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 $\rho$ 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 $\sqrt{nlogn}$ scaling (i) for fixed infinite horizon configurations-letting first $n\to \infty$ and then $\rho \to 0$-studied e.g. by Szász and Varjú (J Stat Phys 129(1):59-80, 2007) and (ii) Boltzmann-Grad type situations-letting first $\rho \to 0$ and then $n\to \infty$-studied by Marklof and Tóth (Commun Math Phys 347(3):933-981, 2016) .