aa r X i v : . [ phy s i c s . h i s t - ph ] J u l The Science of Θ∆ cs Wayne C. MyrvoldDepartment of PhilosophyThe University of Western [email protected] 24, 2020
Abstract
There is a long tradition of thinking of thermodynamics, not asa theory of fundamental physics (or even a candidate theory of fun-damental physics), but as a theory of how manipulations of a physi-cal system may be used to obtain desired effects, such as mechanicalwork. On this view, the basic concepts of thermodynamics, heat andwork, and with them, the concept of entropy, are relative to a classof envisaged manipulations. This view has been dismissed by manyphilosophers of physics, in my opinion too hastily. This paper is asketch and defense of a science of manipulations and their effects onphysical systems. This is, I claim, the best way to make sense of ther-modynamics as it is found in textbooks and as it is practiced. I callthis science thermo-dynamics (with hyphen), or Θ∆ cs , for short, tohighlight that it may be different from the science of thermodynamics,as the reader conceives it. Even if one is not convinced that it is thebest way to make sense of thermodynamics as it is practiced, it shouldbe non-controversial that Θ∆ cs is a legitimate science. An upshot ofthe discussion is a clarification of the roles of the Gibbs and von Neu-mann entropies. Given the definition of statistical thermo-dynamicentropy, it can be proven that, under the assumption of availability ofthermodynamically reversible processes, these functions are the unique(up to an additive constant) functions that represent thermo-dynamicentropy. Light is also shed on the use of coarse-grained entropies. ontents cs . . . . . . 36 Introduction
In what follows I will tell you about a science that I call thermo-dynamics . Following the word of the Lord, when he first bestowedthat word upon us, I retain the hyphen, to emphasize the etymologyof the word: it is formed from the Greek works for heat and power . Following the word of the Laird, I will often abbreviate it as Θ∆ cs (tobe pronounced “thermo-dynamics”), which also emphasizes its Greekroots. The reason I emphasize the etymology is that the science ofthermo-dynamics has at its core a distinction between two modes ofenergy transfer between physical systems: as heat, and as work.The concepts of Θ∆ cs are, I claim, the best way to make sense ofmost of what is called “thermodynamics” in the textbooks, thoughthat content is often obscured in the presentation. Be warned, how-ever: the scope of Θ∆ cs is narrower than thermodynamics as it issometimes conceived. The scope of Θ∆ cs includes the zeroth, first,second, and (in the quantum context) third laws of thermodynamics,all of which were designated laws of thermodynamics by 1914 at thelatest. It does not include a relative late-comer to the family of lawsof thermodynamics, which Brown and Uffink (2001) have dubbed the Minus First Law , which, though it had long been identified as an im-portant principle, was not called by anyone a law of thermodynamicsprior to the 1960s. A thermo-dynamic theory involves treating certain variables as ma-nipulable , in a sense that I will explain in the next section, and has todo with the responses of physical systems to manipulations of those The word’s first appearance is in Part VI of Kelvin’s “On the Dynamical Theory ofHeat” (Thomson 1857, read before the Royal Society of Edinburgh on May 1, 1854). Therehe recapitulates what in 1853 he had called the “Fundamental Principles in the Theory ofthe Motive Power of Heat,” now re-labelled “Fundamental Principles of General Thermo-dynamics.” Maxwell used this and related abbreviations in his correspondence with P. G. Tait.See letter to Tait of Dec. 1, 1873, in Harman (1995, p. 947). The minus first law, which Brown and Uffink also call the
Equilibrium Principle , isgiven by them as,An isolated system in an arbitrary initial state within a finite fixed volume willspontaneously attain a unique state of equilibrium (Brown and Uffink, 2001,p. 528).They point out that a principle of this sort had been recognized as a law of thermodynamicsearlier, by Uhlenbeck and Ford (1963, p. 5). ariables. A designation of certain variables as manipulable is notsomething that appears in, or supervenes on, fundamental physics; itmust be added. For this reason, Θ∆ cs is not and cannot be a compre-hensive or fundamental physical theory. It is nonetheless a perfectlyrespectable theory, a useful one, and, for beings such as us, who are nottranscendent intellects beholding the cosmos from outside but ratheragents embedded in the world and interacting with it, perhaps evenan indispensable theory. Confusion arises when it is mistaken for thesort of theory that could possibly be a fundamental one. Indeed, I willargue that some of the various puzzles and paradoxes that have arisenfrom thermodynamics stem from confusing Θ∆ cs with fundamentalphysics.The idea that thermodynamics should be thought of as a theory ofthis sort is not new; see Appendix for a sampling of quotations fromthe history of Θ∆ cs . A conception of thermodynamics along these linesis rapidly becoming the mainstream view among workers in quantumthermodynamics, who view it as a species of resource theory , akinto quantum information theory (see Goura et al. 2015; Goold et al.2016; Ng and Woods 2018; Lostaglio 2019). This development has sofar attracted little attention from philosophers; a notable exception isWallace (2016).I start by outlining the basic concepts needed to formulate Θ∆ cs . In this section I highlight some routine features of scientific practicethat are so ubiquitous that for the most part we don’t really thinkabout them, and they are passed over without comment. It is worth-while, however, to get clear about what’s going on.Consider a sort of problem that is frequently found in textbooksand in the scientific literature. One is asked to consider a systemsubjected to an external force, or to an external potential, and to cal-culate certain aspects of its behaviour ( e.g. to solve the equations ofmotion, or to find the energy eigenvalues), subject to that externalpotential. The Hamiltonian for such a system consists of its internalHamiltonian, which includes the kinetic energies of its parts and termsinvolving interactions, if any, between its parts, plus the external po-tential. H = H int + V ext . (1) or the purposes of a problem like that, nothing needs to be said aboutthe source of the external potential. It is treated as given . Presumably,the external potential is an interaction potential between the systemin question and some other system, but we are not asked to includethat system in our calculations, and, in particular, we do not considerthe effect of the system under consideration on the system that is thesource of the external potential . This is what it means to treat theexternal potential as given.I will call variables treated in this way exogenous variables . Notethat designation of a variable as exogenous has to do with how it ishandled in a given investigation; the distinction between exogenousand other variables is not intrinsic to the physical nature of thosevariables. The phrase “exogenous variable” should be taken as shortfor “variable treated exogenously.” Were it not for the awkwardnessof language that would ensue, I would eschew adjectival uses of “ex-ogenous” in favour of adverbial.The same variable may be treated exogenously in one investigation,and included as part of the system under consideration in another. Anan example consider the usual pedagogical entry into celestial mechan-ics. First one treats of a body in a fixed external 1 /r potential, andshows that its trajectories take the form of conic sections (ellipses,parabolas, or hyperbolas, depending on the energy), subject to thearea law with respect to the origin of coordinates. This yields a re-spectable first approximation to planetary motion, as the gravitationaleffect of the sun dominates the net force on any planet, and, to a firstapproximation, the effect of the planet on the sun is negligible, and wemay treat the sun as fixed. The next step on the journey to celestialmechanics is from the one-body to the two-body problem, in whichthe sun’s position is treated as dynamical variable.We find this distinction between exogenous and other variablesalso in computer modelling of physical systems. Consider climatemodels. Various aspects of the earth’s climate system are treated insuch a model, and their behaviour subjected to the dynamics writteninto the model. Some variables, such as solar radiation and green-house gas emissions from volcanoes and from anthropogenic sources,are treated as inputs. No attempt is made to include solar dynamicsor the geophysics of volcano eruptions in the dynamics of the model.When dealing with exogenous variables, there is often a range ofpossible values to be considered, and we may be interested in the dif-ferences that changes to the exogenous variable make to the behaviour f the system at hand. Crucially, we treat the exogenous variables asones that can vary independently of the states of the systems underconsideration—that is, they are treated as free variables . This is acrucial aspect of controlled experiments. The systems under consid-eration are subjected to a range of treatments, and a well-designedexperiment is one in which the treatments may be regarded as vary-ing independently of the initial states of the systems to be studied.Often some randomizing device is employed, which is thought of asrendering its outputs effectively free for the purpose at hand . I will say that a variable is being treated as a manipulable vari-able in a given theoretical investigation if (i) it is being treated asexogenous variable, and (ii) there is a range of possible values, or,perhaps, of possible alternative temporal evolutions of that variable,under consideration.One might be tempted to say that treatment of certain variablesas exogenous is a concession to our limited calculational and compu-tational abilities. It might be better, one might think, to include inour climate models solar dynamics, the dynamics of volcanoes, anda sufficiently detailed model of human activities that anthropogenicemissions could be included among the modelled variables. This wouldbe a mistake. For certain purposes, it is essential to treat certain vari-ables as manipulable. These purposes include attribution studies. Touse climate models to estimate the contribution of various inputs toobserved global warming, researchers vary those inputs while holdingothers fixed. It is investigations