0\;{\rm
and}\;\hat{H}^{\prime\prime}(x_{\tau}(t))z=0,\;\forall t\in{\bf R}\\}$
and $Y$ be the orthogonal complement of $X$ in $L_{1}$, i.e., $L_{1}=X\oplus
Y$. Since
$H^{\prime\prime}(x_{\tau}(t))=B_{0}+\hat{H}^{\prime\prime}(x_{\tau}(t))$, by
(H4) it is easy to see that there exists ${\lambda}_{0}>0$ such that
$\displaystyle\int_{0}^{\tau}H^{\prime\prime}(x_{\tau}(t))z_{0}\cdot
z_{0}\,dt\geq{\lambda}_{0}||z_{0}||,\qquad\forall z_{0}\in Y.$
Thus for any $z=z_{-}+z_{0}\in\check{P}_{\tau,m}M^{-}(\check{A}_{\tau})\oplus
Y$ with $||z||=1$, we have
$\displaystyle f_{\tau m}(z)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\langle(\check{A}_{\tau}-\check{B}_{\tau})z,z\rangle=\frac{1}{2}\langle\check{A}_{\tau}z_{-},z_{-}\rangle-\frac{1}{2}\int_{0}^{\tau}H^{\prime\prime}(x_{\tau}(t))z\cdot
z\,dt$ $\displaystyle\leq$
$\displaystyle-\frac{1}{2}||\check{A}_{\tau}^{\\#}||^{-1}||z_{-}||^{2}-\frac{1}{2}\int_{0}^{\tau}H^{\prime\prime}(x_{\tau}(t))z_{0}\cdot
z_{0}\,dt-\int_{0}^{\tau}H^{\prime\prime}(x_{\tau}(t))z_{-}\cdot z_{0}\,dt$
$\displaystyle\leq$
$\displaystyle-\frac{1}{2}||\check{A}_{\tau}^{\\#}||^{-1}||z_{-}||^{2}-\frac{{\lambda}_{0}}{2}||z_{0}||^{2}+\max_{t\in[0,\tau]}||H^{\prime\prime}(x_{\tau}(t))||\,||z_{-}||\,||z_{0}||.$
(4.22)
Since
$\displaystyle||z_{-}||\,||z_{0}||\leq\frac{{\varepsilon}}{4}||z_{-}||^{2}+\frac{1}{{\varepsilon}}||z_{0}||^{2},\quad\forall{\varepsilon}>0.$
By choosing ${\varepsilon}$ suitably one can see that there exists $0<c_{0}<1$
with $|1-c_{0}|$ small enough such that if $||z_{0}||\leq c_{0}$,
$f_{\tau m}(z)\leq-\frac{{\lambda}_{0}}{4}c_{0}^{2}.$ (4.23)
When $||z_{0}||\leq c_{0}$, we have $||z_{-}||^{2}\geq 1-c_{0}^{2}$. By (4)
and (H4)
$\displaystyle f_{\tau
m}(z)\leq-\frac{1}{2}||\check{A}_{\tau}^{\\#}||^{-1}||z_{-}||^{2}\leq-\frac{1}{2}||\check{A}_{\tau}^{\\#}||^{-1}(1-c_{0}^{2}).$
Hence we always have
$f_{\tau m}(z)\leq-c||z||^{2},\qquad\forall
z\in\check{P}_{\tau,m}M^{-}(\check{A}_{\tau})\oplus Y,$ (4.24)
where
$c=\max\\{\frac{{\lambda}_{0}}{4}c_{0}^{2},\frac{1}{2}||\check{A}_{\tau}^{\\#}||^{-1}(1-c_{0}^{2})\\}$
is independent of $m$. Let
$d=\min\\{\frac{1}{4}||(\check{A}_{\tau}-\check{B}_{\tau})^{\\#}||^{-1},\frac{c}{2}\\}.$
By (4.24) and Theorem 2.1 of [31] and Remark 3.1 and the definition of
$i_{L_{1}}({\gamma}(x_{\tau}))$, for $m$ large enough, we have
$\displaystyle mn+n+i_{L_{1}}({\gamma}(x_{\tau}))$ $\displaystyle=$
$\displaystyle\dim
M^{-}_{d}(\check{P}_{\tau,m}(\check{A}_{\tau}-\check{B}_{\tau})\check{P}_{\tau,m})$
(4.25) $\displaystyle\geq$
$\displaystyle\dim(\check{P}_{\tau,m}M^{-}(\check{A}_{\tau})\oplus Y)$
$\displaystyle=$ $\displaystyle mn+n-\dim X,$
which implies that
$i_{L^{1}}({\gamma}(x_{\tau}))\geq-\dim X.$ (4.26)
Since $x_{\tau}$ is a nonconstant brake solution of (1.1), by the definition
of $X$ we have
$\nu_{L^{1}}({\gamma}(x_{\tau}))\geq\dim X+1.$ (4.27)
Hence by (4.26) and (4.27) we have
$i_{L^{1}}({\gamma}(x_{\tau}))+\nu_{L^{1}}({\gamma}(x_{\tau}))\geq 1.$ (4.28)
Step 5. Finish the proof of Theorem 1.3.
By Theorem 2.1 and Theorem 6.2 below (also Theorem 2.6 of [32]) we have
$\displaystyle i_{L_{0}}({\gamma}_{x_{\tau}}^{k})\geq
i_{L_{0}}({\gamma}_{{x_{\tau}}})+\frac{k-1}{2}(i_{1}({\gamma}^{2})+\nu_{1}({\gamma}^{2})-n),\quad{\rm
if}\,k\in 2{\bf N}-1,$ (4.29) $\displaystyle
i_{L_{0}}({\gamma}_{x_{\tau}}^{k})\geq
i_{L_{0}}({\gamma}_{{x_{\tau}}})+i_{\sqrt{-1}}^{L_{0}}({\gamma}_{{x_{\tau}}})+(\frac{k}{2}-1)(i_{1}({\gamma}^{2})+\nu_{1}({\gamma}^{2})-n),\quad{\rm
if}\,k\in 2{\bf N}.$ (4.30)
Since $B_{0}$ is semipositive and $\hat{H}$ satisfies (H4), by Corollary 3.2,
we have
$i_{L_{0}}({\gamma}_{x_{\tau}})+\nu_{L_{0}}({\gamma}_{x_{\tau}})\geq 0.$
(4.31)
By Proposition C of [42] and the definitions of $i_{L_{0}}$ and $i_{L_{1}}$ we
have
$\displaystyle i_{1}({\gamma}^{2})=i_{L_{0}}({\gamma})+i_{L_{1}}({\gamma})+n,$
$\displaystyle\nu_{1}({\gamma}^{2})=\nu_{L_{0}}({\gamma})+\nu_{L_{1}}({\gamma}).$
So by (4.28) and (4.31) we have
$i_{1}({\gamma}^{2})+\nu_{1}({\gamma}^{2})-n\geq 1.$ (4.32)
So by (4.29), (4.30) and (4.32) we have
$\displaystyle i_{L_{0}}({\gamma}_{x_{\tau}}^{k})\geq
i_{L_{0}}({\gamma}_{{x_{\tau}}})+\frac{k-1}{2},\quad{\rm if}\,k\in 2{\bf
N}-1,$ (4.33) $\displaystyle i_{L_{0}}({\gamma}_{x_{\tau}}^{k})\geq
i_{L_{0}}({\gamma}_{{x_{\tau}}})+\frac{k-1}{2},\quad{\rm if}\,k\in 2{\bf N}.$
(4.34)
By (4.20) and the definition of ${\gamma}_{x_{\tau}}$ we have
$i_{L_{0}}({\gamma}_{{x_{\tau}}})^{k})\leq
i_{L_{0}}(B_{0})+\nu_{L_{0}}(B_{0})+1.$ (4.35)
By Corollary 3.2, we have
$i_{L_{0}}({\gamma}_{{x_{\tau}}})\geq-n.$ (4.36)
So by (4.33)-(4.36) we have
$k\leq 2(i_{L_{0}}(B_{0})+\nu_{L_{0}}(B_{0}))+2n+4.$ (4.37)
Claim 2. $k$ can not be $2(i_{L_{0}}(B_{0})+\nu_{L_{0}}(B_{0}))+2n+3$ and
$2(i_{L_{0}}(B_{0})+\nu_{L_{0}}(B_{0}))+2n+4$.
Hence by Claim 2, $k\leq 2(i_{L_{0}}(B_{0})+\nu_{L_{0}}(B_{0}))+2n+2$, and
Theorem 1.3 holds.
Proof of Claim 2. We first show that $k$ can not be
$2(i_{L_{0}}(B_{0})+\nu_{L_{0}}(B_{0}))+2n+3$. Otherwise, we have
$k=2(i_{L_{0}}(B_{0})+\nu_{L_{0}}(B_{0}))+2n+3.$ (4.38)
The equality in (4.29) holds, then by (4.32), in this case there must hold
that
$\displaystyle i_{1}({\gamma}^{2})+\nu_{1}({\gamma}^{2})-n=1$ (4.39)
and
$\displaystyle i_{L_{0}}({\gamma}_{{x_{\tau}}})=-n.$ (4.40)
By Corollary 3.2 again we have that
$\nu_{L_{0}}({\gamma}_{{x_{\tau}}})=n.$ (4.41)
Also by (4.39) we have
$\displaystyle
i_{L^{1}}({\gamma}(x_{\tau})+\nu_{L^{1}}({\gamma}(x_{\tau}))=1.$
Denote by $\nu_{L^{1}}({\gamma}(x_{\tau}))=r$. Then we have
$\displaystyle i_{L^{1}}({\gamma}(x_{\tau}))=1-r,$ (4.42)
$\displaystyle\nu_{L^{1}}({\gamma}(x_{\tau}))\geq 1.$ (4.43)
By (4.40) and (4.42) we have
$i_{L_{0}}({\gamma}_{{x_{\tau}}})-i_{L_{1}}({\gamma}_{{x_{\tau}}})=r-n-1.$
(4.44)
So we can write
${\gamma}_{{x_{\tau}}}(\frac{\tau}{2})=\left(\begin{array}[]{cc}A&0\\\
C&D\end{array}\right)$ with $A,C,D$ to be $n\times n$ real matrices. Hence by
(4.2) of [35] we have
$\displaystyle{\gamma}_{{x_{\tau}}}^{2}(\tau)=N{\gamma}_{{x_{\tau}}}(\frac{\tau}{2})^{-1}N{\gamma}_{{x_{\tau}}}(\frac{\tau}{2})=\left(\begin{array}[]{cc}D^{T}A&0\\\
C^{T}A&A^{T}D\end{array}\right).$ (4.47)
Since ${\gamma}_{x_{\tau}}(\frac{\tau}{2})$ is a symplectic matrix we have
$\displaystyle A^{T}D=D^{T}A=I_{n},\quad C^{T}A=A^{T}C.$
So we have
$\displaystyle{\gamma}_{{x_{\tau}}}^{2}(\tau)=\left(\begin{array}[]{cc}I_{n}&0\\\
C^{T}A&I_{n}\end{array}\right).$ (4.50)
Note that here $C^{T}A$ is a symmetric matrix and $A$ is invertible. So by
(4.43) there exists a orthogonal matrix $Q$ such that
$Q(C^{T}A)Q^{T}={\rm
diag}(0,0,...,0,{\lambda}_{1},{\lambda}_{2},...,{\lambda}_{p},{\lambda}_{p+1},...,{\lambda}_{n-p-r})$
(4.51)
with ${\lambda}_{j}>0$ for $j=1,2,...,p$ and ${\lambda}_{j}<0$, for
$j=p+1,p+2,...,n-p-r$, where $1\leq p\leq n-r$. Then it is easy to check that
$(I_{2})^{\diamond^{r}}\diamond N_{1}(1,-1)^{\diamond^{p}}\diamond
N_{1}(1,1)^{\diamond^{(n-p-r)}}\in{\Omega}^{0}({\gamma}_{{x_{\tau}}})$ with
${\Omega}^{0}({\gamma}_{x\tau})$ to be defined in Section 6 below. Then by
Theorem 6.2 below or Theorem 2.6 of [32], when the equality in (4.29) holds,
there must hold $p=n-r$. Hence we have
$\displaystyle Q(C^{T}A)Q^{T}={\rm
diag}(0,0,...,0,{\lambda}_{1},{\lambda}_{2},...,{\lambda}_{n-r}),$ (4.52)
$\displaystyle{\lambda}_{j}>0,\qquad{\rm for}\;j=1,2,...,n-r.$ (4.53)
Case 1. If ${\rm det}A>0$, then there exists a invertible matrix path
$\rho(s)$ for $s\in[0,\frac{\tau}{2}]$ connecting it and $I_{n}$ such that
$\rho(0)=I_{n}$ and $\rho(1)=A$.
We define a symplectic path $\phi_{1}$ by
$\displaystyle\phi_{1}(s)=\left(\begin{array}[]{cc}\rho(s)^{-1}&0\\\
0&\rho(s)^{T}\end{array}\right)\left(\begin{array}[]{cc}A&0\\\
C&D\end{array}\right),\quad\forall s\in[0,\frac{\tau}{2}].$ (4.58)
Then $\nu_{L_{j}}(\phi_{1}(s)=constant$ for $j=0,1$ and
$s\in[0,\frac{\tau}{2}]$. So by Definition 2.5 and Lemma 2.8 and Proposition
2.11 of [42], for $j=1,2$ we have
$\mu_{F}^{CLM}(V_{j},{\rm Gr}(\phi_{1}),[0,\frac{\tau}{2}])=0.$ (4.59)
Also we have $\phi_{1}(0)=\left(\begin{array}[]{cc}A&0\\\
C&D\end{array}\right)$ and $\phi_{1}(0)=\left(\begin{array}[]{cc}I_{n}&0\\\
A^{T}C&I_{n}\end{array}\right)$.
Note that we can always choose the orthogonal matrix $Q$ in (4.52) such that
${\rm det}Q=1$ (otherwise we replace it by ${\rm diag}(-1,1,...,1)Q$). Then
there exists a invertible matrix path $\rho_{2}(s)$ for
$s\in[0,\frac{\tau}{2}]$ connecting it and $I_{n}$ such that
$\rho_{2}(0)=I_{n}$ and $\rho_{2}(\frac{\tau}{2})=Q$. We define a symplectic
path $\phi_{2}$ by
$\displaystyle\phi_{2}(s)=\left(\begin{array}[]{cc}I_{n}&0\\\
\rho_{2}(s)A^{T}C\rho_{2}(s)^{T}&I_{n}\end{array}\right),\quad\forall
s\in[0,\frac{\tau}{2}].$ (4.62)
Then $\nu_{L_{j}}(\phi_{2}(s)=constant$ and for $j=0,1$ and
$s\in[0,\frac{\tau}{2}]$. So by Definition 2.5 and Lemma 2.8 and Proposition
2.11 of [42] again, for $j=1,2$ we have
$\mu_{F}^{CLM}(V_{j},{\rm Gr}(\phi_{2}),[0,\frac{\tau}{2}])=0.$ (4.63)
Also we have
$\displaystyle\phi_{2}(0)=\left(\begin{array}[]{cc}I_{n}&0\\\
A^{T}C&I_{n}\end{array}\right)$ (4.66)
$\displaystyle\phi_{2}(\frac{\tau}{2})=\left(\begin{array}[]{cc}I_{n}&0\\\
QA^{T}CQ^{T}&I_{n}\end{array}\right)=(I_{2})^{\diamond^{r}}\diamond
N_{1}(1,{\lambda}_{1})\diamond\cdots\diamond N_{1}(1,{\lambda}_{n-r}).$ (4.69)
By the Reparametrization invariance and Path additivity of the Maslov index
$\mu_{F}^{CLM}$ in [11] and (4.59) and (4.63), for $j=1,2$ we have
$\displaystyle\mu_{F}^{CLM}(V_{j},{\rm
Gr}({\gamma}_{x_{\tau}}),[0,\frac{\tau}{2}])=\mu_{F}^{CLM}(V_{j},{\rm
Gr}(\phi_{2}*(\phi_{1}*{\gamma}_{{x_{\tau}}})),[0,\frac{\tau}{2}]),$
where the joint path $\phi_{2}*(\phi_{1}*{\gamma}_{{x_{\tau}}})$ is defined by
(6.1). So by definition for $j=0,1$ we have
$i_{L_{j}}({\gamma}_{{x_{\tau}}})=i_{L_{j}}(\phi_{2}*(\phi_{1}*{\gamma}_{{x_{\tau}}})).$
(4.70)
Then by Theorem 2.3 and (4.69) we have
$\displaystyle
i_{L_{0}}({\gamma}_{{x_{\tau}}})-i_{L_{1}}({\gamma}_{{x_{\tau}}})=\frac{1}{2}{\rm
sgn}M_{\varepsilon}((I_{2})^{\diamond^{r}}\diamond
N_{1}(1,{\lambda}_{1})^{T}\diamond\cdots\diamond
N_{1}(1,{\lambda}_{n-r})^{T}).$ (4.71)
By Remark 2.1 and the computations (2.161)-(2.172) at the end of Section 2,
for ${\varepsilon}>0$ small enough we have
${\rm sgn}M_{\varepsilon}((I_{2})^{\diamond^{r}}\diamond
N_{1}(1,{\lambda}_{1})^{T}\diamond\cdots\diamond
N_{1}(1,{\lambda}_{n-r})^{T})=2(r-n).$ (4.72)
So we have
$i_{L_{0}}({\gamma}_{{x_{\tau}}})-i_{L_{1}}({\gamma}_{{x_{\tau}}})=r-n,$
(4.73)
which contradicts to (4.44).
Case 2. If ${\rm det}A<0$, then there exists a invertible matrix path
$\rho(s)$ for $s\in[0,\frac{\tau}{2}]$ such that $\rho(0)={\rm
diag}(-1,1,1,...,1)$ and $\rho(1)=A$. by similar arguments we can show that
$\displaystyle
i_{L_{0}}({\gamma}_{{x_{\tau}}})-i_{L_{1}}({\gamma}_{{x_{\tau}}})=\frac{1}{2}{\rm
sgn}M_{\varepsilon}((-I_{2})\diamond(I_{2})^{\diamond^{(}r-1)}\diamond
N_{1}(1,{\lambda}_{1})\diamond\cdots\diamond N_{1}(1,{\lambda}_{n-r}))=r-n,$
(4.74)
which still contradicts to (4.44).
Hence we have proved that $k$ can not be
$2(i_{L_{0}}(B_{0})+\nu_{L_{0}}(B_{0}))+2n+3$. By the same argument we can
prove that $k$ can not be $2(i_{L_{0}}(B_{0})+\nu_{L_{0}}(B_{0}))+2n+4$. Thus
Claim 2 is proved and the proof of Theorem 1.3 is complete.
Proof of Theorem 1.1. Note that this is the case $B_{0}=0$ of Theorem 1.3.
Then by Theorem 1.3 and the fact that $i_{L_{0}}(0)=-n$ and
$\nu_{L_{0}}(0)=n$, the minimal period of $x_{T}$ is no less than
$\frac{T}{2n+2}$. In the following we prove that if (1.12) holds then the
minimal period of $x_{T}$ belongs to $\\{T,\frac{T}{2}\\}$.
Let $x_{T}$ is the $k$-time iteration of $x_{\tau}$ with $\tau$ being the
minimal period of $x_{\tau}$ and $\tau=\frac{T}{k}$. Then by the proof of
Theorem 1.3 with $B_{0}=0$ we have (4.28), (4.29) and (4.30) hold. Since
(1.12) holds, by Lemma 3.3 we have
$i_{L_{0}}({\gamma}_{x\tau})\geq 0.$ (4.75)
So by (4.29) if $k$ is odd, we have
$1\geq 0+\frac{k-1}{2}.$ (4.76)
Hence $k\leq 3$. Now we prove that $k$ can not be $3$, other wise we have
$\displaystyle i_{L_{0}}({\gamma}_{x_{\tau}})=0,$ (4.77)
$\displaystyle\nu_{L_{0}}({\gamma}_{x_{\tau}})=0,$ (4.78) $\displaystyle
i_{L_{1}}({\gamma}_{x_{\tau}})+\nu_{L_{1}}({\gamma}_{x_{\tau}})=1.$ (4.79)
And by Theorem 2.1 and Theorem 6.2 we have
$\displaystyle 1\geq
i_{L_{0}}({\gamma}_{x_{\tau}}^{3})=i_{L_{0}}({\gamma}_{x_{\tau}})+i_{e^{2\pi/3}}({\gamma}_{x_{\tau}}^{2})\geq(i_{1}({\gamma}_{x_{\tau}}^{2})-\nu_{1}({\gamma}_{x_{\tau}}^{2})-n)\geq
1.$ (4.80)
Then all the equalities of (4.80) hold. By Lemma 6.2 and 2 of Theorem 6.2
again, there exist $p\geq 0$, $q\geq 0$ with $p+q\leq n$ and
$0<\theta_{1}\leq\theta_{2}\leq...\leq\theta_{n-(p+q)}\leq 2\pi/3$ such that
$(I_{2})^{\diamond^{p}}\diamond N_{1}(1,-1)^{\diamond^{q}}\diamond
R(\theta_{1})\diamond R(\theta_{2})\diamond...\diamond
R(\theta_{n-p-q})\in{\Omega}^{0}(({\gamma}^{2}_{x_{\tau}})(\tau)),$ (4.81)
where ${\Omega}^{0}(M)$ for a symplectic matrix $M$ is defined in Section 6\.
By (4.81) we have
$-1\notin{\sigma}(({\gamma}^{2}_{x_{\tau}})(\tau)).$ (4.82)
Now we denote by
${\gamma}_{x_{\tau}}(\frac{\tau}{2})=\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right)$ with $A,B,C,D$ are all $n\times n$ matrices.
Claim 1. Both $D$ and $A$ are invertible.
We first prove $D$ is invertible. Otherwise, there exists a $n\times n$
invertible matrix $P$ such that $P^{-1}DP=\left(\begin{array}[]{cc}0&0\\\
0&R\end{array}\right)$ and $R$ is a $(n-r)\times(n-r)$ matrix with $r\geq 1$.
So we have
$\displaystyle\left(\begin{array}[]{cc}P^{T}&0\\\
0&P^{-1}\end{array}\right)\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right)\left(\begin{array}[]{cc}(P^{-1})^{T}&0\\\
0&P\end{array}\right):=\left(\begin{array}[]{cc}\tilde{A}&\tilde{B}\\\
\tilde{C}&\tilde{D}\end{array}\right)$ (4.91)
with $\tilde{D}=\left(\begin{array}[]{cc}0&0\\\ 0&R\end{array}\right)$. Since
$\left(\begin{array}[]{cc}\tilde{A}&\tilde{B}\\\
\tilde{C}&\tilde{D}\end{array}\right)$ is a symplectic matrix, we have
$\tilde{A}^{T}D-\tilde{C}^{T}\tilde{B}=I_{n}.$ (4.92)
Since $\tilde{D}=\left(\begin{array}[]{cc}0&0\\\ 0&R\end{array}\right)$,
$\tilde{B}^{T}\tilde{D}$ and $\tilde{A}^{T}\tilde{D}$ both have form
$\displaystyle\tilde{B}^{T}\tilde{D}=\left(\begin{array}[]{cc}0&*\\\
0&*\end{array}\right),\;\;\tilde{A}^{T}\tilde{D}=\left(\begin{array}[]{cc}0&*\\\
0&*\end{array}\right).$ (4.97)
So by (4.92) and (4.97) we have
$\tilde{A}^{T}\tilde{D}+\tilde{C}^{T}\tilde{B}=2\tilde{A}^{T}\tilde{D}-I_{n}=\left(\begin{array}[]{cc}-I_{r}&*\\\
0&*\end{array}\right).$ (4.98)
By direct computation and (4.97) and (4.98) we have
$\displaystyle N\left(\begin{array}[]{cc}\tilde{A}&\tilde{B}\\\
\tilde{C}&\tilde{D}\end{array}\right)^{-1}N\left(\begin{array}[]{cc}\tilde{A}&\tilde{B}\\\
\tilde{C}&\tilde{D}\end{array}\right)$ (4.103) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}P^{T}&0\\\
0&P^{-1}\end{array}\right)N\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right)^{-1}N\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right)\left(\begin{array}[]{cc}(P^{-1})^{T}&0\\\
0&P\end{array}\right)$ (4.112) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\tilde{D}^{T}\tilde{A}+\tilde{B}^{T}\tilde{C}&2\tilde{B}^{T}\tilde{D}\\\
2\tilde{A}^{T}\tilde{C}&\tilde{A}^{T}\tilde{D}+\tilde{C}^{T}\tilde{B}\end{array}\right)$
(4.115) $\displaystyle=$ $\displaystyle\left(\begin{array}[]{cccc}*&*&0&*\\\
*&*&0&*\\\ *&*&-I_{r}&*\\\ *&*&0&*\end{array}\right)$ (4.120)
Since by (4.2) o f[35] we have
$\displaystyle{\gamma}^{2}_{x_{\tau}}(\tau)=N\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right)^{-1}N\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right),$ (4.125)
by (4.112) and (4.120) we have
$\displaystyle-1\in{\sigma}({\gamma}^{2}_{x_{\tau}}(\tau)),$
which contradicts to (4.82). Thus we have proved that $D$ is invertible.
Similarly we can prove $A$ is invertible, and Claim 1 is proved.
Claim 2. There exists a invertible $n\times n$ real matrix $Q$ with ${\rm
det}Q>0$ such that
$Q^{-1}(B^{T}C)Q={\rm
diag}({0,0,...,0,{\lambda}_{1},{\lambda}_{2},...{\lambda}_{n-r}})$ (4.126)
with $r=\nu_{L_{1}}({\gamma}_{x_{\tau}})$ and ${\lambda}_{i}\in(-1,0)$ for
$i=1,2,...,n-r$.
In fact
$\displaystyle{\gamma}^{2}_{x_{\tau}}(\tau)$ $\displaystyle=$ $\displaystyle
N\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right)^{-1}N\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right)$ (4.131) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}D^{T}&B^{T}\\\
C^{T}&A^{T}\end{array}\right)\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right)$ (4.136) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}I+2B^{T}C&2B^{T}D\\\
2A^{T}C&I+2C^{T}B\end{array}\right).$ (4.139)
Since $B$ and $D$ are both invertible, for any ${\omega}\in{\bf C}$, we have
$\displaystyle\left(\begin{array}[]{cc}I_{n}&0\\\
-\frac{1}{2}(I_{n}+2C^{T}B-{\omega}I_{n})D^{-1}(B^{T})^{-1}&I_{n}\end{array}\right)\left(\begin{array}[]{cc}I+2B^{T}C-{\omega}I_{n}&2B^{T}D\\\
2A^{T}C&I+2C^{T}B-{\omega}I_{n}\end{array}\right)$ (4.144) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}I+2B^{T}C-{\omega}I_{n}&2B^{T}D\\\
-\frac{1}{2}(I_{n}+2C^{T}B-{\omega}I_{n})D^{-1}(B^{T})^{-1}(I+2B^{T}C-{\omega}I_{n})+2A^{T}C&0\end{array}\right).$
(4.147)
So we have
$\displaystyle{\rm det}({\gamma}^{2}_{x_{\tau}}(\tau)-{\omega}I_{2n})$
$\displaystyle=$ $\displaystyle{\rm det}(B^{T}D){\rm
det}((I_{n}+2C^{T}B-{\omega}I_{n})D^{-1}(B^{T})^{-1}(I+2B^{T}C-{\omega}I_{n})-4A^{T}C)$
(4.148) $\displaystyle=$ $\displaystyle{\rm
det}(D[(I_{n}+2C^{T}B-{\omega}I_{n})D^{-1}(B^{T})^{-1}(I+2B^{T}C-{\omega}I_{n})-4A^{T}C]B^{T})$
$\displaystyle=$ $\displaystyle{\rm
det}(D[I_{n}+2C^{T}B-{\omega}I_{n})D^{-1}(B^{T})^{-1}(I+2B^{T}C-{\omega}I_{n}]B^{T}-4DA^{T}CB^{T})$
$\displaystyle=$ $\displaystyle{\rm
det}((I+2B^{T}C-{\omega}I_{n})^{2}-4(1+CB^{T})CB^{T})$ $\displaystyle=$
$\displaystyle{\rm det}({\omega}^{2}I_{n}-2{\omega}(I+2CB^{T})+I).$
By (4.81) we have
${\sigma}({\gamma}^{2}_{x_{\tau}}(\tau))\subset{\bf U}.$ (4.149)
So for ${\omega}\in{\bf U}$ by (4.148) we have
${\rm
det}({\gamma}^{2}_{x_{\tau}}(\tau)-{\omega}I_{2n})=(-4)^{n}{\omega}^{n}{\rm
det}(CB^{T}-\frac{1}{2}({\rm Re}\,{\omega}-1)).$ (4.150)
Hence by (4.81) again we have ${\sigma}(CB^{T})\subset(-1,0]$, moreover there
exists a invertible $n\times n$ matrix $S$ such that
$S^{-1}CB^{T}S={\rm
diag}(0,0,...,0,{\lambda}_{1},{\lambda}_{2},...,{\lambda}_{n-r}).$ (4.151)
with $r=\nu_{L_{1}}({\gamma}_{x_{\tau}})$ and ${\lambda}_{i}\in(-1,0)$ for
$i=1,2,...,n-r$. Since $S^{-1}CB^{T}S=(B^{T}S)^{-1}B^{T}C(B^{T}S)$, let
$Q=B^{T}S$, if ${\rm det}Q<0$ we replace it by $B^{T}S{\rm
diag}(-1,1,1,...,1)$, Claim 2 is proved.
Continue the proof of Theorem 1.1.
If ${\rm det}B>0$, there is a continuous symplectic matrix path joint
$\left(\begin{array}[]{cc}B^{-1}&0\\\ 0&B^{T}\end{array}\right)$ and $I_{2n}$.
Since
$\displaystyle\left(\begin{array}[]{cc}B^{-1}&0\\\
0&B^{T}\end{array}\right)\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right)=\left(\begin{array}[]{cc}B^{-1}A&I_{n}\\\
B^{T}C&B^{T}D\end{array}\right).$ (4.158)
By Lemma 2.2, for ${\varepsilon}>0$ small enough, we have
${\rm sgn}M_{\varepsilon}\left(\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right)\right)={\rm
sgn}M_{\varepsilon}\left(\left(\begin{array}[]{cc}B^{-1}A&I_{n}\\\
B^{T}C&B^{T}D\end{array}\right)\right).$ (4.159)
If ${\rm det}B<0$, there is a continuous symplectic path joint
$\left(\begin{array}[]{cc}B^{-1}&0\\\ 0&B^{T}\end{array}\right)$ and
$(-I_{2})\diamond I_{2(n-1)}$. By direct computation we have
$\displaystyle{\rm sgn}M_{\varepsilon}\left(\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right)\right)={\rm
sgn}M_{\varepsilon}\left(\left((-I_{2})\diamond
I_{2(n-1)}\right)\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right)\right).$ (4.164)
So by Lemma 2.2 again we have (4.159) holds. So whenever ${\rm det}(B)>0$ or
not, (4.159) always holds.
Denote by $\left(\begin{array}[]{cc}P^{T}&0\\\
0&P^{-1}\end{array}\right)\left(\begin{array}[]{cc}B^{-1}A&I_{n}\\\
B^{T}C&B^{T}D\end{array}\right)\left(\begin{array}[]{cc}P&0\\\
0&(P^{-1})^{T}\end{array}\right)=\left(\begin{array}[]{cc}\tilde{A}&I_{n}\\\
\tilde{C}&\tilde{D}\end{array}\right)$. By Claim 2, we have
$\tilde{C}={\rm
diag}(0,0,..,0,{\lambda}_{1},{\lambda}_{2},...,{\lambda}_{n-r}).$ (4.165)
Since $\left(\begin{array}[]{cc}\tilde{A}&I_{n}\\\
\tilde{C}&\tilde{D}\end{array}\right)$ is a symplectic matrix, we have
$\tilde{A}$ and $\tilde{D}$ are both symmetric and have the follow forms:
$\displaystyle\tilde{A}=\left(\begin{array}[]{cc}A_{11}&0\\\
0&A_{22}\end{array}\right),\quad\tilde{D}=\left(\begin{array}[]{cc}D_{11}&0\\\
0&D_{22}\end{array}\right),$ (4.170)
where $A_{11}$ and $D_{11}$ are $r\times r$ invertible matrices, $A_{22}$ and
$D_{22}$ are $(n-r)\times(n-r)$ invertible matrices. So we have
$\left(\begin{array}[]{cc}\tilde{A}&I_{n}\\\
\tilde{C}&\tilde{D}\end{array}\right)=\left(\begin{array}[]{cc}A_{11}&I_{r}\\\
0&D_{11}\end{array}\right)\diamond\left(\begin{array}[]{cc}A_{22}&I_{n-r}\\\
{\Lambda}&D_{22}\end{array}\right),$ (4.171)
where ${\Lambda}={\rm diag}({\lambda}_{1},{\lambda}_{2},...,{\lambda}_{n-r})$.
Since $N\left(\begin{array}[]{cc}A_{11}&I_{r}\\\
0&D_{11}\end{array}\right)^{-1}N\left(\begin{array}[]{cc}A_{11}&I_{r}\\\
0&D_{11}\end{array}\right)=\left(\begin{array}[]{cc}I_{r}&2D_{11}\\\
0&I_{r}\end{array}\right)$, by (4.81) $D_{11}$ is negative definite. So we can
joint it to $-I_{r}$ by a invertible symmetric matrix path. Then by Lemma 2.2,
Remark 2.1, and computations below Remark 2.1 in Section 2, we have
$\displaystyle{\rm
sgn}M_{\varepsilon}\left(\left(\begin{array}[]{cc}A_{11}&I_{r}\\\
0&D_{11}\end{array}\right)\right)$ $\displaystyle=$ $\displaystyle{\rm
sgn}M_{\varepsilon}\left(\left(\begin{array}[]{cc}-I_{r}&I_{r}\\\
0&-I_{r}\end{array}\right)\right)$ $\displaystyle=$ $\displaystyle r\,{\rm
sgn}M_{\varepsilon}(N_{1}(-1,1))$ $\displaystyle=$ $\displaystyle 2r.$ (4.177)
Since $M_{\varepsilon}\left(\left(\begin{array}[]{cc}A_{22}&I_{n-r}\\\
{\Lambda}&D_{22}\end{array}\right)\right)$ is invertible for
${\varepsilon}=0$, for ${\varepsilon}>0$ small enough, we have
$\displaystyle{\rm
sgn}M_{\varepsilon}\left(\left(\begin{array}[]{cc}A_{22}&I_{n-r}\\\
{\Lambda}&D_{22}\end{array}\right)\right)={\rm
sgn}M_{0}\left(\left(\begin{array}[]{cc}A_{22}&I_{n-r}\\\
{\Lambda}&D_{22}\end{array}\right)\right)$ (4.182) $\displaystyle=$
$\displaystyle{\rm sgn}\left\\{\left(\begin{array}[]{cc}A_{22}&{\Lambda}\\\
I_{n-r}&D_{22}\end{array}\right)\left(\begin{array}[]{cc}0&-I_{n-r}\\\
-I_{n-r}&0\end{array}\right)\left(\begin{array}[]{cc}A_{22}&I_{n-r}\\\
{\Lambda}&D_{22}\end{array}\right)+\left(\begin{array}[]{cc}0&I_{n-r}\\\
I_{n-r}&0\end{array}\right)\right\\}$ (4.191) $\displaystyle=$
$\displaystyle{\rm
sgn}\left\\{2\left(\begin{array}[]{cc}-A_{22}{\Lambda}&-{\Lambda}\\\
-{\Lambda}&-D_{22}\end{array}\right)\right\\}$ (4.194) $\displaystyle=$
$\displaystyle{\rm sgn}\left(\begin{array}[]{cc}-A_{22}{\Lambda}&-{\Lambda}\\\
-{\Lambda}&-D_{22}\end{array}\right).$ (4.197)
Since $\left(\begin{array}[]{cc}A_{22}&I_{n-r}\\\
{\Lambda}&D_{22}\end{array}\right)$ is a symplectic matrix, we have
$\displaystyle A_{22}D_{22}-{\Lambda}=I_{n-r},$ $\displaystyle
A_{22}{\Lambda}={\Lambda}A_{22}.$ (4.198)
Hence
$\displaystyle
A_{22}^{-1}{\Lambda}-D_{22}=A_{22}^{-1}({\Lambda}-A_{22}D_{22})=-A_{22}^{-1}.$
So we have
$\displaystyle\left(\begin{array}[]{cc}I_{n-r}&0\\\
-A_{22}^{-1}&I_{n-r}\end{array}\right)\left(\begin{array}[]{cc}-A_{22}{\Lambda}&-{\Lambda}\\\
-{\Lambda}&-D_{22}\end{array}\right)\left(\begin{array}[]{cc}I_{n-r}&-A_{22}^{-1}\\\
0&I_{n-r}\end{array}\right)$ (4.205) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}-A_{22}{\Lambda}&0\\\
0&A_{22}^{-1}{\Lambda}-D_{22}\end{array}\right)$ (4.208) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}-A_{22}{\Lambda}&0\\\
0&-A_{22}^{-1}\end{array}\right).$ (4.211)
By (4.198), there exist invertible matrix $R$ such that
$\displaystyle R^{-1}A_{22}R={\rm
diag}(\alpha_{1},\alpha_{2},...,\alpha_{n-r}),\quad\alpha_{i}\in{\bf
R}\setminus\\{0\\},\;i=1,2,...,n-r,$ (4.212) $\displaystyle
R^{-1}{\Lambda}R={\rm
diag}({\lambda}_{i_{1}},{\lambda}_{i_{2}},...,{\lambda}_{i_{n-r}}),\;\;\\{i_{1},i_{2},...,i_{n-r}\\}=\\{1,2,...,n-r\\}.$
(4.213)
So we have
$\displaystyle R^{-1}(-A_{22}{\Lambda})R={\rm
diag}(-{\lambda}_{i_{1}}\alpha_{1},-{\lambda}_{i_{2}}\alpha_{2},...,-{\lambda}_{i_{n-r}}\alpha_{n-r}),$
(4.214) $\displaystyle R^{-1}(-A_{22}^{-1})R={\rm
diag}(-\frac{1}{\alpha_{1}},-\frac{1}{\alpha_{2}},...,\frac{1}{\alpha_{n-r}}).$
(4.215)
Since ${\lambda}_{i}\in(-1,0)$ for $i=1,2,...,n-r$, by (4.212)-(4.215) we have
${\rm sgn}(-A_{22}{\Lambda})+{\rm sgn}(-A_{22}^{-1})=0.$ (4.216)
Hence by (4.197), (4.211) and (4.216) we have
$\displaystyle{\rm
sgn}M_{\varepsilon}\left(\left(\begin{array}[]{cc}A_{22}&I_{n-r}\\\
{\Lambda}&D_{22}\end{array}\right)\right)={\rm sgn}(-A_{22}{\Lambda})+{\rm
sgn}(-A_{22}^{-1})=0.$ (4.219)
Since ${\rm det}Q>0$ we can joint it to $I_{n}$ by a invertible matrix path.
Hence by Lemma 2.2 and Remark 2.1, (4.171), (4.177) and (4.219), we have
$\displaystyle{\rm
sgn}M_{\varepsilon}\left(\left(\begin{array}[]{cc}B^{-1}A&I_{n}\\\
B^{T}C&B^{T}D\end{array}\right)\right)$ $\displaystyle=$ $\displaystyle{\rm
sgn}M_{\varepsilon}\left(\left(\begin{array}[]{cc}A_{11}&I_{r}\\\
0&D_{11}\end{array}\right)\right)+{\rm
sgn}M_{\varepsilon}\left(\left(\begin{array}[]{cc}A_{22}&I_{n-r}\\\
{\Lambda}&D_{22}\end{array}\right)\right)$ $\displaystyle=$ $\displaystyle
2r+0$ $\displaystyle=$ $\displaystyle 2r.$ (4.227)
Then by Theorem 2.3, (4.159) and (4.227) we have
$i_{L_{0}}({\gamma}_{x_{\tau}})-i_{L_{1}}({\gamma}_{x_{\tau}})=r.$ (4.228)
However by (4.77), (4.79) and $\nu_{L_{1}}({\gamma}_{x_{\tau}})=r$ we have
$\displaystyle
i_{L_{0}}({\gamma}_{x_{\tau}})-i_{L_{1}}({\gamma}_{x_{\tau}})=r-1,$
which contradicts to (4.228).
Thus we have prove that $k$ can not be $3$. So if $k$ is odd, it must be $1$.
By the same proof we have if $k$ is even, it must be $2$. Then
$\tau\in\\{T,\frac{T}{2}\\}$. The proof of Theorem 1.1 is complete.
Proof of Corollary 1.2. Since $0<T<\frac{\pi}{||B_{0}||}$, there is
$\varepsilon>0$ small enough such that
$\displaystyle 0\leq
B_{0}\leq||B_{0}||I_{2n}<(\frac{\pi}{T}-\varepsilon)I_{2n}.$
It is easy to see that
$\displaystyle{\gamma}_{(\frac{\pi}{T}-\varepsilon)I_{2n}}(t)={\rm
exp}(({\frac{\pi}{T}-\varepsilon})tJ)\quad\forall t\in[0,\frac{T}{2}].$
So we have
$\displaystyle\nu_{L_{0}}({\gamma}_{(\frac{\pi}{T}-\varepsilon)I_{2n}})=0,$
$\displaystyle i_{L_{0}}((\frac{\pi}{T}-\varepsilon)I_{2n})=0.$
Then by (5.40) and Lemma 3.1 and Corollary 3.1 we have
$\displaystyle 0\leq i_{-1}(B_{0})+\nu_{-1}(B_{0})\leq
i_{-1}((\frac{\pi}{T}-\varepsilon)I_{2n})=0.$
So we have
$\displaystyle i_{-1}(B_{0})+\nu_{-1}(B_{0})=0.$
Hence by the same proof of Theorem 1.1, the conclusions of Corollary 1.2
holds.
Remark 4.1. Under the same conditions of Theorem 1.3, if
$\int_{0}^{\frac{T}{2}}H^{\prime\prime}_{22}(x_{T}(t))\,dt>0$, by the same
proof of Theorem 1.1, we have
$\displaystyle\tau\geq\frac{T}{2(i_{L_{0}}(B_{0})+\nu_{L_{0}}(B_{0}))+2}.$
Moreover, if $0<T<\frac{\pi}{||B_{0}||}$ or
$i_{L_{0}}(B_{0})+\nu_{L_{0}}(B_{0})=0$, we have $\tau\in\\{T,\frac{T}{2}\\}$.
Proof of Theorem 1.2. This is the case $n=1$ and $B_{0}=0$ of Theorem 1.3, by
the proof Theorem 1.3, for any $T>0$ we obtain an T-periodic brake solution
$x_{T}$ satisfies
$\displaystyle i_{L_{0}}({\gamma}_{x_{T}})\leq 1.$ (4.229)
If it’s minimal period is $\tau=T/k$ for some $k\in{\bf N}$, we denote
$x_{\tau}=x_{T}|_{[0,\tau]}$. Then by the proof of Theorem 1.3 we have
$i_{1}({\gamma}_{x\tau}^{2})+\nu_{1}({\gamma}_{{x_{\tau}}}^{2})\geq 2.$
(4.230)
In the following we prove Theorem 1.2 in 2 steps.
Step 1. For $k=2p+1$ for some $p\geq 0$, we prove that $p=0$.
Firstly by the proof of Theorem 1.3 we have
$1\geq i_{L_{0}}({\gamma}_{{x_{\tau}}}^{2p+1})\geq
p(i_{1}({\gamma}_{{x_{\tau}}}^{2})+\nu_{1}({\gamma}_{{x_{\tau}}}^{2})-1)+i_{L_{0}}({\gamma}).$
(4.231)
We divide the argument into three cases.
Case 1. $i_{1}({\gamma}_{x\tau}^{2})+\nu_{1}({\gamma}_{{x_{\tau}}}^{2})=2$. If
$\nu_{1}({\gamma}_{{x_{\tau}}}^{2})=1$, then
$i_{1}({\gamma}_{{x_{\tau}}}^{2})=1\in 2{\bf Z}+1$. By Lemma 6.3, we have
$N_{1}(1,1)\in{\Omega}^{0}({\gamma}_{{x_{\tau}}}^{2}(\tau))$. Since
$1=i_{1}({\gamma}_{{x_{\tau}}}^{2})=i_{L_{0}}({\gamma}_{{x_{\tau}}})+i_{L_{1}}({\gamma}_{x\tau})+1.$
(4.232)
By Corollary 2.1 we have
$|i_{L_{0}}({\gamma}_{{x_{\tau}}})-i_{L_{1}}({\gamma}_{{x_{\tau}}})|\leq 1.$
(4.233)
Then by (4.232) and (4.233) we have
$i_{L_{0}}({\gamma}_{{x_{\tau}}})=i_{L_{1}}({\gamma}_{{x_{\tau}}})=0.$ (4.234)
So by Theorem 2.1, Lemma 6.2, and (6.21), we have
$\displaystyle i_{L_{0}}({\gamma}_{{x_{\tau}}}^{3})$ $\displaystyle=$
$\displaystyle
i_{L_{0}}({\gamma}_{{x_{\tau}}})+i_{e^{2\pi\sqrt{-1}/3}}({\gamma}_{x\tau}^{2})$
(4.235) $\displaystyle=$ $\displaystyle
i_{L_{0}}({\gamma}_{{x_{\tau}}})+i_{1}({\gamma}_{{x_{\tau}}}^{2})+S_{N_{1}(1,1)}(1)$
$\displaystyle=$ $\displaystyle 0+1+1$ $\displaystyle=$ $\displaystyle 2>1\geq
i_{L_{0}}({\gamma}_{x\tau}^{2p+1}).$
Then by Theorem 3.3 we have
$\displaystyle 2p+1<3.$
Hence $p=0$.
If $\nu_{1}({\gamma}_{{x_{\tau}}}^{2})=2$, then
$i_{1}({\gamma}_{{x_{\tau}}}^{2})=0$. But now
${\gamma}_{{x_{\tau}}}^{2}(\tau)=I_{2}$, by Lemma 6.3
$i_{1}({\gamma}_{{x_{\tau}}}^{2})\in 2{\bf Z}+1$, which yields a
contradiction. So this case can not happen. So in Case 1, we have proved
$p=0$.
Case 2. $i_{1}({\gamma}_{x\tau}^{2})+\nu_{1}({\gamma}_{{x_{\tau}}}^{2})=3$.
If $\nu_{1}({\gamma}_{{x_{\tau}}}^{2})=1$, then
$\displaystyle i_{1}({\gamma}_{x\tau}^{2})=2\in 2{\bf Z}.$ (4.236)
By Lemma 6.3 we have
$N_{1}(1,-1)\in{\Omega}^{0}(({\gamma}_{{x_{\tau}}}^{2})(\tau))$. So if $p\geq
1$, by Theorem 3.3, Theorem 2.1, Lemma 6.2 and (6.21), we have we have
$\displaystyle 1\geq i_{L_{0}}({\gamma}_{{x_{\tau}}}^{2p+1})$
$\displaystyle\geq$ $\displaystyle i_{L_{0}}({\gamma}_{{x_{\tau}}}^{3})$
(4.237) $\displaystyle=$ $\displaystyle
i_{L_{0}}({\gamma}_{{x_{\tau}}})+i_{e^{2\pi\sqrt{-1}/3}}({\gamma}_{{x_{\tau}}}^{2})$
$\displaystyle=$ $\displaystyle
i_{L_{0}}({\gamma}_{{x_{\tau}}})+i_{1}({\gamma}_{x\tau}^{2})+S_{N_{1}(1,-1)}(1)$
$\displaystyle\geq$ $\displaystyle-1+2+0$ $\displaystyle=$ $\displaystyle 1.$
So there must hold
$\displaystyle i_{L_{0}}({\gamma}_{x\tau})=-1.$
Then by Corollary 2.1 we have
$\displaystyle i_{L_{1}}({\gamma}_{{x_{\tau}}})\leq 0.$
So we have
$\displaystyle
i_{1}({\gamma}_{{x_{\tau}}}^{2})=i_{L_{0}}({\gamma}_{{x_{\tau}}}+i_{L_{1}}({\gamma}_{{x_{\tau}}})+1\leq-1+0+1=0,$
which contradicts (4.236). Thus we have $p=0$.
If $\nu_{1}({\gamma}_{{x_{\tau}}}^{2})=2$, then
$i_{1}({\gamma}_{{x_{\tau}}}^{2})=1,\quad{\gamma}_{x_{\tau}}^{2}(\tau)=I_{2}.$
(4.238)
If $p\geq 1$, by Theorem 3.3, Theorem 2.1, Corollary 3.2, Lemma 6.2 and
(6.21), we have we have
$\displaystyle 1\geq i_{L_{0}}({\gamma}_{{x_{\tau}}}^{2p+1})$
$\displaystyle\geq$ $\displaystyle i_{L_{0}}({\gamma}_{{x_{\tau}}}^{2+1})$
$\displaystyle=$ $\displaystyle
i_{L_{0}}({\gamma}_{{x_{\tau}}})+i_{e^{2\pi\sqrt{-1}/3}}({\gamma}_{{x_{\tau}}}^{2})$
$\displaystyle=$ $\displaystyle
i_{L_{0}}({\gamma}_{{x_{\tau}}})+i_{1}({\gamma}_{{x_{\tau}}}^{2})+S_{I_{2}}(1)$
$\displaystyle\geq$ $\displaystyle-1+1+1$ $\displaystyle=$ $\displaystyle 1.$
So there must hold
$\displaystyle i_{L_{0}}({\gamma}_{x\tau})=-1.$
Then by Corollary 2.1 we have
$\displaystyle i_{L_{1}}({\gamma}_{{x_{\tau}}})\leq 0.$
So we have
$\displaystyle
i_{1}({\gamma}_{x\tau}^{2})=i_{L_{0}}({\gamma}_{{x_{\tau}}}+i_{L_{1}}({\gamma}_{{x_{\tau}}})+1\leq-1+0+1=0,$
which contradicts (4.238). Thus we have $p=0$.
Case 3. $i_{1}({\gamma}_{{x_{\tau}}}^{2})+\nu_{1}({\gamma}_{x_{\tau}}^{2})\geq
4$.
In this case
$i_{1}({\gamma}_{{x_{\tau}}}^{2})+\nu_{1}({\gamma}_{{x_{\tau}}}^{2})-1\geq 3$.
By Corollary 3.2 we have $i_{L_{0}}\geq-1$. So by (4.231) we have
$p\leq 2/3,$ (4.239)
which yields $p=0$. So we finish Step 1.
Step 2. For $k=2p+2$ for some $p\geq 0$, we prove that $p=0$.
In fact, apply Bott-type iteration formula of Theorem 2.1 to the the case of
the iteration time equals to 4 and note that by Corollary 3.1
$i_{\sqrt{-1}}({\gamma}_{{x_{\tau}}})\geq 0$. Then by the same argument of
Step 1, we can prove that $p=0$.
Thus by Steps 1 and 2, Theorem 1.2 is proved.
A natural question is that can we prove the minimal period is $T$ in this way?
We have the following remark.
Remark 4.2. Only use the Maslov-type index theory to estimate the iteration
time of the $T$-periodic brake solution $x_{T}$ obtained by the first 4 steps
in the proof of Theorem 1.3 with $B_{0}=0$, we can not hope to prove $T$ is
the minimal period of $x_{T}$. Even $H^{\prime\prime}(z)>0$ for all $z\in{\bf
R}^{2n}\setminus\\{0\\}$. For $n=1$ and $T=4\pi$, we can not exclude the
following case:
$\displaystyle x_{T}(t)=\left(\begin{array}[]{c}\sin t\\\ \cos
t\end{array}\right),$ (4.242) $\displaystyle H^{\prime}(x_{T}(t))=x_{T}(t),$
$\displaystyle H^{\prime\prime}(x_{T}(t))\equiv I_{2n}.$
It is easy to check that ${\gamma}_{x_{T}}(t)=R(t)$ for $t\in[0,2\pi]$. Hence
by Lemma 5.1 of [30] or the proof of Lemma 3.1 of [42] we have
$\displaystyle
i_{L_{0}}({\gamma}_{x_{T}})=\sum_{0<s<2\pi}\nu_{L_{0}}({\gamma}_{x_{T}})(s)=1.$
In this case the minimal period of $x_{T}$ is $\frac{T}{2}$. Similarly for
$n>1$ we can construct examples to support this remark.
## 5 Proof of Theorems 1.4-1.5 and Corollary 1.4
In this section we study the minimal period problem for symmeytric brake orbit
solutions of the even reversible Hamiltonian system (1.1) and complete the
proof of Theorems 1.4-1.5 and Corollary 1.4.
For $T>0$, let $E_{T}=\\{x\in W^{1/2,2}(S_{\tau},{\bf
R}^{2n})|\,x(-t)=Nx(t)\;a.e.\;t\in{\bf R}\\}$ with the usual $W^{1/2,2}$ norm
and inner product. Correspondingly $\hat{E}$ and $\tilde{E}$ are defined to be
the symmetric ones and the $\frac{T}{2}$-periodic ones in $E_{T}$
respectively. Also $\\{P_{T,m}\\}$ and $\\{\hat{P}_{m}\\}$ are the Galerkin
approximation scheme w.r.t. $A_{T}$ and $\hat{A}$ respectively, where
$\\{P_{T,m}\\}$, $\\{\hat{P}_{m}\\}$, $A_{T}$, and $\hat{A}$ are defined by
the same way as in Section 2, we only need to replace $\tau$ by $T$.
For $z\in E_{T}$, we define
$f(z)=\frac{1}{2}\langle A_{T}z,z\rangle-\int_{0}^{T}H(z)dt.$ (5.1)
For $z\in\hat{E}$, we define
$\hat{f}(z)=\frac{1}{2}\langle\hat{A}z,z\rangle-\int_{0}^{T}H(z)dt.$ (5.2)
We have the following lemma.
Lemma 5.1. Let $z\in\hat{E}$. If $\hat{f}^{\prime}(z)=0$, then
$f^{\prime}(z)=0$.
Proof. Let $z\in\hat{E}$ and $\hat{f}^{\prime}(z)=0$. So for any $y\in\hat{E}$
we have
$\langle\hat{f}^{\prime}(z),y\rangle=\int_{0}^{T}J\dot{z}(t)\cdot
y(t)\,dt-\int_{0}^{T}H^{\prime}(z(t))\cdot y(t)\,dt=0,\quad\forall
y\in\hat{E}.$ (5.3)
Since $H$ is even and $z\in\hat{E}$, we have
$H^{\prime}(z(t+\frac{T}{2}))=H^{\prime}(-z(t))=-H^{\prime}(z(t)).$ (5.4)
So $H^{\prime}(z)\in\hat{E}$ and
$\langle f^{\prime}(z),y\rangle=\int_{0}^{T}J\dot{z}(t)\cdot
y(t)\,dt-\int_{0}^{T}H^{\prime}(z(t))\cdot y(t)\,dt=0,\quad\forall
y\in\tilde{E}.$ (5.5)
By (5.4) and (5.5), we have
$\langle f^{\prime}(z),y\rangle=\int_{0}^{T}J\dot{z}(t)\cdot
y(t)\,dt-\int_{0}^{T}H^{\prime}(z(t))\cdot y(t)\,dt=0,\quad\forall y\in
E_{T}.$ (5.6)
Hence $f^{\prime}(z)=0$
By Lemma 5.1 and arguments in the proof of Theorem 1.3 in Section 4, to look
for the $T$-period symmetric solutions of (1.1) is equivalent to look for
critical points of $\hat{f}$.
Proof of Theorem 1.5. For any given $T>0$, we prove the existence of
$T$-periodic symmetric brake orbit solution of (1.1) whose minimal period
satisfies the inequalities in the conclusion of Theorem 1.5. Since the proof
of existence of $T$-periodic symmetric brake orbit solution $x_{T}$ of (1.1)
is similar to that of the proof of Theorem 1.3, we will only give the sketch.
We divide the proof into several steps.
Step 1. Similarly as Step 1 in the proof of Theorem 1.3, for any $K>0$ we can
truncate the function $\hat{H}$ suitably and evenly to $\hat{H}_{K}$ such that
it satisfies the growth condition (4.4). Correspondingly we obtain a new even
and reversible function $H_{K}$ satisfies condition (4.4).
Set
$\hat{f}_{K}(z)=\frac{1}{2}\langle\hat{A}z,z\rangle-\int_{0}^{T}H_{K}(z)dt,\qquad\forall
z\in\hat{E}.$ (5.7)
Then $\hat{f}_{K}\in C^{2}(\hat{E},{\bf R})$ and
$\hat{f}_{K}(z)=\frac{1}{2}\langle(\hat{A}-\hat{B}_{0})z,z\rangle-\int_{0}^{T}\hat{H}_{K}(z)dt,\qquad\forall
z\in\hat{E},$ (5.8)
where $\hat{B}_{0}$ is the selfadjoint linear compact operator on $\hat{E}$
defined by
$\langle\hat{B}_{0}z,z\rangle=\int_{0}^{T}B_{0}z(t)\cdot z(t)\,dt.$ (5.9)
Step 2. For $m>0$, let $\hat{f}_{Km}=\hat{f}|\hat{E}_{m}$, where
$\hat{E}_{m}=\hat{P}_{m}\hat{E}$. Set
$\displaystyle X_{m}=M^{-}(\hat{P}_{m}(\hat{A}-\hat{B}_{0})\hat{P}_{m})\oplus
M^{0}(\hat{P}_{m}(\hat{A}-\hat{B}_{0})\hat{P}_{m}),$ $\displaystyle
Y_{m}=M^{+}(\hat{P}_{m}(\hat{A}-\hat{B}_{0})\hat{P}_{m}).$
By the same argument of Step 2 in the proof of Theorem 1.3, we can show that
$\hat{f}_{K}m$ satisfies the hypotheses of Theorem 4.1. Moreover, we obtain a
critical point $x_{Km}$ of $\hat{f}_{Km}$ with critical value $C_{K}m$ which
satisfies
$\displaystyle m^{-}(x_{Km})\leq\dim X_{m}+1.$ (5.10)
and
$\delta\leq C_{Km}\leq\frac{1}{2}||\hat{A}-\hat{B}_{0}||r_{1}^{2},$ (5.11)
where $\delta$ is a positive number depending on $K$ and $r_{1}>0$ is
independent of $K$ and $m$.
Step 3. We prove that there exists a symmetric $T$-periodic brake orbit
solution $x_{T}$ of (1.1) which satisfies
$i_{\sqrt{-1}}^{L_{0}}({\gamma}_{x_{T}})\leq
i_{\sqrt{-1}}^{L_{0}}(B_{0})+\nu_{\sqrt{-1}}^{L_{0}}(B_{0})+1.$ (5.12)
From the proof of Theorem 1.3 we have $f_{K}$ satisfies $(PS)_{c}^{*}$
condition for $c\in{\bf R}$, by the same proof of Lemma 5.1, we have
$\hat{f}_{K}$ satisfies $(PS)_{c}^{*}$ condition for $c\in{\bf R}$, i.e., any
sequence ${z_{m}}$ such that $z_{m}\in\hat{E}_{m}$,
$\hat{f}_{Km}^{\prime}(z_{m})\to 0$ and $\hat{f}_{Km}(z_{m})\to c$ possesses a
convergent subsequence in $\hat{E}$. Hence in the sense of subsequence we have
$x_{Km}\to
x_{K},\qquad\hat{f}_{K}(x_{K})=c_{K},\qquad\hat{f}^{\prime}_{K}(x_{K})=0.$
(5.13)
By similar argument as in [44], $x_{K}$ is a classical nonconstant symmetric
$T$-periodic solution of
$\dot{x}=JH_{K}^{\prime}(x),\quad x\in{\bf R}^{2n}.$ (5.14)
Set $B_{K}(t)=H^{\prime\prime}_{K}(x_{K}(t))$, Then $B_{K}\in
C(S_{T/2},\mathcal{L}_{s}({\bf R}^{2n}))$. Let $\hat{B}_{K}$ be the operator
defined by the same way of the definition of $\hat{B}_{0}$. It is easy to show
that
$||\hat{f}^{\prime\prime}(z)-(\hat{A}-\hat{B}_{K})||\to 0\qquad{\rm
as}\;\;||z-x_{K}||\to 0.$ (5.15)
So for $0<d<\frac{1}{4}||(A_{T}-B_{K_{T}})^{\\#}||^{-1}$, there exists
$r_{2}>0$ such that
$||\hat{f}_{Km}^{\prime\prime}(z)-\hat{P}_{m}(\hat{A}-\hat{B}_{K})\hat{P}_{m}||\leq||\hat{f}^{\prime\prime}(z)-(\hat{A}-\hat{B}_{K})||<\frac{1}{2}d,\qquad\forall
z\in\\{z\in\hat{E}:||z-x_{K}||\leq r_{2}\\}.$ (5.16)
Then for $z\in\\{z\in\hat{E}:||z-x_{K}||\leq r_{2}\\}\cap\hat{E}_{m}$,
$\forall u\in
M^{-}_{d}(\hat{P}_{m}(\hat{A}-\hat{B}_{T})\hat{P}_{m})\setminus\\{0\\}$, we
have
$\displaystyle\langle\hat{f}_{Km}^{\prime\prime}(z)u,u\rangle$
$\displaystyle\leq$
$\displaystyle\langle\hat{P}_{m}(\hat{A}-\hat{B}_{K})\hat{P}_{m}u,u\rangle+\|\hat{f}_{Km}^{\prime\prime}(z)-\hat{P}_{m}(\hat{A}-\hat{B}_{K})\hat{P}_{m}\|\|u\|^{2}$
$\displaystyle\leq$ $\displaystyle-\frac{1}{2}d\|u\|^{2}.$
So we have
$m^{-}(\hat{f}_{Km}^{\prime\prime}(z))\geq\dim
M^{-}_{d}(\hat{P}_{m}(\hat{A}-\hat{B}_{K})\hat{P}_{m}).$ (5.17)
By Theorem 3.1, Remark 3.1, there is $m^{*}>0$ such that for $m\geq m^{*}$ we
have
$\displaystyle\dim
X_{m}=mn+i_{\sqrt{-1}}^{L_{0}}(B_{0})+\nu_{\sqrt{-1}}^{L_{0}}(B_{0}),$ (5.18)
$\displaystyle\dim
M^{-}_{d}(\hat{P}_{m}(\hat{A}-\hat{B}_{K})\hat{P}_{m})=mn+i_{\sqrt{-1}}^{L_{0}}(B_{K}).$
(5.19)
Then by (5.10), (5.13), and (5.17)-(5.19), we have
$i_{\sqrt{-1}}^{L_{0}}(B_{K})\leq
i_{\sqrt{-1}}^{L_{0}}(B_{0})+\nu_{\sqrt{-1}}^{L_{0}}(B_{0})+1.$ (5.20)
By the similar argument as in the section 6 of [44], there is a constant
$M_{3}$ independent of $K$ such that $||x_{K}||_{\infty}\leq M_{3}$. Choose
$K>M_{3}$. Then $x_{K}$ is a non-constant symmetric $T$-periodic brake orbit
solution of the problem (1.1). From now on in the proof of Theorem 1.2, we
write $B=B_{K}$ and $x_{T}=x_{K}$. Then $x_{T}$ is a non-constant symmetric
$T$-periodic solution of the problem (1.1), and $B$ satisfies
$i_{\sqrt{-1}}^{L_{0}}({\gamma}_{x_{T}})=_{\sqrt{-1}}^{L_{0}}(B)\leq
i_{\sqrt{-1}}^{L_{0}}(B_{0})+\nu_{\sqrt{-1}}^{L_{0}}(B_{0})+1.$ (5.21)
Step 4. Finish the proof of Theorem 1.5.
Since $x_{T}$ obtained in Step 3 is a nonconstant and symmetric $T$-period
brake orbit solution, its minimal period $\tau=\frac{T}{4r+s}$ for some
nonnegative integer $r$ and $s=1$ or $s=3$. We now estimate $r$.
We denote by $x_{\tau}=x_{T}|_{[0,\tau]}$, then it is a symmetric period
solution of (1.1) with the minimal $\tau$ and $X_{T}=x_{\tau}^{4r+s}$ being
the $4r+s$ times iteration of $x_{\tau}$. As in Section 1, let
${\gamma}_{x_{T}}$ and ${\gamma}_{x_{\tau}}$ the symplectic path associated to
$(\tau,x)$ and $(T,x_{T})$ respectively. Then ${\gamma}_{x_{\tau}}\in
C([0,\frac{\tau}{4}],{\rm Sp}(2n))$ and ${\gamma}_{x_{T}}\in
C([0,\frac{T}{4}],{\rm Sp}(2n))$. Also we have
${\gamma}_{x_{T}}={\gamma}_{x_{\tau}}^{4r+s}$, which is the $4r+s$ times
iteration of ${\gamma}_{x_{\tau}}$.
By (5.21) we have
$i_{\sqrt{-1}}^{L_{0}}({\gamma}_{x_{\tau}}^{4r+s})\leq
i_{\sqrt{-1}}^{L_{0}}(B_{0})+\nu_{\sqrt{-1}}^{L_{0}}(B_{0})+1.$ (5.22)
Since $x_{\tau}$ is also a nonconstant symmetric periodic solution of (1.1).
It is clear that
$\displaystyle\nu_{-1}(x_{\tau}^{2})$ $\displaystyle\geq$ $\displaystyle 1.$
(5.23)
Since $\hat{H}$ satisfies condition (H5) and $B_{0}$ is semipositive, by
Corollary 3.1 of [51] (also by Theorem 6.2) we have
$i_{-1}({\gamma}_{x_{\tau}}^{2})\geq 0.$ (5.24)
By Corollary 3.2 of [51] (cf. aslo [29]), we have
$i_{1}({\gamma}_{x_{\tau}}^{2})+\nu_{1}({\gamma}_{x_{\tau}}^{2})\geq n.$
(5.25)
It is easy to see that
${\gamma}_{x_{\tau}}^{4}(\frac{\tau}{2}+t)={\gamma}_{x_{\tau}}^{2}(t)\,{\gamma}_{x_{\tau}}^{2}(\frac{\tau}{2}),\qquad\forall
t\in[0,\frac{\tau}{2}].$ (5.26)
So by Theorem 6.1 of Bott-type iteration formula we have
$\displaystyle
i_{1}({\gamma}_{x_{\tau}}^{4})+\nu_{1}({\gamma}_{x_{\tau}}^{4})$
$\displaystyle=$ $\displaystyle
i_{1}({\gamma}_{x_{\tau}}^{2})+\nu_{1}({\gamma}_{x_{\tau}}^{2})+i_{-1}({\gamma}_{x_{\tau}}^{2})+\nu_{-1}({\gamma}_{x_{\tau}}^{2})$
(5.27) $\displaystyle\geq$ $\displaystyle n+0+1$ $\displaystyle=$
$\displaystyle n+1.$
If $r\geq 1$, then by Theorems 2.2 and 6.2 and (5.27) we have
$\displaystyle i_{-1}({\gamma}_{x_{\tau}}^{4r})$ $\displaystyle=$
$\displaystyle i_{-1}(({\gamma}_{x_{\tau}}^{2})^{2p})$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{r}i_{{\omega}_{2r}^{2j-1}}({\gamma}_{x_{\tau}}^{4})$
$\displaystyle\geq$
$\displaystyle\sum_{j=1}^{r}(i_{1}({\gamma}_{x_{\tau}}^{4})+\nu_{1}({\gamma}_{x_{\tau}}^{4})-n)$
$\displaystyle=$ $\displaystyle
r(i_{1}({\gamma}_{x_{\tau}}^{4})+\nu_{1}({\gamma}_{x_{\tau}}^{4})-n)$
$\displaystyle\geq$ $\displaystyle r,$ (5.29)
where ${\omega}_{2r}=e^{\pi\sqrt{-1}/(2r)}$ as defined in Theorem 2.2.
By Theorem 3.2, we have
$i_{\sqrt{-1}}^{L_{0}}({\gamma}_{x_{\tau}}^{4r+s})\geq
i_{\sqrt{-1}}^{L_{0}}({\gamma}_{x_{\tau}}^{4r}).$ (5.30)
Then (5.22), (5.29) and (5.30) yield
$r\leq i_{\sqrt{-1}}^{L_{0}}(B_{0})+\nu_{\sqrt{-1}}^{L_{0}}(B_{0})+1.$ (5.31)
Thus for $i_{\sqrt{-1}}^{L_{0}}(B_{0})+\nu_{\sqrt{-1}}^{L_{0}}(B_{0})$ is odd,
by (5.31) we have
$4r+s\leq 4r+3\leq
4(i_{\sqrt{-1}}^{L_{0}}(B_{0})+\nu_{\sqrt{-1}}^{L_{0}}(B_{0}))+7.$ (5.32)
Claim 3. For $i_{\sqrt{-1}}^{L_{0}}(B_{0})+\nu_{\sqrt{-1}}^{L_{0}}(B_{0})$ is
even, the equality in (5.31) can not hold.
Otherwise, $r\geq 1$ and the equality in (5) holds i.e.,
$i_{{\omega}_{2r}^{2j-1}}({\gamma}_{x_{\tau}}^{4})=i_{1}({\gamma}_{x_{\tau}}^{4})+\nu_{1}({\gamma}_{x_{\tau}}^{4})-n=1,\quad
j=1,2,...,r.$ (5.33)
By the definition of ${\omega}_{2r}$, we have ${\omega}_{2r}^{2j-1}\neq-1$ for
$j=1,2,...,r$. So by (5.33) and 2 of Theorem 6.2, we have $I_{2p}\diamond
N_{1}(1,-1)^{\diamond q}\diamond
K\in{\Omega}^{0}({\gamma}_{x_{\tau}}^{4}(\tau))$ for some non-negative
integers $p$ and $q$ satisfying $0\leq p+q\leq n$ and $K\in{\rm Sp}(2(n-p-q))$
with ${\sigma}(K)\in{\bf U}\setminus\\{1\\}$ satisfying the condition that all
eigenvalues of $K$ located with the arc between $1$ and ${\omega}_{2r}$ in
${\bf U}^{+}\setminus\\{\pm 1\\}$ possess total multiplicity $n-p-q$. So there
are no eigenvalues of $K$ on the arc between ${\omega}_{2r}^{2j-1}$ and $-1$
except ${\omega}_{2r}^{2r-1}$ with $r=1$. However, whether
${\omega}_{2r}^{2r-1}\in{\sigma}({\gamma}_{x_{\tau}}^{4}(\tau))$ or not, we
always have
$\displaystyle S^{+}_{{\gamma}_{x_{\tau}}^{4}(\tau)}({\omega}_{2r}^{2r-1})=0,$
(5.34) $\displaystyle i_{{\omega}_{2r}^{2r-1}}({\gamma}_{x_{\tau}}^{4})=1.$
(5.35)
So (6.21) and Lemma 6.2, we have
$\displaystyle i_{-1}({\gamma}_{x_{\tau}}^{4})$ $\displaystyle=$
$\displaystyle
i_{{\omega}_{2r}^{2r-1}}({\gamma}_{x_{\tau}}^{4})+S^{+}_{{\gamma}_{x_{\tau}}^{4}(\tau)}({\omega}_{2r}^{2r-1})$
(5.36) $\displaystyle=$ $\displaystyle 1+0=1.$
But by (5.26), Lemma 6.1, and Theorem 6.1, we have
$\displaystyle i_{-1}({\gamma}_{x_{\tau}}^{4r})$ $\displaystyle=$
$\displaystyle i_{-1}(({\gamma}_{x_{\tau}}^{2r})^{2})$ $\displaystyle=$
$\displaystyle
i_{\sqrt{-1}}({\gamma}_{x_{\tau}}^{2r})+i_{-\sqrt{-1}}({\gamma}_{x_{\tau}}^{2r})$
$\displaystyle=$ $\displaystyle 2i_{\sqrt{-1}}({\gamma}_{x_{\tau}}^{2r}).$
Then $i_{-1}({\gamma}_{x_{\tau}}^{4r})$ is an even integer, which yields a
contradiction to (5.36). So Claim 3 holds, and we have
$r\leq i_{\sqrt{-1}}^{L_{0}}(B_{0})+\nu_{\sqrt{-1}}^{L_{0}}(B_{0}).$ (5.37)
Hence
$4r+s\leq 4r+3\leq
4(i_{\sqrt{-1}}^{L_{0}}(B_{0})+\nu_{\sqrt{-1}}^{L_{0}}(B_{0}))+3.$ (5.38)
Theorem 1.5 holds from (5.32) and (5.38).
Proof of Theorem 1.4. This is the case $B_{0}\equiv 0$ of Theorem 1.2. From
Theorem 3.1 it is easy to see that
$i_{\sqrt{-1}}^{L_{0}}(0)=0,\qquad\nu_{\sqrt{-1}}^{L_{0}}(0)=0.$ (5.39)
Then $i_{\sqrt{-1}}^{L_{0}}(0)+\nu_{\sqrt{-1}}^{L_{0}}(0)=0$ and is also even.
So Theorem 1.4 holds from Theorem 1.5.
Proof of Corollary 1.2. Since $0<T<\frac{\pi}{||B_{0}||}$, there is
$\varepsilon>0$ small enough such that
$0\leq B_{0}\leq||B_{0}||I_{2n}<(\frac{\pi}{T}-\varepsilon)I_{2n}.$ (5.40)
It is easy to see that
${\gamma}_{(\frac{\pi}{T}-\varepsilon)I_{2n}}(t)={\rm
exp}(({\frac{\pi}{T}-\varepsilon})tJ)\quad\forall t\in[0,\frac{T}{4}].$ (5.41)
Since
$\nu_{L_{0}}({\rm exp}(({\frac{\pi}{T}-\varepsilon})tJ))=0,\qquad\forall
t\in[0,\frac{T}{2}].$ (5.42)
We have
$i_{L_{0}}({\gamma}_{(\frac{\pi}{T}-\varepsilon)I_{2n}}^{2})=0,\;i_{L_{0}}({\gamma}_{(\frac{\pi}{T}-\varepsilon)I_{2n}})=0.$
(5.43)
So by Theorem 2.1 we have
$\displaystyle
i_{\sqrt{-1}}^{L_{0}}((\frac{\pi}{T}-\varepsilon)I_{2n})=i_{L_{0}}({\gamma}_{(\frac{\pi}{T}-\varepsilon)I_{2n}}^{2})-i_{L_{0}}({\gamma}_{(\frac{\pi}{T}-\varepsilon)I_{2n}})=0.$
(5.44)
Then by (5.40) and Lemma 3.1 and Corollary 3.1 we have
$0\leq i_{-1}(B_{0},\frac{T}{2})+\nu_{-1}(B_{0},\frac{T}{2})\leq
i_{-1}((\frac{\pi}{T}-\varepsilon)I_{2n},\frac{T}{2})=0.$ (5.45)
So we have
$i_{-1}(B_{0},\frac{T}{2})+\nu_{-1}(B_{0},\frac{T}{2})=0.$ (5.46)
Hence by Theorem 1.1 or Corollary 1.1, the conclusion of Corollary 1.2 holds.
Also a natural question is that can we prove the minimal period is $T$ in this
way? We have the following remark.
Remark 5.1. Only use the Maslov-type index theory to estimate the iteration
time of the symmetric $T$-periodic brake solution $x_{T}$ obtained in the
proof of Theorem 1.5 with $B_{0}=0$, we can not hope to prove $T$ is the
minimal period of $x_{T}$. Even $H^{\prime\prime}(z)>0$ for all $z\in{\bf
R}^{2n}\setminus\\{0\\}$. For $n=1$ and $T=6\pi$, we can not exclude the
following case:
$\displaystyle x_{T}(t)=\left(\begin{array}[]{c}\sin t\\\ \cos
t\end{array}\right),$ (5.49) $\displaystyle H^{\prime}(x_{T}(t))=x_{T}(t),$
$\displaystyle H^{\prime\prime}(x_{T}(t))\equiv I_{2n}.$ (5.50)
It is easy to check that ${\gamma}_{x_{T}}(t)=R(t)$ for $t\in[0,3\pi]$. Hence
by Theorem 2.1 and Lemma 5.1 of [30] or the proof of Lemma 3.1 of [42] we have
$i_{\sqrt{-1}}^{L_{0}}({\gamma}_{x_{T}})=\sum_{3\pi/4\leq
s<3\pi}\nu_{L_{0}}({\gamma}_{x_{T}})(s)=1.$ (5.51)
In this case the minimal period of $x_{T}$ is $\frac{T}{3}$. Similarly for
$n>1$ we can construct examples to support this remark.
## 6 Appendix on Maslov-type indices $(i_{\omega},\nu_{\omega})$
We first recall briefly the Maslov-type index theory of
$(i_{\omega},\nu_{\omega})$. All the details can be found in [41].
For any ${\omega}\in{\bf U}$, the following codimension 1 hypersuface in ${\rm
Sp}(2n)$ is defined by:
${\rm Sp}(2n)_{\omega}^{0}=\\{M\in{\rm Sp}(2n)|{\rm
det}(M-{\omega}I_{2n})=0\\}.$
For any two continuous path $\xi$ and $\eta$: $[0,\tau]\to{\rm Sp}(2n)$ with
$\xi(\tau)=\eta(0)$, their joint path is defined by
$\eta*\xi(t)=\left\\{\begin{array}[]{lr}\xi(2t)&{\rm if}\,0\leq
t\leq\frac{\tau}{2},\\\ \eta(2t-\tau)&{\rm if}\,\frac{\tau}{2}\leq
t\leq\tau.\end{array}\right.$ (6.1)
Given any two $(2m_{k}\times 2m_{k})$\- matrices of square block form
$M_{k}=\left(\begin{array}[]{cc}A_{k}&B_{k}\\\ C_{k}&D_{k}\end{array}\right)$
for $k=1,2$, as in [41], the $\diamond$-product of $M_{1}$ and $M_{2}$ is
defined by the following $(2(m_{1}+m_{2})\times 2(m_{1}+m_{2}))$-matrix
$M_{1}\diamond M_{2}$:
$M_{1}\diamond M_{2}=\left(\begin{array}[]{cccc}A_{1}&0&B_{1}&0\\\
0&A_{2}&0&B_{2}\\\ C_{1}&0&D_{1}&0\\\ 0&C_{2}&0&D_{2}\end{array}\right).$
A special path $\xi_{n}$ is defined by
$\xi_{n}(t)=\left(\begin{array}[]{cc}2-\frac{t}{\tau}&0\\\
0&(2-\frac{t}{\tau})^{-1}\end{array}\right)^{\diamond n},\qquad\forall
t\in[0,\tau].$
Definition 6.1. For any ${\omega}\in{\bf U}$ and $M\in{\rm Sp}(2n)$, define
$\nu_{\omega}(M)=\dim_{\bf C}\ker(M-{\omega}I_{2n}).$ (6.2)
For any ${\gamma}\in\mathcal{P}_{\tau}(2n)$, define
$\nu_{\omega}({\gamma})=\nu_{\omega}({\gamma}(\tau)).$ (6.3)
If ${\gamma}(\tau)\notin{\rm Sp}(2n)_{\omega}^{0}$, we define
$i_{\omega}({\gamma})=[{\rm Sp}(2n)_{\omega}^{0}\,:\,{\gamma}*\xi_{n}],$ (6.4)
where the right-hand side of (6.4) is the usual homotopy intersection number
and the orientation of ${\gamma}*\xi_{n}$ is its positive time direction under
homotopy with fixed endpoints.
If ${\gamma}(\tau)\in{\rm Sp}(2n)_{\omega}^{0}$, we let
$\mathcal{F}({\gamma})$ be the set of all open neighborhoods of ${\gamma}$ in
$\mathcal{P}_{\tau}(2n)$, and define
$i_{\omega}({\gamma})=\sup_{U\in\mathcal{F}({\gamma})}\inf\\{i_{\omega}(\beta)|\,\beta(\tau)\in
U\,{\rm and}\,\beta(\tau)\notin{\rm Sp}(2n)_{\omega}^{0}\\}.$ (6.5)
Then $(i_{\omega}({\gamma}),\nu_{\omega}({\gamma}))\in{\bf
Z}\times\\{0,1,...,2n\\}$, is called the index function of ${\gamma}$ at
${\omega}$.
Lemma 6.1. (Lemma 5.3.1 of [41]) For any ${\gamma}\in\mathcal{P}_{\tau}(2n)$
and ${\omega}\in{\bf U}$, there hold
$\displaystyle
i_{\omega}({\gamma})=i_{\bar{{\omega}}}({\gamma}),\qquad\nu_{\omega}({\gamma})=\nu_{\bar{{\omega}}}({\gamma}).$
(6.6)
As in [38], for any $M\in{\rm Sp}(2n)$ we define
$\displaystyle{\Omega}(M)=\\{P\in{\rm Sp}(2n)$ $\displaystyle|$
$\displaystyle{\sigma}(P)\cap{\bf U}={\sigma}(M)\cap{\bf U}$ (6.7)
$\displaystyle{\rm
and}\,\nu_{\lambda}(P)=\nu_{\lambda}(M),\;\;\forall{\lambda}\in{\sigma}(M)\cap{\bf
U}\\}.$
We denote by ${\Omega}^{0}(M)$ the path connected component of ${\Omega}(M)$
containing $M$, and call it the homotopy component of $M$ in ${\rm Sp}(2n)$.
The following symplectic matrices were introduced as basic normal forms in
[41]:
$\displaystyle D({\lambda})=\left(\begin{array}[]{cc}{\lambda}&0\\\
0&{\lambda}^{-1}\end{array}\right),\qquad$ $\displaystyle{\lambda}=\pm 2,$
(6.10) $\displaystyle
N_{1}({\lambda},b)=\left(\begin{array}[]{cc}{\lambda}&b\\\
0&{\lambda}\end{array}\right),\qquad$ $\displaystyle{\lambda}=\pm 1,\,b=\pm
1,\,0,$ (6.13) $\displaystyle
R(\theta)=\left(\begin{array}[]{cc}\cos(\theta)&-\sin(\theta)\\\
\sin(\theta)&\cos(\theta)\end{array}\right),\qquad$
$\displaystyle\theta\in(0,\pi)\cup(\pi,2\pi),$ (6.16) $\displaystyle
N_{2}({\omega},b)=\left(\begin{array}[]{cc}R(\theta)&b\\\
0&R(\theta)\end{array}\right),\qquad$
$\displaystyle\theta\in(0,\pi)\cup(\pi,2\pi),$ (6.19)
where $b=\left(\begin{array}[]{cc}b_{1}&b_{2}\\\
b_{3}&b_{4}\end{array}\right)$ with $b_{i}\in{\bf R}$ and $b_{2}\neq b_{3}$.
For any $M\in{\rm Sp}(2n)$ and ${\omega}\in{\bf U}$, splitting number of $M$
at ${\omega}$ is defined by
$S_{M}^{\pm}=\lim_{\epsilon\to 0^{+}}i_{{\omega}{\rm
exp}(\pm\sqrt{-1}\epsilon)}({\gamma})-i_{\omega}({\gamma})$ (6.20)
for any path ${\gamma}\in\mathcal{P}_{\tau}(2n)$ satisfying
${\gamma}(\tau)=M$.
Splitting numbers possesses the following properties.
Lemma 6.2. (cf. [40], Lemma 9.1.5 and List 9.1.12 of [41]) Splitting number
$S_{M}^{\pm}({\omega})$ are well defined; that is they are independent of the
choice of the path ${\gamma}\in\mathcal{P}_{\tau}(2n)$ satisfying
${\gamma}(\tau)=M$. For ${\omega}\in{\bf U}$ and $M\in{\rm Sp}(2n)$,
$S_{N}^{\pm}({\omega})$ are constant for all $N\in{\Omega}^{0}(M)$. Moreover
we have
(1) $(S_{M}^{+}(\pm 1),S_{M}^{-}(\pm 1))=(1,1)$ for $M=\pm N_{1}(1,b)$ with
$b=1$ or $0$;
(2) $(S_{M}^{+}(\pm 1),S_{M}^{-}(\pm 1))=(0,0)$ for $M=\pm N_{1}(1,b)$ with
$b=-1$;
(3) $(S_{M}^{+}(e^{\sqrt{-1}\theta}),S_{M}^{-}(e^{\sqrt{-1}\theta}))=(0,1)$
for $M=R(\theta)$ with $\theta\in(0,\pi)\cup(\pi,2\pi)$;
(4) $(S_{M}^{+}({\omega}),S_{M}^{-}({\omega}))=(0,0)$ for ${\omega}\in{\bf
U}\setminus{\bf R}$ and $M=N_{2}({\omega},b)$ is trivial i.e., for
sufficiently small $\alpha>0$, $MR((t-1)\alpha)^{\diamond n}$ possesses no
eigenvalues on ${\bf U}$ for $t\in[0,1)$.
(5) $(S_{M}^{+}({\omega}),S_{M}^{-}({\omega})=(1,1)$ for ${\omega}\in{\bf
U}\setminus{\bf R}$ and $M=N_{2}({\omega},b)$ is non-trivial.
(6) $(S_{M}^{+}({\omega}),S_{M}^{-}({\omega})=(0,0)$ for any ${\omega}\in{\bf
U}$ and $M\in{\rm Sp}(2n)$ with ${\sigma}(M)\cap{\bf U}=\emptyset$.
(7) $S_{M_{1}\diamond
M_{2}}^{\pm}({\omega})=S_{M_{1}}^{\pm}({\omega})+S_{M_{2}}^{\pm}({\omega})$,
for any $M_{j}\in{\rm Sp}(2n_{j})$ with $j=1,2$ and ${\omega}\in{\bf U}$.
By the definition of splitting numbers and Lemma 6.2, for
$0\leq\theta_{1}<\theta_{2}<2\pi$ and ${\gamma}\in\mathcal{P}_{\tau}(2n)$ with
$\tau>0$, we have
$\displaystyle i_{{\rm exp}(\sqrt{-1}\theta_{2})}({\gamma})$ $\displaystyle=$
$\displaystyle i_{{\rm
exp}(\sqrt{-1}\theta_{1})}+S^{+}_{{\gamma}(\tau)}(e^{\sqrt{-1}\theta_{1}})$
(6.21) $\displaystyle+$
$\displaystyle\sum_{\theta\in(\theta_{1},\theta_{2})}\left(S^{+}_{{\gamma}(\tau)}(e^{\sqrt{-1}\theta})-S^{-}_{{\gamma}(\tau)}(e^{\sqrt{-1}\theta})\right)-S^{-}_{{\gamma}(\tau)}(e^{\sqrt{-1}\theta_{2}}).$
For any symplectic path ${\gamma}\in\mathcal{P}_{\tau}(2n)$ and $m\in{\bf N}$,
we define its $m$th iteration in the periodic boundary sense
${\gamma}(m):[0,m\tau]\to{\rm Sp}(2n)$ by
${\gamma}(m)(t)={\gamma}(t-j\tau){\gamma}(\tau)^{j}\qquad{\rm for}\,j\tau\leq
t\leq(j+1)\tau,\;j=0,1,...,m-1.$ (6.22)
Definition 6.2.(cf.[40], [41]) For any ${\gamma}\in\mathcal{P}_{\tau}(2n)$ and
${\omega}\in{\bf U}$, we define
$(i_{\omega}({\gamma},m),\nu_{\omega}({\gamma},m))=(i_{\omega}({\gamma}(m)),\nu_{\omega}({\gamma}(m))),\qquad\forall
m\in{\bf N}.$ (6.23)
We have the following Bott-type iteration formula.
Theorem 6.1. (cf. [40], Theorem 9.2.1 of [41]) For any $\tau>0$,
${\gamma}\in\mathcal{P}_{\tau}(2n)$, $z\in{\bf U}$, and $m\in{\bf N}$,
$\displaystyle
i_{z}({\gamma},m)=\sum_{{\omega}^{k}=z}i_{\omega}({\gamma}),\qquad\nu_{z}({\gamma},m)=\sum_{{\omega}^{m}=z}\nu_{\omega}({\gamma}).$
(6.24)
By Theorem 8.1.4 of [41], we have the following Lemma.
Lemma 6.3. For ${\gamma}\in\mathcal{P}_{\tau}(2)$ with $\tau>0$, the following
results hold.
1\. If $N_{1}(1,1)\in{\Omega}^{0}({\gamma}(\tau))$, then
$\displaystyle
i_{1}({\gamma},m)=m(i_{1}({\gamma})+1)-1,\qquad\nu_{1}({\gamma},m)=1,\quad\forall
m\in{\bf N},$ (6.25) $\displaystyle i_{1}({\gamma})\in 2{\bf Z}+1.$ (6.26)
2\. If $N_{1}(1,1)\in{\Omega}^{0}({\gamma}(\tau))$, then
$\displaystyle
i_{1}({\gamma},m)=m(i_{1}({\gamma})+1)-1,\qquad\nu_{1}({\gamma},m)=2,\quad\forall
m\in{\bf N},$ (6.27) $\displaystyle i_{1}({\gamma})\in 2{\bf Z}+1.$ (6.28)
3\. If $N_{1}(1,-1)\in{\Omega}^{0}({\gamma}(\tau))$, then
$\displaystyle
i_{1}({\gamma},m)=m(i_{1}({\gamma}),\qquad\nu_{1}({\gamma},m)=1,\quad\forall
m\in{\bf N},$ (6.29) $\displaystyle i_{1}({\gamma})\in 2{\bf Z}.$ (6.30)
Denote by ${\bf U}^{+}=\\{{\omega}\in{\bf U}|\,Im\,{\omega}\geq 0\\}$ and
${\bf U}^{-}=\\{{\omega}\in{\bf U}|\,Im\,{\omega}\leq 0\\}$. The following
theorem was proved by Liu and Long in [33, 34], which plays a important role
in the proof of our main results in Sections 4-5.
Theorem 6.2. (Theorem 10.1.1 of [41])
1\. For any ${\gamma}\in\mathcal{P}_{\tau}(2n)$ and ${\omega}\in{\bf
U}\setminus\\{1\\}$, it always holds that
$i_{1}({\gamma})+\nu_{1}({\gamma})-n\leq i_{\omega}({\gamma})\leq
i_{1}({\gamma})+n-\nu_{\omega}({\gamma}).$ (6.31)
2\. The left equality in (6.31) holds for some ${\omega}\in{\bf
U}^{+}\setminus\\{1\\}$ (or ${\bf U}^{-}\setminus\\{1\\}$) if and only if
$I_{2p}\diamond N_{1}(1,-1)^{\diamond q}\diamond
K\in{\Omega}^{0}({\gamma}(\tau))$ for some non-negative integers $p$ and $q$
satisfying $0\leq p+q\leq n$ and $K\in{\rm Sp}(2(n-p-q))$ with
${\sigma}(K)\in{\bf U}\setminus\\{1\\}$ satisfying the condition that all
eigenvalues of $K$ located with the arc between $1$ and ${\omega}$ including
${\bf U}^{+}\setminus\\{1\\}$ (or ${\bf U}^{-}\setminus\\{1\\}$)possess total
multiplicity $n-p-q$. If ${\omega}\neq-1$, all eigenvalues of $K$ are in ${\bf
U}\setminus{\bf R}$ and those in ${\bf U}^{+}\setminus{\bf R}$ (or ${\bf
U}^{-}\setminus{\bf R}$) are all Krein-negative (or Krein-positive) definite.
If ${\omega}=-1$, it holds that $(-I_{2s})\diamond N_{1}(-1,1)^{\diamond
t}\diamond H\in{\Omega}^{0}({\gamma}(\tau))$ for some non-negative integers
$s$ and $t$ satisfying $0\leq s+t\leq n-p-q$, and some $H\in{\rm Sp}(2(n-p-q-
s-t))$ satisfying ${\sigma}(H)\subset{\bf U}\setminus{\bf R}$ and that all
elements in ${\sigma}(H)\cap{\bf U}^{+}$ (or ${\sigma}(H)\cap{\bf U}^{-}$) are
all Krein-negative (or Krein-positive) definite.
3\. The left equality of (6.31) holds for all ${\omega}\in{\bf
U}\setminus\\{1\\}$ if and only if $I_{2p}\diamond
N_{1}(1,-1)^{\diamond(n-p)}\in{\Omega}^{0}({\gamma}(\tau))$ for some integer
$p\in[0,n]$. Especially in this case, all the eigenvalues of ${\gamma}(\tau)$
are equal to $1$ and $\nu_{\gamma}=n+p\geq n$.
4\. The right equality in (6.31) holds for some ${\omega}\in{\bf
U}^{+}\setminus\\{1\\}$ (or ${\bf U}^{-}\setminus\\{1\\}$) if and only if
$I_{2p}\diamond N_{1}(1,1)^{\diamond r}\diamond
K\in{\Omega}^{0}({\gamma}(\tau))$ for some non-negative integers $p$ and $r$
satisfying $0\leq p+r\leq n$ and $K\in{\rm Sp}(2(n-p-r))$ with
${\sigma}(K)\in{\bf U}\setminus\\{1\\}$ satisfying the condition that all
eigenvalues of $K$ located with the arc between $1$ and ${\omega}$ including
${\bf U}^{+}\setminus\\{1\\}$ (or ${\bf U}^{-}\setminus\\{1\\}$)possess total
multiplicity $n-p-r$. If ${\omega}\neq-1$, all eigenvalues of $K$ are in ${\bf
U}\setminus{\bf R}$ and those in ${\bf U}^{+}\setminus{\bf R}$ (or ${\bf
U}^{-}\setminus{\bf R}$) are all Krein-positive (or Krein-negative) definite.
If ${\omega}=-1$, it holds that $(-I_{2s})\diamond N_{1}(-1,1)^{\diamond
t}\diamond H\in{\Omega}^{0}({\gamma}(\tau))$ for some non-negative integers
$s$ and $t$ satisfying $0\leq s+t\leq n-p-r$, and some $H\in{\rm Sp}(2(n-p-q-
r-t))$ satisfying ${\sigma}(H)\subset{\bf U}\setminus{\bf R}$ and that all
elements in ${\sigma}(H)\cap{\bf U}^{+}$ (or ${\sigma}(H)\cap{\bf U}^{-}$) are
all Krein-positive (or Krein-negative) definite.
5\. The right equality of (6.31) holds for all ${\omega}\in{\bf
U}\setminus\\{1\\}$ if and only if $I_{2p}\diamond
N_{1}(1,1)^{\diamond(n-p)}\in{\Omega}^{0}({\gamma}(\tau))$ for some integer
$p\in[0,n]$. Especially in this case, all the eigenvalues of ${\gamma}(\tau)$
are equal to $1$ and $\nu_{\gamma}=n+p\geq n$.
6\. Both equalities of (6.31) holds for all ${\omega}\in{\bf
U}\setminus\\{1\\}$ if and only if ${\gamma}(\tau)=I_{2n}$.
Acknowledgements Part of the work was finished during the author’s visit at
University of Michgan, he sincerely thanks Professor Yongbin Ruan for his
invitation and the Department of Mathematics of University of Michigan for its
hospitality.
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|
arxiv-papers
| 2011-10-31T19:43:26 |
2024-09-04T02:49:23.786108
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Duanzhi Zhang",
"submitter": "Duanzhi Zhang",
"url": "https://arxiv.org/abs/1110.6915"
}
|
1111.0026
|
Case I: The General Non-Abelian Case
# Warped Angle-deficit of a 5 Dimensional Cosmic String.
R J Slagter and D Masselink Institute of Physics, University of Amsterdam
and
ASFYON, Astronomisch Fysisch Onderzoek Nederland, Bussum, The Netherlands
info@asfyon.nl
###### Abstract
We present a cosmic string on a warped five dimensional space time in
Einstein-Yang-Mills theory. Four-dimensional cosmic strings show some serious
problems concerning the mechanism of string smoothing related to the string
mass per unit length, $G\mu\approx 10^{-6}$. A warped cosmic string could
overcome this problem and also the superstring requirement that $G\mu$ must be
of order 1, which is far above observational bounds. Also the absence of
observational evidence of axially symmetric lensing effect caused by cosmic
strings could be explained by the warped cosmic string model we present: the
angle deficit of the string is warped down to unobservable value in the brane,
compared to its value in the bulk. It turns out that only for negative
cosmological constant, a consistent numerical solution of the model is
possible.
## 1 Introduction
Recently, there is growing interest in the Randall-Sundrum(RS) warped 5D
geometry[1, 2]. One of the interesting outcomes of this idea is the solution
of the large hierarchy problem between the weak scale and the fundamental
scale of gravity. The predicted Kaluza-Klein particles in the model could be
detected with the LHC at CERN. In the original RS scenario, it was proposed
that our universe is five dimensional, described by the metric
$ds^{2}=e^{-2\mid y\mid ky_{c}}g_{\mu\nu}dx^{\mu}dx^{\nu}+y_{c}dy^{2}.$ (1)
The extra dimension y makes a finite contribution to the 5D volume because of
the exponential warp factor, where $y_{c}$ is the size of the extra dimension.
At low energies, gravity is localized at the brane and general relativity is
recovered. At high energy gravity ”leaks” into the bulk. The 4D Planck scale
will be an effective scale which can become much larger than the fundamental
Planck scale $M_{P}$ if the extra dimension is much larger than $M_{P}^{-1}$.
Further, the self-gravity of the brane must be incorporated. This will protect
the 3 dimensional space from the large extra dimensions by curvature rather
than straightforward compactification. Also matter fields in the bulk can be
incorporated. This will lead to a kind of ”holographic” principle, i.e., the
5D dynamics may be determined from knowledge of the fields on the 4D boundary.
For an overview, see [3]. We will consider here the 5D model with a general
Yang-Mills field, dependent of $r,y$ and $t$. In a following article we
investigate the interplay of the 4D and 5D coupled equations with the junction
conditions.
## 2 The model
We will consider here the RS2 model with two branes at $y=0$, the weak visible
brane and at $y=y_{c}$, the gravity brane (in the RS1 model one let
$y_{c}\rightarrow\infty$). The action of the model under consideration is [4]
$\displaystyle{\cal S}=\frac{1}{16\pi}\int
d^{5}x\sqrt{-^{(5)}g}\Bigl{[}\frac{1}{G_{5}}(^{(5)}R-\Lambda_{5})+\kappa\Bigl{(}^{(5)}R_{\mu\nu\alpha\beta}^{(5)}R^{\mu\nu\alpha\beta}-4^{(5)}R_{\alpha\beta}^{(5)}R^{\alpha\beta}$
(2) $\displaystyle+^{(5)}R^{2}\Bigr{)}-\frac{1}{g^{2}}Tr{\bf
F^{2}}\Bigr{]}+\int
d^{4}x\sqrt{-^{(4)}g}\Bigl{[}\frac{1}{G_{5}}\Lambda_{4}+S_{4}\Bigr{]}$ (3)
with $G_{5}$ the gravitational constant, $\Lambda_{5}$ the cosmological
constant, $\kappa$ the Gauss-Bonnet coupling, $g$ the gauge coupling,
$\Lambda_{4}$ the brane tension and $S_{4}$ the effective 4D Lagrangian, which
is given by a generic functional of the brane metric and matter fields on the
brane and will also contain the extrinsic curvature corrections due to the
projection of the 5D curvature. For the moment we will consider here only the
5D equation in a general setting and with a Yang-Mills matter field. The 4D
induced equations together with the junction conditions will be presented in
part 2 of a next article.
The coupled set of equations of the EYM-GB system will then become( from now
on all the indices run from 0..4)
$\displaystyle\Lambda_{5}g_{\mu\nu}+G_{\mu\nu}-\kappa GB_{\mu\nu}=8\pi
G_{5}T_{\mu\nu},$ (4) $\displaystyle{\cal D}_{\mu}F^{\mu\nu a}=0,$ (5)
with the Einstein tensor
$\displaystyle G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R,$ (6)
and Gauss-Bonnet tensor
$\displaystyle
GB_{\mu\nu}=\frac{1}{2}g_{\mu\nu}\Bigl{(}R_{\gamma\delta\lambda\sigma}R^{\gamma\delta\lambda\sigma}-4R_{\gamma\delta}R^{\gamma\delta}+R^{2}\Bigr{)}-2RR_{\mu\nu}+4R_{\mu\gamma}{R^{\gamma}}_{\nu}$
(7)
$\displaystyle+4R_{\gamma\delta}{{{R^{\gamma}}_{\mu}}^{\delta}}_{\nu}-2R_{\mu\gamma\delta\lambda}{R_{\nu}}^{\gamma\delta\lambda}.$
(8)
Further, with $R_{\mu\nu}$ the Ricci tensor and $T_{\mu\nu}$ the energy-
momentum tensor
$\displaystyle T_{\mu\nu}={\bf
Tr}F_{\mu\lambda}F_{\nu}^{\lambda}-\frac{1}{2}g_{\mu\nu}{\bf
Tr}F_{\alpha\beta}F^{\alpha\beta},$ (9)
and with
$F_{\mu\nu}^{a}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+g\epsilon^{abc}A_{\mu}^{b}A_{\nu}^{c}$,
and ${\cal
D}_{\alpha}F_{\mu\nu}^{a}=\nabla_{\alpha}F_{\mu\nu}^{a}+g\epsilon^{abc}A_{\alpha}^{b}F_{\mu\nu}^{c}$
where $A_{\mu}^{a}$ represents the YM potential.
We will consider the warped axially symmetric space time
$ds^{2}=-F(t,r,y)[dt^{2}-dz^{2}-dr^{2}-A(t,r,y)d\varphi^{2}]+dy^{2},$ (10)
with y the bulk dimension and the YM parameterization
$\displaystyle A_{t}^{(a)}=\Bigl{(}0,0,\Phi(t,r,y)\Bigr{)},\quad
A_{r}^{(a)}=A_{z}^{(a)}=A_{y}^{(a)}=0,$ (11) $\displaystyle
A_{\phi}^{(a)}=\Bigl{(}0,0,W(t,r,y)\Bigr{)}.$ (12)
So the metric and YM components depend t and the two space dimensions r and y.
The set of PDE’s become, for $\kappa=0$ for the time being,
$F_{tt}=F_{rr}+\frac{1}{2}FF_{yy}+\frac{3}{4F}(F_{t}^{2}-F_{r}^{2})+\frac{1}{2}\Lambda
F^{2}-\frac{4\pi G}{A}\Bigl{[}W_{r}^{2}-W_{t}^{2}+FW_{y}^{2}\Bigr{]},$ (13)
$\displaystyle
A_{tt}=A_{rr}+FA_{yy}-\frac{1}{2A}(A_{r}^{2}+A_{y}^{2}-A_{t}^{2})+\frac{1}{F}(F_{r}A_{r}-F_{t}A_{t}+2FF_{y}A_{y})$
(14) $\displaystyle-\frac{16\pi
G}{F}\Bigl{[}W_{t}^{2}-W_{r}^{2}-FW_{y}^{2}\Bigr{]},$ (15)
$W_{tt}=W_{rr}+FW_{yy}+\frac{1}{2A}W_{t}A_{t}+W_{y}(F_{y}-\frac{F}{2A}A_{y})-\frac{1}{2A}W_{r}A_{r}.$
(16)
## 3 The Static case
In the static case the resulting PDE’s become
$A_{rr}+FA_{yy}+2F_{y}A_{y}+\frac{F_{r}A_{r}}{F}-\frac{FA_{y}^{2}+A_{r}^{2}}{2A}+\frac{16\pi
G}{F}\Bigl{(}W_{r}^{2}+FW_{y}^{2}\Bigr{)}=0,$ (17)
$F_{rr}+FF_{yy}+F_{y}^{2}+\frac{F_{r}A_{r}+FF_{y}A_{y}}{2A}+\frac{2}{3}\Lambda
F^{2}-\frac{16\pi G}{3A}\Bigl{(}W_{r}^{2}+FW_{y}^{2}\Bigr{)}=0,$ (18)
$W_{rr}+FW_{yy}-\frac{W_{r}A_{r}}{2A}+W_{y}(F_{y}-\frac{FA_{y}}{2A})=0,$ (19)
$\Phi_{rr}+F\Phi_{yy}+\frac{\Phi_{r}A_{r}}{2A}+\Phi_{y}(F_{y}+\frac{FA_{y}}{2A})=0.$
(20)
We also have the two constraints
$F\Phi_{y}^{2}+\Phi_{r}^{2}=0,\quad W_{r}\Phi_{r}+FW_{y}\Phi_{y}=0.$ (21)
When we substitute the equations for $\Phi$ and W into the conservation
equation $\nabla_{\mu}T^{\nu\mu}=0$, we obtain identically zero, as it should
be. When we introduce the quantities $\theta_{i}$ defined by
$\displaystyle\theta_{1}\equiv\frac{F}{\sqrt{A}}A_{r},\quad\theta_{2}\equiv\sqrt{A}F_{r},\quad\theta_{3}\equiv\frac{F^{2}}{\sqrt{A}}A_{y},\quad\theta_{4}\equiv
F\sqrt{A}F_{y},$ (22)
then the equations can be written as
$\frac{\partial}{\partial r}\theta_{1}+\frac{\partial}{\partial
y}\theta_{3}=-\frac{16\pi G}{\sqrt{A}}(W_{r}^{2}+FW_{y}^{2}),$ (23)
$\frac{\partial}{\partial r}\theta_{2}+\frac{\partial}{\partial
y}\theta_{4}=\frac{16\pi
G}{3\sqrt{A}}(W_{r}^{2}+FW_{y}^{2})-\frac{2}{3}\Lambda F^{2}\sqrt{A},$ (24)
$\Bigl{[}\frac{W_{r}}{\sqrt{A}}\Bigr{]}_{r}+\Bigl{[}\frac{FW_{y}}{\sqrt{A}}\Bigr{]}_{y}=0,$
(25)
$\Bigl{[}\sqrt{A}\Phi_{r}\Bigr{]}_{r}+\Bigl{[}F\sqrt{A}\Phi_{y}\Bigr{]}_{y}=0.$
(26)
The Ricci scalar ${}^{(5)}R$ becomes:
${}^{(5)}R=\frac{8\pi
G}{3AF^{2}}\bigl{(}FW_{y}^{2}+W_{r}^{2}\Bigr{)}+\frac{5}{3}\Lambda_{5}.$ (27)
We now investigate the properties of the static solution for large values of
$r$ and $y$. We will assume that
$\int_{0}^{\infty}\sqrt{A}\sigma dr$ (28)
converges, where $\sigma$ is the energy density $T_{0}^{0}=-4\pi
G\frac{W_{r}^{2}+FW_{y}^{2}}{2AF^{2}}$. Further,
$\lim_{r\rightarrow\infty}\sqrt{A}\sigma=0.$ (29)
## 4 Analysis of the Angle Deficit
The angle deficit can be calculated for a class of static translational
symmetric space times which are asymptotically Minkowski minus a wedge. If we
denote with $l$ the length of an orbit of
$\Bigl{(}\frac{\partial}{\partial\varphi}\Bigr{)}^{a}$ in the brane, then the
angle deficit is given by[5, 6, 7]
$(2\pi-\Delta\varphi)=\lim_{r\rightarrow\infty}\frac{dl}{dr},$ (30)
with
$l=\int_{0}^{2\pi}\sqrt{g_{ab}\Bigl{(}\frac{\partial}{\partial\varphi}\Bigr{)}^{a}\Bigl{(}\frac{\partial}{\partial\varphi}\Bigr{)}^{b}}d\varphi.$
(31)
One better can use the Gauss-Bonnet theorem to obtain the angle deficit by
calculating the integral of the Gaussian curvature over the surface of
$S(t,z)$ = const. If one transports a vector around a closed curve, then the
angle rotation $\alpha$ will be given[7] by the area integral over of the
subsurface of S
$\alpha=\int d^{3}x\sqrt{{}^{(3)}g}^{(3)}K,$ (32)
with
${}^{(3)}K=\frac{1}{2}{{}^{(3)}g}^{ik}{{}^{(3)}g}^{jl}{{}^{(5)}R}_{ijkl}.$
(33)
For our case, we obtain
$\displaystyle\sqrt{{}^{(3)}g}{{}^{(3)}K}=-\frac{1}{2}\Bigl{(}\frac{F_{r}\sqrt{A}}{F}\Bigr{)}_{r}-\frac{1}{2}\Bigl{(}\frac{A_{r}}{\sqrt{A}}\Bigr{)}_{r}-\frac{1}{2}\Bigl{(}\frac{FA_{y}}{\sqrt{A}}\Bigr{)}_{y}-(\sqrt{A}F_{y})_{y}$
(34)
$\displaystyle+\frac{1}{4}\sqrt{A}F_{y}\Bigl{(}\frac{F_{y}}{F}+\frac{A_{y}}{A}\Bigr{)}$
(35)
$\displaystyle=-\frac{1}{4}\Bigl{[}\frac{F_{y}^{2}}{F^{2}}+\frac{F_{r}^{2}}{F^{3}}-\frac{2}{3}\Lambda-\frac{80\pi
G}{3F^{2}A}(W_{r}^{2}+FW_{y}^{2})\Bigr{]}.$ (36)
Then Eq.(28) becomes
$\displaystyle\alpha=-\pi\Bigl{\\{}\Big{[}\Bigl{(}\frac{\sqrt{A}F_{r}}{F}\Bigr{)}+\Bigl{(}\frac{A_{r}}{\sqrt{A}}\Bigr{)}\Bigr{]}^{\infty}_{r=0}+\Bigl{[}\frac{FA_{y}}{\sqrt{A}}+2\sqrt{A}F_{y}\Bigr{]}^{\infty}_{y=0}$
(37)
$\displaystyle-\frac{1}{2}\int_{0}^{\infty}\int_{0}^{\infty}\sqrt{A}F_{y}(\frac{F_{y}}{F}+\frac{A_{y}}{A})drdy\Bigr{\\}}.$
(38)
Or, using the second expression in Eq.(30)
$\displaystyle\alpha=-\frac{1}{2}\pi\int_{0}^{\infty}\int_{0}^{\infty}F\sqrt{A}\Bigl{(}\frac{20}{3}\sigma-\frac{2}{3}\Lambda\Bigr{)}drdy$
(39)
$\displaystyle-\frac{1}{2}\pi\int_{0}^{\infty}\int_{0}^{\infty}F\sqrt{A}\Bigl{[}\frac{F_{y}^{2}}{F^{2}}+\frac{F_{r}^{2}}{F^{3}}\Bigr{]}drdy.$
(40)
If one assumes that in the 4 dimensional case, for
$r\rightarrow\infty:F\rightarrow 1$ and $\sqrt{A}\rightarrow br$ and for
$r\rightarrow 0:F\rightarrow 1$ and $\sqrt{A}\rightarrow r$, than the first
term of Eq.(31) represents the well-known result [6] that $\alpha=2\pi(1-b)$
in de brane, so S is asymptotically a conical surface. In the 5 dimensional
case the results depend on the boundary values of our warp factor F, i.e., the
last two terms in Eq. (31). From Eq.(32) one observes that the first term
represents the proper mass per unit length of the string plus a contribution
from the cosmological constant. The second term is the correction term.
Now we try to obtain for the asymptotic warped metric
$ds^{2}=F_{c}e^{k_{2}y+a_{2}}\Bigl{[}-dt^{2}+dz^{2}+dr^{2}+(k_{1}r+a_{1})^{2}d\varphi^{2}\Bigr{]}+dy^{2}.$
(41)
For $y=0$ we recover de 4D result of a flat space time minus a wedge by the
transformation[6]
$\displaystyle
r^{\prime}=r+\frac{a_{1}}{k_{1}},\qquad\varphi^{\prime}=k_{1}\varphi\quad(0\leq\varphi\leq
2\pi),$ (42)
i.e.,
$ds^{2}=-dt^{2}+dz^{2}+d(r^{\prime})^{2}+(r^{\prime})^{2}d(\varphi^{\prime})^{2}+dy^{2},$
(43)
where now $\varphi^{\prime}$ has a different range then $\varphi$. For $y\neq
0$ we have the warped metric
$ds^{2}=F_{c}e^{k_{2}y+a_{2}}\bigl{[}-dt^{2}+dz^{2}+d(r^{\prime})^{2}+(r^{\prime})^{2}d(\varphi^{\prime})^{2}\Bigl{]}+dy^{2}.$
(44)
The angle deficit is determined by $k_{1}F_{c}e^{k_{2}y+a_{2}}$.
Let us consider now
$\displaystyle\frac{\partial}{\partial
r}(\theta_{1}+\theta_{2})+\frac{\partial}{\partial
y}(\theta_{3}+\theta_{4})=-\frac{32\pi
G}{3\sqrt{A}}\Bigl{(}W_{r}^{2}+FW_{y}^{2}\Bigr{)}-\frac{2}{3}\Lambda\sqrt{A}F^{2}$
(45) $\displaystyle=-\frac{32\pi G}{3}\Bigl{[}\frac{\partial}{\partial
r}\Bigl{(}\frac{WW_{r}}{\sqrt{A}}\Bigr{)}+\frac{\partial}{\partial
y}\Bigl{(}\frac{FWW_{y}}{\sqrt{A}}\Bigr{)}\Bigr{]}-\frac{2}{3}\Lambda\sqrt{A}F^{2},$
(46)
where we used the Eq.’s (19)-(22). After rearranging we then obtain
$\frac{\partial}{\partial r}\Bigl{(}\theta_{1}+\theta_{2}+\frac{32}{3}\pi
G\frac{WW_{r}}{\sqrt{A}}\Bigr{)}=-\frac{\partial}{\partial
y}\Bigl{(}\theta_{3}+\theta_{4}+\frac{32}{3}\pi
G\frac{FWW_{y}}{\sqrt{A}}\Bigr{)}-\frac{2}{3}\Lambda\sqrt{A}F^{2}.$ (47)
So we notice that the $\Phi$-field disappears from the equation. It will have
only a contribution on the brane. It is quite easy to obtain a particular
solution of this equation, Eq.(38). For
$\displaystyle F=F_{c}e^{\pm\sqrt{-\Lambda}y+a_{2}}\qquad
A=A_{c}(k_{1}r+a_{1})^{2},$ (48)
we obtain for $W$ a solution of the form $W(r,y)=W_{1}(r)W_{2}(y)$, where
$W_{1}$ and $W_{2}$ are given by Bessel functions. This oscillatory behavior
of $W$ is not uncommon for gravitating YM vortices.
So it seems to be possible to find the desired asymptotic warped form for the
conical space time, i.e., Eq.(33).
The next task is to obtain from the junction condition and the brane-bulk
splitting, relations between the several constants in the model.
## 5 Numerical solutions
For a given set of initial conditions, these PDE’s determine the behavior of
F, A, W and $\Phi$. We will impose particular asymptotic conditions, in order
to obtain acceptable solutions of the cosmic string. First, we have
$\lim_{r\rightarrow\infty}\Phi(r,y)=1,\lim_{r\rightarrow\infty}W(r,y)=0,F(0,y)=1,A(0,y)=0,\frac{\partial}{\partial
r}A(0,y)=1$. Further,
$\lim_{r\rightarrow\infty}\frac{g_{\varphi\varphi}}{r^{2}}=1$.
We solve the system using the numerical code CADSOL-FIDISOL. The asymptotic
form of the metric component $g_{\varphi\varphi}$ behaves as expected.
Figure 1: Typical solution of F, A, W and $\Phi$ for negative $\Lambda$ with
initial values: $F=e^{(-r+y)},A=r^{2},W=e^{(-r^{2}-y^{2})},\Phi=1-e^{(-r-y)}$
and Dirichlet boundary conditions on the outer boundaries. We also plotted
$g_{\varphi\varphi}$
## 6 Conclusions
In earlier attempts[8, 9, 10], we tried to build a 5-dimensional cosmic string
without a warp factor and investigated the causal structure. Here we
considered a different approach. It seems possible that the absence of cosmic
strings in observational data could be explained by our model, where the
effective angle-deficit resides in the bulk and not in the brane. In this part
we considered the 5D equations in general form, without the splitting of the
energy-momentum tensor in a bulk and brane part. We find a consistent set of
equations in the bulk. The asymptotic behavior of the metric outside the core
of the string seems to have the desired form. This solution must be consistent
with the system of equations obtained by the bulk-brane splitting. It is also
interesting to investigate holographic ideas in our model. These subjects are
under study by the authors and will be published in a followup article.
Figure 2: The 5-D cosmic string
## References
## References
* [1] Randall L and Sundrum R 1999 Phys. Rev. Lett. 83 3370, 4690
* [2] Randall L and Sundrum R ,hep-th/9905221, hep-th/9906064
* [3] Maartens R, gr-qc/0312059
* [4] Okuyama N and Maeda K 2008 arXiv: gr-qc/0212022v2
* [5] Vilenkin A and Shellard E P S 1994 Cosmic Strings and Other Topological Defects Cambridge Monographs
* [6] Garfinkle D Phys. Rev. D 32,6 1323
* [7] Ford L H and Vilenkin A 1981 J. Phys. A: Math. Gen. 14 2353
* [8] Slagter R J 2006 in the proceedings of the conferenceMG11 Berlin page 1434
* [9] Slagter R J 2008 Int.J.Mod.Phys.D 18 613
* [10] Slagter R J 2009 in the procedings of the conference The Invisible Universe Paris
|
arxiv-papers
| 2011-10-31T20:44:00 |
2024-09-04T02:49:23.808384
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Reinoud Jan Slagter, Derk Masselink",
"submitter": "Reinoud Slagter",
"url": "https://arxiv.org/abs/1111.0026"
}
|
1111.0162
|
# The energy injection and losses in the Monte Carlo simulations of a
diffusive shock
###### Abstract
Although diffusive shock acceleration (DSA) could be simulated by some well-
established models, the assumption of the injection rate from the thermal
particles to the superthermal population is still a contentious problem. But
in the self-consistent Monte Carlo simulations, because of the prescribed
scattering law instead of the assumption of the injected function, hence
particle injection rate is intrinsically defined by the prescribed scattering
law. We expect to examine the correlation of the energy injection with the
prescribed multiple scattering angular distributions. According to the
Rankine-Hugoniot conditions, the energy injection and the losses in the
simulation system can directly decide the shock energy spectrum slope. By the
simulations performed with multiple scattering law in the dynamical Monte
Carlo model, the energy injection and energy loss functions are obtained. As
results, the case applying anisotropic scattering law produce a small energy
injection and large energy losses leading to a soft shock energy spectrum, the
case applying isotropic scattering law produce a large energy injection and
small energy losses leading to a hard shock energy spectrum.
WANG ET AL. THE ENERGY INJECTION AND LOSSES IN A DIFFUSIVE SHOCK Xin, Wang,
(wangxin@nao.cas.cn) Yihua, Yan, (yyh@nao.cas.cn)
11affiliationtext: Key Laboratory of Solar Activities of National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100012,
China22affiliationtext: State Key Laboratory of Space Weather, Chinese Academy
of Sciences, Beijing 100080, China
## 1 Introduction
The gradual solar energetic particles with a power-law energy spectrum are
generally thought to be accelerated by the first-Fermi acceleration mechanism
at the interplanetary shocks (IPs) (Axford et al., 1977; Krymsky, 1977; Bell,
1978; Blandford and Ostriker, 1978). It is well known that the diffusive shock
accelerated the particles efficiently by the accelerated particles scattering
off the instability of Alfven waves which are generated by the accelerated
particles themselves (Lagage and Cesarsky, 1983; Gosling et. al. , 1981; Cane
et.al. , 1990; Lee and Ryan, 1986; Pelletier et.al., 2006; Li et al., 2009).
The diffusive shock acceleration (DSA) is so efficient that the back-reaction
of the accelerated particles on the shock dynamics cannot be neglected. So the
theoretical challenge is how to efficiently model the full shock dynamics
(Caprioli et. al., 2010; Zank, 2000; Li et al., 2003; Lee, 2005). To
efficiently model the shock dynamics and the particles’ acceleration, there
are largely three basic approaches: stationary Monte Carlo simulations, fully
numerical simulations, and semi-analytic solutions. In the stationary Monte
Carlo simulations, the particle population with a prescribed scattering law is
calculated based on the particle-in-cell (PIC) techniques (Ellison et al.,
1996; Vladimirov et al., 2006). In the fully numerical simulations, a time-
dependent diffusion-convection equation for the CR transport is solved with
coupled gas dynamics conservation laws (Kang and Jones, 2007; Zirakashvili and
Aharonian, 2010). In the semi-analytic approach, the stationary or quasi-
stationary diffusion-convection equations coupled to the gas dynamical
equations are solved (Blasi et. al., 2007; Malkov et. al., 2000). Since the
velocity distribution of superthermal particles in the Maxwellian tail is not
isotropic in the shock frame, the diffusion-convection equation cannot
directly follow the injection from the non-diffusive thermal pool into the
diffusive CR population. So considering both the quasi-stationary analytic
models and the time-dependent numerical models, the injection of particles
into the acceleration mechanism is based on an assumption of the transparency
function for thermal leakage (Blasi et. al., 2005; Kang and Jones, 2007;
Vainio and Laitinen, 2007) in priori. Thus, the dynamical Monte Carlo
simulations based on the PIC techniques are expected to model the shock
dynamics time-dependently and also can eliminate the suspicion arising from
the assumption of the injection (Knerr et. al., 1996; Wang and Yan, 2011). In
plasma simulation (Monte Carlo model and hybrid model), since the proton’ mass
is very larger than the electron’ mass, the total plasma can be treated as one
species of proton fluid with a massless electronic fluid which just balance
the electric charge state for maintaining a neutral fluid (Leroy et al.,
1982). There is no distinction between thermal and non-thermal particles,
hence particle injection is intrinsically defined by the prescribed scattering
properties, and so it is not controlled with a free parameter (Caprioli et.
al., 2010).
Actually, Wang and Yan (2011) have extended the dynamical Monte Carlo models
invoking multiple scattering angular distributions. Unlike the previous
KJE(Knerr et. al., 1996) dynamical Monte Carlo models invoking a purely
isotropic scattering angular distribution, this multiple scattering law allow
the particles are scattered by angles distributed with Gaussian functions.
According to the simulations using the extended multiple scattering angular
distributions, a series of similar energy spectrums with a little difference
with respect of the power-law tail are obtained. And the results show that the
energy spectral index is effected by the prescribed scattering law.
Specifically, the total shock’s energy spectral index is less than one and
shows an increasing function of the dispersion of the scattering angular
distribution, but the subshock’s energy spectral index is more than one and
shows a decreasing function of the dispersion of the scattering angular
distribution.
In an effort to research why the multiple scattering angular distributions can
produce the difference of the energy spectral index, it is necessary to
analyze the energy injection and the energy losses in the entire simulation
system. Because the energy injection and losses are important factors for
deciding the acceleration efficiency and the energy spectrum slope owing to
the Rankine-Hugoniot relationship based on the energy conservational law.
However, in the Monte Carlo simulation, the particle injection and the energy
loss processes are treated in natural, self-consistent manner and decided by
the prescribed scattering law. In order to obtain the complete energy
injection and loss real-time functions in the entire simulation system, we
perform the simulations by the multiple scattering law considering an improved
simulation system. In this new simulation system, a radial reflective
boundary(RRB) is set for preventing the energy losses via the radial
diffusion. Under these scenarios, the performed simulation cases consist of
four specific standard deviation values of the Gaussian distribution function.
In Section 2, the basic simulation method is introduced with respect to the
Gaussian scattering angular distributions for obtaining the energy injection
and loss functions of time in each case. In Section 3, we present the energy
analysis for all cases with four types of scattering angle distributions.
Section 4 includes a summary and the conclusions.
## 2 Method
The Monte Carlo model is a general model, although it is considerably
expensive computationally, and it is important in many applications to include
the dynamical effects of nonlinear DSA in simulations. Since the prescribed
scattering law can replace the electromagnetical field calculation which is
used in hybrid simulations (Giacalone, 2004; Winske and Omidi, 2011), we
assume that the individual particle scatters elastically off the background
scattering centers with the scattering angles according to a Gaussian
distribution in the local frame. And the particle’s mean free path is
proportional to the local velocities in its local frame with
$\lambda=V_{L}\cdot\tau.$ (1)
Where, $\tau$ is the average scattering time. Under the prescribed scattering
law, the injection is purely correlated with those particles from the “thermal
pool” in the downstream region become into the superthermal particles (Ellison
et al., 2005).
Figure 1: Schematic diagram of the simulation box. The shock is produced by
incoming flow toward the reflective wall at the right boundary of the box
(Xmax=300,Radial distance R=50).
In these simulations, the entire shock is simulated in one-dimensional box as
shown in Figure 1, the initial continually inflow enter into the box from the
left boundary with a supersonic bulk velocity ($U_{0}$), a stationary
reflective wall at the right boundary of the box act to form a piston shock
moving from right to left. After a certain time, a steady compression region
(i.e. downstream region) will be formed in front of the reflective wall. The
bulk velocity in downstream region is become to zero, since the particles
dissipate in the downstream region and their large translational energy is
converted into isotropic, random energy. To model the finite size of system
and the lack of sufficient scattering far upstream to turn particles around
(Mitchell et. al., 1983), the presented simulation includes the escape of the
energetic particles at an upstream “free escape boundary” (FEB). This FEB
moves with the shock front at a shock velocity ($V_{sh}$) and remains a
constant distance in front of the shock position (i.e. $X_{FEB}$=90). This
distance is enough large for majority of the injected particles diffuse
between the foreshock region and the downstream region. The size of the
foreshock is the distance from the shock to the FEB and thus sets a limit on
the maximum energy a particle can obtain. Since the injected particles cross
the shock and diffuse upstream, they negatively contribute to the bulk
velocity, and the bulk velocity become smaller and smaller from the FEB to the
shock position. Holding the length of the foreshock region constant eventually
(when enough time has elapsed to create a larger number of accelerated
particles) produces a steady state with respect to the amount of the energy
entering and exiting the system from the upstream region. In addition, we set
the radial boundary as $R_{y,z}=50$ for preventing particle’s perpendicular
diffusion to the infinity. Simultaneously, The radial reflective boundary can
ensure the particles have efficient diffusive processes in the one-dimensional
system along $\hat{x}$ direction. So we are able to compare the difference of
the energy injection and losses obtained from each case. We can further
investigate the possibility that the cases applying anisotropic scattering
angular distribution would produce a different acceleration efficiency
compared with the case applying an isotropic scattering angular distributions.
So an anisotropic scattering law in the theory of the CR-diffusion is also
needed (Bell, 2004).
According to particle-in-cell (PIC) techniques (Forslund, 1985; Spitkovsky,
2003; Nishikawa et. al., 2008), the total box length in this simulation system
is $X_{max}$=300, and it is divided up into $n_{x}$=600 grids. The initial
number of particles in each grid is $n_{0}$=650. In addition, we use a flux-
weighted inflow to ensure the particles entering into the box with the same
density flow in upstream with the time. This inflow in “preinflow box” (PIB)
is put in the left boundary of the simulation box. The total simulation time
$T_{max}=2400$,and it is divided into the number of time steps $N_{t}=72000$
with a time step $dt=1/30$. The size of the FEB distant from the shock front
is set as $X_{feb}=90$. The radii of the radial reflective boundary is set as
$R_{y,z}=50$. These simulation codes consist of the three substeps. (i)
Individual particles move along the $\hat{x}$, $\hat{y}$, and $\hat{z}$ axis
with their local velocities in each component, respectively.
$x=x_{0}+v_{x}\cdot t\\\ y=y_{0}+v_{y}\cdot t\\\ z=z_{0}+v_{z}\cdot t\\\ $ (2)
Since the magnetic field $B_{0}$ is parallel to the simulated shock’s normal
direction, the fluid quantities only vary in the $\hat{x}$ direction. (ii)
Collect the moments. Summation of particle masses and velocities are collected
on a background computational grid based on PIC techniques. In this substep,
the statistical average bulk speed of each grid represents the velocity of
each scattering center. Once the value of the bulk speed drops to zero, the
position of the shock front is decided by the displacement of the
corresponding grid, and it means that the shock position is moved with an
evolutional velocity $v_{sh}$ far away to the stationary reflected wall.
Simultaneously, the size of the downstream region is extended dynamically with
a constant velocity $v_{sh}$. Similarly, the foreshock region or precursor
with a bulk velocity gradient is formed by the “back pressure” of the backward
diffused particles. The moving of the FEB is also parallel to the shock moving
with the same constant velocity $v_{sh}$. (iii) Applying multiple scattering
laws. According to the scattering rate (i.e. $R_{s}=dt/\tau$, where $R_{s}$ is
the probability of the scattering events in time step $dt$, and $\tau$ is the
average scattering time). These fraction of the particles are chosen to
scatter the background scattering centers with their corresponding scattering
angles obeying to the given Gaussian distributions. The chosen particles
scatter off the collected background with their local velocities and
scattering angles. The scattered particles move along their path until they
have new scatters. In the duration of the time step, if the all chosen
particles have completed their scatters, the background bulk speed is
subsequently changed. In the turn, the varied background bulk speed also will
change the particle’s individual velocity in the local frame in the next time
step. The entire simulation time consists of the number of ($N_{t}=72000$)
time step involving the above three substeps.
These presented simulations are all based on one-dimensional simulation box
and the all simulated parameters has been described in detail elsewhere (Wang
and Yan, 2011). Here we list the simulation parameters in the Table 1.
Upstream supersonic flow $U_{0}$ with an initial Maxwellian thermal velocity
$V_{L}$ in their local frame and the inflow in a “pre-inflow box” (PIB) are
both moving along one-dimensional simulation box from the left to the right.
The parallel magnetic field $B_{0}$ is along the $\hat{x}$ axis direction. FEB
with a constant length $X_{feb}=90$ in front of the shock position. The radial
reflective boundary (RRB) is set as $R_{y,z}=50$. The simulation box is
dynamically consist of three regions: upstream, precursor and downstream. The
bulk fluid speed in upstream region is $U=U_{0}$, the bulk fluid speed in
downstream region is $U=0$, and the bulk fluid speed with a gradient of
velocity in the precursor region is $U_{0}>U>0$. To obtain the detailed
information of the total particles in the simulation processes at any instant
of time, we should build a large database for recording the velocities,
positions, and the elapsed time of the all particles, as well as the indices
and the bulk speeds of the total grids. Then we can obtain the energy
spectrums from the downstream, precursor, and upstream regions. The escaped
particles’ mass, momentum, and energy losses via the FEB can be also obtained.
By analyzing the particle injection in the downstream region and the energy
losses via FEB in the precursor region, we can find that how the prescribed
scattering law to affect the shock compression ratio and the energy spectral
index.
To examine the relationships between the shock energy spectral index and the
prescribed scattering law by the energy injection and loss functions, we
perform the Monte Carlo simulations with multiple scattering angular
distributions using a new simulation system based on Matlab platform. The
simulated cases are presented by Gaussian function with a standard deviation
$\sigma$ and an average value (i.e.,the expect value) $\mu=0$ involving four
cases: (1) Case A: $\sigma=\pi$/4. (2) Case B: $\sigma=\pi$/2. (3) Case C:
$\sigma=\pi$. (4) Case D: isotropic distribution.
Table 1: The Simulation Parameters
Physical parameters | Dimensionless Value | Scaled Value
---|---|---
Inflow velocity | $u_{0}=0.3$ | 403km/s
Thermal speed | $\upsilon_{0}=0.02$ | 26.9km/s
Scattering time | $\tau=0.833$ | 0.13s
Box size | $X_{max}=300$ | $10R_{e}$
Total time | $t_{max}=2400$ | 6.3minutes
Time step size | $dt=1/30$ | 0.0053s
Number of zones | $nx=600$ | …
Initial particles per cell | $n_{0}=650$ | …
FEB distance | $X_{feb}=90$ | $3R_{e}$
Radial distance | $R_{y,z}=50$ | $\sim 1.5R_{e}$
Note: The Mach number M =11.6. The $R_{e}$ is the Earth’s radii. The data
adapt from the Earth bow shock (Knerr et. al., 1996).
## 3 Energy analysis
### 3.1 Shock structures
We present the entire shock evolution with the velocity profiles of the time
sequences in each case as shown in Figure 2. The continuous inflow with a
supersonic velocity $U_{0}$ move from the left boundary ($X=0$) of the
upstream region to the downstream region at the right of the box with the
time. The total bulk speed profiles are consist of three regions with the
time: the upstream region $U=U_{0}$ , precursor region $0<U<U_{0}$, and
downstream region $U=0$. Total profiles of the bulk speed is distinct by two
positions of the FEB and the shock front with the time. From the Cases A, B,
and C to D, the precursor explicitly shows an increasing slope of the bulk
speed, the shock’s position $X_{sh}$ also shows an increasing displacement
increment in the $\hat{X}$ axis at the end of the simulation, respectively.
This means the shock evolutes with an increasing velocity $V_{sh}$ from the
Cases A, B, and C to D, respectively. The simulated results of the dynamical
shock in four cases are listed in the Table 2. In addition, by introducing a
radial reflective boundary (RRB) in the present simulations, we also obtain
the difference of the shock front position $\Delta X_{sh}$ compared with the
previous simulations (Wang and Yan, 2011) with an increasing value of the
$(\Delta X_{sh})_{A}$=-3.5, $(\Delta X_{sh})_{B}$=+6, $(\Delta
X_{sh})_{C}$=+6, and $(\Delta X_{sh})_{D}$=+17.5 from the Cases A, B, and C to
D, respectively. It is obvious to see that the affection of the RRB enhances
this difference of the simulated shock for the four cases using the multiple
scattering angular distributions.
According to the relationships between the upstream and the downstream, we are
able to calculate the total shock compression ratio $r_{tot}$ in the shock
frame in each case as followings.
$r_{tot}=\frac{U_{0}+|V_{sh}|}{|V_{sh}|}$ (3)
Figure 2: The entire evolutional velocity profiles in four cases. The dashed
line denotes the FEB position in each plot. The precursor is located in the
area between the downstream region and the upstream region in each case.
Different shock evolutional velocity $V_{sh}$ in different cases will probably
lead to different dynamical shock structure. To showing this difference, we
present the subtle velocity profiles at the end of simulation in each case in
Figure 3. Evidently, the fluctuation of the velocity between the $V_{sub}$ and
$V_{d}$ with an obliviously increasing value from the Cases A, B, and C to D,
respectively. And the specific structure in each plot consists of three main
parts: precursor, subshock and downstream. The smooth precursor with a large
scale is between the FEB and the subshock’s position $X_{sub}$, where the bulk
velocity gradually drops from $U_{0}$ to $V_{sub}$. The sharp subshock with a
short scale just spans three-grid-length involving a deep drop of the bulk
speed abruptly from $v_{sub}$ to $v_{d}$, where the scale of the three-grid-
length is about the thermal mean free path of the thermalized particles in the
downstream region. So the subshock’s velocity can be defined by the value of
the $V_{sub}$ in each case. The velocity $V_{d}$ represents the downstream
bulk speed at the shock position at the end of the simulation. The bottom
solid line denotes the backward shock evolutional velocity $V_{sh}$ with an
increasing value from the Cases A, B, and C to D, respectively. Because the
subshock is the fraction of the total shock, we can calculate the subshock’s
compression ratio $r_{sub}$ according to the total shock compression ratio
$r_{tot}$ as following.
$r_{sub}=\frac{V_{sub}}{U_{0}}\times r_{tot}$ (4)
Figure 3: The subtle structures of the subshock in four cases at the end of
simulation time. The drops of velocity in the subshcok region are denoted by
the values between $V_{sub}$ and $V_{d}$ in each case.
### 3.2 Energy injection & losses
We have monitored the energy of the total particles over the time in different
regions with respect to all cases. Figure 4 shows all the types of energy
functions with time. The $E_{tot}$ is the energy summation of the total
particles in the total simulation system over the time. The $E_{box}$ is the
energy summation of the actual particles in the simulation box over the time.
The $E_{pib}$ is the energy summation of the continuous new particles enter
into the simulation box from the “preinflow box” over the time. The $E_{dow1}$
is the energy summation of the all particles in the downstream region over the
time. The $E_{dow2}$ is the energy summation of the all particles which their
local velocity over the value of the initial velocity $U_{0}$ in the
downstream region over the time. The $E_{dow3}$ is the energy $E_{dow2}$ minus
the $E_{inj}$, which is the initial individual particle’s energy (i.e.
$\varepsilon_{k}=1/2mU_{0}^{2}+1/2mv_{0}^{2}$) summation of the injected
particles from the “thermal pool” at the local velocity of $V_{L}=U_{0}$ to
the superthermal particles in the downstream region, over the time. The
$E_{feb}$ is the energy summation of the total particles in the precursor
region over the time. The $E_{out}$ is the energy summation of the all
particles escaped from the FEB over the time. Clearly, the total energy
$E_{tot}$ in the simulation system at any instant in time is not equal to the
actual box energy $E_{box}$ at any instant in time in each plot. It is evident
from the real-time functions in Figure 4, the non-linear divergence between
the curves for $E_{box}$ and $E_{tot}$ is produced with a decreasing value
from the Cases A, B, and C to D, respectively. Also, the energy loss function
$E_{out}$ is produced with a decreasing value from the Cases A, B, and C to D,
respectively. Simultaneously, the difference between the energy functions
$E_{dow2}$ and $E_{dow3}$ shows an increasing energy injection $E_{inj}$ from
the Cases A, B, and C to D, respectively.
As shown in Table 3, all the listed results of the particle injection and
losses in each case are calculated at the end of the simulation (i.e.
$T_{max}$=2400). The $M_{loss}$, $P_{loss}$ and $E_{loss}$ are the mass loss,
the momentum loss and the energy loss of the particles escaped via to the FEB,
respectively. The $E_{feb}$, $E_{inj}$, $E_{tot}$, and $E_{dow1}$, with the
unit of an initial box energy $E_{0}$, are all the energy values in their
respective statistical volumes at the end of simulation. The $R_{inj}$
represents the rate of the energy injection $E_{inj}$ with the total
downstream energy $E_{dow1}$ at the end of simulation. And the $R_{loss}$
represents the rate of the energy losses $E_{loss}$ with the total energy in
the system $E_{tot}$ at the end of the simulation. These correlations are
presented as follows.
$E_{inj}=E_{dow2}-E_{dow3}$ (5) $R_{inj}=E_{inj}/E_{dow1}$ (6)
$E_{loss}=E_{out}$ (7) $R_{loss}=E_{out}/E_{tot}$ (8)
Table 2: The results of the shock simulation Items | Case A | Case B | Case C | Case D
---|---|---|---|---
$X_{sh}$ | 199.5 | 165.5 | 146 | 106.5
$X_{FEB}$ | 109.5 | 75.5 | 56 | 16.5
$V_{sub}$ | 0.1075 | 0.1460 | 0.1757 | 0.2525
$V_{d}$ | +0.0207 | -0.0024 | +0.0144 | +0.0045
$V_{sh}$ | -0.0419 | -0.0560 | -0.0642 | -0.0806
$r_{tot}$ | 8.1642 | 6.3532 | 5.6753 | 4.7209
$r_{sub}$ | 2.9258 | 3.0910 | 3.3246 | 3.9734
$\Gamma_{tot}$ | 0.7094 | 0.7802 | 0.8208 | 0.9031
$\Gamma_{sub}$ | 1.2789 | 1.2174 | 1.1453 | 1.0045
$VL_{max}$ | 11.4115 | 14.2978 | 17.2347 | 21.6285
$ErrorBar$ | +0.0017 | -0.0022 | +0.0014 | -0.0025
Table 3: The results of the particle injection and losses
Items | Case A | Case B | Case C | Case D
---|---|---|---|---
$M_{loss}$ | 1037 | 338 | 182 | 38
$P_{loss}$ | 0.0352 | 0.0189 | 0.0123 | 0.0025
$E_{loss}$ | 0.7468 | 0.5861 | 0.4397 | 0.0904
$E_{tot}$ | 3.3534 | 3.4056 | 3.3574 | 3.4025
$E_{feb}$ | 0.8393 | 0.5881 | 0.5310 | 0.3397
$E_{dow1}$ | 2.1451 | 2.5612 | 2.6359 | 2.6903
$E_{inj}$ | 0.1025 | 0.1912 | 0.2873 | 0.3955
$R_{inj}$ | 4.78% | 7.47% | 10.90% | 14.70%
$R_{loss}$ | 22.27% | 17.21% | 13.10% | 2.66%
Notes: The units of mass, momentum, and energy are normalized to the proton
mass $M_{p}$, initial total momentum $P_{0}$ and initial box energy $E_{0}$,
respectively.
Figure 4: Various energy values vs. time (all normalized to the initial total
energy $E_{0}$ in the simulation box) in each case. All quantities are
calculated in the box frame.
Figure 5: The four plots denote the mass losses, momentum losses, energy
losses via the FEB and the injected energies in the downstream region,
respectively. The solid line, dashed line, dash-dotted line and the dotted
line represent the cases A, B, C and D in each plot, respectively. The units
are normalized to the initial box proton mass $M_{p}$, initial box momentum
$P_{0}$ and initial box energy $E_{0}$, respectively.
For the comparison, the mass loss, momentum loss, energy loss and energy
injection functions with the time are calculated in Figure 5. Since the
simulation system are based on the computational calculations, the existence
of the energy losses is inevitable. Figure 5 show that the mass loss, momentum
loss, and the energy loss functions with a decreasing value in any instant of
time from the Cases A, B, and C to D, respectively. Among of theses loss
functions, the energy loss function shows a decreasing values of
$(E_{loss})_{A}=0.7468$, $(E_{loss})_{B}=0.5861$, $(E_{loss})_{C}=0.4397$, and
$(E_{loss})_{D}=0.0904$ at the end of the simulation from the Cases A, B, and
C to D, respectively. On the contrary, the energy injection function show an
increasing values of $(E_{inj})_{A}=0.1025$, $(E_{inj})_{B}=0.1912$,
$(E_{inj})_{C}=0.2873$, and $(E_{inj})_{D}=0.3955$ at the end of the
simulation from the Cases A, B, and C to D, respectively. By of the existence
of the energy losses in the simulation system, the shock compression ratios
are naturally affected according to the Rankine-Hugoniot conditions.
Therefore, the difference of the energy losses or injection produced by the
prescribed scattering angular distributions can directly affect all aspects of
the simulated shock including the subtle shock structures, compression ratios,
maximum energy particles, and the energy spectrums, as well as other aspects.
It is just this self-consistent injection mechanism and PIC techniques which
allow the energy injection and loss functions to be obtained. So the further
energy analysis for the diffusive shock acceleration could be done easily.
### 3.3 Maximum energy
We select some individual particles from the downstream region at the end of
the simulation for obtaining the plots in the coordinates of the phase, space
and time. The trajectories of the selected particles are shown in Figure 6.
Among of these selected particles in each case, one of these trajectories
clearly shows the fully acceleration processes of the maximum energy particle
which undergoes the multiple crossings with the shock front. The maximum value
of the local velocity marked in each plot shows an increasing values of
$(VL_{max})_{A}=11.4115$, $(VL_{max})_{B}=14.2978$, $(VL_{max})_{C}=17.2347$,
and $(VL_{max})_{D}=21.6285$ from the Cases A, B, and C to D, respectively.
And the corresponding statistical error of the local velocity in each case is
listed in the Table 3. Consequently, The cutoff energy at the “power-law” tail
in the energy spectrum is given with an increasing value of
$(E_{max})_{A}$=1.23 MeV, $(E_{max})_{B}$=1.93 MeV, $(E_{max})_{C}$=2.80 MeV
and $(E_{max})_{D}$=4.41 MeV from the Cases A, B, and C to D, respectively. As
for the escaped particles, owing to their energies are higher than the cutoff
energy, they are not available in the system by of their escaping via the FEB
eventually. Since the FEB is a constant distance (i.e. $X_{feb}=90$) in front
of the shock and maintains the parallel moving of the shock front in each
case, once an accelerated particle diffuse beyond the position of the FEB,
this particle will be excluded from the system. The Table 3 shows the numbers
of the escaped particles at the end of the simulation with a decreasing mass
losses of $(M_{loss})_{A}=1037$, $(M_{loss})_{B}=338$, $(M_{loss})_{C}=182$,
and $(M_{loss})_{D}=38$ from the Cases A, B, and C to D, respectively. Also
the energy statistical data exhibit the energy loss rate with a decreasing
value of $(R_{loss})_{A}=22.27\%$, $(R_{loss})_{B}=17.21\%$,
$(R_{loss})_{C}=13.10\%$, and $(R_{loss})_{D}=2.66\%$ from the Cases A, B, and
C to D, at the end of the simulation, respectively.
Figure 6: The individual particles with their local velocities vs their
positions with respect to time in each plot. The shaded area indicates the
shock front, the solid line in the bottom plane denotes the position of the
FEB in each case, respectively. Some irregular curves trace the individual
particle’s trajectories near the shock front with time. The maximum energy of
accelerated particles in each case is marked with the value of the local
velocity, respectively.
Except for the maximum energy particles, there are also common energetic
particles are shown in the plots with some of them obtained finite energy
accelerations from the multiple crossings with the shock and some of them do
not have additional energy gains owing to their lack of probability for
crossing back into the precursor. If the cutoff energy of the simulation
system is not effected by the prescribed scattering angular distribution,
these maximum energy particles in different cases should be identical or at
least be similar equal in the range of error bar. But the actual difference of
the cutoff energy particles in different cases should be contributed by the
different prescribed scattering angular distributions dominating the different
energy injection.
### 3.4 Heating, acceleration & spectrum
As shown in Figure 7, the four energy spectrums with the “power-law” tails
represent the four cases, averaged over the precursor region, at the end of
simulation, respectively. The thin solid curve with a narrow peak is the
initial Maxwellian distribution in the shock frame. The four extended energy
spectrums are all consist of two very different parts: the low energy part and
the high energy part. The low energy part in the left side of the initial
spectrum, range from the low energy to the central peak, shows the “irregular
fluctuation” in each case. The high energy part in the right side of the
initial spectrum, range from the central peak energy to the cutoff energy,
shows the smooth “power-law” tail in each case. The “irregular fluctuation”
indicates that the supersonic upstream fluid slows down in precursor region
and its translational energy begin to convert into the irregular random
energy. The “power-law” tail implies that the injected particles from the
“thermal pool” in the downstream region scatter into the precursor region
crossing the shock front for multiple energy gains and become into the
superthermal particles.
Look at extended curves closely, the low energy part in each case has a
clearly joint point with the high energy part. And the joint point show an
increasing energy value from the Cases A, B, and C to D, respectively.
Consequently, the corresponding cutoff energy at the “power-law” tail in the
precursor region also shows an increasing value from the Cases A, B, and C to
D, respectively. This joint point should be correlated to the average thermal
velocity in the downstream region. As shown in the Figure 8, the four thermal
velocity functions are averaged over the downstream region with the time. And
each curve denotes the evolution of the average thermal velocity with the time
and shows a constant after a certain duration (i.e, $t=500$). Eventually, the
average thermal velocity $V_{th}$ shows an increasing value from the Cases A,
B, and C to D, at any instant of time, respectively. As expected, the energy
injection from the “thermal pool” in the downstream region shows an increasing
value from the Cases A, B, and C to D, respectively. Therefore, as show in the
Figure 7, the energy spectrum in the precursor region shows an increasing hard
spectral slope as the dispersion value $\sigma$ of the Gaussian scattering
angular distribution increases. This correlation of the energy spectrum
averaged over the precursor region with the prescribed scattering law is
consistent with the energy spectrum averaged over the downstream region.
Figure 7: This plot represents the energy spectrums on the precursor region at
the end of the simulation. The thick solid line with a narrow peak at
$E=$1.3105keV represents the initial Maxwell energy distributions. The solid,
dashed, dash-dotted and dotted extended curves with the “power-law” tail
present the energy spectrum corresponding to Cases A, B, C and D,
respectively. All these energy spectrum are calculated in the same shock
frame. Figure 8: This plot denotes the average thermal velocity with the time
in the downstream region in each case.
Generally, we could predict the power-law energy spectral index from diffusive
shock acceleration theory:
$dJ/dE\propto E^{-\Gamma}$ (9)
where $dJ/dE$ is the energy flux and the $\Gamma$ is the energy spectral
index. And the spectrum index can be calculated as following:
$\Gamma_{tot}=(r_{tot}+2)/[2\times(r_{tot}-1)].$ (10)
$\Gamma_{sub}=(r_{sub}+2)/[2\times(r_{sub}-1)].$ (11)
According to Equation 10 and Equation 11, we substitute the corresponding
values of the compression ratio $r$ in each case. Then, the two types of
energy spectral indices $\Gamma_{tot}$ and $\Gamma_{sub}$ in each case are
calculated. As listed in the Table 2, the total shock energy spectral index
shows an increasing value of the $(\Gamma_{tot})A$= 0.7094, $(\Gamma_{tot})B$
=0.7802, $(\Gamma_{tot})C$ =0.8208, and $(\Gamma_{tot})D$=0.9031 from the
Cases A, B, and C to D, respectively. However, the subshock’s energy spectral
index is a decreasing value of the $(\Gamma_{sub})A$= 1.2789,
$(\Gamma_{sub})B$ =1.2174, $(\Gamma_{sub})C$ =1.1453, and
$(\Gamma_{sub})D$=1.0045 from the Cases A, B, and C to D, respectively.
As shown in Figure 9, all of the values of the subshock’s energy spectral
index are more than one (i.e. $\Gamma_{sub}>1$ ), and the solid line denotes
the subshock’s energy spectral index with a decreasing value from the Cases A,
B, and C to D as the energy injection increases, respectively. However, all of
the values of the total shock’s energy spectral index are less than one (i.e.
$\Gamma_{tot}<1$), and the dashed line denotes the total shock’s energy
spectral index with an increasing value from the Cases A, B, and C to D as the
energy injection increases, respectively. Simultaneously, as shown in Figure
10, the solid line denotes the subshock energy spectral index with a
decreasing value from the Cases A, B, and C to D as the energy loss decreases,
respectively. However, the dashed line denotes the total shock’s energy
spectral index with an increasing value from the Cases A, B, and C to D as the
energy loss decreases, respectively. According to the diffusive shock
acceleration theory, if the energy loss is limited to be the minimum, the
simulation models based on the computer will more closely fit the realistic
physical situation. The Figure 9 and Figure 10 indicate that the correlations
of the energy spectral index with the energy injection or the energy losses
are consistent with the energy spectral index is dependent on the prescribed
multiple scattering angular distributions. As seen from the Cases A, B, and C
to D, the subshock’s energy spectral index and the total shock’s energy
spectral index are both approximating to the realistic value one (i.e.
$\Gamma\sim 1$ ) as the energy injection increases or as the energy loss
decreases. As predicted, the Rankine-Hugoniot (RH) jump conditions allow to
derive the relation of the compression ratio with the Mach number as:
$r=(\gamma_{a}+1)/(\gamma_{a}-1+2/M^{2})$. For a nonrelativistic shock, the
adiabatic index $\gamma_{a}$ = 5/3 , if the Mach number $M\gg 1$, then the
maximum compression ratio should be 4. According the Rankie-Hugoniot
conditions, the total shock compression ratio should be less than standard
value 4, and the corresponding total shock’s energy spectral index should be
less than the standard value one for a nonrelativistic shock (Pelletier,
2001). Simultaneously, we can see that if the energy injection achieves to the
enough high level or the energy loss is limited to the enough low level, the
subshock’s energy spectral index will closely approximate the standard value
of one. We present explicitly these relationships between the energy spectral
indices, the energy injection or energy losses, and the prescribed scattering
law. And these relationships will be very helpful to improve simulation models
by the best choice of the prescribed scattering law. Using the prescribed
scattering law instead of the assumption of the transparent function in the
thermal leakage mechanism, as far as the injection problem is concerned, the
dynamical Monte Carlo model based on the PIC techniques is nothing less than a
fully self-consistent and time-dependent model.
Figure 9: The plot shows the correlation of the energy spectral index vs the
energy injection. The triangles represent the total energy spectral index of
the all cases. The circles indicate the subshock’s energy spectral index of
all cases. Figure 10: The plot shows the correlation of the energy spectral
index vs the energy losses. The triangles represent the total energy spectral
index of the all cases. The circles indicate the subshock’s energy spectral
index of all cases.
## 4 Summary and conclusions
In summary, we performed the dynamical Monte Carlo simulations using the
Gaussian scattering angular distributions based on the Matlab platform by
monitoring the particle’s mass, momentum and energy at any instant in time.
The specific energy injection and loss functions with time are presented. We
successfully examine the correlation between the energy spectral index and the
prescribed Gaussian scattering angular distributions by the energy injection
and loss functions in four cases. Simultaneously, this correlation is further
enhanced by using the radial reflective boundary (RRB).
In conclusion, the relationship between the energy injection or energy losses
and the prescribed scattering law verify that the shock energy spectral index
is surely dependent on the prescribed scattering law. As expected, the maximum
energy of accelerated particles is correlated with the particle injection rate
from the “thermal pool” to superthermal population. So we find that the energy
injection rate increases as the standard deviation value of the scattering
angular distribution increases. In these multiple scattering angular
distribution scenario, the prescribed scattering law dominates the energy
injection or the energy losses. So this self-consistent energy injection
mechanism is capable to instead of the assumption of the thermal leakage
injected function. Consequently, the cases applying anisotropic scattering
angular distribution will produce a small energy injection and large energy
losses leading to a soft energy spectrum, the case applying isotropic
scattering angular distribution will produce a large energy injection and
small energy losses leading to a hard energy spectrum. These relationships
will drive us to find a newly plausible prescribed scattering law which making
the simulation model more close to the realistic physics.
###### Acknowledgements.
The authors would like to thank Profs. Hongbo Hu, Siming Liu, Xueshang Feng,
and Gang Qin for many useful and interesting discussions concerning this work.
In addition, we also appreciate Profs. Qijun Fu and Shujuan Wang, as well as
other members of the solar radio group at NAOC. This work was funded in part
by CAS-NSFC grant 10778605 and NSFC grant 10921303 and the National Basic
Research Program of the MOST (Grant No. 2011CB811401).
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|
arxiv-papers
| 2011-11-01T10:28:43 |
2024-09-04T02:49:23.817162
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xin Wang and Yihua Yan",
"submitter": "Xin Wang Mr.",
"url": "https://arxiv.org/abs/1111.0162"
}
|
1111.0169
|
# Abelian monopoles in finite temperature lattice $SU(2)$ gluodynamics: first
study with improved action
V. G. Bornyakov High Energy Physics Institute, 142280 Protvino, Russia
and Institute of Theoretical and Experimental Physics, 117259 Moscow, Russia
A. G. Kononenko Joint Institute for Nuclear Research, 141980, Dubna, Russia
Moscow State University, Physics Department, Moscow, Russia
###### Abstract
The properties of the thermal Abelian color-magnetic monopoles in the
maximally Abelian gauge are studied in the deconfinement phase of the lattice
$SU(2)$ gluodynamics. To check universality of the monopole properties we
employ the tadpole improved Symanzik action. The simulated annealing algorithm
combined with multiple gauge copies is applied for fixing the maximally
Abelian gauge to avoid effects of Gribov copies. We compute the density,
interaction parameters, thermal mass and chemical potential of the thermal
Abelian monopoles in the temperature range between $T_{c}$ and $3T_{c}$. In
comparison with earlier findings our results for these quantities are improved
either with respect to effects of Gribov copies or with respect to lattice
artifacts.
Lattice gauge theory, deconfinement phase, thermal monopoles, Gribov problem,
simulated annealing
###### pacs:
11.15.Ha, 12.38.Gc, 12.38.Aw
## I Introduction
The signatures of the strong interactions in the quark-gluon matter were found
in heavy ion collisions experiments Adams et al. (2005). There are proposals
Liao and Shuryak (2007); Chernodub and Zakharov (2007) suggesting that color-
magnetic monopoles contribution can explain this rather unexpected property.
These proposals inspired a number of publications devoted to the properties
and possible roles of the monopoles in the quark-gluon phase Shuryak (2009);
Ratti and Shuryak (2009); D’Alessandro and D’Elia (2008); Liao and Shuryak
(2008); D’Alessandro et al. (2010); Bornyakov and Braguta (2011, 2012).
Lattice gauge theory suggests a direct way to study fluctuations contributing
to the Euclidean space functional integral, in particular, the color-magnetic
monopoles can be studied. In a number of papers the evidence was found that
the nonperturbative properties of the nonabelian gauge theories such as
confinement, deconfining transition, chiral symmetry breaking, etc. are
closely related to the Abelian monopoles defined in the maximally Abelian
gauge $(MAG)$ ’t Hooft (1981); Kronfeld et al. (1987). This was called a
monopole dominance Shiba and Suzuki (1994). The drawback of this approach to
the monopole studies is that the definition is based on the choice of Abelian
gauge. There are various arguments supporting the statement that the Abelian
monopoles found in the MAG are important physical fluctuations surviving the
cutoff removal: scaling of the monopole density at $T=0$ according to
dimension $3$ for infrared $(percolating)$ cluster Bornyakov et al. (2005);
Abelian and monopole dominance for a number of infrared physics observables
(string tension Shiba and Suzuki (1994); Suzuki and Yotsuyanagi (1990);
Bornyakov et al. (2005), chiral condensate Woloshyn (1995), hadron spectrum
Kitahara et al. (1998)); monopoles in the MAG are correlated with gauge
invariant objects - instantons and calorons Ilgenfritz et al. (2006); Hart and
Teper (1996). It has been recently argued that the MAG is a proper Abelian
gauge to find gauge invariant monopoles since t’Hooft-Polyakov monopoles can
be identified in this gauge by the Abelian flux, but this is not possible in
other Abelian gauges Bonati et al. (2010). Most of these results were obtained
for $SU(2)$ gluodynamics but then confirmed for $SU(3)$ theory and QCD Arasaki
et al. (1997); Bornyakov et al. (2004). Listed above properties of Abelian
monopoles survive the continuum limit and removal of the Gribov copy effects.
It is worth noticing that removal of Gribov copy effects changes numerical
values of monopole characteristics quite substantially Bali et al. (1996).
In this paper we are studying thermal monopoles. It was shown in Ref.
Chernodub and Zakharov (2007) that thermal monopoles in Minkowski space are
associated with Euclidean monopole trajectories wrapped around the temperature
direction of the Euclidean volume. So the density of the monopoles in the
Minkowski space is given by the average of the absolute value of the monopole
wrapping number.
In Liao and Shuryak (2007) another approach to study thermal monopole
properties in the quark-gluon plasma phase based on the molecular dynamics
algorithm was suggested and implemented. The results for parameters of inter-
monopole interaction were found in agreement with lattice results Liao and
Shuryak (2008).
