Power series solution of the inhomogeneous exclusion process

TL;DR

This paper introduces a power series method to analytically solve the inhomogeneous TASEP for nonequilibrium steady states, providing insights into particle currents and higher-order terms relevant to biological translation models.

Contribution

It presents a novel power series approach for inhomogeneous TASEP, enabling analytical computation of steady states and particle currents, including higher-order terms via Young tableaux.

Findings

Analytical expressions for particle current up to cubic order in entry/exit rates.

Method to compute higher-order terms using Young tableaux.

Addresses the long-standing problem of exact steady states in inhomogeneous TASEP.

Abstract

We develop a power series method for the nonequilibrium steady state of the inhomogeneous one-dimensional totally asymmetric simple exclusion process (TASEP) in contact with two particle reservoirs and with site-dependent hopping rates in the bulk. The power series is performed in the entrance or exit rates governing particle exchange with the reservoirs, and the corresponding particle current is computed analytically up to the cubic term in the entry or exit rate, respectively. We also show how to compute higher-order terms using combinatorial objects known as Young tableaux. Our results address the long outstanding problem of finding the exact nonequilibrium steady state of the inhomogeneous TASEP. The findings are particularly relevant to the modelling of mRNA translation in which the rate of translation initiation, corresponding to the entrance rate in the TASEP, is typically small.