How to initialize a second class particle?

TL;DR

This paper characterizes the limit distribution of the second class particle in various symmetric and asymmetric interacting particle systems, extending previous results with a novel signed coupling measure and revealing invariance properties.

Contribution

It introduces a signed coupling measure for initial data, enabling the extension of limit distribution results to a broader class of models with diverse hydrodynamics.

Findings

Limit distribution of second class particle identified in multiple models.

Introduction of signed coupling measure as initial data.

Invariance property of the one-site marginal distribution.

Abstract

We identify the ballistically and diffusively rescaled limit distribution of the second class particle position in a wide range of asymmetric and symmetric interacting particle systems with established hydrodynamic behavior, respectively (including zero-range, misanthrope and many other models). The initial condition is a step profile which, in some classical cases of asymmetric models, gives rise to a rarefaction fan scenario. We also point out a model with non-concave, non-convex hydrodynamics, where the rescaled second class particle distribution has both continuous and discrete counterparts. The results follow from a substantial generalization of P.A. Ferrari and C. Kipnis' arguments ("Second class particles in the rarefaction fan", Ann. Inst. H. Poincar\'e, 31, 1995) for the totally asymmetric simple exclusion process. The main novelty is the introduction of a signed coupling measure as initial data, which nevertheless results in a proper probability initial distribution for the site of the second class particle and makes the extension possible. We also reveal in full generality a very interesting invariance property of the one-site marginal distribution of the process underneath the second class particle which in particular proves the intrinsicality of our choice for the initial distribution. Finally, we give a lower estimate on the probability of survival of a second class particle-antiparticle pair.