aa r X i v : . [ m a t h - ph ] F e b Tangible phenomenological thermodynamics
Philipp Kammerlander and Renato RennerInstitute for Theoretical Physics, ETH Z¨urich, Wolfgang-Pauli-Str. 27, 8093 Z¨urich, SwitzerlandFebruary 24, 2020
Abstract
In this paper, the foundations of classical phenomenological thermodynamics are being thor-oughly revisited. A new rigorous basis for thermodynamics is laid out in the main text andpresented in full detail in the appendix. All relevant concepts, such as work, heat, internalenergy, heat reservoirs, reversibility, absolute temperature and entropy, are introduced on anabstract level and connected through traditional results, such as Carnot’s Theorem, Clausius’Theorem and the Entropy Theorem. The paper offers insights into the basic assumptions onehas to make in order to formally introduce a phenomenological thermodynamic theory. Thiscontribution is of particular importance when applying phenomenological thermodynamicsto systems, such as black holes, where the microscopic physics is not yet fully understood.Altogether, this work can serve as a basis for a complete and rigorous introduction to thermo-dynamics in an undergraduate course which follows the traditional lines as closely as possible. ontents Introduction 3
Motivation . . . . . . . . . . . . . . . 3Outline of the paper . . . . . . . . . . 4
Thermodynamics 5
11 Quasistatic processes 46
12 The temperature of arbitrary sys-tems 5313 Example: The ideal gas 55
14 Scaling 60
Discussion and conclusion 66
Summary . . . . . . . . . . . . . . . . 66Main contributions . . . . . . . . . . . 66
Acknowledgements 69
Appendices 70
A Systems 70B Processes and states 71C Work and work processes 72D The first law 77E Equivalent systems 80F Heat and heat reservoirs 89G Carnot’s Theorem 90H Absolute temperature 91I The temperature of heat flows 94J Clausius’ Theorem and thermody-namic entropy 96
References 99 ntroduction The first steps towards a theoretical formulation of thermodynamics have been made more than200 years ago [1–5] and today we are still applying the theory to systems which are studied in thecontext of extending our currently best microscopic theories. An example are black holes, withwhich we hope to find out how to come up with a theory of quantum gravity [6]. In the pastcenturies the applications of thermodynamics have moved from steam engines and locomotives tosystems of almost all orders of magnitude in size and levels of complexity, from single photons tosolar systems and galaxy clusters. And even with a focus on the traditional foundations, for themoment ignoring the most current investigations, it is worth having a closer look at the basis ofa theory which is so universally applicable.
Motivation
Among the many books that have been written and lectures held about the subject [7–24], thetwo common ways to introduce thermodynamics are the phenomenological and the statistical ap-proaches. In the former, the laws of thermodynamics are postulated and the theory is built onthem. In the latter, the macroscopic thermodynamic properties of a system are derived by inves-tigating its microscopic degrees of freedom using methods from statistical physics, e.g. [25]. Inparticular, when taking the statistical approach, one of the main goals is to derive the fundamen-tal laws from microscopic considerations. The work presented here is solely concerned with thephenomenological paradigm. Microscopic properties of a system may be intuitively used at times,but only to give intuition in certain examples, if at all.Even when restricted to the phenomenological approach there exist many different views onthe foundations of the theory and how it should be introduced. This may be unimportant forapplications e.g. in engineering, but the different and seemingly contradicting alternatives areconfusing and unsatisfactory from a conceptual point of view. Modern literature on systematicapproaches to thermodynamics is scarce. An exception is the approach presented in [22, 26, 27].However, in numerous conversations with fellow physicists the authors realized that their uneasyfeeling about the foundations of phenomenological thermodynamics, and especially about how itis taught in undergraduate courses, is shared by many others. Hence this paper.Besides the educational value, the increasing interest in the new research field of quantumthermodynamics is another good reason for looking more deeply into the basics of classical phe-nomenological thermodynamics. Recently various ideas have been proposed on how to use ther-modynamics for a description of small (quantum) systems. In particular, a considerable amountof effort has gone into the quest for finding thermodynamic applications where quantum systemscan outperform their classical counterparts. Nevertheless, the community is still far from reachingan agreement on the conclusions drawn from a quantum analysis of thermodynamic systems. Anexample is the ongoing controversy about the definition of work in the presence of quantum co-herent states [28–30]. In the authors’ view this is also because a fair comparison between classicaland quantum thermodynamics is bound by the limited agreement on the basics of the classicaltheory itself.In this paper we will introduce a new way of laying out the foundations of thermodynamicsallowing for a derivation of the theory along the general lines of the traditional approach. Startingfrom the very basic concepts such as systems, processes and states, we develop a rigorous languageto formulate the first and second laws of thermodynamics, which can be seen as the core postu-lates of the theory. These are then used to prove powerful standard results like Carnot’s Theoremor Clausius’ Theorem, which in turn allow us to define thermodynamic entropy with all its veryuseful properties. 3esides the achievement of making all basic thermodynamic concepts precise without losingcontact to their intuitive meanings, this paper offers insights into the basic assumptions one hasto make in order to formally introduce a phenomenological thermodynamic theory. For instance,we show that the zeroth law is redundant as a postulate, and we capture precisely what propertiesa heat reservoir must have. We also introduce a definition of quasistatic processes arising frominitially discrete considerations which allows one to use the usual formulas for computing internalenergy and entropy differences over piecewise continuously differentiable paths in the state spaceof a thermodynamic system.
Outline of the paper
In Section 1 the notions of thermodynamic systems , processes and states are introduced. Section 2introduces the work cost of thermodynamic processes and related notions, whereas Section 3 usesall terms introduced up to this point to state the first law and derive the internal energy functionfrom it. In Section 4 we intuitively describe the notion of what it means for two systems to be copies of one another, which is made formally precise in the appendix. Section 5 treats the notions heat and heat reservoirs which are then used in Section 6 to state the second law and briefly discuss itsimmediate consequences. The more complex but also more important consequences of the secondlaw, namely Carnot’s Theorem and absolute temperature , are discussed in Section 7 and Section 8,respectively. Using the definition of absolute temperature for heat reservoirs it is then possible todefine the absolute temperature of a heat flow in Section 9, which finally allows us to state andprove
Clausius’ Theorem and introduce thermodynamic entropy in Section 10.Having introduced the very basics of every thermodynamic theory, the paper then offers severalinsights into more common applications and advances the abstract framework in order to applyit to traditional settings. In Section 11 the discrete considerations are extended to account forquasistatic processes with associated continuous and differentiable quantities. This is followed bythe discussion of the notion of temperature for other systems than heat reservoirs in Section 12.The thoroughly worked out example of the ideal gas is presented in Section 13, while furtherexplorations into topics such as scaling thermodynamic systems and the principle of maximumentropy is presented in Section 14. 4 hermodynamics
Postulates: thermodynamic systems, thermodynamic processes and states, concatenationof processes
New notions: atomic systems, involved, composition of systems, state space
Technical results for this section can be found in Appendices A and B.
Summary:
Through postulates the notions systems, composition of systems, thermody-namic processes and their input and output states are introduced. A further postulatemakes sure that two thermodynamic processes can be concatenated. From these postulatesfurther notions, among them disjoint systems, subsystems and state space are defined. Thestates of composite systems are defined by the states of their subsystems. Basic propertiesof the composition operation are discussed as well as the correspondence of systems andcomposition with finite set theory and set union.We start by introducing all basic principles and postulates from which we will derive theconsequences for the theory step by step. At times it is going to be abstract. Table 1 togetherwith other examples discussed in the text compensates for the technical nature of this section bylisting the main concepts together with different examples for an easier intuitive understanding.This table is recycled from our previous paper on the zeroth law [31].Postulates are in orange. Definitions, lemmas, propositions, theorems and other derived con-cepts appear with black titles.The section on the basic principles is about the notions of thermodynamic systems, ther-modynamic processes and thermodynamic states. Admittedly, we could also call them systems,processes and states at this point, without the specification “thermodynamic”, as there is nothingthermodynamic about them yet. The thermodynamic aspects will come in later. At this point,we are concerned with the very basic notions needed to formulate thermodynamic assumptions,such as the first and second laws.
Sometimes physical systems are said to be a “subset of the universe”. For the purposes presentedhere this view, taken literally, is valid. We think of systems as finite non-empty subsets of the“world”. Composing two systems corresponds to uniting the sets. Singletons are seen as indivisiblesystems.
Postulate 1 (Thermodynamic systems) . The thermodynamic world is a set Ω and the set ofthermodynamic systems consists of finite non-empty subsets of the thermodynamic world, S := { S ⊂ Ω | < | S | < ∞} .The structure of systems as sets is illustrated in Figure 1.1. The natural way to compose two systems is to unite the sets. The system composed from S ∈ S and S ∈ S is denoted by S = S ∨ S . For the composition of systems we introduce the notation ∨ instead of ∪ , which isnormally used to denote the union of sets. However, ∨ is nothing else but the union ∪ applied tofinitely many representing sets of the systems. 5 mole of ideal gas A two ideal gases S = A ∨ A ′ , A ˆ= A ′ water tank, N ≫ R intuitivedescription A container filled with an idealgas. It is possible to read off thepressure inside the container andone can vary the volume by mov-ing a piston. Two such ideal gases composed.They can still be addressed individ-ually but now they could also bethermally connected. The system isdescribed by the composition A ∨ A ′ . A very large water tank may bemicroscopically complex but ishere described in the simplestnon-trivial way, by its energy. Itis an approximate reservoir.state spaceΣ S Σ A = ( R > ) ∋ σ = ( p, V )pressure and volume Σ S = ( R > ) ∋ σ = ( p , V , p , V )both pressures and volumes Σ ˜ R = [ E min , E max ] ∋ σ = E (internal) energy processes(examplesfrom P ) • connecting the gas thermallyto another system (irreversiblein general)( p in , V in ) ( p out , V out = V in ) • isothermal expansion or com-pression (generally reversible)( p in , V in ) ( p out , V out ) s.t. p in V in = p out V out • Carnot cycle with the gas • all processes from the left columnapplied to one of the gases individ-ually • thermally connecting the first gasto the second and the second to areservoir (typically irreversible) • connecting the tank thermallyto another system, thereby let-ting them exchange energy (ir-reversible in general) • using the tank in a thermal en-gine together with a cyclic ma-chine and another tankwork processes(examplesfrom P S ) • shaking the gas, or applyingfriction to it (irreversible)( p in , V in ) ( p out , V out = V in )with p out > p in • expanding or compressingthe otherwise perfectly iso-lated gas (reversible, if doneoptimally)( p in , V in ) ( p out , V out ) s.t. p in V γ in = p out V γ out • all processes from the left column • thermally connecting the twogases (irreversible in general) • isolating the gases and compress-ing one while expanding the other(reversible) • applying friction to one of thegases while thermally connectingthem, thereby heating up theother one, too (irreversible) • raising the internal energy of˜ R by doing work on it: thiscould be done for instance byshaking it, applying friction,doing electrical work by let-ting a current flow through aresistancework function For classical mechanical systemsthe infinitesimal work done is al-ways δW = ~F · d ~s . When shakingthe gas or applying friction ~F isthe applied force and d ~s the linesegment of the trajectory alongwhich the force acts. If the vol-ume is changed without frictionthe infinitesimal work done canbe simplified to δW = − p d V . The total work done on the compos-ite system is the sum of the amountsof work done on each individual sys-tem, δW = δW A + δW A ′ , where δW X is defined as in the left columnfor both gases individually. Depending on the mechanismthat is used to heat up the tankthe work function will look dif-ferent. For instance, in the caseof increasing the internal energyby means of an electrical current I ( t ) flowing through a resistance Z the work done on R can bewritten as W = R ZI d t .Table 1: The basic concepts such as states, state spaces, processes, work processes, compositionof systems, work cost functions and (an approximation of) reservoirs are illustrated with simpleexamples, recycled from a previous paper [31]. We think of a truly ideal gas that remains an ideal gas even if V becomes very small and p very high. In thecase of a real gas one could restrict the state space further, e.g. to Σ S = (0 , p max ) × ( V min , ∞ ). If one wants to use the tank as a reservoir, think of the energy range [ E min , E max ] as a relatively small intervalin a much larger spectrum of the system. As long as the energy is in this interval, the tank can exchange energy withother systems while keeping its properties and behaviour relative to other systems invariant (to good approximation).Intuitively one can always achieve this by taking a large enough system. When taking the limit towards an infinitelylarge water tank, N → ∞ such that NV = const . ( V the volume) R ′ becomes an exact reservoir. For instance, whenchoosing E min / max ∝ ±√ N , in this limit the state space Σ R ′ will be the real line, i.e. E min / max → ∓∞ but stilloccupy only a very small part of the actual spectrum of the tank. Reservoirs should be thought of as infinitelylarge systems that do not change their behaviour when exchanging finite amounts of energy with other systems. A compression or expansion of a perfectly isolated ideal gas leads to the known state changes as given in thetable, where γ = c p c V is the heat capacity ratio. A ∅ S = A ∨ A A A S = S ∨ A = A ∨ A ∨ A A S = A ∨ A Figure 1.1: Abstract illustration of the structure of thermodynamic systems: Systems are finitenon-empty subset of the thermodynamic world Ω. The empty set is not a thermodynamic system.Singletons are called atomic systems and are usually called A i to indicate this in the notation.Just like the elements of a set determine the set, the atomic systems contained in a system (as sub-sets) determine the system. Multi-element subsets of Ω are composed systems. The compositionoperation for systems is nothing else but the union of the sets which the systems represent.Since systems are finite sets, the composition operation applied to finitely many systems willoutput a system again. To simplify the notation for the composition of more than two systems wesometimes write S = S ∨ · · · ∨ S n ≡ _ S ∈{ S ,...,S n } S ≡ _ { S , . . . , S n } . (1.1)The set of singletons of Ω denoted by A := { A ⊂ Ω | | A | = 1 } is called the set of atomicsystems . This set is a subset of the set of thermodynamic systems. Non-atomic systems are thosethat are composed of more than one but finitely many different atomic systems, whereas atomicsystems are considered indivisible, i.e. they are not composed of other systems.However, one should not take this as a statement about physical indivisibility. Rather it isan abstract structure the user of the theory needs to choose. As an example, one user coulddefine a container of gas as an atomic system. Another user might say that there are differentcompartments of gas within this container, which could have different pressures and could beseparated into further subsystems. In the latter case, the system would not be atomic. The twousers would then work with different thermodynamic theories.Another aspect shown by this example is the fact that an atomic system need not be small. Thegas container could contain an arbitrary amount of substance and still be seen as an atomic system.The concept of an atomic system is helpful for the definition and postulates regarding basicconcepts such as work or states. When defining and postulating these for atomic systems and inaddition specifying how they behave under composition, these concepts automatically extend toarbitrary thermodynamic systems.Just like we can compose two systems with ∨ (i.e. by set union) we can as well consider the intersection of two systems. This corresponds to taking the intersection of the sets. referring toFigure 1.1, we see for instance that S ∧ S = A .Postulate 1 requires that the empty set is not a thermodynamic system. To some extent, thisis a matter of taste. We do not want “nothing” to be called a system, as it would be difficult toargue that this “nothing” has states on which processes can act. Without states and processesacting on them, one can not do anything physical with it, hence we can exclude it from the set ofsystem from the beginning. 7evertheless, we write S ∧ S = ∅ to say that the two systems S , S ∈ S are disjoint . Thisequality should be seen as notation only, as ∅ is not a system. As an example, in Figure 1.1 wesee that A ∧ S = ∅ .Given a system S ∈ S , its subsystems are defined as its non-empty subsets, i.e. the set ofsubsystems Sub( S ) := { S ′ ∈ S | S ′ ⊂ S } .The set of subsystems Sub( S ) contains S itself since S ∨ S = S . In particular, it does not onlycontain the proper subsystems and is therefore always non-empty. The subsystems of a system S ∈ S , which are in addition atomic, are called atomic subsystems and are summarized in theset Atom( S ) := { A ∈ A | A ∈ Sub( S ) } . Examples from Figure 1.1 are S ∈ Sub( S ) and A ∈ Atom( S ). Just like elements of a set determine the set, the atomic subsystems of a systemdetermine the system. In a more formal notation that is, for any S ∈ S it holds S = _ Atom( S ) . (1.2)We conclude the abstract introduction of systems with some comments. First, a thermo-dynamic system is interpreted as a specific physical system rather than a type of system. Forinstance, A may stand for the specific box containing one mole of gas standing in front of us,whereas A may be the label for a box filled with half a mole of another gas standing in a friend’slab. Nevertheless, one needs to be able to talk about systems of the same type, for instance whenthe two boxes in the different labs are identical in construction and contain the same amount of thesame gas. The notion of two different systems being of the same type will be introduced later inSection 4 based on their thermodynamic properties. When two systems of the same type are dis-cussed, their labels A and A can be seen as their names, while their type is their thermodynamicDNA. Second, we think of a single thermodynamic system S as closed. Nevertheless, it is not neces-sary to formalize the term “closed system” as the coming postulates make sure that the propertiesof a thermodynamic system are such that the intuition one has about closed systems is capturedalso in this framework.Third, it is important to keep in mind that composing two systems does not mean that theyare thermally or otherwise coupled. So far composition is independent of any thermodynamicproperties. Two systems composed are simply described as one new system. When disjointsystems are composed, we speak of disjoint composition . In this case the composition is akinto the tensor product structure from quantum theory, where H ⊗ H allows for a compositedescription of the two systems with associated Hilbert spaces H and H but does not makeany statements about possible interactions between the subsystems. The composition of disjointsystems is the one that is more familiar with the usual ways of composing two systems in physics.Many concepts concerning composition that will be introduced are about disjoint composition. After having introduced the structure of systems, atomic systems and subsystems, as well as theirinterrelations, we move on to the postulate about thermodynamic processes and states.
Postulate 2 (Thermodynamic processes and states) . The non-empty set of thermodynamic pro-cesses (also simply processes ) that the theory allows for is denoted by P . A thermodynamicprocess p ∈ P specifies the initial and final states of a finite and non-zero number of involved atomic systems A ∈ A by means of the functions ⌊·⌋ A : P → Σ A and ⌈·⌉ A : P → Σ A . If A is notinvolved the two functions are undefined.The notion of being involved is sometimes also phrased the other way around. That is, insteadof saying that A ∈ A is involved in p ∈ P , we may say that p acts on A . In a previous paper [31] the introduction and interpretation of systems was different. Instead thinking ofsystems as types , we now think of them as specific instances. Identical twins need names, too. It is not possible to distinguish them just by their DNA. p ∈ P , theprocess has a well-defined initial and final state captured by the functions ⌊·⌋ A and ⌈·⌉ A , respec-tively. Since these functions are always either both defined or both undefined, we will only talkabout the input state whenever we discuss the question whether a certain atomic system is in-volved in the process or not. For an atomic system A ∈ A the co-domain of the functions ⌊·⌋ A and ⌈·⌉ A is the union of their well-defined outputs and denoted by Σ A . It is called the set of states of A .We want the set of states of different atomic systems to be disjoint. W.l.o.g. if A , A ∈ A aredifferent, A = A , then Σ A ∩ Σ A = ∅ . If this was not the case, simply extend the descriptionsof the states with a unique label for each atomic system, e.g. its name. This assumption willallow us to easily extend the definition of state changes for states of composite systems. It alsosimplifies the analysis of state spaces of equivalent systems in Section 4.We do not explicitly require the states to be what is usually called an “equilibrium state”[19, 20, 22]. The way we introduce the theory, thermodynamics does not rely on a basic notion of“equilibrium”. Without the necessity to define this term, we are able to treat states as thermo-dynamic states which could intuitively be seen as non-equilibrium states. For instance, one couldthink of a system that has fluctuations on the time scale of ∼ ∼ A contains a continuum of states. This may very well be thecase for many examples, but it is not a necessary assumption for the theory. The concept of astate is rather abstract up to here. It will become more comprehensible with further postulatesand results derived from them.Since processes are seen as procedures there may very well exist more than one process withthe same input and output states. The initial and final states of a thermodynamic process donot contain all thermodynamically relevant information associated with it. It is helpful to keep inmind this view of processes in the remainder of the paper.For an arbitrary thermodynamic system S = A ∨ · · · ∨ A n ∈ S with pairwise disjoint atomicsystems { A i } ni =1 the state change under a thermodynamic process p ∈ P is given by the statechanges on its atomic subsystems in terms of the functions ⌊·⌋ S : P → Σ S p
7→ ⌊ p ⌋ S := {⌊ p ⌋ A , . . . , ⌊ p ⌋ A n } (1.3)and likewise for ⌈·⌉ S . States of a non-trivially composite system are called joint states .As a consequence of this definition and the fact that different atomic systems have disjointstate spaces, the state space of a disjointly composite system can be seen as the Cartesian productof its subsystems, up to ordering. Furthermore, the initial state on ⌊ p ⌋ S is defined if and only if ⌊ p ⌋ A is defined for all atomic subsystems A ∈ Atom( S ). This, on the other hand, is the case if9nd only if the final states ⌈ p ⌉ A are defined for all atomic subsystems A ∈ Atom( S ) and thus ⌈ p ⌉ S is defined if and only if ⌊ p ⌋ S is.To simplify the notation we use the same symbol for joint states as we used for composite sys-tems. That is, we write ⌊ p ⌋ S ≡ W A ∈ Atom( S ) ⌊ p ⌋ A ≡ ⌊ p ⌋ A ∨· · ·∨⌊ p ⌋ A n for S = A ∨· · ·∨ A n . For twodisjoint systems S ∧ S = ∅ with σ ∈ Σ S and σ ∈ Σ S this reads σ ∨ σ = σ ∨ σ ∈ Σ S ∨ S .The fact that ∨ is commutative when composing states is no issue as state spaces of differentsystems are disjoint, i.e. it is always clear which symbol ( σ or σ ) describes the state of whichsubsystem ( S or S ).Notice that the terms involved and not involved cannot be directly extended to arbitrary sys-tems. These consist of atomic systems, but it could happen that an atomic subsystem is involved ina process, while another one is not. An arbitrary system containing the two would then intuitivelybe involved. However, its state change is undefined, as there is at least one atomic subsystemwhose state change is undefined.The view of composite systems as independent but collectively described subsystems is manifestin the representation of the state space of composite systems. It allows for all combinations ofsubsystem states, not just for pairs which are in “equilibrium” with each other – a term that isanyway not defined at this point. Consequently, states of subsystems in composite systems canstill be separately treated. The subsystems’ states σ ∈ Σ S and σ ∈ Σ S can be extracted asthe corresponding entries of the “tuple” σ ∨ σ ∈ Σ S ∨ S . In particular, this implies that if ⌊ p ⌋ S is defined for a (composite) system S , then ⌊ p ⌋ S ′ is automatically well-defined for all subsystems S ′ ∈ Sub( S ) as well. Likewise, if all subsystems of S have a well-defined state change in somethermodynamic process, then so does S .On the other hand, the structure of state spaces also makes clear that if a proper subsystem S ′ ∈ Sub( S ) has well-defined state changes in a thermodynamic process p ∈ P this does not implythat the same holds for S . There might exist other subsystems of S with undefined state change.We do not aim at describing correlations between thermodynamic systems in this framework.The Cartesian product of two state spaces does not allow for a representation of correlations.Nevertheless, this is not to say that correlations between thermodynamic systems do not appear.Our description just ignores them by describing the total state of a composite system as thecollection of the reduced states of its subsystems. Consequently, correlations between thermody-namic systems cannot be exploited by thermodynamic processes – they have to function whethercorrelations are present or not. Even though it may sound paradoxical, this view does not auto-matically exclude applying the theory to problems regarding the thermodynamics of correlations inside a thermodynamic system (e.g. [32]). For instance, thinking of two qubits that may becorrelated, these correlations may have a thermodynamic interpretation captured by the theory.However, thermodynamically the two “quantum-subsystems” would then have to be summarizedin a single atomic system. Otherwise the correlations would necessarily have to be neglected inour thermodynamic description of the state.This example shows that the thermodynamic subsystem structure introduced by ∨ must notbe confused with the subsystem structure from an underlying physical theory (such as quantummechanics). We conclude the section with one more postulate. Individual thermodynamic processes are obvi-ously necessary objects in order to formulate the theory. However, individual processes are notenough, as in many protocols it will be crucial to be able to describe what happens if one process isexecuted after another. For this, we postulate the existence of a concatenation operation, similarto concatenating functions. The result of a concatenated process will be another process, namelythe one which consists of the consecutive execution of the two processes that are concatenated.10 ostulate 3 (Concatenation of processes) . Let p, p ′ ∈ P such that for all A ∈ A p ∩ A p ′ it holds ⌈ p ⌉ A = ⌊ p ′ ⌋ A . Then p and p ′ can be concatenated to form a new process denoted by p ′ ◦ p ∈ P ,which represents the consecutive execution of p followed by p ′ . If A p ∩ A p ′ = ∅ , then in addition p ′ ◦ p = p ◦ p ′ , i.e. concatenation commutes. An atomic system A ∈ A is involved in the concatenatedprocess p ′ ◦ p if and only if A ∈ A p ∪ A p ′ . For the involved atomic systems the initial and finalstates are ⌊ p ′ ◦ p ⌋ A = ( ⌊ p ⌋ A , if A ∈ A p ⌊ p ′ ⌋ A , otherwise (1.4)and the final state follows the same rules with swapped roles for p and p ′ .If there is an atomic system for which the initial state of p ′ and the final state of p are de-fined but do not match, it is not possible to concatenate the two processes. Hence, in this sense, ◦ : P × P → P is a partially defined function and P is closed under ◦ .The assignment of input and output states for concatenated thermodynamic processes canintuitively be justified as follows. If input and output states of both p and p ′ are defined for anatomic system A then the input state of the concatenation p ′ ◦ p of the two processes must beequal to the input of p on A and accordingly the output must be equal to the output of p ′ on A .Now suppose both p and p ′ have undefined input and output states on A . This is interpretedas “system A is involved in neither of the processes p and p ′ ”. Thus, the concatenation of the twoprocesses does not act on A either and its input and output states are undefined as well.Turning to the case where only one of the processes, say p , has undefined input state (and thusalso undefined output state), it helps to think of an undefined state as a variable. The input andoutput state of p on A are in this case not determined as A is not involved in p . However, whenconcatenating p with p ′ to p ′ ◦ p the total process does act on A , since p ′ does. Therefore the inputstate on A of the concatenation, which was variable according to p , is set to be the input state of p ′ . In the same way it is argued that for undefined ⌈ p ′ ⌉ A the output state ⌈ p ′ ◦ p ⌉ A is set to beequal to ⌈ p ⌉ A if the latter is defined.Every thermodynamic system is a composition of atomic systems and reduced states deter-mine the global state of a composite system. Therefore, the rules for initial and final states ofconcatenated processes for atomic systems translate to rules for arbitrary systems.This concludes the first section with basic postulates on the structure of the theory. A moreformal discussion of processes and states can be found in Appendix B. We now move on to the“real thermodynamic” interplay of systems, states and processes.11 Work and work processes
Postulates: work for atomic systems, additivity of work, freedom of description
New notions: work function, work process, identity process, cyclic, catalytic andreversible process
Technical results for this section can be found in Appendix C.
Summary:
Two postulates guarantee that for any atomic system there exists a workfunction assigning the work cost of a process on that system and that this function isadditive under concatenation. The notion of a work function is then extended to exist forall systems such that it is additive under composition. The terms work process, identityprocess, cyclic process and catalytic process are introduced. Regarding catalytic processes,a further postulate guarantees that a catalytic system can also be left out of the descriptionof a process. The final postulate of this section essentially says that we have a freedomin where to draw the line between things that we explicitly describe in the theory andmeans that may be used to implement processes, but are not modelled in the theory. Thedefinition and discussion of reversible work processes concludes the section.So far we have discussed many aspects of systems, states and to some extent processes. How-ever, the thermodynamic component in these considerations has been missing. This changes now,as we introduce work cost functions. They open the discussion on energy flows and will lead tothe laws of thermodynamics.
Postulate 4 (Work) . For any atomic system A ∈ A exists a function W A : P → R that mapsa thermodynamic process p to W A ( p ), the work done on system A by performing p . The value W A ( p ) is positive whenever positive work is done on A while executing p . If the system A is notinvolved in p , A / ∈ A p , then W A ( p ) = 0 necessarily.Work is seen as the form of energy which can be controlled perfectly. It is in the operator’scontrol to decide when what amount of work flows into or out of a specific system, and in whatform this work is done on or drawn from it. She does so by deciding which state change is inducedin what manner, i.e. by choosing the process to be implemented. The thermodynamic processcontains the information about the work flows exchanged with the involved systems.How to compute the work W A ( p ) of a certain process p does not follow from the thermodynamictheory. On the contrary, it is something that the theory takes as an input. Typically the workcost is computed using a more fundamental theory such as classical mechanics, electrodynamicsor maybe even quantum mechanics. For examples see Table 1.In some texts, e.g. in [19], work is termed the energy coming from “work reservoirs”. Here,the concept of a work reservoir is not fundamental and the physical system from which the energytermed work comes is usually not modelled explicitly as a thermodynamic system. On the otherhand, an explicit modelling is not excluded either. The choice of what to explicitly include in thethermodynamic description is up to the user of the theory, as long as all postulates are satisfied. Postulate 5 (Additivity of work under concatenation) . If for two processes p, p ′ ∈ P the con-catenation p ′ ◦ p is well-defined, then the work cost of the concatenated process equals the sumof the work costs of the individual processes. That is, for all atomic systems A ∈ A it holds that W A ( p ′ ◦ p ) = W A ( p ) + W A ( p ′ ) is additive . 12or atomic systems that are neither involved in p ′ nor p this statement is trivial, as 0 = 0 + 0always holds. However, for atomic systems that are involved in at least one of the concatenatedprocesses the statement is important to relate the work costs of the individual processes to theone of the concatenated process. Additivity under concatenation supports the interpretation ofwork as a currency. If a quantity of work is “invested” now and some other quantity after that,in total the sum of the two has been invested.Based on the work cost function for atomic systems it is possible to define the work costfunction for arbitrary systems in S in an intuitive way. Definition 2.1 (Work function for arbitrary systems) . Let S ∈ S be an arbitrary thermodynamicsystem. We define its work cost function (also simply work function ) W S : P → R by W S = X A ∈ Atom( S ) W A . (2.1)The work function W A ( p ) for atomic systems yields the work done on A during a process p .Hence, the work done on an arbitrary system S is the sum of the work done on all its atomicsubsystems. By defining the work cost function of an arbitrary system as such we automaticallyobtain additivity for disjoint systems S , S ∈ S (Lemma C.2. That is, for all p ∈ P it holdsthat W S ∨ S ( p ) = W S ( p ) + W S ( p ). Furthermore, additivity under concatenation naturally ex-tends to arbitrary systems (Lemma C.3)). If p ′ ◦ p is defined, then for all S ∈ S it holds that W S ( p ′ ◦ p ) = W S ( p ) + W S ( p ′ ).The existence of work cost functions W S for all thermodynamic systems S ∈ S gives insightsinto the interpretation of the structure of systems as the following example illustrates. Example 2.2 (Different views on the structure of systems) . Consider Figure 2.1, where twodifferent views on a scenario are depicted. The panels (a) and (b) show the same physical situationwith different thermodynamic descriptions.In (a) there are two systems A and A , depicted as cylinders filled with gases, each with apiston through which work can be done on or drawn from the gas. The individual work functions W A and W A sum up to the total work cost on the composite system S = A ∨ A . Importantly,by means of the transmission it is possible to address the cylinders individually, in particularthe individual work functions are both well-defined. A thermodynamic process involving S mustspecify what part of the total work W S goes into which subsystem, e.g. in terms of determining thetransmission ratio. States of S specify the states of the subsystem and have the form σ = σ ∨ σ for σ i ∈ Σ A i .In (b) the physically identical system to (a) is described by a thermodynamic system A whichcannot be decomposed into subsystems, i.e. A is atomic. The user of the framework may notknow about the more complex structure of A or she may just ignore it. There is still a pistonthrough which work can be done on or drawn from A . However, the transmission ratio from (a)is fixed and there are no individual work functions W A i as there are no subsystems A i that couldbe addressed individually.Importantly, both views (a) and (b) lead to valid thermodynamic descriptions within theframework, as long as the chosen view is held on to consistently. Which view to take is up to theuser of the theory.We conclude that in general, by Postulate 4, if a system has subsystems then in any thermo-dynamic process the splitting of the total work is determined and one can compute the work doneon any of the subsystems. Among all thermodynamic processes that act on a system S , there are those that act exactly on S and on no other system. These processes deserve special attention as they are the ones addressedin the first law. 13 a) (b) A W A A W A tr a n s m i ss i o n W S = W A + W A S = A ∨ A A W A A W A W A A Figure 2.1: Two different thermodynamic descriptions of the same physical setting. Both viewsare valid thermodynamic descriptions and the user has to decide which one is taken. The usermust then consistently work with the chosen view. (a) The total system is seen as a compositesystem S = A ∨ A . A process must thus specify what amount of the total work W S goes towhich subsystem, e.g. by determining the transmission ratio. For instance, the total work donecould be zero, but the work W A is drawn from the subsystem A and W A = − W A is done on A . Importantly, each subsystem can individually be addressed. (b) When describing the samesetting as an atomic system A , i.e. a system without further subsystems, there is no transmissionratio to choose. Only the total work W A has a meaning and the internal structure of A , howevercomplex it may be, is neglected in the description. Definition 2.3 (Work process) . For S ∈ S a process p ∈ P is a work process on S if all its atomicsubsystems are involved in p and no other atomic systems are. That is, p is a work process on S if S = W A p . The set of work processes on S is denoted by P S .In simple words the above definition says that a process is called a work process on a system S if S is the biggest involved system in p . It is not difficult to see that the set P S is closed underconcatenation of work processes (Lemma C.5).Any thermodynamic process can be seen as a work process on a large enough system. This isa direct consequence of the fact that for any thermodynamic process p ∈ P the set A p is finiteand non-empty, which is part of Postulate 2 on thermodynamic processes and states. The systemdefined as S := W A p is then such that p ∈ P S . Importantly, as the composition of finitely manyatomic systems, S ∈ A is indeed a system.This observation makes work processes the central tool to understand thermodynamics. Theyare the basic building blocks necessary to consistently describe thermodynamic processes. If weunderstand work processes, we understand all thermodynamic processes. Also it allows us tomake definitions and statements for work processes that, if necessary, can then be extended tonotions and more general statements for thermodynamic processes. For instance, work processesare relevant in the formulation of the first law, but of course the first law has consequences forany thermodynamic process.The view of thermodynamic processes as work processes on a large enough system is reminis-cent of completely positive trace preserving maps and unitary maps in quantum mechanics, wherethe Stinespring dilation makes sure that for any cptp map can be seen as a unitary map on alarger system. However, the analogy does not go as far as telling us which of the thermodynamicprocesses are reversible, as is the case with unitary and non-unitary evolution in quantum me-14hanics.We next consider the special case of two disjoint systems S ∧ S = ∅ . Let p i ∈ P S i for i = 1 , S and S , respectively. Such work processes can always be concate-nated both as p ◦ p and p ◦ p since the sets of involved atomic systems are disjoint. Namelythey are Atom( S ) for p and on Atom( S ) for p . Also, Postulate 3 on concatenation requires p ◦ p = p ◦ p in this case. Furthermore, by construction of the concatenation operation, theatomic systems involved in p ◦ p are exactly Atom( S ) ∪ Atom( S ) = Atom( S ∨ S ). Therefore,the concatenated process p ◦ p = p ◦ p ∈ P S ∨ S is a work process on S ∨ S . For such a jointwork process we use the notation p ∨ p := p ◦ p (Definition C.6). On the other hand, usingthis notation means that we are dealing with work processes and systems of the kind discussed inthis paragraph.This notation implies an embedding (an injective mapping) from P S × P S to P S ∨ S andimages of this mapping ( p , p ) p ∨ p stand for the parallel execution of the work process p on subsystem S and of p on S . Hence, what was achievable by means of work processes onthe individual systems S and S can still be realized as work processes on the composite system S ∨ S . In this sense, composition respects work processes.As a further consequence, the input and output states of a joint work process p ∨ p are givenby ⌊ p ∨ p ⌋ S ∨ S = ⌊ p ⌋ S ∨ ⌊ p ⌋ S ∈ Σ S ∨ S and likewise for ⌈·⌉ . While the state space of acomposite system consists of joint states only, this is not the case with work processes. In general,the set P S ∨ S contains more work processes than just the joint work processes of its subsystems.An example of a more general work process on a composite system is thermally connecting twosubsystems and letting them exchange energy.For a joint work process p ∨ p the total work cost W S ∨ S ( p ∨ p ) is the sum of the localwork costs, i.e. W S ( p ∨ p ) = W S ( p ) + W S ( p ) (Lemma C.7). This is a consequence of boththe additivity of work under concatenation and under composition (for disjoint systems). A special kind of work processes on a system are identity processes . They are defined in thefollowing.
