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

  • * Corresponding author: Karsten Matthies

    * Corresponding author: Karsten Matthies 
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  • 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.

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


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  • Figure 1.  The collision parameter $\nu$

    Figure 2.  Two example trees

    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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