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UBC Theses and Dissertations

Critical comparison of some theories of classical irreversible statistical mechanics Seagraves, Paul Henry 1969

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• A CRITICAL COMPARISON OF SOME THEORIES OF . . CLASSICAL IRREVERSIBLE STATISTICAL MECHANICS BY PAUL HENRY SEAGRAVES E . S c , New Mexico I n s t i t u t e of Mining and Technology, 1963 M . S c , U n i v e r s i t y of B r i t i s h Columbia, 1964 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN THE DEPARTMENT OF PHYSICS We accept t h i s t h e s i s as conforming to the re q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA September, I96S T) Paul Henry Seagraves 1969 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 an 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 Physics  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 V a n c o u v e r 8, C a n a d a D a t e September 9 . 1968~ . .- ABSTRACT The i n f i n i t e o r d e r p e r t u r b a t i o n t h e o r y o f P r i g o g i n e and coworkers i s us e d , w i t h some m o d i f i c a t i o n s , t o d i s c u s s t h e t h e o r i e s o f c l a s s i c a l i r r e v e r s i b l e p r o c e s s e s due t o B o g o l i u b o v , S a n d r i & F r i e m a n , and Mazur & B i e l . The l a t t e r a u t h o r s use t h e BBKGY h i e r a r c h y o f e q u a t i o n s as a s t a r t i n g p o i n t . A c c o r d i n g l y , t o d i s c u s s t h e s e t h e o r i e s t h e i n f i n i t e o r d e r p e r t u r b a t i o n t h e o r y i s w r i t t e n out i n such a way t h a t i t r e l a t e s e a s i l y t o t h e BBKGY h i e r a r c h y . The n a t u r e o f t h e a s s u m p t i o n s i n v o l v e d i n t h e t h e o r i e s o f B o g o l i u b o v and S a n d r i & Friema n become p a r t i c u l a r l y c l e a r when com-pared w i t h t h e i n f i n i t e o r d e r p e r t u r b a t i o n e x p a n s i o n . The r e l a t i o n o f the t h e o r y o f Mazur & B i e l w i t h t h e c l u s t e r e x p a n s i o n o f Green i s a l s o e l u c i d a t e d . i i i TABLE OF CONTENTS A b s t r a c t . . . . . . . • . • . . . . . i i T a b l e o f C o n t e n t s .• . . i i i Acknowledgements i v I . I n t r o d u c t i o n . . . . . . 1 I I . The L i o u v i l l e E q u a t i o n and Reduced D i s t r i b u t i o n F u n c t i o n s . . . . . . . . 6 I I I . The Diagram T e c h n i q u e . . . . . . . . . 10 The BBKGY H i e r a r c h y o f E q u a t i o n s . . . 14 IV. The Dependence o f t h e D i s t r i b u t i o n F u n c t i o n s Upon t h e One Body F u n c t i o n . . . . 18 V. The Short-Range E x p a n s i o n . . . . . . . 31 The H i e r a r c h y o f E q u a t i o n s f o r t h e C l u s t e r E x p a n s i o n . . . . . . . 31 V I , The Markowian A p p r o x i m a t i o n 40 V I I . The Theory o f B o g o l i u b o v . . 4 8 V I I I . The Method o f E x t e n s i o n . . . . . . . . 65 I X . The Theory o f Mazur and B i e l . . . . . . . . 74 X. C o n c l u s i o n 82 B i b l i o g r a p h y . . . • . 83 A p p e n d i x I . . 84 . App e n d i x I I 87 i v ACKNOWLEDGEMENTS I wish to express my appreciation to Dr. Luis de Sobrino f o r suggesting t h i s problem and fo r his continued guidance and advice. F i n a n c i a l assistance i n the form of a graduate fellow-ship from the University of B r i t i s h Columbia and a Student-ship from the National Research Council of Canada i s grate-f u l l y acknowledged. INTRODUCTION. S e v e r a l a t t e m p t s have been made t o u n d e r s t a n d i r r e -v e r s i b l e p r o c e s s e s s t a r t i n g f r om t h e L i o u v i l l e e q u a t i o n . I n (1946) Bogoliubov"*" proposed a n o v e l and i n t e r e s t i n g p e r t u r b a t i o n e x p a n s i o n method. H i s t h e o r y has however remained c o n t r o v e r s i a l . Our c e n t r a l aim i n t h i s work i s t o c l a r i f y B o g o l i u b o v ' s e x p a n s i o n method. To do t h i s we a p p l y i n f i n i t e o r d e r p e r t u r b a t i o n t h e o r y u s i n g t h e more r e c e n t (1959) diagram t e c h n i q u e s d e v e l o p e d by P r i g o g i n e 2 and c o w o r k e r s . " I n p a r t i c u l a r we use w i t h some m o d i f i c a t i o n t h e i n f i n i t e o r d e r p e r t u r b a t i o n t h e o r y as w r i t t e n out by 3 Severne. We d i s c u s s t h e n e c e s s a r y r e s t r i c t i o n s on i n f i n i t e o r d e r p e r t u r b a t i o n t h e o r y i n o r d e r t o o b t a i n Bogoliubov. 1 s e x p a n s i o n . I n t h i s same s p i r i t we a l s o d i s c u s s t h e t h e o r i e s o f S a n d r i & F r i e r n a n ^ ' ^ and Mazur &. B i e l . ^ • E f f o r t s t o e s t a b l i s h p o s s i b l e c o n n e c t i o n s between B o g o l i u b o v ' s t h e o r y and i n f i n i t e o r d e r p e r t u r b a t i o n t h e o r y began a e a r l y as (1961) by P r i g o g i n e and R e s i b o i s . ^ The problem has a l s o been c o n s i d e r e d by S t e c k i & T a y l o r (1965), B r o c a s & R e s i b o i s ^ (1966), and Braun & G a r c i a - C o l i n 1 0 (1966). These a u t h o r s use t h e M a s t e r E q u a t i o n approach and have compared t h e i r r e s u l t s w i t h B o g o l i u b o v 1 s e x p a n s i o n . B o g o l i u b o v s t a r t s by d e r i v i n g t h e w e l l known BBKGY hieracchy^" o f e q u a t i o n s . F o r t h i s r e a s o n we choose t o w r i t e out the i n f i n i t e o r d e r p e r t u r b a t i o n e x p a n s i o n i n such a way t h a t i t r e l a t e s e a s i l y t o t h e BBKGY h i e r a r c h y and i t i s our o p i n i o n 2. that these expansions elucidate Bogoliubov's approximation more clearly than does the Master Equation approach. We w i l l be discussing a classical system of many particles. . We spacialize to consider systems of identi-cal particles which interact through a central two body potential with no external fields on the system. Our start-ing point w i l l be the basic evolution equation of s t a t i s t i c a l mechanics, the Liouville equation. In Chapter II we form-a l l y solve the Liouville equation.and write the answer as a perturbation series. Since i t has shown considerable promise in the literature; we introduce f i r s t - , second-, and higher-order distribution functions defined by integra-ting the solution to Liouville's equation over coordinates and momenta of a l l but the particle one, two, etc. To simplify our method of writing these perturbation series we introduce in Chapter III our definitions of diagrams which are similar to those of Prigogine and coworkers. In this Chapter III we perform some simplifications and.we find that our expansion relates easily to the BBKGY hierarchy of equations which couple the reduced distribution functions. Also in Chapter III we introduce the Cluster expansion"*""*" of the distribution functions/-which we find particularly useful to expand the s t a t i s t i c a l i n i t i a l condition needed for the solution of the Liouville equation. In successful theories of irreversible s t a t i s t i c a l mechanics the dependence of the distribution functions upon t h e one body d i s t r i b u t i o n f u n c t i o n has p l a y e d a d o m i -nant r o l e . I n Chapter IV we work out t h i s t h i s dependence i n d e t a i l ; we o b t a i n t h e dependence by p e r f o r m i n g a p a r t i c -u l a r summation o f d i a g r a m s . I n t h e l a t e r p a r t o f t h e c h a p t e r we show t h a t t h i s summation i s n e c e s s a r y t o d e t e r m i n e t h e l o n g t i m e b e h a v i o r o f t h e system. That i s , t h e l o n g t i m e b e h a v i o r o f a p h y s i c a l system i s d e t e r m i n e d t h r o u g h t h e one body d i s t r i b u t i o n f u n c t i o n . I n C h a p t e r V we s p e c i a l i z e t h e t h e o r y t o c o n s i d e r systems f o r w h i c h o n l y s m a l l c l u s t e r s o f p a r t i c l e s i n t e r a c t s i m u l t a n e o u s l y . Such a s i t u a t i o n o c c u r s f o r d i l u t e systems o r f o r systems where t h e p a r t i c l e s i n t e r a c t t h r o u g h s h o r t range f o r c e s . We c a l l such systems s h o r t - r a n g e systems, and we c a r r y out the. c a l c u l a t i o n s . t o i n c l u d e a l l two and t h r e e p a r t i c l e c o l l i s i o n e f f e c t s . We f i n d t h a t a v a l u a b l e g u i d e f o r s e l e c t i n g and summing diagrams i s t h e h i e r a r c h y o f e q u a t i o n s , s i m i l a r t o t h e BBKGY h i e r a r c h y , t h a t c o u p l e t h e c l u s t e r e x p a n s i o n c o e f f i c i e n t s t h a t were i n t r o d u c e d i n Chapter I I I . We b e g i n Chapter V by d e r i v i n g t h i s h i e r a r c h y o f e q u a t i o n s i n a form which i s u s e f u l f o r our purposes. These e q u a t i o n s have f o r m e r l y been o b t a i n e d by Green. . The e q u a t i o n s o f Chapter IV and V a r e non-Markowian. A method f o r f i n d i n g w e l l behaved Markowian a p p r o x i m a t i o n s has hot t o our knowledge been s a t i s f a c t o r i l y d e termined i n t h e l i t e r a t u r e . I n Chapter VI we produce a Markowian a p p r o x i m a t i o n p r o c e d u r e t h a t r e l a t e s most e a s i l y t o methods fo u n d i n t h e l i t e r a t u r e . We p o i n t out some a s y m p t o t i c 4, d i v e r g e n c e d i f f i c u l t i e s o f t h i s method. I n C hapter V I I we d i s c u s s B o g o l i u b o v ' s t h e o r y . We b e g i n by w r i t i n g a resume o f h i s t h e o r y . Then we d e v e l o p f r o m t h e knowledge o f t h e former c h a p t e r s an e x p a n s i o n method t h a t we f e e l most n e a r l y as p o s s i b l e f o l l o w s : B o g o l i u b o v ' s e x p a n s i o n , and we s o l v e our e q u a t i o n s s i d e by s i d e w i t h t h o s e o f B o g o l i u b o v . What we do i s de t e r m i n e < i f i t i s p o s s i b l e t o time c o a r s e g r a i n t h e d i s t r i b u t i o n f u n c t i o n s by a c e r t a i n a p p r o x i m a t i o n o f t h e f i n e g r a i n b e h a v i o r . We d e t e r m i n e t h e n e c e s s a r y r e s t r i c t i o n s on our e x p a n s i o n s needed t o o b t a i n Bogoliubov's;; t h i s amounts t o a d i s c u s s i o n o f B o g o l i u b o v ' s a s y m p t o t i c boundary c o n d i t i o n s . We f i n d however t h a t a m e a n i n g f u l a s y m p t o t i c l i m i t cannot be f o u n d . I n C h apter V I I I we d i s c u s s t h e t h e o r y o f S a n d r i & Fr i e m a n ^ ' ^ w h i c h t h e y c a l l t h e method o f e x t e n s i o n . We f i n d t h a t t h i e r t h e o r y i s a n o t h e r a p p r o x i m a t i o n t h a t can be o b t a i n e d from our c o a r s e g r a i n i n g p r o c e d u r e i n t r o d u c e d i n Chapter V I I . I n C h a p t e r IX we d i s c u s s t h e t h e o r y o f Mazur & B i e l . ^ The r e l a t i o n o f t h e i r t h e o r y t o t h e C l u s t e r e x p a n s i o n o f Green i s e l u c i d a t e d . I n Chapter X we g i v e some c o n c l u d i n g r e m a r k s . F i g u r e I i s a s y n o p t i c diagram o f t h e o r g a n i z a t i o n o f t h i s t h e s i s . 