First numerical investigations of the wrapping monopole trajectories were
performed in $SU(2)$ Yang-Mills theory at high temperatures in Refs. Bornyakov
et al. (1992) and Ejiri (1996). A more systematic study of the thermal
monopoles was performed in Ref. D’Alessandro and D’Elia (2008). It was found
in D’Alessandro and D’Elia (2008) that the density of monopoles is independent
of the lattice spacing, as it should be for a physical quantity. The
density–density spatial correlation functions were computed in D’Alessandro
and D’Elia (2008). It was shown that there is a repulsive (attractive)
interaction for a monopole–monopole (monopole–antimonopole) pairs, which at
large distances might be described by a screened Coulomb potential with a
screening length of the order of $0.1$ fm. In Ref. Liao and Shuryak (2008) it
was proposed to associate the respective coupling constant with a magnetic
coupling $\alpha_{m}$. In the paper D’Alessandro et al. (2010) trajectories
which wrap more than one time around the time direction were investigated. It
was shown that these trajectories contribute significantly to a total monopole
density at $T$ slightly above $T_{c}$. It was also demonstrated that Bose
condensation of thermal monopoles, indicated by vanishing of the monopole
chemical potential, happens at temperature very close to $T_{c}$. However, the
relaxation algorithm applied in D’Alessandro and D’Elia (2008) to fix the MAG
is a source of the systematic errors due to effects of Gribov copies. It is
known since long ago that these effects are strong in the MAG and results for
gauge noninvariant observables can be substantially corrupted by inadequate
gauge fixing Bali et al. (1996). For the density of magnetic currents at zero
temperature it might be as high as $20\%$.
For nonzero temperature the effects of Gribov copies were not investigated
until recently. In a recent paper Bornyakov and Braguta (2012) this gap was
partially closed. It was shown that indeed gauge fixing with SA algorithm and
$10$ gauge copies per configuration gives rise to the density of the thermal
monopoles $20$ to $30\%$ lower (depending on the temperature) than values
found in D’Alessandro and D’Elia (2008). Large systematic effects due to
effects of Gribov copies found in Ref. Bornyakov and Braguta (2012) imply that
results obtained in earlier papers D’Alessandro and D’Elia (2008); Liao and
Shuryak (2008); D’Alessandro et al. (2010) for the density and other monopole
properties can not be considered as quantitatively precise and need further
independent verification . The quantitatively precise determination of such
parameters as monopole density, monopole coupling and others is necessary, in
particular, to verify the conjecture Liao and Shuryak (2007) that the magnetic
monopoles are weakly interacting (in comparison with electrically charged
fluctuations) just above transition but become strongly interacting at high
temperatures. In this paper we use the same gauge fixing procedure as in Refs.
Bornyakov and Braguta (2011, 2012) to avoid systematic effects due to Gribov
copies.
The careful study of the finite volume and finite lattice spacing effects was
made in D’Alessandro and D’Elia (2008). We fix our spatial lattice size to
$L_{s}=48$ which was shown in Ref. D’Alessandro and D’Elia (2008) to be large
enough to avoid finite volume effects. We check finite lattice spacing effects
comparing results obtained on lattices with $N_{t}=4$ and 6 at two
temperatures. Let us emphasize that our studies are computationally much more
demanding in comparison with studies undertaken in Refs. D’Alessandro and
D’Elia (2008); D’Alessandro et al. (2010), since we produce $10$ Gribov copies
per configuration to avoid Gribov copies effect. For this reason our check of
the continuum limit is not as extensive as it was in Refs. D’Alessandro and
D’Elia (2008); D’Alessandro et al. (2010).
The important contribution of this work to the thermal monopole studies is a
check of universality. In studies of magnetic currents at zero temperature it
was found Bornyakov et al. (2005) that the density of the infrared magnetic
currents is different for different lattice actions with difference as large
as $30\%$. The conclusion was made that the ultraviolet fluctuations
contribute to the infrared density and this contribution has to be removed.
Partial removal was made by the use of the improved action. In present paper
we use the improved lattice action - tadpole improved Symanzik action and
compare our results for the density and other quantities with results obtained
with the Wilson action D’Alessandro and D’Elia (2008); Liao and Shuryak
(2008); D’Alessandro et al. (2010); Bornyakov and Braguta (2012). We find that
the universality holds for monopoles which do not form short range
(ultraviolet) dipoles.
We also want to point out that in this work we use more natural procedure of
computing monopole correlators in comparison with papers D’Alessandro et al.
(2010) and Bornyakov and Braguta (2012). It was mentioned in Ref. D’Alessandro
et al. (2010) that monopole trajectories had a lot of small loops attached to
them which were UV noise. Presence of such loops do not allow to determined
monopole spatial coordinates unambiguously for all time slices. This problem
was bypassed in D’Alessandro et al. (2010); Bornyakov and Braguta (2012) by
using only one time slice. We remove the small loops attached to thermal
monopole trajectories and thus we are able to determine the monopole
coordinates in every time slice unambiguously. Then we use all time slices to
compute the correlators what allows us to decrease the statistical errors
substantially.
## II Simulation details
We studied the $SU(2)$ lattice gauge theory with the tadpole improved Symanzik
action:
$S=\beta_{impr}\sum_{pl}S_{pl}-\frac{\beta_{impr}}{20u_{0}^{2}}\sum_{rt}S_{rt}$
(1)
where $S_{pl}$ and $S_{rt}$ denote plaquette and 1$\times$2 rectangular loop
terms in the action:
$S_{pl,rt}=\frac{1}{2}Tr(1-U_{pl,rt}),$ (2)
parameter $u_{0}$ is the input tadpole improvement factor taken here equal to
the fourth root of the average plaquette P =
$\langle\frac{1}{2}U_{pl}\rangle$. We use the same code to generate
configurations of the lattice gauge field as was used in Ref. Bornyakov et al.
(2005). Our calculations were performed on the asymmetric lattices with
lattice volume $V=L_{t}L_{s}^{3}$, where $L_{t,s}$ is the number of sites in
the time (space) direction. The temperature $T$ is given by:
$T=\frac{1}{aL_{t}}~{},$ (3)
where $a$ is the lattice spacing. To determine the values of $u_{0}$ we used
results of Ref. Bornyakov et al. (2005) either directly or to make
interpolation to necessary values of $\beta$. The critical value of the
coupling constant for $L_{t}=6$ is $\beta_{c}=3.248$ Bornyakov et al. (2007).
For $L_{t}=6$ the ratio $T/T_{c}$ was obtained using data for the string
tension from Ref. Bornyakov et al. (2005) again either directly or via
interpolation. For $L_{t}=4$ ratio $T/T_{c}$ was taken to be equal to the
ratio for $L_{t}=6$ multiplied by factor $1.5$. In Table 1 we provide the
information about the gauge field ensembles and parameters used in our study.
The MAG is fixed by finding an extremum of the gauge functional:
$F_{U}(g)=~{}\frac{1}{4V}\sum_{x\mu}~{}\frac{1}{2}~{}\operatorname{Tr}~{}\biggl{(}U^{g}_{x\mu}\sigma_{3}U^{g\dagger}_{x\mu}\sigma_{3}\biggr{)}\;,$
(4)
with respect to gauge transformations $g_{x}$ of the link variables
$U_{x\mu}$:
$U_{x\mu}\stackrel{{\scriptstyle
g}}{{\mapsto}}U_{x\mu}^{g}=g_{x}^{\dagger}U_{x\mu}g_{x+\mu}\;;\qquad g_{x}\in
SU(2)\,.$ (5)
We apply the simulated annealing (SA) algorithm which proved to be very
efficient for this gauge Bali et al. (1996) as well as for other gauges such
as center gauges Bornyakov et al. (2001) and Landau gauge Bogolubsky et al.
(2007). To further decrease the Gribov copy effects we generated $10$ Gribov
copies per configuration starting every time gauge fixing procedure from a
randomly selected gauge copy of the original Monte Carlo configuration.
$\beta$ | $u_{0}$ | $L_{t}$ | $L_{s}$ | $T/T_{c}$ | $N_{meas}$
---|---|---|---|---|---
3.640 | 0.92172 | 4 | 48 | 3.00 | 300
3.544 | 0.91877 | 4 | 48 | 2.50 | 200
3.480 | 0.91681 | 4 | 48 | 2.26 | 200
3.410 | 0.91438 | 4 | 48 | 2.00 | 200
3.248 | 0.90803 | 4 | 48 | 1.50 | 254
3.640 | 0.92172 | 6 | 48 | 2.00 | 200
3.480 | 0.91681 | 6 | 48 | 1.50 | 200
3.400 | 0.91402 | 6 | 48 | 1.31 | 270
3.340 | 0.91176 | 6 | 48 | 1.20 | 200
3.300 | 0.91015 | 6 | 48 | 1.10 | 203
3.285 | 0.90954 | 6 | 48 | 1.07 | 206
3.265 | 0.90867 | 6 | 48 | 1.03 | 200
Table 1: Values of $\beta$, parameter $u_{0}$, lattice sizes, temperature,
number of measurements used in this paper.
## III Monopole Density
The monopole current is defined on the links $\\{x,\mu\\}^{*}$ of the dual
lattice and take integer values $j_{\mu}(x)=0,\pm 1,\pm 2$. The monopole
currents form closed loops combined into clusters. Wrapped clusters are closed
through the lattice boundary. The wrapping number $N_{wr}\in Z$ of a given
cluster is defined by:
$N_{wr}=\frac{1}{L_{t}}\sum_{j_{4}(x)\in cluster}j_{4}(x)\,$ (6)
The density $\rho$ of the thermal monopoles is defined as follows:
$\rho=\frac{\langle~{}\sum_{clusters}|N_{wr}|~{}\rangle}{L_{s}^{3}a^{3}}\,$
(7)
where the sum is taken over all wrapped clusters for a given configuration.
Following Ref. Chernodub and Zakharov (2007) we distinguish two regions of
temperatures: low temperature region $T\lesssim 2T_{c}$ and high temperatures
$T\gtrsim 2T_{c}$. In Ref. Chernodub and Zakharov (2007) it was proposed that
at low temperatures the density of monopoles is almost insensitive to
temperature thus indicating that the monopoles form a dense liquid while at
high temperature the monopole density has a power dependence on temperature.
These statements were based on the results obtained in Refs Bornyakov et al.
(1992) and Ejiri (1996). The behavior of the density at low temperature was
not discussed in earlier papers D’Alessandro and D’Elia (2008); D’Alessandro
et al. (2010); Bornyakov and Braguta (2012).
The results for a range of temperatures $T_{c}<T\lesssim 1.5T_{c}$ obtained on
lattices with $L_{t}=6$ are presented In FIG. 1. It can be seen that the
thermal monopole density monotonously increases as temperature grows. The
density at $T/T_{c}=1.5$ is $2.3$ times as large as one at $T/T_{c}=1.03$ what
indicates that the monopole density is considerably sensitive to temperature.
This fact allows us to say that previous results on this observable did not
reflect real situation and the conclusion made in Ref. Chernodub and Zakharov
(2007) was incorrect. In FIG. 1 we also show the results from Ref.
D’Alessandro et al. (2010) for comparison. The results demonstrate similar
dependence on temperature, however our results are lower at any temperature.
It is due to Gribov copy effect.
In FIG. 2 we show our data for temperatures $T>1.5T_{c}$. The data of Refs.
D’Alessandro and D’Elia (2008) and Bornyakov and Braguta (2012) are also shown
for comparison. As was concluded in Ref. Bornyakov and Braguta (2012) effects
of Gribov copies in results of Ref. D’Alessandro and D’Elia (2008) are between
20% at $T/T_{c}=2$ and almost $30\%$ for $T/T_{c}=7$. Our results deviate from
those of Ref. Bornyakov and Braguta (2012) by about $10\%$. Thus we observe
violation of universality of the thermal monopole density at given lattice
spacing. We need to check whether the continuum limit is reached in case of
the Symanzik action for our lattice spacings. To answer to this question we
compare results obtained on $L_{t}=4$ and $L_{t}=6$ lattices at at
$T=1.5~{}T_{c}$ and $T=2~{}T_{c}$. One can see from FIG. 2 that the change of
the density with decreasing lattice spacing is small for both temperature
values. Quantitatively it is less than $1\%$ for $T=2~{}T_{c}$ and about $3\%$
for $T=1.5~{}T_{c}$. This allows us to state that in case of Symanzik action
the results for the monopole density obtained on $L_{t}=4$ lattices can be
considered as being close to the continuum limit. Similar conclusion about
closeness to the continuum limit was made for the Wilson action in Ref.
Bornyakov and Braguta (2012). Thus the violation of universality seen in FIG.
2 may persist to continuum limit. Note that at zero temperature the
universality breaking effects were found to be much stronger Bornyakov et al.
(2005), up to $30\%$.
Different values for monopole density in case of Symanzik and Wilson actions
can be explained by a fact that configurations have different number of short
range dipoles at the same temperatures. By short range dipoles we imply paired
monopole trajectories having opposite wrappings and separated by distance of
order of one lattice spacing. Thus, if we calculate monopole density omitting
small distance,it should bring densities closer to each other. We found indeed
that the distance between two densities decreases after we remove dipoles of
size $a$.
We will come back to discussion of the universality for the thermal monopole
density in the next section.
Figure 1: The behavior of the monopole density(normalized by $T_{c}^{3}$) at
low temperatures(blue empty circles). The results from Ref. D’Alessandro et
al. (2010) are presented for comparison(red empty triangles).
The dimensional reduction suggests for the density $\rho$ the following
temperature dependence at high enough temperature:
$\rho(T)^{1/3}=c_{\rho}g^{2}(T)T$ (8)
where the temperature dependent running coupling $g^{2}(T)$ is described at
high temperature by the two-loop expression with the scale parameter
$\Lambda_{T}$:
$g^{-2}(T)=\frac{11}{12\pi^{2}}\ln(T/\Lambda_{T})+\frac{17}{44\pi^{2}}(\ln(2\ln(T/\Lambda_{T}))$
(9)
We fit our data to function determined by equations (8) and (9). The good fit
with $\chi^{2}/dof=0.28$ was obtained for $T\geq 2T_{c}$. The values for fit
parameters were $c_{\rho}=0.160(6)$, $\Lambda_{T}/T_{c}=0.144(3)$. Thus for
high enough temperature the density $\rho(T)$ is well described by the form
which follows from dimensional reduction. The values of parameters $c_{\rho}$
and $\Lambda_{T}$ differ from values obtained in Bornyakov and Braguta (2012)
though the difference in results for the density is small. But note that we
used fit over range of temperatures between $2T_{c}$ and $3T_{c}$ while in
Bornyakov and Braguta (2012) $T\geq 3T_{c}$ were used. The data for the
thermal monopole density are presented in TAB. 5.
Figure 2: The dependence of the thermal monopole density(normalized by
$T^{3}$) on temperature(circles). The line is a fit to eq.(9). The data from
Ref. D’Alessandro and D’Elia (2008)(triangles) and from Ref. Bornyakov and
Braguta (2012)(squares) are presented for comparison. The densities at
$1.5T_{c}$ and $2T_{c}$ with $L_{t}=6$ are labelled by the crosses.
## IV Monopole Interaction
We computed two types of monopole density correlators $g(r)$, for monopoles
having the same charges (MM correlator) and for monopoles having opposite
charges (AM correlator). The correlators are defined as follows:
$g_{MM}(r)=\frac{\langle\rho_{M}(0)\rho_{M}(r)\rangle}{\rho_{M}^{2}}+\frac{\langle\rho_{A}(0)\rho_{A}(r)\rangle}{\rho_{A}^{2}}$
(10)
$g_{AM}(r)=\frac{\langle\rho_{A}(0)\rho_{M}(r)\rangle}{\rho_{A}\rho_{M}}+\frac{\langle\rho_{M}(0)\rho_{A}(r)\rangle}{\rho_{A}\rho_{M}}$
(11)
where $\rho_{M,A}(0)$ and $\rho_{M,A}(r)$ are local densities. It can be
reexpressed as:
$g_{MM}(r)=\frac{1}{\rho_{M}}\frac{1}{4\pi
r^{2}}\langle\frac{dN_{M}(r)}{dr}\rangle+\frac{1}{\rho_{A}}\frac{1}{4\pi
r^{2}}\langle\frac{dN_{A}(r)}{dr}\rangle\,,$ (12)
where $N_{M}(r)$ is the number of monopoles, and similarly for $g_{AM}$.
In our computations following Refs. D’Alessandro and D’Elia (2008); Bornyakov
and Braguta (2012) we take $dN(r)$ to be a number of monopoles in a spherical
shell of finite thickness $dr=a$ at a distance $r$ from a reference particle,
whereas $4\pi r^{2}dr$ is equal to a volume of this shell, i.e. number of
lattice sites in it.
Correlators $g_{MM,AM}(r)$ were calculated for nine temperatures in the range
between $1.03T_{c}$ and $3T_{c}$. Three AM and MM correlators are presented in
FIG. 3 and FIG. 4 respectively.
Figure 3: The correlation function $g_{AM}(r)$ for monopole-antimonopole case
for three values of temperature: $3T_{c}$, $1.5_{c}$ and $1.07T_{c}$ with
$L_{t}=4,4$ and $6$ respectively. Figure 4: The correlation function
$g_{MM}(r)$ for monopole-monopole case for the same values of temperature as
in Fig. 3. The lines are fits for these three cases.
One can see that in AM case the correlators start below 1, reach maximal value
equal to $1.3-1.4$ at distance $r$ between $0.5/T$ and $0.8/T$ and then
decrease down to $1$ at distance between $1.8/T$ and $2.5/T$. In other words
the repulsion at short distances changes to attraction at large distances. In
MM case a picture is different, the correlators start at small value below 1
and smoothly reach 1 from below at same distances as AM correlators do. In
this case we can see only the repulsion. These results are in a good
qualitative agreement with results obtained in Refs. D’Alessandro and D’Elia
(2008); Bornyakov and Braguta (2012).
It should be mentioned that we applied more correct procedure to compute MM
and AM correlators. In Refs. Bornyakov and Braguta (2012) and D’Alessandro et
al. (2010) only one time slice was used to calculate the correlators since the
monopole coordinates could not be determined for all time slices unambiguously
because of small loops attached to monopole trajectory. We used a special
algorithm which cut off all UV loops out of each monopoles trajectory. Then we
could take into consideration all time slices to compute the correlators. This
leads to more precise results for the correlators. For example, at $T=3T_{c}$
the statistical errors decreased by factor 1.7 approximately.
One can also see that correlators in FIG. 3 and FIG. 4 are temperature
dependent. It is possible to fit the data with the temperature dependent
potential:
$g_{MM,AM}(r)=e^{-V(r)/T}$ (13)
where $V(r)$ can be well approximated by a screened potential:
$V(r)=\frac{\alpha_{m}}{r}e^{-m_{D}r}$ (14)
at large distances. In eq. (14) $m_{D}$ is a screening mass and $\alpha_{m}$
is a magnetic coupling.
To decide which data points can be included in the fit range we used the
following method. We plot the dependence of $rV(r)$ on $rT$ in log scale for
all temperatures (see FIG. 5). The linear dependence should be valid at
distances where eq. (14) is satisfied. Thus data at small distances which do
not fall onto a line are to be discarded. Typically we discarded three data
points. Data at large distances were discarded because of high statistical
errors. Our results for the fit parameters are presented in Table 2. Note that
our values for $\chi^{2}/N_{dof}$ shown in Table 2 are in general lower than
values of this quantities characterizing fit quality obtained in Refs. Liao
and Shuryak (2008); Bornyakov and Braguta (2012).
Figure 5: The dependence of $rV(r)$ on $rT$. Note log scale for x axis. The red line is the fit to eq. (14). $T/Tc$ | $\alpha_{m}$ | $m_{D}/T$ | $\chi^{2}$
---|---|---|---
3.00 | 2.84(21) | 1.47(07) | 0.81
2.50 | 3.03(31) | 1.67(08) | 0.89
2.26 | 3.18(24) | 1.79(06) | 0.56
2.00 | 2.61(15) | 1.75(05) | 0.30
1.50 | 2.86(15) | 2.13(05) | 0.16
2.00 | 2.19(18) | 1.43(08) | 0.82
1.50 | 2.47(12) | 1.87(05) | 0.16
1.20 | 2.04(14) | 2.06(08) | 0.73
1.10 | 1.43(08) | 1.81(07) | 0.44
1.07 | 1.53(07) | 1.97(05) | 0.23
1.03 | 1.76(10) | 2.24(7) | 0.26
Table 2: Values of the magnetic coupling $\alpha_{m}$ and the screening mass
$m_{D}$ obtained by fitting of MM correlators to eq. (13). The double line
separates the parameters obtained for $L_{t}=4$(above the line) from those
obtained for $L_{t}=6$(below the line). Figure 6: Comparison of MM and AM
correlators computed at $T=2T_{c}$ on lattices with for $L_{t}=6$ (empty blue
squares) and $L_{t}=4$ (filled red circles).
In order to check finite lattice spacing effects we compared correlators and
respective fit parameters on lattices with $L_{t}=4$ and 6 at $T=1.5T_{c}$ and
$2T_{c}$. The comparison of these correlators for $T=2T_{c}$ is presented in
FIG. 6. One can see that for $g_{MM}(r)$ the finite lattice spacing effects
are small at all distances being maximal at distances $rT\approx 1.2$. For
$g_{AM}(r)$ the good agreement is also observed at distances to the right of
the distance corresponding to the maximum of the correlator while at short
distances the finite lattice spacing effects are large. Results for
$T=1.5T_{c}$ are similar.
Figure 7: Comparison of MM and AM correlators computed at $T=1.5T_{c}$ in
this work(red filled circles) and Ref. Bornyakov and Braguta (2012)(blue empty
squares).
In FIG. 7 we compare the MM and AM correlators computed in this work and in
Ref. Bornyakov and Braguta (2012) at $T\sim 1.5T_{c}$. It can be seen that the
correlators are in good agreement with exception for AM correlator at small
distances. This implies that we observe universality of the correlators apart
from distribution of the small dipoles: with Wilson action we observe more
such dipoles than with improved action. Above we have concluded that the
number of small dipoles is decreasing with decreasing lattice spacing. Thus,
we may expect that in the continuum limit the universality will restore at
small distances as well.
The dependence of the fit parameters, $m_{D}$ and $\alpha_{m}$, on
temperature, obtained for MM correlators, is presented in FIG. 8 and FIG 9,
respectively. One can see that the behavior of the fit parameters at $T$ close
to $T_{c}$ is different from their behavior at high temperature. Close to
$T_{c}$, in the range between $1.03T_{c}$ and $1.1T_{c}$ we observe decreasing
of both $m_{D}$ and $\alpha_{m}$. Such behavior was not observed before. This
indicates that just above the transition the monopole interaction becomes
weaker with increasing temperature. Since near the phase transition the finite
volume effects might be strong this observation should be verified on larger
lattices. At a bit higher temperature, $T=1.2T_{c}$, both parameters jump to
higher values and then their dependence on the temperature becomes different.
While for $m_{D}$ we observe slow decreasing, in agreement with results of
Ref. Bornyakov and Braguta (2012),magnetic coupling $\alpha_{m}$ is slowly
increasing. This is in qualitative agreement with Refs. Liao and Shuryak
(2008); Bornyakov and Braguta (2012). Coming to quantitative comparison we
find that our values for $m_{D}$ are almost within error bars although
systematically lower than values reported in Ref. Bornyakov and Braguta
(2012). The values for $\alpha_{m}$ presented in FIG 9 are substantially lower
than the values obtained in Ref. Bornyakov and Braguta (2012). In the
temperature range $T\geq 2T_{c}$ our results for $\alpha_{m}$ are also lower
than values obtained in Ref. Liao and Shuryak (2008). The data in FIG 9
indicate that $\alpha_{m}$ approaches its maximum value of about $3$ at rather
small temperature $T\sim 2.5T_{c}$. But for temperatures close to $T_{c}$ our
values of $\alpha_{m}$ are higher than values presented in Liao and Shuryak
(2008).
The difference in the behavior of the correlator parameters can be explained
at least partially by the fact that in this work we eliminated all small loops
out of each wrapping cluster. This gave us an opportunity to determine the
monopole location in a given time slice unambiguously and thus, to use all
time slices for correlators computation. This procedure was used in studies of
the thermal monopoles for the first time.
We now can compute the plasma parameter $\Gamma$ which is defined as follows:
$\Gamma=\alpha_{m}\left(\frac{4\pi\rho}{3T^{3}}\right)^{1/3}$ (15)
$\Gamma$ is equal to ratio of the system potential energy to its kinetic
energy. If $\Gamma<<1$ the system is a weakly coupled plasma; if $\Gamma>1$,
it is a strongly coupled plasma. For $1\leq\Gamma\leq\Gamma_{c}\sim 100$, the
system is in a liquid state. The dependence of $\Gamma$ on temperature is
presented in FIG. 10. $\Gamma$ is roughly proportional to $\alpha$ since
$\rho^{1/3}/T$ varies slowly with temperature (see FIG. 1 and 2). Thus at
small temperature we observe in FIG. 10 the nonmonotonic behavior we saw in
FIG 9. At temperatures above $1.5T_{c}$ $\Gamma$ can be well approximated by a
constant. We cannot exclude slight increase or slight decrease of $\Gamma$ for
higher temperatures, though. The independence of $\Gamma$ on temperature was
predicted in Ref. Liao and Shuryak (2008). But quantitatively our result is
rather different: in Ref. Liao and Shuryak (2008) $\Gamma$ was found
substantially higher and approaching a constant value about $5$ at
temperatures above $4T_{c}$. Despite this quantitative difference we confirm
result of Ref. Liao and Shuryak (2008) that the thermal monopoles are in a
liquid state at all temperatures.
Figure 8: The dependence of the screening mass $m_{D}$ on temperature.
Figure 9: The dependence of the magnetic coupling $\alpha_{m}$ on
temperature. Figure 10: The dependence of the Coulomb plasma parameter on
temperature.
## V Monopole Condensation
In this section we consider thermal monopole trajectories which wrap more than
one time in a time direction. It was proposed in D’Alessandro et al. (2010)
that these trajectories can serve as an indicator of Bose-Einstein
condensation of the thermal monopoles when the phase transition is approached
from above. The main idea of this proposal is the following: a trajectory
wrapping $k$ times in a time direction is associated with a set of $k$
monopoles permutated cyclically. Having such an interpretation one can assess
a density of these trajectories assuming that they form a system of non-
relativistic noninteracting bosons. According to Ref. D’Alessandro et al.
(2010) this density can be written as follows:
$\rho_{k}=\frac{e^{-\hat{\mu}k}}{\lambda^{3}k^{5/2}}$ (16)
where $k$ is a number of wrappings, $\hat{\mu}\equiv-\mu/T$ is a chemical
potential, and $\lambda$ is the De Broglie thermal wavelength. The
condensation temperature is determined by the vanishing of the chemical
potential.
To take into account interactions between monopoles it was suggested in
D’Alessandro et al. (2010) to modify eq. (16) to
$\frac{\rho_{k}}{T^{3}}=\frac{Ae^{-\hat{\mu}k}}{k^{\alpha}},$ (17)
with a free parameter $\alpha$. The condensation of monopoles still should be
signalled by vanishing of effective chemical potential $\hat{\mu}$.
Figure 11: Normalized densities for trajectories wrapping more than once in a time direction (empty symbols). Note a log scale for Y-axes. For comparison we show here the data from Ref. D’Alessandro et al. (2010)(filled symbols). | $1.03Tc$ | $1.07Tc$ | $1.1Tc$ | $1.20Tc$
---|---|---|---|---
$\mu(\alpha=0)$ | 1.0(1) | 1.72(5) | 2.4(2) | 2.8(3)
$\mu(\alpha=2)$ | 0.50(9) | 0.99(3) | 1.6(1) | 2.0(3)
$\mu(\alpha=2.5)$ | 0.37(8) | 0.82(4) | 1.4(1) | 1.8(2)
$\mu(\alpha=3)$ | 0.26(7) | 0.65(5) | 1.2(1) | 1.6(2)
$\chi^{2}(\alpha=0)$ | 2.09 | 0.46 | 2.27 | 1.99
$\chi^{2}(\alpha=2)$ | 1.35 | 0.27 | 1.84 | 1.54
$\chi^{2}(\alpha=2.5)$ | 1.22 | 0.41 | 1.73 | 1.42
$\chi^{2}(\alpha=3)$ | 1.14 | 0.64 | 1.61 | 1.30
Table 3: Values of fit parameters $\mu$ obtained by fitting data for
$\frac{\rho_{k}}{T^{3}}$ to eq. (17) for 4 values of parameter $\alpha$.
Values of $\chi^{2}$ are also shown.
One can see from FIG. 11 that our values for $\frac{\rho_{k}}{T^{3}}$ are
systematically lower than values presented in Ref. D’Alessandro et al. (2010).
This is in a qualitative agreement with the fact reported above that our total
density $\rho$ is substantially lower than total density found in Ref.
D’Alessandro et al. (2010). As in the case of the total density this
difference is to be explained mostly by large systematic effects due to Gribov
copies in results of Ref. D’Alessandro et al. (2010).
We fitted our data for $\frac{\rho_{k}}{T^{3}}$ to eq. (17) for temperatures
close to $T_{c}$. Since our data do not allow us to keep free all parameters
we made fits for $4$ fixed values of $\alpha$. The results of the fit are
presented in Table 3. One can see that $\chi^{2}$ is decreasing with
increasing $\alpha$ for three temperature values out of four.
Figure 12: The dependence chemical potential on temperature(blue filled
triangles). The data from ref. D’Alessandro et al. (2010) is presented for
comparison(red empty triangles).
We also computed the monopole thermal mass using the mean squared monopole
fluctuation $\Delta r^{2}$ . The relation between $\Delta r^{2}$ and the mass
of a nonrelativistic free particle is as follows D’Alessandro et al. (2010):
$m=\frac{1}{2T\Delta r^{2}}$ (18)
$\Delta r^{2}$ can be computed on a lattice in the following way D’Alessandro
et al. (2010):
$a^{-2}\Delta r^{2}=\frac{1}{L}\sum_{i=1}^{L}d_{i}^{2}$ (19)
where $L$ \- is a total trajectory lengths, $d_{i}^{2}$ is a squared spatial
distance between the monopole coordinate at $t=0$ and its current coordinate
after $i$ steps along the monopole trajectory.
As in Refs. D’Alessandro et al. (2010); Bornyakov and Braguta (2012) we
computed $\Delta r^{2}$ for trajectories with one wrapping. Furthermore, we
have checked the influence of the small loops on the value of the monopole
mass determined via eq. (18).
Figure 13: The dimensionless ratio $m/T$ as a function of temperature
obtained in our work ($L_{t}=4$ \- red filled squares, $L_{t}=6$ \- red filled
circles), in Ref. Bornyakov and Braguta (2012)(empty triangles) and in Ref.
D’Alessandro et al. (2010), $a=0.047$ (blue empty circles).
The comparison of the thermal monopole mass obtained for the Wilson action
D’Alessandro et al. (2010); Bornyakov and Braguta (2012) with the values
obtained in this paper is presented in FIG. 13 It is seen that the values
obtained for the Symanzik action demonstrates the same dependence on
temperature as for the Wilson case, but are higher at all temperatures than
results of Bornyakov and Braguta (2012) and lower than results of D’Alessandro
et al. (2010).
The monopole mass computed after removal of the loops increases by a factor of
$6\%$ at high temperature in comparison with the mass obtained in case when
all loops are untouched. But the difference between two masses is getting
larger as temperature approaches to $T_{c}$ reaching $16\%$ at $1.03T_{c}$.
Such behavior of the thermal monopole mass is expectable as with decreasing
temperature the number of loops attached to a wrapped cluster increases. This
can be seen from FIG. 14 where the temperature dependence of number of loops
per one wrapped cluster is presented ($r_{lp}$). When temperature decreases
from $3T_{c}$ to $1.03T_{c}$, difference in $r_{lp}$ is one order.
Figure 14: The dependence the average number of loops per one wrapped cluster
on temperature.
It was found in Bornyakov and Braguta (2012) that the thermal monopole mass
can be fitted by the following function:
$\frac{m}{T}=b\ln(\frac{T}{\Lambda_{m}})$ (20)
The comparison of our results for both cases with the results obtained in Ref.
Bornyakov and Braguta (2012) are presented in Tab. 4.
$b$ | $\Lambda_{m}/T_{c}$ | $\chi^{2}$
---|---|---
3.653(6) | 0.718(2) | 0.2
3.80(1) | 0.86(2) | 0.58
3.66(7) | 0.78(2) | 0.26
Table 4: The comparison of the fit parameters for three cases. The Wilson
case (first line) Bornyakov and Braguta (2012), the Symanzik case when loops
are untouched(second line) and the Symanzik case when all loops
eliminated(third line).
## VI Conclusions
We summarize our findings. Using the improved lattice action (1) and the
adequate gauge fixing procedure we completed careful study of the properties
of the thermal color-magnetic monopoles.
Comparing our results for thermal monopole density and parameters of the
monopole interactions with results of Ref. Bornyakov and Braguta (2012) we
find rather small deviations, at the level of $10\%$ for the density. We have
found that this difference decreases even further when small dipoles are not
counted. This implies universality for infrared thermal monopoles determined
in the MAG. Establishing of the universality of the thermal monopole
properties is important to prove that these monopoles determined after the MAG
fixing are fluctuations of the gauge field relevant for infrared physics
rather than artifacts of the gauge fixing.
To study cutoff effects we made computations with two lattice spacings (using
lattices with $L_{t}=4$ and 6) at temperatures $T=1.5T_{c}$ and $2T_{c}$. We
find that results for the thermal monopole density are not depending on the
lattice spacing (see FIG. 2 and Table 5) and thus they are computed in the
continuum limit. Results for parameters of the monopole interaction show
slight dependence on the lattice spacing (see Table 2) and thus they are close
to the continuum limit. The exception is the monopole mass which shows strong
dependence on the lattice spacing.
We confirmed observation made before in Ref. Bornyakov and Braguta (2012) that
without proper gauge fixing the systematic effects due to Gribov copy effects
are large, e.g., up to $30\%$ for the monopole density. We found that these
effects are even more essential for the monopole trajectories with multiple
wrappings.
Studying the correlation functions we obtained the values for the Coulomb
coupling constant $\alpha_{m}$ which are substantially smaller than values
obtained in Ref. Liao and Shuryak (2008). Since our results are close to the
continuum limit and the same is true for results of Ref. Liao and Shuryak
(2008) this difference is to be explained by Gribov copy effects. Furthermore,
we found highly nonmonotonic behavior for both $\alpha_{m}$ and screening mass
$m_{D}$ near the transition temperature $T_{c}$. Both parameters first
decrease with increasing temperature and then jump up at the temperature
$1.2T_{c}$ before monotonous dependence on $T$ is settled. We shall admit that
although our measurements in this range of temperature were done on lattices
with $L_{t}=6$, the finite lattice spacing effects should be checked by
simulations on lattices with larger value of $L_{t}$ to check this effect.
Comparatively small values of $\alpha_{m}$ give rise to small values of the
plasma parameter $\Gamma_{M}$. We find that this parameter flattens at the
value about $2$ at high temperatures in contrast to value of $5$ found in Ref.
Liao and Shuryak (2008). Still we confirm that the monopoles are in a liquid
state at all temperatures considered.
We have repeated the study of the monopole trajectories with multiple wrapping
undertaken in D’Alessandro et al. (2010). Although we obtained the values for
the densities of such trajectories quite different from the values found in
D’Alessandro et al. (2010), our values for the effective chemical potential
$\mu$ are quite close to results of Re. D’Alessandro et al. (2010). Moreover
we confirm that $\mu$ goes to zero, indicating Bose-Einstein condensation, at
the temperature very close to $T_{c}$.
We made first studies of the effects of UV fluctuations on parameters of the
monopole potential and on monopole thermal mass removing contributions from
the closed loops attached to the wrapped monopole loops. Due to removing such
contribution we were able to identify the currents $j_{0}$ of the wrapped loop
unambiguously and thus to use all time-slices of the lattice in the
computation of the correlators, thus decreasing the statistical error
substantially.
### Acknowledgments
We would like to express our gratitude to V.V. Braguta, M.I. Polikarpov and
V.I. Zakharov for very useful and illuminating discussions. We also would like
to thank both E. D. Merkulova and E. E. Kurshev who helped us a lot with the
algorithms used in this work. This investigation has been supported by the
Federal Special-Purpose Programme ’Cadres’ of the Russian Ministry of Science
and Education and by grant RFBR 11-02-01227-a.
### Appendix
In this appendix we present a table of the densities for all studied
temperatures.
$T/Tc$ | $\rho_{1}/T^{3}$ | $\rho_{2}/T^{3}$ | $\rho_{3}/T^{3}$ | $\rho_{4}/T^{3}$ | $\rho_{5}/T^{3}$ | $\rho_{6}/T^{3}$ | $\rho_{7}/T^{3}$ | $\rho_{8}/T^{3}$ | $\rho_{9}/T^{3}$
---|---|---|---|---|---|---|---|---|---
$1.03$ | $0.245(2)$ | $0.89(3)10^{-2}$ | $0.16(1)10^{-2}$ | $0.38(6)10^{-3}$ | $0.24(5)10^{-3}$ | $0.10(3)10^{-3}$ | $0.5(2)10^{-4}$ | $0.3(2)10^{-4}$ | $0.1(1)10^{-4}$
$1.07$ | $0.261(2)$ | $0.74(2)10^{-2}$ | $0.13(1)10^{-2}$ | $0.22(5)10^{-3}$ | $0.6(2)10^{-4}$ | $0.2(1)10^{-4}$ | $0.5(3)10^{-4}$ | $0.1(1)10^{-4}$ | $0.1(1)10^{-4}$
$1.10$ | $0.260(1)$ | $0.61(2)10^{-2}$ | $0.53(6)10^{-3}$ | $0.7(2)10^{-4}$ | $0.5(2)10^{-4}$ | $0.2(1)10^{-4}$ | | |
$1.20$ | $0.249(1)$ | $0.35(2)10^{-2}$ | $0.19(5)10^{-3}$ | $0.4(2)10^{-4}$ | | | | |
$1.31$ | $0.224(3)$ | $0.21(2)10^{-2}$ | $0.4(4)10^{-4}$ | | | | | |
$1.50^{a}$ | $0.193(6)$ | $0.93(5)10^{-3}$ | $0.9(4)10^{-5}$ | | | | | |
$1.50^{b}$ | $0.1886(12)$ | $0.53(7)10^{-3}$ | $0.9(9)10^{-5}$ | | | | | |
$2.00^{b}$ | $0.1390(10)$ | $0.11(3)10^{-3}$ | | | | | | |
$2.00^{a}$ | $0.139(6)$ | $0.14(2)10^{-3}$ | | | | | | |
$2.26$ | $0.123(5)$ | $0.08(2)10^{-3}$ | | | | | | |
$2.5$ | $0.111(1)$ | $0.3(1)10^{-4}$ | | | | | | |
$3.00$ | $0.938(4)10^{-1}$ | $0.21(6)10^{-4}$ | | | | | | |
Table 5: The monopole density (normalized by $T^{3}$) of monopole
trajectories wrapped one and more times in time direction as a function of
$T/T_{c}$. The superscript $a$ and $b$ above the temperature value refers to
two different lattice spacing $4$ and $6$ respectively.
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|
arxiv-papers
| 2011-11-01T11:11:09 |
2024-09-04T02:49:23.826439
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V. G. Bornyakov, A. G. Kononenko",
"submitter": "Kononenko Anton G.",
"url": "https://arxiv.org/abs/1111.0169"
}
|
1111.0181
|
# Revisiting the $B\to\pi\rho$, $\pi\omega$ Decays in the Perturbative QCD
Approach Beyond the Leading Order
Zhou Rui Gao Xiangdong Cai-Dian Lü lucd@ihep.ac.cn Institute of High Energy
Physics and Theoretical Physics Center for Science Facilities, Chinese Academy
of Sciences, Beijing 100049, People’s Republic of China
###### Abstract
We calculate the branching ratios and CP asymmetries of the $B\to\pi\rho$,
$\pi\omega$ decays in the perturbative QCD factorization approach up to the
next-to-leading-order contributions. We find that the next-to-leading-order
contributions can interfere with the leading-order part constructively or
destructively for different decay modes. Our numerical results have a much
better agreement with current available data than previous leading-order
calculations, e.g., the next-to-leading-order corrections enhance the
$B^{0}\rightarrow\pi^{0}\rho^{0}$ branching ratios by a factor 2.5, which is
helpful to narrow the gaps between theoretic predictions and experimental
data. We also update the direct CP-violation parameters, the mixing-induced
CP-violation parameters of these modes, which show a better agreement with
experimental data than many of the other approaches.
###### pacs:
13.25.Hw, 11.10.Hi, 12.38.Bx
## I Introduction
The charmless B meson decays are not only suitable to study CP violations but
also sensitive to new physicsiiba . During the past decade, the B factory
experiments achieved great successes. Furthermore, the current LHC experiments
will provide 2–3 orders more B meson events than the B factories lhcb1 . A
large number of rare $B$ meson decay channels will be measured by the future
super B factories. The research on the charmless decays of $B$ meson is
therefore becoming more interesting than ever before lhcb2 .
The theoretical calculations of color-suppressed decay channels, such as
$B^{0}\to\pi^{0}\pi^{0}$, met a difficulty for a relatively much smaller
branching ratios than the experimental measurements prl831914 ; prd63074009 ;
pdg2010 . The difference between direct CP-asymmetry measurement of $B^{0}\to
K^{+}\pi^{-}$ and $B^{+}\to K^{+}\pi^{0}$ showed a very large discrepancy
between the leading-order (LO) theoretical calculations and experimental data,
which induced a lot of new physics discussions pikpuzzle . One of the standard
model solutions to this puzzle also requires large color-suppressed tree
amplitudes prd114005 . Some of the next-to-leading-order (NLO) QCD
calculations in the perturbative QCD factorization approach (pQCD) prd114005 ;
prd094020 ; 0807 ; prd114001 show that the NLO contributions can
significantly change the LO predictions for some decay modes, especially the
color-suppressed modes. It is therefore necessary to calculate the NLO
corrections to those two-body charmless B meson decays in order to improve the
reliability of the theoretical predictions.
The $B\to\pi\rho$ decays, which are helpful for the determination of the
Cabibbo–Kobayashi–Maskawa(CKM) unitary triangle $\alpha$ angle measurement in
addition to the $B\to\pi\pi$ decays, have a much more complication. Either of
$B^{0}$ or $\bar{B}^{0}$ meson can decay to both the $\pi^{-}\rho^{+}$ and
$\pi^{+}\rho^{-}$ final states, which lead to altogether four decay
amplitudes. Since $B^{0}$ and $\bar{B}^{0}$ meson mix easily, these channels
exhibit unique features of mixing and decay interference in B physics. The
recent B factory measurements indeed show that the interesting phenomenology
with possible large direct CP asymmetry exp . Unlike the branching ratios, the
CP asymmetries are sensitive to high order contributions. Similar to the
color-suppressed $B^{0}\to\pi^{0}\pi^{0}$ mode, the neutral decay modes
$B^{0}\to\pi^{0}\rho^{0}$, $\pi^{0}\omega$ are also expected to receive
considerable NLO contributions. Therefore, it is necessary to calculate NLO
corrections to the $B\to\pi\rho,\pi\omega$ decays in the pQCD approach for the
reason that previous pQCD calculations epjc23275 are already too old with
only LO accuracy. In this paper, we calculate the NLO contributions arising
from the vertex corrections, the quark loops and the chromo-magnetic penguin
operator $O_{8g}$. Combining our results with the NLO accuracy Wilson
coefficients and Sudakov suppression factors, we present a numerical analysis
of $B\to\pi\rho$, $\pi\omega$ decays.
Our paper is organized as follows: we first review the pQCD factorization
approach in Sec. II. Then, in Sec. III, we show our analytical results of NLO
calculations. The numerical results are given in Sec. IV. Finally we close
this paper with a conclusion.
## II Theoretical framework
For the studied $B\to\pi\rho,\pi\omega$ decays, the weak effective Hamiltonian
$\mathcal{H}_{eff}$ for $b\rightarrow d$ transition can be written as
$\displaystyle\mathcal{H}_{eff}=\frac{G_{F}}{\sqrt{2}}[\xi_{u}(C_{1}(\mu)O_{1}^{u}(\mu)+C_{2}(\mu)O_{2}^{u}(\mu))-\xi_{t}\sum_{i=3}^{10}C_{i}(\mu)O_{i}(\mu)]$
(1)
where $\xi_{u}=V_{ub}V^{*}_{ud}$, $\xi_{t}=V_{tb}V^{*}_{td}$ are the CKM
matrix elements. $O_{i}(\mu)$ and $C_{i}(\mu)$ are the four-quark operators
and corresponding Wilson coefficients, respectively. Expressions of $C_{i}$
and $O_{i}$ can be found in Ref.rmp681125 . In the following, we will use this
effective Hamiltonian to calculate decay amplitudes in the pQCD approach. So,
we first give a brief review of pQCD approach and present relevant wave
functions.
### II.1 pQCD factorization approach
In the framework of the pQCD factorization, three scales are involved in the
non-leptonic decays of B mesons: the weak interaction scale $m_{W}$, the hard
subprocess scale $t$ and the transverse momenta of the constituent quark
$k_{T}$. The large logs between W boson mass scale and the hard scale $t$ have
been resummed by the renormalization group equation method to give the
effective Hamiltonian of four-quark operators. In two-body charmless hadronic
B decays, the final state meson masses are negligible compared with the large
B meson mass. Therefore the constituent quarks in the final state mesons are
collinear objects in the rest frame of B meson. The momentum of light quark in
B meson is at the order of $\Lambda_{QCD}$, such that a hard gluon is needed
to transfer energy to make it a collinear quark into the final state meson.
These perturbative calculations meet end-point singularity in dealing with the
meson distribution amplitudes at the end-point. Usually in the collinear
factorization approaches such as QCD factorizationprl831914 and soft-
collinear effective theoryprd63114020 , people parameterize this kind of decay
amplitudes into free parameters to fit the data. While in the perturbative QCD
factorization approach, we take back the parton transverse momentum $k_{T}$ to
regulate this divergence.
In the pQCD approach, the decay amplitude $A(B\rightarrow M_{2}M_{3})$ can be
written conceptually as the convolution prd69094018
$\displaystyle\mathcal{A}(B\rightarrow M_{2}M_{3})=\int
d^{4}k_{1}d^{4}k_{2}d^{4}k_{3}\text{Tr}[C(t)\Phi_{B}(k_{1})\Phi_{M_{2}}(k_{2})\Phi_{M_{3}}(k_{3})H(k_{1},k_{2},k_{3},t)]$
(2)
where $k_{i}$ are momenta of light quarks included in each meson, and Tr
denotes the trace over Dirac and color indices. The hard function
$H(k_{1},k_{2},k_{3},t)$ describes the four-quark operator and the spectator
quark connected by a hard gluon of order $\bar{\Lambda}M_{B}$, which can be
calculated perturbatively. The energy scale $t$ is chosen as the maximal
virtuality of internal particles in a hard amplitude, in order to suppress
higher order correctionsprd074004 . $\Phi_{M_{i}}$ is the wave function of
meson $M_{i}$. The hard kernel $H$ depends on the processes considered, while
the wave functions $\Phi_{M_{i}}$ are process independent that can be
extracted from other well measured processes, so one can make quantitative
predictions here.
It is convenient to work at the B meson rest frame and the light cone
coordinate. The final state meson $M_{2}$ is moving along the direction of
$v=(0,1,\bf 0_{T})$ and $M_{3}$ is along $n=(1,0,\bf 0_{T})$. Here we use
$x_{i}$ to denote the momentum fractions of anti-quarks in mesons, and $\bf
k_{iT}$ to denote the transverse momenta of the anti-quarks. The mass of light
meson ($\pi$) is neglected. After integration over $k_{1}^{-}$, $k_{2}^{-}$
and $k_{3}^{+}$ in Eq.(2), we are led to
$\displaystyle\mathcal{A}$ $\displaystyle=$ $\displaystyle\int
dx_{1}dx_{2}dx_{3}b_{1}db_{1}b_{2}db_{2}b_{3}db_{3}$ (3)
$\displaystyle\text{Tr}[C(t)\Phi_{B}(x_{1},b_{1})\Phi_{M_{2}}(x_{2},b_{2})\Phi_{M_{3}}(x_{3},b_{3})H(x_{i},b_{i},t)S_{t}(x_{i})\exp(-S(t))]$
where $b_{i}$ are the conjugate variables of $\bf k_{iT}$. The jet function
$S_{t}(x_{i})$ arises from the threshold resummation of the large double
logarithms $\ln^{2}(x_{i})$, and the Sudakov exponent $S(t)$ comes from the
double logarithms of collinear and soft divergences.
### II.2 Wave Functions
There are generally two Lorentz structures in the B meson distribution
amplitudes, which can be decomposed as qiaocf
$\displaystyle\int_{0}^{1}\frac{d^{4}z}{(2\pi)^{4}}e^{ik_{1}\cdot z}\langle
0|\bar{b}_{\alpha}(0)d_{\beta}(z)|B(p_{B})\rangle=-\frac{i}{\sqrt{2N_{c}}}[(\hbox
to0.0pt{/\hss}{p_{B}}+m_{B})\gamma_{5}(\phi_{B}(k_{1})-\frac{\hbox
to0.0pt{/\hss}{n}-\hbox to0.0pt{/\hss}{v}}{\sqrt{2}}\bar{\phi}_{B}(k_{1}))].$
(4)
With $N_{c}=3$, they obey the following normalization conditions:
$\displaystyle\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\phi_{B}(k_{1})=\frac{f_{B}}{2\sqrt{2N_{c}}},\quad\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\bar{\phi}_{B}(k_{1})=0.$
(5)
However, the contribution of $\bar{\phi}_{B}$ is numerically
neglectedepjc28515 . Therefore, we will only consider the contributions from
$\phi_{B}$. In b space the B meson wave function can be expressed by
$\displaystyle\Phi_{B}(x,b)=\frac{1}{\sqrt{2N_{c}}}(\hbox
to0.0pt{/\hss}{P_{B}}+m_{B})\gamma_{5}\phi_{B}(x,b).$ (6)
For the light pseudo-scalar mesons $\pi$, the wave function can be defined as
zpc48239
$\displaystyle\Phi(P,x,\xi)=\frac{i}{\sqrt{2N_{c}}}\gamma_{5}[\hbox
to0.0pt{/\hss}{P}\phi^{A}(x)+m_{0}\phi^{P}(x)+\xi m_{0}(\hbox
to0.0pt{/\hss}{n}\hbox to0.0pt{/\hss}{v}-1)\phi^{T}(x)],$ (7)
where $P$ is the momentum of the light meson $\pi$, $m_{0}$ is the chiral mass
which is defined using the meson mass $m_{P}$ and the quark masses as
$m_{0}=m^{2}_{P}/(m_{q_{1}}+m_{q_{2}})$. $x$ is the momentum fraction of the
quark (or anti-quark) inside the meson, respectively. When the momentum
fraction of the quark (anti-quark) is set to be $x$, the parameter $\xi$
should be chosen as $+1$$(-1)$.
For the considered decays, the vector meson $V(\rho,\omega)$ is longitudinally
polarized. The longitudinal polarized component of the wave function is
defined as:
$\displaystyle\Phi_{V}=\frac{1}{\sqrt{2N_{c}}}[\hbox
to0.0pt{/\hss}{\epsilon}(m_{V}\phi_{V}(x)+\hbox
to0.0pt{/\hss}{P_{V}}\phi_{V}^{t}(x))+m_{V}\phi_{V}^{s}(x)],$ (8)
where the polarization vector $\epsilon$ satisfies $P_{V}\cdot\epsilon=0$.
## III Analytical calculations
Figure 1: NLO corrections to the hard kernels. The diagrams (a–f), (g,h) and
(i,j) are commonly called vertex corrections, quark-loop corrections, and
chromo-magnetic penguin corrections, respectively.
.
Our NLO corrections for pQCD approach include the following parts:
* •
The NLO hard kernel $H^{(1)}(x_{i},b_{i},t)$, which includes the vertex
corrections, the quark loops and chromo-magnetic penguins.
* •
The NLO Wilson coefficients $C^{NLO}(t)$, which have been calculated in the
literature rmp681125 .
* •
The exponential Sudakov factor $\exp[-S^{NLO}(t)]$ includes the Sudakov factor
$s(P,b)$ and renormalization group running factor $g_{2}(t,b)$.
So, at the NLO, Eq.(3) can be written as
$\displaystyle\mathcal{A}^{NLO}$ $\displaystyle=$ $\displaystyle\int
dx_{1}dx_{2}dx_{3}b_{1}db_{1}b_{2}db_{2}b_{3}db_{3}\text{Tr}[C^{NLO}(t)\Phi_{B}(x_{1},b_{1})\Phi_{M_{2}}(x_{2},b_{2})\Phi_{M_{3}}(x_{3},b_{3})$
(9)
$\displaystyle(H^{(0)}(x_{i},b_{i},t)+H^{(1)}(x_{i},b_{i},t))S_{t}(x_{i})\exp(-S^{NLO}(t))].$
We will give these calculations in the following of this section.
### III.1 Vertex corrections
The vertex corrections are part of the complete NLO Wilson coefficients for
four-quark operators, which cancel the explicit renormalization scale $\mu$
dependence of the Wilson coefficients. The vertex correction diagrams are
illustrated by Figs.1(a)–1(f), among which Fig.(e) and (f) are new compared to
the QCDF calculation prl831914 . Here, we have introduced transverse momentum
$k_{T}$ in regularizing the infrared divergence. Our results are different
from the QCDF approachprl831914 for different regularization schemes.
The vertex corrections to the $B\rightarrow\pi\rho,\pi\omega$ decays modify
the Wilson coefficients for the emission amplitudes into
$\displaystyle a_{1}(\mu)$ $\displaystyle\rightarrow$ $\displaystyle
a_{1}(\mu)+\frac{\alpha_{s}(\mu)}{4\pi}C_{F}[\frac{C_{1}(\mu)}{N_{c}}V_{1}(M)+C_{2}(\mu)V^{\prime}_{1}(M)],$
$\displaystyle a_{2}(\mu)$ $\displaystyle\rightarrow$ $\displaystyle
a_{2}(\mu)+\frac{\alpha_{s}(\mu)}{4\pi}C_{F}[\frac{C_{2}(\mu)}{N_{c}}V_{2}(M)+C_{1}(\mu)V^{\prime}_{2}(M)],$
$\displaystyle a_{i}(\mu)$ $\displaystyle\rightarrow$ $\displaystyle
a_{i}(\mu)+\frac{\alpha_{s}(\mu)}{4\pi}C_{F}[\frac{C_{i\pm
1}(\mu)}{N_{c}}V_{i}(M)+C_{i}(\mu)V^{\prime}_{i}(M)],\quad i=3-10,$ (10)
where $M$ denotes the meson emitted from the weak vertex, and the upper
(lower) sign applies for odd (even) $i$. When the emitted meson $M$ is a
pseudo-scalar meson, the functions $V_{i}(M)$ and $V^{\prime}_{i}(M)$ are
given by
$\displaystyle V_{i}(M)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{lll}8\ln\frac{m_{b}}{\mu}-18+\frac{\pi^{2}}{3}-6i\pi+\frac{2\sqrt{2N_{c}}}{f_{M}}\int_{0}^{1}dx\phi_{M}^{A}(x)g_{1}(x),&\quad
for\quad$i=1-4,9,10$\\\
-16\ln\frac{m_{b}}{\mu}+6+\frac{\pi^{2}}{3}+\frac{2\sqrt{2N_{c}}}{f_{M}}\int_{0}^{1}dx\phi_{M}^{A}(x)g_{2}(x),&\quad
for\quad$i=5,7$\\\
-16\ln\frac{m_{b}}{\mu}+6+\frac{\pi^{2}}{3}+\frac{2\sqrt{2N_{c}}}{f_{M}}\int_{0}^{1}dx\phi_{M}^{P}(x)h(x)&\quad
for\quad$i=6,8$\end{array}\right.$ (14) $\displaystyle V^{\prime}_{i}(M)$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{lll}-4\ln\frac{m_{b}}{\mu}+\frac{\pi^{2}}{3}-3i\pi,&\quad
for\quad$i=1-4,5,7,9,10$\\\ -16\ln\frac{m_{b}}{\mu}+12+\frac{\pi^{2}}{3}&\quad
for\quad$i=6,8$\end{array}\right.$ (17)
where $m_{b}$ is the mass of b quark. The functions $g_{1}(x)$, $g_{2}(x)$ and
$h(x)$ are given as
$\displaystyle g_{1}(x)$ $\displaystyle=$ $\displaystyle 3\ln
x+3\ln(1-x)+\frac{\ln(1-x)}{x}-2\frac{\ln x}{1-x}$
$\displaystyle+[\ln^{2}x+2\text{Li}_{2}(\frac{x}{x-1})+4i\pi\ln
x-(x\rightarrow 1-x)],$ $\displaystyle g_{2}(x)$ $\displaystyle=$
$\displaystyle-3\ln x-3\ln(1-x)+2\frac{\ln(1-x)}{x}+\frac{\ln x}{1-x}$
$\displaystyle+[\ln^{2}x+2\text{Li}_{2}(\frac{x}{x-1})+4i\pi\ln
x-(x\rightarrow 1-x)],$ $\displaystyle h(x)$ $\displaystyle=$
$\displaystyle\ln^{2}x+2\text{Li}_{2}(\frac{x}{x-1})+\frac{\ln
x}{2(1-x)}+4i\pi\ln x-(x\rightarrow 1-x).$ (18)
When a vector meson $V(V=\rho,\omega)$ is emitted from the weak vertex,
$\phi_{M}^{A}(\phi_{M}^{P})$ is replaced by $\phi_{V}(-\phi_{V}^{s})$, and
$f_{M}$ by $f_{V}^{T}$ in the third line of Eq.(14). Note that, the amplitude
$F^{P}_{e\pi}$ from the operators $O_{5-8}$ vanishes at LO, because neither
the scalar nor the pseudo-scalar density gives contributions to the vector
meson production, i.e. $<V|S+P|0>=0$. On including the vertex corrections, the
NLO piece $a_{VC}$, containing the vertex-correction of $a_{6,8}$ in
Eq.(III.1), contributes through the following additional amplitudesprd094020 :
$\displaystyle f_{V}F^{P}_{e\pi}\rightarrow
a_{VC}f^{T}_{V}F^{P}_{e\pi}+f_{V}F_{e\pi}$ (19)
where $F_{e\pi}$ is the decay amplitude of factorizable emission diagrams with
the structure of $(V-A)\otimes(V-A)$ insertion; while $F^{P}_{e\pi}$ is the
corresponding decay amplitude with $(S-P)\otimes(S+P)$insertion.
### III.2 Quark loops
The contributions from the quark loops are illustrated by Fig.1(g)-1(h). The
quark-loop contributions are generally called the Bander–Silver–Soni mechanism
prl43242 , which plays a very important role in producing the direct CP-
violation strong phase in the QCDF/SCET approaches. We include quark-loop
amplitudes from the up-, charm-, and QCD-penguin-loop corrections, the quark
loops from the electroweak penguins are neglected due to their smallness.
For the $b\rightarrow d$ transition, the contributions from the various quark
loops are described by the effective Hamiltonian
$\mathcal{H}^{(ql)}_{eff}$prd114005 ,
$\displaystyle\mathcal{H}^{(ql)}_{eff}$ $\displaystyle=$
$\displaystyle-\sum_{q=u,c}\sum_{q^{\prime}}\frac{G_{F}}{\sqrt{2}}V_{qb}V^{*}_{qd}\frac{\alpha_{s}(\mu)}{2\pi}C^{(q)}(\mu,l^{2})(\bar{d}\gamma_{\rho}(1-\gamma_{5})T^{a}b)(\bar{q^{\prime}}\gamma^{\rho}T^{a}q^{\prime})$
(20)
$\displaystyle+\sum_{q^{\prime}}\frac{G_{F}}{\sqrt{2}}V_{tb}V^{*}_{td}\frac{\alpha_{s}(\mu)}{2\pi}C^{(t)}(\mu,l^{2})(\bar{d}\gamma_{\rho}(1-\gamma_{5})T^{a}b)(\bar{q^{\prime}}\gamma^{\rho}T^{a}q^{\prime}),$
with
$\displaystyle C^{(q)}(\mu,l^{2})$ $\displaystyle=$
$\displaystyle[G^{(q)}(\mu,l^{2})-\frac{2}{3}]C_{2}(\mu),$ $\displaystyle
C^{(t)}(\mu,l^{2})$ $\displaystyle=$
$\displaystyle[G^{(s)}(\mu,l^{2})-\frac{2}{3}]C_{3}(\mu)+\sum_{q^{\prime\prime}=u,d,s,c}G^{(q^{\prime\prime})}(\mu,l^{2})[C_{4}(\mu)+C_{6}(\mu)],$
(21)
where $l^{2}$ being the invariant mass of the intermediate gluon, which
connects the quark loops with the $\bar{q^{\prime}}q$ pair. Because of the
absence of the end-point singularities associated with $l^{2},l^{\prime
2}\rightarrow 0$, we have dropped the parton transverse momenta $k_{T}$ in
$l^{2},l^{\prime 2}$ for simplicity. The integration function
$G^{(q)}(\mu,l^{2})$ for the loop of the quarks $q=(u,d,s,c)$ is defined as
$\displaystyle
G^{(q)}(\mu,l^{2})=-4\int_{0}^{1}dxx(1-x)\ln\frac{m_{q}^{2}-x(1-x)l^{2}}{\mu^{2}}.$
(22)
Finally, the quark-loop contributions shown in Fig.1(g) and 1(h) to the
considered $B\rightarrow\pi V$ decays with $V=\rho,\omega$ can be written as
$\displaystyle\mathcal{A}^{(ql)}_{V\pi}=\langle
V\pi|\mathcal{H}^{(ql)}_{eff}|\bar{B}\rangle=\sum_{q=u,c,t}\xi^{*}_{q}[\mathcal{M}^{(q)}_{V\pi}+\mathcal{M}^{(q)}_{\pi
V}].$ (23)
The two kinds of topological decay amplitude of the $B\rightarrow V$ or
$B\rightarrow\pi$ transition are written as
$\displaystyle\mathcal{M}^{ql}_{V\pi}$ $\displaystyle=$
$\displaystyle\frac{4}{\sqrt{3}}C_{F}^{2}m_{B}^{4}\int_{0}^{1}dx_{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{1}db_{1}b_{2}db_{2}\phi_{B}(x_{1},b_{1})$
(24)
$\displaystyle\times\left\\{[-(1+x_{2})\phi_{V}(x_{2})\phi^{A}_{\pi}(x_{3})+2r_{\pi}\phi_{V}(x_{2})\phi^{P}_{\pi}(x_{3})\right.$
$\displaystyle\left.-(1-2x_{2})r_{V}\phi^{A}_{\pi}(x_{3})(\phi^{s}_{V}(x_{2})+\phi^{t}_{V}(x_{2}))\right.$
$\displaystyle\left.+2(2+x_{2})r_{\pi}r_{V}\phi^{s}_{V}(x_{2})\phi^{P}_{\pi}(x_{3})-2x_{2}r_{\pi}r_{V}\phi^{t}_{V}(x_{2})\phi^{P}_{\pi}(x_{3})]\right.$
$\displaystyle\left.\times\alpha_{s}^{2}(t_{1})h_{ql}(x_{1},x_{2},b_{1},b_{2})C^{(q)}(t_{1},l^{2})\exp[-S_{ql}(t_{1})]\right.$
$\displaystyle\left.+2r_{V}(2r_{\pi}\phi^{P}_{\pi}(x_{3})-\phi^{A}_{\pi}(x_{3}))\phi^{s}_{V}(x_{2})\right.$
$\displaystyle\left.\times\alpha_{s}^{2}(t_{2})h_{ql}(x_{2},x_{1},b_{2},b_{1})C^{(q)}(t_{2},l^{2})\exp[-S_{ql}(t_{2})]\right\\},$
$\displaystyle\mathcal{M}^{ql}_{\pi V}$ $\displaystyle=$
$\displaystyle\frac{4}{\sqrt{3}}C_{F}^{2}m_{B}^{4}\int_{0}^{1}dx_{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{1}db_{1}b_{2}db_{2}\phi_{B}(x_{1},b_{1})$
(25)
$\displaystyle\left\\{[(1+x_{2})\phi^{A}_{\pi}(x_{2})\phi_{V}(x_{3})+(1-2x_{2})r_{\pi}(\phi^{P}_{\pi}(x_{2})+\phi^{T}_{\pi}(x_{2}))\phi_{V}(x_{3})\right.$
$\displaystyle\left.2r_{V}r_{\pi}(2+x_{2})\phi_{\pi}^{P}(x_{2})\phi_{V}^{s}(x_{3})-2r_{V}x_{2}\phi_{\pi}^{T}(x_{2})\phi_{V}^{s}(x_{3})\right.$
$\displaystyle\left.+2r_{V}\phi_{\pi}^{A}(x_{2})\phi_{V}^{s}(x_{3})]\times\alpha_{s}^{2}(t_{1})h_{ql}(x_{1},x_{2},b_{1},b_{2})C^{(q)}(t_{1},l^{2})\exp[-S_{ql}(t_{1})]\right.$
$\displaystyle\left.+2r_{\pi}[\phi^{P}_{\pi}(x_{2})\phi^{s}_{V}(x_{3})+2\phi^{P}_{\pi}(x_{2})\phi^{s}_{V}(x_{3})]\right.$
$\displaystyle\left.\times\alpha_{s}^{2}(t_{2})h_{ql}(x_{2},x_{1},b_{2},b_{1})C^{(q)}(t_{2},l^{2})\exp[-S_{ql}(t_{2})]\right\\},$
where the ratios $r_{V}=m_{V}/m_{B},r_{\pi}=m_{0}^{\pi}/m_{B}$. The hard
scales and the gluon invariant masses are given by
$\displaystyle t_{1}$ $\displaystyle=$
$\displaystyle\max(\sqrt{x_{2}}m_{B},\sqrt{x_{1}x_{2}}m_{B},\sqrt{x_{3}(1-x_{2})}m_{B},1/b_{1},1/b_{2}),$
$\displaystyle t_{2}$ $\displaystyle=$
$\displaystyle\max(\sqrt{x_{1}}m_{B},\sqrt{x_{1}x_{2}}m_{B},\sqrt{|x_{3}-x_{1}|}m_{B},1/b_{1},1/b_{2}),$
$\displaystyle l^{2}$ $\displaystyle=$ $\displaystyle
x_{3}(1-x_{2})m_{B}^{2},\quad l^{\prime 2}=(x_{3}-x_{1})m_{B}^{2}.$ (26)
The hard functions $h_{ql}$ are included in the appendix.
### III.3 Chromo-magnetic penguins
The chromo-magnetic penguin contributions are of NLO in $\alpha_{s}$ within
the pQCD formalism. They are at the same order in $\alpha_{s}$ as the penguin
contributions.
According to ref.ptp110549 , there are ten chromo-magnetic penguin diagrams
contributing to the $B$ decays, but only two of them are important, as
illustrated by Fig. 1(i)and 1(j), while the other eight diagrams are
negligible. The corresponding weak effective Hamiltonian contains the
$b\rightarrow dg$ transition:
$\displaystyle\mathcal{H}^{(mg)}_{eff}=-\frac{G_{F}}{\sqrt{2}}\xi^{*}_{t}C^{eff}_{8g}O_{8g},$
(27)
with
$\displaystyle
O_{8g}=\frac{g}{8\pi^{2}}m_{b}\bar{d}_{i}\sigma_{\mu\nu}(1+\gamma_{5})T^{a}_{ij}G^{a\mu\nu}b_{j}$
(28)
where $i,j$ being the color indices of quarks. The corresponding effective
Wilson coefficient $C^{eff}_{8g}=C_{8g}+C_{5}$prd114005 .
The decay amplitudes of Fig.1(i) and 1(j) can be written as
$\displaystyle\mathcal{M}^{(mg)}_{V\pi}$ $\displaystyle=$
$\displaystyle\frac{4}{\sqrt{3}}C_{F}^{2}m_{B}^{6}\int_{0}^{1}dx_{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{1}db_{1}b_{2}db_{2}b_{3}db_{3}\phi_{B}(x_{1},b_{1})$
(29)
$\displaystyle\times\left\\{[(1-x_{2})\phi^{A}_{\pi}(x_{3})[2\phi_{V}(x_{2})+r_{V}(3\phi^{s}_{V}(x_{2})+\phi^{t}_{V}(x_{2}))\right.$
$\displaystyle\left.+x_{2}r_{V}(\phi^{s}_{V}(x_{2})-\phi^{t}_{V}(x_{2}))]-r_{\pi}x_{3}(1+x_{2})(3\phi^{P}_{\pi}(x_{3})-\phi^{T}_{\pi}(x_{3}))\phi_{V}(x_{2})\right.$
$\displaystyle\left.-r_{\pi}r_{V}(1-x_{2})(\phi^{s}_{V}(x_{2})-\phi^{t}_{V}(x_{2}))(3\phi^{P}_{\pi}(x_{3})+\phi^{T}_{\pi}(x_{3}))\right.$
$\displaystyle\left.+r_{\pi}r_{V}x_{3}(1-2x_{2})(\phi^{s}_{V}(x_{2})+\phi^{t}_{V}(x_{2}))(\phi^{T}_{\pi}(x_{3})-3\phi^{P}_{\pi}(x_{3}))]\right.$
$\displaystyle\left.\times
C^{eff}_{8g}\alpha_{s}^{2}(t_{1})h_{mg}(A,B,C,b_{1},b_{2},b_{3})S_{t}(x_{2})\exp[-S_{mg}]\right.$
$\displaystyle\left.+2r_{V}[2\phi^{A}_{\pi}(x_{3})+x_{3}r_{\pi}(\phi^{T}_{\pi}(x_{3})-3\phi^{P}_{\pi}(x_{3})]\phi^{s}_{V}(x_{2})\right.$
$\displaystyle\left.\times
C^{eff}_{8g}\alpha_{s}^{2}(t_{2})h_{mg}(A^{\prime},B^{\prime},C^{\prime},b_{2},b_{1},b_{3})S_{t}(x_{1})\exp[-S_{mg}]\right\\},$
$\displaystyle\mathcal{M}^{(mg)}_{\pi V}$ $\displaystyle=$
$\displaystyle\frac{4}{\sqrt{3}}C_{F}^{2}m_{B}^{6}\int_{0}^{1}dx_{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{1}db_{1}b_{2}db_{2}b_{3}db_{3}\phi_{B}(x_{1},b_{1})$
(30)
$\displaystyle\times\left\\{[(1-x_{2})\phi_{V}(x_{3})[2\phi^{A}_{\pi}(x_{2})+r_{\pi}(3\phi^{P}_{\pi}(x_{2})+\phi^{T}_{\pi}(x_{2}))\right.$
$\displaystyle\left.+x_{2}r_{\pi}(\phi^{P}_{\pi}(x_{2})-\phi^{T}_{V}(x_{2}))]+r_{V}x_{3}(1+x_{2})(3\phi^{s}_{V}(x_{3})-\phi^{t}_{V}(x_{3}))\phi^{A}_{\pi}(x_{2})\right.$
$\displaystyle\left.+r_{\pi}r_{V}(1-x_{2})(3\phi^{s}_{V}(x_{3})+\phi^{t}_{V}(x_{3}))(\phi^{P}_{\pi}(x_{2})-\phi^{T}_{\pi}(x_{2}))\right.$
$\displaystyle\left.+r_{\pi}r_{V}x_{3}(1-2x_{2})(3\phi^{s}_{V}(x_{3})-\phi^{t}_{V}(x_{3}))(\phi^{T}_{\pi}(x_{2})+\phi^{P}_{\pi}(x_{2}))]\right.$
$\displaystyle\left.\times
C^{eff}_{8g}\alpha_{s}^{2}(t_{1})h_{mg}(A,B,C,b_{1},b_{2},b_{3})S_{t}(x_{2})\exp[-S_{mg}]\right.$
$\displaystyle\left.+2r_{\pi}[2\phi_{V}(x_{3})-x_{3}r_{V}(\phi^{t}_{V}(x_{3})-3\phi^{s}_{V}(x_{3})]\phi^{P}_{\pi}(x_{2})\right.$
$\displaystyle\left.\times
C^{eff}_{8g}\alpha_{s}^{2}(t_{2})h_{mg}(A^{\prime},B^{\prime},C^{\prime},b_{2},b_{1},b_{3})S_{t}(x_{1})\exp[-S_{mg}]\right\\},$
where
$\displaystyle A$ $\displaystyle=$ $\displaystyle\sqrt{x_{2}}m_{b},\quad
B=B^{\prime}=\sqrt{x_{1}x_{2}}m_{B},\quad C=\sqrt{x_{3}(1-x_{2})}m_{B},$
$\displaystyle A^{\prime}$ $\displaystyle=$
$\displaystyle\sqrt{x_{1}}m_{b},\quad C^{\prime}=\sqrt{|x_{1}-x_{3}|}m_{B}.$
(31)
The hard scales $t_{1},t_{2}$ are the same as in Eq.(III.2). The hard function
$h_{mg}$ and the Sudakov exponent $S_{mg}$ are given in the appendix. The jet
function $S_{t}(x_{i})$ can be found in Ref.0105003 .
## IV Numerical results and discussions
Besides those specified in the text, the following input parameters will also
be used in the numerical calculationspdg2010 :
$\displaystyle m_{B}$ $\displaystyle=$ $\displaystyle
5.28\text{GeV},\quad\tau_{B^{0}}=1.53\text{ps},\quad\tau_{B^{+}}=1.638\text{ps},\quad$
$\displaystyle f_{B}$ $\displaystyle=$ $\displaystyle 0.21\pm
0.01\text{GeV},\quad|V_{ub}|=(3.47^{+0.16}_{-0.12})\times
10^{-3},\quad|V_{ud}|=0.97428,$ $\displaystyle|V_{tb}|$ $\displaystyle=$
$\displaystyle 0.999,\quad|V_{td}|=(8.62^{+0.26}_{-0.20})\times
10^{-3},\quad\alpha=(90\pm 5)^{\circ}.$ (32)
The corresponding values of $\Lambda_{\text{QCD}}$ are derived from
$\alpha_{s}(m_{Z})=0.1184$ using LO and NLO formulas, respectively:
$\displaystyle\text{LO}:\quad\Lambda_{\text{QCD}}^{(5)}$ $\displaystyle=$
$\displaystyle(0.110\pm
0.005)\text{GeV},\quad\Lambda_{\text{QCD}}^{(4)}=0.148\text{GeV};$
$\displaystyle\text{NLO}:\quad\Lambda_{\text{QCD}}^{(5)}$ $\displaystyle=$
$\displaystyle(0.228\pm
0.008)\text{GeV},\quad\Lambda_{\text{QCD}}^{(4)}=0.325\text{GeV}.$ (33)
The $B$ meson distribution amplitude is given by
$\displaystyle\phi_{B}(x,b)=N_{B}x^{2}(1-x)^{2}\exp[-\frac{M_{B}^{2}x^{2}}{2\omega_{b}^{2}}-\frac{1}{2}(\omega_{b}b)^{2}],$
(34)
where the shape parameter $\omega_{b}=0.40\pm 0.04$GeV has been fixed using
the rich experimental data on the $B^{0}_{d}$ and $B^{\pm}$ decaysprd63074009
; prd014019 ; prd074018 .
For the $\pi$ meson, the twist-2 distribution amplitude $\phi^{A}(x)$, and the
twist-3 distribution amplitudes $\phi^{P}(x)$ and $\phi^{T}(x)$ are written as
epjc23275
$\displaystyle\phi^{A}_{\pi}(x)$ $\displaystyle=$
$\displaystyle\frac{3f_{\pi}}{\sqrt{2N_{c}}}x(1-x)[1+a_{2}^{\pi}C^{3/2}_{2}(2x-1)+0.25C^{3/2}_{4}(2x-1)],$
$\displaystyle\phi^{P}_{\pi}(x)$ $\displaystyle=$
$\displaystyle\frac{f_{\pi}}{2\sqrt{2N_{c}}}[1+0.43C^{1/2}_{2}(2x-1)+0.09C^{1/2}_{4}(2x-1)],$
$\displaystyle\phi^{T}_{\pi}(x)$ $\displaystyle=$
$\displaystyle\frac{f_{\pi}}{2\sqrt{2N_{c}}}(1-2x)[1+0.55(10x^{2}-10x+1)]$
(35)
with the pion decay constant $f_{\pi}=0.13\text{GeV}$. The Gegenbauer
polynomials are defined by
$\displaystyle C^{1/2}_{2}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(3t^{2}-1),\quad
C^{1/2}_{4}(t)=\frac{1}{8}(35t^{4}-30t^{2}+3),$ $\displaystyle C^{3/2}_{2}(t)$
$\displaystyle=$ $\displaystyle\frac{3}{2}(5t^{2}-1),\quad
C^{3/2}_{4}(t)=\frac{15}{8}(21t^{4}-14t^{2}+1)$ (36)
whose coefficients correspond to $m^{\pi}_{0}=1.4\text{GeV}$.
The distribution amplitudes for the vector meson are listed below epjc23275 :
$\displaystyle\phi_{V}(x)$ $\displaystyle=$
$\displaystyle\frac{3}{\sqrt{6}}f_{V}x(1-x)[1+a_{V}^{\parallel}C^{3/2}_{2}(2x-1)],$
$\displaystyle\phi_{V}^{t}(x)$ $\displaystyle=$
$\displaystyle\frac{f^{T}_{V}}{2\sqrt{6}}[3(2x-1)^{2}+0.3(2x-1)^{2}(5(2x-1)^{2}-3)$
$\displaystyle+0.21(3-30(2x-1)^{2}+35(2x-1)^{4})],$
$\displaystyle\phi_{V}^{s}(x)$ $\displaystyle=$
$\displaystyle\frac{3}{2\sqrt{6}}f^{T}_{V}(1-2x)[1+0.76(10x^{2}-10x+1)],$ (37)
with the decay constant $f_{\rho}=0.216\text{GeV}$,
$f^{T}_{\rho}=0.165\text{GeV}$, $f_{\omega}=0.195\text{GeV}$ and
$f^{T}_{\omega}=0.145\text{GeV}$prd094020 .
### IV.1 Branching Ratios
The considered NLO contributions can interfere with the LO part constructively
or destructively for different decay modes. In Table 1, we show our pQCD
results for the CP-averaged branching ratios of the seven
$B\rightarrow\pi\rho,\pi\omega$ decays together with the experimental data. In
order to show the effects of the improvement, we use the same updated input
paraments for the LO and NLO calculations, which make the LO-pQCD predictions
larger than the previous pQCD calculations epjc23275 . Apparently, most of the
NLO-pQCD predictions agree with the experimental measured values and better
than the LO results.
For comparison, we also list theoretical predictions based on the traditional
QCD factorization approach (QCDF-I) npb675333 , modified QCD factorization
approach (QCDF-II) 09095229 which include the fitted penguin annihilation
topology and color-suppressed tree amplitudes, and the ones obtained using
SCET 0801 . Comparing with the experimental data pdg2010 , it is easy to see
that the LO-pQCD predictions are worse than the QCDF results, but our NLO-pQCD
results have a better agreement with the experimental data. Our NLO
predictions of the branching ratios for $B\rightarrow\pi^{\pm}\rho^{\mp}$
decays are close to QCDF-II result but larger than those in SCET. Neglecting
the small terms, it is due to the different $B\rightarrow\pi$ and
$B\rightarrow\rho$ form factors: SCET uses the smaller form factors
$F^{B\rightarrow\pi}=0.198$ and $A_{0}^{B\rightarrow\rho}=0.291$; while in our
NLO calculations, $F^{B\rightarrow\pi}=0.23$ and
$A_{0}^{B\rightarrow\rho}=0.30$.
Table 1: Branching ratios $(\times 10^{-6})$ of
$B\rightarrow\pi\rho,\pi\omega$ decays in the pQCD approach, together with
results from the QCDF-I npb675333 , QCDF-II09095229 , the ones obtained from
one solution of SCET 0801 and the experimental data pdg2010 .
Mode | LO-pQCD | NLO-pQCD | QCDF-I npb675333 | QCDF-II 09095229 | SCET 0801 | Data pdg2010
---|---|---|---|---|---|---
$B^{0}/\bar{B}^{0}\rightarrow\pi^{\pm}\rho^{\mp}$ | 41.3 | $25.7^{+7.0+2.4+1.3+1.8}_{-5.5-1.9-2.0-1.6}$ | $36.5^{+18.2+10.3+2.0+3.9}_{-14.7-8.6-3.5-2.9}$ | $25.1^{+1.5+1.4}_{-2.2-1.8}$ | $16.8^{+0.5+1.6}_{-0.5-1.5}$ | $23\pm 2.3$
$B^{+}\rightarrow\pi^{+}\rho^{0}$ | 9.0 | $5.4^{+1.4+0.5+0.6+0.3}_{-1.1-0.3-0.5-0.0}$ | $11.9^{+6.3+3.6+2.5+1.3}_{-5.0-3.1-1.2-1.1}$ | $8.7^{+2.7+1.7}_{-1.3-1.4}$ | $7.9^{+0.2+0.8}_{-0.1-0.8}$ | $8.3\pm 1.2$
$B^{+}\rightarrow\rho^{+}\pi^{0}$ | 14.1 | $9.6_{-2.1-0.7-1.3-0.6}^{+2.5+0.8+0.7+0.7}$ | $14.0^{+6.5+5.1+1.0+0.8}_{-5.5-4.3-0.6-0.7}$ | $11.8^{+1.8+1.4}_{-1.1-1.4}$ | $11.4^{+0.6+1.1}_{-0.6-0.9}$ | $10.9\pm 1.4$
$B^{0}\rightarrow\rho^{0}\pi^{0}$ | 0.15 | $0.37^{+0.09+0.02+0.03+0.08}_{-0.08-0.01-0.05-0.02}$ | $0.4^{+0.2+0.2+0.9+0.5}_{-0.2-0.1-0.3-0.3}$ | $1.3^{+1.7+1.2}_{-0.6-0.6}$ | $1.5^{+0.1+0.1}_{-0.1-0.1}$ | $2.0\pm 0.5$
$B^{+}\rightarrow\pi^{+}\omega$ | 8.4 | $4.6^{+1.2+0.5+0.5+0.1}_{-0.9-0.4-0.4-0.1}$ | $8.8^{+4.4+2.6+1.8+0.8}_{-3.5-2.2-0.9-0.9}$ | $6.7^{+2.1+1.3}_{-1.0-1.1}$ | $8.5^{+0.3+0.8}_{-0.3-0.8}$ | $6.9\pm 0.5$
$B^{0}\rightarrow\pi^{0}\omega$ | 0.22 | $0.32^{+0.06+0.01+0.04+0.04}_{-0.05-0.02-0.07-0.04}$ | $0.01^{+0.00+0.02+0.02+0.03}_{-0.00-0.00-0.00-0.00}$ | $0.01^{+0.02+0.04}_{-0.00-0.01}$ | $0.015^{+0.024+0.002}_{-0.000-0.002}$ | $<0.5$
For the color-suppressed tree dominant mode $B^{0}\rightarrow\pi^{0}\rho^{0}$,
the NLO pQCD contributions enhance its branching ratio by a factor 2.5, which
are helpful to pin down the gap between the pQCD calculations and the
experimental data. This NLO $\mathcal{BR}(B^{0}\rightarrow\pi^{0}\rho^{0})$ is
comparable with the result of QCDF-I, but still smaller than QCDF-II and SCET
results and the experimental data. Soft corrections to $a_{2}$ enhance the
QCDF-II predictions, while in the SCET framework, the hard-scattering form
factor $\zeta_{J}$ is fitted to be relatively large and comparable with the
soft form factor $\zeta$. In a very recent paper prd034023 , the authors show
the existence of residual infrared divergences caused by Glauber gluons in
non-factorizable emission diagrams, which may resolve the large discrepancy
between the theoretical predictions on
$\mathcal{BR}(B^{0}\rightarrow\pi^{0}\rho^{0})$ and the data. For another
color-suppressed tree dominant mode $B^{0}\rightarrow\pi^{0}\omega$, our pQCD
prediction is comparable with the $B^{0}\rightarrow\pi^{0}\rho^{0}$ mode;
while both QCDF and SCET predictions for this mode are less than
$B^{0}\rightarrow\pi^{0}\rho^{0}$ results. This should be clarified by future
experiments.
The theoretical uncertainties of the NLO-pQCD predictions are also shown in
Table 1. The first error comes from the B meson wave function parameters
$\omega_{b}=0.40\pm 0.04$ and $f_{B}=0.21\pm 0.01\text{GeV}$; the second error
arises from the uncertainties of the CKM matrix elements
$|V_{ub}|=(3.47^{+0.16}_{-0.12})\times 10^{-3}$,
$|V_{td}|=(8.62^{+0.26}_{-0.20})\times 10^{-3}$ and the CKM angles
$\alpha=(90\pm 5)^{\circ}$; the third error comes from the uncertainties of
final state meson wave function parameters $a_{2}^{\pi}=0.44^{+0.1}_{-0.2}$,
$a_{\rho}^{\parallel}=0.18\pm 0.1$ zpc48239 ; the fourth error is from the
hard scale $t$ varying from $0.75t$ to $1.25t$ and
$\Lambda^{(5)}_{QCD}=0.228^{+0.008}_{-0.009}\text{GeV}$, which characterizes
the uncertainty of higher order contributions. It is easy to see that the most
important uncertainty in our approach comes from the B meson wave function and
CKM elements $V_{ub}$. The total theoretical error is in general around $30\%$
to $50\%$ in size, which is smaller than the previous leading-order
calculation.
Since both tree and penguin diagrams contribute to these decays, the decay
amplitude for a given decay mode with $\bar{b}\rightarrow\bar{d}$ transition
can be parameterized using CKM unitarity as
$\displaystyle\mathcal{A}=\xi^{*}_{u}T-\xi^{*}_{t}P=\xi^{*}_{u}T[1+ze^{i(\alpha+\delta)}],$
(38)
where the parameter $z=|\xi_{t}/\xi_{u}||P/T|$, the weak phase
$\alpha=\arg[-\xi_{t}/\xi_{u}]$, and $\delta=\arg[P/T]$ is the relative strong
phase between T and P part. The corresponding charge conjugate decay mode is
then
$\displaystyle\mathcal{\overline{A}}=\xi_{u}T-\xi_{t}P=\xi_{u}T[1+ze^{i(-\alpha+\delta)}].$
(39)
The CP-averaged branching ratio is
$\displaystyle\mathcal{B}r(B\rightarrow\pi\rho)=\frac{\tau_{B}}{16\pi
m_{B}}\frac{|\mathcal{A}|^{2}+|\mathcal{\overline{A}}|^{2}}{2}=\frac{\tau_{B}}{16\pi
m_{B}}|\xi_{u}T|^{2}[1+2z\cos\alpha\cos\delta+z^{2}],$ (40)
which shows a clear CKM angle $\alpha$ dependence. This potentially gives a
way to measure the CKM angle $\alpha$ by these decays, if we can really pin
down the large theoretical uncertainties of the branching ratio calculations.
For illustration, we show the LO and NLO results of
$Br(B^{0}/\bar{B}^{0}\rightarrow\pi^{\pm}\rho^{\mp})$ in Fig 2 as a function
of $\alpha$ with the hard scales varied from $0.75t$ to $1.25t$. We observe
that the scale dependence of the NLO branching ratio is significantly smaller
than that of the LO branching ratio, roughly from $\approx 50\%$, reduced to
less than $10\%$.
Figure 2: The scale dependence of
$Br(B^{0}/\bar{B}^{0}\rightarrow\pi^{\pm}\rho^{\mp})$ of the LO(the black
band) and the NLO(the gray band).
### IV.2 CP asymmetries
Using (38) and (39), we can derive the direct CP-violating parameter
$\displaystyle
A^{dir}_{CP}=\frac{\mathcal{|\overline{A}}|^{2}-|\mathcal{A}|^{2}}{|\mathcal{A}|^{2}+\mathcal{|\overline{A}}|^{2}}=\frac{2z\sin\alpha\sin\delta}{1+2z\cos\alpha\cos\delta+z^{2}}.$
(41)
It is clear that the non-zero direct CP asymmetry requires at least two
comparable contributions with different strong phase and different weak phase.
Since $A^{dir}_{CP}$ is proportional to $\sin\alpha$, it can be used to
measure the CKM angle $\alpha$, if we know the strong phase difference between
the tree and penguin diagrams. The CKM angle $\alpha$ dependence of the direct
CP-violating asymmetries of these decays are shown in Fig. 3. The accuracy of
this measurement requires more precise theoretical calculation and more
experimental data.
The numerical results for the direct CP-violating asymmetries of
$B^{\pm}\rightarrow\pi^{\pm}\rho^{0}$, $\rho^{\pm}\pi^{0}$, $\pi^{\pm}\omega$
and $B^{0}\rightarrow\pi^{0}\rho^{0}$, $\pi^{0}\omega$ decays are listed in
Table 2. The direct CP-violation parameters of
$B^{+}\rightarrow\pi^{+}\rho^{0}$ is negative, while the direct CP-violation
parameter of the other modes are positive. The direct CP-violation parameter
of $B^{+}\rightarrow\pi^{+}\omega$ is rather small for the almost canceled
contributions of annihilation diagram, which are the dominant contributions to
the strong phases in pQCD approach. Because the NLO Wilson evolution increases
the penguin amplitudes and dilutes the tree amplitudes, the NLO direct CP-
violation parameters (absolute value) of those decays are slightly enhanced
compared with the LO predictions. However, for the color-suppressed tree
dominant modes $B^{0}\rightarrow\pi^{0}\rho^{0}$ and
$B^{0}\rightarrow\pi^{0}\omega$, the direct CP asymmetry varies from $-50\%$
to $47\%$ and from $52\%$ to $98\%$, respectively. The big changes are
attributed to a huge change of the strong phase of color-suppressed tree
amplitudes caused by the vertex corrections.
Table 2: The pQCD predictions for the direct CP-violating asymmetries of
$B^{\pm}\rightarrow\pi^{\pm}\rho^{0},\rho^{\pm}\pi^{0},\pi^{\pm}\omega$ and
$B^{0}\rightarrow\pi^{0}\rho^{0},\pi^{0}\omega$ decays$(\text{in units of
}\%)$. We cite theoretical results evaluated in QCDF-I npb675333 , QCDF-
II09095229 , SCET 0801 and experimental data pdg2010 for comparison.
Mode | LO | NLO | QCDF-I npb675333 | QCDF-II 09095229 | SCET 0801 | Data pdg2010
---|---|---|---|---|---|---
$B^{\pm}\rightarrow\pi^{\pm}\rho^{0}$ | -26.4 | $-13.2^{+4.8+0.7+6.5+8.5}_{-5.3-0.7-5.5-9.6}$ | $4.1^{+1.3+2.2+0.6+19.0}_{-0.9-2.0-0.7-18.8}$ | $-9.8^{+3.4+11.4}_{-2.6-10.2}$ | $-19.2^{+15.5+1.7}_{-13.4-1.9}$ | $18^{+9}_{-17}$
$B^{\pm}\rightarrow\rho^{\pm}\pi^{0}$ | 20.1 | $34.7^{+4.4+1.6+4.4+8.8}_{-4.1-1.6-4.8-8.2}$ | $-4.0^{+1.2+1.8+0.4+17.5}_{-1.2-2.2-0.4-17.7}$ | $9.7^{+2.1+8.0}_{-3.1-10.3}$ | $12.3^{+9.4+0.9}_{-10.0-1.1}$ | $2\pm 11$
$B^{\pm}\rightarrow\pi^{\pm}\omega$ | 0.4 | $5.3^{+0.3+0.3+1.2+0.8}_{-0.1-0.3-0.5-2.5}$ | $-1.8^{+0.5+2.7+0.8+2.1}_{-0.5-3.3-0.7-2.2}$ | $-13.2^{+3.2+12.0}_{-2.1-10.7}$ | $2.3^{+13.4+0.2}_{-13.2-0.4}$ | $-4\pm 6$
$B^{0}\rightarrow\pi^{0}\rho^{0}$ | -49.8 | $46.5_{-8.2-2.1-2.6-7.0}^{+8.4+2.2+6.2+7.1}$ | $-15.7^{+4.8+12.3+11.0+19.8}_{-4.7-14.0-12.9-25.8}$ | $11.0^{+5.0+23.5}_{-5.7-28.8}$ | $-3.5^{+21.4+0.3}_{-20.3-0.3}$ | $-30\pm 40$
$B^{0}\rightarrow\pi^{0}\omega$ | 51.9 | $97.6_{-1.1-2.1-1.3-3.0}^{+0.0+1.5+0.7+3.0}$ | – | $-17.0^{+55.4+98.6}_{-22.8-82.3}$ | $39.5^{+79.1+3.4}_{-185.5-3.1}$ | –
The theoretical uncertainties of the NLO-pQCD predictions are also shown in
Table 2. The first error shown in the table, comes from the B meson wave
function parameters $\omega_{b}=0.40\pm 0.04$ and $f_{B}=0.21\pm
0.01\text{GeV}$; The second error arises from the uncertainties of the CKM
matrix elements $|V_{ub}|=(3.47^{+0.16}_{-0.12})\times 10^{-3}$,
$|V_{td}|=(9.62^{+0.26}_{-0.2})\times 10^{-3}$ and the CKM angles
$\alpha=(90\pm 5)^{\circ}$; the third error comes from the uncertainties of
final state meson wave function parameters $a_{2}^{\pi}=0.44^{+0.1}_{-0.2}$,
$a_{\rho}^{\parallel}=0.18\pm 0.1$; the fourth error is from the hard scale
$t$ varying from $0.75t$ to $1.25t$ and
$\Lambda^{(5)}_{QCD}=0.228^{+0.008}_{-0.009}\text{GeV}$, characterizing the
uncertainty of higher order contributions. Unlike the CP-averaged branching
ratios, the direct CP asymmetry is not sensitive to the wave function
parameters and CKM factors, since these parameter dependence canceled out in
Eq.(41). In addition, the CKM angles ($\alpha$) uncertainty is quite small
($\sim 5\%$). Therefore, the most important uncertainties here are the scale
dependence, which shows the importance of the NLO calculations.
Figure 3: Direct CP-violation parameters of $B^{0}\rightarrow\pi^{0}\omega$
(the top band), $B^{+}\rightarrow\rho^{0}\pi^{0}$ (the second band),
$B^{+}\rightarrow\rho^{+}\pi^{0}$ (the third band),
$B^{0}\rightarrow\pi^{+}\omega$ (the fourth band),
$B^{+}\rightarrow\pi^{+}\rho^{0}$ (the bottom band), as a function of CKM
angle $\alpha$
.
We also cite results evaluated in QCDF-I npb675333 , QCDF-II 09095229 ,
SCET0801 for comparison in Table 2. Our predictions on direct CP asymmetries
are typically larger in magnitude, most of which have the same sign with SCET
approach. In QCDF framework, the strong phases are either at the order of
$\alpha_{s}$ or power suppressed in $\Lambda_{QCD}/m_{b}$. So predictions in
the QCDF-I approach on these channels are usually small in magnitude, most
have different signs from our pQCD results npb675333 . In fact, the QCDF-II
results 09095229 quoted in Table 2 already included large strong phase coming
from penguin annihilation contributions, so that their results agree well with
our pQCD ones.
For the neutral $B^{0}$ decays, the situation is more complicated due to the
$B^{0}--\bar{B}^{0}$ mixing. The CP asymmetry is time dependentpdg2010 , when
the final states are CP-eigenstates. A time dependent asymmetry can be defined
by
$\displaystyle A_{f}(t)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(\bar{B}^{0}(t)\rightarrow
f)-\Gamma(B^{0}(t)\rightarrow f)}{\Gamma(\bar{B}^{0}(t)\rightarrow
f)+\Gamma(B^{0}(t)\rightarrow f)}$ (42) $\displaystyle=$ $\displaystyle
S_{f}\sin\Delta mt+A_{CP}^{dir}\cos\Delta mt,$ (43)
where $\Delta m$ is the mass difference of the two mass eigenstates of the
neutral B meson. The mixing-induced CP-asymmetry parameter $S_{f}$ is referred
to as
$\displaystyle S_{f}$ $\displaystyle=$
$\displaystyle\frac{2Im(\lambda_{f})}{1+|\lambda_{f}|^{2}},$
$\displaystyle\lambda_{f}$ $\displaystyle=$
$\displaystyle\frac{\xi_{t}}{\xi^{*}_{t}}\frac{\mathcal{\overline{A}}}{\mathcal{A}}=e^{2i\alpha}\frac{1+ze^{i(\delta-\alpha)}}{1+ze^{i(\delta+\alpha)}}.$
(44)
If penguin contribution is suppressed comparing with the tree contribution, we
will have the approximate relation $S_{f}\simeq\sin 2\alpha$ for a negligible
$z$ parameter. From Fig 4, one can see that the $S_{f}$ is not a simple $\sin
2\alpha$ behavior, since the $z\simeq 3.5$ for $\pi^{0}\rho^{0}$ and $z\simeq
1.0$ for $\pi^{0}\omega$, reflecting a very large penguin contribution.
Table 3: The pQCD predictions for the CP-violating parameters $S_{f}$ of
$B^{0}\rightarrow\pi^{0}\rho^{0},\pi^{0}\omega$ (in unit of %), together with
results from the QCDF-II 09095229 , the ones obtained using SCET 0801 and the
experimental data pdg2010 . The errors for these entries correspond to the
uncertainties in the scale dependence and other input parameters,
respectively.
. Mode LO NLO QCDF-II 09095229 SCET 0801 Data pdg2010
$S_{\pi^{0}\rho^{0}}$ 47 $24^{+26+9}_{-19-12}$ $-24^{+15+20}_{-14-22}$
$-19^{+14+10}_{-14-15}$ $10\pm 40$ $S_{\pi^{0}\omega}$ -37
$21_{-10-11}^{+5+13}$ $78^{+14+20}_{-20-139}$ $72^{+36+7}_{-154-11}$ –
Figure 4: Mixing-induced CP-violation parameters $S_{\pi^{0}\rho^{0}}$ ( the
gray band), mixing CP-violation parameters $S_{\pi^{0}\omega}$ (the black
band), as a function of CKM angle $\alpha$.
The pQCD numerical results for the CP-violating parameters $S_{f}$ of
$B^{0}\rightarrow\pi^{0}\rho^{0},\pi^{0}\omega$ are displayed in Table 3,
together with the QCDF-II 09095229 and SCET 0801 results. It can be seen
that the pQCD central value for $S_{\pi^{0}\rho^{0}}$ has a different sign
from the other two approaches, because of the penguin contribution is bigger
than the tree contribution in our approach. Our theoretical errors for these
entries shown in the table correspond to the uncertainties in the scale
dependence and other input parameters, respectively. It is easy to see that
the uncertainty is very large. Currently, no relevant experimental
measurements for the CP-violating asymmetries of these decays are available.
Our predictions for these quantities are different from those in QCDF-II and
SCET. We have to wait for the experimental data to resolve these
disagreements.
### IV.3 Time dependent asymmetry parameters of
$B^{0}(\bar{B}^{0})\to\pi^{\pm}\rho^{\mp}$ decays
Table 4: The LO-and NLO-pQCD predictions for the CP-violating parameters $C$, $S$, $\Delta C$ and $\Delta S$ of $B^{0}/\bar{B}^{0}\rightarrow\pi^{\pm}\rho^{\mp}$ $(\text{in units of }\%)$, together with results from the QCDF-I npb675333 , QCDF-II09095229 , the ones obtained using SCET 0801 and the experimental data pdg2010 . The errors for these entries correspond to the uncertainties in the scale dependence and other input parameters, respectively. Mode | LO | NLO | QCDF-I npb675333 | QCDF-II 09095229 | SCET 0801 | Data pdg2010
---|---|---|---|---|---|---
$A_{CP}$ | -11 | $-17^{+4+4}_{-3-4}$ | $1^{+0+1+0+10}_{-0-1-0-10}$ | $-11^{+0+7}_{-0-5}$ | $-21^{+3+2}_{-2-3}$ | $-13\pm 4$
$C$ | 6 | $15^{+2+2}_{-2-2}$ | $0^{+0+1+0+2}_{-0-1-0-2}$ | $9^{+0+5}_{-0-7}$ | $1^{+9+0}_{-10-0}$ | $1\pm 14$
$S$ | -12 | $-31^{+6+16}_{-3-15}$ | $13^{+60+4+2+2}_{-65-3-1-1}$ | $-4^{+1+10}_{-1-9}$ | $-1^{+6+8}_{-7-14}$ | $1\pm 9$
$\Delta C$ | 17 | $26_{-2-8}^{+2+5}$ | $16^{+6+23+1+1}_{-7-26-2-2}$ | $26^{+2+2}_{-2-2}$ | $12^{+9+1}_{-10-1}$ | $37\pm 8$
$\Delta S$ | -7 | $-7^{+0+2}_{-0-1}$ | $-2^{+1+0+0+1}_{-0-1-0-1}$ | $-2^{+0+3}_{-0-2}$ | $43^{+5+3}_{-7-3}$ | $-5\pm 10$
Both $B^{0}$ and $\bar{B}^{0}$ can decay into both the $\pi^{+}\rho^{-}$ and
$\pi^{-}\rho^{+}$ final states. This is an interesting example of CP asymmetry
in B decays, which is the only measured combination of four channels. $A_{f}$,
$\bar{A}_{f}$, $A_{\bar{f}}$ and $\bar{A}_{\bar{f}}$ are defined as follows
npb361141 :
$\displaystyle A_{f}$ $\displaystyle=$
$\displaystyle\langle\pi^{-}\rho^{+}|H_{eff}|B^{0}\rangle,\quad
A_{\bar{f}}=\langle\pi^{+}\rho^{-}|H_{eff}|B^{0}\rangle,\quad$
$\displaystyle\bar{A}_{f}$ $\displaystyle=$
$\displaystyle\langle\pi^{-}\rho^{+}|H_{eff}|\bar{B}^{0}\rangle,\quad\bar{A}_{\bar{f}}=\langle\pi^{+}\rho^{-}|H_{eff}|\bar{B}^{0}\rangle.$
(45)
The system of four decay modes can define the time- and flavor-integrated
charge asymmetry:
$\displaystyle
A_{CP}=\frac{|A_{f}|^{2}+|\bar{A}_{f}|^{2}-|A_{\bar{f}}|^{2}-|\bar{A}_{\bar{f}}|^{2}}{|A_{f}|^{2}+|\bar{A}_{f}|^{2}+|A_{\bar{f}}|^{2}+|\bar{A}_{\bar{f}}|^{2}}.$
(46)
In the standard approximation, which neglects CP violation in the
$B^{0}-\bar{B}^{0}$ mixing matrix and the width difference of the two mass
eigenstates, the four time dependent widths are given by the following
formulas epjc23275 :
$\displaystyle\Gamma(B^{0}(t)\rightarrow\pi^{-}\rho^{+})$ $\displaystyle=$
$\displaystyle e^{-\Gamma
t}\frac{1}{2}(|A_{f}|^{2}+|\bar{A}_{f}|^{2})[1+C_{f}\cos\Delta mt-
S_{f}\sin\Delta mt],$
$\displaystyle\Gamma(\bar{B}^{0}(t)\rightarrow\pi^{+}\rho^{-})$
$\displaystyle=$ $\displaystyle e^{-\Gamma
t}\frac{1}{2}(|A_{\bar{f}}|^{2}+|\bar{A}_{\bar{f}}|^{2})[1-C_{\bar{f}}\cos\Delta
mt+S_{\bar{f}}\sin\Delta mt],$
$\displaystyle\Gamma(B^{0}(t)\rightarrow\pi^{+}\rho^{-})$ $\displaystyle=$
$\displaystyle e^{-\Gamma
t}\frac{1}{2}(|A_{\bar{f}}|^{2}+|\bar{A}_{\bar{f}}|^{2})[1+C_{\bar{f}}\cos\Delta
mt-S_{\bar{f}}\sin\Delta mt],$
$\displaystyle\Gamma(\bar{B}^{0}(t)\rightarrow\pi^{-}\rho^{+})$
$\displaystyle=$ $\displaystyle e^{-\Gamma
t}\frac{1}{2}(|A_{f}|^{2}+|\bar{A}_{f}|^{2})[1-C_{f}\cos\Delta
mt+S_{f}\sin\Delta mt],$ (47)
where $\Delta m>0$ denotes the mass difference, and $\Gamma$ is the common
total width of the B meson eigenstates. $C_{f}$ and $S_{f}$ are defined as
$\displaystyle C_{f}$ $\displaystyle=$
$\displaystyle\frac{|A_{f}|^{2}-|\bar{A}_{f}|^{2}}{|\bar{A}_{f}|^{2}+|A_{f}|^{2}},\quad
S_{f}=\frac{2Im(\lambda_{f})}{1+|\bar{A}_{f}/A_{f}|^{2}},\quad\lambda_{f}=\frac{\xi_{t}}{\xi^{*}_{t}}\frac{\bar{A}_{f}}{A_{f}},$
(48)
For decays to the CP-conjugate final state, one replaces $f$ by $\bar{f}$ to
obtain the formula for $C_{\bar{f}}$ and $S_{\bar{f}}$. Furthermore, we define
$C\equiv\frac{1}{2}(C_{f}+C_{\bar{f}})$,
$S\equiv\frac{1}{2}(S_{f}+S_{\bar{f}})$, $\Delta
C\equiv\frac{1}{2}(C_{f}-C_{\bar{f}})$ and $\Delta
S\equiv\frac{1}{2}(S_{f}-S_{\bar{f}})$. S is referred to as mixing-induced CP
asymmetry and C is the direct CP asymmetry, while $\Delta C$ and $\Delta S$
are CP-even under CP transformation $\lambda_{f}\rightarrow
1/\lambda_{\bar{f}}$. If $f$ is CP eigenstate there are only two different
amplitudes since $f=\bar{f}$, and $\Delta C$, $\Delta S$ vanish. The
complicated formulas (IV.3) return back to the simpler one in Eq.(43).
According to (38) and (39), we can write Eq.(IV.3) as
$\displaystyle A_{f}$ $\displaystyle=$ $\displaystyle\xi_{u}T-\xi_{t}P,\quad
A_{\bar{f}}=\xi_{u}T^{\prime}-\xi_{t}P^{\prime},$ $\displaystyle\bar{A}_{f}$
$\displaystyle=$
$\displaystyle\xi^{*}_{u}T^{\prime}-\xi^{*}_{t}P^{\prime},\quad\bar{A}_{\bar{f}}=\xi^{*}_{u}T-\xi^{*}_{t}P,$
(49)
where T and P denote the tree diagram amplitude and penguin diagram amplitude
of $B^{0}\rightarrow\rho^{+}\pi^{-}$, respectively; while $\text{T}^{\prime}$
and $\text{P}^{\prime}$ denote the tree diagram amplitude and penguin diagram
amplitude of $B^{0}\rightarrow\pi^{+}\rho^{-}$, respectively. The asymmetries
$\Delta
S\approx\frac{2|T||T^{\prime}|}{|T|^{2}+|T^{\prime}|^{2}}\sin\theta\cos\alpha$
are suppressed by the small penguin-to-tree ratios
($|\text{P}/\text{T}|,|\text{P}^{\prime}/\text{T}^{\prime}|\ll 1$) and the
small relative phase $\theta$ between T and $\text{T}^{\prime}$ ($\theta\simeq
3.4^{\circ}$), hence they are always small in pQCD factorization. This
conclusion is similar to that in QCDF npb675333 ; 09095229 , although the
absolute magnitude of $\Delta S$ are much larger in pQCD than in QCDF. All the
CP-violation parameters of $B^{0}/\bar{B}^{0}\rightarrow\pi^{\pm}\rho^{\mp}$
decays including the LO epjc23275 and NLO results of pQCD, QCDF-I npb675333 ,
QCDF-II 09095229 , SCET 0801 and the experimental data are collected in Table
4. It is clear that the NLO-pQCD prediction for the CP-violation parameter
$A_{CP}$, $\Delta C$ and $\Delta S$ agrees with the experimental results very
well. The predictions of pQCD for CP-violation parameters in Table 4 are
comparable with the QCDF-II, and are better than QCDF-I and SCET predictions,
which is also shown in other B decay channels direct .
## V conclusion
In the framework of the pQCD approach, we calculated the NLO QCD corrections
to the $B\rightarrow\pi\rho$, $\pi\omega$ decays including the vertex
corrections, the quark loops, the magnetic penguin, and the NLO Wilson
coefficients, the Sudakov factor and RG factor. We found that the NLO
corrections improved the scale dependence significantly, and had great effects
on some of the decay channels. Our NLO-pQCD calculations agree well with the
measured values. For example, compared with LO predictions, the NLO
corrections decease (increase) the branching ratio of
$B^{0}/\bar{B}^{0}\rightarrow\pi^{\pm}\rho^{\mp}(B^{0}\rightarrow\pi^{0}\rho^{0})$,
and improve the consistency of the pQCD predictions. The NLO corrections play
an important role in modifying direct CP asymmetries. For the color-allowed
tree dominant modes, the NLO Wilson coefficients enhance the penguin
amplitudes, the larger subdominant penguin amplitudes increase the magnitudes
of the direct CP asymmetries due to the stronger interference with the
dominant tree amplitudes. The predictions of pQCD for CP-violation parameters
are better than QCDF-I and SCET predictions.
###### Acknowledgements.
We thank Yu Fusheng, Hsiang-nan Li, Xin Liu and Wei Wang for helpful
discussions. This work is partially supported by National Natural Science
Foundation of China under the Grant No. 10735080, and 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; and the China Postdoctoral Science Foundation under grant
No. 20100480466.
## Appendix
We show here the hard function $h_{ql}$ and $h_{mg}$ the Sudakov exponents
$S_{ql,mg}(t)$ appearing in the expressions of the decay amplitudes in III,
$\displaystyle h_{ql}(x_{1},x_{2},b_{1},b_{2})$ $\displaystyle=$
$\displaystyle K_{0}(\sqrt{x_{1}x_{2}}m_{B}b_{1})$ (50)
$\displaystyle\times[\theta(b_{1}-b_{2})K_{0}(\sqrt{x_{2}}m_{B}b_{1})I_{0}(\sqrt{x_{2}}m_{B}b_{2})$
$\displaystyle+\theta(b_{2}-b_{1})K_{0}(\sqrt{x_{2}}m_{B}b_{2})I_{0}(\sqrt{x_{2}}m_{B}b_{1})]S_{t}(x_{2}),$
$\displaystyle h_{mg}(A,B,C,b_{1},b_{2},b_{3})$ $\displaystyle=$
$\displaystyle-
K_{0}(Bb_{1})K_{0}(Cb_{3})\times\int_{0}^{\pi/2}d\theta\tan\theta$ (51)
$\displaystyle
J_{0}(Ab_{1}\tan\theta)J_{0}(Ab_{2}\tan\theta)J_{0}(Ab_{3}\tan\theta)$
where $J_{0}$ is the Bessel function and $K_{0}$, $I_{0}$ are modified Bessel
functions with $K_{0}(-ix)=-(\pi/2)Y_{0}(x)+i(\pi/2)J_{0}(x)$.
The Sudakov exponents used in the text are defined by
$\displaystyle
S_{ql}(t)=s(x_{1}m_{B},b_{1})+s(x_{2}m_{B},b_{2})+s((1-x_{2})m_{B},b_{2})+\frac{5}{3}g_{2}(t,b_{1})+2g_{2}(t,b_{2}),$
(52) $\displaystyle S_{mg}(t)$ $\displaystyle=$ $\displaystyle
s(x_{1}m_{B},b_{1})+s(x_{2}m_{B},b_{2})+s((1-x_{2})m_{B},b_{2})+s(x_{3}m_{B},b_{3})$
(53)
$\displaystyle+s((1-x_{3})m_{B},b_{3})+\frac{5}{3}g_{2}(t,b_{1})+2g_{2}(t,b_{2})+2g_{2}(t,b_{3})$
where the functions $s(P,b)$ have been defined in Ref.prd527 . The RG factor
$g_{2}(t,b)$ is given by
$\displaystyle g_{2}(t,b)$ $\displaystyle=$
$\displaystyle-\frac{2}{\beta_{0}}\ln[\frac{\ln(t/\Lambda_{QCD})}{-\ln(b\Lambda_{QCD})}]+\frac{2\beta_{1}}{\beta^{3}_{0}}[\frac{\ln(\ln(\frac{1}{b^{2}\Lambda^{2}_{QCD}}))}{\ln(\frac{1}{b^{2}\Lambda^{2}_{QCD}})}-\frac{\ln(\ln(\frac{t^{2}}{\Lambda^{2}_{QCD}}))}{\ln(\frac{t^{2}}{\Lambda^{2}_{QCD}})}$
$\displaystyle+\frac{1}{\ln(\frac{1}{b^{2}\Lambda^{2}_{QCD}})}-\frac{1}{\ln(\frac{t^{2}}{\Lambda^{2}_{QCD}})}]$
$\displaystyle\beta_{0}$ $\displaystyle=$ $\displaystyle
11-\frac{2}{3}n_{f},\quad\beta_{1}=102-\frac{38}{3}n_{f}$ (54)
where $n_{f}$ is the number of quarks with mass less than the energy scale
$t$.
## References
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* (7) S. Baek, C.-W. Chiang, D. London, Phys. Lett. B 675, 59-63 (2009) and references therein.
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* (13) C.D. Lü, M.Z. Yang, Eur. Phys. J. C 23, 275 (2002).
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* (21) M. Bander, D. Silverman and A. Soni, Phys. Rev. Lett. 43, 242 (1979); J.M. Gerard and W.S. Hou, Phys. Rev. D 43, 2909 (1991).
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* (28) Wei Wang, Yu-Ming Wang, De-Shan Yang and C.D. Lü, Phys. Rev. D 78, 034011 (2008).
* (29) H.-n. Li and S. Mishima, Phys. Rev. D 83, 034023 (2011).
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|
arxiv-papers
| 2011-11-01T12:07:41 |
2024-09-04T02:49:23.835420
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhou Rui, Gao Xiangdong, and Cai-Dian Lu",
"submitter": "Zhou Rui",
"url": "https://arxiv.org/abs/1111.0181"
}
|
1111.0206
|
# Calculation of the Structure Properties of Asymmetrical Nuclear Matter
Gholam Hossein Bordbar1,2111Corresponding author. E-mail:
bordbar@physics.susc.ac.ir and Hamideh Nadgaran1 Department of Physics, Shiraz
University, Shiraz 71454, Iran
and
Research Institute for Astronomy and Astrophysics of Maragha, P.O. Box
55134-441, Maragha 55177-36698, Iran
###### Abstract
In this paper the structure properties of asymmetrical nuclear matter has been
calculated employing $AV_{18}$ potential for different values of proton to
neutron ratio. These calculations have been also made for the case of
symmetrical nuclear matter with $UV_{14}$, $AV_{14}$ and $AV_{18}$ potentials.
In our calculations, we use the lowest order constrained variational (LOCV)
method to compute the correlation function of the system.
## I Introduction
The interpretation of many astrophysical phenomena depends on a profound
understanding of different parts of physics. Nuclear physics has an important
role in determining the energy and evolution of stellar matter. Most of
calculations for asymmetrical nuclear matter has a close relationship with
astrophysics. These studies are also potentially useful for understanding the
effective nucleon-nucleon interactions in dense asymmetrical nuclear matter,
an important ingredient in nuclear structure physics, heavy ion collision
physics as well as compact star physics. Nuclear matter is defined as a
hypothetical system of nucleons interacting without coulomb forces, with a
fixed ratio of protons and neutrons, and can be supposed as an idealization of
matter inside a large nucleus. The aim of a nuclear matter theory is to match
the known experimental bulk properties, such as the binding energy,
equilibrium density, symmetry energy, incompressibility, etc., starting from
the fundamental two-body interactions (Pandharipande & Wiringa rk1 (1979)).
A good many-body theory for nuclear matter can be useful for studying the
details of nucleon-nucleon interactions. The observed phase shifts from
scattering experiments plus the properties of the only bound two-nucleon
system, the deuteron, aren’t enough to obtain a unique nucleon-nucleon
potential. Nuclear matter studies can help us understand better exactly how
the properties of the matter are affected by different elements of a
potential, and what sorts of features are required to produce the observed
saturation. Nuclear matter studies may also indicate whether a potential model
for nuclear forces is workable or not (Pandharipande & Wiringa rk1 (1979)).
The starting point for a microscopic theory of finite nuclei is to solve the
infinite matter problem. A solution of the infinite matter problem would also
be the first step in obtaining the equation of state for dense matter, which
is necessary in the study of neutron stars. At the end, it is simply a very
interesting many-body problem in its own right. Methods developed for it
should be helpful in other dense quantum fluids such as liquid helium
(Pandharipande & Wiringa rk1 (1979)).
The starting point for any nuclear matter calculation is a two-body potential
that models the nucleon-nucleon interaction (Pandharipande & Wiringa rk1
(1979)). The first nuclear matter calculations were done by Euler (rk32
(1937)). Very little was known about the interaction of nucleons at that time
(Pandharipande & Wiringa rk1 (1979)). At the same time Yukawa potential was
formulated as:
$V=\gamma\frac{e^{-\mu r}}{r},$ (1)
where $\gamma$ is a constant and $\mu$ is defined as
$\frac{\hbar}{M_{\pi}C}=\frac{1}{\mu}$ ($C$ is the speed of light and
$M_{\pi}$ is the mass of $\pi$ meson) and $r$ is the relative distance between
two nucleons (Cohen rk2 (1971); Wong rk42 (2004)). Several years later,
Gammel, Christian and Thaler (rk35 (1957)) introduced a potential of the form:
$V=V_{C}(r)+V_{T}(r)S_{12}.$ (2)
In Eq. (2), $V_{C}(r)$ is the central potential, $V_{T}(r)$ is the tensor
potential and
$S_{12}=3(\sigma_{1}\cdot\hat{r})(\sigma_{2}\cdot\hat{r})-\sigma_{1}\cdot\sigma_{2}$
is the usual tensor operator. Then the potential was allowed to depend at most
linearly on the relative momentum p, and a spin-orbit term was added to it,
$V=V_{C}(r)+V_{T}(r)S_{12}+V_{ls}(r)\textbf{L . S}.$ (3)
Where L is the relative angular momentum and S is the total spin of the
nucleon pair. This was the form originally proposed by Wigner and Eisenbud
(rk36 (1941)).
In 1962 the two most widely used potentials were introduced. Both abandoned
the Wigner form. The Hamada and Johnston (rk37 (1962)) model had the form,
$V=V_{C}(r)+V_{T}(r)S_{12}+V_{LS}(r)\textbf{L . S}+V_{LL}(r)L_{12},$ (4)
where
$L_{12}=[\delta_{LJ}+(\sigma_{1}.\sigma_{2})]L^{2}-(\textbf{L.S})^{2}$
and the Yale potential was defined as (Lassila et al. rk38 (1962)),
$V=V_{C}(r)+V_{T}(r)S_{12}+V_{LS}(r)\textbf{L .
S}+V_{q}(r)[(\textbf{L.S})^{2}+\textbf{L.S}-L^{2}].$ (5)
In 1968 another potential was introduced by Reid (rk3 (1968)). This potential
has a central term, $V_{C}(r)$, for uncoupled states (singlet and triplet with
$\textbf{L}=\textbf{J}$) and for coupled states (triplet with
$\textbf{L}=\textbf{J}\pm 1$) has the form of Eq. (3). In 1974, Bethe and
Johnston (rk33 (1974)) introduced a potential that had the general form of the
Reid potential. BJ potential has a very hard core in $(S,T)=(0,0),(1,1)$
channels.
Generally the above potentials are limited to a few operators and don’t fit
the data for all the scattering channels very well. In many-body calculations
of nuclei and nuclear matter, it is suitable to represent the two nucleon
interaction as an operator (Lagaris & Pandharipande rk5 (1981)):
$V_{ij}=\sum_{p}V^{p}(r_{ij})O^{p}_{ij},$ (6)
where $V^{p}(r_{ij})$ are functions of the interparticle distance $r_{ij}$,
and $O^{p}_{ij}$ are suitably chosen operators. The nucleon-nucleon ($NN$)
interaction scattering data uniquely show the occurrence of terms belonging to
the eight operators (Lagaris & Pandharipande rk5 (1981)):
$O^{p=1-8}_{ij}=1,\sigma_{i}.\sigma_{j},\tau_{i}.\tau_{j},(\sigma_{i}.\sigma_{j})(\tau_{i}.\tau_{j}),S_{ij},S_{ij}(\tau_{i}.\tau_{j}),(\textbf{L.S})_{ij},(\textbf{L.S})_{ij}(\tau_{i}.\tau_{j})$
(7)
in the $V_{ij}$. Many nuclear matter calculations have been done with $V_{8}$
potential models (Lagaris & Pandharipande rk5 (1981)). This potential has two
different models. One of them is Reid-$V_{8}$ (Pandharipande & Wiringa rk1
(1979)) and the other is BJ-II $V_{8}$ (Pandharipande & Wiringa rk1 (1979))
model. There is also a $V_{6}$ model. The $V_{i=7,8}$ terms are neglected in
the $V_{6}$ model. The HJ $V_{6}$ model is obtained by neglecting the L.S and
quadratic spin-orbit terms in Hamada and Johnston potential (Pandharipande &
Wiringa rk1 (1979)), while the GT-5200 potential (Pandharipande & Wiringa rk1
(1979)) is itself of a $V_{6}$ form.
Another $NN$ interaction model is $V_{12}$. In this model, in addition to the
$8$ operators of Eq. (7), there is four momentum-dependent terms:
$O^{p=9-12}_{ij}=L^{2},L^{2}(\sigma_{i}.\sigma_{j}),L^{2}(\tau_{i}.\tau_{j}),L^{2}(\sigma_{i}.\sigma_{j})(\tau_{i}.\tau_{j}).$
(8)
The $V_{12}$ potential like the $V_{6}$ model has two different forms, which
are Reid-$V_{12}$ and BJ-II $V_{12}$ (Lagaris & Pandharipande rk5 (1981)).
In 1981 a phenomenologically two-nucleon interaction potential was introduced
by Lagaris and Pandharipande (rk5 (1981)). This potential was obtained by
fitting the nucleon-nucleon phase shifts up to 425 $MeV$ in $S$, $P$, $D$ and
$F$ waves, and the deuteron properties. It has two additional terms other than
the operators in Eqs. (3) and (4) and is called as $V_{14}$ or $Urbana\
V_{14}$ ($UV_{14}$) potential.
$O^{p=13,14}_{ij}=(L.S)^{2},(L.S)^{2}(\tau_{i}.\tau_{j}).$ (9)
In $UV_{14}$ model, the two nucleon interaction is written as:
$V_{ij}=\sum_{p=1,14}\Big{(}V^{p}_{\pi}(r_{ij})+V^{p}_{I}(r_{ij})+V^{p}_{S}(r_{ij})\Big{)}O^{p}_{ij},$
(10)
where $V_{\pi}^{p}(r_{ij})$ is the well known one-pion-exchange interaction,
$V^{p}_{I}(r_{ij})$ is an intermediate range interaction and
$V^{p}_{S}(r_{ij})$ is a purely phenomenological short-range interaction.
There is also another form of $V_{14}$ potential which was proposed by Wiringa
and collaborators (rk6 (1984)). It is called $Argonne$ $V_{14}$ ($AV_{14}$)
potential. It has the general form of $UV_{14}$ potential. The difference
between $AV_{14}$ and $UV_{14}$ models are in how the functions
$V_{\pi}^{p}(r_{ij})$, $V^{p}_{I}(r_{ij})$ and $V^{p}_{S}(r_{ij})$ are
defined.
Traditionally, $NN$ potentials are formed by fitting $np$ data for $T=0$
states and either $np$ or $pp$ data for $T=1$ states. Unfortunately, potential
models which have been fitted only to the $np$ data often give not a good
description of the $pp$ data (Stocks & Swart rk7 (1993)), even after applying
the essential correlations for the coulomb interaction. By the same token,
potentials fit to $pp$ data in $T=1$ states give simply a mediocre description
of $np$ data. Substantially, this problem is due to charge-independence
breaking in the strong interaction. In the present work we use an updated
version of the Argonne potential, $AV_{18}$ model (Wiringa et al. rk8 (1995)),
that fits both $pp$ and $np$ data, as well as low energy $nn$ scattering
parameters and deuteron properties. This potential is written in an operator
format that depends on the values of $S$, $T$ and $T_{Z}$ of the $NN$ pair.
$AV_{18}$ potential includes a charge- independent (CI) part that has 14
operator components (as in $AV_{14}$ model) and a charge-independent breaking
(CIB) part that has three charge- dependent (CD) and one charge-asymmetric
(CA) operators. The four additional operators that break charge-independence
are given by
$O^{p=15-18}_{ij}=T_{ij},(\sigma_{i}.\sigma_{j})T_{ij},S_{ij}T_{ij},(\tau_{zi}+\tau_{zj})$
(11)
where
$T_{ij}=3\tau_{zi}\tau_{zj}-\tau_{i}.\tau_{j}$
is the tensor operator. In between the operators of Eq. (11), the first three
represent charge-dependence while the last one represents charge-asymmetry.
In this paper, we use the lowest-order constrained variational method (LOCV)
to calculate the correlation function of the nuclear matter. Primarily, the
technique of LOCV was used to study the bulk properties of quantal fluids
(Owen et al. rk9 (1977); Modarres & Irvine 1979a ). The method was later
extended to calculate the symmetry coefficient for the semi-empirical mass
formula (Howes et al. 1978a , rk41 (1979); Modarres & Irvine 1979a , 1979b ),
the properties of beta-stable matter (Modarres & Irvine 1979a , 1979b ; Howes
et al. 1978b ), the surface energies of quantal fluids (Howes et al. 1978b )
and the binding energies of finite nuclei (Bishop et al. rk13 (1978); Modarres
rk44 (1984)). The LOCV method was further extended for finite temperature
calculation and it was very successfully applied to neutron, nuclear and
asymmetrical nuclear matter (Modarres rk14 (1993), rk15 (1995), rk16 (1997))
in order to calculate different thermodynamic properties of these systems.
Recently, LOCV calculations have been done for the symmetric nuclear matter
with phenomenological two-nucleon interaction operators (Bordbar & Modarres
rk29 (1997)) and the asymmetrical nuclear matter with $AV_{18}$ potential
(Bordbar & Modarres rk30 (1998)). The incompressibility of hot asymmetrical
nuclear matter have been also investigated within an LOCV approach (Modarres &
Bordbar rk31 (1998)). Very recently, some nucleonic systems such as the spin
polarized neutron matter (Bordbar & Bigdeli 2007a ), symmetric nuclear matter
(Bordbar & Bigdeli 2007b ), asymmetrical nuclear matter (Bordbar & Bigdeli
2008a ), and neutron star matter (Bordbar & Bigdeli 2008a ) at zero
temperature have been studied using LOCV method with the realistic strong
interaction in the absence of magnetic field. The thermodynamic properties of
the spin polarized neutron matter (Bordbar & Bigdeli 2008b ), symmetric
nuclear matter (Bigdeli et al. rk21 (2009)), and asymmetrical nuclear matter
(Bigdeli et al. rk22 (2010)) have been also studied at finite temperature in
absence of the magnetic field. These calculations have been extended in the
presence of magnetic field for the spin polarized neutron matter at zero
temperature (Bordbar et al. rk23 (2011)). The LOCV method is a fully self-
consistent formalism and it does not bring any free parameter into the
calculation. It considers the normalization constraint to keep the higher
order terms as small as possible. The functional minimization procedure
represents an enormous computational simplification over unconstrained methods
(i.e., to parameterize the short-range behavior of correlation functions) that
attempts to go beyond the lowest order (Bordbar & Modarres rk30 (1998)).
In the present work, we intend to calculate the structure function of
asymmetrical nuclear matter using the LOCV method employing $UV_{14}$,
$AV_{14}$ and $AV_{18}$ potentials. So the plan of this article is as follows:
The LOCV method is described in Sec. II. Section III is devoted to a summary
of the pair distribution function and the structure function. Our results and
discussion are presented in Sec. IV. Finally, summary and conclusions are
presented in sec. V.
## II LOCV formalism for asymmetrical nuclear matter
We consider a trial many-body wave function of the form
$\Psi=F\Phi,$ (12)
where $\Phi$ is a slater determinant of plane waves of $A$ independent
nucleons, $F$ is an $A$-body correlation operator which will be replaced by a
Jastrow form. i,e.,
$F=\mathcal{S}\prod_{i>j}f(ij),$ (13)
and $\mathcal{S}$ is a symmetrizing operator. The cluster expansion of the
energy functional is written as
$E([f])=\frac{1}{A}\frac{<\Psi|H|\Psi>}{<\Psi|\Psi>}=E_{1}+E_{2}+E_{3}+\cdots.$
(14)
The one-body term $E_{1}$ for an asymmetrical nuclear matter that consists of
$Z$ protons and $N$ neutrons is
$E_{1}=\sum_{i=1,2}\frac{3}{5}\frac{\hbar^{2}k_{i}^{F^{2}}}{2m_{i}}\frac{\rho_{i}}{\rho}$
(15)
Labels 1 and 2 are used instead of proton and neutron, respectively, and
$k_{i}^{F}=(3\pi^{2}\rho_{i})^{\frac{1}{3}}$ is the Fermi momentum of particle
$i$ ($\rho=\rho_{1}+\rho_{2}$).
The two-body energy $E_{2}$ is
$E_{2}=\frac{1}{2A}\sum_{ij}<ij|\mathcal{V}(12)|ij-ji>$ (16)
and
$\mathcal{V}(12)=-\frac{\hbar^{2}}{2m}[f(12),[\nabla_{12}^{2},f(12)]]+f(12)V(12)f(12).$
(17)
The two-body correlation operator $f(12)$ is defined as follows:
$f(ij)=\sum_{\alpha,p=1}^{3}f^{(p)}_{\alpha}(ij)O^{(p)}_{\alpha}(ij).$ (18)
$\alpha=\\{J,L,S,T,T_{z}\\}$ and the operators $O^{p}_{\alpha}(ij)$ are
written as
$O^{p=1-3}_{\alpha}=1,(\frac{2}{3}+\frac{1}{6}S_{12}),(\frac{1}{3}-\frac{1}{6}S_{12}),$
(19)
where $S_{12}$ is the tensor operator. We choose $p=1$ for uncoupled channels
and $p=2,3$ for coupled channels. The two-body nucleon-nucleon interaction
$V(12)$ has the following form:
$V(12)=\sum_{p=1}^{18}V^{p}(r_{12})O^{p}_{12},$ (20)
where the 18 operators that are defined as before, are denoted by the labels
$c,\sigma,\tau,\sigma\tau,t,$
$t\tau,ls,ls\tau,l2,l2\sigma,l2\tau,l2\sigma\tau,ls2,ls2\tau,T,\sigma T,tT,$
and $\tau z$ (Wiringa rk6 (1984)). By using correlation operators in the form
of Eq. (18) and the two-nucleon potential from Eq. (20), we find the following
equation for the two-body energy (Bordbar & Modarres rk30 (1998)):
$\displaystyle E_{2}$ $\displaystyle=$
$\displaystyle\frac{2}{\pi^{4}\rho}\bigg{(}\frac{\hbar^{2}}{2m}\bigg{)}\sum_{JLSTT_{z}}(2J+1)\frac{1}{2}\Big{[}1-(-1)^{L+S+T}\Big{]}$
$\displaystyle\times$
$\displaystyle\bigg{|}\bigg{\langle}\frac{1}{2}\tau_{z1}\frac{1}{2}\tau_{z2}\bigg{|}TT_{z}\bigg{\rangle}\bigg{|}^{2}\int
dr\bigg{\\{}\bigg{[}\Big{(}f^{(1)^{\prime}}_{\alpha}\Big{)}^{2}a^{(1)^{2}}_{\alpha}(k_{F}r)$
$\displaystyle+$
$\displaystyle\frac{2m}{\hbar}\Big{(}\Big{\\{}V_{c}-3V_{\sigma}+(V_{\tau}-3V_{\sigma\tau})(4T-3)+(V_{T}-3V_{\sigma
T})$ $\displaystyle\times$ $\displaystyle[T(6T_{z}^{2}-4)]+2V_{\tau
z}T_{z}\Big{\\}}a^{(1)^{2}}_{\alpha}(k_{F}r)+\Big{[}V_{l2}-3V_{l2\sigma}$
$\displaystyle+$
$\displaystyle(V_{l2\tau}-3V_{l2\sigma\tau})(4T-3)\Big{]}c^{(1)^{2}}_{\alpha}(k_{F}r)\Big{)}\bigg{]}+\sum_{i=2,3}\bigg{[}\Big{(}f^{(i)^{\prime}}_{\alpha}\Big{)}^{2}a^{(i)^{2}}_{\alpha}$
$\displaystyle+$
$\displaystyle\frac{2m}{\hbar^{2}}\Big{(}\Big{\\{}V_{c}+V_{\sigma}+(-6i+14)V_{t}-(i-1)V_{ls}+[V_{\tau}+V_{\sigma\tau}$
$\displaystyle+$
$\displaystyle(-6i+14)V_{t\tau}-(i-1)V_{ls\tau}](4T-3)+[V_{T}+V_{\sigma
T}(-6i+14)V_{tT}]$ $\displaystyle\times$
$\displaystyle[T(6T^{2}_{z}-4)]+2V_{\tau
z}T_{z}\Big{\\}}a^{(i)^{2}}_{\alpha}(k_{F}r)+[V_{l2}+V_{l2\sigma}+(V_{l2\tau}+V_{l2\sigma\tau})$
$\displaystyle\times$
$\displaystyle(4T-3)]c^{(i)^{2}}_{\alpha}(k_{F}r)+[V_{ls2}+V_{ls2\tau}(4T-3)]d^{(i)^{2}}_{\alpha}(k_{F}r)\Big{)}f^{(i)^{2}}_{\alpha}\bigg{]}$
$\displaystyle+$
$\displaystyle\frac{2m}{\hbar^{2}}\bigg{\\{}V_{ls}+2V_{l2}-2V_{l2\sigma}-3V_{ls2}+[(V_{ls\tau}-2V_{l2\tau}-2V_{l2\sigma\tau}-3V_{ls2\tau})$
$\displaystyle\times$
$\displaystyle(4T-3)]b^{2}_{\alpha}(k_{F}r)f^{(2)}_{\alpha}f^{(3)}_{\alpha}+\frac{1}{r^{2}}\Big{(}f^{(2)}_{\alpha}-f^{(3)}_{\alpha}\Big{)}^{2}b^{2}_{\alpha}(k_{F}r)\bigg{\\}}$
where the coefficients $a^{(1)}_{\alpha}(x)$, etc., are defined as
$\displaystyle a_{\alpha}^{(1)^{2}}(x)$ $\displaystyle=$ $\displaystyle
x^{2}I_{L,T_{z}}(x),$ (22) $\displaystyle a_{\alpha}^{(2)^{2}}(x)$
$\displaystyle=$ $\displaystyle x^{2}[\beta I_{J-1,T_{z}}(x)+\gamma
I_{J+1,T_{z}}(x)],$ $\displaystyle a_{\alpha}^{(3)^{2}}(x)$ $\displaystyle=$
$\displaystyle x^{2}[\gamma I_{J-1,T_{z}}(x)+\beta I_{J+1,T_{z}}(x)],$
$\displaystyle b_{\alpha}^{2}(x)$ $\displaystyle=$ $\displaystyle
x^{2}[\beta_{23}I_{J-1,T_{z}}(x)-\beta_{23}I_{J+1,T_{z}}(x)],$ $\displaystyle
c_{\alpha}^{(1)^{2}}(x)$ $\displaystyle=$ $\displaystyle
x^{2}\nu_{1}I_{L,T_{z}}(x),$ $\displaystyle c_{\alpha}^{(2)^{2}}(x)$
$\displaystyle=$ $\displaystyle
x^{2}[\eta_{2}I_{J-1,T_{z}}(x)+\nu_{2}I_{J+1,T_{z}}(x)],$ $\displaystyle
c_{\alpha}^{(3)^{2}}(x)$ $\displaystyle=$ $\displaystyle
x^{2}[\eta_{3}I_{J-1,T_{z}}(x)+\nu_{3}I_{J+1,T_{z}}(x)],$ $\displaystyle
d_{\alpha}^{(2)^{2}}(x)$ $\displaystyle=$ $\displaystyle
x^{2}[\xi_{2}I_{J-1,T_{z}}(x)+\lambda_{2}I_{J+1,T_{z}}(x)],$ $\displaystyle
d_{\alpha}^{(3)^{2}}(x)$ $\displaystyle=$ $\displaystyle
x^{2}[\xi_{3}I_{J-1,T_{z}}(x)+\lambda_{3}I_{J+1,T_{z}}(x)],$
with
$\displaystyle\beta_{1}$ $\displaystyle=$ $\displaystyle 1\hskip
22.76219pt\beta=\frac{J+1}{2J+1}\hskip 22.76219pt\gamma=\frac{J}{2J+1}\hskip
22.76219pt\beta_{23}=\frac{2J(J+1)}{2J+1}$ (23) $\displaystyle\nu_{1}$
$\displaystyle=$ $\displaystyle L(L+1)\hskip
22.76219pt\nu_{2}=\frac{J^{2}(J+1)}{2J+1}\hskip
22.76219pt\nu_{3}=\frac{J^{3}+2J^{2}+3J+2}{2J+1}$ $\displaystyle\eta_{2}$
$\displaystyle=$ $\displaystyle\frac{J(J^{2}+2J+1)}{2J+1}\hskip
28.45274pt\eta_{3}=\frac{J(J^{2}+J+2)}{2J+1}\hskip 56.9055pt$
$\displaystyle\xi_{3}$ $\displaystyle=$
$\displaystyle\frac{J^{3}+2J^{2}+2J+1}{2J+1}\hskip
28.45274pt\xi_{3}=\frac{J(J^{2}+J+4)}{2J+1}\hskip 56.9055pt$
$\displaystyle\lambda_{2}$ $\displaystyle=$
$\displaystyle\frac{J(J^{2}+J+1)}{2J+1}\hskip
28.45274pt\lambda_{3}=\frac{J^{3}+2J^{2}+5J+4}{2J+1}$
and
$I_{J,T_{Z}}(x)=\int dqP_{T_{Z}}(q)J^{2}_{J}(xq).$ (24)
$P_{T_{Z}}(q)$ is written as [$\tau_{1Z}$ or $\tau_{2Z}=-\frac{1}{2}{}$
(neutron) and $+\frac{1}{2}$ (proton)],
$P_{T_{Z}}=\frac{2}{3}\pi\bigg{[}k_{\tau Z1}^{F^{3}}+k_{\tau
Z2}^{F^{3}}-\frac{3}{2}\Big{(}k_{\tau Z1}^{F^{2}}+k_{\tau
Z2}^{F^{2}}\Big{)}q-\frac{3}{16}\Big{(}k_{\tau Z1}^{F^{2}}-k_{\tau
Z2}^{F^{2}}\Big{)}^{2}+q^{3}\bigg{]}$ (25)
for
$\frac{1}{2}\Big{|}k_{\tau_{Z1}}^{F}-k_{\tau_{Z2}}^{F}\Big{|}<q<\frac{1}{2}\Big{|}k_{\tau_{Z1}}^{F}+k_{\tau_{Z2}}^{F}\Big{|},$
$P_{T_{Z}}(q)=\frac{4}{3}\pi\min\Big{(}k_{\tau Z1}^{F^{3}},k_{\tau
Z2}^{F^{3}}\Big{)}$
for $q<\frac{1}{2}\Big{|}k_{\tau Z1}^{F}-k_{\tau Z2}^{F}\Big{|}$, and
$P_{T_{Z}}(q)=0$
for $q>\frac{1}{2}\Big{|}k_{\tau Z1}^{F}+k_{\tau Z2}^{F}\Big{|}$. The
$J_{J}(x)$ are the familiar Bessel functions.
Now, we can minimize the two-body energy, Eq. (II), with respect to the
variations in the functions $f_{\alpha}^{i}$ but subject to the normalization
constraint (Owen et al. rk9 (1977); Modarres & Irvine 1979a , 1979b ; Bordbar
& Modarres rk30 (1998))
$\frac{1}{A}\sum_{ij}<ij|h^{2}_{T_{Z}}(12)-f^{2}(12)|ij>_{a}=0,$ (26)
where in the case of asymmetrical nuclear matter the function $h_{T_{Z}}(x)$
is defined as
$\displaystyle h_{T_{z}}(r)$ $\displaystyle=$
$\displaystyle\bigg{[}1-\frac{9}{2}\bigg{(}\frac{J_{1}(k_{i}^{F}r)}{k_{i}^{F}r}\bigg{)}^{2}\bigg{]}^{-\frac{1}{2}}\hskip
56.9055ptT_{z}=\pm 1$ $\displaystyle=$ $\displaystyle 1\hskip
167.87108ptT_{z}=0$
In terms of channel correlation functions we can write Eq. (26) as follows:
$\displaystyle\frac{4}{\pi^{4}\rho}\sum_{\alpha,i}(2J+1)\frac{1}{2}\Big{[}1-(-1)^{L+S+T}\Big{]}\bigg{|}\bigg{\langle}\frac{1}{2}\tau_{z1}\frac{1}{2}\tau_{z2}\bigg{|}TT_{z}\bigg{\rangle}\bigg{|}^{2}$
(28)
$\displaystyle\times\int_{0}^{\infty}dr\Big{[}h_{T_{z}}^{2}(k_{F}r)-f^{(i)^{2}}_{\alpha}(r)\Big{]}a^{(i)^{2}}_{\alpha}(k_{F}r)=0\hskip
56.9055pt$
As we will see later, the above constraint introduces a Lagrange multiplier
$\lambda$ through which all of the correlation functions are coupled. From the
minimization of the two-body cluster energy we get a set of coupled and
uncoupled Euler-Lagrange differential equations. The Euler-Lagrange equations
for uncoupled states are
$\displaystyle g^{(1)^{\prime\prime}}_{\alpha}$ $\displaystyle-$
$\displaystyle\bigg{\\{}\frac{a_{\alpha}^{(1)^{\prime\prime}}}{a_{\alpha}^{(1)}}+\frac{m}{\hbar^{2}}\Big{[}V_{c}-3V_{\sigma}+(V_{\tau}-3V_{\sigma\tau})(4T-3)$
$\displaystyle+$ $\displaystyle(V_{T}-3V_{\sigma T})[T(6T^{2}_{z}-4)]+2V_{\tau
z}T_{z}+\lambda\Big{]}+\frac{m}{\hbar^{2}}\Big{[}V_{l2}-3V_{l2\sigma}$
$\displaystyle+$
$\displaystyle(V_{l2\tau}-3V_{l2\sigma\tau})(4T-3)\Big{]}\frac{c_{\alpha}^{(1)^{2}}}{a_{\alpha}^{(1)^{2}}}\bigg{\\}}g_{\alpha}^{(1)}=0,$
while the coupled equations are written as
$\displaystyle g^{(2)^{\prime\prime}}_{\alpha}$ $\displaystyle-$
$\displaystyle\bigg{\\{}\frac{a_{\alpha}^{(2)^{\prime\prime}}}{a_{\alpha}^{(2)}}+\frac{m}{\hbar^{2}}\Big{[}V_{c}+V_{\sigma}+2V_{t}-V_{ls}+(V_{\tau}+V_{\sigma\tau}+2V_{t\tau}$
$\displaystyle-$ $\displaystyle V_{ls\tau})(4T-3)+(V_{T}+V_{\sigma
T}+2V_{tT})[T(6T_{z}^{2}-4)]+2V_{\tau z}T_{z}+\lambda\Big{]}$ $\displaystyle+$
$\displaystyle\frac{m}{\hbar^{2}}\Big{[}V_{l2}+V_{l2\sigma}+(V_{l2\tau}+V_{l2\sigma\tau})(4T-3)\Big{]}\frac{c_{\alpha}^{(2)^{2}}}{a_{\alpha}^{(2)^{2}}}+\frac{m}{\hbar^{2}}\Big{[}V_{ls2}+V_{ls2\tau}$
$\displaystyle\times$
$\displaystyle(4T-3)\Big{]}\frac{d_{\alpha}^{(2)^{2}}}{a_{\alpha}^{(2)^{2}}}+\frac{b^{2}_{\alpha}}{r^{2}a^{(2)^{2}}_{\alpha}}\bigg{\\}}g^{(2)}_{\alpha}+\bigg{\\{}\frac{1}{r^{2}}-\frac{m}{2\hbar^{2}}\Big{[}V_{ls}-2V_{l2}-2V_{l2\sigma}$
$\displaystyle-$ $\displaystyle
3V_{ls2}+(V_{ls\tau}-2V_{l2\tau}-2V_{l2\sigma\tau}-3V_{ls2\tau})(4T-3)\Big{]}\bigg{\\}}$
$\displaystyle\times$
$\displaystyle\frac{b^{2}_{\alpha}}{a^{(2)}_{\alpha}a^{(3)}_{\alpha}}g^{(3)}_{\alpha}=0,$
$\displaystyle g^{(3)^{\prime\prime}}_{\alpha}$ $\displaystyle-$
$\displaystyle\bigg{\\{}\frac{a_{\alpha}^{(3)^{\prime\prime}}}{a_{\alpha}^{(3)}}+\frac{m}{\hbar^{2}}\Big{[}V_{c}+V_{\sigma}-4V_{t}-2V_{ls}+(V_{\tau}+V_{\sigma\tau}-4V_{t\tau}$
$\displaystyle-$ $\displaystyle 2V_{ls\tau})(4T-3)+(V_{T}+V_{\sigma
T}-4V_{tT})[T(6T_{z}^{2}-4)]+2V_{\tau z}T_{z}+\lambda\Big{]}$ $\displaystyle+$
$\displaystyle\frac{m}{\hbar^{2}}\Big{[}V_{l2}+V_{l2\sigma}+(V_{l2\tau}+V_{l2\sigma\tau})(4T-3)\Big{]}\frac{c_{\alpha}^{(3)^{2}}}{a_{\alpha}^{(3)^{2}}}+\frac{m}{\hbar^{2}}\Big{[}V_{ls2}+V_{ls2\tau}$
$\displaystyle\times$
$\displaystyle(4T-3)\Big{]}\frac{d_{\alpha}^{(3)^{2}}}{a_{\alpha}^{(3)^{2}}}+\frac{b^{2}_{\alpha}}{r^{2}a^{(2)^{2}}_{\alpha}}\bigg{\\}}g^{(3)}_{\alpha}+\bigg{\\{}\frac{1}{r^{2}}-\frac{m}{2\hbar^{2}}\Big{[}V_{ls}-2V_{l2}-2V_{l2\sigma}$
$\displaystyle-$ $\displaystyle
3V_{ls2}+(V_{ls\tau}-2V_{l2\tau}-2V_{l2\sigma\tau}-3V_{ls2\tau})(4T-3)\Big{]}\bigg{\\}}$
$\displaystyle\times$
$\displaystyle\frac{b^{2}_{\alpha}}{a^{(2)}_{\alpha}a^{(3)}_{\alpha}}g^{(2)}_{\alpha}=0,$
where
$g_{\alpha}^{(i)}(k_{F}r)=f^{(i)}_{\alpha}(r)a_{\alpha}^{(i)}(k_{F}r).$ (32)
The primes in the above equation means differentiation with respect to $r$. As
we pointed out before, the Lagrange multiplier $\lambda$ is associated with
the normalization constraint, Eq. (28). The constraint is incorporated by
solving the Euler-Lagrange equations only out to certain distances, until the
logarithmic derivative of the correlation functions matches those of
$h_{T_{Z}}(r)$ and then we set the correlation functions equal to
$h_{T_{Z}}(r)$ (beyond these state-dependence healing distances) (Bordbar &
Modarres rk30 (1998)). Finally, by solving the above differential equations
(Eqs. (II), (II) and (II)) numerically, we obtain the correlation functions.
## III Structure function
There are two types of structure functions, dynamic $S(\mathbf{k},w)$, and
static $S(\mathbf{k})$ structure functions. They measure the response of the
system to density fluctuations (Feenberg rkfeenberg (1969)).
The static structure function of a system consisting of $A$ particles is
defined as (Feenberg rkfeenberg (1969)):
$S(\mathbf{k})=1+\frac{1}{A}\int
d^{3}r_{1}d^{3}r_{2}e^{i\mathbf{k}.\mathbf{r}_{12}}\rho_{1}(\mathbf{r}_{1})\rho_{1}(\mathbf{r}_{2})[g(\mathbf{r}_{1},\mathbf{r}_{2})-1],$
(33)
where $\rho_{1}(\mathbf{r})$ is the one-particle density and
$g(\mathbf{r}_{1},\mathbf{r}_{2})$ is the pair distribution function. In
infinite systems, $\rho_{1}(\mathbf{r})$ is constant ($=\rho$) and $g$ is a
function of the interparticle distance
${r}_{12}=|\mathbf{r}_{1}-\mathbf{r}_{2}|$, therefore Eq. (33) takes the
following form,
$S(\mathbf{k})=1+\rho\int
e^{i\mathbf{k}.\mathbf{r}_{12}}[g(r_{12})-1]d^{3}r_{12}.$ (34)
For calculating the pair distribution function, we use the lowest order term
in the cluster expansion of $g(r_{12})$ as follows (Clark rk26 (1979)),
$g(r_{12})=f^{2}(r_{12})g_{F}(r_{12}),$ (35)
where $f(r_{12})$ is the two-body correlation function and $g_{F}(r_{12})$ is
the two-body radial distribution function of the noninteracting Fermi-gas,
$g_{F}(r_{12})=1-\frac{1}{\nu}l^{2}(k_{F}r_{12}).$ (36)
In the above equation, $\nu$ is the degeneracy factor, and $l(x)=3x^{-3}(sinx-
xcosx)$ is the statistical correlation function or the slater factor.
## IV Results and discussion
### IV.1 Correlation function
In Fig. 1, we have plotted our result for the correlation function of
symmetrical nuclear matter versus internucleon distance ($r_{12}=r$) employing
$UV_{14}$, $AV_{14}$ and $AV_{18}$ potentials at density $\rho=0.16\ fm^{-3}$.
Here the correlation functions are calculated from average over all states. We
can see that the correlation function is zero at the internucleon distance
$r<0.06\ fm$ for the three potentials. This distance represents the famous
hard core of the nucleon-nucleon potential. When the internucleon distance
increases, the correlation also increases until approaches to unity,
approximately at $r>3.8\ fm$. This means that at $r$ greater than the above
value, the nucleons are out of the range of nuclear force (correlation
length). The value of correlation for $AV_{18}$ potential has a maximum
greater than unity and then approaches to unity. However, for $UV_{14}$ and
$AV_{14}$ potentials, there is no such a maximum. In Fig. 2, we have plotted
the correlation function of asymmetrical nuclear matter employing $AV_{18}$
potential for different values of proton to neutron ratio ($pnrat=0.2,\ 0.6,\
1.0$) at different isospin channels ($nn$, $np$, $pp$). From this Figure, it
can be seen that for all values of $pnrat$, the correlation functions of $nn$
and $pp$ channels have the maximums greater than unity, whereas at $np$
channel, there is no such a maximum. This means that at $pp$ and $nn$
channels, the nucleon-nucleon potential is more attractive than at $np$
channel. We can see that at $nn$ and $pp$ channels, the maximum values of
correlation function decrease by increasing $pnrat$. We have found that at
$pp$ and $np$ channels, the correlation length decreases as $pnrat$ increases,
while at $nn$ channel, by increasing $pnrat$, the correlation length
increases. In addition, for each $pnrat$, the value of the correlation length
at $pp$ channel is greater than that of $np$ channel, and the correlation
length at $nn$ channel has a greater value than $pp$ channel. These have been
clarified in Table 1 in which the values of the correlation length for
different values of $pnrat$ at different isospin channels have been presented.
### IV.2 Pair distribution function
We know that the pair distribution function, $g(r)$, represents the
probability of finding two particles at the relative distance of $r$. In Fig.
3, we have plotted our results for the pair distribution function of
symmetrical nuclear matter versus internucleon distance with $UV_{14}$,
$AV_{14}$ and $AV_{18}$ potentials at density $\rho=0.16\ fm^{-3}$. Our
results are in a good agreement with those of others calculations employing
the $Reid$ potential (Modarres rk28 (1987)). Figure 3 shows that for $r$ in
the range of $1.1\ fm$ to $3.4\ fm$, the pair distribution function
corresponding to $AV_{18}$ potential is greater than those of $UV_{14}$ and
$AV_{14}$ potentials. This is due to the behavior of two-body correlation as
mentioned in the above discussions. In the Fermi gas model due to the absence
of interaction between nucleons, the pair distribution function is not zero
even in the small internucleon distances as shown in Fig. 3. But in the real
system, in which there is interaction between nucleons, the value of $g(r)$ at
$r<0.06\ fm$ is zero for the three potentials. The same as for the case of
correlation function, this distance represents the hard core of the nuclear
potential. From Fig. 3, it can be seen that the value of $g(r)$ increases as
the internucleon distance increases and finally approaches to unity,
approximately at $r>4\ fm$. In Fig. 4, we have plotted the pair distribution
function of asymmetrical nuclear matter employing $AV_{18}$ potential at
different values of proton to neutron ratio ($pnrat$) for $\rho=0.16\ fm^{-3}$
and different isospin channels ($nn$, $np$, $pp$). We can see that at all
channels, by increasing $pnrat$, the pair distribution function decreases,
corresponding to decreasing of the correlation. Besides, from Fig. 4, it can
be seen that for each $pnrat$, the pair distribution functions of $nn$ and
$pp$ channels have identical behaviors, while at $np$ channel, $g(r)$, behaves
differently compared to the other two channels. These are corresponding to the
behavior of correlation function at these channels.
### IV.3 Structure function
In Fig. 5, we have plotted our results for the structure function of
symmetrical nuclear matter versus relative momentum ($k$) with $UV_{14}$,
$AV_{14}$ and $AV_{18}$ potentials at density $\rho=0.16\ fm^{-3}$. There is
an overall agreement between our results and those of others calculated with
the $Reid$ potential (Modarres rk28 (1987)). From Fig. 5, it is seen that the
nucleon-nucleon interaction leads to the reduction of the structure function
of nuclear matter with respect to that of the non-interacting $Fermi$ $gas$
system. In Fig. 6, we have plotted the structure function of asymmetrical
nuclear matter with $AV_{18}$ potential at different isospin channels ($nn$,
$np$, $pp$) for different values of proton to neutron ratio ($pnrat$) and
$\rho=0.16\ fm^{-3}$. It is seen that similar to the pair distribution
function, the structure function of $nn$ channel is like that of the $pp$
channel, especially at higher values of $k$. We have found that this
similarity becomes more clear as $pnrat$ increases. However, there is a
substantial difference between structure functions of $np$ channel and $pp$
and $nn$ channels.
## V Summary and conclusions
Using the lowest order constrained variational (LOCV) method, we have computed
the correlation function, the pair distribution function and the structure
function of the symmetrical and asymmetrical nuclear matter. In order to
investigate the effect of nucleon-nucleon interaction on the properties of
nuclear matter, we have also computed the pair distribution function and the
structure function of noninteracting Fermi gas. Here, we have used $AV_{18}$
potential to represent the nucleon-nucleon interaction for the asymmetrical
nuclear matter. These calculations have been done at different isospin
channels. In the case of symmetrical nuclear matter, the calculations have
been done with $UV_{14}$, $AV_{14}$ and $AV_{18}$ potentials. There is an
overall agreement between our results and those of others calculated with the
$Reid$ potential. It was seen that the nucleon-nucleon interaction leads to
the reduction of the structure function of nuclear matter with respect to that
of the non-interacting Fermi gas system. We have found that at $np$ and $pp$
channels, the correlation length decreases as the proton to neutron ratio
($pnrat)$ increases, while at $nn$ channel, by increasing $pnrat$, the
correlation length increases. However, the behavior of the pair distribution
function at $np$ channel is considerably different pair from those of other
two channels. This is due to the difference between the behavior of
correlation functions of these channels. It was indicated that for higher $k$
and $pnrat$, the structure functions of $nn$ and $pp$ channels are identical,
corresponding to the similarity between the pair distribution functions of
these channels. We have also shown that the structure function at $np$ channel
was different from those of $nn$ and $pp$ channels.
## Acknowledgements
This work has been supported by Research Institute for Astronomy and
Astrophysics of Maragha. We wish to thank Shiraz University Research Council.
## References
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Table 1: The correlation length of asymmetrical nuclear matter employing $AV_{18}$ potential for different values of proton to neutron ratio at different isospin channels ($nn$, $pp$ and $np$). $pnrat$ | | correlation length$\ (fm)$ |
---|---|---|---
| $nn$ | $np$ | $pp$
0.2 | 2.95 | 2.09 | 2.18
0.6 | 3.36 | 1.97 | 2.11
1.0 | 3.39 | 1.94 | 2.06
Figure 1: The correlation function of symmetrical nuclear matter employing
$UV_{14}$, $AV_{14}$ and $AV_{18}$ potentials. The correlation functions have
been calculated from average over all states.
Figure 2: The correlation function of asymmetrical nuclear matter employing
$AV_{18}$ potential for $\rho=0.16\ fm^{-3}$ and different values of $pnrat$
at different isospin channels ($nn$, $pp$ and $np$).
Figure 3: The pair distribution function for symmetrical nuclear matter
calculated with $UV_{14}$, $AV_{14}$ and $AV_{18}$ potentials at density
$\rho=0.16\ fm^{-3}$. The pair distribution function corresponding to the
$fermi$ $gas$ is also brought for comparison.
Figure 4: As Fig. 2, but for the pair distribution function of asymmetrical
nuclear matter.
Figure 5: The structure function of symmetrical nuclear matter with $UV_{14}$,
$AV_{14}$ and $AV_{18}$ potentials at density $\rho=0.16\ fm^{-3}$.
Figure 6: As Fig. 2, but for the structure function of asymmetrical nuclear
matter.
|
arxiv-papers
| 2011-11-01T14:07:59 |
2024-09-04T02:49:23.848010
|
{
"license": "Public Domain",
"authors": "G. H. Bordbar and H. Nadgaran",
"submitter": "Gholam Hossein Bordbar",
"url": "https://arxiv.org/abs/1111.0206"
}
|
1111.0227
|
# Resonant control of polar molecules in an optical lattice
Thomas M. Hanna Joint Quantum Institute, NIST and University of Maryland, 100
Bureau Drive, Stop 8423, Gaithersburg, MD 20899-8423, USA Eite Tiesinga
Joint Quantum Institute, NIST and University of Maryland, 100 Bureau Drive,
Stop 8423, Gaithersburg, MD 20899-8423, USA William F. Mitchell Applied and
Computational Mathematics Division, National Institute of Standards and
Technology, 100 Bureau Drive Stop 8910, Gaithersburg, Maryland 20899-8910, USA
Paul S. Julienne Joint Quantum Institute, NIST and University of Maryland,
100 Bureau Drive, Stop 8423, Gaithersburg, MD 20899-8423, USA
###### Abstract
We study the resonant control of two nonreactive polar molecules in an optical
lattice site, focussing on the example of RbCs. Collisional control can be
achieved by tuning bound states of the intermolecular dipolar potential, by
varying the applied electric field or trap frequency. We consider a wide range
of electric fields and trapping geometries, showing that a three-dimensional
optical lattice allows for significantly wider avoided crossings than free
space or quasi-two dimensional geometries. Furthermore, we find that dipolar
confinement induced resonances can be created with reasonable trapping
frequencies and electric fields, and have widths that will enable useful
control in forthcoming experiments.
