Maxwell’s Demon and Baron Munchausen: Free Will as a Perpetuum Mobile

Stud. Hist. Phil. Mod. Phys., Vol. 30, No. 3, pp. 347}372, 1999! 1999 Elsevier Science Ltd. All rights reserved

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Maxwell:s Demon and Baron Munchausen:Free Will as a Perpetuum Mobile

Orly R. Shenker*

1. Introduction

&Maxwell's Demon' is the name of a thought experiment devised by J. C.Maxwell in 1867 (Knott, 1911, pp. 213}214). The experiment is meant toillustrate a perpetuum mobile of the second kind. The Demon manipulatesparticles, turning their random thermal #uctuations into ordered movementexploitable as work, thereby violating the Second Law of thermodynamics.Maxwell's perpetuum mobile is extremely elusive. During the 130 years since its"rst presentation many people have attempted to show that it is not a perpetualmotion machine after all. Le! and Rex (1990, p. 2), summarising these attempts,conclude that &Maxwell's Demon lives on. After more than 120 years of uncer-tain life and at least two pronouncements of death, this fanciful character seemsmore vibrant than ever'. This description expresses an ambiguity in the currentapproach towards the Demon. On the one hand, many believe the conundrumwas solved in a series of writings inspired by Szilard's (1929) in#uential paper.On the other hand, there appears to be a vague and seemingly inexplicablefeeling of dissatisfaction among the writers in the "eld. The reason may be, asEarman and Norton (1998, p. 435) suggest, a &lack of any self-re#ection on whatthe goals of the enterprise are and what the rules of the game are'. I propose todescribe these goals and rules by analysing the logical structure of the argumentunderlying the thought experiment. While Szilard's followers believe the argu-ment is sound, I suggest that we should see it as a reductio ad absurdum. This,I believe, is how Maxwell himself saw it. While returning to Maxwell's logicalunderstanding of the experiment, though, we ought to take it one step farther.Maxwell found one contradiction in the argument's premisses, whereas I suggestthat there are two. The argument is not only a reductio, but a double reductio.

(Received 12 October 1998; revised 26 March 1999)* Program in the History and Philosophy of Science, the Hebrew University, Jerusalem 91905,Israel (e-mail: [email protected]).

PII: S 1 3 5 5 - 2 1 9 8 ( 9 9 ) 0 0 0 1 4 - 3

347

The Demon has resisted solution so far because the second contradictionescaped Maxwell as well as Szilard and his followers. Finding this secondcontradiction and removing it is the project undertaken here.

Section 2 describes the thought experiment, focusing on the features commonto all its numerous versions. Section 3 describes the problem created by theDemon, which calls for a solution. Section 4 examines the prevalent solutionsfor the problem. Sections 5}8 present an alternative solution. Section 9 discussesthe quantum mechanical Demon, and Section 10 addresses some possibleobjections.

2. A Variety of Demons and their Common Structure

Following Maxwell's original Demon, numerous related thought experimentshave been created. They vary greatly in details but share some fundamentalfeatures that make them all versions of the same original idea.! This basicstructure is common to all the Demons, from Maxwell's original version ofa Demon creating a temperature di!erence, through Szilard's famous versionfocusing on a pressure di!erence and a single molecule, to Szilard's other versionwhich separates a chemical mixture." It also appears in &demonless' Demons,like those of Von Smoluchowski, Feynman, Feyerabend, Popper and Gordon.#For illustration, I focus on Szilard's single-molecule-with-demon version (here-inafter &Szilard's Demon'), and Feynman's demonless-ratchet-and-pawl version(hereinafter &Feynman's Demon'). These two will now be presented. The general-isation to other Demons is easy.

Szilard's Demon is illustrated in Figure 1.$ At stage 1 a gas consisting ofa single molecule is in thermal equilibrium with a heat bath.% At stage 2 a parti-tion is inserted through a slit in the container wall, and held in place by stoppers.Assume, for simplicity, that the partition divides the container in half. At stage3 a robot measures whether the particle is on the right or the left hand side of the

! The introduction to Le! and Rex (1990) introduces some of these versions." Respectively: Knott (1911, pp. 213}214), with close versions in Demers (1944) and Brillouin (1962,pp. 162}183); Szilard (1929), with a close version in Bennett (1987); Szilard (1929), discussed in Le!and Rex (1994).# Respectively: Von Smoluchowski (1912, 1914); Feynman, Leighton and Sands (1963, Vol. I, Ch.46); Feyerabend (1966); Popper (1974); Gordon (1981, 1983).$ Fig. 1 is in#uenced by Bennett (1987).% The idea that a single molecule can be a &gas' is problematic. Szilard needs it in order to use theideal gas law. Jauch and Baron (1972, pp. 171}172) think that Szilard's version is inadequate forinvestigating the Second Law because a single particle means that the insertion of a partition(stage 2 of Fig. 1) con"nes the gas to half of the container without investing work, thereby violatingGuy}Lussac's law (see also Popper (1974), Ch. 36). Costa de Beauregard and Tribus (1974, p. 179)correctly reply that such a violation of Guy}Lussac's law is forbidden only from a point of viewwhich is internal to thermodynamics, while the Demon is meant for inspecting this theory in light ofprinciples external to it, such as mechanistic atomism. This di$culty is closely related to the use ofthermodynamics in the analysis of the thought experiment, discussed in Section 7 below.

348 Studies in History and Philosophy of Modern Physics

Fig. 1. Szilard+s Version of Maxwell+s Demon

partition and stores the information in its memory. (Only the robot's memoryappears in the illustration.) At stage 4 the robot releases one stopper accordingto the contents of its memory, allowing the partition to move towards the emptyside. The partition becomes a piston that is pushed by the pressure of the gas. Atstage 5 the pressure of the ideal gas produces an amount of work !pd<"k! ln2that is stored in a work reservoir. At stages 6 and 7 the partition is removed andrestored to its original position, and at stage 8 the robot's memory is erased. Theoperation cycle is thereby closed, and the only change is a change of entropy:heat energy has turned into energy available as work.

The system's component that I call a &robot' is called a &demon' in mostdiscussions of the experiment. By using the word &robot' I emphasise that thesystem is not supernatural in any way, as already noted by Maxwell (Knott,1911, p. 214) and Szilard (1929, p. 547). This non-demonic nature of the system isfurther emphasised by versions of the thought experiment that contain neithera demon nor a robot. An example of such a demonless Demon is Feynman'sratchet-and-pawl thought experiment. &Let us say we have a box of gas ata certain temperature, and inside there is an axle with vanes in it. Because of thebombardments of gas molecules on the vane, the vane oscillates and jiggles. Allwe have to do is to hook onto the other end of the axle a wheel which can turnonly one way * the ratchet and pawl. Then when the shaft tries to jiggle oneway, it will not turn, and when it jiggles the other, it will turn. Then the wheelwill slowly turn, and perhaps we might even tie a #ea onto a string hanging froma drum on the shaft, and lift the #ea!'! Once again, heat energy is a source ofwork.

! Feynman, Leighton and Sands (1963, Vol. I, p. 46-1). See illustration there. Feynman's Demon hasrecently been built as a nanoscale Brownian motor, see Musser (1999). The details of the operation ofthis device agree with the approach of Feynman as well as the one advocated in this paper.

