1 Sepember 1978, Volume 201, Number 4358

Time, Structure, and FluctuationsIlya Prigogine

The problem of time (t) in physics andchemistry is closely related to the formu-lation of the second law of thermody-namics. Therefore, another possible titleof this lecture could have been "TheMacroscopic and Microscopic Aspectsof the Second Law of Thermodynam-ics."

It is the main thesis of this lecture thatwe are only at the beginning of a new de-velopment of theoretical chemistry andphysics 'in which thermodynamic con-cepts will play an even more basic role.Because of the complexity of the sub-ject, I shall limit myself here mainly toconceptual problems. The conceptual

Summary. Fundamental conceptual problems that arise from the macroscopic andmicroscopic aspects of the second law of thermodynamics are considered. It is shownthat nonequilibrium may become a source of order and that irreversible processesmay lead to a new type of dynamic states of matter called "dissipative structures."The thermodynamic theory of such structures is outlined. A microscopic definition ofirreversible processes is given, and a transformation theory is developed that allowsone to introduce nonunitary equations of motion that explicitly display irreversibilityand approach to thermodynamic equilibrium. The work of the group at the Universityof Brussels in these fields is briefly reviewed. In this new development of theoreticalchemistry and physics, it is likely that thermodynamic concepts will play an ever-increasing role.

The second law of thermodynamicshas played a fundamental role in the his-tory of science far beyond its originalscope. Suffice it to mention Boltzmann'swork on kinetic theory, Planck's discov-ery of quantum theory, and Einstein'stheory of spontaneous emission, all ofwhich were based on the second law ofthermodynamics.

Copyright K 1978 by the Nobel Foundation.The author is professor of physics and chemistry,

Universite Libre de Bruxelles, Brussels, Belgium;director of the Instituts Internationaux de Physiqueet de Chimie (Solvay), Brussels; and professor ofphysics and chemical engineering and director of theCenter for Statistical Mechanics and Thermodynam-ics, University of Texas, Austin 78712. This articleis the lecture he delivered in Stockholm, Sweden, on8 December 1977 when he received the Nobel Prizein Chemistry. Minor corrections and additions havebeen made by the author. The article is publishedhere with the permission of the Nobel Foundationand will also be included in the complete volume ofLes Prix Nobel en 1977 as well as in the series NobelLectures (in English) published by the Elsevier Pub-lishing Company, Amsterdam and New York.

SCIENCE, VOL. 201, 1 SEPTEMBER 1978

SCIE:NCE

structure we have to show that nonequi-librium may be a source of order. Irre-versible processes may lead to a newtype of dynamic states of matter which Ihave called "dissipative structures." Idiscuss the thermodynamic theory ofsuch structures in the sections on en-tropy production, thermodynamic stabil-ity theory, and application to chemicalreactions.These structures are of special interest

today in chemistry and biology. Theymanifest a coherent, supermolecularcharacter which leads to new, quitespectacular manifestations, for example,in biochemical cycles involving oscilla-tory enzymes.How do such coherent structures ap-

pear as the result of reactive collisions?This question is briefly discussed in thesection on the law of large numbers. Iwould emphasize that conventionalchemical kinetics corresponds to a"mean field" theory very similar to thevan der Waals theory of the equation ofstate or Weiss's theory of ferromagnet-ism. Exactly as in these cases, the meanfield theory breaks down near the insta-bility where the new dissipative struc-tures originate. Here (as in equilibriumtheory) fluctuations play an essentialrole.

In the last two sections I shall turn tothe microscopic aspects and review therecent work done by our group at theUniversity of Brussels in this direction.This work leads to a microscopic defini-tion of irreversible processes. How-ever, this development is only possiblethrough a transformation theory whichallows one to introduce new nonunitaryequations of motion that explicitly dis-play irreversibility and approach to ther-modynamic equilibrium.The inclusion of thermodynamic ele-

ments leads to a reformulation of (classi-cal or quantum) dynamics. This is a mostsurprising feature. Since the beginning ofthis century we were prepared to findnew theoretical structures in the micro-world of elementary particles or in themacroworld of cosmological dimensions.We see now that even for phenomena onour own level the incorporation of ther-modynamic elements leads to new theo-retical structures. This is the price we

problems have both macroscopic and mi-croscopic aspects. For example, fromthe macroscopic point of view classicalthermodynamics has largely clarified theconcept of equilibrium structures such ascrystals.Thermodynamic equilibrium may be

characterized by the minimum of theHelmholtz free energy, usually definedby

F = E-TS (1)

where E is the internal energy, T is theabsolute temperature, and S is the en-tropy. Are most types of "organiza-tions" around us of this nature? Oneneed only ask such a question to see thatthe answer is no. Obviously in a town orin a living system we have a quite dif-ferent type of functional order. To obtaina thermodynamic theory for this type of

0036-8075/78/0901-0777$02.00/0 Copyright © 1978 AAAS 777

on

Oct

ober

16,

200

9 w

ww

.sci

ence

mag

.org

Dow

nloa

ded

from

must pay for a formulation of theoreticalmethods in which time appears with itsfull meaning associated with irreversi-bility or even with "history" and notmerely as a geometric parameter associ-ated with motion.

Entropy Production

At the very core of the second law ofthermodynamics we find the basic dis-tinction between "reversible" and "irre-versible" processes (1). This leads ulti-mately to the introduction of S and theformulation of the second law of ther-modynamics. The classical formulationdue to Clausius refers to isolated sys-tems exchanging neither energy nor mat-ter with the outside world. The secondlaw then merely ascertains the existenceof a function, S, which increases mono-tonically until it reaches its maximum atthe state of thermodynamic equilibrium,

dS-_0 (2)dt-It is easy to extend this formulation tosystems which exchange energy andmatter with the outside world (Fig. 1).We have then to distinguish in the en-tropy change dS two terms: the first, deS,is the transfer of entropy across theboundaries of the system; the second,diS, is the entropy produced within thesystem. The second law assumes that theS production inside the system is posi-tive (or zero)

diS1 0 (3)

The basic distinction here is betweenreversible processes and irreversibleprocesses. Only irreversible processescontribute to entropy production. Obvi-ously, the second law expresses the factthat irreversible processes lead to theone-sidedness of time. The positive timedirection is associated with the increaseof S. Let me emphasize the strong andvery specific way in which the one-sided-ness of time appears in the second law.The formulation of this law implies theexistence of a function having quite spe-cific properties, as expressed by the factthat for an isolated system the functioncan only increase in time. Such functionsplay an important role in the modem the-ory of stability as initiated by the classicwork of Lyapounov. For this reasonthey are called Lyapounov functions (orfunctionals).Entropy is a Lyapounov function for

isolated systems. Thermodynamic po-tentials such as the Helmholtz or Gibbsfree energy are also Lyapounov func-tions for other "boundary conditions"

778

deS

Fig. 1. The exchange of entropy between theoutside and the inside.

