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 $$\rho $$ 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 $$\sqrt{n\log n}$$ n log n scaling (i) for fixed infinite horizon configurations—letting first $$n\rightarrow \infty $$ n → ∞ and then $$\rho \rightarrow 0$$ 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 $$\rho \rightarrow 0$$ ρ → 0 and then $$n\rightarrow \infty $$ n → ∞ —studied by Marklof and Tóth (Commun Math Phys 347(3):933–981, 2016) .