Maxwell+s Demon and Baron Munchausen: Free Will as a Perpetuum Mobile 349

In all the versions of the experiment, including Szilard's and Feynman's, theDemon system includes the following three components. First, there is a particleor particles whose random thermal motion is turned into work. Second, there isa trapper, a device that traps this motion whenever it has the right position andmomentum. In Szilard's version the trapping device is the partition-piston; inFeynman's it is the ratchet. Third, there is a controller, an apparatus thatcontrols the trapper and prevents it from losing the trapped motion. In Szilard'sversion it is the robot; in Feynman's it is the pawl. In some of the versions,a single device serves as both a trapper and a controller. Such is the semi-permeable partition in Szilard's version of the Demon that separates a chemicalmixture (Szilard, 1929; Le! and Rex, 1994). For the sake of clarity it is best,however, to emphasise the conceptual di!erence between these roles, by havingthem undertaken by di!erent components. I shall therefore distinguish threecomponents in every Demon version: particles, trapper and controller. I ignorethe heat bath and work reservoir present in all versions, although their role iscentral to the process. The reason is that adding them complicates the argumentwithout adding insights or altering the conclusions.

Can such a system be a perpetuum mobile?

3. What is the Problem, and What Would Count as its Solution?

Maxwell believed that a perpetuum mobile like that of the Demon is possible.He thought that the Demon did not violate the fundamental scienti"c principlesas he knew them. In this sense, Maxwell thought the experiment was not a&problem' and therefore did not call for a &solution'. Nevertheless, he did not rushto establish a factory for perpetual motion machines. The reason was that hethought it would be impractical to construct them. &[W]e are not clever enough',he said (Knott, 1911, p. 214), meaning that people are not quick and keen-sighted enough to create and operate such machines. Today, when scientistsmanipulate individual atoms, such a pragmatic obstacle is less obvious. Clearly,the Demon raises questions of principle, involving the foundations of statisticalmechanics. It is discussed as such in all the later literature.

In believing that a perpetuum mobile of the Demon's kind is possible, Maxwellwas right in one sense and wrong in another. He was right in stating that theDemon can sometimes reduce entropy, and is therefore a perpetuum mobile inthe thermodynamical sense of the word. The observation that such systems arepossible was a great novelty in its time, but today is almost trivial; any Brownianmovement illustrates it.! But Maxwell concluded that the Demon can do better

! Fluctuations in the pressure which air molecules exert on microscopically visible particles (themost famous example is the pollen observed by Brown) cause the latter to rise against gravity everynow and then, thereby turning heat energy into gravitational potential energy, which has lowerentropy. Einstein (1905, pp. 1}2) sees this phenomenon as vitiating the conceptual supremacy of theSecond Law of thermodynamics, in favour of statistical mechanical atomism. See further discussionin Section 7.


than Brownian movement. Such movement is unpredictable and uncontrollable,and therefore the reduction of entropy it occasionally brings about cannot beexploited to produce work. Clearly, a system that overcame this di$culty wouldbe a perpetuum mobile in the statistical mechanical (not only the thermody-namical) sense of the word. The operation cycle of Maxwell's Demon invariablyends with a negative entropy balance; it produces work predictably and reli-ably.! The Demon seems, therefore, to be a perpetual motion machine in thestatistical mechanical sense of the word.

Earman and Norton (1998, pp. 435}436) ask, &why should one want toexorcise the Demon? And what exactly would count as a legitimate and e!ectiveexorcism? Readers will search the literature in vain for explicit or tacit answers'.This paper's answers are that we want to exorcise the Demon because it seems tobe a perpetuum mobile in the statistical mechanical sense of the term; a legitimateand e!ective exorcism would show that it is not. To show that the Demon is nota statistical mechanical perpetuum mobile is to show that the Demon is subject to#uctuations (thus violating the Second Law in a trivial way), and that there is noway in which it could possibly accumulate these #uctuations and exploit themreliably to produce work. Such an exorcism is undertaken here; I will now showthat existing proposals do not provide exorcism in this sense.

4. Existing Proposals: Szilard:s School

4.1. Szilard+s Principle

Almost all the currently accepted solutions for Maxwell's Demon belong toone school of thought, which emerged from Szilard's (1929) paper, &On theDecrease of Entropy in a Thermodynamic System by the Intervention ofIntelligent Beings' (the few exceptions are mentioned in Section 4.2). Szilard'sbasic idea is as follows. The hypothesis that perpetual motion machines areimpossible is supported by ample empirical evidence. It is, therefore, heuristic-ally promising to assume that the description of Szilard's Demon in Section 2above is either mistaken or incomplete. Szilard chose the incompleteness option:something is missing in that description, he thought. Szilard's idea was that theDemon produces work (at stage 5 of Fig. 1), but one of the stages necessary forthis production is dissipative, by an amount that compensates for the entropyreduction. This dissipation is the detail missing in the above description of theDemon. Adding it solves the problem, since the net entropy balance becomeszero or positive.

! In the terms of Earman and Norton (1998, p. 442), the Demon seems to bring about not onlya Straight Violation of the Second Law (i.e. a case where the entropy of a closed system decreases),but an Embellished Violation of this law as well (where this decrease is exploited reliably to providework).


Szilard believed that the dissipative stage is the measurement (stage 3). Hisargument for this was eliminative. We understand well enough the physics of allother stages, he thought, and have good reasons to believe that there is notheoretical minimum to the dissipation in them. In other words, this dissipationcan, in principle, always be reduced. In this sense we say that these stages aredissipationless. The one operation we do not understand well enough ismeasurement. Therefore, it makes sense to hypothesise that the missing dissipa-tion occurs there. Szilard analysed one speci"c measuring device to support thishypothesis, but did not generalise the results. He o!ered no constructive argu-ment, relying solely on the process of elimination. The search for a constructiveargument was undertaken by his followers.

Szilard's followers di!er from each other regarding the source of the dissipa-tion compensating for the Demon's work. Some advocate dissipation in meas-urement, as Szilard did. Others focus on memory erasure (stage 8 of Fig. 1).! Yetthey all agree to Szilard's Principle, which is the following. Part One: the Demonreduces entropy, in (what seems to be a) violation of the Second Law. Part !wo:one of the Demon's operations, which is necessary for the entropy reduction, isdissipative. The amount of this dissipation compensates for the entropy reduc-tion, thereby saving the Second Law. I call the proponents of this principle&Szilard's school'.

4.2. Part two of Szilard+s Principle: Some Shortcomings

Part Two of Szilard's Principle has (at least) two major shortcomings, whichmake it unacceptable. One is the &sound vs profound dilemma', as Earman andNorton (1998, p. 436; and 1999) call it. The dilemma is as follows. The argumentsof Szilard's followers share the eliminative structure of Szilard's original argu-ment. The eliminative argument allows for two interpretations, each leading toone horn of the dilemma. The "rst interpretation sees the argument as establish-ing that measurement or memory erasure is dissipative. This proof relies,however, on the Second Law, as the key to the eliminative process. Whereasrelying on the Second Law is normally very plausible, this is not the case whereMaxwell's Demon is concerned, since the Demon is meant to serve as a counter-example for this very law. This problem is the soundness horn of the dilemma.The second interpretation of the eliminative argument is that it merely suggeststhat dissipation in measurement is a heuristically promising idea. This inter-pretation does not rely on the Second Law's validity, but sees it as a likelyoutcome of the investigation. In this case we need an independent argument for

! In the classical domain, the main "gures of the "rst camp are Szilard (1929) and Brillouin (1962),while those of the second are Landauer (1961), Bennett (1987) and Zurek (1989). To these one mayadd Fahn (1996), who combines both the ideas and the problems of the two camps. In the quantummechanical domain, Von Neumann (1955) is the leading "gure in the measurement camp, andLubkin (1987) represents the erasure camp. A rich annotated bibliography appears in Le! and Rex(1990).


the dissipation in the operation of the Demon, one that relies neither on theDemon nor on the Second Law. Szilard's school claims that the Demondemonstrates alleged principles in the physics of information, regarding theentropy of measurement and memory erasure. These provide the dissipationthat compensates for the Demon's work. Presently, however, there is no avail-able satisfactory independent justi"cation for any of these ideas, and presenttechnology cannot test them empirically.!" This is Earman and Norton'sprofoundness horn of the dilemma.

