On the mean-field limit of Vlasov–Poisson–Fokker–Planck equations

Abstract

The derivation of effective descriptions for interacting many-body systems is an important branch of applied mathematics. We prove a propagation of chaos result for a system of N particles subject to Newtonian time evolution with or without additional white noise influencing the velocities of the particles. We assume that the particles interact according to a regularized Coulomb-interaction with a regularization parameter that vanishes in the N → ∞ limit. The respective effective description is the so called Vlasov-Poisson-Fokker-Planck (VPFP), respectively the Vlasov-Poisson (VP) equation in the case of no or sub-dominant white noise. To obtain our result we combine the relative entropy method from Jabin and Wang, J. Funct. Anal. 271(12), 3588–3627 (2016) with the control on the difference between the trajectories of the true and the effective description provided in Huang et al., J. Stat. Phys. 181, 1915–1965 (2020) for the VPFP case respectively in Lazarovici and Pickl, Arch. Ration. Mech. Anal. 225(3), 1201–1231 (2017) for the VP case. This allows us to prove strong convergence of the marginals, i.e., convergence in L1.