Chaotic lensed billiards

Figure 1.  On the left, a lensed Sinai-type billiard showing focusing and defocusing of trajectories. On the right, a lensed Bunimovich-like billiard showing initially focusing followed by dispersing trajectories. In both cases, the potential is $ -1 $ on the shaded region and $ 0 $ outside. A parallel beam of trajectories emanates from the vertical right wall, stopping at the first collision with the boundary of the unshaded region of the billiard domain

Figure 2.  Sample trajectories (of equal number of steps) for the lensed Bunimovich billiard. The values $ C $ of the potential on the semi-disc, from top to bottom, left to right: $ 0 $, $ -1 $, $ -10 $, $ -100 $, $ -1000 $, $ -10000 $. Numerical approximation of the positive mean Lyapunov exponent (for the billiard flow) gave the values: $ 1.00\pm 0.02 $ for $ C = 0 $ (top-left) and $ 4.46\pm 0.14 $ for $ C = -10000 $ (bottom-right). See also Figure 12

Figure 3.  In the lensed billiard phase space the refracting boundary is duplicated

Figure 4.  This lensed billiard table involves $ 4 $ parameters: $ \ell_1, \ell_2 $, the signed curvature $ \kappa\in [-2, 2] $ of the interface segment, and the value $ C $ of the potential function to the left of the interface segment. To its right, the potential is $ 0 $. The figure shows one example for $ \kappa<0 $. Initial conditions for billiard trajectories are set at the vertical wall on the right. If $ C $ is sufficiently large ($ C\geq 1 $ if the initial particle speed is set equal to $ \sqrt{2} $) for the system to be a standard billiard on the region to the right of the interface segment, then $ \kappa<0 $ corresponds to a semidispersing billiard, and $ \kappa>0 $ to a focusing billiard

Figure 5.  Singular set for the lensed billiard map of the family of billiard tables shown in Figure 4, restricted to the part of the phase space over the right vertical side of the rectangle. The parameters are $ \ell_1 = 0.5 $, $ \ell_2 = 1 $, $ C = -0.5 $ (top group of ten plots), $ C = 0.5 $ (bottom group); in each group of plots, the values of $ \kappa $ are, clockwise from the top left: $ -2.0 $, $ -1.5 $, $ -1.0 $, $ -0.5 $, $ -0.1 $, $ 0.1 $, $ 0.5 $, $ 1.0 $, $ 1.5 $, $ 2.0 $. Only the singular initial conditions leading to the critical refracting angle are shown, for orbits of length less than $ 16 $

Figure 6.  Phase-portraits for bounded lensed billiard systems. Numbers indicate values of the potential constant. The orbits shown here are for the return map to the right vertical side of the rectangular domain, for the top four systems, and the return map to the outer ellipse for the bottom system, rather than $ \mathcal{T} $ itself. Only the relevant part of the phase space is shown

Figure 7.  Sojourn time distributions obtained by simulating long trajectories of the lensed billiard system for the table shown in the inset. The potential is $ -1 $ on the left half of the table and $ 0 $ on the right, while particle energy is $ 0.01 $. The particle undergoes many more collisions in the left than the right region, although mean sojourn times are the same in both. Corollary 2.4 gives the mean value $ \approx 17.3054 $ (the table has area $ \approx 1.558 $ and height $ 1 $); simulated values are $ 17.317 $ (left) and $ 17.305 $ (right). The dotted line is the graph of the probability density function of the exponential distribution with parameter $ \lambda = 1/17.305 = 0.058 $

Figure 8.  Notation for the calculation of the differential of the lensed billiard. On the left a refraction, and on the right a reflection. $ C_1 $ and $ C_2 $ are the values of the potential function at the indicated regions

Figure 9.  Positive Lyapunov exponent as a function of the signed curvature parameter $ \kappa $ of the refracting boundary. The parameters are $ \ell_2-\ell_1 = 1 $ and $ C = 1 $, so this is a standard (purely reflecting) billiard in the complement of the shaded region (see insert), to the right of the curved line, where $ V = 0 $

Figure 10.  Family of billiard tables of Figure 4 with $ C $ over the interval $ [-3, 1) $. On the left, $ \kappa = -1 $, $ \ell_1 = 1 $, $ \ell_2 = 2 $; on the right, $ \kappa = 1 $, $ \ell_1 = 1, \ell_2 = 4 $. The "fat tail" on the left for $ C\leq 0 $ and high variance on the right for part of the range $ 0\leq C< 1 $ suggest that the billiard system is not ergodic in those ranges of $ C $. The larger $ \ell_2 $ on the right was chosen under the expectation that, for $ C $ sufficiently close to $ 1 $, the defocusing mechanism causing ergodicity will come into play. The somewhat more clearly defined shape of the graph on the right roughly in the range $ 0.8<C<1 $ seems to justify this expectation

Figure 11.  Here, the parameters are: $ \kappa = -1 $, $ C = 0.5 $, $ \ell_2 = \ell_1+0.5 $, and $ \ell_1 $ ranges from $ 0 $ to $ 5 $

Figure 12.  Lensed Bunimovich and Sinai billiards. The system on the left is not ergodic for $ C>0 $, and the one on the right is not ergodic for $ C\leq 0. $

Figure 13.  Half-lensed Bunimovich stadium

Figure 14.  A lensed interpolation between Bunimovich and Sinai

Figure 15.  Left: The time spent by a segment of trajectory inside the lens is $ T(\theta) = 2R\cos\theta_r/\mathcal{s} $, where $ \theta_r $ and $ \theta $ are related by Snell's law and $ \mathcal{s} = \sqrt{2(E-C)/m} $ is particle speed. Right: Large expansion factor when $ E-C $ is small. The circular arc consisting of the intersection of the boundary of the lens and the incident parallel beam has angle $ 2\theta_{\text{ crit}} $. As $ C $ approaches $ E $, this angle approaches $ 0 $, and the directions of trajectories as they exit the lens span an interval of angles approaching $ (0, 2\pi) $

Figure 16.  In the limit $ C\rightarrow -\infty $, the system on the left behaves as if $ C = 1 $, corresponding to a semi-dispersing billiard. For the system on the right, the deep well limit is the composition of a standard Sinai billiard (with reflecting walls) and a rotation by $ \pi $

Figure 17.  A deep well billiard system and the field of thin cones. When the potential in $ \mathcal{A}_0 $ is strongly negative, a trajectory that falls into $ \mathcal{A}_0 $ undergoes many collisions with $ \partial \mathcal{A}_0 $ during a time interval having mean value $ \pi A/a $ until the first return to $ \mathcal{C} $ for which the direction of approach lies in the field of thin cones defined by $ |\sin\theta|< \sqrt{\frac{1}{1+|C|}} $. Here, $ \theta $ is the angle the velocity of incidence makes with a normal vector to $ \mathcal{C} $ (pointing into $ \mathcal{A}_1 $) at the collision point. The trajectory then reemerges into $ \mathcal{A}_1 $ at a point in $ \mathcal{C} $ and velocity having the Liouville measure distribution