Derivation of the Boltzmann equation with moderately soft potentials from a perturbed Nanbu particles system

TL;DR

This paper derives the 3D Boltzmann equation with moderately soft potentials from a particle system, using grazing collision effects to handle singularities and prove convergence.

Contribution

It introduces a novel approach to derive the Boltzmann equation with soft potentials from particle systems by exploiting grazing collision regularization.

Findings

Established a convergence from particle system to Boltzmann equation

Handled singular angular interactions with grazing collision effects

Provided a qualitative convergence proof for the soft potential case

Abstract

We derive the 3D spatially homogeneous Boltzmann's equation with moderately soft potentials and singular angular interaction, from an interacting particles system. The collision kernel is of the form $B(z,\sigma)=|z|^{\gamma}b\left( \frac{z}{|z|}\cdot \sigma\right)$ and for $K>0$, $\sin(\theta)b\left(\cos(\theta)\right)\sim K\theta^{-1-\nu}$, with $\gamma\in (-2,-1)$ and $\nu\in(1,2)$ satisfying $\gamma+\nu>0$. We use at the particle level the regularizing effects of the grazing collisions, in order to control the singularity of the soft potential. This enables to use a classical compactness argument, and provide a qualitative convergence result from the interacting particles system toward the solution of the limit macroscopic equation.