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 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 scaling (i) for fixed infinite horizon configurations—letting first letting the time to go to infnity and then the size of the scaterrer go to zero —studied e.g. by Szász and Varjú (J Stat Phys 129(1):59–80, 2007) and (ii) Boltzmann–Grad type situations—letting first and then —studied by Marklof and Tóth (Commun Math Phys 347(3):933–981, 2016) .