Analytical Approach to the One-Dimensional Disordered Exclusion Process with Open Boundaries and Random Sequential Dynamics

TL;DR

This paper provides an analytical study of a disordered asymmetric exclusion process with open boundaries, revealing phase transitions, stability regions, and effects of disorder on current and density profiles.

Contribution

It introduces a perturbative mean field approach to analyze the disordered ASEP with open boundaries, extending known results and deriving new analytical expressions for phase transitions and disorder effects.

Findings

High-density, low-density, and maximum-current phases identified.

First-order phase transition occurs at unequal injection and extraction rates.

Disorder extends the maximum-current phase stability and induces a cusp in the current-density relation.

Abstract

A one dimensional disordered particle hopping rate asymmetric exclusion process (ASEP) with open boundaries and a random sequential dynamics is studied analytically. Combining the exact results of the steady states in the pure case with a perturbative mean field-like approach the broken particle-hole symmetry is highlighted and the phase diagram is studied in the parameter space $(\alpha,\beta)$, where $\alpha$ and $\beta$ represent respectively the injection rate and the extraction rate of particles. The model displays, as in the pure case, high-density, low-density and maximum-current phases. All critical lines are determined analytically showing that the high-density low-density first order phase transition occurs at $\alpha \neq \beta$. We show that the maximum-current phase extends its stability region as the disorder is increased and the usual $1/\sqrt{\ell}$-decay of the density profile in this phase is universal. Assuming that some exact results for the disordered model on a ring hold for a system with open boundaries, we derive some analytical results for platoon phase transition within the low-density phase and we give an analytical expression of its corresponding critical injection rate $\alpha^*$. As it was observed numerically$^{(19)}$, we show that the quenched disorder induces a cusp in the current-density relation at maximum flow in a certain region of parameter space and determine the analytical expression of its slope. The results of numerical simulations we develop agree with the analytical ones.