Hydrodynamic mean field solutions of 1D exclusion processes with spatially varying hopping rates

TL;DR

This paper studies the steady-state behavior of a one-dimensional exclusion process with spatially varying hopping rates, providing analytical solutions and confirming their accuracy with simulations, revealing phase stability and sensitivity to rate variations.

Contribution

It introduces a mean field approach to analyze 1D exclusion processes with smoothly varying hopping rates, deriving Ricatti equations and solving them analytically and numerically.

Findings

Analytical solutions for steady-state currents are obtained and match simulations.

The phase structure remains stable when the hopping rate asymmetry is consistent across the chain.

Current sensitivity depends on the relative phase between forward and backward hopping rates.

Abstract

We analyze the open boundary partially asymmetric exclusion process with smoothly varying internal hopping rates in the infinite-size, mean field limit. The mean field equations for particle densities are written in terms of Ricatti equations with the steady-state current $J$ as a parameter. These equations are solved both analytically and numerically. Upon imposing the boundary conditions set by the injection and extraction rates, the currents $J$ are found self-consistently. We find a number of cases where analytic solutions can be found exactly or approximated. Results for $J$ from asymptotic analyses for slowly varying hopping rates agree extremely well with those from extensive Monte Carlo simulations, suggesting that mean field currents asymptotically approach the exact currents in the hydrodynamic limit, as the hopping rates vary slowly over the lattice. If the forward hopping rate is greater than or less than the backward hopping rate throughout the entire chain, the three standard steady-state phases are preserved. Our analysis reveals the sensitivity of the current to the relative phase between the forward and backward hopping rate functions.