Maxwell Demon

Hemmo and Shenker Maxwell's Demon

Maxwell's Demon

Meir Hemmo† and Orly Shenker‡

1. Introduction

"Maxwell's Demon", the famous thought experiment of James Clerk Maxwell, has been devised in 1867 as a counter example for the Second Law of thermodynamics. During the 140 years since the Demon was first suggested, numerous attempts have been made to counter Maxwell's argument. The attempts have been to show that Maxwell was wrong, since his Demon cannot work for one reason or another (see Leff and Rex 2003 for details and references). Unlike the previous writers on the subject, Albert (2000, Ch. 5.) argues that Maxwell is basically right, in the sense that a counter example for the Second Law of thermodynamics is compatible with the principles of statistical mechanics. Albert shows this by a general phase space argument given in the framework of a Boltzmannian approach to statistical mechanics. In this paper we argue, following Albert, that Maxwell's Demon is consistent with the laws of statistical mechanics as framed in the Boltzmannian approach.1 We add some necessary details to Albert's argument, solve some difficulties in it, and point out some assumptions that underlie it. Our analysis yields some further conclusions concerning issues that typically arise in the literature on Maxwell's thought experiment. Among them are the question of whether and how macrophysics can be reduced to microphysics, and whether information processing is constrained by the laws of thermodynamics. In particular, we prove by an explicit counter example that the so-called Landauer-Bennett thesis (Landauer 1961, Bennett 2003), according to which erasure of information is necessarily accompanied by a certain minimum amount of entropy increase, is false.

The paper is structured as follows. In Section 2 we define a Demon (or a Demonic universe) and show that it is indeed compatible with the principles of statistical mechanics. Our definition of a Demon resembles Albert's in its main properties, but differs from it in several features which we shall point out, and which are aimed to solve some difficulties in Albert's proposal. In constructing the Demon we draw attention to some important assumptions that are made on the way. Among them is the approach to statistical mechanics we call the liberal stance. We believe that this stance is the main lesson to be learned from the Demon question, and that it teaches us something about the full meaning and scope of statistical mechanics. In Section 3 we show that the laws of statistical mechanics put some restrictions on the efficiency of the Demon. In Section 4 we demonstrate that the cycle of operation of the Demon can be completed. In particular the information stored in the Demon's memory can be erased, in a way that does not increase the total entropy of the universe nor of the environment. This result refutes the thesis of Landauer and Bennett regarding the entropy cost of memory erasure (Landauer 1961, Bennett

† Department of Philosophy, University of Haifa, Haifa 31905, Israel, [email protected]‡ Department of Natural and Life Sciences, The Open University of Israel, 108 Ravutski Street, Raanana, Israel, [email protected]. 1 Justifying the choice of a Boltzmannian framework is beyond the scope of this paper. Some arguments for it are given in Albert (2000), Goldstein (2001), Callender (1999), Lebowitz (1999). See Uffink (2004 for a historical introduction to Boltzmann's work and references. Albert's version of Maxwell's Demon has no counterpart in the Gibbsian framework, since it hinges on the notion of entropy as the phase space volume of a macrostate, as we show below.

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2003). Finally, in Section 5 we draw some conclusions with respect to the scope of statistical mechanics and the project of reducing thermodynamics to it. 2. Why a Demon is compatible with statistical mechanics

2.1 Definition of a Demonic universe. Let's start by defining a Demonic universe. We distinguish between a thermodynamic Demonic universe and a statistical mechanical one. A universe is Demonic in the thermodynamic sense if there can be an operation that reduces the total entropy of the universe (or of some isolated part thereof), thereby violating the Second Law of thermodynamics. Any spontaneous fluctuation in the thermodynamic magnitudes in an isolated system which is initially in equilibrium (relative to a given set of constraints) is an instance of such a violation.2 By contrast, a universe is Demonic in a statistical sense if the probability for a spontaneous decrease3 in the total entropy of the universe4 is higher5 than the probability determined in the usual Boltzmannian way; that is, the probability for entropy decrease is higher than the ratio between the Lebesgue measure of the phase space regions that correspond to the initial and final macrostates of the universe. In what follows we shall consider Demons of this second (i.e. statistical) type.

D

E

I nitial G macrostate Final G macrostate

Possible fi nal macrostates of D.All same size as the initial macrostate.

I nitial D macrostate I

F3

F2

I nitial & fi nal E macrostate

Volume (F) =F1+F2+F3 = Volume (I ) in accordance with Liouville’s theorem.

Accessible region

G

F1

Region F

Figure 1: Maxwell’s Demon – Macrostates

D

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F3

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Accessible region

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Region F

D

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F3

F2



Accessible region

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F1

Region F

Figure 1: Maxwell’s Demon – Macrostates

2.2 Phase space. Consider the phase space Γ of the universe6 illustrated in Figure 1. Each point in Γ describes a microstate of the whole universe. We divide the degrees of freedom of the universe into three sets which we call D, G and E (for Demon, Gas and Environment, respectively). It is crucial to note that these are just names of sets of degrees of freedom of the universe: they are not meant to indicate any properties of

2 This is actually a violation of the so-called Minus First law, as formulated by Uffink and Brown (2001). We do not address the difference between the Minus First and the Second laws of thermodynamics here.3 Or an increase (but we shall not consider this possibility in details here).4 Or of some isolated system of the universe. The idealization of perfect isolation is of course unrealizable. For a discussion of this idealization, see Sklar (2000, Ch. 3).5 Or lower, but we do not consider this possibility here.6 Or of a perfectly isolated part of the universe.

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the corresponding subsystems beyond the (generalized) position and momentum and their functions. All the properties of D, G and E that are relevant for a statistical mechanical analysis, must be explicitly represented in the phase space of the universe.7

Indeed, classical (and quantum) mechanics is egalitarian in the sense that all the degrees of freedom in Γ are to be treated on equal footing (call this the principle of mechanical egalitarianism). Any deviation from it is a deviation from mechanics, and, in a system for which mechanics is the only relevant theory of physics, such a deviation is non-physical. In Figure 1 we distinguish between the subsets D, G and E. only for the purpose of linking our Demonic universe (as defined above) with Maxwell's thought experiment as it appears in the literature. Maxwell himself (1868) and many writers after him have rightly stressed that Maxwell's Demon is not supernatural. We apply this idea in our analysis and make it precise by a purely statistical mechanical treatment of all the systems involved, and by a strictly egalitarian treatment of the mechanical degrees of freedom, regardless of the set (D, G or E) to which they belong.8

In particular, in our description of the D subsystem as a mere set of mechanical degrees of freedom we don't need to refer to capacities for measuring observables, storing information, or manipulating gas molecules. Ever since Szilard (1929) and Landauer (1961), such terms have been correctly taken to have purely statistical mechanical counterparts. In what follows, whenever such terms appear, we shall describe their counterparts in phase space notions, in accordance with the principle of mechanical egalitarianism.

2.3 Accessible region and macrostates. Consider the phase space Γ of the universe consisting of D, G and E (see Figure 1). Γ contains a subspace consisting of all the microstates that are consistent with limitations such as the total energy of the universe.9 This is the universe's accessible region.10 The limitations can change with time; the actual state of the universe at any given time is necessarily confined to the region which is accessible for it at that time.

