The distribution of localization measures of chaotic eigenstates in the stadium billiard

Batistić, Benjamin; Lozej, Črt; Robnik, Marko

doi:10.33581/1561-4085-2020-23-1-17-32

Cited by 17 publications

(25 citation statements)

References 41 publications

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“…There are three main localization measures: A, the information entropy measure, C the correlation localization measure, and nIP R the normalized inverse participation ratio. As recently shown [39,41,42], they are all proportional to each other (linearly related) and thus equivalent. The energy spectra of the localized chaotic eigenstates can be well described by the fractional power law level repulsion, P (S) ∝ S β , for small S, and β ∈ [0, 1]: β = 0 corresponds to the maximal localization and Poissonian statistics, while β = 1 corresponds to the maximal extendedness (delocalization) and the RMT statistics.…”

Section: Introduction and Overviewmentioning

confidence: 80%

“…Namely, it turns out that A has a distribution, described by a probability density. If the classical chaotic region is uniform in the sense of uniform recurrence times, the distribution is universally the beta distribution, as recently shown [41,42]. On the other hand, if the recurrence times over the chaotic region vary widely, due to the stickiness effects, the distribution becomes nonuniversal, being specific of the classical phase space structure and the dynamical properties.…”

Section: Introduction and Overviewmentioning

confidence: 91%

“…The billiard system is ergodic and chaotic at any nonzero value of ε, but the typical time for a chaotic trajectory to fill the entire phase space depends strongly on ε. The details of the diffusion-like motion have been recently studied by Lozej and Robnik [51] and its consequences for quantum chaos in reference [52]. If ε = 1 this time, called diffusion time, or transport time t T , is of order unity, while for very small ε it becomes very large.…”

Section: Quantum Chaos Of Classically Fully Chaotic (Ergodic) Systemsmentioning

confidence: 99%

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A Brief Introduction to Stationary Quantum Chaos in Generic Systems

Robnik

2020

Nonlinear Phenomena in Complex Systems

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We review the basic aspects of quantum chaos (wave chaos) in mixed-type Hamiltonian systems with divided phase space, where regular regions containing the invariant tori coexist with the chaotic regions. The quantum evolution of classically chaotic bound systems does not possess the sensitive dependence on initial conditions, and thus no chaotic behaviour occurs, as the motion is always almost periodic. However, the study of the stationary solutions of the Schrödinger equation in the quantum phase space (Wigner functions or Husimi functions) reveals precise analogy of the structure of the classical phase portrait. In classically integrable regions the spectral (energy) statistics is Poissonian, while in the ergodic chaotic regions the random matrix theory applies. If we have the mixed-type classical phase space, in the semiclassical limit (short wavelength approximation) the spectrum is composed of Poissonian level sequence supported by the regular part of the phase space, and chaotic sequences supported by classically chaotic regions, being statistically independent of each other, as described by the Berry-Robnik distribution. In quantum systems with discrete energy spectrum the Heisenberg time tH = πℏ/ΔE, where ΔE is the mean level spacing (inverse energy level density), is an important time scale. The classical transport time scale tT (transport time) in relation to the Heisenberg time scale tH (their ratio is the parameter α = tH / tT ) determines the degree of localization of the chaotic eigenstates, whose measure A is based on the information entropy. We show that A is linearly related to the normalized inverse participation ratio. We study the structure of quantum localized chaotic eigenstates (their Wigner and Husimi functions) and the distribution of localization measure A. The latter one is well described by the beta distribution, if there are no sticky regions in the classical phase space. Otherwise, they have a complex nonuniversal structure. We show that the localized chaotic states display the fractional power-law repulsion between the nearest energy levels in the sense that the probability density (level spacing distribution) to find successive levels on a distance S goes like ∝ S β for small S , where 0 ≤ β ≤ 1, and β = 1 corresponds to completely extended states, and β = 0 to the maximally localized states. β goes from 0 to 1 when α goes from 0 to ∞, β is a function of <A>, as demonstrated in the quantum kicked rotator, the stadium billiard, and a mixed-type billiard.

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Section: Introduction and Overviewmentioning

confidence: 80%

Section: Introduction and Overviewmentioning

confidence: 91%

Section: Quantum Chaos Of Classically Fully Chaotic (Ergodic) Systemsmentioning

confidence: 99%

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A Brief Introduction to Stationary Quantum Chaos in Generic Systems

Robnik

2020

Nonlinear Phenomena in Complex Systems

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“…As is well known, there are at least three localization measures of chaotic eigenstates, namely the entropy localization measure A, the correlation localization measure C, and the normalized inverse participation ratio R = nIP R. They have been found to be linearly related and equivalent [26][27][28][29][30] The entropy localization measure of a single eigenstate H m (q, p), denoted by A m is defined as…”

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

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%

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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Statistical properties of the localization measure of chaotic eigenstates in the Dicke model

Wang

Robnik

2020

Phys. Rev. E

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The distribution of localization measures of chaotic eigenstates in the stadium billiard

Cited by 17 publications

References 41 publications

A Brief Introduction to Stationary Quantum Chaos in Generic Systems

A Brief Introduction to Stationary Quantum Chaos in Generic Systems

Classical and Quantum Mixed-Type Lemon Billiards without Stickiness

Statistical properties of the localization measure of chaotic eigenstates in the Dicke model

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