MEAN FIELD LIMIT AND QUANTITATIVE ESTIMATES WITH SINGULAR ATTRACTIVE KERNELS

We prove 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 long-standing problem, we take advantage of a new modulated free energy and prove some precise large deviation estimates encoding the competition between diffusion and attraction. Combined with the range of repulsive kernels, already treated in the proceedings of the Séminaire Laurent Schwartz, we provide the full proof of results announced by the authors in 2019.