Proc. Nat. Acad. Sci. USAVol. 68, No. 9, pp. 2102-2107, September 1971

Fluctuations in Nonequilibrium Systems(chemical kinetics/birth and death model/phase-space description/Langevin/Volterra-Lotka)

G. NICOLIS AND I. PRIGOGINE*

FacultM des Sciences, UniversitM Libre de Bruxelles, Brussels, Belgium

Contributed by I. Prigogine, June 25, 1971

ABSTRACT The theory of fluctuations is extended tononlinear systems far from equilibrium. Systems whoseevolution involves two separate time scales, e.g., chemi-cally reacting mixtures near a local equilibrium regime,are studied in detail. It is shown that the usual stochasticdescription of chemical kinetics based on a "birth anddeath" model is inadequate and has to be replaced by amore detailed phase-space description. This enables oneto develop for such systems a plausible mechanism for theemergence of instabilities, in which the departure fromthe steady state is governed by large fluctuations of macro-scopic size, while small thermal fluctuations are stilldescribed by a generalization of Einstein's equilibriumtheory. On the other hand, far from a local equilibriumregime, infinitesimal fluctuations may increase and attainmacroscopic values. In this case the system evolves to astate of "generalized turbulence", in which the distinc-tion between macroscopic averages and fluctuations be-comes meaningless.

Certain types of nonlinear systems maintained beyond acritical distance from thermodynamic equilibrium mayundergo instabilities in their steady-state solutions.Hydrodynamic transitions such as thermal convectionare well known examples of this behavior. Purely dis-sipative systems such as chemically reacting mixturesmay also present instabilities and evolve subsequentlyto new regimes showing spatial, temporal, or functionalorganization (1).

It is important to distinguish between "macroscopic"transitions and phenomena such as plasma instabilities(2) or laser thresholds (3). The origin of the latter is"molecular" in the sense that the momentum distribu-tion function has a highly non-Maxwellian form. As aresult, the system becomes unstable with respect toinfinitesimal thermal fluctuations and may exhibit acritical fluctuation behavior in the neighborhood of theonset of the instability. In contrast, macroscopictransitions in chemical kinetics and fluid dynamicsoccur, as a rule, under conditions that are close to alocal equilibrium state. Yet these systems, whichlocally remain dissipative, and thus tend to dampthermal fluctuations, undergo large-scale transitionsthat change their macroscopic state (appearance ofconvection cells, spatial dissipative structures, andlimit cycles in chemical autocatalytic systems, etc.).

An understanding of the molecular origin of such orga-nization phenomena requires a detailed study of fluc-tuations around the nonequilibrium state that is goingto become unstable. The present paper outlines anapproach to this problem based on the theory of sto-chastic processes (4).The theory of fluctuations has been developed ex-

tensively for systems near thermodynamic equilibrium.The classical approach is based on a combination of theideas of linear thermodynamics of irreversible processesand of equilibrium considerations (5). Equivalent tothis formulation is the Langevin method (6), whichamounts to introducing a priori the fluctuations in theequations of evolution through suitably correlatedstochastic force terms. A potentially more powerfulmethod consists in the derivation of stochastic masterequations based on the assumption that the kineticequations define a markovian process. As a rule, theseequations represent a birth and death process (7)t. Astill more general approach is based on statisticalmechanics, with a kinetic equation such as the Boltz-mann equation as a starting point (8).An important result, recently proved in full general-

ity (9), establishes that around equilibrium all themethods lead to equivalent results. In particular, theone-time fluctuations around equilibrium are shown tosatisfy in the limit of small fluctuations the well-knownEinstein distribution (5, 9). Thus, in an ideal reactingmixture, the relative mean quadratic fluctuations(5X2/Xe2) 1/2, Xe being the equilibrium average, arevery small, of the order of X -l/2. In other terms forsuch systems there exist clearcut distinctions betweenmacroscopic averages and fluctuations.The behavior of fluctuations around steady non-

equilibrium states is much less known (10). It is only inthe domain of linear systems that general conclusionshave been derived. Most of the models studied ex-plicitly refer to current fluctuations (11) or to com-

t The first application of stochastic theory to nonlinear chemicalkinetics seems to be due to Delbruck (1940). Eigen has appliedstochastic theory to the problem of self-organization and evolu-tion of biological macromolecules [Eigen, M., Naturwissen-schaften, in press].

2102

* Also at the Center for Statistical Mechanics and Thermo-dynamics, University of Texas, Austin, Texas.

