In this talk I will present some recent results on mean field limits for interacting diffusions. We study, problems for which the mean field limit exhibits phase transitions, in the sense that the limiting McKean-Vlasov PDE can have more than one stationary states, at a sufficiently strong interaction strength/low temperature. We provide a general characterization of first and second order phase transitions for mean field dynamics on the torus and we study fluctuations around the mean field limit. As a case study, we consider the combined mean field/homogenization limit for noisy Kuramoto oscillators. Applications of this type of dynamics to models for opinion formation and to sampling and optimization algorithms are also discussed.
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