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\title{Microscopic reversibility and thermodynamic fluctuations }
\author{D. Gabrielli}
\address{SISSA, Scuola Internazionale Superiore di Studi Avanzati\\
Via Beirut 2-4, 34014 Trieste, Italia }
\author{G. Jona-Lasinio}
\address{Dipartimento di Fisica, Universit\`a di Roma "La Sapienza"\\
Piazza A. Moro 2, 00185 Roma, Italia}
\author{C. Landim}
\address{IMPA, Estrada Dona Castorina 110\\
J. Botanico, 22460 Rio de Janeiro RJ, Brasil\\
and LAMS de l'Universit\'e de Rouen, Facult\'e de Sciences\\
BP 118,F-76134 Mont-Saint-Aignan Cedex, France}
\author{M.E. Vares}
\address{IMPA, Estrada Dona Castorina 110\\
J. Botanico, 22460 Rio de Janeiro RJ, Brasil}
%\date{\today}
\maketitle
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ABSTRACT
\begin{abstract}
In this paper we show that certain properties of thermodynamic fluctuations
which were derived long ago by Onsager and Onsager-Machlup as a consequence
of microscopic reversibility , can actually hold also if the dynamics deviates
from reversibility. This result is based on the explicit construction of
models which can be analysed rigorously. What happens is that small scale
irreversibilities are washed out at the macroscopic level.
\end{abstract}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PAPER CONTENT
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\narrowtext
\section{Introduction}
Fundamental contributions to the theory of irreversible processes
were the derivation of the reciprocal relations for transport
coefficients in states deviating only slightly from equilibrium
and the calculation of the most probable trajectory creating
a fluctuation near equilibrium. The first result was obtained
by Onsager in 1931 \cite{ON1} and the second one by Onsager
and Machlup \cite{ON2} in 1953. The calculation of the most
probable trajectory relies on the reciprocal relations which
in turn are a consequence of microscopic reversibility. It turns
out that the trajectory in question is just the time reversal
of the most probable trajectory describing relaxation to equilibrium
of a fluctuation. The latter is a solution to the hydrodynamical equations.
In this paper we discuss the following question: is microscopic
reversibility a necessary condition for the validity of the
above results? The answer to this question is far from obvious
because in going from the microscopic to the macroscopic scale
a lot of information is lost and irreversibilities at a small scale
may be erased when taking macroscopic averages. We will show
that this is in fact the case by exhibiting microscopically non reversible
stochastic dynamics which nonetheless fluctuate following the same time-
reversal rule of Onsager-Machlup. Actually our results are not restricted to
situations near equiliblrium and the problem can be discussed rigorously
for arbitrary fluctuations.
The models we shall discuss belong to the category of interacting
particle systems and have been analysed in detail in \cite{LAND}
and \cite{JLVL}. In particular in the last reference we already
made a connection with the Onsager-Machlup theory by showing
that the regression to equilibrium of any fluctuation (even far
from equilibrium) takes place
with highest probability along a trajectory of the hydrodynamic
equation. The models consist in a superposition of an accelerated
symmetric Kawasaki process and a Glauber spin flip process.
The structure of the paper is as follows. In section II
we describe the models and summarize the results of \cite{JLVL}
needed for our purpose.
In section III we discuss under what conditions our dynamics
becomes reversible with respect to the invariant measure, which
will be a Bernoulli product measure.
In section IV we give conditions sufficient to insure
the validity of Onsager-Machlup time-reversal relation
and show that they can be satisfied
by irreversible dynamics. It also turns out that if the fluctuations
are homogeneous in space any dynamics in the class considered
satisfies Onsager-Machlup.
In the present paper we do not supply all the proofs which will
be given in an more extended forthcoming publication.
\narrowtext
\section{Description of the models }
The systems considered consist of particles moving on the sites
of a lattice. There are two basic dynamical processes:
i. a particle can move to a neighbouring site if this is empty
ii. a particle can disappear or be created in a site according
to whether this is occupied or empty.
The first process is clearly
conservative while the second is not.
Mathematically we consider a family of Markov processes whose state space
is $X_N=\{0,1\}^{Z_N}$,
where $N$ is an integer and $Z_N$ denotes the set of integers modulo $N$.
We shall denote with $\eta$ a point in the state space, that is a configuration
of the system. This is therefore given by a function $\eta(i)$ defined
on each site and taking the values $0$ or $1$.
