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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.Sc, M.Sc,  New Mexico I n s t i t u t e o f M i n i n g and Technology, 1963 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 , 1964  A THESIS SUBMITTED I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN THE DEPARTMENT OF PHYSICS  We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d standard  THE UNIVERSITY OF BRITISH COLUMBIA September, I96S  T) Paul Henry Seagraves  1969  In  presenting  advanced  Library  agree  this  degree  shall  that  at  make  thesis  the  it  permission  in  University  freely  for  may  be  granted  tatives.  It  is  understood  financial  gain  Department  of  by  not  Date  the  that  be  Head  British  allowed  for  of  my  of  the  Columbia,  reference  copying  copying  Columbia  S e p t e m b e r 9 . 1968~  fulfilment  available  Physics  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a  of  extensive  purposes  shall  partial  of  this  Department  or  without  requirements  for  I  the  agree  and  study.  thesis  or  publication  my w r i t t e n  by  of  that  I  an  further  for  scholarly  his  represen-  this  thesis  permission.  for  . The  .-  infinite  ABSTRACT order  perturbation  theory  and  coworkers i s used, with  the  theories of classical irreversible  Bogoliubov, Sandri  of Prigogine  some m o d i f i c a t i o n s , t o d i s c u s s p r o c e s s e s due t o  & Frieman, and Mazur & B i e l .  The  latter  a u t h o r s u s e t h e BBKGY h i e r a r c h y  o f equations as a  point.  these t h e o r i e s the i n f i n i t e  order  Accordingly, perturbation  to discuss  theory  i s w r i t t e n o u t i n s u c h 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 . of the assumptions involved and  Sandri  pared with  r e l a t i o n of the theory  order  c l e a r when com-  perturbation  o f Mazur & B i e l w i t h  e x p a n s i o n o f Green i s a l s o  The n a t u r e  i n the theories of Bogoliubov  & F r i e m a n become p a r t i c u l a r l y the i n f i n i t e  starting  elucidated.  expansion. the cluster  The  iii TABLE OF CONTENTS Abstract Table  .  .  .  o f Contents  . . . .  •  .  •  .  .  .  .•  .  .i  .  . i i i  Acknowledgements I.  Introduction.  II.  The L i o u v i l l e  i  iv .  .  .  .  1  .  E q u a t i o n and Reduced Functions  .  .  .  .  .  .  .  .  6  I I I . The D i a g r a m T e c h n i q u e .  .  .  .  .  .  .  .  .  10  Distribution  The  BBKGY H i e r a r c h y o f E q u a t i o n s  . . .  14  I V . The D e p e n d e n c e 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 . V. The S h o r t - R a n g e E x p a n s i o n The  .  .  .  .  .  .  .  .  .  .  31  .  .  .  .  31  Hierarchy of Equations f o r  the C l u s t e r Expansion  .  .  .  40  V I , The M a r k o w i a n A p p r o x i m a t i o n VII. VIII.  . 4 8  The T h e o r y o f B o g o l i u b o v . The M e t h o d o f E x t e n s i o n  .  .  I X . The T h e o r y o f M a z u r a n d B i e l  Bibliography..  .  Appendix I  .  .  .  .  .  .  .  .  . . . . . . . .  65 74 82  X. C o n c l u s i o n  . Appendix I I  18  •  .  83 84 87  iv ACKNOWLEDGEMENTS I wish t o express my a p p r e c i a t i o n t o Dr. L u i s de S o b r i n o f o r s u g g e s t i n g t h i s problem and f o r h i s continued guidance and a d v i c e . Financial assistance  i n the form o f a graduate f e l l o w -  s h i p from t h e U n i v e r s i t y o f B r i t i s h Columbia and a Students h i p from t h e N a t i o n a l Research C o u n c i l o f Canada i s g r a t e fully  acknowledged.  INTRODUCTION. S e v e r a l a t t e m p t s h a v e b e e n made t o u n d e r s t a n d versible In  processes  s t a r t i n g from t h e L i o u v i l l e  remained c o n t r o v e r s i a l . clarify  Bogoliubov's  apply i n f i n i t e recent and  equation.  (1946) Bogoliubov"*" p r o p o s e d a n o v e l a n d 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.  to  irre-  H i s t h e o r y has however  Our c e n t r a l a i m i n t h i s work i s e x p a n s i o n method.  To do t h i s  we  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  (1959) d i a g r a m t e c h n i q u e s d e v e l o p e d b y P r i g o g i n e 2  coworkers."  the i n f i n i t e 3 Severne.  I n p a r t i c u l a r we u s e w i t h some m o d i f i c a t i o n  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 o u t by  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 o n 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. s 1  expansion. of  In this  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  S a n d r i & F r i e r n a n ^ ' ^ a n d M a z u r &. B i e l . ^ •Efforts to establish  possible  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  connections  between  order perturbation theory  b e g a n a e a r l y a s (1961) b y P r i g o g i n e a n d R e s i b o i s . ^  The (1965),  p r o b l e m h a s 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  B r o c a s & R e s i b o i s ^ (1966), a n d B r a u n & G a r c i a - C o l i n These a u t h o r s u s e t h e M a s t e r E q u a t i o n 1  of  (1966).  a p p r o a c h 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 s Bogoliubov  1 0  expansion.  s t a r t s b y d e r i v i n g t h e w e l l known BBKGY h i e r a c c h y ^ "  equations.  F o r t h i s r e a s o n we c h o o s e t o w r i t e o u t t h e  infinite  order p e r t u r b a t i o n expansion  i n s u c h a way t h a t i t  relates  e a s i l y t o t h e BBKGY h i e r a r c h y a n d i t i s o u r o p i n i o n  2.  that these expansions elucidate Bogoliubov's approximation more c l e a r l y than does the Master Equation approach. We w i l l be discussing a c l a s s i c a l system of many p a r t i c l e s . . We s p a c i a l i z e to consider systems of i d e n t i c a l p a r t i c l e s which interact through a central two body p o t e n t i a l with no external f i e l d s on the system.  Our s t a r t -  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 L i o u v i l l e equation.  In Chapter II we form-  a l l y solve the L i o u v i l l e equation.and write the answer as a perturbation s e r i e s .  Since i t has shown considerable  promise i n the l i t e r a t u r e ; we introduce f i r s t - ,  second-,  and higher-order d i s t r i b u t i o n functions defined by integrat i n g the solution to L i o u v i l l e ' s equation over coordinates and momenta of a l l but the p a r t i c l e one, two, etc. To s i m p l i f y our method of writing these perturbation series we introduce i n Chapter III our d e f i n i t i o n s of diagrams which are similar to those of Prigogine and coworkers.  In  t h i s Chapter I I I we perform some s i m p l i f i c a t i o n s and.we f i n d that our expansion relates e a s i l y to the BBKGY hierarchy of equations which couple the reduced d i s t r i b u t i o n functions. Also i n Chapter III we introduce the Cluster expansion"*""*" of the  d i s t r i b u t i o n functions/-which we f i n d p a r t i c u l a r l y useful  to expand the s t a t i s t i c a l i n i t i a l condition needed f o r the  solution of the L i o u v i l l e equation. In successful theories of i r r e v e r s i b l e  statistical  mechanics the dependence of the d i s t r i b u t i o n functions  u p o n t h e one nant r o l e . in detail; ular we  body d i s t r i b u t i o n f u n c t i o n has I n C h a p t e r I V we  we  played a  work o u t t h i s t h i s  dependence  o b t a i n t h e d e p e n d e n c e by p e r f o r m i n g a  summation o f d i a g r a m s .  show t h a t t h i s  In the l a t e r  partic-  part of the chapter  summation i s n e c e s s a r y t o d e t e r m i n e  long time behavior of the system.  one b o d y d i s t r i b u 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 the theory t o consider  where t h e p a r t i c l e s  range  We  a n d we  call  such systems  collision effects.  g u i d e f o r s e l e c t i n g and  systems  i n t e r a c t through short short-range  c a r r y out the. c a l c u l a t i o n s . t o  three particle  interact  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  o r f o r systems forces.  the  function.  f o r which o n l y small c l u s t e r s of p a r t i c l e s  simultaneously.  the  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 determined through  systems  domi-  We  systems,  include a l l  two  and  find that a valuable  summing d i a g r a m s  i s the  hierarchy  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  couple  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 C h a p t e r V by d e r i v i n g t h i s  hierarchy  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.  T h e s e e q u a t i o n s 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.  . The  e q u a t i o n s o f C h a p t e r I V and V a r e n o n - M a r k o w i a n .  A method f o r f i n d i n g w e l l b e h a v e d M a r k o w i a n 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  determined  i n the l i t e r a t u r e .  a Markowian  I n C h a p t e r V I we  produce  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 m e t h o d s found i n the l i t e r a t u r e .  We  p o i n t o u t some a s y m p t o t i c  4, divergence d i f f i c u l t i e s In  Chapter  of this  method.  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 .  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 . from t h e knowledge o f t h e former  Then we  follows:  e x p a n s i o n , a n d we s o l v e o u r e q u a t i o n s  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 . if  develop  c h a p t e r s an e x p a n s i o n  m e t h o d t h a t we f e e l most n e a r l y a s p o s s i b l e Bogoliubov's  We  side  What we do i s d e t e r m i n e <  i ti s p o s s i b l e t o time coarse g r a i n the 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  grain  behavior.  We d e t e r m i n e  the necessary r e s t r i c t i o n s  on o u r  expansions  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 Bogoliubov's asymptotic boundary c o n d i t i o n s . We f i n d h o w e v e r 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 be  cannot  found. In  Chapter  Frieman^'^  which  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 & 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 another approximation t h a t can be o b t a i n e d f r o m o u r c o a r s e g r a i n i n g p r o c e d u r e in  Chapter V I I . In  Chapter  The r e l a t i o n Green  I X we d i s c u s s t h e t h e o r y o f M a z u r &  of their  Biel.^  theory t o the Cluster expansion of  i s elucidated. In  C h a p t e r X we g i v e some c o n c l u d i n g  F i g u r e I i s a s y n o p t i c diagram of  introduced  this  thesis.  remarks.  of theorganization  5. Liouville's and r e d u c e d functions  equation distribution  E x p a n s i o n i n terms of P r i g o g i n e type diagrams  Our d e f i n i t i o n o f d i a g r a m s r e l a t e e a s i l y t o t h e BBKGY hierarchy of equations  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 time behavior i s determined by t h e o n e b o d y f u n c t i o n  S e l e c t i o n a n d summing o f diagrams f o r a system which i n t e r a c t through a s h o r t range i n t e r p a r t i c l e potential  C a l c u l a t i o n o f t h e Markowian equations f o rthe d i s t r i b u t i o n functions  Cluster method  expansion  The t h e o r y o f Mazur and B i e l is similar to the Cluster expansion method  We determine i f i t i s possible to time coarse g r a i n our f u n c t i o n s by a.-certain approximation of the fine grain b e h a v i o r w i t h a subsequent study o f p e r s i s t a n t effects  The t h e o r y o f B o g o l i u b o v approximates t h i s procedure  Figure! I .  The M e t h o d o f E x t e n s i o n approximates t h i s procedure  II.  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 t h e b a s i c  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 equation.  C o n s i d e r 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 . of t h e system i s d e s c r i b e d "(*• , ^'t £1 )<£ • • -j  by a d i s t r i b u t i o n  and v e l o c i t y o f p a r t i c l e [•  (2.1)  state  function  where oc^ or are the r e s p e c t i v e p o s i t i o n  ;  normalized  The s t a t i s t i c a l  The d i s t r i b u t i o n f u n c t i o n i s  by  J " - J ^ «  ^ t f i  F= N  I  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 a symmetric f u n c t i o n of ( ^ ^ j j ( ^ / J . . . f ^ , ^ ) • u t i o n of  F  (2.3)  i n time i s governed by the L i o u v i l l e ~H  (2.2) where  N  K  N  F"  - f K"-I )  and I " are o p e r a t o r s  W  K = £. Ki = i-l w  w  F  The e v o l equation  N  d e f i n e d by  XI-vrhi-  here m i s t h e 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 t h r o u g h a c e n t r a l two body p o t e n t i a l \Jlj (ftfj on the system.  