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The End of

CERTAINTYTime,Chaos,and the

New Laws ofNature

ILYA PRIGOGINEIn collaborationwith IsabelleStengers

THE FREE PRESS

New York London Toronto Sydney Singapore

THE FREE PRESSA Division ofSimon & Schuster Inc.

1230Avenue of the Americas

New York, NY 10020

Copyright \302\251 1996by Editions Odile JacobEnglish language translation copyright \302\251 1997by Editions Odile JacobAll rights reserved,

including the right of reproductionin whole or in part in any form.

First Free PressEdition 1997Published by arrangement with Editions Odile Jacob, Paris, France

THE FREE PRESSand colophon are trademarks

of Simon &Schuster Inc.

Designed by Carla Bolte

Manufactured in the United StatesofAmerica

10 987654321Library ofCongressCataloging-in-Publication Data

Prigogine, I. (Ilya)

[La fin des certitudes, English]

The end of certainty; time, chaos, and the new laws ofnature/Uya

Prigogine; in collaboration with Isabelle Stengers.p. cm.

Includes bibliographical references and index.

1.Science\342\200\224Philosophy, 2. Spaceand time. 3. Chaotic behavior

in systems, 4.Natural history. I. Stengers, Isabelle.

II. Tide.Q175.P75131997530.11\342\200\224dc21 97-3001

CIPISBN 0-684-83705-6

Original French edition entitled La Fin desCertitudes, by Ilya Prigogine in

collaboration with Isabelle Stengers,published by Editions Odile Jacob

CONTENTS

Acknowledgments vii

Author's Note ix

Introduction:A NewRationality? 1

1.Epicurus'Dilemma 9

2.Only an Illusion? 57

3.FromProbability to Irreversibility 73

4.TheLaws ofChaos 89

5.BeyondNewton'sLaws 107

6.A Unified Formulation ofQuantum Theory 129

7. OurDialoguewith Nature 153

8.DoesTimePrecedeExistence? 163

9.A Narrow Path 183

Notes 191Glossary 201Index 207

ACKNOWLEDGMENTS

Thisbookhas had a somewhatunusual history.

Originally,IsabelleStengersand I had intendedto translate

our bookEntre le Temps et I'Etemite(BetweenTimeand

Eternity) into English.1We had already prepared several

versions,oneofwhichappearedin German,and anotherin Russian.2But at the sametime,we weremakingimportant progressin the mathematical formulation ofourapproach. As a result, we abandoned our translation of the

originalbookand decidedto write a new version,whichhas recently appeared in French.3IsabelleStengershasaskednot to bedesignatedas a coauthorofthis newpresentation, but only as my collaborator.Although I felt

obligedto respecther wishes, I would like to stressthat

without her, this bookwould never have beenwritten.Iam most grateful for herassistance.

This textis the result ofdecadesofwork by both theBrusselsand Austin groups.While the physicalideas had

beenclearfor a considerablelengthoftime,their precisemathematical formulation has emergedonly during the

past few years.4I expressmy gratitude to the young,enthusiastic coworkerswho have beeninstrumental in

definingthe new approachto the nature oftime that forms the

basisofthis book,especiallyIoannisAntoniou (Brussels),

Vll

viii Acknowledgments

DeanDriebe(Austin), HiroshiHasegawa(Austin), TomioPetrosky(Austin), and ShuichiTasaki (Kyoto).I wouldalsoliketo mentionmy oldgroup in Brussels,who laid thefoundations that made further progresspossible.My thanks

to Radu Balescu,Michel de Haan, FrancoiseHenin,ClaudeGeorge,Alkis Grecos,and Fernand Mayne.Unfortunately, PierreResiboisand LeonRosenfeldare nolongerwith us.

Thework presentedin this bookcouldnot have beenaccomplishedwithout the support ofa numberoforganizations. I especiallywant to thank the BelgianCom-munaute Francaise,the Belgianfederal government,theInternationalSolvay Institutes(Brussels),the UnitedStates

Departmentof Energy, the EuropeanUnion,and theWelch Foundation (Texas).

English is not my native language,and I am muchobligedto Dr. E. C. GeorgeSudarshan and Dr. DeanDriebe,both from the University ofTexasat Austin, andDavid Lortimer(London),who have read the text with

great care.I also thank my Frenchpublisher,OdileJacob,who encouragedmeto write this new book,and StephenMorrow,my editorin the UnitedStates,as well asJudythSchaubhutSmith, for their help in preparingthe Englishedition.

I believethat we are at an important turning point in

the history ofscience.We have cometo the end of theroad paved by Galileoand Newton,whichpresentedus

with an imageofa time-reversible,deterministicuniverse.We now seethe erosionofdeterminismand the

emergenceofa new formulation ofthe laws ofphysics.

Ilya Prigogine

AUTHOR'SNOTE

I have triedto make this booka readable,self-containedaccountaccessibleto generalreaders.However,

especiallyin chapters5 and 6,I decidedto go into more

technical detail becausethe findings I have presenteddeviate

significantly from traditional views.In spiteofthe fact that

this volume is the result ofdecadesofwork, manyquestions still await answers.But taking into considerationthefinite life span ofeachofus, the fruits ofmy labors areshownsuch as they are today. I invite readersnot on a visit

to an archaeologicalmuseum,but rather on an adventurein sciencein the making.

IX

Introduction

A NEW RATIONALITY?

Earlierthis century in The Open Universe:An

Argument/or Indeterminism, Karl Popperwrote,\"Commonsenseinclines,on the onehand, to assertthat every event is

causedby someprecedingevents, so that every event canbe explainedor predicted....On the other hand, ...commonsenseattributes to mature and sane humanpersons ...the ability to choosefreely betweenalternative

possibilitiesofacting.\"1This \"dilemma ofdeterminism,\"as William Jamescalledit, is closelyrelatedto the meaningof time.2Is the future given, or is it under perpetualconstruction?A profound dilemma for all of mankind,as time is the fundamental dimensionof our existence.It was the incorporationof time into the conceptualschemeof Galilean physics that marked the originsofmodernscience.

This triumph ofhuman thought is also at the rootofthe main problemaddressedby this book:the denial ofwhat has beencalledthe arrow of time. As is well known,Albert Einstein often asserted,\"Timeis an illusion.\"In-

1

2 The End of Certainty

deedtime, as describedby the basiclaws ofphysics,from

classicalNewtoniandynamics to relativity and quantumphysics,doesnot includeany distinctionbetweenpast andfuture. Even today, for many physicistsit is a matter offaith

that as far as the fundamental descriptionofnature isconcerned, thereis no arrow oftime.

Yet everywhere\342\200\224in chemistry, geology,cosmology,biology, and the human sciences\342\200\224past and future playdifferent roles.Howcan the arrow oftime emergefrom what

physicsdescribesas a time-symmetricalworld?This is thetime paradox, oneofthe central concernsofthis book.

Thetime paradoxwas identifiedonly in the secondhalfof the nineteenthcentury after the ViennesephysicistLudwigBoltzmanntriedto emulatewhat CharlesDarwinhad donein biologyin an effort to formulate an

evolutionary approach to physics.The laws ofNewtonian physicshad longsincebeenacceptedas expressingthe idealofobjective knowledge.As they impliedthe equivalencebetween past and future, any attempt to confera fundamental

meaningon the arrow oftime was resistedas a threat tothis ideal.Isaac Newton'slaws were consideredfinal in

theirdomain ofapplication,somewhat the way quantummechanicsis now consideredto be final by manyphysicists. How then can we introduceunidirectionaltimewithout destroying these amazing achievementsof thehuman mind?

SinceBoltzmann,the arrow oftime has beenrelegatedto the realm ofphenomenology.We, as imperfecthuman

observers,are responsiblefor the difference betweenpastand future through the approximationswe introducein

ourdescriptionofnature.This is still the prevailingscientific wisdom.Certainexpertslament that we stand beforean unsolvable mystery for whichsciencecan provide no

Introduction 3

answer.We believethat this is no longerthe casebecauseof two recentdevelopments:the spectaculargrowth ofnonequilibriumphysicsand the dynamicsofunstable

systems, beginningwith the idea ofchaos.Over the past several decades,a new sciencehas been

born,the physics of nonequilibrium processes,and has ledto

conceptssuch as self-organizationand dissipative structures,

whichare widely used today in a largespectrumofdisciplines, includingcosmology,chemistry, and biology, aswell as ecologyand the socialsciences.The physicsofnonequilibriumprocessesdescribesthe effectsofunidirectional time and gives fresh meaningto the termirreversibility. In the past, the arrow of time appeared in

physicsonly through simpleprocessessuch as diffusion orviscosity, whichcouldbe understoodwithout anyextension of the usual time-reversibledynamics. This is nolongerthe case.We now know that irreversibility leadstoa host of novel phenomena,such as vortex formation,chemicaloscillations,and laserlight, all illustrating theessential constructive roleofthe arrow oftime.Irreversibilitycan no longerbe identifiedwith a mereappearancethat

would disappearif we had perfect knowledge.Instead, it

leads to coherence,to effects that encompassbillionsandbillionsofparticles.Figuratively speaking,matter at

equilibrium, with no arrow oftime,is \"blind,\" but with the

arrowoftime,it beginsto \"see.\"Without this newcoherence due to irreversible,nonequilibriumprocesses,life onearth would beimpossibleto envision.Theclaim that thearrow oftime is \"only phenomenological,\"or subjective,is thereforeabsurd.We are actually the childrenof thearrow oftime,ofevolution, not its progenitors.

Thesecondcrucialdevelopmentin revising the conceptof time was the formulation of the physicsof unstable


systems.Classicalscienceemphasizedorderand stability;

now, in contrast, we seefluctuations, instability, multiplechoices,and limited predictabilityat all levels ofobservation. Ideassuchas chaoshave becomequite popular,influencing our thinking in practically all fieldsofscience,from

cosmologyto economics.As we shall demonstrate,we cannow extendclassicaland quantum physicsto includeinstability

and chaos.We are then able to obtain a formulationofthe laws ofnature appropriatefor the descriptionofourevolving universe, a descriptionthat containsthe arrow oftime,sincepast and future no longerplay symmetricalroles.In the classicalview\342\200\224and herewe includequantummechanicsand relativity\342\200\224laws of nature expresscertitudes. When appropriate initial conditionsare given, wecan predictwith certainty the future, or \"retrodict\" the

past.Onceinstability is included,this is no longerthe case,and the meaningof the laws ofnature changesradically,for they now expresspossibilitiesorprobabilities.Herewe

go against oneofthe basictraditions ofWestern thought,the beliefin certainty. As statedby GerdGigerenzeret al.in The Empire of Chance,\"Despitethe upheavals in sciencein the over two millennia separatingAristotle from theParis ofClaudeBernard,they sharedat least oneattitudeoffaith: Sciencewas about causes,not chance.Kant even

promoteduniversal causal determinismto the status ofa

necessaryconditionofall scientificknowledge.\"3Therewere, however, dissenting voices.The great

physicistJamesClerkMaxwell spoke ofa \"new kind ofknowledge\"that would overcomethe prejudiceofdeterminism.4 But, on the whole,the prevailing opinionwasthat probabilitieswerestates ofmind rather than states ofthe world.This is so even today in spite of the fact that

quantum mechanicshas includedstatisticalconceptsin the

Introduction 5

coreof physics.But the fundamental objectofquantum mechanics,the wave function, satisfiesa deterministic,time-reversibleequation.To introduceprobability and

irreversibility, the orthodoxformulation ofquantummechanics requiresan observer.

Throughhis measurements,the observerwould bringirreversibility to a time-symmetricuniverse.Again, as in

the time paradox,we would be responsiblein somesensefor the evolutionary patterns ofthe universe.This roleofthe observer,whichgave quantum mechanicsits subjectiveflavor, was the main reasonthat preventedEinstein from

endorsingquantum mechanics,and it has sinceledto

unending controversies.Theroleofthe observerwas a necessaryconceptin the

introductionof irreversibility, or the flow of time,into

quantum theory. But once it is shown that instabilitybreakstime symmetry, the observeris no longeressential.In solving the time paradox,we also solve the quantumparadox,and obtain a new, realisticformulation ofquantum theory.This doesnot meana return to classicaldeterministic orthodoxy; on the contrary, we go beyond thecertitudesassociatedwith the traditional laws ofquantumtheory and emphasizethe fundamental roleofprobabilities. In both classicaland quantum physics,the basiclawsnow expresspossibilities.We neednot only laws, but alsoevents that bringan elementofradicalnovelty to the

description ofnature. This novelty leadsus to the \"new kindofknowledge\"anticipatedby Maxwell.ForAbraham DeMoivre, one of the founders of the classicaltheory ofprobabilities,chancecan neitherbe defined norunderstood.5 As we shall illustrate, we are now able to includeprobabilities in the formulation of the basic laws ofphysics.Oncethis is done,Newtoniandeterminismfails;


the future is no longerdeterminedby the present,and the

symmetry betweenpast and future is broken.Thisconfronts us with the most difficult questionsofall: What arethe roots oftime? Didtime start with the \"big bang\"?Ordoestime preexistour universe?

Thesequestionsplaceus at the very frontiers ofspaceand time.A detailedexplanationofthe cosmologicalimplications ofour positionwould requirea specialmonograph. Briefly stated,however, we believethat the bigbangwas an event associatedwith an instability within the

mediumthat producedour universe.It markedthe start ofour universe but not the start oftime.Although ouruniverse has an age,the mediumthat producedour universehas none.Timehas no beginning,and probablyno end.

But herewe enterthe world ofspeculation.The main

purposeofthis bookis to presentthe formulation ofthelaws ofnature within the rangeoflow energies.Thisis thedomain ofmacroscopicphysics,chemistry, and biology.Itis the domain in which human existenceactually takes

place.The problemsoftime and determinismhave remained

at the coreofWestern thought sincethe pre-Socratics.How can we conceiveofhuman creativity or ethics in a

deterministicworld?This questionreflects a profoundcontradictionin

Western humanistic tradition, which emphasizesthe

importance ofknowledgeand objectivity, as well as individual

responsibilityand freedomof choiceas impliedby the

ideal ofdemocracy.Popperand many otherphilosophershave pointed out that we are faced with an unsolvable

problemas long as nature is describedsolelyby a

deterministic science.6Consideringourselvesas distinct from

the natural worldwould imply a dualismthat is difficult for

Introduction 7

the modernmind to accept.Our aim in this work is toshow that we can now overcomethis obstacle.If\"the

passion ofthe westernworld is to reunitewith the groundofits being,\"as RichardTarnas has written, perhapsit is nottooboldto say that we are closingin on the objectofourpassion.7

Mankind is at a turning point, the beginningofa new

rationality in whichscienceis no longeridentified with

certitudeand probabilitywith ignorance.We agreecompletely with Yvor Leclercwhenhe writes, \"In the presentcenturywe are suffering from the separationofscienceand

philosophywhich followedupon the triumph ofNewtonian physicsin the eighteenthcentury8JacobBronowski

beautifully expressedthe same thought in this way: \"Theunderstandingofhuman nature and ofthe humancondition within nature is oneof the central themesofsci-

\"9ence.At the endofthis century, it is often askedwhat the

future of sciencemay be.Forsome,such as StephenW.

Hawking in his BriefHistory of Time, we are closeto the

end,the momentwhenwe shallbeableto read the \"mind

ofGod.\"10In contrast, we believe that we are actually at

the beginningofa new scientificera.We are observingthebirth ofa sciencethat is no longerlimited to idealizedand

simplifiedsituations but reflects the complexityofthe real

world, a sciencethat views us and our creativity as part ofa fundamental trend presentat all levels ofnature.

Chapter1

EPICURUS'DILEMMA

I

Isthe universe ruledby deterministiclaws? What is the

nature oftime? Thesequestionswereformulated by the

pre-Socraticsat the very start ofWestern rationality. After

morethan twenty-five hundredyears,they are still with us.However, recentdevelopmentsin physicsandmathematics associatedwith chaosand instability have openedupdifferent avenues ofinvestigation. We are beginningto seethese problems,which deal with the very position ofmankind in nature, in a new light,and can now avoid the

contradictionsofthe past.TheGreekphilosopherEpicuruswas the first to address

a fundamental dilemma.As a follower ofDemocritus,hebelievedthat the world is made of atoms and the void.Moreover,heconcluded,atoms fall through the void at thesame speedand on parallel paths. How then could theycollide?Howcouldnovelty associatedwith combinationsofatoms ever appear?ForEpicurus,the problemsofsci-

9


ence,the intelligibility ofnature, and human destinycouldnot be separated.What couldbe the meaningofhumanfreedomin a deterministicworld ofatoms? As Epicuruswrote to Meneceus,\"Ourwill is autonomousand

independent and to it we can attribute praise or disapproval.Thus,in orderto keepour freedom,it would have beenbetterto remain attached to the beliefin godsrather than

beingslavesto the fate ofthe physicists:Theformergivesus the hopeofwinningthe benevolenceofdeitiesthroughpromiseand sacrifices;the latter, on the contrary, bringswith it an inviolable necessity.\"1How contemporary this

quotationsounds!Again and again, the greatestthinkersin

Western tradition, suchas ImmanuelKant, Alfred NorthWhitehead, and Martin Heidegger,felt that they had tomake a tragic choicebetweenan alienating scienceor anantiscientific philosophy.They attempted to find somecompromise,but noneprovedto besatisfactory.

Epicurus thought that he had found a solutionto this

dilemma,whichhe termedthe clinamen. As expressedbyLucretius,\"While the first bodiesare beingcarrieddownwards by their own weight in straight lines through the

void, at times quite uncertain and at uncertain places,they

deviate slightly from their course,just enoughto bedefinedas

having changeddirection.\"2But no mechanismwas given forthis clinamen.No wonderthat it has always beenconsidered a foreign, arbitrary element.

But do we needthis novelty at all?ForHeraclitus,asunderstood by Popper,\"Truth lies in having grasped theessentialbecoming ofnature, i.e.,having representedit as

implicitly infinite, as a processin itself.\"3 Parmenidestookthe oppositeview. In his celebratedpoemon the uniquereality ofexistence,he wrote,\"Norwas it ever, norwill it

be,sincenow it is, all together.\"4

Epicurus'Dilemma 11

It is amusing that the Epicurus clinamenhas appearedrepeatedlyin the scienceofourcentury. Inhis classicpaperon the emissionofphotonsassociatedwith the transitionsbetweenatomicstates (1916),Einsteinexplicitlyexpressedhis confidencein scientificdeterminism,although heassumed that theseemissionsare ruledby chance.

Greekphilosophy was unable to solve this dilemma.Plato linked truth with being,that is,with the unchangingreality beyond becoming.Yet he was consciousof the

paradoxicalcharacterofthis positionbecauseit woulddebase both life and thought.In The Sophist,he concludedthat we needboth beingand becoming.5

This duality has plaguedWestern thought ever since.As

observedby the FrenchphilosopherJeanWahl, the historyofWestern philosophyis, on the whole,an unhappy one,characterizedby perpetualoscillationsbetweenthe worldas an automaton and a theologyin whichGodgovernstheuniverse.6Bothare forms ofdeterminism.

This debate tooka turn in the eighteenthcentury with

the discoveryofthe \"laws ofnature.\" Theforemost

examplewas Newton'slaw relating forceand acceleration,

whichwas both deterministicand, moreimportant, time

reversible.Oncewe know the initial conditions,we cancalculate all subsequentstates aswellas the precedingones.Moreover,past and future play the same rolebecauseNewton'slaw is invariant with respectto the time

inversion \302\243 \342\200\224> -t.This leads to nightmares such as the demonimaginedby Pierre-Simonde Laplace,capableofobserving

the currentstate ofthe universe and predictingitsevolution.7

As is wellknown, Newton'slaw has beensupersededinthe twentieth century by quantum mechanicsand

relativity.Still, the basiccharacteristicsofhis laws\342\200\224determinism


and time symmetry\342\200\224have survived. It is true that

quantum mechanicsno longerdealswith trajectoriesbut with

wave functions (seeSectionIV ofthis chapterandChapter 6),but it is important to note that the basicequationofquantum mechanics,Schrodinger'sequation,is onceagaindeterministicand time reversible.

By way ofsuch equations,laws ofnature lead tocertitudes. Onceinitial conditionsare given, everything isdetermined. Nature is an automaton, whichwe can control,at least in principle.Novelty, choice,and spontaneousaction are real only from our human point ofview.

Many historiansbelievethat an essentialrolein this

vision ofnature was playedby the Christian Godasconceived in the seventeenth century as an omnipotentlegislator.Theologyand scienceagreed.As Gottfried vonLeibnizwrote,\"In the least ofsubstances,eyesas piercingas thoseofGodcouldread the wholecourseofthings in

the universe, quaesint, quaefuerint, quaemoxfutura trahantur\"

(thosewhichare, whichhave been,and whichshall be in

the future).8 The discoveryofnature'sdeterministiclawswas thus bringinghuman knowledgecloserto the divine,

atemporalpoint ofview.Theconceptofa passivenature subjectto deterministic

and time-reversiblelaws is quite specificto the Westernworld.In Chinaand Japan, nature means \"what is byitself.\" In his excellentbookScienceand Societyin East and

West, JosephNeedhamtells us of the irony with whichChinesemenoflettersgreetedthe Jesuits'announcementof the triumphs ofmodernscience.9Forthem, the ideathat nature is governedby simple,knowablelaws seemedto be a perfect exampleof anthropocentricfoolishness.Accordingto Chinesetradition, nature isspontaneoushar-

Epicurus'Dilemma 13

mony; speakingabout \"laws ofnature\" would thus subjectnature to someexternalauthority.

In a messageto the great Indian poet,Rabindranath

Tagore,Einsteinwrote:

If the moon, in the act of completingits eternal path round

the earth, were gifted with self-consciousness,it would feel

thoroughly convincedthat it would travel its path on its own,in accordancewith a resolutiontaken onceand for all.

Sowould a Being,endowedwith higher insight and more

perfect intelligence, watching man and his doings, smile

about this illusion of his that he was acting accordingto his

own freewill.

This is my belief,although I know well that it is not fully

demonstrable.If one thinks out to the very last consequencewhat one exactlyknows and understands, there would hardly

be any human beingwho could be imperviousto this view,

providedhis self-lovedid not rub up against it. Man defendshimself from being regarded as an impotent object in the

courseof the Universe.But shouldthe lawfulness of

happenings,such as unveils itselfmore and more clearlyin inorganic

nature, ceaseto function in the activities in our brain?10

To Einstein, this appearedto be the only positioncompatible with the achievementsofscience.But this

conclusion is as difficult to acceptnow as it was to Epicurus.Timeis our basic existentialdimension.Sincethe nineteenthcentury, philosophyhas becomemoreand moretime

centered, aswe seein the work ofGeorgWilhelm Hegel,Edmund Husserl,William James,HenriBergson,Martin

Heidegger,and Alfred North Whitehead.ForphysicistssuchasEinstein,the problemhas beensolved.Forphiloso-


phers,it remains the central question ofontology,at the

very basisofthe meaningofhuman existence.In The Open Universe:An Argument for Indeterminism,

Popper wrote, \"I regard Laplaciandeterminism\342\200\224confirmed

as it may seemto beby the prima facie deterministictheories of physics,and by their marvelous success\342\200\224as themost solid and seriousobstacleto our understandingand

justifying the nature ofhuman freedom,creativity, and

responsibility.\" ForPopper,\"Thereality oftime and changeis the cruxofrealism.\"11

In his short essay,\"ThePossibleand the Real,\"Bergsonargued, \"What is the role of time? .. .Timepreventseverything from beinggiven at once....Is it not thevehicle ofcreativity and choice?Is not the existenceoftime

the proofofindeterminismin nature?\"12Forboth Popperand Bergson,we need\"indeterminism.\"But how do we

gobeyonddeterminism?This difficulty is wellanalyzed in

an essayby William Jamesentitled \"TheDilemmaofDeterminism.\"13 In accordwith well-definedmechanisms,determinismis \"mathematizable,\" as shownby the laws ofnature formulated by Newton,Schrodinger,and Einstein.Incontrast, deviations from determinismseemto introduceanthropomorphicconceptssuchas chanceoraccident.

The conflict betweenthe time-reversibleview ofphysics and time-centeredphilosophyhas ledto an openclash.What is the purposeofscienceif it cannotincorporatesomeofthe basicaspectsofhuman experience?The anti-

scientificattitude of Heideggeris well known.AlreadyFriedrichNietzschehad concludedthat thereare no facts,

only interpretations.As statedby JohnR.Searle,postmodern philosophy,with its idea ofdeconstruction,challengesWestern traditions regardingthe nature oftruth, objectiv-

Epicurus'Dilemma 15

ity, and reality.14In addition, the roleof evolution, ofevents, in our descriptionofnature is steadily increasing.How then can we maintain a time-reversibleview ofphysics?

In October1994,thereappeareda specialissueofScientific

American devoted to \"life in the universe.\"15At all

levels\342\200\224cosmology, geology,biology,and humansociety\342\200\224

we seea processofevolution in regardto instabilitiesandfluctuations. We thereforecannotavoid the question:Howare theseevolutionary patterns rootedin the fundamentallaws ofphysics?Only onearticle,written by the eminentphysicistStevenWeinberg,is relevant to this problem.Hewrites, \"As muchas we would like to take a unified viewofnature, we keepencounteringa stubborn duality in the

roleofintelligentlife in the universe, as both subjectandstudent. ...On the one hand, there is the Schrodingerequation,whichdescribesin a perfectly deterministic wayhow the wave function ofany systemchangeswith time.Then, quite separate,there is a set ofprinciplesthat tellshow to use the wave function to calculatethe probabilitiesofvarious possibleoutcomeswhen someonemakes themeasurement.\"16

Doesthis suggest that through our measurements,weourselvesare at the originofcosmicevolution? Weinbergspeaksof a stubborn duality, a point ofview found in

many recentpublicationssuch as StephenW Hawking'sBriefHistory of Time.17HereHawking advocatesa purelygeometricalinterpretation of cosmology.In short, timewouldbean accidentofspace.But heunderstandsthat this

interpretationis not enough.We needan arrow oftime todealwith intelligentlife.Therefore,alongwith many othercosmologists,Hawking introducesthe so-calledanthropic


principle.Nevertheless,this principleis as arbitrary as was

Epicurus'clinamen.Hawking gives no indicationofhowthe anthropic principlecouldever emergefrom a static

geometricaluniverse.As mentionedearlier,Einstein attempted to maintain

the unity ofnature, includingmankind, at the costofreducing us to mereautomata. This was also the view ofBaruchSpinoza.But therewasanotherapproachsuggestedby ReneDescartes,also in the seventeenthcentury, whichinvolved the conceptofdualism:on onesideis matter, res

externa, as describedby geometry,and on the other, the

mind, associatedwith res cogitans.iSIn this way, Descartesdescribedthe striking difference betweenthe behaviorofsimplephysicalsystemssuchas a frictionlesspendulumand

the functioning of the human brain.Curiously, the

anthropic principlebrings us back to Cartesiandualism.In The Emperor'sNew Mind, RogerPenrosestates,\"It is

ourpresentlackofunderstandingofthe fundamental lawsofphysicsthat prevents us from comingto grips with the

conceptof 'mind'in physicalorlogicalterms.\"19We

believe that Penroseis right:We needa new formulation ofthe fundamental laws ofphysics.Theevolutionary aspectsofnature have to beexpressedin termsofthe basiclaws ofphysics.Only in this way can we give a satisfactory answerto Epicurus'dilemma.Thereasons for indeterminism,for

temporalasymmetry, must berootedin dynamics.Formulations that do not containthesefeatures are incomplete,exactly as would be formulations ofphysics that ignoregravitation orelectricity.

Probability plays an essentialrolein most sciences,from

economicsto genetics.Still, the idea that probabilityis

merelya state ofmind has survived. We now have to go a

step farther and show how probabilityentersthe funda-

Epicurus'Dilemma 17

mental laws ofphysics,whetherclassicalor quantum. A

new formulation ofthe laws ofnature is now possible.Inthis way, we obtain a moreacceptabledescriptionin whichthereis roomfor both the laws ofnature and novelty and

creativity.At the beginningof this chapter, we mentionedthe

pre-Socratics.In fact, we owe to the ancientGreekstwoidealsthat have sinceshapedhuman history. The first is the

intelligibility ofnature, or in Whitehead'swords, \"the

attempt to frame a coherent,logical,necessarysystemofgeneralideasin terms ofwhichevery elementofourexperiencecan be interpreted.\"20The secondis the idea ofdemocracybased on the assumptionofhuman freedom,creativity, and responsibility.As longas scienceledto the

descriptionof nature as an automaton, thesetwo idealswerecontradictory.It is this contradictionthat we are

beginning to overcome.

IIIn SectionI,we emphasizedthat the problemsoftime and

determinismform the dividing linebetweenscienceand

philosophy,oralternatively,betweenC.P.Snow'stwo

cultures.21 But scienceis far from beinga monolithicbloc.Infact, the nineteenthcentury left us a doubleheritage:the

laws ofnature, such as Newton'slaw, whichdescribesatime-reversibleuniverse, and an evolutionary descriptionassociatedwith entropy.

Entropy is an essentialpart ofthermodynamics,thescience that dealsspecificallywith irreversible,time-orientedprocesses.Everyoneis to someextentfamiliar with theseprocesses.Think about radioactive decay, orabout

viscosity,whichslowsthe motionofa fluid. In contrast to time-


reversibleprocesses,such as the motionof a frictionless

pendulum,wherefuture and past play the same role(wecan interchangefuture, that is, +t, with past, \342\200\224t),

irreversible processeshave a directionin time.A radioactivesubstancepreparedin the pastwill disappearin the future.Becauseofviscosity, the liquid flow slowsover time.

The primordialroleofthe directionoftime is evidentin the processeswe study at the macroscopiclevel, such aschemicalreactionsor transport processes.We start with

chemicalcompoundsthat may react.As time goeson, theyreachequilibriumand the reactionstops.Similarly, if westart with an inhomogeneousstate, diffusion will tend to

homogenizethe system.Solarradiation is the result ofirreversible nuclearprocesses.No descriptionof the ecos-pherewould bepossiblewithout taking into accounttheinnumerableirreversibleprocessesthat determineweatherand climate.Nature involves both time-reversible and time-

irreversible processes,but it is fair to say that irreversible

processesare the rule and reversibleprocessestheexception. Reversibleprocessescorrespondto idealizations:We

have to ignorefriction to make the pendulummove re-versiblySuchidealizationsare problematicbecausethereis

no absolutevoid in nature. As previouslymentioned,time-reversibleprocessesare describedby equationsofmotion,whichare invariant with respectto time inversion,as is thecase in Newton's equation in classicalmechanicsorSchrodinger'sequationin quantum mechanics.Forirreversible processes,however, we needa descriptionthat

breakstime symmetry.The distinction between reversible and irreversible

processeswas introducedthrough the conceptofentropyassociatedwith the so-calledsecondlaw ofthermodynamics. Entropy had already beendefined by RudolfJulius

Epicurus'Dilemma 19

Clausiusin 1865(in Greek,entropysimply means

\"evolution\.22") Accordingto this law, irreversibleprocessesproduce entropy. In contrast, reversibleprocessesleave the

entropyconstant.We shall comeback repeatedlyto this secondlaw. For

now, let us recallClausius'scelebratedformulation: \"Theenergyofthe universe is constant.Theentropyoftheuniverse is increasing.\"This increasein entropy is due to theirreversibleprocessesthat take placein the universe.Clausius's statement was the first formulation ofan

evolutionaryview ofthe universe basedon the existenceofthese

processes.Arthur Stanley Eddington calledentropy the\"arrow of time.\"23 Nevertheless,accordingto thefundamental laws of physics, there should be no irreversible

processes.We thereforesee that we have inheritedtwo

conflictingviews ofnature from the nineteenthcentury:the time-reversibleview based on the laws ofdynamicsand the evolutionary view based on entropy. How cantheseconflictingviews bereconciled?After so many years,this problemis still with us.

For the Viennesephysicist Ludwig Boltzmann,thenineteenthcentury was the century ofCharlesDarwin,the man who defined life as the result ofa never-endingprocessofevolution and thus placedbecomingat the

center ofour understandingofnature. Still, for most

physicists, Boltzmannis now associatedwith a conclusionquiteoppositeto that ofDarwin; he is creditedwith havingshown that irreversibility is only an illusion.It wasBoltzmann's tragedy to have attempted in physicswhat Darwinhad accomplishedin biology\342\200\224only to reachan impasse.

At first glance,the similaritiesbetweenthe approachesofthesetwo giants ofthe nineteenthcentury are striking.Darwin showedthat if we start with the study ofpopula-


tions,and not individuals, we can understandhowindividual variability, subject to selectionpressure,producesadrift. Correspondingly,Boltzmannarguedthat we cannotunderstand the secondlaw of thermodynamics,and the

spontaneous increasein entropy it predicts,by startingwith individual dynamical trajectories;we must begininstead with largepopulationsofparticles.The increaseinentropy would be the globaldrift resultingfrom the

numerous collisionsbetweentheseparticles.In 1872,Boltzmannpublishedhis famous H-theorem,

whichincludedthe H-function,a microscopicanalogueofentropy24 This theoremtakes into accountthe effects ofcollisionsthat modify the velocitiesofparticlesat eachinstant. It showsthat collisionsbringthe distributionofvelocities ofthe populationofparticlescloserto equilibrium(the so-calledMaxwell-Boltzmanndistribution). As the

population approachesequilibrium,Boltzmann'sH-function decreasesand reachesits minimum value at

equilibrium;this minimum value means that collisionsno longer

modify the distributionofvelocities.ForBoltzmann,the

particlecollisionsare thus the mechanismthat leads the

systemto equilibrium.BothBoltzmannand Darwin replacedthe study of

\"individuals\" with the study ofpopulations,and showedthat

slight variations (the variability of individuals, ormicroscopic collisions)taking placeover a longperiodoftime

can generateevolution at a collectivelevel.(Inlater

chapters, we shall comeback to the roleofpopulations.)Exactly as biologicalevolution cannotbedefinedat the level

ofindividuals, the flow of time is also a globalproperty(seeChapters5 and 6).But whileDarwinattemptedto

explain the appearanceofnew species,Boltzmanndescribedan evolution toward equilibriumand uniformity. Signifi-

Epicurus'Dilemma 21

cantly, thesetwo theorieshave had very different fortunes.Darwin's theory of evolution, whichwas to triumph in

spite of fiercecontroversies,remains the basis for ourunderstandingoflife. On the otherhand, Boltzmann's

interpretation ofirreversibility succumbedto its critics,andhe was gradually forcedto retreat.He couldnot excludethe possibilityof \"antithermodynamical\" evolutions,as aresult of which entropy would diminish and inhomo-geneities,insteadofbeingleveled,would increasespontaneously.

Thesituation confronting Boltzmann was indeeddramatic. Hewas convincedthat in orderto understandnature we have to includeevolutionary features and that

irreversibility, as definedby the secondlaw ofthermodynamics, was a decisivestepin this direction.But hewas alsoheirto the grand tradition ofdynamics,and realizedthat it

stoodin the way of his attempt to give a microscopicmeaningto the arrow oftime.

Fromtoday'svantage point,Boltzmann'sneedto choosebetweenhis convictionthat physicshad to understand

becoming, and his loyalty to its traditional role,seemsparticularly poignant.The fact that his attempt would end in

failure now seemsself-evident.Every student learns that

a trajectory is time reversible,and thus allowsnodistinction betweenfuture and past. As HenriPoincarenoted,explainingirreversibility in terms of trajectoriesthat are

time-reversibleprocesses,however numerous,appears to

bea purelylogicalerror.25Supposethat we invert the signofthe velocity ofall molecules.Thesystemwould then gointo its own \"past.\" Even if entropywas increasingbeforevelocity inversion,it would now decrease.This wasJosephLoschmidt'svelocity-reversal paradox,whichwas thereason why Boltzmanncouldnot excludeantithermodynam-


ical behavior.When faced with severecriticism,hereplaced his microscopicinterpretation of the secondlawwith a probabilistic interpretationbaseduponourlackofinformation.

In a complexsystemformedby hugenumbersofmolecules (onthe orderof1023,orAvogadro's number),suchas a gas or liquid, it is obviousthat we are unable to

computethe behaviorofeachmolecule.Forthis reason,Boltz-

mann introducedthe assumptionthat all microscopicstatesofsucha systemhave the sameaprioriprobability.Thedifference would beassociatedwith the macroscopicstate, asdescribedby temperature,pressure,and otherparameters.Boltzmann defined the probabilityof eachmacroscopicstate by calculating the numberofmicroscopicstates that

give riseto it.Boltzmannwould have us imagine,for instance,a

volume dividedinto two equalcompartmentsthat

communicate with eachother.This volume containsa largenumberofmolecules,whichwe shall call N. Although we areunable to follow the path of each individual molecule,through measuringa macroscopicquantity, such as the

pressure in each compartment,we can determinethenumberof moleculesit contains.We can also preparea starting point,or \"initial state\" as it is generallyreferredto by physicists,whereoneof the two compartmentsisnearly empty.What can we expectto observe?Over the

courseoftime,moleculeswill populate the emptycompartment. Indeed,the greatmajority ofall possiblemicroscopic

states correspondsto a macroscopicsituation whereeachcompartmentcontains the same numberofmolecules. Thesestates correspondto equilibrium,or to

pressures that would beequal in the two compartments.Oncethis state has beenachieved,the moleculeswill continueto

Epicurus'Dilemma 23

passfrom onecompartmentto the other,but on average,the numberofmoleculesgoingto the right and left will beequal.Apart from slight transitory fluctuations, thenumber ofmoleculesin the two compartmentswill remainconstant over time,and equilibriumwill be preserved.However, there is a basic weaknessin this argument.Aspontaneouslong-termdeviation from equilibriumis not

impossible,even if it is, as Boltzmann concluded,\"improbable.\"

Boltzmann'sprobability-basedinterpretationmakesthe

macroscopiccharacterofour observationsresponsibleforthe irreversibility we observe.If we couldfollow theindividual motionof the molecules,we would seea time-reversiblesystemin whicheachmoleculefollows the lawsofNewtonianphysics.Becausewe can only describethenumberofmoleculesin eachcompartment,we concludethat the system evolves toward equilibrium.Accordingto this interpretation,irreversibility is not a basic law ofnature; it is merely a consequenceof the approximate,macroscopiccharacterofour observations.

Ernst Zermeloadded anothercriticismofBoltzmann's

argument to Loschmidt's reversal paradox26 in quotingPoincare'srecurrencetheorem,which shows that if wewereto wait longenough,we couldobservethespontaneous return ofa dynamical systemto a state ascloseto theinitial state as we might wish.As the physicistRomanSmoluchowskiconcluded,\"If we continuedourobservation for an immeasurablylong time,all processeswould

appear to be reversible.\"27This appliesdirectly toBoltzmann's two-compartmentmodel.After a sufficiently longtime,the initially empty compartmentwill again becomeempty. Irreversibility correspondsonly to an appearancethat is devoidofany fundamental significance.

24 TheEnd of Certainty

Let us now return to the situation discussedin SectionI.Throughourown approximations,we wouldberesponsible for the evolutionary characterof the universe. Inorderto makesuch an argument plausible,the first step in

assuring that irreversibility will be the result ofourapproximations is to view the consequencesof the secondlaw as trivial and self-evident.In his recentbook,The

Quark and the Jaguar, Murray Gell-Mannwrites,

The explanation [of irreversibility] is that there are more

ways for nails or pennies to be mixed up than sorted.Therearemoreways for peanut butter and jelly to contaminateeachother's containersthan there are to remain completelypure.And there are more ways for oxygenand nitrogen gasmolecules to be mixed up man segregated.To the extent that

chance is operating, it is likely that a closedsystem that has

someorder will move toward disorder,which offers so many

more possibilities.Howare thosepossibilitiesto be counted?An entire closedsystem, exactlydescribed,can existin a

variety of states,often calledmicrostates.In quantum

mechanics,theseare understoodto bepossiblequantum states of the

system. Thesemicrostatesare groupedinto categories(sometimes calledmacrostates)accordingto the various propertiesthat are beingdistinguished by coarsegraining. The microstatesin a given macrostateare then treated as equivalent, so that

only their number matters. . . .

Entropy and information are very closelyrelated.In fact,

entropy can be regardedas a measureof ignorance.When it

is known only that a system is in a given macrostate,the

entropy of the macrostatemeasuresthe degreeof ignorancethe

microstatesystem is in by counting the number of bits of

additional information neededto specifyit, with all the

microstates in the macrostatetreatedas equally probable.28

Epicurus'Dilemma 25

Similarargumentscan be found in most booksdealingwith the arrow oftime.We believe that theseargumentsare untenable.They imply that it is our own ignorance,our coarsegraining,that leads to the secondlaw. Forawell-informedobserver,such as the demonimaginedby

Laplace, the world would appear as perfectly timereversible. We wouldbethe father oftime,ofevolution, andnot its children.Irreversibility subsists,whatever theprecision ofourexperiments.This meansthat attributing thesepropertiesto incompleteinformation can hardly beconsidered. It is interestingto note that Max Planckhad

already opposedthe idea of incompleteinformation todescribethe secondlaw. In his Treatiseon Thermodynamics

he wrote,

It would be absurd to assumethat the validity of the secondlaw dependsin any way on the skill of the physicist or

chemist in observingor experimenting.The gist of the

second law has nothing to do with experiment; the law asserts

briefly that there exists in nature a quantity which always changes

in the same way in all natural processes.The propositionstatedin

this generalform may be correct or incorrect;but whicheverit may be, it will remain so, irrespectiveof whether thinking

and measuring beingsexist on the earth or not, and whether

or not, assuming they do exist, they are able to measurethe

details of physical or chemicalprocessesmore accuratelyby

one,two, or a hundred decimalplacesthan we can.The

limitation of the law, if any, must lie in the sameprovinceas its

essentialidea, in the observedNature, and not in the

Observer. That man's experienceis calledupon in the deductionof the law is of no consequence;for that is, in fact, our only

way of arriving at a knowledgeof natural law.29


However,Planck'sviews wereto remain isolated.As wehave indicated,most scientistsconsideredthe secondlawthe result ofapproximations,or the intrusionofsubjectiveviews into the exactworldofphysics.In a celebratedstatement, Max Bornassertedthat \"irreversibility is the effectof the introductionof ignoranceinto the basic laws ofphysics.\"30

Our own point ofview is that the laws ofphysics,asformulated in the traditional way, describean idealized,stableworld that is quite different from the unstable,evolving

world in whichwe live.Themain reasonto discardthebanalization ofirreversibility is that we can no longerassociate the arrow oftime only with an increasein disorder.Recent developments in nonequilibriumphysics and

chemistrypoint in the oppositedirection.They show

unambiguously that the arrow of time is a sourceof order.

This is already clearin simpleexperimentssuchas thermal

diffusion, which has beenknown since the nineteenthcentury. Let us considera boxcontainingtwo components(such as hydrogen and nitrogen)where we heat oneboundary and coolthe other(seeFigure1.1).Thesystemevolves to a steady state in whichonecomponentisenriched in the hot part and the other in the coldpart. Theentropyproducedby the irreversibleheat flow leadsto an

orderingprocess,whichwould be impossibleif taken

independently from the heat flow. Irreversibility leads toboth orderand disorder.

The constructive roleof irreversibility is even morestriking in far-from-equilibriumsituations where non-equilibriumleads to new forms ofcoherence.(We shall

comeback to nonequilibriumphysicsin Chapter2.)We

have now learnedthat it is preciselythrough irreversible

processesassociatedwith the arrow of time that nature

achievesits most delicateand complexstructures.Life is

Epicurus'Dilemma 27

Figure1.1ThermalDiffusionAs a result of the differencein temperature betweenthe two

containers, the black moleculeshave a higher concentrationin the left

compartment. Thiscorrespondsto thermal diffusion.

T\\ T2

O\342\200\242

\342\200\242

Oo o

\342\200\242 oo # oo o

possibleonly in a nonequilibriumuniverse.Nonequilib-rium leadsto conceptssuch as self-organizationand dissi-

pative structures,whichwill be describedin moredetailin Chapter2.In From Being to Becoming, we had alreadyformulated the following conclusionsbasedon theremarkable developmentsin nonequilibriumphysicsand

chemistryover the past several decades:

\342\200\242 Irreversible processes(associatedwith the arrow oftime)are as real as reversibleprocessesdescribedby thefundamental laws ofphysics;they do not correspondtoapproximationsadded to the basiclaws.