For each $N$ the dynamics is defined by the action of
the generator $L_N$ of the Markov process on functions $f(\eta)$
\begin{equation}
L_Nf(\eta)=\frac{N^2}{2}\sum_{i\in Z_N}(f(\eta^{i,i+1})-f(\eta))+
\sum_{i\in Z_N}c(i,\eta)(f(\eta^i)-f(\eta))
\label{eqn1}
\end{equation}
where the addition in $Z_N$ means addition modulo $N$
\begin{equation}
\eta^{i,k}(j)= \left\{
\begin{array}{ccl}
\eta(j) &if& j\neq i,k \\
\eta(k) &if& j=i \\
\eta(i) &if& j=k
\end{array}
\right.
\label{eqn2}
\end{equation}
\begin{equation}
\eta^i(j)=\left\{
\begin{array}{ccl}
\eta(j) &if& j\neq i \\
1-\eta(i) &if& j=i
\end{array}
\right.
\label{eqn3}
\end{equation}
The coefficients $c(i,\eta)$ depend on the values of $\eta(j)$ with $j$
within a fixed distance $R$ from the site $i$. They are translation
invariant that is $c(i,\eta)=c(\tau_i\eta)$ where $(\tau_k\eta)(j)=\eta(k+j)$.
Let us consider now the unit interval $S=[0,1)$ with periodic condition at the
boundary and a function $\gamma$ defined on $S$ and taking values in $[0,1]$.
Let $\nu_{\gamma}^N$ the probability measure on the state space of the system
obtained by assigning a Bernoulli distribution to each site, taking
the product over all sites and defined by
\begin{equation}
\nu_{\gamma}^N\{\eta(k)=1\}=\gamma (\frac{k}{N})
\label{eqn4}
\end{equation}
The main object of our study is the empirical density $\mu_t^N$:
\begin{equation}
\mu_t^N(x)=\frac{1}{N}\sum_{k\in Z_N}\eta_t(k)\delta(x-\frac{k}{N})
\label{eqn5}
\end{equation}
If we denote by $Q_{\gamma}^N$ the distribution law
of the tragectories $\mu_t^N(x)$ when the initial measure
is concentrated on a configuration
such that $\mu^N_0(x)\rightarrow \gamma(x)$ as $N\rightarrow \infty$,
it is possible to show that
$Q_{\gamma}^N$ converges weakly as $N$ goes to infinity
to the measure concentrated on
the path $\rho(t,x)$ that is the unique solution of
\begin{equation}
\left\{
\begin{array}{ccl}
\partial_t \rho &=& \frac{1}{2}\partial_x^2\rho +B(\rho)-D(\rho) \\
\rho(0,\cdot) &=& \gamma(\cdot)
\end{array}
\right.
\label{eqn6}
\end{equation}
with
\begin{equation}
B(\rho)=E_{\nu_{\rho}}(c(\eta)(1-\eta(0)))
\label{eqn11}
\end{equation}
\begin{equation}
D(\rho)=E_{\nu_{\rho}}(c(\eta)\eta(0))
\label{eqn12}
\end{equation}
Where $\nu_{\rho}$ is the Bernoulli product distribution with
$\gamma(x)\equiv \rho$.
Typically $B(\rho)$ and $D(\rho)$ are polynomials in the variable $\rho$.
The equilibrium state corresponds to a density $\rho_0$ which is
the solution of the equation $B(\rho)=D(\rho)$ that gives an absolute
minimum of the potential $V(\rho)=\int^{\rho}[D(\rho') - B(\rho')]d\rho'$.
The above result is a law of large numbers that shows that the empirical
density in the limit of large $N$ behaves deterministically according to
equation (\ref{eqn6}).
We can now ask what is the probability that our system follows a trajectory
different from the solution of (\ref{eqn6}) when $N$ is large
but not infinite. This probability is exponentially small in $N$ and
can be estimated using the methods of the theory of large deviations
introduced for the systems of interest in \cite{KOV} and developed in
\cite{LAND},\cite{JLVL}. The main idea consists in introducing a modified
system for which the trajectory of interest (fluctuation) is typical
being a solution of the corresponding hydrodynamic equation, and then
comparing the two evolutions.
