Derivation of a non-autonomous linear Boltzmann equation from a heterogeneous Rayleigh gas

Karsten Matthies; George Stone

doi:10.3934/dcds.2018143

Article Contents

2018, Volume 38, Issue 7: 3299-3355. Doi: 10.3934/dcds.2018143

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Derivation of a non-autonomous linear Boltzmann equation from a heterogeneous Rayleigh gas

Karsten Matthies and
George Stone

Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom

Received: June 2017

Revised: February 2018

Published: July 2018

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Abstract

A linear Boltzmann equation with non-autonomous collision operator is rigorously derived in the Boltzmann-Grad limit for the deterministic dynamics of a Rayleigh gas where a tagged particle is undergoing hard-sphere collisions with heterogeneously distributed background particles, which do not interact among each other. The validity of the linear Boltzmann equation holds for arbitrary long times under moderate assumptions on spatial continuity and higher moments of the initial distributions of the tagged particle and the heterogeneous, non-equilibrium distribution of the background. The empiric particle dynamics are compared to the Boltzmann dynamics using evolution systems for Kolmogorov equations of 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 parameters of a collision between the tagged particle and particle $j$.

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Figure 2. A tree with two collisions, the time of the final collision is $\tau$.

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