\342\200\242 Irreversibleprocessesplay a fundamental constructive

rolein nature.31

What impact do theseconceptshave on currentthinkingabout dynamical systems?Boltzmann was well aware that

nothinganalogousto irreversibility exists in classicaldynamics; he therefore concludedthat irreversibility can bederivedonly from assumptionsabout the initial conditions

O

o\342\200\242


in the early stagesofour universe.We can keepour usual

formulations of dynamics,but we need to supplementthemwith appropriateinitial conditions.In this view, the

originaluniverse was highly organized,and therefore in an

improbablestate\342\200\224a suggestionstill acceptedin a numberofrecentbooks.32Theinitial conditionsprevailing in ouruniverse leadto interestingand largely unsolvedproblems(seeChapter8),but we believethat Boltzmann'sargument is nolongerdefensible.Whatever the past, thereexistat presenttwo typesofprocesses:time-reversibleprocesses,wherethe

applicationofexistingdynamicshas provedto besuccessful

(i.e.,the motionofthe moonin classicalmechanics,orthe

hydrogen atom in quantum mechanics),and irreversible

processeslike heat conditions,where the asymmetrybetween past and future is obvious.Our objectiveis to devisea new formulation ofphysicsthat explains,independentlyofany cosmologicalconsiderations,the difference betweenthesebehaviors.This can indeedbe achievedfor unstableand thermodynamic systems.We can overcomewhat lookedlike an apparent contradictionbetweenthe time-reversiblelaws of dynamics and the evolutionary view of naturebasedon entropy. But let us not getaheadofourselves.

Nearly two hundred years ago,Joseph-LouisLagrangedescribedanalytical mechanicsbasedon Newton'slaws asa branch ofmathematics.33 In the Frenchscientificliterature, one often speaks of \"rational mechanics.\"In this

sense,Newton'slaws would define the laws ofreasonand

representa truth ofabsolutegenerality. Sincethe birth ofquantum mechanicsand relativity, we know that this is not

the case.Thetemptation is now strong to ascribea similar

status ofabsolute truth to quantum theory. In The Quarkand the Jaguar, Gell-Mannasserts,\"Quantummechanicsis

Epicurus'Dilemma 29

not itselfa theory; rather it is the framework into whichall

contemporaryphysicaltheory must fit.\"34 Is this really so?As statedby my late friend LeonRosenfeld,\"Every theoryis basedonphysicalconceptsexpressedthroughmathematical idealizations.They are introducedto give an adequaterepresentationof the physical phenomena.No physical

conceptis sufficiently defined without the knowledge of its domain

ofvalidity\"35

It is this \"domainofvalidity\" requiredfor the basic

conceptsofphysics,such as trajectoriesin classicalmechanics

orwave functions in quantum theory, that we are

beginningto delineate.Theselimits are associatedwith

instabilityand chaos,whichwe shall briefly introducein the next

section.Oncewe includetheseconcepts,we cometo a

new formulation of the laws of nature, one that is nolongerbuilt on certitudes,as is the casefor

deterministic laws, but rather on possibilities.Moreover,in this

probabilisticformulation, time symmetry is destroyed.Theevolutionary characterofthe universe has to bereflectedwithin the contextof the fundamental laws of physics.Rememberthe idealofthe intelligibility ofnature asformulated by Whitehead (seeSectionI): Every elementofour experiencehas to beincludedin a coherentsystemofgeneralideas.Basedon this rewritingofthe lawsofnature,we can now completethe work pioneeredby Boltzmannmorethan a century ago.

It is interestingthat great mathematicians, suchas EmileBorel,alsounderstoodthe needto overcomedeterminism.Borelnotedthat considerationsofisolatedsystems,suchas

moon-earth,are always idealizations,and that determinismmay fail whenwe leave this reductionistview.36That is

indeed what our own researchshows.


III

Everyoneis to someextentfamiliar with the differencebetween stableand unstable systems.Considera pendulum,for example.Supposethat it is originally at equilibrium,wherethe potentialenergyis at a minimum.If a smallperturbation is followedby a return to equilibrium(seeFigure1.2),this systemrepresentsa stableequilibrium.In contrast,if we put a pencilon its head, the smallestperturbationwill causeit to fall to the left orright,giving us a modelofunstable equilibrium.

Thereis a basicdistinctionbetweenstableand unstablemotions.In short, stable dynamical systemsare those in

whichslight changesin the initial conditionsproducecorrespondingly slight effects.But for a largeclassofdynamical systems,small perturbationsin the initial conditionsare

amplified over the courseoftime.Chaoticsystemsare an

extremeexampleofunstable motionbecausetrajectoriesidentified by distinct initial conditions,no matter howclose,diverge exponentiallyover time.This is known as

\"sensitivity to initial conditions.\"A classicillustration ofamplification through chaosis the \"butterfly effect\"; by

Figure1.2Stableand UnstableEquilibrium

kpot kpot

Stable

Unstable

Epicurus'Dilemma 31

just flapping its wings, a butterfly in Amazonia may affectthe weather in the UnitedStates.We shall seeexamplesofchaoticsystemslater on in Chapters3 and 4.

Thetermdeterministic chaoshas also enteredthediscussion ofchaoticsystems.Indeed,the equationsofmotionremain deterministic,as is the casein Newtoniandynamics,even if a particular outcomeappearsto be random.Thediscoveryofthe important roleofinstability has led to arevival ofclassicaldynamics,previously considereda closedsubject.In fact, until recentlyit was thought that all systemsdescribedby Newton'slaws are alike.Ofcourse,everyoneknew that the trajectory of a falling stone was easiertosolve than a \"three-bodyproblem,\"suchas the oneinvolving

the sun, Earth, and Jupiter.But this was consideredtobemerelya questionofcomputation.It was only at the endof the nineteenthcentury that Poincareshowed that this

was not the case.Theproblemsare fundamentally different

dependingon whetherornot a dynamical systemis stable.We have mentionedchaoticsystems,but thereare other

typesofinstability to beconsidered.Let us first describeinqualitative terms in what senseinstability leads to an

extension ofthe laws ofdynamics.In classicaldynamics,the

initial state is determinedby the positionsq and velocitiesv (ormomentump).*Oncetheseare known, we can

determine the trajectory by using Newton'slaws (or anyother equivalent formulation ofdynamics).We can then

representthe dynamical state by a point q0,p0 in a spaceformedby the coordinatesand momenta.This is known asthe phasespace(Figure1.3).Insteadofexamininga singlesystem,we can also study a collectionofsystems\342\200\224an \"en-

*For the purpose of simplification, we have used a single letter even when we

are discussing a system formed by many particles.


Figure1.3Trajectory in PhaseSpaceThe dynamical state is represented by a point in the phase spaceq, p.The time evolution is describedby a trajectory starting at the initial

point q0, p0.

Po'lo

semble,\"as it has beencalledsincethe pioneeringwork ofAlbert EinsteinandJosiahWillard Gibbsat the beginningofthis century.

At this point, it would be helpful to reproducepart ofGibbs'sfamous prefaceto his Elementary PrinciplesinStatistical Mechanics:

We may imagine a great number of systems of the same

nature, but differing in the configurations and velocitieswhich

they have at a given instant, and differing not merely infini-

tesimally, but it may be so as to embrace every conceivablecombination of configurations and velocities.And here we

may set the problem, not to follow a particular system

through its successionof configurations, but to determine

how the whole number of systems will be distributed amongthe various conceivableconfigurations and velocitiesat any

requiredtime, when the distribution has beengiven for someone time. . . .

Epicurus' Dilemma 33

The laws of thermodynamics,as empirically determined,

expressthe approximate and probable behaviorof systems of a

great number of particles,or, more precisely,they expressthe

laws of mechanicsfor such systems as they appear to beingswho have not the finenessof perception to appreciatequantities of the order of magnitude of those 'which relate to

single particles,and who cannot repeat their experimentsoften

enough to obtain any but the most probableresults,37

Gibbs introducedpopulation dynamicsinto physicsby

using an ensembleapproach.An ensembleis representedby a cloudofpoints in phasespace(seeFigure1.4).Thecloudis describedby a function p(q,p,t),whichhas a

simple physicalinterpretation:the probability offinding at time

t, a point in the small regionofphase spacearound the

point q,p. A trajectory correspondsto a specialcasein

whichp is vanishing everywhereexceptat the point q0,p0-This situation is describedby a specialform ofp.Functions that have the property ofvanishing everywhereexcept at a singlepoint are calledDiracdelta functions d(x).The function d(x \342\200\224

xQ) is vanishing for all points x 5* xQ.Therefore,for a singletrajectoryat time zero,the

distribution function p takesthe form p= 8(q-q0)${p\342\200\224

p0)-*We

shall comeback to the propertiesofdelta functions later.

*When we take x= x0, the function 8(x-xQ) diverges to infinity. The8-func-

tion therefore has \"abnormal\" properties as compared to a continuous function

such as xor sin x. It is called a generalized junction or distribution (not to beconfused with probability distribution p). Generalized functions are used in

conjunction with test functions <f(x), which are continuous functions (i.e.,J dx(p(x)B(x- x())

=<p(x0)). Also note that at time t we have for a free particle

moving at the speedm the probability p = 8(p-p0)B(q-

qQ-

m ), as the

momentum remains constant and the coordinate varies linearly with time.


Figure1.4Ensemblesin PhaseSpaceGibbs'sensembleis representedby a cloudof particlesdiffering

according to their initial conditions.The shapeof the cloudchangesover time.

As was clearly statedby Gibbs, however,the ensembleapproachwas merelya convenientcomputationaltoolforhim whenexactinitial conditionswerenot available.In his

opinion,probabilitiesexpressignorance,or lack ofinformation. Moreover,it has always beenacceptedthat fromthe dynamical point ofview, individual trajectoriesand

probabilitydistributionspresent equivalent problems.We

can start with individual trajectoriesand then derive theevolution of probability functions, and vice versa. Theprobabilitypcorrespondssimplyto a superpositionoftrajectories, and leadsto no new properties.Thetwo levelsofdescription,the individual level (correspondingto singletrajectories)and the statistical level (correspondingtoensembles),would beequivalent.

Is this always the case?Forsimplestablesystems,wherewe donot expectany irreversibility, this is indeedtrue.Gibbs and Einstein were right. The individual point ofview (in terms oftrajectories)and the statistical point ofview (in terms ofprobabilities)are then equivalent.This

Epicurus'Dilemma 35

canbeeasilyverified, and we shall comeback to this pointin Chapter5.However, is this also true for unstablesystems? Howis it that all theoriesdealingwith irreversible

processeson the molecularlevel, such as Boltzmann'skinetictheory, involve probabilitiesand not trajectories?Is this again becauseof our approximations,our coarsegraining? Howcan we then explainthe successofkinetictheory, the quantitative predictionsofmany propertiesofdilutegases,suchas thermal conductivity and diffusion, all

ofwhichhave beenverified by experimentation?Poincarewas so impressedby the successofkinetic

theorythat hewrote,\"Perhapsthe kinetictheory ofgaseswill

serveas a model.. . Physical laws will then take on a

completely new form; they will take on a statistical character.\"3*

Thesewereindeedpropheticwords.In an extraordinarilydaring move, Boltzmannintroducedprobabilityas an

empirical tool.Now, morethan onehundred years later, weare beginningto understand how probabilisticconceptsemergewhenwe go from dynamicsto thermodynamics.Instability destroysthe equivalencebetweenthe individual

and statistical levels ofdescription.Probabilitiesthen

acquire an intrinsical dynamical meaning.This knowledgehas ledto a new kind ofphysics,the physicsofpopulations, whichis the basicsubjectofthis book.

By way ofexplanation,let us considera simplifiedexample ofchaos.Supposethat we have two typesofmotiondenotedas + or \342\200\224 (i.e.,motion\"up\" or \"down\") within

the phase spaceillustrated in Figure1.4.This leadsus tothe two typesofsituations representedby Figures1.5and1.6.In Figure1.5,thereare two different regionsin phasespace,onecorrespondingto the motion

\342\200\224,

the otherto themotion+.If we discardthe regioncloseto the boundary,each\342\200\224 is surroundedby \342\200\224,

and each+ by +.This corre-


Figure1.5StableDynamicalSystemThe motions denotedas + or-lie in distinct regionsofphasespace.

spondsto a stablesystem.Slightchangesin the initial

conditions do not alter the result.In Figure1.6,instead,each+ is surroundedby \342\200\224,

andvice versa. Theslightestchangein initial conditionsisamplified, and the systemis thereforeunstable.A primary re-

Figure1.6UnstableDynamicalSystemEach motion + is surrounded by

-and viceversa.

Epicurus'Dilemma 37

suit ofthis instability is that trajectoriesnow becomeidealizations. We can no longerprepare a singletrajectory, asthis would imply infinite precision.Forstablesystems,this

is without significance,but for unstable systems,with their

sensitivity to initial conditions,we can only prepareprobability distributions,includingvarious typesofmotion.

Is this difficulty merelya practicalone?Yes,if weconsider that trajectorieshave now becomeuncomputable.But there is more:Probabilitydistribution permits us to

incorporatewithin the framework of the dynamical

description the complexmicrostructureofthe phasespace.Itthereforecontainsadditional information that is lackingat

the level ofindividual trajectories.As we shall seein

Chapter 4, this has fundamental consequences.At the level ofdistributionfunctions p,we obtain a new dynamicaldescription that permits us to predict the future evolution ofthe ensemble,includingcharacteristictime scales.This is

impossible at the level of individual trajectories.Theequivalencebetweenthe individual and statistical levels is

indeedbroken.We obtain new solutionsfor the

probabilitydistributionp that are irreducible becausethey do not

apply to singletrajectories.The laws ofchaoshave to beformulated at the statisticallevel.That is what we meant in

the precedingsectionwhenwe spokeabout ageneralization ofdynamicsthat cannotbeexpressedin terms oftrajectories. This leads to a situation that has never beenencounteredin the past.Theinitial conditionis no longera point in the phasespacebut someregiondescribedby pat the initial time (= zero.We thus have a nonlocal

description. Thereare still trajectories,but they are the outcomeofa stochastic,probabilisticprocess.No matter how

precisely matchedour initial conditionsare, we obtaindifferent trajectoriesfrom them.Moreover,as we shall see,time


symmetry is broken,as past and future play different rolesin the statistical formulation. Ofcourse,for stable systems,we revert to the usual descriptionin terms ofdeterministic trajectories.

Why has it taken so longto arrive at a generalizationofthe laws ofnature that includesirreversibility and

probability? One of the reasons is ideological\342\200\224the desiretoachieve a quasi-divinepoint ofview in our descriptionofnature. But therehas also beena technical,mathematical

probleminvolved. Our work is basedon recentprogressinfunctional analysis,a field ofmathematics that has cometothe forefront only in recentdecades.As we shall see,ourformulation requiresan extendedfunctional space.Thisnew field ofmathematics,which uses generalizedfunctions orfractals, as BenoitMandelbrotcalledthem,is now

playing a criticalrolein the understandingofthe laws ofnature.39We needa \"divine\" point ofview to retain theidea of determinism.But no human measurements,notheoreticalpredictions,can give us initial conditionswith

infinite precision.It is interestingto contemplatewhat becomesof the

Laplacedemonin the world ofdeterministicchaos.Hecan no longerpredictthe future unlessheknowsthe initial

conditionswith infinite precision.Only then can hecontinue to use a trajectory description.But there is an evenmorepowerful instability that leads to the destructionoftrajectories,whatever the precisionof the initial description.This form ofinstability is offundamental importancebecause it appliesto both classicaland quantum mechanics.

Our story actually beginsat the end ofthe nineteenth

century with the work ofJules-HenriPoincare.Accordingto Poincare,a dynamical systemis characterizedin termsofthe kineticenergyofits particlesplus the potentialen-

Epicurus'Dilemma 39

ergy dueto their interaction.40A simpleexamplewouldbefree, noninteractingparticles,wherethere is no potentialenergy,and the calculationoftrajectoriesis trivial. Suchsystemsare by definition integrable.Poincarethen askedthe question:Are all systemsintegrable?Can we choosesuitable variables to eliminatepotential energy?Byshowing

that this was generallyimpossible,he proved that

dynamical systemswerelargely nonintegrable.

It is worthwhile to pause for a momentand reflect onPoincares conclusions.Supposehe had proved that all

dynamical systemsare integrable.This would mean that all

dynamical motionsare isomorphicto free noninteractingparticles.Therewould beno placefor the arrow oftime,for self-organization,or life itself. Integrablesystemsdescribea static, deterministicworld.Poincarenot onlydemonstratednonintegrability, but alsoidentifiedthe

reason for it: the existenceofresonancesbetween the degreesoffreedom. As we shall seein greaterdetail in Chapter5,thereis

a frequency that correspondsto eachmodeofmotion.Thesimplestexampleofthis is the harmonicoscillator,inwhicha particleand central point are given.Theparticleis

heldby a forceproportionalto its distancefrom that point.If we displacethe particlefrom the center,it will oscillatewith a well-definedfrequency.It is through thesefrequencies that we arrive at the notionof resonance,which is

crucialto Poincare'stheorem.We are all moreorlessfamiliar with the conceptof

resonance. When we forcea spring to deviate from its

equilibrium position,it vibrates with a characteristicfrequency.Now let us subject this spring to an externalforcewith a

frequency that can bevaried.When the two frequencies,that of the spring and that of the externalforce,have a

simplenumericalratio (that is, whenoneofthe frequen-


ciesis either equal to the other, or two, three,four . . .

times larger),the amplitudeof the motionofthe springincreasesdramatically. The same phenomenon occurswhenwe play a noteon a musicalinstrument.We heartheharmonics.Resonance\"couples\"sounds.

Nowconsiderthe caseofa systemcharacterizedby two

frequencies.By definition,wheneverthe sum n10)1+n2<CQ2

= 0,wheren, and \302\253, are nonvanishing integers,we have9h \342\200\224 _ Hiresonance.This means that \302\2532

\342\200\224

\302\273i;the ratio ofthe

frequencies is then a rational number.As Poincarehas shown,in dynamicsresonanceslead to terms with \"dangerous\"denominatorssuch as

\342\200\236,,,

.\342\200\236 ,\342\200\236

Whenever thereare res-11 \"2 2\"

onances(i.e.,points in phase spacewheren1a>1 + n2a>2=

zero),theseterms diverge.As a result,we encounterobstacles wheneverwe try to calculatetrajectories.

This is the origin of Poincare'snonintegrability. The\"problemof small denominators\"was already known by

eighteenth-centuryastronomers,but Poincare'stheoremshowed that this difficulty, whichhe called the \"generalproblemofdynamics,\" is sharedby the great majority ofdynamical systems.For a considerablelengthoftime,however, the importanceofPoincare'sfindings was overlooked.

Max Bornwrote, \"It would indeedbe remarkable if

Nature had fortified herselfagainst further advances in

knowledgebehindthe analytical difficulties ofthe many-

bodyproblem.\"41It was hard to believethat a technicaldifficulty (divergencesdue to resonances)could alter the

conceptualstructureofdynamics.We now seethis

problem in a different way. Forus,Poincare'sdivergencesare an

opportunity.Indeed,we can go beyond his negativestatement and show that nonintegrabilitypaves the way, as doeschaos,for a new statistical formulation ofthe laws ofdy-

Epicurus'Dilemma 41

namics.It tooksixty years after Poincare,through the workofAndrei N.Kolmogorov,continuedby Vladimir Igore-vich Arnold andJiirgenKurt Moser(the so-calledKAM

theory), for nonintegrabilityto be understoodnot as the

frustrating manifestation ofsomeresistanceofnature againstthe advances ofknowledge,to paraphraseBorn,but as anewstarting point for dynamics.42

The KAM theory deals with the influenceofresonances on trajectories.ThefrequenciesCO dependin

general on the values ofdynamic variables suchas coordinatesand momenta.They thereforetake on different values at

different points in the phasespace.Theresult is that somepointswill becharacterizedby resonances,and otherswill

not.Again, for chaosthis leads to an extraordinarycomplexity in the phasespace.Accordingto the KAM theory,we observetwo typesoftrajectories:\"nice\" deterministictrajectoriesand \"random\" trajectoriesassociatedwith

resonances, which wander erratically through regionsofphasespace.

Another important result ofthis theory is that whenweincreasethe value ofenergy,we increasethe regionswhererandomness prevails. Forsomecriticalvalue of energy,chaosappears:over time we observe the exponentialdivergence of neighboringtrajectories.Furthermore,for

fully developedchaos,the cloudofpoints generatedbya trajectory leads to diffusion. But diffusion is associatedwith the approachto uniformity in ourfuture. It is an

irreversible processthat createsentropy (seeSectionI).Although we started with classicaldynamics,we can nowobservethe breakingoftime symmetryHow this is

possible is the main problemwe have to solve in ordertoovercome the timeparadox.


Poincareresonancesplay a fundamental rolein physics.Emissionor absorption of light is due to resonances,asis the approachto equilibrium in a systemof interactingparticles.Interactingfields again lead to resonances.It is

difficult to identify an important problemin classicalorquantum physicswhereresonancesdo not play a significantpart.But how can we overcomethe divergencesassociatedwith resonances?Heresomeessentialprogresshas beenmade.As in SectionIII,we have to distinguishtheindividual level (trajectories)from the statisticallevel (ensembles,as

describedby the probabilitydistributionp).At theindividual level we have divergences,but thesecan be solvedat

a statistical level (seeChapters 5 and 6), whereresonances producea couplingofevents looselyanalogousto the

couplingofsoundsby resonance.This leadsto new, non-Newtonianterms that are incompatible with a trajectory

description and instead requirea statistical,probabilisticdescription.This is not astonishing.Resonancesare not localevents,inasmuch as they do not occurat a given point or instant.

They imply a nonlocaldescriptionand therefore cannot beincludedin the trajectory descriptionassociatedwith

Newtonian dynamics.As we shall see,they lead to diffusive

motion. When we start at a point PQin phasespace,we can nolongerpredictwith certainty its positionPx after a time

period X. In short, the initial point PQleadsto many possiblepoints\342\200\224Pv P2, P3\342\200\224with

well-definedprobabilities.In Figure1.7,eachpoint in the domainD has a nonva-

nishingorwell-definedtransition probabilityofappearingat time X. This situation is similar to that ofthe \"random

walk,\" or \"Brownian motion.\"In the simplestcase,this

conditionmay beillustrated by a particleon a

one-dimensional lattice that makesa one-steptransition at regular time

intervals (seeFigure1.8).

Epicurus'Dilemma 43

Figure1.7Diffusive MotionAfter a time t, the system may producea result at any point, suchas Pj, P2, P3, in the domain D.

At every step, the probabilityis i that the particlewill

go to the left and 2 that it will go to the right.At every

step, the future is uncertain.Fromthe very beginning,itis impossible to speak of trajectories.Mathematically,Brownianmotionis describedby diffusion-type equations(the so-calledFokker-Planckequations).Sincediffusion is

Figure1.8A RandomWalk

Brownian motion on a one-dimensionallattice.At every step,the

probability is\\

that the particle will go to the left and\\

that it will

go to the right.

1 12 2


time oriented,if we start with a cloudofpoints, all ofwhichare situatedat the sameorigin,as time goeson thecloudwill disperse.Someparticleswill be found fartherfrom the origin,otherscloser.It is quite remarkablethat,

starting with classicaldynamics,resonanceslead preciselyto diffusive terms,which is to say, resonancesintroduceuncertainty even within the framework of classicalmechanics, and so break time symmetry.

Forintegrablesystems,when thesediffusive

contributions are absent,we comeback to the trajectorydescription, but in generalthe laws of dynamics have to beformulated at the level of probabilitydistributions.Thebasicquestion is therefore:In whichsituations can we

expect the diffusive terms to beobservable?When this is so,probabilitybecomesa basicpropertyofnature. Thisquestion, whichinvolves defining the limits ofthe validity ofNewtoniandynamics(or the validity ofquantum theory,whichwe shall considerin the next section),is nothingshort of revolutionary. For centuries,trajectorieshavebeenconsideredthe basic, primitive objectof classical

physics.In contrast,we now considerthemoflimited

validity for resonant systems.We shall return repeatedlytothis question in Chapter5, and to a parallel question for

quantum mechanicsin Chapter6.Forthe moment,however, let us presentsomeprovisional answers.Fortransient

interactions(a beamofparticlescollideswith an obstacleand escapes),diffusive terms are negligible.But forpersistent interactions(a steady flow ofparticlesfalls onto the

obstacle),they becomedominant.In computersimulations, as in the real world, we can reproducebothsituations and thereforetest our predictions.Theresultsshow

unambiguouslythe appearanceofdiffusive terms for per-

Epicurus'Dilemma 45

sistent interactions,and thereforethe breakdown of the

Newtonian,as wellas the orthodox,quantum mechanicaldescriptions.In both thesecases,we obtain \"irreducible\"

probabilisticdescriptions,as in deterministicchaos.But thereis yet anothersituation that is even more

remarkable. Macroscopicsystemsare generally defined in

terms ofthe thermodynamic limit, accordingto whichboththe numberN of particles and the volume V becomelarge.We shall study this limit in Chapters 5 and 6.In theobservationofphenomenaassociatedwith this limit, thenew propertiesofmatter becomeobvious.

As longas we considermerelya few particles,wecannot say if they form a liquidorgas.Statesofmatter as wellas phase transitions are ultimately definedby the

thermodynamiclimit.The existenceofphase transitions shows

that we have to becareful whenwe adopta reductionistattitude. Phase transitions correspondto emergingproperties. They are meaningful only at the level ofpopulations,and not ofsingleparticles.This contentionis somewhat

analogousto that whichis based on Poincareresonances.Persistentinteractionsmeanthat we cannottake a part ofthe systemand considerit in isolation.It is at this globallevel, at the level ofpopulations, that the symmetrybetween past and future is broken,and sciencecan recognizethe flow oftime.This solvesa long-standingpuzzle.It is

indeedin macroscopicphysicsthat irreversibility and

probabilityare the most conspicuous.

Thermodynamics applies to non-integrablesystems.This means that we cannot solve the dynamical problemin terms of trajectories,but we can solve it in terms ofprobabilities.Therefore,as is the casefor deterministicchaos,the new statisticalformulation ofclassicalmechanics


leadsto an extensionofthe mathematical framework.Tosomeextent,this isreminiscentofgeneralrelativity. As

Einstein showed,we have to move from Euclideangeometryto Riemanniangeometryto includegravitation. Infunctional calculus,a special roleis played by the so-calledHilbertspace,which extends Euclideangeometry tosituations involving an infinite number of dimensions(the \"function space\.")Traditionally, quantum mechanicsand statistical mechanicshave utilized Hilbertspace.Toobtain our new formulation, which is valid for unstable

systemsand the thermodynamic limit, we have to movefrom Hilbertspaceto moregeneralfunctional spaces.Thisobservation will be explainedin detail in Chapters 4

through 6.Sincethe beginningof this century, we have become

used to the ideathat classicalmechanicshas to beextendedwhen we considermicroscopicobjects,such as atoms orelementaryparticles,or when we deal with astrophysicaldimensions.Surprisingly,instability alsorequiresan

extension ofclassicalmechanics.Thesituation in quantummechanics, to whichwe now turn, is quite similar.Instabilitydriven by resonancesplays a fundamental rolein changingthe formulation ofquantum theory.

IVIn quantum mechanics,we encountera rather strangesituation. As is well known, this theory has beenremarkablysuccessfulin all its predictions.Still,morethan sixty yearsafter its formulation, discussionsabout its meaningand

scopeare as heatedas ever.This is uniquein the history ofscience.43In spiteofall its successes,most physicistsshare

Epicurus'Dilemma 47

somefeelingof uneasiness.RichardFeynman onceremarked that nobody really \"understands\"quantum theory.

Here,the basicquantity is the wave function*\302\245,

whichplays somewhat the roleofthe trajectory in classicalmechanics. Indeed,the fundamental equationof quantumtheory, the Schrodingerequation,describesthe timeevolution ofthe wave function.It transforms the wavefunction ^(tf), as given at the initial time

tQ, into the wavefunction

*\302\245(t)at time t, exactlyas trajectoriesin classical

mechanicslead from onephasepoint to another.Like Newton'sequation,Schrodinger'sequationis

deterministic and time reversible. Again, as in classical

dynamics,thereappearsa gap betweenthe dynamicaldescription of quantum mechanicsand the evolutionary

descriptionassociatedwith entropy.Thephysicalinterpretation ofthe wave function *\302\245 is that ofa probability

amplitude. This implies that the square | 4*12 = W* (4* hasboth a real and imaginary part; 4** is the complexconjugate

of4*) is a probability, whichwe shall again denoteby

p.Thereare moregeneralforms of probabilitycorresponding to ensemblesobtained by the superimposingofvarious wave functions.Theseare calledmixtures,as

opposed to pure casesthat obtain from a singlewave

function.

Thebasicassumptionofquantum theory is that everydynamical problemcan be solvedat the level ofprobability amplitudes exactly as every dynamical problemin

classical mechanicswas traditionally associatedwith trajectorydynamics.But strangely, in orderto attribute well-definedpropertiesto matter, we have to gobeyondprobabilityamplitudes; we needprobabilitiesthemselves.To understandthis difficulty, let us considera simple example.Suppose


that energycan take on two values, Ex and Er Thecorresponding wave function is ux or uT Nowconsiderthelinear superimposition*\302\245

= ciui + c2u2.Thewave functionthen \"participates\"at both levels.Thesystemis neitherat

level 1nor level 2,but rather in a kind of intermediatestate.Let us now measurethe energy associatedwith 4*.

Accordingto quantum mechanics,we then find eitherExorE2 with probabilitiesgiven by the squaresofthe

probability amplitudes I q I

2 and I c212.Initially we startedwith a singlewave function 4*, but

we still end up with a mixtureoftwo wave functions, ux

and wr This is often calledthe \"reduction,\"or \"collapse,\"ofthe wave function.We needto move from potentialitiesdescribedby the wave function *\302\245 to actualities that we canmeasure.In the traditional languageofquantum theory,we move from a pure state (the wave function) to an

ensemble, ormixture.But how is this possible?As mentionedearlier,Schrodinger'sequationtransforms a wave functioninto anotherwave function, and not into an ensemble.This has often beencalledthe quantum paradox.It has beensuggestedthat the transition from potentiality to actualityis dueto ourown measurements.This is the point ofview

expressedby StevenWeinbergin SectionI ofthis chapterand in a considerablenumberoftextbooks.It is the same

type ofexplanationas waspresentedfor the time paradoxin classicalmechanics.In that caseas well, it is difficult tounderstand how a human action, such as observation,couldbe made responsiblefor the transition frompotentialities to actualities.Would the evolution ofthe universe

be different in the absenceofhumankind? In his

Introduction to The New Physics:ASynthesis, Paul C.W Davieswrites:

Epicurus'Dilemma 49

At rockbottom, quantum mechanicsprovidesa highly

successful procedurefor predictingthe results of observationsof

microsystems,but when we ask what actually happenswhenan observation takes place, we get nonsense!Attempts to

breakout of this paradoxrange from the bizarre,such as the

many universes interpretation of Hugh Everett, to the

mystical ideasofJohn von Neumann and EugeneWigner, whoinvoke the observer'sconsciousness.After half a century of

argument, the quantum observationdebate remains as lively

as ever.The problemsof the physics of the very small and the

very largeareformidable,but it may be that this frontier\342\200\224the

interfaceof mind and matter\342\200\224will turn out to be the most

challenginglegacyof the NewPhysics.44

This \"interfacebetweenmind and matter\" is also at thecoreofthe time paradox.Ifthe arrow oftime existedonlybecauseour human consciousnessinterferedwith a worldotherwiseruledby time-symmetricallaws, the veryacquisition ofknowledgewould becomeparadoxical,sinceany

measure already implies an irreversible process.If we wish tolearn anything at all about a time-reversibleobject,wecannotavoid the irreversibleprocessesinvolved in

measurement, whetherat the level ofan apparatus or ofourown sensorymechanisms.Thus,in classicalphysics,whenwe askhow we can understand\"observation\"in terms offundamental time-reversiblelaws, we get \"nonsense,\"asDaviesputs it.In classicalphysics,this intrusion ofirreversibility was perceivedas a minor problem.The greatsuccessofclassicaldynamicsleft no doubt about its

objective character.The situation is quite different in quantumtheory.Herethe needto includemeasurementin ourfundamental descriptionofnature is explicitlyassertedin the


very structure of the theory. It thereforeseemsthat wehave an irreducibleduality: on the onehand, thetime-reversible Schrodingerequation,and on the other,the

collapse ofthe wave function.This dualistic nature ofquantum mechanicswas

repeatedly emphasizedby the great physicistWolfgang Pauli.Ina letterto Markus Fierz in 1947,he wrote, \"Somethingonly really happenswhen an observationis made,and in

conjunctionwith that. ..entropy necessarilyincreases.Betweenobservations,nothingat all happens.45Still, the

paper on which we write ages and becomesyellow,whetherornot we observeit.

Howcan this paradoxbesolved?Therehave beenmany

proposalsput forth in addition to the extremepositionsmentionedby Davies, includingNielsBohr's\"Copenhagen interpretation.\"*Bohrconcludedthat themeasurement apparatus has to be treated classically.It is as if we,who belongto the macroworld,needan intermediary tocommunicatewith the microworld,just as in somereligions we needa priestorshaman to communicatewith theother world.

But this hardly solvesthe problem,as the Copenhageninterpretationdoesnot lead to any prescriptionofwhatshould characterizethe physicalsystemswe may use as ameasurementdevice.Bohravoids the basicquestion:What

kind ofdynamical processesare responsiblefor the collapseof the wave function? LeonRosenfeld,Bohr'sclosestcoworker,was quite consciousof the limitations of the

Copenhageninterpretation.He consideredit only a first

step, the nextbeingto give a dynamical interpretationof

*Wehighly recommend Rae, Quantum Physics, and A. Shimony, \"Conceptual

Foundations of Quantum Mechanics,\" in Davies' New Physics.

Epicurus'Dilemma 51

the roleofthe apparatus.His convictionledto a numberofpublicationsin commonwith our own researchgroup,whichanticipatedourpresentapproach.46

Otherphysicistshave proposedidentifying the

measuringinstrument with some\"macroscopic\"device.In their

minds, the conceptofsuch a deviceis associatedwith

approximations. Forpracticalreasons,we wouldbeunabletomeasurethe quantum propertiesofthe apparatus.Furthermore, it has often beensuggestedthat we shouldconsiderthe apparatus as an \"open\"quantum systemconnectedtothe entireworld.47Contingentperturbationsandfluctuations stemmingfrom the environment would beresponsible for our ability to perform measurements.But what ismeant by \"environment\"?Who makesthe distinctionbetween an objectand its environment? This distinctionis

only a modified version of the von Neumannproposal,whichstates that through ouractionsand observations,it iswe who producethe collapseofthe wave function.

Theneedto eliminatethe subjectiveelementassociatedwith the observer has beenstressedby John Bellin his

excellentbook,Speakableand Unspeakablein QuantumMechanics.^ It is also an important considerationin the recentwork of Murray Gell-Mannand JamesB.Hartle,who

arguethat the appeal to an observerbecomeseven moreobscurein connectionwith cosmology.49Who measuresthe universe? This is not the placefor a detaileddiscussionoftheir approach;nevertheless,a briefdescriptionoftheir

latest findings would seemto be in order.Gell-Mannand othersintroducea coarse-grained

description ofthe quantum mechanicalhistoriesoftheuniverse that transforms the structureofquantum mechanics,leadingfrom a theory ofprobabilityamplitudesto a theoryofprobabilitiesproper.As an example,let us again con-


siderthe wave function 4* =cxux + c2u2 obtainedby the su-

perimpositionofthe wave functions ux and uT Ifwe thentake the square (for purposesof simplification,we may

suppose*\302\245 is real) we have *\302\2452=

clxulx + ^2^2+ ^c\\c2u\\ut

Let us now presume that we can ignore the doubleproductcalledthe \"interferenceterm.\" All the mystery ofquantum theory then disappears.The probability *\302\2452 is

\"simply\" the sum ofprobabilities.Thereis no longeranyneedto speakofthe transition from potentiality to

actuality,and we can work directlywith probabilities.But how is

this possible?Interferencetermsplay a centralrolein many

applications ofquantum theory.Still, suppressingtheinterference termis preciselywhat Gell-Mannand his

colleagues propose.Why then,in somesituations,do we needexact,fine-grainedquantum descriptions,includinginterference, and in others,coarse-grainedones suppressinginterferences?Again, who actually doesthe coarsegraining?

Is it in any way reasonableto discussthe solution offundamental problemsin terms ofapproximations?How is

this consistentwith Gell-Mann'sown statement, alreadyquoted in SectionII, that quantum mechanicsis theframework into whichall theory must fit?

Still othersin the field hopeto solve the quantummechanical puzzle by reintroducingthe Epicurus clinamenin a modernform.Indeed,GiancarloGhirardi,EmanueleRimini,and Tullio Weber supposethat at sometime,forsomeunknown reason,a spontaneouscollapseofthe wave

function occurs.50Herethe conceptofchanceentersthe

discussion,but without any deeperjustification as a deus exmachina. Why doesthis new clinamenapply to somesituations and not to others?

What is especiallyunsatisfactory about all theseattemptsto elucidatethe conceptualfoundations ofquantum the-

Epicurus'Dilemma 53

ory is that they makeno new predictionsthat can actuallybetested.

Our own conclusioncoincideswith that ofmany otherspecialistssuchas AbnerShimonyin the UnitedStatesand

Bernardd'Espagnatin France.51Accordingto them,radical innovations have to bemade that wouldpreserveall theachievementsof quantum mechanics,but eliminatethedifficulties related to the theory'sdualistic structure.Notethat the measurementproblemis not isolated.As

emphasized by LeonRosenfeld,measurementis associatedwith

irreversibility.But in quantum mechanics,thereis noplacefor irreversibleprocesses,whetherornot they are involvedwith measurement.The difficulty of introducingirreversibility into quantum theory was already establisheddecadesago (in the contextofergodictheory) by von

Neumann, Pauli, and Fierz.52As in classicalmechanics,theytriedto solve the problemby coarsegraining,but their

attempts remainedunsuccessful.Thismay bethe reasonthat

von Neumanneventually adopted a dual formulation: the

Schrodingerequation on oneside,and the collapseofthewave function on the other.53But this is hardly satisfactoryas longas the collapseis not describedin dynamical terms.This is preciselywhat our own theory achieves.Thecentral roleis again playedby instability. However,deterministic chaosguidedby exponentiallydivergingtrajectoriesis

not applicablehere.In quantum mechanics,thereare notrajectories.Therefore,we have to considerinstability in

terms ofPoincareresonances.We can incorporatePoincareresonancesinto a statistical

descriptionand derive diffusive terms that lieoutside the

rangeofquantum mechanicsin terms ofwave functions.Thedescriptionis onceagain basedon the level ofprobability p (also calledthe density matrix in quantum me-


chanics;seeChapter6) and no longeron wave functions.

ThroughPoincareresonances,we achieve the transitionfrom probabilityamplitudesto probabilityproperwithout

drawingon any nondynamicalassumptions.As in classicaldynamics,the basicquestionis,When are

these diffusive terms observable?What are the limits

of traditional quantum theory? The answeris similar tothat for classicaldynamics(seeSectionIII).In short, it is in

persistent interactionsthat the diffusive termsbecomedominant (seeChapter7).As in classicalmechanics,this

prediction has beenverified by numericalsimulations.Onlyby goingbeyond a reductionistdescriptioncan we give arealisticinterpretationofquantum theory.Thereis nocollapse ofthe wave function, as the dynamical laws are nowat the level ofp,the densitymatrix, and not ofwavefunctions *\302\245. Moreover,the observerno longerplays anyspecial role.Themeasurementdevicehas to presenta brokentime symmetry. Forthesesystems,thereis a privilegeddirection oftime,exactlyas thereis a privilegeddirectionoftime in ourperceptionofnature. It is this common arrow oftime that is the necessaryconditionofourcommunicationwith the physicalworld; it is the basisofourcommunication with our fellow human beings.

Thus,instability plays a centralrolein bothclassicaland

quantum mechanics,and as such,obligesus to extendthe

scopeofboth disciplines.In so doing,we have to leave thefield of simple integrablesystems.The possibilityof aunified formulation ofquantum theory is particularly

exciting becausethis problemhas beenso hotly debatedover

the pastdecades,but the needfor an extensionofclassical

theory is even moreunexpected.We recognizethat this

meansa break with a rational tradition that harksbackto

the very foundationsofWestern scienceas conceivedby

Epicurus'Dilemma 55

Galileoand Newton.But it is no merecoincidencethat

the applicationofrecentmathematical methodstounstable systemsleadspreciselyto the extensionsdefinedin this

book.They allow us to includea descriptionofthe

evolutionary characteristicsofour universe basedon aprobabilistic descriptionofnature.In a recentarticle,I.Bernard Cohenspokeof the probabilistic revolution as arevolution in applications.Hewrote,\"Even if the decades18QD\342\200\2241930 do not showa singlerevolution in the domainofprobability,they provideevidenceofa probabilizingrevolution, that is, of a true revolution of fantastic

consequences attendant on the introductionofprobabilityandstatistics into areas that have undergonerevolutionarychangesas a result.\"54 This \"probabilizingrevolution\" isstill goingon.

V

We now cometo the closeofthis chapter.We beganwith

Epicurusand Lucretius,and their invention ofthe clina-men to permitthe appearanceofnovelty. After twenty-five hundred years,we can at last give a precisephysicalmeaningto this concept,which originatesininstabilities identifiedby the moderntheory ofdynamical systems.If the world wereformed by stabledynamical systems,itwould be radically different from the one we observearound us.It would bea static, predictableworld, but wewould not behereto make the predictions.In our world,we discoverfluctuations, bifurcations, and instabilitiesat all

levels.Stablesystemsleadingto certitudescorrespondonlyto idealizations,or approximations.Curiously,this insightwas anticipatedby Poincare.In discussingthe laws ofthermodynamics he wrote,


Theselaws canhave only one significance,which is that there

is a property common to all possibilities;but in the

deterministic hypothesis there is only a singlepossibility, and the

laws no longer have any meaning.In the indeterministic

hypothesis, on the other hand, they would have meaning, evenif they were taken in an absolutesense;they would appearas

a limitation imposedupon freedom.But thesewords remind

me that I am digressingand am on the point of leaving the

domains of mathematics and physics.55

Today we are not afraid ofthe \"indeterministichypothesis.\" It is the natural outcomeof the moderntheory ofinstability and chaos.Oncewe have an arrow oftime,weunderstand immediately the two main characteristicsofnature:its unity and its diversity: unity, becausethe arrowoftime is commonto all parts ofthe universe (your future

is my future; the future ofthe sun is the future ofany otherstar); diversity, as in the roomwhereI write,becausethereis air, a mixtureofgasesthat has moreorlessreachedthermal equilibriumand is in a state ofmoleculardisorder,andthereare the beautiful flowers arrangedby my wife, whichare objectsfar from equilibrium,highly organizedthanks

to temporal,irreversible,nonequilibriumprocesses.Noformulation ofthe laws ofnature that doesnot take intoaccountthis constructiveroleoftime can ever besatisfactory.

Chapter2

ONLY AN ILLUSION?

I

Theresultspresentedin this bookhave matured slowly.

It is now morethan fifty years sinceI publishedmy

first paper on nonequilibriumthermodynamics,in whichI pointedout the constructive roleof irreversibility.1 To

my knowledge,this was also the first paperthat dealt with

self-organizationas associatedwith distancefromequilibrium. After so many years,I often wonderwhy I wasfascinated with the problemoftime,and why it tookso verylongto establishits relationshipwith dynamics.While this

is not the placeto discussthe history ofthermodynamicsand statistical mechanicsover the past half century, I dowant to explainmy own motivations, and indicatesomeofthe main difficulties I encounteredalongthe way.

I have always consideredscienceto be a dialoguewith

nature. As in a real dialogue,the answersare often

unexpected\342\200\224andsometimesastonishing.