Fluctuations in Nonequilibrium Systems 2103

position fluctuations in open, chemically reacting mix-tures (12). In all cases, the equivalence of the Langevin,birth and death, and kinetic equation approach isconfirmed. Moreover, small fluctuations are describedby the following generalization of Einstein's formula(12):

P({aX}) c exp [WS2W]

with5X2 = X = XO. (1.1)

=6X X - Xo} denotes the fluctuation of a set ofstate variables around the steady state, k is Boltz-mann's constant, and 1/2(62s)o is the second-order excessentropy evaluated around the nonequilibrium state.As we saw above, in order to understand the onset of

a macroscopic transition, it is necessary to studyfluctuations in nonlinear systems very far from thermo-dynamic equilibrium. In this domain the validity of theLangevin type of approach is no longer guaranteed (7).This leaves us with the master equations based on birthand death processes and with the more general kineticequation approach. In Sec. 2 the birth and death processapproach is applied to a simple nonlinear chemicalkinetic model. § It is found that this formulation leads toresults which, far from equilibrium, are different from(1.1). On the other hand, a number of general argu-ments (see Sec. 2) indicate that in the whole range oflocal equilibrium theory in which local thermodynamicvariables may be used to describe irreversible processes(1, 4), Eq. (1.1) should remain valid for small fluctu-ations. In Sec. 3, we show that in order to express con-sistently the local equilibrium condition it is necessaryto adopt a phase-space description based on a Boltz-mann-type kinetic equation. Assuming that this equa-tion defines a markovian process in the complete phasespace, we then show that Eq. (1.1) is recovered in thelimit of small fluctuations. This unexpected result im-plies that for nonlinear systems far from equilibrium,the usual birth and death stochastic approach isgenerally inadequate. Sec. 4-6 are devoted to the studyof oscillatory systems without asymptotic stability.In Sec. 4, the Volterra-Lotka model is considered in itschemical kinetic and ecological versions. We show thatthe usual stochastic analysis predicts that the steadystate is unstable with respect to infinitesimal fluctu-ations. On the other hand, the phase-space approach,outlined in Sec. 5, predicts that small thermal fluctu-ations are stable and obey Eq. (1.1). In Sec. 6, theimplications of these results in chemical kinetics andecology are discussed. An analogy is drawn between the

behavior of fluctuations, which in the Volterra-Lotkamodel attain a macroscopic level, and the well-knownphenomenon of turbulence in fluid dynamics. A plausi-ble mechanism of setting up a macroscopic instabilityin a dissipative system is also outlined.

2. A SIMPLE NONLINEAR MODEL

The main ideas of this Section will be illustrated on thesimple bimolecular scheme

kiA+M - X+M

2X-22X -*E+D. (2.1)

A slightly different version of this scheme was consid-ered by Babloyantz and Nicolis (12). A comparisonbetween their results and the conclusions of this sectionis given in ref. 4. The system is open to large reservoirsof A, M, D, E. The inverse reaction rates are neglected:the system thus operates automatically far from ther-modynamic equilibrium.Assuming the reaction occurs in ideal mixture con-

ditions and that the system remains spatially homo-genous, one can write the usual conservation of massequations which determine the time evolution of X.These equations admit a single steady-state

Xo = (k1AM/2k2) 1/2 (2.2)

that is asymptotically stable with respect to arbitraryperturbations.

In order to study the fluctuations around state (2.2),we make the usual assumption that the state of thesystem is described in terms of a probability functionP(A ,M,D,E,X,t). The evolution of this function isgiven in terms of a birth and death type of masterequation (4, 12), which in reduced form (summed overall reservoir variables) reads:

dP(Xjt) = kAMP(X - 1,t) - k1AMP(X,t)dt

+ k2(X + 1)(X + 2)P(X + 2,t)- k2X(X - 1)P(Xt). (2.3)

AM are averages over the reservoir states (i.e., knownquantities) and are seen to appear as parameters.Eq. (2.3) admits a steady-state solution, which can be

computed exactly in the thermodynamic limit

X0-) a, V- a, (X0/V) = finite (2.4)

where V is the volume of the system. The mean quad-ratic fluctuation corresponding to this distribution caneasily be computed (4). The result is a value differentfrom the Poisson variance:

+ The models treated in refs. 10-12 are all stable with respectto arbitrary perturbations.§ Chemical kinetic models are chosen due to the simplicity of theformalism (the stochastic variables are integers) and since thesemodels may give rise to oscillatory behavior and to instabilitiesfar from equilibrium.

