The Transition to Chaos

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1 Linda E. Reichl The Transition to Chaos Conservative Classical Systems and Quantum Manifestations Second Edition With 180 Illustrations v I.,,-,,t,...,* ', Springer

2 Dedication Acknowledgements v vii 1 Overview Introduction Historical Overview Plan of the Book References 12 2 Fundamental Concepts Introduction Conventional Perturbation Theory Integrability Noether's Theorem Hidden Symmetries Poincare Surface of Section Nonlinear Resonance and Chaos Single-Resonance Hamiltonians Two-Resonance Hamiltonian KAM Theory The Definition of Chaos ' Lyapounov Exponent KS Metric Entropy and if-flows 43

3 2.7 Time-Dependent Hamiltonians Conclusions., Problems References 56 Area-Preserving Maps Introduction Twist Maps Derivation of a Twist Map from a Torus Generating Functions Birkhoff Fixed Point Theorem The Tangent Map Homoclinic and Heteroclinic Points Melnikov Distance Whisker Maps The Standard Map Rational and Irrational Orbits Accelerator Modes ' Scaling Behavior of Noble KAM Tori Rational Approximates Scaling Properties of Twist Maps Renormalization in Twist Maps Integrable Twist Map Nonintegrable Twist Map The Universal Map Bifurcation of M-Cycles Some General Properties The Quadratic Map Scaling in the Quadratic de Vogelaere Map Cantori Diffusion in Two-Dimensional Twist Maps Effect of Cantori Diffusion in the Standard Map Conclusions Problems References 130 Global Properties Introduction Important Models Delta-Kicked Rotor (Standard Map) The Duffing Oscillator Driven Particle in Infinite Square-Well Potential Driven One-Dimensional Hydrogen Renormalization Map 146

4 xi The Paradigm Hamiltonian The Renormalization Map Fixed Points of the Renormalization Map Application of Renormalization Predictions Driven Square-Well System Duffing Oscillator Scattering Chaos Stochastic Tiling Delta-Kicked Harmonic Oscillator Two Primary Resonances Model Arnol'd Diffusion Resonance Networks Numerical Observations Diffusion Along Separatrix Layers Diffusion "Coefficient Some Applications Conclusions Problems 186? 4.10 References 187 Random Matrix Theory Introduction Ensembles Gaussian Ensembles Circular Ensembles Cluster Functions Cluster Expansion of the Probability Densities Probability Densities and Quaternion Determinants Cluster Functions for Gaussian Ensembles Cluster Functions for Circular Ensembles Eigenvalue Number Density^ Eigenvalue Number Density for Gaussian Ensembles Eigenvalue Number Density for Circular Ensembles Eigenvalue Correlations - A3-Statistic A3-Statistic - General Expressions A3-Statistics for Gaussian Ensembles A 3 -Statistics for Circular Ensembles^ Eigenvalue Nearest Neighbor Spacing Distribution (GOE) ^Eigenvalue Spacing Distributions (N=2) Nearest Neighbor Spacing Distribution (N >oc) Approximate Nearest Neighbor Spacing Distributions for GOE (N-HX) Eigenvector Statistics - Gaussian Ensembles General Properties Distribution of Eigenvector Components (GOE). 231

5 xii Distribution of Eigenvector Components (GUE) Gaussian Symplectic Ensemble Conclusions Problems References Bounded Quantum Systems Introduction Quantum Integrability Symmetries and Degeneracy Quantum Billiards The Rectangular Billiard The Stadium The Sinai Billiard The Ripple Billiard The Quantized Baker's Map Time Average as an Invariant Integrable and Nonintegrable Spin Systems f Classical Spin Models ' Quantum Spin Models Two-Dimensional Clusters with N Spin \ Objects Anharmonic Oscillators Polynomial Anharmonicity Coupled Morse Oscillators Conclusions Problems References Manifestations of Chaos in Quantum Scattering Processes Introduction Scattering Theory Hamiltonian Energy Eigenstates The Reaction Matrix The Scattering Matrix Wigner-Smith and Partial Delay Times^ Delay Time of a Wave Packet Delay Times for Multichannel Scattering Delay Times and Complex Poles Scattering Theory and GOE The Average S-Matrix When Does a GOE Hamiltonian Yield a COE S- Matrix? S-Matrix Correlation Function (GOE) 323