###### pacs:
03.65.Nk, 34.10.+x, 34.50.Cx
Ultracold gases of polar molecules are of interest for their long-range
dipolar interactions, which give them unique applications in areas such as
many-body phases Baranov (2008), quantum information DeMille (2002), and
precision measurement Sandars (1967); DeMille et al. (2000); Hudson et al.
(2002). Cold gases of LiCs Deiglmayr et al. (2008) and RbCs Sage et al. (2005)
have been formed with temperatures $T\lesssim 1\,$mK, and work continues to
produce degenerate gases Lercher et al. ; Cho et al. . Since the creation of a
near-degenerate gas of 40K87Rb with $T\lesssim 1\,\mu$K K.-K. Ni et al.
(2008), a number of studies have been done of its collision properties K.-K.
Ni et al. (2010); Ospelkaus et al. (2010, 2010); de Miranda et al. (2011). KRb
has an exothermic reaction producing $\mathrm{K}_{2}+\mathrm{Rb}_{2}$, which
occurs with almost unit probability when two molecules are sufficiently close.
This allows a simple description of the collision properties in terms of
universal physics Quéméner and Bohn (2010a, b); Micheli et al. (2010);
Idziaszek et al. (2010); Gao (2010); Kotochigova (2010); Quéméner and Bohn
(2011); Julienne et al. . Such techniques should also apply to other species
with reactive collisions, and to the quenching of any vibrationally excited
molecule. For many studies it is therefore desirable to keep molecules
separated, for example by confining the gas in a three-dimensional (‘3D’)
lattice cho or in a quasi-two dimensional (‘2D’) geometry with the molecules
polarized perpendicular to the plane (‘side-by-side’) de Miranda et al.
(2011); Quéméner and Bohn (2010a, b); Micheli et al. (2010); Ticknor (2010);
D’Incao and Greene (2011). This reduces the likelihood of molecules
approaching each other along the attractive ‘head-to-tail’ path which is
available in 3D.
In contrast to reactive molecules such as KRb, ground state NaK, NaRb, NaCs,
KCs and RbCs are not reactive, and so are available for experiments on longer
timescales and at higher densities where control of elastic collisions is
useful. The long range dipole-dipole interaction between two molecules
produces an anisotropic potential which is capable of supporting bound states
Kanjilal and Blume (2008). Tuning these bound states around a collision
threshold with an electric field allows resonant control of the interactions
Ticknor and Bohn (2005); Roudnev and Cavagnero (2009); Idziaszek et al.
(2010), in analogy to the magnetic and optical control that has been so useful
for neutral atoms Chin et al. (2010). Because three-body recombination can
still occur Ticknor and Rittenhouse (2010), isolating a pair of molecules in
an optical lattice site provides an ideal, loss-free environment for studying
the two-body energy spectrum. Such a scenario is analogous to several
experiments performed on atom pairs Syassen et al. (2007); Ospelkaus et al.
(2006); Thalhammer et al. (2006); Volz et al. (2006). Optical lattices have
also been used to tune atomic collisions through confinement induced
resonances Olshanii (1998); Petrov et al. (2000); sal ; Haller et al. (2010);
Fröhlich et al. (2011) (CIRs), which depend on the scattering length being
comparable to the characteristic length of the confinement. With the
interactions of polar molecules having an even longer range, it is reasonable
to anticipate easy creation of a CIR.
In this paper we study the states and control possibilities of two polar
molecules isolated in an optical lattice site, focussing on the specific
example of RbCs. We examine in detail the effects of tuning the lattice
parameters and electric field. We show that the optical lattice can be used to
increase the resonance width past what is possible in free space or 2D
geometries. We compare the eigenenergies obtained for a quasi-2D lattice site
to scattering calculations for a system with confinement in only one
direction, which accurately reproduce the avoided crossings and show their
utility for resonant control. Our studies show that tuning the confinement has
a significant effect on the collisional and bound state properties of the pair
of molecules, allowing the creation of useful CIRs.
Quantity | Definition | Value ($a_{0}$)
---|---|---
Mean scattering length | $\bar{a}=\frac{2\pi}{[\Gamma(1/4)]^{2}}(2m_{\textrm{r}}C_{6}/\hbar^{2})^{1/4}$ | 233.5
Confinement length | $\ell_{\mathrm{ho}}=\sqrt{\hbar/(2m_{\textrm{r}}\omega)}$ |
$\omega/2\pi=1$ kHz | | 5728
$\omega/2\pi=50$ kHz | | 810.5
Dipole length | $a_{\mathrm{\mu}}=m_{\textrm{r}}\mu^{2}/\hbar^{2}$ | $3.1\times 10^{4}$
Table 1: Characteristic length scales for the interaction of ultracold
molecules in an optical lattice, with values given for RbCs in parameter
regimes used in the present work. For the van der Waals coefficient, we use
$C_{6}=142129E_{h}a_{0}^{6}$ Kotochigova (2010), where $E_{h}=4.3597\times
10^{-18}$ J is the Hartree energy and $a_{0}=52.918$ pm is the Bohr radius. We
give the confinement length for optical lattice sites with frequencies
$\omega/2\pi=1\,$kHz and 50 kHz. The dipole length is given for RbCs molecules
with a dipole moment of $\mu=1.0\,$D, where
$\textrm{D}=0.39343ea_{0}=3.336\times 10^{-30}$ Cm is the Debye and $e$ is the
charge of an electron. Here, $m_{\textrm{r}}$ is the reduced mass.
We consider two ground state 87Rb133Cs molecules in a cylindrically symmetric
optical lattice site. In Table 1 we list length scales relevant to ultracold
molecular collisions and give representative values for the parameter regimes
used in this work. Typical van der Waals coefficients for polar molecules are
of order $10^{5}\,E_{h}a_{0}^{6}$ – $10^{7}\,E_{h}a_{0}^{6}$, much larger than
those for pairs of alkali atoms ($10^{3}\,E_{h}a_{0}^{6}$ –
$10^{4}\,E_{h}a_{0}^{6}$). However, the mean scattering length scales as
$\bar{a}\propto C_{6}^{1/4}$, giving a similar characteristic length to the
van der Waals part of the potential. We take the dipole moment
$\mu=\langle\hat{\mu}\rangle_{\textrm{z}}$ to be the expectation value of the
electric dipole operator $\hat{\mu}$ for the molecular ground state in the
electric field direction. We calculate this electric-field dependent quantity
according to the method of Ref. boh . The dipole length, tunable with an
electric field, is typically the largest length scale in the problem. For a
trapping frequency $\omega/2\pi=50\,$kHz, $a_{\mu}=\ell_{\textrm{ho}}$ for
$\mu=0.16\,$D. We note that the use of a strong dipole moment takes us beyond
the region of validity of pseudopotential approaches such as those of Refs.
Kanjilal et al. (2007); Derevianko (2003).
The molecules are assumed to be rigid rotors, aligned in the axial direction
by an applied electric field. We approximate the lattice site with a harmonic
trap and consider only the relative motion of the molecules. The combined
interaction and trapping potential is given by
$\displaystyle V(\rho,z)$
$\displaystyle=\frac{\mu^{2}}{r^{3}}\left(1-\frac{3z^{2}}{r^{2}}\right)+\frac{C_{12}}{r^{12}}-\frac{C_{6}}{r^{6}}+\frac{\hbar^{2}}{2m_{\textrm{r}}}\frac{m^{2}}{\rho^{2}}$
$\displaystyle+\tfrac{1}{2}m_{\textrm{r}}(\omega_{\rho}^{2}\rho^{2}+\omega_{z}^{2}z^{2})\,.$
(1)
Here, $\rho$ and $z$ are the relative radial and axial coordinates,
respectively. Also, $\omega_{\rho,\textrm{z}}=2\pi f_{\rho,\mathrm{z}}$, where
$f_{\rho,\textrm{z}}$ are the corresponding trapping frequencies. The
intermolecular separation is given by $r=\sqrt{\rho^{2}+z^{2}}$, and the
projection of the relative motion along the axis of symmetry is given by $m$.
We impose a repulsive short-range potential, $C_{12}/r^{12}$, setting the
$C_{12}$ coefficient such that the potential $C_{12}/r^{12}-C_{6}/r^{6}$
contains six bound states and gives a scattering length of 100$a_{0}$. We
neglect the anisotropic $C_{6}$ coefficient. While arbitrary, setting the
short range part of the potential in this way allows us to conveniently study
the important long range effects.
Although the collisions under consideration involve four atoms, the approach
described above is justified by the separation in energy scale between the
chemical bonds within the ground state dimers ($\sim$THz) and the collision
energy or bond between them ($\lesssim\,$MHz). We also note that the van der
Waals coefficient between polar molecules has contributions from the rotation
of the molecules as well as the induced dipole moments of the electron clouds.
An electric field polarizes the molecules and changes the rotational
contribution. We have calculated the extent of this change and checked that it
does not noticeably change the results presented here, as was the case in Ref.
Julienne et al. . Consequently, we neglect this effect. Because we confine
ourselves to a single collision channel, the resonances we find correspond to
shape resonances Chin et al. (2010), in which the potential experienced by a
colliding pair supports a near-degenerate quasi-bound state. We note that
Feshbach resonances, in which a colliding pair is coupled to a near-degenerate
bound state of a different spin configuration, are possible for the general
case of coupling between states of different molecular spin and rotational
quantum number.
We first study the two-body energy spectrum. We solve for eigenstates and
eigenvalues of the Hamiltonian with the potential of Eq. (1) using PHAML
Version 1.8.0 pha ; Mitchell and Tiesinga (2005), a parallel two-dimensional
finite element code for elliptic boundary value and eigenvalue problems. PHAML
features adaptive grid refinement of the discretized spatial coordinates to
concentrate the grid in areas where the wave function varies rapidly, and high
order elements to obtain an accurate solution. For these computations we used
eighth degree elements. Within PHAML, ARPACK leh was used to solve the
discrete eigenvalue problem, using the shift-and-invert spectral
transformation to compute interior eigenvalues, and MUMPS Amestoy et al.
(2001) to solve the resulting linear system of equations. The parallel
computations were performed on two nodes of a Linux cluster. A particular
advantage of the two-dimensional solver is its ability to readily account for
the anisotropic interaction and trapping potential. By contrast, an expansion
in spherical harmonics or non-interacting trap states will struggle to
accurately resolve the wavefunction without a very large basis set. However,
for analysis of the wavefunctions we calculate projections onto these
functions. We solve for the function $F(\rho,z)$, where the full wavefunction
is given by $\psi(\rho,z,\phi)=F(\rho,z)e^{im\phi}$. Our bound state
calculations consider only $m=0$, but in the scattering calculations described
below we will consider the effects of nonzero $m$. We study spherically
symmetric ($\omega_{z}=\omega_{\rho}$) and quasi-2D
($\omega_{z}\gg\omega_{\rho}$) geometries, with the dipoles always aligned
along the $z-$axis.
Figure 1: (a) Eigenenergies for two RbCs molecules in an optical lattice site
with $f_{z}=f_{\rho}=25\,$kHz. Trap states are adiabatically converted to
bound states as the dipolar interaction is increased. Points marked ‘b’ and
‘c’ correspond to the wavefunctions shown in the lower panels, which
illustrate the head-to-tail configuration at two different dipole moments. For
the three lowest energy trap states, the partial wave at $\mu=0$ is indicated.
We also give the partial waves into which these states have a significant
projection after being converted to bound states as $\mu$ is increased. Red
crosses indicate the non-interacting trap state energies. In (b) and (c), we
plot $\rho|\psi|^{2}$, and scale all lengths by the confinement length in the
$z$-direction, $\ell_{\textrm{ho}}^{\textrm{z}}$.
The eigenenergies of two RbCs molecules in a spherically symmetric lattice
site with $f_{z}=f_{\rho}=25\,$kHz are shown as a function of dipole moment in
Fig. 1a. We use the term bound states to refer to those that are bound when
the trap is adiabatically turned off. States close to the non-interacting trap
level energies,
$(n_{\textrm{z}}+1/2)\hbar\omega_{\textrm{z}}+(2n_{\rho}+|m|+1)\hbar\omega_{\rho}$,
are called trap states. Here, Bose symmetry allows the trap state quantum
numbers to be $n_{\textrm{z}}=0,2,4\ldots$, $n_{\rho}=0,1,2,\ldots$ and
$m=0,\pm 2,\pm 4,\ldots$. At zero dipole moment, the trap state energies are
affected by the van der Waals interactions. Dipole moments above approximately
0.1 D cause a significant change to these energies. A large number of avoided
crossings occur as trap states are brought into the potential by the
increasing dipolar attraction. All crossings are avoided, although some are
too narrow to be visible on the scale shown. States of different $\ell$ are
mixed by the dipolar interactions, with the broadest crossings occuring
between states of low $\ell$. For example, the first two trap levels converted
to bound states as $\mu$ is increased are primarily of mixed $s$\- and
$d$-wave symmetry, whereas the steeply descending state with a series of
narrow crossings near $\mu=0.31\,$D is concentrated in $\ell=10$. Figures 1b
and 1c illustrate the wavefunction near $E/h=-50\,$kHz for dipole moments of
$\mu=0.17\,$D and $\mu=0.498\,$D, respectively. We plot the function
$\rho|\psi|^{2}$, to more clearly illustrate both short and large length
scales. For the state at $\mu=0.17\,$D, vibrational nodes of the bound states
of the van der Waals potential can be seen as rings of constant
$r<0.25\ell_{\textrm{ho}}^{\textrm{z}}$. A $d$-wave component at long range
provides the head-to-tail configuration, with the wavefunction concentrated in
the region $\rho<|z|$. This makes $E$ decrease as $\mu$ increases. The state
at $\mu=0.498\,$D is much more strongly coupled between partial waves, and has
a correspondingly more complicated configuration.
Figure 2: Elastic collision rate as a function of $\mu$ for a trapping
frequency of $f_{\textrm{z}}=50$ kHz and collision energy of
$k_{\textrm{B}}\times 200$ nK, showing the contributions of the $m=0$, $\pm 2$
and $\pm 4$ partial waves (upper panel), and the corresponding $m=0$ bound
state calculation with $f_{z}=50\,$kHz and $f_{\rho}=1\,$kHz (lower panel).
The $z$ trap states with a substantial admixture are indicated for each bound
state not concentrated solely in $n_{\textrm{z}}=0$.
We now compare these results to the case of a quasi-2D optical lattice site,
with $f_{\textrm{z}}=50\,$kHz and $f_{\rho}=1\,$kHz. Eigenvalues are plotted
as a function of $\mu$ in the lower panel of Fig. 2. A quasi-continuum of
radial trap levels is found, spaced by $2f_{\rho}$, instead of the strongly
mixed states of Fig. 1. Only the $n_{\textrm{z}}=0$ trap state is within the
energy range shown, although some bound states have substantial admixture in
higher $z$ states, as indicated in the lower panel of Fig. 2. The quasi-2D
configuration has the effect of making the avoided crossings narrower than in
the 3D lattice case. An intuitive explanation of this effect is that a
sufficiently strong dipole moment allows molecules to overcome the confinement
which holds them side-by-side, and move to the attractive head-to-tail
configuration. The overlap between the asymptotic states representing these
two configurations is reduced by a higher trap aspect ratio.
It is desirable to attach meaningful quantum numbers to describe the states
that are observed. This is made difficult at large $\mu$ by the strong
anisotropic interactions; however, some approximate quantum numbers can be
used. States can be described by their projections onto the non-interacting
trap levels with quantum numbers $n_{\textrm{z}}$ and $n_{\rho}$, although
both van der Waals and dipole-dipole interactions mix these levels. We
indicate in Fig. 2 the states which are concentrated in higher $z$ trap
levels. As we discuss below, it is these states that provide the possibility
of creating CIRs. States can also be described by their vibrational quantum
number and projections onto spherical harmonics described by the orbital
angular momentum $\ell$, which are good quantum numbers in the zero-dipole
limit. Different $\ell$ also become strongly mixed at sufficient $\mu$, as
discussed above for the states shown in Fig. 1.
We now make the link between our calculated bound state energies and
scattering properties in a quasi-2D system. Our scattering calculations use
the coupled channels technique discussed in Julienne et al. , adapted for
elastic boundary conditions at short range. We use the potential of Eq. (1)
with $\omega_{\rho}$ set to 0, and propagate the scattering matrix in a basis
of spherical harmonics to a value of $r$ large enough to match onto the
$n_{\textrm{z}}=0$ trap state. The chosen collision energy of
$k_{\textrm{B}}\times 200$ nK, corresponding to $h\times 4$ kHz, is such that
higher $z$ trap states are not significantly populated at this separation.
Here, $k_{\textrm{B}}$ is the Boltzmann constant. We then propagate outwards
in $\rho$, extracting the scattering properties at long range using the
conventional tools of scattering theory Taylor (1972). The results are shown
in the upper panel of Fig. 2, and agree well with the bound state
calculations. The resonances at low dipole moment are widest and most isolated
from other features, making them the most useful for resonant control.
Contributions to the elastic collision rate coefficient from collisions with
higher $m$ make the resonance minima nonzero.
Figure 3: Confinement induced resonances created by tuning of
$f_{\textrm{z}}$. The top panel shows the elastic collision rate for a
quasi-2D trap with $f_{\textrm{z}}=50$ kHz. We have summed the contributions
of partial waves from $m=0$ to $|m|=4$. Solid lines correspond to
$\mu=0.3048\,$D and collision energies of $k_{\textrm{B}}\times 1\,$nK,
$k_{\textrm{B}}\times 100\,$nK and $k_{\textrm{B}}\times 200\,$nK, as
labelled. The dashed line ($\mu=0.3058\,$D) shows the sensitivity of the
resonance location to electric field variation. The bottom panel shows the
eigenenergies of the Schrödinger equation with the potential of Eq. (1), using
$\mu=0.3048$ D and trapping frequencies of $f_{\mathrm{z}}=50$ kHz and
$f_{\rho}=1$ kHz. The dashed green line shows the perturbative result of Eq.
(2). Dotted red lines and arrows show the intersections of this calculation
with scattering states of the given kinetic energies. These intersections
agree well with the calculated resonance locations of the upper panel.
Optical lattices have been used to control scattering lengths in neutral gases
Haller et al. (2010); Fröhlich et al. (2011) by making a state corresponding
to an excited trap level near degenerate with the colliding atoms. This
depends on the scattering length being of the same order as the characteristic
length scale of the confinement. With the large dipole length characteristic
of interactions between polar molecules, it is reasonable to expect that
similar confinement induced effects should occur. In the lower panel of Fig. 3
we show this effect for RbCs molecules, calculating eigenenergies as a
function of $f_{\textrm{z}}$ while maintaining $f_{\rho}=1$ kHz. We assume
$\mu=0.3048$ D, corresponding to an easily accessible electric field of 0.67
kV/cm. The figure shows six trap levels with $n_{\textrm{z}}=0$ and a range of
$n_{\rho}$, and a single bound state crossing these levels. As shown in Fig.
2, this bound state has substantial admixture in higher $z$ trap states. The
energy of the bound state therefore increases with $f_{\textrm{z}}$ faster
than the trap levels.
We have perturbatively calculated the change $\delta E$ in an eigenenergy from
changing $\omega_{\textrm{z}}$ to
$\omega_{\textrm{z}}+\delta\omega_{\textrm{z}}$, which results in
$\displaystyle\delta
E=\frac{1}{2}\left\langle\left(\frac{z}{\ell_{\textrm{ho}}^{\textrm{z}}}\right)^{2}\right\rangle\hbar\delta\omega_{\textrm{z}}\,.$
(2)
Here $\langle\cdots\rangle$ indicates calculating the expectation value with
respect to the selected wavefunction. The result for the bound state of Fig.
3, evaluated with the numerically obtained wavefunction at
$f_{\textrm{z}}=40\,$kHz, is shown as a green dashed line. The red dotted
lines correspond to the energy of a non-interacting pair of molecules in the
ground trap state, with $k_{\textrm{B}}\times 1$ nK and $k_{\textrm{B}}\times
200$ nK of relative kinetic energy. The points at which these cross the
perturbative calculation correspond well with the scattering resonances shown
in the upper panel of Fig. 3. For a relative kinetic energy of
$k_{\textrm{B}}\times 200$ nK, the feature has a width with respect to
$f_{\textrm{z}}$ variation of approximately 5 kHz, although accurate electric
field control will be necessary due to the resonance location being strongly
dependent on the dipole moment. This is shown by the dashed line in the top
panel, for a dipole moment of 0.3058 D, which corresponds to a change in the
applied electric field of approximately 2.5 V/cm. The temperature is also of
significance, as shown by the calculations for collision energies of
$k_{\textrm{B}}\times 100$ nK and $k_{\textrm{B}}\times 1$ nK, which produce
narrower peaks at lower trapping frequencies. This result illustrates that the
location and properties of the resonances can be controlled by manipulating
both the electric field and the confinement.
In conclusion, we have studied the role that an optical lattice can play in
controlling the collisional properties of nonreactive polar molecules. We have
shown that tight confinement allows for much broader avoided crossings, giving
a greater resonance width than is available in free space. We have also shown
that confinement induced resonances can be easily created, with the caveat
that their location is sensitive to the dipole moment. Measurements of
resonance locations would constrain the short range potential, for which we
studied just one example with a scattering length of 100 $a_{0}$. However,
this should not significantly alter the density of states or our finding that
RbCs will have several accessible resonances for dipole moments less than 0.5
D and trapping frequencies on the order of tens of kHz. These results will
therefore be of significance for upcoming experiments using non-reactive polar
molecules.
We acknowledge funding from an AFOSR MURI on ultracold molecules (T.M.H. and
P.S.J.) and partial funding from the ONR (P.S.J.). We thank Z. Idziaszek for
stimulating discussions.
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|
arxiv-papers
| 2011-11-01T15:48:56 |
2024-09-04T02:49:23.855461
|
{
"license": "Public Domain",
"authors": "Thomas M. Hanna, Eite Tiesinga, William F. Mitchell and Paul S.\n Julienne",
"submitter": "Thomas Hanna",
"url": "https://arxiv.org/abs/1111.0227"
}
|
1111.0337
|
Adrián Yanes
OpenWeather:
A Peer-to-Peer Weather Data Transmission Protocol
Faculty of Electronics, Communications and Automation
Department of Communications and Networking (Comnet)
Thesis submitted for examination for the degree of Master of Science in
Technology.
Otaniemi, Espoo, 31.08.2011
Thesis supervisor and instructor: Prof. Jörg Ott
Aalto University
School of Electrical Engineering | Abstract of the
Master’s thesis
---|---
Author: Adrián Yanes Title: OpenWeather: a peer-to-peer weather data
transmission protocol Date: 31.08.2011 Language: English Number of pages: 115
+ 23
---
Faculty of Electronics, Communications and Automation Department of
Communication and Networking Professorship: Networking Technology Code: S-38
Supervisor: Prof. Jörg Ott
The study of the weather is performed using instruments termed _weather
stations_. These weather stations are distributed around the world, collecting
the data from the different phenomena. Several weather organizations have been
deploying thousands of these instruments, creating big networks to collect
weather data. These instruments are collecting the weather data and delivering
it for later processing in the collections points. Nevertheless, all the
methodologies used to transmit the weather data are based in protocols non
adapted for this purpose. Thus, the weather stations are limited by the data
formats and protocols used in them, not taking advantage of the real-time data
available on them. We research the weather instruments, their technology and
their network capabilities, in order to provide a solution for the mentioned
problem. OpenWeather is the protocol proposed to provide a more optimum and
reliable way to transmit the weather data. We evaluate the environmental
factors, such as location or bandwidth availability, in order to design a
protocol adapted to the requirements established by the automatic weather
stations. A peer to peer architecture is proposed, providing a functional
implementation of OpenWeather protocol. The evaluation of the protocol is
executed in a real scenario, providing the hints to adapt the protocol to a
common automatic weather station.
Keywords: P2P, peer to peer, weather stations, real-time, protocol
standardization, embedded system, IETF, RFC
_If you want to accomplish something in the world, idealism is not enough, you
need to choose a method that works to achieve the goal. In other words, you
need to be “pragmatic”._
Richard Matthew Stallman
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Preface
Before I started this thesis, my knowledge about weather stations and the
technologies behind them was pretty limited. Nevertheless, in some way the
weather data transmission got my attention. Probably my preference for open
systems, libre software and my passion for network protocols, was the trigger
to look for a topic that combines all of these areas.
During this thesis my main goal has been to show how a modern instrument as an
automatic weather station, can be improved using concepts brought from open
and standard technologies.
Furthermore, the impact that the weather has in our everyday deserves a deeper
attention from the engineering point of view. Although the scientists are
doing a great job finding new ways to understand the weather, they really need
improvements in the technology field, to achieve even more better results.
OpenWeather looks for a digression. This research is looking for the attention
of the scientists and the industry; for those vendors that are manufacturing
instruments without a common standard, and for those scientists that are
experimenting issues with the weather data acquisition. Both communities must
find an agreement to standardize the methods and the technologies to transmit
the weather data.
I truly think that if we start using protocols designed having in
consideration the characteristics of the weather data, the result of their use
will change completely the vision about what is the weather, what causes it
and how it can be predicted.
Acknowledgements
Too many people have been involved directly or indirectly in this thesis,
however, nothing of this would have happened without the support of my
parents, Luisa Pilar and José Emilio. Thanks for your support during all these
years; from the beginning you have trusted me blindly, thanks for teaching me
the value of the knowledge, I love you.
Thanks to the Aalto University, and especially to professor Jörg Ott, for
supervising my thesis.
Thank you to Antti Lauri from the SMEAR project & University of Helsinki, for
inviting me to spend a weekend in the premises of the SMEAR project in
Tampere, having the possibility to discuss with some engineers and scientists,
the issues and particularities of one of the biggest weather stations in the
world. Also thanks to Pasi P. Aalto and Erkky Siivola, from the University of
Helsinki. Your patience and experience with weather stations have been really
useful to me.
I really need to say a big thanks to Vaisala corporation, especially to Pekka
Korhonen and Jing Lin, thanks for providing me with all the necessary hardware
-included an amazing last generation weather station- to perform my research
and for the support offered.
A big thank you to Gonzalo Mariscal, for supporting me and my constant
requests during two years.
Several friends have been involved in all the OpenWeather matters, thanks to
the guys of the Polyteknikkojen Radiokerho (Radio Club), to provide me the
materials and the place to install the weather station. Thanks to Jose Azeredo
Lima, for his support with some of the figures. Thanks to those friends that
probably read this thesis even more times than me: Borja Tarraso, Sergey
Vetrogronov and David Fernández, thanks for your dedication helping me, your
support has been really important for me.
Finally, I want to dedicate this thesis not to one, two or three persons, I
want to do it to a community, the community of the libre and open source
software. To all of those developers that are working so hard to make the
world a better place, without your work this thesis would never exist:
respect.
Otaniemi, Espoo, August 2011.
Adrián.
###### Contents
1. 1 Introduction
1. 1.1 Background
2. 1.2 Problem statement
3. 1.3 Research objectives and scope
4. 1.4 Motivations
5. 1.5 Outline of the thesis
2. 2 The impact of the weather data
1. 2.1 Weather data collection and diffusion
1. 2.1.1 Governmental organizations
2. 2.1.2 Corporations
3. 2.1.3 Individuals
4. 2.1.4 Weather data publication
2. 2.2 Summary
3. 3 Infrastructure for the weather data
1. 3.1 A meteorological instrument
1. 3.1.1 Industrial design
2. 3.1.2 Electronics and data handling
3. 3.1.3 Software
4. 3.1.4 Networking
2. 3.2 Meteorological data networks
1. 3.2.1 Common architectures
2. 3.2.2 Data distribution
3. 3.3 Summary
4. 4 State of the art in the weather data transmission
1. 4.1 The evolution of the digital interfaces in a weather instrument
2. 4.2 The absence of a protocol
3. 4.3 The missing standard
4. 4.4 Data transmission and Automatic Weather Stations
5. 4.5 Summary
5. 5 Introduction to OpenWeather
1. 5.1 Overview and goals
1. 5.1.1 Improvements in the current technology
2. 5.1.2 The role of OpenWeather and data spreading
3. 5.1.3 Contribution to the current methodologies for weather data acquistion
4. 5.1.4 Impact on weather instrument industry
2. 5.2 Basic functionality of OpenWeather
1. 5.2.1 Peer to Peer Architecture
2. 5.2.2 Service Oriented Architecture in nodes
3. 5.3 Summary
6. 6 Protocol specification
1. 6.1 Definitions
2. 6.2 Architecture
1. 6.2.1 Standards used for data units
2. 6.2.2 Nodes
3. 6.3 Protocol operations
1. 6.3.1 Session establishment - Peer handshake
2. 6.3.2 Service discovery
3. 6.3.3 Real-time data retrieval
4. 6.3.4 Data on demand
4. 6.4 Data messages
1. 6.4.1 Header
2. 6.4.2 Types of data messages
3. 6.4.3 Protocol codes
4. 6.4.4 MetaInfo data field
5. 6.4.5 Data field
6. 6.4.6 Internal protocol data
5. 6.5 Protocol considerations
6. 6.6 Summary
7. 7 Experimental evaluation setup
1. 7.1 Scenario
2. 7.2 Prototype implementation
1. 7.2.1 Technologies used
2. 7.2.2 Software Architecture
3. 7.3 Testing
1. 7.3.1 Test 1: Handshake between nodes
2. 7.3.2 Test 2: Service discovery
3. 7.3.3 Test 3: Real-time data retrieval
4. 7.4 Summary
8. 8 Conclusions
9. References
## Glossary
ACL
Access Control List
API
Application Programming Interface
APRS
Automatic Position Reporting System
ASCII
American Standard Code for Information Interchange
ASOS
Automated Surface Observing System
AWOS
Automated Weather Observing System
AWS
Automatic Weather Station
AX.25
Link Access Protocol for Amateur Packet Radio
BSON
Binary-JSON
CPU
Central processing unit
CSV
Comma-Separated Values
CWOP
Citizen Weather Observer Program
DDoS
Distributed denial-of-service
DNS
Dynamic Name Server
DoS
Denial-of-service
ECMWF
European Centre for Medium-Range Weather Forecasts
FMI
Finnish Meteorological Institute
FTP
File Transfer Protocol
GDPFS
Global Data-processing and Forecasting System
GOS
Global Observing System
GPRS
General Packet Radio Service
GSM
Global System for Mobile Communications
GTS
Global Telecommunication System and WMO Information System
GUI
Graphical User Interface
HTTP
Hyper Transfer Text Protocol
ICAO
International Civil Aviation Organization
IETF
Internet Engineering Task Force
IO
in / out
IP
Internet Protocol
ISO
International Standard Organization
JSON
JavaScript Object Notation
kB
Kilobyte
kbit
kilobits
MB
Megabyte
Mbits
Megabits
METAR
Meteorological Service For International Air Navigation
MHz
Megahertz
NAT
Network address translation
NMEA-0183
National Marine Electronics Association 0183
NOAA
National Oceanic and Atmospheric Administration
NTP
Network Time Protocol
OS
Operating System
P2P
Peer to peer
PROM
Programmable Read-Only Memory
PTH
Pressure, Temperature, Humidity
PTU
Pressure, Temperature and Humidity
RAM
Random-access memory
RFC
Request for Comments
RS-232
Recommended Standard 232
RS-422
Recommended Standard 422
RS-485
Recommended Standard 485
RTT
Round-trip time
SDI-12
Serial Data Interface at 1200 Baud
SHA
Secure Hash Algorithm
SI
Système international d’unités - International System of Units
SMB
Server Message Block
SOA
Service-oriented architecture
TCP
Transmission Control Protocol
TLS
Transport Layer Security
TSV
Tab-separated values
UML
Unified Modeling Language
UMTS
Universal Mobile Telecommunications System
URI
Uniform Resource Identifier
URL
Uniform Resource Locator
USB
Universal Serial Bus
UTC
Coordinated Universal Time
UTF
Universal Character Set - Transformation Format
UTM
Universal Transverse Mercator
WMO
World Meteorological Organization
XML
eXtensible Markup Language
###### List of Figures
1. 1.1 Common scenario to collect, transmit, manipulate and storage data in a weather station.
2. 2.1 Layers abstracted in the weather collection data workflow.
3. 2.2 Weather data collection workflow. World Climate Data and Monitoring Programme.22footnotemark: 2
4. 2.3 Meteoclimat screenshot showing weather forecasts.55footnotemark: 5
5. 2.4 FMI website [23] spreading local weather observations.
6. 3.1 NOAA weather buoy [32], example of a complex an robustness AWS.
7. 3.2 Generic AWS with different instruments and materials combination.
8. 3.3 Abstracted electronic schema of an AWS reading data from one sensor.
9. 3.4 Types of storages available in an AWS.
10. 3.5 Wiring schema showing how to re-wire the AWS to use RS-422.
11. 3.6 Location of the datalogger in an AWS.
12. 3.7 Screenshots of some popular desktop applications for AWS.
13. 3.8 AWS es connectivity schema.
14. 3.9 Comparison between pure star-topology against star-topology and the connectivity technologies used in AWS es.
15. 3.10 Example of an AWS using APRS at Helsinki area99footnotemark: 9.
16. 3.11 Weather data message using APRS [24].
17. 4.1 Weather data workflow, normal AWS VS METAR’s AWS.
18. 4.2 Example of an AWS transmitting weather data.
19. 4.3 Example of a AWS and a datalogger transmitting weather data.
20. 4.4 Workstation taking the role of the weather data transmission.
21. 5.1 Comparison of the currently centralized architecture provided by the industry against OpenWeather architecture.
22. 5.2 Example of a OpenWeather’s JSON object inside of data message.
23. 5.3 Example of an API call through HTTP and OpenWeather.
24. 5.4 Middle-layer for data normalization.
25. 5.5 OpenWeather stack over TCP/IP.
26. 5.6 Uses cases available in OpenWeather via SOA.
27. 6.1 Session establishment sequence diagram.
28. 6.2 Service discovery sequence diagram.
29. 6.3 Real-time data sequence diagram.
30. 6.4 On demand data sequence diagram.
31. 6.5 OpenWeather data message structure.
32. 6.6 OpenWeather MetaInfo data field with data array elements.
33. 6.7 OpenWeather’s MetaInfo data field with the data array elements.
34. 7.1 AWS installed to simulate a real scenario.
35. 7.2 Network topology used in the evaluation setup.
36. 7.3 Software prototype conceived.
37. 7.4 Unified Modeling Language (UML) diagram of the prototype.
38. 7.5 Prototype use case diagram.
39. 7.6 UML diagram of the library.
40. 8.1 GUI of the OpenWeather prototype -AWS control-.
41. 8.2 GUI of the OpenWeather prototype -Node control-.
42. 8.3 GUI of the OpenWeather prototype -Data message visualizer-.
###### List of Tables
1. 3.1 Comparison between standards and bandwidth offered.
2. 4.1 Example of data format used in a specific AWS to communicate the barometric pressure.
3. 4.2 Another example of data format used in a specific AWS to communicate different data as temperature or barometric pressure.
4. 4.3 Some acronyms used in METAR format [36].
5. 4.4 Example of command configuring the baud rate of the digital interface in an AWS.
6. 4.5 Example of command asking for Pressure, Temperature, Humidity (PTH) data.
7. 5.1 Comparison of one vendor format against OpenWeather JSON format.
8. 6.1 Data units implicit on the data fields.
9. 6.2 Example of CWOP’s AWS identification.
10. 6.3 ID partially based on CWOP notation.
11. 6.4 ID’s partially based in CWOP’s identification system.
12. 6.5 IDs based in the SHA-256 result of the CWOP notation.
13. 6.6 Header field (Header object) in a data message of OpenWeather.
14. 6.7 MetaInfo field in a data message of OpenWeather protocol.
15. 6.8 Bandwidth field in a data message of OpenWeather
16. 6.9 Bandwidths equivalency in Bandwidth data field.
17. 6.10 ID’s field in a data message of OpenWeather protocol.
18. 6.11 Keep-Alive field in a data messages of OpenWeather protocol.
19. 6.12 Location field in a data messages of OpenWeather protocol.
20. 6.13 Peer-IP & Port fields in a data message of OpenWeather protocol.
21. 6.14 Peers-Requested field in a data messages of OpenWeather protocol.
22. 6.15 Timestamp field in a data message of OpenWeather.
23. 6.16 Update-Interval field in a data messages of OpenWeather protocol.
24. 6.17 Version field in a data message of OpenWeather.
25. 6.18 MetaInfo data field (MetaInfo object) in a data message of OpenWeather.
26. 6.19 Data field in a data message of OpenWeather protocol.
27. 6.20 PTU real-time data in the raw format used by the AWS.
28. 6.21 PTU data field in a data message of OpenWeather protocol.
29. 6.22 PTU data field with real-time data in a data message of OpenWeather protocol.
30. 6.23 Wind data field in a data message of OpenWeather protocol.
31. 6.24 Wind data field with real-time in a data message of OpenWeather protocol.
32. 6.25 Precipitation data field in a data message of OpenWeather protocol.
33. 6.26 Precipitation data field with real-time in a data message of OpenWeather protocol.
34. 6.27 Real-time data message of OpenWeather protocol.
35. 6.28 Real-time data message of OpenWeather protocol.
36. 6.29 Peer’s list exchange in OpenWeather protocol.
37. 6.30 Services list availability request.
38. 6.31 Peer’s list exchange in OpenWeather protocol.
39. 7.1 Hardware and OS specifications of the evaluation setup.
40. 7.2 Data messages transmitted between _Node 1_ and _Node 2_.
41. 7.3 TCP flow sequence between _Node 1_ and _Node 2_.
42. 7.4 Data messages transmitted between _Node 3_ and _Node 4_.
43. 7.5 TCP flow sequence between _Node 3_ and _Node 4_.
44. 7.6 Data messages sent between _Node 3_ and _Node 4_.
45. 7.7 TCP flow sequence between _Node 1_ and _Node 2_.
## Chapter 1 Introduction
From the beginning of the time, the weather has been an important factor in
the human life. Its impact of it in our everyday, gives as result that during
centuries we have been trying to understand and predict it as much as
possible.
We all are familiar with some weather concepts, because it really has an
impact on how we proceed in our life. For instance, it is really common to
check the forecast before we start some outdoor activity or even without any
special reason, only to know which kind of atmospherical conditions we are
going to experiment the following days; this is possible by the meteorology.
The science of meteorology takes the role of the scientific study of the
atmosphere, this implies to know certain phenomena behave and which kind of
predictions can be made based on them, and of course the impact of them in our
lives. To achieve this goal, the science of meteorology has been developing
different techniques and methods to measure and collect the necessary data to
make these predictions. The human history is full of inventions of different
instruments designed to make this possible. In the past, these instruments
were based just in mechanical principles with a high limited accuracy.
Nowadays, we can find a huge set of alternatives based in digital mechanisms
which allow us to predict the weather and understand the atmosphere phenomena
with high precision and accuracy; giving us a better knowledge of our
environment and at the end making our life easier. Even if in the last years
the transition from pure mechanical instruments to the digital technology has
been really fast, certain parts still have not been renovated or are under
development.
The purpose of this thesis is to study some possible improvements of these
parts, more specifically in the protocols used to transmit the weather data
collected in different instruments to the places in which the data is
processed for its broadcasting.
When I started researching some weather instruments their technology caught my
attention, mainly in all the aspects of measuring a phenomenon with precision
and feasibility, and at the same time I was confused about how the protocols
used in them are full of legacy and low efficiency, in terms of data
transmission and real time data availability.
Nowadays, we have functional and reliable weather data systems to study the
different phenomena, however, the potential of the real time data gets blocked
by the methods used on the weather data collection. Even if at the end, we
have the capability to process and interpret the data, a huge amount of effort
is needed to make this happen, due to the methods and technology used for the
collection. This fact got my attention when I was trying to find some research
area in which the protocols and the information theory could help to make this
process more useful, faster and reliable.
After understanding and verifying how the weather instruments work, I found
really important to ask some meteorological scientists what the state of the
art is, concerning atmosphere data collection. I had the great opportunity to
visit the SMEAR [40] project for a weekend, study how the data is collected,
transmitted, processed and stored. At the same time, the scientists that are
using this data to study the atmosphere, could confirmed that some huge
improvements can be made in order to improve the data transmission (this
affirmation is mostly based on the technical issues that they are
experimenting in their research).
This fact and the interest in peer to peer protocols and the real data
transmission, were the final trigger to start this thesis and try to find a
possible solution to improve the speed and reliability of the weather data
transmission.
Applying the concept of "peer" to any group of sensors which are collecting
weather data and assuming that also the scientist is a peer that fetches and
exchanges data with other scientists (also considered peers), it was the
foundation to research, looking for a protocol that allows the weather
stations to exchange and route data with other weather stations and at the
same time provides a infrastructure to access data collected in real time.
### 1.1 Background
A weather instrument is an artifact which main task consists in the data
collection from one to multiple atmosphere phenomena. These instruments are
designed thinking in a specific use case: a particular natural phenomenon, and
at the same time with a well defined goal: the collection of data that helps
to study and predict phenomenon.
Nowadays, we can find several solutions to achieve this goal. Science has
found different ways to measure the same phenomenon in different ways and with
different reliability. However, common techniques are used around world to
measure the same phenomenon. Sometimes the reasons for using a certain
technique can go from the complexity and reliability of it, to the cost of it.
The standard way to measure a particular phenomenon is developing a specific
instrument (also named _sensor_) for it, this instrument is able to measure
and understand it better.
Some popular concept to refer these sensors is _"weather stations"_ ,
nevertheless, this term is not correct at all due to the amount of instruments
in a weather station can be barely different compare with other vendors’
instruments. Notwithstanding, this term is accepted as common to refer the
group of sensors used to collect weather data (we will use this concept from
now on to refer to a group of sensors creating an identity named "weather
station").
The following list111These sensors are an example based on the market’s offer,
notwithstanding the amount of different sensors to measure the phenomena
increases really fast, being difficult to track them all. enumerates some
common instruments in a weather station:
* •
Thermometer for measuring air and sea surface temperature
* •
Barometer for measuring atmospheric pressure
* •
Hygrometer for measuring humidity
* •
Anemometer for measuring wind speed
* •
Wind vane for measuring wind direction
* •
Rain gauge for measuring precipitation
* •
Disdrometer for measuring drop size distribution
* •
Transmissometer for measuring visibility
* •
Ceiling projector for measuring cloud ceiling
All of these instruments have a defined mechanism to measure a specific
phenomenon and collect the data to be processed later. These instruments or
sensors are applying some physic principle to get this data and converted it
into digital information for future transmission.
After the data is collected in the instrument222A device named datalogger is
involved in this process., it is transmitted to some organization, such as a
meteorological institute, to interpret the data and get some conclusions
concerning the current status of the weather and future predictions.
With information collected in different instruments around the world, we can
know the status of the weather and how it will be in the future, all of these
weather stations around the world are "weather data pickers", and the success
of the final weather prediction resides in the efficiency and reliability in
which this data is collected, transmitted and processed.
For a while, all of this process has been optimized in several ways, like
creating better instruments, infrastructures and organizations focused only in
this field. However, the standardization process only impacted on measure
techniques and data units, putting in a secondary plane, other parts of the
process such as communication protocols, digital interfaces used, etc.
### 1.2 Problem statement
The nature of the data collected in the weather stations involves to place
them in different locations around the world. It is creating a trickier
scenario for the data collection. Multiple weather stations are located in
inaccessible places, but their location is mandatory to deploy feasible models
for weather predictions. Commonly, these instruments are placed in different
locations in which sometimes the environment is not friendly at all to be
combined with digital technology; some examples of these are isolated places
such as mountains, roads or forests. These environmental conditions bring
issues as lower bandwidth availability, difficulties to get enough energy
24x365 and the variable weather conditions in which some instruments are
subdue with the implications of these in terms of lifetime.
The industry has been developing different instruments to achieve this
objective and avoid the mentioned issues. However, the main effort has been to
develop instruments with high accuracy, low power consumption, resistance, and
small size; resting importance to the methods used in the transmission
efficiency of the data collected.
It is a fact that these instruments are getting more complex, reliable and
tiny with the time. Nevertheless, there is non defined standard to transmit
and process the data collected from the instruments to the locations in which
this data is useful (meteorological organizations, computation centers,
databases, etc). The common practice is that the vendors choose their own data
format / protocol for this purpose, and depending on the manufacturer the
instrument formats and transfers the information using some standard for
peripheral devices such as Recommended Standard 232 (RS-232), Recommended
Standard 422 (RS-422), Recommended Standard 485 (RS-485), or Universal Serial
Bus (USB). At the same time one of the following serial communications
protocol is commonly used to transmit the collected data:
* •
RAW American Standard Code for Information Interchange (ASCII) 333The concept
of RAW refers to a serial communication in which is not used any special data
format, just data formatted using ASCII as character-encoding scheme.
* •
Serial Data Interface at 1200 Baud (SDI-12)
* •
National Marine Electronics Association 0183 (NMEA-0183)
These are the standards that the industry established to transmit the data
from the instruments that they are manufacturing. However, the mentioned
standards are generic for serial communications data transmission, without any
direct or indirect relation or adaptation to the weather data. That means that
the industry chooses only to take care of the data transmission for their own
instruments, creating their own data formats, timings of transmission, data
definition and so on. This common practice between vendors is causing the non-
existence of an international standard and by default the incompatibility of
these instruments with others brands, plus the possibility to combine the
output data of different instruments from different brands.
The use of a non-adapted protocol for the data transmission decreases the
efficiency and the possibility of a easy manipulation of the data. Even
assuming that the industry chose this way to transmit the data based in the
mainstream digital solutions, in terms of serial communications, is possible
and feasible to deploy a standard to format and transmit this information in a
more optimized and reliable way; this will imply the participation of
different vendors to standardize this format. The process of standardization
is a well-known practice in different fields of the industry due to the
advantages that it brings in terms of compatibility, interoperability, safety,
repeatability, or quality; at the same time standardization is supported in
multiple cases (depending of the industry) for international laws.
Thus, the choice made by the industry makes the optimization of the data
manipulation really painful, in addition, it is rare to have one point of
weather collection with only one brand of instruments. It entails that at the
end of the data transmission, the data collection scenario must be combined
with different software from different vendors and different parameters. This
makes the process of the weather data collection even more arduous, since the
original format in which the data is transmitted is completely useless and
must be converted to be combined with other data.
The absence of a standard is forcing to pre-process the weather data after its
transmission, even if this is something needed in any network data
transmission at some point; the format used in the process can save a lot of
CPU cycles, memory and bandwidth. This absence forces the weather data
collection centers to convert the data in a useful format for future
computation, and this is happening through custom software developed by the
vendor’s instruments or in some cases, custom software developed by the
organization itself. As an example of this, the SMEAR project[40] has
developed several parsers and scripts to manipulate this data before it can be
processed, wasting time and resources that can be easily solve through a
standardization.
We needed to highlight that most of the end users of this software are
scientists that need the data to get some conclusions about the weather. It
means that at the end of the data collection workflow, it is manipulated
through software focused in mathematics computation like MathLab, which does
not support any data format used by the weather instruments, forcing the
scientists to have the data in dummy formats as Comma-Separated Values (CSV),
Tab-separated values (TSV), or just plain text, to be able to use it.
The following figure shows an example of how the data is processed and where
the conflictive points are:
Figure 1.1: Common scenario to collect, transmit, manipulate and storage data
in a weather station.
As it is observable in the figure 1.1 the parsing and the implication of
specific software in the process, is causing the implementation of unnecessary
subprocess as parsing, packaging and data conversion. At the same time, the
process described is decreasing the possibilities to have easily accessible
information in real time.
Though meteorology needs big amounts of data collected in different places and
the analysis of this data is made using different times frequencies, we can
not ignore how useful the data of our environment can be if its accessible in
real time. As example of this can be that industry has been focusing, in the
last years, on developing technologies that allow the users to get information
on demand and in real time, this is supported by the principle that with more
detailed and updated information we can act with more precision and
feasibility.
The absence of a weather data transmission protocol is impeding us to know how
powerful can be the combination of multiple weather data sources in real time.
It can provide the mechanisms to deploy different models and perform analysis
of the data based in the real current situation of the weather, regardless the
location or brand of the weather station. Even if nowadays we have enough
precision understanding the atmosphere phenomena to predict future weather
conditions, we still need to advance in the physics to deeper understand the
impact of these phenomena and how they work, providing us a better knowledge
of our environment, and at the end, improving our quality of life.
Currently the weather data information is collected in real-time (because the
sensors are taking samples of the current environment), notwithstanding the
technology used in the process of the data transmission does not take this in
consideration, non using a standardized and optimized process for this
specific data.
This fact is, at some point, blocking the possibility to explore how useful
this data could be for us, but it is not accessible at all because engineering
issues. On the other hand, the absence of a common protocol even to exchange
non real-time data, generates a big amount of issues in terms of data
combination and comparison; causing several problems of incompatibility
between the organizations focus in the weather study, and forcing the use of
extra resources in operations such as data normalization (something that can
be fixed through a common data format).
### 1.3 Research objectives and scope
The purpose of this thesis is to identify the points in the weather data
transmission in which the process is not optimized according to the nature of
the data. At the same time, a protocol is proposed as proof of concept,
showing how the weather data transmission can be improved without too much
effort from the vendor’s side.
The foundation of this research is to find a path having in mind a real
scenario as the SMEAR project[40], in which the process of the data
transmission and manipulation can be improved offering new use cases for the
data, in terms of real time acquisition, manipulation and storage.
The following points identify the approach of the research briefly:
* •
Identify the blocker points in data transmission concerns
* •
Study how the weather data transmission and manipulation can be improved
* •
Develop a protocol prototype specification that provides an improvement in the
current scenario
As final objective the author is looking forward to motivate the vendors to
start a standardization process to improve the mentioned problems. Based on
the opinions shared with atmospherical scientists, the absence of accessible
real-time comes from the engineering side, and it is needed to develop some
technology that ensures an easy a feasible method to access this data.
### 1.4 Motivations
After working with weather instruments, understanding how they work and how
they transmit the data, I noticed the issues previously exposed. However, my
vision was not enough to be sure about the key-problem treated (because it was
only based in end-user weather instruments). When I had the opportunity to see
how the biggest weather station in the world (concerning gas measurements) is
fetching, transmitting and manipulating data, and at the same time, talk with
some scientists about my suspicions were confirmed by them: this process can
be optimized.
In addition, the absence of a standard in something so important as the
weather data transmission, gave me enough reasons to perform this research,
based in the idea that maybe some conclusions can be directly applied to the
industry.
Finally, my vision about certain user rights is implied in this research as
well. I am convinced that a society informed has always more possibilities to
have a better quality of life. In the last years several misunderstandings and
confusions have been happening concerning the current situation of the climate
in our planet. Unfortunately, the absence of accessible and understandable
information generates confusion in our society. Although this is an issue in
which the science has been leading from the beginning of the time, I support
that the improvement of the methods used in the science, are always helping us
to make the information more accessible, hence to have the possibility to
spread the knowledge with less effort. In this case, I think this research can
contribute to improve how we transmit and understand the weather data
paradigm. It is a good moral reason to me to perform this study.
History shows how the proper use of technologies adapted to specific
scenarios, promotes the advance of linear sciences as Maths or Physics, and
always these new findings are supported by new technologies. To find these new
technologies, it is needed to analyze from the engineering point of view,
which things can be improved and how; this philosophy turns this thesis in an
exercise to find how a science as meteorology can benefit from communications
technologies around it if they are optimized for its needs.
### 1.5 Outline of the thesis
This thesis is structured as follows: the second chapter gives a general
overview about how the weather data collection is structured, and which
organizations are interacting on this activity. The third chapter explains
briefly how a weather instrument works and what kind of technologies are
involved in the process, after that, it is analyzed how the meteorological
networks composed by these instruments work. The fourth exposes the technical
deficiencies found by the author on the weather data transmission. In chapter
five the OpenWeather protocol is presented, a prototype protocol developed by
the author, adapted to the needs exposed in the previous chapters. Chapter six
specifies from the technical point of view how the protocol works, its
operations and architecture, accompanied with justification of the technical
decisions taken on the thesis, concerning its implementation. Chapter seven
evaluates the implementation of the protocol in a real scenario based in a
specific weather instrument. Finally, chapter eight summarizes the conclusions
of this thesis.
## Chapter 2 The impact of the weather data
Even if it is obvious for all of us, weather is one of the most important
factors of the environment, with a high impact in our life. At the same time
most of us are not familiar with the repercussions of the weather, what is
causing different phenomena and the implications of them. Finally, our needs
concerning the weather are limited by the availability of the data that is
given to us. The role of the weather forecast broadcasting resides in
different organizations. However, the advantages of the technology are
bringing us the capability to have a more frequent and reliable access. The
following sections analyzes how the weather data is spread and in which points
of its diffusion can be improved.
### 2.1 Weather data collection and diffusion
Depending of the region of the world, we can find more or less geographical
locations in which a weather station has been placed to collect information
about different phenomena. It is important to clarify they are several
categories of phenomena with different needs in terms of data collection
requirements. In addition, we have different units and time frequencies to
make this data useful.
Fortunately, nowadays, most of the known phenomena have a solid basement of
understanding, meaning this that we can measure them and get some conclusions
and to act in consequence. The Système international d’unités - International
System of Units (SI) is used as the recognized standard of units for these
measurements111Some countries as Burma, Liberia, the United States or the
United Kingdom, have other local standards coexisting with the SI. This
implies some adaptions concerning the weather data. Due to the local units it
is necessary to include unit conversions in the data manipulation process..
The figure 2.1 shows the scenario abstracting the data to a generic input:
Figure 2.1: Layers abstracted in the weather collection data workflow.
As we can see the scenario gives as an abstract input of data from the
different environmental phenomena. After that, the data is sent to the data
processing center (commonly a governmental & scientist organizations). At the
end, the data is interpreted and the conclusions are spread. The Physics are
giving us the possibility to understand these phenomena based in the
observation and correlation of them; for this it is needed to establish direct
dependencies between the phenomena.
Figure 2.2: Weather data collection workflow. World Climate Data and
Monitoring Programme.333The World Climate Data and Monitoring Programme
(WCDMP) is a programme of the World Climate Programme that facilitates the
effective collection and management of climate data and the monitoring of the
global climate system, including the detection and assessment of climate
variability and changes.[44]
Commonly, we can find several governmental and scientist organizations around
the world, focused in the weather data collection. As example of this,in
Finland we have the Finnish Meteorological Institute(FMI) [23], or different
example can be a worldwide organizations such as the World Meteorological
Organization (WMO) [44], in charge of the coordination of the exchange and
collection of weather data between organizations around the world. These
organizations are the official source of information for weather data. Even
so, they are not the only ones.
Thousands of individuals are helping with the weather data collection as well.
Those individuals in possession of some weather instruments can collaborate
transmitting the data to some governmental organization, for instance the
program Citizen Weather Observer Program (CWOP) [18] has over 20,000 members
in 149 countries. This is possible using technologies like Automatic Position
Reporting System (APRS) [24] system, which is mentioned in CWOP website[18] as
the following:
_"The Automatic Position Reporting System (APRS) is a part of ham radio that
provides an ideal way for weather station operators to distribute their
weather data much further than the regions within their transmitter range.
APRS was originally intended for position information data but actually
provides a means for automatic transmission of all sorts of digital data. This
is especially true now that the original APRS packet radio concept has been
enhanced to include the capabilities of the Internet. The reporting of citizen
weather data is a particularly useful application of the APRS Internet Service
(APRS-IS)."_
#### 2.1.1 Governmental organizations
Denominated as meteorological institutes or meteorological agencies, it is
possible to find a big group of organizations around the world, which purpose
is to study the weather. Almost all of these organizations are funded by the
governments, moreover of these state and local organizations, other country-
region organizations exist to coordinate the study of the weather in a bigger
extension area. As an example, the Finnish Meteorological Institute (FMI) [23]
is in charge of studying the weather in the region of Finland. At the same
time the FMI is member of the European Centre for Medium-Range Weather
Forecasts (ECMWF) [19], organization in charge _"to provide operational
medium- and extended-range forecasts and a state-of-the-art super-computing
facility for scientific research."_.The same scenario can be found in
different continents as America with organizations as National Oceanic and
Atmospheric Administration (NOAA) [32].
These worldwide organizations are creating the infrastructure to collect the
weather data around the world. It is necessary to highlight that the study of
the weather is an expensive activity, involving a big amount of resources such
as high-tech instruments, installation of these instruments in different
locations (with the extra cost that it implies) and use of computation centers
to evaluate the data. Due to these facts, we can find that the amount of
weather stations around the world and the effort or size of these
organizations can vary significantly depending of the economy of the region.
This means that the weather infrastructure in the occidental world is well
designed, implemented and functional. However, in other areas like Africa, the
amount of available weather stations decrease for economical reasons. In
addition, and due to the nature of the weather, organizations like NOAA and
ECMWF are installing weather collection points outside their official
operation areas444Both organizations are restricted to America and Europe,
nevertheless, these organizations have permission to place collection points
out of their area to improve the quality of the studies and to encourage the
international cooperation., thus getting better samples to evaluate the global
weather conditions.
These state-region organizations have a huge cooperation between them.
Scientists are pretty conscious about the need to get samples of weather data
from different regions to evaluate it, thus, they are fomenting the
cooperation of the weather data exchange. The WMO defined the proceedings of
measurement for meteorological variables[46], providing a common basement to
perform the measurements related with the weather. Furthermore, the WMO is
conscious about the issue of data exchange, in chapter four the process of
standardization that WMO is supporting and the issues of it are analyzed
deeply.
#### 2.1.2 Corporations
As it was mentioned previously, weather has a big impact in our life. It
implies that not only practical advantages can be extracted from the study of
it, also the study of the weather is generating a big range of economical
activities.
Industries like construction or military, have even more interest in know
which phenomena are occurring and the future predictions of them. This
interest have fomented a whole parallel industry of services of weather data
reports.
At the same time, some professional forecast services have appeared as an
alternative for independent studies in particular regions of the world.
Although this economical activity is mainly deployed by private corporations
some governmental organizations are offering also private services.
#### 2.1.3 Individuals
The program CWOP mentioned in the section 2.1, is a perfect example about how
individuals can help to collect and to study the weather data. Furthermore,
non official programmes have been appearing around the world; using the
Internet as foundation, different communities of weather observers are
contributing to create individual networks of data exchange, in which a user
can access the data of different weather stations around the world.
Figure 2.3: Meteoclimat screenshot showing weather forecasts.666This data is
collected by individuals that have installed a specify software in their
computers to send the data to Meteoclimatic servers.
Meteoclimatic[29], is a good example of this:" a big network of automatic non
professional weather stations", in which hundreds of users share the data
collected from their weather stations without any commercial purpose. Often,
these communities share efforts with governmental organizations in programs as
CWOP, however, the turn up of theses communities are supported by the demand
of the users to have a system in which their data is useful for other
individuals, and at the same time give them some independency from
governmental organizations, in terms of data availability.
#### 2.1.4 Weather data publication
The previous sections mention which organizations are involved on the process
of data collection. However, the process does not end here; after the
collection and evaluation of the data, the final step is to spread and make it
useful. The implications of the broadcasting concerning the weather forecast
are multiple and they are out of the scope of this thesis. Even so, the
spreading of the data is limited for the protocols used in the acquisition of
it. As mentioned in section 2.1.3, some communities of individuals appeared,
taking the role of data availability disposal for the end user. Proving this
the fact that the way in which the information is managed by the governmental
and private organizations, sometimes does not fit with the end user’s wishes.
In the past, the weather forecast was delivered through traditional methods as
newspapers, radio and TV. Nevertheless, nowadays the Internet has taken this
role in several aspects. Almost, all the governmental weather organizations
mentioned in this chapter have a web site in which they publish -in different
quantities and formats-, the information collected and extracted from their
meteorological networks. Although traditional media still report the daily
forecast, the tendency points to the Internet as the future mainstream channel
of this information.
In addition, other commercial web sites offer this information partially free
of charge. This practice caused the appearance of several sites offering
Application Programming Interface (API) services to fetch weather data, giving
the possibility to the developers to get some storage data to perform some
operations. Due to this API availability, some organizations non related
directly with the weather data collection workflow, have published some web
sites that are exposing data fetched from different APIs and providing a
different range of alternatives to the users.
Figure 2.4: FMI website [23] spreading local weather observations.
The author could not find any API offering the capability to connect directly
to the weather instruments to fetch RAW data streams; all the APIs available
are offering pre-processed data.
### 2.2 Summary
In this chapter we have given general background information in order to make
the scope of the thesis more familiar, in terms of which organizations are in
charge of the weather collection and the structure and collaboration between
them. Also, it has been analyzed how different organizations of the same field
coexist.
We discussed how the same activity is performed in different layers, being
involved in the process from official organizations to individuals. Some
schemas have been presented, giving a global vision about how the weather data
workflow works.
We know now that there is even a global organization named WMO. This
organization is only dictating some guidelines to perform the measurements.
The next chapter introduces a general overview of a weather instrument, to
understand how it works, its technologies and limitations.
In the next chapter some concepts and scenarios are explained to understand
how a weather instrument works, the technologies that are conforming it, and
giving us a global vision of the technologies to have in consideration when we
are implementing a protocol for a weather instrument.
## Chapter 3 Infrastructure for the weather data
History is full of attempts to understand the weather. From the very
beginning, humans have been focusing their attention in the weather, putting a
lot of effort trying to understand and predict it. The first treatise
concerning weather observations was _Meteorologica_ , written by Aristotle
(340 B.C.). Despite of this, _"the birth of meteorology as a genuine natural
science did not take place until the invention of weather instruments, such as
the thermometer at the end of the sixteenth century, the barometer (for
measuring air pressure) in 1643, and the hygrometer (for measuring humidity)
in the late 1700s"_[1]. It was with the invention of the telegraph, in 1843,
when the weather observations started to be useful owing to the capability to
transmit the weather reports to different locations. Since this time elapsed,
the industry has been developing and improving the weather instruments to
achieve better measurements. Furthermore, the networks for weather data
collection have been maturing. This chapter introduces the technology that is
composing a modern weather instrument, its role in the weather’s collection
infrastructure and shows us some concepts to understand the conflicts of this
setup exposed in chapter four.
### 3.1 A meteorological instrument
The purpose of a weather instrument is to measure a particular phenomenon
under certain conditions, to collect some data that can be processed to obtain
some conclusions (in terms of understanding and predictability). The success
of the prediction and understanding comes supported by the accuracy that these
instruments can provide. The industry has been creating new instruments based
on new techniques discovered in Physics, to measure the phenomena; in
addition, the advance of the digital technology, is providing to the
physicians a great scenario in which physical principles can be combined
easily with digital technology, producing as result modern instruments with
the ability to transform the result of these physical principles in digital
data.
Despite their size and appearance, the weather instruments are complex
artifacts. The materials used to build them are a combination between plastic
and metal, this combination provides the necessary robustness to place the
weather instruments at isolated places with all kind of degradation
conditions. Furthermore, these instruments must have a low power consumption
in order to fit the requirements of their locations. That forces the
manufacturers to use more tiny and efficient technologies for measuring the
phenomena without sacrificing energy and accuracy.
It is not possible to discuss all these instruments in this thesis. For this
reason the following subsections of this chapter are focused on automatic
weather stations(Automatic Weather Station (AWS) es). The WMO defines an AWS
as: _meteorological station at which observations are made and transmitted
automatically_[46], at the same time this concept comes with other nuances as
Automated Weather Observing System (AWOS) and Automated Surface Observing
System (ASOS): _a combined system of instruments, interfaces and processing
and transmission units is usually called an automated weather observing system
AWOS or automated surface observing system ASOS. It has become common practice
to refer to such a system as an AWS_.
The focus on the AWS es is supported by the popularity of these weather
stations as main tools to measure the weather. The author considers more
useful to focus on this technology because a wide range of AWS es is available
for the end-non professional user; meaning this that is possible to experiment
with a new protocol using this scenario without affecting the current setups
used for scientific purposes. In addition, later migration of the protocol to
professional instruments should not be difficult because the manufacturers are
using mostly the same technologies in the data transmission interfaces for
both brands (professional and end-user).
#### 3.1.1 Industrial design
Depending of the type of phenomenon to measure, the physical principle needed
will require an instrument with certain sizes, materials and lifetime. It is
rarely possible to measure multiple phenomena with the same instrument, this
fact causes the creation of instruments focused only on one phenomenon 111We
refer here to high-tech and professional instruments for scientific purposes.
It is possible to find several sensors giving an output for different
phenomena in one instrument. However, this is not common in the instruments
used for scientist observations; at the same time this configuration should be
considered as a weather station not as an ”individual” weather instrument. and
even in only one specify and tiny part of it.
The industrial design of an instrument is one of the keys for the success of
the observations; the ability to put available the required technical
conditions to perform the measurement through a digital interface, reside on
it. To avoid conflicts in the study of the phenomenon, the materials should be
chosen very carefully based on a complex equation between: robustness,
durability, impact, impact assessment, etc. Furthermore, the shapes and sizes
depend on the environment in which the instrument is going to be placed and
the requirements needed for the physical principle used.
Figure 3.1: NOAA weather buoy [32], example of a complex an robustness AWS.
We can find in the market dozens of instruments for the same purpose, using in
some cases the same principles to measure the phenomenon and even with some
strong differences concerning the industrial design. Though, the instruments
from different manufacturers have similar dimensions and they are build with
similar materials, there is non available standard concerning all these
characteristics, only some general guidelines are provided by the WMO[44]
suggesting dimensions and sizes for some instruments, an example of this
recommendation is the following:
_Wind-measuring systems can be designed in many different ways; […] The first
system consists of an anemometer with a response length of 5 m, a pulse
generator that generates pulses at a frequency proportional to the rotation
rate of the anemometer (preferably several pulses per rotation), a counting
device that counts the pulses at intervals of 0.25 s, and a microprocessor
that computes averages and standard deviation over 10 min intervals on the
basis of 0.25 s samples._[46]
Figure 3.2: Generic AWS with different instruments and materials combination.
The figure 3.2 shows a generic schema in which we can see different
combinations of materials as plastic and metal, at the same time the
instruments are placed in different heights due to technical requirements for
the techniques used to perform the measurements.
Most of the instruments available at the market are the result of the
coordination between the requirements requested by the physicists and the
possibilities that the technology developed by the industry. Notwithstanding,
the instruments industry and their industrial design, is something really big
and complex and it is out of the scope of the thesis. Furthermore, we need to
be conscious about the industrial design of the instruments, because it is
strong-linked to the electronics that they can house, conditioning this the
digital interfaces for data transmission that we can install in them.
#### 3.1.2 Electronics and data handling
The electronics of a weather instrument are barely different irrespective of
the phenomenon to measure. The industry is producing a wide range of
instruments with a complete different set of sensors. Nevertheless, as
embedded systems, all these instruments have a common need to conform these
type of systems. The WMO gives again some general guidelines with respect to
electronics and weather instruments. The following paragraphs summarize them.
##### CPU
As other electronic device in charge of process data, an AWS has a Central
processing unit (CPU) running at clock frequency of a few Megahertz (MHz).
This CPU is microprocessor based with 8-bit wide.222Nowadays some
manufacturers are introducing progressively new microprocessors using 32-bit
wide. Despite the low bit wide of these microprocessors, an AWS does not
needed more calculation power because the amount of data generated by the
sensors will be rarely up of 1 Kilobyte (kB), meaning this that frequencies
oscillating between 8-33 MHz will fit perfectly in the requirements to process
the data.
##### Volatile Memory
Often 32-64 kB is the maximum amount of volatile memory available on an AWS,
it makes the instrument non capable to keep too much data on a Random-access
memory (RAM) at all. Forcing to the manufacturers to design the instruments
with fast and reliable mass storages, ready to transfer the data from the
volatile memory to the persistent storage.
Figure 3.3: Abstracted electronic schema of an AWS reading data from one
sensor.
The figure LABEL:3.3 shows the workflow data of an abstract sensor. In the
first step the sensor generates the data from the phenomenon, based on the
observation of some physical principle; the data acquired is processed by the
microprocessor in the the second step, placing the data on the volatile
memory. When the data is placed on RAM the in / out (IO) operations start,
transferring the data from the volatile memory to the mass storage (persistent
memory). According to the Guide to Meteorological Instruments and Methods of
Observation [46] published by the WMO, it is highly recommended to equip the
AWS with a battery backup dedicated to the volatile memory to avoid data loss
due to some power fails. This non common feature in generic computers can be
an advantage to have in mind when a protocol is implemented, because it
enables the possibility to have some methodology in the protocol to recover
the session after one power failure.
##### Mass storage
Typically, an AWS, will have mass storage device to save the data collected
from the sensors. The storage of data in the AWS has been changing in the last
years due to the continuously decreasing price of flash memories. It is common
to find very different architectures in terms of data storage in the AWS.
Figure 3.4: Types of storages available in an AWS.
The number of sensors and the frequency in which the information is
transferred to the data centers, determines the size of available memory in an
AWS. Based on the market, the mainstream option in terms of memory size for
mass storage is around 1 Megabyte (MB), that space is more than enough to save
thousands of samples in case that the AWS has not send the data to the
collecting point.
##### Sensors
The sensors are the digital interfaces that make an AWS different from other
embedded devices. As explained in section 1.1, a sensor is a digital interface
using some physical principle to measure a particular phenomenon. Their
principles, implementation and complexity are out of the scope of this thesis.
Even so, we need to consider the sampling frequency of them because they are
involved in the frequency in which the data is produced.
The sampling frequency of the sensor depends on the data required to
understand the phenomenon. A big range of sampling frequencies are used to
measure different phenomena. Nevertheless, the author is not assuming this
frequencies as a need for the protocol.
A correct behavior of the sensors requires a high-accurate calibration of
them. The manufacturers have been developing several methodologies and
mechanisms to calibrate the instruments and verify their correct behavior.
These calibrations are not considered as part of the problem statement of this
thesis because they are unrelated to the methods of the data transmission.
##### Digital interfaces
As mentioned in the section 1.2, an AWS is equipped with at least one
peripheral device to provide data interaction. These interfaces offer the
possibility to configure the AWS and transfer data from it. The type of device
is a serial communication physical interface, and depending on the type and
vendor of the instrument, it will be one the following333Other types of
interfaces can be found in the instruments. However, the industry stablished
—with non-written agreement— the use of the mentioned interfaces as
mainstream.:
* •
RS-232
* •
RS-422
* •
RS-485
* •
USB
These four types are well-known in the industry. They are available in almost
all the modern computers, however the relation of them with this thesis is
focus mainly in the bandwidth that they offer. The table 3.1 shows a
comparison between these physical digital interfaces and their bandwidth.
Standard | Bandwidth | Bytes/s | kB
---|---|---|---
TIA/EIA-232-F[2] | 116 kilobits (kbit)/s | 14848 | 14.5 kB
TIA/EIA-422-F[3] | 200 kbit/s | 25600 | 25 kB
TIA/EIA-485-F[4] | 35 Megabits (Mbits)/s | 4587520 | 4.375 MB
USB[12]444Referencing the USB in low power mode (specification 1.0) | 1.5 Mbits/s | 196680 | 192 kB
Table 3.1: Comparison between standards and bandwidth offered.
Even so, as the table 3.1 shows, the minimum bandwidth provided by theses
interfaces (RS-232) should be enough. As described in the sensors section, the
total amount of data generated by the sensors of one AWS rarely exceeds 1kB;
fitting perfectly this in the bandwidth offered by the RS-232.
Due to the constant renovation in digital interfaces that the industry does,
we do not consider other old interfaces in the analysis, assuming that the
protocol will work with instruments manufactured in the last 10 years555Those
should be equipped with the interfaces mentioned in the Table 3.1..
Although the interfaces are not conditioning our protocol implementation, it
is necessary to highlight that most of the vendors offer the possibility to
re-wire the AWS to make it work with different physical interfaces.
Figure 3.5: Wiring schema showing how to re-wire the AWS to use RS-422.
##### Datalogger
The datalogger is one the most critical parts of an AWS. It is in charge of
the data logging produced by the sensors and deliver by the operating system.
Its main task is to keep track of the data collected by the AWS. This
component plays an important role in the implementation of the protocol,
because of the data of the protocol must be originated in this part.
Depending on the architecture of the AWS, the datalogger can be an external
embedded system with serial communication capabilities, able to send data
through a network and with multi-station capability666Some dataloggers are
able to track and to operate several AWS at the same time.. Small AWS es can
have datalogging capabilities, keeping the data in a persistent memory for a
short period. To have the datalogger implemented internally implies increasing
the complexity of the AWS, converting it in a more complex embedded system
with features as data delivery through a network, long-term data storage, etc.
Often, the architecture chosen for AWS es is an external device connected
through the physical interface. These devices are equipped with some kind of
connectivity such as Global System for Mobile Communications (GSM), General
Packet Radio Service (GPRS) or Universal Mobile Telecommunications System
(UMTS) modems, using them to deliver the data to the collection point.
Figure 3.6: Location of the datalogger in an AWS.
#### 3.1.3 Software
As it is common in the embedded systems, an AWS has a tiny internal software.
The programming languages used to develop this software have no relevance in
this topic. We assume that the internal operating system of the AWS will offer
us the data collected from the sensors, moreover of some set of options to
configure and calibrate the AWS.
We need to differentiate between the software embedded in the AWS and the
software at the end of the peripheral device.
##### AWS’s Operating System
The operating system installed in an AWS resides in a Programmable Read-Only
Memory (PROM). Its architecture is based in a real-time clock implemented on
the mother board of the AWS. The OS provides a limited set of options to
interact with the AWS, most of these options are focused in data acquisition,
calibration and hardware configuration. This software is in charge of the
formatted data of the AWS, in other words, it gets the data from the sensors,
applies the necessary formulas to extract a meaningful result and formats the
data in one of the following serial communication protocols777We need to
distinguish between the data format used to communicate with the interface
(ASCII, NMEA-0183, etc) and the format in which the data is formatted, this is
explained the section 4.2.:
* •
RAW ASCII
* •
SDI-12
* •
NMEA-0183
After the data is formatted, it is transmitted through the peripheral device
to the the datalogger.
##### External software used for datalogging / data distribution
As explained in 3.1.2, an AWS needs a datalogger device to track the data
collected from the sensors. Irrespective of the type of datalogger, at the end
of it, we will find some computer in charge of the data manipulation and
storage. The software installed on the computers can be really differently
implemented and designed depending of the vendor, but its main task is to
understand the data format chosen by the vendor to transmit information and
take use of it.
The market offer concerning software for AWS es is too big, even some
companies not related with the manufacturing of the instruments, are releasing
software for datalogging purposes. It is common that the AWS is provided from
the factory with its own set of software, nevertheless due to the serial
communications protocols used by the AWS, is simple to implement a software
that interprets and takes advantage of the data format chosen by the vendor to
implement new capabilities.
Figure 3.7: Screenshots of some popular desktop applications for AWS.
#### 3.1.4 Networking
As mentioned in the datalogger subsection, the connectivity capabilities in an
AWS resides on it. The industry offers multiple options to provide
connectivity in an AWS, nevertheless, most of these options are limited for
bandwidth, energy and geographical limitations. It is possible to find AWS es
directly connected to a computer via USB, providing this the connectivity, or
we can find an isolated AWS in the middle of a mountain connected through a
radio-link to the closest place.
The common technologies to provide connectivity to an AWS are:
* •
GSM
* •
GPRS
* •
UMTS
In places with better geographical location and energy availability, it is
possible to find the following technologies offering connectivity:
* •
Ethernet
* •
USB
* •
802.11b/g
Whatever the connectivity on the AWS is, the common pattern is that this
connectivity is reliable but offers a rather low bandwidth.
### 3.2 Meteorological data networks
The previous section gave a general overview of AWS es, the relation between
them and this thesis, is how they behave in terms of networking communication,
which kind of topologies are used and in which points this communication can
be improved.
To understand the workflow of the data in terms of weather data collection, we
should see an AWS as an individual node without interaction with other nodes,
except the collection point.
The collection point is the place in which different data from different AWS
is received. It is not mandatory that this collection point is the end of the
weather data workflow, for instance it is possible to find an intermediate
collection point that has been stablished for geographical reasons to improve
the connectivity888Some AWS are located at inaccessible places, sometimes this
implies to establish a collection point close to them to avoid issues such as
lack of connectivity (GSM, GPRS, UMTS).. Even so, we consider the collection
point, the place in which the data has been received and it is ready to be
processed.
Figure 3.8: AWS es connectivity schema.
When the data arrives at the collection point, different mechanisms get
activated to process it. As described in section 1.2, rarely, the data
received comes from the same brand of instruments, meaning this that the data
will be received in different formats and different time frequencies; this
fact forces to implement these mechanisms to homogenize the data and make it
understandable on the collection point. The collection point is the hop in
which to have a standard protocol to communicate with the AWS will have a
bigger benefit, because it is in this hop in which the most effort is made, it
in terms of data parsing, power calculation and data homogenization.
#### 3.2.1 Common architectures
The definition of star topology fits in the methodology used to collect data
from different AWS es. The nodes have a strong dependency with the collection
point, without it, an AWS will have a high limited time to save data before it
is fetched manually. Furthermore, the meteorological networks are not
following the pure definition of star topology because different nodes are
transmitting data with different connectivity technologies. Nevertheless,
seems the nodes are not interacting between them, the network is not affected
by bandwidth limitations. This topology is chosen by weather organizations
based in the geographical limitations. However, the possibility to
interconnect AWS es between them has not been study deeply. The assumption for
this is that the utility of the data is based on the availability of it, for
this reason the data delivered with big delays is not considered at all in the
weather data collection workflow. Interconnect the nodes of the meteorological
networks it not feasible with the current technology at all for different
factors such as bandwidth, geographical locations or absence of a common
protocol.
Figure 3.9: Comparison between pure star-topology against star-topology and
the connectivity technologies used in AWS es.
Not only star-topology is used in the meteorological networks, the combination
of different instruments can end in different topologies depending of the
datalogger configuration. For instance, it is possible to have some local
network of sensors connected to a datalogger that is part of a star topology,
commonly, this topology will be a combination between bus-topology and star-
topology. These combinations will not affect a common protocol in anyway, due
to its implementation should happen on the datalogger’s side, not mattering
the combination of topologies behind it.
##### APRS
APRS is using unnumbered Link Access Protocol for Amateur Packet Radio (AX.25)
frames[43]. AX.25 is a data link layer protocol without too many capabilities
in terms of bandwidth’s offer, error correction and data integrity. Though it
is used in some weather stations to spread the data, it is not a good choice
because it is not warranting a constant visibility and connection of the node.
The AWS using the APRS technology are spreading the data based radio
technologies. It is allowing to any node with a radio equipment to receive the
information produced in the weather station. Furthermore this topology does
not offer any warranty in data delivery because it does not use the collection
point model.
Figure 3.10: Example of an AWS using APRS at Helsinki area101010Source:
http://aprs.fi.
APRS has gained popularity inside the radio amateur community and programs as
CWOP due to the simplicity and technical requirements that it implies. The
_Weather Station Siting, Performance, and Data Quality Guide_[25] explains how
to setup an AWS to get integrated in the CWOP using APRS.
Figure 3.11: Weather data message using APRS [24].
Nevertheless, APRS is not used in scientific installations. Although it is not
possible to re-implement APRS to adapt it to OpenWeather, it will be possible
to use the same data format as it used in OpenWeather under AX.25. Thus, it
will offer compatibility between applications using OpenWeather. To provide
this capability, will involve modifying the way in which APRS is used, one way
to do it can be to send the same data beacon with different formats: standard
APRS messages for weather reports and after it a data message based in
OpenWeather format.
Even with these incompatibilities the data provided by the APRS data message
can be transformed to OpenWeather’s data format in a middle point having to
modify the APRS protocol.
The author assumes that the AWS es will behave as nodes with connectivity to a
common point, being able to interact between them, through the collection
point or point to point.
#### 3.2.2 Data distribution
Data distribution is the ultimate’s reason for weather data collection. We can
identify at least fours levels of different data in the process for weather
data collection.
* •
RAW data, produced in the sensors’ instruments
* •
Network data, used in the transmission from the instruments to the collection
point
* •
Operational data, result of the scientific’s practices
* •
Informational data, mainly focus in the general public (forecasts, climate
reports, etc)
After the data is collected and processed, the conclusions made by the
scientists must be spread to inform the society. It is necessary to highlight
that only a few conclusions get to the general public, some of them are known
as forecast or climate reports. Most of the data processed is not useful for
non scientists, because the complexity or amount of information on it. At the
end of the work flow we have the data in two categories, the data that will be
minimized to make it understandable to a general public, known as
informational data 111111An example of this is the weather forecast shown
every day in newspapers, TV, radio, etc. and the data that must be shared
between different international and local governmental organizations, known as
operational data.
As part of the problem statement, the data distribution is one of the big
efforts that these organizations need to do to make the data that they collect
understandable. In 2002 the WMO started a standardization process to create a
metadata standard to fix part of this problem, however nowadays this
standardization process is still on progress without any draft available[45].
A standard protocol to communicate with the AWS will help the development of a
common data format between organizations because all of them will be fetching
the data with the same methods and mechanisms.
### 3.3 Summary
We have now introduced the elements and process involved in measuring and
collecting weather data and the technologies related with them. Some topics
have been explained to provide a general understanding of how an AWS works.
We have highlighted the limitations the AWS es, concerning data storage and
CPU calculation; at the same time the maximum bandwidth available for the
digital interfaces has been analyzed. The role of the datalogger has been
exposed and its implications of it in the implementation of OpenWeather’s
format.
In addition, the connectivity technologies available in an AWS have been
enumerated, analyzing the bandwidth offered and concluding that only the
interruption of the connection and not the bandwidth’s offer can be an issue.
Finally, the topologies used in the meteorological networks haven analyzed
briefly, clarifying that the AWS are behaving as nodes without interaction
between them, only sending data to a common point named "collection point"
(the node that interacts with all the AWS es). The APRS protocol and its
topology have been explained, taking in consideration the possibility to be
compatible with the implementation of OpenWeather.
The next chapter describes the technical issues related with the data
transmission in the AWS es.
## Chapter 4 State of the art in the weather data transmission
The previous chapters we have introduced a general overview of the basics
needed to understand how weather data are collected and how a weather
instrument is designed to undertake its function. Even though the purpose of
this thesis is to analyze the issues found in the weather data transmission
and to provide an alternative to fix these problems. Nowadays, the way in
which a weather instrument is transmitting the data can be classified as
generic, because the methodologies used in this task have not been optimized
thinking in the data implied in the process. This practice limits the
possibility to acquire data without the implementation of intermediary hops in
which the data is parsed and converted to a useful data format. This results
in an unnecessary investment of CPU cycles, delays in the data delivery,
incompatibility between difference brand of instruments, and at the end
causing the investment of more resources and effort to exchange data between
organizations. This chapter analyzes the technical points that are causing
this issues in the weather data transmission.
### 4.1 The evolution of the digital interfaces in a
weather instrument
As mentioned in chapter three, the meteorology did not advance until the
invention of the telegraph. The value of the weather data resides in the
ability to combine it with other sources to get some conclusions to make
predictions. Nevertheless, this combination involves having the possibility to
transmit this data fast and far enough. The telegraph brought this
possibility, and with this new chance scientists had the opportunity to
understand concepts as wind flow and storm movement[1] among others. During
the 19th and 20th century the industry has been developing new improvements in
the instruments manufactured; all of these improvements come supported for the
new methods found by the physics to measure the phenomena, and the conversion
of them to digital instruments.
In 1969 the RS-232-C standard was published; this interface has been the
mainstream technology used in the weather instruments for more than thirty
years; only in the last decade some updates have been introduced in the
industry, migrating to new standards as ANSI/EIA/TIA-232-F[2],
ANSI/EIA/TIA-422-F[3], ANSI/EIA/TIA-485-F[4] or USB.
As far as we can judge this slow transition in as of the digital interfaces
used in a weather instrument come supported for the fact of the wide use of
RS-232-C in different fields of the industry, at the same time these
interfaces fit perfectly in the needs of the weather data transmission: enough
bandwidth, low cost and they are an international standard. If some updates
have been introduced in the industry of the weather’s instruments, they come
supported by the need to adapt these interfaces to the hardware ports
available at the moderns computers, seldom by the requirement of more
bandwidth111In some big AWS es in which have been placed many sensors and
complex instruments, exists the possibility to need a bigger bandwidth, even
so this is a specific case out of the mainstream setups..
It is an observable fact that the industry performs some updates in the
technology to make it compatible with the moderns computers despite that the
is not needed in terms of data delivery. Moreover, the new standards are
offering more capabilities a part of more bandwidth, for example, technologies
as USB, bring the opportunity to plug an AWS to a computer and have it working
without previous configurations as bit-rate, parity, etc.222Interfaces based
in ANSI/EIA/TIA-232-F, ANSI/EIA/TIA-422-F, ANSI/EIA/TIA-485-F require to adapt
the software to certain bit-rates, flow controls and other parameters.
These interfaces provided by the industry are generic as in other
technologies, not mattering the type of data transmitted through them; a well-
known process of standardization has been performed to develop these
interfaces. Though does not exist any standard specifying which type of
interface should provide an AWS, the WMO recognizes the universality of the
interfaces mentioned, and establishes them as requirement for the AWS es
performing official measurements for governmental organizations[46]. Based on
this we assume that a protocol implemented in an AWS must work under these
technologies; because these interfaces are generic, they have not any
requirement for the data transmitted, giving complete freedom to us to
implement any protocol over them.
As mentioned in section 3.1.2, the bandwidth offered for the different
interfaces available in a weather instrument, are offering even more bandwidth
than the amount of data that an AWS’s CPU can process. Hence, a weather
instrument has not limitations (concerning bandwidth) in the data interfaces
that would prevent the possibility to implement a protocol to afford the needs
of the data delivery.
Based on this retrospective we assume that the digital interfaces provided by
the industry are well know and tested standards, providing mechanisms to
achieve the goal of the data transmission. However, as it is explained in
section 4.2 no weather data transmission protocol has been defined for them.
We identified this as the first deficiency in the weather data transmission
because of the potential offered by these digital interfaces is not used in
the weather instruments.
### 4.2 The absence of a protocol
The goal of the Internet Engineering Task Force (IETF) [26] is to make the
Internet work better. One of its multiple task implies to take care about the
standardization process of the new Internet standards. A protocol is
considered as standard when the IETF publishes a memorandum333This memorandums
are named as Request for Comments (RFC) for historical reasons., specifying
all the aspects of the protocol and assigning a number in the STD series of
it[7].
A research performed by the author in the RFC s available at IETF’s website
[26]444The searched has been performed over all the content of the RFC
published: ftp://ftp.rfc-editor.org/in-notes/tar/RFC-all.tar.gz . Retrieved:
28-03-2011., looking for the following terms: "weather", "meteorology",
"weather station", "atmosphere", "weather data", gave as result the following
number of mentions. Only 9 RFC s do direct or indirect mention to the weather
data.
The first RFC mentioning a protocol related with the weather data is the RFC
765 [38] File Transfer Protocol (FTP):
_3.4.2. BLOCK MODE The file is transmitted as a series of data blocks preceded
by one or more header bytes. The header bytes contain a count field, and
descriptor code. The count field indicates the total length of the data block
in bytes, thus marking the beginning of the next data block (there are no
filler bits). The descriptor code defines: last block in the file (EOF) last
block in the record (EOR), restart marker (see the Section on Error Recovery
and Restart) or suspect data (i.e., the data being transferred is suspected of
errors and is not reliable). This last code is NOT intended for error control
within FTP. It is motivated by the desire of sites exchanging certain types of
data (e.g., seismic or weather data) to send and receive all the data despite
local errors (such as "magnetic tape read errors"), but to indicate in the
transmission that certain portions are suspect). Record structures are allowed
in this mode, and any representation type may be used._
Nevertheless, this reference of weather data is just an example (as the other
references) that disappeared in later updates of the File Transfer Protocol
(FTP).
The industry has focused its effort in improving the measure methodologies,
the robustness of the instruments or other features as power consumption or
life-time. Thus, the methodologies utilized to transmit weather data have been
developed independently by the vendors, choosing their own data formats and
techniques.
Nevertheless, the WMO initialized different programs as Global Observing
System (GOS), Global Telecommunication System and WMO Information System
(GTS), Global Data-processing and Forecasting System (GDPFS) [44] among
others, in which the weather data exchange is a key-component of the systems
to archive the goals of these programs. In addition, as mentioned in the
section 3.2.2 the WMO started a process of standardization 9 years ago.
Even assuming that the industry focused its attention on prioritizing
measurements methods and product quality, the technologies related to the
weather data transmission are outdated. The proof of this is that only a few
governmental organizations have access to real-time information 555All of
these instruments are generating by default real-time data. collected from the
AWS es666Note that these organizations can have this capability due to they
invest a big effort in to develop custom systems for their weather
instrument’s setup., at the same time programs as CWOP still depend of
technologies such as FTP or APRS, that they do not contemplate scenarios in
which scalability, data on demand or real-time data is needed. Finally, as a
real example, the SMEAR project[40] is experimenting the issues of not having
a standard protocol for the AWS, producing as result the implementation of
intermediary points to parse and normalize the data, incompatibility between
different sources of data from the same phenomenon collected with different
instruments and scalability of the system among others.
Based in these facts, we can say that during the last 40 years the industry
unattended the communication’s side of the AWS, adapting the instruments to be
capable to use protocols as FTP to transmit the data from the AWS to the
collection point; focalizing the effort transmiting the data not mattering at
all the technologies used or if they are or not optimized for that purpose.
This practice gave as result multiple data formats implemented by the vendors
without any common agreement, creating a huge incompatibility between the
instruments and several bottlenecks in the data transmissions.
The following subsections expose some data format used by the vendors to
archive the data transmission and analyze why these data formats are causing
bottlenecks.
##### Data formats used by the vendors
As mentioned in previous chapters, the format in which the data is produced by
AWS is formatted is up to the vendors. Nowadays the only standards used or
involved in this process is ASCII as character-encoding scheme or NMEA-0183.
Depending on the digital interface different control characters can be used,
for instance is a common practice to generate one line of data follow by the
carriage return (CR) or carriage return followed by line feed (CR+LF)777CR
hexadecimal value: 0x0D. LF hexadecimal value: 0x0A. CR+LF: hexadecimal value
0x0D 0x0A..
>"BARDATA"<LF>
---
<<LF><CR>"OK"<LF><CR>
<"BAR 29775"<LF><CR>
<"ELEVATION 27"<LF><CR>
<"DEW POINT 56"<LF><CR>
<"VIRTUAL TEMP 63"<LF><CR>
<"C 29"<LF><CR> <"R 1001"<LF><CR>
<"BARCAL 0"<LF><CR> <"GAIN 1533"<LF><CR>
<"OFFSET 18110"<LF><CR>
Table 4.1: Example of data format used in a specific AWS to communicate the
barometric pressure.
Depending the AWS’s brand the data’s format is completely different from other
brands and vendors. In most of the cases the data format is implemented based
in the vendor’s wishes. These wishes can be supported by technical reasons or
not. Some vendors used acronyms to refer the data values returned by the
sensors, others use the whole word to refer the phenomenon; not mattering the
technique used in the data format, is a fact that they do not exist any
compatibility of formats between vendors.
0r2,Ta=10.6C,Tp=10.8C,Ua=74.6P,Pa=1006.0HKHK
---
Table 4.2: Another example of data format used in a specific AWS to
communicate different data as temperature or barometric pressure.
A part of these big differences between the formats used in the digital
interfaces, is needed to highlight that also the field’s value used in CSV or
TSV files producted by the AWS are unique and incompatible between vendors.
Thus, two levels of incompatibility exist, first the original data is
delivered in a custom formatted untill the software’s side. In the software’s
side this data is converted to a CSV or TSV format with the custom fields
chosen by the vendors; this causes that even having the final data in a
standard format as CSV or TSV, the order of the fields and their denomination
will be different, forcing to the scientist to add an extra layer to the
workflow to normalize this data and make it ready to be combined.
##### Data formats used by governmental organizations
Despite the fact that vendors used privative and non standard formats for the
data, the WMO has defined some specific data representation for certain users.
An example of this is the Meteorological Service For International Air
Navigation (METAR) format. Approved by the International Civil Aviation
Organization (ICAO), this format is the only one considered as official to
communicate weather forecasting to the aviation and at the same it is widely
use for other purposes as general weather forecasting.
Phenomenon | METAR’s acronym
---|---
cumulonimbus clouds | CB
thunderstorm | TS
moderate or severe turbulence | MOD TURB, SEV TURB
wind shear | WS
hail | GR
Table 4.3: Some acronyms used in METAR format [36].
However, this format has not relationship with the formats used by the
vendors. Only a few AWS es have the ability to product the METAR format by
default. The AWS doing this are only focus in product data useful for the
aviation, wasting the opportunity to provide the data in other formats for
different use.
Figure 4.1: Weather data workflow, normal AWS VS METAR’s AWS.
METAR format is just an example of the multiple data formats invented for a
specific purpose. The point to highlight is that often the weather data can be
represented in a complete different format compare with the original format
used for it. Nevertheless, the optimization of the data format until the point
in which it is transformed marks a big difference in terms of data
manipulation.
With the current technology the weather data arrives in different formats and
with difference times frequencies, forcing to implement customized and
particular mechanisms to transform this data to the format required. The
complexity of this task resides in the requirement de facto requested by the
AWS es: they need intermediary points to convert the data because by default
the data provided is useless for the required result.
In conclusion, it does not matter if the vendors provide a well known
documented data format of their instruments. Because the observation of the
weather is performed with different instruments, the data must be normalized
to make it understandable. Thus, at the end of the data workflow (when we take
data from different sources and instruments), an intermediary layer to
translate the vendor’s data format to a common format is required.
##### Mainstream architecture used for the weather data transmission
To understand where are located the bottlenecks in the weather data
transmissions is needed to understand the current architecture used by the
vendors to archive this goal. As explained in section 3.1.2, an AWS is an
embedded system collecting information produced by the sensors attached to it.
As embedded system, it has small capabilities to perform big CPU calculations,
massive data storage or data delivery, however moderns systems are pretty
balanced in terms of hardware and software to archive this goal. Although the
AWS have been optimized to collect and delivery the data, the protocols used
for it are generic an non-specific. As explained in previous chapters the
quality of the weather predictions reside in the ability to collect and
process the atmosphere data with efficiency, reliability and fast delivery.
Despite of this, the methodologies for network communications are not
optimized for this purpose. The following figure shows how the data is
delivery.
Figure 4.2: Example of an AWS transmitting weather data.
In the figure 4.2 we can appreciate an example of the methodology used to
transmit the weather data. In the hardware’s level the data is delivered
through a digital interface as explained in section 3.1.2, using some custom
vendor’s data format, commonly based in abbreviations as "Tmp (Temperature)",
"Bp (Barometric Pressure )" "Ws (Wind Speed)", among others. These
abbreviations are understood by the software. Depending of the AWS’s setup
this process can happen all together between the AWS and the datalogger:
Figure 4.3: Example of a AWS and a datalogger transmitting weather data.
If the AWS/ datalogger has not network capabilities, a third entity can enter
in the workflow. This entity is commonly a modern computer with the peripheral
devices needed to interact with the AWS. The computer takes the role of the
weather data transmission, due to the possibilities that it offers, one
computer can manage several AWS es at the same time. Nevertheless, it does not
introduce new protocols to send the data, it stills using protocols as FTP or
in some setups just shared folders using Server Message Block (SMB):
Figure 4.4: Workstation taking the role of the weather data transmission.
##### FTP, the mainstream protocol in the weather data transmission
Disregarding the setup used to send the data to the collection point, the
protocol used will be generic and in most of the cases based in FTP. Although
FTP has the capability to operate under stream mode [39], the author could not
find any vendor offering the capability to deliver the data through stream FTP
connections. Even being this possible, it will involve to use the image mode
(commonly known as binary mode, thus, involving byte ordering choices) to
transmit the data, however this choice will subject the data transmission to
problems with the endianness888 _”Endianness describes how multi-byte data is
represented by a computer system and is dictated by the CPU architecture of
the system. Unfortunately not all computer systems are designed with the same
Endian- architecture. The difference in Endian-architecture is an issue when
software or data is shared between computer systems. An analysis of the
computer system and its interfaces will determine the requirements of the
Endian implementation of the software.”[15]_..
This setup can fill the requirements to delivery weather data collected over
different time frequencies, however, it can not offer real-time capabilities,
because the FTP is not designed for this purpose. The author identifies the
use of FTP as a deficiency in the weather data transmission 999All the AWS
checked by the author are offering the data delivery based in ASCII files
using the FTP ASCII mode and sending the data using the FTP block mode. Though
is possible to find some AWS using different methodologies as Hyper Transfer
Text Protocol (HTTP) get methods or email delivery, the FTP choice is
mainstream overall the industry., the reasons for this are based in the fact
that the protocol is designed to provide network capabilities to delivery data
streams based in files. Notwithstanding, the AWS es are producing data streams
based in real-time data; the use of FTP involves an intermediary step to
convert these data streams to files, to continue after this sending theses
these files to the collection point. Even though to track this data in files
is needed for storage and backup reasons, the data streams generated in real-
time by the AWS are not used at all to send them directly to the collection
point. In addition, the use of this methodology is forcing extra IO operations
required by the FTP, that are not required in other protocols in which the
data transfer does not involve the use of files.
Thus, it is not available any protocol taking advantage of all the
capabilities offered by the AWS es and its sensors, instead generic protocols
as FTP or SMB have been chosen to transmit data. These protocols are widely
and accepted as the mainstream solutions for data transmission available on
the weather instruments.
### 4.3 The missing standard
One of the important factors of an implemented protocol, is to know how is
going to be represented the data transmitted at the end of the transmission.
This helps to design the best representation required by the data; for
instance a protocol implementing real-time capabilities should be focus in
fast data delivery and data integrity, among others.
In addition, to know the final representation of the data helps to implement a
protocol optimized for the data that is transporting, this gives as result a
better software for the protocol, besides it provides the capability to
implement different protocols giving the same data result101010A good example
of this are the peer to peer networks, in which the protocol’s designers know
that at end of the process the data must be a file..
Nevertheless, the weather data has certain particularities; the WMO defines a
set of methods to perform different measurements, notwithstanding theses
methods are changing based in the advance of the physics, and these changes
are causing an instability concerning what is the best way to measure a
phenomenon, thus the data representation can get affected easily. Furthermore,
the correlation between phenomena generates certain scenarios in which the
data results can change completely if a new method is found to measure the
phenomenon. This fact determines to which point we can have or not standards
for these particular data. The WMO defines which system of units must be used
to represent the data for scientific purposes, in addition several guidelines
are provided by the WMO to perform the measurements under standard procedures.
However, these guidelines are not enough to specify the final format of the
data.
The WMO started a process of standardization in 2002, the goal is to create a
data format to fit the requirements of the GOS, in other words to provide a
common basement to represent the data of the weather’s observations. This is
an arduous task, not only for the amount of data that is needed to manage,
also for the big a mount of different phenomena in the atmosphere that are
producing different data and their particularities. It is expected that in
some point, the WMO will publish a standard for weather’s metadata
representation, nevertheless, after 9 years this process still under
development.
The absence of a standard for weather data representation is one of the key-
issues of the current situation. Without knowing how must be formatted the
data at the end of the collection workflow, is understandable that vendors
ended implementing their own formats without compatibility.
This is an open issue that unfortunately can not be treated in this thesis.
The author recognizes that the implementation of a protocol to transmit the
weather data without to know the final format of the data is a risky but an
interesting feature. In chapter eight, an exposition of the solution chose (a
software library to normalize the data) is explained.
We identify the absence of a common format for data representation as one of
the major technical deficiencies in the weather data transmission. In
addition, the absence of a common data format in the collection point as well,
forces to convert the weather data multiple times to the final format.
OpenWeather considers this issue and provides some mechanisms to implement
smoothly and mostly transparent the conversion from OpenWeather’s format to a
future data standard.
### 4.4 Data transmission and Automatic
Weather Stations
As embedded systems the AWS have more limitations that moderns computers, not
having capabilities to perform complex CPU operations or to manipulate a
considerable amount of data. Most of the modern AWS es offer the possibility
to interact with them in a small scale. Commonly, this interact is focused in
three tasks:
* •
AWS configuration
* •
Sensor’s calibration
* •
Data retrieval
Even so in most of the cases the AWS es behave as "broadcasters" of weather
data. The tasks of configuration and sensor’s calibration are performed only a
few times in the instrument, happening this at the beginning of the AWS’s
installation and in some periodical calibrations during the life-time of the
instrument; both operations are performed in most of the cases through
command’s line parameters or some Graphical User Interface (GUI) developed for
this purposed. As it was explained in section 3.2.2, the data transmission
with an AWS is performed through digital interfaces based in serial
communications standards, it means that at the end all the data transmitted
and received in an AWS goes through some data format implemented by the vendor
that provides a set of custom instructions.
>"BAUD 9600"<LF>
---
<<LF><CR>"OK"<LF><CR>
Table 4.4: Example of command configuring the baud rate of the digital
interface in an AWS.
Even if this practice is something understandable111111The author recognize
that to have a proprietary set of instructions can be a method to keep some
industrial’s secret of the instruments, however this practice difficulties the
implementation of standard methodologies to interact with multiple the
instrument., an exception should be made in the data retrieval operation.
Most of the AWS es offer the possibility to retrieve particular data if a
specific command is sent to them. Again the method to obtain this data is up
to the vendor, not being compatible these instructions between vendors, and
even sometimes even not between the products of the same manufacturer.
The mechanisms to retrieve data from the AWS es are critical in order to
implement a protocol with real-time capabilities. We need to differentiate two
use cases on an AWS. The first use case involves the data broadcasting that
the AWS is performing by default if it is configured as "automatic
mode"121212This is the default configuration used in almost all the
scenarios.. The AWS just send the data through the digital interface in the
time frequency configured, for this case is not required interaction with the
AWS; to read the data from the digital interface is enough to use it in the
protocol. Nevertheless the second use case involves the retrieval of
particular data. One example of this is a user interested in to know the
average of temperature recorded by the AWS in the last week. This data is not
sent by default because it is not part of the information collected in real-
time for the AWS, to get the data the user must send a command asking for it
to the AWS:
Command: aR2<cr><lf>
---
Response: 0R2,Ta=23.6C,Ua=14.2P,Pa=1026.6H<cr><lf>
Table 4.5: Example of command asking for PTH data.
This second use case introduces much more complexity. If a particular data not
send by default is needed, the interaction with the AWS is mandatory, however,
to interact with it implies to do it using the methodology specify by the
vendor. To implement a protocol that takes this use case in consideration
involves to implement a command-translator between the AWS and the protocol
implementation. We identify this issue as another technical deficiency in the
weather transmission in order to enable the capability to retrieve specific
data on demand.
### 4.5 Summary
In this chapter we described the state of art in the weather data
transmission. We have been analyzing the different interfaces available in an
AWS, focusing on their bandwidth, and based in the bit-rate that they offer,
concluding that the AWS are not taking advantages of all the capabilities
offered by the digital interfaces. This fact is enough reason to claim that
the AWS es are capable to manage more amount of data that the current
quantities that they do.
Data formats used by the vendors and data format requested for the
governmental organizations have been compared; finding that is not any
relation between the original format used in the AWS s and the final format in
which the weather data is represented, being this one of the reasons that
forces the implementation of intermediary points to translate the data to
different data formats.
The absence of a protocol dedicated to the weather data transmission has been
studied; the use of the FTP has been explained and the limitations that it can
involve to transmit data in real-time have been analyzed. We conclude that FTP
is chose by the industry as non-optimal solution that fix partially the issue
of the weather data transmission. In addition, the key issues of FTP has been
exposed in order to implement a system that use this protocol to delivery data
in real-time.
We analyzed the implications of a missing standard to represent the weather
data, concluding that without a consensus of the international community about
how the weather data should be represented, is really complex to implement a
protocol to fit all the requirements needed.
Despite the absence of a protocol and the use of multiple protocols and data
formats, the industry and weather organizations are using these methodologies
to acquire weather data in their weather data networks. Although projects such
as GOS or GDPFS, are looking for technologies to optimize and standardize the
weather data transmission, the current status of weather data acquisition is
based on the methodologies that the industry provided without previous
agreement. These methodologies have been accepted by the weather organizations
as the standards for the weather data transmission, achieving until today
their purpose.
At the end of the chapter we exposed how to retrieve particular data from the
AWS involves user interaction, adding complexity to the data workflow and
requiring an intermediary step to translate the data requests to the native
format used in the AWS to retrieve the data. We identify this as an impediment
in order to implement a protocol that provides data on demand.
The next chapter explains in which consists OpenWeather, its architecture and
how it can fix the issues explained in this chapter.
## Chapter 5 Introduction to OpenWeather
The previous chapter summarized the issues found by the author in the
protocols used for weather data. It has been analyzed how the weather
instruments use protocols as FTP or SMB to transmit data. Nevertheless, these
protocols are not designed to be used in a scenario in which the data is
generated based on real-time inputs. In addition, the current methodologies
provided by the industry, are not efficient enough to interact with the AWS
without additional effort in performing data normalization or data delivery.
This chapter gives a general overview of OpenWeather, the protocol developed
by the author, in order to provide a solution to problems that weather
instruments encounter during data transmission.
### 5.1 Overview and goals
OpenWeather is an application layer protocol based on Transmission Control
Protocol (TCP)/Internet Protocol (IP). It assumes a reliable transport layer
(TCP), in order to achieve a successful data delivery, based on such
mechanisms as error detection, flow control, congestion control, etc.
The protocol is built assuming three principles:
* •
Every AWS is considered to be a node
* •
A node accepts incoming sessions from peering hosts and initiates outgoing
sessions to peering hosts as well
* •
An AWS must have the capability to provide and to request services from other
nodes.
These principles are supported by assumptions that an AWS is an embedded
system with networking capabilities, able to interact via TCP/IP to deliver
the data produced by its sensors. The sensors’ output are considered to be
services offered by the AWS (node) to other nodes.
In addition, the star topology explained in section 3.1.4, disappears to give
way to a decentralized topology based on a peer to peer architecture.
OpenWeather provides the capability to dispense a unique collection point.
Instead, all nodes can be collection point and at the same time to be part of
other collections points. In addition, the protocol offers a service oriented
model (Service-oriented architecture (SOA)), to provide an easy way to
interact with the nodes and retrieve or send data to them.
Figure 5.1: Comparison of the currently centralized architecture provided by
the industry against OpenWeather architecture.
From the perspective of portability and data delivery, the protocol has been
designed to avoid problems with the endianness and data normalization; to
achieve this goal, JavaScript Object Notation (JSON) [16] has been chosen as
data interchange format between nodes.
JSON allows OpenWeather to use data streams based on parsable objects,
facilitating the data manipulation and normalizing the data to one common
format. Additionally, JSON is well supported by several libraries[17],
bringing the possibility to easily create applications based on OpenWeather
format.
Figure 5.2: Example of a OpenWeather’s JSON object inside of data message.
#### 5.1.1 Improvements in the current technology
OpenWeather provides a new paradigm for weather data collection. Based on a
Peer to peer (P2P) architecture, it allows the users to interact between
multiple nodes, retrieving and sending information inside of the network
independently of the brand’s instruments used. At the same time, it brings the
possibility to combine the real-time data streams obtained from the nodes,
providing a stack to build applications using multiple data sources without
requiring extra resources on the data manipulation.
In addition, the protocol is designed to be extensible, adaptable to new types
of data, while maintaining compatibility with future formats. Furthermore, the
service oriented model (SOA) of the nodes, allows the users to develop
applications that only want to obtain some specific data from a particular
service.
Finally, the protocol brings new opportunities to be operated under
distributed models and to provide implementational basis for future standards
of the weather data categorization. Because the data interchange format is
text-based and human-readable, it provides the capability to combine the
protocol with database applications without the need to develop extra API s,
facilitating even more possibilities to take advantage of the data.
#### 5.1.2 The role of OpenWeather and data spreading
OpenWeather is designed to fix deficiencies in weather data transmission,
while helping with the tasks of spreading data to the end users. Though most
of the phenomena require scientific analysis to make the data understandable,
some phenomena as atmospheric temperature, pressure or wind speed, are simple
enough and known to be spread across them directly to the end users without
the need of additional processing. OpenWeather allows to connect to an AWS
111Through a intermediary layer implemented through software., to retrieve
this type of data in real-time and —host to host— based, not needing more than
a computer with software supporting the OpenWeather protocol and network
connectivity.
In addition, the technologies used in OpenWeather can facilitate the creation
of new API s for web services oriented on weather’s forecasts. Some websites
offer the possibility for calling API s to obtain weather data. However, these
API calls are completely different between websites, which leads with extra
development time of web applications which utilizes different web resources
for data extraction. This problem can be easily handled with OpenWeather,
creating standard API calls according to the protocol specification. This
enables the use of such encapsulated protocols methods as HTTP for creating
for an intermediary bridge between the web application and the end nodes.
Figure 5.3: Example of an API call through HTTP and OpenWeather.
#### 5.1.3 Contribution to the current methodologies for weather data
acquistion
Even if OpenWeather is a proof of concept of an adapted protocol for AWS, it
proves how the problems exposed in chapter four can be resolved. The
feasibility of migration of scientific installations for production, will be
deemed feasible as the principles applied in OpenWeather, just adopting the
P2P architecture or the use of a human-readable lightweight format as JSON, it
will be enough to observe improvements in data delivery and acquisition. In
chapter seven is analyzed the results of use OpenWeather.
As it was mentioned in chapter four, the WMO has several worldwide projects,
such as GOS, in which different weather organizations around the world are
involved in the process of creation of future basis for weather data
processing. As described on WMO’s website[44], one of the purposes of the
project is: _’The coordinated system of methods and facilities for making
meteorological and other environmental observations on a global scale in
support of all WMO Programmes”_. OpenWeather, as scalable and extensible
protocol, can proven useful in certain areas of projects as GOS or SMEAR[40],
concerning data availability.
#### 5.1.4 Impact on weather instrument industry
As it was analyzed in chapter four, the industry has not started the process
of standardization for their instruments. Despite the issues that this
practice causes, OpenWeather aims to be the first solution that tries to fix
the absence of such protocol and at the same time provides a basis to be
adapted for the future data standard format, providing better archiving
mechanisms for a more efficient exchange of weather data.
Furthermore, the P2P architecture brings such a new industry paradigm,
allowing to develop new products in which real-time data retrieval will be put
to use.
### 5.2 Basic functionality of OpenWeather
Considering any AWS a node, the implementation of OpenWeather should be done
inside of the AWS’s software itself. Nevertheless, the author can not
implement a fully functional prototype, because it is not available any open
source / libre software version of AWS’s Operating System (OS). Instead, an
intermediary layer has been created for the evaluation setup, to normalize
data from vendor format into OpenWeather format. 222The removal of this layer
depends on cooperation between vendors in order to implement a protocol inside
of the AWS’s OS..
Figure 5.4: Middle-layer for data normalization.
This layer provides the conversion from native vendor format explained in
section 4.2, to an operational format in which OpenWeather can work. When the
data is pulled through a digital interface, the middle-layer recognizes the
vendor format and converts it according with OpenWeather requirements.
This middle-layer is located between the hardware and the network level,
giving as a result formatted data ready to be used in the protocol. With the
introduction of this layer, the steps mentioned in previous
chapters333Concerning data parsing. disappear. The data normalization occurs
only once at a time, instead of multiple times along the data workflow.
Original sender AWS data:0r2,Ta=10.6C,Tp=10.8C,Ua=74.6P,Pa=1006.0HKHK
---
OpenWeather’s format:
"Data" : {
"PTU" : {
"Air-Temperature" : "23.6",
"Relative-Humidity" : "14.2",
"Air-Pressure": "1026.6"
}
Table 5.1: Comparison of one vendor format against OpenWeather JSON format.
When data is normalized by this intermediary layer, the AWS is ready to
operate inside of OpenWeather network. This intermediary layer will not be
needed if the vendors establish a process of standardization.
#### 5.2.1 Peer to Peer Architecture
As mentioned in section 5.1, OpenWeather is designed based on a P2P
architecture. The RFC 5694 (Peer-to-Peer (P2P) Architecture: Definition,
Taxonomies, Examples, and Applicability)[8], defines a P2P system as the
following:
_[…] We consider a system to be P2P if the elements that form the system share
their resources in order to provide the service the system has been designed
to provide. The elements in the system both provide services to other elements
and request services from other elements. […]_
OpenWeather is according with the definition established by the RFC 5694 [8].
The protocol is thought to share the resources available in an AWS and at the
same time request services from others. In order to function properly the
OpenWeather network requires a minimum activity that must be performed by the
nodes (as peers’s list exchange).
Note that user itself is considered to be a node. It is not necessary to have
an AWS in order to be considered a node. A node is part of OpenWeather
network, interacting with other nodes, sending and retrieving data, while time
offering services to them444Thus, a user without an AWS can interact with
other nodes offering for example peer list exchange..
An OpenWeather node possesses the following properties:
* •
A node has a unique ID within OpenWeather’s network
* •
The geographical location of a node is essential to its connection in order to
OpenWeather’s network
* •
A node of the OpenWeather network can require the use of Network address
translation (NAT) [21] 555As described in RFC 5128 (State of Peer-to-Peer
(P2P) Communication across Network Address Translators (NATs)[41], will be
recommendable to implement the TCP/UDP Hole Punching technique in
OpenWeather’s software, in order to avoid peer connectivity issues.
Opposed to other P2P networks, OpenWeather does not use the P2P architecture
to archive a better performance transmitting big amounts of data666In fact, as
explained in section 3.1.2, the amount of data generated by a node is
insignificantly small.; the justification of use of P2P architecture in
OpenWeather is based on the distribution of the nodes and for better
interaction with them. The centralized model, fails to utilize its ability to
use weather data from different collections points without a pre-normalizing
data. In addition, the P2P architecture enables scaling of the network as
well, as giving the advantage of not being restricted by the limitations of a
central node.
#### 5.2.2 Service Oriented Architecture in nodes
As explained in section 3.1.2, an AWS produces real-time data collected by its
sensors. At the same time some AWS are able to store specific data in
persistent memory such as averages figures, daily reports, etc. These features
provide two data use cases for OpenWeather:
* •
Data becomes available in real-time
* •
Data can be retrieved on demand without the need to be real-time specific
OpenWeather handles these use cases providing an extra layer based on SOA. In
order to achieve this, OpenWeather provides a mechanism to discover which
services being available in a particular node, being possible after the
initialization of the session, to interact with these services.
Figure 5.5: OpenWeather stack over TCP/IP.
The fundamental reasons of choice of SOA for OpenWeather, is to facilitate the
accessibility of the data. A user can be both interested in receiving only
real-time data or in to retrieving a particular chunk of data. To provide this
capability, the protocol must be SOA oriented, in order to alleviate data
access through these services.
Figure 5.6: Uses cases available in OpenWeather via SOA.
Real-time data messages flow is considered to be as a continuos service
offered by the AWS via OpenWeather. Additionally, the possibility to retrieve
saved data in the AWS exits. Both real-time data and data on demand, is sent
and retrieved through OpenWeather data message system, using JSON. Thus,
OpenWeather offers the same possibilities as the common methodologies
currently used by the vendors explained in chapter four, moreover the chance
to get real-time data through a reliable and efficient way.
### 5.3 Summary
In this chapter we gave an introduction of OpenWeather, highlighting the
general guidelines applied in its design. We exposed some of the principles
used in OpenWeather nodes we considered some possible examples of future
applications using OpenWeather.
We introduced the areas in which OpenWeather can have a contribution or
impact. Projects as GOS or SMEAR[40] seeking for new technologies for data
acquisition, could get a positive use of OpenWeather concepts.
In addition, the basic functionality of the protocol, such as its
architectural principles or software model implementation have been introduced
as well.
## Chapter 6 Protocol specification
In this chapter, the OpenWeather protocol specifications are explained.
### 6.1 Definitions
The following subsections summarized the role of the elements involved in the
protocol. Some of the definitions are widely used in other protocols.
##### OpenWeather network
The nodes used in OpenWeather protocol conform to OpenWeather network
standards. Inside of this network a node is able to interact with other nodes,
requesting and delivering services to other nodes. These services are oriented
to provide weather data. Because OpenWeather is based in a P2P architecture,
its topology is decentralized. This topology makes the nodes independent of
central nodes in order to interact between them.
##### OpenWeather node
A node is an active or passive element connected to OpenWeather network. One
node can offer none to multiple services. An element is considered a node when
it has a working implementation of OpenWeather protocol and is connected to
the network.
##### Peer
Every node is considered a peer of OpenWeather network. All nodes in
OpenWeather network are able to be clients and servers at the same time. This
is establishing the basis of the P2P architecture used in OpenWeather. A peer
must be able to offer services to others peers, however it is not mandatory to
offer a service111Doing reference here to high-level services related with the
data delivery, de facto, a peer is always offering a minimum amount of
services integrated within the protocol, needed to interact in the network. in
order to be connected to OpenWeather’s network.
##### Weather data
The purpose of OpenWeather is to create a network in which the data exchange
comes from the weather data sources. To obtain this data the nodes can be
connected to an AWS or other system of weather data collection. OpenWeather
does not differentiate between the original source of the instrument’s brand,
because data normalization222As it was explained in chapter 3 and chapter 4,
this step is required because it is not possible to modify AWS’s OS without
the vendor’s collaboration. is required in order to make the data network
available.
### 6.2 Architecture
As it is mentioned in section 5.2.1, the architecture used in OpenWeather
matches the requirements mentioned in the RFC 5694 [8], with OpenWeather
containing nodes offering and requesting services between them.
The technical reasons why a P2P architecture is a better network solution for
a topology as define by default by the AWS, are supported in the following
points:
* •
An AWS is an individual entity being part of a bigger network that does not
need a centralized model except for data processing.
* •
The process executed over the weather data in order to extract meaningful
conclusions does not posses a technical requirement to be linked to the
network layer.
* •
The collection point model forces the node to depend exclusively on one node
in the network, adding unnecessary risks to the data flow.
* •
The common architecture used in the weather data flow, is forced by the legacy
of the protocols used within it.
The P2P architecture is chosen by OpenWeather because it brings autonomy and
robustness to the nodes. In addition, it provides the network the capability
to scale and to share resources without single dependencies. Moreover, the
geographical situation of the nodes, is suitable for developing models in
which the nodes can collaborate to distribute the data. Finally, the P2P
architecture provides the capability to retrieve data directly from the node,
without going through a common point that can be collapsed or not available.
#### 6.2.1 Standards used for data units
OpenWeather does not provide the weather date measurement units. The protocol
is designed to deliver weather data formatted according to the data units
specified in International Standard Organization (ISO) 80000 [27] family and
the _Guide to Meteorological Instruments and Methods of Observation_ [46].
Table 6.1 provides the data units used in the prototype:
Data field | Data unit | Acronym
---|---|---
Air-Temperature | Celsius | C
Relative-Humidity | Percentage | % RH
Air-Pressure | Hectopascals | hPa
Wind direction | Degress | degrees
Wind speed | Meters per second | $m\over s$
Rain accumulation | Millimeters | mm
Rain duration | Seconds | s
Rain intensity | Millimeters per hour | $mm\over h$
Rain peak | Millimeters per hour | $mm\over h$
Hail accumulation | Hits per square centimeter | $Hits\over cm^{2}$
Hail duration | Seconds | s
Hail intensity | Hits per square centimeter per hour | $Hits\over cm^{2}h$
Hail peak | Hits per square centimeter per hour | $Hits\over cm^{2}h$
Table 6.1: Data units implicit on the data fields.
Since the data units have a known standard, the author considers that it is
not necessary to increase data messages sizes and data fields, but only to
provide the data units. Instead, it is more pragmatic and efficient to assume
that weather data will be supplied with appropriate data units. It is
necessary to highlight that despite the absence of network protocol for
weather data, the vendors maintain a strict control of data units used in the
AWS, facilitating this the implementation of OpenWeather across vendors.
#### 6.2.2 Nodes
A node connected to a OpenWeather network behaves as a deterministic finite
automaton, not executing without a clear definition operations or a definite
result. All the operations performed by the nodes are identified by codes
placed in the MetaInfo data field. Any data message delivered in OpenWeather
protocol contains all information333Through the protocol code. required to
identify the type of operation to be performed by the software when the data
message is received / delivered.
Any data requested or delivered by a node using OpenWeather is based on a
request and a confirmation of it. With this mechanism the nodes are notified
of status of the operations of execution in the application layer are
successful or not. This same mechanism is implemented in protocols as HTTP
[22] in order to control the status of retrieval and delivery operations.
A node is able to interact with multiple nodes, being only limited by the
bandwidth and system resources availability. OpenWeather does not define a
minimum or maximum of connections needed, however a node requires a >=1 number
of peers on its internal list in order to interact with OpenWeather network.
##### Automatic Weather Stations as individual nodes
The section 3.1.2 explains how the AWS are categorized as embedded systems. By
the definition, an embedded system has certain limitations in data processing
and data delivery. Nevertheless the AWS are still able to do some networking
operations and data processing when the size of them is small. OpenWeather has
been designed to work around these limitations.
Taking this as a basis, OpenWeather transforms the centralized model currently
used by the industry, to a decentralized model taking advantage of a P2P
architecture. OpenWeather considers every AWS as a node using SOA. Because the
AWS are under constant connection and deliver data to collection points, the
only modification needed in the equipment is to change the network protocols
used to deliver this data444An adaption of the AWS’s OS will be required in
order to integrate the OpenWeather’s stack inside of the AWS..
Instead of using an architecture in which the AWS plainly sends the data over
the network without any further interaction, OpenWeather provides the
mechanisms to convert the AWS to an entity able to respond to the data
requests made by the user in real-time. Although all of this process can be
handled through the centralized model, the independence of nodes from the
collection point is mandatory in order to achieve scalability and data
accessibility. For instance, a user located outside of a specific network of
AWS, can not access the data produced by them without the need to go through
the collection point555If the AWS work but the collection point is down, the
data will not be accessible., this use case avoids any possibility to combine
data in real-time from different AWS in different geographical locations,
restricting any possibility to interact directly with the AWS.
Enabling the AWS to behave as a nodes, the protocol provides the basis to take
advantage of the real-time data and at the same time fix the issues exposed in
chapter four. Though this thesis sets an ambitious goal: the transition from a
centralized model to a decentralized model, it has to be noted that the
industry has been using the same technologies for decades, not taking
advantage of the improvements made in the networking technologies, concerning
data delivery and acquisition. The decentralized models have a proven
successful track, offering scalability and robustness.
As any other network protocol, OpenWeather has a defined set of operations.
These operations provide the core principles to deliver and retrieve data from
nodes. However, these principles do not need to contain the whole data flow.
##### Super-nodes
OpenWeather refers to super-nodes to those nodes with static IP/ hostname,
which are always available to exchange peer lists. Unlike other P2P
applications, an OpenWeather super-node does not have any other extra
property, except its bandwidth availability 666It must be higher than average
so that it may process higher network traffic. and an updated list of peers,
to deliver to the other nodes. The role of a super-node is to be always
available and to provide updated peer lists to those nodes without one. This
is enough to guarantee that the nodes will be able to connect to it if they
can not find other nodes available.
##### Peer list calculation algorithm
One of the biggest challenges of the P2P architecture is to identify which
peers are superior to others. This issue is mostly found in those
architectures in which the purpose of the network is to transfer data based on
user reputation777Meaning the amount of data shared and uploaded to other
peers.. Since all nodes are consider peers containing unique data, OpenWeather
does not make distinction amount them.
Even so, for practical reasons, it is necessary to develop an algorithm to
calculate which peers are better than others in terms of connectivity and
bandwidth availability, to provide a list to the nodes to guarantee the
connection to OpenWeather network.
The author considers that due to the nature of the data and the main factor of
its importance is availability. Thus, the algorithm shall be a node bandwidth,
network latency and geographical location.
Bandwidth and latency are two obvious and common used factors in other P2P
architectures. However in this case is important to note that most of these
nodes are going to have better network visibility with nodes are located in
proximity. The geographical location of the node, available in the MetaInfo
data field through the "Location" data field, can be used to calculate the
closest peers.
The algorithm to calculate the best peers to keep on the internal list, is too
a vast and complex topic to be analyzed in this thesis. In the prototype
created, the author used random peers in order to verify the protocol
specifications. It is necessary to highlight that the peer list calculation
must be analyzed deeply in order to implement OpenWeather in production
scenario.
##### Node identification
In section 5.2.1 is mentioned that a node has a unique ID. This ID is used to
identify the node and at the same time by the user/software to recognize which
node is currently active. The value of this ID is based on Secure Hash
Algorithm (SHA)-256[34]. Nevertheless, the length of it and its alphanumeric
composition make it really difficult to remember the node ID, even when using
some mnemonic techniques. However, it can be easily fixed with a proper
algorithm, based on a standardized AWS system for identification and use of
the RFC 3986 _Uniform Resource Identifier (URI): Generic Syntax_ [6].
As example, the CWOP uses different parameters[31] to identify the AWS; some
of them are:
* •
Block number 2 digits representing the WMO-assigned block
* •
Station number 3 digits representing the WMO-assigned station
* •
Place name: common name of station location
* •
Country name: country name is ISO short English form
The block number refers to the geographical region888Extracted from station
index numbers database, CWOP Meteorological Station Location Information [31].
of the AWS, and the station number is assigned base on _the nearest 10 degree
meridian which is numerically lower than the station longitude_[31]. The place
name and country name are values assigned based on the geographical location
of the AWS. Although CWOP also provides the latitude and the longitude, their
introduction in Uniform Resource Locator (URL) generation, will cause greater
complexity.
02;974;EFHK;Helsinki-Vantaa;;Finland;6;60-19N;024-58E;60-19N;024-58E;51;56;P
Table 6.2: Example of CWOP’s AWS identification.
The table 6.2 shows all data used by CWOP to identify an AWS, the the
following syntax is used to generate the URL:
owp://Country Name/Place Name/Block number + Station number
---
Table 6.3: ID partially based on CWOP notation.
Based on the data used in the table 6.2 the output will be:
owp://finland/helsinki-vantaa/02974
---
Table 6.4: ID’s partially based in CWOP’s identification system.
The scheme is denominated as owp (OpenWeather Protocol), the authority field
is used for country name, the absolute path is based on the place name and the
station number assigned by the WMO. This combination is enough to guarantee
the uniqueness of the node accessed through the URL.
The value of the ID used in the OpenWeather data message will be the resulting
hash of the data "02;974;Helsinki-Vantaa;;Finland" generated with SHA-256.
{
"OpenWeatherMessage": {
...
"ID" :"a88a9b6b4c0381e0509ce36cadb5fd06e5446ab23881020b9f212db24b16ee75",
...
},
---
Table 6.5: IDs based in the SHA-256 result of the CWOP notation.
### 6.3 Protocol operations
The protocol allows the following operations:
* •
Session establishment
* •
Service discovery
* •
Real-time data retrieval
* •
Data on demand
Note that all of these operations have an implicit internal functional
workflow, based on the requests and retrievals and their results. The
following sections analyze the functioning of these operations.
#### 6.3.1 Session establishment - Peer handshake
The first operation needed for OpenWeather is session establishment. The
elements involved in this operation can go from 2..n nodes. Thus, a node can
execute the operation to establish session with multiple nodes at the same
time, nevertheless, the session establishment is always an isolated process
between two nodes.
These nodes must offer the basic services integrated in the protocol, as peer
exchange information or peers-list exchange. The session establishment between
nodes is denominated peer handshake. At this point the nodes exchange their
information in order to identify each other, sending a data message with the
parameters mentioned in section 6.2.4. This operation is categorized as an
internal protocol requirement, using the code 100 as type of data message.
Figure 6.1: Session establishment sequence diagram.
When the nodes establish a session, two operations are performed
* •
Peers-list exchange.
* •
Alive verification.
The first operation —peers-list exchange— is performed in order to verify if
the nodes can update their internal list of peers available.
The second operation performed is alive verification. The peers send a data
message after the exchange of the peer list, in order to verify that the nodes
are ready to request weather data999Note that this check is realized to ensure
the availability of the node twice.. If the alive verification is not
successful, the node executing it will close the TCP connection with the node
that is not responding to it.
#### 6.3.2 Service discovery
OpenWeather assumes that when two nodes establish a session, the purpose of it
is to exchange certain data, even if it is just for protocol requirements. As
it is explained in section 5.2.2, OpenWeather is designed according to a
service oriented architecture SOA. All data sent or receive by a node goes
through services provided in the OpenWeather software implementation.
Figure 6.2: Service discovery sequence diagram.
The nodes involved in the session must exchange the type of data messages, in
order to be aware of services available to the nodes. Note that this operation
informs the nodes which sensors are available to other nodes and which kind of
weather data can be retrieved from them. After the nodes communicate through
the services available, other operations as real-time data retrieval or data
on demand, can be performed.
#### 6.3.3 Real-time data retrieval
When the nodes establish the session and service discovery operations is
performed successfully, they are consider to be ready to send and receive
weather data between them. As it is explained in section 5.2.2, the data can
be real-time data or data on demand. In case of real time data, the node
requesting it, must send a type of data message with code 200, immediately
after, the other node involved in the session, must start to delivery real-
time data messages.
Figure 6.3: Real-time data sequence diagram.
The real-time data will be deliver until the node requesting it decides to
stop the data stream101010The data streams can be interrupted by other
exceptions as connectivity or software issues.. This data stream provides the
real-time data generated in the remote AWS. As in any other network solution
the delay that the nodes can experience can affect the delivery of the data.
Nevertheless, all the data messages are timestamped when the data was
assembled within them. Because this timestamp is available, it will be
feasible to implement an algorithm on the software side, applying a correction
factor to the timestamp based on the latency of the nodes, to fix this issue.
#### 6.3.4 Data on demand
Apart from the the real-time data, a user can request data on demand. When a
user requests data on demand, it creates individual requests with a specific
timestamp. Based on these requests, the remote node will deliver an individual
data message timestamped with the date and time provided, the requests and the
weather data on that time.
Figure 6.4: On demand data sequence diagram.
OpenWeather does not support the capability to request a range of dates or
times on protocol level, meaning that it is not possible to retrieve isolated
weather data samples form the node during a certain period of time. Instead,
it is possible to implement on the software side the functionality to process
a group of data messages with a certain timestamp. The justification of this
limitation is based on the bandwidth availability in an AWS. In contrast with
individual weather data samples, a range of them can have a considerable size
and this can cause significant obstacles for the AWS: heavy CPU load,
bandwidth consumption, etc.
### 6.4 Data messages
A data message refers to the data transmitted using the OpenWeather protocol.
A data message can contain multiple informational values, referring to weather
data or data needed for protocol maintenance.
All data fields contained in an OpenWeather data message are considered to be
encapsulated data represented through JSON objects using Universal Character
Set - Transformation Format (UTF)-8[13][47] as character encoding. According
with the RFC 4627[16], the definition of an JSON’s object is:
_[…] An object is an unordered collection of zero or more name/value pairs,
where a name is a string and a value is a string, number, boolean, null,
object, or array.[…]_
Therefore, any data field contained in an OpenWeather data message, is an
individual or group of JSON objects or values. These objects are optimized
according to the data that they contain. For instance, some data fields are
JSON objects containing other objects at the same time. The data optimization
made in the protocol using these data structures, allows data encapsulation
which makes enables a fast data the data processing from the network to
software levels.
All OpenWeather data messages are formatted using JSON syntax. Type of data
contained in the data message is insignificant as it is structured in one JSON
object composed for different sub-objects. These objects are represented as
data fields in terms of networking architecture.
Figure 6.5: OpenWeather data message structure.
The parent object is denominated OpenWeatherMessage; this object is present in
all the data messages inside of OpenWeather network. This parent contains two
sub-objects; the MetaInfo object and the Data object or Info object. The
MetaInfo object is a data field acting as the header of the data message in
OpenWeather protocol111111This object is added an individual data field named
”Type”, explained in the next section.. Furthermore an OpenWeather data
message contains the Data object or the Info object. The Data object is a data
field containing all data related to the weather data that the data message
transports. The Info object contains the information used internally by the
OpenWeather protocol.
#### 6.4.1 Header
OpenWeather uses a fixed121212In terms of data fields provided. header data
field in all the data messages, in order to guarantee its functioning. The
function of this header is to provide all necessary data parameters needed by
the OpenWeather protocol in every data message. Though it requires some data
repetition, its insignificant size of this header, compensates the
disadvantages of its repetition during transmission.
Table 6.6 shows the fields contained in the header:
{
"OpenWeatherMessage": {
"Type" : "",
"MetaInfo" : {
"ID" : "",
"Peer-IP": "",
"Port": "",
"Location": "",
"Update-Interval": "",
"Peers-request":"",
"Keep-Alive":"",
"Bandwidth": "",
"Timestamp" : "",
"Version" : "",
},
---
Table 6.6: Header field (Header object) in a data message of OpenWeather.
As the table 6.6 exposes all data messages start with the term
”OpenWeatherMessage”, building JSON parent object of the data message. Any
data contained within the data message will belong to this parent object.
Although this hierarchy does not impact the data message size, it provides
significant assistance to the post processing of the data on the software
side. This design is inspired by the same concepts use in eXtensible Markup
Language (XML) and XML Schemas [14], concerning the metadata fields.
Nevertheless, OpenWeather does not providing any extra fields for metadata
definition, meaning that the software utilizing OpenWeather, should recognize
the expected format beforehand131313XML allows data type provision in the data
itself. However, this practice increases the size of the data considerably..
With this practice speed up and simplifies the parsing compare to XML.
#### 6.4.2 Types of data messages
The second field contained in an OpenWeather’s message is denominated Type.
This field indicates which type of data is located within a data message
through a numerical code and if it is related with weather data, peers
exchange, protocol itself, etc.
Depending on the type of data message it will be in one of the following
categories:
* •
Data messages for protocol maintenance only.
* •
Data messages use to transport weather data only.
* –
Real-time data.
* –
Data on demand.
#### 6.4.3 Protocol codes
The "Type" field can contain a numerical value from 1..n. This numerical value
is known as the protocol code associated with the type of messages. The codes
used are divided in categories and subcategories:
* •
Codes assigned to data messages used for protocol maintenance.
* –
Protocol codes (From: 1..1xx)
* *
Requests
* *
Retrievals
* *
Status
* ·
Success
* ·
Error
* •
Codes assigned to data messages for weather data exchange between peers:
* –
Peer codes
* *
Requests
* ·
Real-time data: 200
* ·
Data on demand: 201
* *
Retrievals
* ·
Real-time data: 300
* ·
Data on demand: 301
* *
Status
* ·
Success: 500..599
* ·
Error: 600..699
The numerical value is used by the software in order to recognize the data
processing procedure.
All the protocol codes used in the prototype are available in the appendix.
#### 6.4.4 MetaInfo data field
The MetaInfo data field (MetaInfo JSON object) defines fixed data fields
transmitted in every data message. The purpose of these fields is to provide
all information needed, in order to identify the peer’s ID, its geographical
location, IP address, among other data. The use of this data throughout all
data messages makes allows for easier implementation and extensibility of the
P2P architecture, as it enables the software to be aware properties and status
of a specific peer at all times.
The MetaInfo field contains the following data fields:
* •
Bandwidth
* •
ID
* •
Keep-Alive
* •
Location
* •
Peer-IP
* •
Peers-Request
* •
Port
* •
Timestamp
* •
Update-Interval
* •
Version
The MetaInfo data field is structure as a JSON object containing an array of
elements141414Note: in the following figures the expression ”ARRAY DATA
ELEMENTS” is used to refer the MetaInfo data fields.. These elements are the
fields mentioned above. Every element does reference to an specific parameter
needed by OpenWeather protocol.
{
"OpenWeatherMessage": {
"Type" : 1,
"MetaInfo" : {
¯ ARRAY DATA ELEMENTS
},
},
---
Table 6.7: MetaInfo field in a data message of OpenWeather protocol.
Figure 6.6: OpenWeather MetaInfo data field with data array elements.
##### Bandwidth
As any other network oriented software, the amount of bandwidth is a critical
factor in its proper functionality. Most software solutions using P2P
architecture offer a dedicated section to control the bandwidth parameters.
OpenWeather informs others nodes of the amount of bandwidth that a node has
available while giving full control of the amount of bandwidth and connections
and remote connections allow. As opposed to mainstream solutions, in which the
node is only controls the amount of connections and bandwidth locally, the
bandwidth control in OpenWeather can be managed both locally and remotely. To
achieve this, the field "Bandwidth" is provided in every data message,
informing the nodes what is the capacity of the node whereby are operating.
{
"OpenWeatherMessage": {
...
"Bandwidth" : "4", // Correspondency 1 Megabit/s
...
},
---
Table 6.8: Bandwidth field in a data message of OpenWeather
The user must provide this parameter to configure its node. Due to the a big
amount of possibilities for bandwidth quality, this data field contains a
numeric value that should be translated by the software to bits per second.
Nevertheless, if the user considers that its bandwidth does not fit in the
categories provided, it is possible to provide an integer number that will be
translated by the software to bits per seconds. Thus, if the "Bandwidth" data
field contains a numeric value higher than 6, the value will be translated for
the software to bits per second. This feature allows the user to use a custom
parameter.
Numeric value | Bandwidth equivalency
---|---
0 | 56 kbit/s
1 | 128 kbit/s
2 | 256 kbit/s
3 | 512 kbit/s
4 | 1 Mbits/s
5 | 10 Mbits/s
6 | 100 Mbits/s
Table 6.9: Bandwidths equivalency in Bandwidth data field.
##### ID
As explained in sections 5.2.1, every peer has an unique ID throughout
OpenWeather’s network. In fact, its properties make it theoretically unique in
the world.The ID is generated based in the AWS identification. The ID data
field is thought to be representation of the AWS, such representation is the
result of the hash applied over some identification system for AWS
es151515Several weather organizations provide this identification..
If the AWS is not part of some identification system, its ID can be generated
randomly by the software, however is highly recommended to provide an ID
assigned for some organization as the CWOP or NOAA.161616In the evaluation
setup, the author uses a random ID..
{
"OpenWeatherMessage": {
...
"ID" :"4f9a67e8496d69b8707858576ec12b8aa3fa5519c23a79ea071dc7dbc0c9b2e3",
...
},
---
Table 6.10: ID’s field in a data message of OpenWeather protocol.
##### Keep-Alive
Due to possible node connection instability, it is necessary to implement a
mechanism to identify the current connection status with a specific node is,
on the application layer level. OpenWeather implements the field "Keep-Alive".
This field provides the amount of time that the software must wait until the
connection is close.
{
"OpenWeatherMessage": {
...
"Keep-Alive" : "120000",
...
},
---
Table 6.11: Keep-Alive field in a data messages of OpenWeather protocol.
When a node stops sending data to other node/s, the connection will be closed
when the sum of the timestamps of the last data messages received and the
"Keep-Alive" value, is less than the current date and time.
The protocol assumes that if the node is not delivering data, is not useful to
keep a connection with it. The same principle is applied in a number of
network oriented software solutions. The value of this field is expressed in
milliseconds, and by the default has a timeout of 120000 milliseconds (2
minutes). Though possible, the customization this parameter is not
recommended, as it assumes responsibility between nodes when necessary.
##### Location
The "Location" field does reference to the geographical coordinates of the
node, expressed in the Universal Transverse Mercator (UTM) system. This data
field has two different functions:
* •
Identify the geographical location of the node.171717Mandatory due to the
nature of the data.
* •
Provide identificational information to other peers, does providing them with
the updated information which store in the node’s internal list. 181818This is
explained deeper in section 7.2.
{
"OpenWeatherMessage": {
...
"Location" : "4597807 269999 30T",
...
},
---
Table 6.12: Location field in a data messages of OpenWeather protocol.
This field should be filled manually by the user. It is highly recommended to
provide this parameter with as much accuracy as possible.
##### Peer’s IP address & port
The MetaInfo’s field contains two fields dedicated to TCP/IP:
* •
Peer-IP
* •
Port
The field "Peer-IP" contains the public IP address assigned to the computer’s
network interface that is running the software supporting OpenWeather’s
protocol.This field can be an IP address using 32-bit number (IP v4) or 128
bit number (IP v6). The introduction of this field is based on the requirement
of the protocol to possess an updated address of the peer in order to able to
connect to it. Though the field is labeled as "Peer-IP" not necessarily must
be the numeric address. It is possible to implement the OpenWeather protocol
to use hostname resolution based on Dynamic Name Server (DNS)
requests191919However as it is implicit in the use of DNS, it will be required
to have the hostname of the peers recorded in the name servers., with a few
modifications on the software’s side.
The field "Port" contains the port used in the TCP to establish a connection
with the peer. The default TCP port number is 62535202020Port number choose
according with the range of ports available for dynamic and/or private use
published by IANA[5].212121We assume fixed ports and port forwarding
techniques for this. The functioning of OpenWeather behind firewalls or/and
NAT is out of the scope of this thesis.nevertheless any port can be used
inside of TCP’s range always that it does not conflict with other ports.
{
"OpenWeatherMessage": {
...
"Peer-IP" : "140.186.70.148",
"Port": "62535",
...
},
---
Table 6.13: Peer-IP & Port fields in a data message of OpenWeather protocol.
Both fields, "Peer-IP" and "Port", are present in others P2P
architectures222222Often denominated with different terms and syntax., the
reason for this is that these fields facilitate a significant part of the
software implementation and the network functionality. Adding these fields to
all data messages, enables the software to keep the peer list updated and
working between nodes and at the same time it facilitates the protocol session
establishment.
##### Peers-Requested
The "Peers-Requested" field provides the number of peers that the node
requests to other nodes in order to fill its internal list of peers. In a P2P
architecture it is critical to keep an updated list of peers to guarantee
successful delivery of the data throughout the network. By default this field
is set to 20, with a possible range of 1..100.
{
"OpenWeatherMessage": {
...
"Peers-Requested" : "20",
...
},
---
Table 6.14: Peers-Requested field in a data messages of OpenWeather protocol.
##### Timestamp
As explained in chapter two, the success of weather prediction depends on
different factors. One of the most important variables are the geographical
location and the time and date, in which the weather data samples were
collected. OpenWeather provides the field "Timestamp" to supply a solution for
this condition. Every data message contains the timestamp in which the data
was assembled. This provides a feasible mechanism to know when the weather
data sample by the data message received was collected.
The data format used by OpenWeather protocol follows the RFC 3339 (Date and
Time on the Internet: Timestamps)[28] and it follows the guidelines
established by the ISO 8601:2004[20] as well. All data messages are
timestamped using the Coordinated Universal Time (UTC).232323The conversion to
the original timezone of the data message can be managed through software.
{
"OpenWeatherMessage": {
...
"Timestamp" : "2011-05-29T12:10:23Z",
...
},
---
Table 6.15: Timestamp field in a data message of OpenWeather.
Note that OpenWeather protocol does not use the timestamp value for any
purpose related with protocol operations. This Timestamp field is provided in
order to fit the requirements of the weather data. Because the weather data
requires precise stamping of the time in which it was acquire, this field is
introduced. In addition, as in other real-time data systems, it is recommended
to sync the time of the node using protocol such as Network Time Protocol
(NTP), to guarantee the quality of the data. Such synchronization must be
managed independently of OpenWeather.
##### Update-Interval
The "Update-Interval" field contains the time value, expressed in
milliseconds, that other peers should wait before to requesting protocol
information. This field can be used to manage data availability provided
absence of network congestion.
{
"OpenWeatherMessage": {
...
"Update-Interval" : "120000",
...
},
---
Table 6.16: Update-Interval field in a data messages of OpenWeather protocol.
By default this field is set to 120000 milliseconds (2 minutes), however this
parameter that can be customize by the user.
##### Protocol versioning
Following the same principles as HTTP and other protocols, OpenWeather uses
<major>.<minor> numbering scheme to indicate the versions of the protocol. The
versioning is indicated in the "Version" field of the data header, adding the
term ”OpenWeather” and the character ’/’ before the numbering.
{
"OpenWeatherMessage": {
...
"Version" : "OpenWeather/1.0",
...
},
---
Table 6.17: Version field in a data message of OpenWeather.
##### MetaInfo data field summary
The table 6.18 shows the structure of the MetaInfo data field with all array
elements already filled in with data:
{
"OpenWeatherMessage": {
"Type" : 1,
"MetaInfo" : {
"ID" :"4f9a67e8496d69b8707858576ec12b8aa3fa5519c23a79ea071dc7dbc0c9b2e3",
"Peer-IP" : "140.186.70.148",
"Port": "62535",
"Location" : "4597807 269999 30T",
"Update-Interval" : "120000",
"Peers-Request" : "20",
"Keep-Alive" : "120000",
"Bandwidth" : "4",
"Timestamp" : "2011-05-29T12:10:23Z",
"Version" : "OpenWeather/1.0",
},
...
}
---
Table 6.18: MetaInfo data field (MetaInfo object) in a data message of
OpenWeather.
All the OpenWeather data messages will contain a header as the shown in the
table 6.18, fill in with the particular data of the node.
#### 6.4.5 Data field
As part of the MetaInfo data field (MetaInfo object), OpenWeather data
messages can contain a field named Data (Data object). This data field is a
JSON object composed from different sub-objects. The values or sub-objects
having this object as a parent, are dedicated to transport weather data.
The Data field is necessary in order to complement the MetaInfo data field.
The MetaInfo data field only provides information about the node itself. The
data field contains the data that the node retrieves or request from others
nodes. The type of data available in this data field can be:
* •
Real-time weather data
* •
Data requested/delivery in demand (non real-time)
{
"OpenWeatherMessage": {
"Type" : 1,
"MetaInfo" : {
¯ ARRAY DATA ELEMENTS
},
"Data" : {
¯ ARRAY DATA OBJECTS
}
},
---
Table 6.19: Data field in a data message of OpenWeather protocol.
All phenomena data transmitted in OpenWeather uses the data units, specify in
the _Guide to Meteorological Instruments and Methods of Observation_[46],
published by the WMO [44]. The author assumes that the protocol must follow
this standard, because it is adopted by the major number of
countries242424Exceptions: United States, Liberia and Myanmar (Burma).. Though
some countries still keep local units for measurements, OpenWeather protocol
does not take in consideration these use cases, nevertheless the
implementation of the conversion between units, can easily be done on the
software side.
All values or sub-objects containing information about weather data will
always have the Data object as a parent.. The following sections explain how
these different types of data are assembled in OpenWeather.
##### Real-time data messages
The section 6.4.4 introduced the persistent data provided in every data
message of OpenWeather. However, this data is provided in order to guarantee
the protocol’s functioning. A part of the header field, the data messages can
contain weather data. This section explains how a real-time message is
assembled. Note that the prototype used in the experimental setup only
supports data extracted from the following phenomena:
* •
PTU -Pressure, Temperature, Humidity
* –
Air temperature
* –
Relative humidity
* –
Air pressure
* •
Wind
* –
Direction (minimum, average, maximum)
* –
Speed (minimum, average, maximum)
* •
Precipitation
* –
Rain (accumulation, duration, intensity, peak)
* –
Hail (accumulation, duration, intensity, peak)
These data have been chosen because it is available in most of the AWS of
semi-professional / end-user range. In addition, the data used in OpenWeather
provides a functional prototype adapted to this thesis. The author highlights
that none of these data fields (concerning weather data) are used claiming
them to be a standard or a suggestion of it. As mentioned in section 4.3, only
a process of standardization can provide the correct data fields to use.
Nevertheless, the use of these weather data fields is enough to develop a
prototype.
Note that some data objects contain values in the data fields such as
"minimum", "maximum" or "accumulation" among others, that are representing
data collected in time intervals. Depending of the phenomenon these time
intervals can be completely different. The recommend intervals of measurement
are described in the _"Guide to Meteorological Instruments and Methods of
Observation"_ [46], and theoretically they must be always the same
independently of the brand of the weather instrument used.
##### Pressure, temperature and humidity data
The Pressure, Temperature and Humidity (PTU), are the most common data
available in an AWS, due to the close relation between the phenomena and the
ease of its acquirability. Any modern AWS will is equipped with necessary
sensors to measure these phenomena.
The AWS es collect this data in real-time, transforming the raw input data
from the sensors into digital data. The workflow of this data is described in
section 3.1.4. As other data in OpenWeather, it will be normalized in the
layer implemented between the hardware layer and OpenWeather252525Explained in
section 5.2..
0r2,Ta=18.7C,Ua=77.4P,Pa=1002.1H
---
Table 6.20: PTU real-time data in the raw format used by the AWS.
When the PTU data is transformed to OpenWeather’s format, it has the following
format:
{
"OpenWeatherMessage": {
"Type" : 1,
"MetaInfo" : {
¯ ARRAY DATA ELEMENTS
},
"Data" : {
"PTU" : {
"Air-Temperature" : "",
"Relative-Humidity" : "",
"Air-Pressure": ""
},
},
---
Table 6.21: PTU data field in a data message of OpenWeather protocol.
The three data fields contained in the Data object are:
* •
Air-Temperature: expressed in degree Celsius (°C)
* •
Relative-Humidity: expressed in percentage in base of relative humidity
* •
Air-Pressure: expressed in Hectopascals (hPa)
These data fields are encapsulated as an array of data elements inside of the
JSON object PTU. The table 6.22 shows an example of the PTU object filled with
real-time data:
"Data" : {
"PTU" : {
"Air-Temperature" : "20.0", // Celsius: ºC
"Relative-Humidity" : "59.5", // %RH
"Air-Pressure": "1002.1" // Hectopascals: hPa
},
---
Table 6.22: PTU data field with real-time data in a data message of
OpenWeather protocol.
The frequency of reporting this data will depend on the configuration of the
AWS. Most of AWS offer a time interval between 1 second and 3 seconds, to
generate this data.
##### Wind data
The wind is other of the most popular phenomena to measure in AWS es. The wind
data contains two sub-objects: direction and speed.
{
"OpenWeatherMessage": {
"Type" : 1,
"MetaInfo" : {
¯ ARRAY DATA ELEMENTS
},
"Data" : {
...
"WIND" : {
"Direction" : {
"min" : "",
"ave" : "",
"max" : ""
},
"Speed" : {
"min" : "",
"ave" : "",
"max" : ""
}
},
---
Table 6.23: Wind data field in a data message of OpenWeather protocol.
At the same time these two objects are composed by three array data elements:
* •
Direction
* –
Minimum (min): expressed in degrees
* –
Maximum (max): expressed in degrees
* –
Average (avg): expressed in degrees
* •
Speed
* –
Minimum (min): expressed in meters per second $m\over s$
* –
Maximum (max): expressed in meters per second $m\over s$
* –
Average (avg): expressed in meters per second $m\over s$
{
"OpenWeatherMessage": {
"Type" : 1,
"MetaInfo" : {
¯ ARRAY DATA ELEMENTS
},
"Data" : {
...
"WIND" : {
"Direction" : {
"min" : "217", // Degrees
"ave" : "217",// Degrees
"max" : "218"// Degrees
},
"Speed" : {
"min" : "4.2",// m/s
"ave" : "4.2",// m/s
"max" : "4.5"// m/s
}
},
---
Table 6.24: Wind data field with real-time in a data message of OpenWeather
protocol.
##### Precipitation data
Precipitation is the last phenomena that typically all the AWS es measure.
Inside of the concept of precipitations encompasses two different classes,
rain and hail. Thus, the precipitation is structure two sub-objects containing
an array of four data elements.
{
"OpenWeatherMessage": {
"Type" : 1,
"MetaInfo" : {
¯ ARRAY DATA ELEMENTS
},
"Data" : {
...
"PRECIPITATION" : {
"Rain" : {
"accumulation" : ""
"duration" : "",
"intensity" : "",
"peak" : ""
},
"Hail" : {
"accumulation" : "",
"duration" : "",
"intensity" : "",
"peak" : ""
}
}
---
Table 6.25: Precipitation data field in a data message of OpenWeather
protocol.
Both of them are measured with the same data fields:
* •
Rain
* –
Accumulation (accumulation): expressed in millimeters
* –
Duration (duration): expressed in seconds
* –
Intensity (intensity): expressed in millimeters per hour
* –
Peak (peak): expressed in millimeters per hour
* •
Hail
* –
Accumulation (accumulation): expressed in hits per $cm^{2}$
* –
Duration (duration): expressed in seconds
* –
Intensity (intensity): expressed in hits per $cm^{2}$
* –
Peak (peak): expressed in hits per $cm^{2}$
Compared with the PTU or wind, precipitation may be absent in the current
weather. It means that the measurement of these phenomena will happen only
when it is present. Despite this, OpenWeather always delivers the
precipitation data field in the real-time data messages262626A zero value is
assigned to the data fields when the phenomena are not present..
{
"OpenWeatherMessage": {
"Type" : 1,
"MetaInfo" : {
¯ ARRAY DATA ELEMENTS
},
"Data" : {
...
"PRECIPITATION" : {
"Rain" : {
"accumulation" : "12" // mm
"duration" : "34", // seconds
"intensity" : "12", // mm/h
"peak" : "9" // mm/h
},
"Hail" : {
"accumulation" : "2", //hits/cm^2
"duration" : "78", //seconds
"intensity" : "1", // hits/cm^2h
"peak" : "1" // hits/cm^2h
}
}
---
Table 6.26: Precipitation data field with real-time in a data message of
OpenWeather protocol.
##### Data field overview
The table 6.27 indicates the Data field structure with all objects and their
array elements filled with data:
{
"OpenWeatherMessage": {
"Type" : 1,
"MetaInfo" : {
ARRAY DATA ELEMENTS
},
"Data" : {
"PTU" : {
"Air-Temperature" : "20.0", // Celsius: C
"Relative-Humidity" : "59.5", // %RH
"Air-Pressure": "1002.1" // Hectopascals: hPa
},
"WIND" : {
"Direction" : {
"min" : "217", // Degrees
"ave" : "217",// Degrees
"max" : "218"// Degrees
},
"Speed" : {
"min" : "4.2",// m/s
"ave" : "4.2",// m/s
"max" : "4.5"// m/s
}
},
"PRECIPITATION" : {
"Rain" : {
"accumulation" : "12" // mm
"duration" : "34", // seconds
"intensity" : "12", // mm/h
"peak" : "9" // mm/h
},
"Hail" : {
"accumulation" : "2", //hits/cm^2
"duration" : "78", //seconds
"intensity" : "1", // hits/cm^2h
"peak" : "1" // hits/cm^2h
}
}
}
}
}
---
Table 6.27: Real-time data message of OpenWeather protocol.
##### Data on demand
As highlighted in section 5.2.2, OpenWeather possesses the capability to
transport data on demand (not being the data generated in real-time)272727This
data can be stored in the AWS itself.. In this use case, this data is only
delivered by the nodes when the user requests it. To achieve this operation,
OpenWeather uses the object’s hierarchy, to know which kind of data the user
is requesting. The protocol encodes such data in OpenWeather data header,
after that it is interpreted by the software to localize the data requested
from the AWS.
Note that the levels of hierarchy can be as deep as it is required.
Nevertheless, the prototype only offers the possibility to retrieve the same
data as in real-time.282828Mark with a different timestamp inside of anAWS or
datalogger.
Figure 6.7: OpenWeather’s MetaInfo data field with the data array elements.
Through the different levels established in the object’s hierarchy, it is easy
to find the information that the user expects.
As explained in section 6.4.2, OpenWeather uses numerical codes to identify
the types of data messages. In this case the data on demand must be requested
by a user (node), thus the protocol’s code will be 201292929Review the
protocol codes reference..
The data message will contain a JSON object containing an array of data
elements. The data field is named "Retrieve", it contains the data requested,
indicated by the letter ’D’ as a variable to be reference for data objects
requested (PTU, wind or precipitation). In addition a timestamp303030This
variable follows exactly the same standards used in the Timestamp field used
in the MetaInfo data field. is added to the request in order to specify in
which sample is interested the user.313131It is possible to change this field
value in order to adapt it to request samples from a range of time.
{
"OpenWeatherMessage": {
"Type" : 201,
"MetaInfo" : {
ARRAY DATA ELEMENTS
},
"Data" : {
"Retrive" : {
["D":"PTU","D":"WIND","D":"PRECIPITATION"],
"Timestamp": "2011-05-29T12:10:23Z"
}
}
}
}
---
Table 6.28: Real-time data message of OpenWeather protocol.
This request will return the PTU, wind and precipitations recorded in the
timestamp provided. The next data message received by the node in response of
this will have exactly the same format as a real-time data message, except the
code and the timestamp in the header; they will provide referencing to the
response for the data on demand in the date and time specified.
#### 6.4.6 Internal protocol data
As any other P2P architecture, OpenWeather needs a certain amount of internal
data to keep working. Commonly, this data is focused in peer’s information as
hostnames and ports used by the nodes. OpenWeather uses a mechanism to
exchange list of peers between nodes, to guarantee the well-functioning of
OpenWeather network. The information provided in these data messages can have
different purposes. The author reserves this type of data for future
implementations, nevertheless the protocol has been implemented to be able to
transfer list of peers and keep updated the nodes with them.
The data messages used for this purpose are categorized as protocol dedicated,
as explained in section 6.4.2 these data messages can be requests, retrievals
or status information.
Opposed to weather data messages, the internal data messages do not have a
Data object, but instead are composed by an Info object. This info object
contains the data fields referencing the information required by the protocol.
The type of data message —code 101—, notifies to the node that it must return
a list of peers. Because this message also contains the MetaInfo object, the
receiver is inform of all the information necessary to deliver the best peer
list to the node in the same requests.
In the case of a list of peers, the Info object will contain a list of
variables composed by an array of data elements with the IP address of the
nodes, the port and the bandwidth available in it:
{
"OpenWeatherMessage": {
"Type" : 101,
"MetaInfo" : {
ARRAY DATA ELEMENTS
},
"Info": {
"Peer-ID" : ["Peer-IP":"226.134.231.73","Port": "62535","Bandwidth":"2"],
"Peer-ID" : ["Peer-IP":"116.234.231.13","Port": "62535","Bandwidth":"1"],
"Peer-ID" : ["Peer-IP":"186.214.211.53","Port": "62535","Bandwidth":"5"],
"Peer-ID" : ["Peer-IP":"182.124.221.23","Port": "62535","Bandwidth":"6"],
"Peer-ID" : ["Peer-IP":"190.144.231.13","Port": "62535","Bandwidth":"1"]
}
}
---
Table 6.29: Peer’s list exchange in OpenWeather protocol.
