Scaling Limit of Two-component Interacting Brownian Motions

TL;DR

This paper investigates the large-scale behavior of a two-component interacting Brownian system, showing that the empirical densities converge to a solution of the Maxwell-Stefan equation, a key PDE in multi-species diffusion.

Contribution

It establishes the hydrodynamic limit of a two-component Brownian motion system with singular interactions, deriving the Maxwell-Stefan equation as the limiting PDE.

Findings

Empirical densities converge to Maxwell-Stefan PDE

System exhibits singular interaction behavior

Hydrodynamic limit proven for two-component system

Abstract

This paper presents our study of the asymptotic behavior of a two-component system of Brownian motions undergoing certain singular interactions. In particular, the system is a combination of two different types of particles and the mechanical properties and interaction parameters depend on the corresponding type of particles. We prove that the hydrodynamic limit of the empirical densities of two types is the solution of a certain quasi-linear parabolic partial differential equation known as the Maxwell-Stefan equation.