Content-Type: multipart/mixed; boundary="-------------0507280549594" This is a multi-part message in MIME format. ---------------0507280549594 Content-Type: text/plain; name="05-257.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-257.keywords" Fluctuation theorem, nonequilibrium, chaotic motions, statistical mechanics ---------------0507280549594 Content-Type: application/x-tex; name="bggz10.bbl" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="bggz10.bbl" \begin{thebibliography}{39} \expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi \expandafter\ifx\csname bibnamefont\endcsname\relax \def\bibnamefont#1{#1}\fi \expandafter\ifx\csname bibfnamefont\endcsname\relax \def\bibfnamefont#1{#1}\fi \expandafter\ifx\csname citenamefont\endcsname\relax \def\citenamefont#1{#1}\fi \expandafter\ifx\csname url\endcsname\relax \def\url#1{\texttt{#1}}\fi \expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi \providecommand{\bibinfo}[2]{#2} \providecommand{\eprint}[2][]{\url{#2}} \bibitem[{\citenamefont{Ruelle}(1999)}]{Ru99} \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Ruelle}}, \bibinfo{journal}{Journal of Statistical Physics} \textbf{\bibinfo{volume}{95}}, \bibinfo{pages}{393} (\bibinfo{year}{1999}). \bibitem[{\citenamefont{G.Gallavotti}(1999)}]{Ga99} \bibinfo{author}{\bibnamefont{G.Gallavotti}}, \bibinfo{journal}{Open Systems and Information Dynamics} \textbf{\bibinfo{volume}{6}}, \bibinfo{pages}{101} (\bibinfo{year}{1999}). \bibitem[{\citenamefont{Gallavotti}(2000)}]{Ga00} \bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Gallavotti}}, \emph{\bibinfo{title}{Statistical Mechanics. 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name="bggz10.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="bggz10.tex" \documentclass[pre,twocolumn,showpacs,superscriptaddress,floatfix]{revtex4} %\documentclass[pre,preprint,showpacs,superscriptaddress,floatfix]{revtex4} \newcount\tipo\tipo=1 %1 \usepackage{amsmath} \usepackage{graphicx} \voffset1.3truecm \let\a=\alpha\let\b=\beta \let\g=\gamma \let\d=\delta \let\e=\varepsilon \let\z=\zeta \let\h=\eta \let\th=\vartheta\let\k=\kappa \let\l=\lambda \let\m=\mu \let\n=\nu \let\x=\xi \let\p=\pi \let\r=\rho \let\s=\sigma \let\t=\tau \let\iu=\upsilon \let\f=\varphi\let\ch=\chi \let\ps=\psi \let\o=\omega \let\y=\upsilon \let\G=\Gamma \let\D=\Delta \let\Th=\Theta \let\L=\Lambda\let\X=\Xi \let\P=\Pi \let\Si=\Sigma \let\F=\Phi \let\Ps=\Psi \let\O=\Omega \let\U=\Upsilon %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \let\dpr=\partial\let\0=\noindent\let\fra=\frac \def\media#1{\langle{#1}\rangle}\let\==\equiv \def\*{\vskip3mm}\def\eg{{\it e.g. }}\def\ie{{\it i.e.\ }} \def\V#1{{\bf#1}}\let\io=\infty\let\ig=\int \def\otto{\,{\kern-1.truept\leftarrow\kern-5.truept\to\kern-1.truept}\,} \let\wt=\widetilde\def\Defi{\,{\buildrel def\over=}\,} \def\Onlinecite#1{[\onlinecite{#1}]} \def\tende#1{\ \vtop{\ialign{##\crcr\rightarrowfill\crcr \noalign{\kern-1pt\nointerlineskip} \hglue3.pt${\scriptstyle% #1}$\hglue3.pt\crcr}}\,} \newcommand\defi{\,{\buildrel def \over =}\,} \newcommand\revtex{{R\kern-1mm\lower0.5mm\hbox{E}\kern-0.6mm V\kern-0.5mm% \lower0.5mm\hbox{T}\kern-0.5mm E\kern-.5mm \lower0.5mm\hbox{X}}} % \begin{document} \preprint{FM 04-04} \title{Chaotic Hypothesis, Fluctuation Theorem and singularities} \author{F.