Classical and quantum chaos in the generalized parabolic lemon-shaped billiard

Lopac, Vjera; Mrkonjić, Ivana; Radić, Danko

doi:10.1103/physreve.59.303

Cited by 30 publications

(20 citation statements)

References 45 publications

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“…29 In case (ii), two examples are the Bunimovich stadium 30 and the Sinai billiard, 31 in such a cases, the time evolution of a single initial condition is enough to fill the entire phase space. Finally, case (iii), mixed type systems, in such a case, chaotic seas are generally surrounding KAM islands and invariant curves are observed [32][33][34][35][36][37][38][39][40] (and references in therein). If a time dependent perturbation is introduced in the boundary, @Q ¼ @QðtÞ, the system exchanges energy/velocity with the particle upon collision.…”

Section: Introductionmentioning

confidence: 99%

In-flight and collisional dissipation as a mechanism to suppress Fermi acceleration in a breathing Lorentz gas

Oliveira¹,

Leonel²

2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Some dynamical properties for a time dependent Lorentz gas considering both the dissipative and non dissipative dynamics are studied. The model is described by using a four-dimensional nonlinear mapping. For the conservative dynamics, scaling laws are obtained for the behavior of the average velocity for an ensemble of non interacting particles and the unlimited energy growth is confirmed. For the dissipative case, four different kinds of damping forces are considered namely: (i) restitution coefficient which makes the particle experiences a loss of energy upon collisions; and in-flight dissipation given by (ii) F ¼ ÀgV 2 ; (iii) F ¼ ÀgV l with l 6 ¼ 1 and l 6 ¼ 2 and; (iv) F ¼ ÀgV, where g is the dissipation parameter. Extensive numerical simulations were made and our results confirm that the unlimited energy growth, observed for the conservative dynamics, is suppressed for the dissipative case. The behaviour of the average velocity is described using scaling arguments and classes of universalities are defined. We revisit the problem of non-interacting particles in a time dependent Lorentz gas. We describe the model by using a four dimensional nonlinear map. As it is known, the phase space of the Lorentz gas with static scatterers is fully chaotic and the velocity of the particle is constant. However, when a time dependent perturbation is introduced to the boundary, the energy is no longer conserved and the unlimited energy growth is observed for the conservative case. On the other hand, our results show that when dissipation is introduced into the system, either by collisional dissipation or dissipation during the flight, the unlimited energy growth is no longer observed. Depending on the strength of the dissipation and considering short time, the average velocity can either grows and reaches a constant plateau or decays until the particle reaches the state of rest. For the cases, when the dynamics does not stop between the scatterers, the behaviour of the average velocity is described by using scaling arguments and once the scaling exponents are known, classes of universalities are defined.

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Section: Introductionmentioning

confidence: 99%

In-flight and collisional dissipation as a mechanism to suppress Fermi acceleration in a breathing Lorentz gas

Oliveira¹,

Leonel²

2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…In our previous work we analyzed several types of billiards with noncircular arcs (parabolic, hyperbolic, elliptical and generalized power-law), exhibiting mixed dynamics [30,31,32]. Next we investigated, in the full parameter space [33], the elliptical stadium billiards (ESB), first introduced by Donnay [7].…”

Section: Introductionmentioning

confidence: 99%

“…In our further description we refer to the impact points T(x, y) in the first quadrant, with no loss of generality for the obtained results. In the Poincaré sections the points P(x, v x ) are obtained by plotting the slope of the velocity direction v x = cos φ versus the x-coordinate of the intersection point with the x-axis, as explained in [30,31,32,33].…”

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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“…An interesting case to confirm that µ(χ) can be used to detect the transition from integrable to chaotic behavior is the lemon-shaped billiard [4]. Two instances of the lemon-shaped billiard are shown in Fig.4.…”

Section: Transition To Chaosmentioning

confidence: 99%

“…The most studied issues in this context are the statistical properties of nuclear and atomic spectra [1] and of two-(recently three) dimensional billiards [2][3][4][5]. In the mid-eighties, a conjecture (which became known as the BGS conjecture [6]) was made that the quantal energy level spacing of (ergodic) systems whose classical equivalent exhibit chaotic behavior obeys the Wigner GaussianOrthogonal-Ensemble statistics, known from random matrix theory, whereas for classically integrable systems the Poisson distribution applies.…”

Section: Introductionmentioning

confidence: 99%

Morphological image analysis of quantum motion in billiards

2000

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Morphological image analysis is applied to the time evolution of the probability distribution of a quantum particle moving in two and three-dimensional billiards. It is shown that the timeaveraged Euler characteristic of the probability density provides a well defined quantity to distinguish between classically integrable and non-integrable billiards. In three dimensions the time-averaged mean breadth of the probability density may also be used for this purpose.

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Classical and quantum chaos in the generalized parabolic lemon-shaped billiard

Cited by 30 publications

References 45 publications

In-flight and collisional dissipation as a mechanism to suppress Fermi acceleration in a breathing Lorentz gas

In-flight and collisional dissipation as a mechanism to suppress Fermi acceleration in a breathing Lorentz gas

Chaotic properties of the truncated elliptical billiards

Morphological image analysis of quantum motion in billiards

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