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classical and quantum chaos predrag cvitanovi´ ­ roberto artuso ­ freddy christiansen ­ per c dahlqvist ­ ronnie mainieri ­ hans henrik rugh ­ gregor tanner ­ g´bor vattay ­ niall whelan ­ andreas wirzba a printed august 24 2000 version 7.0.1 aug 6 2000 www.nbi.dk/chaosbook comments to predrag@nbi.dk

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contents contributors 1 overture 1.1 why this book 1.2 chaos ahead 1.3 a game of pinball 1.4 periodic orbit theory 1.5 evolution operators 1.6 from chaos to statistical mechanics 1.7 semiclassical quantization 1.8 guide to literature guide to exercises resum´ e exercises 2 trajectories 2.1 flows 2.2 maps 2.3 infinite-dimensional flows exercises ii 1 2 3 4 12 17 20 21 23 25 26 30 31 31 36 40 47 51 51 56 56 60 63 64 65 66 68 72 73 3 local stability 3.1 flows transport neighborhoods 3.2 linear stability of maps 3.3 billiards exercises 4 transporting densities 4.1 measures 4.2 density evolution 4.3 invariant measures 4.4 evolution operators resum´ e exercises i .

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ii 5 averaging 5.1 dynamical averaging 5.2 evolution operators resum´ e exercises contents 77 77 83 87 89 91 91 98 100 101 6 trace formulas 6.1 trace of an evolution operator 6.2 an asymptotic trace formula resum´ e exercises 7 qualitative dynamics 7.1 temporal ordering itineraries 7.2 3-disk symbolic dynamics 7.3 spatial ordering of stretch fold 7.4 unimodal map symbolic dynamics 7.5 spatial ordering symbol plane 7.6 pruning 7.7 topological dynamics resum´ e exercises 8 fixed points and how to get them 8.1 one-dimensional mappings 8.2 d-dimensional mappings 8.3 flows 8.4 periodic orbits as extremal orbits resum´ e exercises 9 counting 9.1 counting itineraries 9.2 topological trace formula 9.3 determinant of a graph 9.4 topological zeta function 9.5 counting cycles 9.6 infinite partitions resum´ e exercises flows 103 103 106 110 111 116 122 123 129 134 141 142 145 146 151 156 159 167 167 169 171 175 176 180 184 187 10 spectral determinants 10.1 spectral determinants for maps 10.2 spectral determinant for flows 10.3 dynamical zeta functions 10.4 the simplest of spectral determinants a single fixed point 195 196 198 199 203

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contents 10.5 false zeros 10.6 all too many eigenvalues 10.7 more examples of spectral determinants resum´ e exercises 11 cycle expansions 11.1 pseudocycles and shadowing 11.2 cycle formulas for dynamical averages 11.3 cycle expansions for finite alphabets 11.4 stability ordering of cycle expansions 11.5 dirichlet series exercises iii 204 205 206 210 212 219 219 227 230 231 235 239 245 246 249 254 258 259 265 267 267 271 272 274 276 277 278 279 281 284 286 12 why does it work 12.1 curvature expansions geometric picture 12.2 analyticity of spectral determinants 12.3 hyperbolic maps 12.4 on importance of pruning resum´ e exercises 13 getting used to cycles 13.1 escape rates 13.2 flow conservation sum rules 13.3 lyapunov exponents 13.4 correlation functions 13.5 trace formulas vs level sums 13.6 eigenstates 13.7 why not just run it on a computer 13.8 ma-the-matical caveats 13.9 cycles as the skeleton of chaos resum´ e exercises 14 thermodynamic formalism 14.1 r´nyi entropies e 14.2 fractal dimensions resum´ e exercises 289 289 294 298 299 303 303 308 311 315 15 discrete symmetries 15.1 preview 15.2 discrete symmetries 15.3 dynamics in the fundamental domain 15.4 factorizations of dynamical zeta functions .

