Stationary measures and hydrodynamics of zero range processes with several species of particles

TL;DR

This paper investigates zero range processes with multiple particle types, establishing conditions for stationary measures, deriving hydrodynamic limits as hyperbolic conservation laws, and analyzing stationary profiles with open boundaries.

Contribution

It provides necessary and sufficient conditions for stationary product measures and derives the hydrodynamic limit as a system of hyperbolic conservation laws.

Findings

Characterized stationary product measures for multi-species zero range processes.

Proved hydrodynamic limit as a system of hyperbolic conservation laws.

Analyzed stationary density profiles with open boundary conditions.

Abstract

We study general zero range processes with different types of particles on a d-dimensional lattice with periodic boundary conditions. A necessary and sufficient condition on the jump rates for the existence of stationary product measures is established. For translation invariant jump rates we prove the hydrodynamic limit on the Euler scale using Yau's relative entropy method. The limit equation is a system of conservation laws, which are hyperbolic and have a globally convex entropy. We analyze this system in terms of entropy variables. In addition we obtain stationary density profiles for open boundaries.