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UBC Theses and Dissertations

A comparison and criticism of two axiomatic systems for classical thermodynamics Erickson, Arvon Donald 1975

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A COMPARISON AND CRITICISM OF TWO AXIOMATIC SYSTEMS FOR CLASSICAL THERMODYNAMICS by ARVON DONALD ERICKSON B.Sc, University of B r i t i s h Columbia, 1972 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF " ,,THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of PHYSICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1975 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e H e a d o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook Place Vancouver, Canada V6T 1W5 ABSTRACT This thesis i s a comparison and c r i t i c i s m of two approaches to c l a s s i c a l thermodynamics, that due to P. R a s t a l l i n the Journal of  Mathematical Physics, Volume 11, Number 10, October 1970, page 2955, and that due to H.A. Buchdahl and W. Greve i n Z e i t s c h r i f t fur Physik 168, 1962, page 316 and page 386. Rastall's approach i s somewhat more general and appeals because of his generally careful attention to the e x p l i c i t , orderly statement of the assumptions and rigorous proofs based only on these assumptions. However, h i s approach i s s i g n i f i c a n t l y improved by substituting the main assumption of the Buchdahl-Greve paper for two of his assumptions. The Buchdahl-Greve assumption i s fundamentally appealing and as w e l l as leading to a more natural proof of a c r u c i a l theorem, i t also allows a straightforward proof for the continuity of the entropy function, which R a s t a l l assumes ad hoc. The proof given i n the Buchdahl-Greve paper for the continuity of the entropy function i s shown to be c i r c u l a r but a correct proof based on the Buchdahl-Greve assumption i s possible and one i s given i n an appendix. Suggestions are also made concerning how one might improve t h i s combined Buchdahl-Rastall approach. i i TABLE OF CONTENTS Page ABSTRACT • i INTRODUCTION 1 GENERAL CRITICISMS OF RASTALL'S APPROACH 2 GENERAL CRITICISMS OF BUCHDAHL'S APPROACH 2 DETAILED CRITIQUE 4 Adiabatic A c c e s s i b i l i t y 4 Absolute Zero A c c e s s i b i l i t y . . 6 The Relationship Between "Anergic" and Adiabatic Processes. . . . . . 9 The Increase of Entropy 22 Developing the Concept of Temperature 22 The Continuity of the Entropy Function 31 SUMMARY 35 CONCLUSIONS 36 SPECULATIONS 37 BIBLIOGRAPHY 39 APPENDIX A: A Proof that the Entropy Function i s Continuous . . 42 APPENDIX B: A L i s t i n g of the Axioms Discussed , 44 o 1 INTRODUCTION This thesis i s a comparison and c r i t i c i s m of two approaches to c l a s s i c a l thermodynamics. The two approaches chosen, due to Ra s t a l l and Buchdahl, are well suited for such a discussion because i n spite of the fact that s u p e r f i c i a l l y they seem to be quite d i f f e r e n t , fundamentally they are not too di s s i m i l a r and therefore a very detailed comparison i s possible. The reason the two approaches are not es s e n t i a l l y d i s s i m i l a r i s that the two key points of Rastall's approach can also be found i n the Buchdahl approach. The discussion of these two approaches begins with three general c r i t i c i s m s , one having to do with the formalism of the Ras t a l l approach and the other two having to do with Buchdahl's presentation of his approach and a (rather common) attitude towards thermodynamics. More particular c r i t i c i s m s follow. I t i s pointed out that one of Rastall's assumptions seems to imply that absolute zero i s accessible. I t i s also suggested that this assumption and two others that R a s t a l l makes, should.be replaced by the p r i n c i p a l assumption of the Buchdahl approach* Next the symmetry between Rastall's "anergic" processes and adiabatic processes i s emphasized. Since Rastall's AX. XII"*" i s the main linkage between anergic and adiabatic processes, t h i s leads i n a natural way to consideration of the strength and importance of Rastall's AX. XII . Some of i t s many ramifications are discussed. Rastall's rather novel approach to temperature i s the next important topic. Two main c r i t i c i s m s are put forward. The f i r s t i s "*"For a l i s t i n g of the axioms and theorems, see Appendix B. 2 that the axiom through which temperature i s introduced into the theory i s superfluous. The second i s that his approach to temperature lacks assumptions of s u f f i c i e n t strength to ensure that we end up with the ordinary notion of (absolute) temperature. F i n a l l y the Buchdahl-Greve claim for the continuity of the entropy function i s considered. The proof for the continuity of this function i s shown to be c i r c u l a r . GENERAL CRITICISMS OF RASTALL'S APPROACH Ra s t a l l works with s i x domains i n his formulation of c l a s s i c a l 2 thermodynamics: E, S, ir, N, A and dom q . I find the use of a large number of domains aesthetically displeasing i f i t i s possible to use a smaller number of domains. (For example, R a s t a l l introduces E, the set of states, as we l l as S, the set of equilibrium states, but then proceeds to l i m i t himself to S i n the important axioms and theorems). Moreover, i n general, the more domains i n a theory, the more d i f f i c u l t i t i s to see the ramifications of the axioms. Thus there are p r a c t i c a l as well as aesthetic considerations behind one of the main themes of this 3 thesis, namely that S i s the only essential thermodynamic domain. GENERAL CRITICISMS OF BUCHDAHL'S APPROACH In his introduction to the f i r s t paper Buchdahl makes what seems 2 These domains are defined i n Rastall's paper but i t w i l l not be necessary to refer to his definitions to understand what follows. 3 Processes can be introduced by mapping the unit i n t e r v a l of the r e a l numbers onto S . More w i l l be said l a t e r on this point. 3 to me to be a rather s l i g h t i n g reference to thermodynamics, saying: "The strength of such [continuity] assumptions and the precise context i n which they are introduced i s perhaps largely a matter of taste i n a phenomeno-l o g i c a l theory." But every physical theory i s a phenomenological theory; i f the theory i s not phenomenological i t i s not physical. I t strikes me as naive to think that s t a t i s t i c a l mechanics, for example, i s somehow physically more fundamental than thermodynamics. Thermodynamics i s physics. S t a t i s t i c a l mechanics i s physics. But s t r i c t l y speaking, an attempt to derive the principles of the former from those of the l a t t e r i s not physics but i s rather about physics. Even i f such an attempt were e n t i r e l y successful t h i s would be a proof only of the l o g i c a l precedence of s t a t i s t i c a l mechanics, not i t s physical precedence. We would s t i l l have two physical theories; i t would s t i l l be both possible and correct to learn thermodynamics without any reference to s t a t i s t i c a l mechanics because thermodynamics would s t i l l be able to explain on i t s own a naturally cohesive body of experimental facts. The presentation of the Buchdahl, Buchdahl-Greve papers i s very poor. This i s not a t r i v i a l fiaullt because i t e a s i l y leads to duplication of e f f o r t . Rastall's paper possibly provides an example, as i t w i l l turn out that both of the key ideas of Rastall.'s approach, (Rastall TH. 5 and AX. X I I ) , are to be found i n the work of Buchdahl and Greve and Ra s t a l l apparently f a i l e d to notice t h i s . The (1962) paper by Buchdahl i s a c l a s s i c example of argument based on physical i n t u i t i o n rather than on e x p l i c i t l y assumed axioms. This sort of argument i s necessary at an early stage i n the development of 4 4 a theory but i t e a s i l y leads to mistakes. The "hiatus" of his 1958 paper, (essentially the counting of an uncountable s e t ) , i s just the sort of mistake so eas i l y made i f too much reliance i s placed on physical i n t u i t i o n . In the other 1962 paper, coauthored with the mathematician Greve, more attention i s paid to making assumptions e x p l i c i t . However the quality of the mathematical arguments based on these assumptions i s very poor indeed. - FF©r*-example,"as we shaMasee^eeone i O f othenproofs concerned with showing the continuity of the entropy function i s so poorly stated and proved as to force one to guess at i t s precise intent. And the other proof i s e a s i l y seen to be c i r c u l a r . DETAILED CRITIQUE Ra s t a l l notes i n the second section of his paper that time does not appear i n the formal theory. This i s as i t should be. There are dangers involved i n introducing time into c l a s s i c a l thermodynamics and they show up c l e a r l y i n the controversies concerning the d i f f e r i n g results obtained by the thermodynamic approach to fluctuations as opposed to that of s t a t i s t i c a l mechanics. Adiabatic A c c e s s i b i l i t y As R a s t a l l notes, his AXIOM I I I : given states a and b, b i s a d i a b a t i c a l l y accessible from a or vice versa, (a b V b -*• a) , l i m i t s h is formulation to physical systems i n which f r i c t i o n i s n e g l i g i b l e . 1 H.A. Buchdahl, Z..Physik 152, 425(1958). 5 He does suggest a replacement axiom: given states a and b, 3 state c 3 (a + c V c + a) and (b c V c b) . By AX. 