Queuing Transitions in the Asymmetric Simple Exclusion Process

TL;DR

This paper investigates a phase transition in the asymmetric simple exclusion process, revealing how queues form and scale around an obstacle, with implications for interface growth and polymer localization.

Contribution

It numerically confirms a dynamic queuing phase transition and establishes the universal scaling behavior of the queue profile near the obstacle.

Findings

Below the transition, the queue length scales linearly with system size.

Above the transition, the queue follows a power-law density profile with exponent 1/3.

Fast bonds produce depletion queues with a different, possibly 2/3, exponent.

Abstract

Stochastic driven flow along a channel can be modeled by the asymmetric simple exclusion process. We confirm numerically the presence of a dynamic queuing phase transition at a nonzero obstruction strength, and establish its scaling properties. Below the transition, the traffic jam is macroscopic in the sense that the length of the queue scales linearly with system size. Above the transition, only a power-law shaped queue remains. Its density profile scales as $\delta \rho\sim x^{-\nu}$ with $\nu={1/3}$, and $x$ is the distance from the obstacle. We construct a heuristic argument, indicating that the exponent $\nu={1/3}$ is universal and independent of the dynamic exponent of the underlying dynamic process. Fast bonds create only power-law shaped depletion queues, and with an exponent that could be equal to $\nu={2/3}$, but the numerical results yield consistently somewhat smaller values $\nu\simeq 0.63(3)$. The implications of these results to faceting of growing interfaces and localization of directed polymers in random media, both in the presence of a columnar defect are pointed out as well.