5. L i o u v i l l e ' s e q u a t i o n and r e d u c e d d i s t r i b u t i o n f u n c t i o n s E x p a n s i o n i n terms o f P r i g o g i n e t y p e d i a g r a m s Our d e f i n i t i o n o f diagrams r e l a t e e a s i l y t o t h e BBKGY h i e r a r c h y o f e q u a t i o n s The f u n d a m e n t a l r o l e o f t h e one body d i s t r i b u t i o n f u n c t i o n i t h e l o n g t i m e b e h a v i o r i s d e t e r m i n e d by t h e one body f u n c t i o n S e l e c t i o n and summing o f diagrams f o r a system w h i c h i n t e r a c t t h r o u g h a s h o r t range i n t e r p a r t i c l e p o t e n t i a l C a l c u l a t i o n o f t h e Markowian e q u a t i o n s f o r t h e d i s t r i b u t i o n f u n c t i o n s We d e t e r m i n e i f i t i s p o s s i b l e t o t i m e c o a r s e g r a i n our f u n c t i o n s by a . - c e r t a i n a p p r o x i m a t i o n o f t h e f i n e g r a i n b e h a v i o r w i t h a subsequent s t u d y o f p e r s i s t a n t e f f e c t s C l u s t e r e x p a n s i o n method The t h e o r y o f Mazur and B i e l i s s i m i l a r t o t h e C l u s t e r e x p a n s i o n method The t h e o r y o f B o g o l i u b o v a p p r o x i m a t e s t h i s p r o c e d u r e The Method o f E x t e n s i o n a p p r o x i m a t e s t h i s p r o c e d u r e F i g u r e ! I . I I . THE LIOUVILLE EQUATION AND REDUCED DISTRIBUTION FUNCTIONS. The s t a r t i n g p o i n t of t h i s work i s the basic equation of c l a s s i c a l s t a t i s t i c a l mechanics, the L i o u v i l l e equation. Consider a system, of volume V, formed by N i d e n t i c a l p a r t i c l e s w i t h known i n t e r a c t i o n s . The s t a t i s t i c a l s t a t e of the system i s described by a d i s t r i b u t i o n f u n c t i o n "(*• , ^ 't £1 )<£;• • -j where oc^ or are the r e s p e c t i v e p o s i t i o n and v e l o c i t y of p a r t i c l e [• The d i s t r i b u t i o n f u n c t i o n i s normalized by (2.1) J " - J ^ « ^ t f i FN= I a and, f o r a system of i d e n t i c a l p a r t i c l e s , F^ must be symmetric f u n c t i o n of ( ^ ^ j j ( ^ / J . . . f ^ , ^ ) • The e v o l -u t i o n of F N i n time i s governed by the L i o u v i l l e equation (2.2) ~ H N F " - f K " - I w ) FN where K W and I " are operators defined by (2.3) K w = £. Ki = X I - v r h i -i-l here m i s the mass of a p a r t i c l e . We have presupposed that the p a r t i c l e s i n t e r a c t through a c e n t r a l two body p o t e n t i a l \Jlj (ftfj and that there are no e x t e r n a l f o r c e s a c t i n g on the system. And, since we are i n t e r e s t e d only i n the bulk p r o p e r t i e s of the system,, the w a l l or container 7. . p o t e n t i a l has been n e g l e c t e d . The f o r m a l s o l u t i o n of L i o u v i l l e ' s e q u a t i o n i s (2.5) F"« e-'K"-1")t F-^ O) We f i n d t h a t a more u s e f u l form of the s o l u t i o n i s the p e r t u r b a t i o n s e r i e s e x p a n s i o n (2.6) F w = e-K"*F-fO.) + r K N * p t i e K " * ' i V K " * ' ^ ( 0 ) +• e" The reduced s - p a r t i c l e d i s t r i b u t i o n f u n c t i o n s (2 . - -0 F'(i.,«,...,*„«;t).J-/^'"^w, • • • 4 f » ^ N F" are a l s o i n t r o d u c e d . I n (2.7) the p o s i t i o n and v e l o c i t y i n t e g r a l s are t o be always u n d e r s t o o d t o extend over the f u l l range of t h e i n t e g r a t i o n v a r i a b l e s . These f u n c t i o n s g i v e the p r o b a b i l i t y d e n s i t y f o r the dynamic s t a t e s of the s p a r t i c l e s c o n s i d e r e d b e i n g l o c a t e d , r e s p e c t i v e l y , i n the i n f i n i t e s i m a l phase volume elements dockets,... around .the p o i n t s «I}AT . . t j tts,tr<, a t the time t . U s i n g t he d e f i n i t i o n (2.7) and the p e r t u r b a t i o n s e r i e s ( 2 . 6 ) , t o g e t h e r w i t h (2.4) we have (2.8) . . ( c o n t i n u e d ) (2 .8 ) F s = j ^ - ' ^ S 8 ' t < j •> ^ • ' " C l t , , * " * . £ I^-t^Cit^^H I e-K"4'F"(0) A t t h i s p o i n t i t i s c o n v e n i e n t t o i n t r o d u c e a s i m p l i -f y i n g r e l a t i o n . L i o u v i l l e ' s e q u a t i o n (2.2) assumes the system t o be i n s e n s i t i v e t o i t s b o u n d a r i e s and thus t o the s h i f t i n g of i t s b o u n d a r i e s . Under t h i s c o n d i t i o n we have ( 2 . 9 ) J <t*i e~Hlt &(<*;) = j d*c G o~y, ^ eM U L t J l G fe) ( 2 . 1 0 ) J </*j e' K IJ ' f G f e - ^ j ) '- -<*j) because (2.11) K- - Yi'^. , K ; - ^ . f / T j i n the e x p o n e n t i a l s produce a T a y l o r e x p a n s i o n (2.12) e'HliG(*;) G ( Vi-ATii) and the i n t e g r a l s s i m p l i f y by s h i f t i n g the b o u n d a r i e s of the system. Note t h a t (2.13) e K w t I:j e " K w f = e ^ I j . e ^ ' J * and t h a t u s i n g (2.9) the p e r t u r b a t i o n s e r i e s ( 2 . 8 ) may be w r i t t e n (2.14) . . . ( c o n t i n u e d ) 9. (2.14)F s = j ^ S i ' ^ l . . . ^ < / t f f t •  • I where (2.15) K S- £ We. w i l l w r i t e the terms of the e x p a n s i o n (2.14) i n a form c o n v e n i e n t f o r the diagram t e c h n i q u e t o be i n t r o d u c e d i n t he f o l l o w i n g s e c t i o n . T h i s p o i n t w i l l be i l l u s t r a t e d by an example. C o n s i d e r the c o n t r i b u t i o n to F (2.16) J * i t 3 e K « 9 * » f ' K i ? t 3 P w ( o ) The s p a t i a l i n t e g r a t i o n s are commuted from l e f t to r i g h t as f a r as p o s s i b l e and used to p e r f o r m any r e d u c t i o n on the F w(0) t h u s our example (2.16) w i l l be w r i t t e n (2.17) e - ^ k K ' 2 t a i 2 e ^ ^ i t 2 ( 7 $ ^ 8 e ^ i 2 I , ? e - ^ ^ I I I . THE DIAGRAM TECHNIQUE. As an a i d i n w o r k i n g w i t h the s e r i e s (2.14) the diagram 2 t e c h n i q u e of P r i g o g i n e and coworkers - w i l l be used w i t h some m o d i f i c a t i o n . F i r s t we w r i t e the terms of (2.14) i n the form s i m i l a r t o (2 . 1 7 ) . We see. t h a t each term s t a r t s w i t h -/< sf -K*t 6 and we w i l l w r i t e t h i s , f i r s t e x p l i c i t l y . F o r th e s s e t s of p a r t i c l e c o o r d i n a t e s of F s we draw and l a b e l s l i n e s r u n n i n g from l e f t t o r i g h t . F o r example F i s a f u n c t i o n of t h e c o o r d i n a t e s of two p a r t i c l e s ' and, t h e r e f o r e , the diagram f o r each term c o n t r i b u t i n g t o F s t a r t s on the l e f t w i t h -Kzt (3.1) e z We f i n d t h a t t h e m a t h e m a t i c a l b e h a v i o r of a g e n e r a l term can be broken down i n t o t h r e e d i s t i n c t i v e b u i l d i n g b l o c k s . Each of t h e s e c o n t a i n s an e x p r e s s i o n of the form \« „ , c J-ijC which i s made to c o r r e s p o n d t o a v e r t e x . The t h r e e t y p e s of v e r t i c e s c o r r e s p o n d i n g t o the t h r e e d i s t i n c -t i v e b u i l d i n g b l o c k s w i l l be d e f i n e d as f o l l o w s : ( a ) . i f t o the l e f t of the e x p r e s s i o n j e / < « , e ^ ^ I - j e l) m the l a b e l s i and j have a l r e a d y been used, l i n e s l a b e l e d i and j are a v a i l a b l e and we c r o s s them d e f i n i n g a v e r t e x (3.2) D C ' U t , C K , ' J * " r i ; e - K , J t B j j J ° J As an example, f o r the term ( 2 . 1 7 ) , u s i n g the s t a r t i n g p r o c e d u r e ( 3 . 1 ) , we draw ( 3 . 3 ) e - ^ D C T J ^ i ^ ^ e ^ r , , ^ * * 2 • 2 o  J ( b ) . i f t o the l e f t o f the e x p r e s s i o n j dtM e "Iij e o n l y one of the l a b e l s i o r j (say i ) has appeared, we draw a branch on the a v a i l a b l e l i n e ( i ) and d e f i n e the v e r t e x t o i n c l u d e the i n t e g r a t i o n o v er the new l a b e l ( 3 . 4 ) 1 — C Z - A f ^ i ^ l X r ^ * - ! , . * - ^ ' -The symbol A w i l l be e x p l a i n e d s h o r t l y . F o r example the second b u i l d i n g b l o c k of (2.17) i s of t h i s t y p e and. we draw (3.5) e'K *pC== ] < * S ) V - ^ ^ J-y" *-e ^ 12 ( c ) . i f to the l e f t of the expression cli^e } I i; C *" neither i nor j has appeared, we introduce two new lines running to the right through a vertex defined by (3.6) <± - \ % * * t \ ! ± U v i j t / ' i ^ j e The example (2.17) can be represented now by (3.7) e" K 3 C _ l F f i m (o) It is noted that time labels are not needed i f the vertices are ordered from l e f t to right to indicate the respective limits of the time integrations t ^ * ( ^ tx >/t} • - - ^  0 The expansion, in terms of diagrams simplifies greatly because diagrams containing a type (c) vertex integrate to zero or one writes (3.8) < H = O j We have relegated this calculation to our Appendix I . . Also in Appendix I we find that for a diagram with a (b) type vertex, where a particle label j f i r s t appears, the J^. J-Vj part of the I t-j in the vertex also integrates to zero, or one writes 13. . We w i l l now explain the symbol A. Working from an example, l e t us consider a l l the contributions to F that have the form There are (N-2)(N-3)(N-4) diagrams of thi s form corresponding to the d i f f e r e n t ways of choosing the dummy labels }> ]'> j" and we see that these diagrams are indistinguishable. In general, this type of redundance occurs wherever (b) type verti c e s appear. And, one sees that diagrams that d i f f e r only.in t h e i r dummy p a r t i c l e labels can be summed by writing in the (b) type vertices ' . (3.11) A=N-r r= number of l i n e s to the l e f t of the vertex (b). This concludes the d e f i n i t i o n s of our. basic v e r t i c e s which we summarize here, [ I -U-i (3.12) (a) ZXZ = J i * ^ * " Ir e-K'J*" j j }° J (3.13) where r i s the number of lines at the l e f t of the vertex (b). The BBKGY H i e r a r c h y of Equations. 14. The aim of t h i s work i s to d i s c u s s the t h e o r i e s of 1 4 5 6 Bogoliubov , Sand r i & Frieman ' ' and Mazur & B i e l . These authors use or d e r i v e the BBKGY chain of equations as a f i r s t step. I t seems a p p r o p r i a t e to d e r i v e t h i s chain of equations at t h i s p o i n t . I t a l s o s u p p l i e s us a good example of a c a l c u l a t i o n using diagrams and diagram language. For tr 5 a given r we separate out the f i r s t v e r t e x only of the c o n t r i b u t i n g diagrams. T h i s g i v e s , being c a r e f u l to r e t a i n the l e f t m o s t e f o r each term of the p e r t u r b a t i o n s e r i e s (2.14), (3.14) F s = e - ^ F < ( o U e - ^ H ^ " ^ H i<jiS i l l . i i S 5 :" 5 Here, the p o i n t s i n d i c a t e the l i n e s make a l l p o s s i b l e v e r t e x . connections to the r i g h t i n c l u d i n g p o s s i b l e i n i t i a l v a l u e s . We re c o g n i z e t h a t i n the second term, except f o r the operator <°" 1 , a l l p o s s i b l e connections d e f i n e s F (*,) and i n the t h i r d term, except f o r the o p e r a t o r e ' they d e f i n e T h e r e f o r e , i f the f i r s t v e r t e x i s w r i t t e n i n d e t a i l one has (3.15) ...(continued) IS. ; -(3.15) F s - e~KH F*(o) + e-*5± JZ (d*. c K l J * ' I , . e ^ ' i <<[e F s(r,)l n • i • Kr;"t|7- . ^ ' ' I T a~Kst, i K; S + I ^ I T -.-ft sti^i-By w r i t i n g J I;;? = e e a net e 1->S+I lt- s t | <? -X£J^^FI f'and using the r e l a t i o n (2.9) we obtain (3.16) F ^ e ^ F ^ . e - ^ J A e ^ ' l 'JZ I £ JF'(f.j This i s an i n t e g r a l form of the BBKGY chain ( 3 . 1 7 ) - ^ W F ' - Z I r £ I F S * ( ^ ^ I ^ F ^ ' Often a short e r n o t a t i o n w i l l be used i n f o l l o w i n g chapters (3.18) + Vs Fs~" = LSF5-*' H5'- Ks-I5 where (3.19) I 5 = H IL: (3.20) L 5 = H L- L. - ^ L [ d l s „ ^ , ^ - & . Return now to the general p e r t u r b a t i o n expansion. We f i n d that one key to making the expansion u s e f u l i s to introduce the c l u s t e r expansion"^ (3.21) F'= F'i*,^,) (3.22) F ^ f f ? ^ ) ^ ^ ) ^ ^ ? ! / ^ ! / * ) I 6 . (3.23) F3= F/(<X,,4r.)F^ of>,'iy;)F,7?jJ(f,)+ F / t e . ^ G ^ f * , , ^ , l i ^ i ) + ) G* y,, +• F/fe,#)G* f4T, f %t, F 4 - •• • -5 The b are i n t e r p r e t e d as depending on i n t e r p a r t i c l e d i s t a n c e s and are c a l l e d c o r r e l a t i o n s . The c l u s t e r e x p a n s i o n i s used here to expand the v a r i o u s F (OJthat appear of the r i g h t of each term i n the g e n e r a l e x p a n s i o n of . FS(t). C o n s i d e r f o r example the c o n t r i b u t i o n t o F' -K'i i ' (3.24) e < , i F 3(0) • 2 I 3 A c l u s t e r e x p a n s i o n of FVo)is made and the i n i t i a l c o r r e l -a t i o n s are r e p r e s e n t e d by d o t t e d l i n e s a t the r i g h t of the diagrams. D e f i n i n g (3.25) F'(O) ~- 1 f o r the i n i t i a l v a l u e of the one body f u n c t i o n the example (3.24) may be w r i t t e n f o l l o w i n g the d e c o m p o s i t i o n (3.23) 17. Our method o f r e p r e s e n t i n g the g e n e r a l p e r t u r b a t i o n s e r i e s i n terms o f diagrams i s now complete. In a l l the f o l l o w i n g s e c t i o n s of t h i s work the so-c a l l e d thermodynamic l i m i t w i l l be t a k e n (3.27) N — > ° = j V — > Q ° such t h a t ^ - n remains c o n s t a n t . T h i s has t h e e f f e c t t h a t i n each (b) type v e r t e x (3.28) $ = ^  — « The thermodynamic l i m i t s h o u l d not a f f e c t the bu l k p r o p e r t i e s of the system which a l l o w s us to assume convergence under t h i s l i m i t . IV. THE DEPENDENCE OF THE DISTRIBUTION FUNCTIONS UPON THE ONE BODY FUNCTION. I n " t h e s u c c e s s f u l t h e o r i e s of I r r e v e r s i b l e s t a t i s t i c a l mechanics the dependence of the d i s t r i b u t i o n f u n c t i o n s F, F , F upon the one b o d y , d i s t r i b u t i o n f u n c t i o n F' has p l a y e d a dominant r o l e . In t h i s c h a p t e r we work out the dependence of the d i s t r i b u t i o n f u n c t i o n s F s upon F' i n . d e t a i l . The o n l y . d e p a r t u r e from a g e n e r a l t h e o r y w i l l be t h a t the thermodynamic l i m i t (3.27) w i l l be assumed v a l i d and t a k e n . We f i n d the dependence on F 1 by p e r f o r m i n g a p a r t i c u l a r sum of diagrams and i n the l a t e r p a r t of t h i s c h a p t e r we p o i n t out t h a t t h i s summation i s n e c e s s a r y t o de t e r m i n e the l o n g time b e h a v i o r of a p h y s i c a l system. That i s , the l o n g time b e h a v i o r of a p h y s i c a l system i s . d e t e r m i n e d t h r o u g h the one body f u n c t i o n . C o n s i d e r the e x p a n s i o n of the two body f u n c t i o n ^ F2= e - ^ i ; + e - K * Z D + c - ^ X l + f ^ X 3 (4.1) <a> (fe) (c) (<{) A  3.  