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Kinetic equation for a classical gas with a long range attraction. Elliott, Richard Amos 1966

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KINETIC EQUATION FOR A CLASSICAL GAS WITH A LONG RANGE ATTRACTION by RICHARD AMOS ELLIOTT B.A. Queen's University, 1960 B.Sc.(Honours) Queen's University, 1961 M.Sc. Queen's University, 1963 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of PHYSICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1966 In presenting this thesis i n pa r t i a l fulfilment of the requirements f o r ' an advanced degree at the University of Bri t i s h Columbia,, I agree that the Library shall make i t freely aval]able for reference and study, I further agree that permission, for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of PMYSTCS The University of Br i t i s h Columbia Vancouver 8, Canada The University of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of RICHARD AMOS ELLIOTT B.A., Queen's University, i960 B.Sc, (Honours), Queen's University, 1961 M.Sc, Queen's University, 1963 THURSDAY, SEPTEMBER 15, 1966 at 10.30 A.M. I IH ROOM;301, HENNINGS BUILDING COMMITTEE IN CHARGE Chairman: I. McT. Cowan F. A. Kaerapffer L. de Sobrino R. Barrie R. F. Snider Ro Howard J . M. McMillan External Examiner: M. K. Sundaresan Professor of Physics Department of Physics Carleton University Research Supervisor: L. de Sobrino KINETIC EQUATION FOR A CLASSICAL GAS WITH A LONG RANGE ATTRACTION ABSTRACT A c l a s s i c a l gas whose p a r t i c l e s interact through a weak long range a t t r a c t i o n and a strong short range repulsion i s studied* The L i o u v i l l e equation i s solved as an i n f i n i t e order perturbation expansion* The terms i n t h i s series are c l a s s i f i e d by Prigogine type diagrams according to t h e i r order i n the r a t i o of the range of the inte r a c t i o n to the average in.ter= p a r t i c l e distance„ I t i s shown that., provided the range of the short range force i s much less than the average i n t e r p a r t i c l e distance which i n turn i s much less than the range of the long range force s the terms can be grouped into two classes* The one class, represented by chain diagrams., constitutes the s i g n i f i -cant contributions of the short range interaction; the other, represented by r i n g diagrams, makes up*, apart from a self"consistent f i e l d term, the s i g n i f i c a n t contributions from, the long range force* These con-tributions are summed to y i e l d a k i n e t i c equation* The orders of magnitude of the terms i n t h i s equation are compared for various ranges of the parameters of the system* Retaining only the dominant terms then produces a set of eight k i n e t i c equations each of which i s v a l i d f o r a d e f i n i t e range of the parameters of the system. The short-time s t a b i l i t y of the system i s examined and a c r i t e r i o n f o r s t a b i l i t y obtained. The e q u i l i -brium two-particle correlation function and an equation of state are determined, the l a t t e r being compared to the Van de Waals equation of state. PUBLICATION Kinetic Equation of a Van Der Waals Gas, B u l l . Am. Phys* S o c , 11 322 (1966). GRADUATE STUDIES F i e l d of Study: Physics Electromagnetic Theory Plasma Physics Special R e l a t i v i t y Theory S t a t i s t i c a l Mechanics Advanced Quantum Mechanics G. M. Volkoff Luis de Sobriao H. Schmidt Robert Barrie F. A. Kaempffer i i ABSTRACT A c l a s s i c a l gas whose p a r t i c l e s interact through a weak long range a t t r a c t i o n and a strong short range repulsion i s studied. The L i o u v i l l e equation i s solved as an i n f i n i t e order perturba-t i o n expansion. The terras i n t h i s series are c l a s s i f i e d by Prigogine type diagrams according to t h e i r order i n the r a t i o of the range of the i n t e r a c t i o n to the average i n t e r p a r t i c l e distance. It i s shown that, provided the range of the short range force i s much less than the average i n t e r p a r t i c l e distance which i n turn i s much less than the range of the long range force, the terms can be grouped into two classes. The one cla s s , represented by chain diagrams, constitutes the s i g n i f i -cant contributions of the short range i n t e r a c t i o n ; the other, represented by ring diagrams, makes up, apart from a s e l f -consistent f i e l d term, the s i g n i f i c a n t contributions from the long range force. These contributions are summed to y i e l d a ki n e t i c equation. The orders of magnitude of the terms i n t h i s equation are compared for various ranges of the parameters of the system. Retaining only the dominant terms then produces a set of eight k i n e t i c equations each of which i s v a l i d f or a d e f i n i t e range of the parameters of the system. The short-time s t a b i l i t y of the system i s examined and a c r i t e r i o n f or s t a b i l i t y obtained. The equilibrium two-particle c o r r e l a t i o n function and an equation of state are determined, the l a t t e r being compared to the Van der Waals equation of state. i i i TABLE OF CONTENTS Page A b s t r a c t . i i Table of Contents i i i L i s t of T a b l e s v L i s t of F i g u r e s v i Acknowledgements v i i Chapter I - I n t r o d u c t i o n 1 Chapter I I - General Equations of E v o l u t i o n 5 S e c t i o n 1: The Formal S o l u t i o n of the L i o u v i l l e E q u a t i o n 5 S e c t i o n 2: Reduced D i s t r i b u t i o n F u n c t i o n s 11 S e c t i o n 3: Diagram R e p r e s e n t a t i o n 16 S e c t i o n 4: E q u a t i o n of E v o l u t i o n of the Reduced F o u r i e r C o e f f i c i e n t s 26 Chapter I I I - E v o l u t i o n of a C l a s s i c a l Gas with a Long Range A t t r a c t i o n . 35 S e c t i o n 1: The System and I n t e r a c t i o n s 35 S e c t i o n 2: Ring and Chain Diagrams 42 S e c t i o n 3: The Asymptotic C o l l i s i o n , C r e a t i o n and D e s t r u c t i o n Terms 56 S e c t i o n 4: Summation of the Rings 61 S e c t i o n 5: Summation of the Chains , 69 S e c t i o n 6: The Long-time Eq u a t i o n s . H-theorem ... — . 77 S e c t i o n 7: The Short-time Eq u a t i o n . I n s t a b i l i t y .... 90 S e c t i o n 8: Two P a r t i c l e C o r r e l a t i o n s at E q u i l i b r i u m . . 99 S e c t i o n 9: The Equ a t i o n of S t a t e 107 iv Page Chapter IV - Discussion HO Bibliography 1 1 6 Appendix A Appendix B 1 2 2 V LIST OF TABLES Page Table I - Relative Order of Magnitude of the S e l f -Consistent F i e l d and C o l l i s i o n Terms 80 Table I I - parameters for Argon and Ethane 113 LIST OP FIGURES Page 0 , 18 Figure 2.1 • Figure 2.2 1 9 Figure 2.3 ^ Figure 2.4 - * * 2 * 0 _ 23 Figure 2.5 0 - 24 Figure 2.6 28 Figure 2.7 44 Figure 3.1 45 Figure 3.2 „ „ . 5 5 Figure 3.3 62 Figure 3.4 « 0 _ 69 Figure 3.5 • . , 118 Figure A . l A 0 120 Figure A, ^  • * v i i ACKNOWLEDGEMENTS I wish to express my appreciation to Dr. Luis de Sobrino for suggesting this problem and for his continued guidance and valuable advice. Financial assistance in the form of a Queen Elizabeth Scholarship (Dr. H.R. MacMillan - donor) and research grants from the National Research Council of Canada is gratefully acknowledged. - 1 -CHAPTER I INTRODUCTION A model of a gas which took account of the long range a t t r a c -t i v e forces between molecules as well as t h e i r repulsive cores was f i r s t studied by Van der Waals"*". The semi-quantitative success enjoyed by the model which he proposed stimulated an intensive study, e s p e c i a l l y i n connection with the problem of condensation. Recent progress has been made i n the understanding of the e q u i l i -brium properties of the Van der Waals gas, notably by: Kac, 2 3 4 Uhlenbeck and Hemmer , Van Kampen j and Lebowitz and Penrose . The non-equilibrium problem, which i s of importance i n understand-ing the dynamics of condensation has been considered by Sobrino . He began with a k i n e t i c equation that takes account of the long range forces only through a s e l f - c o n s i s t e n t f i e l d approximation and assumes that p a r t i c l e s before making a strong c o l l i s i o n are correlated only because of the reduction i n the avail a b l e amount of phase space due to the non-zero s i z e of the p a r t i c l e s . In the l i g h t of these recent developments i t was considered desirable to examine the problem of the non-equilibrium behaviour of gases with strong short range repulsive and weak long range a t t r a c t i v e forces, the aim being to derive, i n as rigorous a manner as possible, k i n e t i c equations describing the development of such systems and to determine the range of v a l i d i t y of these equations. In thi s way i t was hoped that the conditions under which Sobrino's assumptions were j u s t i f i e d could be determined - 2 -and at the same time to see i f the correlations due to the long range forces could in some measure explain the behaviour of gases near the c r i t i c a l point. The study of non-uniform gases initiated by Maxwell6 culmina-ted in the famous Boltzmann equation? which describes the evolution of the one-particle distribution function of a gas as due to a flow and a collision term. The collision term has usually been approximated by considering only interactions of a few particles at a time. If the particles interact through a long range force and the gas is not extremely dilute this is clearly an unjustified assumption. The addition of a self-consistent field approximation is a decided improvement but a complete treatment of the long range part of the force is desirable. The Prigogine theory of non-equilibrium statistical mechanics** has been applied by Balescu®>*0 to a problem involving long range forces, namely a plasma, but a kinetic treatment of systems involving both long and short range forces apparently does not exist. The emphasis of Balescu's work was on spatially homogeneous plasmas although he also considered non-uniform systems. Severne-^ has developed the Prigogine theory for non-uniform systems in a consistent and elegant manner. This paper is the starting point of this inves-tigation. We begin in Chapter II by obtaining a formal solution of the Liouville equation as a perturbation expansion in the interaction. Reduced distribution functions and correlation functions are defined and their equations of evolution determined from this - 3 -f o r m a l s o l u t i o n . The time r a t e of change of the one p a r t i c l e d i s t r i b u t i o n f u n c t i o n i s g i v e n by an e q u a t i o n c o n s i s t i n g of a f l o w t e r m , a s e l f - c o n s i s t e n t f i e l d te rm, a d e s t r u c t i o n term and a c o l l i s i o n t e r m . The l a t t e r two of these are 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 s e r i e s w i t h the d e s t r u c t i o n term a r i s i n g s o l e l y from the i n i t i a l s t a t e c o r r e l a t i o n s . The c o r r e l a t i o n f u n c t i o n s obey e q u a t i o n s i n v o l v i n g a d e s t r u c t i o n term and a c r e a t i o n t e r m , the l a t t e r b e i n g due to c o r r e l a t i o n s c r e a t e d t h r o u g h the i n t e r -a c t i o n of the p a r t i c l e s . The d e r i v a t i o n of these e q u a t i o n s i s 8 9 f a c i l i t a t e d by the use of p r i g o g i n e type diagrams ' . A s i d e from a s l i g h t m o d i f i c a t i o n o f the diagram t e c h n i q u e the d e r i v a -t i o n i s t h a t of S e v e r n e . In C h a p t e r I I I the development of C h a p t e r I I i s a p p l i e d to the s p e c i a l case of a gas whose 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 s t r o n g s h o r t range r e p u l s i o n and a weak l o n g range a t t r a c t i o n , the l e n g t h s c a l e b e i n g the average i n t e r p a r t i c l e d i s t a n c e . In s e c t i o n 2 the terms i n the e x p a n s i o n of the c o l l i s i o n term are c l a s s i f i e d by means o f diagrams a c c o r d i n g t o t h e i r o r d e r i n the r a t i o of the range o f the i n t e r a c t i o n to the average i n t e r p a r t i c l e d i s t a n c e . T h i s a l l o w s the s e l e c t i o n of two t y p e s of c o n t r i b u t i o n s , one r e p r e s e n t e d by r i n g diagrams f o r the l o n g range p a r t of the i n t e r a c t i o n and the o t h e r r e p r e s e n t e d by c h a i n diagrams f o r the s h o r t range i n t e r a c t i o n . T h i s p r o c e d u r e i s a l s o a p p l i e d to the c r e a t i o n t e r m s . In s e c t i o n 3 the b e h a v i o u r o f the c o l l i s i o n , c r e a t i o n and d e s t r u c t i o n terms f o r a s y m p t o t i c a l l y l o n g t imes i s e x a m i n e d . I t i s shown t h a t the d e s t r u c t i o n terms can be i g n o r e d - 4 -in the long-time development and the terms in the expansion of the creation and collision terms can be simplified. This sim-plification allows the contributions to the collision term represented by rings and chains to be summed exactly by techni-ques developed by Balescu and Taylor 1 2 and Prigogine and Henin 1 3 respectively. These summations are performed in sections 4 and 5. In section 6 the results of the preceeding sections of Chapter III are collected and a set of kinetic equations produced together with conditions under which they are applicable. An H-Theorem is proved for these equations. Section 7 deals with the question of the i n i t i a l stability of the system. In section 8 a calculation of the equilibrium two body correlations is performed and this is used in section 9 to obtain the equation of state of the system. The results are discussed in Chapter IV. - 5 -CHAPTER II GENERAL EQUATIONS OF EVOLUTION 1. The L i o u v i l l e Equation and Its Formal Solution The s t a t i s t i c a l mechanical state of a system of a large number of p a r t i c l e s i s s p e c i f i e d by the density of i t s representa-t i v e s t a t i s t i c a l ensemble i n phase s p a c e 1 4 This phase density i s a function of N positions, N v e l o c i t i e s and time, N being the number of p a r t i c l e s i n the system. Accordingly t h i s function i s c a l l e d the N-pa r t i c l e d i s t r i b u t i o n function and i s written ; v,, ; t) « I* *-s *° t>e interpreted as the p r o b a b i l i t y density that the system w i l l be i n a microstate such that p a r t i c l e 1 have po s i t i o n i , and v e l o c i t y <&r, p a r t i c l e 2, p o s i t i o n 2-^ and v e l o c i t y ^ and so on fo r a l l p a r t i c l e s i n the system. The values of macroscopically observable quantities are calculated by taking the ensemble average of the quantity. That i s , i f &(l2f]>l&l) i s a function of the p o s i t i o n and v e l o c i t y co-ordinates the observable value of 6 i s found by inte g r a t i n g G(t*l, dr}) weighted with FN over a l l positions and v e l o c i t i e s : (2.1) <G(T)> ~ jcdxf&vr G(t*l>L*l) %(*.~-**>J4r,-- *~;0 Quantities c a l c u l a t e d i n t h i s way are accurate f o r very large N and the method i s thus applicable only to very large systems. This i s no hardship as any systems of i n t e r e s t to us consist of the order of 1 0 2 3 p a r t i c l e s and one can take the so c a l l e d thermodynamic l i m i t (2.2) N —*o°, . f l — * oo , N / S l = c = f i n i t e - 6 -where l l i s the volume of the system and c i s then the average concentration. We consider a system of N i d e n t i c a l i n t e r a c t i n g p a r t i c l e s whose Hamiltonian i s (2.3) (2.4) This presupposes that the interactions between p a r t i c l e s are ce n t r a l two body forces. Also i t i s assumed that the range of the interactions i s much l e s s than the siz e of the container. 15 The well known L i o u v i l l e theorem of c l a s s i c a l mechanics states that the phase density develops i n time according to the equation N (2.5) l!k - _ J ,O,. ^3. Or defining the L i o u v i l l e operator as (2.6a) (2.6b) we have (2.7) L = L 0 + 1SL A/ + * fz* ^ a t L ^ The formal s o l u t i o n of the L i o u v i l l e equation i s accomplished by Laplace transforming (2.7) to get - 7 -(2.8) - i / r ( t s o ) + z fate'** fr„(t) = L U*e1** f„Ct) or (2.9) j d t e 1 * * F„(t) - -i (L-2)' F„(o) It i s e a s i l y shown8 that the L i o u v i l l e operator L i s Hermitean and thus has only r e a l eigenvalues, The resolvent operator (L- i s therefore bounded for a l l non-real z and has s i n g u l a r i t i e s only on the r e a l z-axis . Application of the inverse Laplace transform to (2.9) gives oof e.i (2.10) frCt) ^ ~L J d i e ' l i t (L-i)" F „ ( 0 ) ; S > 0 The exponential factor Q''2-' , with t >c?„ allows us to close the path of i n t e g r a t i o n , indicated i n (2.10), along a semicircle of i n f i n i t e radius i n the lower half-plane.