Abstract

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.