Hydrodynamic Limit of Brownian Particles Interacting with Short and Long Range Forces

TL;DR

This paper derives a diffusive integro-differential equation as the hydrodynamic limit for a system of Brownian particles with short and long-range interactions, under diffusive scaling, on a torus.

Contribution

It provides a rigorous derivation of the macroscopic density evolution equation from microscopic particle dynamics with combined short and long-range forces.

Findings

Derived the hydrodynamic limit as a diffusive integro-differential equation.

Established the scaling limits for particles with Kac-type long-range interactions.

Connected microscopic dynamics to macroscopic PDE in a rigorous framework.

Abstract

We investigate the time evolution of a model system of interacting particles, moving in a $d$-dimensional torus. The microscopic dynamics are first order in time with velocities set equal to the negative gradient of a potential energy term $\Psi$ plus independent Brownian motions: $\Psi$ is the sum of pair potentials, $V(r)+\gamma^d J(\gamma r)$, the second term has the form of a Kac potential with inverse range $\gamma$. Using diffusive hydrodynamical scaling (spatial scale $\gamma^{-1}$, temporal scale $\gamma^{-2}$) we obtain, in the limit $\gamma\downarrow 0$, a diffusive type integro-differential equation describing the time evolution of the macroscopic density profile.