Statistical properties of the system of two falling balls

Abstract

We consider the motion of two point masses along a vertical half-line that are subject to constant gravitational force and collide elastically with each other and the floor. This model was introduced by Wojtkowski who established hyperbolicity and ergodicity in case the lower ball is heavier. Here, we investigate the dynamics in discrete time and prove that, for an open set of the external parameter (the relative mass of the lower ball), the system mixes polynomially-modulo logarithmic factors, correlations decay as O(1/n(2))-and satisfies the Central Limit Theorem.

The flow of two falling balls mixes rapidly

“…By detailed analysis of the geometry of the system we identify special periodic points and show that the ratio of certain periods in continuous time is Diophantine for almost every value of the mass parameter in an interval. Using results of Melbourne ([13]) and our previous achievements [1] we conclude that for these values of the parameter the flow mixes faster than any polynomial. Even though the calculations are presented for the specific physical system, the method is quite general and can be applied to other suspension flows, too.…”

“…It can be shown that τ is piecewise C 2 with the same discontinuities as the discrete time dynamics T (see [1], subsection 3.7 for details). The flow is then isomorphic to the following suspension.…”

“…In this section we introduce the system and recall the necessary notations and results from our earlier paper [1]. The exposition is self contained, for further details about the dynamics and its properties we refer to our previous work.…”

“…However, for three or more particles ergodicity is still an open question. In the ergodic regime of the two falling balls a detailed geometric description of the system and a quantitative analysis of the discrete time map lead to the verification of fine statistical properties in [1], in particular polynomial decay of correlations and the central limit theorem for Hölder observables.…”

“…It is shown in [1] that none of these sets are empty. They are of course disjoint, moreover, even the closures of any two of them are disjoint provided their indices differ by more than 1.…”