such as these that, in part, underwriteconclusions that most, all, or perhaps more than all of the observedwarming can be attributed to anthropogenic greenhouse gases. And,of course, this is relevant to policy decisions (or would be, if anyonewere making informed policy decisions); one can make projections bymodelling future climate under a variety of emissions scenarios.All of this is, of course, meant to be consistent with the concept ofmanipulability as it appears in the causal modelling literature (Pearl,2000; Woodward, 2003, 2017).When speaking of manipulable variables, and a set of alternativemanipulations, one almost inevitably begins to talk of choices of ma-nipulations. This carries with it a suggestion that human agency iscentral to the concept, which in turn raises the suspicion that sub-jectivity is being brought in. This is not the case; a variable treated Borrowing the apt phrase of Bell (1977). s manipulable need not be manipulable by us (see above, re vol-canoes). Nevertheless, some who have developed a conception veryclose to what I am calling Θ∆ cs have lapsed into talk that suggeststhat its concepts are subjective. This is an error, in my view. Itstems, I think, from overextension of the familiar subjective/objectivedichotomy. Objective features of a physical system are supposed tobelong to that system, in and of itself; they are features that cannotchange without change of its physical state. The concepts of Θ∆ cs arerelative to a specification of manipulable variables and a set of alter-native manipulations of those variables, and as such are not there inthe physical states of things. It does not follow that they are sub-jective, although, if all one had at hand was the objective/subjectivedichotomy, it is understandable that one might lapse into saying thatthey are. An equilibrium thermo-dynamic state of a system A may be specifiedby its total internal energy E and the values of one or more manipula-ble variables λ = { λ , . . . , λ n } . As a running example, you can thinkof a gas confined to a container with a moveable piston, whose wallsare represented as an external potential that strongly repels mole-cules that get too close. We consider a family of such potentials,corresponding to different positions of the piston.It is often assumed that, besides changes to the variables λ , thereare other manipulations that may be performed. For example, thesystem A may be coupled to other systems regarded as heat reservoirsat various temperatures. This coupling may be applied or removed;that is, the interaction Hamiltonian between A and the heat reservoiris being treated as a manipulable variable. A heat reservoir is a systemwith which is associated a definite temperature, from which no work isextracted and on which no work is done; its only exchanges of energywith other systems are as heat. What it means to count a systemas a heat reservoir at a given temperature will be discussed a bitmore in the next section. Often, one imagines heat reservoirs availablefor arbitrary temperatures. But one can also consider the thermo-dynamic theory of an adiabatically isolated system, or a theory onwhich there is access to only one heat reservoir, or some other limitedset. orresponding to any manipulation is a transformation of the stateof the system. A small change dλ i in one of the manipulable variables,with the others held fixed, and no heat exchange, may result in achange dE in the internal energy of the system. We define, A i ( E, λ ) = ∂E∂λ i , (2)where it is understood that the other variables are held fixed, andthere is no exchange of energy with any heat reservoir or anythingelse. In standard thermodynamics, the quantities A i are usually as-sumed to have steady, time-independent values. We can take thiscondition (which will be modified in section 5) as a criterion of ther-mal equilibrium of the system. In any process involving a small changein the variables λ , we define work done on the system as d ¯ W = X i A i dλ i . (3)The convention in play is that work done on the system, increasingits energy, counts as positive. If the only other changes to the internalenergy of the system A are due to interactions with heat reservoirs,we have a neat partitioning of any change in the energy of A into awork component and a heat component. Changes in energy of A dueto changes in the manipulable variables counts as work; exchanges ofenergy with heat reservoirs, as heat. As with work, we count heattransfer into the system A as positive.A thermo-dynamic theory consists of a system A , a class of Hamil-tonians H λ that depend on manipulable variables λ , and a set M of possible manipulations of those variables. The class might includemanipulations that go beyond what can feasibly be achieved by us; wecan very well consider how a system would react to more fine-grainedmanipulations than we can achieve, or to manipulations that proceedso slowly that we would not have the patience to see them through.What one needs to know about the effects of these manipulations isgiven by the dependence of the generalized forces A i on the values ofthe parameters ( E, λ ) specifying the state. The structure of the setof manipulations may vary from theory to theory. One thing that Iwill assume in what follows is that manipulations can be composed:if there is a manipulation that takes a state a to a state b , and a ma-nipulation that takes a state b to a state c , these manipulations canbe performed in succession, forming a manipulation that takes state a to b and then to c . e will not be assuming that thermodynamically reversible pro-cesses, or even processes that approximate thermodynamic reversibil-ity arbitrarily closely, are always available. Dropping the assumptionof the availability of reversible processes requires revision of the fa-miliar framework of thermodynamics, as it means dropping the as-sumption of the availability of an entropy function. In its place wewill define quantities S M ( a → b ), defined relative to a class of avail-able manipulations M , to be thought of as analogues, in the currentcontext, of entropy difference between states a and b . These will berepresentable as differences in the values of some state function only in the limiting case in which all states can be connected reversibly.For any two thermo-dynamic states a , b , let M ( a → b ) be the setof manipulations in M that lead from a to b . These may involve heatexchanges with one or more heat reservoirs { B i } with temperatures T i . For any manipulation M in M ( a → b ), let Q i ( a → b ) M be the heattransferred over the course of M into A from the reservoir B i (positiveif there is energy flow from B i to A , negative if there is energy flowthe other way). We define, σ M ( a → b ) = X i Q i ( a → b ) M T i . (4)We define, as analogues of entropies (which we will henceforth justcall “entropies”), S M ( a → b ) = l.u.b. { σ M ( a → b ) | M ∈ M ( a → b ) } . (5)where “l.u.b.” stands for “least upper bound,” that is, the smallestreal number that is at least as large as all members of the set. Via theobvious extension of this definition we also define quantities such as S M ( a → b → c ) for processes with any number of intermediate steps.It follows from the assumption about composition of manipulationsand the definition of the entropies that S M ( a → b → c ) = S M ( a → b ) + S M ( b → c ) , (6) The word, appropriately, is formed from the Greek ἡ τρωπὴ , transformation, for whatClausius called the transformational content ( Verwandlungsinhalt ) of a body (Clausius,1865, p. 390). If b cannot be reached from a via any manipulation in M , or if the set considered hasno upper bound, S M ( a → b ) is undefined. To avoid qualifying every formula involvingentropies with a proviso that all quantities mentioned therein are defined, we can, if welike, allow S M ( a → b ) to take values in the extended reals, which supplement the realswith ±∞ . Then, if b cannot be reached from a , S M ( a → b ) = −∞ . nd similarly for processes with longer chains of intermediate states.One version of the second law of thermodynamics says that, if asystem undergoes a cyclic transformation, returning it to its originalstate, the sum of Q/T over all heat exchanges in the process cannotbe positive. We can write this as:
The second law of thermo-dynamics . For any state a , S M ( a → a ) ≤ . It follows from the second law that, for any states a , b , S M ( a → b → a ) ≤ , (7)and similarly for cycles consisting of larger numbers of states.By the second law, S M ( a → b → a ) cannot exceed zero. If it isequal to zero, then there is no harm in adding to the list of possiblemanipulations a fictitious reversible process that can be run in eitherdirection, from a to b , or, with signs of heat exchanges reversed, from b to a . We don’t expect any actual process to satisfy this condition;as John Norton (2016) has emphasized, any process will involve somedissipation of energy, and fail to be completely reversible. If one tooktalk of reversible processes too literally, one would end up ascribingabsurd properties to them; they would be processes that take placeinfinitely slowly and yet somehow manage to get completed. Nortonargues that talk of reversible processes should be regarded as short-hand for talk of limiting properties of sets of actual processes. Ourdefinition of entropy makes this explicit.In what follows, take the statement that a and b can be reversiblyconnected as no more than a convenient way of saying that S M ( a → b → a ) is equal to 0. On the macroscopic scale, it may be the casethat, for all a , b , S M ( a → b → a ) is close enough to zero that we canneglect the fact that it is not exactly zero. In standard thermody-namics, which is usually meant to apply at the macroscopic scale, it isnormally assumed that any two states of a system can be connectedby a reversible process. If this holds—that is, if, for all states a , b , S M ( a → b → a ) = 0—it follows from the second law that there isa function S on the set of thermodynamic states, defined up to anadditive constant, such that S M ( a → b ) = S M ( b ) − S M ( a ) . (8) Proof left as an exercise for the reader.) If, however, we want to pushΘ∆ cs down to the nanoscale, on which departures from reversibilityare non-negligible, we need not assume this.Call a transformation from a thermo-dynamic state a to a state b adiabatic if no exchanges of energy occur that are not due to manipu-lation of the variables λ ; no heat is exchanged with any heat reservoir.The following is a simple consequence of the definition of entropy. Proposition 1.
