ilya prigogine - time dynamics chaos

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TIME, DYNAMICS AND CHAOS*)Integrating Poincare's "Non-lntegrable Systems"

Ilya Prigogine,'> ,*,>

Time has alway s haunted man. Time is indeed our fundamentalexistential dimension. It has fascinated philosophers as well asscien tists. It has often been stated that scien ce has solve d the problemof time. Is this really true? Indeed, a very fundamental property of

r \ ^ the basic equations of physics, be it classical physics or quantum3 ^ physics, i s time reversibility. We may, in these equations, replace t by

-t without changing the form of these equations. In contrast, in themacroscopic world, we deal with irreversible processes: +t and -t do notplay the same role. There exists an "arrow of time." We come,therefore, to the strange conclusion that in the microscopic dynamic

^ world, there would be no natural time ordering in contrast of whati ' happens in the macroscopic world. For example, if we consider two

^s positio ns of a pendulum, as represented in Figure I, we cannot sayM ) which position comes earlier.

Figure I

*> Prcscntaiion al Nobel Conference XXVI, Gusiavus Adolplms College. SJIIIIPcicr. Minnesota. Ociohcr T anfl J. 1990: **' The Solvay Insmutes***) Ccnicr tor Studies in Sialisiical Mechanics and Complex Systems, TheUniversity of Texas al AuMm

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DISCLAIMERThis report was prepared as an account of work sponsored by an agency of the UnitedI Statesr X I NeTher the United States Government nor any agency thereof, nor any of theirwmMmimand opinions of authors expressed herein do not necessarily state orUnited States Government or any agency thereof.

With classical dynamics, time has lost its direction. Similarly, inquantum mechanics, we cannot speak about "older" or "younger" wavefunctions. But can this be the whole story? How can the time emergefrom a time-reversible world? This conflict has become quite evidentsince the formulation by Clausius in 1863 of the well-known second lawof thermodynamics. Clausius stated, "The entropy of the universe isincreasing." This was the birth of evolutionary cosmolo gy.

Figure 2

For every isolated system, entropy can only increase. Entropyexpre sses, as Eddington used to say, "the arrow of time." [Theformulation of thermodynamics was the result of the work of engineersand physical chemists. The'gr eat mathematicians and phys icists' of thistime considered it as the best as a useful practical tool however withoutany functional significan ce. The first to ask the question of the relationbetween entropy and the microscopic equations of motion w asBoltzmann . Boltzmann was one of (he main founders of kinetic theory. ; .He tried to explain the increase of entropy as the result of molecular "'':-collisions leading to molecular disorder (the Maxwell velocitydistribution law ). Boltzmann 's approach is still of great importancetoday as it leads for dilute gases to results that are in excellent

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agreement with experiment. Still, Boltzmann was defeated, as people fS ^J *were quick to point out to him that his results clashed with dynamiclime reversibility (see i .e. ')) , Boltzmann was like a man in love withtwo women. He could not choose between his conviction that time \ n

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irreversible evolution was an essential aspect of nature, and hisconfidence in the classical equations of motion which seem to preventthe existence of a privileged direction of time. 1 cannot go into detailsabout this question, but let me stress that one of the aims of this lectureis to show that Boltzmann was right, but this involves quite recentresults in which modern chaos theory plays an essential role. For

i,Boltzmann's generation, as well as for the generations that followed, the-, , ' l .' i

conclusion of this debate was that the arrow of time was not in nature,but in our mind. Einstein's saying, time [as irreversibility] is an"illusion," is well known.')

I always found it curious that this conclusion did not trigger acrisis in science. How can we deny the existence of a privilegeddirection of lime? As Popper wrote "this would brand uni-directionalchange as an illusion. This would make our world an illusion and withit, alt our attempts to find more about our world." The ambition ofclassical science was to describe the behavior of nature in terms ofuniversal, time-reversible taws. It is interesting to reflect on therelation between this ambition and the theological concepts thatprevailed in the 17th century. For God, there is, of course, no distinctionbetween past, present, and future. Is science not bringing us closer toGod's like view of the universe?2) This ambition of classical science wasnever realized. Often, science seemed close lo this goal, and every limesomething failed. This gives a dramatic form to the history of wesiernscience. As you know, quantum mechanics is based on Schrodingcr'sequation, which is time reversible, but it had to introduce themeasurement process and with it to attribute a fundamental role to theobserver to obtain a consislem description. General relativity started as

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a geometrical, "timeless" theory, lo discover the need for some initialsingularity or instability to obtain a consistent description of thecosmological evolution of our universe. To describe nature, we needboth laws and events, and this, in turn, implies a temporal clement,which was missing- in the- traditional presentation of dynamics includingquantum theory and relativity. Last year's Nobel conference inMinnesota had the provocative title, "The End of Science? " I don'tbelieve we can speak about the end of science, but indeed we come tothe end of a certain form of rationality associated to the classicalideology of science. As I want lo show here, in the building up of thisnew scientific rationality, non-equilibrium physics and "chaos" certainlywill play an essential role.

