The flow of two falling balls mixes rapidly

TL;DR

This paper analyzes the mixing properties of a suspension flow modeling two falling balls, showing that for most mass ratios, the flow mixes faster than any polynomial, with a method applicable to other systems.

Contribution

It introduces a detailed geometric analysis of the two falling balls system and demonstrates a general method for proving super-polynomial mixing in suspension flows.

Findings

Flow mixes faster than any polynomial for almost all mass ratios.

The method applies to other suspension flows beyond the specific system.

Identifies special periodic points influencing mixing rates.

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 (\cite{M}) and our previous achievements \cite{BBNV} 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.