karl popper - entropy

8/13/2019 Karl Popper - Entropy

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The ritish Society for the Philosophy of Science

Irreversibility; Or, Entropy since 1905Author(s): Karl R. PopperSource: The British Journal for the Philosophy of Science, Vol. 8, No. 30 (Aug., 1957), pp.151-155

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NOTES AND COMMENTS

Irreversibility or, Entropysince 190o5

I To avoid misunderstandingswhich might ariseotherwise, I wish to

make clear at the outset that I do not intend, n this note,to suggest hat the

second aw of thermodynamicss infactfalse. What I wish to suggest is that

the logical situation in connection with the entropy aw (as I shall here

call the 'second law of thermodynamics') is highly unsatisfactory,and

that it has been so ever since 1905.

That the law is in need of a kind of phenomenological formulation

hadbeen realised ong before thatdate; and Planckformulated t in analogyto the following formulation of the 'first law' which, as is well known,can be formulated as the ' law of the excluded erpetualmotionmachineofthe irst order'

(I) There does not exist a perpetualmotion machine of the first

order; that is to say, an otherwiseclosed physicalsystem which eitherconstantly pumps heat into its physical environment, or which con-

stantlyabsorbsheat from its physicalenvironment.

The entropy law (second law) may be formulated by analogy, asfollows:

(II) There does not exist a perpetualmotion machine,of the second

order, that is to say,a physicalsystem,immersedin a heat bath, which,

by cooling down (or, which is the same, by absorbingheat from the

surroundingheat bath), can move a heavy body againsta force, thus

increasing ts potential energy; or in terrestrialerms,a machinewhich,

by cooling down, can lift a weight.

I am not aware that a better phenomenological formulation of the

entropy law exists.

The explanation f this law in terms of the kinetic theory of gases is

well known. It should be observed that this was developed by Maxwelland Boltzmann at a time when molecules and atoms were considered

either as mere nstrumentsf thought(useful fictions, mathematical hypo-

theses); or else as real,but as so small that we would never obtain directexperimentalevidence of their individual existence.

Accordingly, few physicistswere shocked by an unintended result of

the kinetic or statistical explanation' of the entropy law: I am alludingto the result that the entropy law is not strictly true, but holds in the

overwhelming majority of cases. It was felt to be quite satisfactorythat

ISI







K. R. POPPER

no deviations from the law were expected to be observable,ccording to

the statisticaltheory; for in this way, the phenomenologicalaw would be

fully explained.z The purpose of this paper is to draw attention to the fact that this

attitude became untenable the moment it was accepted that deviations

from the law canbe in fact observed. This was accepted n 1905, followingEinstein'sfirstpaperson the Brownian movement.'

Accordingto Einstein's heorywe are forcedto say thatin the Brownian

movement, observable heavy particles are sometimes lifted against the

gravitational ield of the earth,at the expenseof a (slight)cooling down of

the liquid. Of course, ust as many particlesare sinking, thus restoringthe

lost heat. But thisdoesnot matter : the letterof Planck's aw is undoubtedlyviolated, as Einstein in fact stated.

This situation is a little more serious than is usuallyseen. For the now

generally accepted theory of the Brownian movement clearly impliesthat, precisely as small fluctuations affect small suspended bodies, bigfluctuationswill affect big suspendedbodies; and it implies, moreover,

that big fluctuations,although very improbable and therefore very rare,

must occur, preciselyas small fluctuationsmust occur.

All this is of coursevery well known to every physicist; and I believe

that most physicistswill accordingly agree that the entropy law, in Planck's

formulation, is simply falsifiedby the Brownian movement, as interpreted

by Einstein. However, they may say, perhaps, hat the Maxwell-Boltzmann

law is certainlynot falsifiedbut supportedby Einstein. This is quitecorrect.

But the following consequenceshave not, apparently,been drawn fromthe situation which I have described.

(i) If we give up Planck's aw of the excludedperpetualmotion machine

of second order, without restatingit in some form, then there is nothingleft in our physical laws to tell us that we cannot build such a perpetualmotion machine. (The appealto the exclusion of Maxwell's demon does

not help; for the impossibilityof the demon is derived from the fact that

otherwise the phenomenologicalntropy law-in a form similar to Planck's

-would become untrue.)