For this purpose we consider the Markov process defined by the generator
\begin{equation}
\left.\begin{array}{ccl}
L_{N,t}^Hf(\eta)&=&\frac{N^2}{2}\sum_{|i-j|=1}\eta(i)(1-\eta(j))
e^{H(t,\frac{j}{N})-H(t,\frac{i}{N})}[f(\eta^{i,j})-f(\eta)] \\
&+& \sum_i c(i,\eta)[(1-\eta(i))e^{H(t,\frac{i}{N})}+\eta(i)
e^{-H(t,\frac{i}{N})}][f(\eta^i)-f(\eta)]
\end{array}\right.
\label{eqn9}
\end{equation}
with $c$, $\eta^{k,j}$, $\eta^i$ as previously defined and
$H$ can be interpreted as an external field.
The deterministic equation satisfied by the empirical density is now
\begin{equation}
\left\{
\begin{array}{ccl}
\partial_t \rho &=& \frac{1}{2}\partial_x^2\rho-\partial_x(\rho(1-\rho)
\partial_xH)+B(\rho)e^H-D(\rho)e^{-H} \\
\rho(0,\cdot) &=& \gamma(\cdot)
\end{array}
\right.
\label{eqn10}
\end{equation}
We remark that while equation (\ref{eqn6}) is of gradient type,
equation (\ref{eqn10}) does not and
this is due to the asymmetry of the exchange dynamics.
Given a function $\rho(x,t)$ twice differentiable with respect to $x$ and once
with respect to $t$ this equation determines uniquely the field $H$.
The probability that the original system
follows a trajectory different from a solution of (\ref{eqn6})
can now be expressed in terms of the
field $H$ and the polynomials $B$ and $D$. We introduce the large deviation
functional
\begin{equation}
\left.\begin{array}{ccl}
I(\rho) &=& \frac{1}{2}\int_0^{t_0}\int_0^1dtdx\rho_t(1-\rho_t)
(\partial_xH_t)^2 \\
&+& \int_0^{t_0}\int_0^1dtdxB(\rho_t)(1-e^{H_t}+H_te^{H_t}) \\
&+& \int_0^{t_0}\int_0^1dtdxD(\rho_t)(1-e^{-H_t}-H_te^{-H_t})
\end{array}\right.
\label{eqn14}
\end{equation}
Let $G$ be a set of trajectories in the interval $[0,t_0]$.
The large fluctuation estimate asserts that
\begin{equation}
Q_{\gamma}^N(G)\simeq e^{-NI(G)}
\label{lf}
\end{equation}
where
\begin{equation}
I(G)=\inf_{\rho \in G}I(\rho)
\label{inf}
\end{equation}
The sign $\simeq$ has to be interpreted as asymptotic equality of the
logarithms.
>From the equations (\ref{lf}), (\ref{inf}), one sees that to find the most
probable trajectory that creates a certain state $\gamma(x)$ one has to
find the $\rho(x,t)$ that minimizes $I(\rho)$ in the set $G$ of all
trajectories that connect the equilibrium state to $\gamma(x)$.
\section{Reversibility}
Reversibility means that a principle of detailed
balance holds for the microscopic dynamics. Mathematically this
is expressed by the self-adjointness
of the generator of the process with respect to the
scalar product defined by the measure.
A reversible measure for a process with generator of the form
(\ref{eqn1}) exists only if we impose some restrictions on the
functions $c$.
The condition of reversibility is
\begin{equation}
(g,L_Nf)_{\mu}=(L_Ng,f)_{\mu}
\label{eqn21}
\end{equation}
for all functions $f$,$g$ on $X_N$. In our case this condition
reads
\begin{equation}
\left.\begin{array}{ccl}
& & \sum_{\eta}[g(\eta)(\frac{N^2}{2}\sum_i(f(\eta^{i,i+1})-f(\eta)) \\
& & +\sum_ic(i,\eta)(f(\eta^i)-f(\eta))]\mu(\eta)= \\
& & \sum_{\eta}[(\frac{N^2}{2}\sum_i(g(\eta^{i,i+1})-g(\eta)) \\
& & +\sum_ic(i,\eta)(g(\eta^i)-g(\eta))f(\eta)]\mu(\eta)
\end{array}\right.
