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| [[Image:BunimovichStadium.svg|thumb|The Bunimovich stadium is a chaotic dynamical billiard]]
| | Sports Centre Manager Roman Carmouche from Aylmer, has interests which include lawn darts, [http://I-Vls.com/blogs/post/7183 new launching property] developers in singapore and sleeping. Gains inspiration by paying a visit to St Augustine's Abbey. |
| A '''billiard''' is a [[dynamical system]] in which a particle alternates between motion in a straight line and [[specular reflection]]s from a boundary. When the particle hits the boundary it reflects from it without loss of [[speed]]. Billiard dynamical systems are [[Hamiltonian mechanics|Hamiltonian]] idealizations of the [[billiards|game of billiards]], but where the region contained by the boundary can have shapes other than rectangular and even be multidimensional. Dynamical billiards may also be studied on [[non-Euclidean geometry|non-Euclidean geometries]]; indeed, the very first studies of billiards established their [[ergodic theory|ergodic motion]] on [[surface]]s 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.
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| The motion of the particle in the billiard is a straight line, with constant energy, between reflections with the boundary (a [[geodesic]] if the [[Riemannian metric]] of the billiard table is not flat). All [[Reflection (physics)|reflections]] are [[specular reflection|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 described by the '''billiard map''' that completely characterizes the motion of the particle.
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| Billiards capture all the complexity of Hamiltonian systems, from [[Integrable system|integrability]] to [[chaos theory|chaotic motion]], without the difficulties of integrating the [[equations of motion]] to determine its [[Poincaré map]]. [[George David Birkhoff|Birkhoff]] showed that a billiard system with an [[ellipse|elliptic]] table is integrable.
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| == Equations of motion ==
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| The [[Hamiltonian (quantum mechanics)|Hamiltonian]] for a particle of mass ''m'' moving freely without friction on a surface is:
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| :<math>H(p, q) = \frac {p^2}{2m} + V(q)</math> | |
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| where <math>\scriptstyle V(q)</math> is a potential designed to be zero inside the region <math>\scriptstyle \Omega</math> in which the particle can move, and infinity otherwise:
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| :<math>V(q) =
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| \begin{cases}
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| 0 &q \in \Omega \\
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| \infty &q \notin \Omega
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| \end{cases}
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| </math>
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| 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:
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| :<math>H(p, q) = \frac{1}{2m}p^i p^j g_{ij}(q) + V(q)</math>
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| where <math>\scriptstyle g_{ij}(q)</math> is the [[metric tensor]] at point <math>\scriptstyle q \;\in\; \Omega</math>. Because of the very simple structure of this Hamiltonian, the [[equations of motion]] for the particle, the [[Hamilton–Jacobi equation]]s, are nothing other than the [[geodesic equation]]s on the manifold: the particle moves along [[geodesic]]s.
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| == Notable billiard tables ==
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| ===Hadamard's billiards===
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| {{main|Hadamard's dynamical system}}
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| 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.
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| ===Artin's billiard===
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| {{main|Artin billiard}}
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| Artin's billiard considers 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. It is notable for being exactly solvable, and yet not only [[ergodic]] but also [[mixing (mathematics)|strongly mixing]]. It is an example of an [[Anosov flow|Anosov system]]. This system was first studied by [[Emil Artin]] in 1924.
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| === Sinai billiard ===
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| [[Image:SinaiBilliard.svg|thumb|A trajectory in the Sinai billiard]]
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| 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.
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| The billiard was introduced by [[Yakov G. Sinai]] as an example of an interacting [[Hamiltonian system]] that displays physical thermodynamic properties: all of its possible trajectories are [[ergodic]] and it has a positive [[Lyapunov exponent]]. As a model of a classical gas, the Sinai billiard is sometimes called the '''Lorentz gas'''.
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| 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.
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| === Bunimovich stadium ===
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| The table called the '''Bunimovich stadium''' is a rectangle capped by semicircles. Until it was introduced by [[Leonid Bunimovich]], billiards with positive [[Lyapunov exponent]]s 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.
