Content-Type: multipart/mixed; boundary="-------------9901301419309" This is a multi-part message in MIME format. ---------------9901301419309 Content-Type: text/plain; name="99-37.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-37.comments" This is an update of 97-457 and appears in Physics Reports 310, 1-96 (1999). A summary appears in the Notices of the AMS 45}, 571-581 (1998), mp_arc 98-339. ---------------9901301419309 Content-Type: text/plain; name="99-37.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-37.keywords" Second Law, Thermodynamics, Entropy ---------------9901301419309 Content-Type: application/x-tex; name="secondlaw_esi_99.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="secondlaw_esi_99.tex" %%%%%%%%%%%%%%%%%%%% %%THIS IS A PLAIN TEX FILE. IT IS SELF CONTAINED %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%CORRECTIONS OF THE VERSION OF AUG. 06 1998 %%MADE IN ACCORD WITH THE GALLEY PROOFS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \magnification=\magstephalf %\magnification=\magstep1 \baselineskip=3ex %\baselineskip=4ex \raggedbottom \overfullrule=0pt \font\fivepoint=cmr5 %\headline={\hfill{\fivepoint EHLJY 05/Jan/99}} \input epsf.sty \def\d{{\rm d}} \def\N{{\cal N}} \def\T{{\cal T}} \def\D{{\cal D}} \def\I{{\cal I}} \def\sr{{\cal R}} \def\R{{\bf R}} \def\S{{\cal S}} \def\simt{\mathrel{\rlap{\hbox{$\sim$}}\raise.9ex\hbox{{\fivepoint $\,$T}}}} \def\sima{\mathrel{\rlap{\hbox{$\sim$}}\raise.95ex\hbox{{\fivepoint $\,$A}}}} \def\lanbox{\hbox{$\, \vrule height 0.25cm width 0.25cm depth 0.01cm \,$}} \def\uprho{\raise1pt\hbox{$\rho$}} \def\mfr#1/#2{\hbox{${{#1} \over {#2}}$}} \def\boun{$\partial A_X$} \font\subsubt=cmtt10 scaled \magstep1 \font\subt=cmbx10 scaled \magstep1 \font\tit=cmbx10 scaled \magstep2 \font\eightpoint=cmr8 \font\fivepoint=cmr5 \font\sixpoint=cmr6 \font\ninepoint=cmr9 \font\sevenpoint=cmr7 \catcode`@=11 \def\eqalignii#1{\,\vcenter{\openup1\jot \m@th \ialign{\strut\hfil$\displaystyle{##}$& $\displaystyle{{}##}$\hfil& $\displaystyle{{}##}$\hfil\crcr#1\crcr}}\,} \catcode`@=12 %this is for automatic equation numbering \def\eqlbl#1{\global\advance\equno by 1 \global\edef#1{{\number\chno.\number\equno }} (\number\chno.\number\equno )} %\eqno\eqlbl\fermat This is an example of usage \newcount\chno \chno=0 \newcount\equno \equno=0 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \centerline{\tit THE PHYSICS AND MATHEMATICS OF} \medskip \centerline{\tit THE SECOND LAW OF THERMODYNAMICS} \bigskip \bigskip \centerline{Elliott H. Lieb\footnote{$^*$}{\sixpoint Work partially supported by U.S. National Science Foundation grant PHY95-13072A01.} } \centerline{\it Departments of Physics and Mathematics, Princeton University} \centerline{\it Jadwin Hall, P.O. Box 708, Princeton, NJ 08544, USA} \bigskip \centerline{Jakob Yngvason \footnote{$^{**}$}{\sixpoint Work partially supported by the Adalsteinn Kristjansson Foundation, University of Iceland.} } \centerline{\it Institut f\"ur Theoretische Physik, Universit\"at Wien,} \centerline{\it Boltzmanngasse 5, A 1090 Vienna, Austria} \footnote{}{\baselineskip=0.6\baselineskip\hskip -\parindent\sixpoint \copyright 1997 by the authors. Reproduction of this article, by any means, is permitted for non-commercial purposes.\par} \bigskip \bigskip {\narrower\smallskip\noindent \bigskip\bigskip\noindent {\subt Abstract:} The essential postulates of classical thermodynamics are formulated, from which the second law is deduced as the principle of increase of entropy in irreversible adiabatic processes that take one equilibrium state to another. The entropy constructed here is defined only for equilibrium states and no attempt is made to define it otherwise. Statistical mechanics does not enter these considerations. One of the main concepts that makes everything work is the comparison principle (which, in essence, states that given any two states of the same chemical composition at least one is adiabatically accessible from the other) and we show that it can be derived from some assumptions about the pressure and thermal equilibrium. Temperature is derived from entropy, but at the start not even the concept of `hotness' is assumed. Our formulation offers a certain clarity and rigor that goes beyond most textbook discussions of the second law. } \bigskip \bigskip \bigskip 1998 PACS: \ 05.70.-a \smallskip Mathematical Sciences Classification (MSC) 1991 and 2000: 80A05,\ 80A10 %\centerline{\tit (DRAFT)} \bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip This paper is scheduled to appear in Physics Reports {\bf 310}, 1-96 (1999) \vfill\eject {\vbox {\ninepoint \baselineskip=0.9\baselineskip \noindent I. INTRODUCTION \item{A.} The basic Questions\dotfill 3 \item{B.} Other approaches\dotfill 6 \item{C.} Outline of the paper\dotfill 10 \item{D.} Acknowledgements\dotfill 11\break \noindent II. ADIABATIC ACCESSIBILITY AND CONSTRUCTION OF ENTROPY \item{A.} Basic concepts\dotfill 12 \item\item{1.} Systems and their state spaces\dotfill 13 \item\item{2.} The order relation\dotfill 16 \item{B.} The entropy principle\dotfill 18 \item{C.} Assumptions about the order relation\dotfill 20 \item{D.} The construction of entropy for a single system\dotfill 23 \item{E.} Construction of a universal entropy in the absence of mixing\dotfill 27 \item{F.} Concavity of entropy\dotfill 30 \item{G.} Irreversibility and Carath\'eodory's principle\dotfill 32 \item{H.} Some further results on uniqueness\dotfill 33\break \noindent III. SIMPLE SYSTEMS \item{{\phantom{A.}}} Preface\dotfill 36 \item{A.} Coordinates for simple systems\dotfill 37 \item{B.} Assumptions about simple systems\dotfill 39 \item{C.} The geometry of forward sectors\dotfill 42\break \noindent IV. THERMAL EQUILIBRIUM \item{A.} Assumptions about thermal contact\dotfill 51 \item{B.} The comparison principle in compound systems\dotfill 55 \item\item{1.} Scaled products of a single simple system\dotfill 55 \item\item{2.} Products of different simple systems\dotfill 56 \item{C.} The role of transversality\dotfill 59\break %\vfill\eject \noindent V. TEMPERATURE AND ITS PROPERTIES \item{A.} Differentiability of entropy and the definition of temperature\dotfill 62 \item{B.} The geometry of isotherms and adiabats\dotfill 68 \item{C.} Thermal equilibrium and the uniqueness of entropy\dotfill 69\break \noindent VI. MIXING AND CHEMICAL REACTIONS \item{A.} The difficulty in fixing entropy constants\dotfill 72 \item{B.} Determination of additive entropy constants\dotfill 73\break \noindent VII. SUMMARY AND CONCLUSIONS\dotfill 83\break \noindent LIST OF SYMBOLS \dotfill 88\break \noindent INDEX OF TECHNICAL TERMS \dotfill 89\break \noindent REFERENCES\dotfill 91\break }} \vfill\eject %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent {\tit I. INTRODUCTION} \bigskip The second law of thermodynamics is, without a doubt, one of the most perfect laws in physics. Any {\it reproducible} violation of it, however small, would bring the discoverer great riches as well as a trip to Stockholm. The world's energy problems would be solved at one stroke. It is not possible to find any other law (except, perhaps, for super selection rules such as charge conservation) for which a proposed violation would bring more skepticism than this one. Not even Maxwell's laws of electricity or Newton's law of gravitation are so sacrosanct, for each has measurable corrections coming from quantum effects or general relativity. The law has caught the attention of poets and philosophers and has been called the greatest scientific achievement of the nineteenth century. Engels disliked it, for it supported opposition to dialectical materialism, while Pope Pius XII regarded it as proving the existence of a higher being (Bazarow, 1964, Sect. 20). \bigskip\noindent {\subt A. The basic questions} \bigskip In this paper we shall attempt to formulate the essential elements of {\it classical } thermodynamics of equilibrium states and deduce from them the second law as the principle of the increase of entropy. `Classical' means that there is {\it no mention of statistical mechanics here} and `equilibrium' means that we deal only with states of systems in equilibrium and do not attempt to define quantities such as entropy and temperature for systems not in equilibrium. This is not to say that we are concerned only with `thermostatics' because, as will be explained more fully later, arbitrarily violent processes are allowed to occur in the passage from one equilibrium state to another. Most students of physics regard the subject as essentially perfectly understood and finished, and concentrate instead on the statistical mechanics from which it ostensibly can be derived. But many will admit, if pressed, that thermodynamics is something that they are sure that someone else understands and they will confess to some misgiving about the logic of the steps in traditional presentations that lead to the formulation of an entropy function. If classical thermodynamics is the most perfect physical theory it surely deserves a solid, unambiguous foundation free of little pictures involving unreal Carnot cycles and the like. [For examples of `un-ordinary' Carnot cycles see (Truesdell and Bharatha 1977, p.~48).] There are two aims to our presentation. One is frankly pedagogical, i.e., to formulate the foundations of the theory in a clear and unambiguous way. The second is to formulate equilibrium thermodynamics as an `ideal physical theory', which is to say a theory in which there are well defined mathematical constructs and well defined rules for translating physical reality into these constructs; having done so the mathematics then grinds out whatever answers it can and these are then translated back into physical statements. The point here is that while `physical intuition' is a useful guide for formulating the mathematical structure and may even be a source of inspiration for constructing mathematical proofs, it should not be necessary to rely on it once the initial `translation' into mathematical language has been given. These goals are not new, of course; see e.g., (Duistermaat, 1968), (Giles, 1964, Sect. 1.1) and (Serrin, 1986, Sect. 1.1). Indeed, it seems to us that many formulations of thermodynamics, including most textbook presentations, suffer from mixing the physics with the mathematics. Physics refers to the real world of experiments and results of measurement, the latter quantified in the form of numbers. Mathematics refers to a logical structure and to rules of calculation; usually these are built around numbers, but not always. Thus, mathematics has two functions: one is to provide a transparent logical structure with which to view physics and inspire experiment. The other is to be like a mill into which the miller pours the grain of experiment and out of which comes the flour of verifiable predictions. It is astonishing that this paradigm works to perfection in thermodynamics. (Another good example is Newtonian mechanics, in which the relevant mathematical structure is the calculus.) Our theory of the second law concerns the mathematical structure, primarily. As such it starts with some axioms and proceeds with rules of logic to uncover some non-trivial theorems about the existence of entropy and some of its properties. We do, however, explain how physics leads us to these particular axioms and we explain the physical applicability of the theorems. As noted in I.C below, we have a total of 15 axioms, which might seem like a lot. We can assure the reader that any other mathematical structure that derives entropy with minimal assumptions will have at least that many, and usually more. (We could roll several axioms into one, as others often do, by using sub-headings, e.g., our A1-A6 might perfectly well be denoted by A1(i)-(vi).) The point is that we leave nothing to the imagination or to silent agreement; it is all laid out. It must also be emphasized that our desire to clarify the structure of classical equilibrium thermodynamics is not merely pedagogical and not merely nit-picking. If the law of entropy increase is ever going to be derived from statistical mechanics---a goal that has so far eluded the deepest thinkers---then it is important to be absolutely clear about what it is that one wants to derive. Many attempts have been made in the last century and a half to formulate the second law precisely and to quantify it by means of an entropy function. Three of these formulations are classic (Kestin, 1976), (see also Clausius (1850), Thomson (1849)) and they can be paraphrased as follows: \smallskip {\sl Clausius:\/} No process is possible, the sole result of which is that heat is transferred from a body to a hotter one. {\sl Kelvin (and Planck):\/} No process is possible, the sole result of which is that a body is cooled and work is done. {\sl Carath\'eodory:\/} In any neighborhood of any state there are states that cannot be reached from it by an adiabatic process. \smallskip The crowning glory of thermodynamics is the quantification of these statements by means of a precise, measurable quantity called entropy. There are two kinds of problems, however. One is to give a precise meaning to the words above. What is `heat'? What is `hot' and `cold'? What is `adiabatic'? What is a `neighborhood'? Just about the only word that is relatively unambiguous is `work' because it comes from mechanics. The second sort of problem involves the rules of logic that lead from these statements to an entropy. Is it really necessary to draw pictures, some of which are false, or at least not self evident? What are all the hidden assumptions that enter the derivation of entropy? For instance, we all know that discontinuities can and do occur at phase transitions, but almost every presentation of classical thermodynamics is based on the differential calculus (which presupposes continuous derivatives), especially (Carath\'eodory, 1925) and (Truesdell-Bharata, 1977, p.xvii). We note, in passing, that the Clausius, Kelvin-Planck and Carath\'eodory formulations are all assertions about {\it impossible} processes. Our formulation will rely, instead, mainly on assertions about {\it possible} processes and thus is noticeably different. At the end of Section VII, where everything is succintly summarized, the relationship of these approaches is discussed. This discussion is left to the end because it it cannot be done without first presenting our results in some detail. Some readers might wish to start by