* Designating t h i s closed contour as P and integ r a t i n g around i t i n a counter clockwise manner produces the following expression for FN(t) The operator i d e n t i t y (2.12) ft''-B" = ft'(8-A) 5"' permits us to write, using the d e f i n i t i o n of L given i n (2.6a), (2.13) ( L e - ? r ' - ( L - 0 " ' =0..-2)"'a«$L ( L - * r ; * If (L-i)'1 Fw(o) has a di s c o n t i n u i t y across the r e a l axis i t s a n a l y t i c continuation i n t o the lower half-plane i s to be used i n the c a l c u l a t i o n of the contour i n t e g r a l . - 8 -By i t e r a t i o n one obtains (2.14) ( L - * ) - = f (rlT a.-0'' [SL(L.-'T'f 7)=0 Patting t h i s into (2.11) y i e l d s a formal perturbation s o l u t i o n of the L i o u v i l l e equation: (2.15) fr(t)= < j | £ ( L - * f [-*iL(L.-*r'T ^ ( 0 ) We s h a l l now proceed to expand FN i n eigenfunctions of the unperturbed L i o u v i l l e operator L 0 . As only asymptotically large systems are of i n t e r e s t (see ( 2 . 2 ) ) i t can be argued that the e f f e c t on any l o c a l quantity by p a r t i c l e s near the surface of the container i s n e g l i g i b l e . This allows freedom of choice f o r the boundary conditions which we take to be periodic. The container i s assumed cubic f o r s i m p l i c i t y . The eigenfunctions of the operator L e = -<Z ^ * jb^ a r e c l e a r l y of the form H i A. - Zj. (2.16) <p - IX & ' and have the eigenvalues (2.17) 3 U J = | V * > Imposing the periodic boundary conditions (2.18) ^ a ^ ) - ^ ( f * + ^ ? ) = + requires that - 9 -(2.19) A. = % + + > being integers. The eigenvalue spectrum i s there-fore d i s c r e t e . Expanding ( * , - • • O i n the eigenfunc-tions of Lo, which form a complete set i n configuration space, one obtains the Fourier s e r i e s (2.20) F„(*.s„; Si" L / ° 4 . . . 4 ( * • . . . . * • „ ; * ) e with the Fourier c o e f f i c i e n t s f£ being given by C2.21) /%,... ± J*. - J(U) F„U,-^~>* e The formal s o l u t i o n of the L i o u v i l l e equation (2.15) now can be written with the d e f i n i t i o n of the matrix element being (2.23) <CiilfiU*l> = J l ' l ^ f e ^ ' ' ^ ' ' " ^ ' ^ The matrix elements of the operator ( L Q - Z ) " 1 follow immedi-at e l y from (2.17) and (2.23). They are (2,24.) <[*'3/tt.-2)"7^*J> - ( J L ^ • ^ - Z ) /I - 10 -f K i -wi th 64 being the Kronecker d e l t a . Obviously t h i s matrix i s diagonal. The i n t e r a c t i o n matrix elements are also e a s i l y calculated. One defines the Fourier transform of the in t e r a c t i o n p o t e n t i a l to be (2.25) V d ^ - ^ J ) = £ Vk e - ' so that also (2.26) /14L = i f 2- ^ * . ( | - - \ - . \ Substituting t h i s i n t o (2.23) one obtains: (2.27) <w]\-^L\[r]> £ I 0>7*;-4) C*r-4r*-' -fL where (2.28) = | V ^ ' ( ? y k ) It i s c l e a r that <SL has no diagonal matrix elements but determines t r a n s i t i o n s where only two wave vectors are modified. This i s a consequence of the interactions being two body. The sum of the wave vectors remains i n v a r i a n t , Z >^ - 2_ 27 • This i s due to the f a c t that depends only on the distance between p a r t i c l e s j and n and i s thus invariant with respect to tr a n s l a t i o n . In fac t the conservation of wave vectors i s equiva-^ lent to the conservation of momentum. - 11 -2. Reduced D i s t r i b u t i o n Functions A p r e s c r i p t i o n f o r c a l c u l a t i n g macroscopic observables was given i n equation (2.1). However these quantities depend not on a l l N v e l o c i t i e s and positions but on only a few of them 1 7 This means that some of the information contained i n FN i s redundant and what i s needed i s a weighting function involving only those positions and v e l o c i t i e s on which the quantity to be averaged depends. We accordingly define reduced d i s t r i b u t i o n functions as follows: (2.29) - r V s , i ^ - . •• > t ) In the l i m i t of a large system, (2.2), t h i s can be written (2.30) ^te.-**;*.'"**'.*; = N ifezF suvf*' FN(x,-z»' and as FN i s normalized to unity i t follows that f s^< i s normalized to Ns , The following abbreviated notations w i l l frequently be used. (2.31) 4s0.->s) = 4 i S(x,-..¥ w ^--.Vi) and <f>l#t) = These reduced d i s t r i b u t i o n functions allow us to calcu l a t e values f o r any macroscopic quantity i n which we may be interested. For example the l o c a l number density at ^  i s given by (2.32) 7 7 U , t ) = fJls.ctv, £(*-*,) -f,(x, >v,}t) - 12 -and the pressure tensor b y 1 7 (2.33) Pte'*^ - M-Z.)™** Two conditions are imposed on the reduced d i s t r i b u t i o n functions at the i n i t i a l time. A) There ex i s t s no v e l o c i t y correlations independent of the r e l a t i v e position of the p a r t i c l e s . This implies the following f a c t o r i z a t i o n property fo r the reduced d i s t r i b u t i o n functions -fss, > s > s ; (2.34) 5 ' and i s generally c a l l e d the 'weak* molecular chaos assumption. B) At the i n i t i a l time the reduced d i s t r i b u t i o n functions remain f i n i t e i n passing to the l i m i t of an i n f i n i t e system (2.2), thus ensuring that a l l intensive proper-t i e s of the system are i n i t i a l l y f i n i t e . Substituting the Fourier expansion of the N-particle d i s t r i b u t i o n function (2.20) into the d e f i n i t i o n of the reduced d i s t r i b u t i o n functions we have (2.35) I = J [ consistent with t h i s we define reduced Fourier c o e f f i c i e n t s - 13 -( 2 * 3 6 ) ) - 4 , - A S such that (2.37) ^ s , ( X ( - - . - s ; - ; ' - ' ^ 0 = cs Z /f^i^-'^e The weak molecular chaos hypothesis (A) imposes a factoriza-t i o n condition on the i n i t i a l reduced Fourier c o e f f i c i e n t s (2.38) /^(v,".*Wo) = -*>0) JX, The passage to the l i m i t of an i n f i n i t e system implies that the wave vector spectrum becomes continuous. A summation over a wave vector then goes over to an i n t e g r a l (2.39) T —> r This introduces divergent volume factors into (2.37) so that i n order for the reduced d i s t r i b u t i o n functions to s a t i s f y condition fc) /~ /<\ (B) the must be related to c o e f f i c i e n t s defined by (2.40, = ( & ) ' / % < * • - * * ) the c o e f f i c i e n t s /^^j , which we c a l l regular, remaining f i n i t e i n passing to the l i m i t (2.2) of an i n f i n i t e system. The s p a t i a l c o r r e l a t i o n s between p a r t i c l e s can be expressed by the c o r r e l a t i o n functions 3 S ( V ; 5 ) defined by the c l u s t e r 18 decomposition - 14 --RO) - 3,(0 f ao.z; = 3.(0 +• e j ' >2) (2.41) -f3(',2>2) - ^ ( O ^ W 9 , 1 2 ) + 9 , 0 ) 9 a ( 2 > 3 ) -r %tW*>D + "3,(3) 5t('p2) + g 3 ( i J 2; 3) s ' where the summation i s over a l l d i s t i n c t p a r t i t i o n s of the set s into subsets s* to each of which i s associated a function °\ 0 > " > S ' ) i n the product 7T . s' s' The d e f i n i t i o n s (2.41) along with (2.37) define a cl u s t e r decomposition of the reduced Fourier c o e f f i c i e n t s . (2.42) / ? i 4 K , t f O = AJ*,) * Y 4 ( 4 l U . , * . ) 4 , . . - t f s ) = £ 7T s ' The 0 ^ thus defined are the Fourier c o e f f i c i e n t s of the c o r r e l a t i o n functions (2.43) - C I v , ) e ' - ' ^ — * * « - * - ) - 15 -and associated with the are regular coefficients (see (2.40)) (2 . 4 4 ) = ( ^ 4 V « - * J that remain finite as JT —> e° . If the particles of the system are uncorrelated then f s ( V ' " s ) = 7T f , 0 ) and also This we term the correlation vacuum and rewrite (2.42) separat-ing out the vacuum term explicitly (2.45) . ^ W - ^ ) = ^Jc-v... - V S ) 4- f ^ U - V c ) with (2.46) ^ J ) ( V i . . . 4 r 4 ) - 7T / ^ * ) and (2 . 4 7 ) r f * ' ^ - « i ) = i ' 7 T yf;;''<v,•• •*•..) where ^  means the sum over a l l distinct partitions of the set s into subsets s' except the completely factorized vacuum term A1*} > (all s» = 1 ) . - 16 -3. Diagram Representation The N-particle d i s t r i b u t i o n function, as stated e a r l i e r contains more information than can be used. Thus the formal s o l u t i o n f o r FN(t) or equivalently the time development of the N-body Fourier c o e f f i c i e n t s i s not act u a l l y required. What i s needed i s an expression for the evolution of the reduced Fourier c o e f f i c i e n t s , /f^ . The d e f i n i t i o n of the reduced c o e f f i c i -ents (2.36) together with the perturbation expansion for a s £ i v e n * n (2.22) y i e l d s the following expression for : (2.48) •••*•,•;*> = -=L j>rf*e jM r z i <^\(Lr^r[^L(L^)'Tmy &...jifr-*"'*0) Subsequently we s h a l l consider only those reduced c o e f f i c i e n t s which have s = s'. This merely serves to si m p l i f y the notation since one can obtain f£® ( < r , . ^ s . ) , s < s ' from («r, • >• -tr s , ) simply by s e t t i n g the wave vectors l a b e l l e d s+-i,- '-> equal to zero. Also <r s /) , s > s ' i s obtained by integrating / ^ ^ ( o r . . . - w s ) over the v e l o c i t i e s ><f s V i , -4/s . The notation (2.49) ft%(M)t) = w i l l be used. We rewrite equation (2.48) to read (2.50) = ~ ^ « - ' " •I f < £ * 5 | ( L . - « - ' [ - « L a . - i r ^ | r t v > / i - . . . A - ' * - , , f - ; < , ) - 17 -We now proceed to represent the terms i n t h i s expansion by Prigogine type diagrams8,9. These diagrams are constructed i n the following manner: I) With each matrix element of the propagator (Lo-z)"" 1, < f&Jl(La-zV' | Ck'\y > i s associated a set of l i n e s running from r i g h t to l e f t - one l i n e f or each non-zero wave vector which i s l a b e l l e d with the index of that wave vector. II) With each matrix element of the in t e r a c t i o n SL?^ , | f £ ' 5 ) > » i s associated a vertex consisting of the confluence of l i n e s l a b e l l e d by j and n i n the set lil and the s i m i l a r i l y l a b e l l e d l i n e s i n the set w . As an example consider a p a r t i c u l a r f i r s t order contribution to a p a r t i c u l a r reduced c o e f f i c i e n t , say the contribution from an i n t e r a c t i o n between p a r t i c l e s 1 and 2 to which we write \ the superscript i n d i c a t i n g the f i r s t order and the subscripts the p a r t i c l e s involved. -3 Using the d e f i n i t i o n s of the matrix elements (2.24) and (2.27) t h i s becomes - 18 -. -i it [, , SN- 3 Because of the f a c t o r i z a t i o n condition (2.38) and the fac t that /°o(^;j0) i s normalized to unity we have (2,52) f>kh kU,^,^t) 0) ' A The diagram representing t h i s contribution i s shown i n F i g . 2,1. 3 ( 3 F i g . 2.1 Our diagrams d i f f e r from those used by the Prigogine school i n that we do not separate out and represent by a separate diagram the value k = o i n the summations over the wave vectors. That i s , using the above example we do not draw a separate diagram to represent the contribution when A° or h\ are zero. However, since we are looking for contributions to s p e c i f i e d reduced c o e f f i c i e n t s the values of a l l the wave vectors at the l e f t of a diagram are f i x e d and therefore the d i s t i n c t i o n between diagrams which have one and two l i n e s on the l e f t i s maintained. We then have two basic types of diagrams, type A and type B. - 19 -These are displayed as F i g . 2.2a and F i g . 2.2b. f - i - -t (a) (b) (c) F i g . 2.2 The diagram of F i g . 2.2c would also appear to be important. However, i t can be shown3 that because of the integration over N-s v e l o c i t i e s t h i s type of diagram does not contribute to f c t l ^ ' ' u n l e s s one of i t s indices j or n i s one of the set of indices (>,••• < s') . And since we are concerned only with reduced c o e f f i c i e n t s with s = s'and a l l wave vectors l a b e l l e d from 1 to s are non zero, t h i s type of diagram does not contribute. Each term i n the perturbation expansion ultimately connects on the r i g h t to an i n i t i a l reduced Fourier c o e f f i c i e n t /f^ (tel)O) (c.f.(2.52)). If the c l u s t e r decomposition of t h i s c o e f f i c i e n t as given by (2.45) with I^ written i n the f u l l y expanded form (2.47) i s inserted f o r /^vj ( r V ] j 0) a set of contributions each of which a r i s e s from a d i s t i n c t c o r r e l a t i o n pattern ( i . e . a term i n the se r i e s (2.47)) at the time t = o r e s u l t s . Each of t h i s set of contributions i s represented by a d i s t i n c t diagram i n which the s o l i d l i n e s representing wave vectors of correlated p a r t i c l e s are connected by v e r t i c a l dotted l i n e s . For example the diagram of - 20 -F i g . 2 a3 represents a f i f t h order contribution to a r i s i n g from the i n i t i a l c o r r e l a t i o n pattern feiv,;o) If^.K^.; o) (v v;o) The same diagram without the dotted l i n e between the s o l i d l i n e s l a b e l l e d 2 and 3 F i g . 2„3 would represent a f i f t h order contribution to jO a r i s i n g from the completely f a c t o r i z e d c o r r e l a t i o n vacuum fc«t.;o) f^**.') A.(tf3J<>) ^ ; ^ ^ ° ) • The v e r t i c e s of the diagrams are considered to be ordered from r i g h t to l e f t . Breaking the diagram of F i g . 2.3 at the arrows t 3 r e s u l t s i n two fragments with the one on the r i g h t representing a contribution to the "intermediate state" 43' '> e ) where t > t ' > o „ Diagrams are termed non-connex provided they are made up of two or more component structures which are unconnected to each other, Dotted l i n e s representing the i n i t i a l c o r r e l a t i o n s are to be included i n determining the connexity of the diagram c The - 21 -permutation c l a s s of a non-connex diagram i s formed by changing the r e l a t i v e ordering of the component structures„ The permuta-t i o n c l a s s being a l l such possible permutations„ F i g . 2.4a displays a simple non-connex diagram. This together with the diagram of Fig„ 2.4b forms the permutation cl a s s of t h i s diagram c (a) (b) F i g . 2.4 The following f a c t o r i z a t i o n theorem i s due to Balescu but a more general proof has been given by Resi b o i s 1 ? A non-connex diagram completed by i t s permutation c l a s s represents a sum of contributions which i s equal to the product of the contributions represented by each of the component structures, A simple example of t h i s theorem i s worked out i n Appendix A, The consequence of t h i s theorem i s that contributions to the Fourier c o e f f i c i e n t can be written as a product of f a c t o r s , the number of factors being determined by the number of unconnected component structures of the representative diagram. - 22 -Moreover, each component structure of the diagram represents a term in the development of some Fourier coefficient, /f^jCt)-Since these component structures are connex they represent non-factorizable contributions to the (*•) and when these are compared to the factorization properties of the terms in the cluster decomposition (2.45) i t is clear that they must be contributions to the (*) 6 This means that the time develop-ment of each term in the cluster decomposition of / f ^ is represented by a class of diagrams whose factorization property as determined from the connexity of the class is the same as the factorization property of that term in the cluster decomposition, Equivalently: i f a set of particles is uncorrelated at t = o (no vertical dotted lines between the solid lines labelled by this set) and there are no interaction matrix elements involving the members of this set (no vertices labelled by the set) then the set of particles is uncorrelated at time t; but, i f the members of the set are i n i t i a l l y correlated (vertical dotted lines connecting the solid lines labelled by the set) or i f there are interaction matrix elements involving the particles of the set (vertices labelled by the set) then the set is correlated at time t c For example, contributions to the two particle coefficient ftt,*. (*J,&X\*) c a n be separated into two classes,, One, which is represented entirely by connex diagrams, is non-factorizable and contributes only to ^^(^,>^rx)t) „ The otherg which is represented by non-connex diagrams containing two component structures (each component representing a contribution to either ^ ( v ^ r ) or f\£<¥x;+) J is factorizable and contributes only to - 23 -/°H, A ( ^  > ; ~t) • A t y p i c a l non-connex diagram r e p r e s e n t i n g a c o n t r i b u t i o n to fiH ^ and a connex diagram r e p r e s e n t i n g a con-t r i b u t i o n t o are d i s p l a y e d i n F i g . 2 .5a and F i g . 2 .5b r e s p e c t i v e l y . s (a) (b) F i g . 