THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Nuclear Engineering Technical Report IRREVERSIBLE STATISTICAL MECHANICS OF MODERATELY DENSE FLUIDS Ralph J. Nossal ORA Project 03712 under contract with: U. S. ATOMIC ENERGY COMMISSION CHICAGO OPERATIONS OFFICE CONTRACT NO. AT(1-1)-917 ARGONNE, ILLINOIS administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR July 1963

This report was also a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan, 1963.

TABLE OF CONTENTS Page LIST OF TABLES v LIST OF FIGURES vii ABSTRACT ix CHAPTER I. INTRODUCTION 1 II. REVIEW 7 III. DERIVATION OF: KINETIC EQUATIONS 20 IV. REDUCTION OF THE KINETIC EQUATION FOR THE SINGLET DENSITY 37 V. SOLUTION OF THE KINETIC EQUATION FOR fl.l(Xl;t1xlo); MOMENTUM DENSITIES AND CORRELATION FUNCTIONS 61 VI. DETERMINATION OF Gs(r,t) 86 VII. BOGOLIUBOV FORMULATION; EXPANSIONS IN DENSITY 103 VIII. CALCULATIONS OF Gd(r,t) 119 IX. SUMMARY AND COMMENTARY 127 APPENDIX A. NOTES ON PROBABILITY THEORY 137 B. REDUCTION OF COLLISION INTEGRALS 146 C. RELATIONSHIPS BETWEEN SYSTEM CONSTANTS 152 D. CALCULATIONAL AIDS TO CHAPTER VI 154 E. THE FUNCTIONAL CALCULUS 157 F. THE BBGKY HIERARCHY 162 BIBLIOGRAPHY 165 iii

LIST OF TABLES Table Page 4-1 Definitions of system constants appearing in the collision integral expansion 60 5-1 Second order terms of momentum density 70 D-l Some useful integrals 154 (t2) D-2 Contributions to G 2)(r,t) 155 s.9 15V

LIST OF FIGURES Figure Page 5-1. Momentum auto-correlation functions 76 5-2. Typical intermolecular potential 77 5-3. Typical static pair correlation function 77 vii

ABSTRACT Primary attention is given to the calculation of time-relaxed probability densities and associated correlation functions for particles moving through an equilibrium assembly of like particles. An integro-differential equation describing the temporal evolution of the probability densities is obtained from the Liouville equation. In contrast to the BBGKY hierarchy, the derived equation contains all the information necessary for its solution; i.e., the kinetic equation for a reduced probability density does not contain integrals over unknown higher order densities. The kinetic equation for the one-particle density is studied in detail. An expansion for the collision integral appearing in this equation is obtained such that effects of interparticle forces are represented by terms of increasingly higher order. Collision integral terms through the "fourth power of the forces" are obtained explicitly. Solution of the kinetic equation is obtained by an iteration procedure in which the zeroth order terms are those which would be the only ones appearing were the assembly an ideal gas. Higher order terms are coupled with constants which are, effectively, increasingly higher powers of the ratio of potential to kinetic energy of the assembly. The time-relaxed momentum density is obtained from the full probability density by integrating over spatial coordinates. Corresponding expansions are obtained, also, for some related correlation functions (e.g., the momentum auto-correlation function). An expansion for the "self" part of the spatial two-time density Gs(r,t), is also found in direct correspondence with the expansion obtained for the full u-space density. The general functional form of this expression is derived and explicit evaluation is made of the low order terms of the expansion. By taking Fourier transforms with respect to space and time, expressions are obtained for the classical "scattering functions" for incoherent scattering of slow neutrons. Corrections for the purpose of obtaining the proper quantum mechanical scattering functions are discussed Bogoliubov type equations are also investigated in order to obtain kinetic equations pertinent to a density expansion of the one particle probability function. Methods previously developed for model systems interacting solely through repulsive forces are extended to systems for which attractive terms are also present in the interparticle potentials. The asymptotic collision integrals present in the kinetic equation appear as coefficients of a double expansion in,C and E~, where cis the density, and is a coupling constant ix

ABSTRACT (Concluded) related to the ratio of potential to kinetic energies of the assembly. Approximate expressions for the two-particle probability density, consistent with the assumptions employed in the derivation of the kinetic equation for the one-particle density, are also derived. Finally, investigation of the "distinct" part of the two-time spatial density, Gd(r,t), is initiated. A relationship between the "convolution approximation" for Gd(r,t) and the superposition approximation for the twoparticle density is explicitly demonstrated. x

CHAPTER I INTRODUCTION This investigation was undertaken with two goals in mind. One was the expectation of obtaining a better understanding of the manner in which the cross section for scattering of slow neutrons from matter is dependent upon the detailed properties of the scattering medium. The other was to study the possibilities of calculating, from first principles, certain probablity densities arising in theories of non-equilibrium statistical mechanics.. The emphasis in this thesis probably falls upon the latter. Specifically, major effort is directed towards the derivation and subsequent solution of equations providing the temporal evolution of conditional probability densitites for locating particles moving within an equilibrium assembly. A particular example of these functions is G(r,t), the probability density for finding a particle in the neighborhood of a point of configuration space at time t, conditioned upon the presence of a particle being located at a different specified point at some specified earlier time. This quantity is of particular interest and importance because of its relation to the scattering cross-section. The work which follows is presented in subdivisions of eight chapters. Chapter two is primarily a brief review of various techniques employed in other studies of non1

2 equilibrium statistical mechanics where the goal is to obtain kinetic equations from the Liouville equation. Included in this section, also, is a statement of the relationship between the neutron scattering cross-section and the timerelaxed spatial density, G(rt). In chapter three, kinetic equations are obtained from the Liouville equation to describe the temporal evolution of reduced time-relaxed probability densities. Although the work was inspired by a method used by Zwanzige in his studies of the statistical mechanics of non- quilibrium systems, the equations obtained differ quite significantly from any which have previously appeared in the literature. The kinetic equation for the one-particle time-relaxed density is further investigated in chapter four. Considerable effort is expended in obtaining pertinent expansions for the collision integral appearing in the equation. An iteration scheme for the solution of this equation consistant with the expansion of the collision integral, is developed in chapter five. The resulting expressions have the form of expansions in terms related, roughly, to powers of the ratio of potential to kinetic energies of the assembly. From the full /U-space density, momentum densities and related time-relaxed expectations and correlation functions are obtained. Some brief remarks are made concerning the calculation of the dielectric constant for partially ionized gases.

3 In chapter six, major effort is expended in calculating the "self" part of the time-relaxed spatial density, Gs(r,t). Corresponding expressions for the incoherent part of the neutron scattering cross-section are determined and comparison is made with some expressions based upon various proposed approximation schemes. Relationships between Gs(r t) (classical) and the exact quantum-mechanical cross-section are stated. The work of chapters 4-6 provides expressions for conditional densities and related correlation functions which are, essentially, expansions in inverse powers of the temperature. As an alternative, in chapter seven Bogoliubov type kinetic equations are developed to provide density expansions for these quantities. Equations pertinent to the study of the evolution of the time-relaxed densities are developed and, by employing a double expansion, the Bogoliubov methods originally developed for model systems interacting solely through repulsive forces are extended to systems for which attractive terms are also present in the interparticle potentials. An approximation for the two particle probability density is also obtained. Chapter eight deals mainly with calculations of the distinct part of the conditional spatial density, Gd(r,t), ** e.g., the "Gaussian approximation" for determination of scattering cross-sections.

4 and associated contributions to the neutron scattering crosssection. The main results of this study are summarized in chapter nine. Much of the material appearing on the following pages is new, at least in detail, if not in basic concept The original Zwanzig kinetic equations have been modified and extended for the purpose of obtaining equations appropriate to this particular non-equilibrium problem. Expansions of the collision integrals appearing in these equations are also obtained for the first time. The kinetic equations have been developed without the imposition of frequently employed approximations such as time smoothing or termination of a hierarchy. From the kinetic equations, expressions are obtained for various probability densities and correlation functions. These expressions are also new, although expansions for the dielectric constant and the resulting functions which are obtained from the correlation functions have already appeared in similar form elsewhere. However, particularly for the scattering functions it is felt that the discussion in this thesis more clearly demonstrates the effect of variation in temperature. The studies of the Bogoliubov methods which appear in chapter 7 are somewhat of a deviation from the main pattern of this investigation. However, they are included because

5 they are an interesting continuation of the investigations which have previously appeared in the literature. The extension of the theory to systems with attractive forces, while requiring some new calculation, is quite straightforward. However, when one tries to obtain expressions for the two particle densities, modifications are necessary because the usual BBGKY hierarchy (which underlies the theory) does not apply here. After making necessary corrections, expansions for the two particle time-relaxed density are obtained consistent with assumptions used to obtain a kinetic equation for the one-particle density A major problem in the field of non-equilibrium statistical mechanics is that there are almost as many approaches to a problem as there are researchers investigating it. This chaos is probably more apparent than real, and the equivalence of various theories is being quite rapidly elucidated. The particular methods used in this thesis were chosen simply because they seemed best for the particular problem at hand. This point is discussed at greated length in the next chapter. Note that most of the calculations of this report fall within the context of classical statistical mechanics, In many studies of phenomena investigated by statistical mechanics, classical theory has preceded and suggested analogous quantum theories. Although quantum theories already exist, explicit calculations are few; perhaps, in this sense,

6 the studies appearing here can be considered preliminary to a more "basic" quantum calculation. However, it must be pointed out that many physical phenomena of macroscopic nature seem to be adequately represented by the classical theories, and the calculations appearing in this study certainly have meaning with relation to such investigations. A limitation that should be acknowledged here is that, mainly for calculational convenience, in chapter four ( and ff. ) is imposed the requirement that the internal potential energy of the assembly is to be represented by purely additive pair potentials. Furthermore, at various points it is assumed that the medium is isotropic, As a result, the investigation is restricted to a study of fluid assemblies. These assumptions are commonly employed in non-equilibrium studies of statistical mechanics of course, it is to be expected that as our understanding develops, extensions of the theory will be made to somewhat more complicated systems.