The second problem with Szilard's school is that it sacri"ces basic ideas ofstatistical mechanics in order to save the Second Law of thermodynamics.Szilard and his school claim that if we add the dissipation in measurement ormemory erasure, then the Demon never reduces the entropy of the universe. Thealleged dissipation compensates for the entropy reduction in every operationcycle.!! This way the Second Law is invariably obeyed. The principles ofstatistical mechanics, however, are violated. According to these principles,entropy can decrease as well as increase, with some non-zero probability. HereI employ Boltzmann's approach to statistical mechanics. For the present dis-cussion, this approach is more convenient than Gibbs', because it emphasisesthe role of #uctuations in individual systems. The predictions of both ap-proaches agree in the case of ideal gases, on which we shall focus in any case.!#From a Boltzmannian statistical mechanics point of view, a Demon that neverreduces entropy is no less problematic than one that always produces work.Both violate the statistical predictions of the theory. A non-problematic Demonshould reduce entropy sometimes, with the probability given by the theory (seeSection 3 above).

Some proposals for solving the Demon problem do not belong to Szilard'sschool, and do not share the above di$culties. Such are the proposals ofFeynman and Von Smoluchowski.!$ These solutions invariably focus on de-monless Demons. The solution for Maxwell's Demon proposed in this paperagrees in spirit with these solutions. However, although fundamentally correct,they are not general, and, in particular, do not show the way to solve Demonswith demons. As we shall see below, the fact that the present solution agreesmore with demonless discussions than with those using demons is not a coinci-dence.

!" The proposal of Bennett (1982, 1987) relies on the thesis of Landauer (1961) regarding theentropy of logically irreversible operations. Landauer's thesis, however, relies on the Second Law.See further criticism in Shenker (1997) and Earman and Norton (1999).!! It may be that Szilard himself intended to adhere to the theory's statistical nature, but hisfollowers almost immediately understood him as making assertions in clear violation of theseprinciples. See Earman and Norton (1998, pp. 451}461).!# Neither of the two approaches is without foundational problems, and by choosing Boltzmann'sapproach for clarity in the present discussion I do not claim that it is less problematic than Gibbs'.For the di!erences between the two formalisms of the theory see, for example, Ehrenfest andEhrenfest (1912); Tolman (1938, Ch. VI and XIII); Sklar (1993, Ch. 2, 5}7, 9.II).!$ Feynman, Leighton and Sands (1963, Vol. I, Ch. 46), also discussed in Musser (1999); VonSmoluchowski (1912, 1914), also discussed in Earman and Norton (1998, pp. 444}451).


4.3. Part One of Szilard's Principle: Some Implicit Assumptions

Rejecting Part Two of Szilard's Principle seems to leave us with a perpetuummobile, provided, of course, that we accept Part One of this principle, statingthat the Demon can produce work. This paper focuses on Part One; Sections5}10 below advocate rejecting it. If Part One is rejected, Maxwell's Demon doesnot threat statistical mechanics to begin with, and so there is no need to turn to(the dubious) Part Two of Szilard's Principle. The rejection of Part One is basedon an analysis of some premisses underlying it. Without these premisses PartOne cannot be adequately supported. In this sense one may say that thesepremisses are necessary for the argument of Szilard's school. I now indicate whatthey are; a more detailed account appears in Sections 6, 7, and 8 (see also Table 1of Section 5).

The basic idea of the Demon is its ability to exploit #uctuations to providework. In most Demons molecules #uctuate, occasionally colliding with a mov-able macroscopic object, which then collects one component of their momentum(e.g. the Demons of Szilard and Feynman, see Section 2).!" The basic idea of theDemon relies, then, on the atomistic hypothesis, that matter is made of particles(i.e. is not continuous); on the mechanistic hypothesis, that the particles obey thelaws of mechanics; and on the principles of statistical mechanics. Atomismentails the molecular structure of matter; mechanics contributes the notion ofmomentum and the idea of collisions as the way in which heat is exploitable toprovide work; and the frequency and magnitude of the #uctuations and colli-sions are (presumably) accounted for by statistical mechanics. These threepremisses appear in all the versions of the Demon argument (Szilard's school aswell as Feynman's approach and the one advocated here).

A fourth premiss in the argument of Szilard's school is the universality ofthermodynamics. This premiss enables Szilard to apply the ideal gas law tocalculate the amount of work produced by the Demon,!# from which, bya process of elimination, he deduces the amount of compensating dissipation.Moreover, the very notions of heat and temperature come from thermo-dynamics.

These four premisses!$ are needed in order to deduce that the Demon (in itsvarious versions) can produce work reliably and predictably, in seeming viola-tion of thermodynamics as well as statistical mechanics. Szilard's school solvesthe problem in Part Two of their argument; this paper shows that the solutionought to take the form of rejecting Part One to begin with, so that the need forPart Two never arises. This is the topic of the next section.

!" In some Demons this mechanism is less obvious. Such is, for instance, Szilard's (1929, pp.543}545; Le! and Rex, 1994) version, which separates a chemical mixture. To produce work whenmixing the chemicals again, some mechanical process like collisions is required.!# See footnote 5 above.!$ A "fth one is discussed later, in Section 8.


5. Maxwell:s Demon as a Double Reductio ad Absurdum: A Summary of theArgument

Recall the problem confronting us (see Section 3). The Demon seems to be notonly a thermodynamical perpetuum mobile, but a statistical mechanical perpetuummobile as well, for it seems to accumulate #uctuations reliably and predictably.In Section 4 we saw that the (alleged) solution of Szilard's school for thisproblem is unacceptable because, while agreeing with thermodynamics, it dis-agrees with statistical mechanics. According to Szilard's school, the Demonnever reduces entropy, because of the compensating dissipation. Statisticalmechanics demands, however, that the Demon be subject to entropy #uctua-tions, yet be unable to accumulate them systematically. As we have seen inSection 3, to &solve' the Demon problem is to show precisely this.

The solution for Maxwell's Demon advocated here arises from a method ofinvestigation uncommon in modern studies of the Demon: the logical method.In this method the thought experiment is viewed as the topic of an argument,and the investigation focuses on its logical structure. The logical investigationreveals that di!erent solutions for the Demon have di!erent underlying logicalstructures; they aim at di!erent logical goals. In the light of these goals it ispossible to assess the merits of the di!erent approaches. The logical structure ofthe three existing approaches to the Demon and its solution are summarised inTable 1.

The premisses of rows 1}4 were discussed in Section 4 (I leave the explicationof premiss 5 till later, see Section 8). All three approaches (columns I, II and III)accept the premisses of rows 1, 2 and 3. The main disagreement among them,which gives rise to most other di!erences, involves the additional premisses ofrows 4 and 5. This entails the disagreement regarding the lemma of row 6, whichis Part One of Szilard's Principle (see Sections 4.1 and 4.3). Part Two of Szilard'sPrinciple (Sections 4.1 and 4.2) appears as the premise of row 7. The disagree-ment regarding row 6 combines with the discordance regarding row 7, to bringabout the di!erence in the conclusions of rows 8, 9 and 10.

The order of the columns in Table 1 (Szilard, then Maxwell, and "nally thepresent proposal) re#ects an argumentative evolution, which di!ers from thechronological one (in which Maxwell of course comes "rst). In this orderSzilard's approach is the farthest from the presently advocated one. Szilard,I claim, understood the Demon in a misleading way. We should "rst return toMaxwell and then take his ideas one step farther.