As an adolescent,I was enchantedwith archaeology,

57


philosophy,and especiallymusic.My motherused to saythat I couldread musicbeforereadingbooks.When I

entered the university, I spent muchmoretime at the pianothan in lecturehalls. In all the subjects I enjoyed, time

playedan essentialrole,whetherin the gradual emergenceofcivilizations, the ethical problemsassociatedwith human

freedom,or the temporalorganization ofsoundsin music.Thencamethe threat ofwar. It seemedmoreappropriateto undertake a careerin hard sciences,and so I startedto study physicsand chemistry at the FreeUniversity ofBrussels.

I often questionedmy teachersabout the meaningoftime,but their answerswereconflicting.Forthe

philosophers, this was the most difficult problemofall, closelyrelated to ethicsand the very nature ofhuman existence.Thephysicistsfound my question somewhatnaive, as theanswerhad already beengiven by Newton,and later

improved uponby Einstein.As a consequence,I felt bothastonished and frustrated. In science,time was consideredameregeometricalparameter.In 1796,more than onehundred years beforeAlbert Einstein and HermannMinkowski,Joseph-LouisLagrangehad calleddynamicsa\"four-dimensionalgeometry\"2Einstein went on to say,

\"Time[as associatedwith irreversibility] is an illusion.\"With my own background,thesestatementswereimpossible for meto accept.Nevertheless,the tradition ofspatial-izedtime is still very muchalive today, as witnessedby the

work of scientistssuch as StephenW Hawking.3In his

Brief History of Time, Hawking introduces\"imaginarytime\" to eliminateany distinctionbetweenspaceand time,a conceptwe shall examinein greaterdepth in Chapter8.

I am certainly not the first to have felt that the spatial-ization of time is incompatiblewith both the evolving

Only an Illusion? 59

universe, which we observe around us, and our ownhuman experience.This was the starting point for theFrenchphilosopherHenriBergson,for whom\"time isinvention ornothingat all.\"4 In Chapter1,I mentionedoneofBergson'slater articles,\"ThePossibleand the Real,\"

published on the occasionof his NobelPrizein 1930,wherehe expressedhis feelingthat human existenceconsists of \"the continual creationofunpredictablenovelty,\"

concludingthat time provesthat thereis indetermination in

nature.5 Theuniverse around us is only oneofa numberofpossibleworlds.Bergsonwould have beenquite amazedto read HenriPoincare'squotation at the endofChapterl.6Curiously, though, their conclusionspointed in thesame direction.I also quoted Alfred North Whiteheadfrom Processand Reality, for whomthe ultimate goalwas toreconcilepermanenceand change,to conceiveofexistence as a process.Accordingto him, classicalscience,whichoriginatedin the seventeenthcentury, was an

example ofmisplacedconcretenessunableto expresscreativity

as the basic property of nature, \"wherebythe actualworld has its characterof temporal passageto novelty.\"

Whiteheads conceptionofthe actual worldwasobviouslyincompatiblewith any deterministicdescription.7

I couldgoon by quotingMartin Heideggerand others,includingArthur Stanley Eddington,who wrote,\"In any

attempt to bridgethe domainsofexperiencebelongingtothe spiritual and physicalsidesofournature, time occupiesthe keyposition.\"8But insteadofbuildingthis bridge,timehas remaineda controversial issuefrom the pre-Socraticstothe present day. As mentionedfor classicalscience,the

problemoftime had beensolvedby Newtonand Einstein,but for mostphilosophers,this solutionwas incomplete.In

their opinion,we had to turn to metaphysics.


My personalconvictionwas quite different.

Abandoningscienceappearedto be tooheavy a priceto pay.After

all, sciencehad ledto a uniqueand fruitful dialoguebetween mankindand nature.Perhapsclassicalsciencecouldindeedlimit timeto a geometricalparameter becauseitwas dealingonly with simpleproblems.Therewas noneedto extendthe conceptof time when we dealt with africtionless pendulum, for instance.But once scienceencounteredcomplexsystems,it would have to modify its

approachto time.An examplethat often cameto mind wasassociatedwith architecture.Thereis not muchdifferencebetweenan Iranian brickfrom the fifth century beforeChrist and a neogothicbrickfrom the nineteenthcentury, but the results\342\200\224the palacesof Persepolisand the

neogothicchurches\342\200\224are in striking contrast.Timewouldthen be an \"emerging\"property.But what couldbe theroots oftime? I becameconvincedthat macroscopicirreversibility was the manifestation of the randomness ofprobabilisticprocesseson a microscopicscale.What thenwas the originofthis randomness?

With thesepreoccupations,it was only natural that I

turn to thermodynamics, especiallybecausein Brusselsthere was already an establishedschoolin the subjectfounded by TheophileDeDonder(1870-1957).

IIIn Chapter1,we mentionedthe classicalformulation ofthe secondlaw ofthermodynamicsattributed to Clausius.This law is basedon an inequality:Theentropy, S, ofan

isolatedsystemincreasesmonotonicallyuntil it reachesitsmaximum value at thermodynamic equilibrium.We

thereforehave dS > 0 for the changein entropy over the


courseoftime.Howcan we extendthis statement tosystems that are not isolated,but whichexchangeenergyandmatter with the outsideworld?We must then distinguishtwo terms in the entropy change,dS:the first, d S, is thetransfer ofentropyacrossthe boundariesofthe system;the

second,rf.S, is the entropyproducedwithin the system.As

a result,we have dS= d S + d.S.We can now expressthesecondlaw by stating that whatever the boundaryconditions, the entropyproductiond.Sis positive,that is, d.S>0.Irreversible processesare creating entropy. DeDonderwenteven farther: Heexpressedthe productionofentropyperunit time P = -j^ in terms of the rates ofvariousirreversible processes(chemicalreactionrates, diffusion, etc.)and thermodynamic forces.In fact, he consideredonlychemicalreactions,but further generalizationwas easy.9

DeDonderhimselfdid not go very far alongthis road.He was concernedmainly with equilibriumand the

neighborhoodofequilibrium.Limitedas it was, his work

representedan important step in the formulation ofnon-equilibriumthermodynamics,even if it seemedto leadnowherefor a considerablelengthoftime.I still rememberthe hostility with whichDeDonder'swork was met.Forthe vast majority ofscientists,thermodynamicshad to belimited strictly to equilibrium.

That was the opinionofJ.Willard Gibbs, as well as ofGilbertN.Lewis,the mostrenownedthermodynamicistofhis day. Forthem, irreversibility associatedwith

unidirectional time was anathema. Lewiswent so far as to write,\"We shall see that nearly everywhere the physicisthas

purgedfrom his sciencethe useofone-way time ...aliento the idealsofphysics.\"10

I myself experiencedthis type of hostility in 1946,when I organizedthe first Conferenceon Statistical


Mechanicsand Thermodynamics under the auspicesofthe InternationalUnion for Pure and Applied Physics(IUPAP).Thesemeetingshave sincebeenheldon a

regular basisand continueto attract largecrowds,but at that

time we were a small group of approximatelythirty to

forty people.After I had presentedmy own lectureonirreversible thermodynamics,the greatestexpertin the fieldofthermodynamicsmadethe following comment:\"I amastonishedthat this young man is so interestedin nonequi-librium physics.Irreversibleprocessesare transient.Whynot wait and study equilibriumas everyoneelsedoes?\"Iwas so amazedat this responsethat I did not have thepresence ofmind to answer:\"But we are all transient.Is it notnatural to be interestedin our commonhuman

condition?\"

Throughoutmy entirelife I have encounteredhostilityto the conceptofunidirectionaltime.It is still the

prevailingview that thermodynamicsas a disciplineshould

remain limited to equilibrium.In Chapter1,I mentionedthe attempts to banalize the secondlaw that are so mucha

part ofthe credoofa numberoffamous physicists.Icontinue to beastonishedby this attitude. Everywherearoundus we seethe emergenceofstructuresthat bearwitnesstothe \"creativity ofnature,\" to useWhitehead'sterm.I have

always felt that this creativity had to beconnectedin someway to the distancefrom equilibrium,and was thus the

result ofirreversibleprocesses.Compare,for example,a crystal and a town.A crystal is

an equilibriumstructure that can bemaintainedin a

vacuum, but if we isolatedthe town, it would diebecauseitsstructure dependson its function.Functionand structureare inseparablein that the latter expressesthe interactionsofthe town with its environment.


In Erwin Schrodingers beautiful bookWhat Is Life?hediscussesthe metabolismofa living body in terms ofentropy productionand entropy flow. Ifan organismis in a

steady state, its entropy remains constant over time,andthereforedS= 0.As a result,the entropyproductiond.Siscompensatedby the entropy flow d S + d S = 0,or d S =d.S< 0.Life, concludesSchrodinger,feedson a \"negativeentropy flow\"11The moreimportant point, however, isthat life is associatedwith entropy productionand

therefore with irreversibleprocesses.But how can structure, as in living systemsor towns,

emergein nonequilibriumconditions?Hereagain, as in

dynamics,the problemofstability playsan essentialrole.At

thermodynamic equilibrium,entropy has a maximumvalue whenthe systemis isolated.Fora systemmaintainedat temperatureT, we have a similar situation.We thenintroduce \"freeenergy,\" F=E \342\200\224 TS,a linearcombinationofenergyE and entropyS.As shown in all texts onthermodynamics, free energy,F, is at its minimum at equilibrium(seeFigure2.1).Consequently,perturbations orfluctuations have no effect becausethey are followedby a return

to equilibrium.Thesituation is not unlike that ofthe

stable pendulumconsideredin Chapter1,SectionIII.What happens in a steadystate correspondingto non-

equilibrium?We saw such an examplein a discussionofthermal diffusion in Chapter1,SectionII.Isa

nonequilibrium steady state truly stable? In near-equilibriumsituations (known as \"linear\" nonequilibriumthermodynamics), the answeris yes.As shown in 1945,the steadystate correspondsto a minimum ofentropyproductionperunit time P= -^.12At equilibriumP= 0,entropyproduction vanishes,while in the linear regimearoundequilibrium, P is minimum (seeFigure2.2).13


Figure2.1Minimum ofFFreeenergy is minimum at equilibrium (X

=X\342\200\236

Again fluctuations dieout.But therealready appearsaremarkablenew characteristic:A nonequilibriumsystemmay evolve spontaneouslyto a state ofincreasedcomplexity.The orderingwe observe is the outcomeof irreversible

processes,and couldnot beachievedat equilibrium.Thisis clearin the exampleofthermal difFusion mentionedin

Chapter1,wherethe temperaturegradient leadsto apartial separationofthe compounds.Many other caseshavesincebeenstudied in which complexity has consistentlybeenassociatedwith irreversibility. Theseresultsbecamethe guidelinesfor our future research.

But can we extrapolatethe resultsoffar-from-equilib-rium situations from thoseat near-equilibrium?Mycolleague Paul GlansdorfF and I investigated this problemfor

many years,14and arrived at a surprisingconclusion:Contrary to what happensat equilibrium,ornearequilibrium,systemsfar from equilibriumdo not conformto anyminimum principlethat is valid for functions offree energyorentropyproduction.As a consequence,thereis noguarantee that fluctuations are damped.We can only achieve


Figure2.2Minimum ofPEntropy production P =

d\\S/dt is minimum in a steady state (X=

A.J().

a formulation of sufficient conditions for stability, whichwe call the \"generalevolution criterion.\"This requiresspecifyingthe mechanismofirreversibleprocesses.Near-equilibriumlaws ofnature are universal, but whenthey arefar from equilibrium,they becomemechanismdependent.We thereforebeginto perceivethe originofthe variety in

nature we observearound us.Matter acquiresnewproperties whenfar from equilibriumin that fluctuations andinstabilities are now the norm. Matter becomesmore\"active.\" Although thereis at presentan enormousliterature surroundingthis subject,15for the momentwe shall

consideronly a simpleexample.Supposethat we have achemicalreaction{A} =^{X} -^ {F}in which{A}is a

set ofinitial products,{X}a set ofintermediateones,and

{F}a set offinal ones.At equilibrium,we have a detailedbalancewherethereare as many transitions from {^4}to

{X}as from {X}to {^4},with the sameapplyingto {X}and {F}.The ratio of initial to final products {^4}/{F}takeson a well-definedvalue correspondingto maximumentropy if the systemis isolated.Now consideran open


system,such as a chemicalreactor.By controllingthe flowofmatter, we may fix the values ofboth the initial andfinal products {A}and {F}.We progressivelyincreasetheratio {A}/{F},starting from its equilibriumvalue. What

will happen to the intermediateproducts {X}when wemove away from equilibrium?

Chemicalreactionsare generallydescribedby nonlinearequations.Thereare many solutionsfor the intermediateconcentrations{X}for given values of{A}and {F},but

only onecorrespondsto thermodynamicequilibriumandmaximumentropy.This solution,whichwe call the

\"thermodynamic branch,\" may be extendedto the domain ofnonequilibrium.The unexpectedresult is that this branch

generallybecomesunstable at somecriticaldistancefrom

equilibrium(seeFigure2.3).Thepoint wherethis occursis known as the bifurcation point.

Beyond the bifurcation point,a set ofnew phenomenaarises;we may have oscillatingchemicalreactions,non-equilibriumspatial structures,orchemicalwaves.We have

given the name dissipative structures to thesespatiotemporalorganizations.Thermodynamicsleads us to theformulation of two conditionsfor the occurrenceofdissipativestructuresin chemistry:(1)far-from-equilibriumsituationsdefinedby a criticaldistance;and (2) catalytic steps,suchas

the productionof the intermediatecompoundY from

compoundX together with the productionofX from Y.

It is interestingto note that theseconditionsare satisfiedin all living systems:Nucleotidescodefor proteins,whichin turn codefor nucleotides.

We wereextremelyfortunate in that soonafter we had

predictedthesevarious possibilities,the experimentalresults of the Belousov-Zhabotinskireaction\342\200\224a spectacularexampleofchemicaloscillations\342\200\224became widelyknown.16I rememberouramazement whenwe sawthe reactingso-


Figure2.3ThermodynamicBranchThe two steady-state solutions th and d are functions of the ratio

A/F. At the bifurcation point, the thennodynamic branch th

becomesunstable, and another branch d becomesstable.

X .i**'

Equilibrium

YA/F

Point ofbifurcation

lution becomeblue,and then red,and then blue again.Today, many other oscillatoryreactionsare known,17but

the Belousov-Zhabotinskireactionremains historicbecause it proved that matter far from equilibriumacquiresnew properties.Billionsofmoleculesbecomesimultaneously blue, and then red. This entails the appearanceoflong-rangecorrelationsin far-from-equilibriumconditions that are absent in a state ofequilibrium.Again, wecan say that matter at equilibriumis \"blind,\" but far from

equilibriumit beginsto \"see.\"We have observed that at

nearequilibrium,dissipationassociatedwith entropyproduction is at a minimum.Far from equilibrium,it is justthe opposite.New processesset in and increasetheproduction ofentropy.

Therehas beensteadyprogressin far-from-equilibriumchemistry.In recentyears, nonequilibriumspatial struc-


tures have beenobserved.18Thesewerefirst predictedbyAlan MathisonTuring in the contextofmorphogenesis.19

When we push the systemfarther into nonequilibrium,new bifurcations typical of chaoticbehavior may arise.Neighboringtrajectoriesdivergeexponentiallyas in

deterministic chaosakin to the dynamical systemsweconsidered in Chapter1,SectionIII.

In short, distancefrom equilibriumbecomesan essential

parameterin describingnature much like temperature in

equilibriumthermodynamics.When we lower the

temperature, we observe a successionof phase transitions

through various states ofmatter. But in nonequilibriumphysics,the variety ofbehaviorsis muchgreater.We haveconsideredchemistryfor the purposesof this discussion,but similar processesassociatedwith nonequilibriumdissi-

pative structureshave beenstudied in many other fields,includinghydrodynamics,optics,and liquid crystals.

Let us now lookmorecloselyat the critical effect offluctuations. As we have seen,near-equilibriumfluctuations are harmless,but far from equilibrium,they play acentralrole.Not only dowe needirreversibility, but wealso have to abandon the deterministicdescriptionassociated with dynamics.Thesystem\"chooses\"oneofthepossible branchesavailable when far from equilibrium.But

nothingin the macroscopicequationsjustifies thepreference for any onesolution.This introducesan irreducibleprobabilisticelement.Oneof the simplestbifurcations isthe so-called\"pitchfork bifurcation\" representedin Figure2.4,whereX = 0correspondsto equilibrium.

The thermodynamic branch is stable from X =0 toX = X.BeyondX, it becomesunstable,and a symmetricalpair ofnew stable solutionsemerges.It is the fluctuationsthat decidewhichbranch will be selected.If we wereto

suppressfluctuations, the systemwould maintain itself in


Figure2.4Pitchfork BifurcationConcentrationX is a function of the parameter X, which measuresthe distance from equilibrium. At the bifurcation point, the

thermodynamic branch becomesunstable, and the two new solutions

bx and b2 emerge.

X

Thermodynamic branch (\\

\\

Stable ,

~\342\200\224

\342\226\240

b2

Multiple solutions X

an unstablestate.Attempts have beenmade to decreasethefluctuations so that we can subject the unstableregionto

experiment;nevertheless,sooneror later, fluctuations ofinternal or externalorigintake over and bringthe systemto oneofthe branchesb1 or b2.

Bifurcations are a sourceofsymmetry breaking.In fact,the solutionsof the equationbeyond X generallyhave a

lowersymmetry than the thermodynamicbranch.20Bifurcations are the manifestation ofan intrinsicdifferentiationbetweenparts of the systemitself and the systemand its

environment.Oncea dissipative structure is formed,the

homogeneityoftime (as in oscillatorychemicalreactions)orspace(as in nonequilibriumTuring structures),orboth,is broken.

In general,we have a successionofbifurcations asrepresented schematicallyin Figure2.5.The temporaldescription of such systems involves both deterministicprocesses(betweenbifurcations) and probabilisticprocesses(in the choiceofthe branches).Thereis also a historical


dimensioninvolved. If we observe that the system is in

state d2, that means that it has gonethrough the statesbx

and q (seeFigure2.5).Oncewe have dissipative structures,we can speak of

self-organization.Even if we know the initial values and

boundaryconstraints,thereare still many states availabletothe systemamongwhichit \"chooses\"as a result offluctuations. Suchconclusionsare ofinterestbeyond the realmsofphysicsand chemistry.Indeed,bifurcations can beconsidered the sourceof diversification and innovation.21Theseconceptsare now appliedto a wide group ofproblems in biology,sociology,and economicsat

interdisciplinarycentersthroughoutthe world.In Western Europe

alone,therehave beenmorethan fifty centersfornonlinear processesfounded over the past ten years.

Freudwrote that the history ofscienceis the history ofalienation.Copernicusshowed that the earth is not at the

centerof the planetary system,Darwin that we are onespeciesofanimal amongmany others,and Freud that our

Figure2.5SuccessiveBifurcations with IncreasingDistancefrom

Equilibrium

X


rational activity is only part of the unconscious.We can

now invert this perspective:We seethat human creativityand innovation can beunderstoodas the amplification oflaws ofnature already present in physicsorchemistry.

IllThe resultspresentedthus far show that the attempts totrivialize thermodynamics mentionedin Chapter 1 are

necessarilydoomedto failure. The arrow oftime plays an

essentialrolein the formation of structures in both the

physicalsciencesand biology.But we are only at the

beginning ofourquest.Thereis still a gap betweenthe most

complexstructureswe can producein nonequilibriumsituations in chemistryand the complexitywe find in

biology.This is not only a problemfor pure science.In a recent

report to the European Communities,Christof Karl

Biebracher,GregoireNicolis,and PeterSchusterwrote,

The maintenanceof organization in nature is not\342\200\224and

cannot be\342\200\224achieved by central management; order can only bemaintained by self-organization. Self-organizing systems

allow adaptation to the prevailing environment, i.e.,they

react to changesin the environment with a thermodynamic

responsewhich makes the systems extraordinarily flexibleand

robust against perturbations from outside conditions. We

want to point out the superiority of self-organizingsystems

over conventionalhuman technologywhich carefully avoids

complexity and hierarchically managesnearly all technical

processes.For instance,in synthetic chemistry, different

reaction stepsare usually carefully separatedfrom each other and

contributions from the diffusion of the reactants are avoided

by stirring reactors.An entirely new technologywill have to

be developedto tap the high guidanceand regulation poten-


tial of self-organizingsystems for technicalprocesses.The

superiority of self-organizingsystems is illustrated by biologicalsystems where complexproductscan be formed with

unsurpassed accuracy,efficiencyand speed!\"22

The results of nonequilibriumthermodynamics arecloseto the views expressedby Bergsonand Whitehead.Nature is indeedrelated to the creationofunpredictablenovelty, wherethe possibleis richerthan the real.Ouruniverse has followeda path involving a successionofbifurcations. While other universes may have followed otherpaths,we are fortunate that ourshas ledto life, culture,andthe arts.

The dreamofmy youth was to contributeto theunification ofscienceand philosophyby resolvingthe enigmaof time.*Nonequilibriumphysicsshows that this is

entirely possible.The resultsdescribedin this chaptergavemethe impetusto explorethe conceptoftime on the

microscopic level.I have emphasized the roleoffluctuations\342\200\224but what is their origin?How can we reconciletheir behavior with the deterministicdescriptionbased

upon the traditional formulation of the laws of nature?Were we to doso,we would losethe distinctionbetweennear-and far-from-equilibriumprocesses.Moreover,wewould becalling into questionsuch unique and marvelousconstructionsofthe human mind as classicaland quantummechanics.

I must confessthat thesethoughts ledto many sleeplessnights.Without the supportofmy colleaguesand students,I would most certainly have given up.

*I expressed this dream in three short essays written for a student journal as

early as 1937!

Chapter3

FROM PROBABILITYTOIRREVERSIBILITY

I

Aswe saw in Chapter2,irreversibleprocessesdescribefundamental features ofnature leadingto nonequi-

librium dissipativestructures.Suchprocesseswould not bepossible in a world ruled by the time-reversiblelaws ofclassicaland quantum mechanics.Dissipativestructures

require an arrow oftime.Furthermore,there is no hopeofexplainingthe appearanceofsuch structuresthroughapproximations that would be introducedby theselaws.

I have always beenconvincedthat an understandingofthe dynamical origin of dissipative structures, and moregenerallyof complexity, is one of the most fascinatingconceptualproblemsofcontemporaryscience.As alreadystated in Chapter1,for unstable systemswe have toformulate the laws of dynamicsat the statistical level.This

73


changesourdescriptionofnature in a radicalway In sucha formulation, the basic objectsofphysicsare no longertrajectoriesor wave functions; they are probabilities.We

have thus cometo the end of the \"probabilisticrevolution\" that could already be found in areas other than

physicsby the eighteenthcentury However,when facedwith the implicationsofthis radicalconclusion,I hesitatedfor sometime,reachingfor lessextremesolutions.In From

Being to Becoming, I wrote,\"In quantum mechanics,thereare observationswhosenumericalvalue cannotbedetermined simultaneously, i.e.,coordinatesand momentum.(This is the essenceofHeisenberg'suncertainty relationsand Bohr'scomplementarityprinciple.)Herewe also havea complementarity\342\200\224one betweendynamical and thermo-dynamical descriptions.\"1This would have beena muchless extremeapproach to the conceptualproblemassociated with irreversibility.

In retrospect,I regretthis statement in my earlierbook.If there is morethan a single description,who wouldchoosethe right one?The existenceofthe arrow oftimeis not a matter ofconvenience.It is a fact imposedbyobservation. However,it is only in recentyears that the resultswe obtainedby studyingthe dynamicsofunstable systemsforcedus to reformulate dynamicsat the statistical level,and to concludethat this formulation leadsto an extensionofclassicaland quantum mechanics.In this chapter, I

describe someofthe stepsinvolved.Forapproximatelyonehundred years,we have known

that even simpleprobabilisticprocessesare time oriented.In Chapter1,we mentionedthe \"randomwalk.\" Another

exampleis the \"urn model\"proposedby Paul and TatianaEhrenfest(seeFigure3.1).2

ConsiderN objects(such as balls) distributedbetween

From Probability to Irreversibility 75

Figure3.1TheEhrenfestUrnModelN balls are distributed between two urns, A and B.At time n,there are k balls in A and N-k balls in B.At regular time

intervals, a ball is removedat random from one urn and placedinto

the other.

pottery

N-k

k-1 \\\"^ ^*V N-k+1

or /^ ^\\ or

fe+1 / \\iV-fe-l.

two urns A and B.At regular time intervals (for example,every second)a ball is chosenat random and moved fromoneurn to the other.Supposethat at time n, thereare k

balls in A, and thereforeN \342\200\224 k in B.At time n + 1therecan be eitherk \342\200\224 1or k + 1balls in A. Theseare well-defined transition probabilities.But let us go on with the

game.We expectthat as a result ofthe exchangeofballs,we shall reach a situation where there will beapproximately ~2

balls in eachurn.However,fluctuations willcontinue. We might even end up in the situation at time n

wherethereare again k balls in urn A. It is at the level ofprobability distribution that we seean irreversibleapproachto


equilibrium.Whatever the starting point, it can beshownthat the probabilitypn(k) offinding k balls in oneurn aftern moves as n \342\200\224>

\302\260o tends to the binomial distributionN\\ ... M

fe!(N-fe)! .This expressionhas a maximumvalue ofk = ^,but also takesinto accountfluctuations in distribution.Inthe Boltzmannmodel,the maximumentropycorrespondspreciselyto the binomialdistribution.

The Ehrenfest modelis an exampleof a \"Markov

process\"(or\"Markovchain\,")namedafter the greatRussian mathematician, Andrei Markov, who was the first todescribesuch processes.Oncewe have a probabilisticdescription, it is often possibleto derive irreversibility. Buthow do we relate theseprobabilisticprocessesto dynamics?That is the fundamental problem.

We have seenthat a basicstep in this directionwastaken

by the fathers ofstatisticalphysics,or the physicsofpopulations. Maxwell,Boltzmann,Gibbs, and Einstein all

emphasized the roleofensemblesdescribedby a probabilitydistributionp.An important questionthen is, What is theform of this distribution function onceequilibriumis

reached?Let qv . . . ,qs be the coordinatesand pl, \342\226\240 \342\226\240 \342\226\240,psthe momentaof the particles forming this system. In

Chapter1,the phasespacewas definedby the coordinatesand momenta.We also introducedthe probabilitydistributions p(q,p, t) (seeChapter1,SectionIII).We shall nowuse the singleletterq for all coordinatesand p for all

momenta. Equilibrium is reachedwhenp becomestime

independent. In every textbook,it is shown that this occurswhenp dependsonly on the total energy.As mentionedin

Chapter1,SectionIII,the total energy is the sum ofthekineticenergy (dueto the motionofthe particles)and the

potential energy (due to interactions).When expressedinterms ofq and p, this energy,whichis calledthe Hamilton-

ian H(p,q),remainsconstant over time.This is the princi-


pie ofconservationofenergy,the first principleofthermodynamics. It is therefore natural that at equilibrium,p is

a function ofthe Hamiltonian H.An important exceptioncaseis that of ensemblesin

whichall systemshave the same energy E. Thedistribution function then vanishes throughout the phase space,save on the surface H (p, q) = E, where the distributionfunction is constant.This is calledthe \"microcanonicalensemble.\" Gibbsshowedthat suchensemblesdo indeedsatisfy

the laws of equilibriumthermodynamics.He alsoconsideredotherensemblessuch as the \"canonicalensemble,\"

in whichall systemsinteract with a reservoirat

temperature T. This leads to a distribution function that

dependsexponentiallyon the Hamiltonian,p now beingproportionalto exp (\342\200\224 ^f), whereT is the temperatureofthe reservoirand k the Boltzmannconstant,whichmakesthe exponentdimensionless.

Oncethe equilibriumdistributionis given, we cancalculate all thermodynamic equilibriumpropertiessuch as

pressure,specificheat, etc.We can even gobeyondmacroscopic thermodynamicsbecausewe are able to includefluctuations. It is generallyacceptedthat in the vast field ofequilibriumstatistical thermodynamics,thereare noconceptual difficulties left, only computational ones,whichcan be solvedlargely through numericalsimulations.Theapplicationofensembletheory to equilibriumsituationshas undoubtedlybeenquite successful.Notethat the

dynamical interpretationofequilibriumthermodynamicsbyGibbs is in terms ofensembles,and not in terms oftrajectories. It is this approachthat we have to extendin ordertoincludeirreversibility.

This is quite natural, as thereis no time orderingat thelevel oftrajectories(orwave functions) becausefuture and

past play the same rolein accordancewith classicaland


quantum physics.However,what happens at the level ofstatistical description,in terms ofdistribution functions?Let us lookat a glassofwater.In this glass,thereis a hugenumberofmolecules,a quantity on the orderof 1023.Fromthe dynamical point ofview, this is a nonintegrablePoincaresystem,as defined in Chapter1,sincethereareinteractionsbetweenthe moleculesthat we cannoteliminate. We may visualize theseinteractionsas leadingtocollisions betweenthe molecules(the term\"collision\"will bedefined moreprecisely in Chapter5), and describethewater containingthemin terms ofthe statistical ensemblep.Is the water aging? Certainlynot, if we considertheindividual water molecules,whichare stableover geologicaltime.Still,thereis a natural time orderin this systemfromthe point ofview ofthe statistical description.Aging is a

property ofpopulations,exactlyas it is in the Darwinian

theory ofbiologicalevolution.It is the statisticaldistribution that approachesthe equilibriumdistribution,such asthe canonicaldistribution defined above.To describethis

approachto equilibrium,we needthe idea ofcorrelation.

Considera probabilitydistributionp(xpx2),

dependingon two variables xp x2.If xt and x2 are independent,

we have the factorization p(xp x2)=p1(x1)p2(x2).Theprobabilityp(xpx2) is then the productoftwoprobabilities. In contrast,if p(xpx2) cannot befactorized,x; and x2are correlated.Now let us return to the moleculesin the

glassofwater.Thecollisionsbetweenthesemoleculeshavetwo effects: They make the velocity distribution moresymmetrical, and they producecorrelations(seeFigure3.2).But two correlatedparticleswill eventually collidewith a third one.Binary correlationsare then transformedinto ternary ones,and so on (seeFigure3.3).

We now have a flow ofcorrelationsthat are orderedin


Figure3.2Collisionsand CorrelationsThe collisionof two particles createsa correlationbetween them

(representedby a wavy line).

oo

o

o oBeforecollision After collision

time.A valuable and provocative analogy to this flowwould be human communication.When two peoplemeet,they converse,and consequendymodify their

thinkingto someextent.Thesemodificationsare brought to

subsequentmeetings,and modifiedfurther. The word forthis phenomenonis dissemination. Thereis a flow ofcommunication in society,just as thereis a flow ofcorrelationsin matter. Of course,we may also conceiveof inverse

processesthat make the velocity distributionlesssymmetrical by destroyingcorrelations(seeFigure3.4).

We thereforeneedan elementthat will validate the

processesthat make the velocity distribution moresymmetrical over the courseof time.As we shall see,this is

preciselythe roleofPoincareresonances.We now beginto

geta glimpseofa statistical descriptionthat includesirreversibility. This descriptionwill be a dynamics ojcorrelations

leadingto the equilibriumdistribution.Theexistenceofa flow ofcorrelationsorderedin time,

asrepresentedin Figure3.3,has beenverified by computersimulations.3We can also reproduceprocessessuchas those


Figure3.3FlowofCorrelationsSuccessivecollisionsleadto binary, ternary, ...correlations.

o o^oo

o -o

\342\200\224\342\226\272 o o -o

\342\200\224\342\226\272

oo

representedin Figure3.4through time inversion,wherewe invert the velocity ofthe particles.But we can achievethis inverted flow ofcorrelationsonly for briefperiodsoftime and for a limitednumberofparticles,after whichwe

again have a directedflow of correlationsinvolving an

ever-increasingnumberofparticlesleadingthe systemto

equilibrium.

Figure3.4DestructionofCorrelationsIn (a) the particles (representedby black points) interact with the

obstacle(representedby the circle).Initially all particles have the

same velocity. The collisionvaries the velocitiesand createscorrelations betweenthe particles and the obstacle.In (b) we representthe oppositeprocess.We considerthe effect of a velocityinversion; as a result of the inverted collision,correlationswith the

obstacle aredestroyed,and the initial velocity is recovered.

o \342\200\224\342\226\272 Ct^*-^ \"*^~>D \342\200\224\342\226\272 o

(a) (b)


Theseresults,whichgive meaningto irreversibility at

the statistical level, wereobtained nearly thirty years ago.4At that time,however, certainbasic questions still

remained unanswered:Howcan irreversibility appearat thestatistical level ofdescription,and not whenwe describedynamicsin terms oftrajectories?Is this due to ourapproximations? Moreover,is the successionofcorrelationsthat we observe, for examplein computerexperiments,perhaps the result of the limitations of computertime?

Obviously,a shorterprogramis requiredto prepareuncor-related particles that producecorrelationsthroughcollisions than to prepareensemblesthat couldlead to inverse

processesin whichcorrelationsare destroyed.But why start at all with probabilitydistributions?Such

distributionsdescribethe behaviorofbundlesoftrajectories,

or ensembles.Dowe use ensemblesbecauseofour\"ignorance,\"or is there,as argued in Chapter1,a deeperreasoninvolved? Forunstablesystems,ensemblesindeeddisplay new propertiesas comparedwith individualtrajectories. This is what we shall now demonstratewith several

simpleexamples.

IIIn this section,we shall be concernedwith deterministicchaos,as well as an especiallysimple type ofchaos,both

correspondingto chaotic maps. Contrary to what occursinordinary dynamics,time in maps acts only at discreteintervals, as is the casein the Ehrenfesturn modelwestudied in SectionI.Mapsthereforerepresenta simplifiedformofdynamicsthat makesit easy for us to comparetheindividual level ofdescription(the trajectories)with thestatistical description.We shall considertwo maps; the first


charts simpleperiodicbehavior, the seconddeterministicchaos.

In the first instance,we shall considerthe \"equationsofmotion\" x , , =x + ?,modulo1,whichmeansthat we are

n + 1 n *\342\226\240''

dealing only with numbers between0 and 1.After two1 3

shifts, we are back to the initial point (i.e.,xo = 4, xt =4, x23 2 5 1=4+4=4 =4).This situation is representedin Figure3.5.Insteadofconsideringindividual points locatedby

trajectories, it is worth examiningensemblesdescribedby the

probabilitydistributionp(x).A trajectorycorrespondsto a

specificsetofensembleswherethe coordinatex takeson a

well-definedvalue xn, and the distribution function p is

then reducedto a singlepoint.As mentionedin Chapter1,SectionIII,this can bewritten as Pn(x) =8(x\342\200\224 x ). (Deltais a symbolfor a function that vanishes for all values ofx

Figure3.5PeriodicMapThere is a simple geometrical construction that moves from the

initial point P0to the next point Px accordingto the map xn + j \342\200\224>

x\342\200\236

+ l /2.We go from P0to P',then to P\" on the bisector,and fromthere to Px.Obviously, if we start with Plt wecomeback to P0.

X\342\200\236


exceptx= x .)By usingdistributionfunction p, the

mappingcan be expressedas a relation betweenp +l(x) and

Pn(x).We may then write Pn + l(x)= U p (x). Formally,p + j

is obtained through the operatorU, known as thePerron-Frobeniusoperator,acting onPn{x).5At this point,although its explicitform is not important to us, it is

interesting to note that no new element(in addition to the

equationofmotion)entersinto the constructionof U.

Obviously,the ensembledescriptionmust allow the

trajectory descriptionas a specialcase;we thereforehave 8(x\342\200\224

xn + i) = U8(x\342\200\224 xj.This is simplya way ofrewriting the

equationofmotion,as xn becomesxn + after one shift.

Themain questionis,however, 75 this the only solution, or arethere new solutionsfor the evolution ofensembles,as describedby

the Perron-Frobeniusoperator, which cannot beexpressedin terms

of trajectories?In ourexampleofa periodicmap, the answeris no.Thereis not any difference betweenthe behaviorofindividual trajectoriesand ensemblesfor stablesystems.It is

this equivalencebetween the individual point of view

(correspondingto trajectoriesor wave functions) and thestatistical point ofview (correspondingto ensembles)that

is brokenfor unstable dynamical systems.Thesimplestexampleofa chaoticmap is the Bernoulli

map. Herewe doublethe value ofa numberbetween0and1every second.Theequationofmotionis now xn+l

= 2x(modulo1).This map is representedin Figure3.6.Theequationofmotionis again deterministicin that onceweknow x , the numberx ,,is determined.Herewe have anrv \302\253+l

exampleofdeterministicchaos,so calledbecauseif wefollow a trajectory through numericalsimulations,we seethat it becomeserratic.As the coordinatex is multipliedbytwo at eachstep,the distancebetweentwo trajectorieswill


Figure3.6BernoulliMapIn this exampleof deterministic chaos,we start from point P0and

go to point Pu as the value of x doubles(modulo 1).

be (2\") = exp(n log2), again modulo1.In terms ofcontinuous time t, this can bewritten asexp(tX), with A = log2,whereX is calledthe Lyapunov exponent.This showsthat trajectoriesdivergeexponentially,and it is this

divergencethat is the signature ofdeterministicchaos.Ifwe

wait longenough,any arbitrarily selectedpoint between0and 1will eventually be approachedby the trajectory (seeFigure3.7).Herewe have a dynamical processleadingto

randomness.In the past, this apparent flow in thedeterministic universe was repeatedlyinvestigated by greatmathematicians such as LeopoldKronecker(1884) and

HermannWeyl (1916).Accordingto Jan von Plato, similar

resultshad beenobtainedas early as medieval times,so this

is certainly not a new problem.6What is new, however, is

the statistical formulation of the Bernoullimap, whichlinks randomnessto operatortheory.


Figure3.7NumericalSimulationsofTrajectoriesfor theBernoulliMapThe initial conditions are slightly different for each simulation.

This differenceis amplified as time goes on. (These numericalsimulations are the work of DeanDriebe).

1

0.8

0.6

0.4

0.2

10 20 30 40


Figure3.8Simulationofpn(x) for the BernoulliMapNumerical simulation of the evolution of the probabilitydistribution. In contrast with the trajectory description,the probabilities

rapidly reachthe asymptotic uniform distribution. (Thesenumerical simulations are the work ofDeanDriebe.)

0 0.2 0.4 0.6 0.8 1 X 0 0.2 0.4 0.6 O.i 1 X

PW

2 -

1.5-1

0.5-

t = 2

-J l_

0 0.2 0.4 0.6 O.i 1 X

PW

21.5-

1 -

0.5-_J l l l_

0 0.2 0.4 0.6 O.i 1 X

PW

21.5

1

0.5

( = 4

_i l l i_

0 0.2 0.4 0.6 0.8 1 X


We now turn to the statistical descriptionin terms ofthe Perron-Frobeniusoperator.In Figure3.8,we seetheeffect ofthe operatorU on the distributionfunction.Thedifferencefrom the trajectory descriptionis strikingbecause the distributionfunction p (x) leadsrapidly to a

constant. We may thereforeconcludethat theremust bea basicdifference betweenthe descriptionin terms oftrajectorieson the onehand and in terms ofensembleson the other.In short, instability at the level oftrajectoriesleadsto

stability at the level ofstatistical descriptions.How is this possible?The Perron-Frobeniusoperator

still admits a trajectory description8(x\342\200\224 x +1)=U8(x\342\200\224

x ), but the unexpectedfeature is that it also allows newsolutions that are applicableonly to statistical ensembles,and not to individual trajectories.The equivalencebetween the individual point ofview and the statistical

description is broken.This remarkablefact leads to a new chapter in

mathematics and theoreticalphysics.7Although the problemofchaoscannotbe solvedat the level ofindividual

trajectories,it can besolvedat the level ofensembles.We can now

speakofthe laws ofchaos.sAs we shall seein Chapter4, we

may even predict the speedat whichthe distributionpapproaches equilibrium(which for the Bernoullimap is a

constant),and establishthe relationshipbetweenthis speedand the Lyapunov exponent.

How can we understand the difference betweenindividual descriptionand statisticaldescription?We shall

analyzethis situation in moredetail in Chapter4, wherewe

shall seethat thesenew solutionsrequiresmoothnessin thedistribution functions.This is the reasonthat suchsolutions are not applicableto individual trajectories.A

trajectory representedby 8(x\342\200\224 x) is not a smoothfunction; it is


different from zeroonly for d vanishes if xdiffers

at all from x .n

The descriptionin terms of distribution functions is

thereforericherthan that derivedfrom individualtrajectories. This agreeswith the conclusionswe arrived at in

Chapter1,SectionIII.Trajectoriesare merelyspecialsolutions ofthe Perron-Frobeniusequation for unstable maps.This also appliesto systemswith Poincareresonances(seeChapters5 and 6).Thetime-orientedflow ofcorrelationsis an essentialelementin the new solutionsfor the

probability distribution,while no time-orientedprocessesexistat the level ofindividual trajectories.

This break in the equivalencebetweenthe individualand the statisticaldescriptionis the fundamental inspirationofour approach.In the next chapter, we shall discussin

greaterdetail the new solutionsthat arise in chaoticmapsat the statisticallevel.

Thesituation whichwe now find ourselvesin is

reminiscent ofthe onewe encounteredin thermodynamics(Chapter 2). The very successof equilibriumthermodynamicshas retarded the discovery ofnew propertiesofmatter in

nonequilibriumsituations wheredissipative structures and

self-organization appear.In parallel, the successofclassical

trajectory theory and quantum mechanicshas hamperedthe

extensionofdynamicsto the statisticallevel in which

irreversibility can beincorporatedinto the basicdescriptionofnature.

Chapter4

THE LAWS OF CHAOS

I

Inthe precedingchapter, we formulated the principal

factor that makesit possiblefor us to extendclassicaland

quantum mechanicsfor unstable dynamical systems:the

breaking of the equivalencebetweenthe individual

description (in terms of trajectories)and the statistical

description (in termsofensembles).We now wish to analyzethis inequivalencemorecloselyfor simple chaoticmapsand illustrate how this observationrelatesto recentdevelopments in mathematics.1Let us first return to theBernoullimap, whichwe have already introducedas an

exampleofdeterministicchaos.We seefrom the equationofmotionxn+x

= 2xn (mod1)that we may calculatexn for arbitrary n oncewe know theinitial conditionxQ.However,an essentialelementofrandomness still appearsto bepresent.An arbitrary numberxbetween0and 1can berepresentedin a binary digital sys-

89


tern:x =\"f + ~a + IT \342\200\242 \342\200\242 \342\200\242

>wherew. = 0or 1(we are

using the negative indicesu_v u_2 to introducethe bakertransformation, whichwe shall study in SectionIII.Eachnumberx is thus representedby a seriesofdigits.We can

easily verify that the Bernoullimap leadsto the shift w' =u . (for instance,u' ,= u ,) as it moves the numbers u. tothe left. Becausethe value ofeachdigit in the seriesw_pu _2,...is independentof the others,the result ofeachsuccessiveshift is as randomas flipping a coin.This systemis calleda \"Bernoullishift,\" in memory ofthe pioneeringwork in gamesofchancedoneby the great eighteenth-century mathematician, JakobBernoulli.Herewe can alsoobserve a sensitivity to initial conditions:Two numbers

differing only slightly (for example,by w_40, whichmeanslessthan 2~39)will differ by i after 40 steps.As we have

already explained,this sensitivity correspondsto a positiveLyapunov exponentwhosevalue is log2 as x doublesat

eachstep (seeChapter3,SectionII).From the outset,the Bernoullimap introducesan arrow

oftime that can only point in onedirection.If, insteadofxn+1

=2xn (mod1),we considerthe map xn+l

=hxn, we

find a single-pointattractor at x=0.Thetime symmetry is

brokenat the level of the equationofmotion,which is

thus not invertible. This is in contrast to the dynamicalsystems describedby Newton,whoseequations ofmotionare invariant with respectto time inversion.