~Bonetto} \affiliation{School of Mathematics, Georgia Institute of Technology, Atlanta Georgia 30332} \affiliation{Dipartimento di Matematica, Universit\`a di Roma {\em Tor Vergata}, V.le della Ricerca Scientifica, 00133, Roma, Italy} \author{G.~Gallavotti}\affiliation{INFN, Universit\`a di Roma {\em La Sapienza}, P.~A.~Moro 2, 00185, Roma, Italy} \affiliation{Dipartimento di Fisica, Universit\`a di Roma {\em La Sapienza}, P.~A.~Moro 2, 00185, Roma, Italy} \author{A.~Giuliani} \affiliation{Dipartimento di Matematica, Universit\`a di Roma {\em Tor Vergata}, V.le della Ricerca Scientifica, 00133, Roma, Italy} \affiliation{INFN, Universit\`a di Roma {\em La Sapienza}, P.~A.~Moro 2, 00185, Roma, Italy} \author{F.~Zamponi} \affiliation{Dipartimento di Fisica, Universit\`a di Roma {\em La Sapienza}, P.~A.~Moro 2, 00185, Roma, Italy} \affiliation{SOFT-INFM-CNR, Universit\`a di Roma {\em La Sapienza}, P.~A.~Moro 2, 00185, Roma, Italy} %\date{\today} \relax \begin{abstract} Recently the ``Fluctuation theorem'' has been criticized and incorrect contents have been attributed to it. Here we reestablish and comment the original statements. We also discuss aspects of the extension of the chaotic hypothesis and of the fluctuation relation to singular systems. \end{abstract} \pacs{47.52.+j, 05.45.-a, 05.70.Ln, 05.20.-y} \maketitle %\keywords{Nonequilibrium Thermodynamics, Entropy, Temperature} \section{Anosov systems} \0{\bf Fluctuations} \vskip3mm Mathematically the {\it Fluctuation theorem} is a property of the phase space contraction of a {\it time reversible} Anosov map $S$, modeling a {\it time evolution}. The possible connection between the fluctuation theorem and Physics is a different matter that we will not discuss here: there are many places where this is attempted, \Onlinecite{Ru99,Ga99,Ga00}. We shall denote by $\O$ the {\it phase space} (a smooth finite boundaryless Riemannian manifold) and by $\s(x)$ the volume {\it contraction} \begin{equation}\s(x)=-\log |\det \dpr_xS(x)|\label{1}\end{equation} {\it Time reversal} is defined as an isometry $I: \O\otto\O$ with \begin{equation}I S = S^{-1}I, \qquad \s(Ix)=-\s(x)\label{2}\end{equation} For Anosov maps, existence of a unique invariant probability distribution $\m$, called the {\it SRB distribution} and describing the long--time statistics of the motions whose initial data are chosen randomly with respect to the volume measure, is established, \cite{Ru95,GBG04}. It has the property that, with the exception of points $x\in\O$ in a set of $0$--volume, it is \begin{equation}\lim_{\t\to\infty} \fra1\t \sum_{t=0}^{\t-1} F(S^tx)\defi \media{F}=\int_\O F(y)\m(dy)\label{3}\end{equation} for all smooth observables $F$ defined on phase space. It is intuitive that ``phase space cannot expand''; this is expressed by the following result of Ruelle, \Onlinecite{Ru96}, \* {\bf Proposition}: {\it If $\s_+\defi\media{\s}$ it is $\s_+\ge0$} \* Clearly if $S$ is volume preserving it is $\s_+=0$. If $\s_+>0$ the system does not admit any stationary distribution of the form $\m(dx)=\r(x)dx$, with density with respect to the volume measure $dx$ (often called {\it absolutely continuous} with respect to the volume). This motivates calling systems for which $\media{\s}>0$ {\it dissipative} and {\it conservative} the others. For Anosov systems which are {\it transitive} ({\it i.e.