's I and I I the r e l a t i o n "->" i s r e f l e x i v e and t r a n s i t i v e ; i t i s therefore what i s known as a preordering. Thus Rastall's replacement axiom i s e s s e n t i a l l y that under the preordering every p a i r i of states i n £ has either an upper bound or a lower bound. R a s t a l l does not follow up on the suggestion, leaving that to Boyling.^* However, Boyling's approach to thermodynamics i s e s s e n t i a l l y Caratheodory's and so i t i s not at a l l obvious that one could combine Rastall's overall approach with the suggested replacement axiom i n a reasonably natural way. Caratheodory's approach r e l i e s heavily on d i f f e r e n t i a l geometry; Boyling's approach combines this heavy reliance on d i f f e r e n t i a l geometry with a s i g n i f i c a n t amount of topology. Thus i t i s not surprising that Boyling can make use of the weaker replacement axiom since he has two strong and we l l developed mathematical theories to f a l l back on. I f you don't have th i s and i f a good deal of the subsequent theory i s to be based d i r e c t l y or i n d i r e c t l y on the prior existence of adiabatic processes, i t naturally follows that the axioms for the adiabatic processes have to be strong. R a s t a l l remarks that the weaker replacement axiom would "considerably complicate" his theory and I think this i s decidedly an understatement. Buchdahl also assumes what Ra s t a l l c a l l s AX. I l l . He says that an 6 argument due to Wilson which purports to derive t h i s axiom i s c i r c u l a r ^J.B. Boyling, P. Roy. Soc. A 329, 35(1972). 6 A.H. Wilson, Thermodynamics and S t a t i s t i c a l Mechanics, pp. 72, 73, 82. 6 and t h i s appears to be true. However the best way to avoid the c i r c u l a r i t y i s not to assume the axiom that R a s t a l l c a l l s AX. I l l . A far better way arises when one realizes that since AX. I l l l i m i t s the theory to systems i n which f r i c t i o n i s neglible and since these systems nevertheless have an i n t e r n a l energy, i t i s inappropriate to make the existence of the i n t e r n a l energy function dependent on an AX. I l l or some stronger axiom.^ AX. I l l can be- derived without c i r c u l a r i t y and t h i s w i l l be done l a t e r . Absolute Zero A c c e s s i b i l i t y In his t h i r d section R a s t a l l defines A(a) = {b E s|a b} ; i n words, A(a) i s the set of a l l states i n S ad i a b a t i c a l l y accessible from a . He assumes that A(a) i s closed but does not, as I think he 8 should, include i t i n the l i s t of axioms, saying that the assumption " i s made purely for mathematical convenience". This assumption means that A(a) contains i t s boundary, which i s equivalent to saying that A(a) includes a l l i t s l i m i t points. Now absolute zero can be regarded as the l i m i t of an i n f i n i t e sequence of states ad i a b a t i c a l l y accessible from a . Thus, assuming that A(a) i s closed amounts to assuming that absolute zero i s (adiabatically) accessible from a, i f we decide to include H For example, Giles i n Mathematical Foundations of Thermodynamics, p. 27, makes use of a stronger axiom. And note'that R a s t a l l l a t e r assumes AX. VII which considerably strengthens AX. I l l . However, nothing more than AX. I l l i s needed to prove the mere existence of an i n t e r n a l energy function. 8 This question i s discussed quite well i n the f i r s t section of the Buchdahl-Greve paper. 7 absolute zero i n S . Therefore I see two p o s s i b i l i t i e s . We could simply decide to exclude 9 absolute zero from S; Boyling chooses this alternative i n his approach. But suppose we do not. Unless one (incorrectly) thinks of a point as necessarily being " i n f i n i t e l y small" i n some sense, ^there does not seem to be any a p r i o r i reason for thinking that i t i s never possible to experimentally determine whether or not a l i m i t point of a set i s actually i n the set. More precisely, i f we have a theoretical set defined i n terms of a c c e s s i b i l i t y and i f there i s some experimental procedure for determining a c c e s s i b i l i t y then there does not seem to be any a p r i o r i reason for thinking that i t would never be possible to experimentally determine whether or not a l i m i t point of the set i s accessible, and therefore contained i n the set. But i f there i s no a p r i o r i reason for thinking that the a c c e s s i b i l i t y of a l i m i t state, such as absolute zero, can't be experimentally determined, then i t i s inadmissible to a r b i t r a r i l y decide the question on the basis,of "mathematical convenience". In any case, absolute zero i s usually assumed to be inaccessible. An assumption apparently contradicting this deserves, on t h i s basis alone, more comment than the statement that i t i s being made "purely for mathematical convenience and usually has no physical content". Rastall's AX. VII can be considered to show a certain "continuity" of "9 J.B. Boyling, Caratheodory's P r i n c i p l e and the Existence of Global Integrating Factors, Communications i n Math. Phys., v o l . 10, 1968, p. 52 . 10 A point i n a theory i s something beyond which one does not distinguish —-i n the theory! What the point i s made to correspond with physically may or may not be " i n f i n i t e l y small", may or may not be composed of (physically) distinguishable parts. 8 the r e l a t i o n of adiabatic a c c e s s i b i l i t y . Buchdahl remarks that his assumption (3.3), namely the assumption that i f the state c' i s adia b a t i c a l l y inaccessible from the state c then there exists E > 0 such that d(c,c^) < e and d(c',c|) < e imply that c| i s adiab a t i c a l l y inaccessible from c^, could be regarded as an expression of the continuity of the r e l a t i o n of adiabatic a c c e s s i b i l i t y . This suggests that some connection exists between the two assumptions. In fact, I suggest that i t would be best to replace Rastall's AX. VII and his assumption that the I ' l l A(a) are closed, by the Buchdahl-Greve assumption (3.3). We have already seen that the l a t t e r of Rastall's assumptions i s , at the lea s t , rather contrived i n that i t implies the a c c e s s i b i l i t y of absolute zero. But actually, both of these assumptions that R a s t a l l makes seem rather contrived and a r t i f i c i a l . They appear to be purely ad hoc assumptions introduced only to prove the c r u c i a l TH. 5, which states that the front i e r s of a l l the A(a) p a r t i t i o n S . Now i t i s well known that for every p a r t i t i o n of a set there i s a corresponding equivalence r e l a t i o n and vice versa. By Ax. 's I arid I I itsisleiearl^thatL'mutua-l^adiabatiecaecessibility i s an 12 equivalence r e l a t i o n . But using (3.3) Buchdahl proves' that for every a e S the fr o n t i e r of A(a) i s the set of a l l states i n S which are adiaba t i c a l l y mutually accessible from a . Therefore, for every a e S ^Of course R a s t a l l does not introduce a distance function d but this d i f f i c u l t y i s ea s i l y surmounted. See appendix B . "Actually his proof i s incorrect as i t stands, since i t i s s i g n i f i c a n t l y incomplete. I t can be completed by adding parts (i) and ( i i ) of Rastall's TH. 4 . 9 the f r o n t i e r of A(a) i s an equivalence class and the set of a l l these equivalence classes i s the p a r t i t i o n of Rastall's TH. 5 . Thus we can s t i l l prove TH. 5 after replacing AX. VII and the assumption that the A(a) are closed, by the Buchdahl-Greve (3.3) . Moreover, (3.3) could not be considered an ad hoc assumption, introduced only to prove TH. 5, because i t can also be used to prove the continuity of the entropy, function, something that R a s t a l l assumes ad hoc. I t i s clear then that we should make the suggested substitution since we can s t i l l prove TH. 5 and we are replacing three ad hoc assumptions by one fundamental continuity assumption. I t i s a very natural assumption, i n no way ad hoc. Boyling makes use of i t i n his attempt to make Caratheodory's approach to c l a s s i c a l thgrmodynamics more rigorous. The Relationship Between "Anergic" and Adiabatic Processes R a s t a l l defines i n his fourth section a new set of processes 13 which he c a l l s "anergic processes". The way that these new processes are introduced i s noticably different from the way that adiabatic processes are introduced. The former are made to seem much less important than the l a t t e r because the introduction of the new processes leans heavily on the prior existence of the adiabatic processes. This bias i s not wrong because Ra s t a l l i s defining something new and the new should be based on the old. However the biased introduction of anergic processes does mask the natural __ R a s t a l l says his anergic processes are a " s l i g h t generalization" of Buchdahl's isometric processes. However Buchdahl does not require that the work done i n an isometric process be zero. Thus the conditions imposed on isometric processes are, i n fact, less severe than those that R a s t a l l imposes for his anergic processes. 