Two-Particle Circular Billiards Versus Randomly Perturbed One-Particle Circular Billiards

TL;DR

This study compares two-particle circular billiards with randomly perturbed one-particle billiards to understand chaotic dynamics, showing that simpler models can approximate complex two-particle behavior, especially for small particle sizes.

Contribution

It demonstrates that randomly perturbed one-particle billiards can serve as effective proxies for analyzing the chaotic dynamics of two-particle circular billiards, reducing computational complexity.

Findings

Recurrence-rate coefficients align between two-particle and perturbed one-particle models.

Finite-time Lyapunov exponents show similar chaotic behavior in both models.

Autocorrelation coefficients suggest comparable temporal dynamics.

Abstract

We study a two-particle circular billiard containing two finite-size circular particles that collide elastically with the billiard boundary and with each other. Such a two-particle circular billiard provides a clean example of an "intermittent" system. This billiard system behaves chaotically, but the time scale on which chaos manifests can become arbitrarily long as the sizes of the confined particles become smaller. The finite-time dynamics of this system depends on the relative frequencies of (chaotic) particle-particle collisions versus (integrable) particle-boundary collisions, and investigating these dynamics is computationally intensive because of the long time scales involved. To help improve understanding of such two-particle dynamics, we compare the results of diagnostics used to measure chaotic dynamics for a two-particle circular billiard with those computed for two types of one-particle circular billiards in which a confined particle undergoes random perturbations. Importantly, such one-particle approximations are much less computationally demanding than the original two-particle system, and we expect them to yield reasonable estimates of the extent of chaotic behavior in the two-particle system when the sizes of confined particles are small. Our computations of recurrence-rate coefficients, finite-time Lyapunov exponents, and autocorrelation coefficients support this hypothesis and suggest that studying randomly perturbed one-particle billiards has the potential to yield insights into the aggregate properties of two-particle billiards, which are difficult to investigate directly without enormous computation times (especially when the sizes of the confined particles are small).