2 .5 A c o n t r i b u t i o n t o the r e d u c e d s - p a r t i c l e c o e f f i c i e n t ^=>^([/ri)t) i s s a i d to be r e d u c i b l e i f i t s r e p r e s e n t a t i v e diagram can be d i v i d e d i n t o two f r a g m e n t s , w i t h a t l e a s t one v e r t e x i n e a c h , by a v e r t i c a l c u t such t h a t the fragment on the r i g h t r e p r e s e n t s a c o n t r i b u t i o n to the c o m p l e t e l y f a c t o r i z e d c o r r e l a t i o n vacuum of the i n t e r m e d i a t e s t a t e T h a t i s , i f the fragment on the r i g h t i s non-connex and has as many u n c o n n e c t e d component s t r u c t u r e s as i t has l i n e s a t t h e v l e f t , the complete d iagram i s termed r e d u c i b l e . O t h e r w i s e i t i s s a i d t o be i r r e d u c i b l e and i f b e s i d e s b e i n g i r r e d u c i b l e i t a l s o r e p r e s e n t s a c o n t r i b u t i o n to the s - p a r t i c l e c o r r e l a t i o n c o e f f i c i e n t - 24 -^Cii J ^) * s s a i d to be s t r i c t l y i r r e d u c i b l e . It should be noted that a l l zero and f i r s t order contributions are inherently s t r i c t l y i r r e d u c i b l e since one cannot divide the representative diagrams in t o two fragments with one vertex i n each. Examples of diagrams representing the three types of contributions to /°£^lLi^l)' b) a r e & i v e n i n F i S » 2.6. T n e diagram F i g . 2.6a represents a reducible contribution and F i g . 2 c6b a s t r i c t l y (3) i r r e d u c i b l e contribution to while F i g . 2,6c represents an i r r e d u c i b l e contribution to ^ ^ h 3 3 ( C ) F i g . 2.6 - 25 -This completes the exposition of the general properties of the diagramso The only restriction that has been made is that the interactions be central two body forces whose range is much less than the dimensions of the system as a whole0 In the next section i t will be shown how the concept of irreducibility can be used to derive equations of evolution for the reduced Fourier coefficientso This will be applied to the particular case of a classical gas with a long range attraction in Chapter III 0 - 26 -4 e Equations of Evolution of the Reduced Fourier C o e f f i c i e n t s F i r s t l e t us consider the reduced one-particle Fourier c o e f f i c i e n t . According to equation (2.50) i t evolves i n time as (2.53) ft,v, . t ) = - j i , j j , ^ " " fcdaf •L I < 4 1 l ( L . - 0 - ' [ - > « - ( L . - ) - T l f « > / ? f 5 < * - * - i 0 > This i s equivalent to the two equations (2.54) = f/**1" /%,(*)*) and (2.55) /%,(*.}£) ^ Following Severne^we separate out the contributions up to a r b i t r a r y order m i n /\ and rewrite (2.55) as (2.56) = -i IW"~'[Z f- <4,|(L.-*) _ ,f-^L(L . - * r ,f|fA'I> with /^k'fe^ljz) s a t i s f y i n g an equation analogous to (2.55). The c l u s t e r decomposition (2.45) can of course be applied to these Laplace transformed c o e f f i c i e n t s . We use t h i s and rewrite (2.56) as the sum of contributions from i n i t i a l state c o r r e l a t i o n s - 27 -and from intermediate c o r r e l a t i o n vacuum (completely factorized) states. This i s accomplished i n the following way. We write (2.56) for m = o and r e t a i n i n the second term only the contribu-t i o n from the c o r r e l a t i o n vacuum: -i <4 . | (L.-*rU.> /°4,(*.;* = o) I k^f" <±.|cu-v"(oiL)| z) ik'i j t-i For m ^ l we write i n the f i r s t term the i r r e d u c i b l e contributions of order m from the i n i t i a l state correlations and i n the second term the i r r e d u c i b l e contributions of order m+1 from the i n t e r -mediate state c o r r e l a t i o n vacuum. Thus » - 1 : <-*1|(L.-*)-(-^oa.-*)",i^'i> r m ^ ' ^ ° ) + ^ -f'-'X" <4 ,|[ ( L .-.r'<-«L)T|H'[> ,?£W*) Ch ] m - 2= - i i ^ r - j?™ < i , | f t . - » r , ( - » L ) ] , ( L . - . r | ^ j > r m ( ^ ^ = ° j • 1 m : . * . i t i r , < 4 . i r(L . -*r , ( -«ura . -»r , u^> w^^<o (4*1 where ^means that only i r r e d u c i b l e contributions are included. - 28 -This condition of i r r e d u c i b i l i t y ensures that no contribution written above for a given ra = mi has already been included i n a c o r r e l a t i o n vacuum contribution of order m<mi. That the expressions written above, when summed over a l l m, also c o n s t i -tute an exhaustive r e c l a s s i f i c a t i o n of the terms i n the expan-sion (2.55) i s f a i r l y obvious. For, ( i ) a l l i r r e d u c i b l e contributions from i n i t i a l state correlations are written as such and ( i i ) a l l i r r e d u c i b l e contributions from the i n i t i a l state c o r r e l a t i o n vacuum as well as a l l reducible contributions are included i n the contributions from the intermediate state. As an example to i l l u s t r a t e the second statement consider the contributions to represented by the diagrams displayed i n F i g . 2.7a, that i s , an ir r e d u c i b l e contribution from the i n i t i a l c o r r e l a t i o n vacuum, and F i g . 2.7b, that i s , a reducible contribution from the i n i t i a l state/. 3 2 2. (a) (b) F i g . 2.7 The term i n equation (2.55) corresponding to F i g . 2.7a i s - 29 -This can be rewritten as with which i s the zero order contribution to f^f&l;^ . S i m i l a r i l y the term i n equation (2.55) represented by F i g . 2.7b i s fat' I <i\ [(L ^ ^ - A 4 L)] a(L.-*;-'(-UL)(Lr * r , | v t f ^ > A/ < =H This can be rewritten as with - 30 -^ (2) which i s a f i r s t order contribution to C f•y?; ?) Summing over a l l m we now have: (2.57) Writing the matrix element of the f i n a l state propagator e x p l i c i t -l y y i e l d s (2-58) r t ! % „ B _ L _ U,,,.,,.) + i j ^ f - £ a ,K -^o i f i ' j > / f *Vi^ ) ' IS J i | s i under a p p l i c a t i o n of the inverse Laplace transform r equation (2.58) becomes T , z e. - 31 -(2.59) +i4,-">/%>(*>*) = fer £ $Wm)fi*\tel,t) where we have used the d e f i n i t i o n s (2.60) <f>(±\te'l) = ' < ^ . U - a S L ) ' ^ > (2.61) £M(*,)*) = ^ f f * e J W I J and (2.62) yte.Wit)_= - r , J; . £ c < 4 , J [ - ^ L (L . -er ' J 'Y 3 4 L ) | ^ ' i ^ _ , •» = » Since there are no correlations in the intermediate state the i r r e d u c i b i l i t y property i s a c h a r a c t e r i s t i c of the tr a n s i t i o n s represented by the matrix elements of (2.62), Hence the notation When we write out the matrix element of (2,60) e x p l i c i t l y , using (2.27) and (2.28) we have (2.63) M f(* , l f f i )^(«; f j = ' ^ 2 J * . - 32 -The q u a n t i t i e s we are a c t u a l l y i n t e r e s t e d i n are not the F o u r i e r c o e f f i c i e n t s but the r e d u c e d d i s t r i b u t i o n f u n c t i o n s . A p p l y i n g the i n v e r s e F o u r i e r t r a n s f o r m to (2 .63) y i e l d s the c o n t r i b u t i o n from t h a t term to the time development of the o n e - p a r t i c l e d i s t r i b u t i o n f u n c t i o n . The r e s u l t i s : (2 .64) - I jjxjt*^ J L y i x , - * j ) . ^ t y z , * ^ ) - ? , ^ , ^ * ) which i s a s e l f - c o n s i s t e n t f i e l d or V l a s s o v t e r m . The q u a n t i t y (v*,?), the " d e s t r u c t i o n t e r m " , d e s c r i b e s the p r o p a g a t i o n of the i n i t i a l c o r r e l a t i o n s t h r o u g h the s y s t e m . I t w i l l be seen l a t e r that i t s e f f e c t i s g e n e r a l l y t r a n s i e n t and may be n e g l e c t e d f o r l o n g t i m e s . The o p e r a t o r i s to be i n t e r p r e t e d as a g e n e r a l c o l l i s i o n o p e r a t o r . I t s form i s i n t i m a t e l y r e l a t e d to t h a t o f the i n t e r a c t i o n s and i t s p r o p e r t i e s w i l l be d i s c u s s e d i n d e t a i l l a t e r . An e n t i r e l y s i m i l a r method may be u s e d t o o b t a i n the t ime development of the s - p a r t i c l e F o u r i e r c o e f f i c i e n t s , A c c o r d i n g t o the d i s c u s s i o n i n S e c t i o n 3 , the c o n t r i b u t i o n s to the c o r r e l a t i o n c o e f f i c i e n t <3 '^ must not f a c t o r i z e . The diagrams r e p r e s e n t i n g these c o n t r i b u t i o n s are t h e n connex . S e p a r a t i n g out the i r r e d u c i b l e c o n t r i b u t i o n s from the i n i t i a l s t a t e c o r r e l a t i o n s and the i n t e r m e d i a t e s t a t e c o r r e l a t i o n vacuum we can w r i t e - 33 -(2,65) = -I faf Z " " I < V • * J ( L . - * / ' [ - ^ L ^ / f ( < r v / > The superscript ( i r s ) means that s t r i c t i r r e d u c i b i l i t y i s to be applied to ensure that the contributions determine ^ and not some other term of the c l u s t e r expansion of . Dia-grams contributing to have s l i n e s on the l e f t and since each vertex involves at most two l i n e s no connex diagram connect-ing to the intermediate c o r r e l a t i o n vacuum can be constructed with fewer than s-1 v e r t i c e s . We define the creation operators as (2.66a) CCV'^U^;*) = Z <^--is|[(L.^)-'(->^)r|f^> , which under inverse Laplace transformation becomes (2.66b) C(v-* .U* ' J;2) = t i£^ e " ; 2 ' r Z < ^ " ' ^ | [ a ^ ) - , ( - ^ o r i ^ , s ; The f i r s t term of (2.65) when the inverse Laplace transform has been applied we write as This enables us to write the following equation for the s - p a r t i c l e c o r r e l a t i o n c o e f f i c i e n t : - 34 -(2.68) 1 c % ( w } t ) = $gu»y,t) Here again the term represents the e f f e c t of the i n i t i a l state c o r r e l a t i o n s and the creation operator C i s c l o s e l y related to the c o l l i s i o n operator If. Equations (2.59) and (2.68) are exact within the l i m i t s imposed by the N-body formalism. Summation of the i n f i n i t e s e r i e s of (2.61), (2.62), (2.66a) and (2.67 ) would be equivalent to solving the N-body problem exactly. An attempt at an exact summation i s therefore quite useless. In the next chapter we s h a l l develop c r i t e r i a for r e t a i n i n g those diagrams which y i e l d s i g n i f i c a n t contributions to the development of a c l a s s i c a l gas with a weak long range a t t r a c t i o n . These r e s t r i c t e d classes of terms are then summed to y i e l d the equations of evolution. - 35 -CHAPTER III EVOLUTION OF A CLASSICAL GAS WITH A LONG RANGE ATTRACTION 1. The System and Interactions In this chapter we shall apply the methods outlined in Chapter II to the particular case of a gas of identical particles whose interactions are derived from a potential which is the sum of a strong short range repulsion and a weak long range attrac-tion. The scale to which the ranges of the two potentials are compared is the average interparticle distance while the strength of the interaction is compared to the average kinetic energy of a particle. To be more precise, we write the inter-action between two particles separated by a distance r, 3 V( , as (3 .1 ) MM =yu ULr) - V WW The functions U(r) and W(r) having the properties (3.2) <j (r) ~ I ; y- o r ' — 0 , r > X~' and (3 .3 ) W ( r ) ^ j t K-' < r < ^ - i with the inverse ranges, X and <x, satisfying the inequalities - 36 -(3 .4 ) y 3 » c » * 2 Here c i s the c o n c e n t r a t i o n of the g a s , c " " 1 ^ then b e i n g the average i n t e r p a r t i c l e d i s t a n c e . S i n c e the range of the i n t e r -a c t i o n s i s assumed to be much l e s s than the s i z e o f the c o n -t a i n e r we have a l s o t h a t I f the average k i n e t i c energy per p a r t i c l e i s 1/B then the s t r e n g t h parameters o f the i n t e r a c t i o n s , ^ and ^ , s a t i s f y the i n e q u a l i t i e s ( 3 ' 5 ) yu » ^ » V At the same t ime we assume t h a t the volume i n t e g r a l s of the i n t e r a c t i o n s are such t h a t - 3 (3 .6) z/c JdrW(y) = v e t o ~ ^  *~ — V(. and (3 .7 ) yufdlUiy) = y» VL ~ * Z «~*%0 No f u r t h e r r e s t r i c t i o n s are p l a c e d on the i n t e r a c t i o n s . However we d i s t i n g u i s h between the f o l l o w i n g two c a s e s which r e l a t e the range of the l o n g range i n t e r a c t i o n to the l e n g t h s c a l e o f the i n h o m o g e n e i t i e s of the s y s t e m . T h a t i s , i f [_/, i s the s h o r t e s t w a v e l e n g t h of the s p a t i a l F o u r i e r t r a n s f o r m of the one p a r t i c l e d i s t r i b u t i o n f u n c t i o n , then we c o n s i d e r the two c a s e s - 37 -(3.8a) L k > y ^-1 and (3.8b) ~ In what follows we s h a l l f i r s t work under the assumption that (3.8a) i s v a l i d . After c r i t e r i a have been established for obtaining a l l s i g n i f i c a n t contributions to the c o l l i s i o n term of equation (2.59) and these contributions summed we s h a l l consider what happens when the r e s t r i c t i o n i s relaxed to that of (3.8b). The nature of the Fourier transform i s such that i f the o r i g i n a l function extends over a range R i n coordinate space then the spectral breadth of the image function i n k space i s KcrR-"1. This means, since the one p a r t i c l e d i s t r i b u t i o n function f^ varies over a distance L n , with L n very large, that the one body reduced Fourier c o e f f i c i e n t , fA , i s sharply peaked about kj = o with a width ^ L h " 1 . Writing and Wk as the Fourier transforms of the potentials U(r) and W(r), we have that the range of i n k space is«>* and that the spectral breadth of Wk i s cs <x . Since X » o < » L h " 1 , by (3.4) and (3.8a), Ufc i s a very broad function of k i n comparison with the sharply peaked behaviour of W^  and the even more sharply peaked /% • According to equations (2.59) and (2.60) the contributions to the time development of the Fourier c o e f f i c i e n t s are of two types: one which arises from the completely f a c t o r i z e d c o r r e l a -t i o n vacuum of an intermediate state and one from the i n i t i a l - 38 -s t a t e c o r r e l a t i o n s . The s p e c i f i c a t i o n o f these i n i t i a l c o r r e l a -t i o n s i s n e c e s s a r y b e f o r e one can s e l e c t the dominant c o n t r i b u -t i o n s t o the e v o l u t i o n o f the s y s t e m . I t i s d e s i r a b l e not to be too r e s t r i c t i v e but r a t h e r to s p e c i f y a g e n e r a l c l a s s of i n i t i a l c o n d i t i o n s i n o r d e r t h a t the e q u a t i o n s d e r i v e d be more w i d e l y a p p l i c a b l e . We s h a l l assume t h a t the i n i t i a l t ime does not p l a y a p r i v i l e g e d r o l e , the c o r r e l a t i o n s a t t ime t = o b e i n g o f the same n a t u r e as those a t l a t e r t i m e s . T h i s amounts to assuming t h a t the i n i t i a l c o r r e l a t i o n s are of the same n a t u r e as a r i s e from the i n t e r a c t i o n s . Under t h i s a s s u m p t i o n i t i s o b v i o u s t h a t these c o r r e l a t i o n s between p a r t i c l e s become v a n i s h i n g l y s m a l l when the p a r t i c l e s a re s e p a r a t e d by some d i s t a n c e L c and more-o v e r (3 . 9 ) L c £ oT' F o r example , c o n s i d e r the two p a r t i c l e c o r r e l a t i o n 3^(2,,*J T h i s can be r e w r i t t e n i n terms of the c o o r d i n a t e s of the c e n t r e o f mass of the two p a r t i c l e s and t h e i r r e l a t i v e s e p a r a t i o n , (3.10) R= ) £ = *, -a, 2. t o g i v e <3i(S>i:) • S i n c e we i n s i s t on l o n g wavelength inhomo-g e n e i t i e s the f u n c t i o n 9*(!?j£) w i l l be a s l o w l y v a r y i n g f u n c t i o n o f R and the f a c t t h a t the p a r t i c l e s a re u n c o r r e l a t e d when w i d e l y s e p a r a t e d means t h a t <3j(S>Jr) must become z e r o when m > L c • The r e g u l a r F o u r i e r c o e f f i c i e n t "ftjzkls g i v e n by - 39 -(3 .11) or writing A ^ A + g , ^ , = - A (3 .12) Clearly Y is a sharply peaked function of ci about q = 0 with a width ~- and at the same time due to the finite range It is not necessary, as will be seen later, to make any assumption about the size of the i n i t i a l correlations, other than that they be finite, in order to derive an equation of evolution valid for long times. However, not too surprisingly, the size of the correlations at time zero is crucially important for the short-time equations. If i t is assumed that the corre-lations at t = 0 are of the same order of magnitude as at equilibrium then the size of the correlations can be estimated in a manner similar to the following for the two body correla-tion. The equilibrium N-body distribution function is given by of the correlations it is peaked about k = 0 with a width ~ L ~ ' ^ o i ' where the partition function - 40 -and Kg i s B o l t z m a n n ' s c o n s t a n t , T b e i n g the t e m p e r a t u r e . The i n d i c a t e d v e l o c i t y i n t e g r a t i o n s can be i m m e d i a t e l y performed to g i v e w i t h the c o n f i g u r a t i o n i n t e g r a l QN = j f y * / e " w v From the d e f i n i t i o n of the r e d u c e d d i s t r i b u t i o n f u n c t i o n s (2 .29) we have A g a i n the v e l o c i t y i n t e g r a l s are s t r a i g h t f o r w a r d , each g i v i n g a f a c t o r ( ~ ^ ^ ) / z which c a n c e l s w i t h one o f the f a c t o r s i n Z J J to g i v e The f u n c t i o n F(x ( , x ^) can be expanded i n terms of the c l u s t e r 20 i n t e g r a l s . To f i r s t o r d e r t h i s g i v e s F(x,>*i.) ~ e~'^rV and we w r i t e (3.i3) ~ c x ^ r v , ) ^ ) e ' ^ r U \ 1 + ^ w ' 2 + > • • - 41 -By (2.41) F o r f X,-jc t| ^ x~* » (3.13) s t a t e s t h a t ^-o and t h e r e f o r e (3.14) 3>.