Spontaneous symmetry breaking: exact results for a biased random walk model of an exclusion process

TL;DR

This paper rigorously analyzes a biased random walk model of an exclusion process, demonstrating spontaneous symmetry breaking and exponential divergence of transition times between symmetry-broken states.

Contribution

It provides an exact solution for the stationary measure and transition times in a biased random walk limit of the exclusion process, confirming symmetry breaking phenomena.

Findings

Stationary measure concentrates around two symmetric points.

Average transition time diverges exponentially with system size.

Supports the existence of spontaneous symmetry breaking in the model.

Abstract

It has been recently suggested that a totally asymmetric exclusion process with two species on an open chain could exhibit spontaneous symmetry breaking in some range of the parameters defining its dynamics. The symmetry breaking is manifested by the existence of a phase in which the densities of the two species are not equal. In order to provide a more rigorous basis to these observations we consider the limit of the process when the rate at which particles leave the system goes to zero. In this limit the process reduces to a biased random walk in the positive quarter plane, with specific boundary conditions. The stationary probability measure of the position of the walker in the plane is shown to be concentrated around two symmetrically located points, one on each axis, corresponding to the fact that the system is typically in one of the two states of broken symmetry in the exclusion process. We compute the average time for the walker to traverse the quarter plane from one axis to the other, which corresponds to the average time separating two flips between states of broken symmetry in the exclusion process. This time is shown to diverge exponentially with the size of the chain.