10 symmetry between these processes and adiabatic (reversible) processes. Perhaps the most fundamental symmetry i s that for adiabatic processes, (and therefore adiabatic reversible processes), the work done i s path independent while for anergic processes the heat absorbed i s path independent. I t i s es s e n t i a l l y this path independence that makes i t worthwhile to single out the adiabatic processes for the special attention they t r a d i t i o n a l l y receive. Therefore the fact that anergic processes share t h i s property i s a clear indication of the fundamental importance of these processes and I think t h i s should be stressed. After a l l , the importance of path independence for adiabatic processes i s usually acknowledged by c a l l i n g t h i s fact the F i r s t Law of Thermodynamics. A clearcut symmetry between anergic processes and adiabatic reversible processes shows up i n r e l a t i o n to AX. XII . The inherent symmetry of AX. XII i s a clear indication of i t s great strength and importance but Ra s t a l l apparently f a i l e d to notice this symmetry. Before discussing AX. XILjI need to introduce two terms which w i l l be i n frequent use from t h i s point on. The f i r s t term i s "adiabatically equivalent". As we saw previously, AX. I and AX. I I indicate that mutual adiabatic a c c e s s i b i l i t y i s an equivalence r e l a t i o n on S . Two states are said to be adiab a t i c a l l y equivalent i f and only i f they are adiabatically mutually accessible. The second term to be introduced i s "anergically equivalent" which for convenience w i l l replace "anergically mutually accessible". To show that anergic a c c e s s i b i l i t y i s an equivalence r e l a t i o n on S i t i s necessary to exhibit r e f l e x i v i t y , symmetry and t r a n s i t i v i t y . These 11 properties follow from Rastall's AX. VII I , TH. 7 and AX. IX i n that order. The notation for adiabatic and anergic equivalence w i l l be [b]^ and respectively. That i s , [b]^ i s the set of a l l states i n S which are adiabatically equivalent to the state b . A corresponding statement holds for [ b ] 2 • We are now i n a position to deal with AX. XII of Rastall's paper. In words this axiom says the following: given any state c e S and any complete set of adiab a t i c a l l y equivalent states, there i s a unique state d i n that set such that this state d i s also i n the set of a l l states anergically equivalent to c . The marked symmetry inherent i n AX. XII becomes obvious when we note that by interchanging the words "ad i a b a t i c a l l y " and "anergically" we get a new theorem. In the notation just introduced AX. XII says that given [ b ] ^ V c e S 3ld e S 3d e [b] ] L and d e [ c ] 2 . But of course every c e S i s contained i n [a]^ for some a E S . Therefore "Yc e S" i s equivalent to "given an arbitrary [a]^, Yc e [a]^", Thus AX. XII can be taken as the following statement. AX. XII (restated): given [a]^ and [b]^, Yc e [ a ^ 3'- d e S 3d e [ c ] 2 and d e [ b ^ . The new theorem i s obtained simply by interchanging everywhere the subscripts 1 and 2 . TH. A : given [ a ] 2 and [ b ] 2 , Yc E [a]23! d e S 9 d e [ c ^ and d e [tl2 • Proof: given [ a ] 2 and [b]2> consider c e [ a ] 2 . We must show J \ d e S 3d e [ c ] ^ and d e [ b ] 2 . But consider [b]^ and [ c ] ^ ; since b £ [b]^ then by AX. XII, corresponding to b3! d e S B d e [ c ] ^ 11 a and d e [ b ] 2 which was what we had to prove. The new theorem implies another statement which i s si m i l a r to Rastall's TH. 6 : TH. B : given [ a ^ and [b']2 3 d e S 9 {d} = [ a ^ f ) [ b ] 2 . Proof: to prove t h i s using TH. A, consider [a]^ and [ b ] 2 • Now ia^2 a n e ( l u i v a l e n c e class and therefore contains a . Thus we may set c = a i n TH. A and therefore conclude that ^! d e S 9 d e [ b ] 2 and d e [ a ] 1 . 12 Having proved that AX. XII implies TH. A and TH. A implies TH. B i t remains only to show that TH. B implies AX. XII to obtain the equivalence of these three statements. Therefore, given [a]^ and [b]^ consider a fixed but arbitrary c e t a l ^ • Then for [c]^ and [b]^ by TH. B 3! d e S 3 d e [ c ] 2 and d e [b]^ and so we are finished. I now wish to prove a corollary to TH. B which i s a very clear indication of i t s importance. COR. B : an arbitrary [a]^ and [ b ] 2 act as a two dimensional coordinate system for S . Proof: note that the given [a]^ and [ b ] 2 intersect i n a unique state d . By the properties of equivalence classes [a]^ = [d]^ and [ b ] 2 = f ^ ] 2 " (Thus, d w i l l be the o r i g i n of the coordinate system and t^]-^ a n d t* i e a x e s ^ * ^ o w c o n s i d e r a fixed but arbitrary state c i n S . This state i s a member of [°]^ a n <^ ^ c^2 " BY ^ [d]^ and [ c ] 2 intersect i n a unique state x ; s i m i l a r l y f d ] 2 a n <^ [c ] ^ intersect i n the unique state y . Thus the state c has a unique representation i n the ordered pair (x,y) with x e [d]^ = [a]^ and y e [ d ] 2 = [ b ] 2 . Moreover, having (x,y), consider [ x ] 2 and [y]^ • By. TH. B [ x ] 2 and [y]^ intersect i n a unique state c' e S . But c' = c because [ x ] 2 = [ c ] 2 > [y] = [c] and by TH. B [ c ] 1 and [ c ] 2 intersect i n the unique state c e S . In summary, given [a]^ and [b]2» every re e S i s uniquely and f a i t h f u l l y represented by (x,y), an ordered pair of states i n S 13 such that x e [a]^ and y e [ b ^ • In t h i s sense [a]^ and [b]^ act as a two dimensional coordinate system for S. I t should be clear that there i s nothing very unusual i n what has been done here. Consider the Euclidean plane E^ • Here we take two d i s t i n c t Tines £ and which meet i n a point; we c a l l the two l i n e s axes and the point of intersection d we c a l l the o r i g i n . Then any point c £ E^ can be uniquely and f a i t h f u l l y represented by an ordered pair of points x,y e E^ such that x i s on £^ and y i s on . For given c we consider the l i n e which i s p a r a l l e l to £^ a n d passes through c . This l i n e intersects £^ i n the unique point x . Next we consider the l i n e which i s p a r a l l e l to £^ and passes through c . This l i n e intersects £^ i - n the unique point y . Thus, c can be uniquely represented by the ordered pair (x,y) where x i s on £^ and y i s on • Moreover the representation i s f a i t h f u l . There i s only one l i n e which"is p a r a l l e l to £ 2 a n d passes through x and by construction this l i n e passes through c as w e l l ; a similar statement holds for £^ and y . Therefore, having (x,y) we retrieve the state c e E^ that we started with. In summary, given two d i s t i n c t intersecting l i n e s £^ and Z^, every c e E^ i s uniquely and f a i t h f u l l y represented by (x,y), an ordered pair of points i n E^ such that x i s on £^ and y i s on . In th i s sense £^ and £^ act as a two dimensional coordinate system for E^ Thus we see that an arbitrary [a]^ and [b]^ act as a two dimensional coordinate system for S i n a close analogy to the way that a pair of l i n e s £^ and £^ act as a two dimensional coordinate system for E 0 . I t i s not d i f f i c u l t to see that what allows the analogy i s the 14 fact that the property of par a l l e l i s m i s an equivalence r e l a t i o n just as mutual adiabatic a c c e s s i b i l i t y and anergic a c c e s s i b i l i t y are. Thus, loosely speaking, [g]-^ i s p a r a l l e l to [h]^ (and [g]^ i s p a r a l l e l to [h^) for every g,h e S . By the properties of any equivalence r e l a t i o n , the two equivalence classes [g]^ and [h]^ cannot have any points i n common: that i s , the p a r a l l e l l i n e s [g]-^ and [h]^ do not intersect. Also AX. XII i n the form of TH, B ensures that any two non-p a r a l l e l l i n e s [g]^ ai u ^ [hr:].l,l wMl'rintefsecfcninaone-and only one point. Roughly speaking, one can characterize anergic processes as no-work processes and adiabatic processes as no-heat processes. Thus i t appears that the reason the dimension of the coordinate system guaranteed by COR. B, i s always two, no matter how complicated the corresponding physical system, i s that there are always i n thermodynamics precisely two ways to change the inte r n a l energy of the system, namely by heat or by work. Since the law of conservation of energy for thermodynamics can be interpreted as expressing t h i s fact, we see that AX. XII, TH. A, TH. B and COR. B have a very fundamental source. One might therefore expect s i g n i f i c a n t advantages from working with S i t s e l f , using t h i s very natural coordinate system, rather than taking the usual course and working with an associated R n . This may perhaps be more c l e a r l y seen i f we think of the situ a t i o n as being concerned with the equation dU = dQ + dW and the equation dU = TdS + £ X.dx. . Both of these equations are quite i t 1 1 general. And when i t comes to applying the theory, when i t comes to considering s p e c i f i c thermodynamic systems, I think i t i s d e f i n i t e l y the second equation that i s more important. But I think the most appropriate 15 course would be to assume the f i r s t and develop the second. That i s , when we are concerned with developing thermodynamics, rather than applying i t , the f i r s t equation i s more important. The second equation does not provide as good a s t a r t i n g point for the development of thermodynamics; we should s t a r t at a simpler and more basic l e v e l with a simpler and more basic d i s t i n c t i o n . Now the simplest and most basic d i s t i n c t i o n f o r thermodynamics i s surely the d i s t i n c t i o n between the two types of energy, heat and work. Therefore i f , as I have suggested, i t i s e s s e n t i a l l y the d i s t i n c t i o n between heat and work which gives r i s e to the two dimensional coordinate system, the above considerations suggest that when we are concerned with developing thermodynamics as a formal theory we should work with t h i s two dimensional coordinate system, rather than the n dimensional system n a t u r a l l y associated with the equation flU = TdS + E X.dx. . i In p a r t i c u l a r , the two dimensional system might prove useful i n showing that the entropy function i s r e a l valued. Buchdahl claims to have shown t h i s , proving that T, the range of the entropy function he defines, i s "isomorphic" with the r e a l numbers. S t r i c t l y speaking, though, a l l he a c t u a l l y proves i s that r i s order-isomorphic with the r e a l numbers; that i s , r and the r e a l numbers are i n d i s t i n g u i s h a b l e as far as t h e i r ordering properties are concerned. However, there i s of course a greal deal more to the set of r e a l numbers than j u s t i t s ordering properties. One can add and subtract, multiply and divide r e a l numbers: they constitute a f i e l d . Thus one might t r y to show that the range of the 14 entropy function also constitutes a f i e l d . Now there i s a standard method by which one can show that the two coordinate l i n e s of an a f f i n e plane are 14 ' See page 101 of Foundations of P r o j e c t i v e Geometry, by Robin Hartshorne. 