3 N a) -f»; 1 2 if) ' .1 9-For convenience we define a diagram —O f o r the one body f u n c t i o n ( 4 . 2 ) / + F ' = - ^ = H ^ - C ; + - o + - o c ; • 3, 3 4 - o c + - c - X + - c H 3 —^; + -< 27 -f — C ~ T " : + -C i I i + HI; + ~TI; "TE 4 CD + U i + • • -3 3, I t w i l l be proved p r e s e n t l y that the two body f u n c t i o n may be w r i t t e n ( 4 . 3 ) •F2 =e- K * -K2t 2 0 . where i t i s u n d e r s t o o d t h a t i f a — O connects to a v e r t e x i t i s a f u n c t i o n of time t h r o u g h the i n t e g r a t e d time v a r i a b l e of t h a t v e r t e x . In (4.1) the terms ( a , e , f , j , m , . . . ) have gone t o c o n t r i b u t e t o the term (a') above and the term (o) -has gone t o c o n t r i b u t e to ( c ' ) above. I n the n e x t p a r a g r a p h we g i v e a s y s t e m a t i c p r o c e d u r e f o r w r i t i n g out the e x p a n s i o n ( 4 . 3 ) , and f o r the s i m i l a r e x p a n s i o n s of h i g h e r o r d e r d i s t r i b u t i o n f u n c t i o n s . The s y s t e m a t i c p r o c e d u r e i s as follows. The diagrams are drawn as b e f o r e r e p l a c i n g — ( b y — o where the s e r i e s e x p a n s i o n f o r — o i s t a k e n i n t o account by o m i t t i n g diagrams w i t h f r a g m e n t s t h a t connect i n by o n l y one l i n e . For example the p a t t e r n -^(4.4) e-* l f - y ~ ~ X f > , 1 \ , i s not k e p t because we f i n d i t c o n t r i b u t e s t o (4.5) i T h i s method of e x p a n s i o n can be p r o v e d v a l i d by use of the " f a c t o r i z a t i o n " theorem. F o r a g e n e r a l p r o o f see R e s i b o i s . The f a c t o r i z a t i o n theorem d e a l s w i t h diagrams t h a t are i d e n t i c a l e x c e p t f o r the l e f t t o r i g h t o r " t i m e " o r d e r i n g of t h e i r v e r t i c e s . The s e t of a l l such diagrams i s c a l l e d a p e r m u t a t i o n c l a s s . For example (4.6) ~ C j J and fi / i s a p e r m u t a t i o n c l a s s . The f a c t o r i z a t i o n theorem r e a d s : (4.7) The sum of a complete p e r m u t a t i o n c l a s s i s e q u a l to the p r o d u c t of the c o n t r i b u t i o n s r e p r e s e n t e d by the component s t r u c t u r e s . F o r our example (4.6) we w r i t e (4.8) I ~ < ~ I ^ C i L-CU J J * J J - / I J z z x z z x n z z x z z »ti f> m Oi M m where the v e r t i c a l bar i n d i c a t e s the v e r t i c e s have the same l i m i t s on t h e i r time i n t e g r a t i o n s . I n d e t a i l (4.8) i s w r i t t e n and s i n c e o p e r a t o r s w i t h l a b e l s i and j commute w i t h those w i t h l a b e l s 1 and m, the component s t r u c t u r e s f a c t o r . To get (4.3) from (4.1) we c o n s i d e r a l l diagrams of (4.1) t h a t have fragments t h a t connect i n by one l i n e , f o r example (4.4) We t a k e the p e r m u t a t i o n c l a s s formed by t h e s e fragments w i t h the o t h e r s t r u c t u r e s i n the diagram, f o r the example (4.4) we c o n s i d e r 11. When we sum o v e r the p e r m u t a t i o n c l a s s e s the fragments f a c t o r , f o r the example (4.10) the sum of the diagrams g i v e ( 4 . 1 1 ) i r — , where t h e . a r r o w s i n d i c a t e the v e r t i c e s are f u n c t i o n s of the same time v a r i a b l e . The i n f i n i t y of diagrams i d e n t i c a l t o (4.4) except f o r d i f f e r e n t fragments t h a t connect to the v e r t e x (a) on the branch i g i v e s , by summing over t h e ' p e r m u t a t i o n c l a s s e s , a l l p o s s i b l e ways of drawing a t (a) diagrams t h a t b e g i n w i t h one l i n e . The sum of t h e s e d e f i n e s •—o and the e x p a n s i o n s.cheme o u t l i n e d above i s j u s t i f i e d . W i t h an a l t e r a t i o n we can use t h i s e x p a n s i o n scheme to o b t a i n , e x c e p t f o r i n i t i a l c o r r e l a t i o n s , a c l o s e d e q u a t i o n f o r the one body d i s t r i b u t i o n f u n c t i o n . I t i s i n s t r u c t i v e to r e c a l l the c o u p l i n g o f the one body f u n c t i o n to the two body f u n c t i o n t h r o u g h the BBKGY c h a i n (3.18) (4.12) j f+K'F'zil'F 2-U s i n g (3-. 14 o r 3.15) we f i n d t h a t the s o l u t i o n to (4.12) can be w r i t t e n as f o l l o w s (4.13) F'- e*'*—, 4- e-^ J - C Z e F 2 ( t j e 2 Except f o r i n i t i a l c o r r e l a t i o n s a c l o s e d e q u a t i o n f o r F' i s o b t a i n e d by s u b s t i t u t i n g the e x p a n s i o n (4.3) f o r F i n t o ( 4 . 1 3 ) , one has 2 3. (4.14) F ' ^ e ^ n +e-K,i 4e-' f i t -<j + e * - O C I t i s r e c o g n i z e d t h a t the expansion here i n terms of the one body f u n c t i o n f o l l o w s the same scheme as f o r the many p a r t i c l e f u n c t i o n s except we allow the l e f t m o s t (b) type v e r t e x , the one i n equation (4.13) to be r e t a i n e d . An e x p r e s s i o n e q u i v a l e n t to (4.14) has f o r m e r l y been obtained by Severne. H i s approach to the problem i s very s i m i l a r to ours. A s l i g h t d i f f e r e n c e i s found i n h i s summation of terms to o b t a i n the dependence of the d i s t r i b u t i o n f u n c t i o n upon the one body f u n c t i o n . The answers d i f f e r only i n the term with i n i t i a l c o r r e l a t i o n s . Where we have a diagram that terminates on the r i g h t with some product of i n i t i a l c o r r e l a t i o n with one or more — o , Severne has an i n f i n i t e s e r i e s of terms where t h e — O ' s are. expanded as i n (4.2). 24. In the f o r e g o i n g p a r a g r a p h we r i d o u r s e l v e s of diagrams w i t h f r a g m e n t s t h a t connect i n by one l i n e , by summing them i n t o the v a r i o u s — O . Through some a p p r o x i m a t i o n s we w i l l " i n d i c a t e " t h a t t h i s summation of diagrams i s n e c e s s a r y t o d e t e r m i n e the l o n g time b e h a v i o r of the system. We w i l l n e g l e c t terms w i t h i n i t i a l c o r r e l a t i o n s , assume the system f o r g e t s i t s p a s t h i s t o r y and we w i l l t a k e the a s y m p t o t i c b e h a v i o r . From ( 4 . 1 4 ) , by n e g l e c t i n g i n i t i a l c o r r e l a t i o n s , we have a c l o s e d e q u a t i o n f o r — O ; and here we c o n s i d e r a c o u p l e o f r e p r e s e n t a t i v e terms (4.15) —O = -H + ...+ — O C ! + 25. By t a k i n g t h e time d e r i v a t i v e , one f i n d s (4.16) y r - - n f ^ ^ e ^ I a e - ^ t { - + + E q u a t i o n (4.16) i s s a i d t o be non-Markowian because the e v o l u t i o n of the one body d i s t r i b u t i o n a t t i m e t depends on i t s v a l u e s f o r a l l t i m e s T < t . For our purposes here we wi assume the system " f o r g e t s i t s p a s t h i s t o r y " by making a crude Markowian a p p r o x i m a t i o n (4.17) o=eKtt*?i{Q e ^ ' M = ~ « so t h a t (4.16) i s approx i m a t e d by (4.18) ^ ^ n j ^ ^ e ^ l ^ e ^ ^ f ... + X . : v ; x i . . . . ' ; As a f u r t h e r a p p r o x i m a t i o n we t a k e the a s y m p t o t i c b e h a v i o r (4.19) jf7"l^it^{--* © where (4.20) X Jo »6 i l J e (4.21) i < 4 -_ w p + M p q ^ J e K i Jf M J : - e - , r : J f ' ' (4.22) e W For c o n v e n i e n c e we a l s o d e f i n e (4.23) so t h a t we can w r i t e (4.19) as f o l l o w s (4-24) if-.- + -cDc:*-<62S+-F i n a l l y we t a k e the T a y l o r e x p a n s i o n of —& (4.25) , - £ { . . . + _ < x ; + . - < ^ 2 ^ 4 . . (4.25) has r e p l a c e d the .expansion of (4.15) which i s by-i t e r a t i o n (4.26) _ 0 r _ 1 + | . : . + ^ c x ; + - < ^ ^ + . . . ] where i n the terms (x) and (y) the v e r t i c a l l i n e i n d i c a t e s a complete p e r m u t a t i o n c l a s s . By comparing (4.25) w i t h (4.26) i t i s found t h a t each fragment of (4.26) t h a t connects i n by one l i n e g i v e s a time d i v e r g e n c e o r o r d e r t . (We s h o u l d r e -c a l l t h a t terms w i t h i n i t i a l c o r r e l a t i o n s have been n e g l e c t e d and the same may not be t r u e f o r those terms.) In o r d e r t o d i s c u s s the l o n g time b e h a v i o r we must sum over the d i v e r g e n c e s t h a t we ob s e r v e d i n the former p a r a g r a p h . And we see t h a t t h i s i s done by s e p a r a t i n g out 28. and summing, where i t a p p e a r s , t h e i n f i n i t e s e r i e s — o -e F T h i s i l l u s t r a t e s , t h a t t h e l o n g time b e h a v i o r i s : d e t e r m i n e d by t h e one p a r t i c l e d i s t r i b u t i o n f u n c t i o n . I t i s not known now i f an a r b i t r a r y s e l e c t i o n , say ( 4 . 1 5 ) , o f terms converges i n t i m e . T h i s has o n l y been v e r i f i e d f o r s p e c i a l c a s e s t h r o u g h an H t y p e theorem. We w i l l d i s c u s s t h e Markowian a p p r o x i m a t i o n i n more d e t a i l l a t e r . T h i s b r i n g s us t o our n e x t t o p i c which i s an o r d e r o f magnitude c a l c u l a t i o n . We r o u g h l y a p p r o x i m a t e t h e s t r e n g t h and range o f t h e i n t e r p a r t i c l e p o t e n t i a l . Each v e r t e x c o n t a i n s t h e i n t e r p a r t i c l e p o t e n t i a l and t h u s g i v e s a measure £ o f t h e s t r e n g t h o f t h e p o t e n t i a l . Each (b) t y p e v e r t e x c o n t a i n s an i n t e g r a t i o n o v er t h e . i n t e r p a r t i c l e p o t e n t i a l , w h i c h i s l i m i t e d by range o f t h e range o f t h e p o t e n t i a l , and a l s o s i n c e t h e d e n s i t y n i s a f a c t o r e a c h i ( b ) t y p e v e r t e x g i v e s a measure A o f t h e number o f p a r t i c l e s w i t h i n t h e i n t e r a c t i o n s p h e r e . F o r t h e v e r t i c e s we have th e o r d e r s o f magnitude (4 .27) ZXZ. ~ C ( £ ) - < Z ~0(£X) F o r t h e r e m a i n d e r o f t h i s t h e s i s we villi be w o r k i n g w i t h t h e e x p a n s i o n i n s m a l l A which we c a l l the s h o r t - r a n g e t h e o r y . V. THE SHORT-RANGE EXPANSION In t h i s c h a p t e r we s p e c i a l i z e t o c o n s i d e r those systems f o r w h i c h o n l y s m a l l c l u s t e r s of p a r t i c l e s i n t e r a c t s i m u l t a n e o u s l y . Such a s i t u a t i o n o c c u r s f o r d i l u t e systems o r f o r systems i n which the p a r t i c l e s i n t e r a c t t h r ough s h o r t -range f o r c e s . From the o r d e r of magnitude c a l c u l a t i o n (4.27) we w i l l be w o r k i n g w i t h the e x p a n s i o n i n s m a l l A The t a s k of t h i s c h a p t e r i s t o s e l e c t diagrams f o r the ex p a n s i o n i n s m a l l A, and to o b t a i n r e a s o n a b l y compact answers we w i l l sum the v a r i o u s i n f i n i t e s e r i e s t h a t appear. We demonstrate the c a l u c l a t i o n s f o r the e x p a n s i o n of F ; and we c a r r y out the c a l c u l a t i o n s t o i n c l u d e a l l two and t h r e e p a r t -i c l e c o l l i s i o n , o r c o r r e l a t i o n , e f f e c t s . . The H i e r a r c h y of E q u a t i o n s f o r the C l u s t e r E x p a n s i o n We f i n d t h a t a v a l u a b l e guide f o r s e l e c t i n g and summing diagrams i s the h i e r a r c h y of e q u a t i o n s , s i m i l a r t o the BBKGY h i e r a r c h y , t h a t c o u p l e the c l u s t e r c o e f f i c i e n t s (3.21 to 3.23). Our purpose i n t h i s p a r a g r a p h i s t o d e r i v e t h i s h i e r a r c h y of e q u a t i o n s . We w i l l i d e n t i f y by diagrams the v a r i o u s c l u s t e r c o e f f i c i e n t s G (5.1) A « I (5.2) -K st 2 3 0 . where the v e r t i c a l bar indicates the sum of a l l diagrams that have the s li n e s connected in some way.. We see that t h i s i s a possible way to write the G$, for example for F3 we write (5.3) F'-e-"'*^**'"3*^ *V* 3*T-° + e - ^ i ^ + e " ^ T The f a c t o r i z a t i o n theorem (4.7) has been used extensively here.. Further, (5.2) i s the only selection of diagrams for G $ because the system of equations (3.21 to 3.23) yiel d s a unique solution for the G 5 in terms'of the FS . A word of