Totally Asymmetric Zero-Range process in the Rarefaction Fan

TL;DR

This paper analyzes the behavior of second class particles in a totally asymmetric zero-range process, deriving their asymptotic distribution and law of large numbers in the context of hydrodynamic limits and rarefaction fans.

Contribution

It provides explicit calculations of joint probabilities and the law of large numbers for second class particles under specific initial conditions in the zero-range process.

Findings

Convergence of weighted sums of joint probabilities for second class particles.

Explicit limit formulas for second class particle distributions.

Law of Large Numbers for particle positions in specified initial states.

Abstract

We consider the one-dimensional totally asymmetric zero-range process starting from a step decreasing profile leading in the hydrodynamic limit to the rarefaction fan of the associate hydrodynamic equation. Under that initial condition, we show that the weighted sum of joint probabilities for second class particles sharing the same site, is convergent and we compute its limit. We derive the Law of Large Numbers for the position of a second class particle initially at the origin under the initial state in which all positive sites are empty and all negative sites are occupied and also for a slight perturbation of the invariant state.