Proc. Nati. Acad. Sci. USAVol. 76, No. 8, pp. 3607-3611, August 1979Physics

From deterministic dynamics to probabilistic descriptions(Markov semigroups/Bernoulli systems/I-theorem/internal time and entropy operator)

B. MISRA, I. PRIGOGINEt, AND M. COURBAGEFacult6 des Sciences Universit6 Libre de Bruxelles, Campus Plaine, Boulevard du Triomphe, 1050 Bruxelles, Belgium

Contributed by I. Prigogine, April 9, 1979

ABSTRACT The present work is devoted to the followingquestion: What is the relationship between the deterministiclaws of dynamics and probabilistic description of physicalprocesses? It is generally accepted that probabilistic processescan arise from deterministic dynamics only through a processof "coarse graining" or "contraction of description" that inev-itably involves a loss of information. In this work we present analternative point of view toward the relationship betweendeterministic dynamics and probabilistic descriptions. Speakingin general terms, we demonstrate the possibility of obtaining(stochastic) Markov processes from deterministic dynamicssimply through a "change of representation" that involves noloss of information provided the dynamical system under con-sideration has a suitably high degree of instability of motion.The fundamental implications of this finding for statisticalmechanics and other areas of physics are discussed. From amathematical point of view, the theory we present is a theoryof invertible, positivity-preserving, and necessarily nonunitarysimilarity transformations that convert the unitary groups as-sociated with deterministic dynamics to contraction semigroupsassociated with stochastic Markov processes. We explicitlyconstruct such similarity transformations for the so-calledBernoulli systems. This construction illustrates also the con-struction of the so-called Lyapounov variables and the operatorof "internal time," which play an important role in our approachto the problem of irreversibility. The theory we present can alsobe viewed as a theory of entropy-increasing evolutions and theirrelationship to deterministic dynamics.

1. IntroductionAccording to both classical and quantum mechanics, thetime-evolution of states obeys deterministic laws that aresymmetric with respect to inversion of time. Irreversibility ofphysical processes, on the otherhand, is expressed by the secondlaw of thermodynamics. For isolated systems, it affirms theexistence of a physical quantity, the entropy, that increasesmonotonically with time until it reaches its maximum atequilibrium. To elucidate the connection between the dy-namical description with its reversible and deterministic lawsand the thermodynamical description with its law of monotonicincrease of entropy is a fundamental problem of statisticalmechanics.

This problem is closely related to the question of the possiblerelations that might exist between the deterministic andprobabilistic descriptions of physical processes. Indeed, thestochastic Markov processes provide the best possible modelsto represent irreversible evolution obeying the law of increasingof entropy. As is well known, the usual expression for en-tropy

S pt In pt dM [1.1]

(and, in fact, any convex function of the distribution functionsp on the phase space r) is a Lyapounov functional ("h-func-

tion") for such processes (1). One would thus achieve a dy-namical interpretation of the second law if one could establisha satisfactory link between deterministic dynamical evolutionsand probabilistic Markov processes.The interest of the problem of the possible connections be-

tween probabilistic and deterministic descriptions is, however,not confined to the domain of statistical mechanics. It concernsthe problem of the meaning of probability in naturalscience.

It is generally believed that probabilistic processes can arisefrom deterministic dynamics only as a result of some form of"coarse-graining" or approximations. The main purpose of thispaper is to develop an alternative viewpoint toward the relationbetween deterministic and probabilistic descriptions. Morespecifically, we develop a theory of "equivalence," mediatedby nonunitary similarity transformations, between deterministicevolution and stochastic (Markov) processes.The viewpoint toward the relation between deterministic

evolution and stochastic Markov processes developed here isclosely related to the theory of irreversibility developed byPrigogine et al. (2). The main feature of this theory is that theproblem of reconciling dynamical evolution and irreversible(thermodynamical) evolution is viewed in terms of establishingan "equivalence" between them via a nonunitary similaritytransformation. In essence, the approach thus consists in thedetermination of a suitable nonunitary transformation A actingon the distribution function p so that the original deterministicLiouville equation