caution here; the expression (3.11) w i l l not allow the i d e n t i f i c a t i o n (5.2) unless the thermodynamic l i m i t (3.27) has been taken. It i s convenient here to point out an important property of the b . We term as factorable any function J of s p a r t i c l e coordinates that can be written in a form (5.4) J^*,^...,*,^) = X ^ . & . t i JTr('*i>,yi<, zyj^y,..) We f i n d that the diagrams that contribute to G sare not in general factorable in thi s form; since the s starting l i n e s , being connected in some way, always have an overlap of the i n t e r p a r t i c l e potentials and i n i t i a l c o r relations. We proceed to derive the hierarchy of equations for t h e G ; we do th i s by example for the case of G? . We separate out in the expansion for C? the f i r s t vertex only of the contrib-uting diagrams to obtain Where a p r o d u c t of diagrams means the diagrams formed by c o n n e c t i n g the l i n e s as i n d i c a t e d by t h e i r l a b e l s . A f t e r s l i g h t , rearrangement we s u b s t i t u t e i n (5.5) the d e f i n i t i o n s of the diagrams t o o b t a i n (5.6) G ^ e - ^ ^ f e - ^ f ^ e ^ ' I 3 ^ ^ , ) t A W f £ W C (ij + F4'(VJ M + G»ft) ft)+(0 ^ ft)] 4. + + A L + A I3 G«(*,)} 32-Or by taking the time derivative one has (5.7) $W-I*G'-- InfcGh + FiGl] t A LH [ F3 G^y- F 4'£^ 4 G,j C"44 C ^ ] f A fc4 And t h i s may be written in short hand, as (5.8) V F ^ X i T ^ where I 3 F J i s the produce of I 3 * a n d the cluster expansion of F from which we retain only the non-factorable, in the sense (5.4), terms excepting X G which i s written separately on the l e f t hand side of the equation. S i m i l a r l y L? F * Is the product of L3- YL ^-i/i and the cluster expansion of F4" i( $ retaining only the non-factorable terms. Our derivation of (5.8) reveals the generalization (5.9) ^ f - S 4 K s G 5 - r s G s = r F ' + A p F * ' Though these equations are more complicated in the way they are written than the BBKGY hierarchy (3.18), (5.10) *n* K S F S - I 5 F 5 = A L 5 F UJ- 1 i / S R s T S r J > 1 5 r - s+ 1 3 3 . t h e e q u a t i o n s ( 5 . 9 ) a r e a c t u a l l y much s i m p l e r b e c a u s e we s ee t h a t t h e y may-be o b t a i n e d f r o m t h e BBKGY h i e r a r c h y by c a n c e l l i n g t h e r e d u n d a n t f a c t o r a b l e b e h a v i o r . The e q u a t i o n s ( 5 . 9 ) h a v e f o r m e r l y b e e n o b t a i n e d by Green." '"" ' ' F o r ' c o n v e n i e n t r e f e r e n c e we t a b u l a t e h e r e t h e f i r s t t h r e e o f t h e s e e q u a t i o n s , ( 5 . 1 2 ) ^ K V - I ' G S I ' F / F i f A L . j F ' G i + F i&a ) ( 5 . 1 3 ) ^ K ^ - r G ^ I . j F / G ^ f F ' G J ] L e t us b e g i n o u r s e l e c t i o n and summing o f d i a g r a m s • f o r t h e s h o r t - r a n g e t h e o r y . We w i l l d e m o n s t r a t e t h e c a l -c u l a t i o n s f o r t h e e x p a n s i o n o f F t o i n c l u d e a l l two and t h r e e p a r t i c l e c o l l i s i o n , o r c o r r e l a t i o n , e f f e c t s . By ( 5 . 1 1 ) _ , ... one h a s ( 5 . 1 4 ) F ^ F ( ' F a f G 2 O u r p r o b l e m h a s r e d u c e d t o e x p a n d i n g G ; ( 5 . 1 2 ) t h e s o l u t i o n we h a v e f r o m 3 4 . ( 5 . i o ) G2-~ e-^*GVo) + e-^*j[Jftc f^/"i2GYt) + 1 2 4AL ) 3 [F;WG 2V0 4 F J('iJG^)] Or i n terms of diagrams one has (5.16) c - ^ Q : e ^ : i + ^ D a + e ^ X ^ ± 1 ( 7 ] ' 3 / T h e - e x p r e s s i o n e"(< r l X ] i s an i n f i n i t e s e r i e s of terms; as a f i r s t s t e p i n f i n d i n g an ex p a n s i o n i n o r d e r s of A we are i n t e r e s t e d i n s e p a r a t i n g out the z e r o t h o r d e r c o n t r i b u t i o n s . To do t h i s we i t e r a t e (5.16) where the ZU appears, i n z e r o t h o r d e r , and we s u b s t i t u t e t h i s answer i n t o (5.14) t o o b t a i n an e x p r e s s i o n f o r F 2 ( 5 > 1 7 ) F 2 = e - ^ Z £ 4 e - ^ { / + D C + X X l + D O O C - f . . } 35". We r e c o g n i z e t h a t t h i s e x p a n s i o n of r f o l l o w s t he r e c i p e of C h a p t e r IV f o r expanding a g e n e r a l F$ In terms of the — o . In'( 5 . 1 7 ) one sees t h a t the z e r o t h o r d e r terms g i v e a l l the p o s s i b l e b e h a v i o r where "two p a r t i c l e s f e e l the i n f l u e n c e of each o t h e r o n l y . To o b t a i n the e f f e c t of a t h i r d p a r t i c l e upon t h e s e p a r t i c l e s we w i l l need to s e p a r a t e out a l l terms o f f i r s t o r d e r i n X[. . To t h i s end we d e f i n e diagrams of the type (5.18) W\^CKi -o -a -0 where the v e r t i c a l l i n e i n d i c a t e s a sum of a l l diagrams each h a v i n g a l l t he l i n e s c o nnected i n some way and t e r m i n a t i n g on the r i g h t as i n d i c a t e d . T h i s w i l l always mean t h a t t h e r e are no (b) type v e r t i c e s i n the c o n t r i b u t i n g diagrams; f o r example we w r i t e (5.19) . M^e-KHZC--e-^DC + e-ft (5.20) r ^ E . - ^ ^ e - ^ X i + e ^ X X l 4 e ^ I X X X j + • • • N o t i c e i n (5.20) t h a t the diagram w i t h i t s l i n e s connected by i n i t i a l c o r r e l a t i o n s a l o n e i s i n c l u d e d i n the sum. With the s e d e f i n i t i o n s we can w r i t e F 2 to f i r s t o r d e r i n X as f o l l o w s 36. (5.21) Fz = e^2iZ°Q + A ~d(iC'i£ 4iE + tr;) Our task now i s to sum the i n f i n i t e s e r i e s that appear i n (5.21). The hie r a r c h y of equations ( 5 . 9 ) makes t h i s task an easy one. In p a r t i c u l a r we w i l l use the equations ( 5 . 9 ) i n the i n t e g r a l form ( 5 . 2 2 ) G ^ - ^ V f o ^ e " ^ e ^ f I ^ ^ O ^ A ^ " Y ' . ) } . H5 = K5-IS By s u b s t i t u t i n g t h i s equation f o r s=2, (5.23) G*-.e^G2{o) + eH2l(^ eHl''I2F,WW 4 A e" fl ,3 f F,' ft) Gl, (i.) f Fi (t.) i n t o (5.14); we sum the i n f i n i t e s e r i e s s t r u c t u r e of the equation (5.17), or one has 37 (5.24) F 2--e-^ 4A<? where a s u b s c r i p t on a p a r e n t h e s i s means the i n c l u d e d f u n c t i o n s , o r o p e r a t i o n s y i e l d s a f u n c t i o n of t h a t v a r i a b l e . The e x p r e s s i o n (5.25) g i v e s us the sum of the z e r o t h o r d e r terms, o r by comparison w i t h (5.17) one has w i t h the d e f i n i t i o n s (5.19 and 5.20) And, by r e t a i n i n g from (5.24) o n l y the f i r s t o r d e r terms we have i n p l a c e of (5.21) the e x p r e s s i o n (5.25) M2 r e-W ZJ°0 e-H2*/Jl, ?" H< V F, (5.26) e ^ I O - e - ^ Z l 3 8 . •(5.27) F 2 = e - ^ Z t 4 e - f 2 i •tft , i 1 3 ' 3 4 L » < " f a ' f c ' i n ' i i l 3 ' 1 + i f r f The r e m a i n i n g unsummed s t r u c t u r e s i n (5.27) are the z e r o t h o r d e r c o n t r i b u t i o n s t o C? . To sum t h e s e we use the e x p r e s s i o n (5.22) f o r G 3 and r e t a i n o n l y the z e r o t h o r d e r terms; and, a f t e r some f u r t h e r i t e r a t i o n w i t h the e x p r e s s i o n (5.22) f o r G> , we o b t a i n the f o l l o w i n g sums of diagrams. I j-O -| —\-0 I i m, Ho/*, h° 4 T ° \ ^—6 \ i, (5.29) ?-i<n 2_ - e TJj/f ( cyclic •3 a (5.30) e~Hii T h i s c o n c l u d e s our d i s c u s s i o n f o r the s e l e c t i o n and summing of diagrams f o r the s h o r t - r a n g e t h e o r y . The meaning of our e q u a t i o n s w i l l become more c l e a r i n the f o l l o w i n g c h a p t e r . V I . THE MARKOWIAN APPROXIMATION To- examine i f the f u n c t i o n s of the former c h a p t e r s d e s c r i b e an approach t o e q u i l i b r i u m we must c o n s i d e r t h e i r a s y m p t o t i c b e h a v i o r . Any a s y m p t o t i c c o n s i d e r a t i o n i s c o m p l i c a t e d because i n C h a p t e r IV we showed t h a t the l o n g time b e h a v i o r was d e t e r m i n e d t h r o u g h the one body d i s t r i b u t i o n f u n c t i o n and t h i s f u n c t i o n i s i t s e l f v e r y c o m p l i c a t e d . In t h i s c h a p t e r , we w i l l d i s c u s s the c o m p l i c a t i o n due to the f a c t t h a t the d i s t r i b u t i o n f u n c t i o n s , i n c l u d i n g the one body f u n c t i o n , at a time ~t depends on the one body f u n c t i o n t h r o u g h v a r i o u s t i m e s ±m tZ-tm >/ 0 of the p a s t . In the l i t e r a t u r e t h i s d i f f i c u l t y has to some e x t e n t been overcome by s e e k i n g approx-i m a t i o n s t o t h e two p a r t i c l e and h i g h e r o r d e r d i s t r i b u t i o n f u n c t i o n s w h i c h depend on time o n l y t h r o u g h the one p a r t i c l e f u n c t i o n . (6.1) F 5 * F S f o ^ . . . ; ^ r . j m ) and w i t h t h e s e f u n c t i o n s the e v o l u t i o n of the. one body f u n c t i o n i s d e t e r m i n e d , t h r o u g h the BBKGY c h a i n , to be approximated by \ pi (6.2) — +ff'F'^ \LFi(i,lv,i<3ttj*r3.F'(t)) Theseapproximations, i f they e x i s t , d e f i n e a Markowian p r o c e s s because t h e b e h a v i o r a t a time t i s c o m p l e t e l y d e t e r m i n e d by F'and ^f . a t the time t ; t h a t i s , the system " f o r g e t s i t s p a s t h i s t o r y " . A method for f i n d i n g w e l l behaved Markowian approx-i m a t i o n s has not to our knowledge been s a t i s f a c t o r i l y d e t e r m i n e d . I n t h i s c h a p t e r we produce a Markowian a p p r o x i m a t i o n p r o c e d u r e t h a t we f i n d r e l a t e s most e a s i l y to those found i n the 41. l i t e r a t u r e . At.-the end of the c h a p t e r we p o i n t out some a s y m p t o t i c d i v e r g e n c e d i f f i c u l t i e s of t h i s method. The Markowian a p p r o x i m a t i o n p r o c e d u r e we use to r e l a t e our t h e o r y t o those of the l i t e r a t u r e i n v o l v e s two s t e p s . F i r s t we i l l u s t r a t e a way f o r w r i t i n g w i t h o u t making any a p p r o x i m a t i o n s , the d i s t r i b u t i o n f u n c t i o n s i n a form ( f o r 5>| ) (6.3) F s r f*(*i)Vlr..i*t<ri.t \F'(i)) And second we w i l l assume the e x p l i c i t , i e : o t h e r than t h r o u g h F'(i), time b e h a v i o r t o be s h o r t l i v e d so t h a t we can o b t a i n Markowian e q u a t i o n s by t a k i n g the a s y m p t o t i c l i m i t . (6.4) F^^v .«LTi. llr, F 5 ^ , / , . . . ^ ^ r l F ' ^ ) ) J J } r-5><*> To c a s t the d i s t r i b u t i o n f u n c t i o n s i n t o the form (6.3) we c o n s i d e r more c a r e f u l l y the r e p l a c e m e n t (4.17) (6.5) . . . — o = e K ^ » F-(^) ^ cKli Fl (i) =•••--* One has from (4.14) and d e f i n i t i o n s (5.2), (5.18) t h a t the one body f u n c t i o n s a t i s f i e d . • H l r (6.6) _ ^ =-H + A - C j +A - O 1 ' - n a - d ( Z ^ - f ~J0o + T i ] f j 4 o(f) = -H + A l j i , e M > l a e-^ K f X + -Ti), + 0 (f) And; i t f o l l o w s t h a t f o r a j — -oV which depends on time t h r o u g h r/x> j ^ "^^Vi <2 we can w r i t e 42. (6 . - 0 (_o:., _ H F A\jtl^Lri.c,<*t<{Z?>tX<J3l * OW) By combining the e q u a t i o n s (6.6 & 6.7) we f i n d a c o n n e c t i o n between \ —o/ and — s (6.8) {-o} = ^ - A f ^ e ^ ' l . z e - ^ ' f Z j + Z : ^ ) * W ) Then by i t e r a t i o n we o b t a i n j — o r as an ex p a n s i o n i n terms of — £ , one has •(6-9) {H ( >r-*--Ai^ OM = - * - A \\ e * *>U e • O ) •* 0 (X2) W i t h t h i s r e l a t i o n we can w r i t e the g e n e r a l F oy the former c h a p t e r s i n the form ( 6 . 3 ) . L e t us work w i t h the example f o r r ; one has by s u b s t i t u t i o n o f (6.9) i n t o (5.27) the f o l l o w i n g e x p r e s s i o n f o r F2" '(6.10) . . . ( c o n t i n u e d ) 4 . 3 . (6.10) F^-e 2_ C-KH + e i 0 ii '- 3 . 2 2 ' 2-3 ~ .1 3 - ' . 3 2 . -i . 2 . 3 ^ Vi* ~—' %J •*•» — .... ll P / ' 1 2. ^ ' : i i i l l : 3 The terms w i t h a minus s i g n i n (6.10) are new and we w i l l c a l l t h e s e and s i m i l a r terms, t h a t a r i s e t h r o u g h t h e use of the e q u a t i o n ( 6 . 9 ) , Markowian c o r r e c t i o n terms. We are now i n a p o s i t i o n to c o n s i d e r the a s y m p t o t i c l i m i t ( 6 . 4 ) . We co n s i d e r f i r s t t he z e r o t h o r d e r c o n t r i b u t i o n s t o F ^ 2 r e ^ ^ ) ^ W ^ e - w 2 r G 2 ( o ) } + 0(\) And f o r the e v o l u t i o n of the one body f u n c t i o n we have from (6.11) F 2 ~ M» \ e (6.2) (6.12) i f + K ' F ' S r A ' / ^ i„ .