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

We will revisit the classical questions of understanding the statistics of various deterministic dynamics of N hard spheres of diameter ε with random initial data in the Boltzmann-Grad scaling as ε 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.