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

摘要

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