Exact probability function for bulk density and current in the asymmetric exclusion process

TL;DR

This paper derives an exact probability distribution for bulk density and current in the asymmetric exclusion process, revealing non-Gaussian fluctuations and phase transition behaviors in a nonequilibrium steady state.

Contribution

It provides a novel exact form of the joint probability function for finite and infinite systems, using an improved operator algebra approach.

Findings

Distribution is non-Gaussian

Density fluctuations are discontinuous at phase transitions

Current fluctuations are continuous across phase transitions

Abstract

We examine the asymmetric simple exclusion process with open boundaries, a paradigm of driven diffusive systems, having a nonequilibrium steady state transition. We provide a full derivation and expanded discussion and digression on results previously reported briefly in M. Depken and R. Stinchcombe, Phys. Rev. Lett. {\bf 93}, 040602, (2004). In particular we derive an exact form for the joint probability function for the bulk density and current, both for finite systems, and also in the thermodynamic limit. The resulting distribution is non-Gaussian, and while the fluctuations in the current are continuous at the continuous phase transitions, the density fluctuations are discontinuous. The derivations are done by using the standard operator algebraic techniques, and by introducing a modified version of the original operator algebra. As a byproduct of these considerations we also arrive at a novel and very simple way of calculating the normalization constant appearing in the standard treatment with the operator algebra. Like the partition function in equilibrium systems, this normalization constant is shown to completely characterize the fluctuations, albeit in a very different manner.