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This paper proves the mean field limit and quantitative estimates formany-particle systems with singular attractive interactions between particles.As an important example, a full rigorous derivation (with quantitativeestimates) of the Patlak-Keller-Segel model in optimal subcritical regimes isobtained for the first time. To give an answer to this longstanding problem, wetake advantage of a new modulated free energy and we prove some precise largedeviation estimates encoding the competition between diffusion and attraction.Combined with the range of repulsive kernels, already treated in thes{\'e}minaire Laurent Schwartz proceeding[https://slsedp.centre-mersenne.org/journals/SLSEDP/ ], we provide the fullproof of results announced by the authors in [C. R. Acad. Sciences SectionMaths (2019)].

Mean-field limit and quantitative estimates with singular attractive kernels