From microscopic dynamics to kinetic equations

Starting from a system of N particles at a microscopic scale, we describe different scaling limits which lead to kinetic equations in a macroscopic regime: the low-density limit, the weak-coupling limit, the grazing collision limit and the mean-field limit. A particular relevance is given to the rigorous derivation of the Boltzmann equation (starting from a system of N particles interacting via a short range potential) and to a consistency result concerning the Landau equation. A Kac's model for the Landau equation is presented as well. The last part of the work is dedicated to the Vlasov-Poisson system, in particular we discuss the Cauchy problem related to this equation in presence of a point charge.