Divergences and the Approach to Equilibrium in the Lorentz and the Wind-Tree Models

We prove that for wide class of physically interesting initial states the time evolution of the wind particles's correlation functions can be described at any finite time by a convergent power series in the density of the tree particles, provided this density is small enough. We show that all the coefficients (except the lowest ones) of this power series contain terms diverging as $t\to +\infty$. Nevertheless, we prove that the radius of convergence does not shrink to zero as $t\to +\infty$ and that the divergent terms can be resummed into a cutoff, thereby constructing a new series for the correlation functions having each term bounded as $t\to +\infty$. Although divergence free, this series does not converge uniformly in time; however, it can be used to show that equilibrium cannot be reached if the tree-tree interaction allows over-lapping and to study the limiting case of vanishing tree size but nonvanishing free path; in this last case we find an exact expression for the Green's functions showing that the approach to equilibrium is described by a diffusion process which rigorously verifies the Boltzmann equation.