glancing at Section VII. Of course we are neither the first nor, presumably, the last to present a derivation of the second law (in the sense of an entropy principle) that pretends to remove all confusion and, at the same time, to achieve an unparalleled precision of logic and structure. Indeed, such attempts have multiplied in the past three or four decades. These other theories, reviewed in Sect. I.B, appeal to their creators as much as ours does to us and we must therefore conclude that ultimately a question of `taste' is involved. It is not easy to classify other approaches to the problem that concerns us. We shall attempt to do so briefly, but first let us state the problem clearly. Physical systems have certain states (which always mean equilibrium states in this paper) and, by means of certain actions, called {\it adiabatic processes}, it is possible to change the state of a system to some other state. (Warning: The word `adiabatic' is used in several ways in physics. Sometimes it means `slow and gentle', which might conjure up the idea of a quasi-static process, but this is certainly not our intention. The usage we have in the back of our minds is `without exchange of heat', but we shall avoid defining the word `heat'. The operational meaning of `adiabatic' will be defined later on, but for now the reader should simply accept it as singling out a particular class of processes about which certain physically interesting statements are going to be made.) Adiabatic processes do not have to be very gentle, and they certainly do not have to be describable by a curve in the space of equilibrium states. One is allowed, like the gorilla in a well-known advertisement for luggage, to jump up and down on the system and even dismantle it temporarily, provided the system returns to some equilibrium state at the end of the day. In thermodynamics, unlike mechanics, not all conceivable transitions are adiabatic and it is a nontrivial problem to characterize the allowed transitions. We shall characterize them as transitions that have no {\it net} effect on other systems except that energy has been exchanged with a mechanical source. The truly remarkable fact, which has many consequences, is that for every system there is a function, $S$, on the space of its (equilibrium) states, with the property that one can go adiabatically from a state $X$ to a state $Y$ if and only if $S(X) \leq S(Y)$. This, in essence, is the `entropy principle' (EP) (see subsection II.B). The $S$ function can clearly be multiplied by an arbitrary constant and still continue to do its job, and thus it is not at all obvious that the function $S_1$ for system $1$ has anything to do with the function $S_2$ for system $2$. The second remarkable fact is that the $S$ functions for all the thermodynamic systems in the universe can be simultaneously calibrated (i.e., the multiplicative constants can be determined) in such a way that the entropies are {\it additive}, i.e., the $S$ function for a compound system is obtained merely by adding the $S$ functions of the individual systems, $S_{1,2} = S_1+S_2$. (`Compound' does not mean chemical compound; a compound system is just a collection of several systems.) To appreciate this fact it is necessary to recognize that the systems comprising a compound system can interact with each other in several ways, and therefore the possible adiabatic transitions in a compound are far more numerous than those allowed for separate, isolated systems. Nevertheless, the increase of the function $S_1+S_2$ continues to describe the adiabatic processes exactly---neither allowing more nor allowing less than actually occur. The statement $S_1(X_1)+S_2(X_2)\leq S_1(X'_1)+S_2(X'_2)$ does not require $S_1(X_1)\leq S_1(X'_1)$. The main problem, from our point of view, is this: What properties of adiabatic processes permit us to construct such a function? To what extent is it unique? And what properties of the interactions of different systems in a compound system result in additive entropy functions? The existence of an entropy function can be discussed in principle, as in Section II, without parametrizing the equilibrium states by quantities such as energy, volume, etc.. But it is an additional fact that when states are parametrized in the conventional ways then the derivatives of $S$ exist and contain all the information about the equation of state, e.g., the temperature $T$ is defined by $\partial S(U,V)/ \partial U|_V^{\phantom Y} = 1/T$. In our approach to the second law temperature is never formally invoked until the very end when the differentiability of $S$ is proved---not even the more primitive relative notions of `hotness' and `coldness' are used. The priority of entropy is common in statistical mechanics and in some other approaches to thermodynamics such as in (Tisza, 1966) and (Callen, 1985), but the elimination of hotness and coldness is not usual in thermodynamics, as the formulations of Clausius and Kelvin show. The laws of thermal equilibrium (Section V), in particular the zeroth law of thermodynamics, do play a crucial role for us by relating one system to another (and they are ultimately responsible for the fact that entropies can be adjusted to be additive), but thermal equilibrium is only an equivalence relation and, in our form, it is not a statement about hotness. It seems to us that temperature is far from being an `obvious' physical quantity. It emerges, finally, as a derivative of entropy, and unlike quantities in mechanics or electromagnetism, such as forces and masses, it is not vectorial, i.e., it cannot be added or multiplied by a scalar. Even pressure, while it cannot be `added' in an unambiguous way, can at least be multiplied by a scalar. (Here, we are not speaking about changing a temperature scale; we mean that once a scale has been fixed, it does not mean very much to multiply a given temperature, e.g., the boiling point of water, by the number 17. Whatever meaning one might attach to this is surely not independent of the chosen scale. Indeed, is $T$ the right variable or is it $1/T$? In relativity theory this question has led to an ongoing debate about the natural quantity to choose as the fourth component of a four-vector. On the other hand, it does mean something unambiguous, to multiply the pressure in the boiler by 17. Mechanics dictates the meaning.) Another mysterious quantity is `heat'. No one has ever seen heat, nor will it ever be seen, smelled or touched. Clausius wrote about ``the kind of motion we call heat", but thermodynamics---either practical or theoretical---does not rely for its validity on the notion of molecules jumping around. There is no way to measure heat flux directly (other than by its effect on the source and sink) and, while we do not wish to be considered antediluvian, it remains true that `caloric' accounts for physics at a macroscopic level just as well as `heat' does. The reader will find no mention of heat in our derivation of entropy, except as a mnemonic guide. To conclude this very brief outline of the main conceptual points, the concept of {\it convexity} has to be mentioned. It is well known, as Gibbs (Gibbs 1928), Maxwell and others emphasized, that thermodynamics without convex functions (e.g., free energy per unit volume as a function of density) may lead to unstable systems. (A good discussion of convexity is in (Wightman, 1979).) Despite this fact, convexity is almost invisible in most fundamental approaches to the second law. In our treatment it is {\it essential} for the description of simple systems in Section III, which are the building blocks of thermodynamics. The concepts and goals we have just enunciated will be discussed in more detail in the following sections. The reader who impatiently wants a quick survey of our results can jump to Section VII where it can be found in capsule form. We also draw the readers attention to the article (Lieb-Yngvason 1998), where a summary of this work appeared. Let us now turn to a brief discussion of other modes of thought about the questions we have raised. \bigskip\bigskip\noindent {\subt B. Other approaches} \bigskip The simplest solution to the problem of the foundation of thermodynamics is perhaps that of Tisza (1966), and expanded by Callen (1985) (see also (Guggenheim, 1933)), who, following the tradition of Gibbs (1928), postulate the existence of an additive entropy function from which all equilibrium properties of a substance are then to be derived. This approach has the advantage of bringing one quickly to the applications of thermodynamics, but it leaves unstated such questions as: What physical assumptions are needed in order to insure the existence of such a function? By no means do we wish to minimize the importance of this approach, for the manifold implications of entropy are well known to be non-trivial and highly important theoretically and practically, as Gibbs was one of the first to show in detail in his great work (Gibbs, 1928). Among the many foundational works on the existence of entropy, the most relevant for our considerations and aims here are those that we might, for want of a better word, call `order theoretical' because the emphasis is on the derivation of entropy from postulated properties of adiabatic processes. This line of thought goes back to Carath\'eodory (1909 and 1925), although there are some precursors (see Planck, 1926) and was particularly advocated by (Born, 1921 and 1964). This basic idea, if not Carath\'eodory's implementation of it with differential forms, was developed in various mutations in the works of Landsberg (956), Buchdahl (1958, 1960, 1962, 1966), Buchdahl and Greve (1962), Falk and Jung (1959), Bernstein (1960), Giles (964), Cooper (1967), Boyling, (1968, 1972), Roberts and Luce (1968), Duistermaat (1968), Hornix (1968), Rastall (1970), Zeleznik (1975) and Borchers (1981). The work of Boyling (1968, 1972), which takes off from the work of Bernstein (1960) is perhaps the most direct and rigorous expression of the original Carth\'eodory idea of using differential forms. See also the discussion in Landsberg (1970). Planck (1926) criticized some of Carath\'eodory's work for not identifying processes that are not adiabatic. He suggested basing thermodynamics on the fact that `rubbing' is an adiabatic process that is not reversible, an idea he already had in his 1879 dissertation. {}From this it follows that while one can undo a rubbing operation by some means, one cannot do so adiabatically. We derive this principle of Planck from our axioms. It is very convenient because it means that in an adiabatic process one can effectively add as much `heat' (colloquially speaking) as one wishes, but the one thing one cannot do is subtract heat, i.e., use a `refrigerator'. Most authors introduce the idea of an `empirical temperature', and later derive the absolute temperature scale. In the same vein they often also introduce an `empirical entropy' and later derive a `metric', or additive, entropy, e.g., (Falk and Jung, 1959) and (Buchdahl, 1958, et seq., 1966), (Buchdahl and Greve, 1962), (Cooper, 1967). We avoid all this; one of our results, as stated above, is the derivation of absolute temperature directly, without ever mentioning even `hot' and `cold'. One of the key concepts that is eventually needed, although it is not obvious at first, is that of the comparison principle (or hypothesis), (CH). It concerns classes of thermodynamic states and asserts that for any two states $X$ and $Y$ within a class one can either go {\it adiabatically} from $X$ to $Y$, which we write as $$ X \prec Y, $$ (pronounced ``$X$ precedes $Y$" or ``$Y$ follows $X$") or else one can go from $Y$ to $X$, i.e., $Y \prec X$. Obviously, this is not always possible (we cannot transmute lead into gold, although we {\it can} transmute hydrogen plus oxygen into water), so we would like to be able to break up the universe of states into equivalence classes, inside each of which the hypothesis holds. It turns out that the key requirement for an equivalence relation is that if $X\prec Y$ and $Z\prec Y$ then either $X\prec Z$ or $Z\prec X$. Likewise, if $Y\prec X$ and $Y\prec Z$ by then either $X \prec Z$ or $Z\prec X$. We find this first clearly stated in Landsberg (1956) and it is also found in one form or another in many places, see e.g., (Falk and Jung, 1959), (Buchdahl, 1958, 1962), (Giles, 1964). However, all authors, except for Duistermaat (1968), seem to take this postulate for granted and do not feel obliged to obtain it from something else. One of the central points in our work is to {\it derive} the comparison hypothesis. This is discussed further below. The formulation of the second law of thermodynamics that is closest to ours is that of Giles (Giles, 1964). His book is full of deep insights and we recommend it highly to the reader. It is a classic that does not appear to be as known and appreciated as it should. His derivation of entropy from a few postulates about adiabatic processes is impressive and was the starting point for a number of further investigations. The overlap of our work with Giles's is only partial (the foundational parts, mainly those in our section II) and where there is overlap there are also differences. To define the entropy of a state, the starting point in both approaches is to let a process that by itself would be adiabatically impossible work against another one that is possible, so that the total process is adiabatically possible. The processes used by us and by Giles are, however, different; for instance Giles uses a fixed external calibrating system, whereas we define the entropy of a state by letting a system interact with a copy of itself. ( According to R.\ E.