On the Macroscopic limit of Brownian Particles with local interaction

TL;DR

This paper reviews the derivation of macroscopic PDE limits from systems of interacting Brownian particles with local interactions, providing new heuristic and numerical insights into the form of the nonlinear term.

Contribution

It offers new heuristic and numerical insights into the explicit form of the nonlinear PDE arising from microscopic particle interactions, which is largely unknown.

Findings

Heuristic methods suggest the form of the nonlinearity.

Numerical simulations support the proposed PDE structure.

The review clarifies existing gaps in rigorous derivations.

Abstract

An interacting particle system made of diffusion processes with local interaction is considered and the macroscopic limit to a nonlinear PDE is investigated. Few rigorous results exists on this problem and in particular the explicit form of the nonlinearity is not known. The paper reviews this subject, some of the main ideas to get the limit nonlinear PDE and provides both heuristic and numerical informations on the precise form of the nonlinearity which are new with respect to the literature and coherent with the few known informations.