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There seems to be a consensus among physicists that there is a connection between information processing and thermodynamics. In particular, Landauer’s Principle (LP) is widely assumed as part of the foundation of information theoretic/computational reasoning in diverse areas of physics including cosmology. It is also often appealed to in discussions about Maxwell’s demon and the status of the Second Law of Thermodynamics. However, LP has been challenged. In his 2005, Norton argued that LP has not been proved. LPSG offered a new proof of LP. Norton argued that the LPSG proof is unsound and Ladyman and Robertson defended it. However, Norton’s latest work also generalizes his critique to argue for a no go result that he purports to be the end of the thermodynamics of computation. Here we review the dialectic as it currently stands and consider Norton’s no go result.

A Maxwell Demon violates the Second Law of Thermodyanmics (SL). The form of the latter with which we will operate says that it is not possible to construct a system that will reliably produce a process whose sole effect is the conversion of heat into work (Kelvin version). The caveat about reliability is necessary since once the SL was thought about in the context of statistical mechanics it became widely accepted that such processes can occur, albeit very rarely (See for example [

Maxwell’s demon made his machine reliable by using its intelligence to manipulate micro features of the thermodynamic system so that they accumulated to violate SL. Hence, Maxwell’s Demon is often thought of as an information processing device. The Demon must gain information about the thermodynamical system and perform operations accordingly, and, if it is then to return to its initial state so as to perform a cyclic process the

In [

Here we review the dialectic as it currently stands and consider Norton’s no go result. In Section 1 we discuss what it means to say that a physical system implements a computation and review the analysis of implementation due to LPSG. In Section 2 we describe the structure of the LPSG proof of LP, in Section 3 we outline Norton’s critique of it and Ladyman’s and Robertson’s defence of it. In Section 4 we introduce Norton’s more general no-go result for the thermodynamics of computation, explicate his argument and raise some problems with it. In Section 6 we briefly offer some conclusions.

Precise formulation and assessment of LP requires an account of what it means to say that physical processes, which are subject to the laws of thermodynamics, implement computations or logical transformations which are abstract mathematical entities. (We shall from now on speak of “computation” but we have in mind a general notion that would include paradigmatic computations like finding prime factors or sorting lists as well as logical operations such as AND or COPY). In this section we summarize the model of implementation that LPSG developed and applied in their proof of LP. However, the following features must belong to any adequate model of implementation for the reasons given in each case.

Implementation means that a physical system is taken to represent abstract mathematical or logical states. There is a representation relation between physical degrees of freedom and the input and output states of the computation. The same physical degrees of freedom may or may not represent both input and output logical states. The representation relation is either a matter of stipulation and thus completely anthropocentric or it is natural and independent of our choices. The analysis of implementation below is neutral about this matter but in at least some cases it seems completely unnatural and conventional. (Whether or not information is always in the eye of the beholder is a matter of controversy. There are those who would regard everything including the universe itself as a computer, and others who merely claim that there is natural, in the sense of non-anthropocentric, computation in some physical systems such as DNA and associated molecules in cells.)

Computations cannot be implemented by single processes but only by families of them. This is because computations have multiple inputs and outputs, but physical processes are particular sequences of physical states. For example, RESET is the map that takes 0 to 0 and 1 to 0, but any given physical process can only instantiate one or other of the two possible initial to final state transformations. Hence to say that a physical system implements a computation requires not only that its initial and final states are those associated with the input and output states of the computation on some particular occasion, but also that had the initial state been that corresponding to one of the other input states of the computation, the resultant physical final state would have been that associated with the appropriate output state of the computation. In this sense implementation, like information, is modal (See [

The physical system must be such that it can be prepared in any of the physical states that represent the logical states, and there must be a time evolution that can be set in motion and which will bring about the physical process that brings about the appropriate final physical state whatever the given initial state. This amounts to the insistence that everything involved in the computation must be included in the operation of the device, ruling out an agent choosing what to do with the device depending on which initial state it is in. LPSG argued that it is trivial that LP is false without this requirement. Norton’s schemes to violate LP all involve such an external agent (see [

In the case of RESET, the physical system must not have another copy of the input state and use it to perform a different choice of operation. If this is done that the system is not implementing a logically irreversible computation it is effectively implementing UNCOPY (which might be loosely called the erasure of known data).

