Chaos: Classical and Quantum

Chaos: Classical and QuantumVolume I: Deterministic ChaosPredrag Cvitanovic, Roberto Artuso, Ronnie Mainieri,Gregor Tanner and Gabor Vattayformerly of CATS

GONE WITH THE WIND PRESS

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ATLANTA

ChaosBook.org version13.5, Sep 7 2011

This is dedicated to our long-suering partners.

ContentsContributorsAcknowledgements1 Overture1.1 Why ChaosBook?1.2 Chaos ahead1.3 The future as in a mirror1.4 A game of pinball1.5 Chaos for cyclists1.6 Change in time1.7 From chaos to statistical mechanics1.8 Chaos: what is it good for?1.9 What is not in ChaosBook

xiiixvii112371015171820

resume 21 further reading 23 guide to exercises 25 exercises 26 references 26

I Geometry of chaos

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2 Go with the flow2.1 Dynamical systems2.2 Flows2.3 Computing trajectories

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resume 39 further reading 39 exercises 41 references 42

3 Discrete time dynamics3.1 Poincare sections3.2 Constructing a Poincare section3.3 Maps

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4 Local stability4.1 Flows transport neighborhoods4.2 Linear flows4.3 Stability of flows4.4 Neighborhood volume4.5 Stability of maps

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5 Cycle stability5.1 Stability of periodic orbits

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viii Contents

5.25.35.4

Floquet multipliers are invariantStability of Poincare map cyclesThere goes the neighborhood

838485


6 Go straight6.1 Changing coordinates6.2 Rectification of flows6.3 Collinear helium6.4 Rectification of maps6.5 Rectification of a periodic orbit6.6 Cycle Floquet multipliers are metric invariants

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7 Hamiltonian dynamics7.1 Hamiltonian flows7.2 Stability of Hamiltonian flows7.3 Symplectic maps7.4 Poincare invariants

103103105107109

further reading 110 exercises 111 references 111

8 Billiards8.1 Billiard dynamics8.2 Stability of billiards

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9 World in a mirror9.1 Discrete symmetries9.2 Symmetries of solutions9.3 Relative periodic orbits9.4 Dynamics reduced to fundamental domain

121121128131133


10 Relativity for cyclists10.1 Continuous symmetries10.2 Symmetries of solutions10.3 Stability10.4 Reduced state space10.5 Method of images: Hilbert bases

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11 Charting the state space11.1 Qualitative dynamics11.2 From d-dimensional flows to 1-dimensional maps11.3 Temporal ordering: itineraries11.4 Spatial ordering11.5 Kneading theory11.6 Symbolic dynamics, basic notionsresume 188 further reading 189 exercises 190 references 191

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Contents ix

12 Stretch, fold, prune12.1 Going global: stable/unstable manifolds12.2 Horseshoes12.3 Symbol plane12.4 Prune danish12.5 Recoding, symmetries, tilings

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13 Fixed points, and how to get them13.1 Where are the cycles?13.2 One-dimensional mappings13.3 Multipoint shooting method13.4 Flows

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II Chaos rules

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14 Walkabout: Transition graphs14.1 Matrix representations of topological dynamics14.2 Transition graphs: wander from node to node14.3 Transition graphs: stroll from link to link

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15 Counting15.1 How many ways to get there from here?15.2 Topological trace formula15.3 Determinant of a graph15.4 Topological zeta function15.5 Topological zeta function for an infinite partition15.6 Shadowing15.7 Counting cycles

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16 Transporting densities16.1 Measures16.2 Perron-Frobenius operator16.3 Why not just leave it to a computer?16.4 Invariant measures16.5 Density evolution for infinitesimal times16.6 Liouville operator

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17 Averaging17.1 Dynamical averaging17.2 Evolution operators17.3 Lyapunov exponentsresume 296 further reading 297 exercises 298 references 299

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18 Trace formulas18.1 A trace formula for maps18.2 A trace formula for flows18.3 An asymptotic trace formula

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19 Spectral determinants19.1 Spectral determinants for maps19.2 Spectral determinant for flows19.3 Dynamical zeta functions19.4 False zeros19.5 Spectral determinants vs. dynamical zeta functions19.6 All too many eigenvalues?

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20 Cycle expansions20.1 Pseudocycles and shadowing20.2 Construction of cycle expansions20.3 Cycle formulas for dynamical averages20.4 Cycle expansions for finite alphabets20.5 Stability ordering of cycle expansions20.6 Dirichlet series

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21 Discrete factorization21.1 Preview21.2 Discrete symmetries21.3 Dynamics in the fundamental domain21.4 Factorizations of dynamical zeta functions21.5 C 2 factorization21.6 D3 factorization: 3-disk game of pinball

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III Chaos: what to do about it?

365

22 Why cycle?22.1 Escape rates22.2 Natural measure in terms of periodic orbits22.3 Flow conservation sum rules22.4 Correlation functions22.5 Trace formulas vs. level sums

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23 Why does it work?23.1 Linear maps: exact spectra23.2 Evolution operator in a matrix representation23.3 Classical Fredholm theory23.4 Analyticity of spectral determinants

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23.5 Hyperbolic maps23.6 The physics of eigenvalues and eigenfunctions23.7 Troubles ahead

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24 Intermittency24.1 Intermittency everywhere24.2 Intermittency for pedestrians24.3 Intermittency for cyclists24.4 BER zeta functions

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25 Deterministic diusion25.1 Diusion in periodic arrays25.2 Diusion induced by chains of 1d maps25.3 Marginal stability and anomalous diusion

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IV The rest is noise

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26 Noise26.1 Deterministic transport26.2 Brownian diusion26.3 Weak noise26.4 Weak noise approximation

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27 Relaxation for cyclists27.1 Fictitious time relaxation27.2 Discrete iteration relaxation method27.3 Least action method

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Index

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V ChaosBook.org web appendices

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A A brief history of chaosA.1 Chaos is bornA.2 Chaos with usA.3 Death of the Old Quantum Theory

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further reading 491 references 492

B Linear stabilityB.1 Linear algebraB.2 Eigenvalues and eigenvectorsB.3 Eigenspectra: what to make out of them?

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B.4 Stability of Hamiltonian flowsB.5 Monodromy matrix for Hamiltonian flows

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exercises 507 references 507

C Discrete symmetries of dynamicsC.1 Preliminaries and definitionsC.2 Invariants and reducibilityC.3 Lattice derivativesC.4 Periodic latticesC.5 Discrete Fourier transformsC.6 C4v factorizationC.7 C2v factorizationC.8 Henon map symmetriesfurther reading 531 exercises 531 references 533

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ContributorsNo man but a blockhead ever wrote except for moneySamuel Johnson

This book is a result of collaborative labors of many people over a span ofseveral decades. Coauthors of a chapter or a section are indicated in the bylineto the chapter/section title. If you are referring to a specific coauthored sectionrather than the entire book, cite it as (for example):C. Chandre, F.K. Diakonos and P. Schmelcher, section Discrete cyclistrelaxation method, in P. Cvitanovic, R. Artuso, R. Mainieri, G. Tanner and G. Vattay, Chaos: Classical and Quantum (Niels Bohr Institute,Copenhagen 2010); ChaosBook.org/version13.

Do not cite chapters by their numbers, as those change from version to version.Chapters without a byline are written by Predrag Cvitanovic. Friends whosecontributions and ideas were invaluable to us but have not contributed writtentext to this book, are credited in the acknowledgments.Roberto Artuso16 Transporting densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26918.2 A trace formula for flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30622.4 Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37124 Intermittency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40125 Deterministic diusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427Ronnie Mainieri2 Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 The Poincare section of a flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 Local stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.1 Understanding flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9111.1 Temporal ordering: itineraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173Appendix 28: A brief history of chaos . . . . . . . . . . . . . . . . . . . . . . . . . 477Gabor VattayGregor Tanner24 Intermittency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .401Appendix B.5: Jacobians of Hamiltonian flows . . . . . . . . . . . . . . . . . 505Arindam BasuRossler flow figures, tables, cycles in chapters 11, 13 and exercise13.10Ofer Biham27.1 Cyclists relaxation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454Daniel Borrero Oct 23 2008, soluCycles.texSolution 13.15