CHAPTER II REVIEW It is well known that the scattering cross-section for neutrons is strongly dependent upon the dynamical behavior of the scattering medium. For low energy neutrons (for which s-wave scattering is predominant), for a system in which the directions of nuclear spins are uncorrelated and where the distribution of isotopic species is random, if one uses a Fermi-pseudopotential to represent neutron-nuclei interactions, the first Born approximation provides the following expression for the differential scattering cross-section:' -32a 1 aK ( a s (w i) + 2 S ( ) ) (21) 3S^i 4x5 F coh coh inc incc In the above, 0000 (2.2) E- = the energy of the incident neutron; E'= the energy of the scattered neutron; K and K' are the associated wave0 4. +4 numbers; = hl(E -E8); it (Ko-K); and, Qco and ~ 1~ coh inc. are the (energy independent) coherent and incoherent scattering lengths, respectively. For monatomic systems, the "scattering functions" are defined by: -=i[Wt- kor ] S o(k,w) E dk dt e G(rt) oooo (203a) and Sinco(kw) - t- lor ] S. (kw)) J dk dt e G (r,t) ooooo (2.3b) 7

8 G(r,t) and Gs(r,t) are defined by: 0 o (2.4a) G(r,t) 1/n < 6(q ()-') (t) )' and G (r,t) = l/n r< |6(r+qj(O)-r') (a'-q^(t) ) dP' > (2.4b) n = the number of particles in the scattering assembly. qk(0) and j.(t) are the positions of the kth nucleus at t = 0,.th and of the j nucleus at time t, respectively. The expectation values denoted by < oooooooooo > are to be taken with respect to the distribution of initial states of the scattering system. Thus, it is seen that the dependence of the cross-section on the scattering medium is manifest through the G functions. It is also to be noted that the G functions are determined only by the properties of the scattering medium. In the classical limit, these expressions have well understood meaning: 0ooo (205a) G(r,t) = the probability density for finding a particle in the neighborhood of r at time t, given that there was a particle at r = 0 at t = 0; oooo (265b) Gs(r,t) = the probability density for finding a particle at r at time t, given that the same particle was at r = 0 at t = 00

It is quickly seen that, in the interpretation of classical mechanics, 2.5a follows from 2.4a. Viz., by definition, G(r,t) E l/n < A d' [ 6(+qk>(O)->) 6(Ptj(t) ) ] >. J J = /n C udr < 6(r+qk(O) -) ('-q (t)) > But, the classical joint expectation of any quantity, < Jl(Xt)J2(Xo) > is defined by r r (2.6) < J (X)J C(Xo) > j JdX J1(X)J2(XO) f(Xo) f(X;t|Xo) (2 it 2 0... 1 o where J(X) is a function of the phase variable X = (X",..,x). f(Xo) is the probability density for X at t = 0; f(X;t|X ) is the conditional density for X @ t, given that at t = 0,X0. Thus, G(rt) = 1/n d d, dqkdj 6(,r+qk()-r')6(r'-q(t) )f(j;tl )f(qk = l/n idrI dqk, 6(r+q(0)-O)) f(PiVk;tt) 1/2 where the integration has been performed over qj(t). It has been noticed that, because the scattering system is assumed to be in thermal equilibrium** fo(q ) = l/0, where L is the volume occupied by the assemblyo Next, performing the integration over r8 **Here, we assume also that the scattering system is isotropic.

10 t) k fj,k ko k But, f( r+qk[qk;t) f( I t ) Recognizing this, and summing over k and j after performing the integration over q I 0 G(rt) fslf( rio; t ) + (n-l) fdstinct( r; t kk jk In the above equation, f self (r IO;t) is the probability k,k density for finding the kth designated particle in the neighborhood of r, given that it was at r = 0 @ t = 0; fj,k(rlO;t) is the probability density for finding the jth particle at r @ time t, given that the kh designated particle (j # k ) was located at r 0 @ t O 0o Thus, G (r,t) = fself( riO t ).oo.00 (2.7a) s Frequently, there is also defined: ** (2.7b) Gd(rt) = (nl-) fdito rio, t ) **Anticipating, somewhat, we note that Gd(rO) = c g2(r) Gd(r.O) =c g2(r), with g (r) defined by eqno 4.19. c is the average density of thb assembly: c = (n-1)/ Q

11 A number of investigators have considered the possibility of expressing the neutron scattering cross-section in terms of the classical G functions given by 2o5 and 2.7. Of course, the actual cross-section requires calculation of the quantum function defined by 2.4. However, theories have been devise-1 for the purpose of relating the exact cross section to scattering functions, S(k, w ), determined from the'classical' G functions. Clearly, there is some advantage to proceeding in this way, if for no other reason than to provide the investigator of eqn. 2.4 an opportunity to utilize some of the results of the extensive inquiry already made into similar problems which have been studied by the methods of classical statistical mechanics. However, the calculation of G(r,t) is just a part of a much larger program. The problem is basically as follows. Given the initial positions and momenta of a group of classical particles which are a subset of a collection of like particles, we wish to know the probability density for finding the subset in the neighborhood of a given point in their phase space at some later time, Furthermore, let it be postulated that the complete collection is at thermal equilibrium and has a canonical distribution of positions and momenta in r -space. Thus, we wish to find some way of calculating f Ss (X,;tIX ) ** which, when multiplied by dxloood.s is the **Xs is the six-s dimensional vector, (xl o,X) xi is the 6-dimensional vector, (agiRi)~

12 probability of finding particle #1 in a differential volume element dx1 about xl, etc., at time t, given that at zero time the particles were located at x1,...,x o f (Xs;tXs ) o o s s So is often referred to, in the following, as the conditional or time-relaxed s-particle probability density. But, how to calculate such quantities? All that is really known is that the temporal dependence of the density for the totality of particles in the assembly, n, is given by the classical Liouville equation, viz., nf, (X;tXs = [ H (X); f ] (2.8) nat 0 n n n n Hn(Xn) is the total Hamiltonian for the system; fnn(Xn;tlXso) is the n-particle probability density; the bracket notation is the usual Poisson bracket, viz., [H;f 3 = ( qHoV fn,n H V f;, ~n 1 q Pi Pi P i n n One needs to obtain, from 208, a kinetic equation providing the temporal evolution of the s-particle density of interest, fs s(Xs; t Xso). Then, one must solve the equation' The deduction of appropriate kinetic equations for reduced densities is a central problem in non-equilibrium statistical mechanics. Although most efforts have been directed towards obtaining kinetic equations for "distribution fucntions" other than the conditional densities of interest here, it is pertinent and instructive to review some of the

13 techniques employed in those studies as a prelude to calculations carried on later in this report. A first advance in the general theory of non-equilibrium 1P-14 statistical mechanics came with the deduction, from the Liouville equation, of a hierarchy of coupled integro-differential equations giving the evolution of many particle, reduced distribution functions. This set of equations, often referred to as the BBGKY hierarchy, provides the temporal evolution of an (s-l) particle distribution function in terms of the next highest, iLeo, (s) particle distribution~ Frequently,** the set may be written as follows: o.o.co.e (29) Unfortunately, this is an almost empty scheme, because to determine the sth order distribution one must know, first, higher order distributions which are themselves to be determined from similarly open equations, Of course, one might try to find some way of terminating the chain of kinetic equations at low ordero Commonly employed is the "superposition approximation", which consists of assuming that the s + 1st order distribution is proportional to products of s order distributionso E ogo, to termi15 nate the BBGKY hierarchy at second order, it is assumed that o o.but, not always of 0, appendix Fo

