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The foundations of thermodynamics
JOURNAL
OF MATHEMATICAL
ANALYSIS
The Foundations
AND
APPLICATIONS
17, 172-193 (1967)
of Thermodynamics*
J. L. B. COOPER Department
of Mathematics, California Institute and University of Toronto
of Technology
Submitted by F. V. Atkinson
1.
INTRODUCTION
A central result of classical macroscopic thermodynamics is the deduction of the existence and properties of entropy from forms of the second law derived from experience. The arguments involve, in addition to the first and second laws, assumptions about heat and temperature and the properties of perfect gases (see, for example, the excellent account in Zemansky [l]). They have been criticized by mathematical physicists (see Born [2] and [3]) on the grounds that these assumptions involve redundancies, and a development free from these criticisms was made by Caratheodory [4], whose treatment does not involve heat and temperature as initial concepts, but is based on development of thermodynamic concepts in terms of mechanical concepts. Since Caratheodory’s work numerous attempts have been made to generalize his approach by deducing the laws of thermodynamics from considerations of relations of equilibrium and accessibility between states. Some of these attempts (e.g., Falk and Jung [5], Buchdahl [6], Landsberg [7]) involve ad hoc assumptions which differ only trivially from assuming existence of entropy ab initio. A consequential account of thermodynamics based on a concept of accessibility is given by Giles [S]. While the proof of the existence of entropy in this system is not open to logical objection, difficulties arise for the more elementary quantities such as pressure and temperature: it is possible to find a model obeying the axioms in Giles development in which temperature and pressure are not uniquely defined for any state; the problem, in other words, is that of the relation between entropy and the quantities of classical physics. The present work is based firstly on an attempt to investigate how far the existence of an entropy functions, continuous in terms of the mechanical quantities defining the state of a system, is a consequence of relations of * I wish to express my gratitude to the National Science Foundation whose grant GP-4070 in part supported the work for this paper. 172
of the U.S.A.
THE
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inaccessibility between states. It is also based on an observation that there is a flaw, apparently overlooked up to now, in the treatment of absolute temperature in the Caratheodory system. The second section of this article contains a critique of the classical approaches; it is shown that the forms of the second law of thermodynamics due to Caratheodory, and that due to Kelvin, are not adequate for the deduction of the Kelvin scale of absolute temperature, while that of Clausius is adequate. The inadequacy is overcome in classical physics by assuming the existence of ideal gases: but this assumption is not made by Caratheodory or by the later expositors of his theory [3-51 so that there is here a logical gap; and it is in any case illogical to base a physical theory on the assumption of existence of substances which do not exist in nature. The third section proves the existence of entropy from the assumption of relations of inaccessibility between states, provided that the state space obeys some very general topological conditions and the accessibility relations obey continuity conditions. The assumption is then made that systems can interact to form composite systems with accessibility assumptions related to those of the original system by natural rules; in that case the existence of an entropy function whose value for the composed system is the sum of its values for the original systems can be proved. The fifth section introduces the concept of temperature, and presents the form of the second law, which can be stated as follows: No state of a pair of thermodynamic systems in which they are not in thermal equilibrium is accessible from a state in which the systems are in thermal equilibrium and are in the same conjigurations as in the jirst mentioned state.
It is shown that the properties of absolute temperature follow from this statement, except that of continuity as a function of state. This form of the second law is in fact equivalent to that of Clausius. It seems impossible to prove more without assuming the existence of systems for which the differential form for the work done in small displacements (or for equilibrium conditions) is of a sufficient degree of smoothness; and this is discussed in the last section which demonstrates the role of the inverse of absolute temperature as an integrating factor. The axioms used in this article may be divided into groups concerning, respectively: - The structure of the state space s E - Energy interchanges - Accessibility of states Act Temp - Temperature - Interactions of states Int
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The axioms are numbered S I,2 ,... E 1,2 ,... Act I,2 ,... according to these groups. As is the case in the classical treatments, thermodynamics is treated here as a part of physics developed in the light of preliminary theories such as those of mechanics, electromagnetism, etc. We call these theories the ground theories. The axioms used in this article have been given a mathematical form and no attempt at description of the nature of the interactions between systems has been made; but it is appropriate to motivate these axioms by an account of physical models which obey them. These physical models consist of bodies which have two possible types of interaction: those governed exclusively by the laws of the ground theories (which we call adiabatic interactions) and those in which energy interchanges take place between systems which are in equilibrium as far as the ground theories are concerned (systems in diathermal contact). For a system to obey the axioms laid down here it needs to be capable of having any of its parts brought into diathermal contact, but to be in only adiabatic contact with the external world. The theory presented here is a theory of systems in equilibrium. For this reason it has appeared logical to avoid such concepts of classical theory such as the work done in quasistatic processes, which are either a contradiction in terms or limits of processes through nonequilibrium states which cannot be described in terms of equilibrium theory.
2.
CRITIQUE
OF CLASSICAL THEORIES
The three classical formulations of the second law of thermodynamics are as follows: L 1. CLAUSIUS’ LAW. No process is possible in nature, the sole results of which are that heat is transferredfrom a colder body to a hotter one.
L 2. KELVIN’S LAW (The impossibility of a perpetual motion machine of the second kind). No process is possible, the sole result of which is that a body is cooled and work is done.l L 3. CARATHBODORY’S LAW. In any neighborhood of any state s of an isolated thermodynamic system there exist states which cannot be reached from s by any possible processes. ’ The formulation due to Planck is equivalent: no process is possible, result of which is that a body is cooled and a weight raised.
the sole
THE
To this assumption and assumptions:
FOUNDATIONS
OF THERMODYNAMICS
classical theory adds (inter a&a) the following
175 definitions
(a) To any state s of a thermodynamic system corresponds a number r(s) which is such that two states si , s2 of two systems in thermal equilibrium are if and only if the values of T(S) for these two states are the same. Any such function is called an empirical temperature. (b) The work done in a small quasistatic adiabatic change of a thermodynamic system is given by a differential form dQ. From these assumptions, together with some other less important assumptions which need not be specified here, the following form of the second law is deduced.
L 4. THE ENTROPY LAW. There is a universal function of empirical temperature T(r), which is itserf an empirical temperature and a function of state q~such that dW = T dv. T is itself called the absolute temperature. We shall now discuss the relationship between these propositions; in doing so we take for granted the normal implicit assumptions (differentiability of functions involved etc.) made in the classical theories. To begin with, it is trivial that both Clausius’ Law and Kelvin’s Law imply Caratheodory’s Law, which was indeed intended by Caratheodory as a more general statement than either of them. For any thermodynamic system either contains pairs of heat reservoirs and mechanical systems on which work can be done or can be put into equilibrium with systems which contain such parts to form a larger thermodynamic system. (It is indeed an essential assumption of the theory that any two thermodynamic systems can interact to form a larger system.) Given any state s of the larger system, there exist states arbitrarily near it which differ from it only in that a small amount of heat has been transferred from a colder body to a hotter one, or in that a small amount of heat has been converted into mechanical work, and transition to these states is prohibited by L 1 or L 2, respectively. Thus: A.
EitherofLl
orL2impliesL3.
We now examine the reverse implications and the question of whether these laws imply L 4. To do so, we shall take it for granted that
B. L 3 implies that there is a universal function g(r) of empirical temperature such that dW = g(T) a$ for somefunction p. Subject to some reservations concerning his definition of empirical temperature and implicit assumptions about differentiability, this deduction
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COOPER
is carried through by Caratheodory. The point at which his argument and that of his followers breaks down is in the implicit assumption that g(r) is itself a temperature, that is to say that rr # ra implies that g(Ti) # g(7,). In fact, this assumption cannot be deduced from Caratheodory’s axiom L 3. For L 3 would hold if dW were itself a perfect differential and quasistatic processes were those with dW > 0; and in that case we would have g(r) a constant. We can use similar arguments to study the relations of L 1 and L 2 with the other axioms. Because of A, either of these implies the conclusion B. Let us now consider a system which passes through a Carnot cycle at empirical temperatures 71 and ~a , ra C 7i . It follows from B that if the change of entropy at 71 is Ap that at ~a is -Apl and the quantities of work done at these temperatures areg(rJAq, and -g(T.JAq, respectively. Clausius’ law asserts that these cannot be equal: otherwise the cycle would result only in a transfer of heat; and therefore g(rr) # g(r,) so that L 1 does imply L 4. Kelvin’s principle asserts only that it is impossible that g(r) # 0. Conversely, g(ri) = g(r,) implies L 1 and g(r) f 0 implies L 2. We conclude that: L 1 implies L 4. L 2 does not imply L 4: L 2 is equivalent to L 3 with the addition assumption that the work done in any isothermal change is non-zero.
of the
In classical thermodynamics it is possible to ignore the theoretical inequivalence of L 1 and L 2 because of the assumption that a perfect gas exists. Since g(r) can be proved to be a universal function, independent of the thermodynamic system, it is necessary to know only of one substance for which g(r) is a strictly monotonic function of absolute temperature in order to be able to infer that g(r) is an absolute temperature; and to this extent the difficulty can be overcome; however, in view of the fact that perfect gases do not exist it would appear preferable to use only Clausius’ form of the second law in a classical treatment since this needs no such additional assumptions. The inequivalence of the forms of the second law may conceivably have physical importance in the extension of thermodynamics to negative temperatures, which has been made in nuclear physics. (See Ramsay [9].) For such an extension of thermodynamics no equivalent of a perfect gas exists; in view of this, Ramsay’s statement that the three forms of the second law are equivalent for negative temperatures seems difficult to justify: for example, it is theoretically conceivable that a perpetual motion machine of the second kind could be universally impossible but that Clausius law does not hold for negative temperatures.
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177
3. ACCESSIBILITY CONDITIONS AND ENTROPY FUNCTIONS This section discusses the existence of an entropy function on the basis of accessibility relations. No assumption about temperature, etc. are made here: the sole assumptions in this section concern the nature of the phase space of a thermodynamic system and the existence and continuity of the accessibility relation. We assume then that we are concerned with a thermodynamic system 6 which has a set of states S, for which: S. 1. The state spaceS of a thermodynamic system 6 is a separabletopological space. By this is meant that 5’ is a topological space in which there is a countable set of points which is everywhere dense. This axiom is obeyed by almost all the state spaces in use in physical theories: the normal state spaces, in addition, have topologies based on a metric; but we do not need to assume this here. Later it will be shown that if the existence of a differential form for energy changes is assumed, obeying certain properties, the hypothesis of separability can be dropped. We assume further that there is a relationship which we write s12 s2 among the states of 6; this relationship may be read “a transition from s, to sa is possible”. We assume that transition between states is always possible in at least one direction, and that transition is continuous. We write s1 ++ sa for the negation of the statement s12 sa , and we write s, +-+ s2 if both s1F s2 and sa* si . If s1t+ s2 we say that s2 is reversibly accessible from s1, or that the two states are mutually accessible. If s, --t sa and sa -H s, we write si + sa. Formally, the assumptions made are firstly the normal axioms for a linear preorder: Act 1. (a) For any two states s1 and s2 either s1r=?rs2 OY s22 s. (b)
If s1=t sz and s2--t s3 then s12 s, .