\label{eqn22}
\end{equation}
that with some algebra,using the periodic boundary condition,
can be trasformed into
\begin{equation}
\left.\begin{array}{ccl}
& & \sum_{\eta}\sum_i\frac{N^2}{2}g(\eta)f(\eta^{i,i+1})(\mu(\eta)
-\mu(\eta^{i,i+1})) \\
& & +\sum_{\eta}\sum_ig(\eta)f(\eta^i)(c(i,\eta)\mu(\eta)-
c(i,\eta^i)\mu(\eta^i))=0
\end{array}
\right.
\label{eqn23}
\end{equation}
Since this equality must hold for every $g$ and $f$, this condition is
equivalent to
\begin{equation}
\left\{
\begin{array}{ccl}
\mu(\eta)-\mu(\eta^{i,i+1}) &=& 0 \\
c(i,\eta)\mu(\eta)-c(i,\eta^i)\mu(\eta^i) &=& 0
\end{array}
\right.
\label{eqn24}
\end{equation}
for every $\eta$ and $i$. The first condition imposes that the measure
$\mu$ be of the form
\begin{equation}
\mu(\eta)=\mu(\sum_{i=1}^N\eta(i))
\label{eqn25}
\end{equation}
that is to say $\mu$ must assign an equal weight to configurations
with the same number of $1$. The second condition, with a $\mu$ of this
type, is a restriction for the functions $c$. The most general form
of $c(i,\eta)$ that satisfies this condition is:
\begin{equation}
c(i,\eta)=c_1(1-\eta(i))h(i,\eta)+c_2\eta(i)h(i,\eta)
\label{eqn26}
\end{equation}
with $c_1$ and $c_2$ arbitrary positive constant and $h(i,\eta)$
a function that does not depend on the variable $\eta(i)$ and such that
$h(i,\eta)=h(\tau_i\eta)$. For processes of this type it is possible
to compute explicity the unique reversible measure that is a
Bernoulli measure with parameter $p=\frac{c_1}{c_1+c_2}$.
We emphasize that periodic boundary conditions are crucial for the validity of
(\ref{eqn26}) with a nontrivial $h$.
\section{The minima of $I(\rho)$ }
Let us consider a fluctuation that can be connected to the equilibrium
density by a trajectory solution of
the hydrodynamical equation (\ref{eqn6}). Then
from the form of $I(\rho)$ it is obvious that such a fluctuation
relaxes most likely following this trajectory. In
fact the corresponding $H$ is zero which implies $I=0$.
We want to
investigate now the trajectory that creates the non-equilibrium
state $\gamma(x)$ with highest probability, that is
to say the trajectory $\rho(x,t)$ with the
boundary conditions
\begin{equation}
\lim_{t\rightarrow -\infty}\rho(x,t)=\rho_0
\label{eqn31}
\end{equation}
\begin{equation}
\rho(x,0)=\gamma(x)
\label{eqn32}
\end{equation}
that minimizes the functional
$I$, with $\rho_0$ the equilibrium state
We consider polynomials $B$ and $D$ of the form
\begin{equation}
B(\rho)=c_1A(\rho)(1-\rho)
\label{eqn33}
\end{equation}
\begin{equation}
D(\rho)=c_2A(\rho)\rho
\label{eqn34}
\end{equation}
with $c_1$ and $c_2$ arbitrary positive constant and $A(\rho)$ a generic
strictly positive polynomial.
Note that
the potential that generates the polynomial part of the hydrodynamic
equation with $B$ and $D$ of this type is
always a single well potential with only one stable equilibrium point.
In this case it is possible
to prove (see appendix) that the unique solution of
our variational problem is the function $\rho^*(x,t)$ defined by
\begin{equation}
\rho^*(x,t)=\rho(x,-t)
\label{eqn35}
\end{equation}
where $\rho(x,t)$ is the solution of the hydrodynamic equation which relaxes
to equilibrium. $\rho^*(x,t)$ is therefore a solution of the hydrodynamic
equation with inverted drift
\begin{equation}
\partial_t\rho=-\frac{1}{2}\partial_x^2\rho +D(\rho)-B(\rho)
\label{eqn36}
\end{equation}
Equation (\ref{eqn35}) is the Onsager-Machlup time-reversal relation.
All reversible processes generate
hydrodynamic equations with coefficient $B(\rho)$ and $D(\rho)$ of the
form (\ref{eqn33}) and (\ref{eqn34}), so for all these systems (\ref{eqn35})
holds.