14 f3(x x2 x3) = f2(xlx2)f2(xlax3)f2(x2x3)/f (x1)f 1(x21 Unfortunately, the applicability of this approximation is 16 still controversial; it is known that, for the equilibrium case, the superposition approximation fails to provide the correct virial coefficients for a gas beyond the third coefficiento17 Also, there is lacking in this theory a logical prescription for demonstrating the progressively 18 more important contributions of higher order collisions an effect which is demonstrated in certain other theories.* 3,12 For example, Bogoliubov, using an elegant functional analysis, obtained a temporal equation for the one particle distribution which not only contains terms involving higher order collisions ad infinitum, but remarkably demonstrates that the higher order collision effects are associated with a power series expansion of the density. Unfortunately, the kinetic equation is provided in a functional form which, although formally correct, is in practise limiting in that manybody determanistic mechanical problems must first be solved before the theory is applicable to dense gaseso However, the Bogoliubov theory is of considerable help in understanding the validity of commonly employed kinetic equations such as the Vlasov and Boltzmann gas equationso 3,20,21 Although ** On the other hand, it has been shown that the superposition approximation can be used to terminate the BBGKY hierarchy so long as one doesn"t demand information about the distribution functions inconsistent wiat the order of the approximation used in the termination o

15 kinetic equations obtained by other methods from the basis of the analysis of most of this report, Bogoliubov type equations are considered in some detail in chapter 7, below.** Motivated by the success of theories of Brownian motion, in which the detailed motion of a particle is replaced by a sum of average motions over small periods of time, there have been developed various theories for "course grained" distribution functionso Notable among these are the theories of Kirkwood and his associates ## Important work has been performed, using these theories, in obtaining expressions for linear transport coefficients in dense fluids. The origins and validity of the Boltzmann equation have also been clarified to a great extent The distribution functions investigated by these methods are "time smoothed" functions, f ss(X;t) E 1/TJd dx 0dx f (X;t X -ps+l~ ~ ~+ n n,n n defined for small but finite T. Unfortunately, when calculating the neutron scattering cross-sections the short time behavior of the G-functions (eqno 2o5) is quite important and it is to be expected that course graining would introduce **In addition to the great difficulty in obtaining solutions to equations of this type, their derivation requires the imposition of some restrictive assumptions concerning the functional dependence of the higher order distributions upon those of lower order0 (Hollinger 22 has indicated that these assumptions imply a lack of correlations between particles before collision, an assumption akin the Strosszahlansatz of Bo ltzmann 18 ##For an excellent review of Kirkwood's work and the subsequent work of Rice, et alo, cfo Rice and Frisch. 23

16 erroro Also, the introduction of course-graining requires approximation at the earliest step in the analysis; one would prefer to defer making approximations until it is demonstrated that no further progress is possible without them. The same objections pertain to theories in which other types of course graining (eogo, in phase space) are introduced. 24,25,26 Prigogine and collaborators have developed an elegant scheme to obtain equations describing the temporal behavior of systems close to equilibrium. Their scheme involves a Fourier expansion in configuration space of the general n-particle distribution function, after which the temporal evolution of the expansion coefficients is investigated. An elaborate diagram scheme, analogous to those used in the quantum theory of fields, has been developed to expedite evaluation of contributions to the kinetic equations for coefficients; in effect, the BBGKY hierarchy is replaced by a hierarchy in momentum space onlyo Unfortunately, this work is not directly applicable to the study of short time evolution of a distribution function differing significantly from the equilibrium distributiono Although the Prigogine methods could, undoubtedly, be modified so as to be applicable to the study of such problems, the methods which are used in this thesis for that purpose seem, at the outset, to be somewhat simpler and are, for that reason, preferred. It is interesting to note that the Prigogine methods

17 and the Bogoliubov methods, while seemingly very different and each involving a great deal of specialized difficult analysis, have both been used to obtain an identical equation for the evolution of the one particle distribution 21,26 function for an ionized gas The schemes for obtaining kinetic equations, described on the last few pages, have been applied to investigations other than the calculation of the conditional probability densities defined on pc 11, Relatively little attention has been paid to the latter problem. An attempt to derive integro-differential equations giving the temporal evolution of, specifically, the conditional densities was made by 27 Vineyard 7 Starting with the Liouville equation, he derived a double set of equations (which reduce to a single set for a stationary assembly) analogous to the BBGKY hierarchy of the usual non-equilibrium theory0 But, as before, there remains the problem of terminating the chaino A phenomenological probabilistic calculation of timerelaxed pair correlation functions has been performed for 28 dilute gases by Christovo However, there is little clue furnished concerning the actual calculation of the kernels appearing in this theoryo A knowledge of the latter is necessary before application of the theory can be madeo Thus, the calculation of time-relaxed densities is still very much an open question~ Each of the methods discussed above is in some way not appropriate to the task

18 of obtaining kinetic equations applicable to this study. But, we have not yet mentioned a procedure only relatively recently suggested. By defining the projection operator, Pr -1/,n [dq1o dq and manipulating the Liouville equation in a clever way, Zwanzig derived 2,9 the following equation for the one particle non-equilibrium momentum density: r.~'~~; ~ I- s(l-Pr)L 1 (p1;t) fd+ ood (PrL ds e (i-Pr)L f (P l o pn;t-s) +, pl;t) j dP2 ~dPn,n n t( 1Pr)L + Pr L e (1-Pr) f (X;0)) ** nn n Although fo1 (EL;t) is dependent upon the n-particle density, fon(PEl,o,En;t), there is no dependence upon any unknown spatial probability densityo As such, this equation is already an improvement over the BBGKY hierarchyo Also, the equation is valid for all times and does not require course grainingo We find that similar equations are very useful in the study of the conditional probability densities, fs,(Xs;tIX, ); Zwanziges original suggestion for using projection operators is amended and extended in the next chapter, providing the equations we desire. It is to be noted that Gs(r~t) may be calculated once fl,l(X1; t IXso) is knowno Gd(r,t) may be similarly obtained from f2,2(X1,X2 t IX1 )o Although the major part of this report deals with the derivation and solution of equations for probability densities, considerable attention is paid, also, to obtaining expressions for the cross-sections~ It is necessary to point out that *' L is the Liouville operatoro For definitions, see (3o4), belowo

19 expansions for the latter, similar to the ones obtained in this report, have already been reported elsewhere.** Expressions for the scattering functions (and their inverse spatial fourier transforms) have been obtained in various forms of perturbation series. A notable characteristic of these expressions, however, is that they have been obtained without first performing calculations for G(r,t). On the other hand, one of the reasons for performing both theoretical and experimental studies of neutron scattering is the hope of better understanding the functional form of the G-functions. Because we study these functions directly, it is felt that the calculations presented in this thesis contribute to such understanding beyond that afforded by the expressions for the transforms of G(r,t). In addition, the effects on the cross-section caused by varying the temperature are perhaps better demonstrated by this theory. Finally, it should be pointed out that the conditional probability densities are of great significance if one wishes to calculate transport coefficients, It is well 23,31,32 known that the linear transport coefficients can be related to temporal integrals of appropriate time-relaxed correlation functions, An example is the proportionality of the dielectric constant to the Laplace transform of the momentum auto-correlation function. Since this point is elaborated upon in chapter five, we shall not discuss it further here. **For a review, cf., Nelkin 30or Schofield. o ~~~ or Schofield.

CHAPTER 3 DERIVATION OF KINETIC EQUATIONS A central problem of irreversible statistical mechanics has been the derivation of kinetic equations necessary for the description of the temporal behavior of pertinent reduced probability densities. Here, this question is discussed in detail for the particular case of time-relaxed probability densities in equilibrium assemblies. Interest in these functions is not only intrinsic; as has been indicated in the previous chapter, various quantities of physical importance such as the linear transport coefficients and the slow neutron scattering cross-section may be related to temporal integrals of appropriate densities and correlation functions Whereas major concern in the following is the derivation of equations for time-relaxed densities in equilibrium systems, it is indicated that under certain conditions closely related equations may be derived for probability densities of a more general natureo The method employed in the following investigation is an extension of a procedure originally suggested by R. W. Zwanzig in his studies of the approach to equilibrium. Let it be agreed that f ( sooo, 9 oo o 0p;t| q 90 ooq P o ~ )'Ps n,n 1 n 1 n o So lo O - f ( X;tlX ) nn n so 20

21 will represent the conditional probability density for finding a specified isolated n-particle system in the neighborhood of a point ( o fo o lsooo1 Pn )EX of r -space, at time t, given that at t = 0, X = X. ( X designates a s S0 s point in a pertinent 6s dimensional reduced phase space.) Marginal densities** are obtained from fn ( X;tlX ) in the following manner: f (q1 9s.a;p QOp;t X ) =.. C(3.2) a,b - a b P1 a 1 b o = dq )dq:dp. dP f (X;t|Xe,~ 1 a+l n b+l n n,n( n; s is the marginal conditional density, with respect to:f ( X;tX ) ), for finding the set of position coorn,n so0 dinates { q,..., ga} and momentum coordinates {R...*,b} in the neighborhood of the point ( ql,'.a'P' i'''Pb) of the appropriate reduced r -space, at time t, given that at t = 0: X = X s It is noted that f o and that B.b a, b > 0 an h. " for all a,b. As is well known, once the initial state f ( X;oIX ) n So is specified, fn ( X nt I X ) is determined for all time by the L iouville equation. The latter may be written in the following convenient form: af (X;tIX ) = L f (X;tIX ) oooo 0o (3o3) _.nn n So nn n so at ** For a review of some of the concepts and terminology of probability theory see appendix AX below.

