Partially Asymmetric Simple Exclusion Model in the Presence of an Impurity on a Ring

TL;DR

This paper analyzes a generalized two-species exclusion model on a ring with an impurity, incorporating backward hopping for particles, and uses algebraic methods to explore its phase structure and shock phenomena.

Contribution

It introduces a model with backward hopping and impurity dynamics, extending previous asymmetric exclusion models and providing exact solutions for stationary states.

Findings

Derived quadratic algebra using Matrix Product Ansatz

Computed stationary bulk density and impurity speed

Identified phase structure and shock formation in the model

Abstract

We study a generalized two-species model on a ring. The original model [1] describes ordinary particles hopping exclusively in one direction in the presence of an impurity. The impurity hops with a rate different from that of ordinary particles and can be overtaken by them. Here we let the ordinary particles hop also backward with the rate q. Using Matrix Product Ansatz (MPA), we obtain the relevant quadratic algebra. A finite dimensional representation of this algebra enables us to compute the stationary bulk density of the ordinary particles, as well as the speed of impurity on a set of special surfaces of the parameter space. We will obtain the phase structure of this model in the accessible region and show how the phase structure of the original model is modified. In the infinite-volume limit this model presents a shock in one of its phases.