>*,;*,.*) * -c*f{<4) f(«t) ; |x,-*J<*-' F o r ! « , - * , / > X" cxfl*)(ftot) { » + £ r W 0 * , - * . l ) J and (3.15) ^ , ^ ; ^ * 0 2: £pW(lx.-Z<l) c>fte)0V<f,) ) |!f 1-^|>yf-» T r e a t i n g (3.14) and (3.15) as exac t and u s i n g (3.12) y i e l d s < 3 . i . ) y ; . ^ „ * ) * 4 (_L, + _ ^ w 4 In o b t a i n i n g the second term o f (3.16) the i n t e g r a t i o n o v e r r-has been extended i n t o the r e g i o n t < i t b e i n g n e g l i g i b l y s m a l l . S i n c e the s p h e r i c a l B e s s e l f u n c t i o n behaves a s y m p t o t i c a l l y as (/(.and a l s o ^ / f ) I" as Y~ -+ o the f i r s t term of (3.16) i s ^  Vyt? • ' r n e s e c o n ( i term i s v e r y s m a l l because 1// « / . The one p a r t i c l e c o e f f i c i e n t i s of o r d e r u n i t y and t h e r e f o r e we m a i n t a i n t h a t the e q u i l i b r i u m c o r r e l a t i o n c o e f f i -c i e n t s s a t i s f y the i n e q u a l i t y « • • » > * 3 « = & rti A c a l c u l a t i o n of the e q u i l i b r i u m two p a r t i c l e c o r r e l a t i o n f u n c t i o n w i l l be performed i n s e c t i o n 8 of t h i s c h a p t e r . I t w i l l be seen then t h a t (3.14) and (3.15) a re c o n s i s t e n t w i t h the a p p r o x i m a t i o n s u s e d throughout t h i s work. - 42 -2. R i n g and C h a i n Diagrams In t h i s s e c t i o n the c l a s s e s of diagrams r e p r e s e n t i n g s i g n i -f i c a n t c o n t r i b u t i o n s t o the time development of the system w i l l be d e t e r m i n e d . F i r s t l e t us c o n s i d e r the e v o l u t i o n of the o n e -p a r t i c l e F o u r i e r c o e f f i c i e n t which i s c o n t r o l l e d by e q u a t i o n ( 2 . 5 9 ) . The c o n t r i b u t i o n s a r i s i n g from the i n t e r m e d i a t e s t a t e c o r r e l a t i o n vacuum are g i v e n by the c o l l i s i o n term (3 .18) j ^ i * ; = fey>s~' H fa'Ti'fam)*-*') W i t h the c o l l i s i o n o p e r a t o r \th!\) t't') b e i n g d e f i n e d by ( 2 . 6 2 ) . The L a p l a c e t r a n s f o r m of t h i s o p e r a t o r as d e t e r m i n e d by t r a n s f o r m i n g e q u a t i o n (2 .62) i s (3 .19) f t e l t f j i * ) = t K ^ I C - ^ L a o - ^ T c - ^ n l f ^ , . ^ The diagram t e c h n i q u e can be more c o n v e n i e n t l y a p p l i e d when we d e f i n e a new o p e r a t o r (3 .20a) i f (hMVl)*) = (*-.-*>"' ^(^\m^) (io'^" (3 .20b) - £ i <kt\(La-i)-''[-16LU*-*r'T\^'iy.„ Each term i n the e x p a n s i o n o f (3 .20b) i s r e p r e s e n t e d by a d i a g r a m . R e f e r r i n g back to C h a p t e r I I . 3 and n o t i n g t h a t the f i n a l s t a t e i n v o l v e s o n l y one n o n - z e r o w a v e - v e c t o r , k j , i t i s c l e a r t h a t each diagram b e g i n s on the l e f t w i t h a type B v e r t e x (see F i g . 2 . 2 b ) . The r e q u i r e m e n t t h a t the c o n t r i b u t i o n s be i r r e d u c i b l e i s e q u i v a l e n t to r e q u i r i n g t h a t i t be i m p o s s i b l e t o d i v i d e the d i a g r a m , by a v e r t i c a l c u t , such t h a t the - 43 -fragment on the r i g h t be non-connex and have as many component structures as i t has l i n e s on the l e f t . This means, since there are no correlations among the p a r t i c l e s of the set t**'l and thus no dotted l i n e s connecting the s o l i d l i n e s l a b e l l e d by the [ h ! \ , that the f i r s t vertex on the r i g h t of a diagram must be a type A vertex (see F i g . 2.2a). Each diagram represent-ing a term i n (3.20b) therefore consists of a type B vertex on the l e f t connected through combinations of ver t i c e s of types A and B to a type A vertex on the r i g h t . The diagrams with fewest v e r t i c e s which can be constructed i n t h i s manner are displayed i n F i g . 3.1. The simplest possible diagram which represents a term i n the expansion of the c o l l i s i o n operator i s F i g . 3.1a. The diagram F i g . 3.1b i s t y p i c a l of diagrams which are c a l l e d 'chains'. They are made up of a single B type vertex and a number of type A v e r t i c e s . F i g . 3.1c and F i g . 3.If are examples of diagrams c a l l e d 'rings'. These diagrams are constructed from a single A type vertex together with a number of type B v e r t i c e s . The e x p l i c i t form of the term i n the expansion (3.20) corresponding to the diagram F i g . 3.1a, as determined from (2.24), (2.27) and (2.28) i s (3.21, < (|£f I L A V„ h) j w o ( 9 . . 2 - ) ! The e f f e c t of the Kronecker deltas has been i m p l i c i t l y included at each vertex and the wave vectors l a b e l l e d according to the - 45 -c o n v e n t i o n i l l u s t r a t e d i n F i g . 3 . 2 . F i g . 3 .2 A l s o , the l a b e l l i n g of the l i n e s i n F i g . 3 . 1 obeys the c o n v e n -t i o n t h a t numbered l i n e s c o r r e s p o n d to a s i n g l e p a r t i c l e and a r e not summed o v e r w h i l e l e t t e r e d l i n e s i m p l y a summation over a l l p a r t i c l e s o f the system except those a l r e a d y f i x e d . F o r example , the i n d i c e s l a b e l l i n g the l i n e s of F i g . 3 .1e are 1, j , m, and n . T h i s i m p l i e s the summations: £ o v e r a l l p a r t i c l e s of the system except t h a t l a b e l l e d 1: £ over a l l p a r t i c l e s except t h o s e l a b e l l e d 1 or j : and X over a l l p a r t i c l e s except those l a b e l l e d by 1, j o r m. The d i s t r i b u t i o n f u n c t i o n s and hence the r e d u c e d F o u r i e r c o e f f i c i e n t s are symmetric w i t h r e s p e c t to the p e r m u t a t i o n o f the p a r t i c l e s . T h i s means t h a t each term i n the summations o v e r the i n d i c e s l a b e l l i n g the l i n e s i s e q u i v a -l e n t to e v e r y o t h e r t e r m . T h e r e f o r e the j summation can be r e p l a c e d by ( N - l ) t imes one of the t e r m s , the m summation by N-2 t imes one of the terms and the n summation by N-3 t imes one of the t e r m s . - 46 -In the thermodynamic l i m i t ( 2 „ 2 ) these f a c t o r s N-s can be r e p l a c e d by N„ A l s o , the J summation of (3 ,21) goes over to an i n t e g r a l (see(2 .39)) and (3 .21) becomes (3 .22) i in* (lf(^7ir~ fdl V,x_,i K ^ + where (3 .23a) <3 = and (3 .23b) <3a = fe. The terms i n the e x p a n s i o n (3.19) of the c o l l i s i o n o p e r a t o r c o r r e s p o n d i n g to those i n the e x p a n s i o n (3 .20) are i m m e d i a t e l y o b t a i n a b l e by d r o p p i n g the p r o p a g a t o r s from e i t h e r end o f the l a t t e r e x p r e s s i o n s . Thus (3 .22) c o r r e s p o n d s to the f o l l o w i n g terra i n the e x p a n s i o n of the c o l l i s i o n o p e r a t o r (3 .24a) l f*3c ( g f j j , V ( ^ ( ( A ; . , ) . I where the K r o n e c k e r d e l t a has been r e p l a c e d to make e x p l i c i t the c o n s e r v a t i o n of wave v e c t o r s . S i m i l a r c a l c u l a t i o n s y i e l d the f o l l o w i n g e x p r e s s i o n s f o r the terms of (3 .19) c o r r e s p o n d -i n g to the diagrams F i g . 3 . 1 b - 3° - 47 -(3 .24b) i 1*>c (lYJdlJA!' Vf,,. , , , , 3 [ ^ y , r . U / * W W , « ^ ' J - ? ( a (3 .24c) itin'cy&Yjit V^u'.nl^S-*)^ Vl£+ <fj- U ' '3'-' ^  I 3,,. + ^ / . y ; 4/)- V 4 - « (3 .24d ) ^ wnkYlU Vk,tu. n (fr*;-*) ?.t -j-—-r, 777 - 7 (3 .24e) cMO'aVjJl V , ^ / ^ , , , % % ^ ^ . ^ , g (3 .24f ) t ^ c ) 3 ( 1 ) ^ / < 4 . ' j . ^ ^ ^ U ^ ) - ^ -(3 .24g) ^ O ^ y ^ + ^ ^ / ; . 23 - 48 -The i n e q u a l i t i e s g i v e n i n (3 .8a) and the subsequent d i s c u s -s i o n e n a b l e s us to p l a c e c e r t a i n r e s t r i c t i o n s on the magnitude of the w a v e - v e c t o r s a p p e a r i n g i n these e x p r e s s i o n s . Because of the l o n g wavelength n a t u r e o f the i n h o r a o g e n e i t i e s a l l the primed wave v e c t o r s a re v e r y s m a l l , A- ^ L^' . The wave v e c t o r s , J , which are i n t e g r a t i o n v a r i a b l e s are not so r e s t r i c t e d . The average v a l u e of these wave v e c t o r s i s of the o r d e r of the i n v e r s e of the range of the i n t e r a c t i o n . T h a t i s , </> > < * » L ^ ~ A / . To a good a p p r o x i m a t i o n we can n e g l e c t the ^ i n c o m p a r i s o n w i t h the J s and w r i t e the f o l l o w i n g approximate e x p r e s s i o n s f o r those of ( 3 . 2 4 ) : (3 .25a) - i f r t (D'JJl V / i . 2 , t T - i _ V, i - 2ia. (3 .25b) (3 .25c ) Utn'cfdYjjgy £.§,z 1-1,3 (3 .25d) - 49 -(3 .25e) Ufa*) 3(iffJJ V, V, / - 5 „ ( 3 . 2 5 f ) -UWiD'JM ^ " 7 ^ 7 ^ J - i ^ T l ^ i V J ^ L (3 .25g) i - 2 , a V r T l ^ i T F 7 Comparing (3 .25c ) and (3 .25d) w i t h p a r t i c u l a r r e f e r e n c e t o the f a c t o r s u n d e r l i n e d w i t h a wavy l i n e we see t h a t the r a t i o o f (3 .25d) t o (3 .25c) i s ~ / 0 > . We t h e r e f o r e c o n s i d e r a l l terms i n the e x p a n s i o n of the c o l l i s i o n o p e r a t o r (3 .20) which are a s s o c i a t e d w i t h diagrams of the type F i g . 3 . 1 d to be n e g l i -g i b l e i n c o m p a r i s o n to those a s s o c i a t e d w i t h diagrams of the type F i g . 3 . 1 c . A s i m i l a r argument can be used to show t h a t (3 .25e) i s n e g l i g i b l e ( ~ - ^ y ' / < ,e> ) i n compar ison w i t h ( 3 . 2 5 f ) . The o n l y d i f f e r e n c e between these diagrams i s i n the l a b e l l i n g of the l i n e s . In f u t u r e the l a b e l s may be dropped w i t h the u n d e r s t a n d i n g t h a t the diagram i s a c t u a l l y of the type F i g . 3 .1c and F i g . 3 . I f . Diagrams of the type F i g . 3 . 1 d and F i g . 3 .1e are e x c l u d e d from the c l a s s of diagrams termed ' r i n g s ' . - 50 -A c u r s o r y e x a m i n a t i o n o f the e x p r e s s i o n s (3 .25) and the diagrams o f F i g . 3 . 1 r e v e a l s a remarkable p r o p e r t y of these d i a g r a m s . Namely, the a d d i t i o n of each v e r t e x o f type A ( ~I>C ) e n t a i l s an a d d i t i o n a l i n t e g r a t i o n over wave v e c t o r s w h i l e the a d d i t i o n o f each B type v e r t e x ( —CL ) g i v e s a new p a r t i c l e i n d e x and a f a c t o r c but no i n t e g r a t i o n o v e r wave v e c t o r s . F o r example b o t h (3 .25b) and (3 .25c ) are of o r d e r 2 but the former i s o f f i r s t o r d e r i n c and has i n t e g r a t i o n s over / a n d Jl' whereas o the l a t t e r has o n l y an Jt i n t e g r a t i o n and i s of o r d e r c „ i t i s easy to see t h a t t h i s i s t r u e i n g e n e r a l . I t w i l l be shown l a t e r t h a t the o n l y v a l u e o f z t h a t i s i m p o r t a n t i n the e x p r e s s i o n s of (3 .25) i s H -o . I f we now assume f o r purposes o f i l l u s t r a t i o n t h a t (3 .26) v, ~ V , Jf< hr i n s e r t t h i s i n the e x p r e s s i o n s o f (3 .25) and e s t i m a t e the o r d e r of magnitude o f these q u a n t i t i e s t h e r e r e s u l t s , r e s p e c t i v e l y , (3 .27a) ^ ^ I f f c ^ (3 .27b) ~{\V?ctf (3 .27c ) ~ ( W ) 3 c * / T V ( 3 . 2 7 f ) - 51 -(3 .2?g) - ( i ? / ; V / r 7 In g e n e r a l i t can be a s s e r t e d t h a t a diagram c o n t a i n i n g m type A v e r t i c e s and n type B v e r t i c e s i s of the o r d e r of (3 .28) ( W ) c" /T 3 C l e a r l y , i f /Y « c , the dominant c o n t r i b u t i o n of o r d e r p i n the i n t e r a c t i o n , p = m+n , s i n c e on ^ / and D > ; , i s of the o r d e r o f (3 .29) (22/ / , c p - ' h-* w = » I t i s e q u a l l y o b v i o u s t h a t i f |T 3 ^>cthe dominant c o n t r i b u t i o n of o r d e r p i n the i n t e r a c t i o n i s of the o r d e r o f 3P-2 (3 .30) (*AWc/r (•e- , TT} = p -1 , 7-1=1 The c l a s s o f diagrams f o r which n = 1 a r e those w i t h a s i n g l e type B v e r t e x . These we have c a l l e d c h a i n s . The c l a s s w i t h m = 1 a re those which have a s i n g l e type A v e r t e x and are c a l l e d r i n g s . Of c o u r s e , diagrams of the type F i g . 3 „ l d and F i g . 3 .1e a re e x c l u d e d from the c l a s s of r i n g s as they are deemed to be n e g l i g i b l e . I t i s now e v i d e n t t h a t so f a r as the s h o r t range i n t e r a c t i o n i s c o n c e r n e d the o n l y s i g n i f i c a n t terms i n the e x p a n s i o n o f the c o l l i s i o n o p e r a t o r are those c o r r e s p o n d i n g to the c h a i n d i a g r a m s „ A l s o , the o n l y s i g n i f i c a n t c o n t r i b u t i o n s due to the l o n g range i n t e r a c t i o n a re those r e p r e s e n t e d by r i n g s . These s ta tements f o l l o w d i r e c t l y from the i n e q u a l i t y (3 .4 ) ^ » c » ^3 - 52 -and the fact that the Fourier transforms of the short and long range interactions have spectral widths of the order of X and respectively. This i s not to say that either the strong short range or weak long range part .of the in t e r a c t i o n may not be completely overshadowed by the other. This would depend on the magnitude of the parameters of the int e r a c t i o n s . Since, as we s h a l l show l a t e r , the c o l l i s i o n operator causes the system to approach equilibrium, one would expect that both parts of the in t e r a c t i o n would be equally e f f e c t i v e i f the relaxation times for either part separately were of the same order of magnitude. The relaxa-t i o n time of a gas of hard spheres of diameter X '« c 3 i s * (3.31) t{SU tgfafi)* while that of a gas whose interactions are described by the long range potential |p r > o < 3 « c f i s (3.32) Equating (3.31) and (3.32) yiel d s (3.33) v& £ ~ ) as the condition that the two parts of the in t e r a c t i o n play roughly equal roles i n d r i v i n g the system to equilibrium. If 373 ^ > > 1 the long range i n t e r a c t i o n should dominate whereas i f z/fi,¥-<< I the short range part should. Retaining both the * See reference 9, section 9 - 53 -c h a i n s and r i n g s e n s u r e s t h a t no s i g n i f i c a n t c o n t r i b u t i o n s are d i s c a r d e d a l t h o u g h one c l a s s or the o t h e r may be n e g l i g i b l e . When the F o u r i e r t r a n s f o r m of the p o t e n t i a l A VJ, i s r e p l a c e d b y / M t ^ - z ^ i / ^ the p r o d u c t s i n v o l v e d i n the e x p r e s s i o n s (3 .25) n e c e s s a r i l y i n t r o d u c e c r o s s terms of the form J^U;,)*(vW^ ) h I t might be e x p e c t e d i n the l i g h t of the p r e c e d i n g d i s c u s s i o n t h a t terras of t h i s type w i t h a = m , ( the number o f type A v e r t i c e s ) and b = n , ( the number of type B v e r t i c e s ) would g i v e s i g n i f i c a n t c o n t r i b u t i o n s . T h i s p o s s i b i l i t y can be r u l e d out on the b a s i s o f the f o l l o w i n g argument . The average v a l u e s of and i n the i n t e g r a t i o n s o f (3 .25) are %L and lo , r e s p e c t i v e l y , where these are d e f i n e d by (3 .6 ) and ( 3 . 7 ) . The c o n t r i b u t i o n of a c r o s s terra o f the above type i s , s i n c e the presence of e f f e c t i v e l y c u t s o f f the i n t e g r a t i o n a t J~-<x i s T h i s i s to be compared w i t h an (m + n ) t h o r d e r r i n g which i s (3 .35) ~ [TW) C <x and an (im + n ) t h o r d e r c h a i n which i s (3 .36) ~~ (/*U) C W The r a t i o of (3 .34) t o (3 .36) i s - 54 -N o t i n g t h a t K 3 a n d t6>~°<~5and u s i n g (3 .6) t h i s becomes ( 3 . 3 8 , which by (3 .4 ) and (3 .5) i s c l e a r l y « | and thus (3 .34) i s n e g l i g i b l e i n c o m p a r i s o n w i t h (3 .36) f o r a l l v\ ^ ' and a l l -w > I . The r a t i o of (3 .34) to (3 .35) i s (3 .39) which by (3 .4 ) and (3 .7 ) i s « | f o r a l l m. T h i s shows t h a t these c r o s s terms may be n e g l e c t e d . E x t e n d i n g t h i s development t o the F o u r i e r c o e f f i c i e n t s of the c o r r e l a t i o n f u n c t i o n s p r e s e n t s no new d i f f i c u l t i e s . The r e q u i r e m e n t of s t r i c t i r r e d u c i b i l i t y f o r the c o n t r i b u t i o n s t o the c r e a t i o n o p e r a t o r (2 .66a) means t h a t the diagrams have no s t r u c t u r e s which are not c o n n e c t e d t o g e t h e r and t h a t the f i r s t v e r t e x on the r i g h t of the diagram i s o f type A . The l o n g range i n t e r a c t i o n a g a i n c o n t r i b u t e s s i g n i f i c a n t l y o n l y t h r o u g h terms r e p r e s e n t e d by diagrams which c o n t a i n the minimum p o s s i b l e number o f type A v e r t i c e s w h i l e the s h o r t range i n t e r a c t i o n c o n t r i b u t e s o n l y when the r e p r e s e n t a t i v e diagrams c o n t a i n the minimum p o s s i b l e number o f type B v e r t i c e s . A p p l y i n g these r u l e s to the diagrams c o n t r i b u t i n g to the two body c o r r e l a t i o n c o e f f i c i e n t ^fh,hL we see t h a t they a re i d e n t i c a l t o the r i n g s and c h a i n s w i t h the l e f t m o s t v e r t e x s p l i t o f f . T y p i c a l o f these diagrams are F i g . 