\ Barieau (quoted in (Hornix, 1967-1968)) Giles was unaware of the fact that predecessors of the idea of an external entropy meter can be discerned in (Lewis and Randall, 1923).) To be a bit more precise, Giles uses a standard process as a reference and counts how many times a reference process has to be repeated to counteract some multiple of the process whose entropy (or rather `irreversibility') is to be determined. In contrast, we construct the entropy function for a single system in terms of the amount of substance in a reference state of `high entropy' that can be converted into the state under investigation with the help of a reference state of `low entropy'. (This is reminiscent of an old definition of heat by Laplace and Lavoisier (quoted in (Borchers, 1981)) in terms of the amount of ice that a body can melt.) We give a simple formula for the entropy; Giles's definition is less direct, in our view. However, when we calibrate the entropy functions of different systems with each other, we do find it convenient to use a third system as a `standard' of comparison. Giles' work and ours use very little of the calculus. Contrary to almost all treatments, and contrary to the assertion (Truesdell-Bharata, 1977) that the differential calculus is the appropriate tool for thermodynamics, we and he agree that entropy and its essential properties can best be described by maximum principles instead of equations among derivatives. To be sure, real analysis does eventually come into the discussion, but only at an advanced stage (Sections III and V in our treatment). In Giles, too, temperature appears as a totally derived quantity, but Giles's derivation requires some assumptions, such as differentiability of the entropy. We prove the required differentiability from natural assumptions about the pressure. Among the differences, it can be mentioned that the `cancellation law', which plays a key role in our proofs, is taken by Giles to be an axiom, whereas we derive it from the assumption of `stability', which is common to both approaches (see Section II for definitions). The most important point of contact, however, and at the same time the most significant difference, concerns the comparison hypothesis which, as we emphasized above, is a concept that plays an essential role, although this may not be apparent at first. This hypothesis serves to divide the universe nicely into equivalence classes of mutually accessible states. Giles takes the comparison property as an axiom and does not attempt to justify it from physical premises. The main part of our work is devoted to just that justification, and to inquire what happens if it is violated. (There is also a discussion of this point in (Giles, 1964, Sect 13.3) in connection with hysteresis.) To get an idea of what is involved, note that we can easily go adiabatically from cold hydrogen plus oxygen to hot water and we can go from ice to hot water, but can we go either from the cold gases to ice or the reverse---as the comparison hypothesis demands? It would appear that the only real possibility, if there is one at all, is to invoke hydrolysis to dissociate the ice, but what if hydrolysis did not exist? In other examples the requisite machinery might not be available to save the comparison hypothesis. For this reason we prefer to derive it, when needed, from properties of `simple systems' and not to invoke it when considering situations involving variable composition or particle number, as in Section VI. Another point of difference is the fact that convexity is central to our work. Giles mentions it, but it is not central in his work perhaps because he is considering more general systems than we do. To a large extent convexity eliminates the need for explicit topological considerations about state spaces, which otherwise has to be put in `by hand'. Further developments of the Giles' approach are in (Cooper, 1967), (Roberts and Luce, 1968) and (Duistermaat, 1968). Cooper assumes the existence of an empirical temperature and introduces topological notions which permits certain simplifications. Roberts and Luce have an elegant formulation of the entropy principle, which is mathematically appealing and is based on axioms about the order relation, $\prec$, (in particular the comparison principle, which they call conditional connectedness), but these axioms are not physically obvious, especially axiom 6 and the comparison hypothesis. Duistermaat is concerned with general statements about morphisms of order relations, thermodynamics being but one application. A line of thought that is entirely different from the above starts with Carnot (1824) and was amplified in the classics of Clausius and Kelvin (cf.\ (Kestin, 1976)) and many others. It has dominated most textbook presentations of thermodynamics to this day. The central idea concerns cyclic processes and the efficiency of heat engines; heat and empirical temperature enter as primitive concepts. Some of the modern developments along these lines go well beyond the study of equilibrium states and cyclic processes and use some sophisticated mathematical ideas. A representative list of references is Arens (1963), Coleman and Owen (1974, 1977), Coleman, Owen and Serrin (1981), Dafermos (1979), Day (1987, 1988), Feinberg and Lavine (1983), Green and Naghdi (1978), Gurtin (1975), Man (1989), Owen (1984), Pitteri (1982), Serrin (1979, 1983, 1986), Silhavy (1997), Truesdell and Bharata (1977), Truesdell (1980, 1984). Undoubtedly this approach is important for the practical analysis of many physical systems, but we neither analyze nor take a position on the validity of the claims made by its proponents. Some of these are, quite frankly, highly polemical and are of two kinds: claims of mathematical rigor and physical exactness on the one hand and assertions that these qualities are lacking in other approaches. See, for example, Truesdell's contribution in (Serrin, 1986, Chapter 5). The chief reason we omit discussion of this approach is that it does not directly address the questions we have set for ourselves. Namely, using only the existence of equilibrium states and the existence of certain processes that take one into another, when can it be said that the list of allowed processes is characterized {\it exactly} by the increase of an entropy function? Finally, we mention an interesting recent paper by Macdonald (1995) that falls in neither of the two categories described above. In this paper \lq heat\rq\ and \lq reversible processes\rq\ are among the primitive concepts and the existence of reversible processes linking any two states of a system is taken as a postulate. Macdonald gives a simple definition of entropy of a state in terms of the maximal amount of heat, extracted from an infinite reservoir, that the system absorbs in processes terminating in the given state. The reservoir thus plays the role of an entropy meter. The further development of the theory along these lines, however, relies on unstated assumptions about differentiability of the so defined entropy that are not entirely obvious. %\vfill\eject \bigskip\noindent {\subt C. Outline of the paper} \bigskip In Section II we formally introduce the relation $\prec$ and explain it more fully, but it is to be emphasized, in connection with what was said above about an ideal physical theory, that $\prec$ has a well defined mathematical meaning independent of the physical context in which it may be used. The concept of an entropy function, which characterizes this accessibility relation, is introduced next; at the end of the section it will be shown to be unique up to a trivial affine transformation of scale. We show that the existence of such a function is {\it equivalent} to certain simple properties of the relation $\prec$, which we call axioms A1 to A6 and the `hypothesis' CH. Any formulation of thermodynamics must implicitly contain these axioms, since they are equivalent to the entropy principle, and it is not surprising that they can be found in Giles, for example. We do believe that our presentation has the virtue of directness and clarity, however. We give a simple formula for the entropy, entirely in terms of the relation $\prec$ without invoking Carnot cycles or any other gedanken experiment. Axioms A1 to A6 are highly plausible; it is CH (the comparison hypothesis) that is not obvious but is {\it crucial} for the existence of entropy. We call it a hypothesis rather than an axiom because our ultimate goal is to derive it from some additional axioms. In a certain sense it can be said that the rest of the paper is devoted to {\it deriving} the comparison hypothesis from plausible assumptions. The content of Section II, i.e., the derivation of an entropy function, stands on its own feet; the implementation of it via CH is an independent question and we feel it is pedagogically significant to isolate the main input in the derivation from the derivation itself. Section III introduces one of our most novel contributions. We {\it prove } that comparison holds for the states inside certain systems which we call {\it simple systems}. To obtain it we need a few new axioms, S1 to S3. These axioms are mainly about {\it mechanical} processes, and not about the entropy. In short, our most important assumptions concern the continuity of the generalized pressure and the existence of irreversible processes. Given the other axioms, the latter is equivalent to Carath\'eodory's principle. The comparison hypothesis, CH, does not concern simple systems alone, but also their products, i.e., compound systems composed of possibly interacting simple systems. In order to compare states in different simple systems (and, in particular, to calibrate the various entropies so that they can be added together) the notion of a {\it thermal join} is introduced in Section IV. This concerns states that are usually said to be in thermal equilibrium, but we emphasize that temperature is not mentioned. The thermal join is, by assumption, a simple system and, using the zeroth law and three other axioms about the thermal join, we reduce the comparison hypothesis among states of {\it compound systems} to the previously derived result for simple systems. This derivation is another novel contribution. With the aid of the thermal join we can prove that the multiplicative constants of the entropies of all systems can be chosen so that entropy is additive, i.e., the sum of the entropies of simple systems gives a correct entropy function for compound systems. This entropy correctly describes all adiabatic processes in which there is no change of the constituents of compound systems. What remains elusive are the additive constants, discussed in Section VI. These are important when changes (due to mixing and chemical reactions) occur. Section V establishes the continuous differentiability of the entropy and defines inverse temperature as the derivative of the entropy with respect to the energy---in the usual way. No new assumptions are needed here. The fact that the entropy of a simple system is determined uniquely by its adiabats and isotherms is also proved here. In Section VI we discuss the vexed question of comparing states of systems that differ in constitution or in quantity of matter. How can the entropy of a bottle of water be compared with the sum of the entropies of a container of hydrogen and a container of oxygen? To do so requires being able to transform one into the other, but this may not be easy to do reversibly. The usual theoretical underpinning here is the use of semi-permeable membranes in a `van't Hoff box' but such membranes are usually far from perfect physical objects, if they exist at all. We examine in detail just how far one can go in determining the {\it additive} constants for the entropies of different systems in the the real world in which perfect semi-permeable membranes do not exist. In Section VII we collect all our axioms together and summarize our results briefly. %\vfill\eject \bigskip\noindent {\subt D. Acknowledgements} \bigskip We are deeply indebted to Jan Philip Solovej for many useful discussions and important insights, especially in regard to Sections III and VI. Our thanks also go to Fredrick Almgren for helping us understand convex functions, to Roy Jackson, Pierluigi Contucci, Thor Bak and Bernhard Baumgartner for critically reading our manuscript and to Martin Kruskal for emphasizing the importance of Giles' book to us. We thank Robin Giles for a thoughtful and detailed review with many helpful comments. We thank John C. Wheeler for a clarifying correspondence about the relationship between adiabatic processes, as usually understood, and our definition of adiabatic accessibility. Some of the rough spots in our story were pointed out to us by various people during various public lectures we gave, and that is also very much appreciated. A significant part of this work was carried out at Nordita in Copenhagen and at the Erwin Schr\"odinger Institute in Vienna; we are grateful for their hospitality and support. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \vfill\eject %%%%%%%%%%%%%%%%%%%% \noindent \leftline {\tit II. ADIABATIC ACCESSIBILITY } \smallskip \leftline {\tit \phantom{Ix}\enspace AND CONSTRUCTION OF ENTROPY } \bigskip Thermodynamics concerns systems, their states and an order relation among these states. The order relation is that of {\bf adiabatic accessibility}, which, physically, is defined by processes whose only net effect on the surroundings is exchange of energy with a mechanical source. The glory of classical thermodynamics is that there always is an {\it additive} function, called {\bf entropy}, on the state space of any system, that {\it exactly} describes the order relation in terms of the increase of entropy. Additivity is very important physically and is certainly not obvious; it tells us that the entropy of a compound system composed of two systems that can interact and exchange energy with each other is the sum of the individual entropies. This means that the pairs of states accessible from a given pair of states, which is a far larger set than merely the pairs individually accessible by the systems in isolation, is given by studying the sum of the individual entropy functions. This is even more surprising when we consider that the individual entropies each have undetermined multiplicative constants; there is a way to adjust, or calibrate the constants in such a way that the sum gives the correct result for the accessible states---and this can be done once and for all so that the same calibration works for all possible pairs of systems. Were additivity to fail we would have to rewrite the steam tables every time a new steam engine is invented. The other important point about entropy, which is often overlooked, is that entropy not only increases, but entropy also tells us exactly which processes are adiabatically possible in any given system; states of high entropy in a system are {\it always } accessible from states of lower entropy. As we shall see this is generally true but it could conceivably fail when there are chemical reactions or mixing, as discussed in Section VI. In this section we begin by defining these basic