A lower bound for the number of elastic collisions

TL;DR

This paper demonstrates that the maximum number of elastic collisions among equal balls in higher dimensions can surpass previous quadratic bounds, reaching cubic order for certain configurations.

Contribution

It provides the first explicit example showing the number of elastic collisions can grow faster than quadratic, establishing a new lower bound of order n^3.

Findings

Number of elastic collisions can exceed n^3/27 for n≥3 and d≥2.

Previous lower bounds were of order n^2.

The result applies to equal mass and size balls in multi-dimensional space.

Abstract

We prove by example that the number of elastic collisions of $n$ balls of equal mass and equal size in $d$-dimensional space can be greater than $n^3/27$ for $n\geq 3$ and $d\geq 2$. The previously known lower bound was of order $n^2$.