The rigorous derivation of the Linear Landau equation from a particle system in a weak-coupling limit

Abstract

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 \begin{document}$N$\end{document} goes to infinity and the range of the potential \begin{document}$e$\end{document} goes to zero in such a way that \begin{document}$Ne^{2} = α$\end{document} , with \begin{document}$α$\end{document} 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.