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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A comparison and criticism of two axiomatic systems for classical thermodynamics Erickson, Arvon Donald 1975
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Title | A comparison and criticism of two axiomatic systems for classical thermodynamics |
Creator |
Erickson, Arvon Donald |
Publisher | University of British Columbia |
Date Issued | 1975 |
Description | This thesis is a comparison and criticism of two approaches to classical thermodynamics, that due to P. Rastall in the "Journal of Mathematical Physics", Volume 11, Number 10, October 1970, page 2955, and that due to H.A. Buchdahl and W. Greve in "Zeitschrift für Physik" 168, 1962, page 316 and page 386. Rastall's approach is somewhat more general and appeals because of his generally careful attention to the explicit, orderly statement of the assumptions and rigorous proofs based only on these assumptions. However, his approach is significantly improved by substituting the main assumption of the Buchdahl-Greve paper for two of his assumptions. The Buchdahl-Greve assumption is fundamentally appealing and as well as leading to a more natural proof of a crucial theorem, it also allows a straightforward proof for the continuity of the entropy function, which Rastall assumes ad hoc. The proof given in the Buchdahl-Greve paper for the continuity of the entropy function is shown to be circular but a correct proof based on the Buchdahl-Greve assumption is possible and one is given in an appendix. Suggestions are also made concerning how one might improve this combined Buchdahl-Rastall approach. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-01-28 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085270 |
URI | http://hdl.handle.net/2429/19310 |
Degree |
Master of Science - MSc |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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