The Many Faces of Irreversibility

8/9/2019 The Many Faces of Irreversibility

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The ritish Society for the Philosophy of Science

The Many Faces of IrreversibilityAuthor(s): K. G. DenbighSource: The British Journal for the Philosophy of Science, Vol. 40, No. 4 (Dec., 1989), pp. 501-518Published by: Oxford University Press on behalf of The British Society for the Philosophy ofScienceStable URL: http://www.jstor.org/stable/687738 .Accessed: 11/09/2014 06:41

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Brit. i. Phil. Sci. 40 (1989), 501-518 Printed n Great Britain

The Many Faces of IrreversibilityK. G. DENBIGH

ABSTRACT

Irreversibility, t is claimed, s a much broader oncept han s entropy ncrease, s

is shown by the occurrence f certain processes which are irreversible ithoutseeming o involve any intrinsic ntropy change. These processes nclude hespreading utwards nto space of particles, r of radiation, nd they also ncludecertain biological and mental phenomena. For instance, he irreversible ndtreelike branching which is characteristic f natural evolution s not entropicwhen t is considered n itself-i.e. in abstraction romaccompanying iochemicaland physiological ctivity. What appears o be the common eature fall forms firreversibility s the fanning out of trajectories, new entities or new states, in thetemporal direction towards the future.

1 Introduction2 Classes of Irreversible rocesses3 Irreversibility Defined4 T-invariant heories5 The Temporal Reference Direction6 Irreversibility as no Spatial Analogue7 Does Irreversibility Necessarily Involve Entropy?8 Fanning Out Towards The Future

I INTRODUCTION

In the Timaeus, lato spoke of time as 'revolving' nd t may be that he believedthat 'time tself' s cyclic n some sense. Even so he did not suppose hat mostsequences of events are anything other than irreversible. o be sure 'life cycle'is a commonly used expression; ut clearly t does not mean either that theevents of a person's ifecan occur again in the reverse order, or that a personcan be reborn and then lead exactly the same life over again. Plato's view, if Ihave understood it correctly, was that there are cyclic motions in the heavensbut irreversible sequences on Earth.

So too in Asiatic cultures, even in those which have been widely regarded as

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502 K. G. Denbigh

having adopted the notion of cyclic time. For example Schipper and WangHsiu-Huei [1987] have pointed out that, although Taoist ritual used a conceptof 'cycles' nested within each other, this nesting was in a 'time' which, in itself,was taken as linear. Similarly in Indian thought; Anindita Balslev [19871recently pointed out that the supposed recurrence of 'world cycles' does notinvolve 'any idea of exact repetition of the particular, and that instead theemphasis is on the similarity of the generic features'.

As we know, Judeo-Christian thought uses the notion of a linear andprogressive time. But 'progressive' n what sense? Science has been widely seenas indicating a universe-wide process of 'running down', an approach to the'heat death'. However, I have argued elsewhere [Denbigh, 1989] that theentropy law is a good deal less restrictive than is commonly supposed.Something quite distinct from a running down may also be taking place if wecan but get a lead on it.

Towards this end it is useful to give consideration to the concept ofirreversibility. This has a much wider field of application than has the conceptof entropy increase, as may be seen from the fact that there exists a widevariety of processes which are undoubtedly irreversible whilst seemingly notgiving rise to any closely correlated entropy change.

Thus the object of this paper is not at all concerned with the current theorieseither of entropy increase or of 'chaos', but rather to consider 'one waytemporal development in a much broader context, one which will include theirreversibility of biological and mental processes. This essay will therefore notattempt the mathematical sophistication of Harold Grad's famous paper 'TheMany Faces of Entropy' [1961] but instead will be entirely qualitative andphenomenological.

Before proceeding let me repeat the truism that one cannot talk about eithertime or irreversibility without using temporal words. If one were to say, forexample, that some particular sequence of events does not occur in the reverseorder, the understanding of 'sequence', 'events' and 'occur' depends on a prioracceptance of certain temporal presuppositions which are deeply embedded nlanguage. Even the using of the present tense in a 'tenseless' (i.e. timelesslytrue) manner does not always eliminate the presupposing of 'time's arrow'.The same applies to the use of many substantives. For instance to speak of 'anexpansion' or of a 'light source' is tacitly to adopt a particular direction of timeand not its reverse. The question how that direction is chosen will be deferredto Section 5.

2 CLASSES OF IRREVERSIBLE PROCESSES

The class which comes first to the thoughts of most scientists is that group ofprocesses, occurring in physico-chemical systems, which may be called thethermodynamic lass. Typical examples are the flow of heat from hotter to cooler



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The Many Faces of Irreversibility 503

bodies, frictional and viscous phenomena, inelastic collisions, processes ofmixing and diffusion, and the immense number of chemical and nuclearreactions.

