On the theory of kinetic equations for interacting particle systems with long range interactions

89 pagesIn this paper we review the formal derivation of different classes of kinetic equations for long range potentials. We consider suitable scaling limits for Lorentz and Rayleigh gases as well as interacting particle systems whose dynamics can be approximated by means of kinetic equations. The resulting kinetic equations are the Landau and the Balescu-Lenard equations. In the derivation of the kinetic equations particular emphasis is made in the fact that all the kinetic regimes can be obtained approximating the dynamics of the interacting particle systems by the evolution of a single particle in a random force field with a friction term which is due to the interaction with the surrounding particles. The case of particles interacting by means of Coulombian potentials as well as the cutoffs which yield the so-called Coulombian logarithm are discussed in detail