Derivation of kinetic and diffusion equations from a hard-sphere Rayleigh gas using collision trees and semigroups

Karsten Matthies; Theodora Syntaka

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arXiv · 2405.04449

Derivation of kinetic and diffusion equations from a hard-sphere Rayleigh gas using collision trees and semigroups

Karsten Matthies·Theodora Syntaka

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Abstract

We will revisit the classical questions of understanding the statistics of various deterministic dynamics of $N$ hard spheres of diameter $\varepsilon$ with random initial data in the Boltzmann-Grad scaling as $\varepsilon$ tends to zero and $N$ tends to infinity. The convergence of the empiric particle dynamics to the Boltzmann-type dynamics is shown using semigroup methods to describe probability measures on collision trees associated to physical trajectories in the case of a Rayleigh gas. As an application we derive the diffusion equation by a further rescaling.

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Karsten Matthies, Theodora Syntaka. 2024-05-07. Derivation of kinetic and diffusion equations from a hard-sphere Rayleigh gas using collision trees and semigroups. https://arxiv.org/abs/2405.04449

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