The table 6.29 shows the response of the data message, providing a list of
peers. Note that the "Peer-ID" will contain the unique ID of the peers. After
the requester gets this data message, the software should update the internal
list of peers with the new data and to deliver a status data message to the
node that provides the list of peers to confirm the correct retrieval of the
data.
##### Services availability
OpenWeather offers a mechanism to know which services are available in an AWS.
A node requesting data from these services, must send a data message with code
102, to obtain a response with the services remotely available in the node.
{
"OpenWeatherMessage": {
"Type" : 102,
"MetaInfo" : {
ARRAY DATA ELEMENTS
}
}
}
---
Table 6.30: Services list availability request.
After this data message is received by the remote node, it will reply with
another data message, providing the list of the services:
{
"OpenWeatherMessage": {
"Type" : 101,
"MetaInfo" : {
ARRAY DATA ELEMENTS
},
"Info": {
"Services" : { "PTU":"RO","WIND": "RO","PRECIPITATION":"RO"}
}
}
---
Table 6.31: Peer’s list exchange in OpenWeather protocol.
One array of data is delivered in the reply:
* •
Services array: indicating the type of service and its availability.323232R is
equal to ”real time data” and O to ”data on demand”. Both can be present or
isolated.
With this information the software knows which services can be checked on the
remote node and which kind of data —real-time or on demand— can be retrieved
from them.
### 6.5 Protocol considerations
The following sections describe some aspects of OpenWeather related with other
protocols or future features of it.
##### OpenWeather and other protocols
We can find dozens of protocols available, using P2P architectures and/or
optimizations in the data delivery. Nevertheless, the author could not find
any protocol suitable enough to fit in the characteristic required by the AWS
es. Protocols as Bittorrent[10], have a proven track delivering large amount
of data and scaling their networks properly. FastTrack[42] has been successful
achieving similar results as Bitorrent. However, almost all the P2P protocols
are oriented to transfer files or real-time data with a big size (such as
video or voice streams). In addition, these protocols are designed focusing in
nodes with common computational capabilities (such as desktops or small
servers), not considering embedded system inside of their purpose (being
difficult to handle the necessary resources to implement these protocols on an
embedded system).
Other alternatives as HTTP, were considered by the author as solutions for
this thesis. Nevertheless, HTTP still has a big dependency of the centralized
model. At the same time, HTTP works under synchronously mode, something that
will limit the real-time capabilities needed for the AWS es.
Finally, because the use of FTP (a generic protocol) is under use for weather
instruments, the author considered much more interesting to research a custom
solution for the AWS es.
Nevertheless, several concepts have been taken from the mentioned protocols.
OpenWeather uses the same philosophy as HTTP, providing in the header of the
data message all the information needed. The same approach as HTTP has been
chosen to identify the type of data messages. Through protocol codes the data
message is identified in a category / purpose, being simple to extend the
amount of protocol operations, just creating new identifiers through the
codes. Moreover, the protocol uses JSON as data format, being text-based as
HTTP. Concepts such node ID, peers-requested or update-interval have been
taken from protocols as Bitorrent[10] or FastTrack[42] . These properties
allow OpenWeather to implement methodologies tested in other P2P networks with
successful results.
##### Aggregation of data between nodes
As in other P2P networks, the scalability of the OpenWeather network can be an
issue. Although OpenWeather does not implement an aggregation technique
between the nodes, it is ready to be adapted to it. The nodes conforming to
OpenWeather protocol could require the capability to request and retrieve data
using indirect paths to the end node. These paths could be found using the
connections already established with other nodes.
The aggregation of the data will be executed using the same data format as
common on OpenWeather protocol, thus, the data messages will use JSON format
plus the required fields in the data message to provide such functionality.
The same operations of the protocol will be available through aggregation. In
addition, the protocol will require the implementation of new operations for
internal use.
We need to consider the nature of the weather data networks when we chose the
aggregation technique. As it is described in section 3.1.2, the amount of
bandwidth is commonly limited in an AWS. Several techniques have been
developed to aggregate information from different sources having in
consideration connectivity and bandwidth availability issues. These techniques
are classified based in how they aggregate and route the data [35].
In case that the aggregation is required in OpenWeather, it should be a
combination of gossiping and tree-based methods, in order to provide a
feasible way to aggregate data between nodes. The reason for this combination
is that both methodologies have one specified purpose. Gossiping techniques
are focused into offer robust communications, meanwhile, tree-based techniques
are focused in to have better performance transferring data. Because a weather
network needs to guarantee the flow of the data and at the same time the
availability of the data as soon as possible, a research combining both
techniques must be performed in order to find suitable solutions for such
environment.
Notwithstanding, the OpenWeather specification available in this thesis
provides the capability to request and retrieve the list of peers of a remote
node. The combination of this list of peers and the Keep-Alive value of them,
can be used to build a tree-based structure with the nodes that have a
established session. Through this tree, OpenWeather can be able to find new
paths to other nodes. This will require the implementation of a internal
operation of the protocol, providing the capability to make queries to other
nodes, in order to build new paths. In addition, the tree-based structure will
not be enough to guarantee the robustness necessary for the weather data
transmission. Hence, it will be required to find the compatibility of this
technique with gossiping methodologies, implementing an algorithm inside the
protocol that periodically and randomly tries to update the table of nodes
available, and the paths of them.
Finally, we need to highlight that the aggregation of data is a complicated
area, not being possible to treat it in this thesis.
##### Compatibility with centralized models
In chapters two and three we introduced the different techniques and
topologies used by weather organizations to acquire weather data. The
centralized model was explained, showing how the nodes have a strong
dependency of one common point. This setup is the current solution chosen by
weather organizations, and almost all big weather data networks are builded
based on such infrastructure. 3 Thus, it is needed to consider the
compatibility of OpenWeather with this topology. Although OpenWeather is
designed to have the AWS es as independent nodes, infrastructures using the
centralized model, can provide a node doing a bridge between the collection
point and OpenWeather network. It will be required to develop the methods to
retrieve the data from the subnet of the collection point. As it was mentioned
in section 5.1.2 and the example of HTTP and OpenWeather, it is possible to
encapsulate data to other protocols with the proper adaption.
Because every weather organization has their own setups and methodologies, an
independent study will be required in order to design a bridge from the
collection point models to the P2P architecture of OpenWeather.
### 6.6 Summary
In this chapter the core architecture of OpenWeather was presented. The
definitions establish by the protocol have been explained. The roles of nodes
and their identification is presented to justify how OpenWeather can be
adapted for future use in a different system for AWS identification.
We have explained the architecture of OpenWeather. Justifying the use of the
AWS es as indivudal nodes conforming a P2P network. The protocol functionality
is analyzed, explaining how the different operations perform. The main
characteristics of OpenWeather have been exposed.
The structure used in OpenWeather data messages has been analyzed, explaining
how JSON is used as syntax to encapsulate the data. In addition, the
application of object hierarchy on data has been explained. All data fields,
which compose data messages were defined technically.
The protocol codes and their categories have been described, justifying their
numeration and purposes.
The differences between real-time data messages, data messages on demand and
internal data messages, have been justified, putting attention in how the
different data messages have a common structure and use. Finally an example of
all the types of data messages implemented in the protocol are explained,
providing enough information to implement a functional prototype of it.
## Chapter 7 Experimental evaluation setup
In this chapter, the experimental setup used to implemented the proof of
concept of the OpenWeather protocol is explained. A generic AWS has been setup
to test the protocol with real-time data. The software architecture
implementing the functionality of the protocol is introduced as well. The
purpose of this chapter is to introduce the general guidelines followed by the
implementation of a prototype of OpenWeather protocol and to analyze the tests
cases performed using it.
### 7.1 Scenario
The AWS utilized in the experimental setup is the model WXT520, manufactured
by Vaisala Oyj. Along with other AWS es sharing these characteristics, it is
able to measure the following phenomena:
* •
Liquid Precipitation
* –
Hail
* –
Rain
* •
Relative Humidity
* •
Wind
* –
Direction
* –
Speed
* •
Air Temperature
* •
Barometric Pressure
The geographical location of the AWS is N 60º 11’ 15.6” E 024º 50’
14.8”111UTM: Zone: 35 Easting: 380076 Northing: 6674276. Municipality of
Otaniemi, Espoo, Finland.. The AWS has been connected to a computer, in which
the software developed to implement OpenWeather protocol is installed. The AWS
has been configured following the manufacturer suggestions, emulating a normal
installation environment. The digital interface configured in the AWS is a
RS-232 port, offering a maximum amount of bit rate of 116 kbit/s.
Figure 7.1: AWS installed to simulate a real scenario.
The AWS is plugged in continually 24 hours and installed on a mast of 2 meters
length. The RS-232 port provides the data acquired in the AWS to computer a
that operates an implementation of OpenWeather protocol.
Thus, the AWS used to implement the protocol has not been modified to adapt it
to OpenWeather, all the adaptions realized have been made through a software
implementation. This fact allows the verification of the adaptability of the
protocol to the current technology without no major modifications to the AWS.
##### Evaluation setup
The evaluation setup consists four nodes. All of them run a copy of the
prototype, thus acting as nodes. Nevertheless, only one node is connected to a
functional AWS, the other three simulate the weather data input222Generated
randomly based on the same patterns as a normal AWS..
The table 7.1 shows nodes specifications:
CPU | Memory | Network connection | Operating system | Hostname
---|---|---|---|---
2.4GHz | 4GB | 100Mbps | GNU/Linux | Node 1
2.2GHz | 1GB | 100Mbps | GNU/Linux | Node 2
900GHz | 1GB | 128Kbps | GNU/Linux | Node 3
1GHz | 1GB | 56Kbps | GNU/Linux | Node 4
Table 7.1: Hardware and OS specifications of the evaluation setup.
Figure 7.2: Network topology used in the evaluation setup.
All the nodes posses network visibility among them, with maximum network
latency less than 75 milliseconds. The bandwidth in node number two and four
has been limited (Round-trip time (RTT)) to 128kbit/s and 56kbit/s
respectively. These restrictions emulate the network limitations mentioned in
chapter three.
The purpose of this setup is to create an environment that simulates common
conditions experience during weather data acquisition. All the nodes use
OpenWeather protocol to exchange data between them. This environment provides
the necessary resources to test and verify the characteristics of the
protocol, such data message size, times of response, etc.
### 7.2 Prototype implementation
In order to verify the feasibility and the functionality of OpenWeather, the
author developed a proof of concept of the protocol, to test and verify its
feasibility as alternative protocol for weather data transmission. This
implementation provides the necessary data to independently evaluate the
protocol.
#### 7.2.1 Technologies used
OpenWeather is designed to have an emphasis on the data structures used in the
software implementation. In addition, the object hierarchy used to structure
the data makes the implementation of the protocol easier by using an object
oriented programming language.
Thus, C++ has been chosen as the primary language used in the prototype. The
C++ standard library is used to write the intermediary layer (in combination
with some Python scripts). Because the target of this protocol can have end
users which are not familiar with command-line applications, a functional GUI
has been implemented. The Qt framework[33] has been chosen to implement the
GUI, together with QJson[9] for the data representation.
#### 7.2.2 Software Architecture
The prototype requires to be implemented supporting the functionality
described in the P2P architecture taxonomies[8]. Thus, the nodes should posses
the capability to request services, and at the same time, offer services to
others nodes. This requirement conditions the node to behave as a client and
server at the same time.
To realize this architecture, the concept of local peer is introduced. The
local peer refers to the node itself; representing the AWS entity in the
network; nevertheless, as described in section 5.2.1, a node without an AWS
can be part of the network as well.
Because the node behaves as client and server at the same time, the software
implementation is designed to maximize the utilization of the common resources
between both modalities. Thus, the implementation of the classes have been
done using abstract interfaces, not mattering if the data to process has been
received through the client or server module.
##### Common implementation
The prototype implementing OpenWeather protocol has been optimized for the
data structures and object hierarchy explained in chapter six. The handling of
JSON data through TCP sockets is the basis of the implementation. The
prototype focuses its core functionality in to take advance of the most
optimal way of sockets management and data manipulation.
Figure 7.3: Software prototype conceived.
The prototype is structured in three parts:
* •
The GUI providing access to certain functionalities of OpenWeather protocol.
* •
The network level implementation of OpenWeather protocol.
* •
The intermediary middle layer adapting the WXT520 to OpenWeather protocol.
Despite this modularity in the components, everything is assembled in one
application. The prototype implements the client and the server modules
internally. Both modules have access to the core implementation of OpenWeather
protocol, and at the same time the application is linked with the OpenWeather
parser (_libopenweatherparser_).
The implementation of the protocol has been made based on the objects
hierarchy explained in chapter six. Thus, the representation of the
OpenWeather data only involves the transformation of JSON objects into
primitive data types.
Figure 7.4: UML diagram of the prototype.
The figure 7.4 shows a general overview of the classes implemented in the
prototype in order to make functional the OpenWeather protocol. All the
classes developed are able to manage the data in both modes (client and
server), being possible to retrieve and delivery data using the same internal
software mechanisms, with complete transparency for the end user.
##### Client module
The software implements certain parts fully pertaining to client operations.
Client operations are identified those that involve the data request to other
nodes. When the software is using OpenWeather to retrieve data from other
nodes, we denominate that it is working under client mode.
The client module of the software allows the following operations:
* •
Request session establishment - peer handshake
* •
Request real-time and/or data on demand
* •
Request the service availability in remote node/s
##### Server module
As requirement of the RFC 5694[8], an application implementing a P2P
architecture must be able to offer services. To achieve this, the prototype
implements one part that provides the server functionality. A socket listening
to the TCP port used in OpenWeather is created when the prototype software is
executed. Thus, the software allows other peers to connect to it, providing
exactly the same features that client mode is able to request. Because the
OpenWeather protocol is designed to not distinguish between the nodes and the
services that they offer, the implementation of the server module is nearly
identical to the client mode.
The server module o allows the following operations:
* •
Session establishment
* •
Delivery of real-time and/or on data on demand
* •
Delivery of the service available on the local node
##### GUI
The graphical interface aims to provide the possibility to use the
protocol333A set of screenshots took from the GUI is available in the
appendix.. The GUI allows fully utilization of the AWS data interface to check
the data received, to connect to OpenWeather, and to perform the operations
described in the chapters six (connect to other peers, delivery real-time data
samples or retrieve the services available in the remote peers).
Figure 7.5: Prototype use case diagram.
The GUI has single instances of the ConnectionsManager and the MessagesManager
classes. Both classes provide the functionality required to handle peers and
connections. In addition, the library _libopenweatherparser_ , provides the
middle layer explained in section 5.2. This library is linked to the AWS,
providing the RAW data collected from its digital interface, and converting it
from the vendor’s format to OpenWeather’s format.
##### Connections manager
The ConnectionsManager class is in charge of handling the sockets, managing
all the connections of the node. In addition, this class controls the socket
used to allow remote nodes to connect to the local peer (server module).
All sockets are handled using threads, thus, all the connections are managed
independently in a secondary plane, not blocking the GUI or not interfering
with other connections. This implementation allows the prototype to manage
multiple connections with multiple peers without performance issues.
##### Peers manager
The PeersManager class is in charge of the peers. The purpose of this class is
to provide a control system of the peers that the node can connect to and
their information; at the same time this class manages the local peer and the
services that it offers to the remote peers.
This class gets updated information when a the data messages received contain
data related with the peers (protocol internal traffic), for instance if some
peer updates its metainformation or just confirms the receival of message.
##### Messages manager
The MessagesManager class handles the OpenWeather data messages. This class is
able to generate data messages based on the specifications of OpenWeather
protocol. Every connection containing a data message is able to access it.
This class provides the core functionality of the protocol, being able to
understand the protocol codes and based on them, executing the operations
needed in order to achieve the expected result.
All data messages are assembled and disassembled in this class, because as
OpenWeather requires JSON as its primary data format, this class provides
mechanisms to generate and validate the data format of the messages.
##### Libopenweatherparser
This library has been developed in order to create a bridge between the AWS
and the prototype. The data format used by the vendor in the AWS has been
implemented in the library, creating the functionality to convert the vendor’s
format to the OpenWeather data format. This library is thought to normalize
the data from one to multiple vendors, offering primitive data types ready to
be assembled in JSON objects as output.
Figure 7.6: UML diagram of the library.
The library acts as an intermediary layer. Should the vendors choose to
implement the OpenWeather format, the requirement of this library will be
dropped. However, since the source code the AWS operating system is not
available, it is not currently possible to implement the OpenWeather protocol
integrated with the vendors software without their cooperation.
### 7.3 Testing
The prototype provides the capability to perform the operations described on
chapter six. The main goal of the testing is to analyze if the implementation
of the protocol achieves its purpose and the results that its generates.
The scenario used for the testing is described in the previous section. The
following sections explain the utilization of different nodes used to transmit
weather data using the OpenWeather protocol.
The methodologies followed to evaluate the behavior of the protocol are based
in the analysis of the network traffic between nodes and the verification of
the protocol operations. The tool used to capture the data messages is
Wireshark[11]. This tool provides enough information to verify the operations
of the protocol in the network layer.
The following protocol operations have been implemented in the prototype:
* •
Session establishment - peer handshake
* •
Service discovery
* •
Real-time data retrieval
##### Implementation considerations
Although the chapter six specifies more operations, as data on demand or peer
list exchange, they have not be implemented due to their similarity in the
architecture and data messages size, with the test cases executed.
The Keep-Alive functionality has not being implemented in the prototype,
because this feature is just an extra check performed for OpenWeather to
double assure the connectivity and the response of the node in the application
layer, and it does not influence the functionality of the prototype.
All nodes have been synchronized according with date-time through NTP [30]
with the ntp1.funet.fi server before to execute any operation. This
synchronization has been performed in order to guarantee the accuracy of the
measurements. Nevertheless, as it is explained in the Timestamp section, it is
highly recommended to sync the clock of the nodes to guarantee the quality of
the weather data.
The RAW ASCII representation of the data messages appears in different order
compare to the specifications. This is due to the software re-orders the data
elements by alphabetical order (always inside of the objects hierarchy).
All the data messages are keeping similar space constrains between the data
elements inside the JSON object. This is causing a known additional increase
of the data message size, this size can be reduced even more, suppressing
theses spaces. In addition, the migration to a binary representation of the
data messages using Binary-JSON (BSON) [37] should be straightforward444Though
will cause conflicts with the endianess..
The execution of the test has been done 50 times, extracting the average from
it. The times of the sequence are including the execution of the software
operations in both sides.
#### 7.3.1 Test 1: Handshake between nodes
The purpose of this test is to validate the operation described in section
6.3.1 —Session establishment & Peer handshake—. In this test the peers
involved are exchanging information about themselves, in order to establish
the session.
The _Node 1_ will send a handshake data message to the _Node 2_. This data
message contains all the MetaInfo data field filled with the data of the _Node
1_ , the protocol code used is 100.
##### Sequence
The scenario assumes that the _Node 1_ knows the IP address, TCP port and ID
of the _Node 2_ , because it was obtained from some list of peers received
from other nodes.
The following sequence happens in the network layer:
1. 1.
_Node 1_ sends a data message containing all its metainformation to the remote
_Node 2_ , connected through the port specified and requests session
establishment.
2. 2.
_Node 2_ receives a data message delivery by _Node 1_ , containing all its
metainformation and requesting the session establishment.
3. 3.
_Node 2_ sends a data message to _Node 1_ , providing all its meta-information
and confirming the session establishment with the protocol code 101.
4. 4.
The session is established between both nodes.
The following sequence happens in the software layer:
1. 1.
The button session-establishment generates the connection sequence to the node
chosen (_Node 2_).
2. 2.
A thread is created, establishing a TCP connection to the chosen node. The
messages manager assembles data message with all metainformation of the local
node and with the protocol code 100.
3. 3.
The data message is delivery through the socket managed by the thread.
4. 4.
The messages manager in the _Node 2_ receives a data message and creates a
thread to handle it.
5. 5.
The messages manager called by the thread, parses the data message and
identifies its protocol code.
6. 6.
A response is generated based on the protocol code of the data message, and is
deliver to _Node 2_.
7. 7.
Because the operation is the session establishment the peers manager gets
executed in both sides, updating the peer information (if needed) of the
peers.
##### Analysis
* •
The data session captured with Wireshark involves 7 TCP segments.
* •
The data message (OpenWeather) generated by _Node 1_ , has a size of 375
bytes.
* •
The data message (OpenWeather) generated by _Node 2_ , has a size of 375
bytes.
* •
The total size of the OpenWeather data message is 750 bytes.
* •
The total size of the sequence (TCP/IP and OpenWeather) is 1227 bytes.
The RAW ASCII representation of the data message data capture is shown in
table 7.2.
_Node 1_
---
{ "OpenWeatherMessage" : { "MetaInfo" : { "Bandwidth" : 6, "ID" :
"33c11957579d1093e931bd540536b40e90339dbded8e2a2ce4e
64c480c8132bc", "Keep-Alive" : 120000, "Location" : "6672224
385565 35V", "Peer-IP" : "172.21.25.16", "Peers-Requested" :
20, "Port" : 62535, "Timestamp" : "2011-07-20T16:51:29", "Update
-Interval" : 120000, "Version" : "OpenWeather/1.0" }, "Type" : 100 }
}
_Node 2_
{ "OpenWeatherMessage" : { "MetaInfo" : { "Bandwidth" : 6, "ID" :
"11f1cb9fb5bc57cf7905dc26c3ef045ae7b54d5ff1c7e233ff2d31be
4977bd18", "Keep-Alive" : 120000, "Location" : "6672224 385565
35V", "Peer-IP" : "172.21.25.20", "Peers-Requested" : 20, "Port"
: 62535, "Timestamp" : "2011-07-20T16:51:29", "Update-Interval"
: 120000, "Version" : "OpenWeather/1.0" }, "Type" : 101 }
}
Table 7.2: Data messages transmitted between _Node 1_ and _Node 2_.
The TCP flow between both nodes using OpenWeather is the following:
| 172.21.25.16 172.21.25.20 |
| SYN | |Seq = 0 Ack = 1303623571
|(39239) ------------------> (62535) |
| SYN, ACK | |Seq = 0 Ack = 1
|(39239) <------------------ (62535) |
| ACK | |Seq = 1 Ack = 1
|(39239) ------------------> (62535) |
| PSH, ACK - Len: 375 |Seq = 1 Ack = 1
|(39239) ------------------> (62535) |
| ACK | |Seq = 1 Ack = 376
|(39239) <------------------ (62535) |
| PSH, ACK - Len: 375 |Seq = 1 Ack = 376
|(39239) <------------------ (62535) |
| ACK | |Seq = 376 Ack = 376
|(39239) ------------------> (62535) |
Table 7.3: TCP flow sequence between _Node 1_ and _Node 2_.
The time of execution of this TCP sequence is 65 milliseconds on average.
Both nodes have delivered the data successfully, achieving the session
establishment as result of the sequence.
##### Discussion
The measurements show that OpenWeather requires a small amount of data for the
session establishment. In addition, a low response time is needed to complete
the operation. It achieves the goal to provide a mechanism to establish
session even with really low bandwidth availability. This small size of data
can be easily handled by the memory and processor of an AWS. As the protocol
specification requires, the session establishment provides all the necessary
information to both nodes, to proceed requesting other data, after the peer
registration happens in the software side.
#### 7.3.2 Test 2: Service discovery
The purpose of this test is to validate the operation described in section
LABEL:7.3.2 —Service discovery—. In this test the peers involved are
exchanging information about service availability, in order to know which
services could be requested.
The _Node 3_ will send a service discovery data message to _Node 4_. This data
message contains all the MetaInfo data field filled with the data of the _Node
3_ , in addition the protocol code used is 102.
##### Sequence
The scenario assumes that _Node 3_ and _Node 4_ have established the session,
following exactly the same steps than mentioned in section 6.3.1.
The following sequence happens in the network layer:
1. 1.
_Node 3_ sends a data message containing all its metainformation to the remote
host of the _Node 4_ , using the session already established between them.
2. 2.
_Node 4_ receives a data message delivered by _Node 3_ , containing all its
metainformation and requesting the services available on it.
3. 3.
_Node 4_ sends a data message to the _Node 3_ , providing all its
metainformation and delivering a data message with all the services available
on it through the protocol code 103.
4. 4.
_Node 3_ receives the list of services available in the _Node 4_.
The following sequence happens in the application layer:
1. 1.
The button services discovery generates the connection sequence to the node
chosen (_Node 4_).
2. 2.
The thread previously created by the session, uses the TCP connection
established to the chosen node. The messages manager assembles a data message
with all the metainformation of the local node and sends through connection
with the protocol code 102.
3. 3.
The data message is delivery through the socket managed by the thread.
4. 4.
The connections manager in _Node 3_ receives a data message and creates a
thread to handle it.
5. 5.
The messages manager is called by the thread, parses the data messages and
identifies its protocol code.
6. 6.
A response is generated based on the protocol code of the data message, and is
deliver to the _Node 4_.
7. 7.
Due to the operation being service discovery, the peers manager gets executed
in _Node 4_ , checking the services available on it and providing their
information into the data message.
##### Analysis
* •
The data session captured with Wireshark involves 7 TCP segments.
* •
The data message (OpenWeather) generated by the _Node 3_ , has a size of 375
bytes.
* •
The data message (OpenWeather) generated by the _Node 4_ , has a size of 458
bytes.
* •
The total size of the OpenWeather data message is 833 bytes.
* •
The total size of the sequence (TCP/IP and OpenWeather) is 1310 bytes.
The RAW ASCII representation of the data message captured is shown table 7.4
_Node 3_
---
{ "OpenWeatherMessage" : { "MetaInfo" : { "Bandwidth" : 1, "ID" :
"654b7b521acc7549bf6854b1113d44e6433bf94a1b4caf4327e33
e9bc89b4025", "Keep-Alive" : 120000, "Location" : "6672224 385
565 35V", "Peer-IP" : "172.21.25.35", "Peers-Requested" : 20,
"Port" : 62535, "Timestamp" : "2011-07-24T12:04:09", "Update-
Interval" : 120000, "Version" : "OpenWeather/1.0" }, "Type" : 102 }
}
_Node 4_
{ "OpenWeatherMessage" : { "Info" : { "Services" : { "PRECIPITATION"
: "RO", "PTU" : "RO", "WIND" : "RO" } }, "MetaInfo" : { "Bandwidth" :
0, "ID" : "3b1f665e0d622aab7b2e71b29d966dd2a22c5d427f337585
09d4205720de9d2e", "Keep-Alive" : 120000, "Location" : "6672224
385565 35V", "Peer-IP" : "172.21.25.40", "Peers-Requested" : 20,
"Port" : 62535, "Timestamp" : "2011-07-24T12:04:09", "Update-
Interval" : 120000, "Version" : "OpenWeather/1.0" }, "Type" : 103 }
}
Table 7.4: Data messages transmitted between _Node 3_ and _Node 4_.
The TCP flow between both nodes using OpenWeather is the following:
| 172.21.25.35 172.21.25.40 |
| SYN | |Seq = 0 Ack = 2259331907
|(50550) ------------------> (62535) |
| SYN, ACK | |Seq = 0 Ack = 1
|(50550) <------------------ (62535) |
| ACK | |Seq = 1 Ack = 1
|(50550) ------------------> (62535) |
| PSH, ACK - Len: 375 |Seq = 1 Ack = 1
|(50550) ------------------> (62535) |
| ACK | |Seq = 1 Ack = 376
|(50550) <------------------ (62535) |
| PSH, ACK - Len: 458 |Seq = 1 Ack = 376
|(50550) <------------------ (62535) |
| ACK | |Seq = 376 Ack = 459
|(50550) ------------------> (62535) |
Table 7.5: TCP flow sequence between _Node 3_ and _Node 4_.
The time of execution of this TCP sequence is 84 milliseconds on average.
Both nodes have delivered the data successfully, achieving the service
discovery as the result of the sequence.
##### Discussion
The service discover operation has bigger data message size than the session
establishment. Nevertheless, this operation considered fairly small in size,
and it fitting to the environment with low bandwidth available. As the session
establishment, the service discovery is a common operation inside of the
protocol. Its fast delivery is critical, thus, in order to provide the
services available as soon as possible to the requester.
#### 7.3.3 Test 3: Real-time data retrieval
The purpose of this test is to validate the operation described in section
7.3.3 —Real-time data retrieval—. In this test the peers involved are
exchanging real-time weather data.
The _Node 4_ will send a real-time data request to _Node 1_. This data message
contains all the MetaInfo data field filled with the data of _Node 4_ , in
addition the protocol code used is200.
##### Sequence
The scenario assumes that _Node 4_ and _Node 1_ , have established the
session, following exactly the same steps mentioned in section 6.3.1.
The following sequence happens in the network layer:
1. 1.
_Node 4_ sends a data message containing all its meta-information to the
remote host of the _Node 1_ , using the session already established between
them.
2. 2.
_Node 1_ receives a data message delivered by _Node 4_ , containing all its
metainformation and requesting real-time data.
3. 3.
_Node 1_ sends a data message to _Node 4_ , providing all its metainformation
and delivering a data message with the latest weather data available on its
AWS, assigning the protocol code 201.
4. 4.
_Node 4_ receives the latest real-time data sample available in _Node 1_.
The following sequence happens in the application layer:
1. 1.
The button assign to request real-time data, generates the connection sequence
to the node chosen (_Node 1_).
2. 2.
The thread previously created by the session, uses the TCP connection
established to the chosen node. The messages manager assembles a data message
with all the metainformation of the local node and assigning the protocol code
200.
3. 3.
The data message is delivered through the socket managed by the thread.
4. 4.
The connections manager in _Node 1_ receives a data message and creates a
thread to handle it.
5. 5.
The messages manager called by the thread, parses the data messages and
identifies its protocol code.
6. 6.
A response is generated based on the protocol code of the data message. Since
this response involves real-time weather data, a call is made to the
libopenweatherparser, to obtain the latest real-time data available in the
AWS. After that, the data is deliver to _Node 4_.
##### Analysis
* •
The data session captured with Wireshark involves 7 TCP segments.
* •
The data messages (OpenWeather) generated by _Node 4_ , has a size of 375
bytes.
* •
The data messages (OpenWeather) generated by _Node 1_ , has a size of 814
bytes.
* •
The total size of the OpenWeather data messages is 1189 bytes.
* •
The total size of the sequence (TCP/IP and OpenWeather) is 1666 bytes.
The RAW ASCII representation of the data messages is shown in table 7.6.
_Node 4_
---
{ "OpenWeatherMessage" : { "MetaInfo" : { "Bandwidth" : 0,
"ID" : "3b1f665e0d622aab7b2e71b29d966dd2a22c5d427
f33758509d4205720de9d2e", "Keep-Alive" : 120000, "
Location" : "6672224 385565 35V", "Peer-IP" : "172.21.
25.40", "Peers-Requested" : 20, "Port" : 62535, "Timest
amp" : "2011-07-25T14:15:35","Update-Interval" :
120000, "Version" : "OpenWeather/1.0" },"Type" : 200 } }
_Node 1_
{ "OpenWeatherMessage" : { "Data" : { "PRECIPITATION" : {
"Hail" : { "accumulation" : "0", "duration" : "0", "intensity" : "0"
, "peak" : "0" }, "Rain" : { "accumulation" : "0", "duration" : "0",
"intensity" : "0", "peak" : "0" } }, "PTU" : { "Air-Pressure" : "10
14.1", "Air-Temperature" : "19.1", "Relative-Humidity" : "69.4"
}, "WIND" : { "Direction" : { "ave" : "160", "max" : "160", "min"
: "160" }, "Speed" : { "ave" : "1.7", "max" : "1.8", "min" : "1.7"
} } }, "MetaInfo" : { "Bandwidth" : 6, "ID" :"33c11957579d10
93e931bd540536b40e90339dbded8e2a2ce4e64c480c8132
bc", "Keep-Alive" : 120000, "Location" : "6672224 385565 35V
", "Peer-IP" : "172.21.25.16", "Peers-Requested" : 20, "Port"
: 62535, "Timestamp" : "2011-07-25T14:15:35", "Update-Inter
val" : 120000, "Version" : "OpenWeather/1.0" }, "Type" : 300 }
}
Table 7.6: Data messages sent between _Node 3_ and _Node 4_.
The TCP flow between both nodes using OpenWeather is the following:
| 172.21.25.20 172.21.25.40 |
| SYN | |Seq = 0 Ack = 1015394402
|(49983) ------------------> (62535) |
| SYN, ACK | |Seq = 0 Ack = 1
|(49983) <------------------ (62535) |
| ACK | |Seq = 1 Ack = 1
|(49983) ------------------> (62535) |
| PSH, ACK - Len: 374 |Seq = 1 Ack = 1
|(49983) ------------------> (62535) |
| ACK | |Seq = 1 Ack = 375
|(49983) <------------------ (62535) |
| PSH, ACK - Len: 814 |Seq = 1 Ack = 375
|(49983) <------------------ (62535) |
| ACK | |Seq = 375 Ack = 8153
|(49983) ------------------> (62535) |
Table 7.7: TCP flow sequence between _Node 1_ and _Node 2_.
The time of execution of this TCP sequence is 96 milliseconds on average.
Both nodes have delivered the data successfully, achieving the transmission of
a real-time weather sample as result of the sequence.
##### Discussion
Though this real-time data sample does not contain rain or hail data (both are
delivered with a 0 value), we can observe how the PTU and the wind data
(together with the MetaInfo data field) are up to 1.5 kB. Even with this data
size, it will fit in the memory available in an AWS described in section
3.1.2. Assuming that an AWS has between 32-64 kB of volatile memory, and
taking half of its memory for internal use of AWS operating system, there is
still enough memory to handle real-time data samples using the OpenWeather
protocol.
### 7.4 Summary
In this chapter the scenario and software architecture used to evaluate
OpenWeather has been introduced. We tested three different use cases of the
protocol with the prototype developed.
In all the use cases executed, the protocol is taking advantage of its
properties and achieving a successful result.
Though the prototype implements the partial functionality of the OpenWeather
protocol, it shows how the P2P can be implemented in applications oriented to
weather transmission. In addition, the small sizes of the data messages and
the robustness of the data transmission offered by TCP, provide enough
confidence to confirm that the protocol can be implemented in the environments
with low bandwidth availability.
Our goal was to verify a feasible implementation of the OpenWeather protocol
and verify its functionality with a real scenario. Both purposes have been
achieved.
Finally, the use of a real scenario and the integration of the prototype with
it, proves how a modern AWS can be adapted to OpenWeather protocol with a few
modifications through a software adaption. This fact supports that the current
technology can be adapted to new methodologies to transmit the weather data,
without a modification in the electronics or industrial design of the AWS.
## Chapter 8 Conclusions
In this thesis we exposed the basis of weather observation, how different
organizations around the world are collecting and studying enormous amounts
data of different phenomena. From the very beginning the industry has been
building really complex instruments to measure these phenomena. Many people,
from individuals to scientists, are spending their time and resources to part
take in the worldwide observation of weather. It is a fact that we need to
understand the weather in order to better understand our planet and
implicitly, to increase our quality of life.
We have analyzed how the instruments used for such purposes and their limits
restrict our knowledge expansion. We described how the industry has been
improving these instruments in many different ways. Areas such as the
industrial design of the instruments or their internal electronics, have been
experiencing tremendous improvements during the last decades, thus allowing
the industry to offer weather measure instruments of strong robustness and
high accuracy.
Based on the study of these instruments and the scientific discussion of those
using them, such the SMEAR project[40], we have come to a conclusion that
methods used in them can be improved significantly concerning real-time
weather data transmission.
Through the analysis of the different architectures used to collect the
weather data, we found several points related to technologies used on network
level that need to be changed in order to achieve a successful delivery of
real-time data.
We explained how the industry have been introducing new digital interfaces in
order to adapt the AWS to the new standards. Nevertheless, although the
digital interfaces have been upgraded, the protocols used to transmit the data
through them have certain particularities such the use of vendor data specific
formats.
In addition, the analysis performed in different instruments and the network
technologies that they use, has indicated that the data format and the
protocol standards used are of low compatibility with capabilities such real-
time data acquisition or data exchange.
The mainstream methodologies currently used to transmit the weather data, such
the FTP or the use of CSV as data formats, are limiting the possibility to
deliver data with frequency and accuracy high enough to consider it real-time
data. Nevertheless, these methodologies are currently considered the state of
the art and thought to be sufficient for performing in current architectures
used to acquire data.
Though some organizations as NOAA or ICAO, have been creating some data
formats for certain purposes (such air navigation or CWOP),nowadays , the
global standard still not adapted for the weather industry. The WMO, conscious
of this situation, started a process of standardization for weather data
representation in 2002. At the moment, this process still under development
without any official standard published.
The absence of a standard data format and a protocol to transmit it, is
avoiding the possibility to take advantage of all the capabilities that an AWS
can offer, more specifically the real-time data acquisition. Although the
weather organizations have access to weather data samples updated with small
frequencies of time, programs as GOS or GDPFS, are seeking to establish the
basis of future systems for weather observation, providing features as real-
time capabilities and compatibility between data formats.
All the issues mentioned previously have been considered during the
development of OpenWeather. As solution for the problem statement, OpenWeather
aims to provide all the features necessary to take advantage of the weather
instruments concerning their capabilities to accomplish weather data
transmission in real-time.
Based on the architecture used to collect weather data, we use its topology to
adapt it to the P2P architecture. Thus, we transform any AWS in a node
offering services to other nodes. To achieve such behavior, we developed the
OpenWeather protocol from scratch, conceiving it will all the necessary
properties to make it P2P and at the same time, adapting its core
functionality to the weather data requirements. Being conscious of the absence
of standards in such area, OpenWeather has been designed adopting as much
standards as possible into its architecture, such the use of standard
measurement units or date-time format.
As a result, OpenWeather provides a new way to transmit weather data and to
interact with the AWS es.
The implementation of the protocol in a software prototype and its posteriorly
use, verify its feasibility in order to translate the protocol specifications
to a functional software implementation to be tested in a more complex
scenario.
In the experimental setup we verify that OpenWeather —in its implementation as
prototype— works in a scenario using the same technologies that are currently
common among weather observation experts. The prototype implemented gives us
the possibility to communicate with other nodes, executing the protocol
operations designed to achieve the weather data transmission. In addition, the
P2P functionality of the protocol has been tested, verifying that the AWS es
can be treated as independent nodes, requesting and offering services at the
same time, and still achieving a successful weather data transmission without
a centralized collection point.
We identify as requirement the adaption of the intermediary layer developed to
other vendor’s data formats, in order to make compatible OpenWeather with
different weather instruments from different brands.
Although we described how nodes using the OpenWeather protocol could be able
to gather data between them, such functionality has not being implemented in
the prototype. Hence, future research should be performed in order to evaluate
the capabilities of the protocol to scale in large networks. In addition, the
implementation of weather data networks using scalable methodologies, should
be study together with their connectivity technologies. Thus, the possibility
to use other protocols on the AWS es to transport data instead of TCP, should
be considered, looking for protocols more optimized for low bandwidth
availability.
Through the execution of the test cases, we analyzed the results of the
protocol in the scenario given. These results show how the protocol can fit in
the technical specifications of an AWS, making possible to use it in future
adaptations.
The main goal of this thesis has been to study state of the affairs in weather
observation systems, their technologies and methodologies, trying to find ways
of their improvement. OpenWeather fits that goal. Through the prototype we can
show how the weather data transmission can be improved in several aspects from
network topology to data structure use.
This topic suggests deeper research, as it could provide a solid basis for
future implementation of a global real-time weather observation with a high
capability in data exchange operations. In addition, in this thesis we have
not treated security matters related with the weather data transmission.
Despite the nature of the weather data, a complete solution has to consider
security threats. Thus, an independent study is required to evaluate how the
weather data transmission can be protected. Although, it would be possible to
use cryptographic protocols such as Transport Layer Security (TLS) together
with OpenWeather, such combination will have an impact on the bandwidth used
to transmit weather data. In addition, Access Control List (ACL) mechanisms
could be considered to assure the identity of the nodes and their locations,
in order to guarantee their legitimacy. Moreover, weather data networks can be
an objective of Denial-of-service (DoS) or Distributed denial-of-service
(DDoS) attacks. Although this should be treated independently of OpenWeather
protocol, future adaptions of it should have these threats in consideration to
provide methodologies to lead with them.
The involvement of organizations such WMO and the vendors, is critical to make
this happen, possibly in cooperation with standardization organizations for
communication protocols such the IETF.In addition, any adaption of the
industry to protocols designed and adapted for a most efficient use of
resources available, will provide an improvement in their products, providing
new ways to use their instruments to understand the weather phenomena.
Finally the author believes that the understanding of the weather phenomena
will be accompanied by open and scalable network technologies. Thus, the
OpenWeather protocol could be a first step to make it happen.
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Appendix I
Protocol code | Description | Category
---|---|---
100 | HANDSHAKE | Protocol codes - Requests
101 | HANDSHAKE-S | Protocol codes - Status
102 | SERVICES-AVAILABLE | Protocol codes - Requests
103 | SERVICES-AVAILABLE-R | Protocol codes - Retrievals
104 | SERVICES-AVAILABLE-S | Protocol codes - Status
104 | LIST-PEERS | Protocol codes - Requests
105 | LIST-PEERS-R | Protocol codes - Retrievals
106 | LIST-PEERS-S | Protocol codes - Status
200 | REAL-TIME-DATA | Peer codes - Requests
201 | ON-DEMAND-DATA | Peer codes - Requests
300 | REAL-TIME-DATA-R | Peer codes - Retrievals
301 | ON-DEMAND-DATA-R | Peer codes - Retrievals
500 | REAL-TIME-DATA-S | Peer codes - Status
501 | ON-DEMAND-DATA-S | Peer codes - Status
Appendix II
Figure 8.1: GUI of the OpenWeather prototype -AWS control-.
Figure 8.2: GUI of the OpenWeather prototype -Node control-.
Figure 8.3: GUI of the OpenWeather prototype -Data message visualizer-.
|
arxiv-papers
| 2011-11-01T23:02:45 |
2024-09-04T02:49:23.869957
|
{
"license": "Public Domain",
"authors": "Adrian Yanes",
"submitter": "Adrian Yanes",
"url": "https://arxiv.org/abs/1111.0337"
}
|
1111.0343
|
# Application of metasurface description for multilayered metamaterials and an
alternative theory for metamaterial perfect absorber
Jiangfeng Zhou∗† Center for Integrated Nanotechnologies, Los Alamos National
Laboratory, Los Alamos, New Mexico 87545, USA Hou-Tong Chen∗ Center for
Integrated Nanotechnologies, Los Alamos National Laboratory, Los Alamos, New
Mexico 87545, USA Thomas Koschny Ames Laboratory and Department of Physics
and Astronomy, Iowa State University, Ames, Iowa 50011, USA Abul K. Azad
Center for Integrated Nanotechnologies, Los Alamos National Laboratory, Los
Alamos, New Mexico 87545, USA Antoinette J. Taylor Center for Integrated
Nanotechnologies, Los Alamos National Laboratory, Los Alamos, New Mexico
87545, USA Costas M. Soukoulis Ames Laboratory and Department of Physics and
Astronomy, Iowa State University, Ames, Iowa 50011, USA John F. O’Hara∗‡
Center for Integrated Nanotechnologies, Los Alamos National Laboratory, Los
Alamos, New Mexico 87545, USA
###### Abstract
We analyze single and multilayered metamaterials by modeling each layer as a
metasurface with effective surface electric and magnetic susceptibility
derived through a thin film approximation. Employing a transfer matrix method,
these metasurfaces can be assembled into multilayered metamaterials to realize
certain functionalities. We demonstrate numerically that this approach
provides an alternative interpretation of metamaterial-based perfect
absorption, showing that the underlying mechanism is a modified Fabry-Perot
resonance. This method provides a general approach applicable for decoupled or
weakly coupled multilayered metamaterials.
###### pacs:
78.67.Pt,81.05.Xj,41.20.Jb
††footnotetext: ∗Correspondence should be addressed to J. F. Zhou
(jfengz@gmail.com), H.-T. Chen (chenht@lanl.gov) or J. F. O’Hara
(oharaj@okstate.edu)
†Current address: Department of Physics, University of South Florida
‡Current address: Department of Electrical and Computer Engineering, Oklahoma
State University
Electromagnetic (EM) metamaterials are artificial materials that can be
engineered to exhibit controlled optical properties not found in nature over
most of the EM spectrum Soukoulis et al. (2007); Shalaev (2007). They usually
consist of multiple identical layers of periodically arranged artificial
structures, and are considered bulk homogeneous media with constitutive
parameters obtained by a retrieval based on effective medium theory Smith et
al. (2002); Koschny et al. (2004). Recently, heterogeneous metamaterials
consisting of two or more _distinct_ layers were used to realize
functionalities such as perfect absorption at THz Landy et al. (2008) and
infrared frequencies Liu et al. (2010a, b); Hao et al. (2010), and EM wave
tunneling Zhou et al. (2005). In that work, each layer of the metamaterial
Zhou et al. (2005) or all the layers as an entirety Landy et al. (2008); Liu
et al. (2010a, b) were considered as a homogeneous medium. The effective
permittivity and permeability, were calculated using an established retrieval
procedure Smith et al. (2002); Koschny et al. (2004). However, the
metamaterials in these systems consist of only _one_ functional layer of
artificial structures (meta-atoms), which is analogous to a single molecular
layer in natural materials. It is challenging to define bulk effective
permittivity and permeability for such single-“meta-atom”-layer systems, since
these macroscopic material properties typically result from averaging fields
over many molecular layers. In addition the thickness of the effective bulk
material in these systems is not uniquely defined, which therefore likewise
renders the effective material properties arbitrary Zhou et al. (2008);
Holloway et al. (2009). Further complications arise because metamaterials
consisting of _distinct_ layers, such as perfect absorbers Landy et al.
(2008); Liu et al. (2010a, b), are inhomogeneous in the wave propagation
direction, and cannot be strictly considered homogeneous bulk media.
In this paper, we use an effective medium model that treats each layer of the
metamaterial as a metasurface with unique effective surface electric and
magnetic susceptibility, $\chi_{se}$ and $\chi_{sm}$. Through a thin film
approximation, we obtain the same equations of $\chi_{se}$ and $\chi_{sm}$ as
previous metasurface work Holloway et al. (2009), and also reveal the
relations between surface effective susceptibilities and bulk effective
material parameters. We then use a transfer matrix method to analyze the
overall EM properties of multilayered metamaterials using the effective
material parameters (surface susceptibilities) of each layer. We find that the
overall properties of multilayered metamaterials can be determined by their
individual layer properties, in the absence of inter-layer resonance coupling.
We also find that such individual layer properties are responsible for
metamaterial perfect absorbers. This contrasts with previous explanations
based on bulk effective medium theory Landy et al. (2008); Liu et al. (2010a,
b). To wit, in previous work, the _entire_ metamaterial was considered as a
homogeneous medium with independently engineered effective permittivity and
permeability to reach the condition
$\epsilon_{\mathrm{eff}}=\mu_{\mathrm{eff}}$, both having large imaginary part
resulting in the effective refractive index,
$n=n^{{}^{\prime}}+\mathrm{i}n^{{}^{\prime\prime}}$ and
$n^{{}^{\prime\prime}}\gg n^{{}^{\prime}}$. The EM wave thus propagates
through the first interface (air-metamaterial) without reflection and the
strength decays rapidly to zero inside the metamaterial before reaching the
second interface (metamaterial-air). However, our results show that the
interaction (or the assumed magnetic resonance) between the two metallic
layers has a negligible effect on the absorption. Instead, the functional
mechanism is the Fabry-Perot interference resulting from the multiple
reflections in the cavity bounded by two metamaterial layers. Finally, we also
find that the metamaterial EM tunneling Zhou et al. (2005) and the
metamaterial anti-reflection Chen et al. (2010) can be explained very well by
our approach. Our approach is generally applicable for decoupled or weakly
coupled multilayered metamaterials Zhou et al. (2009), where the coupling due
to evanescent modes is inconsequential.
Figure 1: (a) A schematic of a single layer metamaterial considered as a
homogeneous thin film and the electric and magnetic field across it.
$\textbf{E}^{\mathrm{i}}$, $\textbf{E}^{\mathrm{r}}$ and
$\textbf{E}^{\mathrm{t}}$ represent transverse electric field of the incident,
reflected and transmitted EM wave under normal incidence, respectively;
$\textbf{E}_{t-}$, $\textbf{E}_{t+}$, $\textbf{H}_{t-}$ and $\textbf{H}_{t+}$
are the total transverse electric and magnetic fields at each boundary of the
film; $\textbf{E}^{av}_{t}$ and $\textbf{H}^{av}_{t}$ are the average
transverse electric and magnetic fields inside the film; $\chi_{\mathrm{se}}$
and $\chi_{\mathrm{sm}}$ represent the effective surface electric and magnetic
susceptibilities; $\epsilon_{\mathrm{eff}}$ and $\mu_{\mathrm{eff}}$ represent
the effective permittivity and permeability. (b) Schematic of a multilayered
metamaterial consisting of N layers separated by N-1 layers of dielectric
spacers.
We begin with Fig. 1(a), where a single-layer metamaterial is considered as a
homogeneous thin film with thickness, $d$, same as the actual thickness of a
single-layered of metamaterial structure and approaching zero as compared to
the incident wavelength. Transmission and reflection occurs as a plane EM wave
propagates normally through the thin film, and generally leads to
discontinuities of the transverse electric and magnetic fields, which can be
described by the following boundary conditions Tretyakov (2003):
$\displaystyle\textbf{n}\times(\textbf{E}_{t+}-\textbf{E}_{t-})$
$\displaystyle=$
$\displaystyle\mathrm{i}\omega\mu_{0}\mu_{\mathrm{eff}}d\textbf{H}^{av}_{t}$
(1) $\displaystyle\textbf{n}\times(\textbf{H}_{t+}-\textbf{H}_{t-})$
$\displaystyle=$
$\displaystyle-\mathrm{i}\omega\epsilon_{0}\epsilon_{\mathrm{eff}}d\textbf{E}^{av}_{t}$
(2)
where $\textbf{E}_{t-}=(1+R)\textbf{E}^{\mathrm{i}}$,
$\textbf{E}_{t+}=T\textbf{E}^{\mathrm{i}}$,
$\textbf{H}_{t-}=(1-R)\textbf{H}^{\mathrm{i}}$,
$\textbf{H}_{t+}=T\textbf{H}^{\mathrm{i}}$, $\textbf{E}^{av}_{t}$ and
$\textbf{H}^{av}_{t}$ are defined in the Fig. 1(a) caption. The average
electric and magnetic fields inside the thin film, can be approximately
defined as
$\textbf{E}^{av}_{t}=(\textbf{E}_{t-}+\textbf{E}_{t+})/2=(1+R+T)\textbf{E}^{\mathrm{i}}/2$
and
$\textbf{H}^{av}_{t}=(\textbf{H}_{t-}+\textbf{H}_{t+})/2=(1-R+T)\textbf{H}^{\mathrm{i}}/2$
for very thin films, i.e., $d\ll\lambda$. $\epsilon_{\mathrm{eff}}$ and
$\mu_{\mathrm{eff}}$ are the effective permittivity and permeability of the
thin film. The right-hand side of equations (1) and (2) contains the bulk
magnetic and electric current densities,
$J_{m}=-\mathrm{i}\omega\mu_{\mathrm{eff}}\textbf{H}^{av}_{t}$ and
$J_{e}=-\mathrm{i}\omega\epsilon_{\mathrm{eff}}\textbf{E}^{av}_{t}$. In the
limit $d\ll\lambda$, the thin film can also be equivalently considered as a
single interface (metasurface) with surface current density, $J_{se}=\int
J_{e}\mathrm{d}z=J_{e}d$ and $J_{sm}=\int J_{m}\mathrm{d}z=J_{m}d$, resulting
from the discontinuity of transverse electric and magnetic fields across the
thin film, respectively. The surface electric and magnetic current densities
can be characterized by effective surface electric and magnetic
susceptibilities, $J_{se}=-i\omega\epsilon_{0}\chi_{se}\textbf{E}^{av}_{t}$
and $J_{sm}=-i\omega\mu_{0}\chi_{sm}\textbf{H}^{av}_{t}$. We can obtain
$\chi_{se}=(\epsilon_{\mathrm{eff}}-1)d$, $\chi_{sm}=(\mu_{\mathrm{eff}}-1)d$,
where the constant $1$ results from the permittivity or permeability of vacuum
when replacing a finite thickness slab by a zero thickness surface. Using the
previous equations for average fields, $\chi_{se}$ and $\chi_{sm}$ can now be
expressed as functions of the complex transmission and reflection coefficients
$T$ and $R$:
$\displaystyle\chi_{se}$ $\displaystyle=$
$\displaystyle\frac{2\mathrm{i}}{k_{0}}\frac{1-R-T}{1+R+T}$ (3)
$\displaystyle\chi_{sm}$ $\displaystyle=$
$\displaystyle\frac{2\mathrm{i}}{k_{0}}\frac{1+R-T}{1-R+T}$ (4)
where $k_{0}$ is the wavevector in vacuum. Equations (3) and (4) are in
consistent with previous work Holloway et al. (2009), except for a sign
reversal for $\chi_{sm}$ in Ref. 10, which we believe is a misprint.
Since the transmission and reflection coefficients are independent from the
thickness, $d$, the surface susceptibilities, $\chi_{se}$ and $\chi_{sm}$, are
well-defined parameters describing the properties of single-layered
metamaterial in isolation. Hence they are distinct from the effective
parameters of bulk metamaterials, $\epsilon_{\mathrm{eff}}$ and
$\mu_{\mathrm{eff}}$, which are non-unique and depend on the effective
metamaterial thickness $d$. Despite this, we also find that the effective
permittivity and permeability of individual metamaterial layers calculated
using a usual retrieval procedure Smith et al. (2002) show good consistency
with $\epsilon_{\mathrm{eff}}=\chi_{se}/d+1$, and
$\mu_{\mathrm{eff}}=\chi_{sm}/d+1$ obtained from equations (3) and (4). The
main exception is the anti-resonance artifacts obtained from the retrieval
procedure and due to periodicity effect are absent in the effective surface
susceptibilities. This means common retrieval procedures may be used to obtain
the surface susceptibilities of single-layer metamaterials with some accuracy.
Employing a transfer matrix method, we can determine the overall transmission
and reflection of a decoupled multilayered metamaterial from the effective
material parameters derived for each layer. In the following, we use a
metamaterial perfect absorber in Ref. Liu et al. (2010a) as an example to
demonstrate how to apply this approach and reveal its underlying mechanism.
Figure 2(a) shows a perfect absorber metamaterial Liu et al. (2010a)
consisting of two layers of metallic structures separated by a dielectric
spacer. The first metallic structure is an array of cross-wire resonators and
the second is a metallic ground plane. Each can be modeled as a metasurface
respectively. The whole metamaterial is then considered as a three-layered
system consisting of two metasurfaces separated by a dielectric spacer.
In the perfect absorber metamaterial, the cross-wire structure exhibits
electric resonance modes with resonance frequencies determined by its
structural parameters. To obtain the effective material parameters of
metamaterials, we performed numerical simulations with CST Microwave Studio
(Computer Simulation Technology GmbH, Darmstadt, Germany), which uses a
finite-difference time-domain method to determine $R$ and $T$ of the metallo-
dielectric structures. The unit cell used in the simulation for the first
layer, $\mathrm{MM}_{1}$, is schematically shown as the inset in Fig. 2(b). It
consists of a gold cross-wire with thickness, $d_{1}=0.1\ \mu m$, width
$w=0.4\ \mu m$, length $l=1.7\ \mu m$ and period $a=2\ \mu m$, on the spacer
layer with thickness, $s=0.185\ \mu m$, and dielectric constant,
$\epsilon_{s}=2.28(1+0.04\mathrm{i})$. Gold is modeled as a Drude metal with a
plasma frequency, $f_{p}=2181$ THz, and damping frequency, $f_{\tau}=6.5$ THz
Ordal et al. (1985). Then the effective material parameters of the
metamaterial, $\mathrm{MM}_{1}$, bounded by vacuum, are calculated using
equations (3) and (4) with slight modification to handle the substrate
surrounding the metamaterial structure Zhao et al. (2010). Similarly, the gold
plate with thickness, $d_{2}=0.2\hskip 2.84526pt\mu m$, was modeled as the
second layer using a unit cell shown as $\mathrm{MM}_{2}$ in the inset of Fig.
2(b). The calculated effective surface electric susceptibility of the
metamaterial is shown in Fig. 2(b), where the cross-wires exhibit electric
resonances at wavelengths of 4.85 $\mu m$ and 1.68 $\mu m$, and the gold plate
exhibits a plasmonic response with large negative permittivity over the entire
wavelength range from 1.5 to 8 $\mu m$. Importantly, the effective magnetic
susceptibility (not shown here) for $\mathrm{MM}_{1}$ and $\mathrm{MM}_{2}$
was calculated to be a constant close to zero over the entire wavelength
range. Figure 2(c) shows two absorption peaks at the wavelength of 1.86 and
6.18 $\mu m$, obtained by full EM simulations of the entire multilayered
metamaterial (solid curve). The latter peak corresponds to the absorption
reported in Ref. Liu et al. (2010a).
Figure 2: (a) The perfect absorber metamaterial is modeled as a stack of three
layers, the cross-wire metamaterial, a dielectric spacer, and the metallic
plate. (b) The real (solids curves) and imaginary (dashed curves) parts of the
effective surface electric susceptibility of the metasurface representing
cross-wires (red) and metallic plate (blue). The inset $\mathrm{MM}_{1}$ and
$\mathrm{MM}_{2}$ shows the unit cells used in numerical simulations to obtain
the metasurface parameters. (c) The solid curves show the reflectance (blue),
$R$, and absorptance (red), $A$, obtained from direct simulations, while the
dashed curves show the calculated values, $R_{c}$ (blue), $A_{c}$ (red), using
a 3-layer metamaterial model.
To determine the behavior of the whole absorber we first derive the transfer
matrix of the individual layers. The transfer matrix of a metasurface, bounded
by vacuum on each side, can be determined from the relation between the
transfer matrix and S-parameter matrix,
$M=\left(\matrix{M_{11}&M_{12}\cr
M_{21}&M_{22}}\right)=\left(\matrix{S_{12}-S_{11}S_{22}S_{21}^{-1}&S_{11}S_{21}^{-1}\cr-
S_{21}^{-1}S_{22}&S_{21}^{-1}}\right)$ (5)
where $S_{21}$, $S_{12}$ are forward and backward transmission coefficients,
and $S_{11}$, $S_{22}$ are reflection coefficients at front and back sides of
the metasurface, respectively. Equation (5) also applies to the dielectric
spacer. All of the individual transfer matrices are now multiplied to obtain
the total transfer matrix of the whole metamaterial,
$M^{\mathrm{tot}}$=$M_{\mathrm{MM}_{1}}M_{\mathrm{S}}M_{\mathrm{MM}_{2}}$.
This now constitutes a full description of the metamaterial perfect absorber
based on the effective parameters of the individual metasurfaces. Using the
relation between the transfer matrix and S-parameter matrix again, we can
obtain the transmission and reflection coefficients of the whole metamaterial,
$\widetilde{T}=S_{21}=1/M_{22}^{\mathrm{tot}}$ and
$\widetilde{R}=S_{11}=M_{12}^{\mathrm{tot}}/M_{22}^{\mathrm{tot}}$. As shown
in Fig. 2(c), the calculated reflectance, $R_{c}=|\widetilde{R}|^{2}$, and
absorptance, $A_{c}=1-|\widetilde{R}|^{2}-|\widetilde{T}|^{2}$, agree very
well with the corresponding $R$ and $A$ obtained from a direct simulation of
the whole metamaterial. Since the transfer matrix calculations only take
account of the transmissions and reflections between individual layers, and
since each layer’s properties were determined in isolation, this shows that
any inter-layer resonance coupling occurring between the cross-wire and the
metallic plate layers is inconsequential. Hence the magnetic response reported
in previous absorber work Landy et al. (2008); Liu et al. (2010a, b), is
unlikely to have any significant functional effect, since it relies on strong
coupling between two metallic layers in the form of anti-parallel resonance
currents. To further understand the magnetic response, we examined the double-
fishnet structure Dolling et al. (2006), where the magnetic resonance mode
exists due to the strong coupling between two metallic layers. As we expected,
the transfer matrix calculation failed to reproduce the magnetic resonance in
the direct simulations of whole double-fishnet structure since it violates the
decoupling or weakly coupling assumption.
To better understand the mechanism of the perfect absorber, we derived the
overall reflection coefficient, $\widetilde{R}$, using the mathematical form
of a slightly modified Fabry-Perot cavity, in terms of the transmission and
reflection coefficients of metasurfaces:
$\widetilde{R}=\frac{R_{12}+\alpha
R_{23}e^{2\mathrm{i}\beta}}{1-R_{21}R_{23}e^{2\mathrm{i}\beta}}$ (6)
where, $T_{21}$, $T_{12}$, $R_{12}$ and $R_{21}$, are transmission and
reflection coefficients of the metasurfaces, $\mathrm{MM}_{1}$, regarded as an
interface bounded by semi-infinite media. They are generally functions of
effective surface electric and magnetic susceptibilities of $\mathrm{MM}_{1}$.
They can also be obtained by numerical simulation of $\mathrm{MM}_{1}$ using
the structure shown in Fig. 2(b). $R_{23}=-1$ is the reflection coefficients
from the gold ground plane, $\mathrm{MM}_{2}$; $\beta=n_{s}kd_{s}$ is the
propagating phase term in the spacer; and $\alpha=T_{21}T_{12}-R_{12}R_{21}$.
At the perfect absorbing wavelength, the reflection coefficient,
$\widetilde{R}=0$, which requires the following conditions:
$\displaystyle|R_{12}|=|\alpha|$ (7)
$\displaystyle\phi(R_{12})-\phi(\alpha)-2\beta=2m\pi,\quad|m|=0,1,2,...$ (8)
Figure 3: (a) The amplitude of $R_{12}$ (blue) and $\alpha$ (red). (b) The
phase terms in equation (8), $\theta=\phi(R_{12})-\phi(\alpha)-2\beta$ (red),
the phase of $R_{12}$ (blue) and $\alpha$ (green), and the propagation phase
$2\beta$ (black) are shown respectively.
To understand these conditions, similar to recent work on anti-reflection
metamaterials Chen et al. (2010), we calculated the amplitude and phase terms
shown in equations (7) and (8) using the effective material parameters of the
metasurfaces. As shown in Fig. 3(a), in the strongly absorbing regions
(shaded) centered at the wavelengths of 6.18 and 1.86 $\mu m$, the amplitudes
of $R_{12}$ and $\alpha$ are almost equal, roughly satisfying the amplitude
condition. In Fig. 3(b), the phase term, $\theta$ crosses zero and 2$\pi$ at
wavelengths of 6.18 and 1.86 $\mu m$, respectively, indicating the phase
condition in equation (8) is perfectly fulfilled at the absorption peaks.
Several other absorption peaks (not shown here) can be observed at shorter
wavelengths when $\theta$ reaches 4$\pi$, 6$\pi$ etc. Figure 3 also shows that
equations (7) and (8) are not simultaneously satisfied at the same
wavelengths, which explains why the reflectance $|\widetilde{R}|^{2}$ (blue
dashed curve in Fig. 2(b)) does not reach zero. Equations (6-8) and Fig. 3
indicate that the nature of the absorber is Fabry-Perot-like resonance modes
resulting from multiple wave reflections between metasurfaces
$\mathrm{MM}_{1}$ and $\mathrm{MM}_{2}$. The metasurface $\mathrm{MM}_{1}$ and
ground plane $\mathrm{MM}_{2}$ form a Fabry-Perot cavity filled with a lossy
dielectric spacer. Strong absorption occurs as the EM wave propagates through
the lossy dielectric spacer multiple times. The metasurface $\mathrm{MM}_{1}$
provides the proper surface susceptibility to fulfill the conditions presented
in equations (7) and (8). Practically, the understanding of this mechanism
helps us to improve the metamaterial designs. For instance, to achieve a
perfect absorption, we need to optimize the design of the metamaterial,
$\mathrm{MM}_{1}$, to reach the conditions in equation (7) and (8)
simultaneously. Adjusting the slope of $|R_{12}|$, $|\alpha|$ and $\theta$
curves shown in Fig. 3 can also optimize the absorbing bandwidth of the whole
metamaterial.
In conclusion, we have proposed an effective medium model for individual
metasurfaces. In our model, each layer of the multilayered metamaterial is
considered as a metasurface. These metasurfaces may be combined to determine
the properties of multilayered metamaterials using the transfer matrix method.
This alternate interpretation resolves the problems of defining a multilayered
metamaterial as a single-“meta-atom”-layered bulk medium. This method provides
a general approach applicable for any decoupled or weakly coupled multilayered
metamaterials. We applied this method to the recently demonstrated perfect
absorber metamaterials and identified the underlying mechanism as Fabry-Perot
type resonance modes in contrast to the previously reported mechanism of
independent engineering of the bulk effective permittivity and permeability.
We have also found that this model accurately reproduces the previously
reported EM wave tunneling effects Zhou et al. (2005).
We acknowledge support from the Los Alamos National Laboratory LDRD Program.
This work was performed, in part, at the Center for Integrated
Nanotechnologies, a US Department of Energy, Office of Basic Energy Sciences
Nanoscale Science Research Center operated jointly by Los Alamos and Sandia
National Laboratories. Work at Ames Laboratory was supported by the Department
of Energy (Basic Energy Sciences) under contract No. DE-AC02-07CH11358. This
was partially supported by the U.S. Office of Naval Research, Award No.
N000141010925. We thank Christopher Holloway, Willie Padilla and Richard
Averitt for helpful discussions.
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* Tretyakov (2003) S. Tretyakov, _Analytical Modeling in Applied Electromagnetics_ (Artech House, 2003), ISBN 1-58053-367-1.
* Ordal et al. (1985) M. A. Ordal, R. J. Bell, J. R. W. Alexander, L. L. Long, and M. R. Querry, Appl. Opt. 24, 4493 (1985), URL http://ao.osa.org/abstract.cfm?URI=ao-24-24-4493.
* Zhao et al. (2010) R. Zhao, T. Koschny, and C. M. Soukoulis, Opt. Express 18, 14553 (2010), URL http://www.opticsexpress.org/abstract.cfm?URI=oe-18-14-14553.
* Dolling et al. (2006) G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, Science 312, 892 (2006).
|
arxiv-papers
| 2011-11-01T23:20:03 |
2024-09-04T02:49:23.901164
|
{
"license": "Public Domain",
"authors": "Jiangfeng Zhou, Hou-Tong Chen, Thomas Koschny, Abul K. Azad,\n Antoinette J. Taylor, Costas M. Soukoulis, John F. O'Hara",
"submitter": "Jiangfeng Zhou",
"url": "https://arxiv.org/abs/1111.0343"
}
|
1111.0426
|
# Development and Performance of spark-resistant Micromegas Detectors
National Technical Univ. of Athens, Greece
E-mail Konstantinos Karakostas
National Technical Univ. of Athens, Greece
E-mail Konstantinos.Karakostas@cern.ch Matthias Schott
CERN, Switzerland
E-mail Matthias.Schott@cern.ch
###### Abstract:
The Muon ATLAS MicroMegas Activity (MAMMA) focuses on the development and
testing of large-area muon detectors based on the bulk-Micromegas technology.
These detectors are candidates for the upgrade of the ATLAS Muon System in
view of the luminosity upgrade of Large Hadron Collider at CERN (sLHC). They
will combine trigger and precision measurement capability in a single device.
A novel protection scheme using resistive strips above the readout electrode
has been developed. The response and sparking properties of resistive
Micromegas detectors were successfully tested in a mixed (neutron and gamma)
high radiation field, in a X-ray test facility, in hadron beams, and in the
ATLAS cavern. Finally, we introduced a 2-dimensional readout structure in the
resistive Micromegas and studied the detector response with X-rays.
## 1 Introduction
The Micromegas (Micro-MEsh Gaseous Structure) detectors have been invented for
the detection of ionizing particles in experimental physics, in particular in
particle and nuclear physics. It was first proposed in 1996 [1]; its basic
operation principle is illustrated in Fig. 1. A planar drift electrode is
placed few mm above a readout electrode. The gap is filled with ionization
gas. In addition, a metal mesh is placed $\sim 0.1$ mm above the readout
electrode. The region between drift electrode and mesh is called the drift
region, while the region between mesh and readout electrodes is called the
amplification region. Both the mesh and the drift electrode are set at
negative high voltage, so that a electric field of $\sim 600$ V/cm is present
in the drift region and a field of $\sim 50$ kV/cm is present in the
amplification region. The readout electrodes are set to ground potential.
Charged particles transversing the drift region ionize the gas. The resulting
ionization electrons drift towards the mesh with a drift velocity of 5
cm/$\mu$s. The mesh itself appears transparent to the ionization electrons
when the electric field in the amplification region is much larger than that
in the drift region. Once reaching the amplification region, the ionization
electrons cause a cascade of secondary electrons (avalanche) leading to a
large amplification factor, which can be measured by the readout electrodes. A
significant step in the development of Micromegas detectors was achieved in
2006 and its known as bulk-Micromegas technology. A detailed description can
be found in [2].
## 2 Resistive Chambers
The thin amplification region together with its high electric field implies a
large risk of sparking. Sparks can cause damage to the detector itself, on the
underlying electronics, but lead also to significant dead-times. This serious
disadvantage was overcome recently, with the development of spark resistant
Micromegas chambers by the MAMMA group [3]. The resistive Micromegas developed
by MAMMA group has separate resistive strips rather than a continuous
resistive layer to avoid charge spreading across several readout strips and to
keep the area affected by a discharge as small as possible. The resistive
strips are separated by an insulating layer from the readout strips and
individually grounded through a large resistance. The Micromegas structure is
built on top of the resistive strips. It employs a woven stainless steel mesh
which is kept at a distance of 128 $\mu$m from the resistive strips by means
of small pillars (Fig. 1). Above the amplification mesh, at a distance of 4 or
5 mm, another stainless steel mesh serves as drift electrode. The signal on
the readout strips is then capacitively coupled to resistive strips. It has
been shown that this design provides a spark-resistant layout for Micromegas
chambers even in very high flux environments [4].
Figure 1: Resistive Micrommegas Layout.
The basic Micromegas design can be easily extended to a two-dimensional
readout. The readout strips in the x-direction are placed parallel to the
resistive strips, while the readout-strips in the y-direction are placed
perpendicular. All strips are separated by isolation material. The signal on
the readout strips is again capacitively coupled to resistive strips. Hence it
is expected that the induced signal on the x-strips is smaller then the signal
on the y-readout strips due to the larger distance to the resistive strips and
screening effects. In order to ensure that the induced charge in both layers
is of similar magnitude the lower readout-strips should be wider.
We present here preliminary results on the performance of spark resistant
Micromegas chambers in a beam of neutrons with a flux of $10^{6}Hz/cm^{2}$.
The detectors have been operated with three Ar:CO2 gas mixtures, with 80:20,
85:15 and 93:7. Fig. 2 shows a comparison of the the high voltage drop in case
of sparks and the current that chamber draws for the bulk Micromegas on the
left and resistive one on the right. Only a few sparks per second were
observed in a beam with 1.5$\cdot$106 neutrons$/$cm${}^{2}/$s. Hence, the
spark signal is reduced by a factor of 1000 compared to a standard Micromegas.
The spark rate was found four times higher with the 80:20 compared to the 93:7
Ar:CO2 gas mixture. The neutron interaction rate was found independent of the
gas.
Figure 2: Performance of standard (left) and resistive (right) Micromegas
chambers.
## References
* [1] I. Giomataris et al.,: Micro-Pattern Gaseous Detectors, Nucl. Instrum. Methods A 376 (1996) 29
* [2] I. Giomataris et al., Micromegas in a bulk; Nucl Instrum. Methods, A560 2006, PP:405
* [3] Alexopoulos, T. et. al: A spark-resistant bulk-Micromegas chamber for high-rate applications, Nucl.Instrum.Meth., A640, 2011, PP:110-118
* [4] Alexopoulos, T. et. al: Development of large size Micromegas detector for the upgrade of the ATLAS muon system, Nucl.Instrum.Meth., A617, 2010, PP: 161-165
|
arxiv-papers
| 2011-11-02T08:57:19 |
2024-09-04T02:49:23.908538
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "George Iakovidis, Kostantinos Karakostas, Matthias Schott",
"submitter": "George Iakovidis Mr",
"url": "https://arxiv.org/abs/1111.0426"
}
|
1111.0521
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2011-167 LHCb-PAPER-2011-014
Measurement of the effective $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ lifetime
The LHCb Collaboration111Authors are listed on the following pages.
A measurement of the effective $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ lifetime
is presented using approximately 37 $\mbox{\,pb}^{-1}$ of data collected by
LHCb during 2010\. This quantity can be used to put constraints on
contributions from processes beyond the Standard Model in the $B^{0}_{s}$
meson system and is determined by two complementary approaches as
$\tau_{KK}=1.440\pm 0.096~{}\mathrm{(stat)}\pm 0.008~{}\mathrm{(syst)}\pm
0.003~{}(\mathrm{model})~{}{\rm\,ps}.$
R. Aaij23, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi42, C. Adrover6, A.
Affolder48, Z. Ajaltouni5, J. Albrecht37, F. Alessio37, M. Alexander47, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr22, S. Amato2, Y. Amhis38, J.
Anderson39, R.B. Appleby50, O. Aquines Gutierrez10, F. Archilli18,37, L.
Arrabito53, A. Artamonov 34, M. Artuso52,37, E. Aslanides6, G. Auriemma22,m,
S. Bachmann11, J.J. Back44, D.S. Bailey50, V. Balagura30,37, W. Baldini16,
R.J. Barlow50, C. Barschel37, S. Barsuk7, W. Barter43, A. Bates47, C. Bauer10,
Th. Bauer23, A. Bay38, I. Bediaga1, S. Belogurov30, K. Belous34, I.
Belyaev30,37, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson46, J.
Benton42, R. Bernet39, M.-O. Bettler17, M. van Beuzekom23, A. Bien11, S.
Bifani12, A. Bizzeti17,h, P.M. Bjørnstad50, T. Blake37, F. Blanc38, C.
Blanks49, J. Blouw11, S. Blusk52, A. Bobrov33, V. Bocci22, A. Bondar33, N.
Bondar29, W. Bonivento15, S. Borghi47, A. Borgia52, T.J.V. Bowcock48, C.
Bozzi16, T. Brambach9, J. van den Brand24, J. Bressieux38, D. Brett50, S.
Brisbane51, M. Britsch10, T. Britton52, N.H. Brook42, H. Brown48, A. Büchler-
Germann39, I. Burducea28, A. Bursche39, J. Buytaert37, S. Cadeddu15, J.M.
Caicedo Carvajal37, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A.
Camboni35, P. Campana18,37, A. Carbone14, G. Carboni21,k, R. Cardinale19,i,37,
A. Cardini15, L. Carson36, K. Carvalho Akiba2, G. Casse48, M. Cattaneo37, M.
Charles51, Ph. Charpentier37, N. Chiapolini39, K. Ciba37, X. Cid Vidal36, G.
Ciezarek49, P.E.L. Clarke46,37, M. Clemencic37, H.V. Cliff43, J. Closier37, C.
Coca28, V. Coco23, J. Cogan6, P. Collins37, A. Comerma-Montells35, F.
Constantin28, G. Conti38, A. Contu51, A. Cook42, M. Coombes42, G. Corti37,
G.A. Cowan38, R. Currie46, B. D’Almagne7, C. D’Ambrosio37, P. David8, I. De
Bonis4, S. De Capua21,k, M. De Cian39, F. De Lorenzi12, J.M. De Miranda1, L.
De Paula2, P. De Simone18, D. Decamp4, M. Deckenhoff9, H. Degaudenzi38,37, M.
Deissenroth11, L. Del Buono8, C. Deplano15, D. Derkach14,37, O. Deschamps5, F.
Dettori15,d, J. Dickens43, H. Dijkstra37, P. Diniz Batista1, F. Domingo
Bonal35,n, S. Donleavy48, F. Dordei11, A. Dosil Suárez36, D. Dossett44, A.
Dovbnya40, F. Dupertuis38, R. Dzhelyadin34, A. Dziurda25, S. Easo45, U.
Egede49, V. Egorychev30, S. Eidelman33, D. van Eijk23, F. Eisele11, S.
Eisenhardt46, R. Ekelhof9, L. Eklund47, Ch. Elsasser39, D.G. d’Enterria35,o,
D. Esperante Pereira36, L. Estève43, A. Falabella16,e, E. Fanchini20,j, C.
Färber11, G. Fardell46, C. Farinelli23, S. Farry12, V. Fave38, V. Fernandez
Albor36, M. Ferro-Luzzi37, S. Filippov32, C. Fitzpatrick46, M. Fontana10, F.
Fontanelli19,i, R. Forty37, M. Frank37, C. Frei37, M. Frosini17,f,37, S.
Furcas20, A. Gallas Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini51, Y.
Gao3, J-C. Garnier37, J. Garofoli52, J. Garra Tico43, L. Garrido35, D.
Gascon35, C. Gaspar37, N. Gauvin38, M. Gersabeck37, T. Gershon44,37, Ph.
Ghez4, V. Gibson43, V.V. Gligorov37, C. Göbel54, D. Golubkov30, A.
Golutvin49,30,37, A. Gomes2, H. Gordon51, M. Grabalosa Gándara35, R. Graciani
Diaz35, L.A. Granado Cardoso37, E. Graugés35, G. Graziani17, A. Grecu28, E.
Greening51, S. Gregson43, B. Gui52, E. Gushchin32, Yu. Guz34, T. Gys37, G.
Haefeli38, C. Haen37, S.C. Haines43, T. Hampson42, S. Hansmann-Menzemer11, R.
Harji49, N. Harnew51, J. Harrison50, P.F. Harrison44, J. He7, V. Heijne23, K.
Hennessy48, P. Henrard5, J.A. Hernando Morata36, E. van Herwijnen37, E.
Hicks48, K. Holubyev11, P. Hopchev4, W. Hulsbergen23, P. Hunt51, T. Huse48,
R.S. Huston12, D. Hutchcroft48, D. Hynds47, V. Iakovenko41, P. Ilten12, J.
Imong42, R. Jacobsson37, A. Jaeger11, M. Jahjah Hussein5, E. Jans23, F.
Jansen23, P. Jaton38, B. Jean-Marie7, F. Jing3, M. John51, D. Johnson51, C.R.
Jones43, B. Jost37, M. Kaballo9, S. Kandybei40, M. Karacson37, T.M. Karbach9,
J. Keaveney12, U. Kerzel37, T. Ketel24, A. Keune38, B. Khanji6, Y.M. Kim46, M.
Knecht38, S. Koblitz37, P. Koppenburg23, A. Kozlinskiy23, L. Kravchuk32, K.
Kreplin11, M. Kreps44, G. Krocker11, P. Krokovny11, F. Kruse9, K.
Kruzelecki37, M. Kucharczyk20,25,37,j, R. Kumar14,37, T. Kvaratskheliya30,37,
V.N. La Thi38, D. Lacarrere37, G. Lafferty50, A. Lai15, D. Lambert46, R.W.
Lambert37, E. Lanciotti37, G. Lanfranchi18, C. Langenbruch11, T. Latham44, R.
Le Gac6, J. van Leerdam23, J.-P. Lees4, R. Lefèvre5, A. Leflat31,37, J.
Lefrançois7, O. Leroy6, T. Lesiak25, L. Li3, L. Li Gioi5, M. Lieng9, M.
Liles48, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, J.H. Lopes2, E. Lopez
Asamar35, N. Lopez-March38, J. Luisier38, F. Machefert7, I.V.
Machikhiliyan4,30, F. Maciuc10, O. Maev29,37, J. Magnin1, S. Malde51, R.M.D.
Mamunur37, G. Manca15,d, G. Mancinelli6, N. Mangiafave43, U. Marconi14, R.
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Sánchez7, D. Martinez Santos37, A. Massafferri1, Z. Mathe12, C. Matteuzzi20,
M. Matveev29, E. Maurice6, B. Maynard52, A. Mazurov16,32,37, G. McGregor50, R.
McNulty12, C. Mclean14, M. Meissner11, M. Merk23, J. Merkel9, R. Messi21,k, S.
Miglioranzi37, D.A. Milanes13,37, M.-N. Minard4, S. Monteil5, D. Moran12, P.
Morawski25, R. Mountain52, I. Mous23, F. Muheim46, K. Müller39, R.
Muresan28,38, B. Muryn26, M. Musy35, J. Mylroie-Smith48, P. Naik42, T.
Nakada38, R. Nandakumar45, J. Nardulli45, I. Nasteva1, M. Nedos9, M.
Needham46, N. Neufeld37, C. Nguyen-Mau38,p, M. Nicol7, S. Nies9, V. Niess5, N.
Nikitin31, A. Nomerotski51, A. Novoselov34, A. Oblakowska-Mucha26, V.
Obraztsov34, S. Oggero23, S. Ogilvy47, O. Okhrimenko41, R. Oldeman15,d, M.
Orlandea28, J.M. Otalora Goicochea2, P. Owen49, K. Pal52, J. Palacios39, A.
Palano13,b, M. Palutan18, J. Panman37, A. Papanestis45, M. Pappagallo13,b, C.
Parkes47,37, C.J. Parkinson49, G. Passaleva17, G.D. Patel48, M. Patel49, S.K.
Paterson49, G.N. Patrick45, C. Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos
Alvarez36, A. Pellegrino23, G. Penso22,l, M. Pepe Altarelli37, S.
Perazzini14,c, D.L. Perego20,j, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35,
P. Perret5, M. Perrin-Terrin6, G. Pessina20, A. Petrella16,37, A.
Petrolini19,i, E. Picatoste Olloqui35, B. Pie Valls35, B. Pietrzyk4, T.
Pilar44, D. Pinci22, R. Plackett47, S. Playfer46, M. Plo Casasus36, G.
Polok25, A. Poluektov44,33, E. Polycarpo2, D. Popov10, B. Popovici28, C.
Potterat35, A. Powell51, T. du Pree23, J. Prisciandaro38, V. Pugatch41, A.
Puig Navarro35, W. Qian52, J.H. Rademacker42, B. Rakotomiaramanana38, M.S.
Rangel2, I. Raniuk40, G. Raven24, S. Redford51, M.M. Reid44, A.C. dos Reis1,
S. Ricciardi45, K. Rinnert48, D.A. Roa Romero5, P. Robbe7, E. Rodrigues47, F.
Rodrigues2, P. Rodriguez Perez36, G.J. Rogers43, S. Roiser37, V. Romanovsky34,
M. Rosello35,n, J. Rouvinet38, T. Ruf37, H. Ruiz35, G. Sabatino21,k, J.J.
Saborido Silva36, N. Sagidova29, P. Sail47, B. Saitta15,d, C. Salzmann39, M.
Sannino19,i, R. Santacesaria22, C. Santamarina Rios36, R. Santinelli37, E.
Santovetti21,k, M. Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M.
Savrie16,e, D. Savrina30, P. Schaack49, M. Schiller11, S. Schleich9, M.
Schmelling10, B. Schmidt37, O. Schneider38, A. Schopper37, M.-H. Schune7, R.
Schwemmer37, B. Sciascia18, A. Sciubba18,l, M. Seco36, A. Semennikov30, K.
Senderowska26, I. Sepp49, N. Serra39, J. Serrano6, P. Seyfert11, B. Shao3, M.
Shapkin34, I. Shapoval40,37, P. Shatalov30, Y. Shcheglov29, T. Shears48, L.
Shekhtman33, O. Shevchenko40, V. Shevchenko30, A. Shires49, R. Silva
Coutinho54, H.P. Skottowe43, T. Skwarnicki52, A.C. Smith37, N.A. Smith48, E.
Smith51,45, K. Sobczak5, F.J.P. Soler47, A. Solomin42, F. Soomro18, B. Souza
De Paula2, B. Spaan9, A. Sparkes46, P. Spradlin47, F. Stagni37, S. Stahl11, O.
Steinkamp39, S. Stoica28, S. Stone52,37, B. Storaci23, M. Straticiuc28, U.
Straumann39, N. Styles46, V.K. Subbiah37, S. Swientek9, M. Szczekowski27, P.
Szczypka38, T. Szumlak26, S. T’Jampens4, E. Teodorescu28, F. Teubert37, C.
Thomas51, E. Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin39, S. Topp-
Joergensen51, N. Torr51, E. Tournefier4,49, M.T. Tran38, A. Tsaregorodtsev6,
N. Tuning23, M. Ubeda Garcia37, A. Ukleja27, P. Urquijo52, U. Uwer11, V.
Vagnoni14, G. Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36, S.
Vecchi16, J.J. Velthuis42, M. Veltri17,g, K. Vervink37, B. Viaud7, I. Videau7,
X. Vilasis-Cardona35,n, J. Visniakov36, A. Vollhardt39, D. Voong42, A.
Vorobyev29, H. Voss10, K. Wacker9, S. Wandernoth11, J. Wang52, D.R. Ward43,
A.D. Webber50, D. Websdale49, M. Whitehead44, D. Wiedner11, L. Wiggers23, G.
Wilkinson51, M.P. Williams44,45, M. Williams49, F.F. Wilson45, J. Wishahi9, M.
Witek25, W. Witzeling37, S.A. Wotton43, K. Wyllie37, Y. Xie46, F. Xing51, Z.
Xing52, Z. Yang3, R. Young46, O. Yushchenko34, M. Zavertyaev10,a, F. Zhang3,
L. Zhang52, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, E. Zverev31, A.
Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
24Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, Netherlands
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Cracow, Poland
26Faculty of Physics & Applied Computer Science, Cracow, Poland
27Soltan Institute for Nuclear Studies, Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
43Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
44Department of Physics, University of Warwick, Coventry, United Kingdom
45STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
46School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
47School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
48Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
49Imperial College London, London, United Kingdom
50School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
51Department of Physics, University of Oxford, Oxford, United Kingdom
52Syracuse University, Syracuse, NY, United States
53CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oInstitució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain
pHanoi University of Science, Hanoi, Viet Nam
## 1 Introduction
The study of charmless $B$ meson decays of the form $B\\!\rightarrow
h^{+}h^{\prime-}$, where $h^{(\prime)}$ is either a kaon, pion or proton,
offers a rich opportunity to explore the phase structure of the CKM matrix and
to search for manifestations of physics beyond the Standard Model. The
effective lifetime, defined as the decay time expectation value, of the
$B^{0}_{s}$ meson measured in the decay channel $B^{0}_{s}\\!\rightarrow
K^{+}K^{-}$ (charge conjugate modes are implied throughout the paper) is of
considerable interest as it can be used to put constraints on contributions
from new physical phenomena to the $B^{0}_{s}$ meson system [1, 2, 3, 4]. The
$B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ decay was first observed by CDF [5, 6].
The decay has subsequently been confirmed by Belle [7].
The detailed formalism of the effective lifetime in $B^{0}_{s}\\!\rightarrow
K^{+}K^{-}$ decay can be found in Refs. [3, 4]. The untagged decay time
distribution can be written as
$\displaystyle\Gamma(t)$ $\displaystyle\propto$ $\displaystyle\left(1-{\cal
A}_{\Delta\Gamma_{s}}\right)e^{-\Gamma_{L}t}+\left(1+{\cal
A}_{\Delta\Gamma_{s}}\right)e^{-\Gamma_{H}t}\,.$ (1)
The parameter ${\cal A}_{\Delta\Gamma_{s}}$ is defined as ${\cal
A}_{\Delta\Gamma_{s}}=-2{\rm Re}(\lambda)/\left(1+|\lambda|^{2}\right)$ where
$\lambda=(q/p)(\overline{A}/A)$ and the complex coefficients $p$ and $q$
define the mass eigenstates of the $B^{0}_{s}$–$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ system in terms of the flavour
eigenstates (see, e.g., Ref. [8]), while $A$ ($\overline{A}$) gives the
amplitude for $B^{0}_{s}$ ($\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$) decay to the $C\\!P$ even
$K^{+}K^{-}$ final state. In the absence of $C\\!P$ violation, ${\rm
Re}(\lambda)=1$ and $\rm{Im(\lambda)=0}$, so that the distribution involves
only the term containing $\Gamma_{L}$. Any deviation from a pure single
exponential with decay constant $\Gamma^{-1}_{L}$ is a measure of $C\\!P$
violation.
When modelling the decay time distribution shown in Eq. 1 with a single
exponential function in a maximum likelihood fit, it converges to the
effective lifetime given in Eq. 2 [9]. For small values of the relative width
difference
$\Delta\Gamma_{s}/\Gamma_{s}=(\Gamma_{L}-\Gamma_{H})/\left((\Gamma_{L}+\Gamma_{H})/2\right)$,
the distribution can be approximated by Taylor expansion as shown in the
second part of the equation [3]
$\tau_{KK}=\tau_{B^{0}_{s}}\frac{1}{1-y_{s}^{2}}\left[\frac{1+2{\cal
A}_{\Delta\Gamma_{s}}y_{s}+y_{s}^{2}}{1+{\cal
A}_{\Delta\Gamma_{s}}y_{s}}\right]=\tau_{B^{0}_{s}}\left(1+{\cal
A}_{\Delta\Gamma_{s}}y_{s}+\mathcal{O}(y_{s}^{2})\right),$ (2)
where $\tau_{B^{0}_{s}}=2/\left(\Gamma_{H}+\Gamma_{L}\right)=\Gamma_{s}^{-1}$
and $y_{s}=\Delta\Gamma_{s}/2\Gamma_{s}$. The Standard Model predictions for
these parameters are ${\cal A}_{\Delta\Gamma_{s}}=-0.97^{+0.014}_{-0.009}$[3]
and $y_{s}=0.066\pm 0.016$[10].
The decay $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ is dominated by loop diagrams
carrying, in the Standard Model, the same phase as the $B^{0}_{s}$–$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mixing amplitude and hence the
measured effective lifetime is expected to be close to $\Gamma_{L}^{-1}$. The
tree contribution to the $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ decay amplitude,
however, introduces $C\\!P$ violation effects. The Standard Model prediction
is $\tau_{KK}=1.390\pm 0.032~{}{\rm\,ps}$ [3]. In the presence of physics
beyond the Standard Model, deviations of the measured value from this
prediction are possible.
The measurement has been performed using a data sample corresponding to an
integrated luminosity of $37~{}\mbox{\,pb}^{-1}$ collected by LHCb at an
energy of $\sqrt{s}=7$ TeV during 2010. A key aspect of the analysis is the
correction of lifetime biasing effects, referred to as the acceptance, which
are introduced by the selection criteria to enrich the $B$ meson sample. Two
complementary data-driven approaches have been developed to compensate for
this bias. One method relies on extracting the acceptance function from data,
and then applies this acceptance correction to obtain a measurement of the
$B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ lifetime. The other approach cancels the
acceptance bias by taking the ratio of the $B^{0}_{s}\\!\rightarrow
K^{+}K^{-}$ lifetime distribution with that of $B^{0}\\!\rightarrow
K^{+}\pi^{-}$.
## 2 Data Sample
The LHCb detector [11] is a single arm spectrometer with a pseudorapidity
acceptance of $2<\eta<5$ for charged particles. The detector includes a high
precision tracking system which consists of a silicon vertex detector and
several dedicated tracking planes with silicon microstrip detectors (Inner
Tracker) covering the region with high charged particle multiplicity and straw
tube detectors (Outer Tracker) for the region with lower occupancy. The Inner
and Outer trackers are placed after the dipole magnet to allow the measurement
of the charged particles’ momenta as they traverse the detector. Excellent
particle identification capabilities are provided by two ring imaging
Cherenkov detectors which allow charged pions, kaons and protons to be
distinguished from each other in the momentum range 2–100
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The experiment employs a multi-level
trigger to reduce the readout rate and enhance signal purity: a hardware
trigger based on the measurement of the energy deposited in the calorimeter
cells and the momentum transverse to the beamline of muon candidates, as well
as a software trigger which allows the reconstruction of the full event
information.
$B$ mesons are produced with an average momentum of around 100
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and have decay vertices displaced from
the primary interaction vertex. Background particles tend to have low momentum
and tend to originate from the primary $pp$ collision. These features are
exploited in the event selection. In the absolute lifetime measurement the
final event selection is designed to be more stringent than the trigger
requirements, as this simplifies the calculation of the candidate’s acceptance
function. The tracks associated with the final state particles of the $B$
meson decay are required to have a good track fit quality ($\chi^{2}$/ndf $<3$
for one of the two tracks and $\chi^{2}$/ndf $<4$ for the other), have high
momentum ($p>13.5$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$), and at least one
particle must have a transverse momentum of more than 2.5
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The primary proton-proton interaction
vertex (or vertices in case of multiple interactions) of the event is fitted
from the reconstructed charged particles. The reconstructed trajectory of at
least one of the final state particles is required to have a distance of
closest approach to all primary vertices of at least 0.25$\rm\,mm$.
The $B$ meson candidate is obtained by reconstructing the vertex formed by the
two-particle final state. The $B$ meson transverse momentum is required to be
greater than 0.9 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and the distance of
the decay vertex to the closest primary $pp$ interaction vertex has to be
larger than 2.4$\rm\,mm$. In the final stage of the selection the modes
$B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ and $B^{0}\\!\rightarrow K^{+}\pi^{-}$
are separated by pion/kaon likelihood variables which use information obtained
from the ring imaging Cherenkov detectors.
The event selection used in the relative lifetime analysis is very similar.
However, some selection criteria can be slightly relaxed as the analysis does
not depend on the exact trigger requirements.
## 3 Relative Lifetime Measurement
Figure 1: Results of the relative lifetime fit. Left: Fit to the time-
integrated $KK$ mass spectrum. Right: Fit to the $KK$ decay time distribution.
The black points show the total number of candidates per picosecond in each
decay time bin, the stacked histogram shows the $B^{0}_{s}\\!\rightarrow
K^{+}K^{-}$ yield in red (dark) and the background yield in grey (light).
This analysis exploits the fact that the kinematic properties of the
$B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ decay are very similar to those of
$B^{0}\\!\rightarrow K^{+}\pi^{-}$. The two different decay modes can be
separated using information from the ring imaging Cherenkov detectors. The
left part of Fig. 1 shows the invariant mass distribution of the
$B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ candidates after the final event
selection. In addition 1,424 $B^{0}\\!\rightarrow K^{+}\pi^{-}$ candidates are
selected. Using a data-driven particle identification calibration method
described in the systematics section, the remaining contamination in the
$B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ sample from other $B\\!\rightarrow
h^{+}h^{\prime-}$ final states in the analysed mass region is estimated to be
3.8%.
$B$ mesons in either channel can be selected using identical kinematic
constraints and hence their decay time acceptance functions are almost
identical. Therefore the effects of the decay time acceptance cancel in the
ratio and the effective $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ lifetime can be
extracted relative to the $B^{0}\\!\rightarrow K^{+}\pi^{-}$ mode from the
variation of the ratio $R(t)$ of the yield of $B$ meson candidates in both
decay modes with decay time :
$R(t)=R(0)e^{-t\left(\tau_{KK}^{-1}~{}-~{}\tau_{K\pi}^{-1}\right)}.$ (3)
The cancellation of acceptance effects has been verified using simulated
events, including the full simulation of detector effects, trigger response
and final event selection. Any non-cancelling acceptance bias on the measured
lifetime is found to be smaller 1$\rm\,fs$.
In order to extract the effective $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$
lifetime, the yield of $B$ meson candidates is determined in bins of decay
time for both decay modes. Thirty bins between -1${\rm\,ps}$ and 35${\rm\,ps}$
are chosen such that each bin contains approximately the same number of $B$
meson candidates. The ratio of the yields is then fitted as a function of
decay time and the relative lifetime can be determined according to Eq. 3.
With this approach it is not necessary to parametrise the decay time
distribution of the background. In order to maximise the statistical
precision, both steps of the analysis are combined in a simultaneous fit to
the $K^{+}K^{-}$ and $K^{+}\pi^{-}$ invariant mass spectra across all decay
time bins. The signal distributions are described by Gaussian functions and
the combinatorial background by first order polynomials. The parameters of the
signal and background probability density functions (PDFs) are fixed to the
results of time-integrated mass fits before the lifetime fit is performed. The
$B^{0}\\!\rightarrow K^{+}\pi^{-}$ yield ($N_{B\rightarrow K\pi}$) is allowed
to float freely in each bin but the $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ yield
($N_{B_{s}\rightarrow KK}$) is constrained to follow
$N_{B_{s}\rightarrow KK}(\bar{t}_{i})=N_{B\rightarrow
K\pi}(\bar{t}_{i})R(0)e^{-\bar{t}_{i}\left(\tau_{KK}^{-1}~{}-~{}\tau_{K\pi}^{-1}\right)},$
(4)
where $\bar{t}_{i}$ is the mean decay time in the $i^{\rm th}$ bin. In total
the simultaneous fit has 94 free parameters and tests using Toy Monte Carlo
simulated data have found the fit to be unbiased to below 1$\rm\,fs$ on the
measured $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ lifetime. Each mass fit used in
the simultaneous fit is unbinned and must be split into mass bins in order to
evaluate the fit $\chi^{2}$. Two mass bins are chosen, one signal dominated
and one background dominated, in order to guarantee a minimum of 5-6
candidates in each bin. Using this appraoch the $\chi^{2}$ per degree of
freedom of the simultaneous fit is found to be 0.82. The right part of Fig. 1
shows the decay time distribution obtained from the fit and the fitted
reciprocal lifetime difference is
$\tau_{KK}^{-1}~{}-~{}\tau_{K\pi}^{-1}=0.013~{}\pm~{}0.045~{}\mathrm{(stat)}~{}{\rm\,ps}^{-1}.$
Taking the $B^{0}\\!\rightarrow K^{+}\pi^{-}$ lifetime as equal to the mean
$B^{0}$ lifetime ($\tau_{B^{0}}=1.519\pm 0.007~{}{\rm\,ps}$) [8], this
measurement can be expressed as
$\tau_{KK}=1.490\pm 0.100~{}\mathrm{(stat)}\pm
0.007~{}\mbox{(input)~{}${\rm\,ps}$}$
where the second uncertainty originates from the uncertainty of the $B^{0}$
lifetime.
## 4 Absolute Lifetime Measurement
The absolute lifetime measurement method directly determines the effective
$B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ lifetime using an acceptance correction
calculated from the data. This method was first used at the NA11 spectrometer
at CERN SPS [12], further developed within CDF [13, 14] and was subsequently
studied and implemented in LHCb [15, 16]. The per event acceptance function is
determined by evaluating whether the candidate would be selected for different
values of the $B$ meson candidate decay time. For example, for a $B$ meson
candidate, with given kinematic properties, the measured decay time of the $B$
meson candidate is directly related to the point of closest approach of the
final state particles to the associated primary vertex. Thus a selection
requirement on this quantity directly translates into a discrete decision
about acceptance or rejection of a candidate as a function of its decay time.
This is illustrated in Fig. 2. In the presence of several reconstructed
primary interaction vertices, the meson may enter a decay-time region where
one of the final state particles no longer fulfills the selection criteria
with respect to another primary vertex. Hence the acceptance function is
determined as a series of step changes. These _turning points_ at which the
candidates enter or leave the acceptance of a given primary vertex form the
basis of extracting the per-event acceptance function in the data. The turning
points are determined by moving the reconstructed primary vertex position of
the event along the $B$ meson momentum vector, and then reapplying the event
selection criteria. The analysis presented in this paper only includes events
with a single turning point. The drop of the acceptance to zero when the final
state particles are so far downstream that one is outside the detector
acceptance occurs only after many lifetimes and hence is safely neglected.
(a)
(b)
Figure 2: Decay-time acceptance function for an event of a two-body hadronic
decay. The light blue (shaded) regions show the bands for accepting the impact
parameter of a track. The impact parameter of the negative track (IP2) is too
small in (a) and lies within the accepted range in (b). The actual measured
decay time lies in the accepted region. The acceptance intervals give
conditional likelihoods used in the lifetime fit.
The distributions of the turning points, combined with the decay-time
distributions, are converted into an average acceptance function (see Fig. 3).
The average acceptance is not used in the lifetime fit, except in the
determination of the background decay-time distribution.
The effective $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ lifetime is extracted by an
unbinned maximum likelihood fit using an analytical probability density
function (PDF) for the signal decay time and a non-parametric PDF for the
combinatorial background, as described below. The measurement is factorised
into two independent fits.
A first fit is performed to the observed mass spectrum and used to determine
the signal and background probabilities of each event. Events with $B^{0}_{s}$
candidates in the mass range $5272-5800$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ were used, hence reducing the
contribution of partially reconstructed background and contamination of
$B^{0}$ decays below the $B^{0}_{s}$ mass peak. The signal distribution is
modelled with a Gaussian, and the background with a linear distribution. The
fitted mass value is compatible with the current world average [8].
The signal and background probabilities are used in the subsequent lifetime
fit. The decay-time PDF of the signal is calculated analytically taking into
account the per-event acceptance and the decay-time resolution. The decay-time
PDF of the combinatorial background is estimated from data using a non-
parametric method and is modelled by a sum of kernel functions which represent
each candidate by a normalised Gaussian function centred at the measured decay
time with a width proportional to an estimate of the density of candidates at
this decay time [17]. The lifetime fit is performed in the decay-time range of
$0.6-15~{}{\rm\,ps}$, hence only candidates within this range were accepted.
The analysis was tested on the $B^{0}\\!\rightarrow K^{+}\pi^{-}$ channel, for
which a lifetime compatible with the world average value was obtained, and
applied to the $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ channel only once the full
analysis procedure had been fixed. The result of the lifetime fit is
$\tau_{KK}=1.440\pm 0.096~{}\mathrm{(stat)}\;\mathrm{ps}$
and is illustrated in Fig. 3.
Figure 3: Left: Average decay-time acceptance function for signal events,
where the error band is an estimate of the statistical uncertainty. The plot
is scaled to 1 at large decay times, not taking into account the total signal
efficiency. Right: Decay-time distribution of the $B^{0}_{s}\\!\rightarrow
K^{+}K^{-}$ candidates and the fitted functions. The estimation of the
background distribution is sensitive to fluctuations due to the limited
statistics. Both plots are for the absolute lifetime measurement.
## 5 Systematic Uncertainties
Table 1: Summary of systematic uncertainties on the $B^{0}_{s}\\!\rightarrow
K^{+}K^{-}$ lifetime measurements.
Source of uncertainty | Uncertainty on | Uncertainty on
---|---|---
| $\tau_{KK}$ (fs) | $\tau_{KK}^{-1}~{}-~{}\tau_{K\pi}^{-1}$(ns-1)
Fit method | 3.2 |
Acceptance correction | 6.3 | 0.5
Mass model | 1.9 |
$B\\!\rightarrow h^{+}h^{\prime-}$ background | 1.9 | 1.4
Partially reconstructed background | 1.9 | 1.1
Combinatorial background | 1.5 | 1.6
Primary vertex association | 1.2 | 0.5
Detector length scale | 1.5 | 0.7
Production asymmetry | 1.4 | 0.6
Minimum accepted lifetime | 1.1 | N/A
Total (added in quadrature) | 8.4 | 2.7
Effective lifetime interpretation | 2.8 | 1.1
$\qquad\qquad\qquad\qquad\Bigg{\\}}$
The systematic uncertainties are listed in Table 1 and discussed below. The
dominant contributions to the systematic uncertainty for the absolute lifetime
measurement come from the treatment of the acceptance correction
($6.3~{}\rm\,fs$) and the fitting procedure ($3.2~{}\rm\,fs$). The systematic
uncertainty from the acceptance correction is determined by applying the same
analysis technique to a kinematically similar high statistics decay in the
charm sector ($D^{0}\\!\rightarrow K^{-}\pi^{+}$ [18]). This analysis yields a
lifetime value in good agreement with the current world average and of better
statistical accuracy. The uncertainty on the comparison between the measured
value and the world average is rescaled by the $B$ meson and charm meson
lifetime ratio. The uncertainty due to the fitting procedure is evaluated
using simplified simulations. A large number of pseudo-experiments are
simulated and the pull of the fitted lifetimes compared to the input value to
the fit is used to estimate the accuracy of the fit. These sources of
uncertainty are not dominant in the relative method, and are estimated from
simplified simulations which also include the systematic uncertainty of the
mass model. Hence a common systematic uncertainty is assigned to these three
sources.
The effect of the contamination of other $B\\!\rightarrow h^{+}h^{\prime-}$
modes to the signal modes is determined by a data-driven method. The
misidentification probability of protons, pions and kaons is measured in data
using the decays $K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$,
$D^{0}\rightarrow K^{+}\pi^{-}$, $\phi\rightarrow K^{+}K^{-}$ and $\mathchar
28931\relax\rightarrow p\pi^{-}$, where the particle type is inferred from
kinematic constraints alone [19]. As the particle identification likelihood
separating protons, kaons and pions depends on kinematic properties such as
momentum, transverse momentum, and number of reconstructed primary interaction
vertices, the sample is reweighted to reflect the different kinematic range of
the final state particles in $B\\!\rightarrow h^{+}h^{\prime-}$ decays. The
effect on the measured lifetime is evaluated with simplified simulations.
Decays of $B^{0}_{s}$ and $B^{0}$ to three or more final state particles,
which have been partially reconstructed, lie predominantly in the mass range
below the $B^{0}_{s}$ mass peak outside the analysed region. Residual
background from this source is estimated from data and evaluated with a sample
of fully simulated partially reconstructed decays. The effect on the fitted
lifetime is then evaluated.
In the absolute lifetime measurement, the combinatorial background of the
decay time distribution is described by a non-parametric function, based on
the observed events with masses above the $B^{0}_{s}$ meson signal region. The
systematic uncertainty is evaluated by varying the region used for evaluating
the combinatorial background. In the relative lifetime measurement, the
combinatorial background in the $hh^{\prime}$ invariant mass spectrum is
described by a first order polynomial. To estimate the systematic uncertainty,
a sample of simulated events is obtained with a simplified simulation using an
exponential function, and subsequently fitted with a first order polynomial.
Events may contain several primary interactions and a reconstructed $B$ meson
candidate may be associated to the wrong primary vertex. This effect is
studied using the more abundant charm meson decays where the lifetime is
measured separately for events with only one or any number of primary vertices
and the observed variation is scaled to the $B$ meson system.
Particle decay times are measured from the distance between the primary vertex
and secondary decay vertex in the silicon vertex detector. The systematic
uncertainty from this source is determined by considering the potential error
on the length scale of the detector from the mechanical survey, thermal
expansion and the current alignment precision.
The analysis assumes that $B^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mesons are produced in equal
quantities. The influence of a production asymmetry for $B^{0}_{s}$ mesons on
the measured lifetime is found to be small.
In the absolute lifetime method both a Gaussian and a Crystal Ball mass model
[20] are implemented and the effect on fully simulated data is evaluated to
estimate the systematic uncertainty due to the modelling of the signal PDF. In
the relative lifetime method this uncertainty is evaluated with simplified
simulations and included in the fitting procedure uncertainty.
In the absolute $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ lifetime measurement a
cut is applied on the minimal reconstructed decay time. As the background
decay time estimation will smear this step in the distribution, a systematic
uncertainty is quoted from varying this cut.
There is an additional uncertainty introduced if the result is interpreted
using Eq. 2, as this expression does not take into account detector resolution
and decay time acceptance. This effect was studied using simplified
simulations modelling the acceptance observed in the data and conservative
values of $\Delta\Gamma_{s}$ = 0.1 ${\rm\,ps}$ and ${\cal
A}_{\Delta\Gamma_{s}}$ = -0.6. The observed bias with respect to the
prediction of Eq. 2 is 3 $\rm\,fs$. This effect is labelled “Effective
lifetime interpretation” in Table 1 and is not a source of systematic
uncertainty on the measurement but is relevant to the interpretation of the
measured lifetime.
## 6 Results and Conclusions
The effective $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ lifetime has been measured
in $pp$ interactions using a data sample corresponding to an integrated
luminosity of $37\rm\,pb^{-1}$ recorded by the LHCb experiment in 2010. Two
complementary approaches have been followed to compensate for acceptance
effects introduced by the trigger and final event selection used to enrich the
sample of $B^{0}_{s}$ mesons. The absolute measurement extracts the per event
acceptance function directly from the data and finds:
$\tau_{KK}=1.440\pm 0.096~{}\mathrm{(stat)}\pm 0.008~{}\mathrm{(syst)}\pm
0.003~{}(\mathrm{model})~{}{\rm\,ps}$
where the third source of uncertainty labelled “model” is related to the
interpretation of the effective lifetime.
The relative method exploits the fact that the kinematic properties of the
various $B\\!\rightarrow h^{+}h^{\prime-}$ modes are almost identical and
extracts the $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ lifetime relative to the
$B^{0}\\!\rightarrow K^{+}\pi^{-}$ lifetime as:
$\tau_{KK}^{-1}~{}-~{}\tau_{K\pi}^{-1}=0.013\pm 0.045~{}\mathrm{(stat)}\pm
0.003~{}\mathrm{(syst)}\pm 0.001~{}(\mathrm{model})~{}{\rm\,ps}^{-1}.$
Taking the $B^{0}\\!\rightarrow K^{+}\pi^{-}$ lifetime as equal to the mean
$B^{0}$ lifetime ($\tau_{B^{0}}=1.519\pm 0.007~{}{\rm\,ps}$) [8], this
measurement can be expressed as:
$\tau_{KK}=1.490\pm 0.100~{}\mathrm{(stat)}\pm 0.006~{}\mathrm{(syst)}\pm
0.002~{}(\mathrm{model})\pm 0.007~{}\mbox{(input)~{}${\rm\,ps}$}.$
where the last uncertainty originates from the uncertainty of the $B^{0}$
lifetime. Both measurements are found to be compatible with each other, taking
the overlap in the data analysed into account.
Due to the large overlap of the data analysed by the two methods and the high
correlation of the systematic uncertainties, there is no significant gain from
a combination of the two numbers. Instead, the result obtained using the
absolute lifetime method is taken as the final result. The measured effective
$B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ lifetime is in agreement with the
Standard Model prediction of $\tau_{KK}=1.390\pm 0.032~{}{\rm\,ps}$ [3].
## 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
| 2011-11-02T14:54:16 |
2024-09-04T02:49:23.914979
|
{
"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, 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, S. Brisbane, M. Britsch, T. Britton, N.H. Brook, H.\n Brown, A. B\\\"uchler-Germann, I. Burducea, A. Bursche, J. Buytaert, S.\n Cadeddu, J.M. Caicedo Carvajal, O. Callot, M. Calvi, M. Calvo Gomez, A.\n Camboni, P. Campana, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, L.\n Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, M. Charles, Ph.\n Charpentier, N. Chiapolini, 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, G. Conti, A. Contu, A. Cook, M.\n Coombes, G. Corti, G.A. Cowan, R. Currie, B. D'Almagne, C. D'Ambrosio, P.\n David, I. De Bonis, S. De Capua, M. De Cian, F. De Lorenzi, J.M. De Miranda,\n L. De Paula, P. De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi, M.\n Deissenroth, L. Del Buono, C. Deplano, D. Derkach, O. Deschamps, F. Dettori,\n J. Dickens, H. Dijkstra, P. Diniz Batista, F. Domingo Bonal, S. Donleavy, F.\n Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R.\n Dzhelyadin, A. Dziurda, S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van\n Eijk, F. Eisele, S. Eisenhardt, R. Ekelhof, L. Eklund, Ch. Elsasser, D.G.\n d'Enterria, D. Esperante Pereira, L. Est\\`eve, A. Falabella, E. Fanchini, C.\n F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, V. Fernandez Albor, M.\n Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana, F. Fontanelli, R.\n Forty, 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, 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, G. Haefeli, C. Haen, S.C. Haines, T.\n Hampson, S. Hansmann-Menzemer, R. Harji, N. Harnew, J. Harrison, P.F.\n Harrison, J. He, V. Heijne, K. Hennessy, P. Henrard, J.A. Hernando Morata, E.\n van Herwijnen, 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, U. Kerzel, T. Ketel, A.\n Keune, B. Khanji, Y.M. Kim, M. Knecht, S. Koblitz, P. Koppenburg, A.\n Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F.\n Kruse, K. Kruzelecki, M. Kucharczyk, R. Kumar, 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, 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.H. Lopes, E. Lopez Asamar, N. Lopez-March, J. Luisier, 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, C.\n Mclean, M. Meissner, M. Merk, J. Merkel, R. Messi, S. Miglioranzi, D.A.\n Milanes, M.-N. Minard, S. Monteil, D. Moran, P. Morawski, R. Mountain, I.\n Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, M. Musy, J.\n Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, J. Nardulli, I. Nasteva, M.\n Nedos, M. Needham, N. Neufeld, C. Nguyen-Mau, M. Nicol, S. Nies, 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, E. Picatoste Olloqui, B. Pie Valls, B. Pietrzyk, T. Pilar, D.\n Pinci, R. Plackett, S. Playfer, M. Plo Casasus, G. Polok, A. Poluektov, E.\n Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, T. du Pree, 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, K. Rinnert, D.A. Roa Romero, P. Robbe, E.\n Rodrigues, F. Rodrigues, P. Rodriguez Perez, G.J. Rogers, S. Roiser, V.\n Romanovsky, M. Rosello, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino, J.J.\n Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, M. Sannino, R.\n Santacesaria, C. Santamarina Rios, R. Santinelli, E. Santovetti, M. Sapunov,\n A. Sarti, C. Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M.\n Schiller, S. Schleich, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper,\n M.-H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov,\n K. Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, B. Shao, M.\n Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O.\n Shevchenko, V. Shevchenko, A. Shires, R. Silva Coutinho, H.P. Skottowe, T.\n Skwarnicki, A.C. Smith, 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, N. Styles, V.K. Subbiah, S. Swientek, M.\n 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, M.T. Tran, A. Tsaregorodtsev, N.\n Tuning, M. Ubeda Garcia, A. Ukleja, P. Urquijo, U. Uwer, V. Vagnoni, G.\n Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J.J. Velthuis, M.\n Veltri, K. Vervink, B. Viaud, I. Videau, X. Vilasis-Cardona, J. Visniakov, A.\n Vollhardt, D. Voong, A. Vorobyev, H. Voss, K. Wacker, S. Wandernoth, J. Wang,\n D.R. Ward, A.D. Webber, D. Websdale, M. Whitehead, D. Wiedner, L. Wiggers, G.\n Wilkinson, M.P. Williams, M. Williams, F.F. Wilson, J. Wishahi, M. Witek, W.\n Witzeling, S.A. Wotton, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R.\n Young, O. Yushchenko, M. Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y.\n Zhang, A. Zhelezov, L. Zhong, E. Zverev, A. Zvyagin",
"submitter": "Lars Eklund",
"url": "https://arxiv.org/abs/1111.0521"
}
|
1111.0722
|
# Multiple brake orbits on compact convex symmetric reversible hypersurfaces
in ${\bf R}^{2n}$
Duanzhi Zhang and Chungen Liu
School of Mathematics and LPMC, Nankai University
Tianjin 300071, People’s Republic of China Partially supported by the NSF of
China (10801078, 11171314) and Nankai University. E-mail:
zhangdz@nankai.edu.cnCorresponding author. Partially supported by the NSF of
China (11071127, 10621101), 973 Program of MOST (2011CB808002). E-mail:
liucg@nankai.edu.cn
###### Abstract
In this paper, we prove that there exist at least
$\left[\frac{n+1}{2}\right]+1$ geometrically distinct brake orbits on every
$C^{2}$ compact convex symmetric hypersurface ${\Sigma}$ in ${\bf R}^{2n}$ for
$n\geq 2$ satisfying the reversible condition $N{\Sigma}={\Sigma}$ with
$N={\rm diag}(-I_{n},I_{n})$. As a consequence, we show that there exist at
least $\left[\frac{n+1}{2}\right]+1$ geometrically distinct brake orbits in
every bounded convex symmetric domain in ${\bf R}^{n}$ with $n\geq 2$ which
gives a positive answer to the Seifert conjecture of 1948 in the symmetric
case for $n=3$. As an application, for $n=4$ and $5$, we prove that if there
are exactly $n$ geometrically distinct closed characteristics on ${\Sigma}$,
then all of them are symmetric brake orbits after suitable time translation.
MSC(2000): 58E05; 70H05; 34C25
Key words: Brake orbit, Maslov-type index, H${\rm\ddot{o}}$rmander index,
Convex symmetric
## 1 Introduction
Let $V\in C^{2}({\bf R}^{n},{\bf R})$ and $h>0$ such that
${\Omega}\equiv\\{q\in{\bf R}^{n}|V(q)<h\\}$ is nonempty, bounded, open and
connected. Consider the following fixed energy problem of the second order
autonomous Hamiltonian system
$\displaystyle\ddot{q}(t)+V^{\prime}(q(t))=0,\quad{\rm for}\;q(t)\in{\Omega},$
(1.1) $\displaystyle\frac{1}{2}|\dot{q}(t)|^{2}+V(q(t))=h,\qquad\forall
t\in{\bf R},$ (1.2) $\displaystyle\dot{q}(0)=\dot{q}(\frac{\tau}{2})=0,$ (1.3)
$\displaystyle q(\frac{\tau}{2}+t)=q(\frac{\tau}{2}-t),\qquad
q(t+\tau)=q(t),\quad\forall t\in{\bf R}.$ (1.4)
A solution $(\tau,q)$ of (1.1)-(1.4) is called a brake orbit in ${\Omega}$.
We call two brake orbits $q_{1}$ and $q_{2}:{\bf R}\to{\bf R}^{n}$
geometrically distinct if $q_{1}({\bf R})\neq q_{2}({\bf R})$.
We denote by $\mathcal{O}({\Omega})$ and $\tilde{\mathcal{O}}({\Omega})$ the
sets of all brake orbits and geometrically distinct brake orbits in ${\Omega}$
respectively.
Let $J_{k}=\left(\begin{array}[]{cc}0&-I_{k}\\\ I_{k}&0\end{array}\right)$ and
$N_{k}=\left(\begin{array}[]{cc}-I_{k}&0\\\ 0&I_{k}\end{array}\right)$ with
$I_{k}$ being the identity in ${\bf R}^{k}$. If $k=n$ we will omit the
subscript $k$ for convenience, i.e., $J_{n}=J$ and $N_{n}=N$.
The symplectic group ${\rm Sp}(2k)$ for any $k\in{\bf N}$ is defined by
${\rm Sp}(2n)=\\{M\in\mathcal{L}({\bf R}^{2k})|M^{T}J_{k}M=J_{k}\\},$
where $M^{T}$ is the transpose of matrix $M$.
For any $\tau>0$, the symplectic path in ${\rm Sp}(2k)$ starting from the
identity $I_{2k}$ is defined by
$\mathcal{P}_{\tau}(2k)=\\{\gamma\in C([0,\tau],{\rm
Sp}(2k))|\gamma(0)=I_{2k}\\}.$
Suppose that $H\in C^{2}({\bf R}^{2n}\setminus\\{0\\},{\bf R})\cap C^{1}({\bf
R}^{2n},{\bf R})$ satisfying
$H(Nx)=H(x),\qquad\forall\,x\in{\bf R}^{2n}.$ (1.5)
We consider the following fixed energy problem
$\displaystyle\dot{x}(t)$ $\displaystyle=$ $\displaystyle JH^{\prime}(x(t)),$
(1.6) $\displaystyle H(x(t))$ $\displaystyle=$ $\displaystyle h,$ (1.7)
$\displaystyle x(-t)$ $\displaystyle=$ $\displaystyle Nx(t),$ (1.8)
$\displaystyle x(\tau+t)$ $\displaystyle=$ $\displaystyle
x(t),\;\forall\,t\in{\bf R}.$ (1.9)
A solution $(\tau,x)$ of (1.6)-(1.9) is also called a brake orbit on
${\Sigma}:=\\{y\in{\bf R}^{2n}\,|\,H(y)=h\\}$.
Remark 1.1. It is well known that via
$H(p,q)={1\over 2}|p|^{2}+V(q),$ (1.10)
$x=(p,q)$ and $p=\dot{q}$, the elements in $\mathcal{O}(\\{V<h\\})$ and the
solutions of (1.6)-(1.9) are one to one correspondent.
In more general setting, let ${\Sigma}$ be a $C^{2}$ compact hypersurface in
${\bf R}^{2n}$ bounding a compact set $C$ with nonempty interior. Suppose
${\Sigma}$ has non-vanishing Guassian curvature and satisfies the reversible
condition $N({\Sigma}-x_{0})={\Sigma}-x_{0}:=\\{x-x_{0}|x\in{\Sigma}\\}$ for
some $x_{0}\in C$. Without loss of generality, we may assume $x_{0}=0$. We
denote the set of all such hypersurface in ${\bf R}^{2n}$ by
$\mathcal{H}_{b}(2n)$. For $x\in{\Sigma}$, let $N_{\Sigma}(x)$ be the unit
outward normal vector at $x\in{\Sigma}$. Note that here by the reversible
condition there holds $N_{\Sigma}(Nx)=NN_{\Sigma}(x)$. We consider the
dynamics problem of finding $\tau>0$ and an absolutely continuous curve
$x:[0,\tau]\to{\bf R}^{2n}$ such that
$\displaystyle\dot{x}(t)$ $\displaystyle=$ $\displaystyle
JN_{\Sigma}(x(t)),\qquad x(t)\in{\Sigma},$ (1.11) $\displaystyle x(-t)$
$\displaystyle=$ $\displaystyle Nx(t),\qquad x(\tau+t)=x(t),\qquad{\rm
for\;\;all}\;\;t\in{\bf R}.$ (1.12)
A solution $(\tau,x)$ of the problem (1.11)-(1.12) is a special closed
characteristic on ${\Sigma}$, here we still call it a brake orbit on
${\Sigma}$.
We also call two brake orbits $(\tau_{1},x_{1})$ and $(\tau_{2},x_{2})$
geometrically distinct if $x_{1}({\bf R})\neq x_{2}({\bf R})$, otherwise we
say they are equivalent. Any two equivalent brake orbits are geometrically the
same. We denote by ${\mathcal{J}}_{b}({\Sigma})$ the set of all brake orbits
on ${\Sigma}$, by $[(\tau,x)]$ the equivalent class of
$(\tau,x)\in{\mathcal{J}}_{b}({\Sigma})$ in this equivalent relation and by
$\tilde{\mathcal{J}}_{b}({\Sigma})$ the set of $[(\tau,x)]$ for all
$(\tau,x)\in{\mathcal{J}}_{b}({\Sigma})$. From now on, in the notation
$[(\tau,x)]$ we always assume $x$ has minimal period $\tau$. We also denote by
$\tilde{\mathcal{J}}({\Sigma})$ the set of all geometrically distinct closed
characteristics on ${\Sigma}$.
Let $(\tau,x)$ be a solution of (1.6)-(1.9). We consider the boundary value
problem of the linearized Hamiltonian system
$\displaystyle\dot{y}(t)=JH^{\prime\prime}(x(t))y(t),$ (1.13) $\displaystyle
y(t+\tau)=y(t),\quad y(-t)=Ny(t),\qquad\forall t\in{\bf R}.$ (1.14)
Denote by ${\gamma}_{x}(t)$ the fundamental solution of the system (1.13),
i.e., ${\gamma}_{x}(t)$ is the solution of the following problem
$\displaystyle\dot{{\gamma}_{x}}(t)$ $\displaystyle=$ $\displaystyle
JH^{\prime\prime}(x(t)){\gamma}_{x}(t),$ (1.15) $\displaystyle{\gamma}_{x}(0)$
$\displaystyle=$ $\displaystyle I_{2n}.$ (1.16)
We call ${\gamma}_{x}\in C([0,\tau/2],{\rm Sp}(2n))$ the associated symplectic
path of $(\tau,x)$.
Let $B^{n}_{1}(0)$ denote the open unit ball ${\bf R}^{n}$ centered at the
origin $0$. In [20] of 1948, H. Seifert proved
$\tilde{\mathcal{O}}({\Omega})\neq\emptyset$ provided $V^{\prime}\neq 0$ on
$\partial{\Omega}$, $V$ is analytic and ${\Omega}$ is homeomorphic to
$B^{n}_{1}(0)$. Then he proposed his famous conjecture:
${}^{\\#}\tilde{\mathcal{O}}({\Omega})\geq n$ under the same conditions.
After 1948, many studies have been carried out for the brake orbit problem. S.
Bolotin proved first in [4](also see [5]) of 1978 the existence of brake
orbits in general setting. K. Hayashi in [10], H. Gluck and W. Ziller in [8],
and V. Benci in [2] in 1983-1984 proved
${}^{\\#}\tilde{\mathcal{O}}({\Omega})\geq 1$ if $V$ is $C^{1}$,
$\bar{{\Omega}}=\\{V\leq h\\}$ is compact, and $V^{\prime}(q)\neq 0$ for all
$q\in\partial{{\Omega}}$. In 1987, P. Rabinowitz in [19] proved that if $H$
satisfies (1.5), ${\Sigma}\equiv H^{-1}(h)$ is star-shaped, and $x\cdot
H^{\prime}(x)\neq 0$ for all $x\in{\Sigma}$, then
${}^{\\#}\tilde{\mathcal{J}}_{b}({\Sigma})\geq 1$. In 1987, V. Benci and F.
Giannoni gave a different proof of the existence of one brake orbit in [3].
In 1989, A. Szulkin in [21] proved that ${}^{\\#}\tilde{{\cal
J}_{b}}(H^{-1}(h))\geq n$, if $H$ satisfies conditions in [19] of Rabinowitz
and the energy hypersurface $H^{-1}(h)$ is $\sqrt{2}$-pinched. E. van Groesen
in [9] of 1985 and A. Ambrosetti, V. Benci, Y. Long in [1] of 1993 also proved
${}^{\\#}\tilde{\mathcal{O}}({\Omega})\geq n$ under different pinching
conditions.
Without pinching condition, in [17] Y. Long, C. Zhu and the second author of
this paper proved the following result: For $n\geq 2$, suppose $H$ satisfies
(H1) (smoothness) $H\in C^{2}({\bf R}^{2n}\setminus\\{0\\},{\bf R})\cap
C^{1}({\bf R}^{2n},{\bf R})$,
(H2) (reversibility) $H(Ny)=H(y)$ for all $y\in{\bf R}^{2n}$.
(H3) (convexity) $H^{\prime\prime}(y)$ is positive definite for all $y\in{\bf
R}^{2n}\setminus\\{0\\}$,
(H4) (symmetry) $H(-y)=H(y)$ for all $y\in{\bf R}^{2n}$.
Then for any given $h>\min\\{H(y)|\;y\in{\bf R}^{2n}\\}$ and
${\Sigma}=H^{-1}(h)$, there holds
${}^{\\#}\tilde{{\cal J}}_{b}({\Sigma})\geq 2.$
As a consequence they also proved that: For $n\geq 2$, suppose $V(0)=0$,
$V(q)\geq 0$, $V(-q)=V(q)$ and $V^{\prime\prime}(q)$ is positive definite for
all $q\in{\bf R}^{n}\setminus\\{0\\}$. Then for ${\Omega}\equiv\\{q\in{\bf
R}^{n}|V(q)<h\\}$ with $h>0$, there holds
${}^{\\#}\tilde{\mathcal{O}}({\Omega})\geq 2.$
Under the same condition of [17], in 2009 Liu and Zhang in [14] proved that
${}^{\\#}\tilde{{\cal J}}_{b}({\Sigma})\geq\left[\frac{n}{2}\right]+1$, also
they proved
${}^{\\#}\tilde{\mathcal{O}}({\Omega})\geq\left[\frac{n}{2}\right]+1$ under
the same condition of [17]. Moreover if all brake orbits on ${\Sigma}$ are
nondegenerate, Liu and Zhang in [14] proved that ${}^{\\#}\tilde{{\cal
J}}_{b}({\Sigma})\geq n+\mathfrak{A}({{\Sigma}}),$ where
$2\mathfrak{A}(\Sigma)$ is the number of geometrically distinct asymmetric
brake orbits on ${\Sigma}$.
Definition 1.1. We denote
$\begin{array}[]{ll}\mathcal{H}_{b}^{c}(2n)=\\{{\Sigma}\in\mathcal{H}_{b}(2n)|\;{\Sigma}\;{is\;strictly\;convex\;}\\},\\\
\mathcal{H}_{b}^{s,c}(2n)=\\{{\Sigma}\in\mathcal{H}_{b}^{c}(2n)|\;-{\Sigma}={\Sigma}\\}.\end{array}$
Definition 1.2. For ${\Sigma}\in\mathcal{H}_{b}^{s,c}(2n)$, a brake orbit
$(\tau,x)$ on ${\Sigma}$ is called symmetric if $x({\bf R})=-x({\bf R})$.
Similarly, for a $C^{2}$ convex symmetric bounded domain $\Omega\subset{\bf
R}^{n}$, a brake orbit $(\tau,q)\in\mathcal{O}(\Omega)$ is called symmetric if
$q({\bf R})=-q({\bf R})$.
Note that a brake orbit $(\tau,x)\in\mathcal{J}_{b}({\Sigma})$ with minimal
period $\tau$ is symmetric if $x(t+\tau/2)=-x(t)$ for $t\in{\bf R}$, a brake
orbit $(\tau,q)\in\mathcal{O}(\Omega)$ with minimal period $\tau$ is symmetric
if $q(t+\tau/2)=-q(t)$ for $t\in{\bf R}$.
In this paper, we denote by ${\bf N}$, ${\bf Z}$, ${\bf Q}$ and ${\bf R}$ the
sets of positive integers, integers, rational numbers and real numbers
respectively. We denote by $\langle\cdot,\cdot\rangle$ the standard inner
product in ${\bf R}^{n}$ or ${\bf R}^{2n}$, by $(\cdot,\cdot)$ the inner
product of corresponding Hilbert space. For any $a\in{\bf R}$, we denote
$E(a)=\inf\\{k\in{\bf Z}|k\geq a\\}$ and $[a]=\sup\\{k\in{\bf Z}|k\leq a\\}$.
The following are the main results of this paper.
Theorem 1.1. For any ${\Sigma}\in\mathcal{H}_{b}^{s,c}(2n)$ with $n\geq 2$, we
have
${}^{\\#}\tilde{{\cal J}}_{b}({\Sigma})\geq\left[\frac{n+1}{2}\right]+1.$
Corollary 1.1. Suppose $V(0)=0$, $V(q)\geq 0$, $V(-q)=V(q)$ and
$V^{\prime\prime}(q)$ is positive definite for all $q\in{\bf
R}^{n}\setminus\\{0\\}$ with $n\geq 3$. Then for any given $h>0$ and
${\Omega}\equiv\\{q\in{\bf R}^{n}|V(q)<h\\}$, we have
${}^{\\#}\tilde{\mathcal{O}}({\Omega})\geq\left[\frac{n+1}{2}\right]+1.$
Remark 1.2. Note that for $n=3$, Corollary 1.1 yields
${}^{\\#}\tilde{\mathcal{O}}({\Omega})\geq 3$, which gives a positive answer
to Seifert’s conjecture in the convex symmetric case.
As a consequence of Theorem 1.1, we can prove
Theorem 1.2. For $n=4,5$ and any ${\Sigma}\in\mathcal{H}_{b}^{s,c}(2n)$,
suppose
${}^{\\#}\tilde{{\cal J}}({\Sigma})=n.$
Then all of them are symmetric brake orbits after suitable translation.
Example 1.1. A typical example of ${\Sigma}\in\mathcal{H}_{b}^{s,c}(2n)$ is
the ellipsoid $\mathcal{E}_{n}(r)$ defined as follows. Let
$r=(r_{1},\cdots,r_{n})$ with $r_{j}>0$ for $1\leq j\leq n$. Define
$\mathcal{E}_{n}(r)=\left\\{x=(x_{1},\cdots,x_{n},y_{1},\cdots,y_{n})\in{\bf
R}^{2n}\;\left|\;\sum_{k=1}^{n}\frac{x_{k}^{2}+y_{k}^{2}}{r_{k}^{2}}=1\right.\right\\}.$
If $r_{j}/r_{k}\notin{\bf Q}$ whenever $j\neq k$, from [7] one can see that
there are precisely $n$ geometrically distinct symmetric brake orbits on
$\mathcal{E}_{n}(r)$ and all of them are nondegenerate.
## 2 Index theories of $(i_{L_{j}},\nu_{L_{j}})$ and
$(i_{\omega},\nu_{\omega})$
Let $\mathcal{L}({\bf R}^{2n})$ denotes the set of $2n\times 2n$ real matrices
and $\mathcal{L}_{s}({\bf R}^{2n})$ denotes its subset of symmetric ones. For
any $F\in\mathcal{L}_{s}({\bf R}^{2n})$, we denote by $m^{*}(F)$ the dimension
of maximal positive definite subspace, negative definite subspace, and kernel
of any $F$ for $*=+,-,0$ respectively.
In this section, we make some preparation for the proof of Theorem 3.1 below.
We first briefly review the index function $(i_{\omega},\nu_{\omega})$ and
$(i_{L_{j}},\nu_{L_{j}})$ for $j=0,1$, more details can be found in [14] and
[16]. Following Theorem 2.3 of [23] we study the differences
$i_{L_{0}}({\gamma})-i_{L_{1}}({\gamma})$ and
$i_{L_{0}}({\gamma})+\nu_{L_{0}}({\gamma})-i_{L_{1}}({\gamma})-\nu_{L_{1}}({\gamma})$
for ${\gamma}\in\mathcal{P}_{\tau}(2n)$ by compute ${\rm
sgn}M_{\varepsilon}({\gamma}(\tau))$. We obtain some basic lemmas which will
be used frequently in the proof of the main theorem of this paper.
For any ${\omega}\in{\bf U}$, the following codimension 1 hypersuface in ${\rm
Sp}(2n)$ is defined by:
${\rm Sp}(2n)_{\omega}^{0}=\\{M\in{\rm Sp}(2n)|{\rm
det}(M-{\omega}I_{2n})=0\\}.$
For any two continuous path $\xi$ and $\eta$: $[0,\tau]\to{\rm Sp}(2n)$ with
$\xi(\tau)=\eta(0)$, their joint path is defined by
$\displaystyle\eta*\xi(t)=\left\\{\begin{array}[]{lr}\xi(2t)&{\rm if}\,0\leq
t\leq\frac{\tau}{2},\\\ \eta(2t-\tau)&{\rm if}\,\frac{\tau}{2}\leq
t\leq\tau.\end{array}\right.$ (2.3)
Given any two $(2m_{k}\times 2m_{k})$\- matrices of square block form
$M_{k}=\left(\begin{array}[]{cc}A_{k}&B_{k}\\\ C_{k}&D_{k}\end{array}\right)$
for $k=1,2$, as in [16], the $\diamond$-product of $M_{1}$ and $M_{2}$ is
defined by the following $(2(m_{1}+m_{2})\times 2(m_{1}+m_{2}))$-matrix
$M_{1}\diamond M_{2}$:
$M_{1}\diamond M_{2}=\left(\begin{array}[]{cccc}A_{1}&0&B_{1}&0\\\
0&A_{2}&0&B_{2}\\\ C_{1}&0&D_{1}&0\\\ 0&C_{2}&0&D_{2}\end{array}\right).$
A special path $\xi_{n}$ is defined by
$\xi_{n}(t)=\left(\begin{array}[]{cc}2-\frac{t}{\tau}&0\\\
0&(2-\frac{t}{\tau})^{-1}\end{array}\right)^{\diamond n},\qquad\forall
t\in[0,\tau].$
Definition 2.1. For any ${\omega}\in{\bf U}$ and $M\in{\rm Sp}(2n)$, define
$\displaystyle\nu_{\omega}(M)=\dim_{\bf C}\ker(M-{\omega}I_{2n}).$
For any ${\gamma}\in\mathcal{P}_{\tau}(2n)$, define
$\displaystyle\nu_{\omega}({\gamma})=\nu_{\omega}({\gamma}(\tau)).$
If ${\gamma}(\tau)\notin{\rm Sp}(2n)_{\omega}^{0}$, we define
$i_{\omega}({\gamma})=[{\rm Sp}(2n)_{\omega}^{0}\,:\,{\gamma}*\xi_{n}],$ (2.4)
where the right-hand side of (2.4) is the usual homotopy intersection number
and the orientation of ${\gamma}*\xi_{n}$ is its positive time direction under
homotopy with fixed endpoints. If ${\gamma}(\tau)\in{\rm
Sp}(2n)_{\omega}^{0}$, we let $\mathcal{F}({\gamma})$ be the set of all open
neighborhoods of ${\gamma}$ in $\mathcal{P}_{\tau}(2n)$, and define
$\displaystyle
i_{\omega}({\gamma})=\sup_{U\in\mathcal{F}({\gamma})}\inf\\{i_{\omega}(\beta)|\,\beta(\tau)\in
U\,{\rm and}\,\beta(\tau)\notin{\rm Sp}(2n)_{\omega}^{0}\\}.$
Then $(i_{\omega}({\gamma}),\nu_{\omega}({\gamma}))\in{\bf
Z}\times\\{0,1,...,2n\\}$, is called the index function of ${\gamma}$ at
${\omega}$.
For any $M\in{\rm Sp}(2n)$ we define
$\displaystyle{\Omega}(M)=\\{P\in{\rm Sp}(2n)$ $\displaystyle|$
$\displaystyle{\sigma}(P)\cap{\bf U}={\sigma}(M)\cap{\bf U}$
$\displaystyle{\rm
and}\,\nu_{\lambda}(P)=\nu_{\lambda}(M),\;\;\forall{\lambda}\in{\sigma}(M)\cap{\bf
U}\\},$
where we denote by ${\sigma}(P)$ the spectrum of $P$.
We denote by ${\Omega}^{0}(M)$ the path connected component of ${\Omega}(M)$
containing $M$, and call it the homotopy component of $M$ in ${\rm Sp}(2n)$.
Definition 2.2. For any $M_{1}$,$M_{2}\in{\rm Sp}(2n)$, we call $M_{1}\approx
M_{2}$ if $M_{1}\in{\Omega}^{0}(M_{2})$.
Remark 2.1. It is easy to check that $\approx$ is an equivalent relation. If
$M_{1}\approx M_{2}$, we have $M_{1}^{k}\approx M_{2}^{k}$ for any $k\in{\bf
N}$ and $M_{1}\diamond M_{3}\approx M_{2}\diamond M_{4}$ for $M_{3}\approx
M_{4}$. Also we have $PMP^{-1}\approx M$ for any $P,M\in{\rm Sp}(2n)$.
The following symplectic matrices were introduced as basic normal forms in
[16]:
$\displaystyle D({\lambda})=\left(\begin{array}[]{cc}{\lambda}&0\\\
0&{\lambda}^{-1}\end{array}\right),\qquad$ $\displaystyle{\lambda}=\pm 2,$
(2.7) $\displaystyle
N_{1}({\lambda},b)=\left(\begin{array}[]{cc}{\lambda}&b\\\
0&{\lambda}\end{array}\right),\qquad$ $\displaystyle{\lambda}=\pm 1,\,b=\pm
1,\,0,$ (2.10) $\displaystyle
R(\theta)=\left(\begin{array}[]{cc}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{array}\right),\qquad$
$\displaystyle\theta\in(0,\pi)\cup(\pi,2\pi),$ (2.13) $\displaystyle
N_{2}({\omega},b)=\left(\begin{array}[]{cc}R(\theta)&b\\\
0&R(\theta)\end{array}\right),\qquad$
$\displaystyle\theta\in(0,\pi)\cup(\pi,2\pi),$ (2.16)
where $b=\left(\begin{array}[]{cc}b_{1}&b_{2}\\\
b_{3}&b_{4}\end{array}\right)$ with $b_{i}\in{\bf R}$ and $b_{2}\neq b_{3}$.
For any $M\in{\rm Sp}(2n)$ and ${\omega}\in{\bf U}$, splitting number of $M$
at ${\omega}$ is defined by
$\displaystyle S_{M}^{\pm}({\omega})=\lim_{\epsilon\to 0^{+}}i_{{\omega}{\rm
exp}(\pm\sqrt{-1}\epsilon)}({\gamma})-i_{\omega}({\gamma})$
for any path ${\gamma}\in\mathcal{P}_{\tau}(2n)$ satisfying
${\gamma}(\tau)=M$.
Splitting numbers possesses the following properties.
Lemma 2.1. (cf. [15], Lemma 9.1.5 and List 9.1.12 of [16]) Splitting number
$S_{M}^{\pm}({\omega})$ are well defined, i.e., they are independent of the
choice of the path ${\gamma}\in\mathcal{P}_{\tau}(2n)$ satisfying
${\gamma}(\tau)=M$. For ${\omega}\in{\bf U}$ and $M\in{\rm Sp}(2n)$,
$S_{Q}^{\pm}({\omega})=S_{M}^{\pm}({\omega})$ if $Q\approx M$. Moreover we
have
(1) $(S_{M}^{+}(\pm 1),S_{M}^{-}(\pm 1))=(1,1)$ for $M=\pm N_{1}(1,b)$ with
$b=1$ or $0$;
(2) $(S_{M}^{+}(\pm 1),S_{M}^{-}(\pm 1))=(0,0)$ for $M=\pm N_{1}(1,b)$ with
$b=-1$;
(3) $(S_{M}^{+}(e^{\sqrt{-1}\theta}),S_{M}^{-}(e^{\sqrt{-1}\theta}))=(0,1)$
for $M=R(\theta)$ with $\theta\in(0,\pi)\cup(\pi,2\pi)$;
(4) $(S_{M}^{+}({\omega}),S_{M}^{-}({\omega}))=(0,0)$ for ${\omega}\in{\bf
U}\setminus{\bf R}$ and $M=N_{2}({\omega},b)$ is trivial i.e., for
sufficiently small $\alpha>0$, $MR((t-1)\alpha)^{\diamond n}$ possesses no
eigenvalues on ${\bf U}$ for $t\in[0,1)$.
(5) $(S_{M}^{+}({\omega}),S_{M}^{-}({\omega})=(1,1)$ for ${\omega}\in{\bf
U}\setminus{\bf R}$ and $M=N_{2}({\omega},b)$ is non-trivial.
(6) $(S_{M}^{+}({\omega}),S_{M}^{-}({\omega})=(0,0)$ for any ${\omega}\in{\bf
U}$ and $M\in{\rm Sp}(2n)$ with ${\sigma}(M)\cap{\bf U}=\emptyset$.
(7) $S_{M_{1}\diamond
M_{2}}^{\pm}({\omega})=S_{M_{1}}^{\pm}({\omega})+S_{M_{2}}^{\pm}({\omega})$,
for any $M_{j}\in{\rm Sp}(2n_{j})$ with $j=1,2$ and ${\omega}\in{\bf U}$.
Let
$\displaystyle F={\bf R}^{2n}\oplus{\bf R}^{2n}$
possess the standard inner product. We define the symplectic structure of $F$
by
$\displaystyle\\{v,w\\}=(\mathcal{J}v,w),\;\forall v,w\in F,\;{\rm
where}\;\mathcal{J}=(-J)\oplus J=\left(\begin{array}[]{cc}-J&0\\\
0&J\end{array}\right).\;$ (2.19)
We denote by ${\rm Lag}(F)$ the set of Lagrangian subspaces of $F$, and equip
it with the topology as a subspace of the Grassmannian of all $2n$-dimensional
subspaces of $F$.
It is easy to check that, for any $M\in{\rm Sp}(2n)$ its graph
${\rm Gr}(M)\equiv\left\\{\left(\begin{array}[]{c}x\\\
Mx\end{array}\right)|x\in{\bf R}^{2n}\right\\}$
is a Lagrangian subspace of $F$.
Let
$\displaystyle V_{1}=\\{0\\}\times{\bf R}^{n}\times\\{0\\}\times{\bf
R}^{n}\subset{\bf R}^{4n},\quad V_{2}={\bf R}^{n}\times\\{0\\}\times{\bf
R}^{n}\times\\{0\\}\subset{\bf R}^{4n}.$
By Proposition 6.1 of [18] and Lemma 2.8 and Definition 2.5 of [17], we give
the following definition.
Definition 2.3. For any continuous path ${\gamma}\in\mathcal{P}_{\tau}(2n)$,
we define the following Maslov-type indices:
$\displaystyle i_{L_{0}}({\gamma})=\mu^{CLM}_{F}(V_{1},{\rm
Gr}({\gamma}),[0,\tau])-n,$ $\displaystyle
i_{L_{1}}({\gamma})=\mu^{CLM}_{F}(V_{2},{\rm Gr}({\gamma}),[0,\tau])-n,$
$\displaystyle\nu_{L_{j}}({\gamma})=\dim({\gamma}(\tau)L_{j}\cap L_{j}),\qquad
j=0,1,$
where we denote by $i^{CLM}_{F}(V,W,[a,b])$ the Maslov index for Lagrangian
subspace path pair $(V,W)$ in $F$ on $[a,b]$ defined by Cappell, Lee, and
Miller in [6]. For any $M\in{\rm Sp}(2n)$ and $j=0,1$, we also denote by
$\nu_{L_{j}}(M)=\dim(ML_{j}\cap L_{j})$.
Definition 2.4. For two paths
$\gamma_{0},\;\gamma_{1}\in\mathcal{P_{\tau}}(2n)$ and $j=0,1$, we say that
they are $L_{j}$-homotopic and denoted by $\gamma_{0}\sim_{L_{j}}\gamma_{1}$,
if there is a continuous map $\delta:[0,1]\to\mathcal{P}(2n)$ such that
$\delta(0)=\gamma_{0}$ and $\delta(1)=\gamma_{1}$, and
$\nu_{L_{j}}(\delta(s))$ is constant for $s\in[0,1]$.
Lemma 2.2.([11]) (1) If $\gamma_{0}\sim_{L_{j}}\gamma_{1}$, there hold
$i_{L_{j}}(\gamma_{0})=i_{L_{j}}(\gamma_{1}),\;\nu_{L_{j}}(\gamma_{0})=\nu_{L_{j}}(\gamma_{1}).$
(2) If $\gamma=\gamma_{1}\diamond\gamma_{2}\in\mathcal{P}(2n)$, and
correspondingly $L_{j}=L_{j}^{\prime}\oplus L_{j}^{\prime\prime}$, then
$i_{L_{j}}(\gamma)=i_{L^{\prime}_{j}}(\gamma_{1})+i_{L_{j}^{\prime\prime}}(\gamma_{2}),\;\nu_{L_{j}}(\gamma)=\nu_{L^{\prime}_{j}}(\gamma_{1})+\nu_{L_{j}^{\prime\prime}}(\gamma_{2}).$
(3) If $\gamma\in\mathcal{P}(2n)$ is the fundamental solution of
$\dot{x}(t)=JB(t)x(t)$
with symmetric matrix function
$B(t)=\left(\begin{array}[]{cc}b_{11}(t)&b_{12}(t)\\\
b_{21}(t)&b_{22}(t)\end{array}\right)$ satisfying $b_{22}(t)>0$ for any $t\in
R$, then there holds
$i_{L_{0}}(\gamma)=\sum_{0<s<1}\nu_{L_{0}}(\gamma_{s}),\;\gamma_{s}(t)=\gamma(st).$
(4) If $b_{11}(t)>0$ for any $t\in{\bf R}$, there holds
$i_{L_{1}}(\gamma)=\sum_{0<s<1}\nu_{L_{1}}(\gamma_{s}),\;\gamma_{s}(t)=\gamma(st).$
Definition 2.5. For any ${\gamma}\in\mathcal{P}_{\tau}$ and $k\in{\bf
N}\equiv\\{1,2,...\\}$, in this paper the $k$-time iteration ${\gamma}^{k}$ of
${\gamma}\in\mathcal{P}_{\tau}(2n)$ in brake orbit boundary sense is defined
by $\tilde{{\gamma}}|_{[0,k\tau]}$ with
$\displaystyle\tilde{{\gamma}}(t)=\left\\{\begin{array}[]{l}{\gamma}(t-2j\tau)(N{\gamma}(\tau)^{-1}N{\gamma}(\tau))^{j},\;t\in[2j\tau,(2j+1)\tau],j=0,1,2,...\\\
N{\gamma}(2j\tau+2\tau-t)N(N{\gamma}(\tau)^{-1}N{\gamma}(\tau))^{j+1},\;t\in[(2j+1)\tau,(2j+2)\tau],j=0,1,2,...\end{array}\right.$
(2.22)
By [17] or Corollary 5.1 of [14]
$\displaystyle\lim_{k\to\infty}\frac{i_{L_{0}}(\gamma^{k})}{k}$ exists, as
usual we define the mean $i_{L_{0}}$ index of ${\gamma}$ by
$\hat{i}_{L_{0}}({\gamma})=\displaystyle\lim_{k\to\infty}\frac{i_{L_{0}}(\gamma^{k})}{k}$.
For any $P\in{\rm Sp}(2n)$ and $\varepsilon\in{\bf R}$, we set
$\displaystyle
M_{\varepsilon}(P)=P^{T}\left(\begin{array}[]{cc}\sin{2{\varepsilon}}I_{n}&-\cos{2{\varepsilon}I_{n}}\\\
-\cos{2{\varepsilon}}I_{n}&-\sin
2{\varepsilon}I_{n}\end{array}\right)P+\left(\begin{array}[]{cc}\sin{2{\varepsilon}}I_{n}&\cos{2{\varepsilon}}I_{n}\\\
\cos{2{\varepsilon}}I_{n}&-\sin 2{\varepsilon}I_{n}\end{array}\right).$ (2.27)
Then we have the following
Theorem 2.1.(Theorem 2.3 of [23]) For ${\gamma}\in\mathcal{P}_{\tau}(2k)$ with
$\tau>0$, we have
$\displaystyle i_{L_{0}}({\gamma})-i_{L_{1}}({\gamma})=\frac{1}{2}{\rm
sgn}M_{\varepsilon}({\gamma}(\tau)),$
where ${\rm
sgn}M_{\varepsilon}({\gamma}(\tau))=m^{+}(M_{\varepsilon}({\gamma}(\tau)))-m^{-}(M_{\varepsilon}({\gamma}(\tau)))$
is the signature of the symmetric matrix $M_{\varepsilon}({\gamma}(\tau))$ and
$0<{\varepsilon}\ll 1$. we also have,
$\displaystyle(i_{L_{0}}({\gamma})+\nu_{L_{0}}({\gamma}))-(i_{L_{1}}({\gamma})+\nu_{L_{1}}({\gamma}))=\frac{1}{2}{\rm
sign}M_{\varepsilon}({\gamma}(\tau)),$
where $0<-{\varepsilon}\ll 1$.
Remark 2.2. (Remark 2.1 of [23]) For any $n_{j}\times n_{j}$ symplectic matrix
$P_{j}$ with $j=1,2$ and $n_{j}\in{\bf N}$, we have
$\displaystyle M_{\varepsilon}(P_{1}\diamond
P_{2})=M_{\varepsilon}(P_{1})\diamond M_{\varepsilon}(P_{2}),$
$\displaystyle{\rm sgn}M_{\varepsilon}(P_{1}\diamond P_{2})={\rm
sgn}M_{\varepsilon}(P_{1})+{\rm sgn}M_{\varepsilon}(P_{2}),$
where ${\varepsilon}\in{\bf R}$.
In the following of this section we will give some lemmas which will be used
frequently in the proof of our main theorem later.
Lemma 2.3. For $k\in{\bf N}$ and any symplectic matrix
$P=\left(\begin{array}[]{cc}I_{k}&0\\\ C&I_{k}\end{array}\right)$, there holds
$P\approx I_{2}^{\diamond p}\diamond N_{1}(1,1)^{\diamond q}\diamond
N_{1}(1,-1)^{\diamond r}$ with $p,q,r$ satisfying
$\displaystyle m^{0}(C)=p,\quad m^{-}(C)=q,\quad m^{+}(C)=r.$
Proof. It is clear that
$\displaystyle P\approx\left(\begin{array}[]{cc}I_{k}&0\\\
B&I_{k}\end{array}\right),$ (2.30)
where $B={\rm diag}(0,-I_{m^{-}(C)},I_{m^{+}(C)})$. Since $J_{1}N_{1}(1,\pm
1)(J_{1})^{-1}=\left(\begin{array}[]{cc}1&0\\\ \mp 1&1\end{array}\right)$, by
Remark 2.1 we have $N_{1}(1,\pm 1)\approx\left(\begin{array}[]{cc}1&0\\\ \mp
1&1\end{array}\right)$. Then
$\displaystyle P\approx I_{2}^{\diamond m^{0}(C)}\diamond N_{1}(1,1)^{\diamond
m^{-}(C)}\diamond N_{1}(1,-1)^{\diamond m^{+}(C)}.$
By Lemma 2.1 we have
$S_{P}^{+}(1)=m^{0}(C)+m^{-}(C)=p+q.$ (2.31)
By the definition of the relation $\approx$, we have
$2p+q+r=\nu_{1}(P)=2m^{0}(C)+m^{+}(C)+m^{-}(C).$ (2.32)
Also we have
$p+q+r=m^{0}(C)+m^{+}(C)+m^{-}(C)=k.$ (2.33)
By (2.31)-(2.33) we have
$\displaystyle m^{0}(C)=p,\quad m^{-}(C)=q,\quad m^{+}(C)=r.$
The proof of Lemma 2.3 is complete.
Definition 2.6. We call two symplectic matrices $M_{1}$ and $M_{2}$ in ${\rm
Sp}(2k)$ are special homotopic(or $(L_{0},L_{1})$-homotopic) and denote by
$M_{1}\sim M_{2}$, if there are $P_{j}\in{\rm Sp}(2k)$ with $P_{j}={\rm
diag}(Q_{j},(Q_{j}^{T})^{-1})$, where $Q_{j}$ is a $k\times k$ invertible real
matrix, and ${\rm det}(Q_{j})>0$ for $j=1,2$, such that
$M_{1}=P_{1}M_{2}P_{2}.$
It is clear that $\sim$ is an equivalent relation.
Lemma 2.4. For $M_{1},\,M_{2}\in{\rm Sp}(2k)$, if $M_{1}\sim M_{2}$, then
$\displaystyle{\color[rgb]{1,0,0}{\rm sgn}M_{\varepsilon}(M_{1})={\rm
sgn}M_{\varepsilon}(M_{2}),\quad 0\leq|{\varepsilon}|\ll 1,}$ (2.34)
$\displaystyle N_{k}M_{1}^{-1}N_{k}M_{1}\approx N_{k}M_{2}^{-1}N_{k}M_{2}.$
(2.35)
Proof. By Definition 2.6, there are $P_{j}\in{\rm Sp}(2k)$ with $P_{j}={\rm
diag}(Q_{j},(Q_{j}^{T})^{-1})$, $Q_{j}$ being $k\times k$ invertible real
matrix, and ${\rm det}(Q_{j})>0$ such that
$M_{1}=P_{1}M_{2}P_{2}.$
Since ${\rm det}(Q_{j})>0$ for $j=1,2$, we can joint $Q_{j}$ to $I_{k}$ by
invertible matrix path. Hence we can joint $P_{1}M_{2}P_{2}$ to $M_{2}$ by
symplectic path preserving the nullity $\nu_{L_{0}}$ and $\nu_{L_{1}}$. By
Lemma 2.2 of [23], (2.34) holds. Since $P_{j}N_{k}=N_{k}P_{j}$ for $j=1,2$.
Direct computation shows that
$N_{k}(P_{1}M_{2}P_{2})^{-1}N_{k}(P_{1}M_{2}P_{2})=P_{2}^{-1}N_{k}M_{2}^{-1}N_{k}M_{2}P_{2}.$
(2.36)
Thus (2.35) holds from Remark 2.1. The proof of Lemma 2.4 is complete.
Lemma 2.5. Let $P=\left(\begin{array}[]{cc}A&B\\\ C&D\end{array}\right)\in{\rm
Sp}(2k)$, where $A,B,C,D$ are all $k\times k$ matrices. Then
(i) $\frac{1}{2}{\rm sgn}M_{\varepsilon}(P)\leq k-\nu_{L_{0}}(P)$, for
$0<{\varepsilon}\ll 1$. If $B=0$, we have $\frac{1}{2}{\rm
sgn}M_{\varepsilon}(P)\leq 0$ for $0<{\varepsilon}\ll 1$.
(ii) Let $m^{+}(A^{T}C)=q$, we have
$\displaystyle\frac{1}{2}{\rm sgn}M_{\varepsilon}(P)\leq k-q,\quad
0\leq|{\varepsilon}|\ll 1.$ (2.37)
Moreover if $B=0$, we have
$\displaystyle\frac{1}{2}{\rm sgn}M_{\varepsilon}(P)\leq-q,\quad
0<-{\varepsilon}\ll 1.$ (2.38)
(iii) $\frac{1}{2}{\rm sgn}M_{\varepsilon}(P)\geq\dim\ker C-k$ for
$0<{\varepsilon}\ll 1$, If $C=0$, then $\frac{1}{2}{\rm
sgn}M_{\varepsilon}(P)\geq 0$ for $0<{\varepsilon}\ll 1$
(iv) If both $B$ and $C$ are invertible, we have
$\displaystyle{\rm sgn}M_{\varepsilon}(P)={\rm sgn}M_{0}(P),\quad
0\leq|{\varepsilon}|\ll 1.$
Proof. Since $P$ is symplectic, so is for $P^{T}$. From $P^{T}J_{k}P=J_{k}$
and $PJ_{k}P^{T}=J_{k}$ we get $A^{T}C,B^{T}D,AB^{T},CD^{T}$ are all symmetric
matrices and
$AD^{T}-BC^{T}=I_{k},\quad A^{T}D-C^{T}B=I_{k}.$ (2.39)
We denote by $s=\sin 2{\varepsilon}$ and $c=\cos 2{\varepsilon}$. By
definition of $M_{\varepsilon}(P)$, we have
$\displaystyle M_{\varepsilon}(P)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}A^{T}&C^{T}\\\
B^{T}&D^{T}\end{array}\right)\left(\begin{array}[]{cc}sI_{k}&-cI_{k}\\\
-cI_{k}&-sI_{k}\end{array}\right)\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right)+\left(\begin{array}[]{cc}sI_{k}&cI_{k}\\\
cI_{k}&-sI_{k}\end{array}\right)$ (2.48) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}A^{T}&C^{T}\\\
B^{T}&D^{T}\end{array}\right)\left(\begin{array}[]{cc}sI_{k}&-2cI_{k}\\\
0&-sI_{k}\end{array}\right)\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right)+\left(\begin{array}[]{cc}sI_{k}&2cI_{k}\\\
0&-sI_{k}\end{array}\right)$ (2.57) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}sA^{T}A-2cA^{T}C-sC^{T}C+sI_{k}&*\\\
sB^{T}A-2cB^{T}C-sD^{T}C&sB^{T}B-2cB^{T}D-sD^{T}D-sI_{k}\end{array}\right)$
(2.60) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}sA^{T}A-2cA^{T}C-sC^{T}C+sI_{k}&sA^{T}B-2cC^{T}B-sC^{T}D\\\
sB^{T}A-2cB^{T}C-sD^{T}C&sB^{T}B-2cB^{T}D-sD^{T}D-sI_{k}\end{array}\right),$
(2.63)
where in the second equality we have used that $P^{T}J_{k}P=J_{k}$, in the
fourth equality we have used that $M_{\varepsilon}(P)$ is a symmetric matrix.
