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I explore the reduction of thermodynamics to statistical mechanics by treating the former as a control theory: A theory of which transitions between states can be induced on a system (assumed to obey some known underlying dynamics) by means of operations from a fixed list. I recover the results of standard thermodynamics in this framework on the assumption that the available operations do not include measurements which affect subsequent choices of operations. I then relax this assumption and use the framework to consider the vexed questions of Maxwell’s demon and Landauer’s principle. Throughout, I assume rather than prove the basic irreversibility features of statistical mechanics, taking care to distinguish them from the conceptually distinct assumptions of thermodynamics proper.

Thermodynamics is misnamed. The name implies that it stands alongside the panoply of other “X-dynamics” theories in physics: Classical dynamics, quantum dynamics, electrodynamics, hydrodynamics, chromodynamics and so forth [

Thermodynamics basically delivers on the state space part of the recipe: Its state space is the space of systems at equilibrium. But it is not in the business of telling us how those equilibrium states evolve if left to themselves, except in the trivial sense that they do not evolve at all: That is what equilibrium means, after all. When the states of thermodynamical systems change, it is because we do things

There is a general name for the study of how a system can be manipulated through external intervention:

This conception of thermodynamics is perfectly applicable to the theory understood phenomenologically: That is, without any consideration of its microphysical foundations. However, my purpose in this paper is instead to use the control-theory paradigm to explicate the relation between thermodynamics and statistical mechanics. That is: I will begin by assuming the main results of non-equilibrium statistical mechanics and then consider what forms of control theory they can underpin. In doing so I hope to clarify both the control-theory perspective itself and the reduction of thermodynamics to statistical mechanics, as well as providing some new ways to get insight into some puzzles in the literature: Notably, those surrounding Maxwell’s Demon and Landauer’s Principle.

In Sections 2 and 3, I review the core results of statistical mechanics (making no attempt to justify them). In Sections 4 and 5 I introduce the general idea of a control theory and describe two simple examples: Adiabatic manipulation of a system and the placing of systems in and out of thermal contact. In Sections 6–8, I apply these ideas to construct a general account of classical thermodynamics as a control theory, and demonstrate that a rather minimal form of thermodynamics possesses the full control strength of much more general theories; I also explicate the notion of a one-molecule gas from the control-theoretic (and statistical-mechanical) perspective). In the remainder of the paper, I extend the notion of control theory to include systems with feedback, and demonstrate in what senses this does and does not increase the scope of thermodynamics.

I develop the quantum and classical versions of the theory in parallel, and fairly deliberately flit between quantum and classical examples. When I use classical examples, in each case (I believe) the discussion transfers straightforwardly to the quantum case unless noted otherwise. The same is

Statistical mechanics, as I will understand it in this paper,

The _{I}_{I}^{N}

The

Given two systems, their

The

The _{I}

For any given system there is some time, the

Now, to be sure, it is controversial at best

There are a variety of responses to offer to this problem, among them:

Perhaps no system can be treated as isolated, and interaction with an external environment somehow makes the dynamics of any realistic system non-Hamiltonian.

Perhaps the probability distribution (or mixed state) needs to be understood not as a property of the physical system but as somehow tracking our ignorance about the system’s true state, and the increase in Gibbs entropy represents an increase in our level of ignorance.

Perhaps the true dynamics is not, after all, Hamiltonian, but incorporates some time-asymmetric correction.

My own preferred solution to the problem (and the one that I believe most naturally incorporates the insights of the “Boltzmannian” approach to statistical mechanics) is that the state

But from the point of view of understanding the reduction of thermodynamics to

The “state which maximises the Gibbs entropy” can be evaluated explicitly. If the initial state _{I}_{I}_{U}_{U}_{i}_{i}_{i}_{i}_{i}

We can define the _{i}_{i}

Now, suppose that the effective spread Δ_{0} is narrow enough that the Gibbs entropy can be accurately approximated simply as the logarithm of _{0}). States of this kind are called _{0}, so that _{0}))_{0})). A generalised equilibrium state can usefully be thought of as a statistical mixture of microcanonical distributions.

