Hydrodynamic limit of condensing two-species zero range processes with sub-critical initial profiles

TL;DR

This paper establishes the hydrodynamic limit for two-species condensing zero range processes with sub-critical initial profiles, using entropy methods and proving global solutions in specific cases.

Contribution

It proves the hydrodynamic limit for two-species condensing ZRPs with bounded jump rates and sub-critical initial profiles, including global existence for species-blind cases.

Findings

Hydrodynamic limit proven for sub-critical initial profiles.

Global existence of solutions in species-blind case.

Hydrodynamic limit valid for all times in specific cases.

Abstract

Two-species condensing zero range processes (ZRPs) are interacting particle systems with two species of particles and zero range interaction exhibiting phase separation outside a domain of sub-critical densities. We prove the hydrodynamic limit of nearest neighbour mean zero two-species condensing zero range processes with bounded local jump rate for sub-critical initial profiles, i.e., for initial profiles whose image is contained in the region of sub-critical densities. The proof is based on H. T. Yau's relative entropy method, which relies on the existence of sufficiently regular solutions to the hydrodynamic equation. In the particular case of the species-blind ZRP, we prove that the solutions of the hydrodynamic equation exist globally in time and thus the hydrodynamic limit is valid for all times.