Combined mean field limit and non-relativistic limit of Vlasov–Maxwell particle system to Vlasov–Poisson system

In this paper, we consider the mean field limit and non-relativistic limit of the relativistic Vlasov–Maxwell particle system to the Vlasov–Poisson equation. With the relativistic Vlasov–Maxwell particle system being a starting point, we carry out the estimates (with respect to N and c) between the characteristic equation of both the Vlasov–Maxwell particle model and the Vlasov–Poisson equation, where the probabilistic method is exploited. In the last step, we take both a large N limit and a non-relativistic limit (meaning c tending to infinity) to close the argument. Deriving such a mean-field limit for interactions with Coulomb singularity is a difficult task; therefore, we introduce an N-dependent mollifier to smoothen the interactions. Note that the length scale of our mollifier is N−α for an α that can be chosen arbitrarily close to but smaller than 1/3, which improves previous results in the literature on deriving Vlasov–Maxwell in the non-relativistic setting.