  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.