(such as imposed values of temperatureand volume).

In all these cases the system evolvesto an equilibrium state characterized bythe existence of a thermodynamic poten-tial. This equilibrium state is an "attrac-tor" for nonequilibrium states. This is anessential aspect that was rightly empha-sized by Planck (1).However, thermodynamic potentials

exist only for exceptional situations. Theinequality in Eq. 3, which does not in-volve the total differential of a function,does not in general permit one to define aLyapounov function. Before we comeback to this question, I should empha-size that 150 years after its formulationthe second law of thermodynamics stillappears to be more a program than awell-defined theory in the usual sense, asnothing precise (except the sign) is saidabout the S production. Even the rangeof validity of this inequality is left un-specified. This is one of the main reasonswhy the applications of thermodynamicswere essentially limited to equilibriumprocesses.To extend thermodynamics to non-

equilibrium processes, we need an ex-plicit expression for the S production.Progress has been achieved along thisline by supposing that even outside equi-librium S depends only on the same vari-ables as at equilibrium. This is the as-sumption of "local" equilibrium (2).Once this assumption is accepted, weobtain for P, the entropy production perunit time,

p diS =EJX 4P ~~~~~~~~~(4)where the Jp are the rates of the variousirreversible processes p involved (chem-ical reactions, heat flow, diffusion) andthe Xp are the corresponding generalizedforces (affinities, gradients of temper-ature, gradients of chemical potential).Equation 4 is the basic formula of macro-scopic thermodynamics of irreversibleprocesses.

I have used supplementary assump-tions to derive the explicit expression

(Eq. 4) for the S production. This for-mula can only be established in someneighborhood of equilibrium [see (3)].This neighborhood defines the region oflocal equilibrium, which I shall discussfrom the point of view of statistical me-chanics in the section on nonunitarytransformation theory.

At thermodynamic equilibrium wehave simultaneously for all irreversibleprocesses

Jp = 0 (5a)and

X= 0 (Sb)It is therefore quite natural to assume, atleast near equilibrium, linear homoge-neous relations between the flows andthe forces. Such a scheme automaticallyincludes empirical laws such as Fourier'slaw, which expresses that the flow ofheat is proportional to the gradient oftemperature, or Fick's law for diffusion,which states that the flow of diffusion isproportional to the gradient of concen-tration. We obtain in this way the linearthermodynamics of irreversible process-es characterized by the relations (4)

Jp= E Lpp,Xp, (6)Pt

Linear thermodynamics of irreversibleprocesses is dominated by two importantresults. The first is expressed by the On-sager reciprocity relations (5), whichstate that

Lpp,= Lp,p (7)

When the flow Jp corresponding to theirreversible process p is influenced bythe force Xp, of the irreversible processp', then the flow Jp, is also influenced bythe force Xp through the same coeffi-cient.The importance of the Onsager rela-

tions resides in their generality. Theyhave been subjected to many experimen-tal tests. Their validity has shown thatnonequilibrium thermodynamics leads,as does equilibrium thermodynamics, togeneral results independent of any spe-cific molecular model. The discovery ofthe reciprocity relations correspondsreally to a turning point in the history ofthermodynamics.A second interesting theorem valid

near equilibrium is the theory of mini-mum S production (6), which states thatfor steady states sufficiently close toequilibrium S production reaches itsminimum. Time-dependent states (corre-sponding to the same boundary condi-tions) have a higher S production. Thetheorem of minimum S production re-quires even more restrictive conditions

SCIENCE, VOL. 201

on

Oct

ober

16,

200

9 w

ww

.sci

ence

mag

.org

Dow

nloa

ded

from

than the linear relations (Eq. 6). It is val-id in the frame of a strictly linear theoryin which the deviations from equilibriumare so small that the phenomenologicalcoefficients Lp,, may be treated as con-stants.The theorem of minimum S production

expresses a kind of "inertial" propertyof nonequilibrium systems. When givenboundary conditions prevent the systemfrom reaching thermodynamic equilibri-um (that is, zero S production), the sys-tem settles down to the state of "leastdissipation."

It has been clear since the formulationof this theorem that this property isstrictly valid only in the neighborhood ofequilibrium. For many years great ef-forts were made to generalize this theo-rem to situations further away from equi-librium. It came as a great surprise whenit was finally shown that far from equilib-rium the thermodynamic behavior couldbe quite different, in fact, even oppositeto that indicated by the theorem of mini-mum S production.

It is remarkable that this new type ofbehavior appears in typical situationsthat have been studied in classical hydro-dynamics. An example that was first ana-lyzed from this point of view is the so-called "Bnard instability." Consider ahorizontal layer of fluid between two in-finite parallel planes in a constant gravi-tational field. Let us maintain the lowerboundary at temperature T1 and the high-er boundary at temperature T2 withT1 > T2. For a sufficiently large value ofthe "adverse" gradient (T1 - T2)/(T, + T2), the state of rest becomes un-stable and convection starts. The en-tropy production is then increased as theconvection provides a new mechanismof heat transport. Moreover, the state offlow, which appears beyond the instabili-ty, is a state of organization as comparedto the state of rest. Indeed, a macroscop-ic number of molecules has to move in acoherent fashion over macroscopic timesto realize the flow pattern.We have here a good example of the

fact that nonequilibrium may be a sourceof order. We shall see in the sections onthermodynamic stability theory and ap-plication to chemical reactions that thissituation is not limited to hydrodynamicsituations but also occurs in chemicalsystems when well-defined conditionsare imposed on the kinetic laws.