II . T he Time Paradon Pwilds U pThe 20th century is characterized by the discovery of quite

unexpected features in which 'the arrow of time is essential. Examplesare the discovery of unstable elementary particles and of evolutionarycosmology. 1 would like, however, first to emphasize in this lectureprocesses involving a macroscopic scale such as studied in non-equilibrium physics. A first remark: Contrary to what Boltzmannbelieved, irreversibility plays a co nsi ruc tiv e role. It not only isinvolved in processes leading to disorder, bill also can lead to orderThis already appears in very simple examples such as presented inFigure 3. Consider two boxes containing two components, say hydrogenand nitrogen. If ihc boxes were at the same temperature, theproportion of these two components would be the same in the two

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compartm ents. If, on the contrary, we establis h a temperaturedifference, we observe that the concentration of one of the components,say hy drogen, bec ome s larger in the, compartment that is at the highertemperature.

Figure 3

The disorder associated with the flow of heat is used in this experimentto create "order". This is quite characteristic. Irreversibility leads bothto order and disorder. A striking example is the case of chem icaloscillations. Suppose we have a chemical reaction that may transform"red" molecules into "blue" ones and vice versa. It has been shown boththeoretically and experimentally that, far from equilibrium, such areaction may present a time-periodic behavior. The reaction vesse lbecomes in succession red, then blue, and so on. Let me emphasize howunexpected the appearance of chemical coherence is. We usuallyimagine chemical reactions as the result of random collisions betweenthe molecules. Obviously, this cannot be the case far from equilibrium.We need long-range correlations lo produce chemical oscillations. Whenwe push such systems further away from equilibrium, the oscillationsmay become quite irregular in time. One then speaks about "dissipaiivechaos"; however, I shall not go into more details about this subject,which is treated adequately in many texts.3)4) What is important isthat irreversibility leads to new spacetime structures (which I havecalled "dissipaiive structures"), and which are essential for theunderstanding of the world around us. Therefore, irreversib ility is"real", it cannot be in our mind, and we have to incorporate it in one

way or another in the frame of microscopic dyna mics . Recently , therehave been many monographs dealing with this problem, but I like tomention here the excellent introduction due to Peter Coveney and RogerHighfield, The Arrow of Time*) In this book, they called this problem,"time's greatest mystery."

But how to go beyond this paradox?In the work done by my colleagues and me, we have followed the

idea that the arrow of time must be associated with dynamicalinsta bilit y. Let me first present a very simple example of an unstabledynamic system, the so-called Baker transformation. See Figure 4. Weconsider a square. We squash it and put the right part on the top of theleft as seen on Figure 4. This leads to a progress ive fragmentation ofthe surface of the square. This is obviously an unstable dynamicsystem, as two points as close as one wants will finally show up indistant stripes. Such a system can be characterized by a Lyapounovexponent:

(Sx), = (8x)0 exp(Xt)

Figure 4

The distance (Sx), between two neighboring trajectories increasesexponentially with time. The coefficient, X, is called the Lyapounovexponent, and is, in the case of the Baker iransformation, equal to Ig2.The existence of a positive Lyapounov exponent is characteristic of"chaotic" systems. There exists, then, a temporal horizon beyond w hichthe concept of trajectory fails, and a probabilistic description is to be

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used. This is all well known. What I want to emphasize, however, isthat in addition to the kinematic time t, we can introduce for suchsystems a "second, internal" time, T, which measures the number ofshifts required to produce a given partition. For exam ple, starting fromthe state [a] of Figure 4, we need two shifts to obtain state [c]. As hasbeen shown by our group, especially in Ihc work of Prof. Misra,6) thisinternal time is represented by an operator that leads to a non-commutative algebra very much like one we are using in quantummechanics. Once you have the internal time, it is easy to construct anentropy and therefore to associate lo the Baker transformation an arrowof time.

It is important to notice that the internal lime refers lo a globalproperty as expressed by the partitions of the square. It is a"topological" property. It is only in considering the square as a wholethat you can associate to it a given internal time or an "age." It is tikewhen you look at some person. The age you will attribute to him doesnot depend on a specific detail of his body, but results from a globaljudgment. A detailed presentation of the Baker iransformation and iisrelation to the arrow of time can be found in my book with Prof.Nicol is .4 ) How ever, the Baker transformation corresponds to a highlyidealized situation, and it is nol clear on this basis why the arrow oftime would be so prevalent in nature as testified by the universalvalidity of ihe second law of thermodynamics. Thai is the problem towhich I want to turn now.

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III. Poincare's Theorem and the Science of Chaos-Large Poincarl Systems

In 1889, Poincarl asked a fundamental question.7) 8> I shouldmention that his question was not formulated in these terms, but this isthe formulation 1 shall use for the sake of the discussion. Poincareasked if Ihe physical universe is isomorphic lo a system of non-interacting units. As it is well known, the energy (the "Hamittonian" ll |is generally formed by Ihe sum of two terms, the kinetic energy of theunits involved and the potential energy corresponding to theirinteractions. Therefore, Poincare's question was. "Can we eliminate theinteractions?"