(4
s1 z? s1.
The physical model for 6 is a system isolated from the external world by barriers impassible to heat but through which mechanical, electromagnetic, gravitational or other interactions with the external world are possible: these interactions will be summed up under the term interactions of the ground theories. The ground theories are the parts of physics established independently of thermodynamics such as mechanics, electromagnetic theory. Within 6, subsystems capable of being isolated by barriers impassible to heat may exist: but it must be assumed that these internal barriers can be removed, otherwise Act 1 (a) would not hold.
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DEFINITION. A function f(s) is an entropy function for reZation 2 if it is real valued and ;f s1 * s2 $ and only iff(sl)
an accessibility < f(sJ.
Attempts have been made (see references in Section 1) to prove the existence of an entropy function on the basis of the Caratheodory axiom L 3, which in this context would take the form: L 3’.
In any neighborhood of a state s there is a state s’ such that s -H s’.
The following example will show that L 3’ is inadequate to establish existence of an entropy function, even in very simple phase spaces. Let the state space be the Euclidean plane of points (x, y) and let (x1 , yi) =t (xa , y2) mean that yr < ya or that yr = ys and x1 < x2 . Suppose that f(s) were an entropy function for this accessibility relation. For any value of y let a(y) = inff(x, y), b(y) = supf(x, y). Then a(y) < b(y); and if yr < ys , b(y,) < a(ya). The open intervals )a(y), b(y)( are therefore nonempty nonoverlapping intervals on the real line; but there can only be a countable number of such intervals, whereas the possible set of values of y is noncountable; so we have a contradiction and conclude that no f(s) can exist. It will be noted that the accessibility relation obeys the Caratheodory axiom L 3’, since any neighborhood of (x, y) in the usual topology of the plane contains a point (x, y’) withy’ < y which is not accessible from (x, y). The existence of an entropy function will be proved here under the assumption that accessibility is continuous: we assume Act 2. If s, + s, then there are neighborhoods N(s,) and N(s,) of s1 , s2 , respectively,
such that if s E N(s,)
then s -+ s2 and if s E N(s,) then s, -+ s.
THEOREM 1. If a system obeys S 1, Act 1 and 2, then there is a continuous entropy function for -+.
Conditions Act I(b) and Act l(c) mean that ti is a preorder, and t) is easily seen to be an equivalence relation. To each a E S corresponds an equivalence class: the set of all s such that s t--* a; we write a” for this class and +.!? for the space of all equivalence classes. We write a < b if and only if there is an a E a^ and b E 6 such that a 3 b. Then it is easy to see that < is linear preorder in s. Introduce into s the interval topology in which a base of open sets is given by the sets of one of the forms )a”,6( = (2 : a^< 4 < 6j, ) C, d( = (2 : 2 < 8), )a”,--f (= (2 : $ > 6}. 3 is a Hausdorff space with this topology. The map I/ : a -+ a” is continuous in this topology: for if 8 is any open set in 3 in the interval topology and #(p) E 8 then there is an interval )S, 6(C 0 which contains g(p). Let a E a”,6 E 6, then by definition a -+ p ---f 6 and so by Act 2 there are neighborhoods N,(p), N,(p) of p such that a -+ N,(p) and N,(p) -+ b. N,(p) n N,(p) is then a neighborhood of p
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OF THERMODYNAMICS
which is accessible from a and from which b is accessible and so is mapped by # into )a, 6(. Since S is separable, there is a sequence (t,J dense in S. The images of these points form a linearly ordered countable set of points and so (Sierpinski [ll]) there is an order preserving map of the sequence onto a subset of the rational numbers. Call this map u, and write u(f) = sup {u($J; 2, < s”} for any s”E S; the definition is consistent in that if s” is a 2, , u(sA)coincides with u(&), as originally defined. If a^ < 6, then ~(a^) < U(J), and if aA,6 are both members of the sequence (&) the inequality is strict. In general, suppose that for some a”, 6 l2<6
u(d) = u(6).
(1)
The interval )6,6) = { 2 : a” < 2 < &} can contain at most one 2; for if there were more than one, say a” < & < !a < 6 then ~(a^) < ~(3,) < u(&) < u(b). If t^ is the sole member of (2,) in )a^, 6), the interval )a^, 2( must be empty because the (2,) are dense. The sets {s : 9(s) > Z}, {s : #(s) < i} are then disjoint open sets which, together, span S; so that this case can occur only if 5’ is disconnected and a” is the supremum of the values of s”corresponding to a component of S. Since each component is an open set and each open set contains a t, there can be at most a countable set of components of S, and therefore at most a countable (&J for which (2) is satisfied. Choose a sequence of positive numbers (q,J whose sum is finite, and write
D(f)= U(Q) + c {vn: dn< f}. Then v(SA)is a real-valued increasing function of s” and v(&) = v(&) implies that si = sa ; so that z, is strictly increasing. It remains to show that the mapping can be taken continuous, as well as order-preserving. Let, then, g be an order preserving map of S to the reals; we can suppose g bounded without loss of generality, since the entire real line has an order preserving map onto )O, I(. If g is discontinuous at k, then, for some 8 > 0, )g(G) - 8, g(4) + S( does not contain the image in g of any open interval of S; and this is possible only if one of the intervals )g(G) - S, g(4)(, )g(k), g(k) + 6( does not meet g(S). For any x put 6, = sup (6 : )g(f) - 6, g@)(n g(S) = m}
73: = sup (8 : l&q, g(q + qn g(Q = @a>. Put
S(i) = 6,
if
gw
-
r](2) = 7jrn
if
A4
+ 7124&%
8, c&,
V)
= 0
otherwise;
d4
= 0
otherwise.
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The set of points f for which any one of 6, , Q is nonzero is at most countable since the corresponding intervals )g(a) - 6, , g(G)(, )g(G), g(R) + v;( are disjoint and lie in a bounded set; and the sum of the nonzero 6; , ya is finite. For any s write h(f) = g(P) - c {S(f) : 2 < f> - c {7&q; 2 < f}. If a” < 6 the interval )g(a”),g(6)( contains those and only those intervals )g($), g(i) + Q( with a” < 2 < 6 and those intervals )g(G) - 6, , g(4)( with a” < 2 < 6 so that
whence
g(6)- c (72: f < 6)- c (6,: f < 6)
- c (7); : 4 < i} - 1 (6, : R < 6).
Equality is possible only if the entire interval )g(G), g(&)( is the union of the intervals )g(4) - 6, , g(f)(, )g($, g(4) + qa(, which lie in it and of the set of points g(4) for which these intervals are both nonempty; this is possible only if all the points g(2) in )g(a^),g(&)( are isolated points. However, if g(R) is isolated and with isolated neighbors, 6, and rla are zero; so that strict inequality holds also in this case; and it follows that h(s) is strictly increasing. If now 6, = S(2) # 0, the construction makes h(2) = sup{h(y) : 5 < a} and h is continuous on the left at w?;if 6, # 0, S(2) = 0, 4 is isolated on the left and continuity of any function on the left at !Zis automatic; continuity on the right follows similarly. Thus, h and so ho I/ are continuous everywhere.2 Theorem 1 applies strictly only to those thermodynamic systems which do not contain adiabatically permanently isolated components, which we have termed simple systems. For other systems the preorder defined by accessibility does not generate a linear order on the set of equivalence classes, but only a partial order. If the system has n components one would expect the partial order to be the natural partial order of Rn, real n-space, with x < y if X, < yr for all Y. To generalize this case we replace Act 1, assumption (a), which states effectively that every finite set of points has an order like that in R, by (a) below, and then we can prove the following theorem: ’ This theorem is clearly parallel to the known result (Sierpifiski [l 11) that a linearly ordered set with an order dense countable subset has an order preserving map to the reals. Direct deduction of it from this theorem does not appear any shorter than the proof given.
THE FOUNDATIONS THEOREM
l(a).
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181
If the state space S is a separable topological space and
if there is de$ned in it a relation + satisfying Act 1 (a) and (c), Act 2, and
Act l(a). Given any finite set A of states S there is a set of maps p, *.*p, of A to R such that Ord a) sr -+ ss if and only ifp,(s,) < p,(sz) for r = 1, 2,... n holds for uny pair of states in A, then there is a set of maps j1 **aj, of S to R such that Ord n holds for all s1 , s2 in S with j replacing p. The other question of interest is how far the assumption of the separability of the phase space is necessary for the truth of the theorem. The example given at the beginning of this section-that of R2 with lexicographic ordering-serves also as an example to show that separability is essential. For if R2 is lexicographically ordered and is given the interval topology associated with this order, then the continuity conditions are satisfied for this topology but, as we have seen, no real valued entropy function generating the order is possible. As this counter example involves a space which is not linear or even arcwise connected, unlike the normal state spaces of physical systems, we give one involving a linear space. Let S be the Hilbert space of all real valued functions of a real variable such that x 1f(x)]” < 00; for any real number a let e,(x) = 1 =0
if
x = a,
if
x#a
and let B, denote the ball of unit radius with centre 2e,. In S we define a preordering 2 by the conditions f 2 g if either (i) fisinnoB,, (ii)
fEB,,gEBbwitha
(iii)
f andg are in the same B, and
j/f - 2e, jl 2 ljg - 2e, /I.
Then it is easily seen that the preorder is continuous in the sense of Act 2, but that no entropy function exists for it.3 However, the following assumption makes it possible to apply Theorem 1 even in the nonseparable case: Act 3. The state space S contains a separable space S’ such that every state s satis$es s t) s’ for somestate in S’. * Dr. S. Rolewicz has constructed an example of a Hilbert Space of dimension & in which the preorder is realized by a continuous map to the ordinals of cardinal less than X1 .
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Clearly Act 1, 2 and 3 imply the conclusion of Theorem 1, since we can apply that theorem to S’. In the physical case a subspace S’ satisfying Act 3 can be found: the set of states having a constant configuration, as defined in Section 4. 4.
