The Transition to Chaos : Conservative Classical Systems and Quantum Manifestations

This book provides a thorough and comprehensive discussion of classical and quantum chaos theory for bounded systems and for scattering processes. Specific discussions include: • Noether’s theorem, integrability, KAM theory, and a definition of chaotic behavior. • Area-preserving maps, quantum billi...

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Main Author: Reichl, Linda
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: New York, NY Springer New York 2004, 2004
Edition:2nd ed. 2004
Series:Institute for Nonlinear Science
Subjects:
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
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245 0 0 |a The Transition to Chaos  |h Elektronische Ressource  |b Conservative Classical Systems and Quantum Manifestations  |c by Linda Reichl 
250 |a 2nd ed. 2004 
260 |a New York, NY  |b Springer New York  |c 2004, 2004 
300 |a XVIII, 675 p. 154 illus  |b online resource 
505 0 |a 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 --  
505 0 |a 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.1 Vandermonde Determinant -- H.2 Gaussian Unitary Ensemble (GUE) -- H.3 Gaussian Orthogonal Ensemble (GOE) --  
505 0 |a 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.3 References -- Author Index 
505 0 |a 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 - 
653 |a Statistical physics 
653 |a Complex Systems 
653 |a Statistical Physics and Dynamical Systems 
653 |a Dynamical systems 
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520 |a This book provides a thorough and comprehensive discussion of classical and quantum chaos theory for bounded systems and for scattering processes. Specific discussions include: • Noether’s theorem, integrability, KAM theory, and a definition of chaotic behavior. • Area-preserving maps, quantum billiards, semiclassical quantization, chaotic scattering, scaling in classical and quantum dynamics, dynamic localization, dynamic tunneling, effects of chaos in periodically driven systems and stochastic systems. • Random matrix theory and supersymmetry. The book is divided into several parts. Chapters 2 through 4 deal with the dynamics of nonlinear conservative classical systems. Chapter 5 and several appendices give a thorough grounding in random matrix theory and supersymmetry techniques. Chapters 6 and 7 discuss the manifestations of chaos in bounded quantum systems and open quantum systems respectively. Chapter 8 focuses on the semiclassical description of quantum systems with underlying classical chaos, and Chapter 9 discusses the quantum mechanics of systems driven by time-periodic forces. Chapter 10 reviews some recent work on the stochastic manifestations of chaos. The presentation is complete and self-contained; appendices provide much of the needed mathematical background, and there are extensive references to the current literature. End of chapter problems help students clarify their understanding. In this new edition, the presentation has been brought up to date throughout, and a new chapter on open quantum systems has been added. About the author: 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