bX2 = 4 X, Xo. (2.5)

The steady-state solution is thus incompatible with thegeneralized Einstein form (1,1), even in the limit of

Proc. Nat. Acad. Sci. USA 68 (1971)

2104 Applied Mathematical Sciences: Nicolis and Prigogine

small fluctuations. The factor 3/4 in front of XO in Eq.(2.5) is model-dependent. It appears, therefore, that fornonlinear systems the stochastic master equations donot admit solutions of a universal form, but rather theypredict a behavior for the fluctuations that dependsstrongly on the detailed kinetic properties of the reac-tion.

This conclusion cannot be valid in gen:ral. Let usrecall that in chemical kinetics one usually deals withsystems described by a local equilibrium theory (1, 4),in which the state functions (entropy, density, etc.)are described, locally, by the same independent vari-ables as in equilibrium. It is natural to expect that inthis case Eq. (1.1) should also apply. A sufficient con-dition for the validity of this theory is that the distribu-tion of momenta deviates only slightly from a localMaxwellian form (1). ¶ However, this implies the ex-istence of two largely separated time scales (4): a shortrelaxation time between frequent elastic collisions,which restore continuously an average Maxwelliandistribution, and a longer, macroscopic scale overwhich the chemical composition changes as a result ofreactive collisions that tend to perturb the Maxwelliandistribution. The effects related to the relaxation timecannot be accounted for in a description of the typeoutlined in this section, where the internal states (e.g.,the values of the momenta) of the molecules are dis-carded. Thus, fluctuations must be discussed on thebasis of a kinetic equation, such as the Boltzmannequation, containing the effect of both reactive andelastic collisions.

3. PHASE-SPACE DESCRIPTION

Let us now illustrate briefly the phase-space theory offluctuations in the simple example considered in Sec. 2.As we deal with dilute mixtures, the starting point ofthe average description will be the Boltzmann equation.We adopt the notation F for the Boltzmann prob-ability density of component y corresponding to an in-ternal state a, and assume for simplicity that the spec-trum of a is discrete. The bar over Fat indicates that inthe Boltzmann equation, Fa7 represents an averagequantity.The Boltzmann equation corresponding to model

(2.1) reads (13)

dP,,XJ k

fdpx\a = ELBjk-lFJAFkM- 2EAajkl aXFjX + ( dtIa)dt jkl d(3.1)

In the right of this equation, (dFaX/dt)ei describes theeffect of elastic collisions; the remaining two terms referto reactive collisions. BIpkz and Aijk, are the transitionprobabilities per unit time for scattering between 2

molecules in states (kl) into 2 molecules in states (ij)for the reactions corresponding to the two steps in (2.1).They satisfy the well-known conditions:

Aijkk = 0

Ajki = Aji,kl = A iik > 0 for (kl) $ (ij)

(3.2)E Ajiki = 0-kl

According to the remarks made in the previous sec-tion, at the limit of very frequent elastic collisions weobtain:

PI_x° = local Maxwellian (3.3a)

and similarly for A, B, D, E, i.e.

(dtPaX)el - 0. (3.3b)

We may thus neglect the explicit effect of elastic colli-sions in the Boltzmann equation. Of course, the influ-ence of these collisions remains implicitly in the reactiveterms through the fact that the molecular speed dis-tributions are now Maxwellian and that distinction ismade between molecules occupying different momen-tum states.

Let us set FaArAp = fa, f being the average numberof molecules in the phase-space volume ArAp. Todescribe the fluctuations around this average occupa-tion number, we assume that Eq. (3.1) defines a Mar-kovian process in the complete phase space. It is theneasy to derive, by a method due to Siegert (14), amaster equation for the probability function P({f}, t).The reduced form of this equation, summed over allreservoir variables, reads (4):

dP({fx} X t) - Z BiJkfMdt ijkl

X [P(fkX - 1 {If'}, t) - P(fkX, {f'}, t) ]

+ EAijkI[(fix + 1)(fX + 1)P(f x + 1f4 + 1,ijkd

fkX lfx - 1, Iff}, t)- fiXfiXP(fiXfjXfkXf X, {f'}, t). (3.4)

{f'} denotes the occupation numbers of the states notimplied in the reaction steps. When Eq. (3.4) is multi-plied by fosx and averaged over all fs, it yields Eq.(3.1), provided a factorization assumption is made on P.The difficulty in solving Eq. (3.4) arises from the in-

finite number of coupled terms contained in the sumsover internal states on the right side. For this reason,we study this equation in the limit of small fluctuations.One obtains a Fokker-Planck-type equation satisfied bythe reduced probability functions for individual leveloccupations.