-«VrF,'(i) iC^rGz(o)} t 0(f) 4 4 . Bogoliubov"'" has f o r m e r l y o b t a i n e d the f i r s t term on the r i g h t hand s i d e and he p r o v e s , f o r a homogeneous system, t h a t t h i s term i s the Boltzman c o l l i s i o n i n t e g r a l . The Boltzman c o l l i s i o n i n t e g r a l t a k e s i n t o account a l l the two p a r t i c l e c o l l i s i o n " e f f e c t s and I t i s hoped t h a t (6.10) w i t h the l i m i t (6.4) w i l l g e n e r a l i z e Boltzman's c o l l i s i o n i n t e g -r a l t o t a k e i n t o account t h r e e p a r t i c l e and h i g h e r o r d e r c o l l i s i o n e f f e c t s . . We f i n d , however, t h a t w i t h the l i m i t (6<,4) the Markowian c o r r e c t i o n terms d i v e r g e . L e t us c o n s i d e r f o r . e x a m p l e from (6.10) the c o r r e c t -Ion term To see the d i v e r g e n c e more e a s i l y l e t us s p e c i a l i z e and -Kst c o n s i d e r a homogeneous system where i n (6.13) the 6 o p e r a t i o n s have no e f f e c t (6.14) - A e - « 4 ^ From t h i s l e t us c o n s i d e r the ii i n t e g r a t i o n , the b e h a v i o r which came d i r e c t l y from (6.9) (6.1 5) f ^ z f a j K ^ . f a - t o * m m m - f / H The o p e r a t o r ? ,|3 d e s c r i b e s the e v o l u t i o n of any f u n c t i o n under the mutual i n f l u e n c e of the p a r t i c l e s one and t h r e e , the o t h e r p a r t i c l e s r e m a i n i n g s t a t i o n a r y . One sees t h a t the i n t e g r a l (6.15) v a n i s h e s except where the o p e r a t o r 4 5 . e 1 5 + 2 moves from the p r o b a b i l i t y d e n s i t i e s F,Yf) F ^ ) t^(i) t h e p a r t i c l e s one and t h r e e w i t h i n range of t h e i r i n t e r p a r t -i c l e p o t e n t i a l When i L i s l a r g e enough so t h a t t h e s e i n t e r a c t i o n s have t a k e n p l a c e , the t £ i n t e g r a n d i s a c o n s t -a n t ; and under the l i m i t (6.4) g i v e s r i s e t o the d i v e r g e n c e (6.16) - r Lue'^F^F^Fid) More g e n e r a l l y , i t has become apparent to us t h a t the e q u a t i o n (6.9) i s e n t i r e l y s i m i l a r t o the o r d i n a r y and d i v e r g e n t p e r t u r b a t i o n e x p a n s i o n (4.2) f o r • [ — 0 } . In (4.2) we have a p e r t u r b a t i o n expansion which, connects j — 0 } to i t s i n i t i a l v a l u e — I ; and t h i s e x p a n s i o n i s s i m i l a r t o the e x p a n s i o n (6.9) which c o n n e c t s j — 0 ] to i t s v a l v e @ of the f u t u r e . In Ch a p t e r IV we summed the s e r i e s — O i n o r d e r to g e t r i d of the a s y m p t o t i c d i v e r g e n c e s of the form ( 4 . 2 5 ) . In t h i s c h a p t e r we have e f f e c t i v e l y d e s t r o y e d , t h i s work because t h e use of the ex p a n s i o n (6.9) f o r 0} w i t h the l i m i t (6.4) r e i n t r o d u c e s d i v e r g e n c e s s i m i l a r to ( 4 . 2 5 ) , i n t he second and. h i g h e r o r d e r d i s t r i b u t i o n f u n c t i o n s , i n the e n t i r e l y s i m i l a r p o s i t i o n s . T h i s d i f f i c u l t y p r e v e n t s us from o b t a i n i n g p h y s i c a l l y i n t e r e s t i n g r e s u l t s . To a v o i d some c o n f u s i o n we s h o u l d d i s c u s s a c e r t a i n p o i n t about t h e way the Boltzman c o l l i s i o n term i n (6.12) was o b t a i n e d . The Boltzman e q u a t i o n s h o u l d be an approx-i m a t i o n t o t h e f i r s t o r d e r c o n t r i b u t i o n to F' ( n e g l e c t i n g terms w i t h i n i t i a l c o r r e l a t i o n s ) 4 6 . . (6.17) F Y O = e ^ - i n e K , f - C ( Z S ^ ) + ^ To o b t a i n t h e Boltzman e q u a t i o n we d i d "not" use the ex p a n s i o n (6.9) i n t h i s i n t e g r a l e q u a t i o n s t o w r i t e F' i n a f orm (6.18) F7*)« e - " ' * - f + A e " K ' * H Z ( Z ^ l C ) + 0 6 f ) w h i c h d i v e r g e s i n e x a c t l y the same way as (6.15) or (6.9) i f we t r i e d t o t a k e an a s y m p t o t i c l i m i t . R a t h e r , we worked from the d i f f e r e n t i a l e q u a t i o n s f o r the one body f u n c t i o n (6.19) ^ i f f ' - - » i ^ t t < I ) t o M and i n t h i s e q u a t i o n we used the e x p a n s i o n (6.9) to o b t a i n (6.20) g t K r ' X U e - M & ^ + OW) *U.,tt'Hlie"liF;(i)fi(i)iOtf) Or, by i n t e g r a t i o n we have r a t h e r than (6.18) the b e h a v i o r ( 6 . 2 1 ) p' r e-W^tXe-MJJi^ Lne-^e^ f/ft) ^ ' f t ) + flf^ where ^ = e™,/:'(*,) And t h i s e x p r e s s i o n i s known to converge because i t g i v e s the Boltzman e q u a t i o n a s y m p t o t i c a l l y . That i s , w o r k i n g w i t h the d i f f e r e n t i a l e q u a t i o n (6.19) r a t h e r than the i n t e g r a l e q u a t i o n (6.17) g i v e s t h i s one convergent term ( 6 . 2 1 ) . 4 5 S a n d r i & Frieman ' ' p o i n t out i n t h e i r t h e o r y t h a t 4 7 . t h e i r s o l u t i o n f o r the t h r e e p a r t i c l e c o l l i s i o n e f f e c t s d i v e r g e . In C h a p t e r V I I I we show the c o n n e c t i o n of t h e i r t h e o r y to o u r s . In C h a p t e r V I I we f i n d t h e s e same d i v e r g e n c e s i n B o g o l i u b o v ' s t h e o r y . A l s o we s h o u l d mention t h a t a s y m p t o t i c d i v e r g e n c e s in B o g o l i u b o v ' s of a d i f f e r e n t n a t u r e has been d i s c u s s e d by Cohen & Dorfman„ In (6.10) the terms which d e s c r i b e t h r e e p a r t i c l e c o l l i s i o n e f f e c t s c o n t a i n s an i n t e -g r a t i o n o v e r the phase space of the p a r t i c l e 3; the f o u r p a r t i c l e c o l l i s i o n terms would c o n t a i n an i n t e g r a t i o n over the phase spaces of the p a r t i c l e s 3 and 4; e t c . Cohen & Dorfman make e s t i m a t e s of the amount of phase space a v a i l a b l e f o r t h e s e i n t e g r a t i o n s . A c c o r d i n g to them the amount of phase space a v a i l a b l e f o r the f o u r p a r t i c l e and h i g h e r o r d e r c o l l i s i o n e f f e c t s d i v e r g e a s y m p t o t i c a l l y . V I I . THE THEORY OF BOGOLIUBOV S i n c e i t s i n t r o d u c t i o n i n (1946) the t h e o r y of Bogoliubov"'" has a t t r a c t e d c o n s i d e r a b l e i n t e r e s t i n the l i t e r a t u r e . We w i l l g i v e here a b r i e f resume of h i s t h e o r y , Then we d e v e l o p from the knowledge of the former c h a p t e r s an ex p a n s i o n method t h a t we f e e l as n e a r l y as p o s s i b l e f o l l o w s B o g o l i u b o v ' s e x p a n s i o n , and we s o l v e our e q u a t i o n s s i d e by s i d e w i t h t h o s e o f B o g o l i u b o v . The c o n n e c t i o n between h i s t h e o r y and the d i v e r g e n t Markowian a p p r o x i m a t i o n p r o c e d u r e of C h a p t e r V I , becomes a p p a r e n t . As a s t a r t i n g p o i n t B o g o l i u b o v d e r i v e s the BBKGY h i e r a r c h y (3.18) which f o r t h e s h o r t - r a n g e t h e o r y , w i t h the thermodynamic l i m i t ( 3 . 2 7 ) , we w r i t e ( 7 . 1 . ) ^VF'-XL'F*' Of p a r t i c u l a r i n t e r e s t i s t h e te m p o r a l change o f the one body d i s t r i b u t i o n f u n c t i o n ( 7 . 2 ) ^ W F + A L FZ L- V B o g o l i u b o v ' s f i r s t major assumption i s t h a t f o r a time s c a l e c o a r s e r than a c o l l i s i o n time the e q u a t i o n (7.2) can be approxim a t e d by (7.3) H-'Mfe.sr/M) 4 9 . where A depends f u n c t i o n a l l y o n F V ^ b u t does h o t depend on time e x p l i c i t l y . H i s second major assumption i s t h a t f o r the c o a r s e g r a i n e d time s c a l e a l l the F depend on time o n l y t h r o u g h F' , s o t h a t one can w r i t e (7.4) F s ( ^ . . . , 5s,*S;t)= F 5 ( ^ , . . . , * s , i s / F ' ) B o g o l i u b o v t r i e s t o o b t a i n (7.3) by s u c c e s i v e a p p r o x i m a t i o n s i n the form (7.5) j f = F'j + X A ' f l , ^ ) F')+ A M 2 ( * , , ^ F') + -~ and f o r t h e many p a r t i c l e f u n c t i o n (7.4)- he t r i e s a p e r t u r b -a t i o n e x p a n s i o n ( 7 . 6 ) F s = F s o f * 0 4 7 , . ^ <i5/F»J + A 1 f « ^ ^ , v ^ 4 5 / F ' J + A a F " ^ . , ^ - -B o g o l i u b o v uses the f o l l o w i n g n o t a t i o n f o r w r i t i n g the time d e r i v a t i v e of a f u n c t i o n ^ (<x^ o/7 ^ I F ' ) w h i c h depends on F t J ( 7 . 7 ) ± 4 f a o r l y . . ^ F ' j \ B ' i $ ' •' where the o p e r a t o r denotes d i f f e r e n t i a t i o n w i t h r e s p e c t t o "t ( ^ depends on ir through F' ) w i t h subsequent s u b s t i t u t -i o n of /f(F') f or i J l ! . He s u b s t i t u t e s the ex p a n s i o n s (7.5) and (7.6) w i t h the o p e r a t i o n s (7.7) i n t o the BBKGY h i e r a r c h y (7.1) t o o b t a i n by e q u a t i n g e q u a l o r d e r s i n A the system of e q u a t i o n s 5 0 . (7.8) A ^ ^ F ^ - K ' F ' (7.9) AxU,,¥hF')= L F20 (7.10) t\z(%¥,/)= LF: •2/ (7.11) &0FZo+\^F*°--0 (7.12) «® 6 F 4 ' + f f F 2 1 = L 2 F 3 o - & ' F t 0 ( 7 . 1 3 ) $ * F S o - r H 5 F s o =0 ( 7 14) ®°Fs£+HsF$i •= <ptl~- L S F S + ' ^ - ' - J ! The problem i s to f i n d the s o l u t i o n to t h i s system of equations. One n o t i c e s that the v a r i a b l e i does not appear e x p l i c i t l y i n these equations, and f o r t h i s reason Bogoliubov says that the problem has been reduced to the determination of the express-ions r and A as f u n c t i o n a l s of the " a r b i t r a r y " f u n c t i o n h This allows him to replace everywhere F by e F where f i s some parameter independent of t . One n o t i c e s the property (7.15) ^ - ^ F ' . - ^ ^ F ' ^ f e ^ f ^ ^ Hence from the d e f i n i t i o n of the o p e r a t o r ^ 0 i t f o l l o w s that (7.16) r f i ; ( / e ~ K V > j V F S l ' ( / < r K V ) S u b s t i t u t i n g these r e s u l t s i n t o the equations f o r the Fu( WKCF) one obtains (7.17) j ^ F s « Y / e ^ T j t HsFJ'"f /e-^F')-- ^ ' Y where ~o 5 1 . F o r the s o l u t i o n o f (7.17) one has (7.18) FSl'( \€^F)r:e-»5TF si( \fx)v F o l l o w i n g B o g o l i u b o v we r e p l a c e the a r b i t r a r y f u n c t i o n a l argument F by e F to o b t a i n ( w i t h a change I n the f ' v a r i a b l e of i n t e g r a t i o n ) (7.19) F''( lFJ:e-»s*Fsi( | e k' rr) S i n c e t h e s e e q u a t i o n s (7.19) h o l d f o r a r b i t r a r y T , and s i n c e the l e f t - h a n d s i d e does not depend on ?" the l i m i t t->x> can be t a k e n . F o r t h i s l i m i t B o g o l i u b o v uses the boundary c o n d i t -i o n s ( 7 . 2 0 ) j L ^ ^ F t ' C le«"tF')-- JL, e-^lTe^F' r-5.oo ?" — > CO ( 7 . 2 1 ) e - ^ S r F i : ( \e"'7F)--0 i 70 These c o n d i t i o n s a re v e r y f a r r e a c h i n g a s s u m p t i o n s , and we w i l l d i s c u s s them l a t e r . W i th th e s e boundary c o n d i t i o n s i n (7.19) one has (7.22) F s ° ( e - W ^ T T e ^ F ' (7.23) FS'( i f ) - Qf^'r'i / f r r ' F 9 By combining (7.22 & 7.23) w i t h the d e f i n i t i o n s of the b «. and ^ 5 ' V B o g o l i u b o v ' s e x p a n s i o n can be w r i t t e n out to any 52. o r d e r i n A . We s h o u l d mention t h a t , as B o g o l i u b o v p r o v e s , the s o l u t i o n f o r A1 f o r a homogeneous system g i v e s Boltzman's e q u a t i o n (7 .24 ) ^ ^  - L. C F, F2_ ^^^pJi^^L^^ ^^aytj^y^ . To e x p l a i n B o g o l i u b o v ' s t h e o r y we f i n d i t e a s i e r to go back and work from our more g e n e r a l t h e o r y of the former c h a p t e r s . We w i l l d e v e l o p an e x p a n s i o n method t h a t we f e e l as n e a r l y as p o s s i b l e resembles B o g o l i u b o v ' s e x p a n s i o n . T h i s e x p a n s i o n w i l l be n o t h i n g more than an a l t e r n a t e method f o r f i n d i n g o r w r i t i n g out the g e n e r a l F S i n the form (6.3) t h a t we d e r i v e d f o r the Markowian a p p r o x i m a t i o n p r o c e d u r e i n Cha p t e r V I . We know a l r e a d y t h a t t h e s e e x p r e s s i o n s d i v e r g e but l e t us i g n o r e t h i s f o r now. We. c o n s i d e r the e x p r e s s i o n s (6.3) w r i t t e n i n the s l i g h t l y d i f f e r e n t form (7.25) V i * ^ . . , * ^ ^ ^ where the l a s t i d e n t i t y i s a s h o r t hand, we o f t e n use. I t seems t h a t the e x p l i c i t time b e h a v i o r ^ d e s c r i b e s the f i n e g r a i n e d e v o l u t i o n i n t i m e , and we w i l l t r y to c o a r s e g r a i n our e q u a t i o n s by p e r f o r m i n g a c e r t a i n a p p r o x i m a t i o n of the e x p l i c i t time b e h a v i o r . And, we have i n c l u d e d i n (7.25) the c o n s t a n t t0 so t h a t our c o a r s e g r a i n i n g p r o c e d u r e i s not r e s t r i c t e d t o the o r i g i n i n time t . We f i n d i t c o n v e n i e n t to d e f i n e the f u n c t i o n s 5 3 . (7.26) FS= F s ( e K , t F , | f o + * 0 which have the same form as (7.25) but here we t r e a t , t h r o u g h a parameter cr independent of t , the e x p l i c i t time b e h a v i o r K'tC> as though 'we c o u l d i g n o r e t he dependence upon S r . Wi t h these f u n c t i o n s (7.26) the BBKGY h i e r a r c h y (7.1) can be w r i t t e n as f o l l o w s (7.27) [(£H^|] . + u'F'(e«r\t) = A t s F 5 " p ' ' f ' W o r one has (7-28) {(f jJ T T R : T'H ! F>(e«fW '\L'F»'(l* ,Vh)-[§t}l From our work of Ch a p t e r VI we c o u l d i n p r i n c i p l e d e t e r m i n e dt 16 the f u n c t i o n I V * W „ J r, - , n (7.29) r'.