Hydrodynamic Limit of a Boundary-Driven Elastic Exclusion Process and a Stefan Problem

TL;DR

This paper establishes the hydrodynamic limit of a boundary-driven elastic exclusion process, modeling reflecting Brownian spheres, and connects it to a Stefan problem, providing insights into particle systems with boundary interactions.

Contribution

It introduces a hydrodynamic limit for an elastic exclusion process with boundary conditions, linking particle systems to a Stefan problem framework.

Findings

Derived the hydrodynamic limit for elastic exclusion processes.

Connected the process to a Stefan problem in boundary-driven systems.

Demonstrated the model's relevance for reflecting Brownian spheres.

Abstract

Burdzy, Pal, and Swanson considered solid spheres of small radius moving in the unit interval, reflecting instantaneously from each other and at x=0, and killed at x=1, with mass being added to the system from the left at constant rate. By transforming to a system with zero-width particles moving as independent Brownian motion, they derived a limiting stationary distribution for a particular initial distribution, as the width of a particle decreases to zero and the number of particles increases to infinity. This space-removing transformation has a direct analogy in the isomorphism between a new unbounded-range exclusion process and a superimposition of random walks with random boundary. We derive the hydrodynamic limit for these isomorphic processes, demonstrating that this elastic exclusion is an appropriate model for the reflecting Brownian spheres in one dimension.