Survival probability of particles inside the Lemon Billiard

TL;DR

This paper investigates escape dynamics in the lemon billiard, a two-parameter open dynamical system defined by the intersection of two identical circles. By simulating particle trajectories with boundary holes of size $h$ placed in chaotic and mixed regions, it reveals a two-stage decay in the survival probability: an initial exponential regime with rate $\kappa$ and a long-time power-law tail $P(n)\sim A n^{-\gamma}$ due to stickiness near regular islands. The study demonstrates scaling laws: $\kappa(h)\sim h^{z}$ with distinct exponents for different hole locations, and in the mixed region, $A(h)\sim h^{z_3}$ while $\gamma$ remains nearly constant; similar scaling with the shape parameter $B$ appears for small $B$ but degrades as $B$ increases. Overall, the work highlights how hole placement and the underlying phase-space structure govern escape statistics in open billiards and demonstrates robust scaling with respect to hole size, with only limited scaling in the geometry parameter $B$.

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.

Figures (7)

• Figure 1: (Color online): Geometry of the lemon billiard. The boundary is formed by the intersection of two identical circles of radius $R$, whose centers are shifted symmetrically by a distance $2B$. The blue and red arcs represent the left- and right-shifted circles, respectively. The cyan area corresponds to the billiard domain. The green dots indicate the points of intersection between the two circles.
• Figure 2: (Color online): An illustration of two consecutive collisions of a particle and the angles involved in the billiard.
• Figure 3: Schematic representation of the calculation of the time to the next collision. From the particle’s current position (black dot), an extended line is constructed along its trajectory (cyan line), intersecting both the left- and right-shifted circles. The intersection points are marked by orange and purple dots. The smaller of the two corresponding time intervals determines the effective time at which the next collision occurs.
• Figure 4: Phase space structure for different values of the parameter $B$. (a) For $B=0$, the billiard is a perfect circle and the system is integrable; phase space is filled with invariant curves. (b) For $B=0.1$, integrability breaks, and chaotic regions emerge alongside stability islands. (c) For $B=0.7$, the central period-2 island becomes dominant. (d) For $B=0.9$, the phase space is mostly chaotic, with a single large regular island.
• Figure 5: (Color online): (a) Phase space for $B = 0.1$, with hole positions indicated at $s_{\text{exit}} = 0.50\mathcal{L}$ (chaotic region) and $s_{\text{exit}} = 0.75\mathcal{L}$ (mixed region). (b) Survival probability curves for different hole sizes $h$ at $s_{\text{exit}} = 0.50\mathcal{L}$. (c) Survival probability curves for different $h$ at $s_{\text{exit}} = 0.75\mathcal{L}$. (d) Log-log plot of the escape rate $\kappa$ as a function of $h$, showing power-law scaling $\kappa(h) \sim h^z$ for both hole positions.