Haake Quantum Signatures of Chaos

3rd Auflage 2010
ISBN: 978-3-642-05428-0
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 54, 576 Seiten, eBook

Reihe: Springer Series in Synergetics

Nine years have passed since I dispatched the second edition, and the book still appears to be in demand. The time may be ripe for an update. As the perhaps most conspicable extension, I describe the understanding of u- versal spectral ?uctuations recently reached on the basis of periodic-orbit theory. To make the presentation of those semiclassical developments selfcontained, I decided to to underpin them by a new short chapter on classical Hamiltonian mechanics. Inasmuch as the semiclassical theory not only draws inspiration from the nonlinear sigma model but actually aims at constructing that model in terms of periodic orbits, it appeared indicated to make small additions to the previous treatment within the chapter on superanalysis. Less voluminous but as close to my heart are additions to the chapter on level dynamics which close previous gaps in that approach to spectral universality. It was a pleasant duty to pay my respect to collegues in our Transregio- Sonderforschungsbereich, Martin Zirnbauer, Alex Altland, Alan Huckleberry, and Peter Heinzner, by including a short account of their beautiful work on nonstandard symmetry classes. The chapter on random matrices has not been expanded in proportion to the development of the ?eld but now includes an up-to-date treatment of an old topic in algebra, Newton’s relations, to provide a background to the Riemann-Siegel loo- like of semiclassical periodic-orbit theory.

Zielgruppe

Research

Autoren/Hrsg.

Haake, Fritz

Inhaltsverzeichnis

Time Reversal and Unitary Symmetries.- Level Repulsion.- Random-Matrix Theory.- Level Clustering.- Level Dynamics.- Quantum Localization.- Dissipative Systems.- Classical Hamiltonian Chaos.- Semiclassical Roles for Classical Orbits.- Superanalysis for Random-Matrix Theory.

Leseproben

"Chapter 4 Random-Matrix Theory (p. 61 -62)

4.1 Preliminaries

A wealth of empirical and numerical evidence suggests universality for local fluctuations in quantum energy or quasi-energy spectra of systems that display global chaos in their classical phase spaces. Exceptions apart, all such Hamiltonian matrices of sufficiently large dimension yield the same spectral fluctuations provided they have the same group of canonical transformations (see Chap. 2). In particular, the level spacing distribution P(S) generally takes the form characteristic of the universality class defined by the canonical group. Most notable among the exceptions barred by the term “untypical” are systems with “localization” that will be discussed in Chap. 7. Conversely, “generic” classically integrable systems with at least two degrees of freedom tend to display universal local fluctuations of yet another type, to be considered in Chap. 5.

The aforementioned universality is the starting point for the theory of random matrices (RMT). After early success in reproducing universal features in spectra of highly excited nuclei, that theory was boosted into even higher esteem when the connection of “integrable” and “chaotic” with different types of universal spectral fluctuations was spelled out by Bohigas, Giannoni, and Schmit [1], with important hints due to Berry and Tabor [2], McDonald and Kaufman [3], Casati, Valz-Gris, and Guarneri [4], and Berry [5]. The classic version of random-matrix theory deals with three Gaussian ensembles of Hermitian matrices, one for each group of canonical transformations. Any member of an ensemble can serve as a model of a Hamiltonian.

Similarly, there are three ensembles of random unitary matrices to represent Floquet or scattering matrices. “Poissonian” ensembles of diagonal matrices with independent, random, diagonal elements are often used to model integrable Hamiltonians. Even systems with localization have recently been accommodated in their own “universality class” of banded random matrices that is to be touched upon in Chap. 11. Random-matrix theory phenomenologically represents spectral fluctuations such as those expressed in the level spacing distribution or in correlation functions of the density of levels by suitable ensemble averages.

The immense usefulness of RMT lies in the fact that it yields closed-from results formany spectral characteristics. The extent to which an individual Hamiltonian or Floquet operator can be expected to be faithful to the RMT averages is open to discussion. A partial answer to that question is provided by a certain ergodicity property of the various ensembles. Explanations of the success of random-matrix theory will be presented in Chap. 6 (level dynamics) and Chap. 10 (periodic-orbit theory)."