Long time validity of the linearized Boltzmann uncut-off and the linearized Landau equations from the Newton Law

Abstract

We provide a rigorous justification of the linearized Boltzmann- and Landau equations from interacting particle systems with long-range interaction. The result shows that the fluctuations of Hamiltonian $N$- particle systems governed by truncated power law potentials of the form $\mathcal{U}(r)\sim |r/\epsilon {|}^{-s}$ (near $r\approx 0$) converge to solutions of kinetic equations in appropriate scaling limits $\epsilon \to 0$ and $N\to \infty$. We prove that for $s\in [0,1)$, the limiting system approaches the uncutoff linearized Boltzmann equation or the linearized Landau equation, depending on the scaling limit. The Coulomb singularity $s=1$ appears as a threshold value. Kinetic scaling limits with $s\in (0,1]$ universally converge to the linearized Landau equation, and we prove the onset of the Coulomb logarithm for $s=1$. To the best of our knowledge, this is the first result on the derivation of kinetic equations from interacting particle systems with long-range power-law interaction.