Aspects of diffusion in the stadium billiard

Lozej, Črt; Robnik, Marko

doi:10.1103/physreve.97.012206

Cited by 13 publications

(30 citation statements)

References 38 publications

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“…This paper is a continuation of our previous recent paper [1] on classical and quantum ergodic billiard (B = 0.5) with strong stickiness effects, from the family of lemon billiards introduced by Heller and Tomsovic in 1993 [2]. In the present paper we study lemon billiards [3] with the shape parameters B = 0.42, 0.55, 0.6, which are mixed-type billiards without stickiness regions and thus serve as ideal examples of systems with a simple divided phase space.…”

Section: Introductionsupporting

confidence: 56%

“…Due to the two kinks the Lazutkin invariant tori (related to the boundary glancing orbits) do not exist. The period-2 orbit connecting the centers of the two circular arcs at the positions (1 − B, 0) and (−1 + B, 0) is always stable (and therefore surrounded by a regular island) except for the case B = 1/2, where it is a marginaly unstable orbit (MUPO), the case being ergodic and treated in our previous paper [1]. L is the circumference of the entire billiard given by…”

Section: The Definition Of the Lemon Billiards And Their Classical Dynamical Propertiesmentioning

confidence: 99%

“…The family of lemon billiards was introduced by Heller and Tomsovic in 1993 [2], and has been studied in a number of works [8][9][10][11][12], most recently by Lozej [3] and Bunimovich et al [13], and in our recent work [1]. The lemon billiard boundary is defined by the intersection of two circles of equal unit radius with the distance between their center 2B being less than their diameters and B ∈ (0, 1), and is given by the following implicit equations in Cartesian coordinates…”

Section: The Definition Of the Lemon Billiards And Their Classical Dynamical Propertiesmentioning

confidence: 99%

“…As is well known the localization measures of a number of consecutive eigenstates over a certain energy interval display a distribution P (A). In classically ergodic systems with no stickiness A obeys the beta distribution [30,31], while in mixed-type systems [29] it has a nonuniversal distribution with typically two peaks, as well as also in ergodic systems with strong stickiness [1]. The beta distribution is…”

Section: The Statistical Properties Of the Entropy Localization Measure Of Ph Functionsmentioning

confidence: 99%

“…For a general introduction to the subjects in quantum chaos related to our work the reader is referred to the previous paper [1]. Here we only mention the introductions to the general quantum chaos in the books by Stöckmann [4] and Haake [5], and the recent review papers on the stationary quantum chaos in generic (mixed-type) systems [6,7].…”

Section: Introductionmentioning

confidence: 99%

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Classical and Quantum Mixed-Type Lemon Billiards without Stickiness

Lozej

Lukman

Robnik

2021

Nonlinear Phenomena in Complex Systems

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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 in Phys. Today 46 38 (1993). This paper is a continuation of our recent paper on classical and quantum ergodic lemon billiard (B = 0:5) with strong stickiness effects published in Phys. Rev. E 103 012204 (2021). Here we study the classical and quantum lemon billiards, for the cases B = 0:42; 0:55; 0:6, which are mixed-type billiards without stickiness regions and thus serve as ideal examples of systems with simple divided phase space. The classical phase portraits show the structure of one large chaotic sea with uniform chaoticity (no stickiness regions) surrounding a large regular island with almost no further substructure, being entirely covered by invariant tori. The boundary between the chaotic sea and the regular island is smooth, except for a few points. The classical transport time is estimated to be very short (just a few collisions), therefore the localization of the chaotic eigenstates is rather weak. The quantum states are characterized by the following universal properties of mixed-type systems without stickiness in the chaotic regions: (i) Using the Poincare-Husimi (PH) functions the eigenstates are separated to the regular ones and chaotic ones. The regular eigenenergies obey the Poissonian statistics, while the chaotic ones exhibit the Brody distribution with various values of the level repulsion exponent β, its value depending on the strength of the localization of the chaotic eigenstates. Consequently, the total spectrum is well described by the Berry-Robnik-Brody (BRB) distribution. (ii) The entropy localization measure A (also the normalized inverse participation ratio) has a bimodal universal distribution, where the narrow peak at small A encompasses the regular eigenstates, theoretically well understood, while the peak at larger A comprises the chaotic eigenstates, and is well described by the beta distribution. (iii) Thus the BRB energy level spacing distribution captures two effects: the divided phase space dictated by the classical Berry-Robnik parameter ρ2 measuring the relative size of the largest chaotic region, in agreement with the Berry-Robnik picture, and the localization of chaotic PH functions characterized by the level repulsion (Brody) parameter β. (iv) Examination of the PH functions shows that they are supported either on the classical invariant tori in the regular islands or on the chaotic sea, where they are only weakly localized. With increasing energy the localization of chaotic states decreases, as the PH functions tend towards uniform spreading over the classical chaotic region, and correspondingly β tends to 1.

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Section: Introductionsupporting

confidence: 56%

Section: The Definition Of the Lemon Billiards And Their Classical Dynamical Propertiesmentioning

confidence: 99%