Correlation Decay in Certain Soft Billiards

Abstract

Motivated by the 2D finite horizon periodic Lorentz gas, soft planar billiard systems with axis-symmetric potentials are studied in this paper. Since Sinai’s celebrated discovery that elastic collisions of a point particle with strictly convex scatterers give rise to hyperbolic, and consequently, nice ergodic behaviour, several authors (most notably Sinai, Kubo, Knauf) have found potentials with analogous properties. These investigations concluded in the work of V. Donnay and C. Liverani who obtained general conditions for a 2-D rotationally symmetric potential to provide ergodic dynamics. Our main aim here is to understand when these potentials lead to stronger stochastic properties, in particular to exponential decay of correlations and the central limit theorem. In the main argument we work with systems in general for which the rotation function satisfies certain conditions. One of these conditions has already been used by Donnay and Liverani to obtain hyperbolicity and ergodicity. What we prove is that if, in addition, the rotation function is regular in a reasonable sense, the rate of mixing is exponential, and, consequently, the central limit theorem applies. Finally, we give examples of specific potentials that fit our assumptions. This way we give a full discussion in the case of constant potentials and show potentials with any kind of power law behaviour at the origin for which the correlations decay exponentially.