(ii) If we wish to retain some form of the principle that a perpetualmotion machine of the second order cannot be built, then we have to re-

formulate a phenomenological law and to prove that the formulation is

consistent with the presentinterpretationof the Brownian movement.

3The

presentsituation is

logically highly unsatisfactory;for in

manyarguments within statistical mechanics (and also in information theory),use is made of the fact that certainevents are impossiblebecause otherwise

the entropy law (in its phenomenological or Planckianform) would be

violated. Clearly,all these argumentsare circular.

1 Einstein,AnnalenderPhysik,I905, 17, 549-560 ; 19, 371-38I

IS2







IRREVERSIBILITY; OR, ENTROPY SINCE 1905

The fact that these argumentsare employed illustratesmy claim that,

although nothing new has been said here, physicists habitually fail in this

point to put two and two together. Moreover, they fail in this pointnot accidentally,but because no satisfactoryphenomenologicalformulation

of the entropy law exists. It seemsto me that the difficulties n the way of

a satisfactory ormulation are formidable.

One reformation which suggests itself would be to insert the word

'constantly' (or 'as often repeatedas we like') into Planck'sformulation

of the entropy law. But this does not seem satisfactorysince, accordingto Boltzmann'stheory, fluctuationsare of courserepetitive; that is to say,

they occur again and again. Admittedly, they are rare. But their rarity

hardlyjustifies us in assertingthat usableluctuations re too rareto be tech-nically utilised. And although this last formulation may express our in-

tuitions and our intentionsquite well, the terms used are much too vague,and they suggest that we know in advance what kind of fluctuationscan

ever be technicallyutilised. (Moreover, unicellularorganismsmay utilise

the Brownian movement.)A satisfactoryphenomenological formulation of the entropy law, and

one which appearsto be in full agreementwith our intentionsappearsto

be in the following:

(F) A gas or a liquid in a closed circulartube, immersed in a heat

bath of any temperatureand fitted with a one-way valve, does not

constantlycirculatethrough the tube, however slowly.An equivalentformulation seems to be:

(F') There does not exist a semi-permeablemembrane with an

asymmetric structure (like a one-way valve) such that the proba-

bilities of passing through it are not equal in both directions.

This formulationis logically very much weaker than the one alludedto by von Neumann,' where we read: ' Our arrangements rathersimilar

to Maxwell's demon , i.e. to a semi-permeablewall which transmits

molecules coming from the right and reflects those coming from the left.'

For von Neumann's wall is supposedby him to work with the transition

probabilitiesI and o, while all I propose to demandis that the probabilitiesare unequal.2

1J.von Neumann,Mathematicaloundationsf Quantum echanics,ranslatedyR.T. Beyer,1955,P. 369

2 Inthe sameplacen vonNeumann's ook,wefindthefollowingunsatisfactoryargument gainsthis wall: '... and therefore,by the secondlaw of thermo-dynamics,uch a wall cannotexist.' The arguments unsatisfactoryecausehe

'second law' hereused as an independent uthoritys Planck'sversion-refutedsince19OS--towhichvon Neumannrefersn note I84 on p. 359. The argumentis thusmuchworse han f it weremerelycircularwhich t also s).

L I53







K. R. POPPER

4 I think that everybody will agree that either of these formulations

(F) or (F'), if not equivalentto the intended entropy law (as I think it is)

must at any rate be derivablefrom it. (My formulationsare intended asa phenomenological version of the assertionthat even a probabilistic,or

fallible, Maxwell demon does not exist.) Accordingly, any theory which

purportsto explain the entropy law would have to be strong enough to

entail the formulae (F) and (F'). I have little doubt that my formulae (F)and (F') cannot be derivedfrom any of the versionsof statisticalmechanics,

as they exist at present,except of course if the non-existenceof a Maxwell

demon is simply assumed ad hoc,in some form or other. And it will be

extremely difficult-I fear even impossible-to prove that my formulae

(F) and (F') areconsistentwith the presenttheory of Brownian movement.My reasonfor holding thispessimisticview is this. In the present theory,'the absolute size of the 'molecules' does not play any r61e, and gases