LPSG’s model of implementation is in terms of “_{L}_{L}_{in}_{out}

It is important to bear in mind this model or some alternative conceptual structure satisfying the above four features when discussing physical implementations of RESET in the context of LP. It is not sufficient just to imagine some particular physical process that could be representing a particular part of a computation such as, for example, 1 → 0.

LPSG offered a new proof of LP based on the construction of a thermodynamic cycle, arguing that if there was a machine that violated LP it would be possible to harness it to produce a violation of the second law of thermodynamics (SL). LPSG thus argued: If not LP then not SL; SL therefore LP. (LPSG thus follow the sound horn of the dilemma posed by Earman and Norton 1999 according to which LP can at most be used to illustrate why certain Maxwell demon’s don’t work but, contrary to the profound horn, cannot be used to demonstrate that there are no Maxwell demons. Bennett 2003 conceded that the profound horn of the dilemma cannot be defended.)

LPSG constructed their thermodynamic cycle using an

LPSG presupposed standard thermodynamics which they in no way sought to revise, though they did use the Kelvin form of SL in a weakened form that prohibits cycles that reliably produce the sole effect of converting heat into work as mentioned above.

In [

The processes LPSG use can be used to construct a counterexample to LP,

It is possible to use LPSG’s processes to violate SL so LPSG use inconsistent assumptions.

LPSG (and LPS [

We will not discuss this final objection here but will focus on the disagreement about the physical processes that LPSG use. LPSG’s processes are part of “a standard repertoire of idealized processes from the literature in the thermodynamics of computation” [

Ladyman and Robertson [

Norton’s counterexample to LP uses a process that is not used in LPSG’s proof and furthermore the process in question is not admissible.

The processes LPSG allow do not violate SL.

Norton illicitly allows a controlled operation of a degree of freedom onto itself [

Norton only considers individual physical processes in his analysis but as argued above, implementation requires consideration of a system with some means of ensuring the time evolution that corresponds to a family of physical processes.

According to Norton, the processes LPSG use, which, as stated above, are standard in the literature, allow for violations of SL and LP. However, the processes Norton uses are importantly different from those LPSG use. For example, one of the processes Norton uses is the insertion of a partition into a one-molecule gas followed by measurement of which side of the partition the one-molecule gas is on, where the result is then used to control an operation that inserts a piston on the relevant side, and the partition removed and the isothermal expansion of the gas to its initial volume. This would be a cycle whose sole effect is the conversion of heat (from the heat bath) to work (moving the piston). Another of Norton’s suggestions is to perform ‘dissipationless erasure’ by removing the partition and re-inserting it. If the molecule is then on the side of the partition that represents 0 leave it, if not repeat until it is.

In these cases, Norton fails to consider the degree of freedom that must be used to control the operation on the gas to extract work and perform RESET respectively. Norton’s schemes require that a controlled operation can be performed from a degree of freedom onto itself (see Section 4.1 in [

Norton also objects to the idea that the insertion and removal of a partition in a one-molecule gas does not change the thermodynamic entropy of the system. This raises subtle issues concerning the connection between information-theoretic and thermodynamic entropies, however, Norton’s objections to the processes LPSG use is much more far-reaching and fundamental. Clearly, the soundness of the LPSG proof depends on the admissibility of the processes they use. While concerns may be raised about dissipationless measurement and unspecified means of controlling operations, the idea that at least some processes such as the isothermal compression and expansion of a one-molecule gas are thermodynamically reversible is orthodox. This is the latest focus of Norton’s assault on the thermodynamics of computation as he now argues that all such microscopic thermodynamically reversible processes are physically unrealizable even in principle. In the next section we explain and assess Norton’s “no go result”.