The phase space Γ can also be partitioned into sub regions which form the set of Boltzmannian macrostates. Figure 1 depicts five such macrostates, denoted I, F1, F2 and F3, and the rest of Γ (the complementary set). In general, the partition of Γ into macrostates can be described by a mapping that determines the region to which any point in Γ belongs. In Figure 1, the partition of Γ into macrostates can be thought of as a mapping from each microstate in Γ to one of the macrostates I, F1, F2, F3 or the rest of Γ.

What are macrostates? In Boltzmannian statistical mechanics macrostates are modes of description. More particularly, they are modes of describing microstates. In order to describe a microstate in this way, one first partitions the phase space into

7 We don't address here the question by which criterion the universe is divided into subsystems such as D, G and E. 8 Notions such as teleology or freedom of choice are among those incompatible with mechanics. For this point see Shenker (1999).9We use the term "limitation" here, and not "constraints", since if the system is taken to be the whole universe, nothing external could constrain it. In classical mechanics the spacetime structure, as well as the total energy, are taken to be given limitation. We do not address their origin, for which we would need to turn to other theories. 10If the dynamics of the universe is such that its accessible region is metrically decomposable into dynamically disjoint regions each with positive measure (as in KAM's theorem; see Walker and Ford 1969), we can consider the effectively accessible region as determined by the univers's initial state.

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regions, and then determines which of these regions contains that microstate. The macrostate of a system at a given time is the region which contains the actual microstate of the system at that time. The partition of the phase space of a system into regions is a way of fixing and selecting a macroscopic mode of description; and, by specifying the macrostate to which the actual microstate of the system belongs we determine the property of the system under that description. The macroscopic description of the actual microstate is then relative to the choice of partition (or a mode of description) of Γ.

Are there any conditions on the ways Γ may be partitioned into macrostates? Classical Boltzmannian statistical mechanics seems to impose only two very weak conditions on what can be counted as a macrostate. We take the minimalist view that the sufficient and necessary conditions for a partition of Γ to count as an acceptable set of macrostates are the following. (a) All the subsets of Γ in this partition be given by some measurable function defined over Γ.11 (This condition is necessary in order to make sense of Boltzmann's idea that the entropy of a system is the measure of its macrostate.) (b) In addition to this, in order for a partition to be acceptable, its measurable subsets have to be disjoint and cover all of Γ. Each point must belong to one, and only one, measurable set of points in Γ. (This condition ensures that the universe has well defined macroscopic properties at all times.)

Since by the minimalist approach (a) and (b) are together necessary and sufficient conditions for the acceptability of a partition into macrostates, any partition of Γ which satisfies these conditions is physically meaningful. We call this the liberal stance: whatever is consistent with the laws of the physical theory is in principle possible. By this we do not mean, of course, that anything that's possible by the theory actually exists, or is practical. But we take it that whatever is compatible with the laws of the theory is a possibility of our world. Take for example a trajectory in the phase space which is a solution of the dynamical equations of motion and which is compatible with everything we know to be true of our world. It would make no sense to rule out this trajectory as impossible. Suppose that the trajectory passes through some finite sequence of macrostates in the phase space. Again, the sequence of these macrostates cannot be rendered physically impossible. A physical process compatible with the principles of the theory can be rendered physically impossible only if it is inconsistent with some other physical law.12 But if that's not true, the process is physically possible and ought to teach us about the nature and scope of the physical theory in question, in particular statistical mechanics including the question of Maxwell's Demon. We believe that this liberal stance in statistical mechanics is not only natural but mandatory.

In particular, the liberal stance means that we need to consider the implication of all conceivable partitions, including the one depicted in Figure 1. Since, as we said, conditions (a) and (b) are both necessary and sufficient for the acceptability of a partition of Γ into macrostates, there are no physical (nor metaphysical) criteria – beyond those that lead to conditions (a) and (b) – for preferring some partitions of Γ over others. In particular, it is crucial to notice that the partition of Γ is independent of the (micro) dynamics of the universe (that is, its Hamiltonian).13,14 This means that for any possible dynamics of the universe, there is an infinite number of possible 11 A measurable function need not be integrable, continuous, or computable or have any other nice mathematical properties. 12Or some other lawlike situations or whatever else we take to be true of our world. The point here doesn't depend on some particular view of laws. 13Here we disagree with Shalizi and Moore (2003) who seem to look for a dynamical basis for the partition of phase space to macrostates.

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partitions, giving rise to different macroscopic descriptions of the universe's micro evolution.15 If, in particular, a given partition gives rise to a Maxwellian Demon, then we shall say that a Demon is possible. We shall say that a Demon is possible even if the partition in question is extremely exotic and even if it is hard to see how any measuring device of the sorts we know could correspond to it. This liberal stance with respect to the choice of partition is very important, not only for our treatment of Maxwell's Demon, but for understanding the meaning and scope of statistical mechanics.16

The liberal stance, with its minimalist requirements (a) and (b) for the acceptability of macrostates, seem to us to conform to the idea that grouping microstates in Γ into subsets should reflect some measurement or resolution capabilities of some observer (human or other).17 In Boltzmann's work, the macrostates in Γ space expresses equivalence groups in μ space, relative to some given resolution power with respect to a molecular state; all the microstates that differ by permutation of molecules only (relative to the given resolution power) belong to the same macrostate. The analysis in terms of μ space is suitable mainly for the explanation of thermodynamic properties of ideal gases and is therefore quite limited. In order to generalize the Boltzmannian approach one needs to be able to divide the microstates in Γ according to any physical magnitudes one is interested in.18 This idea is made precise by the liberal stance. So construed, the introduction of macrostates is in accord with the usual Boltzmannian approach to statistical mechanics.19 (We further discuss the partition that corresponds to thermodynamic magnitudes below.)

2.4 Dynamics. To obtain a Demonic universe it is sufficient that the microdynamics be such that all20 of the phase space points belonging to I are mapped, after a finite time interval ∆t= τ, into the F region (i.e. the union of the three macrostates F1, F2 and F3). This means that if the universe starts out in some microstate in region I at t=0 then, at time t=τ, the microstate will be in one of the regions F1, F2 or F3. If, say, the universe ends up in F1, then F2 and F3 contain only points that belong to counterfactual trajectory segments.

14A similar question arises in quantum mechanics where all Hermeatian operators in the Hilbert space of a system are taken to correspond to real observable properties, regardless of the dynamical implications that this might have (e.g. the measurement problem, or Schrödinger's cat paradox). 15An essentially similar argument has been given by Grunbaum, see Sklar 1993 p. 357: "for any specified ensemble there will plainly be coarse-grainings that make the ensemble's entropy do whatever one likes, at least for finite time intervals."16In fact, the liberal stance in its full implications is much stronger than what we need in order to prove the possibility of a Demon. We don't need that all possible partitions of Γ will be physically meaningful, but only some partitions that are finer than those corresponding to the human measuring capabilities, and not even extremely contrived ones. But, we put forward the present view because we think that it is in the right direction regardless of the Demon question. 17 This idea is expressed for example by Tolman (1938, p. 167), although Tolman usually works in a Gibssian framework rather than a Boltzmannian one.18 A full presentation of a Boltzmannian approach is beyond the scope of this paper and is undertaken elsewhere. 19 We don't suggest that writers on the subject have actually endorsed the liberal stance in any explicit way. We do, however, claim that this stance is not only consistent with the Boltzmannian approach but is necessary and puts on the table the full scope of statistical mechanics.20 There are constraints on Albert's construction entailed by the time reversal invariance of the microdynamics due to which the term "all" should be replaced by "a large part". This will be still sufficient for constructing a Demon (see Section 3).

