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Fluctuation theorem, nonequilibrium, chaotic motions, statistical mechanics
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\begin{thebibliography}{39}
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\providecommand{\bibinfo}[2]{#2}
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\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 $\ -p^*<p<p^*$). 
The function $\z(p)$ is called in probability
theory the {\it rate function} for the large deviations of $p$.

The function $\z(p)$ is analytic in $p$ and {\it convex} in the interval of
definition $(-p^*,p^*)$. Analyticity and convexity
of large deviation rates are general properties, established by Sinai and
valid for the SRB-averages of smooth observables (in Anosov systems),
\cite{Si72,Si77,GBG04}. In fact more can be said for the specific case
of the large deviation rate
of the observable $p$, and one can prove the following {\it fluctuation theorem}:

\*{\bf Proposition}: {\it In transitive time reversible dissipative
Anosov systems the
rate function $\z(p)$ for the dimensionless phase space contraction $p(x)$ 
defined in (\ref{4}) is
analytic and strictly convex in an interval $(-p^*,p^*)$ with
$+\infty>p^*\ge1$ and $\z(p)=-\infty$ for $|p|> p^*$.  Furthermore
\begin{equation}\z(-p)=\z(p)-p\s_+, \qquad {\rm for} \qquad
  |p|<p^*\label{5}
\end{equation}
which is called the ``fluctuation relation''.
}
\*

Strict convexity follows from a theorem of Griffiths and Ruelle which
shows that the only way strict convexity could fail is if
$\s(x)=\f(Sx)-\f(x)+ c$ where $\f(x)$ is a smooth function (typically
a Lipschitz continuous function) and $c$ is a constant, see
propositions (6.4.2) and (6.4.3) in \Onlinecite{GBG04}. The constant $c$
vanishes if time reversal holds and $\s(x)=\f(Sx)-\f(x)$ contradicts
the assumption that $\s_+>0$, 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 $|a|<p^*\s_+$ might be forgotten, see \Onlinecite{Ga04}).

Note that $p^*$ is certainly $<+\infty$ because the variable $\s(x)$
is bounded (being continuous on the bounded manifold on which the
Anosov map is defined).

However no confusion should be made between $p^*\s_+$ and
$\sigma_{max}\defi\max |\s(x)|$: unlike $\sigma_{max}$ the quantity
$p^*$ is a {\it non trivial} dynamical quantity, independent on the
metric used on phase space to measure distances, hence volume. 
This point has not been always understood and confusion has appeared 
in the published literature with nefast consequences. 
In fact it is very easy to build examples of Anosov systems in which 
$p^*\s_+< \sigma_{max}$:
still, this does not mean that fluctuation relation is violated for
such systems. Some explicit examples are discussed in next Section.

\vskip3mm
\0{\bf Conservative systems and the equilibrium limit}
\vskip3mm

Considering more closely the cases $\s_+=0$ it follows that
$\s(x)=\f(Sx)-\f(x)$ (again by the above mentioned result of 
Griffiths and Ruelle), with $\f$ a smooth function of phase space. 
Hence the variable
%
\begin{equation}a=\fra1\t\sum_{j=-\t/2}^{\t/2-1}
\s(S^jx)\=\frac{\f(S^{-\frac{\t}2}x)-
\f(S^{\frac{\t}2}x)}\t
\label{7}\end{equation}
%
is {\it bounded} and tends to $0$ {\it uniformly}. One could repeat
the theory developed for $p$ when $\s_+>0$ 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 $a<a_-=-\s_+$ and $a>a_+$ with straight lines of slope $\pm 1$. 
It does not satisfy the FR for $|a|>\media{a}$. As $h\to0$, $\media{a} \to 0$,
which means that the interval in which the FR is verified shrinks to $0$.
In this limit $a_+\to 0$, so $\wt \z(a)$ approaches $-|a|$ (dashed lines).
Refrasing this in terms of
$p=\fra{a}{\media{a}}$ one obtains that FR remains always valid
for $|p|<1$, even as $h\to0$. The three curves for 
$\wt\z_0(a)$ have the same tangent
on left side. The function $\wt\z_0(a)$ is finite {\it only} in the
interval $[-2h,2h]$ and it is $-\infty$ outside it, while the function
$\wt\z(a)$ is finite for all $a$'a and is a straight line outside $[a_-,a_+]$.}
\label{fig_zeta}
\end{figure}

