Fick law and sticky Brownian motions

TL;DR

This paper analyzes a particle system with boundary reservoirs, proving that its hydrodynamic limit follows a heat equation with boundary conditions involving derivatives, and establishes existence, uniqueness, and chaos propagation.

Contribution

It introduces a new boundary condition for the heat equation derived from a particle system with reservoirs of equal size, extending previous models.

Findings

Hydrodynamic limit is the heat equation with specific boundary conditions.

Existence and uniqueness of weak solutions are proven.

Propagation of chaos property is established.

Abstract

We consider an interacting particle system in the interval $[1,N]$ with reservoirs at the boundaries. While the dynamics in the channel is the simple symmetric exclusion process, the reservoirs are also particle systems which interact with the given system by exchanging particles. In this paper we study the case where the size of each reservoir is the same as the size of the channel . We will prove that the hydrodynamic limit equation is the heat equation with boundary conditions which relate first and second spatial derivatives at the boundaries for which we will prove the existence and uniqueness of weak solutions. The propagation of chaos property can also be derived.