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It is important to realize that the claim that all the I points are mapped to the F region describes a combination of dynamics and partition. We can achieve such a combination in two ways: one is to start out with a given dynamics and select a partition that suits it best according to our purpose (which is to construct a Demonic universe), and the other is to start out with a partition and create or choose a dynamics that suits it best according to our purpose. As we go along we shall use both methods. To clarify this important point we begin with the case of a given dynamics. Suppose, then, that we have some given dynamics; it may be the actual dynamics of the universe, or any other dynamics that we wish to consider. We select a region called I, which we take to be the initial macrostate. We then follow the trajectories that start out in I, and see where they arrive at after τ seconds, given the known dynamics; we call that region F. Note that – by the liberal stance – F may have any shape, and it can be very irregular; it is acceptable as long as it is measurable. (We may then add some points to F and make it larger than I, if we so wish.)

Our choice of partition, such that F will contain all the points that originate in I, is not accidental; we made this choice in view of the microdynamics of the universe and in view of our desire to construct a Demon. But the partition is not dictated by the dynamics: we could choose a different partition, in such a way that the trajectories starting out in I would end up, at τ, in different macrostates. The outcome would simply be that we wouldn't have a Demon. (In fact, the thermodynamic normal case is exactly of this nature, since trajectories starting out in any macrostate spread over all the other macrostates after a short time interval!)

This point is crucial and therefore merits some repeating. In order to construct a Demon in our universe, given its actual dynamics, all we need to do is choose the right partition (but we shall have some dynamical constraints on closing the cycle, see Section 4). In the next section we shall discuss the idea that a partition corresponds to the measuring capabilities of some observer; hence in order to construct a Demon we need to choose the right observer. (Maxwell has emphasized that his demon needs to have sharp eyes, so the focus on measuring capabilities has some connection to traditional accounts of the Demon.)

It is important to distinguish between two conceptually different aspects of this mapping: its microdynamical level and its macrodynamical level. On the microdynamical level the mapping is continuous and its time reverse is also continuous. This is in accordance with the well known properties of the classical mechanical equations of motion. On the macroscopic level, in choosing the partition we take the dynamics into account (as explained above), in order to get from the dynamics the outcome that we want (that is, a Demon). However, the dynamical considerations are used in choosing F, but not in choosing how to divide F into its three sub regions, F1, F2 and F3. This further division is not affected by the dynamics in any way and may, in fact, be arbitrary. One outcome of this independence is of particular importance for us: since F1, F2 and F3 are disjoint and, together, cover up all of F, the macroscopic evolution may be non continuous and unstable in the sense that two neighboring microstates that start out in the same macrostate I may be mapped by the dynamics into two different F macrostates. This outcome is consistent with the microscopic continuity, due to the differences between the micro and macro levels. Note further that in both set ups of Figure 1 and 2, the trajectory of the universe might actually pass from one F state to another (say, at some later time after τ). In this sense too the macroscopic behavior of the universe might be unstable and discontinuous.

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In the set up illustrated by Figure 1, the regions I and F are both topologically connected, and so each point at the boundary between F1 and F2 (and F2 and F3) belongs to one of them. There is an important difference between this set up and the one proposed by Albert (2000). In Albert's set up the three subsets of F are topologically disconnected (see Figure 2). If the volume of F1+F2+F3 is equal to the volume of I (and not larger than I), then this disconnectedness must be a feature of the microdymanics, since in that case the region F, to which all the I points are mapped by the dynamics after time τ, is disconnected; Albert's choice of partition is such that the macrostates F1, F2 and F3 reflect this dynamics. If this is indeed the dynamics of the universe, then the nature of the classical dynamics imposes some constraints on the nature of the I macrostate. Since the dynamical transformation is continuous and time reversal invariant this construction implies that the region I must also consist of three topologically disconnected regions. In fact, if the F regions are topologically disconnected (as in Albert's set up), the whole of phase space is decomposable into dynamically disconnected regions. This also means that the dynamics in Albert's set up must be microscopically unstable in the sense that there must be pairs of trajectories which start in two neighboring points in I and end up far apart, in two different F regions.21 We prefer the partition and the dynamics illustrated in Figure 1 over that illustrated in Figure 2, since in our setup the whole phase space can be topologically connected, and the system may be ergodic in the Birkhoff-Von Neumann sense of the term. Note that both set ups of Figure 1 and 2 imply the macroscopic instability we mentioned before, just because in both, points starting out in a single macrostate I may end up in a number of distinct macrostates in the F region.

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Final D macrostates

I nitial D macrostate

F3

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Macrostates grid coincides with disconnected regions f ormed by dynamics

Accessible region

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I

Figure 2: Albert’s construction

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Macrostates grid coincides with disconnected regions f ormed by dynamics

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Figure 2: Albert’s construction

The constructions above demonstrate that dynamics such as those in Figures 1 and 2 are compatible with the laws of statistical mechanics. As long as there are no no-go theorems which rule out such dynamics on the basis of the laws of statistical

21 Note that the macroscopic instability cannot be avoided by taking the F states to be connected by thin pipes. This is because on our minimalist view (condition (b) in Section 2.3) the partition to macrostates covers also the points on the pipes. So the pipes proposal is a special case of Figure 1.

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mechanics, we have to accept its implications as physically meaningful. This holds even if the dynamics is exotic and hard to realize in practice.

We have applied the liberal stance twice so far. As long as we don't have no-go theorems, we allow all the partitions of the phase space into macrostates, and we allow any dynamics that is compatible with the equations of motion. The amusing and insightful task now is to find interesting pairs of partition and dynamics. One such pair yields Maxwellian Demons (as in Figure 1 and 2), whereas we believe that another yields the usual statistical mechanical predictions. Let's see how this works.

2.5 Thermodynamic magnitudes. By the liberal stance, all the partitions of Γ to macrostates that obey our minimalist view (see requirements (a) and (b) in Section 2.3) are acceptable. Nevertheless, the partition that corresponds to the thermodynamic magnitudes is of special interest to us. We therefore turn to discuss its meaning in the present framework.

So far we have treated partitions of the phase space of the whole universe. Let us now divide the degrees of freedom of the universe into two sets, call them S1 and S2. (S1 and S2 are sub systems of the universe.22 As and illustration, in Figure 1, take D to be the S1 system and G to be the S2 system.) Take some partition of the phase space Γ of the universe into macrostates. Consider the projections of the macrostates in this partition onto each of the two subspaces corresponding to S1 and S2, and call these projections the macrostates of S1 and the macrostates of S2, respectively. The set of the projected macrostates of S1 corresponds (by definition) to the spectrum of some macroscopic observable O1 associated with S1, and the set of the projected macrostates of S2 corresponds to some macroscopic observable O2 of S2. (In the illustration, the projections of I, F1, F2 and F3 on D are the spectrum of O1 and the projections on G are the spectrum of O2.)