\vskip3mm
\0{\bf How to remove singularities}
\vskip3mm

From the discussion above it turns out that singular terms which are
proportional to total derivatives of unbounded functions (like the
term $\frac{dV}{dt}$ that appears in the phase space contraction rate
of thermostatted systems) can induce ``undesired'' (or ``unphysical'')
modifications of the large deviations function $\z(p)$.

On heuristic grounds, when dealing with singular systems, one could
follow the prescription that unbounded terms in $\s(x)$ which are
proportional to total derivatives should be {\it subtracted} from the
phase space contraction rate.  If the resulting $\s_0(x)$ is bounded (as
it is \eg for the isokinetic thermostat models considered) 
or at least if the tails of its distribution decay faster than exponentially,
then its large deviations function should 
verify the FR for $|p|\leq p^*$, $p^*$ being the
intrinsic dynamic quantity defined above.

Note that after the subtraction of the divergent terms the remaining
contraction, in the considered cases, is bounded for isokinetic
thermostats or has a Gaussian tail (\ie faster than
exponential) in the case of Nos\'e--Hoover thermostats.

If the singular terms are not subtracted, the FR {\it will appear to
be valid only for $|p|\leq 1$ even if $p^* > 1$}.  This seems to have
generated statements that the Chaotic Hypothesis does not apply to
isokinetic systems, see \cite{ESR03}.

The heuristic prescription above can be motivated by a careful analysis of
the proof of the fluctuation theorem for Anosov flows.
In the following let us call again $a$ the integral of the total phase space
contraction rate $\s(x)$ (which includes singular terms) and $a_0$ the integral
of the bounded variable $\s_0(x)$ from which singular total derivatives
have been removed.

The fluctuation theorem was proved in \cite{GC95,Ga96,Ru99} {\it for
Anosov maps} and only later it has been extended in \cite{Ge98} to Anosov
flows. Very sketchily, 
the extension of the 
fluctuation theorem to Anosov flows in \cite{Ge98} is proved 
as follows. One reduces the Anosov flow on $\O$ to a map via
a Poincar\'e's section, associated with surfaces on $\O$ transversal
to the flow. The passage of the flow through any one of such
surfaces is called a {\it timing event}. The map between two consecutive
timing events is called a ``Poincar\'e's map''. The union $\O_P$ of the
surfaces represents the phase space of the Poincar\'e's map.
The surfaces in $\O_P$ can be suitably chosen, in such a way that the 
Poincar\'e's map is a chaotic map which although not smooth, hence not
an Anosov map, has (a non trivial fact \cite{Ge98}), 
all the properties necessary to prove the
fluctuation theorem (which therefore applies to systems
more general than the Anosov maps, although there is not a
general characterization of the systems which are not Anosov and to
which it applies). So, for such a map the fluctuation 
theorem holds and this in turn leads to a FR for the flow by the 
theory in \cite{Ge98} {\it under the assumption that the variable $\s(x)$ 
is bounded}. 