COMPOSITION
OF SYSTEMS: ADDITIVITY
OF ENTROPY
The entropy function whose existence is proved in Section 2 is very illdetermined: if p is any continuous strictly increasing map of the real line to itself, and f is any entropy function in the section of par 2, so is pof. It will now be shown that entropy can be determined uniquely up to linear maps on the assumption that a thermodynamic system (at least one particular one) can interact with (at least) three other such systems, and that the entropy is additive for interacting systems. DEFINITIONS. Systems 61 and 62 will be said to be isomorphic if there is a one-one map of the state spaceof G1 onto that of Gz which is a homeomorphism and preserves all thermodynamic relations. At present all that is required is preservation of the relation +. In sections in which more structure is postulated for thermodynamic systems the conditions for isomorphism are correspondingly stronger. A system G is called the composition of systems @, G,..., 6” and is written 6 = {G, G2,..., S%} if there is a homeomorphism of the product space Si X S2 **a x S” onto the state space of 6 which is such that if t s1* s2,--*>P> is the state corresponding to (s’, s2,..., s”) then
Int (a) {sl, s2,..., sr-l, sir, sr+l,..., P} -+ {si, s2,.,., sr-l, sZr,srfl,..., sn} if and only if sir -+ s2r. Int (b) If 6P is isomorphic with 6* the state derived from (sl ,..., sp,..., SQ,...) by permutation of s* and SQis reversibly accessible from it. Int (c) If A, B are complementary subsets of {I, 2,..., n> and GA is the composition (Sal, Gas,..., @D} and GB is {@I ,..., G*+ then B = (GA, V}. The significance of the last condition is that is extends (a). In the physical model, systems will be formed into a composition when they can interact so as to be in adiabatic connection and also so that they can be diathermally connected. Without the possibility of diathermal connection the manifold of states would not be linearly preordered. We now make the following assumption, about interaction of systems. Int 1. There is a class of elementary systems, which have simply connected state spaces. This class contains at least four systemsisomorphic to a system G. The composition of any four elementary systemsexists.
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We avoid the (at first sight natural) assumption that the composition of any two thermodynamic systems exists, because this would lead to an infinite regress and to the conclusion that all thermodynamic systems can be formed into an interacting system. THEOREM 2. Under the assumptions of Act l-3 and Int 1, there is a function which is defined on the state spacesof all elementary systems, which is entropy for any one system, and which is such that the function ~(9) + ~(9) of (9, s”} is an entropy for {G, G}.
v is uniquely determined for any one system by its values for two states of that system and, when so defined for one system, is defined for any other system by its value for one state of that system. possible choices of v are related linearly. The property required for v is that it is an entropy for 6
particular uniquely Any two and that
is1 , s2>-+ {s, ,s4) if and only if v(sl) + v(sJ < v(sJ + q(sJ. Suppose that for 6 (in the sense of the last section). Then p)(s) is a strictly increasing continuous function of p(s); say ~(0) = #(p(a)). The values of p(s) fill a connected interval of the real line, in one-one correspondence with the classes of reversibly accessible states of S; and by considering t/ instead of v we see that it is sufficient to prove the result on the assumption that the set of states S forms a segment of the real line. For the remainder of this proof we shall therefore speak of states when we mean (in the general case) equivalence classes of states, and regard them as represented by real numbers p(s). To begin with, we define ‘p for the system 6, postulated in Int 1, four isomorphs of which can interact. If p(s) is any entropy function for B, then any other entropy function for 6 is a continuous strictly increasing function of p(s), and we may therefore define p in terms of the values of p(s), which fill an interval of the real line. To simplify notation we shall write single letters s, U, v instead of p(s) etc.: this amounts to identifying reversibly accessible states and identifying them with the values of an entropy function.
p is any entropy function
LEMMA 1. If u, v are any two states there is just one state s such that {s, s} = (u, v}. We write s($(u + v)) for this state: s(*(u + v)) is a continuous strictly increasiragfunctzbn of {u, v}.
Let
qu,
4 = is : 694 + {u, VI>,
R(u, v) = {s : {u, v} + {s, s)}. Because of the continuity
of +, L and R are open. Neither
is empty,
since
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184
one of u, v belongs to each. Since S is connected neither L or R, and it must satisfy
there is an s which is in
Suppose now that s satisfies (1) and that p + s + 4. Then {P> PI -+ {s, 4 ++ b4 v> -
Q 41
so that s is unique. Also, we can find u1 , ua so that ui -+ u -+ u2 and {PY P> * @l Yv}-+ (us, V} --+{q, q} and then can find vr, vs so that uul-+ v + ~1sand {p, p} --f {ui , vi> --f {us , va} + {q, q>. Then if ui -+ u1 - us, v, -+ 01 + v2 we have {P, P> -
{u19 v’> + {q,qJ
so that p --f s(+(ul + vi)) -+ q, proving the continuity of s(&(u + v)). We now choose two states, s(O), s(l) say. W e write s(h) for s(&(s(O) + s(1))) and define S(T) for all dyadic rationals in (0, 1) by induction, putting
s((2p + 1)/2”) = s(&(p/29
+ s((P + 1)/2”-7)).
Then we have: LEMMA 2. If a, , a2 , a3 , a4 are dyadic a, + a2 = a3 + a4 then
rationals
in (0, 1) such that
If the a’s are of the form p/2” with n = 0 or 1 this is true by the definitions. Suppose it true if all a’s are of form p/2”-l; then if a, = (2~~ + 1)/2” {WV
s(al)~+4,
W>
f-) (s(PJ29,
S((Pl + 1)/2n-1), s(p,/29,
S((P, + 1)/2”-‘)I
*
{s(P,/29,
s((P, + 1)/2”-l),
s((P, + 1)/2”-9)
-
{W,
@A
44,
s(p,/2”-Y,
+dh
Here the first and third steps follow from Int (a) and the definitions and the second from the inductive hypothesis and Int (a). Now we prove LEMMA 3. Wh
sub
of s(a,),
The values of s(a) for dyadic rationals a are dense in the interval
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OF THERMODYNAMICS
If not, there is an interval (u, V) which contains no s(u), but is such that every interval containing (24,~) properly contains an s(u). Let p = inf{u : z, f-f s(u)}.
a = sup{a : s(u) 4-+ U},
Then 01= p for otherwise (01,/3) would contain a dyadic Y and u + S(Y) + o. The values of s(+(s(p) + s(q))), p < OL< q, are of the form S(U),and tend to 44@ + 4) as P + 01,q -+ 6: thus u < s(u) < v(0) for some a. Here the first and third steps follow from Int (c), and the second from the inductive hypothesis and Int (c). Now for any state u with s(O)+ u + s(l) write v(u) = inf{u : a dyadic rational, IL t
s(u)}
Then if #(u) = sup{u : a dyadic rational, s(u) --t u}, we have #(u) = a(u); for, if not, then the interval ($(u), p)(u)) is nonempty, and so contains a dyadic rational Y for which II -+ S(Y) -+ u, a contradiction. Further, if u ---f v then p)(u) < p)(w). For, if not, y(u) = v(w); we can suppose without loss of generality that u is the state with least and o that with greatest entropy for which v(s) has this common value. For any positive E any neighborhood of u contains a state s(u) with p)(u) - E < a < v(u) and any neighborhood of ‘u a state s(b) with p)(v) < b < v,(v) + E. Since s(Q(u + w)) is continuous in u and v and u --f s(Q(u + v)) + v, the neighborhood can be chosen so that s(Q[.r(u) + s(b)]) is in (u, v); but, by Lemma 2,
so that v(u) < *(a + b) < v(v), contradicting p)(u) = v(o). v(u) is therefore a strictly increasing function of u, hence v-,-‘(u) is continuous. If we write s(x) = u when p)(u) = x, it follows from Lemma 2, by considering approximations of xi by dyadic rationals, that Wl)> 4x2)) t-) w%), 4x4)) if and only if Xl +
x2
=
x3
+
x4
,
and then that x1 + x2 < x3 + x4 implies w%)>
s(x2))
+-+
w%),
G4h
so that {Ul 9u2>
-
‘*)&3
<
(aP(%)
t u4)
if and only if d%) +
du2)
+
?+44)*
186
COOPER
The definition of p)(s) can be extended outside (s(O), s(l)} by defining s(x) for 1 < x < 2 by w,
42 - 41 = {s(l), s(l)1
and, by induction on n, for n < x < (n + 1) by w,
s(n + 1 - 4> f-t {s(l), w>.
The process will end if there is an n such that
for every s: otherwise it will continue for indefinitely definition applies to negative X. Writing x = ~(0) if for an interval of the real line and additivity follows If 61 is any other system, choose an initial state S(x) as m, e9) t-) w, W)l
large X; and a similar s = s(x), y is defined as before. G(O), and then define
and put ~($1) = x if s1 = $(x). The statements concerning uniqueness of p are readily proved. To prove additivity, note that if G1G2 are any two elementary systems wd
s2@2)l
+ W3)~ s2(x4N
if and only if W), @)> sl(xl>, s2(x2N-
W)~ 40), +.a
s2(x4N,
and hence if and only if {4x1),
@2),
W), s2(0)l -+ W3), 4x4), syo>, s2(0))
i.e., Wl),
+ Is(xJ, 4x4))
G2N
thus if and only if Xl +
x2
< x3 +
x4
.
This extends the proof of the additivity of entropy to arbitrary pairs of systems. The assumption that up to four elementary systems can be combined is essential to the proof of additivity of entropy just given: and the result certainly does not follow if only combinations of two systems are supposed to exist. This is shown by the following counter example. Suppose that the states of a system are represented isotonically and continuously by the positive numbers, and suppose that if and only if U12+ 241242 + u22 < 2432+ UQU4+ u42.
THE
FOUNDATIONS
187
OF THERMODYNAMICS
Then
if S(O) = 0, s(1) = 1, we have 3~($)~ = 1, so that ~(a) = l/d; it is easily verified that {s(i), s(S)} and s(i) = g , s(2) = )(4 + 1/3y2: {s(i), s(Q)} are not reversibly accessible.
5.
INTERNAL
ENERGY
AND
TEMPERATURE
Further development depends on more specialized assumptions, and in particular on postulates concerning the types of interaction and energy interchange between systems. DEFINITION. Two systems are in adiabatic interaction sf their equilibrium conditions, changes of state and energy interchanges are completely spec$ied by the laws of the ground theories. A system is adiabatically isolated if its interactions with all other systems are adiabatic.
The law of conservation
of energy takes the following
form:
E 1. A thermodynamic system has a continuous function of state called its internal energy and written E(s). In a change of state of an adiabatically isolated system the increase of E(s) is equal to the total work done on the system by systemsexternal to it according to the ground theories. A thermodynamic systemis simple ifits state spaceis homeomorDEFINITION. phic with a product X x E where X is a normed linear space and E is an interval of the real line. To each state correspondsa point (x, E(s)) of X x E, where x is the configuration of the system according to the ground theories and E(s) is its internal energy. E 2. The internal energy of the composition of two systems is the sum of the internal energies of the component systems. Two thermodynamic systems are in diathermal interaction if DEFINITION. exchanges of internal energy between them are possible without any changes of conjiguration. They are in thermal equilibrium sf their states are such that they remain unaltered when put into diathermal interaction. Thermal
equilibrium
is governed
by the following
assumption.