16 d i v i s i o n rings. I t appears to me that with some modification this method could be used to show that [ f ] 2 °f the two dimensional coordinate system for S i s a d i v i s i o n ring."'""' Since a f i e l d i s a d i v i s i o n ring whose mu l t i p l i c a t i o n property happens to be commutative, this would be quite a 16 large step towards showing that [ f ] 2 i s a f i e l d . Because, as I w i l l l a t e r show, the entropy function can be regarded as a projection function onto the y-coordinate [ f ] 2 of the two dimensional coordinate system, t h i s would also mean that quite a large step had been taken towards showing that the entropy function i s r e a l valued. The strength of Rastall's AX. XII makes i t f r u i t f u l to take another look at adiabatic processes. As-noted.previously, AX.'s I and I I can be seen as assuming the existence of a preordering on S . In part i c u l a r then, we have a preordering on S . TH. C : this preordering on S induces on any a P a r t i a l ordering which AX. I l l strengthens to a t o t a l ordering. Proof: consider a fixed but arbitrary [ f ^ • Since "->" i s a preordering on S, i n part i c u l a r i t i s a preordering on ' Therefore we have the re f l e x i v e property, (x £ x) , and the t r a n s i t i v e property, (x _< y and y < z imply x < z ) , automatically. "^The major d i f f i c u l t y i n doing this i s discussed i n the section e n t i t l e d Speculations. 16 However there seems to be a l i t t l e doubt as to whether one would necessarily want to show that our " m u l t i p l i c a t i o n " i s commutative. This i s because we might want to connect the addition of two states to the anergic processes l i n k i n g them, and the m u l t i p l i c a t i o n of the two states to the adiabatic processes l i n k i n g them. In this case the difference i n commutativity between addition and m u l t i p l i c a t i o n might be an expression of the fact that anergic processes are inherently reversible while adiabatic processes are not. 17 Now suppose we have a,b e [ f ] 2 3 a -> b and b a . Then the states a and b are adiabatically mutually accessible. That i s , a,b e [a]^ . By TH. B, [ f ] 2 ^ ~a\ = ^ f o r S O m e d £ S * N o W s l n c e a e [ f^2 and a e [a]^ then a = d . But s i m i l a r l y b = d . Thus a,b £ [ f ] 2 3 a -> b •and b -* a implies that b = a so that we have antisymmetry, (x. <_ y and y < x imply x = y) . R e f l e x i v i t y , t r a n s i t i v i t y and antisymmetry give us a p a r t i a l ordering of [ f J 2 . B u t by AX. I l l we have trichotomy, (x < y or y < x ) , on Z, therefore on S and therefore on [ f ] ^ • Thus we have r e f l e x i v i t y , t r a n s i t i v i t y , antisymmetry and trichotomy so [ f ] 2 i s t o t a l l y ordered, as claimed. I t i s natural to ask whether or not there i s harmony between the fundamental induced t o t a l orderings of [ f ] ^ and [g]^ for every f,g e S . That i s , consider a,b £ [ f ] 2 . By TH. B [ a ^ intersects [ g ] 2 at the state c and [b]-^ intersects t g ^ a t t n e s t a t e ^ • ^he question then i s t h i s : does a <_ b imply c <_ d ? We should c l e a r l y expect an affirm-ative answer and th i s turns out to be correct. For suppose a <_ b . Now i f a = b then c = d by TH. B } so by r e f l e x i v i t y c <_ d i n th i s case. Therefore suppose a _< b and a ^  b , Now a <_ b means that a ->- b . Assume b -> a also: then a,b e [a]^ = [b]^ • But by TH. B the intersection of [a]^ = [b]^ and ^ s ^ e unique state d . Thus d = a = b which contradicts the assumption that a ^  b . Therefore we have a -> b and b -f> a . Now suppose d _< c so that d •+ c . However, b,d e [b]^ so b d (and d -* b but we don't need t h i s ) : s i m i l a r l y c ->• a . By the t r a n s i t i v i t y of then, b d, d -* c, and c a imply b -> a which i s a contradiction. , Therefore we can conclude that 18 d c . Since the ordering of [ g ] 2 i s t o t a l , t h i s means that c _< d . Thus i t i s the case that a _< b implies c _< d . E s s e n t i a l l y by mapping the [ f ] 2 into R R a s t a l l could have obtained a t o t a l ordering of any [ f ] 2 i n d i r e c t l y through the natural t o t a l ordering of R : V x,y e5 [ f ] 2 we require that x <_ yOq(x,y) = U(y) - U(x) _> 0 where U i s an i n t e r n a l energy for S . I t i s easy to v e r i f y that r e f l e x i v i t y , t r a n s i t i v i t y and trichotomy must hold. To demonstrate the antisymmetry property we must show that U ^ : U ( [ f ] 2 ) [ f ] 2 i s a function. That i s , given x,y e [ f ] 2 5 x _< y and y <^  x, then U(x) = U(y) since R i s t o t a l l y ordered and therefore has the antisymmetry property. Thus U(x) - U(y) = 0 . But x,y e [ f ] 2 by assumption and therefore U(x) - U(y) = 0 implies q(x,y) = 0 . By AX. XI, x •> y . 'Similarly we have y ->• x . Therefore x and y are mutually a d i a b a t i c a l l y accessible but by TH. 6 t h i s implies x = y; thus U i s a function and so the ordering i s antisymmetric, as claimed. R a s t a l l ' s AX. XI can be seen as ensuring harmony between the fundamental d i r e c t t o t a l ordering of an [ f ] 2 > whose existence was proved i n TH. C, and the i n d i r e c t ordering of that [ f ] 2 obtained by going through R . Also, R a s t a l l ' s TH. 9 can be seen as ensuring harmony between the i n d i r e c t t o t a l ordering of any a n < ^ * Thus, whether the t o t a l orderings are obtained d i r e c t l y or i n d i r e c t l y the end r e s u l t i s the same. That i s , we can obtain an i d e n t i c a l end r e s u l t by a d i r e c t method. The d i r e c t method i s n a t u r a l l y to be preferred. B a s i c a l l y we are t r y i n g to formalize the theory of equilibrium states and we should therefore avoid going outside of S, the set of a l l equilibrium states, 19 wherever t h i s i s possible: TH. C shows that i t i s possible here. I next wish to show how i t i s possible to reverse what was done i n the proof of TH. C . Suppose one started off by assuming the existence of two equivalence relations on S and suppose one assumed that they s a t i s f i e d TH. B . Now a weakened form of the set-theoretical Axiom of Choice ensures the existence of at least one t o t a l ordering of a fixed but a r b i t r a r y [f]^ • BY COR. ,B a nY state i n S has a unique represen-tation i n the coordinate system consisting of [ f l ^ a n d '•^ 2 " Given the states c,d e S we define a preordering on S i n the following way. If c = (x^,y^) and d = (x2>Y2) then we w i l l say that c <^  d H Y 2 • I t i s easy to see that t h i s d e f i n i t i o n gives us a trichotomous preordering on S . I f the set S i s i d e n t i f i e d with a l l equilibrium states of a thermodynamic system and i f the preordering i s i d e n t i f i e d with the physical concept of adiabatic a c c e s s i b i l i t y , then c a l l i n g the t o t a l ordering of [f]2 the entropy for S automatically gives us the P r i n c i p l e of the Increase of Entropy and Rastall's AX. I I I . " ^ Since the preordering, which comes from the t o t a l ordering of [ f ^ ' w a s i d e n t i f i e d with adiabatic a c c e s s i b i l i t y we naturally i d e n t i f y [ ]^ with anergic equivalence and [ ]^ with adiabatic'.equivalence. I t i s to be noted that Rastall's AX. I l l , (or Buchdahl's (3.2)), i s a result of t h i s approach. I t i s not i t s e l f needed i n t h i s approach to the P r i n c i p l e even though the P r i n c i p l e i s a fundamental statement concerning adiabatic a c c e s s i b i l i t y . No p r i o r physical assumption l i k e AX. I l l i s needed because the t r a d i t i o n a l bias towards adiabatic processes i n the foundations of thermodynamics i s 17 : This step c l e a r l y suggests the great importance of the direct t o t a l orderings that TH. C dealt with. 20 missing. The two equivalence relations corresponding to anergic and adiabatic equivalence are introduced together and linked by the symmetric TH. B . Work i s not favoured over heat i n the foundations when th i s approach i s used. One can find even more to indicate the strength of AX. XII . In t his section R a s t a l l proves a theorem on the r e l a t i o n between adiabatic and anergic processes, namely that,i.