In his most recent work, Norton [

The thermodynamically irreversible processes that Norton says cannot occur are often called “quasi-static”, implying that they can happen arbitrarily slowly. Not all arbitrarily slow processes are thermodynamically irreversible (as Norton forthcoming b points out). In the kind of processes commonly discussed in the literature in general, and used in the LPSG proof of LP in particular, the point of considering the processes as occurring over long periods of time is to allow the systems undergoing them to stay virtually in thermal equilibrium with a heat bath [

In the limit of infinite time the system is idealized as being perfectly thermodynamically reversible. Examples include the isothermal expansion and contraction of gases. Norton says of such processes that because they are at “perfect equilibrium” [

According to Norton, if such a process was to proceed, and if its final state is not intrinsically more probable than its initial state, then a difference in the free energy between the two states must be introduced to make it more probable. Heat flow between states in thermal equilibrium requires a disequilibrium between them, and Norton argues that, at the microscopic level, this cannot be idealized away, and hence the conclusion of the theorem is that “fluctuations obliterate thermodynamically reversible processes” [

When considering how microscopic reversible processes proceed Norton says “[e]ach computational step is carried out by a thermodynamically reversible process, whose stages are parameterized by _{0} (at equilibrium the mean value is the equilibrium value). Tolman considers the case of the mean value _{0} of the volume _{lim}_{lim}_{lim}_{lim}

where _{n}

which can be re-expressed as

As Tolman says, this formula is not precise, because

_{0} is large) will be exponentially suppressed, as when _{0} is large, _{0}) is very large compared to _{0}) is a minimum. Obviously the most probable fluctuations are those close to the mean, and they contribute most to _{0}) is a minimum is a gaussian centered around the mean value _{0}.

Processes proceed spontaneously by fluctuations so if a process involves an evolution from stage 1 to stage 2 then stage 2 must be more probable than stage 1. Therefore the ratio in _{0} of the section above is relabeled by Norton as stage “_{1}” and the fluctuation value _{2}”.

The probability densities for the system fluctuating between stages _{1} and _{2} are:

(This is the same as

“The process is thermodynamically reversible, hence it is in equilibrium at every stage. Equilibrium requires the vanishing of the generalized thermodynamic force

Integrating over

Therefore, from

Since this holds for all

The crucial point of Norton’s analysis is that the probability of the system being in the final state is the same as that of it being in the initial state. Hence the “process” fails reliably to take the system from the initial state to the final state; indeed the system fluctuates through all stages of the “process” with equal probability. Norton says, “if we try to set up the process in its initial stage, it is as likely to remain there as to fluctuate to any intermediate stage or the final stage. If the process has arrived at its final stage, it is as likely to remain there as to fluctuate back to any earlier stage” [

Hence the conclusion quoted above that “[f]luctuations obliterate thermodynamically reversible processes” [

It is important to note that it has not been assumed thus far that the system in question is microscopic. Given that Norton does not want to deny the validity of phenomenological thermodynamics, or at least its approximate validity, it must be possible for the system’s evolution ‘to overcome fluctuations’, in the sense that the final state can be reliably arrived at from the initial state. However, in his analysis this happens by the introduction of a difference in the free energy of the two states so that the ratio _{2}_{1} increases to the extent that the system will surely evolve from state 1 to state 2, though the associated increase in entropy between the states is negligible at the macroscopic level allowing us to maintain the approximate truth of the claim that the process is thermodynamically reversible. Yet, argues Norton, at the microscopic level this entropy cannot be ignored because it is much greater than the Landauer bound.

The example that he gives is as follows [_{2})_{1}) = 7.2 × 10^{10}. As 25 kT is the thermal energy of 10 oxygen molecules it is negligible for macroscopic processes.

We offer a number of objections to the no-go result below.

