On the Mean-Field Limit for the Vlasov–Poisson–Fokker–Planck System

We rigorously justify the mean-field limit of an N-particle system subject to Brownian motions and interacting through the Newtonian potential in $R^3$. Our result leads to a derivation of the Vlasov–Poisson–Fokker–Planck (VPFP) equations from the regularized microscopic N-particle system. More precisely, we show that the maximal distance between the exact microscopic trajectories and the mean-field trajectories is bounded by $N^{-\frac{1}{3}+\epsilon }$ ($\frac{1}{63}\le \epsilon <\frac{1}{36}$) with a blob size of $N^{-\delta }$ ($\frac{1}{3}\le \delta <\frac{19}{54}-\frac{2\epsilon }{3}$) up to a probability of $1-N^{-\alpha }$ for any $\alpha >0$. Moreover, we prove the convergence rate between the empirical measure associated to the regularized particle system and the solution of the VPFP equations. The technical novelty of this paper is that our estimates rely on the randomness coming from the initial data and from the Brownian motions.