Dynamical billiards

From Wikipedia, the free encyclopedia

The Bunimovich stadium is a chaotic dynamical billiard

A billiard is a dynamical system in which a particle alternates between motion in a straight line and specular reflections off of a boundary. When the particle hits the boundary it reflects from it without loss of speed. Billiard dynamical systems are Hamiltonian idealizations of the game of billiards, but where the region contained by the boundary can have shapes other than rectangular and even be many dimensional. Dynamical billiards may also be studied on non-Euclidean geometries; indeed, the very first studies of billiards established their ergodic motion on surfaces of constant negative curvature. The study of billiards which are kept out of a region, rather than being kept in a region, is known as outer billiard theory.

The motion of the particle in the billiard is a straight line, with constant energy, between reflections with the boundary (a geodesic if the table has any curvature). All reflections are specular: the angle of incidence just before the collision is equal to the angle of reflection just after the collision. The sequence of reflections is called the billiard map and completely characterizes the motion of the particle.

Billiards capture all the complexity of Hamiltonian systems, from integrability to chaotic motion, without the difficulties of integrating the equations of motion to determine its Poincaré map. Birkhoff showed that a billiard system with an elliptic table is integrable.

In one dimension billiards (i.e. hard rods) show deterministic chaos and are ergodic if they have different masses. The mathematical problem of one dimensional billiards of different masses and a single billiard in a flat-sided box are equivalent. The chaotic property means billiards are extremely efficient samplers of their phase space.

1 Equations of motion
2 Notable billiard tables
3 Quantum chaos
4 Applications
5 References for Sinai's billiards
6 References for the Bunimovich stadium
7 External links

[edit] Equations of motion

The Hamiltonian for a particle of mass m moving freely without friction on a surface is:

$H(p,q)=\frac {p^2}{2m}+V(q),$

where $V (q)$ is a potential designed to be zero inside the region $Ω$ in which the particle can move, and infinity otherwise:

$V(q)=\begin{cases} 0 \qquad q \in \Omega \\ \infty \qquad q \notin \Omega \end{cases}.$

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. If the particle is to move on a non-Euclidean manifold, then the Hamiltonian is replaced by:

$H(p,q)=\frac {p^i p^j g_{ij}(q) }{2m}+V(q),$

where $g i j (q)$ is the metric tensor at point $q \in \Omega$ . Because of the very simple structure of this Hamiltonian, the equations of motion for the particle, the Hamilton–Jacobi equations, are nothing other than the geodesic equations on the manifold: the particle moves along geodesics.

[edit] Notable billiard tables

[edit] Hadamard's billiards

See main article Hadamard's billiards.

Hadamard's billiards concern the motion of a free point particle on a surface of constant negative curvature, in particular, the simplest compact Riemann surface with negative curvature, a surface of genus 2 (a two-holed donut). The model is exactly solvable, and is given by the geodesic flow on the surface. It is the earliest example of deterministic chaos ever studied, having been introduced by Jacques Hadamard in 1898.

[edit] Artin's billiards

See main article Artin's billiards.

Artin's billiards concern the free motion of a point particle on a surface of constant negative curvature, in particular, the simplest non-compact Riemann surface, a surface with one cusp. The billiards are notable in being exactly solvable, and being not only ergodic but also strongly mixing. Thus they are an example of an Anosov system. Artin billiards were first studied by Emil Artin in 1924.

[edit] Sinai billiard

A trajectory in the Sinai billiard

The table of the Sinai billiard is a square with a disk removed from its center; the table is flat, having no curvature. The billiard arises from studying the behavior of two interacting disks bouncing inside a square, reflecting off the boundaries of the square and off each other. By eliminating the center of mass as a configuration variable, the dynamics of two interacting disks reduces to the dynamics in the Sinai billiard.

The billiard was introduced by Yakov G. Sinai as an example of an interacting Hamiltonian system that displays physical thermodynamic properties: it is ergodic and has a positive Lyapunov exponent. As a model of a classical gas, the Sinai billiard is sometimes called the Lorentz gas.

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.

[edit] Bunimovich stadium

The table called the Bunimovich stadium is a rectangle capped by semi-circles. Until it was introduced by Leonid Bunimovich, billiards with positive Lyapunov exponents were thought to need convex scatters, such as the disk in the Sinai billiard, to produce the exponential divergence of orbits. Bunimovich showed that by considering the orbits beyond the focusing point of a concave region it was possible to obtain exponential divergence.