3 ,3a and F i g . 3 .3b r e s p e c t i v e l y . Diagrams r e p r e s e n t i n g c o n t r i b u t i o n s to h i g h e r c o r r e l a t i o n c o e f f i c i e n t s are s l i g h t l y more complex . However as we s h a l l o n l y be i n t e r e s t e d i n the two - 55 -body c o r r e l a t i o n we s h a l l not d i s c u s s these f u r t h e r . - 56 -3. Asymptotic Forms of the C o l l i s i o n , Creation and Destruction Terms  In this section the forms taken by the c o l l i s i o n , creation and destruction terms of equations (2.59) and (2.68) for long times w i l l be determined. It i s shown i n Appendix B that the c o l l i s i o n operator (2.62) tends to zero for times much greater than the c o l l i s i o n time, (3.40) 1\>(t) ? 0 f o h tr»tc with the c o l l i s i o n time defined as the time required for a p a r t i c l e of average v e l o c i t y , (<m^)-'/2 , to traverse the range of the in t e r a c t i o n , K _ 1: (3.41) tc - / r " ( w / J ) " 1 Provided then, that we are interested i n times much greater than the c o l l i s i o n time the statement (3.40) allows us to extend the upper l i m i t of integration in (3.18) to i n f i n i t y and write the c o l l i s i o n term as This i s c l e a r l y non-Markof f ian but (?<</l; t-r) can be expanded i n a Taylor s e r i e s , assumed convergent, about t. The c o l l i s i o n term i s then written (3.43) fcf'X U W4,lr<rt;T) £ Uwm',tJ which, since the c o l l i s i o n operator Ifstf^o for T»tc and 2L_ p^'1 tL. , t„ being the relaxation time, i s e s s e n t i a l l y - 57 ~ an expansion i n t c / t r . The r e l a x a t i o n time of the system i s the s h o r t e r of or t[f) which are g i v e n by (3.31) and (3.32) as The c o l l i s i o n times a p p r o p r i a t e to the s h o r t and long range p a r t s of the i n t e r a c t i o n are from (3.41) (3.44) t?~ tf'W/4 and (3.45) tc l)~ ^'(w/S)'* Since (3.46) tf/tf ~ %i « I and (3.47) 4()A"> - *l££~r/s«l and o n l y terms i n the expansion of the c o l l i s i o n o p erator r e p r e s e n t e d by r i n g s need be c o n s i d e r e d i f t r ( ^ « t ( s ) , while o n l y c h a i n s need be c o n s i d e r e d i f t r ^ s ^ <<. t , we have that (3.48) U/t^ « | T h e r e f o r e o n l y the n = 0 term of (3.43) i s r e t a i n e d and the c o l l i s i o n term takes the M a r k o f f i a n form* * Severne, r e f e r e n c e 11,gives an i t e r a t i v e e x p r e s s i o n f o r the c o l l i s i o n term which i n c l u d e s h i g h e r order i n t c / t r , but (3.49) i s c o n s i s t e n t with our r e t a i n i n g o n l y the r i n g s and c h a i n s . - 58 -(3 .49) The t ime i n t e g r a l of the c o l l i s i o n o p e r a t o r i s (3 .50) - = + where z=-t-iO means t h a t the l i m i t i n g v a l u e z=0 i s approached from above the r e a l a x i s (Im z > 0 ) . U s i n g t h i s (3 .49) becomes ( 3 . 5 D s k ^ - t ) = fcf j mt\mi*=+ io)r^im-jt) The d i s c u s s i o n i n C h a p t e r I I I . 2 showed t h a t o n l y r i n g and c h a i n type diagrams need be r e t a i n e d . The f u r t h e r r e s t r i c t i o n o f the c o l l i s i o n o p e r a t o r t o z = + t O # i l l e n a b l e us to sum these two i n f i n i t e s e r i e s e x a c t l y . T h i s w i l l be done i n s e c t i o n s 4 and 5 o f t h i s c h a p t e r . The lowest o r d e r c o n t r i b u t i o n t o the d e s t r u c t i o n term (2 .61) i s (3 .52) ^,^1= i <^/(-^L)(U-£)-'U;4;> Y { AT ,AT. \ t- 0 ) T h i s becomes, on p a s s i n g t o the l i m i t o f an i n f i n i t e system (2 .2 ) and c a l c u l a t i n g the m a t r i x e lements by u s i n g ( 2 . 2 4 ) , (2 .27) and (2 .28) (3 .53) 4 ^ ; ^ = ixjf*-1" j«. ®& ju vti. S n - 59 -The discussion of Chapter I I I . l assures us that ~ L n * and that J> . The inequality of (3.8a) then indicates that we can ignore the wave vector k j i n comparison with i . Thus (3-54) )t)m ~ i ^ fa Vt i,ia F I 7 T \ , - » ( « . ^ - * - - " ) This i s c l o s e l y related to terms i n the c o l l i s i o n operator (see,(3.25a))and an argument l i k e that of Appendix B can be used to show that, provided the range of the correlations i s of the order of or less than the range of the interactions, the destruction term decays to zero for times much greater than the c o l l i s i o n time: (3.55) Jd* (#.)*) £ > \ t c The creation operator defined by (2.66b) and the destruc-t i o n term of (2.67) are c l o s e l y related to the c o l l i s i o n operator (2.62) and the destruction term (2.61) respectively and we assume that they have the same time dependence. That i s (3.56) C ( ^ , " ' ^ U £ ' j ; 0 -> o l,r t » t c and (3.57) < £ P ^ - * ' ' ^ ^ t>~>tc The statement (3.56) allows us to extend the t' integration of (2.68) to i n f i n i t y and perfqrm a Taylor series expansion s i m i l a r to that of (3.43). The r e s u l t i s that the creation - 60 -term of (2.68) i s given by (3.58) C^(M;t) = fcf-> Z C(*,'-iJf^>i =The decay of the destruction terms (3.55) and (3.56) for times much greater than the c o l l i s i o n time means that the long time behaviour of the system i s e s s e n t i a l l y independent of the i n i t i a l c o r r e l a t i o n s provided the range of these i n i t i a l c o r r e l a -tions i s of the order of the range of the interactions. In t h i s case the destruction terms of (2.59) and (2.68) can be ignored for times much greater than the c o l l i s i o n time. - 61 -4. Summation of the Rings We now proceed to obtain an e x p l i c i t form for the contribu-t i o n to the c o l l i s i o n term of (2.59), Jy^^ir,;^), by the long range i n t e r a c t i o n . This i s achieved by summing over a l l possible r i n g diagrams, The complete class of rings can be systematically generated by adding basic v e r t i c e s of type B to the diagram F i g . 3.1a i n the manner i l l u s t r a t e d i n F i g . 3.4. The term of the c o l l i s i o n operator represented by the lower l e f t diagram of F i g . 3,4 i s given by (3.25f). According to (3.51) the contribution from the diagram to the c o l l i s i o n term fo r long times i s obtained by operating on P(s'!([irS; t) with the operator of (3.25f), l e t t i n g z +iO, integrating over s ' - l v e l o c i t i e s and summing over t k ' i . Taking the l i m i t z—*+iO means that the factors must be written as z r Z where (3.59) jrl <£_(*) = / T O T T T ? DO . ~ TTc 4- f>-5C The contribution to the c o l l i s i o n term from this diagram i s then (3.60) ITT fa^^ ( y " t ) 3 ( ^ f pi \tyl>2„. niS.(I^) - 63 -When Green's theorem i s applied to the v e l o c i t y integrals of (3.60) i t i s seen that we can replace 2 „ by 3, , by i 2 and by § 3 where £ ^ i s defined by (3.23) and (3.61) ^ = % In general i t can be asserted* that i f one of the indices j , n, say n, associated with a vertex does not appear to the l e f t of that vertex then the v e l o c i t y derivative operator 3^ can be replaced by 9^. We introduce the following compact notations: (3.62a) d} = * 7 ! 3 * Vv; il-fy (3.62b) d^"= | W, and (3.63) — n These quantities have the following symmetry properties (3.64) = (3.65) {d^f ~ -d^ = and (3.66) In terms of these (3.60) becomes * See reference 9, p.50. - 64 -(3.67) - I I I J fo t ^ ^ y | ^ ^ ^ ^ £ 3 ^ ^ ^ The e x p l i c i t form for the contribution to the c o l l i s i o n term from the long range i n t e r a c t i o n i s obtained by i n s e r t i n g the expressions for the terms of the c o l l i s i o n operator repre-sented by rings into (3.51). The rings are generated i n the manner indicated i n F i g . 3.4, each of the diagrams y i e l d i n g a term s i m i l a r to (3.67). The r e s u l t i s : (3.68) TH(*,M^ = f I C zm [ jtl jdn* U r Jti fa £ +-til fa% cj, a1 c dt s» c «/*&_3T < f i 3 fa fa (%; e%< - ) - • » * where /^ ,. i s understood to mean • - 65 -The quantity of interest is actually the one particle distri-bution function, , rather than its Fourier coefficient f^lv-t) ' Thus we require an expression for (3.69) J t e , , c ? Tjk,l*.:<) e'-''2' Taking the inverse Fourier transform of (3.68) yields (3.70) J(x,^,-t)l^ - fdlfael, *:ZJZ, 40)4(1) where (3.7D = ^ The series (3.70) can be rewritten as (3.72) J k . ' ^ U i ^ ^0 with - 66 -(3.73) R l ) = J^lUJ2l-fO)-fn) + fax S!7Jti(i) ]das hUz f (2) for £ ¥ « / y , -fOJW o r (3.74) FO) = r ct2l«»m + j^e^+d) R\) I n t r o d u c i n g the q u a n t i t i e s (3.75a) £ 0 ) = / V 2 f ( 2 ) - 67 -(3.75b) £(.) = , - i £ ^ W , j</^-«-3..) (3.76a) = l^zClzl U')+<V (3.76b) - - < TT ^  VV^  | ^ cTJ/' i and (3.77a) d(l) = £ 0 ) (3.77b) - - i £Ut, ',*) into (3.74) yields the i n t e g r a l equation (3.78) £OF0) = TO) + i* <-(0Jets/* SJJ-lt) F * 0 The solut i o n of t h i s equation as obtained by Balescu and T a y l o r 1 2 i s (3.79, F( 0 = FU i « ) = ^ + % M*.) JJK L U - A) where (3.80) A ~ i and (3.81) ft*) == Lp.)f(^) It must be noted that equation (3.79) i s v a l i d only i f the quantity £*(&,), when considered as a function of a complex variable, has no zeros i n the upper half-plane. It w i l l be shown l a t e r (Chapter III.8) that t h i s requirement i s equivalent - 68 -to the condition that the system be stable. The fact that the d i s t r i b u t i o n functions are r e a l and thus J(x ^r, ;t) * s a l s o means that only the imaginary part of F ( l ) i s required to obtain J " ( - * / € ( > *)j r i n g s from (3,72). The imaginary part of F ( l ) i s given by (3.82) UFO)= -TJT^ Noting that (3.83) £.(*) = ~il°-k and using (3.75), (3.76) and (3.82) i n (3.72) we obtain (3.84) J(X^,;t)\ = * * ' f t l fat i - i , - ^ l t lungs 1 \£(0\ with £(1) given by (3.75b). This completes the summation of the rings. In e f f e c t , summing over the rings, which involve simul-taneous c o l l i s i o n s between a r b i t r a r y numbers of p a r t i c l e s , has resulted i n a c o l l i s i o n term involving only two p a r t i c l e s which interac t through an e f f e c t i v e potential whose Fourier transform i s ~ We/fd)/ . - 69 -5. The Summation of the Chains The contribution to the c o l l i s i o n term of (2.59), Jjt%(*!);t) , from the short range in t e r a c t i o n w i l l now be determined. Accord-A Cs') ing to (3.51) th i s i s obtained by operating on /O^, t) with the c o l l i s i o n operator, l e t t i n g z-?+iO, integrating over s ' - l v e l o c i t i e s and summing over . The c o l l i s i o n operator to be used consists of a se r i e s , the terms of which are represented by the class of diagrams c a l l e d chains. The f i r s t few terms of t h i s series are represented by the diagrams of F i g . 3.5. The term i n the c o l l i s i o n operator represented by the second of these diagrams i s given by (3.25b). The contribution to the c o l l i s i o n term by t h i s diagram i s determined by i n s e r t -ing t h i s expression into (3.51). The re s u l t i s F i g . 3.5 - 70 -(3.85) _ i | - ; r J * : v ^ , ji^ (r*>c)(%)3 [4[J!'UX1'** xii-U'iJ The corresponding contribution to (3.69) JU„ST,',t) = c z Jk(<r,i*)e'-' i s given by (3.86) -?nxfj4it (^fjjlJjf uA n-hz 7\SJ*>%) Uti„g>) ((*-<£')' Each of the chains represents a contribution to the c o l l i -sion term s i m i l a r to (3.86). The series made up of these c o n t r i -butions i s written (3 . 87) JU, r, ;*) / c K a ; ^ = ' ? ^ f a z ]>/ Ujf U:8.T 7[Ui I,2) U, if S,2 + (ZYfaWfOt, ;i.2„-nl<i'U)U,i-ri The summation of the series of (3.87) i s accomplished by compar-ing i t terra by term with a series describing the scattering of a - 71 -beam of p a r t i c l e s from a f i x e d centre of force.* The Hamiltonian of a p a r t i c l e moving i n a central force f i e l d i s (3.88) H - i - w^ - r - 7\ VU) and the L i o u v i l l e equation appropriate to t h i s simple system i s (3.89) = - i %• || Defining the Fourier c o e f f i c i e n t s of -f(Zj$r;t) by (3.90) {(z.'it) = ^ ^ r ; e £ l -and of the p o t e n t i a l by <3'91> MU) = fe yxea'Z we can write the following equation for the time evolution of the Fourier c o e f f i c i e n t (3.92) fr/i --.fee V ^ ' & e It i s assumed that the system i s i n i t i a l l y homogeneous: (3.93 ) and we consider only the evolution of f0 . Solving (3.92) by i t e r a t i o n with the boundary condition (3.93) we have * See Chapter 6 of reference 8. - 72 -(3.94, = . ^ ) . / ; ^ J [ A V, fe e - * * « - ' ^ Now i f * » (i^T1 (3.95) J ^ ' J / ^ ' = * nlM-e) and thus f o r t imes l o n g i n c o m p a r i s o n w i t h the c o l l i s i o n t ime t c ^ ( i - ^ y ' the s e r i e s o f (3.94) becomes (3.96) fjt) -f>(0) = _t j- Sjutyil^vUMV, ^ % D e f i n i n g the f r e e p a r t i c l e p r o p a g a t o r i n phase space by (3.97a) GU,*T) = ~ jJi iKLflir)^1' ' - 73 -or (3.97b) GCz.tT) = with inverse (3.98) nt-U-e) - jd* and noting that (3.99) < ^ =jfr}<1Z i 7 e we can rewrite (3.96) as (3.100) ^ ^ o , = A [ fey/^ Hvir This i s equivalent to the set of two equations: (3.101) fiM-fiO) = f | . i and (3.102) = ^ J'*'^*"*'^ f?' 'A 3**''^ This l a t t e r equation i n e f f e c t gives the solut i o n of the time independent L i o u v i l l e equation and hence j-C*,^) i s conserved along the trajectory of motion. It i s thus equal to fi(4T'>o) for some As' which can be found by tracing back along the t r a -j e c t o r y . - 74 -When the expression for the propagator i n (3.97b) i s inser-ted into (3.102) there r e s u l t s (3.103) T(*>1t*,<£) -fi(*;o) = I ±[ (&) • ) where the z-axis coincides with . If we l e t ^  0 0 the z' integration then extends from - o o to<« and (3.103) becomes (3-104) E j : v ) = * k 7<*.»*,*) Equation (3.101) can be written (3.105) ttejv-fi&io) - ^IH>fe'(£<l„rh T(w<*) which then i n conjunction with (3.104) yiel d s (3.106) fa iL\o) - £Jd*J<} [T (*>^ ~ o) We express the element of area dxdy i n c y l i n d r i c a l polar coordi-nates (b)<f>>Z) 15 and note that (3,108) bdbdf - <r(&,<f) d&dif where (T(&;(fl) i s the cross section for scat t e r i n g into the element of s o l i d angle x ^ * ? d& of^ at an angle 6 to the o r i g i n a l d i r e c t i o n . Equation (3.107) then becomes - 75 -(3.109) /?(2,r)-fi(#;o) - ~, J lctfot*^d<r&4)^fo^)-fi(#^ with © being the angle between and /tr' . The quantity on the right of (3.109) i s equivalent to the right hand side of (3.96) and we can write the quantity enclosed i n the brackets,£ J, as a f i n i t e displacement operator acting on the v e l o c i t y , v i z . , (3.110) C(if)- j™$fJ6^*Wj)<r^l"^h_ |J A d i r e c t comparison of the series enclosed i n brackets i n (3,96) with that of (3.87) reveals that these are i d e n t i c a l i f we write (3.111) x r - - 9 ( 2 Therefore we can write for the quantity enclosed i n the brackets of (3.87) (3.112) = jk* jdtfcb^e ne>f) %x [e and thus (3.113) or m t ( 3 . i i 4 ) J)*)\ - \<&, Uwo*^* <n&>4) %z 'Chains. ' h /> - 76 -where the angle G is that between ( V / - -^y) and f</;-^ .) . This completes the summation of the chains. The expression (3.114) for the contribution to the c o l l i -sion term by the short range interaction is just the Boltzmann collision term. This is not surprising as the chains essentially represent a l l possible two body collisions. - 77 -6. L o n g - t i m e E q u a t i o n s . H-Theorem The r e s u l t s of the p r e c e e d i n g s e c t i o n s can now be employed to produce an e q u a t i o n d e s c r i b i n g the development of the one p a r t i c l e d i s t r i b u t i o n f u n c t i o n which i s v a l i d f o r t imes l o n g compared t o the c o l l i s i o n t i m e . T h i s e q u a t i o n i s o b t a i n e d by F o u r i e r t r a n s f o r m i n g e q u a t i o n ( 2 . 5 9 ) . A c c o r d i n g t o (3 .55) the d e s t r u c t i o n term can be i g n o r e d . W r i t i n g /t*U(»-^WCY) f o r 2V(r) i n the s e l f - c o n s i s t e n t f i e l d term as g i v e n by (2 .64) we have (3 .115) Z-MW.*) + Aft .2.{tlxtl*,)*) r i n g s = J R b e i n & w i t h J ^ v ^ t ) ) c h a i n s = J C and J O c ^ v ^ t ) g i v e n by (3 .114) and (3 .84) r e s p e c t i v e l y . Not a l l the terms i n t h i s e q u a t i o n a r e s i g n i f i c a n t . The c o n t r i b u t i o n from the s e l f - c o n s i s t e n t f i e l d term to the e v o l u -t i o n o f the one p a r t i c l e F o u r i e r c o e f f i c i e n t i s g i v e n by ( 2 . 