Classical and quantum chaos

Автор(ы):Cvitanovic P. asf.
06.10.2007
Год изд.:2000
Описание: Even though the experimental evidence for the fractal geometry of nature is circumstantial, in studies of probabilistically assembled fractal aggregates we know of nothing better than contemplating such numbers. In deterministic systems we can do much better. Chaotic dynamics is generated by interplay of locally unstable motions, and interweaving of their global stable and unstable manifolds. These features are robust and accessible in systems as noisy as slices of rat brains. Poincare, the first to understand deterministic chaos, already said as much (modulo rat brains). Once the topology of chaotic dynamics is understood, a powerful theory yields the macroscopically measurable consequences of chaotic dynamics, such as atomic spectra, transport coefficients, gas pressures. That is what we will focus on in this book. We teach you how to evaluate a determinant, take a logarithm, stuff like that. Should take 20 pages or so. Well, we fail - so far we have not found a way to traverse this material in less than a semester, or 200-300 pages of text.
Оглавление:
Classical and quantum chaos — обложка книги. Обложка книги.
1 Overture [1]
  1.1 Why this book? [2]
  1.2 Chaos ahead [3]
  1.3 A game of pinball [4]
  1.4 Periodic orbit theory [12]
  1.5 Evolution operators [17]
  1.6 From chaos to statistical mechanics [20]
  1.7 Semiclassical quantization [21]
  1.8 Guide to literature [23]
    Guide to exercises [25]
    Resume [26]
    Exercises [30]
2 Trajectories [31]
  2.1 Flows [31]
  2.2 Maps [36]
  2.3 Infinite-dimensional flows [40]
    Exercises [47]
3 Local stability [51]
  3.1 Flows transport neighborhoods [51]
  3.2 Linear stability of maps [56]
  3.3 Billiards [56]
    Exercises [60]
4 Transporting densities [63]
  4.1 Measures [64]
  4.2 Density evolution [65]
  4.3 Invariant measures [66]
  4.4 Evolution operators [68]
    Resume [72]
    Exercises [73]
5 Averaging [77]
  5.1 Dynamical averaging [77]
  5.2 Evolution operators [83]
    Resume [87]
    Exercises [89]
6 Trace formulas [91]
  6.1 Trace of an evolution operator [91]
  6.2 An asymptotic trace formula [98]
    Resume [100]
    Exercises [101]
7 Qualitative dynamics [103]
  7.1 Temporal ordering: Itineraries [103]
  7.2 3-disk symbolic dynamics [106]
  7.3 Spatial ordering of "stretch & fold" flows [110]
  7.4 Unimodal map symbolic dynamics [111]
  7.5 Spatial ordering: Symbol plane [116]
  7.6 Pruning [122]
  7.7 Topological dynamics [123]
    Resume [129]
    Exercises [134]
8 Fixed points, and how to get them [141]
  8.1 One-dimensional mappings [142]
  8.2 d-dimensional mappings [145]
  8.3 Flows [146]
  8.4 Periodic orbits as extremal orbits [151]
    Resume [156]
    Exercises [159]
9 Counting [167]
  9.1 Counting itineraries [167]
  9.2 Topological trace formula [169]
  9.3 Determinant of a graph [171]
  9.4 Topological zeta function [175]
  9.5 Counting cycles [176]
  9.6 Infinite partitions [180]
    Resume [184]
    Exercises [187]
10 Spectral determinants [195]
  10.1 Spectral determinants for maps [196]
  10.2 Spectral determinant for flows [198]
  10.3 Dynamical zeta functions [199]
  10.4 The simplest of spectral determinants: A single fixed point [203]
  10.5 False zeros [204]
  10.6 All too many eigenvalues? [205]
  10.7 More examples of spectral determinants [206]
    Resume [210]
