Long Range Particle Dynamics and the Linear Boltzmann Equation

TL;DR

This paper rigorously derives the linear Boltzmann equation from a long-range particle interaction model, providing the first full proof of this connection under specific decay and initial distribution assumptions.

Contribution

It offers the first complete proof linking long-range particle dynamics with the linear Boltzmann equation, using novel estimates and a tree-based structure for collision history analysis.

Findings

Weak-$ abla$ convergence of particle density to the linear Boltzmann solution

Explicit error estimates between long-range and short-range dynamics

Validation of the linear Boltzmann equation as a limit for long-range interactions

Abstract

This paper gives the first full proof of the justification of the linear Boltzmann equation from an underlying long range particle evolution. We suppose that a tagged particle is interacting with a background via a two body potential that is decaying faster than $ C\exp\left(-C|x|^{\frac{3}{2}}\right) $, and that the background is initially distributed according to a function in $L^1((\mathbb{R}^3,(1+|v|^2)\mathrm{d}v)$ in velocity and uniformly in space. Under finite mass and energy assumptions on the initial density, the tagged particle density converges weak-$\star$ in $L^\infty$ to a solution of the linear Boltzmann equation. The proof uses estimates on two body scattering and on the relationship between long range dynamics and dynamics with a truncated interaction potential to explicitly estimate the error between densities for long and short range dynamics. To compare the difference between the short range dynamics and the linear Boltzmann equation, we use a tree based structure to encode the collisional history of the tagged particle.