6 3 ) . p u t t i n g xriyuU^-VWjg f o r /I and n o t i n g t h a t f o r d i s t r i b u t i o n s whose h a l f - w i d t h s a re o f the o r d e r of the average v e l o c i t y . (3 .116) ^ ~ K ^ 4 ^ 0 we have t h a t the o r d e r of magnitude of (2 .63) i s , - 78 -( 3 . H 7 ) SCF ^ ^C7/L;'(^P- Z-WLKW/Q* where, s i n c e t h i s w i l l be compared w i t h terms a l s o i n v o l v i n g a f a c t o r , the F o u r i e r c o e f f i c i e n t has been d r o p p e d . A c c o r d i n g to ( 3 . 7 ) # l 0 a n d the f i r s t term of (3 .117) i s t h e r e f o r e n e g l i g i b l e . T h i s means t h a t o n l y the l o n g range i n t e r a c t i o n c o n t r i b u t e s to the s e l f - c o n s i s t e n t f i e l d term and (3.118) SCF = jdSfa' ^W(^'^^^^)t)f,U^t)^-^^ The c o l l i s i o n term a r i s i n g from the s h o r t range i n t e r a c t i o n , JQ i s g i v e n i n ( 3 . 1 1 4 ) . S i n c e the c r o s s s e c t i o n f o r s c a t t e r i n g i s of the o r d e r of X t h i s term i s c l e a r l y of o r d e r (3 .119) and from the d e f i n i t i o n of t ^ ® ^ g i v e n i n (3 .31) we have (3 .120) J C ^ Vti? The c o l l i s i o n term a r i s i n g from the l o n g range i n t e r a c t i o n i s g i v e n by ( 3 . 8 4 ) . T h i s was o b t a i n e d by summing a l l 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 r i n g s . The o r d e r o f magnitude o f an n t h o r d e r r i n g i s a n d , p r o v i d e d f Z O C f i ^ \ , the s e r i e s converges r a p i d l y . ( I t w i l l be seen l a t e r t h a t t h i s i s e q u i v a l e n t to r e q u i r i n g t h a t the system be s t a b l e ) . We t h e r e f o r e e s t i m a t e the o r d e r of magnitude of the c o l l i s i o n t e r m , J R , to be the same as t h a t of - 79 -the contribution represented by the second order r i n g . That i s (3.121a) j ^ 2 1 * c ^ ^ Y " ^ and since ^ ^ - ^ ^ t h i s i s equivalent to (3.121b) ^ ^ ( J ? f ^ V t « . with being defined by (3.32). It i s then clear from (3.120) and (3.121b) that the two c o l l i s i o n terms w i l l be about equally e f f e c t i v e i f *f J~ t(yl) f that i s , i f (3.33) ^ £ ~ / The r e l a t i v e magnitudes of the se l f - c o n s i s t e n t f i e l d term and the c o l l i s i o n terms can be estimated from (3.118), (3.119) and (3.121b). We f i n d that the r a t i o s of J c and J R to the s e l f -consistent f i e l d term are (3.122) J c / s c p ~ & ffl and (3.123) \ / £ C F ~ respectively. We can d i s t i n g u i s h between several p o s s i b i l i t i e s depending on the r e l a t i v e magnitudes of the combinations of parameters ?p, % and (*/.K)"'. These are displayed i n Table I. In the s e l e c t i o n of the ri n g diagrams i t was e x p l i c i t l y assumed that the condition TABLE I R e l a t i v e o r d e r of magnitude of the s e l f - c o n s i s t e n t f i e l d and c o l l i s i o n t e r m s . (°&)~ (°<Lh)"' » 0) 1 j S C P » J C » J R (MO v p . » ( o . L J ' ; j c » JR » sc f S C ^ - J c ~ J R J c - J R » S C F ( i ) ^ « (oiU)" ; SCF » J R » J c (/7,^>> j J " R » S C F » J c JR»S>cf» Jc - 81 -(3.8a) » * was s a t i s f i e d . When i t i s , a l l entries i n Table I are possible. Relaxation of t h i s r e s t r i c t i o n to (3.8b) |_ h ^ o C ' automatically excludes the second two columns and a l l but the f i v e entries marked with arrows of column one i n Table I. This follows from (3.4) and (3.5) It w i l l be seen l a t e r i n the section that the only d i s s i p a -t i v e terms of (3.115) are J c and J R . Although these terms may be small they are very important as they drive the system to l o c a l equilibrium. In order to obtain an i r r e v e r s i b l e equation then, i t i s necessary to r e t a i n at least one of these c o l l i s i o n terms. That i s , even i f both JQ and J R are much less than SCF they cannot be neglected i f the long-time behaviour of the sys-tem i s of i n t e r e s t . Therefore, only i f J C » J R , i . e . >^ <2 « ~ , can J R be neglected i n the long-time equations and vice versa. Of course i f a short-time r e v e r s i b l e equation i s of in t e r e s t the c o l l i s i o n terms may be neglected i f they are very small i n comparison with the other terms. Relaxing the r e s t r i c t i o n (3.8a) to (3.8b) means that the wave vectors 4 / c a n no longer be neglected i n comparison with the wave vector J***- i n the terms i n the expansion of the c o l l i s i o n operator represented by rings (see (3.24a, c, f ) ) . However, these terms w i l l s t i l l be of the same order of magni-tude and, although they cannot be summed exactly, presumably - 82 -y i e l d a r e s u l t of the same o r d e r as J R . E q u a t i o n (3 .115) can t h e n s t i l l be c o n s i d e r e d v a l i d p r o v i d e d J R i s n e g l i g i b l e . T h a t i s , i f J R « J C > w e c a n d r o p J R from (3.115) and have an e q u a t i o n v a l i d f o r l o n g t imes even though the wavelength of the inhomo-g e n e i t i e s i s comparable to the range of the l o n g range i n t e r -a c t i o n . I f on the c o n t r a r y J R ^ JQ, i . e . ^ f i ^ ^ , and L r , ~ ' we do not have an e q u a t i o n v a l i d f o r l o n g t i m e s . The f l o w t e r m , ir. if , i s o f the o r d e r of <<"> L^' I f , as a c c o r d i n g to ( 3 . 6 ) , we see t h a t the s e l f - c o n s i s t e n t f i e l d term (3 .124) scr - u'e»i,iT"2 ~ U' and t h e r e f o r e the f l o w term i s o f the same o r d e r as the s e l f -c o n s i s t e n t f i e l d t e r m . We can now w r i t e the f o l l o w i n g s e t s of e q u a t i o n s which are v a l i d f o r l o n g t i m e s . F i r s t , i f the system i s weakly inhomo- geneous , >><*"', we have s i x d i s t i n c t e q u a t i o n s d e p e n d i n g on the r e l a t i v e s i z e o f the r a t i o of the s t r e n g t h of the weak i n t e r a c t i o n t o the average k i n e t i c e n e r g y , j the r a t i o of the wave-l e n g t h o f the i n h o r a o g e n e i t i e s to the range o f the l o n g range i n t e r a c t i o n , o(l.K \ and the r a t i o of the range of the s h o r t range p a r t t o the range o f the l o n g range p a r t o f the i n t e r a c t i o n , pf . These a r e : . (3 .125a) 4-4f>*£ -h S C F - J c v a l i d f o r -y/i«% • t l k h ^ ^ | , ' * > -y^ ^ ~ 1 - 83 -(3.125b) |£ = Jc v a l i d for ; ^ ° g t > > \ (3.125c) | f +4f,2£ + ^ _ J c + ^ - i v a l i d for (<*L K ) (3.125d) H - Jc -f- JR v a l i d for 2 ^ - ° ^ » ( * L k ) " ' (3.125e) |£ + * . f f + zr v a l i d for ^ « ^ £ U V ' (3.125f) H = va l i d f or ^ » ; >M*' L*)" If the system i s strongly inhomogeneous, i_h~o(., then the following two equations apply: (3.126a) f | -h*£- H J t v a l i d for ^ <*fc ; ^Lb s£ £ (3.126b) I f — . . - T R v a l i d for ) If ffiZ^v/e have no equation which i s v a l i d for long times when the system i s strongly inhomogeneous. It w i l l now be shown that the equations (3.125) and (3.126) - 84 -are i r r e v e r s i b l e . F i r s t i t w i l l be demonstrated that a quantity which i s interpreted as the entropy i s a monotone increasing function of time. The homogeneous equilibrium d i s t r i b u t i o n w i l l then be shown to be a stationary s o l u t i o n of the equations (3.125) and (3.126). The l o c a l entropy density i s defined by (3.127) = - t^Jte ft(Xi1f;t)/«[+,tZ,l£)t)] where Kg i s Boltzmann's constant. The time rate of change of S(x;t) i s (3.128) || = - / r f l p g + with ~ being given by whichever of the equations (3.125), (3.126) i s v a l i d . For example, i f 7p^% £ , then i s determined by (3.125c). A l l the other equations are s i m p l i f i c a t i o n s of (3.125c) and can be obtained by dropping one or more terms from i t . The time rate of change of S, i f (3.125c) i s v a l i d , i s (3.129) | | - f c / ^ j V H + SCP - T c - J * f D + ^ J The f i r s t term i n the curl y bracket gives fo fa - s - ' f l r n - K ] = to f r ^ (3.130) - - i L . r - 8 5 -This expression, the divergence of a vector function Y , has no d e f i n i t e sign. It represents a flow of entropy from one region to another. The se l f - c o n s i s t e n t f i e l d term yields Since f i(x,v_jt) —? 0 as*r->°o, the integration over /tr by Green's theorem, gives zero. Therefore, the self-consistent f i e l d term does not a f f e c t the change of entropy. The c o l l i s i o n term i s displayed i n (3.114). It gives a term (3.131) Qc- -Kzjcffjel&e^e <?&><}) \<*»J ^ %x [ / + ^ ^ / ^ ) J = -fifty Jtft %x LlrJ^kl)] [ki')fW)-Poftz)] This can be rewritten as 21 (3.132) which, since i s positive or negative as x i s greater than or less than y, shows that - 86 -(3.133) 6?c > O The c o l l i s i o n term J R of (3.84) produces (3.134) <3» = - * W | £ ) , / * i [ , + X>tM,t)J 2 A p a r t i a l integration over v e l o c i t i e s gives ( 3.35, = f J „ . ^ M2 J « J a ) £1%™$ and since by d e f i n i t i o n f ( l ) i s posi t i v e (3.136) O It w i l l now be shown that the l o c a l equilibrium d i s t r i b u -t i o n (3.137) f c z , * ) ^ (^r I w - ) e i s both necessary and s u f f i c i e n t for and Q R to be zero. It i s clear from (3.132) that the equality of (3.133) can hold i f and only i f - 87 -(3 .138) {(l')-fn') - 40) i^) 22 and as i s w e l l known t h i s c o n d i t i o n i s met o n l y f o r the l o c a l M a x w e l l i a n , ( 3 . 1 3 7 ) . T h a t (3.137) i s a s u f f i c i e n t c o n d i t i o n f o r the e q u a l i t y of (3 .136) to h o l d i s e s t a b l i s h e d by a s t r a i g h t -f o r w a r d s u b s t i t u t i o n of the l o c a l M a x w e l l i a n i n t o ( 3 . 1 3 5 ) . S i n c e (3 .139) fu = - -mp>(Z) Cir- " ' 2 0 ] 4 ° ( ^ ^ ) the c o m b i n a t i o n • 4(0 4(2) ~J makes the r i g h t hand s i d e of (3 .135) i d e n t i c a l l y z e r o . I t i s a l s o c l e a r t h a t i n o r d e r f o r the r i g h t hand s i d e of (3 .135) t o be i d e n t i c a l l y z e r o we must have (3 .140) 3vUz>&) - - D(x)[<r + * ( * ) ^ where D(x) i s a s c a l a r i n d e p e n d e n t of v_ and _a(x_) i s a v e c t o r . The s o l u t i o n o f t h i s d i f f e r e n t i a l e q u a t i o n i s ( 3 . 4 i ) ^ x , t f ) = cu)e~~ and s i n c e = J4r<r4(*>*> a n d [du Z>V) fid*) = - 88 -D(x) can be i d e n t i f i e d w i t h fi(x) , U(x) w i t h -ac*) \ and C(xJ w i t h (^?Ul^^/z7i(x) . T h i s proves t h a t (3.137) i s n e c e s s a r y f o r Q R to be z e r o . C o l l e c t i n g the r e s u l t s o f ( 3 . 1 3 0 ) , (3 .133) and (3.136) we have f o r the case when (3 .125c) d e s c r i b e s the e v o l u t i o n of the system t h a t (3 .142) § * S C Z ; * ) ± = Q ^ Q a > o T h i s shows t h a t the l o c a l e n t r o p y d e n s i t y may i n c r e a s e o r d e c r e a s e d e p e n d i n g on the f l o w i n t o or out of the r e g i o n . At the same time the c o l l i s i o n terms J^ -. and cause the l o c a l e n t r o p y d e n s i t y t o i n c r e a s e u n t i l such t ime as l o c a l e q u i l i -br ium i s r e a c h e d . T h i s i s not a s t a t i o n a r y s t a t e of the system as the f l o w term i s not n e c e s s a r i l y z e r o . S i n c e the system i s i s o l a t e d t h e r e can be no f l o w a c r o s s the b o u n d a r i e s and hence the volume i n t e g r a l of (3.142) i s (3 .143) j^I" - $c + > O where "s and Q are the volume i n t e g r a l s o f S and Q . T h i s means t h a t the t o t a l e n t r o p y of the system i s monotone i n c r e a s i n g , t h a t i s , the system i s i r r e v e r s i b l e . The f l o w term of (3 .142) i s z e r o f o r the homogeneous e q u i l i b r i u m d i s t r i b u t i o n (3.144 ) f(z, y) = cft<l> = C <?" ^ and t h i s i s t h e r e f o r e a s t a t i o n a r y s o l u t i o n of (3 .125c) The p r o o f t h a t the system approaches l o c a l e q u i l i b r i u m does not - 8 9 -e n t a i l an approach to t h i s a b s o l u t e e q u i l i b r i u m as n o t h i n g has been s a i d about the t r a n s i t i o n from the l o c a l to a b s o l u t e 23 M a x w e l l i a n d i s t r i b u t i o n s . G r a d has shown t h a t systems d e s c r i b e d by a l i n e a r Bol tzmann e q u a t i o n do i n d e e d approach a b s o l u t e e q u i l i b r i u m . I t i s t h e r e f o r e t a k e n as p l a u s i b l e t h a t the system d e s c r i b e d by (3 .125c) does e v o l v e u n t i l the d i s t r i b u -t i o n (3 .144) i s r e a c h e d . T h i s d i s c u s s i o n has been c o n c e r n e d w i t h systems whose e v o l u t i o n i s c o n t r o l l e d by ( 3 . 1 2 5 c ) . However, s i n c e each o f the o t h e r e q u a t i o n s of (3 .125) and (3.126) i n v o l v e one or b o t h of JQ and J R , systems c o n t r o l l e d by them w i l l behave i n the same g e n e r a l manner. That i s such systems w i l l e v o l v e i n such a way t h a t the e n t r o o y i s monotone i n c r e a s i n g . The l o c a l p r o d u c t i o n o f e n t r o p y w i l l be p o s i t i v e u n t i l l o c a l e q u i l i b r i u m i s r e a c h e d . The r a t e a t which t h i s s t a t e i s approached s h o u l d be p r o p o r t i o n a l to the magnitude of JQ a n d / o r J R . A c c o r d i n g to (3 .120) and (3 .121b) these a re — l/^ G) and ~ \/t(t) , r e s p e c t i v e l y . Thus the system approaches l o c a l e q u i l i b r i u m i n t imes of the o r d e r of the s h o r t e r of tfi or - 90 -7. The S h o r t - t i m e E q u a t i o n s . I n s t a b i l i t y I t w i l l now be shown t h a t the l o c a t i o n of the z e r o s of the q u a n t i t y £ * ( / ) d e f i n e d i n e q u a t i o n (3 .75b) i s r e l a t e d to the i n i t i a l s t a b i l i t y o f the s y s t e m . To do t h i s we w i l l r e q u i r e e q u a t i o n s d e s c r i b i n g the s h o r t - t i m e b e h a v i o u r of the s y s t e m . T h a t i s , we c o n s i d e r the development o f the system over t imes s h o r t compared to the r e l a x a t i o n time but l o n g compared t o the t ime a p a r t i c l e of average speed t a k e s to c r o s s the range of the s h o r t range i n t e r a c t i o n , (3 .145) y-'Cmfi)"1 - t?> « t « *r T h i s s t i l l l e a v e s two p o s s i b i l i t i e s ( i ) t » ~ ^''(wfi)^ and ( i i ) t~t® . The f i r s t o f these p r e s e n t s no new d i f f i c u l t i e s i n d e r i v i n g the e q u a t i o n s of e v o l u t i o n as the upper l i m i t o f i n t e g r a t i o n i n (3 .18) can s t i l l be extended to i n f i n i t y and the d e s t r u c t i o n term i g n o r e d . T h i s means t h a t the e q u a t i o n s (3 .125) and (3 .126) s t i l l d e s c r i b e the development of the system f o r s h o r t t imes except t h a t i t i s no l o n g e r n e c e s s a r y to i n s i s t on i r r e v e r s i -b i l i t y . R e f e r r i n g back to T a b l e I. we see t h a t i n a d d i t i o n to the e q u a t i o n s of (3 .125) and (3 .126) we can d i s t i n g u i s h a n o t h e r e q u a t i o n (3 .146) | £ + SCF =0 which i s v a l i d f o r s h o r t t imes i f (a) Tp. « ~ , (21 J 1 « | ; (b) 3 ^ ~ f £ < < ("iUf1 ; o r (c) ^ L ^ ) " ' » ^ f » ~ . T h i s means a l s o t h a t (3 .125a) and (3 .126a) a re v a l i d f o r s h o r t t imes o n l y i f - 91 -9 (3 , 1 2 5 c ) i s now v a l i d o n l y i f f£ ~ ("k)"' 5 and (3 . 125e) i s now v a l i d o n l y i f z-^ ~ (U^)"'. With these a d d i t i o n a l r e s t r i c t i o n s , a l l terras of any p a r t i c u l a r one of the e q u a t i o n s o f (3 . 125 ) or (3 . 126 ) are of the same o r d e r of m a g n i t u d e . T h a t i s , each o f the terms i n any of these e q u a t i o n s i s o f the o r d e r o f the i n v e r s e r e l a x a t i o n time a n d , s i n c e we are i n t e r e s t e d i n much s h o r t e r t i m e s , the s h o r t - t i m e b e h a v i o u r p r e d i c t e d by the e q u a t i o n s (3 . 125) and (3 . 126 ) i s the t r i v i a l one (3 . 1 4 7 ) | | = O T h i s t o g e t h e r w i t h (3 . 146 ) g i v e s the s h o r t - t i m e b e h a v i o u r o f the system f o r a l l p o s s i b l e v a l u e s of the parameters > and (^U) - 1 as l o n g as t imes g r e a t e r t h a n t [ i ] a re of i n t e r e s t . The s e c o n d p o s s i b i l i t y , t ( / } ~ t « £ r > makes the e x t e n s i o n of the upper l i m i t o f i n t e g r a t i o n i n (3 . 1 8 ) i n v a l i d f o r the l o n g range i n t e r a c t i o n and a l s o the d e s t r u c t i o n term does not decay r a p i d l y enough t o n e g l e c t i t . T h i s l a t t e r d i f f i c u l t y i s surmounted by i n v o k i n g (3 . 1 7 ) t o argue t h a t the d e s t r u c t i o n term i s much l e s s t h a n the s e l f - c o n s i s t e n t f i e l d term and i s t h e r e f o r e n e g l i g i b l e f o r s h o r t t i m e s . We can e s t i m a t e the o r d e r o f magnitude o f the l o n g range c o l l i s i o n term f o r s h o r t t imes by c o n s i d e r i n g the c o n t r i b u t i o n to t h i s term r e p r e s e n t e d by the second o r d e r r i n g . T h i s c o n t r i b u t i o n i s g i v e n by s u b s t i -t u t i n g the e x p r e s s i o n f o r the second o r d e r term i n the c o l l i s i o n o p e r a t o r , which i s g i v e n i n e q u a t i o n (B .2) of Appendix B, i n t o ( 3 . 