[(irl\ ^ ' ^ ^ M l ^ ^ ^ ^ . ^ - ^ J M ^ ' ^ ' l ^ + f ) W i t h t h i s f u n c t i o n J we can deduce an e q u a t i o n f o r V <j <r (7.30) ( T F ^ + H S F S - A L S F S 4 ' - J S where J s has the f u n c t i o n a l form (7.31) T- W F ' | * . + r ) , [(0 i 0 + <S-=t The c h a i n of e q u a t i o n s (7.30) we w i l l t r y t o s o l v e by a p e r t u r b a t i o n e x p a n s i o n s i m i l a r t o B o g o l i u b o v ' s , but f i r s t we must d e t e r m i n e p r o p e r boundary c o n d i t i o n s . To determ-i n e them, we w r i t e the Fs i n terms of diagrams as f o l l o w s 5 4 . (7.32) h-s(i): e - H S i L Z + e-* 5 * Z o 3 3 o > where t h e arrows i n d i c a t e the sum of diagrams t h a t have not a l r e a d y been used t o produce the f i r s t term. From our g e n e r a l p r o c e d u r e of Ch a p t e r IV we know t h a t a p o s s i b l e c o n t r i b u t i o n t o the second term must have a t l e a s t two of i t s s l i n e s c o n nected; o t h e r w i s e , the diagram c o n t r i b u t e s to the f i r s t term. As an example c o n t r i b u t i o n t o the second term we c o n s i d e r i (7.33) f ' ^ 1 1 4-S ' We use the r e l a t i o n (6.9) i n (7.33) i n o r d e r t o w r i t e t h i s term i n t h e form (7.25) v 5 , ' ( 7 . 3 4 ) e - ^ C o - * - * " C S + ' e " K - * r ~ * L.L0 LL* L One sees i n (7.34) t h a t each term d e s c r i b e s some c o r r e l a t i o n between the p a r t i c l e s 2 and 3. By a s i m i l a r c a l c u l a t i o n f o r any c o n t r i b u t i o n t o the second term i n (7.32) one always f i n d s t h a t a l l d e s c r i b e some c o r r e l a t i o n among the s p a r t -i c l e s . F o r convenience we d e f i n e a f u n c t i o n f o r the s e c o r r e l a t i o n s . 5 5. (7.35) F s f e*' tFlt)--e-KSiirf:K ,*F'+ Cs(e«'\F'li) where QS i s the second term i n (7 . 3 2 ) . For our f u n c t i o n s (7.26) w i t h the independent v a r i a b l e s <r and t we would have from (7.35) the c o r r e s p o n d i n g f u n c t i o n s (7.36) fs(ei<'l F<) toiS-)-- e-K'(^r)TreK't f< + Cie^F'l'u^) When c i s s e t e q u a l t o z e r o , one has (7.37) F ^ e ^ F ' f g r e - ^ T T e ^ F - ' + C ^ e ^ F ' / f o ) We have i n (7.37) the f u n c t i o n a l form of Fs a t the o r i g i n of cr ; t h i s g i v e s us a boundary c o n d i t i o n w i t h which we c o u l d i n p r i n c i p l e s o l v e t he e q u a t i o n s ( 7 . 3 0 ) . As a s p e c i a l case of (7.37) we w i l l sometimes c o n s i d e r the case where a- and t have the same o r i g i n , , o r where to=0. To b r i n g out what happens when t o = 0 l e t us c o n s i d e r a g a i n the example (7.33 & 7.34) from which we deduce the c o r r e s p o n d i n g c o n t r i b u t i o n s to Ci(eK'tF'\t^<r) (7.38) e - ^ * - f ' ^ T T e ^ F ' W J c J o One sees t h a t (7.38) v a n i s h e s as a c o n t r i b u t i o n to C 5(e krF'lio) f o r to=0. In g e n e r a l we f i n d t h a t the o n l y c o n t r i b u t i o n s to the second term i n (7.32) t h a t l e a d t o c o r r e s p o n d i n g c o n t r i b u t i o n s to C^e^F'/o) are those t h a t do not c o n t a i n any v e r t i c e s , one has 5 , 6 . (7.39) C ^ ^ F ' W O :&(°)*Gltlo) (7.40) c 5 ( e ^ F'/o) 4 4 i j eic. To s o l v e the e q u a t i o n s (7.30) we t r y a p e r t u r b a t i o n e x p a n s i o n f o r the v a r i o u s F s (7.41) F 5 = F K e * V / ^ ^ f ^ ^ I f we are a b l e t o s o l v e f o r the Fil , they w i l l g i v e us i n o r d i n a r y time a s e r i e s e x p a n s i o n f o r F 5 (7.42) ^- F^(e"^rlt) + /\F"(e^F'li) + X2F^(et<,'F,it)i--' In o r d e r t o s o l v e the e q u a t i o n s (7.31) we a l s o need some way to f i n d o r expand the J . F o r t h i s we w r i t e , from the BBKGY h i e r a r c h y , the e v o l u t i o n e q u a t i o n f o r the one p a r t i c l e f u n c t i o n i n the form ( 7.43) i ^ l F ' . - Xt*t, F ' H ) = e^Ate^n*,*:*) F u r t h e r , w i t h the e x p a n s i o n (7.42) f o r Fz we can w r i t e (7.43) i n s u c c e s s i v e a p p r o x i m a t i o n s where we have no z e r o t h o r d e r term. In (7.29) we can p e r f o r m the o p e r a t i o n s j(~ff-)6 as f o l l o w s ; f o r each F ' one can 5.7. ' w r i t e (7.45) ( j P ) ^ A l ' F ^ A ^ F ^ A ^ F ^ . . . where the o p e r a t o r ^ denotes d i f f e r e n t i a t i o n w i t h r e g a r d to ~i (F s'depends on t o n l y i n the c o m b i n a t i o n e*fF') w i t h subsequent s u b s t i t u t i o n of s^AW^'H) f or eH>f F' . For use i n e q u a t i o n (7.30) we must r e w r i t e t h e s e <?k * / T i i n the form e K ' ( U ^ } Ar(^'i F'\t6 + <r) and we denote t h i s by <9** over b a r s ( 7 - 4 6 ) [ ( ^ ) ] f *«--*/ ^ ^ ' ^ ' ^ ' ^ ' F ^ ' + A 9 ^ 3 ^ ^ . . . where denotes d i f f e r e n t i a t i o n w i t h r e g a r d t o ir w i t h subsequent s u b s t i t u t i o n of e K' ( t o + c)^(e*^F\t0 + e ) i 0 r 4 <? K < < F ' L e t us t r y our p e r t u r b a t i o n e x p a n s i o n i n the f o l l o w i n g way. For the one p a r t i c l e f u n c t i o n we t r y an o r d i n a r y e x p a n s i o n (7.42) f o r FZ i n e q u a t i o n (7.43 o r 7.44) and f o r the many p a r t i c l e f u n c t i o n we t r y the ex p a n s i o n (7.41) w i t h . ( 7 . 4 6 ) i n (7.30) to o b t a i n by e q u a t i n g e q u a l o r d e r s i n X' the s e t of e q u a t i o n s (7.47) A ' s L F ^ f e ^ F ' W (7.48) . / » * - - L F*7etf ' * F ' | t ) (7.49) ^ " ^ = 0 58. (7.50) yp 2' + hF Ft0 r Lf F3° Fzo « (7.51) H ' F S O = O '(7.52) H S F « V = < £ f / = L ? F ^ ; - ' - i : c 9 J F S > We are to s o l v e t h i s system of e q u a t i o n s s u b j e c t t o the boundary c o n d i t i o n (7.37) which f o r the p e r t u r b a t i o n expan-s i o n we can w r i t e as f o l l o w s S (7.53) F**(r*o) retire*'* F'(t) + CioleKltF lt0) (7.54) F s ' V - o ) - - c ^ ( e ^ F , M J where we have i n t r o d u c e d f or Ci(e^'F'ji:0) the ex p a n s i o n (7.55) C ^ V / t o ) : C a f t ^ F l ^ j ' U C s Y ^ , t F , | f o ) 4 A l C " ^ K , * F , | t o ) And, l e t us keep i n mind the s p e c i a l case where <r and t have the same o r i g i n , o r where t<= - O • f o r t h i s case we f i n d from our r e l a t i o n s ( 7 . 3 9 ) , ( 7 . 4 0 ) , e t c ' the f o l l o w i n g c o n d i t i o n s (7.56) c*°(eMF'lo) ~- Cs(cl('iF'lo) (7.57) Csl(eKlfF' lo) =o ^ l >o We f i n d t h a t our system of e q u a t i o n s (7.47 to 7.52) w i t h the boundary c o n d i t i o n s (7.53 & 7.54) deco u p l e and we can s o l v e them to any o r d e r . As i s the p l a n of t h i s c h a p t e r , l e t us s o l v e our 59. system of e q u a t i o n s s i d e by s i d e w i i l i B o g o l i u b o v ' s system (7.8 t o 7.14): we denote B o g o l i u b o v ' s s o l u t i o n by B's i n the e q u a t i o n numbers. The c a l c u l a t i o n s go smoothly u n t i l we compute the F2'/a : a t t h a t p o i n t i s where the f i r s t Markowian c o r r e c t i o n term appears. To s o l v e our system of e q u a t i o n s we s t a r t w i t h (7.51) f o r F S o or i n B o g o l i u b o v ' s case (7.13) f o r h i s F*°[ lCK'rF') ( 7 5 8 ) ^ ( ^ I ^ ^ j ^ f ^ ( e ^ F ' I U ^ ) - o (7.58B) xllU^ < Hs F<-( , r ^ ' ) . o We s o l v e (7.58) s u b j e c t t o the boundary c o n d i t i o n (7.53) and s o l v e (7.58B) s u b j e c t t o the boundary c o n d i t i o n (7.20) to ' o b t a i n (7.59) F ^ ' - ' F ' l v ^ e - ^ (7.59B) FSo( IF') = Jb~ e~H$T TTe^F'(t) (7.59B) g i v e s B o g o l i u b o v ' s /V ; to get our A' we must deduce i n o r d i n a r y time our F from (7.59). T h i s i s easy when we c o n s i d e r the s p e c i a l case of (7.59) where <r and t have the same o r i g i n . T a k i n g i n t o account (7.56 and 7.39) one has f o r t o = 0 (7.60) p f e ^ F ' H ^ ^ ^ ^ F ' W + e - ^ G Y o ) and f o r the c o r r e s p o n d i n g one has 6 0 . (7.61) F ^ e ^ F ' j i ] re-H 2t|Te« 4 F'ft) f e ' ^ G Y o ) T h i s o f co u r s e agrees w i t h the z e r o t h o r d e r term i n (6. 1 0 ) . With (7.61) o ur A1 f o l l o w s from (7.47) and we o b t a i n B o g o l i u b o v ' s A'from (7.59B) w i t h (7.9) (7.62) h W f ' l t ) - U - « ^ M F;(i)e^F[(t), L f " J * GY4> (7.62B.) A ' U t j * r)~- J t ~ Le-^e^FK^e^Fld) C ? oo We are i n a p o s i t i o n to s o l v e f o r F s' and B o g o l i u b o v ' s F s l ; however, we w i l l o n l y work out FZI and B o g o l i u b o v ' s F For F 1 1 we use (7.50) t a k i n g i n t o account (7.59) and i n B o g o l i u b o v ' s case we use (7.12) t a k i n g i n t o account (7.59B) (7.63) jr'+ H lF*' = ^e^^e-^TTe^iF](i)i{ze-^a°(e^f'ji) - $ ' j e - W v e fe TTe ^  F' $ + e " H v G 2 *(e *'* F'| t.)} (7.63B) ^ " F ^ t HlF*' = {l1e-H>Tre^F,-£,e-ti>lTe^F'(t)} y£.( -5* S i n c e we have theh'c-, we can pe r f o r m t h e t S ' a n d ^ ' o p e r a t i o n s ; and f o r B o g o l i u b o v ' s e q u a t i o n (7.63B) we r e p l a c e the " a r b i t -r a r y f u n c t i o n a l argument" F' by Q'1* 'F - ' (7.64) . . . ( c o n t i n u e d ) 61. • ( 7 > 6 4 ) J£ ? ' + H l F « » - L x e - H V ^ f r e ^ F 7 ^ H 2 e ^ V C ^ ( e ^ F ' / ^ Tfe^F'it) (7.64B) iE!Llf!!!l£J4^Fll( le-K+FJ-/A - » o o And we s o l v e (7.64) s u b j e c t t o the boundary c o n d i t i o n s (7.54.) and s o l v e (7.64B) s u b j e c t t o the boundary c o n d i t i o n ( 7 . 2 1 ) , or use ( 7 . 2 3 ) , (7.65) F'Ye K , fF7#ptr) = e - i ^ c ^ e ^ F ' / * , ) 6 2 . (7.65B) Fz'( I F ' ) ' ' -lU l/^e-H>[Llbe'H'^-rU,e-H^lfl'e^F'(i) We are not a b l e t o go f u r t h e r w i t h B o g o l i u b o v ' s t h e o r y because c l e a r l y the second term i n (7.65B) d i v e r g e s . To b r i n g out a c o n n e c t i o n t o our t h e o r y , l e t us f i r s t c o n s i d e r the g e n e r a l s o l u t i o n f o r our F S a and F 5 t ; from (7.51) and (7.52) u s i n g the boundary c o n d i t i o n s (7.53) and (7.54) one has (7.66) M« - e~^e-^hTreK'i- F'(i),e~^ C'ie^F'lQ (7.67) F s' e"/ y V C 5 ;(V*" f f ' /to) + J M, e^f^^F'jt^i,) We f i n d t h a t our ex p a n s i o n becomes almost i d e n t i c a l t o B o g o l i u b o v ' s e x p a n s i o n i f we make the f o l l o w i n g two assumpt-i o n s : ( i ) . We assume our e x p a n s i o n i s v a l i d f o r a s y m p t o t i c tf" so t h a t we can co a r s e g r a i n our f u n c t i o n s by t a k i n g the l i m i t s (7.68) i i . FSd> (cK>F'[tl\t0i*:) ff" > CO /A —> 6 3 . (7.69) F ? t (eK> F'hVtotc) where one n o t i c e s t h a t the to i s no l o n g e r i m p o r t a n t i n comparison to &. ( i i ) . We assume t h a t the c o r r e l a t i o n s which e x i s t e d a t the o r i g i n of cr v a n i s h due to the n a t u r a l motion of the system, o r one has (7.70) k~ e - " J , r C ^ F W | t , ) - ' 0 W i t h t h e s e a s y m p t o t i c c o n d i t i o n s i n (7.66 & 7.67) one has (7.71) , e~HS//' TTF'(i-) o- - -> oo jfc —y oo i f (7.72) 1^ = JUL [AeMy'tCeKH'lt.+ O yl< —5» w L e t us c o n s i d e r our s o l u t i o n f o r F 2 / (7.65) w i t h t h e s e a s y m p t o t i c c o n d i t i o n s (7.73) F»~/i.'. r ^ , f f - * " ' f l , e - » , / ' 7 f ^ F W } -/£,( V', e " > f e ' ' / 1 L 1 } e - V t L ^ ^ T T e ^ f « ) And we compare (7.73) w i t h B o g o l i u b o v ' s s o l u t i o n f o r (7.65B) and we see they are the same ex c e p t f o r some o p e r a t o r s G t. The i m p o r t a n t p o i n t i s t h a t F X I d i v e r g e s i f we ta k e the a s y m p t o t i c l i m i t . o f t he e x p l i c i t time b e h a v i o r . From our work i n Chapter VI we of course e x p e c t e d a d i v e r g e n t F*1 because, from ( 6 . 1 0 ) , the f i r s t o r d e r term 64-. of F2" c o n t a i n s Markowian c o r r e c t i o n terms. I t i s not appar-ent t h a t (7.65) w i l l g i v e , us i n o r d i n a r y time the same Fx' as i n ( 6 . 