concepts more precisely, and then we present the entropy principle. Next, we introduce certain axioms, A1-A6, relating the concepts. All these axioms are completely intuitive. However, one other assumption---which we call the {\it comparison hypothesis}---is needed for the construction of entropy. It is not at all obvious physically, but it is an essential part of conventional thermodynamics. Eventually, in Sections III and IV, this hypothesis will be {\it derived} from some more detailed physical considerations. For the present, however, this hypothesis will be assumed and, using it, the existence of an entropy function will be proved. We also discuss the extent to which the entropy function is uniquely determined by the order relation; the comparison hypothesis plays a key role here. The existence of an entropy function is equivalent to axioms A1-A6 in conjunction with CH, neither more nor less is required. The state space need not have any structure besides the one implied by the order relation. However, state spaces parametrized by the energy and work coordinates have an additional, convex structure, which implies concavity of the entropy, provided that the formation of convex combination of states is an adiabatic process. We add this requirement as axiom A7 to our list of general axioms about the order relation. The axioms in this section are so general that they encompass situations where {\it all} states in a whole neighborhood of a given state are adiabatically accessible from it. {\bf Carath\'eodory's principle} is the statement that this does {\it not} happen for physical thermodynamic systems. In contrast, ideal mechanical systems have the property that every state is accessible from every other one (by mechanical means alone), and thus the world of mechanical systems will trivially obey the entropy principle in the sense that every state has the same entropy. In the last subsection we discuss the connection between Carath\'eodory's principle and the existence of irreversible processes starting from a given state. This principle will again be invoked when, in Section III, we derive the comparison hypothesis for simple thermodynamic systems. Temperature will not be used in this section, not even the notion of `hot' and `cold'. There will be no cycles, Carnot or otherwise. The entropy only depends on, and is defined by the order relation. Thus, while the approach given here is not the only path to the second law, it has the advantage of a certain simplicity and clarity that at least has pedagogic and conceptual value. We ask the reader's patience with our syllogisms, the point being that everything is here clearly spread out in full view. There are no hidden assumptions, as often occur in many textbook presentations. Finally, we hope that the reader will not be confused by our sometimes lengthy asides about the motivation and heuristic meaning of our various definitions and theorems. We also hope these remarks will not be construed as part of the structure of the second law. The definitions and theorems are self-contained, as we state them, and the remarks that surround them are intended only as a helpful guide. \bigskip\noindent {\subt A. Basic concepts } \bigskip \noindent {\subsubt 1. Systems and their state spaces} \medskip Physically speaking a thermodynamic {\it system} consists of certain specified amounts of different kinds of matter; it might be divisible into parts that can interact with each other in a specified way. A special class of systems called simple systems will be discussed in the next chapter. In any case the possible interaction of the system with its surroundings is specified. It is a ``black box" in the sense that we do not need to know what is in the box, but only its response to exchanging energy, volume, etc. with other systems. The states of a system to be considered here are {\it always} equilibrium states, but the equilibrium may depend upon the existence of internal barriers in the system. Intermediate, non-equilibrium states that a system passes through when changing from one equilibrium state to another will not be considered. The entropy of a system not in equilibrium may, like the temperature of such a system, have a meaning as an approximate and useful concept, but this is not our concern in this treatment. Our systems can be quite complicated and the outside world can act on them in several ways, e.g., by changing the volume and magnetization, or removing barriers. Indeed, we are allowed to chop a system into pieces violently and reassemble them in several ways, each time waiting for the eventual establishment of equilibrium. Our systems must be macroscopic, i.e, not too small. Tiny systems (atoms, molecules, DNA) exist, to be sure, but we cannot describe their equilibria thermodynamically, i.e., their equilibrium states cannot be described in terms of the simple coordinates we use later on. There is a gradual shift from tiny systems to macroscopic ones, and the empirical fact is that large enough systems conform to the axioms given below. At some stage a system becomes `macroscopic'; we do not attempt to explain this phenomenon or to give an exact rule about which systems are `macroscopic'. On the other hand, systems that are too large are also ruled out because gravitational forces become important. Two suns cannot unite to form one bigger sun with the same properties (the way two glasses of water can unite to become one large glass of water). A star with two solar masses is intrinsically different from a sun of one solar mass. In principle, the two suns could be kept apart and regarded as one system, but then this would only be a `constrained' equilibrium because of the gravitational attraction. In other words the conventional notions of `extensivity' and `intensivity' fail for cosmic bodies. Nevertheless, it is possible to define an entropy for such systems by measuring its effect on some standard body. Giles' method is applicable, and our formula (2.20) in Section II.E (which, in the context of our development, is used only for calibrating the entropies defined by (2.14) in Section II.D, but which could be taken as an independent definition) would allow it, too. (The `nice' systems that do satisfy size-scaling are called `perfect' by Giles.) The entropy, so defined, would satisfy additivity but not extensivity, in the `entropy principle' of Section II.B. However, to prove this would requires a significant enhancement of the basic axioms. In particular, we would have to take the comparison hypothesis, CH, for all systems as an axiom --- as Giles does. It is left to the interested reader to carry out such an extension of our scheme. A basic operation is {\bf composition} of two or more systems to form a new system. Physically, this simply means putting the individual systems side by side and regarding them as one system. We then speak of each system in the union as a {\bf subsystem}. The subsystems may or may not interact for a while, by exchanging heat or volume for instance, but the important point is that a state of the total system (when in equilibrium) is described completely by the states of the subsystems. {}From the mathematical point of view a system is just a collection of points called a {\bf state space}, usually denoted by $\Gamma$. The individual points of a state space are called {\bf states} and are denoted here by capital Roman letters, $X, Y, Z, $ etc. {}From the next section on we shall build up our collection of states satisfying our axioms from the states of certain special systems, called {\it simple systems}. (To jump ahead for the moment, these are systems with one or more work coordinates but with only one energy coordinate.) In the present section, however, the manner in which states are described (i.e., the coordinates one uses, such as energy and volume, etc.) are of no importance. Not even topological properties are assumed here about our systems, as is often done. In a sense it is amazing that much of the second law follows from certain abstract properties of the relation among states, independent of physical details (and hence of concepts such as Carnot cycles). In approaches like Giles', where it is taken as an axiom that comparable states fall into equivalence classes, it is even possible to do without the system concept altogether, or define it simply as an equivalence class of states. In our approach, however, one of the main goals is to derive the property which Giles takes as an axiom, and systems are basic objects in our axiomatic scheme. Mathematically, the composition of two spaces, $\Gamma_1 $ and $\Gamma_2 $ is simply the Cartesian product of the state spaces $\Gamma_1 \times \Gamma_2$. In other words, the states in $\Gamma_1 \times \Gamma_2 $ are pairs $(X_1,X_2)$ with $X_1 \in \Gamma_1 $ and $X_2 \in \Gamma_2 $. {}From the physical interpretation of the composition it is clear that the two spaces $\Gamma_1 \times \Gamma_2 $ and $\Gamma_2 \times \Gamma_1$ are to be identified. Likewise, when forming multiple compositions of state spaces, the order and the grouping of the spaces is immaterial. Thus $(\Gamma_1 \times \Gamma_2)\times \Gamma_3$, $\Gamma_1 \times (\Gamma_2\times \Gamma_3)$ and $\Gamma_1 \times \Gamma_2\times \Gamma_3$ are to be identified as far as composition of state spaces is concerned. Strictly speaking, a symbol like $(X_1,\dots , X_{N})$ with states $X_{i}$ in state spaces $\Gamma_{i}$, $i=1,\dots,N$ thus stands for an equivalence class of $n$-tuples, corresponding to the different groupings and permutations of the state spaces. Identifications of this type are not uncommon in mathematics (the formation of direct sums of vector spaces is an example). A further operation we shall assume is the formation of {\bf scaled copies} of a given system whose state space is $\Gamma$. If $t>0$ is some fixed number (the scaling parameter) the state space $\Gamma^{(t)}$ consists of points denoted $tX$ with $X\in \Gamma$. On the abstract level $tX$ is merely a symbol, or mnemonic, to define points in $\Gamma^{(t)}$, but the symbol acquires meaning through the axioms given later in Sect.\ II.C. In the physical world, and from Sect.\ III onward, the state spaces will always be subsets of some $\R^n$ (parametrized by energy, volume, etc.). In this case $tX$ has the concrete representation as the product of the real number $t$ and the vector $X\in\R^n$. Thus in this case $\Gamma^{(t)}$ is simply the image of the set $\Gamma\subset \R^n$ under scaling by the real parameter $t$. Hence, we shall sometimes denote $\Gamma^{(t)}$ by $t\Gamma$. Physically, $\Gamma^{(t)}$ is interpreted as the state space of a system that has the same properties as the system with state space $\Gamma$, except that the amount of each chemical substance in the system has been scaled by the factor $t$ and the range of extensive variables like energy, volume etc. has been scaled accordingly. Likewise, $tX$ is obtained from $X$ by scaling energy, volume etc., but also the matter content of a state $X$ is scaled by the parameter $t$. {}From this physical interpretation it is clear that $s(tX)=(st)X$ and ${(\Gamma^{(t)})}^{(s)}=\Gamma^{(st)}$ and we take these relations also for granted on the abstract level. The same apples to the identifications $\Gamma^{(1)}=\Gamma$ and $1X=X$, and also $(\Gamma_{1}\times\Gamma_{2})^{(t)}=\Gamma_{1}^{(t)} \times\Gamma_{2}^{(t)}$ and $t(X,Y)=(tX,tY)$. The operation of forming compound states is thus an associative and commutative binary operation on the set of all states, and the group of positive real numbers acts by the scaling operation on this set in a way compatible with the binary operation and the multiplicative structure of the real numbers. The same is true for the set of all state spaces. {}From an algebraic point of view the simple systems, to be discussed in Section III, are a basis for this algebraic structure. While the relation between $\Gamma$ and $\Gamma^{(t)}$ is physically and intuitively fairly obvious, there can be surprises. Electromagnetic radiation in a cavity (`photon gas'), which is mentioned after (2.6), is an interesting case; the two state spaces $\Gamma$ and $\Gamma^{(t)}$ and the thermodynamic functions on these spaces are identical in this case! Moreover, the two spaces are physically indistinguishable. This will be explained in more detail in Section II.B. The formation of scaled copies involves a certain physical idealization because it ignores the molecular structure of matter. Scaling to arbitrarily small sizes brings quantum effects to the fore and macroscopic thermodynamics is no longer applicable. At the other extreme, scaling to arbitrarily large sizes brings in unwanted gravitational effects as discussed above. In spite of these well known limitations the idealization of continuous scaling is common practice in thermodynamics and simplifies things considerably. (In the statistical mechanics literature this goes under the rubric of the \lq thermodynamic limit\rq.) It should be noted that scaling is quite compatible with the inclusion of `surface effects' in thermodynamics. This will be discussed in Section III. A. By composing scaled copies of $N$ systems with state spaces $\Gamma_1, \dots , \Gamma_N$, one can form, for $t_1,\dots,t_N>0$, their {\bf scaled product} $\Gamma^{(t_1)}_1 \times \cdots \times \Gamma^{(t_N)}_N$ whose points are $(t_1 X_1, t_2 X_2, \dots , t_N X_N)$. In the particular case that the $\Gamma_j$'s are identical, i.e., $\Gamma_1= \Gamma_2 = \cdots =\Gamma$, we shall call any space of the the form $\Gamma^{(t_1)} \times \cdots \times \Gamma^{(t_N)}$ a {\bf multiple scaled copy} of $\Gamma$. As will be explained later in connection with Eq.