and t h a t t h e r e  are no e x t e r n a l f o r c e s a c t i n g  And, s i n c e we are i n t e r e s t e d o n l y i n the  b u l k p r o p e r t i e s of t h e system,, the w a l l o r c o n t a i n e r  7. . p o t e n t i a l h a s been n e g l e c t e d . Liouville's  The f o r m a l  solution of  equation i s  (2.5)  F"« e-' "- "  We f i n d  t h a t a more u s e f u l f o r m o f t h e s o l u t i o n i s t h e  K  perturbation  (2.6)  series  1  F-^O)  expansion  F = e- "*F-fO.) w  )t  K  +  r  K  *pt  N  K i  e  "*'iV "*'^(0) K  +• e"  The  reduced  functions  F'(i.,«,...,*„« t).J-/^'"^ , • • • 4 f » ^  (2.--0 are  s-particle distribution  ;  also introduced.  integrals  w  a r e t o be a l w a y s u n d e r s t o o d t o e x t e n d o v e r t h e  range o f t h e i n t e g r a t i o n v a r i a b l e s .  give  thep r o b a b i l i t y density  s particles  considered  infinitesimal  (2.7)  « AT I}  . .  being  t j  tt ,tr<, s  ..(continued)  functions  located, respectively, i n the  a t the time t .  and t h e p e r t u r b a t i o n  These  f o r t h e dynamic s t a t e s o f t h e  phase volume elements  we h a v e (2.8)  F"  I n ( 2 . 7 ) t h e p o s i t i o n and v e l o c i t y  full  points  N  dockets,...  Using  a r o u n d .the the definition  s e r i e s (2.6), together  with (2.4)  (2.8)  F =j ^ - ' ^ s  8  S  t<j  ^•'"Clt,,*"*. £  At fying  this point  relation.  shifting  •>  I^-t^Cit^^H  I  i t i sconvenient to introduce  L i o u v i l l e ' s equation  s y s t e m t o be i n s e n s i t i v e  (2.9)  J <t*i e~  K 4  a simpli-  ( 2 . 2 ) assumes t h e  (2.10)  J </*j e ' J ' G f e - ^ j ) '-  Under t h i s  &(<*;) = j d*c G  K I  e- " 'F"(0)  t o i t s b o u n d a r i e s and t h u s t o t h e  o f i t s boundaries. Hlt  '  c o n d i t i o n we h a v e o~y, ^eM U L t J l G fe) -<*j)  f  because (2.11) in  theexponentials  (2.12) and  K- - Yi'^.  ,  produce a T a y l o r  e' G(*;)  G(  Hli  theintegrals  K ; - ^.f/Tj  simplify  expansion  Vi-ATii)  by s h i f t i n g t h e b o u n d a r i e s o f t h e  system. Note (2.13) and  e  that  K w t  that using  I:j e "  K w f  (2.9) the p e r t u r b a t i o n  written (2.14)  = e ^ I j .e^'J*  ...(continued)  s e r i e s ( 2 . 8 ) may be  9. (2.14)F  s  = j ^  t  •  S  ' ^  i  . . . ^  l  <  /  t  f  f  ••I  where (2.15)  K S  We. w i l l  £ write  t h e terms o f t h e e x p a n s i o n (2.14) i n a  f o r m c o n v e n i e n t f o r t h e d i a g r a m t e c h n i q u e t o be i n t r o d u c e d in  the following  section.  by an e x a m p l e .  This point w i l l  be  Consider the contribution  illustrated  to  F  (2.16)  J*it e «9*»  The  spatial  far  as p o s s i b l e  integrations  e - ^ k  P (o)  K i ? t 3  w  a r e commuted f r o m l e f t  t o r i g h t as  and u s e d t o p e r f o r m any r e d u c t i o n on t h e F ( 0 ) w  t h u s o u r example (2.17)  f'  K  3  K  (2.16) w i l l '  2  t  a  i  2  be w r i t t e n  e ^ ^ i t  2  ( 7  $  ^  8  e ^  i  2  I ,  ?  e - ^ ^  III.  THE DIAGRAM TECHNIQUE. As  an a i d i n w o r k i n g  w i t h t h e s e r i e s (2.14) t h e diagram 2  technique  o f P r i g o g i n e and c o w o r k e r s  modification. form s i m i l a r  be u s e d w i t h  some  F i r s t we w r i t e t h e t e r m s o f ( 2 . 1 4 ) i n t h e to (2.17).  We  see. t h a t e a c h t e r m s t a r t s  -/< f  with  -K*t  s  6  and we w i l l  the  -will  write this, f i r s t  s sets of p a r t i c l e  s l i n e s running  explicitly. of F  coordinates  from l e f t  we d r a w and l a b e l  F o r example F  to right.  function of the coordinates  s  isa  o f two p a r t i c l e s ' a n d , 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 l e f t with -K t  For  s t a r t s on t h e  z  (3.1)  e  We f i n d  t h a t the mathematical  be  z  b r o k e n down i n t o  of these \« „ , c The  three d i s t i n c t i v e  c o n t a i n s an e x p r e s s i o n  J-ijC  of a general building  building (a).  of v e r t i c e s corresponding  blocks w i l l  used, l i n e s  l)  m  j  j  to a vertex.  to the three  the labels  i and j h a v e a l r e a d y  l a b e l e d i and j a r e a v a i l a b l e and we  DC'Ut,C J  °  K  ,  Each  distinc-  of the expression  them d e f i n i n g a v e r t e x (3.2)  blocks.  be d e f i n e d a s f o l l o w s :  i f to the l e f t  je/<«,e^^I-je  term can  of the form  w h i c h i s made t o c o r r e s p o n d  three types  tive  behavior  'J*"ri;eJ  K  ,  J  t  B  been cross  As an e x a m p l e , procedure (3.3)  f o r t h e term  (2.17),  using the s t a r t i n g  ( 3 . 1 ) , we d r a w  e - ^ D C T J ^ i ^ ^ e ^ r , , ^ * * 2  (b).  •2  o  J  i f to the l e f t  j dt e M  "Iij  e  of the expression o n l y one o f t h e l a b e l s  i or j  ( s a y i ) h a s a p p e a r e d , we d r a w a b r a n c h on t h e available the  integration  o v e r t h e new l a b e l  1—CZ-Af^i^lXr^*-!,.*-^'-  (3.4)  The For  l i n e ( i ) and d e f i n e t h e v e r t e x t o i n c l u d e  example  symbol A w i l l  be e x p l a i n e d  t h e second b u i l d i n g  shortly.  b l o c k o f (2.17) i s o f t h i s  t y p e and. we d r a w (3.5)  e'  K  *pC==  ] < * ) V - ^^J-y" S  *-  e  ^  12 (c).  i f to the l e f t of the expression  cli^e we  Ii; C  }  *" neither i nor j has appeared,  introduce two new  l i n e s running to the right  through a vertex defined by <±  (3.6)  - \ % * *  \ ! ± U  t  v  j t / ' i ^ j e  i  The example (2.17) can be represented now  (3.7)  e"  l  3C_  K  F  f  i  m  by  (o)  It i s noted that time labels are not needed i f the v e r t i c e s are ordered from l e f t to right to indicate the respective l i m i t s of the time integrations  t ^ * ^ t >/t • - - ^ 0 (  x  }  The expansion, in terms of diagrams s i m p l i f i e s greatly because diagrams containing a type (c) vertex integrate to zero or one writes (3.8)  <  j  H  = O  We have relegated this calculation to our Appendix I . . Also in Appendix I we f i n d that for a diagram with a (b) type vertex, where a p a r t i c l e label j f i r s t appears, the J^. J-Vj part of the one writes  I -j t  in the vertex also integrates to zero, or  13. We w i l l now e x p l a i n let  the symbol A.  .  Working from an example,  us c o n s i d e r a l l the c o n t r i b u t i o n s  to F  that have the  form  There are (N-2)(N-3)(N-4) diagrams o f t h i s form corresponding to the d i f f e r e n t ways of choosing the dummy l a b e l s  }> ]'> j"  and we see that these diagrams are i n d i s t i n g u i s h a b l e . general,  In  t h i s type of redundance occurs wherever (b) type  v e r t i c e s appear.  And, one sees that diagrams that  differ  o n l y . i n t h e i r dummy p a r t i c l e l a b e l s can be summed by w r i t i n g i n the (b) type v e r t i c e s (3.11)  A=N-r  '  r= number of l i n e s to the l e f t of the  This  .  vertex ( b ) .  concludes the d e f i n i t i o n s of our. b a s i c v e r t i c e s which we  summarize here, [  (3.12)  (a) ZXZ j  I  j  -U-i = }  °  Ji*^*"  Ir  e- 'J*" K  J  (3.13)  where r i s the number of l i n e s at the l e f t of the vertex ( b ) .  The  BBKGY H i e r a r c h y  The 1  , Sandri  These authors  at this  ' ' and Mazur & B i e l .  t h e BBKGY c h a i n  using  I t also  6  of equations  to d e r i v e  this  as  chain of  supplies  us a good  example  d i a g r a m s and d i a g r a m  language.  For  5  tr  r  we s e p a r a t e  contributing the  point.  the t h e o r i e s of  4 5  I t seems a p p r o p r i a t e  a calculation  a given  & Frieman  use o r d e r i v e  step.  equations of  14.  Equations.  aim o f t h i s work i s t o d i s c u s s  Bogoliubov  a first  of  diagrams.  leftmost e  out the f i r s t This  gives,  vertex being  only  of the  careful  f o r each term o f the p e r t u r b a t i o n  to r e t a i n series  (2.14),  (3.14)  F = -^F<(oUe-^H  ^  i<jiS  i l l .  s  e  5  Here,  the p o i n t s  connections We r e c o g n i z e <°"  1  that  except  Therefore,  i n the second  (3.15)  f o r the operatore  i f the f i r s t  ...(continued)  H ii S 5  make a l l p o s s i b l e v e r t e x .  including possible  , a l l p o s s i b l e connections  term,  ^  :"  i n d i c a t e the l i n e s  to the r i g h t  "  vertex  term,  except  initial  values.  f o r the operator  d e f i n e s F (*,) and i n t h e t h i r d ' they  define  i s written  in detail  one has  IS.  (3.15)  F - e~ s  By w r i• ti i• n g  5±  +  I;;?  e  s  a net ie ->K; Sl+ I-^ I T <?  ~K t, s  .^''IT e  =  a  1  -  F (r,)l  K l  S+I  t  'and u s i n g t h e r e l a t i o n  FI  -.-ft sti^i-  -  s t |  (2.9) we o b t a i n  f  £J  (3.16)  F*(o) e-* JZ (d*. c J * ' I , . e ^ ' i <<[e  Kr;"t|7J  n  X ^^  KH  ;  F ^ e ^ F ^ . e - ^ J A e ^ ' l 'JZ  I F'(f.j £J  T h i s i s an i n t e g r a l form o f t h e BBKGY c h a i n (3.17)  -  ^  W  F  '  -  Z  I r  £  I  F  *  S  O f t e n a s h o r t 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 (3.18)  +  V F ~" = L F -*' s  s  S  H '-  5  5  (  ^  ^  chapters  K -I s  5  where (3.19) (3.20)  I = H 5  L =H 5  I: L  L-  L.  -  ^ L [  d  l  s  „ ^ , ^ - & .  Return now t o the g e n e r a l p e r t u r b a t i o n e x p a n s i o n . f i n d t h a t one key t o making the expansion u s e f u l i s t o introduce the c l u s t e r expansion"^ (3.21) (3.22)  F'=  F'i*,^,)  F ^ f f ? ^ ) ^ ^ ) ^ ^ ? ! / ^ ! / * )  We  I  ^  I 6. (3.23)  F = F/(<X,,4r.)F^of>,'iy;)F,7?j(f,)+ F / t e . ^ G ^ f * , , ^ , l i ^ i ) 3  J  +  ) G*  y,,  +• F/fe,#)G* f4T,  f  %, t  F - •• • 4  -5  The  b are interpreted  distances is  a s d e p e n d i n g on i n t e r p a r t i c l e  and a r e 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  expansion  u s e d h e r e t o e x p a n d t h e v a r i o u s F (OJthat a p p e a r o f t h e  r i g h t o f each term i n t h e g e n e r a l e x p a n s i o n o f . C o n s i d e r f o r example t h e c o n t r i b u t i o n t o -K'i i ' (3.24) e < , i F (0) • I 3  F (t). S  F'  3  2  A c l u s t e r e x p a n s i o n o f F V o ) i s made and t h e i n i t i a l ations  are represented  diagrams. (3.25) for  by d o t t e d  correl-  l i n e s at the r i g h t of the  Defining F'(O)  ~-  the i n i t i a l  1 v a l u e o f t h e one body f u n c t i o n  ( 3 . 2 4 ) may be w r i t t e n  following  t h e example  the decomposition  (3.23)  17. Our method o f r e p r e s e n t i n g in  the general  perturbation  series  t e r m s o f d i a g r a m s i s now c o m p l e t e . In a l l t h e f o l l o w i n g  called  thermodynamic  (3.27)  N—>°= V— °  s e c t i o n s o f t h i s work t h e s o -  l i m i t w i l l be t a k e n such t h a t ^ - n  > Q  j  T h i s has t h e e f f e c t t h a t i n each  (3.28) The  $ = ^  thermodynamic  —  limit.  (b) type  constant.  vertex  «  limit  should  of t h e system which a l l o w s this  remains  not a f f e c t the bulk  us t o assume c o n v e r g e n c e  properties under  IV.  THE DEPENDENCE OF THE DISTRIBUTION  FUNCTIONS UPON THE  ONE BODY FUNCTION. In"the  successful theories of I r r e v e r s i b l e  mechanics t h e dependence o f t h e d i s t r i b u t i o n upon t h e one b o d y , d i s t r i b u t i o n dominant r o l e . of  In t h i s  the d i s t r i b u t i o n  functions F  F' i n . d e t a i l .  upon  s  thermodynamic l i m i t  (3.27) w i l l  t h e d e p e n d e n c e on F  be a s s u m e d  that this  summation i s n e c e s s a r y  time behavior of a p h y s i c a l behavior of a p h y s i c a l body  system.  system  and t a k e n .  a particular  the expansion  F= e - ^ i ; e-K*ZD 2  +  <> a  A  +  to determine  the long  That i s , the long  i s . determined  through  (fe)  +  f ^ X 3  (c)  .  (<{)  3  N  a)  f»;  o f t h e two body  c - ^ X l  1  3  -  2  if)  '  sum  c h a p t e r we p o i n t  time  t h e one  function. Consider  ^ (4.1)  valid  by p e r f o r m i n g  1  The  be t h a t t h e  d i a g r a m s and i n t h e l a t e r p a r t o f t h i s  out  F, F , F  c h a p t e r we work o u t t h e d e p e n d e n c e  a general theory w i l l  of  functions  F' h a s p l a y e d a  function  o n l y . d e p a r t u r e from  We f i n d  statistical  function  . 