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| === Generalized billiards ===
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| Generalized billiards (GB) describe a motion of a mass point (a particle) inside a closed domain <math>\scriptstyle \Pi \,\subset\, \mathbb{R}^n</math> with the piece-wise smooth boundary <math>\scriptstyle \Gamma</math>. On the boundary <math>\scriptstyle \Gamma</math> the velocity of point is transformed as the particle underwent the action of generalized billiard law. GB were introduced by [[Lev D. Pustyl'nikov]] in the general case,<ref name="Pustyln1">{{cite journal |first=L. D. |last=Pustyl'nikov |title=The law of entropy increase and generalized billiards |journal=[[Russian Mathematical Surveys]] |volume=54 |issue=3 |pages=650–651 |year=1999 |doi= }}</ref> and, in the case when <math>\scriptstyle \Pi</math> is a parallelepiped<ref name="Pustyln2">{{cite journal |first=L. D. |last=Pustyl'nikov |title=Poincaré models, rogorous justification of the second law of thermodynamics from mechanics, and the Fermi acceleration mechanism |journal=[[Russian Mathematical Surveys]] |volume=50 |issue=1 |pages=145–189 |year=1995 }}</ref> in connection with the justification of the second law of thermodynamics (the law of entropy increase). From the physical point of view, GB describe a gas consisting of finitely many particles moving in a vessel, while the walls of the vessel heat up or cool down. The essence of the generalization is the following. As the particle hits the boundary <math>\scriptstyle \Gamma</math>, its velocity transforms with the help of a given function <math>\scriptstyle f(\gamma,\, t)</math>, defined on the direct product <math>\scriptstyle \Gamma \,\times\, \mathbb{R}^1</math> (where <math>\scriptstyle \mathbb{R}^1</math> is the real line, <math>\scriptstyle \gamma \,\in\, \Gamma</math> is a point of the boundary and <math>\scriptstyle t \,\in\, \mathbb{R}^1</math> is time), according to the following law. Suppose that the trajectory of the particle, which moves with the velocity <math>\scriptstyle v</math>, intersects <math>\scriptstyle \Gamma</math> at the point <math>\scriptstyle \gamma \,\in\, \Gamma</math> at time <math>\scriptstyle t^*</math>. Then at time <math>\scriptstyle t^*</math> the particle acquires the velocity <math>\scriptstyle v^*</math>, as if it underwent an elastic push from the infinitely-heavy plane <math>\scriptstyle \Gamma^*</math>, which is tangent to <math>\scriptstyle \Gamma</math> at the point <math>\scriptstyle \gamma</math>, and at time <math>\scriptstyle t^*</math> moves along the normal to <math>\scriptstyle \Gamma</math> at <math>\scriptstyle \gamma</math> with the velocity <math>\scriptstyle \frac{\partial f}{\partial t} (\gamma,\, t^*)</math>. We emphasize that the ''position'' of the boundary itself is fixed, while its action upon the particle is defined through the function <math>\scriptstyle f</math>.
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| We take the positive direction of motion of the plane <math>\scriptstyle \Gamma^*</math> to be towards the ''interior'' of <math>\scriptstyle \Pi</math>. Thus if the derivative <math>\scriptstyle \frac{\partial f}{\partial t} (\gamma,\, t) \;>\; 0</math>, then the particle accelerates after the impact.
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| If the velocity <math>\scriptstyle v^*</math>, acquired by the particle as the result of the above reflection law, is directed to the interior of the domain <math>\scriptstyle \Pi</math>, then the particle will leave the boundary and continue moving in <math>\scriptstyle \Pi</math> until the next collision with <math>\scriptstyle \Gamma</math>. If the velocity <math>\scriptstyle v^*</math> is directed towards the outside of <math>\scriptstyle \Pi</math>, then the particle remains on <math>\scriptstyle \Gamma</math> at the point <math>\scriptstyle \gamma</math> until at some time <math>\scriptstyle \tilde{t} \;>\; t^*</math> the interaction with the boundary will force the particle to leave it.
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| If the function <math>\scriptstyle f(\gamma,\, t)</math> does not depend on time <math>\scriptstyle t</math>; i.e., <math>\scriptstyle \frac{\partial f}{\partial t} \;=\; 0</math>, the generalized billiard coincides with the classical one.