So
$\displaystyle M_{0}(P)=-2\left(\begin{array}[]{cc}A^{T}C&C^{T}B\\\
B^{T}C&B^{T}D\end{array}\right)=-2\left(\begin{array}[]{cc}C^{T}&0\\\
0&B^{T}\end{array}\right)\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right),$ (2.70)
where we have used $A^{T}C$ is symmetric. So if both $B$ and $C$ are
invertible, $M_{0}(P)$ is invertible and symmetric, its signature is invariant
under small perturbation, so (iv) holds.
If $\nu_{L_{0}}(P)=\dim\ker B>0$, since $B^{T}D=D^{T}B$, for any $x\in\ker
B\subseteq{\bf R}^{k}$, $x\neq 0$, and $0<{\varepsilon}\ll 1$, we have
$\displaystyle M_{\varepsilon}(P)\left(\begin{array}[]{c}0\\\
x\end{array}\right)\cdot\left(\begin{array}[]{c}0\\\
x\end{array}\right)=(sB^{T}B-2cD^{T}B-sD^{T}D-sI_{k})x\cdot x$ (2.75)
$\displaystyle=-s(D^{T}D+I_{k})x\cdot x$ $\displaystyle<0.$ (2.76)
So $M_{\varepsilon}(P)$ is negative definite on $(0\oplus\ker B)\subseteq{\bf
R}^{2k}$. Hence $m^{-}(M_{\varepsilon}(p)\geq\dim\ker B$ which yields that
$\frac{1}{2}{\rm sgn}M_{\varepsilon}(P)\leq k-\dim\ker B=k-\nu_{L_{0}}(P)$,
for $0<{\varepsilon}\ll 1$. Thus (i) holds. Similarly we can prove (iii).
If $m^{+}(A^{T}C)=q>0$, let $A^{T}C$ is positive definite on $E\subseteq{\bf
R}^{k}$, then for $0\leq|s|\ll 1$, similar to (2.76) we have
$M_{\varepsilon}(P)$ is negative on $E\oplus 0\subseteq{\bf R}^{2k}$. Hence
$m^{-}(M_{\varepsilon}(P)\geq q$, which yields (2.37).
If $B=0$, by (2.63) we have
$\displaystyle
M_{\varepsilon}(P)=\left(\begin{array}[]{cc}sA^{T}A-2cA^{T}C-sC^{T}C+sI_{k}&-sC^{T}D\\\
-sD^{T}C&-sD^{T}D-sI_{k}\end{array}\right).$ (2.79)
Since
$\displaystyle\left(\begin{array}[]{cc}I_{k}&-C^{T}D(D^{T}D+I_{k})^{-1}\\\
0&I_{k}\end{array}\right)\left(\begin{array}[]{cc}sA^{T}A-2cA^{T}C-sC^{T}C+sI_{k}&-sC^{T}D\\\
-sD^{T}C&-sD^{T}D-sI_{k}\end{array}\right)\cdot$ (2.84)
$\displaystyle\quad\cdot\left(\begin{array}[]{cc}I_{k}&0\\\
-(D^{T}D+I_{k})^{-1}D^{T}C&I_{k}\end{array}\right)$ (2.87)
$\displaystyle=\left(\begin{array}[]{cc}sA^{T}A-2cA^{T}C-sC^{T}C+sI_{k}+sC^{T}D(D^{T}D+I_{k})^{-1}D^{T}C&0\\\
0&-sD^{T}D-sI_{k}\end{array}\right),$ (2.90)
for $0<-s\ll 1$, we have
$m^{-}(M_{\varepsilon}(P))\geq k+m^{+}(A^{T}C)$ (2.91)
which yields (2.38). So (ii) holds and the proof of Lemma 2.5 is complete.
Lemma 2.6. ([23]) For ${\gamma}\in\mathcal{P}_{\tau}(2)$, $b>0$, and
$0<{\varepsilon}\ll 1$ small enough we have
$\displaystyle{\rm sgn}M_{\pm{\varepsilon}}(R(\theta))=0,\quad{\rm
for}\;\theta\in{\bf R},$ $\displaystyle{\rm sgn}M_{\varepsilon}(P)=0,\quad{\rm
if}\;P=\pm\left(\begin{array}[]{cc}1&b\\\ 0&1\end{array}\right)\;{\rm
or}\;\pm\left(\begin{array}[]{cc}1&0\\\ -b&1\end{array}\right),$ (2.96)
$\displaystyle{\rm sgn}M_{\varepsilon}(P)=2,\quad{\rm
if}\;P=\pm\left(\begin{array}[]{cc}1&-b\\\ 0&1\end{array}\right),$ (2.99)
$\displaystyle{\rm sgn}M_{\varepsilon}(P)=-2,\quad{\rm
if}\;P=\pm\left(\begin{array}[]{cc}1&0\\\ b&1\end{array}\right).$ (2.102)
## 3 Proofs of Theorems 1.1 and 1.2.
In this section we prove Theorems 1.1 and 1.2. The proof mainly depends on the
method in [14] and the following
Theorem 3.1. For any odd number $n\geq 3$, $\tau>0$ and ${\gamma}\in{\cal
P}_{\tau}(2n)$, let $P={\gamma}(\tau)$. If $i_{L_{0}}\geq 0$, $i_{L_{1}}\geq
0$, $i({\gamma})\geq n$, ${\gamma}^{2}(t)={\gamma}(t-\tau){\gamma}(\tau)$ for
all $t\in[\tau,2\tau]$, and $P\sim(-I_{2})\diamond Q$ with $Q\in{\rm
Sp}(2n-2)$, then
$i_{L_{1}}({\gamma})+S_{P^{2}}^{+}(1)-\nu_{L_{0}}({\gamma})>\frac{1-n}{2}.$
(3.1)
Proof. If the conclusion of Theorem 3.1 does not hold, then
$i_{L_{1}}({\gamma})+S_{P^{2}}^{+}(1)-\nu_{L_{0}}({\gamma})\leq\frac{1-n}{2}.$
(3.2)
In the following we shall obtain a contradiction from (3.2). Hence (3.1) holds
and Theorem 3.1 is proved.
Since $n\geq 3$ and $n$ is odd, in the following of the proof of Theorem 3.1
we write $n=2p+1$ for some $p\in{\bf N}$. We denote by
$Q=\left(\begin{array}[]{cc}A&B\\\ C&D\end{array}\right)$, where $A,B,C,D$ are
$(n-1)\times(n-1)$ matrices. Then since $Q$ is a symplectic matrix we have
$A^{T}C=C^{T}A,\;B^{T}D=D^{T}B,\;AB^{T}=BA^{T},\;CD^{T}=DC^{T},$ (3.3)
$AD^{T}-BC^{T}=I_{n-1},\quad A^{T}D-C^{T}B=I_{n-1},$ (3.4) $\dim\ker
B=\nu_{L_{0}}({\gamma})-1,\quad\dim\ker C=\nu_{L_{1}}({\gamma})-1.$ (3.5)
Since ${\gamma}^{2}(t)={\gamma}(t-\tau){\gamma}(\tau)$ for all
$t\in[\tau,2\tau]$ we have ${\gamma}^{2}$ is also the twice iteration of
${\gamma}$ in the periodic boundary value case, so by the Bott-type formula
(cf. Theorem 9.2.1 of [16]) and the proof of Lemma 4.1 of [17] we have
$\displaystyle i({\gamma}^{2})+2S_{P^{2}}^{+}(1)-\nu({\gamma}^{2})$ (3.6)
$\displaystyle=$ $\displaystyle
2i({\gamma})+2S_{P}^{+}(1)+\sum_{\theta\in(0,\pi)}(S_{P}^{+}(e^{\sqrt{-1}\theta})$
$\displaystyle-(\sum_{\theta\in(0,\pi)}(S_{P}^{-}(e^{\sqrt{-1}\theta})+(\nu(P)-S_{P}^{-}(1))+(\nu_{-1}(P)-S_{P}^{-}(-1)))$
$\displaystyle\geq$ $\displaystyle 2n+2S_{P}^{+}(1)-n$ $\displaystyle=$
$\displaystyle n+2S_{P}^{+}(1)$ $\displaystyle\geq$ $\displaystyle n,$
where we have used the condition $i({\gamma})\geq n$ and
$S^{+}_{P^{2}}(1)=S^{+}_{P}(1)+S^{+}_{P}(-1)$,
$\nu(\gamma^{2})=\nu(\gamma)+\nu_{-1}(\gamma)$. By by Proposition C of [17]
and Proposition 6.1 of [14] we have
$i_{L_{0}}({\gamma})+i_{L_{1}}({\gamma})=i({\gamma}^{2})-n,\quad\nu_{L_{0}}({\gamma})+\nu_{L_{1}}({\gamma})=\nu({\gamma}^{2}).$
(3.7)
So by (3.6) and (3.7) we have
$\displaystyle(i_{L_{1}}({\gamma})+S_{P^{2}}^{+}(1)-\nu_{L_{0}}({\gamma}))+(i_{L_{0}}({\gamma})+S_{P^{2}}^{+}(1)-\nu_{L_{1}}({\gamma}))$
$\displaystyle=i({\gamma}^{2})+2S_{P^{2}}^{+}(1)-\nu({\gamma}^{2})-n$
$\displaystyle\geq n-n$ $\displaystyle=0.$ (3.8)
By Theorem 2.1 and Lemma 2.6 we have
$\displaystyle(i_{L_{1}}({\gamma})+S_{P^{2}}^{+}(1)-\nu_{L_{0}}({\gamma}))-(i_{L_{0}}({\gamma})+S_{P^{2}}^{+}(1)-\nu_{L_{1}}({\gamma}))$
$\displaystyle=i_{L_{1}}({\gamma})-i_{L_{0}}({\gamma})-\nu_{L_{0}}({\gamma}))+\nu_{L_{1}}({\gamma})$
$\displaystyle=-\frac{1}{2}{\rm sgn}M_{\varepsilon}(Q)-\frac{1}{2}{\rm
sgn}M_{\varepsilon}(-I_{2})$ $\displaystyle=-\frac{1}{2}{\rm
sgn}M_{\varepsilon}(Q)$ $\displaystyle\geq 1-n.$ (3.9)
So by (3.8) and (3.9) we have
$i_{L_{1}}({\gamma})+S_{P^{2}}^{+}(1)-\nu_{L_{0}}({\gamma})\geq\frac{1-n}{2}.$
(3.10)
By (3.2), the inequality of (3.10) must be equality. Then both (3.6) and (3.9)
are equality. So we have
$i({\gamma}^{2})+2S_{P^{2}}^{+}(1)-\nu({\gamma}^{2})=n.$ (3.11)
$i_{L_{1}}({\gamma})+S_{P^{2}}^{+}(1)-\nu_{L_{0}}({\gamma})=\frac{1-n}{2}.$
(3.12)
$i_{L_{0}}({\gamma})+\nu_{L_{0}}({\gamma})-i_{L_{1}}({\gamma})-\nu_{L_{1}}({\gamma})=n-1.$
(3.13)
Thus by (3.6), (3.11), Theorem 1.8.10 of [16], and Lemma 2.1 we have
$\displaystyle P\approx(-I_{2})^{\diamond p_{1}}\diamond N_{1}(1,-1)^{\diamond
p_{2}}\diamond N_{1}(-1,1)^{\diamond p_{3}}\diamond R(\theta_{1})\diamond
R(\theta_{2})\diamond\cdots\diamond R(\theta_{p_{4}}),$
where $p_{j}\geq 0$ for $j=1,2,3,4$, $p_{1}+p_{2}+p_{3}+p_{4}=n$ and
$\theta_{j}\in(0,\pi)$ for $1\leq j\leq p_{4}$. Otherwise by (3.6) and Lemma
2.1 we have $i({\gamma}^{2})+2S_{P^{2}}^{+}(1)-\nu({\gamma}^{2})>n$ which
contradicts to (3.11). So by Remark 2.1, we have
$P^{2}\approx I_{2}^{\diamond p_{1}}\diamond N_{1}(1,-1)^{\diamond
p_{2}}\diamond R(\theta_{1})\diamond R(\theta_{2})\diamond\cdots\diamond
R(\theta_{p_{3}}),$ (3.14)
where $p_{i}\geq 0$ for $1\leq i\leq 3$, $p_{1}+p_{2}+p_{3}=n$ and
$\theta_{j}\in(0,2\pi)$ for $1\leq j\leq p_{3}$.
Note that, since ${\gamma}^{2}(t)={\gamma}(t-\tau){\gamma}(\tau)$, we have
$\displaystyle{\gamma}^{2}(2\tau)={\gamma}(\tau)^{2}=P^{2}.$ (3.15)
By Definition 2.5 we have
$\displaystyle{\gamma}^{2}(2\tau)=N{\gamma}(\tau)^{-1}N{\gamma}(\tau)=NP^{-1}NP.$
(3.16)
So by (3.15) and (3.16) we have
$\displaystyle P^{2}=NP^{-1}NP.$ (3.17)
By (3.17), Lemma 2.4, and $P\sim(-I_{2})\diamond Q$ we have
$\displaystyle P^{2}$ $\displaystyle=$ $\displaystyle NP^{-1}NP$ (3.18)
$\displaystyle\approx$ $\displaystyle N((-I_{2})\diamond
Q)^{-1}N((-I_{2})\diamond Q)$ $\displaystyle=$ $\displaystyle
I_{2}\diamond(N_{n-1}Q^{-1}N_{n-1}Q).$
So by (3.14), we have
$\displaystyle p_{1}\geq 1.$ (3.19)
Also by (3.18) and Lemma 2.5, we have
$\displaystyle P^{2}\approx
I_{2}\diamond(N_{n-1}Q^{\prime-1}N_{n-1}Q^{\prime}),\quad\forall\,Q^{\prime}\sim
Q\;{\rm where}\;Q^{\prime}\in{\rm Sp}(2n-2).$ (3.20)
By (3.14) it is easy to check that
${\rm tr}(P^{2})=2n-2p_{3}+2\sum_{j=1}^{p_{3}}\cos\theta_{j}.$ (3.21)
By (3.11), (3.14) and Lemma 2.1 we have
$\displaystyle
n=i({\gamma}^{2})+2S_{P^{2}}^{+}(1)-\nu({\gamma}^{2})=i({\gamma}^{2})-p_{2}\geq
i({\gamma}^{2})-n+1.$
So
$i({\gamma}^{2})\leq 2n-1.$ (3.22)
By (3.7) we have
$i({\gamma}^{2})=n+i_{L_{0}}({\gamma})+i_{L_{1}}({\gamma}).$ (3.23)
Since $i_{L_{0}}({\gamma})\geq 0$ and $i_{L_{1}}({\gamma})\geq 0$, we have
$n\leq i({\gamma}^{2})\leq 2n-1$. So we can divide the index $i({\gamma}^{2})$
into the following three cases.
Case I. $i({\gamma}^{2})=n$.
In this case, by (3.7), $i_{L_{0}}({\gamma})\geq 0$, and
$i_{L_{1}}({\gamma})\geq 0$, we have
$i_{L_{0}}({\gamma})=0=i_{L_{1}}({\gamma}).$ (3.24)
So by (3.13) we have
$\nu_{L_{0}}({\gamma})-\nu_{L_{1}}({\gamma})=n-1.$ (3.25)
Since $\nu_{L_{1}}({\gamma})\geq 1$ and $\nu_{L_{0}}({\gamma})\leq n$, we have
$\nu_{L_{0}}({\gamma})=n,\quad\nu_{L_{1}}({\gamma})=1.$ (3.26)
By (3.7) we have
$\nu({\gamma}^{2})=\nu(P^{2})=n+1.$ (3.27)
By (3.12), (3.24) and (3.26) we have
$S_{P^{2}}^{+}(1)=\frac{1-n}{2}+n=\frac{1+n}{2}=p+1.$ (3.28)
So by (3.14), (3.27), (3.28), and Lemma 2.1 we have
$P^{2}\approx I_{2}^{\diamond(p+1)}\diamond
R(\theta_{1})\diamond\cdots\diamond R(\theta_{p}),$ (3.29)
where $\theta_{j}\in(0,2\pi)$. By (3.5) and (3.26) we have $B=0$. By (3.18),
(3.3), and (3.4), we have
$\displaystyle P^{2}$ $\displaystyle=$ $\displaystyle NP^{-1}NP\approx
I_{2}\diamond(N_{n-1}Q^{-1}N_{n-1}Q)$ (3.34) $\displaystyle=$ $\displaystyle
I_{2}\diamond\left(\begin{array}[]{cc}D^{T}&0\\\
C^{T}&A^{T}\end{array}\right)\left(\begin{array}[]{cc}A&0\\\
C&D\end{array}\right)$ $\displaystyle=$ $\displaystyle
I_{2}\diamond\left(\begin{array}[]{cc}D^{T}A&0\\\
2C^{T}A&AD^{T}\end{array}\right)$ (3.37) $\displaystyle=$ $\displaystyle
I_{2}\diamond\left(\begin{array}[]{cc}I_{2p}&0\\\
2A^{T}C&I_{2p}\end{array}\right).$ (3.40)
Hence ${\sigma}(P^{2})=\\{1\\}$ which contradicts to (3.29) since $p\geq 1$.
Case II. $i({\gamma}^{2})=n+2k$, where $1\leq k\leq p$.
In this case by (3.7) we have
$\displaystyle i_{L_{0}}({\gamma})+i_{L_{1}}({\gamma})=2k.$
Since $i_{L_{0}}({\gamma})\geq 0$ and $i_{L_{1}}({\gamma})\geq 0$ we can write
$i_{L_{0}}({\gamma})=k+r$ and $i_{L_{1}}({\gamma})=k-r$ for some integer
$-k\leq r\leq k$. Then by (3.13) we have
$n-1\geq\nu_{L_{0}}({\gamma})-\nu_{L_{1}}({\gamma})=n-2r-1.$ (3.41)
Thus $r\geq 0$ and $0\leq r\leq k$.
By Theorem 2.1 and (i) of Lemma 2.5 we have
$2r=i_{L_{0}}({\gamma})-i_{L_{1}}({\gamma})=\frac{1}{2}M_{\varepsilon}(P)\leq
n-\nu_{L_{0}}(P)$ (3.42)
which yields that $\nu_{L_{0}}({\gamma})\leq n-2r$. So by (3.41) and
$\nu_{L_{1}}({\gamma})\geq 1$ we have
$\nu_{L_{0}}({\gamma})=n-2r,\quad\nu_{L_{1}}({\gamma})=1.$ (3.43)
Then by (3.12) we have
$S_{P^{2}}^{+}(1)=(n-2r)+\frac{1-n}{2}-(k-r)=\frac{1+n}{2}-k-r=p+1-k-r.$
(3.44)
Then by (3.14) and $\nu(P^{2})=n-2r+1$ and Lemma 2.1 we have
$P^{2}\approx I_{2}^{\diamond(p+1-k-r)}\diamond N_{1}(1,-1)^{\diamond
2k}\diamond R(\theta_{1})\diamond\cdots\diamond R(\theta_{q}),$ (3.45)
where $q=n-(p+1-k-r)-2k=p+r-k\geq 0$. Then we have the following three
subcases (i)-(iii).
(i) $q=0$.
The only possibility is $k=p$ and $r=0$, in this case $P^{2}\approx
I_{2}\diamond N_{1}(1,-1)^{\diamond 2p}$ and $B=0$. By direct computation we
have
$\displaystyle N_{1}(1,-1)^{\diamond 2p}\approx
N_{2p}Q^{-1}N_{2p}Q=\left(\begin{array}[]{cc}I_{n-1}&0\\\
2A^{T}C&I_{n-1}\end{array}\right).$ (3.48)
Then by Lemma 2.3 we have
$\displaystyle m^{+}(A^{T}C)=2p.$
By (ii) of Lemma 2.5 we have
$\frac{1}{2}{\rm sgn}M_{\varepsilon}(Q)\leq 2p-2p=0,\qquad 0<-{\varepsilon}\ll
1.$ (3.49)
Thus by (3.49) and Theorem 2.1, for $0<-{\varepsilon}\ll 1$ we have,
$\displaystyle(i_{L_{0}}({\gamma})+\nu_{L_{0}}({\gamma}))-(i_{L_{1}}({\gamma})+\nu_{L_{1}}({\gamma}))$
$\displaystyle=$ $\displaystyle\frac{1}{2}{\rm sgn}M_{\varepsilon}(P)$
$\displaystyle=$ $\displaystyle\frac{1}{2}{\rm
sgn}M_{\varepsilon}(I_{2})+\frac{1}{2}M_{\varepsilon}(Q)$ $\displaystyle=$
$\displaystyle 0+\frac{1}{2}M_{\varepsilon}(Q)$ $\displaystyle\leq$
$\displaystyle 0$
which contradicts (3.13).
(ii) $q>0$ and $r=0$.
In this case $\nu_{L_{0}}({\gamma})=n$ and $\nu_{L_{1}}({\gamma})=1$, also we
have $B=0$. By the equality of (3.48) we have
$\displaystyle{\rm tr}\,(P^{2})=2n$
which contradicts to (3.21) with $p_{3}=q>0$.
(iii) $q>0$ and $r>0$.
In this case, by (3.44) we have $r<p$ (otherwise, then $p=r=k$. From (3.19)
there holds $S^{+}_{P^{2}}(1)\geq 1$, so from (3.44) we have $1\leq
S^{+}_{P^{2}}(1)=1-p\leq 0$ a contradiction). Here it is easy to see ${\rm
rank}B=2r$. Then there are two invertible $2p\times 2p$ matrices $U$ and $V$
with ${\rm det}U>0$ and ${\rm det}V>0$ such that
$\displaystyle UBV=\left(\begin{array}[]{cc}I_{2r}&0\\\
0&0\end{array}\right).$ (3.52)
So there holds
$Q\sim\,{\rm diag}(U,(U^{T})^{-1})Q{\rm
diag}((V^{T})^{-1},V)=\left(\begin{array}[]{cccc}A_{1}&B_{1}&I_{2r}&0\\\
C_{1}&D_{1}&0&0\\\ A_{3}&B_{3}&A_{2}&B_{2}\\\
C_{3}&D_{3}&C_{2}&D_{2}\end{array}\right):=Q_{1},$ (3.53)
where for $j=1,2,3$, $A_{j}$ is a $2r\times 2r$ matrix, $D_{j}$ is a
$(2p-2r)\times(2p-2r)$ matrix for $j=1,2,3$, $B_{j}$ is a $2r\times(2p-2r)$
matrix, and $C_{j}$ is $(2p-2r)\times 2r$ matrix. Since $Q_{1}$ is still a
symplectic matrix, we have $Q_{1}^{T}J_{2p}Q_{1}=J_{2p}$, then it is easy to
check that
$C_{1}=0,\;B_{2}=0.$ (3.54)
So
$Q_{1}=\left(\begin{array}[]{cccc}A_{1}&B_{1}&I_{2r}&0\\\ 0&D_{1}&0&0\\\
A_{3}&B_{3}&A_{2}&0\\\ C_{3}&D_{3}&C_{2}&D_{2}\end{array}\right).$ (3.55)
So for the case (iii) of Case II, we have the following 3 subcases 1-3.
Subcase 1. $A_{3}=0$.
In this case since $Q_{1}$ is symplectic, by direct computation we have
$\displaystyle
N_{2p}Q_{1}^{-1}N_{2p}Q_{1}=\left(\begin{array}[]{cccc}I_{2r}&*&*&*\\\
*&I_{2p-2r}&*&*\\\ *&*&I_{2r}&*\\\ *&*&*&I_{2p-2r}\end{array}\right).$ (3.60)
Hence we have
$\displaystyle{\rm tr}(N_{2p}Q_{1}^{-1}N_{2p}Q_{1})=4p.$
Since $Q_{1}\sim Q$, we have
$P\sim(-I_{2})\diamond Q_{1}.$ (3.61)
Then by the proof of Lemma 2.4 we have
$\displaystyle{\rm tr}P^{2}$ $\displaystyle=$ $\displaystyle{\rm
tr}(NP^{-1}NP)$ (3.62) $\displaystyle=$ $\displaystyle{\rm
tr}N((-I_{2})\diamond Q_{1})^{-1}N((-I_{2})\diamond Q_{1})$ $\displaystyle=$
$\displaystyle{\rm tr}\,I_{2}\diamond((N_{2p}Q_{1}^{-1}N_{2p}Q_{1})$
$\displaystyle=$ $\displaystyle 4p+2=2n.$
By (3.21) and $p_{3}=q>0$ we have
${\rm tr}(P^{2})<2n.$ (3.63)
(3.62) and (3.63) yield a contradiction.
Subcase 2. $A_{3}$ is invertible.
By $Q_{1}$ is symplectic we have
$\left(\begin{array}[]{cc}A^{T}_{1}&0\\\
B_{1}^{T}&D_{1}^{T}\end{array}\right)\left(\begin{array}[]{cc}A_{2}&0\\\
C_{2}&D_{2}\end{array}\right)-\left(\begin{array}[]{cc}A_{3}^{T}&C_{3}^{T}\\\
B_{3}^{T}&D_{3}^{T}\end{array}\right)\left(\begin{array}[]{cc}I_{2r}&0\\\
0&0\end{array}\right)=I_{2p}.$ (3.64)
Hence
$D_{1}^{T}D_{2}=I_{2p-2r}.$ (3.65)
By direct computation we have
$\displaystyle\left(\begin{array}[]{cccc}A_{1}&B_{1}&I_{2r}&0\\\
0&D_{1}&0&0\\\ A_{3}&B_{3}&A_{2}&0\\\
C_{3}&D_{3}&C_{2}&D_{2}\end{array}\right)\left(\begin{array}[]{cccc}I_{2r}&-A_{3}^{-1}B_{3}&0&0\\\
0&I_{2p-2r}&0&0\\\ 0&0&I_{2r}&0\\\
0&0&B_{3}^{T}(A_{3}^{T})^{-1}&I_{2p-2r}\end{array}\right)=\left(\begin{array}[]{cccc}A_{1}&\tilde{B_{1}}&I_{2r}&0\\\
0&D_{1}&0&0\\\ A_{3}&0&A_{2}&0\\\
C_{3}&\tilde{D}_{3}&\tilde{C}_{2}&D_{2}\end{array}\right).$ (3.78)
So by (3.65) we have
$\displaystyle\left(\begin{array}[]{cccc}I_{2r}&-\tilde{B}_{1}D_{2}^{T}&0&0\\\
0&I_{2p-2r}&0&0\\\ 0&0&I_{2r}&0\\\
0&0&D_{2}\tilde{B}_{1}^{T}&I_{2p-2r}\end{array}\right)\left(\begin{array}[]{cccc}A_{1}&\tilde{B_{1}}&I_{2r}&0\\\
0&D_{1}&0&0\\\ A_{3}&0&A_{2}&0\\\
C_{3}&\tilde{D}_{3}&\tilde{C}_{2}&D_{2}\end{array}\right)$ (3.87)
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cccc}A_{1}&0&I_{2r}&0\\\
0&D_{1}&0&0\\\ A_{3}&0&A_{2}&0\\\
\tilde{C}_{3}&\tilde{D}_{3}&\hat{C}_{2}&D_{2}\end{array}\right):=Q_{2}.$
(3.92)
Then we have
$Q_{2}\sim Q_{1}\sim Q.$ (3.93)
Since $Q_{2}$ is a symplectic matrix, we have $Q_{2}^{T}J_{2p}Q_{2}=J_{2p}$,
then it is easy to check that
$\tilde{C}_{3}=0,\;\hat{C}_{2}=0.$ (3.94)
Hence we have
$Q_{2}=\left(\begin{array}[]{cc}A_{1}&I_{2r}\\\
A_{3}&A_{2}\end{array}\right)\diamond\left(\begin{array}[]{cc}D_{1}&0\\\
\tilde{D}_{3}&D_{2}\end{array}\right).$ (3.95)
Since
$N_{2p-2r}\left(\begin{array}[]{cc}D_{1}&0\\\
\tilde{D}_{3}&D_{2}\end{array}\right)^{-1}N_{2p-2r}\left(\begin{array}[]{cc}D_{1}&0\\\
\tilde{D}_{3}&D_{2}\end{array}\right)=\left(\begin{array}[]{cc}I_{2p-2r}&0\\\
2D_{1}^{T}\tilde{D}_{3}&I_{2p-2r}\end{array}\right),$ (3.96)
by (3.93), (3.20), and Lemma 2.4, there is a symplectic matrix $W$ such that
$P^{2}\approx I_{2}\diamond W\diamond\left(\begin{array}[]{cc}I_{2p-2r}&0\\\
2D_{1}^{T}\tilde{D}_{3}&I_{2p-2r}\end{array}\right).$ (3.97)
Then by (3.14) and Lemma 2.3, $D_{1}^{T}\tilde{D}_{3}$ is semipositive and
$\displaystyle 1+m^{0}(D_{1}^{T}\tilde{D}_{3})\leq S_{P^{2}}^{+}(1).$
So by (3.44) we have
$m^{0}(D_{1}^{T}\tilde{D}_{3})\leq p+1-k-r-1=p-k-r=(2p-2r)-(p+k-r)\leq
2p-2r-1.$ (3.98)
Since $D_{1}^{T}\tilde{D}_{3}$ is a semipositive $(2p-2r)\times(2p-2r)$
matrix, by (3.98) we have $m^{+}(D_{1}^{T}\tilde{D}_{3})>0$. Then by Theorem
2.1, (ii) of Lemma 2.5 and Lemma 2.6, for $0<-{\varepsilon}\ll 1$ we have
$\displaystyle(i_{L_{0}}({\gamma})+\nu_{L_{0}}({\gamma}))-(i_{L_{1}}({\gamma})+\nu_{L_{1}}({\gamma}))$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(M_{\varepsilon}(-I_{2})+M_{\varepsilon}\left(\left(\begin{array}[]{cc}A_{1}&I_{2r}\\\
A_{3}&A_{2}\end{array}\right)\right)+M_{\varepsilon}\left(\left(\begin{array}[]{cc}D_{1}&0\\\
\tilde{D}_{3}&D_{2}\end{array}\right)\right)\right)$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}(0+4r+2(2p-2r-1))$ $\displaystyle=$ $\displaystyle
2p-1$ $\displaystyle=$ $\displaystyle n-2$ (3.104)
which contradicts to (3.13).
Subcase 3. $A_{3}\neq 0$ and $A_{3}$ is not invertible.
In this case, suppose ${\rm rank}A_{3}={\lambda}$, then $0<{\lambda}<2r$.
There is a invertible $2r\times 2r$ matrix $G$ with ${\rm det}G>0$ such that
$GA_{3}G^{-1}=\left(\begin{array}[]{cc}{\Lambda}&0\\\ 0&0\end{array}\right),$
(3.105)
where ${\Lambda}$ is a ${\lambda}\times{\lambda}$ invertible matrix. Then we
have
$\displaystyle\left(\begin{array}[]{cccc}(G^{T})^{-1}&0&0&0\\\
0&I_{2p-2r}&0&0\\\ 0&0&G&0\\\
0&0&0&I_{2p-2r}\end{array}\right)\left(\begin{array}[]{cccc}A_{1}&B_{1}&I_{2r}&0\\\
0&D_{1}&0&0\\\ A_{3}&B_{3}&A_{2}&0\\\
C_{3}&D_{3}&C_{2}&D_{2}\end{array}\right)\left(\begin{array}[]{cccc}(G)^{-1}&0&0&0\\\
0&I_{2p-2r}&0&0\\\ 0&0&G^{T}&0\\\ 0&0&0&I_{2p-2r}\end{array}\right)$ (3.118)
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}\tilde{A_{1}}&\tilde{B}_{1}&I_{2r}&0\\\
0&D_{1}&0&0\\\ GA_{3}G^{-1}&\tilde{B}_{3}&\tilde{A}_{2}&0\\\
\tilde{C}_{3}&D_{3}&\tilde{C}_{2}&D_{2}\end{array}\right):=Q_{3}.$ (3.123)
By (3.105) we can write $Q_{3}$ as the following block form
$\displaystyle
Q_{3}=\left(\begin{array}[]{cccccc}U_{1}&U_{2}&F_{1}&I_{\lambda}&0&0\\\
U_{3}&U_{4}&F_{2}&0&I_{2r-{\lambda}}&0\\\ 0&0&D_{1}&0&0&0\\\
{\Lambda}&0&E_{1}&W_{1}&W_{2}&0\\\ 0&0&E_{2}&W_{3}&W_{4}&0\\\
G_{1}&G_{2}&D_{3}&K_{1}&K_{2}&D_{2}\end{array}\right).$ (3.130)
Let $R_{1}=\left(\begin{array}[]{ccc}I_{\lambda}&0&0\\\
0&I_{2r-{\lambda}}&0\\\ -G_{1}{\Lambda}^{-1}&0&I_{2p-2r}\end{array}\right)$
and $R_{2}=\left(\begin{array}[]{ccc}I_{\lambda}&0&-{\Lambda}^{-1}E_{1}\\\
0&I_{2r-{\lambda}}&0\\\ 0&0&I_{2p-2r}\end{array}\right)$. By (3.130) we have
$\displaystyle{\rm diag}((R_{1}^{T})^{-1},R_{1})Q_{3}{\rm
diag}(R_{2},(R^{T}_{2})^{-1})=\left(\begin{array}[]{cccccc}U_{1}&U_{2}&\tilde{F}_{1}&I_{\lambda}&0&0\\\
U_{3}&U_{4}&\tilde{F}_{2}&0&I_{2r-{\lambda}}&0\\\ 0&0&D_{1}&0&0&0\\\
{\Lambda}&0&0&W_{1}&W_{2}&0\\\ 0&0&E_{2}&W_{3}&W_{4}&0\\\
0&G_{2}&\tilde{D}_{3}&\tilde{K}_{1}&\tilde{K}_{2}&D_{2}\end{array}\right):=Q_{4}.$
(3.137)
Since $Q_{4}$ is a symplectic matrix we have
$\displaystyle Q_{4}^{T}JQ_{4}=J.$
Then by (3) and direct computation we have $U_{2}=0$, $U_{3}=0$, $W_{2}=0$,
$W_{3}=0$, $\tilde{F}_{1}=0$, $\tilde{K_{1}}=0$, and $U_{1}$, $U_{4}$,
$W_{1}$, $W_{4}$ are all symmetric matrices, and
$\displaystyle U_{4}W_{4}=I_{2r-{\lambda}},$ (3.138) $\displaystyle
D_{1}D_{2}^{T}=I_{2p-2r},$ (3.139) $\displaystyle
U_{4}\tilde{E}_{2}=G_{2}^{T}D_{1},$ (3.140)
So
$\displaystyle Q_{4}=\left(\begin{array}[]{cccccc}U_{1}&0&0&I_{\lambda}&0&0\\\
0&U_{4}&\tilde{F}_{2}&0&I_{2r-{\lambda}}&0\\\ 0&0&D_{1}&0&0&0\\\
{\Lambda}&0&0&W_{1}&0&0\\\ 0&0&\tilde{E}_{2}&0&W_{4}&0\\\
0&G_{2}&\tilde{D}_{3}&0&K_{2}&D_{2}\end{array}\right).$ (3.147)
By (3.138)-(3.140), we have both $\tilde{E}_{2}$ and $G_{2}$ are zero or
nonzero. By definition 2.3 we have $Q_{4}\sim Q_{3}\sim Q$. Then by (3.43),
$\left(\begin{array}[]{ccc}{\Lambda}&0&0\\\ 0&0&\tilde{E}_{2}\\\
0&G_{2}&\tilde{D}_{3}\end{array}\right)$ is invertible. So both
$\tilde{E}_{2}$ and $G_{2}$ are nonzero.
Since $Q_{4}$ is symplectic, by (3.140) we have
$\left(\begin{array}[]{ccc}U_{1}&0&0\\\ 0&U_{4}&\tilde{F}_{2}\\\
0&0&D_{1}\end{array}\right)^{T}\left(\begin{array}[]{ccc}{\Lambda}&0&0\\\
0&0&\tilde{E}_{2}\\\
0&G_{2}&\tilde{D}_{3}\end{array}\right)=\left(\begin{array}[]{ccc}U_{1}{\Lambda}&0&0\\\
0&0&U_{4}\tilde{E}_{2}\\\
0&(U_{4}\tilde{E}_{2})^{T}&D_{1}^{T}\tilde{D}_{3}+\tilde{B}_{2}^{T}\tilde{E}_{2}\end{array}\right)$
(3.148)
which is a symmetric matrix.
Denote by $F=\left(\begin{array}[]{cc}0&U_{4}\tilde{E}_{2}\\\
(U_{4}\tilde{E_{2}})^{T}&D_{1}^{T}\tilde{D}_{3}+\tilde{B}_{2}^{T}\tilde{E}_{2}\end{array}\right)$.
Since $U_{4}\tilde{E}_{2}$ is nonzero, in the following we prove that
$m^{+}(F)\geq 1$.
Note that here $U_{4}\tilde{E}_{2}$ is a $(2r-{\lambda})\times(2p-2r)$ matrix
and $D_{1}^{T}\tilde{D}_{3}+\tilde{B}_{2}^{T}\tilde{E}_{2}$ is a
$(2p-2r)\times(2p-2r)$ matrix. Denote by $U_{4}\tilde{E}_{2}=(e_{ij})$ and
$D_{1}^{T}\tilde{D}_{3}+\tilde{B}_{2}^{T}\tilde{E}_{2}=(d_{ij})$, where
$e_{ij}$ and $d_{ij}$ are elements on the $i$-th row and $j$-th column of the
corresponding matrix. Since $U_{4}\tilde{E}_{2}$ is nonzero, there exist an
$e_{ij}\neq 0$ for some $1\leq i\leq 2r-{\lambda}$ and $1\leq j\leq 2p-2r$.
Let $x=(0,..,0,e_{ij},0,...0)^{T}\in{\bf R}^{2r-{\lambda}}$ whose $i$-th row
is $e_{ij}$ and other rows are all zero, and
$y=(0,...,0,\rho,0,...,0)^{T}\in{\bf R}^{2p-2r}$ whose $j$-th row is $\rho$
and other rows are all zero. Then we have
$\displaystyle F\left(\begin{array}[]{c}x\\\
y\end{array}\right)\cdot\left(\begin{array}[]{c}x\\\ y\end{array}\right)=2\rho
e_{ij}^{2}-\rho^{2}d_{jj}>0$ (3.153)
for $\rho>0$ is small enough. Hence the dimension of positive definite space
of $F$ is at least 1, thus $m^{+}(F)\geq 1$. Then
$m^{+}\left(\left(\begin{array}[]{ccc}U_{1}{\Lambda}&0&0\\\
0&0&U_{4}\tilde{E}_{2}\\\
0&(U_{4}\tilde{E}_{2})^{T}&D_{1}^{T}\tilde{D}_{3}+\tilde{B}_{2}^{T}\tilde{E}_{2}\end{array}\right)\right)=m^{+}({\Lambda})+m^{+}(F)\geq
1.$ (3.154)
Then by (3.148), (3.154) and (ii) of Lemma 2.5, we have
$\frac{1}{2}{\rm sgn}M_{\varepsilon}(Q_{4})\leq 2p-1=n-2,\quad
0<-{\varepsilon}\ll 1.$ (3.155)
Since $Q\sim Q_{4}$, by (3.155) and Lemma 2.4 we have
$\frac{1}{2}{\rm sgn}M_{\varepsilon}(Q)\leq 2p-1,0<-{\varepsilon}\ll 1.$
(3.156)
Then since $P\sim(-I_{2})\diamond Q$, by Theorem 2.1, Remark 2.2 and Lemma 2.4
we have
$\displaystyle(i_{L_{0}}({\gamma})+\nu_{L_{0}}({\gamma}))-(i_{L_{1}}({\gamma})+\nu_{L_{1}}({\gamma}))$
(3.157) $\displaystyle=$ $\displaystyle\frac{1}{2}M_{\varepsilon}(P)$
$\displaystyle=$ $\displaystyle\frac{1}{2}{\rm
sgn}M_{\varepsilon}((-I_{2})\diamond Q)$ $\displaystyle=$
$\displaystyle\frac{1}{2}{\rm sgn}M_{\varepsilon}(-I_{2})+\frac{1}{2}{\rm
sgn}M_{\varepsilon}(Q)$ $\displaystyle=$ $\displaystyle 0+\frac{1}{2}{\rm
sgn}M_{\varepsilon}(Q)$ $\displaystyle\leq$ $\displaystyle n-2.$
Thus (3.13) and (3.157) yields a contradiction. And in Case II we can always
obtain a contradiction.
Case III. $i({\gamma}^{2})=n+2k+1$, where $0\leq k\leq p-1$.
In this case by (3.7) we have
$i_{L_{0}}({\gamma})+i_{L_{1}}({\gamma})=2k+1.$ (3.158)
Since $i_{L_{0}}({\gamma})\geq 0$ and $i_{L_{1}}({\gamma})\geq 0$ we can write
$i_{L_{0}}({\gamma})=k+1+r$ and $i_{L_{1}}({\gamma})=k-r$ for some integer
$-k\leq r\leq k$. Then by (3.13) we have
$n-1\geq\nu_{L_{0}}({\gamma})-\nu_{L_{1}}({\gamma})=n-2r-2.$ (3.159)
Thus $r\geq 0$ and $0\leq r\leq k$.
By Theorem 2.1 and (i) of Lemma 2.5 we have
$2r+1=i_{L_{0}}({\gamma})-i_{L_{1}}({\gamma})=\frac{1}{2}M_{\varepsilon}(P)\leq
n-\nu_{L_{0}}({\gamma})$ (3.160)
which yields $\nu_{L_{0}}({\gamma})\leq n-2r-1$. Then by (3.159) and
$\nu_{L_{1}}({\gamma})\geq 1$ we have
$\nu_{L_{0}}({\gamma})=n-2r-1,\quad\nu_{L_{1}}({\gamma})=1.$ (3.161)
Then by (3.12) we have
$S_{P^{2}}^{+}(1)=(n-2r-1)+\frac{1-n}{2}-(k-r)=\frac{1+n}{2}-k-r-1=p-k-r\geq
1.$ (3.162)
Then by (3.14) and
$\nu(P^{2})=\nu_{L_{0}}({\gamma})+\nu_{L_{1}}({\gamma})=n-2r$ and Lemma 2.1 we
have
$\displaystyle P^{2}\approx I_{2}^{\diamond(p-k-r)}\diamond
N_{1}(1,-1)^{\diamond(2k+1)}\diamond R(\theta_{1})\diamond\cdots\diamond
R(\theta_{q}),$
where $q=n-(p-k-r)-(2k+1)=p+r-k\geq p-k\geq 1$.
Since in this case ${\rm rank}B=2r+1\leq n-2$, by the same argument of (iii)
in Case II, we have
$\displaystyle Q\sim
Q_{1}=\left(\begin{array}[]{cccc}A_{1}&B_{1}&I_{2r+1}&0\\\ 0&D_{1}&0&0\\\
A_{3}&B_{3}&A_{2}&0\\\ C_{3}&D_{3}&C_{2}&D_{2}\end{array}\right).$ (3.167)
Then by the same argument of Subcases 1, 2, 3 of Case II, we can always obtain
a contradiction in Case III. The proof of Theorem 3.1 is complete.
Now we are ready to give a proof of Theorem 1.1. For
${\Sigma}\in\mathcal{H}_{b}^{s,c}(2n)$, let
$j_{\Sigma}:{\Sigma}\rightarrow[0,+\infty)$ be the gauge function of
${\Sigma}$ defined by
$\displaystyle j_{{\Sigma}}(0)=0,\quad{\rm and}\quad
j_{\Sigma}(x)=\inf\\{\lambda>0\mid\frac{x}{\lambda}\in C\\},\quad\forall
x\in{\bf R}^{2n}\setminus\\{0\\},$
where $C$ is the domain enclosed by ${\Sigma}$.
Define
$\displaystyle H_{\alpha}(x)=(j_{\Sigma}(x))^{\alpha},\;\alpha>1,\quad
H_{\Sigma}(x)=H_{2}(x),\;\forall x\in{\bf R}^{2n}.$ (3.168)
Then $H_{\Sigma}\in C^{2}({\bf R}^{2n}\backslash\\{0\\},{\bf R})\cap
C^{1,1}({\bf R}^{2n},{\bf R})$.
We consider the following fixed energy problem
$\displaystyle\dot{x}(t)$ $\displaystyle=$ $\displaystyle
JH_{\Sigma}^{\prime}(x(t)),$ (3.169) $\displaystyle H_{\Sigma}(x(t))$
$\displaystyle=$ $\displaystyle 1,$ (3.170) $\displaystyle x(-t)$
$\displaystyle=$ $\displaystyle Nx(t),$ (3.171) $\displaystyle x(\tau+t)$
$\displaystyle=$ $\displaystyle x(t),\quad\forall\,t\in{\bf R}.$ (3.172)
Denote by $\mathcal{J}_{b}({\Sigma},2)\;(\mathcal{J}_{b}({\Sigma},\alpha)$ for
$\alpha=2$ in (3.168)) the set of all solutions $(\tau,x)$ of problem
(3.169)-(3.172) and by $\tilde{\mathcal{J}}_{b}({\Sigma},2)$ the set of all
geometrically distinct solutions of (3.169)-(3.172). By Remark 1.2 of [14] or
discussion in [17], elements in $\mathcal{J}_{b}({\Sigma})$ and
$\mathcal{J}_{b}({\Sigma},2)$ are one to one correspondent. So we have
${}^{\\#}\tilde{\mathcal{J}}_{b}({\Sigma})$=${}^{\\#}\tilde{\mathcal{J}}_{b}({\Sigma},2)$.
For readers’ convenience in the following we list some known results which
will be used in the proof of Theorem 1.1. In the following of this paper, we
write
$(i_{L_{0}}(\gamma,k),\nu_{L_{0}}(\gamma,k))=(i_{L_{0}}(\gamma^{k}),\nu_{L_{0}}(\gamma^{k}))$
for any symplectic path $\gamma\in\mathcal{P}_{{\tau}}(2n)$ and $k\in{\bf N}$,
where ${\gamma}^{k}$ is defined by Definition 2.5. We have
Lemma 3.1. (Theorem 1.5 and of [14] and Theorem 4.3 of [18]) Let
${\gamma}_{j}\in\mathcal{P}_{{\tau_{j}}}(2n)$ for $j=1,\cdots,q$. Let
$M_{j}={\gamma}^{2}_{j}(2\tau_{j})=N{\gamma}_{j}(\tau_{j})^{-1}N{\gamma}_{j}(\tau_{j})$,
for $j=1,\cdots,q$. Suppose
$\displaystyle\hat{i}_{L_{0}}({\gamma}_{j})>0,\quad j=1,\cdots,q.$
Then there exist infinitely many $(R,m_{1},m_{2},\cdots,m_{q})\in{\bf
N}^{q+1}$ such that
(i) $\nu_{L_{0}}({\gamma}_{j},2m_{j}\pm 1)=\nu_{L_{0}}({\gamma}_{j})$,
(ii)
$i_{L_{0}}({\gamma}_{j},2m_{j}-1)+\nu_{L_{0}}({\gamma}_{j},2m_{j}-1)=R-(i_{L_{1}}({\gamma}_{j})+n+S_{M_{j}}^{+}(1)-\nu_{L_{0}}({\gamma}_{j}))$,
(iii) $i_{L_{0}}({\gamma}_{j},2m_{j}+1)=R+i_{L_{0}}({\gamma}_{j})$.
and (iv) $\nu({\gamma}_{j}^{2},2m_{j}\pm 1)=\nu({\gamma}_{j}^{2})$,
(v)
$i({\gamma}_{j}^{2},2m_{j}-1)+\nu({\gamma}_{j}^{2},2m_{j}-1)=2R-(i({\gamma}_{j}^{2})+2S_{M_{j}}^{+}(1)-\nu({\gamma}_{j}^{2}))$,
(vi) $i({\gamma}_{j}^{2},2m_{j}+1)=2R+i({\gamma}_{j}^{2})$,
where we have set
$i({\gamma}_{j}^{2},n_{j})=i({\gamma}_{j}^{2n_{j}},[0,2n_{j}\tau_{j}])$,
$\nu({\gamma}_{j}^{2},n_{j})=\nu({\gamma}_{j}^{2n_{j}},[0,2n_{j}\tau_{j}])$
for $n_{j}\in{\bf N}$.
Lemma 3.2 (Lemma 1.1 of [14]) Let $(\tau,x)\in\mathcal{J}_{b}({\Sigma},2)$ be
symmetric in the sense that $x(t+\frac{\tau}{2})=-x(t)$ for all $t\in{\bf R}$
and ${\gamma}$ be the associated symplectic path of $(\tau,x)$. Set
$M={\gamma}(\frac{\tau}{2})$. Then there is a continuous symplectic path
$\displaystyle\Psi(s)=P(s)MP(s)^{-1},\quad s\in[0,1]$
such that
$\displaystyle\Psi(0)=M,\qquad\Psi(1)=(-I_{2})\diamond\tilde{M},\;\;\;\;\tilde{M}\in{\rm
Sp}(2n-2),$
$\displaystyle\nu_{1}(\Psi(s))=\nu_{1}(M),\quad\nu_{2}(\Psi(s))=\nu_{2}(M),\quad\forall\;s\in[0,1],$
where $P(s)=\left(\begin{array}[]{cc}\psi(s)^{-1}&0\\\
0&\psi(s)^{T}\end{array}\right)$ and $\psi$ is a continuous $n\times n$ matrix
path with ${\rm det}\psi(s)>0$ for all $s\in[0,1]$.
For any $(\tau,x)\in\mathcal{J}_{b}({\Sigma},2)$ and $m\in{\bf N}$, as in [14]
we denote by $i_{L_{j}}(x,m)=i_{L_{j}}({\gamma}_{x}^{m},[0,\frac{m\tau}{2}])$
and $\nu_{L_{j}}(x,m)=\nu_{L_{j}}({\gamma}_{x}^{m},[0,\frac{m\tau}{2}])$ for
$j=0,1$ respectively. Also we denote by
$i(x,m)=i({\gamma}_{x}^{2m},[0,m\tau])$ and
$\nu(x,m)=\nu({\gamma}_{x}^{2m},[0,m\tau])$. If $m=1$, we denote by
$i(x)=i(x,1)$ and $\nu(x)=\nu(x,1)$. By Lemma 6.3 of [14] we have
Lemma 3.3. Suppose ${}^{\\#}\tilde{\mathcal{J}}_{b}({\Sigma})<+\infty$. Then
there exist an integer $K\geq 0$ and an injection map $\phi:{\bf
N}+K\mapsto\mathcal{J}_{b}({\Sigma},2)\times{\bf N}$ such that
(i) For any $k\in{\bf N}+K$, $[(\tau,x)]\in\mathcal{J}_{b}({\Sigma},2)$ and
$m\in{\bf N}$ satisfying $\phi(k)=([(\tau\;,x)],m)$, there holds
$i_{L_{0}}(x,m)\leq k-1\leq i_{L_{0}}(x,m)+\nu_{L_{0}}(x,m)-1,$
where $x$ has minimal period $\tau$.
(ii) For any $k_{j}\in{\bf N}+K$, $k_{1}<k_{2}$,
$(\tau_{j},x_{j})\in\mathcal{J}_{b}({\Sigma},2)$ satisfying
$\phi(k_{j})=([(\tau_{j}\;,x_{j})],m_{j})$ with $j=1,2$ and
$[(\tau_{1}\;,x_{1})]=[(\tau_{2}\;,x_{2})]$, there holds
$m_{1}<m_{2}.$
Lemma 3.4. (Lemma 7.2 of [14]) Let ${\gamma}\in{\cal P}_{\tau}(2n)$ be
extended to $[0,+\infty)$ by ${\gamma}(\tau+t)={\gamma}(t){\gamma}(\tau)$ for
all $t>0$. Suppose ${\gamma}(\tau)=M=P^{-1}(I_{2}\diamond\tilde{M})P$ with
$\tilde{M}\in{\rm Sp}(2n-2)$ and $i({\gamma})\geq n$. Then we have
$\displaystyle i({\gamma},2)+2S_{M^{2}}^{+}(1)-\nu({\gamma},2)\geq n+2.$
Lemma 3.5 (Lemma 7.3 of [14]) For any $(\tau,x)\in\mathcal{J}_{b}({\Sigma},2)$
and $m\in{\bf N}$, we have
$\displaystyle i_{L_{0}}(x,m+1)-i_{L_{0}}(x,m)$ $\displaystyle\geq$
$\displaystyle 1,$ $\displaystyle i_{L_{0}}(x,m+1)+\nu_{L_{0}}(x,m+1)-1$
$\displaystyle\geq$ $\displaystyle
i_{L_{0}}(x,m+1)>i_{L_{0}}(x,m)+\nu_{L_{0}}(x,m)-1.$
Proof of Theorem 1.1. By Theorem 1.1 of [14] we have ${}^{\\#}\tilde{{\cal
J}}_{b}({\Sigma})\geq\left[\frac{n}{2}\right]+1$ for $n\in{\bf N}$. So we only
need to prove Theorem q.q for the case $n\geq 3$ and $n$ is odd. The method of
the proof is similar as that of [14].
It is suffices to consider the case
${}^{\\#}\tilde{\mathcal{J}}_{b}({\Sigma})<+\infty$. Since
$-{\Sigma}={\Sigma}$, for $(\tau,x)\in\mathcal{J}_{b}({\Sigma},2)$ we have
$\displaystyle H_{\Sigma}(x)=H_{\Sigma}(-x),$ $\displaystyle
H_{\Sigma}^{\prime}(x)=-H_{\Sigma}^{\prime}(-x),$ $\displaystyle
H_{\Sigma}^{\prime\prime}(x)=H_{\Sigma}^{\prime\prime}(-x).$ (3.173)
So $(\tau,-x)\in\mathcal{J}_{b}({\Sigma},2)$. By (3.173) and the definition of
${\gamma}_{x}$ we have that
$\displaystyle{\gamma}_{x}={\gamma}_{-x}.$
So we have
$\displaystyle(i_{L_{0}}(x,m),\nu_{L_{0}}(x,m))=(i_{L_{0}}(-x,m),\nu_{L_{0}}(-x,m)),$
$\displaystyle(i_{L_{1}}(x,m),\nu_{L_{1}}(x,m))=(i_{L_{1}}(-x,m),\nu_{L_{1}}(-x,m)),\quad\forall
m\in{\bf N}.$ (3.174)
So we can write
$\tilde{\mathcal{J}}_{b}({\Sigma},2)=\\{[(\tau_{j},x_{j})]|j=1,\cdots,p\\}\cup\\{[(\tau_{k},x_{k})],[(\tau_{k},-x_{k})]|k=p+1,\cdots,p+q\\}.$
(3.175)
with $x_{j}({\bf R})=-x_{j}({\bf R})$ for $j=1,\cdots,p$ and $x_{k}({\bf
R})\neq-x_{k}({\bf R})$ for $k=p+1,\cdots,p+q$. Here we remind that
$(\tau_{j},x_{j})$ has minimal period $\tau_{j}$ for $j=1,\cdots,p+q$ and
$x_{j}(\frac{\tau_{j}}{2}+t)=-x_{j}(t),\;t\in{\bf R}$ for $j=1,\cdots,p$.
By Lemma 3.3 we have an integer $K\geq 0$ and an injection map $\phi:{\bf
N}+K\to\mathcal{J}_{b}({\Sigma},2)\times{\bf N}$. By (3.174),
$(\tau_{k},x_{k})$ and $(\tau_{k},-x_{k})$ have the same
$(i_{L_{0}},\nu_{L_{0}})$-indices. So by Lemma 3.3, without loss of
generality, we can further require that
$\displaystyle{\rm
Im}(\phi)\subseteq\\{[(\tau_{k},x_{k})]|k=1,2,\cdots,p+q\\}\times{\bf N}.$
(3.176)
By the strict convexity of $H_{\Sigma}$ and (6.19) of [14]), we have
$\displaystyle\hat{i}_{L_{0}}(x_{k})>0,\quad k=1,2,\cdots,p+q.$
Applying Lemma 3.1 to the following associated symplectic paths
${\gamma}_{1},\;\cdots,\;{\gamma}_{p+q},\;{\gamma}_{p+q+1},\;\cdots,\;{\gamma}_{p+2q}$
of
$(\tau_{1},x_{1}),\;\cdots,\;(\tau_{p+q},x_{p+q}),\;(2\tau_{p+1},x_{p+1}^{2}),\;\cdots,\;(2\tau_{p+q},x_{p+q}^{2})$
respectively, there exists a vector $(R,m_{1},\cdots,m_{p+2q})\in{\bf
N}^{p+2q+1}$ such that $R>K+n$ and
$\displaystyle i_{L_{0}}(x_{k},2m_{k}+1)=R+i_{L_{0}}(x_{k}),$ $\displaystyle
i_{L_{0}}(x_{k},2m_{k}-1)+\nu_{L_{0}}(x_{k},2m_{k}-1)$ $\displaystyle=$
$\displaystyle R-(i_{L_{1}}(x_{k})+n+S_{M_{k}}^{+}(1)-\nu_{L_{0}}(x_{k})),$
(3.178)
for $k=1,\cdots,p+q,$ $M_{k}={\gamma}_{k}^{2}(\tau_{k})$, and
$\displaystyle i_{L_{0}}(x_{k},4m_{k}+2)=R+i_{L_{0}}(x_{k},2),$ $\displaystyle
i_{L_{0}}(x_{k},4m_{k}-2)+\nu_{L_{0}}(x_{k},4m_{k}-2)$ $\displaystyle=$
$\displaystyle
R-(i_{L_{1}}(x_{k},2)+n+S_{M_{k}}^{+}(1)-\nu_{L_{0}}(x_{k},2)),$ (3.180)
for $k=p+q+1,\cdots,p+2q$ and
$M_{k}={\gamma}_{k}^{4}(2\tau_{k})={\gamma}_{k}^{2}(\tau_{k})^{2}$.
By Lemma 3.1, we also have
$\displaystyle i(x_{k},2m_{k}+1)$ $\displaystyle=$ $\displaystyle
2R+i(x_{k}),$ (3.181) $\displaystyle i(x_{k},2m_{k}-1)+\nu(x_{k},2m_{k}-1)$
$\displaystyle=$ $\displaystyle 2R-(i(x_{k})+2S_{M_{k}}^{+}(1)-\nu(x_{k})),$
(3.182)
for $k=1,\cdots,p+q,$ $M_{k}={\gamma}_{k}^{2}(\tau_{k})$, and
$\displaystyle i(x_{k},4m_{k}+2)$ $\displaystyle=$ $\displaystyle
2R+i(x_{k},2),$ (3.183) $\displaystyle i(x_{k},4m_{k}-2)+\nu(x_{k},4m_{k}-2)$
$\displaystyle=$ $\displaystyle
2R-(i(x_{k},2)+2S_{M_{k}}^{+}(1)-\nu(x_{k},2)),$ (3.184)
for $k=p+q+1,\cdots,p+2q$ and
$M_{k}={\gamma}_{k}^{4}(2\tau_{k})={\gamma}_{k}^{2}(\tau_{k})^{2}$.
From (3.176), we can set
$\displaystyle\phi(R-(s-1))=([(\tau_{k(s)},x_{k(s)})],m(s)),\qquad\forall s\in
S:=\left\\{1,2,\cdots,\left[\frac{n+1}{2}\right]+1\right\\},$
where $k(s)\in\\{1,2,\cdots,p+q\\}$ and $m(s)\in{\bf N}$.
We continue our proof to study the symmetric and asymmetric orbits separately.
Let
$\displaystyle S_{1}=\\{s\in S|k(s)\leq p\\},\qquad S_{2}=S\setminus S_{1}.$
We shall prove that ${}^{\\#}S_{1}\leq p$ and ${}^{\\#}S_{2}\leq 2q$, together
with the definitions of $S_{1}$ and $S_{2}$, these yield Theorem 1.1.
Claim 1. ${}^{\\#}S_{1}\leq p$.
Proof of Claim 1. By the definition of $S_{1}$,
$([(\tau_{k(s)},x_{k(s)})],m(s))$ is symmetric when $k(s)\leq p$. We further
prove that $m(s)=2m_{k(s)}$ for $s\in S_{1}$.
In fact, by the definition of $\phi$ and Lemma 3.3, for all
$s=1,2,\cdots,\left[\frac{n+1}{2}\right]+1$ we have
$\displaystyle i_{L_{0}}(x_{k(s)},m(s))$ $\displaystyle\leq$
$\displaystyle(R-(s-1))-1=R-s$ (3.185) $\displaystyle\leq$ $\displaystyle
i_{L_{0}}(x_{k(s)},m(s))+\nu_{L_{0}}(x_{k(s)},m(s))-1.$
By the strict convexity of $H_{\Sigma}$ and Lemma 2.2, we have
$i_{L_{0}}(x_{k(s)})\geq 0$, so there holds
$\displaystyle i_{L_{0}}(x_{k(s)},m(s))\leq R-s<R\leq
R+i_{L_{0}}(x_{k(s)})=i_{L_{0}}(x_{k(s)},2m_{k(s)}+1),$ (3.186)
for every $s=1,2,\cdots,\left[\frac{n+1}{2}\right]+1$, where we have used (3)
in the last equality. Note that the proofs of (3.185) and (3.186) do not
depend on the condition $s\in S_{1}$.
By Lemma 3.2, ${\gamma}_{x_{k}}$ satisfies conditions of Theorem 3.1 with
$\tau=\frac{\tau_{k}}{2}$. Note that by definition
$i_{L_{1}}(x_{k})=i_{L_{1}}({\gamma}_{x_{k}})$ and
$\nu_{L_{0}}(x_{k})=\nu_{L_{0}}({\gamma}_{x_{k}})$. So by Theorem 3.1 we have
$i_{L_{1}}(x_{k})+S_{M_{k}}^{+}(1)-\nu_{L_{0}}(x_{k})>\frac{1-n}{2},\quad\forall
k=1,\cdots,p.$ (3.187)
Also for $1\leq s\leq\left[\frac{n+1}{2}\right]+1$, we have
$-\frac{n+3}{2}=-\left(\left[\frac{n+1}{2}\right]+1\right)\leq-s.$ (3.188)
Hence by (3.185),(3.187) and(3.188), if $k(s)\leq p$ we have
$\displaystyle
i_{L_{0}}(x_{k(s)},2m_{k(s)}-1)+\nu_{L_{0}}(x_{k(s)},2m_{k(s)}-1)-1$ (3.189)
$\displaystyle=$ $\displaystyle
R-(i_{L_{1}}(x_{k(s)})+n+S_{M_{k(s)}}^{+}(1)-\nu_{L_{0}}(x_{k(s)}))-1$
$\displaystyle<$ $\displaystyle R-\frac{1-n}{2}-1-n=R-\frac{n+3}{2}\leq R-s$
$\displaystyle\leq$ $\displaystyle
i_{L_{0}}(x_{k(s)},m(s))+\nu_{L_{0}}(x_{k(s)},m(s))-1.$
Thus by (3.186) and (3.189) and Lemma 3.5 of [14] we have
$2m_{k(s)}-1<m(s)<2m_{k(s)}+1.$ (3.190)
Hence
$m(s)=2m_{k(s)}.$ (3.191)
So we have
$\phi(R-s+1)=([(\tau_{k(s)},x_{k(s)})],2m_{k(s)}),\qquad\forall s\in S_{1}.$
(3.192)
Then by the injectivity of $\phi$, it induces another injection map
$\phi_{1}:S_{1}\rightarrow\\{1,\cdots,p\\},\;s\mapsto k(s).$ (3.193)
There for ${}^{\\#}S_{1}\leq p$. Claim 1 is proved.
Claim 2. ${}^{\\#}S_{2}\leq 2q$.
Proof of Claim 2. By the formulas (3.181)-(3.184), and (59) of [13] (also
Claim 4 on p. 352 of [16]), we have
$m_{k}=2m_{k+q}\quad{\rm for}\;\;k=p+1,p+2,\cdots,p+q.$ (3.194)
We set
$\mathcal{A}_{k}=i_{L_{1}}(x_{k},2)+S_{M_{k}}^{+}(1)-\nu_{L_{0}}(x_{k},2)$ and
$\mathcal{B}_{k}=i_{L_{0}}(x_{k},2)+S_{M_{k}}^{+}(1)-\nu_{L_{1}}(x_{k},2)$,
$p+1\leq k\leq p+q$, where
$M_{k}={\gamma}_{k}(2\tau_{k})={\gamma}(\tau_{k})^{2}$. By (3.7), we have
$\mathcal{A}_{k}+\mathcal{B}_{k}=i(x_{k},2)+2S_{M_{k}}^{+}(1)-\nu(x_{k},2)-n,\;\;\;p+1\leq
k\leq p+q.$ (3.195)
By similar discussion of the proof of Lemma 3.2, for any $p+1\leq k\leq p+q$
there exist $P_{k}\in{\rm Sp}(2n)$ and $\tilde{M}_{k}\in{\rm Sp}(2n-2)$ such
that
$\displaystyle{\gamma}(\tau_{k})=P_{k}^{-1}(I_{2}\diamond\tilde{M}_{k})P_{k}.$
Hence by Lemma 3.4 and (3.195), we have
$\mathcal{A}_{k}+\mathcal{B}_{k}\geq n+2-n=2.$ (3.196)
By Theorem 2.1, there holds
$\displaystyle|\mathcal{A}_{k}-\mathcal{B}_{k}|$ $\displaystyle=$
$\displaystyle|(i_{L_{0}}(x_{k},2)+\nu_{L_{0}}(x_{k},2))-(i_{L_{1}}(x_{k},2)+\nu_{L_{1}}(x_{k},2))|\leq
n.$ (3.197)
So by (3.196) and (3.197) we have
$\mathcal{A}_{k}\geq\frac{1}{2}((\mathcal{A}_{k}+\mathcal{B}_{k})-|\mathcal{A}_{k}-\mathcal{B}_{k}|)\geq\frac{2-n}{2},\quad
p+1\leq k\leq p+q.$ (3.198)
By (3.180), (3.185), (3.188), (3.194) and (3.198), for $p+1\leq k(s)\leq p+q$
we have
$\displaystyle
i_{L_{0}}(x_{k(s)},2m_{k(s)}-2)+\nu_{L_{0}}(x_{k(s)},2m_{k(s)}-2)-1$ (3.199)
$\displaystyle=$ $\displaystyle
i_{L_{0}}(x_{k(s)},4m_{k(s)+q}-2)+\nu_{L_{0}}(x_{k(s)},4m_{k(s)+q}-2)-1$
$\displaystyle=$ $\displaystyle
R-(i_{L_{1}}(x_{k(s)},2)+n+S_{M_{k(s)}}^{+}(1)-\nu_{L_{0}}(x_{k(s)},2))-1$
$\displaystyle=$ $\displaystyle R-\mathcal{A}_{k(s)}-1-n$ $\displaystyle\leq$
$\displaystyle R-\frac{2-n}{2}-1-n$ $\displaystyle=$ $\displaystyle
R-(2+\frac{n}{2})$ $\displaystyle<$ $\displaystyle R-\frac{n+3}{2}$
$\displaystyle\leq$ $\displaystyle R-s$ $\displaystyle\leq$ $\displaystyle
i_{L_{0}}(x_{k(s)},m(s))+\nu_{L_{0}}(x_{k(s)},m(s))-1.$
Thus by (3.186), (3.199) and Lemma 3.5, we have
$\displaystyle 2m_{k(s)}-2<m(s)<2m_{k(s)}+1,\qquad p<k(s)\leq p+q.$
So
$\displaystyle m(s)\in\\{2m_{k(s)}-1,2m_{k(s)}\\},\qquad{\rm
for}\;\;p<k(s)\leq p+q.\\}$
Especially this yields that for any $s_{0}$ and $s\in S_{2}$, if
$k(s)=k(s_{0})$, then
$\displaystyle
m(s)\in\\{2m_{k(s)}-1,2m_{k(s)}\\}=\\{2m_{k(s_{0})}-1,2m_{k(s_{0})}\\}.$
Thus by the injectivity of the map $\phi$ from Lemma 3.3, we have
${}^{\\#}\\{s\in S_{2}|k(s)=k(s_{0})\\}\leq 2$
which yields Claim 2.
By Claim 1 and Claim 2, we have
${}^{\\#}\tilde{\mathcal{J}}_{b}({\Sigma})=^{\\#}\tilde{\mathcal{J}}_{b}({\Sigma},2)=p+2q\geq^{\\#}S_{1}+^{\\#}S_{2}=\left[\frac{n+1}{2}\right]+1.$
The proof of Theorem 1.1 is complete.
Proof of Theorem 1.2. By [13], there are at least $n$ closed characteristics
on every $C^{2}$ compact convex central symmetric hypersurface ${\Sigma}$ of
${\bf R}^{2n}$. Hence by Example 1.1 the assumption of Theorem 1.2 is
reasonable. Here we prove the case $n=5$, the proof of the case $n=4$ is the
same.
We call a closed characteristic $x$ on ${\Sigma}$ a dual brake orbit on
${\Sigma}$ if $x(-t)=-Nx(t)$. Then by the similar proof of Lemma 3.1 of [22],
a closed characteristic $x$ on ${\Sigma}$ can became a dual brake orbit after
suitable time translation if and only if $x({\bf R})=-Nx({\bf R})$. So by
Lemma 3.1 of [22] again, if a closed characteristic $x$ on ${\Sigma}$ can both
became brake orbits and dual brake orbits after suitable translation, then
$x({\bf R})=Nx({\bf R})=-Nx({\bf R})$, Thus $x({\bf R})=-x({\bf R})$.
Since we also have $-N{\Sigma}={\Sigma}$, $(-N)^{2}=I_{2n}$ and
$(-N)J=-J(-N)$, dually by the same proof of Theorem 1.1, there are at least
$[(n+1)/2]+1=4$ geometrically distinct dual brake orbits on ${\Sigma}$.
If there are exactly 5 closed characteristics on ${\Sigma}$. By Theorem 1.1,
four closed characteristics of them must be brake orbits after suitable time
translation, then the fifth, say $y$, must be brake orbits after suitable time
translation, otherwise $Ny(-\cdot)$ will be the sixth geometrically distinct
closed characteristic on ${\Sigma}$ which yields a contradiction. Hence all
closed characteristics on ${\Sigma}$ must be brake orbits on ${\Sigma}$. By
the same argument we can prove that all closed characteristics on ${\Sigma}$
must be dual brake orbits on ${\Sigma}$. Then by the argument in the second
paragraph of the proof of this theorem, all these five closed characteristics
on ${\Sigma}$ must be symmetric. Hence all of them bust be symmetric brake
orbits after suitable time translation. Thus we have proved the case $n=5$ of
Theorem 1.2 and the proof of Theorem 1.2 is complete.
## References
* [1] A. Ambrosetti, V. Benci, Y. Long, A note on the existence of multiple brake orbits. Nonlinear Anal. T. M. A., 21 (1993) 643-649.
* [2] V. Benci, Closed geodesics for the Jacobi metric and periodic solutions of prescribed energy of natural Hamiltonian systems. Ann. I. H. P. Analyse Nonl. 1 (1984) 401-412.
* [3] V. Benci, F. Giannoni, A new proof of the existence of a brake orbit. In “Advanced Topics in the Theory of Dynamical Systems”. Notes Rep. Math. Sci. Eng. 6 (1989) 37-49.
* [4] S. Bolotin, Libration motions of natural dynamical systems. Vestnik Moskov Univ. Ser. I. Mat. Mekh. 6 (1978) 72-77 (in Russian).
* [5] S. Bolotin, V.V. Kozlov, Librations with many degrees of freedom. J. Appl. Math. Mech. 42 (1978) 245-250 (in Russian).
* [6] S. E. Cappell, R. Lee, E. Y. Miller, On the Maslov-type index. Comm. Pure Appl. Math., 47 (1994) 121-186.
* [7] I. Ekeland, Convexity Methods in Hamiltonian Mechanics. Spring-Verlag. Berlin, 1990.
* [8] H. Gluck, W. Ziller, Existence of periodic solutions of conservtive systems. Seminar on Minimal Submanifolds, Princeton University Press(1983), 65-98.
* [9] E. W. C. van Groesen, Analytical mini-max methods for Hamiltonian brake orbits of prescribed energy. J. Math. Anal. Appl. 132 (1988) 1-12.
* [10] K. Hayashi, Periodic solution of classical Hamiltonian systems. Tokyo J. Math. 6(1983), 473-486.
* [11] C. Liu, Maslov-type index theory for symplectic paths with Lagrangian boundary conditions. Adv. Nonlinear Stud. 7 (2007) no. 1, 131–161.
* [12] C. Liu, Asymptotically linear Hamiltonian systems with Lagrangian boundary conditions. Pacific J. Math. 232 (2007) no.1, 233-255.
* [13] C. Liu, Y. Long, C. Zhu, Multiplicity of closed characteristics on symmetric convex hypersurfaces in ${\bf R}^{2n}$. Math. Ann. 323 (2002) no. 2, 201–215.
* [14] C. Liu and D. Zhang, Iteration theory of $L$-index and Multiplicity of brake orbits. arXiv: 0908.0021vl [math. SG].
* [15] Y. Long, Bott formula of the Maslov-type index theory. Pacific J. Math. 187 (1999) 113-149.
* [16] Y. Long, Index Theory for Symplectic Paths with Applications. Birkhäuser. Basel. (2002).
* [17] Y. Long, D. Zhang, C. Zhu, Multiple brake orbits in bounded convex symmetric domains. Advances in Math. 203 (2006) 568-635.
* [18] Y. Long and C. Zhu, Closed characteristics on compact convex hypersurfaces in ${\bf R}^{2n}$. Ann. Math., 155 (2002) 317-368.
* [19] P. H. Rabinowitz, On the existence of periodic solutions for a class of symmetric Hamiltonian systems. Nonlinear Anal. T. M. A. 11 (1987) 599-611.
* [20] H. Seifert, Periodische Bewegungen mechanischer Systeme. Math. Z. 51 (1948) 197-216.
* [21] A. Szulkin, An index theory and existence of multiple brake orbits for star-shaped Hamiltonian systems. Math. Ann. 283 (1989) 241-255.
* [22] D. Zhang, Brake type closed characteristics on reversible compact convex hypersurfaces in ${\bf R}^{2n}$. Nonlinear Anal. T. M. A. 74 (2011) 3149-3158.
* [23] D. Zhang, Minimal period problems for brake orbits of nonlinear autonomous reversible semipositive Hamiltonian systems. arXiv: 1110.6915vl [math. SG].
|
arxiv-papers
| 2011-11-03T03:59:32 |
2024-09-04T02:49:23.927898
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Duanzhi Zhang and Chungen Liu",
"submitter": "Duanzhi Zhang",
"url": "https://arxiv.org/abs/1111.0722"
}
|
1111.0735
|
# Using Automated Dependency Analysis To Generate Representation Information
Andrew N. Jackson
Andrew.Jackson@bl.uk
###### Abstract
To preserve access to digital content, we must preserve the representation
information that captures the intended interpretation of the data. In
particular, we must be able to capture performance dependency requirements,
i.e. to identify the other resources that are required in order for the
intended interpretation to be constructed successfully. Critically, we must
identify the digital objects that are only referenced in the source data, but
are embedded in the performance, such as fonts. This paper describes a new
technique for analysing the dynamic dependencies of digital media, focussing
on analysing the process that underlies the performance, rather than parsing
and deconstructing the source data. This allows the results of format-specific
characterisation tools to be verified independently, and facilitates the
generation of representation information for any digital media format, even
when no suitable characterisation tool exists.
## 1 Introduction
When attempting to preserve access to digital media, keeping the bitstreams is
not sufficient - we must also preserve information on how the bits should be
interpreted. This need is widely recognised, and this data is referred to as
Representation Information (RI) by the Open Archival Information System (OAIS)
reference model [4]. The reference model also recognises that software can
provide valuable RI, expecially when the source code is included. However,
software is not the only dynamic dependency that must be captured in order to
preserve access. The interpretation of a digital object may inherit further
information from the technical environment as the performance proceeds, such
as passwords or licenses for encrypted resources, default colour spaces, page
dimensions or other rendering parameters and, critically, other digital
objects that the rendering requires. This last case can include linked items
that, while only referenced in the original data, are included directly in the
performance. In the context of hypertext, the term ‘transclusion’ has been
coined to describe this class of included resource [5].
The classic example of a transcluded resource is that of fonts. Many document
formats (PDF, DOC, etc.) only reference the fonts that should be used to
render the content via a simple name (e.g. ‘Symbol’), and the confusion and
damage that these potentially ambiguous references can cause has been well
documented [1]. Indeed, this is precisely why the PDF/A standard [2] requires
that all fonts, even the so-called ‘Postscript Standard Fonts’ (e.g.
Helvetica, Times, etc.), should be embedded directly in archival documents
instead of merely referenced. Similarly, beyond fonts, there are a wide range
of local or networked resources that may be transcluded, such as media files
and plug-ins displayed in web pages, documents and presentations, or XSD
Schema referenced from XML. We must be able to identify these different kinds
of transcluded resources, so that we can either include them as explicit RI or
embed them directly in the target item (as the PDF/A standard dictates for
fonts).
Traditionally, this kind of dependency analysis has been approached using
normal characterisation techniques. Software capable of parsing a particular
format of interest is written (or re-used and modified) to extract the data
that indicates which external dependencies may be required. Clearly, creating
this type of software requires a very detailed understanding of the particular
data format, and this demands that a significant amount of effort be expended
for each format of interest. Worse still, in many cases, direct deconstruction
of the bitstream(s) is not sufficient because the intended interpretation
deliberately depends on information held only in the wider technical
environment, i.e. the reference to the external dependency is implicit and
cannot be drawn from the data.
This paper outlines a complementary approach, developed as part of the SCAPE
project111http://www.scape-project.eu/, which shifts the focus from the data
held in the digital file(s) to the process that underlies the performance.
Instead of examining the bytes, we use the appropriate rendering software to
walk-through or simulate the required performance. During this process we
trace certain operating system operations to determine which resources are
being used, and use this to build a detailed map of the additional RI required
for the performance, including all transcluded resources. Critically, this
method does not require a detailed understanding of file format, and so can be
used to determine the dependencies of a wide range of media without the
significant up-front investment that developing a specialised characterisation
tool requires.
## 2 Method
Most modern CPUs can run under at least two operating modes: ‘privileged’ mode
and ‘user’ mode. Code running in privileged mode has full access to all
resources and devices, whereas code running in user mode has somewhat limited
access. This architecture means only highly-trusted code has direct access to
sensitive resources, and so attempts to ensure that any badly-written code
cannot bring the whole system to a halt, or damage data or devices by misusing
them. However, code running in user space must be able to pass requests to
devices, e.g. when saving a file to disk, and so a bridge must be built
between the user and the protected modes. It is the responsibility of the
operating system kernel to manage this divide. To this end, the kernel
provides a library of system calls that implement the protected mode actions
that the user code needs.
Most operating systems come with software that allows these ‘system calls’ to
be tracked and reported during execution, thus allowing any file system
request to be noted and stored without interfering significantly with the
execution process itself222The tracing does slow the execution down slightly,
mostly due to the I/O overhead of writing the trace out to disk, but the
process is otherwise unaffected. . The precise details required to implement
this tracing approach therefore depend only upon the platform, i.e. upon the
operating system kernel and the software available for monitoring processes
running on that kernel.
This monitoring technique allows all file-system resources that are ‘touched’
during the execution of any process to be identified, and can distinguish
between files being read and files being written to. This includes software
dependencies, both directly linked to the original software and executed by
it, as well as media resources.
Of course, this means the list of files we recover includes those needed to
simply run the software as well as those specific to a particular digital
media file. Where this causes confusion, we can separate the two cases by, for
example, running the process twice, once without the input file and once with,
and comparing the results. Alternatively, we can first load the software
alone, with no document, and then start monitoring that running process just
before we ask it to load a particular file. The resources used by that process
can then be analysed from the time the input file was loaded, as any
additional resource requirements must occur in the wake of that event.
### 2.1 Debian Linux
On Linux, we can make use of the standard system call tracer ‘strace’, which
is a debugging tool capable of printing out a trace of all the system calls
made by another process or program333http://sourceforge.net/projects/strace/.
This tool can be compiled on any operating system based on a reasonably recent
Linux kernel, and is available as a standard package on many distributions. In
this work, we used Debian Linux 6.0.2 and the Debian strace
package444http://packages.debian.org/stable/strace. For example, monitoring a
process that opens a Type 1 Postscript (PFB) font file creates a trace log
that looks like this:
> 5336 open("/usr/share/fonts/type1/gsfonts/
>
> n019004l.pfb", O_RDONLY) = 4
>
> 5336 read(4, "\200\1\f\5\0\0%!PS-
>
> AdobeFont-1.0: Nimbus"…, 4096) = 4096
>
> …more read calls…
>
> 5336 read(4, "", 4096) = 0
>
> 5336 close(4) = 0
Access to software can also be tracked, as direct dependencies like dynamic
linked libraries (e.g. ‘/usr/lib/libMag-ickCore.so.3’) appear in the system
trace in exactly the same way as any other required resource. As well as
library calls, a process may launch secondary ‘child’ processes, and as
launching a process also requires privileged access, these events be tracked
in much the same way (via the ‘fork’ or ‘execve’ system calls). The strace
program can be instructed to track these child processes, and helpfully
reports a brief summary of the command-line arguments that we passed when a
new process was launched.
### 2.2 Mac OS X
On OS X (and also Solaris, FreeBSD and some others) we can use the DTrace tool
from Sun/Oracle555http://opensolaris.org/os/community/dtrace/. This is similar
in principle to strace, but is capable of tracking any and all function calls
during execution (not just system calls at the kernel level). DTrace is a very
powerful and complex tool, and configuring it for our purposes would be a
fairly time-consuming activity. Fortunately, DTrace comes with a tool called
‘dtruss’, which pre-configures DTrace to provide essentially the same
monitoring capability as the strace tool. The OS X kernel calls have slightly
different names, the format of the log file is slightly different, and the OS
X version of DTrace is not able to log the arguments passed to child
processes, but these minor differences do not prevent the dependency analysis
from working.
### 2.3 Windows
Windows represents the primary platform for consumption of a wide range of
digital media, but unfortunately (despite the maturity of the operating
system) it was not possible to find a utility capable of reliably assessing
file usage. The ‘SysInternals Suite’666http://technet.microsoft.com/en-
gb/sysinternals/bb842062 has some utilities that can identify which files a
process is currently accessing (such as Process Explorer or Handle) and
similar utilities (ProcessActivityView, OpenedFilesView) have been published
by a third-party called Nirsoft777http://www.nirsoft.net/. These proved
difficult to invoke as automated processes, and even when this was successful,
the results proved unreliable. Each time the process was traced, a slightly
different set of files would be reported, and files opened for only brief
times did not appear at all. Sometimes, even the source file itself did not
appear in the list, proving that important file events were being missed. This
behaviour suggests that these programs were rapidly sampling the usage of file
resources, rather than monitoring them continuously.
An alternative tool called StraceNT888https://github.com/ipankajg/ihpublic/
provides a more promising approach, as it can explicitly intercept system
calls and so is capable of performing the continuous resource monitoring we
need. However, in its current state it is difficult to configure and,
critically, only reports the name of the library call, not the values of the
arguments. This means that although it can be used to tell if a file was
opened, it does not log the file name and so the resources cannot be
identified. However, the tool is open source, so might provide a useful basis
for future work.
One limited alternative on Windows is to use the Cygwin UNIX-like environment
instead of using Windows tools directly. Cygwin comes with its own strace
utility, and this has functionality very similar to Linux strace.
Unfortunately, this only works for applications built on top of the Cygwin
pseudo-kernel (e.g. the Cygwin ImageMagick package). Running Windows software
from Cygwin reports nothing useful, as the file system calls are not being
handled by the Cygwin pseudo-kernel.
## 3 Results
In this initial investigation, we looked at two example files, covering two
different media formats that support transcluded resources: a PDF document and
a PowerPoint presentation.
### 3.1 PDF Font Dependencies
The fonts required to render the PDF test file (the ‘ANSI/NISO Z39.87 - Data
Dictionary - Technical Metadata for Digital Still Images’ standards document
[3]) were first established by using a commonly available tool,
pdffonts999Part of Xpdf: http://foolabs.com/xpdf/, which is designed to parse
PDF files and look for font dependencies. This indicated that the document
used six fonts, one of which was embedded (see Table 1 for details).
The same document was rendered via three different pieces of software,
stepping through each page in turn either manually (for Adobe Reader or Apple
Preview) or automatically. The automated approach simulated the true rendering
process by rendering each page of the PDF to a separate image via the
ImageMagick101010http://www.imagemagick.org/ conversion commmand ‘convert
input.pdf output.jpg’. This creates a sequence of numbered JPG images called
‘output-###.jpg’, one for each page.
All system calls were traced during these rendering processes, and the files
that the process opened and read were collated. These lists were then further
examined to pick out all of the dependent media files - in this case, fonts.
The reconstructed font mappings are shown in Table 1.
Tool | Operating System | List of Fonts
---|---|---
pdffonts 3.02 | OS X 10.7 | Arial-BoldMT. ArialMT, Arial-ItalicMT, Arial-BoldItalicMT TimesNewRomanPSMT, BBNPHD+SymbolMT (embedded)
Apple Preview 5.5 | OS X 10.7 | /Library/Fonts/Microsoft/… Arial Bold.ttf, Arial.ttf, Arial Italic.ttf, Arial Bold Italic.ttf, Times New Roman.ttf
Adobe Reader X (10.1.0) | OS X 10.7 | /Library/Fonts/Microsoft/… Arial Bold.ttf, Arial.ttf, Arial Italic.ttf, Arial Bold Italic.ttf
Adobe Reader 9.4.2 | Debian Linux 6.0.2 | /usr/share/fonts/truetype/ttf-dejavu/… DejaVuSans.ttf, DejaVuSans-Bold.ttf /opt/Adobe/Reader9/Resource/Font/ZX______.PFB
ImageMagick 6.7.1 | OS X 10.7 via MacPorts | /opt/local/share/ghostscript/9.02/Resource/Font/… NimbusSanL-Bold, NimbusSanL-Regu, NimbusSanL-ReguItal, NimbusSanL-BoldItal, NimbusRomNo9L-Regu
ImageMagick 6.6.0 | Debian Linux 6.0.2 | /usr/share/fonts/type1/gsfonts/… n019004l.pfb, n019003l.pfb, n019023l.pfb, n019024l.pfb, n021003l.pfb
ImageMagick 6.4.0 | Cygwin on WinXP | /usr/share/ghostscript/fonts/… n019004l.pfb, n019003l.pfb, n019023l.pfb, n019024l.pfb, n021003l.pfb
Table 1: Font dependencies of a specific PDF document, as determined via a
range of tools.
The two manual renderings on OS X gave completely identical results, with each
font declaration being matched to the appropriate Microsoft TrueType font. The
manual rendering via Adobe Reader on Debian was more complex. The process
required three font files, but comparing the ‘no-file’ case with the ‘file’
case showed that the first two (DejaVuSans and DejaVuSans-Bold) were involved
only in rendering the user interface, and not the document itself. The third
file, ‘ZX______.PFB’, was supplied with the Adobe Reader package and upon
inspection was found to be a Type 1 Postscript Multiple Master font called
‘Adobe Sans MM’, which contains all the variants of a typeface that Adobe
Reader uses to render standard or missing fonts. Adobe have presumably taken
this approach in order to ensure the standard Postscript fonts are rendered
consistently across platforms, without depending on any external software
packages that are beyond their control.
Although the precise details and naming conventions differed between the
platforms, each of the ImageMagick simulated renderings pulled in the
essentially the same set of Type 1 PostScript files, which are the open source
(GPL-compatible license) versions of the Adobe standard fonts. This is not
immediately apparent due to the different naming conventions using on
different installations, but manual inspection quickly determined that, for
example, NimbusSanL-Bold and n019004l.pfb were essentially the same font, but
from different versions of the gsfonts package. The information in the system
trace log made it easy to determine how ImageMagick was invoking GhostScript,
and to track down the font mapping tables that GhostScript was using to map
the PDF font names into the available fonts.
Interestingly, as well as revealing that these apparently identical
performances depend on different versions of different files in two different
formats (TrueType or Type 1 Postscript fonts), the results also show that
while Apple Preview and ImageMagick indicate that Times New Roman is a
required font (in agreement with the pdffonts results) this font is not
actually brought in during the Adobe Reader rendering processes. A detailed
examination of the source document revealed that while Times New Roman is
declared as a font dependency on one page of the document, this appears to be
an artefact inherited from an older version of the document, as none of the
text displayed on the page is actually rendered in that font.
### 3.2 PowerPoint with Linked Media
A simple PowerPoint presentation was created in Microsoft PowerPoint for Mac
2011 (version 14.1.2), containing some text and a single image. When placing
the image, PowerPoint was instructed to only refer to the external file, and
not embed it, simulating the default behaviour when including large media
files. The rendering process was then performed manually, looking through the
presentation while tracing the system calls. As well as picking up all the
font dependencies, the fact that the image was being loaded from an external
location could also be detected easily.
The presentation was then closed, and the referenced image was deleted. When
re-opening the presentation, the system call trace revealed that PowerPoint
was hunting for the missing file, guessing a number of locations based on the
original absolute pathname. This approach can therefore be used to spot
missing media referenced by PowerPoint presentations.
## 4 Conclusions
Process monitoring and system call tracing is a valuable analysis technique,
complementary to the more usual format-oriented approach. It enables us to
perform detailed quality assurance of existing characterisation tools, using a
completely independent approach to validate the identification of the
resources required to render a digital object. Furthermore, because the
tracing process depends only on standard system functionality, and not on the
particular software in question, it can work for all types of digital media
without developing software for each format. As the PowerPoint example shows,
the only requirement for performing this analysis is the provision of suitable
rendering software.
Before using this approach in a production setting, it will be necessary to
test it over a wider range of documents and types of transclusion, e.g.
embedded XML Schema. In particular, the monitoring should be extended to track
network requests for resources as well as local file or software calls.
Although all network activity is visible via kernel system calls, the raw
socket data is at such a low level that it is extremely difficult to analyse.
Fortunately, tools like netstat111111http://en.wikipedia.org/wiki/Netstat and
WireShark121212http://www.wireshark.org/ have been designed to solve precisely
this problem, and could be deployed alongside system call tracing to supply
the necessary intelligence on network protocols. Beyond widening the range of
resources, extending this approach to the Windows platform would be highly
desirable. The current lack of a suitable call tracing tool is quite
unfortunate, and means that this approach cannot be applied to software that
only runs on Windows. Hopefully, StraceNT can provide a way forward.
Beyond the direct resource dependencies outlined here, this approach could be
combined with knowledge of the platform package management system in order to
build an even richer model of the representation information network a digital
object requires. For example, Debian has a rigorous package management
processes, and by looking up which packages provide the files implicated in
the rendering, we can validate not only the required binary software packages,
but also determine the location of the underlying open source software, and
even the identities of the developers and other individuals involved. This
allows very rich RI to be generated in an automated fashion. Furthermore, as
the Debian package management infrastructure also tracks the development and
discontinuation of the various software packages, this information could be
leveraged to help build a semi-automatic preservation watch system.
## 5 Acknowledgments
This work was partially supported by the SCAPE Project. The SCAPE project is
co-funded by the European Union under FP7 ICT-2009.4.1 (Grant Agreement number
270137).
## References
* [1] G. Brown and K. Woods. Born Broken : Fonts and Information Loss in Legacy Digital Documents. International Journal of Digital Curation, 6(1):5–19, 2011.
* [2] International Standardization Organization. ISO 19005-1:2005 Document management – Electronic document file format for long-term preservation – Part 1: Use of PDF 1.4 (PDF/A-1), 2005.
* [3] National Information Standards Organization. ANSI/NISO Z39.87 - Data Dictionary - Technical Metadata for Digital Still Images, 2006.
* [4] The Consultative Committee for Space Data. Reference Model For An Open Archival Information System (OAIS), 2009\.
* [5] Theodor Holm Nelson and Robert Adamson Smith. Back to the Future, 2007.
|
arxiv-papers
| 2011-11-03T06:26:29 |
2024-09-04T02:49:23.938525
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Andrew N. Jackson",
"submitter": "Andrew Jackson",
"url": "https://arxiv.org/abs/1111.0735"
}
|
1111.0782
|
DESY 11-191
BFKL equation for the adjoint representation of the gauge group
in the next-to-leading approximation at $N=4$ SUSY †
V.S. Fadin ∗, L.N. Lipatov ∗∗
Universität Hamburg,
II. Institut für Theoretische Physik,
Luruper Chaussee, 149, D-22761 Hamburg,
* Budker Nuclear Physics Institute
and Novosibirsk State University,
630090, Novosibirsk, Russia
** Petersburg Nuclear Physics Institute
and St.Petersburg State University,
Gatchina, 188300, St.Petersburg, Russia
Abstract
We calculate the eigenvalues of the next-to-leading kernel for the BFKL
equation in the adjoint representation of the gauge group $SU(N_{c})$ in the
N=4 supersymmetric Yang-Mills model. These eigenvalues are used to obtain the
high energy behavior of the remainder function for the 6-point scattering
amplitude with the maximal helicity violation in the kinematical regions
containing the Mandelstam cut contribution. The leading and next-to-leading
singularities of the corresponding collinear anomalous dimension are
calculated in all orders of perturbation theory. We compare our result with
the known collinear limit and with the recently suggested ansatz for the
remainder function in three loops and obtain the full agreement providing that
the numerical parameters in this anzatz are chosen in an appropriate way.
†The work was supported in part by grant 14.740.11.0082 of Federal Program
“Personnel of Innovational Russia,” in part by RFBR grants 10-02-01238,
10-02-01338-a.
## 1 Introduction
In the Regge pole model scattering amplitudes at large energies $\sqrt{s}$ and
fixed momentum transfers $\sqrt{-t}$ have the form [1]
$A_{Regge}^{p}(s,t)=\xi_{p}(t)\,s^{1+\omega_{p}(t)}\,\gamma^{2}(t)\,,\,\,\xi_{p}(t)=e^{-i\pi\omega_{p}(t)}-p\,,$
(1)
where $p=\pm 1$ is the signature of the reggeon with the trajectory
$\omega_{p}(t)$ and $\gamma^{2}(t)$ represents the product of reggeon
vertices. The Pomeron is the Regge pole of the $t$-channel partial wave
$f_{\omega}(t)$ with vacuum quantum numbers and the positive signature
describing an approximately constant behaviour of total cross-sections for the
hadron-hadron scattering. S. Mandelstam demonstrated, that the Regge poles
generate cut singularities in the $\omega$-plane [2].
In the leading logarithmic approximation (LLA) the scattering amplitude at
high energies in QCD has the Regge form [3]
$M_{AB}^{A^{\prime}B^{\prime}}(s,t)=M_{AB}^{A^{\prime}B^{\prime}}(s,t)|_{Born}\,s^{\omega(t)}\,,$
(2)
where $M_{Born}$ is the Born amplitude and the gluon Regge trajectory is given
below
$\omega(-|q|^{2})=-\frac{\alpha_{s}N_{c}}{4\pi^{2}}\,\int
d^{2}k\,\frac{|q|^{2}}{|k|^{2}|q-k|^{2}}\approx-\frac{\alpha_{s}N_{c}}{2\pi}\,\ln\frac{|q^{2}|}{\lambda^{2}}\,.$
(3)
Here $\lambda$ is the infrared cut-off. In the multi-Regge kinematics, where
the pair energies $\sqrt{s_{k}}$ of the produced gluons are large in
comparison with momentum transfers $|q_{i}|$, the production amplitudes in LLA
are constructed from products of the Regge factors $s_{k}^{\omega(t_{k})}$ and
effective reggeon-reggeon-gluon vertices $C_{\mu}(q_{r},q_{r+1})$ [3]. The
amplitudes satisfy the Steinmann relations and the $s$-channel unitarity
incorporated in bootstrap equations [4].
The knowledge of $M_{2\rightarrow 2+n}$ allows one to construct the BFKL
equation for the Pomeron wave function using analyticity, unitarity,
renormalizability and crossing symmetry [3]. The integral kernel of this
equation has the property of the holomorphic separability [5] and is invariant
under the Möbius transformations [6]. The generalization of this equation to a
composite state of several gluons [7] in the multi-color QCD leads to an
integrable XXX model [8] having a duality symmetry [9].
The next-to-leading correction to the color singlet kernel in QCD is also
calculated [10]. Its eigenvalue contains non-analytic terms proportional to
$\delta_{n,0}$ and $\delta_{n,2}$, where $n$ is the conformal spin of the
Möbius group. But in the case of the $N=4$ extended supersymmetric gauge model
these Kronecker symbols are canceled leading to an expression having the
properties of the hermitian separability [11] and maximal transcendentality
[12]. The last property allowed to calculate the anomalous dimensions of
twist-two operators up to three loops [13, 14]. It turns out, that evolution
equations for the so-called quasi-partonic operators are integrable in $N=4$
SUSY at the multi-color limit [15]. The $N=4$ four-dimensional conformal field
theory according to the Maldacena guess is equivalent to the superstring model
living on the anti-de-Sitter 10-dimensional space [16, 17, 18]. Therefore the
Pomeron in N=4 SUSY is equivalent to the reggeized graviton in this space. The
equivalence gives a possibility to calculate the intercept of the BFKL Pomeron
at large coupling constants [14, 19]. The Möbius invariance of the BFKL kernel
was demonstrated also in two loops [20]. For next-to-leading calculations one
can use the effective field theory for reggeized gluons [21]. The generalized
bootstrap equation gives a possibility to prove the multi-Regge form of
production amplitudes in the next-to-leading approximation [22].
Another application of the BFKL approach is a verification of the BDS ansatz
[23] for the inelastic amplitudes in $N=4$ SUSY. It was demonstrated [24, 25],
that the BDS amplitude $M^{BDS}_{2\rightarrow 4}$ should be multiplied by the
factor containing the contribution of the Mandelstam cuts [2] in LLA. In the
two-loop approximation this factor can be found also from properties of
analyticity and factorization [26] or directly from recently obtained exact
result [27] for $M_{2\rightarrow 4}$ (see [28]). In a general case the wave
function in LLA for the composite $n$-gluon state in the adjoint
representation satisfies the Schrödinger equation for an open integrable
Heisenberg spin chain [29].
In this paper we shall calculate the eigenvalues $\omega(t)$ of the kernel $K$
for the BFKL equation in the adjoint representation of the gauge group at
$N=4$ SUSY in the next-to-leading approximation. The Green function of this
equation allows one to find the asymptotic behavior of the inelastic amplitude
in the Regge kinematics. There is a hypothesis [30, 31], that the inelastic
amplitude with the maximal helicity violation in a planar approximation is
factorized in the product of the BDS amplitude $M^{BDS}$, containing in
crossing channels the Regge factors with corresponding infrared divergencies,
and the remainder function $R$ depending on the anharmonic ratios
$M=R\,M^{BDS}\,.$ (4)
In an accordance with this hypothesis the $q^{2}$-dependence of the
eigenvalues of the octet BFKL equation is given by the expression (cf. [25] in
LLA)
$\omega(-q^{2})=\omega_{g}(-q^{2})+\omega_{0}\,,$ (5)
where $\omega_{g}(t)$ is the gluon Regge trajectory, which can be expressed in
all orders of the perturbation theory of $N=4$ SUSY in terms of two functions
entering in the expression for the BDS amplitude [24]. The ”intercept”
$\omega_{0}$ does not depend on $q^{2}$ due to the conformal invariance of
$N=4$ SUSY and can be written in terms of the ”energy” $E=-\omega_{0}$ being
the eigenvalue of the BFKL kernel discussed in the next section.
## 2 Integral kernel in the adjoint representation
The homogeneous BFKL equation can be written in the form
$\omega_{0}\phi=\hat{K}\phi\,,$ (6)
where $\hat{K}$ is the integral operator from which the gluon Regge trajectory
is subtracted. In the momentum representation it has the form
$\hat{K}\phi(\vec{q}_{1},\vec{q}_{2})=\int\frac{d^{2}q_{1}^{\;\prime}}{|q_{1}^{\;\prime}|^{2}|q_{2}^{\;\prime}|^{2}}\,K(\vec{q}_{1},\vec{q}_{1}^{\;\prime};\vec{q})\,\phi(\vec{q}_{1}^{\;\prime},\vec{q}_{2}^{\;\prime})\,,\,\,\vec{q}=\vec{q}_{1}+\vec{q}_{2}=\vec{q}^{\;\prime}_{1}+\vec{q}^{\;\prime}_{2}\,.$
(7)
The integral kernel for $N=4$ SUSY can be presented as follows (cf. [3, 22]
for the QCD case)
$K(\vec{q}_{1},\vec{q}_{1}^{\;\prime};\vec{q})=\delta^{2}(\vec{q}_{1}-\vec{q}_{1}^{\;\prime})\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;2}\left(\omega_{g}(-\vec{q}_{1}^{\;2})+\omega_{g}(-\vec{q}_{2}^{\;2})-\omega_{g}(-\vec{q}^{\;2})\right)+K_{r}(\vec{q}_{1},\vec{q}_{1}^{\;\prime};\vec{q})\,,$
(8)
where the first term corresponds to virtual corrections with the gluon regge
trajectory subtraction (see (5)) and the second term appears from the real
intermediate states in the $s$-channel. The total contribution does not
contain infrared divergencies. Using results of Refs. [32] it can be written
in the form
$K(\vec{q}_{1},\vec{q}_{1}^{\;\prime};\vec{q})=\frac{1}{2}\delta^{2}(\vec{q}_{1}-\vec{q}_{1}^{\;\prime})\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;2}\left(\omega_{g}(-\vec{q}_{1}^{\;2})+\omega_{g}(-\vec{q}_{2}^{\;2})-2\omega_{g}(-\vec{q}^{\;2})\right)+K^{ns}(\vec{q}_{1},\vec{q}_{1}^{\;\prime};\vec{q}),$
(9)
where
$\omega_{g}(-\vec{q}_{1}^{\;2})+\omega_{g}(-\vec{q}_{2}^{\;2})-2\omega_{g}(-\vec{q}^{\;2})=-\frac{\alpha\,N_{c}}{2\pi}\left(1-\zeta(2)\,\frac{\alpha\,N_{c}}{2\pi}\right)\,\ln\left(\frac{\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;2}}{\vec{q}^{\;4}}\right)$
(10)
and
$K^{ns}(\vec{q}_{1},\vec{q}_{1}^{\;\prime};\vec{q})=-\delta^{2}(\vec{q}_{1}-\vec{q}^{\;\prime}_{1})\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;2}\frac{\alpha\,N_{c}}{8\pi^{2}}\Biggl{(}\left(1-\zeta(2)\,\frac{\alpha\,N_{c}}{2\pi}\right)\int
d^{2}k\;\left(\frac{2}{\vec{k}^{\;2}}+2\frac{\vec{k}(\vec{q}_{1}-\vec{k})}{\vec{k}^{\;2}(\vec{q}_{1}-\vec{k})^{2}}\right)$
$-3\alpha\,N_{c}\zeta(3)\Biggr{)}+\frac{\alpha\,N_{c}}{8\pi^{2}}\left\\{\Biggl{(}1-\zeta(2)\,\frac{\alpha\,N_{c}}{2\pi}\Biggr{)}\left(\frac{\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;\prime\;2}+\vec{q}_{1}^{\;\prime\;2}\vec{q}_{2}^{\;2}}{\vec{k}^{\;2}}-\vec{q}^{\;2}\right)+\right.$
$\frac{\alpha\,N_{c}}{4\pi}\Biggl{[}\frac{\vec{q}^{\,2}}{2}\left(\ln\left(\frac{\vec{q}_{1}^{\;2}}{\vec{q}^{\;2}}\right)\ln\left(\frac{\vec{q}_{2}^{\;2}}{\vec{q}^{\;2}}\right)+\ln\left(\frac{\vec{q}_{1}^{\;\prime\;2}}{\vec{q}^{\;2}}\right)\ln\left(\frac{\vec{q}_{2}^{\;\prime
2}}{\vec{q}^{\;2}}\right)+\ln^{2}\left(\frac{\vec{q}_{1}^{\;2}}{\vec{q}_{1}^{\;\prime\;2}}\right)\right)-\frac{\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;\prime\;2}+\vec{q}_{2}^{\;2}\vec{q}_{1}^{\;\prime\;2}}{\vec{k}^{\;2}}\ln^{2}\left(\frac{\vec{q}_{1}^{\;2}}{\vec{q}_{1}^{\;\prime\;2}}\right)$
$-\frac{1}{2}\,\frac{\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;\prime\;2}-\vec{q}_{2}^{\;2}\vec{q}_{1}^{\;\prime\;2}}{\vec{k}^{\;2}}\ln\left(\frac{\vec{q}_{1}^{\;2}}{\vec{q}_{1}^{\;\prime\;2}}\right)\,\ln\left(\frac{\vec{q}_{1}^{\;2}\vec{q}_{1}^{\;\prime\;2}}{\vec{k}^{\;4}}\right)+\biggl{[}\vec{q}^{\;2}(\vec{k}^{\;2}-\vec{q}_{1}^{\;2}-\vec{q}_{1}^{\;\prime\;2})$
$\left.+2\vec{q}_{1}^{\;2}\vec{q}_{1}^{\;\prime\;2}-\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;\prime\;2}-\vec{q}_{2}^{\;2}\vec{q}_{1}^{\;\prime\;2}+\frac{\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;\prime\;2}-\vec{q}_{2}^{\;2}\vec{q}_{1}^{\;\prime\;2}}{\vec{k}^{\;2}}(\vec{q}_{1}^{\;2}-\vec{q}_{1}^{\;\prime\;2})\biggr{]}I(\vec{q}_{1}^{\;2},\vec{q}_{1}^{\;\prime\;2},\vec{k}^{\;2})\Biggr{]}\\!\\!\right\\}$
$+\left(\vec{q}_{1}\leftrightarrow\vec{q}_{2},\;\;\vec{q}_{1}^{\;\prime}\leftrightarrow\vec{q}_{2}^{\;\prime}\right)~{},$
(11)
where $\vec{k}=\vec{q}_{1}-\vec{q}_{1}^{\;\prime}$ and the function $I$ is
given below
$I(\vec{q}_{1}^{\;2},\vec{q}_{1}^{\;\prime\;2},\vec{k}^{\;2})=\int_{0}^{1}\frac{dx}{\vec{q}_{1}^{\;2}(1-x)+\vec{q}_{1}^{\;\prime\;2}x-\vec{k}^{\;2}x(1-x)}\ln\left(\frac{\vec{q}_{1}^{\;2}(1-x)+\vec{q}_{1}^{\;\prime\;2}x}{\vec{k}^{\;2}x(1-x)}\right)~{}.$
(12)
Note that $I(a,b,c)$ is a totally symmetric function of the variables $a,\;b$
and $c$.
One could expect, that the BFKL kernel in $N=4$ SUSY is Möbius invariant in
the momentum representation, which would lead to the following simple form of
its eigenfunctions (cf. [25])
$\phi_{\nu
n}(\vec{q}_{1},\vec{q}_{2})=\left|\frac{q_{1}}{q_{2}}\right|^{2i\nu}\,e^{i\,n\,\phi}\,,$
(13)
where $\phi$ is the azimuthal angle of the complex number constructed from
transverse components of the vectors $\vec{q}_{1}$ and $\vec{q}_{2}$
$\frac{q_{1}}{q_{2}}=\left|\frac{q_{1}}{q_{2}}\right|\,e^{i\phi}\,.$ (14)
However, in the existing form the kernel is not Möbius invariant and in future
one should construct the similarity transformation to the invariant form (cf.
[20]). Such transformation exists because the remainder function $R$,
corresponding to the correction factor for the BDS expression, should be
invariant under four-dimensional dual conformal transformations and the Green
function obtained from the BFKL equation in the adjoint representation allows
to find the asymptotic behavior of the remainder function in the Mandelstam
kinematical regions [25].
## 3 Eigenvalues of the kernel
It is important, that the eigenvalues of the BFKL kernel do not depend on its
representation and can be found from our expression (8). To calculate these
eigenvalues we consider the BFKL equation in the limit (cf. [25])
$|q_{1}|\sim|q^{\prime}_{1}|\ll|q|\approx|q_{2}|\approx|q^{\prime}_{2}|\,.$
(15)
Denoting the two dimensional vectors $\vec{q}_{1}$ and
$\vec{q}^{\;\prime}_{1}$ by $\vec{p}$ and $\vec{p}^{\;\prime}$, respectively,
we write the BFKL equation in the form
$\int\frac{d^{2}p^{\;\prime}}{|p^{\;\prime}|^{2}}\,K(\vec{p},\vec{p}^{\;\prime})\,\Phi(\vec{p}^{\;\prime})=\omega_{0}\,\Phi(\vec{p})\,.$
(16)
Its kernel is given below
$K(\vec{p},\vec{p}^{\;\prime})=-\delta^{2}(\vec{p}-\vec{p}^{\;\prime})\,|p|^{2}\,\frac{\alpha
N_{c}}{4\pi^{2}}\,\left(\left(1-\frac{\alpha N_{c}}{2\pi}\zeta(2)\right)\,\int
d^{2}p^{\;\prime}\,\left(\frac{2}{|p^{\;\prime}|^{2}}+\frac{2(p^{\;\prime},p-p^{\;\prime})}{|p^{\;\prime}|^{2}|p-p^{\;\prime}|^{2}}\right)-3\alpha\,\zeta(3)\right)$
$+\frac{\alpha N_{c}}{4\pi^{2}}\,\left(1-\frac{\alpha
N_{c}}{2\pi}\zeta(2)\right)\,\left(\frac{|p|^{2}+|p^{\;\prime}|^{2}}{|p-p^{\;\prime}|^{2}}-1\right)+\frac{\alpha^{2}N_{c}^{2}}{32\,\pi^{3}}\,R(\vec{p},\vec{p}^{\;\prime})\,.$
(17)
Here $\vec{p}$ and $\vec{p}^{\;\prime}$ are momenta of the same reggeized
gluon before and after its scattering in the $t_{2}$-channel (momenta of
another gluon tend to infinity together with $q$). The reduced kernel
$R(\vec{p},\vec{p}^{\;\prime})$ is given below
$R(\vec{p},\vec{p}^{\;\prime})=\left(\frac{1}{2}-\frac{|p|^{2}+|p^{\;\prime}|^{2}}{|p-p^{\;\prime}|^{2}}\right)\,\ln^{2}\frac{|p|^{2}}{|p^{\;\prime}|^{2}}-\frac{|p|^{2}-|p^{\;\prime}|^{2}}{2|p-p^{\;\prime}|^{2}}\,\ln\frac{|p|^{2}}{|p^{\;\prime}|^{2}}\,\ln\frac{|p|^{2}|p^{\;\prime}|^{2}}{|p-p^{\;\prime}|^{4}}$
(18)
$+\left(-|p+p^{\;\prime}|^{2}+\frac{(|p|^{2}-|p^{\;\prime}|^{2})^{2}}{|p-p^{\;\prime}|^{2}}\right)\,\int_{0}^{1}dx\,\frac{1}{|(1-x)p+xp^{\;\prime}|^{2}}\,\ln\frac{|(1-x)p+xp^{\;\prime}|^{2}}{x(1-x)|p-p^{\;\prime}|^{2}}\,.$
(19)
From the rotational and dilatational invariance of the kernel we obtain its
eigenfunctions in the simple form
$\Phi_{\nu n}(\vec{p})=|p|^{2i\nu}e^{i\phi n}\,,$ (20)
where $\phi$ is the angle of the transverse vector $\overrightarrow{p}$ with
respect to the axis $x$. Note, that $\nu$ is real and $n$ is integer.
The orthonormality condition for this set of functions is obvious
$\frac{1}{2\pi^{2}}\int\frac{d^{2}p}{|p|^{2}}\,\Phi^{*}_{\mu
m}(\vec{p})\,\Phi_{\nu
n}(\vec{p}^{\;\prime})=\delta(\mu-\nu)\,\delta_{m,n}\,.$ (21)
The corresponding eigenvalues can be calculated with the action of the BFKL
kernel on the eigenfunctions and are given below
$\omega(\nu,n)=-a\left(E_{\nu n}+a\,\epsilon_{\nu
n}\right)\,,\,\,a=\frac{\alpha N_{c}}{2\pi}\,,$ (22)
where $E_{\nu n}$ is the ”energy” in the leading approximation [25]
$E_{\nu
n}=-\frac{1}{2}\,\frac{|n|}{\nu^{2}+\frac{n^{2}}{4}}+\psi(1+i\nu+\frac{|n|}{2})+\psi(1-i\nu+\frac{|n|}{2})-2\psi(1)\,,\,\,\psi(x)=(\ln\Gamma(x))^{\prime}$
(23)
and the next-to-leading correction $\epsilon_{\nu n}$ can be written as
follows
$\epsilon_{\nu
n}=-\frac{1}{4}\left(\psi^{\prime\prime}(1+i\nu+\frac{|n|}{2})+\psi^{\prime\prime}(1-i\nu+\frac{|n|}{2})+\frac{2i\nu\left(\psi^{\prime}(1-i\nu+\frac{|n|}{2})-\psi^{\prime}(1+i\nu+\frac{|n|}{2})\right)}{\nu^{2}+\frac{n^{2}}{4}}\right)$
$-\zeta(2)\,E_{\nu
n}-3\zeta(3)-\frac{1}{4}\,\frac{|n|\,\left(\nu^{2}-\frac{n^{2}}{4}\right)}{\left(\nu^{2}+\frac{n^{2}}{4}\right)^{3}}\,.$
(24)
Here the $\zeta$-functions are expressed in terms of polylogarithms
$Li_{n}(x)=\sum_{k=1}^{\infty}\frac{x^{k}}{k^{n}}\,,\,\,\zeta(n)=Li_{n}(1)\,.$
(25)
Note, that $\omega(\nu,n)$ has the important property
$\omega(0,0)=0\,.$ (26)
It is in an agreement with the existence of the eigenfunction $\Phi=1$ with a
vanishing eigenvalue, which is a consequence of the bootstrap relation [3,
22].
## 4 Corrections to the remainder function
One can easily construct the Green function for the conformally invariant BFKL
kernel in terms of its eigenvalues. This Green function allows us to calculate
the remainder functions $R_{n}$ for an arbitrary number of external legs in
the regions, where there are Mandelstam’s cuts corresponding to the composite
states of two reggeized gluons. For simplicity we consider the remainder
function $R_{6}$ for the gluon transition $2\rightarrow 4$ depending on three
anharmonic ratios (cf. [28])
$u_{1}=\frac{ss_{2}}{s_{012}s_{123}}\,,\,\,u_{2}=\frac{s_{1}t_{3}}{s_{012}t_{2}}\,,\,\,u_{3}=\frac{s_{3}t_{1}}{s_{123}t_{2}}\,.$
(27)
In the multi-regge kinematics one obtains
$s\gg s_{012},s_{123}\gg s_{1},s_{2},s_{3}\gg t_{1},t_{2},t_{3}\,,$ (28)
which corresponds to the following restrictions on the variables $u_{k}$
$1-u_{1}\rightarrow 0\,,\,\,\tilde{u}_{2}=\frac{u_{2}}{1-u_{1}}\sim
1\,,\,\,\tilde{u}_{3}=\frac{u_{3}}{1-u_{1}}\sim 1\,.$ (29)
It is convenient also to introduce the complex variable $w$ [28]
$w=|w|e^{i\phi_{23}}\,,\,\,|w|^{2}=\frac{u_{2}}{u_{3}}\,,\,\,\cos\phi_{23}=\frac{1-u_{1}-u_{2}-u_{3}}{2\sqrt{u_{2}u_{3}}}$
(30)
expressed in terms of transverse momenta of produced particles $k_{1}$,
$k_{2}$ and momentum transfers $q_{1},q_{2},q_{3}$
$w=\frac{q_{3}k_{1}}{k_{2}q_{1}}\,.$ (31)
In this case the remainder function $R$ in the Mandelstam region, where
$s,s_{2}\rightarrow+\infty\,,\,\,s_{1},s_{3}\rightarrow-\infty\,,$ (32)
can be presented in the form of a dispersion-like relation [26]
$R\,e^{i\pi\delta}=cos\,\pi\omega_{ab}+i\,\frac{a}{2}\sum_{n=-\infty}^{\infty}(-1)^{n}\left(\frac{w}{w^{*}}\right)^{\frac{n}{2}}\int_{-\infty}^{\infty}\frac{|w|^{2i\nu}d\nu}{\nu^{2}+\frac{n^{2}}{4}}\,\Phi_{Reg}(\nu,n)\left(-\frac{1}{\sqrt{u_{2}u_{3}}}\right)^{\omega(\nu,n)},$
(33)
where
$\delta=\frac{\gamma_{K}}{8}\,\ln(\tilde{u}_{2}\tilde{u}_{3})=\frac{\gamma_{K}}{8}\,\ln\frac{|w|^{2}}{|1+w|^{4}}\,,\,\,\omega_{ab}=\frac{\gamma_{K}}{8}\,\ln\frac{\tilde{u}_{2}}{\tilde{u}_{3}}=\frac{\gamma_{K}}{8}\,\ln|w|^{2}$
(34)
and the cusp anomalous dimensions
$\gamma_{K}=4a-4\,a^{2}\,\zeta(2)+22\,\zeta(4)\,a^{3}+...$ (35)
is known in all orders of perturbation theory [33].
Further, instead of the traditional variable $1/(1-u_{1})$ (see [24, 25]) we
used in eq. (33) the following energy invariant
$\frac{1}{\sqrt{u_{2}u_{3}}}=s_{2}\,\frac{|q_{2}|^{2}}{\sqrt{|k_{1}|^{2}|q_{1}|^{2}}\,|k_{2}|^{2}|q_{3}|^{2}}=\frac{1}{1-u_{1}}\,\frac{|1+w|^{2}}{|w|}\,,$
(36)
because according to the Regge theory the amplitude should be factorized in
the $t_{2}$-channel. As a result, by expanding this expression for $R$ in the
perturbation theory
$R=1+i\,a^{2}\left(b_{1}\ln\frac{1}{1-u_{1}}+b_{2}\right)+a^{3}\left(ic_{1}\ln^{2}\frac{1}{1-u_{1}}+(d_{1}+ic_{2})\ln\frac{1}{1-u_{1}}+d_{2}+ic_{3}\right)+...=$
$1+i\,a^{2}\left(\widetilde{b}_{1}\ln\frac{1}{\sqrt{u_{2}u_{3}}}+\widetilde{b}_{2}\right)+a^{3}\left(i\widetilde{c}_{1}\ln^{2}\frac{1}{\sqrt{u_{2}u_{3}}}+(\widetilde{d}_{1}+i\widetilde{c}_{2})\ln\frac{1}{\sqrt{u_{2}u_{3}}}+\widetilde{d}_{2}+i\widetilde{c_{3}}\right)+...\,,$
(37)
we obtain [25, 28]
$\widetilde{b}_{1}=b_{1}=-\frac{\pi}{2}\,\ln|1+w|^{2}\ln\frac{|1+w|^{2}}{|w|^{2}}\,,$
(38)
$\widetilde{b_{2}}=b_{2}-b_{1}\ln\frac{|1+w|^{2}}{|w|}\,,\,\,\frac{1}{\pi}\,b_{2}=\frac{1}{2}\ln|w|^{2}\ln^{2}|1+w|^{2}$
$-\frac{1}{3}\ln^{3}|1+w|^{2}+\ln|w|^{2}\left(Li_{2}(-w)+Li_{2}(-w^{*})\right)-2\left(Li_{3}(-w)+L_{3}(-w^{*})\right)\,,$
(39)
and (see ref. [28])
$\frac{4}{\pi}\,\widetilde{c}_{1}=\frac{4}{\pi}\,c_{1}=\ln|w|^{2}\,\ln^{2}|1+w|^{2}-\frac{2}{3}\ln^{3}|1+w|^{2}-\frac{1}{4}\ln^{2}|w|^{2}\ln|1+w|^{2}$
$+\frac{1}{2}\,\ln|w|^{2}\left(Li_{2}(-w)+Li_{2}(-w^{*})\right)-Li_{3}(-w)-Li_{3}(-w^{*})\,,$
(40)
$\frac{4}{\pi^{2}}\,\widetilde{d}_{1}=\frac{4}{\pi^{2}}\,d_{1}=-\ln|w|^{2}\,\ln^{2}|1+w|^{2}+\frac{2}{3}\ln^{3}|1+w|^{2}+\frac{1}{2}\ln^{2}|w|^{2}\ln|1+w|^{2}$
$+\ln|w|^{2}\left(Li_{2}(-w)+Li_{2}(-w^{*})\right)-2Li_{3}(-w)-2Li_{3}(-w^{*})\,.$
(41)
Note, that in the second order the real contribution to $R$ is absent [26].
The product of two impact factors $\Phi_{Reg}(\nu,n)$ can be obtained with the
use of the Fourier transformation of the function $\widetilde{b}_{2}$
$\Phi_{Reg}(\nu,n)=1+\Phi_{Reg}^{(1)}(\nu,n)\,a+\Phi_{Reg}^{(2)}(\nu,n)\,a^{2}+...\,,$
(42)
$\Phi_{Reg}^{(1)}(\nu,n)=\Phi^{(1)}(\nu,n)+\Delta\Phi(\nu,n)=-\frac{1}{2}E_{\nu
n}^{2}-\frac{3}{8}\,\frac{n^{2}}{\left(\nu^{2}+\frac{n^{2}}{4}\right)^{2}}-\zeta(2)\,,$
(43)
where $\Delta\Phi(\nu,n)$ is the contribution of the term
$-b_{1}\ln\frac{|1+w|^{2}}{|w|}$ in $\widetilde{b}_{2}$ (39) and the
contribution $\Phi^{(1)}(\nu,n)$ appearing from the term $b_{2}$ was
calculated in ref. [28] 111In the reference [28] the quantity
$\Phi^{(1)}(\nu,n)$ was found for the remainder function, but here we need it
for the full amplitude. According to (33) they differ by the term appearing
from the expansion of $\exp(i\pi\delta)$ and proportional to the second order
contribution to the anomalous dimension $\gamma_{K}$ (35).
$\Phi^{(1)}(\nu,n)=E_{\nu
n}^{2}-\frac{1}{4}\,\frac{n^{2}}{\left(\nu^{2}+\frac{n^{2}}{4}\right)^{2}}-\zeta(2)\,.$
(44)
The knowledge of eigenvalues (24) in the next-to-leading approximation gives a
possibility to calculate the coefficients $\widetilde{c}_{2}$ and
$\widetilde{d}_{2}$ from expression (33)
$\frac{1}{\pi}\,\widetilde{c}_{2}=-\frac{1}{4}\,\ln|w|^{2}\left(S_{1,2}(-w)+S_{1,2}(-w^{*})+\ln(1+w)\,Li_{2}(-w)+\ln(1+w^{*})\,Li_{2}(-w*)\right)$
$+\frac{\zeta(3)}{2}\,\ln|1+w|^{2}-\ln\frac{|1+w|^{2}}{|w|}\left(Li_{3}(-w)+Li_{3}(-w^{*})-\frac{1}{2}\ln|w|^{2}(Li_{2}(-w)+Li_{2}(-w^{*}))\right)$
$+\frac{1}{4}\,\ln|1+w|^{2}(Li_{3}(-w)+Li_{3}(-w^{*}))+\frac{1}{16}\,\ln^{2}|w|^{2}\ln|1+w|^{2}\ln\frac{|1+w|^{2}}{|w|^{2}}$
$+\frac{1}{8}\,\ln^{2}|1+w|^{2}\ln^{2}\frac{|1+w|^{2}}{|w|^{2}}+\frac{1}{8}\ln^{2}|w|^{2}\ln(1+w)\,\ln(1+w^{*})+\zeta(2)\,\ln|1+w|^{2}\ln\frac{|1+w|^{2}}{|w|^{2}}\,,$
(45)
$\widetilde{d}_{2}=\pi\left(\widetilde{c}_{2}-\ln\frac{|1+w|^{2}}{|w|}\,\widetilde{b}_{2}+2\,\zeta(2)\,\widetilde{b}_{1}\right)\,.$
(46)
In the above expression the function $S_{1,2}(-x)$ has the following
representation
$S_{1,2}(-x)=\int_{0}^{x}\frac{dx^{\prime}}{2x^{\prime}}\,\ln^{2}(1+x^{\prime})=Li_{3}(\frac{x}{1+x})+Li_{3}(-x)-\ln(1+x)Li_{2}(-x)-\frac{1}{6}\ln^{3}(1+x)\,.$
(47)
One can verify with the use of the known relations among polylogarithms
$Li_{n}(x)$, that the coefficients $\widetilde{c}_{2}$ and $\widetilde{d}_{2}$
are single-valued functions on the two-dimensional plane $\overrightarrow{w}$
and are symmetric to the inversion $w\rightarrow 1/w$. We can calculate also
the coefficients $c_{2}$ and $d_{2}$ in (37) using the relations
$c_{2}=\widetilde{c}_{2}+2\widetilde{c}_{1}\ln\frac{|1+w|^{2}}{|w|}\,,\,\,d_{2}=\widetilde{d}_{2}+\widetilde{d}_{1}\,\ln\frac{|1+w|^{2}}{|w|}\,.$
(48)
Note, that recently the authors of ref. [35] suggested an anzatz for the
remainder function $R_{6}$ in three loops based on the theory of symbols. They
calculated its high energy behavior in our Mandelstam region in the form of
the polynomial expansion in $\log(1-u_{1})$. It turns out, that up to three
loops their results are completely coincides with our perturbative expansion
(37). In particular, one can derive the expressions (58) and (66) from the
paper [35] using the fact, that the corresponding functions
$g_{1}^{(2)}(w,w^{*})$ and $h_{0}^{(3)}(w,w^{*})$ are related with our
coefficients $c_{2}$ and $d_{2}$ in (37) as follows
$g_{1}^{(2)}(w,w^{*})=-\frac{c_{2}}{2\pi}\,,\,\,h_{0}^{(3)}(w,w^{*})=-\frac{d_{2}}{(2\pi)^{2}}\,.$
(49)
It gives a possibility to fix the parameters $\gamma^{\prime}$ and
$\gamma^{\prime\prime\prime}$ appearing in ref. [35] in the form
$\gamma^{\prime}=-\frac{9}{2}\,,\,\,\gamma^{\prime\prime\prime}=0\,.$ (50)
In expression (63) of the paper [35] also the additional function
$g_{0}^{(3)}(w,w^{*})$ was calculated. This function contains three unknown
parameters appearing in the last line of (63). Our coefficients $c_{3}$ and
$\widetilde{c}_{3}$ in (37) can be expressed in terms of it
$c_{3}=2\pi\,g_{0}^{(3)}(w,w^{*})\,,\,\,\widetilde{c}_{3}=c_{3}-\ln\frac{|1+w|^{2}}{|w|}\,c_{2}+\ln^{2}\frac{|1+w|^{2}}{|w|}\,c_{1}\,.$
(51)
It gives a possibility to construct the following function
$\rho(w,w^{*})=\frac{\widetilde{c}_{3}}{\pi}+\pi\,\widetilde{c}_{1}+\ln\frac{|1+w|^{2}}{|w|}\,\left(\zeta(2)\,\ln^{2}\frac{|1+w|^{2}}{|w|}-\frac{11}{2}\,\zeta(4)\right),$
(52)
where the term proportional to $\zeta(4)$ appears from the third order
contribution to $\gamma_{K}$ (35) which was calculated firstly in ref. [13].
The important next-to-next-to-leading corrections to the product of impact-
factors $\Phi_{Reg}(\nu,n)$ (42) can be expressed through $\rho(w,w^{*})$
$\Phi^{(2)}_{Reg}(\nu,n)=(-1)^{n}\left(\nu^{2}+\frac{n^{2}}{4}\right)\int\frac{d^{2}w}{\pi}\,\rho(w,w^{*})\,|w|^{-2i\nu-2}\,\left(\frac{w^{*}}{w}\right)^{\frac{n}{2}}\,.$
(53)
We are going to calculate $\Phi^{(2)}_{Reg}(\nu,n)$ in future.
Similar results can be obtained for the remainder function describing the
$3\rightarrow 3$ transitions in the corresponding Mandelstam regions [34].
## 5 Collinear limit
It is well known, that the BFKL equation for the Pomeron wave function gives a
possibility to predict the leading singularities of the anomalous dimensions
$\gamma$ of the twist-2 operators at $\omega\rightarrow 0$ in all orders of
perturbation theory [3, 10]. In particular, for the case of $N=4$ SUSY the
predictions of Ref. [11] are in a full agreement with the direct calculations
of $\gamma$ up to 5 loops [13, 36, 37]. As it follows from the previous
section, the BFKL kernel for the adjoint representation of the gauge group
allows one to find the high energy corrections to the remainder functions. On
the other hand, in the collinear limit the remainder functions obey the
renormalization group-like equations [38, 39]. The analytic continuation of
the collinear expressions for $R$ to the Mandelstam regions was performed in
Ref. [40]. The leading asymptotics corresponds to the unit conformal spin
$|n|=1$. The anomalous dimensions $\gamma_{col}$ for the collinear limit in
the Euclidean region were constructed [39] and the relation between the Regge
and collinear limits was investigated [40].
To calculate $\gamma_{col}$ in the Mandelstam region we present expression
(33) in the following form with the use of the Fourier transformation
$R\,e^{i\pi\delta}=cos\,\pi\omega_{ab}+i\,\frac{a}{2}\sum_{n=-\infty}^{\infty}(-1)^{n}\left(\frac{w}{w^{*}}\right)^{\frac{n}{2}}\int_{-\infty}^{\infty}d\nu\,|w|^{2i\nu}\,L_{\nu
n}\left(-\frac{1}{1-u_{1}}\right),$ (54)
where
$L_{\nu
n}\left(-\frac{1}{1-u_{1}}\right)=\sum_{n^{\prime}=-\infty}^{\infty}(-1)^{n^{\prime}-n}\int_{-\infty}^{\infty}\frac{\Phi_{reg}(\nu^{\prime},n^{\prime})d\nu^{\prime}}{\nu^{\prime
2}+\frac{n^{\prime 2}}{4}}\,S_{\nu^{\prime}n^{\prime}}^{\nu
n}\,\left(-\frac{1}{1-u_{1}}\right)^{\omega(\nu^{\prime},n^{\prime})}$ (55)
and
$S_{\nu^{\prime}n^{\prime}}^{\nu
n}=\int\frac{d^{2}w}{2\pi^{2}}\,|w|^{2i(\nu^{\prime}-\nu)-2}\,\left(\frac{w}{w^{*}}\right)^{\frac{n^{\prime}-n}{2}}\,\left(\frac{|1+w|^{2}}{|w|}\right)^{\omega(\nu^{\prime}n^{\prime})}\,.$
(56)
The collinear limit $w\rightarrow 0$ or $w\rightarrow\infty$ of the remainder
function (54) should be performed at fixed $1-u_{1}$ [39, 40]. Generally
expressions (54) and (55) correspond to the collinear renormalization with an
infinite number of the multiplicatively renormalizable operators (cf. [40]).
But in the case, when we take into account only the asymptotic terms at
$|w|\rightarrow\infty$ with the conformal spin $|n|=1$, we can obtain for $R$
the simple expression
$R\,e^{i\pi\delta}\approx
cos\,\pi\omega_{ab}-ia\,\cos\phi_{23}\,|w|^{-1}\int_{-i\infty}^{i\infty}d\omega\,\frac{\Phi^{Reg}(\nu,1)}{\left(\nu^{2}+\frac{1}{4}\right)\,\frac{d\omega}{d\nu}}\,|w|^{2\gamma_{col}(\omega)}\,\left(-\frac{1}{1-u_{1}}\right)^{\omega}\,,$
(57)
where the contour of integration goes to the right of the BFKL singularity
$\nu\sim\sqrt{\omega-\omega(0,1)}$ present in the integrand in an accordance
with the fact, that the functions
$\gamma=\gamma_{col}(\omega),\,\nu=\nu(\omega)$ satisfy the set of equations
222Note, that our definition of the collinear anomalous dimension
$\gamma_{col}$ differs with the factor $-1/2$ from that used in ref. [39].
$\gamma=\frac{1}{2}+i\nu+\frac{\omega}{2}\,,\,\,\omega=\omega(\nu,1)\,.$ (58)
For finding $\gamma_{col}$ in perturbation theory the function $\omega(\nu,1)$
(22) should be expanded near the point $\nu=i/2$
$\lim_{\nu\rightarrow\frac{i}{2}}\omega(\nu,1)=\frac{a}{2}\,f_{1}\left(i\nu+\frac{1}{2}\right)-\frac{a^{2}}{8}\,f_{2}\left(i\nu+\frac{1}{2}\right)\,,$
(59)
where
$f_{1}(x)=\frac{1}{x}-1-x-x^{2}(1-4\zeta(3))-x^{3}+O(x^{4})\,,$ (60)
$f_{2}(x)=\frac{1}{x^{3}}+\frac{1}{x^{2}}+\frac{4\zeta(2)}{x}-8\zeta(3)-4\zeta(2)-2+O(x)\,,$
(61)
Thus, we obtain the following equation for $\gamma=\gamma_{col}(\omega)$
$\omega=\frac{a}{2}\,f_{1}(\gamma)-\frac{a^{2}}{8}\,\left(f^{\prime}_{1}(\gamma)f_{1}(\gamma)+f_{2}(\gamma)\right)\,.$
(62)
Its perturbative solution is given below
$\gamma_{col}(\omega)=\frac{a}{2}\,\left(\frac{1}{\omega}-1\right)-\frac{a^{2}}{4}\left(\frac{1}{\omega^{2}}+2\,\frac{\zeta(2)}{\omega}\right)+\frac{a^{3}}{4\,\omega^{2}}\left(1+2\zeta(2)+\zeta(3)\right)+O(a^{4})\,.$
(63)
The above approach is similar to that for the singlet BFKL kernel, but in that
case one obtained the main contribution to the Bjorken limit from $n=0$ [10].
The collinear anomalous dimension $\gamma_{col}(\omega)$ in the Mandelstam
region $s,s_{2}>0,s_{1},s_{3}<0$ can be found in one loop using the results of
the paper [40]. We start with the perturbative expansion of the remainder
function in the collinear limit $|w|\rightarrow\infty$ in LLA of the Operator
Product Expansion (OPE) [38]
$R_{OPE}\approx
a\cos\phi\,\frac{e^{-\sigma}}{2|w|}\,\sum_{k=0}^{\infty}\frac{(-a\ln|w|)^{k}}{k!}\,h_{k}(\sigma)\,,\,\,\sigma=\frac{1}{2}\,\ln\frac{u_{1}}{1-u_{1}}\,,$
(64)
where we expressed the world sheet coordinates $\tau$ and $\sigma$ in terms of
our variables $w$ and $u_{1}$ (see eqs (76)-(79) from ref. [40]) and included
one loop contribution contained in the BDS amplitude. The analytic
continuation of the two loop remainder function calculated in ref. [27] to the
Mandelstam region $s,s_{2}>0,\,s_{1},s_{3}<0$ gives the result (see eqs. (51),
(C.12)-(C.16) from ref. [40])
$\cos\phi\,\rightarrow\cos\phi_{23};\;\;h_{k}(\sigma)\rightarrow-
h_{k}(\sigma)+\Delta_{k}(\sigma)\,,\,\,\frac{\Delta_{0}(\sigma)}{2\pi
i}=-2\,e^{\sigma}\,,\,\,$ $\frac{\Delta_{1}(\sigma)}{2\pi
i}=4\left(\cosh\sigma\ln(1+e^{2\sigma})-e^{\sigma}\right).$ (65)
Here the functions $h_{k}(\sigma)$ for $k=0,1$ in the right hand side of the
first relation are known from ref. [39]. They are not essential for the
calculation of $\gamma_{col}$ because they are real and fall at large
$\sigma$. The contributions $\Delta_{k}(\sigma)$ appear from the analytic
continuation of the corresponding discontinuities of the functions
$h_{k}(\sigma)$ on the cut $-1<\widetilde{s}_{2}<0$, where
$\widetilde{s}_{2}=\exp(2\sigma)$ [40]. After the continuation we can write
this discontinuity using the collinear renormalization group in the form
$\Delta
R_{OPE}=-\,a\,\cos\phi_{23}\,\frac{1}{|w|}\,\int_{-i\infty}^{i\infty}\frac{d\omega}{\omega(\omega+1)}\,|w|^{2\gamma_{col}(\omega)}e^{2\omega\sigma}\,,$
(66)
where
$\gamma_{col}(\omega)=a\,\omega\,(1+\omega)\,\int_{0}^{\infty}d(2\sigma)\,e^{-2\sigma\omega}\left(e^{-\sigma}\cosh(\sigma)\ln\left(1+e^{2\sigma}\right)-1\right)=$
$\frac{a}{2}\left(\frac{1}{\omega(\omega+1)}-2\omega+(\omega+1)\left(\psi(\omega+1)-\psi(\frac{\omega+2}{2})\right)+\omega\,\psi(\omega+2)-\omega\,\psi(\frac{\omega+3}{2})\right).$
(67)
As one can see from expression (63), the BFKL approach reproduces correctly
the first two terms of $\gamma_{col}$ at $\omega\rightarrow 0$.
## 6 Conclusion
In this paper we solved the BFKL equation for the channel with color octet
quantum numbers in the next-to-leading approximation. The eigenvalues of its
integral kernel were used to calculate in the next-to-leading logarithmic
approximation the remainder function for the production amplitude
$2\rightarrow 4$ in the multi-Regge kinematics at the Mandelstam channels. The
obtained result in three loops is in an agreement with the recently suggested
anzatz [35] for the remainder function. This anzatz allowed us to construct
the product of corresponding impact-factors in the next-to-next-to-leading
approximation. The collinear anomalous dimension in the Mandelstam region was
calculated explicitly in one loop. Its leading and next-to-leading
singularities are found in all loops.
Acknowledgments. We thank J. Bartels, A. Prygarin and G. Vacca for helpful
discussions, the Hamburg University and DESY for the warm hospitality and
support. This work was done in the framework of the program LEXI ”Connecting
Particles with the Cosmos”.
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|
arxiv-papers
| 2011-11-03T10:44:01 |
2024-09-04T02:49:23.945690
|
{
"license": "Public Domain",
"authors": "V. S. Fadin and L. N. Lipatov",
"submitter": "Victor Fadin",
"url": "https://arxiv.org/abs/1111.0782"
}
|
1111.0791
|
# Entanglement and Subsystem Particle Numbers in Free Fermion Systems
Y. F. Zhang1 Huichao Li1 L. Sheng1 shengli@nju.edu.cn R. Shen1 D. Y. Xing1
1National Laboratory of Solid State Microstructures and Department of Physics,
Nanjing University, Nanjing 210093, China
###### Abstract
We study the relationship between bipartite entanglement and subsystem
particle number in a half-filled free fermion system. It is proposed that the
spin-projected particle numbers can distinguish the quantum spin Hall state
from other states, and be linked to the topological invariant of the system.
It is also shown that the subsystem particle number fluctuation displays
behavior very similar to the entanglement entropy. It provides a lower-bound
estimation for the entanglement entropy, which can be measured experimentally.
###### pacs:
73.22.Pr, 03.65.Vf, 03.65.Ud, 65.40.gd, 05.40.Ca
## I Introduction
Topological phases of matter are usually distinguished by using some global
topological properties, such as topological invariants and topologically
protected gapless edge modes, rather than certain local order parameters. The
integer quantum Hall effect iqhe , fractional quantum Hall effect fqhe , and
band Chern insulators ci can be characterized by Chern numbers or Berry
phases tknn . The quantum spin Hall (QSH) effect qshe1 ; qshe2 and the three-
dimensional topological insulators 3d1 ; 3d2 are characterized by the $Z_{2}$
invariant z2 or spin Chern number spinch1 ; spinch2 . Recently, quantum
entanglement entangle1 , which reveals the phase information of the quantum-
mechanical ground-state wavefunction, has been used as a tool to characterize
the topological phases. As shown by Levin and Wen wen and also by Kitaev and
Preskill kitaev , the existence of topological entanglement entropy in a fully
gapped system, such as fractional quantum Hall and the gapped integer spin
systems teefqh ; teeqsl , indicates existence of long-range quantum
entanglement (topological order toporder in equivalent parlance). Another
important progress is the demonstration that the entanglement spectrum (ES)
es1 reveals the gapless edge spectrum for fractional quantum Hall systems,
Chern insulators, and topological insulators es1 ; es2 ; es3 .
Supposing $A$ and $B$ to be two blocks of a large system in a pure quantum
state, the reduced density matrix (RDM) $\rho_{A}$ can be obtained by tracing
over degrees of freedom of $B$. Then the Von Neumann entanglement entropy (EE)
can be computed
$S_{ent}=-\mbox{tr}(\rho_{A}\ln\rho_{A})=-\mbox{tr}(\rho_{B}\ln\rho_{B})\ .$
(1)
It has been shown that for bipartite subsystems $A$ and $B$ with a smooth
boundary, $S_{ent}$ has the form of $S_{ent}=\alpha L-S_{top}$, where $L$ is
the length of the boundary, $\alpha$ is a non universal coefficient, and
$-S_{top}$ is a universal constant called the _topological entanglement
entropy_ wen ; kitaev . Moreover, if we write the RDM in the form of
$\rho_{A}=\exp(-H_{ent})/Z$, where $Z$ is a normalization constant, and
$H_{ent}$ is known as the _entanglement Hamiltonian_ , the eigenvalue spectrum
$\\{\varepsilon_{i}\\}$ of $H_{ent}$ is called the ES, which stores more
information about the quantum entanglement than the EE es1 .
(a) cylinder geometry
(b) torus geometry
Figure 1: (Color online) Schematic view of a cylinder and a torus. The
entanglement cuts divide the system into two equal parts $A$ and $B$. For the
cylinder geometry, the entanglement cut leads to one interface (a); and for
the torus geometry, the cuts lead to two interfaces (b).
In this paper, we study the relationship between bipartite entanglement and
subsystem particle number in half-filled free fermion systems. It was proposed
in Ref. trace , for systems with translational invariance in one dimension,
the discontinuity in the subsystem particle number as a function of the
conserved momentum indicates whether or not the ES has a spectral flow, which
is determined by the topological invariant of the system es3 . Nevertheless,
this approach has an exceptional case for a half-filled QSH system with two-
dimensional inversion symmetry. To overcome the inadequacy, we define spin-
projected particle numbers, based on which spin trace indices can be well
defined, for the QSH system with or without $s_{z}$ conservation. Spin trace
indices are univocally related to the topological invariant of QSH system,
i.e., the $Z_{2}$ index. We further investigate the relationship between the
EE and subsystem particle number fluctuation. The latter is also dominated by
the boundary excitations of the system, and satisfies a similar area law as
the EE. It gives a lower-bound estimation of the EE, and can be measured
experimentally.
In the next section, we introduce the model Hamiltonian, and explain the
procedure to calculate the ES and EE. In Sec. III, numerical calculation of
the ES is carried out, and the connection between the subsystem spin-projected
particle numbers and the topological invariants in different phases is
established. In Sec. IV, the relationship between the EE and subsystem
particle number fluctuation is discussed. The final section is a summary.
## II MODEL HAMILTONIAN
We begin with the tight-binding model Hamiltonian for the QSH system
introduced by Kane and Mele qshe1 ; z2 , plus an additional exchange field
yang
$\begin{split}H&=-\sum\limits_{\langle\bm{i},\bm{j}\rangle}c_{\bm{i}}^{\dagger}c_{\bm{j}}+iv_{so}\sum\limits_{\ll\bm{i},\bm{j}\gg}c_{\bm{i}}^{\dagger}\sigma_{z}v_{ij}c_{\bm{j}}\\\
&+iv_{r}\sum\limits_{\langle\bm{i},\bm{j}\rangle}c_{\bm{i}}^{\dagger}(\bm{\sigma}\times\bm{d}_{\bm{i}\bm{j}})_{z}c_{\bm{j}}+\sum_{\bm{i}}m_{i}c_{\bm{i}}^{\dagger}c_{\bm{i}}+g\sum\limits_{\bm{i}}c_{\bm{i}}^{\dagger}\sigma_{z}c_{\bm{i}}\
.\end{split}$ (2)
Here, the first term is the usual nearest neighbor hopping term with
$c_{\bm{i}}^{\dagger}=({c^{\dagger}_{\bm{i},\uparrow}},{c^{\dagger}_{\bm{i},\downarrow}})$
as the electron creation operator on site $\bm{i}$, where the hopping integral
is set to be unity. The second term is the intrinsic spin-orbit coupling (SOC)
with
$v_{ij}=(\bm{d}_{kj}\times\bm{d}_{ik})_{z}/|(\bm{d}_{kj}\times\bm{d}_{ik})_{z}|$,
where $\bm{k}$ is the common nearest neighbor of $\bm{i}$ and $\bm{j}$, and
vector $\bm{d}_{ik}$ points from $\bm{k}$ to $\bm{i}$. The third term stands
for the Rashba SOC with $\bm{\sigma}$ the Pauli matrix. The fourth term stands
for a staggered sublattice potential $(m_{i}=\pm m)$, and the last term
represents a uniform exchange field with strength $g$.
We consider systems with cylinder or torus boundary conditions, consisting of
$N_{x}$ ($N_{x}$ to be even) zigzag chains along the circumferential direction
($y$ direction). The size of the sample will be denoted as $N=N_{x}\times
N_{y}$ with $N_{y}$ as the number of atomic sites on each chain. We perform
the entanglement cut along the $y$ direction, which results in one or two
interfaces between the two equal parts $A$ and $B$, respectively, for the
cylinder or torus geometry, as shown in Fig. 1. In order to examine the EE and
ES, an Schmidt decomposition on the ground-state wavefunction or calculation
of the RDM is usually needed. For non-interacting fermion systems, however,
the necessary information of the entanglement can also be obtained from the
following two-point correlators dmrg
$c_{\tau_{1},\tau_{2}}(\bm{i},\bm{j})=\langle
c^{\dagger}_{\bm{i},\tau_{1}}{c_{\bm{j},\tau_{2}}}\rangle\ .$ (3)
Here, $\langle\cdot\rangle$ means the ground-state expectation of an operator.
$\tau$ can be an index of spin, pseudospin, or orbital degree of freedom.
Using the Fourier transformation (FT) along the $y$ direction, the Hamiltonian
can be rewritten as
$H=\sum_{k_{y},i,j}c^{\dagger}_{i}(k_{y})h_{i,j}(k_{y})c_{j}(ky)$, where
$c^{\dagger}_{i}(k_{y})=(c^{\dagger}_{i,\uparrow}(k_{y}),c^{\dagger}_{i,\downarrow}(k_{y}))$
are the electron creation operators. After performing the entanglement cut, we
treat part $A$ as the subsystem, and trace out the degrees of freedom of $B$.
It should be noted that any of the correlators
$c_{\tau_{1},\tau_{2}}(\bm{i},\bm{j})$ with $\bm{i}$ and $\bm{j}$ confined in
$A$ is unchanged by the tracing. When carrying out the FT on the correlators,
we can get
$c_{\tau_{1},\tau_{2}}(\bm{i},\bm{j})=\frac{1}{N_{y}}\sum_{k_{y}}e^{ik_{y}\cdot(i_{y}-j_{y})}\langle
c^{\dagger}_{i,\tau_{1}}(k_{y})c_{j,\tau_{2}}(k_{y})\rangle\ ,$ (4)
where $i$ and $j$ discriminate the zigzag chains. We use $\langle
c^{\dagger}_{i,\tau_{1}}(k_{y})c_{j,\tau_{2}}(k_{y})\rangle$ to form a
hermitian matrix ${\cal C}(k_{y})$. Then the entanglement Hamiltonian is given
by dmrg
$H_{ent}=\ln({\cal C}^{-1}-1)\ .$ (5)
The spectrum $\\{\zeta_{i}\\}$ of ${\cal C}$ is related to spectrum
$\\{\varepsilon_{i}\\}$ of $H_{ent}$ by $\zeta_{i}=1/(e^{\varepsilon_{i}}+1)$,
where $\zeta_{i}$ acts as the average fermion number in the entanglement
energy level $\varepsilon_{i}$ at “temperature” $T=1$. By using the spectrum
of ${\cal C}$, the EE at each $k_{y}$ sector is given by
$s_{ent}(k_{y})=\sum_{i}s_{i}$, with
$s_{i}=-\zeta_{i}\ln\zeta_{i}-(1-\zeta_{i})\ln(1-\zeta_{i})\ .$ (6)
Figure 2: (a-c) Entanglement spectrum in the cylinder geometry (left panels)
and torus geometry (right panels) for the QSH phase with $v_{so}=m=0.2$,
$v_{r}=0.1$, $g=0$ (upper row), the insulator phase with $v_{r}=-0.3$,
$m=0.3$, $v_{so}=g=0$ (middle row), and the quantum anomalous Hall phase with
$v_{r}=g=-0.3$, $v_{so}=m=0$ (lower row). (g) Phase diagram in the $m$ versus
$g$ plane for $v_{so}=0$ and $v_{r}\neq 0$. Points $A$ and $B$ correspond to
the parameter values used (c,d) and (e,f), respectively.
From the viewpoint of probability theory, $s_{i}$ in Eq. (6) can be regarded
as the Shannon (information) entropy of the Bernoulli distribution, i.e., the
$i$-th entanglement level $\varepsilon_{i}$ has probability $\zeta_{i}$ of
being occupied while $(1-\zeta_{i})$ of being unoccupied. As a result,
$S_{ent}$ is the Shannon entropy of a series of such independent Bernoulli
distributions. In the following, we will perform systematic numerical
simulations to study various phases of Hamiltonian (2) in terms of the ES and
the subsystem particle number.
## III Entanglement spectrum and subsystem particle number
At $g=0$, Hamiltonian (2) is the standard Kane-Mele model qshe1 , which is
invariant under time reversal symmetry. The system is in a QSH phase when
$|m/v_{so}|<[9-\frac{3}{4}(v_{r}/v_{so})^{2}]$, and is an insulator when
$|m/v_{so}|>[9-\frac{3}{4}(v_{r}/v_{so})^{2}]$. On the other hand, if we set
$v_{so}=0$, $v_{r}$ and $g$ nonzero, a middle band gap opens when
$|g|\neq|m|$. The system is in a quantum anomalous Hall phase with Chern
number $C=\pm 2$ yang for $|g|<|m|$, and is an insulator for $|g|>|m|$. The
band gap closes at the transition point $|g|=|m|$. The phase diagram for
$v_{so}=0$ and $v_{r}\neq 0$ is plotted in Fig. 2(g).
Figs. 2(a) and (b) show the ES for the QSH phase, Figs. 2(c) and (d) for the
insulator phase, and Figs. 2(e) and (f) for the quantum anomalous Hall phase.
Here, it should be emphasized that the nontrivial topological phases exhibit
gapless ES [Figs. 2(a), (b), (e), and (f)], corresponding to physical gapless
edge modes, and this property is named as the _spectral flow_ es3 . However,
the spectral flow is broken for the topologically trivial phase [Figs. 2(c)
and (d)], which is also consistent with the property of the correspondent edge
states.
In a recent work trace , the authors proposed a new characteristic quantity
called the ”trace index” to describe topological invariants, which is defined
through a subsystem particle number operator $N_{A}(k_{y})=\sum_{i\in
A}c^{\dagger}_{i,k_{y}}c_{i,k_{y}}$. The expectation of $N_{A}(k_{y})$ is
given by
$\displaystyle\langle N_{A}(k_{y})\rangle=\langle GS|\sum_{i\in
A}c^{\dagger}_{i}(k_{y})c_{i}(k_{y})|GS\rangle=\mbox{Tr}{\cal C}\ .$ (7)
In Fig. 3, we plot the expectation of $N_{A}(k_{y})$ for the three different
phases mentioned above. In the cylinder geometry, $N_{A}(k_{y})$ is
discontinuous at some discrete momenta in the nontrivial topological phases,
as shown in Figs. 3(a) and (c). This is in contrast to the normal insulator
phase [see Fig. 3(b)], where $N_{A}(k_{y})$ is a continuous function of
$k_{y}$. In the torus geometry, $N_{A}(k_{y})$ is exactly equal to half of the
total particle number in the $k_{y}$ sector, without showing any
discontinuity, because the change of the particle number in $A$ around
interface $I$ is just canceled by that around interface $II$ due to the
rotation invariance of the torus. In the cylinder geometry, the _trace index_
was defined as the total discontinuities of $\langle N_{A}(k_{y})\rangle$ with
varying momentum. Alexandradinata, Hughes, and Bernevig trace presented a
detailed analysis and proved that the trace index is equivalent to the Chern
number (or $Z_{2}$ invariant) for the Chern ($Z_{2}$) insulators. Therefore,
the subsystem particle number provides a new alternative tool to reveal the
topological invariants.
However, as mentioned in Ref. trace , there is an exceptional case in which
the subspace of the occupied bands at the symmetric momenta is not closed
under time reversal in the ground state. If at the symmetric momenta the
Kramers’ doublet that extends along the edge of $A$ is singly-occupied,
$\langle N_{A}(k_{y})\rangle$ is continuous, even when the system is in a
nontrivial topological phase. For the half-filled system under consideration,
an exception still happens. While the two-dimensional inversion symmetry
remains unchanged ($m=0$), $N_{A}(k_{y})$ becomes continuous, as shown in Fig.
3(d). This is because the Kramers’ partners extending along the edge
simultaneously cross the Fermi level at the symmetric momentum ($k_{y}=\pi$)
and have opposite contributions to the discontinuities of $\langle
N_{A}(k_{y})\rangle$.
Figure 3: (Color online) The subsystem particle number in the cylinder
geometry and torus geometry for the QSH phase (a), the insulator phase (b),
and the quantum anomalous Hall phase (c) with all the parameters same as those
in Fig. 2. (d) is also for the QSH phase with $v_{so}=v_{r}=0.2$, where the
two-dimensional inversion symmetry is retained. Discontinuities in the
expectation of the particle number can be observed only in the cylinder
geometry.
To overcome this difficulty, enlightened by the definition of the spin Chern
number spinch2 , we define spin trace indices. Considering that $s_{z}$ is not
necessarily conserved, it is an adaptable way that we choose operator
$Ps_{z}P$ to split the fiber bundle of the occupied states into two bundles
with nontrivial Chern numbers, where $P$ is the ground state projector. At
half filling and in the presence of time reversal symmetry ($g=0$), $Ps_{z}P$
is always a time-odd operator ($TPs_{z}PT^{-1}=-Ps_{z}P$), so that the
spectrum of $Ps_{z}P$ is symmetric in respect to the origin. As a result, the
spectral spaces of $Ps_{z}P$ can provide a splitting of the Hilbert space
spanned by the wave functions of the occupied states, resulting in a smooth
decomposition $P(k_{y})$ into
$P(k_{y})=P^{+}(k_{y})\oplus P^{-}(k_{y})\ ,$ (8)
with $\alpha=\pm$ corresponding to the positive and negative sectors of the
spectrum of $Ps_{z}P$ for all $k_{y}\in(0,2\pi]$. Straightforwardly, the two-
point correlator matrix can also be decomposed into ${\cal C}(k_{y})={\cal
C}^{+}(k_{y})\oplus{\cal C}^{-}(k_{y})$. In this way, the Chern numbers
$C_{\pm}$ of the two spin sectors can be well defined on these new wave
functions. It has been proved that $C_{\pm}$ are topological invariants
protected by the energy and spin spectrum gaps spinch2 . It will be shown
below that the traces of ${\cal C}^{\pm}$, called the _spin-projected
subsystem particle numbers_ , are related to the topological invariants.
Figure 4: (Color online) Subsystem spin-projected particle numbers in the
cylinder geometry for the QSH phase (a-c) with $g=0$ and different parameters:
(a) $v_{so}=0.2$ and $v_{r}=m=0$, (b) $v_{so}=v_{r}=0.2$ and $m=0$, (c)
$v_{so}=m=0.2$ and $v_{r}=0.1$, and for the insulator phase (d) with
$v_{so}=v_{r}=0.05$, $m=0.5$ and $g=0$.
We plot $\mbox{Tr}{\cal C}^{\alpha}$ $(\alpha=\pm)$ as functions of $k_{y}$ in
Fig. 4. Both $\mbox{Tr}{\cal C}^{\alpha}(k_{y})$ are discontinuous in the QSH
phase [Figs. 4(a-c)], in contrast to the continuous functions in the normal
insulator phase [Fig. 4(d)]. If $\mbox{Tr}{\cal C}^{\alpha}(k_{y})$ is
discontinuous at some discrete momenta $\\{k_{dis}\\}$ with
$k_{dis}\in(0,2\pi]$, we can define the _spin trace indices_ as the total
discontinuity, i.e., difference between the limits of $\mbox{Tr}{\cal
C}^{\alpha}(k_{y})$ from right and left,
$A^{\alpha}\equiv\sum_{k_{dis}}(\lim_{k\rightarrow k_{dis+}}\mbox{Tr}{\cal
C}^{\alpha}(k)-\lim_{k\rightarrow k_{dis-}}\mbox{Tr}{\cal C}^{\alpha}(k))\ ,$
(9)
in the thermodynamic limit. As shown in Figs. 4(a) and (b), no matter whether
$s_{z}$ is conserved, both $\mbox{Tr}{\cal C}^{+}(k_{y})$ and $\mbox{Tr}{\cal
C}^{-}(k_{y})$ show discontinuities at momentum $k_{y}=\pi$ with $A^{+}=1$ and
$A^{-}=-1$ in the QSH phase, where the two-dimensional inversion symmetry is
present ($m=0$). Figure 4(c) shows the discontinuities of $\mbox{Tr}{\cal
C}^{+}(k_{y})$ and $\mbox{Tr}{\cal C}^{-}(k_{y})$ in the QSH phase in which
$s_{z}$ is not conserved ($v_{r}\neq 0$) and the two-dimensional inversion
symmetry is broken ($m\neq 0$). In this case, the spin trace indices are equal
to 1 and $-1$, respectively, contributed by two different momentum points. It
is noteworthy that in analogy with the Laughlin gauge experiment, $A^{\alpha}$
can be regarded as the number of particles with spin $\alpha$ pumped from one
edge to the other when a unit flux is inserted adiabatically, and so
$A^{\alpha}$ is equivalent to the Chern numbers $C_{\alpha}$. On the other
hand, the $Z_{2}$ index can be defined as the parity of $A^{\alpha}$,
$A_{Z_{2}}\equiv A^{\alpha}mod\ 2,$ (10)
for any $\alpha$. As shown in Figs. 4, $A_{Z_{2}}=1$ for QSH phase [Figs.
