Modulated Free Energy and Mean Field Limit

We proves the mean field limit and quantitative estimates for many-particle systems with singular attractive interactions between particles.   As an important example,  a full rigorous derivation (with quantitative estimates) of the  Patlak-Keller-Segel model in optimal subcritical regimes is obtained for the first time. To give an answer to this longstanding problem, we take advantage of a new modulated free energy and we prove some precise large deviation estimates encoding the competition between diffusion and attraction. This modulated free energy approach can also treat the systems with a wide range of repulsive kernels, including the vanishing viscosity case. Based on joint works with D. Bresch and P.-E. Jabin.