} with a dense orbit), reversible and dissipative one can define the dimensionless phase space contraction, a quantity often related to entropy creation rate (see \Onlinecite{Ga04b}), averaged over a time interval of size $\t$. This is \begin{equation}p(x)= \fra1{\s_+ \t }\sum_{-\t/2}^{\t/2-1} \s(S^k x)\label{4}\end{equation} provided {\it of course} $\s_+>0$. Then for such systems the probability with respect to the stationary state, {\it i.e.} to the SRB distribution $\m$, that the variable $p(x)$ takes values in $\D=[p,p+\d p]$ can be written as $\P_\t(\D)=e^{\t \max_{p\in \D}\z(p) + O(1)}$, where $\z(p)$ is a suitable function and, for any fixed choice of $\D$ contained in an open interval $(-p^*,p^*)$, $p^*\ge 1$, the correction term at the exponent is $O(1)$ with respect to $\t^{-1}$, as $\t\to\io$ (this is often informally expressed as $\lim_{\t\to\infty}\fra1\t \log \P_\t(p)=\z(p)$ for $\ -p^*
p^*\ge1$ and $\z(p)=-\infty$ for $|p|> p^*$. Furthermore \begin{equation}\z(-p)=\z(p)-p\s_+, \qquad {\rm for} \qquad |p|
0$, because $\t^{-1}\sum_{-\t/2}^{\t/2-1} \s(S^k x)=\t^{-1}\big[\f(S^{\t/2-1}x)-\f(S^{-\t/2}x)\big]\to 0$ as $\t\to\infty$. The value of $p^*$ must be $p^*\ge1$ otherwise the average of $p$ could not be $1$ (as it is by its very definition): it is defined, adopting as in \cite{GC95,GC95b} and in the proposition above the natural convention that $\z(p)=-\io$ for the values of $p$ whose probability goes to $0$ with $\tau$ faster than exponentially, as the infimum of the $p>0$ for which $\z(p)=-\io$. Alternatively $\pm p^*$ are the asymptotic slopes as $\l\to\pm\io$ of the Laplace transform $\log \media{e^{\l p}}_{SRB}$ \cite{Ga95b}. The fluctuation relation was discovered in a numerical experiment, \Onlinecite{ECM93}, dealing with a non smooth system (hence not Anosov). The formulation and proof of the above proposition is in \Onlinecite{GC95} and in the context of Anosov systems the relation (\ref{5}) is properly called the {\it fluctuation theorem} (a name later given to other very different relations with remarkable confusion, \Onlinecite{CG99}). The theorem can be extended to Anosov flows ({\it i.e.} to systems evolving in continuous time), \cite{Ge98}. \* \0{\bf Alternative formulations} \vskip3mm Sometimes, {\it e.g.} in \cite{SE00,ESR03}, rather than the above $p$ the quantity $a=\t^{-1}\sum_{j=-\t/2}^{\t/2-1} \s(S^j x)$ is considered and eq.(\ref{5}) becomes \begin{equation}\widetilde\z(-a)=\widetilde\z(a)-a,\qquad {\rm for}\ |a|< a^*\= p^*\s_+\label{6}\end{equation} where $\widetilde\z(a)$ is trivially related to $\z(p)$. This form dangerously suggests that in systems with $\s_+=0$ the distribution of $a$ is asymmetric (because the extra condition $|a|
0$ but one would reach the
conclusion that $\widetilde \z(a)=-\infty$ for $|a|>0$ and we see that
the result is trivial. In fact in this case it follows that the
system admits an absolutely continuous SRB distribution. The
distribution of $a$ is symmetric (trivially by time reversal
symmetry) and becomes a delta function around $0$ as $\t\to\infty$.