If _{I}_{I}

A microcanonical distribution is completely characterised (up to details of the precise energy width _{I}

The

At the risk of repetition, it is not (or should not be!) controversial that these probability distributions are empirically correct as regards predictions of measurements made on equilibrated systems, both in terms of statistical averages and of fluctuations around those averages. It is an important and urgent question

Given this understanding of statistical mechanics, we can proceed to the control theory of systems governed by it. We will develop several different control theories, but each will have the same general form, being specified by:

A

A set of

A set of

A set of

Our goal is to understand the range of transitions between states of the controlled object that can be induced. In this section and the next I develop two extremely basic control theories intended to serve as components for thermodynamics proper in Section 6.

The first such theory,

The controlled object is a statistical-mechanical system which is parameter-stable and initially at microcanonical equilibrium.

The control operations consist of (a) smooth modifications to the external parameters of the controlled object over some finite interval of time; (b) leaving the controlled object alone for a time long compared to its equilibration timescale.

There are no feedback measurements: The control operations are applied without any feedback as to the results of previous operations.

The control processes are sequences of control operations ending with a leave-alone operation.

Because of parameter stability, the end state is guaranteed to be not just at generalised equilibrium but at microcanonical equilibrium. The control processes therefore consist of moving the system’s state around in the space of microcanonical equilibrium states. Since for any value of the parameters the controlled object’s evolution is entropy-nondecreasing, one result is immediate: The only possible transitions are between states _{G}_{G}

To answer this, consider the following special control processes: A process is

A crucial feature of quasi-static processes is that the increase in Gibbs entropy in such a process is extremely small, tending to zero as the length of the process tends to infinity. To see this [_{eq}

The system’s dynamics is determined by some equation of the form
_{eq}

Now the rate of change of Gibbs entropy in such a process is given by
_{e}q_{eq}

But since _{eq}

(To see intuitively what is going on here, consider a very small change

To summarise:

In at least some cases, the result that quasi-static adiabatic processes are isentropic does not rely on any explicit equilibration assumption. To be specific: If the Hamiltonian has the form

In any case, we now have a complete solution to the control problem. By quasi-static processes we can move the controlled object’s state around arbitrarily on a given constant-entropy hypersurface; by applying a

A little terminology: The

Following the conventions of thermodynamics, we write _{J}_{I}_{I}^{I}δV

Our second control theory,

The result of this joint equilibration can be calculated explicitly. If two systems each confined to a narrow energy band are allowed to jointly equilibrate, the energies of one or other may end up spread across a wide range. For instance, if one system consists of a single atom initially with a definite energy

There is a well-known result that characterises systems that equilibrate with thermally stable systems which is worth rehearsing here. Suppose two systems have density-of-state functions _{1}, _{2} and are initially in microcanonical equilibrium with total energy _{1}, _{2} is then
_{1} is

Assuming that the second system is thermally stable, we express the second term on the right hand side in terms of its Gibbs entropy and expand to first order around

Since the partial derivative here is just the inverse of the microcanonical temperature

In any case, so long as we assume thermal stability then systems placed into thermal contact may be treated as remaining separately at equilibrium as they evolve towards a joint state of higher entropy.

We can now state thermal contact theory:

The controlled object is a fixed, finite collection of mutually isolated thermally stable statistical mechanical systems.

The available control operations are (i) placing two systems in thermal contact; (ii) breaking thermal contact between two systems; (iii) waiting for some period of time.

There are no feedback measurements.

The control processes are arbitrary sequences of control operations.

