Classical and quantum chaos in a circular billiard with a straight cut

Ree, Suhan; Reichl, L. E.

doi:10.1103/physreve.60.1607

Cited by 65 publications

(49 citation statements)

References 29 publications

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“…Rigorous investigations were concentrating on methods for producing fully chaotic billiards and on specific properties (Bernoulli, K-property, mixing and hyperbolicity) expressing differences between chaotic systems [5,6,7,8,9]. Various aspects of billiard dynamics have been extensively examined during last decades [10,11,12,13,14,15,16,17,18,19]. In recent years, properties of classical billiards and their quantum-mechanical counterparts were used to explain and improve performances of devices in microelectronics and nanotechnology, especially of optical microresonators in dielectrical and polymer lasers [20,21,22,23,24,25].…”

Section: Introductionmentioning

confidence: 99%

Chaotic properties of the truncated elliptical billiards

Lopac¹,

Šimić²

2011

Communications in Nonlinear Science and Numerical Simulation

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Chaotic properties of symmetrical two-dimensional stadium-like billiards with elliptical arcs are studied numerically and analytically. For the two-parameter truncated elliptical billiard the existence and linear stability of several lowest-order periodic orbits are investigated in the full parameter space. Poincaré plots are computed and used for evaluation of the degree of chaoticity with the box-counting method. The limit of the fully chaotic behavior is identified with circular arcs. Above this limit, for flattened elliptical arcs, mixed dynamics with numerous stable elliptic islands is present, similarly as in the elliptical stadium billiards. Below this limit the full chaos extends over the whole region of elongated shapes and the existing orbits are either unstable or neutral. This is conspicuously different from the behavior in the elliptical stadium billiards, where the chaotic region is strictly bounded from both sides. To examine the mechanism of this difference, a generalization to a novel three-parameter family of boundary shapes is proposed and suggested for further evaluation.

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Section: Introductionmentioning

confidence: 99%

Chaotic properties of the truncated elliptical billiards

Lopac¹,

Šimić²

2011

Communications in Nonlinear Science and Numerical Simulation

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“…The first peak to appear is located at γ = 0 and is already apparent in the analytical expression for ρ 4 (γ 4 ). These peaks are associated to periodic orbits in the vesicle, already discussed in [8]. Between two arrivals at the border, the angle θ changes as ∆θ = π − 2γ.…”

Section: Discussion Of the Results And Comparison With Simulationsmentioning

confidence: 99%

“…Since two quantities are conserved (kinetic energy and angular momentum), the system is integrable and therefore non-ergodic [8]. A recent study of the open version of this system was undertaken by Vicentini and Kokshenev [5], and the algebraic long time of the survival probability was studied in detail, in the limit of a very small opening (weakly open billiard), using a coarse-grained description of the system.…”

Section: Introductionmentioning

confidence: 99%

Particle–wall collision statistics in the open circular billiard

Stilck

2007

Physica A: Statistical Mechanics and its Applications

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In the open circular billiard particles are placed initially with a uniform distribution in their positions inside a planar circular vesicle. They all have velocities of the same magnitude, whose initial directions are also uniformly distributed. No particle-particle interactions are included, only specular elastic collisions of the particles with the wall of the vesicle. The particles may escape through an aperture with an angle 2δ. The collisions of the particles with the wall are characterized by the angular position and the angle of incidence. We study the evolution of the system considering the probability distributions of these variables at successive times n the particle reaches the border of the vesicle. These distributions are calculated analytically and measured in numerical simulations. For finite apertures δ < π/2, a particular set of initial conditions exists for which the particles are in periodic orbits and never escape the vesicle. This set is of zero measure, but the selection of angular momenta close to these orbits is observed after some collisions, and thus the distributions of probability have a structure formed by peaks. We calculate the marginal distributions up to n = 4, but for δ > π/2 a solution is found for arbitrary n. The escape probability as a function of n −1 decays with an exponent 4 for δ > π/2 and evidences for a power law decay are found for lower apertures as well.PACS numbers: 45.50.-j,05.20.-y,02.70.-c I. INTRODUCTIONIn the theory of dynamical systems, the study of the decay of simple hamiltonian systems [1]- [5] is of much interest. These studies are an extension of earlier research concerning the closed version of such systems, where the main question is their ergodicity. The pioneering work in this area is centered on the system called Sinai billiard, a circular billiard with a smaller circular exclusion area in its interior [6], which was shown to be ergodic. Other two-dimensional ballistic billiards are known to be ergodic as well, such as the Bunimowitch stadium [4,7]. One main interest in the studies of open billiards is the decay dynamics for long times. Bauer and Bertsch [1] found an exponential decay in an chaotic dynamics and a power law decay for a system with regular dynamics. These first results concerning the integrable case where later questioned by Legrand and Sornette [2], but it became clear that the difficulty in settling this question using numerical experiments is related to the high sensitivity of the results to initial conditions [3]. In a more detailed simulational study for the chaotic two-dimensional Bunimovich stadium [4], algebraic tails were found at sufficiently long times, but the weight of the algebraic tail tends to zero in the limit where the size of the aperture vanishes.In the classical circular billiard the (non-interacting) particles undergo elastic specular collisions with the wall. * Electronic address: jstilck@if.uff.brSince two quantities are conserved (kinetic energy and angular momentum), the system is integrable and therefor...