22 The Liouville operator appearing in eqn. 3.3 may be expressed as the sum of three terms: L = L +L+L with, n 1n n + L 4 - I.o v 0oo0o (304a) L 1 i av(Jq Iq ) (V - V ) 0ooo (3.4b) 1 j k q j Pj Pk n L E [ aU[i] V 0 0.0. (3.4c) 2 1 aq Pi To obtain 3.4a-3.4c, it is required that all forces be assumed to be velocity independent. Also, the internal potential energy is assumed to be well represented as the sum of two-particle central force interactions. In the above expressions, V(lqi - qj ) is the potential between the i and j particles and u[ is the external potential acting upon the it particle. It has been assumed that all particles are alike. Kinetic equations for the lower order (marginal) probability densities are obtained directly from the Liouville equation. As has been mentioned in the previous chapters, a quantity of special interest to us is the one-particle time-relaxed function, f,1 ( pl'ql;t IE,1q1 ), defined as the probability density for finding particle # 1 in the neighborhood of the phase point (,1',l ) at time t, given that it was at ( g1',1 ) at t = 0. Let us obtain o o an equation describing the temporal evolution of this function. To obtain a kinetic equation for fl, it is convenient 1,1

23 to introduce the following operator. Define: 8 E f ( x,..x;0o|x, ); I*I d6x2o~od6x.oo (3.5) n[il 2'n 12 n where f (0) designates the probability density at nil t = 0 for finding the n-l particle subsystem in the neighborhood of phase point ( x2.....,x ), given that the coordinates of particle #1 are xl = ( l',q1 ) Note that, once given f ( x,..... x 0) one n,n -n0 easily finds fnll by the relationship f ) (0) f (0) / ( where f 1 (0)= If dx ooodx f (0) #-, \ j J 2 n n,n (unconditioned one-particle density.) Note, also, that 6 is a projection operator; i.e., Fcn(X;t) = Fcn(X;t), since fdx200dxn f 2 (xx.o;0[x9) 1. Using 3.5, the following two identities are obtained from the Liouville equation: 9f (X itlx ) = eL(lf + ( 1-) f (3.6) atLal,n n0,n "g^~n v n,n n,n The second of these equations is easily solved (formally) to yield: t s(f-~)L ( 1-6 ) f n ds e( (14)L fnil(O) f (1;t-sXlo) + + et(l-P)L(l- ) fn,n(Xn;oxlo) 0oooo (308)

24 Placing (3o8) into (3.6), one obtains: at rll-B~L =[9f (X;tjx1 )] = eLief ] + ~ dTLe (1 )n(1-~)L f (tT) + ~taT n,n n6 n' n; n l 0 + (L e (l1-)f (X;0 jx ) (3 9) n,n n 1~ ~v Let us at this point assume that there are no external fields** How are the initial conditions specified? For the case of interest (viz., a set of particles moving through an equilibrium assembly of like particles) the initial condition for the full n - particle density, conditioned upon the positions of a designated subset of s particles, is given by: f (X;oXx ) = av(q1 2 S e,,s,qs+l,o qne 2m.n 6()e (P;- PO Z ( 2 -ffmkT)Tn/2 i (q$0Op^),qn (mk)/ eqo 1o' 1' so Here, c oooo (30 10) Z = tdq ooodq e V (ql~~'qn) ( 1) qn I 1 n and o In] f (q, o c ) fdq,dq dp o dp f (X ) = s.. co (3012). 2 m- 3 sV/ 2 S + ) A n T2^ 3s/2 j s+l " - qn It is evident that we have assumed that the equilibrium state of the assembly can be specified by a canonical distribution. The proof that (3o10) is correct is as follows. Let xi = ( g Pi) ** i oeo, set L2 = 0. This condition will be relaxed later.

25 It is certainly true that f (;OlXn ) 6(X- o) n, n no t i I But, the quantities of interest here are of the form: f ( X;O Ix ), i.e., the probability density for n,n n so finding the system in the neighborhood of X given the n exact position of the subset of particles #' s (1 - s) and no knowledge of the positions of the other particles except that the probability density for the entire system is a canonical distribution We recognize the following relationship for conditional probability densities:** f (xy) = f y,) f (z'ly) dz' XIY XY,Z ZIY The generalization of this relationship leads us to the conclusion: f (X;OIX ) = 0 Jf (X;o ) f (x,00x X n,n n s0 n' i nn n n I n o e But, because X^ A fXiy(Xly) = f(X;Y)/f(X) we have f (x,ooox IX ) = f[n]X )/ f[S](X ) eq s+lo no so eq no eq so Therefore: f ( X;o0x ) = n n n So = | dx o dx ( 6( x oo(x -x ) f (X ) S+ lo 1 lo n no eq no eq so = f Ofl](X s 0o09X_ [eX Si f[S](x,00oX ) eq lo So **For proof of this statement refer to appendix A.

26 In particular, as a special case of (3.10): f (X;O|x ) eBV(ql, qn e, Pt /2m ql1ql 1 ( po) nn n;:-l Z /l (2 rmkT)'fln 001 (3013) qn Note how this differs from f (0), the latter being given njl by: A f (x,o 00X;Ol ) = e V(qX 0,qn) eP i2/2m nll 2 l 1 q/S (27rmkT)3(n-l1/2 n... (3.14) With these considerations in mind, however, the following relationship is noted: qf (X;ox ) f (x,,,,x n;01 )of(x o1x. n i n n'"*^ nil 2 = f (X;olx l)...(3.15) n,n n;"**^0 Thus, referring to equation 3.9, it is seen that the inhomogeneous term vanishes due to the fact that ( 1- ) f (X;0Ix1) = 0 In this way, the simplification of the kinetic equation has been initiated. Further simplification ensues upon noting that: (3,016) QL(3f = f {-c } f C(x;tlx( )o n,n nil r ql 1, 1 1o This statement is true because ~1) QB L (3 f >cf f f )f (xl;tlxo) o n, n m q ql 1,1 which follows due to the fact that, in the limit Q+ X- fd d0adood~ q yields boundary terms which' 2 n qi 0 for i # 1; 2) n L ff = 0 if fn (X2,o,x;OJ ) 1 n,n nl 2 n is an even function of the variables (4.-q ), i = 2,3,o0ono This is due to the fact a typical contribution will be

27 proportional to d200d F( qqJ ) f i dn Flj( iqj-q )o If f ap te which, in turn, will be equal to zero because the intermolecular potential, under the assumptions already stated, is an even function of the intermolecular coordinates (hence, the forces are odd functions and the integration is over an odd function.)** Of course, these arguments are valid for any isolated system for which the initial n-particle density is symmetric in the interparticle coordinates. We need not concern ourselves only with particles moving through equilibrium assemblies. Other non-equilibrium problems might be attacked this way (e.g., upon the assumption of binary isotropic interactions, (cf. 3.4b), an approximate kinetic equation from which hydrodynamical transport equations might be deduced could be obtained were the local flow velocity of the fluid a slowly varying function of the configurational coordinates.) In **It is true that for some of the particles, the effective configurational integration space will not be symmetric because of proximity to the boundaries of the system. However, as the following argument demonstrates, these particles give a vanishingly small contribution. Notice that, for the case of specific interest here, | dq2dS 1 (qlqv2)p0 f ( f )1 dq gq ) f 1 d2. 12 1 2 ~p1 ni 1,1 9 2 gP 2 2 1 1,1 where Q is the volume of the assembly. g (q -q ) (cf.,also, eqn. 4.19 & ff.) depends on the density ut is not strongly volume dependent in the limit Q + -. Thus, each particle near the wall of the container contributes, at most, a quantity of the order of 1/Q. However, it seems clear that the number of such particles will be proportional to the surface area of the container. For this reason the total contribution from these terms will c Q- - Q13 Thus, in the limit, n - *, n / c = constant, such contributions become vanishingly small.