    Exercises [212]
11 Cycle expansions [219]
  11.1 Pseudocycles and shadowing [219]
  11.2 Cycle formulas for dynamical averages [227]
  11.3 Cycle expansions for finite alphabets [230]
  11.4 Stability ordering of cycle expansions [231]
  11.5 Dirichlet series [235]
    Exercises [239]
12 Why does it work? [245]
  12.1 Curvature expansions: geometric picture [246]
  12.2 Analyticity of spectral determinants [249]
  12.3 Hyperbolic maps [254]
  12.4 On importance of pruning [258]
    Resume [259]
    Exercises [265]
13 Getting used to cycles [267]
  13.1 Escape rates [267]
  13.2 Flow conservation sum rules [271]
  13.3 Lyapunov exponents [272]
  13.4 Correlation functions [274]
  13.5 Trace formulas vs. level sums [276]
  13.6 Eigenstates [277]
  13.7 Why not just run it on a computer? [278]
  13.8 Ma-the-matical caveats [279]
  13.9 Cycles as the skeleton of chaos [281]
    Resume [284]
    Exercises [286]
14 Thermo dynamic formalism [289]
  14.1 Renyi entropies [289]
  14.2 Fractal dimensions [294]
    Resume [298]
    Exercises [299]
15 Discrete symmetries [303]
  15.1 Preview [303]
  15.2 Discrete symmetries [308]
  15.3 Dynamics in the fundamental domain [311]
  15.4 Factorizations of dynamical zeta functions [315]
  15.5 (?) factorizations [317]
  15.6 (?) factorization: 3-disk game of pinball [319]
    Resume [323]
    Exercises [325]
16 Deterministic diffusion [329]
  16.1 Diffusion in periodic arrays [330]
  16.2 Diffusion induced by chains of 1-d maps [334]
    Resume [343]
    Exercises [345]
17 Why doesn't it work? [347]
  17.1 Escape, averages and periodic orbits [348]
  17.2 Know thy enemy [352]
  17.3 Defeating your enemy: Intermittency resummed [359]
  17.4 Marginal stability and anomalous diffusion [364]
  17.5 Probabilistic or BER zeta functions [368]
    Resume [372]
    Exercises [374]
18 Semiclassical evolution [377]
  18.1 Quantum mechanics: A brief review [378]
  18.2 Semiclassical evolution [382]
  18.3 Semiclassical propagator [391]
  18.4 Semiclassical Green's function [395]
    Resume [401]
    Exercises [404]
19 Semiclassical quantization [409]
  19.1 Trace formula [409]
  19.2 Semiclassical spectral determinant [414]
  19.3 One-dimensional systems [416]
  19.4 Two-dimensional systems [417]
    Resume [418]
    Exercises [423]
20 Semiclassical chaotic scattering [425]
  20.1 Quantum mechanical scattering matrix [425]
  20.2 Krein-Friedel-Lloyd formula [428]
    Exercises [433]
21 Helium atom [435]
  21.1 Classical dynamics of collinear helium [436]
  21.2 Semiclassical quantization of collinear helium [448]
    Resume [458]
    Exercises [460]
22 Diffraction distraction [463]
  22.1 Quantum eavesdropping [463]
  22.2 An application [470]
    Exercises [479]
23 Irrationally winding [481]
  23.1 Mode locking [482]
  23.2 Local theory: "Golden mean" renormalization [488]
  23.3 Global theory: Thermodynamic averaging [490]
  23.4 Hausdorff dimension of irrational windings [492]
  23.5 Thermodynamics of Farey tree: Farey model [494]
    Resume [500]
    Exercises [502]
24 Statistical mechanics [503]
  24.1 The thermodynamic limit [503]
  24.2 Ising models [506]
  24.3 Fisher droplet model [509]
  24.4 Scaling functions [515]
  24.5 Geometrization [519]
    Resume [527]
    Exercises [529]
Summary and conclusions [533]
A Linear stability of Hamiltonian flows [537]
  A.1 Symplectic invariance [537]
  A.2 Monodromy matrix for Hamiltonian flows [539]