Theycan all be

brought togetherunder the umbrella of the Second

Law since they are characterized by the non-decrease of entropy, so long as allconsequential changes in the environment are allowed for as well as thechanges in the system in question.

There is also a group of physical processes which are irreversible withoutbeing entropic-or at least not with any certainty. As long ago as 1932, E. A.Milne pointed out (Whitrow [1980], p. 10) that a swarm of noncollidingparticles tend, in due course, to move further and further apart in space, andcontinue to do so forever, even if initially their vector velocities were such that

theywere

movingtowards each other. A similar

irreversibilitymanifests itself

in the case of radiation, for the wave fronts tend to expand rather than tocontract.' In terms of Maxwell's theory one speaks of using retarded, and not ofadvanced, potentials in the solution of the equations.

This theme was developed further by Popper [1956, 1957, 1958] in a seriesof short papers where he discussed the example of circular waves movingoutwards on the surface of a pond, due to a disturbance at its centre; a cine filmof this process run backwards would show an entirely 'unphysical' process ofwaves being generated at the pond's periphery and subsequently converging othe central point. Popper argued that the normal tendency of the waves tomove outwards is a non-entropic orm of irreversibility. For although it is truethat the water waves are damped by viscosity, this is an adventitious factorsince the same phenomenon of an exclusively outwards motion would occurin the idealized situation of a non-viscous liquid. It can also be argued thatMilne's example of particles becoming more and more separated from eachother in free space is probably non-entropic. There are two opposing effects. Onthe one hand there is the familiar fact that the adiabatic expansion of a perfectgas from one closed volume to another gives rise to an entropy increase in thegas which goes up linearly with the logarithm of its volume. On the otherhand, in the situation where the same gas expands freely into unconfined pacethen, as it becomes more dilute, there occurs a progressive separation of thefastest moving molecules from the slowest moving molecules. This separationeffect would seem to imply a reduction of entropy, thereby reducing, orcompletely cancelling, the entropy increase due to the expansion. Currenttheory does not provide the means for settling the matter, due to uncertaintyabout whether or not a statistical entropy can properly be attributed toparticles within an unbounded space.

For further discussion on these forms of irreversibility he reader is referred

Of course under special conditions, e.g. by use of spherical mirrors, one can obtain a contractionof wave fronts. Similarly if a large circular hoop were dropped on to the surface of a pond, itwould result, at least for a short time, in inwards moving waves.



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to papers by Hill and Grtinbaum [195 7] and by Penrose and Percival [1962].For my present purposes it is sufficient to notice that there are physical

phenomenawhich are irreversible without

necessarily being entropic.Another very important kind of irreversibility s displayed n biology where itis familiar enough that the evolutions of the various species of organisms donot normally occur in reverse. The feathered birds do not return to being scalyreptiles, nor do the reptiles revert to their own parent genera. To be sure thereare individual instances of regression2 or of simplification of function. Simpson[1950] pointed out that the notion that evolution is invariably accompaniedby increase of 'complexity' is very difficult to substantiate. What does seemcertain, however, is that, in the temporal direction we call 'future', thereoccurs a

branching-whatDarwin called a

divergence. Throughoutthe

periodsince life first appeared on Earth new species have branched off from existingspecies,3 with the consequence that the overall evolutionary scheme has atemporal structure resembling the above-ground structure of a tree. Biologistswould find it quite unacceptable, I think, to suppose that at some future datethis structure would start to regress, resulting in all the 'advanced' organismsreturning to their ancestral states, and leading eventually to all life beingin the form of unicellular organisms, before these too vanish into a lifelessEarth.

A somewhat similar 'branching towards the future' occurs in eachindividual organism. Ontogeny, it was said by Ernest Haeckel, recapitulatesphylogeny No doubt this is an oversimplification, but nevertheless it is true, inthe present context, that the bodily development of each individual traces outan irreversible path, just as does the development of the biosphere as a whole.The cells of an embryo have the capacity to divide and to differentiate, givingrise to a large number of different orts of cells which go to form the tissues andorgans of the adult organism; it would appear contra natura o suppose that thisbranching process could ever occur in the opposite direction whereby an adultorganism would gradually lose the differentiation of its cells and tissues, andwould eventually revert to a single ovum and a spermatozoon.

To be sure, the specifically biological kinds of irreversibility are necessarilyaccompanied by the ordinary physiological and biochemical processes of theliving body.These are processes of fluid flow, of heat transfer and of chemicalreaction, and, as such, they are entropy producing in the normal way. What Ihave argued is that living things display their own distinctive kinds of

2 It is known that minor forms of adaptive hange, such as the colouring of moths, can be reversedif all ancestral environments are retraced. See, for example, Harvey and Partridge [198 7].