Szilard and his followers see the Demon argument as almost-sound (col-umn I). All its premisses (rows 1}5) are true, they think, and therefore theconclusion ought to be true as well. But the thought experiment seemsto contradict the Second Law, yielding a result which is presumably false(row 6, Part One of Szilard's Principle). Szilard and his school amend theconclusion by adding a &missing' assumption, namely, the dissipation inmeasurement or memory erasure (row 7, Part Two of Szilard's Principle). Thisaddition makes the conclusion accord with the Second Law (row 8). It is,


Table 1. !he ¸ogical Structures underlying three approaches to Maxwell1s Demon

Line Logical structure Contents of propositions I II IIIno. Szilard's Maxwell Present

school proposal

1 Premisses Atomistic hypothesis Accepted Accepted Accepted2 Mechanistic hypothesis Accepted Accepted Accepted3 Principles of statistical

mechanicsAccepted Accepted Accepted

4 Other premisses,which contradictpremisses 1}3

Universality ofthermodynamics

Accepted Rejected Rejected

5 Free will (free choice ofone's own mechanicalstate)

Accepted Accepted Rejected

6 Lemma (based onrows 1}3 & 4}5)

The Demon produceswork

Always(Part Oneof Szilard'sPrinciple)

Always Sometimes

7 An additionalpremiss

Compensating dissipation Always(Part Twoof Szilard'sPrinciple)

None Irrelevant

8 Conclusions fromrows 6 & 7

Accordance with theSecond Law ofthermodynamics

Yes No No

9 Accordance with theprinciples of statisticalmechanics

No No Yes

10 Contradiction between a conclusion (row 9)and a premiss (row 1); hence also acontradictory conclusion (&p and not p').

Yes Yes No

however, unacceptable, for the reasons mentioned in Section 4 above (see rows9 and 10 of column I).

Maxwell's original argument uses a strategy which di!ers signi"cantly fromthat of Szilard's school (see column II of Table 1). Maxwell interprets his Demonas the topic of a reductio ad absurdum argument.!" The Demon argumentlogically entails propositions that contradict its premisses: without the lateraddition (by Szilard, 1929) of row 7, row 6 entails a &No' at row 8, thereby!" This is a natural understanding of Maxwell's writings. See especially Knott (1911, pp. 213}214);Maxwell (1878a, pp. 645}646, 1878b, pp. 669}671); Strutt (1924, pp. 47}48). Such an opinion wouldcohere with Maxwell's general philosophical stance at the time of the Demon's inception, asinterpreted by De Regt (1996). For background, see Klein (1970). This interpretation is very close toEarman and Norton (1998, pp. 438}442).


contradicting the premiss of row 4. The result is a contradictory conclusion, ofthe form &p and not p', indicating a contradiction among the argument'spremisses, which must be removed. Maxwell's solution is a rejection of premiss4 (see Section 7). In this way, the reductio argument examines the consistency ofthe theories that are used in its description and analysis. This examination wasMaxwell's aim in devising the thought experiment.

The present paper follows Maxwell's aim as well as his reductio strategy,rejecting Szilard's approach. Szilard, believing that the argument was sound, wasunaware of any inconsistency in the premisses. Maxwell was awareof one such inconsistency (that between row 4 and row 1; see Section 7).Taking Maxwell's approach a step farther, I claim that the argument underly-ing the Demon is a double reductio ad absurdum, since there are two con-tradictions in its premisses (between row 5 and row 2 as well as between row 4 androw 1; see Section 8). Maxwell removed one contradiction from the premisses byrejecting premiss 4 (see Section 7). Our task is to locate and remove the other.

This step is not straightforward, since a reductio argument need not involveany connection between the contents of the contradictory conclusion and thecontradicting premisses; contradicting premisses can lead to any conclusion.And indeed, the contents of the contradicting premisses and the contradictoryconclusion in the Demon's argument are very di!erent. The conclusion iscontradictory in that it both accepts and violates, at one and the same time,some principles of statistical mechanics (rows 3, 9, and 10 of columns I and II;see Section 6). The contradicting premisses, on the other hand, do not deal withdissipation at all. Instead, they involve two basic postulates of physics: theatomistic and mechanistic hypotheses. The atomistic hypothesis, that matter ismade of atoms, is contradicted by the thermodynamical premiss that matter iscontinuous (rows 1 and 4; see Sections 4 and 7). The mechanistic hypothesis, thatthe atoms are subject to the laws of mechanics, is contradicted by the Demon'sability to manipulate particles in a &free' way which is not allowed by mechanics(rows 2 and 5; see Sections 4 and 8). To remove the contradictions from thepremisses we must give up one of each contradicting pair. The choice is not toodi$cult in this case, for modern science clearly prefers the atomistic andmechanistic hypotheses, and rejects the premisses contradicting them.

The contradiction in the conclusion is explicit and easy to detect. Thecontradictions in the premisses, in contrast, are hidden and implicit. Maxwellshows us where to look for one of them; the aim of the following analysis is touncover the other. This is a bene"t of the very process of understanding theDemon argument as a reductio one. This understanding enables us to search fora solution in places inaccessible to Szilard's school. The latter, believing that thepremisses are true, had to "nd a solution in terms of dissipation. We are free ofthis constraint.

Once we remove the contradictions from the premisses, the conclusion tooceases to be contradictory (compare rows 4, 5 and 10 of column III with those ofcolumns I and II). The reason is that without the rejected premisses the Demoncan no longer perform its operations successively, and therefore cannot produce


net work. For instance, Szilard's Demon cannot perform the stages of Fig. 1 inthe order from 1 to 8; Feynman's pawl is released at the wrong times, letting thewheel turn backwards.!" A Demon unable to follow its planned operationsconsecutively cannot produce work to begin with. It can no more produce workthan Brownian movement can. Therefore, the compensating dissipation ad-vocated by Szilard's school is no longer needed. Since no compensating dissipa-tion is needed, the Demon is not a perpetuum mobile even if there is no minimumentropy change associated with information processing (see rows 6 and 7 ofcolumn III).

Removing the two contradictions from the premisses neutralises the argumentand eliminates the problem. Since the Demon is no longer able to reduce entropyto begin with, it ceases to threaten the foundations of statistical mechanics. Itbecomes an unproblematic aspect of the standard understanding of the theory,and no longer teaches us anything new. In particular, it does not teach usanything about the physics of information.

6. The Contradictory Conclusion

Some Demons, like Szilard's, start at equilibrium. Others start with a sourceof low entropy, such as the temperature di!erence between the pawl and the gasin Feynman's Demon. In the former case it is invariably assumed that the systemremains in equilibrium, unless the specially devised component (Szilard's de-mon) is operated. In the latter case it is invariably assumed that the total entropywill increase, unless the specially devised component (Feynman's ratchet andpawl) is operated. If these assumptions are not made, there is no point inincluding these components in the system, since the system is a perpetuum mobileeven without them. In this sense, all versions of the Demon agree that theprinciples of statistical mechanics hold in the general case. In other words, theseprinciples are premisses in the Demon argument. All the approaches to theDemon agree on this point (see row 3 in all columns).

At the same time, the approach of Szilard's school, as well as that of Maxwell,entail that, once we add the special components, the principles of statisticalmechanics cease to hold in the system. Maxwell believes that the Demon beginsto do better than Brownian movement, while Szilard's followers believe it beginsto do worse. Both results disagree with statistical mechanics (row 9 of columnsI and II). In short, Szilard's school and Maxwell think that, whereas thepremisses include the principles of statistical mechanics, the conclusion is aninstance of their violation. This leads to a contradictory conclusion, of the form&p and not p' (row 10 of columns I and II). This indicates a contradiction amongthe premisses; restoring consistency will turn the reductio argument into a soundone (row 10 of column III).

!" This has been shown experimentally, see Musser (1999).