Given the previous discussion, thermal contact theory shares with adiabatic control theory the feature of inducing transitions between systems at equilibrium, and we can characterise the evolution of the systems during the control process entirely in terms of the energy flow between systems. The energy flow between two bodies in thermal contact is called

The quantitative

But since the thermodynamical temperature

So heat will flow from

If we define two systems as being in

Two systems each in thermal equilibrium with a third system are at thermal equilibrium with one another; hence, thermal equilibrium is an equivalence relation. (The

There exist real-valued

Returning to control theory, we can now see just what transitions can and cannot be achieved via thermal contact theory. Specifically, the only transitions that can be induced are the heating and cooling of systems, and a system can be heated only if there is another system available at a higher temperature. The exact range of transitions thus achievable will depend on the size of the systems (if I have bodies at temperatures 300 K and 400 K, I can induce

A useful extreme case involves

We are now in a position to do some non-trivial thermodynamics. In fact, we can consider two different thermodynamic theories that can thought of as two extremes. To be precise:

The controlled object is a fixed, finite collection of mutually isolated statistical mechanical systems, assumed to be both thermally and parameter stable.

The control operations are (i) arbitrary entropy-non-decreasing transition maps on the combined states of the system; (ii) leaving the systems alone for a time longer than the equilibration timescale of each system.

There are no feedback measurements.

The control processes are arbitrary sequences of control operations terminating in operation (ii) (that is, arbitrary sequences after which the systems are allowed to reach equilibrium).

The only constraints on this control theory are that control operations do not actually

By contrast, here is

The controlled object is a fixed, finite collection of mutually isolated statistical mechanical systems, assumed to be both thermally and parameter stable.

The control operations are (i) moving two systems into or out of thermal contact; (ii) making smooth changes in the parameters determining the Hamiltonians of one or more system over some finite interval of time; (iii) leaving the systems alone for a time longer than the equilibration timescale of each system.

There are no feedback measurements.

The control processes are arbitrary sequences of control operations terminating in operation (iii) (that is, arbitrary sequences after which the systems are allowed to reach equilibrium).

The control theory for maximal thermodynamics is straightforward. The theory induces transitions between equilibrium states; no such transition can decrease entropy; transitions are otherwise totally arbitrary. So we can induce a transition

To begin a demonstration, recall that in the previous sections we defined the

The reader will probably recognise this result as another form of the First Law of Thermodynamics. In this context, it is a fairly trivial result: Its content, insofar as it has any, is just that there is a useful decomposition of energy changes by their various causes. In phenomenological treatments of thermodynamics the First Law gets physical content via some independent understanding of what “work done” is (in the axiomatic treatment of [

The concept of a quasi-static transition also generalises from adiabatic control theory to minimal thermodynamics. If d

Putting our results so far together, we know that

Any given system can be induced to make any entropy-nondecreasing transition between states.

Any given system’s entropy may be reduced by allowing it to exchange heat with a system at a lower temperature, at the cost of increasing that system’s temperature by a greater amount.

The total entropy of the controlled object may not decrease.

The only remaining question is then: Which transitions between collections of systems that do not decrease the total entropy can be induced by a combination of (1) and (2)? So far as I know there is no

The ideal gas is an example of a Carnot system: Informally, it is clear that its temperature can be arbitrarily increased or decreased by adiabatically changing its volume. More formally, from its equation of state (

In any case, given a Carnot system we can transfer entropy between systems with arbitrarily little net entropy increase. For given two systems at temperatures _{A}_{B}_{A} > T_{B}_{A}_{B}

We then have a complete solution to the control problem for minimal thermodynamics: The possible transitions of the controlled object are exactly those which do not decrease the total entropy of all of the components. So “minimal” thermodynamics is, indeed, not actually that minimal.

The major loophole in all this—feedback—will be discussed from Section 9 onwards. Firstly, though, it will be useful to make a connection with the Second Law of Thermodynamics in its more phenomenological form.

While “the Second Law of Thermodynamics” is often read simply as synonymous with “entropy cannot decrease”, in phenomenal thermodynamics it has more directly empirical statements, each of which translates straightforwardly into our framework. Here’s the first:

_{A}_{B}

_{B}

_{A}

And the second:

In both cases the “leaving the states of all other systems unchanged” clause is crucial. It is trivial to move heat from system

Specifically, let’s define

The controlled object consists of (a) a collection of heat baths at various initial temperatures; (b) another finite collection of statistical-mechanical systems, the

The control operations are (a) moving one or more systems in the auxiliary object into or out of thermal contact with other auxiliary-object systems and/or with one or more heat baths; (b) applying any desired smooth change to the parameters of the systems in the auxiliary object over some finite period of time; (c) inducing one or more systems in the auxiliary object to evolve in an arbitrary entropy-nondecreasing way.