Definition 2.4 (Identity process) . An identity process on S , where S ∈ S is an arbitrary ther-modynamic system, is a work process id S ∈ P S on S with ⌊ id S ⌋ S = ⌈ id S ⌉ S and zero work cost onall atomic subsystems of S , W A (id S ) = 0 for all A ∈ Atom( S ).We sometimes write id σS for an identity process on S with initial and final state σ ∈ Σ S , toindicate on which state the identity process acts. This notation manifests that there is not a singleidentity map that can be applied to an arbitrary input state. An identity process, just like anyother thermodynamic process, determines its input and output state. Identity processes will bediscussed further after the first law.Besides identity processes, there are other ways in which a thermodynamic process can acttrivially on a system. Definition 2.5 (Cyclic and catalytic process) . Given a system C ∈ S an arbitrary thermodynamicprocess p ∈ P is called cyclic on C if ⌈ p ⌉ C = ⌊ p ⌋ C .The process is called catalytic on C if it is cyclic on C and in addition W C ( p ) = 0. The definition of a cyclic process on S which is in addition a work process on S differs from aidentity process on S by the missing requirement on the work costs on atomic subsystems. Notice that the work costs of p for subsystems of C do not have to be zero, only the total work done on C does.
15s a consequence of Definition 2.5, if a process p is cyclic on two systems S and S then it isalso cyclic on their composition S ∨ S . This holds independently of whether the two systems aredisjoint or not. If the systems are in addition disjoint, then the same holds for catalytic processes.Consider now the case of a work process p ∈ P S ∨ C on a disjointly composite system S ∨ C . Ifthe process is cyclic on C it is still possible that non-zero work flows into C occur. Hence, leaving C out of the description would in this case leave open the question where these work flows go.However, if the process is catalytic on C , i.e. W C ( p ) = 0 in addition, one may wonder why sucha process is not considered a work process on S alone. On the one hand technically it is not awork process on S according to Definition 2.3. On the other hand, even though it acts on C , itdoes so in the most trivial sense, and could hence intuitively be considered a work process. Weresolve this issue with an additional postulate on “where to draw the line”, stating that whenevera process is catalytic on a part of a system, there exists a corresponding work process on rest withthe same thermodynamic properties, but without acting on the catalytic part. Postulate 6 (Freedom of description) . For
S, C ∈ S disjoint, let p ∈ P S ∨ C be such that p iscatalytic on C , i.e. p is cyclic on C and fulfils W C ( p ) = 0. Then there exists a work process˜ p ∈ P S on S alone such that ⌊ ˜ p ⌋ S = ⌊ p ⌋ S and ⌈ ˜ p ⌉ S = ⌈ p ⌉ S as well as W A (˜ p ) = W A ( p ) for all A ∈ Atom( S ).This postulate automatically implies that W C ′ (˜ p ) = 0 for all subsystems C ′ ∈ Sub( C ) of C in the new process, since neither of them is involved in ˜ p . Likewise, W S ′ ˜ p ) = W S ′ ( p ) for al lsubsystems S ′ ∈ Sub( S ) since their work costs on general subsystems are computed through thework costs on atomic subsystems. Any other system that was not acted on by p neither is by ˜ p .Also, since the state of a composite system determines the states of all possible subsystems andvice versa, it holds that ⌊ ˜ p ⌋ S ′ = ⌊ p ⌋ S ′ and ⌈ ˜ p ⌉ S ′ = ⌈ p ⌉ S ′ for all S ′ ∈ Sub( S ).Postulate 6 is about where to draw the line between objects that thermodynamics explicitlydescribes and such that are not part of the theory but may nevertheless be used when executing aprocess. More specifically, whether a catalytic system C is mentioned explicitly in the descriptionof the process p , or whether it is suppressed, does not matter. The postulate states that if thereis a process in which C is made explicit, then there is also one in which it is not. The importantpoint about this is that the thermodynamic properties of both p and ˜ p regarding the system S arethe same.This section is concluded with the definition and the discussion of reversible work processes. Definition 2.6 (Reversible processes) . A work process p ∈ P S on a system S ∈ S is called reversible if there exists another work process p rev ∈ P S on S , the reverse work process, such that p rev ◦ p is an identity process.As the definition suggests, given a specific reversible work processes p ∈ P S , there may be morethan one reverse work process. Nevertheless, their thermodynamic effect on the system S are allthe same. In the following we may also just write reversible process and reverse process insteadof emphasizing that we talk about work processes on specified systems. In general, whenever it isclear from the context whether the object is a work process or a thermodynamic process, we willjust write process.Reversibility of arbitrary thermodynamic processes does not require a separate definition. Sinceevery thermodynamic process is a work process on some large enough system, the thermodynamicprocess is called reversible if it is reversible as a work process.While identity processes have a zero work cost on any involved system, the work cost of areverse work process is the negative forward process (Lemma C.11). More precisely, if p ∈ P S is areversible work process on the system S with a reverse process p rev ∈ P S , then W A ( p rev ) = − W A ( p )16or all atomic subsystems A ∈ Atom( S ). This is easily seen since p rev ◦ p is an identity processand thus fulfils 0 = W A ( p rev ◦ p ) = W A ( p ) + W A ( p rev ).In Appendix C it is shown in detail that if a concatenated process p = p ◦ p is reversible, thenso are the processes p and p (Proposition C.14). This result holds for any processes p , p ∈ P as long as their concatenation is defined. In particular, they need not be work processes on thesame or on disjoint systems. 17 The first law
Postulates: the first law
New notions: preorder on state space, internal energy
Technical results for this section can be found in Appendix D.
Summary:
The first law is stated as a postulate. It makes sure that for any two statesthere is a work process transforming one into the other. Furthermore, the total work cost ofwork process must not depend on anything except for its input and output states. Throughthe existence of work processes connecting two states a preorder on the state space isintroduced, which will come in handy later. Due to this preordered structure it follows thatidentity processes exist for any state on any system. Finally, the first law and its immediateconsequences are used to defined the internal energy of a system, which is then shown tobe additive.
The first law will imply that every change in “internal energy” of a system is equal to the “sumof work and heat”. However, in order to state this properly we first need definitions saying what“internal energy” and “heat” is. The first law lays the basics for these definitions.
Postulate 7 (The first law) . For any system S ∈ S the following two statements hold:(i) For any pair of states σ , σ ∈ Σ S there is a work process p ∈ P S on S with ⌊ p ⌋ S = σ and ⌈ p ⌉ S = σ or there is a work process p ′ ∈ P S on S with ⌊ p ′ ⌋ S = σ and ⌊ p ′ ⌋ S = σ .(ii) The total work cost of a work process p ∈ P S on S , W S ( p ), only depends on ⌊ p ⌋ S and ⌈ p ⌉ S and not on any other details of the process.By the first law, the set of states of a system is preordered. A preordered set is a set M together with a relation → such that the relation is (i) reflexive, i.e. ∀ m ∈ M : m → m , and (ii)transitive, i.e. if both m → m ′ and m ′ → m ′′ , then m → m ′′ . Definition 3.1 (Preordered states) . For any system S ∈ S the preorder → on Σ S is establishedby the reachability via a work process, i.e. for σ, σ ′ ∈ Σ S define σ → σ ′ : ⇔ ∃ p ∈ P S s . t . ⌊ p ⌋ S = σ, ⌈ p ⌉ S = σ ′ . (3.1)Processes p ∈ P S can be seen as labels of the preordered pairs. In this sense, if one wants toprecisely state which work process is responsible for the preordering of two states, one can write σ p → σ ′ if the work process p is such that ⌊ p ⌋ S = σ and ⌈ p ⌉ S = σ ′ . As mentioned before, theremay be more than one label for a preordered pair.The relation introduced in Eq. 3.1 is reflexive since for all systems S and all states σ ∈ Σ S there exists work process p ∈ P S such that σ p → σ . This is a consequence of Postulate 7 (i). Fur-thermore, if σ p → σ ′ and σ ′ p ′ → σ ′′ then σ p ′ ◦ p −→ σ ′′ is preordered too by means of the concatenatedprocess p ′ ◦ p . Hence the relation is also transitive, which makes it a preorder.Restricting our considerations to atomic systems A ∈ A for the moment, the reflexivity of thepreorder → offers further insights. Let q ∈ P A be a process which relates σ q −→ σ . Since thisprocess only acts on A and initial and final states match, it can be concatenated with itself. Thestate change of the process in which q is applied twice is obviously the same as the state changeunder q itself. Therefore Postulate 7 (ii) requires W A ( q ) (ii) = W A ( q ◦ q ) = W A ( q ) + W A ( q ) = 2 W A ( q ) , (3.2)18hich is to say that the work cost of such processes is zero. Obviously, this makes it an identityprocess on A for the state σ (Lemma D.3).According to the first law, such processes exists for any state on any atomic system. We deducethat identity processes exist for all atomic subsystems and all states. Using the decomposition ofan arbitrary system into its atomic subsystems, it is then possible to construct identity processfor all states of arbitrary systems. To see this, let S = A ∨ · · · ∨ A n with different atomic systems A i = A j and consider an arbitrary joint state σ = σ ∨ · · · ∨ σ n . For the atomic states we knowthat corresponding identity processes id σ i A i exist. Therefore, the joint process id σS := id σ A ∨ id σ n A n is a cyclic work process on S . But this process also fulfils W A i (id σS ) = 0 for all i = 1 , . . . , n byconstruction, thus it is an identity process. We conclude that identity processes exist for all statesof all thermodynamic systems (Lemma D.4).The existence of identity processes for any state in the theory suggests that what is calleda state here can intuitively be seen as an “equilibrium state”. It implies that it is possible toessentially “do nothing” while the state does not change. Certainly, this is a property one wouldexpect of an object called equilibrium state. Notice that this is a consequence of the first law ratherthan an additional postulate on or definition of equilibrium. Hence, we remain in the position toclaim that no further assumptions are made regarding what can be called a thermodynamic state.As an example of what is usually considered a non-equilibrium state, take the current positionand momentum of an oscillating pendulum as its state. Since both position and momentum areparameters evolving in time even when no action is taken from the outside, such a definition ofthe state would not work here intuitively. Even if we “do nothing” the state would oscillate, i.e.change, and would thus not be considered an “equilibrium state”.In the previous section we mentioned the embedding of P S × P S for two disjoint systems S and S in the set of work processes P S ∨ S . Together with the existence of identity processesfor all states this observation is strengthened. We can now conclude that not only the Cartesianproduct but also the individual sets P S and P S are represented in the set of work processes onthe composite system. For any state σ ∈ Σ S and any work process p ∈ P S the embedding( p , id σ S ) p ∨ id σ S allows for a representation of p in P S ∨ S .We also mentioned that the set of work processes of a disjointly composite system S ∨ S contains more than just joint work processes. With the first law we can now argue for thisstatement more precisely. Consider two systems S and S with states σ , σ ′ ∈ Σ S which arepreordered such that σ → σ ′ but σ ′ σ and likewise for σ , σ ′ ∈ Σ S . I.e. the work processesthat label the preorderings of σ i → σ ′ i are irreversible. Then the joint work processes on S ∨ S will make sure that the joint states ( σ , σ ) → ( σ ′ , σ ), ( σ , σ ′ ) → ( σ ′ , σ ′ ), ( σ , σ ) → ( σ , σ ′ )and ( σ ′ , σ ) → ( σ ′ , σ ′ ) are all preordered, in particular comparable. But then, since the workprocesses labelling these preorderings are irreversible, the joint states ( σ , σ ′ ) and ( σ ′ , σ ) cannotbe preordered in either direction with joint work processes. Therefore, if any two states of anysystem must be comparable, as the first law requires, there must be more than just joint workprocesses in P S ∨ S . The first law eventually guarantees that every system S has a well-defined internal energy func-tion. The existence of such a function relies on the fact that the total work cost of a work processon S only depends on the initial and final state of the process, and not on any other property ofit. This is not true for arbitrary thermodynamic processes on S , which allow for different workcosts on S even though they induce the same state transfer. In general, the work cost depends onhow exactly the process is carried out.We are now in the position to define the internal energy function of a system.19 efinition 3.2 (Internal energy) . For a system S ∈ S fix an arbitrary reference state σ ∈ Σ S and an arbitrary reference energy U S ∈ R . The internal energy of a state σ ∈ Σ S is defined as U S ( σ ) := U S + W S ( p ) , where p ∈ P S is s.t. ⌊ p ⌋ S = σ and ⌈ p ⌉ S = σ . (3.3) U S ( σ ) := U S − W S ( p ′ ) , where p ′ ∈ P S is s.t. ⌊ p ′ ⌋ S = σ and ⌈ p ′ ⌉ S = σ . (3.4)Since only differences ∆ U S of internal energies physically matter, the choice of U S is arbitraryfor now. Likewise, the reference state σ ∈ Σ S is arbitrary independently for each system S . Definition 3.3 (State function) . A state function on a system S ∈ S (also state variable ) is afunction Z : Σ S → Z from the state space Σ S to a target space Z . The co-domain Z is typically R n , most often n = 1.When a system S undergoes a process p ∈ P we denote the change in any state function Z S using an abbreviated notation by ∆ Z S ( p ) := Z S ( ⌈ p ⌉ S ) − Z S ( ⌊ p ⌋ S ). On the left hand side thedependence of ∆ Z S on p may be omitted if the context makes clear which process is meant.The internal energy function U S is a well-defined state function on S for any system S ∈ S (Lemma D.7). Investigating this function further, it follows in Proposition D.8 that it inheritsthe additivity property for composite systems from the work functions. That is, for two disjointsystems S ∧ S = ∅ we always have that ∆ U S = ∆ U S + ∆ U S . Notice that we only talk aboutdifferences in internal energies here since the absolute values depend on the constants U S i of thetwo systems. In general, if Z A is a state variable but A / ∈ A p is not involved in the process p , wewrite ∆ Z A ( p ) = 0. This of course only works if a “minus operation‘” is defined on the co-domain Z . For all practical purposesconsidered here this is the case. Equivalent systems
Postulates: arbitrarily many copies of atomic systems
New notions: thermodynamic isomorphism, equivalence of atomic and arbitrarysystems
Technical results for this section can be found in Appendix E.
Summary:
The notion of equivalent systems (copies of systems) is introduced and shownto be sensible. The intuitive results one would expect from such a notion are explained hereand justified with technical results in Appendix E. Finally, we postulate the existence ofarbitrarily many copies of atomic systems.In the beginning we have emphasized that elements of S are seen as specific physical instancesrather than types of systems. Nevertheless, it will be crucial to be able to talk about copies ofsystems or, in other words, systems of the same type. These are systems that can be interchangedwithout any noticeable thermodynamic differences, even though they are different systems. It ispossible to turn this intuition into a mathematical concept through the notion of a thermodynamicisomorphism. Some ideas presented here are inspired by T. Kriv´achy’s master’s thesis [33] whichthe authors supervised. Definition 4.1 (Thermodynamic isomorphism) . The pair of bijective maps ϕ : P → P , ϕ A : A →A is called a thermodynamic isomorphism if for any thermodynamic processes p, p ′ ∈ P and anyatomic system A ∈ A it holds(i) ϕ ( p ′ ◦ p ) = ϕ ( p ′ ) ◦ ϕ ( p ) whenever the concatenation p ′ ◦ p or ϕ ( p ′ ) ◦ ϕ ( p ) is defined,(ii) A is involved in p if and only if ϕ A ( A ) is involved in ϕ ( p ), and(iii) W ϕ A ( A ) ( ϕ ( p )) = W A ( p ).The requirements on an isomorphism reveal the fundamental structures behind the discussedthermodynamic concepts. Namely these are (i) the thermodynamic processes with concatenation ,(ii) atomic systems , linked to processes through the notion of an atomic system being involved ina process , and (iii) work .Remember that in the beginning we already established that input and output states are al-ways either both defined or undefined. Hence (ii) is equivalent to saying that: ⌊ ϕ ( p ) ⌋ ϕ A ( A ) isdefined ⇔ ⌊ p ⌋ A is defined.Typically when defining isomorphisms of algebraic structures one first introduces the notionof a homomorphism. One can do so, but it would go beyond the purposes presented here.Both mappings, ϕ and ϕ A , are part of the definition of a thermodynamic isomorphism. How-ever, they are not independent degrees of freedom. In Appendix E, where the details of this sectionare discussed, we prove in Lemma E.2 that if both ϕ, ϕ A and ϕ, ϕ ′A are thermodynamic isomor-phisms, then ϕ A = ϕ ′A . In this sense, ϕ A is determined by ϕ and one could think of coming upwith a Definition 4.1 such that it only talks about ϕ , while ϕ A is derived from it afterwards. Eventhough this is possible it would make the definition much less readable and intuitive. Thereforewe do not go further into this. 21he mapping of atomic systems can naturally be extended to arbitrary systems S ∈ S bydefining ϕ S ( S ) := _ A ∈ Atom( S ) ϕ A ( A ) . (4.1)A thermodynamic isomorphism maps all thermodynamic processes and systems to possibly otherthermodynamic processes and systems, while preserving the structure of these sets. Consequently,the two mappings ϕ and ϕ S (or ϕ A , to use the primitive) induce a mapping of states ϕ Σ : S S ∈S Σ S → S S ∈S Σ S by means of ϕ Σ ( ⌊ p ⌋ S ) := ⌊ ϕ ( p ) ⌋ ϕ S ( S ) , (4.2)which can be shown to be bijective, too (Lemma E.6).As it turns out, an isomorphism ϕ maps work processes on a system S to work processes onthe system ϕ S ( S ), the properties of being reversible and being an identity process are preservedunder ϕ , and internal energy changes fulfil U S ( σ ′ ) − U S ( σ ) = U ϕ S ( S ) ( ϕ Σ ( σ ′ )) − U ϕ S ( S ) ( ϕ Σ ( σ )).In short, it can be proved that any thermodynamic structure introduced so far is preserved byisomorphisms, showing that it is indeed a sensible way of defining the concept. It is now possible to consider special isomorphisms to define when two atomic systems A , A ∈ A are copies of each other. Intuitively, this is the case when it is possible to swap A with A bymeans of an isomorphism such that nothing else changes in the theory. The notion of isomorphismsallows us to to make precise what is meant by “nothing else changes”. Definition 4.2 (Equivalence of atomic systems) . Two atomic systems A , A ∈ A are called equiv-alent , and we write A ˆ= A , if there exists a thermodynamic isomorphism ϕ, ϕ A which additionallyfulfils(iv) ϕ A ( A ) = A , ϕ A ( A ) = A and ϕ A ( A ) = A for all A ∈ A r { A , A } , and(v) for A ∈ A r { A , A } it holds ⌊ ϕ ( p ) ⌋ A = ⌊ p ⌋ A and ⌈ ϕ ( p ) ⌉ A = ⌈ p ⌉ A .The points (iv) and (v) in this definition formally capture the additional intuitive requirementson an isomorphism that only swaps A with A . The part of the isomorphism acting on the setof atomic systems shall only swap the two, and all processes shall induce the same state changeas before on any atomic system apart from A and A .Using the results already known from isomorphisms in general, it can then be shown that ˆ=is indeed an equivalence relation on A . Having introduced a notion of equivalence for atomicsystems, we use it to extend the concept of equivalence to arbitrary thermodynamic systems. Definition 4.3 (Equivalence of systems) . Let S , S ∈ S be two arbitrary systems. They are equivalent , and we write S ˆ= S , if there exists a bijection between Atom( S ) and Atom( S ) whichrespects the equivalence classes of ˆ= for atomic systems.Definition 4.3 can be rephrased as: The two systems are equivalent if | Atom( S ) | = | Atom( S ) | =: n and there exists a labelling { A ( i ) k } i =1 ,..,n = Atom( S k ) for k = 1 , A ( i )1 ˆ= A ( i )2 for i = 1 , . . . , n . (4.3)Also for equivalent thermodynamic systems an isomorphism can be defined which fulfils anal-ogous properties to the ones in Definition 4.1 and Definition 4.2. This isomorphism is essentiallythe concatenation of the individual isomorphisms for the atomic equivalences A ( i )1 ˆ= A ( i )2 . Just like22 A A A S A A A Figure 4.1: An example for two arbitrary thermodynamic systems S ˆ= S with n = 3 (Defi-nition 4.3). The equivalent atomic systems are connected with red arrows. They are A ˆ= A , A ˆ= A and the trivial A ˆ= A .before, one has to check that this definition preserves all kinds of thermodynamic properties ofsystems in order to justify calling the related systems equivalent . In particular, now one also hasto take composition of systems into account, which can be done. These technical results are nottricky to derive but cumbersome at times.Having done so in Appendix E we can move on to the main postulate, for which the machinery ofequivalent systems has been introduced. As it turns out, the power of the laws of thermodynamicspartly relies on the assumption that in every situation it is possible to extend any particular settingof systems by “duplicating” certain subsystems, i.e. by adding some other copies of systems to thesetting. For instance, this is important in the proof of Carnot’s Theorem. The proof of Carnot’sTheorem is constructive and uses the fact that systems of the type of those already consideredcan be added to the setting. Postulate 8 (Arbitrarily many copies of (atomic) systems) . Given an atomic system A ∈ A and n ∈ N one may always assume that there exist n different equivalent atomic system { A i } ni =1 ⊂ A , A ˆ= A ˆ= · · · ˆ= A n ˆ= A .This implies that there are in principle infinitely many copies available of any type of atomicsystem. As a consequence, the same must hold for arbitrary systems because they consist ofatomic systems: For any S ∈ S there exist arbitrarily many different systems S ′ ∈ S with S ∧ S ′ = ∅ such that S ˆ= S ′ . However, it does not mean that infinitely many such systems must be present. Rather, it should be interpretedas saying that it is thinkable to have as many copies of an atomic system as one wants. In this sense, there is noupper limit to the number of thermodynamic systems that are thinkable. Heat and heat reservoirs
Postulates: - New notions: heat, heat reservoirs
Technical results for this section can be found in Appendix F.
Summary:
The notion of heat is defined based on the previously introduced notions ofwork and internal energy. Heat is shown to be additive in the same way as work is, i.e.under concatenation and under composition. As a preparation for the second law, heatreservoirs are defined as thermodynamic systems with special properties.In the previous sections we have introduced work and derived the internal energy function fromit. Having these quantities it is possible to define heat.
As the first law (Postulate 7) requires, the total work cost of a work processes on a system S mayonly depend on initial and final state of the process. A general thermodynamic process acting on S may affect more systems than just S and the work cost on S does not only depend on initialand final states. It depends on how exactly the state change is induced. In a typical formulationof thermodynamics this statement amounts to saying that the differential δW S is not exact. Onthe other hand, we have seen that the internal energy U S of a system S is a state function, whichwould be the same as saying that the differential d U S is exact. Hence, the difference of the two,internal energy minus work, gives rise to another quantity that generally depends on the executedprocess and not just on the input and output states. This is what we call heat. It represents thechange in internal energy that is not a consequence of work done on or drawn from the system,i.e. the change in internal energy that is due to less controlled energy flows. Definition 5.1 (Heat) . For an arbitrary thermodynamic process p ∈ P the heat flowing to asystem S ∈ S under p is defined as Q S ( p ) := ∆ U S ( p ) − W S ( p ) . (5.1)Here, ∆ U S ( p ) is defined as ∆ U S ( p ) := X Atom( S ) ∆ U A ( p ) (5.2)which is a natural extension to the usual definition of the additive function ∆ U S for processes inwhich not necessarily all atomic subsystems of S are involved. It follows automatically that if no atomic subsystem of S is involved in p , then Q S ( p ) = 0.Also, Q S inherits additivity under composition (for disjoint systems) from ∆ U S and W S . The sameholds for additivity under concatenation, which is shown by considering atomic systems under aconcatenated process p ′ ◦ p ∈ P and distinguishing the three cases where (i) A is neither involvedin p nor p ′ , (ii) A is involved in one of them, and (iii) A is involved in both of them. Furthermore,heat flows in reverse processes simply change their signs, just like work and internal energy do.This is a direct consequence of the definition of heat. And finally, heat flows of equivalent systemsin equivalent processes are identical, as is shown in Lemma F.2 in Appendix F, together with Recall that if A is not involved in p we denote ∆ U A ( p ) = 0. p in this notation and rearrange Eq. (5.1) we immediately obtain a standard formof the first law [1] stating that the internal energy of a system S may change in a thermodynamicprocess due to work and heat only by means of∆ U S = W S + Q S . (5.3)Hence, the way the first law is introduced in this work through Postulate 7 implies the commonfirst law as in Eq. (5.3). In addition, following the arguments in this framework provides signif-icant advantages over the standard statements. It does not suffer from undefined terms, whichwould be the case when stating Eq. (5.3) directly. Furthermore, the different roles of work andheat as energy contributions to the change in internal energy is not just stated but imprinted inthe previous postulates, definitions and derived results. Work must be the energy the user of thetheory controls, as it must be known how to compute it and how to reuse it a priori . Heat, onthe other hand, is the energy the user is aware of, but he does not control it directly – hence it isdefined as “everything else”.Consider now a work process p ∈ P S ∨ S on a disjointly composite system. Instead of sayingthat the heat Q S ( p ) flows into system S during the process p one can also say that Q S ( p ) flowsout of S or, in other words, that − Q S ( p ) flows into S . For this statement to be compatiblewith Definition 5.1 it is necessary that Q S ( p ) = − Q S ( p ), otherwise the definition applied to thesystem S would lead to a different notion of heat than the intuition explained above suggests.This is in fact the case, since for a work process p ∈ P S ∨ S it holds that W S ( p ) + W S ( p ) = W S ∨ S ( p ) = ∆ U S ∨ S = ∆ U S + ∆ U S . (5.4)Using Definition 5.1 for system S now implies Q S ( p ) = ∆ U S − W S ( p ) = − ∆ U S + W S ( p ) = − Q S ( p ) . (5.5)It is worth emphasizing that Definition 5.1 tells us what amount of heat flows into S , butnot necessarily from which other system. Only in a bipartite setting such as the one discussedin the previous paragraph a statement about where the heat is coming from is possible. As aconsequence, in a more complex composite system it does not make sense to ask what amountof heat flows from one specific (atomic) subsystem to another, unless these are the only involvedones in the process. Nevertheless, the intuition of heat flows between specific subsystems can andwill be used, first and foremost when illustrating processes such as the Carnot process, which actson at least three subsystems – two reservoirs and a cyclic machine. Whenever this intuition isused it will be made clear in what sense. The first example of this shows up in Figure 7.1, whereinternal arrows are used to mark heat flows in a composite system with up to three subsystems.The final paragraphs of this section discuss how these arrows must be interpreted.In order to clarify the distinction between work and heat we here continue and extend Exam-ple 2.2. Example 5.2 (Work and heat flows in a bipartite system) . Consider Figure 5.1, where a compositesystem S = A ∨ A is shown. The subsystems are depicted as cylinders filled with gases and thetotal work W S is distributed over A and A via the transmission mechanism. Furthermore, a heatconductor can be put in place between the gases such that they can directly exchange energy. Athermodynamic process involving S must specify what part of the total work W S goes into whichsubsystem, e.g. in terms of determining the transmission ratio. It also specifies whether, when,and how a thermal contact is established between A and A .Suppose work W > A , e.g. by moving the piston back and forth veryfast, and part of this energy is “passed on” to A . Does the energy flow to A count as work orheat? 25 W A A W A tr a n s m i ss i o n W S = W A + W A S = A ∨ A heat conductor Figure 5.1: Similar to Figure 2.1 the composite system S consists of two subsystems A and A .The total work done on S is distributed to A and A through a transmission ratio specified bythe thermodynamic process that is executed. The process also determines whether and when athermal contact is established between the subsystems.The answer depends on what is meant by “passed on” and can be illustrated by consideringtwo different ways of implementing such a state change. Consider the process p ′ ∈ P S in whichthe work W S ( p ′ ) = W A ( p ′ ) = W is done on A followed by the process p ∈ P S transferring partof this work via the transmission to A , i.e. W S ( p ) = 0 and W A ( p ) = − W A ( p ) >
0. In thiscase, the energy exchanged between A and A is called work and for the total process we find W A ( p ◦ p ′ ) = W − W A ( p ) and W A ( p ◦ p ′ ) = 0 + W A ( p ).On the other hand, if after p ′ the process q ∈ P S is applied, in which the energy flow from A to A happens through the heat conductor (in particular it does not use the controlling mechanismof the transmission), then W A ( q ) = W A ( q ) = 0 and we find a non-zero heat flow Q A ( q ) =∆ U A ( q ) − W A ( q ) = ∆ U A ( q ) − > W A ( p ) = Q A ( q ). We conclude that it depends on the actual process whether energyexchanged between thermodynamic systems is termed heat or work. This is in accordance withthe first law (Postulate 7) which only requires the total work W S to be independent of anythingexcept the state changes. And indeed, in the case when both p ◦ p ′ and q ◦ p ′ induce the samestate change we find W S ( p ◦ p ′ ) = W A ( p ◦ p ′ ) + W A ( p ◦ p ′ )= W A ( p ′ ) + W A ( p ) + W A ( p ′ ) + W A ( p )= W − W A ( p ) + 0 + W A ( p )= W , and W S ( q ◦ p ′ ) = W A ( p ′ ) + W A ( q ) + W A ( p ′ ) + W A ( q )= W + 0 + 0 + 0= W . (5.6)
Definition 5.1 can be applied to an arbitrary system. However, in thermodynamics one oftenmakes use of special systems providing or taking up heat, namely heat reservoirs (also heat baths ).These systems play a central role, not least in the second law and Carnot’s Theorem.A heat reservoir is thought of as a large but simple system. Large here stands for the fact thatits behaviour does not change significantly when moderate amounts of energy are drawn from orgiven to it. Alternatively, this means that it essentially does not matter whether a finite amount26f energy is supplied by one or more copies of the same reservoir. In this sense heat reservoirs areregarded as infinite systems.On the other hand, a heat reservoir is simple to the extent that there is only one macroscopicparameter that defines its state. A heat reservoir’s only purpose is to provide or take up heat.In particular, one should not be able to extract work from such a system in any process.Formally, the set of heat reservoirs of a thermodynamic theory is characterized as follows.
Definition 5.3 (Heat reservoirs) . An atomic system R ∈ A is called a heat reservoir if:(i) For all R ∈ R the internal energy function U R is injective.(ii) For all R ∈ R and all p ∈ P it holds W R ( p ) ≥
0, i.e. there is no thermodynamic process thatextracts work from a reservoir.(iii) For any thermodynamic process p ∈ P that acts on a reservoir R ∈ R , and for any energydifference ∆ U ∈ R , there exists a corresponding process p ′ ∈ P acting on the states of R shifted by ∆ U . That is, W A ( p ′ ) = W A ( p ) for all A ∈ A , ⌊ p ′ ⌋ A = ⌊ p ⌋ A and ⌈ p ′ ⌉ A = ⌈ p ⌉ A forall A ∈ A r { R } , and U R ( ⌊ p ′ ⌋ R ) = U R ( ⌊ p ⌋ R ) + ∆ U as well as U R ( ⌈ p ′ ⌉ R ) = U R ( ⌈ p ⌉ R ) + ∆ U .The set R := { R ∈ A | R is heat reservoir } ⊂ A is called set of heat reservoirs .It follows that if R is a heat reservoir and R ˆ= R ′ , then R ′ is a heat reservoir, too. All re-quirements (i)-(iii) hold either for both R and R ′ or neither of them since they are statementsabout the existence or inexistence of some process and processes of equivalent systems are in 1-1correspondence.The first and the second points state that the description of thermodynamic states of heatreservoirs are simple. The states must be in one-to-one correspondence with the internal energy,i.e. no other macroscopic quantity is needed to describe it, and one cannot extract work from areservoir in any thermodynamic process. In some introductions to thermodynamics this is capturedby saying that heat reservoirs have no “work coordinates” [22, 26] (or that these stay constant,for that matter). This does not yet exclude that one could use a cyclic machine to extract workindirectly from a reservoir by using a heat flow coming from the reservoir. Only the second lawguarantees that even a more sophisticated setting does not allow one to extract work from a singlereservoir.The fact that a heat reservoir cannot produce positive work is an important difference incomparison with work reservoirs. Work reservoirs are sometimes used in traditional approacheswhen a system is needed to explicitly model work and work flows. In the case of heat reservoirs itis excluded that energy done on it as work can later again be used as such.Definition 5.3 (ii) also guarantees that for any two states σ, σ ′ ∈ Σ R with U R ( σ ′ ) ≥ U R ( σ )there is a work process p ∈ P R with ⌊ p ⌋ R = σ and ⌈ p ⌉ R = σ ′ . This can be seen by Postulate 7(i), stating that either σ → σ ′ or σ ′ → σ or both. If σ ′ → σ , the work process on R inducing thisstate change would extract work, which is forbidden. Hence the work process with positive workmust exist.We note that for any reversible process p ∈ P it must hold W R ( p ) = 0 for all R ∈ R . Thisfollows due to the fact that in the reverse process the work done is the negative of the one fromthe forward process. Hence a non-zero work cost for reservoirs in a reversible process would haveto violate (ii) either in the forward or the reverse process.Point (iii) formally captures the statement that a heat reservoir is invariant under translationsof its internal energy. Keeping in mind that internal energy is injective for heat reservoirs, (iii) Since heat reservoirs are large systems they are of course not simple in a microscopic sense. On the contrary,we know that the larger the system, the more complex it gets when one tries to describe the interplay betweenits microscopic degrees of freedom. However, when saying simple, we here mean that the used thermodynamicdescription is simple. In our previous paper [31], which focussed on the zeroth law, this point was phrased in very different terms.In the remainder of this work it will become clear that the previous requirement can be derived from this lesscomplicated and more intuitive one. S S S W W W Q Q (b) S S S W W W Q ′ Q ′′ Q ′′′ Figure 5.2: The three systems S , S , S ∈ S undergo a process p ∈ P S ∨ S ∨ S with work flows W S i ( p ) =: W i and heat flows Q S i ( p ) =: Q i . Since the internal energy is a state variable, thesum of work and heat flows into a cyclic systems must be zero, i.e. W + Q = 0. While externalwork arrows are not arbitrary since they are determined by the work functions W S i , internal heatarrows have a freedom. This is illustrated by the difference between (a) and (b). (a) suggests thatthe heat flows into S and S are coming from S directly, while (b) says that the heat Q ′′′ flowsbetween S and S . Both representations are valid as long as the sums of the internally exchangedheat flows satisfy Q ′ + Q ′′ = Q ≡ − Q − Q , Q ′′′ − Q ′ = Q , − Q ′′ − Q ′′′ = Q . Internal arrowscould only be argued to be unique in this setting if p was seen as a concatenated process p = p ◦ p with e.g. p i ∈ P S i ∨ S . In this case, the heat flows could be split up into the ones flowing duringthe individual processes p i .can be read as “What is possible with some initial state is possible with any.”An obvious one is that reservoirs are infinitely large systems in the sense that their spectrum is( −∞ , ∞ ). Even though this seems unphysical at first sight, this is how we think of heat reservoirs.They are seen as infinitely big systems that do not change their behaviour under finite changesof energy. Of course, for all practical purposes, a system with approximate characteristics (i)-(iii)which is much larger than all other involved systems can serve as a heat reservoir. But in thetheoretical modelling the rigorous treatment asks for an “infinite” system.A further important observation is that requirement (iii) for heat reservoirs asks for the exis-tence of processes independent of the initial states, i.e. for all initial states. This means that theactual state of a reservoir, and hence its actual internal energy, does not have an influence on whatcan be done with it. The reservoir’s characteristics are in this sense independent of its currentstate. This is the reason why, from now on when using reservoirs, we will not discuss their statesin any more depth.Arguably, the formulated assumptions on heat reservoirs are neither new nor surprising. Theyare mostly standard assumptions, see e.g. [26], that may not even be spelled out in certain textson thermodynamics. However, here they are central for the precise arguments given in the comingsections.In particular, stating the requirements on heat reservoirs explicitly is also necessary in orderto formulate the second law according to Kelvin [3], as is done in the next section.We close this section with a comment on the pictorial representation of heat flows. Whenillustrating a process on a composite system we use thick lines to border reservoirs and thin linesfor other systems (see e.g. Figure 7.1). Circles border cyclic systems while squares leave openwhether the system involved undergoes cyclic evolution or not. Directed arrows mark positivework (external) and heat (internal) flows.Importantly, arrows showing heat flows of single processes in more complex structures haveno direct mathematical meaning since heat flows are not uniquely defined except when exchangedbetween exactly two subsystems. Only the sum of all internal arrows associated to a subsystemdoes have a mathematical meaning. The internal arrows nevertheless help to map the abstractprocesses to well-known situations such as Carnot engines. For an example, see Figure 5.2. Bothillustrations show the same process p ∈ P S ∨ S ∨ S on the composite system S ∨ S ∨ S . While28a) suggests that only the heat Q i flows from S to S i , (b) says that the heat Q ′′′ from S to S . The mathematical formalism leaves open which of the possibilities actually happen – they areconsidered thermodynamically equivalent as long as for each system ∆ U i = W i + Q i is satisfied.This ambiguity is not a problem. On the contrary, it is an asset of our framework that it ispossible to make the usual thermodynamic statements without ever having to refer to a technicalnotion of a “heat flow from S to S ” in a composite system involving other systems ( S ) as well.When we use such a wording nevertheless, it is either a non-technical statement appealingto the reader’s intuition, or we talk about concatenated processes. In the latter case, heat flowsbetween subsystems may have a mathematical meaning even though other subsystems are presentas well. This is the case if the process can be split up into parts, each of which involves only twosubsystems. Thus the framework is not restricted by the fact that we cannot in general give amathematical meaning to the heat flow arrows. If one wants to say that heat flows from or to aspecific system, one can (at least sometimes) do it by splitting the process into appropriate parts.We will explicitly comment on this when the situation shows up.29 The second law
Postulates: the second law
New notions: - Summary:
The second law is postulated in the form of the Kelvin-Planck statement andits immediate consequences are discussed.Considered to be the core postulate of phenomenological thermodynamics by many, the secondlaw certainly takes a central role in a theoretical introduction to the theory. The most prominentformulations are due to Carnot [7], Clausius [4], Kelvin [3] and Planck [8]. The similar statementsby Kelvin and Planck, sometimes called the Kelvin-Planck statement, can be summarized by “Itis impossible to devise a cyclically operating device, the sole effect of which is to absorb energyin the form of heat from a single thermal reservoir and to deliver an equivalent amount of work.”It shares the problem with the other versions by Clausius and Carnot that some terms used (e.g.“sole effect”, “heat” or “reservoir”) are ambiguous if no further specifications are made. Withthe notions and definitions we have made up until here we are now able to formally state theKelvin-Planck version of the second law.