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iv 15.5 c2 factorizations 15.6 c3v factorization resum´ e exercises 3-disk game of pinball contents 317 319 323 325 16 deterministic diffusion 16.1 diffusion in periodic arrays 16.2 diffusion induced by chains of resum´ e exercises 1-d maps 329 330 334 343 345 347 348 352 359 364 368 372 374 17 why doesn t it work 17.1 escape averages and periodic orbits 17.2 know thy enemy 17.3 defeating your enemy intermittency resummed 17.4 marginal stability and anomalous diffusion 17.5 probabilistic or ber zeta functions resum´ e exercises 18 semiclassical evolution 18.1 quantum mechanics a brief review 18.2 semiclassical evolution 18.3 semiclassical propagator 18.4 semiclassical green s function resum´ e exercises 19 semiclassical quantization 19.1 trace formula 19.2 semiclassical spectral determinant 19.3 one-dimensional systems 19.4 two-dimensional systems resum´ e exercises 377 378 382 391 395 401 404 409 409 414 416 417 418 423 20 semiclassical chaotic scattering 425 20.1 quantum mechanical scattering matrix 425 20.2 krein-friedel-lloyd formula 428 exercises 433 21 helium atom 21.1 classical dynamics of collinear helium 21.2 semiclassical quantization of collinear helium resum´ e exercises 435 436 448 458 460 .

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contents v 22 diffraction distraction 463 22.1 quantum eavesdropping 463 22.2 an application 470 exercises 479 23 irrationally winding 23.1 mode locking 23.2 local theory golden mean renormalization 23.3 global theory thermodynamic averaging 23.4 hausdorff dimension of irrational windings 23.5 thermodynamics of farey tree farey model resum´ e exercises 24 statistical mechanics 24.1 the thermodynamic limit 24.2 ising models 24.3 fisher droplet model 24.4 scaling functions 24.5 geometrization resum´ e exercises summary and conclusions 481 482 488 490 492 494 500 502 503 503 506 509 515 519 527 529 533 a linear stability of hamiltonian flows 537 a.1 symplectic invariance 537 a.2 monodromy matrix for hamiltonian flows 539 b symbolic dynamics techniques b.1 symbolic dynamics basic notions b.2 topological zeta functions for infinite subshifts b.3 prime factorization for dynamical itineraries b.4 counting curvatures exercises 543 543 546 555 559 560 563 563 568 575 577 577 582 585 c applications c.1 evolution operator for lyapunov exponents c.2 advection of vector fields by chaotic flows exercises d discrete symmetries d.1 c4v factorization d.2 c2v factorization d.3 symmetries of the symbol plane .

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vi contents e convergence of spectral determinants 587 e.1 estimate of the nth cumulant 587 f infinite dimensional operators f.1 matrix-valued functions f.2 trace class and hilbert-schmidt class f.3 determinants of trace class operators f.4 von koch matrices f.5 regularization g trace of the scattering matrix index 589 589 591 593 597 599 603 606 ii material available on 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 i.1 chaos is born i.2 chaos grows up i.3 chaos with us i.4 death of the old quantum theory 611 611 615 616 620 623 j solutions k projects 645 k.1 deterministic diffusion zig-zag map 647 k.2 deterministic diffusion sawtooth map 654

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contents vii contributors no man but a blockhead ever wrote except for money samuel johnson predrag cvitanovi´ c 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 poincar´ section of a flow 39 e 3 local stability 51 understanding flows 7.1 temporal ordering itineraries .103 24 statistical mechanics 503 appendix i a brief history of chaos 611 g´bor vattay a 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

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viii contents appendix b.2.1 periodic points of unimodal maps .553 carl p dettmann 11.4 stability ordering of cycle expansions 231 mitchell j feigenbaum appendix a.1 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 pr¨ gel-bennet u 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

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contents niall whelan ix 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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x contents

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chapter 1 overture if i have seen less far than other men it is because i have stood behind giants edoardo specchio rereading classic theoretical physics textbooks leaves a sense that there are holes large enough to steam a eurostar train through them here we learn about harmonic oscillators and keplerian ellipses but where is the chapter on chaotic oscillators the tumbling hyperion we have just quantized hydrogen where is the chapter on helium we have learned that an instanton is a solution of fieldtheoretic equations of motion but shouldn t a strongly nonlinear field theory have turbulent solutions how are we to think about systems where the middle does not hold everything continuously falls apart every trajectory is unstable 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 i 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 unstable dynamical flows poincar´ sections smale horseshoes symbolic dynamics pruning discrete e symmetries periodic orbits averaging over chaotic sets evolution operators dynamical zeta functions spectral determinants cycle expansions quantum trace formulas and zeta functions and so on to the semiclassical quantization of helium this chapter is a quick par-course of the main topics covered in the book throughout the book indicates that the section is probably best skipped on first reading fast track points you where to skip to 1