3 The number of species among the animals and plants alone is now believed to exceed 107. Ofcourse it is not to be supposed that the increase in the number of species goes on withoutinterruption. Indeed it appears that, at very big intervals of time, there may occur considerableextinctions of species. However, the fossil record indicates that the 'niches' made vacant arequickly filled, and thereafter the normal increase in the number of species is resumed.



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50o6 K. G. Denbigh

provided by human consciousness, the same ordering s achieved by using an'entropy clock' (involving some chosen physico-chemical process), or byjudging the order relative to the outwards flow of radiation or of particles froma source, or again by judging the order relative to some biological growthprocess of the type of rings in a tree trunk. Of course there will occur exceptionsto this statement if fluctuation phenomena are significant, but this is veryunlikely except in very small systems.

3 IRREVERSIBILITY DEFINED

The notion of irreversibility as been taken so far as tactily understood, ut itneeds now to be expressed more exactly. In fact irreversibility s a matter ofdegree and for this reason it is best defined as the negation of reversibilitywhich is an idealized nd limiting kind of process, not capable of being fullyrealized.

Reversibility nd its negation are characteristics either of 'things' nor oftheories, but only of the processes which can occur in 'things'. Let usconcentrate attention, at least for the present, on those macroscopic andinanimate 'things' which can be specified in terms of their temperature,volume and chemical composition, together with the intensities of anyprevailing fields. Such specifications are sufficient to fix the momentarymacroscopic tate of the entity ('system') in question; a process is a temporalsuccession of such states due to the changing of one state into another.4

A process is said to be reversible if, and only if, the system which undergoesthat process, together with all parts of its environment which are affected, an berestored reproducibly5 o their original states. For example, let the system gofrom an initial state A, through states B, C, etc., to a final state X. Thecorresponding simultaneous states of the affected environment are oc, fl, y,etc., up to a final state w. There is reversibility f it is possible not only for thesystem to be restored from X to A, but for this reversal to be accompanied by a

simultaneous reversal of the affected parts of the environment from (w o oc. Inshort all relevant parts of the universe must be capable of being put back tohow they were

Although the foregoing definition of reversibility can be applied to all theclasses of Section 2, it is easiest to apply, in a precise and mathematical sense,to the thermodynamic class. Some of the authors of textbooks dealing onlywith that class use an alternative definition; namely that a process is said to be

4 Quantum theory shows that the states of an ideally isolated physico-chemical system constitute

a discrete set. And of course the notion of 'state' tacitly supposes that physical entities cannot bein two or more states simultaneously.

s The condition of reproducibility-i.e. the attainability of reversal whenever it is desired-isnecessary because a momentary restoration of an original state can occur, in principle, byspontaneous fluctuation. This is discussed in Section 4.



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reversible if it can be made to proceed in the opposite direction by aninfinitesimal change in the system's environment. Yet a third definition saysthat a process is reversible if it passes

througha continuous

sequenceof states

none of which departs more than infinitesimally from an equilibrium state. Inmost physico-chemical situations there are no important differences betweenthe three definitions.6

Complete reversibility s not actually attainable in the real world. Irreversibi-lity is the natural state of affairs, although the concept of reversibility remains auseful idealization for purposes of theory. In the case of physico-chemicalsystems the matter is, of course, closely related to the Second Law ofthermodynamics: the overall entropy change is zero only in the limiting case ofa reversible process; in all real situations the entropy of system plusenvironment increases. Thus in the former case the application of theoryresults in mathematical equalities; in the latter case it yields only inequalities.

Irreversibility can, of course, be minimized under carefully controlledlaboratory conditions; for example by using mechanical systems which arealmost frictionless, or, in the case of electrical processes, by using superconduc-tors. A familiar example is the vaporization of a liquid by means of a piston andcylinder together with a heat reservoir. The liquid is vaporized by drawing outthe piston so slowly that the pressure above the liquid is only very slightlylower than the equilibrium vapour pressure. The liquid takes in heat from thereservoir which is at a temperature only very slightly higher. Subsequentrecompression at a pressure minimally greater than the vapour pressureresults in recondensation, and the amount of heat restored to the reservoir isthen only minimally greater than the amount withdrawn during the originalvaporization. Thus a cycle has been completed on the substance in questionand, at the same time, the 'external world', namely the heat reservoir, has beenput back almost to what it was.

