Breakdown of the Quantum Distinction of Regular and Chaotic Classical Dynamics in Dissipative Systems

A new measure for statistical properties of the wave function components of quantum systems, the distribution of the product of two partial widths, is introduced. It is tested with data obtained in analog experiments with microwave billiards, where the product of two partial widths equals the resonance strengths in the microwave spectra. The billiards are from the family of the Limaçons, one with chaotic and two with mixed classical dynamics. For completely chaotic systems the partial widths generically obey a Porter-Thomas distribution. We show that in this case the distribution of their product equals a K0 distribution. While we find deviations of the experimental strength distribution from the K0 distribution for the billiards with mixed dynamics, the distributions agree perfectly for the chaotic billiard, when taking into account the experimental threshold of detection in the theoretical description. Hence, the strength distribution provides another stringent test for the connection between statistical properties of systems with classical chaotic dynamics and random matrix theory.

Since the first attempts to calculate the helium ground state in the early days of Bohr-Sommerfeld quantization, two-electron atoms have posed a series of unexpected challenges to theoretical physics. Despite the seemingly simple problem of three charged particles with known interactions, it took more than half a century after quantum mechanics was established to describe the spectra of two-electron atoms satisfactorily. The evolution of the understanding of correlated two- electron dynamics and its importance for doubly excited resonance states is presented here, with an emphasis on the concepts introduced. The authors begin by reviewing the historical development and summarizing the progress in measuring the spectra of two-electron atoms and in calculating them by solving the corresponding Schrödinger equation numerically. They devote the second part of the review to approximate quantum methods, in particular adiabatic and group- theoretical approaches. These methods explain and predict the striking regularities of two-electron resonance spectra, including propensity rules for decay and dipole transitions of resonant states. This progress was made possible through the identification of approximate dynamical symmetries leading to corresponding collective quantum numbers for correlated electron-pair dynamics. The quantum numbers are very different from the independent particle classification, suitable for low-lying states in atomic systems. The third section of the review describes modern semiclassical concepts and their application to two-electron atoms. Simple interpretations of the approximate quantum numbers and propensity rules can be given in terms of a few key periodic orbits of the classical three-body problem. This includes the puzzling existence of Rydberg series for electron-pair motion. Qualitative and quantitative semiclassical estimates for doubly excited states are obtained for both regular and chaotic classical two-electron dynamics using modern semiclassical techniques. These techniques set the stage for a theoretical investigation of the regime of extreme excitation towards the three- body breakup threshold. Together with periodic orbit spectroscopy, they supply new tools for the analysis of complex experimental spectra.

Quantization of a generic chaotic 3D billiard with smooth boundary. I. Energy level statistics

Antidot arrays consisting of a lattice of nanometer scale holes etched through a two-dimensional electron gas display distinct resistance anomalies which reflect the complex electron motion in the lattice. While in conventional conductors (at low temperatures) the electrons are predominantly scattered by randomly distributed defects the situation is quite different in antidot arrays: here, the electrons predominantly collide with the periodically arranged antidots. The electrons in the lattice probe a potential landscape which can be viewed as a physical realization of a Sinai type of billiard, known for its chaotic classical dynamics.

Complex and quantized electron motion in antidot arrays

Random matrix theory is used to show that the proximity to a superconductoropens a gap in the excitation spectrum of an electron gas confined to abilliard with a chaotic classical dynamics. In contrast, a gaplessspectrum is obtained for a non-chaotic rectangular billiard, and it isargued that this is generic for integrable systems.

The purpose of this article is a brief overview of the results of research on the use of noise-like signals in broadband radio systems conducted under the leadership of Yuri Vasilyevich Gulyaev. Methods. The conducted research was based on the previous experience of the V.A. Kotelnikov IRE RAS research team related to the development of analog noise-like devices (shumotrons) based on the concept of dynamic chaos. The continuation of these studies was associated with the development of digital chaos based on integer generating algorithms that could be easily reproduced on any digital technology. Results. Promising directions of using information technologies using dynamic chaos for the transmission, processing, storage and protection of information are considered. Broadband information transmission systems using complex signals with a large base, built on the basis of systems with chaotic dynamics, are presented. Finite-dimensional mathematical algorithms for calculating chaotic signals by reconstructing nonlinear dynamics in dissipative systems with a delay are proposed. Conclusion. It is shown that a digital information transmission system with spectrum expansion and dynamic change of chaotic codes has high noise immunity, secrecy, electromagnetic compatibility and ensures reliable and confidential transmission of messages in a complex electromagnetic environment. Schemes for masking, protecting, processing, and transmitting information are implemented based on original chaotic algorithms. An experimental study of the noise radar layout in laboratory conditions demonstrated a sufficiently high accuracy of radar range measurements over the entire measurement range with dual spectral signal processing, as well as a high range resolution of 15 cm (with an effective bandwidth of 800- 900 MHz).

We determine numerically the full complex Lyapunov spectrum of SU(2) Yang-Mills fields on a 3-dimensional lattice from the classical chaotic dynamics. The equation of state, S(E), is determined from the Kolmogorov-Sinai entropy extrapolated to the large size limit.

