THE RIGOROUS DERIVATION OF THE LINEAR LANDAU EQUATION FROM A PARTICLE SYSTEM IN A WEAK-COUPLING LIMIT

We consider a system of N particles interacting via a short-range smooth potential, in a weak-coupling regime. This means that the number of particles N goes to infinity and the range of the potential epsilon goes to zero in such a way that N epsilon(2) = alpha, with alpha diverging in a suitable way. We provide a rigorous derivation of the Linear Landau equation from this particle system. The strategy of the proof consists in showing the asymptotic equivalence between the one-particle marginal and the solution of the linear Boltzmann equation with vanishing mean free path. This point follows [3] and makes use of technicalities developed in [16]. Then, following the ideas of Landau, we prove the asympotic equivalence between the solutions of the Boltzmann and Landau linear equation in the grazing collision limit.