This is indeed a very important question. If Poincard's answerhad been yes, there could be no coherence in the universe. There wouldbe no life, and no Nobel Conferences. So it is very fortunate that heproved thai you cannot, in general, eliminate interactions; moreover, hegave the reason for this result. The reason is the existence ofresonances between the various units.

Figure 3

Everybod y's familiar with Ihe idea of resonance. This i s the way thechildren learn to swing . Let's formulate more precisely Poincare'squestion. We start with a Hamilionian of ihe form,

H = H(p,q) (3 .1)

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where p,q are the momenta and the coordinates. We then ask thequestion if we can reduce it to the form

H = H(J) (3.2 )

where J are the new momenta (the so-called action variables). In thisform, the Hamiltonian depends only on the momenta. To perform thetransformation from (3.1) to (3.2) Poincare considered the class oftransformations which conserve Ihe structure of the Hamiltpnian theory(so-called canonical or unitary transformations). More precisely.Poincare considered Hamiltonians of the form

H = HoO) + X V(J.o) (3.3)

where X is the coupling constant and V is the potential energy whichdepends both on the momenta J and the coordinates a (called theangle variables ). For two degrees of freedom the potential can beexpanded in a Fourier series

V< J 1.J 2.a 1.a 2) = I V . ^ J p I j ) j^l+W) (3 .4)n, ,n 2

where n, ,! ^ arc integers. The application of perturbation techniquesleads then to expressions of the form

V n ' n 2 (3 .5)n|Q>|+n2b>29

with the frequencies u> l defined as to, =3 ll 0 /3 J i . Here we see thedangerous role of resonances (or "small denominators")

n|Ci)| + n22 = 0 ( 3 . 6 )

Obviously we expect difficulties when (3.6) vanishes while thenumerator in (3.5) does nol. This has been called by Poincare 7) the"fundamental difficulty of dynamics". We come in this way toPoincare's classification of dynamical systems.7)8)' 0) If there are"enough" resonances, Ihe system is "non-integrable". A decis iveprogress in our understanding of the role of the resonances has beenachieved in the* SO's by Kolmogorov, Arnold, and M oser [the so-calle dKAM theory]. (See, i.e., ').) They have shown thai if the couplingconstant in (3.3), X, is small enough, [and also other conditions which I

.shall not discuss here are satisfied,"! "most* trajectories remain periodicas in integrable systems. This is not astonishing. Formula (3 .6) can bewritten as

112= - a rational numberW 2 n,Now rationals are "rare" as compared to irrationals. How ever, w hateverthe value of the coupling constant X , there appear now in addition,random trajectories characterized by a positive Lyapounov exponentand therefore by "chaos". This is indeed a fundamental result, since it isquite unexpected to find randomness at Ihe heart of dynamics, which

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was always considered to be ihe stronghold of a deterministicdescription. However, it should be emphasized thai the KAM theory hasnot solved the problem of the integration of Poincare's non-integrablesystems. The statement by Arnold that dynamical systems with evenonly two degrees of freedom, lie beyond our present mathematics hasbeen widely quoted.

But there is a class of dynamical systems we call large Poincaresystems [LPS], for which we may indeed eliminate entirely Poincare'sdivergences and therefore indeed "integrate" a class of Poincare's "non-integrable systems. This result is the outcome of years of research withmy colleagues in Brussels and Austin9); however, it is only recently thaithe problem of Ihe integration of ,large Poincare syste ms has beensolved . I want to acknowledge from Ihe start the fundamentalcontributions of Tomio Petrosky as well as of Hiroshi Hasegawa andShuichi Tasaki.10 )- '4)

First, what is a large Poincare system? It is a system with a"continuous" spectrum. For example, Ihe Fourier series in formula (3. 4)has now to be replaced by a Fourier integral. The resonance cond itionstakes then a new form. The resonance conditions for a small sys temwith an arbitrary number of degrees of freedom are (see 3.4)

n,*)! + n2W2 + ^013 + ... = 0 (3 .7)

where the n, are integers. As mentioned, Ihe resonance cond itionsexpress the existence of rational relations between frequencies. Forlarge Poincare systems, condition (3.4) has to be replaced by

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where a>k are frequencies associated to the radiation and C0| is theenergy level associa ted to the unstable state. This is an example ofintegration of resonances. All many-body systems involving "collisions"are LPS (see Fig. 7) as collisions also involve resonances (see Section V).

Figure 7

Large Poincare systems are not integrable in the usual sense because ofthe Poincare resonances, but what we want to point out in this lecture isthat we can integrate Ihem through new methods eliminating allPoincare divergences. This leads to a new "global" formulation ofdynamics [classical or quantum]. As we deal here with chaotic systems,we may expect new features in this formulation of dynam ics. Indeed,we shall find, as compared with the dynamics of integrable systems, anincreased role of randomness, and above all a breaking of timesymmetry and therefore the emergence of irreversibility al the heart ofthis new dynam ics. We, in a sense, invert ihe usual formulation of thetime paradox. The usual attempt was to try to deduce Ihe arrow of timefrom a dynam ics based on time reversible equa tions. In contrast, wenow generalize dynamics to include irreversibility.

