Statistics of the total number of collisions and the ordering time in a freely expanding hard-point gas

TL;DR

This paper analytically investigates the distributions of collision counts and ordering times in a one-dimensional gas of particles, revealing universal features in these statistical properties.

Contribution

It provides the first analytical derivation of the distributions of total collisions and ordering time in a free-expanding hard-point gas.

Findings

Distributions of total collisions are derived analytically.

Ordering time distribution exhibits universal features.

Results apply to a broad class of initial conditions.

Abstract

We consider a Jepsen gas of $N$ hard-point particles undergoing free expansion on a line, starting from random initial positions of the particles having random initial velocities. The particles undergo binary elastic collisions upon contact and move freely in-between collisions. After a certain ordering time $T_{o}$, the system reaches a ``fan'' state where all the velocities are completely ordered from left to right in an increasing fashion and there is no further collision. We compute analytically the distributions of (i) the total number of collisions and (ii) the ordering time $T_{o}$. We show that several features of these distributions are universal.