A rigorous derivation of Haff's law for a periodic two-disk fluid

TL;DR

This paper rigorously derives Haff's law for a periodic two-disk fluid by modeling it as a point particle in a Sinai billiard, extending to general dispersing billiards with finite horizon.

Contribution

It provides a rigorous derivation of Haff's law for a class of dispersing billiards, connecting fluid dynamics with billiard models.

Findings

Haff's law is derived for a periodic two-disk fluid.

Results apply to general dispersing billiards with finite horizon.

The derivation bridges fluid dynamics and billiard systems.

Abstract

We derive Haff's cooling law for a periodic fluid consisting of two hard disks per unit cell by reducing it to a point particle moving inside a Sinai billiard with finite horizon with an inelastic collision rule. Indeed, our results also apply to general dispersing billiards with piece-wise smooth boundary with finite horizon and no cusps.