= Lpt [1.2]

is transformed by it to a dissipative evolution equation de-scribing the irreversible approach of the system to equilibrium.The transformation pt -P lit = Apt converts the Liouvilleequation into the equation

i-at= ]Pt, 1 = ALA-'. [1.3]

The above-mentioned requirement on A is thus the require-ment that the functional

Q= I "ptI2du [1.4]

is a Lyapounov functional (f-function) for the evolutionobeying Eq. 1.3. This is the main condition on A. Naturally,there are other physically motivated conditions to be fulfilledby A. But we need not discuss them here because they aretreated in refs. 2 and 3. For our purpose, the main point to retainis the important notion of "equivalence;" via a nonunitary A,between the dynamical and thermodynamical descriptions. Inconformity with this idea, we seek to determine the conditionson dynamics so that it becomes equivalent, via a similaritytransformation A, to a stochastic Markov process.

t Also at the Center for Statistical Mechanics at the University of Texas,Austin, TX.

3607

The publication costs of this article were defrayed in part by pagecharge payment. This article must therefore be hereby marked "ad-vertisement" in accordance with 18 U. S. C. §1734 solely to indicatethis fact.

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It is easy to show that the existence of such a transformationconnecting the dynamical evolution to a stochastic Markovprocess entails the existence of an operator M (acting on thedistribution function) with the property that

(Pt, Mpt) [1.5]decreases monotonically with t as Pt evolves according to theLiouville equation. Here we have used the inner productnotation (p,zp) to denote the integral

pe dM. [1.6]

The monotonic decrease of the expression 1.5 can be succinctlyexpressed by the commutation relation

i[L,M] D < 0 [1.7]with (p,Dp) = 0 only for the equilibrium ensemble.

In this way, the possibility of relating the dynamical evolutionto a stochastic Markov process turns out to be closely linked withthe possibility of introducing a new dynamical variable Msatisfying 1.7. These operators, called Lyapounov variables,have been studied by Misra to display the intrinsic irreversibilityof appropriate classes of dynamical systems (4).

If one could introduce a new observable representing non-equilibrium entropy, it would be given by such a Lyapounovvariable M with a change of sign. In this sense, then, the exis-tence of M expresses the intrinsic irreversibility of the dy-namical evolution.

Naturally, one does not expect M to exist for all dynamicalsystems. Moreover, one expects that (even if it exists) it can bedefined only in an extended frame of dynamics. In fact, animportant observation of Poincare (5) shows that the operatorM cannot correspond to multiplication by a phase function. Itcan be defined only as acting on the distribution functions andnot on individual phase points. One thus expects the existenceof M to be associated with some physical mechanism thatrenders the description of dynamical evolution in terms of phasespace trajectories an unobservable idealization, thus forcing theuse density functions.

Recent developments in classical mechanics show that sucha mechanism is the phenomenon of instability of dynamicalmotion. Many forms of instability have been discovered, andthey are found to be more common than originally believed (6).The common feature of dynamical systems having a suitablyhigh degree of instability is that each finite region of phasespace, no matter how small, contains phase points that movealong rapidly diverging or qualitatively distinct types of tra-jectories. Obviously, in this situation, the concept of deter-ministic evolution along phase space trajectories cannot bedefined operationally and, hence, constitutes a physically un-realizable idealization. Therefore, in dealing with dynamicallyunstable systems, classical mechanics seems to have reached thelimit of the applicability of some of its own concepts. Thislimitation on the applicability of the classical concept. of phasespace trajectories is-it seems to us-of a fundamental char-acter. It forces upon us the necessity of a new approach to thetheory of dynamical evolution of such systems that involves theuse of distribution functions in an essential manner.

It is shown in ref. 4 that the mixing property is necessary andthe condition of K-flow is sufficient for the existence of a Ly-apounov variable M. As is well known, mixing flows and (afortiori) K-flows are unstable to a high degree: arbitrarily closephase points move along widely diverging trajectories.

It is also found in ref. 4 that for K-flows there exists an op-erator time T satisfying the commutation relation

i[L,T] = I. [1.8]

The existence of an operator of time or "age" satisfying 1.8seems to express in a compact manner the inherent (but hidden)stochastic and nondeterministic character of the evolution.Once T has been constructed, it is easy to proceed further.

