Chaos. Classical and Quantum

Contributors 3 Acknowledgements 11 Overture 17 Why ChaosBook? 18 Chaos ahead 19 The future as in a mirror 20 A game of pinball 25 Chaos for cyclists 30 Evolution 35 From chaos to statistical mechanics 38 A guide to the literature 39 Go with the flow 47 Dynamical systems 47 Flows 51 Computing trajectories 55 Do it again 61 Poincaré sections 61 Constructing a Poincaré section 64 Maps 66 Local stability 73 Flows transport neighborhoods 73 Linear flows 76 Stability of flows 80 Stability of maps 83 Newtonian dynamics 89 Hamiltonian flows 89 Stability of Hamiltonian flows 91 Symplectic maps 93 Billiards 101 Billiard dynamics 101 Stability of billiards 104 Get straight 111 Changing coordinates 111 Rectification of flows 113 Classical dynamics of collinear helium 114 Rectification of maps 118 Cycle stability 123 Stability of periodic orbits 123 Cycle stabilities are cycle invariants 126 Stability of Poincaré map cycles 128 Rectification of a 1-dimensional periodic orbit 128 Smooth conjugacies and cycle stability 130 Neighborhood of a cycle 130 Transporting densities 135 Measures 135 Perron-Frobenius operator 137 Invariant measures 139 Density evolution for infinitesimal times 142 Liouville operator 145 Averaging 153 Dynamical averaging 153 Evolution operators 160 Lyapunov exponents 162 Why not just run it on a computer? 166 Qualitative dynamics, for pedestrians 173 Qualitative dynamics 173 A brief detour; recoding, symmetries, tilings 178 Stretch and fold 180 Kneading theory 185 Markov graphs 187 Symbolic dynamics, basic notions 189 Qualitative dynamics, for cyclists 199 Going global: Stable/unstable manifolds 200 Horseshoes 201 Spatial ordering 204 Pruning 206 Counting 219 Counting itineraries 219 Topological trace formula 222 Determinant of a graph 224 Topological zeta function 227 Counting cycles 230 Infinite partitions 234 Shadowing 235 Trace formulas 247 Trace of an evolution operator 247 A trace formula for maps 249 A trace formula for flows 251 An asymptotic trace formula 254 Spectral determinants 259 Spectral determinants for maps 259 Spectral determinant for flows 261 Dynamical zeta functions 263 False zeros 267 Spectral determinants vs. dynamical zeta functions 267 All too many eigenvalues? 269 Why does it work? 277 Linear maps: exact spectra 278 Evolution operator in a matrix representation 282 Classical Fredholm theory 285 Analyticity of spectral determinants 287 Hyperbolic maps 292 The physics of eigenvalues and eigenfunctions 294 Troubles ahead 297 Fixed points, and how to get them 303 Where are the cycles? 304 One-dimensional mappings 306 Multipoint shooting method 307 d-dimensional mappings 310 Flows 310 Cycle expansions 321 Pseudocycles and shadowing 321 Construction of cycle expansions 324 Cycle formulas for dynamical averages 328 Cycle expansions for finite alphabets 332 Stability ordering of cycle expansions 333 Dirichlet series 336 Why cycle? 345 Escape rates 345 Natural measure in terms of periodic orbits 348 Flow conservation sum rules 349 Correlation functions 350 Trace formulas vs. level sums 352 Thermodynamic formalism 357 Rényi entropies 357 Fractal dimensions 362 Intermittency 369 Intermittency everywhere 370 Intermittency for pedestrians 373 Intermittency for cyclists 385 BER zeta functions 391 Discrete symmetries 401 Preview 402 Discrete symmetries 406 Dynamics in the fundamental domain 408 Factorizations of dynamical zeta functions 412 C2 factorization 414 C3v factorization: 3-disk game of pinball 416 Deterministic diffusion 425 Diffusion in periodic arrays 426 Diffusion induced by chains of 1-d maps 430 Marginal stability and anomalous diffusion 438 Irrationally winding 447 Mode locking 448 Local theory: ``Golden mean" renormalization 454 Global theory: Thermodynamic averaging 456 Hausdorff dimension of irrational windings 458 Thermodynamics of Farey tree: Farey model 460 Prologue 471 Quantum pinball 472 Quantization of helium 474 Quantum mechanics, briefly 477 WKB quantization 483 WKB ansatz 483 Method of stationary phase 486 WKB quantization 487 Beyond the quadratic saddle point 489 Semiclassical evolution 495 Hamilton-Jacobi theory 495 Semiclassical propagator 504 Semiclassical Green's function 507 Noise 521 Deterministic transport 522 Brownian difussion 523 Weak noise 524 Weak noise approximation 526 Semiclassical quantization 531 Trace formula 531 Semiclassical spectral determinant 536 One-dof systems 538 Two-dof systems 539 Relaxation for cyclists 545 Fictitious time relaxation 546 Discrete iteration relaxation method 552 Least action method 554 Quantum scattering 561 Density of states 561 Quantum mechanical scattering matrix 565 Krein-Friedel-Lloyd formula 566 Wigner time delay 569 Chaotic multiscattering 575 Quantum mechanical scattering matrix 576 N-scatterer spectral determinant 579 Semiclassical 1-disk scattering 583 From quantum cycle to semiclassical cycle 590 Heisenberg uncertainty 593 Helium atom 595 Classical dynamics of collinear helium 596 Chaos, symbolic dynamics and periodic orbits 597 Local coordinates, fundamental matrix 602 Getting ready 604 Semiclassical quantization of collinear helium 605 Diffraction distraction 619 Quantum eavesdropping 619 An application 625 Epilogue 635 Index 639 A brief history of chaos 655 Chaos is born 655 Chaos grows up 659 Chaos with us 660 Death of the Old Quantum Theory 664 Infinite-dimensional flows 667 Stability of Hamiltonian flows 671 Symplectic invariance 671 Monodromy matrix for Hamiltonian flows 672 Implementing evolution 675 Koopmania 675 Implementing evolution 677 Symbolic dynamics techniques 683 Topological zeta functions for infinite subshifts 683 Prime factorization for dynamical itineraries 691 Counting itineraries 697 Counting curvatures 697 Finding cycles 701 Newton-Raphson method 701 Hybrid Newton-Raphson / relaxation method 702 Applications 705 Evolution operator for Lyapunov exponents 705 Advection of vector fields by chaotic flows 710 Discrete symmetries 717 Preliminaries and definitions 717 C4v factorization 722 C2v factorization 727 Hénon map symmetries 729 Symmetries of the symbol square 730 Convergence of spectral determinants 731 Curvature expansions: geometric picture 731 On importance of pruning 734 Ma-the-matical caveats 735 Estimate of the nth cumulant 736 Infinite dimensional operators 739 Matrix-valued functions 739 Operator norms 741 Trace class and Hilbert-Schmidt class 742 Determinants of trace class operators 744 Von Koch matrices 748 Regularization 749 Statistical mechanics recycled 753 The thermodynamic limit 753 Ising models 755 Fisher droplet model 759 Scaling functions 764 Geometrization 768 Noise/quantum corrections 781 Periodic orbits as integrable systems 781 The Birkhoff normal form 785 Bohr-Sommerfeld quantization of periodic orbits 786 Quantum calculation of corrections 788 Solutions 797 Projects 843 Deterministic diffusion, zig-zag map 845 Deterministic diffusion, sawtooth map 852