# Studies in History and Philosophy of Modern Physics

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3 186 J.D. Norton / Studies in History and Philosophy of Modern Physics 42 (2011) work kt ln 2 work kt ln 2 expansion heat kt ln 2 compression heat kt ln 2 Fig. 1. Reversible isothermal expansion and contraction of a one-molecule gas. L R removal or L R L R insertion Fig. 2. Removal and insertion of the partition. This expansion and contraction may be achieved more rapidly by two further processes shown in Fig. 2: Process 2a. Removal of the partition. A partition at the midpoint that traps the molecule on one or other side is removed. The removal involves no work or heat. Process 2b. Insertion of the partition. A partition, inserted at the midpoint of the cylinder, traps the molecule on one or other side with equal probability. The insertion involves no work or heat. Processes 1a and 2a are analogous in that they both double the volume of the gas; and Process 1b and 2b both halve the volume of the gas Thermodynamic entropy and the second law We would normally say that the expansion Processes 1a and 2a increase the thermodynamic entropy of the gas by k ln 2; and that the compression Processes 1b and 2b decrease the thermodynamic entropy by k ln 2. This follows from an application to Processes 1a and 1b of the Clausius definition of thermodynamic entropy: the difference in thermodynamic entropy DS therm for two states of a system is defined by Z DS therm ¼ dq rev =T ð1þ where q rev is the heat gained by the system during a reversible process connecting the states along which the integration proceeds. However the circumstances are not normal and we must proceed cautiously. The Clausius formula (1) can only identify thermodynamic entropy S therm as a property of the initial and final states themselves, if every reversible process that connects the two states gives the same entropy difference through formula (1). This sameness can fail if the second law of thermodynamics fails. The applicable version of the law is the Thomson form that prohibits any process whose net effect is just the full conversion of a quantity of heat into work. If we take the special case of two states at the same temperature T connected by isothermal, reversible processes, one easily sees that this law is necessary if the entropy change of (1) is to depend only on the properties of the initial and final states, so that every isothermal, reversible process connecting them returns the same entropy difference. For, imagine otherwise, that the law fails for a cycle of isothermal, reversible processes on which the initial and final states lie as shown on the left in Fig. 3. Then the heats q 1 and q 2 passed to the system in the two sections of the cycle, process 1 and process 2, must satisfy q 1 þq 2 o0. Therefore q 1 cannot equal ( q 2 ). Now consider the initial and final states connected by two processes, process 1 and the reverse of process 2, as shown in the figure on the right. The two processes pass unequal heats q 1 and ( q 2 ) to the system. Recalling that these are isothermal processes, it follows that the Clausius formula (1) would assign a different entropy difference between the states according to the process that connects them A possible fluctuation-derived violation of the second law Caution is warranted in asserting the second law of thermodynamics, for the random motions of the single molecule, understood as density fluctuations, do constitute continuing violations of the second law. When the molecule moves to one side or other of the chamber, that motion is a spontaneous recompression of the one-molecule gas to one side. All that is needed, it would seem, is the insertion of the partition to make one of the fleeting violations permanent. More carefully, combining the spontaneous compression of Process 2b with the reversible expansion of Processes 1a produces a cycle that assuredly violates the second law of thermodynamics. The first step realizes a process forbidden by the second law. The second step converts it into a form that explicitly breaches the Thomson form of the law: kt ln 2 of heat is drawn from the heat bath and converted fully to work, while the gas is returned to its original state. The sole effect of the cycle is to convert heat to work. It is a simple, vivid illustration of the sort of process a Maxwell demon seeks to realize.

