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

AbstractWe 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 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→∞. The probability of microscopic configurations corresponding to some other profile ρ(x), x=i/N, has the asymptotic form exp[−N $$F$$ ({ρ})]; $$F$$ is the large deviation functional. In contrast to equilibrium systems, for which $$F$$ eq({ρ}) is just the integral of the appropriately normalized local free energy density, the $$F$$ we find here for the nonequilibrium system is a nonlocal function of ρ. This gives rise to the long range correlations in the SNS predicted by fluctuating hydrodynamics and suggests similar non-local behavior of $$F$$ in general SNS, where the long range correlations have been observed experimentally.