xiv Contributors

Cristel Chandre27.1 Cyclists relaxation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45427.2 Discrete cyclists relaxation methods . . . . . . . . . . . . . . . . . . . . . . 458Freddy Christiansen13.2 One-dimensional mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21913.3 Multipoint shooting method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .220Per Dahlqvist24 Intermittency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40127.3 Orbit length extremization method for billiards . . . . . . . . . . . . 460Carl P. Dettmann20.5 Stability ordering of cycle expansions . . . . . . . . . . . . . . . . . . . . . 337Fotis K. Diakonos27.2 Discrete cyclists relaxation methods . . . . . . . . . . . . . . . . . . . . . . 458G. Bard ErmentroutExercise 5.1Mitchell J. FeigenbaumAppendix B.4: Symplectic invariance . . . . . . . . . . . . . . . . . . . . . . . . . 504Jonathan HalcrowExample 3.5: Sections of Lorenz flow . . . . . . . . . . . . . . . . . . . . . . . . . . 50Example 4.7: Stability of Lorenz flow equilibria . . . . . . . . . . . . . . . . . 67Example 4.8: Lorenz flow: Global portrait . . . . . . . . . . . . . . . . . . . . . . 69Example 9.10: Desymmetrization of Lorenz flow . . . . . . . . . . . . . . . 129Example 11.4: Lorenz flow: a 1d return map . . . . . . . . . . . . . . . . . . 178Exercises 9.9 and figure 2.5Kai T. Hansen11.3 Unimodal map symbolic dynamics . . . . . . . . . . . . . . . . . . . . . . . 17915.5 Topological zeta function for an infinite partition . . . . . . . . . . . 25611.5 Kneading theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184figures throughout the textRainer KlagesFigure 25.5Yueheng LanSolutions 1.1, 2.1, 2.2, 2.3, 2.4, 2.5, 9.6, 12.6, 11.6, 16.1, 16.2, 16.3,16.5, 16.7, 16.10, 17.1 and figures 1.9, 9.2, 9.8 11.5,Joachim Mathiesen17.3 Lyapunov exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293Rossler flow figures, tables, cycles in Section 17.3 and exercise 13.10Yamato MatsuokaFigure 12.4Radford Mitchell, Jr.Example 3.6Rytis Paskauskas4.5.1 Stability of Poincare return maps . . . . . . . . . . . . . . . . . . . . . . . . . . 735.3 Stability of Poincare map cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Exercises 2.8, 3.1, 4.4 and solution 4.1Adam Prugel-BennetSolutions 1.2, 2.10, 8.1, 19.1, 20.2 23.3, 27.1.Lamberto Rondoni16 Transporting densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

xv

13.1.1 Cycles from long time series . . . . . . . . . . . . . . . . . . . . . . . . . . . 21722.2.1 Unstable periodic orbits are dense . . . . . . . . . . . . . . . . . . . . . . 370Table 15.2Juri RolfSolution 23.3Per E. Rosenqvistexercises, figures throughout the textHans Henrik Rugh23 Why does it work? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377Peter Schmelcher27.2 Discrete cyclists relaxation methods . . . . . . . . . . . . . . . . . . . . . . 458Evangelos SiminosExample 3.5: Sections of Lorenz flow . . . . . . . . . . . . . . . . . . . . . . . . . . 50Example 4.7: Stability of Lorenz flow equilibria . . . . . . . . . . . . . . . . . 67Example 4.8: Lorenz flow: Global portrait . . . . . . . . . . . . . . . . . . . . . . 69Example 9.10: Desymmetrization of Lorenz flow . . . . . . . . . . . . . . . 129Example 11.4: Lorenz flow: a 1d return map . . . . . . . . . . . . . . . . . . 178Exercise 9.9Solution 10.21Gabor SimonRossler flow figures, tables, cycles in chapters 2, 13 and exercise 13.10Edward A. Spiegel2 Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3116 Transporting densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269Luz V. Vela-Arevalo7.1 Hamiltonian flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Exercises 7.1, 7.3, 7.5R. WilczakFigure 10.1, Fig. 10.4Exercise 10.26Solutions 10.1, 10.5, 10.6, 10.7, 10.8, 10.10, 10.14, 10.15, 10.16, 10.17,10.18, 10.19, 10.20, 10.22, 10.23

AcknowledgementsI feel I never want to write another book. Whats the good! I can ekeliving on stories and little articles, that dont cost a tithe of the outputa book costs. Why write novels any more!D.H. Lawrence

This book owes its existence to the Niels Bohr Institutes and Norditas hospitable and nurturing environment, and the private, national and cross-nationalfoundations that have supported the collaborators research over a span of several decades. P.C. thanks M.J. Feigenbaum of Rockefeller University; D. Ruelle of I.H.E.S., Bures-sur-Yvette; I. Procaccia of the Weizmann Institute;P.H. Damgaard of the Niels Bohr International Academy; G. Mazenko of U. ofChicago James Franck Institute and Argonne National Laboratory; T. Geiselof Max-Planck-Institut fur Dynamik und Selbstorganisation, Gottingen; I. Andric of Rudjer Boskovic Institute; P. Hemmer of University of Trondheim; TheMax-Planck Institut fur Mathematik, Bonn; J. Lowenstein of New York University; Edificio Celi, Milano; Fundacao de Faca, Porto Seguro; and Dr. Dj. Cvitanovic, Kostrena, for the hospitality during various stages of this work, and theCarlsberg Foundation, Glen P. Robinson, Humboldt Foundation and NationalScience Fundation grant DMS-0807574 for partial support.The authors gratefully acknowledge collaborations and/or stimulating discussions with E. Aurell, M. Avila, V. Baladi, D. Barkley, B. Brenner, A. de Carvalho, D.J. Driebe, B. Eckhardt, M.J. Feigenbaum, J. Frjland, S. Froehlich,P. Gaspar, P. Gaspard, J. Guckenheimer, G.H. Gunaratne, P. Grassberger,H. Gutowitz, M. Gutzwiller, K.T. Hansen, P.J. Holmes, T. Janssen, R. Klages,Y. Lan, B. Lauritzen, J. Milnor, M. Nordahl, I. Procaccia, J.M. Robbins, P.E. Rosenqvist, D. Ruelle, G. Russberg, B. Sandstede, M. Sieber, D. Sullivan, N. Sndergaard,T. Tel, C. Tresser, R. Wilczak, and D. Wintgen.We thank Dorte Glass for typing parts of the manuscript; D. Borrero, B. Lautrup, J.F Gibson and D. Viswanath for comments and corrections to the preliminary versions of this text; M.A. Porter for lengthening the manuscript bythe 2013 definite articles hitherto missing; M.V. Berry for the quotation onpage 477; H. Fogedby for the quotation on page 386; J. Greensite for the quotation on page 4; Ya.B. Pesin for the remarks quoted on page 491; M.A. Porterfor the quotations on page 15 and page 483; E.A. Spiegel for quotation onpage 1; and E. Valesco for the quotation on page 18.F. Haakes heartfelt lament on page 306 was uttered at the end of the firstconference presentation of cycle expansions, in 1988. G.P. Morriss adviceto students as how to read the introduction to this book, page 3, was oerred

xviii Acknowledgements

during a 2002 graduate course in Dresden. K. Huangs C.N. Yang interviewquoted on page 275 is available on ChaosBook.org/extras. T.D. Lee remarks on as to who is to blame, page 31 and page 216, as well as M. Shubshelpful technical remark on page 396 came during the Rockefeller UniversityDecember 2004 Feigenbaum Fest . Quotes on pages 31, 103, and 272 aretaken from a book review by J. Guckenheimer [0.1].Who is the 3-legged dog reappearing throughout the book? Long ago, whenwe were innocent and knew not Borel measurable to sets, P. Cvitanovicasked V. Baladi a question about dynamical zeta functions, who then askedJ.-P. Eckmann, who then asked D. Ruelle. The answer was transmitted back:The master says: It is holomorphic in a strip. Hence His Masters Voicelogo, and the 3-legged dog is us, still eager to fetch the bone. The answer hasmade it to the book, though not precisely in His Masters voice. As a matter offact, the answer is the book. We are still chewing on it.Profound thanks to all the unsung heroesstudents and colleagues, too numerous to list herewho have supported this project over many years in manyways, by surviving pilot courses based on this book, by providing invaluableinsights, by teaching us, by inspiring us.

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OvertureIf I have seen less far than other men it is because I have stood behindgiants.Edoardo Specchio

R

ereading classic theoretical physics textbooks leaves a sense that thereare holes large enough to steam a Eurostar train through them. Herewe learn about harmonic oscillators and Keplerian ellipses - but whereis the chapter on chaotic oscillators, the tumbling Hyperion? We have justquantized hydrogen, where is the chapter on the classical 3-body problem andits implications for quantization of helium? We have learned that an instantonis a solution of field-theoretic equations of motion, but shouldnt a stronglynonlinear field theory have turbulent solutions? How are we to think aboutsystems where things fall apart; the center cannot hold; every trajectory isunstable?This chapter oers a quick survey of the main topics covered in the book.We start out by making promiseswe will right wrongs, no longer shall yousuer the slings and arrows of outrageous Science of Perplexity. We relegatea historical overview of the development of chaotic dynamics to Appendix 28,and head straight to the starting line: A pinball game is used to motivate andillustrate most of the concepts to be developed in ChaosBook.This is a textbook, not a research monograph, and you should be able to follow the thread of the argument without constant excursions to sources. Hencethere are no literature references in the text proper, all learned remarks andbibliographical pointers are relegated to the Further reading section at theend of each chapter.

1.1 Why ChaosBook?It seems sometimes that through a preoccupation with science, weacquire a firmer hold over the vicissitudes of life and meet them withgreater calm, but in reality we have done no more than to find a wayto escape from our sorrows.Hermann Minkowski in a letter to David Hilbert

The problem has been with us since Newtons first frustrating (and unsuccessful) crack at the 3-body problem, lunar dynamics. Nature is rich in systemsgoverned by simple deterministic laws whose asymptotic dynamics are complex beyond belief, systems which are locally unstable (almost) everywherebut globally recurrent. How do we describe their long term dynamics?