We may summarize the situation as follows.

Diagram I

On top we have the class of integrable systems (classical or quantum).This is the main field explored by dynam ics. The basic structure of thedynamic integrable systems is expressed by celebrated laws (or

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principles) such as the action principle which states that the trajectoryis such that some functional (the action) is minimum. This structure hasbeen the starting point both for quantum mechanics and generalrelativity.

But Poincare's theorem limits the class of integrable syste ms. Thebasic question is then: what happens next? How nature solves theproblem of the small denominators. Computer calcula tions do not leadlo infinites!

As mentioned, a first step was the KAM theory. The physicaleffect of resonances is the appearance of random motio n. For LPSalmost all motions are random. The remarkable fact is that we thenagain can "integrate" the equations of motion. But now the structure ofdynamics becomes radically different from thai of integrable systems.It is fascinating that we now have to deviate from ihe structureinherent in the dynamic scheme as associated with the classicaltradition.

IV. Poincare's Theorem and the Quantum Mechanica lEigenvalue Problem

The first example I want to consider refers to quantum mechanics.As is well known, quantum mechanics has led to a kind of revolution inour thinking. In class ical mech anics, "observables" are represented bynumbers. The new point of view , taken by quantum m echanics, is torepresent observables by operators. For exam ple, the Hamiltonian Hnow becomes the Hamiltonian operator llUp . To this operator weassociate eigenfunclions un and eigenvalues en

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Hlu n> = e , , ^ ( 4 .1 )

The operator, H op , acting on the eigen function, lun>, reproduces thisfunction multiplied by the eigenvalue, e n . The eigenvalues correspondto the numerical values of the physical quantity associated to theoperator, H,,. . Once we have a complete set of eigenfunclions andeigenvalues, we have ihe "spectral" representation (we drop thesubscript "op") associated to H

H = S e ^ x u , , ( 4.2 )

Finding the spectral representation (or solving the eigenvalue problem)is the central problem of quantum mechanics; however, this problemhas only been solved in a few simple situations and most of Ihe time wehave to resort to perturbation techniques. We may start, as in thePoincare theorem, with a Hamiltonian of the form

H = HQ+XV (4 .3)

where we suppose thai the eigenvalue problem can be solved for the"unperturbed" Hamiltonian HQ . We look then for eigenstates andeigenvalues of H , which we could expand in powers of the couplingconstant X. It is here that contact with Poincare's classification can bemade. For non-integrable Poincare system s, the expansion ofeigenfunclions and eigenvalues in powers of the coupling constant leadsto the Poincare catastrophe due to Ihe divergence associated lo Ihe

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small denominators. The relation between Poincare's theorem and thequantum eigenvalue problem has been studied in a recent paper by T.T. Petrosky and the author10 ).

Lei us again consider Ihe problem of quantum transitions (see Fig6).* When


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meaning to Poincare's denominators, we have to time-order thedynamical states the unstable atomic state first, the emission ofradiation later. This corresponds to Bohr's picture in which theradiation emitted by the atom corresponds to a retarded wave. Moreprecisely , to the transition I - k we associate the denominator

1i i , - u ik -

(4.5)

and to the transition k -> 1 the denominatorIcoi-iOfc+ie (4 .6)

As we have shown 11 )' 2), this simple rule leads to the elimination of allPoincare divergence when we integrate over the wave lengths of theradiation. It is a standard procedure in theoretical phys ics to expressthe difference between past and future through "analytic continuation".The general reader may just accept that we modify the Poincaredenominators differently according to ihe type of process to which theyare associated. We obtain in this way complex solutions ofSchrodinger's situation. The eigenv alues contain now an imaginary partcorresponding to damping and the eigenstates have a broken-timesymmetry. We have chosen this simple example because in this casethere exists a standard solution which is, however, not analytic in thecoupling constant (the particle disappears from the spectrum'3).

We can therefore compare our results with the standardtreatment and see if our approach makes sense. We indeed recover all17

known results, but in addition, as we have states with broken timesymmetry, we can introduce a functional which plays the role ofBoltzman's H function and which decreases monotonously when theparticle emits the radiation and decays to ihe ground slate. (See F ig. 9 .)

Figure 9

We can also, at least as a thought experiment, perform a time-inv ersionat time tg after the start of the decay. The result is representedschematically on Figure 10. At the time of the inversion tg , the H -quantity has a jump _ exp(y tg), where l/y is the life-lime of theunstable state. Then H starts to decrease again, at 2tg we haveK(2t0> = H (t=0) and Ihe decrease of H continues until the particle hasdecayed.

Figure 10

As we can associate an H-function to the particle decay the decaybecomes an irreversible process.