Lyapounov variables M can be constructed as monotonicallydecreasing positive operator functions of T

M = M(T), [1.9]and the nonunitary A connecting the given dynamical evolutionwith a dissipative irreversible evolution can be constructed asa square root of M:

A = MI/2 [1.10]In this way, we see that, at least for a class of abstract dynamicalsystems, the K-flows, the dynamical evolution is equivalent toa dissipative irreversible evolution.

However, let us keep in view that it might be possible to es-tablish the desired "equivalence" with a Markov process, fora suitably restricted class of initial conditions, even for systemsthat are not mixing, but present other types of instabilities (suchas Poincare's catastrophe). This would correspond to allowingM and A to be more singular objects than those we have con-sidered in this paper (7). Although this situation could be ofconsiderable interest in statistical mechanics, we do not considerit further in this paper.

Obviously, the central question of our approach is: Whatform of instability ensures the existence of an equivalence be-tween the dynamical evolution and a stochastic Markov pro-cess? As explained before, for K-flows, both M and A can beconstructed by a "canonical" procedure as functions of an op-erator time T. It is tempting to conjecture that in this case thetransformation not only converts the dynamical evolution ofthe K-flow to a dissipative process but also converts it to a sto-chastic Markov process. At present, we are not able to decidethis conjecture in its full generality. However, we show (andthis is the main purpose of the rest of the paper) that this con-jecture is true for an important subclass of K-systems-the so-called Bernouilli systems (8, 9).

It seems to us that the significance of this result extends farbeyond its immediate application in statistical physics. It proves,of course, the Boltzmann's k-theorem for the Bernouilli sys-tems without invoking the questionable "molecular chaos"hypothesis or any form of "coarse-graining." But more im-portantly, it confirms our view of how probabilities could arisein physics other than as a result of approximation.From a mathematical point of view, the theory we present

is a theory of positivity preserving similarity transformationsthat connect unitary groups describing deterministic dynamicalevolution with dissipative semigroups associated with Markovprocesses. This theory is, evidently, in its infancy. In fact, thevery possibility of such a connection between deterministic andprobabilistic descriptions (which is established here in thespecific case of Bernouilli systems) is rather an unexpected resultthat puts the entire problem of the'role of probability in physicsin a new perspective.To keep the article brief, we shall omit most of the details of

proofs in the following. A more complete version of this paperwill appear in a forthcoming publication (10).2. Notion of "equivalence" between deterministicdynamics and probabilistic ProcessesWe now proceed to formulate and discuss in greater detail thenotion of "equivalence" between deterministic evolution andprobabilistic Markov processes. Let F denote the state space (aconstant energy surface of the phase space) of the system.

Deterministic evolution of the system is described by aone-parameter group Tt of one-one transformations that mapr onto itself:

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TtT = Tt+8.In the case of Hamiltonian systems, Tt will be a group of ca-nonical transformations generated by the Hamiltonian.We shall suppose that there is a measure ,u (defined on a

a-algebra 3 of subsets of r) that is invariant under the dy-namical group Tt. For Hamiltonian systems, the existence ofsuch an invariant measure is guaranteed by Liouville's theorem.Furthermore, the invariant measure will be supposed to be afinite and, for convenience, normalized measure.To define a probabilistic process (within the same state space

r), one needs to specify the transition probabilities P(t,W,A)that the system starting initially from the state w will reach thesubset A of r in time t. It is evident that the function P shouldsatisfy the following conditions:

(i) P(t,W,A)>(ii) P(t,wP) =(iii) For t, w fixed the function P(tW,A) defines a proba-

bility measure on r. [Moreover, one imposes the tech-nical condition that for fixed t and A, P(t,w,A) is ameasurable function of co.] The probabilistic processdefined by the transition probabilities is called a Markovprocess if P satisfies the Chapman-Kolmogorov equa-tion:

P(t + s,w,.A) = fP(t,wv,dw')P(S,co',A). [2.2]

