Bethe Ansatz Solution for a Defect Particle in the Asymmetric Exclusion Process

TL;DR

This paper uses the Bethe ansatz to exactly analyze the long-term behavior of a defect particle in the asymmetric exclusion process, providing new insights into its diffusion and large deviation properties.

Contribution

It introduces an exact Bethe ansatz solution for the defect particle's dynamics, extending previous matrix product results and deriving new diffusion and large deviation characteristics.

Findings

Diffusion constant grows as system size to the power 1/2.

Distribution of travel distance is Gaussian in certain regimes.

Velocity and diffusion expressions match previous infinite system results.

Abstract

The asymmetric exclusion process on a ring in one-dimension is considered with a single defect particle. The steady state has previously been solved by a matrix product method. Here we use the Bethe ansatz to solve exactly for the long time limit behaviour of the generating function of the distance travelled by the defect particle. This allows us to recover steady state properties known from the matrix approach such as the velocity, and obtain new results such as the diffusion constant of the defect particle. In the case where the defect particle is a second class particle we determine the large deviation function and show that in a certain range the distribution of the distance travelled about the mean is Gaussian. Moreover the variance (diffusion constant) grows as L to the power 1/2 where is the system size. This behaviour can be related to the superdiffusive spreading of excess mass fluctuations on an infinite system. In the case where the defect particle produces a shock, our expressions for the velocity and the diffusion constant coincide with those calculated previously for an infinite system by Ferrari and Fontes.