4(a-c)] and $A_{Z_{2}}=0$ for insulator phase [Figs. 4(d)]. Therefore, the
subsystem particle number expectation can be used to characterize the
topological invariants. Especially, for the QSH systems, the spin trace
indices are well-defined quantities that can reveal the $Z_{2}$ invariant and
distinguish different quantum phases.
## IV Entanglement entropy and subsystem particle number fluctuation
We have shown that topological properties of the ground state can be extracted
from the expectation of subsystem particle number. Now we turn to the variance
of $N_{A}(k_{y})$. In the past two years, extensive works have been devoted to
the study of the relation between the EE and subsystem particle fluctuation
for non-topological systems fluc1 . In this section, we will show that the
relation is rather general, it does apply to non-interacting electron systems
with a nontrivial band topology. We start from the definition of the variance
$\displaystyle\triangle N^{2}_{A}(k_{y})=\langle
N^{2}_{A}(k_{y})\rangle-\langle N_{A}(k_{y})\rangle^{2}\ .$ (11)
Substituting Eq. (7) into Eq. (11) and using the Wick’s theorem to expand all
the four-point correlators, one can obtain
$\displaystyle\triangle N^{2}_{A}(k_{y})$ $\displaystyle=\sum_{i,j\in
A}\langle c^{\dagger}_{i,k_{y}}c_{j,k_{y}}\rangle\langle
c_{j,k_{y}}c^{\dagger}_{i,k_{y}}\rangle$ $\displaystyle=\mbox{Tr}[{\cal
C}(1-{\cal C})]\ ,$ (12)
yielding $\triangle N^{2}_{A}(k_{y})=\sum_{i}\zeta_{i}(1-\zeta_{i})$, which is
in keeping with the variance formula of the Bernoulli distributions. It then
follows that the variance is also dominated by the low-energy boundary
excitations ($0<\zeta_{i}<1$). Moreover, each maximally entangled state with
$\varepsilon_{m}=0$ $(\zeta_{m}=1/2)$ contributes a maximal value to the
subsystem particle number fluctuation and the EE, which cannot be eliminated
by adiabatic continuous deformation.
Figure 5: (Color online) Entanglement entropy in comparison with subsystem
particle number fluctuation for the QSH phase (upper row), the insulator phase
(middle row), and the quantum anomalous Hall phase (lower row), in the
cylinder geometry (left panels) and torus geometry (right panels). All the
parameters are same as those in Fig. 2.
In order to find a definite relationship between the EE and the variance, one
can construct a concave function $f(x)=-\ln x/(1-x)$ for $x\in[0,1]$, and
apply the Jensen’s inequality
$-x\ln x-(1-x)\ln(1-x)\geqslant(4\ln 2)\cdot x(1-x)\ .$ (13)
The equality is taken if and only if $x=1/2$. Equation (13) enables us to make
a lower-bound estimation of the EE
$s_{ent}(k_{y})\geqslant(4\ln 2)\cdot\triangle N^{2}_{A}(k_{y})\ .$ (14)
Thus a lower bound of the EE is given by $s_{0}(k_{y})\equiv(4\ln
2)\cdot\triangle N^{2}_{A}(k_{y})$, which is directly proportional to the
particle number fluctuation of subsystem. In Fig. 5 we plot $s_{ent}(k_{y})$
and $s_{0}(k_{y})$ in the QSH phase, insulator phase, and quantum anomalous
Hall phase. In all the cases, the curves for the particle number fluctuation
behave somewhat similarly, and are very close to the corresponding EE. This
similarity was observed in the non-topological systems lately fluc1 , and here
we find that the similarity remains to hold for the topologically nontrivial
system.
Furthermore, one can use $N_{A}(k_{y})=\sum_{i\in
A}c^{\dagger}_{i,k_{y}}c_{i,k_{y}}$ to verify $\triangle
N^{2}_{A}=\sum_{k_{y}}\triangle
N^{2}_{A}(k_{y})\rightarrow\frac{L_{y}}{2\pi}\int dk_{y}N^{2}_{A}(k_{y})$,
indicating that $\triangle N^{2}_{A}(k_{y})$ satisfies a area law area ,
similar to the EE,
$S_{ent}=\sum_{k_{y}}s_{ent}(k_{y})\rightarrow\frac{L_{y}}{2\pi}\int
dk_{y}s_{ent}(k_{y})$. Therefore, the subsystem particle number fluctuation
shares several common characteristics with the EE, and so can be utilized to
detect the EE experimentally.
## V Summary
To conclude, we have investigated the relationship between the quantum
entanglement and subsystem particle number. The spin trace indices can reveal
the topological invariants and be used to classify different phases in QSH
systems. This new tool always works well even though $s_{z}$ is not conserved.
As to the subsystem particle number fluctuation, it shares several common
properties with the EE. They both satisfy the same area law, and are dominated
by the boundary excitations with each zero mode having a maximal contribution.
The connection between the two quantities is universal, regardless of whether
the system has a nontrivial band topology. As a result, the subsystem particle
number fluctuation, as an observable quantity, can be used to detect the EE
experimentally fluc1 .
## VI ACKNOWLEDGMENTS
This work is supported by the State Key Program for Basic Researches of China
under Grants Nos. 2009CB929504 (LS), 2011CB922103, and 2010CB923400 (DYX), the
National Natural Science Foundation of China under Grant Nos. 11074110 (LS),
11074111 (RS), 11174125, 11074109, 91021003 (DYX), and a project funded by the
PAPD of Jiangsu Higher Education Institutions.
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|
arxiv-papers
| 2011-11-03T11:01:45 |
2024-09-04T02:49:23.952983
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Y. F. Zhang, L. Sheng, R. Shen, Rui Wang, D. Y. Xing",
"submitter": "Yi Fu Zhang PhD",
"url": "https://arxiv.org/abs/1111.0791"
}
|
1111.0825
|
# Scattering of Scalar Waves by Schwarzschild Black Hole Immersed in Magnetic
Field
Juhua Chen jhchen@hunnu.edu.cn Hao Liao Yongjiu Wang College of Physics and
Information Science, Key Laboratory of Low Dimensional Quantum Structures and
Quantum Control of Ministry of Education, Hunan Normal University, Changsha,
Hunan 410081
###### Abstract
The magnetic field is one of the most important constituents of the cosmic
space and one of the main sources of the dynamics of interacting matter in the
universe. The astronomical observations imply the existence of a strong
magnetic fields of up to $10^{4}-10^{8}G$ near supermassive black holes in the
active galactic nuclei and even around stellar mass black holes. In this
paper, with the quantum scattering theory, we analysis the Schröedinger-type
scalar wave equation of black hole immersed in magnetic field and numerically
investigate its absorption cross section and scattering cross section. We find
that the absorption cross sections oscillate about the geometric optical value
in the high frequency regime. Furthermore in low frequency regime, the
magnetic field makes the absorption cross section weaker and this effect is
more obviously on lower frequency brand. On the other hand, for the effects of
scattering cross sections for the black hole immersed in magnetic field, we
find that the magnetic field makes the scattering flux weaker and its width
narrower in the forward direction. We find that there also exists the glory
phenomenon along the backforward direction. At fixed frequency, the glory peak
is higher and the glory width becomes narrower due to the black hole immersed
in magnetic field.
Keywords: absorption cross section, scattering cross section, magnetic field.
PACS numbers: 04.70.-s, 04.40.-b,04.62.+v
## I Introduction
It is well known that general relativity and quantum mechanics are
incompatible in their current form. However, after Hawking found that black
holes can emit, as well as scatter, absorb, and that the evaporation rate is
proportional to the total absorption cross section. A lot of scholars are
interest in the absorption of quantum fields by black hole since 1970s. By
using numerical methods, Sanchez Sanchez1 ; Sanchez found that the absorption
cross section of massless scalar wave exhibits oscillation around the
geometry-optical limit characteristic of diffraction patterns by Schwarzschild
black hole. Unruh Unruh showed that the scattering cross section for the
fermion is exactly 1/8 of that for the scalar wave in the low-energy limit. By
numerically solving the single-particle Dirac equation in Painlevé-Gullstrand
coordinates, Chris Doran et al Doran studied the absorption of massive spin-
half particle by a small Schwarzschild black hole and they found oscillations
around the classical limit whose precise form depends on the particle mass.
Crispino et al Crispino1 have computed numerically the absorption cross
section of electromagnetic waves for arbitrary frequencies and have found that
its high-frequency behavior is very similar to that for massless scalar field
by Schwarzschild black hole. In last several years, Oliveria et al Oliveira
extended to study the absorption of planar in a draining bathtub, the
absorption cross section of sound waves with arbitrary frequencies in the
canonical acoustic hole spacetime Crispino and electromagnetic absorption
cross section from Reissner-Nordström black holes Crispino2 . Recently,
absorption cross section (or gray body factors) has been of interest in the
context of higher-dimensional using standard field theory in curved spacetimes
Das ; Das1 ; Crispino4 and effective string model Gubser .
The magnetic field is one of the most important constituents of the cosmic
space and one of the main sources of the dynamics of interacting matter in the
universe. In addition some other theories Tyulbashev ; Zhang ; Han imply the
existence of a strong magnetic fields of up to $10^{4}-10^{8}G$ near
supermassive black holes in the active galactic nuclei and even around stellar
mass black holes. In order to make estimations of possible influence of the
magnetic field on the supermassive black holes, we need the two parameters at
hand: the magnetic field parameter $B$ and the mass of the black hole $M$.
Interaction of a black hole and a magnetic field can happen in a lot of
physical situations: when an accretion disk or other matter distribution
around black hole is charged; when taking into consideration galactic and
intergalactic magnetic fields, and, possibly, if mini-black holes are created
in particle collisions in the brane-world scenarios. So astrophysics have
highly interest to investigate the magnetic fields around black holes
Konoplya1 . A magnetic field is important as a background field testing black
hole geometry. A magnetic field near a black hole leads to a number of
processes, such as extraction of rotational energy from a black hole, known as
the Blandford-Znajek effect Blandford , negative absorption (masers) of
electrons Aliev . At the classical level, the magnetic perturbation can also
be described by its damped characteristic modes, which called the quasinormal
modes (QNMs) Kokkotas ; Kokkotas1 ; Noller ; Konoplya2 which could be
observed in experiments, and by the scattering properties, which are encoded
in the S-matrix of the perturbation. All of these effects are usually called
the ”fingerprints” of a black hole. In recent few years, we all know that
quasinormal modes of black holes has gained considerable attention because of
their applications in string theory through the AdS/CFT correspondence.
In this paper we mainly focus on the scalar scattering process of black hole
immersed in magnetic field and how the interaction of black hole and strong
magnetic field effects on scalar absorption and scattering cross sections. The
outline of this paper is as follows: In Sec.II, we set up scalar field
equation black holes immersed in a magnetic field and analysis effective
potential. In the Sec. III and IV, we concentrate on the absorption and
scattering cross section of the scalar wave by black holes immersed in a
magnetic field. In the last section, a brief conclusion is given.
## II Scalar Field Equation and Effective Potential
The Diaz and Ernst solution Ernst describing the black holes immersed in a
magnetic field takes the follow form:
$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\Lambda^{2}[(1-\frac{2M}{r})dt^{2}+(1-\frac{2M}{r})^{-1}dr^{2}$
(1) $\displaystyle-$ $\displaystyle
r^{2}d\theta^{2}]-\frac{r^{2}sin^{2}\theta}{\Lambda^{2}}d\phi^{2},$
where the external magnetic field is determined by the parameter $B$
$\displaystyle\Lambda=1+\frac{1}{4}B^{2}r^{2}sin^{2}\theta,$ (2)
and the unit magnetic field measured in $Gs$ is $B_{M}=1/M=2.4\times
10^{19}\frac{M_{Sun}}{M}$.
The general perturbation equation for the massless scalar field $\Psi$ in the
curve spacetime is given by
$\displaystyle\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu})\Psi=0.$
(3)
For very strong magnetic fields in centres of galaxies or in colliders,
corresponds to $M\ll M$ in our units, so that one can safely neglect terms
higher than $B^{2}$ in Eq.(3). Indeed, in the expansion of $\Lambda^{4}$ in
powers of B, the next term after that proportional to $B^{2}r^{2}$, is $\sim
B^{4}r^{4}$ and, thereby, is very small in the region near the black hole. The
term $B^{4}r^{4}$ is growing far from black hole, and, moreover the potential
in the asymptotically far region is diverging, what creates a kind of
confining by the magnetic field of the Ernst solution. This happens because
the non-decaying magnetic field is assumed to exist everywhere in the
universe. Therefore it is clear that in order to estimate a real astrophysical
situation, one needs to match the Ernst solution with a Schwarzschild solution
at some large r. Fortunately we do not need to do this for the scattering
problem: the scattering properties of astrophysical interest is stipulated by
the behavior of the effective potential in some region near black hole, while
its behavior far from black hole is insignificant Konoplya . In this way we
take into consideration only dominant correction due-to magnetic field to the
effective potential of the Schwarzschild black hole. By neglecting terms
$B^{4}$ and higher order terms and separating the angular variables, we reduce
the wave equation (3) to the Schröedinger wave equation The Klein-Gordon
equation can be written in the spacetime (1) as
$\displaystyle\frac{1}{1-\frac{2M}{r}}\frac{\partial^{2}\Psi}{\partial
t^{2}}-\frac{1}{r^{2}}\frac{\partial}{\partial
r}[(1-\frac{2M}{r})r^{2}\frac{\partial\Psi}{\partial
r}]+\frac{1}{r^{2}}\nabla^{2}\Psi=0.$ (4)
The positive-frequency solutions of Eq.(4) take as follows
$\displaystyle\Psi_{\omega lm}=[\psi_{\omega l}(r)/r]Y_{lm}e^{-i\omega t},$
(5)
where $Y_{lm}$ are scalar spherical harmonic functions and $l$ and $m$ are the
corresponding angular momentum quantum numbers. In this case, the functions
$\psi_{lm}(r)$ satisfy the follow differential equation
$\displaystyle(1-\frac{2M}{r})\frac{d}{dr}[(1-\frac{2M}{r})\frac{d\psi_{\omega
l}}{dr}]+[\omega^{2}-V^{(l)}_{eff}(r)]\psi_{\omega l}=0,$ (6)
where
$\displaystyle V^{(l)}_{eff}(r)$ $\displaystyle=$
$\displaystyle(1-\frac{2M}{r})[\frac{l(l+1)}{r^{2}}+\frac{2M}{r^{3}}+4B^{2}m^{2}].$
(7)
Figure 1: (color online). The effective scattering potential $V_{eff}(r)$
given by Eq.7 for scalar waves by the black hole immersed in magnetic field
with $l=0,1,2$ for $B=0$ (red solid line, i.e. Schwarzschild case) and
$B=0.08$ (blue dashed line), and $B=0.12$ (black dotted line).
Figure 2: (color online). The effective scattering potential $V_{eff}(x)$
given by Eq.7 for scalar waves by the black hole immersed in magnetic field in
tortoise coordinate with $l=0,1,2$ for $B=0$ (red solid line, i.e.
Schwarzschild case) and $B=0.08$ (blue dashed line), and $B=0.12$ (black
dotted line). From this figure, we can see the effective scattering potential
$V_{eff}(x)$ act as the typical scattering barrier in quantum mechanics
theory.
The effective potential $V^{(l)}_{eff}(r)$ is plotted in Fig.1 for $l=0,1,2$.
From this figure, we can see that the effective potential $V^{(l)}_{eff}(r)$
depends only on the values of $r$, angular quantum number $l$, ADM mass $M$,
magnetic field $B$, respectively, and that the peak value of potential barrier
gets upper and the location of the peak point ($r=r_{p}$) moves along the
right when the angular momentum $l$ increases. We can find that the the height
of the effective scattering potential increases as the angular momentum $l$
increases. If we introduce the tortoise coordinate
$\displaystyle x=\int{(1-\frac{2M}{r})^{-1}dr},$ (8)
The effective potentials $V^{(l)}_{eff}(r)$ are changed into
$V^{(l)}_{eff}(x)$, which are showed in Fig.2 for $l=0,1,2$, it’s obvious that
they act as the typical scattering barriers in quantum mechanics theory. We
see that the peak value of potential barrier gets upper and the location of
the peak point ($x=x_{p}$) moves along the right when the angular momentum $l$
increases. We also find that the height of the effective scattering barrier
increases as the magnetic field $B$ increases, at the same time we can see
that the height of the effective scattering barrier, affecting by the magnetic
field,becomes higher than that of Schwarzschild black hole.
After introducing this coordinate transition, we can obtain the following
Schrödinger-type equation
$\displaystyle\frac{d^{2}\psi_{\omega
l}}{dx^{2}}+[\omega^{2}-V^{(l)}_{eff}(x)]\psi_{\omega l}=0.$ (9)
The perturbation must be purely ingoing at the black hole event horizon
$r=r_{+}$. So while $r\rightarrow r_{+}$ i.e. $x\rightarrow-\infty$, we impose
the boundary condition
$\displaystyle\psi_{\omega l}=A^{tr}_{\omega l}e^{-i\omega x},$ $\displaystyle
for$ $\displaystyle x\rightarrow-\infty.$ (10)
It is straightforward to check that in the original coordinate system (1) the
ingoing solution $e^{-i\omega x}$ is well defined at $r=r_{+}$, whereas the
out going solution $e^{+i\omega x}$ is divergent. Towards spatial infinity,
the asymptotic form of the solution is
$\displaystyle\psi_{\omega l}$ $\displaystyle=$ $\displaystyle\omega
x[A^{in}_{\omega l}(-i)^{l+1}h^{(1)\ast}_{l}(\omega x)+A^{out}_{\omega
l}(i)^{l+1}h^{(1)}_{l}(\omega x)]$ (11) $\displaystyle for$ $\displaystyle
x\rightarrow+\infty,$
where $h^{(1)}_{l}(\omega x)$ are spherical Bessel functions of the third kind
Abrammowitz , at the same time $A_{in}$ and $A_{out}$ are complex constants.
We note that $h^{(1)}_{l}(\omega x)\approx(-i)^{l+1}e^{ix}/x$ as
$x\rightarrow\infty$ and that the effective potential goes to zero as
$x\rightarrow-\infty$, so we obtain
$\displaystyle\psi_{\omega
l}\approx\bigg{\\{}\begin{array}[]{rrrr}A^{tr}_{\omega l}e^{-i\omega
x},&for&x\rightarrow-\infty;\\\ A^{in}_{\omega l}e^{-i\omega
x}+A^{out}_{\omega l}e^{+i\omega x},&for&x\rightarrow+\infty.\end{array}$ (14)
with the conserved relation
$\displaystyle|A^{tr}_{\omega l}|^{2}+|A^{out}_{\omega l}|^{2}=|A^{in}_{\omega
l}|^{2}$ (15)
The phase shift $\delta_{l}$ is defined by
$\displaystyle e^{2i\delta_{l}}=(-1)^{l+1}A_{out}/A_{in}.$ (16)
In order investigate the absorption cross section and scattering cross
section, we must numerically solve the radial equation (9) under the boundary
conditions Eq.(10) and Eq.(11), then compute the ingoing and outgoing
coefficients $A^{in}_{\omega l}$ and $A^{out}_{\omega l}$ by matching onto
Eq.(16) to give out the numerical phase shift.
## III Absorption cross section
Base on the quantum mechanics theory, we know that the total absorption cross
section is
$\displaystyle\sigma_{abs}=\frac{\pi}{\omega^{2}}\sum_{l=0}^{\infty}(2l+1)(1-|e^{2i\delta_{l}}|^{2}),$
(17)
so we can define the partial absorption cross section as
$\displaystyle\sigma^{(l)}_{abs}=\frac{\pi}{\omega^{2}}(2l+1)(1-|e^{2i\delta_{l}}|^{2}),$
(18)
and the absorption cross section have relation
$\displaystyle\sigma_{abs}(\omega)=\sum_{l=0}^{\infty}\sigma^{(l)}_{abs}(\omega)=\frac{\pi}{\omega^{2}}\sum_{l=0}^{\infty}(2l+1)|T_{\omega
l}|^{2}.$ (19)
By using mathematica program, we straightforwardly compute values of ingoing
and outgoing coefficients $A^{in}_{\omega l}$ and $A^{out}_{\omega l}$. Then
from Eq.(16), Eq.(17)and Eq.(18), we can simulate the partial absorption cross
sections and their total absorption cross sections of the scalar field from
the black hole immersed in magnetic field.
In Fig.3 we show the partial absorption cross sections $\sigma_{abs}^{(l)}$,
i.e. $l=0,1,2$, by the black hole immersed in magnetic field for different
magnetic parameters $B=0.1$ and $\Lambda=0,0.08$ and $0.12$. We find that the
S-wave $(l=0)$ contribution is responsible for the nonvanishing cross section
in the zero-energy limit. Furthermore, by comparing different $l$ partial
absorption cross section curves, we find that the larger the value of $l$ is,
the smaller the corresponding value of $\sigma_{abs}^{(l)}$ is. This is
compatible with the fact that the scattering barrier $V_{eff}$ is bigger or
larger values of $l$, which is showed in Fig.1 and Fig.2. These properties are
similar to other black hole scattering systemSanchez ; Crispino ; Crispino3 .
On the other hand with phase-integral method, Andersson Andersson had gotten
very similar results (see Fig.7 therein).
In order to consider effects of magnetic field on the partial absorption cross
section. In Fig.4 we plot the partial absorption cross section for $l=0,1,2$
with $B=0$ ( i.e. Schwarzschild black hole case), $B=0.08$ and $B=0.12$. We
see that the magnetic field make the absorption weaker, even for low frequency
mode. This is agree with the fact that the magnetic field is stronger, the
higher value of the effective scattering barrier peak is for a fixed value of
$l$, which can be seen in Fig.1 and Fig.2. But for high enough values of the
frequency, the magnetic field does not effect the partial absorption cross
section obviously. From Eq.(19), we know the absorption cross section have
relation with the transmission coefficients Decanini1 . This feature can be
tested the transmission coefficients in Fig.5, where we find that high enough
values of the frequency all transmission coefficients with fixed $l$ tend to
the unity. These properties help us understand the absorption process better.
Figure 3: (color online). The behavior of the partial absorption cross section
$\sigma_{abs}^{(l)}$, from $l=0$ to $l=5$ for scalar waves by the black hole
immersed in magnetic field.
Figure 4: (color online). The behavior of the partial absorption cross section
$\sigma_{abs}^{(l)}$, from $l=0,1,2$ for scalar waves by the black hole
immersed in magnetic field with $B=0$ (red solid line, i.e. Schwarzschild
black hole case) and $B=0.08$ (blue dashed line), and $B=0.12$ (black dotted
line).
Figure 5: (color online). The transmission coefficients with $l=1,l=2$ are
showed for different magnetic field with $B=0$ (red solid line, i.e.
Schwarzschild black hole case) and $B=0.08$ (blue dashed line), and $B=0.12$
(black dotted line).
Figure 6: (color online). The behavior of the partial absorption cross section
$\sigma_{abs}^{(l)}$ and $\sigma_{abs}^{total}$ by the black hole immersed in
magnetic field with $B=0$ (top-left i.e. Schwarzschild black hole case),
$B=0.08$ (top-right), $B=0.12$ (bottom-left), and their corresponding total
absorption cross sections $\sigma_{abs}^{total}$ (bottom-right).
Figure 7: (color online). The behavior of the partial scattering cross
sections $\sigma^{(l)}_{sca}$, from $l=1$ to $l=6$, at $M\omega=1$ for the
scalar wave is scattered by the black hole immersed in magnetic field with
$B=0.2$.
In Fig.6 we plot total absorption cross sections $\sigma_{abs}$ which
contribute from $l=0$ to $l=5$ by the black hole immersed in magnetic field
with fixed parameters $B=0$ ( i.e. Schwarzschild case), $B=0.08$ and $B=0.12$.
We can see that I) between the intermediate regime $\omega M\sim(0.4,1)$, the
contributions from the partial absorption sections create a regular
oscillatory pattern. Each maximum in the oscillation of the total absorption
cross section is linked to the maximum of a particular partial wave. II) If
the wavelength of the incoming wave is much smaller than the black hole
horizon (i.e. $\omega M>>1$), the absorption cross section tends to the
geometry-optical limit of $\sigma_{abs}^{hf}=\pi b^{2}_{c}$. This is verified
by the total absorption cross section for the massless scalar field which was
computed by Sanchez Sanchez in last century. At the same time, these
properties are also found for electromagnetic wave absorption cross section
Crispino1 and for Fermion absorption cross section in the Schwarzschild black
hole Doran .
In bottom-left position of Fig.6, we plot total absorption cross sections for
different values of magnetic parameters. We also consider the contributions of
the angular momentum from $l=0$ to $l=5$ in Eq.(18). We can see that big
values of the magnetic parameter $B$ correspond to low total absorption cross
section which is consistent with the fact of the partial section in Fig.4. and
the scattering barrier which is showed in Fig.1 and 2. But we can find that
the absorption cross sections oscillate about the geometric optical value in
the high frequency regime. However in low frequency regime, the magnetic field
makes the absorption cross section weaker, i.e. the magnetic makes obvious
effect on lower frequency brand, not on high frequency brand. We note that
this is a general result for massless scalar waves in Reissner-Nordström black
hole Crispino3 and for the minimally-coupled massless scalar wave in
stationary black hole spacetimes Higuchi . There are similar properties for
total absorption section from the charged black hole coupling to Born-Infeld
electrodynamics chen and dark energy Liao .
## IV Scattering cross section
From the quantum mechanics theory, it’s well known that the scattering
amplitude is expressed as
$\displaystyle
f(\theta)=\frac{1}{2i\omega}\sum_{l=0}^{\infty}(2l+1)[e^{2i\delta_{l}}-1]P_{l}(cos\theta).$
(20)
From this scattering amplitude, we can give the differential scattering cross
section immediately
$\displaystyle\frac{d\sigma}{d\Omega}=|f(\theta)|^{2}.$ (21)
At last we can define the scattering and absorption cross sections Gotfried ;
Dolan
$\displaystyle\sigma_{sca}$ $\displaystyle=$
$\displaystyle\int\frac{d\sigma}{d\Omega}d\Omega=\frac{\pi}{\omega^{2}}\sum_{l=0}^{\infty}(2l+1)|e^{2i\delta_{l}}-1|^{2},$
(22)
so the partial scattering cross section is
$\displaystyle\sigma^{(l)}_{sca}=\frac{\pi}{\omega^{2}}(2l+1)|e^{2i\delta_{l}}-1|^{2}.$
(23)
In order to simulate the scattering cross sections (22)-(23), we must
numerically solve differential equation (9) under boundary conditions (10) and
(11), to obtain numerical values for the phase shifts via Eq.(16).
Figures 7 show the partial scattering cross section a function of angle for
six different partial waves from $l=1$ to $l=6$. By comparing these figures,
we can see that, when the $L$ increases, the flux is preferentially scattered
in the forward direction, i.e. the scattering angle width become narrower. At
the time a more complicated pattern arises and we find a damping oscillation
pattern. The similar properties are observed for black hole scattering Dolan ;
Futterman ; Dolan1 . The explanation for the physical origin of the
oscillations can be found in Ref.Matzner .
Figures 8, 9 and 10 compare the scattering cross sections for the Scharzschild
black hole with the black hole immersed in magnetic field with $B=0.2$ and
$0.3$. We find that the magnetic field makes the scattering flux weaker and
its width narrower in the forward direction. In the other words, the scalar
field scattering becomes more diffusing due to the black hole immersed in
magnetic field. In Fig.10 we can see that there exists the glory phenomenon
along the backforward direction Dolan ; Crispino3 . At fixed frequency, the
glory peak is higher and the glory width becomes narrower due to the black
hole immersed in magnetic field. So we can find that even the scalar field
scattering becomes more diffusing due to the black hole immersed in magnetic
field, but the glory phenomenon along the backforward direction becomes better
for astronomy observation.
Figure 8: (color online). The behavior of the total scattering cross sections
$\sigma^{(l)}_{sca}$ at $M\omega=1$ between ($-180^{\circ}$-$180^{\circ}$) for
the scalar wave is scattered by the black hole immersed in magnetic field with
$B=0$ (red solid line, i.e. Schwarzschild case) and $B=0.2$ (blue dashed
line), and $B=0.3$ (black dotted line).
Figure 9: (color online). The behavior of the total scattering cross sections
$\sigma^{(l)}_{sca}$ at $M\omega=1$ between ($0^{\circ}$-$180^{\circ}$) for
the scalar wave is scattered by the black hole immersed in magnetic field with
$B=0$ (red solid line, i.e. Schwarzschild case) and $B=0.2$ (blue dashed
line), and $B=0.3$ (black dotted line).
Figure 10: (color online). The behavior of the total scattering cross sections
$\sigma^{(l)}_{sca}$ at $M\omega=1$ between ($60^{\circ}$-$180^{\circ}$) for
the scalar wave is scattered by the black hole immersed in magnetic field with
$B=0$ (red solid line, i.e. Schwarzschild case) and $B=0.08$ (blue dashed
line), and $B=0.12$ (black dotted line).
## V Conclusions
In this paper we have investigated the scattering and absorption cross section
of the scalar wave by the black hole immersed in magnetic field. We found that
the magnetic parameter $B$ makes the absorption cross section lower which is
consistent with the fact of the scattering barrier which is showed in Fig.1
and 2. We also found that the absorption cross sections oscillate about the
geometric optical value in the high frequency regime. However in low frequency
regime, the magnetic field makes the absorption cross section weaker and this
effect is more obviously on lower frequency brand. For the effects of the
scattering cross sections for the black hole immersed in magnetic field, we
found that the magnetic field makes the scattering flux weaker and its
scattering width narrower in the forward direction. At the same time we found
that there also exists the glory phenomenon along the backforward direction.
At fixed frequency, the glory peak is higher and the glory width becomes
narrower due to the black hole immersed in magnetic field. So the glory
phenomenon along the backforward direction becomes better for astronomy
observation.
Just as the Brazil physicist Crispino et al Crispino3 have pointed out: ” In
principle, highly accurate measurements of, for example, the gravitational
wave flux scattered by a black hole could one day be used to estimate the
black hole s charge. A more immediate possibility is that scattering and
absorption patterns may be observed with black hole analog systems created in
the laboratory. Even if experimental verification is not forthcoming, we hope
that studies of wave scattering by black holes will continue to improve our
understanding of how black holes interact with their environments.”
## VI Acknowledgments
This project is supported by the National Natural Science Foundation of China
under Grant No.10873004, the State Key Development Program for Basic Research
Program of China under Grant No.2010CB832803 and the Program for Changjiang
Scholars and Innovative Research Team in University, No. IRT0964.
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|
arxiv-papers
| 2011-11-03T12:57:21 |
2024-09-04T02:49:23.960214
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Juhua Chen, Hao Liao and Yongjiu Wang",
"submitter": "Juhua Chen",
"url": "https://arxiv.org/abs/1111.0825"
}
|
1111.0841
|
# A non explicit counterexample to a problem of quasi-normality
Shahar Nevo Shahar Nevo
Department of Mathematics
Bar-Ilan University, 52900 Ramat-Gan, Israel nevosh@macs.biu.ac.il and
Xuecheng Pang Xuecheng Pang
Department of Mathematics, East China Normal University,Shanghai 200062, P. R.
China xcpang@euler.math.ecnu.edu.cn
###### Abstract.
In 1986, S.Y. Li and H. Xie proved the following theorem: Let $k\geq 2$ and
let $\mathcal{F}$ be a family of functions meromorphic in some domain $D,$ all
of whose zeros are of multiplicity at least $k.$ Then $\mathcal{F}$ is normal
if and only if the family
$\mathcal{F}=\left\\{\frac{f^{(k)}}{1+(f)^{k+1}}:f\in\mathcal{F}\right\\}$ is
locally uniformly bounded in $D.$
Here we give, in the case $k=2,$ a counterexample to show that if the
condition on the multiplicities of the zeros is omitted, then the local
uniform boundedness of $\mathcal{F}_{2}$ does not imply even quasi-normality.
In addition, we give a simpler proof for the Li-Xie theorem that does not use
Nevanlinna’s Theory which was used in the original proof.
###### Key words and phrases:
Quasi-normal family, Zalcman’s Lemma, Differential inequality, Interpolation
theory
###### 2010 Mathematics Subject Classification:
30A10, 30D45
## 1\. Introduction
Marty’s Theorem characterizes normality by using the first derivative and it
has an obvious geometrical meaning.
H.L. Royden, [3], extended one direction of Marty’s Theorem and proved
###### Theorem 1.
Let $\mathcal{F}$ be a family of meromorphic functions in $D,$ with the
property that for each compact set $K\subset D,$ there is a positive
increasing function $h_{K}$ such that
(1) $|f^{\prime}(z)|\leq h_{K}(|f(z)|)$
for all $f\in\mathcal{F}$ and $z\in K$. Then $\mathcal{F}$ is normal in $D.$
This result was extended further in various directions. In [1], (1) is limited
to only 5 values. In [4, Thm.2], $h_{K}$ is replaced by a nonnegative function
that needs to be bounded in a neighborhood of some $x_{0},$ $0\leq
x_{0}<\infty.$ Then, in [7] it was shown that it is enough that $h_{K}$ be
finite only in a single point $x_{0},$ $x_{0}>0<\infty.$ Moreover, in [4,
Thm.3], this result is extended further to higher derivatives, i.e., (1) is
replaced by $|f^{(\ell)}(z)|\leq h_{K}(|f(z)|)$, $f\in\mathcal{F},$ $z\in K,$
where $\ell\geq 2$ and the members of $\mathcal{F}$ have zeros of multiplicity
$\geq l.$ The following generalization of Marty’s Theorem also deals with
higher derivatives.
###### Theorem 2.
[2] Let $\mathcal{F}$ be a family of functions meromorphic on $D$ such that
each $f\in\mathcal{F}$ has zeros only of multiplicity $\geq k$. Then
$\mathcal{F}$ is normal in $D$ if and only if the family
(2)
$\mathcal{F}_{k}=\left\\{\dfrac{f^{(k)}}{1+|f^{k+1}|}:f\in\mathcal{F}\right\\}\quad\text{is
locally uniformly bounded in $D$.}$
The direction $(\Rightarrow)$ in Theorem 2 is true even without the assumption
that the zeros of each $f\in\mathcal{F}$ are of multiplicity at least $k$. In
Section 2, we give a simpler proof for Theorem 2, without using Nevanlinna’s
Theory. The condition on the multiplicities of $f\in\mathcal{F}$ is essential
in the direction $(\Leftarrow)$. Indeed, let $\hat{\mathcal{F}}_{k}$ be the
family of all polynomials of degree at most $k-1$ in some domain
$D\subset\mathbb{C}.$ Then $\frac{f^{(k)}}{1+|f|^{k+1}}=0$ for each
$f\in\hat{\mathcal{F}}_{k}$, but $\hat{\mathcal{F}}_{k}$ is not normal in $D.$
However, $\hat{\mathcal{F}}_{k}$ is a quasi-normal family in $D$ (of order
$k-1).$ The question that naturally arises is whether the condition (2)
implies quasi-normality.
The conjecture that (2) implies quasi-normality (without the assumption on the
multiplicities of the zeros) gets support also from another direction.
First let us set some notation. For $z_{0}\in\mathbb{C}$ and $r>0,$
$\Delta(z_{0},r)=\\{z:|z-z_{0}|<r\\}.$ We write
$f_{n}\overset{\chi}{\Longrightarrow}f$ on $D$ to indicate that the sequence
$\\{f_{n}\\}$ converges to $f$ in the spherical metric uniformly on compact
subsets of $D$ and $f_{n}\Rightarrow f$ on $D$ if the convergence is in the
Euclidean metric.
Let us recall the well-known result of L. Zalcman.
###### Lemma 1 (Zalcman’s Lemma).
[6] A family $\mathcal{F}$ of functions meromorphic in some domain $D$ is not
normal at $z_{0}\in D$ if and only if there exist points $z_{n}$ in $D,$
$z_{n}\to z_{0};$ numbers $\rho_{n}\to 0^{+}$, and functions
$f_{n}\in\mathcal{F}$ such that
(3)
$f_{n}(z_{n}+\rho_{n}\zeta)\overset{\chi}{\Rightarrow}g(\zeta)\quad\text{in}\quad\mathbb{C},$
where $g$ is a nonconstant meromorphic function in $\mathbb{C}.$
Now, suppose that $g$ is a limit function from (3), and we have some $C>0$ and
$r>0$ such that
(4) $\frac{|f_{n}^{(k)}(z)|}{1+|f_{n}(z)|^{k+1}}\leq C\quad\text{for
every}\quad z\in\Delta(z_{0},r)\quad\text{and}\quad n\in\mathbb{N}.$
Let us denote the poles of $g$ (if any) by $P_{g}.$ Then
(5) $f_{n}(z_{n}+\rho_{n}\zeta)\Rightarrow
g(\zeta)\quad\text{on}\quad\mathbb{C}\setminus P_{g}.$
(Here we substitute “$\overset{\chi}{\Rightarrow}$” by “$\Rightarrow$” since
in every compact subset of $C\setminus P_{g}$, $f_{n}(z_{n}+\rho_{n}\zeta)$ is
holomorphic for large enough $n).$
Differentiating (5) $k$ times given
$\rho_{n}^{k}f_{n}^{(k)}(z_{n}+\rho_{n}\zeta)\Rightarrow
g^{(k)}(\zeta)\quad\text{in}\quad\mathbb{C}\setminus P_{g}.$
But then by (3) and (4), we get that $g^{(k)}\equiv 0$ in $\mathbb{C}\setminus
P_{g}$ and so $g^{(k)}\equiv 0$ in $\mathbb{C}.$ This implies that $g$ is a
polynomial of degree at most $k-1.$ Hence, we get that the collection of all
limit functions obtained by (3) is a quasi-normal family.
However, it turns out that without the condition on the multiplicities of the
zeros, the family $\mathcal{F}$ of Theorem 2 is not quasi-normal.
We suffice to construct a detailed counterexample for the case $k=2.$ This is
the content of Section 3.
## 2\. Proof of Theorem 2
Assume first that $\mathcal{F}$ is locally uniformly bounded in $D,$ and
suppose by negation that $\mathcal{F}_{k}$ is not normal at some $z_{0}\in D.$
Then similarly to (3) we get the existence of $f_{n},z_{n},\rho_{n}$ and $g$
such that $f_{n}(z_{n}+\rho_{n}\zeta)\overset{\chi}{\Rightarrow}g(\zeta)$ in
$\mathbb{C}.$ With the same reasoning, we deduce that $g$ is a polynomial of
degree at most $k-1.$ But now according to the condition on the multiplicities
of the zeros of each $f_{n},$ we get that the zeros of $g$ also must be of
multiplicity at least $k.$ This implies that $g$ has no zeros and thus $g$ is
a constant function, a contradiction.
For the proof of the opposite direction, we need the following lemma.
###### Lemma 2.
Let $\\{f_{n}\\}_{n=1}^{\infty}$ be a sequence of meromorphic functions in a
domain $D,$ satisfying $f_{n}\overset{\chi}{\Rightarrow}\infty$ in $D.$ Then
for every $\ell\in\mathbb{N}$,
$\frac{f_{n}^{(\ell)}}{f_{n}^{\ell+1}}\Rightarrow 0$ in $D.$
###### Proof.
We apply induction. Since $\frac{1}{f_{n}(z)}\Rightarrow 0$ in $D,$ we can
differentiate it and obtain that
$\frac{f_{n}^{\prime}(z)}{f_{n}^{2}(z)}\Rightarrow 0$ in $D,$ and this proves
the case $\ell=1.$
∎
Assume that the lemma holds for $m\leq\ell$. We prove it now for the case
$m=\ell+1.$ We have $\frac{f_{n}^{(\ell)}}{f_{n}^{\ell+1}}(z)\Rightarrow 0$ in
$D,$ and hence, since $f_{n}(z)\Rightarrow\infty$ in $D,$ also
$\frac{f_{n}^{(\ell)}(z)}{f_{n}(z)^{\ell+2}}\Rightarrow 0$ in $D.$
Differentiating the last convergence gives
$\frac{f_{n}^{(\ell+1)}(z)}{f_{n}^{\ell+2}}-(\ell+2)\frac{f_{n}^{\prime}}{f_{n}^{2}}\frac{f_{n}^{(\ell)}}{f_{n}^{\ell+1}}(z)\Rightarrow
0\quad\text{in}\quad D.$
The induction assumption for $m=1$ and $m=\ell$ implies that the right term in
the left hand above converges uniformly to 0 on compacta of $D$, and thus also
$\frac{f_{n}^{(\ell+1)}}{f_{n}^{\ell+2}}(z)\Rightarrow 0$ in $D,$ as required.
Let us prove now the opposite direction of Theorem 2. Assume that
$\mathcal{F}$ is normal in $D$, and suppose by negation that $\mathcal{F}_{k}$
is not locally uniformly bounded in any neighborhood of some $z_{0}\in D.$
Thus, there exist functions $f_{n}\in\mathcal{F},$ and points $z_{n}\to z_{0}$
such that
(6)
$\frac{f_{n}^{(k)}{(z_{n})}}{1+|f_{n}^{k+1}(z_{n})|}\underset{n\to\infty}{\rightarrow}\infty.$
By the normality of $\mathcal{F}$, $\\{f_{n}\\}_{n=1}^{\infty}$ has a
subsequence that, without loss of generality, we also denote by
$\\{f_{n}\\}_{n=1}^{\infty}$, such that $f_{n}\overset{\chi}{\Rightarrow}f$ in
$D.$
We separate now into cases according to the nature of $f.$
Case 1.1 $f(z_{0})\in\mathbb{C}.$
For small enough $r>0$, $f_{n}^{(k)}(z)\Rightarrow f^{(k)}(z)$ in
$\Delta(z_{0},r),$ and also $1+|f_{n}^{k+1}(z)|\Rightarrow 1+|f(z)|^{k+1}$ in
$\Delta(z_{0},r).$ Since $1+|f_{n}(z)|^{k+1}\geq 1,$ we get that
$\frac{f_{n}^{(k)}(z)}{1+|f_{n}(z)|^{k+1}}\Rightarrow\frac{f^{(k)}(z)}{1+|f(z)|^{k+1}}$
in $\Delta(z_{0},r),$ a contradiction to (6).
Case 1.2 $f(z_{0})=\infty.$
Here, for small enough $r>0$, $f$ is holomorphic in $\Delta^{\prime}(z_{0},r)$
and in addition $|f_{n}(z)|\geq 2$ and $|f(z)|\geq 2$ for large enough $n.$
Thus $\frac{f_{n}(z)}{1+f_{n}(z)^{k+1}}$ are holomorphic in $\Delta(z_{0},r)$
for large enough $n.$ We then get by the maximum principle that
$\frac{f_{n}^{(k)}(z)}{1+f_{n}(z)^{k+1}}\Rightarrow\frac{f^{(k)}(z)}{1+f(z)^{k+1}}\quad\text{in}\quad\Delta(z_{0},r)$
and then for large enough $n,$
$\max_{|z-z_{0}|\leq
r/2}\frac{|f_{n}^{(k)}(z)|}{1+|f_{n}(z)|^{k+1}}\leq\max_{|z-z_{0}|\leq
r/2}\frac{|f_{n}^{(k)}(z)|}{|1+f_{n}(z)^{k+1}|}\leq\max_{|z-z_{0}|\leq
r/2}\frac{|f^{(k)}(z)|}{|1+f(z)^{k+1}|}+1.$
The last expression is a positive constant, that does not depend on $n$ and
this is a contradiction to (6).
Case 2 $f=\infty.$
In this case, we get by Lemma 2 that
$\frac{f_{n}^{(k)}(z)}{f_{n}(z)^{k+1}}\Rightarrow 0$ in $D,$ and this is a
contradiction to (6).
## 3\. Constructing the counterexample
We construct a sequence of holomorphic functions $\\{f_{n}\\}_{n=1}^{\infty}$,
such that for every $n\geq 1$ and $z\in\Delta(0,2)$,
$\frac{|f_{n}^{\prime\prime}(z)|}{1+|f_{n}(z)|^{3}}\leq 1$ and
$\\{f_{n}\\}_{n=1}^{\infty}$ is not quasi-normal in $\Delta(0,2).$
Let $g_{n}(z)=z^{n}-1,$ $n\geq 1.$ The zeros of $g_{n}$ are all simple,
$g_{n}(z_{\ell}^{(n)})=0,$ $0\leq\ell\leq n-1,$ where $z_{\ell}^{(n)}$ is the
$\ell$-th root of unity of order $n.$ Define for every $n\geq 1$,
$h_{n}=g_{n}e^{p_{n}},$ where $p_{n}$ is a polynomial to be determined. We
have $h_{n}^{\prime}=(g_{n}^{\prime}+g_{n}p_{n}^{\prime})e^{p_{n}},$ and
$g_{n}^{\prime}(z_{\ell}^{(n)})\neq 0,$ $0\leq\ell\leq n-1.$ We want that
(7)
$p_{n}^{\prime}(z_{\ell}^{(n)})=-g_{n}^{\prime\prime}(z_{\ell}^{(n)})/2g_{n}^{\prime}(z_{\ell}^{(n)}),\quad
0\leq\ell\leq n-1$
to get that $h_{n}^{\prime\prime}(z_{\ell}^{(n)})=0.$
We have
$h_{n}^{(3)}=e^{p_{n}}\big{(}g_{n}^{(3)}+3g_{n}^{\prime\prime}p_{n}^{\prime}+\boldsymbol{3g_{n}^{\prime}p_{n}^{\prime\prime}}+g_{n}p_{n}^{(3)}+3g_{n}^{\prime}p_{n}^{\prime}{{}^{2}}+3g_{n}p_{n}^{\prime}p_{n}^{\prime\prime}+g_{n}p_{n}^{\prime}{}^{3}\big{)}$
We want that
(8)
$p_{n}^{\prime\prime}(z_{\ell}^{(n)})=-(g_{n}^{(3)}+3g_{n}^{\prime\prime}p_{n}^{\prime}+3g_{n}^{\prime}p_{n}^{\prime}{}^{2})/3g_{n}^{\prime}\Big{|}_{z=z_{\ell}^{(n)}},\quad
0\leq\ell\leq n-1$
to get $h_{n}^{(3)}(z_{\ell}^{(n)})=0.$
Observe that when (7) is satisfied to determine
$p_{n}^{\prime}(z_{\ell}^{(n)})$, then as in (7), condition (8) is in fact a
condition that depends only on the values of $g_{n}$ and its derivatives at
the points $z_{\ell}^{(n)},$ $0\leq\ell\leq n-1.$
We have
$\displaystyle h_{n}^{(4)}$
$\displaystyle=e^{p_{n}}\big{(}g_{n}^{(4)}+4g_{n}^{(3)}p_{n}^{\prime}+6g_{n}^{\prime\prime}p_{n}^{\prime\prime}+\boldsymbol{4g_{n}^{\prime}p_{n}^{(3)}}+g_{n}p_{n}^{(4)}+6g_{n}^{\prime\prime}p_{n}^{\prime}{}^{2}+12g_{n}^{\prime}p_{n}^{\prime}p_{n}^{\prime\prime}+3g_{n}p_{n}^{\prime\prime}{}^{2}$
$\displaystyle\quad+2g_{n}p_{n}^{\prime}p_{n}^{(3)}+4g_{n}^{\prime}p_{n}^{\prime}{}^{3}+6g_{n}p_{n}^{\prime}{}^{2}p_{n}^{\prime\prime}+g_{n}p_{n}^{\prime}{}^{4}\big{)},$
we want that
(9) $\displaystyle p_{n}^{(3)}(z_{\ell}^{(n)})$
$\displaystyle=-\big{(}g_{n}^{(4)}+4g_{n}^{(3)}p_{n}^{\prime}+6g_{n}^{\prime\prime}p_{n}^{\prime\prime}+6g_{n}^{\prime\prime}p_{n}^{\prime}{}^{2}+12g_{n}^{\prime}p_{n}^{\prime}p_{n}^{\prime\prime}+4g_{n}^{\prime}p_{n}^{\prime}{}^{3}\big{)}/4g_{n}^{\prime}\Big{|}_{z=z_{\ell}^{(n)}},$
$\displaystyle\quad\quad 0\leq\ell\leq n-1$
to get $h_{n}^{(4)}(z_{\ell}^{(n)})=0.$ Observe that when (7) and (8) are
satisfied to determine $p_{n}^{\prime}(z_{\ell}^{(n)})$ and
$p_{n}^{\prime\prime}(z_{\ell}^{(n)})$, then also (9) is in fact a condition
that depends only on the values of $g_{n}$ and its derivatives at the points
$z_{\ell}^{(n)}$, $0\leq\ell\leq n-1.$ By the theory of interpolation [5, p.
52], for every $n\geq 1$ the conditions (7), (8) and (9) can be achieved with
a polynomial $p_{n}$ of degree at most $4n-1.$
Now, by our construction, for every $n\geq 1,$ $h_{n}^{\prime\prime}$ has a
zero of multiplicity at least 3 at each point $z_{\ell}^{(n)},$ $0\leq\ell\leq
n-1$, and so $\frac{h_{n}^{\prime\prime}}{h_{n}^{3}}$ is holomorphic (in fact,
entire) in $\Delta(0,2).$ Thus we have
$\max\limits_{z\in\overline{\Delta}(0,2)}|h_{n}^{\prime\prime}(z)/h_{n}^{3}(z)|=c_{n}>0.$
Define now for every $n\geq 1,$ $f_{n}:=a_{n}\cdot h_{n},$ where $|a_{n}|$ is
a large enough constant such that $\left|\frac{c_{n}}{a_{n}^{2}}\right|\leq 1$
and such that every subsequence of $\\{f_{n}\\}_{n=1}^{\infty}$ is not normal
at any point of $\partial\Delta=\\{z:|z|=1\\}.$ In fact, we can take $|a_{n}|$
to be so large such that $f_{n}\to\infty$ locally uniformly in
$\mathbb{C}\setminus\partial\Delta.$
Now, for $z=z_{\ell}^{(n)},$ $0\leq\ell\leq n-1$,
$f_{n}^{\prime\prime}(z_{\ell}^{(n)})=0$ and thus the left hand side of (2) is
zero. If $z\neq z_{\ell}^{(n)},$ $z\in\Delta(0,2),$ then $f_{n}(z)\neq 0$ and
$\frac{|f_{n}^{\prime\prime}(z)|}{1+|f_{n}(z)|^{3}}\leq\frac{|f_{n}^{\prime\prime}(z)|}{|f_{n}(z)|^{3}}=\frac{1}{|a_{n}|^{2}}\,\frac{|h_{n}^{\prime\prime}(z)|}{|h_{n}(z)|^{3}}\leq\frac{c_{n}}{|a_{n}|^{2}}\leq
1$
and (2) is satisfied (uniformly in $\Delta(0,2)).$ This completes the proof
that $\\{f_{n}\\}_{n=1}^{\infty}$ has the desired properties to be a
counterexample.
## 4\. Some Remarks
###### Remark 1.
We have not obtained an explicit formula for $f_{n},$ and this explains the
title of this paper.
###### Remark 2.
We have shown in fact a stronger counterexample: The condition that
$\left\\{\frac{f^{\prime\prime}}{f^{3}}:f\in\mathcal{F}\right\\}$ is locally
uniformly bounded does not imply quasi-normality of the family $\mathcal{F}.$
###### Remark 3.
An interesting open problem is to find a differential inequality (maybe of the
sort that was mentioned in this paper) that implies quasi-normality and does
not imply normality.
## References
* [1] A. Hinkkanen, Normal families and Ahlfors’s Five Island Theorem , New Zealand J. Math. 22 (1993), 39-41.
* [2] S.Y. Li and H. Xie On normal families of meromorphic functions, Acta Math. Sin.4 (1986), 468-476.
* [3] H.L. Royden, A criterion for the normality of a family of meromorphic functions, Ann. Acad. Sci. Fenn. Ser. A.I. 10 (1985), 499-500.
* [4] W. Schwick, On a normality criterion of H.L. Royden, New Zealand J. Math. 23 (1994), 91-92.
* [5] J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Springer-Verlag, New York, 1980.
* [6] L. Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly 82 (1975), 813-817.
* [7] L. Zalcman, Normal families: new perspectives, Bull. Amer. Math. Soc. (N.S.) 35 (1998), 215-230.
|
arxiv-papers
| 2011-11-03T14:05:10 |
2024-09-04T02:49:23.966457
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shahar Nevo and Xuecheng Pang",
"submitter": "Shahar Nevo",
"url": "https://arxiv.org/abs/1111.0841"
}
|
1111.0844
|
# Differential inequalities, normality and quasi-normality
Xiaojun Liu, Shahar Nevo and Xuecheng Pang Xiaojun Liu, Department of
Mathematics, University of Shanghai for Science and Technology, Shanghai
200093, P.R. China Xiaojunliu2007@hotmail.com Shahar Nevo, Department of
Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel
nevosh@macs.biu.ac.il Xuecheng Pang, Department of Mathematics, East China
Normal University, Shanghai 200241, P.R.China xcpang@math.ecnu.edu.cn
###### Abstract.
We prove that if $D$ is a domain in $\mathbb{C}$, $\alpha>1$ and $C>0$, then
the family $\mathcal{F}$ of functions $f$ meromorphic in $D$ such that
$\frac{|f^{\prime}(z)|}{1+|f(z)|^{\alpha}}>C\quad\text{for every }z\in D$
is normal in $D$. For $\alpha=1$, the same assumptions imply quasi-normality
but not necessarily normality.
###### Key words and phrases:
Normal family, quasi-normal family, differential inequality
###### 2010 Mathematics Subject Classification:
30A10, 30D35
Research of first author supported by the NNSF of China Approved No.11071074
and also supported by the Outstanding Youth Foundation of Shanghai No.
slg10015.
Research of third author supported by the NNSF of China Approved No.11071074.
## 1\. Introduction
Throughout we use the following notation $D$ denotes a domain in $\mathbb{C}$.
For $z_{0}\in\mathbb{C}$ and $r>0$, $\Delta(z_{0},r)=\\{z:|z-z_{0}|<r\\}$,
$\Delta^{\prime}(z_{0},r)=\\{z:0<|z-z_{0}|<r\\}$,
$\overline{\Delta}(z_{0},r)=\\{z:|z-z_{0}|\leq r\\}$,
$\Gamma(z_{0},r)=\\{z:|z-z_{0}|=r\\}$ and
$R(z_{0},R_{1},R_{2})=\\{z:R_{1}<|z-z_{0}|<R_{2}\\}$. We write
$f_{n}(z)\overset{\chi}{\Rightarrow}f(z)$ on $D$ to indicate that the sequence
$\\{f_{n}\\}$ converges to $f$ in the spherical metric, uniformly on compact
subsets of $D$, and $f_{n}\Rightarrow f$ on $D$ if the convergence is in the
Euclidean metric. The spherical derivative is denoted by $f^{\\#}(z)$. We
shall also use the notion of $Q_{m}-$ normality. For this recall that given a
set $E\subset D$, then the derived set of order $m$ of $E$ with respect to $D$
is defined by induction: $E^{(1)}_{D}$ is the set of accumulation points of
$E$ in $D$. $E^{(m)}_{D}=\left(E^{(m-1)}_{D}\right)^{(1)}_{D}$. A family
$\mathcal{F}$ of functions meromorphic in $D$ is said to be $Q_{m}-$normal in
$D$ if every subsequence $\\{f_{n}\\}^{\infty}_{n=1}$ of functions from
$\mathcal{F}$ has a subsequence that converges uniformly with respect to
$\chi$ on $D\backslash E$, where $E^{(m)}_{D}=\emptyset$ (Here if $m=0$, then
$\mathcal{F}$ is in fact normal family and if $m=1$, then $\mathcal{F}$ is
quasi-normal family). If, in addition there exists some $\nu\in\mathbb{N}$,
such that $E$ can always be taken to satisfy
$\left|E^{(m-1)}_{D}\right|\leq\nu$, then $\mathcal{F}$ is said to be
$Q_{m}-$normal family of order at most $\nu$.
For more about $Q_{m}-$normality see [1]. This paper deals with the meaning of
some differential inequalities. A natural point of departure is the following
famous criterion of normality due to F. Marty.
###### Marty’s Theorem.
[6, p. 75] A family $\mathcal{F}$ of functions meromorphic in a domain $D$ is
normal if and only if $\\{f^{\\#}(z):f\in\mathcal{F}\\}$ is locally uniformly
bounded in $D$.
Following Marty’s Theorem, L. Royden proved the following generalization.
###### Theorem R.
[5] Let $\mathcal{F}$ be a family of meromorphic functions in $D$, with the
property that for each compact set $K\subset D$, there is a positive
increasing function $h_{K}$, such that
$|f^{\prime}(z)|\leq h_{K}(|f(z)|)$
for all $f\in\mathcal{F}$ and $z\in K$, then $\mathcal{F}$ is normal in $D$.
This result was significantly extended further in various directions, see [3],
[7] and [9].
In [2], J. Grahl and the second author proved a counterpart to Marty’s
Theorem.
###### Theorem GN.
Let $\mathcal{F}$ be a family of functions meromorphic in $D$ and let
$\varepsilon>0$. If $f^{\\#}(z)\geq\varepsilon$ for every $f\in\mathcal{F}$
and $z\in D$, then $\mathcal{F}$ is normal in $D$.
It is equivalent to say that local uniform boundedness of the spherical
derivatives from zero implies normality.
The proof uses mainly Gu’s criterion to normality, Zalcman’s Lemma and Pang-
Zalcman Lemma. N. Steinmetz [8] gave shorter proof of Theorem GN, using the
Schwarzian derivative and some well-known facts on linear differential
equations. Here in this paper, we prove a generalization of Theorem GN (with
much simpler proof) and also, for the first time we present a differential
inequality that distinguish between normality to quasi-normality.
###### Theorem 1.
Let $0\leq\alpha<\infty$ and $C>0$. Let $\mathcal{F}_{\alpha,C}(D)$ be the
family of all meromorphic functions $f$ in $D$, such that
$\frac{|f^{\prime}(z)|}{1+|f(z)|^{\alpha}}>C\quad\text{for every }z\in D.$
The the following hold:
1. (1)
If $\alpha>1$, then $\mathcal{F}_{\alpha,C}(D)$ is normal in $D$;
2. (2)
If $\alpha=1$, then $\mathcal{F}_{\alpha,C}(D)$ is quasi-normal in $D$, but
not necessarily normal.
In section 2, we prove Theorem 1. In section 3, we show that
$\mathcal{F}_{1,C}(D)$ can be of infinite order and discuss the validity of
Theorem 1 for $\alpha<1$. In section 4, we discuss the reverse inequality
$\frac{\displaystyle|f^{\prime}(z)|}{\displaystyle 1+|f(z)|^{\alpha}}<C$.
## 2\. Proof of Theorem 1
We first state explicity the famous lemma of Pang and Zalcman (that was
already mentioned). Observe that this lemma is “if and only if”.
###### Lemma 1.
[4] Let $\mathcal{F}$ be a family of meromorphic functions in a domain $D$,
all of whose zeros have multiplicity at least $m$, and all of whose poles have
multiplicity at least $p$, and let $-p<\alpha<m$. Then $\mathcal{F}$ is not
normal at some $z_{0}\in D$ if and only if there exist sequences
$\\{f_{n}\\}^{\infty}_{n=1}\subset\mathcal{F}$,
$\\{z_{n}\\}^{\infty}_{n=1}\subset D$,
$\\{\rho_{n}\\}^{\infty}_{n=1}\subset(0,1)$, such that $\rho_{n}\to 0^{+}$,
$z_{n}\to z_{0}$ and
$g_{n}(\zeta):=\rho^{-\alpha}_{n}f_{n}(z_{n}+\rho_{n}\zeta)\overset{\chi}{\Rightarrow}g(\zeta)\quad\text{on}\quad\mathbb{C},$
where $g$ is a nonconstant meromorphic function in $D$.
Here the “if” direction holds if $g_{n}(\zeta)$ converges in some open set
$\Omega\subset\mathbb{C}$ to a nonconstant meromorphic function $g$ in
$\Omega$. For a full proof of this lemma see [2].
### 2.1. Proof of (1) of Theorem 1
Let $\\{f_{n}\\}^{\infty}_{n=1}$ be a sequence of functions in
$\mathcal{F}_{\alpha,C}(D)$. Let $z_{0}\in D$ and assume by negation that
$\\{f_{n}\\}$ is not normal at $z_{0}$. Suppose that there exist $r>0$, such
that each $f_{n}$ is holomorphic in $\Delta(z_{0},r)$. We take
$\beta>\frac{\displaystyle 1}{\displaystyle\alpha-1}>0$. By Lemma 1, there
exist a subsequence of $\\{f_{n}\\}$, that without loss of generality will
also be denoted by $\\{f_{n}\\}^{\infty}_{n=1}$ and sequences $\rho_{n}\to
0^{+}$, $z_{n}\to z_{0}$, such that
(1)
$g_{n}(\zeta):=\rho^{\beta}_{n}f_{n}(z_{n}+\rho_{n}\zeta)\overset{\chi}{\Rightarrow}g(\zeta)\quad\text{on}\quad\mathbb{C},$
where $g$ is a nonconstant entire function in $\mathbb{C}$.
Taking $\zeta_{0}\in\mathbb{C}$, such that
(2) $g(\zeta_{0})\neq 0,\ \infty.$
Then by (1), (2) and the value of $\beta$, we get that for large enough $n$,
$\displaystyle\rho^{\beta+1}_{n}\left|f^{\prime}_{n}(z_{n}+\rho_{n}\zeta_{0})\right|=\rho^{1+\beta-\beta\alpha}_{n}\left|\frac{f^{\prime}_{n}(z_{n}+\rho_{n}\zeta_{0})}{f^{\alpha}_{n}(z_{n}+\rho_{n}\zeta_{0})}\right||\rho^{\beta}_{n}f_{n}(z_{n}+\rho_{n}\zeta_{0})|^{\alpha}$
$\displaystyle>\rho^{1+\beta-\beta\alpha}_{n}\cdot
C\frac{|g(\zeta_{0})|^{\alpha}}{2}\underset{n\to\infty}{\longrightarrow}\infty.$
We thus got a contradiction and the holomorphic case is proven.
Suppose now that there is no $r>0$, such that for infinitely many indices $n$,
$f_{n}$ is holomorphic in $\Delta(z_{0},r)$. Hence we deduce the existence of
some subsequence of $\\{f_{n}\\}^{\infty}_{n=1}$, that without loss of
generality will also be denoted by $\\{f_{n}\\}^{\infty}_{n=1}$, and a
sequence $z_{n}\to z_{0}$, such that $f_{n}(z_{n})=\infty$ (otherwise we are
again in the holomorphic case and we are done). We can also assume (after
moving to subsequence of $\\{f_{n}\\}^{\infty}_{n=1}$…) that there exist a
sequence $\widetilde{z}_{n}\to z_{0}$, such that $f_{n}(\widetilde{z}_{n})=0$.
Indeed, otherwise, for some $\delta>0$ and large enough $n$, $f_{n}\neq 0$ in
$\Delta(z_{0},\delta)$ and $|f^{\prime}_{n}|>C$ there. Then by Gu’s criterion
we deduce that $\\{f_{n}\\}$ is normal.
###### Claim.
$\left\\{\frac{\displaystyle f_{n}}{\displaystyle
f^{\prime}_{n}}\right\\}^{\infty}_{n=1}$ is normal in $D$.
Proof of Claim. If $|f_{n}(z)|\leq 1$, then $\left|\frac{\displaystyle
f^{\prime}_{n}(z)}{\displaystyle f_{n}(z)}\right|\geq|f^{\prime}_{n}(z)|>C$.
If $|f_{n}(z)|>1$, then $\left|\frac{\displaystyle
f^{\prime}_{n}(z)}{\displaystyle
f_{n}(z)}\right|\geq\frac{\displaystyle|f^{\prime}_{n}(z)|}{\displaystyle
1+|f_{n}(z)|}>\frac{\displaystyle|f^{\prime}_{n}(z)|}{\displaystyle
1+|f_{n}(z)|^{\alpha}}>C$. Thus in any case, $\left|\frac{\displaystyle
f^{\prime}_{n}(z)}{\displaystyle f_{n}(z)}\right|>C$ for every $n$ and every
$z\in D$. Hence $\left\\{\frac{\displaystyle f^{\prime}_{n}}{\displaystyle
f_{n}}\right\\}^{\infty}_{n=1}$ is normal and so is
$\left\\{\frac{\displaystyle f_{n}}{\displaystyle
f^{\prime}_{n}}\right\\}^{\infty}_{n=1}$.
According to the claim, we can assume (after moving to subsequence of
$\\{f_{n}\\}^{\infty}_{n=1}$…) that
$\frac{f_{n}(z)}{f^{\prime}_{n}(z)}\overset{\chi}{\Rightarrow}H(z)\quad\text{in}\quad
D.$
Since $\frac{\displaystyle f_{n}}{\displaystyle f^{\prime}_{n}}$ vanish at the
zeros and at the poles of $f_{n}$, we deduce that $H$ is holomorphic in $D$.
We have
$\left(\frac{\displaystyle f_{n}}{\displaystyle
f^{\prime}_{n}}\right)^{\prime}=1-\frac{f_{n}f^{\prime\prime}_{n}}{f^{{}^{\prime}2}_{n}}.$
Thus we have
$\left(\frac{\displaystyle f_{n}}{\displaystyle
f^{\prime}_{n}}\right)^{\prime}\Bigg{|}_{z=\widetilde{z}_{n}}=1.$
At the poles $z_{n}$ of $f_{n}$, the situation is different. Each $z_{n}$ is a
pole of order $k=k_{n}$ of $f_{n}$. This means that in some neighborhood of
$z_{n}$, we have
$f_{n}(z)=\frac{a_{-k}}{(z-z_{n})^{k}}+\frac{a_{-k+1}}{(z-z_{n})^{k-1}}+\cdots\quad\quad(a_{-k}\neq
0).$
Thus
$f^{\prime}_{n}(z)=\frac{-ka_{-k}}{(z-z_{n})^{k+1}}+\cdots,\quad\text{and}\quad
f^{\prime\prime}_{n}(z)=\frac{k(k+1)a_{-k}}{(z-z_{n})^{k+2}}+\cdots.$
We then get that
$\frac{f_{n}f^{\prime\prime}_{n}}{f^{\prime
2}_{n}}\Bigg{|}_{z=z_{n}}=\frac{k(k+1)a^{2}_{-k}}{(ka_{-k})^{2}}=\frac{k+1}{k}=1+\frac{1}{k},$
and so
$\left(\frac{\displaystyle f_{n}}{\displaystyle
f^{\prime}_{n}}\right)^{\prime}\Bigg{|}_{z=z_{n}}=1-\left(1+\frac{1}{k}\right)=-\frac{1}{k}.$
Since $z_{n}\to z_{0}$ and also $\widetilde{z}_{n}\to z_{0}$, we get a
contradiction to any possible value of $H^{\prime}(0)$. This completes the
proof of (1).
### 2.2. Proof of (2) of Theorem 1
The family $\\{nz:\ n\in\mathbb{N}\\}$ which is not normal at $z=0$, shows
that local uniform boundedness of
$\left\\{\frac{\displaystyle|f^{\prime}|}{\displaystyle 1+|f|}:\
f\in\mathcal{F}\right\\}$ does not imply in general normality. In order to
prove quasi-normality, observe first that for every
$f\in\mathcal{F}_{1,C}(D)$, we have $\left|\frac{\displaystyle
f^{\prime}}{\displaystyle f}\right|>C$ and also $|f^{\prime}|>C$. Thus both
$\\{f^{\prime}:\ f\in\mathcal{F}_{1,C}(D)\\}$ and $\\{f^{\prime}/f:\
f\in\mathcal{F}_{1,C}(D)\\}$ are normal in $D$.
Let us take now a sequence $\\{f_{n}\\}^{\infty}_{n=1}$ of functions from
$\mathcal{F}_{1,C}(D)$. If, by negation $\\{f_{n}\\}_{n}$ is not normal at
some $\widehat{z}_{0}\in D$, then we can assume (after moving to subsequence…)
that there exist $z_{n}\to\widehat{z}_{0}$, and $\rho_{n}\to 0^{+}$ and a
nonconstant function $g$, meromorphic in $\mathbb{C}$ such that
$g_{n}(\zeta)=\rho^{-\frac{1}{2}}_{n}f_{n}(z_{n}+\rho_{n}\zeta)\overset{\chi}{\Rightarrow}g(\zeta)\quad\text{on}\quad\mathbb{C}.$
Let $P_{g}$ denotes the set of poles of $g$ in $\mathbb{C}$. If $g$ is not of
the form $g(\zeta)=a\zeta+b$, then we get by differentiation,
$g^{\prime}_{n}(\zeta)=\rho^{\frac{1}{2}}_{n}f^{\prime}_{n}(z_{n}+\rho_{n}\zeta)\Rightarrow
g^{\prime}(\zeta)\quad\text{on}\quad\mathbb{C}\backslash P_{g}.$
The derivative $g^{\prime}$ is nonconstant, and thus by Lemma 1,
$\\{f^{\prime}_{n}\\}_{n=1}^{\infty}$ is not normal at $\widehat{z}_{0}$, a
contradiction.
Thus, we must have $g(\zeta)=a\zeta+b$ $(a\neq 0)$ and by Rouche’s Theorem,
for any neighborhood $U$ of $\widehat{z}_{0}$, $f_{n}$ has for large enough
$n$ a zero in $U$. This means that we can assume (after moving to
subsequence…) that there exists a sequence $z^{\ast}_{n}\to\widehat{z}_{0}$,
such that $f_{n}(z^{\ast}_{n})=0$.
Now, suppose by negation that $\\{f_{n}\\}^{\infty}_{n=1}$ is not quasinormal
at some $z_{0}\in D$. After moving to subsequence, that will also be called
$\\{f_{n}\\}^{\infty}_{n=1}$, we can assume that there exist a sequence
$\\{z_{k}\\}^{\infty}_{k=1}$ of distinct points in $D$, such that
$z_{k}\underset{k\to\infty}{\longrightarrow}z_{0}$ and each subsequence of
$\\{f_{n}\\}^{\infty}_{n=1}$ is not normal at each $z_{k}$. According to the
previous discussion, for every $k=1,2,\cdots$, there exists $n_{k}$ and a
sequence $\\{z_{k,n}\\}^{\infty}_{n=n_{k}}$,
$z_{k,n}\underset{n\to\infty}{\longrightarrow}z_{k}$, such that
$f_{n}(z_{k,n})=0$ for every $n\geq n_{k}$.
Hence for every $\delta>0$, and for every $N\in\mathbb{N}$, $f_{n}$ has in
$\Delta(z_{0},\delta)$ at least $N$ zeros for large enough $n$. Now, since
$\left\\{\frac{\displaystyle f_{n}}{\displaystyle
f^{\prime}_{n}}:n\in\mathbb{N}\right\\}^{\infty}_{n=1}$ is normal, we can also
assume (after moving to subsequence…) that
$\frac{f_{n}(z)}{f^{\prime}_{n}(z)}\Rightarrow H(z)\quad\text{in}\quad D,$
where $H$ is holomorphic in $D$. Each zero of $f_{n}$ is also a zero of
$f_{n}/f^{\prime}_{n}$, so by the above discussion the number of zeros of
$f_{n}$ in any neighborhood of $z_{0}$ tends to $\infty$, as $n\to\infty$, and
thus we conclude that $H\equiv 0$. Hence we have
$\left(\frac{f_{n}}{f^{\prime}_{n}}\right)^{\prime}\Rightarrow
0\quad\text{in}\quad D.$
But on the other hand,
$\left(\frac{f_{n}}{f^{\prime}_{n}}\right)^{\prime}\Bigg{|}_{z=z_{k,n}}=1.$
This is a contradiction and (2) of Theorem 1 is proven.
## 3\. Some remarks
### 3.1. The order of quasi-normality of $\mathcal{F}_{1,C}(D)$
We shall show now that the order of quasi-normality of $\mathcal{F}_{1,C}(D)$
can general be large as we we like. Since we can make a linear change of the
variable, it is enough if we construct in some specific domain $D$, a sequence
$\\{f_{n}\\}^{\infty}_{n=1}$ of functions such that every subsequence of
$\\{f_{n}\\}^{\infty}_{n=1}$ has the same infinite set of points of non-
normality in $D$, and
$\inf\limits_{z\in D}\frac{|f^{\prime}_{n}(z)|}{1+|f_{n}(z)|}\geq
C\quad\text{for some}\quad C>0.$
So let $D=\left\\{z:|\operatorname{Im}z|<1,\ |z-\pi k|>\frac{\displaystyle
1}{\displaystyle 2},\ k\in\mathbb{Z}\right\\}$, and define for every $n\geq
1$, $f_{n}(z)=n\cos z$. It is obvious that every subsequence of
$\\{f_{n}\\}^{\infty}_{n=1}$ is not normal exactly at the points
$z_{k}=\frac{\displaystyle\pi}{\displaystyle 2}+\pi k$, $k\in\mathbb{Z}$. Thus
$\\{f_{n}\\}^{\infty}_{n=1}$ is quasi-normal of infinite order in $D$. Because
of the periodicity of $\cos z$, there exist some $C>0$, such that
$\frac{|f^{\prime}_{n}(z)|}{1+|f_{n}(z)|}\geq C\quad\text{for every}\ n\
\text{and for every}\ z\in D.$
Hence $\mathcal{F}_{1,C}(D)$ is quasi-normal of infinite order in $D$. We
deduce that for every domain $D$, and for every $\nu\in\mathbb{N}$ there
exists $C_{D,\nu}>0,$ such that $\mathcal{F}_{1,C_{D,\nu}}(D)$ is quasi-normal
in $D$, but not quasi-normal of order at most $\nu$.
### 3.2. The case $0\leq\alpha<1$
In this case for every bounded domain $D$ and every $C>0$,
$\mathcal{F}_{\alpha,C}(D)$ has no degree of normality. To be more precise we
have the following theorem.
###### Theorem 2.
Let $0\leq\alpha<1$, $m\geq 0$, $C>0$ and $D$ a bounded domain in $D$. Then
$\mathcal{F}_{\alpha,C}(D)$ is not $Q_{m}-$normal in $D$.
###### Proof.
For a given $0\leq\alpha<1$, let us first prove the theorem for some specific
domain. Let $1<\varepsilon<3^{\frac{1-\alpha}{1+\alpha}}$. Consider the
polynomial functions $P_{n}(z)=z^{n}-3^{n}$ defined on the ring
$D_{\varepsilon}:=R\left(0,\frac{\displaystyle
3}{\displaystyle\varepsilon},3\varepsilon\right)$. Clearly every subsequence
of $\\{P_{n}\\}^{\infty}_{n=1}$ is not normal exactly at any point of
$\Gamma(0,3)$ ($\Gamma(0,3)$ is of power $\aleph$ and of course
$\left(\Gamma(0,3)\right)^{(m)}_{D_{\varepsilon}}=\Gamma(0,3)$ for every
$m\geq 1$).
###### Claim.
$\inf\limits_{z\in
D_{\varepsilon}}\frac{\displaystyle|P^{\prime}_{n}(z)|}{\displaystyle
1+|P_{n}(z)|^{\alpha}}\underset{n\to\infty}{\longrightarrow}\infty$.
Proof of Claim. For every $z\in D_{\varepsilon}$, we have
$\frac{|P^{\prime}_{n}(z)|}{1+|P_{n}(z)|^{\alpha}}=\frac{n|z|^{n-1}}{1+|z^{n}-3^{n}|^{\alpha}}>\frac{n\cdot\left(\frac{3}{\varepsilon}\right)^{n}\cdot\frac{\varepsilon}{3}}{1+(2\cdot(3\varepsilon)^{n})^{\alpha}}>\frac{n\cdot\left(\frac{3}{\varepsilon}\right)^{n}\cdot\frac{\varepsilon}{3}}{2(2\cdot(3\varepsilon)^{n})^{\alpha}}=\frac{n\varepsilon}{6\cdot
2^{1+\alpha}}\left(\frac{3^{1-\alpha}}{3^{1+\alpha}}\right)^{n}.$
Since $\varepsilon^{1+\alpha}<3^{1-\alpha}$, the last expression tends to
$\infty$, as $n\to\infty$, and this proves the claim.
Now, give $C>0$, we have by the claim that there exists $N$, such that
$\inf\limits_{z\in
D_{\varepsilon}}\frac{\displaystyle|P^{\prime}_{n}(z)|}{\displaystyle
1+|P_{n}(z)|^{\alpha}}>C\quad\text{for}\quad n\geq N,$
and thus
$\\{P_{n}\\}^{\infty}_{n=N}\subset\mathcal{F}_{\alpha,C}(D_{\varepsilon})$.
Since $\\{P_{n}\\}^{\infty}_{n=N}$ is not $Q_{m}-$normal in $D_{\varepsilon}$,
we proved the theorem for $D=D_{\varepsilon}$.
Now, let $D$ be some bounded domain. There is a ring $R(z_{0},R_{1},R_{2})$,
together with a linear transformation $\varphi$, $\varphi(z)=az+b$, such that
$\varphi:$ $R(z_{0},R_{1},R_{2})\longrightarrow D_{\varepsilon}$ is one to one
and onto (that is, $R(z_{0},R_{1},R_{2})$ and $D_{\varepsilon}$ are
conformally equivalent) and such that $\varphi^{-1}(\Gamma(0,3))\cap D$
contain an arc of a circle. Every subsequence of
$\\{P_{n}\circ\varphi\\}^{\infty}_{n=1}$ is not $Q_{m}-$normal in $D$, for
every $m\geq 1$. Also for every $C>0$, there exists $N$, such that
$\\{P_{n}\circ\varphi\\}^{\infty}_{n=N}$ is contained in
$\mathcal{F}_{\alpha,C}(D)$ and thus also $\mathcal{F}_{\alpha,C}(D)$ is not
$Q_{m}-$normal in $D$ for every $m\geq 1$. This completes the proof of the
theorem. ∎
### 3.3. The case $\alpha<0$
Consider the family $\mathcal{F}_{\alpha,C}(D)$. If $\alpha<0$ and $f(z)=0$,
then $|f(z)|^{\alpha}$ is not well-defined, so if we require in addition that
$f\neq 0$, then since $|f^{\prime}|>C$, we get by Gu’s criterion that
$\mathcal{F}_{\alpha,C}(D)$ is normal.
If we permit that $f(z)=0$, then consider $z_{0}$, such that $f(z_{0})=0$, we
have
$\lim\limits_{z\to z_{0}}\frac{|f^{\prime}(z)|}{1+|f(z)|^{\alpha}}=0,$
and so the condition
$\frac{|f^{\prime}(z)|}{1+|f(z)|^{\alpha}}>C$
cannot be satisfied.
## 4\. The reverse inequality $\frac{\displaystyle|f^{\prime}|}{\displaystyle
1+|f|^{\alpha}}<C$
Let $\alpha>0$, and let $\mathcal{F}$ be a family of functions meromorphic in
a domain $D$. By Theorem R, if
$\mathcal{F}_{\alpha}:=\left\\{\frac{\displaystyle f^{\prime}}{\displaystyle
1+|f|^{\alpha}}:f\in\mathcal{F}\right\\}$ is locally uniformly bounded in $D$,
then $\mathcal{F}$ is normal.
For $0\leq\alpha\leq 1$, the converse is false. Consider the family
$\mathcal{F}=\\{z^{n}:n\in\mathbb{N}\\}$, in $D=\Delta(3,1)$. Obviously,
$f_{n}(z)\Rightarrow\infty$ in $D$, but
$\frac{|f^{\prime}_{n}(z)|}{1+|f_{n}(z)|^{\alpha}}=\frac{\displaystyle
n|z|^{n-1}}{\displaystyle 1+|z|^{n\alpha}}.$
Thus, since $\alpha<1$, we get that
$\inf\limits_{z\in
D}\frac{|f^{\prime}_{n}(z)|}{1+|f_{n}(z)|^{\alpha}}\underset{n\to\infty}{\longrightarrow}\infty,$
and thus $\mathcal{F}_{\alpha}$ is not locally uniformly bounded.
For $\alpha\geq 2$, the converse holds.
Indeed, assume that $\mathcal{F}$ is normal in $D$. We have for every
$f\in\mathcal{F}$ and $z\in D$
$\frac{|f^{\prime}(z)|}{1+|f(z)|^{\alpha}}=\frac{|f^{\prime}(z)|}{1+|f(z)|^{2}}\cdot\frac{1+|f(z)|^{2}}{1+|f(z)|^{\alpha}}.$
By Marty’s Theorem, $\mathcal{F}_{2}$ is locally uniformly bounded in $D$. In
addition, $h(x)=\frac{\displaystyle 1+x^{2}}{\displaystyle 1+x^{\alpha}}$ is
bounded in $[0,+\infty)$, and there is some $M>0$, such that
$\frac{\displaystyle 1+|f(z)|^{2}}{\displaystyle 1+|f(z)|^{\alpha}}\leq M$
for every $f\in\mathcal{F}$, $z\in D$. We then deduce that
$\mathcal{F}_{\alpha}$ is locally uniformly bounded in $D$.
We are left with the case $1<\alpha<2$. We show now that for meromorphic
functions, normality does not imply local uniform boundedness, for every
$1<\alpha<2$. Take $\mathcal{F}=\left\\{\frac{\displaystyle 1}{\displaystyle
z}\right\\}$ (only a single function) in $\Delta$. We have
$\frac{|f^{\prime}(z)|}{1+|f(z)|^{\alpha}}\underset{z\to
0}{\longrightarrow}\infty.$
For holomorphic functions, we can approve the converse:
###### Theorem 3.
Let $1<\alpha<2$. Suppose that $\mathcal{F}$ is a normal family of holomorphic
functions in $D$. Then
$\mathcal{F}_{\alpha}=\left\\{\frac{|f^{\prime}(z)|}{1+|f(z)|^{\alpha}}:f\in\mathcal{F}\right\\}$
is locally uniformly bounded in $D$.
###### Proof.
Suppose to the contrary that $\mathcal{F}_{\alpha}$ is not locally uniformly
bounded in $D$. Then there exist $z_{0}\in D$, $z_{n}\to z_{0}$ and
$f_{n}\in\mathcal{F}$, such that
(3)
$\frac{|f^{\prime}_{n}(z_{n})|}{1+|f_{n}(z_{n})|^{\alpha}}\underset{n\to\infty}{\longrightarrow}\infty.$
The sequence $\\{f_{n}\\}^{\infty}_{n=1}$ has a uniform convergent subsequence
in $D$, that without loss of generality we also call
$\\{f_{n}\\}^{\infty}_{n=1}$. So we assume that
$f_{n}\Rightarrow f\quad\text{in}\quad D.$
Let us separate into two cases, according to the behavior of $f$.
Case (1) $f$ is holomorphic in $D$.
Then $f^{\prime}_{n}\Rightarrow f^{\prime}$ in $D$, and we easily get a
contradiction to (3).
Case (2) $f\equiv\infty$.
In particular, we have
$f_{n}(z_{0})\underset{n\to\infty}{\longrightarrow}\infty.$
We take $R>0$, such that $\overline{\Delta}(z_{0},R)\subset D$ and
(4) $0<\rho<R\frac{\sqrt{1+\alpha}-\sqrt{2}}{\sqrt{1+\alpha}+\sqrt{2}}.$
By Harnack’s inequality, for large enough $n$, we have for every
$z\in\overline{\Delta}(z_{0},\rho)$
(5)
$|f_{n}(z_{0})|^{\frac{R-\rho}{R+\rho}}\leq|f_{n}(z)|\leq|f_{n}(z_{0})|^{\frac{R+\rho}{R-\rho}}.$
By (4) we get that
(6) $\frac{R+\rho}{R-\rho}<\sqrt{\frac{1+\alpha}{2}}\quad\bigg{(}\text{and
thus}\quad\frac{R-\rho}{R+\rho}>\sqrt{\frac{2}{1+\alpha}}\,\bigg{)}.$
Now, by (5),(6) and Cauchy’s integral formula, we get that for every
$z\in\overline{\Delta}(z_{0},\rho/2)$ and large enough $n$,
$|f^{\prime}_{n}(z)|=\frac{1}{2\pi}\left|\displaystyle\int\limits_{|\zeta-
z_{0}|=\rho}\frac{f_{n}(\zeta)}{(\zeta-z)^{2}}d\zeta\right|\leq\frac{\rho}{(\rho/2)^{2}}\max\limits_{|\zeta-
z_{0}|=\rho}|f_{n}(\zeta)|\leq\frac{4}{\rho}|f_{n}(z_{0})|^{\sqrt{\frac{1+\alpha}{2}}}.$
Thus, by the last inequality, (4) and (5), we have for large enough $n$,
$\displaystyle\frac{|f^{\prime}_{n}(z_{n})|}{1+|f_{n}(z_{n})|^{\alpha}}$
$\displaystyle\leq\frac{\displaystyle\frac{4}{\rho}|f_{n}(z_{0})|^{\sqrt{\frac{1+\alpha}{2}}}}{\displaystyle
1+|f_{n}(z_{0})|^{\frac{\alpha}{\sqrt{(1+\alpha)/2}}}}\leq\frac{4}{\rho}|f_{n}(z_{0})|^{\sqrt{\frac{1+\alpha}{2}}-\frac{\alpha}{\sqrt{(1+\alpha)/2}}}$
$\displaystyle=\frac{4}{\rho}|f_{n}(z_{0})|^{\frac{1-\alpha}{2}\big{/}\sqrt{(1+\alpha)/2}}\underset{n\to\infty}{\longrightarrow}0.$
This is a contradiction to (3) and thus the Theorem follows. ∎
## References
* [1] Chi-Tai Chuang, Normal families of meromorphic functions, World scientific, 1993.
* [2] J. Grahl, and S. Nevo, Spherical derivatives and normal families, to appear in J. d’ Anal. Math., arXiv: 1010.4654.
* [3] A. Hinkkanen, Normal families and Ahlfor’s Five Island Theorem, New Zealand J. Math. 22 (1993), 39–41.
* [4] X.C. Pang, and L.Zalcman, Normal families and shared values, Bull. London Math. Soc. 32 (2000), 325–331.
* [5] H. L. Royden, A criterion for the normality of a family of meromorphic functions, Ann. Acad. Sci. Fenn. Ser. A. I. 10 (1985), 499–500.
* [6] J. Schiff, Normal families, Springer, New-York, 1993.
* [7] W. Schwick, On a normality criterion of H. L. Royden, New Zealand J. Math, 23 (1994), 91–92.
* [8] N. Steinmetz, Normal families and linear differential equation, to appear in J. Anal. Math.
* [9] L. Zalcman, Normal families: new perspectives, Bull. Amer. Math. Soc. (N.S.)35 (1998), 215–230.
|
arxiv-papers
| 2011-11-03T14:18:41 |
2024-09-04T02:49:23.971860
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiaojun Liu, Shahar Nevo and Xuecheng Pang",
"submitter": "Shahar Nevo",
"url": "https://arxiv.org/abs/1111.0844"
}
|
1111.0903
|
# Modified Friedmann Equations From Debye Entropic Gravity
A. Sheykhi1,2 111 sheykhi@mail.uk.ac.ir and Z. Teimoori1 1Department of
Physics, Shahid Bahonar University, P.O. Box 76175, Kerman, Iran
2 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O.
Box 55134-441, Maragha, Iran
###### Abstract
A remarkable new idea on the origin of gravity was recently proposed by
Verlinde who claimed that the laws of gravitation are no longer fundamental,
but rather emerge naturally as an entropic force. In Verlinde derivation, the
equipartition law of energy on the holographic screen plays a crucial role.
However, the equipartition law of energy fails at the very low temperature.
Therefore, the formalism of the entropic force should be modified while the
temperature of the holographic screen is very low. Considering the Debye
entropic gravity and following the strategy of Verlinde, we derive the
modified Newton’s law of gravitation and the corresponding Friedmann equations
which are valid in all range of temperature. In the limit of strong
gravitational field, i.e. high temperature compared to Debye temperature,
$T\gg T_{D}$, one recovers the standard Newton’s law and Friedmann equations.
We also generalize our study to the entropy corrected area law and derive the
dynamical cosmological equations for all range of temperature. Some limits of
the obtained results are also studied.
## I Introduction
Although gravity is the most universal force of nature, however, the origin of
it in quantum level is still unclear. This is due to the fact that it is
remarkably hard to combine gravity with quantum mechanics compared with all
the other forces, and hence the final theory of the quantum gravity has not
been established yet. The universality of gravity suggests that its emergence
should be understood from general principles that are independent of the
specific details of the underlying microscopic theory.
According to Einstein’s theory of general relativity, the concept of gravity
has strongly connected to the spacetime geometry. Indeed, Einstein field
equations tell us that the presence of energy or stress causes the deformation
of the spacetime geometry. In 1970’s Bekenstein and Hawking HB discovered
black holes thermodynamics. With combination of quantum mechanics and general
relativity they predicted that a black hole behaves like a black body,
emitting thermal radiations, with a temperature proportional to its surface
gravity at the black hole horizon and with an entropy proportional to its
horizon area. The Hawking temperature and the horizon entropy together with
the black hole mass obey the first law of thermodynamics, $dM=TdS$. Since the
discovery of black hole thermodynamics in 1970’s, physicists have been
speculating that there should be a direct relation between thermodynamics and
Einstein equations. This is expected because the geometrical quantities like
horizon area and surface gravity are proportional to entropy and horizon
temperature, respectively. After Bekenstein and Hawking a lot of works have
been done to disclose the connection between themodynamics and gravity B ; D .
In 1995 Jacobson Jac put forward a great step and showed that the Einstein
field equation is just an equation of state for the spacetime and in
particular it can be derived from the proportionality of entropy and the
horizon area together with the fundamental relation $\delta Q=TdS$. Following
Jacobson, however, an overwhelming flood of papers has appeared which attempt
to show that there is indeed a deeper connection between gravitational field
equations and horizon thermodynamics. It has been shown that, not only in
Einstein gravity but also in a wide variety of theories, the gravitational
field equations for the spacetime metric has a predisposition to thermodynamic
behavior. This result, first pointed out in Pad , has now been demonstrated in
various theories including f(R) gravity Elin and cosmological setups Cai2 ;
Cai3 ; CaiKim ; Wang ; Cai33 ; Shey0 ; Shey1 ; Shey2 . For a recent review on
the thermodynamical aspects of gravity and complete list of references see
Pad0 .
Recently, Verlinde Ver has invented a conceptual theory that gravity is no
longer fundamental, but is emergent. According to Verlinde, one can start from
the first principles, and gravity appears as an entropic force naturally and
unavoidably in a theory in which space is emergent through a holographic
scenario. Similar discoveries are also made by Padmanabhan Padm who observed
that the equipartition law for horizon degrees of freedom combined with the
Smarr formula leads to the Newton’s law of gravity. In addition, Verlinde’s
arguments reveal a fact that the key to understanding gravity is information
(or entropy). In Verlinde derivation the holographic principle and the
equipartition law of energy play a crucial role. The holographic principle was
originally proposed by ’t Hooft Hooft and then developed in cosmology by
Susskind and others Sus . According to this principle the combination of
quantum mechanics and gravity requires the three dimensional world to be an
image of data that can be stored on a two dimensional projection much like a
holographic image. The studies on the entropic gravity scenario have arisen a
lot attention recently Cai4 ; Smolin ; Li ; Tian ; Vancea ; Modesto ; Sheykhi1
; BLi ; Sheykhi2 ; Gu ; other ; mann .
It is well-known that the equipartition law of energy for a system of
particles only valid for the situation where the kinetic energy of the
particles is much larger than the effective interacting potential between
particles. This means that the equipartition law of energy break down at very
low temperatures. It is found that Debye model, which modified the
equipartition law of energy, is in good agreement with experimental results
for most solid objects. According to Verlinde, we know that the gravity can be
explained as an entropic force, it means that the gravity may have a
statistical thermodynamics explanation. Therefore, the formalism of the
entropic force should be modified while the temperature of the holographic
screen is very low. This means that Newtonian gravity takes a different form
in the background of an extreme weak gravitational field. In the present work,
inspired by the Debye’s model in statistical thermodynamics, we generalize the
formalism of the entropic gravity to the very law temperatures.
The outline of the present paper is as follows. The modified Newton’s law of
gravitation and the corresponding Friedmann equations which are valid in all
range of temperature are extracted in the next section. Sec. III is devoted to
the derivation of the Entropy corrected Friedmann equations in Debye entropic
force scenario. The paper ends with a conclusion, which appears in Sec. IV.
## II Debye Entropic Gravity and Friedmann Equation
Consider a closed holographic screen and a free particle of mass $m$ near it
on the side that spacetime has already emerged. In Verlide’s picture when the
particle has an entropic reason to be on one side of the screen and the screen
carries a temperature, it will experience an effective macroscopic force due
to the statistical tendency to increase its entropy. This is described by
$F=T\frac{\triangle S}{\triangle x}.$ (1)
where $\triangle x$ is the displacement of the particle from the holographic
screen, while $T$ and $\triangle S$ are the temperature and the entropy change
on the screen, respectively. According to the Unruh formula, the temperature
in Eq. (1) associated with the holographic screen is
$k_{B}T=\frac{\hbar g}{2\pi c}.$ (2)
where $g$ represents the proper gravitational acceleration on the screen which
is produced by the matter distribution inside the screen. Suppose we have a
mass distribution $M$ which induces a holographic screen $\mathcal{S}$ at some
distance $R$ that has encoded on it gravitational information. Suppose we have
also a test mass $m$ which is assumed to be very close to the holographic
screen as compared to its reduced Compton wavelength
$\lambda_{m}=\frac{\hbar}{mc}$. Assuming the holographic screen forms a closed
surface. The key statement is that we need to have a temperature in order to
have a force. One can think about the boundary as a storage device for
information. Assuming that the holographic principle holds, the maximal
storage space, or total number of bits, is proportional to the area $A$. Let
us denote the number of used bits by $N$. It is natural to assume that this
number will be proportional to the area $A=4\pi R^{2}$. Thus we write
$A=NQ,$ (3)
where $Q$ is a fundamental constant which should be specified later. Note that
$N$ is the number of bits and thus for one unit change we have $\triangle
N=1$, hence from (3) we find $\triangle A=Q$. Motivated by Bekenstein’s area
law of black hole entropy, we assume the entropy associated with the
holographic screen obey the area law, namely
$S=\frac{A}{4\ell_{p}^{2}},$ (4)
where $\ell_{p}^{2}=G\hbar/c^{3}$ is the Planck length. Following Ver we also
assume the energy on the holographic screen is proportional to the mass
distribution $M$ that would emerge in the part of space enclosed by the screen
$E=Mc^{2}.$ (5)
According to statistical thermodynamics the equipartition law of energy for
free particle only valid for the situation in which the kinetic energy of
particle is much larger than the effective interacting potential between them.
Therefore, the equipartition law of energy fails in the very low temperatures.
It is found that Debye model, which modified the equipartition law of energy,
is in good agreement with experimental results for most solid objects.
Following Verlinde’s scenario, the laws of gravitation are no longer
fundamental, but rather emerge naturally as an entropic force. It means that
the gravity may have a statistical thermodynamics origin. Thus, any
modification of statistical mechanics should modify the laws of gravity
accordingly. Motivated by this point, we modify the equipartition law of
energy as
$E=\frac{1}{2}Nk_{B}T\mathcal{D}(x),$ (6)
where the Debye function is defined by
$\mathcal{D}(x)\equiv\frac{3}{x^{3}}\int_{0}^{x}\frac{y^{3}}{e^{y}-1}dy.$ (7)
Here $x$ is related to the temperature $T$ as follows
$x\equiv\frac{T_{D}}{T}=\frac{\hbar\omega_{D}}{Tk_{B}},$ (8)
where $T_{D}$ is the Debye critical temperature and $\omega_{D}$ is the Debye
frequency. Combining Eqs. (3), (5) and (6), we obtain the temperature of the
holographic screen as
$T=\frac{2Mc^{2}Q}{4\pi R^{2}k_{B}\mathcal{D}(x)}.$ (9)
Substituting Eqs. (9) and (4) in (1), and using relation $\triangle A=Q$, we
get
$F=-\frac{Mm}{R^{2}}\frac{1}{\mathcal{D}(x)}\left(\frac{Q^{2}c^{3}}{8\pi\ell_{p}^{2}k_{B}\hbar}\right)$
(10)
where we have taken $\triangle x=-\frac{\hbar}{mc}$ for one fundamental unit
change in the entropy and the entropy gradient points radially from the
outside of the surface to inside. In order to derive the Newton’s law of
gravitation we must define $Q^{2}=8\pi k_{B}\ell_{p}^{4}$. Finally we reach
$F=-G\frac{Mm}{R^{2}}\frac{1}{\mathcal{D}(x)}.$ (11)
The corresponding gravitational acceleration is obtained as
$g=\frac{GM}{R^{2}}\frac{1}{\mathcal{D}(x)}.$ (12)
Using relation (2) we can define the Debye acceleration $g_{D}$ which is
related to the Debye temperature as
$T_{D}=\frac{\hbar g_{D}}{2\pi k_{B}c},\ \ \
x=\frac{T_{D}}{T}=\frac{g_{D}}{g}$ (13)
Eq. (11) is the Newton’s law of gravitation resulting from Debye entropic
force. Let us study two different limits of Eq. (11). In the strong
gravitational field limit, i.e. at high temperature, $T\gg T_{D}$ ($x\ll 1$),
the Debye function reduces to
$\mathcal{D}(x)\approx 1.$ (14)
As a result in this limit, the usual Newtonian gravity is restored. In the
weak gravitational field limit, i.e. at very law temperature, $T\ll T_{D}$
($x\gg 1$) we have
$\mathcal{D}(x)=\frac{\pi^{4}}{5x^{3}}=\frac{\pi^{4}}{5}\left(\frac{g}{g_{D}}\right)^{3}$
(15)
In this limit, the Newton’s law is modified as
$F=-\frac{5GMm}{R^{2}}\frac{g_{D}^{3}}{\pi^{4}g^{3}},$ (16)
while the gravitational acceleration becomes
$g=\left(\frac{5GMg_{D}^{3}}{\pi^{4}}\right)^{\frac{1}{4}}\frac{1}{\sqrt{R}}\
\ \Rightarrow\ g\propto\frac{1}{\sqrt{R}}$ (17)
Therefore in this limit the gravitational field differs from Newtonian
gravity. Let us then consider the cosmological implications of the presented
model. We assume the background spacetime is spatially homogeneous and
isotropic which is described by the line element
$ds^{2}={h}_{\mu\nu}dx^{\mu}dx^{\nu}+R^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (18)
where $R=a(t)r$, $x^{0}=t,x^{1}=r$, the two dimensional metric
$h_{\mu\nu}$=diag $(-1,a^{2}/(1-kr^{2}))$. Here $k$ denotes the curvature of
space with $k=0,1,-1$ corresponding to flat, closed, and open universes,
respectively. The dynamical apparent horizon, a marginally trapped surface
with vanishing expansion, is determined by the relation
$h^{\mu\nu}\partial_{\mu}R\partial_{\nu}R=0$. A simple calculation gives the
apparent horizon radius for the Friedmann-Robertson-Walker (FRW) universe
$R=ar=\frac{1}{\sqrt{H^{2}+k/a^{2}}}.$ (19)
The matter source in the FRW universe is assumed as a perfect fluid with
stress-energy tensor
$T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu}.$ (20)
Now we are in a position to derive the dynamical equation for Newtonian
cosmology. Consider a compact spatial region $V$ with a compact boundary
$\mathcal{S}$, which is a sphere with physical radius $R=a(t)r$. Note that
here $r$ is a dimensionless quantity which remains constant for any
cosmological object partaking in free cosmic expansion. Combining the second
law of Newton for the test particle $m$ near the surface, with gravitational
force (11) we get
$m\ddot{R}=m\ddot{a}r=-G\frac{Mm}{R^{2}}\frac{1}{\mathcal{D}(x)}.$ (21)
We also assume $\rho=M/V$ is the energy density of the matter inside the the
volume $V=\frac{4}{3}\pi a^{3}r^{3}$. Thus, Eq. (21) can be rewritten as
$\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}\rho\frac{1}{\mathcal{D}(x)}$ (22)
This is the dynamical equation for Newtonian cosmology which is valid in all
range of the temperature. For strong gravitational field
($\mathcal{D}(x)\simeq 1$) we reach the well-known formula
$\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}\rho.$ (23)
Next we want to derive the Friedmann equations of FRW universe. For this
purpose we need to employ the concept of the active gravitational mass
$\mathcal{M}$ Pad3 , since this quintity produces the acceleration in general
relativity. From Eq. (22) with replacing $M$ with $\mathcal{M}$ we have
$\mathcal{M}=-\frac{\ddot{a}a^{2}r^{3}}{G}\mathcal{D}(x)$ (24)
On the other side, the active gravitational mass is defined as Cai4
$\mathcal{M}=2\int_{V}{dV\left(T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu}\right)u^{\mu}u^{\nu}}.$
(25)
A simple calculation gives
$\mathcal{M}=(\rho+3p)\frac{4\pi}{3}a^{3}r^{3}.$ (26)
Equating Eqs. (24) and (26) we obtain
$\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}(\rho+3p)\frac{1}{\mathcal{D}(x)}.$ (27)
This is the modified acceleration equation for the dynamical evolution of the
FRW universe. Multiplying $\dot{a}a$ on both sides of Eq. (27), and using the
continuity equation
$\dot{\rho}+3H(\rho+p)=0,$ (28)
after integrating we find
$H^{2}+\frac{k}{a^{2}}=\frac{8\pi G}{3a^{2}}\int\frac{d(\rho
a^{2})}{\mathcal{D}(x)}.$ (29)
This is the first Friedmann equation resulting from Debye entropic force. Eqs
(29) and (28) together with the equation of state $p=w\rho$ govern the
evolution of the universe. It is important to note that since $x=x(T)$ and the
temperature is also a function of scale factor namely, $T=T(a)$, thus in
general we cannot integrate Eq. (29) and derive the simplified result. When
$x\ll 1\Rightarrow\mathcal{D}(x)\approx 1$, the well-known Friedmann equation
in standard cosmology is recovered. For $x\gg 1$, using Eq. (15) we find
$H^{2}+\frac{k}{a^{2}}=\frac{8\pi
G}{3}\rho\frac{5}{\pi^{4}}\left(\frac{g_{D}}{g}\right)^{3}.$ (30)
In order to derive the second Friedmann equation (29), we have to combine the
first Friedmann equation with continuity equation (28). Let us put $k=0$ for
simplicity, which has been confirmed by recent observations. Differentiating
Eq. (29) we find
$\displaystyle 2HdH=\frac{8\pi G}{3}\left[-\frac{2}{a^{2}}\frac{da}{a}\int
d(\rho
a^{2})\frac{1}{\mathcal{D}(x)}+d\rho\frac{1}{\mathcal{D}(x)}+2\rho\frac{da}{a}\frac{1}{\mathcal{D}(x)}\right]$
(31)
Multiplying $\frac{3}{2}$ on both sides of Eq. (31) and dividing by $dt$, we
find
$\displaystyle 3H\dot{H}=-\frac{8\pi G}{a^{2}}\frac{\dot{a}}{a}\int d(\rho
a^{2})\frac{1}{\mathcal{D}(x)}+4\pi G\dot{\rho}\frac{1}{\mathcal{D}(x)}+8\pi
G\rho\frac{\dot{a}}{a}\frac{1}{\mathcal{D}(x)}.$ (32)
Using the continuity equation (28), the above equation can be written as
$\displaystyle-\left[\dot{H}\mathcal{D}(x)+\frac{8\pi
G}{3a^{2}}\mathcal{D}(x)\int d(\rho a^{2})\frac{1}{\mathcal{D}(x)}-\frac{8\pi
G}{3}\rho\right]=4\pi G(\rho+p).$ (33)
For $x\ll 1$ we have $D(x)\approx 1$, and the well-known second Friedmann
equation of FRW universe in flat spacetime is recovered, namely
$\displaystyle-\dot{H}=4\pi G(\rho+p).$ (34)
It is worth noting that the Friedmann equation in Debye entropic force
scenario was first studied in Gao . Let us stress the difference between our
derivation in this section and that of Gao . The author of Gao has derived
Friedmann equations, following the method of FW , by applying the
equipartition law of energy, $E=NT/2$, to the apparent horizon of a FRW
universe with the assumption that the apparent horizon has the temperature
$T=\hbar/(2\pi R)$, where $R$ is the apparent horizon radius. Thus the total
energy change of the system is obtained as FW
$\displaystyle dE=\frac{1}{2}NdT+\frac{1}{2}TdN=\frac{dR}{G},$ (35)
during the infinitesimal time interval $dt$, where the apparent horizon radius
evolves from $R$ to $R+dR$. Indeed, the above equation is just the first law
of thermodynamics in the form $dE=TdS$ on the apparent horizon, where
$T=\hbar/(2\pi R)$ and $S=A/(4\hbar G)$ is the entropy of the system which
assumed to obey the area-law and $A=4\pi R^{2}$ is the apparent horizon area.
While in the present work we have not employed the first law of thermodynamics
for deriving the modified Friedmann equations. Therefore, our result is
independent of the definition of the temperature in a dynamical spacetime.
## III Entropic Corrected Friedmann Equation in Debye entropic gravity
In this section we would like to consider the effects of the quantum
correction terms to the entropy expression, on the laws of gravity in Debye
model of entropic gravity. The result we will obtain are valid in all range of
temperature. The correction terms to the entropy expression originate from the
loop quantum gravity (LQG). The quantum corrections provided to the entropy-
area relationship leads to the curvature correction in the Einstein-Hilbert
action and vice versa Zhu ; Suj . In the presence of quantum corrections the
entropy takes the following form Zhang
$S=\frac{A}{4\ell_{p}^{2}}-\beta\ln{\frac{A}{4\ell_{p}^{2}}}+\gamma\frac{\ell_{p}^{2}}{A}+\mathrm{const},$
(36)
where $\beta$ and $\gamma$ are dimensionless constants of order unity. These
corrections arise in the black hole entropy in LQG due to thermal equilibrium
fluctuations and quantum fluctuations Rovelli . We will show that these
corrections modify the Newton’s law of gravitation as well as the Friedmann
equations. First of all we rewrite Eq. (36) in the following form
$S=\frac{A}{4\ell_{p}^{2}}+{s}(A),$ (37)
where $s(A)$ represents the correction terms in the entropy expression. In
this case the entropy change is obtained as
$\triangle S=\frac{\partial S}{\partial A}\triangle
A=\left(\frac{1}{4\ell_{p}^{2}}+\frac{\partial{s}(A)}{\partial
A}\right)\triangle A.$ (38)
Substituting Eqs. (9) and (38) in Eq. (1) and using relations $\triangle
x=-\frac{\hbar}{mc}$ and $\triangle A=Q$, one can easily find
$F=-\frac{Mm}{R^{2}\mathcal{D}(x)}\left(\frac{Q^{2}c^{3}}{2\pi
k_{B}\hbar}\right)\left[\frac{1}{4\ell_{p}^{2}}+\frac{\partial{s}}{\partial
A}\right]_{A=4\pi R^{2}}.$ (39)
If we define $Q^{2}\equiv 8\pi k_{B}\ell_{p}^{4}$, as before, we immediately
derive the modified Newton’s law of gravity in Debye entropic gravity
$F=-G\frac{Mm}{R^{2}}\frac{1}{\mathcal{D}(x)}\left[1-\frac{\beta}{\pi}\frac{\ell_{p}^{2}}{R^{2}}-\frac{\gamma}{4\pi^{2}}\frac{\ell_{p}^{4}}{R^{4}}\right].$
(40)
In the absence of correction terms ($\beta=\gamma=0$), the above equation
reduces to the result of the previous section. Let us study two different
limit of the above equation. In the strong gravitational limit, i.e. at high
temperature, $T_{D}\ll T$ ($\mathcal{D}(x)\approx 1)$ we have
$F=-G\frac{Mm}{R^{2}}\left[1-\frac{\beta}{\pi}\frac{\ell_{p}^{2}}{R^{2}}-\frac{\gamma}{4\pi^{2}}\frac{\ell_{p}^{4}}{R^{4}}\right],$
(41)
which is exactly the result obtained in Sheykhi1 . When $\beta=\gamma=0$, one
recovers the well-known Newton’s law. on the other hand, at very law
temperature $T_{D}\gg T$ we have $\mathcal{D}(x)=\frac{\pi^{4}}{5x^{3}}$ and
Eq. (40) reduces to
$F=-G\frac{Mm}{R^{2}}\frac{5}{\pi^{4}}\frac{g_{D}^{3}}{g^{3}}\left[1-\frac{\beta}{\pi}\frac{\ell_{p}^{2}}{R^{2}}-\frac{\gamma}{4\pi^{2}}\frac{\ell_{p}^{4}}{R^{4}}\right].$
(42)
To derive Friedmann equation we follow the method of the previous section.
Combining the second law of Newton for the test particle $m$ near the screen
with gravitational force (42) we obtain
$F=m\ddot{R}=m\ddot{a}r=-\frac{MmG}{a^{2}r^{2}}\frac{1}{\mathcal{D}(x)}\left[1-\frac{\beta}{\pi}\frac{\ell_{p}^{2}}{R^{2}}-\frac{\gamma}{4\pi^{2}}\frac{\ell_{p}^{4}}{R^{4}}\right]$
(43)
which from it we can derive the acceleration equation
$\frac{\ddot{a}}{a}=-\frac{4\pi
G}{3}\rho\frac{1}{\mathcal{D}(x)}\left[1-\frac{\beta}{\pi}\frac{\ell_{p}^{2}}{R^{2}}-\frac{\gamma}{4\pi^{2}}\frac{\ell_{p}^{4}}{R^{4}}\right].$
(44)
With the entropic corrections terms, the active gravitational mass
$\mathcal{M}$ will be modified accordingly. The active gravitational mass
$\mathcal{M}$ in this case is obtained as
$\mathcal{M}=-\frac{\ddot{a}a^{2}}{G}r^{3}\mathcal{D}(x)\left[1-\frac{\beta}{\pi}\frac{\ell_{p}^{2}}{R^{2}}-\frac{\gamma}{4\pi^{2}}\frac{\ell_{p}^{4}}{R^{4}}\right]^{-1}.$
(45)
Equating the above equation with Eq. (26) yields
$\frac{\ddot{a}}{a}=-\frac{4\pi
G}{3}(\rho+3p)\frac{1}{\mathcal{D}(x)}\left[1-\frac{\beta}{\pi}\frac{\ell_{p}^{2}}{R^{2}}-\frac{\gamma}{4\pi^{2}}\frac{\ell_{p}^{4}}{R^{4}}\right].$
(46)
Next we multiply the both sides of the above equation by $a\dot{a}$, after
using the continuity equation (28) and integrating we find
$\displaystyle H^{2}+\frac{k}{a^{2}}$ $\displaystyle=$
$\displaystyle\frac{8\pi G}{3a^{2}}\left[\int\frac{d(\rho
a^{2})}{\mathcal{D}(x)}-\frac{\beta}{\pi}\frac{\ell_{p}^{2}}{r^{2}}\int\frac{d(\rho
a^{2})}{a^{2}\mathcal{D}(x)}-\frac{\gamma}{4\pi^{2}}\frac{\ell_{p}^{4}}{r^{4}}\int\frac{d(\rho
a^{2})}{a^{4}\mathcal{D}(x)}\right].$ (47)
Where $k$ is an integration constant. Unfortunately, the above equation cannot
be integrated in general for an arbitrary $\mathcal{D}(x)$. In the limiting
case $D(x)\approx 1$, the integrations can be done following the method
developed in Sheykhi1 . We find ( see Sheykhi1 for details)
$\displaystyle\left(H^{2}+\frac{k}{a^{2}}\right)+\frac{\beta\ell_{p}^{2}(1+3\omega)}{3\pi(1+\omega)}\left(H^{2}+\frac{k}{a^{2}}\right)^{2}+\frac{\gamma\ell_{p}^{4}(1+3\omega)}{4\pi^{2}(5+3\omega)}\left(H^{2}+\frac{k}{a^{2}}\right)^{3}=\frac{8\pi
G}{3}\rho.$ (48)
Again we see that in the absence of correction terms $(\beta=0,\gamma=0)$ the
well-known Friedmann equation is recovered. For $x\gg 1$
($\mathcal{D}(x)=\frac{\pi^{4}}{5x^{3}})$ Eq. (47) can be written
$\displaystyle H^{2}+\frac{k}{a^{2}}$ $\displaystyle=$
$\displaystyle\frac{8\pi G}{3}\left[\frac{1}{a^{2}}\int d(\rho
a^{2})\frac{5x^{3}}{\pi^{4}}\right.$ (49)
$\displaystyle\left.-\frac{\beta}{\pi}\frac{\ell_{p}^{2}}{a^{2}r^{2}}\int\frac{d(\rho
a^{2})}{a^{2}}\frac{5x^{3}}{\pi^{4}}-\frac{\gamma}{4\pi^{2}}\frac{\ell_{p}^{4}}{a^{2}r^{4}}\int\frac{d(\rho
a^{2})}{a^{4}}\frac{5x^{3}}{\pi^{4}}\right].$
## IV conclusion
Verlinde proposal on the entropic origin of the gravity is based strongly on
the assumption that the equipartition law of energy holds on the holographic
screen induced by the mass distribution of the system. However, from the
theory of statistical mechanics we know that the equipartition law of energy
does not hold in the limit of very low temperature. By low temperature, we
mean that the temperature of the system is much smaller than Debye
temperature, i.e. $T\ll T_{D}$. It was demonstrated that the Debye model is
very successful in interpreting the physics at the very low temperature. Since
the discovery of black holes thermodynamics, physicist have been thought that
the gravitational systems such as black hole and our universe can also be
regarded as a thermodynamical system. Hence, it is expected that the
equipartition law of energy for the gravitational system should be modified in
the limit of very low temperature (or very weak gravitational field).
In this paper inspired by the Verlinde proposal and following the Debye model
of equipartition law of energy in statistical thermodynamics, we modified the
entropic gravity. First, we studied the Debye entropic gravity and derived the
modified Newton’s law of gravitation and the corresponding Friedmann equations
which are valid in all range of temperature. We found that the modified
entropic force returns to the Newton’s law of gravitation while the
temperature of the holographic screen is much higher than the Debye
temperature. Then we extended our study to the case where there are correction
terms such as logarithmic correction in the entropy expression. In this case
we again reproduced the gravitational equations for all range of temperature.
Our study shows a deep connection between Debye entropic gravity and modified
Friedmann equation. The microscopic statistical thermodynamical model of
spacetime may shed light on the origin of the Debye entropic gravity and the
microscopic origin for the Newton’s law of gravity and also Friedmann
equations in cosmology.
###### Acknowledgements.
This work has been supported financially by Research Institute for Astronomy
and Astrophysics of Maragha (RIAAM), Iran.
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|
arxiv-papers
| 2011-11-03T16:26:04 |
2024-09-04T02:49:23.979190
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Sheykhi and Z. Teimoori",
"submitter": "Ahmad Sheykhi",
"url": "https://arxiv.org/abs/1111.0903"
}
|
1111.0956
|
# Structural and Electronic properties of the 2D Superconductor CuS with
1$\frac{1}{3}$-valent Copper
I.I. Mazin Code 6390, Naval Research Laboratory, Washington, DC 20375, USA
###### Abstract
We present first principle calculations of the structural and electronic
properties of the CuS covellite material. Symmetry-lowering structural
transition is well reproduced. However, the microscopic origin of the
transition is unclear. The calculations firmly establish that the so far
controversial Cu valency in this compound is 1.33. We also argue that recently
reported high-temperature superconductivity in CuS is unlikely to occur in the
stoichiometric defect-free material, since the determined Cu valency is too
close to 1 to ensure proximity to a Mott-Hubbard state and superexchange spin
fluctuations of considerable strength. On the other hand, one can imagine a
related system with more holes per Cu in the same structural motif ($e.g.$,
due to defects or O impurities) in which case combination of superexchange and
an enlarged compared to CuS Fermi surface may lead to unconventional
superconductivity, similar to HTSC cuprate, but, unlike them, of an $f$-wave
symmetry.
Copper sulfide in the so-called covellite structure (Fig. 1) has recently
attracted attention due to a new report about possible superconductivity at 40
KRaveau . This report has been met with understandable skepticism, because
previous researchessc1 ; sc2 reported reproducible superconductivity at
rather low temperatures, around 1-2 K. On the other hand, inspection of the
literature reveals that reported physical properties of covellite are
drastically different in different papers. For instance, one paper reported a
well defined Curie-Weiss magnetic susceptibility1st , while others observed a
nearly constant behavior consistent with the Pauli susceptibility in absence
of any local moments.
Figure 1: (color online) Crystal structure of covellite in the high-symmetry
phase. Large yellow spheres indicate sulfure, and blue spheres copper. The
dark spheres form the planar CuS layers (Cu1), and the light spheres form the
warped Cu2S2 bilayers (Cu2). Fat yellow sticks indicate strong covalent bonds
inside the S2-S2 dumbells.
Structural properties of the covellite are also intriguing. At room
temperature it consists of triangular layers of Cu and S, stacked as follows,
using standard hexagonal stacking notation: Cu1 and S1 form layers $A$ and
$B$, at the same height, so that Cu1 has coordination of three and no direct
overlap. Cu2 and S2 form layers $B$ and $C$, so that Cu2 is directly above S1
and bonds with it, too, albeit more weakly than to S2. Thus, compared to the
S2 layer, the Cu2 layer is closer to the Cu1+S1 one, and Cu2 appears to be
inside a tetrahedron, closer to its base. The next layer is again $C$, so that
two S2 atoms are right on top of each other and form a strongly covalent bond,
the shortest bond in this system, essentially making up an S2 molecule.
At $T=55$ K the system spotaneously undergoes a transition from a hexagonal
structure to a lower symmetry orthorhombic structure. To a good approximation,
the transition amounts to sliding the Cu2-S2 plane with respect to the Cu1-S1
plane by 0.2 Å, and the two neighboring Cu2-S2 planes by 0.1 Å with respect to
each other, in the same direction. The bond lengths change very little, one of
the three Cu2-S2 bonds shortens by 0.04 Å, and the S2-S2 bonds lengthens by
0.05 Å, and all other bonds remain essentially unchanged. Note that such
transitions are quite uncommon for metals, but rather characteristic of
insulating Jahn-Teller systems. Transport properties are hardly sensitive to
this transition, which is however clearly seen in the specific heat.
Thus there are three questions to be asked. First, what is the nature of the
low temperature symmetry-lowering? Second, why some experiments indicate pure
Pauli susceptibility, while others observe local moments (through Curie-Weiss
behavior)? Third, why one particular experiment sees indications of high
temperature superconductivity, while others do not? Of, course, there is
always a chance the “outliers” experiments are simply incorrect, but it is
always worth asking the question, whether some sample issues may
possibly account for such discrepancies.
In order to adress the first question, we have performed density functional
calculations (DFT) of both hexagonal (H) and orthorhombic (O) structure.
First, we optimized the crystal structure using the standard VASP program with
default settings (including gradient corrections), and starting from the
experimental structure as reported in Ref. 1st, .
After that, all calculations in the determined crystal structures were
performed using the standard all-electron LAPW code WIEN2k. We have also
verified that the calculated forces in the optimized structures are small
enough. As a technical note, to obtain full convergences in the energy
differences we had to go up to $RK_{\max}=9$.
Table 1: The calculated total energy (meV/cell) of the low-temperature orthorhombic and the high-temperature hexagonal structure, using either the experimental or the calculated optimized parameters. Structural parameters used, as well as selected bond length (Å) are also shown. Note that one unit cell includes 6 formula units. The cell volume is given in $\AA^{3}$ The last column corresponds to the orthorhombic structure with internal coordinates optimized, while keeping the experimental unit cell. | H-exp | O-exp | H-calc | O-calc | O-c.o.
---|---|---|---|---|---
$a$ | 3.789 | 3.760 | 3.807 | 3.793 | 3.760
$b$ | 3.789 | 6.564 | 3.807 | 6.623 | 6.564
$c$ | 16.321 | 16.235 | 16.496 | 16.453 | 16.235
$z_{Cu2}$ | 0.1072 | 0.1070 | 0.1069 | 0.1077 | 0.1083
$z_{S2}$ | 0.0611 | 0.0627 | 0.0639 | 0.0646 | 0.0651
$y_{Cu1}$ | n/a | 0.6377 | n/a | 0.6227 | 0.6077
$y_{Cu2}$ | n/a | 0.3372 | n/a | 0.3410 | 0.3413
$y_{S1}$ | n/a | 0.3068 | n/a | 0.2917 | 0.2760
$y_{S2}$ | n/a | 0.0008 | n/a | 0.0064 | 0.0069
Cu1-S1 | 3$\times$2.19 | 2$\times 2.18$ | 3$\times 2.20$ | 2$\times 2.20$ | 2$\times 2.20$
| | 2.17 | | 2.19 | 2.19
Cu2-S1 | 2.33 | 2.33 | 2.36 | 2.33 | 2.34
Cu2-S2 | 3$\times 2.31$ | 2$\times 2.30$ | 3$\times 2.31$ | 2$\times 2.30$ | 2$\times 2.30$
| | 2.28 | | 2.33 | 2.33
S2-S2 | 1.99 | 2.04 | 2.11 | 2.13 | 2.12
Volume | 202.9 | 200.3 | 207.0 | 206.7 | 207.3
Energy | 0 | -85 | -258 | -265 | -189
The results are shown in Table 1. Even though there is some discrepancy
between the calculated and the experimental low-temperature structures (mostly
in terms of an overall overestimation of the equilibrium volume), the correct
symmetry lowering is well reproduced. In fact, given that only one paper has
reported internal positions for the orthorhombic structre, and the same paper
found a Curie-Weiss law, suggesting, as discussed below, crystallographic
defects in their sample, it is fairly possible that the calculations predict
the structure of an ideal material better than this one experiment has
measured.
A more important question now is, what the mechanism for this well-reproduce
symmetry lowering can be? Ionic symmetry-lowering mechanisms (such as Jahn-
Teller) are excluded in a wide band metal like CuS. Typically, a lower
symmetry is stabilized in a metal if it results in a reduced density of states
at the Fermi level (“quasinesting mechanism”). However, the density of states
at the Fermi level does not change at this transition (Fig. 2), and the states
below Fermi actually shift slightly upward. Thus, one-electron energy is not
the reason for the transition. A look at the calculated Fermi surfaces (Fig.
4) shows that while they become more 2D in the orthorhombic structure (the in-
plane plasma frequency remains the same, $\approx 4.0$ eV, while that out of
plane, 1.36 eV, drops by 12%), there is no shrinkage in their size.
Figure 2: (color online) Calculated density of states in the high-temperature
(“hex”) and low temperature (“ortho”) structures, using in both cases
optimized parameters.
We cannot say definitively what causes the low-temperature symmetry lowering
in CuS, but we can say confidently that it is not van der Waals interaction as
conjectured in Ref. W, (for one, it would not be reproduced in LGA/GGA
calculations with requested accuracy, and, also, as dicussed below, the
interplanar bonding is covalent, and not van der Waals), and not a typical
metallic mechanism driven by Fermi surface changes. One candidate is ionic
Coulomb interaction. Indeed the calculated Ewald energy is noticeably lower in
the orthorhombic structure. However the Ewald energy is only part of total
electrostatic energy, so from this fact alone one cannot derive definitive
conclusions.
Figure 3: Calculated band dispersions in the hexagonal structure. The points
$\Gamma$, M and K are in the central plane ($k_{z}=0$) and A, L and H in the
basal plane ($k_{z}=\pi/c$). In the top panel, the width of the lines is
proportional to the amount of the S2-$p_{z}$ character in the corresponding
states.
Let us now discuss the electronic structure. Since the differences between the
two structures are very small, we shall limit our discussion by the high-
temperature hexagonal structure. The calculated band structure is shown in
Fig. 3. Note two sets of bands, one at -7 eV and the other at 1 eV, of strong
S2-$p_{z}$ character. These are bonding and antibonding bands of the S2-S2
dumbells. Historically, there has been a heated discussion of the Cu valency
in this compound, and what is an appropriate ionic model. Both
(Cu${}^{+1})_{3}$(S${}_{2}^{-2})($S${}^{-1})$ (Ref. 1st, ) and
(Cu${}^{+1})_{3}($S${}_{2}^{-1})($S${}^{-2})$ (Ref. W, ) have been discussed,
assuming monovalent copper. On the other hand, XPSXPS and NQRIrek data
indicated Cu valency larger than 1, but smaller than 1.5. From our
calculations it is immediately obvious that S1 is divalent, while S2 is
monovalent (the antibonding $p_{z}$ band of the S2-S2 dimer is 1 eV above the
Fermi level, while all S1-derived bands are below the Fermi level), so that Cu
has valency 1.33, and the appropriate ionic model is
(Cu${}^{+4/3})_{3}$(S${}_{2}^{-2})($S${}^{-2}).$
This means that the Cu $d-$band has 1/3 hole per Cu ion, 2.5 times fewer than
in the high-Tc cuprates (optimal doping corresponds to 0.8-0.85 holes). This
may be too far from half filling for strong correlation effects, but it is
nevertheless suggestive of possible spin fluctuations. We will return to this
point later.
Figure 4: (color online) Calculated Fermi surfaces in the hexagonal (top) and
orthorhombic (bottom) structure, viewed along the $c$-axis. Note reduced
$k_{z}$ dispersion in the bottom panel.
In order to understand the Cu $d$ bands near the Fermi level, let us consider
a simple tight binding model with two $d$ orbitals, with $m=\pm 2$
(corresponding to combinations of the $x^{2}-y^{2}$ and $xy$ cubic harmonics,
which belong to the same representation in the hexagonal group). Since these
orbitals are the ones spread most far in the plane, their hybridization with S
is the strongest and they form the highest antibonding states near the
$\Gamma$ point, crossing the Fermi level. Integrating out the S $p_{x,y}$
orbitals we arrive at the following model band structure:
$E_{k}=\frac{1}{2(\varepsilon_{d}-\varepsilon_{p})}\left[(3t_{pd\sigma}^{2}+4t_{pd\pi}^{2})\sum_{i}\cos(\mathbf{k\cdot
R}_{i})\pm(3t_{pd\sigma}^{2}-4t_{pd\pi}^{2})\sqrt{\sum_{i}\cos^{2}(\mathbf{k\cdot
R}_{i})-\sum_{i>j}\cos(\mathbf{k\cdot R}_{i})\cos(\mathbf{k\cdot
R}_{j})}\right],$ (1)
where $t$ are the Cu-S hopping amplitudes, and Ri are the three standard
triangular lattice vectors, $\sum_{i}\mathbf{R}_{i}=0.$ Note that these bands
are degenerate at $\Gamma,$ unless spin-orbit coupling is taken into account.
Near the top of the band the dispersion is isotropic, and away from it the
Fermi surface develops a characteristic hexagonal rosette shape (Fig. 4).
Let us now look at the calculated bands (Fig. 3). It is more instructive to
concentrate on the righ hand side of Fig. 3, where the $k_{z}$ dispersion does
not obscure the states degeneracy. We see, as predicted by the model, three
sets of nearly parabolic bands, each four times degenerate at the point
A=(0,0,$\pi/c).$ One of them is below the Fermi level and two above, forming
the eight FS sheets we see in Fig. 4. The middle bands are predominantly
formed by the Cu1 and the lower (fully occupied) and upper ones by the Cu2,
although there is substantial mixture of all three Cu orbitals. The average
occupation of Cu d orbitals, as described above, is 1/3 hole per Cu, too small
to form a magnetic state, even in LDA+U with $U\sim 8$ eV (as verified by
direct calculations). Formation of an ordered magnetic state is additionally
hindered by the fact that supexchange is this case is antiferromagnetic, and
frustrated, as it should be on a triangular lattice. One may think that
additional hole doping, achieved through Cu vacancies, broken S-S bonds or
interstitial oxygen (note that this structures includes large pores, one per
formula unit, in each Cu-S layer) should bring the $d-$bands closer to half-
occupancy and promote local magnetic moments. Note that in at least one
experimental paper a Curie-Weiss behavior was reported, corresponding to 0.28
$\mu_{B}/$Cu1st , and in another a weak, but inconsistent with the Pauli law,
temperature dependence was foundNMR , while other authors reported
temperature-independent susceptibility.
Figure 5: (color online) A model Fermi surface, calculated using Eq. 1,
overlapped with the wave vectors corresponding to superexchange on a
triangular lattice. The signs show a possible $f-$wave pairing state,
consistent with superexchnage-induced spin fluctuations.
One can speculate that the unexpected high-temperature superconductivity
observed by Raveau $et$ $al$Raveau is a phenomenon of the same sort, namely
that this superconductivity forms in a portion of a sample, the same portion
where some previous researchers observed local magnetic moments. As discussed
above, it is highly unlikely that a stoichiometric, defectless CuS sample
would support either local moments or unconventional superconductivity.
However, it is of interest to consider a hypothetical situation that would
take place if such moments were present. Indeed in that case one can write
down the superexchange interaction between the nearest neighbors as
antiferromagnetic Heisenberg exchange, in which case in the reciprocal space
it will have the following functional form:
$J(\mathbf{q)=}J\sum_{i}\cos(\mathbf{k\cdot R}_{i}).$ (2)
In Fig. 5 we show an example of a Fermi surface generated for the model band
structure (Eq. 1), for the simplest case of $t_{pd\pi}=0$. The wave vectors
corresponding to the peaks of the superexchange interaction (2) are shown by
arrows. An interesting observation is that for this particular doping this
superexchange interaction (or, better to say, spin fluctuations generated by
this superexchange) would be pairing for a triplet $f-$state shown in the same
picturef . Indeed, the superexchange vectors always span the lobes of the
order parameter with the same sign. Since in a triplet case spin-fluctuations
generate an attractive interaction, it will be pairing for the geometry shown
in Fig. 5. Note that this is opposite to high-$T_{c}$ cuprates, where the
superexchange interaction spans parts with the opposite parts of the $d-$wave
order parameter, but in a singlet channel this interaction is repulsive, and
therefore pairing when the corresponding parts of the Fermi surface have
opposite signs of the order parameter.
While the model Fermi surface shown in Fig. 5 is roughly similar to that
calculated in the stoichiometric CuS, the system at this doping is too far
from the ordered magnetism to let us assume sizeable superexchange-like
magnetic fluctuations. As mentioned, our attempts to stabilize an
antiferromagnetic (more precisely, ferrimagnetic, since we only tried
collinear magnetic patterns) using a triple unit cell failed, even in LDA+U.
One may think of a hole doped system, where superexchange is operative and the
inner Fermi surfaces (albeit not the outer ones) have geometry similar to that
featured in Fig. 5.
Of course, it may not be possible to stabilize a system at sufficient hole
doping and retain the required crystallography. We prefer to think about the
model discussed in the perevious paragraphs as inspired by the CuS covellite,
bit not necessary applicable to actual materials derived from this one. The
reason we paid so much attention to it is that this is a simple generic model,
describing any triangular planar structure with transition metals and ligands
in the same plane, as in the covellite, in case where correlations are
sufficiently strong to bring about spin fluctuations controlled by
superexchange. It is quite exciting that, compared to the popular spin-
fluctuation scenario of superconductivity in cuprates, to which it is
conceptually so similar, this simple generic model results in a completely
different superconducting state, triplet $f$, as opposed to singlet $d$. This
finding may have implications far beyound this particular material and (yet
unconfirmed) superconductivity in it.
## References
* (1) B. Raveau, T. Sarkar, Solid State Sciences 13, 1874 (2011)
* (2) A.P. Gonçalves, E.B. Lopes, A. Casaca, M. Dias, M. Almeida, J. Cryst. Growth 310, 2742 (2008).
* (3) Y. Takano, N. Uchiyama, S. Ogawa, N. Môri, Y. Kimishima, S. Arisawa, A. Ishii, T. Hatano, K. Togano, Physica C 341, 739 (2000).
* (4) H. Fjellvag, F. Gronvold, S. Stolen, A.F. Andresen, R. Mueller-Kaefu, A. Simon, Z. für Kristallogr. 184, 111 (1988).
* (5) W. Liang, M.H. Whangbo, Solid State Comm. 85, 405 (1993).
* (6) C. I. Pearce, R. A. D. Pattrick, D. J. Vaughn, C. M. B. Henderson, and G. van der Laan, Geochim. Cosmochim. Acta 70, 4635 (2006).
* (7) R. R. Gainov, A. V. Dooglav, I. N. Pen’kov, I. R. Mukhamedshin, N. N. Mozgova, I. A. Evlampiev, and I. A. Bryzgalov, Phys. Rev. B79, 075115 2009.
* (8) Y. Itoh, A. Hayashi, H. Yamagata, M. Matsumura, K. Koga and Y. Ueda, J. Phys. Soc. Jpn. 65, 1953 (1996).
* (9) To be specific, this states is $D_{6h}(\Gamma^{-}_{4})$, using the notations of Sigrist and UedaSU , with the basis functions ${\bf\hat{z}}(k_{y}(k_{y}^{2}-3k_{x}^{2})$ or $k_{z}[(k_{y}^{2}-k_{x}^{2}){\bf\hat{y}}-2k_{x}k_{y}{\bf\hat{x}}]$.
* (10) M. Sigrost and K. Ueda, Rev. Mod. Phys. 63, 239 (1991).
|
arxiv-papers
| 2011-11-03T19:31:21 |
2024-09-04T02:49:23.986857
|
{
"license": "Public Domain",
"authors": "I. I. Mazin",
"submitter": "Igor Mazin",
"url": "https://arxiv.org/abs/1111.0956"
}
|
1111.1018
|
arxiv-papers
| 2011-11-04T00:15:50 |
2024-09-04T02:49:23.992891
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Javier Segura",
"submitter": "Javier Segura",
"url": "https://arxiv.org/abs/1111.1018"
}
|
|
1111.1031
|
# Electronic structure and symmetry of valence states of epitaxial NiTiSn and
NiZr0.5Hf0.5Sn thin films by hard x-ray photoelectron spectroscopy.
Xeniya Kozina Institut für Anorganische und Analytische Chemie, Johannes
Gutenberg - Universität, 55099 Mainz, Germany. Tino Jaeger Institut für
Physik, Johannes Gutenberg - Universität, 55099 Mainz, Germany. Siham Ouardi
Institut für Anorganische und Analytische Chemie, Johannes Gutenberg -
Universität, 55099 Mainz, Germany. Andrei Gloskowskij Institut für
Anorganische und Analytische Chemie, Johannes Gutenberg - Universität, 55099
Mainz, Germany. Gregory Stryganyuk Institut für Anorganische und Analytische
Chemie, Johannes Gutenberg - Universität, 55099 Mainz, Germany. Gerhard Jakob
Institut für Physik, Johannes Gutenberg - Universität, 55099 Mainz, Germany.
Takeharu Sugiyama Japan Synchrotron Radiation Research Institute (JASRI),
SPring-8, Hyogo 679-5198, Japan Eiji Ikenaga Japan Synchrotron Radiation
Research Institute (JASRI), SPring-8, Hyogo 679-5198, Japan Gerhard H. Fecher
fecher@uni-mainz.de Institut für Anorganische und Analytische Chemie, Johannes
Gutenberg - Universität, 55099 Mainz, Germany. Max Planck Institute for
Chemical Physics of Solids, 01187 Dresden, Germany. Claudia Felser Institut
für Anorganische und Analytische Chemie, Johannes Gutenberg - Universität,
55099 Mainz, Germany. Max Planck Institute for Chemical Physics of Solids,
01187 Dresden, Germany.
###### Abstract
The electronic band structure of thin films and superlattices made of Heusler
compounds with NiTiSn and NiZr0.5Hf0.5Sn composition was studied by means of
polarization dependent hard x-ray photoelectron spectroscopy. The linear
dichroism allowed to distinguish the symmetry of the valence states of the
different types of layered structures. The films exhibit a larger amount of
”in-gap” states compared to bulk samples. It is shown that the films and
superlattices grown with NiTiSn as starting layer exhibit an electronic
structure close to bulk materials.
Thermoelectric materials, Superlattice, Electronic structure, Dichroism in
photoemission, Photoelectron spectroscopy
††preprint: Kozina et al, NiTiSn
The progressively growing interest in exploration and design of the materials
exhibiting thermoelectric properties is mediated by their potential
applications in new environment friendly industrial technologies for power
generation and refrigeration Sootsman, Chung, and Kanatzidis (2009). As the
efficiency of a thermoelectric device solely depends on the dimensionless
figure of merit $ZT=S^{2}\sigma\kappa^{-1}$ at operating temperature, the most
interesting materials are those with high $ZT$, which is, in turn, defined by
thermopower $S$, electric conductivity $\sigma$ and thermal conductivity
$\kappa$. Due to the unique tunability of properties, thermal and chemical
stability, non-toxicity and ease in synthesis, among other half-Heusler
compounds the Ni$X$Sn based family of compounds and their solid solutions have
become the most perspective ones for reaching high $ZT$ values Aliev _et al._
(1990); Sakurada and Shutoh (2005); Shutoh and Sakurada (2005); Chaput _et
al._ (2006). Many attempts were made towards optimization of $ZT$ via
enlarging either $S$ or $\sigma$ Shutoh and Sakurada (2005); Schwall and Balke
(2011). Alternetively a reduction of $\kappa$ allows significantly to rise
$ZT$ values, as it was demonstrated for YX0.5X’0.5Z family of half-Heusler
compounds Hohl _et al._ (1999); Shen _et al._ (2001); Sakurada and Shutoh
(2005).
Boundary scattering of electrons and phonons play a major role in further
suppression of the thermal conductivity in polycrystalline materials Savvides
and Goldsmid (1980) and thin film superlattices. In the latter case the
phonons are scattered at the superlattice interfaces when their mean free path
is shorter than the period of the superlattice leading to low values of the
cross-plane $\kappa$ Yanga _et al._ (2002). Improvement of the quality of
such multilayer stacks as it was previously demonstrated for epitaxial
NiTiSn/NiZr0.5Hf0.5Sn superlattices Jaeger _et al._ (2011) will create new
options for producing high performance thermoelectric devices.
To improve the transport properties of the materials it is necessary to
understand and explore their electronic structure close to the Fermi energy
($\epsilon\rm_{F}$). Hard x-ray photoelectron spectroscopy (HAXPES) is a
powerful method to probe both chemical states and electronic structure of bulk
materials and buried layers in a non-destructive way Fecher _et al._ (2008);
Kozina _et al._ (2010). The combination of HAXPES with polarized radiation
for excitation significantly extends its applicability. The use of linearly
$s$ and $p$ polarized light in HAXPES enables the analysis of the symmetry of
bulk electronic states Ouardi _et al._ (2011). In the present study the
valence band electronic structure of NiTiSn/NiZr0.5Hf0.5Sn superlattices were
investigated by means of HAXPES and linear dichroism.
For the present study, multilayer stacks consisting of alternating NiTiSn and
NiZr0.5Hf0.5Sn layers were deposited by means of dc-sputtering. The details of
fabrication and characterization of the samples are described in Reference
Jaeger _et al._ (2011). Sketches of the investigated thin film, bilayer, and
superlattice samples are shown in Fig. 1. The topmost AlOX layer serves as a
protective cap preventing the oxidation and degradation of the thin films.
Figure 1: Sketch of the sample structures. The layers in (a), (b) and (c)
correspond to the 30-nm-thick films of NiTiSn and NiZr0.5Hf0.5Sn compounds
grown on different buffer layers. (d) presents a bilayer sample and (e) shows
the superlattice.
The HAXPES experiment was performed at BL47XU of Spring-8 (Japan) using 7.940
keV linearly polarized photons for excitation. Vertical ($s$) direction of
polarization was achieved by means of a in-vacuum phase retarder based on a
600-${\mu}$m-thick diamond crystal with a degree of polarization above 90 %.
Horizontal ($p$) polarization was obtained directly from the undulator without
any additional polarization optics. The energy resolution was set to 250 meV
and was verified by spectra of the Au valence band at the $\epsilon\rm_{F}$.
Gracing incidence – normal emission geometry was used ($\theta$=2∘) that
ensures that the polarization vector was nearly parallel ($p$) or
perpendicular ($s$) to the surface normal. For further details on HAXPES
experiment see Ouardi _et al._ (2011); Kozina _et al._ (2011).
Fig. 2(a) presents the valence band spectra of NiTiSn and NiZr0.5Hf0.5Sn
30-nm-thick films grown on different buffer layers (NiTiSn or NiZr0.5Hf0.5Sn
(see Fig. 1 (a), (b) and (c))). The spectra of the materials grown on a NiTiSn
buffer reveal clearly narrow structures originating from the band structure of
the Heusler compounds. The structure at lower binding energies corresponds to
the $d$\- states. They are separated by the intrinsic Heusler
$sp$-hybridization gap (at about -6 eV) from the $s$-states. In the range
above -6 eV the 4-peak- structure peculiar to Ni$X$Sn compounds is clearly
resolved. Such shape of the energy distribution curve is formed mainly by the
partial density of Ni-$3d$ states as was shown previously (see References
Ouardi _et al._ (2011); Tobola _et al._ (1998); Pierre _et al._ (1997) for
the calculated density of states (DOS)). The contribution of the Sn $s$ states
gives rise to the broad peak at -8.26 eV (peak F). Apparently the intensity of
$s$-states becomes comparable with that of $d$-states at about 8 keV
excitation energy. Such a behavior is a direct consequence of different cross
sections for $s$ and $d$ states.
Figure 2: Valence band spectra of the single NiTiSn and NiZr0.5Hf0.5Sn films
grown on different buffer layers (a) compared to polycrystalline NiTiSn and
NiZr0.5Hf0.5Sn bulk samples (b). (Note that the additional intensity at below
-10 eV seen in a) emerges from the AlOx cap layer.)
The peaks positions of NiTiSn and NiZr0.5Hf0.5Sn films grown on a NiTiSn
buffer agree well with those of polycrystalline NiTiSn and NiZr0.5Hf0.5Sn bulk
samples shown in Fig. 2(b). As it was demonstrated before Miyamoto _et al._
(2008); Ouardi _et al._ (2011), the intensity of peaks B (-1.21 eV) and C
(-2.41 eV) undergoes drastic changes. When going from NiTiSn to
NiZr0.5Hf0.5Sn, i. e. by substitution of Ti atoms by (Zr,Hf), peaks B and C
are increased. This follows from the fact that the Ti 3$d$ partial DOS
contributes significantly to the total DOS in this energy range along with the
Ni 3$d$ states. Larger cross sections for Zr 4$d$ and Hf 5$d$ states compared
to the Ti 3$d$ states enhance the peaks. Moreover, feature D shifts towards
higher binding energies by 0.21 eV in the spectra of both bulk samples and
thin films when Ti is substituted by (Zr,Hf). This correlation in the spectra
of the epitaxially grown thin films and the pure polycrystalline samples
together with the agreement with previously reported results implies the
formation of a well ordered crystalline C1b structure in the films of both
compounds when grown on a NiTiSn buffer layer.
For both compounds one observes the appearance of ”in-gap” states close to
$\epsilon_{\rm F}$ (feature A). Substitution of Ti atoms with (Zr,Hf) leads to
an increase of ”in-gap” states in both thin films and bulk samples that is in
a good agreement with recent work Miyamoto _et al._ (2008). Such states are
attributed to the disorder at the Ti-site, viz. formation of the antisites of
Ti atoms with the vacancies Ouardi _et al._ (2010). They are responsible for
the remarkable thermoelectric properties of the materials. The relatively high
amount of ”in-gap” states in the thin films compared with the bulk materials
can be explained by the presence of additional crystalline defects in the thin
films induced by lattice strain, interface states with broken symmetry, or
interdiffusion of atoms in conjugated layers. NiZr0.5Hf0.5Sn grown on a
NiZr0.5Hf0.5Sn buffer (Fig. 2(a)) has obviously a high degree of disorder as
is revealed from both smeared out valence band and completely closed band gap.
Further investigations were performed on bilayers and superlattices (Fig. 1
(d), (e)). Both, $p$\- and $s$-polarized, hard x-rays were used for
excitation. The photoelectron spectra of both samples (Fig. 3) are typical for
the electronic structure of the compounds, as described above. The high
probing depth in the order of tens of nanometers allows to obtain the
information from several 1.5-nm-thick layers of the superlattice. Their
contribution to the total signal is nonequivalent as is seen in Fig. 3. One
notices a relative redistribution of peaks B, C, and D when comparing the
spectra taken with both orthogonal polarization. A clear enhancement of the
signals from the B and C states – similar to the NiZr0.5Hf0.5Sn sample – is
explained by the presence of the topmost 1.5 nm-thick NiZr0.5Hf0.5Sn layer in
the superlattice. Here, most of the obtained signal is attributed to the
NiZr0.5Hf0.5Sn layer whereas the intensity from the underlying and other
layers is damped due to increased inelastic scattering probability for
electrons passing larger distances through the upper layers of the structure.
Figure 3: Polarization-dependent valence band spectra of a
NiZr0.5Hf0.5Sn/NiTiSn bilayer (a) and the NiTiSn/NiZr0.5Hf0.5Sn superlattice
(b). The spectra obtained with $s$ and $p$ polarized x-rays are shown together
with the difference curves.
The spectra shown in Fig. 3 were normalized to the secondary electron
background at about -14 eV to account for different intensities for different
kind of polarization (see also Ouardi _et al._ (2011)). Substantial changes
of the spectra from both samples are quite obvious when the polarization is
switched from $p$ to $s$. In both cases the peak at -8.31 eV arising from Sn
$s$ ($a1$) states is enhanced with $p$-polarized photons, while the intensity
of the $d$-part of the spectra is lowered. Namely the features originating
from $e$ and $t_{2}$ states (-2.36 eV) and $t_{2}$ states (-3.06 eV) of Ni as
well as $e$ and $t_{2}$ states (-1.3 eV) of Ti are enhanced when using
$p$-polarized photons for excitation Ouardi _et al._ (2010). The relative
change in the intensity of peak E arising from $t_{1}$ states of Ni and Ti is
larger in the superlattice sample (see difference curve in Fig. 3(b)). This is
due to the different overlying material in the two samples and therefore a
increased contribution of states from Zr and Hf. In the bilayer sample the
enhancement of the relative change in peak D at -3.06 eV giving a sharper
feature in the difference curve (Fig. 3(a)) is caused mainly by changes of the
cross sections for $t_{2}$ states of Ni similarly as it was observed
previously for polycrystalline NiTiSn Ouardi _et al._ (2011). This is in a
good agreement with the present case as the 30 nm overlying layer mostly
contributes to the overall signal obtained from the bilayer structure. From
the polarization dependence it is also concluded that the in-gap states have
$d$-type character.
summary, the investigation of electronic properties of thin films as well as
superlattices of NiTiSn and NiZr0.5Hf0.5Sn thermoelectric materials were
performed by means of HAXPES. The polarization dependent HAXPES investigation
allowed clearly to distinguish the states of different symmetry contributing
to the total DOS in the valence band region in the pure NiTiSn and
NiZr0.5Hf0.5Sn thin films. The impact of the different materials could even be
resolved in the complex multilayered structures. Utilizing of NiTiSn as buffer
layer for epitaxial growth of the different thin films and superlattices of
both materials results in a high quality of the crystalline structure. The
studies showed the appearance of ”in-gap” $d$-states in both compounds that
may be mediated by disorder at the interfaces and possible strain effects
common for thin film structures. The ”in-gap” states can serve as a tool for
artificial tuning of the thermoelectric properties in thin films – in
particular an increase of the conductivity –, as was shown already for bulk
materials.
Financial support by the DFG (Fe633/8-1 and Ja821/4-1 in SPP 1386) is
gratefully acknowledged. HAXPES was performed at BL47XU of SPring-8 with
approval of JASRI (Proposals No. 2011A1464, 2010A0017).
## References
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* Shen _et al._ (2001) Q. Shen, L. Chen, T. Goto, T. Hirai, J. Yang, G. P. Meisner, and C. Uher, Appl. Phys. Lett. 79, 25 (2001).
* Savvides and Goldsmid (1980) N. Savvides and H. J. Goldsmid, J. Phys. C: Solid St. Phys. 13, 4657 (1980).
* Yanga _et al._ (2002) B. Yanga, W. L. Liu, J. L. Liu, K. L. Wang, and G. Chen, Appl. Phys. Lett. 81, 3588 (2002).
* Jaeger _et al._ (2011) T. Jaeger, C. Mix, M. Schwall, X. Kozina, J. Barth, B. Balke, M. Finsterbusch, Y. U. Idzerda, C. Felser, and G. Jakob, Thin Solid Films XX, XX (2011).
* Fecher _et al._ (2008) G. H. Fecher, B. Balke, A. Gloskowskii, S. Ouardi, C. Felser, T. Ishikawa, M. Yamamoto, Y. Yamashita, H. Yoshikawa, S. Ueda, and K. Kobayashi, Appl. Phys. Lett. 92, 193513 (2008).
* Kozina _et al._ (2010) X. Kozina, S. Ouardi, B. Balke, G. Stryganyuk, G. H. Fecher, C. Felser, S. Ikeda, H. Ohno, and E. Ikenaga, Appl. Phys. Lett. 96, 072105 (2010).
* Ouardi _et al._ (2011) S. Ouardi, G. H. Fecher, X. Kozina, G. Stryganyuk, B. Balke, C. Felser, E. Ikenaga, T. Sugiyama, N. Kawamura, M. Suzuki, and K. Kobayashi, Phys. Rev. Lett. 107, 036402 (2011).
* Kozina _et al._ (2011) X. Kozina, G. H. Fecher, G. Stryganyuk, S. Ouardi, B. Balke, C. Felser, G. Schönhense, E. Ikenaga, T. Sugiyama, N. Kawamura, M. Suzuki, T. Taira, T. Uemura, M. Yamamoto, H. Sukegawa, W. Wang, K. Inomata, and K. Kobayashi, Phys. Rev. B 84, 054449 (2011).
* Tobola _et al._ (1998) J. Tobola, J. Pierre, S. Kaprzyk, R. V. Skolozdra, and M. A. Kouacou, J. Phys.: Condens. Matter 10, 1013 (1998).
* Pierre _et al._ (1997) J. Pierre, R. V. Skolozdra, J. Tobola, S. Kaprzyk, C. Hordequin, M. A. Kouacou, I. Karla, R. Currat, and E. Leliévre-Berna, J. Alloys Comp. 262 - 263, 101 (1997).
* Miyamoto _et al._ (2008) K. Miyamoto, A. Kimura, K. Sakamoto, M. Ye, Y. Cui, K. Shimada, H. Namatame, M. Taniguchi, S. i. Fujimori, Y. Saitoh, E. Ikenaga, K. Kobayashi, J. Tadano, and T. Kanomata, Appl. Phys. Exp. 1, 081901 (2008).
* Ouardi _et al._ (2010) S. Ouardi, G. H. Fecher, B. Balke, X. Kozina, G. Stryganyuk, C. Felser, S. Lowitzer, D. Ködderitzsch, H. Ebert, and E. Ikenaga, Phys. Rev. B 82, 085108 (2010).
|
arxiv-papers
| 2011-11-04T02:41:26 |
2024-09-04T02:49:23.996883
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xeniya Kozina and Tino Jaeger and Siham Ouardi and Andrei Gloskowskij\n and Gregory Stryganyuk and Gerhard Jakob and Takeharu Sugiyama and Eiji\n Ikenaga and Gerhard H. Fecher and Claudia Felser",
"submitter": "Gerhard H. Fecher Dr.",
"url": "https://arxiv.org/abs/1111.1031"
}
|
1111.1183
|
11institutetext: Code 7210, Naval Research Laboratory, 4555 Overlook Ave., SW,
Washington DC 20375-5320 tom.wilson@nrl.navy.mil
# Techniques of Radio Astronomy
T. L. Wilson 11
## Abstract
This chapter provides an overview of the techniques of radio astronomy. This
study began in 1931 with Jansky’s discovery of emission from the cosmos, but
the period of rapid progress began fifteen years later. From then to the
present, the wavelength range expanded from a few meters to the sub-
millimeters, the angular resolution increased from degrees to finer than milli
arc seconds and the receiver sensitivities have improved by large factors.
Today, the technique of aperture synthesis produces images comparable to or
exceeding those obtained with the best optical facilities. In addition to
technical advances, the scientific discoveries made in the radio range have
contributed much to opening new visions of our universe. There are numerous
national radio facilities spread over the world. In the near future, a new era
of truly global radio observatories will begin. This chapter contains a short
history of the development of the field, details of calibration procedures,
coherent/heterodyne and incoherent/bolometer receiver systems, observing
methods for single apertures and interferometers, and an overview of aperture
synthesis.
keywords: Radio Astronomy–Coherent Receivers–Heterodyne Receivers–Incoherent
Receivers–Bolometers–Polarimeters–Spectrometers–High Angluar
Resolution–Imaging–Aperture Synthesis
## 1 Introduction
Following a short introduction, the basics of simple radiative transfer,
propagation through the interstellar medium, polarization, receivers,
antennas, interferometry and aperture synthesis are presented. References are
given mostly to more recent publications, where citations to earlier work can
be found; no internal reports or web sites are cited. The units follow the
usage in the astronomy literature. For more details, see Thompson et al.
(2001), Gurvits et al. (2005), Wilson et al. (2008), and Burke & Graham-Smith
(2009).
The origins of optical astronomy are lost in pre-history. In contrast radio
astronomy began recently, in 1931, when K. Jansky showed that the source of
excess radiation at $\nu=$20.5 MHz ($\lambda=$14.6 m) arose from outside the
solar system. G. Reber followed up and extended Jansky’s work, but the most
rapid progress occurred after 1945, when the field developed quickly. The
studies included broadband radio emission from the Sun, as well as emission
from extended regions in our galaxy, and later other galaxies. In wavelength,
the studies began at a few meters where the emission was rather intense and
more easily measured (see Sullivan 2005, 2009). Later, this was expanded to
include centimeter, millimeter and then sub-mm wavelengths. In Fig. 1 a plot
of transmission through the atmosphere as a function of frequency $\nu$ and
wavelength, $\lambda$ is presented. The extreme limits of the earth-bound
radio window extend roughly from a lower frequency of $\nu\cong 10$ MHz
($\lambda\cong 30$ m) where the ionosphere sets a limit, to a highest
frequency of $\nu\cong 1.5$ THz ($\lambda\cong 0.2$ mm), where molecular
transitions of atmospheric H2O and N2 absorb astronomical signals. There is
also a prominent atmospheric feature at $\sim 55$ GHz, or 6 mm, from O2. The
limits shown in Fig. 1 are not sharp since there are variations both with
altitude, geographic position and time. Reliable measurements at the shortest
wavelengths require remarkable sites on earth. Measurements at wavelengths
shorter than $\lambda$=0.2 mm require the use of high flying aircraft,
balloons or satellites. The curve in Fig. 1 allows an estimate of the height
above sea level needed to carry out astronomical measurements.
Figure 1: A plot of transmission through the atmosphere versus wavelength,
$\lambda$ in metric units and frequency, $\nu$, in Hertz. The thick curve
gives the fraction of the atmosphere (left vertical axis) and the altitude
(right axis) needed to reach a transmission of 0.5. The fine scale variations
in the thick curve are caused by molecular transitions (see Townes & Schawlow
1975). The thin vertical line on the left ($\sim 10$MHz) marks the boundary
where ionospheric effects impede astronomical measurements. The labels above
indicate the types of facilities needed to measure at the frequencies and
wavelengths shown. For example, from the thick curve, at $\lambda$=100 $\mu$m,
one half of the astronomical signal penetrates to an altitude of 45 km. In
contrast, at $\lambda$=10 cm, all of this signal is present at the earth’s
surface. The arrows at the bottom of the figure indicate the type of atomic or
nuclear process that gives rise to the radiation at the frequencies and
wavelengths shown above (from Wilson et al. 2008).
The broadband emission mechanism that dominates at meter wavelengths has been
associated with the synchrotron process. Thus although the photons have
energies in the micro electron volt range, this emission is caused by highly
relativistic electrons (with $\gamma$ factors of more than 103) moving in
microgauss fields. In the centimeter and millimeter wavelength ranges, some
broadband emission is produced by the synchrotron process, but additional
emission arises from free-free Bremsstrahlung from ionized gas near high mass
stars and quasi-thermal broadband emission from dust grains. In the mm/sub-mm
range, emission from dust grains dominates, although free-free and synchrotron
emission may also contribute. Spectral lines of molecules become more
prominent at mm/sub-mm wavelengths (see Rybicki & Lightman 1979, Lequeux 2004,
Tielens 2005).
Radio astronomy measurements are carried out at wavelengths vastly longer than
those used in the optical range (see Fig. 1), so extinction of radio waves by
dust is not an important effect. However, the longer wavelengths lead to lower
angular resolution, $\theta$, since this is proportional to $\lambda$/D where
D is the size of the aperture (see Jenkins & White 2001). In the 1940’s, the
angular resolutions of radio telescopes were on scales of many arc minutes, at
best. In time, interferometric techniques were applied to radio astronomy,
following the method first used by Michelson. This was further developed,
resulting in Aperture Synthesis, mainly by M. Ryle and associates at Cambridge
University (for a history, see Kellermann & Moran 2001). Aperture synthesis
has allowed imaging with angular resolutions finer than milli arc seconds with
facilities such as the Very Long Baseline Array (VLBA).
Ground based measurements in the sub-mm wavelength range have been made
possible by the erection of facilities on extreme sites such as Mauna Kea, the
South Pole and the 5 km high site of the Atacama Large Millimeter/sub-mm Array
(ALMA). Recently there has been renewed interest in high resolution imaging at
meter wavelengths. This is due to the use of corrections for smearing by
fluctuations in the electron content of the ionosphere and advances that
facilitate imaging over wide angles (see, e.g., Venkata 2010). With time, the
general trend has been toward higher sensitivity, shorter wavelength, and
higher angular resolution.
Improvements in angular resolution have been accompanied by improvements in
receiver sensitivity. Jansky used the highest quality receiver system then
available. Reber had access to excellent systems. At the longest wavelengths,
emission from astronomical sources dominates. At mm/sub-mm wavelengths, the
transparency of the earth’s atmosphere is an important factor, adding both
noise and attenuating the astronomical signal, so both lowering receiver noise
and measuring from high, dry sites are important. At meter and cm wavelengths,
the sky is more transparent and radio sources are weaker.