¶ This does not exclude the existence of large overall con-straints such as chemical affinities that may drive the systemvery far from thermodynamic equilibrium.

Proc. Nat. Acad. Sci. USA 68 (1971)

Fluctuations in Nonequilibrium Systems 2105

The final result reads (4):

at) -2(Z A~jklfjX) X XaPl(Xat)

+ ap( ) 2fax Z Aajk fjX.bXa2 jkl

We have set

faX = faX + Xa-

(3.5)

(3.6)

At the steady state, this equation reduces to a form thatno longer contains explicitly the transition probabilitiesA ijkl and Bijkl, and which admits the solution:

Pi(x.) = (2rjaX) 1/2 exp [- (3.7)

for whichX= = X (3.8)

Integrating this relation over positions and momentaand recalling that Fx remains locally Maxwellian, werecover the extended Einstein relation (1.1). We see,therefore, that the phase-space description gives rise toresults that for small thermal fluctuations assume auniversal form compatible with the extended Einsteinformula, substantiating the qualitative arguments ad-vanced in the previous section. Comparing the twomaster Eq. (2.3) and (3.4), one can see that the inade-quacies of the usual birth and death type stochasticformulation are due to the fact that in this method theinternal states of the system are treated incorrectly.For instance, Eq. (2.3) implies that there is a finiteprobability that two molecules of X [see term contain-ing P(X + 2,t) ] be at the same state. In the more com-plete phase-space description, the first of the conditions(3.2) implies that, in the thermodynamic limit, theprobability of this event is vanishingly small. Neces-sarily then, a bimolecular step introduces in the generalmaster equation (3.4) 2 molecules of X belonging totwo different internal states.

In the more general case of systems far from localequilibrium, although Eq. (3.8) is not expected to bevalid, there is no reason to believe that the birth anddeath process description will again become adequate.It is only for systems that do not admit a statisticalmechanical formulation (e.g. social or ecological prob-lems) that this description can be applied, althoughagain there can be no objective criteria assuring thatthe process is Markovian.

4. THE VOLTERRA-LOTKA MODEL

In the model discussed in the previous two sections,the macroscopic steady, stateXo or AX was asymptoti-cally stable with respect to arbitrary perturbations. As apreliminary to the problem of fluctuations around non-equilibrium states in the neighborhood of instabilities,we now study a system whose steady state lacks asymp-totic stability. The particular model we choose is the

Volterra-Lotka model, which originally was conceivedto describe the competition between a number ofpredator and prey biological species (15).

Let X, Y denote the populations of two interactingpredator-prey species. The Volterra-Lotka equationsread:

dt

dY= k2XY 63Y

dt

where el, E3, k2 are positive. Let us set

el= k1A

E3 = k3D

(4.1)

(4.2)

Eq. (4.1) are then the conservation of mass equationsof the following set of irreversible autocatalytic chem-ical reactions (in the limit of an ideal mixture):

kiA + X 2X

k2

X + Y -> 2Yka

Y+ D-E + D. (4.3)

This chemical analog of the Volterra-Lotka model isvery useful in understanding the mechanism of fluctu-ations in the vicinity of the onset of oscillations and in-stabilities.The macroscopic behavior of system (4.1) to (4.3) has

been analyzed by Volterra (15). The main results are asfollows:

(i) The system admits one nontrivial steady-statesolution

k3D k1AXo= , O =k2 k2

(4.4)

(ii) Small perturbations around (X0, Y0) exhibitundamped oscillations with a universal frequency

Co = (kik3AD) /2 (4.5)

(iii) For arbitrary perturbations, Eq. (4.1) admit aconstant of motion (15, 16). Thus, finite perturbationsare also periodic in time with periods depending on theinitial conditions. The trajectories of the system in thespace (X, Y) consist of a dense net of closed curves thatare all orbitally stable (but not asymptotically stable).

