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

In a flat 2-torus with a disk of diameter $r$ removed, let ${\mathrm{\Phi }}_r(t)$ 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 ${\mathrm{\Phi }}_r(t\mathrm{/}r)$ behaves like $\frac{2}{{\pi }^{2}t}$ 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.