Time-reversal-invariant hexagonal billiards with a point symmetry

Lima, Tiago Araújo; Carmo, Ricardo Batista do; Terto, K; Aguiar, F. M. de

doi:10.1103/physreve.104.064211

Cited by 2 publications

(9 citation statements)

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“…These facts are in agreement with the intermediate statistics well fitted with eq. ( 6) as in [23,27,40,42]. The same occurs for the singlets in the E-C 3 B family, where the condition k max k c is equivalent to the 70,000th level.…”

Section: A the Singlets Casementioning

confidence: 87%

“…Quantitative characterization of these mixed-phase spaces can be made through the chaotic (regular) fraction ρ c (ρ r ) of each phase portrait with ρ c + ρ r = 1 and 0 ρ c 1. The phase plane is partitioned into N c small disjoint cells to measure these quantities [27,29,[35][36][37]. For a given orbit, let N (n) be the number of different cells in the phase space, which are visited up to n impacts in the cross-section.…”

Section: The Bi-parametric Billiards Families and Classical Dynamicsmentioning

confidence: 99%

“…Thus, it is necessary to use new intermediate formulas to study the distribution of doublets in billiards with mixed classical phase space. The following formulas we derived in [27]. Following the same steps in [39] led to the eq.…”

Section: B the Doublets Casementioning

confidence: 99%

“…The BT [20], BGS [21][22][23] and LSS conjectures have been investigated in the literature, but there have been comparatively fewer studies on the LSS findings [24][25][26][27]. Until now, little has been said about the situations of C 3 symmetric billiards with mixed classical phase space, where chaotic sea and stable KAM-islands coexist.…”

Section: Introductionmentioning

confidence: 99%

“…We start the analysis by presenting the billiards, discussing their classical dynamics, showing some mixed phase spaces, and calculating the fraction of the chaotic sea on these phase spaces. Finally, we follow with the quantized billiards' spectral properties, investigating the nns distribution p(s) with formulas for intermediate quantum statistics derived for the doublets recently [27].…”

Section: Introductionmentioning

confidence: 99%

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Classical and Quantum Elliptical Billiards: Mixed Phase Space and Short Correlations in Singlets and Doublets

Araújo¹

2023

Preprint

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Billiards are flat cavities where a particle is free to move between elastic collisions with the boundary. In chaos theory these systems are simple prototypes, their conservative dynamics of a billiard may vary from regular to chaotic, depending only on the border. The results reported here seek to shed light on the quantization of classically chaotic systems. We present numerical results on classical and quantum properties in two bi-parametric families of Billiards, Elliptical Stadium Billiard (ESB) and Elliptical-C3 Billiards (E-C3B). Both are elliptical perturbations of chaotic billiards with originally circular sectors on their borders. Our numerical calculations show evidence that the elliptical families can present a mixed classical phase space, identified by a parameter ρc < 1, which we use to guide our analysis of quantum spectra. We explored the short correlations through nearest neighbor spacing distribution p(s), which showed that in the mixed region of the classical phase space, p(s) is well described by the Berry-Robnik-Brody (BRB) distributions for the ESB. In agreement with the expected from the so-called ergodic parameter α = tH/tT, the ratio between the Heisenberg time and the classical diffusive-like transport time signals the possibility of quantum dynamical localization when α < 1. For the E-C3B family, the eigenstates can be split into singlets and doublets. BRB describes p(s) for singlets as the previous family in the mixed region. However, the p(s) for doublets are described by new distributions recently introduced in the literature but only tested in a few cases for ρc < 1. We observed that as ρc decreases, the p(s)'s tend to move away simultaneously from the GOE (singlets) and GUE (doublets) distributions.

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Section: A the Singlets Casementioning

confidence: 87%