It is most interesting that (\ref{eqn35}) can hold
for irreversible models too; namely if we consider processes with
functions $c$ of the form
\begin{equation}
c(i,\eta)=c_1(1-\eta(i))h_1(i,\eta)+c_2\eta(i)h_2(i,\eta)
\label{eqn41}
\end{equation}
with $h_2$ different from $h_1$, we obtain a irreversible
process, but if we choose $h_2$ in such a way that
\begin{equation}
E_{\nu_{\rho}}(h_2(\eta))=E_{\nu_{\rho}}(h_1(\eta))
\label{eqn42}
\end{equation}
the polynomials $B$ and $D$ that we obtain are of the requested
form for the validity of (\ref{eqn35}).
An illuminating example is:
\begin{equation}
c(i,\eta)=c_1(1-\eta(i))\eta(i+1)\eta(i-1)+c_2\eta(i)\eta(i+1)\eta(i+2)
\label{exe}
\end{equation}
The microscopic irreversibility of this model is evident,
but the polynomials $B$ and $D$
are of the wanted form (\ref{eqn33}), (\ref{eqn34}):
\begin{equation}
B(\rho)=c_1(1-\rho)\rho^2
\end{equation}
\begin{equation}
D(\rho)=c_2\rho^3
\end{equation}
If we consider only spatially homogeneous fluctuations we can solve explicity
the equation (\ref{eqn10}) for the field $H$
\begin{equation}
H=log\frac{\dot{\rho}+\sqrt{\dot{\rho}^2+4B(\rho)D(\rho)}}{2B(\rho)}
\label{camp}
\end{equation}
and we obtain an expression of the functional $I$ in terms of the
trajectories $\rho$ only:
\begin{equation}
\left.\begin{array}{ccl}
I(\rho)&=&\int (B(\rho)+D(\rho)-\sqrt{\dot{\rho}^2+4B(\rho)D(\rho)} \\
& &+\dot{\rho}log\frac{\dot{\rho}+
\sqrt{\dot{\rho}^2+4B(\rho)D(\rho)}}{2B(\rho)})dt
\label{fuo}
\end{array}\right.
\end{equation}
One can show quite generally that for a fluctuation which can be connected
to equilibrium by a solution of (\ref{eqn6}) the minimizing trajectory
satisfies (\ref{eqn35}) ( $\rho$ depends
now only on $t$) for all polynomials $B$ and $D$. Therefore in this case any
dynamics reversible or irreversible satisfies the
time-reversal relation of Onsager-Machlup. A similar argument applies
also to the case in which the fluctuation cannot be directly connected
to equilibrium by a solution of (\ref{eqn6}). This can happen for example if
the potential has local minima.
\section{Concluding remarks}
The models we have considered are rather special and the
periodic boundary conditions play a crucial role for the validity of our
argument. It is necessary to study to what extent the result can be
generalized.
However an important principle has been demonstrated: microscopic
reversibility is not a necessary condition for the validity of certain
macroscopic reversibility properties.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ACKNOWLEDGMENTS
\acknowledgments
We thank Errico Presutti for his interest in this work and for
many discussions which were instrumental in clarifying basic
issues. One of us (GJ-L) thanks S.R.S. Varadhan for a useful
conversation and the Centro Linceo Interdisciplinare where
part of this work was done.
%
\appendix
\section*{Minimization of $I(\rho)$}
The basic point is that for polynomials of the form
(\ref{eqn33}),(\ref{eqn34}), it
is possible to write explicitely the field $H$ that generates the
solutions of equation (\ref{eqn36}):
\begin{equation}
H=\log\frac{c_2\rho^*}{c_1(1-\rho^*)}
\label{camp}
\end{equation}
For this reason it is possible to obtain on the solutions
of (\ref{eqn36}) an expression of the functional $I$
in terms of the
trajectory $\rho(x,t)$ only. Using (\ref{eqn36}), integrating by parts
and remembering the periodic boundary condition, we obtain the expression:
\begin{equation}
I(\rho^*)=\int_{-\infty}^{0}\int_0^1\partial_t\rho^*\
log(\frac{c_2\rho^*}{c_1(1-\rho*)})dtdx
\label{funin}
\end{equation}
The value
of this functional can be immediatly calculated and depends only on the values
of $\rho^*(x,t)$ at $t=0$ and $t=-\infty$:
\begin{equation}
\left.\begin{array}{ccl}
I(\rho)&=&\left\{\int_0^1\rho^*(x,t)\log(\frac{c_1}{c_2})dx+
\int_0^1\rho^*(x,t)\log\rho^*(x,t)dx\right. \\
&+&\left.\int_0^1(1-\rho^*(x,t))\log(1-\rho^*(x,t))dx\right\}|_{t=-\infty}^{t=0}
\end{array}\right.
