Bridging bulk and surface: An interacting particle system towards the field-road diffusion model

TL;DR

This paper derives a macroscopic field-road diffusion model from a microscopic particle system, connecting two different dimensional spaces through a novel hydrodynamic limit approach.

Contribution

It introduces a new hydrodynamic limit framework for bridging different dimensional spaces in particle systems, specifically linking microscopic lattice models to macroscopic PDEs.

Findings

Derived the field-road diffusion PDEs from the particle system

Established the hydrodynamic limit for a two-dimensional coupled system

Demonstrated the novel connection between microscopic and macroscopic models

Abstract

We recover the so-called field-road diffusion model as the hydrodynamic limit of an interacting particle system. The former consists of two parabolic PDEs posed on two sets of different dimensions (a "field" and a "road" in a population dynamics context), and coupled through exchange terms between the field's boundary and the road. The latter stands as a Symmetric Simple Exclusion Process (SSEP): particles evolve on two microscopic lattices following a Markov jump process, with the constraint that each site cannot host more than one particle at the same time. The system is in contact with reservoirs that allow to create or remove particles at the boundary sites. The dynamics of these reservoirs are slowed down compared to the diffusive dynamics, to reach the reactions and the boundary conditions awaited at the macroscopic scale. This issue of bridging two spaces of different dimensions is, as far as we know, new in the hydrodynamic limit context, and raises perspectives towards future related works.