The history of radio astronomy is replete with major discoveries. The first
was implicit in the data taken by Jansky. In this, the intensity of the
extended radiation from the Milky Way exceeded that of the quiet Sun. This
remarkable fact shows that radio and optical measurements sample fundamentally
different phenomena. The radiation measured by Jansky was caused by the
synchrotron mechanism; this interpretation was made more than 15 years later
(see Rybicki & Lightman 1979). The next discovery, in the 1940’s, showed that
the active Sun caused disturbances seen in radar receivers. In Australia, a
unique instrument was used to associate this variable emission with sunspots
(see Dulk 1985, Gary & Keller 2004). Among later discoveries have been: (1)
discrete cosmic radio sources, at first, supernova remnants and radio galaxies
(in 1948, see Kirshner 2004), (2) the 21 cm line of atomic hydrogen (in 1951,
see Sparke & Gallagher 2007, Kalberla et al. 2005), (3) Quasi Stellar Objects
(in 1963, see Begelman & Rees 2009), (4) the Cosmic Microwave Background (in
1965, see Silk 2008), (5) Interstellar molecules (see Herbst & Dishoeck 2009)
and the connection with Star Formation, later including circumstellar and
protoplanetary disks (in 1968, see Stahler & Palla 2005, Reipurth et al.
2007), (6) Pulsars (in 1968, see Lyne & Graham-Smith 2006), (7) distance
determinations using source proper motions determined from Very Long Baseline
Interferometry (see Reid 1993) and (8) molecules in high redshift sources (see
Solomon & Vanden Bout 2005). These areas of research have led to
investigations such as the dynamics of galaxies, dark matter, tests of general
relativity, Black Holes, the early universe and gravitational radiation (for
overviews see Longair 2006, Harwit 2006). Radio astronomy has been recognized
by the physics community in that four Nobel Prizes (1974, 1978, 1993 and 2006)
were awarded for work in this field. In chemistry, the community has been made
aware of the importance of a more general chemistry involving ions and
molecules (see Herbst 2001). Two Nobel Prizes for chemistry were awarded to
persons actively engaged in molecular line astronomy.
Over time, the trend has been away from small groups of researchers
constructing special purpose instruments toward the establishment of large
facilities where users propose projects carried out by specialized staffs.
These large facilities are in the process of becoming global. Similarly, the
evolution of data reduction has been toward standardized packages developed by
large teams. In addition, the demands of the interpretation of astronomical
phenomena have led to multi-wavelength analyses interpreted with the use of
detailed models.
Outside the norm are projects designed to measure a particular phenomenon. A
prime example is the study of the cosmic microwave background (CMB) emission
from the early universe. CMB data were taken with the COBE and WMAP
satellites. These results showed that the CMB is is a Black Body (see Eq. 6)
with a temperature of 2.73 K. Aside from a dipole moment caused by our motion,
there is angular structure in the CMB at a very low level; this is being
studied with the PLANCK satellite. Much effort continues to be devoted to
measurements of the polarization of the CMB with ground-based experiments such
as BICEP, CBI, DASI and QUIET. For details and references to other CMB
experiments, see their websites. In spectroscopy, there have been extensive
surveys of the 21 cm line of atomic hydrogen, H I (see Kalberla et al. 2005)
and the rotational $J=1-0$ line from the ground state of carbon monoxide (see
Dame et al. 1987). These surveys have been extended to external galaxies (see
Giovanelli & Haynes 1991). During the Era of Reionization (redshift $z\sim$10
to 15), the H I line is shifted to meter wavelengths. The detection of such a
feature is the goal of a number of individual groups, under the name HERA
(Hydrogen Epoch of Reionization Arrays).
### 1.1 A Selected List of Radio Astronomy Facilities
There are a large number of existing facilities; a selection is listed here.
General purpose instruments include the largest single dishes: the Parkes
64-m, the Robert C. Byrd Green Bank Telescope, hereafter GBT, the Effelsberg
100 meter, the 15-m James Clerk Maxwell Telescope (JCMT), the IRAM 30-m
millimeter telescope and the 305-m Arecibo instrument. All of these have been
in operation for a number of years. Interferometers form another category of
instruments. The Expanded Very Large Array, the EVLA, is now in the test phase
with ′′shared risk′′ observing. Other large interferometer systems are the
VLBA, the Westerbork Synthesis Radio Telescope in the Netherlands, the
Australia Telescope, the Giant Meter Wave Telescope in India, the MERLIN array
a number of arrays at Cambridge University in the UK and the MOST facility in
Australia. In the mm range, CARMA in California and Plateau de Bure in France
are in full operation, as is the Sub-Millimeter Array of the Harvard-
Smithsonian CfA and ASIAA on Mauna Kea, Hawaii. At longer wavelengths, the Low
Frequency Array, LOFAR, has started the first measurements and will expand by
adding stations throughout Europe. The Square Kilometer Array, the SKA, is in
the planning phase as is the FASR solar facility, while the Australian SKA
Precursor (ASKAP), the South African SKA precuror, (MeerKAT), the Murchison
Widefield Array in Western Australia and Long Wavelength Array in New Mexico
are under construction. A portion of the Allen Telescope Array, ATA, is in
operation. A number of facilities are under construction, being commissioned
or have recently become operational. At sub-mm wavelengths, the Herschel
Satellite Observatory has been delivering data. The Five Hundred Meter
Aperture Spherical Telescope, FAST, a design based on the Arecibo instrument,
is being planned in China. This will be the world’s largest single aperture.
The Large Millimeter Telescope, LMT, a joint Mexican-US project, will soon
begin science operations as will the Stratospheric Far-Infrared Observatory
(SOFIA) operated by NASA and the German DLR organization. Descriptions of
these instruments are to be found in the internet. Finally, the most ambitious
ground based astronomy project to date is ALMA which will start early science
operations in late 2011 (for an account of the variety of ALMA science goals,
see Bachiller & Cernicharo 2008).
## 2 Radiative Transfer and Black Body Radiation
The total flux of a source is obtained by integrating Intensity (in Watts m-2
Hz-1 steradian-1) over the total solid angle $\Omega_{\rm s}$ subtended by the
source
$S_{\nu}=\int\limits_{\Omega_{\rm s}}I_{\nu}(\theta,\varphi)\cos\theta\,{\rm
d}\Omega.$ (1)
The flux density of astronomical sources is given in units of the Jansky
(hereafter Jy), that is, $1\,{\rm Jy}=10^{-26}\,{\rm W\,m}^{-2}{\rm Hz}^{-1}$.
The equation of transfer is useful in interpreting the behavior of
astronomcial sources, receiver systems, the effect of the earth’s atmosphere
on measurements. Much of this analysis is based on a one dimensional version
of the general expression as (see Lequeux 2004 or Tielens 2005):
$\framebox{$\displaystyle\frac{{\rm d}I_{\nu}}{{\rm
d}s}=-\kappa_{\nu}I_{\nu}+\varepsilon_{\nu}$}\quad.$ (2)
The linear absorption coefficient $\kappa_{\nu}$ and the emissivity
$\varepsilon_{\nu}$ are independent of the intensity $I_{\nu}$. From the
optical depth definition ${\rm d}\tau_{\nu}=-\kappa_{\nu}\,{\rm d}s$, the
Kirchhoff relation $\varepsilon_{\nu}/\kappa_{\nu}=B_{\nu}$ (see (Eq. 6)) and
the assumption of an isothermal medium, the result is:
$\framebox{$\displaystyle I_{\nu}(s)=I_{\nu}(0)\,{\rm
e}^{-\tau_{\nu}(s)}+B_{\nu}(T)\,(1-\,{\rm e}^{-\tau_{\nu}(s)})$}\quad.$ (3)
For a large optical depth, that is for $\tau_{\nu}(0)\rightarrow\infty$, (Eq.
3) approaches the limit
$I_{\nu}=B_{\nu}(T)\thinspace.$ (4)
This is case for planets and the 2.73 K CMB. From (Eq, 3), the difference
between $I_{\nu}(s)$ and $I_{\nu}(0)$ gives
$\Delta
I_{\nu}(s)=I_{\nu}(s)-I_{\nu}(0)=(B_{\nu}(T)-I_{\nu}(0))(1-{\rm\,e}^{-\tau})\;.$
(5)
this represents the result of an on-source minus off-source measurement, which
is relevant for discrete sources.
The spectral distribution of the radiation of a black body in thermodynamic
equilibrium is given by the Planck law
$\framebox{$\displaystyle B_{\nu}(T)=\frac{2h\nu^{3}}{c^{2}}\frac{1}{{\rm
e}^{h\nu/kT}-1}$}\quad.$ (6)
If $h\nu\ll kT$, the Rayleigh-Jeans Law is obtained:
$\framebox{$\displaystyle B_{\rm RJ}(\nu,T)=\frac{2\nu^{2}}{c^{2}}kT$}\quad.$
(7)
In the Rayleigh-Jeans relation, the brightness and the thermodynamic
temperatures of Black Body emitters are strictly proportional (Eq. 7). This
feature is useful, so the normal expression of brightness of an extended
source is brightness temperature $T_{\rm B}$:
$T_{\rm
B}=\frac{c^{2}}{2k}\frac{1}{\nu^{2}}\,I_{\nu}=\frac{\lambda^{2}}{2k}\,I_{\nu}\thinspace.$
(8)
If $I_{\nu}$ is emitted by a black body and $h\nu\ll kT$ then (Eq. 8) gives
the thermodynamic temperature of the source, a value that is independent of
$\nu$. If other processes are responsible for the emission of the radiation
(e.g., synchrotron, free-free or broadband dust emission), $T_{\rm B}$ will
depend on the frequency; however (Eq. 8) is still used. If the condition
$\nu(\rm GHz)\ll 20.84\left(T(\rm K)\right)$ is not valid, (Eq. 8) can still
be applied, but $T_{\rm B}$ will differ from the thermodynamic temperature of
a black body. However, corrections are simple to obtain.
If (Eq. 8) is combined with (Eq. 5), the result is an expression for
brightness temperature:
$\displaystyle
J(T)=\frac{c^{2}}{2k\nu^{2}}(B_{\nu}(T)-I_{\nu}(0))(1-{\rm\,e}^{-\tau_{\nu}(s)})\;.$
The expression $J(T)$ can be expressed as a temperature in most cases. This
quantity is referred to as $T^{*}_{\rm R}$, the radiation temperature in the
mm/sub-mm range, or the brightness temperature, $T_{\rm B}$ for longer
wavelengths. In the Rayleigh-Jeans approximation the equation of transfer is:
$\framebox{$\displaystyle\frac{{\rm d}T_{\rm B}(s)}{{\rm d}\tau_{\nu}}=T_{\rm
bk}(0)-T(s)$}\quad,$ (9)
where $T_{\rm B}$ is the measured quantity, $T_{\rm bk}(s)$ is the background
source temperature and $T(s)$ is the temperature of the intervening medium If
the medium is isothermal, the general (one dimensional) solution becomes
$\framebox{$\displaystyle T_{\rm B}=T_{\rm bk}(0)\,{\rm
e}^{-\tau_{\nu}(s)}+T\,(1-\,{\rm e}^{-\tau_{\nu}(s)})$}\quad.$ (10)
### 2.1 The Nyquist Theorem and Noise Temperature
This theorem relates the thermodynamic quantity temperature to the electrical
quantities voltage and power. This is essential for the analysis of noise in
receiver systems. The average power per unit bandwidth, $P_{\nu}$ (also
referred to as Power Spectral Density, PSD), produced by a resistor $R$ is
$P_{\nu}=\langle iv\rangle=\frac{\langle v^{2}\rangle}{2R}=\frac{1}{4R}\langle
v_{\rm N}^{2}\rangle\thinspace,$ (11)
where $v(t)$ is the voltage that is produced by $i$ across $R$, and
$\langle\cdots\rangle$ indicates a time average. The first factor
$\frac{1}{2}$ arises from the condition for the transfer of maximum power from
$R$ over a broad range of frequencies. The second factor $\frac{1}{2}$ arises
from the time average of $v^{2}$. Then
$\langle v_{\rm N}^{2}\rangle=4R\,k\,T\;.$ (12)
When inserted into (Eq. 11), the result is
$P_{\nu}=k\,T\;.$ (13)
(Eq. 13) can also be obtained by a reformulation of the Planck law for one
dimension in the Rayleigh-Jeans limit. Thus, the available noise power of a
resistor is proportional to its temperature, the noise temperature $T_{\rm
N}$, independent of the value of $R$ and of frequency.
Not all circuit elements can be characterized by thermal noise. For example a
microwave oscillator can deliver 1 $\mu$W, the equivalent of more than
$10^{16}$ K, although the physical temperature is $\sim$300 K. This is an
example of a very nonthermal process, so temperature is not a useful concept
in this case.
### 2.2 Overview of Intensity, Flux Density and Main Beam Brightness
Temperature
Temperatures in radio astronomy have given rise to some confusion. A short
summary with references to later sections is given here. Power is measured by
an instrument consisting of an antenna and receiver. The power input can be
calibrated and expressed as Flux Density or Intensity. For very extended
sources, Intensity (see (Eq. 8)) can be expressed as a temperature, the main
beam brightness temperature, TMB. To obtain TMB, the measurements must be
calibrated (Section 5.3) and corrected using the appropriate efficiencies (see
Eq. 37 and following). For discrete sources, the combination of (Eq. 1) with
(Eq. 8) gives:
$\framebox{$\displaystyle S_{\nu}=\frac{2\,k\,\nu^{2}}{c^{2}}T_{\rm
B}\,\mbox{\greeksym{D}}\Omega$}\quad.$ (14)
For a source with a Gaussian spatial distribution, this relation is
$\left[\frac{S_{\nu}}{\rm Jy}\right]=0.0736\,T_{\rm
B}\,\left[\frac{\theta}{\rm arc\,seconds}\right]^{2}\left[\frac{\lambda}{\rm
mm}\right]^{-2}$ (15)
if the flux density $S_{\nu}$ and the actual (or ′′true′′) source size are
known, then the true brightness temperature, $T_{\rm B}$, of the source can be
determined. For Local Thermodynamic Equilibrium (LTE), $T_{\rm B}$ represents
the physical temperature of the source. If the apparent source size, that is,
the source angular size as measured with an antenna is known, (Eq. 15) allows
a calculation of TMB. For discrete sources, TMB depends on the angular
resolution. If the antenna beam size (see Fig. 3 and discussion) has a
Gaussian shape $\theta_{\rm b}$, the relation of actual $\theta_{\rm s}$ and
apparent size $\theta_{\rm o}$ is:
$\theta_{\rm o}^{2}=\theta_{\rm s}^{2}+\theta_{\rm b}^{2}\,.$ (16)
then from (Eq. 14), the relation of TMB and TB is:
$T_{\rm MB}\left(\theta_{\rm s}^{2}+\theta_{\rm b}^{2}\right)=T_{\rm
B}\,\theta_{\rm s}^{2}$ (17)
Finally, the PSD entering the receiver (Eq. 13) is antenna temperature, TA;
this is relevant for estimating signal to noise ratios (see (Eq. 39) and (Eq.
42)). To establish temperature scales and relate received power to source
parameters for filled apertures, see Section 5.3. For interferometry and
Aperture Synthesis, see Section 6.
### 2.3 Interstellar Dispersion and Polarization
Pulsars emit radiation in a short time interval (see Lorimer & Kramer 2004,
Lyne & Graham-Smith 2006). If all frequencies are emitted at the same instant,
the arrival time delay of different frequencies is caused by the ionized
Interstellar Medium (ISM). This is characterized by the quantity
$\int_{0}^{L}N(l){\rm\,d}l$, which is the column density of the electrons to a
distance $L$. Since distances in astronomy are measured in parsecs it has
become customary to express the dispersion measure as:
${\mathrm{DM}}=\int\limits_{0}^{L}\left(\frac{N}{\mathrm{cm}^{-3}}\right){\rm\,d}\left({\frac{l}{\mathrm{pc}}}\right)$
(18)
The lower frequencies are delayed more in the ISM, so the relative time delay
is:
$\displaystyle\frac{\Delta\tau_{\mathrm{D}}}{\mbox{\greeksym{m}}\rm
s}=1.34\times
10^{-9}\left[\frac{\mathrm{DM}}{\mathrm{cm}^{-2}}\right]\left[\displaystyle\frac{1}{\left(\displaystyle\frac{\nu_{1}}{\mathrm{MHz}}\right)^{2}}-\frac{1}{\left(\displaystyle\frac{\nu_{2}}{\mathrm{MHz}}\right)^{2}}\right]$
(19)
Since both time delay $\Delta\tau_{\mathrm{D}}$ and observing frequencies
$\nu_{1}<\nu_{2}$ can be measured with high precision, a very accurate value
of DM for a given pulsar can be determined. Provided the distance to the
pulsar, $L$, is known, a good estimate of the average electron density between
observer and pulsar can be found. However since $L$ is usually known with
moderate accuracy, only approximate values for $N$ can be obtained. Often the
opposite procedure is used: From reasonable values for $N$, a measured DM
provides information on the unknown distance $L$ to the pulsar.
Broadband linear polarization is caused by non-thermal processes (see Rybicki
& Lightman 1979) including Pulsar radiation, quasi-thermal emission from
aligned, non-spherical dust grains (see Hildebrand 1983) and scattering from
free electrons. Faraday rotation will change the position angle of linear
polarization as the radiation passes through an ionized medium; this varies as
$\lambda^{2}$, so this effect is larger for longer wavelengths. It is usual to
characterize polarization by the four Stokes Parameters, which are the sum or
difference of measured quantities. The total intensity of a wave is given by
the parameter $I$. The amount and angle of linear polarization by the
parameters $Q$ and $U$, while the amount and sense of circular polarization is
given by the parameter $V$. Hertz dipoles are sensitive to a single linear
polarization. By rotating the dipole over an angle perpendicular to the
direction of the radiation, it is possible to determine the amount and angle
of linearly polarized radiation. Helical antennas or arrangements of two Hertz
dipoles are sensitive to circular polarization. Generally, polarized radiation
is a combination of linear and circular, and is usually less than 100%
polarized, so four Stokes parameters must be specified. The definition of the
sense of circular polarization in radio astronomy is the same as in Electrical
Engineering but opposite to that used in the optical range; see Born & Wolf
(1965) for a complete analysis of polarization, using the optical definition
of circular polarization. Poincaré introduced a representation that permits an
easy visualization of all the different states of polarization of a vector
wave. See Thompson et al. (2001), Crutcher (2008), Thum et al. (2008) or
Wilson et al. (2008) for more details.
## 3 Receiver Systems
### 3.1 Coherent and Incoherent Receivers
Receivers are assumed to be linear power measuring devices, i. e. any non-
linearity is a small quantity. There are two types of receivers: coherent and
incoherent. Coherent receivers are those which preserve the phase of the input
radiation while incoherent do not. Heterodyne (technically
′′superheterodyne′′) receivers are those which those which shift the frequency
of the input but preserve phase. The most commonly used coherent receivers
employ heterodyning, that is, frequency shifting (see Section 4.2.1). The most
commonly used incoherent receivers are bolometers (Section 4.1); these are
direct detection receivers, that is, operate at sky frequency. Both coherent
and incoherent receivers add noise to the astronomical input signal; it is
assumed that the noise of both the input signal and the receiver follow
Gaussian distributions. The noise contribution of coherent receivers is
expressed in Kelvins. Bolometer noise is characterized by the Noise Equivalent
Power, or NEP, in units of Watts Hz-1/2 (see Section 3.1.1 and Section 5.3.3).
NEP is the input power level which doubles the output power. More extensive
discussions of receiver properties are given in Rieke (2002) or Wilson et al.
(2008).
To analyze the performance of a receiver, the commonly accepted model is an
ideal receiver with no internal noise, but connected to two noise sources, one
for the external noise (including the astronomical signal) and a second for
the receiver noise. To be useful, receiver systems must increase the input
power level. The noise contribution is characterized by the Noise Factor, $F$.
If the signal-to-noise ratio at the input is expressed as $(S_{1}/N_{1})$ and
at the output as $(S_{2}/N_{2})$, the noise factor is:
$F=\frac{S_{1}/N_{1}}{S_{2}/N_{2}}\;.$ (20)
A further step is to assume that the signal is amplified by a gain factor $G$
but otherwise unchanged. Then $S_{2}=G\,S_{1}$ and:
$F=\frac{N_{2}}{G\,N_{1}}\;.$ (21)
For a direct detection system such as a bolometer, $G=1$. For coherent
receivers, there must be a minimum noise contribution (see Section 4.2.4), so
$F>1$. For coherent receivers $F$ is expressed in temperature units as $T_{R}$
using the relation
$T_{R}=(F-1)\cdot 290{\rm K}\;.$ (22)
#### 3.1.1 Receiver Calibration
Heterodyne receiver noise performance is usually expressed in degrees Kelvin.
In the calibration process, a power scale (the PSD) is established at the
receiver input. This is measured in terms of the noise temperature. To
calibrate a receiver, the noise temperature increment $\Delta T$ at the
receiver input must be related to a given measured receiver output increment
$\Delta z$ (this applies to coherent receivers which have a wide dynamic range
and a total power or ′′DC′′ response). Usually resistive loads at two known
(thermodynamic) temperatures $T_{\rm L}$ and $T_{\rm H}$ are used. The
receiver outputs are $z_{\rm L}$ and $z_{\rm H}$, while $T_{\rm L}$ and
$T_{\rm H}$ are the resistive loads at two temperatures. The relations are:
$\displaystyle z_{\rm L}$ $\displaystyle=$ $\displaystyle(T_{\rm L}+T_{\rm
R})\,G\thinspace,$ $\displaystyle z_{\rm H}$ $\displaystyle=$
$\displaystyle(T_{\rm H}+T_{\rm R})\,G\thinspace,$
taking
$y=z_{\rm H}/z_{\rm L}\thinspace.$ (23)
the result is:
$\framebox{$\displaystyle T_{\rm rx}=\frac{T_{\rm H}-T_{\rm
L}\,y}{y-1}$}\quad,$ (24)
This is known as the ′′y-factor′′; the procedure is a ′′hot-cold′′
measurement. The determination of the y factor is calculated in the Rayleigh-
Jeans limit. Absorbers at temperatures of $T_{\rm H}$ and $T_{\rm L}$ are used
to produce the inputs. Often these are chosen to be at the ambient temperature
($T_{\rm H}\cong 293$ K or $20\hbox{${}^{\circ}$}$ C) and at the temperature
of liquid nitrogen ($T_{\rm L}\cong 78$ K or $-195\hbox{${}^{\circ}$}$ C).
When receivers are installed on antennas, such ′′hot-cold′′ calibrations are
done only infrequently. As will be discussed in Section 5.3.2, in the cm and
meter range, calibration signals are provided by noise diodes; from
measurements of sources with known flux densities intensity scales are
established. Any atmospheric corrections are assumed to be small at these
wavelengths. As will be discussed in Section 5.3.3, in the mm/sub-mm
wavelength range, from measurements of an ambient load (or two loads at
different temperatures), combined with measurements of emission from the
atmosphere and models of the atmosphere, estimates of atmospheric transmission
are made.
Bolometer performance is characterized by the Noise Equivalent Power, or NEP,
given in units of Watts Hz-1/2. The expression for NEP can be related to a
receiver noise temperature. For ground based bolometer systems, background
noise dominates. For these, the background noise is given as TBG:
$\framebox{$\displaystyle{\rm NEP}=2\varepsilon\,k\,T_{\rm
BG}\,\sqrt{\Delta\nu}$}\quad.$ (25)
here $\varepsilon$ is the emissivity of the background and $\Delta\nu$ is the
bandwidth. Typical values for ground-based mm/sub-mm bolometers are
$\varepsilon=0.5$, T${}_{\rm BG}=300$ K and $\Delta\nu=100$ GHz. For these
values, NEP$=1.3\times 10^{-15}$ Watts Hz-1/2. With the collecting area of the
IRAM 30 m or the JCMT telescopes, sources in the milli-Jansky (mJy) range can
be measured.
Usually bolometers are ′′A. C.′′ coupled, that is, the output responds to
differences in the input power, so hot-cold measurements are not useful for
characterizing bolometers. The response of bolometers is usually determined by
measurements of sources with known flux densities, followed by measurements
at, for example, elevations of 20o, 30o, 60o and 90o to determine the
atmospheric transmission (see Section 5.3.4).
#### 3.1.2 Noise Uncertainties due to Random Processes
The noise contributions from source, atmosphere, ground, telescope surface and
receiver are always additive:
$T_{\rm sys}=\sum T_{i}$ (26)
From Gaussian statistics, the Root Mean Square, RMS, noise is given by the
mean value divided by the square root of the number of samples. From the
estimate that the number of samples is given by the product of receiver
bandwidth multiplied by the integration time, the result is:
$\framebox{$\displaystyle\mbox{\greeksym{D}}T_{\rm RMS}=\frac{T_{\rm
sys}}{\displaystyle\sqrt{\mbox{\greeksym{D}}\nu\,\tau}}$}\quad.$ (27)
A much more elaborate derivation is to be found in Chapter 4 of Rohlfs &
Wilson (2004), while a somewhat simpler account is in Wilson et al. (2008).
The calibration process in (Section 3.1.1) allows the receiver noise to be
expressed in degrees Kelvin. The relation of Tsys to Trx is $T_{\rm
sys}=T_{\rm A}+T_{\rm rx}$, where $T_{\rm A}$ represents the power entering
the receiver; at some wavelengths $T_{\rm A}$ will dominate $T_{\rm rx}$. In
the mm/sub-mm range, use is made of T${}_{\rm sys}^{*}$, the system noise
outside the atmosphere, since the attenuation of astronomical radiation is
large. This will be presented in Section 5.3.1 and following.
#### 3.1.3 Receiver Stability
Sensitive receivers are designed to achieve a low value for $T_{\rm rx}$.
Since the signals received are of exceedingly low power, receivers must also
provide large receiver gains, $G$ (of order $10^{12}$), for sufficient output
power. Thus even very small gain instabilities can dominate the thermal
receiver noise. Since receiver stability considerations are of prime
importance, comparison switching was necessary for early receivers (Dicke
1946). Great advances have been made in improving receiver stability since the
1960’s so the need for rapid switching is lessened. In the meter and cm
wavelength range, the time between reference measurements has increased.
However in the mm/sub-mm range, instabilities of the atmosphere play an
important role; to insure that noise decreases following (Eq. 27), the effects
of atmospheric and/or receiver instabilities must be eliminated. For single
dish measurements, atmospheric changes can be compensated for by rapidly
differencing a measurement of the target source and a reference. Such
comparison or ′′Dicke′′ switched measurements are necessary for ground-based
observations. If a typical procedure consists of using a total power receiver
to measure on-source for 1/2 of the total time, then an off-source comparison
for 1/2 of the time and taking the difference of on-source minus off-source
measurements, the $\mbox{\greeksym{D}}T_{\rm RMS}$ will be a factor of 2
larger than the value given by (Eq. 27).
## 4 Practical Aspects of Receivers
This section concentrates on receivers that are currently in use. For more
details see Goldsmith (1988), Rieke (2002), or Wilson et al. (2008).
### 4.1 Bolometer Radiometers
Bolometers operate by use of the effect that the resistance, $R$, of a
material varies with the temperature. In the 1970’s, the most sensitive
bolometers were semiconductor devices pioneered by F. Low. This is achieved
when the bolometer element is cooled to very low temperatures. When radiation
is incident, the characteristics change, so this is a measure of the intensity
of the incident radiation. Because this is a thermal effect, it is independent
of the frequency and polarization of the radiation absorbed. Thus bolometers
are intrinsically broadband devices. It is possible to mount a polarization-
sensitive device before the bolometer and thereby measure the direction and
degree of linear polarization. Also, it is possible to carry out spectroscopy,
if frequency sensitive elements, either filters, Michelson or Fabry-Perot
interferometers, are placed before the bolometer element. Since these
spectrometers operate at the sky frequency, the fractional resolution
($\mbox{\greeksym{D}}\nu/\nu$) is at best $\sim 10^{-4}$. The data from each
bolometer detector element (pixel) must be read out and then amplified.
For single dish (i. e. filled apertures) broadband continuum measurements at
$\lambda<$ 2 mm, multi-beam bolometers are the most common systems and such
systems can have a large number of beams. A promising new development in
bolometer receivers is Transition Edge Sensors referred to as TES bolometers.
These superconducting devices may allow more than an order of magnitude
increase in sensitivity, if the bolometer is not background limited. For
bolometers used on earth-bound telescopes, the improvement with TES systems
may be only $\sim$2–3 times more sensitive than the semiconductor bolometers,
but TES’s will allow readouts from a much larger number of pixels.
A number of large bolometer arrays have produced numerous publications: (1)
MAMBO2 (MAx-Planck-Millimeter Bolometer) used on the IRAM 30-m telescope at
1.3 mm, (2) SCUBA (Submillimeter Common User Bolometer Array; Holland et al.
1999) on the JCMT, (3) the LABOCA (LArge Bolometer CAmera) array on the APEX
12 meter telescope, (4) SHARC (Sub-mm High Angular Resolution Camera) on the
Caltech Sub-mm Observatory 10-m telescope and (5) MUSTANG (MUtiplexed Squid
TES Array) on the GBT. SCUBA will be replaced with SCUBA-2 now being
constructed at the U. K. Astronomy Technology Center, and there are plans to
replace the MUSTANG array by MUSTANG-2, which is a larger TES system.
### 4.2 Coherent Receivers
Usually, coherent receivers make use of heterodyning to shift the signal input
frequency without changing other properties of the input signal; in practice,
this is carried out by the use of mixers (Section 4.2.2). The heterodyne
process is used in all branches of communications technology; use of
heterodyning allows measurements with unlimited spectral resolution. Although
heterodyne receivers have a number of components, these systems have more
flexibility than bolometers.
#### 4.2.1 Noise Contributions in Coherent Receivers
The noise generated in the first element dominates the system noise. The
mathematical expression is given by the Friis relation which accounts for the
effect of cascaded amplifiers on the noise performance of a receiver:
$\framebox{$\displaystyle T_{\rm S}=T_{\rm S1}+\frac{1}{G_{1}}T_{\rm
S2}+\frac{1}{G_{1}G_{2}}T_{\rm S3}+\dots+\frac{1}{G_{1}G_{2}\dots
G_{n-1}}T_{{\rm S}n}$}\quad.$ (28)
Where $G_{1}$ is the gain of the first element, and $T_{\rm S1}$ is the noise
temperature of this element. For $\lambda>$3 mm ($\nu<115$GHz), the best
cooled first elements, High Electron Mobility Transistors (HEMTs), typically
have $G_{1}=10^{3}$ and $T_{\rm S1}=50$K; for $\lambda<$0.8 mm, the best
cooled first elements, superconducting mixers, typically have $G_{1}\leq 1$,
that is, a small loss, and $T_{\rm S1}\leq 500$K. The stage following the
mixer should have the lowest noise temperature and high gain.
#### 4.2.2 Mixers
Mixers have been used in heterodyne receivers since Jansky’s time. At first
these were metal-oxide-semiconductor or Schottky mixers. Mixers allow the
signal frequency to be changed without altering the characteristics of the
signal. In the mixing process, the input signal is multiplied by an intense
monochromatic signal from a local oscillator, LO. The frequency stability of
the LO signal is maintained by a stabilization device in which the LO signal
is compared with a stable input, in recent times, an atomic standard. These
phaselock loop systems produce a pure, highly stable, monochromatic signal.
The mixer can be operated in the Double Sideband (DSB) mode, in which two sky
frequencies, ′′signal′′ and ′′image′′ at equal separations from the LO
frequency (equal to the IF frequency) are shifted into intermediate (IF)
frequency band. For spectral line measurements, usually one sideband is
wanted, but the other not. DSB operation adds both noise and (usually)
unwanted spectral lines; for spectral line measurements, single sideband (SSB)
operation is preferred. In SSB operation, the unwanted sideband is suppressed,
at the cost of more complexity. In the sub-mm wavelength ranges, DSB mixers
are still commonly used as the first stage of a receiver; in the mm and cm
ranges, SSB operation is now the rule.
A significant improvement can be obtained if the mixer junction is operated in
the superconducting mode. The noise temperatures and LO power requirements of
superconducting mixers are much lower than Schottky mixers. Finally, the
physical layout of such devices is simpler since the mixer is a planar device,
deposited on a substrate by lithographic techniques. SIS mixers consist of a
superconducting layer, a thin insulating layer and another superconducting
layer (see Phillips & Woody 1982).
Figure 2: Receiver noise temperatures for coherent amplifier systems compared
to the temperatures from the Milky Way galaxy (at long wavelengths, on left
part of figure) and the atmosphere (at mm/sub-mm wavelengths on the right
side). The atmospheric emission is based on a model of zenith emission for 0.4
mm of water vapor, that is, excellent weather (plot from B. Nicolic (Cambridge
Univ.) using the ′′AM′′ program of S. Paine (Harvard-Smithsonian Center for
Astrophysics)). This does not take into account the absorption of the
astronomical signal. In the 1 to 26 GHz range, the two horizontal lines
represent the noise temperatures of the best HEMT amplifiers, while the solid
line represents the noise temperatures of maser receivers. The shaded region
between 85 and 115.6 GHz is the receiver noise for the SEQUOIA array (Five
College Radio Astronomy Observatory) which consists of monolithic millimeter
integrated circuits (MMIC). The meaning of the other symbols is given in the
upper left of the diagram (SIS’s are Superconductor-Insulator-Superconductor
mixers, HEB’s are Hot Electron Bolometer mixers). The double sideband (DSB)
mixer noise temperatures were converted to single sideband (SSB) noise
temperatures by doubling the receiver noise. The ALMA mixer noise temperatures
are SSB, as are the HEMT values. The line marked ′′10 h$\nu$/kT′′ refers to
the limit described in (Eq. 30). Some data used in this diagram are taken from
Rieke (2002). The figure is from Wilson et al. (2008)
Superconducting Hot Electron Bolometer-mixers (HEB) are heterodyne devices, in
spite of the name. These mixers make use of superconducting thin films which
have sub-micron sizes (see Kawamura et al. 2002).
A number of multi-beam heterodyne cameras are in operation in the cm range,
but only a few in the mm/sub-mm range. The first mm multi-beam system was the
SEQUOIA array receiver pioneered by S. Weinreb; such devices are becoming more
common. In contrast, multibeam systems that use SIS front ends are rare.
Examples are a 9 beam Heterodyne Receiver Array of SIS mixers at 1.3 mm, HERA,
on the IRAM 30-m millimeter telescope, HARP-B, a 16 beam SIS system in
operation at the JCMT for 0.8 mm and the CHAMP+ receiver at the Max-Planck-
Inst. für Radioastronomy on the APEX 12-m telescope.
#### 4.2.3 Square Law Detectors
For heterodyne receivers the input is normally amplified (for $\nu<115$GHz),
translated in frequency and then detected in a device that produces an output
signal $y(t)$ which is proportional to the square of $v(t)$:
$y(t)=a\,v^{2}(t)$ (29)
Once detected, phase information is lost. For interferometers, the output of
each antenna is a voltage, shifted in frequency and then digitized. This
output is brought to a central location for correlation.
#### 4.2.4 The Minimum Noise in a Coherent System
The ultimate limit for coherent receivers or amplifiers is obtained by an
application of the Heisenberg uncertainty principle involving phase and number
of photons. From this, the $minimum$ noise of a coherent amplifier results in
a receiver noise temperature of
$\framebox{$\displaystyle T_{\rm rx}({\rm minimum})=\frac{h\nu}{k}$}\quad.$
(30)
For incoherent detectors, such as bolometers, phase is not preserved, so this
limit does not exist. In the mm wavelength region, this noise temperature
limit is quite small; at $\lambda$=2.6 mm ($\nu$=115 GHz), this limit is 5.5
K. The value for the ALMA receiver in this range is about 5 to 6 times the
minimum. A significant difference between radio and optical regimes is that
the minimum noise in the radio range is small, so that the power from a single
receiver can be amplified and then divided. For example, for the EVLA, the
voltage output of all 351 antenna pairs are combined with little or no loss in
the signal-to-noise ratio. Another example is given in Section 4.3.1, where a
radio polarimeter can produce all four Stokes parameters from two inputs
without a loss of the signal-to-noise ratio.
### 4.3 Back Ends: Polarimeters & Spectrometers
The term ′′Back End′′ is used to specify the devices following the IF
amplifiers. Many different back ends have been designed for specialized
purposes such as continuum, spectral or polarization measurements.
For a single dish continuum correlation receiver, the (identical) receiver
input is divided, amplified in two identical systems and then the outputs are
multiplied. The gain fluctuations are uncorrelated but the signals are, so the
effect on the output is the same as with a Dicke switched system, but with no
time spent on a reference.
#### 4.3.1 Polarimeters
A typical heterodyne dual polarization receiver consists of two identical
systems, each sensitive to one of the two orthogonal polarizations, linear or
circular. Both systems must be connected to the same local oscillator to
insure that the phases have a definite relation. Given this arrangement, a
polarimeter can provide values of all four Stokes parameters simultaneously.
All Stokes parameters can also be measured using a single receiver whose input
is switched from one sense of polarization to the other, but then the
integration time for each polarization will be halved.
#### 4.3.2 Spectrometers
Spectrometers analyze the spectral information contained in the radiation
field. To accomplish this, the spectrometer must be SSB and the frequency
resolution $\Delta\nu$ is usually very good, sometimes in the kHz range. In
addition, the time stability must be high. If a resolution of $\Delta\nu$ is
to be achieved for the spectrometer, all those parts of the system that enter
critically into the frequency response have to be maintained to better than
$0.1\,\Delta\nu$. For an overview of the current state of spectrometers, see
Baker et al. (2007).
Conceptually, the simplest spectrometer is composed of a set of $n$ adjacent
filters, each with a bandwidth $\Delta\nu$. Following each filter is a square-
law detector and integrator. For a finer resolution, another set of $n$
filters must be constructed.
Another approach to spectral analysis is to Fourier Transform (FT) the input,
$v(t)$, to obtain $v(\nu)$ and then square $v(\nu)$ to obtain the Power
Spectral Density. The maximum bandwidth is limited by the sampling rate. From
(another!) Nyquist theorem, it is necessary to sample at a rate equal to twice
the bandwidth. In the simplest scheme, for a bandwidth of 1 GHz, the sampling
must occur at a rate of 2 GHz. After sampling and Fourier Transform, the
output is squared to produce power in an ′′FX′′ autocorrelator. For $10^{3}$
samples, each channel will have a 1 MHz resolution.
For ′′XF′′ systems, the input $v(t)$ is multiplied (the ′′X′′) with a delayed
signal $v(t-\tau)$ to obtain the autocorrelation function $R(\tau)$. This is
then Fourier Transformed to obtain the spectrum. For $10^{3}$ samples, there
will be $10^{3}$ frequency channels. For an XF system the time delays are
performed in a set of serial digital shift registers with a sample delayed by
a time $\tau$. Autocorrelation can also be carried out with the help of analog
devices using a series of cable delay lines; these can provide very large
bandwidths. The first XF system for astronomy was a digital autocorrelator
built by S. Weinreb in 1963.
The two significant advantages of digital spectrometers are: (1) flexibility
and (2) a noise behavior that follows $1/\sqrt{t}$ after many hours of
integration. The flexibility allows the choice of many different frequency
resolutions and bandwidths or even to employ a number of different
spectrometers, each with different bandwidths, simultaneously.
A serious drawback of digital auto and cross correlation spectrometers had
been limited bandwidths. However, advances in digital technology in recent
years have allowed the construction of autocorrelation spectrometers with
several 103 channels covering instantaneous bandwidths of several GHz.
Autocorrelation systems are used in single antennas. The calculation of
spectra makes use of the symmetric nature of the autocorrelation function,
ACF, so the number of delays gives the number of spectral channels.
Cross-correlators are used in interferometers and in some single dish
applications. When used in an interferometer, the cross-correlation is between
different inputs so will not necessarily be symmetric. Thus, the zero delay of
the cross-correlator is placed in channel $N/2$. The number of delays, $N$,
allows the determination of $N/2$ spectral intensities, and $N/2$ phases. The
cross-correlation hardware can employ either an XF or a FX correlator. For
more details about the use of cross-correlation, see Section 6.
Until recently, spectrometers with bandwidths of several GHz often made use of
Acoustic Optical analog techniques. The Acoustic Optical Spectrometer (AOS)
makes use of the diffraction of light by ultrasonic waves: these cause
periodic density variations in the crystal through which it passes. These
density variations in turn cause variations in the bulk constants of the
crystal, so that a plane light wave passing through this medium will be
modulated by the interaction with the crystal. The modulated light is detected
in a charge coupled device. Typical AOS’s have an instantaneous bandwidth of 2
GHz and 2000 spectral channels.
In all cases, the spectra of the individual channels of a spectrometer are
expressed in terms of temperature with the relation:
$T_{i}=\left[\left(S_{i}-R_{i}\right)/R_{i}\right]\cdot T_{\rm sys}$ (31)
where $S_{i}$ is the normalized spectrum of channel $i$ for the on-source
measurement and $R_{i}$ is the corresponding reference for this channel. For
mm/sub-mm spectra, $T_{\rm sys}$ is replaced by $T_{\rm sys}^{*}$ (corrected
for atmospheric losses; see Section 5.3.3). For cross-correlators, as used in
interferometers, the signals from two antennas are multiplied. In this case,
the value of $T_{\rm sys}$ is the square root of the product of the system
noise temperatures of the two systems.
## 5 Antennas
The antenna serves to focus power into the feed, a device that efficiently
transfers power in the electromagnetic wave to the receiver. According to the
principle of reciprocity, the properties of antennas such as beam sizes,
efficiencies etc. are the same whether these are used for receiving or
transmitting. Reciprocity holds in astronomy, so it is usual to
interchangeably use expressions that involve either transmission or reception
when discussing antenna properties. All of the following applies to the far-
field radiation.
### 5.1 The Hertz Dipole
The total power radiated from a Hertz dipole carrying an oscillating current
$I$ at a wavelength $\lambda$ is
$\framebox{$\displaystyle P=\frac{2c}{3}\left(\frac{I\Delta
l}{2\lambda}\right)^{2}$}\quad.$ (32)
For the Hertz dipole, the radiation is linearly polarized with the electric
field along the direction of the dipole. The radiation pattern has a donut
shape, with the cylindrically symmetric maximum perpendicular to the axis of
the dipole. Along the direction of the dipole, the radiation field is zero. To
improve directivity, reflecting screens have been placed behind a dipole, and
in addition, collections of dipoles, driven in phase, are used. Hertz dipole
radiators have the best efficiency when the size of the dipole is
$1/2\,\lambda$ .
### 5.2 Filled Apertures
This Section is a simplified description of antenna properties needed for the
interpretation of astronomical measurements. For more detail, see Baars
(2007). At cm and shorter wavelengths, flared waveguides (′′feed horns′′) or
dipoles are used to convey power focussed by the antenna (i. e.,
electromagnetic waves in free space) to the receiver (voltage). At the longest
wavelengths, dipoles are used as the antennas. Details are to be found in Love
(1976) and Goldsmith (1988, 1994).
#### 5.2.1 Angular Resolution and Efficiencies
From diffraction theory (see Jenkins & White 2001), the angular resolution of
a reflector of diameter $D$ at a wavelength $\lambda$ is
$\framebox{$\displaystyle\theta=k\frac{\lambda}{D}$}\quad.$ (33)
where $k$ is of order unity. This universal result gives a value for $\theta$
(here in radians when $D$ and $\lambda$ have the same units). Diffraction
theory also predicts the unavoidable presence of sidelobes, i. e. secondary
maxima. The sidelobes can be reduced by tapering the antenna illumination.
Tapering lowers the response to very compact sources and increases the value
of $\theta$, i. e. widens the beam.
The reciprocity concept provides a method to measure the power pattern
(response pattern or Point Spread Function, PSF) using transmitters. However,
the distance from a large antenna A (diameter $D\gg\lambda$) to a transmitter
B (small in size) must be so large that B produces plane waves across the
aperture $D$ of antenna A, that is, so B is in the far field of A. This is the
Rayleigh distance; it requires that the curvature of a wavefront emitted by B
is much less than $\lambda$/16 across the geometric dimensions of antenna A.
By definition, at the Rayleigh distance $\mathcal{D}$, the curvature must be
$\gg D^{2}/8\lambda$ for an antenna of diameter $D$.
Often, the normalized power pattern is measured:
$\framebox{$\displaystyle P_{\rm n}(\vartheta,\varphi)=\frac{1}{P_{\rm
max}}\,P(\vartheta,\varphi)$}\quad.$ (34)
For larger apertures, the transmitter is usually replaced by a small diameter
radio source of known flux density (see Baars et al. 1977, Ott et al. 1994).
The flux densities of a few primary calibration sources are determined by
measurements using horn antennas at centimeter and millimeter wavelengths. At
mm/sub-mm wavelengths, it is usual to employ planets, or moons of planets,
whose surface temperatures are known (see Altenhoff 1985, Sandell 1994).
[width=2.6cm,height=5.9cm]wilson-fig3.pdf
Figure 3: A polar power pattern showing the main beam, and near and far
sidelobes. The weaker far sidelobes have been combined to form the stray
pattern
The beam solid angle $\Omega_{\rm A}$ of an antenna is given by
$\framebox{$\displaystyle\Omega_{\rm
A}=\parbox{22.76219pt}{$\vspace*{-2mm}{\displaystyle\int\\!\\!\int\atop{\\!\\!\\!\\!\\!\scriptstyle
4\pi}}$ }P_{\rm
n}(\vartheta,\varphi)\,{\rm\,d}\Omega=\int\limits_{0}^{2\pi}\\!\int\limits_{0}^{\pi}P_{\rm
n}(\vartheta,\varphi)\sin\vartheta{\rm\,d}\vartheta{\rm\,d}\varphi$}$ (35)
this is measured in steradians (sr). The integration is extended over all
angles, so $\Omega_{\rm A}$ is the solid angle of an ideal antenna having
$P_{\rm n}=1$ for $\Omega_{\rm A}$ and $P_{\rm n}=0$ everywhere else. For most
antennas the (normalized) power pattern has much larger values for a limited
range of both $\vartheta$ and $\varphi$ than for the remainder; the range
where $\Omega_{\rm A}$ is large is the main beam of the antenna; the remainder
are the sidelobes or backlobes (Fig. 3).
In analogy to (Eq. 35) the main beam solid angle $\Omega_{\rm MB}$ is defined
as
$\framebox{$\displaystyle\Omega_{\rm
MB}=\mathop{\int\\!\\!\int}\limits_{\scriptstyle\rm main\atop\scriptstyle\rm
lobe}P_{\rm n}(\vartheta,\varphi)\,{\rm\,d}\Omega$}\quad.$ (36)
The quality of a single antenna depends on how well the power pattern is
concentrated in the main beam. The definition of main beam efficiency or beam
efficiency, $\eta_{\rm B}$, is:
$\framebox{$\displaystyle\eta_{\rm B}=\frac{\Omega_{\rm MB}}{\Omega_{\rm
A}}$}\quad.$ (37)
$\eta_{\rm B}$ is the fraction of the power is concentrated in the main beam.
The main beam efficiency can be modified (within limits) for parabolic
antennas by changing the illumination of the main reflector. An
underilluminated antenna has a wider main beam but lower sidelobes. The
angular extent of the main beam is usually described by the full width to half
power width (FWHP), the angle between points of the main beam where the
normalized power pattern falls to $1/2$ of the maximum. For elliptically
shaped main beams, values for widths in orthogonal directions are needed. The
beamwidth, $\theta$ is given by (Eq. 33). If the FWHP beamwidth is well
defined, the location of an isolated source is determined to the accuracy
given by the FWHP divided by the S/N ratio. Thus, it is possible to determine
positions to small fractions of the FWHP beamwidth, if the signal-to-noise
ratio is high and noise is the only limit.
If a plane wave with the power density $\mid\\!\langle\vec{S}\rangle\\!\mid$
in Watts m-2 is intercepted by an antenna, a certain amount of power is
extracted from this wave. This power is $P_{\rm e}$ and the fraction is:
$A_{\rm e}=P_{\rm e}\,/\mid\\!\langle\vec{S}\rangle\\!\mid$ (38)
the effective aperture of the antenna. $A_{\rm e}$ has the dimension of m2.
Compared to the geometric aperture $A_{\rm g}$ an aperture efficiency
$\eta_{\rm A}$ can be defined by:
$\framebox{$\displaystyle A_{\rm e}=\eta_{\rm A}A_{\rm g}$}\quad.$ (39)
If an antenna with a normalized power pattern $P_{\rm n}(\vartheta,\varphi)$
is used to receive radiation from a brightness distribution
$B_{\nu}(\vartheta,\varphi)$ in the sky, at the output terminals of the
antenna the power per unit bandwidth (PSD), in Watts Hz-1, $P_{\nu}$ is:
$P_{\nu}={\textstyle\frac{1}{2}}\,A_{\rm e}\int\\!\\!\int
B_{\nu}(\vartheta,\varphi)\,P_{\rm n}(\vartheta,\varphi)\,{\rm\,d}\Omega\;.$
(40)
By definition, this operates in the Rayleigh-Jeans limit, so the equivalent
distribution of brightness temperature can be replaced by an equivalent
antenna temperature $T_{\rm A}$ (Eq. 13):
$P_{\nu}=k\,T_{\rm A}\,.$ (41)
This definition of antenna temperature relates the output of the antenna to
the power from a matched resistor. When these two power levels are equal, then
the antenna temperature is given by the temperature of the resistor. The
effective aperture $A_{\rm e}$ can be replaced by the the beam solid angle
$\Omega_{\rm A}\cdot\lambda^{2}$. Then (Eq. 40) becomes
$\displaystyle T_{\rm A}(\vartheta_{0},\varphi_{0})=\frac{\int T_{\rm
B}(\vartheta,\varphi)P_{\rm
n}(\vartheta-\vartheta_{0},\varphi-\varphi_{0})\sin\vartheta{\rm\,d}\vartheta{\rm\,d}\varphi}{\int
P_{\rm n}(\vartheta,\varphi){\rm\,d}\Omega}$ (42)
From (Eq. 42), $T_{\rm A}<T_{\rm B}$ in all cases. The numerator is the
convolution of the brightness temperature with the beam pattern of the
telescope (Fourier methods are of great value in this analysis; see Bracewell
1986). The brightness temperature $T_{\rm b}(\vartheta,\varphi)$ corresponds
to the thermodynamic temperature of the radiating material only for thermal
radiation in the Rayleigh-Jeans limit from an optically thick source; in all
other cases $T_{\rm B}$ is a convenient quantity that represents source
intensity at a given frequency. The quantity $T_{\rm A}$ in (Eq. 42) was
obtained for an antenna in which ohmic losses and absorption in the earth’s
atmosphere were neglected. These losses can be corrected in the calibration
process. Since $T_{\rm A}$ is the quantity measured while $T_{\rm B}$ is
desired, (Eq. 42) must be inverted. (Eq. 42) can be solved only if $T_{\rm
A}(\vartheta,\varphi)$ and $P_{\rm n}(\vartheta,\varphi)$ are known exactly
over the full range of angles. In practice this inversion is possible only
approximately, since both $T_{\rm A}(\vartheta,\varphi)$ and $P_{\rm
n}(\vartheta,\varphi)$ are known only for a limited range of $\vartheta$ and
$\varphi$ values, and the measured data are affected by noise. Therefore only
an approximate deconvolution can be performed. If the source distribution
$T_{\rm B}(\vartheta,\varphi)$ has a small extent compared to the telescope
beam, the best estimate for the upper limit to the actual FWHP source size is
1/2 of the FWHP of the telescope beam.
#### 5.2.2 Efficiencies for Compact Sources
For a source small compared to the beam (Eq. 40) and (Eq. 41) give:
$P_{\nu}\,={\textstyle\frac{1}{2}}A_{\rm e}\,S_{\nu}=k\,T_{\rm A}$ (43)
$T_{\rm A}$ is the antenna temperature at the receiver, while $T_{\rm
A}^{\prime}$ is this quantity corrected for effect of the earth’s atmosphere.
In the meter and cm range $T_{\rm A}=T_{\rm A}^{\prime}$, so in the following,
$T_{\rm A}^{\prime}$ will be used:
$\framebox{$\displaystyle T_{\rm A}^{\prime}=\Gamma S_{\nu}$}$ (44)
where $\Gamma$ is the sensitivity of the telescope measured in K Jy-1.
Introducing the aperture efficiency $\eta_{\rm A}$ according to (Eq. 39) we
find
$\framebox{$\displaystyle\Gamma=\eta_{\rm A}\frac{\pi D^{2}}{8k}$}\quad.$ (45)
Thus $\Gamma$ or $\eta_{\rm A}$ can be measured with the help of a calibrating
source provided that the diameter $D$ and the noise power scale in the
receiving system are known. When (Eq. 44) is solved for $S_{\nu}$, the result
is:
$S_{\nu}=3520\,\frac{T_{\rm A}^{\prime}[{\rm K}]}{\eta_{\rm A}[{\rm
D/m}]^{2}}\,.$ (46)
The brightness temperature is defined as the Rayleigh-Jeans temperature of an
equivalent black body which will give the same power per unit area per unit
frequency interval per unit solid angle as the celestial source. Both $T_{\rm
A}^{\prime}$ and TMB are defined in the Rayleigh-Jeans limit, but the
brightness temperature scale has to be corrected for antenna efficiency. The
conversion from source flux density to source brightness temperature for
sources with sizes small compared to the telescope beam is given by (Eq. 15).
For sources small compared to the beam, the antenna and main beam brightness
temperatures are related by the main beam efficiency, $\eta_{\rm B}$:
$\eta_{\rm B}=\frac{T_{\rm A}^{\prime}}{T_{\rm MB}}\,.$ (47)
This is valid for sources where sidelobe structure is not important (see the
discussion after (Eq. 42)). Although a source may not have a Gaussian shape,
fits of multiple Gaussians can be used to obtain an accurate representation.
What remains is a calibration of the temperature scales and a correction for
absorption in the earth’s atmosphere. This is dealt with in Section 5.3
#### 5.2.3 Foci, Blockage and Surface Accuracy
If the size of a radio telescope is more than a few hundred wavelengths,
designs are similar to those of optical telescopes. Cassegrain, Gregorian and
Nasmyth systems have been used. See Fig. 4 for a sketch of these focal
systems. In a Cassegrain system, a convex hyperbolic reflector is introduced
into the converging beam immediately in front of the prime focus. This
reflector transfers the converging rays to a secondary focus which, in most
practical systems is situated close to the apex of the main dish. A Gregorian
system makes use of a concave reflector with an elliptical profile. This must
be positioned behind the prime focus in the diverging beam. In the Nasmyth
system this secondary focus is situated in the elevation axis of the telescope
by introducing another, usually flat, mirror. The advantage of a Nasmyth
system is that the receiver front ends remain horizontal while when the
telescope is pointed toward different elevations. This is an advantage for
receivers cooled with liquid helium, which may become unstable when tipped.
Cassegrain and Nasmyth foci are commonly used in the mm/sub-mm wavelength
ranges.
In a secondary reflector system, feed illumination beyond the edge receives
radiation from the sky, which has a temperature of only a few K. For low-noise
systems, this results in only a small overall system noise temperature. This
is significantly less than for prime focus systems. This is quantified in the
so-called ′′G/T value′′, that is, the ratio of antenna gain of to system
noise. Any telescope design must aim to minimize the excess noise at the
receiver input while maximizing gain. For a specific antenna, this
maximization involves the design of feeds and the choice of foci.
Figure 4: The geometry of parabolic apertures: (a) Cassegrain, (b) Gregorian,
(c) Nasmyth and (d) offset Cassegrain systems (from Wilson et al. 2008).
The secondary reflector and its supports block the central parts in the main
dish from reflecting the incoming radiation, causing some significant
differences between the actual beam pattern and that of an unobstructed
antenna. Modern designs seek to minimize blockage due to the support legs and
subreflector.
The beam pattern differs from a uniformly illuminated unblocked aperture for 3
reasons:
(1) the illumination of the reflector will not be uniform but has a taper by
10 dB, that is, a factor of 10 or more at the edge of the reflector. This is
in contrast to optical telescopes which have no taper.
(2) the side-lobe level is strongly influenced by this taper: a larger taper
lowers the sidelobe level.
(3) the secondary reflector must be supported by three or four support legs,
which will produce aperture blocking and thus affect the shape of the beam
pattern.
Feed leg blockage will cause deviations from circular symmetry. For altitude-
azimuth telescopes these sidelobes will change position on the sky with hour
angle (see Reich et al. 1978). This may be a serious defect, since these
effects will be significant for maps of low intensity regions near an intense
source. The sidelobe response may depend on the polarization of the incoming
radiation (see Section 5.3.6).
A disadvantage of on-axis systems, regardless of focus, is that they are often
more susceptible to instrumental frequency baselines, so-called baseline
ripples across the receiver band than primary focus systems (see Morris 1978).
Part of this ripple is caused by reflections of noise from source or receiver
in the antenna structure. Ripples from the receiver can be removed if the
amplitude and phase are constant in time. Baseline ripples caused by the
source, sky or ground radiation are more difficult to eliminate since these
will change over short times. It is known that large amounts of blockage and
larger feed sizes lead to large baseline ripples. The influence of baseline
ripples on measurements can be reduced to a limited extent by appropriate
observing procedures. A possible solution is an off-axis system such as the
GBT of the National Radio Astronomy Observatory. In contrast to the GBT, the
Effelsberg 100-m has a large amount of blocking from massive feed support legs
and, as a result, show large instrumental frequency baseline ripples. These
ripples might be mitigated by the use of scattering cones in the reflector.
The gain of a filled aperture antenna with small scale surface irregularities
$\varepsilon$ cannot increase indefinitely with increasing frequency but
reaches a maximum at $\lambda_{\rm m}=4\pi\varepsilon$, and this gain is a
factor of 2.7 below that of an error-free antenna of identical dimensions. The
usual rule-of-thumb is that the irregularities should be 1/16 of the shortest
wavelength used. Larger filled aperture radio telescopes are made up of
panels. For these, the irregularities are of two types: (1) roughness of the
individual panels, and (2) misadjustment of panels. The second irregularity
gives rise to an error beam. The FWHP of the error beam is given approximately
by the ratio of wavelength to panel size. In addition, if the surface material
is not a perfect conductor, there will be some loss and consequently
additional noise.
### 5.3 Single Dish Observational Techniques
#### 5.3.1 The Earth’s Atmosphere
For ground–based facilities, the amplitudes of astronomical signals have been
attenuated and the phases have been altered by the earth’s atmosphere. In
addition to attenuation, the receiver noise is increased by atmospheric
emission, the signal is refracted and there are changes in the path length.
These effects may change slowly with time, but there can also be rapid changes
such as scintillation and anomalous refraction. Thus propagation properties
must be taken into account if the astronomical measurements are to be
correctly interpreted. At meter wavelengths, these effects are caused by the
ionosphere. In the mm/sub-mm range, tropospheric effects are especially
important. The various constituents of the atmosphere absorb by different
amounts. Because the atmosphere can be considered to be in LTE, these
constituents also emit radiation.
The total amount of precipitable water (usually measured in mm) is an integral
along the line-of-sight to a source. Frequently, the amount of H2O is
determined by measurements of the continuum emission of the atmosphere with a
small dish at 225 GHz. For a set of measurements at elevations of 20o, 30o,
60o and 90o, combined with models, rather accurate values of the atmospheric
$\tau$ can be obtained. For extremely dry mm/sub-mm sites, measurements of the
183 GHz spectral line of water vapor can be used to estimate the total amount
of H2O in the atmosphere. For sea level sites, the 22.235 GHz line of water
vapor has been used for this purpose. The scale height $H_{\rm H_{2}O}\approx
2\,{\rm km}$, is considerably less than $H_{\rm air}\approx 8\,{\rm km}$ of
dry air. For this reason, sites for submillimeter radio telescopes are usually
mountain sites with elevations above $\approx 3000$ m. For ionospheric
effects, even the highest sites on earth provide no improvement.
The effect on the intensity of a radio source due to propagation through the
atmosphere is given by the standard relation for radiative transfer (from (Eq.
10)):
$\framebox{$\displaystyle T_{\rm B}(s)=T_{\rm B}(0)\,{\rm
e}^{-\tau_{\nu}(s)}+T_{\rm atm}\,(1-\,{\rm e}^{-\tau_{\nu}(s)})$}\quad.$ (48)
Here $s$ is the (geometric) path length along the line-of-sight with $s=0$ at
the upper edge of the atmosphere and $s=s_{0}$ at the antenna, $\tau_{\nu}(s)$
is the optical depth, $T_{\rm atm}$ is the temperature of the atmosphere and
$T_{\rm B}(0)$ is the temperature of the astronomical source above the
atmosphere. Both the (volume) absorption coefficient $\kappa$ and the gas
temperature $T_{\rm atm}$ will vary with $s$. Introducing the mass absorption
coefficient $k_{\nu}$ by
$\kappa_{\nu}=k_{\nu}\cdot\varrho\,,$ (49)
where $\varrho$ is the gas density; this variation of $\kappa$ can mainly be
related to that of $\varrho$ as long as the gas mixture remains constant along
the line-of-sight. This is a simplified relation. For a more detailed
calculations, a multi-layer model is needed.
Models can provide corrections for average effects; fluctuations and detailed
corrections needed for astronomy must be determined from real-time
measurements.
#### 5.3.2 Meter and Centimeter Calibration Procedures
This involves a three step procedure: (1) the measurements must be corrected
for atmospheric effects, (2) relative calibrations are made using secondary
standards and (3) if needed, gain versus elevation curves for the antenna must
be established.
In the cm wavelength range, atmospheric effects are usually small. For steps
(2) and (3) the calibration is carried out with the use of a pulsed signal
injected before the receiver. This pulsed signal is added to the receiver
input. The calibration signal must be stable, broadband and of reasonable
size. Often noise diodes are used as pulsed broadband calibration sources.
These are secondary standards that provide broadband radiation with effective
temperatures $>10^{5}$ K. With a pulsed calibration, the receiver outputs are
recorded separately as: (1) receiver only, (2) receiver plus calibration and
(3) repeat of this cycle. If the calibration signal has a known value and the
zero point of the receiver system is measured, the receiver noise is
determined (see Eq. 24). Most often the calibration value in either Jy/beam or
TMB units is determined by a continuum scan through a non-time variable
compact discrete source of known flux density. Lists of calibration sources
are to be found in Baars et al. (1977), Altenhoff (1985), Ott et al. (1994)
and Sandell (1994).
#### 5.3.3 Millimeter and Sub-mm Calibration Procedures
In the mm/sub-mm wavelength range, the atmosphere has a larger influence and
can change on timescales of seconds, so more complex corrections are needed.
Also,large telescopes may operate close to the limits caused by their surface
accuracy, so that the power received in the error beam may be comparable to
that received in the main beam. In addition, many sources such as molecular
clouds are rather extended. Thus, relevant values of telescope efficiencies
must be used (see Downes 1989). The calibration procedure used in the mm/sub-
mm range is referred to as the chopper wheel method (Penzias & Burrus 1973).
This consists of two steps:
(1) the measurement of the receiver noise (the method is very similar to that
in Section (3.1.1). and
(2) the measurement of the receiver response when directed toward cold sky at
a certain elevation.
In the following it is assumed that the receiver is operated in the SSB mode.
For (1), the output of the receiver while measuring an ambient load, $T_{\rm
amb}$, is denoted by $V_{\rm amb}$:
$V_{\rm amb}=G\,(T_{\rm amb}+T_{\rm rx})\,.$ (50)
where $G$ is the system gain. This is sometimes repeated with a second load at
a different temperature. The result is a determination of the receiver noise
as in Section (3.1.1). For step (2), the load is removed; then the output
refers to noise from a source-free sky ($T_{\rm sky}$), ground ( $T_{\rm
gr}=T_{\rm amb}$) and receiver:
$V_{\rm sky}=G\,[F_{\rm eff}\,T_{\rm sky}+(1-F_{\rm eff})\,T_{\rm gr}+T_{\rm
rx}]\,.$ (51)
where $F_{\rm eff}$ is the forward efficiency. This is the fraction of power
in the forward beam of the feed. This can be interpreted as the response to a
source with the angular size of the Moon (it is assumed that $F_{\rm eff}$ is
appropriate for an extended molecular cloud). Taking the difference between
$V_{\rm amb}$ and $V_{\rm sky}$:
$\Delta V_{\rm cal}=V_{\rm amb}-V_{\rm sky}=G\,F_{\rm eff}\,T_{\rm
amb}{\rm\,e}^{-\tau_{\nu}}\,,$ (52)
where $\tau_{\nu}$ is the atmospheric absorption at the frequency of interest.
If it is assumed that $T_{\rm sky}(s)=T_{\rm atm}\,(1-\,{\rm
e}^{-\tau_{\nu}})$ describes the emission of the atmosphere, and, as in (Eq.
48), $\tau_{\nu}$ in is the same for emission and absorption, emission
measurements can provide the value of $\tau_{\nu}$. If $T_{\rm atm}=T_{\rm
amb}$, the correction is simplified. For more complex situations, models of
the atmosphere are needed (see e.g., Pardo et al. 2009). Once $\tau_{\nu}$ is
known, the signal from the radio source, $T_{\rm A}$, after passing through
the earth’s atmosphere, is
$\displaystyle\Delta V_{\rm sig}=G\,T_{\rm A}^{\prime}\,{\rm e}^{-\tau_{\nu}}$
or
$\displaystyle T_{\rm A}^{\prime}=\frac{\Delta V_{\rm sig}}{\Delta V_{\rm
cal}}\,F_{\rm eff}\,T_{\rm amb}$
where $T_{\rm A}^{\prime}$ is the antenna temperature of the source outside
the earth’s atmosphere. We define
$T_{\rm A}^{*}=\frac{T_{\rm A}^{\prime}}{F_{\rm eff}}=\frac{\Delta V_{\rm
sig}}{\Delta V_{\rm cal}}\,T_{\rm amb}\,$ (53)
The right side involves only measured quantities. $T_{\rm A}^{*}$ is commonly
referred to as the corrected antenna temperature, but it is really a forward
beam brightness temperature. An analogous temperature is $T_{\rm sys}^{*}$,
the system noise correcting for all atmospheric effects:
$T_{\rm sys}^{*}=\left(\frac{T_{\rm rx}+T_{\rm sky}}{F_{\rm eff}}\right)\,{\rm
e}^{\tau}$ (54)
This result is used to determine continuum or line temperature scales (Eq.
31). A typical set of values for $\lambda=3$mm are: $T_{\rm rx}$=40 K, $T_{\rm
sky}$=50 K, $\tau$=0.3. Using these, the $T_{\rm sys}^{*}$=135 K.
For sources $\ll$30′, there is an additional correction for the telescope beam
efficiency, which is commonly referred to as $B_{\rm eff}$. Then
$\displaystyle T_{\rm MB}=\frac{F_{\rm eff}}{B_{\rm eff}}\,T_{\rm A}^{*}$
Typical values of $F_{\rm eff}$ are $\cong 0.9$, and at the shortest
wavelengths used for a telescope, $B_{\rm eff}\cong 0.6$. In general, for
extended sources, the brightness temperature corrected for absorption by the
earth’s atmosphere, $T_{\rm A}^{*}$, should be used.
#### 5.3.4 Bolometer Calibrations
Since most bolometers are A. C. coupled (i. e. respond to differences), so the
D. C. response (i. e. respond to total power) used in ′′hot–cold′′ or
′′chopper wheel′′ calibration methods cannot be used. Instead astronomical
data are calibrated in two steps:
(1) measurements of atmospheric emission at a number of elevations to
determine the opacities at the azimuth of the target source, and
(2) the measurement of the response of a nearby source with a known flux
density; immediately after this, a measurement of the target source is carried
out.
#### 5.3.5 Continuum Observing Strategies
1) Position Switching and Wobbler Switching. Switching against a load or
absorber is used only in exceptional circumstances, such as studies of the
2.73 K cosmic microwave background. For the CMB, Penzias & Wilson (1965) used
a helium cooled load with a precisely known temperature. For compact regions,
compensation of transmission variations of the atmosphere is possible if
double beam systems can be used. At higher frequencies, in the mm/sub-mm
range, rapid movement of the telescope beam (by small movements of the sub-
reflector or a mirror in the path from antenna to receiver) over small angles
is referred to as ′′beam switching′′, ′′wobbling′′ or ′′wobbler switching′′.
This is used to produce two beams on the sky for a single pixel receiver. The
individual telescope beams should be spaced by a distance of 3 FWHP beam
widths.
2) Mapping of Extended Regions and On the Fly Mapping. Multi-beam bolometer
systems are preferred for continuum measurements at $\nu>$ 100 GHz. Usually, a
wobbler system is needed for such arrays. With these, it is possible to
measure a fairly large region and to better cancel sky noise due to weather.
Some details of more recent data taking and reduction methods are given in
e.g., Johnstone et al. (2000) or Motte et al. (2006).
If extended areas are to be mapped, scans are made along one direction (e.g.,
Azimuth or Right Ascension). Then the antenna is offset in the orthogonal
direction by 1/2 to 1/3 of a beamwidth, and the scanning is repeated until the
region is completely mapped. This is referred to as a ′′raster scan′′. There
should be reference positions free of sources at the beginning and the end of
each scan, to allow the determination of zero levels and calibrations should
be made before the scans are begun. For more secure results, the map is then
repeated by scanning in the orthogonal direction (e.g., Elevation or
Declination). Then both sets of results are placed on a common grid, and
averaged; this is referred to as ′′basket weaving′′.
Extended emission regions can also be mapped using a double beam system, with
the receiver input periodically switched between the first and second beam. In
this procedure, there is some suppression of very extended emission. A
summation of the beam switched data along the scan direction has been used to
reconstruct infrared images. More sophisticated schemes can recover most, but
not all, of the information (Emerson et al. 1979; ′′EKH′′). Most mm/sub-mm
antennas employ wobbler switching in azimuth to cancel ground radiation. By
measuring a source using scans in azimuth at different hour angles, then
transforming the positions to an astronomical coordinate frame and combining
the maps it is possible to reduce the effect of sidelobes caused by feed legs
and supress sky noise (Johnstone et al. 2000).
#### 5.3.6 Additional Requirements
for Spectral Line Observations
In addition to the requirements placed on continuum receivers, there are three
additional requirements for spectral line receiver systems.
If the observed frequency of a line is compared to the known rest frequency,
the relative radial velocity of the source and the receiving system can be
determined. But this velocity contains the motion of the source as well as
that of the receiving system, so the velocity measurements are referred to
some standard of rest. This velocity can be separated into several independent
components: (1) Earth rotation with a maximum velocity $v=0.46$ km s-1 and (2)
The motion of the center of the Earth relative to the barycenter of the Solar
System is said to be reduced to the heliocentric system. Correction algorithms
are available for observations of the earth relative to center of mass of the
solar system. The standard solar motion is the motion relative to the mode of
the velocity of the stars in the solar neighborhood. Data where the standard
solar motion has been taken into account are said to refer to the local
standard of rest (LSR). Most extragalactic spectral line data do not include
the LSR correction but are referred to the heliocentric velocity. For high
redshift sources, special relativity corrections must be included.
For larger bandwidths, there is an instrumental spectrum and a ′′baseline′′
must be subtracted from the (on-off)/off spectrum. Often a linear fit to
spectrum is sufficient, but if curvature is present, polynomials of second or
higher order must be subtracted. At high galactic latitudes, more intense 21
cm line radiation from the galactic plane can give rise to artifacts in
spectra from scattering of radiation within the antenna (see Kalberla et al.
2010). This is apparently less of a problem in surveys of galactic carbon
monoxide (see Dame et al. 1987).
#### 5.3.7 Spectral Line Observing Strategies
Astronomical radiation is often only a small fraction of the total power
received. To avoid stability problems, the signal of interest must be compared
with another that contains approximately the same total power and differs only
that it contains no source. The receiver must be stable so that any gain or
bandpass changes occur over time scales long compared to the time needed for
position change. To detect an astronomical source, three observing modes are
used to produce a suitable comparison.
1) Position Switching and Wobbler Switching. The signal ′′on source′′ is
compared with a measurement obtained at a nearby position in the sky. For
spectral lines, there must be little line radiation at the comparison region.
This is referred to as the ′′total power′′ observing mode. A variant of this
method is wobbler switching. This is very useful for compact sources,
especially in the mm/sub-mm range.
2) On the Fly Mapping. This very important observing method is an extension of
method (1). In this procedure, spectral line data is taken at a rate of
perhaps one spectrum or more per second.
3) Frequency Switching. For many sources, spectral line radiation at $\nu_{0}$
is restricted to a narrow band, that is, present only over a small frequency
interval, $\Delta\nu$, for example $\Delta\nu/\nu_{0}\approx 10^{-5}$. If all
other effects vary very little over $\Delta\nu$, changing the frequency of a
receiver on a short time by up to $10\,\Delta\nu$ produces a comparison signal
with the line well shifted. The line is measured all of the time, so this is
an efficient observing mode.
## 6 Interferometers and Aperture Synthesis
From diffraction theory, the angular resolution is given by (Eq. 33). However,
as shown by Michelson (see Jenkins & White 2001), a much higher resolving
power can be obtained by coherently combining the output of two reflectors of
diameter $d\ll B$ separated by a distance $B$ yeilding
$\theta\approx\lambda$/B. In the radio/mm/sub-mm range, from (Eq. 30), the
outputs can be amplified without seriously degrading the signal-to-noise
ratio. This amplified signal can be divided and used to produce a large number
of cross-correlations.
Aperture synthesis is a further development. This is the procedure to produce
high quality images of sources by combining a number of measurements for
different antenna spacings up to the maximum $B$. The longest spacing gives
the angular resolution of an equivalent large aperture. This has become the
method to obtain high quality, high angular resolution images. The first
practical demonstration of aperture synthesis in radio astronomy was made by
M. Ryle and his associates (see Section 3 in Kellermann & Moran 2001).
Aperture synthesis allows the reproduction of the imaging properties of a
large aperture by sampling the radiation field at individual positions within
the aperture. Using this approach, a remarkable improvement of the radio
astronomical imaging was made possible. More detailed accounts are to be found
in Taylor et al. (1999), Thompson et al. (2001) or Dutrey (2001).
The simplest case is a two element system in which electromagnetic waves are
received by two antennas. These induce the voltage $V_{1}$ at $A_{1}$:
$V_{1}\propto E{\rm\,e}^{\,{\rm\,i\,}\omega t}\,,$ (55)
while at $A_{2}$:
$V_{2}\propto E{\rm\,e}^{\,{\rm\,i\,}\omega\,(t-\tau)}\,,$ (56)
where $E$ is the amplitude of the incoming electromagnetic plane wave, $\tau$
is the geometric delay caused by the relative orientation of the
interferometer baseline $\vec{B}$ and the direction of the wave propagation.
For simplicity, receiver noise and instrumental phase were neglected in (Eq.
55) and (Eq. 56). The outputs will be correlated. Today all radio
interferometers use direct correlation followed by an integrator.
Figure 5: A schematic diagram of a two element correlation interferometer.
The antenna output voltages are $V_{1}$ and $V_{2}$; the instrumental delay is
$\tau_{\rm i}$ and the geometric delay is $\tau_{\rm g}$. $\vec{s}$ is the
direction to the source. Perpendicular to $\vec{s}$ is the projection of the
baseline $\vec{B}$. The signal is digitized after conversion to an
intermediate frequency. Time delays are introduced using digital shift
registers (from Wilson et al. 2008).
The output is proportional to:
$\displaystyle
R(\tau)\propto\frac{E^{2}}{T}\int\limits_{0}^{T}{\rm\,e}^{\,{\rm\,i\,}\omega
t}{\rm\,e}^{\,-{\rm\,i\,}\omega(t-\tau)}\,{\rm\,d}t\,.$
If $T$ is a time much longer than the time of a single full oscillation, i.e.,
$T\gg 2\pi/\omega$ then the average over time $T$ will not differ much from
the average over a single full period, resulting in
$\framebox{$\displaystyle
R(\tau)\propto{\textstyle\frac{1}{2}}E^{2}{\rm\,e}^{\,{\rm\,i\,}\omega\tau}$}\quad.$
(57)
The output of the correlator $+$ integrator varies periodically with $\tau$,
the delay. Since $\vec{s}$ is slowly changing due to the rotation of the
earth, $\tau$ will vary, producing interference fringes as a function of time.
The basic components of a two element system are shown in Fig. 5. If the radio
brightness distribution is given by $I_{\nu}(\vec{s})$, the power received per
bandwidth ${\rm\,d}\nu$ from the source element ${\rm\,d}\Omega$ is
$A(\vec{s})I_{\nu}(\vec{s}){\rm\,d}\Omega{\rm\,d}\nu$, where $A(\vec{s})$ is
the effective collecting area in the direction $\vec{s}$; the same
$A(\vec{s})$ is assumed for each of the antennas. The amplifiers are assumed
to have constant gain and phase factors (neglected here for simplicity).
The output of the correlator for radiation from the direction $\vec{s}$ (Fig.
5) is
$r_{12}=A(\vec{s})\,I_{\nu}(\vec{s})\,{\rm\,e}^{{\rm\,i\,}\omega\tau}\,{\rm\,d}\Omega{\rm\,d}\nu$
(58)
where $\tau$ is the difference between the geometrical and instrumental delays
$\tau_{\rm g}$ and $\tau_{\rm i}$. If $\vec{B}$ is the baseline vector between
the two antennas
$\tau=\tau_{\rm g}-\tau_{\rm i}=\frac{1}{c}\,\vec{B}\cdot\vec{s}-\tau_{\rm i}$
(59)
the total response is obtained by integrating over the source $S$
$\framebox{$\displaystyle
R(\vec{B})=\parbox{22.76219pt}{$\vspace*{-2mm}{\displaystyle\int\\!\\!\int\atop{\\!\\!\\!\\!\\!\scriptstyle\Omega}}$
}A(\vec{s})I_{\nu}(\vec{s}){\rm\,e}^{2\pi{\rm\,i\,}\nu\left(\frac{1}{c}\,\vec{B}\cdot\vec{s}-\tau_{\rm
i}\right)}{\rm\,d}\Omega{\rm\,d}\nu$}\quad$ (60)
The function $R(\vec{B})$, the Visibility Function is closely related to the
mutual coherence function (see Born & Wolf 1965, Thompson et al. 2001, Wilson
et al. 2008) of the source. For parabolic antennas, it is usually assumed that
$A(\vec{s})=0$ outside the main beam area so that (Eq. 60) is integrated only
over this region. A one dimensional version of (Eq. 60), for a baseline $B$,
frequency $\nu=\nu_{0}$ and instrumental time delay $\tau_{i}=0$, is
$R(B)=\int
A(\theta)\,I_{\nu}(\theta){\rm\,e}^{2\pi{\rm\,i\,}\nu_{0}\left(\frac{1}{c}\,B\,\theta\right)}{\rm\,d}\theta$
(61)
With $\theta=x$ and $B_{x}/\lambda=u$, this is
$R(B)=\int
A(\theta)\,I_{\nu}(\theta){\rm\,e}^{2\pi{\rm\,i\,}u\,x}{\rm\,d}\theta$ (62)
This form of (Eq. 60) illustrates more clearly the Fourier Transform relation
of $u$ and $x$. This simplified version will be used to provide illustrations
of interferometer responses (see Section 6.2). In two dimensions, (Eq. 60)
takes on a similar form with the additional variables $y$ and
$B_{y}/\lambda=v$. The image can be obtained from the inverse Fourier
transform of Visibilities; see (Eq. 65).
### 6.1 Calibration
Amplitude and phase must be calibrated for all interferometer measurements. In
addition, the instrumental passband must be calibrated for spectral line
measurements. The amplitude scale is calibrated by a determination of the
system noise at each antenna using methods presented for single dish
measurements (see Section 5.3.2 and following). In the centimeter range, the
atmosphere plays a small role while in the mm and sub-mm wavelength ranges,
the atmospheric effects must be accounted for. For phase measurements, a
suitable point-like source with an accurately known position is required to
determine $2\pi\nu\tau_{i}$ in (Eq. 60). For interferometers, the best
calibration sources are usually unresolved or point-like sources. Most often
these are extragalactic time variable sources. To calibrate the response in
units of flux density or brightness temperatures, these amplitude measurements
must be referenced to primary calibrators (see a list of non-variable sources
of known flux densities in Ott et al. 1994 or Sandell 1994).
The calibration of the instrumental passband is carried out by a longer
integration on an intense source to determine the channel-to-channel gains and
offsets. The amplitude, phase and passband calibrations are carried out before
the source measurements. The passband calibration is usually carried out every
few hours or once per observing session. The amplitude and phase calibrations
are made more often; the time between such calibrations depends on the
stability of the electronics and weather. If weather conditions require
frequent measurements of calibrators (perhaps less than once per minute for
′′fast switching′′), integration time is reduced. In case of even more rapid
weather changes, the ALMA project will make use of water vapor radiometers
mounted on each antenna (see Section 5.3.1). These will be used to determine
the total amount of H2O vapor above each antenna, and use this to make
corrections to phase.
### 6.2 Responses of Interferometers
#### 6.2.1 Time Delays and Bandwidth
The instrumental response is reduced if the bandwidth at the correlator is
large compared to the delay caused by the separation of the antennas. For
large bandwidths, the loss of correlation can be minimized by adjusting the
phase delay so that the difference of arrival time between antennas is
negligible. In practice, this is done by inserting a delay between the
antennas so that $\frac{1}{c}\,\vec{B}\cdot\vec{s}$ equals $\tau_{\rm i}$.
This is equivalent to centering the response on the central, or white light
fringe. Similarly, the reduction of the response caused by finite bandwidth
can be estimated by an integration of (Eq. 60) over frequency, taking
$A(\vec{s})$ and $I_{\nu}(\vec{s})$ as constants. The result is a factor,
$\sin(\Delta\nu\tau)/\Delta\nu\tau\,$ in (Eq. 60). This will reduce the
interferometer response if $\Delta\nu\tau\sim 1$ . For typical bandwidths of
100 MHz, the offset from the zero delay must be $\ll 10^{-8}$ s. This
adjustment of delays is referred to as fringe stopping. The exponent in (Eq.
60) has both sine and cosine components, but digital cross-correlators record
both components, so that the entire response can be recovered.
#### 6.2.2 Beam Narrowing
The white light fringe the delay compensation must be set with a high accuracy
to prevent a reduction in the interferometer response. For a finite primary
antenna beamwidth, $\theta_{b}$, this cannot be the case over the entire beam.
For a bandwidth $\Delta\nu$ there will be a phase difference. Converting the
wavelengths to frequencies and using $\sin{\theta}\cong\theta$ the result is
$\Delta\phi=2\pi\,\frac{\theta_{\rm
offset}}{\theta_{b}}\,\frac{\Delta\nu}{\nu}$ (63)
This effect can be important for continuum measurements made with large
bandwidths, but can be reduced if the cross correlation is carried out using a
series of narrow contiguous IF sections. For each of these IF sections, an
extra delay is introduced to center the response at the value which is
appropriate for that wavelength before correlation.
#### 6.2.3 Source Size
From an idealized source, of shape $I(\nu_{0})=I_{0}$ for
$\theta\,<\,\theta_{0}$ and $I(\nu_{0})=0$ for $\theta\,>\,\theta_{0}$; we
take the primary beamsize of each antenna to be much larger, and define the
fringe width for a baseline $B$ $\theta_{b}$ to be $\frac{\lambda}{B}$, The
result is
$R(B)=A\,I_{0}\cdot\theta_{0}\,{\rm\,e}^{{\rm\,i\,}\pi\frac{\theta_{0}}{\theta_{b}}}\,\left[\frac{\sin{(\pi\theta_{0}/\theta_{b})}}{{(\pi\theta_{0}/\theta_{b})}}\right]$
(64)
The first terms are normalization and phase factors. The important term is in
brackets. If $\theta_{0}>>\theta_{b}$, the interferometer response is reduced.
This is sometimes referred to as the problem of ′′missing short spacings′′’.
To correct for the loss of source flux density, the interferometer data must
be supplemented by single dish measurements. The diameter of the single dish
antenna should be larger than the shortest interferometer spacing. This single
dish image must extend to the FWHP of the smallest of the interferometer
antennas. When Fourier transformed and appropriately combined with the
interferometer response, the resulting data set has no missing flux density.
### 6.3 Aperture Synthesis
To produce an image, the integral equation (Eq. 60) must be inverted. A number
of approximations may have to be applied to produce high quality images. In
addition, the data are affected by noise. The most important steps of this
development will be presented.
For imaging over a limited region of the sky rectangular coordinates are
adequate, so relation (Eq. 60) can be rewritten with coordinates $(x,y)$ in
the image plane and coordinates $(u,v)$ in the Fourier plane. The coordinate
$w$, corresponding to the difference in height, is set to zero. Then the
relevant relation is:
$\framebox{$\displaystyle
I^{\prime}(x,y)=A(x,y)\,I(x,y)=\int\limits_{-\infty}^{\infty}V(u,v,0){\rm\,e}^{-2\pi{\rm\,i\,}(ux+vy)}{\rm\,d}u{\rm\,d}v$}\quad$
(65)
where $I^{\prime}(x,y)$ is the intensity $I(x,y)$ modified by the primary beam
shape $A(x,y)$. It is easy to correct $I^{\prime}(x,y)$ by dividing by
$A(x,y)$. Usually data present beyond the half power point is excluded.
The most important definitions are:
(1) Dynamic Range: The ratio of the maximum to the minimum intensity in an
image. In images made with an interferometer array, it is assumed that
corrections for primary beam taper have been applied. If the minimum intensity
is determined by the random noise in an image, the dynamic range is defined by
the signal-to-noise ratio of the brightest feature in the image. The dynamic
range is an indication of the ability to recognize low intensity features in
the presence of intense features. If the minimum noise is determined by
artifacts, i.e., noise in excess of the theoretical value, ′′image improvement
techniques′′ should be applied.
(2) Image Fidelity: This is defined by the agreement between the measured
results and the actual (′′true′′) source structure. A quantitative assessment
of fidelity is:
$\displaystyle F=|(S-R)|/S$
where $F$ is the fidelity, $R$ is the resulting image obtained from the
measurement, and $S$ is the actual source structure. The highest fidelity is
$F=0$. Usually errors can only be estimated using a priori knowledge of the
correct source structure. In many cases, $S$ is a source model, while $R$ is
obtained by processing $S$ with a model of the instrumental response. This
relation can only be applied when the value of $R$ is more than 5 times the
RMS noise.
Figure 6: An artists sketch of ALMA. To date, this is the most ambitious
construction project in ground based astronomy. ALMA is now being built in
north Chile on a 5 km high site. It will consist of fifty-four 12-m and twelve
7-m antennas, operating in 10 bands between wavelength 1 cm and 0.3 mm. In
Early Science, four receiver bands at 3, 1.3, 0.8 and 0.6 mm will be
available. The high ALMA sensitivity is due to the extremely low noise
receivers, the highly accurate antennas, and the high altitude site. At the
largest antenna spacing, and shortest wavelength, the angular resolution will
be $\sim$5 milliarcseconds (courtesy ESO/NRAO/NAOJ).
#### 6.3.1 Interferometric Observations
Usually measurements are carried out in 1 of 4 ways.
1\. Measurements of a single target source. This is similar to the case of
single telescope position switching. Two significant differences with single
dish measurements are that the interferometer measurement may have to extend
over a wide range of hour angles to provide a better coverage of the $(u,v)$
or Fourier plane, and that instrumental phase must be determined also. After
the measurement of a calibration source or reference source, which has a known
position and size, the effect of instrumental phases in the instrument and
atmosphere is removed and a calibration of the amplitudes of the source is
made. Target sources and calibrators are usually observed alternately; the
calibrator should be close to the target source. The time variations caused by
instrumental and weather effects must be slower than the time between
measurements of source and calibrator. If, as is the case for mm/sub-mm
wavelength measurements, weather is an important influence, target and
calibration source must be measured often. For ALMA (see Fig. 6), observing
will follow a two part scheme. For fast switching there will be integrations
of perhaps 10 seconds on a nearby calibrator, then a few minutes on-source.
This method will reduce the amount of phase fluctuations, at the cost of on-
source observing time. For more rapid changes in the earth’s atmosphere,
phases will be corrected using measurements of atmospheric water vapor from
measurements of the 183 GHz line.
2\. Snapshot Mode. A series of short observations (at different hour angles)
of one source after another, and then the measurements are repeated. For
sensitivity reasons, snapshots are usually made in the radio continuum or more
intense spectral lines. As in observing method (1), measurements of source and
calibrator are interspersed to remove the effects of instrumental phase drifts
and to calibrate the amplitudes of the sources in question. The images will
affected by the shape of the synthesized beam since there is sparse coverage
in the $(u,v)$ plane. If the size of the source to be imaged is comparable to
the primary beam of the individual antennas there should be a correction for
the power pattern..
3\. Multi-Configuration Imaging Here the goal is the image of a source either
with high dynamic range or high sensitivity. Measurements with a number of
different interferometer configurations better fill the $uv$ plane. These
measurements are taken at different epochs and after calibration, the
visibilities are entered into a common data set.
4\. Mosaicing An extension of procedure (1) can be used for sources with an
extent much larger than the primary antenna beam. These images require
measurements at adjacent pointings. This is spoken of as mosaicing. In a
mosaic, the antennas are pointed at narby positions. These positions should
overlap at the half power power point. The images can be formed separately and
then combined to produce an image of the larger region. Another method is to
combine the data in the $(u,v)$ plane and then form the image.
### 6.4 Interferometer Sensitivity
The random noise limit to an interferometer system can be calculated following
the method used for a single telescope (Eq. 27). The use of (Eq. 43) provides
a conversion from $\Delta T_{\rm RMS}$ to $\Delta S_{\nu}$. collecting area of
a single antenna. For an array of $n$ identical antennas, there are
$N=n(n-1)/2$ simultaneous pairwise correlations, so the RMS variation in flux
density is:
$\Delta S_{\nu}=\frac{2\,M\,k\,T_{\rm sys}^{*}}{A_{\rm
e}\sqrt{2\,N\,t\,\Delta\nu}}\,.$ (66)
with M$\cong 1$, $A_{\rm e}$ the effective area of each antenna and $T_{\rm
sys}^{*}$ given by (Eq. 54). This relation can be recast in the form of
brightness temperature fluctuations using the Rayleigh-Jeans relation; then
the RMS noise in brightness temperature units is:
$\Delta T_{\rm B}=\frac{2\,M\,k\,\lambda^{2}\,T_{\rm sys}^{*}}{A_{\rm
e}\Omega_{\rm b}\sqrt{2\,N\,t\,\Delta\nu}}\,.$ (67)
For a Gaussian beam, $\Omega_{\rm mb}=1.133\,\theta^{2}$, so the RMS
temperature fluctuations can be related to observed properties of a synthesis
image.
Aperture synthesis is based on discrete samples of the visibility function
$V(u,v)$, with the goal of the densest possible coverage of the $(u,v)$ or
Fourier plane. It has been observed that the RMS noise in a synthesis image
obtained by Fourier transforming the $(u,v)$ data is often higher than given
by (Eq. 66) or (Eq. 67). Possible causes are: (1) phase fluctuations caused by
atmospheric or instrumental instabilities, (2) incomplete sampling of the
$(u,v)$ plane, which gives rise to artifacts such as stripe-like features in
the images, or (3) grating rings around more intense sources; these are
analogous to high sidelobes in single dish diffraction patterns.
### 6.5 Corrections of Visibility Functions
#### 6.5.1 Amplitude and Phase Closure
The relation between the measured $\widetilde{V_{ik}}$ visibility and actual
visibility $V_{ik}$ is considered linear:
$\widetilde{V_{ik}}(t)=g_{i}(t)\,g^{*}_{k}(t)\,V_{ik}+\varepsilon_{ik}(t)\;.$
(68)
Values for the complex antenna gain factors $g_{k}$ and the noise term
$\varepsilon_{ik}(t)$ are determined by measuring calibration sources as
frequently as possible. Actual values for $g_{k}$ are then computed by linear
interpolation. The (complex) gain of the array is obtained by the
multiplication of the gains of the individual antennas. If the array consists
of $n$ such antennas, $n(n-1)/2$ visibilities can be measured simultaneously,
but only $(n-1)$ independent gains $g_{k}$ are needed since one antenna in the
array can be taken as a reference. So in an array with many antennas, the
number of antenna pairs greatly exceeds the number of antennas. For phase, one
must determine $n$ phases. Often these conditions can be introduced into the
solution in the form of closure errors. Defining the phases $\varphi,\theta$
and $\psi$ by
$\begin{array}[]{rcl}\widetilde{V_{ik}}&=&|\widetilde{V_{ik}}|\,{\rm\,e}^{{\rm\,i\,}\varphi_{ik}}\,,\\\
G_{ik}&=&|g_{i}|\,|g_{k}|\,{\rm\,e}^{{\rm\,i\,}\theta_{i}}{\rm\,e}^{-{\rm\,i\,}\theta_{k}}\,,\\\
V_{ik}&=&|V_{ik}|\,{\rm\,e}^{{\rm\,i\,}\psi_{ik}}\,.\\\ \end{array}$ (69)
From (Eq. 68) the visibility phase $\psi_{ik}$ on the baseline $ik$ will be
related to the observed phase $\varphi_{ik}$ by
$\varphi_{ik}=\psi_{ik}+\theta_{i}-\theta_{k}+\varepsilon_{ik}\,,$ (70)
where $\varepsilon_{ik}$ is the phase noise. Then the closure phase
$\Psi_{ikl}$ around a closed triangle of baseline $ik,kl,li$,
$\Psi_{ikl}=\varphi_{ik}+\varphi_{kl}+\varphi_{li}=\psi_{ik}+\psi_{kl}+\psi_{li}+\varepsilon_{ik}+\varepsilon_{kl}+\varepsilon_{li}\,,$
(71)
will be independent of the phase shifts $\theta$ introduced by the individual
antennas and the time variations. With this procedure, phase errors can be
minimized.
If four or more antennas are used simultaneously, then the closure amplitudes
can be formed. These are independent of the antenna gain factors:
$A_{klmn}=\frac{|V_{kl}||V_{mn}|}{|V_{km}||V_{ln}|}\;.$ (72)
Both phase and closure amplitudes can be used to improve the quality of the
complex visibility function.
At each antenna there is an unknown complex gain factor $g$ with amplitude and
phase, the total number of unknowns can be reduced significantly by measuring
closure phases and amplitudes. If four antennas are available, 50 % of the
phase information and 33 % of the amplitude information can thus be recovered;
in a 10 antenna configuration, these ratios are 80 % and 78 % respectively.
#### 6.5.2 Calibrations, Gridding, FFTs, Weighting and Self Calibration
For two antenna interferometers, phase calibration can only be made pair-wise.
This is referred to as ′′baseline based′′ solutions for the calibration. For a
multi-antenna system, ′′antenna based′′ solutions are preferred. These are
determined by applying phase and amplitude closure for subsets of antennas and
then solving for the best fit for each.
Normally the Cooley-Tukey fast Fourier transform algorithm is used to invert
(Eq. 65) To apply the simplest version of the FFT, the visibilities must be
placed on a regular grid with sizes that are powers of two of the sampling
interval. Since the data seldom lie on such regular grids, an interpolation
scheme must be used. From the gridded $(u,v)$ data, an image with a resolution
corresponding to $\lambda/D$, where $D$ is the array size, is obtained.
However, this may still contain artifacts caused by the observing procedure,
especially the limited coverage of the ($u,v$) plane. Therefore the dynamic
range of such so-called dirty maps is rather small. This can be improved by
further analysis.
If the calibrated visibility function $V(u,v)$ is known for the full $(u,v)$
plane both in amplitude and in phase, this can be used to determine the
modified (i.e., structure on angular scales finer than $\lambda/D$ are lost)
intensity distribution $I^{\prime}(x,y)$ by performing the Fourier
transformation (Eq. 65). However, in a realistic situation $V(u,v)$ is only
sampled at discrete points and in some regions of the $(u,v)$ plane, $V(u,v)$
is not measured at all. The visibilities can be weighted by a grading
function, $g$. For a discrete number of visibilities, a version of (Eq. 65)
involving a summation, not an integral, is used to obtain an image with the
use of a discrete Fourier transform (DFT):
$I_{\rm
D}(x,y)=\sum_{k}g(u_{k},v_{k})V(u_{k},v_{k}){\rm\,e}^{-2\pi{\rm\,i\,}(u_{k}x+v_{k}y)}\,,$
(73)
where $g(u,v)$ is a weighting function referred to as the grading or
apodisation. $g(u,v)$ can be used to change the effective beam shape and side
lobe level. There are two widely used weighting functions: uniform and
natural. Uniform weighting uses $g(u_{k},v_{k})=1$, while natural weighting
uses $g(u_{k},g_{k})=1/N_{\rm s}(k)$, where $N_{\rm s}(k)$ is the number of
data points within a symmetric region of the $(u,v)$ plane. Data which are
naturally weighted result in lower angular resolution but give a better
signal-to-noise ratio than uniform weighting. But these are only extreme
cases. Intermediate weighting schemes are referred to as robust weighting.
Often the reconstructed image $I_{\rm D}$ may not be a particularly good
representation of $I^{\prime}$, but these are related by:
$I_{\rm D}(x,y)=P_{\rm D}(x,y)\otimes I^{\prime}(x,y)\,,$ (74)
where $I^{\prime}(x,y)$ is the best representation of the source intensity
modified by the primary beam shape; it contains only those spatial frequencies
$(u_{k},v_{k})$ where the visibility function has been measured. (see (Eq.
65)). The expression for $P_{\rm D}$ is:
$P_{\rm D}=\sum_{k}g(u_{k},v_{k}){\rm\,e}^{-2\pi{\rm\,i\,}(u_{k}x+v_{k}y)}$
(75)
this is the response to a point source, or the point spread function PSF for
the dirty beam. Thus $P_{\rm D}$ is a transfer function that distorts the
image; $P_{\rm D}$ is produced assuming an amplitude of unity and phase zero
at each point sampled. This is the response of the interferometer system to a
point source. The sum in (Eq. 75) extends over the same positions
$(u_{k},v_{k})$ as in (Eq. 73); the sidelobe structure of the beam depends on
the distribution of these points.
Amplitude and phase errors scatter power across the image, giving the
appearance of enhanced noise. This problem can be alleviated to an impressive
extent by the method of self-calibration. This process can be applied if there
is a sufficiently intense compact feature in the field contained within the
primary beam of the interferometer system. If self-calibration can be applied,
the positional information is usually lost. Self-calibration can be restricted
to an improvement of phase alone or to both phase and amplitude. Normally,
self-calibration is carried in the $(u,v)$ plane. If this method is used on
objects with low signal-to-noise ratios, this may lead to a concentration of
random noise into one part of the interferometer image (see Cornwell &
Fomalont 1989). For measurements of weak spectral lines, self-calibration is
carried out using a continuum source in the field. The corrections are then
applied to the spectral line data. In the case of intense lines, one of the
frequency channels containing the emission is used.
#### 6.5.3 More Elaborate Improvements of Visibility Functions: The CLEANing
Procedure
CLEANing is the most commonly used technique to improve single radio
interferometer images (Högbom 1974). In addition to its inherent low dynamic
range, the dirty map often contains features such as negative intensity
artifacts that cannot be real. Another unsatisfactory aspect is that the
solution is quite often rather unstable, in that it can change drastically
when more visibility data are added.
The CLEAN method approximates the intensity distribution that represents the
best image of the source (subject to angular resolution, noise, etc.),
$I(x,y)$, by the superposition of a finite number of point sources with
positive intensity $A_{i}$ placed at positions $(x_{i},y_{i})$. The goal of
CLEAN to determine the $A_{i}(x_{i},y_{i})$, such that
$I^{\prime\prime}(x,y)=\sum_{i}A_{i}\,P_{\rm
D}(x-x_{i},y-y_{i})+I_{\varepsilon}(x,y)\,$ (76)
where $I^{\prime\prime}$ is the dirty map obtained from the inversion of the
visibility function and $P_{\rm D}$ is the dirty beam (Eq. 75).
$I_{\varepsilon}(x,y)$ is the residual brightness distribution after
decomposition. Approximation (Eq. 76) is considered successful if
$I_{\varepsilon}$ is of the order of the noise in the measured intensities.
This decomposition must be carried out iteratively.
The CLEAN algorithm is most commonly applied in the image plane. This is an
iterative method which functions in the following fashion: (1) find the peak
intensity of the dirty image, then subtract a fraction $\gamma$ (the so-called
′′loop gain′′) having the shape of the dirty beam from the image, and (2)
repeat this $n$ times.
This loop gain has values $0<\gamma<1$ while $n$ is often taken to be 104. The
goal is that the intensities of the residuals are comparable to the noise
limit. Finally, the resulting model is convolved with a clean beam of Gaussian
shape with a FWHP given by the angular resolution expected from $\lambda/D$
where $D$ is the maximum baseline length. Whether this algorithm produces a
realistic image depends on the quality of the data and other variables.
#### 6.5.4 More Elaborate Improvements of Visibility Functions: The Maximum
Entropy Procedure
The Maximum Entropy Deconvolution Method (MEM) is commonly used to produce a
single optimal image from a set of separate but contiguous images (Gull &
Daniell 1978). The problem of how to select the ′′best′′ image from many
possible images which all agree with the measured visibilities is solved by
MEM. Using MEM, those values of the interpolated visibilities are selected, so
that the resulting image is consistent with all previous relevant data. In
addition, the MEM image has maximum smoothness. This is obtained by maximizing
the entropy of the image. One definition of entropy is given by
${\cal
H}=-\sum_{i}I_{i}\left[\ln\bigg{(}\frac{I_{i}}{M_{i}}\bigg{)}-1\right]\,,$
(77)
where $I_{i}$ is the deconvolved intensity and $M_{i}$ is a reference image
incorporating all ′′a priori′′ knowledge. In the simplest case $M_{i}$ is the
empty field $M_{i}={\rm const}>0$, or perhaps a lower angular resolution
image.
Additional constraints might require that all measured visibilities should be
reproduced exactly, but in the presence of noise such constraints are often
incompatible with $I_{i}>0$ everywhere. Therefore the MEM image is usually
constrained to fit the data such that
$\chi^{2}=\sum\frac{|V_{i}-V_{i}^{\prime}|^{2}}{\sigma_{i}^{2}}$ (78)
has the expected value, where $V_{i}$ is the measured visibility,
$V_{i}^{\prime}$ is a visibility corresponding to the MEM image and
$\sigma_{i}$ is the error of the measurement.
Acknowledgement: K. Weiler made a thorough review of the text and H. Bond
suggested a number of improvements.
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|
arxiv-papers
| 2011-11-04T16:56:28 |
2024-09-04T02:49:24.006063
|
{
"license": "Public Domain",
"authors": "T. L. Wilson",
"submitter": "Thomas Wilson",
"url": "https://arxiv.org/abs/1111.1183"
}
|
1111.1379
|
# A criterion of normality based on a single holomorphic function II
Xiaojun Liu1 and Shahar Nevo2 Xiaojun Liu, Department of Mathematics
University of Shanghai for Science and Technology, Shanghai 200093, P.R. China
Xiaojunliu2007@hotmail.com Shahar Nevo, Department of Mathematics
Bar-Ilan University, 52900 Ramat-Gan, Israel nevosh@macs.biu.ac.il
###### Abstract.
In this paper, we continue to discuss normality based on a singleholomorphic
function. We obtain the following result. Let $\mathcal{F}$ be a family of
functions holomorphic on a domain $D\subset\mathbb{C}$. Let $k\geq 2$ be an
integer and let $h(\not\equiv 0)$ be a holomorphic function on $D$, such that
$h(z)$ has no common zeros with any $f\in\mathcal{F}$. Assume also that the
following two conditions hold for every $f\in\mathcal{F}$:(a)
$f(z)=0\Longrightarrow f^{\prime}(z)=h(z)$ and (b)
$f^{\prime}(z)=h(z)\Longrightarrow|f^{(k)}(z)|\leq c$, where $c$ is a
constant. Then $\mathcal{F}$ is normal on $D$.
A geometrical approach is used to arrive at the result which significantly
improves the previous results of the authors, A criterion of normality based
on a single holomorphic function, Acta Math. Sinica, English Series (1) 27
(2011), 141–154 and of Chang, Fang, and Zalcman, Normal families of
holomorphic functions, Illinois Math. J. (1) 48 (2004), 319–337. We also deal
with two other similar criterions of normality. Our results are shown to be
sharp.
###### Key words and phrases:
Normal family, holomorphic functions, zero points
###### 2010 Mathematics Subject Classification:
30D35
1 Research supported by the NNSF of China Approved No.11071074 and also
supported by the Outstanding Youth Foundation of Shanghai No. slg10015.
2 Research supported by the Israel Science Foundation Grant No. 395/07
## 1\. Introduction
In [11], X.C. Pang and L. Zalcman proved the following theorem.
###### Theorem PZ.
Let $\mathcal{F}$ be a family of meromorphic functions on a domain
$D\subset\mathbb{C}$, all of whose zeros have multiplicity at least $k$, where
$k\geq 1$ is an integer. Suppose there exist constants $b\neq 0$ and $h>0$
such that, for every $f\in\mathcal{F}$, $f(z)=0\Longleftrightarrow
f^{(k)}(z)=b$ and $f(z)=0\Longrightarrow 0<|f^{(k+1)}(z)|\leq h$. Then
$\mathcal{F}$ is a normal family on $D$.
Then, in [1], J.M Chang, M.L. Fang, and L. Zalcman proved the following
result.
###### Theorem CFZ1.
[1, Theorem 4] Let $\mathcal{F}$ be a family of functions holomorphic on a
domain $D\subset\mathbb{C}$. Let $k\geq 2$ be an integer, and let $h(z)\neq 0$
be a function analytic in $D$. Assume also that the following two conditions
hold for every $f\in\mathcal{F}$:
1. (a)
$f(z)=0\Longrightarrow f^{\prime}(z)=h(z)$; and
2. (b)
$f^{\prime}(z)=h(z)\Longrightarrow|f^{(k)}(z)|\leq c$, where $c$ is a
constant.
Then $\mathcal{F}$ is normal on $D$.
And in [4], we replaced the condition $h(z)\neq 0$ with $h(z)\not\equiv 0$ and
obtained the following result.
###### Theorem LN.
Let $\mathcal{F}$ be a family of functions holomorphic on a domain
$D\subset\mathbb{C}$. Let $k\geq 2$ be an integer, and let $h(z)(\not\equiv
0)$ be a holomorphic function on $D$, all of whose zeros have multiplicity at
most $k-1$, that has no common zeros with any $f\in\mathcal{F}$. Assume also
that the following two conditions hold for every $f\in\mathcal{F}$:
1. (a)
$f(z)=0\Longrightarrow f^{\prime}(z)=h(z)$ and
2. (b)
$f^{\prime}(z)=h(z)\Longrightarrow|f^{(k)}(z)|\leq c$, where $c$ is a
constant.
Then $\mathcal{F}$ is normal on $D$.
We now pose the following question: can the restriction for the zeros of
$h(z)$ with multiplicity at most $k-1$ be dropped? In this paper, we continue
to study the above problem and obtain an affirmative answer.
###### Theorem 1.
Let $\mathcal{F}$ be a family of functions holomorphic on a domain
$D\subset\mathbb{C}$. Let $k\geq 2$ be an integer, and let $h(z)(\not\equiv
0)$ be a holomorphic function on $D$ that has no common zeros with any
$f\in\mathcal{F}$. Assume also that the following two conditions hold for
every $f\in\mathcal{F}$:
1. (a)
$f(z)=0\Longrightarrow f^{\prime}(z)=h(z)$ and
2. (b)
$f^{\prime}(z)=h(z)\Longrightarrow|f^{(k)}(z)|\leq c$, where $c$ is a
constant.
Then $\mathcal{F}$ is normal on $D$.
Also in [1], the case for the $k-$th derivative was considered and the
following result was proved .
###### Theorem CFZ2.
[1, Theorem 1] Let $\mathcal{F}$ be a family of functions holomorphic on a
domain $D\subset\mathbb{C}$, all of whose zeros have multiplicity at least
$k$, where $k\neq 2$ is a positive integer; and let $h(z)\neq 0$ be a function
analytic in $D$. Assume also that the following two conditions hold for every
$f\in\mathcal{F}$:
1. (a)
$f(z)=0\Longrightarrow f^{(k)}(z)=h(z)$; and
2. (b)
$f^{(k)}(z)=h(z)\Longrightarrow|f^{(k+1)}(z)|\leq c$, where $c$ is a constant.
Then $\mathcal{F}$ is normal on $D$.
For the case $k=2$, the following result was obtained.
###### Theorem CFZ3.
[1, Theorem 3] Let $\mathcal{F}$ be a family of functions holomorphic on a
domain $D\subset\mathbb{C}$, all of whose zeros are multiple, where $s\geq 4$
is an even integer; and let $h(z)\neq 0$ be a function analytic in $D$. Assume
also that the following two conditions hold for every $f\in\mathcal{F}$:
1. (a)
$f(z)=0\Longrightarrow f^{\prime\prime}(z)=h(z)$; and
2. (b)
$f^{\prime\prime}(z)=h(z)\Longrightarrow|f^{\prime\prime\prime}(z)|+|f^{(s)}(z)|\leq
c$, where $c$ is a constant.
Then $\mathcal{F}$ is normal on $D$.
In view of the improvement of Theorems CFZ1 and LN via Theorem 1, the question
that naturally arises concerning Theorem CFZ2 and CFZ3, is whether the
condition $h(z)\neq 0$, $z\in D$, can be relaxed to “$h\not\equiv 0$ ”. It
turns out that the answer is negative in both cases. It is negative even if
$h$ has no common zero with any $f\in\mathcal{F}$ (like in Theorem 1). To
construct the first example, concerning Theorem CFZ2, we first need to mention
the following famous result of F. Lucas.
###### Theorem Lu.
[5], [6, p. 22] Let $P(z)$ be a nonconstant polynomial. Then all the zeros of
$P^{\prime}(z)$ lie in the convex hull $H$ of the zeros of $P(z)$. Moreover,
there are no zeros of $P^{\prime}(z)$ on the boundary of $H$, unless this zero
is a multiple zero of $P(z)$ or the zeros of $P(z)$ are colinear.
###### Example 1.
Let $r\geq 1$ and $k\geq 3$ be integers, $D=\Delta$ be the unit disc and
$h(z)=z^{r}$. Define
$f_{n}(z)=a_{n}\left(z^{\ell}-\frac{\displaystyle 1}{\displaystyle
n^{\ell}}\right)^{k},$
where $\ell=k+r$ and $a_{n}=\frac{\displaystyle n^{(k-1)\ell}}{\displaystyle
k!\ell^{k}}$.
We have
$f_{n}(z)=a_{n}\prod\limits_{j=1}^{\ell}\left(z-\alpha^{(n)}_{j}\right)^{k},$
where $\alpha^{(n)}_{j}=\frac{\displaystyle\exp\left(i\frac{2\pi
j}{\ell}\right)}{\displaystyle n}$, for $1\leq j\leq\ell$.
By calculation,
$\displaystyle f_{n}^{(k)}\left(\alpha^{(n)}_{j}\right)$
$\displaystyle=k!a_{n}\prod\limits_{t=1,t\neq
j}^{\ell}\left(\alpha^{(n)}_{j}-\alpha^{(n)}_{t}\right)^{k}=k!a_{n}\left[\left(z^{\ell}-\frac{\displaystyle
1}{\displaystyle
n^{\ell}}\right)^{\prime}\Bigg{|}_{z=\alpha^{(n)}_{j}}\right]^{k}$
$\displaystyle=k!a_{n}\ell^{k}\left(\alpha^{(n)}_{j}\right)^{k(\ell-1)}.$
Thus,
(1)
$\arg\left[f_{n}^{(k)}\left(\alpha^{(n)}_{j}\right)\right]=(\ell-1)k\cdot\frac{\displaystyle
2\pi j}{\displaystyle\ell}=-\frac{\displaystyle 2\pi
kj}{\displaystyle\ell}=\frac{\displaystyle 2\pi
ri}{\displaystyle\ell}=\arg\left[z^{r}\Big{|}_{z=\alpha^{(n)}_{j}}\right].$
Here the equalities are modulo $2\pi$, and we used in the last equality that
$r+k=\ell$.
We have
(2) $\left|f_{n}^{(k)}\left(\alpha^{(n)}_{j}\right)\right|=\frac{\displaystyle
k!\ell^{k}n^{\ell(k-1)}}{\displaystyle k!\ell^{k}}\left(\frac{\displaystyle
1}{\displaystyle n}\right)^{k(\ell-1)}=\left(\frac{\displaystyle
1}{\displaystyle
n}\right)^{r}=\left|z^{r}\right|\Bigg{|}_{z=\alpha^{(n)}_{j}}.$
From (1) and (2) we have that $f_{n}(z)=0\Longrightarrow f^{(k)}_{n}(z)=h(z)$,
i.e., assumption (a) of Theorem CFZ2 holds.
In order to confirm (b) of Theorem CFZ2, set
$\widetilde{f}_{n}(z)=f_{n}(z)-\frac{\displaystyle
z^{\ell}}{\displaystyle\ell(\ell-1)\cdots(r+1)}.$
We have $f^{(k)}_{n}(z)=h(z)\Longleftrightarrow\widetilde{f}^{(k)}_{n}(z)=0$.
Now
(3) $\widetilde{f}_{n}(z)=0\Longleftrightarrow\frac{\displaystyle
n^{k(\ell-1)-r}}{\displaystyle k!\ell^{k}}\left(z^{\ell}-\frac{\displaystyle
1}{\displaystyle n^{\ell}}\right)^{k}=\frac{\displaystyle
z^{\ell}}{\displaystyle\ell(\ell-1)\cdots(r+1)}.$
Suppose by negation that there exist a sequence $\\{z_{n}\\}^{\infty}_{n=1}$
$(z_{n}\to 0)$ and a sequence of natural numbers $\\{k_{n}\\}^{\infty}_{n=1}$
$(k_{n}\underset{n\to\infty}{\longrightarrow}\infty)$, such that
$\widetilde{f}_{k_{n}}(z_{n})=0$. Then since
$\frac{\displaystyle(k_{n}z_{n})^{\ell}-1}{\displaystyle(k_{n}z_{n})^{\ell}}\underset{n\to\infty}{\longrightarrow}1$,
from (3) we get
(4) $\frac{\displaystyle k_{n}^{(k-1)\ell}(k_{n}z_{n})^{k\ell}}{\displaystyle
k_{n}^{k\ell}z^{\ell}_{n}}\underset{n\to\infty}{\longrightarrow}\frac{\displaystyle
k!\ell^{k}}{\displaystyle\ell(\ell-1)\cdots(r+1)}.$
But the left hand side of (4) tends to $\infty$, as $n\to\infty$, a
contradiction.
We deduce that there exists some $0<C_{1}<\infty$, such that every zero
$z_{n}$ of $\widetilde{f}_{n}$ satisfies $|z_{n}|\leq\frac{\displaystyle
C_{1}}{\displaystyle n}$. By Theorem Lu, we have also
$|\widehat{z}_{n}|\leq\frac{\displaystyle C_{1}}{\displaystyle n}$ for every
$\widehat{z}_{n}$, which is a zero of $\widetilde{f}^{(k)}_{n}$. But those
$\\{\widehat{z}_{n}\\}$ are exactly the points where $f^{(k)}_{n}(z)=h(z)$.
Hence $f^{(k)}_{n}(z)=h(z)$ implies that $|z|\leq\frac{\displaystyle
C_{1}}{\displaystyle n}$, and we have only to prove the following claim.
###### Claim 1.
There exists $0<C<\infty$, such that $|z|\leq\frac{\displaystyle
C_{1}}{\displaystyle n}$ implies $|f^{(k+1)}_{n}(z)|\leq C$.
###### Proof.
We have $f_{n}(z)=\frac{\displaystyle n^{(k-1)\ell}}{\displaystyle
k!\ell^{k}}\left(z^{\ell}-\frac{\displaystyle 1}{\displaystyle
n^{\ell}}\right)^{k}=\frac{\displaystyle n^{(k-1)\ell}}{\displaystyle
k!\ell^{k}}\sum\limits_{j=0}^{k}\binom{k}{j}z^{\ell
j}\left(\frac{\displaystyle 1}{\displaystyle n}\right)^{\ell(k-j)}(-1)^{k-j}$.
Thus, since $\ell j\geq k+1$ only for $j\geq 1$, we get that
$f^{(k+1)}_{n}(z)=\frac{\displaystyle n^{(k-1)\ell}}{\displaystyle
k!\ell^{k}}\sum\limits_{j=1}^{k}\binom{k}{j}\left(\frac{\displaystyle
1}{\displaystyle n}\right)^{\ell k-\ell j}(-1)^{k-j}\ell j(\ell
j-1)\cdots(\ell j-k-1)z^{\ell j-k-1}.$
Thus, if $|z|\leq\frac{\displaystyle C_{1}}{\displaystyle n}$, then
$\displaystyle|f^{(k+1)}_{n}(z)|$ $\displaystyle\leq\frac{\displaystyle
n^{(k-1)\ell}}{\displaystyle
k!\ell^{k}}\sum\limits_{j=1}^{k}\binom{k}{j}C^{\ell j-k-1}_{1}\ell j(\ell
j-1)\cdots(\ell j-k-1)n^{k+1-\ell j}\cdot n^{\ell j-\ell k}$
$\displaystyle=\frac{\displaystyle n^{k+1-\ell}}{\displaystyle
k!\ell^{k}}\sum\limits_{j=1}^{k}\binom{k}{j}C^{\ell j-k-1}_{1}\ell j(\ell
j-1)\cdots(\ell j-k-1)\leq C,$
where $C=\frac{\displaystyle 1}{\displaystyle
k!\ell^{k}}\sum\limits_{j=1}^{k}\binom{k}{j}C^{\ell j-k-1}_{1}\ell j(\ell
j-1)\cdots(\ell j-k-1)$. (Here we used that $k+1-\ell\leq 0$.) The Claim is
proved. ∎
Hence, $\\{f_{n}\\}$ with $h$ satisfy (a) and (b) of Theorem CFZ2, but
$\\{f_{n}\\}$ is not normal at $z=0$.
Observe that when $k=1$, then $a_{n}=\frac{\displaystyle
1}{\displaystyle\ell}\not\to\infty$, and we do not get a non-normal family, as
expected by Theorem 1.
The following example shows that the condition $h(z)\neq 0$ is essential also
forTheorem CFZ3.
###### Example 2.
(cf. [1, Ex. 4] Let $s\geq 4$ be an even integer and consider the family
$\mathcal{F}=\\{f_{n}(z)\\}^{\infty}_{n=1}$,
$f_{n}(z)=\frac{\displaystyle n^{s}}{\displaystyle
2s^{2}}\left(z^{s}-\frac{\displaystyle 1}{\displaystyle
n^{s}}\right)^{2}\quad\text{on}\quad\Delta.$
Let $h(z)=z^{s-2}$.
We have that
$f_{n}(z)=\frac{\displaystyle n^{s}}{\displaystyle
2s^{2}}\prod\limits_{j=1}^{s}\left(z-\alpha^{(n)}_{j}\right)^{2},$
where $\alpha^{(n)}_{j}=\frac{\displaystyle\exp(i2\pi j/s)}{\displaystyle n}$,
$1\leq j\leq s$.
By calculation we have
(5) $f^{\prime\prime}_{n}(z)=\frac{\displaystyle n^{s}}{\displaystyle
s}\left((2s-1)z^{s}-\frac{\displaystyle(s-1)}{\displaystyle
n^{s}}\right)z^{s-2},$ (6) $\displaystyle f^{\prime\prime\prime}_{n}(z)$
$\displaystyle=\frac{\displaystyle n^{s}}{\displaystyle
s}\left[(2s-1)(2s-2)z^{s}-\frac{\displaystyle(s-1)(s-2)}{\displaystyle
n^{s}}\right]z^{s-3}$
$\displaystyle=\frac{n^{s}}{s}(s-1)z^{s-3}\left[(4s-2)z^{s}-\frac{s-2}{n^{s}}\right],$
and
(7) $f^{(s)}_{n}(z)=\frac{\displaystyle n^{s}}{\displaystyle
s}\left[(2s-1)(2s-2)\cdots(s+1)z^{s}-\frac{\displaystyle(s-1)!}{\displaystyle
n^{s}}\right].$
Now, if $f_{n}(z)=0$, then $z=\alpha^{(n)}_{j}$ for some $1\leq j\leq s$, and
thus $z^{s}=\frac{\displaystyle 1}{\displaystyle n^{s}}$ and by (5),
$f^{\prime\prime}_{n}(z)=z^{s-2}=h(z)$.
If $f^{\prime\prime}_{n}(z)=z^{s-2}=h(z)$, then by (5), $z=0$ or
$z=\alpha^{(n)}_{j}$, $1\leq j\leq s$. By (6) and (7), we get
(8) $f^{(3)}_{n}(0)=0,\quad
f^{(s)}_{n}(0)=-\frac{\displaystyle(s-1)!}{\displaystyle n^{s}}$
and
(9) $f^{(3)}_{n}\left(\alpha^{(n)}_{j}\right)=3(s-1)\frac{\displaystyle
1}{\displaystyle n^{s-3}},\quad
f^{(s)}_{n}\left(\alpha^{(n)}_{j}\right)=\frac{\displaystyle 1}{\displaystyle
s}\left[\frac{\displaystyle(2s-1)!}{\displaystyle s!}-(s-1)!\right].$
From (8) and (9), we see that the family $\mathcal{F}$ with $h$ satisfy
assumption (a) and (b) of Theorem CFZ3, but $\mathcal{F}$ is not normal at
$z=0$. Indeed, the reason must be that $h(0)=0$.
In Example 1, we have that $f^{(k+1)}(z)\neq 0$ at the zero points of
$f^{(k)}(z)-h(z)$. If we strengthen condition (b) of Theorem CFZ2 to be
$f^{(k)}(z)=h(z)\Longrightarrow f^{(k+1)}(z)=0$, then we can obtain the
following normal criterion.
###### Theorem 2.
Let $\mathcal{F}$ be a family of functions holomorphic on a domain
$D\subset\mathbb{C}$, all of whose zeros have multiplicity at least $k$, where
$k\neq 2$ be a positive integer. Let $h(z)(\not\equiv 0)$ be a holomorphic
function on $D$, that has no common zeros with any $f\in\mathcal{F}$. Assume
also that the following two conditions hold for every $f\in\mathcal{F}$:
1. (a)
$f(z)=0\Longrightarrow f^{(k)}(z)=h(z)$; and
2. (b)
$f^{(k)}(z)=h(z)\Longrightarrow f^{(k+1)}(z)=0$.
Then $\mathcal{F}$ is normal on $D$.
Similarly, if we strengthen the condition (b) of Theorem CFZ3 to
$f^{\prime\prime}(z)=h(z)\Longrightarrow
f^{\prime\prime\prime}(z)=f^{(s)}(z)=0$, then we can also obtain the normality
criterion.
###### Theorem 3.
Let $\mathcal{F}$ be a family of functions holomorphic on a domain
$D\subset\mathbb{C}$, all of whose zeros are multiple, where $s\geq 2$ is an
even integer. Let $h(z)(\not\equiv 0)$ be a holomorphic function on $D$, that
has no common zeros with any $f\in\mathcal{F}$. Assume also that the following
two conditions hold for every $f\in\mathcal{F}$:
1. (a)
$f(z)=0\Longrightarrow f^{\prime\prime}(z)=h(z)$; and
2. (b)
$f^{\prime\prime}(z)=h(z)\Longrightarrow
f^{\prime\prime\prime}(z)=f^{(s)}(z)=0$.
Then $\mathcal{F}$ is normal on $D$.
Before we go to the proofs of the main results, let us set some notation.
Throughout, $D$ is a domain in $\mathbb{C}$. For $z_{0}\in\mathbb{C}$ and
$r>0$, $\Delta(z_{0},r)=\\{z:|z-z_{0}|<r\\}$ and
$\Delta^{\prime}(z_{0},r)=\\{z:0<|z-z_{0}|<r\\}$. The unit disc will be
denoted by $\Delta$ and $\mathbb{C}^{\ast}=\mathbb{C}\setminus\\{0\\}$. We
write $f_{n}(z)\overset{\chi}{\Rightarrow}f(z)$ on $D$ to indicate that the
sequence $\\{f_{n}\\}$ converges to $f$ in the spherical metric, uniformly on
compact subsets of $D$, and $f_{n}\Rightarrow f$ on $D$ if the convergence is
in the Euclidean metric. For a meromorphic function $f(z)$ in $D$ and
$a\in\widehat{\mathbb{C}}$, $\overline{E}_{f}(a):=\\{z\in D:f(z)=a\\}$. The
spherical derivative of the meromorphic function $f$ at the point $z$ is
denoted by $f^{\\#}(z).$
Frequently, given a sequence $\\{f_{n}\\}_{1}^{\infty}$ of functions, we need
to extract an appropriate subsequence; and this necessity may recur within a
single proof. To avoid the awkwardness of multiple indices, we again denote
the extracted subsequence by $\\{f_{n}\\}$ (rather than, say,
$\\{f_{n_{k}}\\})$ and designate this operation by writing “taking a
subsequence and renumbering,” or simply “renumbering”. The same convention
applies to sequences of constants.
The plan of the paper is as follows. In Section 2, we state a number of
preliminary results. Then in Section 3 we prove Theorem 1. Finally, in Section
4 we prove Theorem 2.
## 2\. Preliminary results
The following lemma is the local version of a well-known lemma of X. C. Pang
and L. Zalcman [11, Lemma 2]. For a proof see [4, Lemma 2], also cf.[9, Lemma
2], [14, pp. 216–217], [7, pp. 299–300], [8, p. 4].
###### Lemma 1.
Let $\mathcal{F}$ be a family of functions meromorphic in a domain $D$, all of
whose zeros have multiplicity at least $k$, and suppose that there exists
$A\geq 1$, such that $|f^{(k)}(z)|\leq A$ whenever $f(z)=0$. Then if
$\mathcal{F}$ is not normal at $z_{0}\in D$, there exist, for each
$0\leq\alpha\leq k$,
1. (a)
points $z_{n}\to z_{0}$;
2. (b)
functions $f_{n}\in\mathcal{F}$;and
3. (c)
positive numbers $\rho_{n}\to 0^{+}$
such that
$g_{n}(\zeta):=\rho^{-\alpha}_{n}f_{n}(z_{n}+f_{n}\zeta)\overset{\chi}{\Rightarrow}g(\zeta)$
on $\mathbb{C}$, where $g$ is a nonconstant meromorphic function on
$\mathbb{C}$, such that for every $\zeta\in\mathbb{C}$, $g^{\\#}(\zeta)\leq
g^{\\#}(0)=kA+1$.
###### Lemma 2.
[1, Lemma 5] Let $f$ be a nonconstant entire function of order $\rho$,
$0\leq\rho\leq 1$, all of whose zeros have multiplicity at least $k$, where
$k\neq 2$ is a positive integer. And let $a\neq 0$ be a constant. If
$\overline{E}_{f}(0)\subset\overline{E}_{f^{(k)}}(a)\subset\overline{E}_{f^{(k+1)}}(0)$,
then
$f(z)=\frac{\displaystyle a(z-b)^{k}}{\displaystyle k!},$
where $b$ is a constant.
###### Lemma 3.
[1, Lemma 6] Let $f$ be a nonconstant entire function of order $\rho$,
$0\leq\rho\leq 1$, all of whose zeros are multiple. Let $s\geq 4$ be an even
integer and $a\neq 0$ be a constant. If
$\overline{E}_{f}(0)\subset\overline{E}_{f^{\prime\prime}}(a)\subset\overline{E}_{f^{\prime\prime\prime}}(0)\cap\overline{E}_{f^{(s)}}(0)$,
then
$f(z)=\frac{\displaystyle a(z-b)^{2}}{\displaystyle 2},$
where $b$ is a constant.
###### Lemma 4.
(see [2, pp. 118–119,122–123]) Let $f$ be a meromorphic function on
$\mathbb{C}$. If $f^{\\#}$ is uniformly bounded on $\mathbb{C}$, then the
order of $f$ is at most $2$. If $f$ is an entire function, then the order of
$f$ is at most $1$.
The following lemma is a slight generalization of Theorem CFZ2 for sequences.
###### Lemma 5.
(cf. [4, Lemma 5]) Let $\\{f_{n}\\}$ be a sequence of functions holomorphic on
a domain $D\subset\mathbb{C}$, all of whose zeros have multiplicity at least
$k$, and let $\\{h_{n}\\}$ be a sequence of functions analytic on $D$ such
that $h_{n}{(z)}\Rightarrow h{(z)}$ on $D$, where $h(z)\neq 0$ for $z\in D$
and $k\neq 2$ be a positive integer. Suppose that, for each $n$,
$f_{n}(z)=0\Longrightarrow f^{(k)}_{n}(z)=h_{n}(z)$ and
$f^{(k)}_{n}(z)=h_{n}(z)\Longrightarrow f^{(k+1)}_{n}(z)=0$. Then
$\\{f_{n}\\}$ is normal on $D$.
###### Proof.
Suppose to the contrary that there exists $z_{0}\in D$ such that $\\{f_{n}\\}$
is not normal at $z_{0}$. The convergence of $\\{h_{n}\\}$ to $h$ implies
that, in some neighborhood of $z_{0}$, we have
$f_{n}(z)=0\Rightarrow|f_{n}^{(k)}(z)|\leq|h(z_{0})|+1$ (for large enough
$n$). Thus we can apply Lemma 1 with $\alpha=k$ and $A$ such that
$kA+1>\max\Big{\\{}|h(z_{0})|+1,\frac{\displaystyle|h(z_{0})|}{\displaystyle(k-1)!},\frac{\displaystyle
k\cdot
k!}{\displaystyle|h(z_{0})|}\Big{\\}}=\max\Big{\\{}|h(z_{0})|+1,\frac{\displaystyle
k\cdot k!}{\displaystyle|h(z_{0})|}\Big{\\}}$. So we can take an appropriate
subsequence of $\\{f_{n}\\}$ (denoted also by $\\{f_{n}\\}$ after
renumbering), together with points $z_{n}\to z_{0}$ and positive numbers
$\rho_{n}\to 0^{+}$ such that
$g_{n}(\zeta)=\frac{f_{n}(z_{n}+\rho_{n}\zeta)}{\rho^{k}_{n}}\overset{\chi}{\Longrightarrow}g(\zeta)\quad\text{on}\quad\mathbb{C},$
where $g$ is a nonconstant entire function and $g^{\sharp}(\zeta)\leq
g^{\sharp}(0)=kA+1=k(|h(z_{0})|+1)+1$. We claim that
(10)
$\overline{E}_{g}(0)\subset\overline{E}_{g^{(k)}}(h(z_{0}))\subset\overline{E}_{g^{(k+1)}}(0).$
In fact, if there exists $\zeta_{0}\in\mathbb{C}$, such that $g(\zeta_{0})=0$,
then since $g(\zeta)\not\equiv 0$, there exist $\zeta_{n}$,
$\zeta_{n}\to\zeta_{0}$, such that if $n$ is sufficiently large,
$g_{n}(\zeta_{n})=\frac{f_{n}(z_{n}+\rho_{n}\zeta_{n})}{\rho^{k}_{n}}=0.$
Thus $f_{n}(z_{n}+\rho_{n}\zeta_{n})=0$, so that
$f^{(k)}_{n}(z_{n}+\rho_{n}\zeta_{n})=h_{n}(z_{n}+\rho_{n}\zeta_{n})$, i.e.,
that $g^{(k)}_{n}(\zeta_{n})=h_{n}(z_{n}+\rho_{n}\zeta_{n})$. Since
$g^{(k)}(\zeta_{0})=\lim\limits_{n\to\infty}g^{(k)}_{n}(\zeta_{n})=h(z_{0})$,
we have established the first part of the Claim that
$\overline{E}_{g}(0)\subset\overline{E}_{g^{(k)}}(h(z_{0}))$.
Now, suppose there exists $\zeta_{0}\in\mathbb{C}$, such that
$g^{(k)}(\zeta_{0})=h(z_{0})$. If $g^{(k)}(\zeta)\equiv h(z_{0})$, then
$g^{(k+1)}\equiv 0$ and we are done. Thus we can assume that $g^{(k)}$ is not
constant and since
$f^{(k)}_{n}(z_{n}+\rho_{n}\zeta)-h_{n}(z_{n}+\rho_{n}\zeta)\Rightarrow
g^{(k)}(\zeta)-h(z_{0})$, we get by Hurwitz’s Theorem that there exist
$\zeta_{n}$, $\zeta_{n}\to\zeta_{0}$, such that
$f^{(k)}_{n}(z_{n}+\rho_{n}\zeta_{n})-h_{n}(z_{n}+\rho_{n}\zeta_{n})=g^{(k)}_{n}(\zeta_{n})-h_{n}(z_{n}+\rho_{n}\zeta_{n})=0.$
Thus we have $f^{(k+1)}_{n}(z_{n}+\rho_{n}\zeta_{n})=0$ and
$g^{(k+1)}_{n}(\zeta_{n})=0$. Letting $n\to\infty$, we get that
$g^{(k+1)}(\zeta_{0})=0$. This completes the proof of the Claim. Now, by
Lemmas 4 and 2, we have $g(\zeta)=\frac{\displaystyle
h(z_{0})(\zeta-\zeta_{1})^{k}}{\displaystyle k!}$, where $\zeta_{1}$ is a
constant. Thus
$g^{\sharp}(0)=\frac{\displaystyle|h(z_{0})||\zeta_{1}|^{k-1}/(k-1)!}{\displaystyle
1+|h(z_{0})|^{2}|\zeta_{1}|^{2k}/k!^{2}}.$
Now, if $|\zeta_{1}|\leq 1$, then
$g^{\sharp}(0)\leq\frac{\displaystyle|h(z_{0})|}{\displaystyle(k-1)!}<kA+1$,
and if $|\zeta_{1}|>1$, then
$g^{\sharp}(0)\leq\frac{\displaystyle|h(z_{0})||\zeta_{1}|^{k-1}/(k-1)!}{\displaystyle|h(z_{0})|^{2}|\zeta_{1}|^{2k}/k!^{2}}\leq\frac{\displaystyle
k\cdot k!}{\displaystyle|h(z_{0})|}<kA+1$. In either case we get a
contradiction. ∎
Similarly, we can get a slight generalization of Theorem CFZ3 for sequences.
###### Lemma 6.
Let $\\{f_{n}\\}$ be a sequence of functions holomorphic on a domain
$D\subset\mathbb{C}$, all of whose zeros are multiple and $\\{h_{n}\\}$ be a
sequence of functions analytic on $D$ such that $h_{n}{(z)}\Rightarrow h{(z)}$
on $D$, where $h(z)\neq 0$ for $z\in D$ and $s\geq 2$ be an even integer.
Suppose that, for each $n$, $f_{n}(z)=0\Longrightarrow
f^{\prime\prime}_{n}(z)=h_{n}(z)$ and
$f^{\prime\prime}_{n}(z)=h_{n}(z)\Longrightarrow
f^{\prime\prime\prime}(z)=f^{(s)}_{n}(z)=0$, then $\\{f_{n}\\}$ is normal on
$D$.
The proof is very similar to the proof of Lemma 5. We start to argue the same
(with $2$ instead of $k$), and then instead of proving (10) we prove that
$\overline{E}_{g}(0)\subset\overline{E}_{g^{\prime\prime}}(h(z_{0}))\subset\overline{E}_{g^{(3)}}(0)\cap\overline{E}_{g^{(s)}}(0).$
The left inclusion is proved in the same manner. Concerning the right
inclusion, we now deduce from
$f^{\prime\prime}_{n}(z_{n}+\rho_{n}\zeta_{n})-h_{n}(z_{n}+\rho_{n}\zeta_{n})=0$
that
$f^{(3)}_{n}(z_{n}+\rho_{n}\zeta_{n})=f^{(s)}_{n}(z_{n}+\rho_{n}\zeta_{n})=0$.
Then, since $\rho_{n}f^{(3)}_{n}(z_{n}+\rho_{n}\zeta)\Rightarrow
g^{(3)}(\zeta)$ in $\mathbb{C}$ and
$\rho^{s-2}_{n}f^{(s)}_{n}(z_{n}+\rho_{n}\zeta)\Rightarrow g^{(s)}(\zeta)$ in
$\mathbb{C}$, we conclude that $g^{(3)}(\zeta_{0})=g^{(s)}(\zeta_{0})=0$. To
get the final contradiction, we apply now Lemmas 4 and 3 instead of Lemmas 4
and 2.
The following result will play an essential role in treating transcendental
functions which is used in the proofs of Theorems 2 and 3.
###### Theorem B.
$($[15] see also [2, p. 117]$)$ Let $f(z)$ be a function homomorphic
in$\\{z:R<|z|<\infty\\}$, with essential singularity at $z=\infty$. Then
$\varlimsup\limits_{|z|\to\infty}|z|f^{\\#}(z)=+\infty$.
For the proof of Theorem 2, we need also the following Lemma.
###### Lemma 7.
Let $h$ be a holomorphic function on $D,$ with a zero of order $\ell(\geq 1)$
at $z_{0}\in D.$ Let $\\{f_{n}\\}^{\infty}_{n=1}$ be a sequence of functions
with zeros of multiplicity at least $k$, such that $\\{f_{n}\\}$ and $h$
satisfy conditions (a) and (b) of Theorem 2. Let
$\\{\alpha_{n}\\}^{\infty}_{n=1}$ be a sequence of nonzero numbers such that
$\alpha_{n}\to 0$ as $n\to\infty$. Then
$\\{f_{n}(z_{0}+\alpha_{n}\zeta)/\alpha^{k+\ell}_{n}\\}^{\infty}_{n=1}$ is
normal in $\mathbb{C}^{\ast}$.
###### Proof.
Without loss of generality, we may assume that $z_{0}=0$. In a neighborhood of
the origin we have $h(z)=z^{\ell}b(z)$, where $b(z)$ is analytic, $b(0)\neq
0$. Define $r_{n}(\zeta)=\zeta^{\ell}b(\alpha_{n}\zeta)$. We will show that
the assumptions of Lemma 5 hold in $\mathbb{C}^{\ast}$ for the sequence
$\\{G_{n}(\zeta)\\}^{\infty}_{n=1}$,
$G_{n}(\zeta):=f_{n}(\alpha_{n}\zeta)/\alpha^{k+\ell}_{n}$ and
$\\{r_{n}(\zeta)\\}^{\infty}_{n=1}$. First, we have that
$r_{n}(\zeta)\Rightarrow b(0)\zeta^{\ell}$ on $\mathbb{C}$ and
$\zeta^{\ell}\neq 0$ in $\mathbb{C}^{\ast}$. Assume that $G_{n}(\zeta)=0$.
Then $f_{n}(\alpha_{n}\zeta)=0$ and
$f^{(k)}_{n}(\alpha_{n}\zeta)=(\alpha_{n}\zeta)^{\ell}b(\alpha_{n}\zeta)$, and
we get that $G^{(k)}_{n}(\zeta)=r_{n}(\zeta)$. Suppose now that
$G^{(k)}_{n}(\zeta)=r_{n}(\zeta)$. This means that
$f^{(k)}_{n}(\alpha_{n}\zeta)=h(\alpha_{n}\zeta)$ and thus
$f^{(k+1)}_{n}(\alpha_{n}\zeta)=0$. We have $G^{(k+1)}_{n}(\zeta)=0$, and thus
the assumptions of Lemma 5 hold. Hence we deduce that $\\{G_{n}(\zeta)\\}$ is
normal in $\mathbb{C}^{\ast}$, and the lemma is proved. ∎
The following lemma plays a similar role in the proof of Theorem 3, to the
role of Lemma 7 in the proof of Theorem 2.
###### Lemma 8.
Let $h$ be a holomorphic function on $D,$ with a zero of order $\ell(\geq 1)$
at $z_{0}\in D.$ Let $\\{f_{n}\\}^{\infty}_{n=1}$ be a sequence of functions
whose zeros are multiple, such that $\\{f_{n}\\}$ and $h$ satisfy conditions
(a) and (b) of Theorem 3. Let $\\{\alpha_{n}\\}^{\infty}_{n=1}$ be a sequence
of nonzero numbers such that $\alpha_{n}\to 0$ as $n\to\infty$. Then
$\\{f_{n}(z_{0}+\alpha_{n}\zeta)/\alpha^{2+\ell}_{n}\\}^{\infty}_{n=1}$ is
normal in $\mathbb{C}^{\ast}$.
The proof of this lemma is analogous to the proof of Lemma 7. Of course, we
use Lemma 6 instead of Lemma 5.
## 3\. Proof of Theorem 1
In this section, we do not use any of the preliminary results. The proof is
elementary.
By Theorem CFZ1, $\mathcal{F}$ is normal at every point $z_{0}\in D$ at which
$h(z_{0})\neq 0$(so immediately we get that $\mathcal{F}$ is quasinormal). So
let $z_{0}$ be a zero of $h$ of order $\ell(\geq 1)$. Without loss of
generality, we can assume that $z_{0}=0$, and then $h(z)=z^{\ell}b(z)$. Here
$b$ is an analytic function in $\Delta(0,\delta)$ and $b(z)\neq 0$ there. We
assume that $0<\delta<1$, and by taking a subsequence and renumbering, we can
assume that
(11) $f_{n}\Longrightarrow f\quad\text{in}\quad\Delta^{\prime}(0,\delta).$
Now, if $f$ is holomorphic in $\Delta^{\prime}(0,\delta)$, we deduce by the
maximum principle that $f_{n}\Rightarrow f$ on $\Delta(0,\delta)$, and we are
done. So let us assume that $f_{n}\Rightarrow\infty$ in
$\Delta^{\prime}(0,\delta)$. Fix $\eta$, $0<\eta<\delta$. By the minimum
principle (i.e., the maximum principle for $\\{1/f_{n}\\}$), there exists
$N=N(\eta)$, such that for every $n\geq N$, $f_{n}$ has $k_{n}(k_{n}\geq 1)$
simple zeros in $\overline{\Delta}(0,\eta)-\\{0\\}$, say $\alpha^{(n)}_{1}$,
$\alpha^{(n)}_{2}$, $\cdots$, $\alpha^{(n)}_{k_{n}}$ (otherwise we get that
$f_{n}\Rightarrow\infty$ in $\Delta(0,\eta)$ and we are done). Since
$f_{n}\Rightarrow\infty$ in $\Delta^{\prime}(0,\delta)$, we get that
(12) $\max\limits_{1\leq j\leq k_{n}}\\{|\alpha^{(n)}_{j}|\\}\to
0,\quad\text{as}\quad n\to\infty.$
We can write
$f_{n}(z)=t_{n}(z)\prod\limits_{i=1}^{k_{n}}\left(z-\alpha^{(n)}_{i}\right)$,
where $t_{n}(z)\neq 0$ for $z\in\overline{\Delta}(0,\eta)$ and $n\geq N$.
Since $\eta<1$, we get by (12) that $\frac{\displaystyle
t_{n}(z)}{\displaystyle b(z)}\Rightarrow\infty$ in
$\overline{\Delta}(0,\eta)$. By condition (a) of Theorem 1, we have, for
$n\geq N$,
$f^{\prime}_{n}(\alpha^{(n)}_{j})=\alpha^{(n)\ell}_{j}b(\alpha^{(n)}_{j})$,
$1\leq j\leq k_{n}$. By calculation,
$f^{\prime}_{n}(z)=t^{\prime}_{n}(z)\prod\limits_{i=1}^{k_{n}}\left(z-\alpha^{(n)}_{i}\right)+t_{n}(z)\left[\prod\limits_{i=1}^{k_{n}}\left(z-\alpha^{(n)}_{i}\right)\right]^{\prime},$
and so
(13)
$t_{n}\left(\alpha^{(n)}_{j}\right)\left[\prod\limits_{i=1}^{k_{n}}\left(z-\alpha^{(n)}_{i}\right)\right]^{\prime}\Bigg{|}_{z=\alpha^{(n)}_{j}}=\alpha^{(n)\ell}_{j}b\left(\alpha^{(n)}_{j}\right).$
Define, for $n\geq N$,
$M_{n}(z):=\frac{\displaystyle t_{n}(z)}{\displaystyle
b(z)}\left[\prod\limits_{i=1}^{k_{n}}\left(z-\alpha^{(n)}_{i}\right)\right]^{\prime}-z^{\ell}.$
By (13) we get that $M_{n}\left(\alpha^{(n)}_{j}\right)=0$ for $1\leq j\leq
k_{n}$, and so for $n\geq N,$ $M_{n}$ has at least $k_{n}$ zeros in
$\Delta^{\prime}(0,\eta)$, including multiplicities. Here we use the fact $h$
has no common zero with any $f_{n}.$ Since such a zero must be $z=0$ and would
be a zero of order $m$ (must be $m\geq 2$ by condition (a)) of $f_{n}$, and it
would be a zero of order $m-1$ of $M_{n}$ (if $\ell>m-1$) or even of order
$\ell<m-1$ (if $\ell<m-1$), then we would not know that the number of zeros
(including multiplicities) of $M_{n}$ is at least $k_{n}$. This fact, under
the assumption that there are no common zeros, will lead to the desired
contradiction.
###### Claim 2.
$\frac{\displaystyle t_{n}(z)}{\displaystyle
b(z)}\left[\prod\limits_{i=1}^{k_{n}}\left(z-\alpha^{(n)}_{i}\right)\right]^{\prime}\Rightarrow\infty$
in $\Delta^{\prime}(0,\eta)$.
###### Proof.
We write
(14) $\frac{\displaystyle t_{n}(z)}{\displaystyle
b(z)}\left[\prod\limits_{i=1}^{k_{n}}\left(z-\alpha^{(n)}_{i}\right)\right]^{\prime}=\sum\limits_{j=1}^{k_{n}}\frac{\displaystyle
t_{n}(z)}{\displaystyle b(z)}\prod\limits_{i=1,i\neq
j}^{k_{n}}\left(z-\alpha^{(n)}_{i}\right).$
For any $\varepsilon$, $0<\varepsilon<\eta$, we have that
(15) $\frac{\displaystyle t_{n}(z)}{\displaystyle
b(z)}\prod\limits_{i=2}^{k_{n}}\left(z-\alpha^{(n)}_{i}\right)\Longrightarrow\infty\quad\text{in}\quad\overline{R}_{\varepsilon,\eta}:=\\{z:\varepsilon\leq|z|\leq\eta\\}.$
Indeed, $\frac{\displaystyle t_{n}(z)}{\displaystyle
b(z)}\prod\limits_{i=2}^{k_{n}}\big{(}z-\alpha^{(n)}_{i}\big{)}=\frac{\displaystyle
f_{n}(z)}{\displaystyle b(z)\big{(}z-\alpha^{(n)}_{1}\big{)}}$, and since
$\eta<1$ and by (11) and (12), this term tends uniformly to $\infty$ in
$\overline{R}_{\varepsilon,\eta}$.
Now, for every $j$, $2\leq j\leq k_{n}$, we have that
$\frac{\displaystyle\frac{\displaystyle t_{n}(z)}{\displaystyle
b(z)}\prod\limits_{i=2}^{k_{n}}\left(z-\alpha^{(n)}_{i}\right)}{\displaystyle\frac{\displaystyle
t_{n}(z)}{\displaystyle b(z)}\prod\limits_{i=1,i\neq
j}^{k_{n}}\left(z-\alpha^{(n)}_{i}\right)}=\frac{\displaystyle
z-\alpha^{(n)}_{j}}{\displaystyle z-\alpha^{(n)}_{1}},$
and by (12) this term tends uniformly to $1$ as $n\to\infty$. This means, that
for every $1\leq j\leq k_{n}$, and $z\in\overline{R}_{\varepsilon,\eta}$,
$\frac{\displaystyle t_{n}(z)}{\displaystyle b(z)}\prod\limits_{i=1,i\neq
j}^{k_{n}}\big{(}z-\alpha^{(n)}_{i}\big{)}$ lies in the same quarter plane,
that is,
(16) $\Pi_{n,z}:=\left\\{z:\arg\left[\frac{\displaystyle
t_{n}(z)}{\displaystyle
b(z)}\prod\limits_{i=2}^{k_{n}}\left(z-\alpha^{(n)}_{i}\right)\right]-\frac{\displaystyle\pi}{\displaystyle
4}<\arg z<\arg\left[\frac{\displaystyle t_{n}(z)}{\displaystyle
b(z)}\prod\limits_{i=2}^{k_{n}}\left(z-\alpha^{(n)}_{i}\right)\right]+\frac{\displaystyle\pi}{\displaystyle
4}\right\\},$
for large enough $n$.
Now, if $a$ and $b$ are two complex numbers in the same quarter plane, then
$a+b$ also belongs to that quarter plane and $|a+b|\geq|a|$, $|b|$. We then
conclude by (16) that for each $z\in\overline{R}_{\varepsilon,\eta}$, we have
for large enough $n$,
$\left|\frac{\displaystyle t_{n}(z)}{\displaystyle
b(z)}\left[\prod\limits_{i=1}^{k_{n}}\left(z-\alpha^{(n)}_{i}\right)\right]^{\prime}\right|\geq\left|\frac{\displaystyle
t_{n}(z)}{\displaystyle
b(z)}\prod\limits_{i=2}^{k_{n}}\left(z-\alpha^{(n)}_{i}\right)\right|,$
and by (15) and (14), the Claim is proved.∎
Now, $\frac{\displaystyle t_{n}(z)}{\displaystyle
b(z)}\bigg{[}\prod\limits_{i=1}^{k_{n}}\big{(}z-\alpha^{(n)}_{i}\big{)}\bigg{]}^{\prime}$
has for large enough $n$ exactly $k_{n}-1$ zeros in $\Delta(0,\eta)$ (by
Theorem Lu). Then for large enough $n$ we have, for every $z$, $|z|=\eta$,
$\left|M_{n}(z)-\frac{\displaystyle t_{n}(z)}{\displaystyle
b(z)}\left[\prod\limits_{i=1}^{k_{n}}\left(z-\alpha^{(n)}_{i}\right)\right]^{\prime}\right|=|z^{\ell}|<\left|\frac{\displaystyle
t_{n}(z)}{\displaystyle
b(z)}\left[\prod\limits_{i=1}^{k_{n}}\left(z-\alpha^{(n)}_{i}\right)\right]^{\prime}\right|,$
and by Rouche’s Theorem, we get that $M_{n}$ has $k_{n}-1$ zeros in
$\Delta(0,\eta)$, a contradiction. Theorem 1 is proved. $\square$
## 4\. Proof of Theorem 2
This proof is similar to the proof of Theorem 1 in [4]. By our Theorem 1, we
need only to prove the case that $k\geq 3$. By Theorem CFZ2, $\mathcal{F}$ is
normal at every point $z_{0}\in D$ at which $h(z_{0})\neq 0$ (so that
$\mathcal{F}$ is quasinormal in $D$ ). Consider $z_{0}\in D$ such that
$h(z_{0})=0$. Without loss of generality, we can assume that $z_{0}=0$, and
then $h(z)=z^{\ell}b(z)$, where $\ell(\geq 1)$ is an integer, $b(z)\neq 0$ is
an analytic function in $\Delta(0,\delta)$. We take a subsequence
$\\{f_{n}\\}^{\infty}_{1}\subset\mathcal{F}$, and we want to prove that
$\\{f_{n}\\}$ is not normal at $z=0.$ Suppose by negation that $\\{f_{n}\\}$
is not normal at $z=0.$ Since $\\{f_{n}\\}$ is normal in
$\Delta^{\prime}(0,\delta),$ we can assume (after renumbering) that
$f_{n}\Rightarrow F$ on $\Delta^{\prime}(0,\delta)$. If
$F(z)\not\equiv\infty,$ then it is a holomorphic function; hence by the
maximum principle, $F$ extends to be analytic also at $z=0$, and so
$f_{n}\Rightarrow F$ on $\Delta(0,\delta)$, and we are done. Hence we assume
that
(17)
$f_{n}(z)\Longrightarrow\infty\quad\text{on}\quad\Delta^{\prime}(0,\delta).$
Define $\mathcal{F}_{1}=\left\\{F=\frac{\displaystyle f_{n}}{\displaystyle
h}:n\in\mathbb{N}\right\\}.$ It is enough to prove that $\mathcal{F}_{1}$ is
normal in $\Delta(0,\delta).$ Indeed, if (after renumbering)
$\frac{\displaystyle f_{n}(z)}{\displaystyle h}\Rightarrow H(z)$ on
$\Delta(0,\delta),$ then since $h\neq 0$ in $\Delta^{\prime}(0,\delta)$, it
follows from (17) that $H(z)\equiv\infty$ in $\Delta^{\prime}(0,\delta)$, and
thus $H(z)\equiv\infty$ also in $\Delta(0,\delta).$ In particular,
$\frac{\displaystyle f_{n}}{\displaystyle h}(z)\neq 0$ on each compact subset
of $\Delta(0,\delta)$ for large enough $n.$ Since $h\neq 0$ on
$\Delta^{\prime}(0,\delta)$ and since $f_{n}(0)\neq 0$ for every $n\geq 1$ by
assumptions of the theorem, we obtain $f_{n}(z)\neq 0$ on each compact subset
of $\Delta(0,\delta)$ for large enough $n.$ Then by the minimum principle, it
follows from (17) that $f_{n}(z)\Rightarrow\infty$ on $\Delta(0,\delta)$, and
this implies the normality of $\mathcal{F}.$ So suppose to the contrary that
$\mathcal{F}_{1}$ is not normal at $z=0$. By Lemma 1 and the assumptions of
Theorem 2, there exist (after renumbering) points $z_{n}\to 0$, $\rho_{n}\to
0^{+}$ and a nonconstant meromorphic function on $\mathbb{C}$, $g(\zeta)$ such
that
(18)
$g_{n}(\zeta)=\frac{F_{n}(z_{n}+\rho_{n}\zeta)}{\rho^{k}_{n}}=\frac{f_{n}(z_{n}+\rho_{n}\zeta)}{\rho^{k}_{n}h(z_{n}+\rho_{n}\zeta)}\overset{\chi}{\Longrightarrow}g(\zeta)\quad\text{on}\quad\mathbb{C},$
all of whose zeros have multiplicity at least $k$ and
(19) $\text{for every}\quad\zeta\in\mathbb{C},\quad g^{\sharp}(\zeta)\leq
g^{\sharp}(0)=kA+1,$
where $A>1$ is a constant. Here we have used Lemma 1 with $\alpha=k$. Observe
that $g_{n}(z)=0$ implies $g^{(k)}_{n}(\zeta)=1$ and so $A$ can be chosen to
be any number such that $A\geq 1.$ After renumbering we can assume that
$\\{z_{n}/\rho_{n}\\}^{\infty}_{n=1}$ converges. We separate now into two
cases.
Case (A)
(20) $\frac{z_{n}}{\rho_{n}}\to\infty.$
###### Claim 3.
$(1)$ $g(\zeta)=0\Longrightarrow g^{(k)}(\zeta)=1$; $(2)$
$g^{(k)}(\zeta)=1\Longrightarrow g^{(k+1)}(\zeta)=0$.
###### Proof.
Observe that from (18) and the fact that $h(z)\neq 0$ in
$\Delta^{\prime}(0,\delta),$ it follows that $g$ is an entire function.
Suppose that $g(\zeta_{0})=0$. Since $g(\zeta)\not\equiv 0$, there exist
$\zeta_{n}\to\zeta_{0}$, such that $g_{n}(\zeta_{n})=0$, and thus
$f_{n}(z_{n}+\rho_{n}\zeta_{n})=0$. Since $f_{n}$ and $h$ has no common zeros,
it follows by the assumption that $\zeta_{n}$ is a zero of multiplicity $k$ of
$g_{n}(\zeta)$. By Leibniz’s rule, and condition (a) of Theorem 2, it follows
that $g^{(k)}_{n}(\zeta_{n})=1$ and thus $g^{(k)}(\zeta_{0})=1$.
For the proof of the other part of the Claim, observe first that by (20) we
have
$\frac{\displaystyle
f_{n}(z_{n}+\rho_{n}\zeta)}{\displaystyle\rho_{n}^{k}z^{\ell}_{n}}\Rightarrow
g(\zeta)\quad\text{on}\quad\mathbb{C},$
and thus
$\frac{\displaystyle f^{(k)}_{n}(z_{n}+\rho_{n}\zeta)}{\displaystyle
z^{\ell}_{n}}\Rightarrow g^{(k)}(\zeta)\quad\text{on}\quad\mathbb{C},$
and then again by (19) we get that
$\frac{\displaystyle f^{(k)}_{n}(z_{n}+\rho_{n}\zeta)}{\displaystyle
h(z_{n}+\rho_{n}\zeta)}\Rightarrow
g^{(k)}(\zeta)\quad\text{on}\quad\mathbb{C}.$
Thus, if there exists $\zeta_{0}\in\mathbb{C}$, such that
$g^{(k)}(\zeta_{0})=1$, there exists a sequence $\zeta_{n}\to\zeta_{0}$, such
that $f^{(k)}_{n}(z_{n}+\rho_{n}\zeta_{n})=h(z_{n}+\rho_{n}\zeta)\neq 0$. By
assumption (b) of Theorem 2 we get that
$f^{(k+1)}_{n}(z_{n}+\rho_{n}\zeta_{n})=0$, and letting $n$ tend to $\infty$
we get that $g^{(k+1)}(\zeta_{0})=0$. The Claim is proved.∎
We conclude by Lemma 2 and by Lemma 4 that
$g(\zeta)=\frac{\displaystyle(\zeta-b)^{k}}{\displaystyle k!}$ for some
$b\in\mathbb{C}$ (observe that $g$ is holomorphic by (20)). By calculation we
get that
$g^{\sharp}(0)=\frac{\displaystyle|b|^{k-1}/(k-1)!}{\displaystyle
1+|b|^{2k}/k!^{2}}.$
Then if $|b|\leq 1$, we get that $g^{\sharp}(0)\leq\frac{\displaystyle
1}{\displaystyle(k-1)!}$, and if $|b|\geq 1$, then
$g^{\sharp}(0)\leq\frac{\displaystyle k}{\displaystyle 2}$. In either case, we
get a contradiction to (19).