As explained in many textbooks, this important condition ex-presses the important property of Markov process that the fu-ture statistical behavior of the system depends solely on theinitial condition independently of the past history.A measure ji on r is said to be an invariant measure for the

process [with transition probabilities P(t,w,A)] if

P(t,w,A)dji = q(A) [2.3]

for all t > 0 and A E 13.To formulate the notion of equivalence between determin-

istic evolutions and stochastic Markov processes, it is necessaryto consider how the distribution functions on r transform underrespective evolutions. In the case of deterministic evolution, itis easy to see that the initial distribution p transforms in timet to the distribution pt defined by

Pt(co) = p(T-t(c). [2.4]The transformations Ut mapping p to pt are unitary operatorsof L2 (= the Hilbert space of square integrable functions withrespect to ji):

(Utp)(w) = p(T-tv). [2.4']The group property of Tt passes on to Ut. The generator of

this unitary group is the Liouvillian operator L:

Ut = e-iLt [2.5]

For Hamiltonian systems L is given byLp = i[Hp]PB, [2.6]

in which [,]PB denotes the familiar Poisson bracket.Let us now consider the transformation of distribution

functions under the stochastic evolutions corresponding toMarkov processes. To this end, let us first note that with everyMarkov process [with transition probabilities P(t,w, A)] havingan invariant measure ,u, one can associate a family Wt of op-erators defined by

Wtf(w) = f(c')P(t,wv,dwv') [2.7]

forfeL2.

The Chapman-Kolmogorov equation for P(t,w,A) thenimplies the semigroup property for Wt:

WtW, = Wt+, (for t,s > 0). [2.8]

Moreover, the previously stated properties (i-iii) of the tran-sition probabilities entail the following properties of Wt:

(a) The operation Wt preserves positivity. This means that,if f(w) 2 0 for almost all (a.a.) w e r, then Wtf((co) 2 0for a.a. c too.

(b) Wt Pequ = Pequ, in which Pequ is the uniform distributionPequ 1.

By putting f = A, the characteristic function of the set A c rin 2.7, one finds

P(t,w,A) = (WttPA)(w). [2.9]By using the above relation between Wt and P(t,w,A), the

invariance of the measure ,u for the process (relation 2.3) is easilyseen to be equivalent to the condition:

(c) Wt Pequ = Pequ.To sum up, every Markov process with an invariant measure

defines through formula 2.7 a semigroup Wt of operators actingon L2 and having properties a through c. The converse of thisstatement is also true: every semigroup Wt of operators on L,with properties a through c defines a Markov process (having;z as invariant measure) whose transition probabilities P(t,W,A)are obtained from Wt by formula 2.9 [by a slight variation oftheorem 2-1 given by Dynkin (11)].

It is easy to see that if p denotes the distribution functiondescribing the initial state of the system, the state jt to whichit evolves in time t under the Markov process is given by

lit = W* Pa [2.10]It seems natural now to consider the deterministic evolution

described by the unitary group Ut (induced from Tt) as"equivalent" to the stochastic evolution associated with thesemigroup Wt (see relation 2.7 and 2.9) if there exists a boundedtransformation A on L2 such that

(i) AUtp = WtAp (for t > 0),(ii) A preserves positivity,(iii) fpdu = f ApdM,(iv) A has a densely defined inverse, A-', and(v) Apequ = Pequ.Condition i simply says that the "change of representation"

p o Ap converts the deterministic evolution corresponding toUt to the stochastic evolution associated with Wt. Conditionsii through v are necessary requirements (from a physical pointof view) in order that A.may be interpreted as a "change ofrepresentation." In fact, conditions ii and iii simply express therequirement that A maps "states" into "states"; v says that thecontemplated change of representation leaves the equilibriumstate unchanged; and condition iv expresses the important re-quirement that the passage from deterministic to probabilisticdescription brought about by A involves "no loss of informa-tion." Condition i can be rewritten in the form

W-= AUtA-1 (for t > 0). [2.11]The problem for us now is to determine the class of dynam-

ical groups Ut similar to semigroups Wt that satisfy (in additionto conditions a through c) the following requirement:

11W(p - pequ)II2 = 11Wp- 112 - 0 [2.12]monotonically with t as t a o.