Temp 1. There exits a real-valued function O(s) defined for all states of all simple thermodynamic systems, which is such that two states s1 and s2 of any two systems are in thermal equilibrium if and onb zf e(9) = O(s2).For a fixed configuration x, 0(x, E) is a strictly increasing function of E.
188
COOPER
In some expositions of thermodynamics the existence of temperature is deduced from other axioms. Miller [12] and Landsberg [9] give arguments deducing the existence of temperature from the postulate that thermal equilibrium is an equivalence relationship; it is easily seen that such deductions must be fallacious, since an equivalence relationship could involve see Whaples [13] for a detailed equality of two distinct co-ordinates; discussion. Caratheodory states that existence of temperature can be inferred from axioms that bodies in fixed configurations change their nonconfiguration co-ordinates to those of states in which they are in thermal equilibrium; but this does not allow us to deduce that every state has a temperature. It seems that any reasonable assumptions which allow the existence of temperature to be deduced differ only tautologically from the statement that it does exist. In what follows any function satisfying the conditions laid down for e(s) in the axiom Temp will be called a temperature. THEOREM 3. The composition of two simple systemsin thermal equilibrium is a simple system.
A state (si, sa> in which the systems are in thermal equilibrium is completely determined by the configurations x1, x2 corresponding to s1 and s2 and by the common temperature 0. The internal energies El, E2 of systems are determined by x1, x2 and B because of Temp 1; and so El + E2 is determined by B and is a continuous strictly increasing function of 6 and so xi, x2, El + E2 determine 8 and so El, E2 and the states sl, s2. If 6, is any other system, in a state s3, in diathermal contact with {G’, @}, then it is in diathermal contact with one of G, G2 and so energy interchange between G3 and {Sl, G2} occurs unless 8(S) = O(sl) = 0(s2) = 0. 0 is therefore a temperature for (~1, ~a}.
6. ABSOLUTE TEMPERATURE The last section puts us in a position to state a form of the second law of thermodynamics which is adequate for all subsequent development. Act. 4. If G1 and G2 are any two systemswhich are in thermaf equilibrium an so’, so2,then no other state (9, s2} of the composedsystem (G, G2} in whkh each system is the same configuration is accessiblefrom the state {sol, s,,“}. This could be regarded as a weaker form of the Clausius Law: it states that heat cannot pass between two bodies at the same temperature without change of configuration. We can now prove the existence of an absolute temperature.
THE
FOUNDATIONS
189
OF THERMODYNAMICS
THEOREM 4. For any $xed con$guration of a simple system, the entropy (as normalized in Section 4) is a differentiable function of the internal energy for all save a countable set of points: and at thesepoints it is dz@rentiable on the left and on the right. The derivative, where it exists:
g(x, E) = ii
dx, E + 4 - dx, E) h
,
or more generally the left and right hand derivatives, g*(x, E) = lim
a@, E + h) - dx, 4
h
h+Of
,
depend only on the temperature and are stricUy increasing functions of temperature. Consider first a composition ((3, G2) in which G1 and 6s are isomorphic. If x is any configuration, the states {(x, El), (x, E2)} accessible without change of configuration from a state {(x, &I), (x, EOz)} must satisfy El + E2 = EO1+ E02 by the first law (E 1). According to Act 4, no state ((x, El), (x, E2)) is accessibIe from ((x, E), (x, E)] where E = &(El + E2) and so, because of the additivity of the entropy we have
29)(x, B(El + E2)) > v(x, El) + v(x, E2), so that, because of the continuity of v, -v(x, E) is a convex function of E. The existence of left- and right-hand derivatives everywhere, the existence of a derivative at all but a countable set of points, and the property that the derivatives are strictly decreasing now follow from known theorems on convex functions (e.g. [14]). Now, let 61 and G2 be any two simple systems with composition (~1, ~2). For any configurations x1, x2 the same arguments show that dx’,
El) + dx2, E*) -=c94~2~9
E,‘) + v(x2, Ea2)
when
El + E2 = EOl + Eo2
and
0(x1, Eal) = 0(x2, E02).
Let E be such that av(xl, E)/E and +(x2, E)jE exist at E,, . Then it follows from the usual arguments concerning conditional that g(xl, W) = g(x2, Ea2)
maxima
so that it is proved that the derivatives are equal for equal temperatures when they exist. That the left- and right-hand derivatives are equal follows: in general g,(x, EO) is the supremum of all the values of g(x, E) for those values of E > E, for which g(x, E) exists.
190
COOPER
The quantity g+(x, E) is thus a strictly monotonic function of temperature independent of the system for which it is defined; its reciprocal l/g+(x, E), which increases with temperature is called (up to a constant multiple) the absolute temperature. These postulates leave open the possibility that the absolute temperature may have up to a countable number of discontinuities. Actually no more than this can be proved from the classical forms of the second law: to prove the continuity of absolute temperature (say as a function of an empirical temperature) conditions bearing on the smoothness of the system are needed.
7. AESSOLUTETEMPERATURE AS AN INTEGRATING FACTOR In the classical theory absolute temperature appears as the reciprocal of a multiplier which makes the differential form for the work done integrable. By postulating the existence and the smoothness of this form at any rate for one particular system these results can be deduced and the continuity of absolute temperature proved. A system 6 will be called smooth if it satisfies the following: (a) 6 is simple, and its state space has the form S = X x E, where X is a normed linear space. (b) If x1 is a fixed configuration then if (x1 , E,), (x, E) have the same temperature, E is a differentiable function of El and of x. (c) There is for each state s a linear functional on X, p(s), such that for each h E X, (p(s), h) (the value of p(s) at x) is a continuous function of s and for any E,E, there is a constant K so that
(d) The states reversibly accessible from a given state s lie on a manifold whose tangent vectors at s, (dx, dE), satisfy dE + (p(s), dx) = 0. (e) The manifold of states reversibly accessible from s has no tangent vectors in common with the manifold of states in thermal equilibrium with S. In the physical model (p, dx) is the work done by the system in a small displacement dx of its configuration. (e) means that temperature alters in a quasistatic adiabatic process. More generally, we say that a simple system has a smooth direction h if there is a subset of its configuratian space X, which is in one-one correspondence with the real line (say, X, = (st, + th : t real)) and the postulates
THE
FOUNDATIONS
191
OF THERMODYNAMICS
above are obeyed for displacements in X,,: in particular there is a number ph(s) so that dE + p&) dt = 0 (1) in a reversible displacement (dE, dt). We now postulate: There is a simple system with a smooth direction. Let B be such a system. Then the form (1) has an integrating factor. If h(B) is an integrating factor which depends only on temperature 13,then it is easy to see that h(0) es aE/ae logh(B,) = I h se/at -p(ae/aE)
(2)
de
Vectors (dt, dE) for which 0 is constant satisfy
ae
Ed&O:
atdt+
aE
those representing reversible transitions satisfy p dt + dE = 0. These coincide in direction if and only if
ae
--paE=o, at
ae
and hence, because of (e), this equation does not hold. It follows that the integral in (2) defines a continuous function of es, when h(B,) is chosen arbitrarily, and if G(0) = exp h(B) (3) g(e) is an integrating factor for (1). On the other hand, given any t, , the equation
g+P(t,r)
=o,
-f-W = 4 ,
(4)
determines y uniquely as a function of t because of the continuity conditions (a) of the definition of a smooth system. Let y(0) = E,, for this solution. Then we have for the entropy v, dt, , 4) = j-%,+(0, E) dE + ~(0, 01,
(5)
0
which is due to the results of the last section and because along the curve defined by (4).
Q
is constant
192
COOPER
Let the initial state designated (0, 0) be chosen to have a temperature at which g(0) exists so that g, = g- are continuous in a neighborhood of this state. Then for displacements (AE, , At,) from these states (2) and (1) give respectively
4 = k+P, 0) + O(l)14, W, - A-%1+ [P + O(l)14 = 0, whence ‘VP,) = k+(O, ‘3 + WMAE,
+ ~4).
Thus F is differentiable and dy = g+(O)(dE+ P dt).
(6)
If G is any other integrating factor, then the integration gives a differentiable function of v,, so that G =f(~)g+(e) for some function of ‘p. The function G defined by (3) is a function of B only, say, G(Q) = f(vk+(@) = fh+? ~Ng+(e),
(7)
so that f(p(fJ, t)) is a function of 8 only. Because of condition (e) +/at is nowhere zero: hence varying t alone varies q~.It follows that f is a constant, and so that g+P) = C exp w, where y&J is given by (2). Thus g+(0) 1s . continuous and hence g(B) exists for all 8. Now let S be any smooth system. The same arguments as those above, applied to any direction in S, show that 6 = &WE
+ (P, d+)
(8)
In summary, if there is a system which is smooth in some one direction, then the absolute temperature T is a continuous function of empirical temperature and ag,=L i3E
T’
If 5’ is a smooth system, q~is differentiable and T 6 = (d. + (P, W). REFERENCES
1. 2. 3. 4.
M. ZEMANSKY. “Heat and Thermodynamics.” New York, 1957. M. BORN. Phys. Zeut 22 (1921), 218, 249, 282. M. BORN. “National Philosophy of Cause and Chance.” Oxford, London, C. CARATH~ODORY. Math. Ann. 67 (1909), 35.5.
1949.
THE
FOUNDATIONS
OF THERMODYNAMICS
193
5. FALK AND JUNG. “Handbuch des Physik,” Vol. III, p. 2. Springer, Berlin, 1959. 6. H. A. BUCHDAHL. Z. P/zys. 152 (1958), 425. 7. P. T. LANDSBERG. Physica Status Solidi 1 (1961), 120. Foundations of Thermodynamics.” Pergamon, London, 8. P. GILES. “Mathematical 1964. 9. P. T. LANDSBERG. Rev. Modern Phys. 28 (1959), 389. 10. N. F. RAMSAY. Phys. Rev. 103 (1956), 20. 11. W. SIERPINSKI. “Lecons sur les nombres transfinies.” Paris, 1928. 12. A. R. MILLER. Amer. J. Phys. (1952), 488. 13. G. WHAPLFS. /. Rational Mech. Anal. 1 (1952), 301. 14. HARDY, LITTLEWOOD, AND POLYA. “Inequalities.” Cambridge.