7. The First Pair of Contradicting Premisses: The Atomistic Hypothesis andThermodynamics

Maxwell created his Demon in order to point out the contradiction betweenatomism and (the universality of ) thermodynamics. Indeed, the following featureis common to all versions of Maxwell's Demon. On the one hand, they assumethe atomistic hypothesis, that matter is made of particles. On the other hand,they assume the applicability of the laws of thermodynamics. Both premisses areneeded for the argument, and they contradict each other. Maxwell's solution forthe Demon was rejecting the use of thermodynamics in the argument (see row 4,column II). He saw this theory as no more than an approximation which isuseful in the appropriate circumstances. This is the approach of modern science,and we therefore follow in Maxwell's footsteps (row 4, column III).

Szilard and his followers, on the other hand, write as if they are unaware ofthis contradiction. Their arguments rely heavily on thermodynamics in a waythat cannot be eliminated; the arguments cannot be transformed into non-thermodynamical ones without losing their whole point (row 4, column I). Thisis especially the case where the compensatory dissipation is being advocated,due to the eliminative structure of these arguments (see Section 4). Since theapproach of Szilard's school is currently dominant, it seems desirable to reviewsome details of this stage of the Demon argument.

The atomistic hypothesis is a premiss in the argument because all versions ofthe Demon are based on manipulating individual particles rather than a con-tinuous #uid. The laws of thermodynamics are premisses because it is they thatgive rise to the idea that the Demon is a perpetuum mobile. They are brought inthe moment we speak of heat baths, ideal gases, or quasi-static motions.Without these notions one cannot establish the idea that net work is produced,let alone how much.

The two premisses contradict each other. Prima facie, thermodynamics isneutral with regard to the structure of matter. It never addresses it explicitly, butthe idea that matter can be treated as continuous is a necessary workinghypothesis for applying it. The reason is that its propositions are generalisationsof experience with systems in which matter's atomistic structure is hidden, and itis thus reasonable to approximate it as continuous. The very application ofthermodynamics to some system implies that this system may be treated orapproximated as continuous. Once we arrive at scales where the atomic struc-ture is dominant, these approximations no longer hold. Einstein (1905, pp. 1}2)clearly states this when addressing the phenomenon later identi"ed as Brownianmotion: &If the movement discussed here can actually be observed (together withthe laws relating to it that one would expect to "nd), then classical thermo-dynamics can no longer be looked upon as applicable with precision to bodieseven of dimensions distinguishable in a microscope'.!"

!" Feyerabend (1993, pp. 27}28) insists that the violation of the Second Law by Brownianmovement can only be observed from a statistical mechanical point of view.


Returning to the Demon, consider for instance the notion of quasi-staticmotion and the law of ideal gases, both used in stage 5 of Fig. 1. Thermody-namically, stage 5 is a smooth expansion, during which the law p<"nR! holdsat all times. This is why the expression !pd< is applicable for calculating the workproduced at that stage, and why it equals k!ln2. Mechanically, on the other hand,stage 5 is a series of collisions, and not a smooth process. The applicability of theideal gas law is likewise problematic, once we allow for #uctuations.!"

The laws of thermodynamics are &restricting principles' in Einstein's (Einstein,1949, pp. 33, 53, 57) terms, whereas the atomistic hypothesis is part of a &con-structive theory'. The former encompass and generalise indisputable phe-nomena, while the latter suggests how they could come about. The restrictingprinciples are constraints on the constructive theory; in case of discrepancy, it isthe atomistic hypothesis that has to go. However, such a discrepancy can occuronly where thermodynamics can o!er a phenomenological description. Beingbased on generalisations, it o!ers descriptions in circumstances similar to thegeneralised ones. These are the circumstances where matter can be reasonablyapproximated as continuous. If, in these circumstances, the predictions ofatomistic theory agree with those of thermodynamics, then atomistic theory canand should be applied in wider circumstances as well. But once we are in the newdomain, a discrepancy means that it is thermodynamics that gets it wrong. Theclaim that thermodynamics has to be right in these circumstances as welldeprives its laws of their privileged status as &restricting principles', because theycease to be indisputable generalisations of phenomena.

Despite the virtual triviality of these considerations, Szilard and his followerswrite as if they are unaware of them, applying thermodynamics where atomisticstructure is dominant. Therefore there is little wonder that their conclusionaccords with the Second Law and violates statistical mechanical principles. Wejoin Maxwell in preferring the atomistic hypothesis and rejecting the universa-lity of thermodynamics.

8. The Second Pair of Contradicting Premisses: The Mechanistic Hypothesis and(Statistical Mechanical) 9Free Will:

8.1. Introduction

Once Maxwell removed thermodynamics from his argument, the "rst contra-diction disappeared. Were this the end of the story, Maxwell's demon would

!" The atomistic and mechanistic hypotheses can be conceptually distinguished, yet it is hard to "nda case of a pure e!ect of atomism which is independent of the choice of a mechanistic theory. Forinstance, in Demons where heat energy is turned into a more useful form through the mechanism ofcollisions, one could argue that the very notion of collision, and the idea that it is a mechanism fortransferring the desired component of the momentum, involves not only atomism but also somefundamentals of Newtonian mechanics. In light of this di$culty, whenever one examines speci"ccases, one can include the relevant mechanistic principles in the atomistic hypothesis. It is, after all,Maxwell's intention as well as ours to study the relations between thermodynamics and the wholeatomistic-mechanistic approach. (See more about the mechanistic premisses in Section 8 below.)


have been solved one hundred and thirty years ago, and would not have led tosuch a lively debate, continuing up to the present. But removing the "rstcontradiction is not the end of the story: a second contradiction lurks in theargument's premisses, and the argument remains a reductio. Maxwell did notnotice the second contradiction. He mistakenly believed that the argument wasnow sound. Accordingly, he accepted its conclusion as true, and believed thata perpetuum mobile of the Demon kind is possible.

Since the universality of thermodynamics is not one of Maxwell's premisses,the violation of the Second Law in the conclusion is not problematic from hispoint of view (see rows 4 and 8, column II in Table 1). However, his Demonseems to be a perpetuum mobile in the statistical mechanical sense of the term, asit can predict and exploit the entropy reduction brought about by spontaneous#uctuations. Since the principles of statistical mechanics do appear in thepremisses, their violation is very signi"cant. It means that the conclusion is stillcontradictory (rows 3, 9 and 10 in column II). This contradiction indicates thata second contradiction lurks in the premisses, one that escaped Maxwell'sattention.!" I will now point out this contradiction and remove it, with the resultthat the Demon will turn out not to be a perpetuum mobile in the statisticalmechanical sense of the term.

The second contradiction is the following. All versions of Maxwell's Demonmake the trivial but necessary assumption that the &demon' is not demonic butnatural. The whole system obeys the laws of nature no less than any othersystem we know of. Were the Demon supernatural, it could be a perpetuummobile; everything is possible if magic and miracles are available. The plausibil-ity of the conjecture that the Demon is not a perpetual machine is based, then,on the assumption that it is a natural system. Maxwell, his contemporaries andSzilard emphasise this point.!! Being natural, the Demon and all its componentsare subject to mechanics. This is the &mechanistic hypothesis'. An aspect of themechanistic hypothesis that is particularly relevant for the Demon is the ideathat a change of mechanical state can only be brought about by a mechanicalcause.!# These are two distinct demands: a change of mechanical state cannot beuncaused, and its cause must be mechanical as well. Both are necessary for theDemon's argument and on the face of it they seem to be obeyed.