1.1 Why ChaosBook?

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1.2 Chaos ahead

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1.3 The future as in a mirror

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1.4 A game of pinball

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1.5 Chaos for cyclists

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1.6 Change in time

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1.7 From chaos to statistical mechanics 171.8 Chaos: what is it good for?

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1.9 What is not in ChaosBook

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Resume

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Further reading

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A guide to exercises

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Exercises

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References

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Throughout the bookindicates that the section is ona pedestrian level - you are expected to know/learn this materialindicates that the section is on asomewhat advanced, cyclist levelindicates that the section requiresa hearty stomach and is probablybest skipped on first readingfast track points you where toskip totells you where to go for moredepth on a particular topic[chapter 3] on margin links to arelated chapter[exercise 1.2] on margin links toan exercise that might clarify apoint in the textindicates that a figure is stillmissingyou are urged to fetch itIn the hyperlinked ChaosBook.pdf thesedestinations are only a click away.

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CHAPTER 1. OVERTURE

The answer turns out to be that we have to evaluate a determinant, take alogarithm. It would hardly merit a learned treatise, were it not for the fact thatthis determinant that we are to compute is fashioned out of infinitely manyinfinitely small pieces. The feel is of statistical mechanics, and that is howthe problem was solved; in the 1960s the pieces were counted, and in the1970s they were weighted and assembled in a fashion that in beauty and indepth ranks along with thermodynamics, partition functions and path integralsamongst the crown jewels of theoretical physics.This book is not a book about periodic orbits. The red thread throughout thetext is the duality between the local, topological, short-time dynamically invariant compact sets (equilibria, periodic orbits, partially hyperbolic invarianttori) and the global long-time evolution of densities of trajectories. Chaoticdynamics is generated by the interplay of locally unstable motions, and theinterweaving of their global stable and unstable manifolds. These features arerobust and accessible in systems as noisy as slices of rat brains. Poincare,the first to understand deterministic chaos, already said as much (modulo ratbrains). Once this topology is understood, a powerful theory yields the observable consequences of chaotic dynamics, such as atomic spectra, transportcoecients, gas pressures.That is what we will focus on in ChaosBook. The book is a self-containedgraduate textbook on classical and quantum chaos. Your professor does notknow this material, so you are on your own. We will teach you how to evaluatea determinant, take a logarithmstu like that. Ideally, this should take 100pages or so. Well, we failso far we have not found a way to traverse thismaterial in less than a semester, or 200-300 page subset of this text. Nothingto be done.

1.2 Chaos aheadThings fall apart; the centre cannot hold.W.B. Yeats: The Second Coming

The study of chaotic dynamics is no recent fashion. It did not start with thewidespread use of the personal computer. Chaotic systems have been studiedfor over 200 years. During this time many have contributed, and the field followed no single line of development; rather one sees many interwoven strandsof progress.In retrospect many triumphs of both classical and quantum physics were astroke of luck: a few integrable problems, such as the harmonic oscillator andthe Kepler problem, though non-generic, have gotten us very far. The successhas lulled us into a habit of expecting simple solutions to simple equationsanexpectation tempered by our recently acquired ability to numerically scan thestate space of non-integrable dynamical systems. The initial impression mightbe that all of our analytic tools have failed us, and that the chaotic systemsare amenable only to numerical and statistical investigations. Nevertheless,a beautiful theory of deterministic chaos, of predictive quality comparable tothat of the traditional perturbation expansions for nearly integrable systems,intro - 9apr2009


1.3. THE FUTURE AS IN A MIRROR

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already exists.In the traditional approach the integrable motions are used as zeroth-orderapproximations to physical systems, and weak nonlinearities are then accountedfor perturbatively. For strongly nonlinear, non-integrable systems such expansions fail completely; at asymptotic times the dynamics exhibits amazinglyrich structure which is not at all apparent in the integrable approximations.However, hidden in this apparent chaos is a rigid skeleton, a self-similar treeof cycles (periodic orbits) of increasing lengths. The insight of the modern dynamical systems theory is that the zeroth-order approximations to the harshlychaotic dynamics should be very dierent from those for the nearly integrablesystems: a good starting approximation here is the stretching and folding ofbakers dough, rather than the periodic motion of a harmonic oscillator.So, what is chaos, and what is to be done about it? To get some feelingfor how and why unstable cycles come about, we start by playing a gameof pinball. The reminder of the chapter is a quick tour through the materialcovered in ChaosBook. Do not worry if you do not understand every detail atthe first readingthe intention is to give you a feeling for the main themes ofthe book. Details will be filled out later. If you want to get a particular pointclarified right now, [section 1.4] on the margin points at the appropriate section.

section 1.4

1.3 The future as in a mirrorAll you need to know about chaos is contained in the introduction of[ChaosBook]. However, in order to understand the introduction youwill first have to read the rest of the book.Gary Morriss

That deterministic dynamics leads to chaos is no surprise to anyone who hastried pool, billiards or snookerthe game is about beating chaosso we startour 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 whata pendulum is to integrable systems: thinking clearly about what chaos in agame of pinball is will help us tackle more dicult problems, such as computing the diusion constant of a deterministic gas, the drag coecient of aturbulent boundary layer, or the helium spectrum.We all have an intuitive feeling for what a ball does as it bounces amongthe pinball machines disks, and only high-school level Euclidean geometryis needed to describe its trajectory. A physicists pinball game is the gameof pinball stripped to its bare essentials: three equidistantly placed reflectingdisks in a plane, Fig. 1.1. A physicists pinball is free, frictionless, pointlike, spin-less, perfectly elastic, and noiseless. Point-like pinballs are shot atthe disks from random starting positions and angles; they spend some timebouncing between the disks and then escape.At the beginning of the 18th century Baron Gottfried Wilhelm Leibniz wasconfident that given the initial conditions one knew everything a deterministicsystem would do far into the future. He wrote [1.2], anticipating by a centuryand a half the oft-quoted Laplaces Given for one instant an intelligence whichChaosBook.org version13.5, Sep 7 2011

intro - 9apr2009

Fig. 1.1 A physicists bare bones game ofpinball.

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could comprehend all the forces by which nature is animated...:

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1

3

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Fig. 1.2 Sensitivity to initial conditions: twopinballs that start out very close to each otherseparate exponentially with time.

1Stochastic is derived from Greek stochos, meaning a target, as in shooting arrowsat a target, and not always hitting it; targetedflow, with a small component of uncertainty.Today it stands for deterministic drift + diffusion. Random stands for pure diusion,with a Gaussian profile. Probabilistic mighthave a distribution other than a Gaussian one.Boltzmanns Ergodic refers to the deterministic microscopic dynamics of many collidingmolecules. Chaotic is the same thing, butusually for a few degrees of freedom.

x(t) x(0)x(0)

x(t)

That everything is brought forth through an established destiny is justas certain as that three times three is nine. [. . . ]If, for example, onesphere meets another sphere in free space and if their sizes and their pathsand directions before collision are known, we can then foretell and calculate how they will rebound and what course they will take after theimpact. Very simple laws are followed which also apply, no matter howmany spheres are taken or whether objects are taken other than spheres.From this one sees then that everything proceeds mathematicallythat is,infalliblyin the whole wide world, so that if someone could have a sufficient insight into the inner parts of things, and in addition had remembrance and intelligence enough to consider all the circumstances and totake them into account, he would be a prophet and would see the futurein the present as in a mirror.

Leibniz chose to illustrate his faith in determinism precisely with the type ofphysical system that we shall use here as a paradigm of chaos. His claimis wrong in a deep and subtle way: a state of a physical system can neverbe specified to infinite precision, and by this we do not mean that eventuallythe Heisenberg uncertainty principle kicks in. In the classical, deterministicdynamics there is no way to take all the circumstances into account, and asingle trajectory cannot be tracked, only a ball of nearby initial points makesphysical sense. 1

1.3.1 What is chaos?I accept chaos. I am not sure that it accepts me.Bob Dylan, Bringing It All Back Home

A deterministic system is a system whose present state is in principle fullydetermined by its initial conditions.In contrast, radioactive decay, Brownian motion and heat flow are examplesof stochastic systems, for which the initial conditions determine the future onlypartially, due to noise, or other external circumstances beyond our control: thepresent state reflects the past initial conditions plus the particular realization ofthe noise encountered along the way.A deterministic system with suciently complicated dynamics can fool usinto regarding it as a stochastic one; disentangling the deterministic from thestochastic is the main challenge in many real-life settings, from stock marketsto palpitations of chicken hearts. So, what is chaos?In a game of pinball, any two trajectories that start out very close to eachother separate exponentially with time, and in a finite (and in practice, a verysmall) number of bounces their separation x(t) attains the magnitude of L,the characteristic linear extent of the whole system, Fig. 1.2. This property ofsensitivity to initial conditions can be quantified as|x(t)| et |x(0)|

