Separation versus diffusion in a two species system

TL;DR

This paper studies a two-species particle system on the integer lattice, where particles switch types at the boundaries, and derives a hydrodynamic limit described by coupled PDEs with free boundaries.

Contribution

It introduces a novel two-species particle model with boundary switching and rigorously derives the associated hydrodynamic PDE system with free boundaries.

Findings

Hydrodynamic limit described by coupled PDEs with free boundaries.

Rigorous proof of the PDE system from particle dynamics.

Insight into boundary-driven phase transitions in multi-species systems.

Abstract

We consider a finite number of particles that move in $\mathbb Z$ as independent random walks. The particles are of two species that we call $a$ and $b$. The rightmost $a$ particle becomes a $b$ particle at constant rate, while the leftmost $b$ particle becomes $a$ particle at the same rate, independently. We prove that in the hydrodynamic limit the evolution is described by a non linear system of two PDE's with free boundaries.