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Leaking billiards.

Jan Nagler¹, Moritz Krieger, Marco Linke

¹Institut für Theoretische Physik, Universität Bremen, Otto-Hahn-Allee, D-28334 Bremen, Germany.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|May 16, 2007

Summary

This study analyzes particle leakage in billiard systems, revealing distinct escape dynamics. Stadium billiards show algebraic decay in escape time, while lemon billiards exhibit a "flooding law" for mean escape time.

Area of Science:

Physics
Dynamical Systems
Statistical Mechanics

Background:

Billiard systems model confined particles/waves.
Previous research focused on boundary openings.
Internal leakages in billiards are less studied.

Purpose of the Study:

Investigate classical dynamics of internal leakages in billiard systems.
Analyze the stadium-lemon-billiard family.
Characterize escape times and survival probabilities.

Main Methods:

Continuously tune billiard geometry from stadium to lemon.
Model internal leakages within the billiard.
Analyze classical dynamics and escape phenomena.

Main Results:

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Stadium billiards: algebraic decay of mean escape time (n(esc) ~ ε⁻¹), exponential survival probability decay.
Lemon billiards: mean escape time depends on leakage position and geometry.
Quasi-regular motion in lemon billiards follows a
flooding law
(n(esc) ~ 1-ε).

Conclusions:

Leakage dynamics differ significantly between hyperbolic (stadium) and mixed (lemon) billiards.
Findings align with stochastic map approximations for stadium billiards.
The
flooding law
provides insight into leaking Hamiltonian systems with mixed phase spaces.