It is interesting that Boltzmann's orderprinciple as expressed by the canonicaldistribution would assign almost zeroprobability to the occurrence of Benardconvection. Whenever new coherentstates occur far from equilibrium, thevery concept of probability, as implied inI SEPTEMBER 1978

82S

UniversalFig. 2. Time evolution of second-order excessentropy (82S), around equilibrium.

the counting of the number of com-plexions, breaks down. In the case ofBenard convection, we may imagine thatthere are always small convection cur-rents appearing as fluctuations from theaverage state; but, below a certain criti-cal value of the temperature gradient,these fluctuations are damped and dis-appear. However, above some criticalvalue, certain fluctuations are amplifiedand give rise to a macroscopic current.A new supermolecular order appearswhich corresponds basically to a giantfluctuation stabilized by exchanges ofenergy with the outside world. This isthe order characterized by the occur-rence of dissipative structures.

Before discussing further the possi-bility of dissipative structures, I wouldlike to review some aspects of thermody-namic stability theory in relation to thetheory of Lyapounov functions.

Thermodynamic Stability Theory

The states corresponding to thermody-namic equilibrium, or the steady statescorresponding to a minimum of S pro-duction in linear nonequilibrium ther-modynamics, are automatically stable. Ihave already introduced the concept of aLyapounov function. According to thetheorem of minimum S production, the Sproduction is precisely such a Lyapou-nov function in the strictly linear regionaround equilibrium. If the system is per-turbed, the S production will increasebut the system reacts by coming back tothe minimum value of the S production.

Similarly, closed equilibrium statesare stable when corresponding to themaximum of S. If we perturb the systemaround its equilibrium value, we obtain

S=So+8S+ 82S (8)2

where S0 is the equilibrium entropy.However, because the equilibrium statewas a maximum, the first-order termvanishes and therefore the stability isgiven by the sign of the second-orderterm 825.As Glansdorff and Prigogine have

shown, 825 is a Lyapounov function inthe neighborhood of equilibrium inde-pendently of the boundary conditions(3). With classical thermodynamics it ispossible to calculate explicitly this im-portant expression. One obtains (3)

Th2S = C (8)2 +

X (8V)N2 + I pyt8NySN1y] < 0 (9)Here Cv is the specific heat at constantvolume, p is the density, v = Ilp is thespecific volume (the index N. means thatthe composition is maintained constantin the variation of v), X is the isothermalcompressibility, Ny is the mole fractionof component y, and t,,, is the deriva-tive

= (aNy,)pT(10)

where p is pressure.The basic stability conditions of classi-

cal thermodynamics first formulated byGibbs are as follows:

C, > 0 (thermal stability)

X > 0 (mechanical stability)

E ,,8Ny8Ny, > 0 (stability withrespect to diffusion)

These conditions imply that 82S is a neg-ative quadratic function. Moreover, itcan be shown by elementary calculationsthat the time derivative of 82S is relatedto P through (3) (see Eq. 4)

I a 82S = I JPXP = P > O2 atS>IXP0

(1 1)

It is precisely because of the inequal-ities in Eqs. 9 and 11 that 825 is aLyapounov function. The existence ofthe Lyapounov function ensures thedamping of all fluctuations. That is thereason why near equilibrium a macro-scopic description for large systems issufficient. Fluctuations can only play asubordinate role, appearing as correc-tions to the macroscopic laws which canbe neglected for large systems (Fig. 2).We are now prepared to investigate

the fundamental questions: Can we ex-trapolate this stability property furtheraway from equilibrium? Does 82S playthe role of a Lyapounov function whenwe consider larger deviations from equi-librium but still in the frame of macro-

779

t

on

Oct

ober

16,

200

9 w

ww

.sci

ence

mag

.org

Dow

nloa

ded

from

scopic description? We again calculatethe perturbation 82S but now around anonequilibrium state. The inequality inEq. 9 still remains valid in the range ofmacroscopic description. However, thetime derivative of 825 is no longer relatedto the total S production as in Eq. 11 butis related to the perturbation of this Sproduction. In other words, we nowhave (3)

I a82S = I 8Jp8Xp2 at (12)

The right side of Eq. 12 is whatGlansdorff and Prigogine have called the"excess entropy production." I shouldemphasize that 8Jp and UXp are the de-viations from the values Jp and Xp at thestationary state, the stability of which weare testing through a perturbation. Now,contrary to what happens for equilibriumor near-equilibrium situations, the rightside of Eq. 12 corresponding to the ex-cess S production generally has no well-defined sign. If for all t larger than to,where to is the starting time of the per-turbation, we have

E 8Jp8Xp 2 o (13)p

then 825 iS indeed a Lyapounov functionand stability is ensured (see Fig. 3). Inthe linear range, the excess S productionhas the same sign as the S production it-self and we recover the same result aswith the theorem of minimum S produc-tion. However, the situation changes inthe range far from equilibrium. There theform of chemical kinetics plays an essen-tial role.

In the next section I shall consider afew examples. For appropriate types ofchemical kinetics the system may be-come unstable. This result shows thatthere is an essential difference betweenthe laws of equilibrium and the laws faraway from equilibrium. The laws of equi-librium are universal. However, far fromequilibrium the behavior may becomevery specific. This is, of course, a wel-come circumstance, because it permitsus to introduce a distinction in the be-havior of physical systems that would beincomprehensible in an equilibriumworld.

All these considerations are very gen-eral. They may be extended to systemsin which macroscopic motion may begenerated to problems of surface tensionor the effect of an external field (7). Forexample, in the case in which we includemacroscopic motion, we have to consid-er the expression [see (3)]

82Z = 825 !I Pu2 dV < ° (14)

780

and Y, may change in time. Putting thekinetic constants equal to unity, we ob-tain the system of equations

- A + X2Y - BX - Xdt

andStable

Unstable

SpecificFig. 3. Time evolution of second-order excessentropy (82S) in cases of (asymptotically)stable, marginally stable, and unstable situa-tions.

where Z is a Lyapounov function whichdefines the excess entropy, and u is themacroscopic convection velocity. I haveintegrated over the volume V to take intoaccount the space dependence of all u.We may again calculate the time deriva-tive of 82Z, which now takes a morecomplicated form. As the result may befound elsewhere (3), I shall not repro-duce it here. I would only mention thatspontaneous excitation of internal con-vection cannot be generated from a stateat rest which is at thermodynamic equi-librium. This applies, of course, as a spe-cial case to the Bnard instability.

