Mean-field limits for some Riesz interaction gradient flows

TL;DR

This paper establishes mean-field limit results for particle systems with Riesz interactions in one and two dimensions, using a modulated energy approach inspired by Serfaty's work on Ginzburg-Landau vortices.

Contribution

It extends the mean-field limit analysis to Riesz interaction gradient flows in dimensions 1 and 2, addressing previously open cases.

Findings

Proves mean-field limits for Riesz interactions in 1D and 2D.

Uses a modulated energy method inspired by vortex analysis.

Addresses open problems in the mathematical understanding of these systems.

Abstract

This paper is concerned with the mean-field limit for the gradient flow evolution of particle systems with pairwise Riesz interactions, as the number of particles tends to infinity. Based on a modulated energy method, using regularity and stability properties of the limiting equation, as inspired by the work of Serfaty in the context of the Ginzburg-Landau vortices, we prove a mean-field limit result in dimensions 1 and 2 in cases for which this problem was still open.