^{1,2}and Mason A. Porter

^{3}

### 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).

*billiard system*consists of a point particle confined in some domain (which is usually a subset of ) and colliding perfectly elastically against the boundary of that domain.

^{11,28}Such billiards can have chaotic, regular (i.e., integrable), or mixed dynamics.

^{7,8,26}For example, a finite-size circular particle confined within a circular boundary is integrable, but two circular particles confined in a circular domain yields chaotic dynamics, though regular behavior can persist for extremely long times.

^{21}The chaotic dynamics in two-particle billiards appears via the dispersive mechanism

^{28}as a result of particle-particle collisions, whereas particle-boundary collisions lead to regular dynamics when the boundary is circular. Consequently, although the long-time dynamics is chaotic, the regular transients can become arbitrarily long as one considers confined particles with progressively smaller radii. It is desirable to find means to simplify investigations of the statistical properties arising from long-time transient dynamics in two-particle billiards, whose dynamics are not well understood and which require very long computations to simulate. In this paper, we take a step in this direction by considering one-particle billiards with random perturbations and comparing diagnostics for measuring aggregate levels of chaotic dynamics in two-particle versus perturbed one-particle billiards. In particular, we consider two circular particles confined in a circular table and one-particle circular billiards with two types of random perturbations: one in which random perturbations are applied at times determined via a Poisson process and another in which random perturbations are applied at times given by actual particle-particle collision times from the two-particle system. The two-particle circular billiard considered in the present paper provides a clean example of an “intermittent” system. There continues to be considerable interest in intermittent billiards,

^{10}and this example in particular deserves many future investigations.

We thank Leonid Bunimovich, Carl Dettmann, Steven Lansel, and two anonymous referees for helpful comments. We are particularly grateful to Leonid Bunimovich for suggesting that we examine this problem from this point of view and to Carl Dettmann for several useful discussions and helpful comments on manuscript drafts. S.R. also thanks the Nuffield foundation and St. John's College for financial support during the time of the research, Charles Batty for recommendation letters supporting the applications for the aforementioned funding, and the administration of Dartington House for office space provided during the research. S.R. was affiliated with University of Oxford for most of the duration of this project.

I. INTRODUCTION II. MODELLING A CIRCULAR BILLIARD SYSTEM III. QUANTIFYING CHAOTIC DYNAMICS A. Recurrence plots and recurrence rates B. Autocorrelation C. Lyapunov exponents IV. COMPARISON OF TWO-PARTICLE BILLIARDS AND PERTURBED ONE-PARTICLE BILLIARDS A. Random perturbations in a one-particle billiard B. Poisson process for determining perturbation times C. Results of the comparisons V. CONCLUSIONS AND DISCUSSION

### Key Topics

- Chaotic dynamics
- 23.0
- Poisson's equation
- 14.0
- Statistical properties
- 8.0
- Probability theory
- 6.0
- Chaos
- 5.0

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