Nevertheless the fluctuation relation is {\it non trivial} in cases in
which the map $S$ depends on parameters $\underline
E=(E_1,\ldots,E_n)$ and becomes volume preserving (``conservative'')
as $\underline E\to \underline0$: in this case $\s_+\to0$ as
$\underline E\to\underline 0$ and one has to rewrite the fluctuation
relation in an appropriate way to take a meaningful limit.
The result is that the limit as $\underline E\to\underline 0$
of the fluctuation relation
in which both sides are divided by $\underline E^2$ makes sense and yields (in
the case considered here of transitive Anosov dynamical systems)
relations which are non trivial and that can be interpreted as giving
Green--Kubo formulae and Onsager reciprocity for transport
coefficients, \Onlinecite{Ga96a,GR97}.
In fact the very definition of the duality between currents and fluxes
so familiar in nonequilibrium thermodynamics since Onsager can be set
up in such systems using as generating function the $\s_+$ regarded as
a function of $\underline E$. Note that the fluxes are usually ``currents''
divided by the temperature: therefore via the above interpretation one
can try to define the temperature even in nonequilibrium situations,
\Onlinecite{GC04,Ga04b,ZCK05}.
\*
\section{Singular systems}
The fluctuation relation has been proved only for Anosov
systems.
However, a {\it Chaotic Hypothesis} has been proposed, which
states that, {\it for the purpose of studying the physically
interesting observables},
a chaotic dynamical system can be considered
as an Anosov system, \cite{GC95,GC95b,Ga00}.
In applying the chaotic hypothesis to singular
systems, \eg a system of particles interacting via a
Lennard-Jones potential (which is infinite in the origin),
one might encounter apparent difficulties. We will discuss them in
the following.
\vskip3mm
\0{\bf A simple example: Anosov flows}
\vskip3mm
The simplest example (out of many) is provided by the simplest
conservative system which is strictly an Anosov transitive system and
which has therefore an SRB distribution: this is the geodesic flow $S_t$
on a surface of constant negative curvature, \Onlinecite{BGM98}. We
discuss here an evolution in continuous time because the matter is
considered in the literature for such systems, \Onlinecite{Ga04} (even
simpler examples are possible for time evolution maps).
The phase space $M$ is compact, time reversal is just momentum
reversal and the natural metric, induced by the Lobatchevsky metric
$g_{ij}(q)$ on the surface, is time reversal invariant: the SRB
distribution is the Liouville distribution and $\s(x)\equiv0$. However
one can introduce a function $\F(x)$ on $M$ which is very large in a
small vicinity of a point $x_0$, arbitrarily selected, constant
outside a slightly larger vicinity of $x_0$ and positive everywhere. A
new metric could be defined as $g_{new}(x)=(\F(x)+\F(Ix)) g(x)\defi
e^{-F(x)} g(x)$: it is still time reversal invariant but its volume
elements {\it will no longer be invariant} under the time evolution
$S_t$ associated with the geodesic flow with respect to the
Lobatchevsky metric. The
rate of change of phase space volume in the new metric will be
$\sigma_{new}(x)=\frac{d}{dt}F(x)$. Then the
phase space contraction $\s_{new}(x)$ takes values that not only are
not identically $0$ but which can in general be arbitrarily large,
depending on the specific choice of $\Phi(x)$.
The distribution of $a=\fra1\t \int_0^\t
\s_{new}(S_t x) dt= \t^{-1} \big[F(S_\t x)-F(x)\big]$,
at any finite time, will violate (\ref{6}), simply because it is symmetric
around $0$, by time reversal.
In the limit $\t\to\io$, as long as $F(x)$ is bounded,
$a= \t^{-1} \big[F(S_\t
x)-F(x)\big]\tende{\t\to\infty}0$ uniformly in $x$,
as in the corresponding map case,
and the SRB
distribution of $a$ will tend to a delta function centered in $0$ (hence
$\widetilde\z(a)=-\infty$ for $a\ne0$). However, if $F(x)$ is not bounded
(\eg if it is allowed to become infinite in $x_0$) the distribution of
$a$ could be different from a delta function centered in $0$ also for
conservative systems, yielding a finite distribution $\widetilde
\z(a)$. One might be tempted to apply the fluctuation relation to it,
but misleading results could be obtained, as will be discussed in the
following.