There are no feedback measurements.

A control process is an arbitrary sequence of control operations.

In this framework, a control process is

But perhaps we don’t care about

This offers no real improvement, though. In the Clausius case, any such heat flow is entropy-decreasing on the heat baths: Specifically, if they have temperatures _{A}_{B}_{A} > T_{B}_{A}_{B}

I pause to note that we can turn these entirely negative constraints on heat and work into quantitative limits in a familiar way by using our existing control theory results. (Here I largely recapitulate textbook thermodynamics.) Given two heat baths having temperatures _{A}_{B}_{A} > T_{B}_{A}

Adiabatically transition the Carnot system to the lower temperature _{B}

Place the Carnot system in thermal contact with the lower-temperature heat bath, and modify its parameters quasi-statically so as to cause heat to flow from the heat bath to the system. (That is, carry out modifications which if done adiabatically would decrease the system’s temperature.) Do so until heat _{B}

Adiabatically transition the Carnot system to temperature _{A}

Place the Carnot system in thermal contact with the higher-temperature heat bath, and return its parameters quasi-statically to their initial values.

At the end of this process the Carnot system has the same temperature and parameter values as at the beginning and so will be in the same equilibrium state; the process is therefore cyclic, and the entropy and energy of the Carnot system will be unchanged. But the entropy of the system is changed only by the heat flow in steps 2 and 4. If the heat flow out of the system in step 4 is _{A}_{B}/T_{B}_{A}/T_{A}_{A}/Q_{B}_{A}/T_{B}_{A}_{B}

Since the process consists entirely of quasi-static modifications of parameters (and the making and breaking of thermal contact), it can as readily be run in reverse, giving us the equally-familiar formula for the efficiency of a heat engine: _{B}/T_{A}

The Carnot systems used in our analysis so far have been assumed to be parameter-stable, thermally stable systems that can be treated via the microcanonical ensemble (and thus, in effect, to be macroscopically large). But in fact, this is an overly restrictive conception of a Carnot system, and it will be useful to relax it. All we require of such a system is that for any temperature

As I noted in Section 5, it is a standard result in statistical mechanics that a system of any size in equilibrium with a heat bath of temperature ^{−}^{U/T}

To get some insight into which systems are canonical Carnot systems, assume for simplicity that there is only one parameter

Then if the system begins in canonical equilibrium, its initial state is

By the adiabatic theorem, if

This will itself be in adiabatic form if we can find _{i}

For an ideal gas, elementary quantum mechanics tells us that the energy of a given mode is inversely proportional to the volume of the box in which the gas is confined: (Quick proof sketch: increasing the size of the box by a factor ^{2}. Energy is energy density

So an ideal gas is a canonical Carnot system. This result is independent of the number of particles in the gas and independent of any assumption that the gas spontaneously equilibrates. So in principle, even a gas with a single particle—the famous one-molecule gas introduced by [

For the rest of the paper, I will consider how the account developed is modified when feedback is introduced. The one-molecule gas was introduced into thermodynamics for just this purpose, and will function as a useful illustration.

What happens to the Gibbs entropy when a system with state _{i}

Then _{i}_{i}_{i}

The

But we have
_{i}

But the integral in the first term is just 1 (since the _{i}_{G}_{i}

That is, measurement may decrease entropy for two reasons. Firstly, pure chance may mean that the measurement happens to yield a post-measurement state with low Gibbs entropy. But even the

In the quantum case, the situation is slightly more complicated. We can represent the measurement by a collection of mutually orthogonal projectors Π̂_{i}

Insofar as “the Second Law of Thermodynamics” is taken just to mean “entropy never decreases”, then, measurement is a straightforward counter-example, as has been widely recognised (see, for instance, [

To be precise: Define

The controlled object consists of (a) a collection of heat baths at various initial temperatures; (b) another finite collection of statistical-mechanical systems, the

The control operations are (a) moving one or more systems in the auxiliary object into or out of thermal contact with other auxiliary-object systems and/or with one or more heat baths; (b) applying any desired smooth change to the parameters of the systems in the auxiliary object over some finite period of time; (c) inducing one or more systems in the auxiliary object to evolve in an arbitrary entropy-nondecreasing way.