Postulate 9 (The second law) . Consider a heat reservoir R ∈ R together with an arbitrarysystem S ∈ S and a work process p ∈ P R ∨ S on the composite system. If p is cyclic on S , then W S ( p ) ≥
0, i.e. no work can be drawn from S in such a process.The setting of the second law is shown in Figure 6.1. In the formulation of Postulate 9 theproblems of the above mentioned versions of the second law do not occur. The “sole effect” iscaptured in the fact that the process p ∈ P S ∨ R we talk about is a work process on S ∨ R . In ad-dition, the terms “heat” and “reservoir” have been properly defined beforehand and can thereforebe used now without danger of confusion.A total amount of work W R ∨ S ( p ) = W R ( p ) + W S ( p ) is done on the composite system R ∨ S ,where W R ( p ) ≥ S is assumed to be cyclicunder p , i.e. ⌈ p ⌉ S = ⌊ p ⌋ S , the internal energy of S does not change, ∆ U S ( p ) = 0. Hence the workdone on S is equal to the heat flowing from S to R , which is denoted by Q R ( p ). Therefore, thesecond law can be rephrased as “In a process on R ∨ S that is cyclic on S heat can only flow from S to R and not in the other direction”, which is precisely the Kelvin-Planck statement. RS W R ( p ) Q R ( p ) W S ( p )Figure 6.1: The setting for the Kelvin-Planck statement of the second law, Postulate 9.We consider a special case of the setting by assuming for the moment that the process p isreversible in addition. Let p rev ∈ P R ∨ S be a reverse process of p . Both p and p rev fulfil therequirements stated in Postulate 9, that is, both are cyclic on S . Hence both must be such thatno positive heat flows from R to S , which is equivalent to W S ( p ) ≥ W S ( p rev ) ≥
0. However,this implies that for reversible such processes W R ∨ S ( p rev ) = W R ( p rev ) | {z } ≥ + W S ( p rev ) | {z } ≥ = − W R ∨ S ( p ) = − W R ( p ) | {z } ≥ − W S ( p ) | {z } ≥ (6.1)30ince the total work cost of a reverse process is the negative of the forward process. This equalitycan only be fulfilled if all terms are zero, which implies that also Q R ( p ) = − Q S ( p ) = 0 and Q R ( p rev ) = − Q S ( p rev ) = 0.We conclude that reversible processes on a composite system R ∨ S that are in addition cyclicon S must have a trivial heat flow between R and S , and that if the processes ought to be re-versible, no work can be done on a reservoir.The observation that doing work on a reservoirs is irreversible holds for arbitrary processes,not just the ones that fit the setting of the second law. This meets our intuition, which says thatreservoirs provide or take up heat, but not more than that. In fact, after having proved Carnot’sTheorem in the next section, we will be able to make the even stronger statement that doing workin a heat reservoir is not only non-reversible but inefficient.31 Carnot’s Theorem
Postulates: existence of reversible Carnot engines
New notions:
Carnot’s Theorem
Technical results for this section can be found in Appendix G.
Summary:
Carnot’s Theorem states that any machine operating between two heat reser-voirs has a maximal efficiency which only depends on the reservoirs, and that reversibleprocesses are optimal. This theorem will be explained and proved in this section.
This section is concerned with settings as depicted in Figure 7.1. Two reservoirs interact with asystem under a work process p ∈ P R ∨ S ∨ R such that the system undergoes cyclic evolution. Thegoal is to understand what values the work and heat flows can attain within the boundaries setby the laws of thermodynamics. Therefore we assume that W R i ( p ) = 0 because we already knowthat heat can only be don on but never be drawn from a reservoir. Special interest will be given tothe most efficient setting, in which heat is taken from one reservoir and given to the other, whiledoing as little work as necessary – or put differently, while drawing as much work as possible. Thesystem S is often called (Carnot) engine or machine. Clearly, here it is assumed that all systemsare pairwise disjoint, R ∧ S = ∅ = R ∧ S = ∅ = R ∧ R . If they were not, the figure as well asthe following discussion would make no sense. R R SQ R Q R W S Figure 7.1: A typical Carnot engine. Two reservoirs R and R (not necessarily copies ofeach other) interact with another system S through a (not necessarily reversible) work process p ∈ P R ∨ S ∨ R that is cyclic on S . The process dependence of the work and heat flows is omittedby writing W S := W S ( p ), and Q R i := Q R i ( p ). For reversible processes W R i ( p ) = 0 must hold. Asis shown in Lemma G.1 in a reversible setting the heat flows always fulfil Q R ( p ) > > Q R ( p )or Q R ( p ) > > Q R ( p ) (or Q R ( p ) = Q R ( p ) = 0, but this is the trivial case).The heat flows Q R and Q R are defined such that they are positive when positive heat flowsinto the corresponding reservoir. Hence a negative heat flow means that positive heat is flowinginto the cyclic system S . Consequently, the internal energy changes in terms of the quantitiesdefined in Figure 7.1 read ∆ U R i = Q R i and 0 = ∆ U S = W S − Q R − Q R .Since Carnot engines, and in particular reversible ones, are central for the development of thenotions of absolute temperature and entropy we postulate that between any two reservoirs thereexists an engine and a non-trivial reversible process on the three systems. Postulate 10 (Existence of reversible Carnot engines) . Let R , R ∈ R be two reservoirs and Q ∈ R . Then there exists a system S ∈ S and a reversible work process p ∈ P R ∨ S ∨ R , cyclic on S , with Q R ( p ) = Q . 32he reading of this statement is similar to the one of Postulate 8 on the existence of arbitrarilymany copies of any thermodynamic system. It might be very difficult or only approximatelypossible to build a perfect reversible Carnot engine. However, in principle the existence of such amachine is thinkable and this is what the theoretic considerations to come are based on.In traditional texts this postulate is usually not spelled out but implicitly taken for granted inthe construction of the proof of Carnot’s theorem.Postulate 10 will have important consequences when discussing absolute temperature as itwill be necessary in order to define the temperature of all heat reservoirs. It turns out thatwithout this, multiple incomparable definitions of absolute temperature can exist in parallel. Inthe postulate it is not only asked for the existence of a reversible machine but also for a heat flowthat one can choose to match Q ∈ R . Due to this requirement it is possible that the reversibleheat flow Q R ( p ) (or Q R ( p ) due to the symmetry of the labels 1 ↔
2, but not both at the sametime) can be chosen at will, which will become important for technical reasons.Even though at this point it might appear that Postulate 10 is very strong the discussion ofideal gases in Section 13, and in particular Figure 13.1, show that if the theory of phenomenologicalthermodynamic is powerful enough to describe ideal gases with quasistatic processes (Section 11)then the postulate is easily satisfied.Lemma G.1 prepares the stage for Carnot’s Theorem by showing that in any setting fulfillingthe conditions from Figure 7.1 at least one of the heat flows Q R i is positive, or the process is trivial , by which Q R = Q R = 0 is meant. This result holds independent of whether the processis reversible or irreversible. A reversible process fulfils the stronger property that one of the heatflows is strictly positive while the other one is strictly negative. The proof makes direct use of thesecond law (Postulate 9) and the definition of heat reservoirs (Definition 5.3).Even though this result basically only talks about positive and negative heat flows, it alreadysays a lot about Carnot engines, in particular about reversible ones. This will become clear notleast in the proof of Carnot’s Theorem. In some sense, it strengthens the second law by sayingthat it is impossible to have two reservoirs pumping heat into a cyclic machine that generates workout of it. This is even true if the reservoirs are not copies of each other. On the other hand it isa precursor to Carnot’s Theorem, which is based on Lemma G.1 and goes beyond it by makinga quantitative statement. Carnot’s Theorem can be read as a statement about the efficiency ofsuch machines. Theorem 7.1 (Carnot’s Theorem) . Consider a machine S ∈ S and two reservoirs R , R ∈ R undergoing a reversible work process p ∈ P R ∨ S ∨ R which is cyclic on S . Let S ′ ∈ S be anadditional machine operating between the same reservoirs under the work process p ′ ∈ P R ∨ S ∨ R ,which is cyclic on S ′ , fulfils W R ( p ) = W R ( p ) = 0, but is not necessarily reversible. W.l.o.g. Q R ( p ) > Q R ( p ′ ) > Then(i) the ratio of heat flows satisfies the inequality − Q R ( p ′ ) Q R ( p ′ ) ≤ − Q R ( p ) Q R ( p ) , (7.1)(ii) and the positively valued ratio − Q R ( p ) Q R ( p ) for reversible processes only depends on the reservoirs R and R and not on the machine S nor the details of the process. It is universal in thissense. I.e. the processes are non-trivial and if the signs of Q R are not positive, swap the labels 1 , Q R ( p ′ ) > p to work in the direction in which Q R ( p ) >
0, too. R R SQ R ( p ) Q R ( p ) W S ( p ) (b) R R S ′ Q R ( p ′ ) Q R ( p ′ ) W S ′ ( p ′ ) (c) R R S ∨ S ′ Q tot R Q tot R W tot S Figure 7.2: We compare a reversible machine S in (a) with an arbitrary one S ′ in (b) operatingbetween the same reservoirs. By letting the cycles run many times, we construct a process depictedin (c), for which we analyse the total work and heat flows. Depending on the ratios of the heatflows in p and p ′ the constructed process in (c) may violate Lemma G.1 and thus the second law(Postulate 9). Proof.
We prove (i) by contradiction. Figure 7.2 shows the discussed situations. Later on weargue that (i) implies (ii). Suppose the ratios fulfilled − Q R ( p ′ ) Q R ( p ′ ) > − Q R ( p ) Q R ( p ) . (7.2)Since Q R ( p ) and Q R ( p ) must have opposite signs (Lemma G.1) the right hand side of Eq. (7.2)is strictly positive. Hence this case can only occur for Q R ( p ′ ) < Q R ( p ) Q R ( p ′ ) > Q R ( p ) Q R ( p ′ ) , which again compares positive ratios. Choose positive integers k, l ∈ N suchthat Q R ( p ) Q R ( p ′ ) > kl > Q R ( p ) Q R ( p ′ ) . (7.3)The existence of p and p ′ implies together with Definition 5.3 (iii) that one can apply the respectiveprocess as many times as one wants in a row, in the sense that there exist corresponding processesthat do the same and can be concatenated with p and p ′ respectively. Thus, apply the reverseprocess of p on R ∨ S ∨ R now l times followed by k applications of p ′ on R ∨ S ′ ∨ R . The soconstructed process is cyclic on S ∨ S ′ and has total heat flows to R and R of Q tot R = − l Q R ( p ) + k Q R ( p ′ ) = (cid:18) kl − Q R ( p ) Q R ( p ′ ) (cid:19)| {z } > · l Q R ( p ′ ) | {z } < < , (7.4) Q tot R = − l Q R ( p ) + k Q R ( p ′ ) = (cid:18) kl − Q R ( p ) Q R ( p ′ ) (cid:19)| {z } < · l Q R ( p ′ ) | {z } > < . (7.5)This contradicts Lemma G.1, hence the ratios must fulfil (i).If p ′ is reversible too, the argument also works with exchanged roles of p and p ′ and we obtainthe inequality in the other direction. Therefore, if both p and p ′ are reversible, the ratios must beequal. Obviously the ratios for reversible machines must be positive as the signs of the heat flowsare always different. This proves (ii). 34arnot’s Theorem implies that reversible engines are the most efficient ones. Suppose theengine is such that work is extracted, which means that the total work done should be negative.The internal energy of the cyclic engine undergoes no net change, which is why the extracted workfrom S reads − W S = − Q R − Q R = Q R (cid:18) − Q R Q R − (cid:19) . (7.6)Remember that Q R is positive, which is why the expression for the extracted work − W S is max-imised when the term in brackets is maximal. This, however, is the case for reversible machines,as Carnot’s Theorem states.The argument is also valid if positive work is used such that the machine pumps heat from onereservoir to the other. With the same calculation it follows that the work done is minimal whenthe ratio of interest is maximal.Eq. (7.6) also shows how the theorem limits the maximal efficiency of reversible machines.Given two reservoirs, the work extractable in relation to the heat given to a reservoir is alwaysupper bounded by − Q R Q R −
1. Whatever machine one can come up with, this finite bound cannotbe surpassed for given reservoirs. We conclude that Carnot’s Theorem sets limits to what can beachieved with cyclic machines operating between two reservoirs.A further peculiarity of the theorem as stated in this work is the fact that it could be derivedwithout referring to anything similar to the zeroth law of thermodynamics nor to an a priori no-tion of thermal equilibrium. This is the topic of an earlier paper [31]. In short, the zeroth lawrequires that the relation “being in thermal equilibrium with” is transitive. In many introductionsto thermodynamics it is taken to be a vital ingredient to the foundations of thermodynamics asit paves the way for the notion of an empirical temperature, which relies on the relation “beingin thermal equilibrium with” to be an equivalence relation. In this work neither of these notionsever had to be used up to this point. Consequently, we could not even think of formulating (andmaking use of) the zeroth law. Therefore, up to here (as well as also for the remainder of thiswork, as will become clear) the zeroth law is redundant.The most relevant implication of the theorem, however, is the fact that for reversible pro-cesses, the ratio is universal. This is the crucial statement which is used next when definingabsolute temperature for heat reservoirs, and later in the definition of thermodynamic entropy. Itis the ingredient necessary to make statements for the behaviour of classes of systems rather thanindividual ones, as it says that the ratio is “system-independent”.35
Absolute temperature
Postulates: - New notions: temperature ratio, absolute temperature, equivalence relation ∼ on R Technical results for this section can be found in Appendix H.
Summary:
Based on Carnot’s Theorem the absolute temperature of a reservoir is defined.From this follows an equivalence relation on the set of heat reservoirs which shows thatthe zeroth law, which was not postulated in this framework, can be derived and is henceredundant as a postulate.
As one would expect, the ratio − Q R Q R is not only independent of the actual reversible machine butalso of the representative reservoirs of the equivalence class of ˆ=. This is argued at the beginningof Appendix H, where one can also find the proofs of the other non-trivial technical statements ofthis section.In addition, a reversible Carnot engine operating between reservoirs R ˆ= R fulfils − Q R Q R = 1,as is proved in Lemma H.1. The following definition makes use of the invariance of the ratio ofreversible heat flows under equivalences. Definition 8.1 (Temperature ratio) . The temperature ratio τ of two equivalence classes [ R ] , [ R ] ∈ R / ˆ= is τ : R / ˆ= × R / ˆ= −→ R > ([ R ] , [ R ]) Q R Q R (8.1)where R and R are two different heat reservoirs and Q R and Q R are the heat flows of a(non-trivial) reversible Carnot engine operating between them such that Q R >
0. Extendingthis definition to the set of pairs of heat reservoirs, we use the same symbol τ and write for the temperature ratio of two heat reservoirs τ ( R , R ) := τ ([ R ] , [ R ]) . (8.2)Definition 8.1 is well-defined since the ratio of heat flows only depends on the equivalence classesof ˆ= and not on the specific representative. Furthermore, if the equivalence classes [ R ] = [ R ]are equal, there always exist different representatives. This is a consequence of the postulate onthe existence of arbitrarily many copies of any system, Postulate 8.Carnot’s Theorem 7.1 implies that it is irrelevant which reversible engine is used to determinethe value of τ and that τ is a strictly positive function. Furthermore, due to Postulate 10 on the ex-istence of reversible Carnot engines this definition can be applied to an arbitrary pair of reservoirs.In principle, it is thinkable to formulate a theory of thermodynamics without the assumptionof always having a reversible machine operating non-trivially between any two reservoirs. Withoutit, it could happen that there exist two reservoirs that are incomparable. Consequently, Defini-tion 8.1 would have to be phrased independently for the two (or more) classes of reservoirs thatare comparable with each other. This is not a fundamental problem for thermodynamics, but it A Carnot engine operates between two different heat reservoirs. If R = R , then R ∨ S ∨ R = R ∨ S andthere would be no two heat flows to compare.
36s not the common approach taught in introductions to the field.The name temperature ratio of τ is suggestive. τ will be used to define the absolute temperaturefor heat reservoirs. Before introducing this notion we discuss some properties of τ . First, it holds τ ([ R ] , [ R ]) = 1, which means that if one takes reservoirs of the same type, it is only possible topump heat from one to the other, without investing or drawing work (Lemma H.1).Second, we note that τ ( R , R ) = τ ( R , R ) − by definition. Reversing the order in the argu-ment of τ amounts to reversing the reversible Carnot process used in Definition 8.1 to determinethe value of τ . Since reverse heat flows simply change their signs in the reverse process, the heatflows Q R < Q R > − Q R > − Q R < τ ( R , R ) − = (cid:18) − Q R Q R (cid:19) − = − − Q R − Q R = τ ( R , R ) . (8.3)Finally, in Lemma H.3 we show that for three arbitrary heat reservoirs R , R , R ∈ R it alwaysholds τ ( R , R ) · τ ( R , R ) = τ ( R , R ). The properties of τ allow us to define the absolute temperature of an arbitrary heat reservoir. Forthis, choose an arbitrary but fixed reference heat reservoir R ref ∈ R and a reference temperature T ref ∈ R > . Definition 8.2 (Absolute temperature) . The absolute temperature of a heat reservoir R ∈ R isdefined as T := τ ( R, R ref ) · T ref . (8.4)The temperature of a reservoir is absolute up to the choice of the reference reservoir and thereference temperature. However, once this choice is made, any other reservoir with its temperaturecould serve as a reference, too. This is a consequence of the previous Lemma H.3, as for tworeservoirs R , R ∈ R it holds that T = τ ( R , R ref ) · T ref = τ ( R , R ) · τ ( R , R ref ) · T ref = τ ( R , R ) · T . (8.5)That is what makes τ a temperature ratio, as it is called in Definition 8.1.Typically physicists work with the absolute temperature scale such that a reservoir consist-ing of a big water tank has temperature 273 .
16 K at the tripe point of water. From a practicalperspective it makes sense to fix a temperature scale once and for all so that when comparing tem-peratures no confusion can arise. Nevertheless, for developing the theory making specific choicesaccording to some standard is not necessary. Therefore, we will not discuss this issue further andcontinue, knowing that both R ref and T ref have been fixed before defining absolute temperaturefor heat reservoirs.Having defined absolute temperature for reservoirs, we can investigate the relation “ R and R have equal temperature” on the set of heat reservoirs R , denoted by R ∼ R (Definition H.5).Intuitively, one can also call this relation “being in thermal equilibrium with”. By Definition 8.2 R and R are at the same temperature if and only if τ ( R , R ) = 1. Several conclusions can bedrawn from this observation.First, the temperature of a reservoir is independent of its state. It is rather a property of thesystem. When thinking of general thermodynamic systems, this seems odd. But for reservoirsthis is what one would expect, as the property of not changing its behaviour under finite changesof energy intuitively is a defining property of heat reservoirs.37econd, from the definition of absolute temperature and Lemma H.1 it follows that equivalentreservoirs must have the same absolute temperature. i.e. R ˆ= R ⇒ R ∼ R . Hence, the relationˆ= restricted to the set of heat reservoirs R is a sub-relation of ∼ .Third, it holds(i) R ∼ R (reflexive),(ii) R ∼ R ⇒ R ∼ R (symmetric), and(iii) R ∼ R as well as R ∼ R ⇒ R ∼ R (transitive).Here, (i) and (ii) follow directly from Definition 8.2, while (iii) is a consequence of Lemma H.3.Points (i)-(iii) say that ∼ is an equivalence relation (Lemma H.6).Finally, heat reservoirs fulfilling τ ( R , R ) = 1 always allow for the exchange of an arbitraryamount of heat Q between them during a reversible work process p ∈ P R ∨ R at zero work cost.This follows from the fact that any reversible machine operating between them has a heat flowratio of − Q R Q R = 1 together with Postulate 6 on the freedom of description and Postulate 10. The fact that ∼ (“having equal temperature”) is an equivalence relation makes postulating thezeroth law redundant. The zeroth law typically states that the relation “being in thermal equilib-rium with” is transitive [5,10,11,15]. Notice that for a postulate stating this, a notion of “thermalequilibrium” must be introduced beforehand – something we did not have to do either. Togetherwith (the usually implicitly assumed) reflexivity and symmetry of ∼ , the zeroth law makes it anequivalence relation. Typically, this is then used to say that two systems are at equal temperatureif and only if they are in thermal equilibrium relative to each other. Here we have derived suchan equivalence relation on the set of reservoirs from the postulates without any reference to thezeroth law.One may wonder why we have not yet discussed a definition of absolute temperature forarbitrary systems, not just heat reservoirs. As it turns out, we should not expect that temperatureis a quantity that makes sense for an arbitrary system without further assumptions. Compositesystems consisting of more than two different atomic subsystems are simple counter examples.Temperature for arbitrary systems is thus not a fundamental concept of thermodynamics.Instead of a notion of temperature of arbitrary systems, we introduce the notion of the temper-ature of a heat flow in the next section (Section 9). This is the fundamental concept necessary touse the notion of temperature beyond the purpose of the efficiency of engines and the temperatureof heat reservoirs. In particular, the temperature of heat flows is the one which is used to definethermodynamic entropy based on Clausius’ Theorem.One might have hopes to be able to define absolute temperature for all atomic systems, sincethey are indivisible. However, one should not take the statement about indivisibility of atomicsystems as a statement about physical indivisibility. In particular, one should not confuse the term atomic with the frequently used term “simple system” (see e.g. [26]). Simple systems are usuallyconsidered to be those systems to which a meaningful notion of temperature can be assigned. Thisimplies that they cannot be composed of two independent systems (otherwise at least two temper-atures would be necessary in general). For us on the other hand, the term “indivisible” must beunderstood relative to the structure of thermodynamics systems with composition ∨ , which canbe defined in very abstract terms, not referring to any thermodynamic ideas.The discussion on how to define a notion of absolute temperature for systems beyond heatreservoirs is continued in Section 12, when all other basic concepts have been introduced and atechnical investigation based on them is possible.38 The temperature of heat flows
Postulates: - New notions: temperature of heat flows
Technical results for this section can be found in Appendix I.
Summary:
The absolute temperature for heat reservoirs gives rise to a definition of thetemperature of heat flows. The uniqueness of this temperature is discussed, in particularwith respect to reversible heat flows.Being able to talk about the temperature of heat reservoirs is important but not enough. Inparticular when it comes to defining thermodynamic entropy the temperature of heat flows willbe essential.As will be explained and investigated in this section, not every heat flow occurs at sometemperature. Some do, however, and the reversible ones will be of particular interest.
Definition 9.1 (Heat at temperature T ) . Let S = S ∨ S ∈ S be composed of two disjointsubsystems and undergo an arbitrary work process p ∈ P S ∨ S with Q := Q S ( p ) = 0. Wesay that the heat Q flows at temperature T if there exist two different reservoirs R ∼ R attemperature T with processes p ∈ P S ∨ R and p ∈ P S ∨ R s.t. W A ( p i ) = W A ( p ) for all atomicsystems A ∈ A r Atom( S i +1 ) and the state changes on S i under p i are the same as under p , i.e. ⌊ p i ⌋ S i = ⌊ p ⌋ S i and ⌈ p i ⌉ S i = ⌈ p ⌉ S i .Figure 9.1 (a), (b) and (c) illustrate the processes p , p and p schematically. By definition of p i as work processes on S i ∨ R i it is clear that W S i ( p i +1 ) = 0. Furthermore, it holds W R i ( p i ) = 0by assumption in the definition.The idea behind Definition 9.1 is to refer to the already defined concept of the temperature of areservoir to define the temperature of a heat flow. A heat flow occurs at temperature T if it couldalso be exchanged with a reservoir at temperature T and no thermodynamic properties change.Even though this may make sense intuitively it is a priori not clear that this definition eventuallyleads to the correct notion of temperature that is needed for the definition of thermodynamicentropy. However, as we will show in the following, it does so.It is possible to ‘reconstruct’ the process p from the p i in case p satisfies Definition 9.1 forsome T . Together with Definition 5.3 (i) for heat reservoirs and Postulate 10 on the existence ofreversible Carnot engines it follows that if both p and p exist, then the initial states of the tworeservoirs R and R can be restored by a reversible Carnot engine transferring the heat Q from R to R . The effect of the proper concatenations of these processes is depicted in Figure 9.1 (d).The work cost of reversibly transferring the heat Q from R to R by means of a cyclic engine C is zero. This follows from Carnot’s Theorem and Lemma H.1. We have thereby constructeda process from p and p which thermodynamically does the same as p did on all involved systems.We describe some examples to give an intuition for this definition. Using the tuning of Q R in reversible Carnot processes as required by Postulate 10 it becomes obvious thatfor two equivalent reservoirs R ˆ= R and Q ∈ R there is always a reversible process p ∈ P R ∨ S ∨ R , involving anadequate machine S , that transfers the heat Q R ( p ) = Q = − Q R ( p ) from R to R . In such a situation we have W S ( p ) = 0 because of the exactly opposite heat flows to the reservoirs. Since S is cyclic in any case, in this specialsituation S is even catalytic and can be left out of the description (Postulate 6). That is, there exists a process˜ p ∈ P R ∨ R with the same heat flows and state changes.We conclude that for two equivalent reservoirs there is always a reversible process transferring an arbitraryamount of heat from to the other at no work cost. This observation will be relevant later, when we discuss thetemperature of heat flows. S S QW W (b) S R QW (c) R S Q W (d) S R CQW Q Q R S Q W Figure 9.1: (a) Under the process p work W i := W S i ( p ) is done on S i and the heat Q := Q S ( p )flows from S to S . (b) & (c) In the divided processes p and p the same state changes areinduced on S and S , respectively, but the heat flow is exchanged with reservoirs. The work flows W i = W S i ( p i ) also stay invariant. (d) With the use of a reversible Carnot process between R and R it is possible to transfer the heat Q from R to R at zero work cost since R ∼ R . Theinitial states of R and R will then be restored as they only depend on the internal energies andthe engine itself is cyclic by design. Together with the observation that R ∨ C ∨ R is catalyticthis essentially reconstructs p due to Postulate 6. Example 9.2 (Heat flow exchanged with a reservoir) . The most obvious example is the one inwhich an arbitrary system S exchanges heat Q = 0 with a reservoir R . So let p ∈ P S ∨ R be suchthat Q = Q R ( p ) = 0 and think of S = S and S = R . Then, the heat Q flows at temperature T ,where T is the temperature of the reservoir R . To see this, observe that what is referred to p inDefinition 9.1 can now be chosen as p itself (or an equivalent process carried out on S and a copyof R ). On the other hand, Postulate 10 says that a process that transfers heat Q between R anda copy of it always exists. By choosing such a process as p we have found both p and p withreservoirs that are copies of R and thus have the same temperature. Hence the heat Q flows attemperature T according to the definition.Importantly, Definition 9.1 on the temperature of heat flows principally allows that a generalheat flow can be assigned more than one temperature, see the next Example 9.3. Even when heatis exchanged with a reservoir directly, the temperature of the reservoir does not have to be theunique temperature which can be assigned to the heat flow. Example 9.3 (More than one temperature for the same heat flow) . Let S and S be two idealgases with the same amount of substance in states ( p i , V i ), respectively, such that p V > p V .This essentially means that their gas temperatures T i are different, where the gas temperaturecan in this case be defined according to the equation of state pV = nRT with R the universalgas constant and n the amount of substance. When connecting the two gases thermally, e.g. byputting them in contact with a metal rod, one can observe a positive heat flow from S to S (provided that the reference temperature T ref for the definition of absolute temperature has bechosen accordingly). It will now be observable that reservoirs with temperatures between T > T will fulfil the above definition. In particular, more than one temperature can be assigned to thesame heat flow. A more precise discussion of the concepts mentioned here (equation of state,temperature of the gas) is given in Section 13.It is also possible that no temperature can be assigned to a heat flow at all, as the next exampleshows. Example 9.4 (Heat flows without a temperature) . Consider again an ideal gas denoted by S instate ( p, V ). This time, let S = R ∨ R be a system composed of two reservoirs at temperatures T < T . Assume in addition, that pV = nRT , i.e. that the initial state of the ideal gas is suchthat its gas temperature matches the absolute temperature of reservoir R . This means that areversible isothermal compression, say from V to V , can be achieved by thermally connectingthe ideal gas to R and slowly compressing the gas. Denote the heat flow from S to R by Q .Checking Definition 9.1 it follows that the heat Q flows at temperature T .40ow, the same state change on the ideal gas S could be achieved reversibly in a different way,by exchanging a different amount of heat with the other reservoir R . We first reversibly compressthe ideal gas adiabatically , thereby increasing the product pV to the value nRT . This is followedby a (reversible) isothermal compression (or an expansion, if T is so much hotter than T ) inwhich S isothermally exchanges the heat Q with R . The isothermal compression (expansion) isdone such that in the final state, there exists the reversible adiabatic process that brings the gastemperature back to T while the volume has been brought to V such that the net state changeon S is the same as under process described in the previous paragraph.By computing the amount of heat exchanged using the standard equations for the ideal gas,it is easy to see that for T < T it follows Q < Q . From this observation we learn severalthings. First, it is possible to have the same state change on a system (here S ) having exchangedstrictly different reversible heat flows. Second, the temperature of the involved heat flows do nothave to match either. Third, since the two processes are reversible, we can concatenate the onewith a reverse of the other, which will generally not lead to a net heat flow of zero. Neither willthis non-zero heat flow allow for an assigned temperature. On the contrary, since it is the resultof two heat flows at different temperatures it is obvious that it should be impossible to assign atemperature. Hence there exist reversible heat flows without a proper temperature.As is shown in Lemma I.4, the different reversible heat flows Q i at temperature T i from theprevious example, which induce the same state change on S , fulfil Q T = Q T . This observationwill be crucial for the definition of thermodynamic entropy.Having discussed these cases of Definition 9.1 we come to the more important case of a generalreversible heat flow which can be assigned a temperature. In this case it holds that the temperatureis unique, as is shown in Appendix I. This is done by first showing that for a reversible process p ∈ P S ∨ S inducing a heat flow Q = 0 at temperature T according to Definition 9.1 the twoprocesses p i ∈ P S i ∨ R i are reversible, too (Lemma I.2). Based on this, it is then possible to showthat the assigned temperature T must be unique (Lemma I.3). Hence, non-zero reversible heatflows in a bipartite thermodynamic system can be assigned either a unique temperature or notemperature at all. 41 Postulates: existence of reversible processes with heat flows at well-defined temperatures
New notions:
Clausius’ Theorem, entropy, Entropy Theorem
Technical results for this section can be found in Appendix J.
Summary:
The derivation of Clausius’ Theorem allows one to define thermodynamic en-tropy and show that it is a well-defined state function, which is additive under composition.A postulate guarantees that the entropy difference of any two states of a system can becomputed. The Entropy Theorem establishes that entropy is a monotone for the preorder → (as defined after the first law). So far we have used the second law to prove Carnot’s Theorem which was the starting point todefine the temperature of reservoirs and heat flows. Next, we use these insights to prove Clausius’Theorem, which is the basis for the definition of thermodynamic entropy as a state variable.