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2 chapter 1 overture tells you where to go for more depth on a particular topic indicates an exercise that might clarify a point in the text 1.1 why this book it seems sometimes that through a preoccupation with science we acquire a firmer hold over the vicissitudes of life and meet them with greater calm but in reality we have done no more than to find a way to escape from our sorrows hermann minkowski in a letter to david hilbert the problem has been with us since newton s first frustrating and unsuccessful crack at the 3-body problem lunar dynamics nature is rich in systems governed by simple deterministic laws whose asymptotic dynamics are complex beyond belief systems which are locally unstable almost everywhere but globally recurrent how do we describe their long term dynamics the answer turns out to be that we have to evaluate a determinant take a logarithm it would hardly merit a learned treatise were it not for the fact that this determinant that we are to compute is fashioned of infinitely many infinitely small pieces the feel is of statistical mechanics and that is how the problem was solved in 1960 s the pieces were counted and in 1970 s they were weighted and assembled together in a fashion that in beauty and in depth ranks along with thermodynamics partition functions and path integrals amongst the crown jewels of theoretical physics then something happened that might be without parallel this is an area of science where the advent of cheap computation had actually subtracted from our collective understanding the computer pictures and numerical plots of fractal science of 1980 s have overshadowed the deep insights of the 1970 s and these pictures have now migrated into textbooks fractal science posits that certain quantities lyapunov exponents generalized dimensions can be estimated on a computer while some of the numbers so obtained are indeed mathematically sensible characterizations of fractals they are in no sense observable and measurable on the length and time scales dominated by chaotic dynamics 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 dasbuch/book/chapter/intro.tex 4aug2000 printed august 24 2000

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1.2 chaos ahead 3 rat brains poincar´ the first to understand deterministic chaos already said as e 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 sorry about that 1.2 chaos ahead study of chaotic dynamical systems is no recent fashion it did not start with the widespread use of the personal computer chaotic systems 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 success has lulled us into a habit of expecting simple solutions to simple equations an expectation tempered for many by the recently acquired ability to numerically scan the phase space of non-integrable dynamical systems the initial impression might be that all our analytic tools have failed us and that the chaotic systems are amenable only to numerical and statistical investigations however as we show here we already possess a theory of the deterministic chaos of predictive quality comparable to that of the traditional perturbation expansions for nearly integrable systems in the traditional approach the integrable motions are used as zeroth-order approximations to physical systems and weak nonlinearities are then accounted for perturbatively for strongly nonlinear non-integrable systems such expansions fail completely the asymptotic time phase space exhibits amazingly rich structure which is not at all apparent in the integrable approximations however hidden in this apparent chaos is a rigid skeleton a tree of cycles periodic orbits of increasing lengths and self-similar structure the insight of the modern dynamical systems theory is that the zeroth-order approximations to the harshly chaotic dynamics should be very different from those for the nearly integrable systems a good starting approximation here is the linear stretching and folding of a baker s map rather than the winding of a harmonic oscillator so what is chaos and what is to be done about it to get some feeling for how and why unstable cycles come about we start by playing a game of pinball the reminder of the chapter is a quick tour through the material covered in this printed august 24 2000 dasbuch/book/chapter/intro.tex 4aug2000

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4 chapter 1 overture figure 1.1 physicists bare bones game of pinball book do not worry if you do not understand every detail at the first reading ­ the intention is to give you a feeling for the main themes of the book details will be filled out later if you want to get a particular point clarified right now on the margin points at the appropriate section 1.3 a game of pinball man m° begrænse sig det er en hovedbetingelse for al a nydelse søren kierkegaard forførerens dagbog that deterministic dynamics leads to chaos is no surprise to anyone who has tried pool billiards or snooker ­ that is what the game is about ­ so we start our story about what chaos is and what to do about it with a game of pinball this might seem a trifle but the game of pinball is to chaotic dynamics what a pendulum is to integrable systems thinking clearly about what chaos in a game of pinball is will help us tackle more difficult problems such as computing diffusion constants in deterministic gases or computing the helium spectrum we all have an intuitive feeling for what a ball does as it bounces among the pinball machine s disks and only high-school level euclidean geometry is needed to describe its trajectory a physicist s pinball game is the game of pinball stripped to its bare essentials three equidistantly placed reflecting disks in a plane fig 1.1 physicists pinball is free frictionless point-like spin-less perfectly elastic and noiseless point-like pinballs are shot at the disks from random starting positions and angles they spend some time bouncing between the disks and then escape at the beginning of 18th century baron gottfried wilhelm leibniz was confident that given the initial conditions one knew what a deterministic system would do far into the future he wrote [1 that everything is brought forth through an established destiny is just dasbuch/book/chapter/intro.tex 4aug2000 printed august 24 2000

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