\ (2.11), it is sometimes convenient in calculations to allow $t=0$ as scaling parameter (and even negative values). For the moment let us just note that if $\Gamma^{(0)}$ occurs the reader is asked to regard it as the empty set or 'nosystem'. In other words, ignore it. \smallskip Some examples may help clarify the concepts of systems and state spaces. \smallskip \item{(a)} $\Gamma_a$: 1 mole of hydrogen, H$_2$. The state space can be identified with a subset of $\R^2$ with coordinates $U$ ($=$ energy), $V (=$ volume). \item{(b)} $\Gamma_b$: $\mfr1/2$ mole of H$_2$. If $\Gamma_a$ and $\Gamma_b$ are regarded as subsets of $\R^2$ then $\Gamma_b = \Gamma_a^{(1/2)} = \{(\mfr1/2 U,\mfr1/2 V) : (U,V) \in \Gamma_a \}$. \item{(c)} $\Gamma_c$: 1 mole of H$_2$ and $\mfr1/2$ mole of O$_2$ (unmixed). $\Gamma_c = \Gamma_a \times \Gamma_{(\mfr1/2 \ {\rm mole \ O}_2)}$. This is a compound system. \item{(d)} $\Gamma_d$: 1 mole of H$_2$O. \item{(e)} $\Gamma_e$: 1 mole of H$_2 + \mfr1/2$ mole of O$_2$ (mixed). Note that $\Gamma_e \not= \Gamma_d$ and $\Gamma_e \not= \Gamma_c$. This system shows the perils inherent in the concept of equilibrium. The system $\Gamma_e$ makes sense as long as one does not drop in a piece of platinum or walk across the laboratory floor too briskly. Real world thermodynamics requires that we admit such quasi-equilibrium systems, although perhaps not quite as dramatic as this one. \item{(f)} $\Gamma_f$: All the equilibrium states of one mole of H$_2$ and half a mole of O$_2$ (plus a tiny bit of platinum to speed up the reactions) in a container. A typical state will have some fraction of H$_2$O, some fraction of H$_2$ and some O$_2$. Moreover, these fractions can exist in several phases. \bigskip\noindent {\subsubt 2. The order relation} \medskip The basic ingredient of thermodynamics is the relation $$ \prec $$ of {\bf adiabatic accessibility} among states of a system--- or even different systems. The statement $X\prec Y$, when $X$ and $Y$ are points in some (possibly different) state spaces, means that there is an adiabatic transition, in the sense explained below, that takes the point $X$ into the point $Y$. Mathematically, we do not have to ask the meaning of \lq adiabatic\rq. All that matters is that a list of all possible pairs of states $X$'s and $Y$'s such that $X \prec Y$ is regarded as given. This list has to satisfy certain axioms that we prescribe below in subsection C. Among other things it must be reflexive, i.e., $X\prec X$, and transitive, i.e., $X\prec Y$ and $Y\prec Z$ implies $X\prec Z$. (Technically, in standard mathematical terminology this is called a {\it pre}order relation because we can have both $X\prec Y$ and $Y\prec X$ without $X=Y$.) Of course, in order to have an interesting thermodynamics result from our $\prec$ relation it is essential that there are pairs of points $X,Y$ for which $X\prec Y$ is {\it not} true. Although the physical interpretation of the relation $\prec$ is not needed for the mathematical development, for applications it is essential to have a clear understanding of its meaning. It is difficult to avoid some circularity when defining the concept of adiabatic accessibility. The following version (which is in the spirit of Planck's formulation of the second law (Planck, 1926)) appears to be sufficiently general and precise and appeals to us. It has the great virtue (as discovered by Planck) that it avoids having to distinguish between work and heat---or even having to define the concept of heat; heat, in the intuitive sense, can always be generated by rubbing---in accordance with Count Rumford's famous discovery while boring cannons! We emphasize, however, that other definitions are certainly possible. Our physical definition is the following: \medskip {\bf Adiabatic accessibility:} {\it A state $Y$ is adiabatically accessible from a state $X$, in symbols $X\prec Y$, if it is possible to change the state from $X$ to $Y$ by means of an interaction with some device (which may consist of mechanical and electrical parts as well as auxiliary thermodynamic systems) and a weight, in such a way that the device returns to its initial state at the end of the process whereas the weight may have changed its position in a gravitational field.} Let us write $$ X\prec \prec Y \ \ \ {\rm if}\ \ \ X\prec Y \ \ \ {\rm but}\ \ \ Y\not\prec X . \eqno(2.1) $$ In the real world $Y$ is adiabatically accessible from $X$ only if $X\prec \prec Y$. When $X\prec Y$ and also $Y\prec X$ then the state change can only be realized in an idealized sense, for it will take infinitely long time to achieve it in the manner decribed. An alternative way is to say that the \lq device\rq\ that appears in the definition of accessibility has to return to within \lq$\varepsilon$\rq\ of its original state (whatever that may mean) and we take the limit $\varepsilon \to 0$. To avoid this kind of discussion we have taken the definition as given above, but we emphasize that it is certainly possible to redo the whole theory using only the notion of $\prec \prec $. An emphasis on $\prec \prec $ appears in Lewis and Randall's discussion of the second law (Lewis and Randall, 1923, page 116). {\it Remark:} It should be noted that the operational definition above is a definition of the concept of `adiabatic accessibility' and not the concept of an `adiabatic process'. A state change leading from $X$ to $Y$ can be achieved in many different ways (usually infinitely many), and not all of them will be `adiabatic processes' in the usual terminology. Our concern is not the temporal development of the state change which, in real processes, always leads out of the space of equilibrium states. Only the end result for the system and for the rest of the world interests us. However, it is important to clarify the relation between our definition of adiabatic accessiblity and the usual textbook definition of an adiabatic process. This will be discussed in Section C after Theorem 2.1 and again in Sec. III; cf. Theorem 3.8. There it will be shown that our definition indeed coincides with the usual notion based on processes taking place within an 'adiabatic enclosure'. A further point to notice is that the word \lq adiabatic\rq\ is sometimes used to mean ``slow" or quasi-static, but nothing of the sort is meant here. Indeed, an adiabatic process can be quite violent. The explosion of a bomb in a closed container is an adiabatic process. \smallskip Here are some further examples of adiabatic processes: \smallskip \item{1.} Expansion or compression of a gas, with or without the help of a weight being raised or lowered. \item{2.} Rubbing or stirring. \item{3.} Electrical heating. (Note that the concept of `heat' is not needed here.) \item{4.} Natural processes that occur within an isolated compound system after some barriers have been removed. This includes mixing and chemical or nuclear processes. \item{5.} Breaking a system into pieces with a hammer and reassembling (Fig. 1). \item{6.} Combinations of such changes. In the usual parlance, rubbing would be an adiabatic process, but not electrical `heating', because the latter requires the introduction of a pair of wires through the `adiabatic enclosure'. For us, both processes are adiabatic because what is required is that apart from the change of the system itself, nothing more than the displacement of a weight occurs. To achieve electrical heating, one drills a hole in the container, passes a heater wire through it, connects the wires to a generator which, in turn, is connected to a weight. After the heating the generator is removed along with the wires, the hole is plugged, and the system is observed to be in a new state. The generator, etc. is in its old state and the weight is lower. \centerline{\sevenpoint ---- (Insert Figure 1 here) ----} %\epsfxsize 15truecm %\epsfysize 7.5truecm %\epsffile{figure1.eps} We shall use the following terminology concerning any two states $X$ and $Y$. These states are said to be {\bf comparable} (with respect to the relation $\prec$, of course) if either $X \prec Y$ or $Y\prec X$. If both relations hold we say that $X$ and $Y$ are {\bf adiabatically equivalent} and write $$ X\sima Y . \eqno(2.2) $$ The comparison hypothesis referred to above is the statement that any two states in the {\it same} state space are comparable. In the examples of systems (a) to (f) above, all satisfy the comparison hypothesis. Moreover, every point in $\Gamma_c$ is in the relation $\prec$ to many (but not all) points in $\Gamma_d$. States in different systems may or may not be comparable. An example of non-comparable systems is one mole of H$_2$ and one mole of O$_2$. Another is one mole of H$_2$ and two moles of H$_2$. One might think that if the comparison hypothesis, which will be discussed further in Sects. II.C and II.E, were to fail for some state space then the situation could easily be remedied by breaking up the state space into smaller pieces inside each of which the hypothesis holds. This, generally, is false. What is needed to accomplish this is the extra requirement that {\it comparability is an equivalence relation;} this, in turn, amounts to saying that the condition $X \prec Z$ and $Y \prec Z$ implies that $X$ and $Y$ are comparable and, likewise, the condition $Z \prec X$ and $Z \prec Y$ implies that $X$ and $Y$ are comparable. (This axiom can be found in (Giles, 1964), see axiom 2.1.2, and similar requirements were made earlier by Landsberg (1956), Falk and Jung (1959) and Buchdahl (1962, 1966).) While these two conditions are logically independent, they can be shown to be equivalent if the axiom A3 in Section II. C is adopted. In any case, we do not adopt the comparison hypothesis as an axiom because we find it hard to regard it as a physical necessity. In the same vein, we do not assume that comparability is an equivalence relation (which would then lead to the validity of the comparison hypothesis for suitably defined subsystems). Our goal is to prove the comparison hypothesis starting from axioms that we find more appealing physically. \bigskip\noindent {\subt B. The entropy principle} \bigskip Given the relation $\prec$ for all possible states of all possible systems, we can ask whether this relation can be encoded in an entropy function according to the following principle, which expresses the {\bf second law of thermodynamics} in a precise and quantitative way: {\bf Entropy principle:} {\it There is a real-valued function on all states of all systems (including compound systems), called {\bf entropy} and denoted by $S$ such that \item{a)} \underbar{{\tt Monotonicity:}} When $X$ and $Y$ are comparable states then $$ X\prec Y \hbox{ \ \ {\rm if and only if} \ \ } S(X) \leq S(Y) . \eqno(2.3) $$ (See (2.6) below.) \item{b)} \underbar{{\tt Additivity and extensivity:}} If $X$ and $Y$ are states of some (possibly different) systems and if $(X,Y)$ denotes the corresponding state in the composition of the two systems, then the entropy is additive for these states, i.e., $$ S((X,Y)) = S(X) + S(Y) . \eqno(2.4) $$ $S$ is also extensive, i.e., for each $t>0$ and each state $X$ and its scaled copy $tX$, $$ S(t X)=t S(X) .\eqno(2.5) $$} \noindent[Note: {}From now on we shall omit the double parenthesis and write simply $S(X,Y)$ in place of $S((X,Y))$.] A logically equivalent formulation of (2.3), that does not use the word \lq comparable\rq\ is the following pair of statements: $$ \eqalignno{ X\sima Y &\Longrightarrow S(X) = S(Y) \ \ \ \ {\rm and} \cr X\prec\prec Y &\Longrightarrow S(X) < S(Y).& (2.6)\cr } $$ The last line is especially noteworthy. It says that entropy must increase in an irreversible process. Our goal is to construct an entropy function that satisfies the criteria (2.3)-(2,5), and to show that it is essentially unique. We shall proceed in stages, the first being to construct an entropy function for a single system, $\Gamma$, and its multiple scaled copies (in which comparability is assumed to hold). Having done this, the problem of relating different systems will then arise, i.e., the comparison question for compound systems. In the present Section II (and {\it only} in this section) we shall simply complete the project by {\it assuming} what we need by way of comparability. In Section IV, the thermal axioms (the {\it zeroth law of thermodynamics}, in particular) will be invoked to verify our assumptions about comparability in compound systems. In the remainder of this subsection we discuss he significance of conditions (2.3)-(2.5). The physical content of (2.3) was already commented on; adiabatic processes not only increase entropy but an increase of entropy also dictates which adiabatic processes are possible (between comparable states, of course). The content of additivity, (2.4), is considerably more far reaching than one might think from the simplicity of the notation---as we mentioned earlier. Consider four states $X,X',Y,Y'$ and suppose that $X\prec Y$ and $X'\prec Y'$. Then (and this will be one of our axioms) $(X,X')\prec (Y,Y')$, and (2.4) contains nothing new in this case. On the other hand, the compound system can well have an adiabatic process in which $(X,X')\prec (Y,Y')$ but $X\not\prec Y$. In this case, (2.4) conveys much information. Indeed, by monotonicity, there will be many cases of this kind because the inequality $S(X) + S(X') \leq S(Y) + S(Y')$ certainly does not imply that $S(X) \leq S(Y)$. The fact that the inequality $S(X) + S(X') \leq S(Y) + S(Y')$ tells us {\it exactly } which adiabatic processes are allowed in the compound system (assuming comparability), independent of any detailed knowledge of the manner in which the two systems interact, is astonishing and is at the {\it heart of thermodynamics.} Extensivity, (2.5), is {\it almost} a consequence of (2.4) alone---but logically it is independent. Indeed, (2.4) implies that (2.5) holds for {\it rational} numbers $t$ provided one accepts the notion of recombination as given in Axiom A5 below, i.e., one can combine two samples of a system in the same state into a bigger system in a state with the same intensive properties. (For systems, such as cosmic bodies, that do not obey this axiom, extensivity and additivity are truly independent concepts.) On the other hand, using the axiom of choice, one may always change a given entropy function satisfying (2.3) and (2.4) in such a way that (2.5) is violated for some irrational $t$, but then the function $t\mapsto S(tX)$ would end up being unbounded in every $t$ interval. Such pathological cases could be excluded by supplementing (2.3) and (2.4) with the requirement that $S(t X)$ should locally be a bounded function of $t$, either from below or above. This requirement, plus (2.4), would then imply (2.5). For a discussion related to this point see (Giles, 1964), who effectively considers {\it only} rational $t$. See also (Hardy, Littlewood, Polya 1934) for a discussion of the concept of Hamel bases which is relevant in this context. The extensivity condition can sometimes have surprising results, as in the case of electromagnetic radiation (the `photon gas'). As is well known (Landau and Lifschitz, 1969, Sect. 60), the phase space of such a gas (which we imagine to reside in a box with a piston that can be used to change the volume) is the quadrant $\Gamma=\{(U, V) \ : \ 00$ is the corresponding state in the scaled state space $\Gamma^{(t)}$. \item{{\bf A1)}} {\bf Reflexivity.