28 general, the full knowledge of the initial probability density will be unknown. On the other hand, at least f 11( x;O ) must be known for the problem to have a meaning. A good guess would have to be made for fnll(x2...,x;01Xl). Let us remember that to obtain the simplifications 3.15 and 3.16 the absence of external fields was required (cf. p. 3-4). In the spirit of this derivation, non-zero external fields lead to difficulties: were we to choose the projection operator to include the full initial condition, f ] (0) would not be symmetric in the interparticle coordinates and 3.16 would not hold; on the other hand, were we to retain the operator 3.5 with fnl (0) given by 3.14, then 3.15 would not be valid.** In other words, 3.14 and 3.15 could not simultaneously be satisfied. This difficulty can be circumvented by deriving a different set of equations which, while satisfactorily describing systems with external fields, are not nearly as convenient for solution as the one we are here discussing (let us defer further examination of this point for the time being. cfo eqn. 3.28, below.) As a result of 3.15 & 3016, the kinetic equation for **Although it is true that if the external fields are not rapidly varying functions of the configurational coordinates the full f n1(0) could be used and 3.16:2 would be satisfied to a good degree of approximation. 3.15 would hold rigorously. 3.16 would read: f n f (0) -P1 V + aUext V } fl(X;tl )

29 f 1( P' 1q; t 1 p' )_ may now be written.. o (3 18) af _;tX +.... f - ^1,1 1 l 1 q, 1,1 =B |dT L e (l-6>)Lf (0) fl iX?;t-T| Xlo) 0 In the above equation, (3.19) d ooo (In other words, e = f (0) and f i;tlx )= f (Xt )) ~~nll~~~~n rin 10p n Further simplification of the terms appearing on the right hand side of equation 3o18 is deferred until the next chapter. We wish merely to demonstrate the structural form of the equation. Notice that the left hand side is the usual streaming operator, while on the r.h.s. there appears a complicated collision operator which contains a'memory' of the evolution of the system from its initial state. In contrast to the BBGKY hierarchy this kinetic equation does not require integration over any unknown temporal correlation functions or densities of higher order. It will be seen later that f 1, can be expressed in terms of static correlation functions related to the initial full system density. Whereas in practise these correlations are not necessarily easily evaluated, in principle all the necessary information is providedo The above derivation is easily extended to study the evolution of density functions for motion of groups of par

30 tides. Suppose one desires a kinetic equation for the sparticle probability density. Define: sQ f (x,ooo, x;o0[xi,0oX).. Jd x 0,o, 0 00 (3.20) njs~ n s 1 s J s+l o n nis s+l Then, as before, one obtains (cf. 3.15) 0Q f,(~O) = f (0); = (1 s) f (0) = 0 n n~" n,n s n,n In anology to 3.16, we have: GL3f 0= f (0I{ -. + q1 D f (t) nls i, m qi 2,. q Di s} 0. (3.21) where D {V V } o OOO..o........ O......... (3.22) ij Pi P. This follows because L~f = f dx...dx [f I (0) f L n n nn s s+l[ -iDi]f ns ss f (.0) dx oodx CI I-Pik.fk + I- f f = f (0) V. ij D. ] f (t) nis J q, 1 SS w G vt wdhere ddx dx a f _ by virtue of integras+1 n ik n n tion over an odd function. We have, in the above argument, noticed that f dxodx f (X x o x) 1 s+l~00 n n s s+l'~~~ n 1Xl'~~'Xs As a result, the kinetic eq ation describing the evolution of the s-particle density in the absence of external forces is given by., 0.., (3.23) af (X;tlX ) + i~V f + V O'D. f = s s sI so.^m qj s,s 7 q 13 s,s af - [ H;fs ] = ot -fs SS S o

31 In particular, for particles moving through an equilibrium assembly: - 8v(ql'.,qn) f (O) = S e n n [s Z gs(ql..qs) f Pi) (3.24) where s I dqs dqe -S-V(ql,.. qn ) q s s n.... (3.25) qn Also, f (X;0 )= Xn 5(Q.- ) 6(- ) o. (3.26) SOS s SO vi 1 S 1x As has already been indicated, G (r,t) is related to f2 2(x, 2;t Ixl x2 ). The latter quantity would be obtained 2 2 ('X2 1 2 ) ~ from the ean. 3.23 for which s = 2. It has been already pointed out that whereas equations 3.23 and 3.18 are valid only in the absence of external forces, by starting with different projection operators and performing analysis similar to 3.6 - 3.9 one can obtain kinetic equations pertinent to the more general case. Suppose we had started with a projection operator:., 1 0 0o. a0...0. (3~27) Q = j dq +l oc dqn 2n-s j instead of that defined by 3.5. In a manner completely similar to that already used, one can show that the following equation is obtained for the s particle density:

32 {a/at +ji Uext + V D...}f (X;tx) = oV Qi pi 2 Pi + q i i Pi + 2qi q ij iJ1 s,ss SJ Oo. _ t T(1-Q)L = L dT e (1-Q)Lf (t-T) + O s'n 0ooo (3.28) - t(l-Q)L + L e (1-Q) f (X;oix n,nn n o In the above equation L is the complete Liouville operator given by 3.4a - 3.4c (i.e. external fields are allowed). The other symbols appearing in 3.28 have the same meanings as their prior definitions except that the initial value term must now also demonstrate the presence of external fields.** The memory term of 3.28 can be simplified by noticing that because momentum correlations specified independently of distance between particles must be very small (for systems of many particles) it is not unreasonable to factor f CX.cox,~p,c,p; ) HO n f (pi) f (X;t-T x ) sn 1's sns+l n into + 1 i s,s s' So. In particular, for a set of particles moving through an equilibrium assembly, ~F\ ~~~~~~~~~~~2 8p./2m fs,n f s eq (Pi where f (p) = e /(2mkT) 3/ S nl S, S * eq i. eq i **Assuming the unconditioned n-particle density to be given by 2 with 1E(1 (X ) e 1 n i )e _ - Z (Uet^) dqooe n,-n n,-Yi —5 —--— n1 /2 1 then -B[V+jUex'] -P2 S fe(X ) = e /2 x-x) qZ(2 ( U-) -2rrmkT)'" )f where n e~ Cs] 1 -e fD -BC[V + Uext] feq ] )= e i=J dq ldq e eq! s<' Zq (2irmkT)/2 s~l n

33 This factorization closes equation 3.28 in that the latter no longer contains unknown temporal probability densities of higher order than f. Setting the external forces S, S equal to zero would provide an equation equivalent to 3.23. On the other hand, a posteriori, 3.23 rather than 3.28 is more amenable to solution (...certainly true for the function most intimately studied, i.e., f,1; it should be true in general since 3.23 does not contain an inhomogeneous initial value termo) However, it is to be noted in passing that 3.23, taken with 3o28,provides an integral equation for f ( X;tl X ). s,s S So With the external forces set equal to zero, the left hand side of 3.28 is identical with the l.h.s. of 3.23. Also, the initial conditions for the two equations are equal. Hence0 the r.h. sides of the equations must also be equal. For example, for f,,: Ot T(1- )L fdT L e (1-L)L f,(0) f (x l.t-TJX = nl 1, 1 -lo 00000 ( 29) 0 - ft T(1-Q)L - Le-1 (1-Q)L f (pi)f il;t-TXl + 0 t(1-Q)L + Le (l-Q)f (X;o0|x n,n n The last relationship is stated merely for the sake of presenting a complete story. It is not important to the continuation of this analysis; rather, in the next chapters, 3.18 will be the object of major attention. Before proceeding with the further reduction and solution of 3.18, a few words should be said in comparison of

34 the probability densities about which we are concerned (cf., 3.1 and 3.2) with other non-equilibrium densities and "distribution functions" which occur in similar studies of non-equilibrium statistical mechanics. The probability density functions defined by equation 302 are frequently referred to as "specific" densities because the arguments ( ql19'.0.q P1l. oopb ) pertain to designated members of the assembly.* Closely related are Bogoliubov's "distribution functions"**, which are defined by: F(X ) -rdx ooodx f (X;t) s J 9a n nn Xn it is clear that they are related to our density functions by: f (X;t[X ) = F (X;tlX ) S S S ro There also frequently appear in the literature the socalled "generic distribution functions"*: fgert (' n!. ddx f (Xf1100, S;t) dx,.dx f (X t) s 1 s Tn-s) -t-1 n nn n In this case, it is to be noted that the integral provides the probability density for finding a particular ordered act of particles with a given momentum and configuration distribution, whereas the factor n!/(n-s)! is the number of ways ordered sets of s particles can be chosen without replacement from an assembly of n particles. The generic distribution functions are useful in studies of systems possessing a symmetry or equivalence between particles. However, in our study, where attention is focused * —--— 1 33 27 *See, e.g.; Hill, p.182; Vineyard 3 **Bogoliubov, pp. 2, 51.

