Long Range Particle Dynamics and the Linear Boltzmann Equation

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.