\vskip3mm
\0{\bf The effect of singular boundary terms}
\vskip3mm
One can realize that terms of the form $\t^{-1}\big[F(S_\t x)-F(x)\big]$ with
$F(x)$ not bounded can affect the large fluctuations of $\s(x)$, at least if
the probability of an arbitrarily large value of $F$ is not too small, \ie if
asymptotically for big values of $F$ it is exponentially small in $F$
(or larger), \eg it is
of the form $\sim e^{-\k F}$, for some constant $\k>0$. This
is a valuable and interesting remark brought up for the first time,
and correctly interpreted, already in \cite{CV03a} and in the following papers
\cite{CV03,VCC04}. The analysis of \cite{CV03a,CV03,VCC04} applies
to cases where the unbounded fluctuations are driven by an external
white noise. In the following we extend
the theoretical analysis in \cite{CV03,VCC04} to cases in which the
unbounded fluctuations do not arise from a Gaussian noise but from
a deterministic evolution like the ones in \cite{SE00,ESR03}: this is a simple
extension of the main idea and method of \cite{CV03} and provides an
alternative interpretation to the analysis in \cite{SE00,ESR03}.
Two important examples (also considered in \cite{SE00,ESR03})
to which our analysis can be applied are systems of particles
interacting via an unbounded potential (like a Lennard--Jones or a
Week--Chandler potential), driven by an external field and subject to
an isokinetic or a Nos\'e--Hoover thermostat. To be definite one
can consider system of $N$ particles in $d$ dimensions, described
by evolution equations $\dot{\V p}_i=\V E-\dpr_{\V q_i}\Phi-\a \V p_i$,
$\dot{\V q_i}={\V p}_i$. For an isokinetic thermostat, $\a$ is
a function of $\V p_i$, chosen so to keep the total kinetic energy fixed
to $\sum_i {\bf p}_i^2=Nd\b^{-1}$.
For a Nos\'e--Hoover thermostat $\a(t)$ is a variable independent of
$\V q_i(t),\V p_i(t)$ and satisfying the evolution equation $\dot\a=
\fra1Q\big[\sum_i\V p_i^2-Nd\b^{-1}\big]$, with $Q,\b>0$ parameters.
In both cases the phase space contraction $\s(x)$ has the form
$\s_0(x)-\b\fra{d}{dt}V(x)$,
where $\b$ has the interpretation of inverse temperature.
In the isokinetic case,
$\s_0(x)$ is bounded, and $V = \Phi$.
In the Nos\'e--Hoover case $\s_0(x)$ has, in the SRB distribution,
a fast decaying tail
(Gaussian at equilibrium, and likely to
remain such in presence of external forcing)
and $V = \sum_i\fra{\V p_i^2}{2}+\Phi(\V q)+
Q\fra{\a^2}{2}$, \cite{N84,H85}.
In both cases, {\it in equilibrium}, the SRB probability of
$V$ has an exponential tail $\sim e^{-\b V}$ (possibly with power-law
corrections).
{\it For the purpose of illustration} we assume, from now on, that the
same happens in presence of the force $\V E$.
This is an {\it essential and far from obvious assumption} useful,
as discussed below, to
understand the possible role of the singularities, but it
should not be assumed lightly as it is well known that the SRB
distributions may have very peculiar $E$ dependence and, at the
moment, a not intuitive character, \cite{DLS02,BDGJL01}. Nevertheless,
in preliminary numerical simulations, it seems approximately
correct, at least within the accuracy of the numerical data and for $|\V E|$
not too large; furthermore the analysis that follows can be naturally adapted to
more general assumptions on the tails.