Application to Chemical Reactions

Let us now return to the case of chem-ical reactions. A general result is that toviolate the inequality in Eq. 13 we needautocatalytic reactions. More precisely,autocatalytic steps are necessary (butnot sufficient) conditions for the break-down of the stability of the thermody-namic branch. Let us consider a simpleexample, the so-called "Brusselator,"which corresponds to the scheme of re-action (8),

A-X (15a)2X + Y-- 3X (15b)B+X YY+D (15c)

X E (15d)

The initial reactants and final productsare A, B, D, and E, which are main-tained constant while the concentrationsof the two intermediate components, X

dY= BX - X2Ydt

which admits the steady state

XO = A, Yo = BA

where XO and YO are the concentrationsofX and Y at the steady state. Using thethermodynamic stability criterion or nor-mal mode analysis, we may show thatthe solution (Eq. 17) becomes unstablewhenever

Beyond this critical value of B (B,) wehave a "limit cycle," that is, any initialpoint in the space X, Y tends to the sameperiodic trajectory. The important pointis therefore that, in contrast with oscil-lating chemical reactions of the Lotka-Volterra type, the frequency of oscilla-tion is a well-defined function of the mac-roscopic variables such as concentra-tions and temperatures. The chemical re-action leads to coherent time behavior; itbecomes a chemical clock. In the litera-ture this is often called a Hopf bifurca-tion.When diffusion is taken into account,

the variety of instabilities becomes prop-erly enormous. For this reason the reac-tion scheme in Eq. 15 has been studiedby many investigators over the pastyears. In the presence of diffusion, thereaction scheme in Eq. 15 becomes

axat= A + X2Y - BX -

X + Dx a2

=B BX-XX2Y + Dyd2D (19b)at ar2

where DX and Dy are the diffusion coeffi-cients of components X and Y. In addi-tion to the limit cycle, we now have thepossibility of nonuniform steady states.We may call it the Turing bifurcation, as

Turing was the first to notice the possi-bility of such bifurcations in chemical ki-netics in his classic paper on morphogen-esis in 1952 (9). In the presence of dif-fusion, the limit cycle may also becomespace-dependent and lead to chemicalwaves.Some order can be brought into the re-

SCIENCE, VOL. 201

(16a)

(16b)

(17)

B> Bc = I + A2 (18)

(19a)

82S

on

Oct

ober

16,

200

9 w

ww

.sci

ence

mag

.org

Dow

nloa

ded

from

sults if we consider as the basic solutionthe one corresponding to the thermody-namic branch. Other solutions may thenbe obtained as successive bifurcationsfrom this basic one, or as higher orderbifurcations from a nonthermodynamicbranch, taking place when the distancefrom equilibrium is increased.A general feature of interest is that dis-

sipative structures are very sensitive toglobal features which characterize theenvironment of chemical systems, suchas their size and form, the boundary con-ditions imposed on their surface, and soon. All these features influence in a deci-sive way the types of instabilities thatlead to dissipative structures.Far from equilibrium, there appears an

unexpected relation between chemicalkinetics and the "space-time structure"of reacting systems. It is true that the in-teractions which determine the values ofthe relevant kinetic constants and trans-port coefficients result from short-rangeinteractions (valency forces, hydrogenbonds, van der Waals forces). However,the solutions of the kinetic equations de-pend, in addition, on global character-istics. This dependence, which on thethermodynamic branch, near equilibri-um, is rather trivial, becomes decisive inchemical systems under conditions farfrom equilibrium. For example, the oc-currence of dissipative structures gener-ally requires that the system's size ex-ceed some critical value. The critical sizeis a complex function of the parametersdescribing the reaction-diffusion proc-esses. Therefore, we may say that chem-ical instabilities involve long-range orderthrough which the system acts as awhole.There are three aspects that are al-

ways linked in dissipative structures: thefunction as expressed by the chemicalequations; the space-time structure,which results from the instabilities; andthe fluctuations, which trigger the in-stabilities. The interplay between thesethree aspects leads to most unexpectedphenomena, including "order throughfluctuations," which I shall analyze be-low.

x

Fig. 4. Successive bifurcations.

but multiple solutions for the value X2.It is interesting that bifurcation in-

troduces, in a sense, "history" intophysics. Suppose that observationshows us that the system whose bifurca-tion diagram is represented by Fig. 4 is inthe state C and came there through anincrease in the value of X. The inter-pretation of this state X implies theknowledge of the prior history of the sys-tem, which had to go through the bifur-cation points A and B. In this way weintroduce in physics and chemistry a his-torical element, which until now seemedto be reserved only for sciences dealingwith biological, social, and cultural phe-nomena.Every description of a system which

has bifurcations will imply both determi-nistic and probabilistic elements. As weshall see in more detail in the next sec-tion, the system obeys deterministiclaws, such as the laws of chemical kinet-ics, between two bifurcation points,whereas in the neighborhood of the bi-furcation points fluctuations play an es-sential role and determine the branchthat the system will follow.

I shall not go here into the theory ofbifurcations and its various aspects suchas, for example, the theory of catastro-phes due to Thom (10). These questionsare discussed in the recent monograph ofNicolis and Prigogine (8). I shall also notenumerate the examples of coherentstructures in chemistry and biology thatare known at present. Many examplesmay be found in (8).

ent structures? Obviously, a new featurehas to be introduced. Briefly, this is thebreakdown of the conditions of validityof the law of large numbers; as a result,the distribution of reactive particles nearinstabilities is no longer a random distri-bution.