4 T-INVARIANT THEORIES

The consideration of irreversibility does not arise in those theories of sciencewhich are concerned solely with structure (whether this be the structureof atoms or of living creatures), but it does arise as soon as theory seeks todeal with motion, or with other processes of change. It is of great importancethat all existing theories of the latter kind are 'time-invariant'. That is tosay the replacement of t by - t in the theory's equations makes no alterationto any of its predictions. This applies as strongly to relativity and to

' Remaining close to equilibrium is a necessary condition for reversibility in the first sense, but isnot a sufficient condition. This is nicely shown by an example due to Allis and Herlin [1952]concerning gas expansion into a vacuum when it is made to occur by the successive breaking ofan infinite sequence of membranes.



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quantum mechanics7 as it does to Newtonian mechanics and to electro-magnetism.8

The question arises: How can these t-invariant theories describe theirreversible processes of the real world? The answer, of course, is that theydon't What they can be used to describe is the idealized limiting case whichwas discussed in Section 3. For instance, in the case of Newtonian mechanicsthe theory can be applied to the motions of bodies (such as the planets andstars) which are not surrounded by a resisting medium, and which do notundergo inelastic collisions between themselves. Otherwise corrections or adhoc additions to the theory have to be made in order to achieve agreement withexperiment. To be sure there are a number of philosophers who seem tosuppose that the term 'theories' means the same

thingas 'laws of nature', and

who are thus led to the notion that, because the putative 'laws' are t-invariant,the two directions along the t-coordinate are entirely equivalent. '. .. the onlyplausible way', wrote Mehlberg [1961], 'of accounting for the fact that somany well-established and comprehensive laws of nature somehow concealtime's arrow from us is simply to admit that there is nothing to conceal. Timehas no arrow.'

The effect of such a claim is to make the notion of irreversibility appearentirely foreign to physical science. Yet this is not only contrary to the reality ofirreversibility n human experience but it is also entirely contrary to what isaccepted about the objective world in those sciences-notably biology, geologyand astrophysics-which deal with evolving systems.9

Let us briefly consider the logic of the matter. It is true that if a process can be

7 In the case of quantum mechanics the basic Schridinger equation for the state vector 0 doesnot contain dt as a square but only as a first power. However, it is the square of 0 which issignificant in regard to what is observable and, after allowing for this, it remains the case (as inthe other theories mentioned) that the replacement of t by - t makes no difference to thepredictions. Nevertheless there continues to be lively discussion in the literature on thequestions whether QM is fully t-invariant, and on whether it ought not to be. Phenomenawhich are effectively irreversible certainly occur at the single-particle level--e.g. the decay ofnuclei, the absorption of particles n photographic emulsion, etc. Then again the 'measurementproblem' remains very puzzling and seems to involve irreversibility at the micro-level.Furthermore the decay of neutral K mesons provides apparently good evidence that there areinstances of failure of t-invariance at the atomic level. It may be that QM is 'incomplete'precisely in regard to irreversibility.

8 It should be added that t-invariance requires not only the replacement of t by - t but also theinversion of those vector quantities which relate to the entities in question; for example, particlevelocities and spins must be reversed in direction and, if a magnetic field is present, this toomust be reversed. One then speaks of the system in question as being in its 'time-inverted' state,and these inversions and replacements result in the predicted motions or changes being thesame for 'time towards the past' as for 'time towards the future'.

9 Prigogine and his colleagues are prominent among those scientists who reject Mehlberg's view.Prigogine accepts irreversibility, and the reality of 'time's arrow', from the start and he aims atembedding the existing t-invariant theories within a much wider framework.



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made to occur with close approximation to reversibility then it requires a t-invariant theory for its description. (Equivalently a t-noninvariant theorydescribes rreversible

processes.)On the other hand if a

theoryis t-invariant the

processes it describes may or may not be capable of occurring reversibly; for thet-invariance of a theory is a necessary but not a sufficient condition for thereversibility of a process it purports to describe. " Other factors which areimportant relate to the proneness of a system's internal dynamics to developinstabilities, and to the effects of quite minute disturbances originating in theenvironment. For example, it is a commonplace that t-invarient laws mayapply quite accurately to processes occurring at the molecular level, and yetthat macroscopic systems containing vast numbers of molecules may behaveirreversibly, at least during periods of time much shorter than the Poincarerecurrence period. And of course showing that this is so provides much of thecontent of statistical mechanics. The 'laws' describing the behaviour of suchsystems become probabilistic in character, and also-because of the signifi-cance of acts of 'preparing' the system in question-the probabilities projected'towards the past' may not be symmetric with the probabilities projected'towards the future'.