In such cases the non normalized
variable $a$ (introduced before Eq. (\ref{6}))
has the form $a_0+\fra{\b}\t (V_i-V_f)$ where
$V_i,V_f$ are the values of $V(x)$ at the initial and
final instants of the time interval of size $\t$ on which $a$ is
defined, and $a_0 \Defi \fra1\t\ig_0^\t \s_0(S_tx)dt$:
%
\begin{equation}a=\fra1\t\ig_0^\t \s(S_tx)dt\=a_0+\fra\b\t (V_i-V_f)
\label{8}\end{equation}
%
If the system is chaotic and $\t$ is large, the variables $a_0, V_i,V_f$ can be
regarded as independently distributed and the distribution of $V=V_i$
or $V=V_f$ is essentially $\sim e^{-\b V} dV$ to leading
order as $V\to\io$, as discussed above. Therefore the
rate function of the variable $a$ can be computed as
\begin{equation}\label{9}\begin{split}
&\lim_{\t\to\io}\fra1{\t}\log
\int_{- p^* \s_+}^{p^*\s_+} da_0 \int_0^\io dV_i
\int_0^\io dV_f \, \cdot \\
&\cdot \, e^{\t \wt \z_0(a_0) - \b V_i -\b V_f}
\d[\t(a-a_0)+\b V_i - \b V_f] \\
&=\lim_{\t\to\io}\fra1{\t}\log
\int_{- p^* \s_+}^{p^*\s_+} da_0 \, e^{\t\wt \z_0(a_0) - \t|a-a_0|}
\end{split}\end{equation}
%
where $\wt \z_0(a_0)$ is the rate function of $a_0$; thus
%
\begin{equation}\widetilde \z(a)=\max_{a_0 \in [-p^* \s_+,p^* \s_+]}
\Big[ \widetilde\z_0(a_0) - |a-a_0| \Big]\label{10}\end{equation}
%
Defining $a_\mp$ by $\widetilde \z_0 '(a_\mp) = \pm 1$,
by the strict convexity of $\widetilde \z_0(a_0)$ it follows
%
\begin{equation}
\widetilde \z(a) = \left\{
\begin{array}{ll}
\widetilde \z_0(a_-) - a_- + a \ \ , & a < a_- \\
\widetilde \z_0(a) \ \ , & a \in [a_-,a_+] \\
\widetilde \z_0(a_+) + a_+ - a \ \ , & a > a_+ \\
\end{array} \right.
\end{equation}
%
If we assume that $\wt\z_0(a_0)$ satisfies FR (as expected from the
chaotic hypothesis, see below), then $\widetilde \z_0(a_0) =
\widetilde \z_0(-a_0) + a_0$ and by differentiation it follows that
$a_-=-\s_+$, where $\s_+$ is the location of the maximum of
$\widetilde\z_0$, \ie is the average of $a$.
It follows that, if $\wt\z_0(a_0)$ satisfies FR up to $a=p^*\s_+$, then
$\widetilde \z(a)$ satisfies FR only in the
interval $|a|<|a_-|=\s_+$. Outside this interval $\widetilde\z(a)$
does not satisfy the FR and in particular for $a\ge a_+$ it is
$\widetilde\z(a)-\widetilde\z(-a)=const.$, as
already described in \cite{CV03}. Translated into the normalized
variables $p_0=a_0/\s_+$ and $p = a/\s_+$, this means that,
even if the rate function of $p_0$ satisfies FR up to $p^*>1$,
the rate function of $p$ verifies FR only for
$|p|\leq1$. This is the effect due to the presence of the singular
boundary term.