35 on particular groups of particles, they are of little pertinence. Using the conditioned n-particle density defined by 3.1, one may obtain the conditional number density, n(r;t|lq ), o defined as the expected number of particles in the neighborhood of r at time t, given that particle designated #1 was at q1 at t = 0. In order to do so, one takes the expec0 IN% + tation of the density operator defined by N(Xn) 6(r-qi) i.e., K1 1 n(~;t([l ) - <16( -qi)> = dxl dxn[ 6(-< )] f (X;ti, ) -o ^ i 1 "t 1n n n nlo dq fl^ltlt 6 tdj-) f iif i;tl-) _'f1 1;tl0 + (n-l) ) where f (;tl1l ) is the probability density for find1,0 ing a particle at r (configuration space) at time t, given that it was located at ql at t = 0; fdit0 (itIt ) is 10^ ~1,0 lo the probability density for finding a specified particle at r at time t, averaged over. the initial position of the particle, given only that a different specified particle was at q1 at t = 0. It has been implicitly assumed that the o n-particle density is invariant upon permutation of the indices 2,.....,no From the discussion of the preceding chapter, it is seen that n(d;tlo) = 1 <' d' 6(5+qk(0)-')6(?-qj(t))> = G(i,t) n

36 se f for a system composed of indentical particles Gs = fsel;t10) and Gd(r,t) = (n-l) f dst~(r;tO).** To relate to a more familiar case, had we taken the expectation of N (X ) with respect to an n-particle proban bility density which is symmetric in the configuration coordinates of all particles (rather than only 2,.....,n as per the previous example), we would have obtained n(r,t) = dX fn Xn;t) N(X ) = n f1 r,t). For example, for a system in equilibrium, f = 1/ Q, so that 1,0 n(r,t) = n/ t = c = the macroscopic density. **Anticipating somewhat fse lf;t lql and distinct, f (;t[ ). may be explicitly calculated from -1,0 ~2 tq2 1o (x;t[x ) and f CX2;tX ) by 1,1 1 o 2,2 2 2o fself(+I -= f 1 S /2m f1,0 (q t ) = dpo e f( P1+ q t tlpfo ) X fd;tlqlo ) = dXdp2dx20dPlo o 2 2 )mkT4 2 l X iQ d. d1 2 2d plo g2 2-qlT (27kT) 22 2

CHAPTER IV REDUCTION OF THE KINETIC EQUATION FOR THE SINGLET DENSITY It is shown in the preceding chapter how the Liouville equation may be reduced in order to provide kinetic equations for various conditional (i.e., two-time) probability densities. At first glance, the integro-differential equations so obtained seem to express the interaction between the particles of interest and their surroundings in a very concise form. Unfortunately, as closer examination of the pertinent operators shows, the perturbations of the surrounding medium (which are manifested in the collision integral) enter in a very complex way, To better understand these effects, additional investigation of the integral terms appearing in the equations is required. In the following work, further consideration is given to the equation describing the temporal evolution of the one particle density, fl l(xl; t/l ). Specifically, manipulations are performed upon the collision integral in order to obtain expressions convenient for solution by an iteration procedure later employed~ We consider a system without external forces. In this case, the probability of finding a particle in the neighborhood of (lEPl) at time t, given that it was located at (L1'21) at time t = 0, is provided by the solution of equao o tion 3o18: 37

38 (4.1) o a o o a l(i l;t lo' + f ( lt I, t m ~ ql 1,0 0 -t -TL(1- ) =jdT e L(14)L f n) fl, SP;t- lo 0 with initial condition: f Iq1'Pl1q; 0l Plo) (i-l,) (l-l 1,1 0 10 lo To reiterate, the terms appearing in eqn. 4.1 have the following definitions: _ dq20oodqndpoodpn 0o (4.2a) =N e 1 n f (p..)oooo (4.2b) ^^ ^ / ^ e q 1 qn =- - f (X ) Q oooooo (4.2c) njl n L is the Liouville operator defined previously by eqn. 3.4 (except, of course, L2 = 0.) It is to be remembered that in this definition it has been assumed that the interaction between particles can be described as the sum of central pair potentialso Frequently, the six dimensional variable x shall appear in the following analysis to denote (g,p). Further,

39 if ambiguity will not ensue, arguments of various functions may not appear explicitly. Note, also, that the 6-n dimensional vector has been denoted by: X (x t 000000o,X ) n -~1'n n1 n Let us focus our attention on the collision integral of the kinetic equation. Making repeated use of the following well-known* identity T TL(1-) TL (1-) ) t -T'Lo (1-Q) T1L( 1-) e = e J [ 1 + IdTI e L (l-~)e ] >0 1 0 (4.3) the integral term of eqno 4.1 becomes. - T ~iLo (l ) I =!dTi e l L(1-) f )L X f) f (x;t-Tx, ) + g_ 1~ nil n 1,11 1 o10 rt T Lo l8). T L0(14) +T2Lo(1-61) + fddTeL dT2e L L (1-)e L(1-6T)Lf f t-T1) 0 0 2 L- T l) T2Lo (l -) TL (-) + rddT Te 1 Lde (ll-)e x i 1 u 2 1 0tv T L (1-6) +T L (l-8) x dT e 3 L (1-e)e 3 L(1-3)Lf f t-T 3 3 n 1 n 00000 (40 4 *See, e.g., Van Hove 34,Bernard and Callen 35. 4.3 can be verified by noticing that it holds for T = O and also, the differentials re. T of both sides of the equation are equal.

40 But, the integrals of (4.4) can be quickly simplified. To do so, use is made of the following relationships. o o o(4.5) Identity: _ TeLo (1) (4- 5) (Proof: + T (l()) + 2 1e 1 + T L( 1-) + T L (1-e)Lno(1-) +.] 1, l 0 2! But, because i yields boundary terms qi which - 0 as the volume of the system -> Q ) L L (1-A) m= q( i )(n-3) 0 % 0 C-q1 m q Also, since 3 _ in turn implying ( 1- ) = 0 it is clear that the identity holds.) In a similar way, it is quickly seen that.Lon(1-_) = 0 for n -0; &e (1-6) = 0. o (46 As a consequence of (4.6) we have the following Identity: 2Lo (1-e) T2Lo 0o 0 (4.7) ------ C e (1-e) +0 ] (Proof: T Lo (1-?) e 2 =1 +T Lo(l-4) + T= L (l-25)Lo(1-6) + 2 2 t2 = 1 + T2Lo(1-6) + T2(1-?) + o,oo T L [ e 2(1) +0] qoeod0 ) Also, since 0 is a projection operator, ooo (4o8) TLo TL o (1-~)~= 0; L(1-Ce)[e (1-<S)+] = L (1-))e (1-) 0

41 Hence, in virtue of 4.5 - 4.8, the integral term (4.4) can [r] be written as an infinite series of terms I r the latter being given by: -t [r] =f fdr1" /-T 3)0 ()OTT)L [r] = adTgo d L e o2 3(l (T3-T4)Lo L [e r (1-) ]L [e r(l-)]L(l-)Lf f )f ( ) 0 *.. (4.9) Let us continue with the simplification of (4.9). Notice, in these expressions, the appearance of terms: -BV(q, ioob )^ o * qq'A (1-G)Lf (X ) f e(lq)Le i^ f (p.) f n1 l n 1,1 - Z i2 eq 1,1 But, Q L ee-6V(qlg9sqn) *0 bequse V(qlIo..., ) is an even function of the spatial separations between particles, whereas L, is an odd function of these variables (see the discussion following 3.15). Also, E L f jX r) f (X })(-P1 VQl0 0 f IXn) f(X)(n m qj 1,1 fLj n)l —i q V nfl )fP J f (t-T ) Hence, (l-1)L f (X ) f 1t-T e n1l n 1. 1' n' 1, 3 1 1 (4,10) c [Lf (X ) + f s+PtiV )] f bt-Tl) nl n" nl f qt 11 1i Finally, it is convenient to perform the following coordinate transformation in (409). Let: ql6* ql; ti qi-ql * i* 1 (4 11a) nil n 1 0i0o i,1 o00 \iV~ (^ 1 " Si * 41a) which implies~~~~~

42 The Jacobian of the transformation is unity. Under this transformation, the operators appearing in 4.9 become L(X ) L( Lo(5) + L (-) n.0.0000 a (4.12) where L ( ) and L ( C ) are defined by: 0 1 (4.13) Lo(_) _. 1) 0;1z ql >2. m, ). -| L.(E) E -'Flj( )E (V.j) + "jH -k )o(V-4) 1 P1 P] Pdjj pj Pk Utl s n o te ta pi It is noted that f ( =) is not a function of ql. It will be seen later that this is of some importance. In fact, it is mainly for this reason that the transformation 4.11 is performed. For example, under this transformation, the expression given by 4o10 becomes L( [ qf ] lfj lt-TIx10~) 1' = B (=) fnl ) fix_; _ llX o)( l where Lt) -Lt) L().. (4.13) and I + + Lot() J - l l. In summary, finally, the kinetic equation for f (xl; t/x1 ) may be written as: o