However, a necessary condition for the Demon to be a perpetual motionmachine is a certain change of mechanical state that can only be brought aboutin an acausal manner, or at least through a cause not describable in mechanisticterms (it is indicated in row 5 of Table 1). There is absolutely no way to bringthis change about in a mechanistic way. This premiss is as necessary to thedescription of the Demon as the mechanistic hypothesis, and the two clearly

!" See more on Maxwell's approach in footnote 25 below.!! See the quotation from Maxwell in Knott (1911, p. 214); Strutt (1924, pp. 47}48). Thomson's viewmay be found in Thomson (1874, footnote on p. 12, and 1879). Szilard said this in (1929, p. 547), andhis school relies on this idea.!# Any account of &causality' will do here.


contradict each other. Current science prefers the mechanistic hypothesis overthe hypothesis that mechanical acausality of this kind is possible, and thereforethe latter must be discarded. (Some interpretations of quantum mechanics allowacausality in the selection of the actual measurement result from the possibleones; but this is not the acausality I am talking about. See the remarks onquantum mechanics is Section 9 below.) If we remove the contradiction byadhering to the mechanistic hypothesis and forbidding such acausal moves oroperations, then the Demon cannot produce net work. It cannot do so in the"rst place, and no compensating dissipation (a% la Szilard) is needed. I now turnto the details of these two mutually contradictory premisses.

8.2. The uncontrollability postulates and statistical mechanical *free will+

The Demon system is made of particles, a trapper and a controller (seeSection 2). In Szilard's system (Fig. 1) the trapper is the partition and thecontroller is the robot. For a statistical mechanical analysis of the Demon weneed to represent the system in the mechanical state space. Fig. 2 schematicallyrepresents the space of the Fig. 1 system, allotting one axis to each of its threecomponents. (The arrows and other details of Fig. 2 are explained later.) Thelarge cube represents the system's accessible region. Postulates involving thenotion of &accessible region' play a central part in the discussion below, andtherefore some general remarks are needed.

The term &accessible region' expresses the idea that our ability to manipulatea system's states is limited. The totality of e!ects all the external agents have ona system (plus the system's total energy) still leave a certain amount of freedom,a certain set of states the system can assume. This set makes up the accessibleregion in state space. The region's boundaries &act' in both directions: they keepthe system inside, and they prevent the in#uence of external agents fromentering. That the system's trajectory cannot leave the accessible region is wellknown. The other direction is normally less emphasised but is just as important.By de"nition, the boundaries embody the total e!ect external agents can haveon the system, so external agents cannot a!ect the happenings within this region.

Fig. 2. A Schematic State Space of Maxwell+s Demon


In particular, they cannot determine which, among the possible states andtrajectories in this region, will be the actual one. The actual trajectory isdetermined by the interaction between the system and the external agents, whichtakes place when the accessible region's boundaries are "xed. These agents,however, cannot choose the precise trajectory even then, since the degree ofprecision with which they can act on the system is delineated by the shape andsize of the region's boundaries. For these reasons, the system's trajectory evolvesuncontrollably within the accessible region. Let us call this the &external uncon-trollability postulate'.!"

Not only is there no external control over the trajectory inside the accessibleregion; internal control is impossible as well. A mechanistic system does not&choose' its own trajectory in state space; it is placed on a trajectory throughinteractions with external agents, which occur when the accessible region'sboundaries are "xed. From then on the system is con"ned to its trajectory,passively drifting along it. The trajectory can be changed only by an interactionwith an external agent. These agents are, by de"nition, not part of the systemand are therefore not represented in its state space. Looking at it the other way,the system's components that are represented in its state space are incapable ofchoosing, changing, or otherwise controlling the trajectory of the system ofwhich they are part. If some theory or interpretation says the trajectory canbranch, then the important point here is that the system does not choose thebranch it will take. Let us call this the &internal uncontrollability postulate'.

Consider now the case where a system changes its trajectory, and the possiblecauses of such an event. The two uncontrollability postulates entail that thischange cannot be caused by external agents, nor by internal ones, while the lawsof mechanics entail that it cannot happen otherwise. One might conclude that itcannot happen at all, but suppose that it does happen. Then we must concludethat, although it is a change of mechanical state, it happens with no cause, atleast not one expressible in mechanistic terms. Such an occurrence is free of themechanistic causal chain. Suppose, further, that the acausal jump seems (thoughnot necessarily is) teleologically directed, that is, directed towards a pragmati-cally useful aim. This feature defeats any attempt to describe it in mechanisticterms, even negatively (such as &acausal'). We must describe it in terms thatclearly do not belong in physics. The term &free will' seems adequate here. In ourdiscussion, this term has the following narrow and technical meaning: a system

!" Three remarks are in place regarding this postulate. First, uncontrollability is not lawlessness; thetrajectory is subject to the laws of nature. Second, uncontrollability places no constraints on theprobability distribution of states, and should not be confused with ergodicity and the like. Third,a version of the uncontrollability postulate is consistent with the interventionist approach tostatistical mechanics. In this approach, entropy increase re#ects the interaction of the system ofinterest with unknown external systems whose e!ects, while often dramatic, cannot be screened out.Here, the uncontrollability postulates mean an emphasis on the unknowability of the agents actingon (or, rather, in) the system, and the crucial point is being unable (even in principle) to control anddirect the system's state. (For a presentation of this approach see Ridderbos and Redhead (1998) andSklar (1993, pp. 250}254.)


has a free will if it is capable of choosing or controlling its own trajectory in thestate space. (The problematic notions of choice and control should be under-stood intuitively.)

A famous relevant example is Baron Munchausen's adventure in the swamp(e.g. Raspe, 1785, pp. 10}11). Pulling himself out of the swamp by his pigtail, theBaron exercised free will in this sense. Happenings such as he experienced arenot totally impossible by the principles of statistical mechanics; they are onlyextremely improbable. There are trajectories, or sections thereof, whose manifes-tations are such fortunate occurrences. One of them may, of course, have beenthe Baron's trajectory all along. But this kind of trajectory is so rare that thesmallest perturbation (in the open system) would lead the Baron back totrajectories typical of normal mortals. It seems that the Baron had the ability tochoose or change his trajectory in state space at will, and follow sections ofdi!erent trajectories, giving a combination suitable for the pragmatic purpose ofnot drowning in the swamp.

8.3. The *free will+ of Maxwell+s Demon

Maxwell's Demon, in order to be a perpetual motion machine, has to exercisefree will similar to that of Baron Munchausen. It must control its own trajectoryin the accessible region. To show precisely how and where this occurs, I use theexample of Szilard's Demon. The system appears in Fig. 1 and its state space issketched in Fig. 2. The schematic nature of the latter allows an immediategeneralisation to all versions of the Demon.

Let us "rst focus our attention on the particles. We are not interested in all thedetails of their state, but only a very coarse-grained description. For instance,allotting only one axis to the particle of Szilard's Demon in Fig. 2 is not much ofa sacri"ce, because we only need to know whether the particle is on the left orthe right hand side of the partition. For Feynman's Demon, we would only needto know whether or not there are particles at the positions and momenta thatturn the ratchet in the desired direction. Accordingly, we would divide theaccessible section on the particles' axis into sub-sections representing the sets ofstates that give the di!erent answers to these coarse-grained questions.

The same idea applies to the trapper component: we are not interested in allits details, but only a coarse-grained description of its main operations. InSzilard's version, Fig. 1, we need to know whether the partition is in thecontainer or outside, and whether it is held in the centre or free to move to theright or the left. This is illustrated by the trapper's axis in Fig. 2. In Feynman'sversion, we would like to know whether the ratchet is rotating in the desireddirection, standing still or rotating in the opposite direction. Accordingly, wewould divide the accessible section on the trapper's axis into sub-sectionsrepresenting the sets of states that give di!erent answers to these coarse-grainedquestions.

In the plane formed by the axes of the particles and trapper in Fig. 2, a pointrepresents their combined state. The point's trajectory can be described as the


sequence of rectangles through which it passes. The trajectory's route is uncon-trollable, according to the above two uncontrollability postulates.

