Classical Nonintegrability, Quantum Chaos
Title | Classical Nonintegrability, Quantum Chaos PDF eBook |
Author | Andreas Knauf |
Publisher | Birkhäuser |
Pages | 104 |
Release | 2012-12-06 |
Genre | Science |
ISBN | 3034889321 |
Our DMV Seminar on 'Classical Nonintegrability, Quantum Chaos' intended to introduce students and beginning researchers to the techniques applied in nonin tegrable classical and quantum dynamics. Several of these lectures are collected in this volume. The basic phenomenon of nonlinear dynamics is mixing in phase space, lead ing to a positive dynamical entropy and a loss of information about the initial state. The nonlinear motion in phase space gives rise to a linear action on phase space functions which in the case of iterated maps is given by a so-called transfer operator. Good mixing rates lead to a spectral gap for this operator. Similar to the use made of the Riemann zeta function in the investigation of the prime numbers, dynamical zeta functions are now being applied in nonlinear dynamics. In Chapter 2 V. Baladi first introduces dynamical zeta functions and transfer operators, illustrating and motivating these notions with a simple one-dimensional dynamical system. Then she presents a commented list of useful references, helping the newcomer to enter smoothly into this fast-developing field of research. Chapter 3 on irregular scattering and Chapter 4 on quantum chaos by A. Knauf deal with solutions of the Hamilton and the Schr6dinger equation. Scatter ing by a potential force tends to be irregular if three or more scattering centres are present, and a typical phenomenon is the occurrence of a Cantor set of bounded orbits. The presence of this set influences those scattering orbits which come close.
Classical Nonintegrability, Quantum Chaos
Title | Classical Nonintegrability, Quantum Chaos PDF eBook |
Author | Andreas Knauf |
Publisher | |
Pages | 107 |
Release | 1996 |
Genre | |
ISBN |
Dissipative Quantum Chaos and Decoherence
Title | Dissipative Quantum Chaos and Decoherence PDF eBook |
Author | Daniel Braun |
Publisher | Springer |
Pages | 139 |
Release | 2003-07-01 |
Genre | Science |
ISBN | 3540409165 |
This overview of the state of the art of research in an exciting field mainly emphasizes the development of a semiclassical formalism that allows one to incorporate the effect of dissipation and decoherence in a precise, yet tractable way into the quantum mechanics of classically chaotic systems.
Chaos in Classical and Quantum Mechanics
Title | Chaos in Classical and Quantum Mechanics PDF eBook |
Author | Martin C. Gutzwiller |
Publisher | Springer Science & Business Media |
Pages | 445 |
Release | 2013-11-27 |
Genre | Mathematics |
ISBN | 1461209838 |
Describes the chaos apparent in simple mechanical systems with the goal of elucidating the connections between classical and quantum mechanics. It develops the relevant ideas of the last two decades via geometric intuition rather than algebraic manipulation. The historical and cultural background against which these scientific developments have occurred is depicted, and realistic examples are discussed in detail. This book enables entry-level graduate students to tackle fresh problems in this rich field.
The Transition to Chaos
Title | The Transition to Chaos PDF eBook |
Author | Linda Reichl |
Publisher | Springer Science & Business Media |
Pages | 704 |
Release | 2004-05-13 |
Genre | Science |
ISBN | 9780387987880 |
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.
The Transition to Chaos
Title | The Transition to Chaos PDF eBook |
Author | Linda Reichl |
Publisher | Springer Science & Business Media |
Pages | 566 |
Release | 2013-04-17 |
Genre | Science |
ISBN | 1475743521 |
resonances. Nonlinear resonances cause divergences in conventional perturbation expansions. This occurs because nonlinear resonances cause a topological change locally in the structure of the phase space and simple perturbation theory is not adequate to deal with such topological changes. In Sect. (2.3), we introduce the concept of integrability. A sys tem is integrable if it has as many global constants of the motion as degrees of freedom. The connection between global symmetries and global constants of motion was first proven for dynamical systems by Noether [Noether 1918]. We will give a simple derivation of Noether's theorem in Sect. (2.3). As we shall see in more detail in Chapter 5, are whole classes of systems which are now known to be inte there grable due to methods developed for soliton physics. In Sect. (2.3), we illustrate these methods for the simple three-body Toda lattice. It is usually impossible to tell if a system is integrable or not just by looking at the equations of motion. The Poincare surface of section provides a very useful numerical tool for testing for integrability and will be used throughout the remainder of this book. We will illustrate the use of the Poincare surface of section for classic model of Henon and Heiles [Henon and Heiles 1964].
Quantum Chaos
Title | Quantum Chaos PDF eBook |
Author | Katsuhiro Nakamura |
Publisher | CUP Archive |
Pages | 228 |
Release | 1994-06-02 |
Genre | Mathematics |
ISBN | 9780521467469 |
Past studies on chaos have been concerned with classical systems but this book is one of the first to deal with quantum chaos.