First, I shall indicate what is meant bythe law of large numbers. To do so I con-sider a typical probability description ofgreat importance in many fields of sci-ence and technology, the Poisson distri-bution. This distribution involves a vari-able X which may take integral valuesX = 0, 1, 2, 3, *... According to thePoisson distribution, the probability thatX = <X> is given by

<X>xpr(X) = e-<x> (20)

In Eq. 20, <X> corresponds to the aver-age value of X. This law is found to bevalid in a wide range of situations such asthe distribution of telephone calls, wait-ing time in restaurants, and fluctuationsof particles in a medium of given concen-tration. An important feature of the Pois-son distribution is that <X> is the onlyparameter that enters in the distribution.The probability distribution is entirelydetermined by its mean.From Eq. 20, one obtains easily the

so-called "variance," which gives thedispersion around the mean

<(SX2> = <(X - <X>)2> (21)

The characteristic feature is that, ac-cording to the Poisson distribution, thedispersion is equal to the average itself

(22)

Let us consider a situation in which X isan extensive quantity proportional to thenumber of particles N (in a given vol-ume) or to the volume V. We then obtainfor the relative fluctuations the famoussquare root law

[<(8X)2>]112

<X>

1 1 1<X>1/2 N"12 or V"/2 (23)

Function ' Structure

Fluctuations

Generally we have successive bifur-cations when we increase the valueof some characteristic parameter (suchas the bifurcation paramater B in theBrusselator scheme). In Fig. 4 wehave a single solution for the value X1I SEPTEMBER 1978

The Law of Large Numbers and the

Statistics of Chemical Reactions

Let us now turn to the statistical as-pects of the formation of dissipativestructures. Conventional chemical kinet-ics is based on the calculation of the av-erage number of collisions and more spe-cifically on the average number of reac-tive collisions. These collisions occur atrandom. However, how can such achaotic behavior ever give rise to coher-

The order of magnitude of the relativefluctuation is inversely proportional tothe square root of the average. There-fore, for extensive variables of order Nwe obtain relative deviations of orderN-"2. This is the characteristic feature ofthe law of large numbers. As a result, wemay disregard fluctuations for large sys-tems and use a macroscopic description.For other distributions the mean

square deviation is no longer equal to theaverage as in Eq. 22. But, whenever the

781

<(Mt2> = <X>

on

Oct

ober

16,

200

9 w

ww

.sci

ence

mag

.org

Dow

nloa

ded

from

law of large numbers applies, the orderof magnitude of the mean square devia-tion is still the same, and we have

<(8V2>_ finite for V -X00 (24)

Let us now consider a stochastic mod-el for chemical reactions. As has beendone often in the past, it is natural to as-sociate a Markov chain process of the"birth and death" type to a chemical re-action (11). This leads immediately to amaster equation for the probabilityP(X, t) of finding X molecules of speciesX at time t,

dP(X, t)=

dt

E W(X - r-* X) P(X - r, t) -

E W(X-*X+ r) P(X, t) (25)r

where W is a transition probability corre-sponding to the jump from X - r mole-cules of species X to X molecules. Onthe right side of Eq. 25 we have a com-petition between gain and loss terms. Acharacteristic difference with the classi-cal Brownian motion problem is that thetransition probabilities, W(X - r -k X)or W(X -- X + r), are nonlinear in theoccupation numbers. Chemical gamesare nonlinear, and this leads to importantdifferences. For example, it can be easilyshown that the stationary distribution ofX corresponding to the linear chemicalreaction

A= X= F (26)

is given by a Poisson distribution (forgiven average values of A and F) (12).But it came as a great surprise when Nic-olis and Prigogine showed in 1971 (13)that the stationary distribution of X,which appears as an intermediate in thechain

A + M-*X + M

2X -- E + D

(27a)

(27b)

is no longer given by the Poisson distri-bution. This is very important from thepoint of view of the macroscopic kinetictheory. Indeed, as has been shown byMalek-Mansour and Nicolis (14), themacroscopic chemical equations gener-

ally must be corrected by terms associat-ed with deviations from the Poisson dis-tribution. This is the basic reason whytoday so much attention is being devotedto the stochastic theory of chemical reac-tions.

For example, the Schlogl reaction(15),

A+ 2X 3X (28a)X B (28b)

782

500

0._

CD1

0

Fig. 5. Distance dependence of the spatialcorrelation function GijXx well below the criti-cal value of the bifurcation parameter B;A = 2, d1 = 1, d2 = 4 (d1 and d2 are the dif-fusion coefficients of A and B).

has been studied extensively by Nicolisand Turner (16), who have shown thatthis model leads to a "nonequilibriumphase transition" quite similar to that de-scribed by the classical van der Waalsequation. Near the critical point as wellas near the coexistence curve, the law oflarge numbers as expressed by Eq. 24breaks down, as <(8X)2> becomes pro-

portional to a higher power of the vol-ume. As in the case of equilibrium phasetransitions, this breakdown can be ex-

pressed in terms of critical indices.

|B = 3.51

Fig. 6. As the bifurcation parameter B ap-proaches the critical value, the range of G-XXincreases slightly with respect to the behaviorshown in Fig. 5.

500

x._x -

0

In the case of equilibrium phase transi-tions, fluctuations near the critical pointnot only have a large amplitude but alsoextend over large distances. Lemar-chand and Nicolis (17) have investigatedthe same problem for nonequilibriumphase transitions. To make the calcu-lations possible, they considered a se-quence of boxes. In each box the Brus-selator type of reaction (Eq. 15) is takingplace. In addition, there is diffusion be-tween one box and the other. Using theMarkov method, they then calculatedthe correlation between the occupationnumbers of X in two different boxes.One would expect that chemical inelasticcollisions together with diffusion wouldlead to a chaotic behavior. But that is notthe case. In Figs. 5 through 7 the correla-tion functions for below and near thecritical state are represented graphically.It is clear that near the critical point wehave long-range chemical correlations.Like the earlier systems that I have con-sidered, this system acts as a whole inspite of the short-range character of thechemical interactions. Chaos gives riseto order. Moreover, numerical simula-tions indicate that it is only in the limit ofthe number of particles, N x-* 0, that wetend to "long-range" temporal order.To understand this result at least quali-

tatively, let us consider the analogy withphase transitions. When we cool down aparamagnetic substance, we come to theso-called Curie point, below which thesystem behaves like a ferromagnet.Above the Curie point, all directionsplay the same role. Below the Curiepoint, there is a privileged direction cor-responding to the direction of magnet-ization.Nothing in the macroscopic equation

determines which direction the magnet-ization will take. In principle, all direc-tions are equally likely. If the ferromag-net would contain a finite number of par-ticles, this privileged direction would notbe maintained in time. It would rotate.However, if we consider an infinite sys-tem, then no fluctuations whatsoever canshift the direction of the ferromagnet.The long-range order is established onceand for all.There is a striking similarity between

the ferromagnetic system and the case ofoscillating chemical reactions. When weincrease the distance from equilibrium,the system begins to oscillate. It willmove along the limit cycle. The phase onthe limit cycle is determined by the initialfluctuation and plays the same role as thedirection of magnetization. If the systemis finite, fluctuations will progressivelytake over and perturb the rotation. How-ever, if the system is infinite, then we