Arbitrary feedback measurements may be made.

A control process is an arbitrary sequence of control operations.

In this framework, the auxiliary object can straightforwardly be induced (with high probability) to transition from equilibrium state _{G}_{G}_{i}_{i}

The expected value of the entropy of the post-measurement state will be much less than that of _{i}_{G}_{i}_{G}_{i}_{i}

As such, the scope of controlled transitions of the auxiliary object is total: It can be transitioned between any two states. As a corollary, the Clausius and Carnot versions of the Second Law do not apply to this control theory: energy can be arbitrarily transferred from one heat bath to another, or converted from a heat bath into work.

In fact, the full power of the arbitrary transformations available on the auxiliary system is not needed to produce these radical results. Following Szilard’s classic method, let us assume that the auxiliary system is a one-molecule gas confined to a cylindrical container by a movable piston at each end, so that the Hamiltonian of the gas is parametrised by the position of the pistons. Now suppose that the position of the gas atom can be measured. If it is found to be closer to one piston than the other, the second piston can rapidly be moved at zero energy cost to the mid-point between the two. As a result, the volume of the gas has been halved without any change in its internal energy (and so its entropy has been decreased by ln 2; cf

Now suppose we take a heat bath at temperature

To make this explicit, let’s define

The controlled object consists of (a) a collection of heat baths at various initial temperatures; (b) a one-atom gas as defined above.

The control operations are (a) moving the one-atom gas into or out of thermal contact with one or more heat baths; (b) applying any desired smooth change in the positions of the pistons confining the one-atom gas.

The only possible feedback measurement is a measurement of the position of the atom in the one-atom gas.

A control process is an arbitrary sequence of control operations.

Then the control operations available in Szilard theory include arbitrary cyclic transfers of heat between heat baths and conversion of heat into work.

The use of a

The most famous example of measurement-based entropy decrease, of course, is

Szilard control theory, and demonic control theory, allow thermodynamically forbidden transitions. Big deal, one might reasonably think: So does abracadabra control theory, where the allowed control operations include completely arbitrary shifts in a system’s state. We don’t care about abracadabra control theory because we have no reason to think that it is physically possible; we only have reason to care about entropy-decreasing control theories based on measurement if we have reason to think that

Of course, answering the general question of what is physically possible isn’t easy. Is it physically possible to build mile-long relativistic starships? The answer turns on rather detailed questions of material science and the like. But no

To answer that question, consider again heat-bath control theory. The action takes place mostly with respect to the auxiliary object: The heat baths are not manipulated in any way beyond moving into or out of contact with that object. We can then imagine treating the auxiliary object, and the control machinery, as a single larger system: We set the system going, and then simply allow it to run. It churns away, from time to time establishing or breaking physical contact with a heat bath or perhaps drawing on or topping up an external energy reservoir, and in due course completes the control process it was required to implement.

This imagined treatment of the system can be readily incorporated into our system: We can take the auxiliary object of heat-bath theory with feedback together with its controlling mechanisms, draw a box around both together, and treat the result as a single auxiliary object for a heat-bath theory

But we already know that heat bath theory without feedback does not permit any

This raises an interesting question. From the perspective of the controlling system, control theory with feedback looks like a reasonable idealisation, but from the external perspective, we know that something must go wrong with that idealisation. The resolution of this problem lies in the effects of the measurement process on the controlling system itself: The process of iterated measurement is radically indeterministic from the perspective of the controlling object, and it can have only a finite number of relevantly distinct states, so eventually it runs out of states to use.