We may summarize what we have done as follows: to avoidPoincare's catastrophe we have enlarged the type of transformationswhich lead from the eigenfunclio ns of the unperturbed Hamiltonian H (l(see 4.3 ) to the eigenva lues of the full Hamiltonian. In more technicalterms, the situation is as follow s: Poincare considered only canonica l (orunitary) transformations which among other properties keep theeigenvalues of 11 real. We introduce more general transformations

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leading to complex eigenvalues. The specific choice of thesetransformation follows from our time ordering of ihe dynamical stales.

The result is already of great interest in the perspective of theepistomological problems which plague quantum mechanics. Let us firstremind that the basic equation of quantum mechanics is theSchrodinger equation for ihe wave function H*

ayi ^ = H op 4' (4.7)

The equation as shown in all textbooks on quantum mechanics is limereversible and deterministic. (We exclude some "pathological" casesrelated to weak interactions.) The physical interpretation of is thatit represents a probability "amplitude." In contrast, the probabilityproper is given by

|..4icc ,. m 2 (4 g)

We shall come back in the next section to this transition fromprobability amplitudes to probabilities proper I* I2.

Quantum mechanics is probably the most successful theory ofphysics, and still, the discussions about its conceptual foundation havenever ceased. I recommend a recent book by J.S. Bell, Speakable andUnspeakable in Quantum Mechanics.* 1*) Are they quantum jumps? Thisis quite a controversial problem. Schrodinger's equation (4.7) describesa smooth evolu tion. How then lo include quantum jumps. In addition,(4.7) is time sym metric: if there is spontaneous emission, there should

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be also spontaneous absorption. The conventional Copenhageninterpretation is that quantum jumps result from our measurement.This would be rather strange, since all of chemistry and life are theresult of quantum jumps. How then could life b e as a result of ourmeasurements? Our method solves this problem as il associa tes to thequantum jump an irreversible event which can only occur in (our)future.

Before we come back lo these fascinating problems underlying theclose connection between the conceptual foundations of quantummechanics and dynamical instability, let us make the following remark.

The example we have treated is a very simple one as we couldintroduce a natural lime ordering in the frame of the usual quantumdescription (the so-call ed "Hilbert space"). But in general, this isimpossible (think about scattering where all stales play a symmetricalrole). Then we shall see in ihe next section, we have to introduce anatural time ordering on the level of the statistical description. Thisleads to the integration of LPS in quite general situations and to a newform of dynamics, which breaks radically with the past.

V. Poincare's Theorem and a Statistical Formulation of Dynamics

From the example studied in Section IV, it should be clear how wemay avoid Po incare's catastrophe: It is through introducing into thetheory a time ordering of dynamical stales which leads to well-defined"regularization" procedures for the small denominators (see 4.5-4.6).Bui how to introduce this time ordering? Here as we shall see no w, wehave to turn lo Ihe statistical description. Curiously our approach

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validates the way Boltzmann more than a century ago approachedkinetic theory of gases. But Boltzmann could not guess the emergenceof the chaos theory and did not know that he was studying "non-integrable Poincare systems." (He, as well as Maxwell, placed thereforehis hopes in ergodic theory which is indeed useful for theunderstanding of equilibrium, but not for dynamical purposes.)

In the early days of statistical mechanics, Gibbs introduced a quitefundamental concep t, the "Gibbs ensembles." Instead of consideringsingle dynamical systems, he considered a large number of dynamicalsystems evolving in the phase space associated lo the coordinates q|...q Nand moments pi...pN of the particles forming each dynamical system(see Fig. 11).

Figure 11

The description is then, in terms of the probability distribution p inphase space

pCqi-qN'Pi-PN'') < 5 1 )

This description remains also meaningful for quantum system s. Theprobability distribution p is then called the "density matrix." Once weknow p we can calculate both the velocity distribution of the particlesas well as the correlations existing between the particles.

How then does time enter into this description?Let us consider a classical gas. Particles collide and these

collisions give rise to correlations. Sec Figure 12. First we have binary

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correlations, then ternary correlations, and time going on, correlationsinvolving more and more particles.

Figure 12

The formation of correlations is somewhat reminiscent to that of acouple which has a conversation [This would correspond to a collisi on].Even when the partners go away, the memory of their conversationremains. The information associated to this conversation is time goingon, spreading out to more and more participants.

Suppose we look at a glass of water. In this glass of water, thereis an arrow of time that will, in fact, persist forever and corresponds tothe creation of new correlations involving an ever-increasing number ofparticles. According to the correlations which exist between themolecu les, we can distinguish "young" water from "old!" Computerexperiments have been performed recently that show that binarycorrelations appear very rapidly. Ternary correlations inv olve longertime scales and so on. This time oriented flow of correlations breaks thesymmetry involved in the classical description. Let us go from state A(of a many-body system) with no correlations at 1=0 to a state B, atime t involving multiple correlations (.see fig. 13).