Fig. 1.3 Unstable trajectories separate withtime.section 17.3

where , the mean rate of separation of trajectories of the system, is called theLyapunov exponent. For any finite accuracy x = |x(0)| of the initial data, theintro - 9apr2009


1.3. THE FUTURE AS IN A MIRROR

5

dynamics is predictable only up to a finite Lyapunov time1(1.1)T Lyap ln |x/L| ,despite the deterministic and, for Baron Leibniz, infallible simple laws thatrule the pinball motion.A positive Lyapunov exponent does not in itself lead to chaos. One couldtry to play 1- or 2-disk pinball game, but it would not be much of a game;trajectories would only separate, never to meet again. What is also needed ismixing, the coming together again and again of trajectories. While locally thenearby trajectories separate, the interesting dynamics is confined to a globallyfinite region of the state space and thus the separated trajectories are necessarily folded back and can re-approach each other arbitrarily closely, infinitelymany times. For the case at hand there are 2 n topologically distinct n bouncetrajectories that originate from a given disk. More generally, the number ofdistinct trajectories with n bounces can be quantified as

(a)

N(n) ehnwhere h, the growth rate of the number of topologically distinct trajectories, iscalled the topological entropy (h = ln 2 in the case at hand).The appellation chaos is a confusing misnomer, as in deterministic dynamics there is no chaos in the everyday sense of the word; everything proceedsmathematicallythat is, as Baron Leibniz would have it, infallibly. When aphysicist says that a certain system exhibits chaos, he means that the systemobeys deterministic laws of evolution, but that the outcome is highly sensitiveto small uncertainties in the specification of the initial state. The word chaoshas in this context taken on a narrow technical meaning. If a deterministicsystem is locally unstable (positive Lyapunov exponent) and globally mixing(positive entropy)Fig. 1.4it is said to be chaotic.While mathematically correct, the definition of chaos as positive Lyapunov+ positive entropy is useless in practice, as a measurement of these quantitiesis intrinsically asymptotic and beyond reach for systems observed in nature.More powerful is Poincares vision of chaos as the interplay of local instability(unstable periodic orbits) and global mixing (intertwining of their stable andunstable manifolds). 2 In a chaotic system any open ball of initial conditions,no matter how small, will in finite time overlap with any other finite regionand in this sense spread over the extent of the entire asymptotically accessiblestate space. Once this is grasped, the focus of theory shifts from attemptingto predict individual trajectories (which is impossible) to a description of thegeometry of the space of possible outcomes, and evaluation of averages overthis space. How this is accomplished is what ChaosBook is about.A definition of turbulence is even harder to come by. Can you recognizeturbulence when you see it? The word comes from tourbillon, French forvortex, and intuitively it refers to irregular behavior of an infinite-dimensional dynamical system described by deterministic equations of motionsay,a bucket of sloshing water described by the Navier-Stokes equations. But inpractice the word turbulence tends to refer to messy dynamics which we understand poorly. As soon as a phenomenon is understood better, it is reclaimedand renamed: a route to chaos, spatiotemporal chaos, and so on.ChaosBook.org version13.5, Sep 7 2011

intro - 9apr2009

(b)

Fig. 1.4 Dynamics of a chaotic dynamical system is (a) everywhere locally unstable(positive Lyapunov exponent) and (b) globally mixing (positive entropy).(A. Johansen)section 15.1

2

We owe the appellation chaosas wellas several other dynamics catchwordsto J.Yorke who in 1973 entitled a paper [1.3] thathe wrote with T. Li Period 3 implies chaos.

6

CHAPTER 1. OVERTURE

In ChaosBook we shall develop a theory of chaotic dynamics for low dimensional attractors visualized as a succession of nearly periodic but unstablemotions. In the same spirit, we shall think of turbulence in spatially extendedsystems in terms of recurrent spatiotemporal patterns. Pictorially, dynamicsdrives a given spatially extended system (clouds, say) through a repertoire ofunstable patterns; as we watch a turbulent system evolve, every so often wecatch a glimpse of a familiar pattern:

= other swirls

=

For any finite spatial resolution, a deterministic flow follows approximatelyfor a finite time an unstable pattern belonging to a finite alphabet of admissiblepatterns, and the long term dynamics can be thought of as a walk through thespace of such patterns. In ChaosBook we recast this image into mathematics.

1.3.2 When does chaos matter?In dismissing Pollocks fractals because of their limited magnificationrange, Jones-Smith and Mathur would also dismiss half the publishedinvestigations of physical fractals. Richard P. Taylor [1.4, 5]

When should we be mindful of chaos? The solar system is chaotic, yetwe have no trouble keeping track of the annual motions of planets. The ruleof thumb is this; if the Lyapunov time (1.1)the time by which a state spaceregion initially comparable in size to the observational accuracy extends acrossthe entire accessible state spaceis significantly shorter than the observationaltime, you need to master the theory that will be developed here. That iswhy the main successes of the theory are in statistical mechanics, quantummechanics, and questions of long term stability in celestial mechanics.In science popularizations too much has been made of the impact of chaostheory, so a number of caveats are already needed at this point.At present the theory that will be developed here is in practice applicableonly to systems of a low intrinsic dimension the minimum number of coordinates necessary to capture its essential dynamics. If the system is veryturbulent (a description of its long time dynamics requires a space of high intrinsic dimension) we are out of luck. Hence insights that the theory oersin elucidating problems of fully developed turbulence, quantum field theory ofstrong interactions and early cosmology have been modest at best. Even that isa caveat with qualifications. There are applicationssuch as spatially extended(non-equilibrium) systems, plumbers turbulent pipes, etc.,where the few important degrees of freedom can be isolated and studied profitably by methodsto be described here.Thus far the theory has had limited practical success when applied to thevery noisy systems so important in the life sciences and in economics. Eventhough we are often interested in phenomena taking place on time scales muchintro - 9apr2009


1.4. A GAME OF PINBALL

7

longer than the intrinsic time scale (neuronal inter-burst intervals, cardiac pulses,etc.), disentangling chaotic motions from the environmental noise has beenvery hard.In 1980s something happened that might be without parallel; this is anarea of science where the advent of cheap computation had actually subtractedfrom our collective understanding. The computer pictures and numerical plotsof fractal science of the 1980s have overshadowed the deep insights of the1970s, and these pictures have since migrated into textbooks.By a regrettable oversight, ChaosBook has none, so Untitled 5 of Fig. 1.5 will haveto do as the illustration of the power of fractal analysis. Fractal science positsthat certain quantities (Lyapunov exponents, generalized dimensions, . . . ) canbe estimated on a computer. While some of the numbers so obtained are indeed mathematically sensible characterizations of fractals, they are in no senseobservable and measurable on the length-scales and time-scales dominated bychaotic dynamics.Even though the experimental evidence for the fractal geometry of natureis circumstantial [1.7], in studies of probabilistically assembled fractal aggregates we know of nothing better than contemplating such quantities. In deterministic systems we can do much better.

1.4 A game of pinballFormulas hamper the understanding.S. Smale

We are now going to get down to the brass tacks. Time to fasten your seatbelts and turn o all electronic devices. But first, a disclaimer: If you understand the rest of this chapter on the first reading, you either do not need thisbook, or you are delusional. If you do not understand it, it is not because thepeople who figured all this out first are smarter than you: the most you canhope for at this stage is to get a flavor of what lies ahead. If a statement in thischapter mystifies/intrigues, fast forward to a section indicated by [section ...]on the margin, read only the parts that you feel you need. Of course, we thinkthat you need to learn ALL of it, or otherwise we would not have included itin ChaosBook in the first place.Confronted with a potentially chaotic dynamical system, our analysis proceeds in three stages; I. diagnose, II. count, III. measure. First, we determinethe intrinsic dimension of the systemthe minimum number of coordinates necessary to capture its essential dynamics. If the system is very turbulent we are,at present, out of luck. We know only how to deal with the transitional regimebetween regular motions and chaotic dynamics in a few dimensions. That isstill something; even an infinite-dimensional system such as a burning flamefront can turn out to have a very few chaotic degrees of freedom. In this regimethe chaotic dynamics is restricted to a space of low dimension, the number ofrelevant parameters is small, and we can proceed to step II; we count and classify all possible topologically distinct trajectories of the system into a hierarchywhose successive layers require increased precision and patience on the part ofChaosBook.org version13.5, Sep 7 2011

intro - 9apr2009

Fig. 1.5 Katherine Jones-Smith, Untitled5, the drawing used by K. Jones-Smith andR.P. Taylor to test the fractal analysis of Pollocks drip paintings [1.6].remark 1.6

chapter 11chapter 15

8

chapter 20

CHAPTER 1. OVERTURE

the observer. This we shall do in Section 1.4.2. If successful, we can proceedwith step III: investigate the weights of the dierent pieces of the system.We commence our analysis of the pinball game with steps I, II: diagnose,count. We shall return to step IIImeasurein Section 1.5. The three sectionsthat follow are highly technical, they go into the guts of what the book is about.Is today is not your thinking day, skip them, jump straight to Section 1.7.

