The Transition to Chaos

Conservative Classical Systems and Quantum Manifestations
2. Auflage 2004
ISBN: 978-0-387-98788-0
Verlag: Springer

Buch, Englisch, 675 Seiten, Format (B × H): 164 mm x 242 mm, Gewicht: 1096 g

Reihe: Institute for Nonlinear Science

Based on courses given at the universities of Texas in Austin, and California in San Diego, this book deals with the basic mechanisms that determine the dynamic evolution of classical and quantum systems. 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.

Zielgruppe

Research

Autoren/Hrsg.

Reichl, Linda

Fachgebiete

• Naturwissenschaften Physik Angewandte Physik Statistische Physik, Dynamische Systeme
• Naturwissenschaften Physik Quantenphysik Teilchenphysik
• Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik

Inhaltsverzeichnis

1 Overview.- 2 Fundamental Concepts.- 3 Area-Preserving Maps.- 4 Global Properties.- 5 Random Matrix Theory.- 6 Bounded Quantum Systems.- 7 Manifestations of Chaos in Quantum Scattering Processes.- 8 Semiclassical Theory—Path Integrals.- 9 Time-Periodic Systems.- 10 Stochastic Manifestations of Chaos.- A Classical Mechanics.- A.1 Newton’s Equations.- A.2 Lagrange’s Equations.- A.3 Hamilton’s Equations.- A.4 The Poisson Bracket.- A.5 Phase Space Volume Conservation.- A.6 Action-Angle Coordinates.- A.7 Hamilton’s Principal Function.- A.8 References.- B Simple Models.- B.1 The Pendulum.- B.2 Double-Well Potential.- B.3 Infinite Square-Well Potential.- B.4 One-Dimensional Hydrogen.- B.4.1 Zero Stark Field.- B.4.2 Nonzero Stark Field.- C Renormalization Integral.- C.3 References.- D Moyal Bracket.- D.1 The Wigner Function.- D.2 Ordering of Operators.- D.3 Moyal Bracket.- D.4 References.- E Symmetries and the Hamiltonian Matrix.- E.1 Space-Time Symmetries.- E.1.1 Continuous Symmetries.- E.1.2 Discrete Symmetries.- E.2 Structure of the Hamiltonian Matrix.- E.2.1 Space-Time Homogeneity and Isotropy.- E.2.2 Time Reversal Invariance.- E.3 References.- F Invariant Measures.- F.1 General Definition of Invariant Measure.- F.1.1 Invariant Metric (Length).- F.1.2 Invariant Measure (Volume).- F.2 Hermitian Matrices.- F.2.1 Real Symmetric Matrix.- F.2.2 Complex Hermitian Matrices.- F.2.3 Quaternion Real Matrices.- F.2.4 General Formula for Invariant Measure of Hermitian Matrices.- F.3 Unitary Matrices.- F.3.1 Symmetric Unitary Matrices.- F.3.2 General Unitary Matrices.- F.3.3 Symplectic Unitary Matrices.- F.3.4 General Formula for Invariant Measure of Unitary Matrices.- F.3.5 Orthogonal Matrices.- F.4 References.- G Quaternions.- G.1 References.- H Gaussian Ensembles.- H.1Vandermonde Determinant.- H.2 Gaussian Unitary Ensemble (GUE).- H.3 Gaussian Orthogonal Ensemble (GOE).- H.4 Gaussian Symplectic Ensemble (GSE).- H.5 References.- I Circular Ensembles.- 1.1 Vandermonde Determinant.- 1.2 Circular Unitary Ensemble (CUE).- 1.3 Circular Orthogonal Ensemble (COE).- 1.4 Circular Symplectic Ensemble (COE).- 1.5 References.- J Volume of Invariant Measure for Unitary Matrices.- J.1 References.- K Lorentzian Ensembles.- K.1 Normalization of AOE.- K.2 Relation Between COE and AOE.- K.4 Invariance of AOE under Inversion.- K.4.1 Robustness of AOE under Integration.- K.5 References.- L Grassmann Variables and Supermatrices.- L.1 Grassmann Variables.- L.2 Supermatrices.- L.2.1 Transpose of a Supermatrix.- L.2.2 Hermitian Adjoint of a Supermatrix.- L.2.3 Supertrace of a Supermatrix.- L.2.4 Determinant of a Supermatrix.- L.3 References.- M Average Response Function (GOE).- M.3 Gaussian Integral for Response Function Generating Function.- M.4 Expectation Value of the Generating Function (Part 1).- M.5 The Hubbard-Stratonovitch Transformation.- M.6 Expectation Value of the Generating Function (Part 2).- M.7 Average Response Function Density.- M.7.1 Saddle Points for the Integration over a.- M.7.2 Saddle Points for the Integration over ?.- M.7.4 Wigner Semicircle Law.- M.8 References.- N Average S-Matrix (GOE).- N.1 S-Matrix Generating Function.- N.2 Average S-Matrix Generating Function.- N.3 Saddle Point Approximation.- N.4 Integration over Grassmann Variables.- N.5 References.- O Maxwell’s Equations for 2-d Billiards.- O.1 References.- P Lloyd’s Model.- P.1 Localization Length.- P.2 References.- Q Hydrogen in a Constant Electric Field.- Q.1 The Schrödinger Equation.- Q.1.1 Equation for Relative Motion.- Q.2 One-Dimensional Hydrogen.- Q.3References.- Author Index.

Über Autor(innen)

Linda E. Reichl, Ph.D., is a Professor of Physics at the University of Texas at Austin and has served as Acting Director of the Ilya Prigogine Center for Statistical Mechanics and Complex Systems since 1974. She is a Fellow of the American Physical Society and currently is U.S. Editor of the journal Chaos, Solitons, and Fractals.