\label{val}
\end{equation}
We now compare the value of the functional on a generic
trajectory that connects the equilibrium state to the state
$\gamma(x)$ with the
value of the functional
on the solution of (\ref{eqn36}) connecting the same states. Define
\begin{equation}
I(\rho)-I(\rho^*)=\Delta(\rho)
\end{equation}
we have
\begin{equation}
\left.\begin{array}{ccl}
\Delta(\rho) &=&\int_{-\infty}^{0}\int_0^1(\frac{1}{2}\rho(1-\rho)
(\partial_xH)^2
+ c_1(1-\rho)A(\rho)(1-e^{H}+He^{H}) \\
&+& c_2\rho A(\rho)(1-e^{-H}-He^{-H})
-\partial_t\rho\
log(\frac{c_2\rho}{c_1(1-\rho)}))dtdx
\end{array}\right.
\end{equation}
To obtain this expression we have used (\ref{val}). Using equation (
\ref{eqn10}) and integrating by parts we obtain finally the expression
\begin{equation}
\left.\begin{array}{ccl}
\Delta(\rho) &=&\int_{-\infty}^{0}\int_0^1(\frac{1}{2}\rho(1-\rho)
(\partial_xH-\frac{\partial_x\rho}{\rho(1-\rho)})^2 \\
&+& c_1(1-\rho)A(\rho)(1-e^{H}+He^{H}-e^H\log\frac{c_2\rho}{c_1(1-\rho)}) \\
&+& c_2\rho A(\rho)(1-e^{-H}-He^{-H}+e^{-H}
\log\frac{c_2\rho}{c_1(1-\rho)}))dxdt
\end{array}\right.
\end{equation}
The final step consists in introducing a new field $F$
\begin{equation}
F=\log(\frac{c_2\rho}{c_1(1-\rho)})-H
\end{equation}
This field is constructed in such a way that the value $F=0$ generates a
$\rho(x,t)$ solution of (\ref{eqn36}). The functional $\Delta(\rho)$ in
terms of $F$ becomes
\begin{equation}
\left.\begin{array}{ccl}
\Delta(\rho) &=& \frac{1}{2}\int_{-\infty}^{0}\int_0^1dtdx\rho(1-\rho)
(\partial_xF)^2 \\
&+& \int_{-\infty}^{0}\int_0^1dtdxc_1(1-\rho)A(\rho)(1-e^{F}+Fe^{F}) \\
&+& \int_{-\infty}^{0}\int_0^1dtdxc_2\rho A(\rho)(1-e^{-F}-Fe^{-F})
\end{array}\right.
\end{equation}
This functional is obviously positive and zero only if $F$ is zero.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% REFERENCES LIST
%
\begin{references}
\bibitem{ON1} L. Onsager, {\sl Reciprocal relations in irreversible
processes I and II} Phys. Rev. {\bf 37} (1931) 405; Phys. Rev. {\bf 38} (1931)
2265.
\bibitem{ON2} L. Onsager, S. Machlup {\sl Fluctuations in irreversible
processes I and II} Phys. Rev. {\bf 91} (1953) 1505; Phys. Rev. {\bf 91} (1953)
1512.
\bibitem{LAND} C. Landim, {\sl An overview of large deviations of
the empirical measure of interacting particle systems} Ann. Inst. Henry
Poincare' {\bf 55} (1991) 615.
\bibitem{JLVL} G. Jona-Lasinio, C. Landim, M. E. Vares, {\sl Large
deviations for a reaction diffusion model} Prob. Theory Rel. Fields
{\bf 97} (1993) 339.
\bibitem{KOV} C. Kipnis, S. Olla, S. R. S. Varadhan, Commun. Pure Appl.
Math. {\bf 42} (1989) 115.
\end{references}
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