Survival Probability for the Stadium Billiard

TL;DR

This paper analyzes the survival probability in an open stadium billiard, revealing an initial exponential decay transitioning to a long-time decay proportional to 1/time, with explicit formulas derived for the decay constant.

Contribution

It provides an explicit expression for the long-time decay constant of survival probability in the stadium billiard, combining numerical and analytical approaches.

Findings

Long-time survival probability decays as Constant/time.

Explicit formula for the decay constant derived.

Numerical and analytical methods agree with previous studies.

Abstract

We consider the open stadium billiard, consisting of two semicircles joined by parallel straight sides with one hole situated somewhere on one of the sides. Due to the hyperbolic nature of the stadium billiard, the initial decay of trajectories, due to loss through the hole, appears exponential. However, some trajectories (bouncing ball orbits) persist and survive for long times and therefore form the main contribution to the survival probability function at long times. Using both numerical and analytical methods, we concur with previous studies that the long-time survival probability for a reasonably small hole drops like Constant/time; here we obtain an explicit expression for the Constant.