Mean-Field limits for Coulomb-type dynamics

We consider a system of N particles evolving according to the gradient flow of their Coulomb or Riesz interaction, or a similar conservative flow, and possible added random diffusion. By Riesz interaction, we mean inverse power s of the distance with s between d-2 and d where d denotes the dimension. We present a convergence result as N tends to infinity to the expected limiting mean field evolution equation. We also discuss the derivation of Vlasov-Poisson from Newtonian dynamics in the monokinetic case, as well as related results for Ginzburg-Landau vortex dynamics.