Classical and Quantum Chaos Predrag Cvitanovi´c – Roberto Artuso – Per Dahlqvist – Ronnie Mainieri – Gregor Tanner – G´abor Vattay – Niall Whelan – Andreas Wirzba —————————————————————- version 9.2.3 Feb 26 2002 printed June 19, 2002 www.nbi.dk/ChaosBook/ comments to: predrag@nbi.dk Contents Contributors x 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 13 1.5 Evolution operators 18 1.6 From chaos to statistical mechanics 22 1.7 Semiclassical quantization 23 1.8 Guide to literature 25 Guide to exercises 27 Resum´e 28 Exercises 32 2Flows 33 2.1 Dynamical systems 33 2.2 Flows 37 2.3 Changing coordinates 41 2.4 Computing trajectories 44 2.5 Infinite-dimensional flows 45 Resum´e 50 Exercises 52 3Maps 57 3.1 Poincar´e sections 57 3.2 Constructing a Poincar´e section 60 3.3 H´enon map 62 3.4 Billiards 64 Exercises 69 4 Local stability 73 4.1 Flows transport neighborhoods 73 4.2 Linear flows 75 4.3 Nonlinear flows 80 4.4 Hamiltonian flows 82 i ii CONTENTS 4.5 Billiards 83 4.6 Maps 86 4.7 Cycle stabilities are metric invariants 87 4.8 Going global: Stable/unstable manifolds 91 Resum´e 92 Exercises 94 5 Transporting densities 97 5.1 Measures 97 5.2 Density evolution 99 5.3 Invariant measures 102 5.4 Koopman, Perron-Frobenius operators 105 Resum´e 110 Exercises 112 6 Averaging 117 6.1 Dynamical averaging 117 6.2 Evolution operators 124 6.3 Lyapunov exponents 126 Resum´e 131 Exercises 132 7 Trace formulas 135 7.1 Trace of an evolution operator 135 7.2 An asymptotic trace formula 142 Resum´e 145 Exercises 146 8 Spectral determinants 147 8.1 Spectral determinants for maps 148 8.2 Spectral determinant for flows 149 8.3 Dynamical zeta functions 151 8.4 False zeros 155 8.5 More examples of spectral determinants 155 8.6 All too many eigenvalues? 158 Resum´e 161 Exercises 163 9 Why does it work? 169 9.1 The simplest of spectral determinants: A single fixed point 170 9.2 Analyticity of spectral determinants 173 9.3 Hyperbolic maps 181 9.4 Physics of eigenvalues and eigenfunctions 185 9.5 Why not just run it on a computer? 188 Resum´e 192 Exercises 194 CONTENTS iii 10 Qualitative dynamics 197 10.1 Temporal ordering: Itineraries 198 10.2 Symbolic dynamics, basic notions 200 10.3 3-disk symbolic dynamics 204 10.4 Spatial ordering of “stretch & fold” flows 206 10.5 Unimodal map symbolic dynamics 210 10.6 Spatial ordering: Symbol square 215 10.7 Pruning 220 10.8 Topological dynamics 222 Resum´e 230 Exercises 233 11 Counting 239 11.1 Counting itineraries 239 11.2 Topological trace formula 241 11.3 Determinant of a graph 243 11.4 Topological zeta function 247 11.5 Counting cycles 249 11.6 Infinite partitions 252 11.7 Shadowing 255 Resum´e 257 Exercises 260 12 Fixed points, and how to get them 269 12.1 One-dimensional mappings 270 12.2 d-dimensional mappings 274 12.3 Flows 275 12.4 Periodic orbits as extremal orbits 279 12.5 Stability of cycles for maps 283 Exercises 288 13 Cycle expansions 293 13.1 Pseudocycles and shadowing 293 13.2 Cycle formulas for dynamical averages 301 13.3 Cycle expansions for finite alphabets 304 13.4 Stability ordering of cycle expansions 305 13.5 Dirichlet series 308 Resum´e 311 Exercises 314 14 Why cycle? 319 14.1 Escape rates 319 14.2 Flow conservation sum rules 323 14.3 Correlation functions 325 14.4 Trace formulas vs. level sums 326 Resum´e 329 iv CONTENTS Exercises 331 15 Thermodynamic formalism 333 15.1 R´enyi entropies 333 15.2 Fractal dimensions 338 Resum´e 342 Exercises 343 16 Intermittency 347 16.1 Intermittency everywhere 348 16.2 Intermittency for beginners 352 16.3 General intermittent maps 365 16.4 Probabilistic or BER zeta functions 371 Resum´e 376 Exercises 378 17 Discrete symmetries 381 17.1 Preview 382 17.2 Discrete symmetries 386 17.3 Dynamics in the fundamental domain 389 17.4 Factorizations of dynamical zeta functions 393 17.5 C 2 factorization 395 17.6 C 3v factorization: 3-disk game of pinball 397 Resum´e 400 Exercises 403 18 Deterministic diffusion 407 18.1 Diffusion in periodic arrays 408 18.2 Diffusion induced by chains of 1-d maps 412 Resum´e 421 Exercises 424 19 Irrationally winding 425 19.1 Mode locking 426 19.2 Local theory: “Golden mean” renormalization 433 19.3 Global theory: Thermodynamic averaging 435 19.4 Hausdorff dimension of irrational windings 436 19.5 Thermodynamics of Farey tree: Farey model 438 Resum´e 444 Exercises 447 20 Statistical mechanics 449 20.1 The thermodynamic limit 449 20.2 Ising models 452 20.3 Fisher droplet model 455 20.4 Scaling functions 461 