Partially Asymmetric Exclusion Process with Open Boundaries

TL;DR

This paper analyzes a one-dimensional asymmetric exclusion process with open boundaries, deriving the exact stationary measure, phase diagram, and current expressions, and relates it to the XXZ-Heisenberg chain model.

Contribution

It generalizes the matrix product solution for asymmetric exclusion processes to partially asymmetric cases with open boundaries.

Findings

Exact stationary probability measure derived

Phase diagram of the current obtained

Analytic expressions for current in different phases provided

Abstract

Exclusive diffusion on a one-dimensional lattice is studied. In the model particles hop stochastically into both directions with different rates. At the ends of the lattice particles are injected and removed. The exact stationary probability measure is represented in form of a matrix product as a generalization of the solution given by Derrida et al \cite{dehp} for the fully asymmetric process. The phase diagram of the current on the infinite lattice is obtained. Analytic expressions for the current in the different phases are derived. The model is equivalent to a $XXZ$-Heisenberg chain with a certain type of boundary terms the ground state of which corresponds to the stationary solution of the master equation.