Semigroups satisfying conditions a through c and 2.12 will

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be referred to as strong Markov semigroups. They are associ-ated (through formulas 2.7 and 2.9) with truly stochastic Markovprocesses that display the irreversibility expressed in the secondlaw. In fact, condition 2.12 just says that Pequ is an "attractor"for the process in question and the approach to Pequ proceeds"monotonically." Now not only expression 2.12, but also theusual expression [1.1] for negative entropy (as indeed anyconvex functions of p) decreases monotonically for such pro-cesses provided p $ Pequ.The existence of a similarity transformation A connecting

(through 2.11) the dynamical group Ut to a strong Markovsemigroup Wt seems to express the inherent stochastic andirreversible character of the original dynamical evolution. Onethus expects such a transformation to exist (if at all) only forsystems with a suitably high degree of instability of motion. Thisintuitive idea is confirmed by the following:

PROPOSITION. In order that the dynamical group Ut besimilar to a strong Markov semigroup W* (for t > 0), it isnecessary that the dynamical evolution be mixing in the senseof ergodic theory.The proof of this statement follows from noting that, if

AUtA' = W is a (strong) Markov semigroup (for t 2 0), thenA*A = M is a Lyapounov variable for the evolution Ut. Asshown in ref. 4, the existence of Lyapounov variables impliesthat the Liouvillian (restricted to K = the subspace orthogonalto Pequ) has absolutely continuous spectrum, which in its turnimplies that the system is mixing.3. Time and entropy operators for Bernoulli systemsThis and the following section are devoted to carrying out thisprogram of Section 2 for the class of the so-called Bernoullisystems (8, 9). For the sake of simplicity of exposition, however,we shall limit our consideration to the simplest of the Bernoullisystems, the baker's transformation. But we emphasize thatall the results found in this and the next section generalize toarbitrary Bernoulli systems.

Let us start with a brief description of the baker's transfor-mation. The phase space F is now the unit square in the plane,and the measure A is the usual Lebesgue measure of the square.The baker's transformation B sends a point w = (p,q) of thephase space to the point Bw with

Bw = (2p,q/2) if 0 < p < 1/2 andBw = (2p- 1, q/2 + 1/2) if 1/2 < p <1.

The discrete group Bn, (n = 0, i1, +2,. .) that replaces thecontinuous parameter group Tt of the preceding section maybe thought as describing a discrete deterministic process takingplace at regular (unit) intervals of time.A striking, and indeed the characteristic, property of the

baker's transformation B is that the partition P = JA0o,A of theunit square into the right and left halves is "independent" and"generating" with respect to B. (For the definitions of these twoconcepts, see refs. 8 and 9.) It is the existence of an independentand generating partition that characterizes a general Bernoullisystem. The baker's transformation corresponds to the specialcase that the independent and generating partition can bechosen to consist of exactly two sets AO,A with jt(Ao) = g(Ac)= /2.

Returning to the baker's transformation, the discrete groupBn now induces a discrete unitary group Un on L2: (cf. Eq. 2.4).By a Lyapounov variable (or negative entropy operator) of thebaker's transformation, we mean a bounded operation M onL2 such that

(i) M > 0; i.e., ( p,Mp) > O for all p e L2;(ii) Mp u = Pequ, and(iii) (7UfMUnf) - 0 monotonically as n -0 O for allf in

KO , the orthogonal complement of Pequ.

To construct the Lyapounov variables M for the discretegroup Un, we follow the general scheme described in ref. 4 andconstruct first the operator T representing "internal time" or"age" of the system. In the case of continuous parameter groupUt = e-iLt, the operator T is, by definition, a self-adjoint op-erator that satisfies the canonical commutation relation 1.8 ona suitable dense set of vectors of K -. Expressed in terms of theunitary group e-'Lt, this relation becomes

eiLtTe-iLt = T + tI. [3.1]Thus the operator of "age" for the baker's system is, by defi-nition, a self-adjoint operator T that satisfies

U-nTUn = T + nI [3.2]onKjY

Let En denote the eigen projection of T corresponding to theeigen value n:

co

T = E: nEn-n=-Xo

[3.3]

The Ens being eigen projections of a self-adjoint operator sat-isfy

EnEm = bnmEnand

+ co

E En =I.

