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

TL;DR

This paper derives an exact large deviation functional for the asymmetric exclusion process, revealing non-local, non-convex, and non-Gaussian fluctuation behaviors in a driven diffusive system with open boundaries.

Contribution

It provides the first exact asymptotic expression for the probability of arbitrary macroscopic profiles in ASEP, highlighting complex fluctuation phenomena.

Findings

Large deviation functional is non-local and exact.

Functional can be non-convex and have discontinuities.

Fluctuations near typical profiles are non-Gaussian.

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 $\rho(x)$: $P_N(\{\rho(x)\})\sim\exp[-L{\cal F}_{[a,b]}(\{\rho(x)\});\rho_a,\rho_b]$, so that ${\cal F}$ 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 $\cal F$ 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 ${\cal F}(\{\rho(x)\})$ is not convex and others for which ${\cal F}(\{\rho(x)\})$ 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.