(if we r e s t r i c t ourselves to S) , processes which are both anergic and adiabatic are necessarily c y c l i c . This result i s d i s t i n c t l y weaker than one would expect and the weakness of t h i s theorem r e f l e c t s a weakness i n Rastall's approach. The point i s t h i s : R a s t a l l r e s t r i c t s himself to S, the set of equilibrium states, i n the proof of the theorem. Moreover, as R a s t a l l himself notes, AX. I l l l i m i t s his approach to systems i n which one can neglect f r i c t i o n . Now i f there i s no f r i c t i o n involved and i f we are only concerned with equilibrium states t h i s should cer t a i n l y be s u f f i c i e n t to ensure that there be no p r a c t i c a l difference between any reasonable d e f i n i t i o n of a 18 "process" and the notion of a path. I t seems evident that under these circumstances an anergic process should correspond: to a path whose range i s some subset of [ a ^ where a i s the i n i t i a l state. In p a r t i c u l a r , a process which i s also adiabatic should correspond to a path whose range i s a subset V of [a]^ • Now Rastall's TH. 8 gives an argument to show that the f i n a l state c i s the i n i t i a l state a . But since we are dealing with equilibrium states i t i s clear that this argument must T8" A "path i n S" i s a continuous function which maps the unit i n t e r v a l of R into the topological space S . See for example: W.S. Massey, Algebraic Topology: An Introduction. 21 apply to every state d e V, and not just c, because every d e V can be considered the f i n a l state of an adiabatic anergic process whose i n i t i a l state i s a . Thus, on- the assumptions which go into TH. 8 , V = {a} and therefore an anergic adiabatic process i s not just c y c l i c : such a process i s degenerate. This result i s , of course, just what one would expect. Roughly speaking, anergic processes are no-work processes i n the same way that adiabatic processes are no-heat processes; so given that our process i s both anergic and adiabatic, we naturally expect the system to be limited to simply remaining i n the i n i t i a l state. In summary, Rastall's TH. 8 i s weaker than i t should be and t h i s weakness i s a result of the incompleteness of Rastall's approach i n the area of thermodynamic processes. I t appears that his approach should be strengthened i n t h i s area by the addition of the algebraic topological notion of a path. The use of algebraic topology i n thermodynamics might well be expected to be very f r u i t f u l since algebraic topology deals with such things as the path connectedness of points, the l o c a l path connectedness of points, path components of points, etc. To conclude the discussion of AX. XII i t should be noted that Buchdahl uses the idea which vundefOJiess Rastall's AX. XII i n an argument q i n Section 3(b) of his paper. Although R a s t a l l also makes use of t h i s kind of h e u r i s t i c argument i t i s only as? aipreamble to the formal statement of AX. XII . This s i t u a t i o n i s t y p i c a l and goes a long way towards making Rastall's approach the superior of the two. 22 The Increase of Entropy In his f i f t h section R a s t a l l defines a set of mappings f : J 4 R- 3 f (g) = q(a,b), where J i s the set of a l l f r o n t i e r s of the a a sets A(c) and q(a,b) i s the heat absorbed i n an anergic process between the states a and b . (By AX. XII the state b e g i s unique so the functions f are well defined.) He proves some theorems on the f and a a then says that i t i s "convenient to consider a set of functions which includes the f as a subset." Using the expanded set he goes on to prove 3. TH. 12, the P r i n c i p l e of the Increase of Entropy. Considered by i t s e l f , t h i s theorem i s not d i f f i c u l t . On the other hand i t i s undeniably the longest and most complicated theorem that R a s t a l l proves. This i s surprising when one considers the fact that f (3) = q(a,b) by d e f i n i t i o n and also cL that AX. XI states: Va,b e S 3 3 an anergic process l i n k i n g a and b, q(a,b) _> 0 <=*a -> b . An inspection of the proof of TH. 12 shows that i t s r e l a t i v e d i f f i c u l t y arises solely from the use of the expanded set of functions rather than the set of f . I must admit that I do not see a where he finds the expanded set more "convenient" but i n any case i t should be noted that any convenience attained elsewhere i s being balanced by a somewhat a r t i f i c i a l complication of the proof of a rather important theorem i n thermodynamics, namely the P r i n c i p l e of the Increase of Entropy. Developing the Concept Of Temperature Buchdahl does not r e a l l y deal with temperature i n the two papers 19 under consideration. In the paper preceeding these two his treatment of 1 9 — "H.A. Buchdahl, Z. Physik 152, 425 (1958). 23 temperature i s standard. However R a s t a l l , introducing the concept of temperature i n his s i x t h section, t r i e s to handle the concept i n a rather different way. Temperature i s introduced through AX. XIII which states: there exists an equivalence r e l a t i o n C on S and a state a^ e S such that for every 8 e J there exists a state b e 6 such that a^ and b are related v i a C . However th i s axiom i s superfluous since i t follows t r i v i a l l y from the set - theoretical Axiom of Choice. One form of th i s axiom states that i f the set S i s subdivided by a set of non-void d i s j o i n t subsets then there exists a choice-set containing precisely one element from each subset. But Rastall's TH. 5 states that S i s partitioned by the set J = {8 = FrA(a)|a e S} . By the d e f i n i t i o n of a p a r t i t i o n the Axiom of Choice i s applicable and says that there must exist a choice-set T containing precisely one element from each 8 e J . Having t h i s set T there are many ways to obtain a r e l a t i o n C which s a t i s f i e s the conditions of AX. XII. The simplest way would be to define C as follows: V-a.b e S. a and b are related v i a C i f and only i f a = b or a,b E T . The r e l a t i o n C i s c l e a r l y r e f l e x i v e , symmetric and t r a n s i t i v e so that i t i s an equivalence r e l a t i o n on S . Moreover, by construction i t s a t i s f i e s the conditions of the AX. X^ II'I equivalence r e l a t i o n with a^ being any element of T . I t i s therefore clear that Rastall's AX. XITT i s superfluous since i t i s l i t t l e more than a statement of the set-theoretical Axiom of Choice. I t i s of course quite unnecessary to assume an axiom of standard set theory i n a physical theory. There appears to be another problem i n connection with Rastall's 24 AX. XI I I . R a s t a l l says, "One can imagine a system for which two states have the same temperature i f f they have the same empirical entropy. This we exclude." That i s , AX. XIII has been chosen so that the equivalence r e l a t i o n of mutual adiabatic a c c e s s i b i l i t y i s not a possible empirical temperature. However AX. XIII has not been chosen i n such a way as to exclude the equivalence r e l a t i o n of anergic a c c e s s i b i l i t y as an empirical temperature. Moreover i t appears to be evident that the few remaining assumptions'in Rastall's theory after AX. XIII are not s u f f i c i e n t to exclude t h i s p o s s i b i l i t y . I t must be shown that anergic a c c e s s i b i l i t y , (previously shown to be an equivalence r e l a t i o n ) , s a t i s f i e s the requirements of AX. XI I I . This follows from AX. XII which says that V"a e S, (and not just 3 a^ e S), V- 3 e J 3 ! b e 3, (not just 3 b e 3 ), 3 b i s anergically accessible from a . I t must also be shown that anergic a c c e s s i b i l i t y i s not an equivalence r e l a t i o n that should be ti e d to the notion of temperature. One reason i s that from TH. B we have that [a]^H [ b ^ i s a single state V-a.b e S . Thus, i f anergic equivalence i s i d e n t i f i e d with the r e l a t i o n C of AX. XIII i t would turn out that the intersection of an isothermal hypersurface and an isentropic hypersurface i n the corresponding Euclidean space R n i s necessarily a single state no matter how complicated the physical system that E i s supposed to represent. For a two dimensional system t h i s objection vanishes but even here we would not want to i d e n t i f y anergic equivalence with C because t h i s would imply, for example, that i f you locked the piston and heated the gas i t s temperature would remain constant. 25 In summary, AX. XIII i s too weak. In a sense t h i s i s apparent from the f a c t that as i t stands i t can be derived i n a t r i v i a l way from an axiom of standard set theory. But i n p a r t i c u l a r i t i s too weak because i t allows anergic a c c e s s i b i l i t y as an empirical temperature. This could be permitted i f l a t e r assumptions ruled out t h i s p o s s i b i l i t y but they evidently do not. Later assumptions take the empirical temperature C as given and are r e s t r i c t i v e i n other d i r e c t i o n s . For example, AX. XIV involves the empirical temperature C but i s a condition imposed on the f , and not on C, since the empirical temperature i n the axiom i s taken as given and i s used to r e s t r i c t the f . Indeed, following t h i s axiom cL R a s t a l l comments, "We have stated XIV i n terms of a parameterization f of J and the functions F = f o f \ but one shows e a s i l y that i t i s , a a ' i n f a c t , independent of the choice of f and represents a condition imposed on the f "; (the emphasis i s mine). One might t r y to add some r e s t r i c t i v e assumptions when introducing 20 what i s apparently meant to correspond to the absolute temperature. But as i t stands R a s t a l l j u s t takes a f i x e d but a r b i t r a r y empirical temperature to work with i n obtaining h i s "temperature function" and therefore there i s nothing to prevent him from being "unlucky" and getting an anomalous empirical temperature. As we s h a l l see, an anomalous empirical temperature l i k e anergic equivalence n e c e s s a r i l y r e s u l t s i n an anomalous "temperature ' By doing so the concept of empirical temperature i s greatly expanded beyond what we usually mean by the term but of course i n i t s e l f there i s nothing wrong with t h i s . 