Mean Field Limit for Coulomb-Type Flows

TL;DR

This paper proves the mean-field convergence of Coulomb and Riesz particle systems in arbitrary dimensions, using a modulated energy approach, and extends results to Newtonian and Euler-Poisson systems.

Contribution

First rigorous proof of mean-field limit for Coulomb and Riesz interactions in any dimension, including regularized and mixed flows.

Findings

Established mean-field convergence for Coulomb and Riesz systems in arbitrary dimensions.

Extended the method to Newton's law with Coulomb/Riesz interactions in the monokinetic case.

Developed a modulated energy method using Coulomb or Riesz distances.

Abstract

We establish the mean-field convergence for systems of points evolving along the gradient flow of their interaction energy when the interaction is the Coulomb potential or a super-coulombic Riesz potential, for the first time in arbitrary dimension. The proof is based on a modulated energy method using a Coulomb or Riesz distance, assumes that the solutions of the limiting equation are regular enough and exploits a weak-strong stability property for them. The method can handle the addition of a regular interaction kernel, and applies also to conservative and mixed flows. In the appendix, it is also adapted to prove the mean-field convergence of the solutions to Newton's law with Coulomb or Riesz interaction in the monokinetic case to solutions of an Euler-Poisson type system.