Themost important point to keepin mind at this

juncture is that trajectoriesare inadequate.They are incapableofdescribingthe time evolution ofchaoticsystemseven if

they are governedby deterministicequationsofmotion.As Pierre-MauriceDuhemstatedas early as 1906,the

notion of trajectory is an adequate modeofrepresentationonly if the trajectoryremainsmoreor lessthe samewhen

The Laws of Chaos 91

we slightly modify the initial conditions.2Thedescriptionofchaoticsystemsin terms oftrajectorieslackspreciselythis robustness.This is the very meaningofsensitivity toinitial conditions:Two trajectoriestaking off from pointsas closetogetheras we can imaginewill divergeexponentially

over the courseoftime.On the contrary, there is no difficulty in describing

chaoticsystemsat the statistical level.It is thereforeat this

level that we have to formulate the lawsofchaos.InChapter 3,we introducedthe Perron-FrobeniusoperatorU,which transforms the probability distribution p (x) into

P\342\200\236+i(x), leadingus to concludethat thereexist newsolutions that are not applicableto individual trajectories.It is

thesenovel solutionsthat we wish to identify in this

chapter. Thestudy ofthe Perron-Frobeniusoperator,whichis

a rapidly growingfield, is ofspecialinterestherebecausechaoticmapsare perhapsthe simplestsystemsthat displayirreversibleprocesses.

Boltzmannapplieshis ideas to gasescontaininganimmense numberofparticles(onthe orderof1023).Here,onthe otherhand, we are dealingwith only a fewindependent variables (onefor the Bernoullimap and two for thebaker map, whichwe shall considershortly).Onceagain,we shall have to rejectthe contentionthat irreversibilityexistsonly becauseour measurementsare limited to

approximations. But first let us identify the new classofsolutions associatedwith the statistical description.

IIHow do we solve a dynamical problemat the statisticallevel?First we needto determinethe distributionfunction

p(x)so that we can observethe recurrencerelation p +1(x)


= U p (x).The distribution function Pn+l(x) after (n + 1)mapsis obtainedby the actionofthe operatorU on p (x),which is the distribution function after n maps.We shallmeetthe same type ofproblemin classicaland quantummechanics.Forreasonsthat we shall explainin Chapter6,operatorformalism was first introducedin quantumtheory, and then extendedto otherfields ofphysics,most

notably statistical mechanics.An operatoris simplya prescriptionfor how to act on a

given function; as such,it may involve multiplication,differentiation, orany othermathematical operation.Inorderto definethe operator,we must also specifyits domain.Onwhat types of functions doesthe operatoract? Are theycontinuousorbounded?Dothey have othercharacteristicsas well?Thesepropertiesdefine the function space.

In general,an operatorU actingon a function f(x)transforms it into a different function.(Forinstance,if U is

a derivative operator , then Ux2=2x).However,thereare

specialfunctions, known as the eigenfunctions ofthe

operator,which remain invariant when we apply U; they are

multiplied only by a numberknown as the eigenvalue. Inthe above example,e*x is an eigenfunctionto which the

eigenvalue k corresponds.A fundamental theoremin

operator analysisstates that we can expressan operatorin termsofits eigenfunctionsand eigenvalues,both ofwhichdepend on the function space.Of particular importanceisthe so-called\"Hilbertspace,\"whichhas beencarefullyexplored by theoreticalphysicistsworking in quantummechanics. It contains\"nicefunctions\" such as xorsin x,but

not the singular, generalizedfunctions that we shallneedin

orderto introduceirreversibility into the statistical

description. Every new theory in physicsalso requiresnew math-


ematical tools.Here,the basic novelty is our needto gobeyondHilbertspacefor unstable dynamical systems.

After theseinitial considerations,let us onceagainreturn to the Bernoullimap, wherewe can easily derive the

explicitform ofthe evolution operatorU, therebyobtaining

P\342\200\236+1(x) = Upn(x)=|[p\342\200\236(|)

+ Pjr-T1)]-This equationmeans that after

(\302\253

+ 1) iterations,the probabilitypK +1(x)at point x is determinedby the values ofp (x) at points 2

1-j- v

and\342\200\2242\342\200\224

\342\200\242As a consequenceofthe form ofU, if p is aconstantequalto a,pn + ^

is also equalto a,sinceUOL = 0C.

The uniform distribution p =OL, which correspondsto

equilibrium,is the distribution function reachedthroughiteration ofthe shift, for n \342\200\224> \302\260\302\260.

On the contrary, if p (x) =x,we obtain Pk +1(x)=4 + f.1 XIn otherwords, Ux= 4+2wherethe operatorU

transforms the function x into a different function, 4+2-Butwe can easily find the eigenfunctionsas defined above, in

which the operatorreproducesthe same function multi-

pliedby a constant.In the exampleU(x\342\200\224 2) = 2~(x\342\200\224 2)'tne

1 1

eigenfunctionis thereforex \342\200\224

2 and the eigenvalue 2 \342\200\242Ifwe

repeatthe Bernoullimap n times, we obtain U\"(x\342\200\224 2) =

1 1(2)n(x

\342\200\224 2),whichmoves toward 0for \302\253\342\200\224\302\273<*>. Thecontri-bution (x \342\200\224 2) to p(x) is thereforerapidly dampedat a raterelatedto the Lyapunov exponent.Thefunction x \342\200\224

2

belongs to a family ofpolynomialscalledthe Bernoulli

polynomials, denotedas B (x),whichare eigenfunctionsof Uwith eigenvalues(2)\", wheren is the degreeofthepolynomial.3 When p is written as a superpositionofBernoullipolynomials,the polynomialsofa higherdegreedisappearfirst becausetheir dampingfactor is greater.This is thereason that the distributionfunction moves rapidly toward aconstant.In the end,only BQ(x)= 1survives.


We now needto expressthe distributionfunction p andthe Perron-FrobeniusoperatorU in terms ofBernoullipolynomials.Beforewe describethe result, however, weshould once more emphasize the distinction between\"nice\" functions and \"singular\" functions (alsocalledgeneralized functions or distributions,which are not to beconfusedwith probabilitydistributions),as it playsa crucialrole.The simplestsingular function is the delta function

8(x).As we saw in Chapter1,SectionIII,8(x\342\200\224

x0) is zerofor all values where and infinite wherex = xn.We have already noted that singular functions have to beused in conjunctionwith nicefunctions.Forexample,iffix) is a nicecontinuousfunction, the integral J dxf(x)8(x

\342\200\224

xQ) =f(xQ)has a well-definedmeaning.In contrast, the

integral containinga productofsingular functions, suchas

/ dx8{x\342\200\224 x0)8(x\342\200\224

x0) =8(0)= \302\260\302\260, divergesand is thereforemeaningless.

Our basicmathematical problemis defining theoperator U in terms ofits eigenfunctionsand eigenvalues.Thisis calledthe spectral representationof the operatorU.Oncewe have this representation,we can use it to expressUp,that is, the effect ofthe Perron-Frobeniusoperatoronthe probability distributionp.Herewe find a quiteremarkable situation characteristicof deterministicchaos.We have already found a set ofeigenfunctions,B (x),the

Bernoullipolynomials,whichare nicefunctions, but thereis a secondset,B (x),whichis formedby singular functions

related to derivatives of the 5-function.4To obtain the

spectralrepresentationof U and thereforeU p, we needboth setsofeigenfunctions.As a result, the statisticalformulation for the Bernoullimap is applicableonly to niceprobability functions p and not to single trajectoriesthat correspondto singular distribution functions repre-


sentedby 5-functions.Thespectraldecompositionof Uwhenappliedto a 5-functioncontainsproductsofsingular functions that divergeand are meaningless.Theequivalence betweenthe individual description(in terms oftrajectoriesrepresentedby 5-functions)and the statistical

descriptionis broken.Forcontinuousdistributionp,however, we obtain consistentresultsthat gobeyond trajectorytheory. We can calculatethe rate ofapproachtoequilibrium and thereforeto an explicitdynamical formulation ofirreversibleprocessesthat take placein the Bernoullimap.This outcomeconfirmsthe qualitative discussioninChapter 1,SectionIII.Probability distribution takes intoaccount the complexmicrostructureofthe phasespace.Thedescriptionofdeterministicchaosin terms oftrajectoriescorrespondsto an overidealizationand is unableto expressthe approachto equilibrium.

Herewe already encountersomeofthe mostcriticalissues in modernmathematics.In fact, as we shall see in

Chapters5 and6,the determinationofeigenfunctionsand

eigenvaluesis the central problemofstatistical andquantum mechanics.The aim there,as well as for chaos,is to

expressan operator,such as U, in terms ofits

eigenfunctions and eigenvalues.When we succeedin doingso,weobtain the spectralrepresentationofthe operator.Inquantum mechanics,sucha representationhas beenachievedin

simplesituations in terms ofnicefunctions.We may thenuse Hilbertspace.The associationbetweenquantummechanics and operatorcalculusin Hilbertspaceis so closethat quantum mechanicsis often consideredan operatorcalculusin Hilbertspace.In Chapter6,we shall seethat

this is generallynot the case.Ultimately, to grasp the real world, we must leave

Hilbertspace.In the caseofchaoticmaps,we have to go


out of Hilbertspacebecausewe needboth the B (x),whichare nicefunctions, and the B (x),whichare singularfunctions.We can then speakofriggedHilbertspace,orGelfandspace.Inmoretechnicalterms,we obtain anirreducible spectral representationof the Perron-Frobeniusoperatoras it appliesexclusivelyto niceprobabilitydistributions, and not to individual trajectories.Thesefeaturesare fundamental inasmuchas they are typical ofunstable

dynamical systems.We shall find themagain in ourgeneralization ofclassicaldynamicsin Chapter5 and quantummechanicsin Chapter6.The physicalreasonsfor whichwe have to leave Hilbertspaceare related to the problemofpersistentinteractionsmentionedabove, whichrequiresa holistic,nonlocaldescription.It is only outside Hilbertspacethat the equivalencebetweenindividual andstatistical descriptionis irrevocably broken,and irreversibility is

incorporatedinto the laws ofnature.

IllThe Bernoullimap is not an invertible system.We

mentioned earlierthat an arrow of time already existsat thelevel ofequationsofmotion.As our main problemis todescribethe emergenceofirreversibility in invertibledynamical systems,we shall now considerthe baker map, orbaker transformation, which is a generalizationof theBernoullimap.Let us take a squarewhosesideshave length1.First we flatten the squareinto a rectanglewhoselengthis 2;then we cut it in half and build a new square.Ifweexaminethe lowerpart ofthe square,we seethat after oneiteration of this process(or mapping),it splits into two

bands (seeFigure4.1).Moreover,the transformation is re-


Figure4.1TheBakerTransformation

\342\200\224j ^_j % \342\200\224\342\200\224i

X X

(a) (b) (c)

versible:The inverse transformation, which first reshapesthe squareinto a rectanglewith lengthi and height2,returns eachpoint to its initial position.

For the Bernoullimap, the equations of motion are

very simple:At eachstep, the coordinates (x,y) become(2x,2) for 0< x < 2 and (2x-1,

\342\200\224y~)for 2 < x < 1.To

obtain the inverse baker transformation, we only have to

permutex and y.In the baker map, the two coordinatesplay different

roles.Thehorizontalcoordinatex is the dilatingcoordinate, which correspondsto the coordinatex in theBernoullimap as it is multipliedby 2 (mod1)at eachmapping.

Thearea ofthe square is preservedbecausewe alsohave a contractingcoordinatey; in the directionof thevertical coordinate,the points draw closertogetherwhilethe squareis beingflattened into a rectangle.Sincethedistance betweentwo points alongthe horizontalcoordinatex doubleswith eachtransformation, it will be multipliedby 2n after n transformations. Ifwe rewrite2\" as e\"10^2, asthe numbern oftransformations measurestime,the

Lyapunov exponentis log2,exactly as in the Bernoullimapconsideredin SectionII.Thereis also a secondLyapunov


exponentwith the negative value\342\200\224log

2,whichcorresponds to the contractingdirectiony.

The effect ofsuccessiveiterationsin the bakertransformation is worthy ofthe sameattention we gave to themin

the discussionofthe Bernoullimaps(seeFigure3.7).Herewe start with points localizedin a small portionof the

square (see Figure4.2),where we can clearly see the

stretching effect of the positive Lyapunov exponent.As

the coordinatesxand y are limitedto the interval 0\342\200\224 1,the

points are reinjected,leadingto their uniform distribution

throughoutthe square.By numericalsimulation,we arealso able to verify that if we start with the probabilityp (x,y), the distribution moves rapidly toward unifor-

Figure4.2NumericalSimulationofthe BakerTransformationThe maps are ordered accordingto the number of iterations,which representtime. (Thesenumerical simulations are the workofDeanDriebe.)

7

10

11 12


mity, as in the caseofthe Bernoullishift (seeFigure3.8).We can gain a great dealofinsight into the mechanism

ofthe bakertransformation by representingit as a Bernoullishift, as we did in SectionI.Herewe associatewith eachpoint (x,y) ofthe unit squarethe doubly infinite sequenceofnumbers {u}definedby the binary representation

0 oox = x 2^\302\253\342\200\236, y

= 2^ u\342\200\236>

n= -oo\342\200\236= i

whereeachu can take on the values 0or1.Eachpoint x,yis representedby the series...u_2, u_v uQ, uv u2 ... , in

which.. . u_\342\200\236 u_v uQ correspondsto the dilating

coordinate x and uv u2, ... to the contractingcoordinatey. Forinstance,the point x = i,y

= iwill berepresentedby aserieswith u_ j

= 1,u2= 1,with all otheru beingzero.By

inserting theseexpressionsinto the equationsofmotion,we obtain the shift u'= u .,whichis again the Bernoulli

n n\342\200\224Va

shift. We seethat the information containedin the initial

conditionsincludesthe entirepast and future history ofthe system(Figure4.3).

Successiveiterations ofthe baker transformation lead to

fragmentation ofthe shadedand unshadedareas,producing

an increasingnumberofdisconnectedregions.Notethat the digit uQ determineswhether the representativephasespacepoint is in the left half

(wQ=0) orthe right half

(\302\2530

= 1)ofthe unit square.Sincethe digits u , ...can bedeterminedby tossinga coin,the time iterates ofu , u' =

un-i> u\"\342\200\236

=u\342\200\236-7

W^ have the same random properties.This showsthat the processby whichthe point appearsin

the left or right half of the square can be consideredaBernoullishift.

The baker transformation also shares an importantproperty of all dynamical systems,known as recurrence.

100The End of Certainty

Figure4.3Iterationsofthe BakerTransformation

Starting with the partition 0 (calledthe generating partition), werepeatedlyapply the baker transformation. In moving toward the

future, we generatehorizontal bands. Similarly, by moving into

the past, we generatevertical bands.

10 12Past | Future

Generating partition

Considera point (x,y)for whichthe sequence{ujin the

binary digit representationis finite or infinite but periodic,and x and y are then rational numbers.Sinceall u areshifted in the sameway, every state ofthis kind will recycleidentically after a certain periodoftime.The sameholdstrue for mostotherstates.To illustrate this concept,we shall

considerthe binary representationof an irrational point(x,y),whichcontainsan infinity ofnontrivial,

nonrepeating digits.It can beshown that almost all irrationals contain

a finite sequenceofdigitsrepeatedan infinite numberoftimes.Thus, a given sequenceof2m digitsaroundposition0,whichdeterminesthe state ofthe systemto an errorof2~m, will reappearan infinite numberoftimes underthe

effect ofthe shift. Sincem can bemade as large as desired(although finite), almost every state will arbitrarily

approach any point, including,ofcourse,the initial position,an infinite numberoftimes.In otherwords, most ofthe

trajectorieswill traverse the entirephasespace.This is the

famous Poincarerecurrencetheorem,which, together

TheLaws of Chaos 101

with time reversibility, was long advancedas an essential

argument against the existenceof genuinely dissipativeprocesses.However,this view can no longerbesustained.

In summary, the baker transformation is invertible, time

reversible, deterministic, recurrent, and chaotic.Demonstratingthesepropertiesthrough this exampleis especiallyuseful,sincethesesamepropertiescharacterizemany real-worlddynamical systems.As we shall see,despite theseproperties,

chaosallows us to establishgenuineirreversibility by

settingup a descriptionat the statistical level.The dynamicsofconservative systemsinvolve laws of

motionand initial conditions.Herethe laws ofmotionare

simple,but the conceptofinitial data demandsa moredetailed analysis.The initial conditionsofa singletrajectory

correspondto an infinite set {u} (n = \342\200\224\302\260\302\260 to +

\302\260\302\260).

But in

the real world, we can only lookthrough a finite window.Thismeansthat we are ableto controlan arbitrary but

limited numberofdigitsu .Supposethat this window

correspondsto u_?u_2u.[u0

\342\226\240

uxii2uv all other digits beingunknown (thedot indicatesthe separationbetweenx and

y digits).TheBernoullishift u' = u _t impliesthat at thenext step, the previousseriesis replacedby u_4u_3u_2u_l

'u\342\200\236u.u2,

whichcontainsthe unknown digit u_4. Moreprecisely, owingto the existenceofa positiveLyapunovexponent, we needto know the initial positionofthe pointwith an accuracyofN+ n digitsin orderto beable to

determine its positionwith an accuracyofN digitsafter n

iterations.

As we saw in Chapter 1,the traditional meansofsolving

this problemwould be to introducea coarse-grainedprobability distribution, which is not defined by singlepoints, but rather by regions,as originally proposedbyPaul and Tatiana Ehrenfest.5However, two points on an


expandingmanifold, even if not distinguishablebymeasurements ofa given finite precisionat time 0,will beseparated, and thus observable,over time.Traditional coarsegraining thereforecannotbeappliedto the dynamicalevolution. This is oneof the reasonsfor whichwe needamoresophisticatedmethod.

First, however, we should analyze in moredetail whatthe approachto equilibriummeans in terms ofthe bakertransformation.6 In spite ofthe fact that this

transformation is invertible, as are all dynamical systems,theevolutions for t \342\200\224> + \302\260\302\260 and t \342\200\224>

-\302\260\302\260 are different. For< \342\200\224> +

\302\260\302\260,

we move toward increasinglynarrow horizontal bands (seeFigure4.3).In contrast,for t \342\200\224>

\342\200\224

\302\260\302\260,

we move toward

increasingly narrow vertical bands.We see that for chaoticmaps, dynamics lead to two

typesofevolutions.We thus obtain two independentdescriptions, onecharacterizingthe approachto equilibriumin our future (for t \342\200\224> +

\302\260\302\260),

and the otherin ourpast (for t

\342\200\224>

\342\200\224

\302\260\302\260).

Such dynamical decompositionis possible forboth chaoticmaps and nonintegrableclassicalandquantum systems,as we shall seelater on.Fora simpledynamical system,whethera harmonicoscillatoror a two-bodysystem,suchdecompositiondoesnot exist;future and pastcannotbedistinguished.Which ofthe two descriptionsforchaoticmaps should we retain? We shall comeback

repeatedly to this question.Forthe moment,let us take intoaccountthe inherentuniversality that every irreversible

processhas in common.All arrowsoftime in nature havethe same orientation:They all produceentropy in the

samedirectionoftime,whichis by definition the future.

We thereforehave to retain the descriptioncorrespondingto equilibriumreachedin our future, that is, for t \342\200\224> + \302\260\302\260.

TheLaws of Chaos 103

In Chapter1,we mentionedthe time paradoxassociated with the bakermap:While the dynamicsdescribedbythis map are time reversible,irreversibleprocessesdo

appear at the statistical level.As in the Bernoullimap, we canintroducethe Perron-FrobeniusoperatorU defined by

Pn + \\{x,y)= U pn(x,y).But there is a fundamental

difference. A generaltheoremstates that for invertibledynamical systemsthereexistsa spectralrepresentation,definedonHilbertspace,whichinvolves only nicefunctions.7

Moreover, in this representationthere is no damping, as the

eigenvaluesare modulo1.Sucha representationalso existsfor the baker transformation, but it is not ofinterestto us

becauseit offers no new information regardingtrajectories. We simply comeback to 8(x\342\200\224

xn + j)<5(y\342\200\224

yn + t)=

US(x\342\200\224 x )S(y\342\200\224

y ), a solution that is equivalent to the

trajectory description.8Exactlyas we did for the Bernoullimap, we have to go

out ofHilbertspaceto obtain additionalinformation.Forspectralrepresentationsin generalizedspace,which have

recentlybeenobtained,the eigenvaluesare the same (i)mas for the Bernoullimap.9 Moreover,the eigenfunctionsare singular functions, such as the B (x) for the Bernoullimap. Again, theserepresentationsare irreduciblein that

they apply only to suitable test functions, obligingus tolimit ourselvesto continuousdistributionfunctions.Singletrajectoriesdescribedby singular 5-functionsare excluded.As is the casein the Bernoullimap, the equivalencebetween the individual descriptionand the statisticaldescription is broken.Only the statisticaldescriptionincludesthe

approachto equilibriumand thereforeirreversibility.Forthe bakermap, thereis oneimportant new element

involved, however, in comparisonto the Bernoullimap:


The Perron-Frobeniusequation can be applied to bothfuture and past (pn + 1= Upn and Pn_1=U_1pK; hereU\"1 is

the inverse of U).In the realm ofHilbertspacespectralrepresentations,this makesno difference becauseU\"\\

+n2 =U\"\\ U\"i, whatever the signofn1 and

\302\2532 (rememberingthat

the positive sign refers to the future, and the negative tothe past). Hilbertspacecan be describedas a dynamical

group. In contrast, for irreduciblespectralrepresentations,there is an essentialdifference betweenfuture and past.The eigenvaluesof U\" are expressedas (jm)\"

= e~n(-m log 2).This formula correspondsto damping in the future (n >0),and divergencein the past (n <0).Therenow existtwodifferent spectralrepresentations\342\200\224one for the future, andthe other for the past. Thesetwo time directions,whichare containedin the trajectory description(or Hilbertspace),are now disentangled.The dynamical group is

thereby brokeninto two semigroups.As previouslymentioned, in accordancewith our view that all irreversible

processesare orientedin the samedirection,we have toselect the semigroupin whichequilibriumis reachedin ourown future. Nature itself is describedby a semigroupthat

distinguishesbetweenpast and future. Thereis an arrow oftime.As a result,the traditional conflict betweendynamicsand thermodynamicsis eliminated.

In summary, as longaswe are consideringtrajectories,it

seemsparadoxicalto speakoflaws ofchaosbecausewe are

dealingwith the negative aspectsofchaos,such as the

exponential divergenceof trajectories,which lead to un-computability and apparent lawlessness.The situation

changesdrastically when we introducethe probabilisticdescription,which remains valid and computableat all

times.It is thereforeat the probabilisticlevel that the lawsofdynamicshave to be formulated for chaoticsystems.In


the simpleexamplesstudiedabove, irreversibility is linked

only to Lyapunov time,but ourresearchhas recentlybeenextendedto moregeneralmaps that includesuchirreversible phenomenasuch as diffusion and various othertransportprocesses.10

IVAs mentionedin Chapter1,the successofthe statistical

descriptionwhen applied to deterministicchaosstemsfrom the fact that it takes into accountthe complexmi-crostructureofphasespace.In eachfinite regionofphasespace,thereare exponentiallydiverging trajectories.Thevery definition of the Lyapunov exponentinvolves the

comparisonofneighboringtrajectories.It is remarkablethat irreversibility already emergesin simplesituations

involving only a few degreesoffreedom.This is,ofcourse,a blow to the anthropomorphicinterpretation ofirreversibility based on approximationsthat we ourselvesare

supposed to introduce.Unfortunately this interpretation,whichwas formulated after the defeat ofBoltzmann,continues to bepropagatedtoday.

It is true that thereis still a trajectory descriptionifinitial conditionsare known with infinite precision.But this

doesnot correspondto any realistic situation.Whenever we perform an experiment,whetherby computerorsomeothermeans,we are dealingwith situations in whichthe initial conditionsare given with a finite precisionand

lead, for chaoticsystems,to a breakingoftime symmetry.Similarly, we couldimagineinfinite velocities,andtherefore we would no longerneedrelativity theory, which is

based on the existenceofa maximumvelocity\342\200\224the

velocity oflight c in the vacuum\342\200\224but the assumptionofve-

106TheEnd of Certainty

locitiesgreaterthan ccorrespondsto no known observable

reality.Maps are idealizedmodelsthat cannotcapture time's

true continuity. As we now turn ourattention to morerealistic situations, ofspecialimportanceto us will benon-integrablePoincaresystems,wherethe breakbetweentheindividual description(trajectoriesorwave functions) andthe statistical descriptionis even morestriking.Forthesesystems, the Laplace demonis powerless,whether his

knowledgeofthe present is finite or infinite. Thefuture is

no morea given; it becomesa \"construction,\"to use an

expressionofthe FrenchpoetPaul Valery.

Chapter5

BEYOND NEWTON'S LAWS

I

Havinganalyzed maps that representsimplified models

in Chapter 4, we cometo the questionat the veryheart ofourquest:What is the roleofinstability andpersistent interactions in the framework ofclassicaland quantummechanics?Classicalmechanicsis the scienceupon whichour beliefin a deterministic,time-reversibledescriptionofnature is based.In respondingto this question,we must first

grapplewith Newton'slaws,the equationsthat havedominated theoreticalphysicsfor the past threecenturies.

Quantum mechanicslimits the validity ofclassicalmechanics when applied to atoms and elementaryparticles.Relativity shows that classicalmechanicsalso has to bemodifiedwhen dealingwith high energiesor cosmology.Whatever the situation, we may introduceeitheranindividual description(in termsoftrajectories,wave functions,orfields) ora statisticaldescription.Remarkably,at all

levels, instability and nonintegrabilitybreak the equivalence

107


ofboth descriptions.Consequently,we have to revise theformulation ofthe laws ofphysicsin accordancewith the

open,evolving universe in whichmankindlives.As stated previously, our positionis that classical

mechanics is incompletebecauseit doesnot includeirreversible processesassociatedwith an increasein entropy.Toincludetheseprocessesin its formulation, we must

incorporate instability and nonintegrability.Integrablesystemsare the exception.Starting with the three-bodyproblem,most dynamical systemsare nonintegrable.Forintegrablesystems, the two modesof description\342\200\224the trajectorydescription,basedon Newton'slaws, and the statistical

description, basedon ensembles\342\200\224are equivalent.Fornonintegrable systems,this is not so.Even in classicaldynamics,then,we have to use the Gibbsianstatistical approach(seeChapter1,SectionIII).As we saw in Chapter3,SectionI,it is this approachthat leads to the dynamicalinterpretation ofequilibriumthermodynamics.It is thereforequitenatural that we alsohave to employ the statisticaldescription to includeirreversibleprocessesdriving systemsto

equilibrium.In this way we canincorporateirreversibilityinto dynamics.As a result, there appear non-Newtoniancontributionsthat can beconsistentlyincludedin

dynamics at the level of the statistical description.Moreover,thesenewcontributionsbreak timesymmetry. We

therefore obtain a probabilistic formulation of dynamics bymeans of which we can resolve the conflict betweentime-reversibledynamicsand the time-orientedview ofthermodynamics.

We are well aware that this step representsa radical

departure from the past.Trajectorieshave always beenconsidered primitive, fundamental toolsof the trade. This is

no longerthe case.We shall encountersituations where

Beyond Newton's Laws 109

trajectories\"collapse,\"to borrowa termfrom quantummechanics(seeSectionVII).

In hindsight, it is not surprising that we have had toabandon the trajectory description.As we saw in Chapter1,nonintegrability is due to resonances,which expressconditionsthat must be satisfied by frequencies.They arenot localevents that occurat given points in spaceand at a

given instant in time.As such,they introduceelementsthat

are quite foreign to the localtrajectory description.Instead, we needa statistical descriptionto formulatedynamics in situations wherewe expectirreversibleprocessesand thereforean increasein entropy.Suchsituations, after

all, are what we seein the world around us.Indeterminism,as conceivedby Whitehead, Bergson,

and Popper,now appearsin physics.This is no longerthe

result ofsomea priorimetaphysicalchoice,but rather the

needfor a statistical descriptionofunstable dynamicalsystems. Overthe past decades,many scientistshave proposedreformulations or extensionsofquantum theory.But the

fact that we now needto extendclassicalmechanicsaswellis quite unanticipated.Even moreunexpectedis the

realization that this revision ofclassicalmechanicscan guideus

in extendingquantum theory.

IIBeforewe begin our revision of Newton'slaws, let us

summarizethe fundamental conceptsofclassicalmechanics. Considerthe motionofa point ofmassm. With the

passageoftime,its trajectory is describedby its position,r(t), its velocity, v = dr/dt, and its acceleration,a = d2r/dt2.Newton'sbasic equationrelatesaccelerationa to forceFthrough the formula F = ma. This formula includesthe


classicalprincipleof inertia, that is, where there is noforce,there is no acceleration,and the velocity remainsconstant.Newton'sequationremains invariant when weshift from oneobserver to anotherwho moves at aconstant velocity with respectto the first. This is known as theGalilean invariance, which has beenradically altered by

relativity, as we shall seein Chapter8.Herewe are dealingwith Newtonian,nonrelativistic physics.

We seethat time takes its placein Newton'sequationonly by meansofa secondderivative. Newton'stime,so to

speak, is reversible,and future and past assumethe samerole.Moreover,Newton'slaw is deterministic.

Now considera moregeneralsituation in whichasystem is formedby N particles.In three-dimensionalspace,we have the 3N coordinatesqv . . . ,q3N and the

corresponding velocitiesvy . . . ,v3N. Inmodernformulations ofdynamics,we usually defineboth the coordinatesandvelocities (orbetter,the momentapv . . . ,p3N, wherein

simplecasesp = mv) as independentvariables.As in Chapter1,

the state ofthe dynamical systemis then associatedwith

a point in phase space,and its motionwith a trajectoryin this space.The most important quantity in classical

dynamicsis the Hamiltonian H, which is defined as the

energyofthe systemexpressedin terms ofthe variables qAnd p. In general,Histhe sumofthe kineticenergyEkm(p)and the potential energy V(q), wherep and q signify theentireset ofindependentvariables.

Oncewe have obtainedthe HamiltonianH(p,q), we canderive the equationsofmotionthat determinetheevolution ofcoordinatesand momentaover the courseoftime.This procedureis familiar to all students ofmechanics.Such equations, as derived from the Hamiltonian,are

calledthe canonicalequations of motion.Contrary to


Newton'sequations,whichare ofthe secondorder(that

is, they containthe secondtime derivative), Hamiltonian

equationsare ofthe first order.Fora singlefree particle,H\342\200\224 j^, the momentump is constant over time,and thecoordinate varies linearly when time q =

q0 + ^t. Bydefinition, for integrablesystems, the Hamiltonian can beexpressedonly in terms ofmomenta(if necessary,after an

appropriatechangeofvariables).PoincarestudiedHamil-tonians in the form H=HQ{p)+ \"kV(q), whichis the sumofan integrablecontribution(the \"freeHamiltonian\" HQ)and a potential energy due to interactions (A, is a scalingfactor that will beused later on).Heshowedthat this classofHamiltoniansis generallynot integrable,whichis to saythat we cannoteliminateinteractionsand goback to

independent units. We already mentionedin Chapter1 that

nonintegrabilityis due to divergingdenominatorsassociated with Poincareresonances,as a result ofwhichwecannot solve the equationsofmotion(at least in powersofthe

couplingconstant A,).

In the following pages,we shall concernourselves

primarily with nonintegrablelargePoincaresystems(LPS).As

we have seen,Poincareresonancesare associatedwith

frequencies correspondingto various modesofmotion.Afrequency(0k dependson the wavelength k. (Usinglight asan example,ultraviolet has a higher frequency (0 and

shorterwavelength k than infrared light.)When weconsider nonintegrablesystemsin whichthe frequencyvaries

continuouslywith the wavelength, we arrive at the verydefinition ofLPS.This conditionis metwhenthe volumein whichthe systemis locatedis great enoughfor surfaceeffects to be ignored.This is why we call thesesystemslargePoincaresystems.

A simpleexampleofLPSwould be the interactionbe-


tween an oscillatorwith frequency C0l coupledwith a

given field.In this centuryofradioand television,we haveall heard the termelectromagneticwaves. Theamplitudeofthesewaves is definedby a field describedby a function

<p(x,t) ofpositionand time.As was establishedat the

beginning ofthe century, a field can bethought ofas the

superposing of oscillations with frequenciesft)fewhose

wavelength k varies from the sizeofthe systemitselfto thedimensionsofelementaryparticles.In the oscillator-fieldinteractionthat we are considering,resonancesappeareachtime a field frequencyft)fe

is equalto the oscillatorfrequency C0l.When we try to solve the equationsofmotionofthe oscillatorin interactionwith the field, we encounterPoincareresonancesiw _ w\\, whichcorrespondtodivergences whenever

C0X= ft),. In otherwords,thesetermstend

toward infinity and thereforebecomemeaningless.As weshall see,we can eliminatethesedivergencesin ourstatistical description.

Poincareresonanceslead to a form ofchaos.Indeed,innumerable computersimulations have shown that theseresonanceselicitthe appearanceofrandomtrajectories,asis the casefor deterministicchaos.In this sense,there is acloseanalogy betweendeterministicchaosand Poincarenonintegrability.

IllAs in previouschapters,we shall considerthe probabilitydistributionp(q,p,t), whoseevolution over time can easilybederivedfrom the canonicalequationsofmotion.We are

now in the same situation as we were for chaoticmaps,wherewe replacedthe equationsofmotionwith statistical

descriptionsassociatedwith the Perron-Frobeniusoperator. In classicalmechanics,we also encounteran evolution


operatorknown as the Liouville operatorL, whichdetermines the evolution ofp through the equationi-\302\243

= Lp.Thetime changeofp is obtainedby acting onpwith the

operatorL. If the distribution function is timeindependent

-\302\243

= 0,then Lp = 0.This correspondstothermodynamic equilibrium.As we saw in Chapter3,SectionI,pthen dependson only the energy (or the Hamiltonian),whichis an invariant ofmotion.

The solution of dynamical problems at the statisticallevel requiresdeterminingthe spectralrepresentationofL,as was explainedin Chapter4 for chaoticsystems.We

thereforehave to defineits eigenfunctionsand eigenvalues.We have seenthat spectralrepresentationdependson thefunctions which,as used in the past (and still appropriatefor integrablesystems),are in Hilbertspace,the spaceof\"nice\" functions.Accordingto a fundamental textbooktheorem,operatorL has real eigenvalues/ in Hilbertspace.In this case,evolution over time provesto be a

superposition ofoscillatoryterms.In fact, the formalsolution ofthe Liouvilleequation is p(t) \342\200\224 exp(\342\200\224itL)p(0). Theoscillatoryterm

exp(\342\200\224itl)

= costl \342\200\224 i sin tl is associatedwith eigenvalue /,wherefuture and past play the samerole.In orderto includeirreversibility,we needcomplexeigenvalues such as / = CO

\342\200\224 iT , which lead to exponentialdamping e_)\"' for time evolution.This contributionprogressively diminishesin the future (t>0) but is increasedinthe past (t < 0),and thus time symmetry is broken.

However, obtaining complexeigenvaluesis possibleonly whenwe leave Hilbertspace.Our main objectiveisnow to understandfor whichphysicalreasonswe have todo so.This follows from the inescapablefact that thereare

persistent interactionsin the natural world.1When weconsider the roomin whichwe sit, the moleculesin the

atmosphere are constantly colliding.This is quite different


from transitory interactions,suchas a finite numberofmolecules in a vacuum. Themoleculesthen interact over afinite periodof time,and eventually may escapeinto

infinity. Thedistinctionbetweenpersistentand transitoryinteractionstakeson a crucialimportancein moving fromclassicaldynamicsto thermodynamics.Classicaldynamicsextracts a given numberofparticlesand considerstheirmotion in isolation;irreversibility occurswheninteractions never cease.In short, dynamicscorrespondsto areductionist point of view in the sensethat we considera finite number of moleculesin isolation.Irreversibility emergesfrom a moreholistic approach in whichweconsidersystemsdriven by a largenumberofparticlesas awhole.Inmakingthis distinctionmoreprecise,we shall

indicate why we needsingular distribution functions andmust thereforeleave Hilbertspace.

IVTransient interactionsmay bedescribedby localizeddistribution functions.To describepersistent interactions in a

largespacesuch as the atmosphere,we needdelocalizeddistribution functions. In defining more precisely thedistinctionbetweenlocalizedand delocalizeddistributionfunctions p, let us beginwith a simpleexample.In a one-dimensionalsystem,the coordinatex extendsfrom \342\200\224

\302\260\302\260 to+00.Localizeddistributionfunctions are concentratedon afinite sectionofthe line.A specialcaseis a singletrajectorythat is localizedat a given point and moves alongthe lineover the courseoftime.In contrast,delocalizeddistribution functions extendover the entire line.Thesetwo

classesoffunctions describevarious situations.As an

example, let us considerscattering. In the usual scatteringexperiments, we preparea beamofparticlesthat we shootat


an obstacle(the scattering\"center\.")We then have thethreestagesrepresentedin Figure5.1.

In this experiment,the beamfirst approachesthe

scattering center,then interacts with it, and is finally in freemotionagain.Theimportant point hereis that theinteraction processis transient.Fordelocalizeddistributions,onthe otherhand, the beamextendsover the entireaxis,and

scatteringneitherstarts nor stops.We then have what wecall persistentscattering.

Transient scatteringexperimentshave playedasignificant part in the history ofphysicsby allowingus to studythe interactionsbetweenelementaryparticlessuch asprotons and electrons.Still,in many situations\342\200\224particularly in

macroscopicsystemssuchas gasesorliquids\342\200\224we

havepersistent interactionsbecausecollisionsnever cease.In sum,transient interactionsare related to localizeddistributionfunctions, suchas trajectories,while persistentinteractionsare related to delocalizeddistributions,whichextendoverthe entiresystem.

Thermodynamicsystemsare characterizedby persistent

Figure5.1TheThreeStagesofScattering(a) The beam approachesthe scattering center, (b) The beamintersects the scattering center, (c)The beam is onceagain in free motion.


interactions,and must thereforebe describedby delocal-izeddistributions.In defining thesesystems,we have toconsiderthe thermodynamic limit, wherethe numberofparticles JV and the volume V are increased,while theirratio,the concentrationN/l{remains constant.Although

formally we considerthe limits JV \342\200\224>

\302\260\302\260,

V \342\200\224>

\302\260\302\260,

thereare, ofcourse,no dynamical systems\342\200\224not even the universe\342\200\224

wherethe numberofparticlesis infinite. This limit simplymeansthat surface effectsdescribedby the terms of^orpcanbe ignored.The thermodynamic limit plays a centralrolein all macroscopicphysics.Without this concept,wecouldnot even define states ofmatter such as gases,liquids, or solids,or describethe phase transitions betweenthesestates ofmatter. We would also beunable to

distinguishbetweennear-equilibriumand far-from-equilibrium

situations, whichwerediscussedin Chapter2.We shall now illustrate why the introductionofdelocal-

ized distribution functions forces us to leave the classofnicefunctions and thereforeHilbertspace.In orderto do

so, we have to considerseveral elementarymathematicalnotions.In the first place,every student ofmathematicsis familiar with periodicfunctions such as sin (~^)- Thisfunction remainsinvariant whenwe add to the coordinatex the wavelength A,, as

2kx \342\226\240 2k(x + X)sin -y \342\200\224 sin \342\200\224

j^\342\200\224.

Otherperiodicfunctions are cos-=^,orthe morecomplexcombination

2nx,\342\200\224

2kx . \342\226\240 \342\226\240 2kxe *\342\226\240

= cos-y + i sin -yInsteadofthe wavelength A,, we often use the wave vectork =y.The exponentiale'kx is calleda planewave.

In the secondplace,the classicaltheory ofFourierseries


(orFourierintegrals) demonstratesthat a function ofthecoordinatex, whichwe shall callj(x),can beexpressedasa superpositionof periodicfunctions correspondingtowave vectorsk, or morespecifically,as a superpositionofplane waves e'kx. In this superposition,eachplane wave is

multipliedby an amplitude cp(fe), which is a function ofk.

Thisfunction <p(k) is knownas the Fouriertransform o{j(x).In short, we can go from a function J(x)ofcoordinatex

to a description<p(k) in wave vectorsk. Ofcourse,theinverse transformation is equally possible.It is also importantto note that there is a kind of duality betweenj{x)and

<p(k). KJ{x)extendsover a spatial interval Ax (and vanishes

outside),(p(fe) extendsover the spectral interval Ak ~^ .When the spatial interval Ax increases,the spectralintervalAk decreases,and vice versa.2

In Chapter1,SectionIIIand Chapter3,SectionII,wedefined the singular function 8(x).As we saw, 8(x)differs

from zeroonly at x=0.Thespatial interval Ax is thereforezero,and whenAk ~^r, the spectralinterval is infinite.

Inversely, delocalizedfunctions for whichAx \342\200\224>

\302\260\302\260 leads to

singular functions in k such as 8(fe).Thus,delocalizeddistribution functions are an essentialelementin describingpersistent interactions.At equilibrium,the distributionfunction p is a function ofthe HamiltonianH(seeChapter 3,SectionI).TheHamiltoniancontainsthe kineticenergy that is a function ofthe momentap and not ofthe

coordinates,and thus includesa delocalizedpart that has a

singular Fouriertransform. It is hardly astonishingthat

singular functions play a criticalrolein our dynamicaldescription. Indeed,it is our needfor thesefunctions that

forcesus to leave Hilbertspace.Equilibrium distributionsthat are functions ofthe Hamiltonianare already outsideHilbertspace.


V

Let us now comparethe trajectory descriptionwith thestatistical descriptionin terms of the Liouville operator(seeSectionIII).Herewe are in for quite a surprisebecause the statistical descriptionintroducescompletelydifferent concepts.This is obviouseven in the simplestcasewherewe considerthe motionofa free particlealongaline.As we saw in SectionII,the coordinateq oftheparticle varies linearly over time,whilethe momentumpremains constant.On the contrary, the statistical descriptionis defined in terms ofthe wave vectors k, associatedwith

the Fouriertransform ofq, and the momentump.We areused to dealingwith wave vectorswhenwe studyacoustical or opticalproblems,but herewave vectorsappear in a

problemofdynamics.Thereasonis that for a free particle,the Liouville operatorL is simply a derivative operator,whereL = ^ gx.As we noted in Chapter4, SectionI,the

eigenfunctionsare then exponentialsexp (ikx) and the

eigenvalues .Theeigenfunctionexp (ikx) is a periodicfunction, orplanewave, sinceexp(ikx) =coskx + i sin kx.It extendsover the entirespace,in strikingcontrastwith a

trajectory localizedat a singlepoint.The solutionofthe

equationofmotionfor a free particleis obtainedin thestatistical descriptionthrough a superpositionof planewaves.Ofcourse,in this simpleexample,the twodescriptions are expectedto be equivalent. Usingthe theory ofFouriertransformation, we can reconstructthe trajectorystarting with planewaves (seeFigure5.2).Becausethe

trajectory is concentratedat onepoint,we have to superposeplane waves extendingover the entirelengthofthespectral interval (Ak \342\200\224>

\302\260\302\260).

As a result,for q = q0, the amplitudesofthe planewaves


Figure5.2SuperpositionofPlaneWaves

Trajectories resulting from the superpositionof plane wavesthrough constructive interference lead to a function characterizedby dramatic peaking around q

= 0.

M M

/^aS^>n<(a) (b)

increasethrough constructive interference,while for q ^ qQ,

they vanish through destructive interference.In integrablesystems,the wave vectork is constantover time.Bysuperposing

the plane waves, we can reconstructtrajectoriesat

any moment.But the important point to considerhereis

that the trajectory is no longera primitive concept,but

rather a derivedconceptas a constructofplanewaves. It is

thus conceivablethat resonancesmay threaten theconstructive interferencesleadingto a trajectory. This couldnot beconsideredas longas the trajectorywas treatedas a

primitive, irreducibleconcept.Given that a trajectory is

representedby a point in phasespace,we can seethat the

collapseoftrajectorieswould correspondto a situation in

whicha point decomposesover time into a multiplicity ofpoints, exactly as in the diffusion processwe analyzed in

Chapter1.Thesame initial conditionwould then lead toa multiplicity oftrajectories,as was also the casein thediffusion process.


Theeigenvalues-\302\243

ofthe Liouvilleoperatorcorrespondto the frequenciesappearingin Poincareresonances.Theydependon both k and p, and not on the coordinates.Theuseofwave vectork is thereforea logicalstarting point for

discussingthe roleof these resonances.By using planewaves, we candescribenot only trajectories(whichcorrespond

to transient interactions),but alsodelocalizedsituations. As we have seen,this leads to singular functions in

the wave vectork. Let us now examinethe effect ofinteractions on the statistical descriptionby employingthe

language ofwave vectors.

