Exact solution of a partially asymmetric exclusion model using a deformed oscillator algebra

TL;DR

This paper provides an exact solution for a partially asymmetric exclusion process with open boundaries, utilizing a deformed oscillator algebra and q-Hermite polynomials to analyze different bias regimes and phase behaviors.

Contribution

It generalizes the matrix approach to include all asymmetry parameters using q-deformed algebra, revealing new phase behavior in reverse bias conditions.

Findings

Exact expressions for partition sum and currents for all asymmetry values

Identification of a new phase with exponentially decreasing current in reverse bias

Confirmation of the phase diagram for forward bias

Abstract

We study the partially asymmetric exclusion process with open boundaries. We generalise the matrix approach previously used to solve the special case of total asymmetry and derive exact expressions for the partition sum and currents valid for all values of the asymmetry parameter q. Due to the relationship between the matrix algebra and the q-deformed quantum harmonic oscillator algebra we find that q-Hermite polynomials, along with their orthogonality properties and generating functions, are of great utility. We employ two distinct sets of q-Hermite polynomials, one for q<1 and the other for q>1. It turns out that these correspond to two distinct regimes: the previously studied case of forward bias (q<1) and the regime of reverse bias (q>1) where the boundaries support a current opposite in direction to the bulk bias. For the forward bias case we confirm the previously proposed phase diagram whereas the case of reverse bias produces a new phase in which the current decreases exponentially with system size.