1.4.1 Symbolic dynamics

exercise 1.1section 2.1

chapter 12

Fig. 1.6 Binary labeling of the 3-disk pinballtrajectories; a bounce in which the trajectoryreturns to the preceding disk is labeled 0, anda bounce which results in continuation to thethird disk is labeled 1.section 11.6

With the game of pinball we are in luckit is a low dimensional system, freemotion in a plane. The motion of a point particle is such that after a collisionwith one disk it either continues to another disk or it escapes. If we label thethree disks by 1, 2 and 3, we can associate every trajectory with an itinerary,a sequence of labels indicating the order in which the disks are visited; forexample, the two trajectories in Fig. 1.2 have itineraries 2313 , 23132321respectively. Such labeling goes by the name symbolic dynamics. As theparticle cannot collide two times in succession with the same disk, any twoconsecutive symbols must dier. This is an example of pruning, a rule thatforbids certain subsequences of symbols. Deriving pruning rules is in general adicult problem, but with the game of pinball we are luckyfor well-separateddisks there are no further pruning rules.The choice of symbols is in no sense unique. For example, as at each bouncewe can either proceed to the next disk or return to the previous disk, the above3-letter alphabet can be replaced by a binary {0, 1} alphabet, Fig. 1.6. A cleverchoice of an alphabet will incorporate important features of the dynamics, suchas its symmetries.Suppose you wanted to play a good game of pinball, that is, get the pinball tobounce as many times as you possibly canwhat would be a winning strategy?The simplest thing would be to try to aim the pinball so it bounces many timesbetween a pair of disksif you managed to shoot it so it starts out in the periodicorbit bouncing along the line connecting two disk centers, it would stay thereforever. Your game would be just as good if you managed to get it to keepbouncing between the three disks forever, or place it on any periodic orbit. Theonly rub is that any such orbit is unstable, so you have to aim very accurately inorder to stay close to it for a while. So it is pretty clear that if one is interestedin playing well, unstable periodic orbits are importantthey form the skeletononto which all trajectories trapped for long times cling.

1.4.2 Partitioning with periodic orbits

Fig. 1.7 The 3-disk pinball cycles 1232 and121212313.

A trajectory is periodic if it returns to its starting position and momentum.We shall sometimes refer to the set of periodic points that belong to a givenperiodic orbit as a cycle.Short periodic orbits are easily drawn and enumeratedan example is drawnin Fig. 1.7but it is rather hard to perceive the systematics of orbits from theirconfiguration space shapes. In mechanics a trajectory is fully and uniquelyspecified by its position and momentum at a given instant, and no two distinct state space trajectories can intersect. Their projections onto arbitrary subintro - 9apr2009


9

Fig. 1.8 (a) A trajectory starting out fromdisk 1 can either hit another disk or escape.(b) Hitting two disks in a sequence requires amuch sharper aim, with initial conditions thathit further consecutive disks nested withineach other, as in Fig. 1.9.example 3.2

chapter 81

sin

spaces, however, can and do intersect, in rather unilluminating ways. In thepinball example the problem is that we are looking at the projections of a 4dstate space trajectories onto a 2d subspace, the configuration space.Aclearer picture of the dynamics is obtained by constructing a set of state spacePoincare sections.Suppose that the pinball has just bounced o disk 1. Depending on its position and outgoing angle, it could proceed to either disk 2 or 3. Not muchhappens in between the bouncesthe ball just travels at constant velocity alonga straight lineso we can reduce the 4d flow to a 2d map P that takes thecoordinates of the pinball from one disk edge to another disk edge. The trajectory just after the moment of impact is defined by s n , the arc-length position ofthe nth bounce along the billiard wall, and p n = p sin n the momentum component parallel to the billiard wall at the point of impact, see Fig. 1.9. Suchsection of a flow is called a Poincare section. In terms of Poincare sections,the dynamics is reduced to the set of six maps P sk s j : (sn , pn ) (sn+1 , pn+1 ),with s {1, 2, 3}, from the boundary of the disk j to the boundary of the nextdisk k.Next, we mark in the Poincare section those initial conditions which do notThereinarestrips of survivors, as the trajectories originating from oneescapeonetwobounce.disk can hit either of the other two disks, or escape without further ado. Welabel the two strips M12 , M13 . Embedded within them there are four stripsM121 , M123 , M131 , M132 of initial conditions that survive for two bounces, andso forth, see Figs. 1.8 and 1.9. Provided that the disks are suciently separated,after n bounces the survivors are divided into 2 n distinct strips: the Mi th stripconsists of all points with itinerary i = s 1 s2 s3 . . . sn , s = {1, 2, 3}. The unstablecycles as a skeleton of chaos are almost visible here: each such patch containsa periodic point s 1 s2 s3 . . . sn with the basic block infinitely repeated. Periodicpoints are skeletal in the sense that as we look further and further, the stripsshrink but the periodic points stay put forever.We see now why it pays to utilize a symbolic dynamics; it provides a navigation chart through chaotic state space. There exists a unique trajectory forevery admissible infinite length itinerary, and a unique itinerary labels everytrapped trajectory. For example, the only trajectory labeled by 12 is the 2cycle bouncing along the line connecting the centers of disks 1 and 2; anyother trajectory starting out as 12 . . . either eventually escapes or hits the 3rddisk.

0

12.5

1

0

1

1.4.3 Escape rate

(b)

What is a good physical quantity to compute for the game of pinball? Such asystem, for which almost any trajectory eventually leaves a finite region (thepinball table) never to return, is said to be open, or a repeller. The repellerescape rate is an eminently measurable quantity. An example of such a measurement would be an unstable molecular or nuclear state which can be wellapproximated by a classical potential with the possibility of escape in certaindirections. In an experiment many projectiles are injected into a macroscopicblack box enclosing a microscopic non-confining short-range potential, andtheir mean escape rate is measured, as in Fig. 1.1. The numerical experimentChaosBook.org version13.5, Sep 7 2011

intro - 9apr2009

11111111111111100000000000000000000000000000001111111111111111000000000000000111111111111111000000000000000011111111111111110000000000000001111111111111110000000000000000111111111111111100000000000000011111111111111100000000000000001111111111111111000000000000000111111111111111000000000000000011111111111111110000000000000001111111111111110000000000000000111111111111111100000000000000011111111111111100000000000000001111111111111111000000000000000111111111111111000000000000000011111111111111110000000000000001111111111111110000000000000000111111111111111100000000000000011111111111111100000000000000001111111111111111000000000000000111111111111111000000000000000011111111111111110000000000000001111111111111110000000000000000111111111111111100000000000000011111111111111100000000000000001111111111111111000000000000000111111111111111000000000000000011111111111111110000000000000001111111111111110000000000000000111111111111111100000000000000011111111111111100000000000000001111111111111111000000000000000111111111111111000000000000000011111111111111110000000000000001111111111111110000000000000000111111111111111100000000000000011111111111111100000000000000001111111111111111000000000000000111111111111111000000000000000011111111111111110000000000000001111111111111110000000000000000111111111111111100000000000000011111111111111100000000000000001111111111111111000000000000000111111111111111000000000000000011111111111111110000000000000001111111111111110000000000000000111111111111111112130000000000000001111111111111110000000000000000111111111111111100000000000000011111111111111100000000000000001111111111111111000000000000000111111111111111000000000000000011111111111111110000000000000001111111111111110000000000000000111111111111111100000000000000011111111111111100000000000000001111111111111111000000000000000111111111111111000000000000000011111111111111110000000000000001111111111111110000000000000000111111111111111100000000000000011111111111111100000000000000001111111111111111000000000000000111111111111111000000000000000011111111111111110000000000000001111111111111110000000000000000111111111111111100000000000000011111111111111100000000000000001111111111111111000000000000000111111111111111000000000000000011111111111111110000000000000001111111111111110000000000000000111111111111111100000000000000011111111111111100000000000000001111111111111111000000000000000111111111111111000000000000000011111111111111110000000000000001111111111111110000000000000000111111111111111100000000000000011111111111111100000000000000001111111111111111000000000000000111111111111111000000000000000011111111111111110000000000000001111111111111110000000000000000111111111111111100000000000000011111111111111100000000000000001111111111111111000000000000000111111111111111000000000000000011111111111111110000000000000001111111111111110000000000000000111111111111111100000000000000011111111111111100000000000000001111111111111111

0S

(a)

sin

1.4. A GAME OF PINBALL

2.5

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13100000000000000001111111111111111000000000000000001111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000000111111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000011111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000001111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000000111111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000011111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000001111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000000111111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000011111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000000111111111111111100000000000000000111111111111111110000000000000000111111111111111100000000000000001111111111111111121 10 132100000000000000001111111111111111000000000000000001111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000000111111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000011111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000001111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000000111111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000011111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000001111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000000111111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000011111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000001111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000000111111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000011111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000001111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000000111111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000011111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000001111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000000111111111111111110000000000000000111111111111111100000000000000001111111111111111