1 9For  convenience we d e f i n e  a diagram — O  f o r the one body  function (4.2)  /  +  F ' = - ^  =  +- o  H^-C;  + -oc;  4 - o c  3  3,  •  + -c-X  +-c  H  3  ^—; + -<  27  -f — C ~ T " :  i  + HI; + ~TI; CD  -C  I  i  4  +  + 3  I t w i l l be p r o v e d p r e s e n t l y  Ui  "TE +• • -  3,  t h a t the two body f u n c t i o n  be w r i t t e n (4.3)  •F =e2  K  *  -K t 2  may  20. where i t i s u n d e r s t o o d t h a t i f a — O it  i s a f u n c t i o n of time through the i n t e g r a t e d time  of t h a t v e r t e x .  In (4.1) the terms  gone t o c o n t r i b u t e t o t h e t e r m  f o r the s i m i l a r  distribution The  the term  In the next  give a systematic procedure f o r w r i t i n g  ( 4 . 3 ) , and  out the  systematic procedure  w i t h fragments  taken into  that connect  the p a t t e r n -*  expansion  i s as f o l l o w s .  The  diagrams  series  a c c o u n t by o m i t t i n g  i n by o n l y one  line.  i t contributes  to  diagrams For  example  ^lf  ,  -y~~Xf> 1  is  paragraph  functions.  expansion f o r — o i s  e  (o) -  expansions of h i g h e r o r d e r  a r e drawn as b e f o r e r e p l a c i n g — ( b y — o where t h e  (4.4)  variable  ( a , e , f , j , m , . . . ) have  ( a ' ) above and  h a s gone t o c o n t r i b u t e t o ( c ' ) a b o v e . we  connects to a v e r t e x  not kept because  , \  we  find  (4.5) i  T h i s method o f e x p a n s i o n can be p r o v e d v a l i d "factorization" The  theorem.  factorization  identical  F o r a g e n e r a l p r o o f see d e a l s w i t h diagrams  except f o r the l e f t  their vertices. permutation (4.6)  theorem  ~C  j  The  class. J  by use o f t h e  and  are  to r i g h t or "time" ordering  s e t of a l l such diagrams For  that  Resibois.  example  fi  /  i s called  of a  is  a permutation class.  (4.7)  I  J z  ( 4 . 6 ) we  ~ <~ I ^ J  z  x  *  z  »ti  reads:  on t h e i r  write  Ci  J  z  J x  m  f>  where t h e v e r t i c a l limits  theorem  The sum o f a c o m p l e t e p e r m u t a t i o n c l a s s i s equal to the product 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 t h e component s t r u c t u r e s .  F o r o u r example (4.8)  The f a c t o r i z a t i o n  n  L  -  -CU  /I  J  z x z z  z Oi  m  M  b a r i n d i c a t e s t h e v e r t i c e s h a v e t h e same  time i n t e g r a t i o n s .  In d e t a i l  (4.8) i s  written  and  since operators with labels  i a n d j commute w i t h  those  w i t h l a b e l s 1 and m, t h e component s t r u c t u r e s f a c t o r . get  (4.3) from  have f r a g m e n t s  ( 4 . 1 ) we c o n s i d e r a l l d i a g r a m s t h a t connect  We t a k e t h e p e r m u t a t i o n  i n by one l i n e ,  c l a s s formed  the other s t r u c t u r e s i n the diagram, we c o n s i d e r  To  of (4.1) t h a t  f o r example (4.4)  by t h e s e f r a g m e n t s  with  f o r t h e example (4.4)  11. When we  sum  over the permutation c l a s s e s  factor,  f o r t h e example  (4.11)  i r  ( 4 . 1 0 ) t h e sum  the  fragments  of the diagrams  — ,  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 same t i m e v a r i a b l e .  The  infinity  of diagrams  (4.4) except f o r d i f f e r e n t fragments vertex  ( a ) on t h e b r a n c h i g i v e s ,  permutation diagrams •—o  and  begin with  one  to o b t a i n ,  to r e c a l l  The  can use t h i s  except f o r i n i t i a l body d i s t r i b u t i o n  the c o u p l i n g  defines  expansion  scheme  function.  o f t h e one  equation  It is instructive  body f u n c t i o n t o t h e  two  (3.18)  we  find  t h a t the s o l u t i o n to  F'- e * ' * — , 4- e-^  J  -CZ  ee  one  c o r r e l a t i o n s a closed  has  F (tj 2  2  o b t a i n e d by s u b s t i t u t i n g t h e e x p a n s i o n (4.13),  (4.12)  follows  Except f o r i n i t i a l  into  of these  2  can be w r i t t e n as  is  (a)  jf+K'F'zil'F (3-. 14 o r 3.15)  (4.13)  sum  at  c o r r e l a t i o n s , a closed  body f u n c t i o n t h r o u g h t h e BBKGY c h a i n  Using  connect to the  by summing o v e r t h e '  line.  an a l t e r a t i o n we  t h e one  (4.12)  to  t h e e x p a n s i o n s.cheme o u t l i n e d a b o v e i s j u s t i f i e d . With  for  that  of the  identical  c l a s s e s , a l l p o s s i b l e ways o f d r a w i n g  that  give  equation f o r (4.3) f o r  F  F'  2 3. (4.14)  It  F'^e^n  +e-  i s recognized  +e*-OC  4e-' i -<j f  K,i  t  that the expansion  here  i n t e r m s o f t h e one  body f u n c t i o n f o l l o w s t h e same scheme as f o r t h e many functions one  i n equation  equivalent His  except  we a l l o w t h e l e f t m o s t  (b) t y p e  (4.13) t o be r e t a i n e d .  An  to the problem  similar  slight  difference  obtain  the dependence o f the d i s t r i b u t i o n  one  body f u n c t i o n .  initial on one  i s found  i s very  correlations.  the r i g h t  with  o r more — o ,  where t h e — O ' s  i n h i s summation  The answers d i f f e r  Severne  a r e . expanded  A  o f terms t o  i n t h e term that  correlation  h a s an i n f i n i t e as i n ( 4 . 2 ) .  Severne.  f u n c t i o n upon t h e  only  of i n i t i a l  by  to ours.  Where we have a d i a g r a m  some p r o d u c t  vertex, the  expression  t o (4.14) has f o r m e r l y been o b t a i n e d  approach  particle  series  with  terminates with  o f terms  24. In  the f o r e g o i n g paragraph  with  fragments  into  the v a r i o u s — O  "indicate" determine  t h a t connect  that this  .  the long time  we  From  i n by one  b e h a v i o r of the correlations,  and  we  will  couple of r e p r e s e n t a t i v e terms —O  by  summing them  system.  We  assume t h e  take the  we  will  = -H + ...+ — O C ! +  and  here  to  will system  asymptotic  ( 4 . 1 4 ) , by n e g l e c t i n g i n i t i a l  have a c l o s e d e q u a t i o n f o r — O ;  (4.15)  line,  diagrams  summation of d i a g r a m s i s n e c e s s a r y  i t s past history  behavior.  r i d o u r s e l v e s of  T h r o u g h some a p p r o x i m a t i o n s  n e g l e c t terms w i t h i n i t i a l forgets  we  correlations, we  consider a  25. By t a k i n g (4.16)  t h e time d e r i v a t i v e ,  yr--nf^^e^I e-^  one f i n d s  { -  t  a  +  + Equation evolution its  ( 4 . 1 6 ) i s s a i d t o be n o n - M a r k o w i a n  because t h e  o f t h e one body d i s t r i b u t i o n a t t i m e t depends on  values f o r a l ltimesT<t.  assume t h e s y s t e m " f o r g e t s  F o r o u r p u r p o s e s h e r e we w i  i t s past h i s t o r y "  by m a k i n g a  crude Markowian a p p r o x i m a t i o n o=e *?i{Q  (4.17)  so  that  e ^ ' M  Ktt  =  ~  «  ( 4 . 1 6 ) i s a p p r o x i m a t e d by  (4.18) ^ ^ n j ^ ^ e ^ l ^ e ^ ^ f  ...  X  +  .:v; i....'; x  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 t h e a s y m p t o t i c  jf7"l^it^{--*  (4.19)  © where (4.20) (4.21)  Jo  X i<4  -_  w  »  i  6  p +  M  p q ^  l  J  J e  e  K  i  J  f  M  J : - e -  ,  r  :  J ' ' f  behavior  (4.22) For  eW  c o n v e n i e n c e we a l s o  define  (4.23) so t h a t we c a n w r i t e  (4.19) as f o l l o w s  - if-.- -cDc:*-<62S+-  (4  24)  +  F i n a l l y we t a k e t h e T a y l o r  (4.25)  , - £ { . . .  +  _ < x ;  expansion of  —&  + . - < ^ 2 ^ 4 . .  (4.25) has r e p l a c e d t h e .expansion o f (4.15) w h i c h  i s by-  iteration  (4.26) _  0  r  _  1  +  | .  :  where i n t h e terms  .  +  ^  c  x  ;  +  - < ^ ^  call and  By c o m p a r i n g  i s found t h a t each f r a g m e n t  one  . . .  ]  ( x ) and ( y ) t h e v e r t i c a l  complete permutation c l a s s . it  +  line  (4.25) w i t h  o f (4.26) t h a t  l i n e g i v e s a time divergence or order t . t h a t terms w i t h  initial  c o n n e c t s i n by (We s h o u l d r e -  terms.)  o r d e r t o d i s c u s s t h e l o n g t i m e b e h a v i o r we must  sum o v e r t h e d i v e r g e n c e s t h a t we o b s e r v e d paragraph.  (4.26)  c o r r e l a t i o n s h a v e been n e g l e c t e d  t h e same may n o t be t r u e f o r t h o s e In  indicates a  And we s e e t h a t t h i s  i n the former  i s done by s e p a r a t i n g o u t  28. and  summing, w h e r e i t a p p e a r s , t h e i n f i n i t e  This i l l u s t r a t e s , b y t h e one now in  t h a t the long time behavior i s determined :  particle distribution function.  i f an a r b i t r a r y time.  T h i s has  t h r o u g h an H t y p e We detail  will  selection,  (4.15),  say  cases  d i s c u s s t h e M a r k o w i a n a p p r o x i m a t i o n i n more  We  of the i n t e r p a r t i c l e p o t e n t i a l .  c o n t a i n s an i n t e g r a t i o n i s limited  sphere.  (4.27)  (b) t y p e v e r t e x potential,  of the p o t e n t i a l ,  For the v e r t i c e s  ZXZ.  ~  we  interaction  have t h e o r d e r s o f magnitude  C(£)  - < Z  of t h i s t h e s i s  t h e e x p a n s i o n i n s m a l l A w h i c h we  we  and  eachi(b) type vertex gives  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  For the remainder  theory.  Each  o f the range  since the d e n s i t y n i s a f a c t o r  the strength  Each v e r t e x  over t h e . i n t e r p a r t i c l e  by r a n g e  order  and t h u s g i v e s a m e a s u r e  of the s t r e n g t h of the p o t e n t i a l .  a measure  w h i c h i s an  roughly approximate  contains the i n t e r p a r t i c l e p o t e n t i a l  also  converges  later.  range  which  o f terms  theorem.  of magnitude c a l c u l a t i o n .  £  I t i s n o t known  o n l y been v e r i f i e d f o r s p e c i a l  T h i s b r i n g s us t o our n e x t t o p i c  and  F  s e r i e s — o -e  villi  call  ~0(£X) be w o r k i n g w i t h the  short-range  V.  THE SHORT-RANGE EXPANSION In t h i s chapter  systems f o r which only simultaneously.  small  we s p e c i a l i z e clusters  Such a s i t u a t i o n  range f o r c e s . we w i l l  be w o r k i n g w i t h  systems  through  short-  of magnitude c a l c u l a t i o n  the expansion i n small  task  of t h i s chapter  i s to select  e x p a n s i o n i n s m a l l A, and t o o b t a i n we w i l l  sum t h e v a r i o u s  infinite  demonstrate the c a l u c l a t i o n s carry out the calculations icle The  interact  for dilute  interact  those  (4.27)  A  A « I  (5.1) The  From t h e o r d e r  of p a r t i c l e s  occurs  or f o r systems i n which the p a r t i c l e s  to consider  collision, Hierarchy  (5.2)  t h a t appear.  answers We and we  t o i n c l u d e a l l two a n d t h r e e  of Equations f o rthe C l u s t e r  part-  Expansion  guide f o r s e l e c t i n g  of equations,  that couple the c l u s t e r  similar  coefficients  and summing  t o t h e BBKGY (3.21 t o  Our p u r p o s e 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  hierarchy various  series  compact  o r c o r r e l a t i o n , effects..  diagrams i s the h i e r a r c h y  3.23).  reasonably  f o rthe expansion of F ;  We f i n d t h a t a v a l u a b l e  hierarchy,  diagrams f o r the  of equations.  cluster  We w i l l  coefficients  -K t s  2  G  this  i d e n t i f y by d i a g r a m s t h e  30. where the v e r t i c a l  bar i n d i c a t e s the sum of a l l diagrams  that have the s l i n e s connected i n some way..  We see that  t h i s i s a p o s s i b l e way to w r i t e the G , f o r example f o r F $  3  we w r i t e  (5.3)  The  F'-e-"'*^**'" *^  $  Further,  +  +  (5.2) i s the only s e l e c t i o n of diagrams f o r  because the system of equations (3.21 to 3.23) y i e l d s a  unique s o l u t i o n f o r the G  5  i n terms'of the F . S  c a u t i o n here;  the expression  identification  (5.2) unless  has  been taken.  important property function J in  e - ^ i ^ e"^T  3  f a c t o r i z a t i o n theorem (4.7) has been used e x t e n s i v e l y  here.. G  *V* *T-°  3  A word of  (3.11) w i l l not allow the  the thermodynamic l i m i t  (3.27)  I t i s convenient here to p o i n t out an of the b .  We term as f a c t o r a b l e any  of s p a r t i c l e coordinates  that can be w r i t t e n  a form  (5.4)  J^*,^...,*,^) = X ^ . & . t i  JT ('*i>,yi<, r  zyj^y,..)  We f i n d that the diagrams t h a t c o n t r i b u t e to G s a r e not i n general being  f a c t o r a b l e i n t h i s form;  since the s s t a r t i n g l i n e s ,  connected i n some way, always have an overlap  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 proceed to d e r i v e the h i e r a r c h y  correlations.  