Large deviation of the density profile in the steady state of the open symmetric simple exclusion process
Résumé
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.