43 af + fli V f (x;t x ) I1rl(_;tlx ) - - 00 ^" 00 atl,1 m 0 aql- (4.14) fdt -1 (o with, [r] =( dT2L ) x (4.15) dT=) dT1 T2 0o r 1 (T -T 3r)Lo t() x [e A ( 1 —() )] L x -(T -T) 1 (T -T )Lt (T -T )Lot x [e (T3-T4) )]x o.,,ooxL [e )]x x Ln[e (1-)]L()L$()fn ) f iXl;t-Tl ) is the;-treaming operator (Taylor;s series operator): E se 1l,oo (4.16) T having the property that0 when acting upon a function of 1l' P1' and t: fcn( 1;1;pt ) fcn ( l -2lT/m;j;t ) 0 o (416a) ( is defined by: ( () _ o odtpn odtd ~dPn o. o there being obvious similar definitions for ( ) and f a ( ) Thus, it has been demonstrated how the collision integral may be expressed as an infinite series of integral. -1 I[] =~(E) jdT1 L(2) Lt() f 1(=) f (;t-T ) 0

44 operators As shall be seen below, such arrangement (i.e., in series 4o15) expedites grouping the interaction terms according to decreasing effect on the evolution of f (X; t/x ). 0 What, however, are the criteria for determining the relative contribution of various terms in the collision integral series? The identification of the necessary expansion parameters is, of course, one of the desired results of this study: In this regard, it can be asserted a posteriori that further reduction of the interaction terms will demonstrate the intrinsic importance of constants of the following form: ds ood& k g k(~2',o o k)(;'""'"'~(~ ) ( ) ooo'+(k) ). j 2 k k 29 00 K ^ 2 2 %k." 2 (4.18) 2 2 k where 0(i) is the pair potential energy expressed as a function of the distance between particles and, k-l - ev(2O~E) g (~ io.f ) _ a [d oo 0od& e g2 o0 (4.19) k2 k K +. -n _ fd2 0 oo0cd e-V(2, eiEn) The terms (4018) are averages of derivatives of potentials, weighted by the correlation functions gk( i 2..' k ) In general, the latter are dependent in a complicated way upon the density and strength of the intermolecular interactions Dependence upon the variables i2.... 2 k has

45 been determined only for a few very special cases.** However, in the limit that the density' 0 and/or the temperature, the gk are well approximated by g. k-l -BV(~2'''k) (4.19) 2 2 k 2;^7rr?-V 2..,Sk) d2.dk e It may now be asserted that the criterion for grouping terms according to interaction strength is that M, defined as the sum of the differentiation indexes appearing in 4.18, be the same for all members of the group. In other words, defining M m......... (4.20) Mj ~0i, (4.20) it is asserted that all terms of the form 4.18 for which M is the same are of equal'strength'. Perhaps this determination seems rather arbitrary. However, it is noticed that when the approximation 4.19' is valid, simple partial integrations show the equivalence of the order of magnitude of seemingly different expressions. For example, (~~~[ dB((~e] 3 1~*** f[ e i()0 V d(O) ] d = 1 d3 e V 2~() (The factor of 1/p is always cancelled out when expectations are taken, and is irrelevant to the present discussion.) Also, for high temperatures, it is found that successively **See, for example, Hill, p. 184 and ff. *** This is not true for coulomb forces.

46 higher powers of M signify terms which are successivrely less important in their contribution to the collision integral. In effect, one is performing an expansion in a parameter characterizing the iatio of the average potential energy of the assembly to the average kinetic energy. Thus, the zero'th contribution is a collection of terms for which M -0 the first order correction is that for which M = 2; the second order terms are those for which M = 4, etc. All this will become clearer as we proceed, At this point, some of the terms in the collision integral shall be explicitly evaluatedo All those for which M _ 4 shall be obtained. Unfortunately, the ensuing analysis is necessarily lengthy and complicated; we endeavor to present it as lucidly as possible. First, consider the collision integral I From 4.15: 1l] = ) f tdT L(") Lt(E) f n() f 1(x;t-T lxo (4.20) x (4.20) j~3 1, Inil Isl oooo 0 0 0 It is convenient to break the integral into two parts, {1 and I1], defined by: 2, (~ E13] E(E ) jd L(s)Lo ) fnll() f (t-T) (4.20') IC23 J-(=) dT L(=)L1(=) f n,() f (t-T) (4:20") 1 lnil l I[13 can be quickly simplifiedo (?() yields boundary terms in o,, which are zero in the limit that the volume of the assembly grows large, so that from

47 4.20'a it is seen that: t I[1] = (E) dT [L L o((l )f (]) fl (t- t) 1 1 m q, 1,1 But, for the same reason, S (=)Lo (E) f 1() = O. Hence, I[1] =&(E) dI-T L (-)Lot(E) f f (t-T) Writing the explicit J o i nnIl 1,1 1 form of the Liouville operators: [1:@(E) JdT (d j-F.,(V -V ) + -F.( -V ( )f f (t-T) Pj Pji Pi Pjj 1m n j h 1 But, because Fk. is an odd function in the variables t-t. whereas f (E) is an even function in these variables nill and, whereas (E) P: o for k e 1: Pk =I (E)dJ d Flj (i (-( ) -i l() fl t-). Since all particles are alike, and because (3(). f (E) = 0 (because integration is over an odd function of p. ), we finally have: l= c x(2) dT1 3 + PlV ] f x;t-T 1 ) (4.21)!t -n P1 1 —1 1' 0 m m 1 s where (2) 3 2 X - lim d3t g () a li) e (4.22) n-we~ 1 2 a *to _ x In a similar way, I13 =6() dT[LC()-l L (E)flfl, t-T) E2 1JI* — 1 nl 1,3 — om q But, I (E)[Pl~ L] L (E)f (E)ft-T) = 0 because the Q inteql 1 nil 1, + + gration is over an odd function of the variables ( Si-.j) The latter observation is a special case of the following Lemma: If Q E(S)[CL ()L((E)Ln..,0L L L.LtL...]f (E) is to 1 0be 01 non-zero, the the nteger repreenting the nuber of times be non-zero, then the integer representing the number of times

48 that L1 terms appear when added to the integer signifying the number of times L appears, must yield an even integer. This restriction follows from the requirement that integration act upon an even function of spatial variables. t Thus, rCl] =(-) IdTL () L () f ) f t-T) or, 2 J 1 n[1 1,1 finally, I [13 c in(2) jdT V2 f (t-3) 2 J 1 1P 1 0~ 0,. (4.23) o with (2) 3_ 2 2) fd3 g 2(F) T 2 (4.24) In summary, the first collision integral has been reduced to a form involving simple and well known operators, viz: El13 = c fdT(x2>[3+P- ] + n( 2 ) lf T-) (4.25) 0 x(2and n(2) are defined by 4.22 and 4.24, respectively. (c = n/ f = the average density.) Now, consider the second collision integral. From 4.15, [23 it is seen that I can be written as the sum of two terms, defined by: enient to further designate I[ r]= dT 2 T Le i[ a] E(E) adT d2 L= e L2 ~ )Lt(E)ff jt-Tl) ^23r r C T2L. t o. oo(4o26) 2 9 Considering the first of these contributions, it is convenient to further designate Il2a] = y Io[2a] with vi

49 I[ E d j -T LO2a3 E 1Jf 1T f ll - =1) (4.28a) 0 2 - 1 t [2al = - dTT2 T L(428c) r[2a] _ dJ dT2, L1e 2 -T LOLfnIf i(t-.1)..... (4.28b) i 0 2 ee t t < T t I 2 - F dT dT L e 2- L fL f lf(t-T ) (4.28c) 4 1 2 T 1 1 ilT 1 b I 1I 2 20 ~-TpL~f2 It is immediately obvious that I 2] is rather involved; to continue with the reduction of the collision integrals, we procede as follows. Consider IL a] (4.28a). The exponential operator x Lo e may be expanded T L - e 2 = 1 + T L () + T ~2L(-)L() +..... (4.29) 2 2 so that Ia]J =0I[2a]+lI[2a]+2I[2a]+3I[2a]+9 where 1 1 1 1 n1' a 5 dT dT2 *L'^' T f n lf 1$ (t -T *- ** (4'30a) lI2a] - JdT dT2 L1(T 2 )L0 L0 0 fnl(t-T1) (4.30b) t; I [2a dTT2 L1(fT2)Lott Lh t l 1 (t1) (430 and so forth.

