%\documentstyle[twocolumn,prl,aps]{revtex}
\documentstyle[twocolumn,prl,aps]{revtex}
\oddsidemargin=-1.34cm
\evensidemargin=-1.34cm
\topmargin=-1.5cm
\tighten
\begin{document}
%\preprint{}
%\draft
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TITLE PAGE
%
\title{Onsager reciprocity relations without microscopic
reversibility}
\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}
%\date{\today}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ABSTRACT
%
\begin{abstract}
In this paper we show that
Onsager--Machlup time reversal properties of thermodynamic fluctuations
and Onsager reciprocity relations for transport
coefficients can hold also if the
microscopic dynamics is not reversible.
This result is based on
the explicit construction of a class of conservative models which can
be analysed rigorously.
\end{abstract}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PACS NUMBERS
%
%\pacs{ }
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PAPER CONTENT
%
%\documentstyle[10pt]{article}
%\twocolumn
%\oddsidemargin=-1.34cm
%\evensidemargin=-1.34cm
\def\N{I\!\! N}
\def\Z{Z\!\!\! Z}
\def\R{I\!\! R}
\def\T{{T\!\!\! T}}
%\begin{document}
\narrowtext
\medskip
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.
These topics have received a certain amount of attention in
the physical litterature in the course of the last forty years.
No rigorous results have been however established.
More recently, this subjet has been taken
up in various papers attempting more rigorous approaches~:
\cite{E}, \cite{GJLV}, \cite{ELS}, in the context
of so called interacting particle systems and \cite{G1},
\cite{G2} in a context of deterministic dynamical systems.
In \cite{GJLV} we discussed the following question: is microscopic
reversibility a necessary condition for the validity of the
Onsager and Onsager--Machlup 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.
In \cite{GJLV} we have exhibited a class of
microscopic nonreversible stochastic dynamics for which
the time reversal rule of Onsager--Machlup is still valid
even for fluctuations very far from equilibrium.
This class of dynamics concerns dissipative one component systems with
a hydrodynamic equation of
gradient type. Therefore there is no Onsager reciprocity relation
to verify.
In this paper we present a class of nonreversible multi component
conservative models giving rise at
the macroscopic level to nonlinear purely diffusive equations
in the terminology of \cite{E}.
The equations are of the following form
\begin{equation}
\partial_t \rho = \sum_{i=1}^d \partial_{u_i}
\Big\{ D(\rho) \cdot \partial_{u_i} \rho\Big\}
\end{equation}
where $\rho(u,t) =(\rho_1(u,t),\dots , \rho_n(u,t))$ is
a vector standing for the densities of different kinds
of particles and $D$ is in general a nonsymmetric $n\times n$ matrix.
Associated to our models there is an entropy functional
$S(\rho)$ that is written as the integral of a
density $s(\rho)$~: $S(\rho) = \int s(\rho(u)) du$.
The Onsager coefficients are defined in this context by
\begin{equation}
L (\rho) = D(\rho) \cdot R(\rho)
\end{equation}
where the matrix $R$ is determined by the entropy density
$s(\rho(u))$ in the following way
\begin{equation}
(R^{-1})_{i,j} = \frac{ \partial^2}{\partial \rho_i(u) \partial \rho_j(u)}
s(\rho(u))
\label{rr}
\end{equation}
which is by definition a symmetric matrix.
Onsager's reciprocity relations mean that
$L$ is a symmetric matrix, a property which holds
for our models.
In the physical literature one usually proves the
Onsager--Machlup time reversal property from Onsager
reciprocity relations. In our approach, we follow the opposite
order~: we obtain a large deviation functional from which
we prove the Onsager--Machlup time reversal property and compute
the entropy.
This in turn allows us to prove Onsager reciprocity
relations.
Our results are not restricted
to the neighborhood of the equilibrium.
For simplicity, we shall restrict ourselves to one dimensional
two component models but all analysis can be carried out
for any space dimension or for any number of components.
As in \cite{GJLV}, we consider periodic boundary conditions.
The systems considered in this paper differ from those
of \cite{GJLV} due to the conservative character of the
dynamics.
We consider an interacting particle system that
describes the evolution of two types of particles on a lattice.
The stochastic dynamics can be informally described
as follows. Fix a nonnegative function
$g: \N\rightarrow \R_+$ such that $g(0)=0From the equations (\ref{lf}), (\ref{inf}), one sees that to find
the most probable trajectory that connects the equilibrium
$\bar \rho$ to a certain state $\gamma(u)$
one has to find the $\rho(u,t)$ that minimizes
$I_{-\infty, 0}(\rho)$ in the set
${\cal G}_\gamma$ of all trajectories satisfying the boundary
conditions
\begin{equation}
\lim_{t\rightarrow -\infty}\rho(u,t)=\bar \rho\; ,\quad
\rho(u,0)=\gamma(u)\; .