VI

Supposethat the potential energy V in the Hamiltonianis

the sum ofbinary interactions.It then follows from well-establishedtheoremsthat interactionsbetweenparticlesjand n modify the two wave vectors k. and k , while theirsum is conserved,giving us the conservationlaw k. + k =k'.+ &',wherek\\ and k' are the wave vectorsafter inter-j n> J n

action.3We are able to describedynamical evolution within the

statistical formalism pictoriallyby consideringa successionofevents separatedby free motion.At eachevent, the wavevectors k and momentap are modified;between the

events, they remain constant.Let us now examinethenature oftheseevents in moredetail.

In Chapter3,SectionI,we introducedthe notionofcorrelations,whichwe shall now define with greaterprecision. Thedistributionfunction p(q,p,t) dependson bothcoordinatesand momenta.If we integratethis function

over the coordinates,we loseall information about the

position ofparticles,and thus correlations,in space.We

obtain a function p0(p, t), which offers information only


about momenta.Forthis reason,p0 is known as the vacuum

ofcorrelations.On the otherhand,by integrating over all

coordinates exceptthe coordinatesq.,q.ofparticlesi and j,we retain the information about possiblecorrelationsbetween particlesi andj.This function, p2, is calleda binarycorrelation.We can defineternary correlationsand beyondin a similar way. In the statisticaldescription,it is importantto replacethe coordinates,whichdependon thedistribution functions through theirFouriertransform, with wave

vectorsas they appearin the spectraldecompositionoftheLiouville operator.

We shall now take into accountthe law ofconservationofwave vectors,in whicheachevent is representedby a

point,with two entry lines,k., k , and two exitlines,k'.,k' , wherek. + k = k'.+ k' .Moreover,at eachpoint, then' j n j n rmomentap ofthe interactingparticlesare modified,and a

derivative operatorg- appears.Thesimplestevent ofthis

kind is illustrated in Figure5.3.We call the diagramin Figure5.3a propagationevent,

or propagationdiagram.This correspondsto a

modification ofthe binary correlationp2betweenparticlesj and n.

But we can also start from the vacuum ofcorrelationsp0,in which k. = k = 0, and producea binary correlationp,,, , with k.+k =0 to conservethe sumofthe wave vec-

kj kn j n

Figure5.3PropagationDiagramA dynamical event correspondingto the interaction of two

particles leadsfrom wave vectorski, kn to k',-, k'\342\200\236.


Figure5.4CreationFragmentA dynamical event transforms the vacuum of correlationsinto a

binary correlation/, \342\200\224/.

k'\342\200\236=l

tors (seeFigure5.4).We then have what is known as acreation ofcorrelationdiagram,orcreationfragment. We alsohave destructionfragments, as presentedin Figure5.5,which transform binary correlationsinto the vacuum ofcorrelations.4

We now beginto seedynamicsas a history ofcorrelations.

Figure5.6represents,for example,the emergenceof a

five-particlecorrelationstarting from the vacuum ofcorrelations. Events associatedwith interactionsproducecorrelations.

We can now introducethe effect ofPoincareresonancesinto the statistical descriptionof dynamics.Theseresonances coupledynamical processesexactlyas they couple

Figure5.5DestructionFragmentA dynamical event transforms the binary correlation/, -/ into the

vacuum of correlations.

kj= -l


Figure5.6EvolutionofCorrelationsThe four events at points 0X, 02,03, 04 transform the vacuum ofcorrelationsinto a five-particlecorrelation.

harmonicsin music.In our description,they couplecreation and destructionfragments (seeFigure5.7),whichleads to new dynamical processesthat start from a givenstate ofcorrelations(ofwhichthe vacuum ofcorrelationsis merely oneexample)and eventually return to exactlythe samestate.In Figure5.7,thesedynamical processesare

depictedas bubbles.While the state ofcorrelationsis

preserved, the distributionofmomentais changed(remembering

that eachvortex introducesa derivative operatorgr).Thesebubbles correspondto events that must be

consideredas a whole.They introducenon\342\200\224Newtonian ele-


Figure5.7BubbleDueto PoincareResonancesPoincareresonancescouplethe creationand destruction ofcorrelations, and leadto diffusion.

Destruction ofcorrelations Creation ofcorrelations

ments in that no analogueof such processesexists in

trajectory theory.Suchnew processeshave a dramaticeffect on dynamicsbecausethey break time symmetryIndeed, they lead to the type ofdiffusion that had alwaysbeenpostulated in phenomenologicaltheoriesofirreversible processes,includingBoltzmann'skineticequation.To mark the parallel with the phenomenologicaldescription, we have calledthe new elementscollision operators.They act on the distributionfunctions.*

*Wesaw in Chapter 1,Section III that Poincare resonances between frequencies

lead to divergences with small denominators. Herethe frequency of a particle ofmomentum p is kp/m, where fe is the wave vector (seeSection IV). For LPS, in

which fe is a continuous variable, we can avoid the divergences and express the

resonances in terms of 6-functions. This involves a branch of mathematics

associated with analytical continuation (seethe references in the chapter notes). For

a two-body process, the argument of the 6-function is fe/m(p{

\342\200\224

p2), leading to

contributions whenever the frequencies kpl/m and kp2/m are equal, and

otherwise vanishing. Thewave vector fe = 0 therefore plays an especially important

role wherein the argument of the 6-function vanishes, remembering that S(x)=\302\260\302\260 for x= 0 and S(x)= 0 for x * 0.A vanishing wave vector fe corresponds to an

infinite wavelength, and thus to a process that is delocalized in space.Hence,Poincare resonances cannot be included in the trajectory description.


Our approach includesthe usual kinetic theory, but

only as a specialcase.Traditionally this theory, asintroduced by Maxwell,was centeredaround the evolution ofthe velocity distribution,whereit appearedthat only a fewcollisionswould be sufficient to reestablishequilibriumifdisturbed at the initial time.Our approach, on the

contrary, takesinto accountthe progressivebuildup ofhigherand highercorrelationsinvolving moreand moreparticles.This processrequireslong time scales,in agreementwith

the numericalsimulations that have been available for

many years.5As a result,irreversibility leadsto longmemory

effects that profoundlyalter macroscopicphysics.6Many new resultsthat gobeyond the traditional kinetic

theory have already beenobtained.However,it is outsidethe scopeofthis bookto describethem.They will becovered in greaterdetail in a separatemonograph.7

Suffice it to say that we are beginningto understandwhat irreversibility really means.Let usconsiderthe simpleanalogy ofthe agingprocess.On our time scale,the atomsthat makeup ourbodiesare immortal.What is changingisthe relation between the atoms and molecules.In this

sense,agingis a property ofpopulations,and notindividuals. This is also true ofthe inanimate world.

VIILet us now return to our originalobjective,which is the

solutionofthe dynamical problemat the statistical level in

terms ofthe distribution function p.As was the casefordeterministicchaos,this solution involves the spectralrepresentation of the evolution operator,which in classical

dynamicsis the Liouvilleoperator.First we considerdelo-calizeddistributionfunctions associatedwith persistentinteractions that lead to singular functions (seeSectionsIII


and IV).As a result,we have to leave Hilbertspace,whichis limited to localizednicefunctions.We then introducePoincareresonances,which,as we saw in SectionVI, leadto new dynamical processesconnectedwith diffusion.

Oncewe have includedthesetwo features, we obtain an

irreducible,complexspectral representation.Again,

complex meanstimesymmetry is broken,and irreducible meanswe cannotreturn to a trajectory description.The laws ofdynamicsnow take on new meaning.By incorporatingirreversibility they expressnot certitudesbut possibilities.Only if we relaxour conditionsand considerlocalizeddistribution functions associatedwith a finite numberofparticlescanwe recoverthe Newtoniantrajectorydescription. But in generaldiffusion processesdominate.

Thereare thereforemany situations in whichwe can

expectdeviations from Newtonianphysics,and whereourpredictionshave already beenverified by extensive

computer simulations.In SectionIV, we introducedthe

thermodynamic limit, wherethe numberofparticlesJV \342\200\224>

\302\260\302\260

and volume V \342\200\224>

\302\260\302\260,

while -p = the concentrationthat

remains constant.In this limit, interactionsgo on forever,and only the statistical descriptionapplies.It has beenshownby extensivenumericalsimulationsthat even if westart with a trajectory involving an ever-increasingnumberofparticles,diffusive processestake over, and the trajectory\"collapses\"becauseit is transformedover time into a delo-calizedsingular distributionfunction.8

Our new kinetictheory is ofgreat interestin describingdissipative processesfor all time scales,as observed in the

laboratory or the ecosphere.But this is only oneof its

many novel features.BecauseofPoincareresonances,the

dynamical processesdescribedin this sectionlead to long-rangecorrelations,even if the forcesbetweenthe particles


are short range.Theonly exceptionis the state ofequilibrium, wherethe range ofcorrelationsis determinedby that

ofthe forcesbetweenthe particles.Thisexplainsthe fact, asstated in Chapter2,that nonequilibriumallows for a newcoherence,which is clearly manifested by chemicaloscillations and hydrodynamic flows. We now recognizethat

equilibriumphysicsgave us a false imageofmatter. Onceagain, we are facedwith the fact that matter in equilibriumis \"blind,\" while in nonequilibriumit beginsto \"see.\"

In sum, we are now able to gobeyond Newtonianmechanics. Thevalidity ofthe trajectory descriptionused in

classicalmechanicsis severely limited.Thermodynamicsis

incompatiblewith trajectory description,as it requiresastatisticalapproachboth at equilibriumand out ofequilibrium. The fact that the vast majority ofthe dynamicalsystems correspondingto the phenomenathat surround usare LPS is the reasonwhy thermodynamicsis universallyvalid. Transient dynamical interactionssuchas scattering arenot representativeof the situations that we encounterinthe natural world, where interactionsare persistent.Thecollisionprocessesthat appear in our statistical descriptionas a result ofPoincareresonancesare essentialin that theybreak time symmetry and lead to evolutionary patterns in

accordancewith the thermodynamicdescription.The microscopicdepictionof nature associatedwith

thermodynamics has little to do with the comfortable

time-symmetricaldescriptionscientistshave traditionallytaken from Newtonian principles.Ours is a fluctuating,noisy, chaoticworldmoreakin to what the Greekatomists

imagined.In Chapter1,we describedEpicurus'dilemma.Theclinamenhe envisagedno longerbelongsto a

philosophical dream that is foreign to physics.It is the veryexpression ofdynamical instability.


Of course,dynamical instability provides only thoseconditionsnecessaryto generateevolutionary patterns ofnature.Oncewe have achievedour statistical description,we can also formulate the additional factors we needin

orderto observethe emergenceofcomplexity\342\200\224of dissi-

pative structuresat the macroscopiclevel.We now begintounderstand the dynamical roots oforganization,thedynamics at the rootofcomplexitythat are essentialfor self-

organization and the emergenceoflife.

Chapter6

A UNIFIED FORMULATIONOF QUANTUM THEORY

I

Thereare fundamental differencesbetweenclassical

Newtonian dynamics and quantum theory But in

both casesthereexistan individual descriptionin terms oftrajectoriesorwave functions (seeChapter1,SectionIV)

and a statistical descriptionin terms ofprobabilitydistributions. As we have alreadyseen,Poincareresonancesappearin classicalas wellas quantum theory. We can therefore

anticipate that the resultsobtained in classicalmechanicswill

also apply to quantum theory.In fact, in both instanceswehave achieved a new statisticalformulation applicableto LPSoutsideHilbertspace.This descriptionincludes

time-symmetry breaking,and is irreducibleto the individual

description in termsofquantum wave functions.In spiteofquantum theory'sastonishingsuccess,

discussions about its conceptualfoundations have not abated.After seventy years,they are as lively as ever.

129


Forexample,in his recentbookShadowsof the Mind,

RogerPenrosedistinguishesbetween\"Z mysteries\"(forquantum puzzles) and \"X mysteries\"(forthe quantumparadox) in quantum behavior.1Furthermore,the roleofnonlocalityseemsintenselyproblematic.Given that

localityis a property associatedwith the Newtonian pointwise

trajectory description,it is not surprising that quantumtheory, whichincludesthe wave aspectofmatter, leadstoa form ofnonlocality.2

The \"collapse\"of the wave function, whichseemstorequirea dualistic formulation ofquantum theory,represents a further complication.On the onehand, we havethe basicSchrodingerequationfor wave functions, whichis time reversibleand deterministic,exactly as is Newton'sequation;on the other,we have the measurementprocessassociatedwith irreversibility and the collapseofthe wavefunction.This dualistic structure is the basisofJohn vonNeumann'sargument in his famous book,Mathematical

Foundations of Quantum Mechanics.3This situation is indeedbizarrebecausein addition to the basictime-reversible,deterministic Schrodingerequation,therewouldbea seconddynamical law associatedwith the collapse(or reduction)of the wave function.Until now, however, no one hasbeenable to describethe link betweenthesetwo laws ofquantum theory, nor has anyone succeededin giving arealistic interpretation of the reductionof the wavefunction. This is the quantum paradox.

Thequantum paradox,whichderivesfrom the dualisticstructure ofquantum theory, is closelyrelated to anotherproblem.Ourconclusionis that quantum theory is

incomplete.Like classicaltrajectory theory, it is timesymmetric,

and thereforecannotdescribeirreversibleprocessessuchasthe approachto thermodynamicequilibrium.This is par-

A Unified Formulation of Quantum Theory 131

ticularly curiousbecausequantum theory beganin 1900with Max Planck'ssuccessfuldescriptionofblackbodyradiation in equilibriumwith matter. Even today, in spiteofthe great advances madeby Albert Einsteinand Paul A. M.Dirac,we still have no exactquantum theory describingthe approachto equilibriumwhenradiation interacts with

matter. (As we shallsee,this is relatedto the fact that

quantum theory describesintegrablesystems.We shall comeback to this challengein SectionIV.) We needbothequilibrium and nonequilibriumphysicsto describethe worldaround us.An exampleofan equilibriumsituation is thefamous residualblackbody radiation at

3\302\260K,which

originated at a time closeto the big bang.A largepart ofmacroscopicphysicsalso dealswith equilibriumsystems,whetherthey are solids,liquids,or gases.Thereis thus a

gap between quantum theory and thermodynamics as

deepas that betweenclassicaltheory and thermodynamics.Remarkably, the same methodemployed in extendingclassicalmechanicsin Chapter5 also permits us to unify

quantum theory and thermodynamics.Indeed,ourapproach eliminatesthe dualistic structure ofquantummechanics, and thus eliminates the quantum paradox.We

arrive at a realistic interpretation ofquantum theorybecause the transition from wave functions to ensemblescannow be understoodas the result ofPoincareresonanceswithout the mysteriousintervention ofan \"observer\"orthe introductionofotheruncontrollableassumptions.Incontrast to otherattempts to extendquantum theory, asnotedin Chapter1,ourown approachmakeswell-definedpredictionsthat are testable.Thus far, they have beenconfirmed by every numericalsimulation performed.4

Our thinking constitutesa return to realism,but

emphatically not a return to determinism.On the contrary,


we move even farther away from the deterministicvisionofclassicalphysics.We agreewith Popperwhenhewrites,\"My own point ofview is that indeterminismis

compatible with realism,and that the acceptanceofthis fact allowsus to adopt a coherentobjectiveepistemologyof thewholeofquantum theory, and an objectivistinterpretationofprobability.\" We shall thereforeendeavorto bring intothe realm ofphysicswhat Poppercalledhis metaphysicaldream:\"It is likely that the worldwould bejust as indeter-ministicas it is even if therewereno observingsubjectsto

experimentwith it, and to interfere with it.\"5 Thus we will

show that the quantum theory ofunstabledynamicalsystems with persistentinteractionsleads,as in classicalsystems, to a descriptionthat is both statisticaland realistic.Inthis new formulation, the basic quantity is no longerthewave function correspondingto a probability amplitude,

but probability itself. As in classicalphysics, probabilityemergesfrom quantum mechanicsas a fundamental

concept.In this sense,we are on the eve ofthe triumph ofthe

\"probabilisticrevolution,\" which has beengoingon forcenturies.Probability is no longera state ofmind due toour ignorance,but the result ofthe laws ofnature.

IITheobservation that the interactionbetweenatoms and

light leads to well-definedabsorption and emissionfrequencies was the starting point for the formulation ofquantum mechanics.Theatom was describedby NielsH.D.Bohrin terms ofdiscreteenergy levels.In accordancewith experimentaldata (the Ritz-Rydbergprinciple),the

frequencyofspectrallinesis the difference between two energy

levels.Oncetheselevels are known,we can predictthe fre-


quencyofspectrallines.Theproblemsofspectroscopycanbereducedto the calculationoflevels ofenergy.But howcan we reconcilethe existenceofwell-definedenergylevels, which decisivelyinfluencedthe history of quantumtheory, with the Hamiltonianconceptthat is so importantto classicaltheory?TheclassicalHamiltonianexpressesthe

energy ofa dynamical system in terms ofcoordinatesqand momentap, and thereforetakeson a continuousset ofvalues.It cannotlead to discreteenergylevels.Forthis

reason, the HamiltonianHis replacedin quantum theory bythe HamiltonianoperatorH .r

op

We have repeatedlyusedoperatorformalism (the Perron-Frobeniusoperatorwas introducedin Chapter4, and theLiouvilleoperatorin Chapter5), but it was in quantumtheory that operatorcalculus was first introducedinto

physics.In the situations studied in Chapters 4 and 5,weneededoperatorsto achieve the statistical description.Here,even the individual level ofdescriptioncorresponding

to wave functions requiresoperatorformalism.Thebasicproblemin quantum mechanicsis the

determination ofthe eigenfunctionsua and the eigenvaluesEaofthe HamiltonoperatorH(weshall omit the subscriptopwhereverpossible).TheeigenvaluesEa, whichareidentified with the observedvalues ofthe energy levels, form the

spectrumofH.We speakofa discretespectrum whensuccessive eigenvaluesare separatedby finite distances.If the

spacingbetweenlevelstends toward zero,we then speakofa continuous spectrum. Fora free particlein aone-dimensional boxwith a lengthofL, the spacingofthe energylevel is inversely proportionalto ]_}_ As a consequence,whenL \342\200\224>

\302\260\302\260,this spacingmoves toward zero,and we

obtain a continuousspectrum.By definition, the word\"large\" in largePoincaresystems(LPS)means precisely


that thesesystemshave a continuousspectrum.As in

classical theory, the Hamiltonianis herea function ofcoordinates and momenta.However,becausethe Hamiltonianis

now an operator,thesequantities, and thereforeall

dynamical variables, now have to be treatedalso as operators.For today'sphysicists,the transition from functions to

operatorsthat takesplacein quantum theory seemsperfectly natural. They now manipulate operatorswith theeasewith whichmost ofus manipulate natural numbers.Nonetheless,for classicalphysicistssuchas the great DutchscientistHendrikAntoon Lorentz,the introductionofoperators was barely acceptable,and even repulsive.In anycase,individuals such as Werner Heisenberg,Max Born,Pascual Jordan,Erwin Schrodinger,and Paul Dirac,who

daringly introducedoperatorformalism into physics,deserve our admiration.They drastically changedourdescription ofnature in defining the conceptualdifferencebetweena physicalquantity (representedby an operator)and the numericalvalues this physicalquantity may take on(the eigenvaluesofthe correspondingoperator).Thisradical changein outlookhas had far-reachingand profoundimplicationsfor ourconceptionofreality.

As an exampleofthe sophisticationofoperatorformalism,

considerthe commutationrelationsbetweentwooperators. Theseoperatorscommuteif the orderof their

applicationto a function is immaterial.They do notcommute if the orderoftheir applicationchangesthe result.Forinstance,multiplying a function j{x)by x and then

differentiating it with respectto x doesnot leadto the sameresult as first differentiating^^)and then multiplying it byx. This can easilybeverified.Operatorsthat do notcommute exhibit different eigenfunctions;if they docommute, they have commoneigenfunctions.


Thefamous Heisenberguncertainty principlefollows fromthe fact that the coordinateand momentumoperators,asdefined in quantum theory, do not commute.In all

textbooks on quantum mechanics,it is shown that in the\"coordinate representation,\"the operatorcorrespondingto acoordinateq has eigenvaluesthat are the coordinatesofthe quantum object.The operatorqo may thereforebeidentifiedwith the classicalcoordinateq. In contrast, themomentumoperatorp is definedby the derivativeoperator 7 5:whichis a derivative in respectto q.Thetwooperators q and p thus do not commute,and have no

1op rop

'commoneigenfunctions.6In quantum mechanics,we mayuse various representations.In addition to the coordinaterepresentation,we have the momentumrepresentation,where the momentumoperatoris simplyp, and coordi-nates are representedby derivative operators7 5:.Whatever the representation,the two operatorsdo notcommute.

Thefact that q and p do not commutemeansthat we^op rop

cannotdefine states ofa quantum objectfor whichboththe coordinateand the momentumtake on well-definedvalues.This is the rootofHeisenberg'suncertaintyreaction, which forcesus to abandon the \"naive realism\" ofclassicalphysics.We are ableto measurethe momentumorthe coordinateofa given particle,but we cannot say that

this particle has well-definedvalues for both its

momentum and its coordinates.This conclusionwas reachedsixty

yearsagoby HeisenbergandBorn,amongothers.Even so,discussionsabout the meaningofuncertainty relationsstill

go on, and somescientistshave not yet given up the hopeofrestoringthe traditional deterministicrealismofclassical mechanics.7This was oneofthe reasonsfor Einstein'sdissatisfaction with quantum theory.We should note that


Heisenberg'suncertainty principleis compatiblewith a

deterministictime-symmetricaldescriptionofnature (theSchrodingerequation).

What do we meanwhenwe say that a quantum systemis in a particular \"state\"? In classicalmechanics,the state is

a point in phasespace.Hereit is describedby a wavefunction whose evolution over time is expressedby the

Schrodingerequationih/2n9*F(f)/9f =Ho Wit).

This equationidentifiesthe time derivative ofthe wave

function *P with the actionof the Hamiltonian operatoron *F. It is not derived,but rather is assumedat the start,and can thus bevalidated only by experiment.It is thefundamental law ofnature in quantum theory*Notetheformal analogy with the Liouville equationin Chapter5,SectionIII,wherethe basicdifference is that L (theLiouville operator)acts on distributionfunctions p, while Hacts on wave functions.

We have already mentionedthat a wave function

corresponds to a probabilityamplitude.Theparallel that guidedErwin Schrodingerin formulating his equationwas that ofclassicaloptics.In contrast to the trajectory equations ofclassicalmechanics,the Schrodingerequationis a wave

equation.It is a partial differential equationbecauseinaddition to the time derivative, thereare alsoderivatives with

respectto coordinatesappearingin H (rememberingthat

in the coordinaterepresentation,the momentumoperatoris a derivative with respectto coordinates).But classicaland

quantum equationshave an essentialelementin common:They both correspondto a deterministicdescription.

There are various extensions of the Schrodinger equation and the relativistic

Dirac equation, but they are not necessary to this discussion.


Once*P is known at somearbitrary time tQ, togetherwith

appropriateboundary conditions(suchas *P \342\200\224> 0 at

infinite distances),we may calculate *F for arbitrary times in

the future aswellas in the past.In this sense,we reinstitute

the deterministicview ofclassicalmechanics,but it now

appliesto wave functions, and not trajectories.As in the classicalequationsofmotion,the Schrodinger

equation is time reversible.When we replacet by \342\200\224t,

the

equationremainsvalid. We only have to replace*P with its

complexconjugate*F*. As a consequence,if we observethe transition of*P from *Ft to

XV2at time t2 wheret2

> tpwe can also observe a transition from V|/*2 to \\)/*1 It is

worth remindingourselvesofArthur Stanley Eddington'sremark at an early stagein quantum mechanicsto the

effect that quantum probabilitiesare \"obtainedby

introducingtwo symmetricalsystemsofwaves traveling in opposite

directionsof time.\"8 Indeed,as we have seen, the

Schrodingerequation is a wave equation describingthe

evolution ofprobabilityamplitudes.Ifwe now take the

complexconjugateofthe Schrodingerequation,that is, if

we replace/ by \342\200\224 i, *F by *P* (supposingthat Hg is real),and t by \342\200\224t,

we return to Schrodinger'sequation.As stated

by Eddington,*P* may thereforebe viewed as a wave

function propagatinginto the past.Furthermore,as

mentioned in Chapter 1, probability proper is obtained

through multiplying *P by its complexconjugate*P* (that

is, l^l2).Since*P* may be interpretedas *P evolvingbackwardin time,the definition ofprobabilityimpliesthe

meetingoftwo times,onestemmingfrom the past and theotherthe future. In quantum theory, probabilitiesare thus

time symmetric.We now seethat in spite of their fundamental

differences, both classicaland quantum mechanicscorrespond


to laws ofnature that are deterministicand time reversible.No difference betweenpast and future appears in theseformulations.As we noted in Chapters 1and 2,this leadsto the time paradox.In quantum mechanics,it alsoleadstothe quantum paradox,due to the needto introducea du-alistic formulation of quantum theory. In both classical

theory and quantum theory, the Hamiltonianplays acentral role.In quantum theory, its eigenvaluesdeterminethe

energy levels, while,accordingto the Schrodingerequation, it also determinesthe time evolution of the wavefunction.

As in the precedingchapter, we shall concentrateonsystemsin whichthe HamiltonianH is the sum ofa freeHamiltonian H\342\200\236 and a termproducedby the interactionsA,V, wherebyH= HQ+ A,V. Thetime history ofsuchsystems can then bedescribedby transitions betweeneigen-states ofH0 inducedby theseinteractions.

As longas we remain in Hilbertspace,the eigenvaluesEa ofHare real (likethe Liouville operator,His also \"her-mitian,\" and hermitian operatorshave real eigenvaluesinHilbertspace).Theevolution ofthe wave function is a

superposition ofoscillatingterms such asexp(\342\200\224iE t). There

are, however, irreversibleprocessesin quantum mechanics,such as the quantum leapsin Bohr'stheory, whereexcitedatoms decaythrough the emissionofphotonsor unstable

particles(seeFigure6.1)or the decayofunstable particles.Cantheseprocessesbeincludedin Hilbertspacewithin

the framework of traditional quantum theory? Decayprocessesoccurin largesystems.Ifan excitedatom werekept in a cavity, the emittedelectronwould bounceback,and there would be no irreversibleprocess.As we have

seen,the time evolution ofthe wave function is described


Figure6.1Decayofan ExcitedAtomThe atom \"falls\" from the excitedstate to the ground state with

the emissionof a photon.

r- Excited state

*-L Photon

* Ground state

by a superposition,or sum, ofoscillatoryterms.With thelimit oflargesystems,this sum becomesan integral,and

acquiresnewproperties.In the caseofthe decayofexcitedatoms as describedby Figure6.1,the probabilities l^l2decayalmost exponentiallyover time.Herethe word almost

is essential:As longaswe remain in Hilbertspace,therearedeviations from the exponentialfor both very brieftimes

(the sameorderas the frequencyofoscillationsoftheelectron around the nucleus~ 10~16seconds)and very longtimes (for example,ten to onehundred times the lifetimeofan excitedstate, whichis ~ 10-9).However,in spiteofa

great numberofexperimentalstudies,no deviations from

exponentialbehavior have yet beendetected.This is

indeed fortunate, becauseif they did exist,it would raiseserious questions about the entire theoreticalsystem ofparticlephysics.

Supposethat we preparea beamofunstable particles,letit decay, and later on preparea secondbeam.Imaginethe

strange situation of the two beams prepared at differenttimes having different decaylaws.We couldthen distin-


guish betweenthem just as we do betweenolderand

younger individuals!This fantasy would be a violation ofthe principleof indistinguishability for elementaryparticles, which has led to someof the greatest successesofquantum theory* Thepreciseexponentialbehaviorobserved thus far showsthe inadequacyofHilbertspacedescription. We shall comeback to decay processesin thenext section,but at this point we should note that such

processesoughtnot to beconfusedwith processesdrivingthe system to equilibrium.The decay processasrepresented in Fig.6.1only transfers the energyofthe atom tothe photons.

IllAs we have seen,the main issuein quantum mechanicsisthe solutionofthe eigenvalue for the Hamiltonian.Thereare only a few quantum systemsin whichthis problemhasbeensolvedexactly.In orderto do so,we generallyneedtousea perturbationalapproach.As mentioned,we start with

a Hamiltonianin the form H= H0+ Xl^whereHQcorresponds to a Hamiltonian operatorfor which we havesolvedthe eigenvalue (the \"free\" Hamiltonian)and Kis a

perturbation coupledwith HQ,through the so-calledcoupling constantX. We assumethat we know the solutionofthe eigenvalue Hnu '0)= E ^u '0),and that we wish to solve0 On n n 'the equationHu =E u .The standard procedure,whichis1 n n n r

Schrodinger'sperturbational method,is to expandboth

*These include the explanation of superfluidity and the quantum theory ofsolid state.


the eigenvaluesand eigenfunctionsin terms ofpowersofthe couplingconstant X.

Theperturbationalapproachleadsto a recurrenceprocedure involving equationsfor eachorderin X. Thesolution oftheseequations implies the use of terms such as

1/(23(\302\260) \342\200\224 E(\302\260)),

whichbecomeill defined when the de-nominatorvanishes.This situation again correspondstoresonances,*and oncemorewe encounterthe divergenceproblemthat liesat the very centerofPoincares definitionofnonintegrablesystems.

However,thereis an essentialdifference here.We have

already introducedthe distinction betweendiscreteandcontinuousspectrums.In quantum mechanics,this

difference becomescrucial.In fact, when the spectrumis

discrete, it is generally possible to avoid the divergenceproblemthrough an appropriate choiceof theunperturbed Hamiltonian.^ Since all finite quantum systemshave a discretespectrum,we can then concludethat theyare integrable.

The situation changesdramatically when we turn to

largequantum systemsinvolving excitedatoms,scatteringsystems,and so on.In this case,the spectrumis

continuous,whichbrings us back to LPS.Theexampleofa

particle coupledwith a field, whichwe presentedin Chapter5,SectionV, also appliesto quantum systems.We then haveresonanceswheneverthe frequency00?associatedwith the

particleis equalto a frequencyC0fcassociatedwith the field.

*In quantum mechanics, to each energy E corresponds a frequency CO expressedas E = (/)/2lt)C0.

fin more technical terms, we first raise the degeneracy by an appropriatetransformation.


Theonly difference is that in quantum theory, frequenciesare associatedwith energies.The eigenvalueE corre-sponds to the frequency 2^00a,where h is Planck'sconstant.

Theexamplein Figure6.1,which correspondsto an

LPS,illustrates that we have resonanceeachtime the

energy difference betweenthe two levels is equal to the

energy ofthe photonthat is emitted.As in the caseofdeterministicchaosstudiedin Chapter

4, we can extendthe eigenvalue problemto singularfunctions outside Hilbertspace.The formal solution ofthe Schrodingerequationis ^(t) = U(r)*F(0),whereU(t) =e~'Ht; U(t) is the evolution operatorthat links the value ofthe wave function at time t to that at the initial time t =0.Both future and pastplay the samerole,sinceU(t:)U(t2)=

U(t1 + i2), whatever the sign of tt and tr This propertydefineswhat is calleda dynamical group.OutsideHilbertspace,the dynamical group splits into two semigroups.Thereare then two functions correspondingto the excitedatom:The first, <pv decays exponentially in the future

(<p^~e \"(/r), while the second,q>v decays in the past

(<pt~et/r)- Only oneofthesetwo semigroupsis realizedinnature.Inboth cases,thereis an exactexponentialdecay(incontrast to the approximateonedescribedin the precedingsection).This was the first such examplestudied, notablyby Arno Bohmand GeorgeSudarshan,who showed that

in orderto obtainexactexponentiallaws and avoid thedifficulties mentionedin SectionII,Hilbertspacemust beabandoned.9However,in theirapproach,the central

quantityremainsthe probabilityamplitude,and the basic

paradox of quantum mechanics(the collapseof the wave

function) is not solved.As already mentioned,the decayofexcitedatoms or unstable particlescorrespondsonly to a


transfer ofenergy from onesystem(the excitedatom) tothe other (the photon).The approach to equilibriumrequires a fundamental modificationofquantum theory.As

in classicalmechanics,we have to go from the individual

description,associatedwith wave functions, to thestatistical description,associatedwith ensembles.

IVIn the transition from the individual to the statistical

description, quantum theory introducescertainspecificfeatures as comparedto classicalmechanics.There,as we sawin Chapter5, the statisticaldistributionfunction is a

function ofboth the coordinatesand momenta.A trajectorycorrespondsto the delta function (seeChapter1,SectionIII).In quantum mechanics,the quantum state, asassociated with a wave function, is describedby a continuousfunction ofthe independentvariables.We can eithertakethe coordinatesas independentvariables and consider^(q), or we can take the momentaand consider (p).Heisenberg'suncertainty principleprevents us from takingboth.The definition ofa quantum state thereforeinvolves

only half ofthe variables that are used in the definition ofthe classicalstate.

Thequantum state *P representsa probabilityamplitude

for whichthe correspondingprobabilityp is given by the

productofthe amplitudes*\302\245(q)and

x\302\245*(q'),and is therefore

a function oftwo setsofvariables, q and q'orp and p\\ We

can thus write p(q,q')orp(p,p'),wherethe first expressioncorrespondsto the coordinaterepresentation,and thesecond to the momentumrepresentation,whichwill beespecially

useful to us.In quantum mechanics,the probabilitypis often calledthe \"density matrix\" (matrices,as studied


in algebra,alsohave two indices).We can easily write the

equationofevolution for p becausethe equationfor *P

(the Schrodingerequation)is already known.Theevolution equationfor p is the quantum Liouville equation,whoseexplicitform is ih

(-\302\243)

= Hp \342\200\224 pH,which is the\"commutator\" ofp with H.This showsthat whenp is afunction ofH,we have an equilibriumsituation. Thendpldt = 0,as Hcommuteswith a function ofitself.

Now that we have consideredthe distributionfunction

p, whichcorrespondsto a singlewave function, we canalsoconsidersituations in whichp correspondsto a\"mixture\" ofvarious wave functions.In both cases,theLiouville equationremainsthe same.

For integrablesystems, the statistical formulationintroduces no new features. Supposethat we know the

eigenfunctions<pa(p) and the eigenvaluesEa ofH.TheeigenfunctionsofL are then the productsq>a(p)q>a(p')andthe eigenvaluesthe differencesEa

\342\200\224

E\342\200\236. Theproblemsinvolved in deriving the spectralrepresentationsofHand Lare thus equivalent.

TheeigenvaluesEa\342\200\224

E\342\200\236 ofL corresponddirectlyto the

frequenciesmeasuredin spectroscopy,wherethe timeevolution ofthe distribution function p is a superpositionofoscillatingterms e~'^Ea

~\302\243/3)(. Again, thereis no approachto

equilibrium.Moreover,for thosesituations in whichwecan derive the eigenvaluefor the Hamiltonian,eigenfunctions ofL, such as <Pa(p)<Pa(p),correspondto zeroeigenvalues of the Liouville operator,Ea \342\200\224 Ea = 0, and are

thereforeinvariants ofmotion.As a result the system is

integrable(as is a systemofnoninteractingparticles),andcannotreachequilibrium.This is a form ofthe quantum

paradox.We can now seeclearly why it is not sufficient to extend

wave functions beyond Hilbertspace.Indeed,as indicated


in SectionIII,this leads to complexenergiesin the form

Ea = C0a\342\200\224 i y whereCOaisthe realpast and Ya the life span,

whichdescribethe decayofexcitedatoms orunstable

particles, but this still does not account for irreversible

processesassociatedwith the approach to equilibrium.Inspite ofthe complexelementin E , all diagonalelementsofp,whichare productssuchas q>a(p)(Pa(p'),wouldbeinvariants becausethe eigenvalue Ea \342\200\224 Ea again vanishes, andthe systemremainsintegrableand cannotapproachequilibrium.*

TheexperimentalbasisofBohr'stheory ofatoms andthe subsequentemergenceofquantum theory is basedonthe Ritz-Rydbergprinciple,accordingto whicheachfrequency v, as measuredin spectroscopy,is the differencebetween the two numbers Ea and

E\342\200\236,

whichrepresenttwo

quantum levels.This,however, can no longerbe true for

systemspresentingirreversibleprocessesthat lead thesystem to equilibrium.Quantum theory must thereforebefundamentally revised.

Historically,the roots ofmechanicsliein two branchesofphysics:the thermal equilibriumbetweenmatter andradiation that ledPlanckto introducehis famous constanth in 1900,and spectroscopy,which led from the Ritz-Rydbergprincipleto Bohr'satom, and finally, with Heisen-berg(1926),to quantum theory.However,the relationshipbetweenthese two domains has never beenelucidated.We seethat the Ritz-Rydbergprincipleis incompatiblewith the thermal approach to equilibriumdescribedbyPlanck'swork.Thus we needa new formulation makingthermal physicsand spectroscopycompatible.This can be

*Difficulties arise when Ea\342\200\224

E\342\200\236is replaced by Ea

\342\200\224

E*\342\200\236,where

E*\342\200\236is the

complex conjugate of E\342\200\236. Here, Ea-

E*a=-2fya^ 0, with no equilibrium state.


achieved at the level of probability distributions from

which we may derive observablefrequencies(includingtheir complexpart), but thesefrequenciesare no longerdifferencesin energy levels for the systemswe expecttoapproachequilibrium.We have to solve the quantum Liou-ville eigenvalue problemfor LPS in the contextofmoregeneralfunction spaces.As in classicalmechanics,this will

involve two basicingredients:delocalizeddistribution

functions, which lead to singularities, and Poincareresonances,which lead to new dynamical processes.As in classicaldynamics, there then appear new solutionsat the statisticallevel that cannot bereducedto the traditional wavefunction formalism ofquantum mechanics,and no longersatisfy

the Ritz-Rydbergprinciple.In this sense,we can truly

speakofa new formulation ofquantum theory.

V

With certainmodifications,we canfollow theprobabilistic formulation for classicalsystemsgiven in Chapter5.Theformal solutionofthe Liouvilleequationis i(dp/dt)=Lp,where in quantum theory Lp is the commutatorofthe Hamiltonianwith p (aswe have seen,Lp =Hp \342\200\224 pH).It can be written as eitherp(t) = e~iH'p(0)eHHtor p(t) =e~'Ltp(0).What is the differencebetweentheseequations?In the first formulation, it appearsthat we would have two

independent dynamic evolutions:oneassociatedwith e~tHt

and the otherwith e+,Ht, onemoving toward the \"future\"

and the other toward the \"past\" (as t is replacedby \342\200\224 t). If

this wereso,we couldexpectno time-symmetrybreaking,

and the statistical descriptionwould conservethe

timesymmetry of the Schrodingerequation.But this is

no longerthe casewhenwe includePoincareresonances,


which couplethe two timeevolutions (e~'Ht and e+,Hf).

Thereis now only oneindependenttimeevolution (timehas \"onedimension\.")In orderto study time-symmetrybreaking,we have to begin with the expressionp(t) =e~\"Lp(0),whichdescribesa singletimesequencein the Li-ouville space.In otherwords,we have to orderdynamicalevents accordingto a singletime sequence.*We can thendescribeinteractions,as we did for classicalmechanics,asa successionofevents separatedby free motion.Inclassical mechanics,theseevents changethe values ofthe wavevectork and the momentap.In Chapter5,we introducedvarious events leadingto the creationand destructionofcorrelations,wherewe sawthat the decisivefactor was the

appearance,for LPS,ofnewevents (the bubblesin Figure5.7) that couplecreationand destruction.As such, they

radically changeclassicaldynamicsbecausethey introducediffusion, break determinism,and destroytimesymmetry.We can alsoidentify the sameevents in quantummechanics. To do so,we needto introducevariables that play thesameroleas the wave vectork in classicaltheory'sFourierrepresentation.In classicalmechanics,we start with a

statistical formulation in which the distribution functions

p(q,p) are expressedas functions ofthe coordinatesq andthe momentap. We then proceedto the Fouriertransformation pk{p)involving the wave vector k and themomenta.

In quantum mechanics,we can follow a similar

procedure.10 We start with the densitymatrix p(p,p')in the mo-

*If this is not done, we have to be very careful. Feynman's well-known

statement that an electron propagates toward the future and a position moves toward

the past refers to time as it appears in the Schrodinger equation before ordering

dynamical events according to a single time sequence.


mentumrepresentation,whichis a function oftwo setsofvariables, p and p'.We then introducenew variables, k \342\200\224

p \342\200\224 p'and P= (p + p*)/2;we can now write, as in classicalmechanics,pk(P)-It can then be shown that k plays thesamerolein quantum mechanicsas the wave vectordoesinclassicalmechanics.(Forexample,in interactions,the sumofthe wave vectorsis conserved,that is, k.+ k = k'.+k' .)' ' j n j n I

Again as in classicalmechanics,Poincareresonancesintroduce new dynamical events that couplethe creationanddestructionofcorrelations,and thereforedescribequantum diffusive processes.

The formulation of classicaland quantum theory forLPS is moreor lessparallel.A minordifference appearsin

the roleofthe momentumP. Foreachevent, as introducedin Chapter5, the momentaofthe interactingparticlesarealtered.In quantum mechanics,we use the two variables k

and P, wherethe variable P replacesthe classicalmomentum. As thesevariables interact, the modificationofPinvolves Planck'sconstant h. Forh \342\200\224> 0,however, we comeback to the classicalmomentump. But this difference hasno important effect on formal development,and we shall

not attempt to describeit in further detail.In the previouschapter,we introduceda fundamental

difference betweentransitory and persistent interactions.Persistent interactions are especiallysignificant becausethey appearin all situations wherethermodynamicscan beapplied.As in classicalmechanics,the distributionfunction

p correspondingto persistentinteractionsis describedby

singular functions ofthe variable k. In classicaldynamics,as wellas classicaland quantum mechanics,persistentscattering is typical of the situations describedby statistical

mechanicsand cosmology.For example,in the

atmosphere, particles collidecontinuously,are scattered,and


then recollide.Persistentscatteringis describedby delocal-izeddistributionfunctions, whichare singular functions in

the wave vectorspace.As we saw in Chapter5, the latterforceus to go outsideHilbertspace.

By taking into accountdelocalizedsingular distributionfunctions and Poincareresonances,we obtain, as in

classical mechanics,complex,irreduciblespectralrepresentations for the Liouville operatorL. Again, as in classical

dynamics,irreversibility is associatedwith the appearanceofhigher- and higher-ordercorrelations.As in classicalmechanics,this leadsto new features in kinetictheory and

macroscopicphysics.Thebasicconclusionsofourformulation ofquantum mechanicsare as follows:

\342\200\242 Theeigenvaluesofthe Liouville operatorare no longerdifferencesbetween the eigenvaluesof the Hamilton-ian, which are obtained from the Schrodingerequation. Therefore,the Ritz-Rydbergprincipleis violated,whereby the systemsare no longerintegrableand the

approach to equilibriumis possible.\342\200\242 The quantum superpositionprincipleassociatedwith

the linearity ofthe Schrodingerequationis violated.\342\200\242 The eigenfunctionsof the Liouville operatorare not

expressedin terms of probability amplitudes or wave

functions, but rather in terms ofprobabilitiesproper.

Ourpredictionshave already beenverified in simplesituations wherewe can follow the collapseofwave functionsoutsideHilbertspace.11Moreover,they have ledto

interesting predictionsof the form ofspectrallines,and haveallowed us to accurately describethe approach toequilibrium. We regretthat we cannotgo into greaterdetailabout their specificapplications,but our objectivein this


bookis merely to provide a brieftour ofthe theoreticalbackground.

VIAt the Fifth Solvay Conferenceon Physicsthat tookplacein Brusselsin 1927,therewas an historicdebatebetweenEinsteinand Bohr.In the wordsofBohr:

To introduce the discussionon such points, I was askedat the

conferenceto give a report on the epistemologicalproblemsconfronting us in quantum physics and took the opportunity

to center upon the question of an appropriate terminologyand to stressthe viewpointof complementarity.The main

argument was that unambiguous communication of physicalevidencedemandsthat the experimentalarrangement as well

as the recordingof the observationsbe expressedin common

language, suitably refined by the vocabulary of classical

physics.12

But how can we describean apparatus in classicaltermsin a world dominatedby quantum laws? This is the weak

point in the so-calledCopenhageninterpretation.Nevertheless, there is an important elementof truth containedtherein.Measurementis a meansofcommunication.It is

becausewe are both \"actorsand spectators,\"to useBohr'swords, that we can learn somethingabout nature. Butcommunicationrequiresa commontime.Theexistenceofthis commontime is oneofthe basicconsequencesofourapproach.

