HOME

TheInfoList



OR:

A dynamical billiard is a
dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
in which a particle alternates between free motion (typically as a straight line) and specular reflections from a boundary. When the particle hits the boundary it reflects from it
without Without may refer to: * "Without" (''The X-Files''), an episode in the eighth season of ''The X-Files'' * "without", an English preposition * "Without", a film that premiered at the 2011 Slamdance Film Festival * "Without", a song by Jack Savorett ...
loss of speed (i.e. elastic collisions). Billiards are
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
idealizations of the 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 geometries; indeed, the first studies of billiards established their ergodic motion on surfaces of constant negative
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canon ...
. 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 Riemannian metric of the billiard table is not flat). All reflections are
specular Specular reflection, or regular reflection, is the mirror-like reflection of waves, such as light, from a surface. The law of reflection states that a reflected ray of light emerges from the reflecting surface at the same angle to the surf ...
: the angle of incidence just before the collision is equal to the
angle of reflection Reflection is the change in direction of a wavefront at an interface between two different media so that the wavefront returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves. The ...
just after the collision. The sequence of reflections is described by the billiard map that completely characterizes the motion of the particle. Billiards capture all the complexity of Hamiltonian systems, from integrability to
chaotic motion Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have ...
, without the difficulties of integrating the
equations of motion In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (V ...
to determine its Poincaré map. Birkhoff showed that a billiard system with an
elliptic In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
table is integrable.


Equations of motion

The
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
for a particle of mass ''m'' moving freely without friction on a surface is: :H(p, q) = \frac + V(q) where V(q) is a potential designed to be zero inside the region \Omega in which the particle can move, and infinity otherwise: :V(q) = \begin 0 &q \in \Omega \\ \infty &q \notin \Omega \end 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 In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
, then the Hamiltonian is replaced by: :H(p, q) = \fracp^i p^j g_(q) + V(q) where g_(q) is the
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
at point q \;\in\; \Omega. Because of the very simple structure of this Hamiltonian, the
equations of motion In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (V ...
for the particle, the Hamilton–Jacobi equations, are nothing other than the
geodesic equation In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
s on the manifold: the particle moves along geodesics.


Notable billiards and billiard classes


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 Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to hav ...
ever studied, having been introduced by Jacques Hadamard in 1898.


Artin's billiard

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 strongly mixing. It is an example of an Anosov system. This system was first studied by Emil Artin in 1924.


Dispersing and semi-dispersing billiards

Let ''M'' be complete smooth Riemannian manifold without boundary, maximal
sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a p ...
of which is not greater than ''K'' and with the
injectivity radius This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology. The following articles may also be useful; they either contain specialised vocabulary or prov ...
\rho >0 . Consider a collection of ''n'' geodesically
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polyto ...
subsets (walls) B_i \subset M , i =1, \ldots, n , such that their boundaries are smooth submanifolds of codimension one. Let B = M \ (\bigcup_^n \operatorname(B_i)) , where \operatorname(B_i) denotes the interior of the set B_i . The set B \subset M will be called the billiard table. Consider now a particle that moves inside the set ''B'' with unit speed along a geodesic until it reaches one of the sets ''B''i (such an event is called a collision) where it reflects according to the law “the angle of incidence is equal to the angle of reflection” (if it reaches one of the sets B_i \cap B_j , i \neq j , the trajectory is not defined after that moment). Such dynamical system is called semi-dispersing billiard. If the walls are strictly convex, then the billiard is called dispersing. 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 In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canon ...
does for geodesic flows causing the exponential
instability In numerous fields of study, the component of instability within a system is generally characterized by some of the outputs or internal states growing without bounds. Not all systems that are not stable are unstable; systems can also be mar ...
of the dynamics. It is precisely this dispersing mechanism that gives dispersing billiards their strongest
chaotic Chaotic was originally a Danish trading card game. It expanded to an online game in America which then became a television program based on the game. The program was able to be seen on 4Kids TV (Fox affiliates, nationwide), Jetix, The CW4Kid ...
properties, as it was established by
Yakov G. Sinai Yakov Grigorevich Sinai (russian: link=no, Я́ков Григо́рьевич Сина́й; born September 21, 1935) is a Russian-American mathematician known for his work on dynamical systems. He contributed to the modern metric theory of dy ...
. Namely, the billiards are ergodic, mixing, Bernoulli, having a positive Kolmogorov-Sinai entropy and an
exponential decay A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate ...
of
correlations In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics ...
. Chaotic properties of general semi-dispersing billiards are not understood that well, however, those of one important type of semi-dispersing billiards, hard ball gas were studied in some details since 1975 (see next section). General results of
Dmitri Burago Dmitri Yurievich Burago (Дмитрий Юрьевич Бураго, born 1964) is a Russian mathematician, specializing in geometry. He is the son of the professor of mathematics in Leningrad Yuri Dmitrievich Burago, with whom he also published ...
and
Serge Ferleger Serge may refer to: * Serge (fabric), a type of twill fabric *Serge (llama) (born 2005), a llama in the Cirque Franco-Italien and internet meme *Serge (name), a masculine given name (includes a list of people with this name) *Serge (post), a hitch ...
on the uniform estimation on the number of collisions in non-degenerate semi-dispersing billiards allow to establish finiteness of its
topological entropy In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
and no more than exponential growth of periodic trajectories. In contrast, ''degenerate'' semi-dispersing billiards may have infinite topological entropy.


