Survival probability of particles inside the Lemon Billiard

TL;DR

This study investigates how particles escape from the lemon billiard, revealing a two-stage decay pattern influenced by hole position, size, and billiard shape, with implications for understanding chaotic and mixed phase space dynamics.

Contribution

It provides the first detailed numerical analysis of survival probabilities in lemon billiards, highlighting the effects of hole placement and shape parameters on decay regimes.

Findings

Survival probability shows exponential and power-law decay stages.

Decay rate depends on hole size with different scaling in chaotic and mixed regions.

Scaling behavior observed in decay rate for small shape parameters.

Abstract

We study the escape of particles in the lemon billiard, a two-parameter family of billiard systems defined by the intersection of two identical circles. Using numerical simulations, we explore how the survival probability depends on the position and size of the hole, as well as on the billiard shape parameter. We find that the survival probability exhibits a two-stage decay pattern: an initial exponential regime followed by a long-time power-law tail, a signature of the stickiness effect. Our results show that the short-time exponential decay rate follows a power-law dependence on the hole size, with different scaling exponents for holes placed in chaotic regions versus mixed phase space regions. For holes located in mixed phase space regions, the decay exponent of the long-time power-law tail remains approximately constant, while the amplitude follows a power-law scaling with hole size. We also examine the dependence of short-time exponential decay rate on the billiard shape parameter and observe scaling behavior for small values of this parameter, which breaks down as the parameter increases.