Commutators, mean-field, and supercritical mean-field limits for Coulomb/Riesz gases

TL;DR

This paper discusses recent sharp commutator estimates for Coulomb/Riesz gases and how they enable optimal results in mean-field and supercritical mean-field limits, advancing the understanding of gas dynamics.

Contribution

It provides an accessible overview of new commutator estimates and their application to mean-field limits for Coulomb/Riesz gases, building on recent arXiv works.

Findings

Sharp commutator estimates for modulated energies

Optimal mean-field limit results for Coulomb/Riesz gases

Extension to supercritical regimes

Abstract

This paper is intended as a companion to the author's talk "Commutator estimates and mean-field limits for Coulomb/Riesz gases" at the 2025 Journ\'ees \'equations aux d\'eriv\'ees partielles in Aussois. The goal is to provide a concise, accessible account of sharp commutator estimates recently obtained for modulated energies associated to Coulomb/Riesz interactions and how these estimates lead to optimal results for mean-field and supercritical mean-field limits of Coulomb/Riesz gas dynamics via the modulated-energy method. The exposition centers on the works arXiv:2408.14642, arXiv:2407.15650 with Serfaty and arXiv:2511.13461, arXiv:2601.02326 with Hess-Childs and Serfaty.