Dynamical billiards
When the particle hits the boundary it reflects from it without loss of speed (i.e. elastic collisions).
The sequence of reflections is described by the billiard map that completely characterizes the motion of the particle.
The Hamiltonian for a particle of mass m moving freely without friction on a surface is: where
in which the particle can move, and infinity otherwise: This form of the potential guarantees a specular reflection on the boundary.
The kinetic term guarantees that the particle moves in a straight line, without any change in energy.
It is the earliest example of deterministic chaos ever studied, having been introduced by Jacques Hadamard in 1898.
Let M be complete smooth Riemannian manifold without boundary, maximal sectional curvature of which is not greater than K and with the injectivity radius
The naming is motivated by observation that a locally parallel beam of trajectories disperse after a collision with strictly convex part of a wall, but remain locally parallel after a collision with a flat section of a wall.
Dispersing boundary plays the same role for billiards as negative curvature does for geodesic flows causing the exponential instability of the dynamics.
[1] Namely, the billiards are ergodic, mixing, Bernoulli, having a positive Kolmogorov-Sinai entropy and an exponential decay of correlations.
General results of Dmitri Burago and Serge Ferleger[2] on the uniform estimation on the number of collisions in non-degenerate semi-dispersing billiards allow to establish finiteness of its topological entropy and no more than exponential growth of periodic trajectories.
[3] In contrast, degenerate semi-dispersing billiards may have infinite topological entropy.
The billiard was introduced by Yakov G. Sinai as an example of an interacting Hamiltonian system that displays physical thermodynamic properties: almost all (up to a measure zero) of its possible trajectories are ergodic and it has a positive Lyapunov exponent.
Sinai's great achievement with this model was to show that the classical Boltzmann–Gibbs ensemble for an ideal gas is essentially the maximally chaotic Hadamard billiards.
A particle is subject to a constant force (e.g. the gravity of the Earth) and scatters inelastically on a periodically corrugated vibrating floor.
When the floor is made of arc or circles - in a certain intervall of frequencies - one can give a semi-analytic estimates to the rate of exponential separation of the trajectories.
Bunimovich showed that by considering the orbits beyond the focusing point of a concave region it was possible to obtain exponential divergence.
As a result, the particle trajectory changes from a straight line into an arc of a circle.
Generalized billiards (GB) describe a motion of a mass point (a particle) inside a closed domain
the velocity of point is transformed as the particle underwent the action of generalized billiard law.
We emphasize that the position of the boundary itself is fixed, while its action upon the particle is defined through the function
, acquired by the particle as the result of the above reflection law, is directed to the interior of the domain
Main results: in the Newtonian case the energy of particle is bounded, the Gibbs entropy is a constant,[7][8][9] (in Notes) and in relativistic case the energy of particle, the Gibbs entropy, the entropy with respect to the phase volume grow to infinity,[7][9] (in Notes), references to generalized billiards.
but zero inside it translates to the Dirichlet boundary conditions: As usual, the wavefunctions are taken to be orthonormal: Curiously, the free-field Schrödinger equation is the same as the Helmholtz equation, with This implies that two and three-dimensional quantum billiards can be modelled by the classical resonance modes of a radar cavity of a given shape, thus opening a door to experimental verification.
A particularly striking example of scarring on an elliptical table is given by the observation of the so-called quantum mirage.
Billiards, both quantum and classical, have been applied in several areas of physics to model quite diverse real world systems.
[16] One of their most frequent application is to model particles moving inside nanodevices, for example quantum dots,[17][18] pn-junctions,[19] antidot superlattices,[20][21] among others.
The reason for this broadly spread effectiveness of billiards as physical models resides on the fact that in situations with small amount of disorder or noise, the movement of e.g. particles like electrons, or light rays, is very much similar to the movement of the point-particles in billiards.
Open source software to simulate billiards exist for various programming languages.
From most recent to oldest, existing software are: DynamicalBilliards.jl (Julia), Bill2D (C++) and Billiard Simulator (Matlab).