Figure 13

Obviously, the transition from A to B involves quite different physicalprocesses, then the inverse transition, from B to A.

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The time ordering of correlation has lo be introduced intodynamics lo avoid Poincare's catastrophes: Binary correlations comebefore ternary ones and so on. We have therefore to describe dyna micsin terms of the time evolution of correlations.

This corresponds lo a different point of view from that of classicaldynamics: the question is no more to study the positions and momentsof each particle time going on but lo follow the evolution of the relationsbetween the par ticles.I? ) In this conceptual framework we can avoidPoincare's catastrophe by treating transitions to higher correlations as"future oriented" and transitions to lower correlations as "past-oriented"exactly as we have done il in 4.5-4.6.

Gibbs' ensemble theory leads to an equation for the time evolutionof the density matrix p

i = Lp (5.2)

which is formally quite similar to ihe Schrodinger equation (4.7 ). L isthe so-called Liouville operator, which can be expressed in terms of theHamiltonian both in classica l and quantum mechanics. As wementioned, Poincare's theorem deals with Hamiltonians of Ihe form

i

II = H 0 + XV (5 .3)

This corresponds lo a decomposition of the Liouville operator

L = Lg + XLy (5 .4)

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To solve Liouville'* equation (5.2) we need as in Ihe Schrddinger case tosolve the eigenvalue problem (see 4.1)

L V = tjtm> (5 .5)

For integrable system s, there is no problem. The Liouville's equation(5.2) is then of no special interest as the problem reduces to Ihe usualdynamical problems (finding trajectories or wave functions). However,Ihe problem chang es radically for non-integrable system s. Then theLiouville equation describes the emergence of chaos due to thedestruction of the invariants of motion associated to the unperturbedsystem.

Again, Poincare's theorem prevents us from finding solutions of(5. 5) through unitary transformations (preserving the reality of /)which we could expand in powers of the coupling constant X . Asalready mentioned, we solve this difficulty by introducing asupplementary element into the theory: the lime ordering of thecorrelation. We then obtain a complex eigen value problem that can besolved and which leads to damping and to irreversibility through theoccurrence of K -functions (see Section IV). This new dynamics hassome distinct features which present a basic departure from thefeatures of the dynamics of integrable systems as described in classicalor quantum theory.

To understand in qualitative terms what happens, let us analyzemore closely what is involved in the idea of "collisions." In fact, acoll isio n corresponds already lo a complex process in which particles

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come clo se, exchange energy through resonance and depart. We canvisualize a collision as a succe ssio n of stales bound by resonance (see8)). In a Hamiltonian system (the case of hard spheres as a limitingcase which will not be considered here) a collision is not aninstantaneous point-like event but has an extension both in space andtime.

As has been shown recently by T. Petrosky and the author, thespectrum of the Liouv ille operator L is essen tially determined by thedynamics of the collisions. This implies a radical deviation from theusual methods of dynamics valid for integrable systems where theevolution can be resolved into a succession of instantaneous space-timeevents (remember Feynman diagrams).* For this reason the dynamicsof LPS can only be formulated on the statistical level, as we cannotreduce it nor to trajectories in the classical case nor to wave functions,as in the quantum case. This deviation from the great traditions ofdynamics not so astonishing; we deal here with an aspect of dynamicsthat is totally absent in integrable system s. It is, however, alreadypresent in the KAM theory but there the behavior is so complex that itdefies any quantitative description (we have to use qualitative criteriafor the collapse of resonant tori as Ihe result of the coalescence ofresonance s). It is precisely the main progress realized by the study ofLPS to present a simple description of the physical processes due loresonances and which lead to Poincare's non-integrability.

For Ihc reader familiar with kinetic theory, let me mention that ihetradmonal kinetic equations (Ihc Fokkcr-Planck equations) conlain secondderivatives, this is precisely due to the description of the collision as a iwo-sugc process

Our approach has been confirmed by numerical calculationsperformed on simple examples; of LPS. We may start with a statisticaldistribution (as close as w e warn from a point in phase space). We se ethen the system going to various stages corresponding to iheappearance of Lyapounov instability (see 2.1), folding in phase spaceand then diffusion as due to "collisions."

The main point I want to emphasize again is that instabilitiesdestroy the very notion of trajectory (or of wave function in quantummechanics) as the basic description is now in terms of statisticalensembles .

Let us now present some concluding remarks.

VI. Cwc ltiding Remarks

The integration of Poincare's non-integrable dynamical systemsleads for LPS to a new form of dynamics encompassing irreversibility(broken lime symmetry) and exhibiting an increased role of probabilityboth in classical and quantum mechanics. The lime paradox we havedescribed in Section 2 is in this way eliminated (see Fig. 14).

Figure 14

In the "old" situation, we had to bridge the microscopic timereversible level lo Ihc macroscopic level equipped with an arrow oftime (Fig. 14). But how can time arise from no-time?


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Now (Fig. 15) we have a new microscopic level with broken timesymmetry out of which through averaging procedures emerge themacroscopic dissipaiive level. The "old" microscopic level has becomeunstable.