Theapparatus that performsthe measurements,whethera physical constructor our own sensoryperception,must

follow the extendedlaws of dynamics, includingtime-


symmetry breaking.Theredo exist integrabletime-reversible systems,but we cannot observethem in isolation.As emphasizedby Bohr,we needan apparatus that breakstime symmetry.LPSblur this distinctionin that they alreadybreak time symmetry and therefore, in a sense,measurethemselves.We do not have to describean apparatus in

classical terms.Commontime emergesat the quantum levelfor LPSassociatedwith thermodynamicsystems.

Thesubjectiveaspectofquantum theory, whichattributed an unreasonableroleto the observer,deeplytroubledEinstein.To our way of thinking, through his

measurements the observerno longerplays someextravagant rolein the evolution of nature\342\200\224at least no moreso than in

classicalphysics.We all transform information receivedfrom the outsideworld into actionson a human scale,but

we are far from beingthe demiurge,aspostulatedbyquantum physics,who would be responsiblefor the transition

from nature'spotentiality to actuality.In this sense,our approachrestoressanity. It eliminates

the anthropocentricfeatures implicitin the traditional

formulation of quantum theory. Perhaps this would havemade quantum theory moreacceptableto Einstein.

Chapter7

OUR DIALOGUEWITH NATURE

I

Scienceis a dialoguebetweenmankindand nature, the

resultsofwhichhave beenunpredictable.At the

beginning of the twentieth century, who would havedreamedofunstable particles,an expandinguniverse, self-

organization, and dissipative structures?But what makesthis dialoguepossible?A time-reversibleworld would alsobean unknowableworld.Knowledgepresupposesthat theworld affectsus and our instruments,that thereis an

interaction betweenthe knowerand the known, and that this

interactioncreatesa difference betweenpast and future.

Becomingis the sine qua non ofscience,and indeed,ofknowledgeitself.

The attempt to understandnature remainsoneof the

153


basicobjectivesofWestern thought.It should not,however, be identified with the idea ofcontrol.The masterwho believeshe understandshis slavesbecausethey obeyhis orderswould beblind.When we turn to physics,ourexpectationsare obviously very different, but hereas well,Vladimir Nabokov'sconvictionrings true:\"What can becontrolledis never completelyreal; what is real can neverbecompletelycontrolled.\"1Theclassicalidealofscience,aworldwithout time,memory,and history, recallsthetotalitarian nightmares describedby Aldous Huxley, MilanKundera,and GeorgeOrwell.

In our recentbook,Entre le Temps et I'Eternite, IsabelleStengersand I wrote:

Perhapswe need to start by emphasizing the almost

inconceivable characterof dynamic reversibility. The question oftime\342\200\224of what its flow preserves,createsand destroys\342\200\224has

always beenat the center of human concerns.Muchspeculation has calledthe ideaof novelty into questionand affirmed

the inexorablelinkage between causeand effect.Many forms

of mystical teachinghave deniedthe reality of this changingand uncertain world, and definedan idealexistence

permitting escapefrom life'safflictions. We know how important

the ideaof cyclicaltime was in antiquity. But, like the rhythm

of the seasonsor the generationsof man, this eternal return

to the point of origin is itselfmarked by the arrow of time.No speculation,no teachinghas everaffirmed an equivalencebetweenwhat is doneand what is undone:between a plant

that sprouts, flowers and dies,and a plant that resuscitates,

grows younger and returns to its original seed;between a

man who grows olderand learns, and one who becomesa

child, then an embryo,then a cell.2

Our Dialoguewith Nature 155

In Chapter1,we alludedto Epicurus'dilemmaand theatomisticapproachofthe ancients.Today,the situation has

changedsignificandy in the sensethat the morewe knowabout our universe, the moredifficult it becomestobelieve in determinism.We live in an evolutionary universewhoseroots,whichliein the fundamental laws ofphysics,we are now able to identify through the conceptofinstability

associatedwith deterministicchaosand nonintegra-bility. Chance,or probability, is no longera convenientway ofacceptingignorance,but rather part ofa new,extended rationality. As we have seen,for thesesystems,the

equivalenceis brokenbetweenthe individual description(trajectoriesand wave functions) and the statistical

description (in termsofensembles).At the statisticallevel, we can

incorporateinstability. The laws of nature, which nolongerdeal with certitudesbut possibilities,overrulethe

age-olddichotomy betweenbeingand becoming.Theydescribea world ofirregular, chaoticmotionsmoreakin

to the imageofthe ancientatomists than to the world ofregular Newtonianorbits.This disorderconstitutes the

very foundation ofthe macroscopicsystemsto whichwe

apply an evolutionary descriptionassociatedwith the

second law, the law ofincreasingentropy.We have considereddeterministicchaos,and we have

discussedthe roleofPoincareresonancesin both classicaland quantum mechanics.We have seenthat we needtwoconditionsto obtain our statistical formulations, whichgobeyond the usual onesfor classicaland quantummechanics: the existenceofPoincareresonances,which lead tonew diffusion-type processesthat can beincorporatedintothe statistical description,and extendedpersistentinteractions describedby delocalizeddistribution functions.


Theseconditionslead to a moregeneraldefinition ofchaos.As in the caseofdeterministicchaos,we thenobtain newsolutionsfor the statistical equations that cannotbeexpressedin terms oftrajectoriesorwave functions.Iftheseconditionsare not satisfied, we return to the usualformulations.This is the casein many simple examples,suchas two-body motion(for instance,the sun and earth),and typical scatteringexperiments,wherebeforeand after

scattering,the particlesare free.Theseexamples,however,

correspondto idealizations.Thesun and earth are part ofthe many-body planetary system;scattered particleswill

eventually meetother particles,and are thereforeneverfree.

It is only by isolatinga certainnumberofparticlesand

studying their dynamicsthat we obtain the traditionalformulations. Conversely,time-symmetrybreakingis a globalproperty encompassingHamiltoniandynamical systemsasa whole.In the chaoticmapsstudied in Chapters3 and 4,irreversibility occurseven in systemswith few degreesoffreedomdue to the simplificationsused to describethe

equationsofmotion.A remarkablefeature ofour approach is its application

to both classicaland quantum systems.All othertheoretical proposalsthat we are aware ofattempt to eliminatethe

quantum paradoxthrough an exclusivelyquantummechanism. On the contrary, in our view, the quantum paradoxis only oneaspectofthe time paradox.In the Copenhageninterpretation,the needto introducetwo different typesoftime evolution is engenderedby the measurementprocess.Accordingto Bohrhimself,\"Every atomicphenomenonisclosedin the sensethat its observationis basedona

recordingobtainedby means of suitable amplification devices

with irreversiblefunctions,such as permanentmarkson a


photographicplate.\"3It was this measurementproblemthat ledto the needfor a collapseof the wave function,and forcedus to introducea secondtype of dynamicalevolution into quantum mechanics.It is thereforenot

surprising that the time paradoxand quantum paradoxare so

closelylinked.In solvingthe former,we alsosolve thelatter. As we have seenfor LPS,quantum dynamicscan onlybe describedat the statistical level.Moreover,to learnsomethingabout quantum processes,we again needanLPS actingas an apparatus.It is thus the secondlaw ofquantum time evolution,which includesirreversibility,that becomesthe generalone.

As statedby Alastair Rae,\"A pure quantum process(described by the Schrodingerequation) occursonly in oneor moreparametersthat have becomedetachedfrom therest ofthe universe, and perhapseven from space-timeitself, and leave no traceoftheir behavioron the rest oftheuniverse until a measurementinteractiontakes place.\"4Whatever the process,at somepoint irreversibility has tocomeinto the picture.An almost identicalstatement couldhave beenmade regardingclassicalmechanics!

It has often beensaid that in orderto make progressin

these difficult areas, we needthe inspiration of a truly

crazy idea.Heisenbergwas fond ofaskingwhat thedifference is betweenan abstract painter and a goodtheoreticalphysicist.Inhis opinion,an abstract painterneedsto bejustas originalas a goodtheoreticalphysicistneedsto beconservative.5 We have tried to follow Heisenberg'sadvice.Our lineofreasoningin this bookis certainly lessradicalthan mostotherattempts madein the past to solve the timeor quantum paradox.Perhapsour craziestidea is that

trajectories are not primary objects,but rather the result ofa

superpositionofplane waves. Poincareresonancesdestroy


the coherenceofthesesuperpositions,and leadto anirreducible statisticaldescription.Oncethis is understood,the

generalizationto quantum mechanismsbecomeseasy.

IINumerousreferenceshave beenmadein this textto the

thermodynamic limit, which is defined by the limit N(numberofparticles)\342\200\224>

\302\260\302\260,

andvolume V \342\200\224>

\302\260\302\260,wherethe

concentrationN/Vremainsfinite. This limit simplymeansthat when the numberofparticlesN is sufficiendy large,termssuchas 1/N can beignored.This is true for the usual

thermodynamicsystemswhereN is typically on the orderof1023.However,thereare no systemsthat containaninfinite numberofparticles.

Theuniverse itself is highly heterogeneousand far from

equilibrium.This prevents systemsfrom reachinga state ofequilibrium.Forexample,the flow ofenergy that

originates in the irreversiblenuclearreactionswithin the sunmaintains our ecosystem far from equilibrium,and hasthus made it possiblefor life to developon earth.As wesaw in Chapter2,nonequilibriumleads to new collectiveeffects and to a newcoherence.It is interestingthat theseare precisely the consequencesof the dynamical theorypresentedin Chapters5 and 6.

Thereare two typesofeffects producedbynonequilibrium. If, as in the Benardinstability, we heat a liquid from

below, we producecollectiveflows ofmolecules.Whenwe stop the heatingprocess,the flows disintegrateand

return to the usual thermal motion.In chemistry, the

situation is different; irreversibility leads to the formation ofmoleculesthat cannotbe producedin near-equilibriumconditions.In this sense,irreversibility is inscribedin mat-


ter.This is likely to be the originofself-replicatingbio-molecules.While we shall not pursue this question here,letus merelynote that moleculesofcomparablecomplexity

can indeedbe produced,at least through computersimulations,in nonequilibriumconditions.6In the nextchapter,whichdiscussescosmology,we arguethat matteritselfis the result ofirreversibleprocesses.

In nonrelativistic physics,whetherclassicalorquantum,time is universal, but the flow oftime as associatedwith

irreversible processesis not.It is to the fascinatingimplications ofthis distinctionthat we shall now turn.

IllLet us first considera chemicalmodel.Ifwe start at time

tQ

with two identicalsamplesofmixturesoftwo gases,suchascarbonmonoxide(CO)and oxygen(02),a chemicalreaction leadingto carbondioxide(C02)can becatalyzed bymetallic surfaces.In oneofthe samples,we introducesucha catalyst, and in the other,we do not.Ifwe comparethe

two samplesat a later time t, their compositionwill

therefore bequite different. Theentropyproducedin the

sample containingthe catalytic surface will bemuchgreaterasa result ofthe chemicalreaction.If we associatetheproduction ofentropyto the flow oftime,time itselfwill

appear to vary betweenthe two samples.This observationis

in agreementwith our dynamical description.Theflow oftime is rootedin Poincareresonancesthat dependon the

Hamiltonian,that is, on dynamics.Theintroductionofa

catalyst changesthe dynamics, and thereforealters the

microscopicdescription.In anotherexample,gravitationagain changesthe Hamiltonian,and thereforetheresonances. We then have a kind ofnonrelativistic analogueof


the twin paradoxofrelativity, whichwe shallcomeback toin Chapter8.Forthe moment,supposethat we send twotwins (who are simply two LPS) into space,leaving theearth at

tQand comingbackat tx (seeFigure7.1).Before

their return,one twin goesthrough a gravitational field,and the otherdoesnot.Theentropyproduced(as a resultofPoincareresonances)will be different, and our twins

will comeback with different \"ages,\" leadingus to thebasicconclusionthat the flow oftime,even in aNewtonian universe, may have different effects accordingto the

processesconsidered.Our conclusionis in stark contrastwith the Newtonianview, whichwas basedon a universalflow oftime.But what can a flow oftime meanin a

description ofnature in whichpast and future play the samerole?It is irreversibility that leadsto a flow oftime.Timeevolution is no longerdescribedby groups wherepast and

Figure7.1Effect ofa Gravitational Fieldon the Flowof Time

h

Gravitational field /

<0


future play the samerole,but rather by semigroups that

include the directionoftime.When we introducea timeassociatedwith the productionofentropy(seeChapter2),asthe sign ofentropy productionis positive, entropictime

always points in the samedirection.This is the casein thetwo examplesmentionedabove even thoughentropictimedoesnot keeppacewith clocktime.

We couldintroducean \"average\" entropictime for theentireuniverse, but this would not have a great deal ofmeaningbecauseof the heterogeneityof nature.Irreversible geologicalprocesseshave a time scaledistinctfrom

thoseofbiologicalprocesses.Even moreimportant, thereexistsa multiplicity ofevolutions,whichare particularlyevident in the field of biology.As stated by StephenJ.Gould,bacteriahave remainedbasicallythe samesincethePrecambrianera,while otherspecieshave evolved

dramatically,often over short time scales.7It would thereforebea

mistake to considera simple one-dimensionalevolution.Sometwo hundred million years ago, certain reptilesstarted to fly, while othersremainedon earth.At a later

stage,certainmammals returnedto the sea,while othersremainedon land.Similarly, certainapesevolved into hu-manoids,while others did not.

At the conclusionof this chapter, it is appropriate tociteGould'sdefinition ofthe historicalcharacteroflife:

Tounderstand the events and generalitiesof life'spathway, we

must go beyondprinciplesof evolutionary theory to a pale-ontologicalexamination of the contingent pattern of life's

history on our planet\342\200\224the singleactualizedversionamongmillions of plausible alternatives that happenednot to occur.Such a view of life'shistory is highly contrary both to con-


ventional deterministicmodelsof Western scienceand to the

deepestsocialtraditions and psychologicalhopesof Western

cultures for a history culminating in humans as life'shighest

expressionand intended planetary steward.8

We are in a world of multiple fluctuations, someofwhichhave evolved, while othershave regressed.This is in

completeaccordwith the resultsoffar-from-equilibriumthermodynamicsobtainedin Chapter2.But we can now

go even farther. Thesefluctuations are the macroscopicmanifestations of fundamental propertiesof fluctuations

arising on the microscopiclevel of unstable dynamicalsystems.The difficulties emphasizedby Gould are nolongerpresent in our statistical formulation ofthe laws ofnature. Irreversibility, and thereforethe flow oftime,starts

at the dynamical level.It is amplified at the macroscopiclevel, then at the level of life, and finally at the level ofhuman activity. What drove these transitions from onelevel to the nextremainslargely unknown, but at leastwehave achieved a noncontradictory descriptionof naturerootedin dynamical instability. Thedescriptionsofnatureas presentedby biology and physicsnow begin to

converge.

Why doesa commonfuture exist at all? Why is thearrow oftime always pointedin the samedirection?Thiscan only mean that our universe forms a whole.It hasa commonorigin that already impliedtime-symmetrybreaking.Herewe encountercosmologicalproblems.Indealingwith them,we must embracegravity and enterthe

world ofEinstein'stheory ofrelativity.

Chapter8

DOESTIME PRECEDEEXISTENCE?

I

Severalyears ago, I delivereda physicscolloquiumat

LomonosoffUniversity in Moscow.Afterwards,Professor Ivanenko, oneofthe mostrespectedRussianphysicists,askedme to write a short inscriptionon a particular wall

wherethere werealready many sentimentsexpressedbyfamous scientistssuchas Diracand Bohr.I vaguely rememberthe sentencechosenby Dirac,which was somethinglike:\"Beauty and truth gotogetherin theoreticalphysics.\" After

somehesitation, I wrote:\"Timeprecedesexistence.\"Formany physicists,the acceptanceofthe bigbang

theory as the originofour universe means that time must

have a beginning,and perhapsan end.It seemsmorelikelyto methat the birth ofour universe was only oneevent in

the history of the entirecosmos,and that we therefore

163


have to ascribeto that so-called\"meta-universe\" a time

priorto the birth ofour own.We know that we are living in an expandinguniverse.

Thestandard model, whichdominatesthe field ofcosmology today, assertsthat if we wereto gobackwardin time,we would arrive at a singularity, a point that containsthe

totality ofthe energyand matter in the universe.However,the modeldoesnot enableus to describethis singularitybecausethe laws ofphysicscannot be applied to a pointcorrespondingto an infinite densityofmatter and energy.It is no wonderthat JohnArchibald Wheelerspeaksofthe

big bang as confronting us \"with the greatest crisis in

physics.\"1Canwe acceptthe bigbang as a real event, andhow is it possibleto reconcilethis event with laws ofnature that are time reversibleand deterministic?We comebackto the problemsofmeasurementand irreversibility,but now in the cosmologicalcontext.

Sincethe discoveryofthe bigbang,the scientific

community has reactedto the strangenature ofthis singularityby attempting to eliminatethe big bang entirely (seethe

steady-statetheory in SectionsI and III),orconsideringitas a kind of\"illusion\" arisingfrom the use ofan incorrectconceptoftime (seeHawking'simaginary time in SectionII),oreven viewing it as a sort ofmiracleakin to thebiblical descriptionin Genesis.

As we have already noted,it is impossibleto discuss

cosmology today without referringto the theory ofrelativity,

\"the most beautiful theory in physics,\" accordingto thecelebratedtextbookby Lev Davidovich Landau and

Evgeny Mikhailovich Lifschitz.2 In Newtonian physics,even when extendedby quantum theory, spaceand timeare given onceand for all. Moreover,there is a universaltime commonto all observers.In relativity, this is no

DoesTime PrecedeExistence? 165

longerthe case;spaceand time are now part ofthe picture.What consequencesdoesthis have for ourowninterpretation? In his recentbook,About Time, Paul C.W Daviescommentson the impactofrelativity, \"Thevery divisionoftime into past,presentand future seemsto bephysicallymeaningless.\"3He repeatsHermannMinkowski'sfamousstatement:\"Henceforthspaceby itself, and time by itself,are doomedto fade away into mereshadows.\"4

We have already alluded to Einstein'scelebratedassertion that \"for us convincedphysicists,the distinctionbetween past, present and future is an illusion, although a

persistentone.\"5At the end ofhis life, however, Einsteinseemsto have changedhis mind.In 1949,he was offered acollectionof essaysthat includeda contributionby the

great mathematician Kurt Godel,who had taken quiteseriously Einstein'sstatement that time as irreversibility was

only an illusion.When he presentedEinstein with a cos-mologicalmodelin whichit was possibleto return to one'sown past, Einstein was not enthusiastic.In his answer to

Godel,he wrote that he couldnot believethat he could\"telegraphback to his own past.\" Heeven added that this

impossibilityshouldleadphysiciststo reconsidertheproblem ofirreversibility.6That is preciselywhat we have

attempted to do.In any case,we wish to emphasizethat the revolution

brought about by relativity in no way affects our previousconclusions.Irreversibility, or the flow oftime,remainsas

\"real\" as in nonrelativistic physics.Perhapswe couldarguethat irreversibility playsan even greaterrolewhenwe go to

higherand higherenergies.It has beensuggested,mainly

by Hawking, that in the early universe, spaceand time losetheir distinction,and time becomesfully \"spatialized.\"Butno oneto ourknowledgehas deviseda mechanismfor this


spatialization oftime,ora meansby whichspaceand timecouldemergefrom what is often describedas a \"foamymess.\"

Our positionis quite different from thosestated abovein that we considerthe bigbang an irreversibleprocessparexcellence.We suggest that there would have been anirreversiblephasetransition from a preuniversethat we callthe quantum vacuum. This irreversibility would result froman instability in the preuniverseinducedby the interactionsof gravitation and matter. Clearly we are at the edgeofpositiveknowledge,even dangerouslycloseto sciencefiction.

Nevertheless,we arguethat irreversibleprocessesassociated with dynamical processeshave probablyplayedadecisive rolein the birth ofouruniverse.From ourperspective,time is eternal.We have an age,ourcivilization has an age,our universe has an age, but time itself has neither a

beginningnor an end. This brings closertwo of thetraditional views of cosmology:the steady-statetheoryintroducedby HermannBondi,ThomasGold,and Fred

Hoyle, which may apply morepreciselyto the unstablemediumthat generatesour universe (the meta-orpreuniverse), and the standard bigbang approach.7

Again, speculativeelementscannot be avoided,but wefind it interestingthat views emphasizingthe roleoftimeand irreversibility can be formulated morepreciselythan

before,even thoughthe ultimate truth is still far beyondour reach.We agreeentirely with the Indian cosmologistJayant Vishnu Narlikar, who wrote, \"Astrophysicistsoftoday who hold the view that the 'ultimatecosmologicalproblem'has beenmoreor lesssolvedmay wellbe in for a

few surprisesbeforethis century is out.\"8

Does Time PrecedeExistence? 167

IIAs we proceedwith our investigation, let us considerEinstein's specialrelativity. This theory takesas its starting pointtwo inertial observersmoving at a constant velocity with

respectto oneanother.In prerelativistic, Galilean physics,it was acceptedthat the distancebetweenthe two

observers, / j2=

(x2\342\200\224xl)2+ (y2~yl)2 + (z2~zl)2,would remain

the same as the difference betweenthe two instants, (t2\342\200\224

tj)2. Spatial distance was defined in terms ofEuclideangeometry.This,however, ledto different values ofthe

velocity oflight c in the vacuum as measuredby the twoobservers. In accordancewith our experience,if we assumethat both observersmeasurethe samevalue ofthe velocityoflight, we must introduce(as did Lorentz,Poincare,and

Einstein)the spatiotemporalinterval, s22=

<?{tx\342\200\22412)2\342\200\224l22.

It is

this interval that is conservedwhenwe move from oneinertial observerto the other.In contrast to Euclideangeometry,

we now have the Minkowskispace-timeinterval. Thetransition from onecoordinatesystem,x,y, z, t, to another,x\\ y\\ z\\ t\\ is the famous Lorentztransformation that

combines spaceand time.At no point,however, is thedistinction betweenspaceand time lost; in the spatiotemporalinterval, the minus sign indicatesspacedimensions,andthe plussignindicatestime.

This situation is often illustrated by the spatiotemporaldiagram representedin Figure8.1.On oneaxis there is

time t, and on the othera singlegeometricalcoordinatex.In relativity, the velocity of light c in the vacuum is themaximumspeedat whichsignalscan be transmitted.We

can thereforedistinguishamongdifferent regionsin the

diagram.


In this diagramthe observeris situatedat O.His future

is includedin the \"cone\"BOA, and his past in the coneA'O'B'.Theseconesare determinedby the velocity oflight c in that the velocitiesinside themare smallerthan c,and outside them are greater,and thereforeimpossibletorealize.In this diagram,the event C is simultaneouswith

O, while event DprecedesO.But this conclusionis purelyconventional becausea Lorentz transformation wouldrotate the axis t, x, in whichcaseDmight appearassimultaneous with O, and C posteriorto O. Simultaneity is

modified by the Lorentz transformation, but the coneoflight is not. The directionof time is thus invariant. Theproblemofascertainingwhetherornot the laws ofnatureare time symmetricremainsessentiallythe samein

relativityas in prerelativistic physics,but now this questionis

even morepertinent.At best, Oknows all the events that

occurredin his past, that is, in the coneA 'OB'.As

represented in Figure8.2,events starting in Cor D will reachhim only at later times, tx

and t2, even if they are associated

Figure8.1DistinctionBetweenFuture and Past in SpecialRelativity

/\342\226\240\342\226\240

r Absolute ^

C\302\273- x

x D

A' \"ua\"lult B'past


Figure8.2Events starting at C and D will reach the observerO at future

times fj and t2.

with signals traveling at the velocity oflight.As a result,Ocan collectonly limited data. In an amusing analogy with

deterministicchaosmadeby Baidyanath Misraand IoannisAntoniou, it is said that a relativistic observerhas only a

finite window on the outside world, and herealso a

deterministic descriptioncorrespondsto an overidealization.9This gives us yet anotherreasonto proceedto a statistical

description.Thereare, ofcourse,most interestingnew effects

introduced by relativity, such as the famous twin paradox,whereonetwin remainson earth at point x=0,while theother leaves in a spaceshipthat changesdirectionat

tQ (inthe coordinatesystem in which O is at rest), and comesback to earth at 2t(). The time interval, as measuredby the

moving twin, is greaterthan 2t0.This is Einstein'sremarkable time dilation prediction,whichhas beenverified by


usingunstable particles.The lifetime ofthesetwins

therefore dependson the path as predictedby relativity. InChapter7, we stated that the flow oftime dependson a

history ofevents, but Newtonian time is universal and

independent ofhistory. Now time itselfbecomeshistorydependent.

In his seminalbook,TheTheory ofSpace,Timeand

Gravitation,Vladimir A. Fockemphasizesthat we have to be

extremelycareful when discussingthe twin paradoxinasmuch as the effect ofaccelerationon the clockin the

moving spaceship is neglected.10He shows that when weconsidera moredetailedmodelin whichaccelerationis dueto a gravitational field describedby generalrelativity,

different results are obtained.The signoftime dilation can evenbe changed.Thesepredictionsofgeneralrelativity shouldleadto fascinatingnew experimentsto test their validity.

In his BriefHistory ofTime, Hawking introducesimaginary time,T = it, where all four dimensionsare \"spa-tialized\" in the Minkowski spatiotemporal interval.11Accordingto Hawking, real time may wellbe this

imaginary time,whereby the mathematical formula for theLorentz interval becomessymmetric.Hawking'sproposition doesindeedgo beyond relativity, but it is only onemoreattempt to negatethe reality of time in describingthe universe as a static, geometricalstructure,incontradiction to the rolethat the flow oftime plays at all levels ofobservation.

Let us now comeback to the cruxofourargument andconsiderthe effect ofrelativity on the systemsdescribedbyclassicalHamiltonian dynamics or quantum mechanics.Dirac,and otherswho cameafter him, showed how to

combinethe requirementsof specialrelativity with a

Hamiltoniandescription.12Relativity dictatesthat the laws


Figure8.3TheTwin ParadoxObserverO'is in motion in relation to observerO.

of physics remain the same for all inertial systems.InChapters5 and 6,we assumedimplicitly that the systemsas

a wholeare at rest.But accordingto relativity, a similar

description is valid whetheror not the systemas a wholeis

moving at uniform velocity with respectto someobserver.We have seenthat Poincareresonancesdestroythedynamical group in which past and future play the same role,wherebywe obtain semigroupsthat break time symmetry.Inprerelativistic physics,the groupsand semigroupsmaintain the distance

lf2invariant. In relativistic theory, we can

introduceas wellboth groupsand semigroupswhichleaveinvariant the Minkowskiinterval. Unfortunately, the proofis tootechnicalto begiven here.In any case,this

conclusion shows that the Minkowskispace-timeinterval is noway in contradictionto irreversibleprocesses.It is not truethat relativity impliesthe spatialization oftime.As stated


by Minkowski,spaceand time are no longerindependententities,but this doesnot precludethe existenceof anarrow oftime.

Sucha conclusioncouldbeanticipated.Iftime-symmetry breakingoccursin oneinertial frame, by the verydefinition ofrelativity, it has to appearin all inertial referencesystems.The theory ofirreversibleprocessesis thus quitesimilar (apart from certainformal changes)in both non-relativistic and relativistic systems.Thereis, however, onebasicdifference:Interactionsare no longerinstantaneous;rather, they propagateat the velocity oflight.Forchargedparticleswithin the framework ofquantum theory, for

example, interactionsare transmitted by photons.This leadsto additionalirreversibleprocessessuch as radiation

damping,whichresultsfrom the emissionofphotonsby

particles. In more generalterms,in relativistic physics weconsiderparticlesas associatedwith fields (the photonsarethe particlesassociatedwith the electromagneticfield),and

irreversibility resultsfrom the interactionofthesefields.Until now we have consideredthe Minkowskispace-

time interval as it correspondsto specialrelativity. In orderto completeour discussionofcosmology,we have toinclude gravitation, which first requiresa generalizationofthe space-timeinterval.

IllLet us first return to the questionofthe bigbang.As wementionedabove, by following our expandinguniversebackwardin time,we cometo a singularity in whichdensity, temperature,and curvature all becomeinfinite. Fromthe rate ofrecessionofthe galaxiesas observedtoday, we


can estimate that the birth of the universe occurredapproximately fifteen billionyears ago.This periodoftimethat separatesus from the bigbang is surprisinglyshort.To

expressit in years,we use the rotation of the earth as aclock.Fifteen billionrevolutions is indeeda small numberif we rememberthat in the hydrogenatom, the electronrotates some10,000billiontimespersecond!

Whatever the time scale,the existenceofa primordialevent at the originofour universe is certainly oneofthemost extraordinary suggestionssciencehas ever made.Physicsdealsonly with classesofphenomena,and the bigbangdoesnot seemto belongto any ofthese.At first view,it appearsto have no parallel elsewherein physics.

Many scientistshave beenwilling to explainthis

singularityin terms ofthe \"hand ofGod,\"or the triumph of

the biblicalstory ofcreation,whereby sciencewouldreconstruct the existenceofan act that transcendsphysicalrationality. Othershave triedto avoid what they seeas a

disquietingsituation.Oneremarkableattempt in this senseis the steady-stateuniverse proposedby Bondi,Gold,and

Hoyle.13This modelis basedon the perfect cosmologicalprinciple:Notonly is thereno privilegedplacein the

universe, but there is also no privilegedtime.Accordingtothis principle,every observer,in the past and in the future,is able to attribute to the universe the samevalues ofparameters suchas temperatureand matter density.Thesteady-state universe is characterizedby an exponentialexpansioncompensatedby a permanentcreationofmatter. Thesynchronization betweenexpansionand creationmaintains aconstant density ofmatter-energy,and thus leads to the

imageofan eternaluniverse in a state ofcontinouscreation. In spiteofits appeal,the steady-statemodelimplies


certainmajordifficulties. Inparticular, in orderto maintainthe steadystate, we needa fine-tuningbetweencosmolog-ical evolution (the expansionofthe universe) and

microscopicevents (the creationof matter).As long as no

mechanismfor this is proposed,the hypothesisofcompensation betweenexpansionand creationis highlyquestionable.

It was an experimentalresult that ledthe great majorityofcosmologiststo rejectthe steady-statemodelin favor ofthe bigbang,whichis now consideredthe standard model.This occurredin 1965,when Arno Penziasand RobertWilson identified the now-famous fossil radiation at 2.7K.14Theexistenceofsuch radiation had beenpredictedas

early as 1948by Ralph A. Alpher and RobertHerman,who reasonedthat if the universe was much hotter anddenserin the past than it is today, then it must have been\"opaque,\"with photonspossessingsufficient energy tointeract strongly with matter. It can be shown that at a

temperature of approximately3,000K, the equilibriumbetweenmatter and light is destroyed,and ouruniversebecomestransparent as radiation is \"detached\"from matter.Theonly subsequentchangein the propertiesofthephotons that form the thermal radiation is the changein their

wavelength, whichincreaseswith the sizeofthe universe.

Alpher and Hermanwerethus able to predict that if the

photonsindeedformeda blackbody radiation at 3,000Kat the time when their equilibriumwith matter was

destroyed (that is,some300,000years after the \"origin\,")the

temperatureofthis radiation shouldcorrespondtoday to a

temperatureofabout 3 K.This was a landmarkpredictionthat anticipatedoneofthe greatestexperimentalfindingsofthis century.15

Thestandard modelis very muchat the coreofpresent-


day cosmology,and scientistsgenerallyacceptthat it leadsto a correctdescriptionofthe universe starting onesecondafter the bigbang singularity. But the state ofthe universe

duringits first secondoflife still remainsan openquestion.Why is theresomethingrather than nothing? This

appears to be the ultimate questionbeyond the rangeofpositive knowledge.However,this questioncan beformulatedin physicalterms,and therebylinked to the problemofinstability and time.Onesuch formulation that has becomequite popular today definesthe birth ofour universe as a

free lunch. EdwardTryon presentedthis idea in 1973,but it

seemsto hark back to PascualJordan.In Tryon'sview, ouruniverse can bedescribedas having two forms ofenergy:onerelatedto attractive gravitational forces,whichis

negative, and the otherrelated to mass accordingto Einstein'scelebratedformula E= mc2, whichis positive.16

It is tempting to speculatethat the total energy of theuniverse couldbezero,as is the energyofan emptyuniverse. Thebigbangwould thus beassociatedwith

fluctuations in the vacuum conservingthe energy.This is truly an

appealingidea.Thegenerationofnonequilibriumstructures (such as Benard vortices or chemicaloscillations),whereenergyis conserved,alsocorrespondsto a free lunch,for the priceofnonequilibriumstructuresis entropy, andnot energy.In this context,can we specify the originofnegative gravitational energy and its transformation into

positive matter-energy?This is the question that we shall

now address.

IV

PerhapsEinstein'smostprofoundcontributionwas toassociate gravitation with the curvature ofspace-time.As we


have seenin specialrelativity, the Minkowskispace-timeinterval is ds2= c^di2 \342\200\224 dp. In generalrelativity, the space-time interval becomesds2=

~Lg vdx^dxy, where[i, v take onfour values: 0 (time),and 1,2,3 (space).The ten distinctfunctions obtained (given that g v

=gv ) characterize

space-time,or Riemanniangeometry.A simple examplethat illustrates Riemanniangeometry is a sphereconsidered as a curved two-dimensionalspace.

In the Newtonian view, space-timeis given onceandfor all, independentofthe matter it contains.Nowweunderstand, thanks to the Einsteinian revolution, that theconnectionbetweenspace-timeand matter is expressedbyEinstein's fundamental field equations,which relate two

objects:On the onehand, we have an expressionthat

describes the curvature ofspace-timein terms ofthe g andits derivatives with respectto spaceand time,and on theother an expressionthat defines the material content in

terms ofits matter-energycontentand pressure.Thismaterial contentis the sourceofthe curvature ofspace-time.Einsteinappliedhis equationsto the universe as a wholeas

early as 1917,and in so doing,set the courseofmoderncosmology.To achieve this application, he developedatimeless static modelin accordwith his philosophicalviews.BaruchSpinozawas Einstein'sfavorite philosopher,and we can recognizehis spirit in the choiceofthe model.

Then camea successionofsurprises.AlexanderFried-mann and Georges-HenriLemaitreproved that Einstein'suniverse was so unstable, the smallestfluctuation would

destroy it.17On the experimentalside, Edwin PowellHubbleand his colleaguesdiscoveredthe expansionofouruniverse.18Then in 1965camethe observationofresidualblack body radiation, which led to the present standard

cosmologicalmodel.


In orderto go from the basicequationsofgeneralrelativity to the field ofcosmology,we have to introducesimplifying assumptions.Thestandard modelassociatedwith

AlexanderFriedmann,Georges-HenriLemaitre,HowardRobertson,and Arthur Walker is founded on the cosmo-logicalprinciplethat the universe,whenviewedon a largescale,may beconsideredhomogeneousand isotropic.Themetricsthus take on the far simplerform ds2 = <?dt2 -R(t)2dP (the so-calledFriedmanninterval). Thisexpression differs from Minkowskispace-timein two respects:dp is a spatial elementthat correspondsto eithera zero-spacecurvature (as in the Minkowskispace)or to apositive ornegative curvature (as in a sphereorhyperboloid).R(t), which is usually calledthe radius of the universe,correspondsto the limit ofastronomicalobservationsat

time t. Einstein'sequations relate R(t) and the spacecurvature to the averagedensityand pressureofthe energy-matter. Einstein'scosmologicalevolution is alsoformulated as conservingentropy, and his equationsare

consequently time reversible.It is generallyacceptedthat the standard modelpermits

us to understand at least qualitatively what happenedtoouruniverse a fraction ofa secondafter its birth.This is an

extraordinaryachievement,but we are still left with the

questionofwhat occurredbefore.When we extrapolateback to the past, we cometo a point of infinite density.Canwe extrapolatebeyond this point? To give an idea ofthe rangeof values involved here,it is useful to definePlanck'sscales,which measurethe length, time, and

energy obtained by using three universal constants: h,

Planck'sconstant; G, the gravitational constant;and c, the

velocity of light. We then obtain Planck'slength, / =(G/i/c3)~ 1CT33cm,Planck'stime on the orderof 1CT44


seconds,and Planck'senergy,correspondingto a high

temperature on the orderof1032degrees.It is plausiblethat

thesescalesrelate to the very early universe characterizedby an extraordinarilybrieftime,a minusculegeometricalsize,and an enormousenergy.In this \"Planckera,\"quantum effects are likely to play an essentialrole.19We havenow arrived at the very limits of modern-day physics,wherewe are confrontedwith the fundamental problemofthe quantization ofgravity or,equivalently, ofspace-time.A generalsolutionis still far from our grasp,but we may at

least formulate a modelthat includesthe roleofPoincareresonancesand irreversibility at the very beginningofouruniverse.Let us now describesomeofthe stepsthat ledus

to this model.We have noted that the Friedmannspace-timeinterval

can be written (when we considerthe caseofEuclideanthree-dimensionalgeometry)as ds2=

Q.2(t)(dt2\342\200\224dl2), wheret is the conformal time. This is the Minkowskispace-timeinterval multipliedby the function Q,2,whichis calledthe

conformal factor. Suchconformalspace-timeintervals haveremarkablefeatures, includingtheir conservationof theconeof light, for which ds2 = 0.As Narlikar and othershave stated,they are the natural starting point for quantumcosmologybecausethey includethe Friedmannuniverse asa specialcase.20

Theconformalfactor as a function ofspace-timerelatesto a field in the same way as do other fieldssuch as the

electromagneticfield.(Rememberthat a field is a

dynamical system characterizedby a well-definedenergy and

thereforea Hamiltonian).As shown by RobertBrout andhis coworkers,this factor has a unique quality in that it

correspondsto a negative energy (that is, its energy is

unbounded from below),while the energyofany given mat-


ter field is positive.As a result, the gravitational fielddescribed by the conformalfactor may play the roleof areservoirofnegative energyfrom whichthe energytocreate matter is extracted.21

This is the theoreticalbasisofthe \"freelunch\" model,wherethe total energy (gravitational field plus matter) is

conserved,while the gravitational energy is transformedinto matter. Brout and his colleagueshave proposedamechanismfor this extractionofpositive energy.Inaddition to the conformalfield, they have introduceda matter

field, and demonstrated that Einstein'sequations lead toa cooperativeprocessinvolving the simultaneousappearance ofmatter and a curved space-timestarting from theMinkowskispace-time(containingzerogravitational andmassenergy).Theirmodelshows that such a cooperativeprocesscausesthe exponentialgrowth ofthe radius ofthe

universe over the courseoftime.(Thisis known as the deSitteruniverse.)

Theseconclusionsare intriguinginasmuchas theyindicate the possibilityof an irreversible processtransforming

gravitation into matter. They also focus our attention onthe preuniversestage, the Minkowski vacuum, which is

the starting point for irreversibletransformations. It is

important to note that this modeldoesnot describecreationexnihilo. Thequantum vacuum is already endorsedby the

universal constants,and it is assumedthat we can ascribetothemthe values they have today.

Thebirth ofouruniverse is no longerassociatedwith a

singularity, but rather with an instability that is analogousto a phase transition or bifurcation.However, this theorystill presents a numberofvexingproblems.Brout et al.have useda semiclassicalapproximationin whichthematter field is quantizedwhile the conformalfield is treated


classically.This situation is highly unlikely in Planck'sera,wherequantum effects play an essentialrole.

Edgar Gunzigand PasqualeNardonehave asked whythis processdoesnot occuron a continuousbasis if the

quantum vacuum associatedwith a flat geometricalbackground is indeedunstable in the presenceofgravitationalinteractions.They have demonstratedthat in this semiclas-sical approximation,we needan initial fluctuation of a

cloudofheavy particlesofmass on the orderof50Planckmasses(~50.10~5^)in orderto start the process.22

Theseresultscan be incorporatedinto a macroscopicthermodynamic approach,where the universe has to betreated as an opensystem.Thus,we can observe matterand energy beingcreatedat the expenseofgravitationalenergy (seeFigure8.4).Thiscompelsus to makea numberof modifications to the first law of thermodynamics,wherethereis now a sourceofmatter-energyleadingto a

changein the definition of quantities suchas pressure.*Sinceentropy is specificallyassociatedwith matter, thetransformation ofspace-timeinto matter correspondsto a

dissipative, irreversibleprocessproducingentropy.Theinverse process,whichwould transform matter into space-time,is impossible.The birth ofour universe would thus

be the result ofa burst ofentropy.The interactionof the gravitational and matter fields

leadsto divergencesarisingfrom brieftimesand shortdistances that correspondin quantum theory to high valuesofenergy and momentum.Theseso-called\"ultraviolet\"

divergencesare the objectofa numberofinterestingin-

*The \"creation\" pressure is negative. Therefore, an often-quoted theorem of

Hawking and Penrose showing that the universe starts with a singularity and

involves positive pressure is not applicable.


Figure8.4Matter Is Createdat the Expenseofthe Gravitational FieldIn this simple model, the universe would have no stable ground state.

Creation of matter

Unstable ground state

Gravitation

vestigations that have led to a procedureknown as therenormalizationprogram,which has proved to be quitesuccessful.Still,certaindifficulties remain.Thereis a

striking analogy betweenfield theory and the thermodynamicsituation discussedin earlierchapters.Hereagain, we are

dealingwith persistent interactions that neitherstart norstop,and we thereforehave to go beyondHilbertspace.

Although this new field theory is still in the making, itsmain conclusionis reasonable:Theremay be no stable

groundstate at the cosmologicallevel, sincethe conformalfactor reacheslowerenergiesas it createsmatter. While this

lineofresearchcontinuesto bepursued,the two conceptsemphasized in this book,irreversibility and probability,clearly form an important part ofthis approach.Universes

appear at siteswhere the amplitudesof the gravitationaland matter fields have high values. The places and thetimeswherethis occurshave only a statistical meaning,as

they are associatedwith quantum fluctuations ofthe fields.This descriptionappliesnot only to our universe, but alsoto the meta-universe,the mediumin which individualuniverses are born.In ourview, hereagain we have an ex-

t


ample ofPoincareresonancessimilar to that ofthe decayofan excitedatom.In this case,however,the decayprocesscreatesnot photonsbut universes!Even beforeour universe

was created,therewas an arrow of time,and this arrowwill go on forever.

Of course,thus far we have only a simplifiedmodel.Einstein'sdreamofa unified theory that would includeall

interactionsremainsalive today23Nonetheless,such a

theorywould have to take into accountthe time-oriented

characterof the universe as associatedwith its birth and

subsequentevolution.This can beachievedonly if certainfields (suchas gravitation) play different rolesfrom others(suchas matter).Inotherwords,unification is not enough.We needa moredialecticalview ofnature.

Questionsconcerningthe originsoftime will probablyalways be with us.But the idea that time has nobeginning\342\200\224that

indeedtime precedesthe existenceofouruniverse\342\200\224is becomingmoreand moreplausible.

Chapter9

A NARROW PATH

I

Ithas often beensuggestedthat irreversibility has a cos-

mologicaloriginassociatedwith the birth ofouruniverse. It is true that cosmologyis neededto explainwhythe arrow oftime is universal, but irreversibleprocessesdidnot ceasewith the creationofouruniverse;they still goontoday, on all levelsincludinggeologicaland biologicalevolution. Although the dissipative structures introducedin

Chapter2 are routinely observedin the laboratoryas wellas in large-scaleprocessesoccurringin the biosphere,irreversibility can be fully understoodonly in terms ofa

microscopic descriptionthat was traditionally identifiedwith

classicaland quantum mechanics.This requiresa newformulation ofthe laws ofnature that is no longerbasedoncertitudes,but rather possibilities.In acceptingthat thefuture is not determined,we cometo the endofcertainty. Isthis an admissionofdefeat for the human mind? On the

contrary, we believethat the oppositeis true.

