Exact solution and asymptotic behaviour of the asymmetric simple exclusion process on a ring

TL;DR

This paper derives exact formulas for the asymmetric simple exclusion process on a ring, using hypergeometric functions, and analyzes its asymptotic velocity behavior in large systems.

Contribution

It provides explicit formulas for the partition function and average velocity using hypergeometric functions, and explores their asymptotic behavior in large systems.

Findings

Explicit formulas for partition function and average velocity.

Asymptotic expansion of average velocity in the thermodynamic limit.

Abstract

In this paper, we study an exact solution of the asymmetric simple exclusion process on a periodic lattice of finite sites with two typical updates, i.e., random and parallel. Then, we find that the explicit formulas for the partition function and the average velocity are expressed by the Gauss hypergeometric function. In order to obtain these results, we effectively exploit the recursion formula for the partition function for the zero-range process. The zero-range process corresponds to the asymmetric simple exclusion process if one chooses the relevant hop rates of particles, and the recursion gives the partition function, in principle, for any finite system size. Moreover, we reveal the asymptotic behaviour of the average velocity in the thermodynamic limit, expanding the formula as a series in system size.