The first encounter of two billiard particles of small radius

TL;DR

This paper proves that the time until the first collision between two small-radius particles in a Sinai billiard converges to an exponential distribution when scaled appropriately, advancing understanding of particle interactions in dynamical systems.

Contribution

It establishes the weak convergence of the first collision time to an exponential distribution in the small-radius limit for Sinai billiard systems.

Findings

First collision time converges to exponential distribution

Results apply in the rare interaction limit

Advances understanding of energy evolution in hard ball systems

Abstract

We prove that the time of the first collision between two particles in a Sinai billiard table converges weakly to an exponential distribution when time is rescaled by the inverse of the radius of the particles. This results provides a first step in studying the energy evolution of hard ball systems in the rare interaction limit.