If there is a manipulation that takes state a to state b adiabatically, then, for any state c , S M ( b → c ) ≤ S M ( a → c ) and S M ( c → b ) ≥ S M ( c → a ) . In the special case in which all states are reversibly connectable,this says that an adiabatic transformation cannot lower the entropyof a state.It’s a consequence of all this that, given a physical system A , theremay be several thermo-dynamical theories of that system A , depend-ing on the specification of manipulable variables, and on the set M of possible manipulations. This means that a pair of physical states a , b of the system might be assigned different values of the entropy S M ( a → b ) by different thermo-dynamic theories. This will be illus-trated in the next section. If one thought that the entropy differenceof a pair of states of a system was supposed to be a property of thosephysical states alone, this might seem paradoxical. In the context ofΘ∆ cs , there’s nothing paradoxical about it at all.Once the set M of possible manipulations is chosen, how the sys-tem reacts to those manipulations is a matter of physics. These reac-tions are encoded in the equilibrium values of the generalized forces A i ,defined by (2). It is these that determine the dependence of entropies S M ( a → b ) on the values of the manipulable variables. One may say:we may choose the variables to manipulate, but nature chooses the re-sponse to those manipulations. It would be mistake to say that a viewof this sort makes entropy subjective. Entropy remains a measurablequantity, but what quantity it is that is being measured is determinedby the choice of manipulable variables.What we have presented in this section is almost the same as whatis found in typical thermodynamic textbooks. Almost. It is univer-sally agreed that thermodynamic states are defined relative to someselection of a set of variables that is small, compared to the full set ofvariables needed to specify the precise physical state of a system. Thedifference is that these variables are often described as the macroscopic ariables, the ones whose values can be obtained via a macroscopicmeasurement.What to say about this? First: though this is not always explic-itly said, if one reads any textbook of thermodynamics closely enough,one will find that the extensive variables that define an equilibriumstate are invariably treated as manipulable variables, in the sense dis-cussed in the previous section. Sometimes they are called externalvariables . Second: it should be stressed that the selected variables arenot properties of the system to be studied, but of external constraintsplaced on the system. For example, the quantity V that appears inthe equation of state of a gas is the volume available to the gas. Third:even if there is a correspondence between manipulable variables andmacroscopic extensive variables (as there is a correspondence betweenthe position of the walls of a container and the volume occupied by agas in its equilibrium state), these are conceptually distinct. Fourth:an equilibrium thermo-dynamic state need not be a state in which allmacroscopically observable quantities have stable values. Consider,for example, a particle, visible under a microscope of modest power,undergoing Brownian motion. If—as I think we should—we count itsposition as macroscopically observable, this does not settle down toa stable value. What we have, instead, is a stable pattern of fluctua-tions. This can well count as a state of thermo-dynamic equilibrium. Two examples will help illustrate how Θ∆ cs works, and how it differsfrom the standard way of presenting thermodynamics. Consider the following example, discussed by Gibbs (1875, 227–229;1906, 166-167), which has been the topic of considerable discussion Example, from one of the most widely used textbooks,A description of a thermodynamic systems requires the specification of the“walls” that separate it from its surroundings and that provide its boundaryconditions. It is by means of manipulations of the walls that the extensiveparameters of the system are altered and processes are initiated (Callen, 1985,p. 15). ince that time. We consider a container divided by a partition intotwo subvolumes, each containing samples of gas at the same tempera-ture and pressure. The partition is removed, and the gases interdiffuse,until each is equally distributed within the whole volume. Has therebeen an increase of entropy, or not?The answer found in all the textbooks, given already by Gibbs, isthat if the gases initially in the two subvolumes are of the same type,there has been no change of thermodynamic state, and ipso facto nochange in entropy. If the two subvolumes initially contain gases ofdifferent types, initial and final states of the contents of the containerare distinct thermodynamic states, and the entropy of the final stateis higher than that of the initial state. This entropy increase is knownas the entropy of mixing .But what is the criterion for sameness of thermodynamic state?On the standard textbook account, thermodynamic states are definedwith respect to macroscopic variables. On this account, initial andfinal states are distinct if and only if they macroscopically distin-guishable. On the thermo-dynamic account, initial and final statescount as distinct only if the class M contains manipulations that actdifferentially on the two gases, in such a way that their interdiffusionrepresents a lost opportunity to extract work. A standard textbookdevice, originating with Boltzmann (1878) and popularized by Planck(1897, § s a historical note: conflation of these two notions of thermody-namic state goes back as far as Gibbs’ discussion, as Gibbs gives bothanswers to the question of criterion of distinctness of initial and finalstates. He first gives, as a criterion for restoring the initial state of thegases, the condition that we bring about a state “undistinguishablefrom the previous one in its sensible properties” (1875, p. 228; 1906,p. 166). “It is to states of systems thus incompletely defined,” hesays, “that the problems of thermodynamics relate.” But then, in thefollowing paragraph, he writes,We might also imagine the case of two gases which shouldbe absolutely identical in all the properties (sensible andmolecular) which come into play while they exist as gaseseither pure or mixed with each other, but which should dif-fer in respect to the attractions between their atoms andthe atoms of some other substances, and therefore in theirtendency to combine with other substances. In the mixtureof such gases by diffusion an increase of entropy would takeplace, although the process of mixture, dynamically consid-ered, might be absolutely identical in its minutest details(even with respect to the precise path of each atom) withprocesses which might take place without any increase ofentropy. In such respects, entropy stands strongly con-trasted with energy. (Gibbs 1875, pp. 228–229; 1906, p.167)Here he seems to be acknowledging that the key issue is not whetherthe two gases are the same in their sensible properties, but whetheror not they can be separated by external means.This example has given rise to metaphysical discussions that arecompletely irrelevant. The relevant criterion of distinctness, it is saidin some quarters, is whether the particles of the two gases are identicalin a strong sense, according to which exchange of particles makes nodifference whatsoever to the physical state. On such a view, if all theparticles were distinct—that is, if every particle involved differed insome physical property from all the others—then there would alwaysbe an entropy of mixing when the barrier was removed. As RobertSwendsen (2006; 2018) has argued, this gives the wrong answer whenapplied to a colloidal suspension. A colloid, such as paint, or milk,consists of blobs, called colloidal particles, of some type of materialsuspended in some fluid. The colloidal particles may be large enough hat each contains a large number of molecules, and, though their sizesmay be sufficiently uniform that we are justified in treating the colloidas a collection of identical particles, it might be that no two of themcontain exactly the same number of atoms. Someone committed tothe position that for a collection of distinct particles there is always anentropy of mixing when a partition is removed would be committed tothe position that we can lower and raise the entropy of a can of paintmerely by inserting or removing a partition. This is the wrong answer.In the absence of any means of manipulation that is so sensitive tothe minute differences between colloidal particles that each particlecan be differentially manipulated, there is no entropy of mixing whenone removes a partition separating two samples of the same type ofpaint.The entropy of mixing of two distinct gases depends only on thequantities of gas in each subvolume, and on their initial and finalvolumes. It is independent of the degree of dissimilarity. This struckDuhem (1892) as paradoxical, and, following him, Wiedeberg (1894),who spoke of “Gibbs’ paradox.” The alleged paradox stems from atension between the independence of the entropy of mixing from thenature of the gases (as long as they are distinct), and the idea that aresult valid for identical gases should be obtainable as a limit-case ofdistinct gases of diminishing degree of dissimilarity.If entropy is thought of as an intrinsic property of a system, likeits mass or its total energy, then this does seem puzzling. However,as argued by Denbigh and Redhead (1983), if we recall how entropy isdefined—relative to some set of processes, as a limit of some quantitytaken over all processes in that set—this does not seem surprisingat all. The result of any particular process, taking placing within afixed duration of time, may well depend continuously on the relevantparameters of the system. But entropy involves a limit over a setof processes. As two gases become more and more similar, the timerequired to achieve a given degree of separation may increase, but, ifour set of manipulations contains arbitrarily slow processes, this willnot affect entropy as a limit property.An analogy may help. Consider a collection of immortal ants thatcrawl at different rates towards a hill that is one metre tall. All ofthem, as long as they have a nonzero velocity in the proper direction,eventually reach the top of the hill. The distance crawled, and heightreached, at any given time t , is a continuous function of the speed atwhich the ant crawls. But the maximum height reached by an ant is ne metre for any nonzero speed, and zero for a stationary ant, andso is a discontinuous function of the ant’s speed. Suppose that the class of manipulations to be considered involves ac-cess to only one heat reservoir, at temperature T . We ask: if thesystem starts out in a state a and ends up in state b , what is the mostwork that you can extract from it along the way?Let E a and E b be the internal energy of the system in states a and b , respectively. If work is extracted from the system, this means that W is negative. We obtain from the system a positive amount of work W gain = − W . Conservation of energy requires, E b − E a = Q − W gain . (9)From the definition of entropy S M ( a → b ), S M ( a → b ) ≥ QT , (10)and so W gain ≤ − ( E b − E a − T S M ( a → b )) . (11)If the quantity on the right-hand side of (11) is negative, then no workcan be obtained in a transition from a to b using a heat reservoir attemperature T as a resource; on the contrary, the transition requiresexpenditure of a quantity of work (that is, a positive quantity of energygoing into the system), W cost ≥ E b − E a − T S M ( a → b ) . (12)Call the quantity F M ( a → b ) = E b − E a − T S M ( a → b ) (13)the Helmholtz free energy of b relative to a . If the only available heatsources and sinks are at temperature T , a transition from a to b isachievable without expenditure of a positive quantity of work if andonly if F M ( a → b ) < S available, such that S M ( a → b ) = S M ( b ) − S M ( a ). This allows us to define a function F M = E − T S M (14) uch that F M ( a → b ) = F M ( b ) − F M ( a ) . (15)The quantity F M was called the available energy in the 4th edition(1875), and subsequent editions, of Maxwell’s Theory of Heat (pp.187–192). It was called freie Energie by Hemholtz (1882), whence itscurrent name,