2.5

0

2.5

s

Fig. 1.9 The 3-disk game of pinball Poincaresection, trajectories emanating from the disk1 with x0 = (s0 , p0 ) . (a) Strips of initialpoints M12 , M13 which reach disks 2, 3 inone bounce, respectively. (b) Strips of initialpoints M121 , M131 M132 and M123 whichreach disks 1, 2, 3 in two bounces, respectively. Disk radius : center separation ratioa:R = 1:2.5.(Y. Lan)example 17.4

10

exercise 1.2

chapter 22

CHAPTER 1. OVERTURE

might consist of injecting the pinball between the disks in some random direction and asking how many times the pinball bounces on the average before itescapes the region between the disks.For a theorist, a good game of pinball consists in predicting accurately theasymptotic lifetime (or the escape rate) of the pinball. We now show howperiodic orbit theory accomplishes this for us. Each step will be so simple thatyou can follow even at the cursory pace of this overview, and still the result issurprisingly elegant.Consider Fig. 1.9 again. In each bounce the initial conditions get thinnedout, yielding twice as many thin strips as at the previous bounce. The totalarea that remains at a given time is the sum of the areas of the strips, so that thefraction of survivors after n bounces, or the survival probability is given by 1

=

n

=

|M0 | |M1 |+,|M||M|(n)1 |Mi | ,|M| i

|M00 | |M10 | |M01 | |M11 | 2 =+++,|M||M||M||M|(1.2)

where i is a label of the ith strip, |M| is the initial area, and |M i | is the areaof the ith strip of survivors. i = 01, 10, 11, . . . is a label, not a binary number.Since at each bounce one routinely loses about the same fraction of trajectories,one expects the sum (1.2) to fall o exponentially with n and tend to the limit n+1 / n = en e .

(1.3)

The quantity is called the escape rate from the repeller.

1.5 Chaos for cyclistsEtantdonnees des e quations ... et une solution particuliere quelconque de ces e quations, on peut toujours trouver une solution periodique (dont la periode peut, il est vrai, e tre tres longue), telle que ladierence entre les deux solutions soit aussi petite quon le veut, pendant un temps aussi long quon le veut. Dailleurs, ce qui nous rendces solutions periodiques si precieuses, cest quelles sont, pour ansidire, la seule breche par o`u nous puissions esseyer de penetrer dansune place jusquici reputee inabordable.H. Poincare, Les methodes nouvelles de la mechanique celeste

We shall now show that the escape rate can be extracted from a highly convergent exact expansion by reformulating the sum (1.2) in terms of unstableperiodic orbits.If, when asked what the 3-disk escape rate is for a disk of radius 1, centercenter separation 6, velocity 1, you answer that the continuous time escaperate is roughly = 0.4103384077693464893384613078192 . . ., you do notneed this book. If you have no clue, hang on.intro - 9apr2009


1.5. CHAOS FOR CYCLISTS

11

1.5.1 How big is my neighborhood?Not only do the periodic points keep track of topological ordering of the strips,but, as we shall now show, they also determine their size. As a trajectoryevolves, it carries along and distorts its infinitesimal neighborhood. Letx(t) = f t (x0 )denote the trajectory of an initial point x 0 = x(0). Expanding f t (x0 + x0 )to linear order, the evolution of the distance to a neighboring trajectory x i (t) +xi (t) is given by the Jacobian matrix J:xi (t) =

d

J t (x0 )i j x0 j ,

j=1

J t (x0 )i j =

xi (t).x0 j

(1.4)

A trajectory of a pinball moving on a flat surface is specified by two positioncoordinates and the direction of motion, so in this case d = 3. Evaluation ofa cycle Jacobian matrix is a long exercise - here we just state the result. TheJacobian matrix describes the deformation of an infinitesimal neighborhood ofx(t) along the flow; its eigenvectors and eigenvalues give the directions and thecorresponding rates of expansion or contraction, Fig. 1.10. The trajectories thatstart out in an infinitesimal neighborhood separate along the unstable directions(those whose eigenvalues are greater than unity in magnitude), approach eachother along the stable directions (those whose eigenvalues are less than unityin magnitude), and maintain their distance along the marginal directions (thosewhose eigenvalues equal unity in magnitude).In our game of pinball the beam of neighboring trajectories is defocusedalong the unstable eigen-direction of the Jacobian matrix J.As the heights of the strips in Fig. 1.9 are eectively constant, we can concentrate on their thickness. If the height is L, then the area of the ith strip isMi Lli for a strip of width l i .Each strip i in Fig. 1.9 contains a periodic point x i . The finer the intervals,the smaller the variation in flow across them, so the contribution from the stripof width li is well-approximated by the contraction around the periodic pointxi within the interval,li = ai /|i | ,

(1.5)

where i is the unstable eigenvalue of the Jacobian matrix J t (xi ) evaluated atthe ith periodic point for t = T p , the full period (due to the low dimensionality,the Jacobian can have at most one unstable eigenvalue). Only the magnitude ofthis eigenvalue matters, we can disregard its sign. The prefactors a i reflect theoverall size of the system and the particular distribution of starting values ofx. As the asymptotic trajectories are strongly mixed by bouncing chaoticallyaround the repeller, we expect their distribution to be insensitive to smoothvariations in the distribution of initial points.To proceed with the derivation we need the hyperbolicity assumption: forlarge n the prefactors a i O(1) are overwhelmed by the exponential growthof i , so we neglect them. If the hyperbolicity assumption is justified, we canChaosBook.org version13.5, Sep 7 2011

intro - 9apr2009

section 8.2

x(t)

t x(t) = J x(0)

x(0) x(0)Fig. 1.10 The Jacobian matrix Jt maps aninfinitesimal displacement x at x0 into a displacement Jt (x0 )x finite time t later.

section 16.4

section 18.1.1

12

CHAPTER 1. OVERTURE

replace |Mi | Lli in (1.2) by 1/| i | and consider the sumn =

(n)

1/|i | ,

i

where the sum goes over all periodic points of period n. We now define agenerating function for sums over all periodic orbits of all lengths:(z) =

n zn .

(1.6)

n=1

Recall that for large n the nth level sum (1.2) tends to the limit n en , sothe escape rate is determined by the smallest z = e for which (1.6) diverges:(z)

(ze ) =n

n=1

ze.1 ze

(1.7)

This is the property of (z) that motivated its definition. Next, we devise aformula for (1.6) expressing the escape rate in terms of periodic orbits:(z) =

n=1

=

section 18.3

zn

(n)

|i |1

i

zz2z2z2z2z+++++|0 | |1 | |00 | |01 | |10 | |11 |z3z3z3z3++++...+|000 | |001 | |010 | |100 |

(1.8)

For suciently small z this sum is convergent. The escape rate is nowgiven by the leading pole of (1.7), rather than by a numerical extrapolation ofa sequence of n extracted from (1.3). As any finite truncation n < n trunc of(1.8) is a polynomial in z, convergent for any z, finding this pole requires thatwe know something about n for any n, and that might be a tall order.We could now proceed to estimate the location of the leading singularity of(z) from finite truncations of (1.8) by methods such as Pade approximants.However, as we shall now show, it pays to first perform a simple resummationthat converts this divergence into a zero of a related function.

1.5.2 Dynamical zeta function

exercise 15.2section 4.5

If a trajectory retraces a prime cycle r times, its expanding eigenvalue is rp .A prime cycle p is a single traversal of the orbit; its label is a non-repeatingsymbol string of n p symbols. There is only one prime cycle for each cyclicpermutation class. For example, p = 0011 = 1001 = 1100 = 0110 is prime, but0101 = 01 is not. By the chain rule for derivatives the stability of a cycle is thesame everywhere along the orbit, so each prime cycle of length n p contributesn p terms to the sum (1.8). Hence (1.8) can be rewritten as n p r nptpzzn p(z) =(1.9)np=,tp =| p |1 tp| p |ppr=1intro - 9apr2009


1.5. CHAOS FOR CYCLISTS

13

where the index p runs through all distinct prime cycles. Note that we haveresummed the contribution of the cycle p to all times, so truncating the summation up to given p is not a finite time n n p approximation, but an asymptotic, infinite time estimate based by approximating stabilities of all cycles bya finite number of the shortest cycles and their repeats. The n p zn p factors in(1.9) suggest rewriting the sum as a derivative(z) = z

d ln(1 t p ) .dz p

Hence (z) is a logarithmic derivative of the infinite product1/(z) =

(1 t p ) ,

p

tp =

zn p.| p |

(1.10)

This function is called the dynamical zeta function, in analogy to the Riemannzeta function, which motivates the zeta in its definition as 1/(z). This is theprototype formula of periodic orbit theory. The zero of 1/(z) is a pole of (z),and the problem of estimating the asymptotic escape rates from finite n sumssuch as (1.2) is now reduced to a study of the zeros of the dynamical zeta function (1.10). The escape rate is related by (1.7) to a divergence of (z), and (z)diverges whenever 1/(z) has a zero.Easy, you say: Zeros of (1.10) can be read o the formula, a zero

section 22.1section 19.4

z p = | p |1/n pfor each term in the product. Whats the problem? Dead wrong!

