Mean field limits for charged particles

The aim of this thesis is to provide a rigorous mathematical
derivation of the Vlasov-Poisson equation and the Vlasov-Maxwell
equations in the large N limit of interacting charged particles. We
will extend a method previously proposed by Boers and Pickl to
perform a mean field limit for the Vlasov-Poisson equation with the
full Coulomb singularity and an N-dependent cut-off decreasing as
$N^{-1/3 + \epsilon}$. We will then discuss an alternative
approach, deriving the Vlasov-Poisson equation as a combined mean
field and point-particle limit of an N-particle Coulomb system of
extended charges. Finally, we will combine both methods to prove a
mean field limit for the relativistic Vlasov-Maxwell system in 3+1
dimensions. In each case, convergence of the empirical measures to
solutions of the corresponding mean field equation can be shown for
typical initial conditions. This implies, in particular, the
propagation of chaos for the respective dynamics.