Helmholtz free energy . If all heat exchanges are withreservoirs at temperature T , then a transition from a to b requireswork to be done if F M ( b ) > F M ( a ), and can be a source of work if F M ( b ) < F M ( a ).There is an interesting difference between the uses of this conceptby Maxwell and Helmholtz, respectively. Helmholtz imagines a systemin contact with a heat bath at temperature T . All changes undersuch conditions are isothermal changes, and the free energy differencebetween two states is the work needed to effect a state transitionvia an isothermal process. The use of the concept is to determine theequilibrium state of the system, which is the state in which F takes itsminimum value (that is, work has to be done to move the system awayfrom this state). This is the use to which it is put in most moderntextbooks. This presentation may suggest that the Helmholtz freeenergy is a property of the system itself.Maxwell, on the other hand, imagines transitions between arbi-trary initial and final states; these need not be states of temperature T . The change in available energy is the work needed to effect a statetransition, using a heat reservoir at temperature T as a resource. Onthis way of thinking about it, F is a function both of the state of thesystem, via state functions E and S , and of the heat reservoir, via T . In the previous section it was assumed that the equilibrium values ofthe quantities { A i } , defined by eq. (2), are well-defined as functionsof the energy E and the manipulable variables λ .That this is a substantive assumption can be seen by consideringthe example of a gas confined to a container with a moveable pistonwhose position is taken to be manipulable. The generalized force cor-responding to displacements of the piston is the negative of the pres-sure. For a macroscopic gas in equilibrium, we expect an even andsteady pressure on the walls of the container. If we think about whatis happening on the molecular level, we realize that this is a statistical egularity of the same sort as the observed near-constancy of deathsper capita in a given population from year to year, a regularity aris-ing from aggregation of a large number of individually unpredictableevents. A regularity of this sort is not to be thought of as somethingthat occurs with certainty, but, rather, with high probability. If weask whether we could push on the piston and find ourselves able to di-minish the volume with virtually no resistance, we have to admit thatit is not impossible, but (for a macroscopic gas) so highly improbablethat the possibility may be neglected.This means that probabilistic considerations are in play, even inthe cases where there is a determinate (enough) near-certain amountof work required for a given manipulation. The role of probabilitymay be left implicit in cases where deviation from certainty is negligi-ble. However, since probability is playing a role whether explicitly ac-knowledged or not, it is best to introduce probabilistic considerationsexplicitly. This opens up the possibility of a more general theory thatembraces cases in which statistical fluctuations in generalized forcesare non-negligible, with the quasi-deterministic macroscopic theory asa limiting case.It is a commonplace of the literature on philosophy of probabilitythat the word “probability” is used in more than one sense. Thatraises the question of what probability is to mean in this context.I will defer that question (but see Myrvold 2016 for some options),leaving a gap in the account to be filled in. As long as the usualmachinery of probability theory is applicable, the conclusions we willdraw will be independent of how that gap is filled.One thing should be stressed, however. In the latter half of thenineteenth century, it became increasingly common (spurred, in partby Venn’s The Logic of Chance ) to think of probability statementsas involving veiled reference to frequencies in some actual or hypo-thetical series of similar events. It was in this milieu that Boltzmann(1871, 1898), and, following him, Maxwell (1879) and Gibbs (1902),began to think in terms of an imaginary ensemble consisting of a largenumber of systems with the same external parameters and varyingmicrostates. Frequentism is widely (and rightly, in my opinion) re-jected in the literature on the philosophy of probability. Fortunately,nothing in the approach of Boltzmann and his successors is weddedto it. Any readers who have qualms about talk of probabilities stem-ming from a worry that probabilities cannot be ascribed to individualsystems should rest assured that this is not the case. There is no com- itment to frequentism about probabilities. Feel free to take the talkof ensembles by Boltzmann, Maxwell, Gibbs, and the textbook tradi-tion that followed as a picturesque way of talking about a probabilitydistribution applied to propositions about an individual system.Given a thermo-dynamic state of a system, we want to have proba-bility distributions over the work done and heat exchanged as a resultof a manipulation. The reason that these don’t have determinatevalues is that the thermo-dynamic state of a system drastically un-derspecifies the physical state of the system. This suggests that wesupplement our specification of a thermo-dynamic state, which so farinvolves specification of the internal energy and of values of the ma-nipulable variables, with a specification of a probability distributionover possible physical states of the system. This can be done in thecontext of classical or quantum mechanics. In a classical context, wewill have assignments of probabilities to appropriate subsets of the sys-tem’s phase space; in the quantum context, probability distributionsover the pure states of the system.What now happens to the second law of thermodynamics? In aregime in which statistical fluctuations of the force on a piston are non-negligible, we might in a given cycle of an engine end up expending lesswork than expected in the compression stage, and hence might obtainin that cycle more work than the Carnot limit. But, by the sametoken, we might expend more work than expected. We expect thatwe won’t be able to consistently and reliably violate the Carnot limiton efficiency. This suggests a probabilistic version on the second law,expressed in terms of expectation values of heat and work transfers.The second law will then be, to employ Szilard’s vivid analogy, likea theorem about the impossibility of a gambling system intended tobeat the odds set by a casino.Consider somebody playing a thermodynamical gamble withthe help of cyclic processes and with the intention of de-creasing the entropy of the heat reservoirs. Nature will dealwith him like a well established casino, in which it is pos-sible to make an occasional win but for which no systemexists ensuring the gambler a profit (Szilard 1972, p. 73,from Szilard 1925, p. 757).We will be considering exchanges of energy with heat reservoirs.A heat reservoir is a system from which no work is extracted and onwhich no work is done; its only exchanges of energy with other sys- ems are as heat. When two heat reservoirs of the same temperatureare placed in thermal contact, there is no tendency for heat to betransferred in either direction, and the expectation value of the heatexchange is zero. When two reservoirs are placed in thermal contact,the expectation value of heat flow is from warmer to cooler. Any col-lection of heat reservoirs at the same temperature may be regarded asa larger heat reservoir at the same temperature.From considerations of this sort one can argue (see Maroney 2007for exposition) that an appropriate probability distribution to asso-ciate with a heat reservoir is the one that Gibbs labelled the canonicaldistribution . In the classical context, it is defined as the distributionwith density function, with respect to Liouville measure, τ β ( x ) = Z − e − βH ( x ) , (16)where β is the inverse temperature 1 /kT , and Z is the normalizationconstant required to make the integral of this density over all phasespace unity. This depends both on the Hamiltonian H and on β , andis called the partition function . In the quantum context, the canonicaldistribution is represented by a density operator,ˆ τ β = Z − e − β ˆ H , (17)where, again, Z is the constant required to normalize the state. Wewill henceforth take it that to treat a system as a heat reservoir is torepresent its thermo-dynamic state by a canonical distribution, uncor-related with the rest of the world. In the spirit of Szilard’s analogy, if we seek a statistical-mechanicalanalog of the thermo-dynamic entropy, we may take the definition (5)and replace the heat exchanges mentioned therein with their expecta-tion values.A thermo-dynamical state of a system will be specified by itsHamiltonian H , which depends on manipulable variables λ , togetherwith a probability distribution over its state space. In the classicalcontext the probability distribution may be represented by a density unction ρ ; in the quantum context, the salient aspects of such a distri-bution may be represented by a density operation ˆ ρ . Given a thermo-dynamical state a = ( ρ a , H a ), we consider the effects of some manip-ulation, which may consist of manipulation of the variables λ and ofcouplings to various heat reservoirs { B i } . The probability distributionfor A , together with canonical distributions for the heat reservoirs, de-termines an initial probability distribution over the composite systemconsisting of A and the reservoirs { B i } . This will evolve, in accordancewith the Liouville equation (classical) or Schr¨odinger equation (quan-tum), according to the Hamiltonian of the total system, which may bechanging due to the changes in the manipulable variables. This pro-cess will result in a new thermo-dynamic state b = ( ρ b , H b ). Over thecourse of the process quantities { Q i ( a → b ) } of heat may be exchangedwith the reservoirs; the probability distribution over initial conditions,together with the evolution of the joint system, yields a probabilitydistribution over the heats { Q i ( a → b ) } . Let h Q i ( a → b ) i M be the ex-pectation value of the heat obtained from reservoir B i over the courseof the process. As before, let M ( a → b ) be the set of manipulationsin M that lead from a to b . For any manipulation M in M ( a → b ),define σ M ( a → b ) = X i h Q i ( a → b ) i M T i . (18)Define the statistical-mechanical entropy S M ( a → b ) by S M ( a → b ) = l.u.b. { σ M ( a → b ) | M ∈ M ( a → b ) } . (19)We are entitled to use the same notation for this and the entropies asdefined in section 3, as the latter are really only a special case of theentropy defined here, when the probabilities are such that variance inthe heat exchanges are negligible. We are only making explicit thepreviously implicit dependence on probabilistic considerations.With these definitions in hand, the statistical-mechanical entropies S M ( a → b ) are defined once we have specified a class of manipulations.Of particular interest will be classes of manipulations of the followingsort. • At time t , the heat reservoirs B i have canonical distributionsat temperatures T i , uncorrelated with A , and are not interactingwith A . • During the time interval [ t , t ], the composite system consistingof A and the reservoirs { B i } undergoes Hamiltonian evolution, overned by a time-dependent Hamiltonian H ( t ), which may in-clude successive couplings between A and the heat reservoirs { B i } . • The internal Hamiltonians of the reservoirs { B i } do not change. • At time t , the Hamiltonian of the system A is H b , and, as aresult of Hamiltonian evolution of the composite system, themarginal probability distribution of A is ρ b .The initial state of A is arbitrary. No assumption is made about theform of the Hamiltonian H A , the nature of the manipulable variables λ , or about the manipulations applied to them. These could verywell include fine-grained manipulations at the molecular level that wewould regard as well beyond the range of feasibility. In what fol-lows, we will use M θ to designate some class of this sort. That is,the variable M ranges over arbitrary classes of manipulations, andthe variable M θ ranges over classes of manipulations satisfying theseconditions.A class of manipulations of this sort has the advantage that itaffords a clear distinction between energy changes of the system A that are to be counted as work, and those that are to be counted asheat. Changes in energy of A due to manipulation of the exogenousvariables are work; exchanges of energy with the heat reservoirs arecounted as heat. A more general class of manipulations might includeexchanges of energy between the system A and other systems thatare not treated as heat reservoirs—that is, systems with distributionsother than canonical distributions. With respect to this class of ma-nipulations, we might not have a neat partition of energy changes to A into heat and work; changes due to interactions with other systemsmight be classed as neither.Given some such class of manipulations, the second law comes outas a theorem. That is, it can be proven that S M θ ( a → a ) ≤ . (20)As we saw in section 3, it follows from this that if all states are re-versibly connectable—that is, if, for all a , b , S M θ ( a → b → a ) = 0 , (21)then there is a state function S M θ , defined up to an arbitrary constant,such that S M θ ( a → b ) = S M θ ( b ) − S M θ ( a ) . (22) f we ask what form that state-function takes, it turns out that, in theclassical context, it is the quantity called the Gibbs entropy , and, inthe quantum context, the von Neumann entropy .To show this, we must first define these quantities. Consider aprobability distribution P on a classical state-space Γ, that has density ρ with respect to Liouville measure. ρ itself may be treated as arandom variable: if a point x in Γ is randomly selected according to thedistribution P , there will be a corresponding value of ρ ( x ). Similarly,any measurable function of ρ may be treated as a random variable.We define the Gibbs entropy of the distribution P as proportional tothe expectation value, calculate with respect to P , of the logarithm of ρ . S G [ ρ ] = − k h log ρ i P (23)For a quantum state, represented by a density operator ˆ ρ , we definethe von Neumann entropy , S vN [ˆ ρ ] = − k h log ˆ ρ i ˆ ρ = − k Tr[ˆ ρ log ˆ ρ ] . (24)Most of what we will have to say applies equally in the classical andquantum contexts. In what follows, we will use the intentionally am-biguous notation S [ ρ ] to state results that hold both for Gibbs entropyof a probability distribution on a classical phase space and for vonNeumann entropy of a quantum state.The link between these quantities and the statistical thermo-dynamicentropy is provided by the following theorem. Proposition 2. For any manipulation in the class M θ , X i h Q i i T i ≤ S [ ρ A ( t )] − S [ ρ A ( t )] . Recalling the definition (19) of statistical-mechanical entropies,this gives us,
Proposition 3.
