Leaking billiards

Billiards are idealizations for systems where particles or waves are confined to cavities, or to other homogeneous regions. In billiard systems a point particle moves freely except for specular reflections from rigid walls. However, billiard walls are not always completely reflective and measurements inside can also open the billiard. Since boundary openings have been studied extensively in the literature, we rather model leakages inside the billiard. In particular, we investigate the classical dynamics of a leakage for a continuous family of billiard systems, that is, the stadium-lemon-billiard family. With a single parameter the geometry of the billiard can be tuned from stadium (being fully hyperbolic) over circle (integrable) to the lemon-shaped billiard (mixed chaotic). For the stadium billiard we found an algebraically decaying mean escape time with the linear size $ϵ$ of the leakage $⟨n_{\mathrm{esc}}⟩\sim {ϵ}^{-1}$ together with an exponential decay of the survival probability distribution. The finding is nearly independent of the position and size of the leakage, as long as the leakage is much smaller than the system size, and it is in good agreement with a stochastic map approximation of the dynamics. Due to the mixed phase space for lemon billiards, the mean escape time depends both on the position and geometry of the leakage. For systems where quasiregular motion dominates, we found a linear dependence of the mean escape time, $⟨n_{\mathrm{esc}}⟩\sim 1-ϵ$, which we refer to as flooding law. Our findings are helpful in understanding dynamics of leaking Hamiltonian systems.

Figure 1

(Color online) Schematic representation of stadium $(a>0)$, circle $(a=0)$, and lemon-shaped $(a<0)$ billiard.

Figure 2

(Color online) Schematic view of the leakage positions (disks). Half of the system size is portioned in such a way that five disks are placed equidistantly, disk 1 is centered, and disk 5 is tangent to the boundary.

Figure 3

(Color online) Ranges and color legend for escape time diagrams (ETD). Parameter values for the displayed example with central circular hole: $a=-0.2$, hole radius $ϵ=0.05$, and $N_{\max }=10000$. The color legend displays the normalized escape time $n_{\mathrm{esc}}∕N_{\max }$ where $n_{\mathrm{esc}}$ is the number of reflections before escape, and $N_{\max }$ denotes the maximal number of reflections during the simulation. The lighter the color the longer the particle is trapped in the billiard. White colored phase-space points correspond to $n_{\mathrm{esc}}=N_{\max }$, black ones correspond to $n_{\mathrm{esc}}=1$.

Figure 4

(Color online) Escape time diagrams for the Birkhoff phase space, i.e., the $(s,p)$ plane, and for $a=1$. The color of a point codes the number of reflections before escape through a circular central leakage (upper diagram line), and decentrally placed leakages (lower line). White points represent bounded motion (maximal number of specular reflection is $N_{\max }=10000$), all other points represent starting positions of escape orbits. The lighter the color, the larger the number of reflections (see scale in Fig. 3): black $(n_{\mathrm{esc}}=1)$, white $(n_{\mathrm{esc}}=N_{\max })$. Upper line, from left to right: hole radius $ϵ=0.01$, $ϵ=0.05$, $ϵ=0.1$, and $ϵ=0.2$. Lower line: decentral disk positions 2, 3, 4, and 5 (see Fig. 2) for fixed radius $ϵ=0.1$.

Figure 5

(Color online) Escape time diagrams for the stadium billiard with central circular hole of radius $ϵ=0.05$ for varying values of $a>0$. Values for $a$ from left to right: 0.1, 0.2, 0.3, 0.4 (upper diagram line), 0.5, 0.8, 1.4, 2.1 (lower diagram line). Inset: ETD for $a=0.1$ but for the tangential leakage position 5.

Figure 6

(Color online) Iteration maps for lemon-shaped ($a=-1.0$, $a=-0.5$), circle $(a=0)$, and stadium ($a=0.5$, $a=1.0$) billiard for a centered (position 1 in Fig. 2) circular leakage with radius $ϵ=0.05$. The diagram plane is the (full) phase-space region ($0<s⩽1$, $-1<p<1$) (see also Fig. 3). Black points indicate an escape after a single reflection when the trajectory begins at the corresponding starting position in phase-space $(s,p)$. Red points represent trajectories which end up in the hole after two reflections, green and blue correspond to three and four reflections, respectively. White indicates a survival after four reflections.

Figure 7

(Color online) Escape time diagrams for various parameter values $a<0$ for fixed circular hole radius, $ϵ=0.05$, that is centered. $N_{\max }=10000$. Left to right: $-1.97$, $-1.9$, $-1.8$, $-1.7$ (a), $-1.6$, $-1.5$, $-1.4$, $-1.3$ (b), $-1.2$, $-1.1$, $-1.0$, $-0.9$ (c), $-0.8$, $-0.7$, $-0.6$, $-0.5$ (d), $-0.4$, $-0.3$, $-0.2$, $-0.1$ (e). Insets: ETD for $a=-0.05$, $-0.03$, and $a=-0.01$.

Figure 8

(Color online) Escape time diagrams for four decentral positions of a circular hole. $ϵ=0.05$ is fixed, $N=10000$. Left to right in a diagram line: leakage position 2, 3, 4, and 5. $a=-1.9$ (a), $a=-1.0$ (b), $a=-0.7$ (c), $a=-0.5$ (d), $a=-0.2$ (e).

Figure 9

(Color online) Mean escape time distribution dependence on circular leakage position for $a=0.5$ (solid symbols), and for $a=1$ (open symbols). The bold line is the power law $1∕ϵ$. Curves are arbitrarily normalized to coincide at $⟨n_{\mathrm{esc}}⟩=1$.

Figure 10

(Color online) Mean escape time distribution for small circular leakages in a double-logarithmic plot $(a=1.0)$. For small leakage sizes data are in good agreement with the theoretical curve (power law).

Figure 11

(Color online) Mean escape time distribution for $a=0.1$ and five leakage positions. The bold lines are the theoretical curves $1-ϵ$ and $1∕ϵ$, respectively. Curves are arbitrarily normalized to coincide at $⟨n_{\mathrm{esc}}⟩=1$.

Figure 12

(Color online) Mean escape time distribution. Average escape time vs linear hole size. The bold lines are the theoretical curves $1-ϵ$ and $1∕ϵ$, respectively. Curves are arbitrarily normalized to coincide at $⟨n_{\mathrm{esc}}⟩=1$.