The Transition to Chaos

The classical and quantum dynamics of conservative systems governs the behavior of much of the world around us - from the dynamics of galaxies to the vibration and electronic behavior of molecules and the dynamics of systems formed from or driven by laser radiation. Most conservative dynamical systems contain some degree of chaotic behavior, ranging from a self-similar mixture of regular and chaotic motion, to fully developed chaos. This chaotic behavior has a profound effect on the dynamics.

This book combines mathematical rigor with examples that illuminate the dynamical theory of chaotic systems. The emphasis of the 3rd Edition is on topics of modern interest, including scattering systems formed from molecules and nanoscale quantum devices, quantum control and destabilization of systems driven by laser radiation, and thermalization of condensed matter systems. The book is written on a level accessible to graduate students and to the general research community.

Auteur

Linda E. Reichl, Ph.D., is a Professor of Physics at the University of Texas at Austin and is co-Director of the Center for Complex Quantum Systems. She has authored numerous papers on classical and quantum chaos theory and is author of the textbook "A Modern Course in Statistical Physics", now in its fourth edition. She was elected a Fellow of the American Physical Society in 2000 for "original work in the field of quantum chaos".

Résumé
Based on courses given at the universities of Texas in Austin, and California in San Diego, this book treats an active fields of research that touches upon the foundations of physics and chemistry. It presents, in as simple a manner as possible, the basic mechanisms that determine the dynamical evolution of both classical and quantum systems in sufficient generality to include quantum phenomena. The book begins with a discussion of Noether's theorem, integrability, KAM theory, and a definition of chaotic behavior; it continues with a detailed discussion of area-preserving maps, integrable quantum systems, spectral properties, path integrals, and periodically driven systems; and it concludes by showing how to apply the ideas to stochastic systems. The presentation is complete and self-contained; appendices provide much of the needed mathematical background, and there are extensive references to the current literature. Problems at the ends of chapters help students clarify their understanding.In this new edition, the presentation will be brought up to date throughout, and a new chapter on open quantum systems will be added.

Contenu

1. Overview
2. Fundamental Concepts nonlinear resonance (classical and quantum), symmetries, integrability, definition of chaos, KAM tori, Lyapunov exponent, quantum signatures of chaos.3. Classical Chaos as seen in area preserving maps including twist, non-twist, and scattering maps, maps on NHIMs. Quantum analog of area preserving maps, Husimi functions.4. Global properties of 2DofF (two degrees of freedom) chaotic systems, resonance networks (classical and quantum), unstable manifolds, fixed points. Applications to galaxies, waveguides, molecules5. Global properties of systems with 3 or more DoF. Arnold diffusion in a variety of systems.6. Semiclassical Theory Path Integrals.7. Manifestations of chaos in scattering processes (classical and quantum) with application to molecules, wave guides, nanometer scale quantum devices, condensed matter systems.8. Time periodically driven systems. dynamic Anderson localization, Arnold diffusion, Floquet theory, thermalization, quantum control, scattering channels, quantum renormalization.9. Stochastic manifestations of chaos.Appendices: Random matrix theory, Symmetries and invariant measures.