No doubt those who support Mehlberg's view can claim quite correctly thatPoincare's famous theorem shows that a system containing only a finitenumber of particles must eventually return to a state arbitrarily close to itsinitial state.'1 The expected 'recurrence time' is, of course, immensely long-typically it is of the order 101025years for a system consisting of the Avogadronumber of molecules. When one speaks, as I have done, of the reality ofirreversible processes one is saying that processes occurring in macroscopicsystems are effectively rreversible; nd indeed they are, during all periods of theorder of the age of the universe-say a mere 101" years

Of much greater significance are the fluctuation phenomena whichrepresent small (and usually exceedingly small) deviations from the mostprobable state of a system. These too are very infrequent. For instance, adiminution of the entropy of a gram mole of helium by only a millionth part isnot to be expected more than once, on the average, in each 10'109 years(Denbigh [1981], p. 106). Even so the reality of fluctuations is confirmedexperimentally by such phenomena as the Brownian motion and the bluenessof the day sky.12

1o See also Bunge [1968] and Hobson [1971].1 Poincare's theorem was based on the classical mechanics but a somewhat similar theorem

holds in quantum mechanics (Ono [1949]; Percival [1961, 1962]; Hobson [1971]).12 In the light of fluctuation phenomena the Second Law must be regarded as being probabilistic

rather than absolute in character. Nevertheless its status as an 'impossibility theorem' can berecovered by reformulating the 'law' in such a way as to imply the impossibility of knowingwhen recurrence will occur. With this in view, Jaynes [1963] reformulated the Second Law asfollows: Spontaneous decreases in the entropy, although not absolutely prohibited, cannotoccur in an experimentally reproducible process.



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'rightwards'.'3 This lack of the quality of directedness in a line may becompared with the evident directedness of the sequence of real numbers sincethis

sequenceexhibits the relation of

'greaterthan', and this is intrinsic o the

numbers. Thus if x and y are any two real numbers, negative, zero or positive,there exists the property which may be expressed:

y > x = DF (y # x) (3 z) (x + z2= y).

How does the matter stand in the case of the temporal order? In all thatconcerns our own consciousness we do indeed endow this order with a cleardirection, for we are aware (presumably through the action of short-termmemory) of events as being 'earlier than' or 'later than' each other. (To be surewe are also aware of events as being fleetingly 'now' or 'present', and whetheror not this aspect of 'time' is objective, in the sense of being independent ofconsciousness, has been the subject of much controversy. However, the onlyrelevant issue here is the bearing of irreversibility on the directedness, or'arrow', of the temporal order; 'nowness' is irrelevant.)

Notice that if some particular process A -- X can be reversed, together withall consequential changes in the environment, this would obviously not betaken to mean that 'time itself' had been reversed. Thus if the process A -+ Xbegins at time tAand if the reverse process X -- A is made to complete itself inthe same physico-chemical system at time tA, we would certainly not wish toequate tA to tA, as if time had indeed 'gone back' on itself in some sense. Thegood reason why we reject such a possibility is, of course, that the world is fullof innumerable other spontaneous and distinguishable physico-chemicalprocesses which continue to proceed unidirectionally during the period whenthe reversal of any one of them is made to occur.

What is very significant, although very familiar, is that these multitudinousphysico-chemical processes share a common attribute, namely that theiroverall entropy changes occur in parallel, as is asserted by the Second Law."4Thus

(xSi - xSk) (fSi- PS) >,0,where , S and p S refer respectively to the entropies of any pair, oc and P, ofsystems plus their environments, and i and k refer to any pair of instantsirrespective of which of the instants is chosen by convention as having thecharacter of being later than the other. It is notable that in this version of theSecond Law, Schridinger's version, the subjective aspect of the judgmentabout 'later than' is by-passed. Even so it is convenient in science to adopt the

13 Ordering along a line requires either the use of an external viewpoint, or that the line has anextreme element, a terminus.

14 Ofcourse it is assumed that the processes in question occur on the macroscopic scale. Otherwisefluctuation phenomena could not be neglected.



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512 K. G. Denbigh

convention that moment k is 'later than' moment i if the entropies increasefrom i to k. For in that way we avoid what would otherwise be a silly clash

between 'scientific time' and 'human time'.Returning to the title of this section, it will be seen that the reference

direction can be chosen as based either on a consensus of all macroscopicprocesses or on some one long-lived process, such as the decay of a radio-activeelement, which is used as a standard. Neither of these procedures would havethe effect of relegating the status of the Second Law to that of a tautology sincethe empirical content of the law is the near-universality of the parallelism ofentropy changes, as expressed by the foregoing inequality.