Theorem 10.1 (Clausius’ Theorem) . Let S ∈ S be an arbitrary system and { R i } Ni =1 a set ofreservoirs such that the temperature of R i is T i . For each i = 1 , . . . , N let p i ∈ P S ∨ R i be awork process on S and the reservoir R i with W R i ( p ) = 0 such that the concatenated process p := p N ◦ · · · ◦ p is defined and in total cyclic on S .Then:(i) P i Q S ( p i ) T i ≤ p is reversible, then P i Q S ( p i ) T i = 0. Proof. (i) Let R be another reservoir at temperature T and let { C i } Ni =1 ⊂ S be machines operatingcyclically between R and R i under a reversible process q i ∈ P C i ∨ R i ∨ R . Let the machines C i and processes q i be such that the heat flows Q R i ( q i ) = Q S ( p i ) are provided to R i percycle, and define Q R ( q i ) =: Q i . By Definition 9.1 we know that the heat flows Q S ( p i )between R i and S are at temperature T i . From this, we construct a cyclic machine S asdepicted in Figure 10.1.The machines C i together with the reservoir R are used to provide the heat flows Q S ( p i ) tothe reservoirs R i such that, under this extension, all reservoirs except for R become cyclic.For this, the heat flows Q i , exchanged between the machines C i and R , as well as the workflows W C i are needed. The heat flows Q i occur at temperature T . In total, the system S summarized in the dotted box is then cyclic.The second law (Postulate 9) can now be applied to this situation, which yields Q := P i Q i ≥
0. Together with Carnot’s Theorem (ii) and the definition of absolute temperature,which says that for reversible machines − Q i T = Q S ( p i ) T i (10.1)holds, it follows with T > nd ≥ − Q T .Q = − Q + · · · + Q N T = Q S ( p ) T + · · · + Q S ( p N ) T N = N X i =1 Q S ( p i ) T i . (10.2)42 Q · · · · · ·· · · · · ·· · · · · · R Q S ( p ) Q S ( p ) W C C Q R Q S ( p ) Q S ( p ) W C C Q R N Q S ( p N ) Q S ( p N ) W C N C N Q N S W S W S S Figure 10.1: Extending the system S ∨ R ∨ · · · ∨ R N with the cyclic machines { C i } Ni =1 and theadditional reservoir R one can construct a cyclic machine S interacting with a single reservoir R . The second law (Postulate 9) can be applied to this situation.(ii) If all processes p i are reversible Lemma I.3 says that the temperatures of the associated heatflows are unique and equal to the temperatures in the reverse processes. In the reverseprocesses the heat flows go in the opposite direction. Therefore, with the same argument asin (i), we obtain the opposite inequality, which together with (i) implies N X i =1 Q S ( p i ) T i = 0 . (10.3)Even though Clausius’ Theorem is formulated only for heat flows between a system and a setof reservoirs, it also makes a statement about other heat flows. Due to Definition 9.1 any heat flowexchanged between two arbitrary systems to which a temperature can be assigned, can be reducedto the situation in which this heat flows between one of the systems and a reservoir. Therefore, ifa system undergoes a sequential process in which all heat flows to the system can be assigned atemperature, the theorem holds too.The fact that it is possible that more than one temperature can be assigned to a specific heatflow does not lead to problems. The inequality of Clausius’ Theorem (i) holds for all temperaturesthat fulfil Definition 9.1. In the case of reversible heat flows, where the inequality works in bothdirections and thus equality holds, we are guaranteed that the assigned temperatures are unique. The quantity P Ni =1 Q S ( p i ) T i = 0 discussed in Clausius’ Theorem (Theorem 10.1) applied to a se-quence of processes which does not have to be cyclic will serve as the definition of thermodynamicentropy. More precisely, the entropy difference between two states will be computed by means of If the heat flow was Q = 0 this statement does not make any sense. However, in this case the heat does notcontribute either in Eq. (10.2). Postulate 11 (Reversible processes with heat flows at well-defined temperatures) . Given anySystem S ∈ S and any two states σ , σ ∈ Σ S there always exists a finite sequence of reversibleprocesses { p i } Ni =1 , p i ∈ P S ∨ R i , with R i ∈ R such that p := p N ◦ · · · ◦ p is well-defined andtransforms ⌊ p ⌋ S = σ into ⌈ p ⌉ S = σ .The existence of a connecting sequence of processes as asked for by the postulate for arbitrarychoices of σ and σ does not follow from the first law (Postulate 7), which only asks for a workprocess connecting the two states. The processes in the sequence, however, are not work processeson S . Instead, they need to be reversible and the heat flows to S must have a well-defined tem-perature for all members of the sequence.We are now in the position to define thermodynamic entropy, which is a state variable due toClausius’ theorem. Definition 10.2 (Entropy) . Let S ∈ S be a system and σ ∈ Σ S an arbitrary but fixed state. Let S S ∈ R be an arbitrary real constant. For a state σ ∈ Σ S we define its entropy as S S ( σ ) := X i Q S ( p i ) T i + S S , (10.4)where the sum goes over a sequence of reversible concatenable processes { p i } , each of which fulfils p i ∈ P S ∨ R i with W R i ( p ) = 0, with total initial state σ and final state σ .Lemma J.1 proves that entropy is additive under composition for disjoint systems. It is think-able that one works with a theory that does not fulfil Postulate 11, i.e. there would exist pairs ofstates for which there is no such sequence of processes connecting them. Example 10.3 (A system that does not satisfy Postulate 11) . As an example, consider a mixtureof 1 mole of oxygen and 2 moles of hydrogen undergoing the oxyhydrogen reaction, which mightbe known, whereas the opposite process, electrolysis, is unknown and thus not a process in thistheory. In this case, the state of the initial mixture is connected by the work process “oxyhydrogenreaction” with the final state, 1 mole of water, and thus satisfying the first law. On the other hand,this process is irreversible and there is no other known process that could bring the water backto its initial state, the mixture. Hence, there is no sequence of processes that could serve for thedefinition of entropy and consequently, one has to define two measures of entropy on the different“connected subsets” of the state space which are incomparable. The two reference entropies couldbe chosen arbitrarily.Such a case is thinkable and (so far) allowed by the presented framework. We exclude if fromnow on by postulating that for any two states a sequence of processes as described in Postulate 11exists. Hence, for a given system one needs to define only one entropy measure for all of its statespace. Entropy is a very helpful quantity. This is owed to the so called
Entropy Theorem which saysthat entropy is a monotone for the preorder → . That is, if two states of a system are preordered(in the sense established by Definition 3.1), i.e. there exists a work process that transforms oneinto the other, then the entropy of the final state cannot be smaller than the entropy of the initialstate. One can choose this reference state to be the same as for the internal energy but this is not mandatory. This discussion is reminiscent of the one after Postulate 10 asking for the comparability of any two reservoirs,which lead to a unique absolute temperature, instead of different incomparable ones. As was the case for absolutetemperature, also here we aim for accordance of our framework with standard phenomenological thermodynamicsand solve this issue with the postulate on the existence of reversible processes. heorem 10.4 (Entropy Theorem) . Let S ∈ S be a thermodynamic system and p ∈ P S a workprocess on S . Then S S ( ⌊ p ⌋ S ) ≤ S S ( ⌈ p ⌉ S ) . (10.5)with equality if p is reversible. In particular, since entropy is a state variable, any two states thatare preordered in this sense (Definition 3.1) fulfil this inequality, which is to say that entropy is amonotone for → . Proof.
Let { q i } Ni =1 ⊂ P be a valid sequence to determine the entropy difference between the outputand input state of p . That is, for each i the process q i ∈ P S ∨ R i is reversible and concatenable withits predecessor and successor, such that ⌊ q ⌋ S = ⌈ p ⌉ S and ⌈ q N ⌉ S = ⌊ p ⌋ S . In each process q i theheat Q S ( q i ) is exchanged at temperature T i , or Q S ( q i ) = 0.Extend p with an arbitrary identity process to p ∨ id R for an arbitrary R ∈ R r { R , . . . , R N } .Then it is possible to concatenate q N ◦ · · · ◦ q ◦ ( p ∨ id R ) and Q S ( p ∨ id R ) = 0. The total process q N ◦ · · · ◦ q ◦ ( p ∨ id R ) is cyclic on S and fulfils the conditions in order to apply Clausius’ Theorem,which then states0 ≥ N X i =1 Q S ( q i ) T i + 0 = S S ( ⌈ q N ⌉ S ) − S S ( ⌊ q ⌋ S ) = S S ( ⌊ p ⌋ S ) − S S ( ⌈ p ⌉ S ) . (10.6)This proves the inequality. In the case of a reversible p , Clausius’ Theorem requires equality, whichthen implies S S ( ⌊ p ⌋ S ) = S S ( ⌈ p ⌉ S ).The Entropy Theorem establishes the parallels between what we call work processes and whatis usually called adiabatic processes . In standard phenomenological thermodynamics the EntropyTheorem is usually formulated for adiabatic processes, which are defined to be processes in whichno heat is exchanged with other systems. The fact that our work processes serve as the traditionaladiabatic processes was already suggested by the similarity of Postulate 7 (i) to the “adiabaticaccessibility” statement of Lieb and Yngvason [26]. However, they use very different axioms andtechniques to derive the corresponding monotone, the entropy. In particular, since the term adi-abatic is usually taken to mean “no heat flows”, in our way of introducing thermodynamics wecould not even use this concept as heat is only introduced after the first law is stated and internalenergy is derived. Furthermore, as is explained in an example at the end of Appendix J, in ourframework reversible processes with “no heat flow” can still change the entropy of a system. Hencethe traditional definition of an adiabatic process would fail to fulfil the Entropy Theorem.The Entropy Theorem does not strictly hold in the other direction, i.e. one cannot alwaysdeduce the existence of a work process transforming σ into σ from S S ( σ ) ≥ S S ( σ ). However,it almost holds. Starting from the strict inequality S S ( σ ) > S S ( σ ) it follows due to the EntropyTheorem that there cannot be a work process on S transforming σ into σ . Together with thefirst law it then follows that a work process in the other direction, i.e. taking σ as input andgiving σ as output, must exist. Hence, in this sense, the Entropy Theorem is almost invertible.45 Postulates: quasistatic first law (quasistatic Postulate 7), quasistatic reversible processeswith heat flows at well-defined temperatures (quasistatic Postulate 11)
New notions: quasistatic thermodynamic processes, differential work and heat
Summary:
Based on the existing notions of states and processes we define quasistaticprocesses which leads to the notions of differential work and heat. In order to formallyuse the differential quantities instead of the discrete ones discussed so far, the Postulates 7and 11 must be adjusted. The implications of these adjustments are then discussed.In this section we extend our framework, which talked about discrete quantities up to thispoint, to allow for continuous processes and state spaces. This is coupled to additional assump-tions on the technical structure of state spaces and the set of thermodynamic processes. Arguably,many of the additional assumptions are not on an equal footing with the physical assumptionsdiscussed in the main text. They are technical in nature and not necessary to be fulfilled if onewants to work with the minimal assumptions from the main text to obtain the standard resultsin their discrete form. However, they are necessary in order to have notions like the ones of a quasistatic process or differential work and heat , which are about continuous and differentiablequantities. In order to define a quasistatic process on S the minimal requirement on Σ S is that it is equippedwith a topology. Definition 11.1 (Quasistatic process) . A quasistatic work process on a system S ∈ S is a two-parameter family of work processes { p ( λ, λ ′ ) } ≤ λ ≤ λ ′ ≤ ⊂ P S such that:(i) For λ ≤ λ ′ ≤ λ ′′ ∈ [0 ,
1] it holds p ( λ ′ , λ ′′ ) ◦ p ( λ, λ ′ ) = p ( λ, λ ′′ ).(ii) The curve γ : [0 , → Σ S , γ ( λ ) := ⌊ p ( λ, ⌋ S is continuous in the topology of Σ S .(iii) For all A ∈ Atom( S ) the work function W A ( p ( λ, λ ′ )) is continuous in both λ and λ ′ on therespective domain.A quasistatic process on S is then just a quasistatic work process on a larger system S ′ ∈ S such that S ∈ Sub( S ′ ) is a subsystem. Even if we talk about a quasistatic work process on S wesometimes call it a quasistatic process on S when it helps to improve the readability of the textand the context makes clear what is meant. Lemma 11.2 (Properties of quasistatic processes) . Let { p ( λ, λ ′ ) } λ,λ ′ be a quasistatic process on S . For all λ, λ ′ , λ ′′ ∈ [0 ,
1] with λ ≤ λ ′ ≤ λ ′′ it holds:(i) ⌈ p ( λ, λ ′ ) ⌉ A = ⌊ p ( λ ′ , λ ′′ ) ⌋ A for all A ∈ Atom( S ).(ii) ⌊ p ( λ, λ ′ ) ⌋ S is independent of λ ′ and ⌈ p ( λ, λ ′ ) ⌉ S is independent of λ . In particular, the curve γ from Definition 11.1 (ii) satisfies γ ( λ ) = ⌊ p ( λ, λ ′ ) ⌋ S = ⌈ p ( λ ′′ , λ ) ⌉ S for all λ ′ ≥ λ and λ ′′ ≤ λ .(iii) p ( λ, λ ) is an identity process on S .(iv) For all λ, λ ′ ∈ [0 ,
1] and all partitions λ = λ < λ < · · · < λ m = λ ′ it holds W A ( p ( λ, λ ′ )) = P mi =1 W A ( p ( λ i − , λ i )). 46 roof. Lemma 11.2 (i) follows directly from Definition 11.1 (i). Claim (ii) is then a consequenceof (i). For p ( λ, λ ) we find according to Definition 11.1 (i) that it can always be concatenatedwith itself, and p ( λ, λ ) ◦ p ( λ, λ ) = p ( λ, λ ). Due to the additivity of work under concatenation,it follows that W A ( p ( λ, λ )) = 0 for all A ∈ Atom( S ). Hence p ( λ, λ ) is an identity process on S . Finally, claim (iv) is again a simple consequence of Definition 11.1 and the additivity of workunder concatenation.Quasistatic processes are those which allow for a continuous partition into “smaller” processes,which can be concatenated to from the whole process again. The process p ( λ, λ ′ ) connects thestate γ ( λ ) with γ ( λ ′ ), as Lemma 11.2 shows. The family of processes thereby defines a continuouscurve in the state space and the work functions evaluated on the members of the family mustbe continuous as well in the curve parameter. This has to hold for each atomic subsystem of S individually. The naive approach asking only for continuity of W S would allow for an arbitrarychaotic behaviour of the W A . This is something that should not be the case, as one expects that p ( λ, λ ′ ) for very close λ and λ ′ corresponds to a small “step”.The family of processes defining a quasistatic process on S can also be called quasistatic on asubsystem S ′ ∈ Sub( S ) of S . Consequently, a family of processes which are not necessarily workprocesses on a system S are called quasistatic if they form a quasistatic process on the largersystem on which they are work processes.It is easy to see that two quasistatic processes { p ( λ, λ ′ ) } λ,λ ′ and { q ( λ, λ ′ ) } λ,λ ′ can be con-catenated to a new quasistatic process { ( q ◦ p )( λ, λ ′ ) } λ,λ ′ whenever ⌈ p (0 , ⌉ S = ⌊ q (0 , ⌋ S . Theconcatenated quasistatic process is then defined as( q ◦ p )( λ, λ ′ ) = p (2 λ, λ ′ ) , for λ ≤ λ ′ < ,q (0 , λ ′ ) ◦ p (2 λ, , for λ < ≤ λ ′ ,q (2 λ, λ ′ ) , for ≤ λ ≤ λ ′ . (11.1)In practice one often calls p (0 ,
1) the “quasistatic process”. However, in our framework thisprocess alone does not say anything about the continuous curve in the state space. Hence themathematical structure of a quasistatic process must be richer than that. Nevertheless, we willsometimes call a process p ∈ P S quasistatic on S without explicitly mentioning that formally thefamily { p ( λ, λ ′ ) } λ,λ ′ with p = p (0 ,
1) is meant.
Example 11.3 (Identity processes) . An identity process id σS ∈ P S on a system S , which, concate-nated with itself yields itself again, can always be seen as quasistatic processes (with any topologyon Σ S ). To see this, consider the defining family p ( λ, λ ′ ) = id σS for all λ ≤ λ ′ ∈ [0 , γ ( λ ) = σ for all λ ∈ [0 ,
1] is constantand thus continuous in any topology, which shows (ii). In addition, (iii) is fulfilled since identityprocesses have zero work cost on any atomic subsystem and thus W A ( p ( λ, λ ′ )) = 0 constant.This example also works for discrete spaces Σ S equipped with the discrete topology. In thediscrete topology, such quasistatic processes are the only possible ones owed to the fact that everycontinuous curve in a discrete space is constant.We next comment on the continuity of the internal energy of a system, on which quasistaticprocesses exist. For a quasistatic process { p ( λ, λ ) } λ,λ ′ on S the internal energy of S is continuouson the curve in the curve parameter. That is, U S ( λ ) := U S ( γ ( λ )) is continuous in λ and can beseen as follows.Due to Definition 11.1 (iii) together with the additivity of work under composition and thefact that the sum of continuous functions is continuous, we find U S ( λ ′ ) − U S ( λ ) = U S ( γ ( λ ′ )) − U S ( γ ( λ )) = W S ( p ( λ, λ ′ )) λ → λ ′ −→ , (11.2)for λ ≤ λ ′ ∈ [0 , U S can always be expressed asthe work cost of an appropriate work process on S . The proof does not directly generalize to the47nternal energies of subsystems of S because the second equality only holds when the change ininternal energy can be expressed by a step in a quasistatic process. The existence of a quasistaticprocess on S does not imply this.If for any continuous curve in Σ S there exist quasistatic processes on S that, when puttingthe quasistatic curves together, pass through this curve (not necessarily in one direction, as thedifferent quasistatic curves may have different directions in which they are gone through) it followsthat U S is continuous on Σ S , now in the topology of Σ S . This holds because a function that iscontinuous on any continuous path is continuous on the topological space. Again, the same onlyfollows for subsystems S ′ ∈ Sub( S ) and U S ′ if also in Σ S ′ there exist quasistatic processes for anycontinuous curve.Systems with this property fulfil a stronger first law asking not only for the existence of workprocesses between any two states, but that these are quasistatic in addition. We now know thatfor such systems the internal energy function is continuous. Furthermore, the state space of sucha system is necessarily path-connected. We do not formalize this strengthened first law here, as itwould only be an intermediate step towards the “quasistatic first law” below, which relies on evenricher structure on the state space. Example 11.4 (Continuous internal energy of a gas) . A gas described by σ = ( p, V ) ∈ Σ S = R > is again a good example for a system which allows for quasistatic processes between anytwo states. The state space Σ S = R > shall be equipped with the usual topology on R . Forgases adiabatic compression and expansion and isochoric warming (i.e. heating) can be consideredquasistatic processes if they are carried out slow enough. By considering those types of processesany continuous path in Σ S can be seen as a union of quasistatic curves, where the directions ofthe curves play no role for the purposes of determining the internal energy. We conclude fromthis, that the internal energy of a gas is continuous in σ . We now investigate the differential work and heat, which can be defined if further structure ispresent on the state spaces of the involved systems. Specifically, Σ S must be at least a C manifold. The state spaces Σ A of atomic subsystems A ∈ Atom( S ) are then submanifolds of Σ S and thus also C . The same holds for arbitrary subsystems of S . Definition 11.5 ( C -quasistatic processes and differential work) . A (piecewise) C -quasistaticprocess on S ∈ S is a quasistatic process { p ( λ, λ ′ ) } λ,λ ′ ⊂ P S on S with a (piecewise) C -curve γ in Σ S together with continuous 1-forms δ ( p ) W A defined on the curve for all A ∈ Atom( S ) suchthat for all λ ≤ λ ′ ∈ [0 ,
1] it holds Z γ | [ λ,λ ′ ] δ ( p ) W A = W A ( p ( λ, λ ′ )) . (11.3)Here, the left hand side is a part of the path integral over the quasistatic path. The continuous1-form δ ( p ) W A is called differential work of A and may depend on the quasistatic process, which isindicated by the superscript ( p ) standing for the whole family { p ( λ, λ ′ ) } λ,λ ′ . Through additivityof work under composition we obtain corresponding continuous 1-forms for all subsystems of S and of course S itself.We are not only interested in C -curves but also piecewise C -curves as this is what one gener-ically obtains when concatenating two C -quasistatic processes. In order to be able to integrate apiecewise C -curve is sufficient.At this point it makes sense to strengthen the fist law (Postulate 7) so it also talks about piece-wise C -quasistatic processes. The new version is thought for thermodynamic theories in whichthe systems’ state spaces are C manifolds. In addition to the previous first law the “quasistaticversion” will also make sure that the internal energy U S of any system is a differentiable functionand hence the differential d U S exists. Section 13 discusses the example of the ideal gas in thiscontext in more depth. 48 ostulate 7’ (The quasistatic first law) . For any system S ∈ S the following three statementshold:(o) For all states σ ∈ Σ S there exist m := dimΣ S C -quasistatic processes with curves γ , . . . , γ m passing through σ such that the derivatives of these curves at σ are linearly independent.That is, there exist λ i ∈ (0 ,
1) with γ i ( λ i ) = σ for i = 1 , . . . , m , such that γ ′ ( λ ) , . . . , γ ′ m ( λ m )are linearly independent.(i) For any pair of states σ , σ ∈ Σ S there exists a piecewise C -quasistatic process { p ( λ, λ ′ ) } λ,λ ′ ⊂P S on S with ⌊ p (0 , ⌋ S = σ and ⌈ p (0 , ⌉ S = σ or in the other direction, or both.(ii) The total work cost of a work process p ∈ P S on S , W S ( p ), only depends on ⌊ p ⌋ S and ⌈ p ⌉ S and not on any other details of the process.Postulate 7’ (i) obviously implies point (i) of the previous first law, Postulate 7, while (ii) hasnot changed at all. Therefore, all the machinery relying on the first law can as well be based onthe quasistatic first law. The additional point (o) may seem unnecessary at first, since it requiresmore quasistatic processes passing through a given state than just the one that is asked for in (i).However, this point will be crucial in order to show that the internal energy of systems fulfillingthe quasistatic first law is differentiable.One may wonder whether (o) already implies (i) in the quasistatic formulation. In fact italmost does so, but not quite. One could think of constructing the quasistatic processes askedfor in (i) by means of the ones asked for in (o) step by step. Even if this construction lead toa piecewise C -curve from one state to the other (which may depend on further assumptions onthe manifold Σ S ), it would have the problem that the direction of the quasistatic steps mightchange throughout the curve. If this is the case, the quasistatic processes cannot be composed toa piecewise C -quasistatic process connecting the states. Example 11.4 may also serve as an example of a system which fulfils the quasistatic first law.As is argued before the example, one can show that the internal energy of the gas is continuousin this case. We now prove that for any atomic system A ∈ A satisfying the quasistatic firstlaw (Postulate 7’) the internal energy is actually differentiable. The extension of this result toarbitrary systems S ∈ S follows by the additivity of the work functions.According to the Definition 3.2 of internal energy for any state σ ∈ Σ A of an atomic system A ∈ A we can write U A ( σ ) = Z σσ δW A + U A (11.4)where the integral goes over an appropriate piecewise C -curve arising from a C -quasistaticprocess on A transforming σ to σ or the other way around. Notice that Equation (11.4) holds inboth cases, i.e. it also holds when the piecewise C -quasistatic process on A transforms σ to σ ,and is independent of the choice of quasistatic process used to compute the integral, which herein particular means that it is independent of the path over which the integral is evaluated. In thiscase, one integrates in the opposite direction of the given curve, thereby introducing an additionalminus sign, and thus the equality still holds.The existence of such an appropriate curve is guaranteed by Postulate 7’ (i) while the fact thatthe continuous 1-form δW A is independent of the actual process is the statement of (ii). This iswhy the superscript ( p ) is neglected in the notation. With (o) it is now possible to argue that Thinking of the states of system as the vertices and the work processes connecting them as the edges of a graph,(i) says that this graph must be complete, i.e. there is an edge between two arbitrary vertices. In fact, it possibleto formulate thermodynamics with a weaker first law in which (i) only asks for this graph to be connected. In thisformulation, the existence of an (undirected) C -curve connecting any two states as in the above construction isenough to derive (i), which may follow from (o). However, we do not go deeper into this weaker first law here, as itmakes proofs and technical derivations much more cumbersome while there is no gain over the strong formulationexcept for a weaker postulate. A is differentiable in σ . Since the m C -quasistatic processes passing through σ must exist andtheir derivatives at σ are linearly independent, these derivatives build a basis of the tangent space T σ Σ A . These quasistatic processes together with their C -curve can be used to take the directionalderivative of U A in the m “different directions” of the tangent space. Since U A is defined as thepath integral of a continuous 1-form over a piecewise C -curve, these derivatives all exist and arecontinuous. This is sufficient to deduce that U A is differentiable in σ since a continuous functionon a manifold for which all partial derivatives at a point exist and are continuous is differentiablein this point (see e.g. Theorem 9.21 in [34]). Since this can be deduced for an arbitrary σ ∈ Σ A ,it also shows differentiability of U A on Σ A .For any differentiable function f the corresponding differential d f can be defined. This alsoholds for the now differentiable internal energy U A . Relying on this, the differential heat of A ina quasistatic process on a possibly larger system S with A ∈ Atom( S ) is defined as δ ( p ) Q A := d U A − δ ( p ) W A . (11.5)For arbitrary systems the differential heat is defined via additivity under composition.Again, the superscript ( p ) is in general necessary to make explicit that the 1-form may dependon the quasistatic process. In particular, the 1-forms for differential work and heat are not exact,while d U as the differential of a differentiable function is. This is manifest in the notation of theinexact differentials with δ and the the exact differential with d . Generally, path integrals overexact differentials are independent of the path, while integrals over inexact differentials dependon the path. In our case the integrals over differential work and heat are in general not onlypath-dependent but process-dependent , i.e. they depend on the family of processes constitutingthe quasistatic process on which they are defined. Example 11.6 (Differential work and heat are generally process-dependent) . As an example weconsider the isothermal expansion of a gas. If the constant temperature of the gas is achievedthrough thermal contact with a reservoir, as is normally the case when one talks about isothermalexpansion, the differential work is − p d V . On the other hand, one could achieve a constanttemperature throughout the process by continuously applying friction, i.e. bringing up the properamount of work. The energy that keeps the temperature constant must then be counted as workinstead of heat in the previous case. In this case, the differential work will be different and thetotal amount of work done as well. However, the paths corresponding to the quasistatic processesin the state space will be exactly the same.From now on, whenever we talk about quasistatic processes and differential work and heat,we implicitly assume that the basic structure on the state spaces is given and that the quasistaticfirst law, Postulate 7’, is fulfilled. In particular, by a quasistatic process from now on we mean a(piecewise) C -quasistatic one. Also, as is usually done in the literature, we will only write δW S and δQ S , without manifesting the process-dependence of these forms in the notation explicitly. When using quasistatic processes for computing entropy differences we have to consider reversibleones. A quasistatic process { p ( λ, λ ′ ) } λ,λ ′ on a system is called reversible if all members p ( λ, λ ′ ) ofthe family are reversible work processes on this system.Suppose now a system S ∈ S undergoes a quasistatic process { p ( λ, λ ′ ) } λ,λ ′ ⊂ P S ∨ R ∨···∨ R N ,where { R i } i ⊂ R are reservoirs, such that there is a partition 0 = λ < λ < · · · < λ N = 1 and for λ i − < λ ≤ λ ′ < λ i the process p ( λ, λ ′ ) only acts on S ∨ R i , i.e. it can be written as a work processon S ∨ R i composed with an appropriate identity on the other reservoirs. Denote the initial state50f this process on S by σ and the final state by σ . We find S S ( σ ) − S S ( σ ) = N X i =1 Q S,i T i = N X i =1 Z γ | [ λi − ,λi ] δQ S T i = Z γ δQ S T . (11.6)The first equality is just Definition 10.2, where Q S,i is the heat that S exchanges at temperature T i with reservoir R i between parameters λ i − and λ i . Therefore, Q S,i = R γ | [ λi − ,λi ] δQ S and thesecond equality follows. For the third equality T = T ( λ ) is introduced as a (piecewise constant)function on the curve such that T ( λ ) = T i for λ i − < λ < λ i .Starting from this, one can interpret the integral expression from Equation (11.6) with a con-tinuous function T on the path as the limit of reversible quasistatic process with finer and finerpartitions such that in the limit the piecewise constant temperature functions converge pointwiseto the continuous one and the piecewise C -curves converge to γ . The interpretation that theheat δQ S ( γ ( λ )) d γ ( λ )d λ flows at temperature T ( λ ) is valid in this limit. However, for the operationalmeaning of a continuous temperature function it is important to remember that it arises from alimit of piecewise constant functions. Only then the temperature can be put into relation withDefinition 9.1 for the temperature of heat flows.Just like we strengthened the first law in the presence of quasistatic processes, we can alsostrengthen the postulate on the existence of reversible processes with heat flows at well-definedtemperatures, Postulate 11, which is relevant for defining entropy on all of Σ S . Postulate 11’ (Quasistatic reversible processes with heat flows at well-defined temperatures) . For any system S ∈ S it holds that(o) For all states σ ∈ Σ S there exist m := dimΣ S reversible C -quasistatic processes on S ∨ R i , R i ∈ R for i = 1 , . . . , m , with curves γ , . . . , γ m passing through σ such that the derivativesof these curves at σ are linearly independent. That is, there exist λ i ∈ (0 ,
1) with γ i ( λ i ) = σ for i = 1 , . . . , m , such that γ ′ ( λ ) , . . . , γ ′ m ( λ m ) are linearly independent.(i) For any two states σ , σ ∈ Σ S there exists a piecewise C -quasistatic process { p ( λ, λ ′ ) } λ,λ ′ ⊂P S ∨ R ∨···∨ R N with { R i } Ni =1 ⊂ R such that there is a partition { λ i } Ni =0 of [0 ,
1] and in eachsegment ( λ i − , λ i ) S interacts with a single reservoir, and in total it transforms ⌊ p (0 , ⌋ S = σ into ⌈ p (0 , ⌉ S = σ .In this more restrictive version, (i) corresponds to the previous statement with the additionalrequirement that the process connecting the two states is quasistatic. This again makes surethat the entropy can be computed for any state in Σ S , for any system S now with the integralexpression from Eq. (11.6). Of course, also here the result from Clausius’ Theorem (Theorem 10.1)is important in order to make sure that this way of computing the entropy difference does notdepend on the used process.The purpose of the additional point (o) is the same as the corresponding point (o) in Pos-tulate 7’. And again, one might wonder whether (o) already implies (i). In fact, it looks evenmore plausible here than before since the argument of the direction of the processes can no longerpose a problem for reversible processes. Nevertheless, the existence of finitely many curves in aneighbourhood of a point σ with linearly independent derivatives do not immediately imply theexistence of a curve (coming from a quasistatic process) connecting any two points in the manifold.We do not discuss this technical topic any further here as it does not play a major role from aconceptual viewpoint. It is at this point good enough to have sufficient criteria in order to deducedifferentiability of the entropy. This is what we do next.In practice it is often the case that the differential work for reversible quasistatic processes isindependent of the actual process. In this case the notation δW S without the superscript ( p ) isnot just convenient but also formally correct, and the 1-form is defined on all of the state space51 S . For gases for instance, one can find δW S = − p d V during reversible quasistatic processes.This is not to say that the 1-form is exact, as one can see from the gas example.As a derived 1-form from other 1-forms, which are independent of the process, the differentialheat δQ S then also shares this independence.If now Postulate 11’ holds, it follows that entropy is differentiable in the same way as differ-entiability of U S was derived. In the argument for the differentiability of the entropy Clausius’Theorem 10.1 plays an important role, as it is necessary to be able to express entropy in termsof a path-independent integral. Again it is important that the 1-form over which is integrated iscontinuous. This may not be the case in all quasistatic processes. Namely, at the points wherethe piecewise constant function T jumps, the 1-form δQ S T is not continuous. However, with (o) weknow that there are quasistatic paths through any state during which T is constant, in particularcontinuous. This is guaranteed by the condition that the m reversible quasistatic processes areacting on S and a single reservoir. Hence, when showing differentiability of the entropy, one canjust use these paths.As a consequence, the differential d S S exists and is exact. In the context of a reversible qua-sistatic process in which heat is exchanged at temperature T , the differential reads d S S = δQ S T .Also the proof showing that entropy is differentiable is worked out in more detail in Section 13using the example of the ideal gas.With this we have shown that the presented framework, which was formulated in a discretemanner, can be extended to a continuous (even differentiable) formulation such that the usualdifferentiability properties of the internal energy and the entropy hold. This makes life a loteasier when following the standard arguments from phenomenological thermodynamics, e.g. whenderiving different thermodynamic potentials or computing thermodynamic quantities such as thecompressibility or the heat capacity. 52 Postulates: the temperature of a system
New notions: quasistatic thermodynamic processes, differential work and heat
Summary:
We discuss how to extend the notion of absolute temperature for reservoirs(Section 8) and heat flows (Section 9) to the temperature of a system. In particular, wegive insights into when this is possible and when one cannot expect that such a notion canbe meaningfully defined.The concepts of temperature introduced so far include the temperature of a reservoir and the temperature of a heat flow . However, we have not made a general definition of the temperatureof a system S ∈ S nor the temperature of a state σ ∈ Σ S . The reason for this is that it is notalways possible. As an example, it already suffices to consider a composite system S = S ∨ S ina generic state σ ∨ σ . Even if both S in state σ an S in state σ can be assigned individualtemperatures T and T , these will in general not be equal. Therefore, there is no unique “com-bined” temperature of the composed system S – one would intuitively need two “temperatures”in such a setting. Also for atomic systems it is not clear that a single temperature can alwaysbe assigned to each state. Even though atomic systems are considered “indivisible”, this is onlyintroduced in abstract terms of the composition ∨ , thereby not excluding the possibility that theatomic system could still consist of two compartments with independent properties, just not inthe sense composition speaks about it.Nevertheless, we know from the standard theory of thermodynamics that there are manysystems for which a temperature (of the system) can be defined in any state. We here investigatethe case of a system S with a state space Σ S ⊂ R that is an open subset of R , hence thestates are determined by two real parameters x, Y . We further assume that this system fulfils thequasistatic first law as well as the quasistatic version of the postulate for defining entropy andthat in reversible quasistatic processes the differential work can be written as δW = x d Y . Theseproperties have been introduced and discussed in the previous Section 11. They are formulated ina generic way, but we do not claim that there exists no setting with weaker assumptions in whichnevertheless a temperature can be assigned to the system.We will now define the temperature of such a system as the temperature of a reversible heatflow that is exchanged with the system. This is the basic compatibility requirement we have inmind here. Obviously, in the most general case there might be different possible reversible heatflows with different temperatures coming from a system in a given state. In this case the tem-perature could not be uniquely defined. As argued already in the beginning of this section, thisconstruction of temperature cannot always be applied. With the assumptions made, however, itis possible, and eventually allows us to define the temperature of a system via the temperature ofheat flows, which in turn relies on the definition of the absolute temperature of reservoirs.In general, the differentials of work and heat sum up to d U = δW + δQ . Using this for reversiblequasistatic processes in which heat is exchanged at a well-defined temperature it holds δQ = T d S and with the above assumptions it followsd U = x d Y + T d S . (12.1)Considering now processes during which Y is kept constant, we find T = ∂U∂S (cid:12)(cid:12)(cid:12)(cid:12) Y . (12.2)53his equation needs an explanation. First, the internal energy is a priori a function of x and Y .However, the entropy S is another state variable and one can consider U as a function of S and Y instead. Second, due to the differentiability of U and S , the partial derivative exists. We concludethat the temperature of the (infinitesimal) heat flow in a process, in which Y is kept constant, canbe written as the partial derivative of internal energy U w.r.t. entropy S . But the right hand sideonly depends on state variables, hence T is also a state variable in this particular case. Thus wecall T the temperature of the system in state ( S, Y ).Thinking of a gas and identifying x with the pressure p and Y with the volume V (up to aminus sign) it follows that one can generally define the absolute temperature of a gas in the waysketched here. This example is discussed in Section 13 in more depth.This notion of temperature, just like the temperature of a heat flow, eventually relies on theabsolute temperature of reservoirs and is defined as the temperature of the reversible heat flowexchanged with the system in the state of interest.We close this section with the consideration of two gases with state variables p , V and p , V ,internal energies U , U and entropies S , S . The internal energy of the composed system of thetwo gases, U := U + U then fulfilsd U = − p d V − p d V + T d S + T d S . (12.3)In the same spirit as done above for a single system one can now define the temperature of gas 1as T = ∂U ∂S (cid:12)(cid:12)(cid:12)(cid:12) V ,V ,S . (12.4)As before, it will indeed be a state variable, but it would not be a good idea to call T the “tem-perature of the (total) system”, as it is not the only temperature one can assign to the system.One could have done the same with exchanged indices 1 ↔ T , which will in generalnot yield the same result.We conclude that calling T the “temperature of a system” is always a matter of interpretation.The good thing about this is that one does not have to call anything a temperature of a systemin order to apply the thermodynamic theory. It is sufficient to think of T as the temperature of aheat flow, which is a well-defined concept without space for ambiguities.54 Postulates: - New notions: - Summary:
