Collision location for hard spheres in stationary regime

TL;DR

This paper analyzes the asymptotic behavior of collision locations for two small, symmetrically placed spheres with random velocities, revealing a surprising rotationally symmetric distribution as the sphere radius approaches zero.

Contribution

It provides a novel asymptotic analysis of collision location distribution for small spheres with random velocities, uncovering unexpected symmetry properties.

Findings

Collision probability scales as r^{d-1} when r→0.

As r→0, the collision location distribution converges to a (defective) t-distribution.

The distribution exhibits rotational symmetry about the origin.

Abstract

Consider two balls with radius $r>0$ whose centers are at a distance $2$, positioned symmetrically with respect to the origin in ${\mathbb{R}}^d$. Suppose that the initial velocities are independent standard normal vectors. When $r\to0$, the collision probability goes to 0 as $r^{d-1}$, and the asymptotic collision location distribution is a (defective) $t$-distribution. This distribution is rotationally symmetric about the origin for no apparent reason.