Evolution of collision numbers for a chaotic gas dynamics

TL;DR

This paper investigates the growth and recurrence properties of collision counts in a chaotic gas of hard spheres, using ergodic theory and Langevin dynamics to understand long-term collision patterns and limitations.

Contribution

It introduces a conjecture on recurrence in collision numbers and models their evolution with an effective Langevin dynamics, revealing new recurrence properties in chaotic gas systems.

Findings

Triplets of particles have infinitely recurring moments with equal collision counts.

Differences in collision numbers within triplets repeat indefinitely.

For larger sets of pairs, only finite repetitions occur.

Abstract

We put forward a conjecture of recurrence for a gas of hard spheres that collide elastically in a finite volume. The dynamics consists of a sequence of instantaneous binary collisions. We study how the numbers of collisions of different pairs of particles grow as functions of time. We observe that these numbers can be represented as a time-integral of a function on the phase space. Assuming the results of the ergodic theory apply, we describe the evolution of the numbers by an effective Langevin dynamics. We use the facts that hold for these dynamics with probability one, in order to establish properties of a single trajectory of the system. We find that for any triplet of particles there will be an infinite sequence of moments of time, when the numbers of collisions of all three different pairs of the triplet will be equal. Moreover, any value of difference of collision numbers of pairs in the triplet will repeat indefinitely. On the other hand, for larger number of pairs there is but a finite number of repetitions. Thus the ergodic theory produces a limitation on the dynamics.