u t i n g diagrams to o b t a i n  We  of equations f o r t h e G ; we  do t h i s by example f o r the case of G? . the expansion f o r C? the f i r s t  of the  We separate  out i n  vertex only of the c o n t r i b -  Where a p r o d u c t o f d i a g r a m s means t h e d i a g r a m s f o r m e d by connecting  t h e 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 .  s l i g h t , r e a r r a n g e m e n t we s u b s t i t u t e of  After  i n (5.5) the d e f i n i t i o n s  the diagrams t o o b t a i n  (5.6) G ^ e - ^ ^ f e - ^ f ^ e ^ ' I  3  ^ ^ , )  4.  +  t A W f £ W C (ij F '(VJ +  + A  L  + A I G«(*,)} 3  4  M + G»ft)  ft)+(0  ^ ft)]  32-  Or by t a k i n g the time d e r i v a t i v e one has  (5.7)  $ W * IG ' - InfcGh  t A  L [ F3 G ^ y H  f A  And  + FiGl]  fc  F ' £ ^ 4 G,j C" 4 C ^ ] 4  4  4  t h i s may be w r i t t e n i n s h o r t hand, as V F ^ X i T ^  (5.8)  where I F 3  J  i s the produce of I  3  * a n d  the c l u s t e r  expansion  of F from which we r e t a i n only the n o n - f a c t o r a b l e , i n the sense  (5.4), terms e x c e p t i n g X G which i s w r i t t e n s e p a r a t e l y  on the l e f t  hand side of the equation.  Similarly  the product of L - YL ^-i/i and the c l u s t e r expansion 3  L F * Is ?  of  F" 4  i( $  r e t a i n i n g only the n o n - f a c t o r a b l e terms.  Our d e r i v a t i o n of  (5.8) r e v e a l s the g e n e r a l i z a t i o n (5.9)  ^f- 4K G -r G = s  S  5  s  s  r F ' + ApF*'  Though these equations are more complicated are w r i t t e n than the BBKGY h i e r a r c h y (3.18), (5.10)  UJ*n*  1  i/S s rJ K F -TIS F = >A1L5 F s+ R S  S  5  5  5  r-  1  i n the way they  33. the  equations  (5.9)  see  that  may-be  they  cancelling equations  the (5.9)  these  actually  obtained  redundant  much  factorable  reference  we  simpler  from the  have f o r m e r l y  For' c o n v e n i e n t of  are  been  because  BBKGY h i e r a r c h y  behavior. obtained  tabulate  we  here  The  by  the  by  Green."'""'' first  three  equations,  (5.11) (5.12)  ^ K V - I ' G S  (5.13)  ^ K ^ - r G ^ I . j F / G ^ f F ' G J ]  Let for  the  begin  our  short-range  culations three  us  for  the  particle  theory.  collision,  Our  F ^ F  (  problem has  (5.12)  the  or  one ' F  a  f G  summing o f  F  to  diagrams •  demonstrate include  correlation,  all  + Fi&a)  the two  effects.  caland By  has  2  reduced  solution  and  We w i l l  expansion of  _ , ... (5.14)  selection  I'F/Fif AL.jF'Gi  to  expanding G  ;  we h a v e  from  34. G -~ e-^*GVo) + e-^*j[Jftc^f/"i2GYt)  (5.io)  2  +  1  2  4AL [F;WG V0 4 F J ( ' i J G ^ ) ] 2  )3  Or  i n t e r m s o f d i a g r a m s one h a s  (5.16) - ^ Q : e ^ : i +  ^  c  D  a  +  e  ^  X  ^ ± 1 ( 7 ] ' 3 /  T h e - e x p r e s s i o n e" l X ] (<r  first  step  i s an i n f i n i t e  i n f i n d i n g an e x p a n s i o n i n o r d e r s  interested i n separating To  do t h i s we i t e r a t e  order,  out the zeroth order  ( 5 . 1 6 ) where t h e ZU  a n d we s u b s t i t u t e t h i s  expression  forF  5  >  1  7  )  as a  o f A we a r e contributions.  appears, i n  zeroth  a n s w e r i n t o ( 5 . 1 4 ) t o o b t a i n an  2  F = e - ^ Z £ 4e-^{/+DC 2  (  s e r i e s of terms;  +  XXl+DOOC-f..}  35". We r e c o g n i z e  that t h i s expansion of r  of C h a p t e r I V f o r expanding a g e n e r a l  follows the recipe F  $  I n ' ( 5 . 1 7 ) one s e e s t h a t t h e z e r o t h o r d e r possible behavior only.  upon t h e s e  p a r t i c l e s we w i l l  the  order  terms g i v e a l l t h e  where "two p a r t i c l e s f e e l  each o t h e r  of f i r s t  I n terms o f t h e — o .  theinfluence of  To o b t a i n t h e e f f e c t o f a t h i r d need t o s e p a r a t e  i n X[. . To t h i s  particle  o u t a l l terms  e n d we d e f i n e d i a g r a m s o f  type W\^C  (5.18)  Ki  -o -a -0  where t h e v e r t i c a l having  line  a l l thelines  on t h e r i g h t  i n d i c a t e s a sum o f a l l d i a g r a m s e a c h  c o n n e c t e d i n some way a n d t e r m i n a t i n g  as i n d i c a t e d .  This w i l l  a l w a y s mean t h a t  there  a r e no ( b ) t y p e v e r t i c e s i n t h e c o n t r i b u t i n g d i a g r a m s ; f o r e x a m p l e we w r i t e ( 5 . 1 9 ) . M^e-K ZC--e-^DC H  (5.20)  ft  r ^ E . - ^ ^ e - ^ X i + e ^ X X l  Notice  i n (5.20) t h a t t h e diagram w i t h  initial these  + e-  c o r r e l a t i o n s alone  i s included  d e f i n i t i o n s we c a n w r i t e  follows  F  2  4e ^ I X X X j +• • •  i t slines  c o n n e c t e d by  i n t h e sum.  to f i r s t  order  With  i n X as  36. (5.21)  F  z  =  e^ Z° 2i  Q  +A  ~d(iC'i£ iE tr;) 4  Our t a s k now i s t o sum t h e i n f i n i t e in  (5.21).  The h i e r a r c h y o f e q u a t i o n s  s e r i e s t h a t appear  ( 5 . 9 ) makes t h i s t a s k  I n p a r t i c u l a r we w i l l use t h e e q u a t i o n s ( 5 . 9 )  an easy one. i n the i n t e g r a l (5.22)  +  form  G^-^Vfo^e"^e^fI^^O^A^"Y'.)}  . H5 = K5-IS  By s u b s t i t u t i n g t h i s e q u a t i o n f o r s=2, (5.23) G*-.e^G {o) 2  4 A e"  i n t o (5.14); equation  + e (^  e ''I F,WW  H2l  fl , f F,' 3  Hl  ft)  Gl, (i.) f Fi (t.)  we sum t h e i n f i n i t e  ( 5 . 1 7 ) , o r one has  2  s e r i e s s t r u c t u r e of t h e  37 (5.24)  F --e-^ 2  4A<?  w h e r e a s u b s c r i p t on a p a r e n t h e s i s means t h e i n c l u d e d functions, or operations The e x p r e s s i o n  yields  variable.  ( 5 . 2 5 ) g i v e s us t h e sum o f t h e z e r o t h  t e r m s , o r by c o m p a r i s o n w i t h definitions  a function of that  order  ( 5 . 1 7 ) one h a s w i t h t h e  ( 5 . 1 9 and 5.20)  (5.25)  M r -W  (5.26)  e^IO-e-^Zl  2  e  ZJ°0  A n d , by r e t a i n i n g  e-H */Jl, ?" H< 2  V F,  from (5.24) o n l y t h e f i r s t  have i n p l a c e o f (5.21) t h e e x p r e s s i o n  order  t e r m s we  38.  •(5.27)  F = e-^Zt 2  4  e  -f  2  •tft  i  ,  4 L  i  »<  3  1  '  3  " f a ' f c ' i n ' i i l 3  1  The r e m a i n i n g order  after  ifrf  unsummed s t r u c t u r e s i n ( 5 . 2 7 ) a r e t h e z e r o t h  c o n t r i b u t i o n s t o C? .  (5.22) f o r G  +  '  3  To sum t h e s e we u s e t h e e x p r e s s i o n  and r e t a i n o n l y  the zeroth order  some f u r t h e r i t e r a t i o n w i t h  G> , we o b t a i n  terms;  the expression  and,  (5.22) f o r  t h e f o l l o w i n g sums o f d i a g r a m s .  I  j-O -| —\-0  Ho/*,  i m,  h°  I  4  T ° \ ^—6 \ i,  (5.29)  ?-i<  n  2_  -e TJj/f  (  cyclic  •3 a (5.30)  e~  Hii  This concludes of  our discussion  f o r the s e l e c t i o n  diagrams f o r the short-range  equations w i l l  become more c l e a r  theory.  and summing  The m e a n i n g o f o u r  i n the f o l l o w i n g  chapter.  VI.  THE  MARKOWIAN APPROXIMATION  To- e x a m i n e i f t h e f u n c t i o n s o f t h e f o r m e r d e s c r i b e an a p p r o a c h  t o e q u i l i b r i u m we  asymptotic behavior.  Any  determined  f u n c t i o n and  this  this  c h a p t e r , we  t h a t the d i s t r i b u t i o n at  I V we  tZ-tm >/ 0  m  difficulty  has  body  very complicated.  functions,  body  t h e one  of the p a s t .  f u n c t i o n s which  particle  fact  function,  body f u n c t i o n t h r o u g h v a r i o u s In the l i t e r a t u r e  t o some e x t e n t been o v e r c o m e by  i m a t i o n s t o t h e two  In  to the  including  time  distribution  d i s c u s s t h e c o m p l i c a t i o n due  a t i m e ~t d e p e n d s on t h e one  times ±  showed t h a t t h e l o n g  t h r o u g h t h e one  function i s i t s e l f will  must c o n s i d e r t h e i r  asymptotic consideration i s  c o m p l i c a t e d because i n Chapter b e h a v i o r was  chapters  and h i g h e r o r d e r  d e p e n d on t i m e o n l y t h r o u g h  this  seeking  approx-  distribution  t h e one  particle  function. (6.1)  F  *  5  F fo^... S  ;  ^ r . j  m)  and w i t h t h e s e f u n c t i o n s t h e e v o l u t i o n o f the. one i s determined, \  (6.2)  —  t h r o u g h t h e BBKGY c h a i n , t o be  body  function  approximated  pi  \LF (i, v, <3t *r .F'(t))  +ff'F'^  Theseapproximations,  i  l  tj  i  3  i f they e x i s t ,  d e f i n e a Markowian  process  because the b e h a v i o r at a time t i s c o m p l e t e l y determined F'and ^f . a t t h e t i m e t ; history".  this  t h a t we  t h a t i s , the system  A method f o r f i n d i n g  i m a t i o n s has In  by  "forgets i t s past  behaved Markowian  n o t t o o u r k n o w l e d g e been s a t i s f a c t o r i l y  c h a p t e r we find  well  produce a Markowian approximation  r e l a t e s most e a s i l y  to  by  those found  approxdetermined. procedure  i n the  41. literature.  At.-the e n d o f t h e c h a p t e r we p o i n t o u t some  asymptotic divergence The our  difficulties  Markowian approximation  theory t o those  of t h i s procedure  (6.3)  And  F  second  the d i s t r i b u t i o n  f*(*  r  s  .. * <r .t  i)Vlr  we w i l l  we u s e t o r e l a t e  o f t h e 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 approximations,  method.  i  t  w i t h o u t m a k i n g any  f u n c t i o n s i n a form  ( f o r 5>| )  \F'(i))  i  assume t h e e x p l i c i t , i e : o t h e r t h a n  through  F'(i), t i m e 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 c a n o b t a i n M a r k o w i a n e q u a t i o n s by t a k i n g  .« .  F^^v  (6.4)  To  J  cast the d i s t r i b u t i o n  (6.5)  ... —o  One h a s f r o m one  • (6.6)  = e ^» K  ...^^rlF'^))  5  functions into  t h e form  the replacement  F-(^)  ^  c  Kli  (6.3) we  (4.17)  Fl (i)  (4.14) and d e f i n i t i o n s  body f u n c t i o n  =•••--*  (5.2),  (5.18) t h a t t h e  satisfied.  Hlr _^ 1  =-H + A - C j +A  0  + A lji,eM>  i tfollows  /x> j ^ " ^ ^ V i  - O  ' - n a - d ( Z ^ - f ~J = -H  And;  F ^,/,  limit.  r-5><*>  }  c o n s i d e r more c a r e f u l l y  r  llr,  LTi  J  the asymptotic  l  a  o  e-^  +Ti] K  4  f j  f X  t h a t f o r a j — -oV w h i c h  <2 we c a n w r i t e  o(f) + -Ti),  + 0 (f)  d e p e n d s on t i m e  through  42.  (6.-0 By  (_o:.,  combining  {-o}  (6.8)  F  and —  =  —£  ri  * OW)  >  ( 6 . 6 & 6.7) we f i n d  a connection  s  ^ - A f ^ e ^ ' l . z e - ^ ' f Z j + Z : ^ )  Then by i t e r a t i o n of  l  the equations  \ —o/  between  A\jt ^L .c <*t<{Z? tX<J3l ,  _ H  we o b t a i n  j—or  as an e x p a n s i o n  = - * - A \\ this  former  relation  we c a n w r i t e  •  O )  the general  o f (6.9)  into  F" 2  ...(continued)  •* 0 (X ) 2  F  oy t h e  (6.3).  us work w i t h t h e e x a m p l e f o r r  substitution  '(6.10)  i n terms  OM  e * *>U e  c h a p t e r s i n t h e form Let  for  W )  , one h a s  •(6-9) {H ( > r-*-- A i^ With  *  ;  one h a s by  (5.27) t h e f o l l o w i n g  expression  4.3.  (6.10)  2_  -KH  F^-e C  +e i ii  0  '-  3 . 2 2  '  2-  3 ~  -  3  .1  '. 3  2 . -i . 2 . 3 ^  Vi*  ~—' %J  •*•»  1  2.  '  :  The  terms w i t h  (6.9), Markowian  a p o s i t i o n to consider consider  first  (6.11)  F  2  ^  '  i i i l l :  3  a m i n u s s i g n i n ( 6 . 1 0 ) a r e new and we w i l l  t h e s e and s i m i l a r t e r m s , t h a t equation  — .... ll P /  ~ M»  ^  \ e  a r i s e t h r o u g h t h e use o f t h e  c o r r e c t i o n terms.  the asymptotic l i m i t  the zeroth 2 r  call  We a r e now i n (6.4).  We  order contributions to F  e^^)^W^e-  w 2 r  G ( o ) } + 0(\) 2  And f o r t h e e v o l u t i o n o f t h e one body f u n c t i o n we have  from  (6.2) (6.12) i f + K ' F ' S r A ' / ^  i„  .-«V F,'(i) r  iC^ G (o)} r  z  t 0(f)  4 4 .  Bogoliubov"'" h a s f o r m e r l y o b t a i n e d t h e f i r s t  t e r m on t h e  r i g h t hand s i d e and he p r o v e s , f o r a homogeneous that this  term i s the Boltzman collision  particle  c o l l i s i o n " e f f e c t s and I t i s h o p e d t h a t  limit  ral  to take into  collision  (6.4) w i l l  (6<,4) t h e M a r k o w i a n  Ion  . We f i n d ,  The  a c c o u n t a l l t h e two  g e n e r a l i z e Boltzman's  account three p a r t i c l e  effects.  Let  takes into  integral.  Boltzman  the  integral  collision  system,  (6.10)  collision  with  integ-  and h i g h e r o r d e r  however, t h a t w i t h t h e l i m i t  c o r r e c t i o n terms  diverge.  us c o n s i d e r f o r . e x a m p l e f r o m  (6.10) t h e c o r r e c t -  term  To s e e t h e 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 -K t s  c o n s i d e r a homogeneous s y s t e m w h e r e i n ( 6 . 