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Abstract: The boundary of the lemon billiards is defined by the intersection of two circles of equal unit radius with the distance 2B between their centers, as introduced by Heller and Tomsovic [E. J. Heller and S. Tomsovic, Phys. Today 46, 38 (1993)]. This paper is a continuation of our recent papers on a classical and quantum ergodic lemon billiard (B = 0.5) with strong stickiness effects [C. Lozej ˇ et al., Phys. Rev. E 103, 012204 (2021)], as well as on the three billiards with a simple mixed-type phase space and no stickiness [C. Lozej ˇ et al., Nonlin. Phenom. Complex Syst. 24, 1 (2021)]. Here we study two classical and quantum lemon billiards, for the cases B = 0.1953, 0.083, which are mixed-type billiards with a complex structure of phase space, without significant stickiness regions. A preliminary study of their spectra was published recently [ C. Lozej, D. Lukman, and M. ˇ Robnik, Physics 3, 888 (2021)]. We calculate a very large number (106) of consecutive eigenstates and their Poincaré-Husimi (PH) functions, and analyze their localization properties by studying the entropy localization measure and the normalized inverse participation ratio. We introduce an overlap index, which measures the degree of the overlap of PH functions with classically regular and chaotic regions. We observe the existence of regular states associated with invariant tori and chaotic states associated with the classically chaotic regions, and also the mixed-type states. We show that in accordance with the Berry-Robnik picture and the principle of uniform semiclassical condensation of PH functions, the relative fraction of mixed-type states decreases as a power law with increasing energy, thus, in the strict semiclassical limit, leaving only purely regular and chaotic states. Our approach offers a general phenomenological overview of the structural and localization properties of PH functions in quantum mixed-type Hamiltonian systems.
Keywords: quantum physics, energy, localization, quantum chaos, billiards, chaotic systems
Published in DKUM: 12.10.2023; Views: 288; Downloads: 17
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Abstract: In the dissertation I have dealt with time-dependent (nonautonomous) systems, the conservative (Hamiltonian) as well as dissipative, and investigated their dynamical and statistical properties. In conservative (Hamiltonian) time-dependent systems the energy is not conserved, whilst the Liouville theorem about the conservation of the phase space volume still applies. We are interested to know, whether the system can gain energy, and whether this energy can grow unbounded, up to infinity, and we are interested in the system's behaviour in the mean, as well as its statistical properties. An example of such a system goes back to the 1940s, when Fermi proposed the acceleration of cosmic rays (in the first place protons) upon the collisions with moving magnetic domains in the interstellar medium of our Galaxy, and in other galaxies. He then proposed a simple mechanical one-dimensional model, the so-called Fermi-Ulam Model (FUM), where a point particle is moving between two rigid walls, one being at rest and the other one oscillating. If the oscillation is periodic and smooth, it turned out in a nontrivial way, which is, in the modern era of understanding the chaotic dynamical systems, well understood, namely that the unbounded increasing of the energy (the so-called Fermi acceleration) is not possible, due to the barriers in form of invariant tori, which partition the phase space into regions, between which the transitions are not possible. The research has then been extended to other simple dyanamical systems, which have complex dynamics. The first was so-called bouncer model, in which a point particle bounces off the oscillating platform in a gravitational field. In this simple system the Fermi acceleration is possible. Later the research was directed towards two-dimensional billiard systems. It turned out that the Fermi acceleration is possible in all such systems, which are at least partially chaotic (of the mixed type), or even in a system that is integrable as static, namely in case of the elliptic billiard. (The circle billiard is an exception, because it is always integrable, as the angular momentum is conserved even in time-dependent case.) The study of time-dependent systems has developed strongly worldwide around the 1990s, in particular in 2000s, and became one of the central topics in nonlinear dynamics. It turned out, quite generally, but formal and implicit, in the sense of mathematical existence theorems, that in nonautonomous Hamilton systems the energy can grow unbounded, meaning that the system ``pumps" the energy from the environment with which it interacts. There are many open questions: how does the energy increase with time, in particular in the mean of some representative ensemble of initial conditions (typically the phase space of two-dimensional time-dependent billiards is four-dimensional.) It turned out that almost everywhere the power laws apply, empirically, based on the numerical calculations, but with various acceleration exponents. If the Fermi acceleration is not posssible, like e.g. in the FUM, due to the invariant tori, then after a certain time of acceleration stage the crossover into the regime of saturation takes place, whose characteristics also follow the power laws. One of the central themes in the dissertation is the study of these power laws, their critical exponents, analytical relationships among them, using the scaling analysis (Leonel, McClintock and Silva, Phys. Rev. Lett. 2004). Furthermore, the central theme is the question, what happens, if, in a nonautonomous Hamilton system which exhibits Fermi acceleration, we introduce dissipation, either at the collisions with the walls (collisional dissipation) or during the free motion (in-flight dissipation, due to the viscosity of the fluid or the drag force etc.). Dissipation typically transforms the periodic points into point attractors and chaotic components into chaotic attractors. The Fermi acceleration is always suppressed. We are interested in the phase portraits of
Keywords: nonlinear dynamics, dynamical systems, conservative and dissipative systems, time-dependent systems, Fermi acceleration, billiards, kicked systems, chaos, chaotic and periodic attractors, bifurcations, boundary crisis
Published in DKUM: 19.09.2012; Views: 3259; Downloads: 159
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Abstract: We describe a simple new method that provides instructive insights into the dynamics of chaotic time-continuous systems that yield strange attractors as solutions in the phase space. In particular, we show that the norm of the vector field component that is orthogonal to the trajectory is an excellent quantity for visualizing the attraction of strange attractors, thus promoting the understanding of their formation and overall structure. Furthermore, based on the existence of zero orthogonal field strengths in planes that form low-dimensional strange attractors, we also provide an innovative explanation for the origin of chaotic behaviour. For instructive purposes, we first apply the method to a simple limit cycle attractor, and then analyse two paradigmatic mathematical models for classical time-continuous chaos. To facilitate the use of our method in graduate as well as undergraduate courses, we also provide user-friendly programs in which the presented theory is implemented.
Keywords: dynamic systems, chaotic systems, nonlinear dynamics, attractors, strange attractors
Published in DKUM: 07.06.2012; Views: 2441; Downloads: 99
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Abstract: We show that microeconomic chaotic variations of payoffs in the prisoner's dilemma game maintain cooperation over a broad range of defection temptation values where otherwise economic stalemate reigns. Thus, unpredictability at micro scales impedes mutual defection that inflicts social poverty.
Keywords: chaotic systems, game theory, microeconomic chaos, cooperation, social welfare
Published in DKUM: 07.06.2012; Views: 1917; Downloads: 161
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Abstract: We show that Lévy distributed variations of payoffs in the prisoner's dilemma game impede cooperation as the frequency of rare events increases. Lévy flights thus facilitate defection, but also uphold the evolutionary process, arguably maintaining a healthy level of competitiveness amongst the agents.
Keywords: chaotic systems, game theory, Lévy process, cooperation, social welfare
Published in DKUM: 07.06.2012; Views: 1953; Downloads: 164
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