This section treats the example of the ideal gas in full detail. Starting withthe postulates and empirical observations the internal energy as well as the entropy functionof the ideal gas is formally derived. From there one can define the temperature of an idealgas, a concept that is eventually used to illustrate what happens in an irreversible heat flowbetween two ideal gases of different temperatures.The example of a gas, in particular of the ideal gas, has been mentioned in several instancesin this work. In all cases so far, the details were only introduced to the level at which they neededto be known. In this section we want to consider the example of the ideal gas from the beginningto the end, i.e. discuss the postulates and how they are reflected in this example in detail andfull length. This will culminate in the derivation of the ideal gas law, which connects the statevariable pressure and volume, which are initially assumed to describe the states, with the absolutetemperature T . In order to do so we first need to make clear what can be experimentally observed when dealingwith an ideal gas. This tells us which processes exist and how to compute their work cost. Asemphasized in the first sections of this paper, these are the necessary inputs to the theory ofphenomenological thermodynamics. Once these inputs are formulated, one can check whether thepostulates are satisfied and, if yes, the framework can be used to derive the other concepts ofinterest like thermodynamic entropy.Denote by A the system of the ideal gas. We choose the states to be described by pressure p and volume V . This means that an arbitrary state σ can always be written as σ = ( p, V ) ∈ ( R > ) =: Σ A . (13.1)Notice that for this treatment the amount of substance, usually denoted by n (the number ofmoles) or N (the number of particles), is here not a variable but a characteristic constant for thesystem, i.e. n = n A . One could also consider n to be an additional variable in the tuple describingstates and carry out the steps that follow below. However, in order to see a proper application ofthe theory to the example this is not necessary and would make it unnecessarily complicated. Forthe sake of an accessible example we do not do this here.Now, when working with this system, one can find that any state of the form ( p , V ) canbe transformed into ( p , V ) with p ≥ p by a C -quasistatic work process on A with the curve γ ( λ ) = ((1 − λ ) p + λp , V ). This can for instance be done by slowly applying friction on thecontainer from all sides, thereby heating up the gas (in an intuitive sense). Such processes arehere said to be of type
1. During this quasistatic process, in which the volume is kept constant,the differential work on A is found to be δW A = 32 V d p . (13.2)Next, one can observe that whenever two states ( p , V ) and ( p , V ) fulfil p V = p V thenthere exists a reversible C -quasistatic work process on A transforming one into the other along the55urve with pV = const . This is done by compressing or expanding the gas slowly while keepingit isolated from its surroundings. These processes are said to be of type
2. During reversible C -quasistatic processes on A , i.e. processes of type 2, the differential work can be computed as δW A = − p d V . (13.3)The typical set of work processes P A associated with the ideal gas contains all finite alternatingsequences of processes of type 1 and type 2. We here work with exactly these work processes, i.e.we say that P A contains all these sequences and no other processes. As a consequence, any workprocess can be written as a finite alternating sequence of processes of type 1 and type 2.Third, it turns out that for all states ( p , V ) and ( p , V ) with p V = p V there exists aheat reservoir R ∈ R such that they can be reversibly transformed into each other through areversible C -quasistatic work process on A ∨ R along the curve pV = const . in Σ A . Even more,the temperature T of the involved reservoir is found to depend linearly on pV , i.e. there exists a(system-specific) constant n A such that pV = n A RT , where R is the ideal gas constant. Suchprocesses are type
3. Also for this type of processes, Equation (13.3) gives the differential work. Infact, it turns out that the differential work of any reversible quasistatic process on A has this form.We summarize the necessary assumptions outside of the framework that must be given as aninput to the theory in this example: • the description of states as σ = ( p, V ) ∈ ( R > ) = Σ A • quasistatic work process on A of type 1 along constant V , pressure increasing, with δW A = V d p • quasistatic reversible work processes on A of type 2 along constant pV • work processes on A are exactly those which can be written as a finite alternating sequenceof processes of type 1 and type 2 • quasistatic reversible work processes on A ∨ R for R ∈ R of type 3 along constant pV with pV = n A RT for the temperature T of the reservoir R • for all reversible quasistatic process involving A (in particular type 2 and 3) the differentialwork is given by δW A = − p d V The choice of these three type of processes as our “basic processes” on which the thermody-namic description of the system is based is somewhat arbitrary. One could come up with differenttypes of processes and work processes that allow us to describe state changes on A and theirthermodynamic properties. However, the choice is here made such that the calculations of theinternal energy and the entropy can be carried out easily.In addition, the existence and the form of the processes of type 1, 2, and 3 allow us to concludewithout much effort that the ideal gas A satisfies Postulate 7’ and Postulate 11’. RegardingPostulate 7’ consider Figure 13.1 (a):(o) Obviously, dim Σ A = 2. The work processes of types 1 and 2 can pass through any state.Since the C -curves V = const . and pV = const . pass through any state s.t. the derivativesof the curves are linearly independent, it follows that (o) is satisfied. We tacitly assume here that the ideal gas is monoatomic. This implies that the number of degrees of freedomis f = 3, which yields the prefactor f = for the differential work in constant volume processes. Likewise, thissets the isentropic exponent to κ = . In a more general consideration, one could work with f and κ as variablesthat satisfy κ = f +2 f . We all know that n A will turn out to be the amount of substance (in moles) of the ideal gas A . However, forthe purposes presented here this is not important. p V V p p σ ′ Vp V V p p σ ′ σ ′′ Figure 13.1: The ( p, V )-diagram of the processes discussed in the proof showing that Postulates 7’and 11’ are fulfilled. The quasistatic curves γ are marked with thick lines. (a) A piecewise C -quasistatic work process on A consisting of the concatenation of a process of type 2 followed byone of type 1 transforms σ = ( p , V ) into σ = ( p , V ). The dashed lines are graphs of constant pV . (b) The reversible piecewise C -quasistatic process on A ∨ R ′ is a concatenation of a processof type 2 transforming σ = ( p , V ) into σ ′ = ( p ′ , V ′ ) followed by a process of type 3 turning thestate into σ ′′ = ( p ′′ , V ′′ ), followed by another process of type 2 transforming this into the finalstate σ = ( p , V ). The dashed lines are as in (a), the continuous lines are graphs of constant pV .(i) For two given states, choose the numbering ( p , V ) and ( p , V ) s.t. p V ≤ p V . Then thestate change ( p , V ) → ( p , V ) can always be obtained by process of type 1, transforming( p , V ) → ( p ′ , V ), followed by a process of type 1 raising p ′ to p . In total this is a piecewise C -quasistatic work process on A .(ii) Since any work process on A can be written as a sequence of alternating processes of types1 and 2, it can easily be seen that the total work done on A only depends on the initial andfinal state.In a similar spirit, it follows for Postulate 11’, see Figure 13.1 (b):(o) Reversible C -quasistatic processes of type 2 and 3 can pass through any state and theircurves pV = const . and pV = const . have linearly independent first derivatives in anypoint.(i) Any two states ( p , V ) and ( p , V ) can be reversibly transformed into each other by theconcatenation of processes of type 2, followed by type 3, and again followed by type 2. Infact, the reservoir R ′ needed for executing the process of type 3 can be chosen arbitrarily,in particular, it can have any temperature T ′ . In total, this is a reversible piecewise C -quasistatic process on A ∨ R ′ .The other postulates do not only depend on the system A but on the whole structure of sys-tems and processes. We do not check them explicitly here but with a reasonable choice of Ω (thethermodynamic world, see Postulate 1) and thus S (the set of thermodynamic systems) as well as P they are fulfilled as one intuitively expects.We can now compute the internal energy difference (and hence the internal energy functionup to the constant U A ) of two arbitrary states σ = ( p , V ) and σ = ( p , V ) according toDefinition 3.2. This can be done via the process suggested in the discussion of Postulate 7’ (i) This is a consequence of the fact that any line of constant non-zero pV has exactly one intersection with anyother line of constant non-zero pV . σ ′ = ( p ′ , V ) fulfils p ′ V = p V asit is obtained from σ through a process of type 2. We find:∆ U A = U A ( σ ) − U A ( σ )= Z σ ′ σ − p d V + Z σ σ ′ V d p = − p V (cid:20) − V − (cid:21) V V + 32 V (cid:2) p (cid:3) p p ′ = 32 p V (cid:16) V − − V − (cid:17) + 32 V p − p (cid:18) V V (cid:19) ! = 32 ( p V − p V ) . (13.4)With the reference energy U A we can conclude that for any state σ = ( p, V ) ∈ Σ A the internalenergy is U A ( σ ) = 32 pV + U A . (13.5)The internal energy function is obviously C ∞ on Σ A , in particular C . As proved in Section 11,the latter must be the case.Using the process suggested in the discussion of Postulate 11’ above we can compute theentropy difference of the two states σ and σ according to Definition 10.2. The intermediatestates σ ′ = ( p ′ , V ′ ) and σ ′′ = ( p ′′ , V ′′ ) fulfil p ′ V ′ = n A RT ′ = p ′′ V ′′ as well as p ′ ( V ′ ) = p V and p ′′ ( V ′′ ) = p V , as can easily be seen from Figure 13.1 (b). Also, the only non-trivial heatflow appears in the type 3 process transforming ( p ′ , V ′ ) into ( p ′′ , V ′′ ) at constant temperature T ′ .Hence: ∆ S A = S A ( σ ) − S A ( σ ) = Z σ σ d S A = Z σ σ δQ A T = Z σ ′′ σ ′ δQ A T ′ = Z σ ′′ σ ′ d U A + p d VT ′ = 32 T ′ ( p ′′ V ′′ − p ′ V ′ ) + 1 T ′ Z V ′′ V ′ p d V = 0 + p ′ V ′ T ′ Z V ′′ V ′ V d V = n A R ln (cid:18) V ′′ V ′ (cid:19) = n A R (cid:18)
32 ln (cid:18) p p (cid:19) + 52 ln (cid:18) V V (cid:19)(cid:19) . (13.6)As before, with the reference entropy S A it follows that the entropy of any state σ = ( p, V ) canbe written as S A ( σ ) = n A R (cid:18)
32 ln (cid:18) pp (cid:19) + 52 ln (cid:18) VV (cid:19)(cid:19) + S A , (13.7)where ( p , V ) = σ is the reference state for computing the entropy. As was the case before forthe internal energy, it immediately follows that the entropy is C ∞ (Σ A ), in particular C (Σ A ).Notice that Equations (13.5) and (13.7) are the standard expressions for the internal energy and58he entropy of an ideal gas in the variables pressure p and volume V .We now want to compute the internal energy U ( S, V ) as a function of the entropy and thevolume. This will allow us to determine an expression for the temperature of the ideal gas , as isdiscussed in Section 12. With Equations (13.5) and (13.7) it follows that U ( S, V ) = 32 p V (cid:18) VV (cid:19) − · exp (cid:18) S − S A n A R (cid:19) + U A . (13.8)Taking the derivative of U ( S, V ) at constant V with respect to S we obtain the temperature ofthe ideal gas: T = ∂U∂S (cid:12)(cid:12)(cid:12)(cid:12) V = 32 p V (cid:18) VV (cid:19) − ·
23 1 n A R exp (cid:18) S − S A n A R (cid:19) = 23 1 n A R ( U A − U A ) . (13.9)There are important conclusions one can draw from this. First, the internal energy of an ideal gascan be written as U A ( σ ) − U A = n A RT . Hence, the temperature of the ideal gas in state σ , T , ina state variable and U A can be written as a function of T only. The fact the T defined accordingto Equation (13.9) is a state variable in this setting has already been argued in Section 12 on moregeneral assumptions and can be observed in the explicit example here.Second, combining this result with Equation (13.5), we see that the temperature of a state σ = ( p, V ) is always T = pVn A R , which is the ideal gas law. This is remarkable, as this expressionfor the temperature of the gas is exactly the same as we had for the temperature of the heat reser-voir R with the help of which the state σ can be transformed reversibly (through a quasistaticprocess on A ∨ R ) to any other state with the same temperature. Hence, what we call the temper-ature of the system (or the state of the system, for that matter) is equal to the temperature of aheat reservoir with which the system can reversibly interact while exchanging heat in this state.Again, the fact that both notions of temperature are the same has been shown more generally inSection 12. We apply the gained knowledge about the temperature of an ideal gas to Definition 9.1, thetemperature of a heat flow. We know from Lemma I.3 that the temperature of a reversible heatflow Q = 0 is unique if it exists. In Example 9.3 we have also seen that if the heat flow is notreversible it is very well possible that different temperatures can be assigned to it. With theconcept of the temperature of an ideal gas it is now possible to refine this argument.Consider two ideal gases A , A ∈ A in states σ ∈ Σ A and σ ∈ Σ A , respectively. Supposethat the temperatures according to Equation (13.9) are T > T . When thermally connecting thetwo gases, e.g. by bringing them into physical contact with the same thin metallic rod, we knowfrom experience that a non-zero heat flow Q from the hotter gas to the colder one can be observed.What temperature(s) can be assigned to this heat flow?Using Definition 9.1 we see that this question can be rephrased as: what heat reservoirs canbe put in between the two systems without changing the thermodynamic properties of the processon the systems A and A ? Figure 13.2 depicts this interpretation of the question. Thinking ofa long rod used to establish the contact between the systems, one can in principle cut the rod inhalf at any point and find a reservoir with the right temperature to put in between. This will notchange anything with respect to the thermodynamic properties of A and A under the process.Since the temperature decrease over the rod is continuous one can find places to put reservoirs inbetween at all temperatures in the interval [ T , T ]. Hence, any temperature T with T ≤ T ≤ T can be assigned to the heat flow Q and our framework provides a precise language to talk aboutthese intuitive but usually only vaguely described concepts. This is again a statement that is supported by experience, i.e. experiment, rather than the theoretical framework. A A Q (b) A R Q (c) R A Q (d) A R ′ R ′′ Q Q Q R ′′′ A Q Figure 13.2: (a) When thermally connecting two ideal gases at different temperatures a non-zero heat flow Q can be observed to flow from the warmer to the colder gas. Call this process p ∈ P A ∨ A . (b) & (c) According to Definition 9.1 the heat Q flows at temperature T if thereexist two reservoirs R ∼ R , both at temperature T , and the process can be split up into twoprocesses p i ∈ P A i ∨ R i such that for A i the thermodynamic properties of p and p i are the same.(d) Thinking of a rod between the two gases has the advantage that it becomes manifest that therod could be intercepted at any point with a heat reservoir at a certain temperature. Since thedecrease of temperature over the rod from A to A will be continuous, any temperature in theinterval [ T , T ] is attained at one point. In the figure, three different reservoirs at temperatures T ′ ≥ T ′′ ≥ T ′′ are depicted.
14 Scaling
Postulates: - New notions: scaled system, intensive and extensive state variables, principle ofmaximum entropy
Summary:
The scaling of thermodynamic systems is introduced and used to prove theprinciple of maximum entropy.Often one wants to be able to talk about the same type of systems in different sizes, i.e. aboutscaled systems. Intuitively, the properties of a system 2 S that is “twice as big” as S are clear:the state spaces have the same structure with the difference that “extensive” (i.e. homogeneous)variables of corresponding states are scaled by a factor of 2; likewise the thermodynamic processesthat can be carried out on 2 S are the “scaled” processes p that can be carried out on S .We immediately notice that these intuitive ideas lack a formal clarification of the notions“extensive variable” and “scaled process”. In the example of an ideal gas as discussed in Section 13this is rather easy. Let 2 A be the ideal gas from the previous sections scaled by a factor of 2. Weknow that the variable that should be scaled is the volume, while the pressure is “intensive”, i.e.is not scaled with the system size. Also, the system specific constant n A (corresponding to theamount of substance) scales as n A = 2 n A like the volume. On the other hand the pressure doesnot change under the scaling. It is important to realize that the knowledge about the differentbehaviours of the state variables under scaling cannot come out of the framework but must beseen as an input to it.With this, states 2 σ = ( p, V ) ∈ Σ A ↔ σ = ( p, V ) ∈ Σ A are in a 1-1 correspondence. Likewise,the (quasistatic) processes connecting two states 2 σ and 2 σ fulfil δW A = 2 δW A , as can easily beseen from Eq. (13.2) and (13.3), and are in a 1-1 correspondence with the (quasistatic) processesconnecting σ and σ . It follows that, when choosing U A = 2 U A , the internal energy must Even if one does not make this choice, the physically relevant internal energy differences still scale linearly withthe factor 2. The analogous statement holds for the entropy. U A = 2 U A , which is indeed the case as can easily be checked in Eq. (13.5). In orderto obtain the analogous result for entropy it suffices to choose S A = 2 S A . Again, we see thatEq. (13.7 satisfies S A = 2 S A . The derived concept of the temperature of an ideal gas in state σ ∈ Σ A , Eq. (13.9), does not scale with the system size, as the temperature of 2 σ is the same asthe temperature of the state σ . More generally, we consider the case of a scaling factor λ = n ∈ N > and explain the proper wayof talking about the scaled system nS . Let S (2) ˆ= · · · ˆ= S ( n ) ˆ= S be copies of S and let ϕ ( i ) be thethermodynamic isomorphism describing the equivalences. For processes p, p (2) , . . . , p ( n ) ∈ P andstates σ ∈ Σ S , σ ( i ) ∈ Σ S ( i ) with ϕ ( i ) ( p ) = p ( i ) and ϕ ( i )Σ ( σ ) = σ ( i ) we write p ( i ) ˆ= p and σ ( i ) ˆ= σ . Definition 14.1 (Scaled system by a factor of λ = n ∈ N > ) . The system nS ∈ S is the scaledsystem S ∈ S by a factor of n if there exists an embedding ψ : P → P of nS into S ∨ S (2) ∨ · · · ∨ S ( n ) , where S ˆ= S ( i ) are copies of each other. The embedding ψ is an injective mapping with theproperties(i) for ˜ p ∈ P acting on nS it holds ψ (˜ p ) = p ∨ p (2) ∨ · · · ∨ p ( n ) for some p ∈ P acting on S and p ( i ) ˆ= p , (ii) W nS (˜ p ) = W S ∨ S (2) ∨···∨ S ( n ) ( ψ (˜ p )).The embedding can be understood as the nS = S ∨ S (2) ∨ · · · ∨ S ( n ) with restrictions , therestriction being that the thermodynamic processes that can be applied to nS are exactly thoseof the form p ∨ p (2) ∨ · · · ∨ p ( n ) with p ( i ) ˆ= p . As a consequence, the states nσ ∈ Σ nS correspond tostates of the form σ ∨ σ (2) ∨ · · · ∨ σ ( n ) with σ ( i ) ˆ= σ for σ ∈ Σ S .If a process ˜ p ∈ P acts on nS we write ˜ p =: np , where p ∈ P is the process acting on S suchthat ψ (2 p ) = p ∨ p (2) ∨ · · · ∨ p ( n ) . This notation is well-defined since ψ is injective and it can beapplied to any ˜ p ∈ P acting on nS due to (i).The work function W nS scales with the system’s size. This follows from Definition 14.1 (ii), W nS ( np ) (ii) = W S ∨ S (2) ∨···∨ S ( n ) ( ψ (˜ np )) = W S ( p ) + n X i =2 W S ( i ) ( p ( i ) = nW S ( p ) . (14.1)In the second equality we used the additivity of work, while the third equality holds because S ( i ) ˆ= S are copies of each other with p ( i ) ˆ= p . Consequently, the internal energy and the entropyalso scale with the system size. They inherit the scaling property of the work function since anyentropic and energetic quantity homogeneously depends on the work functions.If λ = ν for some ν ∈ N > the scaled system λS is defined as the system satisfying λS ∨ · · · ∨ λS | {z } ν times ˆ= S , (14.2)which allows us to extend the concept of scaling to at least the positive rational numbers λ = µν ∈ Q > by defining λS = µ (cid:0) ν S (cid:1) . (14.3) Here it is important not to forget that the reference state σ = ( p , V ) ∈ Σ A from which the entropy iscomputed also scales to 2 σ = ( p , V ) ∈ Σ A . Otherwise one could come to the wrong conclusion that theentropy does not scale linearly, i.e. is not extensive. Clearly, if p acts on S , then its copy p ( i ) acts on the copy S ( i ) of S . Being able to scale with rational factors, since the rational numbers are dense in R , we can “approximate” λS with λ ∈ R arbitrarily well. However, making the notion “approximate” precise requires at least a topology on S .We do not go further into this technical discussion and are content with rational factors for now. X S on the system S is called extensive if it satisfies X λS ( λσ ) = λX S ( σ ) , (14.4)where X λS is the corresponding state variable on λS . Similarly, it is called intensive if X λS ( λσ ) = X S ( σ ) . (14.5)With volume, internal energy and entropy (and the amount of substance, for that matter, eventhough we here treat it as a constant state variable) we have already encountered several extensivestate variables of the ideal gas. The intensive state variables were pressure and temperature.Clearly, in practice the scaling factor λ cannot become arbitrarily small or large withoutsurpassing the region of validity of the theory.For instance, if 1 l of an ideal gas A at atmospheric pressure and temperature and is scaled witha factor λ = 10 − we know that the scaled system λA will contain less than a particle. There isan obvious problem with the interpretation of λA as a physical system. Even if there were still afew particles in λA quantum effects may become dominant in the description of the system thatwere averaged out in the description of A . Hence, λA and A will not be in correspondence witheach other as was required above. More specifically, the differential work for reversible quasistaticprocess will most likely no longer read δW λA = − λp d V .On the other hand, when scaling an ideal gas A with a factor λ = 10 , the mass of the system λA will be of the order of a solar mass and effects like self-gravitation will become dominant.Notice that the invalidity of thermodynamics at the very large or very small scale is similar tothe one encountered in classical mechanics. In both cases it is not a shortcoming of the frameworkitself but rather the fact that the underlying theory (here: the one to define systems, states,processes, and to compute work) is only an approximation. Even though we know about thedifferent effects at the very small or the very large scale, we keep considering general rationalscaling factors λ ∈ Q r { } since it is impossible to decide on the region of validity on an abstractlevel. The general mathematical way of formalizing the scaling of systems cannot take these effectsinto account. In this section we apply the scaling of the ideal gas to prove the principle of maximum entropy forideal gases. This principle is a consequence of the Entropy Theorem 10.4. As will become evidentduring the developments in this section, the considerations presented here can be extended toother systems that share basic properties with the ideal gas. These properties are discussed at theend of this section.From Eq. (13.5) we immediately see that instead of describing states with the tuple ( p, V ) wecould also choose (
U, V ) as the basic variables. In this section we will take this description, σ = ( U, V ) (14.6)due to the nice property that under scaling with the factor λ one can write λσ = ( λU, λV ).Consider now the composition of two scaled ideal gases A ′ = λA and A ′′ = (1 − λ ) A for λ ∈ (0 , S = A ′ ∨ A ′′ . States of S are described as σ = σ ′ ∨ σ ′′ for σ ′ ∈ Σ A ′ and σ ′′ ∈ Σ A ′′ andwe may write σ = ( U ′ , V ′ , U ′′ , V ′′ ). As shown in Lemma J.1 the entropy of S can be written asthe sum of the entropies of its subsystems, S S ( σ ) = S A ′ ( σ ′ ) + S A ′′ ( σ ′′ ) . (14.7)It was emphasized in Section 1 that the composition operation does not imply that the composedsubsystems are in physical contact. They are simply described as one system. The state σ ′ ∨ σ ′′ is62hus sometimes called constrained in the sense that the two subsystems (ideal gases here) cannotphysically interact.Typically there exists a work process q ∈ P S on S that lets the subsystem exchange energy andvolume for a sufficient amount of time such that q transforms ⌊ q ⌋ S = ( U ′ , V ′ , U ′′ , V ′′ ) = σ ′ ∨ σ ′′ into the final state ⌈ q ⌉ S = ( λ ( U ′ + U ′′ ) , λ ( V ′ + V ′′ ) , (1 − λ )( U ′ + U ′′ ) , (1 − λ )( V ′ + V ′′ )) , (14.8)i.e. the internal energies and the volumes have changed such that each subsystem has a shareproportional to its size (in terms of λ ). This process has work cost W A ′ ( q ) = W A ′′ ( q ) = 0 on bothsubsystems. We call the final state unconstrained and write ⌈ q ⌉ S =: σ ′ + σ ′′ ≡ ( U ′ + U ′′ , V ′ + V ′′ ).This notation is motivated by our choice of extensive variables and has to be explained.First, we profit from the choice of the extensive variables U and V which give manifest meaningto the addition “+” of two states. Even though this is convenient, it is not a necessary choice inorder to define this addition operation. One could also define it in terms of the variables pressure p and volume V such that when computing the internal energies they satisfy λ ( U ′ + U ′′ ) for A ′ and (1 − λ )( U ′ + U ′′ ) for A ′′ , for instance.Second, technically the 2-tuple ( U ′ + U ′′ , V ′ + V ′′ ) is not a proper description of a state on S ,since the states of S are described by 4-tuples. However, by the definition of σ ′ + σ ′′ as the finalstate of q on S , the states of the subsystems A ′ = λA and A ′′ = (1 − λ ) A can always be obtainedfrom the notation ( U ′ + U ′′ , V ′ + V ′′ ) as the part of it proportional to λ and (1 − λ ), respectively.We will use this notation with its implicit meaning from now on.Third, the “decomposition” of an unconstrained state σ ′ + σ ′′ into summands is not unique.That is, there exists arbitrarily many other states ˜ σ ′ ∈ Σ A ′ and ˜ σ ′′ ∈ Σ A ′′ with ˜ σ ′ + ˜ σ ′′ = σ ′ + σ ′′ .Finally, not every state σ ∈ Σ S is an unconstrained state. This is already clear from theconstruction of the unconstrained states as the final states of special processes.Since q ∈ P S is a work process on S we can use the Entropy Theorem 10.4 to bound theentropy of the unconstrained state σ ′ + σ ′′ from below. It states that S A ′ ( σ ′ ) + S A ′′ ( σ ′′ ) ≡ S S ( σ ′ ∨ σ ′′ ) ≤ S S ( σ ′ + σ ′′ ) (14.9)with equality if q is reversible. Typically, q is reversible if and only if it is an identity process, i.e.if and only if the constrained state was already unconstrained, which means that λU ′′ = (1 − λ ) U ′ and λV ′′ = (1 − λ ) V ′ . That is, the states ˜ σ ′ = ( λ ( U ′ + U ′′ ) , λ ( V ′ + V ′′ )) ∈ Σ A ′ and ˜ σ ′′ =((1 − λ )( U ′ + U ′′ ) , (1 − λ )( V ′ + V ′′ )) ∈ Σ A ′′ satisfy ˜ σ ′ ∨ ˜ σ ′′ = ˜ σ ′ + ˜ σ ′′ . For these states the boundin Eq. (14.9) is saturated.With this the derivation of the principle of maximum entropy for ideal gases is done. Theorem 14.2 (Principle of maximum entropy) . For an ideal gas A ∈ S and a scaling parameter λ ∈ (0 ,
1) let S = A ′ ∨ A ′′ be the composed system of A ′ = λA and A ′′ = (1 − λ ) A . Furthermore,let ( U, V ) ∈ Σ S be an unconstrained state. Then S S ( U, V ) = max U ′ + U ′′ = UV ′ + V ′′ = V S A ′ ( U ′ , V ′ ) + S A ′′ ( U ′′ , V ′′ ) . (14.10)Informally one can interpret the statement of this theorem as: in a closed system the entropyis maximal for unconstrained states. Instead of “unconstrained states” the notion of “equilibriumstates” is used more often. However, removing the constraint is what happens operationally, whichis why we stick to the former notion.Now we can provide a good reason for choosing U and V as the variable to describe thestates of the ideal gas. The principle of maximum entropy states that the entropy function for In practice this means that the subsystems may exchange energy and volume (e.g. by means of a freely movingpiston between them) for a time t that is longer than the typical equilibration time of the system. , S S , restricted to unconstrained states ( U, V ), is concave . This can be seen as follows. The S = A ′ ∨ A ′′ ≡ λA ∨ (1 − λ ) A with the restricted state space containing only unconstrained states(and thus only processes respecting this restriction can be applied) is equivalent to the system A ,since λ + (1 − λ ) = 1. This means in particular, that the principle of maximum entropy can bewritten as S A ( U, V ) = max U ′ + U ′′ = UV ′ + V ′′ = V S λA ( U ′ , V ′ ) + S (1 − λ ) A ( U ′′ , V ′′ ) , (14.11)for any state σ = ( U, V ) on A . Notice that now on the left hand side te entropy does no longerhave to be evaluated for a composed system.For arbitrary internal energies U , U , volumes V , V and any parameter λ ∈ [0 ,
1] it then holds S A ( λU + (1 − λ ) U , λV + (1 − λ ) V ) ≥ S λA ( λU , λV ) + S (1 − λ ) A ((1 − λ ) U , (1 − λ ) V )= λS A ( U , V ) + (1 − λ ) S A ( U , V ) . (14.12)This is the definition of a concave function. For the second equality it is crucial that all threequantities entropy, internal energy and volume, are extensive, which explains the choice of vari-ables.Notice that Eq. (13.7) giving an explicit formula for the entropy of an ideal gas in terms ofinternal energy U and volume V must fulfil the principle of maximum entropy according to ourderivations. Hence it must be concave. Indeed S ( U, V ) fulfils both statements. One could nowask why we did not just state Theorem 14.2 as a mathematical statement and proved it by directcalculation using the explicit formula for S ( U, V ) from Eq. (13.7). Indeed, we could have doneso, if the ideal gas was the only case in which the principle of maximum entropy held. However,with the detailed derivation in terms of concepts of the framework we can now discuss extensionsof this theorem to other systems than just ideal gases.In short, any type of system for which the relevant concepts in the derivation can be definedmeaningfully, fulfil a maximum entropy principle. In particular, the fact the the gas was ideal hasnot been used anywhere in this derivation. More generally, the relevant concepts we needed were: • The total system S can be written as the disjoint composition of two other systems S ′ and S ′′ , S = S ′ ∨ S ′′ . • Arbitrary states σ ′ and σ ′′ of S ′ and S ′′ , respectively, can be “added” such that σ ′ + σ ′′ ∈ Σ S .This is the state the system attains after the constraints are removed, i.e. the two subsystemscan interact and exchange e.g. energy, volume, or even particles. Notice that “+” does nothave to be component wise addition of the entries of a tuple denoting the state. Also, as isalways the case with states of a composite system, the individual states of the subsystemscan be retrieved from the joint state. • There are work processes on S with zero work cost on any subsystem that remove theconstraints, i.e. that transform σ ′ ∨ σ ′′ to σ ′ + σ ′′ for any choice of σ ′ ∈ Σ S ′ and σ ′′ ∈ Σ S ′′ .The principle of maximum entropy in the more general case then reads says that for anyunconstrained state σ ∈ Σ S (i.e. any state which is the final state of a process that removes theconstraint) it holds S S ( σ ) = max σ ′ + σ ′′ = σ S S ′ ( σ ′ ) + S S ′′ ( σ ′′ ) , (14.13)where the maximum goes over all σ ′ ∈ Σ S ′ and σ ′′ ∈ Σ S ′′ that satisfy σ ′ + σ ′′ = σ .Turning to concavity, an additional property of the system becomes relevant: For this argument it is again crucial that the states in the statement of the principle of maximum entropy areunconstrained ones only. The system S restricted to unconstrained states is equivalent to both λ ˜ S and − λ S ′′ for some λ ∈ (0 ,
1) and the description of the states of all system happens in corresponding extensivevariables (which are the ones that are relaxed when removing the constraint discussed todefine “+”).If this is the case, then concavity is implied also in the more general case. For concavity, thestates need to be described in terms of extensive variable such that the addition “+” is just thecomponent-wise addition of the entries of the tuple describing thermodynamic states.Even though concavity of a function is a formal property a given function can or can not have,the argument for a concave entropy function in the general case goes over an operational line ofreasoning involving many of the concepts introduced throughout this paper. In this context thestatement that entropy is concave in two extensive variables (e.g. U and V) is a statement aboutwhat happens when one relaxes a constraint between two subsystems by coupling them (e.g. byletting them exchange energy).The general principle of maximum entropy is the basis to investigate further thermodynamicpotentials such as the free energy or the Gibbs free energy. These will fulfil similar principlesof extremal values with different constraints. It will also be relevant when deriving stabilityconditions which for instance imply that the specific heat capacities C V := ∂U∂T (cid:12)(cid:12)(cid:12)(cid:12) V ≥ C p := ∂U∂T (cid:12)(cid:12)(cid:12)(cid:12) p ≥ C p − C V ≥ iscussion and conclusion Summary
In this paper we have introduced the basic Postulates 1-11 in order to derive the foundations ofphenomenological thermodynamics. From the notion of thermodynamic systems, processes, statesand work (Postulates 1-6), we formulated the first law (Postulate 7) and derived the internalenergy function. After introducing the notion of equivalent systems, culminating in Postulate 8requiring that there is an arbitrary number of copies of any system, we stated the definition of heatreservoirs and formulated the second law (Postulate 9). From there, we were able to rigorouslyfollow the standard lines to prove Carnot’s Theorem and introduce absolute temperature. Usingthe notion of absolute temperature of heat reservoirs it was possible to define the concept of thetemperature of heat flows, which was then used to prove Clausius’ Theorem 10.1 and the En-tropy Theorem 10.4. Along the way we defined entropy, for which it was relevant that there existreversible processes over which the entropy difference between any two states can be computed(Postulate 11).We then used the introduced concepts to extend the framework to quasistatic processes. Thisis a “continuous” version of the a priori discrete concept of a thermodynamic process. Withquasistatic processes it became possible to show that whenever the generalized Postulates 7’ and 11’hold, then the internal energy function as well as the entropy are continuously differentiable statevariables. With this it was possible to discuss the absolute temperature of arbitrary systems.The worked out example of the ideal gas gives insights into how the theory can be appliedto traditional settings. The assumptions that have to be made prior to applying the frameworkfor phenomenological thermodynamics in this example are spelled out and discussed. Also, thevalidity of the postulates is checked and specific expressions for the internal energy as well as theentropy function are derived. This allowed us to revisit the example of an (irreversible) heat flowthat can be assigned a spectrum of temperatures more formally.A further exploration into the formalism for describing the scaling of systems led to the deriva-tion of the maximum entropy principle. Again, the ideal gas was a very helpful example. However,the result holds also for the entropy functions of more general systems. The specific assumptionfor the maximum entropy principle to hold have been discussed.This concluded the comprehensive discussion of “the laws of thermodynamics” in the tradi-tional sense contrasted with the basic postulates of the framework of phenomenological thermo-dynamics presented in this paper.