If, as in the case under analysis, $\s(x)$ is not bounded,
we can interpret the chaotic hypothesis as applying to the map
associated with a Poincar\'e's section which avoids the singularities
of the potential, a very natural prescription which allows us to
apply the theory in \cite{Ge98} and derive a FR for both the map and
the flow. For instance, we can choose as
timing events the instants in which either the potential energy or the 
Nos\'e's ``extended Hamiltonian'' exceed some fixed value $\bar V$. 
If we make this choice, the (discrete) average $\hat a$ of $\s(x)$ over a 
sequence of iterations of the Poincar\'e's map will coincide with 
the (discrete) average $\hat a_0$ of $\s_0(x)$ along the same sequence: 
this simply follows from the remark that by construction the total increment 
of $\s(x)-\s_0(x)$ between two timing events, given by $\b (V_f-V_i)$, is $0$ (by construction $\O_P$ 
is chosen as a subset of $\{x\in\O\,:\, V(x)=\bar V\}$ where $V_f=V_i$). Then, by the same 
argument in \cite{Ge98}, the fact that the rate function of $\hat a_0$ 
satisfies a FR and that $\s_0(x)$ is bounded implies that the rate function 
of the continuous average $a_0$ of $\s_0(x)$ along a trajectory of the flow
will satisfy the fluctuation theorem.

Therefore the distribution of $a_0$ will satisfy the FR
({\it by the chaotic hypothesis}) for $|a_0|<p^* \s_+$.  By the above
maximum argument, the distribution of $a$ {\it will also verify, as
a consequence}, the FR but only for $|a|\leq \s_+$ (\ie in the form
(\ref{5}) or (\ref{6}) with $|p| \leq 1$).

Then the (natural) prescription to study FR for chaotic flows is
to reduce the problem to a chaotic map considering only Poincar\'e's
sections which do not pass through a singularity of $\s(x)$. The
integral of $\s(x)$ over a large number of timing events on such
sections is equal to the time integral of $\s_0(x)$ plus a bounded
term which can be neglected. Thus the prescription on the choice of 
Poincar\'e's sections is equivalent to the heuristic prescription of 
removing from $\s(x)$ all the unbounded total derivatives.

It follows that the chaotic hypothesis leads to a clear prediction on
the outcome of possible numerical simulations of particle systems interacting 
via unbounded potentials and subject to the isokinetic
or the Nos\'e--Hoover thermostat: the FR will hold for all $|a|\leq \s_+$ and,
once the term $\frac{dV}{dt}$ is removed, for all $|a_0| < p^* \s_+$
with $p^* \geq 1$. Note that in the cases under analysis $a_0$ coincides with
the dissipation function of Evans and Searles that was in fact predicted to 
satisfy FR \cite{SE00,ESR03}, even though for different reasons. We believe that 
the correct interpretation of the fact that FR for $\wt\z_0(a_0)$ holds 
for all $|a_0| < p^* \s_+$ is the one given above.

The numerical results of \cite{ZRA03,GZG05} agree with the prediction that 
FR for the rate function of $a_0$ is valid even beyond $a_0=\s_+$. 
The prediction that (at least near equilibrium) the rate function of $a$
should satisfy FR only up to $a=\s_+$ and that should become linear for $a\ge
a_+$ at the moment has been experimentally confirmed only in Gaussian cases 
\cite{CV03a,CV03,VCC04}. 
It would be very interesting to investigate in detail 
the structure of $\wt\z(a)$ even in non Gaussian cases. Note that this is 
far from being an easy task (in particular the analysis in \cite{GZG05}
was not sophisticated enough to study this problem). 
In fact, as discussed in detail in \cite{ZRA03}, 
the presence in the definition of $\s$ of a total derivative of an unbounded 
function may enlarge of $2$ orders of magnitudes the times needed for the 
probability distribution of $a$ to reach its asymptotic shape: even 
in the Gaussian region (small fluctuations of $a$ around $\s_+$)
the convergence times for $\wt\z(a)$ are found to be
of order $1000$ decorrelation times, 
versus a time of order $10$ decorrelation times needed for $\wt\z_0(a_0)$ 
to converge to its asymptotic shape~\cite{ZRA03}. Clearly, for times of order 
$1000$ decorrelation times, it is very hard to observe 
fluctuations of $a$ larger than $a_+-\s_+$. In order to verify the prediction 
for the shape of $\wt\z(a)$ beyond $a=a_+$ an {\it ad hoc} experiment would be 
needed, together with a detailed investigation of the finite time corrections 
to $\wt\z(a)$, along the lines in \cite{GZG05}.

