Random walk of second class particles in product shock measures

TL;DR

This paper demonstrates that in certain stochastic particle systems, shock measures with a second class particle perform a simple random walk, extending previous results and including multiple shocks and nonconserving models.

Contribution

It generalizes the understanding of shock measures and second class particle dynamics across various particle systems, including nonconserving models.

Findings

Shock measures perform a simple random walk under certain conditions.

Previous results are special cases of the general theorem.

Multiple shocks and nonconserving models are also analyzed.

Abstract

We consider shock measures in a class of conserving stochastic particle systems on Z. These shock measures have a product structure with a step-like density profile and include a second class particle at the shock position. We show for the asymmetric simple exclusion process, for the exponential bricklayers' process, and for a generalized zero range process, that under certain conditions these shocks, and therefore the second class particles, perform a simple random walk. Some previous results, including random walks of product shock measures and stationary shock measures seen from a second class particle, are direct consequences of our more general theorem. Multiple shocks can also be handled easily in this framework. Similar shock structure is also found in a nonconserving model, the branching coalescing random walk, where the role of the second class particle is played by the rightmost (or leftmost) particle.