Modulated Free Energy and Mean Field Limit

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.