Whether or not there are correlations between the projected macrostates O1 and O2 depends on the interaction between S1 and S2. This interaction may sometimes be such that there is a one-to-one or one-to-many correlation between the two observables. In such cases we may take the correlation between O1 and O2 to be a measurement of O2 by O1, in which case O1 is usually called the pointer observable, and S1 might be called an observer. If the correlations are one-to-one the roles of O1 and O2 may be interchanged. (In the illustration, D can be taken to measure G, but not vice versa.)

Some partitions of Γ correspond to the quantities described by thermodynamics. A thermodynamic partition is constructed as follows. First, consider two sub systems of the universe: one of them (S1 in the above discussion) is us, as observers; the other (S2 in the above discussion) is the observed system. The partition of Г that corresponds to thermodynamics means that the projected macrostates corresponding to our physical states correspond to the states of our senses and measurement capabilities, which are the pointer observables (O1 in the above discussion); and the projected macrostates of the observed systems are the thermodynamic magnitudes (O2 in the above discussion).

The choice of the thermodynamic partition is not accidental, since it fits the dynamics of the universe. The actual dynamics of the universe happens to be such that if we choose the thermodynamic partition, then there is a correlation at the macro level, between our senses and the thermodynamic magnitudes (that is, between O1 and O2). This correlation is such that that we, with our sense organs (as O1), may be

22 As noted above, we don't address the question what is the criterion for a set of degrees of freedom to be considered a system in the intuitive sense of the term.

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taken to measure the thermodynamic properties of other systems (O2). This matching between our sense organs and the thermodynamic magnitudes can be explained in terms of survival advantage:23 We have evolved in accordance with the dynamics of the universe, and so our sense organs have evolved in a way that enables us to determine, directly and immediately, the phase space region that contains the actual microstate of local systems relative to the thermodynamic partition. This capability is advantageous since – given the actual dynamics of the universe – it is efficient, for instance in terms of amounts of Shannon information needed to identify and predict the state of the universe. (It may be that the actual dynamics of the universe is such that other partitions have an even greater survival advantage than the partition we correspond to. In other words, it is conceivable that other creatures are better equipped to deal with the world than us.)

In accordance with the liberal stance, the mode of description depicted in Figure 1 can be thought of as part of the thermodynamic partition, but it can also be part of a very different partition. This point is utterly inconsequential, as far as the possibility of a Demonic universe is concerned, or – in other words – as far as the idea of a Demon is consistent with the principles of statistical mechanics.

The question posed by Maxwell's Demon can be phrased now in the above terms as follows. Take the universe's phase space Γ and consider some partition into macrostates (possibly the thermodynamic partition). Specify some initial macrostate I and final macrostate F in this partition, such that a dynamical evolution between these states (within some time interval ∆t) is compatible with the equations of motion. As is well known, the Boltzmannian approach to the usual cases that agrees with the Second Law of thermodynamics ("a thermodynamically normal case") is the following: the probability that a system will be found in any of its macrostates (at any time) is determined by the standard (Lebesgue) measure of that macrostate (normalized relative to the measure of the whole accessible region); this ought to be the case regardless of the initial macrostate of the system, and within a short time interval after the initial state. The Demon question is now whether in principle there are Hamiltonians that will take the universe in a finite time from some initial macrostate I to some final macrostate F with probability that is different from the (normalized) standard Lebesgue measure of F.

Take for example, the initial state I and the final states F1, F2 and F3 in Figure 1. Theses states belong to some partition of Γ into macrostates. The volume, as determined by the standard measure, of each of the macrostates F1, F2 and F3 is smaller than the volume of I, but the sum of the volumes F1+F2+F3 is equal to or larger than the volume of I (Figure 1 illustrates the simple case in which the volume of F1+F2+F3 is equal to the volume of I). This last requirement is necessary and sufficient in order for the dynamical evolution to obey Liouville's theorem. If our universe, illustrated in Figure 1, were a thermodynamically normal case, then the probability that a system would evolve from I to F were expected to be proportional to the standard measure of F. The Demon problem, as illustrated in Figure 1, is, whether there is a dynamics according to which the probability that a system will evolve from I to any of the F states (i.e. F1 or F2 or F3) within some finite time interval t, is greater24 than the phase space measure of the F region. If such a dynamics is possible, then we have a Demon that satisfies the laws of statistical mechanics. We argue below (following Albert 2000) that this is indeed the case.

23We do not go into the question to what extent survival advantage is an explanation of a property.24Or smaller.

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Note that Liouville's theorem implies that the initial macrostate I cannot be an equilibrium state in the standard Boltzmannian sense of the term. The volume of a Boltzmannian equilibrium state usually takes up almost the entire accessible region in phase space. So if the macrostate I were a state of equilibrium, the remaining volume in the accessible region would be much smaller than the volume of I, and so it would be impossible to have F macrostates whose volumes add up to the volume of I. However, if I is a non-equilibrium state, or if the system is small enough (i.e. has only a few degrees of freedom), the construction in Figure 1 is possible.

2.6 A Demonic universe. In the Boltzmannian approach, the entropy of a microstate is proportional to the logarithm of the standard measure of the macrostate to which the microstate belongs. By this definition, the entropy of the universe at each of the F macrostates is lower than the entropy of the universe at the initial macrostate I. Thus, given the dynamics described above, if the universe starts out in macrostate I, its entropy will decrease after the time interval τ, with certainty.25 In particular, the probability for entropy decrease is higher than the one determined by the volumes of the I region and the actual F region in which the system ends up. This is, then, a Demonic universe.

The trick that achieves this result is the right combination of dynamics and partition of the phase space into macrostates. The choice of the partition in Figure 1 in accordance with the dynamics has the result that the dynamics that maps all the microstates I to the F macrostates makes up a Demonic universe that obeys the laws of statistical mechanics (the same is true of Albert's original set up in Figure 2). As long as there is no proof of a no-go theorem that blocks this choice, we must see it as part of our physics and consider its meaning and implications. Our liberal stance implies that for any dynamics of the universe there is a partition such that the time evolution will be demonic, and vice versa: for every partition of Γ there is a dynamics of the universe such that the time evolution will be demonic. In particular, our own universe is Demonic, if one only picks out the right partition, that is, if one is fortunate enough to have been created with senses that correspond to such a partition!

By this we have established, by an explicit phase space construction, that a Demonic universe is consistent with the laws of statistical mechanics. A Demonic universe that obeys the laws of statistical mechanics is possible in principle, even if it may be hard to come by in practice (and with some limitations; see Section 3).

A different pair of partition and dynamics gives rise to our experience of thermodynamically normal evolutions and to the usual predictions of statistical mechanics (as we said above). The dynamics of our actual universe appears to be mixing quite strongly relative to the partition that we use, that is, relative to the thermodynamic partition. It is important to realize that the thermodynamic partition yields the usual experience for the dynamics of the universe as it has been perceived so far. But this does not mean that we – with our senses that correspond to the thermodynamic partition – will never experience a Demon. And this is how that can happen: as we said above, in order to have a mapping from I to F1 or F2 or F3 we can either start with a dynamics and then choose a partition, or start with a partition and then find the dynamics that will yield a Demon or make it up. Now, in our actual universe, we have so far not encountered trajectory segments that look like the dynamics that gives rise to a Demonic universe, relative to our (thermodynamic) partition. However, statistical mechanics does not rule out the possibility that the

25 There are constraints entailed by the time reversal invariance of the microdynamics, due to which this certainty will be replaced by some high probability (see Section 3).