\label{eqn32}
\end{equation}
It is now possible
to prove, following the same approach of \cite{GJLV},
that the unique solution of
our variational problem is the function $\rho^*(u,t)$ defined by
\begin{equation}
\rho^*(u,t)=\rho(u,-t)
\label{eqn35}
\end{equation}
where $\rho(u,t)$ is the solution of the hydrodynamic
equation which relaxes
to equilibrium with initial state $\gamma$. $\rho^*(u,t)$
is therefore a solution of the hydrodynamic
equation with inverted drift
\begin{equation}
\partial_t\rho=-(\sigma^2/2) \partial_u
\Big\{ D (\rho) \cdot \partial_u \rho \Big\}\; .
\label{eqn36}
\end{equation}
Equation (\ref{eqn35}) is the Onsager-Machlup time-reversal relation.
Denote by $S(\gamma)$ the functional defined by
\begin{equation}
S(\gamma)=\inf_{\rho\in{\cal G}_\gamma} I_{-\infty, 0}(\rho)\; .
\label{entropy}
\end{equation}
which, by the Boltzmann--Einstein relationship,
has to be identified with the entropy of the system.
By inserting (\ref{eqn35}) in (\ref{entropy}) we obtain
an explicit formula for the entropy~:
$$
S(\gamma) = \int_\T s(\gamma (u)) du
$$
where
$$
\quad s(\gamma) = \sum_{j=1}^2 E(\gamma_j (u)) +
F(\gamma_1(u) + \gamma_2(u) )
$$
and $E(\rho) = \int^\rho \log \rho' d\rho'$,
$F(\rho) = \int^\rho \log b(\rho') d\rho'$.
It is possible to check that $S(\rho(\cdot ,t))$ decreases in time if
$\rho(\cdot,t)$ is a solution of the hydrodynamic
equation (\ref{eqn6}).
Of course, the entropy could also be calculated
from the equilibrium measure (\ref{eqm}) and it is easy to
see that the two expressions coincide up to an additive constant.
This explicit expression for the entropy
$S(\cdot)$ permits to check Onsager's relations
in our model. A simple computation shows that
the matrix $R$ defined by equation (\ref{rr}) is
such that
\begin{equation}
(R^{-1})_{i,j} = \delta_{i,j} \frac{1}{\gamma_i(u)} +
\frac{b' (\gamma (u))}{b(\gamma (u))}
\end{equation}
where $\delta_{i,j}$ stands for the delta of
Kronecker and $s(\gamma)$ is the entropy density.
The product $L=DR$ can now be computed using the
explicit formula for $D$ given in
(\ref{difmat}) and shown to
be a symmetric matrix.
In all the above calculations we never used
the symmetry properties of the transition probability
$p(\cdot)$ so that they are valid both for reversible
and irreversible dynamics.
This provides conclusive evidence that macroscopic reversibility,
in the sense of validity of the above results,
does not require microscopic reversibility.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ACKNOWLEDGMENTS
\acknowledgments
G. J-L. acknowledges a useful correspondence with G. Eyink
and interesting discussions with G. Gallavotti.
C. L., thanks the INFN (sezione di Roma)
and the CNRS-CNR agreement for hospitality and support.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% REFERENCES LIST
%
\begin{references}
\bibitem{ON1} L. Onsager, Phys. Rev. {\bf 37} (1931) 405;
Phys. Rev. {\bf 38} (1931) 2265.
\bibitem{ON2} L. Onsager, S. Machlup, Phys. Rev. {\bf 91} (1953) 1505;
Phys. Rev. {\bf 91} (1953) 1512.
\bibitem{E} G. Eyink, J. Stat. Phys. {\bf 61} (1990) 533.
\bibitem{GJLV} D. Gabrielli, G. Jona-Lasinio, C. Landim
and M. E. Vares, {\sl Microscopic reversibility and
thermodynamics fluctuations} (1995), preprint, to appear
in proceedings of the conference ``Boltzmann's legacy'', Rome
May 1994. mp_arc@math.utexas.edu #95-248
\bibitem{ELS} G. Eyink, J. Lebowitz and H. Spohn,
{\sl Hydrodynamics and fluctuations outside of local
equlibrium: driven diffusive systems} (1995), preprint.
\bibitem{G1} G. Gallavotti {\sl Chaotic principle : some applications
to developed turbulence} (1995), preprint.
\bibitem{G2} G. Gallavotti {\sl Chaotic hypothesis: Onsager reciprocity
and fluctuation--dissipation theorem.} (1995), preprint.
\bibitem{JLVL} G. Jona-Lasinio, C. Landim, M. E. Vares,
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.
\bibitem{DV} M. Donsker and S. R. S. Varadhan,
Commun. Pure Appl. Math. {\bf 42} (1989) 243.
\end{references}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FIGURE CAPTION
%
\end{document}