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Surface billiards and Nonvanishing Lyapunov Exponents

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Acta Mathematica Sinica,English Series 1997年第1期13卷 21-34页

作者： Cheng Jian (Department of Mathematics,Nanjing University,Nanjing 210008,China) 南京大学数学系江苏南京

We study in this paper the billiards on surfaces with mix-valued Gaussian curvature and the condition which gives nonvanishing Lyapunov exponents of the system.We introduce a criterion upon which a small perturbation of the surface will also produce a system with positive Lyapunov exponents.Some examples of such surfaces are given in this article.

关键词： billiards Lyapunov exponents Curvature

Experimental investigation of the fluctuations in nonchaotic scattering in microwave billiards

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Chinese Physics B 2019年第10期28卷 160-171页

作者：张润祖张为华 Barbara Dietz 柴国志黄亮 School of Physical Science and Technology Lanzhou UniversityLanzhou 730000China Key Laboratory for Magnetism and Magnetic Materials of MOE Lanzhou UniversityLanzhou 730000China

We report on the experimental investigation of the properties of the eigenvalues and wavefunctions and the fluctuation properties of the scattering matrix of closed and open billiards, respectively, of which the classical dynamics undergoes a transition from integrable via almost integrable to fully chaotic. To realize such a system, we chose a billiard with a 60° sector shape of which the classical dynamics is integrable, and introduced circular scatterers of varying number, size,and position. The spectral properties of generic quantum systems of which the classical counterpart is either integrable or chaotic are universal and well understood. If, however, the classical dynamics is pseudo-integrable or almost-integrable,they exhibit a non-universal intermediate statistics, for which analytical results are known only in a few cases, e.g., if it corresponds to semi-Poisson statistics. Since the latter is, above all, clearly distinguishable from those of integrable and chaotic systems, our aim was to design a billiard with these features which indeed is achievable by adding just one scatterer of appropriate size and position to the sector billiard. We demonstrated that, while the spectral properties of almostintegrable billiards are sensitive to the classical dynamics, this is not the case for the distribution of the wavefunction components, which was analyzed in terms of the strength distribution, and the fluctuation properties of the scattering matrix which coincide with those of typical, fully chaotic systems.

关键词： wave chaos quantum billiards microwave billiards random matrix theory

Perspectives on relativistic quantum chaos

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Communications in Theoretical Physics 2020年第4期72卷 112-137页

作者： Liang Huang Ying-Cheng Lai School of Physical Science and Technology and Key Laboratory for Magnetism and Magnetic Materials of MOELanzhou UniversityLanzhouGansu 730000China School of Electrical Computerand Energy EngineeringArizona State UniversityTempeAZ 85287United States of America Department of Physics Arizona State UniversityTempeAZ 85287United States of America

Quantum Chaos has been investigated for about a half century.It is an old yet vigorous interdisciplinary field with new concepts and interesting topics emerging constantly.Recent years have witnessed a growing interest in quantum chaos in relativistic quantum systems,leading to the still developing field of relativistic quantum chaos.The purpose of this paper is not to provide a thorough review of this area,but rather to outline the basics and introduce the key concepts and methods in a concise way.A few representative topics are discussed,which may help the readers to quickly grasp the essentials of relativistic quantum chaos.A brief overview of the general topics in quantum chaos has also been provided with rich references.

关键词： quantum chaos relativistic quantum systems Dirac billiards

Universality in Statistical Measures of Trajectories in Classical Billiard Systems

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Applied Mathematics 2015年第8期6卷 1407-1425页

作者： Jean-Franç ois Laprise Ahmad Hosseinizadeh Helmut Krö ger Department of Physics Laval University Qu&eacute bec Canada

For classical billiards, we suggest that a matrix of action or length of trajectories in conjunction with statistical measures, level spacing distribution and spectral rigidity, can be used to distinguish chaotic from integrable systems. As examples of 2D chaotic billiards, we considered the Bunimovich stadium billiard and the Sinai billiard. In the level spacing distribution and spectral rigidity, we found GOE behaviour consistent with predictions from random matrix theory. We studied transport properties and computed a diffusion coefficient. For the Sinai billiard, we found normal diffusion, while the stadium billiard showed anomalous diffusion behaviour. As example of a 2D integrable billiard, we considered the rectangular billiard. We found very rigid behaviour with strongly correlated spectra similar to a Dirac comb. These findings present numerical evidence for universality in level spacing fluctuations to hold in classically integrable systems and in classically fully chaotic systems.

关键词： Classical Chaos Dynamical billiards Random Matrix Theory Level Spacing Fluctuations Universality