OF MATHEMATICAL
ANALYSIS
The Foundations
AND
APPLICATIONS
17, 172-193 (1967)
of Thermodynamics*
J. L. B. COOPER Department
of Mathematics, California Institute and University of Toronto
of Technology
Submitted by F. V. Atkinson
1.
INTRODUCTION
A central result of classical macroscopic thermodynamics is the deduction of the existence and properties of entropy from forms of the second law derived from experience. The arguments involve, in addition to the first and second laws, assumptions about heat and temperature and the properties of perfect gases (see, for example, the excellent account in Zemansky [l]). They have been criticized by mathematical physicists (see Born [2] and [3]) on the grounds that these assumptions involve redundancies, and a development free from these criticisms was made by Caratheodory [4], whose treatment does not involve heat and temperature as initial concepts, but is based on development of thermodynamic concepts in terms of mechanical concepts. Since Caratheodory’s work numerous attempts have been made to generalize his approach by deducing the laws of thermodynamics from considerations of relations of equilibrium and accessibility between states. Some of these attempts (e.g., Falk and Jung [5], Buchdahl [6], Landsberg [7]) involve ad hoc assumptions which differ only trivially from assuming existence of entropy ab initio. A consequential account of thermodynamics based on a concept of accessibility is given by Giles [S]. While the proof of the existence of entropy in this system is not open to logical objection, difficulties arise for the more elementary quantities such as pressure and temperature: it is possible to find a model obeying the axioms in Giles development in which temperature and pressure are not uniquely defined for any state; the problem, in other words, is that of the relation between entropy and the quantities of classical physics. The present work is based firstly on an attempt to investigate how far the existence of an entropy functions, continuous in terms of the mechanical quantities defining the state of a system, is a consequence of relations of * I wish to express my gratitude to the National Science Foundation whose grant GP-4070 in part supported the work for this paper. 172
of the U.S.A.
THE
FOUNDATIONS
OF THERMODYNAMICS
173
inaccessibility between states. It is also based on an observation that there is a flaw, apparently overlooked up to now, in the treatment of absolute temperature in the Caratheodory system. The second section of this article contains a critique of the classical approaches; it is shown that the forms of the second law of thermodynamics due to Caratheodory, and that due to Kelvin, are not adequate for the deduction of the Kelvin scale of absolute temperature, while that of Clausius is adequate. The inadequacy is overcome in classical physics by assuming the existence of ideal gases: but this assumption is not made by Caratheodory or by the later expositors of his theory [3-51 so that there is here a logical gap; and it is in any case illogical to base a physical theory on the assumption of existence of substances which do not exist in nature. The third section proves the existence of entropy from the assumption of relations of inaccessibility between states, provided that the state space obeys some very general topological conditions and the accessibility relations obey continuity conditions. The assumption is then made that systems can interact to form composite systems with accessibility assumptions related to those of the original system by natural rules; in that case the existence of an entropy function whose value for the composed system is the sum of its values for the original systems can be proved. The fifth section introduces the concept of temperature, and presents the form of the second law, which can be stated as follows: No state of a pair of thermodynamic systems in which they are not in thermal equilibrium is accessible from a state in which the systems are in thermal equilibrium and are in the same conjigurations as in the jirst mentioned state.
It is shown that the properties of absolute temperature follow from this statement, except that of continuity as a function of state. This form of the second law is in fact equivalent to that of Clausius. It seems impossible to prove more without assuming the existence of systems for which the differential form for the work done in small displacements (or for equilibrium conditions) is of a sufficient degree of smoothness; and this is discussed in the last section which demonstrates the role of the inverse of absolute temperature as an integrating factor. The axioms used in this article may be divided into groups concerning, respectively: - The structure of the state space s E - Energy interchanges - Accessibility of states Act Temp - Temperature - Interactions of states Int
174
COOPER
The axioms are numbered S I,2 ,... E 1,2 ,... Act I,2 ,... according to these groups. As is the case in the classical treatments, thermodynamics is treated here as a part of physics developed in the light of preliminary theories such as those of mechanics, electromagnetism, etc. We call these theories the ground theories. The axioms used in this article have been given a mathematical form and no attempt at description of the nature of the interactions between systems has been made; but it is appropriate to motivate these axioms by an account of physical models which obey them. These physical models consist of bodies which have two possible types of interaction: those governed exclusively by the laws of the ground theories (which we call adiabatic interactions) and those in which energy interchanges take place between systems which are in equilibrium as far as the ground theories are concerned (systems in diathermal contact). For a system to obey the axioms laid down here it needs to be capable of having any of its parts brought into diathermal contact, but to be in only adiabatic contact with the external world. The theory presented here is a theory of systems in equilibrium. For this reason it has appeared logical to avoid such concepts of classical theory such as the work done in quasistatic processes, which are either a contradiction in terms or limits of processes through nonequilibrium states which cannot be described in terms of equilibrium theory.
2.
CRITIQUE
OF CLASSICAL THEORIES
The three classical formulations of the second law of thermodynamics are as follows: L 1. CLAUSIUS’ LAW. No process is possible in nature, the sole results of which are that heat is transferredfrom a colder body to a hotter one.
L 2. KELVIN’S LAW (The impossibility of a perpetual motion machine of the second kind). No process is possible, the sole result of which is that a body is cooled and work is done.l L 3. CARATHBODORY’S LAW. In any neighborhood of any state s of an isolated thermodynamic system there exist states which cannot be reached from s by any possible processes. ’ The formulation due to Planck is equivalent: no process is possible, result of which is that a body is cooled and a weight raised.
the sole
THE
To this assumption and assumptions:
FOUNDATIONS
OF THERMODYNAMICS
classical theory adds (inter a&a) the following
175 definitions
(a) To any state s of a thermodynamic system corresponds a number r(s) which is such that two states si , s2 of two systems in thermal equilibrium are if and only if the values of T(S) for these two states are the same. Any such function is called an empirical temperature. (b) The work done in a small quasistatic adiabatic change of a thermodynamic system is given by a differential form dQ. From these assumptions, together with some other less important assumptions which need not be specified here, the following form of the second law is deduced.
L 4. THE ENTROPY LAW. There is a universal function of empirical temperature T(r), which is itserf an empirical temperature and a function of state q~such that dW = T dv. T is itself called the absolute temperature. We shall now discuss the relationship between these propositions; in doing so we take for granted the normal implicit assumptions (differentiability of functions involved etc.) made in the classical theories. To begin with, it is trivial that both Clausius’ Law and Kelvin’s Law imply Caratheodory’s Law, which was indeed intended by Caratheodory as a more general statement than either of them. For any thermodynamic system either contains pairs of heat reservoirs and mechanical systems on which work can be done or can be put into equilibrium with systems which contain such parts to form a larger thermodynamic system. (It is indeed an essential assumption of the theory that any two thermodynamic systems can interact to form a larger system.) Given any state s of the larger system, there exist states arbitrarily near it which differ from it only in that a small amount of heat has been transferred from a colder body to a hotter one, or in that a small amount of heat has been converted into mechanical work, and transition to these states is prohibited by L 1 or L 2, respectively. Thus: A.
EitherofLl
orL2impliesL3.
We now examine the reverse implications and the question of whether these laws imply L 4. To do so, we shall take it for granted that
B. L 3 implies that there is a universal function g(r) of empirical temperature such that dW = g(T) a$ for somefunction p. Subject to some reservations concerning his definition of empirical temperature and implicit assumptions about differentiability, this deduction
176
COOPER
is carried through by Caratheodory. The point at which his argument and that of his followers breaks down is in the implicit assumption that g(r) is itself a temperature, that is to say that rr # ra implies that g(Ti) # g(7,). In fact, this assumption cannot be deduced from Caratheodory’s axiom L 3. For L 3 would hold if dW were itself a perfect differential and quasistatic processes were those with dW > 0; and in that case we would have g(r) a constant. We can use similar arguments to study the relations of L 1 and L 2 with the other axioms. Because of A, either of these implies the conclusion B. Let us now consider a system which passes through a Carnot cycle at empirical temperatures 71 and ~a , ra C 7i . It follows from B that if the change of entropy at 71 is Ap that at ~a is -Apl and the quantities of work done at these temperatures areg(rJAq, and -g(T.JAq, respectively. Clausius’ law asserts that these cannot be equal: otherwise the cycle would result only in a transfer of heat; and therefore g(rr) # g(r,) so that L 1 does imply L 4. Kelvin’s principle asserts only that it is impossible that g(r) # 0. Conversely, g(ri) = g(r,) implies L 1 and g(r) f 0 implies L 2. We conclude that: L 1 implies L 4. L 2 does not imply L 4: L 2 is equivalent to L 3 with the addition assumption that the work done in any isothermal change is non-zero.
of the
In classical thermodynamics it is possible to ignore the theoretical inequivalence of L 1 and L 2 because of the assumption that a perfect gas exists. Since g(r) can be proved to be a universal function, independent of the thermodynamic system, it is necessary to know only of one substance for which g(r) is a strictly monotonic function of absolute temperature in order to be able to infer that g(r) is an absolute temperature; and to this extent the difficulty can be overcome; however, in view of the fact that perfect gases do not exist it would appear preferable to use only Clausius’ form of the second law in a classical treatment since this needs no such additional assumptions. The inequivalence of the forms of the second law may conceivably have physical importance in the extension of thermodynamics to negative temperatures, which has been made in nuclear physics. (See Ramsay [9].) For such an extension of thermodynamics no equivalent of a perfect gas exists; in view of this, Ramsay’s statement that the three forms of the second law are equivalent for negative temperatures seems difficult to justify: for example, it is theoretically conceivable that a perpetual motion machine of the second kind could be universally impossible but that Clausius law does not hold for negative temperatures.
THE FOUNDATIONS OF THERMODYNAMICS
177
3. ACCESSIBILITY CONDITIONS AND ENTROPY FUNCTIONS This section discusses the existence of an entropy function on the basis of accessibility relations. No assumption about temperature, etc. are made here: the sole assumptions in this section concern the nature of the phase space of a thermodynamic system and the existence and continuity of the accessibility relation. We assume then that we are concerned with a thermodynamic system 6 which has a set of states S, for which: S. 1. The state spaceS of a thermodynamic system 6 is a separabletopological space. By this is meant that 5’ is a topological space in which there is a countable set of points which is everywhere dense. This axiom is obeyed by almost all the state spaces in use in physical theories: the normal state spaces, in addition, have topologies based on a metric; but we do not need to assume this here. Later it will be shown that if the existence of a differential form for energy changes is assumed, obeying certain properties, the hypothesis of separability can be dropped. We assume further that there is a relationship which we write s12 s2 among the states of 6; this relationship may be read “a transition from s, to sa is possible”. We assume that transition between states is always possible in at least one direction, and that transition is continuous. We write s1 ++ sa for the negation of the statement s12 sa , and we write s, +-+ s2 if both s1F s2 and sa* si . If s1t+ s2 we say that s2 is reversibly accessible from s1, or that the two states are mutually accessible. If s, --t sa and sa -H s, we write si + sa. Formally, the assumptions made are firstly the normal axioms for a linear preorder: Act 1. (a) For any two states s1 and s2 either s1r=?rs2 OY s22 s. (b)
If s1=t sz and s2--t s3 then s12 s, .
