Deterministic Chaos in Infinite Quantum Systems

ISBN-13: 9783540570172 / Angielski / Miękka / 1993 / 225 str.

Fabio Benatti

Opis

The purpose of this volume is to give a detailed account of a series of re sults concerning some ergodic questions of quantum mechanics which have the past six years following the formulation of a generalized been addressed in Kolmogorov-Sinai entropy by A.Connes, H.Narnhofer and W.Thirring. Classical ergodicity and mixing are fully developed topics of mathematical physics dealing with the lowest levels in a hierarchy of increasingly random behaviours with the so-called Bernoulli systems at its apex showing a structure that characterizes them as Kolmogorov (K-) systems. It seems not only reasonable, but also inevitable to use classical ergodic theory as a guide in the study of ergodic behaviours of quantum systems. The question is which kind of random behaviours quantum systems can exhibit and whether there is any way of classifying them. Asymptotic statistical independence and, correspondingly, complete lack of control over the distant future are typical features of classical K-systems. These properties are fully characterized by the dynamical entropy of Kolmogorov and Sinai, so that the introduction of a similar concept for quantum systems has provided the opportunity of raising meaningful questions and of proposing some non-trivial answers to them. Since in the following we shall be mainly concerned with infinite quantum systems, the algebraic approach to quantum theory will provide us with the necessary analytical tools which can be used in the commutative context, too."

Spis treści

1 Introduction.- 2 Classical Ergodic Theory.- 2.1 Irreversibility.- 2.1.1 Coarse-Graining.- 2.1.2 Correlations.- 2.1.3 Abstract Dynamical Systems.- 2.1.4 Spectral Theory.- 2.2 Entropy.- 2.2.1 Randomness and Entropy.- 2.2.2 The Entropy of Kolmogorov and Sinai.- 2.2.3 Kolmogorov Systems.- 2.3 Topological Properties of Dynamical Systems.- 2.3.1 Topological Dynamics.- 2.3.2 Topological Entropy.- 3 Algebraic Approach to Classical Ergodic Theory.- 3.1 Abelian C* Dynamical Systems.- 3.2 Abelian W* Dynamical Systems.- 3.3 W* Algebras: KS-Entropy and K-Systems.- 3.4 C* Algebras: Classical Topological Entropy.- 4 Infinite Quantum Systems.- 4.1 Useful Tools from Finite Quantum Systems.- 4.1.1 Density Matrices and von Neumann Entropy.- 4.1.2 Relative Entropy and Completely Positive Maps.- 4.2 GNS-Construction.- 4.2.1 Fermions, Bosons and Toy Models.- 4.3 Ergodic Properties in Quantum Systems.- 4.3.1 Galilei-Invariant Two-Body Interactions.- 4.4 Algebraic Quantum Kolmogorov Systems.- 5 Connes-Narnhofer-Thirring Entropy.- 5.1 Basic Ideas and Construction 1.- 5.2 Basic Ideas and Construction 2.- 5.3 CNT-Entropy: Applications.- 5.3.1 Dynamical Entropy of Quasi-Free Automorphisms.- 5.3.2 CNT-Entropy and Thermodynamics.- 5.4 Short History of the Topic and Latest Developments.- 5.5 Entropic Quantum Kolmogorov Systems.- 5.6 Ideas for a Non-commutative Topological Entropy.- 6 Appendix.- References.- Index of Symbols.

O autorze

The author presents the classical concept of Kolmogorov-Sinai entropy and its non-commutative analogue for infinite quantum systems. He then compares the irregular behaviour of classical versus quantum systems. These notes address both researchers and graduate students in statistical mechanics and mathematical physics.