4 J.D. Norton / Studies in History and Philosophy of Modern Physics 42 (2011) final final process 1 process 1 heat = q 1 process 2 heat = q 1 reverse process 2 heat = q 2 heat = q 2 initial initial state space state space Fig. 3. A Cycle that violates the Thomson form of the second law. One s initial reaction may well be that that we must abandon the second law of thermodynamics if we are to consider fluctuating physical systems such as these; and that may well be the final decision as well. However a literature that is now eighty years old has refused to accept this decision. It seeks to save the second law by finding a hidden locus of entropy creation in the processes of the demon that operates the cycle, now naturalized as a physical device. The earlier tradition, associated most prominently with Brillouin, located the hidden entropy creation in the physical processes used by the demon to identify the position of the molecule. A later tradition deemed this a mistake. Rather, it urged that the demon must retain a memory of which side of the chamber held the molecule. Completing the cycle requires erasure of this memory and, following Landauer, this erasure process creates the entropy needed to preserve the second law. 3. Direct proofs of Landauer s Principle Applied to the case of one bit of information, Landauer s Principle asserts that its erasure increases the thermodynamic entropy of the surroundings by at least k ln 2. In Norton (2005), I investigated at some length the approaches used in attempts to prove this principle and explained why they fail. This section will review very briefly the modes of their failure and the reader is referred to Norton (2005) for more details. The proofs are direct. That is, they look at the physical process of erasure directly and seek to demonstrate that its implementation must create thermodynamic entropy. The standard model of a memory device is just the one-molecule gas above. A single bit of information is encoded in the trapping of the molecule in the left L or right R side of the chamber. Erasure is the process that assuredly moves the molecule to the left chamber L. While this model may seem excessively idealized, it is taken to capture the essential thermodynamic features of a more realistic one-bit memory device in a heat bath. Here are three related attempts to prove Landauer s Principle, along with the reasons they fail Erasure by thermalization The molecule is trapped on one side of the partition. In the erasure procedure, the partition is removed by Process 2a, thermalizing the molecule. The resulting chamber-filling one-molecule gas is isothermally and reversibly compressed by Process 1b to the reset L state. This step passes heat kt ln 2 to the heat bath at temperature T, increasing its entropy by k ln 2. All this proof shows is that a particular, inefficient erasure procedure creates thermodynamic entropy k ln 2, whose origin lies in the ill-advised initial step of removing the partition. It does not show that all possible erasure processes must create thermodynamic entropy. It is sometimes suggested that the erasure must employ the thermodynamic entropy creating step of removal of the partition, or something like it, for the process cannot know which side holds the molecule on pain of requiring further erasure of that knowledge. Yet the erasure must succeed whichever side holds the molecule. 4 However what this consideration does not show is that thermalization is the only way of satisfying this robustness condition. Indeed, in light of the analysis below, it cannot be shown in the standard framework, since its repertoire of processes admits other robust erasure procedures that do not employ thermalization Compression of phase volume Boltzmannian statistical physics relates the thermodynamic entropy S of a system to the accessible volume of the phase space it occupies according to S ¼ k ln ðaccessible phase volumeþ ð2þ Prior to the erasure, the memory device may contain either L or R data. Therefore, this approach asserts, the molecule is associated with a phase volume that spans both chambers. After erasure, the molecule is assuredly in the L half of the chamber, so that the phase volume has been halved. The resulting entropy reduction is k ln 2. That reduction must be compensated by at least a doubling of the phase volume of the surroundings and, correspondingly, an increase in its entropy of at least k ln 2. 5 The error of the proof is that the molecule, prior to the erasure, is not associated with an accessible phase volume that spans the entire chamber. It will assuredly be in one half only. Which that half is will vary from occasion to occasion, but it will always be one half. As a result, the erasure operation does not need to reduce accessible phase volume at all; it merely needs to relocate the part of phase space accessible to the molecule. 4 For an example of this approach, see Leff & Rex (2003, p. 21). 5 Writing some three decades after his initial proposal, concerning a figure that shows information degrees of freedom vertically and environmental degrees of freedom horizontally, Landauer (1993, p. 2) glosses the result in this way: The erasure process we are considering must map the 1 space down into the 0 space. Now, in a closed conservative system phase space cannot be compressed, hence the reduction in the vertical spread [of phase volume] must be compensated by a lateral phase space expansion, i.e. a heating of the horizontal irrelevant degrees of freedom, typically thermal lattice vibrations.