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| This generalized reflection law is very natural. First, it reflects an obvious fact that the walls of the vessel with gas are motionless. Second the action of the wall on the particle is still the classical elastic push. In the essence, we consider infinitesimally moving boundaries with given velocities.
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| It is considered the reflection from the boundary <math>\scriptstyle \Gamma</math> both in the framework of classical mechanics (Newtonian case) and the theory of relativity (relativistic case).
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| Main results: in the Newtonian case the energy of particle is bounded, the Gibbs entropy is a constant,<ref name="Pustyln2" /><ref name="Pustyln3">{{cite journal |first=L. D. |last=Pustyl'nikov |title=Generalized Newtonian periodic billiards in a ball |journal=UMN |volume=60 |issue=2 |pages=171–172 |year=2005 }} English translation in [[Russian Mathematical Surveys]], 60(2), pp. 365-366 (2005).</ref><ref name="Pustyln7">{{cite journal |first=Mikhail V. |last=Deryabin |first2=Lev D. |last2=Pustyl'nikov |title=Nonequilibrium Gas and Generalized Billiards |journal=Journal of Statistical Physics |volume=126 |issue=1 |pages=117–132 |year=2007 |doi=10.1007/s10955-006-9250-4 }}</ref> (in Notes) and in relativistic case the energy of particle, the Gibbs entropy, the entropy with respect to the phase volume grow to infinity,<ref name="Pustyln2" /><ref name="Pustyln7" /> (in Notes), references to generalized billiards.
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| == Quantum chaos ==
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| 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]] <math>\scriptstyle H\psi \;=\; E\psi</math> or, more precisely,
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| :<math>-\frac{\hbar^2}{2m}\nabla^2 \psi_n(q) = E_n \psi_n(q)</math>
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| where <math>\scriptstyle \nabla^2</math> is the [[Laplacian]]. The potential that is infinite outside the region <math>\scriptstyle \Omega</math> but zero inside it translates to the [[Dirichlet boundary conditions]]:
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| :<math>\psi_n(q) = 0 \quad\mbox{for}\quad q\notin \Omega</math>
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| As usual, the wavefunctions are taken to be [[orthonormal]]:
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| :<math>\int_\Omega \overline{\psi_m}(q)\psi_n(q)\,dq = \delta_{mn}</math>
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| Curiously, the free-field Schrödinger equation is the same as the [[Helmholtz equation]],
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| :<math>\left(\nabla^2 + k^2\right)\psi = 0</math>
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| with
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| :<math>k^2 = \frac{1}{\hbar^2}2mE_n</math>
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| 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).
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| The semi-classical limit corresponds to <math>\scriptstyle \hbar \;\to\; 0</math> which can be seen to be equivalent to <math>\scriptstyle m \;\to\; \infty</math>, the mass increasing so that it behaves classically.
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| 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]].
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| A particularly striking example of scarring on an elliptical table is given by the observation of the so-called [[quantum mirage]].
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| == Applications ==
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| The most practical application of theory of quantum billiards is related with [[double-clad fiber]]s.
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| In such a [[fiber laser]], the small core with low [[numerical aperture]] confines the signal, and the wide cladding confines the multi-mode
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| pump. In the [[paraxial approximation]], the complex field of pump in the cladding behaves like a wave function in the quantum billiard.
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| The modes of the cladding with scarring may avoid the core, and symmetrical configurations enhance this effect.
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| The chaotic fibers<ref name="Doya">
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| {{cite journal
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| | title=Modeling and optimization of double-clad fiber amplifiers using chaotic propagation of pump
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| | author= Leproux, P.
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| | coauthors=S. Fevrier, V. Doya, P. Roy, and D. Pagnoux
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| | journal=[[Optical Fiber Technology]]
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| | url=http://www.ingentaconnect.com/content/ap/of/2001/00000007/00000004/art00361
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| | volume=7
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| | year=2003
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| | issue=4
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| | pages=324–339
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| | doi=10.1006/ofte.2001.0361
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| | bibcode = 2001OptFT...7..324L
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| }}</ref> provide good coupling; in the first approximation, such a fiber can be described with the same equations as an idealized billiard.