6 xiii Delay Time Density S-Matrix Correlation Functions (COE) Green's Function and S-Matrix The Green's Function Green's Function for Quantum Waveguide Transmission Amplitudes and the Green's Function Absorption Spectrum and Green's Function Experimental Observation of RMT Predictions Experimental Nuclear Spectral Statistics Experimental Molecular Spectral Statistics Conclusions Problems References 345 Semiclassical Theory Path Integrals Introduction Green's Function and Density of States The Path Integral The General Case, H = f + V Semiclassical Approximation Method of Stationary Phase The Semiclassical Green's Function Conjugate Points Energy Green's Function General Expression Density of States A3-Statistic for a Rectangular Billiard Energy Green's Function for a Rectangular Billiard Density of States for the Rectangular Billiard Semiclassical Expression for the A3-Statistic Gutzwiller Trace Formula. ' Response Function for Chaotic Systems Anisotropic Kepler System Diamagnetic Hydrogen*** The Model Absorption Cross Section Experiment / Semiclassical Cross Section Conclusions Problems References 399 Time-Periodic Systems Introduction Floquet Theory. 403

7 xiv Floquet Matrix Floquet Hamiltonian Nonlinear Quantum Resonances Two Primary Resonance Model Floquet Eigenstates Quantum Resonance Overlap Floquet Eigenvalue Nearest Neighbor Spacing Eigenvalue Avoided Crossings and Wave Function Delocalization Classical Driven Square-Well System Quantum Driven Square-Well System Avoided Crossings and Delocalization High Harmonic Radiation Dynamical Tunneling in Atom Optics Experiments Hamiltonian for Atomic Center-of-Mass Average Momentum of Cesium Atoms Floquet Analysis of Tunneling Oscillations Quantum Renormalization Paradigm Schrodinger Equation Higher-Order Resonances Quantum Renormalization Map Stable Manifold Scaling Functions Scaling of Localization Lengths Quantum Delta-Kicked Rotor The Schrodinger Equation for the Delta-Kicked Rotor KAM-like Behavior of the Quantum Delta-Kicked Rotor The Floquet Map Spectral Statistics \ Dynamic Anderson Localization: Delta-Kicked Rotor Tight-Binding Model for the Delta-Kicked Rotor Diffusion Coefftcient and Localization Length Atom Optics Realization of the Delta-Kicked Rotor Microwave-Driven Hydrogen Experimental Apparatus One-Dimensional Approximation Dynamic Anderson Localization - Microwave-Driven Hydrogen Diffusion in Microwave-Driven Hydrogen Experimental Observation of Dynamic Localization Conclusions Problems References 471

8 xv 10 Stochastic Manifestations of Chaos Introduction Brownian Motion in Two Space Dimensions Random Walk in Two Space Dimensions Time-Periodically Driven Brownian Motion in One Space Dimension Schrodinger-like Equation Conclusion References 485 A Classical Mechanics 486 A.I Newton's Equations 486 A.2 Lagrange's Equations 487 A.3 Hamilton's Equations 487 A.4 The Psisson Bracket 488 A.5 Phase Space Volume Conservation 489 A.6 Action-Angle Coordinates 489 A.7 Hamilton's Principal Function 491 A.8 References 492 B Simple Models 493 B.I The Pendulum 493 B.I.I Libration Trapped Orbits (E o < g) 494 B.1.2 Rotation Untrapped Orbits (E o > g) 495 B.2 Double-Well Potential 496 B.2.1 Trapped Motion (E o < 0) 497 B.2.2 Untrapped Motion (E o > 0) 498 B.3 Infinite Square-Well Potential 499 B.4 One-Dimensional Hydrogen 501 B.4.1 Zero Stark Field. 501 B.4.2 Nonzero Stark Field I. 503 C Renormalization Integral 505 C.I v = N = Integer 505 C.2 v = ^Integer 507 C.3 References 509 D Moyal Bracket 510 D.I The Wigner Function 510 D.2 Ordering of Operators 512 D.3 Moyal Bracket 513 D.4 References 514 E Symmetries and the Hamiltonian Matrix 515 E.I Space-Time Symmetries 516

