Periodic Lorentz gas with small scatterers

Péter Bálint; Henk Bruin; Dalia Terhesiu

doi:10.1007/s00440-023-01197-6

Periodic Lorentz gas with small scatterers

Probab Theory Relat Fields. 2023;186(1-2):159-219. doi: 10.1007/s00440-023-01197-6. Epub 2023 Mar 15.

Authors

Péter Bálint^{1

2}, Henk Bruin³, Dalia Terhesiu⁴

Affiliations

¹ MTA-BME Stochastics Research Group, Budapest University of Technology and Economics, Műegyetem rkp. 3., Budapest, 1111 Hungary.
² Department of Stochastics, Institute of Mathematics, Budapest University of Technology and Economics, Műegyetem rkp. 3., Budapest, 1111 Hungary.
³ Faculty of Mathematics, University of Vienna, Oskar Morgensternplatz 1, 1090 Vienna, Austria.
⁴ Institute of Mathematics, University of Leiden, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands.

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

Keywords: Billiards; Limit theorems; Lorentz gas; Nagaev–Guivarc’h method; Small scatterers.