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Large Deviation of the Density Profile in the Steady State of the Open 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 {\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 $ρ(x)$, $x = i/N$, has the asymptotic form $\exp[-N {\cal F}(\{ρ\})]$; $\cal F$ is the {\it large deviation functional}. In contrast to equilibrium systems, for which ${\cal F}_{eq}(\{ρ\})$ 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 $ρ$. 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.