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future segment of the trajectory actually followed by our universe will turn out as a matter of fact to be Demonic relative to the thermodynamic partition.

2.7 The traditional Demon. Our phase space construction of a Demonic universe (following Albert 2000) looks quite different from the picturesque scenarios suggested by Maxwell and many other writers. The argument for the consistency of a Demonic universe with statistical mechanics does not depend on these traditional presentations; still, it may be interesting to see how these accounts may be connected to our general argument. In order to see the connection between our Demonic universe and the traditional Maxwellian Demons we shall draw attention to the distinction between the three sub systems of the universe: the demon, the gas and the environment (D, G and E in Figure 1). Thus far we have ignored this distinction, in accordance with the principle of mechanical egalitarianism (see Section 2.1). We shall continue to apply this principle, in the sense that we shall not ascribe to the systems D, G and E any properties beyond those expressed in the phase space.

In our construction the partition into the macrostates I and F1, F2 and F3 implies that there is a difference in the way that entropy changes in the subsystems D, G and E, by the dynamics that takes the universe from I to F1 or to F2 or to F3. Consider the subspace of the universe consisting of the G degrees of freedom, which is represented in Figure 1 by the G axis. Take the projection of the macrostate of the whole universe onto this subspace; and call the measure of this projection, relative to this subspace,26 the entropy of the gas; and similarly for the D and E subspaces and entropies. The projection of I on G is larger than the projection of any of the F regions (F1 or F2 or F3) on G, and this means that the entropy of G decreases with certainty, and the gas ends up in a certain predictable low entropy macrostate. The entropy of D, by contrast, is unchanged by this dynamics: The projections of I, F1, F2 and F3 on D all have the same measure. The macrostate of E is also unchanged throughout the evolution. In traditional terms concerning Maxwell's Demon, the entropy of the gas has decreased whereas the entropy of the demon and of the environment have been conserved; and since this outcome has been perfectly predictable, it violates the (statistical version of the) Second Law of thermodynamics.

2.6 Predictability and control. In the literature, Maxwell's Demon is often seen as having the capabilities of prediction and control. It can measure the state of the gas with resolution power much finer than that described by the thermodynamic partition, and hence it is capable of predicting the evolution of the gas and of controlling this evolution. How does this idea of the Demon fit in with our Demonic universe?

Recall our account of measurements and macroscopic resolution power (Section 2.5 above). By the partition described in Figure 1, the D system can measure the macrostates of G, since the partition is such that to each state of G there corresponds one, and only one, macrostate of D. Can D use this measurement in order to predict the macro evolution of G? In our discussion it is essential that all the terms like measurement and prediction will be described in statistical mechanical terms. For instance, we can think of prediction as a sort of computation done by a Turing machine. (We make no claims about computability etc., but only wish to illustrate the kind of descriptions that are acceptable in our discussion.) The machine states and the contents of its tape27 are macrostates of D and of G; the evolution between the states and along the tape is determined by the projection on D and E of the dynamics of the

26 Relative to the whole universe this measure is zero.27The tape will not be infinite, but long enough for all practical purposes of D.

11


universe. Whenever we mention prediction we shall refer to a construction such as this, and whenever we speak of one system knowing the state of another we shall refer to measurements in the above sense.

Consider, then, the partition described in Figure 1, and call it partition A. Consider D with its measuring capabilities (relative to G) as described by partition A. For clarity of the following discussion let's call system D in this case (case A) DAlice. Suppose (as before) that the initial macrostate of the universe is I, and that – due to the structure of DAlice (described by partition A) and due to the projection onto DAlice and onto G of the dynamics of the universe – DAlice can predict that the microdynamics will map all28 the microstates in I to the F region. In this case, DAlice can predict that the universe will evolve into one of the F macrostates after the time interval τ, but it cannot predict to which of the F states the universe will evolve at τ, because – by partition A in Figure 1 – all the microstates in I are indistinguishable for DAlice. Therefore, relative to partition A, DAlice can gain a reliable entropy decrease, but she is incapable of making macroscopic predictions. For DAlice , the universe is genuinely Demonic, but it is macroscopically unpredictable; this is the trade off (emphasized by Albert 2000).

To realize in what sense this is a trade off, consider a different partition (call it B) in which the region I is partitioned into three sub regions corresponding to some macrostates I1, I2 and I3, each with measure not larger than the measure of the F1, F2 and F3 states. The choice of partition B takes into account the dynamics of the universe: we choose I1, I2 and I3 in such a way that the dynamics maps them into F1, F2 and F3, respectively. (This choice takes the dynamics into account in the way that partition A took into account the dynamical mapping of I to F, as discussed above. This choice is not necessary; we could choose another partition; but we chose this one due to our purposes. Region I remains topologically connected.)

Replacing partition A by partition B means that the D system is different from DAlice ; call the new system DBob. DBob's measuring capabilities (relative to G) correspond to the partition B. Suppose (as before) that the initial macrostate of the universe is I, and that the microdynamics maps the microstates in I to the F region, and suppose that DBob knows these two facts. But DBob knows also which, among I1, I2 and I3, is the initial macrostate, and that I1, I2 and I3 are mapped to F1, F2 and F3 respectively. A measurement of which state I1, I2 or I3 is the initial macrostate will allow DBob to predict to which of the F1, F2 and F3 states the universe will evolve at τ.

Since the measure of the I1-I3 states is not larger than the measure of the corresponding F1-F3 states, for DBob the evolution results in no decrease of entropy. Therefore, relative to B, DBob gains macroscopic predictability but loses entropy decrease. This is again the trade off. For DBob, the universe is macroscopically predictable, but it is not Demonic.

Here we can make contact with the literature concerning Maxwell's Demon. We can think of the A partition as corresponding to the human capabilities whereas the B partition is ascribed to a 'demon'; DAlice is human and DBob is a 'demon'. The demon's measurement capabilities may certainly surpass the human ones, but are still not supernatural.

Suppose now that DBob (the demon) acts in the world, and some DAlice, who is human, observes the world. In this case we may think of the universe as divided in such a way that there are four sub systems: DAlice, DBob, G and E. For DAlice the universe

28Again, in the next section we shall show that the term "all" is problematic and will be replaced by "a large part of".

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is Demonic and unpredictable; for DBob the universe is predictable but not Demonic. None of them can have a universe which is both Demonic and predictable. It is important to see that predictability does not entail control. DBob can predict the future macrostate of the universe but cannot choose it. The final state of the universe, at any level of precision, is determined by the initial state of the universe; and since DBob is part of the physical universe (and is actually nothing but a set of degrees of freedom), its own state is part of the state of the universe, and its actions are a mere projection of the evolution of the trajectory of the whole universe. Since in classical mechanics there are no jumps from one trajectory to another, there is no way of selecting one's future. Any impression to the contrary is a sad illusion. This almost truism is easily overlooked, and indeed is often overlooked, in the literature on Maxwell's Demon (see Shenker 1999). The introduction of the principle of mechanical egalitarianism (see Section 1) is meant precisely to fix that idea.

