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

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

2003-01-25

arxiv.org/abs/math/0301300v1

Mathematics

Dynamical Systems

26 pages, 5 figures, to be published in Commun. Math. Phys

Scientific paper

10.1007/s00220-003-0825-5

In a flat 2-torus with a disk of diameter $r$ removed, let $\Phi_r(t)$ 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 $\Phi_r(t/r)$ 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.

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