Discussion of main contributions
It may be surprising how much technical work was necessary to make these intuitively straightfor-ward basic foundations precise. This tells us two things: it is possible to equip the intuitively clearphysical theory of phenomenological thermodynamics with a mathematically rigorous and precisebackground; but it seems that this can only be done with considerable effort on the technical side.After working on this project for a long time, we have good reason to believe that the compli-cated derivations are intrinsically necessary. Having a closer look at the sections which requiredmore technical proofs than others, we see that most of these talk about very intuitive conceptswhile the less technical sections are conceptually more relevant. For example, coming up with asuitable definition for when two systems are considered to be copies of each other took severalpages of definitions and lemmas with proofs while, on the other end of the spectrum, the proof ofCarnot’s Theorem could be distilled to half a page.It is worth mentioning here that the intuition of all concepts can be made precise enough tointroduce thermodynamics according to this framework in a undergraduate course without toomany technicalities. The authors of this paper have reworked the script of the course
Theory of eat held by Gian Michele Graf at ETH in 2005 for the same course in 2019 taught by RenatoRenner [35]. In the script from 2019, the main ideas of this paper are presented on a more intuitivelevel.Nevertheless, we believe it is one of the main achievements of this paper that we handle termssuch as equivalent systems, heat reservoirs or the temperature of a heat flow with great care,thereby crystallising the definitions to the essence. Especially the definition of heat reservoirs issomething the authors have not seen elsewhere in any introduction to phenomenological thermo-dynamics. Even though this term is omnipresent in thermodynamics it seems rather difficult toformalise it. Heat reservoirs should be infinite systems, without using the term “infinite‘” in thedefinition. They should be very simple in terms of what one can do with them. Basically theyshould just provide or take up heat. On the other hand, they should be translation invariant intheir internal energy so the current energy does not dramatically affect what one can do with them.The three points in our Definition 5.3 for heat reservoirs carefully capture these characteristicsin mathematical statements that are sufficient to formally work with the defined notion. Eachpoint could be justified on an intuitive basis and they are simple and clear enough in order tosee that they are necessary conditions. As the derivations using the term heat reservoirs and thesecond law show, they are also sufficient for introducing all basic concepts of phenomenologicalthermodynamics, in particular thermodynamic entropy.Due to the systematic introduction of the eleven postulates, from which all results are derived,it is now possible to investigate which of the usually made assumptions are unnecessary and whatmay be missing in traditional texts.An example for a postulate that is usually missing is, as we would argue, the assumptionthat any two states of any system can be transformed into each other by means of reversibleprocesses with heat flows at well-defined temperatures, Postulate 11. Example 10.3 emphasizesthat, without this postulate, it is thinkable to have systems in the set of thermodynamic systemsin a general formulation of phenomenological thermodynamics that do not fulfil it. This showsthat the postulate is necessary.Thermodynamic equilibrium, on the other hand, is a notion that we did not have to use for-mally. We did not have to define in the beginning what we meant by this. Neither did we requirethat the states we worked with are thermodynamic equilibrium states. Arguably, the result ofLemma D.4 (for any state of any system there exists an identity process) captures already a greatdeal of what is usually considered an equilibrium state. But this lemma came as a consequence ofthe first law (Postulate 7) rather than any requirement on what is called a state. Hence we showthat phenomenological thermodynamics can be phrased as a physical theory without this term.We consider the zeroth law as the most prominent victim of the postulates that are usuallyunnecessarily made. This was the main topic of our previous paper [31], but nevertheless deservesmentioning again. The zeroth law can be phrased in many different ways [5, 10, 11, 15]. Almost allvariations share the core statement that “being in two thermal equilibrium with” is a transitiverelation on the set of systems. That is, if A and B are in thermal equilibrium with each other andlikewise B and C , then so are A and C . Together with the usually implicitly assumed reflexivityand symmetry, the zeroth law then implies that this relation is an equivalence relation.This equivalence relation is then relevant for defining a sensible notion of empirical temper-ature, long before the absolute temperature is introduced with the help of Carnot’s Theorem.More precisely, one can say that two systems have the same empirical temperature if they are inequilibrium with each other.The zeroth law is usually postulated on the same level as the first law and second law. Ourwork now shows that this is in fact not necessary. While the first and second law are corepostulates also in our considerations, the zeroth law was never assumed. Even more, we haveshown that the notion of absolute temperature can nevertheless be defined and the corresponding Other notions instead of systems used in [15] are bodies [11] or assemblies [10]. ≻ . This order relationand its properties are introduced axiomatically in precise mathematical terms. Entropy is then theresult of the Entropy Principle, a theorem they prove within the axiomatic basis that guaranteesthe existence, uniqueness and monotonicity and additivity of the entropy function. From there allother concepts follow. For instance, absolute temperature is derived from the entropy function. Likewise, different versions of the second law as well as Carnot’s Theorem are results from theseconsiderations.In our framework the definition of thermodynamic entropy relies on the traditional ratio ofreversible heat divided by temperature. This way of introducing entropy requires that the notionof absolute temperature and thus also the proof of Carnot’s theorem are there beforehand. On theother hand, our Entropy Theorem, which is similar albeit not equivalent to Lieb and Yngvason’sEntropy Principle, is derived from the thermodynamic definition of entropy and thus eventuallyrelies on the concept of absolute temperature and the second law. We conclude that which concepts As Lieb and Yngvason put it: “Temperature [...] is a corollary of entropy; it is epilogue rather than prologue.”[26]. ostulatesfor ≻ basic ass. first law second lawEntropyPrinciple def. entropyas monotone Carnot’sTheoremdef. absolutetemperature postulates first law second lawEntropyTheorem def. entropyas ∆ S = QT Carnot’sTheoremdef. temperatureof heat flows def. absolutetemperature
Figure 14.1: We contrast the main line of argument presented in the modern prototype of Cara-th´eodory-based introductions to thermodynamics [26] (left) with the paper at hand (right). Thecolor code is definitions in red, assumptions and postulates in orange, and derived implicationsin green. Left: Lieb and Yngvason define an order relation ≻ and postulate several propertiesof it. With this, they are able to prove a theorem called the Entropy Principle, from whichentropy is obtained as the monotone of ≻ . Based on this definition of entropy they show that thesecond law holds as well as Carnot’s theorem. Other approaches in the spirit of Carath´eodoy’s [9]proceed in a similar way. Right: In this paper we capture all basic assumptions in postulates,from which the first and second law are two (they are depicted separately picture the comparisonmore clearly). After introducing absolute temperature via Carnot’s Theorem the temperature ofheat flows between arbitrary systems is defined. The usual definition of thermodynamic entropycan then be used and results such as Clausius’ Theorem or the Entropy Theorem can be proved.are considered “more fundamental” than others heavily depends on the basis one formulates. Andthe method proposed by Lieb and Yngvason, as beautiful as it is, is not the only way to formalizethe main concepts of phenomenological thermodynamics. Acknowledgements
A particular source of inspiration for this work was the script to the course
Theory of heat taught byGian Michele Graf at the ETH in 2005 [24]. Many of the ideas presented here arose from discussionsabout this script. We thank Jakob Yngvason and David Jennings for fruitful discussions on ourprevious paper [31] and Stefan Wolf and Tam´as Kriv´achy for comments on an earlier version ofthis framework. Both authors acknowledge support from the “COST Action MP1209” as well asthe Swiss National Science Foundation through SNSF project No. 200020 165843 and through thethe National Centre of Competence in Research
Quantum Science and Technology (QSIT).69 ppendices
A Systems
For completeness and an easier reading we repeat the postulate introducing thermodynamic sys-tems.
Postulate 1 (Thermodynamic systems) . The thermodynamic world is a set Ω and the the setof thermodynamic systems is consists of finite non-empty subsets of the thermodynamic world, S := { S ⊂ Ω | < | S | < ∞} .In other words, the structure of thermodynamic systems is given by finite set theory. Hencethe following intuitive definition. Definition A.1 (Composition and intersection) . For two systems S , S ∈ S we define their composition as the union, S ∨ S := S ∪ S , (A.1)and their intersection as S ∧ S := S ∩ S , (A.2)where in the latter definition the case of disjoint systems must be seen as notation only, since ∅ isnot a system.Composing and intersecting n > ∨ and ∧ n times. The setof thermodynamic systems is obviously closed under composition of finitely many systems. It isalso closed under intersection except if the systems are disjoint. When two disjoint systems arecomposed we sometimes use the term disjoint composition . Definition A.2 (Atomic systems) . An atomic system A ∈ S is a thermodynamic system whichis represented by a singleton, | A | = 1. The set of atomic systems is denoted by A .Atomic systems are indivisible, i.e. they cannot be written as a composition of two differentthermodynamic systems. Non-atomic systems can always be seen as compositions of finitely manyatomic systems. Definition A.3 (Subsystems and atomic subsystems) . Given a system S ∈ S the set of subsystemsof S is defined as the set of non-empty subsets of S ,Sub( S ) := { S ′ ∈ S | S ′ ⊂ S } . (A.3)The set of atomic subsystems is denoted byAtom( S ) := { A ∈ A | A ∈ Sub( S ) } . (A.4)As such, the set of subsystems does not only contain proper subsystems but also S itself. Bothsets, Sub( S ) and Atom( S ), are always non-empty.By the definitions of composition, subsystems and atomic subsystems, both S = _ Sub( S ) and S = _ Atom( S ) (A.5)hold for any thermodynamic system S ∈ S . However, only in the second equality we have disjointcomposition. Every system can be uniquely composed into its (different, i.e. disjoint) atomicsubsystems. Definition A.4 (Disjoint complement) . Given a system S ∈ S and a proper subsystem S ′ ∈ Sub( S ), S ′ = S , the disjoint complement of S ′ w.r.t. S is the unique system S ′′ ∈ S which fulfils S ′ ∧ S ′′ = ∅ and S ′ ∨ S ′′ = S . It is denoted by S ′′ := S r S ′ .The fact the such a unique disjoint complement always exists is a simple consequence from settheory. 70 Processes and states
Again, we repeat the postulates in which the main concepts, here states and processes, are intro-duced.
Postulate 2 (Thermodynamic processes and states) . The non-empty set of thermodynamic pro-cesses (also simply processes ) that the theory allows for is denoted by P . A thermodynamicprocess p ∈ P specifies the initial and final states of a finite and non-zero number of involved atomic systems A ∈ A by means of the functions ⌊·⌋ A : P → Σ A and ⌈·⌉ A : P → Σ A . If A is notinvolved the two functions are undefined.The notion of being involved is sometimes also phrased the other way around. That is, insteadof saying that A ∈ A is involved in p ∈ P , we may say that p acts on A .The co-domain Σ A of the two function ⌊·⌋ A and ⌈·⌉ A is called the state space or set of states of A . For two unequal atomic systems A = A we assume w.l.o.g. that the corresponding statespaces are disjoint, Σ A ∩ Σ A = ∅ . This assumption basically says that a state contains a labelthat indicates to which system is belongs. It will allow a convenient notation for the states ofgeneral thermodynamic systems and simplifies the analysis of state spaces of equivalent systemsin Section 4. Definition B.1 (Set of involved atomic systems) . For a thermodynamic process p ∈ P the set ofinvolved atomic systems is defined as A p := _ { A ∈ A | A involved in p } . (B.1)As stated in the postulate, in any process p ∈ P at least one and at most finitely many atomicsystems are involved. In terms of A p this reads 0 < |A p | < ∞ for all p ∈ P . Definition B.2 (State space) . For an arbitrary thermodynamic system S = A ∨ · · · ∨ A n , where { A i } ni =1 are pairwise disjoint atomic systems, the state change of S under a thermodynamic process p ∈ P is given by the state changes on its atomic subsystems in terms of the functions ⌊·⌋ S : P → Σ S p
7→ ⌊ p ⌋ S := {⌊ p ⌋ A , . . . , ⌊ p ⌋ A n } (B.2)and likewise for ⌈·⌉ S . States of a non-trivially composite system are called joint states and Σ S iscalled the set of states of S .The input (or output) state of an arbitrary process on a general thermodynamic system S isdefined if and only if the function ⌊·⌋ A are defined for all A ∈ Atom( S ). Otherwise it is undefined.For a simple notation we write ⌊ p ⌋ S ≡ W A ∈ Atom( S ) ⌊ p ⌋ A ≡ ⌊ p ⌋ A ∨ · · · ∨ ⌊ p ⌋ A n for S = A ∨· · · ∨ A n , thereby exchanging the symbol ∪ for set union with the composition symbol ∨ . Thiswill not lead to confusion in the use of the notation as the arguments used with the symbol ∨ willmake clear what is meant exactly.For two disjoint systems S ∧ S = ∅ with σ ∈ Σ S and σ ∈ Σ S this notation reads σ ∨ σ = σ ∨ σ ∈ Σ S ∨ S . The fact that ∨ is commutative when composing states is no issueas state spaces of different systems are disjoint, i.e. it is always clear which symbol ( σ or σ )describes the state of which subsystem ( S or S ).The subsystems’ states σ ∈ Σ S and σ ∈ Σ S can be extracted as the corresponding entriesof the “tuple” σ ∨ σ ∈ Σ S ∨ S . In particular, this implies that if ⌊ p ⌋ S is defined for a (com-posite) system S , then ⌊ p ⌋ S ′ is automatically well-defined for all subsystems S ′ ∈ Sub( S ) as well.Likewise, we already know that if all subsystems of S have a well-defined state change in somethermodynamic process, then so does S . 71n the other hand, the structure of state spaces also makes clear that if a proper subsystem S ′ ∈ Sub( S ) has well-defined state changes in a thermodynamic process p ∈ P this does not implythat the same holds for S . There might exist other subsystems of S with undefined state change.This is the reason why the terms involved and not involved cannot be directly extended to ar-bitrary systems. These consist of atomic systems, but it could happen that an atomic subsystemis involved in a process, while another one is not. An arbitrary system containing the two wouldthen intuitively be involved. However, its state change is undefined, as there is at least one atomicsubsystem whose state change is undefined.Turning to the postulate introducing the partially defined concatenation operation for thermo-dynamic processes, ◦ : P × P → P , we note that it implies that P is closed under ◦ . Postulate 3 (Concatenation of processes) . Let p, p ′ ∈ P such that for all A ∈ A p ∩ A p ′ it holds ⌈ p ⌉ A = ⌊ p ′ ⌋ A . Then p and p ′ can be concatenated to form a new process denoted by p ′ ◦ p ∈ P ,which represents the consecutive execution of p followed by p ′ . If A p ∩ A p ′ = ∅ , then in addition p ′ ◦ p = p ◦ p ′ , i.e. concatenation commutes. An atomic system A ∈ A is involved in the concatenatedprocess p ′ ◦ p if and only if A ∈ A p ∪ A p ′ . For the involved atomic systems the initial and finalstates are ⌊ p ′ ◦ p ⌋ A = ( ⌊ p ⌋ A , if A ∈ A p ⌊ p ′ ⌋ A , otherwise (B.3)and the final state follows the same rules with swapped roles for p and p ′ .This postulate introduces a technical basis to work with consecutively applied thermodynamicprocesses and captures the minimal intuitive standards such a composition operation should have.The technical use of this postulate will become evident in the coming section, in particular whenwork is discussed in detail, as is done in Appendix C.To make the technical requirements more accessible we provide a generic example of twoprocesses which can be concatenated. Example B.3 (Concatenation.) . Let S = A ∨ A and S = A ∨ A be two systems, where A , A , A ∈ A are different atomic systems. Consider two processes p , p ∈ P with A p i =Atom( S i ) for i = 1 , ⌈ p ⌉ A = ⌊ p ⌋ A . Then we know that p ◦ p is defined and the statechanges on the involved subsystems are ⌊ p ◦ p ⌋ A i = ⌊ p ⌋ A , i = 1 , ⌊ p ⌋ A , i = 2 , ⌊ p ⌋ A , i = 3 , (B.4)and ⌈ p ◦ p ⌉ A i = ⌈ p ⌉ A , i = 1 , ⌈ p ⌉ A , i = 2 , ⌈ p ⌉ A , i = 3 . (B.5)The atomic system A is involved in both processes, i.e. it changes its state under both p and p .Thus the total state change on A under p ◦ p is from the initial state of p to the final state of p . The other two atomic systems are only involved in one of the two processes. Thus the initialstate on A is taken to be ⌊ p ⌋ A and the final state on A to be ⌈ p ⌉ A . C Work and work processes
The postulate introducing work states the following.72 ostulate 4 (Work) . For any atomic system A ∈ A exists a function W A : P → R that mapsa thermodynamic process p to W A ( p ), the work done on system A by performing p . The value W A ( p ) is positive whenever positive work is done on A while executing p . If system A is notinvolved in p , A / ∈ A p , then W A ( p ) = 0 necessarily.Technically, the work function is simply a function assigning a real number for any atomicsystem to any processes such that this number is zero whenever the atomic system is not involvedin the process.This function does not follow from a thermodynamic theory but is an input the user has todecide on before formulating the theory. Postulate 5 (Additivity of work under concatenation) . If for two processes p, p ′ ∈ P the con-catenation p ′ ◦ p is well-defined, then the work cost of the concatenated process equals the sumof the work costs of the individual processes. That is, for all atomic systems A ∈ A it holds that W A ( p ′ ◦ p ) = W A ( p ) + W A ( p ′ ) is additive .For processes which can be concatenated, the total work cost of the sequential execution of theprocesses must be equal to the sum of the individual work costs.Based on the work cost function for atomic systems it is possible to define the work costfunction for arbitrary systems in S . Definition C.1 (Work function for arbitrary systems) . Let S ∈ S be an arbitrary thermodynamicsystem. We define its work cost function (also simply work function ) W S : P → R by W S = X A ∈ Atom( S ) W A . (C.1)The following two lemmas state that the work functions of any system fulfil two different typesof additivity, namely additivity under composition and additivity under concatenation. Lemma C.2 (Additivity under composition.) . Let S , S ∈ S be two disjoint systems, S ∧ S = ∅ .Then the work function W S ∨ S : P → R of the composite system S ∨ S is additive : for all p ∈ P it holds W S ∨ S ( p ) = W S ( p ) + W S ( p ). Proof.
Disjoint systems S ∧ S = ∅ fulfil Atom( S ) ∩ Atom( S ) = ∅ . On the other hand, byDefinition A.1 we have that Atom( S ∨ S ) = Atom( S ) ∪ Atom( S ). For any p ∈ P this implies W S ∨ S ( p ) = X A ∈ Atom( S ∨ S ) W A ( p ) = X A ∈ Atom( S ) W A ( p ) + X A ∈ Atom( S ) W A ( p )= W S ( p ) + W S ( p ) . (C.2)With this, it follows that additivity under concatenation naturally extends from atomic sys-tems, for which it is postulated, to arbitrary systems. Lemma C.3 (Additivity under concatenation for arbitrary systems) . If for two processes p, p ′ ∈ P the concatenation p ′ ◦ p is defined, then for all S ∈ S additivity under concatenation holds: W S ( p ′ ◦ p ) = W S ( p ) + W S ( p ′ ) . (C.3) Proof.
We compute W S ( p ′ ◦ p ) = X A ∈ Atom( S ) W A ( p ′ ◦ p ) = X A ∈ Atom( S ) W A ( p ) + W A ( p ′ )= X A ∈ Atom( S ) W A ( p ) + X A ∈ Atom( S ) W A ( p ′ )= W S ( p ) + W S ( p ′ ) . (C.4)73he sets of thermodynamic processes which act exactly on the system S deserve special atten-tion. They are addressed in the first law. Definition C.4 (Work process) . For S ∈ S a process p ∈ P is a work process on S if all its atomicsubsystems are involved in p and no other atomic systems are. That is, p is a work process on S if S = W A p . The set of work processes on S is denoted by P S .Any process is a work process on some system. More precisely, let p ∈ P . Then p is a workprocess on the system S = W A p ∈ S . In this sense the set of thermodynamic processes P is infact the set of work processes on some system , while the set P S is the set of work processes on thesystem S . The latter set is closed under ◦ in the sense specified by the next lemma. Lemma C.5 ( P S closed under ◦ ) . For any system S ∈ S the set of work processes P S is closedunder concatenation. That is, if p ′ ◦ p is defined for p, p ′ ∈ P S , then p ′ ◦ p ∈ P S . Proof.
The fact that p, p ′ ∈ P S means that ⌊ p ⌋ S ′ , ⌈ p ⌉ S ′ and ⌊ p ′ ⌋ S ′ , ⌈ p ′ ⌉ S ′ are defined if and onlyif S ′ ∈ Sub( S ). Since by assumption p ′ ◦ p is defined, ⌊ p ′ ◦ p ⌋ S ′ = ⌊ p ⌋ S ′ and ⌈ p ′ ◦ p ⌉ S ′ = ⌈ p ′ ⌉ S ′ forall S ′ ∈ Sub( S ) by the postulate on concatenation. For systems S ′ ∈ S r Sub( S ) the input andoutput states ⌊ p ′ ◦ p ⌋ S ′ and ⌈ p ′ ◦ p ⌉ S ′ are undefined.Hence, ⌊ p ′ ◦ p ⌋ S ′ and ⌈ p ′ ◦ p ⌉ S ′ are defined if and only if S ′ ∈ Sub( S ) and thus p ′ ◦ p ∈ P S is awork process on S , too. Definition C.6 (Joint work processes.) . For two disjoint systems S ∧ S = ∅ and two workprocesses p i ∈ P S i for i = 1 , p ◦ p the joint work process of p and p and denote it by p ∨ p := p ◦ p ∈ P S ∨ S . (C.5)This definition is well-defined since for two work processes on disjoint system their concate-nation is always defined. Furthermore, according to the postulate on concatenation in this case p ◦ p = p ◦ p since A p ∩ A p = ∅ . Also, the joint work process of p and p as in the abovedefinition is again a work process on S ∨ S since the involved atomic systems are exactly A p ∪A p .As a consequence, the input and output states of a joint work process p ∨ p are given by ⌊ p ∨ p ⌋ S ∨ S = ⌊ p ⌋ S ∨ ⌊ p ⌋ S ∈ Σ S ∨ S and likewise for ⌈·⌉ .Such joint work processes imply an embedding (an injective mapping) from P S ×P S to P S ∨ S and images of this mapping ( p , p ) p ∨ p stand for the parallel execution of the work process p on subsystem S and of p on S . Hence, what was achievable by means of work processes onthe individual systems S and S can still be realized as work processes on the composite system S ∨ S . In this sense, composition respects work processes.While the state space of a composite system consists of joint states only, this is not the casewith work processes. In general, the set P S ∨ S contains more work processes than just the jointwork processes of its subsystems. An example of a more general work process on a compositesystem is thermally connecting two subsystems and letting them exchange energy. Lemma C.7 (Work cost of a joint work process) . Let S = S ∨ S be a composite system withdisjoint subsystems S and S and let p ∈ P S and p ∈ P S be work processes on the subsystems.Then the total work cost of the joint work process p ∨ p ∈ P S on S is given by the sum of thelocal work costs, W S ( p ∨ p ) = W S ( p ) + W S ( p ) . (C.6)74 roof. Additivity of the work cost functions under concatenation and composition imply W S ( p ∨ p ) def . = W S ( p ◦ p ) (C.7) conc . = W S ( p ) + W S ( p ) (C.8) comp . = W S ( p ) + W S ( p ) + W S ( p ) + W S ( p ) (C.9)= W S ( p ) + W S ( p ) , (C.10)for any two work processes p ∈ P S and p ∈ P S . In the last equality it was used that if a system S i is not involved in a thermodynamic process p i +1 , then W S i ( p i +1 ) = 0.Another special kind of work processes on a system are identity processes . Definition C.8 (Identity process) . An identity process on S , where S ∈ S is an arbitrary ther-modynamic system, is a work process id S ∈ P S on S with ⌊ id S ⌋ S = ⌈ id S ⌉ S and zero work cost onall atomic subsystems of S , W A (id S ) = 0 for all A ∈ Atom( S ). For an identity process on S ∈ S acting on the state σ ∈ Σ S the notation id σS is used.A thermodynamic process can act trivially on a system without being a work process on thatsystem, too. This is captured by the notions introduced next. Definition C.9 (Cyclic and catalytic process) . Given a system C ∈ S an arbitrary thermody-namic process p ∈ P is called cyclic on C if ⌈ p ⌉ C = ⌊ p ⌋ C .The process is called catalytic on C if it is cyclic on C and in addition W C ( p ) = 0. The definition of an cyclic process on S which is in addition a work process on S differs froma identity process on S by the missing requirement on the work costs on atomic subsystems.Incorporating a catalytic system explicitly in the thermodynamic description of a process ispossible. However, our theory should be such that it is not mandatory, i.e. that the explicitmentioning of the catalytic system could also be left out. Technically this is captured by the finalpostulate in this section. Postulate 6 (Freedom of description) . For
S, C ∈ S disjoint, let p ∈ P S ∨ C be such that p iscatalytic on C , i.e. p is cyclic on C and fulfils W C ( p ) = 0. Then there exists a work process˜ p ∈ P S on S alone such that ⌊ ˜ p ⌋ S = ⌊ p ⌋ S and ⌈ ˜ p ⌉ S = ⌈ p ⌉ S as well as W A (˜ p ) = W A ( p ) for all A ∈ Atom( S ).Since ˜ p ∈ P S and S ∧ C = ∅ it holds automatically that W C ′ (˜ p ) = 0 for all subsystems C ′ ∈ Sub( C ) of C in the new process. Likewise, W S ′ ˜ p ) = W S ′ ( p ) for all subsystems S ′ ∈ Sub( S )since their work costs on general subsystems are computed through the work costs on atomicsubsystems.Postulate 6 is about where to draw the line between objects that thermodynamics explicitlydescribes and such that are not part of the theory but may nevertheless be used when executinga process.In the final part of this section we discuss the definition of and results on reversible workprocesses. Definition C.10 (Reversible processes) . A work process p ∈ P S on a system S ∈ S is called reversible if there exists another work process p rev ∈ P S on S , the reverse work process, such that p rev ◦ p is an identity process.Considering the work costs of reverse processes we find the intuitive result that they simplychange their signs relative to the forward process. Notice that the work costs of p for subsystems of C do not have to be zero, only the total work done on C does. emma C.11 (Work cost of reverse work processes) . Let p ∈ P S be a reversible work process on S ∈ S with reverse process p rev ∈ P S . Then W A ( p rev ) = − W A ( p ) (C.11)for all A ∈ Atom( S ). Proof.
By definition, p rev ∈ P S is a reverse process for p ∈ P S if and only if p rev ◦ p is an identityprocess. Therefore, by additivity of the work functions under concatenation and the Definition C.8of identity processes 0 = W A ( p rev ◦ p ) = W A ( p ) + W A ( p rev ) (C.12)for all atomic subsystems A ∈ Atom( S ), and the claim follows.We next investigate the reversibility of a concatenated process q ◦ p in relation to the reversibilityof p and q . Clearly, if both p and q are reversible, then so is q ◦ p whenever the concatenation isdefined. This follows from the fact that the reverse processes p rev and q rev can be concatenatedto a reverse process p rev ◦ q rev . But the implication in the opposite direction, i.e. that if q ◦ p isreversible, then so are p and q , needs a bit more work to obtain. Lemma C.12 (Reversibility of p, q ∈ P S relative to q ◦ p ) . Let p, q ∈ P S for some S ∈ S suchthat r := q ◦ p is defined. Then, if r is reversible, both p and q must also be reversible. Proof.
Let r rev ∈ P S be a reverse process for r and consider r rev ◦ q . This process is well-defined,as r rev ◦ r is defined and the output state of r is equal to the output state of q . Also, it can beconcatenated with p from the left (since q can be concatenated with p ) and from the right (since r rev is a reverse process for q ◦ p ). Furthermore, it holds that r rev ◦ q ◦ p is a cyclic work processon S and thus so is p ◦ r rev ◦ q . Finally, we compute W A ( r rev ◦ q ◦ p ) = W A ( r rev ) + W A ( q ) + W A ( p ) = W A ( r rev ) + W A ( r ) = 0 , (C.13)for all A ∈ Atom( S ).Together this implies that r rev ◦ q ◦ p is an identity process and thus p rev := r rev ◦ q ∈ P S isindeed a reverse process for p , which means that p is reversible. To show reversibility of q , weproceed analogously with q rev := p ◦ r rev . Lemma C.13 (Reversibility of p ∨ id ) . Let S , S ∈ S be two disjoint systems. Let further p ∈ P S be an arbitrary work process on S and id ∈ P S an arbitrary identity process on S .Then, if p := p ∨ id is reversible, so is p . Proof.
Let p rev ∈ P S ∨ S be a reverse process. Then W S ( p rev ) = − W S ( p ) = − W S (id ) = 0 dueto Lemma C.11. On the other hand, p rev is cyclic on S , since the state of S did not changeunder p and hence neither under its reverse process.Therefore, p rev is catalytic on S which, together with Postulate 6 on the freedom of thermo-dynamic description, implies that there exists a work process p rev1 ∈ P S on S alone such thatthe state change as well as the work flows under p rev1 on S are the same as the ones change under p rev . Obviously, p rev1 is a reverse process for p , which proves the claim. Proposition C.14 (Reversibility of general p ◦ p ) . Let S , S ∈ S be two arbitrary systems(not necessarily disjoint) and p i ∈ P S i such that p := p ◦ p is defined. Then, if p is reversible, soare p and p . Proof.
We first note that according to the postulate introducing concatenation, we know that p ∈ P S ∨ S is a work process on the composite systems S ∨ S . Also, we know that the disjointcomplements S r S and S r S exist, and are unique systems whenever the one system is notcompletely contained in the other. Let us for the moment assume that this is the case.76hen, extend the processes p i with identities so that they induce defined state changes onall of S ∨ S , i.e. let id S r S and id S r S be identities such that p ′ := p ′ ◦ p ′ is defined, where p ′ := id S r S ∨ p and p ′ := p ∨ id S r S . For i = 1 , p ′ i ∈ P S ∨ S are work processes on S ∨ S . Furthermore, p ′ induces the same state change with the same work costs as p on anyinvolved atomic system. Thus p ′ is also reversible and has the same reverse processes as p .By Lemma C.12 we know that both p ′ ∈ P S ∨ S and p ′ ∈ P S ∨ S must be reversible. Butthen, according to the previous Lemma C.13, both p ∈ P S and p ∈ P S are reversible, whichconcludes the proof for the case when neither S ∈ Sub( S ) nor vice versa.The case S ∈ Sub( S ) follows in the very same way, where we just define p ′ := p and likewise,if S ∈ Sub( S ), define p ′ := p . D The first law
The first law of thermodynamics states the following.
Postulate 7 (The first law) . For any system S ∈ S the following two statements hold:(i) For any pair of states σ , σ ∈ Σ S there is a work process p ∈ P S on S with ⌊ p ⌋ S = σ and ⌈ p ⌉ S = σ or there is a work process p ′ ∈ P S on S with ⌊ p ′ ⌋ S = σ and ⌊ p ′ ⌋ S = σ .(ii) The total work cost of a work process p ∈ P S on S , W S ( p ), only depends on ⌊ p ⌋ S and ⌈ p ⌉ S and not on any other details of the process.In particular, (ii) implies that if p ′ ∈ P S is another work process on S with ⌊ p ⌋ S = ⌊ p ′ ⌋ S and ⌈ p ⌉ S = ⌈ p ′ ⌉ S , then W S ( p ) = W S ( p ′ ).The first law implies a relation on the set of states of any system which will turn out to bea preorder. A preordered set is a set M together with a relation → such that the relation is (i)reflexive, i.e. ∀ m ∈ M : m → m , and (ii) transitive, i.e. if both m → m ′ and m ′ → m ′′ , then m → m ′′ . Definition D.1 (Preordered states) . For any system S ∈ S the preorder → on Σ S is establishedby the reachability via a work process, i.e. for σ, σ ′ ∈ Σ S define σ → σ ′ : ⇔ ∃ p ∈ P S s . t . ⌊ p ⌋ S = σ, ⌈ p ⌉ S = σ ′ . (D.1)Processes p ∈ P S can be seen as labels of the preordered pairs. In this sense, if one wants toprecisely state which work process is responsible for the preordering of two states, one can write σ p → σ ′ if the work process p on S is such that ⌊ p ⌋ S = σ and ⌈ p ⌉ S = σ ′ . As mentioned before,there may be more than one label for a preordered pair. Lemma D.2 (Preordered states) . The relation → in Definition D.1 is a preorder. Proof.
The relation introduced in Eq. 3.1 is reflexive since for all systems S and all states σ ∈ Σ S there exists work process p ∈ P S such that σ p → σ . This is a consequence of Postulate 7 (i).Furthermore, if σ p → σ ′ and σ ′ p ′ → σ ′′ then σ p ′ ◦ p −→ σ ′′ is preordered too by means of the concatenatedprocess p ′ ◦ p . Hence the relation is also transitive, which makes it a preorder.Due to the reflexivity of the preorder induced by the first law it follows that for any state onan atomic system there is an identity process. Lemma D.3 (Existence of identity process for all atomic states) . For any atomic system A ∈ A and any state σ ∈ Σ A there exists an identity process id σA ∈ P A with ⌊ id σA ⌋ A = σ = ⌈ id σA ⌉ A . The fact that such identity processes always exist has not yet been established. It is a consequence of Postulate 7stated in Lemma D.4 in Appendix D. We use this fact here already. roof. Since → is a preorder, in particular reflexive, we know that for any σ ∈ Σ A there is a workprocess q ∈ P A on A such that σ q −→ σ . This process acts on A alone and initial and final statesmatch. Thus it can be concatenated with itself. The state change of the process in which q isapplied twice is obviously the same as the state change under q itself. Therefore Postulate 7 (ii)requires W A ( q ) (ii) = W A ( q ◦ q ) = W A ( q ) + W A ( q ) = 2 W A ( q ) , (D.2)which is to say that the work cost of such processes is zero. Obviously, this makes it an identityprocess on A for the state σ which can rightfully be called id σS ⌉ A .Using Lemma D.3 it is then possible to show that identity processes exist for all states on anythermodynamic system S ∈ S . Lemma D.4 (Existence of identity process for all states) . For any atomic system S ∈ S and anystate σ ∈ Σ S there exists an identity process id σS ∈ P S with ⌊ id σS ⌋ S = σ = ⌈ id σS ⌉ S . Proof.
Using the decomposition of an arbitrary system into its atomic subsystems it is possibleto construct identity process for all states of arbitrary thermodynamic systems. To see this, let S = A ∨ · · · ∨ A n with different atomic systems A i = A j and consider an arbitrary joint state σ = σ ∨ · · · ∨ σ n . For the atomic states we know that corresponding identity processes id σ i A i exist.Therefore, the joint process id σS := id σ A ∨ id σ n A n is a cyclic work process on S . But this processalso fulfils W A i (id σS ) = 0 for all i = 1 , . . . , n by construction, thus it is an identity process. Weconclude that identity processes exist for all states of all thermodynamic systems.As was stated before the first law guarantees that every system S has a well-defined internalenergy function. We are now in the position to define this function. Definition D.5 (Internal energy) . For a system S ∈ S fix an arbitrary reference state σ ∈ Σ S and an arbitrary reference energy U S ∈ R . The internal energy of a state σ ∈ Σ S is defined as U S ( σ ) := U S + W S ( p ) , where p ∈ P S is s.t. ⌊ p ⌋ s = σ and ⌈ p ⌉ S = σ . (D.3) U S ( σ ) := U S − W S ( p ′ ) , where p ′ ∈ P S is s.t. ⌊ p ′ ⌋ s = σ and ⌈ p ′ ⌉ S = σ . (D.4)Only differences ∆ U S of internal energies physically matter. Thus the choice of U S is arbitraryat this point. Likewise, the reference state σ ∈ Σ S is arbitrary independently for each system S . Definition D.6 (State function) . A state function on a system S ∈ S (also state variable ) is afunction Z : Σ S → Z from the state space Σ S to a target space Z . The co-domain Z is typically R n , most often n = 1. When a system S undergoes a process p ∈ P we denote the change inany state function Z S using an abbreviated notation by ∆ Z S ( p ) := Z S ( ⌈ p ⌉ S ) − Z S ( ⌊ p ⌋ S ). Onthe left hand side the dependence of ∆ Z S on p may be omitted if the context makes clear whichprocess is meant.It must now be proven that U S is a state function on S . Lemma D.7 (Internal energy is a state function) . Internal energy as defined in Definition D.5 isa well-defined state function.