(4
s1 z? s1.
The physical model for 6 is a system isolated from the external world by barriers impassible to heat but through which mechanical, electromagnetic, gravitational or other interactions with the external world are possible: these interactions will be summed up under the term interactions of the ground theories. The ground theories are the parts of physics established independently of thermodynamics such as mechanics, electromagnetic theory. Within 6, subsystems capable of being isolated by barriers impassible to heat may exist: but it must be assumed that these internal barriers can be removed, otherwise Act 1 (a) would not hold.
COOPER
178
DEFINITION. A function f(s) is an entropy function for reZation 2 if it is real valued and ;f s1 * s2 $ and only iff(sl)
an accessibility < f(sJ.
Attempts have been made (see references in Section 1) to prove the existence of an entropy function on the basis of the Caratheodory axiom L 3, which in this context would take the form: L 3’.
In any neighborhood of a state s there is a state s’ such that s -H s’.
The following example will show that L 3’ is inadequate to establish existence of an entropy function, even in very simple phase spaces. Let the state space be the Euclidean plane of points (x, y) and let (x1 , yi) =t (xa , y2) mean that yr < ya or that yr = ys and x1 < x2 . Suppose that f(s) were an entropy function for this accessibility relation. For any value of y let a(y) = inff(x, y), b(y) = supf(x, y). Then a(y) < b(y); and if yr < ys , b(y,) < a(ya). The open intervals )a(y), b(y)( are therefore nonempty nonoverlapping intervals on the real line; but there can only be a countable number of such intervals, whereas the possible set of values of y is noncountable; so we have a contradiction and conclude that no f(s) can exist. It will be noted that the accessibility relation obeys the Caratheodory axiom L 3’, since any neighborhood of (x, y) in the usual topology of the plane contains a point (x, y’) withy’ < y which is not accessible from (x, y). The existence of an entropy function will be proved here under the assumption that accessibility is continuous: we assume Act 2. If s, + s, then there are neighborhoods N(s,) and N(s,) of s1 , s2 , respectively,
such that if s E N(s,)
then s -+ s2 and if s E N(s,) then s, -+ s.
THEOREM 1. If a system obeys S 1, Act 1 and 2, then there is a continuous entropy function for -+.
Conditions Act I(b) and Act l(c) mean that ti is a preorder, and t) is easily seen to be an equivalence relation. To each a E S corresponds an equivalence class: the set of all s such that s t--* a; we write a” for this class and +.!? for the space of all equivalence classes. We write a < b if and only if there is an a E a^ and b E 6 such that a 3 b. Then it is easy to see that < is linear preorder in s. Introduce into s the interval topology in which a base of open sets is given by the sets of one of the forms )a”,6( = (2 : a^< 4 < 6j, ) C, d( = (2 : 2 < 8), )a”,--f (= (2 : $ > 6}. 3 is a Hausdorff space with this topology. The map I/ : a -+ a” is continuous in this topology: for if 8 is any open set in 3 in the interval topology and #(p) E 8 then there is an interval )S, 6(C 0 which contains g(p). Let a E a”,6 E 6, then by definition a -+ p ---f 6 and so by Act 2 there are neighborhoods N,(p), N,(p) of p such that a -+ N,(p) and N,(p) -+ b. N,(p) n N,(p) is then a neighborhood of p
THE
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179
OF THERMODYNAMICS
which is accessible from a and from which b is accessible and so is mapped by # into )a, 6(. Since S is separable, there is a sequence (t,J dense in S. The images of these points form a linearly ordered countable set of points and so (Sierpinski [ll]) there is an order preserving map of the sequence onto a subset of the rational numbers. Call this map u, and write u(f) = sup {u($J; 2, < s”} for any s”E S; the definition is consistent in that if s” is a 2, , u(sA)coincides with u(&), as originally defined. If a^ < 6, then ~(a^) < U(J), and if aA,6 are both members of the sequence (&) the inequality is strict. In general, suppose that for some a”, 6 l2<6
u(d) = u(6).
(1)
The interval )6,6) = { 2 : a” < 2 < &} can contain at most one 2; for if there were more than one, say a” < & < !a < 6 then ~(a^) < ~(3,) < u(&) < u(b). If t^ is the sole member of (2,) in )a^, 6), the interval )a^, 2( must be empty because the (2,) are dense. The sets {s : 9(s) > Z}, {s : #(s) < i} are then disjoint open sets which, together, span S; so that this case can occur only if 5’ is disconnected and a” is the supremum of the values of s”corresponding to a component of S. Since each component is an open set and each open set contains a t, there can be at most a countable set of components of S, and therefore at most a countable (&J for which (2) is satisfied. Choose a sequence of positive numbers (q,J whose sum is finite, and write
D(f)= U(Q) + c {vn: dn< f}. Then v(SA)is a real-valued increasing function of s” and v(&) = v(&) implies that si = sa ; so that z, is strictly increasing. It remains to show that the mapping can be taken continuous, as well as order-preserving. Let, then, g be an order preserving map of S to the reals; we can suppose g bounded without loss of generality, since the entire real line has an order preserving map onto )O, I(. If g is discontinuous at k, then, for some 8 > 0, )g(G) - 8, g(4) + S( does not contain the image in g of any open interval of S; and this is possible only if one of the intervals )g(G) - S, g(4)(, )g(k), g(k) + 6( does not meet g(S). For any x put 6, = sup (6 : )g(f) - 6, g@)(n g(S) = m}
73: = sup (8 : l&q, g(q + qn g(Q = @a>. Put
S(i) = 6,
if
gw
-
r](2) = 7jrn
if
A4
+ 7124&%
8, c&,
V)
= 0
otherwise;
d4
= 0
otherwise.
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COOPER
The set of points f for which any one of 6, , Q is nonzero is at most countable since the corresponding intervals )g(a) - 6, , g(G)(, )g(G), g(R) + v;( are disjoint and lie in a bounded set; and the sum of the nonzero 6; , ya is finite. For any s write h(f) = g(P) - c {S(f) : 2 < f> - c {7&q; 2 < f}. If a” < 6 the interval )g(a”),g(6)( contains those and only those intervals )g($), g(i) + Q( with a” < 2 < 6 and those intervals )g(G) - 6, , g(4)( with a” < 2 < 6 so that
whence
g(6)- c (72: f < 6)- c (6,: f < 6)
- c (7); : 4 < i} - 1 (6, : R < 6).
Equality is possible only if the entire interval )g(G), g(&)( is the union of the intervals )g(4) - 6, , g(f)(, )g($, g(4) + qa(, which lie in it and of the set of points g(4) for which these intervals are both nonempty; this is possible only if all the points g(2) in )g(a^),g(&)( are isolated points. However, if g(R) is isolated and with isolated neighbors, 6, and rla are zero; so that strict inequality holds also in this case; and it follows that h(s) is strictly increasing. If now 6, = S(2) # 0, the construction makes h(2) = sup{h(y) : 5 < a} and h is continuous on the left at w?;if 6, # 0, S(2) = 0, 4 is isolated on the left and continuity of any function on the left at !Zis automatic; continuity on the right follows similarly. Thus, h and so ho I/ are continuous everywhere.2 Theorem 1 applies strictly only to those thermodynamic systems which do not contain adiabatically permanently isolated components, which we have termed simple systems. For other systems the preorder defined by accessibility does not generate a linear order on the set of equivalence classes, but only a partial order. If the system has n components one would expect the partial order to be the natural partial order of Rn, real n-space, with x < y if X, < yr for all Y. To generalize this case we replace Act 1, assumption (a), which states effectively that every finite set of points has an order like that in R, by (a) below, and then we can prove the following theorem: ’ This theorem is clearly parallel to the known result (Sierpifiski [l 11) that a linearly ordered set with an order dense countable subset has an order preserving map to the reals. Direct deduction of it from this theorem does not appear any shorter than the proof given.
THE FOUNDATIONS THEOREM
l(a).
OF THERMODYNAMICS
181
If the state space S is a separable topological space and
if there is de$ned in it a relation + satisfying Act 1 (a) and (c), Act 2, and
Act l(a). Given any finite set A of states S there is a set of maps p, *.*p, of A to R such that Ord a) sr -+ ss if and only ifp,(s,) < p,(sz) for r = 1, 2,... n holds for uny pair of states in A, then there is a set of maps j1 **aj, of S to R such that Ord n holds for all s1 , s2 in S with j replacing p. The other question of interest is how far the assumption of the separability of the phase space is necessary for the truth of the theorem. The example given at the beginning of this section-that of R2 with lexicographic ordering-serves also as an example to show that separability is essential. For if R2 is lexicographically ordered and is given the interval topology associated with this order, then the continuity conditions are satisfied for this topology but, as we have seen, no real valued entropy function generating the order is possible. As this counter example involves a space which is not linear or even arcwise connected, unlike the normal state spaces of physical systems, we give one involving a linear space. Let S be the Hilbert space of all real valued functions of a real variable such that x 1f(x)]” < 00; for any real number a let e,(x) = 1 =0
if
x = a,
if
x#a
and let B, denote the ball of unit radius with centre 2e,. In S we define a preordering 2 by the conditions f 2 g if either (i) fisinnoB,, (ii)
fEB,,gEBbwitha
(iii)
f andg are in the same B, and
j/f - 2e, jl 2 ljg - 2e, /I.
Then it is easily seen that the preorder is continuous in the sense of Act 2, but that no entropy function exists for it.3 However, the following assumption makes it possible to apply Theorem 1 even in the nonseparable case: Act 3. The state space S contains a separable space S’ such that every state s satis$es s t) s’ for somestate in S’. * Dr. S. Rolewicz has constructed an example of a Hilbert Space of dimension & in which the preorder is realized by a continuous map to the ordinals of cardinal less than X1 .
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Clearly Act 1, 2 and 3 imply the conclusion of Theorem 1, since we can apply that theorem to S’. In the physical case a subspace S’ satisfying Act 3 can be found: the set of states having a constant configuration, as defined in Section 4. 4.
