An improved upper bound on the number of billiard ball collisions

TL;DR

This paper establishes a new upper bound on the maximum number of elastic collisions among n identical spheres in d-dimensional space, improving understanding of collision limits in billiard systems.

Contribution

It introduces a refined upper bound on collision counts, specifically showing that the logarithm of the maximum number of collisions grows proportionally to n log n.

Findings

New upper bound on collision number: log K_+ ≤ c(d) n log n

Applicable to systems of identical spheres in any dimension

Advances theoretical understanding of collision dynamics

Abstract

We give a new upper bound $K_+$ on the number of totally elastic collisions of $n$ hard spheres with equal radii and equal masses in $R^d$. Our bound satisfies $\log K_+ \leq c(d) n \log n$.