Spherical billiards with almost complete escape

Carl P Dettmann *, M R Rahman

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

1 Citation (Scopus)
42 Downloads (Pure)

Abstract

A dynamical billiard consists of a point particle moving uniformly except for mirror-like collisions with the boundary. Recent work has described the escape of the particle through a hole in the boundary of a circular or spherical billiard, making connections with the Riemann Hypothesis. Unlike the circular case, the sphere with a single hole leads to a non-zero probability of never escaping. Here we study variants in which almost all initial conditions escape, with multiple small holes or a thin strip. We show that equal spacing of holes around the equator is an efficient means of ensuring almost complete escape, and study the long time survival probability for small holes analytically and numerically. We find that it approaches a universal function of a single parameter, hole area multiplied by time.
Original languageEnglish
Article number123119
Number of pages11
JournalChaos: An Interdisciplinary Journal of Nonlinear Science
Volume31
DOIs
Publication statusPublished - 17 Dec 2021

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