Finally a few words about whether irreversibility mplies that 'time itself'

has a direction. In my view time cannot be regarded as an existent; t is not areal 'something' (although it is based on a real relationship). For the essentialcharacteristic of 'things' is their persistence in time and it would clearly bevacuous to say that time persists in time. Although it would be out of place inthis article to discuss absolute v. relational views of time, I believe that 'time', atany one location, is nothing more than a metric based on the relationship of'later than' as it pertains to events at that location.15 This Aristotelian view is,of course, entirely consistent with special relativity whose function is to relatethe temporal orders at different locations, using an assumption about the

maximum speed of signalling. It may be noted too that it is a bad linguisticusage to speak of events (as distinct from 'things') as being in time, since it isevents which are constitutive of time.16

6 IRREVERSIBILITY HAS NO SPATIAL ANALOGUE

Before dealing with this issue one must first ask: In terms of what items oflanguage should the issue be expressed? Clearly not by saying that one can 'goalong' time in one direction only, and that an analogous restriction does not

apply to space. For 'going along' is itself a temporal notion and therefore thecorrect spatial analogue could not be 'going along' space.What is required, I think, is the making of a comparison between a sequence

of events, E1, E2, ., and a series of locations L1,L2, .. along a straight line. Wehave to inquire what property s conferred on the event sequence by the fact ofits irreversibility which has no analogue in the case of the location series. Ofcourse the ordering relation for the one is 'later than' and for the other it is 'tothe right of', but this is not the relevant distinction. What is relevant is the

1S As noted by Bohm [1987] 'the concept of time must involve both irreversible process andrecurrent (cyclical) process', for it is the latter which provides a reliable measure.

16 The issues discussed in this section are dealt with more thoroughly in my book Three Concepts fTime [1981].



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existence of entropy, and the fact that entropy is a function of a body's state, butis not dependent on its location. (That state is, of course, an event n the body's

history.)What is also

very importantis the Second Law

which,as was

seen inthe previous Section, establishes a parallelism between the entropy change ofany one body and the entropy change of any other. (The term 'body' is hereintended to include all relevant parts of the environment.) In short the eventsequence El, E2, . ... relating to the body can be re-expressed as an entropysequence S1, S2, .., and one then obtains the relationship, as already quotedin Section 5, between the entropy changes of any two bodies.

The significant point in the present context is that there is no analogousparallelism involving the locations of two or more bodies. Thus there is no

irreversibilityn

spaceand there is no

spatial counterpartof

entropy.Thermodynamics thus goes far beyond special relativity in pointing up thedistinction between time and space. (Remember that Einstein himself acceptedthat one cannot telegraph into the past ) Whiteheadian philosophy makesthe same claim, although in very different erms. The world is seen as creative,and the temporal process of producing what is new is the fundamentalreality.

7 DOES IRREVERSIBILITY NECESSARILY INVOLVE ENTROPY?

As was seen in Section 2 there is uncertainty about whether or not theirreversible outwards flow of radiation, or of particles, into space is character-ized by entropy increase, as are the other familiar physico-chemical processes.Below I raise the same question about phylogeny, ontogeny and mentalactivities-i.e. the question about whether the irreversibility of these pro-cesses, considered in themselves and in abstraction from metabolism and otherphysiological action, is an entropic kind. This issue is, of course, closely boundup with the project of reductionism, and it is also bound up with the mind/bodyproblem.

Leaving these latter questions aside, it needs first to be said that, becauseentropy is a physico-chemical quantity, it requires the attribution to it of aspatial location. For example, if it were claimed that biological evolution is, initself, an entropic process it would need to be asked whether the entropyincrease in question is held to be located in the total mass of all livingorganisms, or in the whole eco-system or in the Earth's biosphere. Clearly theanswer cannot be obtained by experiment, but it is conceivable that it might beobtained by theoretical argument.

However, this proves to be a mirage. For one reason because living creaturesare 'open' systems, in the sense that they exchange material, as well as energy,with their environment. An estimation of their rate of entropy productionwould therefore have to take account of all consequential changes in their



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514 K. G. Denbigh

surroundings, and this would be exceedingly difficult. For another reasonbecause they are such complicated systems anyway, and this defeats the

possibilities of calculation which might otherwise be achievable by themethods of statistical mechanics. Faced with these difficulties, a number ofbiologists have leaned rather heavily on information theory but, in my view,the value of what they have claimed is vitiated by the confusion which existsconcerning the significance of the familiar expression p1i npi. As I haveargued elsewhere [Denbigh and Denbigh, 1985], this measure does not meanat all the same thing in Shannon's information theory as it does in statisticalthermodynamics. Identity of mathematical form is not sufficient for this to bethe case, for it is a matter of what the pi, and the summation, refer to. Thus

what Shannon dubbed 'entropy' is not the established entropy of thermodyna-micsYet another difficulty in answering the question raised in the section