An example of $\widetilde \z(a)$ is reported in Fig.~\ref{fig_zeta}:
it is a simple stochastic model for the FT (taken from Sect. 5 in
\cite{BGG97}, see also the extensions in \cite{LS99,Ma99}). The
example is the Ising model without interaction in a field $h$, \ie a
Bernoulli scheme with symbols $\pm$ with probabilities
$p_\pm=\frac{e^{\pm h}}{2\cosh h}$. Defining
$a_0=\fra1\t\sum_{i=0}^{\t-1}2 h\s_i$, so that
$\s_+ = \langle a_0 \rangle = 2h \tanh h$,
and setting $x\defi\fra{1+a_0/(2 h)}2$,
and $s(x)=-x\log x-(1-x) \log (1-x)$, one computes
$\wt\z_0(a_0)=s(x)+\fra12 a_0+ const$ which is {\it not Gaussian} and it is
defined in the interval $[-a^*,a^*]$ with $a^*=2 h$. In
this case the large deviation function $\wt\z_0(a_0)$ satisfies FR for
$|a_0|\le a^*$. If a singular term
$V=-\log (\sum_{i=0}^\io 2^{-i-1}\frac{\s_i+1}2)$ is added to $a_0$,
defining $a=a_0+\b(V_i-V_f)$
(with $\b=\log_2 (1+e^{2h})$ so that the probability distribution of $V$ is
$\sim e^{-\b V}$ for large $V$),
the resulting $\wt \z(a)$ does not verify FR for
$a > \media{a} = 2h \tanh h$. In particular, for $h\to 0$, the interval
in which the FR is satisfied vanishes.
\begin{figure}[t]
\includegraphics[width=.5\textwidth]{figura.eps}
\caption{An example in a stochastic model of FR. The graph gives
the two functions $\wt\z_0(a)$ and $\wt\z(a)$ for $h=0.,0.25,0.5$.
The average of $a$ is $\media{a}=\s_+ = 2h \tanh h$,
$a_+ = 2h \tanh 3h$ and $a^* = 2 h$. The
function $\wt\z(a)$ is obtained from $\wt\z_0(a)$ by continuing it
for $a p^*\s_+$,
it follows that the assumption
$\s_+>0$ {\it is, instead, essential} for the proof of fluctuation theorem.
The necessity of the assumption $\s_+>0$ is stressed in
the early paper \cite{Ga95b} which the Authors of \cite{ESR03} quote;
it is stressed also in the paper \cite{Ru99} which also makes clear that
$\widetilde\z(-a)=\widetilde\z(a)-a$ can only hold under the
assumption that $|a|$ does not exceed a maximum value.
\0(3) In the same reference \cite{ESR03} the point is made that the systems
that they consider have an infinite upper bound for $\s(x)$ and this
should explain why their numerical check of the FR fails if one looks
for it in terms of the variable $a$ (rather than in terms of $p$, which
in their equilibrium cases would not even be defined). They attribute
the failure to the fact that the system in equilibrium ({\it i.e.}
with $\s_+=0$) is ``not Anosov''. While this is mathematically obvious
in their case (as the system they consider is not even smooth) the
apparent failure of the FR must be interpreted differently.
As remarked above, at equilibrium a correct repetition of the proof of
fluctuation theorem yields $\widetilde\z(a)=
\widetilde\z(-a)$ instead of $\widetilde\z(a)= \widetilde\z(-a)+a$:
{\it irrespective of whether the chaotic hypothesis holds and irrespective
of whether the system is Anosov or not}. For $\s_+>0$ and small, by the
arguments above, one expects to see the FR for $\wt\z(a)$ only for values
of $a$ with $|a|\leq \s_+$, while, for the variable $a_0$ obtained by
subtracting from $a$ the singular contributions due to the the singular term
$\frac{dV}{dt}$, one expects the FR to hold up to $a_0=p^*\s_+$. In both cases
the FR is expected to hold up to a quantity proportional to $\s_+$. Moreover
the asymptotic shape of the probability distribution for $a$ or $a_0$
will be visible at best
on a time scale proportional to $\s_+^{-1}$.
As remarked above the time scales
needed to see the distributions of $a$ or of $a_0$ reaching their asymptotic
shapes are different: the one for $a$ might be
about $2$ orders of magnitudes larger than the one for $a_0$, see \cite{ZRA03}.