3. Some constraints

By the above construction, a Demonic universe is possible. However, classical dynamics imposes two restrictions on the efficiency of the Demonic universe, as follows.

D

E

G

x - x I

F3

F2

F1Region F

F3

F2

F1y- y

Figure 3: Efficiency of demon

D

E

G

x - x I

F3

F2

F1Region F

F3

F2

F1y- y

Figure 3: Efficiency of demon

3.1 Efficiency. Consider one of the microstate points in the I macrostate, call it x (see Figure 3). By the dynamics of our Demonic universe as described above, this point must sit on a trajectory that takes it to a microstate (call it y) in one of the F macrostates (say, F1) after τ seconds. Consider now the microstates which are the velocity reversals of x and y, call them –x and –y respectively. In many interesting cases (but certainly not all cases29), microstates which are the velocity reversals of

29 Consider, for example, the macrostate in which half of the gas molecules move to the right, something we might feel as wind blowing in the right direction. Relative to the phase space partition that corresponds to our sense, say, this macrostate is easily distinguishable from the case where the

13


each other belong to the same macrostate. Suppose, then that both x and –x belong to the same I macrostate, and similarly, both y and –y belong to the F1 macrostate. This puts quite a strong constraint on the efficiency of the Demonic universe. Let's see why.

Since the dynamics is time reversal invariant, if the trajectory starting out in x in I takes the universe to the microstate y in F1, then the trajectory that starts out in –y in F1 takes the universe back to the microstate –x in I after τ seconds. As the mapping from x in I to y in F1 reduces the entropy of the universe, the reversed evolution from –y in F to –x in I increases the entropy of the universe. If the universe starts off in I and evolves to, say, F1 (thereby decreasing its entropy), we presumably want it to remain in the low entropy state F1, avoiding points like –y which take the universe back to the high entropy state I after τ seconds. To make the universe remain in F1, therefore, we need to increase the volume of F1, so that it will include longer trajectory segments which map F1 to itself. But the larger the volume of F1, the smaller is the entropy difference between I and F1.

This means that there is a trade-off between the amount of entropy that can be decreased and the stability of the low entropy state. If we prefer amount over stability, we choose a partition by which the sum of the volumes of the F macrostates will be equal to the volume of I. Under this description, the trajectory of the universe will oscillate between the I and F states with frequency 1/2τ (this is the case illustrated in Figure 3). Note that the (standard Lebesgue) measure of the x-type points is equal to the measure of the –x-type points, since the time reversal operation is measure preserving. If we prefer stability over amount, we will choose a partition in which the volume of each of F1, F2 and F3 is equal to the volume of I. Under this description, the trajectory will remain in the F states for some time, but the entropy of the universe will decrease by only a small fraction.

3.2 Preparation. In order to display its Demonic nature, the universe must start out in macrostate I. Once it reaches the state I it will evolve spontaneously in the anti-thermodynamic way we spelled out above. But note that the universe will behave Demonically only if and when it reaches macrostate I. How can we bring the universe to this initial macrostate? As we said in Section 3.1, the principle of mechanical egalitarianism entails that there is no genuine choice of states and hence no genuine control over states; the state of the universe – and hence its projection on the various sub systems – is an outcome of the dynamics of the universe and its initial state. In this sense, there is only one way that the universe can arrive at macrostate I, namely, spontaneously.

Our experience tells us that control is possible and that subsystems of the universe can be brought to initial states of our choice. But, according to the principles of mechanics, this is an illusion. The evolutions that appears to us as cases of 'control' are merely cases of correlations between the macro evolutions of our states and of the states of other systems, and that evolution is determined by the dynamics of the universe and the partition into macrostates.

If it happens to be the case that the dynamics of the universe is thermodynamically normal (apart from the evolution from I to F) in the sense that the probabilities for macroscopic behavior are proportional to phase space volume (given by the Lebesgue measure), then the chance of observing a Demonic behavior will correspond to the measure of I relative to the measure of the whole accessible region

wind blows to the left. However, in many cases (consider the air in the room) it is extremely plausible that x is indistinguishable from –x.

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in phase space. In large systems this measure will be quite small since, as we said earlier, I cannot be an equilibrium state. However, our universe is in a low entropy state right now! Therefore, the possibility that our partition is somewhat similar to Figure 1, and that the dynamics of the universe is such that it will map I to F, and that – as a result – we shall observe a Demonic behavior, cannot be ruled out.

4. Completing the operation cycle

4.1 The requirements. We have then established the possibility of a Demonic universe and some limitations on its efficiency. By the definition we gave earlier, a universe is Demonic if its entropy decreases with probability higher than that determined by the standard measures of the initial and final states. However, some writers argue that this is not sufficient: they add the requirement that a Maxwellian Demon be considered a counter example of the Second Law of thermodynamics only if the cycle of operation is completed.30 We don't want to go into the question of whether or not this requirement is justified. Instead we will show now how it can be satisfied by our construction.

What is a completion of an operation cycle? Surely, once the cycle is completed we don't want the universe to return exactly to its initial state, since in particular we want the entropy of the gas to remain low. Instead, the idea is that at the end of the cycle the situation will be as follows: part of the universe (G) will end up with entropy lower than its initial entropy, while the rest of the universe (D and E) will end up with entropy not higher than their initial entropy. Of that latter part of the universe, some subsystems (here, D) will end up at their original initial state; and the rest of the universe (here, E) will end up in a state that may be different from its initial state, but with entropy equal to its initial entropy. Moreover, the overall final state (at the end of the operation cycle) must be such that a subsequent entropy reducing operation cycle can start off, and once the second operation is completed, another one can start off, and then another, perpetually. These requirements are often stated in terms of three properties that the final state of the universe (at the end of the operation cycle) should have: (i) Low entropy; (ii) Return of the D subsystem (the 'demon' in traditional accounts) to its ready state; (iii) Erased memory. Let us explain these in turn using our construction (Figure 1).

(i) The requirement of low entropy means that the total entropy of the universe at the end of the operation cycle must be lower than the entropy in the initial macrostate I.

(ii) The requirement of return means that at the end of the operation cycle D will return to its initial ready macrostate, so that another new cycle of operation can start off.

(iii) The requirement of erased memory means that the final macrostate of the universe at the end of the operation cycle must be macroscopically uncorrelated to the F macrostates. In other words, at the end of the cycle there should be no macroscopic records of whatever sort that will allow retrodicting which state among F1, F2 and F3 was the actual macrostate of the universe prior to the erasure. Obviously, the requirement of erasure refers to the macroscopic level, since the classical microdynamics is incompatible with erasure at the microscopic level, because it is deterministic and time reversal invariance.31 (Note that this requirement

30 For more details concerning the cyclic nature of the Second Law, see Uffink (2001).31 By contrast, the quantum microdynamics is consistent with memory erasure (requirement (iii). The information carried by the value of a quantum mechanical observable can be erased by measuring a

15


is different from the requirement (ii), since the memory could be stored in systems other than D.)

Before we proceed to showing how all this can be achieved, it will be instructive to consider two attempts that don't work. The first attempt doesn't obey Liouville's theorem and the second attempt increases entropy. Consider a dynamics which first takes the system from I to one of the F states (as before; see Figure 1). In the first "failed" attempt, the universe evolves to a macrostate such that D is back to its initial state (requirement ii) while leaving G in its low entropy state (requirement i) and E is unchanged. Such a process erases memory (requirement iii) since from the final macrostate it is impossible to retrodict which among F1-F3 was the previous macrostate of the universe. However, the process violates Liouville's theorem since it maps the volume F1+F2+F3 into a volume equal to F1 or F2 or F3. Therefore such a process is impossible.


