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Semigroup methods for the derivation of Boltzmann-type equations from deterministic particle dynamics

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FKT - Frontiers in kinetic theory: connecting microscopic to macroscopic scales - KineCon 2022

In this talk we will revisit the classical questions of understanding the statistics of various deterministic dynamics of n hard spheres of diameter a with random initial data in the Boltzmann-Grad scaling as a tends to zero and n tends to infinity. I will present some ideas to replace the analysis of the BBGKY hierarchy. The convergence of the empiric dynamics to the Boltzmann-type dynamics is shown using semigroup methods to analyse Kolmogorov equations for associated probability measures on genuine collision histories.   Details are given for a Rayleigh gas where a tagged particle is undergoing 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 the initial distributions of the tagged particle and the possibly non-equilibrium distribution of the background. I aim to give various current and future questions. Based on work with George Stone, Theodora Syntaka and Florian Theil.

This talk is part of the Isaac Newton Institute Seminar Series series.

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