1 0 ) ; i t can be shown t h a t i t does and we have r e l e g a t e d t h i s c a l u c l a t i o n t o our Appendix' I I . Due t o the s t r o n g resemblence of our cr to B o g o l i u b o v ' s T and s i n c e our er d e s c r i b e s , e x c e p t f o r the e^'* a f f i x e d t o the F'i- , the e x p l i c i t time b e h a v i o r we f e e l j u s t i f i e d i n s a y i n g t h a t B o g o l i u b o v ' s T d e s c r i b e s the e x p l i c i t time b e h a v i o r i n h i s t h e o r y . By comparison w i t h our e x p a n s i o n we know t h a t B o g o l i u b o v ' s e x p a n s i o n i s d i v e r g e n t , and i n p a r t i c u l a r i t i s not v a l i d t o t a k e the l i m i t f >CD . I f the l i m i t of the e x p l i c i t time b e h a v i o r were v a l i d , we f e e l j u s t i f i e d i n s a y i n g t h a t B o g o l i u b o v ' s boundary c o n d i t i o n s (7.20) and (7.21) have the same meaning as the assumption ( i i ) ; t h a t i s , the c o r r e l a t i o n s which e x i s t e d , a t the ' a r b i t r a r y ' o r i g i n of ? v a n i s h f o r a s y m p t o t i c V . T h i s c o n c l u d e s our d i s c u s s i o n of B o g o l i u b o v ' s . t h e o r y . the n e x t c h a p t e r we w i l l d i s c u s s the method of e x t e n s i o n due Ar 5 to S a n d r i & F r i e m a n . ' T h i s method i s s i m i l a r t o B o g o l i u b o v ' s t h e o r y . V I I I . THE'METHOD OF EXTENSION The method of e x t e n s i o n due to S a n d r i & Frieman ' w i l l now be d i s c u s s e d . We d i s c u s s t h e i r method f o r the s h o r t - r a n g e t h e o r y and f o r s i m p l i c i t y we assume the system t r e a t e d t o be homogeneous and i n i t i a l l y u n c o r r e l a t e d . These a u t h o r s use as a s t a r t i n g p o i n t t he BBKGY c h a i n (3.18) which f o r t he s h o r t range t h e o r y we w r i t e (8.1) ^ ^ F 5 - ^ ^ ^ We f i n d the method of e x t e n s i o n i s b e t t e r u n d e r s t o o d i f we work f i r s t from an o r d i n a r y p e r t u r b a t i o n e x p a n s i o n . L a t e r we w i l l f o l l o w S a n d r i & Frieman's work more c l o s e l y . L e t us i n t r o d u c e i n t o (8.1) the exp a n s i o n s (8.2) F '=- f ° + Af f ; , 2 - f ? - + - • (8.3) F* = FZ0 + X F" + ,\* F " f • (8.4) F 5 T Fso iXF" i xLFS*+--We o b t a i n by e q u a t i n g e q u a l o r d e r s i n X the s e t of e q u a t i o n s (8.5) £r = O (8.6) $ = L F  z>' "*' c >o L = L' (8.7) 7 T % H s f s o = o (8.8) 4 r f ' *• H 5F Sc*= V F s t'^"' 6 6 . S i n c e t he system i s assumed i n i t i a l l y u n c o r r e l a t e d , f o r i n i t i a l c o n d i t i o n s one has (8.9) $°(o) - - 1 (8.10) fc (o)- O t >o (8.11) FS6(o) , f r f°(o) (8.12) Fsi(o) - O We p r o c e e d t o s o l v e the s e t of e q u a t i o n s (8.5) to (8.8) s t a r t i n g w i t h t he z e r o t h o r d e r and w o r k i n g t o h i g h e r o r d e r , From (8.5 w i t h 8.9) we f i n d t h a t -f" i s time independent (8.13) i0(o)=f = A n d , f o r the many p a r t i c l e f u n c t i o n we o b t a i n from (8.7) (8.14) e TV fc To s o l v e (8.6) f o r -f 1 we w i l l be i n t e r e s t e d i n the two body b e h a v i o r (8.15) F ^ e - ^ W : * * - * 1 * ^ < Z X j i O X ; + 3 0 C < t - ' - } By s u b s t i t u t i o n of (8.15) i n t o (8.6) one f i n d s the s o l u t i o n (8.16) • - c j . < x < - o o c ; * - o o c c ; F o r a s y m p t o t i c time we f i n d t h a t (8.16) d i v e r g e s , o r we have 6 7. . . (8.17) V ~ i t u Le-^ftfi where we have r e c a l l e d the diagram d e f i n i t i o n s (4.20) and (4. 2 3 ) . F or the a s y m p t o t i c b e h a v i o r of the one body f u n c t i o n we now have (8.18) Fl=f°+xt jt&r, Le-W{?fl + o ( J i * ) S a n d r i & Frieman use a t r i c k t o t a k e care of t h e s e d i v e r g e n t or s e c u l a r terms. For t h i s case (8.18) we f i n d t h a t they r e d e f i n e the z e r o t h o r d e r a p p r o x i m a t i o n by the f o l l o w i n g r e l a t i o n s (8.19) F' - f° f O + oJk (kz) (8.20) Jz^Le I* il ( t M y ^ j^tQ (.8.21) ^ - Xi One sees t h a t the terms of (8.18) appear now i n the T a y l o r e x p a n s i o n of -f6 (8.22) f'sf+e-jfa Le-Wf;fS*<ri ^ t i l * f S a n d r i & Frieman assume t h a t the g e n e r a l e x p a n s i o n c o n t a i n s s e r i e s o f the type (8.22) and t h a t t h e s e are t h e s i g n i f i c a n t ones. From our c a l c u l a t i o n (4.15) through (4.25) we know t h a t the g e n e r a l e x p a n s i o n does c o n t a i n s e r i e s of the type 68. (8.22) and t h i s i s the r e a s o n t h e i r method seems to work. The s e r i e s (8.22) comes from the b e h a v i o r (8.23) j L p , ^ A _ < ^ + A _ < c o A - O O C + - - - -A l t h o u g h (8.20) i s Boltzman's e q u a t i o n , the way i t was found was no more than a guess. We w i l l now f o l l o w S a n d r i & Fireman's method more c l o s e l y and f i n d t h ey make o t h e r a s s u m p t i o n s and, as they do, d i v e r g e n c e d i f f i c u l t i e s . I n the l a s t p a r a g r a p h we p r e s e n t e d what we c o n s i d e r the b a s i c r e a s o n the method o f S a n d r i & Frieman works. We w i l l have to go much,much deeper however, to b r i n g out a l l t h e assumptions and meanings of t h e i r t h e o r y . L e t us s t a r t by w r i t i n g the g e n e r a l r i n the form (8.24) F = F * ( * ^ y . K ^ ^ . . . ) where we have i n t r o d u c e d the parameters (8.25) «i--f + C, , «7 = A* + C,, tf-j, = XH+Ci} crl= tft i C 3 We have done n o t h i n g i n (8.24) but a s s e r t t h a t we can w r i t e , i n y e t some undetermined way, the v a r i o u s r i n terms of the parameters (8.25). At t h i s p o i n t we b e g i n t o f o l l o w S a n d r i 8, Frieman's method of e x t e n s i o n . F o l l o w i n g them, we imagine now a f u n c t i o n t h a t has the same form as (8.24) 69. (8.26) F 5 R F 5 ( ^ ^ , r - - - ) ^ ^ s \ r 0 j t , r , r i j . . . ) but here TCifl)'YljTlJ... are independent v a r i a b l e s . With t h i s f u n c t i o n (8.26) the BBKGY c h a i n (8.1) can be w r i t t e n as f o l l o w s . ' (8.27) R&>v^M&>..^ • H ' F ' = A L ' F " L e t us assume, as S a n d r i & Frieman do, t h a t the f o l l o w i n g equ-a t i o n i s a l s o t r u e (8.28) The t r a n s i t i o n from (8.27) t o (8.28) c l e a r l y p u t s some r e s t r i c t -i o n s on the p o s s i b l e s p e c i f i c a t i o n s of the f ^ T , ^ ^ . . . a n d we sh o u l d check t h i s whenever a s p e c i f i c a t i o n i s made. F o l l o w -i n g S a n d r i & Frieman we i n t r o d u c e i n t o (8.28) the p e r t u r b a t i o n e x p a n s i o n s (8.28) F' - f*+A?'4 l ^ P f - -(8.29) p = F ^ f A P + A ' F ^ . - -(8.30) F « = F S o + A F S l +A* F 5 i + - - • And we o b t a i n by e q u a t i n g e q u a l o r d e r s i n A the s e t of eq u a t i o n s 7 0 . (8.31) 7 ^ - " 0 (8.32) ^ = L F « (8-33) i | > ^ > ^ - : L ? -(8.34) g"*B*F"-0 (8.36) j r . f ^ F ° = ° a (8.37) 5To ' [ F ; jf} « ^ S a n d r i & Frieman o f t e n use the i n i t i a l c o n d i t i o n s (8.38) f * ( r o = 0 ) = asd&ay y U c ^ ^ r , ^ , ^ ^ . . . (8.39) P(?tzo) - 6 t' >o s _ (8.40) F S 0 M : TT ^  ^ 0 r ^ J (8.41) ? s « 7 ^ = o) = O These c o n d i t i o n s are me a n i n g l e s s a t t h i s p o i n t because the dependence of r upon Vo?,^,'•• has not .yet been s p e c i f i e d . We w i l l have t o study t he i n i t i a l c o n d i t i o n s l a t e r . The s o l u t i o n t o the z e r o t h o r d e r e q u a t i o n s are the same as b e f o r e , or one has (8.42) r ' ^ r , , V«. M 0 ^ 0 ^ ; ^ , — ) (8.43) F 5 * * e-//s^ ]7 f° 7 1 . From (8.32) w i t h ("8.43) 4' s a t i s f i e s (8.44) Le-^f;;;-0 o r , f o r the s o l u t i o n of ( 8 . 4 4 ) , c o n s i d e r i n g o n l y the a s y m p t o t i c b e h a v i o r , o n e has To o b t a i n the o r d i n a r y p e r t u r b a t i o n . e x p a n s i o n , where the tr, i s never i n t r o d u c e d , one would put ( 8 . 46 ) j - ^  ( tfA^c^ux^ jp^s\^£lv^S-cJ~£^ My^^-O-^S^''^) i n which case f ° i s s i m p l y the i n i t i a l v a l u e of the one body f u n c t i o n . However; i t i s c o n v e n i e n t t o assume, and t h i s i s what S a n d r i & Frieman do, t h a t the g e n e r a l e x p a n s i o n c o n t a i n s the s e r i e s (8.22) by s p e c i f y i n g (8.4.7) .4°-- Jk» Le-»*1 p £ which c a n c e l s the d i v e r g e n c e i n ( 8 . 4 5 ) . W i t h o u t the s u p p o r t of a more g e n e r a l t h e o r y , (8.47) i s no more than a guess. L e t us compute F 1 ' ; from (8.35) t a k i n g i n t o account (8.43 and 8.47) one has (8.48) - L ! e » , r - f t ' ^ t f , ' t [ l 1 J f * M , ! A » ] m i o r , w i t h the i n i t i a l v a l u e (8.91) one has the s o l u t i o n When we c o n s i d e r the a s y m p t o t i c b e h a v i o r , f d i v e r g e s (8 . 50 ) F « ~ J ^ ( % , e - H % L V H i V M & • And t h i s i s the same b e h a v i o r (7.73) we o b t a i n i n Chapter V I I . L e t us suppose t h a t we c o u l d go f u r t h e r w i t h the method of e x t e n s i o n . In t h a t case r e p e a t e d i d e n t i f i c a t i o n s of the type ( 8 . 4 7 ) , or summing of s e r i e s of the type (8.22) i n t o i°, would y i e l d an i° t h a t we can i d e n t i f y as the a s y m p t o t i c one body f u n c t i o n . W i t h t h i s u n d e r s t a n d i n g t h a t ~P i s the one body f u n c t i o n ( w r i t t e n i n terms of the parameters ? MT,^ ... ) and by comparing the method of e x t e n s i o n w i t h our p e r t u r b a t i o n e x p a n s i o n of Chapter V I I , i n p a r t i c u l a r comparing T0 w i t h c~ t h e r e ; we f i n d t h a t . t h e method of e x t e n s i o n i s j u s t a n o ther a p p r o x i m a t i o n t h a t can be o b t a i n e d from our c o a r s e g r a i n i n g p r o c e d u r e . And, we know t h a t t h e s e e x p a n s i o n s d i v e r g e q u i t e b a d l y . To t e r m i n a t e this C h apter we w i l l d i s c u s s the boundary c o n d i t i o n s (8.38 to 8.41) of the method of e x t e n s i o n . Due to the a r b i t r a r y c o n s t a n t s c 0 i n 6~0 (8.25) and th u s an a r b i t r a r y c o n s t a n t i n the c o r r e s p o n d i n g T6 , the p o i n t or o r i g i n from which the p e r t u r b a t i o n e x p a n s i o n i s a t t e m p t e d , i s a r b i t r a r y . In o r d e r to s a t i s f y 7 3 . the boundary c o n d i t i o n s (8.38 t o 8.41) and t o agree w i t h the assumption t h a t the p a r t i c l e s are i n i t i a l l y u n c o r r e l a t e d one must s t a t e t h a t e~0 <r,} <rX).. . o r r a t h e r the c o r r e s p o n d i n g Xo/f, j T , i . . . have the same o r i g i n . O t h e r w i s e , t h e s e boundary c o n d i t i o n s (8.38 to 8.41) s t a t e t h a t r e g a r d l e s s o f where the o r i g i n of T 0 i s the p a r t i c l e s are u n c o r r e l a t e d and t h i s doesn't make sense. I n o r d e r to make the o r i g i n of a r b i t r a r y , w hich i s a more m e a n i n g f u l way to t r y t o coa r s e g r a i n the e q u a t i o n s by a p e r t u r b a t i o n e x p a n s i o n , we must i n t r o d u c e the c o r r e l a t i o n s t h a t e x i s t e d a t the c h o i c e of o r i g i n of To (8.51) F S ( r „ = «) - IT F° + cs( ^ r l ; r , . . . ) o r by c l u s t e r e x p a n s i o n we can w r i t e (8.52) C ' t t , - ^ , . . . ) j ( r t ) r x > r . i r . . ) (8.53) a f:3lv(r,M,-'-h^^(vhr%Jvsy..) •A. S a n d r i & Fireman, see S a n d r i , d i s c u s s t h e i r expansion.method w i t h boundary c o n d i t i o n s s i m i l a r t o (8 . 5 2 ) , ( 8 . 5 3 ) , e t c . We would l i k e t o s t r e s s t h a t t h e s e c o r r e l a t i o n s are not i n i t i a l c o r r e l a t i o n s and i n g e n e r a l a g i v e n <j.s depends upon the one 2° body f u n c t i o n t . I X . THE THEORY OF MAZUR AND BIEL 6 Mazur and B i e l , MB f o r s h o r t , work from t h e BBKGY c h a i n (3.18) f o r the s h o r t - r a n g e system ( 9,1)' 4! S+H $ F S - - A L S F " ' where ( 9 . 2 ) L S = £ L-^- w y A t t h i s s t a g e ; as shown i n the LT'5+/ ; the thermodynamic l i m i t (3.27) has not been t a k e n . L e t us i n t r o d u c e MB's reduced momentum ( v e l o c i t y ) d i s t r i b u t i o n f u n c t i o n s (9.3) ^ ( v v ^ s ; ^ r"v"s ["'p^t ••• ^ ^ f a ' / V - - ^ * ^ and t o s i m p l i f y n o t a t i o n MB d e f i n e s the o p e r a t o r s ( 9 . 4 ) ' " P ' * = ^ ['•'[</*...'