Case (B)
(21) $\frac{z_{n}}{\rho_{n}}\to\alpha\in\mathbb{C}.$
As in Case (A), it follows that $g(\zeta_{0})=0\Longrightarrow
g^{(k)}(\zeta_{0})=1$. Now set
$G_{n}(\zeta)=\frac{f_{n}(\rho_{n}\zeta)}{\rho^{k+\ell}_{n}}.$
From (18) and (21) we have
(22) $G_{n}(\zeta)\Longrightarrow
G(\zeta)=g(\zeta-\alpha)\zeta^{\ell}b(0)\quad\text{on}\quad\mathbb{C}.$
Indeed,
$\frac{f_{n}(\rho_{n}\zeta)}{\rho_{n}^{k+\ell}}=\frac{f_{n}(\rho_{n}\zeta)}{\rho^{k}_{n}h(\rho_{n}\zeta)}\cdot\frac{h(\rho_{n}\zeta)}{\rho_{n}^{\ell}}=\frac{f_{n}\left(z_{n}+\rho_{n}\left(\zeta-\frac{\displaystyle
z_{n}}{\displaystyle\rho_{n}}\right)\right)}{\rho^{k}_{n}h\left(z_{n}+\rho_{n}\left(\zeta-\frac{\displaystyle
z_{n}}{\displaystyle\rho_{n}}\right)\right)}\frac{(\rho_{n}\zeta)^{\ell}b(\rho_{n}\zeta)}{\rho_{n}^{\ell}}$
(cf. [12, p. 7]). Since $g$ has a pole of order $\ell$ at $\zeta=-\alpha$
(here we use the fact that for every $n$, $h$ has no common zeros with
$f_{n}$) and since $\\{G_{n}\\}$ are analytic, we have
(23) $G(0)\neq 0,\ \infty.$
We now consider several subcases, depending on the nature of $G$.
Case (BI) $G$ is a polynomial.
Since $\\{f_{n}\\}$ is not normal at $z=0$, there exist (after renumbering) a
sequence $z^{\ast}_{n}\to 0$ such that
(24) $f_{n}(z^{\ast}_{n})=0.$
Otherwise, there is some $\delta^{\prime}$, $0<\delta^{\prime}<\delta$ such
that (before renumbering) $f_{n}(z)\neq 0$ in $\Delta(0,\delta^{\prime})$, and
since $f_{n}(z)\Rightarrow\infty$ on $\Delta^{\prime}(0,\delta)$ we would have
by the minimum principle that $f_{n}(z)\Rightarrow\infty$ on
$\Delta(0,\delta)$, a contradiction to the non-normality of $\\{f_{n}\\}$ at
$z=0$. We have that all the zeros of $g$ are of multiplicity exactly $k$. Then
by (22) and (23), it follows that all the zeros of $G$ are also of
multiplicity exactly $k$. We consider now two possibilities.
Case (BI1) $\operatorname{deg}(G)=0$.
We can assume that $z^{\ast}_{n}$ from (24) is the closest zero of $f_{n}$ to
the origin. Then we have
(25)
$\frac{f_{n}(\rho_{n}\zeta)}{\rho^{k+\ell}_{n}b(\rho_{n}\zeta)}\Longrightarrow\frac{G(0)}{b(0)}\quad\text{on}\quad\mathbb{C}.$
By (25) we have
(26) $\frac{z^{\ast}_{n}}{\rho_{n}}\to\infty.$
Define $t_{n}(\zeta)=f_{n}(z^{\ast}_{n}\zeta)/\left(z^{\ast
k+\ell}_{n}b(z^{\ast}_{n}\zeta)\right)$. We want to show that
$\\{t_{n}(\zeta)\\}$ is normal in $\mathbb{C}^{\ast}$. For this purpose set
$\tilde{t}_{n}(\zeta)=f_{n}(z^{\ast}_{n}\zeta)/z^{\ast k+\ell}_{n}$. Since
$b(0)\neq 0$, $\infty$ and $z^{\ast}_{n}\to 0$, the normality of $\\{t_{n}\\}$
is equivalent to the normality of $\\{\tilde{t}_{n}\\}$, and the latter
follows by Lemma 7. Now, if $\\{t_{n}\\}$ is not normal at $\zeta=0$, then we
can write (after renumbering) $t_{n}(\zeta)\Rightarrow\infty$ on
$\mathbb{C}^{\ast}$; but $t_{n}(1)=0$, so this is not possible. Hence
$\\{t_{n}(\zeta)\\}$ is normal at $\zeta=0$. By (25) and (26), $t_{n}(0)\to 0$
as $n\to\infty$; and thus since $t_{n}(\zeta)\neq 0$ in $\Delta(0,1/2),$ we
get by Hurwitz’s Theorem that $t_{n}(\zeta)\Rightarrow 0$ on $\mathbb{C}$. But
$t_{n}(1)=0$; so by assumption (b) of Theorem 2, we get that
$t^{(k)}_{n}(1)=1$, a contradiction.
Case (BI2) $G^{(k)}\equiv b(0)\zeta^{\ell}$.
Then we have $G^{(k-1)}(\zeta)=\frac{\displaystyle
b(0)\zeta^{\ell+1}}{\displaystyle\ell+1}+C$ and
$G^{(k-2)}(\zeta)=\frac{\displaystyle
b(0)\zeta^{\ell+2}}{\displaystyle(\ell+1)(\ell+2)}+C\zeta+D$, where $C$ and
$D$ are two constants. Since all zeros of $G$ have multiplicity exactly $k$,
then for any zero $\widehat{\zeta}$ of $G$, we have
$G^{(k-2)}(\widehat{\zeta})=G^{(k-1)}(\widehat{\zeta})=0$. So
(27)
$\frac{\displaystyle\widehat{\zeta}^{\ell+1}}{\displaystyle\ell+1}+C=0,\quad\text{and}\quad\frac{\displaystyle\widehat{\zeta}^{\ell+2}}{\displaystyle(\ell+1)(\ell+2)}+C\widehat{\zeta}+D=0.$
By calculation, we have
$\frac{\displaystyle(\ell+1)C}{\displaystyle\ell+2}\widehat{\zeta}=-D$. If
$CD=0$, then by (27), $\widehat{\zeta}=0$, a contradiction. If $CD\neq 0$,
then $\widehat{\zeta}=-\frac{\displaystyle(\ell+2)D}{\displaystyle(\ell+1)C}$,
which implies that $G$ has only one zero $\zeta_{0}$, and then
$G(\zeta)=\frac{\displaystyle
b(0)\zeta_{0}^{\ell}(\zeta-\zeta_{0})^{k}}{\displaystyle k!}.$
This contradicts $G^{(k)}\equiv b(0)\zeta^{\ell}$.
Case (BI3) $G$ is a nonconstant polynomial and $G^{(k)}\not\equiv
b(0)\zeta^{\ell}$.
Since all zeros of $G$ have multiplicity exactly $k$, we may assume that
$G=A\prod\limits_{j=1}^{t}(\zeta-\zeta_{j})^{k}.$
where $A\neq 0$ is a constant and $\zeta_{j}\neq 0$, $j=1,2,\cdots,t$.
###### Claim 4.
$G(\zeta)=0\Longrightarrow G^{(k)}(\zeta)=b(0)\zeta^{\ell}\Longrightarrow
G^{(k+1)}(\zeta)=0.$
###### Proof.
Suppose first that $G(\zeta_{0})=0$. Then there exists a sequence,
$\zeta_{n}\to\zeta_{0}$, such that $f_{n}(\rho_{n}\zeta_{n})=0$, and thus
$f_{n}^{(k)}(\rho_{n}\zeta_{n})=(\rho_{n}\zeta_{n})^{\ell}b(\rho_{n}\zeta_{n})$,
that is, $\frac{\displaystyle
f_{n}^{(k)}(\rho_{n}\zeta_{n})}{\displaystyle\rho^{\ell}_{n}}=\zeta^{\ell}_{n}b(\rho_{n}\zeta_{n})$.
In the last equation, the left hand side tends to $\zeta^{\ell}_{0}b(0)$ as
$n\to\infty$. This proves the first part of the Claim.
Suppose now that $G^{(k)}(\zeta_{0})=b(0)\zeta^{\ell}_{0}$. Since
$G^{(k)}(\zeta)\not\equiv b(0)\zeta^{\ell}$, there exists a sequence
$\zeta_{n}\to\zeta_{0}$, such that $\frac{\displaystyle
f_{n}^{(k)}(\rho_{n}\zeta_{n})}{\displaystyle\rho^{\ell}_{n}}=\zeta^{\ell}_{n}b(\rho_{n}\zeta_{n})$,
that is,
$f_{n}^{(k)}(\rho_{n}\zeta_{n})=(\rho_{n}\zeta_{n})^{\ell}b(\rho_{n}\zeta_{n})$,
and thus $f_{n}^{(k+1)}(\rho_{n}\zeta_{n})=0$. Since $\frac{\displaystyle
f^{(k+1)}_{n}(\rho_{n}\zeta)}{\displaystyle\rho^{\ell-1}_{n}}\Rightarrow
G^{(k+1)}(\zeta)$, we deduce that $G^{(k+1)}(\zeta_{0})=0$, and this completes
the proof of the Claim. ∎
It follows from Claim 4 that $G^{(k+1)}(\zeta_{j})=0$, for $1\leq j\leq t$.
If $t\geq 2$, we know that for every $1\leq j\leq t$,
$\displaystyle G^{(k+1)}(\zeta)$
$\displaystyle=A\left[\prod\limits_{j=1}^{t}(\zeta-\zeta_{j})^{k}\right]^{(k+1)}$
$\displaystyle=A\left\\{\sum\limits_{\mu=0}^{k+1}\binom{k+1}{\mu}\left[(\zeta-\zeta_{j})^{k}\right]^{(k+1-\mu)}\left[\prod\limits_{i=1,i\neq
j}^{t}(\zeta-\zeta_{i})^{k}\right]^{(\mu)}\right\\}$
$\displaystyle=A\left\\{(k+1)k!\left[\prod\limits_{i=1,i\neq
j}^{t}(\zeta-\zeta_{i})^{k}\right]^{\prime}+(\zeta-\zeta_{j})P_{j}(\zeta)\right\\},$
where $P_{j}$ is a polynomial. Thus, by Claim 4 we have
(28) $\left[\prod\limits_{i=1,i\neq
j}^{t}(\zeta-\zeta_{i})^{k}\right]^{\prime}\Bigg{|}_{\zeta_{j}}=0,\quad 1\leq
j\leq t.$
This means that for every $1\leq j\leq t$,
$\sum\limits_{i=1\atop i\neq
j}^{t}(\zeta-\zeta_{j})^{k-1}\prod\limits_{\ell=1\atop\ell\neq
i,j}^{t}(\zeta-\zeta_{\ell})^{k}\Bigg{|}_{\zeta_{j}}=0.$
Dividing in $\prod\limits_{\ell\neq j}(\zeta_{j}-\zeta_{\ell})^{k-1}$ gives
$\sum\limits_{i=1\atop i\neq j}^{t}\prod\limits_{\ell=1\atop\ell\neq
i,j}^{t}(\zeta_{j}-\zeta_{\ell})=0.$
Thus $T^{\prime\prime}(\zeta_{j})=0$ for $1\leq j\leq t$, where
$T(\zeta)=\prod\limits_{i=1}^{t}(\zeta-\zeta_{i})$.
Now, if $t\geq 3$, then $T^{\prime\prime}$ is of degree $t-2$, and vanishes at
$t$ different points, a contradiction. If $t=2$, we get from (28) that
$\left[(\zeta-\zeta_{2})^{k}\right]^{\prime}\Bigg{|}_{\zeta_{1}}=0$ and this
is also a contradiction. So $t=1$ and $G$ has only one zero $\zeta_{0}\
(\zeta_{0}\neq 0)$, which means that $G(\zeta)=\frac{\displaystyle
b(0)\zeta_{0}^{\ell}(\zeta-\zeta_{0})^{k}}{\displaystyle k!}.$
By Hurwitz’s Theorem, there exists a sequence $\zeta_{n,0}\to\zeta_{0}$, such
that $G_{n}(\zeta_{n,0})=0$. If there exists $\delta^{\prime}$,
$0<\delta^{\prime}<\delta$, such that for every $n$ (after renumbering),
$f_{n}(z)$ has only one zero $z_{n,0}=\rho_{n}\zeta_{n,0}$ in
$\Delta(0,\delta^{\prime})$.
Set
$H_{n}(z)=\frac{f_{n}(z)}{(z-z_{n,0})^{k}}.$
Since $H_{n}(z)$ is a nonvanishing holomorphic function in
$\Delta(0,\delta^{\prime})$ and $H_{n}(z)\Rightarrow\infty$ on
$\Delta^{\prime}(0,\delta)$, we can deduce as before by the minimum principle
that $H_{n}(z)\Rightarrow\infty$ on $\Delta(0,\delta^{\prime})$. But
$H_{n}(2z_{n,0})=\frac{f_{n}(2z_{n,0})}{z^{k}_{n,0}}=\frac{\displaystyle\rho^{\ell}_{n}G_{n}(2\zeta_{n,0})}{\displaystyle\zeta^{k}_{n,0}}\to
0,$
a contradiction. Thus, we can assume, after renumbering, that for every
$\delta^{\prime}>0$, $f_{n}$ has at least two zeros in
$\Delta(0,\delta^{\prime})$ for large enough $n$. Thus, there exists another
sequence of points $z_{n,1}=\rho_{n}\zeta_{n,1}$, tending to zero, where
$z_{n,1}$ is also a zero of $f_{n}(z)$ and $\zeta_{n,1}\to\infty$, as
$n\to\infty$. We can also assume that $z_{n,1}$ is the closest zero to the
origin of $f_{n}$, except $z_{n,0}$. Now set $c_{n}=z_{n,0}/z_{n,1}$ and
define $K_{n}(\zeta)=f_{n}(z_{n,1}\zeta)/z^{k+\ell}_{n,1}$. By Lemma 7,
$\\{K_{n}(\zeta)\\}$ is normal in $\mathbb{C}^{\ast}$. Now, if $\\{K_{n}\\}$
is normal at $\zeta=0$, then after renumbering we can assume that
$K_{n}(\zeta)\Longrightarrow K(\zeta)\quad\text{on}\quad\mathbb{C}.$
If $K(\zeta)\not\equiv$ const., then consider
$L_{n}(\zeta):=\frac{K_{n}(\zeta)}{(\zeta-c_{n})^{k}}.$
Since $c_{n}\underset{n\to\infty}{\longrightarrow}0$, then the sequence
$\\{L_{n}\\}^{\infty}_{1}$ is normal in $\mathbb{C}^{\ast}$. It is also normal
at $\zeta=0$. Indeed, $K_{n}(c_{n})=0$ (a zero of order $k$) and so $L_{n}$ is
a nonvanishing holomorphic function in $\Delta(0,1)$. Thus (after renumbering)
$L_{n}(\zeta)\Longrightarrow\frac{K(\zeta)}{\zeta^{k}}\quad\text{on}\quad\mathbb{C}.$
But
$L_{n}(0)=\frac{K_{n}(0)}{(-c_{n})^{k}}=\frac{G_{n}(0)}{\zeta^{\ell}_{n,1}(-\zeta_{n,0})^{k}}\underset{n\to\infty}{\longrightarrow}0,\quad(\text{since}\quad\zeta_{n,1}\underset{n\to\infty}{\longrightarrow}\infty),$
and $L_{n}(\zeta)\neq 0$ in $\Delta(0,1/2)$; thus $K(\zeta)/\zeta^{k}\equiv 0$
in $\mathbb{C}$, a contradiction. If, on the other hand, $K(\zeta)\equiv$
const., then $K(\zeta)\equiv 0$ and $K^{(k)}(1)=0$. But
$K^{(k)}(1)=\lim\limits_{n\to\infty}K^{(k)}_{n}(1)=\lim\limits_{n\to\infty}\frac{\displaystyle
f^{(k)}_{n}(z_{n,1})}{\displaystyle
z_{n,1}^{\ell}}=\lim\limits_{n\to\infty}\frac{\displaystyle
h(z_{n,1})}{\displaystyle
z_{n,1}^{\ell}}=\lim\limits_{n\to\infty}b(z_{n,1})=b(0)$, a contradiction.
Hence we can deduce that $\\{K_{n}\\}$ is not normal at $\zeta=0$, and since
$K_{n}(\zeta)$ is holomorphic in $\Delta$, then
$K_{n}(\zeta)\Longrightarrow\infty\quad\text{on}\quad\mathbb{C}^{\ast}.$
But $K_{n}(1)=0$, a contradiction.
Case (BII) $G(\zeta)$ is a transcendental entire function.
Consider the family
$\mathcal{F}(G)=\left\\{t_{n}(z):=\frac{G(2^{n}z)}{2^{n(k+\ell)}}:n\in\mathbb{N}\right\\}.$
By Claim 4, we deduce
1. (i)
$t_{n}(z)=0\Longrightarrow t^{(k)}_{n}(z)=z^{\ell}$; and
2. (ii)
$t^{(k)}_{n}(z)=z^{\ell}\Longrightarrow t^{(k+1)}_{n}(z)=0$.
We then get by Theorem CFZ2 that $\mathcal{F}(G)$ is normal in
$\mathbb{C}^{\ast}$. Thus there exists $M>0$ such that for every $z\in
R_{1,2}:=\\{z:1\leq|z|\leq 2\\}$
$t^{\\#}_{n}(z)=\frac{2^{n(k+\ell+1)}|G^{\prime}(2^{n}z)|}{2^{2n(k+\ell)}+|G(2^{n}z)|^{2}}\leq
M.$
Set $r(\zeta):=G(\zeta)/\zeta^{k+\ell}$. Then $r$ is a transcendental
meromorphic function, whose only pole is $\zeta=0$. For every $\zeta$,
$|\zeta|\geq 2$ there exists $n\geq 1$ and $z\in R_{1,2}$, such that
(29) $\zeta=2^{n}z.$
Calculation gives
$r^{\sharp}(\zeta)=\frac{|G^{\prime}(\zeta)\zeta^{k+\ell}-(k+\ell)\zeta^{k+\ell-1}G(\zeta)|}{|\zeta|^{2(k+\ell)}+|G(\zeta)|^{2}}.$
Thus, if $|\zeta|\geq 2$ satisfies (29) then
(30) $\displaystyle|\zeta r^{\sharp}(\zeta)|$
$\displaystyle=|2^{n}z|\frac{|G^{\prime}(2^{n}z)(2^{n}z)^{k+\ell}-(k+\ell)(2^{n}z)^{k+\ell-1}G(2^{n}z)|}{|2^{n}z|^{2(k+\ell)}+|G(2^{n}z)|^{2}}$
$\displaystyle\leq\frac{2^{k+\ell+1}\cdot
2^{n(k+\ell+1)}|G^{\prime}(2^{n}z)|}{2^{2n(k+\ell)}+|G(2^{n}z)|^{2}}+\frac{(k+\ell)2^{(n+1)(k+\ell)}|G(2^{n}z)|}{2^{2n(k+\ell)}+|G(2^{n}z)|^{2}}.$
By separating into two cases, depending on $|G(2^{n}z)|>2^{(n+1)(k+\ell)}$ or
$|G(2^{n}z)|\leq 2^{(n+1)(k+\ell)}$, we see that the last expression in (30)
is less or equal to
$2^{k+\ell+1}t^{\sharp}_{n}(z)+(k+\ell)2^{2(k+\ell)}.$
Thus, to every $|\zeta|\geq 2$,
$|\zeta r^{\sharp}(\zeta)|\leq M\cdot 2^{k+\ell+1}+(k+\ell)2^{2(k+\ell)}.$
But, according to Theorem B,
$\varlimsup\limits_{\zeta\to\infty}|\zeta|r^{\sharp}(\zeta)=\infty$, and we
thus have a contradiction (cf. [3, pp. 19-21]). Theorem 2 is proved. $\square$
## 5\. Proof of Theorem 3
By Theorem CFZ3, $\mathcal{F}$ is normal at every point $z_{0}\in D$ at which
$h(z_{0})\neq 0$ (so that $\mathcal{F}$ is quasinormal in $D$). Consider
$z_{0}\in D$ such that $h(z_{0})=0$. Without loss of generality, we can assume
that $z_{0}=0$, and then $h(z)=z^{\ell}b(z)$, where $\ell(\geq 1)$ is an
integer, $b(z)\neq 0$ is an analytic function in $\Delta(0,\delta)$. We take a
subsequence $\\{f_{n}\\}^{\infty}_{1}\subset\mathcal{F}$, and we only need to
prove that $\\{f_{n}\\}$ is not normal at $z=0.$
Define $\mathcal{F}_{2}=\left\\{F=\frac{\displaystyle f_{n}}{\displaystyle
h}:n\in\mathbb{N}\right\\}.$ It is enough to prove that $\mathcal{F}_{2}$ is
normal in $\Delta(0,\delta).$ Suppose to the contrary that $\mathcal{F}_{2}$
is not normal at $z=0$. By Lemma 1 and the assumptions of Theorem 3, there
exist (after renumbering) points $z_{n}\to 0$, $\rho_{n}\to 0^{+}$ and a
nonconstant meromorphic function on $\mathbb{C}$, $g(\zeta)$ such that
(31)
$g_{n}(\zeta)=\frac{F_{n}(z_{n}+\rho_{n}\zeta)}{\rho^{2}_{n}}=\frac{f_{n}(z_{n}+\rho_{n}\zeta)}{\rho^{2}_{n}h(z_{n}+\rho_{n}\zeta)}\overset{\chi}{\Longrightarrow}g(\zeta)\quad\text{on}\quad\mathbb{C},$
all of whose zeros are multiple and
(32) $\text{for every}\quad\zeta\in\mathbb{C},\quad g^{\sharp}(\zeta)\leq
g^{\sharp}(0)=2A+1,$
where $A>1$ is a constant. After renumbering we can assume that
$\\{z_{n}/\rho_{n}\\}^{\infty}_{n=1}$ converges. We separate now into two
cases.
Case (A) $\frac{\displaystyle z_{n}}{\displaystyle\rho_{n}}\to\infty$.
Similar to the proof of Theorem 2, we can prove that
$g(\zeta)=0\Longrightarrow g^{\prime\prime}(\zeta)=1$ and that
$g^{\prime\prime}(\zeta)=1\Longrightarrow
g^{\prime\prime\prime}(\zeta)=g^{(s)}(\zeta)=0$. Then by Lemmas 4 and 3, we
have
$g(\zeta)=\frac{\displaystyle(\zeta-b)^{2}}{\displaystyle 2},$
for some $b\in\mathbb{C}$. Thus
$g^{\sharp}(0)=\frac{\displaystyle|b|}{\displaystyle 1+|b|^{4}/4}$ and then
$g^{\sharp}(0)\leq 1$, which contradicts (32).
Case (B)
(33) $\frac{z_{n}}{\rho_{n}}\to\alpha\in\mathbb{C}.$
As in the proof of Theorem 2, we have $g(\zeta_{0})=0\Longrightarrow
g^{\prime\prime}(\zeta_{0})=1$. Now set $G_{n}(\zeta)=\frac{\displaystyle
f_{n}(\rho_{n}\zeta)}{\displaystyle\rho^{2+\ell}_{n}}.$ From (31) and (33) we
have
$G_{n}(\zeta)\Longrightarrow
G(\zeta)=b(0)g(\zeta-\alpha)\zeta^{\ell}\quad\text{on}\quad\mathbb{C}.$
Since $g$ has a pole of order $\ell$ at $\zeta=-\alpha$, $G(0)\neq 0,\
\infty.$
We now consider several subcases, depending on the nature of $G$.
Case (BI) $G$ is a polynomial.
By a similar method of proof used in the proof of Theorem 2 (and using Lemma 8
instead of Lemma 7 in the appropriate places), we can get
$G(\zeta)=\frac{\displaystyle
b(0)\zeta_{0}^{\ell}(\zeta-\zeta_{0})^{2}}{\displaystyle 2},$
and also we can arrive at a contradiction.
Case (BII) $G(\zeta)$ is a transcendental entire function.
Consider the family
$\mathcal{F}(G)=\left\\{t_{n}(z):=\frac{G(2^{n}z)}{2^{n(2+\ell)}}:n\in\mathbb{N}\right\\}.$
We have
1. (i)
$t_{n}(z)=0\Longrightarrow t^{\prime\prime}_{n}(z)=z^{\ell}$; and
2. (ii)
$t^{\prime\prime}_{n}(z)=z^{\ell}\Longrightarrow
t^{\prime\prime\prime}_{n}(z)=t^{(s)}_{n}(z)=0$.
We then get by Theorem CFZ3 that $\mathcal{F}(G)$ is normal in
$\mathbb{C}^{\ast}$. Set $r(\zeta):=G(\zeta)/\zeta^{2+\ell}$, and we have
that, for every $\zeta$, $|\zeta|\geq 2,$ there exists $n\geq 1$ and $z\in
R_{1,2}$, such that
$|\zeta r^{\sharp}(\zeta)|\leq M\cdot 2^{2+\ell+1}+(2+\ell)2^{2(2+\ell)}.$
But, according to Theorem B,
$\varlimsup\limits_{\zeta\to\infty}|\zeta|r^{\sharp}(\zeta)=\infty$, and we
thus have a contradiction (cf. [3, pp. 19-21]). Theorem 3 is proved. $\square$
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* [14] L. Zalcman, Normal families: new perspectives, Bull. Amer. Math. Soc. (N.S.) 35 (1998), 215–230.
* [15] G.M. Zhang, W. Sun, and X.C. Pang, On the normality of certain kind of holomorphic functions, Chin. Ann. Math. Ser. A (6) 26 (2005), 765–770.
|
arxiv-papers
| 2011-11-06T07:09:59 |
2024-09-04T02:49:24.029615
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiaojun Liu and Shahar Nevo",
"submitter": "Shahar Nevo",
"url": "https://arxiv.org/abs/1111.1379"
}
|
1111.1383
|
# Gravitational wave in Lorentz violating gravity
Xin Li lixin@ihep.ac.cn Zhe Chang changz@ihep.ac.cn Institute of High Energy
Physics
and
Theoretical Physics Center for Science Facilities,
Chinese Academy of Sciences, 100049 Beijing, China
###### Abstract
By making use of the weak gravitational field approximation, we obtain a
linearized solution of the gravitational vacuum field equation in an
anisotropic spacetime. The plane-wave solution and dispersion relation of
gravitational wave is presented explicitly. There is possibility that the
speed of gravitational wave is larger than the speed of light and the
casuality still holds. We show that the energy-momentum of gravitational wave
in the ansiotropic spacetime is still well defined and conserved.
###### pacs:
04.50.Kd,04.30.-w,04.25.Nx
## I Introduction
Lorentz Invariance is one of the foundations of the Standard model of particle
physics. The constraints on possible Lorentz violating phenomenology are quite
severe, see for example, the summary tables that provided by Kostelecky et
al.Kostelecky1 . The gravitational interaction is far more weak, compare to
other fundamental interactions. This allows one to study the possible Lorentz
violating effects on certain gravity theories, such as Einstein-aether theory
Jacobson and Horava-Lifshitz theory Horava . One feature of Lorentz
invariance violation is that the speed of light differ from the one in special
relativity. The gravity theories with Lorentz violation could have the feature
that the speed of graviton or the speed of gravitational wave differ from the
one in general relativity. Studying the speed of gravitational wave in a
Lorentz violating gravity theory will give different perspective on quantum
gravitational phenomena.
One of the most important prediction of Einstein’s general relativity is
gravitational radiation. Many pioneer works Braginsky ; Thorne ; Weiss have
discussed the gravitational radiation in both theoretical properties and
experimental approaches of detections. Currently, the most sensitive
measurement is provided by ground-based Laser Interferometer Gravitational-
Wave Observatory (LIGO) detector LIGO . Another sensitive measurement, which
is in progress, is the Laser Interferometer Space Antenna (LISA) that detect
and accurately measure gravitational waves from astronomical sources. The
primordial gravitational waves Krauss ; Grishchuk could be of interest to
cosmologists as they provide a new and unique window on the earliest moments
in the history of the universe and on possible new physics at energies many
orders of magnitude beyond those accessible at particle accelerators.
In general relativity, the effects of gravitation are ascribed to spacetime
curvature instead of a force. However, up to now, general relativity still
faces problems.
First, the recent astronomical observations Riess found that our universe is
accelerated expanding. This result can not be obtained directly from
Einstein’s gravity and his cosmological principle. Since normal matters only
provide attractive force. The most widely adopted way to resolve it is
involving the so called dark energy which provides the repulsive force.
Second, the flat rotation curves of spiral galaxies violate the prediction of
Einstein’s gravity Zwicky . The most widely adopted way to resolve it is
involving the so called dark matter which provides enough attractive force
such that the discrepancy is restored.
The above astronomical phenomena occur at very large cosmological scale. The
following anomalies occur in solar system which imply the Newton’s inverse-
square law of universal gravitation and general relativity need modifications.
The third one, two Pioneer spacecrafts suffer an anomalous constant sunward
acceleration, $a_{p}=(8.74\pm 1.33)\times 10^{-10}{\rm m/s^{2}}$ Anderson .
The fourth one, it has been observed at various occasions that satellites
after an Earth swing-by possess a significant unexplained velocity increase by
a few mm/s Anderson1 .
The fifth one, from the analysis of radiometric measurements of distances
between the Earth and the major planets including observations from Martian
orbiters and landers from 1961 to 2003 a secular trend of the Astronomical
Unit of $15\pm 4$ m/cy has been reported Krasinsky .
The sixth one, a recent orbital analysis of Lunar Laser Ranging (LLR) Williams
shows an anomalous secular eccentricity variation of the Moon’s orbit
$\rm(0.9\pm 0.3)\times 10^{-11}/yr$.
All the facts imply that the Einstein’s theory should be modified. By
mimicking Einstein, we have proposed that the modified gravitational theory
should correspond to a new geometry which involves Riemann geometry as its
special case. Finsler geometry Book by Bao as a nature extension of Riemann
geometry is a good candidate to solve the problems mentioned above. A new
geometry (Finsler geometry) involves new spacetime symmetry. The Lorentz
violation is intimately linked to Finsler geometry. Kostelecky Kostelecky
have studied effective field theories with explicit Lorentz violation in
Finsler spacetime.
Finsler geometry really gives better description for the nature of gravity:
the flat rotation curves of spiral galaxies can be deduced naturally without
invoking dark matter Finsler DM ; a Finlerian gravity model could account for
the accelerated expanding university without invoking dark energy Finsler DE ;
a special Finsler space-Randers space Finsler PA could account for the
anomalous acceleration Anderson in solar system observed by Pioneer 10 and 11
spacecrafts; the Finsler spacetime could give a modification on the
gravitational deflection of light Finsler BL , which may account to these
observations without adding dark matter in Bullet Cluster Clowe ; the result
based on the kinematics with a special Finsler spacetime is in good agreement
with secular trend of the Astronomical Unit and secular eccentricity variation
of the Moon’s orbit Finsler AU .
It is interest to investigate the gravitational wave in Finsler spacetime. It
is well known that the gravitational wave propagates with the speed of light
in general relativity. This is due to the fact that the spacetime metric is
close to the Minkowski metric in the weak gravitational field approximation,
and the causal speed of Minkowski spacetime is just the speed of light.
However, in Finsler spacetime the causal speed is generally different with the
speed of light Pfeifer .
In this paper, we will present the solution of linearized gravitational vacuum
field equation in Finsler spacetime. It is shown that there is possibility
that the causal speed of it is larger than the speed of light.
## II Vacuum field equation in Finsler spacetime
Instead of defining an inner product structure over the tangent bundle in
Riemann geometry, Finsler geometry is based on the so called Finsler structure
$F$ with the property $F(x,\lambda y)=\lambda F(x,y)$ for all $\lambda>0$,
where $x$ represents position and $y\equiv\frac{dx}{d\tau}$ represents
velocity. The Finsler metric is given as Book by Bao
$g_{\mu\nu}\equiv\frac{\partial}{\partial y^{\mu}}\frac{\partial}{\partial
y^{\nu}}\left(\frac{1}{2}F^{2}\right).$ (1)
Finsler geometry has its genesis in integrals of the form
$\int^{r}_{s}F(x^{1},\cdots,x^{n};\frac{dx^{1}}{d\tau},\cdots,\frac{dx^{n}}{d\tau})d\tau~{}.$
(2)
The Finsler structure represents the length element of Finsler space.
The parallel transport has been studied in the framework of Cartan connection
Matsumoto ; Antonelli ; Szabo . The notation of parallel transport in Finsler
manifold means that the length $F\left(\frac{dx}{d\tau}\right)$ is constant.
The geodesic equation for Finsler manifold is given as Book by Bao
$\frac{d^{2}x^{\mu}}{d\tau^{2}}+2G^{\mu}=0,$ (3)
where
$G^{\mu}=\frac{1}{4}g^{\mu\nu}\left(\frac{\partial^{2}F^{2}}{\partial
x^{\lambda}\partial y^{\nu}}y^{\lambda}-\frac{\partial F^{2}}{\partial
x^{\nu}}\right)$ (4)
is called geodesic spray coefficient. Obviously, if $F$ is Riemannian metric,
then
$G^{\mu}=\frac{1}{2}\tilde{\gamma}^{\mu}_{\nu\lambda}y^{\nu}y^{\lambda},$ (5)
where $\tilde{\gamma}^{\mu}_{\nu\lambda}$ is the Riemannian Christoffel
symbol. Since the geodesic equation (3) is directly derived from the integral
length
$L=\int F\left(\frac{dx}{d\tau}\right)d\tau,$ (6)
the inner product
$\left(\sqrt{g_{\mu\nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}}=F\left(\frac{dx}{d\tau}\right)\right)$
of two parallel transported vectors is preserved.
In Finsler manifold, there exists a linear connection - the Chern connection
Chern . It is torsion freeness and almost metric-compatibility,
$\Gamma^{\alpha}_{\mu\nu}=\gamma^{\alpha}_{\mu\nu}-g^{\alpha\lambda}\left(A_{\lambda\mu\beta}\frac{N^{\beta}_{\nu}}{F}-A_{\mu\nu\beta}\frac{N^{\beta}_{\lambda}}{F}+A_{\nu\lambda\beta}\frac{N^{\beta}_{\mu}}{F}\right),$
(7)
where $\gamma^{\alpha}_{\mu\nu}$ is the formal Christoffel symbols of the
second kind with the same form of Riemannian connection, $N^{\mu}_{\nu}$ is
defined as
$N^{\mu}_{\nu}\equiv\gamma^{\mu}_{\nu\alpha}y^{\alpha}-A^{\mu}_{\nu\lambda}\gamma^{\lambda}_{\alpha\beta}y^{\alpha}y^{\beta}$
and $A_{\lambda\mu\nu}\equiv\frac{F}{4}\frac{\partial}{\partial
y^{\lambda}}\frac{\partial}{\partial y^{\mu}}\frac{\partial}{\partial
y^{\nu}}(F^{2})$ is the Cartan tensor (regarded as a measurement of deviation
from the Riemannian Manifold). In terms of Chern connection, the curvature of
Finsler space is given as
$R^{~{}\lambda}_{\kappa~{}\mu\nu}=\frac{\delta\Gamma^{\lambda}_{\kappa\nu}}{\delta
x^{\mu}}-\frac{\delta\Gamma^{\lambda}_{\kappa\mu}}{\delta
x^{\nu}}+\Gamma^{\lambda}_{\alpha\mu}\Gamma^{\alpha}_{\kappa\nu}-\Gamma^{\lambda}_{\alpha\nu}\Gamma^{\alpha}_{\kappa\mu},$
(8)
where $\frac{\delta}{\delta x^{\mu}}=\frac{\partial}{\partial
x^{\mu}}-N^{\nu}_{\mu}\frac{\partial}{\partial y^{\nu}}$.
The gravity in Finsler spacetime has been investigated for a long time Takano
; Ikeda ; Tavakol1 ; Bogoslovsky1 . In this paper, we introduce vacuum field
equation by the way discussed first by Pirani Pirani ; Rutz . In Newton’s
theory of gravity, the equation of motion of a test particle is given as
$\frac{d^{2}x^{i}}{dt^{2}}=-\eta^{ij}\frac{\partial\phi}{\partial x^{i}},$ (9)
where $\phi=\phi(x)$ is the gravitational potential and $\eta^{ij}$ is
Euclidean metric. For an infinitesimal transformation $x^{i}\rightarrow
x^{i}+\epsilon\xi^{i}$($|\epsilon|\ll 1$), the equation (9) becomes, up to
first order in $\epsilon$,
$\frac{d^{2}x^{i}}{dt^{2}}+\epsilon\frac{d^{2}\xi^{i}}{dt^{2}}=-\eta^{ij}\frac{\partial\phi}{\partial
x^{i}}-\epsilon\eta^{ij}\xi^{k}\frac{\partial^{2}\phi}{\partial x^{j}\partial
x^{k}}.$ (10)
Combining the above equations(9) and (10), we obtain
$\frac{d^{2}\xi^{i}}{dt^{2}}=\eta^{ij}\xi^{k}\frac{\partial^{2}\phi}{\partial
x^{j}\partial x^{k}}\equiv\xi^{k}H^{i}_{k}.$ (11)
In Newton’s theory of gravity, the vacuum field equation is given as
$H^{i}_{i}=\bigtriangledown^{2}\phi=0$. It means that the tensor $H^{i}_{k}$
is traceless in Newton’s vacuum.
In general relativity, the geodesic deviation gives similar equation
$\frac{D^{2}\xi^{\mu}}{D\tau^{2}}=\xi^{\nu}\tilde{R}^{\mu}_{~{}\nu},$ (12)
where
$\tilde{R}^{\mu}_{~{}\nu}=\tilde{R}^{~{}\mu}_{\lambda~{}\nu\rho}\frac{dx^{\lambda}}{d\tau}\frac{dx^{\rho}}{d\tau}$.
Here, $\tilde{R}^{~{}\mu}_{\lambda~{}\nu\rho}$ is Riemannian curvature tensor,
$D$ denotes the covariant derivative alone the curve $x^{\mu}(\tau)$. The
vacuum field equation in general relativity gives
$\tilde{R}^{~{}\lambda}_{\mu~{}\lambda\nu}=0$Weinberg . It implies that the
tensor $\tilde{R}^{\mu}_{~{}\nu}$ is also traceless,
$\tilde{R}\equiv\tilde{R}^{\mu}_{~{}\mu}=0$.
In Finsler spacetime, the geodesic deviation gives Book by Bao
$\frac{D^{2}\xi^{\mu}}{D\tau^{2}}=\xi^{\nu}R^{\mu}_{~{}\nu},$ (13)
where
$R^{\mu}_{~{}\nu}=R^{~{}\mu}_{\lambda~{}\nu\rho}\frac{dx^{\lambda}}{d\tau}\frac{dx^{\rho}}{d\tau}$.
Here, $R^{~{}\mu}_{\lambda~{}\nu\rho}$ is Finsler curvature tensor defined in
(8), $D$ denotes covariant derivative
$\frac{D\xi^{\mu}}{D\tau}=\frac{d\xi^{\mu}}{d\tau}+\xi^{\nu}\frac{dx^{\lambda}}{d\tau}\Gamma^{\mu}_{\nu\lambda}(x,\frac{dx}{d\tau})$.
Since the vacuum field equations of Newton’s gravity and general relativity
have similar form, we may assume that vacuum field equation in Finsler
spacetime hold similar requirement as the case of Netwon’s gravity and general
relativity. It implies that the tensor $R^{\mu}_{~{}\nu}$ in Finsler geodesic
deviation equation should be traceless, $R\equiv R^{\mu}_{~{}\mu}=0$. We have
proved that the analogy from the geodesic deviation equation is valid at least
in Finsler spacetime of Berwald type Finsler DM . For this reason, we may
suppose that this analogy is valid in general Finsler spacetime.
It should be noticed that $H$ is called the Ricci scaler, which is a
geometrical invariant. For a tangent plane $\Pi\subset T_{x}M$ and a non-zero
vector $y\in T_{x}M$, the flag curvature is defined as
$K(\Pi,y)\equiv\frac{g_{\lambda\mu}R^{\mu}_{~{}\nu}u^{\nu}u^{\lambda}}{F^{2}g_{\rho\theta}u^{\rho}u^{\theta}-(g_{\sigma\kappa}y^{\sigma}u^{\kappa})^{2}},$
(14)
where $u\in\Pi$. The flag curvature is a geometrical invariant that
generalizes the sectional curvature of Riemannian geometry. It is clear that
the Ricci scaler $R$ is the trace of $R^{\mu}_{~{}\nu}$, which is the
predecessor of flag curvature. Therefore, the value of Ricci scaler $R$ is
invariant under the ordinate transformation. Furthermore, the predecessor of
flag curvature could be written in terms of the geodesic spray coefficient
$R^{\mu}_{~{}\nu}=2\frac{\partial G^{\mu}}{\partial
x^{\nu}}-y^{\lambda}\frac{\partial^{2}G^{\mu}}{\partial x^{\lambda}\partial
y^{\nu}}+2G^{\lambda}\frac{\partial^{2}G^{\mu}}{\partial y^{\lambda}\partial
y^{\nu}}-\frac{\partial G^{\mu}}{\partial y^{\lambda}}\frac{\partial
G^{\lambda}}{\partial y^{\nu}}.$ (15)
Thus, the Ricci scaler $R$ is insensitive to connection that one is using, it
only depends on the length element $F$. The gravitational vacuum field
equation $R=0$ is universal in any types of theories of Finsler gravity.
Pfeifer et al. Pfeifer1 have constructed gravitational dynamics for Finsler
spacetimes in terms of an action integral on the unit tangent bundle. Their
researches also show that the gravitational vacuum field equation in Finsler
spacetime is $R=0$.
## III Gravitational wave in Finslerian vacuum
It is hard to find a non trivial solution of the gravitational vacuum field
equation ($R=0$) in Finsler spacetime. Here, we study the weak field radiative
solution of the Finslerian vacuum field equation $R=0$. It is well known that
the Minkowski spacetime is a trivial solution of Einstein’s vacuum field
equation. In the Finsler spacetime, the trivial solution of Finslerian vacuum
field equation is called locally Minkowski spacetime. A Finsler spacetime is
called a locally Minkowshi spacetime if there is a local coordinate system
$(x^{\mu})$, with induced tangent space coordinates $y^{\mu}$, such that $F$
depends only on $y$ and not on $x$. Using the formula (15), one knows obvious
that locally Minkowski spacetime is a solution of Finslerian vacuum field
equation.
We suppose that the metric is close to the locally Minkowski metric
$\eta_{\mu\nu}(y)$,
$g_{\mu\nu}=\eta_{\mu\nu}(y)+h_{\mu\nu}(x,y),$ (16)
where $|h_{\mu\nu}|\ll 1$. In the rest of this section, the lowering and
raising of indices are carried out by $\eta_{\mu\nu}$ and its matrix inverse
$\eta^{\mu\nu}$. To first order in $h$, the geodesic spray coefficient is
$G^{\mu}=\frac{1}{4}\eta^{\mu\nu}\left(2\frac{\partial h_{\alpha\nu}}{\partial
x^{\lambda}}y^{\alpha}y^{\lambda}-\frac{\partial h_{\alpha\beta}}{\partial
x^{\nu}}y^{\alpha}y^{\beta}\right).$ (17)
We have already used the Euler’s theorem for homogeneous functions to obtain
the above equation. And the Ricci scaler is
$\displaystyle R=R^{\mu}_{~{}\mu}$ $\displaystyle=$ $\displaystyle
2\frac{\partial G^{\mu}}{\partial
x^{\mu}}-y^{\theta}\frac{\partial^{2}G^{\mu}}{\partial x^{\theta}\partial
y^{\mu}},$ (18)
where
$2\frac{\partial G^{\mu}}{\partial
x^{\mu}}=\frac{1}{2}\eta^{\mu\nu}\left(2\frac{\partial^{2}h_{\alpha\nu}}{\partial
x^{\lambda}\partial
x^{\mu}}y^{\alpha}y^{\lambda}-\frac{\partial^{2}h_{\alpha\beta}}{\partial
x^{\mu}\partial x^{\nu}}y^{\alpha}y^{\beta}\right)$ (19)
and
$\displaystyle-y^{\theta}\frac{\partial^{2}G^{\mu}}{\partial
x^{\theta}\partial
y^{\mu}}=-\frac{y^{\theta}}{4}\eta^{\mu\nu}\frac{\partial}{\partial
x^{\theta}}\left(2\frac{\partial h_{\mu\nu}}{\partial
x^{\lambda}}y^{\lambda}+2\frac{\partial h_{\alpha\nu}}{\partial
x^{\mu}}y^{\alpha}-2\frac{\partial h_{\alpha\mu}}{\partial
x^{\nu}}y^{\alpha}\right)-\frac{y^{\theta}}{4}\frac{\partial\eta^{\mu\nu}}{\partial
y^{\mu}}\frac{\partial}{\partial x^{\theta}}\bigg{(}2\frac{\partial
h_{\alpha\nu}}{\partial x^{\lambda}}y^{\alpha}y^{\lambda}-\frac{\partial
h_{\alpha\beta}}{\partial x^{\nu}}y^{\alpha}y^{\beta}\bigg{)}.$ (20)
Since the value of Ricci scaler $R$ is invariant under the coordinate
transformation, we must fix the gauge symmetry to yield unique solution. Under
a coordinate transformation
$\bar{x}^{\mu}=x^{\mu}+\epsilon^{\mu}(x),$ (21)
the metric $h_{\mu\nu}$ transforms as
$\bar{h}^{\mu\nu}=h^{\mu\nu}-\frac{\partial\epsilon^{\mu}}{\partial
x^{\lambda}}\eta^{\lambda\nu}-\frac{\partial\epsilon^{\nu}}{\partial
x^{\lambda}}\eta^{\lambda\mu}.$ (22)
By performing the coordinate transformation with
$\eta^{\mu\lambda}\frac{\partial^{2}\epsilon_{\nu}}{\partial x^{\mu}\partial
x^{\lambda}}=\frac{\partial h^{\mu}_{~{}\nu}}{\partial
x^{\mu}}-\frac{1}{2}\frac{\partial h^{\mu}_{~{}\mu}}{\partial x^{\nu}},$ (23)
we find that $\bar{h}_{\mu\nu}$ satisfies
$\frac{\partial\bar{h}^{\mu}_{~{}\nu}}{\partial
x^{\mu}}=\frac{1}{2}\frac{\partial\bar{h}^{\mu}_{~{}\mu}}{\partial x^{\nu}}.$
(24)
This choice of gauge (24) has the same form with the Lorentz gauge in general
relativity, due to the fact that the locally Minkowshi metric $\eta_{\mu\nu}$
does not depend on $x$.
By making use of the Finslerian gauge (24), and noticing that $\eta_{\mu\nu}$
does not depend on $x$, we rewrite the Ricci scaler as
$\displaystyle
R=-\frac{\eta^{\mu\nu}}{2}\frac{\partial^{2}h_{\alpha\beta}}{\partial
x^{\mu}\partial
x^{\nu}}y^{\alpha}y^{\beta}+\frac{1}{4}\frac{\partial\eta^{\mu\nu}}{\partial
y^{\mu}}\frac{\partial^{2}h_{\alpha\beta}}{\partial x^{\nu}\partial
x^{\lambda}}y^{\lambda}y^{\alpha}y^{\beta}.$ (25)
We find from (25) that the solution of $R=0$ has following properties
$\displaystyle
h_{\mu\nu}(x,y)=e_{\mu\nu}\exp(ik_{\lambda}x^{\lambda})+h.c.~{}~{}~{},$ (26)
where
$k_{\mu}k_{\nu}\eta^{\mu\nu}-\frac{1}{2}\frac{\partial\eta^{\mu\nu}}{\partial
y^{\mu}}k_{\nu}k_{\lambda}y^{\lambda}=0,$ (27)
$k=k(y)$ is function of $y$ and $e_{\mu\nu}$ is the polarization tensor. The
term $\frac{\partial\eta^{\mu\nu}}{\partial y^{\mu}}$ could be written as
$\frac{\partial\eta^{\mu\nu}}{\partial
y^{\mu}}=-2A^{~{}\mu\nu}_{\mu}/\tilde{F}=-\eta^{\nu\lambda}\frac{\partial\ln{\rm|det}(\eta)|}{\partial
y^{\lambda}},$ (28)
where $\tilde{F}^{2}=\eta_{\mu\nu}y^{\mu}y^{\nu}$. Substituting the equation
(28) into (27), we obtain
$k_{\mu}k_{\nu}\eta^{\mu\nu}=-\eta^{\nu\lambda}\frac{\partial\ln\sqrt{{\rm|det}(\eta)|}}{\partial
y^{\lambda}}k_{\nu}k_{\mu}y^{\mu}~{}.$ (29)
It is obvious that $k_{\mu}k_{\nu}\eta^{\mu\nu}\neq 0$ while the Finsler
spacetime $\eta_{\mu\nu}$ is not Minkowskian. It implies that the wave vectors
$k_{\mu}$ of gravitational waves is not null in Finsler spacetime
$\eta_{\mu\nu}$.
The Randers spacetime Randers is a special kind of Finsler geometry with
Finsler structure
$\tilde{F}(x,y)\equiv\alpha+\beta,$ (30)
where
$\displaystyle\alpha$ $\displaystyle\equiv$
$\displaystyle\sqrt{\bar{a}_{\mu\nu}y^{\mu}y^{\nu}},$ (31)
$\displaystyle\beta$ $\displaystyle\equiv$
$\displaystyle\bar{b}_{\mu}y^{\mu},$ (32)
and $\bar{a}_{\mu\nu}$ is Riemannian metric. The indices on certain objects
that decorated with a bar are lowered and raised by $\bar{a}_{\mu\nu}$ and its
matrix inverse $\bar{a}^{\mu\nu}$. Substituting the Randers-Finsler structure
$\tilde{F}$ into the dispersion relation of gravitational wave (29) and
supposing the Randers spacetime is very close to Minkowski spacetime
$\bar{a}_{\mu\nu}$, to first order in $\bar{b}$, we obtain
$\displaystyle k_{\mu}k_{\nu}\eta^{\mu\nu}$ $\displaystyle=$
$\displaystyle-\frac{5(k\cdot\bar{l})}{2}\left((k\cdot\bar{b})-\frac{\beta}{\alpha}(k\cdot\bar{l})\right),$
(33) $\displaystyle k\cdot k$ $\displaystyle=$
$\displaystyle-\frac{(k\cdot\bar{l})}{2}\left((k\cdot\bar{b})-\frac{3\beta}{\alpha}(k\cdot\bar{l})\right),$
(34)
where $`\cdot^{\prime}$ denotes the inner product on Minkowski spacetime
$\bar{a}_{\mu\nu}$ and $\bar{l}^{\mu}\equiv y^{\mu}/\alpha$. The causality
should holds in Finsler spacetime $\eta_{\mu\nu}$, thus
$k_{\mu}k_{\nu}\eta^{\mu\nu}>0$ while the signature of Minkowski metric
$\bar{a}_{\mu\nu}$ is of the form $(+~{}-~{}-~{}-)$. If $k\cdot k<0$, it means
that the speed of gravitational wave is larger than speed of light. It implies
that the speed of gravitational wave could larger than speed of light and
causality still holds. The inequalities $k_{\mu}k_{\nu}\eta^{\mu\nu}>0$ and
$k\cdot k<0$ satisfy if
$\frac{3\beta}{\alpha}<\frac{k\cdot\bar{b}}{k\cdot\bar{l}}<\frac{\beta}{\alpha}<0,$
(35)
so that the speed of gravitational wave in the anisotropic spacetime is larger
than the speed of light and the causality still holds.
The sketch figure of the causal structure of Finsler spacetime
($\eta_{\mu\nu}$) is shown in Fig.1. It is clear from Fig.1 that the null
vectors on Finsler spacetime ($\eta_{\mu\nu}$) are spacelike vectors on
Minkowski spacetime. The causal speed of Finsler spacetime could be larger
than the one of Minkowski spacetime.
Figure 1: The solid line denotes the null structure on Finsler spacetime
($\eta_{\mu\nu}$). The dot line denotes the null structure on Minkowski
spacetime. It is obvious that the null structure of Finsler spacetime is
larger than the one in Minkowski spacetime
## IV Conclusions
In this paper, we used the weak gravitational field approximation to get a
linearized solution of the gravitational vacuum field equation in Finsler
spacetime. The plane-waves solution (26) of gravitational wave in an
anisotropic spacetime was presented. It is shown that the gravitational wave
is propagating in locally Minkowski spacetime ($\eta_{\mu\nu}$). The Killing
vectors of locally Minkowski spacetime ($\eta_{\mu\nu}$) are investigated in
Ref.Finsler PF . It was shown that Finsler spacetime admits less symmetry than
Minkowski spacetime, and the translation symmetry in preserved in locally
Minkowski spacetime ($\eta_{\mu\nu}$). Based on the Noether theorem, the
spacetime translational invariance implies that the energy-momentum is well
defined and conserved in locally Minkowski spacetime ($\eta_{\mu\nu}$). The
dispersion relation of gravitational wave in Finsler spacetime (29) was
presented. The speed of gravitational wave could larger than the speed of
light in Randers spacetime and the casuality of gravitational wave still
holds, if the condition (35) is satisfied. Since the wave vector $k_{\mu}$ of
gravitational wave is timelike in locally Minkowski spacetime
($\eta_{\mu\nu}$), it would not lose energy via the gravitational Cherenkov
radiation.
###### Acknowledgements.
We would like to thank M. H. Li and S. Wang for useful discussions. One of us
X. Li thanks Prof. C. Pfeifer for usefol discussions. The work was supported
by the NSF of China under Grant No. 10875129 and 11075166 and 11147176.
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|
arxiv-papers
| 2011-11-06T08:02:54 |
2024-09-04T02:49:24.038224
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xin Li and Zhe Chang",
"submitter": "Xin Li",
"url": "https://arxiv.org/abs/1111.1383"
}
|
1111.1384
|
# On Riemann’s Theorem About
Conditionally Convergent Series
Jürgen Grahl Department of Mathematics, University of Würzburg, Würzburg,
Germany grahl@mathematik.uni-wuerzburg.de and Shahar Nevo Department of
Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel
nevosh@macs.biu.ac.il
###### Abstract.
We extend Riemann’s rearrangement theorem on conditionally convergent series
of real numbers to multiple instead of simple sums.
###### Key words and phrases:
Conditionally convergent series, Fubini’s theorem, symmetric group.
###### 2010 Mathematics Subject Classification:
40A05
This research is part of the European Science Foundation Networking Programme
HCAA and was supported by Israel Science Foundation Grant 395/07.
## 1\. Introduction and statement of results
By a well-known theorem due to B. Riemann, each conditionally convergent
series of real numbers can be rearranged in such a way that the new series
converges to some arbitrarily given real value or to $\infty$ or $-\infty$
(see, for example, [1, § 32]). As to series of vectors in ${\mathbb{R}}^{n}$,
in 1905 P. Lévy [2] and in 1913 E. Steinitz [5] showed the following
interesting extension (see also [3] for a simplified proof).
###### Theorem A.
(Lévy-Steinitz Theorem) The set of all sums of rearrangements of a given
series of vectors in ${\mathbb{R}}^{n}$ is either the empty set or a translate
of a linear subspace (i.e., a set of the form $v+M$ where $v$ is a given
vector and $M$ is a linear subspace).
Here, of course, $M$ is the zero space if and only if the series is absolutely
convergent. For a further generalization of the Lévy-Steinitz theorem to
spaces of infinite dimension, see [4].
In this paper, we extend Riemann’s result in a different direction, turning
from simple to multiple sums which provides many more possibilities of
rearranging a given sum. First of all, we have to introduce some notations.
By ${\rm Sym}\,(n)$ we denote the symmetric group of the set
$\left\\{1,\dots,n\right\\}$, i.e., the group of all permutations of
$\left\\{1,\dots,n\right\\}$.
If $(a_{m})_{m}$ is a sequence of elements of a non-empty set $X$, $J$ is an
infinite subset of ${\rm I\\!N}^{n}$ and if $\tau:J\longrightarrow{\rm I\\!N}$
is a bijection and
$b(j_{1},\dots,j_{n}):=a_{\tau(j_{1},\dots,j_{n})}\qquad\mbox{ for each
}(j_{1},\dots,j_{n})\in J,$
then we say that the mapping $b:J\longrightarrow
X,\quad(j_{1},\dots,j_{n})\mapsto b(j_{1},\dots,j_{n})$ is a rearrangement of
$(a_{m})_{m}$. We write
$\left(b(j_{1},\dots,j_{n})\;\Bigm{|}\;(j_{1},\dots,j_{n})\in J\right)$
for such a rearrangement (which is a more convenient notation for our purposes
than the notation $\left(b_{j_{1},\dots,j_{n}}\right)_{(j_{1},\dots,j_{n})\in
J}$ one would probably expect). Instead of
$\big{(}b(j_{1},\dots,j_{n})\;\big{|}\;(j_{1},\dots,j_{n})\in{\rm
I\\!N}^{n}\big{)}$, we also write
$\big{(}b(j_{1},\dots,j_{n})\;\big{|}\;j_{1},\dots,j_{n}\geq 1\big{)}$ and
also use notations like $\big{(}b(j_{1},\dots,j_{n})\;\big{|}\;j_{1}\geq
k_{1},\dots,j_{n}\geq k_{n}\big{)}$ which should be self-explanatory now.
With these notations, we can state our main result as follows.
###### Theorem 1.
Let $n\geq 2$ be a natural number and let $\sum_{m=1}^{\infty}a_{m}$ be a
conditionally convergent series of real numbers $a_{m}$. For each
$\sigma\in{\rm Sym}\,(n)$, let $\big{(}s_{k}^{(\sigma)}\big{)}_{k\geq 1}$ be a
sequence of real numbers. Then there exists a rearrangement
$\left(b(j_{1},\dots,j_{n})\;|\;j_{1},\dots,j_{n}\geq 1\right)$ of
$(a_{m})_{m}$ such that for each $\sigma\in{\rm Sym}\,(n)$ and each $k\geq 1$,
one has
$\sum_{j_{1}=1}^{k}\sum_{j_{2}=1}^{\infty}\dots\sum_{j_{n}=1}^{\infty}b(j_{\sigma(1)},\dots,j_{\sigma(n)})=s_{k}^{(\sigma)}.$
###### Corollary 2.
Let $n\geq 1$ be a natural number and let $\sum_{m=1}^{\infty}a_{m}$ be a
conditionally convergent series of real numbers $a_{m}$. For each
$\sigma\in{\rm Sym}\,(n)$, let $s^{(\sigma)}$ be a real number or $\pm\infty$.
Then there exists a rearrangement
$\left(b(j_{1},\dots,j_{n})\;|\;j_{1},\dots,j_{n}\geq 1\right)$ of
$(a_{m})_{m}$ such that for each $\sigma\in{\rm Sym}\,(n)$, one has
$\sum_{j_{1}=1}^{\infty}\sum_{j_{2}=1}^{\infty}\dots\sum_{j_{n}=1}^{\infty}b(j_{\sigma(1)},\dots,j_{\sigma(n)})=s^{(\sigma)}.$
###### Proof.
For $n=1$, this is just Riemann’s theorem. For $n\geq 2$, it is an immediate
consequence of Theorem 1. ∎
By moving to continuous functions on ${\mathbb{R}}^{n}$, we can construct an
example of a continuous function in the “positive part” $Q:=[0,\infty)^{n}$ of
${\mathbb{R}}^{n}$ whose iterated integrals exist for each order of
integration, but all of them have different values. This is a kind of
“ultimate” counterexample to show that the assumptions in Fubini’s theorem are
inevitable.
###### Corollary 3.
Let $n\geq 2$ be a natural number. For each $\sigma\in{\rm Sym}\,(n)$, let
$s^{(\sigma)}$ be a real number or $\pm\infty$. Then there exists a function
$f\in C^{\infty}(Q)$ such that
$\int_{0}^{\infty}\dots\int_{0}^{\infty}f(x_{1},\dots,x_{n})\;dx_{\sigma(1)}\;dx_{\sigma(2)}\dots\;dx_{\sigma(n)}=s^{(\sigma)}\quad\mbox{
for each }\sigma\in{\rm Sym}\,(n).$ (1.1)
###### Proof.
Let $\sum_{m=1}^{\infty}a_{m}$ be some conditionally convergent series. By
Corollary 2, there exists a rearrangement
$\left(b(j_{1},\dots,j_{n})\;|\;j_{1},\dots,j_{n}\geq 1\right)$ of
$(a_{m})_{m}$ such that for each $\sigma\in{\rm Sym}\,(n)$, one has
$\sum_{k_{n}=1}^{\infty}\dots\sum_{k_{1}=1}^{\infty}b(k_{\sigma^{-1}(1)},\dots,k_{\sigma^{-1}(n)})=s^{(\sigma)}.$
We set $I=[-0.49\,;\,0.49]^{n}$ and define the function
$\varphi:{\mathbb{R}}^{n}\longrightarrow{\mathbb{R}}$ by
$\varphi(x):=\left\\{\begin{array}[]{rl}Ae^{-1/(0.49-||x||)^{2}}&\mbox{ for
}||x||<0.49\\\ 0&\mbox{ for }||x||\geq 0.49,\end{array}\right.$
where $A>0$ and $||.||$ is the Euclidean norm on ${\mathbb{R}}^{n}$. Then
$\varphi\in C^{\infty}({\mathbb{R}}^{n})$, and $\varphi$ vanishes outside the
compact set $I$. So $\varphi$ is integrable with respect to the Lebesgue
measure $\lambda$, and by choosing an appropriate $A$ we can obtain
$\int_{{\mathbb{R}}^{n}}\varphi(x)\;d\lambda(x)=1.$
In particular, by Fubini’s theorem the last integral can be written in any
order of integration, i.e.
$\int_{-\infty}^{\infty}\dots\int_{-\infty}^{\infty}\varphi(x_{1},\dots,x_{n})\;dx_{\sigma(1)}\;dx_{\sigma(2)}\dots\;dx_{\sigma(n)}=1\qquad\mbox{
for each }\sigma\in{\rm Sym}\,(n).$
Since $\varphi$ vanishes outside $I$, for any $j_{1},\dots,j_{n}\geq 1$, we
also have
$\int_{-j_{n}}^{\infty}\dots\int_{-j_{1}}^{\infty}\varphi(x_{1},\dots,x_{n})\;dx_{\sigma(1)}\;dx_{\sigma(2)}\dots\;dx_{\sigma(n)}=1\qquad\mbox{
for each }\sigma\in{\rm Sym}\,(n).$ (1.2)
Now we define $f:Q\longrightarrow{\mathbb{R}}$ by
$f(x_{1},\dots,x_{n}):=\sum_{j_{1}=1}^{\infty}\sum_{j_{2}=1}^{\infty}\dots\sum_{j_{n}=1}^{\infty}b(j_{1},\dots,j_{n})\cdot\varphi(x_{1}-j_{1},\dots,x_{n}-j_{n}).$
For each $x=(x_{1},\dots,x_{n})\in Q$, at most one of the terms
$\varphi(x_{1}-j_{1},\dots,x_{n}-j_{n})$ is non-zero, so the multiple sum in
the definition of $f$ reduces to just one term, and we conclude that $f\in
C^{\infty}(Q)$. Let $\sigma\in{\rm Sym}\,(n)$ be given. Then we obtain by
(1.2)
$\displaystyle\int_{0}^{\infty}\dots\int_{0}^{\infty}f(x_{1},\dots,x_{n})\;dx_{\sigma(1)}\;dx_{\sigma(2)}\dots\;dx_{\sigma(n)}$
$\displaystyle\qquad=\sum_{j_{\sigma(n)}=1}^{\infty}\dots\sum_{j_{\sigma(1)}=1}^{\infty}b(j_{1},\dots,j_{n})\int_{0}^{\infty}\dots\int_{0}^{\infty}\varphi(x_{1}-j_{1},\dots,x_{n}-j_{n})\;dx_{\sigma(1)}\dots\;dx_{\sigma(n)}$
$\displaystyle\qquad=\sum_{k_{n}=1}^{\infty}\dots\sum_{k_{1}=1}^{\infty}b(k_{\sigma^{-1}(1)},\dots,k_{\sigma^{-1}(n)})\cdot
1$ $\displaystyle\qquad=s^{(\sigma)},$
hence (1.1). ∎
Observe that in the case $n=2$, by Corollary 3 we get the existence of a
function $f\in C^{\infty}([0,\infty)^{2})$ such that
$\int_{0}^{\infty}\int_{0}^{\infty}f(x,y)\;dx\,dy=+\infty\qquad\mbox{ and
}\qquad\int_{0}^{\infty}\int_{0}^{\infty}f(x,y)\;dy\,dx=-\infty.$
For the functions $f$ from Corollary 3, in general, the improper integral
$\int_{Q}f(x_{1},\dots,x_{n})d(x_{1},\dots,x_{n})$ (in the sense of Riemann)
does not exist in the extended sense111We say that the improper integral
$\int_{Q}f(x_{1},\dots,x_{n})\;d(x_{1},\dots,x_{n})$ exists in the extended
sense if for arbitrary exhaustions $(K_{m})_{m}$ of $Q$ with compact sets
$K_{m}$, the limits
$\lim_{m\to\infty}\int_{K_{m}}f(x_{1},\dots,x_{n})\;d(x_{1},\dots,x_{n})$
exist and are equal.. A necessary condition for the existence of this integral
is that $s^{(\sigma)}=s^{(\tau)}$ for every $\sigma,\tau\in{\rm Sym}\,(n)$.
However, it can be shown that this condition is not sufficient for the
convergence of the improper integral.
It is obvious that, by modifying the definition of $f$ (such that its “peaks”
are at the points $\left(\frac{1}{2^{j_{1}}},\dots\frac{1}{2^{j_{n}}}\right)$
rather than at the points $(j_{1},\dots,j_{n})$), one can replace $Q$ by
$(0,1]^{n}$ in Corollary 3, i.e., we can find a function $f\in
C^{\infty}((0,1]^{n})$ whose iterated integrals exist for every order of
integration, but each time give different values. Of course, this is not
possible for the compact cube $[0,1]^{n}$, since continuous functions on
compact sets are Lebesgue-integrable, so by Fubini’s Theorem their integrals
are independent of the order of integration.
## 2\. Proofs
It is well known that a convergent series $\sum_{m=1}^{\infty}a_{m}$ of real
numbers is conditionally convergent if and only if
$\sum_{a_{m}>0}a_{m}=\infty\qquad\mbox{ and
}\qquad\sum_{a_{m}<0}a_{m}=-\infty.$ (2.1)
This property is a bit more general than the property of conditional
convergence: It may also hold for series which are not convergent themselves.
It turns out that this is the property we actually deal with in the proof of
our main result. This gives rise to the following definition.
###### Definition.
We say that a series $\sum_{m=1}^{\infty}a_{m}$ of real numbers is
conditionally convergable if $\lim_{m\to\infty}a_{m}=0$ and if (2.1) holds.
As the proof of Riemann’s theorem shows, a series is conditionally convergable
if and only if it has some rearrangement which is conditionally convergent.
The main advantage of this newly introduced notion is the following:
Conditional convergability is invariant under rearrangements while conditional
convergence is not.
###### Lemma 4.
Let $\sum_{m=1}^{\infty}a_{m}$ be a conditionally convergable series of real
numbers $a_{m}$. Then there is a disjoint partition ${\rm
I\\!N}=\bigcup_{t=1}^{\infty}I_{t}$ of ${\rm I\\!N}$ into infinite subsets
$I_{t}$ such that for each $t\in{\rm I\\!N}$ the series $\sum_{m\in
I_{t}}a_{m}$ is conditionally convergable222In notations like $\sum_{j\in
I_{t}}a_{j}$, the order of summation is of course understood to be in the
natural order of increasing indices $j$. On the other hand, since conditional
convergability is invariant under rearrangements, we do not have to specify
the order of summation at all, at least not for the purpose of Lemma 4..
###### Proof.
I. Let $(\beta_{m})_{m}$ be a sequence of non-negative numbers such that
$\sum_{m=1}^{\infty}\beta_{m}=\infty.$
Then it is evident that one can decompose ${\rm I\\!N}$ into two infinite
disjoint subsets $I_{1},I^{(2)}$ such that $1\in I_{1}$ and
$\sum_{m\in I_{1}}\beta_{m}=\infty\qquad\mbox{ and }\qquad\sum_{m\in
I^{(2)}}\beta_{m}=\infty.$
Let us assume that we have already found subsets
$I_{1},\dots,I_{t},I^{(t+1)}\subseteq{\rm I\\!N}$ such that ${\rm
I\\!N}=I_{1}\cup\dots\cup I_{t}\cup I^{(t+1)}$ is a disjoint union,
$\sum_{m\in I_{s}}\beta_{m}=\infty\quad(s=1,\dots,t)\qquad\mbox{ and
}\qquad\sum_{m\in I^{(t+1)}}\beta_{m}=\infty$
and such that $\min({\rm I\\!N}\setminus(I_{1}\cup\dots\cup I_{s-1}))\in
I_{s}$ for $s=1,\dots,t$. Then we can find a disjoint decomposition
$I^{(t+1)}=I_{t+1}\cup I^{(t+2)}$ such that
$\sum_{m\in I_{t+1}}\beta_{m}=\infty\qquad\mbox{ and }\qquad\sum_{m\in
I^{(t+2)}}\beta_{m}=\infty$
and such that $\min({\rm I\\!N}\setminus(I_{1}\cup\dots\cup I_{t}))\in
I_{t+1}$.
In this way, inductively we construct subsets $I_{t}\subseteq{\rm I\\!N}$ such
that $\sum_{m\in I_{t}}\beta_{m}=\infty$ for all $t$. It is evident that
$\bigcup_{t=1}^{\infty}I_{t}={\rm I\\!N}$ and that this union is disjoint.
(Observe that it is crucial to put the smallest element from ${\rm
I\\!N}\setminus(I_{1}\cup\dots\cup I_{t-1})$ into $I_{t}$ in each step, in
order to guarantee that each natural number appears in some $I_{t}$, i.e.,
that it is not forgotten “forever”.)
II. Let $\sum_{m=1}^{\infty}a_{m}$ be a conditionally convergable series of
real numbers and let
$P:=\left\\{m\in{\rm I\\!N}\;|\;a_{m}\geq 0\right\\},\qquad
N:=\left\\{m\in{\rm I\\!N}\;|\;a_{m}<0\right\\}.$
Then we have
$\sum_{m\in P}a_{m}=+\infty,\qquad\sum_{m\in N}a_{m}=-\infty.$
By I. there exist disjoint decompositions $P=\bigcup_{t=1}^{\infty}P_{t}$ and
$N=\bigcup_{t=1}^{\infty}N_{t}$ of $P$ and $N$ into infinite subsets
$P_{t},N_{t}$ such that
$\sum_{m\in P_{t}}a_{m}=\infty\qquad\mbox{ and }\qquad\sum_{m\in
N_{t}}a_{m}=-\infty$
for all $t$. If we set
$I_{t}:=P_{t}\cup N_{t},$
then for every $t$ the series $\sum_{m\in I_{t}}a_{m}$ is conditionally
convergable, and ${\rm I\\!N}=\bigcup_{t=1}^{\infty}I_{t}$ is a disjoint
decomposition. This proves the assertion. ∎
Since the proof of the general case of Theorem 1 is quite abstract, we start
with a discussion of the case $n=2$ to give the reader an idea of what is
really going on.
###### Proof of the Case $n=2$ of Theorem 1..
Here, ${\rm Sym}\,(2)$ consists of two elements $\sigma=(1\quad
2)=id_{\left\\{1,2\right\\}}$ and $\tau=(2\quad 1)$.
According to Lemma 4, there exists a disjoint partition ${\rm
I\\!N}=\bigcup_{t=1}^{\infty}I_{t}$ of ${\rm I\\!N}$ into infinite subsets
$I_{t}$ such that for each $t\in{\rm I\\!N}$ the series $\sum_{m\in
I_{t}}a_{m}$ is conditionally convergable. By Riemann’s theorem, we can find a
rearrangement $(b(1,k)\;|\;k\in{\rm I\\!N})$ of $(a_{m})_{m\in I_{1}}$ such
that
$\sum_{k=1}^{\infty}b(1,k)=s_{1}^{(\sigma)}.$
In the same way, we can find a rearrangement $(b(j,1)\;|\;j\geq 2)$ of
$(a_{m})_{m\in I_{2}}$ such that
$\sum_{j=2}^{\infty}b(j,1)=s_{1}^{(\tau)}-b(1,1).$
Next, we choose a rearrangement $(b(2,k)\;|\;k\geq 2)$ of $(a_{m})_{m\in
I_{3}}$ such that
$\sum_{k=2}^{\infty}b(2,k)=s_{2}^{(\sigma)}-s_{1}^{(\sigma)}-b(2,1)$
and a rearrangement $(b(j,2)\;|\;j\geq 3)$ of $(a_{m})_{m\in I_{4}}$ such that
$\sum_{j=3}^{\infty}b(j,2)=s_{2}^{(\tau)}-s_{1}^{(\tau)}-b(1,2)-b(2,2),$
and so on. Proceeding in this way, for each $j\geq 2$ we find a rearrangement
$(b(j,k)\;|\;k\geq j)$ of $(a_{m})_{m\in I_{2j-1}}$ such that
$\sum_{k=j}^{\infty}b(j,k)=s_{j}^{(\sigma)}-s_{j-1}^{(\sigma)}-\sum_{k=1}^{j-1}b(j,k),$
and for each $k\geq 2$ we find a rearrangement $(b(j,k)\;|\;j\geq k+1)$ of
$(a_{m})_{m\in I_{2k}}$ such that
$\sum_{j=k+1}^{\infty}b(j,k)=s_{k}^{(\tau)}-s_{k-1}^{(\tau)}-\sum_{j=1}^{k}b(j,k).$
In this way, $b(j,k)$ is uniquely defined for all $j,k\in{\rm I\\!N}$,
$(b(j,k)\;|\;j,k\in{\rm I\\!N})$ is a rearrangement of $(a_{m})_{m}$, and the
$b(j,k)$ satisfy the equations
$\sum_{j=1}^{N}\sum_{k=1}^{\infty}b(j,k)=s_{1}^{(\sigma)}+\sum_{j=2}^{N}\left(s_{j}^{(\sigma)}-s_{j-1}^{(\sigma)}\right)=s_{N}^{(\sigma)},$
$\sum_{k=1}^{N}\sum_{j=1}^{\infty}b(j,k)=s_{1}^{(\tau)}+\sum_{k=2}^{N}\left(s_{k}^{(\tau)}-s_{k-1}^{(\tau)}\right)=s_{N}^{(\tau)}$
for all $N\in{\rm I\\!N}$, as asserted. ∎
Now we turn to the general case.
###### Proof of Theorem 1.
We prove the theorem by induction. It suffices to show that for each $n\geq
2$, the validity of Corollary 2 for $n-1$ implies the validity of the theorem
for $n$. (Here it is important to note that the corollary also holds for $n=1$
in view of Riemann’s theorem.)
So let some $n\geq 2$ be given and assume that Corollary 2 is valid for $n-1$
instead of $n$. Let $(a_{m})_{m}$ be a sequence of real numbers such that
$\sum_{m=1}^{\infty}a_{m}$ is conditionally convergent.
According to Lemma 4, there exists a disjoint partition ${\rm
I\\!N}=\bigcup_{t=1}^{\infty}I_{t}$ of ${\rm I\\!N}$ into infinite subsets
$I_{t}$ such that for each $t\in{\rm I\\!N}$ the series $\sum_{m\in
I_{t}}a_{m}$ is conditionally convergable.
For an integer $d\geq 0$, we consider the following assumption.
Assumption $A_{d}$. The quantities $b(j_{1},\dots,j_{n})$ are already defined
for all $j_{1},\dots,j_{n}\in{\rm I\\!N}$ with
$\left\\{j_{1},\dots,j_{n}\right\\}\cap\left\\{1,\dots,d\right\\}\neq\emptyset$
such that
$\left(b(j_{1},\dots,j_{n})\;|\;\left\\{j_{1},\dots,j_{n}\right\\}\cap\left\\{1,\dots,d\right\\}\neq\emptyset\right)$
is a rearrangement of $\left(a_{m}\;|\;m\in\bigcup_{t=1}^{nd}I_{t}\right)$ and
such that for all $k\in\left\\{1,\dots,d\right\\}$, all
$\nu\in\left\\{1,\dots,n\right\\}$ and all $\sigma\in{\rm Sym}\,(n)$ with
$\sigma(\nu)=1$ one has
$\sum_{j_{2}=1}^{\infty}\dots\sum_{j_{n}=1}^{\infty}b(j_{\sigma(1)},\dots,j_{\sigma(\nu-1)},k,j_{\sigma(\nu+1)},\dots,j_{\sigma(n)})=s_{k}^{(\sigma)}-s_{k-1}^{(\sigma)};$
(2.2)
here, $s_{0}^{(\sigma)}=0$ for all $\sigma\in{\rm Sym}\,(n)$.
Here, for $\nu=1$, the quantity
$b(j_{\sigma(1)},\dots,j_{\sigma(\nu-1)},k,j_{\sigma(\nu+1)},\dots,j_{\sigma(n)})$
is of course understood to be just $b(k,j_{\sigma(2)},\dots,j_{\sigma(n)})$. A
similar comment applies to several other notations in the sequel.
We note that this is trivially satisfied for $d=0$ since in this case the
assumption is empty.
Now let some integer $d\geq 0$ be given and assume that $A_{d}$ is satisfied.
We want to show that also $A_{d+1}$ is satisfied. This is done by induction
once again: For given $\mu\in\left\\{1,\dots,n+1\right\\}$, we consider the
following assumption.
Assumption $B_{d,\mu}$. The quantities $b(j_{1},\dots,j_{n})$ are already
defined for all $j_{1},\dots,j_{n}\in{\rm I\\!N}$ with
$d+1\in\left\\{j_{1},\dots,j_{\mu-1}\right\\}$ such that
$\left(b(j_{1},\dots,j_{n})\;|\;j_{1},\dots,j_{n}\geq
d+1,\;d+1\in\left\\{j_{1},\dots,j_{\mu-1}\right\\}\right)$
is a rearrangement of
$\left(a_{m}\;|\;m\in\bigcup_{t=nd+1}^{nd+\mu-1}I_{t}\right)$ and such that
for all $\nu\in\left\\{1,\dots,\mu-1\right\\}$ and all $\sigma\in{\rm
Sym}\,(n)$ with $\sigma(\nu)=1$, one has
$\sum_{j_{2}=1}^{\infty}\dots\sum_{j_{n}=1}^{\infty}b(j_{\sigma(1)},\dots,j_{\sigma(\nu-1)},d+1,j_{\sigma(\nu+1)},\dots,j_{\sigma(n)})=s_{d+1}^{(\sigma)}-s_{d}^{(\sigma)}.$
(2.3)
Again we note that for $\mu=1$ the assumption $B_{d,\mu}$ is empty, hence
trivially true.
So we let some $\mu\in\left\\{1,\dots,n\right\\}$ be given and assume that
$B_{d,\mu}$ holds. For $\sigma\in{\rm Sym}\,(n),$ we set
$\delta(\sigma,\nu):=\left\\{\begin{array}[]{ll}d+2&\mbox{ if
}\nu\in\left\\{\sigma(1),\dots,\sigma(\mu-1)\right\\},\\\ d+1&\mbox{ if
}\nu\in\left\\{\sigma(\mu+1),\dots,\sigma(n)\right\\}.\end{array}\right.$
It is not needed to define $\delta(\sigma,\sigma(\mu))$ as we will see in the
sequel.
###### Claim.
For all $l=2,\dots,n$ and all $\sigma\in{\rm Sym}\,(n)$ with $\sigma(\mu)=1$,
the series
$\sum_{j_{2}=1}^{\infty}\dots\sum_{j_{l-1}=1}^{\infty}\sum_{j_{l}=1}^{\delta(\sigma,l)-1}\sum_{j_{l+1}=\delta(\sigma,l+1)}^{\infty}\dots\sum_{j_{n}=\delta(\sigma,n)}^{\infty}b(j_{\sigma(1)},\dots,j_{\sigma(\mu-1)},d+1,j_{\sigma(\mu+1)},\dots,j_{\sigma(n)})$
(2.4)
is (well-defined and) convergent.
###### Proof.
Let some $l\in\left\\{2,\dots,n\right\\}$ and some $\sigma\in{\rm Sym}\,(n)$
with $\sigma(\mu)=1$ be given. In view of $l\neq 1=\sigma(\mu)$ we have to
consider only the following two cases.
Case 1: $l\in\left\\{\sigma(1),\dots,\sigma(\mu-1)\right\\}.$
Then $\delta(\sigma,l)-1=d+1$ and there is some
$\lambda\in\left\\{1,\dots,\mu-1\right\\}$ such that $l=\sigma(\lambda)$. Now
we define a permutation $\tau\in{\rm Sym}\,(n)$ as follows:
$\tau(i):=\sigma(i)\quad\mbox{ for
}i\neq\lambda,\mu,\qquad\tau(\lambda):=\sigma(\mu)=1,\qquad\tau(\mu):=\sigma(\lambda).$
(2.5)
The series (2.4) is the sum of the $\delta(\sigma,l)-1=d+1$ series
$\sum_{j_{2}=1}^{\infty}\dots\sum_{j_{l-1}=1}^{\infty}\sum_{j_{l+1}=\delta(\sigma,l+1)}^{\infty}\dots\sum_{j_{n}=\delta(\sigma,n)}^{\infty}b(j_{\tau(1)},\dots,j_{\tau(\lambda-1)},j_{l},j_{\tau(\lambda+1)},\dots,j_{\tau(\mu-1)},d+1,j_{\tau(\mu+1)},\dots,j_{\tau(n)})$
where $j_{l}=1,\dots,d+1$. This series is convergent by assumption
$B_{j_{l}-1,\lambda+1}$ (see (2.3)). This shows the convergence of the series
in (2.4).
Case 2: $l\in\left\\{\sigma(\mu+1),\dots,\sigma(n)\right\\}.$
Then $\delta(\sigma,l)-1=d$ and there is some
$\lambda\in\left\\{\mu+1,\dots,n\right\\}$ such that $l=\sigma(\lambda)$. Now
we define $\tau$ as in (2.5). The series (2.4) is the sum of the
$\delta(\sigma,l)-1=d$ series
$\sum_{j_{2}=1}^{\infty}\dots\sum_{j_{l-1}=1}^{\infty}\sum_{j_{l+1}=\delta(\sigma,l+1)}^{\infty}\dots\sum_{j_{n}=\delta(\sigma,n)}^{\infty}b(j_{\tau(1)},\dots,j_{\tau(\mu-1)},d+1,j_{\tau(\mu+1)},\dots,j_{\tau(\lambda-1)},j_{l},j_{\tau(\lambda+1)},\dots,j_{\tau(n)})$
where $j_{l}=1,\dots,d$. This latter series is convergent by assumption
$A_{j_{l}}$ (see (2.2)). So the series in (2.4) is convergent as well. This
proves our claim. ∎
According to Corollary 2, one can choose
$\left(b(j_{1},\dots,j_{n})\;|\;j_{1},\dots,j_{\mu-1}\geq
d+2,j_{\mu}=d+1,j_{\mu+1},\dots,j_{n}\geq d+1\right)$
as a rearrangement of $I_{nd+\mu}$ such that for all $\sigma\in{\rm Sym}\,(n)$
with $\sigma(\mu)=1,$ one has
$\displaystyle\sum_{j_{2}=\delta(\sigma,2)}^{\infty}\dots\sum_{j_{n}=\delta(\sigma,n)}^{\infty}b(j_{\sigma(1)},\dots,j_{\sigma(\mu-1)},d+1,j_{\sigma(\mu+1)},\dots,j_{\sigma(n)})$
$\displaystyle=s_{d+1}^{(\sigma)}-s_{d}^{(\sigma)}$
$\displaystyle\quad-\sum_{l=2}^{n}\sum_{j_{2}=1}^{\infty}\dots\sum_{j_{l-1}=1}^{\infty}\sum_{j_{l}=1}^{\delta(\sigma,l)-1}\sum_{j_{l+1}=\delta(\sigma,l+1)}^{\infty}\dots\sum_{j_{n}=\delta(\sigma,n)}^{\infty}b(j_{\sigma(1)},\dots,j_{\sigma(\mu-1)},d+1,j_{\sigma(\mu+1)},\dots,j_{\sigma(n)}).$
Here we have used the claim above (see (2.4)) and the fact that we can
identify the subset $\left\\{\sigma\in{\rm
Sym}\,(n)\;|\,\sigma(\mu)=1\right\\}$ with ${\rm Sym}\,(n-1)$.
Then one can see that
$\sum_{j_{2}=1}^{\infty}\dots\sum_{j_{n}=1}^{\infty}b(j_{\sigma(1)},\dots,j_{\sigma(\mu-1)},d+1,j_{\sigma(\mu+1)},\dots,j_{\sigma(n)})=s_{d+1}^{(\sigma)}-s_{d}^{(\sigma)}$
for all $\sigma\in{\rm Sym}\,(n)$ with $\sigma(\mu)=1$.
In this way, we have defined $b(j_{1},\dots,j_{n})$ for all
$j_{1},\dots,j_{n}\in{\rm I\\!N}$ with
$d+1\in\left\\{j_{1},\dots,j_{\mu}\right\\}$ such that
$\left(b(j_{1},\dots,j_{n})\;|\;j_{1},\dots,j_{n}\geq
d+1,\;d+1\in\left\\{j_{1},\dots,j_{\mu}\right\\}\right)$
is a rearrangement of
$\left(a_{m}\;|\;m\in\bigcup_{t=nd+1}^{nd+\mu}I_{t}\right)$ and such that for
all $\nu\in\left\\{1,\dots,\mu\right\\}$ and all $\sigma\in{\rm Sym}\,(n)$
with $\sigma(\nu)=1,$ one has
$\sum_{j_{2}=1}^{\infty}\dots\sum_{j_{n}=1}^{\infty}b(j_{\sigma(1)},\dots,j_{\sigma(\nu-1)},d+1,j_{\sigma(\nu+1)},\dots,j_{\sigma(n)})=s_{d+1}^{(\sigma)}-s_{d}^{(\sigma)}.$
Hence $B_{d,\mu+1}$ holds.
By induction we deduce that $B_{d,n+1}$ holds. But this (together with
assumption $A_{d}$) just means that $A_{d+1}$ holds. So by induction, we
obtain the validity of $A_{d}$ for all $d\geq 0$. This proves our theorem. ∎
## References
* [1] H. Heuser, Lehrbuch der Analysis. Teil 1, Teubner, Stuttgart, 1980.
* [2] P. Lévy, Sur les séries semi-convergentes, Nouv. Ann. d. Math. 64 (1905), 506-511.
* [3] P. Rosenthal, The remarkable theorem of Lévy and Steinitz, Amer. Math. Monthly 94 (1987), 342-351.
* [4] M.A. Sofi, Lévy-Steinitz theorem in infinite dimension, New Zealand J. Math. 38 (2008), 63-73.
* [5] E. Steinitz, Bedingt konvergente Reihen und konvexe Systeme, J. f. Math. 143 (1913), 128-175.
|
arxiv-papers
| 2011-11-06T08:26:47 |
2024-09-04T02:49:24.044713
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jurgen Grahl and Shahar Nevo",
"submitter": "Shahar Nevo",
"url": "https://arxiv.org/abs/1111.1384"
}
|
1111.1497
|
# An IR-based Evaluation Framework for
Web Search Query Segmentation
Rishiraj Saha Roy and Niloy Ganguly
Monojit Choudhury and
Srivatsan Laxman
Indian Institute of Technology Kharagpur Kharagpur, West Bengal, India -
721302. {rishiraj, niloy}@cse.iitkgp.ernet.in Microsoft Research India
Bangalore, Karnataka, India - 560025. {monojitc, slaxman}@microsoft.com
###### Abstract
This paper presents the first evaluation framework for Web search query
segmentation based directly on IR performance. In the past, segmentation
strategies were mainly validated against manual annotations. Our work shows
that the goodness of a segmentation algorithm as judged through evaluation
against a handful of human annotated segmentations hardly reflects its
effectiveness in an IR-based setup. In fact, state-of the-art algorithms are
shown to perform as good as, and sometimes even better than human annotations
– a fact masked by previous validations. The proposed framework also provides
us an objective understanding of the gap between the present best and the best
possible segmentation algorithm. We draw these conclusions based on an
extensive evaluation of six segmentation strategies, including three most
recent algorithms, vis-à-vis segmentations from three human annotators. The
evaluation framework also gives insights about which segments should be
necessarily detected by an algorithm for achieving the best retrieval results.
The meticulously constructed dataset used in our experiments has been made
public for use by the research community.
###### category:
H.3.3 Information Search and Retrieval Query formulation, Retrieval models
###### keywords:
Query segmentation, IR evaluation, Evaluation framework, Test collections,
Manual annotation
††terms: Measurement, Experimentation, Human Factors
## 1 Introduction
Query segmentation is the process of dividing a query into individual semantic
units [3]. For example, the query singular value decomposition online demo can
be broken into singular value decomposition and online demo. All documents
containing the individual terms singular, value and decomposition are not
necessarily relevant for this query. Rather, one can almost always expect to
find the segment singular value decomposition in the relevant documents. In
contrast, although online demo is a segment, finding the phrase or some
variant of it may not affect the relevance of the document. Hence, the
potential of query segmentation goes beyond the detection of multiword named
entities. Rather, segmentation leads to a better understanding of the query
and is crucial to the search engine for improving Information Retrieval (IR)
performance.
There is broad consensus in the literature that query segmentation can lead to
better retrieval performance [2, 3, 7, 9, 13]. However, most automatic
segmentation techniques [3, 4, 7, 9, 13, 15] have so far been evaluated only
against a small set of $500$ queries segmented by human annotators. Such an
approach implicitly assumes that a segmentation technique that scores better
against human annotations will also automatically lead to better IR
performance. We challenge this approach on multiple counts. First, there has
been no systematic study that establishes the quality of human segmentations
in the context of IR performance. Second, grammatical structure in queries is
not as well-understood as natural language sentences where human annotations
have proved useful for training and testing of various Natural Language
Processing (NLP) tools. This leads to considerable inter-annotator
disagreement when humans segment search queries. Third, good quality human
annotations for segmentation can be difficult and expensive to obtain for a
large set of test queries. Thus, there is a need for a more direct IR-based
evaluation framework for assessing query segmentation algorithms. This is the
central motivation of the present work.
We propose an IR-based evaluation framework for query segmentation that
requires only human relevance judgments (RJs) for query-URL pairs for
computing the performance of a segmentation algorithm – such relevance
judgments are anyway needed for training and testing of any IR engine. A
fundamental problem in designing an IR-based evaluation framework for
segmentation algorithms is to decouple the effect of segmentation accuracy
from the way segmentation is used for IR. This is because a query segmentation
algorithm breaks the input query into, typically, a non-overlapping sequence
of words (segments), but it does not prescribe how these segments should be
used during the retrieval and ranking of the documents for that query. We
resolve this problem by providing a formal model of query expansion for a
given segmentation; the various queries obtained can then be issued to any
standard IR engine, which we assume to be a black box.
We conduct extensive experiments within our framework to understand the
performance of several state-of-the-art query segmentation schemes [7, 9, 11]
and segmentations by three human annotators. Our experiments reveal several
interesting facts such as: (a) Segmentation is actively useful in improving IR
performance, even though submitting all segments (detected by an algorithm) in
double quotes to the IR engine degrades performance; (b) All segmentation
strategies, including human segmentations, are yet to reach the best
achievable limits in IR performance; (c) In terms of IR metrics, some of the
segmentation algorithms perform as good as the best human annotator and better
than the average/worst human annotator; (d) Current match-based metrics for
comparing query segmentation against human annotations are only weakly
correlated with the IR-based metrics, and cannot be used as a proxy for IR
performance; and (e) There is scope for improvement for the matching metrics
that compare segmentations against human annotations by differentially
penalizing the straddling, splitting and joining of reference segments. In
short, the proposed evaluation framework not only provides a formal way to
compare segmentation algorithms and estimate their effectiveness in IR, but
also helps us to understand the gaps in human annotation-based evaluation. The
framework also provides valuable insights regarding the segmentations that can
be used for improvement of the algorithms.
The rest of the paper is organized as follows. Sec. 2 introduces our
evaluation framework and its design philosophy. Sec. 3 presents the dataset
and the segmentation algorithms compared on our framework. Sec. 4 discusses
the experimental results and insights derived from them. In Sec. 5, we discuss
a few related issues, and the next section (Sec. 6) gives a brief background
of past approaches to evaluate query segmentation and their limitations. We
conclude by summarizing our contributions and suggesting future work in Sec.
7.
## 2 The evaluation framework
In this section we present a framework for the evaluation of query
segmentation algorithms based on IR performance. Let $\mathbf{q}$ denote a
search query and let $\mathbf{s}^{\mathbf{q}}=\langle
s^{\mathbf{q}}_{1},\ldots,s^{\mathbf{q}}_{n}\rangle$ denote a segmentation of
$\mathbf{q}$ such that a simple concatenation of the $n$ segments equals
$\mathbf{q}$, i.e., we have
$\mathbf{q}=(s^{\mathbf{q}}_{1}+\cdots+s^{\mathbf{q}}_{n})$, where +
represents the concatenation operator. We are given a segmentation algorithm
$\mathcal{A}$ and the task is to evaluate its performance. We require the
following resources:
1. 1.
A test set $\mathcal{Q}$ of unquoted search queries.
2. 2.
A set $\mathcal{U}$ of documents (or URLs) out of which search results will be
retrieved.
3. 3.
Relevance judgments $r(\mathbf{q},u)$ for query-URL pairs
$(\mathbf{q},u)\in\mathcal{Q}\times\mathcal{U}$. The set of all relevance
judgments are collectively denoted by $\mathcal{R}$.
4. 4.
An IR engine that supports quoted queries as input.
The resources needed by our evaluation framework are essentially the same as
those needed for the training and testing of a standard IR engine, namely,
queries, a document corpus and set of relevance judgments. Akin to the
training examples required for an IR engine, we only require relevance
judgments for a small and appropriate subset of $\mathcal{Q}\times\mathcal{U}$
(each query needs only the documents in its own pool to be judged) [14].
It is useful to separate the evaluation of segmentation performance, from the question of how to best exploit the segments to retrieve the most relevant documents. From an IR perspective, a natural interpretation of a segment could be that it consists of words that must appear together, in the same order, in documents where the segment is deemed to match [3]. This can be referred to as ordered contiguity matching. While this can be easily enforced in modern IR engines through use of double quotes around segments, we observe that not all segments must be used this way (see [10] for related ideas and experiments in a different context). Some segments may admit more general matching criteria, such as unordered or intruded contiguity (e.g., a segment a b may be allowed to match b a or a c b in the document). The case of unordered intruded matching may be restricted under linguistic dependence assumptions (e.g., a b can match a of b or b in a). Finally, some segments may even play non-matching roles (e.g., when the segment specifies user intent, like how to and where is). Thus, there may be several different ways to exploit the segments discovered by a segmentation algorithm. Even within the same query, different segments may need to be treated differently. For instance, in the query cannot view | word files | windows 7, the first one might be matched using intruded ordered occurrence (cannot properly view), the second segment may be matched under a linguistic dependency model (files in word) and the last one under ordered contiguity.
Intruded contiguity and linguistic dependency may be difficult to implement
for the broad class of general Web search queries. Identifying how the various
segments of a query should be ideally matched in the document is quite a
challenging and unsolved research problem. On the other hand, an exhaustive
expansion scheme, where every segment is expanded in every possible way, is
computationally expensive and might introduce noise. Moreover, current
commercial IR engines do not support any syntax to specify linguistic
dependence or intruded or unordered occurrence based matching. Hence, in order
to keep the evaluation framework in line with the current IR systems, we focus
on ordered contiguity matching which is easily implemented through the use of
double quotes around segments. However, we note that the philosophy of the
framework does not change with increased sophistication in the retrieval
system – only the expansion sets for the queries have to be appropriately
modified.
Table 1: Example of generation of quoted versions for a segmented query.
Segmented query | Quoted versions
---|---
| we are the people song lyrics
| we are the people "song lyrics"
| we are "the people" song lyrics
we are | the people | song lyrics | we are "the people" "song lyrics"
| "we are" the people song lyrics
| "we are" the people "song lyrics"
| "we are" "the people" song lyrics
| "we are" "the people" "song lyrics"
We propose an evaluation framework for segmentation algorithms that generates
all possible quoted versions of a segmented query (see Table 1) and submits
each quoted version to the IR engine. The corresponding ranked lists of
retrieved documents are then assessed against relevance judgments available
for the query-URL pairs. The IR quality of the best-performing quoted version
is used to measure performance of the segmentation algorithm. We now formally
specify our evaluation framework that computes what we call a Quoted Version
Retrieval Score (QVRS) for the segmentation algorithm given the test set
$\mathcal{Q}$ of queries, the document pool $\mathcal{U}$ and the relevance
judgments $\mathcal{R}$ for query-URL pairs.
#### Quoted query version generation
Let the segmentation output by algorithm $\mathcal{A}$ be denoted by
$\mathcal{A}(\mathbf{q})=\mathbf{s}^{\mathbf{q}}=\langle
s^{\mathbf{q}}_{1},\ldots,s^{\mathbf{q}}_{n}\rangle$. We generate all possible
quoted versions of the query $\mathbf{q}$ based on the segments in
$\mathcal{A}(\mathbf{q})$. In particular, we define
$\mathcal{A}_{0}(\mathbf{q})=(s^{\mathbf{q}}_{1}+\cdots+s^{\mathbf{q}}_{n})$
with no quotes on any of the segments,
$\mathcal{A}_{1}(\mathbf{q})=(s^{\mathbf{q}}_{1}+\cdots+\mathrm{``}s^{\mathbf{q}}_{n}\mathrm{"})$
with quotes only around the last segment $s^{\mathbf{q}}_{n}$, and so on.
Since there are $n$ segments in $\mathcal{A}(\mathbf{q})$, this process will
generate $2^{n}$ versions of the query, $\mathcal{A}_{i}(\mathbf{q})$,
$i=0,\ldots,2^{n}-1$. We note that if $b_{i}=(b_{i1},\ldots,b_{in})$ be the
$n$-bit binary representation of $i$, then $\mathcal{A}_{i}(\mathbf{q})$ will
apply quotes to the $j^{\mathrm{th}}$ segment $s^{\mathbf{q}}_{j}$ iff
$b_{ij}=1$. We deduplicate this set, because
$\\{\mathcal{A}_{i}(\mathbf{q})\>:\>i=0,\ldots,2^{n}-1\\}$ can contain
multiple versions that essentially represent the same quoted query version
(when single words are inside quotes). For example, the query versions "harry
potter" "game" and "harry potter" game are equivalent in terms of the input
semantics of an IR engine. The resulting set of unique quoted query versions
is denoted $\mathcal{Q}_{\mathcal{A}}(\mathbf{q})$.
#### Document retrieval using IR engine
For each $\mathcal{A}_{i}(\mathbf{q})\in\mathcal{Q}_{\mathcal{A}}(\mathbf{q})$
we use the IR engine to retrieve a ranked list $\mathcal{O}_{i}$ of documents
out of the document pool $\mathcal{U}$ that matched the given quoted query
version $\mathcal{A}_{i}(\mathbf{q})$. The number of documents retrieved in
each case depends on the IR metrics we will want to use to assess the quality
of retrieval. For example, to compute an IR metric at the top $k$ positions,
we would require that at least $k$ documents be retrieved from the pool.
#### Measuring retrieval against relevance judgments
Since we have relevance judgments ($\mathcal{R}$) for query-URL pairs in
$\mathcal{Q}\times\mathcal{U}$, we can now compute IR metrics such as
normalized Discounted Cumulative Gain (nDCG), Mean Average Precision (MAP) or
Mean Reciprocal Rank (MRR) to measure the quality of the retrieved ranked list
$\mathcal{O}_{i}$ for query $\mathbf{q}$. We use $@k$ variants of each of
these measures which are defined to be the usual metrics computed after
examining only the top-$k$ positions. For example, we can compute
$\mathrm{nDCG@}k$ for query $\mathbf{q}$ and retrieved document-list
$\mathcal{O}_{i}$ using the following formula:
$\mathrm{nDCG@}k(\mathbf{q},\mathcal{O}_{i}\>,\>\mathcal{R})=r(\mathbf{q},\mathcal{O}_{i}^{1})+\sum_{j=2}^{k}\frac{r(\mathbf{q},\mathcal{O}_{i}^{j})}{\log_{2}j}$
(1)
where $\mathcal{O}_{i}^{j}$, $j=1,\ldots,k$, denotes the $j^{\mathrm{th}}$
document in the ranked-list $\mathcal{O}_{i}$ and
$r(\mathbf{q},\mathcal{O}_{i}^{j})$ denotes the associated relevance judgment
from $\mathcal{R}$.
#### Oracle score using best quoted query version
Different quoted query versions $\mathcal{A}_{i}(\mathbf{q})$ (all derived
from the same basic segmentation $\mathcal{A}(\mathbf{q})$ output by the
segmentation algorithm $\mathcal{A}$) retrieve different ranked lists of
documents $\mathcal{O}_{i}$. As discussed earlier, automatic apriori selection
of a good (or the best) quoted query version is a difficult problem. While
different strategies may be used to select a quoted query version, we would
like our evaluation of the segmentation algorithm $\mathcal{A}$ to be agnostic
of the version-selection step. To this end, we select the best-performing
$\mathcal{A}_{i}(\mathbf{q})$ from the entire set
$\mathcal{Q}_{\mathcal{A}}(\mathbf{q})$ of query versions generated and use it
to define our oracle score for $\mathbf{q}$ and $\mathcal{A}$ under the chosen
IR metric [8]. For example, the oracle score for nDCG@$k$ is as defined below:
$\Omega_{\mathrm{nDCG@}k}(\mathbf{q},\mathcal{A})=\max_{\mathcal{A}_{i}(\mathbf{q})\in\mathcal{Q}_{\mathcal{A}}(\mathbf{q})}\mathrm{nDCG@}k(\mathbf{q},\mathcal{O}_{i}\>,\>\mathcal{R})$
(2)
where $\mathcal{O}_{i}$ denotes the ranked list of documents retrieved by the
IR engine when presented with $\mathcal{A}_{i}(\mathbf{q})$ as the input. We
note that $\mathcal{Q}_{\mathcal{A}}(\mathbf{q})$ always contains the original
unsegmented version of the query. We refer to such an
$\Omega_{\cdot}(\cdot,\cdot)$ as the Oracle.
This forms the basis of our evaluation framework. We note that there can also
be other ways to define this oracle score. For example, instead of seeking the
best IR performance possible across the different query versions, we could
also seek the minimum performance achievable by $\mathcal{A}$ irrespective of
what version-selection strategy is adopted. This would give us a lower bound
on the performance of the segmentation algorithm. However, the main drawback
of this approach is that the minimum performance is almost always achieved by
the fully quoted version (where every segment is in double quotes) (see Table
7). Such a lower bound would not be useful in assessing the comparative
performance of segmentation algorithms.
#### QVRS computation
Once the oracle scores are obtained for all queries in the test set
$\mathcal{Q}$, we can compute the average oracle score achieved by
$\mathcal{A}$. We refer to this as the Quoted Version Retrieval Score (QVRS)
of $\mathcal{A}$ with respect to test set $\mathcal{Q}$, document pool
$\mathcal{U}$ and relevance judgments $\mathcal{R}$. For example, using the
oracle with the nDCG@$k$ metric, we can define the QVRS score as follows:
$QVRS(\mathcal{Q},\mathcal{A},{\mathrm{nDCG@}k})=\frac{1}{|\mathcal{Q}|}\sum_{\mathbf{q}\in\mathcal{Q}}\Omega_{\mathrm{nDCG@}k}(\mathbf{q},\mathcal{A})$
(3)
Similar QVRS scores can be computed using other IR metrics such as MAP@$k$ and
MRR@$k$. In our experiments section, we report results using nDCG@$k$,
MAP@$k$, and MRR@$k$, for $k=5$ and $k=10$ as most Web users examine only the
first five or ten search results.
## 3 Dataset and algorithms
In this section, we describe the dataset used and briefly introduce the
algorithms compared on our framework.
### 3.1 Test set of queries ($\mathcal{Q}$)
We selected a random subset of $500$ queries from a slice of the query logs of
Bing Australia111http://www.bing.com/?cc=au containing $16.7$ million queries
issued over a period of one month (May $2010$). We used the following criteria
to filter the logs before extracting a random sample: (1) Exclude queries with
non-ASCII characters, (2) Exclude queries that occurred fewer than 5 times in
the logs (rarer queries often contained spelling errors), and (3) Restrict
query lengths to between five and eight words. Shorter queries rarely contain
multiple multiword segments, and when they do, they are mostly named entities
that can be easily detected using dictionaries. Moreover, traditional search
engines usually give satisfactory results for short queries. On the other
hand, queries longer than eight words (only $3.24\%$ of all queries in our
log) are usually error messages, complete NL sentences or song lyrics, that
need to be addressed separately.
We denote this set of $500$ queries by $\mathcal{Q}$, the test set of
unsegmented queries needed for all our evaluation experiments. The average
length of queries in $\mathcal{Q}$ (our dataset) is $5.29$ words. The average
query length was $4.31$ words in the Bergsma and Wang $2007$
Corpus222http://bit.ly/xoyT2c (henceforth, BWC07) [3]. Each of these $500$
queries were independently segmented by three human annotators (who issue
around $20$-$30$ search queries per day) who were asked to mark a contiguous
chunk of words in a query as a segment if they thought that these words
together formed a coherent semantic unit. The annotators were free to refer to
other resources and Web search engines during the annotation process,
especially for understanding the query and its possible context(s). We shall
refer to the three sets of annotations (and also the corresponding annotators)
as $H_{A}$, $H_{B}$ and $H_{C}$.
It is important to mention that the queries in $\mathcal{Q}$ have some amount
of word level overlap, even though all the queries have very distinct
information needs. Thus, a document retrieved from the pool might exhibit good
term level match for more than one query in $\mathcal{Q}$. This makes our
corpus an interesting testbed for experimenting with different retrieval
systems. There are existing datasets, including BWC07, that could have been
used for this study. However, refer to Sec. 5.1 for an account of why building
this new dataset was crucial for our research.
### 3.2 Document pool ($\mathcal{U}$) and RJs ($\mathcal{R}$)
Each query in $\mathcal{Q}$ was segmented using all the nine segmentation
strategies considered in our study (six algorithms and three humans). For
every segmentation, all possible quoted versions were generated (total
$4,746$) and then submitted to the Bing API333http://msdn.microsoft.com/en-
us/library/dd251056.aspx and the top ten documents were retrieved. We then
deduplicated these URLs to obtain $14,171$ unique URLs, forming $\mathcal{U}$.
On an average, adding the $9^{th}$ strategy to a group of the remaining eight
resulted in about one new quoted version for every two queries. These new
versions may or may not introduce new documents to the pool. We observed that
for $71.4\%$ of the queries there is less than $50\%$ overlap between the top
ten URLs retrieved for the different quoted versions. This indicates that
different ways of quoting the segments in a query does make a difference in
the search results. By varying the pooling depth (ten in our case), one can
roughly control the number of relevant and non-relevant documents entering the
collection.
For each query-URL pair, where the URL has been retrieved for at least one of
the quoted versions of the query (approx. $28$ per query), we obtained three
independent sets of relevance judgments from human users. These users were
different from annotators $H_{A}$, $H_{B}$ and $H_{C}$ who marked the
segmentations, but having similar familiarity with search systems. For each
query, the corresponding set of URLs was shown to the users after
deduplication and randomization (to prevent position bias for top results),
and asked to mark whether the URL was irrelevant (score = $0$), partially
relevant (score = $1$) or highly relevant (score = $2$) to the query. We then
computed the average rating for each query-URL pair (the entire set forming
$\mathcal{R}$), which has been used for subsequent nDCG, MAP and MRR
computations. Please refer to Table 8 in Sec. 5.3 for inter-annotator
agreement figures and other related discussions.
### 3.3 Segmentation algorithms
Table 2: Segmentation algorithms compared on our framework. Algorithm | Training data
---|---
Li et al. [9] | Click data, Web $n$-gram probabilities
Hagen et al. [7] | Web $n$-gram frequencies, Wikipedia titles
Mishra et al. [11] | Query logs
[11] \+ Wiki | Query logs, Wikipedia titles
PMI-W [7] | Web $n$-gram probabilities (used as baseline)
PMI-Q [11] | Query logs (used as baseline)
Table 2 lists the six segmentation algorithms that have been studied in this
work. Li et al. [9] use the expectation maximization algorithm to arrive at
the most probable segmentation, while Hagen et al. [7] show a simple
frequency-based method produces a performance comparable to the state-of-the-
art. The technique in Mishra et al. [11] uses only query logs for segmenting
queries. In our experiments, we observed that the performance of Mishra et al.
[11] can be improved if we used Wikipedia titles. We refer to this as “[11] \+
Wiki" in our experiments (see Appendix A for details). The Point-wise Mutual
Information (PMI)-based algorithms are used as baselines. The thresholds for
PMI-W and PMI-Q were chosen to be 8.141 and 0.156 respectively, that maximized
the Seg-F (see Sec. 4.2) on our development set.
### 3.4 Public release of data
The test set of search queries along with their manual and some of the
algorithmic segmentations, the theoretical best segmentation output that can
serve as an evaluation benchmark ($BQV_{BF}$ in Sec. 4.1), and the list of
URLs whose contents serve as our document corpus is available for public
use444http://cse.iitkgp.ac.in/resgrp/cnerg/qa/querysegmentation.html. The
relevance judgments for the query-URL pairs have also been made public which
will enable the community to use this dataset for evaluation of any new
segmentation algorithm.
## 4 Experiments and Observations
Table 3: Results of IR-based evaluation of segmentation algorithms using
Lucene (mean oracle scores).
Metric | Unseg. | [9] | [7] | [11] | [11] + | PMI-W | PMI-Q | $H_{A}$ | $H_{B}$ | $H_{C}$ | $BQV_{BF}$
---|---|---|---|---|---|---|---|---|---|---|---
| query | | | | Wiki | | | | | |
nDCG@5 | 0.688 | 0.752* | 0.763* | 0.745 | 0.767* | 0.691 | 0.766* | 0.770 | 0.768 | 0.759 | 0.825
nDCG@10 | 0.701 | 0.756* | 0.767* | 0.751 | 0.768* | 0.704 | 0.767* | 0.770 | 0.768 | 0.763 | 0.832
MAP@5 | 0.882 | 0.930* | 0.942* | 0.930* | 0.945* | 0.884 | 0.932* | 0.944 | 0.942 | 0.936 | 0.958
MAP@10 | 0.865 | 0.910* | 0.921* | 0.910* | 0.923* | 0.867 | 0.912* | 0.923 | 0.921 | 0.916 | 0.944
MRR@5 | 0.538 | 0.632* | 0.649* | 0.609 | 0.650* | 0.543 | 0.648* | 0.656 | 0.648 | 0.632 | 0.711
MRR@10 | 0.549 | 0.640* | 0.658* | 0.619 | 0.658* | 0.555 | 0.656* | 0.665 | 0.656 | 0.640 | 0.717
The highest value in a row (excluding the $BQV_{BF}$ column) and those with no
statistically significant difference with the highest value are marked in
boldface. The values for algorithms that perform better than or have no
statistically significant difference with the minimum of the human
segmentations are marked with *. The paired $t$-test was performed and the
null hypothesis was rejected if the $p$-value was less than $0.05$.
Table 4: Matching metrics for different segmentation algorithms and human
annotations with $BQV_{BF}$ as reference.
Metric | Unseg. | [9] | [7] | [11] | [11] + | PMI-W | PMI-Q | $H_{A}$ | $H_{B}$ | $H_{C}$ | $BQV_{BF}$
---|---|---|---|---|---|---|---|---|---|---|---
| query | | | | Wiki | | | | | |
Qry-Acc | 0.044 | 0.056 | 0.082* | 0.058 | 0.094* | 0.046 | 0.104* | 0.086 | 0.074 | 0.064 | 1.000
Seg-Prec | 0.226* | 0.176* | 0.189* | 0.206* | 0.203* | 0.229* | 0.218* | 0.176 | 0.166 | 0.178 | 1.000
Seg-Rec | 0.325* | 0.166* | 0.162* | 0.210* | 0.174* | 0.323* | 0.196* | 0.144 | 0.133 | 0.154 | 1.000
Seg-F | 0.267* | 0.171* | 0.174* | 0.208* | 0.187* | 0.268* | 0.206* | 0.158 | 0.148 | 0.165 | 1.000
Seg-Acc | 0.470 | 0.624 | 0.661* | 0.601 | 0.667* | 0.474 | 0.660* | 0.675 | 0.675 | 0.663 | 1.000
The highest value in a row (excluding the $BQV_{BF}$ column) and those with no
statistically significant difference with the highest value are marked in
boldface. The values for algorithms that perform better than or have no
statistically significant difference with the minimum of the human
segmentations are marked with *. The paired $t$-test was performed and the
null hypothesis was rejected if the $p$-value was less than $0.05$.
Table 5: Performance of PMI-Q and [9] with respect to matching (mean of
comparisons with $H_{A}$, $H_{B}$ and $H_{C}$ as references) and IR metrics.
Metric | nDCG@10 | MAP@10 | MRR@10 | Qry-Acc | Seg-Prec | Seg-Rec | Seg-F | Seg-Acc
---|---|---|---|---|---|---|---|---
PMI-Q | 0.767 | 0.912 | 0.656 | 0.341 | 0.448 | 0.487 | 0.467 | 0.810
[9] | 0.756 | 0.910 | 0.640 | 0.375 | 0.524 | 0.588 | 0.554 | 0.810
The highest values in a column are marked in boldface.
In this section we present experiments, results and the key inferences made
from them.
### 4.1 IR Experiments
For the retrieval-based evaluation experiments, we use the
Lucene555http://lucene.apache.org/java/docs/index.html text retrieval system,
which is publicly available as a code library. In its default configuration,
Lucene does not perform any automatic query segmentation, which is very
important for examining the effectiveness of segmentation algorithms in an IR-
based scheme. Double quotes can be used in a query to force Lucene to match
the quoted phrase (in Lucene terms) exactly in the documents. Starting with
the segmentations output by each of the six algorithms as well as the three
human annotations, we generated all possible quoted query versions, which
resulted in a total of $4,746$ versions for the $500$ queries. In the notation
of Sec. 2, this corresponds to generating
$\mathcal{Q}_{\mathcal{A}}(\mathbf{q})$ for each segmentation method
$\mathcal{A}$ (including one for each human segmentation) and for every query
$\mathbf{q}\in\mathcal{Q}$. These quoted versions were then passed through
Lucene to retrieve documents from the pool. For each segmentation scheme, we
then use the oracle described in Sec. 2 to obtain the query version yielding
the best result (as determined by the IR metrics – nDCG, MAP and MRR computed
according to the human relevance judgments). These oracle scores are then
averaged over the query set to give us the QVRS measures.
The results are summarized in Table 3. Different rows represent the different
IR metrics that were used and columns correspond to different segmentation
strategies. The second column (marked “Unseg. Query") refers to the original
unsegmented query. This can be assumed to be generated by a trivial
segmentation strategy where each word is always a separate segment. Columns
3-8 denote the six different segmentation algorithms and 9-11 (marked $H_{A}$,
$H_{B}$ and $H_{C}$) represent the human segmentations. The last column
represents the performance of the best quoted versions (denoted by $BQV_{BF}$
in table) of the queries which are computed by brute force, i.e. an exhaustive
search over all possible ways of quoting the parts of a query ($2^{l-1}$
possible quoted versions for an $l$-word query) irrespective of any
segmentation algorithm. The results are reported for two sizes of retrieved
URL lists ($k$), namely five and ten. Since we needed to convert our graded
relevance judgments to binary values for computing MAP@k, URLs with ratings of
$1$ and $2$ were considered as relevant (responsible for the generally high
values) and those with $0$ as irrelevant. For MRR, only URLs with ratings of
$2$ were considered as relevant.
The first observation we make from the results is that human as well as all
algorithmic segmentation schemes consistently outperform unsegmented queries
for all IR metrics. Second, we observe that the performance of some
segmentation algorithms are comparable and sometime even marginally better
than some of the human annotators. Finally, we observe that there is
considerable scope for improving IR performance through better segmentation
(all values less than $BQV_{BF}$). The inferences from these observations are
stated later in this section.
### 4.2 Performance under traditional matching metrics
In the next set of experiments we study the utility of traditional matching
metrics that are used to evaluate query segmentation algorithms against a gold
standard of human segmented queries (henceforth referred to as the reference
segmentation). These metrics are listed below [7]:
1. 1.
Query accuracy (Qry-Acc): The fraction of queries where the output matches
exactly with the reference segmentation.
2. 2.
Segment precision (Seg-Prec): The ratio of the number of segments that overlap
in the output and reference segmentations to the number of output segments,
averaged across all queries in the test set.
3. 3.
Segment recall (Seg-Rec): The ratio of the number of segments that overlap in
the output and reference segmentations to the number of reference segments,
averaged across all queries in the test set.
4. 4.
Segment F-score (Seg-F): The harmonic mean of Seg-Prec and Seg-Rec.
5. 5.
Segmentation accuracy (Seg-Acc): The ratio of correctly predicted boundaries
and non-boundaries in the output segmentation with respect to the reference,
averaged across all queries in the test set.
We computed the matching metrics for various segmentation algorithms against
$H_{A}$, $H_{B}$ and $H_{C}$. According to these metrics, “Mishra et al. [11]
\+ Wiki" turns out to be the best algorithm which agrees with the results of
IR evaluation. However, the average Kendall-Tau rank correlation
coefficient666This coefficient is $1$ when there is perfect concordance
between the rankings, and $-1$ if the trends are reversed. between the ranks
of the strategies as obtained from the IR metrics (Table 3) and the matching
metrics was only $0.75$. This indicates that matching metrics are not perfect
predictors for IR performance. In fact, we discovered some costly flaws in the
relative ranking produced by matching metrics. One such case was rank
inversions between Li et al. [9] and PMI-Q. The relevant results are shown in
Table 5, which demonstrate that while PMI-Q consistently performs better than
Li et al. [9] under IR-based measures, the opposite inference would have been
drawn if we had used any of the matching metrics.
In Bergsma and Wang [3], human annotators were asked to segment queries such
that segments matched exactly in the relevant documents. This essentially
corresponds to determining the best quoted versions for the query. Thus, it
would be interesting to study how traditional matching metrics would perform
if the humans actually marked the best quoted versions. In order to evaluate
this, we used the matching metrics to compare the segmentation outputs by the
algorithms and human annotations against $BQV_{BF}$. The corresponding results
are quoted in Table 4. The results show that matching metrics are very poor
indicators of IR performance with respect to the $BQV_{BF}$. For example, for
three out of the five matching metrics, the unsegmented query is ranked the
best. This shows that even if human annotators managed to correctly guess the
best quoted versions, the matching metrics would fail to estimate the correct
relative rankings of the segmentation algorithms with respect to IR
performance. This fact is also borne out in the Kendall-Tau rank correlation
coefficients reported in Table 6. Another interesting observation from these
experiments is that Seg-Acc emerges as the best matching metric with respect
to IR performance, although its correlation coefficient is still much below
one.
Table 6: Kendall-Tau coefficients between IR and matching metrics with
$BQV_{BF}$ as reference for the latter.
Metric | Qry-Acc | Seg-Prec | Seg-Rec | Seg-F | Seg-Acc
---|---|---|---|---|---
nDCG@10 | 0.432 | -0.854 | -0.886 | -0.854 | 0.674
MAP@10 | 0.322 | -0.887 | -0.920 | -0.887 | 0.750
MRR@10 | 0.395 | -0.782 | -0.814 | -0.782 | 0.598
The highest value in a row is marked in boldface.
### 4.3 Inferences
Segmentation is helpful for IR. By definition, $\Omega_{\cdot}(\cdot,\cdot)$
(i.e., the oracle) values for every IR metric for any segmentation scheme are
at least as large as the corresponding values for the unsegmented query.
Nevertheless, for every IR metrics, we observe significant performance
benefits for all the human and algorithmic segmentations (except for PMI-W)
over the unsegmented query. This indicates that segmentation is indeed helpful
for boosting IR performance. Thus, our results validate the prevailing notion
and some of the earlier observations [2, 9] that segmentation can help improve
IR.
Human segmentations are a good proxy, but not a true gold standard. Our
results indicate that human segmentations perform reasonably well in IR
metrics. The best of the human annotators beats all the segmentation
algorithms, on almost all the metrics. Therefore, evaluation against human
annotations can indeed be considered as the second best alternative to an IR-
based evaluation (though see below for criticisms of current matching
metrics). However, if the objective is to improve IR performance, then human
annotations cannot be considered a true gold standard. There are at least
three reasons for this:
First, in terms of IR metrics, some of the state-of-the-art segmentation
algorithms are performing as well as human segmentations (no statistically
significant difference). Thus, further optimization of the matching metrics
against human annotations is not going to improve the IR performance of the
segmentation algorithms. Thus, evaluation on human annotations might become a
limiting factor for the current segmentation algorithms.
Second, the IR performance of the best quoted version of the queries derived
through our framework is significantly better than that of human annotations
(last column, Table 3). This means that humans fail to predict the correct
boundaries in many instances. Thus, there is scope for improvement for human
annotations.
Third, IR performance of at least one of the three human annotators ($H_{C}$)
is worse than some of the algorithms studied. In other words, while some
annotators (such as $H_{A}$) are good at guessing the “correct" segment
boundaries that will help IR, not all annotators can do it well. Therefore,
unless the annotators are chosen and guided properly, one cannot guarantee the
quality of annotated data for query segmentation. If the queries in the test
set have multiple intents, this issue becomes an even bigger concern.
Matching metrics are misleading. As discussed earlier and demonstrated by
Tables 4 and 6, the matching metrics provide unreliable ranking of the
segmentation algorithms even when applied against a true gold standard,
$BQV_{BF}$, that by definition maximizes IR performance. This counter-
intuitive observation can be explained in two ways. Either the matching
metrics or the IR metrics (or probably both) are misleading. Given that IR
metrics are well-tested and generally assumed to be acceptable, we are forced
to conclude that the matching metrics do not really reflect the quality of a
segmentation with respect to a gold standard. Indeed, this can be illustrated
by a simple example.
_Example._ Let us consider the query the looney toons show cartoon network,
whose best quoted version turns out to be "the looney toons show" "cartoon
network". The underlying segmentation that can give rise to this and therefore
can be assumed to be the reference is:
Ref: the looney toons show | cartoon network
The segmentations
(1) the looney | toons show | cartoon | network
(2) the | looney | toons show cartoon | network
are equally bad if one considers the matching metrics of Qry-Acc, Seg-Prec,
Seg-Rec and Seg-F (all values being zero) with respect to the reference
segmentation. Seg-Acc values for the two segmentations are $3/5$ and $1/5$
respectively. However, the BQV for (1) ("the looney" "toons show" cartoon
network) fetches better pages than the BQV of (2) (the looney toons show
cartoon network). So the segmentation (2) provides no IR benefit over the
unsegmented query and hence performs worse than (1) on IR metrics. However,
the matching metrics, except for Seg-Acc to some extent, fail to capture this
difference between the segmentations.
Figure 1: Distribution of multiword segments in queries across segmentation
strategies.
Distribution of multiword segments across queries gives insights about
effectiveness of strategy. The limitation of the matching metrics can also be
understood from the following analysis of the multiword segments in the
queries. Fig. 1 shows the distribution of queries having a specific number of
multiword segments (for example, $1$ in the legend indicates the proportion of
queries having one multiword segment) when segmented according to the various
strategies. We note that for Hagen et al. [7], $H_{B}$, $H_{A}$ and “Mishra et
al. [11] \+ Wiki", almost all of the queries have two multiword segments. For
$H_{C}$, Li et al. [9], PMI-Q and Mishra et al. [11], the proportion of
queries that have only one multiword segment increases. Finally, PMI-W has
almost negligible queries with a multiword segment. $BQV_{BF}$ is different
from all of them and has a majority of queries with one multiword segment. Now
given that the first group generally does the best in IR, followed by the
second, we can say that out of the two multiword segments marked by these
strategies, only one needs to be quoted. PMI-W as well as unsegmented queries
are bad because these schemes cannot detect the one crucial multiword segment
quoting which improves the performance. Nevertheless, these schemes do well
for matching metrics against $BQV_{BF}$ because both have a large number of
single word segments. Clearly this is not helpful for IR. Finally, Mishra et
al. [11] performs poorly despite being able to identify a multiword segment in
most of the cases because it is not identifying the one that is important for
IR.
Hence, the matching metrics are misleading due to two reasons. First, they do
not take into account that splitting a useful segment (i.e., a segment which
should be quoted to improve IR performance) is less harmful than joining two
unrelated segments. Second, matching metrics are, by definition, agnostic to
which segments are useful for IR. Therefore, they might unnecessarily penalize
a segmentation for not agreeing on the segments which should not be quoted,
but are present in the reference human segmentation. While the latter is an
inherent problem with any evaluation against manually segmented datasets, the
former can be resolved by introducing a new matching metric that
differentially penalizes splitting and joining of segments. This is an
important and interesting research problem that we would like to address in
the future. However, we would like to emphasize here that with the IR system
expected to grow in complexity in the future (supporting more flexible
matching criteria), the need for an IR-based evaluation like ours’ becomes
imperative.
Based on our new evaluation framework and corresponding experiments, we
observe that “Mishra et al. [11] \+ Wiki" has the best performance.
Nevertheless, the algorithms are trained and tested on different datasets, and
therefore, a comparison amongst the algorithms might not be entirely fair.
This is not a drawback of the framework and can be circumvented by
appropriately tuning all the algorithms on similar datasets. However, the
objective of the current work is not to compare segmentation algorithms;
rather, it is to introduce the evaluation framework, gain insights from the
experiments and highlight the drawbacks of human segmentation-based
evaluation.
## 5 Related issues
In this section, we will briefly discuss a few related issues that are
essential for understanding certain design choices and decisions made during
the course of this research.
### 5.1 Motivation for a new dataset
TREC data has been a popular choice for conducting IR-based experiments
throughout the past decade. Since there is no track specifically geared
towards query segmentation, the queries and qrels (query-relevance sets) from
the ad hoc retrieval task for the Web Track would seem the most relevant to
our work. However, $74\%$ of the $50$ queries in the $2010$ Web track ad hoc
task had less than three words. Also, when these $50$ queries were segmented
using the six algorithms, half of the queries did not have a multiword
segment. As discussed earlier, query segmentation is useful but not
necessarily for all types of queries. The benefit of segmentation may be
observed only when there are multiple multiword segments in the queries. The
TREC Million Query Track, last held in $2009$, has a much larger set of
$40,000$ queries, with a better coverage of longer queries. But since the goal
of the track is to test the hypothesis that a test collection built from
several incompletely judged topics is a better tool than a collection built
using traditional TREC pooling, there are only about $35,000$ query-document
relevance judgments for the $40,000$ queries. Such a sparse qrels is not
suitable here – incomplete assessments, especially for documents near the top
ranks, could cause crucial errors in system comparisons. Yet another option
could have been to use BWC07 as $\mathcal{Q}$and create the corresponding
$\mathcal{U}$and $\mathcal{R}$. However, this query set is known to suffer
from several drawbacks [7]. A new dataset for query
segmentation777http://bit.ly/xIhSur containing manual segment markups
collected through crowdsourcing has been recently made publicly available
(after we had completed construction of our set) by Hagen et al. [7], but it
lacks query-document relevance judgments. These factors motivated us to create
a new dataset suitable for our framework, which has been made publicly
available (see Sec. 3.4).
### 5.2 Retrieval using Bing
Table 7: IR-based evaluation using Bing API.
Metric | Unseg. | All quoted for | Oracle for
---|---|---|---
| query | [11] \+ Wiki | [11] \+ Wiki
nDCG@10 | 0.882 | 0.823 | 0.989*
MAP@10 | 0.366 | 0.352 | 0.410*
MRR@10 | 0.541 | 0.515 | 0.572*
The highest value in a row is marked bold. Statistically significant ($p$ <
0.05 for paired $t$-test) improvement over the unsegmented query is marked
with *.
Bing is a large-scale commercial Web search engine that provides an API
service. Instead of Lucene, which is too simplistic, we could have used Bing
as the IR engine in our framework. However, such a choice suffers from two
drawbacks. First, Bing might already be segmenting the query with its own
algorithm as a preprocessing step. Second, there is a serious replicability
issue. The document pool that Bing uses, i.e. the Web, changes dynamically
with documents added and removed from the pool on a regular basis. This makes
it difficult to publish a static gold standard dataset with relevance
judgments for all appropriate query-URL pairs that the Bing API may retrieve
even for the same set of queries. In view of this, the main results were
reported in this paper using the Lucene text retrieval system.
However, since we used Bing API to construct $\mathcal{U}$and corresponding
$\mathcal{R}$, we have the evaluation statistics using the Bing API as well.
For paucity of space, in Table 7 we only present the results for nDCG@10,
MRR@10 and MAP@10 for “Mishra et al. [11] \+ Wiki". The table reports results
for three quoted version-selection strategies: (i) Unsegmented query only
(equivalent to each word being within quotes) (ii) All segments quoted and
(iii) QVRS (oracle for “Mishra et al. [11] \+ Wiki"). For all the three
metrics, QVRS is statistically significantly higher than results for the
unsegmented query. Thus, segmentation can play an important role towards
improving IR performance of the search engine. We note that the strategy of
quoting all the segments is, in fact, detrimental to IR performance. This
emphasizes the point that how the segments should be matched in the documents
is a very important research challenge. Instead of quoting all the segments,
our proposal here is to assume an oracle that will suggest which segments to
quote and which are to be left unquoted for the best IR performance.
Philosophically, this is a major departure from the previous ideas of using
quoted segments, because re-issuing a query by quoting all the segments
implies segmentation as a way to generate a fully quoted version of the query
(all segments in double quotes). This definition severely limits the scope of
segmentation, which ideally should be thought of as a step forward better
query understanding.
Table 8: Inter-annotator agreement on features as observed from our
experiments.
Feature | Pair 1 | Pair 2 | Pair 3 | Mean
---|---|---|---|---
Qry-Acc | 0.728 | 0.644 | 0.534 | 0.635
Seg-Prec | 0.750 | 0.732 | 0.632 | 0.705
Seg-Rec | 0.756 | 0.775 | 0.671 | 0.734
Seg-F | 0.753 | 0.753 | 0.651 | 0.719
Seg-Acc | 0.911 | 0.914 | 0.872 | 0.899
Rel. judg. | 0.962 | 0.959 | 0.969 | 0.963
For relevance judgments, only pairs of (0, 2) and (2, 0) were considered
disagreements.
### 5.3 Inter-annotator agreement
Inter-annotator agreement (IAA) is an important indicator for reliability of
manually created data. Table 8 reports the pairwise IAA statistics for
$H_{A}$, $H_{B}$ and $H_{C}$. Since there are no universally accepted metrics
for IAA, we report the values of the five matching metrics when one of the
annotations (say $H_{A}$) is assumed to be the reference and the remaining
pair ($H_{B}$ and $H_{C}$) is evaluated against it (average reported). As is
evident from the table, the values of all the metrics, except for Seg-Acc, is
less than $0.78$ (similar values reported in [13]), which indicates a rather
low IAA. The value for Seg-Acc is close to $0.9$, which to the contrary,
indicates reasonably high IAA (as in [13]). The last row of Table 8 reports
the IAA for the three sets of relevance judgments (therefore, the actual pairs
for this column are different from that of the other rows). The agreement in
this case is quite high.
There might be several reasons for low IAA for segmentation, such as lack of
proper guidelines and/or an inherent inability of human annotators to mark the
correct segments of a query. Low IAA raises serious doubts about the
reliability of human annotations for query segmentation. On the other hand,
high IAA for relevance judgments naturally makes these annotations much more
reliable for any evaluation, and strengthens the case for our IR-based
evaluation framework which only relies on relevance judgments. We note that
ideally, relevance judgments should be obtained from the user who has issued
the query. This has been referred to as gold annotations, as opposed to silver
or bronze annotations which are obtained from expert and non-expert annotators
respectively who have not issued the query [1]. Gold annotations are
preferable over silver or bronze ones due to relatively higher IAA. Our
annotations are silver standard, though very high IAA essentially indicates
that they might be as reliable as gold standard. The high IAA might be due to
the unambiguous nature of the queries.
## 6 Related work
Since its inception in 2003 [12], many algorithms have been proposed for
automatic segmentation of Web queries. The approaches vary from purely
supervised [3] to fully unsupervised [7, 11] machine learning techniques. They
differ widely in terms of resources usage (Table 2) and the underlying
algorithmic techniques (e.g., expectation maximization [13] and eigenspace
similarity [15]).
### 6.1 Evaluation on manual annotations
Despite the diversity in approaches to the task, till date there has been only
one standard approach for evaluation of query segmentation algorithms, which
is to compare the machine output against a set of queries segmented by humans
[3, 4, 7, 9, 11, 13, 15]. The basic assumption underlying this evaluation
scheme is that humans are capable of segmenting a query in a “correct" or “the
best possible" way, which, if exploited appropriately, will result in maximum
benefits in IR performance. This is probably motivated by the extensive use of
human judgments and annotations as the gold standard in the field of NLP
(e.g., parts-of-speech labeling, phrase boundary identification, etc.).
However, this idea has several shortcomings, as pointed out in Sec. 4.3. Among
those who validate query segmentation against human-labeled data, most [3, 4,
6, 7, 9, 13, 15] report accuracies on BWC07 [3]. The popularity of the BWC07
dataset is partly because it was one of the first human annotated datasets
created for query segmentation, and partly because it is the only publicly
available dataset of its kind. While BWC07 has provided a common benchmark for
comparing various query segmentation algorithms, there are several limitations
of this specific dataset. BWC07 only contains noun phrase queries and there is
a non-trivial amount of noise in the annotations. See [7] for a detailed
criticism of this dataset.
### 6.2 IR-based evaluation
There has been only a handful of studies that explore some initial ideas about
IR-based evaluation [2, 7, 9] for query segmentation. Bendersky et al. [2]
were the first to study the effects of segmentation from an IR perspective.
They wanted to see if retrieval quality could be improved by incorporating
knowledge of query chunks into an MRF-based retrieval system [10]. Their
experiments on different TREC collections using popular IR metrics like MAP
indicate that query segmentation can indeed boost IR performance. Li et al.
[9] examined the usefulness of query segmentation when built into language
models for retrieval, in a Web search setting. However, none of these studies
propose an objective IR-based evaluation framework for query segmentation.
Their scope is limited to the demonstration of one particular strategy for
exploiting segmentations for improving IR, instead of evaluating and comparing
a set of algorithms.
As an excursus to their main work, Hagen et al. [7] examined if submitting
fully quoted queries (generated from algorithm outputs) results in fetching
better pages by the search engines. They study the top fifty retrieved
documents when the following versions of the queries – unsegmented, manually
quoted, quoted by the technique in Bergsma and Wang [3], and by their own
method – are submitted to Bing. Assuming the pages retrieved by manual
quotation as relevant, it was observed that the technique in Bergsma and Wang
[3] achieves the highest average recall. However, the authors also state that
such an assumption need not hold good in reality and emphasized the need for
an in-depth retrieval-based evaluation.
We would like to emphasize here that the aim of a segmentation technique is
not to come up with the best quoted version of a query. While some past works
have explicitly or implicitly assumed this definition, there are also other
works that view segmentation as a purely structural analysis of a query that
identifies chunks or sequences of words that are semantically connected as a
unit [9, 11]. By quoting all the segments we would be penalizing the latter
philosophy of segmentation, which is a more productive and practically useful
view.
There have been a few studies on detection of noun phrases from queries [5,
16]. This task is similar to query segmentation in the sense that the phrase
can be considered as a single unit in the query. Zhang et al. [16] has shown
that such phrase detection schemes can actually help in retrieval, and
therefore, is along the lines of the philosophy of the present evaluation
framework. Nevertheless, as far as we know, this is the first time that a
formal conceptual framework for an IR-based evaluation of query segmentation
has been proposed. Our study, also for the first time, compares the
effectiveness of human segmentation and related matching metrics to an IR-
based evaluation.
## 7 Conclusions and future work
End-user of query segmentation is the retrieval engine; hence, it is essential
that any segmentation algorithm should be evaluated in an IR-based framework.
In this research, we overcome several conceptual challenges to design and
implement the first such scheme of evaluation for query segmentation. Using a
carefully selected query test set and a group of segmentation strategies, we
show that it is possible to have a fair comparison of the relative goodness of
each strategy as measured by standard IR metrics. The proposed framework uses
resources which are essential for any IR system evaluation, and hence does not
require any special input. Our entire dataset – complete with queries,
segmentation outputs and relevance judgments – has also been made publicly
available to facilitate further research by the community.
Moreover, we gain several useful and non-intuitive insights from the
evaluation experiments. Most importantly, we show that human notions of query
segments may not be the best for maximizing retrieval performance, and
treating them as the gold standard limits the scope for improvement for an
algorithm. Also, the matching metrics extensively used till date for comparing
against gold standard segmentations can often be misleading. We would like to
emphasize that in the future, the focus of IR will mostly shift to tail
queries. In such a scenario, an IR-based evaluation scheme gains relevance
because validation against a fixed set of gold standard segmentation may often
lead to overfitting of the algorithms without yielding any real benefit.
A hypothetical oracle has been shown to be quite useful, but we realize that
it will be a much bigger contribution to the community if we could implement a
context-aware oracle that can actually tell the search engine which version of
a segmented query should be chosen at runtime.
## 8 Acknowledgments
We would like to thank Bo-June (Paul) Hsu and Kuansan Wang (Microsoft
Research, Redmond), for providing us with the code for Li et al. [9]. We also
thank Matthias Hagen (Bauhaus Universität Weimar), for providing us with the
segmentation output of Hagen et al. [7] on our test set at a very short
notice. The first author was supported by Microsoft Corporation and Microsoft
Research India under the Microsoft Research India PhD Fellowship Award.
## References
* [1] P. Bailey, N. Craswell, I. Soboroff, P. Thomas, A. P. de Vries, and E. Yilmaz. Relevance assessment: are judges exchangeable and does it matter. In SIGIR ’08, pages 667–674. ACM, 2008.
* [2] M. Bendersky, W. B. Croft, and D. A. Smith. Two-stage query segmentation for information retrieval. In SIGIR ’09, pages 810–811. ACM, 2009.
* [3] S. Bergsma and Q. I. Wang. Learning noun phrase query segmentation. In EMNLP-CoNLL’07, pages 819–826, 2007.
* [4] D. J. Brenes, D. Gayo-Avello, and R. Garcia. On the fly query segmentation using snippets. In CERI ’10, pages 259–266, 2010.
* [5] A. L. da Costa Carvalho, E. S. de Moura, and P. Calado. Using statistical features to find phrasal terms in text collections. JIDM, 1(3):583–597, 2010.
* [6] M. Hagen, M. Potthast, B. Stein, and C. Bräutigam. The power of naive query segmentation. In SIGIR ’10, pages 797–798. ACM, 2010.
* [7] M. Hagen, M. Potthast, B. Stein, and C. Bräutigam. Query segmentation revisited. In WWW ’11, pages 97–106, 2011.
* [8] M. Lease, J. Allan, and W. B. Croft. Regression rank: Learning to meet the opportunity of descriptive queries. In Proceedings of the 31th European Conference on IR Research on Advances in Information Retrieval, ECIR ’09, pages 90–101, Berlin, Heidelberg, 2009. Springer-Verlag.
* [9] Y. Li, B.-J. P. Hsu, C. Zhai, and K. Wang. Unsupervised query segmentation using clickthrough for information retrieval. In SIGIR ’11, pages 285–294. ACM, 2011.
* [10] D. Metzler and W. B. Croft. A markov random field model for term dependencies. In SIGIR’05, pages 472–479, 2005.
* [11] N. Mishra, R. Saha Roy, N. Ganguly, S. Laxman, and M. Choudhury. Unsupervised query segmentation using only query logs. In WWW ’11, pages 91–92. ACM, 2011.
* [12] K. M. Risvik, T. Mikolajewski, and P. Boros. Query segmentation for web search. In WWW (Posters), 2003.
* [13] B. Tan and F. Peng. Unsupervised query segmentation using generative language models and wikipedia. In WWW ’08, pages 347–356. ACM, 2008.
* [14] E. M. Voorhees. Variations in relevance judgments and the measurement of retrieval effectiveness. Inf. Process. Manage., 36:697–716, September 2000.
* [15] C. Zhang, N. Sun, X. Hu, T. Huang, and T.-S. Chua. Query segmentation based on eigenspace similarity. In ACL/AFNLP (Short Papers)’09, pages 185–188, 2009.
* [16] W. Zhang, S. Liu, C. Yu, C. Sun, F. Liu, and W. Meng. Recognition and classification of noun phrases in queries for effective retrieval. In CIKM ’07, pages 711–720. ACM, 2007.
## APPENDIX A: WIKI-BOOST
Algorithm 1 Wiki-Boost($Q^{\prime}$, $W$)
1: $W^{\prime}\leftarrow\emptyset$
2: for all $w\in W$ do
3: $w^{\prime}\leftarrow Seg\mathchar 45\relax Phase\mathchar 45\relax 1(w)$
4: $W^{\prime}\leftarrow W^{\prime}\cup w^{\prime}$
5: end for
6: $W^{\prime}\mathchar 45\relax scores\leftarrow\emptyset$
7: for all $w^{\prime}\in W^{\prime}$ do
8: $w^{\prime}\mathchar 45\relax score\leftarrow
PMI(w^{\prime})\;based\;on\;Q^{\prime}$
9: $W^{\prime}\mathchar 45\relax scores\leftarrow W^{\prime}\mathchar 45\relax
scores\cup w^{\prime}\mathchar 45\relax score$
10: end for
11: $U\mathchar 45\relax scores\leftarrow\emptyset$
12: for all $unique\;unigrams\;u\in Q^{\prime}$ do
13: $u\mathchar 45\relax score\leftarrow probability(u)\;in\;Q^{\prime}$
14: $U\mathchar 45\relax scores\leftarrow U\mathchar 45\relax scores\cup
u\mathchar 45\relax score$
15: end for
16: $W^{\prime}\mathchar 45\relax scores\leftarrow W^{\prime}\mathchar
45\relax scores\cup U\mathchar 45\relax scores$
17: return $W^{\prime}\mathchar 45\relax scores$
In this appendix, we explain how to augment the output of an $n$-gram score
aggregation based segmentation algorithm with Wikipedia
titles888http://dumps.wikimedia.org/enwiki/latest/, accessed April 6, 2011.
Input to Wiki-Boost is a list of queries $Q^{\prime}$ already segmented by the
algorithm in Mishra et al. [11] (or any algorithm that meets the above
criterion) (say, Seg-Phase-1) and $W$, the list of all stemmed Wikipedia
titles ($4,508,386$ entries after removing one-word entries and those with
non-ASCII characters). We compute the PMI-score of an $n$-segment Wikipedia
title $w^{\prime}$ (segmented by Seg-Phase-1) by taking the higher of the PMI
scores of the first $(n-1)$ segments with the last segment and the first
segment and the last $(n-1)$ segments. The frequencies of all $n$-grams are
computed from $Q^{\prime}$. Scores for unigrams are defined to be their
probabilities of occurrence. Thus, the output of the Wiki-Boost is a list of
PMI-scores for each Wikipedia title in $W$.
Following this, we use a second segmentation strategy (say, Seg-Phase-2) that
takes as input $q^{\prime}$ (the query $q$ segmented by Seg-Phase-1) and tries
to further join the segments of $q^{\prime}$ such that the product of scores
of the candidate output segments, computed based on the output of Wiki-Boost,
is maximized. A dynamic programming approach is found to be helpful in
searching over all possible segmentations in Seg-Phase-2\. The output of Seg-
Phase-2 is the final segmentation output.
|
arxiv-papers
| 2011-11-07T07:26:27 |
2024-09-04T02:49:24.053738
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rishiraj Saha Roy, Niloy Ganguly, Monojit Choudhury and Srivatsan\n Laxman",
"submitter": "Rishiraj Saha Roy",
"url": "https://arxiv.org/abs/1111.1497"
}
|
1111.1636
|
# An alternative solution to the $\gamma$-ray Gradient problem
D. Gaggero INFN Pisa and Pisa University, Largo B. Pontecorvo 3, I-56127
Pisa, Italy C. Evoli Institut für Theoretische Physik, Universität Hamburg,
Luruper Chaussee 149, D-22761 Hamburg, Germany. D. Grasso INFN Pisa, Largo
B. Pontecorvo 3, I-56127 Pisa, Italy L. Maccione Ludwig-Maximilians-
Universität, Fakultät für Physik, Theresienstraße 37, D-80333 München, Germany
Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, Föhringer Ring 6,
D-80805 München, Germany
###### Abstract
The Fermi-LAT collaboration recently confirmed EGRET finding of a discrepancy
between the observed longitudinal profile of $\gamma$-ray diffuse emission
from the Galaxy and that computed with GALPROP assuming that cosmic rays are
produced by Galactic supernova remnants. The accurate Fermi-LAT measurements
make this anomaly hardly explainable in terms of conventional diffusion
schemes. Here we use DRAGON numerical diffusion code to implement a physically
motivated scenario in which the diffusion coefficient is spatially correlated
to the source density. We show that under those conditions we are able to
reproduce the observed flat emissivity profile in the outer Galaxy with no
need to change the source term, the diffusion halo height, or the CO-${\rm
H_{2}}$ conversion factor (${\rm X_{CO}}$) with respect to their preferred
values/distributions. We also show that our models are compatible with gamma-
ray longitudinal profiles measured by Fermi-LAT, and still provide a
satisfactory fit of all observed secondary-to-primary ratios, such as B/C and
antiprotons/protons.
## I Introduction
It has been known since the EGRET era that, if one computes the cosmic ray
(CR) Galactocentric radial distribution adopting a source function deduced
from pulsar or supernova remnant (SNR) catalogues, the result appears much
steeper than the profile inferred from the $\gamma$-ray diffuse emission along
the Galactic plane: the latter appears flatter, with a high contribution from
large Galactic radii. This discrepancy is known as the $\gamma$-ray gradient
problem. A sharp rise of the conversion factor between CO emissivity and ${\rm
H_{2}}$ density (the so called ${\rm X_{CO}}$) with the Galactocentric radius
was invoked at the time to fix the problem (Strong et al., 2004): a larger gas
density at large radii compensates for the decreasing CR population and is
able to explain the $\gamma$-ray flux detected at high Galactic longitudes.
Fermi-LAT confirmed the existence of such a problem (Ackermann et al., 2011).
Moreover, the high spatial resolution of the LAT permitted to disentangle the
emission coming from the interaction of CRs with the molecular gas (whose
modelling is strongly affected by the uncertainty on the ${\rm X_{CO}}$) from
the emission originated by the interaction of the Galactic CRs with the atomic
gas (whose density is better known from its 21 cm radio emission). An analysis
based on $\gamma$-ray maps of the third Galactic quadrant (Ackermann et al.,
2011) pointed out that the $\gamma$-ray emissivity from neutral gas (tracing
the actual CR density) is indeed flatter than the predicted one confirming the
gradient problem independently of the ${\rm X_{CO}}$. This result led the
authors of (Ackermann et al., 2011) to look for alternative explanations of
the problem, e.g. invoking a thick CR diffusion halo or a source term that
becomes flatter at large radii. Both solutions, however, do not appear
completely satisfactory: a thick halo is disfavoured both from 10Be/9Be and
synchrotron data; a smooth source distribution is in contrast with SNR
catalogues.
Here we consider a different interpretation based on relaxing the
approximation of isotropic and spatially uniform diffusion.
## II Inhomogeneous and anisotropic diffusion
Nearly all CR diffusion models presented in the literature adopt an isotropic
and spatially uniform diffusion coefficient throughout all the Galaxy. This is
the case, for example, of GALPROP numerical package on which the predictions
of (Ackermann et al., 2011) are based. It is reasonable, however, to expect
that CR diffusion is not isotropic in the Galaxy. This could be the
consequence either of Galactic winds (Gebauer and de Boer, 2009) or just of
the anisotropy of the regular component of the Galactic magnetic field which
is oriented almost azimutally along the Galactic plane. The former possibility
was suggested as a possible solution of the $\gamma$-ray gradient problem
originated by EGRET observations (Breitschwerdt et al., 2002). Here we
consider the latter option and extend also to the recent Fermi-LAT data the
arguments we developed in [Evoli et al. 2008] to interpret earlier EGRET
measurements. Our approach is based on the consideration that, for geometrical
reasons, CRs should escape from the Galaxy almost perpendicularly to the
Galactic plane: their density, therefore, should be determined by the
perpendicular component of the diffusion coefficient $D_{\perp}$. We know –
both from quasi linear diffusion theory and from more realistic numerical
simulations (DeMarco et al., 2007) – that $D_{\perp}$ should increase with
increasing strength of the Galactic magnetic field turbulent component; from a
physical point of view, such behaviour can be understood in terms of magnetic
field line random walk becoming stronger when the turbulence strength
increases. As a consequence, the regions where CR injection is more intense
should also be those characterized by a stronger MHD turbulence and hence a
faster CR escape along the $z$ axis: this should smooth the CR gradient, and
hence the $\gamma$-ray profile, in a rather natural way.
In the next section we will show this effect by means of dedicated numerical
computations.
## III Our method and results
Figure 1: Two different CR proton distribution maps in arbitrary units
computed with DRAGON at $10$ GeV are shown as functions of the Galactic
cylindrical coordinates $R$ and $z$. Panel a) The proton distribution is
computed with no radial dependence of diffusion coefficient. Panel b) Here the
diffusion coefficient is correlated to the source term: $D\propto Q^{\tau}$,
with $\tau=0.8$. The model shows a significant flattening in the CR profile
along $R$. The normalization is fixed at $R_{\rm Sun}=8.5$ kpc in both cases;
notice how the maximum proton density is reduced by a factor $\simeq 2$ in the
second panel.
Figure 2: Two $gamma$-ray longitudinal profiles along the Galactic plane
computed with DRAGON and GammaSky and compared to preliminary Fermi-LAT data
extracted from the talk by A.W.Strong at the Workshop on Indirect Dark Matter
Searches, DESY, Hamburg, June 2011 (http://www.mpe.mpg.de/ aws/talks/). Data
are integrated over the latitude interval $-5^{\circ}<b<+5^{\circ}$ and in
energy between 1104 and 1442 MeV. Red line: IC. Green line: Bremsstrahlung.
Blue line: $\pi^{0}$ decay. Purple line: contribution from unresolved sources.
Grey line: $\pi^{0}$ \+ IC + Bremsstrahlung. Black line: total. Panel a) The
profile is computed with no radial dependence of diffusion coefficient. Panel
b) Here the diffusion coefficient follows the source term: $D\propto
Q^{\tau}$, with $\tau=0.8$. Figure 3: Here the effect of the parameter $\tau$
defined by equation 1 is explored. Dotted line: no radial dependence of
diffusion coefficient ($\tau=0$). Dot-dashed line: $\tau=0.2$. Dashed line:
$\tau=0.5$. Solid lines: $\tau=0.7$ – $0.8$ – $1.0$. The values corresponding
to the solid lines within the grey band match the observed gradient.
In this section we use DRAGON numerical diffusion package111The DRAGON code
for cosmic-ray transport and diffuse emission production is available online
at http://www.desy.de/~maccione/DRAGON/ to solve the diffusion equation in the
presence of a diffusion coefficient spatially correlated to the CR source term
$Q(r,z)$. We perform our analysis using a Plain Diffusion (PD) setup with no
convection and no reacceleration in order to better highlight the effects of
inhomogeneous diffusion; similar results may be obtained with different
choices of the diffusion parameters, and a more detailed study on the effects
of another setup will be performed in a forthcoming paper. The CR propagation
model adopted here is basically the same as the PD model described in (Di
Bernardo et al., 2011); the astrophysical parameters (in particular the source
term, gas distribution and ${\rm X_{CO}}$) are also the same used in that
analysis. Only the normalization of the proton injection spectrum is slightly
tuned to match the recently released proton spectrum measured by the PAMELA
collaboration (Adriani et al., 2011). The model is also compatible with most
other CR data sets. Only a little excess in the antiproton spectrum must be
pointed out which, however, is still compatible with data if astrophysical and
particle physics uncertainties are taken into account. As we mentioned, the CR
distribution is computed with DRAGON: this numerical package is suitable for
our purpose since, differently from GALPROP, it implements the possibility to
vary the diffusion coefficient through the Galaxy. The CR distributions is
then used as an input to compute the $\gamma$-ray longitude profile along the
Galactic plane; the $\gamma$-ray map is evaluated with a separate package
called GammaSky.
The result of a combined DRAGON and GammaSky computation in the case of a
uniform diffusion coefficient and a PD setup is shown in Fig. 2 (panel a). It
is clear from that plot that the predicted longitude profile is too steep
compared to the observations: in the Galactic center region the model
prediction overshoots the data and in the anti-center region the model is
lower than the observations by several $\sigma$. Tuning the ${\rm X_{CO}}(R)$
could help in principle: assuming a lower value of this parameter in the bulge
and a high value at large $R$ could smooth the $\gamma$-ray profile (as done
in several previous works such as (Strong et al., 2004)). Unfortunately, as
pointed out in the introduction, the gradient problem is present especially in
the emissivity profile, and this quantity is independent of the molecular gas:
it only traces the actual CR distribution222The emissivity is the number of
$\gamma$ photons emitted by each gas atom per unit time and unit energy.
So we apply our previous considerations and adopt a diffusion coefficient
correlated to the radial dependence of the source term $Q(R)$ by the following
expression:
$D(R)\,\propto\,Q(R)^{\tau}$ (1)
This is the parametrization we already used in (Evoli et al., 2008) to
interpret EGRET data. The parameter $\tau$ is tuned against data. In Fig. 3 we
show the emissivity profile for different values of $\tau$ in the range
${\rm[0\div 1]}$. It is evident from that figure that an increasing value of
$\tau$ yields a much smoother behaviour of the emissivity as function of $R$.
Values in the range ${\rm[0.7\div 1]}$ allow a good match of Fermi-LAT data
((Ackermann et al., 2011), (Abdo et al., 2010)).
With this result at hand, we considered a modified version of the Plain
Diffusion CR propagation setup with $D(R)=Q^{\tau}$ and $\tau=0.8$. The
smoothing in the CR distribution corresponding to such a value of $\tau$ is
shown in Fig. 1. As shown in Fig. 2 (panel b), the $\gamma$ ray longitude
profile along the Galactic plane is nicely reproduced with no tuning at all of
the ${\rm X_{CO}}$. It is remarkable that a simple CR propagation setup, with
only the addition of the radial dependence of $D$ and no ad hoc tuning,
permits to reproduce the $\gamma$-ray profile with such accuracy. Noticeably,
the modified model is still compatible with most relevant CR data set, most
importantly the B/C. Furthermore, we checked that the $\gamma$-ray spectrum
measured by Fermi-LAT along the Galactic plane is also correctly reproduced
under those conditions (see Fig. 4).
Figure 4: The gamma-ray spectrum corresponding to the plain diffusion model
with varying diffusion coefficient described in the text ($\tau=0.8$). The
spectrum was computed with DRAGON and GammaSky. The data points measured by
Fermi-LAT are taken from the same reference as Fig. 2
## IV Conclusions
In this paper we presented an alternative solution to the well known
$\gamma$-ray gradient problem. Our approach is based on the physically
motivated hypothesis that the CR diffusion coefficient is spatially correlated
to the source density: regions in which star, hence SNR, formation is stronger
are expected to present a stronger turbulence level and therefore a larger
value of the perpendicular diffusion coefficient. This effect favours CR
escape from most active regions helping to smooth their density through the
Galaxy hence also the $\gamma$-ray gradient. We used DRAGON package to
implement this scenario and to check that CR data are still reproduced under
those conditions. In spite of being purely phenomenological (as a self-
consistent theory/computation of non-linear CR - MHD turbulence interaction in
the Galaxy is far from being developed) our approach provides a remarkably
good description of the spectrum and longitude distribution of the diffuse
$\gamma$-ray emission measured by the Fermi-LAT collaboration.
###### Acknowledgements.
D. Gaggero would like to thank the LAPTH (Laboratoire d’Annecy-le-Vieux de
Physique Théorique) for hosting him during the last part of the work presented
in this paper.
## References
* Strong et al. (2004) A. W. Strong, I. V. Moskalenko, O. Reimer, S. Digel, and R. Diehl, Astronomy and Astrophysics 422, L47 (2004), eprint http://arxiv.org/abs/astro-ph/0405275.
* Ackermann et al. (2011) M. Ackermann, M. Ajello, L. Baldini, J. Ballet, G. Barbiellini, D. Bastieri, K. Bechtol, R. Bellazzini, B. Berenji, E. D. Bloom, et al., Astrophysical Journal 726, 81 (2011), eprint http://arxiv.org/abs/1011.0816.
* Gebauer and de Boer (2009) I. Gebauer and W. de Boer (2009), * Brief entry *, eprint http://arxiv.org/abs/0910.2027.
* Breitschwerdt et al. (2002) D. Breitschwerdt, V. A. Dogiel, and H. J. Völk, Astronomy and Astrophysics 385, 216 (2002), eprint http://arxiv.org/abs/astro-ph/0201345.
* DeMarco et al. (2007) D. DeMarco, P. Blasi, and T. Stanev, Journal of Cosmology and Astroparticle Physics 6, 27 (2007), eprint http://arxiv.org/abs/0705.1972.
* Di Bernardo et al. (2011) G. Di Bernardo, C. Evoli, D. Gaggero, D. Grasso, L. Maccione, and M. N. Mazziotta, Astroparticle Physics 34, 528 (2011), eprint http://arxiv.org/pdf/1010.0174v2.
* Adriani et al. (2011) O. Adriani, G. C. Barbarino, G. A. Bazilevskaya, R. Bellotti, M. Boezio, E. A. Bogomolov, L. Bonechi, M. Bongi, V. Bonvicini, S. Borisov, et al., Science 332, 69 (2011), eprint http://arxiv.org/abs/1103.4055.
* Evoli et al. (2008) C. Evoli, D. Gaggero, D. Grasso, and L. Maccione, Journal of Cosmology and Astroparticle Physics 10, 18 (2008), eprint http://arxiv.org/abs/0807.4730.
* Abdo et al. (2010) A. A. Abdo, M. Ackermann, M. Ajello, L. Baldini, J. Ballet, G. Barbiellini, D. Bastieri, B. M. Baughman, K. Bechtol, R. Bellazzini, et al., Astrophysical Journal 710, 133 (2010), eprint http://arxiv.org/abs/0912.3618.
|
arxiv-papers
| 2011-11-07T16:35:19 |
2024-09-04T02:49:24.074652
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Daniele Gaggero and Carmelo Evoli and Dario Grasso and Luca Maccione",
"submitter": "Daniele Gaggero",
"url": "https://arxiv.org/abs/1111.1636"
}
|
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