heading lies in the matter of whether it is indeed legitimate to considerprocesses such as phylogeny, ontogeny, and also the mental processes, inabstraction from the biochemical and physiological processes which necessar-ily accompany them. These latter processes can be identified and studiedseparately in the laboratory, and they are undoubtedly entropy pro-ducing when they occur spontaneously. To suppose that phylogeny,

etc., are something distinct is uncertain-and yet that is clearly whatwe have

in mind when we regard, say, evolution as being its own kind of naturalprocess

Perhaps it is most reasonable to think of the processes of phylogeny, etc., asbeing linked in no more than a contingent manner to the underlying bodilyprocesses. By this I mean that there is no i :I relationship--e.g. in regard todependence on mass. For example a speciation event, occurring over a longperiod, is just as much the coming into existence of a new species whether itoccurs in small creatures or in much larger ones having greater metabolism.

Ontogeny,too, remains the same distinctive

processof cell differentiation, and

of the development of tissues and organs, quite irrespective of the size of theparticular organism, and is thus independent of the overall entropy produc-tion. Similarly again in the case of mental activity: different people differimmensely in regard to the amount of profound new thought they canproduce, even though the amount of physical energy dissipated n their brainsdoes not vary very much from one of them to another.

One is at least safe in saying that it is an entirely open question whetherthese forms of irreversibility are entropic or not. For certainly the occurrence ofentropy increase can be established with reliability only in the case of what, inSection 2, I called the thermodynamic class of processes--e.g. diffusion,mixing, etc-since it is only the members of this class which can be made tooccur in closed laboratory systems.



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516 K. G. Denbigh

What I now want to show is that the ordinary thermodynamic irreversi-bility, entropic irreversibility, can also be understood as a divergence towardsthe future; it is a branching towards an increased number of states of a

givenmacroscopic entity, and not a branching into different entities or trajectories.It will help to clarify the significance of divergence in the thermodynamic

context by first posing the question: Why do spontaneous physico-chemicalprocesses ever occur? This involves two separate issues:

(i) Why is it possible to extract from the environment (or to prepareartificially) a system which is capable of spontaneous change?

(ii) Having obtained such a system, and having isolated it as completely aspossible, why does it continue to change up to a final equilibrium state, and

why do all such processes have an attribute in common, namely a veryhigh probability of entropy increase?

The answer to the first question must be referred back to the conditionsprevailing at the Big Bang. Perhaps this may sound a little pretentious Butconsider what may appear at first sight to be a trivial question, an instance of(i) above: How is it possible to prepare a laboratory system which is not atequilibrium? For instance, let it be the system consisting of a block of hot metallying on top of another block which is cold. It will be clear that the preparationof any system in which there is a temperature

differericerequires the

availability of an energy input, and that the very possibility of having this inputmust be traced back to the Earth's resources of coal, oil or uranium, and thatthese resources, in their turn, have their origins in the early history of theuniverse as a whole. Even the trivial act of placing the one block on top of theother requires muscular effort, and beyond that the intake of foodstuffs, theoccurrence of photosynthesis in plants and of nuclear reactions in the Sun, ...Considerations of this sort make it clear, I think, that all possibility of physicalchange is an inheritance, o to say, from the vast potentiality for change whichexisted in the primitive universe. This view of the matter is well supported bythe existing Big Bang theory.19

The second issue above is very familiar. For present purposes it will besufficient to summarise how the answer to it relates to the concept ofirreversibility as a divergence. Consider some macroscopic system, isolated aswell as can be achieved and thus of nearly constant energy, and let W be thenumber of energy eigenstates accessible to the system when it has that energyand is at equilibrium. All of the W states are assumed to be equally probable-i.e. equally likely to be occupied by the system at any instant. Let SBPbe aquantity related to W by the equation SBP= k InW where k is Boltzmann's

constant. This quantity was shown by Boltzmann and Planck to behave in amanner closely similar to the thermodynamic entropy S. To the extent that" For further discussion on the cosmological understanding of the Second Law, see for example

Gold [1958, 1967, 1974], Gal-Or [1974, 1975), and Davies [1974].



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this is true the change of entropy, S2- Si, between an equilibrium condition, 1,and a later equilibrium condition, 2, due to the lifting of a constraint on thesystem, is given by

S2 -S1 = k ln(W2/W1).

Now by the Second Law, for any isolated system the entropy change, S2- S1,can only be positive or zero. The former case, where W2 > W1, corresponds tothe situation where the transition from the equilibrium condition 1 to the newequilibrium condition 2 can only occur irreversibly (i.e. they are differentequilibria).