Proof.
By the definition and the fact that identity processes exist for any state and have a totalwork cost of zero it holds U S ( σ ) = U S . This is in agreement with both Eq. (D.3) and Eq. (D.4).For an arbitrary state σ , Postulate 7 (i) guarantees the existence of at least one of the workprocesses p and p ′ used to define U S , hence U S ( σ ) is never undefined.Furthermore, if both p in one direction and p ′ in the other direction exist, then there is agreementbetween the two possibilities of computing U S due to the fact that the work cost of reverse processes This of course only works if a “minus operation‘” is defined on the co-domain Z . For all practical purposesconsidered here this is the case. σ and output state σ , then theirwork cost must be the same due to Postulate 7 (ii). Hence, it does not matter which one is usedto compute U S ( σ ) by Eq. (D.3). The same argument also works if σ and σ play exchanged roles,as in Eq. (D.4).To conclude this section, we show that the internal energy function is additive for any system.This is a consequence of the additivity of the work cost functions discussed in the last section. Proposition D.8 (Additivity of U ) . For a disjointly composite system S = S ∨ S , S ∧ S = ∅ ,that undergoes a work process p ∈ P S the change in internal energy of S equals the sum of thechanges of the individual subsystems,∆ U S = ∆ U S + ∆ U S . (D.5) Proof.
We denote initial and final states of p by σ in1 ∨ σ in2 ∈ Σ S ∨ S and σ out1 ∨ σ out2 ∈ Σ S ∨ S .Hence, the short notation from Eq. D.5 can be extended as U S ( σ out1 ∨ σ out2 ) − U S ( σ in1 ∨ σ in2 ) = (cid:0) U S ( σ out1 ) − U S ( σ in1 ) (cid:1) + (cid:0) U S ( σ out2 ) − U S ( σ in ) (cid:1) . (D.6)There are essentially two cases to distinguish: (i) there exist work processes p ∈ P S and p ∈ P S with ⌊ p ⌋ S = σ in1 , ⌈ p ⌉ S = σ out1 and ⌊ p ⌋ S = σ in2 , ⌈ p ⌉ S = σ out2 , i.e. two workprocesses in the same direction; (ii) there exist work processes p ∈ P S and p ∈ P S in differentdirections, w.l.o.g. with ⌊ p ⌋ S = σ out1 , ⌈ p ⌉ S = σ in1 and ⌊ p ⌋ S = σ out2 , ⌈ p ⌉ S = σ in2 .The case of two work processes both in the opposite direction of (i) can be treated just like (i)with − ∆ U S instead of ∆ U S . The first law guarantees that one of these cases always applies.(i) In this case the joint work process p ∨ p ∈ P S gives rise to exactly the same state transfer as p . Hence the work done during each of these work processes must also be the same. By definitionof the internal energy it follows∆ U S = U S ( σ out1 ∨ σ out2 ) − U S ( σ in1 ∨ σ in2 ) = W S ( p ) = W S ( p ∨ p )= W S ( p ) + W S ( p ) = (cid:0) U S ( σ out1 ) − U S ( σ in1 ) (cid:1) + (cid:0) U S ( σ out2 ) − U S ( σ in2 ) (cid:1) = ∆ U S + ∆ U S , (D.7)where Lemma C.7 was used to decompose the total work in the joint work process.(ii) If p ∈ P S exists with ⌊ p ⌋ S = σ in1 , ⌈ p ⌉ S = σ out1 and p ∈ P S ” with ⌊ p ⌋ S = σ out2 , ⌈ p ⌉ S = σ in2 , then p can be concatenated with the joint work process id σ in1 S ∨ p . Furthermore,the concatenated work process p ◦ (id σ in1 S ∨ p ) induces the same state transfer as the joint workprocess p ∨ id σ in2 S . Hence, due to Postulate7, their work costs are equal and we obtain∆ U S − ∆ U S = (cid:0) U S ( σ out1 ∨ σ out2 ) − U S ( σ in1 ∨ σ in2 ) (cid:1) − (cid:0) U S ( σ out2 ) − U S ( σ in2 ) (cid:1) = W S ( p ) + W S ( p ) = W S ( p ) + W S (id σ in1 S ∨ p )= W S ( p ◦ (id σ in1 S ∨ p )) = W S ( p ∨ id σ in2 S ) = W S ( p )= U S ( σ out1 ) − U S ( σ in1 ) = ∆ U S . (D.8)We have proved that when a composite system undergoes a work process, then the total changein internal energy on the composite system is equal to the sum of the internal energy changes onthe individual subsystems. However, this result automatically extends to arbitrary processes on acomposite system, not just work processes, because internal energy U S is a state function.79 Equivalent systems
In this section we explain the technical background of the formalisms introducing thermodynamicisomorphisms and equivalent systems. These are important concepts in order to be able to talkabout copies of systems. Some ideas presented here are inspired by the Master’s Thesis [33] whichthe authors supervised.In the beginning we have emphasized that elements of S are seen as specific physical instancesrather than types of systems. With a notion of copies (or equivalent systems) it will neverthelessbe possible to talk about types of systems. Two systems are copies of each other if they can beinterchanged without any noticeable thermodynamic differences. To make this more precise weintroduce a type of map that called a “thermodynamic isomorphism”. It maps thermodynamicprocesses to other thermodynamic processes while preserving the thermodynamic structure thathas been introduced in the previous sections. The mapping should be such that the two specificsystems are swapped and basically nothing else happens.Before defining equivalent systems we discuss the less restrictive notion of a thermodynamicisomorphism and its properties. Definition E.1 (Thermodynamic isomorphism) . The pair of bijective maps ϕ : P → P , ϕ A : A → A is called a thermodynamic isomorphism if for any thermodynamic processes p, p ′ ∈ P andany atomic system A ∈ A it holds(i) ϕ ( p ′ ◦ p ) = ϕ ( p ′ ) ◦ ϕ ( p ) whenever the concatenation p ′ ◦ p or ϕ ( p ′ ) ◦ ϕ ( p ) is defined,(ii) A is involved in p if and only if ϕ A ( A ) is involved in ϕ ( p ), and(iii) W ϕ A ( A ) ( ϕ ( p )) = W A ( p ).The requirements on an isomorphism emphasize the fundamental structure behind the basicthermodynamic concepts. These are (i) the thermodynamic processes with concatenation , (ii) atomic systems , linked to processes through the notion of an atomic system being involved in aprocess , and (iii) work .We have already established that input and output states are always either both defined orboth undefined. Hence (ii) is equivalent to saying that: ⌊ ϕ ( p ) ⌋ ϕ A ( A ) is defined ⇔ ⌊ p ⌋ A is defined.We directly start with the notion of an isomorphism without introducing homomorphismsfirst, as one might expect when discussing algebraic structures. This is because the notion of anisomorphisms is exactly what we will need for defining equivalences while a detailed discussionof homomorphisms would go beyond the scope of this work. Investigations of less restrictivemappings than an isomorphism may be the topic of future work.Even though both mappings ϕ and ϕ A are part of the definition of a thermodynamic isomor-phism they are not independent degrees of freedom, as the next lemma shows. In this sense, ϕ A is determined by ϕ and one could think of coming up with a more minimal Definition E.1 suchthat it only talks about ϕ , while ϕ A is derived from it afterwards. Even though this is possible itwould make the definition much less readable and intuitive. Therefore we do not go further intothis. Nevertheless it is good to know that the fundamental mapping is the one on processes andthe other concepts depend on this. Lemma E.2 ( ϕ and ϕ A not independent) . Let ϕ, ϕ A and ϕ, ϕ ′A be thermodynamic isomorphisms.Then ϕ A = ϕ ′A . Proof.
Consider an arbitrary atomic system A ∈ A and a work process p ∈ P A on A . ByDefinition E.1 (ii) we know that ⌊ ϕ ( p ) ⌋ ϕ A ( A ′ ) is defined if and only if A ′ = A and likewise, ⌊ ϕ ( p ) ⌋ ϕ ′A ( A ′ ) is defined if and only if A ′ = A . In fact, these statements are equivalent.Since the maps ϕ A and ϕ ′A are bijective the statements can rephrased as ⌊ ϕ ( p ) ⌋ A ′′ is definedif and only if A ′′ = ϕ A ( A ) and ⌊ ϕ ( p ) ⌋ A ′′ is defined if and only if A ′′ = ϕ ′A ( A ). From this itimmediately follows that ϕ ′A ( A ) = ϕ A ( A ). 80otice that we only used (ii) of Definition E.1 in this proof (together with bijectivity of themaps). This is also the point that would become more complicated in a version of the definitionthat does not make use of ϕ A but allows one to derive it.The mapping of atomic systems under a thermodynamic isomorphism can be naturally ex-tended to a mapping on all systems as ϕ S ( S ) := _ A ∈ Atom( S ) ϕ A ( A ) . (E.1)By construction this mapping is bijective, too.We now aim at showing that thermodynamic isomorphisms preserve all thermodynamic prop-erties that have been introduced so far. Lemma E.3 (Work processes under thermodynamic isomorphisms) . Let ϕ, ϕ A be a thermody-namic isomorphism and p ∈ P . Then for all A ∈ A and all S ∈ S :(i) p ∈ P S ⇐⇒ ϕ ( p ) ∈ P ϕ S ( S ) , and in particular p ∈ P A ⇐⇒ ϕ ( p ) ∈ P ϕ A ( A ) ,(ii) p ∈ P S is an identity process ⇐⇒ ϕ ( p ) ∈ P ϕ S ( S ) is an identity process,(iii) p ∈ P S is reversible ⇐⇒ ϕ ( p ) ∈ P ϕ S ( S ) is reversible Proof.
Let A ′ ∈ A and S ′ ∈ S arbitrary.(i) By Definition 2.3 p ∈ P S if and only if ⌊ p ⌋ A ′ is defined for A ′ ∈ Atom( S ) only. UsingDefinition E.1 (ii) this implies ⌊ p ⌋ A ′ def. iff A ′ ∈ Atom( S ) ⇔ ⌊ ϕ ( p ) ⌋ ϕ A ( A ′ ) def. iff A ′ ∈ Atom( S ) ⇔ ⌊ ϕ ( p ) ⌋ ϕ A ( A ′ ) def. iff ϕ A ( A ′ ) ∈ Atom( ϕ S ( S )) ⇔ ⌊ ϕ ( p ) ⌋ A ′′ def. iff A ′′ ∈ Atom( ϕ S ( S )) (E.2)In the second line we used A ′ ∈ Atom( S ) ⇔ ϕ A ( A ′ ) ∈ Atom( ϕ S ( S )), while the third lineholds because ϕ A is bijective. The third line, however, is equivalent to ⌊ ϕ ( p ) ⌋ A ′′ beingdefined for A ′′ ∈ Atom( ϕ S ( S )) and thus true if and only if ϕ ( p ) ∈ P ϕ S ( S ) .(ii) Let p ∈ P S be an identity process. By Definition 2.4 this is the case if and only if it iscyclic on S , i.e. if and only if p can be concatenated with itself, and W A ( p ) = 0 for all A ∈ Atom( S ). By Definition E.1 (i) it holds p ◦ p def. ⇔ ϕ ( p ) ◦ ϕ ( p ) def. , (E.3)which means that ϕ ( p ) is a cyclic work process on ϕ S ( S ) if and only if p is a cyclic workprocess on S . Furthermore, for A ∈ Atom( S ) we have ϕ S ( A ) ∈ Atom( ϕ S ( S )) and W ϕ S ( A ) ( ϕ ( p )) = W A ( p ) (E.4)for all A ∈ Atom( S ). Hence one side of this equation is zero if and only if the other is.(iii) By Definition 2.6 p ∈ P S is reversible if there exists a reverse work process on the samesystem. Thus p ∈ P S is reversible ⇔ ∃ p rev ∈ P S s.t. p rev ◦ p is identity ⇒ ϕ ( p rev ) ◦ ϕ ( p ) is identity and ϕ ( p rev ) ∈ P ϕ S ( S ) ⇒ ϕ ( p rev ) ∈ P ϕ S ( S ) is reversible . (E.5)We used the previously proved (i) and (ii). Since ϕ is bijective by Definition E.1, theargument also works in the other direction.81 emma E.4 (States under thermodynamic isomorphisms) . Let p, q ∈ P and A ∈ A . Furthermore,let ϕ, ϕ A be a thermodynamic isomorphism. Then:(i) ⌊ p ⌋ A = ⌊ q ⌋ A ⇐⇒ ⌊ ϕ ( p ) ⌋ ϕ A ( A ) = ⌊ ϕ ( q ) ⌋ ϕ A ( A ) ,(ii) ⌊ p ⌋ A = ⌈ q ⌉ A ⇐⇒ ⌊ ϕ ( p ) ⌋ ϕ A ( A ) = ⌈ ϕ ( q ) ⌉ ϕ A ( A ) ,(iii) ⌈ p ⌉ A = ⌈ q ⌉ A ⇐⇒ ⌈ ϕ ( p ) ⌉ ϕ A ( A ) = ⌈ ϕ ( q ) ⌉ ϕ A ( A ) . Proof.
We prove (i). The remaining points (ii) and (iii) follow in the very same way. Again wemake use of Definition E.1 (i) and (ii), where in particular(i) is important. We also use Lemma E.3(i).Let σ := ⌊ p ⌋ A = ⌊ q ⌋ A and consider id σA ∈ P A . Both p ◦ id σA and q ◦ id σA are defined and hence byDefinition E.1 (i) so are ϕ ( p ) ◦ ϕ (id σA ) and ϕ ( q ) ◦ ϕ (id σA ). Furthermore, ϕ (id σA ) is an identity processon ϕ A ( A ), has in particular defined input and output states on this system, and we know thatby Definition E.1 (ii) both ⌊ ϕ ( p ) ⌋ ϕ A ( A ) and ⌊ ϕ ( q ) ⌋ ϕ A ( A ) are defined. Hence, they must be equal,as Postulate 3 (concatenation of processes) requires ⌊ ϕ ( p ) ⌋ ϕ A ( A ) = ⌊ ϕ ( q ) ⌋ ϕ A ( A ) . The oppositedirection can be argued in the same way with the inverse maps.With Lemma E.4 it is possible to define a mapping of the states induced by a thermodynamicisomorphism. Definition E.5 (States under thermodynamic isomorphisms) . Given a thermodynamic isomor-phism ϕ, ϕ A together with its corresponding map ϕ S we define the corresponding mapping ofatomic states ϕ Σ : S A ∈A Σ A → S A ∈A Σ A by ϕ Σ ( ⌊ p ⌋ A ) := ⌊ ϕ ( p ) ⌋ ϕ A ( A ) (E.6)and its extension to arbitrary states ϕ Σ : S S ∈S Σ S → S S ∈S Σ S by ϕ Σ ( ⌊ p ⌋ S ) := ⌊ ϕ ( p ) ⌋ ϕ S ( S ) . (E.7)Remember that we assumed that for two different systems, in particular for two different atomicsystems, their state spaces are disjoint. This means that the unions S A ∈A Σ A and S S ∈S Σ S aredisjoint unions. It is immediate that the two functions agree on the intersection of their domains,namely on S A ∈A Σ A . This is a consequence of the fact that any system’s state can be writtenas the composition of its atomic subsystems’ states, i.e. ⌊ p ⌋ S = W A ∈ Atom( S ) ⌊ p ⌋ S for any p ∈ P .In particular, every property of ϕ Σ proved for atomic systems automatically extends to ϕ Σ ingeneral.Lemma E.4 proves that Definition E.5 is well-defined. It says that even for two different pro-cesses with equal input states (or output states, or input state on one process and output state onthe other) the mapping of the states under ϕ Σ is unique. Hence the choice of defining ϕ Σ usingthe input state function ⌊·⌋ was arbitrary – we could have done it with the output state function ⌈·⌉ as well. Furthermore, by Definition E.1 the output of this function is defined if and only if theinput is defined.We remark that by definition ϕ Σ (Σ A ) ⊂ Σ ϕ A ( A ) for any atomic system A ∈ A . In fact, ϕ Σ isbijective, as the next lemma shows. In particular, this implies that ϕ Σ (Σ A ) = Σ ϕ A ( A ) since thedomain (and the codomain, which are equal sets here) is a disjoint union of such sets. Lemma E.6 ( ϕ Σ is bijective) . The mapping of states ϕ Σ associated with a thermodynamicisomorphisms ϕ, ϕ A defined in Definition E.5 is bijective. Proof.
We show that the map ϕ Σ is surjective and injective.For surjectivity consider S ∈ S and σ ∈ Σ S together with the corresponding identity process82d σS ∈ P S . Since both ϕ and ϕ S are bijective, we know that there exist S ′ ∈ S such that ϕ S ( S ′ ) = S and id σ ′ S ′ ∈ P S ′ such that ϕ (id σ ′ S ′ ) = id σS . Therefore, ϕ Σ ( σ ′ ) = ϕ Σ ( ⌊ id σ ′ S ′ ⌋ S ′ ) = ⌊ ϕ (id σ ′ S ′ ) ⌋ ϕ S ( S ′ ) = ⌊ id σS ⌋ S = σ . (E.8)For injectivity, let σ ∈ Σ S and σ ∈ Σ S such that σ = σ . Consider again the identity processesid σ i S i ∈ P S i for i = 1 ,
2. Then we can write ϕ Σ ( σ i ) = ϕ Σ ( ⌊ id σ i S i ⌋ S i ) = ⌊ ϕ (id σ i S i ) ⌋ ϕ S ( S i ) . (E.9)If S = S , ϕ Σ ( σ i ) ∈ Σ ϕ S ( S i ) are contained in disjoint sets (because ϕ S is bijective and statespaces of different systems are disjoint). Hence ϕ Σ ( σ ) = ϕ Σ ( σ ).If S = S , σ = σ implies that id σ S ◦ id σ S is not defined (Postulate 3). By Definition E.1 (i) thisimplies that ϕ (id σ S ) ◦ ϕ (id σ S ) is not defined either and hence also in this case ϕ Σ ( σ ) = ϕ Σ ( σ ).So far we have discussed the mapping of processes, systems and states with their properties.For all of this Definition E.1 (iii) has not been touched. When showing that the internal energyfunction also transforms naturally, this point will become important. Lemma E.7 (Internal energy under thermodynamic isomorphisms) . Let ϕ, ϕ A be a thermo-dynamic isomorphism with associated mappings ϕ S and ϕ Σ for arbitrary systems and states,respectively. Then for all S ∈ S and p ∈ P ∆ U ϕ S ( S ) ( ϕ ( p )) = ∆ U S ( p ) . (E.10) Proof.
We use the previous results together with Definition E.1 (iii), which is the only part of thedefinition on equivalent atomic systems that makes a statement about work costs. In addition, wedirectly apply the first law (Postulate 7). Let S = A ∈ A be an atomic system for the moment.If A is not involved in p the neither is ϕ A ( A ) involved in ϕ ( p ) (due to Definition E.1 (ii)). Hencein this case ∆ U A ( p ) = 0 = ∆ U ϕ A ( A ) ( ϕ ( p )) . (E.11)If A is involved in p , suppose there exists a work process q ∈ P A with ⌊ q ⌋ A = ⌊ p ⌋ A and ⌈ q ⌉ A = ⌈ p ⌉ A .By definition of U A it follows ∆ U A ( p ) = W A ( q ). We now consider the image of q under ϕ . Itfollows ⌊ ϕ ( q ) ⌋ ϕ A ( A ) = ϕ Σ ( ⌊ q ⌋ A ) = ϕ Σ ( ⌊ p ⌋ A ) and ⌈ ϕ ( q ) ⌉ ϕ A ( A ) = ϕ Σ ( ⌈ q ⌉ A ) = ϕ Σ ( ⌈ p ⌉ A ) togetherwith W A ( q ) = W ϕ A ( A ) ( ϕ ( q )). Hence, using again the definition of internal energy,∆ U ϕ A ( A ) ( ϕ ( p )) = W ϕ A ( A ) ( ϕ ( q )) = W A ( q ) = ∆ U A ( p ) . (E.12)If such a q ∈ P A does not exist, then there exists one in the other direction, as the first lawguarantees, and the proof works analogously (with a minus sign).Finally, the proof extends to arbitrary S ∈ A due to the additivity of internal energy undercomposition.In Lemma E.7 we only talk about differences of internal energies as the reference energies of S and ϕ S ( S ) can be chosen independently according to Definition 3.2. However, one can alwaysadjust the reference energies of the two atomic systems such that equality of internal energies forcorresponding states also holds for absolute values of internal energies.We conclude that a thermodynamic isomorphism ϕ, ϕ A , gives rise to corresponding mappings ϕ S on the set of systems and ϕ Σ on the set of all states. These mappings are compatible in anatural way and preserve all concepts introduced so far, as discussed in the results above. Thedefinition of a thermodynamic isomorphism is hence a sensible way of introducing such a concept.It is now possible to further restrict the attention to isomorphisms with special properties.In particular, if it only swaps two atomic system A , A ∈ A we use it to define the notion ofequivalent atomic systems. 83 efinition E.8 (Equivalence of atomic systems) . Two atomic systems A , A ∈ A are called equivalent , and we write A ˆ= A , if there exists a thermodynamic isomorphism ϕ, ϕ A which addi-tionally fulfils(iv) ϕ A ( A ) = A , ϕ A ( A ) = A and ϕ A ( A ) = A for all A ∈ A r { A , A } , and(v) for A ∈ A r { A , A } it holds ⌊ ϕ ( p ) ⌋ A = ⌊ p ⌋ A and ⌈ ϕ ( p ) ⌉ A = ⌈ p ⌉ A .We say that ϕ, ϕ A is a thermodynamic isomorphism for A ˆ= A .Essentially, such an isomorphism swaps the thermodynamic roles of the two atomic systemswith all their belongings such as states, work processes, work functions and so on, such that“nothing else changes”. The notion of a thermodynamic isomorphisms allows us to make thisprecise.Looking at the mapping of states ϕ Σ for an equivalence we see that (iv) and (v) imply thefollowing. For all atomic states σ ∈ S A ∈A r { A ,A } Σ A it holds ϕ Σ ( σ ) = σ . That is, ϕ Σ is equalto the identity map outside Σ A ∪ Σ A . On the other hand, ϕ Σ | Σ A and ϕ Σ | Σ A are bijective andimply a 1-1 correspondence of states in Σ A with states in Σ A .In order to prove that ˆ= is an equivalence relation on A (i.e. in order to justify the name givento A ˆ= A ) we have to work a little more. Lemma E.9 (Inverse of thermodynamic isomorphism) . Let ϕ, ϕ A be a thermodynamic isomor-phism for the atomic systems A ˆ= A ∈ A . Then its inverse ϕ − , ϕ − A is also a thermodynamicisomorphism for A ˆ= A and the corresponding maps for systems and states fulfil( ϕ − ) S = ( ϕ S ) − and ( ϕ − ) Σ = ( ϕ Σ ) − . (E.13) Proof.
First of all, since ϕ − and ϕ − A are the inverses of bijective maps, They are bijectivethemselves. Let now p, p ′ ∈ P and define their images under ϕ − to be q := ϕ − ( p ) and q ′ := ϕ − ( p ′ ). Likewise, let A ∈ A and define A ′ := ϕ − A ( A ). We check Definitions E.1 and E.8 pointby point.(i) If p ′ ◦ p is defined, then ϕ − ( p ′ ◦ p ) = ϕ − (cid:0) ϕ ( q ′ ) ◦ ϕ ( q ) (cid:1) = ϕ − (cid:0) ϕ ( q ′ ◦ q ) (cid:1) = q ′ ◦ q = ϕ − ( p ′ ) ◦ ϕ − ( p ) , (E.14)where Definition E.1 (i) was used for ϕ in the second equality. Writing the equation in termsof q and q ′ it follows that ϕ − satisfies (i).(ii) Using (ii) for ϕ, ϕ A , we immediately obtain ⌊ p ⌋ A def. ⇔ ⌊ ϕ ( q ) ⌋ ϕ A ( A ′ ) def. ⇔ ⌊ q ⌋ A ′ def. ⇔ ⌊ ϕ − ( p ) ⌋ ϕ − A ( A ) def. . (E.15)(iii) We directly check W ϕ − A ( A ) ( ϕ − ( p )) = W A ′ ( q ) = W ϕ A ( A ′ ) ( ϕ ( q )) = W A ( p ) .(iv) Since ϕ A is the simple map that swaps A with A it follows that its inverse does exactlythe same.(v) For A = A , A we compute ⌊ ϕ − ( p ) ⌋ A = ⌊ q ⌋ A = ⌊ ϕ ( q ) ⌋ A = ⌊ p ⌋ A and similarly for ⌈·⌉ · .Finally, we consider the corresponding maps on systems and states. Since ϕ − , ϕ − A is a ther-modynamic isomorphism for the same two systems as ϕ we obtain ( ϕ − ) S = ϕ S , and because ϕ S ( ϕ S ( S )) = S for all S ∈ S this implies that ( ϕ − ) S = ( ϕ S ) − .As for the mapping of states, by Definition E.5 for every S ∈ S and p ∈ P it holds ϕ Σ ( ⌊ p ⌋ S ) = ⌊ ϕ ( p ) ⌋ ϕ S ( S ) and ( ϕ − ) Σ ( ⌊ p ⌋ S ) = ⌊ ϕ − ( p ) ⌋ ϕ − S ( S ) . (E.16)84herefore ϕ Σ (cid:0) ( ϕ − ) Σ ( ⌊ p ⌋ S ) (cid:1) = ϕ Σ (cid:16) ⌊ ϕ − ( p ) ⌋ ϕ − S ( S ) (cid:17) = ⌊ ϕ ( ϕ − ( p )) ⌋ ϕ S ( ϕ − S ( S )) = ⌊ p ⌋ S (E.17)and likewise ϕ − ( ϕ Σ ( ⌊ p ⌋ S )) = ⌊ p ⌋ S , which means that ( ϕ − ) Σ = ( ϕ Σ ) − . Lemma E.10 (Conjugating thermodynamic isomorphisms) . Let A , A , A ∈ A be three differentatomic systems. If ϕ, ϕ A is a thermodynamic isomorphism for A ˆ= A and ψ, ψ A is one for A ˆ= A ,then ϕ − ◦ ψ ◦ ϕ, ϕ − A ◦ ψ A ◦ ϕ A is a thermodynamic isomorphism for A ˆ= A . Furthermore, thecorresponding mapping of systems and states transform analogously,( ϕ − ◦ ψ ◦ ϕ ) S = ϕ − S ◦ ψ S ◦ ϕ S and ( ϕ − ◦ ψ ◦ ϕ ) Σ = ϕ − ◦ ψ Σ ◦ ϕ Σ . (E.18) Proof.
As before, the fact that ϕ − ◦ ψ ◦ ϕ and ϕ − A ◦ ψ A ◦ ϕ A are bijective is trivial to see. Let p, p ′ ∈ P . The proof uses several times the fact that ϕ − is a thermodynamic isomorphism for A ˆ= A , established by Lemma E.9. It is important that the three atomic systems are all different.Otherwise the intuition that the conjugated isomorphism essentially swaps A with A is wrong. We will write ϕ − ( ψ ( ϕ ( · ))) instead of ϕ − ◦ ψ ◦ ϕ in order not to confuse the concatenation ofthese mapping with the concatenation of thermodynamic processes.Let p, p ′ ∈ P and A ∈ A .(i) Suppose p ′ ◦ p is defined, then: ϕ − ( ψ ( ϕ ( p ′ ◦ p ))) = ϕ − ( ψ ( ϕ ( p ′ ) ◦ ϕ ( p ))) = ϕ − ( ψ ( ϕ ( p ′ )) ◦ ψ ( ϕ ( p )))= ϕ − ( ψ ( ϕ ( p ′ ))) ◦ ϕ − ( ψ ( ϕ ( p ))) . (E.19)If the right hand side is defined, the equation must hold, too.(ii) Since all involved maps are thermodynamic isomorphisms and fulfil (ii) individually, it easyto see shell by shell that ⌊ p ⌋ A def. ⇔ · · · ⇔ ⌊ ϕ − ( ψ ( ϕ ( p )))) ⌋ ϕ − A ( ψ A ( ϕ A ( A ))) def. . (E.20)(iii) With the same logic as in (i) and (ii) it follows W ϕ − A ( ψ A ( ϕ A ( A ))) ( ϕ − ( ψ ( ϕ ( p )))) = W ψ A ( ϕ A ( A )) ( ψ ( ϕ ( p ))) = W ϕ A ( A ) ( ϕ ( p )) = W A ( p ) . (E.21)(iv) We check ϕ − A ( ψ A ( ϕ A ( A ))) = ϕ − A ( ψ A ( A )) = ϕ − A ( A ) = A ,ϕ − A ( ψ A ( ϕ A ( A ))) = ϕ − A ( ψ A ( A )) = ϕ − A ( A ) = A ,ϕ − A ( ψ A ( ϕ A ( A ))) = ϕ − A ( ψ A ( A )) = ϕ − A ( A ) = A . (E.22)For all other atomic systems A ∈ A r { A , A , A } it follows ϕ − A ( ψ A ( ϕ A ( A ))) = A sincethey are mapped to themselves under all three mappings.(v) For A ∈ A r { A , A , A } this follows shell by shell. For A we shortly check ⌊ ϕ − ( ψ ( ϕ ( p ))) ⌋ A = ⌊ ψ ( ϕ ( p )) ⌋ ϕ A ( A ) = ⌊ ψ ( ϕ ( p )) ⌋ A = ⌊ ϕ ( p ) ⌋ ψ − A ( A ) = ⌊ ϕ ( p ) ⌋ A = ⌊ p ⌋ ϕ − A ( A ) = ⌊ p ⌋ A . (E.23)The same follows for ⌈·⌉ · . If A = A , for instance, then ψ would not do anything (except for maybe relabelling some processes). Hencethe total map would not do anything either as everything done by ϕ would be undone by ϕ − and A would notbe swapped with A = A . ϕ − ◦ ψ ◦ ϕ ) S = ϕ − S ◦ ψ S ◦ ϕ S is trivial after (iv). For the mapping of states let S ∈ S and p ∈ P and it follows ϕ − ( ψ Σ ( ϕ Σ ( ⌊ p ⌋ S ))) = ϕ − ( ψ Σ ( ⌊ ϕ ( p ) ⌋ ϕ S ( S ) )) = ϕ − ( ⌊ ψ ( ϕ ( p )) ⌋ ψ S ( ϕ S ( S )) )= ⌊ ϕ − ( ψ ( ϕ ( p ))) ⌋ ϕ − S ( ψ S ( ϕ S ( S ))) = ( ϕ − ◦ ψ ◦ ϕ ) Σ ( ⌊ p ⌋ S ) . (E.24) Proposition E.11 ( ˆ= is equivalence relation on A ) . The relation ˆ= is reflexive, symmetric andtransitive, hence it is an equivalence relation on the set of atomic systems A . Proof.
Reflexive: Consider the identity ϕ ( p ) = p for all p ∈ P . This map clearly fulfils allrequirements in Definitions E.1 and E.8 for the atomic system(s) A and A and thus A ˆ= A always.Symmetric: By Definition E.1 the roles of A and A are symmetric. Hence if A ˆ= A then also A ˆ= A .Transitive: Lemma E.10 states that if A ˆ= A and A ˆ= A holds for three different atomic systems,then the two associated thermodynamic isomorphisms can be conjugated to a new thermodynamicisomorphism for A ˆ= A , which proves that in this case also A ˆ= A . If two or all of the threesystems are the same, then transitivity follows from reflexivity and symmetry.A comment on the repeated application of a thermodynamic isomorphism for A ˆ= A : bydefinition of ϕ S it is idempotent, i.e. applying this function twice yields the identity, ϕ S ( ϕ S ( S )) = S for all S ∈ S . One could thus expect that something similar holds for ϕ and ϕ Σ . This is notthe case in general.For ϕ , the mapping of thermodynamic processes, idempotency is in general wrong becausethere may be two thermodynamic process with exactly the same thermodynamic properties, i.e.they act on the same systems, induce the same state changes, have the same work cost, and soon. Under ϕ ( ϕ ( · )) these may be interchanged, which makes ϕ ( ϕ ( · )) different from the identitymapping, even though thermodynamically nothing has changed. As for the mapping of states ϕ Σ , it does not have to be idempotent either. Finding an easy example where this is the case butintuition does not fail is a bit trickier than in the case of processes. However, the point is that ϕ Σ must map thermodynamic states to a “thermodynamically equivalent” states, i.e. a state withexactly the same thermodynamic properties. This can in principle also happen when ϕ Σ ( ϕ Σ ( · ))is not an identity mapping.These observation are the reason why we had go via Lemmas E.9 and E.10 to prove that ˆ= isan equivalence relation.We can now extend the definition of equivalent systems from atomic to arbitrary systems.Namely, two arbitrary systems are equivalent if their atomic subsystems are pairwise equivalent.The extended relation will keep the status of an equivalence relation, as we show below. Definition E.12 (Equivalence of systems) . Let S , S ∈ S be two arbitrary systems. They are equivalent , and we write S ˆ= S , if there exists a bijection between Atom( S ) and Atom( S ) whichrespects the equivalence classes of ˆ= for atomic systems.Definition E.12 can be rephrased as: The two systems are equivalent if | Atom( S ) | = | Atom( S ) | =: n and there exists a labelling { A ( i ) k } i =1 ,..,n = Atom( S k ) for k = 1 , A ( i )1 ˆ= A ( i )2 for i = 1 , . . . , n . (E.25)Again we need to prove that ˆ= is an equivalence relation, now on S . We do not use the ◦ notation for the concatenation of ϕ with itself on purpose in order not to confuse thereader, as ◦ is already used for the concatenation of thermodynamic processes. Instead, we used the notation ϕ ( · )to still be able to denote a concatenation of such functions. roposition E.13 ( ˆ= is equivalence relation on S ) . The relation ˆ= on S is an equivalence relation. Proof.
Reflexive: If S = S , i.e. Atom( S ) = Atom( S ), then a possible labelling is A ( i )1 = A ( i )2 for i = 1 , . . . , | Atom( S ) | =: n .Symmetry: Again, the roles of S and S in Definition E.12 are interchangeable.Transitivity: Suppose S ˆ= S and S ˆ= S . This means that there are two labellings { A ( i ) k } i =1 ,..,n =Atom( S k ) for k = 1 , { ˜ A ( j ) l } j =1 ,..,n = Atom( S l ) for l = 2 , A ( i )1 ˆ= A ( i )2 for i = 1 , . . . , n , ˜ A ( j )2 ˆ= ˜ A ( j )3 for j = 1 , . . . , n . (E.26)If the two labellings do not agree on Atom( S ), one can relabel the j ’s such that ˜ A ( j )2 = A ( j )2 for j = 1 , . . . , n . By transitivity of ˆ= for atomic systems, we then find A ( j )1 ˆ= ˜ A ( j )3 for j = 1 , . . . , n (E.27)and thus S ˆ= S . Lemma E.14 (Equivalence and composition) . Let A , A ∈ A and S ∈ S an arbitrary systemdisjoint with A and A , Atom( S ) ∩ { A , A } = ∅ . Then: A ˆ= A ⇐⇒ A ∨ S ˆ= A ∨ S . Proof.