[edit] Quantum chaos

The quantum version of the billiards is readily studied in several ways. The classical Hamiltonian for the billiards, given above, is replaced by the stationary-state Schrödinger equation $H ψ = E ψ$ or, more precisely,

$-\frac{\hbar^2}{2m}\Delta \psi_n(q) = E_n \psi_n(q),$

where $Δ$ is the Laplacian. The potential that is infinite outside the region $Ω$ but zero inside it translates to the Dirichlet boundary conditions:

$\psi_n(q)=0 \quad\mbox{for}\quad q\notin \Omega.$

As usual, the wavefunctions are taken to be orthonormal:

$\int_\Omega \overline{\psi_m}(q)\psi_n(q)\,dq = \delta_{mn}.$

Curiously, the free-field Schrödinger equation is the same as the Helmholtz equation,

$\left(\Delta+k^2\right)\psi = 0,$

with

$k^2=\frac{2mE_n}{\hbar^2}.$

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. (The study of radar cavity modes must be limited to the transverse magnetic (TM) modes, as these are the ones obeying the Dirichlet boundary conditions).

The semi-classical limit corresponds to $\hbar\to 0$ which can be seen to be equivalent to $m\to\infty$ , the mass increasing so that it behaves classically.

As a general statement, one may say that whenever the classical equations of motion are integrable (e.g. rectangular or circular billiard tables), then the quantum-mechanical version of the billiards is completely solvable. When the classical system is chaotic, then the quantum system is generally not exactly solvable, and presents numerous difficulties in its quantization and evaluation. The general study of chaotic quantum systems is known as quantum chaos.

A particularly striking example of scarring on an elliptical table is given by the observation of the so-called quantum mirage.

[edit] Applications

The most practical applicaiton of theory of quantum billiards is related with double-clad fibers. In such a fiber laser, the small core with low Numerical Aperture confines the signal, and the wide cladding confines the multi-mode pump. In the paraxial approximation, the complex field of pump in the cladding behaves like a wave function in the quantum billiard. The modes of the cladding with scarring may avoid the core, and symmetrical configurations are not good. Especially poor the coopling is in the fiber with circular symmetry. The spiral-shaped fiber, with the core close to the chunk of the spiral, shows good coupling properties. The small spiral deformation forces all the scars to be coupled with the core.

[edit] References for Sinai's billiards

Ya. G. Sinai, "On the Foundations of the Ergodic Hypothesis for a Dynamical System of Statistical Mechanics", Dokl. Acad. Nauk., 153 (1963) No. 6. (in English, Sov. Math Dokl. 4 (1963) pp.1818-1822).
Ya. G. Sinai, "Dynamical Systems with Elastic Reflections", Russian Math. Surveys, 25, (1970) pp. 137-191.
V. I. Arnold and A. Avez, Théorie ergodique des systèms dynamiques, (1967), Gauthier-Villars, Paris. (English edition: Benjamin-Cummings, Reading, Mass. 1968). (Provides discussion and references for Sinai's billiards.)
D. Heitmann, J.P. Kotthaus, "The Spectroscopy of Quantum Dot Arrays", Physics Today (1993) pp. 56-63. (Provides a review of experimental tests of quantum versions of Sinai's billiards realized as nano-scale (mesoscopic) structures on silicon wafers.)
S. Sridhar and W. T. Lu, "Sinai Billiards, Ruelle Zeta-functions and Ruelle Resonances: Microwave Experiments", (2002) Journal of Statistical Physics, Vol. 108 Nos. 5/6, pp. 755-766.
Linas Vepstas, Sinai's Billiards, (2001). (Provides ray-traced images of Sinai's billiards in three-dimensional space. These images provide a graphic, intuitive demonstration of the strong ergodicity of the system.)

[edit] References for the Bunimovich stadium

L.A.Bunimovich, "On the Ergodic Properties of Nowhere Dispersing Billiards", Commun Math Phys, 65 (1979) pp. 295-312.
L.A.Bunimovich and Ya. G. Sinai, "Markov Partitions for Dispersed Billiards", Commun Math Phys, 78 (1980) pp. 247-280.
Flash animation illustrating the chaotic Bunimovich Stadium