183


The Italian author Italo Calvinohas written a delightfulcollectionofstories,Cosmocomics,in whichindividuals

livingin a very early stageofour universe gathertogetherto

rememberthe terribletime whenthe universe was so smallthat their bodiesfilled it completely1What would havebeenthe history ofphysicsif Newtonhad beena memberofthis community? Hewould have observedthe birth and

decayofparticles,the mutual annihilation ofmatter andantimatter. Fromthe start, the universe would have

appeared as a thermodynamicsystemfar from equilibrium,with instabilitiesand bifurcations.

It is true that today we can isolatesimpledynamicalsystems and verify the laws ofclassicaland quantummechanics. Still, they correspondto idealizationsapplicable tostabledynamical systemswithin a universe that is a giantthermodynamic system far from equilibrium,where wefind fluctuations, instabilities,and evolutionary patterns at

all levels.On the otherhand, certainty has longbeenassociated with a denialoftime and creativity. It is interestingto considerthis conundrumin its historicalcontext.

IIHow can we reachcertainty? This is the question that

lies at the heart of the work ofReneDescartes.In his

thought-provokingbookCosmopolis,StephenToulmin

attempts to clarify the circumstancesthat ledDescartesonthis quest.2Hedescribesthe tragic situation ofthe

seventeenth century, a time ofpolitical instability and war

between Catholicsand Protestants in the nameofreligiousdogma.It was in the midst of this strife that Descartesbeganhis searchfor a different kind of certainty that all

humans, independentoftheirreligions,couldshare.This

A Narrow Path 185

ledhim to his famous cogito, the foundation ofhis

philosophy,as wellas his convictionthat sciencebasedon

mathematics was the only way to reachsuchcertainty. Descartes'views, which proved to be immenselysuccessful,influenced Leibniz'sconceptofthe laws ofnature discussedin

Chapter1.(Leibnizalso wanted to createa languagethat

would heal the divisionsamongreligionsand bringaboutthe endofreligiouswars.)Descartes'pursuit ofcertaintyfound its concreterealization in Newton'swork, whichhasremainedthe modelfor physicsfor threecenturies.

Toulmin'sanalysisreveals a remarkableparallel betweenthe historicalcircumstancessurroundingDescartes'questfor certainty and thoseofEinstein's.ForEinstein as well,sciencewas a meansofavoiding the turmoilofeverydayexistence.Hecomparedscientificactivity to the \"longingthat irresistiblypullsthe town-dwelleraway from his noisy,crampedquartersand toward the silenthigh mountains.\"3

Einstein'sview ofthe human conditionwas profoundlypessimistic.He had lived through a particularly tragicperiod in human history spanning the riseof fascism andanti-Semitismand two world wars.His vision ofphysicshas beendefinedas the ultimate triumph ofhuman reasonover a violent world, separatingobjectiveknowledgefromthe domainofthe uncertainand the subjective.

But is scienceas conceivedby Einstein\342\200\224an escapefromthe vagariesofhuman existence\342\200\224still the scienceoftoday?We cannotdesertthe pollutedtownsand citiesfor the highmountains.We have to participate in the building oftomorrow's society.In the wordsofPeterScott,\"Theworld,our world, tries ceaselesslyto extendthe frontiers oftheknowableand the valuable, to transcend the givennessofthings, to imaginea new and betterworld.\"4

Sciencebeganwith the Prometheanaffirmation ofthe


powerof reason,but it seemedto end in alienation\342\200\224a

negationofeverything that gives meaningto human life.Our beliefis that our own age can be seenas oneof a

quest for a new type ofunity in our vision ofthe world,and that sciencemust play an important rolein definingthis new coherence.

As we mentionedin Chapter8,at the end ofhis life,Einsteinwas offered a collectionofessaysthat includedacontributionby the greatmathematician Kurt Godel.Inhis answer to Godel,he rejectedhis idea of a possibleequivalencebetweenpast and future. ForEinstein,nomatter how greatthe temptationofthe eternal,acceptingtheidea of traveling backin time was a denial of the realworld.Hecouldnot endorseGodel'sradicalinterpretationofhis very own views.3

As Carl Rubino has noted,Homer'sIliad revolvesaround the problemof time as Achilles embarks on asearchfor somethingpermanentand immutable:

The wisdomof the Iliad, a bitter lessonthat Achilles, its hero,learns too late, is that such perfection can be gained only at

the cost of one'shumanity: he must losehis life in order to

gain this new degreeof glory. For human men and women,for us, immutability, freedomfrom change,total security,

immunity from life'smaddening ups and downs,will comeonly

when we depart this life, by dying, or becominggods:the

gods,Horacetells us, are the only living beingswho leadsecure lives, free from anxiety and change.6

Homer'sOdysseyappearsas the dialecticalcounterpartto the Iliad.Odysseusis fortunate enoughto begiven the

choicebetween immortality, by remainingforever the

lover ofCalypso,and a return to humanity and ultimately

A Narrow Path 187

oldageand death.In the end,he choosestime over

eternity,human fate over the fate ofthe gods.

SinceHomer,time has beenthe central themeofliterature. We find a reactionquite similar to that ofEinsteinin

an essayby the great writerJorgeLuis Borgesentitled\"A New Refutation ofTime.\"After describingthedoctrines that make time an illusion,he concludes:\"And yet,and yet...denyingtemporalsuccession,denying the self,

denying the astronomicaluniverse, are apparentdesperations and secretconsolations... .Timeis the substance I

am madeof.Timeis a river whichsweepsmealong,but I

am the river; it is the tigerwhichdestroysme,but I am the

tiger;it is a fire whichconsumesme;but I am the fire. Theworld, unfortunately, is real; I,unfortunately, am Borges.\"7Timeand reality are irreduciblylinked.Denyingtimemayeitherbea consolationora triumph ofhuman reason.It is

always a negationofreality.The denialoftime was a temptation for both Einstein

the scientistand Borgesthe poet.Einsteinrepeatedlystatedthat he had learnedmorefrom Fyodor Dostoyevskythan

from any physicist.In a letterto Max Bornin 1924,hewrotethat if he wereforcedto abandonstrict causality,he\"wouldrather bea cobbler,oreven an employeein a

gaming house,than a physicist.\"8In orderto beofany value at

all, physicshad to satisfy his needto escapethe tragedy ofthe human condition.\"And yet, and yet,\" whenconfronted by Godelwith the extremeconsequencesof his

quest, the denialofthe very reality that physicistsendeavorto describe,Einsteinrecoiled.

We can certainly understandEinstein'srefusing chanceas the only answerto ourquestions.What we have triedtofollow is indeeda narrow path betweentwo conceptionsthat both lead to alienation:a world ruledby deterministic


laws,whichleaves no placefor novelty, and a world ruledby a dice-playingGod,whereeverything is absurd,acausal,and incomprehensible.

We have attempted to make this booka journeyalongthe narrow path, and thereby illustrate the roleofhuman

creativity in science.Strangelyenough, this creativity is

often undervalued. We all realizethat if Shakespeare,Beethoven,orvan Goghhad diedsoonafter birth, no oneelsewould ever have achievedwhat they did.Is this alsotrue for scientists?Would someoneelsenot havediscovered the classicallaws of motionif there had beennoNewton?Didthe formulation ofthe secondlaw ofthermodynamics dependentirely on Clausius?Thereis sometruth in the contrast betweenartistic and scientific

creativity. Scienceis a collectiveenterprise.In orderto beacceptable,the solutionto a scientificproblemmust satisfy

exactingcriteriaand demands.Theseconstraints,however,do not eliminatecreativity.They provokeit.

Theformulation ofthe time paradox is itself an

extraordinary feat ofhuman creativity and imagination.Ifscience had beenrestrictedto empiricalfacts, how couldit

ever have dreamedof denying the arrow of time? Theelaborationof time-symmetrical laws was not achieved

merely by introducingarbitrary simplifications.Itcombined empiricalobservationswith the creationoftheoretical structures.This is why the resolutionof the time

paradoxcouldnot beaccomplishedby a simpleappealtocommonsenseor ad hoc modificationsofthe laws ofdynamics. It was not even a matter ofsimplyidentifying the

weaknessesin the classicaledifice.In orderto makefundamental progress,we neededto introducenew physicalconcepts,such as deterministicchaos and Poincareresonances, and new mathematical tools to turn theseweak-

A Narrow Path 189

nesses into strengths. In our dialoguewith nature, wetransform what first appearas obstaclesinto originalconceptual structuresprovidingfresh insights into the

relationshipbetweenthe knowerand the known.

What is now emergingis an \"intermediate\"descriptionthat liessomewherebetweenthe two alienating imagesofadeterministicworld and an arbitrary world ofpure chance.Physical laws lead to a new form of intelligibility as

expressed by irreducibleprobabilisticrepresentations.Whenassociatedwith instability, whetheron the microscopicormacroscopiclevel, the new laws ofnature deal with the-

possibilityof events, but do not reducetheseevents to

deductible,predictableconsequences.This delimitation ofwhat can and cannot bepredictedand controlledmay wellhave satisfiedEinstein'squestfor intelligibility.

As we follow alongthe narrowpath that avoids thedramatic alternatives ofblind laws and arbitrary events, wediscoverthat a largepart ofthe concreteworld around us

has until now \"slippedthrough the meshesofthe scientificnet,\" to use Alfred North Whitehead's expression.9We

face new horizonsat this privilegedmomentin the historyofscience,and it is our hopethat we have beenable tocommunicatethis convictionto our readers.

NOTES

Acknowledgments

1.I.Prigogineand I.Stengers,Entre le Temps et I'Eternite (Paris:Librairie Artheme Fayard, 1988(2nd ed.,Paris, Flammarion,1992).

2.I.Prigogineand I.Stengers,DasParadox der Zeit (Munich: R.Piper& Co.Verlag, 1993);I.Prigogineand I.Stengers,Time,Chaosand Quantum Theory (Moscow:Ed.Progress,1994).

3.I.Prigogine,LaFin desCertitudes (Paris:OdileJacob,1996).4. I.Prigogineand I.Stengers,Order Out of Chaos (New York:

Bantam Books,1984);I.Prigogine,From Being to Becoming (SanFrancisco:W. H.Freeman, 1980).

Introduction

1.K.R.Popper,The OpenUniverse: An Argument for Indeterminism

(Cambridge:Routledge,1982),p. xix.2.W.James, \"TheDilemma of Determinism,\" in TheWill to

Believe (NewYork: Dover,1956).3.G.Gigerenzer,Z.Swijtink, T. Porter, J.Daston,J.Beatty, and

L.Kriiger, TheEmpire of Chance (Cambridge:CambridgeUniversity Press,1989),p.xiii.

4. SeeL.Kriiger, J.Daston,and M.Heidelberger,eds.,TheProbabilistic Revolution (Cambridge,Mass.:MIT Press,1990),1:80.

5.Gigerenzeret al., Empire of Chance.6.Popper,OpenUniverse.

7.R. Tarnas, The Passion of the Western Mind (New York:

Harmony, 1991),p. 443.

191

192Notes

8.I.Leclerc,The Nature of Physical Existence (London:Allen and

Unwin; New York: Humanities Press,1972).9.J.Bronowski, A Senseof the Future (Cambridge,Mass.:MIT

Press,1978),p.ix.10.S.Hawking, A Brief History ofTime:From the Big Bang to Black

Holes(New York: Bantam Books,1988).

Chapter1.Epicurus' Dilemma

1.For Epicurus, seeJ.Barnes,ThePresocratic Philosophers (London:Routledge,1989).He probably had in mind the Stoics,whobelievedin a kind of determinism.

2.For Lucretius, seeTitus Lucretius Carus,DeNatura Rerum, ed.C.Bailey (Oxford:OxfordUniversity Press,1947).

3.K.R.Popper,TheOpenSociety and Its Enemies (Princeton,N.J.:PrincetonUniversity Press,1963).

4. For Parmenides,seeBarnes,Presocratic Philosophers.5.Plato, The Sophist (New York: Garland, 1979).6.J.Wahl, Traite deMetaphysique (Paris:Payot, 1968).7. P. S.Laplace,Oeuvres Completes de Laplace(Paris: Gauthier-

Vilars, 1967).8.G.von Leibniz, Discourse on Metaphysics and Other Essays, ed.

D.Garberand R.Ariew (Indianapolis: Hackett, 1991).9.J.Needham,Scienceand Society in East and West: The Grand

Titration (London:Allen and View, 1969).10.For the Einstein-Tagorecorrespondence,translated by A.

Robinson, seeK.Dutta and A. Robinson,Rabindranath Tagore(London: Bloomsbury, 1995).11.Popper,OpenUniverse, loc.cit.

12.H.Bergson,Oeuvres (Paris: PressesUniversitaires de France,1959),p.1331.

13.James,Dilemma ofDeterminism, loc.cit.14.J.Searle,\"Is There a Crisisin American Higher Education?\"

Bulletin of the American Academy ofArts and Sciences46, no. 4(January 1993):24.

15.Scientific American 271,no.4 (October1994).16.S.Weinberg, in ibid.,p. 44.17.Hawking, Brief History ofTime, loc.cit.18.R.Descartes,Meditations metaphysiques (Paris:J.Vrin, 1976).19.R.Penrose,TheEmperor's New Mind (Oxford:Oxford

University Press,1990),pp.4-5.

Notes 193

20.A. N.Whitehead, Process and Reality, ed.D.Griffin and D.Sherborne, correcteded.(New York: Macmillan, 1978).

21.C.P.Snow,TheTwo Cultures and the Scientific Revolution. TheTwo

Cultures and a SecondLook.(Cambridge:CambridgeUniversityPress,1964).

22.R. J. Clausius, Ann. Phys. 125(1865):353; Prigogine and

Stengers,OrderOut of Chaos,loc.cit.23.A. S.Eddington, The Nature of the Physical World (Ann Arbor:

University of Michigan Press,1958).24.SeePrigogine,From Being to Becoming.25.H.Poincare,\"La Mecaniqueet l'experience,\"in Revue deMeta-

physique et Morale 1 (1893):534\342\200\224537, and LeconsdeThermody-

namique, ed.J.Blondin (Paris:Herman, 1923).26.For Zermelo,seeS.Brush, Kinetic Theory (New York: Perga-

mon Press,1962),vol. 2.27.R. Smoluchowski, \"Vortrage iiber die kinetische Theorieder

Materieund Elektrizitat,\" 1914,quotedin H.Weyl, Philosophy

of Mathematics and Natural Science (Princeton,N.J.:PrincetonUniversity Press,1949).

28.M. Gell-Mann, The Quark and the Jaguar (London: Little,

Brown, 1994),pp.218-220.29.M. Planck, Treatise on Thermodynamics (New York: Dover,

1945).30.M.Born, The ClassicalMechanics ofAtoms (New York: Ungar,

1960);quoted in M.Tabor,Chaosand Integrability in Nonlinear

Dynamics (NewYork: Wiley, 1969).31.Prigogine,From Being to Becoming, p.213.32.SeeH.Price, Time'sArrow and Archimedes' Point: New Directions

for the Physics ofTime (Oxford:OxfordUniversity Press,1996).33.J.L.Lagrange, Theorie desfonctions analytiques (Paris:Imprimerie

de la Republique,1796).34.Gell-Mann,Quark and the Jaguar.35.L. Rosenfeld,\"Unphilosophical Considerationson Causality

in Physics,\" in SelectedPapers ofLion Rosenfeld, ed.R.S.Cohenand J.J.Stachel,Boston Studies in the Philosophy ofScience, vol.21(Dordrecht:Reidel,1979),pp. 666-690.

36.Borel,quoted in L. Kriiger, J.Daston,and M. Heidelberger,Probabilistic Revolution.

37'.J.W Gibbs,Elementary Principles in Statistical Mechanics (NewYork: Scribner's,1902).

38.H.Poincare,TheValue ofScience(New York: Dover,1958).

194 Notes

39.B.Mandelbrot, The Fractal Geometry of Nature (San Francisco:W.H. Freeman, 1983).

40.H.Poincare,New Methods of Celestial Mechanics, ed.D.GorofF(American Institute of Physics, 1993).

41.M.Born, quotedin M.Tabor,Chaosand Integrability in

Nonlinear Dynamics, p. 105.42.Tabor,Chaosand Integrability.

43.M.Jammer, The Philosophy of Quantum Mechanics (New York:

Wiley-Interscience,1974);A. I.M.Rae, Quantum Physics:Illusion or Reality? (Cambridge:CambridgeUniversity Press,1986).

44.P.Davies,TheNew Physics:A Synthesis (Cambridge:CambridgeUniversity Press,1989),p. 6.

45.Quotedby K.V Laurikainen, Beyond the Atom:The Philosophical

Thought ofWolfgang Pauli (Berlin:SpringerVerlag, 1988),p. 193.46.CI.George,I.Prigogine,and L.Rosenfeld,\"TheMacroscopic

Levelof Quantum Mechanics,\" Kong. Danske Viden. Selskab

Matematisk-fysiske Medd.38 (1972):1-44.47.See,e.g.,W. G. Unruh and W. H.Zurek, \"Reduction of a

Wavepacket in Quantum Brownian Motion,\" Phys. Rev. 40(1989):1070.

48.J. S. Bell, Speakable and Unspeakable in Quantum Mechanics

(Cambridge:CambridgeUniversity Press,1989).49.Gell-Mann,Quark and the Jaguar.50.G. C.Ghirardi, A. Rimini, and T Weber, Phys. Rev. D34

(1986):470.51.B.d'Espagnat, Conceptual Foundations of Quantum Theory,

Benjamin, California, 1976.52.SeeI.Farquhar, Ergodic Theory (London:Interscience

Publishers, 1964.)53.J.von Neumann, Mathematical Foundations of Quantum

Mechanics (Princeton,N.J.:PrincetonUniversity Press,1955).54.Cohen, Probabilistic Revolution.

55.H.Poincare,Scienceand Hypothesis (NewYork: SciencePress,1921).

Chapter2.Only an Illusion?

1.I.Prigogine,Bull. Acad. Roy. Belgique 31(1945):600.SeealsoEtude thermodynamique desphenomenes irreversibles (Liege:De-soer,1947).

Notes 195

2.Lagrange, Theorie desfonctions analytiques.

3.Hawking, Brief History ofTime.4. Bergson,L'Evolution creatrice, in Oeuvres, p. 784.5.Ibid.,p. 1344.6.Poincare,Scienceand Hypothesis.7.Whitehead, Process and Reality.

8.Eddington, Nature of the Physical World.

9.T.DeDonder and P.Van Rysselberghe,Affinity (MenloPark,Calif.:Stanford University Press,1967);I.Prigogine,Introduction to Thermodynamics ofIrreversible Processes, 3rd ed.(New York:

Wiley, 1967).10.G.N Lewis,Science 71(1930):570.11.E.Schrodinger,What Is Life?(Cambridge:Cambridge

University Press,1945).12.I.Prigogine,Bull. Acad. Roy. Belgique 3, (1945):600.13.L. Onsager,Phys. Rev. 37 (1931):405; 38 (1931):2265.The

proof of this theorem involves the celebratedOnsagerreciprocity relations.

14.P.Glandsdorff and I.Prigogine,Thermodynamic Theory ofStructure, Stability and Fluctuations (New York: Wiley-Interscience,1971).

15.G.Nicolisand I.Prigogine,Exploring Complexity (SanFrancisco: Freeman, 1989).

16.Ibid.17.For a review of oscillatory reactions,seeChemical Waves and

Patterns, ed. R. Kapral and K. Showalter (Newton, Mass:Kluwer, 1995).

18.For a review of nonequilibrium spatial structures, seeSpecialIssueof Physica A 213,nos.

1\342\200\2242, \"Inhomogeneous Phasesand

Pattern Formation,\" ed.J. Chanau and R. Lefever(North-Holland, 1995).

19.A. M.Turing, Phil. Trans. Roy. Soc.London, Ser.B, 237 (1952):37.

20.Nicolisand Prigogine,Self-Organization and Exploring

Complexity.

21.Nicolisand Prigogine,Exploring Complexity; Prigogine,From

Being to Becoming.22.C.K.Biebracher,G.Nicolis,and P.Schuster, Self-Organization

in the Physico-Chemical and Life Sciences, Report EUR 16546(European Commission,1995).

196Notes

Chapter3.From Probability to Irreversibility

1.Prigogine,From Being to Becoming.2.P.and T.Ehrenfest, Conceptual Foundations ofStatistical

Mechanics (Ithaca, N.Y.:CornellUniversity Press,1959).3.A. Bellemanns and J.Orban, Phys. Letters 24A (1967):620.4. I.Prigogine,Nonequilibrium Statistical Mechanics (New York:

Wiley, 1962);R.Balescu,Equilibrium and Non EquilibriumStatistical Mechanics (New York: Wiley, 1975);P.Resiboisand M.DeLeener,ClassicalKinetics ofFluids (New York: Wiley, 1977).

5.A. Lasotaand M.Mackey, Probabilistic Properties ofDeterministic

Systems (Cambridge:CambridgeUniversity Press,1985).6.Jan von Plato, Creating Modern Probability: Its Mathematics,

Physics, and Philosophy in Historical Perspective (Cambridge,Mass:CambridgeUniversity Press,1994).

7.D. Ruelle, Phys. Rev. Letters 56 (1986):405; Commun. Math

Phys. 125(1989):239;H.Hasegawaand W C.Saphir, Phys.

Rev. A 46 (1992):7401;H.Hasegawaand D.Driebe,Phys. Rev.

E 50 (1994):1781;P. Gaspard,J.of Physics A 25 (1992):L483;I.Antoniou and S.Tasaki,J.ofPhysics A: Math. Gen.26 (1993):73; PhysicaA 190(1992):303.

8.I.Prigogine,LesLoisdu Chaos(Paris:Flammarion, 1994),and

Le leggi del caos(Rome:Laterza, 1993).

Chapter4:TheLawsof Chaos

1.Hasegawaand Saphir, Phys. Rev. A 46 (1992):7401;Hasegawaand Driebe,Phys. Rev. E 50 (1994):1781;P.Colletand J.Eck-man, Iterated Maps on the Interval as Dynamical Systems (Boston:Birckhauser, 1980);R Shields,The Theory of Bernoulli Shifts

(Chicago:University of ChicagoPress,1973).2.P. Duhem, La theorie physique. Son objet. Sa structure (reprint,

Paris:Vrin, 1981),vol. 2.3.Hasegawaand Saphir, Phys. Rev. A 46 (1992):7401;Hasegawa

and Driebe,Phys. Rev. E 50 (1994):1781;Gaspard,Journal ofPhysics 25 (1992):L483;Antoniou and Tasaki, Journal ofPhysics

A: Math. Gen.26 (1993):73.4. Ibid.5.Mandelbrot, TheFractal Geometry ofNature; P.and T.Ehrenfest,

Conceptual Foundations ofStatistical Mechanics.

Notes 197

6.Nicolisand Prigogine,Exploring Complexity; Prigogine,From

Being to Becoming.1.See,e.g.,F. Riesz and B. Sz-Nagy, Functional Analysis (NewYork: Dover,1991).

8.Prigogine,From Being to Becoming; V. Arnold and A. Avez,

Ergodic Problems of Classical Mechanics (New York: Benjamin,1968).

9.Hasegawaand Saphir, Phys. Rev. A 46 (1992):7401;Hasegawaand Driebe,Phys. Rev. E 50 (1994):1781;Gaspard,Journal ofPhysics 25 (1992):L483;Antoniou and Tasaki, Journal ofPhysicsA: Math. Gen.26 (1993):73.

10.P.Gaspard,Physics Utters A 168(1992):13,and Chaos3 (1993):427;H.Hasegawaand D.Driebe,Physics Utters A 168(1992):18,and Phys. Rev. E 50 (1994):1781;H.Hasegawaand E.Luschei,\"Exact PowerSpectrum for a System of Intermittent

Chaos,\" Physics Utters A 186(1994):193.

Chapter5:Beyond Newton's Laws

1.T.Petrosky and I.Prigogine,\"Alternative Formulation ofClassical and Quantum Dynamics for Non-IntegrableSystems,\"

Physica A 175(1991);T. Petrosky and I.Prigogine,\"PoincareResonancesand the Limits of TrajectoryDynamics,\" PNAS90 (1993):9393;T.Petrosky and I.Prigogine,\"Poincare

Resonances and the Extension of ClassicalDynamics,\" Chaos,Soli-tons and Fractals 5 (1995).

2.Seeany text on Fourier series.3.Prigogine,Nonequilibrium Statistical Mechanics.

4. SeePetrosky and Prigogine,\"PoincareResonances.\"5.SeeS.G.Brush, KineticTheory (Oxford:Pergamon Press,1972),

vol. 3.6. SeeY Pomeauand P.Resibois,Physics Reports 19,2 (1975):63.7. T.Petrosky and I.Prigogine,\"New Methodsin Dynamics and

Statistical Physics\" (forthcoming).8.Prigogine,Nonequilibrium Statistical Mechanics; seealso the

citations in note 1 of this chapter.

198Notes

Chapter6:A Unified Formulation of Quantum Theory

1.R. Penrose,Shadows of the Mind (Oxford:OxfordUniversity

Press,1994),chap.5.2.P.Davies,TheNew Physics; Rae, Quantum Physics.3.J.C.von Neumann, Mathematical Foundations ofQuantum Theory.4. T. Petrosky and I.Prigogine, \"Quantum Chaos, Complex

Spectral Representationsand Time-Symmetry Breaking,\"

Chaos,Solitons and Fractals 4 (1994):311;TPetrosky and I.Prigogine, Physics Letters A 182(1993):5;T.Petrosky, I.Prigogine,and Z.Zhang (forthcoming).

5.K. R. Popper,Quantum Theory and the Schism in Physics (To-towa, N.J.:Rowman and Littlefield, 1982).

6.The standard text isby P.A. M.Dirac,ThePrinciples ofQuantumMechanics (Oxford:OxfordUniversity Press,1958).

7.M.Jammer, The Philosophy of Quantum Mechanics (NewYork:

John Wiley, 1974).8.A. Eddington, The Nature of the Physical World (Ann Arbor:

University of Michigan Press,1958).9.A. Bohm, Quantum Mechanics (Berlin: Springer, 1986);A.

Bohm and M. Gadella,Dirac Sets, Gamov Vectors and Gelfand

Triplets (Berlin:Springer, 1989);G.Sudarshan, Symmetry

Principlesat High Energies, ed.A. Perlmutter et al. (San Francisco:

Freeman, 1966);G.Sudarshan, C.B.Chiu, and V. Gorini,Physical Review D 18(1978):2914.

10.Petrosky and Prigogine,\"Quantum Chaos;\"T Petrosky and

Z.Zhang (forthcoming).11.Petrosky and Prigogine,\"Quantum Chaos\"and Physics Letters;

Petrosky, Prigogine,and Zhang (forthcoming).12.N. Bohr, \"The Solvay Meeting and the Developmentof

Quantum Physics,\" in La Theorie quantique des champs (NewYork: Interscience,1962).

Chapter7:Our Dialoguewith Nature

1.V. Nabokov, Look at the Harlequins (New York: McGraw-Hill,1974).

2.Prigogineand Stengers,Entre leTemps et I'Eternite.

3.N. Bohr, Atomic Physics and Human Knowledge (New York:

Wiley, 1958).

Notes 199

4. A. I.M.Rae, Quantum Physics.

5.W. Heisenberg,The Physical Principles of the Quantum Theory

(Chicago:University of ChicagoPress,1930).6.SeeNicolisand Prigogine,Exploring Complexity.7.S.J.Gould,Scientific American 271,no.4 (October1994):84.8.Ibid.

Chapter8:DoesTime PrecedeExistence?

1.J.Wheeler, quoted in H.Pagels,Perfect Symmetry (NewYork:

Bantam Books,1986),p.165.2.L.D.Landau and E.M.Lifschitz, The ClassicalTheory ofFields

(London:Pergamon Press,1959).3.P.Davies,About Time (London:Viking, 1995).4. H.Minkowski, The Principle ofRelativity: Original Papers

(Calcutta: University of Calcutta, 1920).5.A. Einstein, Correspondence Einstein-Michele Besso 1903\342\200\2241955

(Paris:Hermann, 1972).6.Albert Einstein: Philosopher-Scientist, ed.R A. Schlipp (Evanston,

III:Library of Living Philosophers,1949).7.H. Bondi, Cosmology (Cambridge:Cambridge University

Press,1960).8. SeeJ.V Narlikar and T. Padmanabhan, Gravity, Gauge Theory

and Quantum Cosmology (Dordrecht:Reidel,1986).9. I.Antoniou and B.Misra,Journal ofTheoretical Physics 31(1992):

119.10V Fock, TheTheory of Space,Time and Gravitation (New York:

Pergamon Press,1959).11.Hawking, Brief History ofTime.12.P.A. M.Dirac,Rev. Mod.Phys., 21(1949):392;D.J.Currie,T.

F.Jordan, and E.C.G. Sudarshan, Rev. Mod.Phys., 35 (1962):350;R.Balescuand T.Kotera,Physica 33 (1967):558;U BenYa'acov,Physica.

13.Bondi,Cosmology.14.Seethe excellentaccountby S.Weinberg, TheFirst Three

Minutes: A Modern View of the Origin of the Universe (New York:

BasicBooks,1977).15.SeeAlpher and Herman, in Nature 162(1948):774,and

Physical Review 75, no.7 (1949):1089.16.SeeE.P.Tryon, in Nature 266 (1973):396.

200 Notes

17.See, for a general account, S. Weinberg, Gravitation and

Cosmology: Principles and Applications of the General Theory ofRelativity (New York; Wiley, 1972).

18.Ibid.19.SeeJ.V. Narlikar and T.Padmanabham, Gravity.

20.Narlikar and Padmanabhan, Gravity.21.R.Brout, F. Englert, and E.Gunzig, Ann. Phys. 115(1978):78;

General Relativity and Gravitation 10(1979):1;R.Brout et al,Nuclear Physics B 170(1980):228;E. Gunzig and P.Nardone,Physics Letters B 188(1981):412,and also in Fundamentals ofCosmicPhysics 11(1987):311.

22.E. Gunzig, J.Geheniau,and I.Prigogine,Nature 330 (1987):621;I.Prigogne,J.Geheniau,E.Gunzig, and P.Nardone,Proc.Nat.Acad. Sci.USA 85 (1988):1428.

23.S.Weinberg, Dreams of a Final Theory (New York: PantheonBooks,1992).

Chapter9:A Narrow Path

1.I.Calvino, Cosmicomics,trans. W Weaver (NewYork: Harcourt,Brace& World, 1969).

2.S. Toulmin, Cosmopolis (Chicago:Chicago University Press,1990).

3.A. Einstein, Ideas and Opinions (New York: Crown, 1954),p. 225.

4. P. Scott, Knowledge, Culture and the Modern University, 75th

Jubileeof the Rijksuniversiteit (Groningen, Holland, 1984).5.Albert Einstein: Philosopher-Scientist.6.CarloRubino, unpublished.7.J.L.Borges,\"A New Refutation of Time,\" Labyrinths, Penguin

Modern Classics(Harmondsworth: Penguin Books,1970),p.269.

8.A. Einstein and M.Born, The Born-Einstein Letters (New York:

Walker, 1971),p. 82.9.A. N.Whitehead, Process and Reality.

GLOSSARY

anthropicprincipleThe ideathat the conditions of the universe

are explainedby the fact that we are hereto observethem.

bifurcation The branching of a solution into multiple solutions asa system parameter is varied.

big bangThe initial event of our universe, describedas an

explosive creationof matter and energy from a point.

chaosThe behavior of systems in which closetrajectoriesseparateexponentially in time.

clinamenThe idea,due to Epicurus, that an elementof chanceisneededto accountfor the deviation of material motion from rigid

predeterminedevolution.

coarsegrainingThe averaging of dynamics over finite regionsofphasespace.

collapseof the wave function The extradynamical elementneededin orthodoxquantum theory for the wave function,

representing potentialities, to yield an actual state.

degreesof freedom The number of independent variablesneededto specifythe configurational state of a system. A singleparticle in three-dimensionalspacehas three degreesof freedom.

determinismThe viewpoint that evolution isgovernedby a setofrules that, from any particular initial state, can generateone and

only one sequenceof future states.

201

202 Glossary

deterministicchaosChaoticbehavior arising from an entirelydeterministic evolution law.

Diracdelta function The mathematical object, introduced byDirac,which may be considereda function definedas infinity at

one point and zeroeverywhere else.

dissipative structureSpatiotemporal structures that appearin far-

from-equilibrium conditions, such as oscillating chemicalreactionsor regular spatial structures.

eigenstateA state that when actedon by a given operatoryieldsthe same state multiplied by a number.

eigenvalueThe number that an eigenstate is multiplied by after it

is actedupon by the correspondingoperator.

ensembleAn imagined collectionof identical systems with

different initial conditions.

entropy A function of the state of the system that increasesmo-notonically for isolatedsystems and reachesa maximum at

thermodynamic equilibrium.

fractal The term coined by BenoitMandelbrot for mathematical

objectsof noninteger dimension. For example,the length of the

irregular coastlineof a country increasesas the scaleusedto measureit decreases,and so the coastlinehas a dimension betweenone and

two.

Friedmannuniverse A cosmologicalmodelof an expandinguniverse basedon the assumption of homogeneity and isotropy of the

universe on large scales.

Gelfand spaceThe function spacecontaining both the

generalized functions and the well-behavedfunctions they act on.

generalizedfunction The classof mathematical objectsto which

the Diracdelta function belongs.A generalizedfunction is not aregular mathematical function but is defined by how it acts on regularfunctions.

Glossary 203

H-theoremBoltzmann's finding that a function (the H-function)involving the one-particledistribution function appearsunidirectional in time behavior under evolution of a dilute gas ofinteracting particles.

Hamiltonian The energy of a dynamical system expressedin

terms of its coordinatesand momenta.

Heisenberguncertainty principleThe product of the

accuracies by which the position and momentum of a quantum particlemay bedeterminedas limited by Planck'sconstant. Completeaccuracy of either the position or the momentum implies completeindeterminacy of the other one.

HilbertspaceThe spaceof functions for which the integral of the

square of the functions is well definedand finite. This is the

function spacethat was usedas the setting for orthodoxquantummechanics. It has been subsequently appliedto classicalmechanicsand

statistical mechanics.

KAM theory Describesthe dynamical behavior of classesof non-integrable systems. As the energy of a system is increased,chaoticbehavior becomesmore prevalent.

kinetic theory The study of the thermodynamic and transport

propertiesof fluid and gassystems in terms of interparticleinteractions.

largePoincaresystem (LPS)A nonintegrable system due toPoincareresonancestaken in the thermodynamic limit so that its

energy spectrum is continuous.

LaplacedemonThe entity imagined by Laplacethat would beable,given the exactinitial conditions, to calculate the precisefuture evolution of our universe.

Loschmidt'sreversal paradoxThe argument, raisedagainst the

conclusionsof Boltzmann, that sincethe equations of motion in

an interacting particle system are reversible,one can considerreversing all the velocitiesin a system so that any time-orientedfunctions of the state of the system would then behave in an

opposite manner.

204 Glossary

LyapunovexponentThe rate of exponential separationof nearby

trajectoriesin a chaoticsystem.

map A discrete-timedynamical process.Markov processA processwherein the future evolution of a state

dependsonly on the present state.For a continuous time system this

means that the processis localin time, that is, there are no memoryeffects.

Newtonian dynamicsThe rules of evolution that form the coreof classicalphysics and that, in pre-quantum eradeterminism, werebelievedto underlie all physical reality.

nonintegrablesystemAn interacting system that cannot betransformed to noninteracting parts. If such a transformation canbeperformed, the system is integrable and the equations of motion canbe trivially solved.

Perron\342\200\224Frobenius operatorThe time evolution operator for

probability distributions in discrete-timesystems (maps).

phasespaceThe abstract spaceof points in which the coordinatesare the positions and velocitiesof the particles in an evolvingsystem.

Planck era The universe just after the big bang characterizedbythe Planck scales,involving three fundamental constants of nature,

h, c,and G.

PoincarerecurrencetheoremThe finding that the state of aclosedsystem, as definedby the values of the positions and

velocities of all the particles,will recurarbitrarily closelyunder timeevolution of the system.

PoincareresonancesCouplingof degreesof freedomthat leadto

divergent expressionsdue to small denominators if there isresonance betweenthem. The resonancesmay prohibit the solution ofthe equations of motion.

probabilitydistributionfunction The function representing the

relative weights of the systems or initial conditions distributed in an

ensemble.

Glossary 205

resonanceThe constructive interferencethat appearswhen two

frequenciesin a system are rationally related.

Ritz\342\200\224Rydberg principleThe frequency of spectral linesrepresenting

the differencebetweentwo energy levels.

secondlaw ofthermodynamicsThe principle that the entropyof an isolatedsystem may only increaseor remain constant under

time evolution.

self-organizationThe choice between solutions appearing at

a bifurcation point, determinedby probabilistic laws. Far-from-

equilibrium self-organizationleadsto increasedcomplexity.

spectraldecompositionThe expressionof an operatorin termsof its eigenstatesand eigenvalues in a given function space.

steady-stateuniverse A cosmologicalmodelwherein the

expansion of the universe is compensatedby a continuous creationofmatter.

thermodynamiclimit Theprocedureof consideringthe numberN of particlesand the size17of a system becomingarbitrarily largewhile the concentration,c =N/ V, remains finite and constant.

thermodynamicsThe study of the macroscopicpropertiesof asystem and their relations without regard to the underlying dynamics.

Turing structuresPatterns in chemicalsystems arising from an

interplay of reactionand diffusion processes;thesearean exampleofdissipative structure.

INDEX

aging, 78, 125Alper, Ralph A., 174anthropic principle, 15-16Antoniou, Ioannis, 169approximation: dissipative

structures not explained by, 73;evolution as due to, 23,24-25;fundamental problems solved in

terms of, 52;irreversibility as

due to, 23,24,81,91,105architecture, 60Arnold, Vladimir Igorevich, 41arrow of time: all having same

orientation, 102,162;Bernoulli

maps introducing, 90,96;constructive role of, 3; for dealingwith intelligent life, 15;denial

of, 1-2;dissipative structures

requiring, 73;entropy as, 19;as

eternal, 182;in evolving

universe, 4; as fact imposed by

observation, 74;hostility to

conceptof, 61-62;in

macroscopic processes,18;in non-equilibrium physics, 3;nonintegrable systems requiredfor, 39;physics' denial of, 2; in

realistic interpretation ofquantum mechanics, 54;relegated to

phenomenology, 2, 3; as sourceoforder, 26;space-time asconsistent with, 172;in structure

formation, 71;subjective

interpretation of, 49atomism, 9-10,127atoms, Bohr's theory of, 132\342\200\22433,

145

bacteria, 161baker transformations (maps),

96-105,91;approach toequilibrium in terms of, 102;and

Bernoulli map, 90,91;Bernoulli

map compared with, 97-98,103-4;Bernoulli shift for

representing, 99;aschaotic and

deterministic, 101;eigenfunctionsand eigenvalues for, 103;equivalence of individual and statistical

description broken with, 103;as

invertible, 101,102;numerical

simulation of, 98;Perron-Frobenius operator with, 103;recurrence in, 99-101;spectral

representation in, 103;successiveiterations of, 98-99,100;time

paradox associated with, 103;astime reversible, 101,103

207

208 Index

becoming, 10Bell,John, 51Belousov-Zhabotinski reaction,

66-67Benard instability, 158,175Bergson, Henri, 13,14,59,72Bernoulli maps, 83-88,84;arrow

of time introduced in, 90,96;baker transformation asgeneralization of, 96;bakertransformation compared with, 97\342\200\22498,

103-4;as describing chaotic

systems, 89\342\200\22490; evolution operator,93;as not invertible, 96;simulation of distribution function for,

86;simulation of trajectoriesfor, 85

Bernoulli polynomials, 93-94Bernoulli shift, 90,99,101Biebracher, C.K.,71bifurcation point, 66,61,69bifurcations: in chaotic behavior,

68;our universe involving

successive, 72;pitchforkbifurcation, 68,69;as sourceofdiversification and innovation,70;as sourceof symmetry

breaking, 69;successive, 69\342\200\22470,

10big bang, 172-75;as beginning of

the universe, 6, 163-64;birth ofthe universe as a free lunch, 175,179;first secondafter, 175;instability associated with, 6; as

irreversible, 166;as occurringfifteen billion years ago, 173;residual black body radiation,

131,174,176binary correlations, 121,122,

122biology: arrow of time in structure

formation in, 71;Darwinian

evolution, 19,20,183;

multiplicityof evolutions in, 161-62;

self-replicating biomolecules,159.Seealso life

black body radiation, 131,145,174,176

Bohm, Arno, 142Bohr, Niels: atom describedin

terms of energy levels, 132,145;complementarity principle, 74,150;Copenhagen interpretationof quantum mechanics, 50,150,156-57;on quantum leaps, 138;on vocabulary for quantum

physics, 150Boltzmann, Ludwig: on ensembles,

76; evolutionary approach to

physics, 2, 19-21;H-functionand H-theorem, 20;on

irreversibility and dynamical

systems, 27;on irreversibility as

illusory, 2, 19,21;on secondlaw of thermodynamics as

probabilistic, 20,22;two-compartment model, 22-23,76, 91

Bondi, Hermann, 166,173Borel,Emile, 29Borges,JorgeLuis, 187Born, Max, 26,40,134,135,187Brief History ofTime (Hawking), 7,

15,170Bronowski, Jacob,7Brout, Robert, 178,179Brownian motion, 42,43,43butterfly effect, 30-31

Calvino, Italo, 184canonical ensemble, 77canonical equations of motion,

110-11,112causality, 4, 187certainty: coming to end of, 183;

denial of time and creativityassociated with, 184;Descartes'

Index 209

quest for, 184\342\200\22485; resonances

introducing uncertainty, 44chance.Seeprobabilitieschaos:baker transformation as

chaotic, 101;bifurcations

associated with, 68;chaotic systems,30-31;in classical physics, 4;conditions for general definition

of, 156;indeterminism as due

to, 56;and limits of physical

concepts,29;in nonequilibriumphysics, 3;probabilistic laws ofdynamics for, 104;problem of assolvable at ensemble level, 87;resonances'influence on, 41;resonances leading to, 112;simplified example of, 35-37;statistical nature of laws of, 37;time

symmetry broken in, 105;trajectories as inadequate for

describing, 90-91,105.Seealsodeterministic chaos; laws ofchaos

chaotic maps, 81-88;two types ofevolutions with, 102.Seealsobaker transformations; Bernoulli

mapschemical reactions: at equilibrium,

65-66;flow of time varying in,159;irreversibility in, 158-59,183;nonlinear equations for

describing, 66;oscillating

reactions, 66,127,175;synthetic

chemistry, 71;time

directionalityin, 18

chemistry, nonequilibrium, 26,27,67-68

classical physics: chaos included in,4; as deterministic, 136-37;extending to unstable systems, 89;extensions of, 46,109;fundamental concepts of, 109-10;Hamiltonian H in, 133;as

incomplete, 108;instability in, 4,54,107;irreversibility in, 49;laws of nature in, 4, 138,184;naive realism of, 135;predictability in, 4;probabilities in,

5; quantum mechanics limiting

validity of, 107;as reductionist,

114;relativity showing limits of,107;resonances introducinguncertainty into, 44;statistical

descriptions in, 108;time in, 59,60;time-reversible processesin,28.Seealso Newtonian physics;trajectories

Clausius, Rudolf Julius, 18-19clinamen, 10-11,52,55,127coarsegraining, 24,51,52,53,

101-2Cohen,I.Bernard, 55collapse of wave function. See

reduction (collapse)of wave

function

collision operators, 124collisions: in Boltzmann's H-

theorem, 20;and correlations,

78, 19,80;of molecules, 78; in

persistent interactions, 115communication: correlations and

human, 79;measurement as

means of, 150complementarity principle, 74,

150complexity: ofbiological and

chemical structures, 71;emergence of, 128;evolving in non-equilibrium systems, 64;irreversibility associated with, 64

Conferenceon Statistical

Mechanics and Thermodynamics, 61\342\200\22462

conformal factor, 178-79,181conformal time, 178conservation of energy, principle

of, 76-77

210Index

conservation of wave vectors, law

of, 121conservative systems, dynamics of,

101constructive interference, 119continuous spectrum, 133-34,141contracting coordinate, 97,99control, 154Copenhagen interpretation, 50,

150,156-57correlations: binary correlations,

121,122,122;and collision, 78,19,80;communication

compared with,. 79;creation of, 122,147;defined, 78, 120-21;destruction of, 79, 80,122,147;dynamics as a history of, 122;dynamics of, 79;evolution of,