D macrostateaf ter erasure

I

Final E macrostate?

D

E

G

F3

F2

F1

Region F

A

Figure 4: Dissipative erasure



I

Final E macrostate?

D

E

G

F3

F2

F1

Region F

A

Figure 4: Dissipative erasure

The second "failed" attempt maps the F1, F2 and F3 macrostates to region A in Figure 4. Region A has the following properties: it contains all the microstates to which the trajectories leaving F will arrive after τ' seconds; G retains its low entropy state and D returns to its initial ready state (requirement ii). Memory is erased since from the information that the universe is in macrostate A it is impossible to infer the F state in which it has been before (requirement iii). However, due to Liouville's theorem, the entropy of the environment increases, and so the final entropy is the same as the initial entropy at macrostate I. The achievement of reducing the entropy by the transformation from I to one of the macrostates F1 or F2 or F3 is lost, contrary to requirement (i). (The choice of A involves some further problems that we address below.)

We now turn to show, by way of construction, how requirements (i), (ii) and (iii) can be achieved without violating any principle of statistical mechanics.

non commuting observable. However, the quantum dynamics cannot satisfy both requirements (ii) and (iii) without violating unitarity. See an example of a quantum erasure in Herzog et at (1995). Here we only consider a classical erasure.

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4.2 Entropy Decrease. We begin with requirements (i) of low entropy and (ii) return to the ready state. The way to get this is by repeating a trick we have already used: choosing the right pair of partition and dynamics (or: choosing the right partition that will fit the given dynamics of the universe). Consider Figure 5. We choose a partition in two steps (the two steps resemble the process by which we chose the F regions above). In the first step we choose region A to fit the dynamics of the universe, as in the second "failed" attempt above: we follow the trajectories that leave the F1, F2 and F3 regions and call "A" the region into which they are mapped after τ' seconds.


D macrostateaf ter erasure I

Possible fi nal E macrostates

Volume (A1+A2+A3) = Volume (F1+F2+F3) = Volume (I )

D

E

G

F3

F2

F1

Region F

A1A1A2

A3

A1Region A

Figure 5: Entropy Conserving Erasure I


D macrostateaf ter erasure I


Volume (A1+A2+A3) = Volume (F1+F2+F3) = Volume (I )

D

E

G

F3

F2

F1

Region F

A1A1A2

A3

A1Region A

Figure 5: Entropy Conserving Erasure I

Now a delicate point needs to be noticed: if we want to have a Demonic universe with a closed operation cycle, the dynamics must be such that in the state A at the end of the cycle G will retain its low entropy (requirement i) and D will return to its initial ready state (requirement ii). Not every dynamics will be like that; for instance, there is no guarantee that the actual dynamics of our universe is like that (although it might be, especially if we choose the right I macrostate to begin with). And here there is a role to the liberal stance: all we need is a combination of dynamics and partition that is possible in the sense that it does not contradict any law of physics (nor any fact in the world). If we can show that a dynamics and partition which are consistent with the laws of physics give rise to a Demon, then a Demon is possible. We do not have to show that our actual universe is Demonic with a closed cycle. So: the dynamics which we consider from now on is such, that region A which contains all the points mapped from F after τ second is also such that D returns to its initial state and G retains its low entropy.

The problem – as we said above – is that requirement (iii) is not yet fulfilled. Let us see how to solve this problem.

Consider a partition of region A into three disjoint macrostates A1, A2 and A3 such that the union of their volumes is at least as large as the union of the volumes of F1, F2 and F3. In the simplest case, illustrated in Figure 5, the volumes of A1, A2 and A3 are all the same and are equal to the volumes of F1, F2 and F3. By the way that we have chosen A, that is by the dynamics, the F states are mapped (after a certain time

17


interval) to the A states.32 The actual final state of the universe will be one of the A macrostates, and the volume of that macrostate is, by construction, equal to the volume of each of the F macrostates and smaller than the volume of the initial macrostate I. This means that the total entropy of the universe during the evolution from F to A does not change and in particular it does not increase. So the evolution satisfies the requirement of low entropy.33

Let us see now what this entropy conserving transformation implies for the three subsystems separately: G, D and E. The A macrostates are chosen such that the projection along the G axis is the same as in the F macrostates, and so G retains its low entropy. The projection along the D axis is the same as in the initial macrostate I, and so D is back to its initial ready state. Its entropy has not changed throughout the process.

Along the E axis, there are by construction three regions corresponding to three possible final macrostates of E. The entropy of E in each of these states is the same as it was in the initial state I, although its final macrostate is different from its initial one. Is this a problem? Do we need E to end up in the same macrostate it started out in? We think that the answer is No, and indeed this is not required in the literature as well. For example, Bennett (1973, 1987) and Szilard (1929) argue that completing the cycle of operation involves dissipation in the environment, and therefore the environment's final macrostate is a fortiori different from its initial macrostate. (For them not only the macrostate of the environment changes, but the entropy of the environment increases; we have a different macrostate, but entropy is conserved.) Moreover, we can construct the evolution from F to A such that the entropy of E will decrease by taking a partition of A into more numerous and smaller subsets. In this case, obviously, E not only need not but cannot return to its initial macrostate. So requiring that it will return is absolutely out of the question.

4.3 Memory erasure. We will now show, by explicit and general phase space construction, that it is possible to construe the A macrostates in such a way that a genuine memory erasure can be carried out in accordance with the classical dynamics, in particular with Liouville's theorem, while the total entropy of the universe does not increase.

So far, nothing in our construction corresponds to memory erasure since it is possible that the A1, A2 and A3 macrostates are 1:1 correlated to the F1, F2 and F3 macrostates, so that from the final A macrostate it is possible to retrodict the F macrostate. However such a correlation can be easily avoided, as follows. Consider a dynamics such that 1/3 of the points in each of the regions F1, F2 and F3 are mapped into each of the regions A1, A2 and A3. Conversely, by this dynamics, in each A region, 1/3 of the points originate in each of the F regions. Is such dynamics possible? As before, we adopt the liberal stance. As long as we don't see a no-go theorem which prohibits such dynamics, we take it to be possible in principle, albeit perhaps hard to realize in practice.

By this dynamics, the A macrostates are not macroscopically correlated to the F macrostates, and in this sense they bear no information about their macroscopic history. Given the final A macrostate, it is impossible to retrodict the F macrostate. 32If the regions F1, F2 and F3 are topologically disconnected (as in Albert's set up), then so will be the regions A1, A2 and A3. This will put some constraints on the dynamics of the erasure; see below. Since in our set up the F regions are connected, this problem does not arise.33Incidentally, there is a partition into smaller and more numerous A macrostates such that the entropy of the final state at the completion of the cycle would be even smaller than it was at t= τ; but this is more than we need right now.