SCIENCE, VOL. 201

on

Oct

ober

16,

200

9 w

ww

.sci

ence

mag

.org

Dow

nloa

ded

from

may obtain a long-range temporal ordervery similar to the long-range space or-der in the ferromagnetic system. We see,therefore, that the appearance of a peri-odic reaction is a process that breakstime symmetry exactly as ferromagnet-ism is a process that breaks space sym-metry.

The Dynamic Interpretation of the

Lyapounov Function

I shall now consider more closely thedynamic meaning of the entropy andmore specifically the Lyapounov func-tion 82S that I have used above.

Let us start with a very brief summaryof Boltzmann's approach to this prob-lem. Even today, Boltzmann's work ap-pears as a milestone. It is well knownthat an essential element in Boltzmann'sderivation of the e-theorem was the re-placement of the exact dynamic equa-tions (as expressed by the Liouvilleequation to which I shall return later) bythe kinetic equation for the velocity dis-tribution functionf of the molecules,

df +vdf =|dw Alof'f -f]ataX ~~~~~~~~(29)

where dw is the effective solid angle inthe collision, C- is the cross section, and vis the velocity.Once this equation is admitted, it is

easy to show that Boltzmann's '-quanti-ty,

k={dvflogf (30)

satisfies the inequality

ddMe s O (31)dt -

and thus plays the role of a Lyapounovfunction.The progress achieved through Boltz-

mann's approach is striking. Still, manydifficulties remain (18). First, we havepractical difficulties, as, for example, thedifficulty of extending Boltzmann's re-sults to more general situations (for ex-ample, dense gases). Kinetic theory hasmade striking progress in the last fewyears; yet, when one examines recenttexts on kinetic theory or nonequilibriumstatistical mechanics, one does not findanything similar to Boltzmann's X-theo-rem, which remains valid in more gener-al cases. Therefore, Boltzmann's resultremains quite isolated, in contrast to thegenerality we attribute to the second lawof thermodynamics.

In addition, we have theoretical diffi-culties. The most serious is probably1 SEPTEMBER 1978

104

5x103

0C.,£.-

Fig. 7. Critical behavior of the spatial correla-tion function GuXX for the same values of pa-rameters as in Fig. 5. The correlation functiondisplays both linear damping with distanceand spatial oscillations with wavelength equalto that of the macroscopic concentration pat-tern.

Loschmidt's reversibility paradox. Inbrief, if we reverse the velocities of themolecules, we come back to the initialstate. During this approach to the initialstate Boltzmann's k-theorem (Eq. 31) isviolated. We have "antithermodynamicbehavior." This conclusion can be veri-fied, for example, by computer simula-tions.The physical reason for the violation

of Boltzmann's 9t-theorem lies in thelong-range correlations introduced bythe velocity inversion. One would like toargue that such correlations are ex-ceptional and may be disregarded. How-ever, how should one find a criterion todistinguish between abnormal correla-tions and normal correlations, especiallywhen dense systems are considered?The situation becomes even worse

when we consider, instead of the veloc-ity distribution, a Gibbs ensemble corre-sponding to phase density p. Its timeevolution is given by the Liouville equa-tion,

i dP = Lpat (32)

where Lp is the Poisson bracket i{H,p}in classical dynamics and is the commu-tator [H,p] in quantum mechanics (H isthe Hamiltonian). If we consider positiveconvex functionals such as

Q = {p2 dp dq > 0 (33)

where q is the coordinate and p is themomentum conjugate to q, or in quan-tum mechanics

Q = trp+p > 0 (34)

it is easily shown that, as a consequenceof Liouville's equation (Eq. 32),

d-= 0dt (35)

Therefore, Ql as defined in Eqs. 33 or 34is not a Lyapounov function, and thelaws of classical or quantum dynamicsseem to prevent us from constructing aLyapounov functional that would playthe role of the entropy.For this reason it has often been stated

that irreversibility can only be in-troduced into dynamics through a sup-plementary approximation such ascoarse-graining added to the laws of dy-namics (19). I have always found it dif-ficult to accept this conclusion, especial-ly because of the constructive role ofirreversible processes. Can dissipativestructures be the result of mistakes?We obtain a hint about the direction in

which the solution of this paradox maylie by inquiring why Boltzmann's kinet-ics permits one to derive an I-theoremwhereas the Liouville equation does not.Liouville's equation (Eq. 32) is obvious-ly Lt-invariant. If we reverse both thesign of the Liouville operator L (this canbe done in classical dynamics by velocityinversion) and the sign of t, the Liouvilleequation remains invariant. On the otherhand, it can be easily shown (18) that thecollision term in the Boltzmann equationbreaks the Lt-symmetry as it is even inL. We may therefore rephrase our ques-tion by asking: How can we break the Lt-symmetry inherent in classical or quan-tum mechanics? Our point of view hasbeen that the dynamic and thermo-dynamic descriptions are, in a certainsense, "equivalent" representations ofthe evolution of the system connected bya nonunitary transformation. Let mebriefly indicate how we may proceed.The method that I follow has been devel-oped in close collaboration with my col-leagues in Brussels and Austin (20-22).