1 3 ) t h e 6 o p e r a t i o n s h a v e no (6.14)  - A e - «  From t h i s  4  effect  ^  l e t us c o n s i d e r t h e ii i n t e g r a t i o n ,  w h i c h came d i r e c t l y  (6.1 ) 5  the behavior  from (6.9)  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  under t h e m u t u a l  d e s c r i b e s t h e e v o l u t i o n o f any f u n c t i o n influence of the p a r t i c l e s  the  other p a r t i c l e s  the  integral  remaining stationary.  one and t h r e e , One  sees  that  ( 6 . 1 5 ) v a n i s h e s e x c e p t where t h e o p e r a t o r  45. e  15+2  moves f r o m t h e p r o b a b i l i t y  F,Yf) F ^ ) t^(i)  densities  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 r a n g e o f t h e i r icle potential  When i  L  i s l a r g e enough so t h a t  i n t e r a c t i o n s have t a k e n p l a c e , t h e t ant;  and u n d e r t h e l i m i t  £  -  (6.9) i s e n t i r e l y  similar  d i v e r g e n t p e r t u r b a t i o n expansion  the  its initial expansion  In Chapter  and t h i s  In t h i s  expansion  the l i m i t in the  divergences  c h a p t e r we h a v e e f f e c t i v e l y  similar positions.  us f r o m o b t a i n i n g p h y s i c a l l y  0} w i t h to (4.25),  functions, in  This d i f f i c u l t y  interesting  in  destroyed, t h i s  similar  t h e s e c o n d and. h i g h e r o r d e r d i s t r i b u t i o n  @  of the form  (6.9) f o r  (6.4) r e i n t r o d u c e s divergences  entirely  i s similar to  j — 0 ] to i t s valve  w o r k b e c a u s e t h e use o f t h e e x p a n s i o n  In  which, c o n n e c t s j — 0 }  I V we summed t h e s e r i e s — O  to get r i d of the asymptotic  (4.25).  t o us t h a t t h e  (4.2) f o r • [ — 0 } .  (6.9) which connects  of the f u t u r e . order  value —I ;  Lue'^F^F^Fid)  t o t h e o r d i n a r y and  ( 4 . 2 ) we h a v e a p e r t u r b a t i o n e x p a n s i o n to  to the divergence  r  M o r e g e n e r a l l y , i t h a s become a p p a r e n t  these  integrand i s a const-  (6.4) g i v e s r i s e  (6.16)  equation  interpart-  prevents  results.  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 a b o u t t h e way t h e B o l t z m a n c o l l i s i o n was o b t a i n e d .  The B o l t z m a n e q u a t i o n  imation to the f i r s t terms w i t h i n i t i a l  order  term  should  i n (6.12)  be an a p p r o x -  c o n t r i b u t i o n t o F' ( n e g l e c t i n g  correlations)  46.. (6.17)  F Y O = e ^ - i n e  K  ,  f  - C ( Z S ^ )  ^  +  To o b t a i n t h e B o l t z m a n e q u a t i o n we d i d " n o t " u s e t h e expansion  (6.9) i n t h i s  integral  equations  t o w r i t e F' i n a  f orm  F7*)«  (6.18)  e-"'*-f  +  A e "  K  ' * H Z  which diverges i n exactly if  we t r i e d t o t a k e  from  + 06f)  ( Z ^ l C )  t h e same way a s ( 6 . 1 5 ) o r ( 6 . 9 )  an a s y m p t o t i c  the d i f f e r e n t i a l equations  limit.  R a t h e r , we w o r k e d  f o r t h e one body f u n c t i o n  (6.19)  ^ i f f ' - - » i ^ t t < I ) t o M  and  e q u a t i o n we u s e d t h e e x p a n s i o n  i n this  (6.20)  (6.9) t o o b t a i n  + OW)  g t K r ' X U e - M & ^  *U.,tt' e" F;(i)fi(i)iOtf) Hli  Or, (  6  .  li  by i n t e g r a t i o n we h a v e r a t h e r t h a n  2  1  L e-^ ^  p' r e-W^tXe-MJJi^  )  n  where ^ And  this  f/ft) ^ ' f t ) +  e  =  flf^  ™,/:'(*,)  e  e x p r e s s i o n i s known t o c o n v e r g e b e c a u s e i t g i v e s  the Boltzman e q u a t i o n  asymptotically.  the d i f f e r e n t i a l e q u a t i o n equation  (6.18) t h e b e h a v i o r  (6.17) g i v e s t h i s  That i s , working  (6.19) r a t h e r than one c o n v e r g e n t  with  the integral  term  (6.21).  4 5 Sandri & Frieman  ' ' point out i n t h e i r  theory  that  47.  their  solution  In Chapter ours.  VIII  f o r the we  In Chapter  Bogoliubov's  three p a r t i c l e c o l l i s i o n  show t h e V I I we  theory.  f i n d these  A l s o we  d i v e r g e n c e s in B o g o l i u b o v ' s discussed  connection  particle  same d i v e r g e n c e s  of a d i f f e r e n t nature  by Cohen & Dorfman„  over  of t h e i r t h e o r y  should mention t h a t  describe three p a r t i c l e c o l l i s i o n gration  In  effects  integrations.  phase space a v a i l a b l e collision  asymptotic  has  been  c o n t a i n s an the  4;  etc.  According  t o them t h e  f o r t h e f o u r p a r t i c l e and  effects diverge  asymptotically.  inte-  four over  Cohen &  D o r f m a n make e s t i m a t e s o f t h e amount o f p h a s e s p a c e these  to in  t e r m s w o u l d c o n t a i n an i n t e g r a t i o n  t h e p h a s e s p a c e s o f t h e p a r t i c l e s 3 and  for  diverge.  (6.10) the terms which  t h e p h a s e s p a c e o f t h e p a r t i c l e 3;  collision  effects  available  amount o f higher  order  VII.  THE THEORY OF BOGOLIUBOV Since i t s introduction  Bogoliubov"'" h a s a t t r a c t e d literature.  We w i l l  Then we d e v e l o p expansion  considerable interest  g i v e here  a brief  expansion,  As hierarchy  V I , becomes a starting  The c o n n e c t i o n b e t w e e n h i s  (7.1.)  Of  point Bogoliubov  distribution  f o r the short-range theory, with the  ( 3 . 2 7 ) , we w r i t e  i n t e r e s t i s t h e temporal  first  F  Z  L- V  major assumption  coarser than a c o l l i s i o n approximated  change o f t h e one body  function  ^ W F + A L  Bogoliubov's  (7.3)  d e r i v e s t h e BBKGY  ^VF'-XL'F*'  particular  (7.2)  procedure  apparent.  (3.18) which  thermodynamic l i m i t  follows  a n d we s o l v e o u r e q u a t i o n s s i d e by  t h e o r y and t h e d i v e r g e n t Markowian a p p r o x i m a t i o n Chapter  c h a p t e r s an  a s n e a r l y as p o s s i b l e  side with those o f Bogoliubov.  of  i n the  resume o f h i s t h e o r y ,  from t h e knowledge o f t h e former  method t h a t we f e e l  Bogoliubov's  i n (1946) t h e t h e o r y o f  by  H-'Mfe.sr/M)  i st h a t f o r a time  time t h e equation  scale  ( 7 . 2 ) c a n be  49.  w h e r e A d e p e n d s f u n c t i o n a l l y o n F V ^ b u t d o e s 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  the  time s c a l e a l l t h e F  coarse  grained  d e p e n d on t i m e  for only  t h r o u g h F' , s o t h a t one c a n w r i t e (7.4)  F ( ^ . . . , 5s,*S;t)= F ( ^ , . . . , * , i s / F ' ) s  5  s  Bogoliubov t r i e s in  to obtain  (7.3)  by s u c c e s i v e  approximations  t h e form jf=  (7.5)  and  F'j + X A ' f l , ^ ) F')+ A M ( * , , ^ F') + -~ 2  f o r t h e many p a r t i c l e  ation  expansion  (7.6)  F = F s  s o  f*  0  function  4 7 , . ^ <i5/F»J + A  1  (7.4)- he t r i e s  a perturb-  f « ^ ^ , ^ 4 5 / F ' J +A F"^.,^-a  v  Bogoliubov uses t h e 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 t h e time derivative of a function  Ft  J  (7.7) ± 4 f a o r  l  y  . . ^ F ' j  where t h e o p e r a t o r "t  (^  o f /f(F') f o r i J l !  and  (7.6)  of  \ B ' i$  with  to obtain  equations  '  •'  denotes d i f f e r e n t i a t i o n w i t h  d e p e n d s on ir t h r o u g h  ion  (7.1)  ^ I F ' ) w h i c h d e p e n d s on  ^ (<x^o/7  .  F' ) w i t h  respect to  subsequent s u b s t i t u t -  He s u b s t i t u t e s t h e e x p a n s i o n s ( 7 . 5 )  the operations by e q u a t i n g  (7.7)  equal  i n t o t h e BBKGY h i e r a r c h y  orders  in A  t h e system  50. A ^ ^ F ^ - K ' F '  (7.8) (7.9)  A U,,¥ F')=  (7.10)  t\ (%¥,/)=  L  x  h  0  Zo  «® F ' + f f F  (  $*F  (  7  1  3  )  14)  :  & F +\^F*°--0  (7.12)  .  20  •2/  LF  z  (7.11)  7  F  6  S o  4  -r H F 5  ®°F +H F s£  s  21  s o  = L F 2  -&'F  t 0  =0 •= <p ~-  $i  3 o  tl  L F '^-'S  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  One  n o t i c e s t h a t 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 e q u a t i o n s , problem has ions  r  and f o r t h i s reason B o g o l i u b o v says t h a t  been reduced to the d e t e r m i n a t i o n  and A as f u n c t i o n a l s o f the  T h i s a l l o w s him  express-  where f property  ^ - ^ F ' . - ^ ^ F ' ^ f e ^ f ^ ^  r f  i  ;  (  / e ~  K  V > j V F  S l  ' (  / < r  K  0  obtains  (7.17) j ^ F s « Y  / e ^ T j t H F '"f / e - ^ F ' ) - - ^ ' Y s  J  where  i t follows that  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 e q u a t i o n s one  F  One n o t i c e s the  Hence from the d e f i n i t i o n of the o p e r a t o r ^ (7.16)  o f the  the  "arbitrary" function h  to r e p l a c e everywhere F by e  i s some parameter independent of t . (7.15)  equations.  ~o  f o r the F ( u  W F) KC  51.  For  t h e s o l u t i o n o f ( 7 . 1 7 ) one h a s  F '(  (7.18)  \€^F)r:e-» F (  Sl  Following  5T  Bogoliubov  a r g u m e n t F by e  \f )  si  x  we r e p l a c e  F to obtain  v  the a r b i t r a r y f u n c t i o n a l  ( w i t h a change I n t h e f '  variable  of i n t e g r a t i o n )  (7.19) F''(  lFJ:e-» *F (  Since  equations  the  s  these  |e ' r)  si  k  r  (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  l e f t - h a n d s i d e d o e s n o t d e p e n d on ?" t h e  taken.  For this  l i m i t Bogoliubov  limit  t->x>  uses t h e boundary  c a n be condit-  ions (7.20)  j L ^ ^ F t ' C  le«"tF')-- JL,  r-5.oo  (7.21)  ?" — > CO  e-^  S r  F ( i :  \e"' F)--0  discuss  i 70  7  These c o n d i t i o n s a r e v e r y will  e-^lTe^F'  them l a t e r .  f a r reaching  a s s u m p t i o n s , a n d we  With these  boundary c o n d i t i o n s i n  (7.19) one h a s  (7.22)  F °(  (7.23)  F'( i f ) - Qf^'r'i  By and  s  S  combining  e-W^TTe^F' /f  r r  'F9  ( 7 . 2 2 & 7.23) w i t h t h e d e f i n i t i o n s o f t h e b «.  ^ ' V B o g o l i u b o v ' s e x p a n s i o n c a n be w r i t t e n o u t t o any 5  52.  order i n A .  We  the s o l u t i o n  should mention t h a t ,  as B o g o l i u b o v  f o r A f o r a homogeneous s y s t e m 1  gives  proves, Boltzman's  equation (7 .24 )  ^^ -  L. C  F, F2_  To e x p l a i n B o g o l i u b o v ' s and w o r k f r o m  t h e o r y we f i n d  develop  an e x p a n s i o n  as p o s s i b l e r e s e m b l e s  expansion w i l l finding  or  i  L  ^^aytj^y^ .  i t easier  t o go back  o u r more g e n e r a l t h e o r y o f t h e f o r m e r  We w i l l nearly  ^^^pJ ^^ ^^  chapters.  method t h a t we f e e l  Bogoliubov's  as  expansion.  This  be n o t h i n g more t h a n an a l t e r n a t e method f o r  writing  out the  general F i n S  the form  (6.3) that  we d e r i v e d f o r t h e M a r k o w i a n a p p r o x i m a t i o n p r o c e d u r e i n Chapter  VI.  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  b u t l e t us i g n o r e t h i s (6.3) w r i t t e n (7.25)  f o r now.  i n the s l i g h t l y  identity  seems t h a t t h e e x p l i c i t  our  evolution  time  constant t  restricted to  form  i s a s h o r t hand, we o f t e n u s e . time  i n t i m e , and we w i l l  behavior. 0  t r y to coarse  define the f u n c t i o n s  grain  approximation of the  A n d , we h a v e i n c l u d e d i n ( 7 . 2 5 )  so t h a t o u r c o a r s e g r a i n i n g  to the o r i g i n  It  behavior ^ d e s c r i b e s the f i n e  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  explicit the  different  V i * ^ . . , * ^ ^ ^  where t h e l a s t  grained  We. c o n s i d e r t h e e x p r e s s i o n s  i n time t .  procedure  We f i n d  i s not  i t convenient  53. (7.26)  FS= F ( e K t F | f o + * 0 s  ,  ,  w h i c h have t h e same f o r m a s ( 7 . 2 5 ) b u t h e r e we t r e a t , a parameter  cr i n d e p e n d e n t o f t , t h e e x p l i c i t  through  time behavior K't > C  as t h o u g h 'we c o u l d these f u n c t i o n s  r .  t h e d e p e n d e n c e upon S  ignore  ( 7 . 2 6 ) t h e BBKGY h i e r a r c h y  With  ( 7 . 1 ) c a n be  w r i t t e n as f o l l o w s  (7.27)  [(£H^|] .  u'F'( «r\t)  +  =At F "p''f'W s  e  5  o r one h a s (7-28)  {(fjJ  '\L'F»'( * Vh)-[§ }l  'H F>(e«fW !  TTR:T  l  From o u r work o f C h a p t e r V I we c o u l d the  function  ,  i nprinciple  t  determine  I Vdt * W16 „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 c a n d e d u c e an e q u a t i o n f o r V <j <r (7.30) where J  ( T F ^+H s  S  F  S  - A L  S  has t h e f u n c t i o n a l W F ' | * . + r),  (7.31) T-  F  S  4  ' - J  S  form [(0 i + <S-=t 0  The  chain  o f e q u a t i o n s ( 7 . 