scholarly journals Quantum Chaos in Mixed Phase Space and the Julia Set

2003 ◽  
Vol 150 ◽  
pp. 267-280 ◽  
Author(s):  
Akira Shudo ◽  
Kensuke S. Ikeda
Keyword(s):  
Phase Space ◽  
Quantum Chaos ◽  
Julia Set ◽  
Mixed Phase

scholarly journals Illustration of microphysical processes in Amazonian deep convective clouds in the Gamma phase space: Introduction and potential applications

2017 ◽  
Author(s):  
Micael A. Cecchini ◽  
Luiz A. T. Machado ◽  
Manfred Wendisch ◽  
Anja Costa ◽  
Martina Krämer ◽  
...  
Keyword(s):  
Phase Space ◽  
Cloud Droplet ◽  
Mixed Phase ◽  
Relative Dispersion ◽  
Gamma Phase ◽  
Liquid Droplets ◽  
Phase Layer ◽  
Cloud Properties ◽  
Common Framework ◽  
Space Approach

Abstract. The behavior of tropical clouds remains a major open scientific question, given that the associated phys-ics is not well represented by models. One challenge is to realistically reproduce cloud droplet size dis-tributions (DSD) and their evolution over time and space. Many applications, not limited to models, use the Gamma function to represent DSDs. However, there is almost no study dedicated to understanding the phase space of this function, which is given by the three parameters that define the DSD intercept, shape, and curvature. Gamma phase space may provide a common framework for parameterizations and inter-comparisons. Here, we introduce the phase-space approach and its characteristics, focusing on warm-phase microphysical cloud properties and the transition to the mixed-phase layer. We show that trajectories in this phase space can represent DSD evolution and can be related to growth processes. Condensational and collisional growth may be interpreted as pseudo-forces that induce displacements in opposite directions within the phase space. The actually observed movements in the phase space are a result of the combination of such pseudo-forces. Additionally, aerosol effects can be evaluated given their significant impact on DSDs. The DSDs associated with liquid droplets that favor cloud glaciation can be delimited in the phase space, which can help models to adequately predict the transition to the mixed phase. We also consider possible ways to constrain the DSD in two-moment bulk microphysics schemes, where the relative dispersion parameter of the DSD can play a significant role. Overall, the Gamma phase-space approach can be an invaluable tool for studying cloud microphysical evolution and can be readily applied in many scenarios that rely on Gamma DSDs.

Quantum chaotic scattering with a mixed phase space: The three-disk billiard in a magnetic field

2000 ◽  
Vol 61 (1) ◽  
pp. 382-389 ◽  
Author(s):  
Markus Eichengrün ◽  
Walter Schirmacher ◽  
Wolfgang Bregmann
Keyword(s):  
Magnetic Field ◽  
Phase Space ◽  
Mixed Phase ◽  
Chaotic Scattering

scholarly journals Semiclassical Spectra from Periodic-Orbit Clusters in a Mixed Phase Space

1997 ◽  
Vol 79 (6) ◽  
pp. 1022-1025 ◽  
Author(s):  
Henning Schomerus ◽  
Fritz Haake
Keyword(s):  
Phase Space ◽  
Periodic Orbit ◽  
Mixed Phase

scholarly journals Effective dynamics in Hamiltonian systems with mixed phase space

2005 ◽  
Vol 71 (3) ◽  
Author(s):  
Adilson E. Motter ◽  
Alessandro P. S. de Moura ◽  
Celso Grebogi ◽  
Holger Kantz
Keyword(s):  
Phase Space ◽  
Hamiltonian Systems ◽  
Mixed Phase ◽  
Effective Dynamics

Tilted-hat mushroom billiards: Web-like hierarchical mixed phase space

2020 ◽  
Vol 91 ◽  
pp. 105440
Author(s):  
Diogo Ricardo da Costa ◽  
Matheus S. Palmero ◽  
J.A. Méndez-Bermúdez ◽  
Kelly C. Iarosz ◽  
José D. Szezech Jr ◽  
...  
Keyword(s):  
Phase Space ◽  
Mixed Phase

scholarly journals Diffusion phenomena in a mixed phase space

2020 ◽  
Vol 30 (1) ◽  
pp. 013108 ◽  
Author(s):  
Matheus S. Palmero ◽  
Gabriel I. Díaz ◽  
Peter V. E. McClintock ◽  
Edson D. Leonel
Keyword(s):  
Phase Space ◽  
Mixed Phase ◽  
Diffusion Phenomena

Semi-focusing billiards: ergodicity

2008 ◽  
Vol 28 (5) ◽  
pp. 1377-1417 ◽  
Author(s):  
LEONID A. BUNIMOVICH ◽  
GIANLUIGI DEL MAGNO
Keyword(s):  
Phase Space ◽  
Three Dimensional ◽  
Mixed Phase ◽  
Convex Domains ◽  
Math Phys ◽  
Standing Question

AbstractIn Bunimovich and Del Magno [Semi-focusing billiards: hyperbolicity. Comm. Math. Phys.262 (2006), 17–32], we proved that billiards in certain three-dimensional convex domains are hyperbolic. In this paper, we continue the study of these systems, and prove that they enjoy the Bernoulli property. This result answers affirmatively a long-standing question on the existence of ergodic billiards in convex domains in dimensions greater than two. Besides, it shows that the chaotic components of the first rigorously investigated three-dimensional billiards with mixed phase space (mushroom billiards), introduced in Bunimovich and Del Magno, are in fact Bernoulli.

scholarly journals A Brief Introduction to Stationary Quantum Chaos in Generic Systems

2020 ◽  
Vol 23 (2) ◽  
pp. 172-191 ◽  
Author(s):  
Marko Robnik
Keyword(s):  
Phase Space ◽  
Time Scale ◽  
Quantum Chaos ◽  
Mixed Type ◽  
Localized States ◽  
Quantum Evolution ◽  
Density Level ◽  
Level Spacing ◽  
Transport Time ◽  
Classical Phase Space

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.

Probing quantal dynamics of mixed phase space systems with noise

1993 ◽  
Vol 71 (18) ◽  
pp. 2895-2898 ◽  
Author(s):  
L. Sirko ◽  
M. R. W. Bellermann ◽  
A. Haffmans ◽  
P. M. Koch ◽  
D. Richards
Keyword(s):  
Phase Space ◽  
Mixed Phase ◽  
Space Systems
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