On the distribution of free-path lengths for the periodic Lorentz gas III

Abstract

In a flat 2-torus with a disk of diameter $r$ removed, let $Φ_r(t)$ be the distribution of free-path lengths (the probability that a segment of length larger than $t$ with uniformly distributed origin and direction does not meet the disk). We prove that $Φ_r(t/r)$ behaves like $\frac{2}{π^2 t}$ for each $t>2$ and in the limit as $r\to 0^+$, in some appropriate sense. We then discuss the implications of this result in the context of kinetic theory.