3 0 ) we w i l l  perturbation  t r y t o solve  by a  expansion s i m i l a r to Bogoliubov's, but f i r s t  we must d e t e r m i n e p r o p e r b o u n d a r y i n e them, we w r i t e  the F  s  conditions.  i n terms o f diagrams  To determas f o l l o w s  5 4 .  (7.32)  h- (i):  e -  s  H  S  i  L Zo  + e-* *  Z  5  3  3  o  >  w h e r e t h e a r r o w s i n d i c a t e t h e sum o f d i a g r a m s the f i r s t  term.  t h a t have n o t  already  been u s e d t o p r o d u c e  From o u r  general  p r o c e d u r e o f C h a p t e r I V 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 t h e s e c o n d t e r m must h a v e a t l e a s t two o f its  s lines  connected;  to t h e f i r s t  term.  s e c o n d t e r m we  otherwise,  the diagram  contributes  As an e x a m p l e c o n t r i b u t i o n t o t h e  consider  i  (7.33)  f'^  1  1 4S  '  We u s e t h e r e l a t i o n term i n t h e form  (6.9) i n (7.33) i n o r d e r t o w r i t e  this  (7.25) v5, '  (7.34)  e - ^ C o - *-*"CS ' " -*r~* +  L.L  0  e  K  LL*  One s e e s i n ( 7 . 3 4 ) t h a t  L  each term d e s c r i b e s  b e t w e e n t h e p a r t i c l e s 2 and 3. any  some c o r r e l a t i o n  By a s i m i l a r c a l c u l a t i o n f o r  c o n t r i b u t i o n t o t h e s e c o n d t e r m i n ( 7 . 3 2 ) one a l w a y s  finds that a l l describe icles.  some c o r r e l a t i o n among t h e s p a r t -  F o r c o n v e n i e n c e we d e f i n e  correlations  a f u n c t i o n f o r these  . F f e*' Flt)--e-K irf:K *F'+  (7.35)  s  where Q  S  t  C (e«'\F'li)  ,  Si  5 5.  s  i s t h e second term i n (7.32).  For our functions  ( 7 . 2 6 ) w i t h t h e i n d e p e n d e n t v a r i a b l e s <r a n d t we w o u l d have from (7.35) t h e c o r r e s p o n d i n g (7.36)  fs( i<'l  F<) t -)--  e  oiS  c i s s e t equal  When  F^e^F'fg  (7.37)  e  functions  -K'(^r)Tr K't  f<  e  Cie^F'l'u^)  +  t o z e r o , one h a s re-^TTe^F-'  + C^e^F'/fo)  We have i n ( 7 . 3 7 ) t h e f u n c t i o n a l f o r m o f F o f cr ; t h i s g i v e s in principle of  us a b o u n d a r y  c o n d i t i o n w i t h w h i c h we c o u l d  solve theequations  ( 7 . 3 7 ) we w i l l  sometimes  at the o r i g i n  s  (7.30).  consider  As a s p e c i a l  t h e c a s e where a- and t  h a v e t h e same o r i g i n , , o r where t =0.  To b r i n g o u t w h a t  happens when t = 0 l e t us c o n s i d e r  t h e example  o  o  7.34)  f r o m w h i c h we d e d u c e  again  the corresponding  (7.33 &  contributions  C (e ' F'\t^<r)  to  i  (7.38)  K t  e-^*- '^  TTe^F'W  f  J  One s e e s t h a t for  case  t =0. o  (7.38) v a n i s h e s In general  J  c  o  as a c o n t r i b u t i o n t o  we f i n d  that the only  C 5(e k F'lio) r  contributions  to t h e 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 t o C^e^F'/o) any v e r t i c e s , one h a s  a r e t h o s e t h a t do n o t c o n t a i n  5,6.  (7.39)  C ^ ^ F ' W O  (7.40)  c  5  :&(°)*G lo) lt  ( e ^ F'/o)  4  4  i  j  eic.  To for  s o l v e t h e e q u a t i o n s ( 7 . 3 0 ) 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  the various  s  F = F K e * V / ^ ^ f ^ ^  (7.41)  5  I f we a r e a b l e t o s o l v e f o r t h e F ordinary (7.42)  time a s e r i e s  ^-  F^(e"^rlt)  il  , they w i l l  expansion f o r F  + /  \F"(e^F'li)  X F^(e 'F it)i--' 2  +  o r d e r t o s o l v e t h e e q u a t i o n s ( 7 . 3 1 ) we a l s o  to  find  hierarchy, function  ( .43) 7  the evolution  .  F o r t h i s we w r i t e ,  ,  need  some way  f r o m t h e BBKGY  e q u a t i o n f o r t h e one p a r t i c l e  i ^ l  F  ' . -  Xt  e^Ate^n*,*:*)  F'H) =  *t,  t h e e x p a n s i o n (7.42) f o r F  z  we c a n w r i t e  (7.43)  successive approximations  w h e r e we have no z e r o t h o r d e r t e r m . the  t<,  i n the form  Further, with in  the J  us i n  5  In  o r expand  give  o p e r a t i o n s j(~ff-)  6  as f o l l o w s ;  I n ( 7 . 2 9 ) we c a n p e r f o r m f o r each  F ' one c a n  '  5.7.  write (jP)^  (7.45)  A l ' F ^ A ^ F ^ A ^ F ^ . . .  where t h e o p e r a t o r ^  denotes d i f f e r e n t i a t i o n with  t o ~i (F 'depends on t o n l y  i n t h e c o m b i n a t i o n e* F')  s  subsequent s u b s t i t u t i o n use i n e q u a t i o n form  K e  '  ( U  r  o f s^AW^'H)  e  for  F'  H>f  with . For  ( 7 . 3 0 ) we must r e w r i t e t h e s e <? * / T i i n t h e k  ^ A (^' }  f  regard  F'\t + <r)  i  a n d we d e n o t e t h i s  6  by  <9** o v e r  bars  ( 7  -  4 6 )  [(^)]  where  f  *«--*/ ^ ^ ' ^ ' ^ ' ^ ' F ^ '  of e ' K  L e t us t r y o u r p e r t u r b a t i o n F o r t h e one p a r t i c l e (7.42) f o r F  in  Z  3  ^ ^ ...  ( t o +  c)  ^(e*^F\t  0  t o ir w i t h  +e)i r 4 0  <?  K<<  F'  e x p a n s i o n i n t h e f o l l o w i n g way.  ( 7 . 4 3 o r 7.44) a n d f o r t h e many  f u n c t i o n we t r y t h e e x p a n s i o n ( 7 . 4 1 ) w i t h . ( 7 . 4 6 ) by e q u a t i n g  of e q u a t i o n s A'sLF^fe^F'W  (7.47)  (7.49)  ^  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  i n equation  (7.30) t o o b t a i n  (7.48) .  9  denotes d i f f e r e n t i a t i o n with regard  subsequent s u b s t i t u t i o n  particle  +A  /»*--LF*7etf'*F'|t)  ^  "  ^  =0  e q u a l o r d e r s i n X' t h e s e t  58. (7.50)  y p ' + hF F 2  t0  Lf F3°  r  F  zo  «  H ' F  (7.51)  '(7.52)  S  O  = O  H F«V=<£ S  We a r e t o s o l v e  this  boundary c o n d i t i o n  =  f /  L  ?  F ^  ;  c 9 F > J  - ' - i :  S  system o f equations s u b j e c t  to the  (7.37) which f o r t h e p e r t u r b a t i o n  expan-  s i o n we c a n w r i t e a s f o l l o w s S  (7.53)  F**(r*o) retire*'*  F'(t) + C leK F io  lt  lt ) 0  F 'V-o)--c^( ^F MJ  (7.54)  s  ,  e  w h e r e we h a v e i n t r o d u c e d (7.55)  f o r C (e^'F'ji: ) 0  C ft^Fl^j'UC Y^  C ^ V / t o ) :  a  s  A n d , l e t us k e e p i n m i n d t h e s p e c i a l h a v e t h e same o r i g i n , from o u r r e l a t i o n s  the expansion  i  , t  F |fo)4A C"^ ,  c a s e where  o r where t<= - O • f o r t h i s  (7.39),  (7.40),  l  <r  K ,  *F |to) ,  and t  c a s e we  find  e t c ' the f o l l o w i n g  conditions (7.56)  c*°(eMF'lo) ~-  (7.57)  C (e F'  We f i n d  t h a t o u r system o f e q u a t i o n s  the  sl  Klf  C (c ' F'lo) s  lo) =o  boundary c o n d i t i o n s  solve  l(  ^  i  l >o ( 7 . 4 7 t o 7.52) w i t h  ( 7 . 5 3 & 7.54) d e c o u p l e and we c a n  them t o any o r d e r . As i s t h e p l a n  of t h i s  chapter,  l e t us s o l v e o u r  59. system  o f 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  (7.8 t o 7 . 1 4 ) :  we d e n o t e  t h e e q u a t i o n numbers. compute t h e F '/a :  Bogoliubov's  The c a l c u l a t i o n s go s m o o t h l y  c o r r e c t i o n term appears. start with  for his ( 7  5  8  F*°[  .  F  case  (7.13)  K r  ^ f ^ ( e ^ F ' I U ^ )  < s <-( H  -  o  , r ^ ' ) . o  F  We s o l v e ( 7 . 5 8 ) s u b j e c t t o t h e b o u n d a r y c o n d i t i o n solve  Markowian  of equations  or i nBogoliubov's  S o  u n t i l we  lC ' F')  xllU^  (7 58B)  To s o l v e o u r s y s t e m  (7.51) f o r  ^ ( ^ I ^ ^ j  )  by B's i n  a t t h a t p o i n t i s where t h e f i r s t  2  we  solution  system  (7.58B) s u b j e c t t o t h e b o u n d a r y c o n d i t i o n  (7.53) a n d  (7.20) t o '  obtain F ^ ' - ' F ' l v ^ e - ^  (7.59)  F (  (7.59B)  So  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 ; t o g e t o u r A' we must d e d u c e i n o r d i n a r y time o u r F  from  consider the special  c a s e o f (7.59) where <r a n d t have t h e  same o r i g i n . for  account  T h i s i s e a s y when we  (7.56  a n d 7.39) one h a s  t =0 o  (7.60) and  Taking into  (7.59).  p f e ^ F ' H ^ ^ ^ ^ F ' W  f o r the corresponding  + e-^GYo) one h a s  60. (7.61)  F ^ e ^ F ' j i ] re-H t|Te« 2  This o f course With  agrees  (7.61) o u r A  1  F'ft) f e ' ^ G Y o )  4  w i t h t h e z e r o t h order term  f o l l o w s from  i n (6.10).  ( 7 . 4 7 ) and we o b t a i n  B o g o l i u b o v ' s A'from ( 7 . 5 9 B ) w i t h ( 7 . 9 )  (7.62)  h W f ' l t )  (7.62B.)  A'U  t j  *  - U-«^M  r)~-  For  F  1 1  ? oo  t o s o l v e f o r F ' and B o g o l i u b o v ' s s  into  j r ' + H F*' =  ;  F  into  account  (7.59B)  1  y£.(  S i n c e we h a v e theh'c-,  TTe ^ F' $ + e "  fe e  ]  H v  z  G *(e *'* F'| t.)} 2  {l e-H>Tre^F -£ e-ti>lTe^F'(t)}  l  rary functional  s l  (7.59) and i n  e  ( 7 . 6 3 B ) ^ " F ^ t H F*' =  f o r Bogoliubov's  F  ^e^^e-^TT ^iF (i)i{ e-^a°(e^f'ji)  l  v  (7.64)  account  c a s e we u s e ( 7 . 1 2 ) t a k i n g  - $' je- W  and  and B o g o l i u b o v ' s  ZI  we u s e ( 7 . 5 0 ) t a k i n g  Bogoliubov's (7.63)  Le-^e^FK^e^Fld)  o n l y work o u t F  h o w e v e r , we w i l l  J  Jt~ C  We a r e i n a p o s i t i o n  L f " * GY4>  F;(i)e^F[(t),  ,  ,  -5*  we c a n p e r f o r m equation  thetS'and ^ ' o p e r a t i o n s ;  ( 7 . 6 3 B ) we r e p l a c e t h e " a r b i t -  a r g u m e n t " F' by Q' * ' F '  ...(continued)  1  -  61. (  7  >  6  4  )  J£?'  + H  l  F«»-  •  L e - H V ^ f r e ^ F 7 ^ H e ^ C ^ ( e ^ F ' / ^ x  V  2  Tfe^F'it)  (7.64B)  iE!Llf!!!l£J ^F ( ll  le-K+FJ-  4  /A - » o o And we s o l v e and or  solve  (7.64) s u b j e c t  (7.64B) s u b j e c t  t o t h e boundary  use ( 7 . 2 3 ) ,  (7.65)  F'Ye  K,f  t o t h e boundary  F7#ptr) =  e-i^c^e^F'/*,)  conditions  condition  (7.54.)  (7.21),  6 2 .  F '(  (7.65B)  IF')''  z  l/^e- >[L e' '^-rU,e- ^lfl'e^F'(i)  -lU  H  lb  H  H  We a r e n o t 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  t h e s e c o n d t e r m i n (7.65B) d i v e r g e s .  b r i n g o u t a c o n n e c t i o n t o o u r t h e o r y , l e t us f i r s t solution f o r our F  the  general  and  (7.52) u s i n g t h e boundary  and F  S a  conditions  5 t  ; from  To  consider (7.51)  ( 7 . 5 3 ) and ( 7 . 5 4 )  one h a s - e~^e-^ TreK'i-  (7.66)  M«  (7.67)  F '  We f i n d  h  s  e"  /yV  F  '(i),e~^  e^f^^F'jt^i,)  C ( V * " f ' /to) + J M, 5 ;  C'ie^F'lQ  f  t h a t o u r e x p a n s i o n becomes a l m o s t 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 t h e f o l l o w i n g  two a s s u m p t -  ions: (i).  We assume o u r 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 c a n c o a r s e g r a i n o u r f u n c t i o n s by t a k i n g t h e limits (7.68)  i i . ff"  > CO  /A —>  F  Sd>  (c >F'[tl\t i* ) K  0  :  6 3 .  F  (7.69)  ?  t  (e >  F'hVtotc)  K  w h e r e one n o t i c e s t h a t t h e to comparison  i s no l o n g e r i m p o r t a n t i n  t o &.  (ii).  We assume t h a t t h e c o r r e l a t i o n s w h i c h  existed  a t t h e o r i g i n o f cr v a n i s h due t o t h e n a t u r a l m o t i o n system,  o r one h a s k~  (7.70) With  of the  e  -"  J , r  C^FW|t,)-'0  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 . 6 6 & 7.67) one h a s  ,  (7.71) o- - > oo  e~ ' jfc —y oo  TTF'(i-)  HS//  if (7.72)  1^  =  JUL  [AeMy'tCeKH'lt.+  O  yl< —5» w  L e t us c o n s i d e r o u r s o l u t i o n f o r F  (7.65) w i t h  2 /  these  asymptotic conditions (7.73)  F»~/i.'.  r^,ff-*"'fl e-» ,  -/£,( V', e " > f e ' ' / And  1  L  1  }  , /  '7f^FW}  e - V t L ^ ^ T T e ^  we compare (7.73) w i t h B o g o l i u b o v ' s  f«)  solution f o r  (7.65B) a n d we s e e t h e y a r e t h e same e x c e p t f o r some o p e r a t o r s G t. if  The i m p o r t a n t p o i n t i s t h a t  we t a k e t h e a s y m p t o t i c l i m i t . o f  behavior.  theexplicit  From o u r work i n C h a p t e r  a d i v e r g e n t F*  1  because,  from  F  XI  diverges  time  V I we o f c o u r s e  (6.10), thef i r s t  order  expected term  64-. of F " c o n t a i n s M a r k o w i a n c o r r e c t i o n t e r m s .  I t i s n o t appar-  2  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 t i m e t h e same  F' x  as i n ( 6 . 1 0 ) ;  i t c a n be shown t h a t i t d o e s and we h a v e  relegated  c a l u c l a t i o n t o o u r Appendix' I I .  this  strong resemblence  o f o u r cr t o B o g o l i u b o v ' s T and s i n c e o u r  er d e s c r i b e s , e x c e p t f o r t h e e^'* explicit  time  b e h a v i o r we f e e l  By c o m p a r i s o n  justified  i n saying that  time  with our expansion  Bogoliubov's  expansion  is  not v a l i d  to take the l i m i t f  of  the e x p l i c i t  in  saying that Bogoliubov's  time  t o t h e F'i- , t h e  affixed  Bogoliubov's T describes the e x p l i c i t theory.  behavior i n h i s we know t h a t  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 >  CD  .  