Hence in the near to equilibrium case the FR holds for $a$ in a tiny interval
around $a=0$ and becomes visible only on a very long time scale
$\sim\s_+^{-1}$.
The quotations of numerical simulations in \cite{ESR03} fail to take into
account these aspects and this could explain why the FR ``is not seen'' in numerical
experiments which look at moderate time scales and, furthermore, at
values of $a$ which deviate by several multiples of the width of the
distribution of $a$ which (by the FR) is precisely the (small) value
of $\s_+$. For instance in \cite{DK04} the fluctuations are studied over
an interval of variation of $a$ which is an order of magnitude larger than
the average phase space contraction. The experiment in \cite{GC05} is a
very convincing evidence of the relevance of the general theory in \cite{CV03}
and of our analysis in Sect. 2 (which is an extension of \cite{CV03} to non
Gaussian cases).
\0(4) The analysis in Sect. 2 above shows that the probability
distribution describing isokinetic systems near equilibrium {\it
are SRB distributions} (contrary to what is claimed in \cite{ESR03}):
this is mathematically obvious by the very definition of SRB
distribution
in the case of Anosov systems ({\it even if isokinetic}, see \eg
the geodesic flow discussed above)
and it appears to be true also in non Anosov systems that
have so far been considered.
\0(5) If the term $\frac{dV}{dt}$ is removed (as in Ref.~\cite{ZRA03,GZG05}),
the resulting quantity $\s_0(x)$ is bounded and its distribution
verifies the
FR also for $|p|>1$. This prescription, as discussed above, is equivalent
to the very reasonable prescription that the Poincar\'e's section used for
mapping the flow into a map does not pass through a singularity of $\s(x)$.
\0(6) In the case of the thermostatted particle systems considered in
\cite{ESR03} the unbounded derivative $\frac{dV}{dt}$ is also the
contraction rate of the volume in equilibrium, \ie with ${\bf E}=0$.
Thus, for $\V E\neq \V 0$,
one can remove the total derivative from $\s(x)$ simply considering
the contraction {\it with respect to the equilibrium invariant
distribution} $e^{-\b V}$, as stated in \cite{ESR03, SE00}.
However, this
observation does not provide a general prescription to remove the
singular part from the phase space contraction rate because it rests on
the very special fact that the singularities of the function $V(x)$
(\ie of the potential $\Phi$)
do not depend on ${\bf E}$.
The prescription that the phase space contraction should be computed
on non singular Poincar\'e's sections, instead, does not require
any other assumption.
\0(7) The analysis in Sect. 2 above applies as well to understand how to
apply the FR to systems with Gaussian (or unbounded) noise and the
compatibility between the general theory of \cite{Ku98,LS99,Ma99}
with the works \cite{CV03} and \cite{VCC04,GC05}.
\0(8) In other words our analysis suggests that the Chaotic Hypothesis
is an appropriate characterization of thermostatted systems, even
for isokinetic or Nos\'e--Hoover thermostatted systems,
see concluding remark of \cite{ESR03}.
\acknowledgments
We are indebted to E.G.D. Cohen, and R. Van Zon for many enlightening
discussions. F.Z. wish to thank G.Ruocco for many useful discussions.
G.G is indebted to Rutgers University, I.H.E.S and
\'Ecole Normale Sup\'erieure, where he spent periods of leave while
working on this project.
\bibliography{nth2}
\bibliographystyle{apsrev}
\*
\def\revtexz{{\bf
R\lower1mm\hbox{E}V\lower1mm\hbox{T}E\lower1mm\hbox{X}}}
\0e-mail: {\tt
bonetto@math.gatech.edu\\
giovanni.gallavotti@roma1.infn.it\\
alessandro.giuliani@roma1.infn.it\\
francesco.zamponi@phys.uniroma1.it}\\
web: {\tt http://ipparco.roma1.infn.it}\\
Dip. Fisica, U. Roma 1,\\
00185, Roma, Italia
\revtex
\end{document}
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