COMPOSITION
OF SYSTEMS: ADDITIVITY
OF ENTROPY
The entropy function whose existence is proved in Section 2 is very illdetermined: if p is any continuous strictly increasing map of the real line to itself, and f is any entropy function in the section of par 2, so is pof. It will now be shown that entropy can be determined uniquely up to linear maps on the assumption that a thermodynamic system (at least one particular one) can interact with (at least) three other such systems, and that the entropy is additive for interacting systems. DEFINITIONS. Systems 61 and 62 will be said to be isomorphic if there is a one-one map of the state spaceof G1 onto that of Gz which is a homeomorphism and preserves all thermodynamic relations. At present all that is required is preservation of the relation +. In sections in which more structure is postulated for thermodynamic systems the conditions for isomorphism are correspondingly stronger. A system G is called the composition of systems @, G,..., 6” and is written 6 = {G, G2,..., S%} if there is a homeomorphism of the product space Si X S2 **a x S” onto the state space of 6 which is such that if t s1* s2,--*>P> is the state corresponding to (s’, s2,..., s”) then
Int (a) {sl, s2,..., sr-l, sir, sr+l,..., P} -+ {si, s2,.,., sr-l, sZr,srfl,..., sn} if and only if sir -+ s2r. Int (b) If 6P is isomorphic with 6* the state derived from (sl ,..., sp,..., SQ,...) by permutation of s* and SQis reversibly accessible from it. Int (c) If A, B are complementary subsets of {I, 2,..., n> and GA is the composition (Sal, Gas,..., @D} and GB is {@I ,..., G*+ then B = (GA, V}. The significance of the last condition is that is extends (a). In the physical model, systems will be formed into a composition when they can interact so as to be in adiabatic connection and also so that they can be diathermally connected. Without the possibility of diathermal connection the manifold of states would not be linearly preordered. We now make the following assumption, about interaction of systems. Int 1. There is a class of elementary systems, which have simply connected state spaces. This class contains at least four systemsisomorphic to a system G. The composition of any four elementary systemsexists.
THE
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183
OF THERMODYNAMICS
We avoid the (at first sight natural) assumption that the composition of any two thermodynamic systems exists, because this would lead to an infinite regress and to the conclusion that all thermodynamic systems can be formed into an interacting system. THEOREM 2. Under the assumptions of Act l-3 and Int 1, there is a function which is defined on the state spacesof all elementary systems, which is entropy for any one system, and which is such that the function ~(9) + ~(9) of (9, s”} is an entropy for {G, G}.
v is uniquely determined for any one system by its values for two states of that system and, when so defined for one system, is defined for any other system by its value for one state of that system. possible choices of v are related linearly. The property required for v is that it is an entropy for 6
particular uniquely Any two and that
is1 , s2>-+ {s, ,s4) if and only if v(sl) + v(sJ < v(sJ + q(sJ. Suppose that for 6 (in the sense of the last section). Then p)(s) is a strictly increasing continuous function of p(s); say ~(0) = #(p(a)). The values of p(s) fill a connected interval of the real line, in one-one correspondence with the classes of reversibly accessible states of S; and by considering t/ instead of v we see that it is sufficient to prove the result on the assumption that the set of states S forms a segment of the real line. For the remainder of this proof we shall therefore speak of states when we mean (in the general case) equivalence classes of states, and regard them as represented by real numbers p(s). To begin with, we define ‘p for the system 6, postulated in Int 1, four isomorphs of which can interact. If p(s) is any entropy function for B, then any other entropy function for 6 is a continuous strictly increasing function of p(s), and we may therefore define p in terms of the values of p(s), which fill an interval of the real line. To simplify notation we shall write single letters s, U, v instead of p(s) etc.: this amounts to identifying reversibly accessible states and identifying them with the values of an entropy function.
p is any entropy function
LEMMA 1. If u, v are any two states there is just one state s such that {s, s} = (u, v}. We write s($(u + v)) for this state: s(*(u + v)) is a continuous strictly increasiragfunctzbn of {u, v}.
Let
qu,
4 = is : 694 + {u, VI>,
R(u, v) = {s : {u, v} + {s, s)}. Because of the continuity
of +, L and R are open. Neither
is empty,
since
COOPER
184
one of u, v belongs to each. Since S is connected neither L or R, and it must satisfy
there is an s which is in
Suppose now that s satisfies (1) and that p + s + 4. Then {P> PI -+ {s, 4 ++ b4 v> -
Q 41
so that s is unique. Also, we can find u1 , ua so that ui -+ u -+ u2 and {PY P> * @l Yv}-+ (us, V} --+{q, q} and then can find vr, vs so that uul-+ v + ~1sand {p, p} --f {ui , vi> --f {us , va} + {q, q>. Then if ui -+ u1 - us, v, -+ 01 + v2 we have {P, P> -
{u19 v’> + {q,qJ
so that p --f s(+(ul + vi)) -+ q, proving the continuity of s(&(u + v)). We now choose two states, s(O), s(l) say. W e write s(h) for s(&(s(O) + s(1))) and define S(T) for all dyadic rationals in (0, 1) by induction, putting
s((2p + 1)/2”) = s(&(p/29
+ s((P + 1)/2”-7)).
Then we have: LEMMA 2. If a, , a2 , a3 , a4 are dyadic a, + a2 = a3 + a4 then
rationals
in (0, 1) such that
If the a’s are of the form p/2” with n = 0 or 1 this is true by the definitions. Suppose it true if all a’s are of form p/2”-l; then if a, = (2~~ + 1)/2” {WV
s(al)~+4,
W>
f-) (s(PJ29,
S((Pl + 1)/2n-1), s(p,/29,
S((P, + 1)/2”-‘)I
*
{s(P,/29,
s((P, + 1)/2”-l),
s((P, + 1)/2”-9)
-
{W,
@A
44,
s(p,/2”-Y,
+dh
Here the first and third steps follow from Int (a) and the definitions and the second from the inductive hypothesis and Int (a). Now we prove LEMMA 3. Wh
sub
of s(a,),
The values of s(a) for dyadic rationals a are dense in the interval
THE
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185
OF THERMODYNAMICS
If not, there is an interval (u, V) which contains no s(u), but is such that every interval containing (24,~) properly contains an s(u). Let p = inf{u : z, f-f s(u)}.
a = sup{a : s(u) 4-+ U},
Then 01= p for otherwise (01,/3) would contain a dyadic Y and u + S(Y) + o. The values of s(+(s(p) + s(q))), p < OL< q, are of the form S(U),and tend to 44@ + 4) as P + 01,q -+ 6: thus u < s(u) < v(0) for some a. Here the first and third steps follow from Int (c), and the second from the inductive hypothesis and Int (c). Now for any state u with s(O)+ u + s(l) write v(u) = inf{u : a dyadic rational, IL t
s(u)}
Then if #(u) = sup{u : a dyadic rational, s(u) --t u}, we have #(u) = a(u); for, if not, then the interval ($(u), p)(u)) is nonempty, and so contains a dyadic rational Y for which II -+ S(Y) -+ u, a contradiction. Further, if u ---f v then p)(u) < p)(w). For, if not, y(u) = v(w); we can suppose without loss of generality that u is the state with least and o that with greatest entropy for which v(s) has this common value. For any positive E any neighborhood of u contains a state s(u) with p)(u) - E < a < v(u) and any neighborhood of ‘u a state s(b) with p)(v) < b < v,(v) + E. Since s(Q(u + w)) is continuous in u and v and u --f s(Q(u + v)) + v, the neighborhood can be chosen so that s(Q[.r(u) + s(b)]) is in (u, v); but, by Lemma 2,
so that v(u) < *(a + b) < v(v), contradicting p)(u) = v(o). v(u) is therefore a strictly increasing function of u, hence v-,-‘(u) is continuous. If we write s(x) = u when p)(u) = x, it follows from Lemma 2, by considering approximations of xi by dyadic rationals, that Wl)> 4x2)) t-) w%), 4x4)) if and only if Xl +
x2
=
x3
+
x4
,
and then that x1 + x2 < x3 + x4 implies w%)>
s(x2))
+-+
w%),
G4h
so that {Ul 9u2>
-
‘*)&3
<
(aP(%)
t u4)
if and only if d%) +
du2)
+
?+44)*
186
COOPER
The definition of p)(s) can be extended outside (s(O), s(l)} by defining s(x) for 1 < x < 2 by w,
42 - 41 = {s(l), s(l)1
and, by induction on n, for n < x < (n + 1) by w,
s(n + 1 - 4> f-t {s(l), w>.
The process will end if there is an n such that
for every s: otherwise it will continue for indefinitely definition applies to negative X. Writing x = ~(0) if for an interval of the real line and additivity follows If 61 is any other system, choose an initial state S(x) as m, e9) t-) w, W)l
large X; and a similar s = s(x), y is defined as before. G(O), and then define
and put ~($1) = x if s1 = $(x). The statements concerning uniqueness of p are readily proved. To prove additivity, note that if G1G2 are any two elementary systems wd
s2@2)l
+ W3)~ s2(x4N
if and only if W), @)> sl(xl>, s2(x2N-
W)~ 40), +.a
s2(x4N,
and hence if and only if {4x1),
@2),
W), s2(0)l -+ W3), 4x4), syo>, s2(0))
i.e., Wl),
+ Is(xJ, 4x4))
G2N
thus if and only if Xl +
x2
< x3 +
x4
.
This extends the proof of the additivity of entropy to arbitrary pairs of systems. The assumption that up to four elementary systems can be combined is essential to the proof of additivity of entropy just given: and the result certainly does not follow if only combinations of two systems are supposed to exist. This is shown by the following counter example. Suppose that the states of a system are represented isotonically and continuously by the positive numbers, and suppose that if and only if U12+ 241242 + u22 < 2432+ UQU4+ u42.
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OF THERMODYNAMICS
Then
if S(O) = 0, s(1) = 1, we have 3~($)~ = 1, so that ~(a) = l/d; it is easily verified that {s(i), s(S)} and s(i) = g , s(2) = )(4 + 1/3y2: {s(i), s(Q)} are not reversibly accessible.
5.
INTERNAL
ENERGY
AND
TEMPERATURE
Further development depends on more specialized assumptions, and in particular on postulates concerning the types of interaction and energy interchange between systems. DEFINITION. Two systems are in adiabatic interaction sf their equilibrium conditions, changes of state and energy interchanges are completely spec$ied by the laws of the ground theories. A system is adiabatically isolated if its interactions with all other systems are adiabatic.
