Diffusion and Scattering of Shocks in the Partially Asymmetric Simple Exclusion Process

Abstract

We study the behavior of shocks in the asymmetric simple exclusion process on $Z$ whose initial distribution is a product measure with a finite number of shocks. We prove that if the particle hopping rates of this process are in a particular relation with the densities of the initial measure then the distribution of this process at any time is a linear combination of shock measures of the structure similar to that of the initial distribution. The structure of this linear combination allows us to interpret this result by saying that the shocks of the initial distribution perform continuous time random walks on $Z$ interacting by the exclusion rule. We give explicit expressions for the hopping rates of these random walks. The result is derived with a help of quantum algebra technique. We made the presentation self-contained for the benefit of readers not acquainted with this approach, but interested in applying it in the study of interacting particle systems.