Exact Large Deviation Functional of a Stationary Open Driven Diffusive System: The Asymmetric Exclusion Process

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We consider the asymmetric exclusion process (ASEP) in one dimension on sites i = 1,..., N, in contact at sites i=1 and i=N with infinite particle reservoirs at densities ρₐ and ρb. As ρₐ and ρb are varied, the typical macroscopic steady state density profile ρ(x), x∈[a,b], obtained in the limit N=L(b-a)→∞, exhibits shocks and phase transitions. Here we derive an exact asymptotic expression for the probability of observing an arbitrary macroscopic profile ρ(x): Pₙ({ρ(x)}) [-L F[ₐ,b]({ρ(x)});ρₐ,ρb], so that F is the large deviation functional, a quantity similar to the free energy of equilibrium systems. We find, as in the symmetric, purely diffusive case q=1 (treated in an earlier work), that F is in general a non-local functional of ρ(x). Unlike the symmetric case, however, the asymmetric case exhibits ranges of the parameters for which F({ρ(x)}) is not convex and others for which F({ρ(x)}) has discontinuities in its second derivatives at ρ(x) = ρ(x); the fluctuations near ρ(x) are then non-Gaussian and cannot be calculated from the large deviation function.