26 function". In his seventh section R a s t a l l defines a "temperature function" to be a mapping T : S -> R 1 3 (1)Y"a,b e S, aCb-* T(a) = T(b), (2) V" a e J,y-a,b e a, z = f ( a ) - * T(b) = T(a) • F'(z)/F'(z), b a (3) 3 a e S 3 T(a) > 0 . S ince the equivalence r e l a t i o n C was "has the same temperature as" and since the temperature function induces on S another equivalence r e l a t i o n for temperature one might expect that these two relations are i d e n t i c a l . This i s not the case. The former implies the l a t t e r by condition (1) but we are given nothing to obtain the reverse implication. Therefore, i n general we must expect that more than one C-class of states having the same temperature w i l l be mapped onto a given 1 21 temperature i n R . A solution to t h i s problem which naturally comes to mind i s to replace by i n (1) . However we must be careful since strengthening (1) w i l l make i t harder to prove the existence of a temperature function. With the weaker condition (1) i n the d e f i n i t i o n R a s t a l l i s able to show the existence of a temperature function. In fact, AX. XIV i s equivalent to the statement that a temperature function e x i s t s , since i t i s also possible to show that i f a temperature function exists then AX. XIV must hold. For suppose we are given the hypothesis of AX. XIV, namely that we have an ar b i t r a r y pair of r e a l numbers z,w e f(J) along with a,c e f "*"(z) and b,d e f *(w) so that aCb and cCd . In 21 1 I t i s therefore clear that mapping the C-classes into R i n t h i s way cannot possibly extricate us from the d i f f i c u l t i e s incurred i n the choice of anergic equivalence for our C r e l a t i o n . I f anything, the problem i s made worse since even more states may be caused by the mapping to have the same temperature. 27 addition suppose that a temperature function i s known to ex i s t . Since a and c are i n f ~*~(z) = a e J then by condition (2) i n the d e f i n i t i o n of a temperature function we have T(a) = T(c)»F'(z)/F'(z) . S i m i l a r l y a c T b = Td«F^(w)/F^(w) . But by hypothesis we have aCb and cCd so that by the (unstrengthened) condition (1) we have T(a) = T(b) and T(c) = T(d) . Therefore F/(w)/F'(w) = F'(z)/F'(z) or F'(z)/F'(w) = F'(z)/F'(w) , b d a c c d a b which i s the conclusion of AX. XIV . So before strengthening (1) the s i t u a t i o n i s that AX. XIV i s equivalent to the existence of a temperature function. However when condition (1) i s strengthened i n the way suggested then Rastall's proof that a temperature function must exist i s no longer v a l i d because something 22 more must be proved now, namely that T(a) = T(b)-> aCb . But t h i s cannot be proved as things stand because the only relevant axiom i s AX. XIV and the implication i n this axiom i s i n the wrong di r e c t i o n for showing that T(a) = T(b)-> aCb . Therefore i t seems that the most natural way out of the problem would be to replace '•*'!" by i n AX. XIV, as well as i n condition (1) of the d e f i n i t i o n of a temperature function. Even given the changes suggested above, (and assuming an AX. XIII strengthened i n one way or another so as to exclude anomalous empirical temperatureslMke anergic equivalence!) , t h i s method of obtaining an "absolute" temperature does hot appeal to me because i t involves leaving S and going to R . When the theory comes to be applied, one naturally would ^^lote that proving the existence of a temperature function (and an entropy function) i s essential to Rastall's approach, as i s evident from the introductory section of his paper. Therefore the d i f f i c u l t y under discussion i s f a i r l y serious and must be corrected. 28 want to l i n k the temperature of a state to a r e a l number. However one wonders i f the fundamental nature of the temperature of a state of S might not be more cl e a r l y revealed i f the concept were to be introduced 23 into the formal theory without recourse to R . In his f i r s t section R a s t a l l makes a clear d i s t i n c t i o n between formal theory and experiment; However I think that R a s t a l l does not carry out the separation of formal theory and experiment far enough and this i s a good example of that. In his t h i r d section he c l e a r l y demonstrates his awareness that there need be no linkage to R n i n the formal theory: "In most applications of the theory, one can la b e l the states of a system by sets of n co-ordinates, that i s by points i n R n . ... Although coordinates are probably necessary to specify the topology i n applications of the theory, they need not be introduced into the formal structure." I t seems to me that i n the formal theory one should also avoid linkage of R and the temperature which, after a l l , can be thought of as one of the coordinates R a s t a l l mentions. Up to TH. 11 Rastall's theory i s ess e n t i a l l y algebraic i n character. When he then attempts to add some calculus he gets a rather unnatural j o i n t . One aspect of this j o i n t shows up i n AX. XIV and i n condition (1) of his d e f i n i t i o n of a temperature function. Fong seems to suggest something of this nature i n the introduction to his book, Foundations of Thermodynamics, when he emphasizes the importance of ordering over metric i n dealing with the concept of "temperature. 29 I suggest that a more natural approach to temperature would be to s t a r t out with a preordering on S rather than an equivalence r e l a t i o n . Physically this means star t i n g out with the idea of natural heat flow, that i s , a flow of heat not caused by work done on the system. Formally, we assume the existence of a preordering which s a t i s f i e s the following conditions: (1) V"a e S, a <_ a (2) Va,b,c e S, a <_ b and b <^  c - ^a <_ c (3) Va,beS, a _< b or b < ^ a . This r e l a t i o n i s a preordering on S and as such induces an equivalence r e l a t i o n on S . That i s , given a,b e S then a = b a £ b and b _< a . This equivalence r e l a t i o n i s then i d e n t i f i e d with the physical concept of equal temperature. Since the preordering which induces t h i s r e l a t i o n i s to be i d e n t i f i e d with natural heat flow, p a r t i c u l a r care must be taken to ensure that t h i s equivalence r e l a t i o n i s distinguishable from that of mutual adiabatic a c c e s s i b i l i t y . That such care should be necessary i s not too surprising since the concepts of temperature and entropy are intertwined to the extent that the i r units can only be such that the product of the units must result i n the units of energy. In fact, this i s the only s i g n i f i c a n t r e s t r i c t i o n that the units of temperature and entropy must s a t i s f y . This fact suggests a way by which one could perhaps distinguish equal entropy, (mutual adiabatic a c c e s s i b i l i t y ) , and equal temperature; one might use the fact that the most e f f i c i e n t way to change the temperature i s by keeping the entropy fixe d . This requirement, plus the requirement that 24 the temperature preordering must agree with entropy ordering of any t 0 ^ ' -jS :  For example, i f you lock the piston and heat the gas the temperature certainly cannot f a l l . 30 would appear to be s u f f i c i e n t to characterize the concept of temperature for the purposes of a formal theory. An important point that I wish to make concerning the material of section 7 has to do with Rastall's proposed generalization of his theory to allow for the p o s s i b i l i t y of negative temperatures. He simply introduces S and suggests assuming a l l over again for S, the axioms for S . Linkage axioms r e l a t i n g the states of S and S w i l l also be needed. Of course one can do t h i s and i t has the advantage of s i m p l i c i t y . On the other hand this way of obtaining negative temperatures seems rather i n e f f i c i e n t . I have already declared myself for another approach to temperature. However, given Rastall's approach up to t h i s point, i t might be possible to obtain negative temperatures by a way based on the way one obtains the set of a l l r e a l numbers from the set of positive r e a l numbers when one i s formally and axiomatically constructing the r e a l 25 numbers. D i f f i c u l t i e s might arise from the fact that one wishes negative temperatures to be hotter than positive temperatures. Also, temperature i s an equivalence r e l a t i o n on S rather than an equality. These d i f f i c u l t i e s do not appear serious and i f , i n fact, t h i s way of obtaining negative temperatures could be made to work i t would have the advantage of being formally more e f f i c i e n t than Rastall's method. This i s because Rastall's additional axioms would tend to be replaced by __ R.L. Wilder, Introduction to the Foundations of Mathematics, page 161 of the second ed i t i o n . 