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Therefore, the F to A transformation is a memory erasure. This memory erasure is a result of a choice of a pair of partition and dynamics. The partition that we use here can be seen as an extension of the partition in Figure 1 that also contains the macrostates I, F1, F2 and F3. This partition corresponds to the observer DAlice (see Section 2.6), who now has measuring capabilities corresponding to the extended partition and hence knows whether the universe is in A1 or A2 or A3. But DAlice cannot (by definition) distinguish between microstates that belong to the same macrostate in that partition. In particular, since the dynamics maps sub regions of the F macrostates to sub regions of the A macrostates, even if DAlice detects that the universe is in, say, A1, it is unable to tell which part of A1 the universe is in, and therefore is unable to reconstruct the actual historical F macrostate of the universe. Therefore, relative to this partition, the F to A mapping by this dynamics is a memory erasure.



I


D

E

G

Region F

A1A1 Region A

F11F12 F13

F21F22 F23

F31F32F33

11 12 1321 22 23

31 32 33

Figure 6: Entropy Conserving Erasure II



I


D

E

G

Region F

A1A1 Region A

F11F12 F13

F21F22 F23

F31F32F33

11 12 1321 22 23

31 32 33

Figure 6: Entropy Conserving Erasure II

But consider now a more refined partition of the F and A regions into macrostates (see Figure 6), which is an extension of the B partition above (corresponding to the observer DBob). Instead of macrostate F1, for example, we have three macrostates, F11, F12 and F13; and instead of the A1 macrostate we have A11, A12 and A13; and so on, such that F11, F12 and F13 are mapped to A1; F21, F22, F23 are mapped to A2; and F31, F32, F33 are mapped to A3. Conversely, A11, A21, and A31 are mapped to F1, etc.34 Relative to this extended B partition, the dynamics described above is not a memory erasure, since DBob, has measuring capabilities corresponding to the B partition, and if he knows the complete dynamics, he will be able to retrodict the macroscopic history of the universe given the actual Aij state. DBob can know in which part of A1 the universe is, and this information is enough in order to infer from the dynamics the previous F macrostate of the universe.

34 In our set up, the macrostates F11, F12, F13, etc. need not correspond to topologically disconnected regions (for the same reason we have argued before; see section 2). However, if F1, F2 and F3 happen to be topologically disconnected (as in Albert's set up), then so will be the sub regions of F1, F2 and F3 (i.e. F11, F12, F13, etc.) and the regions A1, A2, A3 and their sub regions A11, A12, A13, etc.

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But an erasure dynamics can be constructed relative to this partition, in essentially a similar way to the one above. And there is a third partition, which in turn corresponds to a third observer, DCharlie, such that the erasure relative to DBob is not an erasure relative to DCharlie; and so on. More generally, relative to any given partition, there is an erasing dynamics which is perfectly compatible with Liouville's theorem and with the requirement of low entropy.35

We see then that a memory erasure, similarly to the Demon, is relative to a partition of the phase space. However, there is no universal erasure (that is, an erasure applicable to all possible partitions, however refined), since it would require a dynamics that is maximally mixed in a finite time interval. This is impossible because it is impossible that after a finite time interval every set of positive measure in every macrostate contains end points that arrived from all the other macrostates.

By this construction we have demonstrated that the cycle of operation in a Demonic universe can be completed, in the right sense of completion. The initial and final macrostates of the universe are indeed different, namely, by the end of each cycle the number of macrostates of E is (in our set up) tripled. But this is absolutely irrelevant to the questions of Maxwell's Demon and memory erasure. More generally, our construction shows that the exponential increase in the number of macrostates is perfectly compatible with a reliable and regular and repeatable entropy decrease and genuine memory erasure.

According to the Landauer-Bennett thesis, memory erasure is necessarily accompanied by a compensating entropy increase of kln2 per bit of lost information. Landauer and Bennett base their thesis on Liouville's theorem (see Bennett 2003, Landauer 1961, and Leff and Rex 2003). Our F to A dynamics is a dissipationless memory erasure, and therefore it is a counterexample of the Landauer-Bennett thesis. We take this to be a direct refutation of the Landauer-Bennett thesis in the Boltzmannian framework of statistical mechanics.

Conclusion

We have shown, following Albert (2000), that Maxwell's Demon is compatible with classical statistical mechanics and that both are compatible with memory erasure that conserves (and even decreases) entropy.36 Maxwell intended his Demon thought experiment to be a counter example of the Second Law of thermodynamics. Our setup (and Albert's) is not only a thermodynamic Demon (i.e., an operation that reduces the total entropy of the universe, see Section 2.1) but also a statistical mechanical Demon (i.e., the probability for a spontaneous decrease in the total entropy of the universe is higher than the probability determined by the measure of the macrostates). This result surely has implications with respect to the status of thermodynamics and to the relation between thermodynamics and statistical mechanic. Can we still say that statistical mechanics corresponds to thermodynamics by reproducing a statistical version of its Second Law?

We believe that the answer is: yes, statistical mechanics reproduces a statistical version of the Second Law, but statistical mechanics is much more general than thermodynamics. Thermodynamics is a very special case for which statistical

35 And of course, it is also true that relative to any eraser dynamics (given some partition), there is another, more refined, partition relative to which this dynamics is not an erasure.36It seems to us plausible that Albert's Demon and our memory erasure at the macroscopic level can be set up also if the classical dynamics is replaced with quantum mechanical dynamics. We address the quantum mechanical case elsewhere.

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mechanics is certainly applicable, and which is of special interest to us. But in order to understand the full scope of statistical mechanics, we have to shed our prejudices, and allow physical possibilities that are different from the world we experience.

The full generality of statistical mechanics is brought out by the liberal stance, that whatever is not inconsistent with the laws of mechanics is physically possible, and therefore ought to teach us about the nature of the theory. By the liberal stance, the basic principles of statistical mechanics apply to every evolution which agrees with classical mechanics, and to every measurable partition of the phase space; not only to the actual dynamics of our universe, and not only to the thermodynamic partition that corresponds to the human senses. Statistical mechanics applies to universes in which the dynamics – albeit mechanical – is very different from our own; and it applies to creatures endowed with sense organs that correspond to partitions that are very different from the thermodynamic-human partition.

One of the pairs of dynamics and partition to which statistical mechanics applies gives rise to the Boltzmannian account of our thermodynamic experience. Another pair of partition and dynamics to which statistical mechanics equally applies is the pair that brings about a Demonic universe, as described above. The question of which, among these pairs, corresponds to our universe, is a question of facts: one fact is about the actual dynamics of the universe (concerning which we have some conjectures), and another fact is about the partition: our senses correspond to one out of infinitely many possible partitions.

Given that a Demonic universe is compatible with statistical mechanics, can we construct a Demon relative to the thermodynamic partition, in some subsystems of the universe? This paper shows that Demons in subsystems of the universe cannot be ruled out by the principles of statistical mechanics. The difficulties that we can think about in actually building a Demonic system seem to involve practical issues such as controlling a large number of degrees of freedom, interventionist considerations, etc.

So the answer to the question whether statistical mechanics corresponds to thermodynamics is: yes, given the appropriate pair of dynamics and partition. But statistical mechanics is more general than thermodynamics and therefore applies to other possibilities as well.

Acknowledgement. We thank David Albert, Tim Maudlin, Itamar Pitowsky, and especially Dan Drai for very helpful comments. We also thank the Israel Science Foundation (grant number 240/06) for supporting this research.

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