Lorentz gas with small scatterers

TL;DR

This paper establishes limit laws, including a non-standard Central Limit Theorem and Local Limit Theorem, for a Lorentz gas with small scatterers, bridging two previously studied regimes with different scaling behaviors.

Contribution

It provides the first results on an intermediate regime where scatterer size tends to zero at a controlled rate as time goes to infinity.

Findings

Proves a non-standard Central Limit Theorem for the displacement.

Establishes a Local Limit Theorem in the intermediate regime.

Bridges the gap between two well-studied asymptotic regimes.

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 $\sqrt{n\log n}$ scaling (i) for fixed infinite horizon configurations -- letting first $n\to \infty$ and then $\rho \to 0$ -- studied e.g.~by Sz\'asz \& Varj\'u (2007) and (ii) Boltzmann-Grad type situations -- letting first $\rho \to 0$ and then $n \to \infty$ -- studied by Marklof \& T\'oth (2016).