Periodic Lorentz gas with small scatterers

Abstract

We prove limit laws for infinite horizon planar periodic Lorentz gases when, as time n tends to infinity, the scatterer size ρ may also tend to zero simultaneously at a sufficiently slow pace. In particular we obtain a non-standard Central Limit Theorem as well as a Local Limit Theorem for the displacement function. To the best of our knowledge, these are the first results on an intermediate case between the two well-studied regimes with superdiffusive nlogn−−−−−√ scaling (i) for fixed infinite horizon configurations—letting first n→∞ and then ρ→0—studied e.g. by Szász and Varjú (J Stat Phys 129(1):59–80, 2007) and (ii) Boltzmann–Grad type situations—letting first ρ→0 and then n→∞—studied by Marklof and Tóth (Commun Math Phys 347(3):933–981, 2016) .