Scaling properties of the asymmetric exclusion process with long-range hopping

TL;DR

This paper investigates how long-range hopping with a power-law decay affects the scaling and density profiles in the asymmetric exclusion process, revealing fractional diffusion behavior and mean-field exponents for certain parameter ranges.

Contribution

It introduces a coarse-grained fractional diffusion equation for the process and analyzes its implications for density profiles and scaling in open systems with asymmetric long-range hopping.

Findings

Fractional diffusion replaces the usual diffusion term for 1<sigma<2.

Density profile decay in the maximum-current phase follows a sigma-dependent algebraic law.

The decay exponent in this regime is of the mean-field type.

Abstract

The exclusion process in which particles may jump any distance l>=1 with the probability that decays as l^-(1+sigma) is studied from coarse-grained equation for density profile in the limit when the lattice spacing goes to zero. For 1<sigma<2, the usual diffusion term of this equation is replaced by the fractional one, which affects dynamical-scaling properties of the late-time approach to the stationary state. When applied to an open system with totally asymmetric hopping, this approach gives two results: first, it accounts for the sigma-dependent exponent that characterizes the algebraic decay of density profile in the maximum-current phase for 1<sigma<2, and second, it shows that in this region of sigma the exponent is of the mean-field type.