Statistical entropies defined with respect to M θ sat-isfy S M θ ( a → b ) ≤ S [ ρ b ] − S [ ρ a ] . The classical version of this is found in Gibbs (1902, pp. 160–164), and the quantumversion, in Tolman (1938, § hough not a difficult theorem, Proposition 2 is of sufficient im-portance that it may be called the Fundamental Theorem of StatisticalThermo-dynamics . To get a feel for what it means, consider a heat en-gine operating in a cycle between a hot heat reservoir at temperature T h and a cooler heat sink at temperature T c . It extracts a positiveamount of heat Q h from the hot reservoir, performs work W , and dis-cards a positive amount Q h − W into the sink. To say that it operatesin a cycle means that its initial thermo-dynamic state is restored atthe end of this process (it may have built up some correlations withthe reservoirs along the way, but these don’t matter; the final state isspecified by the restriction of the joint probability distribution to thesystem A ). Proposition 2 tells us that the expectation values of workobtained, heat extracted and heat discarded satisfy (recalling that aquantity of heat counts as positive if it is going into the engine andnegative if it is going out), h Q h i T h − h Q h i − h W i T c ≤ . (25)This gives us, for the expectation value of the work obtained: h W i ≤ (cid:18) − T c T h (cid:19) h Q h i . (26)Thus, the Carnot bound on the efficiency of a cyclical engine operatingbetween these two reservoirs becomes a bound on expectation value ofwork obtained. It should be stressed that we have not presumed thatthe actual values of heat exchanges will be or even will probably beclose to their expectation values. No assumption has been made thatthe probability distributions for these quantities are tightly focussednear the expectation values. These expectation values satisfy the givenrelations even if the variance of their distributions is large.From Proposition 3 the second law of thermo-dynamics is an im-mediate corollary. Corollary 3.1.
For manipulations M θ , S M θ ( a → a ) ≤ for any thermo-dynamic state a . Another immediate corollary of Proposition 3 is, orollary 3.2. If S M θ ( a → b → a ) = 0 , then S M θ ( a → b ) = S [ ρ b ] − S [ ρ a ] . Thus, the state function whose existence is guaranteed by the sec-ond law plus reversibility is, up to an additive constant, the Gibbs orvon Neumann entropy. A probability distribution may encode a lot of details about themicrostate of the system that are irrelevant to the results of availablemanipulations. Consider, for example, a gas consisting of a macro-scopic number of molecules initially confined to the left side of a con-tainer. A partition is removed, and the gas is allowed to expand freelyinto the whole volume of the container. Imagine (as is common in theliterature on the philosophy of statistical mechanics) that it can doso while isolated from its environment. Any probability distributionwith support in the set of states in which all molecules are one sidewill evolve into a distribution with support on a set that is a minus-cule fraction of the available phase space. However, this set will sofinely distributed that only very fine-grained manipulations could dis-tinguish this probability distribution from an equilibrium distributionuniform in the accessible region of phase space. If the only availablemanipulations involve pistons and couplings to heat reservoirs, therewill be no difference, in terms of expected reactions to these manipu-lations, between a probability distribution corresponding to a recentisolated expansion from one side of the box and one on which the gashad been in equilibrium with a heat reservoir for a long time. Theconsiderable knowledge about the state of the gas that comes fromknowing it was in the left half of the box an hour ago is irrelevant toresults of ham-handed interventions.With these considerations in minds, we define an equivalence-relation between thermo-dynamic states.Any two thermo-dynamic states ( ρ, H λ ), ( ρ ′ , H λ ) havingthe same values of the manipulable variables λ , are thermo-dynamically equivalent with respect to M if and only if,for every manipulation M ∈ M , ρ and ρ ′ yield the sameexpectation values for work, h W i , and for heat exchanges, h Q i i , over the course of the manipulation M . We will write a ∼ M a ′ for thermo-dynamic equivalence. It should be stressed that we are not defining the statistical mechanical entropy S M ( a → b ) in terms of S [ ρ b ] and S [ ρ a ]; it is defined by (19). e could, of course, define a stronger notion on which equivalencerequires, not just equality of expectation values, but equality of theprobability distributions for work and heat, but at the moment I seeno need for this. One could also relax the condition a bit, and require,not exact equality, but equality within a certain tolerance (in whichcase the relation will not be strictly speaking an equivalence relation).Define coarse-grained entropies,¯ S M [ a ] = l.u.b { S [ a ′ ] | a ∼ M a ′ } . (27)Obviously, for any state a , ¯ S M [ a ] ≥ S [ a ] . (28)If, for some thermo-dynamic state a , there is another state a ′ that isthermo-dynamically equivalent to it and which maximizes the entropyamong states equivalent to a , we will say that a ′ is a coarse-graining of a . We will say that a is a coarse-grained state if and only if ¯ S M [ a ] = S [ a ]. Note, however, that the coarse-grained entropy is well-definedwhether or not for every state there is a corresponding coarse-grainedstate.With the concept of coarse-grained entropy in hand, we have astrengthening of Proposition 3. Proposition 4.
For any class of manipulations M θ , and any pair ofthermo-dynamic states a , b , S M θ ( a → b ) ≤ S [ ρ b ] − ¯ S M θ [ ρ a ] . The upper bound on S M θ ( a → b ) in Proposition 4 is a differ-ence between two different state-functions, S and ¯ S M θ , depending onwhether the state is the initial or final state of the manipulation. Wemay call ¯ S M θ the departure entropy , and S , the arrival entropy .This sheds light on a move that has routinely been made, since thetime of Gibbs: the use of a coarse-grained entropy (usually obtainedvia a coarse-graining of the state) to track approach to equilibrium ofan isolated system. If a system is isolated, the Gibbs/von Neumannentropy is a constant of the motion. The state can, however, evolveinto a state in which the result of any manipulation would be the sameas would obtain if the state were one with a higher entropy ¯ S M θ . Thequantity ¯ S M θ , rather than S , is the one relevant to bounds on thevalue of the state for obtaining work, and so is the relevant quantity o track, if one is interested in tracking loss of such value as the systemapproaches equilibrium. This is not, as some have suggested, an adhoc move that is made for the sole purpose of finding a quantity thatincreases on the way to equilibrium.From the second law, Corollary 3.1, for any a , b , S M θ ( a → b → a )cannot be positive. It follows from Proposition (4) that the differencebetween the Gibbs/von Neumann entropies of the states a and b , andthe corresponding coarse-grained versions, puts a bound on how closeto zero S M θ ( a → b → a ) can be. Corollary 4.1. − S M θ ( a → b → a ) ≥ (cid:0) ¯ S M θ ( ρ a ) − S [ ρ a ] (cid:1) + (cid:0) ¯ S M θ ( ρ b ) − S [ ρ b ] (cid:1) . An immediate consequence of this is that only coarse-grained statescan be reversibly connected.
