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

Karsten Matthies, George Russell Stone, Florian Theil

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Abstract

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.
Original languageEnglish
Article number14450
Pages (from-to)137-177
Number of pages41
JournalKinetic and Related Models
Volume11
Issue number1
Early online date16 Aug 2017
DOIs
Publication statusPublished - 1 Feb 2018

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Keywords

  • math.AP
  • math-ph
  • math.MP
  • Derivation
  • Boltzmann equation
  • Semigroups
  • Rayleigh gas

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