Exact solution of the zero-range process: fundamental diagram of the corresponding exclusion process

TL;DR

This paper derives exact solutions for the zero-range process's steady state, enabling precise computation of the fundamental diagram of the associated exclusion process using hypergeometric functions.

Contribution

It introduces a novel method to compute expectation values in the zero-range process via an exact partition function expressed with Lauricella hypergeometric functions.

Findings

Exact fundamental diagrams for parallel and sequential updates

Partition function expressed with Lauricella hypergeometric functions

Results consistent with previous grand canonical ensemble studies

Abstract

In this paper, we propose a general way of computing expectation values in the zero-range process, using an exact form of the partition function. As an example, we provide the fundamental diagram (the flux-density plot) of the asymmetric exclusion process corresponding to the zero-range process.We express the partition function for the steady state by the Lauricella hypergeometric function, and thereby have two exact fundamental diagrams each for the parallel and random sequential update rules. Meanwhile, from the viewpoint of equilibrium statistical mechanics, we work within the canonical ensemble but the result obtained is certainly in agreement with previous works done in the grand canonical ensemble.