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

Abstract

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 ρ _a and ρ _b . As ρ _a and ρ _b 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 $\rho (x):{\text{ }}P_N (\{ \rho (x)\} ) \sim \exp [ - L\mathcal{F}_{[a,b]} (\{ \rho (x)\} ;\rho _a ,\rho _b )]$, so that $\mathcal{F}$ 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 $\mathcal{F}$ 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 $\mathcal{F}(\{ \rho (x)\} )$ is not convex and others for which $\mathcal{F}(\{ \rho (x)\} )$ has discontinuities in its second derivatives at ρ(x)=¯ρ(x). In the latter ranges the fluctuations of order $1/\sqrt N$ in the density profile near ¯ρ(x) are then non-Gaussian and cannot be calculated from the large deviation function.