31 def i n i t i o n s and theorems. Also i t should be noted that the suggested method gives a way of obtaining an absolute zero as we l l as the negative temperatures. The p o s s i b i l i t y of excluding absolute zero from S was previously raised. In this way of obtaining negative temperatures we would also have a natural way of obtaining an absolute zero i f we did decide to have no such state i n S . Assuming the above suggestion could be made to work, i t i s s t i l l quite complicated. With the approach to temperature that I recommend, negative temperatures need not be added l a t e r . One observes that saying the temperature of one state i s less than that of another i s not so much a statement about temperature as a statement about heat; only when one says that two states have the same temperature i s one making a statement about temperature, as such. This suggests that we can think of the assumed preordering i n terms of g = -1/kT rather than T, the absolute temperature. In this way, states of "negative" temperature need not be added as an afterthought. The Continuity of the Entropy Function In his seventh section R a s t a l l follows his usual practice, of obtaining by ad hoc assumption any necessary continuity or d i f f e r e n t i a -b i l i t y property. This approach has the advantage of s i m p l i c i t y and after a l l , physics i s not calculus. Nevertheless, the Buchdahl-Greve attempt to derive such properties i s appealing; an ad hoc assumption i s always a l i t t l e j a r r i n g . However, their derivation given for the continuity of s i s very poorly done. The f i r s t proof seems d e f i n i t e l y c i r c u l a r : 32 "To prove the continuity of s one merely has to show that the inverse image s~ (A) of A i s open for every A e I (K, p. 86). Suppose on the contrary that for some A e I, s (A) i s not open. This means that there exists c e s, (A) and a sequence cn £ S (n = 1, 2, ...) such that d(c, c n) < 1/n, c n i s'-^CA), (n = 1, 2, ...) . This, however, i s i n direct c o n f l i c t with (3; 3) since the l a t t e r implies that for s u f f i c i e n t l y large n s ( c n ) e A, so that the required proof has been achieved." Now to ensure that the l a s t statement of the 'proof i s n ' t a t r i v i a l non sequitur, we evidently must assume that, given a state b e S and a neighbourhood U of s(b), then there exists a neighbourhood M of b such that s(M) i s a subset of U . For i f we don't know that this i s so then the l a s t statement of the 'proof simply does not follow. But-it i s a well known fact that this assumption i s equivalent to the assumption that s i s continuous. Thus, Buchdahl's "proof" i s obviously c i r c u l a r . Buchdahl goes on to give a proof of the "ordinary" continuity of s . But as a reason for doing so, he says, " i n general, the continuity of a r e a l function f (which might be defined on a metric space) with respect to the i n t e r v a l topology of i t s range does not imply i t s "ordinary" 26 continuity." Now this i s a very strange statement. I f i t means what i t appears to say then i t ' s f a l s e : for i t ' s a very well known fact that what 27 he defines as "continuity" and what he defines as "ordinary continuity" In a footnote Buchdahl explains that the "Ordinary continuity of f(c) i s to be understood as meaning that i f lim c = c then lim f ( c ) = f ( c ) . " n n f f i s continuous i f f f "'"(A) i s open whenever A i s open. 33 are completely equivalent. I presume that what Buchdahl means i s t h i s ; i f s has been proved to be continuous for a given topology on i t s range space and i f we change the topology of the range space then a new proof for the continuity of s i s needed. Therefore i f we decide to regard T as an i n t e r v a l of the r e a l numbers so that a (possibly) new topology i s induced on r by the natural topology of R, then the continuity of s must be reaffirmed. Whether or not this i s what Buchdahl i s trying to do, there can be no question that the attempted statement and proof are very badly handled. Moreover i t should be noted that this extra proof i s not independent of the f i r s t continuity proof, which I showed to be c i r c u l a r . Since the second proof assumed the v a l i d i t y of the f i r s t , i t s t i l l remains to show that s i s continuous (with respect to the o r i g i n a l topology on T) . A straightforward proof that s i s continuous, based on the Buchdahl-Greve assumption (3.3), i s given i n APPENDIX A. Although this proof i s e n t i r e l y straightforward i t i s somewhat lengthy and i t i s natural to wonder i f there might not be a simpler path to the re s u l t . The basic d i f f i c u l t y i s that s i s defined i n an unnecessarily complicated way. There i s no need to introduce a separate space T and define s as mapping S onto T . I t i s simpler to define s so that s : S -> S father thanedefifiingngi.ttsoottfatf: s : S •> T . Using TH. B we can achieve t h i s s i m p l i f i c a t i o n . Instead of mapping the state b e S onto [b]^ e T, we make s map b onto the unique y e S such that {y} = [ f ^ O [b]^, where the set i s t* i e " y ~ a x i s " of a coordinate system for S . The existence of such a coordinate system, consisting of t ^ ] ^ a n (^ ^ 2 an arbitrary f e S, was the essence of COR. B . Loosely speaking, [b]-34 can be regarded as a l i n e p a r a l l e l to the "x-axis", [f]^> and perpendic-u l a r to the "y-axis", [ f ] ^ • Therefore what we are suggesting i s that a state b e S, should be mapped onto the state y e S, which i s the i n t e r -section of the l i n e that i s p a r a l l e l to the x-axis and passes through b, and the l i n e [ f ^ which i s the y-axis; that i s , the state b e S can be represented by the ordered p a i r (x,y) and what we are suggesting i s that b = (x,y) should be mapped onto i t s y-coordinate rather than onto the e n t i r e l i n e [b]"L • Thus we are suggesting that the entropy function i s e s s e n t i a l l y a p r o j e c t i o n function onto one of the coordinates of the coordinate system whose existence i s guaranteed by COR. B . Now TH. B ensures the existence of a one-to-one map between [ f ] 2 and T . Since we have already shown that a natural t o t a l ordering of [ f ] 2 e x i s t s , we can define a topology for ^ n ^ e s a m e w a Y that Buchdahl and Greve defined a topology for T and thereby obtain the foundation needed to prove for t n e r e s u l t s Buchdahl and Greve obtain for Y . But we must go further than t h i s to obtain a simpler proof for the continuity of s . We have a topology for a n c* o n e could e a s i l y be found for * ^ o r e x a m P l e > w e could l e t open sets i n [ f ] ^ °e the in t e r s e c t i o n s of [^3^ a n ^ open sets i n S . But although COR. B ensures the existence of a one-to-one map between [ f ] ^ x [f] and S, i t i s also necessary that the topology of S be the product topology of [ f ] ^ x [f]^ • The way around t h i s problem i s to begin with two t o p o l o g i c a l spaces, X and Y corresponding to [ f ] ^ and [ f ] ^ , and then introduce the cross-product X x Y to correspond to S, the set of equilibrium states. Then s i s defined as the p r o j e c t i o n of 35 X x Y onto Y and i s therefore t r i v i a l l y continuous. For i f A i s open i n Y then s "*"(A) = A x Y which i s open by d e f i n i t i o n of the product topology. SUMMARY The most important ideas i n c l a s s i c a l thermodynamics are heat and work, temperature and entropy. Given the fundamental importance of heat and work, and keeping i n mind the conservation of energy law, i t i s natural to expect that no-work processes and no-heat processes should have a deep significance. In his (no-work) anergic processes R a s t a l l tightens and formalizes Buchdahl's notion of an isometric process. These anergic processes have a fundamental importance equal to that of the (no-heat) adiabatic processes. The c r i t i c a l axiom l i n k i n g these two types of processes i s Rastall's AX. XII . Although Buchdahl uses the idea which under^lies^thisiaxiomyeven^less than Ra s t a l l does he appear to r e a l i z e i t s strength and importance. The other c r i t i c a l point i n this area i s the result that the fron t i e r s of the sets of states adia b a t i c a l l y accessible from i n i t i a l s&'abes form a p a r t i t i o n of the set of a l l equilibrium states. Buchdahl and Greve deal with this point i n a more convincing fashion. Although R a s t a l l also proves the r e s u l t , his proof requires two assumptions which seem arbitrary and a r t i f i c i a l . In contrast, the proof of Buchdahl and Greve rests on an assumption which appeals by virtue of i t s direct physical connections. The way that R a s t a l l chooses to introduce the concept of entropy into his theory d i f f e r s s i g n i f i c a n t l y from that of Buchdahl and Greve. 