1 8 K W r i t i n g f o r the l o n g range p o t e n t i a l f u n c t i o n - 92 --oO-(3 .148) W ( r ) = - — J h > X"' h < x. we have (3 .149) W, = — e~°^ Qt^tk*' A0**^ F o r i £ . K (3 .150) W J I ^  1 ^ 7 ^ and for^>>{ i t o s c i l l a t e s r a p i d l y . T h i s means t h a t the i n t e g r a -t i o n s , w i t h VV^ w r i t t e n f o r V j , i n A p p e n d i x B e f f e c t i v e l y e x t e n d o v e r a r e g i o n i £ x" . We t h e r e f o r e take (3 .151) W. = — - O ; Ji > * When t h i s i s s u b s t i t u t e d i n t o ( B . M ) and (B.15) we o b t a i n (3 .152) 9(,(3;£J = 71 c ( ^ J ^ e + e ~ ^ e J and (3 .153) 9 ( e ( ? ^ i - 2nc[~^f K€ where , s i n c e M » ° < , we have w r i t t e n | =x . The c o n t r i b u t i o n t o the c o l l i s i o n term from t h i s second o r d e r terra i s - 93 -(3 .154) 4^ ;*^ - fa I fir ln r^)T) A,(*,;*-•>-) ft* (Ui)t-r) The ^ (As- ±-r) c a n be expanded i n a T a y l o r s e r i e s about t as was done i n (3 .43) and a g a i n o n l y the z e r o o r d e r term i n the e x p a n s i o n need be c o n s i d e r e d . We must then c o n s i d e r the i n t e g r a l o v e r t of X , and The f i r s t term o f ^ and 7 2 g i v e (3.155) ^ c ^ M < - a The second two terms o f X g i v e where E\(z) i s the e x p o n e n t i a l i n t e g r a l . Now o<gt ~ ^ ( T ^ ) ^ ^ ^ and . T h e r e f o r e , s i n c e | E . U ) | £ . I f o r x>. | , we have t h a t t h i s term i s (3 .156) 6 c(#JV«/S)"' and (3 .157) \{V,&fl ~ t&*T('»ti''1 »>/! - 94 -T h i s i s p r e c i s e l y the same r e s u l t as was o b t a i n e d e a r l i e r f o r the o r d e r o f magnitude of J R (see (3.121b) ) , t h a t is,^\/t<*> . T h i s means t h a t , f o r t imes o f the o r d e r of tc , the c o l l i s i o n term due to the l o n g range i n t e r a c t i o n w i l l be n e g l i g i b l e and the b e h a v i o u r o f the system i s d e s c r i b e d by e i t h e r (3 .146) or ( 3 . 1 4 7 ) . The former b e i n g v a l i d i f ( a ) ^ < < ; ^ , 1 ^ << j j (b) -yfi~g « ( * L J ~ ' or (c) ( < x L K ) " ' » > ^ » J » a n d the l a t t e r b e i n g a p p l i c a b l e o t h e r w i s e . E q u a t i o n (3 .147) y i e l d s n o t h i n g of i n t e r e s t and we c o n s i d e r o n l y (3 .146) which when the s e l f - c o n s i s t e n t f i e l d term i s w r i t t e n i n i t s e x p l i c i t form becomes (3.158) ^fa^,*) T h i s i s the f a m i l i a r V l a s s o v e q u a t i o n . In the form (3.158) i t i s i n t r a c t a b l e and t h e r e f o r e we s h a l l c o n s i d e r the l i n e a r i z e d form of t h i s e q u a t i o n . (3 .159) f ^ f a , * ; * ) + *r, |^(*^;^> 4- *g fe'JJ? §-wo*-?'/)• ^(pte;*) -f-M'^'j*) - O F o r t imes much l e s s than the r e l a x a t i o n time the v e l o c i t y d i s t r i -b u t i o n i s e s s e n t i a l l y c o n s t a n t i n t i m e , ~&)EE . U s i n g t h i s , and F o u r i e r and L a p l a c e t r a n s f o r m i n g e q u a t i o n (3 .159) r e s u l t s i n - 95 -(3 .160) Z) - UJkLv { PM The F o u r i e r - L a p l a c e t r a n s f o r m of the number d e n s i t y i s o b t a i n e d by i n t e g r a t i n g ,z) o v e r the v e l o c i t y (3 .161) ^ffefewiz) I n t e g r a t i n g (3 .1Q0) o v e r the v e l o c i t y , v_, i t e r a t i n g the r e s u l t i n g e q u a t i o n and summing the s e r i e s o b t a i n e d one f i n d s * t h a t (3 .162) V e ) = _ ± _ J ? F _ f>^}t,o) where (3 .163) n I + 4^ ( f e ) I ™ Z > 0 When (3.163) i s compared w i t h (3 .75b) we see t h a t , i f f o r f ( x j , y 9 ; t ) we w r i t e £ (^(^fz) t h e n the q u a n t i t y t h e r e d e f i n e d i s j u s t L a p l a c e t r a n s f o r m i n g (3 .162) y i e l d s (3 .164 , „ 4 = ^ ^ e - « * _ J _ ^ . * See C h a p t e r 3 o f r e f e r e n c e 9. Note t h a t , because the i n t e r -a c t i o n here c o n s i d e r e d i s a t t r a c t i v e w h i l e B a l e s c u i s c o n c e r n e d w i t h a r e p u l s i v e i n t e r a c t i o n , a t r i v i a l change of s i g n i s r e q u i r e d . - 96 -w i t h i t b e i n g u n d e r s t o o d t h a t the a n a l y t i c c o n t i n u a t i o n of the i n t e g r a n d i n t o the lower h a l f - p l a n e must be u s e d i n e v a l u a t i n g the i n t e g r a l . The b e h a v i o u r of fl^lt) i s d e t e r m i n e d by the l o c a t i o n o f the p o l e s of the i n t e g r a n d of (3 .164) e n c l o s e d w i t h i n P . S i n c e the p a r t o f the c o n t o u r a n t i p a r a l l e l to the r e a l a x i s l i e s above a l l s i n g u l a r i t i e s o f the i n t e g r a n d and the c o n t o u r i s c l o s e d a t i n f i n i t y i n the lower h a l f - p l a n e i t e n c l o s e s a l l the s i n g u l a r i -t i e s of the i n t e g r a n d . We have then (3 .165) 7\k(t) - - t Z e~'*>* { RI+JM* where Z- i s a p o l e o f Tl^d) . T h i s means t h a t 'YLit.(t) o s c i l l a t e s w i t h f r e q u e n c i e s Oj^-ReZj . The a m p l i t u d e of these modes i s g i v e n by the r e s i d u e of 71^(1) a t Z j and they a r e e i t h e r damped, s t e a d y or e x p o n e n t i a l l y growing a c c o r d i n g as Y-mZ , - ° o r > 0 The q u a n t i t y (3 .166) j ^ - _ L _ fttett'-c) i s a Cauchy i n t e g r a l . I t i s r e g u l a r f o r Ijm2>0but has s i n g u l a r ! t i e s l o c a t e d i n the lower h a l f - p l a n e . The p r e c i s e l o c a t i o n o f these s i n g u l a r i t i e s depends on ftel^t-o) but they always r e s u l t i n damped o s c i l l a t i o n s . The net e f f e c t of these modes i s t o d i s s i -pate an i n i t i a l d i s t u r b a n c e and make the system homogeneous. The i n t e r e s t i n g b e h a v i o u r a r i s e s from t h e s i n g u l a r i t i e s o f Yl/e(Z) l o c a t e d a t the z e r o s o f < £ + 0 ) £ ) . I f t h i s f u n c t i o n has a z e r o i n the upper h a l f - p l a n e , an e x p o n e n t i a l l y growing mode - 97 ~ r e s u l t s and the system i s s a i d to be u n s t a b l e . I t can be shown* t h a t t h i s f u n c t i o n has a z e r o i n the upper h a l f - p l a n e and the system i s u n s t a b l e i f and o n l y i f (3.167) int-ycfiWjt > I S i n c e Wjj i s a s h a r p l y peaked f u n c t i o n of k o n l y s m a l l v a l u e s of k w i l l be i m p o r t a n t . In p a r t i c u l a r the v a l u e k=0 i s of i n t e r e s t as (3.167) then a p p a r e n t l y p r e d i c t s a c r i t i c a l c o n c e n t r a t i o n above which a homogeneous system becomes u n s t a b l e . T h i s t r e a t m e n t of the s h o r t time b e h a v i o u r o f the system p r e d i c t s t h a t f o r some s i t u a t i o n s the number d e n s i t y can o s c i l l a t e w i t h e x p o n e n t i a l l y growing a m p l i t u d e . I t must be emphasized t h a t t h i s t rea tment i s v a l i d o n l y f o r s h o r t t imes and t h a t a l t h o u g h the system may be i n i t i a l l y u n s t a b l e i t w i l l not remain so as e v e n t u -a l l y the d i s s i p a t i v e e f f e c t s of t h e c o l l i s i o n s w i l l become i m p o r t a n t . The F o u r i e r t r a n s f o r m o f the i n t e r a c t i o n W ^ ^ ^ and thus the system s h o u l d be i n i t i a l l y s t a b l e i f (3 .168) -ycfi<=<1 < i T h i s s t a t e s t h a t the average i n t e r a c t i o n energy per p a r t i c l e due to the l o n g range i n t e r a c t i o n s h o u l d be l e s s t h a n the average k i n e t i c energy ( c f . ( 3 . 6 ) ) f o r the system to be i n i t i a l l y s t a b l e . The i n e q u a l i t y o f (3 .167) i s of importance i n the summation * See S e c t i o n 19, r e f e r e n c e 9 and the r e f e r e n c e s c i t e d t h e r e . - 98 -of the r i n g s d i s c u s s e d i n C h a p t e r I I I . 4 . I t was p o i n t e d out t h e r e t h a t the s o l u t i o n g i v e n f o r the i n t e g r a l e q u a t i o n o b t a i n e d by summing the r i n g s i s v a l i d o n l y i f the z e r o s £ * ( \ ) , d e f i n e d by ( 3 . 7 5 b ) , l i e i n the lower h a l f - p l a n e . T h i s means t h a t the c o l l i -s i o n term i s v a l i d o n l y i f the i n e q u a l i t y of (3 .167) i s  not s a t i s f i e d o r e q u i v a l e n t l y i f the i n e q u a l i t y o f (3 .168) i s s a t i s f i e d . - 99 -8. E q u i l i b r i u m Two P a r t i c l e C o r r e l a t i o n s The g e n e r a l e q u a t i o n s d e s c r i b i n g the e v o l u t i o n of the F o u r i e r c o e f f i c i e n t s of the s - p a r t i c l e c o r r e l a t i o n f u n c t i o n were g i v e n i n e q u a t i o n ( 2 . 6 8 ) . I t was c o n c l u d e d i n C h a p t e r I I I . 3 t h a t the d e s t r u c t i o n term tB^Ct'Jl)^) d e f i n e d i n (2 .67) c o u l d be i g n o r e d f o r t imes l o n g compared w i t h the c o l l i s i o n t i m e . The o n l y s i g n i -f i c a n t c o n t r i b u t i o n s then a r i s e o n l y from the c r e a t i o n term d e f i n e d i n (3 .58) i f t>>tc. The d i s c u s s i o n o f C h a p t e r I I I . 2 has p r o v i d e d us w i t h the means of s e l e c t i n g the s i g n i f i c a n t terms i n the e x p a n s i o n of the c r e a t i o n o p e r a t o r . In p a r t i c u l a r F i g . 3 . 3 a and F i g . 3 .3b are t y p i c a l o f the two c l a s s e s of diagrams which r e p r e s e n t the s i g n i f i c a n t c o n t r i b u t i o n s to the development of the two p a r t i c l e r e d u c e d F o u r i e r c o e f f i c i e n t . However , even t h i s s i m p l i f i e d s e r i e s cannot i n g e n e r a l be summed and as i t was shown i n s e c t i o n 6 of t h i s c h a p t e r t h a t the one p a r t i c l e d i s t r i b u t i o n f u n c t i o n approaches the e q u i l i b r i u m d i s t r i b u t i o n f o r t imes of the o r d e r of the r e l a x a t i o n t ime we s h a l l c o n s i d e r o n l y two p a r t i c l e c o r r e l a t i o n s at e q u i l i b r i u m . The two c l a s s e s o f c o n t r i b u t i o n s to CC^AiU&'J,  ? - + LO) r e p r e s e n t e d by the m o d i f i e d r i n g s (see F i g . 3 .3a ) and the m o d i f i e d c h a i n s (see F i g . 3 .3b) are summed s e p a r a t e l y . F i r s t the m o d i f i e d r i n g s . These diagrams are j u s t the o r d i n a r y r i n g s w i t h the l e f t most v e r t e x s p l i t o f f . Hence the c o n t r i b u t i o n t o the c r e a t i o n term from the m o d i f i e d r i n g s i s v e r y s i m i l a r to the c o l l i s i o n term Ji((/{\t)\ g i v e n i n ( 3 . 6 8 ) . In f a c t , d r o p p i n g the A and y_2 i n t e g r a t i o n s and the o p e r a t o r cclt from the b e g i n n i n g of t h i s e x p r e s s i o n and c h a n g i n g the K r o n e c k e r d e l t a to - YCA'J g i v e s the s e r i e s f o r )f± ^ '}t)\ . Because t h e system i s - 100 -assumed to be at equilibrium the k' wave vectors are a l l zero and the f i r s t few terms of the serie s are: (3.169) n,.(*^ )jR = \ c <twf<"-<•) . , . . . j where the e f f e c t of the Kronecker delta has been taken into account by writing ^ ^ as YkrA . The series (3.169) can be rewritten i n terms of the quantity F ( j ) defined by (3.73) and passing to the thermodynamic l i m i t (2.2) we have (3.170) ^Ark(^^)j - C [d(zl fte,)(f)7^) + c ^ ^ s j FO) + ca/*f/sj,)[ra)J with F°(l) being the expression (3.73) when C(f"f^) i s written for f ( j ) . It s a t i s f i e s an equation obtained by s u b s t i t u t i n g the equilibrium d i s t r i b u t i o n for the one p a r t i c l e d i s t r i b u t i o n function i n (3.79). The r e s u l t i s (3.171) - 101 -This integral equation has the solution (3.172) F'(#<) - F I  , — - ffa) Ayr * i - w c X . ^ r Putting (3.172) into (3.170) and using (3.62) and (3.63) yields (3.173) X^-^'^lt - i« &.L±-4,-A*t) 4 te-<r)f(«,)fr&) Since the two particle correlation function rather than the Fourier transform is of interest the inverse Fourier transform of (3.173) must be taken and we can use the fact that (3 .174 ) | x /i t <S- (Jf) = I to write (3.175) f ; , v < r - * ) | R = &Wt[l-r*&VILj~'<pr*><ft") In this calculation we restrict ourselves to the particular form of the potential (3.176) v W ) = — ^ ; ^>*" in which case * See Section 51, Reference 9 - 102 -(3.177) — —- <°~* A ^ ^ +• k oxb* It can be argued that correlations between p a r t i c l e s separated by less than the range of the short range i n t e r a c t i o n are immaterial and since t h i s corresponds to values of h £ we are permitted to write for the Fourier transform of the potential (3.178) W A = - i - I When t h i s i s substituted into (3.175) there r e s u l t s (3 .179) rhj.t = j^rr *&]'f«Jfa) The two p a r t i c l e c o r r e l a t i o n function due to the long range i n t e r a c t i o n i s given by (3.180) = *ltee' *} %ru^>^)\rz or writing (3.181) = x . - X t <#+>«»*\ = * fa * f ~ r ,>*>)\* The c o r r e l a t i o n functions must be everywhere f i n i t e . This means that the value r = o must be excluded i n using (3.182). Actually, the passage from (3.177) to (3.178), e n t a i l s the exclusion of the region rot1 and we therefore write - 103 -where the Heaviside function (3.184) ^(X) - / ; X >o -0 ) * <o The contribution to the creation term at equilibrium from the modified chains can be determined in the same way as the contribution to the collision term by the chains. The result is (3.185) V 4 ^ ^ ) ( c = [niSAk-U) g ? 4>%i 4- JdlM/ m'^a-U)^^U^t)(k-^ ^UlU) On passing to the thermodynamic limit (2 .2 ) and applying the inverse Fourier transform we obtain - 104 -(3.186) fz(r,^,,^)l = * jdk efk* { 7rUA>U)%U±ti-2* 4- {A^i'nm u,*-n #•'-'>'*« This expression s i m p l i f i e s to (3.187) 9 t - o - , T . , * o / c = * r . ^ t m * i L&ut»]x Combining t h i s with (3.183) we have the following expression for the equilibrium two p a r t i c l e c o r r e l a t i o n : (3.188) 3> ; ^ J = c f f t / f e ) - | + ^ - ' ' ) " * j When the expression for Wk given i n (3.178) i s substituted into the s t a b i l i t y c r i t e r i o n (3.169) i t i s seen that the system i s stable i f - 105 -(3.189, Jg%. < | It i s cl e a r , from examination of the l a s t term i n (3.188), that i f the parameters of (3.188) are varied so that the system approaches i n s t a b i l i t y , then the range of the correlations becomes i n f i n i t e . Provided the inequality of (3.189) i s s a t i s -f i e d , the range of the correlations w i l l be f i n i t e and of the order of oi'1 . In that case we can write (3.190) ~ cfarfrvr) | e - I e For t < (3.191) ~-cz</H)(f^z) and for t~ > K~' (3.192) ~ %r ^ c-(flU,)ff^) These follow from the fact that (3.193) g~ 1 ^ | . Y > K " The two statements, (3.191) and (3.192), are precisely the same as those given i n (3.14) and (3.15) for g| calculated by retain-ing only the f i r s t order term i n the cl u s t e r expansion. - 106 -It should be noted that even i f the system approaches i n s t a b i l i t y and the range of the correlations becomes i n f i n i t e the order of magnitude of the correlations due to the long range i n t e r a c t i o n i s s t i l l of order Vfo « { . In section 7 of thi s chapter the e f f e c t of the correlations for short times was neglected on the basis that they were of the same order as the correlations at equilibrium. If t h i s i s so, i t i s clear that the cor r e l a t i o n s are of order i/{2«\ and can be neglected. It i s important to note that t h i s c a l c u l a t i o n of the equilibrium correlations i s good only i f the system i s stable. If i t i s unstable then £ ^ ( k ; z ) has zeros i n the upper h a l f -plane. The so l u t i o n given i n (3.172) f o r the i n t e g r a l equation (3.171) i s then no longer v a l i d . - 107 -9. The Equation of State In t h i s section we s h a l l determine the equation of state of a system whose interactions are given by the following forms for the two parts of the potent i a l ; (3.194) u(y) = I ; t-and (3.195) W ( j . ) ~ o • >-<*-<* r ) The equilibrium two p a r t i c l e c o r r e l a t i o n function for th i s form of i n t e r a c t i o n i s given by (3.188). Using t h i s and the d e f i n i -t i o n (2.41) of the c o r r e l a t i o n functions we have the following expression for the two-body d i s t r i b u t i o n function at equilibrium. (3.196) J^{z,,x,)%>^) - cxj(r-yr') f°<4f,)(f4fa) An expression for the pressure tensor i s given i n equation (2.32) and since the hydrostatic pressure i s j u s t 1/3 TrP we have (3.197) pU}t) ~ -L jJ(*J€i <r»»2f,(Zt)€r>t) - 108 -At equilibrium and writing £ for x - x we have (3.198) po - i w c ] ^ ^ ' ^ ) C o With the use of (3.196) and noting that d Y~ and thi s becomes : — E <~-+ (3.199) p. = cKeT - ^ H ^ <f(-*1 + v ( e - + ^ /} Defining and employing the fact that *<r<tf the integrations i n (3.199) give (3.200) P o = cfcT I + f ^ + f ^ - 109 -We know from ( 3 . 