В Symbolic dynamics techniques [543]
  B.1 Symbolic dynamics, basic notions [543]
  B.2 Topological zeta functions for infinite subshifts [546]
  B.3 Prime factorization for dynamical itineraries [555]
  B.4 Counting curvatures [559]
    Exercises [560]
С Applications [563]
  C.1 Evolution operator for Lyapunov exponents [563]
  C.2 Advection of vector fields by chaotic flows [568]
    Exercises [575]
D Discrete symmetries [577]
  D.1 C(?) factorization [577]
  D.2 C(?) factorization [582]
  D.3 Symmetries of the symbol plane [585]
E Convergence of spectral determinants [587]
  E.1 Estimate of the nth cumulant [587]
F Infinite dimensional operators [589]
  F.1 Matrix-valued functions [589]
  F.2 Trace class and Hilbert-Schmidt class [591]
  F.3 Determinants of trace class operators [593]
  F.4 Von Koch matrices [597]
  F.5 Regularization [599]
G Trace of the scattering matrix [603]
    Index [606]
II Materialavailableon www.nbi.dk/ChaosBook [607]
H What reviewers say [609]
  H.1 N. Bohr [609]
  H.2 R.P. Feynman [609]
  H.3 Professor Gatto Nero [609]
I A brief history of chaos [611]
  1.1 Chaos is born [611]
  1.2 Chaos grows up [615]
  1.3 Chaos with us [616]
  1.4 Death of the Old Quantum Theory [620]
J Solutions [623]
К Projects [645]
  K.1 Deterministic diffusion, zig-zag map [647]
  K.2 Deterministic diffusion, sawtooth map [654]

Predrag Cvitanovic
  most of the text
Roberto Artuso
  4 Transporting densities [63]
  6.1.4 A trace formula for flows [96]
  13.4 Correlation functions [274]
  17 Intermittency [347]
  16 Deterministic diffusion [329]
  23 Irrationally winding [481]
Ronnie Mainieri
  2 Trajectories [31]
  2.2.2 The Poincare section of a flow [39]
  3 Local stability [51]
  ?? Understanding flows ??
  7.1 Temporal ordering: itineraries [103]
  24 Statistical mechanics [503]
  Appendix I: A brief history of chaos [611]
Gabor Vattay
  14 Thermodynamic formalism [289]
  18 Semiclassical evolution [377]
  19 Semiclassical trace formula [409]
Ofer Biham
  8.4.1 Relaxation for cyclists [151]
Freddy Christiansen
  8 Fixed points, and what to do about them [141]
Per Dahlqvist
  8.4.2 Orbit length extremization method for billiards [154]
17 Intermittency [347]
  Appendix В.2.1: Periodic points of unimodal maps [553]
Carl P. Dettmann
  11.4 Stability ordering of cycle expansions [231]
Mitchell J. Feigenbaum
  Appendix A.I: Symplectic invariance [537]
Kai T. Hansen
  7.4 Unimodal map symbolic dynamics [111]
  7.4.2 Kneading theory [114]
  ?? Topological zeta function for an infinite partition ??
  figures throughout the text
Adam Priigel-Bennet
  Solutions 11.2, 10.1, 1.2, 2.7, 8.18, 2.9, 10.16
Lamberto Rondoni
  4 Transporting densities [63]
  13.1.1 Unstable periodic orbits are dense [270]
Juri Rolf
  Solution 10.16
Per E. Rosenqvist
  exercises, figures throughout the text
Hans Henrik Rugh
  12 Why does it work? [245]
Edward A. Spiegel
  2 Trajectories [31]
  3 Local stability [51]
  4 Transporting densities [63]
Gregor Tanner
  13.8 Ma-the-matical caveats [279]
  21 The helium atom [435]
  Appendix A.2: Jacobians of Hamiltonian flows [539]
Niall Whelan
  22 Diffraction distraction [463]
  G Trace of the scattering matrix [603]
Andreas Wirzba
  20 Semiclassical chaotic scattering [425]
  Appendix F: Infinite dimensional operators [589]
Unsung Heroes: too numerous to list
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