\vskip3mm
\0{\bf Meaning of the chaotic hypothesis}
\vskip3mm

It has to be stressed that the chaotic hypothesis concerns physical
systems: {\it it is very easy to find dynamical systems for which it does
not hold}. As it is easy (actually even easier) to find systems in
which the ergodic hypothesis does not hold (\eg harmonic lattices or
black body radiation). However, if suitably interpreted, the ergodic 
hypothesis leads even for these systems to
physically correct results (the specific heats at high temperature).
Moreover the failures of the ergodic hypothesis 
have led to new scientific paradigms (like quantum
mechanics from the specific heats at low temperature and Planck's
law).

Since physical systems are almost always not Anosov systems it is very
likely that probing motions in extreme regimes (\eg when particles go
undisturbed through infinite potential walls, as in the
Week-Chandler's isokinetic models) will make visible the features that
distinguish Anosov systems from non Anosov systems: for instance the 
fact that isokinetic or Nos\'e--Hoover 
thermostatted particle systems satisfy the FR only for $|p|\leq 1$ if the 
singular term
$\frac{dV}{dt}$ is not removed from $\s(x)$. Note that these results
{\it can be derived} from the chaotic hypothesis as discussed above:
this is a quite remarkable property and a confirmation of CH rather than an
argument against it.

The results in \cite{ZRA03,GZG05} although not designed to
probe the peculiarities of the Week--Chandler's potential show that
the FR for the rate function of $a_0$ holds even beyond $p=1$ and this 
provides further numerical evidence for the
chaotic hypothesis even in situations in which the non-Gaussian nature
of the large deviations function $\z(p)$ is manifest. As discussed above,
an even stronger confirmation of CH would follow from a possible 
numerical observation of the scenario discussed above for the rate function 
of $a$ in systems in which $a_0$ admits a non Gaussian distribution function; 
it would be very interesting to investigate explicitly this 
phenomenon in thermostatted driven particle systems. The situation
looks much more promising in the case of stochastic systems: the
simple case discussed in Fig. 1 already gives evidence in this direction.

\section{Remarks}

We conclude by comparing more closely our discussion and our interpretation
of apparent violations to the FR coming from singularities in the 
interaction potential with the corresponding discussion 
in \cite{ESR03}, where a different interpretation is given (in particular
it is argued that FR and CH do not hold for thermostatted systems near 
equilibrium). \\

\0(1) As stressed above it is dangerous (and wrong) to consider (\ref{6}) 
without the restriction $|a|\le p^*\s_+$ as the prediction of fluctuation 
theorem. However in \cite{ESR03} this is exactly what the authors do, 
by saying, in a somewhat unusual form, that ``{\it it is convenient 
to consider {\rm [the relation Eq.(\ref{6}) without the condition 
$|a|<p^*\s_+$]} as 
the prediction of the FT}''. The attribution is not correct but it is
convenient to produce arguments, in most of the rest of the paper, 
that show that (\ref{6}) without the restriction $|a|\le p^*\s_+$ 
is incorrect (an {\it a priori} obvious fact). 

\0(2) An argument in \cite{ESR03}
is supposed to prove that the relation $\widetilde\z(-a)=\widetilde\z(a)-a$
(without the condition $|a|<p^*\s_+$) holds for reversible Anosov
systems for all $a$'s, also for $\s_+=0$ (in particular they say that 
``{\it the division by $\s_+$ does not seem to be necessary for the
proof in [37]}''). 
Since at equilibrium as well as near equilibrium, as remarked above and as
illustrated also by \cite{BGM98}, {\it there are examples of Anosov systems
which, although reversible, do not satisfy}
$\widetilde\z(-a)=\widetilde\z(a)-a$ in the sense of \cite{ESR03}
(\ie with $\wt\z(a)$ finite) 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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