Impurity-induced shocks in the asymmetric exclusion process with long-range hopping

TL;DR

This paper studies a long-range hopping variant of the TASEP with a defect, revealing phase transitions and shock behaviors similar to the short-range case, with analytical insights into decay exponents and critical points.

Contribution

It introduces a long-range hopping TASEP model with a defect, analytically characterizing phase transitions and shock profiles, extending understanding beyond short-range models.

Findings

Shock width scales as L^(1/2) or L^(1/3) depending on symmetry.

Density profiles follow a power-law decay with a sigma-dependent exponent.

Critical sigma value for phase transition is identified as 4/3.

Abstract

We consider the totally asymmetric simple exclusion process (TASEP) on the periodic chain in the presence of a single impurity site that is inaccessible to other particles and therefore acts as a static defect. Particles are allowed to advance any distance l \geq 1 on the right with the probability that decays as l^-(1+sigma), where sigma > 1. Despite the long range of hopping, we find the same type of phase transition that occurs in the standard short-range TASEP with a defect site where defect induces a macroscopic shock in the stationary state. In particular, our model displays two main features characteristic of the short-range TASEP with defect site: a growth of the shock width with system size L as L^(1/2) or L^(1/3), depending on the existence of the particle-hole symmetry, and the power-law decay in density profiles of the shock phase. However, unlike the profiles in the short-range case, we find that the latter are well reproduced by the mean-field approximation, which enables us to derive the analytical expression for sigma-dependent exponent nu = sigma-1 of this power-law decay and the point sigma_c = 4/3 at which the transition takes place.