Nonunitary Transformation Theory

As Eq. 34 has proved inadequate, westart with a Lyapounov function of theform

Q = trp+Mp 2 0 (36)(where M is a positive operator) with anonincreasing time derivative

dft (37)

This is certainly not always possible. Insimple dynamic situations when the mo-tion is periodic in either classical or

783

on

Oct

ober

16,

200

9 w

ww

.sci

ence

mag

.org

Dow

nloa

ded

from

quantum mechanics, no Lyapounovfunction may exist as the system returnsafter some time to its initial state. Theexistence of M is related to the type ofspectrum of the Liouville operator. Inthe frame of classical ergodic theory thisquestion has recently been studied byMisra (23). I shall pursue here certainconsequences of the possible existenceof the operator M in Eq. 36, which maybe considered as a "microscopic repre-sentation of entropy." As this quantity ispositive, a general theorem permits us torepresent it as a product of an operator,say A-1, and its hermitian conjugate(A-1)+ (this corresponds to taking the"square root" of a positive operator)

M = (A-')+A-' (38)

Inserting this in Eq. 36, we obtain

l= tr pi+pj (39)

with=A-lp (40)

This is a most interesting result, becauseEq. 39 is precisely the type of equationwe were looking for in the first place. Butwe see that this expression can only existin a new representation related to thepreceding by the transformation in Eq.40.

First let us write the new equations ofmotion. Taking into account Eq. 40, weobtain

i = d1j (41)at

with(D = A-1LA (42)

Now let us use the solution of the equa-tions of motion (Eq. 32). We may replaceEqs. 36 and 37 by the more explicit in-equalities

Q(t) = tr p+(o)e"-'M eiL'p(o) < 0

and

dfld = -trp+(o)eil' xdt

i (ML - LM)e-i1'p(o) < 0 (44)

The microscopic "entropy operator" Mcan therefore not commute with L. Thecommutator represents precisely whatcould be called the "microscopic en-tropy production."We are, of course, reminded of Hei-

senberg's uncertainty relations andBohr's complementarity principle. It ismost interesting to find here also a non-commutativity, but now between dy-namics as expressed by the operator Land "thermodynamics" as expressed byM. We therefore have a new and most

784

interesting type of complementarity be-tween dynamics, which implies theknowledge of trajectories or wave func-tions, and thermodynamics, which im-plies entropy.When the transformation to the new

representation is performed, we obtainfor the entropy production (Eq. 44)

dt - -tr A+(o)eib+'i(¢ - ¢+)e-i 'p( 0 (45)

This result implies that the difference be-tween .t and its hermitian adjoint ¢+does not vanish,

W! - 4r+) 2 0 (46)

Therefore, we reach the important con-clusion that the new operator of motionwhich appears in the transformed Liou-ville equation (Eq. 41) can no longer behermitian as was the Liouville operatorL. This shows that we have to leave theusual class of unitary (or antiunitary)transformations and proceed to an exten-sion of the symmetry of quantum me-chanical operators. Fortunately, it iseasy to determine the class of transfor-mations that we have to consider now.Average values can be calculated both inthe old and in the new representation.The result should be the same; in otherwords, we require that

(A) = tr A+p = tr A+p (47)

Moreover, we are interested in transfor-mations which will depend explicitly onthe Liouville operator. This is indeed thevery physical motivation of the theory.We have seen that the Boltzmann typeequations have a broken Lt-symmetry.We want to realize precisely this newsymmetry through our transformation(20). This can only be done by consid-ering L-dependent transformations A(L).Using finally the fact that the density pand the observables have the same equa-tions of motion, but with L replaced by-L, we obtain the basic condition

A-'(L) = A+(-L) (48)which replaces here the usual conditionof unitarity imposed on quantum me-chanical transformations.

It is not astonishing that we do indeedfind a nonunitary transformation law.Unitary transformations are very muchlike changes in coordinates, which donot affect the physics of the problem.Whatever the coordinate system, thephysics of the system remains unaltered.But here we are dealing with quite a dif-ferent problem. We want to go from onetype of description, the dynamic one, toanother, the "thermodynamic" one.

This is precisely the reason why we needa more drastic type of change in repre-sentation as expressed by the new trans-formation law (Eq. 48).

I have called this transformation a"star-unitary" transformation and in-troduced the notation

A*(L) = A+(-L) (49)

I shall call A* the "star-hermitian" oper-ator associated with A (star alwaysmeans the inversion L -- -L). Then Eq.48 shows that for star-unitary transfor-mations the inverse of the transforma-tion is equal to its star-hermitian con-jugate.

Let us now consider Eq. 42. Using thefact that L as well as Eqs. 48 and 49 arehermitian, we obtain

D* = ID+(-L) =-¢(L) (50)or

(51)The operator of motion is a "star-hermi-tian," a most interesting result. To be astar-hermitian, an operator may be eithereven under L-inversion (that is, it' doesnot change sign when L is replaced by-L) or antihermitian and odd (oddmeans that it changes sign when L is re-placed by -L). A general star-hermitianoperator can therefore be written as

(52)Here the superscripts e and o refer re-spectively to the even and the odd part ofthe new time evolution operator (D. Thecondition of dissipativity (Eq. 46), whichexpresses the existence of a Lyapounovfunction Ql, now becomes

i > O (53)

It is the even part which gives the "en-tropy production."

Let me summarize what has beenachieved. We obtain a new form of themicroscopic equation (as is the Liouvilleequation in classical or quantum me-chanics), which displays explicitly a partwhich may be associated with a Lyapou-nov function. In other words, the equa-tion

i dP = (1 + $)o (54)

contains a "reversible" part, ¢, and an"irreversible" part, ¢. The symmetry ofthis new equation is exactly that ofBoltzmann's phenomenological kineticequation, as the flow term is odd and thecollision term is even in L-inversion.The macroscopic thermodynamic dis-

tinction between reversible and irrevers-ible processes has in this way been trans-

SCIENCE, VOL. 201

y(i)* = isD

i(D = (A) + (4)

on

Oct

ober

16,

200

9 w

ww

.sci

ence

mag

.org

Dow

nloa

ded

from

posed into the microscopic description.We have obtained what could be consid-ered as the "missing link" between mi-croscopic reversible dynamics and mac-roscopic irreversible thermodynamics.The scheme is as follows:

Nonunitarytransformation

Microscopic "reversible"description

Microscopic description? displaying irreversible

processesMacroscopic irreversibledescription

Averaging procedures

The effective construction of theLyapounov function Q (Eq. 36), throughthe transformation A, involves a carefulstudy of the singularities of the resolventcorresponding to the Liouville operator(21).For small deviations from thermo-

dynamical equilibrium it can be shown,as has been done recently by Theodo-sopulu et al. (24), that the Lyapounovfunctional fl (Eq. 36) reduces preciselyto the macroscopic quantity 82S (Eq. 9)when in addition only the time evolutionof conserved quantities is retained. Wetherefore have now established in fullgenerality the link between nonequilibri-um thermodynamics and statistical me-chanics at least in the linear region. Thisis the extension of the result that was ob-tained long ago in the frame of Boltz-mann's theory, valid for dilute gases(25).