CONTENTS v 20.5 Geometrization 465 Resum´e 473 Exercises 475 21 Semiclassical evolution 479 21.1 Quantum mechanics: A brief review 480 21.2 Semiclassical evolution 484 21.3 Semiclassical propagator 493 21.4 Semiclassical Green’s function 497 Resum´e 505 Exercises 507 22 Semiclassical quantization 513 22.1 Trace formula 513 22.2 Semiclassical spectral determinant 518 22.3 One-dimensional systems 520 22.4 Two-dimensional systems 522 Resum´e 522 Exercises 527 23 Helium atom 529 23.1 Classical dynamics of collinear helium 530 23.2 Semiclassical quantization of collinear helium 543 Resum´e 553 Exercises 555 24 Diffraction distraction 557 24.1 Quantum eavesdropping 557 24.2 An application 564 Resum´e 571 Exercises 573 Summary and conclusions 575 24.3 Cycles as the skeleton of chaos 575 Index 580 II Material available on www.nbi.dk/ChaosBook/ 595 A What reviewers say 597 A.1 N. Bohr 597 A.2 R.P. Feynman 597 A.3 Divakar Viswanath 597 A.4 Professor Gatto Nero 597 vi CONTENTS B A brief history of chaos 599 B.1 Chaos is born 599 B.2 Chaos grows up 603 B.3 Chaos with us 604 B.4 Death of the Old Quantum Theory 608 C Stability of Hamiltonian flows 611 C.1 Symplectic invariance 611 C.2 Monodromy matrix for Hamiltonian flows 613 D Implementing evolution 617 D.1 Material invariants 617 D.2 Implementing evolution 618 Exercises 623 E Symbolic dynamics techniques 625 E.1 Topological zeta functions for infinite subshifts 625 E.2 Prime factorization for dynamical itineraries 634 F Counting itineraries 639 F.1 Counting curvatures 639 Exercises 641 G Applications 643 G.1 Evolution operator for Lyapunov exponents 643 G.2 Advection of vector fields by chaotic flows 648 Exercises 655 H Discrete symmetries 657 H.1 Preliminaries and Definitions 657 H.2 C 4v factorization 662 H.3 C 2v factorization 667 H.4 Symmetries of the symbol square 670 I Convergence of spectral determinants 671 I.1 Curvature expansions: geometric picture 671 I.2 On importance of pruning 675 I.3 Ma-the-matical caveats 675 I.4 Estimate of the nth cumulant 677 J Infinite dimensional operators 679 J.1 Matrix-valued functions 679 J.2 Trace class and Hilbert-Schmidt class 681 J.3 Determinants of trace class operators 683 J.4 Von Koch matrices 687 J.5 Regularization 689 CONTENTS vii K Solutions 693 L Projects 723 L.1 Deterministic diffusion, zig-zag map 725 L.2 Deterministic diffusion, sawtooth map 732 viii CONTENTS Viele K¨oche verderben den Brei No man but a blockhead ever wrote except for money Samuel Johnson Predrag Cvitanovi´c most of the text Roberto Artuso 5 Transporting densities 97 7.1.4 A trace formula for flows 140 14.3 Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 16 Intermittency 347 18 Deterministic diffusion 407 19 Irrationally winding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 Ronnie Mainieri 2 Flows 33 3.2 The Poincar´e section of a flow 60 4 Local stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.3.2 Understanding flows 43 10.1 Temporal ordering: itineraries 198 20 Statistical mechanics 449 Appendix B: A brief history of chaos 599 G´abor Vattay 15 Thermodynamic formalism 333 ?? Semiclassical evolution ?? 22 Semiclassical trace formula 513 Ofer Biham 12.4.1 Relaxation for cyclists 280 Freddy Christiansen 12 Fixed points, and what to do about them 269 Per Dahlqvist 12.4.2 Orbit length extremization method for billiards . . . . . . . . . . . . . . 282 16 Intermittency 347 CONTENTS ix Appendix E.1.1: Periodic points of unimodal maps 631 Carl P. Dettmann 13.4 Stability ordering of cycle expansions . . . . . . . . . . . . . . . . . . . . . . . . . . 305 Mitchell J. Feigenbaum Appendix C.1: Symplectic invariance 611 KaiT.Hansen 10.5 Unimodal map symbolic dynamics 210 10.5.2 Kneading theory 213 ?? Topological zeta function for an infinite partition . . . . . . . . . . . . . . . . . ?? figures throughout the text Yueheng Lan figures in chapters 1,and17 Joachim Mathiesen 6.3 Lyapunov exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 R¨ossler system figures, cycles in chapters 2, 3, 4 and 12 Adam Pr¨ugel-Bennet Solutions 13.2, 8.1, 1.2, 3.7, 12.9, 2.11, 9.3 Lamberto Rondoni 5 Transporting densities 97 14.1.2 Unstable periodic orbits are dense 323 Juri Rolf Solution 9.3 Per E. Rosenqvist exercises, figures throughout the text Hans Henrik Rugh 9 Why does it work? 169 G´abor Simon R¨ossler system figures, cycles in chapters 2, 3, 4 and 12 Edward A. Spiegel [...]