Corollary 4.2. If S M θ ( a → b → a ) = 0 , then ¯ S M θ [ a ] = S [ a ] and ¯ S M θ [ b ] = S [ b ] . We can summarize the relations between the thermo-dynamic en-tropies S M θ ( a → b ) and the Gibbs/von Neumann entropies as follows.1. If the states a and b can be connected reversibly, then the thermo-dynamic entropy S M θ ( a → b ) is equal to the difference of theGibbs/von Neumann entropies of the two states. That is, S M θ ( a → b ) = S [ ρ b ] − S [ ρ a ] . This is not an arbitrary or whimsical choice, but a theorem.2. This relation between thermo-dynamic entropy and the Gibbs/vonNeumann entropy can hold for both S M θ ( a → b ) and S M θ ( b → a ) only if a and b can be connected reversibly. If they cannot,then either S M θ ( a → b ) is strictly less than S [ ρ b ] − S [ ρ a ], or S M θ ( b → a ) is strictly less than S [ ρ a ] − S [ ρ b ] (or both).3. If a is not a coarse-grained state, then S M θ ( a → b ) is never equalto S [ b ] − S [ a ] for any state b that can be reached from a , but isalways strictly less.To get a feel for this, suppose that a and b can be connectedadiabatically, that is, purely Hamiltonian evolution can take ρ a to ρ b .One can think of free expansion of an adiabatically isolated gas; ρ b is then distribution that has support on a small but highly fibrillated et that is stretched out throughout the available phase space. Then,because Hamiltonian evolution preserves S , S [ ρ b ] is equal to S [ ρ a ]. Itwould simply be a gross error to conclude from this that a and b areentropically on a par, and that, for some state c that can be reachedfrom both, S M ( a → c ) is equal to S M ( b → c ). Unless the expansioncan be undone adiabatically (which would require fantastically fine-grained control over the evolution of the system), S M ( b → c ) is strictlyless than S M ( a → c ). In any process M that takes a state a to a state b , some of the workdone, or heat discarded into a reservoir, may be recovered by someprocess that takes b back to a . If the process can be reversed with thesigns of all h Q i i reversed, then full recovery is possible. If full recoveryis not possible, and cannot even be approached arbitrarily closely, wewill say that the process is dissipatory . A manipulation M ′ that takes b to a and recovers work done and heat discarded would be one suchthat σ M ( a → b ) + σ M ′ ( b → a ) = 0 . (29)There might be a limit to how closely this can be approached. Definethe dissipation associated with the process of M taking a to b as thedistance between this limit and perfect recovery. δ M ( a → b ) = g.l.b. {− ( σ M ( a → b ) + σ M ′ ( b → a )) | M ′ ∈ M ( b → a ) } = − S M ( b → a ) − σ M ( a → b ) . (30)It follows from the second law that this is non-negative.If there is no limit to how much the dissipation associated withprocesses that connect a to b can be diminished, S M ( a → b → a ) isequal to zero. This is the condition that we earlier called reversibility.It is easy to see that the negative of this places a bound on the minimaldissipation associated with any manipulation that takes a to b . Forany M in M ( A → b ), δ M ( a → b ) ≥ − S M ( a → b → a ) . (31)It follows from this and Corollary 4.1 that the difference between thecoarse-grained and non-coarse grained versions of the Gibbs/von Neu-mann entropies of the states a and b place bounds on the minimaldissipation associate with a process that takes a to b . orollary 4.3. For any states a , b , and any manipulation M in M θ , δ M ( a → b ) ≥ (cid:0) ¯ S M θ ( ρ a ) − S [ ρ a ] (cid:1) + (cid:0) ¯ S M θ ( ρ b ) − S [ ρ b ] (cid:1) . As noted above, in Proposition 1, no adiabatic transformation candecrease the entropy of a state. This is a consequence of the definitionof the entropies S M ( a → b ). One could also consider transformationsof a system A that involve manipulation of A and an auxiliary system C that can couple to it. No adiabatic transformation can decrease theentropy of the joint system AC .These entropies are, of course, defined relative to a class of ma-nipulations. This dependence of the question of whether a given pro-cess involves an atio increase of entropy on the class of manipulationsconsidered was illustrated by Maxwell via a thought experiment, inwhich we imagine a “very observant and neat-fingered being” ca-pable of performing manipulations that are “at present impossible tous” (Maxwell, 1871, p. 308).Suppose we have a class M of manipulations, and supplement itwith some manipulation not in the class, to form a new class M + .It could happen that some state-transformation effected adiabaticallyvia manipulations in M + could lower the entropy of a state, relativeto M . That is, there might be an adiabatic transformation a → b , achievable via manipulations in M + , such that, for some state c , S M ( b → c ) > S M ( a → c ). Someone confused about the dependence ofentropy on a set of manipulations might take this to be a violation ofthe principles of thermo-dynamics, which dictate that, if an adiabaticprocess can take a to b , S M ( b → c ) ≤ S M ( a → c ). There is no suchviolation, because S M + ( b → c ) ≤ S M + ( a → c ).This can be vividly illustrated by imagining a stock M of physi-cally possible manipulations to be supplemented by a magical instan-taneous velocity-reversing operation, yielding an enhanced set M + .Consider our stock example of a container of gas, and let M be theusual sorts of manipulations, consisting of manipulations of the posi-tion of the piston and heat exchanges with various heat reservoirs. Let M + be this stock of operations, supplemented by a magical velocity-reversal. Consider a container of gas initially confined to a subvolume, Letter to P. G. Tait, 11 Dec. 1867, in Harman (1995, p. 332). hich expands to fill the whole container. With respect to M , thisexpansion counts as an entropy increase. An irreversible expansion isa lost opportunity to obtain work. But, since, with respect to M + ,the expansion is adiabatically reversible, there is no entropy increase,no lost opportunity to obtain work, as one can apply the reversal op-eration and wait for the gas to return to its original subvolume. Anapplication of the velocity reversal operation to the expanded gas re-sults in an entropy decrease with respect to M but not M + . Since theoperation preserves phase-space volume (or, in the quantum context,the absolute value of the inner product of any two state-vectors), theproof of Proposition 2 still goes through, and the statistical version ofthe second law holds even for the set M + of manipulations. A demoncapable of performing a velocity reversal could undo the process ofequilibration, but could not operate an engine in a cycle to violate theCarnot bound on efficiency of a heat engine.This may seem paradoxical to some. Surely , it will be said, a gasthat is initially spread out throughout a container and subsequentlyretreats to a corner must be decreasing its entropy. This cannot besustained, however, if one attends to the definition of thermodynamicentropy. If the expansion of a gas can be can be reversed adiabatically,then, by the definition of thermodynamic entropy—not just the defi-nition we have given but by the definitions found in all textbooks ofthermodynamics—it is not an entropy-increasing process. The processof returning to the initial subvolume may be a diminution of Boltz-mann entropy, but this only illustrates that the connection betweenBoltzmann entropy and thermodynamic entropy is somewhat tenuous.Earman and Norton (1998) distinguish between straight and em-bellished violations of the second law of thermodynamics. A straightviolation decreases the entropy of an adiabatically isolated system,without compensatory increase of entropy elsewhere. An embellishedviolation exploits such decreases in entropy reliably to provide work.In a similar vein, David Wallace (2018) distinguishes between twotypes of demon. Adapting the distinction to our terminology, a demonof the first kind decreases entropy defined with respect to some class M of manipulations, by utilizing a manipulation outside the class. Ademon of the second kind violates the Carnot bound on efficiency ofa heat engine over a cycle that restores the state of the demon plusany auxiliary system utilized to its original thermo-dynamic state. ByProposition 2, a demon of the second kind cannot exist without a de- arture from Hamiltonian dynamics. A demon of the first kind onlyillustrates the dependency of entropy on the class of manipulationsconsidered.Maxwell’s purpose in introducing the demon was to illustrate thedependence of thermodynamic concepts on the class of manipulationsconsidered. He was quite explicit about what the point of the thought-experiment was: to emphasize the built-in limitation of conclusionsdrawn from standard thermodynamics to situations in which bodiesconsisting of a large number of molecules are dealt with in bulk. Theseconclusions, he says, may be found to be inapplicable to situationsinvolving manipulation of individual molecules (Maxwell, 1871, pp.308–309). Despite this, the point, a fairly simple one, has been widelymisunderstood, resulting in a vast and largely confused literature onthe physical possibility or impossibility of a Maxwell demon.
The Fundamental Theorem of Statistical Thermo-dynamics, Proposi-tion 2, follows from elementary properties of the Gibbs and von Neu-mann entropies and of Hamiltonian evolution. It is not temporallysymmetric. We consider a transformation that takes state a into state b , and the order matters, because the right hand side of the inequal-ity displayed is not invariant under interchange of a and b . No suchasymmetry is present in the underlying dynamics. Where, then, doesthe temporal asymmetry come in?The mathematical result on which the Fundamental Theorem de-pends is the following (stated here, and proven in the Appendix).Consider a joint system composed of subsystems A and B , which un-dergoes Hamiltonian evolution between times t and t . The totalHamiltonian H AB may change during the process; changes may bemade to H A , corresponding to work done on the system, and to theHamiltonian of interaction between the two systems. We assume that It is essential to the theorem that the dynamics preserve phase-space volume. That thiscondition is required to underwrite the second law is illustrated by Earman and Norton(1999), who, building on the work of Skordos (1993) and Zhang and Zhang (1992), ex-hibit a fictitious system with non-Hamiltonian, energy-conserving, time-reversal invariantdynamics that completely converts heat drawn from a heat reservoir into work. t times t and t the total Hamiltonian is just the sum of the in-ternal Hamiltonians H A and H B , and that H B ( t ) = H B ( t ). Theexpectation value of the energy received by A from B is h Q i = − ( h H B i t − h H B i t ) . Suppose that the state ρ AB at time t is one on which (i) B hascanonical distribution τ β , and (ii) A and B are uncorrelated. Thedistribution of A at t is arbitrary. Proposition 5.