The schematic representation in Fig. 2 is simple, but not realistic. It does notrepresent the Fig. 1 system correctly. For instance, as the state space is depictedin Fig. 2, the trajectory can move directly from the rectangle !particle on right,partition inside centre" to the rectangle !particle on left, partition insidecentre", a clear impossibility in classical physics. Between these two states thecombined system must assume a third state, where the partition is outside. Togeneralise: while all the states in the accessible region are, by de"nition, access-ible, this access may sometimes be indirect, through intermediate states. InFig. 2, the restriction imposed on the routes of access is represented by arrows.When the trajectory is in some given state, it has direct access only to statesconnected to it by arrows. Access to all other states is indirect.

The use of arrows may create the impression that there is some internalstructure to the accessible region, violating the uncontrollability postulates. Thisis not the case, however. The impression is the result of the di$culty ofvisualising, let alone drawing, a more realistic picture. In the realistic accessibleregion there are no arrows, walls, valves or bridges. It is merely a very complexlyshaped multidimensional region surrounded by impenetrable boundaries. Everypoint in it is accessible from every other along a continuous trajectory. Allneighbouring states are directly inter-accessible. The relation of arrow-connec-tedness in Fig. 2 represents the relation of being in the same neighbourhood inthe realistic state space. Unfortunately, we have no choice but to use a repre-sentation of the Fig. 2 type, with all its shortcomings. I shall therefore continue tospeak in terms of cubes and arrows, bearing in mind that the routes constrainedby the arrows are nevertheless consistent with the uncontrollability postulates.

The arrows leave the trajectory a lot of freedom. They allow for a great varietyof rectangle sequences. In particular, for every given sequence there is a reversedone, in which the rectangles appear in the reversed order. For instance, theoperation stages of Fig. 1, in the illustrated order from 1 to 8, are represented byone sequence of rectangles. This sequence is obtained if one follows the horizon-tal black arrowheads, plus the vertical arrows. The reversed sequence, fromstage 8 to 1, is obtained using the horizontal white arrowheads plus the verticalarrows. In particular, at stage 5, following the horizontal white arrows meansthat the partition pushes the particle, turning work back into heat. Bothdirections are possible. Moreover, if we add the heat and work reservoirs to ourconsideration, then the reversed direction becomes more likely than the forwardone; we leave the reservoirs out of the present considerations, however, forsimplicity's sake.

If we want the system of Fig. 1 to act as a perpetuum mobile, we must make itoperate forward, from 1 to 8, more often than backwards. It is, therefore,a necessary condition that it follows the black horizontal arrowheads more oftenthan the white ones. Whenever the system follows another sequence ofrectangles, it must be forced to return to the black arrow sequence. In this sensethe trajectory must be controlled. This demand does not (yet) violate the


uncontrollability postulates, because the particles and trapper are not the wholesystem; the controller may perhaps be able to control them. Indeed, the control-ler's role is precisely this, namely, to make the system follow the black arrowsmore often than the white ones. This demand is necessary for the system to act asa perpetuum mobile. If the controller cannot force the system to move along theblack arrows, then we do not have a perpetuum mobile.

Let us represent the control relations in terms of state space. Adding thecontroller to our system means adding one more dimension to the schematicstate space of Fig. 2 (and adding many more dimensions to more realisticspaces). Each cube in Fig. 2 represents a possible state of the particles plustrapper plus controller. The allowed trajectories can be represented by arrowsanalogous to those indicated in Fig. 2. By designing the system's structure andmanaging the external agents which act on it we can determine the allowedsequences of states, represented by the arrows. It is possible to choose theallowed sequences so that whenever the controller is in a state meant to a!ectthe trapper in some way, the trapper enters the desired state. Observing thesystem in its evolution along such an allowed sequence we get the impressionthat the controller indeed controls the trapper; we have the appearance ofcontrol. That this can easily be achieved is seen from the pervasiveness of controlphenomena in animals and machines. Of course, the appearance of controlemerges only when an allowed sequence of states is followed in a certain order ordirection, as time proceeds.

Constructing a controller that appears to control a trapper involves, then,putting the system on a very special kind of trajectory in state space, that evolvesalong the allowed sequence of states in a certain order or direction, at least asa net e!ect. This, however, is impossible, due to the uncontrollability postulates.There is no (mechanical) way of preventing the system from following a trajec-tory that, say, traces the allowed sequences in the reversed order so that therobot appears to have no control whatsoever over the trapper. In particular, therobot may appear completely unable to prevent the system from transformingwork back into heat at stage 5 of Fig. 1. Consequently, adding the robot to thesystem does not make it a perpetuum mobile.

Yet phenomena of control are abundant. The prima facie di$culty in thisrespect disappears once we notice that these phenomena invariably involvesources of low entropy. Since the aim of Maxwell's Demon is to test thermo-dynamics and statistical mechanics, we assume that the system has no internalnor external source of low entropy. Lacking these, to control the trapper thecontroller must "rst control itself. It must change its own state in the forward(stage 1 to stage 8) direction. Such self-control is in clear contradiction to theinternal uncontrollability postulate.

To emphasise the import of this postulate, suppose we omit the titles of theaxes in Fig. 2. In this case, looking at the state space, we can detect nofundamental di!erence between the status of the particles or trapper and thestatus of the controller. They are all axes representing mechanical degrees offreedom. The state space representation reveals states correlations, but gives no


indication of the fact that the controller controls the partition rather than theother way around. Mechanics is egalitarian as well as time-symmetric. If oneobject can act on another in some way, then so can the second act on the "rst.No mechanical object, subject to mechanical interactions, has priority. Nonehas control over the others. Indeed, if the controller is not supernatural, there isno reason why adding it to the particles and trapper should lead to anyqualitative change in the system's behaviour. In particular, there is no (mechan-ical) reason to assume that a system consisting of particles and a trapper willbehave qualitatively di!erently when another mechanical system, called a &con-troller', is added to it.

Consequently, the point in the three-dimensional state space of Fig. 2, repre-senting the combined state of the particles, trapper and controller, driftspassively along a route in three dimensions, tracing a sequence of cubes. Theuncontrollability postulates prohibit forcing it to follow pragmatically bene"cialtrajectories and avoid disadvantageous ones. Thus no controller, be it a pawl,a robot or a demon, can force the system to produce net work in the long run.

!he problem of Maxwell's Demon is thus reduced to the question of whether onecomponent of a system can force the system of which it is a part to follow onetrajectory rather than another in state space. The answer is: no, it cannot, becauseof the internal uncontrollability postulate. Such an ability would amount toa display of free will in the above sense, analogous to that of Baron Mun-chausen, and would thus clearly contradict the mechanistic hypothesis. This isthe second contradiction in the premisses of the Demon argument, which leadsto the contradictory conclusion whereby a statistical mechanical perpetuummobile seems possible (rows 9 and 10 of Table 1; for the notion of a &statisticalmechanical perpetuum mobile' see Section 3). The choice between the mutuallycontradictory premisses is not hard, since free will (in the above sense) has surelyentered the discussion unnoticed, and is almost certainly in con#ict with thescienti"c opinions of all those writers who unconsciously assume it whendiscussing the Demon.!" Once we give up the Demon's free will, it is no longerable to produce net work; it cannot move forward (in Fig. 1, mainly stage 5)more often than backwards. No compensating dissipation is needed.