... qualitative and numerical analysis of dynamical systems for short times (trajectories, fixed points, bifurcations) and familiarize them with Cantor sets and symbolic dynamics for chaotic dynamics With this, and graduate level exposure to statistical mechanics, partial differential equations and quantum mechanics, the stage is set for any of the one-semester advanced courses based on this book The courses... 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 Poincar´, the first to understand deterministic chaos, already said as e /chapter/intro.tex 15may2002 printed June 19, 2002 1.2 CHAOS AHEAD 3 much (modulo rat brains) Once the topology of chaotic... have been studied for over 200 years During this time many have contributed, and the field followed no single line of development; rather one sees many interwoven strands of progress In retrospect many triumphs of both classical and quantum physics seem a stroke of luck: a few integrable problems, such as the harmonic oscillator and the Kepler problem, though “non-generic”, have gotten us very far The... parts of things, and in addition had remembrance and intelligence enough to consider all the circumstances and to take them into account, he would be a prophet and would see the future in the present as in a mirror Leibniz chose to illustrate his faith in determinism precisely with the type of physical system that we shall use here as a paradigm of chaos His claim is wrong in a deep and subtle way:... 1.7 chapter 16 sect ?? Semiclassical quantization So far, so good – anyone can play a game of classical pinball, and a skilled neuroscientist can poke rat brains But what happens quantum mechanically, that is, if we scatter waves rather than point-like pinballs? Were the game of pinball a closed system, quantum mechanically one would determine its stationary eigenfunctions and eigenenergies For open... along the r1 axis and escapes to infinity along the r2 axis 0 0 2 4 6 8 r1 taught us that from the classical dynamics point of view, helium is an example of the dreaded and intractable 3-body problem Undaunted, we forge ahead and consider the collinear helium, with zero total angular momentum, and the two electrons on the opposite sides of the nucleus - ++ - We set the electron mass to 1, and the nucleus... graduate) A good start is the textbook by Strogatz [5], an introduction to flows, fixed points, manifolds, bifurcations It is probably the most accessible introduction to nonlinear dynamics - it starts out with differential equations, and its broadly chosen examples and many exercises make it favorite with students It is not strong on chaos There the textbook of Alligood, Sauer and Yorke [6] is preferable:... in the book We start out by making promises - we will right wrongs, no longer shall you suffer the slings and arrows of outrageous Science of Perplexity We relegate a historical overview of the development of chaotic dynamics to appendix B, and head straight to the starting line: A pinball game is used to motivate and illustrate most of the concepts to be developed in this book Throughout the book indicates... the NavierStokes equations) But in practice “turbulence” is very much like “cancer” it is used to refer to messy dynamics which we understand poorly As soon as a phenomenon is understood better, it is reclaimed and renamed: “a route to chaos , “spatiotemporal chaos , and so on Confronted with a potentially chaotic dynamical system, we analyze it through a sequence of three distinct stages; diagnose, count,... OVERTURE Figure 1.4: Some examples of 3-disk cycles: (a) 12123 and 13132 are mapped into each other by σ23 , the flip across 1 axis; this cycle has degeneracy 6 under C3v symmetries (C3v is the symmetry group of the equilateral triangle.) Similarly (b) 123 and 132 and (c) 1213, 1232 and 1323 are degenerate under C3v (d) The cycles 121212313 and 121212323 are related by time reversal but not by any C3v . 507 22 Semiclassical quantization 513 22 .1 Trace formula 513 22 .2 Semiclassical spectral determinant 518 22 .3 One-dimensional systems 520 22 .4 Two-dimensional systems 522 Resum´e 522 Exercises 527 23 . to get them 2 69 12. 1 One-dimensional mappings 27 0 12. 2 d-dimensional mappings 27 4 12. 3 Flows 27 5 12. 4 Periodic orbits as extremal orbits 2 79 12. 5 Stability of cycles for maps 28 3 Exercises 28 8 13. 473 Exercises 475 21 Semiclassical evolution 4 79 21 .1 Quantum mechanics: A brief review 480 21 .2 Semiclassical evolution 484 21 .3 Semiclassical propagator 493 21 .4 Semiclassical Green’s function 497 Resum´e