Lorentz gas, aka Sinai billiard

The table of the Lorentz gas (also known as 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 Yakov Grigorevich Sinai (russian: link=no, Я́ков Григо́рьевич Сина́й; born September 21, 1935) is a Russian-American mathematician known for his work on dynamical systems. He contributed to the modern metric theory of dy ...
as an example of an interacting
Hamiltonian system A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can b ...
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 In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with ini ...
. 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.


Bunimovich stadium

The table called the Bunimovich stadium is a rectangle capped by semicircles, a shape called a stadium. Until it was introduced by
Leonid Bunimovich Leonid Abramowich Bunimovich (born August 1, 1947) is a Soviet and American mathematician, who made fundamental contributions to the theory of Dynamical Systems, Statistical Physics and various applications. Bunimovich received his bachelor's degr ...
, billiards with positive
Lyapunov exponent In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with ini ...
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.


Magnetic billiards

Magnetic billiards represent billiards where a ''charged'' particle is propagating under the presence of a perpendicular magnetic field. As a result, the particle trajectory changes from a straight line into an arc of a circle. The radius of this circle is inversely proportional to the magnetic field strength. Such billiards have been useful in real world applications of billiards, typically modelling nanodevices (see Applications).


Generalized billiards

Generalized billiards (GB) describe a motion of a mass point (a particle) inside a closed domain \Pi \,\subset\, \mathbb^n with the piece-wise smooth boundary \Gamma. On the boundary \Gamma 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, and, in the case when \Pi is a parallelepiped in connection with the justification of the second law of thermodynamics. 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 \Gamma, its velocity transforms with the help of a given function f(\gamma,\, t), defined on the direct product \Gamma \,\times\, \mathbb^1 (where \mathbb^1 is the real line, \gamma \,\in\, \Gamma is a point of the boundary and t \,\in\, \mathbb^1 is time), according to the following law. Suppose that the trajectory of the particle, which moves with the velocity v, intersects \Gamma at the point \gamma \,\in\, \Gamma at time t^*. Then at time t^* the particle acquires the velocity v^*, as if it underwent an elastic push from the infinitely-heavy plane \Gamma^*, which is tangent to \Gamma at the point \gamma, and at time t^* moves along the normal to \Gamma at \gamma with the velocity \textstyle\frac (\gamma,\, t^*). We emphasize that the ''position'' of the boundary itself is fixed, while its action upon the particle is defined through the function f. We take the positive direction of motion of the plane \Gamma^* to be towards the ''interior'' of \Pi. Thus if the derivative \textstyle\frac (\gamma,\, t) \;>\; 0, then the particle accelerates after the impact. If the velocity v^*, acquired by the particle as the result of the above reflection law, is directed to the interior of the domain \Pi, then the particle will leave the boundary and continue moving in \Pi until the next collision with \Gamma. If the velocity v^* is directed towards the outside of \Pi, then the particle remains on \Gamma at the point \gamma until at some time \tilde \;>\; t^* the interaction with the boundary will force the particle to leave it. If the function f(\gamma,\, t) does not depend on time t; i.e., \textstyle\frac \;=\; 0, the generalized billiard coincides with the classical one. 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. It is considered the reflection from the boundary \Gamma both in the framework of classical mechanics (Newtonian case) and the theory of relativity (relativistic case). Main results: in the Newtonian case the energy of particle is bounded, the Gibbs entropy is a constant, (in Notes) and in relativistic case the energy of particle, the Gibbs entropy, the entropy with respect to the phase volume grow to infinity, (in Notes), references to generalized billiards.