I f the l i m i t  b e h a v i o r were v a l i d , we f e e l  the c o r r e l a t i o n s which  justified  b o u n d a r y c o n d i t i o n s ( 7 . 2 0 ) and  ( 7 . 2 1 ) h a v e t h e same m e a n i n g a s t h e a s s u m p t i o n is,  Due t o t h e  (ii);  existed, at the ' a r b i t r a r y '  that origin  of ? v a n i s h f o r a s y m p t o t i c V . This concludes  our d i s c u s s i o n of Bogoliubov's.theory.  t h e n e x t c h a p t e r we w i l l Ar  to  Sandri & Frieman.  Bogoliubov's  theory.  d i s c u s s t h e method o f e x t e n s i o n due  5  '  T h i s method i s s i m i l a r t o  THE'METHOD OF EXTENSION  V I I I .  The method o f e x t e n s i o n will  now be d i s c u s s e d .  short-range  theory  due t o S a n d r i  We d i s c u s s  & Frieman '  t h e i r method f o r t h e  and f o r s i m p l i c i t y we assume t h e s y s t e m  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 .  authors  p o i n t t h e BBKGY c h a i n  for  use as a s t a r t i n g  t h e s h o r t range theory  (8.1)  ^ ^ F  We f i n d  we w i l l us  - ^ ^ ^  5  follow Sandri into  (8.2)  F'=-  f ° + Af  (8.3)  F* =  F  (8.4)  F  F  5  We o b t a i n  T  + X  Z0  (8.6)  $  (8.7)  7 T % H  & F r i e m a n ' s work more  i  = L F >' "*' z  f  s  o  x F *+-L  equal  S  orders  Sc  c >o  =o  (8.8) 4 r ' *• H F *= V 5  Let  F " + ,\* F " f •  i nX the set of  O  s  closely.  Later  f ;,2-f?-+- •  by e q u a t i n g  £r =  expansion.  (8.1) t h e expansions  iXF"  so  (8.5)  f  i s b e t t e r u n d e r s t o o d i f we  f r o m an o r d i n a r y p e r t u r b a t i o n  introduce  (3.18) w h i c h  we w r i t e  t h e method o f e x t e n s i o n  work f i r s t  These  F '^"' s t  L = L'  equations  6 6 .  Since  t h e s y s t e m i s assumed i n i t i a l l y  initial  uncorrelated, f o r  c o n d i t i o n s one h a s  (8.9)  $°(o) -  (8.10)  f (o)- O  (8.11)  F (o)  , f r f°(o)  (8.12)  F (o)  - O  -  t >o  c  S6  si  1  We p r o c e e d t o s o l v e t h e s e t o f e q u a t i o n s starting with From  the zeroth order  (8.5 w i t h  (8.13)  8.9) we f i n d  i (o)=f 0  To  solve  that  to higher  -f" i s t i m e  f u n c t i o n we o b t a i n  TV  e  and w o r k i n g  (8.8) order,  independent  =  A n d , f o r t h e many p a r t i c l e (8.14)  (8.5) t o  from (8.7)  f  c  ( 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 t h e two body  behavior (8.15)  F^e-^W:**-* *^ 1  By s u b s t i t u t i o n (8.16)  For  o f (8.15) i n t o  •-cj.  asymptotic  <Z X j i O X ;  < x <  +  30C<t-'-}  ( 8 . 6 ) one f i n d s  the solution  -ooc; *-oocc;  t i m e we f i n d  that  (8.16) d i v e r g e s ,  o r we h a v e  6  (8.17)  V ~ i t u  .  .  Le-^ftfi  w h e r e we have r e c a l l e d (4.23).  7.  the diagram d e f i n i t i o n s  ( 4 . 2 0 ) and  F o r t h e a s y m p t o t i c b e h a v i o r o f t h e one body  f u n c t i o n we now h a v e  F =f°+xt l  (8.18)  S a n d r i & Frieman use a t r i c k or  secular terms.  jt&r,  + o(Ji*)  Le-W{?fl  t o take care o f these d i v e r g e n t  F o r t h i s c a s e ( 8 . 1 8 ) we f i n d t h a t t h e y  r e d e f i n e t h e 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 t h e  following  relations (8.19) (8.20) (.8.21) One  F' - f° f O + oJk (k ) z  Jz^Le  I* il  ( t  M  y  j^tQ  ^  ^ - Xi  s e e s t h a t t h e t e r m s o f ( 8 . 1 8 ) a p p e a r now i n t h e T a y l o r  e x p a n s i o n o f -f (8.22)  f'sf+e-jfa  6  Le-Wf;fS*<ri  ^ t i l *  f  S a n d r i & F r i e m a n assume t h a t t h e 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 t h e t y p e (8.22) and t h a t t h e s e a r e t h e s i g n i f i c a n t ones.  From o u r c a l c u l a t i o n  ( 4 . 1 5 ) t h r o u g h ( 4 . 2 5 ) we know  t h a t t h e 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 o f t h e t y p e  68. (8.22) and t h i s The s e r i e s  (8.23)  j  Although  i s t h e r e a s o n t h e i r method  seems t o w o r k .  (8.22) comes f r o m t h e b e h a v i o r  L  p  ,  ^  A  _  <  ^  +  _  A  <  c  (8.20) i s B o l t z m a n ' s  was no more t h a n a g u e s s .  e q u a t i o n , t h e way  We w i l l  F i r e m a n ' s method more c l o s e l y assumptions  A-OOC+----  o  now  i t was  found  follow Sandri &  and f i n d t h e y make o t h e r  a n d , as t h e y do, d i v e r g e n c e  difficulties.  I n t h e 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 t h e b a s i c r e a s o n t h e method o f S a n d r i & F r i e m a n w o r k s . will  We  h a v e t o go much,much d e e p e r h o w e v e r , t o b r i n g o u t a l l t h e  assumptions writing  and m e a n i n g s o f t h e i r  the general r  F  (8.24)  =  F  *  (  theory.  L e t us s t a r t  by  i n the form  *  ^  y  .  K  ^  ^  .  .  .  )  w h e r e we h a v e i n t r o d u c e d t h e p a r a m e t e r s (8.25)  We  «i--f + C, , «7 = A* + C,,tf-j,= XH+C  i}  l=  tft  i  C  3  h a v e done n o t h i n g i n (8.24) b u t a s s e r t t h a t we  i n y e t some u n d e t e r m i n e d parameters  (8.25).  way,  the v a r i o u s r  A t t h i s p o i n t we  8, F r i e m a n ' s method o f e x t e n s i o n . now  cr  can w r i t e ,  i n terms of t h e  begin to f o l l o w Sandri  Following  a f u n c t i o n t h a t has t h e same f o r m as  them, we  (8.24)  imagine  69. (8.26) but  F  5  R  F (^^, --- ^^s\r 5  r  h e r e T f 'Y T ... Ci  l)  lj  lJ  )  ( 8 . 2 6 ) t h e BBKGY c h a i n  follows  .  Let  t,r,r  i  j  ...) With  this  ( 8 . 1 ) c a n be w r i t t e n a s  '  R&>v^M&>..^  us assume, a s S a n d r i  ation  j  are independent v a r i a b l e s .  function  (8.27)  0  i salso  •H'F'=AL'F"  & F r i e m a n d o , t h a t t h e f o l l o w i n g equ-  true  (8.28)  The  transition  f r o m ( 8 . 2 7 ) t o ( 8 . 2 8 ) c l e a r l y p u t s some  restrict-  i o n s o n t h e p o s s i b l e s p e c i f i c a t i o n s o f t h e f ^ T , ^ ^ . . . a n d we should ing  check t h i s whenever a s p e c i f i c a t i o n  Sandri  & F r i e m a n we i n t r o d u c e  into  i s made.  (8.28) t h e p e r t u r b a t i o n  expansions (8.28)  F' - f*+A?'4  (8.29)  p  (8.30)  F « = F S o + A F S l +A* F 5 i + - - •  equations  l^Pf--  = F ^ f A P + A'F^.--  And we o b t a i n  by e q u a t i n g  equal  orders  Follow-  i nA the set of  7 0 .  (8.31)  7^-"0  (8.32)  ^  = L F«  (8-33)  i|>^>^- L?-  (8.34)  g"*B*F"-0  :  a  (8.36)  jr.  (8.37)  5To  Sandri  ^  f  °  F  °  =  '  [  F  jf} « ^  ;  & Frieman o f t e n use t h e i n i t i a l  (8.38)  f * ( r = 0 ) = asd&ay  (8.39)  P(? zo)  (8.40) (8.41)  yUc^  o  t  F  S  0  M  - 6 s _  ^  conditions r  ,  ^  ,  ^  ^  .  .  .  ' >o  t  TT ^ ^ r ^ J  :  0  ? s « 7 ^ = o) = O  These c o n d i t i o n s a r e m e a n i n g l e s s a t t h i s p o i n t because t h e dependence o f r We w i l l solution  upon Vo?,^,'•• h a s n o t .yet been  have t o study  the i n i t i a l  to the zeroth order  conditions later.  equations  r'^r,,  (8.43)  F ** 5  V«. M ^ ^ ; ^ , — ) 0  e-// ^ ] 7 f° s  0  The  a r e t h e same a s b e f o r e ,  o r one h a s (8.42)  specified.  71. ("8.43) 4'  From ( 8 . 3 2 ) w i t h  Le-^f;;;-  (8.44) or,  satisfies  0  f o r the s o l u t i o n of (8.44),considering only the  asymptotic  behavior,one has  To o b t a i n t h e 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 , is  )  j  (  - ^  jp^s\^£lv^S-cJ~£^  tfA^c^ux^  which case f ° i s simply  function.  My^^-O-^S^''^)  thei n i t i a l  However; i t i s c o n v e n i e n t  value  series  (8.4.7)  expansion  contains  ( 8 . 2 2 ) by s p e c i f y i n g  .4°--  p £  Le-»*1  Jk»  which cancels the divergence o f a more g e n e r a l us  o f t h e one body  t o assume, and t h i s i s  what S a n d r i & F r i e m a n do, t h a t t h e g e n e r a l the  compute F ' ; 1  theory, from  i n (8.45).  Without the  ( 8 . 4 7 ) i s no more t h a n  (8.35) t a k i n g i n t o  account  support  a guess. L e t (8.43 and  8.47) one h a s (8.48)  or,  tr,  n e v e r i n t r o d u c e d , one w o u l d p u t  ( 8 . 46 in  where t h e  - L e» !  with thei n i t i a l  , r  -ft'^tf ' [l  value  ,  t  1 J  f*M, A»] !  ( 8 . 9 1 ) one h a s t h e s o l u t i o n  m i  When we (8.50)  c o n s i d e r the asymptotic  F « ~ J ^ ( % , e -  And  this  Let  us  i s the  would y i e l d  an  one  body f u n c t i o n .  one  body f u n c t i o n  and  by  With  we  find  procedure.  this  can  identify  of  VII.  the  (8.22)  as t h e  into  asymptotic  t h a t ~P  understanding  i s the  i n t e r m s o f t h e p a r a m e t e r s ? T , ^ ... ) M  badly.  VII, in particular  t h a t can  To  we  be o b t a i n e d f r o m  know t h a t t h e s e  Due  (8.38  w i t h c~  T  i s arbitrary.  coarse  0  another  graining  diverge  will  discuss  o f t h e method  constants  constant i n the  from which  we  t o 8.41)  to the a r b i t r a r y  the p o i n t or o r i g i n  our  expansions  t e r m i n a t e this C h a p t e r  t h u s an a r b i t r a r y  attempted,  comparing  t h a t . t h e method o f e x t e n s i o n i s j u s t  And,  extension.  is  identifications  s e r i e s of the type  (written  the boundary c o n d i t i o n s  and  o b t a i n i n Chapter  f u r t h e r w i t h t h e method o f  repeated  i° t h a t we  of Chapter  approximation  quite  ( 7 . 7 3 ) we  c o m p a r i n g t h e method o f e x t e n s i o n w i t h o u r p e r t u r b a t i o n  expansion there;  i  summing o f  diverges  •  L V H V M &  c o u l d go  In t h a t case  (8.47), or  i°,  %  same b e h a v i o r  s u p p o s e t h a t we  extension. type  H  f  behavior,  c  0  i n 6~  corresponding  of  (8.25)  0  T  6  ,  the p e r t u r b a t i o n expansion In order to  satisfy  73.  the  boundary c o n d i t i o n s  ( 8 . 3 8 t o 8.41) and t o a g r e e  the  assumption that the p a r t i c l e s  one  must s t a t e t h a t e~ <r, <r .. . o r r a t h e r 0  Xo/f, j T , . . .  }  X)  h a v e t h e same o r i g i n .  i  boundary c o n d i t i o n s where t h e o r i g i n  arbitrary,  origin (8.51) or  corresponding  Otherwise,  of T  0  i s the p a r t i c l e s I n order  these  are u n c o r r e l a t e d and  t o make t h e o r i g i n o f  w h i c h i s a more m e a n i n g f u l way t o t r y t o c o a r s e  t h e e q u a t i o n s by a p e r t u r b a t i o n  introduce  the  uncorrelated  ( 8 . 3 8 t o 8.41) s t a t e t h a t r e g a r d l e s s o f  t h i s d o e s n ' t make s e n s e .  grain  are i n i t i a l l y  with  the c o r r e l a t i o n s that  e x p a n s i o n , we must  e x i s t e d a t the choice o f  o f To FS(r„  = «) -  IT F°  ^r  + c( s  l ;  r,...)  by c l u s t e r e x p a n s i o n we c a n w r i t e  (8.52)  C'tt,-^,...)  j(r  t )  r  x >  r.  i r  (8.53) a  ..)  f:3l ( ,M,-'-h^^( r v ..) v  r  v  h  %J  sy  •A.  Sandri with  & Fireman,  boundary  would l i k e  see S a n d r i  conditions  to stress that  correlations  similar  their  t o (8.52),  expansion.method (8.53),  e t c . We  these c o r r e l a t i o n s are not i n i t i a l  and i n g e n e r a l  2° body f u n c t i o n t .  , discuss  a given  <j. d e p e n d s upon t h e one s  IX.  THE THEORY OF MAZUR AND B I E L 6  Mazur and B i e l , chain  MB f o r s h o r t , work f r o m t h e BBKGY  (3.18) f o r t h e s h o r t - r a n g e  system  (9,1)' 4 ! + H F - - A L F " ' S  $  S  S  where (9.2)  At  L  this  limit  S  =  £  L-^-  stage;  and  L' T  ( 3 . 2 7 ) h a s n o t been t a k e n .  ^ ( v v ^ s ; ^  to simplify  (9.4)  y  a s shown i n t h e  r e d u c e d momentum ( v e l o c i t y ) (9.3)  w  r  5+/  ; t h e thermodynamic  L e t us i n t r o d u c e MB's  distribution  "v"s [ " ' p ^ t ••• ^  n o t a t i o n MB d e f i n e s  ' " P ' * = ^ ['•'[</*...'.  functions ^ f a ' / V - - ^ * ^  the operators  J  so t h a t t h e - f S may be w r i t t e n (9.5) The  4**  P  s  distribution  (9.6)  F= s  F  5  functions  P* F  5  + (l-P )F 5  r  c a n now be w r i t t e n  =  s  r + A  where (9.7)  hs  (l-P ) s  F  s  -  F -f s  5  5  75. F o l l o w i n g MB we a p p l y t h e o p e r a t o r s P respectively  t o t h e BBKGY c h a i n  and  $  (l-P ) s  (9.1) t o o b t a i n t h e c o u p l e d  sets of equations (9.8)  jf-t  p*H f*  + PHh  s  (9.9)  6  s  - A ?l  $  s  s  f " + A P s  ?  is taken.  range.of  in  s  s  s  l  S  [N -**'jV-> j~-zn 0O  the I n t e r p a r t i c l e p o t e n t i a l ,  P H*F =-fe['''\ *'-' ** s  s  J  s  W  Jl  since the {  |< f =o  ,  s  t h e second  itive  t  finite)  the following  vanishes  One h a s a l s o ; (9.11)  s  U n d e r t h i s l i m i t MB c l a i m t h a t , " due t o t h e  expression  (9.10)  L H  (!