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| The coupling is especially poor in fibers with circular symmetry while 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.<ref name="diri">{{cite journal
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| |title=Boundary behavior of modes of Dirichlet Laplacian
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| |author= Kouznetsov, D.| coauthors=Moloney, J.V.| journal=[[Journal of Modern Optics]]
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| |url=http://www.metapress.com/content/be0lua88cwybywnl/?p=5464d03ba7e7440f9827207df673c804&pi=6
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| |volume=51 | year=2004 | issue=13 | pages=1955–1962 | |
| |ref=http://www.ils.uec.ac.jp/~dima/TMOP102136.pdf|bibcode = 2004JMOp...51.1955K |doi = 10.1080/09500340408232504
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| }}</ref>
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| == See also ==
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| * [[Fermi–Ulam model]] (billiards with oscillating walls)
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| * [[Lubachevsky-Stillinger algorithm]] of compression simulates hard spheres colliding not only with the boundaries but also among themselves while growing in sizes<ref>B. D. Lubachevsky and F. H. Stillinger, Geometric properties of random disk packings, J. Statistical Physics 60 (1990), 561-583 http://www.princeton.edu/~fhs/geodisk/geodisk.pdf</ref>
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| ==Notes==
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| {{reflist}}
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| == References ==
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| === Sinai's billiards ===
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| * {{cite journal| last=Sinai | first=Ya. G. | year=1963 | title=[On the foundations of the ergodic hypothesis for a dynamical system of statistical mechanics] | language=Russian | journal=[[Doklady Akademii Nauk SSSR]] | volume=153 | issue=6 | pages=1261–1264}} (in English, ''Sov. Math Dokl.'' '''4''' (1963) pp. 1818–1822).
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| * Ya. G. Sinai, "Dynamical Systems with Elastic Reflections", ''[[Russian Mathematical Surveys]]'', '''25''', (1970) pp. 137–191.
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| * 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.)''
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| * 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.)''
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| * S. Sridhar and W. T. Lu, "[http://sagar.physics.neu.edu/preprints/sinai-ruelle-jsp2002.pdf Sinai Billiards, Ruelle Zeta-functions and Ruelle Resonances: Microwave Experiments]", (2002) ''Journal of Statistical Physics'', Vol. '''108''' Nos. 5/6, pp. 755–766.
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| * Linas Vepstas, ''[http://www.linas.org/art-gallery/billiards/billiards.html 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.)''
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| *N. Chernov and R. Markarian, "Chaotic Billiards", 2006, Mathematical survey and monographs nº 127, AMS.
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| === Strange billiards ===
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| * T. Schürmann and I. Hoffmann, ''The entropy of strange billiards inside n-simplexes.'' J. Phys. A28, page 5033ff, 1995. [http://arxiv.org/abs/nlin/0208048 PDF-Document]
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| === Bunimovich stadium ===
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| * L.A.Bunimovich, "On the Ergodic Properties of Nowhere Dispersing Billiards", ''Commun Math Phys'', '''65''' (1979) pp. 295–312.
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| * L.A.Bunimovich and Ya. G. Sinai, "Markov Partitions for Dispersed Billiards", ''Commun Math Phys'', '''78''' (1980) pp. 247–280.
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| *[http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Flash/Chaos/Bunimovich/Bunimovich.html Flash animation illustrating the chaotic Bunimovich Stadium]
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| === Generalized billiards ===
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| * M. V. Deryabin and L. D. Pustyl'nikov, "Generalized relativistic billiards", ''Reg. and Chaotic Dyn.'' 8(3), pp. 283–296 (2003).
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| * M. V. Deryabin and L. D. Pustyl'nikov, "On Generalized Relativistic Billiards in External Force Fields", ''Letters in Mathematical Physics'', 63(3), pp. 195–207 (2003).
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| * M. V. Deryabin and L. D. Pustyl'nikov, "Exponential attractors in generalized relativistic billiards", ''Comm. Math. Phys.'' 248(3), pp. 527–552 (2004).
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| == External links ==
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| * {{MathWorld|urlname=Billiards|title=Billiards}}
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| * [http://xweb.geos.ed.ac.uk/~stephan/mod_SinaiBilliard.en.html Simulation of the Sinai Billiard] (Stephan Matthiesen)
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| {{Chaos theory}}
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| {{DEFAULTSORT:Dynamical Billiards}}
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| [[Category:Dynamical systems]]
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