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

Karsten Matthies; George Stone; Florian Theil

doi:10.3934/krm.2018008

Article Contents

2018, Volume 11, Issue 2: 1-41. Doi: 10.3934/krm.2018008

This issue Previous Article Global regularity for a model of Navier-Stokes equations with logarithmic sub-dissipation Next Article Mathematical modeling of a discontinuous solution of the generalized Poisson--Nernst--Planck problem in a two-phase medium

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

1.
Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom
2.
Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

* Corresponding author: Karsten Matthies
* Corresponding author: Karsten Matthies

Received: February 29, 2016

Revised: February 28, 2017

Published: March 2018

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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.

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Mathematics Subject Classification: Primary:82C40;Secondary:35Q20, 37L05, 60K35, 76P05, 82C22.

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Figure 1. The collision parameter $\nu$

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Figure 2. Two example trees

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Figure 3. In the case $v'=0$ we are calculating the volume of $\Delta$, since we know the background particle cannot start in $\Delta$. For $v'\neq 0$ the cylinders get shifted but the principle is the same. (Diagram not to scale)

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