36 Both ways have something to recommend them. Rastall's way has an essential s i m p l i c i t y . On the other hand R a s t a l l makes ad hoc continuity and d i f f e r e n t i a b i l i t y assumptions. Buchdahl and Greve try to avoid t h i s . They base thei r proof of the continuity of the entropy function on the previously mentioned physical assumption. Although thei r proof i s not v a l i d , a correct proof based on this physical assumption i s possible. R a s t a l l attempts to approach the concept of temperature by a new route whereas Buchdahl makes no such attempt. Unfortunately Rastall's approach f a i l s at a c r u c i a l point through lack of s u f f i c i e n t l y strong conditions on the equivalence relations which may be called empirical temperatures. His approach i s such that an anomalous empirical temperature results i n an anomalous "temperature function". Therefore his "temperature function" cannot be i d e n t i f i e d with the usual absolute temperature scale even though, as he shows, many properties are shared. R a s t a l l also suggests a method by which negative temperatures might be introduced. The method has s i m p l i c i t y but i s formally i n e f f i c i e n t . CONCLUSIONS A number of suggestions have been made concerning ways of improving or avoiding d i f f i c u l t i e s i n the two approaches to thermodynamics which we have been discussing. They are gathered here. One begins with the topological spaces X and Y corresponding physically to states which are adiabatically equivalent and anergically equivalent, respectively. S, the product space of X and Y, i s i d e n t i f i e d with the set of a l l equilibrium states of a system. The entropy 37 function i s defined to be the projection of S onto Y and i s therefore continuous. By a weakened form of the Axiom of Choice there exists a t o t a l ordering of the space Y . This ordering induces a preordering on S which i s linked to the physical concept of adiabatic a c c e s s i b i l i t y . A state c = (x^,y^) e S i s said to have antentropy which i s not less than that of d = (x 2,y 2> e S i f f s(d) = y 2 <_ s(c) = y^ . The P r i n c i p l e of the Increase of Entropy follows"automatically. The concept of temperature i s introduced through the preordering of S which corresponds to the physical idea of natural heat flow. An induced equivalence r e l a t i o n gives us temperature equality for the states of S . SPECULATIONS Portions of Rastall's approach remind one of plane geometry. We can interpret the [ f ] ^ a n d ^ 2 a S •'••'•nes "*"n a P ± a n e such that i s p a r a l l e l to every [g]^ and perpendicular to every " "*"n p a r t i c u l a r , AX. XII i s then a simple geometric statement. The question of including the state of absolute zero i n S seems to be connected to the question of r e l a t i n g the theory to a projective geometry rather than some other type of geometry, such as an affine geometry. (A projective geometry includes i t s i n f i n i t y points; an affine geometry doesn't.) This question i s int e r e s t i n g for the reason that i n projective geometry there i s a duality p r i n c i p l e : any v a l i d statement of projective geometry should remain v a l i d when the words "point" and " l i n e " are i n t e r -changed everywhere. Thus i f we were to i d e n t i f y the state of absolute zero with a point of a projective geometry, a physical statement concerning 38 absolute zero might have to be true i f absolute zero were also to be i d e n t i f i e d with a l i n e i n the geometry, that i s , with a set of states. Just what physical requirements such a set of states might therefore have to s a t i s f y i s an interesting question. The most important d i f f i c u l t y i n r e l a t i n g S to a geometry i s that we must be able to s a t i s f y the c r u c i a l geometric axiom which says that given two points there i s at least one l i n e containing both. Physically this seems to amount to the following question: given two equilibrium states, i s i t always the case that there i s some reversible process l i n k i n g them? There doesn't seem to be any obvious objection to such an assumption. The only opinions i n the l i t e r a t u r e that I am aware of are more or less 28 posxtive. Thus we might assume that given any two states of S there i s an equivalence r e l a t i o n on S, (corresponding to some reversible physical process), such that the two states are equivalent. Then the set of a l l states equivalent under t h i s r e l a t i o n might perhaps be considered a l i n e i n the same way that any o r ^^1 w e r e considered to be l i n e s . I f th i s could be done then every pair of points would be contained i n at least one l i n e . Stuart, Gal-Or and Brainard (Editors): A C r i t i c a l Review of Thermo-dynamics, page 208. 39 BIBLIOGRAPHY Boyling, J.B.: Caratheodory's P r i n c i p l e and the Existence of Global Integrating Factors. Communications i n Mathematical Physics 10, 52(1968). ' : An Axiomatic Approach to C l a s s i c a l Thermodynamics. Proceedings of the Royal Society A 329, 35(1972). Bridgman, P.W.: The Nature of Thermodynamics. Harper, New York 1961. Buchdahl, H.A.: A Formal Treatment of the Consequences of the Second Law  Of Thermodynamics i n Caratheodory's Formulation. Z e i t c h r i f t fur Physik 152, 425(1958). : Entropy Concept and Ordering of States. I . Z e i t c h r i f t fur Physik 168, 316(1962). , and W. Greve: Entropy Concept and Ordering of States. I I . Z e i t c h r i f t fur Physik 168, 386(1962). Fong, P.P.: Foundations of Thermodynamics. Oxford University Press, New York, 1963. Giles, R.A.: Mathematical Foundations of Thermodynamics. Macmillan, New York, 1964. Hartshorne, R.: Foundations of Projective Geometry. Benjamin, New York, 1967. Kelley, J.L.: General Topology. D. van Nostrand Company, New York, 1955. Massey, W.S.: » Algebraic Topology: An Introduction. Harcourt, Brace and World, New York, 1967. Pippard, A.B.: Elements of C l a s s i c a l Thermodynamics. Cambridge University Press, Cambridge 1964. R a s t a l l , P.: C l a s s i c a l Thermodynamics Simplified. Journal of Mathematical Physics 11, 2955, October 1970. Stuart, E.B., B. Gal-Or and A.J. Brainard, (Editors): A C r i t i c a l Review of  Thermodynamics. Mono Book Corporation, Baltimore, 1970. Tisza, L.: Generalized Thermodynamics. M.I.T. Press, Cambridge, 1966. 40 Wilder, R.L.: Introduction to the Foundations of Mathematics. Wiley, New York, 1965. Wilson, A.H.: Thermodynamics and S t a t i s t i c a l Mechanics. Cambridge University Press, Cambridge, 1957. 41 APPENDICES 42 APPENDIX A A necessary and s u f f i c i e n t condition for the continuity of s i s that given b e S and a neighbourhood U of s(b) then 3 a 29 neighbourhood M of b 3 s ( M)CZU . Since U i s a neighbourhood of s(b) -J an open N 3 s(b) e N and N C U . By the d e f i n i t i o n of a base for the T-topology 3 N' 9 N'C N, s(b) e N' and N 1 has one of the three following forms: (1) Suppose N' i s of the form {y e Y \ y < a} for some a e P . Choose c e s "'"(a) . Now s(b) e N' so s(b) < s(c) = a . By d e f i n i t i o n of the T-ordering t h i s implies that b i s adiab a t i c a l l y inaccessible from c . Therefore by (3.3) 3 £ > 0 ^ every state i n M , an open e-ball centered at b, i s adiabatically inaccessible from c . By d e f i n i t i o n of the T-ordering, s(d) < s(c) = a V'd e M . The form of N' then implies that s ( M ) C N ' . Since N" C N and N C U , then s (M) C U . (2) Suppose that N' i s of the form {y e Y | a < y] for some a e T . Choose c e s """(a) . Now s(b) e N' so s(c) = a < s(b) . By d e f i n i t i o n of the P-ordering t h i s implies that c i s adiab a t i c a l l y inaccessible from b . Therefore by (3.3) 3 e > 0 3 c i s adiabatically inaccessible from every state i n M , an open e-ball centered at b . By d e f i n i t i o n of the T-ordering, s(c) = a < S(d) V"d e M . The assumed form of N' then implies that s ( M ) C N ' , that i s , S ( M ) C N ' C N C T U . J.L. Kelley, General Topology, p. 86 43 (3) Suppose N' i s of the form {y e T \ a < y < g} for some a, 8 £ r . Choose h e s ^ B) and g e s "'"(a) . Now s(b) £ N' so a < s(b) and s(b) < 6 . Using the method of (1) and (2) we obtain M^ , an open b a l l centered at b such that g i s adiab a t i c a l l y inaccessible from every state i n M^ , and M^ , an open b a l l centered at b such that every state i n i s adiab a t i c a l l y inaccessible from h . Let M = M, o • Then M i s an open b a l l (either or , centered at b such that V d E M, d i s adiabatically inaccessible from h and g i s adiab a t i c a l l y inaccessible from d . By d e f i n i t i o n of the T-ordering this implies that s(g) = a < s(d) < s(h) = BVd e M . The assumed form of N' then implies that s ( M ) C N ' , that i s , s(M) C N ' t C N C O . Therefore there does i n a l l cases exist M, a neighbourhood of b, which s maps into U, the given neighbourhood of s(b), and so the proof i s complete. 44 APPENDIX B The following are the important axioms and theorems that have been discussed. They have been translated into a common notation, based on R a s t a l l ' s , to make comparison easier. R a s t a l l Buchdahl-Greve I: V"a e S ; a -* a :Y"a e S; a -* a I I : V"a,b,c e S ; a -> b, b -> c implies a c ^ a j b . c e S; a -»• b, b -> c implies a -»- c I I I : V"a,b e S ; a b or b -> a (3.2): a,b e S; a-/»b implies b •> a VI: ¥"a e S ; a e FrA(a) (3.1): ¥"a,b e S ; given the neighbourhood U of b, then 3 c e U 3 b-f>- c VII:;V-a,b e S; b e FrA(a) impli-implies a e FrA(b) (Auxiliary axiom: FrA(a) CT A(a) ) > j(3.3) : V"a,b e S; a-/*- b implies ^M, a neighbourhood of a, and N, a neighbourhood of b 3 V c e M and V"d e N, c-f*- d . TH.5: (FrA(a) | a e S} part i t i o n s S TH. : Ya e S; FrA(a) = [a] 45 "XI: V"a,b e S; heat absorbed i n an anergic process from a to b i s >_ 0 i f f a -* b XII: given a e S then V [ c ] ^ 3! b e [ c ] 1 3 b e [ a ] 2 no formal assumption but a comparable idea i s used TH.8: V"a,b e S; i f 3 a n anergic adiabaticapfpcesstfrom ahefco b =then b = a . TH. 12: V a , b € S; a-> b i f f the empirical entropy of a i s <_ that of b XI I I : 3 an equivalence r e l a t i o n Z C on S and a state a Q e S 3 V [ c ] x 3 b e [ c ] 1 a aQCb XIV: Yz ,w e f ( J ) , |^a,c e f _ 1 ( z ) V*b,d e f ''"(w) ; aCb and cCd imply F' (z)/F'(w) = a b F ' (z)/F ' (w) c d 

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