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“…Many aspects of classical and quantum chaos have been widely studied by means of billiards with different shapes [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. We offer a brief review about some commonly used methods to characterize the classical and quantum motion of particles inside billiards.…”

Section: Introductionmentioning

confidence: 99%

“…Billiards are one of the most used systems to analyse the quantum signatures of classical chaotic motion [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. Some advantages of the billiards are their extreme simplicity, their straightforward quantization and the possibility to measure many of the relevant quantities in laboratory experiments [18,19,20,21,22,23,24,25].…”

Section: Introductionmentioning

confidence: 99%

Chaos in the Diamond-Shaped Billiard with Rounded Crown

Salazar

Téllez

Jaramillo

et al. 2015

Rev. Acad. Colomb. Cienc. Ex. Fis. Nat.

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We analyse the classical and quantum behaviour of a particle trapped in a diamond-shaped billiard with rounded crown. We defined this billiard as a half stadium connected with a triangular billiard. A parameter ξ smoothly changes the shape of the billiard from an equilateral triangle (ξ = 1) to a diamond with rounded crown (ξ = 0). The parameter ξ controls the transition between the regular and chaotic regimes. The classical behaviour is regular when the control parameter ξ is one; in contrast, the system is chaotic when ξ = 1 even for values of ξ close to one. Several quantities such as Lyapunov exponent and the entropy of the distribution of the incident angle are used to characterize the chaotic behaviour of the classical system. The average information preserved by the classical trajectories increases rapidly as ξ is decreased from 1 and the Lyapunov exponent remains positive for ξ < 1. The Finite Difference Method was implemented in order to solve the quantum counterpart of the billiard. The energy spectrum and eigenstates were numerically computed for different values of ξ < 1. The spacing distribution between adjacent eigenvalues is analysed as a function of ξ, finding a Poisson and a Gaussian Orthogonal Ensemble (GOE) distribution for regular and chaotic regimes respectively. Several scars and bouncing ball states are shown with their corresponding classical periodic orbits. On the other hand, the results found for the quantum billiard are in agreement with the Bohigas-Giannoni-Schmit conjecture and exhibits the standard features of chaotic billiards such as the scarring of the wavefunction.Keywords: quantum chaos, quantum billiards, random matrices, finite difference method.Caos en el billar de forma de diamante y corona redondeada Resumen Se estudia el comportamiento de una partícula en el interior de un billar triangular donde uno de sus lados toma de medio estadio que se llamó billar diamante con corona redondeada o DSRC por su siglas en inglés. Se definió un parámetro ξ que cambia suavemente la forma la frontera partiendo de un billar triangular ξ = 1 a un billar DSRC ξ = 1. Dicho parámetro controla la transición entre el régimen regular y caótico. Clásicamente, el sistema es regular cuando ξ = 1. Por otro lado, el sistema se torna caótico para ξ = 1 incluyendo valores próximos a 1. Se calcula el coeficiente de Lyapunov y la entropía media de la distribución de losángulos de incidencia para caracterizar el comportamiento caótico del sistema. Se observó un rápido crecimiento de la información de las trayectorias hasta saturar la entropía al cambiar levemente la frontera del billar triangular original. A su vez el coeficiente de Lyapunov se mantuvo positivo durante este proceso una vez que ξ se alejaba de 1. Se implementó el método de diferencias finitas FDM para obtener el espectro y los estados propios de la contraparte cuántica del sistema. La distribución de espaciamiento entre primeros vecinos para varios valores de ξ fue construida numéricamente para diferentes valores de ξ encontrando u...

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Classical and quantum chaos in a circular billiard with a straight cut

Cited by 65 publications

References 29 publications

Chaotic properties of the truncated elliptical billiards

Chaotic properties of the truncated elliptical billiards

Particle–wall collision statistics in the open circular billiard

Chaos in the Diamond-Shaped Billiard with Rounded Crown

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