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\psi \;=\; E\psi or, more precisely, :-\frac\nabla^2 \psi_n(q) = E_n \psi_n(q) where \nabla^2 is the Laplacian. The potential that is infinite outside the region \Omega but zero inside it translates to the
Dirichlet boundary conditions In the mathematical study of differential equations, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differenti ...
: :\psi_n(q) = 0 \quad\mbox\quad q\notin \Omega As usual, the wavefunctions are taken to be orthonormal: :\int_\Omega \overline(q)\psi_n(q)\,dq = \delta_ Curiously, the free-field Schrödinger equation is the same as the Helmholtz equation, :\left(\nabla^2 + k^2\right)\psi = 0 with :k^2 = \frac2mE_n 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 In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
(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 Quantum chaos is a branch of physics which studies how chaotic classical dynamical systems can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is: "What is the relationship between quantum mech ...
. A particularly striking example of scarring on an elliptical table is given by the observation of the so-called quantum mirage.


Applications

Billiards, both quantum and classical, have been applied in several areas of physics to model quite diverse real world systems. Examples include ray-optics,
laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The firs ...
s,
acoustics Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...
, optical fibers (e.g. double-clad fibers ), or quantum-classical correspondence. One of their most frequent application is to model particles moving inside nanodevices, for example
quantum dot Quantum dots (QDs) are semiconductor particles a few nanometres in size, having optical and electronic properties that differ from those of larger particles as a result of quantum mechanics. They are a central topic in nanotechnology. When the ...
s, pn-junctions, antidot superlattices, 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. In addition, the energy conserving nature of the particle collisions is a direct reflection of the energy conservation of Hamiltonian mechanics.


Software

Open source software to simulate billiards exist for various programming languages. From most recent to oldest, existing software are
DynamicalBilliards.jl
(Julia)
Bill2D
(C++) an
Billiard Simulator
(Matlab). The animations present on this page were done with DynamicalBilliards.jl.


See also

* Fermi–Ulam model (billiards with oscillating walls) *
Lubachevsky–Stillinger algorithm Lubachevsky-Stillinger (compression) algorithm (LS algorithm, LSA, or LS protocol) is a numerical procedure suggested by F. H. Stillinger and B.D. Lubachevsky that simulates or imitates a physical process of compressing an assembly of hard particl ...
of compression simulates
hard spheres Hard spheres are widely used as model particles in the statistical mechanical theory of fluids and solids. They are defined simply as impenetrable spheres that cannot overlap in space. They mimic the extremely strong ("infinitely elastic bouncing" ...
colliding not only with the boundaries but also among themselves while growing in sizesB. 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 * Arithmetic billiards * Illumination problem


Notes


References


Sinai's billiards

* (in English, ''Sov. Math Dokl.'' 4 (1963) pp. 1818–1822). * Ya. G. Sinai, "Dynamical Systems with Elastic Reflections", ''
Russian Mathematical Surveys ''Uspekhi Matematicheskikh Nauk'' (russian: Успехи математических наук) is a Russian mathematical journal, published by the Russian Academy of Sciences and Moscow Mathematical Society and translated into English as ''Russia ...
'', 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,

', (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.)'' *N. Chernov and R. Markarian, "Chaotic Billiards", 2006, Mathematical survey and monographs nº 127, AMS.


Strange billiards

* T. Schürmann and I. Hoffmann, ''The entropy of strange billiards inside n-simplexes.'' J. Phys. A28, page 5033ff, 1995
PDF-Document


Bunimovich stadium

* *


Generalized billiards

* M. V. Deryabin and L. D. Pustyl'nikov, "Generalized relativistic billiards", ''Reg. and Chaotic Dyn.'' 8(3), pp. 283–296 (2003). * 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). * M. V. Deryabin and L. D. Pustyl'nikov, "Exponential attractors in generalized relativistic billiards", ''Comm. Math. Phys.'' 248(3), pp. 527–552 (2004).


External links

*
Scholarpedia entry on Dynamical Billiards
(Leonid Bunimovich)
Introduction to dynamical systems using billiards
Max Planck Institute for the Physics of Complex Systems {{DEFAULTSORT:Dynamical Billiards Dynamical systems