Condition 3.2 is now equivalent to the conditionU-'EmU = Em-i.

[3.4]

[3.5]Thus the problem before us is to construct a family En (n

integers from -o to +co) satisfying 3.4 and 3.5. To this end, wemake use of a special basis of K j constructed below.

Let P = IAO,Ac = A 1 be the partition of the unit square intoleft and right halves. Define

Xo = 1 - 2po and

Xn = UnXo (n= +1, +2,....). [3.6]For any finite set S = (n , n2, . . . n,) of (positive or negative)integers, put

Xs = XniXn2. Xnr [3.7]Making use of the independence of the partition P, one can

verify that the collection of functions Ixs , with S finite subsetsof integers, is an orthonormal set and the generating propertyof P entails the completeness of the set (in K -L). Moreover, fromthe definitions of Xn and Xs, it follows that the unitary operatorU induced from the baker's transformation B acts as a shift onthe orthonormal basis:

Uxs = Xs+ 1. [3.8]Here S + 1 denotes the subset (n1 + 1, n2 + 1,... n, + 1) ifS stands for (n , . . nr). In view of 3.8, the eigen projectionEn(n = 0, ±1, ±2,. ..) of the operator time T can now be de-fined as the projection onto the subspace spanned by all Xscorresponding to subsets S such that n e S and all other integersin S are less than n. The property 3.4 of the EnS thus definedfollows from the fact that IXsI is a complete orthonormal basisin K I and condition 3.5 is a consequence of 3.8.The construction of Lyapounov variables M now follows

directly. They are monotonically decreasing operator functionsof T. Corresponding to every (two-sided) sequence X2 of non-negative numbers bounded by 1 and decreasing monotonicallyto 0 as n +o+ the operator

+c[M = a, X En + P [3.91

_co

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(with En eigen projections of T and Po the projection onto Pequ)is easily verified to be a Lyapounov variable for the discretegroup Un.To conclude this section, let us mention how the operator

time T allows us to associate an "age" or "internal time" withwell-defined distribution functions p, or rather with the excess

distribution functions -p- P equ If is an eigen function ofT, the corresponding eigen value is the "age" associated withthe distribution function p. For instance, the excess functionfor the distribution function (1 - Xn) is Xn, which is an eigenvector of T for eigen value n. Condition 3.5 or 3.8 makes thisassociation a "consistent" one in the sense that it ensures thatthe change in "internal time" or "age" of the system broughtabout by the dynamical evolution matches with the progressof external (or observer's) time that serves to label the dynamicalgroup. Existence of a consistent "internal time" operator T inthis sense is, of course, not allowed for all dynamical systems.

If the excess distribution function is not an eigen functionof T but a combination of eigen functions corresponding to twoor more distinct eigen values, then one cannot associate a

well-defined age to p. But one can still associate an "averageage" T(p) to p by the formula

T(p) = 1[2 (,Tp), [3.10]

just as in quantum mechanics.

4. "Equivalence" of Bernoulli systems with stochasticMarkov processes

Let 0 < Xn < 1 (n = 0, ±1, .. .) be any sequence of positivenumbers decreasing monotonically as n increases. If

+ 0

A= E XnEn +Po, [4.1]n=-X

in which Ens are the eigen projections of the operator timeT.

It can be shown (10) that this transformation preserves thepositivity and the normality. However, A-1 is not positivitypreserving. Therefore, to make AUnA-1 positivity preservingwe require that Xn be such that the sequence vn = Xtn+ /Xnalso decreases monotonically as n increases [for instance, takeXn = (1 + en)-1]. With this requirement, it can be shown thatAUnA-I is a semigroup of Markov process and that A has allthe properties listed in Section 2, for n > 0. The transformedgroup AUnA-' is of course defined for both positive andnegative n. But it is important to note that it is only for n > 0

that AUnA-' preserves positivity. We may of course defineanother transformation A such that AUnA'- preserves positi-vity for n < 0. The important point is that the same transformedgroup AUnA-I cannot correspond to probabilistic process forboth positive and negative time n. This breaks the symmetrybetween the positive and negative direction of time and causes

the physical evolution to be described by a semigroup ratherthan a group.