} $X \sima X$. \item{{\bf A2)}} {\bf Transitivity.} {\it $X \prec Y$ and $Y \prec Z$ implies $X \prec Z$.} \item{{\bf A3)}} {\bf Consistency.} {\it $X \prec X^\prime$ and $Y \prec Y^\prime$ implies $(X,Y) \prec (X^\prime, Y^\prime)$.} \item{{\bf A4)}} {\bf Scaling invariance.} {\it If $X\prec Y$, then $tX \prec tY$ for all $t>0$.} \item{{\bf A5)}} {\bf Splitting and recombination.} {\it For $0 < t < 1$ $$X \sima (t X, (1-t) X). \eqno(2.7)$$} (If $X \in \Gamma$, then the right side is in the scaled product $\Gamma^{(t)}\times \Gamma^{(1-t)}$, of course.) \item{{\bf A6)}} {\bf Stability.} {\it If, for some pair of states, $X$ and $Y$, $$(X, \varepsilon Z_0) \prec (Y, \varepsilon Z_1)$$ holds for a sequence of $\varepsilon$'s tending to zero and some states $Z_0$, $Z_1$, then $$X \prec Y.$$} {\it Remark:} `Stability' means simply that one cannot increase the set of accessible states with an infinitesimal grain of dust. Besides these axioms the following property of state spaces, the `comparison hypothesis', plays a crucial role in our analysis in this section. It will eventually be established for all state spaces after we have introduced some more specific axioms in later sections. \item{{\bf CH)}} {\bf Definition:} {\it We say the {\bf comparison hypothesis} (CH) holds for a state space if any two states $X$ and $Y$ in the space are comparable, i.e., $X\prec Y$ or $Y\prec X$.} In the next subsection we shall show that, for every state space, $\Gamma$, assumptions A1-A6, and CH for all two-fold scaled products, $(1-\lambda) \Gamma \times \lambda \Gamma$, not just $\Gamma$ itself, are in fact {\it equivalent} to the existence of an additive and extensive entropy function that characterizes the order relation on the states in {\it all} scaled products of $\Gamma$. Moreover, for each $\Gamma$, this function is unique, up to an affine transformation of scale, $S(X) \rightarrow a S(X)+B$. Before we proceed to the construction of entropy we derive a simple property of the order relation from assumptions A1-A6, which is clearly necessary if the relation is to be characterized by an additive entropy function. \medskip {\bf THEOREM 2.1 (Stability implies cancellation law).} {\it Assume properties A1-A6, especially A6---the stability law. Then the {\bf cancellation law} holds as follows. If $X,Y$ and $Z$ are states of three (possibly distinct) systems then $$(X,Z) \prec (Y,Z) \ \ \ {\rm implies} \ \ \ X \prec Y \qquad {\rm (Cancellation \ Law)}. $$} \medskip {\it Proof:} Let $\varepsilon = 1/n$ with $n = 1,2,3, \dots$. Then we have $$ \eqalignii{(X,\varepsilon Z) &\sima ((1-\varepsilon) X, \varepsilon X, \varepsilon Z) \quad &\hbox{(by A5)} \cr &\prec ((1-\varepsilon) X, \varepsilon Y, \varepsilon Z) \quad &\hbox{(by A1, A3 and A4)} \cr &\sima ((1-2 \varepsilon) X, \varepsilon X, \varepsilon Y, \varepsilon Z) \quad &\hbox{(by A5)} \cr &\prec ((1-2 \varepsilon) X, 2 \varepsilon Y, \varepsilon Z) \quad &\hbox{(by A1, A3, A4 and A5).} \cr} $$ By doing this $n = 1/\varepsilon$ times we find that $(X, \varepsilon Z) \prec (Y, \varepsilon Z)$. By the stability axiom A6 we then have $X \prec Y $. \hfill\lanbox {\it Remark:} Under the additional assumption that $Y$ and $Z$ are comparable states (e.g., if they are in the same state space for which CH holds), the cancellation law is logically equivalent to the following statement (using the consistency axiom A3): $$ {\sl If} \ X \prec\prec Y \ {\sl then}\ (X,Z) \prec\prec (Y,Z) \ {\sl for \ all}\ Z. $$ The cancellation law looks innocent enough, but it is really rather strong. It is a partial converse of the consistency condition A3 and it says that although the ordering in $\Gamma_1 \times \Gamma_2 $ is {\it not} determined simply by the order in $\Gamma_1$ and $\Gamma_2$, there are limits to how much the ordering can vary beyond the minimal requirements of A3. It should also be noted that the cancellation law is in accord with our physical interpretation of the order relation in Subsection II.A.2.; a ``spectator'', namely $Z$, cannot change the states that are adiabatically accessible from $X$. \bigskip {\it Remark about `Adiabatic Processes':\ \ } With the aid of the cancellation law we can now discuss the connection between our notion of adiabatic accessibility and the textbook concept of an `adiabatic process'. One problem we face is that this latter concept is hard to make precise (this was our reason for avoiding it in our operational definition) and therefore the discussion must necssearily be somewhat informal. The general idea of an adiabatic process, however, is that the system of interest is locked in a thermally isolating enclosure that prevents `heat' from flowing into or out of our system. Hence, as far as the system is concerned, all the interaction it has with the external world during an adiabatic process can be thought of as being accomplished by means of some mechanical or electrical devices. Our operational definition of the relation $\prec$ appears at first sight to be based on more general processes, since we allow an auxilary thermodynamical system as part of the device. We shall now show that, despite appearances, our definition coincides with the conventional one. Let us temporarily denote by $\prec^*$ the relation between states based on adiabatic processes, i.e., $X \prec^* Y$ if and only if there is a mechanical/electrical device that starts in a state $M$ and ends up in a state $M'$ while the system changes from $X$ to $Y$. We now assume that the mechanical/electrical device can be restored to the initial state $M$ from the final state $M'$ by adding or substracting mechanical energy, and this latter process can be reduced to the raising or lowering of a weight in a gravitational field. (This can be taken as a definition of what we mean by a 'mechanical/electrical device'. Note that devices with 'dissipation' do not have this property.) Thus, $X\prec^*Y$ means there is a process in which the mechanical/electrical device starts in some state $M$ and ends up in the same state, a weight moves from height $h$ to height $h'$, while the state of our system changes from $X$ to $Y$. In symbols, $$ (X,M,h)\longrightarrow (Y,M,h').\eqno(2.8) $$ In our definition of adiabatic accessibility, on the other hand, we have some {\it arbitrary} device, which interacts with our system and which can generate or remove heat if desired. There is no thermal enclosure. The important constraint is that the device starts in some state $D$ and ends up in the same state $D$. As before a weight moves from height $h$ to height $h'$, while our system starts in state $X$ and ends up in state $Y$. In symbols, $$ (X,D,h) \longrightarrow (Y,D,h') \eqno(2.9) . $$ It is clear that (2.8) is a special case of (2.9), so we conclude that $X\prec^*Y$ implies $X\prec Y$. The device in (2.9) may consist of a thermal part in some state $Z$ and electrical and mechanical parts in some state $M$. Thus $D=(Z,M)$, and (2.9) clearly implies that $(X,Z)\prec^*(Y,Z)$. It is natural to assume that $\prec^*$ satisfies axioms A1-A6, just as $\prec$ does. In that case we can infer the cancellation law for $\prec^*$, i.e., $(X,Z) \prec^*(Y,Z,)$ implies $X \prec^* Y$. Hence, $X\prec Y$ (which is what (2.9) says) implies $X\prec^*Y$. Altogether we have thus shown that $\prec$ and $\prec^*$ are really the same relation. In words: {\it adiabatic accessibility can always be achieved by an adiabatic process applied to the system plus a device and, furthermore, the adiabatic process can be simplified (although this may not be easy to do experimentally) by eliminating all thermodynamic parts of the device, thus making the process an adiabatic one for the system alone.} \vfill\eject \bigskip \noindent {\subt D. The construction of entropy for a single system} \bigskip Given a state space $\Gamma$ we may, as discussed in Subsection I.A.1., construct its {\it multiple scaled copies}, i.e., states of the form $$ Y=(t_1Y_1,\dots,t_NY_N) $$ with $t_i>0$, $Y_i\in\Gamma$. It follows from our assumption A5 that if CH (comparison hypothesis) holds in the state space $\Gamma^{(t_1)} \times \cdots \times \Gamma^{(t_N)}$ with $t_1,...,t_N$ fixed, then any other state of the same form, $Y'=(t_1'Y_1',\dots,t_M'Y_M')$ with $Y_i'\in\Gamma$ , is comparable to $Y$ provided $\sum_i t_i=\sum_jt'_j$ (but not, in general, if the sums are not equal). This is proved as follows for $N=M=2$; the easy extension to the general case is left to the reader. Since $t_1+t_2 = t_1'+t_2'$ we can assume, without loss of generality, that $t_1-t_1' = t_2'-t_2 >0$, because the case $t_1-t_1' =0$ is already covered by CH (which was assumed) for $\Gamma^{(t_1)} \times \Gamma^{(t_2)}$. By the splitting axiom, A5, we have $(t_1Y_1,t_2Y_2) \sima (t_1'Y_1, (t_1-t_1')Y_1, t_2Y_2)$ and $(t_1'Y_1',t_2'Y_2')\sima (t_1'Y_1', (t_1-t_1')Y_2', t_2Y_2')$. The comparability now follows from CH on the space $\Gamma^{(t_1')} \times \Gamma^{(t_1-t_1')} \times \Gamma^{(t_2)}$. The entropy principle for the states in the multiple scaled copies of a single system will now be derived. More precisely, we shall prove the following theorem: \medskip {\bf THEOREM 2.2 (Equivalence of entropy and assumptions A1--A6, CH).} {\it Let $ \Gamma$ be a state space and let $\prec$ be a relation on the multiple scaled copies of $\Gamma$. The following statements are equivalent.\hfill \item{(1)} The relation $\prec$ satisfies axioms A1--A6, and CH holds for all multiple scaled copies of $\Gamma$. \item{(2)} There is a function, $S_\Gamma$ on $\Gamma$ that characterizes the relation in the sense that if \noindent$t_1+\cdots+t_N=t'_1+\cdots +t_M'$, (for all $N\geq 1$ and $M\geq 1$) then $$ (t_1Y_1,...,t_NY_N) \ \prec \ (t_1^{\prime}Y_1^{\prime}, ...,t_M^{\prime}Y_M^{\prime}) $$ holds if and only if $$ \sum_{i=1}^N t_i S_\Gamma(Y_i) \ \leq \ \sum_{j=1}^M t_j^{\prime} S_\Gamma(Y_j')\ . \eqno (2.10) $$ The function $S_\Gamma$ is uniquely determined on $\Gamma$, up to an affine transformation, i.e., any other function $S_\Gamma^*$ on $\Gamma$ satisfying (2.10) is of the form $S_\Gamma^*(X)=aS_\Gamma(X)+B$ with constants $a>0$ and $B$.} \medskip {\bf Definition.} A function $S_\Gamma$ on $\Gamma$ that characterizes the relation $\prec$ on the multiple scaled copies of $\Gamma$ in the sense stated in the theorem is called an {\bf entropy function on} $\Gamma$. \smallskip We shall split the proof of Theorem 2.2 into Lemmas 2.1, 2.2, 2.3 and Theorem 2.3 below. At this point it is convenient to introduce the following notion of {\bf generalized ordering}. While $(a_1 X_1, a_2 X_2, \dots, a_N X_N)$ has so far only been defined when all $a_i > 0$, we can {\it define} the meaning of the relation $$ (a_1 X_1, \dots , a_N X_N) \prec (a^\prime_1 X^\prime_1, \dots , a^\prime_M X^\prime_M) \eqno(2.11) $$ for arbitrary $a_i \in \R$, $a^\prime_i \in \R$, $N$ and $M$ positive integers and $X_i \in \Gamma_i$, $X^\prime_i \in \Gamma^\prime_i$ as follows. If any $a_i$ (or $a^\prime_i$) is zero we just ignore the corresponding term. Example: $(0X_{1},X_{2})\prec (2X_{3},0X_{4})$ means the same thing as $X_{2}\prec 2X_{3}$. If any $a_i$ (or $a^\prime_i$) is negative, just move $a_i X_i$ (or $a^\prime_i X^\prime_i$) to the other side and change the sign of $a_i$ (or $a^\prime_i$). Example: $$ (2 X_1, X_2) \prec (X_3, - 5 X_4, 2X_5, X_6) $$ means that $$ (2X_1, 5 X_4, X_2) \prec (X_3, 2 X_5, X_6) $$ in $\Gamma_1^{(2)} \times \Gamma_4^{(5)} \times \Gamma_2$ and $\Gamma_3 \times \Gamma_5^{(2)} \times \Gamma_6$. (Recall that $\Gamma_a \times \Gamma_b = \Gamma_b \times \Gamma_a)$. It is easy to check, using the cancellation law, that {\it the splitting and recombination axiom A5 extends to nonpositive scaling parameters}, i.e., axioms A1-A6 imply that $X\sima (aX,bX)$ for all $a,b\in\R$ with $a+b=1$, if the relation $\prec$ for nonpositive $a$ and $b$ is understood in the sense just decribed. For the definition of the entropy function we need the following lemma, which depends crucially on the stability assumption A6 and on the comparison hypothesis CH for the state spaces $\Gamma^{(1-\lambda)}\times\Gamma^{(\lambda)}$. \medskip {\bf LEMMA 2.1} {\it Suppose $X_0$ and $X_1$ are two points in $\Gamma$ with $X_0\prec\prec X_1$. For $\lambda\in\R$ define $$ \S_\lambda = \{ X \in \Gamma : ((1 - \lambda) X_0, \lambda X_1) \prec X \}.\eqno(2.12) $$ Then (i) For every $X \in \Gamma$ there is a $\lambda \in \R$ such that $X \in \S_\lambda$. (ii) For every $X \in \Gamma$, $\sup \{ \lambda : X \in \S_\lambda \} < \infty$. } \medskip {\it Remark.} Since $X\sima ((1-\lambda)X,\lambda X)$ by assumption A5, the definition of $\S_\lambda$ really involves the order relation on double scaled copies of $\Gamma$ (or on $\Gamma$ itself, if $\lambda=0$ or 1.) {\it Proof of Lemma 2.1.} (i) If $X_0 \prec X$ then obviously $X \in \S_0$ by axiom A2. For general $X$ we claim that $$ (1 + \alpha) X_0 \prec (\alpha X_1,X) \eqno(2.13) $$ for some $\alpha \geq 0$ and hence $((1 -\lambda) X_0, \lambda X_1) \prec X$ with $\lambda = - \alpha$. The proof relies on stability, A6, and the comparison hypothesis CH (which comes into play for the first time): If (2.13) were not true, then by CH we would have $$(\alpha X_1,X) \prec (1 + \alpha) X_0$$ for all $\alpha >0$ and so, by scaling, A4, and A5 $$ \left (X_1,\, {1 \over \alpha}X\right) \prec \left( X_0,\, {1 \over \alpha} X_0\right). $$ By the stability axiom A6 this would imply $X_1 \prec X_0$ in contradiction to $X_0 \prec\prec X_1$. (ii) If $\sup \{ \lambda : X \in \S_\lambda \} = \infty$, then for some sequence of $\lambda$'s tending to infinity we would have $((1-\lambda)X_0,\lambda X)\prec X$ and hence $(X_0, \lambda X_1) \prec (X, \lambda X_0)$ by A3 and A5. By A4 this implies $\left( {1 \over \lambda} X_0, X_1 \right) \prec \left( {1 \over \lambda} X, X_0 \right)$ and hence $X_1 \prec X_0$ by stability, A6. \hfill\lanbox We can now state our {\bf formula for the entropy function}. If all points in $\Gamma $ are adiabatically equivalent there is nothing to prove (the entropy is constant), so we may assume that there are points $X_0$, $X_1\in\Gamma$ with $X_0\prec\prec X_1$. We then define for $X\in\Gamma$ $$ S_\Gamma(X):=\sup\{\lambda:\ ((1-\lambda)X_0,\lambda X_1)\prec X\}. \eqno (2.14) $$ (The symbol $a:=b$ means that $a$ is defined by $b$.) This $S_\Gamma$ will be referred to as the {\bf canonical entropy} on $\Gamma$ with {\bf reference points} $X_0$ and $X_1$. This definition is illustrated in Figure 2. \centerline{\sevenpoint ---- Insert Figure 2 here ----} By Lemma 2.1 $S_\Gamma(X)$ is well defined and $S_\Gamma(X)<\infty$ for all $X$. (Note that by stability we could replace $\prec$ by $\prec\prec$ in (2.14).) We shall now show that this $S_\Gamma$ has all the right properties. The first step is the following simple lemma, which does not depend on the comparison hypothesis. \medskip {\bf LEMMA 2.2 ($\prec$ is equivalent to $\leq$).