(iv) Oscillations can only occur in the limit whenthe system is displaced far from the state of thermo-dynamic equilibrium (1, 17).We now study fluctuations around state (4.4) or

around a periodic trajectory. We first assume, as in Sec.2, that the macroscopic equations (4.1) define a birth anddeath process in (X, Y) space. The master equationfor the reduced probability distribution

Proc. Nat. Acad. Sci. UTSA 68 (1971)

2106 Applied Mathematical Sciences: Nicolis and Prigogine

P(X, Y, t) reads (4):

dP(X, Y, ) A(X 1)P(X-1, Y t)dt

-AXP(X, Y, t) + (X + 1)(Y- 1)P(X + 1, Y-1,t)

-XYP(X, Y, t) + D(Y + 1) P(X, Y + 1, t)

- DYP(X, Y, t). (4.6)

A detailed analysis of this equation (4) shows that, asfor the model examined in Sec. 2, the distribution offluctuations is not given by the Poisson law. Thus, smallfluctuations contradict Eq. (1.1). Moreover, the dis-tribution function is always time-dependent and themean square correlations 5X2, 6y2 are slowly-increasingfunctions of time, even for infinitesimal fluctuations.Stochastically, therefore, the steady state (4.4) is un-stable and the system exhibits, for long times, abnormalcritical fluctuations. Ecologically, this situation has aclear interpretation: the steady-state prey distributionis never stable because there is no internal mechanismthat reestablishes equilibrium, once the latter is per-turbed by the predator.

5. PHASE-SPACE DESCRIPTION OF THEVOLTERRA-LOTKA MODEL

The chemical analog (4.3) of the Volterra-Lotka modeladmits a phase-space description which, according tothe results of Secs. 2 and 3, is the only correct descrip-tion to be adopted for chemical kinetics in the localequilibrium regime. Assuming again that the system ismaintained uniform in space, one obtains, for a dilutemixture, two coupled Boltzmann equations for theaverage number densities of constituents X and Y in agiven internal state (4, 13):

= -E AajkZ aXPJ + 2 E Ajka. jXFkdt Akl jkl

- E BaJklaXFjY (5.1a)Akl

dPr - E BjjkLFIXF7 + 2 E BjkilFjXFkydt kl Al

- ZCfjklFoYFgJ. (5.1b)ilk

The transition probabilities per unit time for reactivecollisions A jkl, Bikl, Cikl satisfy conditions (3.2).Elastic collisions terms have not been written ex-plicitly, on the assumption that the system has attaineda local equilibrium regime.We observe that the structure of Eq. (5.1) is quite

different from Eq. (4.1) or their generalization tomany components (15, 16) in spite of the fact that, onaveraging Eq. (5.1) over internal states, one obtains(4.1). In particular, (5.1) cannot give rise to a constantof motion as in the Volterra (15) or Kerner (16) analysis.

Before we proceed to the study of fluctuations, wewish to emphasize that the two levels of description

given by Eq. (4.1) and (5.1) correspond to two largelydifferent, but interesting, ecological situations. In thesystem described by Eq. (4.1), it is assumed that theindividuals of the prey population are consumed in-differently by the predator. When the predator popula-tion is comparable to or larger than the prey, it isreasonable to expect that this is indeed the most proba-ble situation. However, in the more realistic case ofsmall predator versus prey ratio, the most naturalcompetition consumes preferentially those prey indi-viduals having small values of some "fitness" parameterthat measures the ability to resist to or escape from thepredator. It is then natural to expect that in such sys-tems, in addition to the effect of the predators, theinternal processes determining the fitness distributionwithin the prey species should play an important role.This situation is well described by a set of equations ofthe type (5.1), provided one reinterprets suitably theparameter a determining the internal state. One of theconsequences of these processes is to permit an evolutionof the prey species in which the unfit individuals areeliminated continuously.

Let us now study the fluctuations around the steady-state solution of Eq. (5.1), and assume that the latterdefine a Markovian process in the complete phase space.Using the same method as in Sec. 3, one can derive amaster equation for the reduced probability distribu-tion P({fx}, {fy}, t). In the limit of small fluctuationsthis gives rise to two Fokker-Planck equations of theform (4):

aPi (Xa,t) -Xa XPi (X., t)

XF[2 f-) EZAi.J2iXJ.A~_ __btn] + a Pi(Xa" t)jlX2E jaf

-)X2 2t

Lf Xijil Jt (Xa2

E<[AjafXjA (5.2)

and a similar equation for PI(yp, t). The fluctuations xa,ye have been defined as in Eq. (3.6). Strictly speaking,Eq. (5.2) is coupled to the equation for PI(yp, t) throughthe average values fX, fy, which must satisfy the self-consistency conditions

faX = E faXP({fX}X {fY} X t)[fX},IVY)

