On the asymmetric zero-range in the rarefaction fan

TL;DR

This paper analyzes the behavior of second class particles in asymmetric zero-range processes with step decreasing initial profiles, deriving convergence results, laws of large numbers, and random characteristic selection under various asymmetry conditions.

Contribution

It provides new results on the asymptotic behavior and characteristics of second class particles in zero-range processes, including convergence of joint probabilities and random characteristic selection.

Findings

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

Law of Large Numbers for second class particle positions.

Random selection among infinite characteristics based on hydrodynamic solutions.

Abstract

We consider the one-dimensional asymmetric zero-range process starting from a step decreasing profile. In the hydrodynamic limit this initial condition leads to the rarefaction fan of the associated hydrodynamic equation. Under this initial condition and for totally asymmetric jumps, we show that the weighted sum of joint probabilities for second class particles sharing the same site is convergent and we compute its limit. For partially asymmetric jumps we derive the Law of Large Numbers for the position of a second class particle under the initial configuration in which all the positive sites are empty, all the negative sites are occupied with infinitely many first class particles and with a single second class particle at the origin. Moreover, we prove that among the infinite characteristics emanating from the position of the second class particle, this particle chooses randomly one of them. The randomness is given in terms of the weak solution of the hydrodynamic equation through some sort of renormalization function. By coupling the zero-range with the exclusion process we derive some limiting laws for more general initial conditions.