5 ) , (3 .6) and (3 .7) t h a t 27? « j and Here. /S --r~ , and i t i s i m m e d i a t e l y o b v i o u s that the second and 1 >TGT f i f t h terms of (3 .200) a re n e g l i g i b l e . T h e r e f o r e Whether o r not the second term of t h i s e q u a t i o n i s s i g n i f i c a n t The e q u a t i o n of s t a t e , (3 .201) i s o f c o u r s e o n l y v a l i d f o r those v a l u e s of the c o n c e n t r a t i o n and temperature which make the system s t a b l e . T h i s i s because the two p a r t i c l e c o r r e l a t i o n f u n c t i o n i s good o n l y f o r s t a b l e s y s t e m s . The c o n c e n t r a t i o n must a l s o be much l e s s t h a n }(3 o r the s e l e c t i o n o f the c h a i n diagrams as r e p r e s e n t i n g the dominant c o n t r i b u t i o n s from the s h o r t range p a r t o f the i n t e r a c t i o n i s i n v a l i d a t e d . (3 .201) depends on the parameters i n v o l v e d . I f ^ £ ^ i t i s n e g l i g i b l e . I f y/S»~ t h e n i t can p l a y an i m p o r t a n t r o l e p r o v i d e d - n o -CHAPTER IV DISCUSSION A c l a s s i c a l gas whose p a r t i c l e s i n t e r a c t through a weak l o n g range a t t r a c t i o n and a s t r o n g s h o r t range r e p u l s i o n has been s t u d i e d . I t has been d e t e r m i n e d t h a t , p r o v i d e d the range of the s h o r t range i n t e r a c t i o n , ){~' , i s much l e s s than the average i n t e r p a r t i c l e d i s t a n c e , i . e . X 3 » C , and the range of the l o n g range i n t e r a c t i o n , <^-1 , i s much g r e a t e r t h a n the average i n t e r -p a r t i c l e d i s t a n c e , i . e . o<V< c , the c o n t r i b u t i o n s to the d e v e l -opment of the system s e p a r a t e i n t o two c l a s s e s . The one c l a s s , which can be r e p r e s e n t e d by c h a i n diagrams d e t e r m i n e s the e f f e c t of the s h o r t range i n t e r a c t i o n to f i r s t o r d e r i n . The o t h e r c l a s s , which can be r e p r e s e n t e d by r i n g diagrams d e t e r m i n e s to f i r s t o r d e r i n uVc the e f f e c t o f the l o n g range i n t e r a c t i o n , a p a r t from a s e l f - c o n s i s t e n t f i e l d c o n t r i b u t i o n . T o g e t h e r these c o n t r i b u t i o n s y i e l d a s e t of e i g h t e q u a t i o n s , ( 3 . 1 2 5 a - f ) and ( 3 . 1 2 6 a , b ) each of which i s v a l i d f o r a d e f i n i t e range of the r a t i o of the s t r e n g t h of the weak i n t e r a c t i o n to the average k i n e t i c energy per p a r t i c l e , the r a t i o of a t y p i c a l wavelength of the i n h o m o g e n e i t i e s t o the range of the l o n g range i n t e r a c t i o n , and the r a t i o of ranges of the two p a r t s of the i n t e r a c t i o n . T h r e e i m p o r t a n t r e s t r i c t i o n s b e s i d e s t h a t on the ranges of the i n t e r a c t i o n were p l a c e d on the system i n o r d e r t o d e r i v e these e q u a t i o n s : ( i ) t h a t the average k i n e t i c energy be much g r e a t e r than the s t r e n g t h o f the weak i n t e r a c t i o n but much l e s s than the s t r e n g t h o f the s t r o n g i n t e r a c t i o n , ( 3 . 5 ) ; ( i i ) t h a t - I l l -the volume i n t e g r a l of the s h o r t range p o t e n t i a l be much l e s s than t h a t of the l o n g range p o t e n t i a l , ( 3 . 7 ) j and ( i i i ) t h a t the system be s t a b l e ( f o l l o w i n g ( 3 . 8 1 ) ) . I t was a l s o r e q u i r e d that the wavelength of the i n h o m o g e n e i t i e s be of the same o r d e r or g r e a t e r than the range of the l o n g range i n t e r a c t i o n s which i s not a s e r i o u s r e s t r i c t i o n . The c o r r e l a t i o n s were assumed to be of the same range as the i n t e r a c t i o n s . T h i s was shown t o be c o n s i s t e n t by the c a l c u l a t i o n of the two body c o r r e l a t i o n f u n c -t i o n i n C h a p t e r I I I . 8 , p r o v i d e d t h a t the system i s s t a b l e . The e q u a t i o n of s t a t e d e r i v e d i n C h a p t e r I I I . 9 , can be compared w i t h the Van der Waals e q u a t i o n of s t a t e , namely , The f i r s t term i n b r a c k e t s of (4 .1) i s a r r i v e d at by c o n s i d e r i n g the r e d u c t i o n i n the phase space a v a i l a b l e to a p a r t i c l e due to the n o n - z e r o d i a m e t e r of the m o l e c u l e s . In t h i s i n v e s t i g a t i o n we have a s s e r t e d t h a t C << tf1 i n which case the Van der Waals e q u a t i o n of s t a t e i s In d e r i v i n g the two p a r t i c l e c o r r e l a t i o n f u n c t i o n a t e q u i l i b r i u m , and i n d e e d throughout t h i s work, we have n e g l e c t e d terms o f o r d e r c/yt} and t h e r e f o r e a term l i k e the second one of (4 .2) does not appear i n ( 3 . 2 0 1 ) . The second term of (3.201) a r i s e s from the two p a r t i c l e c o r r e l a t i o n s due to the l o n g range i n t e r a c t i o n . (3 .201) (4 .1 ) (4 .2 ) - 112 -These a re c o m p l e t e l y n e g l e c t e d i n the d e r i v a t i o n of the Van der Waals e q u a t i o n of s t a t e . The l a s t term i n b o t h e q u a t i o n s (3 .201) and (4 .1 ) a re i d e n t i c a l . They are due to the mutual a t t r a c t i o n of the m o l e c u l e s and i n v o l v e no e f f e c t s due t o c o r r e l a t i o n s . T h a t the r e s t r i c t i o n s mentioned above a re not u n r e a s o n a b l e can be seen by c o n s i d e r i n g two r e a l g a s e s , Argon and Ethane as e x a m p l e s . We take the i n t e r a c t i o n p o t e n t i a l to be of the form g i v e n by (3.194) and ( 3 . 1 9 5 ) . T h a t i s , and Z " W ( r ) - 0 f h - is <2 . - I — ; A more r e a l i s t i c i n t e r a c t i o n i s the L e n n a r d - J o n e s p o t e n t i a l •AVM = HE [($)"-&'] which f i t s q u i t e a c c u r a t e l y a v a i l a b l e d a t a such as the v i s c o s i t y of g a s e s . However, s i n c e t h i s does not have a w e l l d e f i n e d range f o r the l o n g range p a r t of the i n t e r a c t i o n , we choose the s i m p l e r form a b o v e . T h i s i s f i t t e d to the L e n n a r d - J o n e s p a r a -meters by t a k i n g X ' - > e q u a t i n g the volume i n t e g r a l s , )r>a - 113 -and s e t t i n g the f o r c e between p a r t i c l e s s e p a r a t e d by a d i s t a n c e )0/^ e q u a l f o r the two forms of the p o t e n t i a l . The ' h e i g h t ' of the r e p u l s i v e c o r e i s r e s t r i c t e d to v a l u e s such t h a t /AK<<1>20, -lo i.e./t<^<^<^ . T h i s means, because /to i s f i n i t e , t h a t p a r t i c l e s c o l l i d i n g w i t h r e l a t i v e v e l o c i t y > [ ^ ) ^ can approach to d i s -tances l e s s than X " ' from each o t h e r . T a b l e I I l i s t s the v a l u e s of the parameters i/> oi and >^  f o r Argon and Ethane and the range of v a l u e s s p e c i f i e d f o r T and C by the r e s t r i c t i o n y ^ » / r g T > > 2 / and the s t a b i l i t y c r i t e r i o n 4* far TABLE II parameters f o r Argon and Ethane Argon Ethane X c m " 1 2 . 9 x l 0 7 2 .3x10? o< c m " 1 4 . 8 x l 0 5 3 . 7 x l 0 5 Z / e r g s 6 .7x10-20 1 . 3 x l 0 " 1 9 T°K 3 3 0 » T » 5 x l 0 ~ 4 6 1 0 » T » 9 x l O ~ 4 C / T ( ° K c m 3 ) " 1 < 1 . 8 x l 0 1 9 < 4 . 5 x l 0 1 8 - 114 -21 —3 The c r i t i c a l c o n c e n t r a t i o n f o r Argon i s 8x10 cm and f o r 21 —3 Ethane i s 4 .2x10 cm . Thus f o r c o n c e n t r a t i o n s w e l l below the c r i t i c a l c o n c e n t r a t i o n s K J » c and p r o v i d e d c » l O * * 5 the c o n d i t i o n cxJ«c i s a l s o s a t i s f i e d . The r e q u i r e m e n t s on C / T can then be s a t i s f i e d w i t h i n the ranges s p e c i f i e d f o r T and C . The v a l u e s f o r the parameters above s p e c i f y which of the e q u a t i o n s of (3 .125) and (3 .126) are a p p l i c a b l e . F o r a homogene-- 1 o;is s y s t e m , ( ° < L h ) =0, the system w i l l be d e s c r i b e d by ( 3 . 1 2 5 b ) , p r o v i d e d T » e x K f ^ K i n the case of Ethane or T>> 3x10 i n the case of A r g o n . I f the system i s s t r o n g l y inhomogeneous, c < L h ~ l , the system w i l l be d e s c r i b e d by (3 .126a) i f T « 0 . 3 ° K (Ethane) or T « . 0 . 6 ° K (Argon) and by (3 .126b) o t h e r w i s e . In n e i t h e r system i s the c o l l i s i o n term J R o f any s i g n i f i c a n c e . T h i s o f c o u r s e assumes t h a t the system i s s t a b l e . The o r d e r e s t i m a t e of the term J f t , ( 3 . 1 2 1 ) , i s v a l i d o n l y f o r s t a b l e s y s t e m s . I f the system approaches i n s t a b i l i t y the q u a n t i t y £ ( 1 ) d e f i n e d i n (3 .75b) tends to z e r o f o r some v a l u e of A and the e x p r e s s i o n f o r J R blows up making the o r d e r e s t i m a t e i n v a l i d . The second term of the e q u a t i o n o f s t a t e , ( 3 . 2 0 1 ) , i s c o m p l e t e l y n e g l i g i b l e f o r the above v a l u e s of the parameters even i n c o m p a r i s o n to the c o r r e s p o n d i n g term i n the Van der Waals e q u a t i o n of s t a t e . T h i s i n d i c a t e s t h a t i n r e a l gases f o r the s t a b l e r e g i o n the c o r r e l a t i o n s due to the l o n g range f o r c e s can be i g n o r e d . Moreover i t i s apparent t h a t the weak l o n g range i n t e r a c t i o n i s i m p o r t a n t o n l y i n the s e l f - c o n s i s t e n t f i e l d t e r m . T h i s means t h a t no m o d i f i c a t i o n to take i n t o account the c o r r e -l a t i o n s due to the l o n g range f o r c e s i s r e q u i r e d and the k i n e t i c - 115 -e q u a t i o n used by S o b r i n o i s adequate i n t h a t s e n s e . The most o b v i o u s e x t e n s i o n of t h i s work would be to c o n s i d e r h i g h e r o r d e r s i n c/)&z . T h i s would i n v o l v e summing over diagrams of the type F i g . 3 . 1 g which on the s u r f a c e appears to be v e r y d i f f i c u l t . C o n s i d e r a t i o n c o u l d a l s o be g i v e n t o systems which have v e r y s h o r t w a v e l e n g t h i n h o m o g e n e i t i e s , t h a t i s , wavelengths much l e s s than the range of the l o n g range f o r c e . T h i s would e n t a i l summing the inhomogeneous c h a i n s , e . g . (3 .24b) r a t h e r than the homogeneous c h a i n s , e . g . ( 3 . 2 5 b ) . - 116 -BIBLIOGRAPHY 1. Van der W a a l s , J . D . , D i s s e r t a t i o n , L e i d e n , (1873) . 2 . K a c , M. , U h l e n b e c k , G . E . , and Hemmer, P , C , J . M a t h . Phys.4_, 216, 229, (1963) ; i b i d ^ 5, 60, (1964) . 3 . Van Kampen, N . G . , P h y s , Rev . 135A, 362, (1964) . 4 . L e b o w i t z , J . L . and P e n r o s e , 0 . , J . M a t h , P h y s . 7, 98, (1966) . 5. S o b r i n o , L . G . , U n p u b l i s h e d . 6. M a x w e l l , J . C , P h i l . T r a n s , Roy. S o c . 157, 49 , (1867) . 7. Bol tzraann, L . , W i e n . B e r . 66, 275, (1872) . 8. P r i g o g i n e , I . , 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 M e c h a n i c s , I n t e r -s c i e n c e , New Y o r k , (1962) , 9 . B a l e s c u , R . , S t a t i s t i c a l Mechanics of Charged P a r t i c l e s , I n t e r s c i e n c e , New Y o r k , (1963) . 10. B a l e s c u , R . , P h y s . F l u i d s 3 , 52, (1960) . 11 . S e v e r n e , G . , P h y s i c a 31 , 877, (1965) . 12. B a l e s c u , R. and T a y l o r , H „ S . , P h y s . F l u i d s 4 , 85, (1961) . 13. P r i g o g i n e , I . and H e n i n , F . , P h y s i c a 24_, 214, (1958) . 14. T o l m a n , R . C , The P r i n c i p l e s of S t a t i s t i c a l M e c h a n i c s , C l a r e n d o n P r e s s , O x f o r d , (1938) . 15. G o l d s t e i n , H . , C l a s s i c a l M e c h a n i c s , A d d i s o n - W e s l e y , R e a d i n g , Mass , (1953) . 16. S t o n e , M . H . , L i n e a r T r a n s f o r m a t i o n s i n H i l b e r t Space and t h e i r A p p l i c a t i o n s to A n a l y s i s , A m e r i c a n M a t h e m a t i c a l S o c i e t y , New Y o r k , (1932) . 17. K i r k w o o d , J . G . , J . Chem, P h y s . 14, 180, (1946) ; i b i d 15, 72, (1947) . 18. C o h e n , E . G . D . , P h y s i c a 28, 1025, (1962) . 19. R e s i b o i s , P . , P h y s . F l u i d s 6, 817, (1963) . 20 . S a l p e t e r , E . E , , A n n . P h y s . j>, 183, (1958) . - 117 -21 . Chapman, S. and C o w l i n g , T . G . , The M a t h e m a t i c a l T h e o r y o f N o n - U n i f o r m G a s e s , Cambridge U n i v e r s i t y P r e s s , Cambridge , (1939) . 22. Sommerfeld , A . , Thermodynamics and S t a t i s t i c a l M e c h a n i c s , C h a p t e r V , Academic P r e s s , New Y o r k , (1956) . 23 . G r a d , H . , J . S o c . I n d u s t . A p p l . M a t h . 13, 259, (1965) . 24. J a h n k e , E . and Emde, F . , T a b l e s of F u n c t i o n s , D o v e r , New Y o r k , (1945) . - 118 -APPENDIX A A simple demonstration of the f a c t o r i z a t i o n theorem stated i n Chapter 11.3 w i l l be given here. Consider the contributions to / ^ ' ^ ( t ^ l i ^  ' represented by the diagrams of F i g . A . l . These diagrams constitute (a) (b) F i g . A . l the complete permutation class„ The contribution of these diagrams i s (see (2.50) |jL fete, I [<4,^J|(Lo-^,(-^L2^0-^7-^L'J)a.-^"l|m> - 119 -Us i n g the d e f i n i t i o n s of the matrix elements of <£L g i v e n i n (2.27) t o g e t h e r w i t h (2.17) and (2.28) t h i s becomes / A • d > ' A ' ft I +-(2 0 Since the o p e r a t o r s (2.28) Qt«(k) = £ V 4 - ^ J commute with f u n c t i o n s of v e l o c i t i e s other than j or n we can r e w r i t e ( A . l ) as F i g . A.2 The contribution corresponding to the diagram of F i g . A.2 from (2.50) i s -hldie"'*% l<i|a.-»i-c-^n(L.-*r\m> /°t™m,o) m [dh Z <«,| eiL'Ct-Tt) (-ML") e'Ut' \my CmiMi *) - 121 -Comparing this with (A.2) i t is clear that the sum of the contri-butions due to diagrams (a) and (b) of Fig. A.l is just the product of the contributions of the component structures of these diagrams. This is what we wished to demonstrate. A general proof of the factorization theorem is given by Resibois in refer-ence 19. - 122 -APPENDIX B The second o r d e r term i n the e x p a n s i o n o f the c o l l i s i o n o p e r a t o r i s o b t a i n e d by a p p l y i n g the i n v e r s e L a p l a c e t r a n s f o r m to ( 3 . 2 5 a ) . The r e s u l t i s ( B . l ) ^ Lt) ^ ^ ^e^L-^HlTjdi * l x y i * - t .a,, i' 2,2. ~ ^ ( B . 2 ) = 2,z-7(l^).d^ w i t h the t e n s o r f u n c t i o n 7( d e f i n e d by (B .3 ) f(V,t) = ^- Ue' 19* }*) and i JJ ( B . 4 ) A ^ , w _ - o , v- j - i ^ F o r p u r p o s e s of i l l u s t r a t i o n we assume t h a t the p o t e n t i a l i s o f the form (B .5 ) V ( y ) = £ ^ K r The F o u r i e r t r a n s f o r m of t h i s f u n c t i o n i s (see (2 .25 ) ) ( B . 6 ) y ^ _ L _ L_ When t h i s i s i n s e r t e d i n (B .4 ) we o b t a i n J i I f we choose the c o o r d i n a t e system such t h a t 9 and the J2 - a x i s are p a r a l l e l i t i s i m m e d i a t e l y e v i d e n t t h a t the t e n s o r X i s d i a g o n a l and moreover the and ^ i z components are e q u a l . - 123 -That i s , ( B.8) f ( 2,s) = K,(V,z) + ? * e * j + 2^2;*) where Q, , ez and £ 3 are unit vectors along the /,, A and / 3 axes. Transforming from rectangular to c y l i n d r i c a l polar coordinates ( k>)<fj 5 ) w e n a v e and ( B.10) = , . - i e J ^ U / J ? J 5 v 5 r T ' The i n t e g r a l over b i n ( B.9) diverges l o g a r i t h m i c a l l y as h-^^ . The evaluation of the i n t e g r a l i s accomplished by taking the upper l i m i t to be L, a large but f i n i t e number. Performing the integra-tions we obtain* and ( B.12) ^(3;*) ^ _ __I ' where ( B.13) £ z = + Equations ( B . l l ) and ( B.12) y i e l d under inverse Laplace transfor-mation •See Section 28, reference 9. - 124 -( B . i 4 ) yr/t) = i z h ^ ' z c i i ? ) and C l e a r l y , s i n c e J">fr, a l l terras i n (B.14) and (B.15) decay to z e r o f o r t imes £ » ( / r 3 y ' • Thus (B.16) ; t » t c and t h e r e f o r e (B .17) ^ J t ) — ^ o U Y t » t c In g e n e r a l i t can be a s s e r t e d t h a t except f o r p a t h o l o g i c a l cases the s i n g u l a r i t i e s of are not r e a l and a r e l o c a t e d a d i s t a n c e ~ tf ' , the i n v e r s e o f the range o f the i n t e r a c t i o n , f r o m the r e a l a x i s and t h a t the c o l l i s i o n o p e r a t o r t h e r e f o r e tends t o z e r o f o r t imes much g r e a t e r than ~K '(mp>Yl . 

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