5 188 J.D. Norton / Studies in History and Philosophy of Modern Physics 42 (2011) Illicit ensembles and information theoretic entropy For so-called random data, the molecule is equally likely to be in the L or R chamber. That is, we have probabilities P(L)¼P(R)¼1/2. After erasure, the molecule is assuredly in the L chamber. That is, P(L)¼1; P(R)¼0. In this approach, the random data state is treated as if it was the same as a thermalized data state, whose erasure does create k ln 2 in thermodynamic entropy. We arrive at the thermalized data state from the random data state by Process 2a, removal of the partition, so that the molecule can move freely between the two chambers. It is supposed that both states are thermodynamically equivalent, since in both states the molecule is equally probably in either chamber. The thermodynamic equation of the two states is sometimes justified by considering a large set of memory devices containing random data. This set is presented as a canonical ensemble, supposedly thermodynamically equivalent to a canonical ensemble of thermalized memory devices. The approach fails because the random data state and the thermalized state are not thermodynamically equivalent. In the random state, only half the phase space is accessible. In the thermalized state, the full phase space of the chamber is accessible to molecule. This difference of accessibility gives the two states different thermodynamic properties. They differ by k ln 2 in thermodynamic entropy. We do not know which half of the phase space is accessible in the case of random data is irrelevant to the device s thermodynamic properties. (For further discussion, see Norton, 2005, Section 3.) A development of this approach calls on information theory. There one assigns an entropy S info to a probability distribution that assigns probabilities P i to outcomes i¼1, y, n according to S info ¼ k X P i lnp i ð3þ i where units of k are chosen arbitrarily. Applying this formula 6 to the probability distributions before and after erasure, we recover a change in entropy of k ln 2. Where this argument fails is that it only establishes a change in information theoretic entropy S info. Without further assumption, it does not establish a change in thermodynamic entropy S therm as defined by the Clausius formula (1). Thermodynamic and information theoretic entropies have been shown to coincide in the special case in which the probability distribution is a canonical distribution over a volume of phase space that is everywhere accessible to the system point. (The demonstration is given in Norton, 2005, Section 2.2.) This condition of accessibility fails for the probability distribution of random data. There is a second difficulty with this approach that has not, to my knowledge, been taken up in the literature. The input to an erasure procedure will be a memory device containing an indeterminate configuration of data. In the one bit case, the data to be erased may be L or R. This is routinely modeled by describing the data as random, thereby adding a probability distribution to the possible data. This introduction of a probability distribution is essential if formula (3) is to be used, for the quantity it computes is a function of that distribution. The trouble is that the introduction of a probability distribution is not justified by the logical specification of the erasure function; the specification makes no mention of probabilities. It just indicates a function on the set {L,R} that cannot be inverted. This introduction of probability is routinely assumed benign in physical analysis when some variable has an indeterminate value. It is not benign since it adds non-trivial structure to the indeterminateness of a variable and can induce egregious inductive fallacies, as shown in Norton (2010). The real seat of the entropy of formula (3) is this probability distribution and, absent cogent justification of the introduction of the probability distribution, 7 the entropy change associated with erasure by formula (3) is merely an artifact of a misdescription of the indeterminateness of data. Attempts at direct proofs of Landauer s Principle in the literature employ one or more of these three approaches and, as a result, fail. Two such failed attempts from the more recent literature are reviewed in the Appendix. 4. An indirect proof of Landauer s Principle The erasure process considered above takes a memory device that may hold either L or R as data and resets it to L. This is a physical implementation of a logical transformation that maps either of the symbols L or R always to L. It is logically irreversible in the sense that the function is not invertible. Informally, knowing the result is L does not tell us whether an L or an R was reset. Landauer s Principle asserts that the associated physical erasure process must create k ln 2 of thermodynamic entropy. As a result, LPSG call the process thermodynamically irreversible. LPSG seek to establish a generalized form of Landauer s Principle according to which all physical implementations of logically irreversible processes are thermodynamically irreversible. For present purposes, there is no need to recount LPSG s result and proof in all generality. It will be sufficient to rehearse it for the simple case of one bit erasure seen so far. The proof employs two systems: a memory device M and a one-molecule gas G, used as a randomizer. We will use a second one-molecule gas as the memory device M. Its initial state will be the molecule in the left side of the chamber, state M L, and it will record one bit of information according to the final position of the molecule, M L or M R, with the latter state defined analogously. The process of interest is the erasure process that returns the memory device to its initial state M L. The proof proceeds by embedding that erasure process in the following thermodynamic cycle, where the step numbering coincides with the step numbering of LPSG s more general proof: Step 1. The one-molecule gas of G occupies the full chamber. A partition is inserted at the midpoint so the molecule is trapped on one or other side of the chamber with equal probability. Step 2. The location of the molecule is measured and the memory device M is set to L or R according to whether the molecule of G is found in the L or R side of its chamber. (Since the initial state is M L, action is only triggered if the molecule is found to be on the right, in which case it will be shifted to the left.) The shift is performed by a reversible thermodynamic process. Since the thermodynamic entropies of M L and M R are the same, no heat passes to or from the surroundings. Step 3. According to whether the memory device is in state M L or M R, a piston is introduced into the chamber of G and a reversible isothermal expansion in direction R or L performed, returning the system G to its initial state. Heat kt ln 2 is drawn from the heat bath and work kt ln 2 is recovered. Step 4. The erasure process is performed. It transforms the memory device from the probabilistically mixed state of M L or M R with equal probability, to the initial state of M L. 6 Invoking this information theoretic entropy, Leff & Rex (2003, p. 21) writeofa memory device containing random L and R data: ythe initial ensemble entropy per memory associated with the equally likely left and right states is S LR (initial)¼k ln 2. 7 Those tempted to call upon a principle of indifference should see Norton (2008, 2010).