Quantum chaotic transport in double barrier tunnel structures

We discuss application of contemporary methods of the theory of dynamical systems with regular and chaotic hyperbolic dynamics to investigation of topological structure of magnetic fields in conducting media. For substantial classes of magnetic fields, we consider well-known physical models allowing us to reduce investigation of such fields to study of vector fields and Morse-Smale diffeomorphisms as well as diffeomorphisms with nontrivial basic sets satisfying the A axiom introduced by Smale. For the point-charge magnetic field model, we consider the problem of separator playing an important role in the reconnection processes and investigate relations between its singularities. We consider the class of magnetic fields in the solar corona and solve the problem of topological equivalency of fields in this class. We develop a topological modification of the Zeldovich funicular model of the nondissipative cinematic dynamo, constructing a hyperbolic diffeomorphism with chaotic dynamics that is conservative in the neighborhood of its transitive invariant set.

The response of dissipative systems to multi-chromatic fields exhibits generic properties which follow from the discrete time-translation symmetry of each driving component. We derive these properties and illustrate them with paradigmatic examples of classical and quantum dissipative systems. In addition, some computational aspects, in particular a matrix continued-fraction method, are discussed. Moreover, we propose possible implementations with quantum optical settings.

Isoscalar and isovector monopole oscillations that correspond to giant resonances in spherical nuclei are described in the framework of time-dependent relativistic mean-field theory. Time-dependent and self-consistent calculations that reproduce experimental data on monopole resonances in 208Pb, show that the motion of the collective coordinate is regular for isoscalar oscillations, and that it becomes chaotic when initial conditions correspond to the isovector mode. Regular collective dynamics coexists with chaotic oscillations on the microscopic level. Time histories, Fourier spectra, state-space plots, Poincare sections, autocorrelation functions, and Lyapunov exponents are used to characterize the nonlinear system and to identify chaotic oscillations. Analogous considerations apply to higher multipolarities.

Autoparametric Vibrations of a Nonlinear System with a Pendulum and Magnetorheological Damping, by Jerzy Warminski and Krzysztof Kecik: Introduction to Autoparametric Vibrations. - Model of the Nonlinear Oscillator with an Attached Pendulum. - Approximate Analytical Solutions and Their Stability. - Regular and Chaotic Dynamics Near the Main Parametric Resonance. - Experimental Setup of the System with Active MR Damper. - Dynamics of an Autoparametric System with MR Damper. - Influence of a Nonlinear System's Suspension on the InstabilityRegions. - Conclusions and Remarks. - References. - On the Dynamics of Pedestrians-Induced Lateral Vibrations of Footbridges, by Stefano Lenci and Laura Marcheggiani: Introduction and Literature Review. - A Continuous-Time Model: the SAMEO Model. - A Discrete-Time Model. - Conclusions. - References. - Applications for Shape Memory Alloys in Structural and Machine Dynamics, by Matthew P. Cartmell, Arkadiusz J. Z ak and Olga A. Ganilova: Review of the Literature and Introduction. - Modelling of the Shape Memory Effect. - Dynamics of Composite Beams and Plates with Integrated SMA Elements. - Applications to Flexible Rotors. - Antagonistic Actuation Control of Vibration in Plates. - Conclusions. - References. - Theoretical and Experimental Nonlinear Vibrations of Sagged Elastic Cables, by Giuseppe Rega: Introduction. - Cable Modelling and Theoretical Analysis. - Nonlinear Phenomena in Forced Dynamic Response. - Experimental Characterisation of Cable Nonlinear Dynamics. - Further Developments and Research Topics. - References. - Engineering Applications of Non-Smooth Dynamics, by Marian Wiercigroch and Ekaterina Pavlovskaia Non-Smooth Dynamical Systems in Engineering Dynamics. - Drifting Oscillator as an Effective Model of Vibro-Impact Moling. - Nonlinear Dynamics Caused by Fatigue Crack Growth. - Regular and Chaotic Dynamics of a Rotor System with a Bearing Clearance. - Conclusions. - References.

A method for constructing general null Lagrangians and their higher harmonics is presented for dynamical systems with one degree of freedom. It is shown that these Lagrangians can be used to obtain non-standard Lagrangians, which give equations of motion for the law of inertia and some dissipative dynamical systems. The necessary condition for deriving equations of motion by using null Lagrangians is presented, and it is demonstrated that this condition plays the same role for null Lagrangians as the Euler–Lagrange equation plays for standard and non-standard Lagrangians. The obtained results and their applications establish a novel role of null Lagrangians in classical dynamics.

Effect of demagnetization field on chaotic and deterministic nonlinear magnetization dynamics regimes formation in the presence of oscillating and constant external magnetic fields is studied using the Landau–Lifshitz–Gilbert approach. The uniformly magnetized sample is considered to be an axially symmetric particle described by demagnetization factors and to have uniaxial crystallographic anisotropy formed some angle with an applied field direction. The equations of magnetic moment motion are considered to be the synergetic equations and the system behavior is examined on the phase plane. It is investigated as to how the change in particle shape and its orientation with respect to the external field (system’s configuration) can cause the transition between chaos and regular magnetization dynamics. Moreover, the way to completely suppress chaotic dynamics of magnetic moments in such sample is proposed. To produce a regular study, all results are presented in the form of bifurcation diagrams for all sufficient dynamics regimes of the considered system. Our results suggest that varying the particle’s shape and fields’ geometry may provide a useful way of magnetization dynamics control in complex magnetic systems.