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

TL;DR

This paper rigorously derives the linearized Boltzmann and Landau equations from long-range interacting particle systems, establishing convergence in various scaling limits and identifying the Coulomb singularity as a threshold.

Contribution

It provides the first rigorous derivation of kinetic equations from particle systems with long-range power-law interactions, including the Coulomb case.

Findings

Convergence to kinetic equations in appropriate limits.

Identification of Coulomb singularity as a threshold.

Onset of Coulomb logarithm at s=1.

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 $\mathcal{U}(r)\sim |r/\varepsilon|^{-s}$ (near $r \approx 0$) converge to solutions of kinetic equations in appropriate scaling limits $\varepsilon \rightarrow 0$ and $N\rightarrow \infty$. 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.