consisting of molecules of'a very large mass' 2 are freely used in the

discussions. But we know that for sufficientlybig 'molecules', we can

and do in fact build the forbidden valves and membranes. (These of

courseconsume energy, but this does not mattersince it is replacedby the

heat bath.) Thus the logical situation is highly unsatisfactory. This has

led to attemptsto rescue t by draggingin the subjectiveprobabilitytheory,i.e. statesof our knowledge. But neither Planck'sformulation nor those

suggestedhere can ever be validly derived in this way.3

5 A typical example of a discussionof theseproblems which from the

point of view here outlined is completely unsatisfactory is Szillard's

discussionof Maxwell's demon (cf. footnote 4, above).

(a) He first gives four non-equivalent formulations of the entropylaw clearly implying that they are all equivalent. His first formulation

which operateswith the concept of a 'lowest temperature' is hardly clear

enoughand

hardly phenomenological;moreover it demands constant r

persistentoutput of work, in contradistinction to his second ormulation

which is practicallythe same as is Planck's. His third ormulation is very

interesting, because it is an attempt to incorporate Einstein's results of

1905; but for this very reason it contradicts the second. Also, it is not

phenomenological. It runs as follows : 'If we attempt to use fluctuations

in order to extract energy at the expense of heat, then we embark upon

a game of chancein which the expectationof a gain is either equal to zero

1 See, for example,von Neumann,op. cit., pp. 361 sqq.; A. Einstein,Ver-

handlungen. deutschenhysikalischenesellschaft,914,12, pp.840sqq.; L. Szillard,

Zeitschriftfurhysik, 925,32, pp. 753sqq.,and1929,53,pp.840sqq.2 FollowingEinstein,onNeumann, p.cit.,usesgases onsistingf' molecules'

whichareboxeseachof which' musthaveaverylargemass .

3Fora criticism f thesubjectiveheoryof probability,eethePostscripto my

Logicof Scientific iscovery,957.

I54







or less than zero.' Thus we may gain-and refute Planck--even thoughwe shall laterprobably lose our gain (and more). Thefourthformulation

simply does not touch the problem.(b) Szilardthen assumes the axiomatic validity of the entropy law so

defined.

(c) He then proves that a Maxwell demon (which he takes to be an

intelligent being ratherthan a valve, or an asymmetric membrane)would

have to pay for his informationconcerningthe oncoming moleculesby an

increaseof entropy; which is a trivialconsequence f any ormof the entropylaw is to be preservedintact; and which therefore follows if the entropylaw is assumedas a premiss.

(d) He therebyshows that informationor knowledge can be measuredin terms of negative entropy, or entropy in terms of negative knowledge;which immediately leads to the interpretationof statisticalprobability in

terms of subjectiveignorance.But I think that hot air would continue to escapeentropicallyeven if

therewere no intelligentbeingsabout to provide the correspondingamountof nescience.

KARLR. POPPER

A NOTE ON MULTI-DIMENSIONAL TIME

A Note on Multi-Dimensional ime*

H. A. C. DOBBS,n a note contributedto thisJournal,'hashinted at a theoryin which time figuresas a complex variable with an imaginary component.I have been enquiring into the topological requirementsof hypothesesabout multi-dimensionaltime. A brief statement of some of my results

may be of interest to the readers of this Journal. To demand a multi-

dimensional time is to conceive, in the case of time, something analogous

to the extension of metrical geometry to the EuclideanEs-space, the in-finitely many-dimensionalHilbertspace,E0, etc. It is not difficult o exhibit

the set of points in Euclideanspacesof two (or more) dimensionsas a case

of a partiallyorderedset, since the orderingrelationis given by

(a,b)< (c, d) if a < c; b< d.

Partial ordering in topology is a generalisationof the familiar linear or

total ordering. Variousforms of a multi-dimensionaltheory of time can

be extracted from Dunne's well-known hypothesis2 if this is regarded

* I have to thank he Syndicate f theMadrasUniversityorpermissiono usematerialn my doctoraldissertationntitled On SomeSpatialRepresentationsfTime '.

1H. A. C. Dobbs,' The Time of Psychology ndof Physics, thisJournal,953,4, I62

2J. W. Dunne, An Experimentwith Time, London, 1939

I55

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