Figure 15

This leads to a better understanding of the role of chao s. In fact,there exist tw o quite different maniifestalions of chaos . When we studymacroscopic equation which include dissipation, such as the reaction-diffusion equation, or the Stokes-Navier equation for fluids, we arealready facing situations for which the basic microscopic descriptionbelongs to LPS. In other words, Ihe very existence of such equationspresupposes "dynamic chaos". This is not astonishing; indeed,properties such as friction or diffusion involve exchange of energythrough collisions. These macroscopic equations may lead lo chaos(chemical chaos as turbulence). This dissipaiive chaos lies "on top" ofthe dynamic chaos. As we mentioned, dissipaiive chaos is pari of self-organization as it appears in nonequilibrium and non-linear systems.Examples of chemical coherence are oscillating chemical reactions. Inshort, therefore, macroscopic order as manifested in non-equilibrium isthe outcome of dynamical chaos. Even ihe approach lo equilibriumbecomes the result of dynamical chaos. In all these cases, therefore, wehave "Order out of Chaos" (see')).

27

I

Figure 16

Let us also mention that LPS are evolving systems. Once initialconditions are given, they go through various stages such as describedby Lyapounov exponents, diffusional processes ... However,irreversibility is not related to Newtonian Time (or its Einsteiniangeneralization) but to an "internal" time as expressed in terms of therelations between the various units which form the system (such as thecorrelation between the particles). We cannot stop the flow ofcorrelations, as we cannot prevent the decay of unstable atomic states.

Nabokov has written: What is real cannot be controlled, what cantoe controlled is not real. This is also true here. In addition to solvingthe time paradox, the dynamical laws obtained through the integrationof LPS lead to a number of consequences which go far beyond our initialmotivation. We have already mentioned some relations with theepistomological problems of quantum mechanics in Section 4. We cannow go further. As is well known, ihe basic quantity in quantummechanics is the probability amplitude 4* which satisfies Schrodinger'sequation (4.7 ) but we measure probabilities! Therefore, we need anadditional mechanism to go from "potentialities" as described by thewave function to "actualities" as described by probabilities. In hisintroduction to "The New Physics," Paul Davies'9) wrole, "At the rockbottom, quantum mechanics provides a highly successful procedure forpredicting the results of observations on microsystems, but when weask what actually happens when an observation lakes place, we getnons ense! Attempts to break out of this paradox range from the

i


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bizarre, such as the many universes interpretation of Hugh Everett, tothe mystical ideas of John von Neumann and Eugene Wigner, whoinvoke the observer's consciousness. After half a century of argument,the quantum observation debate remains as lively as ever. Theproblems of the physics of the very small and the very large areformidable, but it may be that this frontier - the interface. Of mind andmatter - will turn out lo be the most challenging legacy of the NewPhy sics." It is interesting that the solution to this fundamental problemmay come from dynamical instability and chaos as in our newdynamical description we deal directly with probabilities. In this case,the breaking down of the superposition principle of quantum mechanicsin LPS is due to dynamic instability without any appeal to esotericconsiderations, such as the many-world theory or the existence of thenew universal constant leading to a collapse of the wave function formacroscopic systems. We come to a realistic formulation of quantummechanics eliminating the appeal to any observer situated outsidephysics.

This century has been dominated by two new conceptualframeworks: quantum mechanics and relativity . As has been oftenemphasized, (see, i.e., M. Sachs^O)) the intrusion of subjectivisticelements through the measurement process leads to difficulties whenwe want to combine quantum theory and relativity. However, non-integrable dynamical systems are likely also to alter relativity as thebasic dynamical events (the collisions) do no more correspond toinstantaneous and localized space-time events.

I believe thai we are therefore indeed at the beginning of a"New Physics." Until now, our view of nature was dominated by the

2 9

theory of integrable systems, both in classical and quantummechanics . This corresponds to an undue simpli fication. The worldaround us involves instabilities and chaos, and this requires a drasticrevision of some of the basic concepts of physics.

Let me conclude by expressing my conviction that, in thefuture, the non-integrability theorem of Poincare will be consideredas a turning point somewhat similar lo the discovery that classicalmechanics lead to divergences when applied lo the black-bodyradiation. These divergences had lo be cured by quantum theory.Similarly, Poincare's divergences have to be cured by a newformulation of dynamics in the sense I have tried to describe in aqualitative way in this presentation.

Acknowledgments

This work is the outcome of many years of work of theBrussels-Austin group. It is impossible lo quote all those who havecontributed, bul my special gratitude goes to CI. George, F. Mayne andT. Pelrosky.

We want also to thank The University of Texas at Austin, TheSolvay Institutes, the Department of Energy (DOE grant DE-FG05-88ER13897), the Welch Foundation (grant WELCH FDN F-365), and theEuropean Communities Commission (contract no. PSS*0I43/B).