123;flow of, 78-80,80,88;vacuum of correlations, 121,122,122,123,123

cosmologicalprinciple, 177

cosmology:anthropic principle,15\342\200\22416;

birth of universeassociated with instability, 179;birthof universe resulting from burst

of entropy, 180;cosmologicalprinciple, 177; in Einstein, 176,177;a meta-universe, 164,181;observer in, 51;originaluniverse as highly organized, 28;possible worlds, 59,72; standard

model, 164,174-75,177;succession ofbifurcations in our

universe, 72; the universe as

evolutionary, 4, 155;why isthere something rather than

nothing, 175.Seealso big bang;steady-state theory

coupling constant, 140creation fragments, 122,122,124creativity: as amplification of laws

of nature, 71;certainty and

denial of, 184;democracy as based

on, 17;and determinism, 6; anddistance from equilibrium, 62;of nature, 62;in science,188

crystals, 62

Darwin, Charles, 2, 19-21Davies, Paul C.W., 48-49,

165decay processes:in beams of

unstable particles, 139-40;of excited

atoms, 138-39,139;exponential decay, 139,142;radioactive

decay, 17,18;universes created

from, 182deconstruction, 14DeDonder,Theophile, 60,61delocalized distribution functions:

defined, 114;and going outsideHilbert space,116,117;forpersistent interactions, 114,115,117,125,148,155-56;

persistent scattering describedby, 149;in quantum physics, 146

delta functions, 33,94-95,117,124n,143

democracy, scienceas conflicting

with, 17Democritus, 9DeMoivre, Abraham, 5denominators, problem of small,

40density matrix, 47,53-54,143-44,

147-48Descartes,Rene,16,184-85description: as idealized in

traditional laws of physics, 26;intermediate description ofnature,

189;nonlocal, 37, 42,96.Seealso statistical level ofdescription; individual level ofdescription

de Sitter universe, 179

Index 211

destruction fragments, 122,122,124

destructive interference, 119determinism: baker transformation

as deterministic, 101;as basedon idealizations, 29;in classicaland quantum physics, 136\342\200\22437;

creativity and ethics and, 6;dilemma of, 1,6,14;divine

viewpoint required for, 38;as

marhematizable, 14;moving

away from, 131-32;Newton's

relation of acceleration andforceas deterministic, 110;in

pre-Socraticphilosophy, 9-10;in Western philosophy, 11

deterministic chaos:Bernoulli mapfor, 83\342\200\22488; in equations ofmotion, 31;exponential divergenceas signature of, 84;as

inapplicable in quantum mechanics, 53;individual and statistical

descriptions not equivalent in, 94\342\200\22495;

irreducible probabilisticdescriptions for, 45;Laplace demon in

world of, 38;and Poincare non-integrability, 112;statistical

description of, 105deterministic trajectories, 41diffusion: diffusive term in

persistent interactions, 44-45,54;diffusive terms in quantum

mechanics, 53\342\200\22454; entropyassociated with, 41;as irreversible,3, 105;resonances leading to,

42-44,43,126,155;thermal

diffusion, 26,21,35,64;trajectories leading to in chaos,41

dilating coordinate, 97,99dilemma of determinism, 1,6,

14\"Dilemma of Determinism\"

(James), 14

Dirac, Paul, 134,163Diracdelta functions, 33,94-95,

117,124n,143discrete spectrum, 133,141disorder, as constituting foundation

of microscopic systems, 155dissemination, 79dissipative structures: arrow of

time required by, 73; defined,66;emergenceof, 128;and

equilibrium, 67;homogeneityof spaceand time broken by, 69;and irreversibility, 73, 183;newkinetic theory for describing,126;in nonequilibrium physics,3, 27;and Poincare's recurrence

theorem, 101;self-organizationin, 70

distribution functions: additional

information provided by, 37;and Bernoulli maps, 86,87-88;as density matrices, 143-44;ensembles represented by, 33;at

equilibrium, 76-77,117;evolution over time, 112;integrating,120-21;Liouville operatordetermining evolution of, 113;localized, 114,115;microstructureof phase spaceaccounted for in,95;and Perron-Frobenius

operator, 87-88,91,94;in quantumstate representation, 143-44;smoothness in solutions of,87-88;in statistical descriptionof dynamical problems, 91\342\200\22496,

125-28;uniform distribution,93;written as Bernoulli

polynomials,93-94.Seealso delocalized

distribution functionsdistributions. Seegeneralized

functions

domain of validity,29

212Index

dualism: of Descartes,16;inquantum mechanics, 50,53,130,131;Weinberg on, 15

Duhem, Pierre-Maurice, 90dynamical decomposition, 102dynamical groups, 104,142,

171dynamical systems: dynamics as

history of correlations, 122;general problem of dynamics,40;instability in, 55,127-28;integrable systems, 39,44,54,108,131,144;and

irreversibility,27-28,126;as largely non-

integrable, 39;majority as

nonintegrable large Poincare

systems, 127;past and future in,102;phase spacerepresentation,31,32,110;Poincare on,

38-41;recurrence in, 99-101;returning to initial state in

immeasurable time, 23;solvingproblems at statistical level,91-96,113,125-28;stable,30-31,36,55;within athermodynamic system, 184.Seealso

nonintegrable systems; unstable

systems

Eddington, Arthur Stanley, 19,59,137

Ehrenfest, Paul and Tatiana, 74,81,101

eigenfunctions: with baker

transformation, 103;as central in

statistical and quantum mechanics,

95;defined, 92;of evolution

operator, 93;of Hamiltonian

operator, 133;of Liouville

operator, 113,118;of operators,134\342\200\22435;

in spectralrepresentation of an operator, 94-95;instatistical formulation ofquantum physics, 144

eigenvalues: with baker

transformation, 103;as central in

statistical and quantum mechanics,95;defined, 92;of evolution

operator, 93;of Hamiltonian

operator, 133,138,140-42;ofLiouville operator, 113,118,120,149;in spectral

representation of an operator, 94\342\200\22495;in

statistical formulation ofquantum physics, 144

Einstein, Albert: cosmology of,176,177; on ensembles,32,34,76;on freedom, 13;fundamental field equations, 176;generalrelativity, 46;on Godel'scosmo-logical model, 165,186,187;on

gravitation as curvature ofspace-time, 175\342\200\22476;

mass-

energy equation, 175;onquantum mechanics, 5, 135,151;on

science,185;on scientific

determinism, 11;special relativity,

167;and Spinoza, 176;on timeas an illusion, 1,58,165,187;time dilation prediction, 169;and unified theory, 182;on

unity ofnature, 16electromagnetic waves, 112elementary particles, 115,140Empire of Chance,The (Gigerenzer

et al.),4energy: chaos arising at critical

value of, 41;conservation of,76\342\200\22477; Einstein's mass-energy

equation, 175;free energy in

equilibrium conditions, 63,64;gravitational energy transformedinto matter, 174,179;total

energy, 76, 175energy levels:in Bohr description

of the atom, 132-33,145;eigenvalues determining,138

Index 213

ensembles:canonical ensemble,77;chaos problem solved at level of,87;defined, 31-32;distribution

function at equilibrium, 76-77;representation in phase space,33,34;trajectories comparedwith, 82,83,87; transition fromwave functions to, 131;forunstable systems, 81

Entre le Tempset VEternite (Pri-gogine and Stengers), 154

entropic time, 161entropy: as the arrow of time, 19;

birth of universe resulting fromburst of, 180;in Boltzmann

model, 76;diffusion associated

with, 41;entropic time, 161;flow of time correlated with,

159;and information, 24;and

irreversibility, 17, 18\342\200\22419;

irreversible processesas creating, 61;life as feeding on negative

entropy flow, 63;matter associated

with, 180;in nonrelativistic

twin paradox, 160;observation

increasing, 50;ordering comingfrom, 26;in secondlaw ofthermodynamics, 18-19,60

Epicurus: atomism of, 9-10;clina-men concept,10-11,52,55,127;on human freedom, 10

equations of motion: Bernoulli

maps for, 83-90;canonical

equations of motion, 110-11,112;as deterministic, 31;forfree particle, 118;Newton's law

relating forceand acceleration,

11,109-10;of oscillatorinteraction with field, 112;periodicmaps for, 82-83;time-reversible

processesdescribedby, 18equilibrium: approach to with

baker transformation, 102;black

body radiation in equilibrium

with matter, 131,145,174;inBoltzmann's two-compartmentmodel, 22\342\200\22423; calculating rate

of approach to, 95;and

correlation, 78, 80;creativity and

distance from, 62;and dissipation,67;distance from as parameterfor describing nature, 68;distribution function of ensembles at,

76-77;distribution functions at,

117;free energy and, 63,64;limiting thermodynamics to,61\342\200\22462;

matter acquiring new

properties when far from, 65,67;matter as blind in, 127;stable and unstable, 30,30-31;in

system of particles, 20;thermodynamic equilibrium, 60,63,66,77, 113,130;uniform

distribution corresponding to, 93;the

universe as far from, 158.Seealso nonequilibrium processes

equilibrium statistical

thermodynamics, 77

equilibrium thermodynamics, 88,108

Espagnat, Bernard, 53ethics: and determinism, 6; time

associated with, 58events, 5Everett, Hugh, 49evolution: approximation as

responsible for, 23,24-25;Boltzmann's evolutionary approach to

physics, 2, 19-21;ofcorrelations, 123;Darwinian, 19,20,183;described in terms of

probabilities, 55;of distribution

functions, 112;dynamical instabilityas condition of, 128;as multiplein biology, 161-62;in

reformulated laws of physics, 16;two

types with chaotic maps, 102;the universe as evolutionary, 155

214Index

evolution operator, 93,125-26,142

excitedatoms, decay of, 138-39,139,142-43

exponential decay, 139,142

Feynman, Richard, 47, 147nFierz, Markus, 50,53Fifth Solvay Conferenceon

Physics, 150first law of thermodynamics, 180flow of time: depending on history

of events, 170;as globalproperty, 20;gravitational fields

affecting, 159-60,160;introduced in quantum physics,5; in nonrelativistic physics, 159;progressing up levels oforganization, 162;resonances as sourceof, 159;as universal in

Newtonian physics, 160,164,170;as

varying, 159fluctuations: in big bang, 175,180;

in equilibrium conditions, 63;as

multiple, 162;in nonequilib-rium systems, 64,68-70;originof, 72; in urn model, 75,76

Fock, Vladimir A., 170Fokker-Planck equations, 43Fourier series,116-17Fourier transform, 117,118,147fractals, 38freedom: democracy as based on,

17;and determinism in Western

tradition, 6; Einstein's denial of,13;Epicurus on, 10;timeassociated with, 58

free energy, in equilibrium

conditions, 63,64free Hamiltonians, 111,138,140\"free lunch\" model, 175,179free particle, motion of, 111,118,

156

Freud, Sigmund, 70Friedmann, Alexander, 176,177Friedmann space-time interval,

177, 178From Being to Becoming (Prigogine),

27,74function, and structure, 62functional analysis, 38functional spaces,38,46,92future: in chaotic and simple

dynamical systems, 102,104;the

common future, 162;as aconstruction, 106;in formal

solution of Schrodinger equation,142;interaction betweenknower and known creating,153;in Newton's relation ofacceleration and force, 110;as not

determined, 183;as orientationof arrows of time in nature,

102;and past as asymmetrical in

irreversible processes,28;and

past meeting in probabilities,137;and past not distinguishedin physics, 2, 138;predicting in

classical science,4; in specialrelativity, 168

Galilean invariance, 110Galilean physics, 1,167gases, kinetic theory of.Seekinetic

theory of gasesGelfand space,96Gell-Mann, Murray, 24,28-29,

51,52general evolution criterion, 65generalized functions, 33n; Dirac

delta functions, 33,94-95,117,124n,143;in functional analysis,

38;and going outside Hilbert

space,117;nice functionscontrasted with, 94;as not included

in Hilbert space,92

Index 215

general problem of dynamics,40

general relativity: extension ofclassical mechanics in, 46;goingto cosmology from equations of,177;space-time interval in, 176

geological processes,time scaleof,161

Ghirardi, Giancarlo, 52Gibbs, Josiah Willard, 32-34,61,

76Gigerenzer, Gerd,4Glansdorff, Paul, 64God:and the big bang, 173;dice

playing by, 188;divine

viewpoint required for determinism,38;asgoverning the universe

deterministically, 11,12;scienceas reading the mind of, 7

Godel,Kurt, 165,186,187Gold,Thomas, 166,173Gould, Stephen J.,161gravitation: as curvature of space-

time, 175-76;quantization of,178

gravitational fields: matter as

created at expenseof, 179,180,181;time flow affected by,

159-60,160groups, dynamical, 104,142,171Gunzig, Edgar, 180

Hamiltonian H:in classical physics,133;defined, 110;distributionfunction as function of, 76-77,117;equations of motion

derived from, 110-11;flow oftime dependent on, 159;as her-mitian, 138;potential energy in

as sum of binary interactions,120;in quantum physics, 138;special relativity and descriptionby, 170-72

Hamiltonian operator H:eigenvalues of, 133,138,140-42;inquantum theory, 133;and

Schrodinger equation, 136;instatistical formulation ofquantum physics, 144

harmonic oscillators, 39,102harmonics, 40,123Hartle, JamesB.,51Hawking, Stephen W.: on an-

thropic principle, 15-16;on

future of science,7; on imaginary

time, 58,164,170;on spatializa-tion of time, 165;on universe

starting with a singularity, 180nHegel,GeorgWilhelm Friedrich,

13Heidegger, Martin, 10,13,14Heisenberg, Werner, 134,157Heisenberg uncertainty principle,

74, 135-36,143Heraclitus, 10Herman, Robert, 174hermitian operators, 138H-function of Boltzmann, 20Hilbert space:as dynamical group,

104;dynamical groups outside,142;eigenvalues of Hamiltonian

in, 138;eigenvalues of Liouville

operator in, 113;equivalence ofindividual and statistical

description breaking down outside, 96;extending wave functions

beyond, 144\342\200\22445;in functional

calculus, 46;generalized functions

not included in, 92;going

beyond, 93,95-96,114,116,117,126,181;quantum mechanics as

operator calculus in, 95;riggedHilbert space,96;and spectral

representation with baker

transformation, 103,104Homer, 186-87

216Index

Hoyle, Fred, 166,173H-theorem ofBoltzmann,

20Hubble, Edwin Powell, 176Husserl, Edmund, 13

Iliad (Homer), 186imaginary time, 58,164,170indeterminism: as

anthropomorphic,14;as compatible with

realism, 132;as due to instabilityand chaos,56;in fundamental

laws of physics, 16;statistical

description of unstable systemsrequiring, 109

indistinguishability of elementaryparticles, 140

individual level of description:breaking equivalence with

statistical description, 35,83,87,89,94-95,96,103,106,155;inclassical and quantum physics,129;for integrable systems, 108;operators required for, 133;statistical description in terms ofLiouville operators comparedwith, 118-19;statistical level as

equivalent to, 34-35,42,81,108;thermodynamics as

incompatible with, 127;transition tostatistical level in quantum

physics, 143-46;validity as

limited, 127inertia, 110infinite velocities, 105\342\200\2246

information: and entropy, 24;probabilities as expressing lack

of, 34;probability distributions

providing additional, 37;secondlaw of thermodynamics as dueto lack of, 25-26

initial conditions, sensitivity to, 30,37,90

instability: big bang associated

with, 6; birth of the universeassociated with, 179;and classical

dynamics' revival, 31;in classical

physics, 4, 54,107;classical

physics requiring extension for,46;and distance from

equilibrium, 66;in dynamical systems,55,127-28;equivalence ofindividual and statistical descriptionsdestroyed by, 35;indeterminismas due to, 56;in laws ofnature,4, 155,189;and limits ofphysical concepts,29;in quantum

physics, 4, 53-54,107;at

statistical level of description, 155;time linked to, 175;time

symmetry broken by, 5, 37-38;at

trajectory level leading to

stabilityat statistical level, 87.Seealso

unstable systems

integrable systems, 39,44,54,108,131,144

interactions: defined, 147;betweenknower and known, 153;in rel-ativistic systems, 172;statistical

description affected by, 120-25.Seealso persistent interactions;transient interactions

interference, constructive and

destructive, 119interference terms, 52inverse Fourier transform, 117irreversibility: all processes

oriented in same direction, 102,104\342\200\2245;

as appearance only in

immeasurably long time, 23;approximation as responsible for,

23,24,81,91,105;big bang as

irreversible, 166;and birth ofthe universe, 181,183;inchemical reactions, 158-59,183;constructive roleof, 3, 26,27, 57;

Index 217

and dissipative structures, 73,183;and dynamical systems,27-28,126;emergence of,96,105;and entropy, 17,18-19;entropy created by, 61;gravitation-energytransformation as irreversible, 179;as

illusory, 2, 19,21,165;in laws ofnature, 38,96;leading to longmemory effects, 125;lifeassociated with, 63;and Lyapunov

time, 105;in macroscopicphysics, 45;matter as result of,159;in measurement, 49,53;innature, 18;in nonequilibriumphysics, 3; novel phenomenafollowing from, 3; and the

observer, 5; order and disorder in,

26;past and future asasymmetrical in, 28;in persistentinteractions, 114;probabilities for

describing, 35;progressing uplevels of organization, 162;inquantum physics, 53,138;reality of, 3, 25,27, 165;andsecond law of thermodynamics, 21;statistical description for, 81,108,109;thermodynamics as

scienceof, 17;as transient, 62

James, William, 1,13,14Jordan, Pascual, 134,175

KAM theory, 41Kant, Immanuel, 4, 10kinetic theory ofgases:Boltz-

mann's two-compartmentmodel, 22-23,76,91;Poincare

on, 35;probability in, 35;as

special caseof new approach,125

Kolmogorov, Andrei N., 41Kronecker, Leopold,84

Lagrange, Joseph-Louis, 28,58Landau, Lev Davidovich, 164Laplace, Pierre-Simon de, 11,14,

25,38,106large Poincare systems. Seenonin-

tegrable large Poincare systemslaw of conservation of wave

vectors, 121laws of chaos, 89-106;possibility

of speaking of, 87, 104laws of nature: in classical physics,

4, 138,184;creativity asamplification of, 71;eighteenth-century laws as deterministicand time reversible, 11;fundamental law of quantum physics,136;idealized world describedby, 26,184;and instability, 4,155,189;irreversibility in, 38,96;Poincare on laws ofthermodynamics, 55-56;probabilitiesin, 5, 29,35,38,44,132,189;in quantum physics, 4, 138,184;within the range of low

energies,6; reformulating

fundamental laws ofphysics, 16-17,108;statistical formulation of, 162;time's constructive role for, 56;when far from equilibrium, 65

Leclerc,Yvor, 7Leibniz, Gottfried von, 12,185Lemaitre, Georges-Henri,176,177Lewis, Gilbert N., 61life: and dissipative structures, 66;

duality in accounting for, 15;dynamical instability requiredfor, 128;as feeding on negative

entropy flow, 63;historicalcharacter of, 161-62;irreversibilityassociated with, 3, 63;nonequilibrium processesrequired for, 3,26-27;nonintegrable systemsrequired for, 39

218Index

Lifichitz, Evgeny Mikhailovich,164

light, velocity of, 105-6,167-68linear nonequilibrium

thermodynamics,63

Liouville equation, 113,136,144,146

Liouville operator: defined, 113;eigenvalues of, 113,118,120,149;for free particle, 118;Hamiltonian operator comparedwith, 136;spectralrepresentation of, 113,125,149;statistical

description in terms of, 118-19literature, time as theme of, 187localized distribution functions:

defined, 114;for transient

interactions, 114,115Lorentz, Hendrik Antoon, 134Lorentz transformation, 167,168Loschmidt, Joseph, 21,23LPS.Seenonintegrable large Poin-

caresystemsLucretius, 10,55Lyapunov exponent: in baker

transformation, 97-98,101;in

Bernoulli map, 84,87,90,93;comparison of neighboring

trajectories in, 105Lyapunov time, 105

macroscopic systems, 6, 45,115,128,162

macrostates, 24Mandelbrot, Benoit, 38Markov process,76matter: black body radiation in

equilibrium with, 131,145,174;as blind in equilibrium,127;as blind without arrow oftime, 3; as created at expenseofgravitational fields, 179,180,181;entropy associated with,

180;mind and, 16,49;new

properties acquired when far

from equilibrium, 65,67;permanent creation in steady-statetheory, 173;phase transitions,45,116;probabilities required tounderstand properties of, 47-48;as result of irreversibility, 159;and space-time for Einstein,176;states of matter, 45,116;transformation of space-timeinto, 180

Maxwell, James Clerk:on

ensembles, 76;and kinetic theory, 125;on new kind of knowledge, 4, 5

Maxwell-Boltzmann distribution,20

measurement: as actualizing

potentiality, 48;in Copenhageninterpretation, 156-57;infundamental description ofnature, 49;irreversibility in, 49,53;as means of communication,150;as probabilistic, 15

measuring instruments, 51,54,150-51

Meneceus,10meta-universe, 164,181microcanonical ensemble, 77microstates, 24mind and matter, 16,49Minkowski, Hermann, 165Minkowski space-time interval,

167,171,172,176,177, 178Minkowski vacuum, 179Misra, B.,169mixtures of wave functions, 47,

48,144morphogenesis, 68Moser,Jiirgen Kurt, 41motion, equations of.See

equations of motion

Nabokov, Vladimir, 154Nardone, Pasquale, 180

Index 219

Narlikar, Jayant Vishnu, 166,178nature: as automaton, 12,17;

Chinese and Japanese view of,

12-13;creativity of, 62;dialectical view of required, 182;distance from equilibrium as

parameter in describing, 68;duality in, 15,16;intelligibility of,

17,29;intermediate descriptionof, 189;mankind's position in,9; measurement included in

fundamental description of, 49;microscopic depiction of, 127;nineteenth-century views on as

conflicting, 17,19;probabilityas property of, 44;reversible andirreversible processesin, 18;science as dialogue with, 57,60,153;as semigroup that

distinguishesfuture and past, 104;

unity and diversity of, 56;unpredictable novelty in, 72.Seealso laws ofnature

Needham, Joseph, 12New Physics, The:A Synthesis

(Davies),48\"New Refutation ofTime, A\"

(Borges),187Newtonian physics: as absolute, 2,

28;deviations from, 126;equations of motion invariant with

respect to time inversion, 90;flow of time as universal in, 160,164,170;going beyond, 127;law relating forceand

acceleration, 11,109-10;limits ofvalidity of, 44,107-8;quantum

physics compared with, 129;asrealization ofDescartes'questfor certainty, 185;spaceandtime as given onceand for all,

164,176;unidirectional timedenied in, 2.Seealso classical

physics

nicefunctions, 92,94,95,96Nicolis, Gregoire, 71Nietzsche, Friedrich, 14nonequilibrium chemistry, 26,27,

67-68nonequilibrium processes:

complexity evolving in, 64;effects

produced by, 158\342\200\22459;

fluctuations in nonequilibrium systems,64,68-70;generation of as free

lunch, 175;irreversibility asconstructive in, 26-27;non-equilibrium physics required to

describethe world, 131;structure in nonequilibriumconditions, 63;unidirectional time in,3

nonequilibrium thermodynamics:DeDonder'swork on, 61;linear

nonequilibriumthermodynamics, 63;and views of Bergsonand Whitehead, 72

nonintegrable large Poincare

systems (LPS):continuous spectrain, 133-34;defined, 111;and

deterministic chaos, 112;example of, 111-12;formulation in

quantum mechanics, 148;individual and statistical descriptionnot equivalent for, 106;majorityof dynamical systems as, 127;as

measuring themselves, 151;resonances in, 141-42

nonintegrable systems, 39-41;arrow of time requiring, 39;dynamical decomposition in, 102;glass ofwater as, 78;KAM

theory of, 41;resonances as reason

for, 39-40,109,111;as rule not

exception, 108;thermodynamiclimit corresponding to, 45.Seealso nonintegrable large Poincare

systemsnonlocal description, 37,42,96

220 Index

nonlocality, in quantum theory,130

observer: in cosmology, 51;entropy increasing with

observation, 50;indeterminism asindependent of, 132;macroscopic

character of observation,23;in quantum physics, 5,48-55,131,151;secondlaw ofthermodynamics based on

ignorance of, 25;in special relativity,

167,169Odyssey (Homer), 186Open Universe,The:An Argument for

Indeterminism (Popper), 1,14operator formalism, 92,133,134operators: collision operators, 124;

description requiring, 133;eigenfunctions of, 134\342\200\22435;

evolution operator, 93,125-26,142;hermitian operators, 138;introduction into physics, 134;operator formalism, 92,133,134;quantum physics as

operator calculus in Hilbert space,95.Seealso Hamiltonian operatorH; Liouville operator; Perron-Frobenius operator; spectralrepresentation of an operator

order:arrow of time as sourceof,

26;disorder as constitutingfoundation ofmicroscopicsystems, 155;entropy and, 26;self-organization maintaining,71-72.Seealso entropy; self-organization

oscillating chemical reactions, 66,127,175

oscillators, harmonic, 39,102

Parmenides, 10past: in chaotic and simple

dynamical systems, 102,104;complex

conjugate of wave functions

propagating into, 137;in formal

solution of Schrodingerequation, 142;and future as

asymmetrical in irreversible processes,28;and future meeting in

probabilities, 137;and future not

distinguished in physics, 2, 138;interaction between knower andknown creating, 153;inNewton's relation of acceleration and

force, 110;retrodicting in

classical science,4; in specialrelativity,

168Pauli, Wolfgang, 50,53pendulum, 30Penrose, Roger, 16,130,180nPenzias, Arno, 174periodicfunctions, 116,117periodicmaps, 82,82-83Perron-Frobenius equation, 88,

104Perron-Frobenius operator: in

baker transformations, 103;inBernoulli maps, 83,87-88,91,94,96

persistent interactions, 113-14;de-localized distribution functionsfor describing, 114,115,117,125,148,155-56;diffusive

terms in, 44\342\200\22445, 54;leavingHilbert spacedue to, 96,114;inmacroscopic systems, 115;persistent scattering, 148-49;inthermodynamic systems,115-16,148

perturbational approach to solvingfor eigenvalues, 140-42

phase space:defined, 31;distribution functions accounting for

microstructure of, 95;dynamicalstate represented in, 31,32,110;ensemblesrepresented in, 33,34,76; resonances'influence

Index 221

on, 41;state of a classical systemin, 136;statistical descriptionaccounting for microstructure of,105

phase transitions, 45,116phenomenology, arrow of time

relegated to, 2, 3philosophy: scienceas separated

from, 7, 14,72;as time

centered, 13;time for, 58;unhappy

history ofWestern, 11photons, 172,174physics: Boltzmann's evolutionary

approach to, 2, 19-21;domainof validity, 29;Galilean physics,1,167;idealized worlddescribed by, 26;operatorformalism in, 92;probabilities as basic

objectsof, 74; reformulatingfundamental laws of, 16-17,108;resonance playingfundamental role in, 42;statistical

mechanics, 46,92,95;time for, 58.Seealso classical physics;cosmology;

Newtonian physics;quantum physics; relativity;

thermodynamicsphysics of populations, 35pitchfork bifurcation, 68,

69Planck, Max, 131,145Planck era, 178Planck's constant, 142,145,148,

177Planck's energy, 178Planck's length, 177Planck's scales,177-78Planck's time, 177plane waves: defined, 116;

superposition of, 117,118-19,119,157-58;trajectory as construct

of, 119Plato, 11Plato, Jan von, 84

Poincare, Henri: on distinguishingbetween dynamical systems, 31;on dynamical systems, 38-41;on explaining irreversibility in

terms of trajectories, 21;on free

Hamiltonians, 111;on generalproblem of dynamics, 40;onkinetic theory of gases, 35;onlaws of thermodynamics, 55-56

Poincare resonances.Seeresonances

Poincare's recurrence theorem, 23,39,100-101

Popper, Karl, 1,6, 14,132population dynamics, 33populations: aging as property of,

78, 125;Boltzmann and Darwin

studying, 20;phase transitions

meaningful only at level of, 45;physics of, 35

possibilities, 5, 29\"Possible and the Real,The\"

(Bergson), 14,59possible worlds, 59,72postmodern philosophy, 14predictability: in classical science,

4, 11;with deterministic chaos,38;predictive successofquantum theory, 46;unpredictablenovelty in nature, 72

pre-Socratics,9-10,17principle of conservation of

energy, 76-77principle of indistinguishability of

elementary particles, 140probabilities: as basic objectsof

physics, 74;as basic property ofnature, 44;and birth of the

universe, 181;for diffusive motions,42,43;evolutionarycharacteristics describedin terms of, 55;as

expressing ignorance, 34;asextended form of rationality, 155;in fundamental laws of physics,

222 Index

probabilities (cont.)16-17;in laws of nature, 5, 29,35,38,44,132,189;inmacroscopic physics, 45;inmeasurement, 15;in nonequilibriumsystems, 68;past and future

meeting in, 137;probability

amplitudes giving way to, 54,132,149;the probabilizingrevolution, 55,74, 132;in quantum

physics, 5, 51-52,54,132,149;secondlaw of thermodynamicsas probabilistic, 20,22;subjective interpretation of, 4, 16;astime symmetric, 137;trajectories and, 34;transition

probabilities, 75; for unstable systems, 35.Seealso statistical level ofdescription

probability amplitudes: dynamical

problems solved in terms of, 47;giving way to probabilities, 54,132,149;in physical

interpretation of wave function, 47;quantum state representing, 143;Schrodinger equationdescribing, 137;wave function

corresponding to, 136probability distributions. See

distribution functions

problem of small denominators, 40propagation event (diagram), 121,

121

quantum leaps, of excitedatoms,138-39,139

quantum paradox, 48,130;andstatistical formulation ofquantum physics, 144;time paradoxand, 5, 48,138,156,157

quantum physics: as absolute,28-29;anthropomorphicfeatures of traditional, 151;basic

assumption of, 47;basicconclusions of, 149;basic problem of,133,140;chaos introduced in,4; classical mechanics limited by,

107;clinamen conceptintroduced into, 52;complementarity principle, 74, 150;Copenhagen interpretation, 50,150,156-57;as deterministic,136-37;discrete and continuous

spectrums in, 133,141;dualism

in, 50,53,130,131;eigenfunc-tions and eigenvalues in, 95,133;extending to unstable

systems, 89,131;Feynman on

nobody really understanding, 47;fundamental law of nature in,136;Hamiltonian in, 138;Heisenberg uncertainty

principle, 74, 135-36,143;as

incomplete, 130-31;instability in, 4,53-54,107;irreversibility in,

53,138;laws of nature in, 4,136,138,184;limits of validity

of, 44,54;meaning and scopedebated, 46-47,129-30;mi-crostates and macrostates, 24;Newtonian dynamics comparedwith, 129;nonlocality in, 130;observer's rolein, 5, 48-55,131,151;as operator calculus in

Hilbert space,95;operatorformalism in, 92,133,134;predictive successof, 46;probabilitiesin, 5, 51-52,54,132,149;realistic interpretation of, 54,131;reformulating, 46-55,129-51;resonances in, 53-54,146,148;state of a quantum system, 136,143;and thermodynamics, 131;time flow introduced in, 5;time-reversible processesin, 28;transition from individual to sta-

Index 223

tistical description in, 143\342\200\22446;

unidirectional time denied in, 2.Seealso quantum paradox; wave

functions

quantum vacuum, 175,179,180

radiation: black body, 131,145,174,176;solar, 18

radiation damping, 172radioactive decay, 17,18Rae,Alastair, 157random trajectories, 41random walk, 42,43,74realism: indeterminism as

compatible with, 132;naive realism ofclassical physics, 135;realistic

interpretation of quantum theory,54,131;realistic interpretationof reduction of wave function,130;time and change as crux of,14

recurrence:in baker

transformation, 99-101;Poincare'srecurrence theorem, 23,39,100-101

recurrence relation, 91-92reduction (collapse)of wave

function: with clinamen version ofquantum mechanics, 52;defined, 48;measurement problemleading to, 157;and quantum

paradox, 130;in realistic

interpretation of quantum

mechanics, 54relativity: classical mechanics' limits

shown by, 107;and infinite

velocities, 105;time for, 164-66;unidirectional time denied in, 2.Seealso general relativity; specialrelativity

resonances, 39-44;bubbles due to,123-24,124;in classical and

quantum physics, 129;constructive interference threatened by,

119;coupling creation anddestruction of correlations, 124;and diffusive motion, 42-44,43,126,155;dynamical groupsaffected by, 171;expressing in

terms of delta functions, 124n;flow of time rooted in, 159;fundamental role in physics, 42;leading to chaos, 112;leading toterms with dangerousdenominators, 40,124n;in LPS,141-42;and nonintegrability,

39-40,109,111;as nonlocal,42;in oscillator interaction with

field, 112;and perturbationalapproach, 141;in quantum

physics, 53-54,146,148;sounds

coupled by, 40;statistical

description affected by, 122-25;superpositions of plane wavesaffected by, 157-58;time

symmetry broken by, 41,44,146-47;trajectories influenced by, 41;intransition from wave functionsto ensembles, 131;and velocitydistribution over time, 79

retrodiction, 4reversible processes.Seetime-

reversible processesRiemannian geometry, 176rigged Hilbert space,96Rimini, Emanuele, 52Ritz-Rydberg principle, 132,145,

146,149Robertson, Howard, 177Rosenfeld, Leon,29,50,53Rubino, Carl, 186

scattering, 114\342\200\22415, 115;as not

representative of natural world,127;persistent scattering,148-49;typical experiments as

idealizations, 156

224 Index

scattering center, 115,115Schrodinger, Erwin, 63,134Schrodinger equation: complex

conjugate of, 137;asdeterministic and time reversible, 12,15,47, 137,146;formal solution of,142;as partial differential

equation, 136;probability amplitudesdescribedby, 137;and reductionof the wave function, 48,130;trajectories compared with, 47;wave function evolutiondescribed by, 15,47, 136

Schrodinger's perturbationalmethod, 140-41

Schuster, Peter, 71science:classical ideal of, 154;

creativity in, 188;democracy as

conflicting with, 17;Descarteson, 185;as dialogue with nature,57,60,153;Einstein on, 185;Freud on history of, 70;Hawking

on future of, 7; as not

monolithic, 17;philosophy as

separated from, 7, 14,72.Seealso biology; laws of nature;

physicsscientific laws. Seelaws of nature

Scott, Peter, 185Searle,JohnR.,14secondlaw of thermodynamics,

19;classical formulation of, 19,60;entropy associated with,

18-19,60;and irreversibility,

21;observer's ignorance as basis

of, 25-26;as probabilistic, 20,22;for systems that are not

isolated, 61self-organization: in dissipative

structures, 70;and distance from

equilibrium, 57; in nonequilib-rium physics, 3, 27;noninte-

grable systems required for, 39;

order maintained by, 71-72;technology compared with,

71-72;time orientation

required for, 128self-replicating biomolecules, 159semigroups, 104,142,171sensitivity to initial conditions, 30,

37,90Shimony, Abner, 53simultaneity, 168singular functions. Seegeneralized

functionssmall denominators, problem of,

40Smoluchowski, Roman, 23smoothness, statistical descriptions

requiring, 87\342\200\22488

Snow, C.P.,17solar radiation, 18space-time:arrow of time

consistent with, 172;conformalintervals in, 178;Friedmann

space-time interval, 177, 178;gravitation associated with

curvature of, 175-76;and matterfor Einstein, 176;Minkowski

space-time interval, 167,171,172,176,177,178;quantization

of, 178;Riemannian geometryas characterizing, 176;in specialrelativity, 167;transformed into

matter, 180spatialized time, 58-59,165-66,

171-72special relativity, 167-72;future

and past in, 168;and Hamilton-ian description, 170-72;spatialized time not implied by,

171-72;time dilation, 169,170;twin paradox, 169-70,111

spectral interval, 117spectral representation of an

operator: of evolution operator,

Index 225

125-26;of Liouville operator,113,125,149;ofPerron-Frobenius operator, 94\342\200\22495, 96,103,104

spectroscopy, 133,144,145spectrums, continuous and

discrete, 133,141Spinoza, Baruch, 16,176stability: instability at trajectory

level leading to at statistical level,87;stable dynamical systems,30-31,36,55;and structure in

nonequilibrium conditions, 63standard model (cosmology),164,

174-75,177state of a dynamical system, 31,32state ofa quantum system, 136,

143states of matter, 45,116statistical level of description:

breaking equivalence with

individual description, 35,83,87,89,94-95,96,103,106,155;for chaotic systems, 104;inclassical and quantum physics, 129;in classical dynamics, 108;fordeterministic chaos, 105;dynamical problems solved at,91-96,113,125-28;individual

level as equivalent to, 34\342\200\22435, 42,81,108;instability at, 155;interactions' effect on, 120-25;irreversibility given meaning at,81;for irreversible processes,81,108,109;of laws of nature, 162;of molecules in glass of water,78;operators required for, 133;replacing coordinates with wave

vectors, 121;resonances'effect

on, 122-25;in terms of Perron-Frobenius operator, 87-88;thermodynamics requiring, 127;trajectory description compared

with, 118-19;transition to in

quantum physics, 143-46;for

unstable systems, 109statistical mechanics, 46,92,95steady-state theory, 173-74;

difficulties with, 174;as unified with

big bang, 166Stengers, Isabelle, 154structure: arrow of time in

formation of, 71;and function, 62;innonequilibrium conditions, 63.Seealso dissipative structures

Sudarshan, George,142superimposition, 48,52superposition: of periodic

functions, 116;of plane waves,118-19,119,157-58;principleof, 149;of wave functions,138-39

symmetry breaking. Seetime

symmetry

synthetic chemistry, 71

Tagore, Rabindranath, 13Tarnas, Richard, 7

technology, self-organization

compared with, 71-72ternary correlations, 121test functions, 33n

thermal diffusion, 26,21,35,64thermodynamic branch: beyond

bifurcation point, 66,61;defined, 66;in pitchforkbifurcation, 68,69

thermodynamic equilibrium, 60,63,66,77, 113,130

thermodynamic limit, 45,116,126,158

thermodynamics: equilibriumstatistical thermodynamics, 77;equilibrium thermodynamics,88,108;first law of, 180;as

incompatible with trajectory

226 Index

thermodynamics (cont.)

description, 127;limiting to

equilibrium conditions, 61-62;nonequilibriumthermodynamics, 61,63,72;Poincare on laws

of, 55-56;and quantum theory,131;as scienceof the

irreversible, 17;traditional conflictwith dynamics eliminated, 104.Seealso entropy; secondlaw ofthermodynamics

thermodynamic systems, 115-16,148,158,184

three-body problem, 31,108time: as basic existential

dimension, 13;beginning of, 6,163-64;certainty and denial of,184;in classical science,59,60;communication requiringcommon, 150;constructive roleof,56;as crux of realism, 14;anddilemma of determinism, 1;as

emerging property, 60;entropictime, 161;as eternal, 166;ethicsassociated with, 58;freedomassociated with, 58;geologicaltime scale,161;as history

dependent, 170;as illusory, 1,58,165,187;imaginary time, 58,164,170;instability linked to,175;as literary theme, 187;inmaps, 81;of a meta-universe,164;in Newton's relation ofacceleration and force, 110;philosophers on, 58;philosophybecoming time centered, 13;physicists on, 58;as precedingexistence, 163,182;and realityas linked, 187;in relativity

theory, 164\342\200\22466; simultaneity, 168;spatialized time, 58-59,165-66,171-72;as universal in

Newtonian physics, 160,164,170.See

also arrow of time; evolution;flow of time; future; past; space-time; time paradox; time

symmetry

time dilation, 169,170time paradox: and baker

transformation, 103;defined, 2;formulation as creative act, 188;mind-matter interface at coreof,49;quantum paradox solved

along with, 5, 48,138,156,157time-reversible processes:as almost

inconceivable, 154;baker

transformation as, 101,103;inclassical and quantum mechanics, 28;classical physics as basis ofbeliefin, 107;and entropy, 18-19;irreversibility required for

studying, 49;irreversible processescompared with, 17-18;ofphysics versus time-centered

philosophy, 14;Schrodingerequation as, 137;time-reversibleworld as unknowable, 153;trajectory as, 21;wave function

satisfying time-reversible

equation, 5time symmetry: Bernoulli maps as

breaking, 90;breaking as global

property, 156;chaotic systemsbreaking, 105;complex spectralrepresentation as breaking, 126;dissipative structures breaking,69;instability breaking, 5,37-38;measuring deviceas

breaking, 54,150-51;non-Newtonian processesbreaking,108,124,129;probabilities as

time symmetric, 137;resonances

breaking, 41,44,146-47;semigroups breaking, 171

total energy, 76, 175Toulmin, Stephen, 184-85

Index 227

towns, 62trajectories: and Bernoulli maps,

83-88,85;collapse of, 109,119,126;as construct of planewaves, 119,157-58;and delta

functions, 33,143;ensembles

compared with, 82,83,87;as

idealizations, 37;as inadequatefor describing chaotic systems,90-91,105;laws of chaosparadoxical with, 104;limited

validity of, 44;in Lyapunov

exponent definition, 105;as not

smooth, 87\342\200\22488;as primitive,

108;and probability

distributions, 34;resonances' influence

on, 41;secondlaw ofthermodynamics and individual, 20;as

special solutions of Perron-Frobenius equation, 88;three-

body problem compared with,31;as time reversible, 21;wave

function compared with,

47

trajectory description. Seeindividual level of description

transient interactions: defined, 114;diffusive terms as negligible in,44;localized distributionfunctions for describing, 114,115;scattering as, 115,127;asunrepresentative of nature, 127

transition probabilities, 75

transport processes,18,105Tryon, Edward, 175Turing, Alan Mathison, 68,69twin paradox, 169-70,111;non-

relativistic analogue of, 160two-body motion, 102,156

\"ultraviolet\" divergences, 180-81uncertainty, resonances introducing

into classical mechanics, 44

unidirectional time. Seearrow oftime

unified theory, 182uniform distribution, 93universe, the. Seecosmologyunstable systems: contradiction

between reversible and irreversible

processesovercome in, 28;dynamics formulated at statistical

level for, 73-74;ensembles

deploying new properties for, 81;extending classical and quantummechanics to, 89;going beyondHilbert spacefor, 93,95-96,114;as probabilistic, 35,37;quantum theory of, 132;sensitivity

to initial conditions in, 37;statistical description of, 109;and unidirectional time, 2-3;unstable equilibrium, 30,30.Seealso instability

urn model, 74-76,75, 81

vacuum of correlations, 121,122,122,123,123

Valery, Paul, 106velocities, infinite, 105\342\200\2246

velocity inversion, 80velocity of light, 105-6,167-68velocity-reversal paradox, 21,23viscosity, 3, 17-18von Neumann, John, 49,51,53,

130

Wahl, Jean, 11Walker, Arthur, 177wave functions: as deterministic

and probabilistic, 15;deterministic, time-reversible equationsatisfied by, 5; evolution as

superposition of oscillating terms,138-39;extending beyondHilbert space,144-45;mixture

228 Index

wave functions (cont.)of, 47,48,144;operatorformalism required for descriptionof, 133;physical interpretationof, 47;and probability

amplitudes, 136;resonancesintroduced in terms of, 53-54;stateof quantum system describedby,

136,143;trajectory comparedwith, 47; transition to

ensembles,131.Seealso reduction

(collapse)of wave function;

Schrodinger equation;superposition

wave vectors: defined, 116;andFourier transform, 117;andinteractions' effect on statistical

description, 120-25;law ofconservation of, 121;quantummechanical counterpart for, 147,148;replacing coordinates

with, 121;in statistical

description of free particle motion,

118;vanishing wave vector,124n

Weber, Tullio, 52Weinberg, Steven, 15,48Weyl, Hermann, 84Wheeler, John Archibald, 164Whitehead, Alfred North:

compromise between scienceand

freedom, 10;on creativity ofnature, 62;on existence as

process,59;on intelligibility ofnature, 17,29;and nonequilib-rium thermodynamics, 72;philosophy as time centered, 13;on

slipping through the scientific

net, 189Wigner, Eugene, 49Wilson, Robert, 174

X mysteries, 130

Zermelo, Ernst, 23Z mysteries, 130

ALSOBY ILYA PRIGOGINE

Order Out of Chaos

From Being to Becoming

Exploring Complexity

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