On the number of hard ball collisions

TL;DR

This paper presents a new elementary proof that the number of elastic collisions among a finite number of equal balls in Euclidean space is finite, and provides bounds on the total number of collisions based on initial configurations.

Contribution

It introduces a novel elementary proof for finiteness of collisions and establishes bounds depending on initial distances and configurations.

Findings

Number of collisions is finite for finite balls.

Bound on total collisions depends on initial distances and number of balls.

Large collision counts imply tight configurations among some balls.

Abstract

We give a new and elementary proof that the number of elastic collisions of a finite number of balls in the Euclidean space is finite. We show that if there are $n$ balls of equal masses and radii 1, and at the time of a collision between any two balls the distance between any other pair of balls is greater than $n^{-n}$, then the total number of collisions is bounded by $n^{(5/2+\varepsilon)n}$, for any fixed $\varepsilon>0$ and large $n$. We also show that if there is a number of collisions larger than $n^{cn}$ for an appropriate $c>0$, then a large number of these collisions occur within a subfamily of balls that form a very tight configuration.