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References1) Ilya Prigogine and Isabelle Stengers, Order out of Cha os. BantamBooks. New York, 1984.2) This question is discussed in Ilya Prigogine and Isabelle Stengers,Entre Ie Temps et Eiernite. Fayard, Paris, 1988.3 ) See the excellent introduction by M. Tabor, Cha os andIniegrabiliiv jn Nonlinear Dynamics. J. W iley, N ew York, 1989.4 ) G. Nicolis and Ilya Prigogine, Exploring Com plexity . W.H. Freeman,New York. 1989.5) P. Coveney and R. Highfield, The Arrow of Tim e. Allen, London,1 9 9 0 .6) For a summary, see ref.4).7) H. Poincare. Methodes Nouvelles de la Mecanique Celeste, GauthierVillars. Paris, 1892 , reprinted by Dover. New York. 1957.8) For a simple presentation, see Ilya Prigogine, }^o^ EquilibriumStatistica l Mech anics. Wiley Inlerscience, New Ybrk." 1^62.9) Let us only mention a few references in addition to **:Ilya Prigogine, From Being to pecoming. W.H. Freeman, NewYork. 1980.R. Balescu, Equilibrium and Non Equilibrium StatislicalMechanics. Wiley, New York. 1985.

CI. George, F. Mayne and I. Prigogine. Adv. in Chemical Physics,61(223), 1985.10 ) Tomio Petrosky and Ilya Prigogine. Physica I47A (439). 1988.11 ) Hiroshi Hasegawa, Tomio Petrosky, Ilya Prigogine, and SuichiTasaki, lo appear in Foundations of Physics, 1990.12 ) Tomio Petrosky, Ilya Prigogine and Suichi Tasaki, to appear infhysica A. 1990.13 ) Ilya Prigogine, Tomio Petrosky and Suichi Tasaki, lo be published.14 ) Tomio Pelrosky, Ilya Prigogine and Suichi Tasaki. lo appear inPhysica A, 199015) See, e.g., G. Ballon. Advanced Field Theo ry. W iley Inlerscience,New York. 1963.16) J S. Bell. Speakable ;ind Unspeakable in Quantum M echanics.Cambridge University Press, 1987.

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17 ) The idea of "dynamics of correlations" was introduced in ref.8).18 ) H. Hasegawa and B. Saphir, lo be published.19 ) "The New P hysics," edited by Paul Davis, Cambridge UniversityPress, 1989.20 ) M. Sachs. Einslein versus Bohr. Open Court, Illinois, 1988.

Legends

Fig. 1: No 'tim e ordering in classical or quantum dynamics.Fig. 2: Clausius formulation of the second law of thermodynamicsFig. 3:' Thermal diffusion experiment (see text)Fig. 4: Baker transformation (s ee tex t)Fig. 5: Resonance between coupled oscillatorsFig. 6: Quantum transitionFig. 7: Collisions (see text)Fig. 8: Exam ple of temporal orderingFig. 9: K-function associated to the decay of an unstable particle (seetext)Fig, 10: Time inversion experiment (see text)Fig. II: G ibbs ensemb leFig. 12: Flow of correlationsFig. 13: Breaking of time symmetry (see text)Fig. 14: The time paradox (see text)Fig. 15: Elimination of the time paradox (see text)Fig. 16: Chaos and DissipationDiagram I

M


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Classical Mechanics

Clausius (1865)No time ordering

Quantum Mechanics

*F wave function

No time ordering

d,S

d S = d e S + d j S


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c_KJ

Suoci1mU0

TH

Oo

oo"

o0

o

0

oo

o

o

O

9

ooo

o00

o

H

H


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Resonances among coupled oscillators

JSZ- 0

Quantum transition

Integration over resonances(~ Fermi golden rule)


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Collisions among gas molecules

Collision ~ resonance

stone fallingioutgoing waves

incoming waves

Istone escaping


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^/"-function n


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Statistical description- Gibbs ensemblesdescription in terms ofprobability distribution

P

P C V - V P r - P N ' t )

correlations between velocitypanicles distribution

Flow of correlations

B

B : Binary correlationsC: Ternary correlations


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Trajectories andflowof coirelations

0 t

0 /

A >B : "Dispersion" of correlationsB >A : "Reconcentration" of correlations

Old Situation (Integ rable syste ms )

macroscopiclevel broken t ime symmetry

T?m i c r o s c o p i c

l e v e l . t i m e r e v e r si b l e


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New Situation (Chaos. LPS)

macroscopiclevelT

statisticallevel

Tmicroscopiclevel

broken time symmetry

irreducible, brokentime symmetry

unsta ble .(trajectories,wave functions)

A Self-OrganizationDissipative Chaos

fPhenomenological equation

Fourier, Stokes and Navier.A

Dynamic chaos(LPS)

Order out of chaos

I


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Diagram I

IntegrableSystemsiclassical orquantum mechanicsPoincare'stheorem (1892)

KAM (1954) chaos. LP Sintegrable in anextended new sense

ilya prigogine - time dynamics chaos

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