Increase of entropy, due to irreversible passage from one equilibriumcondition to another, can thus be interpreted20 as an increase in the number ofquantum states accessible to that system at constant energy. Physico-chemical irreversibility thus shows itself as a branching into a larger numberof possible states of existence; it is a spreading or dispersal of the system overthose of its eigenstates which are available for occupation when the system'senergy has a fixed amount.21

This completes my phenomenological survey of the different kinds ofirreversibility, biological and mental as well as physico-chemical. If I am rightin thinking that their common feature is a branching or divergence towardsthe future, this would seem to entail increasing richness and diversity in theworld. My view is thus not unrelated to Bohm's concept of an unfolding. It alsohas an affinity with certain much older insights-notably that the future isopen and that whatever can possibly occur will occur.

I am greatly indebted to Dr Harmke Kamminga for many corrections to themanuscript, and for valuable suggestions for its improvement.

Department of the History and Philosophy of Science

King's CollegeLondon

2() There are other attempted interpretations of entropy, e.g. as disorder, disorganization, lack ofinformation, etc. but counter-examples can be brought against all of these. See, for example,Denbigh, K. G. and Denbigh, J.S. [1985] and Denbigh [1989].

21 It will be appreciated that it has not been necessary for me to deal with the vast field known as'non-equilibrium thermodynamics' which is concerned with giving a significance to entropyduring the temporal period when a physico-chemical system is actually undergoing a processof change.

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West. University of Massachusetts Press.BLUM,H. F. (1968): Time's Arrow and Evolution. Princeton University Press.BOHM, . [1980]: Wholeness and The Implicate Order. Routledge and Kegan Paul.



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BOHM, D. [1987]: Foundations f Physics, 17, 667.BUNGE, M. [1968]: Philosophy of Science, 35, 355.

DAVIES, P. C. W. [1974]: The Physics of Time Asymmetry. Surrey University Press.DEBEAUREGARD, .C. [1963]: LeSecond Principe de la Science du Temps. Editions du Seuil.DENBIGH, K. G. [1981]: Three Concepts f Time. Springer-Verlag.DENBIGH, K. G. [1989]: 'Note on Entropy, Disorder and Disorganization', British Journal

for the Philosophy of Science, 40, 323-332.DENBIGH, K. G. and DENBIGH, J. S. [1985]: Entropy n Relation o Incomplete Knowledge.

Cambridge University Press.GAL-OR, . [1974]: Modern Developments n Thermodynamics. Wiley.GAL-OR, . [1975]: in L. Kubat & J. Zeman, (eds.) Entropy and Information. Academia,

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GOLD, . [1958]: 11th International olvay Congress, Stoops, Brussels.GOLD, T. [1967]: in T. Gold, (ed.) The Nature of Time. Cornell University Press.GOLD, T. [1974]: in B. Gal-Or ed.) Modern Developments n Thermodynamics. Wiley.GRAD, H. [1961]: 'The Many Faces of Entropy', Communications n Pure and Applied

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GRi)NBAIJM, . [1973]: Philosophical Problem of Space and Time. Reidel.HARVEY, P. H. and PARTRIDGE, L. [1987]: Nature, 326, 128; ibid. 329, 397.HILL, E. L. and GRi4NBAUM, A. [1957]: Nature, 179, 1296.HOBSON, A. [1971]: Concepts in Statistical Mechanics. Gordon & Breach.

JAYNES, E. T. [1963]: in K. W. Ford (ed.) Information Theory and Statistical Mechanics.

Benjamin, New York.MEHLBERG, H. [1961]: in H. Feigl and G. Maxwell (eds.) Current ssues n the Philosophy f

Science. Holt, Rinehart & Winston.

ONO,S. [1949]: Mem. Fac. Eng. Kyusho Univ., 11, 12 5.PENROSE, O. & PERCIVAL, I. C. [1962]: Proceedings f the Physics Society, 79, 605.PERCIVAL, I. C. [1961, 1962]: J. Math. Phys., 2, 235; ibid. 3, 386.POPPER, . R. [1956]: Nature, 177, 538; 178, 382.POPPER, . R. [1957]: Nature, 179, 1297.POPPER, K. R. [1958]: Nature, 181, 402.PRIGOGINE, I. [1980]: From Being to Becoming. W. H. Freeman & Co.SCHIPPER, K. & WANG HSlu-HUEI [1987] in J. T. Fraser et al. (eds.) Time, Science & Society

in China and the West. University of Massachusetts Press.

SIMPSON, G. G. [1950]: The Meaning of Evolution. Oxford University Press.THOMPSON, J. M. T. [1989]: Proceedings f the Royal Society, A421, 195-22 5.WHITROW, G. J. [1980]: The Natural Philosophy of time, p. 10, Oxford University Press.

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The Many Faces of Irreversibility