Under the stated conditions, h Q i T ≤ S [ ρ A ( t )] − S [ ρ A ( t )] . Proposition 5 holds for any Hamiltonian dynamics satisfying thespecified conditions, and so does not depend on any time-asymmetryin the underlying dynamics. In fact, it holds regardless of whether t isto the future or past of t . The two times do not enter symmetricallyinto the statement of the theorem, however. It is assumed that thesystems A and B are uncorrelated at t , and this is not required tohold at t . That is the relevant difference between starting point andending point of the process considered.It is sometimes said that the rationale for taking the initial stateof system + heat reservoir to be one without correlations betweenthem is that this has the status of a default assumption: statisticalor probabilistic independence is to be assumed in the absence of anyinteraction that would create correlations. This is too quick. Amongthe things that can create correlations between systems are events inthe common past of two systems. When we couple a system to aheat reservoir, we are not assuming that there are no events in theircommon past that could potentially lead to correlations.What we are assuming is that the reservoir has thermalized, hasundergone a process of equilibration in the course of which detailsof its past history, including previous interactions with the rest ofthe world, have been effectively effaced. A detailed microdescriptionmight reveal some of these details, but it is expected that these willbe irrelevant at the macroscopic scale. To treat a system as a heatreservoir is to treat the fine details of past interactions it might havehad with its environment as irrelevant to its subsequent behaviour.The task of explaining how and why this happens is an interestingand important one. The process produces thermal systems that the cience of Θ∆ cs can take to be available as resources for manipulations.The study of equilibration is not, however, the province of Θ∆ cs . The is a tendency to conflate the second law of thermodynamicswith the tendency of systems to relax to a state of thermal equilib-rium, and this has encouraged the idea that the study of equilibrationdoes fall within the scope of thermodynamics. These are not the samething, however. The distinction can be made vivid by consideringthe impact on the laws of thermo-dynamics of a “Loschmidt demon”that could magically perform a velocity-reversal. Such a demon couldreverse equilibration of an isolated system, but its operations never-theless fall within the scope of Proposition 2, and the second law ofΘ∆ cs holds even if the stock of manipulations is expanded to includedvelocity-reversal.
10 Conclusion
The chief differences between the theory whose outlines have beensketched here, which I am calling thermo-dynamics, or Θ∆ cs , and theusual textbook presentations of thermodynamics, are two-fold. Oneis that we have not assumed that all states are reversibly connectible.Without this assumption, we do not have available a state-function S such that S M ( a → b ) = S M ( b ) − S M ( a ). This is a relatively minorpoint; with a little care, it is fairly easy to see that much of thermo-dynamics goes through without this, and the advantage is that thetheory applies in regimes in which the inevitable dissipation involvedin every process is not taken to be negligible.The more important difference is that, whereas the usual treat-ments say that thermodynamic states are defined relative to a set ofvariables deemed macroscopic , we have defined them in terms of a setof variables deemed manipulable . I maintain that this is the best wayto make sense of the usual treatments, and that one will find, if onereads closely, that the relevant variables are indeed being treated asmanipulable. For the most part, for the purposes of textbook exposi-tion, as long as attention is confined to the macroscopic domain, andwe are not bent on pushing application of the theory into the meso-scopic, it is perfectly acceptable to leave the class of manipulationsunder consideration implicit. The danger of this, however, is that itmight tend to give the impression that entropy is a property of a sys- See Myrvold (2020) for further discussion of these points. em, something that it has in and of itself, rather than being definedrelative to a class of manipulations.Whether or not the reader agrees that Θ∆ cs is the best way tomake sense of textbook presentations of thermodynamics and of ap-plication of its concepts to the physical world, it should be noncontro-versial that it is a legitimate subject. The usual objections to invokingconcepts such as manipulability tend to be of two (related) sorts. Oneis that it brings in excessive subjectivity. The other is that conceptsof that sort are out of place in the study of equilibration. I hope thatI have satisfactorily addressed the former, in the preliminary discus-sion of manipulability. The latter is met by a delimitation of scope.Though Θ∆ cs presumes the availability of systems that can be treatedas heat reservoirs, study of the process of thermalization does not fallwithin its scope.
11 Acknowledgments
I am grateful to a number of people with whom I have discussed thesematters over the years. In particular, I thank Owen Maroney fordrawing my attention to what I have called the Fundamental Theorem,Norton for discussions of reversible processes, David Wallace for urgingthe considerations of §
12 Appendix
In this appendix we prove Proposition 2.As before, ρ is used ambiguously for either a density function withrespect to Liouville measure on classical phase space, or a quantumdensity operator. Hamiltonian evolution is, in the classical context,evolution according to Hamilton’s equations of motion, and, in thequantum context implemented by a family of unitary operators U ( t ). he letter S , without subscript, denotes either the Gibbs entropy orthe von Neumann entropy.In the classical context, the salient fact about Hamiltonian evolution—and, indeed, the only fact that we will use—is that Liouville measureis invariant under evolution of that type. As a consequence, the expec-tation value, with respect to Liouville measure Λ, of any measurablefunction on phase space is invariant under Hamiltonian evolution; thisincludes in particular the Gibbs entropy S G [ ρ ] = − k h ρ log ρ i Λ . (32)In the quantum context, the salient fact about Hamiltonian evo-lution is that it conserves the inner product of two vectors in Hilbertspace. As a consequence, the trace of any operator is invariant; thisincludes in particular the von Neumann entropy S vN [ˆ ρ ] = − k Tr[ˆ ρ log ˆ ρ ] . (33)As conservation of phase space volume (classical) and absolute magni-tude of inner product (quantum) are the only features of Hamiltonianevolution used, we could expand our repertoire of operations to in-clude fictitious operations, such as an instantaneous velocity reversal,that retain these features, and the theorem would still go through.The relevant facts about the Gibbs and von Neumann entropiesare:1. Subadditivity.
For a composite system AB , S [ ρ AB ] ≤ S [ ρ A ] + S [ ρ B ] , with equality if and only if the subsystems are probabilisticallyindependent.2. For any T >
0, let β = 1 /kT . The canonical distribution τ β minimizes h H i ρ − T S [ ρ ] . With these facts in hand, the proof of the theorem is easy. Forbrevity, we will write S AB ( t ) for S [ ρ AB ( t )], etc. . We will consideronly interactions with a single heat reservoir, as the extension to suc-cessive interactions with multiple heat reservoirs is merely a matter ofrepeated application of the theorem. he evolution from t to t does not change the joint entropy S AB .At t , since the systems are uncorrelated, S A + S B is at a minimumfor the value of S AB that obtains at both t and t . Therefore, S A ( t ) + S B ( t ) ≤ S A ( t ) + S B ( t ) , (34)or, ∆ S A + ∆ S B ≥ . (35)Since B has canonical distribution τ β at time t , h H B i t − T S B ( t ) ≤ h H B i t − T S B ( t ) , (36)or, ∆ h H B i − T ∆ S B ≥ . (37)This gives us, h Q i = − ∆ h H B i ≤ − T ∆ S B . (38)From (35), − ∆ S B ≤ ∆ S A , (39)which, combined with (38), yields, h Q i ≤ T ∆ S A , (40)or, h Q i T ≤ ∆ S A , (41)which is the desired result. Θ∆ cs The science that I am calling Θ∆ cs is not a new idea. This under-standing of the basic concepts of thermodynamics has been presentfrom the very beginning of the subject. In this appendix I providesome relevant quotations, with no pretense to exhaustiveness. Josiah Willard Gibbs (1875, pp. 228–229; in Gibbs 1906, pp. 166–167). Part of this has already been quoted above; here is a fullerquotation. hen we say that when two different gases mix by diffusion aswe have supposed, the energy of the whole remains constant, and theentropy receives a certain increase, we mean that the gases could beseparated and brought to the same volume and temperature whichthey had at first by means of a certain change in external bodies, forexample, by the passages of a certain amount of heat from a warmerto a colder body. But when we say that when two gas-masses of thesame kind are mixed under similar circumstances there is no changeof energy or entropy, we do not mean that the gases which have beenmixed can be separated without change to external bodies. On thecontrary, the separation of the gases is entirely impossible. We callthe energy and entropy of the gas-masses when mixed the same aswhen they were unmixed, because we do not recognize any differencein the substance of the two masses. So when gases of different kindsare mixed, if we ask what changes in external bodies are necessaryto bring the system to its original state, we do not mean a statein which each particle shall occupy more or less exactly the sameposition as at some previous epoch, but only a state which shall beundistinguishable from the previous one in its sensible properties. Itis to states of systems thus incompletely defined that the problems ofthermodynamics relate.But if such considerations explain the mixture of gas-masses of thesame kind stands on a different footing from the mixture of gas-massesof different kinds, the fact is not less significant that the increase ofentropy due to the mixture of gases of different kinds, in such a caseas we have supposed, is independent of the nature of the gases.Now we may say without violence to the general laws of gases whichare embodied in our equations suppose other gases to exist than suchas actually do exist, and there does not appear to be any limit tothe resemblance which there might be between two such kinds of gas.But the increase of entropy due to the mixing of given volumes of thegases at a given temperature and pressure would be independent ofthe degree of similarity or dissimilarity between them. We might alsoimagine the case of two gases which should be absolutely identical inall the properties (sensible and molecular) which come into play whilethey exist as gases either pure or mixed with each other, but whichshould differ in respect to the attractions between their atoms andthe