P. B\'alint, I. P. T\'oth
Correlation decay in certain soft billiards
(414K, postscript)
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 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.