  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).