(5.3)Gar = E f,3Yp({fX}I {fY}, I).UX) I IfY)

To a first approximation (consistent with the Fokker-Planck limit), one is allowed to identify the fs with themacroscopic averages appearing in the Botzmann Eq.(5.1). It follows that Eq. (5.2) admit steady-state solu-tions of the form (3.7) and that the steady referencestate fanx, f,,y is always stable with respect to small thermalfluctuations. Similarly, the extended Einstein relation(3.8) is valid in the domain of small fluctuations (4).One has, therefore, an a posteriori justification, in the

Proc. Nat. Acad. Sci. USA 68 (1971)

Fluctuations in Nonequilibrium Systems 2107

domain of small fluctuations, of the results obtained byKerner (16) and Montroll et al. (18) who postulatedrelations implying the validity of a canonical distribu-tion of fluctuations.

In conclusion, we see that in the Volterra-Lotkamodel the analysis based on the birth and death type offormalism is not compatible with the more completephase-space description. By comparing the two masterequations, we see that the inadequacy of the usualstochastic treatment is due to an incorrect account ofthe internal states.We have not yet analyzed completely the behavior of

large fluctuations of macroscopic size. It appears from apreliminary study (4) that such fluctuations are alwaystime-dependent and may drive the system, for suffi-ciently long times, to a new regime far from the initialreference state.

6. CONCLUDING REMARKS. "GENERALIZEDTURBULENCE"

We have shown that in nonlinear systems far fromequilibrium, it is generally necessary to adopt a phase-space stochastic description in order to describe cor-rectly the distribution of thermal fluctuations aroundsteady nonequilibrium states. As a consequence, forsystems in a local equilibrium regime, small fluctuationsalways behave according to the generalized Einsteinrelations (1.1) or (3.7), (3.8).While this result is quite natural for models of the

type discussed in Secs. 2 and 3, it may appear contra-dictory when applied to the Volterra--Lotka model orto any system which is at or slightly beyond a state ofmarginal stability. We believe that the resolution of thisapparent paradox, which is related to the very natureof the onset of oscillations and instabilities, may be thefollowing. The lack of asymptotic stability observed innonlinear dissipative systems (chemical instabilities,etc.) is a purely macroscopic phenomenon that has nomolecular counterpart as long as the system (includingfluctuations) is maintained in a local equilibrium regime.Thus, it is reasonable to expect that systems under-going such macroscopic instabilities cannot evolve froma given macroscopic reference state by a mechanism ofthermal fluctuations of usual size (i.e., very small).This explains why in the phase-space description ofSecs. 3 and 5 small fluctuations are damped. Again, thedifference with instabilities in the velocity distributionmentioned in the introduction should be emphasized.A change in the macroscopic state of a system in a

local equilibrium regime can therefore arise only from amechanism of large thermal fluctuations of macroscopicsize. Macroscopic instabilities seem, thus, to bear someanalogies with first-order phase transitions.

In order to substantiate this conjecture, it would benecessary to solve the complete master equations inphase space for model (4.3) in a way that takes intoaccount, self-consistently, the simultaneous evolution

must study the time-dependent solutions of the masterequation for arbitrary fluctuations, imposing at eachstage the self-consistency conditions (5.3) for anymacroscopic state. This study, which should providesuch information as, e.g., the critical size of fluctuationsbeyond which the system starts to evolve, and the timerequired for the formation of this evolving mode, ispresently in progress.

In conclusion, the separation between macroscopicbehavior and fluctuations is related to the distancefrom thermodynamic equilibrium. In a far from equi-librium regime, corresponding to the onset of oscilla-tions or instabilities, this separation is not possible ingeneral and the evolution of the average values dependsexplicitly on the fluctuations. The situation brings tomind the familiar phenomenon of fluid dynamicsarising beyond instability of the laminar flow and maybe appropriately called "generalized turbulence".We gratefully acknowledge that a first stage of this investiga-

tion was done in collaboration with Dr. A. Babloyantz. We areindebted to Prof. M. Kac for stimulating discussions and fruitfulsuggestions. I. P. would like to express his appreciation to Drs.A. Butterworth and R. Herman for their interest in this work andfor stimulating discussions during his stay at the General MotorsResearch Laboratories.

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Proc. Nat. Acad. Sci. USA 68 (1971)

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