Even if it is appropriate to treat the processes in question as proceeding by fluctuations (which we argue below it is not), it is inconsistent to say both that the final state _{2} has arisen by a fluctuation (premise 1) and that it is an equilibrium state

If fluctuations to any of the “stages” were equally likely, then the probability distribution would be a constant function. This is not compatible with the results of the Einstein-Tolman method because the latter assumes that the free energy is a minimum at equilibrium from which it follows that the probability distribution of the fluctuations will be a Gaussian centered on the mean [

The no go result applies to all systems not just molecular scale processes. If Norton is right the only way to recover macroscopic thermodynamically reversible processes is to introduce an entropy increase that cannot be idealized away as standard thermodynamics demands. Consequently, whilst the target of the no-go result is the thermodynamics of computation, if successful it actually undermines the orthodox way of thinking about thermodynamical process. Whilst this consequence might be palatable for Norton, it is important to note that this is the price of his result. In particular, if the only way to overcome the fluctuations is to make a sufficient increase in the free energy then we cannot take the limit of infinite time discussed above in which the process is supposed to be thermodynamically reversible. Rather than show that reversible processes cannot proceed at the microscale without violating the Landauer bound, Norton’s result is unrestricted. If the argument is sound it shows that all supposedly thermodynamically reversible processes cannot proceed without

The Einstein-Tolman method is for fluctuations. Processes do not proceed by fluctuations. In the next subsection we argue that premise 1 should be rejected.

When Norton [

In Norton’s model the transition probability for the process is determined solely by the free energies of the different stages and therefore synchronically. But if we are considering, say, isothermal compression and isothermal expansion the same two states must be passed through in opposite temporal directions. If the processes were driven solely by properties of each state, then we would either except to only see isothermal expansion or isothermal compression. Furthermore, there would be no way to control the rate at which processes happened which undermines the idea of reversible/quasi-static processes explained above.

The processes in question are not supposed to happen spontaneously but are driven by the dynamics of the computational device. Processes do not proceed only by fluctuations but by fluctuations together with some driving potential/under the operation of a Hamiltonian [

In sum, for the reasons given above we argue that Norton’s way of modeling the processes implementing computations as a series of fluctuations from equilibrium is unjustified. Norton may be right that fluctuations prevent thermodynamically reversible processes occurring at the microscopic scale but we have argued that the above argument for the no go result is not sound.

Even if the only way we can prove LP is by assuming SL, using the former to show that a Maxwell Demon is impossible is still of interest as it shows the overall coherence of physics and information theory. In any case, LP is the formal foundation of the thermodynamics of computation and it is important to consider its precise formulation and justification. Norton [

Behind the no-go result is a more general worry about the illegitimacy of neglecting fluctuations when describing microscopic processes with thermodynamics. We have argued that the no go result is not established, but we have not sought to dismiss Norton’s concerns reasoning with one-molecule gases and other microscopic systems that assumes they can undergo thermodynamically reversible processes.

We are very grateful to John Norton for discussion.

The authors declare no conflict of interest.

A controlled operation of a degree of freedom onto itself is one where the form of the operation depends on the state of the degree of freedom and nothing else. As Ladyman and Robertson argue, it is trivial that LP is false if such operations are admitted, and controlled operations such as CNOT require a control system and a target system that consist of different physical degrees of freedom so that the state of the former can vary independently of the state of the latter prior to the operation being implemented.

Norton’s arguments for the no-go result in this volume do not make reference to the Einstein-Tolman method. Some but not all of our criticisms here apply only to his argument in his 2011 and forthcoming a. We do not consider Norton forthcoming b here.

Hence, sometimes “quasi-static” is used to refer to processes that involve a sequence of states each of which is almost at equilibrium (see for example [

P(being in a state compatible with the control)= P(fluctuating to the state at the control value) is approximately valid when the control value is near the mean value.

In any computational device the physical states that represent computational states are likely to be coarse-grained with respect to the underlying fundamental physical state. For example, in a standard logic gate the computational states are represented by potential differences, and all that matters is whether they are either approximately 0

Berut