} {\it Suppose $X_0 \prec\prec X_1$ are states and $a_0, a_1, a^\prime_0, a^\prime_1$ are real numbers with $a_0 + a_1 = a^\prime_0 + a^\prime_1$. Then the following are equivalent. \item{(i)} $(a_0 X_0, a_1 X_1) \prec (a^\prime_0 X_0, a^\prime_1 X_1)$ \item{(ii)} $a_1 \leq a^\prime_1$ (and hence $a_0 \geq a^\prime_0$). \smallskip\noindent In particular, $\sima$ holds in (i) if and only if $a_1 = a^\prime_1$ and $a_0 = a^\prime_0$.} \smallskip {\it Proof:} We give the proof assuming that the numbers $a_0, a_1, a^\prime_0, a^\prime_1$ are all positive and $a_0 + a_1 = a^\prime_0 + a^\prime_1=1$. The other cases are similar. We write $a_1=\lambda$ and $a_1'=\lambda'$. (i) $\Rightarrow$ (ii). If $\lambda > \lambda^\prime$ then, by A5 and A3, $((1 - \lambda) X_0, \lambda^\prime X_1, (\lambda - \lambda^\prime) X_1) \prec ((1 - \lambda) X_0, (\lambda -\lambda^\prime) X_0, \lambda^\prime X_1)$. By the cancellation law, Theorem 2.1, $((\lambda - \lambda^\prime) X_1) \prec ((\lambda - \lambda^\prime) X_0)$. By scaling invariance, A5, $X_1 \prec X_0$, which contradicts $X_0 \prec\prec X_1$. \hfill\break (ii) $\Rightarrow$ (i). This follows from the following computation. $$ \eqalignii {((1-\lambda)X_0, \lambda X_1) &\sima ((1-\lambda^\prime)X_0, (\lambda^\prime - \lambda)X_0, \lambda X_1) \quad &\hbox{(by axioms A3 and A5)} \cr &\prec ((1-\lambda^\prime)X_0, (\lambda^\prime - \lambda)X_1, \lambda X_1) \quad &\hbox{(by axioms A3 and A4)} \cr &\sima ((1-\lambda^\prime)X_0, \lambda^\prime X_1) \quad &\hbox{(by axioms A3 and A5).} \cr} $$ \hfill\lanbox The next lemma will imply, among other things, that entropy is unique, up to an affine transformation. \medskip {\bf LEMMA 2.3 (Characterization of entropy).} {\it Let $S_\Gamma$ denote the canonical entropy (2.14) on $\Gamma$ with respect to the reference points $X_0\prec\prec X_1$. If $X \in \Gamma$ then the equality $$ \lambda = S_\Gamma (X) $$ is equivalent to $$ X \sima ((1 - \lambda) X_0, \lambda X_1). $$} \smallskip {\it Proof:} First, if $\lambda = S_\Gamma(X)$ then, by the definition of supremum, there is a sequence $\varepsilon_1 \geq \varepsilon_2 \geq \dots \geq 0$ converging to zero, such that $$ ((1 - (\lambda - \varepsilon_n)) X_0, (\lambda - \varepsilon_n) X_1) \prec X$$ for each $n$. Hence, by A5, $$((1 - \lambda) X_0, \lambda X_1, \varepsilon_n X_0) \sima ((1 - \lambda + \varepsilon_n) X_0, (\lambda - \varepsilon_n) X_1, \varepsilon_n X_1) \prec (X, \varepsilon_n X_1),$$ and thus $((1 - \lambda) X_0, \lambda X_1) \prec X$ by the stability property A6. On the other hand, since $\lambda$ is the supremum we have $$X \prec ((1 - (\lambda + \varepsilon) X_0, (\lambda + \varepsilon) X_1) $$ for all $\varepsilon > 0$ by the comparison hypothesis CH. Thus, $$ (X, \varepsilon X_0) \prec ((1 - \lambda) X_0, \lambda X_1, \varepsilon X_1), $$ so, by A6, $X \prec ((1 - \lambda) X_0, \lambda X_1)$. This shows that $X \sima ((1 - \lambda) X_0, \lambda X_1)$ when $\lambda = S_\Gamma(X)$. Conversely, if $\lambda^\prime \in [0,1]$ is such that $X \sima ((1 - \lambda^\prime) X_0, \lambda^\prime X_1)$, then $((1 - \lambda^\prime) X_0, \lambda^\prime X_1) \sima ((1 - \lambda) X_0, \lambda X_1)$ by transitivity. Thus, $\lambda = \lambda^\prime$ by Lemma 2.2. \hfill\lanbox \smallskip %%%%%%%% {\it Remark 1:} Without the comparison hypothesis we could find that $S_\Gamma(X_0)= 0$ and $S_\Gamma(X) = 1$ for all $X$ such that $X_0 \prec X$. \smallskip {\it Remark 2:} {}From Lemma 2.3 and the cancellation law it follows that the canonical entropy with reference points $X_0\prec\prec X_1$ satisfies $0\leq S_\Gamma (X)\leq 1$ if and only if $X$ belongs to the {\bf strip} $\Sigma (X_0, X_1)$ defined by $$ \Sigma (X_0, X_1) := \{ X \in \Gamma : X_0 \prec X \prec X_1 \} \subset \Gamma.$$ Let us make the dependence of the canonical entropy on $X_0$ and $X_1$ explicit by writing $$ S_\Gamma(X)=S_\Gamma(X\vert X_0,X_1) \ . \eqno(2.15) $$ For $X$ outside the strip we can then write $$ S_\Gamma (X \vert X_0, X_1)= S_\Gamma (X_1 \vert X_0, X)^{-1} \qquad\hbox{if\ }X_1\prec X$$ and $$ S_\Gamma (X \vert X_0, X_1)= -{S_\Gamma ( X_0\vert X, X_1)\over 1-S_\Gamma (X_0 \vert X, X_1)} \qquad\hbox{if\ }X\prec X_0. $$ \smallskip %\vfill\eject {\tt Proof of Theorem 2.2:} {\it (1) $\Longrightarrow$ (2):} Put $\lambda_i=S_\Gamma(Y_i)$, $\lambda_i'=S_\Gamma(Y_i')$. By Lemma 2.3 we know that $Y_i\sima ((1-\lambda_i)X_0,\lambda_i X_1)$ and $Y_i'\sima ((1-\lambda_i')X_0,\lambda_i' X_1)$. By the consistency axiom A3 and the recombination axiom A5 it follows that $$ (t_1Y_1,\dots,t_NY_N)\sima (\sum_i t_i(1-\lambda_i)X_0, \sum_i t_i\lambda_i X_1) $$ and $$ (t_1'Y_1',\dots,t_N'Y_N')\sima (\sum_i t_i'(1-\lambda_i')X_0, \sum_i t_i'\lambda_i' X_1) \ . $$ Statement (2) now follows from Lemma 2.2. The implication (2) $\Longrightarrow$ (1) is obvious. The proof of Theorem 2.2 is now complete except for the uniqueness part. We formulate this part separately in Theorem 2.3 below, which is slightly stronger than the last assertion in Theorem 2.2. It implies that an entropy function for the multiple scaled copies of $\Gamma$ is already uniquely determined, up to an affine transformation, by the relation on states of the form $((1-\lambda)X,\lambda Y)$, i.e., it requires only the case $N=M=2$, in the notation of Theorem 2.2. \medskip {\bf THEOREM 2.3 (Uniqueness of entropy)} {\it If $S_\Gamma^*$ is a function on $\Gamma$ that satisfies $$ ((1-\lambda)X,\lambda Y)\prec((1-\lambda)X',\lambda Y') $$ if and only if $$ (1-\lambda)S_\Gamma^*(X)+\lambda S_\Gamma^*(Y)\leq(1-\lambda)S_\Gamma^*(X')+\lambda S_\Gamma^*(Y'), $$ for all $\lambda\in\R$ and $X,Y,X',Y'\in\Gamma$, then $$ S_\Gamma^*(X)=aS_\Gamma(X)+B $$ with $$ a=S_\Gamma^*(X_1)-S_\Gamma^*(X_0)>0,\qquad B=S_\Gamma^*(X_0). $$ Here $S_\Gamma$ is the canonical entropy on $\Gamma$ with reference points $X_0\prec\prec X_1$.} \medskip {\it Proof:} This follows immediately from Lemma 2.3, which says that for every $X$ there is a unique $\lambda$, namely $\lambda=S_\Gamma(X)$, such that $$X\sima ((1-\lambda)X,\lambda X)\sima ((1-\lambda)X_0,\lambda X_1).$$ Hence, by the hypothesis on $S_\Gamma^*$, and $\lambda=S_\Gamma(X)$, we have $$ S_\Gamma^*(X)=(1-\lambda)S_\Gamma^*(X_0)+ \lambda S_\Gamma^*(X_1) = [S_\Gamma^*(X_1)-S_\Gamma^*(X_0)]S_\Gamma(X) +S_\Gamma^*(X_0). $$ The hypothesis on $S_\Gamma^*$ also implies that $a:= S_\Gamma^*(X_1)-S_\Gamma^*(X_0) >0$, because $X_0\prec\prec X_1$.\hfill\lanbox \medskip {\it Remark:} Note that $S_\Gamma^*$ is defined on $\Gamma$ and satisfies $S_\Gamma^*(X) = aS_\Gamma(X)+B$ there. On the space $\Gamma^{(t)}$ a corresponding entropy is, {\it by definition}, given by $S_{\Gamma^{(t)}}^*(tX) = tS_\Gamma^*(X)= atS_\Gamma(X) + tB = aS_\Gamma^{(t)}(tX) + tB$, where $S_\Gamma^{(t)}(tX)$ is the canonical entropy on $\Gamma^{(t)}$ with reference points $tX_0, tX_1$. Thus, $S_{\Gamma^{(t)}}^*(tX)\neq aS_\Gamma^{(t)}(tX) +B$ \ (unless $B=0$, of course). \bigskip It is apparent from formula (2.14) that the definition of the canonical entropy function on $\Gamma$ involves only the relation $\prec$ on the double scaled products $\Gamma^{(1-\lambda)}\times \Gamma^{(\lambda)}$ besides the reference points $X_0$ and $X_1$. Moreover, the canonical entropy uniquely characterizes the relation on all multiple scaled copies of $\Gamma$, which implies in particular that CH holds for all multiple scaled copies. Theorem 2.3 may therefore be rephrased as follows: \medskip {\bf THEOREM 2.4 (The relation on double scaled copies determines the relation everywhere).} {\it Let $\prec$ and $\prec^*$ be two relations on the multiple scaled copies of $\Gamma$ satisfying axioms A1-A6, and also CH for $\Gamma^{(1-\lambda)}\times \Gamma^{(\lambda)}$ for each fixed $\lambda\in[0,1]$. If $\prec$ and $\prec^*$ coincide on $\Gamma^{(1-\lambda)}\times \Gamma^{(\lambda)}$ for each $\lambda\in[0,1]$, then $\prec$ and $\prec^*$ coincide on all multiple scaled copies of $\Gamma$, and CH holds on all the multiple scaled copies.} \medskip The proof of Theorem 2.2 is now complete. \bigskip\noindent {\subt E. Construction of a universal entropy in the absence of mixing} \bigskip In the previous subsection we showed how to construct an entropy for a single system, $\Gamma$, that exactly describes the relation $\prec$ within the states obtained by forming multiple scaled copies of $\Gamma$. It is unique up to a multiplicative constant $a>0$ and an additive constant $B$, i.e., to within an affine transformation. We remind the reader that this entropy was constructed by considering just the product of two scaled copies of $\Gamma$, but our axioms implied that it automatically worked for {\it all} multiple scaled copies of $\Gamma$. We shall refer to $a$ and $B$ as {\bf entropy constants} for the system $\Gamma$. Our goal is to put these entropies together and show that they behave in the right way on products of arbitrarily many copies of {\it different} systems. Moreover, this \lq universal\rq\ entropy will be unique up to {\it one} multiplicative constant---but still many additive constants. The central question here is one of {\it \lq calibration\rq\ }, which is to say that the multiplicative constant in front of each elementary entropy has to be chosen in such a way that the additivity rule (2.4) holds. It is not even obvious yet that the additivity can be made to hold at all, whatever the choice of constants. Let us note that the number of additive constants depends heavily on the kinds of adiabatic processes available. The system consisting of one mole of hydrogen mixed with one mole of helium and the system consisting of one mole of hydrogen mixed with two moles of helium are different. The additive constants are independent {\it unless} a process exists in which both systems can be unmixed, and thereby making the constants comparable. In nature we expect only 92 constants, one for each element of the periodic table, unless we allow nuclear processes as well, in which case there are only two constants (for neutrons and for hydrogen). On the other hand, if un-mixing is not allowed uncountably many constants are undetermined. In Section VI we address the question of adiabatic processes that unmix mixtures and reverse chemical reactions. That such processes exist is not so obvious. To be precise, the principal goal of this subsection is the proof of the following Theorem 2.5, which is a case of the entropy principle that is special in that it is restricted to processes that do not involve mixing or chemical reactions. It is a generalization of Theorem 2.2. \medskip {\bf THEOREM 2.5 (Consistent entropy scales). } {\it Consider a family of systems fulfilling the following requirements: \item{(i)} The state spaces of any two systems in the family are disjoint sets, i.e., every state of a system in the family belongs to exactly one state space. \item{(ii)} All multiple scaled products of systems in the family belong also to the family. \item{(iii)} Every system in the family satisfies the comparison hypothesis. For each state space $\Gamma$ of a system in the family let $S_{\Gamma}$ be some d efinite entropy function on $\Gamma$. Then there are constants $a_{\Gamma}$ and $B_{\Gamma}$ such that the function $S$, defined for all states in all $\Gamma$'s by $$ S(X)= a_{\Gamma} S_{\Gamma} (X)+ B_{\Gamma} $$ for $X\in \Gamma$, has the following properties: \smallskip \item{a).} If $X$ and $Y$ are in the same state space then $$ X\prec Y \quad\quad \hbox{\rm if and only if} \quad\quad S(X)\leq S(Y). $$ \smallskip \item{b).} $S$ is additive and extensive, i.e., $$ S(X,Y) = S(X)+S(Y). \eqno (2.4) $$ and, for $t>0$, $$ S(tX) = tS(X). \eqno (2.5) $$ } \medskip {\it Remark.\/} Note that $\Gamma_1$ and $\Gamma_1\times \Gamma_2$ are disjoint as sets for any (nonempty) state spaces $\Gamma_1$ and $\Gamma_2$. \medskip {\it Proof:} Fix some system $\Gamma_0$ and two points $Z_0\prec \prec Z_1$ in $\Gamma_0$. In each state space $\Gamma$ choose some fixed point $X_{\Gamma} \in \Gamma$ in such a way that the identities $$\eqalignno{ X_{\Gamma_1 \times \Gamma_2}&= (X_{\Gamma_1}, X_{\Gamma_2}) &(2.16)\cr \noalign{\smallskip} X_{t\Gamma} &= tX_{\Gamma}&(2.17)\cr } $$ hold. With the aid or the axiom of choice this can be achieved by considering the formal vector space spanned by all systems and choosing a Hamel basis of systems $\{\Gamma_{\alpha}\}$ in this space such that every system can be written uniquely as a scaled product of a finite number of the $\Gamma_{\alpha}$'s. (See Hardy, Littlewood and Polya, 1934). The choice of an arbitrary state $X_{\Gamma_{\alpha}}$ in each of these `elementary' systems $\Gamma_{\alpha}$ then defines for each $\Gamma$ a unique $X_{\Gamma}$ such that (2.17) holds. (If the reader does not wish to invoke the axiom of choice then an alternative is to hypothesize that every system has a unique decomposition into elementary systems; the simple systems considered in the next section obviously qualify as the elementary systems.) For $X\in \Gamma$ we consider the space $\Gamma \times \Gamma_0$ with its canonical entropy as defined in (2.14), (2.15) relative to the points $(X_{\Gamma}, Z_0)$ and $(X_{\Gamma}, Z_1)$. Using this function we define $$ S(X)= S_{\Gamma \times \Gamma_0}((X,Z_0) \, \, \vert \, \, (X_{\Gamma} , Z_0),(X_{\Gamma} , Z_1)). \eqno(2.18) $$ Note: Equation (2.18) fixes the entropy of $X_{\Gamma}$ to be zero. Let us denote $S(X) $ by $\lambda$ which, by Lemma 2.3, is characterized by $$ (X,Z_0) \sima ( (1-\lambda ) (X_{\Gamma} , Z_0) , \lambda (X_{\Gamma} , Z_1)). $$ By the cancellation law this is equivalent to $$ (X,\lambda Z_0)\sima (X_{\Gamma}, \lambda Z_1)). \eqno(2.19) $$ By (2.16) and (2.17) this immediately implies the additivity and extensivity of $S$. Moreover, since $X\prec Y$ holds if and only if $(X, Z_0) \prec (Y,Z_0) $ it is also clear that $S$ is an entropy function on any $\Gamma$. Hence $S$ and $S_{\Gamma}$ are related by an affine transformation, according to Theorem 2.3. \hfill \lanbox \medskip {\bf Definition (Consistent entropies).