Bethe ansatz solution of zero-range process with nonuniform stationary state

TL;DR

This paper exactly solves the eigenfunctions and eigenvalues of a zero-range process with nonuniform stationary states using the Bethe ansatz, revealing connections to known models and universal scaling behaviors.

Contribution

It extends the Bethe ansatz method to zero-range processes with nonuniform stationary states and analyzes their scaling properties.

Findings

Exact eigenfunctions and eigenvalues obtained

Universal scaling form identified for non-zero interaction

Connections established with exclusion and drop-push models

Abstract

The eigenfunctions and eigenvalues of the master-equation for zero range process with totally asymmetric dynamics on a ring are found exactly using the Bethe ansatz weighted with the stationary weights of particle configurations. The Bethe ansatz applicability requires the rates of hopping of particles out of a site to be the $q$-numbers $[n]_q$. This is a generalization of the rates of hopping of noninteracting particles equal to the occupation number $n$ of a site of departure. The noninteracting case can be restored in the limit $q\to 1$. The limiting cases of the model for $q=0,\infty$ correspond to the totally asymmetric exclusion process, and the drop-push model respectively. We analyze the partition function of the model and apply the Bethe ansatz to evaluate the generating function of the total distance travelled by particles at large time in the scaling limit. In case of non-zero interaction, $q \ne 1$, the generating function has the universal scaling form specific for the Kardar-Parizi-Zhang universality class.