Disordered exclusion process revisited: some exact results in the low-current regime

TL;DR

This paper develops a method to calculate exact low-order coefficients of steady-state weights in disordered exclusion processes, revealing how the current depends on the slowest sites in the low-current regime.

Contribution

It introduces a general polynomial-based approach to compute steady-state weights and identifies the dominant role of minimal hopping rate subsets in determining current.

Findings

Exact calculation of steady-state current up to quadratic order in binary disorder.

In low-current regime, current depends only on the slowest sites, regardless of other rates.

Method applicable to other driven diffusive systems with unidirectional hopping.

Abstract

We study steady state of the totally asymmetric simple exclusion process with inhomogeneous hopping rates associated with sites (site-wise disorder). Using the fact that the non-normalized steady-state weights which solve the master equation are polynomials in all the hopping rates, we propose a general method for calculating their first few lowest coefficients exactly. In case of binary disorder where all slow sites share the same hopping rate r<1, we apply this method to calculate steady-state current up to the quadratic term in r for some particular disorder configurations. For the most general (non-binary) disorder, we show that in the low-current regime the current is determined solely by the current-minimizing subset of equal hopping rates, regardless of other hopping rates. Our approach can be readily applied to any other driven diffusive system with unidirectional hopping if one can identify a hopping rate such that the current vanishes when this rate is set to zero.