7 190 J.D. Norton / Studies in History and Philosophy of Modern Physics 42 (2011) M R M L Fig. 4. Shifting a memory device from state M R to state M L. repertoire of processes. Here is another. The goal is to reset the memory device, provided in state M R or state M L, back to its neutral state M L. The process detects whether the molecule is trapped in the right side of the chamber. If it is, the partition is removed and replaced. This detect-remove-replace process is repeated until the molecule is no longer found in the right side. The probability rapidly approaches one that the erasure is complete upon repetition of the detect-remove-replace process. (There is a probability greater than of shifting of the molecule from R to L after 10 repetitions.) 6. Dissipationless erasure and the no-erasure demon The second law violating process comprised of Steps 1, 2, 3, 4 n is not new. It just a version of the no-erasure demon described in Earman and Norton (1999, pp ) and Norton (2005, pp ). (The label no erasure was used to indicate only that the process does not perform a dissipative erasure of the type traditionally described.) Proponents of Landauer s Principle have challenged this demon. However, I believe that the challenges have failed. The dissipationless erasure processes described here and in the no-erasure demon are not intended as positive proposals. Rather they are introduced as the final stages of a reductio argument. That argument seeks to establish the pointlessness of trying to establish Landauer s Principle or the necessary failure of Maxwell demon in a system that employs processes listed here. For those processes admit construction of both dissipationless erasure routines and a Maxwell demon Challenge from an augmented Landauer principle Bennett (2003) suggested that the no-erasure demon is subject to an extended form of Landauer s Principle. The no-erasure demon must merge computational paths in order to restore the gas and memory device to their original states. This merging, the augmented principle states, is accompanied by a compensating entropy increase in the surroundings. In Norton (2005, Section 5), I have explained in more detail why this augmentation of Landauer s Principle fails to compromise the no-erasure demon. Briefly, the notion of computational path and the computational space it suggests is vague, making any precise determination of the grounding of the claim unclear. The intention seems to be that this merging must create entropy in the same way as the failed reduction of phase volume argument of Section 3.2 for the unaugmented Landauer s Principle. There is no reason to expect this vaguer rendering of a failed argument to fare any better. Finally, the no-erasure demon is constructed from the inventory of admissible processes routinely presumed in the thermodynamics of computation. Bennett s response does not dispute that. Hence his challenge merely worsens the mismatch of Landauer s Principle with the standard inventory by expanding the range of processes admitted by the inventory but prohibited by the now extended Landauer s Principle Challenge from the notion of a control bit LPSG (p. 72, n. 8) have suggested the no-erasure demon fails for different reasons. They write [the no-erasure demon] does not really achieve the implementation of a logically irreversible process because, since the same bit cannot be both the control and the target of a controlled operation, the system must have a memory (our M) of which program it has run, which amounts to the system keeping a copy of the original bit; hence, in our terminology it implements [a logically reversible transformation] L 2. The claim, apparently, is that a detection process conducted on a memory device cannot be used to trigger a process that alters that same memory device, unless some record is left elsewhere of the content of the memory device. The claim is unsustainable. The triggered process can proceed without the continued existence the triggering data; all that it needs is for the data to exist at the time of the triggering and the presumption that the processes can proceed independently once triggered. If it helps, imagine that the process triggered is carried out by a physically distinct robotic machine. The device s sole function is to perform this one process without needing any further data input; it operates autonomously once triggered; and it is programmed to return itself to its unique ready state as its last step. 7. Arbitrariness of the standard inventory In Section 5, I assembled what I called the standard inventory of process employed in this literature. I argued that this inventory admits composite processes that violate the second law of thermodynamics and also effect dissipationless erasure. My point is not to urge that such processes are possible. Rather my point is one that has been central to my earlier papers on the subject: that the literature in the thermodynamics of computation is incoherent. Its basic principles versions of the second law of thermodynamics and Landauer s Principle contradict the processes it admits. I have no interest in defending the standard inventory. My view is that it has been assembled myopically in order to enable a few favored composite processes to be constructed, but without proper attention to the fuller ramifications of the selection. It can be challenged and should be Neglect of fluctuations The principal difficulty for this inventory comes from its selective neglect of fluctuation phenomena. This neglect reflects the discarding of an earlier tradition in which fluctuations played a central role. When the molecular basis of thermodynamics was accepted over a century ago, fluctuation phenomena were recognized to be small violations of the second law of thermodynamics. The open question was whether these small violations could be accumulated into larger ones. The early consensus was