1.5.3 Cycle expansionsHow are formulas such as (1.10) used? We start by computing the lengthsand eigenvalues of the shortest cycles. This usually requires some numericalwork, such as the Newton method searches for periodic solutions; we shallassume that the numerics are under control, and that all short cycles up to givenlength have been found. In our pinball example this can be done by elementarygeometrical optics. It is very important not to miss any short cycles, as thecalculation is as accurate as the shortest cycle droppedincluding cycles longerthan the shortest omitted does not improve the accuracy (unless exponentiallymany more cycles are included). The result of such numerics is a table of theshortest cycles, their periods and their stabilities.Now expand the infinite product (1.10), grouping together the terms of thesame total symbol string length1/

chapter 13

section 27.3

= (1 t0 )(1 t1 )(1 t10 )(1 t100 ) = 1 t0 t1 [t10 t1 t0 ] [(t100 t10 t0 ) + (t101 t10 t1 )][(t1000 t0 t100 ) + (t1110 t1 t110 )+(t1001 t1 t001 t101 t0 + t10 t0 t1 )] . . .

(1.11)

The virtue of the expansion is that the sum of all terms of the same total lengthChaosBook.org version13.5, Sep 7 2011

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chapter 20

14

CHAPTER 1. OVERTURE

n (grouped in brackets above) is a number that is exponentially smaller than atypical term in the sum, for geometrical reasons we explain in the next section.The calculation is now straightforward. We substitute a finite set of theeigenvalues and lengths of the shortest prime cycles into the cycle expansion(1.11), and obtain a polynomial approximation to 1/. We then vary z in (1.10)and determine the escape rate by finding the smallest z = e for which (1.11)vanishes.

1.5.4 Shadowing

section 20.2.2

Fig. 1.11 Approximation to a smooth dynamics (left frame) by the skeleton of periodicpoints, together with their linearized neighborhoods, (right frame). Indicated are segments of two 1-cycles and a 2-cycle that alternates between the neighborhoods of the two1-cycles, shadowing first one of the two 1cycles, and then the other.

When you actually start computing this escape rate, you will find out that theconvergence is very impressive: only three input numbers (the two fixed points0, 1 and the 2-cycle 10) already yield the pinball escape rate to 3-4 significantdigits! We have omitted an infinity of unstable cycles; so why does approximating the dynamics by a finite number of the shortest cycle eigenvalues workso well?The convergence of cycle expansions of dynamical zeta functions is a consequence of the smoothness and analyticity of the underlying flow. Intuitively, one can understand the convergence in terms of the geometrical picturesketched in Fig. 1.11; the key observation is that the long orbits are shadowedby sequences of shorter orbits.A typical term in (1.11) is a dierence of a long cycle {ab} minus its shadowing approximation by shorter cycles {a} and {b}

ab ,(1.12)tab ta tb = tab (1 ta tb /tab ) = tab 1 a b where a and b are symbol sequences of the two shorter cycles. If all orbits areweighted equally (t p = zn p ), such combinations cancel exactly; if orbits of similar symbolic dynamics have similar weights, the weights in such combinationsalmost cancel.This can be understood in the context of the pinball game as follows. Consider orbits 0, 1 and 01. The first corresponds to bouncing between any twodisks while the second corresponds to bouncing successively around all three,tracing out an equilateral triangle. The cycle 01 starts at one disk, say disk 2.It then bounces from disk 3 back to disk 2 then bounces from disk 1 back todisk 2 and so on, so its itinerary is 2321. In terms of the bounce types shown inFig. 1.6, the trajectory is alternating between 0 and 1. The incoming and outgoing angles when it executes these bounces are very close to the correspondingangles for 0 and 1 cycles. Also the distances traversed between bounces aresimilar so that the 2-cycle expanding eigenvalue 01 is close in magnitude tothe product of the 1-cycle eigenvalues 0 1 .To understand this on a more general level, try to visualize the partition ofa chaotic dynamical systems state space in terms of cycle neighborhoods asa tessellation (a tiling) of the dynamical system, with smooth flow approximated by its periodic orbit skeleton, each tile centered on a periodic point,and the scale of the tile determined by the linearization of the flow aroundthe periodic point, as illustrated by Fig. 1.11.intro - 9apr2009


section 20.1

1.6. CHANGE IN TIME

15

The orbits that follow the same symbolic dynamics, such as {ab} and apseudo orbit {a}{b}, lie close to each other in state space; long shadowingpairs have to start out exponentially close to beat the exponential growth inseparation with time. If the weights associated with the orbits are multiplicative along the flow (for example, by the chain rule for products of derivatives)and the flow is smooth, the term in parenthesis in (1.12) falls o exponentiallywith the cycle length, and therefore the curvature expansions are expected tobe highly convergent.

chapter 23

1.6 Change in timeThe above derivation of the dynamical zeta function formula for the escaperate has one shortcoming; it estimates the fraction of survivors as a functionof the number of pinball bounces, but the physically interesting quantity isthe escape rate measured in units of continuous time. For continuous timeflows, the escape rate (1.2) is generalized as follows. Define a finite state spaceregion M such that a trajectory that exits M never reenters. For example, anypinball that falls of the edge of a pinball table in Fig. 1.1 is gone forever. Startwith a uniform distribution of initial points. The fraction of initial x whosetrajectories remain within M at time t is expected to decay exponentially

dxdy (y f t (x))M

et .(t) =dxMThe integral over x starts a trajectory at every x M. The integral over y testswhether this trajectory is still in M at time t. The kernel of this integral

Lt (y, x) = y f t (x)(1.13)is the Dirac delta function, as for a deterministic flow the initial point x mapsinto a unique point y at time t. For discrete time, f n (x) is the nth iterate of themap f . For continuous flows, f t (x) is the trajectory of the initial point x, and itis appropriate to express the finite time kernel L t in terms of A, the generatorof infinitesimal time translationsLt = etA ,very much in the way the quantum evolution is generated by the HamiltonianH, the generator of infinitesimal time quantum transformations.As the kernel L is the key to everything that follows, we shall give it a name,and refer to it and its generalizations as the evolution operator for a d-dimensional map or a d-dimensional flow. 3The number of periodic points increases exponentially with the cycle length(in the case at hand, as 2 n ). As we have already seen, this exponential proliferation of cycles is not as dangerous as it might seem; as a matter of fact, all ourcomputations will be carried out in the n limit. Though a quick look atlong-time density of trajectories might reveal it to be complex beyond belief,this distribution is still generated by a simple deterministic law, and with someluck and insight, our labeling of possible motions will reflect this simplicity.ChaosBook.org version13.5, Sep 7 2011

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section 16.6

3If you are still in Kansas, please place asticker with words change in time over theoending word, whenever you encounter it.ChaosBook expands, indeed, upon a theory,not a fact.

16

CHAPTER 1. OVERTURE

If the rule that gets us from one level of the classification hierarchy to the nextdoes not depend strongly on the level, the resulting hierarchy is approximatelyself-similar. We now turn such approximate self-similarity to our advantage,by turning it into an operation, the action of the evolution operator, whoseiteration encodes the self-similarity.

1.6.1 Trace formulaIn physics, when we do not understand something, we give it a name.Matthias Neubert

Recasting dynamics in terms of evolution operators changes everything. Sofar our formulation has been heuristic, but in the evolution operator formalismthe escape rate and any other dynamical average are given by exact formulas, extracted from the spectra of evolution operators. The key tools are traceformulas and spectral determinants.The trace of an operator is given by the sum of its eigenvalues. The explicitexpression (1.13) for L t (x, y) enables us to evaluate the trace. Identify y with xand integrate x over the whole state space. The result is an expression for tr L tas a sum over neighborhoods of prime cycles p and their repetitionstr Lt =

Tp

p

Fig. 1.12 The trace of an evolution operatoris concentrated in tubes around prime cycles,of length T p and thickness 1/|p |r for the rthrepetition of the prime cycle p.section 18.2

chapter 18

r=1

(t rT p ) ,det 1 M r p

(1.14)

where T p is the period of prime cycle p, and the monodromy matrix M p isthe flow-transverse part of Jacobian matrix J (1.4). This formula has a simple geometrical interpretation sketched in Fig. 1.12. After the rth return toa Poincare section, the initial tube M p has been stretched out along the expandingwith the overlap with the initial volume given by eigen-directions,1/ det 1 M rp 1/| p |, the same weight we obtained heuristically in Section 1.5.1.The spiky sum (1.14) is disquieting in the way reminiscent of the Poisson resummation formulas of Fourier analysis; the left-hand side is the smooth

eigenvalue sum tr e At = e s t , while the right-hand side equals zero everywhere except for the set t = rT p .A Laplace transform smooths the sumover Dirac delta functions in cycle periods and yields the trace formula for theeigenspectrum s 0 , s1 , of the classical evolution operator:

0+

dt est tr Lt

=0

1s s

==

1=sA er(A p sT p ) Tp .r

pr=1 det 1 M p tr

(1.15)

The beauty of trace formulas lies in the fact that everything on the right-handsideprime cycles p, their periods T p and the eigenvalues of M p is an invariantproperty of the flow, independent of any coordinate choice.intro - 9apr2009


1.7. TO STATISTICAL MECHANICS

17

1.6.2 Spectral determinantThe eigenvalues of a linear operator are given by the zeros of the appropriatedeterminant. One way to evaluate determinants is to expand them in terms oftraces, using the identitiesd1dln det (s A) = trln(s A) = tr,dsdssA

exercise 4.1

(1.16)

and integrating over s. In this way the spectral determinant of an evolutionoperator becomes related to the traces that we have just computed: esT p r1 .det (s A) = exp (1.17)r det 1 M r p r=1

chapter 19

p

The 1/r factor is due to the s integration, leading to the replacement T p T p /rT p in the periodic orbit expansion (1.15).We have now retraced the heuristic derivation of the divergent sum (1.7)and the dynamical zeta function (1.10), but this time with no approximations:formula (1.17) is exact. The computation of the zeros of det (s A) proceedsvery much like the computations of Section 1.5.3.