The law of conservation
of energy takes the following
form:
E 1. A thermodynamic system has a continuous function of state called its internal energy and written E(s). In a change of state of an adiabatically isolated system the increase of E(s) is equal to the total work done on the system by systemsexternal to it according to the ground theories. A thermodynamic systemis simple ifits state spaceis homeomorDEFINITION. phic with a product X x E where X is a normed linear space and E is an interval of the real line. To each state correspondsa point (x, E(s)) of X x E, where x is the configuration of the system according to the ground theories and E(s) is its internal energy. E 2. The internal energy of the composition of two systems is the sum of the internal energies of the component systems. Two thermodynamic systems are in diathermal interaction if DEFINITION. exchanges of internal energy between them are possible without any changes of conjiguration. They are in thermal equilibrium sf their states are such that they remain unaltered when put into diathermal interaction. Thermal
equilibrium
is governed
by the following
assumption.
Temp 1. There exits a real-valued function O(s) defined for all states of all simple thermodynamic systems, which is such that two states s1 and s2 of any two systems are in thermal equilibrium if and onb zf e(9) = O(s2).For a fixed configuration x, 0(x, E) is a strictly increasing function of E.
188
COOPER
In some expositions of thermodynamics the existence of temperature is deduced from other axioms. Miller [12] and Landsberg [9] give arguments deducing the existence of temperature from the postulate that thermal equilibrium is an equivalence relationship; it is easily seen that such deductions must be fallacious, since an equivalence relationship could involve see Whaples [13] for a detailed equality of two distinct co-ordinates; discussion. Caratheodory states that existence of temperature can be inferred from axioms that bodies in fixed configurations change their nonconfiguration co-ordinates to those of states in which they are in thermal equilibrium; but this does not allow us to deduce that every state has a temperature. It seems that any reasonable assumptions which allow the existence of temperature to be deduced differ only tautologically from the statement that it does exist. In what follows any function satisfying the conditions laid down for e(s) in the axiom Temp will be called a temperature. THEOREM 3. The composition of two simple systemsin thermal equilibrium is a simple system.
A state (si, sa> in which the systems are in thermal equilibrium is completely determined by the configurations x1, x2 corresponding to s1 and s2 and by the common temperature 0. The internal energies El, E2 of systems are determined by x1, x2 and B because of Temp 1; and so El + E2 is determined by B and is a continuous strictly increasing function of 6 and so xi, x2, El + E2 determine 8 and so El, E2 and the states sl, s2. If 6, is any other system, in a state s3, in diathermal contact with {G’, @}, then it is in diathermal contact with one of G, G2 and so energy interchange between G3 and {Sl, G2} occurs unless 8(S) = O(sl) = 0(s2) = 0. 0 is therefore a temperature for (~1, ~a}.
6. ABSOLUTE TEMPERATURE The last section puts us in a position to state a form of the second law of thermodynamics which is adequate for all subsequent development. Act. 4. If G1 and G2 are any two systemswhich are in thermaf equilibrium an so’, so2,then no other state (9, s2} of the composedsystem (G, G2} in whkh each system is the same configuration is accessiblefrom the state {sol, s,,“}. This could be regarded as a weaker form of the Clausius Law: it states that heat cannot pass between two bodies at the same temperature without change of configuration. We can now prove the existence of an absolute temperature.
THE
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189
OF THERMODYNAMICS
THEOREM 4. For any $xed con$guration of a simple system, the entropy (as normalized in Section 4) is a differentiable function of the internal energy for all save a countable set of points: and at thesepoints it is dz@rentiable on the left and on the right. The derivative, where it exists:
g(x, E) = ii
dx, E + 4 - dx, E) h
,
or more generally the left and right hand derivatives, g*(x, E) = lim
a@, E + h) - dx, 4
h
h+Of
,
depend only on the temperature and are stricUy increasing functions of temperature. Consider first a composition ((3, G2) in which G1 and 6s are isomorphic. If x is any configuration, the states {(x, El), (x, E2)} accessible without change of configuration from a state {(x, &I), (x, EOz)} must satisfy El + E2 = EO1+ E02 by the first law (E 1). According to Act 4, no state ((x, El), (x, E2)) is accessibIe from ((x, E), (x, E)] where E = &(El + E2) and so, because of the additivity of the entropy we have
29)(x, B(El + E2)) > v(x, El) + v(x, E2), so that, because of the continuity of v, -v(x, E) is a convex function of E. The existence of left- and right-hand derivatives everywhere, the existence of a derivative at all but a countable set of points, and the property that the derivatives are strictly decreasing now follow from known theorems on convex functions (e.g. [14]). Now, let 61 and G2 be any two simple systems with composition (~1, ~2). For any configurations x1, x2 the same arguments show that dx’,
El) + dx2, E*) -=c94~2~9
E,‘) + v(x2, Ea2)
when
El + E2 = EOl + Eo2
and
0(x1, Eal) = 0(x2, E02).
Let E be such that av(xl, E)/E and +(x2, E)jE exist at E,, . Then it follows from the usual arguments concerning conditional that g(xl, W) = g(x2, Ea2)
maxima
so that it is proved that the derivatives are equal for equal temperatures when they exist. That the left- and right-hand derivatives are equal follows: in general g,(x, EO) is the supremum of all the values of g(x, E) for those values of E > E, for which g(x, E) exists.
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COOPER
The quantity g+(x, E) is thus a strictly monotonic function of temperature independent of the system for which it is defined; its reciprocal l/g+(x, E), which increases with temperature is called (up to a constant multiple) the absolute temperature. These postulates leave open the possibility that the absolute temperature may have up to a countable number of discontinuities. Actually no more than this can be proved from the classical forms of the second law: to prove the continuity of absolute temperature (say as a function of an empirical temperature) conditions bearing on the smoothness of the system are needed.
7. AESSOLUTETEMPERATURE AS AN INTEGRATING FACTOR In the classical theory absolute temperature appears as the reciprocal of a multiplier which makes the differential form for the work done integrable. By postulating the existence and the smoothness of this form at any rate for one particular system these results can be deduced and the continuity of absolute temperature proved. A system 6 will be called smooth if it satisfies the following: (a) 6 is simple, and its state space has the form S = X x E, where X is a normed linear space. (b) If x1 is a fixed configuration then if (x1 , E,), (x, E) have the same temperature, E is a differentiable function of El and of x. (c) There is for each state s a linear functional on X, p(s), such that for each h E X, (p(s), h) (the value of p(s) at x) is a continuous function of s and for any E,E, there is a constant K so that
(d) The states reversibly accessible from a given state s lie on a manifold whose tangent vectors at s, (dx, dE), satisfy dE + (p(s), dx) = 0. (e) The manifold of states reversibly accessible from s has no tangent vectors in common with the manifold of states in thermal equilibrium with S. In the physical model (p, dx) is the work done by the system in a small displacement dx of its configuration. (e) means that temperature alters in a quasistatic adiabatic process. More generally, we say that a simple system has a smooth direction h if there is a subset of its configuratian space X, which is in one-one correspondence with the real line (say, X, = (st, + th : t real)) and the postulates
THE
FOUNDATIONS
191
OF THERMODYNAMICS
above are obeyed for displacements in X,,: in particular there is a number ph(s) so that dE + p&) dt = 0 (1) in a reversible displacement (dE, dt). We now postulate: There is a simple system with a smooth direction. Let B be such a system. Then the form (1) has an integrating factor. If h(B) is an integrating factor which depends only on temperature 13,then it is easy to see that h(0) es aE/ae logh(B,) = I h se/at -p(ae/aE)
(2)
de
Vectors (dt, dE) for which 0 is constant satisfy
ae
Ed&O:
atdt+
aE
those representing reversible transitions satisfy p dt + dE = 0. These coincide in direction if and only if
ae
--paE=o, at
ae
and hence, because of (e), this equation does not hold. It follows that the integral in (2) defines a continuous function of es, when h(B,) is chosen arbitrarily, and if G(0) = exp h(B) (3) g(e) is an integrating factor for (1). On the other hand, given any t, , the equation
g+P(t,r)
=o,
-f-W = 4 ,
(4)
determines y uniquely as a function of t because of the continuity conditions (a) of the definition of a smooth system. Let y(0) = E,, for this solution. Then we have for the entropy v, dt, , 4) = j-%,+(0, E) dE + ~(0, 01,
(5)
0
which is due to the results of the last section and because along the curve defined by (4).
Q
is constant
192
COOPER
Let the initial state designated (0, 0) be chosen to have a temperature at which g(0) exists so that g, = g- are continuous in a neighborhood of this state. Then for displacements (AE, , At,) from these states (2) and (1) give respectively
4 = k+P, 0) + O(l)14, W, - A-%1+ [P + O(l)14 = 0, whence ‘VP,) = k+(O, ‘3 + WMAE,
+ ~4).
Thus F is differentiable and dy = g+(O)(dE+ P dt).
(6)
If G is any other integrating factor, then the integration gives a differentiable function of v,, so that G =f(~)g+(e) for some function of ‘p. The function G defined by (3) is a function of B only, say, G(Q) = f(vk+(@) = fh+? ~Ng+(e),
(7)
so that f(p(fJ, t)) is a function of 8 only. Because of condition (e) +/at is nowhere zero: hence varying t alone varies q~.It follows that f is a constant, and so that g+P) = C exp w, where y&J is given by (2). Thus g+(0) 1s . continuous and hence g(B) exists for all 8. Now let S be any smooth system. The same arguments as those above, applied to any direction in S, show that 6 = &WE
+ (P, d+)
(8)
In summary, if there is a system which is smooth in some one direction, then the absolute temperature T is a continuous function of empirical temperature and ag,=L i3E
T’
If 5’ is a smooth system, q~is differentiable and T 6 = (d. + (P, W). REFERENCES
1. 2. 3. 4.
M. ZEMANSKY. “Heat and Thermodynamics.” New York, 1957. M. BORN. Phys. Zeut 22 (1921), 218, 249, 282. M. BORN. “National Philosophy of Cause and Chance.” Oxford, London, C. CARATH~ODORY. Math. Ann. 67 (1909), 35.5.
1949.
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FOUNDATIONS
OF THERMODYNAMICS
193
5. FALK AND JUNG. “Handbuch des Physik,” Vol. III, p. 2. Springer, Berlin, 1959. 6. H. A. BUCHDAHL. Z. P/zys. 152 (1958), 425. 7. P. T. LANDSBERG. Physica Status Solidi 1 (1961), 120. Foundations of Thermodynamics.” Pergamon, London, 8. P. GILES. “Mathematical 1964. 9. P. T. LANDSBERG. Rev. Modern Phys. 28 (1959), 389. 10. N. F. RAMSAY. Phys. Rev. 103 (1956), 20. 11. W. SIERPINSKI. “Lecons sur les nombres transfinies.” Paris, 1928. 12. A. R. MILLER. Amer. J. Phys. (1952), 488. 13. G. WHAPLFS. /. Rational Mech. Anal. 1 (1952), 301. 14. HARDY, LITTLEWOOD, AND POLYA. “Inequalities.” Cambridge.