Large Deviation of the Density Profile in the Symmetric Simple Exclusion Process

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 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[−NF({ρ})]; F is the large deviation functional. In contrast to equilibrium systems, for which Feq({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.