Examples of exponentially many collisions in a hard ball system

TL;DR

This paper demonstrates that in a system of identical elastic hard balls in three-dimensional space, certain initial conditions can lead to an exponentially large number of collisions as the number of balls increases.

Contribution

It provides the first explicit examples showing exponential growth in the number of collisions in a hard ball system.

Findings

Existence of initial conditions with exponential collisions

Number of collisions can grow exponentially with the number of balls

Highlights complexity of hard ball dynamics in 3D

Abstract

Consider the system of $n$ identical hard balls in $\mathbb R^3$ moving freely and colliding elastically. We show that there exist initial conditions such that the number of collisions is exponential in $n$.