Modulated Free Energy and Mean Field Limit

TL;DR

This paper introduces a modulated free energy in mean-field theory, enabling convergence rate analysis for systems with Riesz and Coulomb kernels under viscosity, and discusses extensions to more general kernels and related particle systems.

Contribution

It presents a new modulated free energy approach that achieves convergence rates in mean-field limits for repulsive kernels, extending previous methods to more general kernels via Fourier analysis.

Findings

Established convergence rates for Riesz and Coulomb kernels with viscosity.

Extended the approach to general kernels using Fourier transform techniques.

Discussed particle approximation of the Patlak-Keller-Segel system.

Abstract

This is the document corresponding to the talk the first author gave at IH{\'E}S for the Laurent Schwartz seminar on November 19, 2019. It concerns our recent introduction of a modulated free energy in mean-field theory in BrJaWa [4]. This physical object may be seen as a combination of the modulated potential energy introduced by S. Serfaty [See Proc. Int. Cong. Math. (2018)] and of the relative entropy introduced in mean field limit theory by P.-E. Jabin, Z. Wang [See Inventiones 2018]. It allows to obtain, for the first time, a convergence rate in the mean field limit for Riesz and Coulomb repulsive kernels in the presence of viscosity using the estimates in Du [8] and Se1 [20]. The main objective in this paper is to explain how it is possible to cover more general repulsive kernels through a Fourier transform approach as announced in BrJaWa [4] first in the case $\sigma$N $\rightarrow$ 0 when N $\rightarrow$ +$\infty$ and then if $\sigma$ > 0 is fixed. Then we end the paper with comments on the particle approximation of the Patlak-Keller-Segel system which is associated to an attractive kernel and refer to [C.R.