5. Concluding remarksThe most striking conclusion, to emerge from our discussion isthat the deterministic and probabilistic descriptions are not as

radically different as it has been thought in the past and"coarse-graining" or "contraction" is not the only way of re-

lating them. We have demonstrated the possibility of linkingprobabilistic descriptions and deterministic descriptions bysimply a "change of representation" that involves "no loss ofinformation."

Looked at from a slightly different point of view, the presentwork could be considered as a part of a theory of entropy-in-creasing evolutions and their relations to deterministic dy-

namics. Historically, Boltzmann's kinetic equation representsthe first example of an entropy-increasing evolution. To arriveat this equation, Boltzmann had to introduce probability intodynamics from outside. In this work we have demonstrated thatentropy-increasing evolutions can arise from deterministicdynamics simply as a result of "change of representation"brought about by invertible (nonunitary) similarity transfor-mations A. This finding is in conformity with, and concretelyillustrates, the point of view towards the problem of irrevers-ibility developed in ref. 2.A specially attractive feature of the theory of irreversibility

emerging from the considerations of refs. 2 and 4 and thepresent work is the close links it establishes between instability(expressed in terms of mixing and other ergodic properties), theinherent irreversibility (expressed in terms of the existence ofLyapounov variables M), and the intrinsic randomness (ex-pressed in terms of the existence of an "equivalence" with astochastic Markov process) of dynamical motion. We haveshown that the class of systems exhibiting instability containsthe class featuring inherent irreversibility (for instance, theK-systems), which in its turn contains the class (e.g., Bernoullisystems) displaying intrinsic randomness of motion. The preciseboundary between these classes is, however, at present notknown. It will be an interesting problem to determine theprecise extent of these classes. Let us also note that the notionof intrinsic randomness of dynamical systems formulated herediffers from-and seems to refer to a more intrinsic form of therandomness of dynamical evolution than-that expressed bystrict positivity of Kolmogorov "entropy."The concepts of instability, inherent irreversibility, and in-

trinsic randomness are formulated and studied here in theframe of classical mechanics. It obviously will be interesting andimportant to extend and study these concepts for quantumsystems as well as for gravitational systems requiring generalrelativity for their description. We plan to come back to thisquestion in subsequent communications.

Let us note that for an arbitrary system, the average (T) ofthe internal time operator in a state p given by 3.10 can be easilyshown to be equal to the change in the ordinary time dt. Theinternal time operator (when it exists) contains, however, ad-ditional information about the physical system that concernsthe fluctuation or dispersion of the "internal age" around theaverage value. In both classical and quantum dynamics, timeappears simply as an external parameter to label the dynamicalgroup. In contrast, the internal time operators considered hereare new physical observables associated with the irreversibleevolution of the system. From this point of view the concept ofinternal time operator is closer to the concepts of thermody-namic and biological time and may serve as the "microscopic"counterpart of the latter phenomenological concepts.1. Yosida, K. (1974) Functional Analysis (Springer, New York).2. Prigogine, I., George, C., Henin, F. & Rosenfeld, L. (1973) Chem.

Scr. 4, 5-32.3. George, C., Henin, F., Mayne, F. & Prigogine, I. (1978) Hadronic

J. 1, 520-573.4. Misra, B. (1978) Proc. Natl. Acad. Sci. USA 75, 1627-1631.5. Poincare, H. (1889) C. R. Hebd. Seances Acad. Sci. 108, 550-

553.6. Moser, J. (1974) Stable and Random Motions in Dynamical

Systems (Princeton Univ. Press, Princeton, NJ).7. Prigogine, I., Mayne, F., George, C. & DeHaan, M. (1977) Proc.

Natl. Acad. Sci. USA 74, 4152-4156.8. Ornstein, D. S. (1974) Ergodic Theory, Randomness and Dy-

namical Systems (Yale Univ. Press, New Haven, CT).9. Shields, P. (1973) The Theory of Bernoulli Shifts (Chicago Univ.

Press, Chicago).10. Misra, B., Prigogine, I. & Courbage, M. (1979) Physica, in

press.11. Dynkin, E. B. (1965) Markov Processes (Springer, New York),

Vol. 1.

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