9 xvi E.I.I Continuous Symmetries 516 E.I.2 Discrete Symmetries 518 E.2 Structure of the Hamiltonian Matrix 520 E.2.1 Space-Time Homogeneity and Isotropy 520 E.2.2 Time Reversal Invariance 521 E.3 References 524 F Invariant Measures 525 F.I General Definition of Invariant Measure 525 F.I.I Invariant Metric (Length) 526 F.I.2 Invariant Measure (Volume) 526 F.2 Hermitian Matrices 527 F.2.1 Real Symmetric Matrix 527 F.2.2 Complex Hermitian Matrices 530 F.2.3 Quaternion Real Matrices 532 F.2.4 General Formula for Invariant Measure of Hermitian Matrices 534 F.3 Unitary Matrices 534 F.3.1 Symmetric Unitary Matrices 536 F.3.2 General Unitary Matrices 537 F.3.3 Symplectic Unitary Matrices 538 F.3.4 General Formula for Invariant Measure of Unitary Matrices 539 F.3.5 Orthogonal Matrices 539 F.4 References 540 G Quaternions 541 G.I References 545 H Gaussian Ensembles 546 H.I Vandermonde Determinant H.2 Gaussian Unitary Ensemble (GUE) 549 H.3 Gaussian Orthogonal Ensemble (GOE) 550 H.4 Gaussian Symplectic Ensemble (GSE) 558 H.5 References 563 I Circular Ensembles Vandermonde Determinant Circular Unitary Ensemble (CUE) Circular Orthogonal Ensemble (COE) Circular Symplectic Ensemble (COE) References 576 J Volume of Invariant Measure for Unitary Matrices 577 J.I References 582

10 xvii K Lorentzian Ensembles 583 K.I Normalization of AOE 583 K.2 Relation Between COE and AOE 584 K.3 Equivalence of COE and AOE When N->oo 585 K.4 Invariance of AOE under Inversion '. 586 K.4.1 Robustness of AOE under Integration 587 K.5 References 587 L Grassmann Variables and Supermatrices 588 L.I Grassmann Variables 588 L.2 Supermatrices 590 L.2.1 Transpose of a Supermatrix 590 L.2.2 Hermitian Adjoint of a Supermatrix 591 L.2.3 Supertrace of a Supermatrix 591 L.2.4 Determinant of a Supermatrix 592 L.3 References 594 M Average Response Function (GOE) 595 M.I Gaussian Integral for (Det[eljv - HN})' M.2 Gaussian Integral for Det[eljv -Hiv] 597 M.3 Gaussian Integral for Response Function Generating Function 598 M.4 Expectation Value of the Generating Function (Part 1). 600 M.5 The Hubbard-Stratonovitch Transformation 601 M.6 Expectation Value of the Generating Function (Part 2). 604 M.7 Average Response Function Density 608 M.7.1 Saddle Points for the Integration over a 610 M.7.2 Saddle Points for the Integration over w 612 M.7.3 Asymptotic Expression (TV >oo) for the Average Response Function Density 614 M.7.4 Wigner Semicircle Law 616 M.8 References 617 N Average S-Matrix (GOE) 618 N.I S-Matrix Generating Function 618 N.2 Average S-Matrix Generating Function 619 N.3 Saddle Point Approximation 622 N.3.1 Case I: f < /j 623 N.3.2 Case II: f > /x 624 N.4 Integration over Grassmann Variables 624 N.5 References 627 O Maxwell's Equations for 2-d Billiards 628 O.I References 631

11 xviii P Lloyd's Model 632 P.I Localization Length 632 P.2 References 637 Q Hydrogen in a Constant Electric Field 638 Q.I The Schrodinger Equation 638 Q.I.I Equation for Relative Motion 639 Q.1.2 Solution for A o = Q.2 One-Dimensional Hydrogen 642 Q.3 References 644 Subject Index 645 Author Index 667

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