Concluding Remarks

The inclusion of thermodynamic irre-versibility through a nonunitary transfor-mation theory leads to a deep alterationin the structure of dynamics. We are ledfrom groups to semigroups, from trajec-tories to processes. This evolution is inline with some of the main changes in ourdescription of the physical world duringthis century.One of the most important aspects of

Einstein's theory of relativity is that wecannot discuss the problems of space

and time independently of the problem ofthe velocity of light which limits thespeed of propagation of signals. Similar-ly, the elimination of "unobservables"has played an important role in the basicapproach to quantum theory initiated byHeisenberg.The analogy between relativity and

thermodynamics has often been empha-sized by Einstein and Bohr. We cannotpropagate signals with arbitrary speed,and we cannot construct a perpetuummobile forbidden by the second law.From the microscopic point of view

this last interdiction means that quan-tities that are well defined from the pointof view of mechanics cannot be observ-ables if the system satisfies the secondlaw of thermodynamics. For example,the trajectory of the system as a wholecannot be an observable. If it were, wecould at every moment distinguish twotrajectories and the concept of thermalequilibrium would lose its meaning. Dy-namics and thermodynamics limit eachother.

It is interesting that there are otherreasons which at the present time seemto indicate that the relation between dy-namic interaction and irreversibility mayplay a deeper role than was conceiveduntil now. In the classical theory of in-tegrable systems, which has been so im-portant in the formulation of quantummechanics, all interactions can be elimi-nated by an appropriate canonical trans-formation. Is this really the correct pro-totype of dynamic systems to consider,especially when situations involving ele-mentary particles and their interactionsare included? Must we not first go to anoncanonical representation, which per-mits us to disentangle reversible and irre-versible processes on the microscopiclevel and then only to eliminate the re-versible part to obtain well-defined butstill interacting units?These questions will probably be clari-

fied in the coming years. But already thedevelopment of the theory permits us todistinguish various levels of time: time asassociated with classical or quantum dy-namics, time associated with irreversibil-

ity through a Lyapounov function, andtime associated with "history" throughbifurcations. I believe that this diversifi-cation of the concept of time permits abetter integration of theoretical physicsand chemistry with disciplines dealingwith other aspects of nature.

References and Notes

1. M. Planck, Vorlesungen uber Thermodynamik(Teubner, Leipzig, 1930; English translation,Dover, New York).

2. I. Prigogine, Etude thermodynamique desphinomenes irreversibles, thesis, University ofBrussels (1945).

3. P. Glansdorffand I. Prigogine, Thermodynamicsof Structure, Stability and Fluctuations (Wiley-Interscience, New York, 1971).

4. The standard reference for the linear theory ofirreversible processes is the monograph by S. R.de Groot and P. Mazur, Non-EquilibriumThermodynamics (North-Holland, Amsterdam,1969).

5. L. Onsager, Phys. Rev. 37, 405 (1931).6. I. Prigogine, Bull. Cl. Sci. Acad. R. Belg. 31, 600

(1945).7. R. Defay, I. Prigogine, A. Sanfeld, J. Colloid In-

terface Sci. 58, 598 (1977).8. G. Nicolis and I. Prigogine, Self-Organization in

Nonequilibrium Systems (Wiley-Interscience,New York, 1977), chap. 7.

9. A. M. Turing, Phil. Trans. R. Soc. London Ser.B 237, 37 (1952).

10. R. Thom, Stabilite Structurelle et Morpho-gense (Benjamin, New York, 1972).

11. A standard reference is A. T. Barucha-Reid,Elements of the Theory of Markov Processesand Their Applications (McGraw-Hill, NewYork, 1960).

12. G. Nicolis and A. Babloyantz, J. Chem. Phys.51, 2632 (1969).

13. G. Nicolis and I. Prigogine, Proc. Natl. Acad.Sci. U.S.A. 68, 2102 (1971).

14. M. Malek-Mansour and G. Nicolis, J. Stat.Phys. 13, 197 (1975).

15. F. Schlogl, Z. Phys. 248, 446 (1971); ibid. 253,147 (1972).

16. G. Nicolis and J. W. Turner, Physica 89A, 326(1977).

17. H. Lemarchand and G. Nicolis, ibid. 82A, 521(1976).

18. I. Prigogine, in The Boltzmann Equation, E. G.D. Cohen and W. Thirring, Eds. (Springer-Ver-lag, New York, 1973), pp. 401-449.

19. An elegant presentation of this point of view iscontained in G. E. Uhlenbeck, in The Phys-icist's Conception of Nature, J. Mehra, Ed.(Reidel, Dordrecht, Holland, 1973), pp. 501-513.

20. I. Prigogine, C. George, F. Henin, L. Rosenfeld,Chem. Scr. 4, 5 (1973).

21. A. Grecos, T. Guo, W. Guo, Physica 80A, 421(1975).

22. I. Prigogine, F. Mayne, C. George, M. de Haan,Proc. Natl. Acad. Sci. U.S.A. 74, 4152 (1977).

23. B. Misra, ibid. 75, 1629 (1978).24. M. Theodosopulu, A. Grecos, I. Prigogine,

ibid., in press; A. Grecos and M. Theodosopulu,Physica, in press.

25. I. Prigogine, Physica 14, 172 (1949); ibid. 15, 272(1949).

26. This lecture gives a survey of results that havebeen obtained in close collaboration with mycolleagues in Brussels and Austin. It is impos-sible to thank them all individually. I want, how-ever, to express my gratitude to Prof. G. Nicolisand Prof. J. Mehra for their help in the prepara-tion of the final version of this lecture.

I SEPTEMBER 1978 785

on

Oct

ober

16,

200

9 w

ww

.sci

ence

mag

.org

Dow

nloa

ded

from