Large Deviation of the Density Profile in the Steady State of the Open Symmetric Simple Exclusion Process

Details Large Deviation of the Density Profile in the Steady State of the Open Symmetric Simple Exclusion Process Large Deviation of the Density Profile in the Steady State of the Open Symmetric Simple Exclusion Process

2001-09-19

arxiv.org/abs/cond-mat/0109346v3

J. Stat Phys. 107, 599--634 (2002).

Physics

Condensed Matter

37 pages, in Latex, e-mail addresses: derrida@lps.ens.fr, lebowitz@sakharov.rutgers.edu, speer@math.rutgers.edu This is a revi

Scientific paper

We consider an open one dimensional lattice gas on sites $i=1,...,N$, with particles jumping independently with rate 1 to neighboring interior empty sites, the {\it simple symmetric exclusion process}. The particle fluxes at the left and right boundaries, corresponding to exchanges with reservoirs at different chemical potentials, create a stationary nonequilibrium state (SNS) with a steady flux of particles through the system. The mean density profile in this state, which is linear, describes the typical behavior of a macroscopic system, i.e., this profile occurs with probability 1 when $N \to \infty$. The probability of microscopic configurations corresponding to some other profile $\rho(x)$, $x = i/N$, has the asymptotic form $\exp[-N {\cal F}(\{\rho\})]$; $\cal F$ is the {\it large deviation functional}. In contrast to equilibrium systems, for which ${\cal F}_{eq}(\{\rho\})$ is just the integral of the appropriately normalized local free energy density, the $\cal F$ we find here for the nonequilibrium system is a nonlocal function of $\rho$. This gives rise to the long range correlations in the SNS predicted by fluctuating hydrodynamics and suggests similar non-local behavior of $\cal F$ in general SNS, where the long range correlations have been observed experimentally.

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