The flow of two falling balls mixes rapidly

Abstract

In this paper we study the system of two falling balls in continuous time. We modell the system by a suspension flow over a two dimensional, hyperbolic base map. 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.

Superpolynomial and polynomial mixing for semiflows and flows

“…It would suffice to show that we can take ψ k = 1 in (3.2) as is the case for uniformly hyperbolic systems [48]. (In particular, the proof of rapid mixing for two falling balls in [18] seems to rely on such an improvement. Alternatively, one could try to verify the good asymptotics condition [50] described in section 5.2; this would also lead to a stronger conclusion (robust rapid mixing rather than almost sure) in [18].)…”