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 ${\rho }_a$ and ${\rho }_b$. As ${\rho }_a$ and ${\rho }_b$ are varied, the typical macroscopic steady state density profile $\bar{\rho }(x)$, $x\in [a,b]$, obtained in the limit $N=L(b-a)\to \infty$, exhibits shocks and phase transitions. Here we derive an exact asymptotic expression for the probability of observing an arbitrary macroscopic profile $\rho (x)$: $P_N(\{\rho (x)\})\sim \exp [-L{\mathcal{F}}_{[a,b]}(\{\rho (x)\});{\rho }_a,{\rho }_b]$, 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 $\rho (x)$. 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 $\rho (x)=\bar{\rho }(x)$; the fluctuations near $\bar{\rho }(x)$ are then non-Gaussian and cannot be calculated from the large deviation function.

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