} A collection of entropy functions $S_\Gamma$ on state spaces $\Gamma$ is called {\it consistent} if the appropriate linear combination of the functions is an entropy function on all multiple scaled products of these state spaces. In other words, the set is consistent if the multiplicative constants $a_{\Gamma}$, referred to in Theorem 2.5, can all be chosen equal to 1. \smallskip \underbar{{\it Important Remark:}} {}From the definition, (2.14), of the canonical entropy and (2.19) it follows that the entropy (2.18) is given by the formula $$ S(X) = \sup \{ \lambda \, \, : \, \, (X_{\Gamma}, \lambda Z_1) \prec (X , \lambda Z_0) \} \eqno (2.20) $$ for $X\in\Gamma$. The auxiliary system $\Gamma_0$ can thus be regarded as an `entropy meter' in the spirit of (Lewis and Randall, 1923) and (Giles, 1964). Since we have chosen to define the entropy for each system independently, by equation (2.14), the role of $\, \Gamma_0$ in our approach is solely to calibrate the entropy of different systems in order to make them consistent. \medskip {\it Remark about the photon gas:\/} As we discussed in Section II.B the photon gas is special and there are two ways to view it. One way is to regard the scaled copies $\Gamma^{(t)}$ as distinct systems and the other is to say that there is only one $\Gamma$ and the scaled copies are identical to it and, in particular, must have exactly the same entropy function. We shall now see how the first point of view can be reconciled with the latter requirement. Note, first, that in our construction above we cannot take the point $(U,V)=(0,0)$ to be the fiducial point $X_{\Gamma}$ because $(0,0)$ is not in our state space which, according to the discussion in Section III below, has to be an open set and hence cannot contain any of its boundary points such as $(0,0)$. Therefore, we have to make another choice, so let us take $X_{\Gamma}= (1,1)$. But the construction in the proof above sets $S_{\Gamma} (1,1)= 0$ and therefore $S_{\Gamma}(U,V) $ will not have the homogeneous form $S^{\rm hom}(U,V)= V^{1/4}U^{3/4}$. Nevertheless, the entropies of the scaled copies will be extensive, as required by the theorem. If one feels that all scaled copies should have the same entropy (because they represent the same physical system) then the situation can be remedied in the following way: With $S_{\Gamma}(U,V) $ being the entropy constructed as in the proof using $(1,1)$, we note that $S_{\Gamma}(U,V) = S^{\rm hom}(U,V) + B_{\Gamma}$ with the constant $B_{\Gamma}$ given by $B_{\Gamma}= -S_{\Gamma}(2,2)$. This follows from simple algebra and the fact that we know that the entropy of the photon gas constructed in our proof must equal $S^{\rm hom}$ to within an additive constant. (The reader might ask how we know this and the answer is that the entropy of the `gas' is unique up to additive and multiplicative constants, the latter being determined by the system of units employed. Thus, the entropy determined by our construction must be the `correct entropy', up to an additive constant, and this `correct entropy' is what it is, as determined by physical measurement. Hopefully it agrees with the function deduced in (Landau and Lifschitz, 1969).) Let us use our freedom to alter the additive constants as we please, provided we maintain the extensivity condition (2.5). It will not be until Section VI that we have to worry about the additive constants {\it per se} because it is only there that mixing and chemical reactions are treated. Therefore, we redefine the entropy of the state space $\Gamma$ of the photon gas to be $S^*(U,V) := S_{\Gamma}(U,V) + S_{\Gamma}(2,2)$. which is the same as $S^{\rm hom}(U,V)$. We also have to alter the entropy of the scaled copies according to the rule that preserves extensivity, namely $S_{\Gamma^{(t)}}(U,V) \rightarrow S_{\Gamma^{(t)}}(U,V) +tS_{\Gamma}(2,2) =S_{\Gamma^{(t)}}(U,V) + S_{\Gamma^{(t)}}(2t,2t) = S^{\rm hom}(U,V)$. In this way, all the scaled copies now have the same (homogeneous) entropy, but we remind the reader that the same construction could be carried out for any material system with a homogeneous (or, more exactly an affine) entropy function---if one existed. {}From the thermodynamic viewpoint, the photon gas is unusual but not special. \bigskip \bigskip \bigskip\noindent {\subt F. Concavity of entropy} \bigskip Up to now we have not used, or assumed, any geometric property of a state space $\Gamma$. It is an important stability property of thermodynamical systems, however, that the entropy function is a {\it concave} function of the state variables ---a requirement that was emphasized by Maxwell, Gibbs, Callen and many others. Concavity also plays an important role in the definition of temperature, as in section V. In order to have this concavity it is first necessary to make the state space on which entropy is defined into a convex set, and for this purpose the choice of coordinates is important. Here, we begin the discussion of concavity by discussing this geometric property of the underlying state space and some of the consequences of the {\it convex combination axiom} A7 for the relation $\prec$, to be given after the following definition. {\bf Definition:} By a {\bf state space with a convex structure}, or simply a {\bf convex state space}, we mean a state space $\Gamma$, that is a convex subset of some linear space, e.g., $\R^n$. That is, if $X$ and $Y$ are any two points in $\Gamma$ and if $0 \leq t \leq 1$, then the point $tX + (1-t)Y$ is a well-defined point in $\Gamma$. A {\it concave function}, $S$, on $\Gamma$ is one satisfying the inequality $$ S(tX + (1-t)Y) \geq tS(X) + (1-t)S(Y). \eqno(2.21) $$ Our basic convex combination axiom for the relation $\prec$ is the following. \medskip \item{\bf A7)} {\bf Convex combination.} Assume $X$ and $Y$ are states in the same {\it convex} state space, $\Gamma$. For $t \in [0,1]$ let $tX$ and $(1-t)Y$ be the corresponding states of their $t$ scaled and $(1-t)$ scaled copies, respectively. Then the point $(t X, (1-t) Y)$ in the product space $\Gamma^{(t)}\times \Gamma^{(1-t)}$ satisfies $$ (t X, (1-t) Y) \prec t X + (1-t)Y\ . \eqno(2.22) $$ Note that the right side of (2.22) is in $\Gamma$ and is defined by ordinary convex combination of points in the convex set $\Gamma$. \medskip The physical meaning of A7 is more or less evident, but it is essential to note that the convex structure depends heavily on the choice of coordinates for $\Gamma$. A7 means that if we take a bottle containing $1/4$ moles of nitrogen and one containing $3/4$ moles (with possibly different pressures and densities), and if we mix them together, then among the states of one mole of nitrogen that can be reached adiabatically there is one in which the energy is the sum of the two energies and, likewise, the volume is the sum of the two volumes. Again, we emphasize that the choice of energy and volume as the (mechanical) variables with which we can make this statement is an important assumption. If, for example, temperature and pressure were used instead, the statement would not only not hold, it would not even make much sense. The physical example above seems not exceptionable for liquids and gases. On the other hand it is not entirely clear how to ascribe an operational meaning to a convex combination in the state space of a solid, and the physical meaning of axiom A7 is not as obvious in this case. Note, however, that although convexity is a global property, it can often be inferred from a local property of the boundary. (A connected set with a smooth boundary, for instance, is convex if every point on the boundary has a neighbourhood, whose intersection with the set is convex.) In such cases it suffices to consider convex combinations of points that are close together and close to the boundary. For small deformation of an isotropic solid the six strain coordinates, multiplied by the volume, can be taken as work coordinates. Thus, A7 amounts to assuming that a convex combination of these coordinates can always be achieved adiabatically. See, e.g., (Callen, 1985). If $X \in \Gamma$ we denote by $A_X$ the set $\{ Y \in \Gamma : X \prec Y \}$. $A_X$ is called the {\bf forward sector } of $X$ in $\Gamma$. More generally, if $\Gamma^\prime $ is another system, we call the set $$ \{Y\in \Gamma': X\prec Y\}, $$ the forward sector of $X$ in $\Gamma^\prime $. Usually this concept is applied to the case in which $\Gamma$ and $\Gamma^\prime $ are identical, but it can also be useful in cases in which one system is changed into another; an example is the mixing of two liquids in two containers (in which case $\Gamma $ is a compound system) into a third vessel containing the mixture (in which case $\Gamma^\prime $ is simple). The main effect of A7 is that forward sectors are convex sets. \medskip {\bf THEOREM 2.6} {\bf (Forward sectors are convex).} {\it Let $\Gamma$ and $\Gamma'$ be state spaces of two systems, with $\Gamma'$ a convex state space. Assume that A1--A5 hold for $\Gamma$ and $\Gamma'$ and, in addition, A7 holds for $\Gamma'$. Then the forward sector of $X$ in $\Gamma'$, defined above, is a {\it convex\/} subset of $\Gamma'$ for each $X\in \Gamma$.} \smallskip {\it Proof:\/} Suppose $X\prec Y_1$ and $X\prec Y_2$ and that $0 0$ and $B$ such that $S^*(X) = a S(X) + B$ for all $X \in \Gamma$. In particular, $S^*$ must satisfy condition (ii). } \medskip {\it Proof:} In general, if $F$ and $G$ are any two real valued functions on $\Gamma \times \Gamma$, such that $F(X,Y)\leq F(X',Y')$ if and only if $G(X,Y)\leq G(X',Y')$, it is an easy logical exercise to show that there is a monotone increasing function $K$ (i.e., $x\leq y$ implies $K(x)\leq K(y)$) defined on the range of $F$, so that $G = K \circ F$. In our case $F(X,Y)=S(X) + S(Y)$. If the range of $S$ is the interval $L$ then the range of $F$ is $2L$. Thus $K$, which is defined on $2L$, satisfies $$ K(S(X) + S(Y)) = S^* (X) + S^* (Y) \eqno(2.23) $$ for all $X$ and $Y$ in $\Gamma$ because both $S$ and $S^*$ satisfy condition (i). For convenience, define $M$ on $L$ by $M(t) = \mfr1/2 K (2t)$. If we now set $Y = X$ in (1) we obtain $$ S^* (X) = M (S(X)), \quad X \in \Gamma \eqno(2.24) $$ and (2.23) becomes, in general, $$ M \left( {x+y \over 2} \right) = \mfr1/2 M(x) + \mfr1/2 M(y)\eqno(2.25) $$ for all $x$ and $y$ in $L$. Since $M$ is monotone, it is bounded on all finite subintervals of $L$. Hence (Hardy, Littlewood, Polya 1934) $M$ is both concave and convex in the usual sense, i.e., $$ M (t x + (1- t) y) = t M(x) + (1 - t) M(y) $$ for all $0 \leq t \leq 1$ and $x,y \in L$. {}From this it follows that $M(x) = a x + B$ with $a\geq 0$. If $a$ were zero then $S^*$ would be constant on $\Gamma$ which would imply that $S$ is constant as well. In that case we could always replace $a$ by 1 and replace $B$ by $B-S(X)$. \hfill\lanbox {\it Remark:} It should be noted that Theorem 2.10 does not rely on any structural property of $\Gamma$, which could be any abstract set. In particular, continuity plays no role; indeed it cannot be defined because no topology on $\Gamma$ is assumed. The only residue of ``continuity" is the requirement that the range of $S$ be an interval. That condition (ii) is not superfluous for the uniqueness theorem may be seen from the following simple counterexample. {\bf EXAMPLE:} Suppose the state space $\Gamma$ consists of 3 points, $X_0$, $X_1$ and $X_2$, and let $S$ and $S^*$ be defined by $S(X_0)= S^*(X_0)=0$, $S(X_1)=S^*(X_1)=1$, $S(X_2)$=3, $S^*(X_2)$=4. These functions correspond to the same order relation on $\Gamma\times \Gamma$, but they are not related by an affine transformation. The following sharpening of Theorem 2.4 is an immediate corollary of Theorem 2.10 in the case that the convexity axiom A7 holds, so that the range of the entropy is connected. \medskip {\bf THEOREM 2.11 (The relation on $\Gamma\times\Gamma$ determines the relation everywhere)} {\it Let $\prec$ and $\prec^*$ be two relations on the multiple scaled copies of $\Gamma$ satisfying axioms A1-A7, and CH for $\Gamma^{(1-\lambda)}\times \Gamma^{(\lambda)}$ for each fixed $\lambda\in[0,1]$. If $\prec$ and $\prec^*$ coincide on $\Gamma\times \Gamma$, i.e., $$ (X,Y) \prec (X^{\prime}, Y^{\prime}) \ \ \ {\it if \ and \ only \ if}\ \ \ (X,Y) \prec^* (X^{\prime}, Y^{\prime}) $$ for $X,X',Y,Y'\in\Gamma$, then $\prec$ and $\prec^*$ coincide on all multiple scaled copies of $\Gamma$.} \bigskip %%%%% As a last variation on the theme of this subsection let us note that uniqueness of entropy does even not require knowledge of the order relation $\prec$ on all of $\Gamma \times \Gamma$. The knowledge of $\prec$ on a relatively thin ``diagonal" set will suffice, as Theorem 2.12 shows. \medskip {\bf THEOREM 2.12 (Diagonal sets determine entropy).} {\it Let $\prec$ be an order relation on $ \Gamma \times \Gamma$ and let $S$ be a function on $\Gamma$ satisfying conditions (i) and (ii) of Theorem 2.10. Let ${\cal D}$ be a subset of $ \Gamma \times \Gamma$ with the following properties: \item{(i)} $(X,X) \in {\cal D}$ for every $X \in \Gamma$. \item{(ii)} The set $D= \{(S(X),S(Y))\in {\bf R}^2 \, : \, (X,Y)\in {\cal D} \}$ contains an open subset of ${\bf R}^2$ (which necessarily contains the set $\{(x,x) : x\in {\rm Range}\, S\}$). \medskip Suppose now that $ \prec^*$ is another order relation on $ \ \Gamma \times \Gamma$ and that $S^*$ is a function on $ \Gamma$ satisfying condition (i) of Theorem 2.10 with respect to $ \prec^*$ on $ \Gamma \times \Gamma$. Suppose further, that $ \prec$ and $ \prec^*$ agree on ${\cal D}$, i.e., $$ (X,Y) \prec (X^{\prime}, Y^{\prime}) \ \ \ {\it if \ and \ only \ if}\ \ \ (X,Y) \prec^* (X^{\prime}, Y^{\prime}) $$ whenever $(X,Y)$ and $(X^{\prime}, Y^{\prime})$ are both in ${\cal D}$. Then $ \prec$ and $ \prec^*$ agree on all of $\Gamma \times \Gamma$ and hence, by Theorem 2.10, $S$ and $S^*$ are related by an affine transformation. } {\it Proof:} By considering points $(X,X) \in {\cal D}$, the consistency of $S$ and $S^*$ implies that $S^*(X) = M(S(X))$ for all $X \in \Gamma$, where $M$ is some monotone increasing function on $L \subset {\bf R}$. Again, as in the proof of Theorem 2.10, $$ \mfr1/2 M(S(X)) + \mfr1/2 M(S(Y)) = M\Bigl({S(X)) + S(Y)\over 2}\Bigr) \eqno(2.26) $$ for all $(X,Y)\in {\cal D}$. (Note: In deriving Eq.\ (2.25) we did not use the fact that $ \Gamma \times \Gamma$ was the Cartesian product of two spaces; the only thin