. J so t h a t the - f S may be w r i t t e n (9.5) 4** Ps F 5 The d i s t r i b u t i o n f u n c t i o n s r can now be w r i t t e n (9.6) Fs= P* F5 + (l-P5)Fs = r + A 5 where (9.7) hs- (l-Ps) Fs - F s - f 5 75. F o l l o w i n g MB we a p p l y the o p e r a t o r s P $ and ( l - P s ) r e s p e c t i v e l y t o the BBKGY c h a i n (9.1) to o b t a i n the c o u p l e d s e t s o f e q u a t i o n s (9.8) jf-t p*H sf* + P6H sh$ - A ?sls f s " + A P 5 L H s t l (9.9) ( ! - P ? ; H s f s + ( / - P s J f i S ^ -At t h i s p o i n t the thermodynamic l i m i t [N -**'jV->0Oj~-zn f i n i t e ) i s t a k e n . Under t h i s l i m i t MB c l a i m that," due to the f i n i t e r a n g e . o f the I n t e r p a r t i c l e p o t e n t i a l , the f o l l o w i n g e x p r e s s i o n v a n i s h e s (9.10) P sH*F s=-fe['''\ J*'-' Jl** W S F S ^ 0 One has a l s o ; s i n c e the { s do not depend on ; ^ j (9.11) |< sf s=o , L** 5 +' = o i n the second of t h e s e we have assumed the system i s i n s e n s -i t i v e t o the w a l l s of the c o n t a i n e r . With (9.11), and the thermodynamic l i m i t the c o u p l e d e q u a t i o n s (9.8 & 9.9'"') reduce to (9.12) i i S = \ psL5hs" (9.13) $j + H sh'5*r{U\iL (i-p*)L sh s*' T h i s s e p a r a t i o n of the BBKGY c h a i n i n t o t h e s e coupled, e q u a t i o n s i s MB's major r e s u l t and here we be g i n our d i s c u s s -7 6 . i o n of th e t h o e r y . L e t us b e g i n by w r i t i n g the g e n e r a l F s i n terms of diagrams. F o r s m a l l s, i n comparison to N, i t - i s a good a p p r o x i m a t i o n t o w r i t e the F S i n the form (9.14) F*-e-K$i 7 ^ 4 e-K^tZ here the arrows i n d i c a t e the sum of diagrams where a t l e a s t •. two o f the s l i n e s are connected: (9.14) i s good enough f o r our p u rposes because soon we w i l l c o n s i d e r the thermodynamic l i m i t and we know t h a t the g e n e r a l Fs approaches the form (9.14) under t h a t l i m i t . I f we assume t h a t i n i t i a l c o r r e l -a t i o n s are o f f i n i t e range, each diagram i n the second term of (9.14) depends on i n t e r p a r t i c l e d i s t a n c e t h rough an o v e r -l a p of the i n t e r p a r t i c l e p o t e n t i a l s and. i n i t i a l c o r r e l a t i o n s . Under the o p e r a t i o n P s w i t h the thermodynamic l i m i t the second term i n (9.14) v a n i s h e s and one has f o r f S the l i m i t (9.15) I s - jp. {--jet*,... J«s F s ^> TT-f 1 T h i s l i m i t , r a t h e r i d e n t i f i c a t i o n , i s made by MB but a t a f a i r l y advanced s t a g e . o f t h e i r work. L e t us check the s i m p l i f y i n g l i m i t ( 9 . 1 0 ) ; we f i n d a n o n - v a n i s h i n g answer (9.16) P $ H S F $ ^ ^ PSK5TT F' = j t i » PsK5Vi(f'+h') " y -> c» V — > s v-><*> 7 7. S i n c e t h i s - l i m i t v a n i s h e s f o r a homogeneous system, MB's cou p l e d e q u a t i o n s (9.12 & 9.13) are v a l i d o n l y f o r homog-eneous systems. For an inhomogeneous system one s h o u l d have the c o u p l e d e q u a t i o n s ( 9.1?) ^ ' ^ - . L P ' W ' u i ; p'i'A* ( 9 . 1 8 ) ^ ' + r / . ' = i c ^ p ' m ' + A i ^ d-p'jL'h1 (9.19) ~ t f / 5 r = i ^ P 5 ^ 7 T h " 4 l s T T r ' -Ui.™ 0-P*)Lsks+l because o f t h e redundance (9.15) we need no l o n g e r w r i t e out the f o r m u l a f o r 4s . The e x t r a c o m p l i c a t i o n t h a t appears i n t he c o u p l e d e q u a t i o n s (9.17, 9.18 & 9.19) f o r an inhomog-eneous system r e s t r i c t s t h e i r u s e f u l n e s s f o r t h a t case. We p r e f e r t o w r i t e out an a l t e r n a t e to MB's t h e o r y which i s s i m p l e r f o r an inhomogeneous system and i n t e r c h a n g e a b l e w i t h t h e i r t h e o r y f o r a homogeneous system. From t h i s p o i n t on we assume the thermodynamic l i m i t i s v a l i d and has been taken i n the g e n e r a l F s . Our b a s i c s e p a r a t i o n of the w i l l be (9.14) f o r which we w r i t e (9.20) F s = C T C S + C S where (9.21) ^ s = I f F ' 78. We s u b s t i t u t e (9.20) i n t o the BBKGY c h a i n (9.1) to o b t a i n (9.22) ^ + IL\ Ks<s+K5Cs-Is«fs-IsCs^ A I s ^ fA L 5 C S f / and s e p a r a t e t h i s i n t o two e q u a t i o n s ; . one where each term i s f r e e of c o r r e l a t i o n s (9.23) ^ + K s < 5 = A L s < f f f ' + A L i C s + ' and one where each term c o n t a i n s a c o r r e l a t i o n (9.24) VO-VC^ I ^ s f A L S c C $ + l Here, we have s p l i t l / C S + ' i n t o two terms (9.25) LsCS+> = [ 5 0 C S " ' f [ S c C S + ' ' one term t h a t i s f r e e of c o r r e l a t i o n s and one term t h a t c o n t a i n s c o r r e l a t i o n s . I t i s e a s i l y seen, when we draw on the c l u s t e r e x p a n s i o n (3.21 t o 3.23) to expand C S, t h a t t h i s s e p a r a t i o n (9.23 & 9.24) i s v a l i d . F i r s t of a l l Cz and &'~ have the same d e f i n i t i o n (9.26) G*=Ca = C J 2 and one can show by the c l u s t e r e x p a n s i o n of C s + I t h a t 79, (9.27) L,?C S + l s f l - T T F;'(?"*- i) * J "* With (9.27) and the d e f i n i t i o n . o f ^ s (9.21) one f i n d s t h a t (9.23) i s s i m p l y an s f o l d redundance of the e q u a t i o n f o r (9.28) + K ' F ' --\Llz (VK* C2) F o r comparison to MB 1s t h e o r y l e t us w r i t e o u r c o u p l e d e q u a t i o n s (9.23 & 9.24) f o r the s p e c i a l case o f a homogeneous system (9.29) ^p--Xi5uCs+' (9.30) ^f\hlsCs-= I s ^ s f \ L s c C s + l We compare t h e s e e q u a t i o n s w i t h MB's (9.12 & 9.13) and we r e c a l l the e f f e c t of the o p e r a t o r s P S. Our e q u a t i o n s are s l i g h t l y more g e n e r a l because we can w i t h o u t l o s s of g e n e r a l -i t y remove the r e s t r i c t i o n s t h a t the i n t e r p a r t i c l e p o t e n t i a l and i n i t i a l c o r r e l a t i o n s are of f i n i t e range. For an inhomogeneous system our method i s s i m p l e r than MB's because our C'-O: i n the MB e q u a t i o n s the need f o r the b' causes e x c e s s i v e c o m p l i c a t i o n , , We w i l l now compare our e q u a t i o n s here w i t h the c l u s t e r e x p a n s i o n method we u s e d i n Chapter V to s e l e c t and sum diagrams. To be g i n l e t us w r i t e down the e q u a t i o n s f o r P C Z and 30. (9.31) •3f+K'F,« niz(rr;F± =AL t zF* (9.32) ^ + ^  C 2-! 2 C 2 = I 2 F/F' + A L 2 C C 2 (9.33) 4f + K 3 C 5 - I 3 C 3 = I 3 F/Fi F3 + A L 3 C C 4 : These e q u a t i o n s are t o be compared w i t h the " s i m p l e r " e q u a t i o n s (5.11, 5.12 & 5.13). We o b t a i n here the two p a r t i c l e c o l l i s i o n e f f e c t s w i t h o u t d i f f i c u l t y by s u b s t i t -u t i o n of. the s o l u t i o n f o r CZ i n t o (9.31) (9.34) jfc +|f'F'-- A L i a.F l •Jo T h i s e q u a t i o n i s t o be compared w i t h the way we o b t a i n e d our former e q u a t i o n ( 5 . 2 4 ) . L e t us w r i t e out the c o r r e s p o n d i n g MB r e s u l t f o r (9.34) f o r the s p e c i a l case of a homogeneous system. ( 9 . 3 5 ) j f ^ [ J f f ^ V w t e - ^ v ' ' 1 * ' i > f : i * , ) t i a ) ] 0 y-^ 00 We c o u l d extend the s o l u t i o n of (9.34) o r (9.35) to any o r d e r i n A . Though th e s e s o l u t i o n s are c o r r e c t we f i n d them i n c o n v e n i e n t due to the s p e c i a l o p e r a t o r s l?c or the 81. c o r r e s p o n d i n g (l-Ps) which appear i n the s o l u t i o n s . Working from the c l u s t e r e x p a n s i o n e q u a t i o n s ( 5 . 9 ) g i v e s a more t r a n s p a r e n t answer. T h i s c o n c l u d e s our remarks about Mazur & B i e l . ' s . t h e o r y . X. CONCLUSION The aim o f t h i s t h e s i s was to d i s c u s s B o g o l i u b o v ' s t h e o r y of c l a s s i c a l i r r e v e r s i b l e s t a t i s t i c a l mechanics. To do t h i s " we d e v e l o p e d a t h e o r y t h a t .stems from the diagram t e c h n i q u e s o f P i g o g i n e and. c o w o r k e r s . We have found our t h e o r y s u f f i c i e n t l y g e n e r a l f o r us t o o b t a i n the t h e o r y o f B o g o l i u b o v as a s p e c i a l c a s e . We have shown t h a t B o g o l i u b o v ' s e x p a n s i o n d i v e r g e s q u i t e b a d l y . Even to o b t a i n the Boltzman equation.Markowian c o r r e c t i o n terms must be i n t r o d u c e d , and t h e s e terms d i v e r g e . T h i s e v i d e n c e g i v e s us s u f f i c i e n t doubt i f B o g o l i u b o v ' s d e r i v a t i o n of the Boltzman e q u a t i o n i s s i g n i f i c a n t . These r e s u l t s and doubts c a r r y over to our d i s c u s s i o n o f the t h e o r y of S a n d r i & Frieman; t h e i r t h e o r y i s s i m i l a r to B o g o l i u b o v ' s . To show the v e r s a t i l i t y of our e x p a n s i o n t e c h n i q u e s we have d i s c u s s e d the t h e o r y of Mazur & B i e l ; a t h e o r y t h a t has l i t t l e resemblance to B o g o l i u b o v ' s . 83. BIBLIOGRAPGY 1. N.N. Bogoliubov, Stud. S t a t . Mech., 1, 11 (1962). 2. I . P r i g o g i n e , "Non-Equilibrium S t a t i s t i c a l Mechanics", I n t e r s c i e n c e , New York, (1962). 3. G. Severne,.Physica, 11, 377 (1965). 4 . G. S a n d r i , Ann. Phys., 24_, 332 (1963); gj±> 3^0 (1963). 5. E.A. Frieman, J . Math. Phys., i±, 410 (I963). 6. P. Mazur & J . B i e l , P h y s i c a , 3_2, 1633 (1966). 7. I . Pri g o g i n e & P. R e s i b o i s , Physica, 22, 629 (1961). g. J . S t e c k i & H.S. Ta y l o r , Rev. Mod. Phys., 3J7_, 762 (I965). 9. J . Brocas & P. R e s i b o i s , Physica, 32, 1050 (1966). 10. E. Braun & L.S. G a r c i a - C o l i n , Phys. Let., 2J_, 460 (I966). 11. M.S. Green, J . Chem. Phys., 2£, ^36 (1956). 12. P. R e s i b o i s , Phys. F l u i d s , 6, gl7 (I963). 13. E.G.D. Cohen & J.R. Dorfman, J . Math. Phys., 232 (1967). 84. Appendix I . In t h i s appendix we show t h a t diagrams w i t h a-type (c) v e r t e x (3.6) i n t e g r a t e t o z e r o . L e t us c o n s i d e r a diagram w i t h a type (b) o r (c) v e r t e x where the l i n e l a b e l l e d j f i r s t a p p e a r s . Look a t t h e e x p r e s s i o n ji?'f^-.- i n ^ v and a t the i n t e g r a l ( A l . l ) D . . . . J ( f * , , . } ) t e ^ ^ v ' » ^ . J i : G({•*,*•}) Here, the s e t - j r ] r e p r e s e n t s the p a r t i c l e l a b e l s used t o the l e f t of the v e r t e x c o n s i d e r e d . T h i s e x p r e s s i o n becomes, w i t h the h e l p of the r e l a t i o n (2.9) ( A 1.2) D--jr{ 3. )y r } e l f'M^ J V,j^ ; j . £ G({*,*}) Use of Green's theorem i n the v e l o c i t y i n t e g r a l g i v e s (A1.3) IT S The i n t e g r a l then v a n i s h e s because the v e l o c i t y d i s t r i b u t i o n v a n i s h e s s u f f i c i e n t l y f a s t a t i n f i n i t y . We have thus a s i m p l i f i c a t i o n of (b) type v e r t i c e s and by p e r f o r m i n g the c a l c u l a t i o n t w i c e we see t h a t diagrams w i t h a (c) type v e r t e x v a n i s h . 85. Appendix I I -In t h i s appendix we show t h a t the F o b t a i n e d i n (7.65) g i v e s i n o r d i n a r y time the f i r s t o r d e r term o b t a i n e d i n ( 6 . 1 0 ) . To o b t a i n i n o r d i n a r y time a s o l u t i o n f o r F 1 ' from F 1 ' we c o n s i d e r the s p e c i a l case o f (7.65) where & arid t have the same o r i g i n , o r where i o = 0 . One has t a k i n g i n t o a ccount (7.56 & 7.57 w i t h 7.39 & 7.40) f o r the f u n c t i o n s C ^ ^ F ' l o ) and the d e f i n i t i o n o f the o p e r a t o r the f o l l o w i n g e x p r e s s i o n f o r F ^ e ^ F ' k ) And f o r the c o r r e s p o n d i n g F we have, w i t h some changes i n the v a r i a b l e s o f i n t e g r a t i o n , the e x p r e s s i o n 86. W i t h n o t a t i o n o f the type (5.18, 5.19, 5.20, e t c . ) we can w r i t e (A2.2)as f o l l o w s (A2.3) F 2 1 - e - * 1 ^ , e w * * ' ( L „ * L j * ° r 1 A i . 2 3 2 - !_ 2 , 3 £ 0 - — r - G r-tf ID, L e t us expand the f i r s t term i n (A2.3) c o m p l e t e l y , o r by comparison of (5.24) w i t h (5.17) one has (A2.4) F2'= e - ^ { M 'DC* D O C + X X X *•••}* a -— 6 2 3 2 f i E f n : f i i : + t n ' n i + i D 7 + TIT * 2 87. One sees t h a t (A2.4) c o n t a i n s diagrams w i t h fragments t h a t connect i n by one l i n e . We know from C h a p t e r IV t h a t such b e h a v i o r always c o n t r i b u t e s t o lower o r d e r terms; i n t h i s case to F 2 0 . Thus we s h o u l d , w i t h the second term i n (A2.4), be a b l e t o c a n c e l diagrams w i t h fragments t h a t connect In by one l i n e . F u r t h e r , as a remainder i n the second term we s h o u l d o b t a i n the Markowian c o r r e c t i o n terms shown i n (6.10) L e t us i n t r o d u c e i n the second term of (A2.3 or A2.4) the i d e n t i t y so t h a t t h i s second term may be w r i t t e n i n the f o l l o w i n g way W i t h t h i s e x p r e s s i o n f o r the second term i n (A2.3) or (A2.4) •we o b t a i n the d e s i r e d c a n c e l l a t i o n of diagrams w i t h fragments t h a t connect i n by one l i n e ; one has By changing the o r d e r o f the i , and iz i n t e g r a t i o n s i n the second term, we see t h a t (A2.8) i s the same e x p r e s s i o n as the f i r s t o r d e r term i n ( 6 . 1 0 ) . 

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