-P ;H f +(/-P Jfi ^ -  At t h i s p o i n t t h e thermodynamic l i m i t  finite  5  S  F  ^  S  0  do n o t d e p e n d on  s  ;  L** ' = o 5 +  o f t h e s e we have assumed t h e s y s t e m  to the w a l l s of the container.  thermodynamic l i m i t  ^ j  the coupled  With  equations  i s insens-  ( 9 . 1 1 ) , and t h e  (9.8 & 9.9'"') r e d u c e  to (9.12)  ii = \  (9.13)  $j  pLh"  S  s  + H h' *r{U\iL s  5  5  s  (i-p*)L h *'  T h i s s e p a r a t i o n o f t h e BBKGY c h a i n i n t o  s  s  these  coupled,  e q u a t i o n s i s MB's m a j o r r e s u l t and h e r e we b e g i n o u r d i s c u s s -  7 6 .  ion  ofthe thoery. Let  u s b e g i n by w r i t i n g  diagrams.  For small  F*-e-K  here the arrows i n d i c a t e two o f t h e s l i n e s our  a r e c o n n e c t e d : ( 9 . 1 4 ) i s good enough f o r c o n s i d e r t h e thermodynamic  and we know t h a t t h e g e n e r a l F  s  ations are off i n i t e  lap  i n t h e form  t h e sum o f d i a g r a m s w h e r e a t l e a s t •.  (9.14) under t h a t l i m i t .  of  S  p u r p o s e s b e c a u s e soon we w i l l  limit  correl-  range, each diagram i n t h e second term  ofthe i n t e r p a r t i c l e the operation  approaches t h e form  I f we assume t h a t i n i t i a l  ( 9 . 1 4 ) d e p e n d s on i n t e r p a r t i c l e  Under  i n terms o f  e-K^tZ  7^ 4  $i  s  s , i n c o m p a r i s o n t o N, i t - i s a good  approximation to write the F  (9.14)  F  the general  d i s t a n c e t h r o u g h an o v e r -  potentials  and. i n i t i a l  P w i t h t h e thermodynamic  correlations.  l i m i t the  s  s e c o n d t e r m i n ( 9 . 1 4 ) v a n i s h e s and one h a s f o r f S t h e l i m i t (9.15)  I  s  This l i m i t , fairly  P  F  s  stage.of t h e i r  limit $  J«  s  ^> T T - f  rather identification,  advanced  simplifying (9.16)  - jp. {--jet*,...  H  S  F  (9.10); $  ^ ^  y  P K TT S  -> c»  5  i s made by MB b u t a t a  work.  we f i n d  1  L e t us c h e c k t h e  a n o n - v a n i s h i n g answer F' = j t i »  V  P K Vi(f'+h') s  —  v-><*>  5  >  s  "  7 7. Since  t h i s - l i m i t vanishes  coupled  equations  ( 9 . 1 2 & 9.13) a r e v a l i d  eneous s y s t e m s . the  coupled  f o r a homogeneous s y s t e m ,  o n l y f o r homog-  F o r an i n h o m o g e n e o u s s y s t e m one s h o u l d  p'i'A*  ^ ' ^ - . L  (9.18)  ^ ' +r/.'=ic^  p ' m ' + A i ^  (9.19)  ~ t f /  P  P ' W ' u i ;  r = i ^  5  5  ^ 7 T  d-p'jL'h  because o f t h e redundance for 4  formula  the coupled  s  .  1  h"4l TTr' s  0-P*)L k  -Ui.™  in  s  equations  s+l  ( 9 . 1 5 ) we need no l o n g e r w r i t e o u t  The e x t r a c o m p l i c a t i o n t h a t  eneous system r e s t r i c t s  (9.17, their  appears  9.18 & 9.19) f o r an i n h o m o g -  usefulness f o rthat  We p r e f e r t o w r i t e o u t an a l t e r n a t e t o MB's  case.  theory which i s  s i m p l e r f o r an i n h o m o g e n e o u s s y s t e m and i n t e r c h a n g e a b l e their  theory  f o r a homogeneous s y s t e m .  From t h i s  we assume t h e t h e r m o d y n a m i c l i m i t i s v a l i d taken will  i n the general  F . s  F = s  C  TC + S  C  where  (9.21)  ^  s  = IfF'  S  with  p o i n t on  and h a s been  Our b a s i c s e p a r a t i o n o f t h e  be ( 9 . 1 4 ) f o r w h i c h we w r i t e  (9.20)  have  equations  (9.1?)  the  MB's  78. We s u b s t i t u t e  (9.20) i n t o  (9.22)  IL\  ^  +  t h e BBKGY c h a i n  K < +K C -I «f -I C ^ s  s  5  s  A I  and  (9.1) to obtain  s  s  ^  s  s  s  fA L C 5  S f /  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 w h e r e e a c h t e r m  is free  (9.23)  of correlations  ^+K < =AL < s  5  s  f f f  '+ALi C ' s+  and one w h e r e e a c h t e r m c o n t a i n s a c o r r e l a t i o n  VO-VC^  (9.24)  H e r e , we have  (9.25)  one  s p l i t l/C  L C +> s  =[  S  term t h a t  I ^  5  0  S +  ' into  C "' f [ S  i sfree  cluster  separation have  c  f A L  S  c  C  $+ l  two t e r m s  C  S  +  ''  of correlations  contains correlations. the  S  s  a n d one t e r m  that  I t i s e a s i l y s e e n , when we draw on  e x p a n s i o n ( 3 . 2 1 t o 3.23) t o e x p a n d C , t h a t S  ( 9 . 2 3 & 9.24) i s v a l i d .  First  of a l l C  z  t h e same d e f i n i t i o n  (9.26)  G*=C = C a  J2  and one c a n show by t h e c l u s t e r  expansion of C  s + I  that  this  and &'~  79,  L,?C  (9.27)  With  S+ls  f l -  T T F;'(?"*- i)  *  J "*  ( 9 . 2 7 ) and t h e d e f i n i t i o n . o f ^  (9.23) i s s i m p l y an s f o l d  + K ' F ' --\L  (9.28) For  comparison  equations  t o MB s  that  redundance of the e q u a t i o n f o r  (VK*  lz  ( 9 . 2 1 ) one f i n d s  s  C) 2  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  1  ( 9 . 2 3 & 9.24) f o r t h e s p e c i a l  case  o f a homogeneous  system  (9.29)  ^p--Xi C ' 5  ^f\hl C -=  (9.30) We  s  I  s  compare t h e s e  recall  s+  u  s  ^  s  c  C  s  +  l  e q u a t i o n s w i t h MB's  (9.12 & 9.13) and we S  remove t h e r e s t r i c t i o n s  and  initial  excessive  that the i n t e r p a r t i c l e  c o r r e l a t i o n s are o ff i n i t e  inhomogeneous system C'-O:  Our e q u a t i o n s a r e  more g e n e r a l b e c a u s e we c a n w i t h o u t l o s s o f g e n e r a l -  ity  complication,,  F o r an  t h e n e e d f o r t h e b'  We w i l l  expansion  s e l e c t and sum d i a g r a m s .  the equations  range.  potential  o u r method i s s i m p l e r t h a n MB's  i n t h e MB e q u a t i o n s  here w i t h the c l u s t e r to  s  the e f f e c t of the operators P .  slightly  our  f \ L  for P C  Z  and  because  causes  now compare o u r e q u a t i o n s  method we u s e d i n C h a p t e r V  To b e g i n l e t us w r i t e down  30. (9.31)  •3f+K'F «  (9.32)  ^  +^  (9.33)  4f  +K C5-I C  ,  particle  =AL F*  iz  C -! 2  C  2  3  These e q u a t i o n s equations  n (rr;F± = I  2  3  t z  3  =I  F/F' + A L  2  C  C  F/Fi F3 + A L  3  3  C  2  C  4  :  a r e t o be c o m p a r e d w i t h t h e " s i m p l e r "  ( 5 . 1 1 , 5.12 collision  & 5.13).  We  o b t a i n here  effects without d i f f i c u l t y  u t i o n of. t h e s o l u t i o n f o r C  into  Z  (9.34)  2  jfc |f'F'-- A L .F +  t h e two  by  substit-  (9.31)  l  ia  •Jo  This equation former MB  i s t o be compared w i t h t h e way we o b t a i n e d o u r  equation  (5.24).  L e t us w r i t e o u t t h e  r e s u l t f o r (9.34) f o r t h e s p e c i a l  case  corresponding  o f a homogeneous  system. (9.35)  jf^[Jff^Vwte-^v'' *'i>f:i*,)tia)] 1  0  We  could extend  order i n A .  y-^  00  t h e s o l u t i o n o f ( 9 . 3 4 ) o r ( 9 . 3 5 ) t o any  Though  these  s o l u t i o n s a r e c o r r e c t we  i n c o n v e n i e n t due t o t h e s p e c i a l o p e r a t o r s  l?  c  or the  find  them  81. corresponding  (l-P ) s  which appear i n the  solutions. (5.9)  Working from the  cluster  expansion  more t r a n s p a r e n t  answer.  This  Mazur & Biel.'s. t h e o r y .  equations  concludes  our  gives a  remarks  about  X.  CONCLUSION  The  aim  of t h i s  theory of c l a s s i c a l To  do  t h i s " we  techniques theory  expansion  developed  of Pigogine  Bogoliubov's  statistical  mechanics.  a t h e o r y t h a t .stems f r o m t h e We  case.  We  have found  terms d i v e r g e .  expansion  Even t o o b t a i n t h e  Boltzman  This evidence  introduced,  g i v e s us  & Biel;  to Bogoliubov's.  a theory  and  sufficient  d e r i v a t i o n of the Boltzman e q u a t i o n  These r e s u l t s  techniques  of  Bogoliubov's  and  doubts c a r r y over  d i s c u s s i o n of the t h e o r y of S a n d r i & Frieman; similar  our  have shown t h a t  c o r r e c t i o n t e r m s must be  doubt i f B o g o l i u b o v ' s  is  diagram  g e n e r a l f o r us t o o b t a i n t h e t h e o r y  diverges quite badly.  significant.  to d i s c u s s  and. c o w o r k e r s .  as a s p e c i a l  equation.Markowian these  irreversible  sufficiently  Bogoliubov  t h e s i s was  we  To  to  their  show t h e v e r s a t i l i t y  is  our theory of  our  have d i s c u s s e d the t h e o r y of Mazur  t h a t has  little  resemblance to  Bogoliubov's.  83. BIBLIOGRAPGY 1.  N.N. B o g o l i u b o v , S t u d . S t a t . Mech., 1, 11 (1962).  2.  I . P r i g o g i n e , " N o n - E q u i l i b r i u m 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. S e v e r n e , . P h y s i c a , 11, 377 (1965).  4.  G. S a n d r i , Ann. Phys., 24_, 332 (1963); gj±> 3^0  5.  E.A. F r i e m a n , J . Math. Phys., i±, 410 (I963).  6.  P. Mazur & J . B i e l , P h y s i c a , 3_2, 1633  7.  I . P r i g o g i n e & P. R e s i b o i s , P h y s i c a , 22, 629 (1961).  g.  J . S t e c k i & H.S. T a y l o r , Rev. Mod. Phys., 3J7_, 762 (I965).  9.  J . B r o c a s & P. R e s i b o i s , P h y s i c a , 32, 1050 (1966).  (1963).  (1966).  10.  E. Braun & L.S. G a r c i a - C o l i n , Phys. L e t . , 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, g l 7 (I963).  13.  E.G.D. Cohen & J.R. Dorfman, J . Math. Phys., (1967).  232  84. Appendix I . In  this  (c) v e r t e x  a p p e n d i x we show t h a t d i a g r a m s w i t h  (3.6) i n t e g r a t e t o zero.  diagram with  a type  Look a t t h e e x p r e s s i o n  ji?'f^-.-  i  labelled n  ^  v  at the i n t e g r a l  (Al.l)  D....J(f*,,.})te^^v'»^. i J  Here, the s e t - j r ] left  L e t us c o n s i d e r a  (b) o r ( c ) v e r t e x where t h e l i n e  j f i r s t appears. and  a-type  represents  of the vertex  with the help (A1.2)  G({•*,*•})  :  the p a r t i c l e l a b e l s  considered.  This  expression  l f  3  )  r  £  Use o f G r e e n ' s t h e o r e m i n t h e v e l o c i t y  vanishes  then  simplification  vertex  vanishes  sufficiently fast  calculation  G({*,*}) integral  gives  S  IT  The i n t e g r a l  becomes,  of the r e l a t i o n (2.9)  D--jr{ . y }e 'M^JV,j^;j.  (A1.3)  used t o t h e  o f (b) type  because t h e v e l o c i t y at i n f i n i t y . vertices  We h a v e t h u s a  and by p e r f o r m i n g  t w i c e we s e e t h a t d i a g r a m s w i t h  vanish.  distribution  the  a (c) type  85. Appendix I I In gives To  this  (7.56  And  time the f i r s t  obtain i n ordinary  order  obtained  time a s o l u t i o n f o r F  o r where i = 0 .  1  ' from  One h a s t a k i n g i n t o  o  i n (7.65)  term obtained  t h e s p e c i a l c a s e o f ( 7 . 6 5 ) where & arid t  same o r i g i n ,  for  a p p e n d i x we show t h a t t h e F  i n ordinary  consider  the  -  i n (6.10).  F ' we 1  have t h e account  & 7.57 w i t h 7.39 & 7.40) f o r t h e f u n c t i o n s C ^ ^ F ' l o ) a n d  definition  of the operator  the f o l l o w i n g  expression  F^e^F'k)  f o r the corresponding  F  we h a v e , w i t h  some c h a n g e s 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. With notation write (A2.3)  (A2.2)as F  2 1  of the type  ( 5 . 1 8 , 5.19, 5.20, e t c . ) we c a n  follows  - e - * ^ , ew**'(L„*Lj* 2 ° r1 A i . 2 1  -  !_  2 ,  £  3  3  0  -—r-G  r-tf  ID, Let  us e x p a n d t h e f i r s t  t e r m i n (A2.3) c o m p l e t e l y , o r by  comparison o f (5.24) w i t h ( A 2 . 4 ) F '= e - ^ { M  'DC*  2  a 2 — 6  ( 5 . 1 7 ) one h a s  DOC  2  f  + X X X  *•••}*  3  iE n: ii: tn'ni iD f  7  f  +  TIT *  +  2  +  87. One  sees t h a t  (A2.4) c o n t a i n s diagrams w i t h f r a g m e n t s  c o n n e c t i n by one  line.  We  know f r o m C h a p t e r I V t h a t  behavior always c o n t r i b u t e s to lower order terms; case to F . 2 0  Thus we  should,  that such  in this  w i t h the second term i n (A2.4),  be a b l e t o c a n c e l d i a g r a m s w i t h f r a g m e n t s t h a t c o n n e c t I n by one  line.  F u r t h e r , as a r e m a i n d e r  should  o b t a i n the Markowian c o r r e c t i o n terms Let  of  (A2.3  o r A2.4)  so t h a t t h i s  With t h i s •we  the  us i n t r o d u c e  we  shown i n (6.10)  i n the second  term  identity  s e c o n d t e r m may  expression  i n the second term  be w r i t t e n i n t h e f o l l o w i n g way  f o r t h e second term i n (A2.3) o r  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  t h a t c o n n e c t i n by one  line;  one  has  (A2.4)  fragments  By  c h a n g i n g t h e o r d e r o f t h e i , and  s e c o n d t e r m , we first  see t h a t  i  z  integrations  i n the  ( A 2 . 8 ) i s t h e same e x p r e s s i o n as t h e  o r d e r term i n (6.10).  

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