This point (though controversial; cf [

The perspective we will discuss uses what might be called a _{0}, _{1} project onto the 0 and 1 subspaces and _{0} and _{1} are unitary operations on the controlled system. The combined process of _{0} on it if one result is obtained and _{1} if the other is. Measurements with 2^{N}

The problem with this process is that eventually, the system runs out of unused bits. (Note that the procedure described above only works if the bit is guaranteed to be in the 0 subspace initially. To operate repeatably, the system will then have to reset some bits to the initial state. But

Specifically, let’s define a

If the function ^{N}

But now suppose that the function ^{M}^{M} ×V^{N}^{−}^{M}

If the computer is to carry out the reset operation repeatably, its own entropy cannot increase without limit. So a

A more realistic feedback-based control theory, then, might incorporate Landauer’s Principle explicitly, as in the following (call it

The controlled object consists of (a) a collection of heat baths at various initial temperatures; (b) another finite collection of statistical-mechanical systems, the

The control operations are (a) moving one or more systems in the auxiliary object into or out of thermal contact with other auxiliary-object systems and/or with one or more heat baths; (b) applying any desired smooth change to the parameters of the systems in the auxiliary object over some finite period of time; (c) inducing one or more systems in the auxiliary object to evolve in an arbitrary entropy-nondecreasing way; (d) erasing

Arbitrary feedback measurements may be made (including the memory bits) provided that: (a) they have finitely many results; (b) the result of the measurement is faithfully recorded in the state of some collection of bits which initially each have probability 1 of being in the 0 state.

A control process is an arbitrary sequence of control operations.

At first sight, measurement in this framework is in the long run entropy-^{M}_{1}, . . . _{2}^{M} will reduce the entropy by Δ_{i} p_{i}_{i}_{i} p_{i}_{i}

This strategy of using Landauer’s principle to explain why Maxwell demons cannot repeatably violate the Second Law has a long history (see [

Responses to Earman and Norton (see, e. g. [

The results of my exploration of control theory can be summarised as follows:

In the absence of feedback, physically possible control processes are limited to inducing transitions that do not lower Gibbs entropy.

That limit can be reached with access to very minimal control resources: Specifically, a single Carnot system and the ability to adiabatically control and put it in thermal contact with other systems.

Introducing feedback allows arbitrary transitions.

If we try to model the feedback process as an internal dynamical process in a larger system, we find that feedback does not increase the power of the control process.

(3) and (4) can be reconciled by considering the physical changes to the controlling system during feedback processes. In particular, on a computation model of control and feedback, the entropy cost of resetting the memory used to record the result of measurement at least cancels out the entropy reduction induced by the measurement.

I will end with a more general moral. As a rule, and partly for pedagogical reasons, foundational discussions of thermal physics tend to begin with thermodynamics and continue to statistical mechanics. The task of recovering thermodynamics from successfully grounded statistical mechanics is generally not cleanly separated from the task of understanding statistical mechanics itself, and the distinctive requirements of thermodynamics blur into the general problem of understanding statistical-mechanical irreversibility. Conversely, foundational work on thermodynamics proper is often focussed on thermodynamics understood phenomenologically: A well-motivated and worthwhile pursuit, but not one that obviates the need to understand thermodynamics from a statistical-mechanical perspective.

The advantage of the control-theory way of seeing thermodynamics is that it permits a clean separation between the foundational problems of statistical mechanics itself and the reduction problem of grounding thermodynamics in statistical mechanics. I hope to have demonstrated: (a) These really are distinct problems, so that an understanding of (e.g.) why systems spontaneously approach equilibrium does not in itself suffice to give an understanding of thermodynamics; but also (b) that such an understanding, via the interpretation of thermodynamics as the control theory of statistical mechanics, can indeed be obtained, and can shed light on a number of extant problems at the statistical-mechanics/ thermodynamics boundary.

In writing this paper I benefitted greatly from conversations with David Albert, Harvey Brown, Wayne Myrvold, John Norton, Jos Uffink, and in particular Owen Maroney. I also wish to acknowledge comments from an anonymous referee.