!" Schweber (1982) suggests that Maxwell's Demon re#ects ideas, common among Maxwell'sBritish contemporaries, regarding free will and its e!ect on the course of events. Maxwell andThomson were clearly aware of attributing free will to the Demon (see Thomson, 1874, note at p. 12,and 1879; Brush, 1976, pp. 587}593; Daub, 1970, p. 225; Heimann, 1970, p. 66). Yet, none of themproceeded to make explicit the incompatibility of free will with mechanics in the way advocated here.Maxwell clearly thought that (once we remove thermodynamics from the premisses) the idea of thesystem is no longer based on a contradiction, and the system is therefore not impossible. This issuggested by his saying that the reason that a Demon cannot be built is that &we are not cleverenough' (Knott, 1911, p. 214), meaning that people are (merely) not quick and keen-sighted enoughto create and operate such a system. This attitude of Maxwell and his contemporaries may haveinspired Von Smoluchowski's later allusion to &intelligence' as a possible source of anti-thermody-namical behaviour (Earman and Norton, 1998, pp. 448}451), which inspired Szilard and gave rise tohis Principle.


This solution is general. It covers all the versions of Maxwell's Demon. Theproposals of Von Smoluchowski and Feynman!" agree with its spirit, thoughthey never generalise their arguments. The impression that their Demons doproduce work for free at the initial phase of their operations is misleading. TheseDemons begin at non-equilibrium states, and this low entropy is a non-problem-atic source of work. Once the systems reach equilibrium they cease to producework. It is no coincidence that their versions are demonless Demons; this waythey were able to avoid the pitfall of unconsciously ascribing free will to thecontroller.

9. A Remark Concerning Quantum Mechanics

The discussion so far has been purely classical. In a system that manipulatesindividual molecules, quantum mechanical e!ects are dominant. Some of theproposals coming from Szilard's school are indeed quantum mechanical.!#Since the Demon was interpreted as a reductio argument testing the consistencyof classical ideas, relating it to the quantum domain is not methodologicallymandatory. Still, it may be instructive to see that the spirit of the proposedsolution can be carried over to the quantum domain, as follows. A system will becalled a quantum mechanical perpetuum mobile if, in consecutive measurements,the work reservoir turns out to be richer each time, with high probability oron the whole.!$ Using semi-classical intuition, this outcome will be found in casethe following outcome is also found: consecutive measurements of the Demon'sstate will, with high probability, yield eigenvalues whose coarse-grained descrip-tions are the cubes of Fig. 2, and whose order will, on the whole, evolve along theblack arrowheads. This means that obtaining some stage of Fig. 1 as themeasurement's outcome will make the immediately following stage signi"cantlymore probable than all the others. Such a Hamiltonian is not a theoreticalabsurdity, in the same sense that the correlated classical state space trajectory isnot impossible. But both are hard to obtain, in analogous ways, and it is equallydi$cult to eliminate the e!ect of perturbations. A quantum system neither

!" Von Smoluchowski (1912, 1914); Feynman, Leighton and Sands (1963, Vol. I, Ch. 46).!# Mainly Von Neumann (1955) and Lubkin (1987). It is sometimes thought that Brillouin (1962),whose solution is based on Demers (1944, 1945), is also quantum mechanical. I disagree. True,Brillouin mentions photons, but he treats them partly as thermodynamical heat #ows (in the "rstpart of his argument) and partly as classical particles (in the second part). Wave functions,superpositions and their like do not appear in this argument. I would say his argument is notquantum mechanical, but simply confused.!$ These measurements are formally represented by tracing out. Lubkin (1978) claims that measure-ments represented by tracing out increase the entropy. However, tracing out amounts to neglectingsome of the information without changing the system in any way. Therefore this entropy change isnot physical. (Moreover, it is questionable whether Tr(! ln!) is genuine entropy, equivalent tothermodynamical entropy; I address this question in Shenker (1999).)


chooses nor changes its own Hamiltonian, thereby obeying a postulate analo-gous to the classical internal uncontrollability postulate. A system that doescontrol its own Hamiltonian is justi"ably described as having free will in thetechnical sense. A quantum mechanical Baron Munchausen is as unlikely asa classical one.

10. Possible Objections

I now address two possible objections to the proposed solution for Maxwell'sDemon. These objections may be viewed as a dilemma saying that the proposedsolution is either redundant or a case of petitio principii. Suppose we agree toreject the proposals of Szilard's school, so that the Demon remains unsolved.Then the Demon violates the Second Law and its statistical mechanical corre-lates, as well as mechanical causality. There are, then, two physical principlesthat the Demon both assumes and violates. Relying on each of them as the basisfor a solution is equally legitimate, the objection goes. The contradiction of theSecond Law is far more straightforward than the contradiction of mechanicalcausality. Why, then, do we focus our attention on the latter rather than theformer as a logical solution to the Demon? Either both are acceptable, in whichcase the presently proposed solution is redundant; or both are equally unaccept-able due to being circular. I begin by addressing the redundancy horn of thedilemma.

Von Neumann (1955, p. 359) says: &In the sense of phenomenological thermo-dynamics, each conceivable process constitutes valid evidence, provided that itdoes not con#ict with the two fundamental laws of thermodynamics'. Thisremark is not dogmatic, although it has sometimes been interpreted as such.!"That an imaginary process seems to be a perpetuum mobile should not, in itself,vitiate it as a thought experiment. Rather, it should be a heuristically promisingindication of a #aw or incompleteness in the description of that process. Thecontradictions of the Second Law and its statistical correlates appear only in theconclusion of the Demon argument, whereas the contradiction to the hypothesisof mechanical causality already appears in the premisses. The former indicatesthat a problem lurks somewhere; the latter points out the speci"c problem.A non-redundant solution for the Demon must point out the detail of thesystem's operation that initially escaped our eye, creating the misleading impres-sion of perpetual motion. Szilard's school (to which Von Neumann belongs)thinks the crucial detail pertains to the physics of information; I opt fora di!erent solution. The proposals of Szilard's school and mine di!er in theaspect of the system to which the solution is ascribed. But they are the same in

!" This spirit is expressed by Jauch and Baron (1972, pp. 230}231), with respect to Szilard's Demon.In a di!erent way it is applied by Fahn (1996), when he says that after the partition is inserted(stage 2 of Fig. 1) both sides of the partition are still accessible, because otherwise Guy}Lussac's lawis violated. See footnote 5 above.


trying not to be dogmatic in the above sense.!" Therefore, although any of themmay be mistaken, none of them is redundant.

Now the second horn of the dilemma: petitio principii. In Section 4 aboveI discussed the &sound vs profound dilemma' objection to solutions of Szilard'sschool. The &sound' horn was that these solutions are circular, as they rely on theSecond Law while attempting to solve an experiment meant to be a counter-example for this law. The objection to my solution is that relying on thepostulate of mechanical causality is no less dogmatic than relying on the SecondLaw, in the following sense. My presentation, in terms of state space and so on, isbased on accepting the atomistic and mechanistic hypotheses as postulates; the&solution' lies in stating that these are violated. This, the objection goes, is notdi!erent from accepting the Second Law as a postulate and &solving' the Demonby stating that it is violated. To this objection I disagree for the following reason.

The Demon is not a phenomenon. It only exists in our description, and thisdescription cannot but be given in some theoretical framework. Therefore, thethought experiment cannot be used to establish the empirical adequacy ofstatistical mechanics, nor of the atomistic and mechanistic hypotheses. It canonly test their internal consistency. This is why its interpretation as a reductio adabsurdum argument is both natural and useful, and agrees with (what seems tobe) Maxwell's own view. Under this interpretation, it is not only legitimate, butabsolutely mandatory, that we assume these postulates, and see whether, whileholding them, we can still derive a contradiction, by showing the possibility ofa statistical mechanical perpetuum mobile. Maxwell thought we can; I haveshown that we cannot.

Acknowledgement*I wish to thank an anonymous referee for valuable comments.

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Maxwell’s Demon and Baron Munchausen: Free Will as a Perpetuum Mobile

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