atoms of some other substances, and therefore in their tendencyto combine with other substances. In the mixture of such gases bydiffusion an increase of entropy would take place, although the process f mixture, dynamically considered, might be absolutely identical inits minutest details (even with respect to the precise path of eachatom) with processes which might take place without any increaseof entropy. In such respects, entropy stands strongly contrast withenergy. Rudolf Clausius (1877, p. 32). Responding to P. G. Tait’s (unfair)charge that the fact that the possibility of a demon that could, with-out expenditure of work, cool a body below the temperature of itssurroundings “is absolutely fatal to Clausius’ reasoning,” (Tait 1876,pp. 118-120; see also Tait 1877, p. 37), Clausius wrote,Dieses kann ich in keiner Weise zugeben. Wenn die W¨arme alseine Molecularbewegung betrachtet wird, so ist dabei zu bedenken,dass die Molec¨ule so kleine K¨orpertheilchen sind, dass es f¨ur unsunm¨oglich ist, sie einzeln wahrzunehmen. Wir k¨onnen daher nicht aufeinzelne Molec¨ule f¨ur sich allein wirken, oder die Wirkungen einzelnerMolec¨ule f¨ur sich allein erhalten, sondern haben es bei jeder Wirk-ung, welche wir auf einen K¨orper aus¨uben oder von ihm erhalten,gleichzeitig mit einer ungeheuer grossen Menge von Molec¨ulen zu thun,welche sich nach allen m¨oglichen Richtungen und mit allein ¨uberhauptbei den Molec¨ulen vorkommenden Geschwindigkeiten bewegen, undsich an der Wirkung in der Weise gleichm¨assig betheiligen, dass nurzuf¨allige Verschiedenheiten vorkommen, die den allgemeinen Gesetzender Wahrscheinlichkeit unterworfen sind. Dieser Umstand bildet ger-ade die charakteristische Eigenth¨umlichkeit derjenigen Bewegung, welchewir W¨arme nennen, und auf ihm beruhen die Gesetze, welche das Ver-halten der W¨arme von dem anderer Bewegungen unterscheiden.Wenn nun D¨amonen eingreifen, und diese charakteristische Eigen-th¨umlichkeit zerst¨oren, indem sie unter den Molec¨ulen einen Unter-schied machen, und Molec¨ulen von gewissen Geschwindigkeiten denDurchgang durch eine Scheidewand gestatten, Molec¨ulen von anderenGeschwindigkeiten dagegen den Durchgang verwehren, so darf mandas, was unter diesen Umst¨anden geschieht, nicht mehr als eine Wirk-ung der W¨arme ansehen und erwarten, dass es mit den f¨ur die Wirk-ungen der W¨arme geltenden Gesetzen ¨ubereinstimmt.This I can in no way concede. If heat is regarded as a molecularmotion, it should be remembered that the molecules are parts of bodiesthat are so small that it is impossible for us to perceive them indi-vidually. We can therefore not act on single molecules by themselves, r obtain effect from individual molecules by themselves, but rather,in every action that we exert on a body or receive from it, we havesimultaneously to do with an immensely large collection of molecules,which move in all possible directions and with all the speeds occurringamong the molecules, and participate in the action uniformly, in sucha way that there occur only random differences, which are subject tothe general laws of probability. This circumstance forms precisely thecharacteristic property of that motion which we call heat, and on itdepends the laws that distinguish the behavior of heat from that ofother motions.If now demons intervene, and disturb this characteristic propertyby distinguishing between the molecules, and molecules of certainspeeds are permitted passage through a partition, molecules of otherspeeds refused passage, then one may no longer regard what happensunder these conditions as an action of heat and expect it to agree withthe laws valid for the action of heat. James Clerk Maxwell
Available energy is energy which we can direct into any desiredchannel. Dissipated energy is energy we cannot lay hold of and directat pleasure, such as the energy of the confused agitation of moleculeswhich we call heat. Now, confusion, like the correlative term order, isnot a property of material things in themselves, but only in relationto the mind which perceives them. A memorandum-book does not,provided it is neatly written, appear confused to an illiterate person,or to the owner who understands thoroughly, but to any other personable to read it appears to be inextricably confused. Similarly thenotion of dissipated energy could not occur to a being who could notturn any of the energies of nature to his own account, or to one whocould trace the motion of every molecule and seize it at the rightmoment. It is only to a being in the intermediate stage, who can layhold of some forms of energy while others elude his grasp, that energyappears to be passing inevitably from the available to the dissipatedstate (1877, p. 221, in Niven 1890, p. 646).The second law relates to that kind of communication of energywhich we call the transfer of heat as distinguished from another kindof communication of energy which we call work. According to themolecular theory the only difference between these two kinds of com-munication of energy is that the motions and displacements which re concerned in the communication of heat are those of molecules,and are so numerous, so small individually, and so irregular in theirdistribution, that they quite escape all our methods of observation;whereas when the motions and displacements are those of visible bod-ies consisting of great numbers of molecules moving altogether, thecommunication of energy is called work.Hence we have only to suppose our senses sharpened to such adegree that we could trace the motions of molecules as easily as wenow trace those of large bodies, and the distinction between work andheat would vanish, for the communication of heat would be seen tobe a communication of energy of the same kind as that which we callwork. (1878, p. 279, in Niven 1890, p. 669). John von Neumann (1929)If we take into account that the observer can measure only macro-scopically then we find different entropy values (in fact, greater ones,as the observer is now less skilful and possibly can therefore extractless mechanical work from the system) . . . . (von Neumann 2010, p.214, from von Neumann 1929, p. 47).
Harold Grad (1961, pp. 326–27).Whether or not a diffusion occurs when a barrier is removed de-pends not on a difference in physical properties of the two substancesbut on a decision that we are or are not interested in such a differ-ence (which is what governs the choice of an entropy function) . . . Avery illuminating example is given by the “spin-echo” effect. In thisexperiment, it is found that it is possible to produce a highly orderedmicroscopic state and, at a later time, effectively reverse all veloci-ties. To a person who has access to such equipment, a very high level“reversible” entropy will be appropriate; to one who has not, a lowerorder entropy will properly describe all phenomena.
Nicolaas Godfried van Kampen (1984, pp. 306–307). In regardsto the difference in expression of entropies for a uniform sample of gasand a system composed of two different gases, van Kampen wrote,The origin of the difference is that two different processes had tobe chosen for extending the definition of entropy. They are mutuallyexclusive; the first one cannot be used for two different gases and the econd one does not apply to a single gas. But suppose that A and B are so similar that the experimenter has no physical way of distin-guishing between them. Then he does not have the semi-permeablewalls needed for the second process, but on the other hand the firstwill look reversible to him. . . . The point is, that this is perfectly jus-tified and that he will not be led to any wrong results. If you tellhim that ‘actually’ the entropy increased when he opened the channelhe will answer that this is a useless statement since he cannot utilizethe entropy increase for running a machine. The entropy increase isno more physical to him than the one that could be manufactured bytaking a single gas an mentally tagging the molecules by A or B .In fact, this still holds when the experimenter would be able todistinguish between A and B , by means of a mass spectrograph forinstance, but is not interested in the difference because it is not rel-evant for his purpose. This is precisely what engineers do when theymake tables of the entropy of steam, ignoring the fact that it is actuallya mixture of normal and heavy water. Thus, whether such a process isreversible or not depends on how discriminating the observer is. Theexpression for the entropy (which one constructs by one or the otherprocesses mentioned above) depends on whether he is able and will-ing to distinguish between the molecules A and B . This is a paradoxonly for those who attach more physical reality to the entropy than isimplied by its definition. Edward T. Jaynes (1992, p. 5).In the first place, it is necessary to decide at the outset of a prob-lem which macroscopic variables or degrees of freedom we shall mea-sure and/or control; and within the context of the thermodynamicsystem thus defined, entropy will be some function S ( X , . . . , X n ) ofwhatever variables we have chosen. We expect this to obey the sec-ond law T dS ≥ dQ only as long as all experimental manipulationsare confined to that chosen set. If someone, unknown to us, were tovary a macrovariable X n +1 outside that set, he could produce whatwould appear to us as a violation of the second law, since our en-tropy function S ( X , . . . , X n ) might decrease spontaneously, while his S ( X , . . . , X n , X n +1 ) increases. John Goold, Marcus Huber, Arnau Riera, L´ıdia del Rio, andPaul Skrzypczyk (2016, pp. 1–2). f physical theories were people, thermodynamics would be thevillage witch. Over the course of three centuries, she smiled quietlyas other theories rose and withered, surviving major revolutions inphysics, like the advent of general relativity and quantum mechan-ics. The other theories find her somewhat odd, somehow different in nature from the rest, yet everyone comes to her for advice, andno-one dares to contradict her. Einstein, for instance, called her ‘theonly physical theory of universal content, which I am convinced, thatwithin the framework of applicability of its basic concepts will neverbe overthrown.’Her power and resilience lay mostly on her frank intentions: ther-modynamics has never claimed to be a means to understand the mys-teries of the natural world, but rather a path towards efficient ex-ploitation of said world. She tells us how to make the most of someresources, like a hot gas or a magnetized metal, to achieve specificgoals, be them moving a train or formatting a hard drive. Her uni-versality comes from the fact that she does not try to understand themicroscopic details of particular systems. Instead, she only cares toidentify which operations are easy and hard to implement in thosesystems, and which resources are freely available to an experimenter,in order to quantify the cost of state transformations. eferences Bell, J. S. (1977). Free variables and local causality.
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