8 J.D. Norton / Studies in History and Philosophy of Modern Physics 42 (2011) Fig. 5. Smoluchowski trapdoor. that they could not. The operation of devices that sought to accumulate them, it was decided, would be disrupted fatally by fluctuations within their components. This view was elaborated in the 1910s by Marian Smoluchowski through many examples. One is well known. One might try to realize something close to the demon system Maxwell originally imagined by placing a spring-loaded trapdoor over a hole in a wall separating two gases, each initially at the same temperature and pressure. Molecular collisions can open the trapdoor in one direction only, as shown in Fig. 5, to allow molecules to pass from left to right only, creating a disequilibrium in pressure that violates the second law. The proposal fails because the trapdoor will itself carry thermal energy, fluctuating around the equipartition mean of kt/2 per degree of freedom. Since it must be light-weight enough for a collision with a single molecule to open it, its thermal energy will lead the trapdoor to flap about wildly and allow molecules to pass in both directions. (For a short survey, see Earman and Norton, 1998, pp ) 7.2. Reversible, isothermal expansion and contractions Fluctuations will disrupt processes in the inventory of Section 5.1. Consider the pair, Process 1a. Reversible isothermal expansion to double volume and Process 1b. Reversible isothermal compression to half volume. They are carried out by a piston connected by linkages to a weight. Work is extracted from or converted to the thermal energy of the molecule when its collisions with the piston lead it to raise or lower the weight. If the process is to approximate a reversible process, the mean force exerted by the molecular collisions must almost exactly be matched by the mean force of the piston and weight, so the system is as close as possible to a delicate equilibrium of forces. Now each component of the system will have its own mean thermal energy of kt/2 per degree of freedom, comparable in magnitude to the energy of the molecule. That includes the piston, the weight and each component of the linkages that connects them. As a result the entire system of molecule, piston and weight will be bouncing wildly to and fro. If the very slight disequilibrium of forces favors expansion and the raising of the weight, that expansion will not be realized stably. For fluctuation motions will be superimposed upon it, so that the statistical equilibrium state will consist of random motions through some portion of the cylinder. The same holds for compression. A simple but revealing implementation of the process arises when the cylinder, carrying the single molecule, is oriented vertically with the piston resting on the gas pressure, as shown in Fig. 6. The work of expansion and compression arises in the Fig. 6. A piston reversibly compressing a one-molecule gas. raising and lowering of the weight of the piston by the gas pressure. If the process is to be reversible, the piston must have just enough mass so that, absent fluctuations, the expansion or compression sought is only just favored by the balance of forces. 12 That means that we will have a light molecule repeatedly colliding with a very light piston and that very light piston will have thermal motions comparable to those of the molecule. As a result, the equilibrium state for the piston will consist of random motions spread over the cylinder. That thermal fluctuations in the piston s position will be of the size of the cylinder as a whole follows from a rough estimate of the size of the thermal fluctuations in the piston s position. Consider the piston as a thermal system at temperature T, independently of the single molecule of the gas. Its energy will be canonically distributed in conformity with the Boltzmann distribution. The distribution of its energy of height E¼Mgh, for 12 This condition will require adjustments to the mass of the piston as the one-molecule gas expands and contracts and the gas pressure changes. While machinery that adjusts the mass is possible, it will greatly complicate analysis. A better approach is to replace the gravitational field by another force field that pulls the piston down with a force 2kT/h, as developed in Section 7.5 below. We also presume that this force does not act upon the molecule, thereby avoiding the complication of a gravitationally induced pressure gradient.

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