The derivation of the linear Boltzmann equation from a Rayleigh gas particle model

Karsten Matthies, George Russell Stone, Florian Theil

OAI: oai:purehost.bath.ac.uk:publications/524aefae-d386-4450-b283-3511089f6ab2 • DOI: https://doi.org/10.3934/krm.2018008

A linear Boltzmann equation is derived in the Boltzmann-Grad scaling for the deterministic dynamics of many interacting particles with random initial data. We study a Rayleigh gas where a tagged particle is undergoing hard-sphere collisions with background particles, which do not interact among each other. In the Boltzmann-Grad scaling, we derive the validity of a linear Boltzmann equation for arbitrary long times under moderate assumptions on higher moments of the initial distributions of the tagged particle and the possibly non-equilibrium distribution of the background. The convergence of the empiric dynamics to the Boltzmann dynamics is shown using Kolmogorov equations for associated probability measures on collision histories.

math.AP math-ph math.MP derivation Boltzmann equation Semigroups Rayleigh gas