Let { A (1) S , . . . , A ( n ) S } = Atom( S ). By assumption, A and A are not in this list, i.e. allatomic systems we deal with are different. The direction A ˆ= A ⇒ A ∨ S ˆ= A ∨ S followsimmediately with the labelling A ˆ= A ,A ( j ) S ˆ= ˜ A ( j ) S for j = 1 , . . . , n . (E.28)Let now A ∨ S ˆ= A ∨ S . Consider the labelling of atomic system in this equivalence. W.l.o.g. thenumbering can be chosen such that A ˆ= A (1) S ˆ= · · · ˆ= A ( k ) S . It holds k ≥ A is equivalent to at least one atomic system in Atom( S ) ∪ A and it is not A . If k = n , it meansthat all atomic systems at hand are equivalent to each other and consequently A ˆ= A must hold.If k < n then the labelling must be such that members of { A ( k +1) S , . . . , A ( n ) S } are equivalent to oneor more other members of this set but not to any members of { A (1) S , . . . , A ( k ) S } . I.e. the labelling issuch that the “sublabelling” of the subset { A ( k +1) S , . . . , A ( n ) S } of atomic systems is closed. But thismeans that there is no space for A in this part of the labelling. Hence A is must be equivalentto at least one member of { A (1) S , . . . , A ( k ) S } and thus by transitivity also A ˆ= A .This section of the appendix was opened with a definition thermodynamic isomorphisms onwhich the concept of equivalent atomic is based. Having now defined when two arbitrary sys-tems are called equivalent it may strike us that for this definition an explicit notion of a moregeneral thermodynamic isomorphism associated with A ˆ= S was not necessary. This is becausethe definition of equivalence of arbitrary systems relies on the equivalences of the atomic subsys-tems. However, if we want to generalize the specific results derived for atomic equivalences (e.g.Lemma E.3, Lemma E.4, Lemma E.6, Lemma E.7) it is necessary to explicitly talk about thecorrespondence of processes.For this, we construct a thermodynamic isomorphism for the equivalence of two arbitraryequivalent systems. Definition E.15 (Thermodynamic isomorphism for S ˆ= S ) . Let S ˆ= S according to Defini-tion E.12 and let n := | Atom( S ) | = | Atom( S ) | . Furthermore, let l := | Atom( S ) ∩ Atom( S ) | be the number of shared atomic subsystems. Choose a labelling of the atomic subsystems of eachsystem such that A (1)1 = A (1)2 , . . . , A ( l )1 = A ( l )2 , A ( l +1)1 ˆ= A ( l +1)2 , . . . , A ( n )1 ˆ= A ( n )2 (E.29)87this implies that we can choose the isomorphisms for the first l atomic equivalences as identities, ϕ ( i ) ( p ) = p and ϕ A ( A ) = A for i = 1 , . . . , l ). Denote the remaining (non-identity) thermodynamicisomorphisms by ϕ ( i ) , ϕ ( i ) A for i = l + 1 , . . . , n . Then define the thermodynamic isomorphism ϕ, ϕ A for S ˆ= S by ϕ ( p ) := ϕ ( l +1) ( ϕ ( l +2) ( · · · ϕ ( n ) ( p ) · · · )) ,ϕ A ( A ) := ϕ ( l +1) A ( ϕ ( l +2) A ( · · · ϕ ( n ) A ( A ) · · · )) . (E.30)Having defined a thermodynamic isomorphism for two equivalent atomic systems we must nowcheck whether this generalization fulfils the expected generalized properties (i)-(v) from Defini-tions E.1 and E.8. Proposition E.16.
A thermodynamic isomorphism ϕ, ϕ A for S ˆ= S as in Definition E.15 isindeed a thermodynamic isomorphism (i.e. fulfils Definition E.1 (i)-(iii)) and(iv) for A ∈ A r (cid:0) Atom( S ) ∪ Atom( S ) (cid:1) we have ϕ A ( A ) = A , while ϕ A ( A ( i )1 ) = A ( i )2 and ϕ A ( A ( i )2 ) = A ( i )1 , and(v) for all p ∈ P and A ∈ A r (cid:0) Atom( S ) ∪ Atom( S ) (cid:1) we have ⌊ ϕ ( p ) ⌋ A = ⌊ p ⌋ A and the samefor ⌈·⌉ · . Proof.
Any thermodynamic isomorphism for two equivalent atomic systems is bijective by defini-tion, hence so is a finite subsequent application of such maps. For the remainder of the proof wenote that the construction of ϕ is such that the non-identity thermodynamic isomorphisms actnon-trivially on disjoint pairs { A ( i )1 , A ( i )2 } of atomic systems. For instance, ϕ ( n ) acts non-triviallyon A ( n )1 and A ( n )2 , but any A ( i ) k with i = n is untouched by ϕ ( n ) according to Definition E.8 (samefor ϕ ( n ) A ). This observation is key for proving the five points characterizing thermodynamic iso-morphisms for equivalences.Let p, p ′ ∈ P be thermodynamic processes and A ∈ A am atomic system.(i) If p ′ ◦ p is defined we obtain ϕ ( p ′ ◦ p ) = ϕ ( l +1) ( · · · ϕ ( n ) ( p ′ ◦ p ) · · · )= ϕ ( l +1) ( · · · ϕ ( n − ( ϕ ( n ) ( p ′ ) ◦ ϕ ( n ) ( p )) · · · )= . . . = ϕ ( l +1) ( · · · ϕ ( n ) ( p ′ ) · · · ) ◦ ϕ ( l +1) ( · · · ϕ ( n ) ( p ) · · · )= ϕ ( p ′ ) ◦ ϕ ( p ) , (E.31)by using Definition E.1 (i) for ϕ ( i ) for i = l + 1 , . . . , n subsequently. If instead ϕ ( p ′ ) ◦ ϕ ( p )is defined the equation still holds, thus concluding the proof of (i).(ii) We use that the ϕ ( i ) fulfil (ii) individually: ⌊ ϕ ( p ) ⌋ ϕ A ( A ) = ⌊ ϕ ( l +1) ( · · · ϕ ( n ) ( p ) · · · ) ⌋ ϕ ( l +1) A ( ··· ϕ ( n ) A ( A ) ··· ) def. ⇔ ⌊ ϕ ( l +2) ( · · · ϕ ( n ) ( p ) · · · ) ⌋ ϕ ( l +2) A ( ··· ϕ ( n ) A ( A ) ··· ) def. ⇔ · · ·⇔ ⌊ p ⌋ A def. (E.32)(iii) Like in the previous points we compute step by step, using the fact that the ϕ ( i ) , ϕ ( i ) A fulfil88iii) individually: W ϕ A ( A ) ( ϕ ( p )) = W ϕ ( l +1) A ( ··· ϕ ( n ) A (( A ) ··· ) ( ϕ ( l +1) ( · · · ϕ ( n ) ( p ) · · · ))= W ϕ ( l +2) A ( ··· ϕ ( n ) A (( A ) ··· ) ( ϕ ( l +2) ( · · · ϕ ( n ) ( p ) · · · ))= · · · = W A ( p ) . (E.33)(iv) Since all atomic systems A ∈ A r (cid:0) Atom( S ) ∪ Atom( S ) (cid:1) are untouched by the ϕ ( i ) A , i.e.mapped to themselves, it immediately follows that ϕ A ( A ) = A for those.For i = 1 , . . . , l we see with the same argument that ϕ A ( A ( i )1 ) = A ( i )1 = A ( i )2 = ϕ A ( A ( i )2 ),where we used that the labelling is chosen such that the first l atomic subsystems are pairwiseequal.For i = l +1 , . . . , n on the other hand, we know that the the A ( i ) k are mapped to themselves bythe ϕ ( j ) A except for j = i , in which case ϕ ( i ) A ( A ( i )1 ) = A ( i )2 and vice versa. Thus ϕ A ( A ( i )1 ) = A ( i )2 and ϕ A ( A ( i )2 ) = A ( i )1 .Proposition E.16 establishes the natural generalizations of the properties listed in Definition E.8for thermodynamic isomorphisms of atomic equivalences to thermodynamic isomorphisms forgeneric equivalent systems. Since ϕ, ϕ A from Definition E.15 is in particular a thermodynamic iso-morphism we can define associated bijective mappings ϕ S and ϕ Σ of arbitrary systems and states(Definition E.5, Lemma E.6). Also, the results about properties of work processes (Lemma E.3),state changes (Lemma E.4), and internal energy (Lemma E.7) under thermodynamic isomorphismscan be directly applied. They yield invariance of the quantities and properties under ϕ, ϕ A .These observations conclude the construction of thermodynamic isomorphisms describing equiv-alences for general systems. We have shown that all relevant concepts introduced so far are in-variant under equivalences. The concepts that are introduced from here on will also have thisproperty, as will be shown for each. F Heat and heat reservoirs
The heat flow into a system S due to a process p ∈ P , Q S ( p ), denoted by Q s in the following,is defined as the difference of the change in internal energy and work done during a thermody-namic process (Definition 5.1). Heat Q S defined as such inherits all thermodynamically relevantproperties from the work function W S and the internal energy U S . If no atomic subsystem of S isinvolved in p , then Q S ( p ) = 0; likewise, Q S (id) = 0 for identity processes; Q S ( p ) = 0 for all workprocesses p ∈ P S on S ; heat flows in reverse processes change their signs; Q S is additive undercomposition (for disjoint systems); it is additive under concatenation (Lemma F.1); and finally,heat flows are invariant under thermodynamic isomorphisms (Lemma F.2).While the first five statements are direct to see, the latter two need a bit more discussion.They are states in the following two lemmas. Lemma F.1 (Heat under concatenation) . Let p, p ′ ∈ P be arbitrary processes such that p ′ ◦ p isdefined and consider an atomic system A ∈ A . Then Q A ( p ′ ◦ p ) = Q A ( p ) + Q A ( p ′ ) . (F.1)This statement naturally extends to arbitrary systems by additivity under composition.89 roof. We distinguish three cases. First, assume that A is neither involved in p nor in p ′ . Thisimplies that both Q A ( p ) = Q A ( p ′ ) = 0, and that A is not involved in p ′ ◦ p either. But then Q A ( p ′ ◦ p ) = 0 = Q A ( p ) + Q A ( p ′ ) is obviously fulfilled.Second, if A is involved in p but not in p ′ , only Q A ( p ′ ) = 0 and W A ( p ′ ) = 0 necessarily. In thiscase, according to the postulate introducing concatenation, A is involved in p ′ ◦ p and undergoesthe same state change (thus also the same change in internal energy) as it does under p alone. Wefind Q A ( p ′ ◦ p ) = ∆ U A ( p ′ ◦ p ) − W A ( p ′ ◦ p ) = ∆ U A ( p ) − W A ( p ′ ) − W A ( p ) = Q A ( p ) = Q A ( p ) + Q A ( p ′ ) . (F.2)If A is involved in p ′ but not in p the argument works analogously.Third, assume that A is involved in both p and p ′ . In this case we argue directly that both∆ U A and W A are additive under concatenation, the former by the fact that it is a state variable,the latter by the additivity postulate for the work function. With Definition 5.1 for heat thisimplies that also Q A is additive under concatenation. Lemma F.2 (Heat under isomorphisms) . Let ϕ, ϕ A be a thermodynamic isomorphism. Then forany process p ∈ P it holds Q ϕ A ( A ) ( ϕ ( p )) = Q A ( p ) . (F.3) Proof. If A is involved in p we can make use of Lemma E.7 implying that ∆ U ϕ A ( A ) ( ϕ ( p )) =∆ U A ( p ). Together with Definition 4.1 (iii) it immediately follows Q ϕ A ( A ) ( ϕ ( p )) = ∆ U ϕ A ( A ) ( ϕ ( p )) − W ϕ A ( A ) ( ϕ ( p ))= ∆ U A ( p ) − W A ( p )= Q A ( p ) . (F.4)By Definition 4.2 (ii) the atomic system A is involved in p if and only if ϕ A ( A ) is involved in ϕ ( p ).Hence, if A is not involved in p , Q A ( p ) = 0 = Q ϕ A ( A ) ( ϕ ( p )).Definition 5.1 says what amount of heat flows into a specific system S , but it does not specifywhere this heat comes from. In a bipartite setting such as the one discussed in the main text,where a work process p ∈ P S ∨ S is considered, it is possible to say that heat Q S ( p ) flows from S to S . Likewise in the opposite view, one can say that the heat Q S ( p ) ≡ − Q S ( p ) flows from S to S . However, in a more complex composite system such a statement is not necessarily possible,unless one splits it up into two subsystems to be considered, in which case we end up with thebipartite setting again. This becomes relevant in the pictorial representation of work and heatflows, as is discussed in the main text. G Carnot’s Theorem
As a direct consequence of the second law (Postulate 9) and Definition 5.3 for heat reservoirs,we prove the following result on the signs of heat flows to cyclic machines operating between tworeservoirs.
Lemma G.1 (Direction of heat flows in Carnot engines) . Consider a (not necessarily reversible)process p ∈ P R ∨ S ∨ R operating on a cyclic system S and two reservoirs R , R ∈ R . If p isnon-trivial, i.e. if not both heat flows Q R ( p ) and Q R are zero, then(i) at least one of the heat flows is strictly positive, and(ii) if p is reversible, one of the heat flows is strictly positive and the other one strictly negative.90 roof. We prove (i) and then show that (ii) is a consequence of it. Suppose by contradiction that Q R ( p ) ≤ Q R ( p ) ≤
0. Excluding the trivial case, w.l.o.g. we can assume that Q R ( p ) < ↔
2. Since ∆ U R ( p ) = Q R ( p ) ≤ q ∈ P R with W R ( q ) = − ∆ U R ( p ) ≥ q ◦ p . This is a consequence ofDefinition 5.3 (ii). By construction we find that under q ◦ p ∆ U R ( q ◦ p ) = ∆ U R ( q ) + ∆ U R ( p ) = 0 (G.1)implying that not only S but S ∨ R is cyclic, and Q R ( q ◦ p ) = Q R ( p ) < . (G.2)This contradicts the second law, requiring Q R ( q ◦ p ) ≥
0, and proves (i).For a reversible p w.l.o.g. Q R ( p ) >
0. If p rev is a reverse process, we know that Q R ( p rev ) = − Q R ( p ) < p rev it must be that Q R ( p rev ) >
0. Going back to p , thisimplies that Q R ( p ) <
0, which concludes the proof.
H Absolute temperature
We investigate the reversible heat flows between equivalent reservoirs and a cyclic machine.Lemma H.1 proves that reversible heat flows to reversible engines from equivalent reservoirs areexactly opposite and hence − Q R Q R = 1. In addition, we know from Lemma F.2 that swappingequivalent system leaves all heat flows invariant and thus the same holds for the ratio of heat flowsin Carnot engines. Lemma H.1 (Reversible engine with equivalent reservoirs.) . Consider a reversible p ∈ P R ∨ S ∨ R ,cyclic on S , with equivalent heat reservoirs R ˆ= R ∈ R . Then the heat flows to R and R mustbe exactly opposite, Q R ( p ) = − Q R ( p ). Proof. If p is trivial, i.e. Q R ( p ) = Q R ( p ) = 0, we are done. So we assume that it is non-trivial. W.l.o.g. Q R ( p ) >
0, which implies that Q R ( p ) < R ˆ= R there exists a corresponding thermodynamic isomorphism ϕ . Define q := ϕ ( p ) ∈P R ∨ S ∨ R . By definition, q fulfils Q R ( q ) = Q R ( p ) and Q R ( q ) = Q R ( p ). Suppose now bycontradiction that − Q R ( p ) Q R ( p ) = 1. W.l.o.g. − Q R ( p ) Q R ( p ) > p andswap the labels 1 ↔ k, l ∈ N such that − Q R ( p ) Q R ( p ) > kl > p l times followed by k applications of q . The total process is still cyclic on S and theheat flows sum up to Q tot R = l Q R ( p ) + k Q R ( q ) = l Q R ( p ) + k Q R ( p ) = (cid:18) kl − (cid:18) − Q R ( p ) Q R ( p ) (cid:19)(cid:19)| {z } < · l Q R ( p ) | {z } > < , (H.2) Q tot R = l Q R ( p ) + k Q R ( q ) l Q R ( p ) + k Q R ( p ) = (cid:18) kl − (cid:18) − Q R ( p ) Q R ( p ) (cid:19)(cid:19)| {z } > · l Q R ( p ) | {z } < < , (H.3)where we used in the second line that − Q R ( p ) Q R ( p ) < < kl . This contradicts Lemma G.1, and thusconcludes the proof. As in the proof of Carnot’s Theorem, by “ k applications of p ” we mean that one applies p , then the corre-sponding process to p for the new initial states of the reservoirs which exists due to Definition 5.3 (iii) and so on, k times in total. Then apply q or, if the initial states of the reservoirs do not match, a translated version of q , l times in the same manner. τ as follows. Definition H.2 (Temperature ratio) . The temperature ratio τ of two equivalence classes [ R ] , [ R ] ∈ R / ˆ= is τ : R / ˆ= × R / ˆ= −→ R > ([ R ] , [ R ]) Q R Q R (H.4)where R and R are two different heat reservoirs and Q R and Q R are the heat flows of a(non-trivial) reversible Carnot engine operating between them such that Q R >
0. Extendingthis definition to the set of pairs of heat reservoirs, we use the same symbol τ and write for the temperature ratio of two heat reservoirs τ ( R , R ) := τ ([ R ] , [ R ]) . (H.5) τ is well-defined, as discussed in the main text. Furthermore, it obviously fulfils τ ([ R ] , [ R ]) = 1and τ ([ R ] , [ R ]) = τ ([ R ] , [ R ]) − .In the coming lemma about the temperature ratio τ it is proven that it actually behaves as aratio, meaning that multiplying τ ( R , R ) with τ ( R , R ), where R first shows up in the secondargument and then in the first, is equal to τ ( R , R ), independently of the reservoir R . Lemma H.3 ( τ as a ratio) . Let R , R , R ∈ R be three arbitrary heat reservoirs. Then τ ( R , R ) · τ ( R , R ) = τ ( R , R ). Proof.
W.l.o.g. the reservoirs R = R = R = R are different representatives of their respectiveequivalence class. If this was not the case, take equivalent but different reservoirs (this is alwayspossible due to the postulate on the existence of arbitrarily many copies). For the constructiveproof, we need two copies of R in the beginning. Call them R and R ′ and choose them to bedifferent from each other and all the others, too.Let S be a Carnot engine operating between R and R (all systems pairwise disjoint) through areversible work process p ∈ P R ∨ S ∨ R with Q R := Q R ( p ) < Q R := Q R ( p ) >
0. Likewise,let S ′ and p ′ ∈ P R ∨ S ′ ∨ R be an analogous machine and reversible process for the reservoirs R ′ and R with Q R ′ := Q R ′ ( p ′ ) = − Q R < Q R := Q R ( p ′ ) >
0. Such a machine togetherwith the reversible work process on R ′ ∨ S ∨ R exists due to Postulate 10. The described settingis depicted in Figure H.1 (a). It follows that τ ( R , R ) = − Q R Q R and τ ( R ′ , R ) = − Q R ′ Q R . (H.6)We now use an additional machine S ′′ with a reversible process p ′′ ∈ P R ∨ S ′′ ∨ R ′ that transfersthe heat Q R from R to R ′ . Such a machine exists due to Postulate 10 and has no work costaccording to Lemma H.1. Due to the translation invariance of reservoirs (Definition 5.3 (iii)) it ispossible to find p, p ′ and p ′′ which can be concatenated to p ′′ ◦ p ′ ◦ p .Under the concatenated process the reservoirs R and R ′ are cyclic by construction, while themachines S , S ′ and S ′′ are also cyclic. Hence, under this extension the situation changes to the onedepicted in Figure H.1 (b), where the reservoirs R and R interact with a cyclic machine whiletaking up the heat flows Q R and Q R , respectively. The construction could be made analogously A Carnot engine operates between two different heat reservoirs. If R = R , then R ∨ S ∨ R = R ∨ S andthere would be no two heat flows to compare. Notice that this is the first time where we use the fact that we can tune one of the heat flows. In all previousproofs we only used the fact that non-trivial machines exist between any two reservoirs. R R SQ R Q R W S R ′ R S ′ Q R ′ = − Q R Q R W S ′ (b) R Q R S ∨ R ∨ R ′ ∨ S ′ ∨ S ′′ W S + W S ′ R Q R Figure H.1: (a) The two reversibly operating cyclic machines S and S ′ between R , R and R ′ , R , respectively, are such that Q R = − Q R ′ > Q R < Q R >
0. (b) Aftermaking use of Postulate 10 the copies R and R ′ we obtain a reversibly operating cyclic machine S ∨ R ∨ R ′ ∨ S ′ ∨ S ′′ between R and R .for the reverse processes and as a consequence the constructed process is reversible. We concludethat τ ( R , R ) = − Q R Q R , which implies τ ( R , R ) · τ ( R ′ , R ) = (cid:18) − Q R Q R (cid:19) · (cid:18) − Q R ′ Q R (cid:19) = (cid:18) − Q R Q R (cid:19) · (cid:18) Q R Q R (cid:19) = − Q R Q R = τ ( R , R ) . (H.7)Together with the observation that heat flows do not change when exchanging equivalent systems(Lemma F.2), and hence τ ( R ′ , R ) = τ ( R , R ), this concludes the proof.It is then possible to define the absolute temperature of a heat reservoir up to the free choiceof a reference temperature T ref and a reference reservoir R ref . Definition H.4 (Absolute temperature) . The absolute temperature of a heat reservoir R ∈ R isdefined as T := τ ( R, R ref ) · T ref . (H.8)It follows that absolute temperature, just like the heat flows in reversible Carnot engines, isa property of the reservoir as a whole and independent of its state. This is what one expects forreservoirs, which are systems that should not change its properties and how they interact withother systems, after finite amounts of energy have been exchanged.Systems at equal temperature are considered to be in thermal equilibrium. For reservoirs wecan now make this definition precise. Definition H.5 (Thermal equilibrium for reservoirs, ∼ ) . Let R , R ∈ R be heat reservoirs. Wesay that they are in thermal equilibrium if τ ( R , R ) = 1 and write R ∼ R .ˆ=, restricted to the set of heat reservoirs R , is a sub-relation of ∼ . In addition, just like ˆ=, ∼ is an equivalence relation. Lemma H.6 ( ∼ is equivalence relation) . The relation ∼ defined in Definition H.5 is an equivalencerelation. 93 roof. We need to show that it is reflexive, symmetric, and transitive. Reflexivity follows directlyfrom Lemma H.1, while symmetry holds due to τ ([ R ] , [ R ]) = τ ([ R ] , [ R ]) − for all R , R ∈ R .Finally, transitivity is a consequence of Lemma H.3.A thorough discussion of absolute temperature, the relation for thermal equilibrium and itimplication is done in the main text. I The temperature of heat flows
We shortly repeat the definition of the temperature of a heat flows.
Definition I.1 (Heat at temperature T ) . Let S = S ∨ S ∈ S be composed of two disjointsubsystems and undergo an arbitrary work process p ∈ P S ∨ S with Q := Q S ( p ) = 0. Wesay that the heat Q flows at temperature T if there exist two different reservoirs R ∼ R attemperature T with processes p ∈ P S ∨ R and p ∈ P S ∨ R s.t. W A ( p i ) = W A ( p ) for all atomicsystems A ∈ A r Atom( S i +1 ) and the state changes on S i under p i are the same as under p , i.e. ⌊ p i ⌋ S i = ⌊ p ⌋ S i and ⌈ p i ⌉ S i = ⌈ p ⌉ S i .In this section we technically investigate this definition. The case of reversible heat flows willbe of particular interest. Lemma I.2 (Reversible p i for reversible p ) . Consider the setting described in Definition I.1. If p is reversible, then so are the p i . Proof.
Let p rev ∈ P S ∨ S be a reverse process for p . Furthermore let p C ∈ P R ∨ R be the processthat reversibly transports the heat Q from R to R starting from the states ⌊ p ⌋ R and ⌈ p ⌉ R .The process p C exists according to Postulate 10. We then claim that the process p rev2 := ( p rev ∨ p C ) ◦ ( p ∨ id S ∨ R ) ∈ P S ∨ R ∨ S ∨ R is well-defined, where id S ∨ R is the identity process onthe corresponding initial states of p C and p rev for R and S , respectively. See Figure I.1 foran illustration of the actions of the relevant processes. Furthermore, this process is a reverseprocess for p ∨ id S ∨ R on the composite system S ∨ R ∨ S ∨ R that is catalytic on S ∨ R .Hence, according to the catalysis postulate there exists a process ˜ p rev2 ∈ P S ∨ R with the samethermodynamic properties, which then must be a reverse process for p . Thus p (and if we dothe same for p also this one) must be reversible.We are left with the task to show the above claims about p rev2 . The fact that it is well-defined (i.e.the concatenation of the two processes exists) follows from checking that the input and outputstates of the two processes match. Regarding the cyclicity on S ∨ R the consideration is equallyeasy. Finally, we check the net work costs of S and R W S ( p rev2 ) = W S ( p rev ) + W S ( p ) = W S ( p rev ) + W S ( p ) = 0 , (I.1) W R ( p rev2 ) = W R ( p C ) = 0 . (I.2)In the last equality of Eq. (I.1) we used the fact that the work flows in p rev are exactly oppositeto the ones during p . Hence, p rev2 is indeed catalytic on S ∨ R .Based on the previous lemma it is possible to show that non-zero reversible heat flows anassigned temperature is unique. Lemma I.3 (Unique temperature for reversible heat flows) . Consider again the setting describedin Definition I.1, in particular the heat flow is non-zero, Q = 0. If p is reversible, then the heatflow in the reverse process has the same temperature. Furthermore, this temperature is unique. Proof.
We know from Lemma I.2 that for a reversible process p fulfilling Definition I.1 the cor-responding divided processes p i are also reversible. Hence, for the reverse process, p rev , thecorresponding reverse processes of p i will fulfil the definition as well, now of course for the reverseheat flow − Q . Since the reservoirs in the reverse processes of the p i are the same, we have shown94a) Q W R S (b) S R QW − W QQ − W R S (c) S R Q − W R S Figure I.1: (a) Under p ∈ P S ∨ R the heat Q flows from R to S , according to Definition I.1.(b) In the construction of p rev2 ∈ P S ∨ R ∨ S ∨ R the heat Q flows from S via S and R to R andthe state change is exactly the opposite, as well as the work done on S . The systems S and R are catalytic. (c) Essentially, under p rev2 the heat Q flows from S to R while nothing is done onthe systems S and R . Formally, Postulate 6 on the freedom of description then guarantees thatthe process can also be seen as a work process ˜ p rev2 ∈ P S ∨ R alone. Since the state changes areindeed opposite to the ones under p , we have found a reverse process for it.that for the particular reverse process p rev the reverse heat flow also has temperature T .However, this alone does not exclude that more than one temperature could be assigned to thereverse heat flow (as well as the forward heat flow). So let us assume that the heat flow Q ex-changed under the reversible process p ∈ P S ∨ S can be assigned the two temperatures T and T ′ .According to Definition I.1 this means that there exist processes p i ∈ P S i ∨ R i and p ′ i ∈ P S i ∨ R ′ i that divide the process p , where R ∼ R and R ′ ∼ R ′ . Since p is reversible, all four processes p , p , p ′ , p ′ are reversible, too. Call the reverse processes p rev i and p rev i ′ , respectively. If we nowconcatenate the processes p and p rev1 ′ to a new process (id R ∨ p rev1 ′ ) ◦ ( p ∨ id R ′ ), as depicted inFigure I.2, we can construct a reversible engine S operating between R and R ′ . Reversibilityfollows from the fact that all processes used to construct the engine are reversible. According toCarnot’s Theorem, which applies to this situation, we find that TT ′ = − − QQ = 1 , (I.3)i.e. T = T ′ . This implies that R ∼ R ∼ R ′ ∼ R ′ . Remember that for this conclusion it isimportant that Q = 0, which is a condition in Definition I.1. Thus, for reversible processes, if atemperature can be assigned to a heat flow, this temperature is unique.We conclude that a non-zero reversible heat flow can either be assigned a unique temperatureor no temperature at all.Referring to Example 9.4 from the main text, one can investigate the relation between differentreversible heat flows inducing the same state change on a system. Lemma I.4 (Different reversible heat flows inducing the same state change) . Consider the settingas described in Definition I.1 with a reversible process p ∈ P S ∨ S . In addition, consider a differentreversible process p ′ ∈ P S ∨ S with ⌊ p ′ ⌋ S = ⌊ p ⌋ S and ⌈ p ′ ⌉ S = ⌈ p ⌉ S , i.e. both p and p ′ reversiblyinduce the same state change on S . If the heat Q under p is exchanged at temperature T and Q ′ under p ′ at temperature T ′ , and both Q = 0 and Q ′ = 0, then: QT = Q ′ T ′ . (I.4)95a) S R QW (b) S R ′ Q − W (c) R R ′ S − QQ W − W Figure I.2: (a) Under p ∈ P S ∨ R the state change in system S is exactly the opposite comparedto the state change under p rev1 ′ ∈ P S ∨ R ′ depicted in (b). If we concatenate the two reversibleprocesses the total process will be cyclic on S and reversible, since the construction with p rev1 and p ′ will yield a reverse process. Hence Carnot’s Theorem can be directly applied to the situationin (c). Proof.
Let p ∈ P S ∨ R and p ′ ∈ P S ∨ R ′ be the divided processes for p and p ′ , respectively, andlet p rev1 and p rev1 ′ be reverse processes for them. These exist since p and p ′ are reversible accordingto Lemma I.2.Just like in the proof of Lemma I.3 we now construct a reversible process ( p rev1 ∨ id R ′ ) ◦ ( p ′ ∨ id R ),which is essentially a Carnot process on S between the reservoirs R and R ′ . The concatenationis well-defined since the final states of p and p ′ match on S . During this process, heat Q flowsfrom R to S and heat Q ′ flows from S to R ′ . S itself will end up in its initial state since theinitial states of p and p ′ also match on S . Hence, S indeed acts as a cyclic machine betweenthe two reservoirs. Finally, the constructed process is reversible since all its parts are. Accordingto Carnot’s Theorem, we must have QQ ′ = TT ′ , (I.5)which concludes the proof.Importantly, even though Definition I.1 explicitly talks about both systems S and S , theratio QT only depends on the state change on S . Thus, Lemma I.4 does not make any referenceto the state changes on S under p and p ′ . Only the state change on S needs to match. This isan important observation which, after all, is key to be able to use the ratio QT to define a statefunction (the entropy). J Clausius’ Theorem and thermodynamic entropy
We here prove that entropy is additive under composition of disjoint systems.
Lemma J.1 (Additivity of entropy) . Let S , S ∈ S be disjoint systems and write S = S ∨ S .Furthermore, let σ, σ ′ ∈ Σ S be such that σ = σ ∨ σ and σ ′ = σ ′ ∨ σ ′ with σ i , σ ′ i ∈ Σ S i . Write∆ S S := S S ( σ ′ ) − S S ( σ ) and ∆ S S i := S S i ( σ ′ i ) − S S i ( σ ). Then∆ S S = ∆ S S + ∆ S S , (J.1)i.e. entropy differences are additive under composition of disjoint systems. Proof.
Let { p i } Ni =1 ⊂ P be a sequence of processes that allows us to compute the entropy difference∆ S S , i.e. the processes p i ∈ P S ∨ R (1) i in the sequence are concatenable and reversible, and under p i the heat Q S ( p i ) is exchanged at temperature T i . The total initial and final states of the sequenceare σ and σ ′ . We assume that such sequences exist between any two states of a system. Let96 q j } Mj =1 be an analogous sequence for system S and the states σ and σ ′ , where the temperatureof the heat flows are denoted ˜ T j . By the conditions on { p i } i and { q j } j ⊂ P it holds∆ S S = X i Q S ( p i ) T i and ∆ S S = X j Q S ( p j )˜ T j . (J.2)W.l.o.g. we can assume that the sequences of processes { p i } i and { q j } j act on disjoint systems˜ S and ˜ S which include S and S , respectively. This can be done due to Postulate 8 on theexistence of copies of systems. If a reservoirs was acted on by both sequences, replace it in oneof them with a copy. We now construct p ∈ P ˜ S ∨ ˜ S as the sequence of processes consisting of thejoined sequences { p i } i and { q j } j , p = (( p N ∨ id S ) ◦ · · · ◦ ( p ∨ id S )) ◦ (( q M ∨ id S ) ◦ · · · ◦ ( q ∨ id S )) , (J.3)where id S i is a fitting identity on S i . This process is well-defined. The total initial and finalstates on S = S ∨ S of p are σ = σ ∨ σ and σ ′ = σ ′ ∨ σ ′ . The sequence defining p fulfils allrequirements necessary for it to be used to determine the entropy difference of its initial and finalstates on S (reversibility, heat flows at well-defined temperature) and thus we find∆ S S = S S ∨ S ( σ ′ ) − S S ∨ S ( σ )= X i Q S ∨ S ( p i ∨ id S ) T i ! + X j Q S ∨ S ( q j ∨ id S )˜ T j = X i Q S ( p i ) T i ! + X j Q S ( q j )˜ T j = ∆ S S + ∆ S S . (J.4)In the second to last equality we used that Q S ∨ S ( p i ∨ id S ) = Q S ( p i ) since id S is an identityon S , and likewise for exchanged roles of S and S . Due to Clausius’ Theorem we know that∆ S S = ∆ S S + ∆ S S also holds for any other valid way of computing ∆ S S . Therefore, entropydifferences must be additive.By choosing the reference entropies of the systems accordingly, one can even achieve that notjust entropy differences but entropy as a state variable is additive.We conclude this section with a discussion of reversible processes with a net heat flow of zero.Consider a reversible process p ∈ P S ∨ S ′ with Q S ( p ) = 0. If S ′ ∈ R is a reservoir and W S ′ ( p ) = 0,as is the case for the processes in Clausius’ Theorem (Theorem 10.1), then p is essentially a workprocess. In this case S ′ is a catalytic system and according to Postulate 6 on the freedom ofdescription there exists a work process on S that does exactly the same, ˜ p ∈ P S , and acts on S alone. For a work processes ˜ p on S , we truly have Q S (˜ p ) T = 0.If on the other hand S ′ ∈ S is an arbitrary thermodynamic system this is not true, as thefollowing example shows. Example J.2 (Reversible net heat flow of zero) . Consider an ideal gad S undergoing a process p = p ◦ p ′ ◦ p , with two reservoirs R and R , where the temperatures of the reservoirs aredifferent, say T > T . Let p ∈ P S ∨ R be an isothermal compression of the gas, in which theheat Q S ( p ) > S to R . This heat flows at temperature T . Next, the workprocess p ′ ∈ P S is a reversible expansion of the gas, such that the final temperature of the gas is T < T . Now, apply p ∈ P S ∨ R , which is an isothermal expansion at temperature T , which issuch that the heat Q S ( p ) = − Q S ( p ) < S to R .In total, the heat Q S ( p ) = Q S ( p ) + Q S ( p ) = 0 flows into S , i.e. we have a net heat flows ofzero between S and R ∨ R . Nevertheless, according to Definition 10.2, the entropy change under97 is Q S ( p ) T + Q S ( p ) T < . (J.5)This example allows us to observe that it would not be a clever idea to define “adiabaticprocesses” on S as those thermodynamic processes with Q S = 0. Naively, having the traditionaldefinition of an adiabatic process in mind, this sounds plausible. However, as the example shows,during such processes the entropy of S can decrease, i.e., such processes fail to fulfil Theorem 10.4(Entropy Theorem), while adiabatic processes in the traditional sense are usually thought of thosewhich fulfil the Entropy Theorem.The above example also shows why the definition of entropy, Definition 10.2, asks the sequenceof processes { p i } i connecting two states on S to be such that p i ∈ P S ∨ R i and W R i ( p i ) = 0, where R i is a heat reservoir. Only in this setting a net heat flow of zero during p i implies that QT = 0is actually true. 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