1.7 From chaos to statistical mechanicsUnder heaven, all is chaos. The situation is excellent! Chairman Mao Zedong, a letter to Jiang Qing

The replacement of individual trajectories by evolution operators which propagate densities feels like a bit of mathematical voodoo. Nevertheless, something very radical and deeply foundational has taken place. Understanding thedistinction between evolution of individual trajectories and the evolution of thedensities of trajectories is key to understanding statistical mechanicsthis is theconceptual basis of the second law of thermodynamics, and the origin of irreversibility of the arrow of time for deterministic systems with time-reversibleequations of motion: reversibility is attainable for distributions whose measurein the space of density functions goes exponentially to zero with time.Consider a chaotic flow, such as the stirring of red and white paint by somedeterministic machine. If we were able to track individual trajectories, the fluidwould forever remain a striated combination of pure white and pure red; therewould be no pink. What is more, if we reversed the stirring, we would returnto the perfect white/red separation. However, that cannot bein a very fewturns of the stirring stick the thickness of the layers goes from centimeters tongstroms, and the result is irreversibly pink.A century ago it seemed reasonable to assume that statistical mechanics applies only to systems with very many degrees of freedom. More recent is therealization that much of statistical mechanics follows from chaotic dynamics,and already at the level of a few degrees of freedom the evolution of densitiesis irreversible. Furthermore, the theory that we shall develop here generalizes notions of measure and averaging to systems far from equilibrium, andChaosBook.org version13.5, Sep 7 2011

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section 19.5

18

chapter 25

section 25.1

CHAPTER 1. OVERTURE

transports us into regions hitherto inaccessible with the tools of equilibriumstatistical mechanics.By going to a description in terms of the asymptotic time evolution operatorswe give up tracking individual trajectories for long times, but trade in the uncontrollable trajectories for a powerful description of the asymptotic trajectorydensities. This will enable us, for example, to give exact formulas for transportcoecients such as the diusion constants without any probabilistic assumptions. The classical Boltzmann equation for evolution of 1-particle density isbased on stosszahlansatz, neglect of particle correlations prior to, or after a2-particle collision. It is a very good approximate description of dilute gas dynamics, but a dicult starting point for inclusion of systematic corrections. Inthe theory developed here, no correlations are neglected - they are all includedin the cycle averaging formulas

such as the cycle expansion for the diusionconstant 2dD = limT x(T )2 /T of a particle diusing chaotically across aspatially-periodic array,D =

chapter 24

(n p + + n pk )21 1 ,(1)k+1 12d T| p1 pk |

(1.18)

where n p is a translation along one period of a spatially periodic runawaytrajectory p. Such formulas are exact; the issue in their applications is whatare the most eective schemes of estimating the infinite cycle sums requiredfor their evaluation. Unlike most statistical mechanics, here there are no phenomenological macroscopic parameters; quantities such as transport coecients are calculable to any desired accuracy from the microscopic dynamics.The concepts of equilibrium statistical mechanics do help us, however, tounderstand the ways in which the simple-minded periodic orbit theory falters.A non-hyperbolicity of the dynamics manifests itself in power-law correlationsand even phase transitions.

1.8 Chaos: what is it good for?Happy families are all alike; every unhappy family is unhappy in itsown way. Anna Karenina, by Leo Tolstoy

With initial data accuracy x = |x(0)| and system size L, a trajectory is predictable only up to the finite Lyapunov time (1.1), T Lyap 1 ln |L/x| . Beyond that, chaos rules. And so the most successful applications of chaos theory have so far been to problems where observation time is much longer thana typical turnover time, such as statistical mechanics, quantum mechanics,and questions of long term stability in celestial mechanics, where the notion oftracking accurately a given state of the system is nonsensical.So what is chaos good for? Transport! Though superficially indistinguishable from the probabilistic random walk diusion, in low dimensional settingsthe deterministic diusion is quite recognizable, through the fractal dependence of the diusion constant on the system parameters, and perhaps throughnon-Gaussion relaxation to equilibrium (non-vanishing Burnett coecients).intro - 9apr2009


1.8. CHAOS: WHAT IS IT GOOD FOR?

19

(a)

(b) sin( t)frequency

(c)

Fig. 1.13 (a) Washboard mean velocity, (b)cold atom lattice diusion, and (c) AFM tipdrag force.(Y. Lan)

velocity

Several tabletop experiments that could measure transport on macroscopicscales are sketched in Fig. 1.13 (each a tabletop, but an expensive tabletop).Figure 1.13 (a) depicts a slanted washboard; a particle in a gravity field bouncing down the washboard, losing some energy at each bounce, or a charged particle in a constant electric field trickling across a periodic condensed-matterdevice. The interplay between chaotic dynamics and energy loss results ina terminal mean velocity/conductance, a function of the washboard slant orexternal electric field that the periodic theory can predict accurately. Figure 1.13 (b) depicts a cold atom lattice of very accurate spatial periodicity,with a dilute cloud of atoms placed onto a standing wave established by stronglaser fields. Interaction of gravity with gentle time-periodic jiggling of the EMfields induces a diusion of the atomic cloud, with a diusion constant predicted by the periodic orbit theory. Figure 1.13 (c) depicts a tip of an atomicforce microscope (AFM) bouncing against a periodic atomic surface movingat a constant velocity. The frictional drag experienced is the interplay of thechaotic bouncing of the tip and the energy loss at each tip/surface collision,accurately predicted by the periodic orbit theory. None of these experimentsChaosBook.org/projectshave actually been carried out, (save for some numerical experimentation), butare within reach of what can be measured today.Given microscopic dynamics, periodic orbit theory predicts observable macroscopic transport quantities such as the washboard mean velocity, cold atom lattice diusion constant, and AFM tip drag force. But the experimental proposalis sexier than that, and goes into the heart of dynamical systems theory.remark A.1Smale 1960s theory of the hyperbolic structure of the non-wandering set(AKA horseshoe) was motivated by his structural stability conjecture, whichChaosBook.org version13.5, Sep 7 2011

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20

section 12.2

section 25.2

CHAPTER 1. OVERTURE

- in non-technical terms - asserts that all trajectories of a chaotic dynamicalsystem deform smoothly under small variations of system parameters.Why this cannot be true for a system like the washboard in Fig. 1.13 (a) iseasy to see for a cyclist. Take a trajectory which barely grazes the tip of oneof the groves. An arbitrarily small change in the washboard slope can result inloss of this collision, change a forward scattering into a backward scattering,and lead to a discontinuous contribution to the mean velocity. You might holdout hope that such events are rare and average out, but not so - a loss of ashort cycle leads to a significant change in the cycle-expansion formula for atransport coecient, such as (1.18).When we write an equation, it is typically parameterized by a set of parameters by as coupling strengths, and we think of dynamical systems obtained bya smooth variation of a parameter as a family. We would expect measurablepredictions to also vary smoothly, i.e., be structurally stable.But dynamical systems families are families only in a name. That thestructural stability conjecture turned out to be badly wrong is, however, not ablow for chaotic dynamics. Quite to the contrary, it is actually a virtue, perhapsthe most dramatic experimentally measurable prediction of chaotic dynamics.As long as microscopic periodicity is exact, the prediction is counterintuitive for a physicist - transport coecients are not smooth functions of systemparameters, rather they are non-monotonic, nowhere dierentiable functions.Conversely, if the macroscopic measurement yields a smooth dependence ofthe transport on system parameters, the periodicity of the microscopic lattice isdegraded by impurities, and probabilistic assumptions of traditional statisticalmechanics apply. So the proposal is to by measuring macroscopic transportconductance, diusion, drag observe determinism on nanoscales, and forexample determine whether an atomic surface is clean.The signatures of deterministic chaos are even more baing to an engineer:a small increase of pressure across a pipe exhibiting turbulent flow does notnecessarily lead to an increase in the mean flow; mean flow dependence onpressure drop across the pipe is also a fractal function.Is this in contradiction with the traditional statistical mechanics? No - deterministic chaos predictions are valid in settings where a few degrees of freedomare important, and chaotic motion time and space scales are commensuratewith the external driving and spatial scales. Further degrees of freedom act asnoise that smooths out the above fractal eects and restores a smooth functional dependence of transport coecients on external parameters.

1.9 What is not in ChaosBookThere is only one thing which interests me vitally now, and that isthe recording of

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