Open interacting particle systems and Ising measures

TL;DR

This paper surveys open questions in stochastic interacting particle systems with open boundaries, introduces a generalized asymmetric exclusion process, proves invariance of the Ising measure, and discusses phase transitions and current reversal phenomena.

Contribution

It introduces a generalized asymmetric exclusion process with open boundaries, proves invariance of the Ising measure, and analyzes phase transitions and current reversal in this context.

Findings

Invariance of the one-dimensional Ising measure is established.

Explicit computation of the stationary current reveals current reversal at certain densities.

The phase diagram for boundary-induced phase transitions is conjectured based on the extremal-current principle.

Abstract

We first survey some open questions concerning stochastic interacting particle systems with open boundaries. Then an asymmetric exclusion process with open boundaries that generalizes the lattice gas model of Katz, Lebowitz, and Spohn (KLS) is introduced and invariance of the one-dimensional Ising measure is proved. The stationary current is computed in explicit form and is shown to exhibit current reversal at some density. Based on the extremal-current principle for one-dimensional driven diffusive systems with one conservation law, the phase diagram for boundary-induced phase transitions is conjectured for this case. There are two extremal-current phases, unlike in the conventional open asymmetric simple exclusion process, which exhibits only one extremal-current phase or the previously considered conventional open KLS model with one or three extremal-current phases.