50 Writing the Liouville operators explicitly, I a] is seen to be: k C, OI[2a ]=_() dTlfdt2(&im D + - d 2 411 ij Dij + J-Flk Dlk) _T X o O2 x (-I(??~)- -pi)-P(? V )~V)%|ft-T) ) ( Pi-P -P 1 n 1, t1 However, since Fij (l)i-Sj Ls an odd function of the variable ( ti-Sj ), whereas fn ( ) is an even function of this variable, and whereas the Q operator integrates over these variables, upon a little reflection it is seen that the only non-zero terms in 0 [2a] are ~^.i(d 2( lk VP1,oT2 1 - ni+, O =8I()|jd ^ = d d L) (-Tl. L: f t -~L* ~ (4 3 2~()d d. L 1 Lo f n <f.2 m2 1) in the above equation yields E2aI=()f T1 L=. L: m f - (t-T 0L0+ L1 Lnf f() Q0 c 0 0 0 (4.31) Ae 1 i nil i i g 4.31 is just the negative of I as indicated at the top of pg. 47. It is thus logical to define a term $1 CJ[ O 12a1 (2) fdt+[4~fio ] 0I~~~~ 0 o (Lt(E) and LT commute) 0000 4.32 As we shall see, C 1 is one of nine basic integrals into

51 which the collision integral, to the fourth power of the forces, shall be reducedo The next step in the program is the reduction of the 1 2za' contribution given by 4.30b: 1. Since it provides an example of the increasing complexity of the arguments required in this endeavor, this too is to be presented in some detail. (However, rather than burden the reader with details, only the results shall be supplied for the other terms. If one is further interested, he may refer to appendix B for some additional comments.) Thus, upon writing the Liouville operators in explicit form and after dropping some zero contributions: tD o ~ 00 dr jdn 2(-T 2 1,1`,! 1 ~ ~ ~ (~ r I Pi - Consider %(z)(-F (EJ.) -fn ()). The requireX I ~ m n) ment that e act upon an even function of leaves only the following non-zero terms: () (-FXj ([ 3a3 3(pjy P1 )2xElxa a232 x j I -j -y 3(Li - + 3(Pjz-plz)2(pjplxa3 ) fnl1()3 However, (C) must act upon an even function of pj. Thus, the above expression is:

52 B6~ -)=F- )((-3[PJiPlxi]plxa/aplx 3[pj2+p2)3 + +3(p+p 2)-p )a3 + 3(p?-p 2)(-1-p, 2 a 3, L"A-S a J L =-(-S)F,) ( ((3lP 1 ] +3[m+p12]) a3 x x +3(m+p)(l+pl a ) 33 +3(mr+p2)(l+p ) a3 )f 8 a p ~ 8 z 1-.' a&2a n ] xx yz z x Next, adding the similar contributions from F and F and y z noting that: _(E)F a3 fn () =(S(E) F a2 f () etc., -2 x2 -x9y nl x ntc &Y x z we obtain for [2al. 1[2, OOO(OOf 20o0000 o o(4. 34) -I2a3 = -C/m3X (4) dil d (-T 3f(3m+P12).+( —, +p a ) t-T1) i i ap, 2 -3c X2 IdT dT2(T (T2m(3+p V )+2p2+(p p - L ) f2l 1 — 3 j 2P 1 1 2 1, 1 where the system constants are defined by X( 4 f Ed 3i g23() ea 00 0 0) (4.35a) Xy and X(2,2) =a3 p~de 2(ria 2 s ~(. ) (4.35b) J It is easily deconstrated that lI Ea] is closely related 1to to 2~~~ E~~aJ

53 Note that, in light of the Lemma 4.23, 4.30c can be written: |= dT f 2o Lo (-P o V )Lot f f (t-) (4 36) ~1.~1. 2 12 0. m 0 nIl.1,1 0 o But, integrate the above expression by parts to obtain: t 2I[2a3 2 -)Iot t.1[2a3 22a((i- 1 L.4 L LL L f onl1,1(t.1 )) 1- a Thus, logically, one defines a term consting of the sum of 1 [2a] and 2 12a]. and 1 i- 11[2a] + 2 [2a] 1 +11 -b2 8 ap1'1 ~1 {3P 1 — -'IHowever, further simplification is possible upon noting (h) (2t2) that x'4 =3x.** Thus, one obtains, finally 4t, ~ ~ ~ ~ ~~ooooo (o37) ^3^.1 s ^ )t(4) 121 2- 1 t 3 = C X( IdT 12 (15m + 5p2 + 5p+ 1 2 % ) f it_ 39 z *z 3 p z1 T - 1 T z 1 z, z In this way, another of the'basic' reductions of the interaction integral has been accomplished. Next, one considers 3 2a]. It is seen that this lo term is proportional to system constants of the order of the "sixth power of the forces"; i.e., constants for which M, as iFor a proof, see appendix C.

54 defined by 4.20, is equal to 6. Since we are here concerned with calculating contributions through the "fourth power of the forces", only, it need not be evaluated. In this spirit, all termscontributing to the second order (M =4) contribution of the collision integral are evaluated. In all cases, the resultant integral operators are found to be of the general form: Jc.0. (4.38) f JdTl Op Tl;p19,2.~~, 32,0 l(t-T ) It is found that all non-zero terms to order M = 4 are contained in the collision integrals (of 4.15) for which r < 4o These terms are best expressed as the sum of nine basic integral operators: I =[ + o( M6) 00000o0.a00000. (4.39) where:** = ( cx d,ll T t-r 1 V o o oo (4.32) = c ) T (V2 + T V2 fl t-T ) (4.40) 2 1 Pl mPq, -1 1. 23 -mc P1 Pif l 1 d 3" K (15^ + 2 jt — ) C) 000oo (4o 37) -= c (xx dT T2 (-20 + 10 m ( V2 + T V2 ) + 16TP + Additional detals & typical calculation are to be found 2(2in Appendix Bo ) P 3 M 3 -I I M I in Appendix B o

55 The system constant X(4,xx is defined in table 4-1, below.,..... (4.42) f, 1 = a(4 fdT (Sm + d 3V2 +4 rV2 ) + 20 + r~~l ~10 c= 9 P - Pl ^15 i 2 318 3 1 m Pq Iq + p 1 PPT PV +44p + 4p +2(p +4(p p prp p7 )\ f(t-Tr) 31 P ~l1 q 1 gl Pi m q1 1 -3,1 31 PI 1 PI P 9 q! 1 PI 1 t I -y -* +^^f>, +c X *XX 3 dT T2(m(3V2 +4T72 ) + 12 + lOzpV V (t-T) i3? PI ^j P9l s I*51q^^ +2^xy _T 2 2n+P2 V2+ 4V2 )+6+5V + 2y V + ^3 j 1 P, pq Pp 1 1 +c bX X' d'T +2(( D 4T+3 )+6+5p +2 (_.p +p )) oP 13 PI 1 3 P q q p -Z in 2 8 P13pj1 1 4,~~~~~~~~~Cz +c 4 pdT 2(233+2 +14T2pT +p 3p ^+ p-* {2xx TXr18+Q +>~P +3CV pV — )22 -, +2 m3 fd 2 L1 1P M P q q p t t4 2 +C [d t(3+2pV + 14rp41 + 3($"' )+2~(4p p m3 3 J0 2 P 3m P 3m p q q p 0 00000 (4.43) (x,x)fr f d= dT.TV2 +4T2v2 +4TV2 f,(t-T) 6e11 I JtV2 T2 q1 (xY t +c'Y dTT28V2 +2T 2 +8TV2 +T P (v2 +T72 )+ 4 J -,O q PI i 1 q Pig Pl1'1j Cx2y) M p1,, 3m 1 ~r 1q 18~~~~~i +4 (3V72 +4TV2 +T2V2)f (t-T) 1 pl P1; Pf,1 2 Tf11 (xy,xy) + + +C V ) dT T2[Q-4V2 +p V (672 +14TV2 +lT2V2 ) + i 4 io? p lp PI"Mi Plql -3-2 q f2 1 P 3mPPlq1 2L +2 T2V2 + TP V (4V2 +7 TV2 )) (t-T). *3 12 li ^61 qi I ^Pill;) 8^f jti (3) fat t2(57 2 ^4^7 2 - (2+2172 -1t272 )+ n,2 P1;p1q1 ) 2q1 p p^pq'.

56 t (2) f _C2X(2) P1 23V 2 1I + n EV2 52 + +22V f (t-t) P - pcV lq ^ T 19 1 3m p Uq'2 q1 A.. ( 4.44) t (4,xx) (4) r 2 8 Cf ='CC3/ +3 +5'c ] dT 2 )+.819 1 = /3 C3). P l Plql 2 ql +TV2 (3V2 +5TV2 +3T2V2 )) fl t-t) m P 1ql PI 2 P1 sq 4m L (x,x) (4) t _ +cc. +CT' I3 dT | 2(V2 (9+3p V +TpV ) + (3) OEM Pl 2 1 ql (. 32 2( 3+p +T7 s (6+2p v +1 )+T2 v2(3+ f(t-T)?l'q1 1 P1 2' 1 ql m2 ql 1 P,1..... ( 4.45 ) Finally: (.. (4.46) (2) (2) + ( f 1 =-2 C dT T2(n V2 [ X (3+p Vp +2Tp V )+ " 9 1*1 2 pi m 1 Pl 3m 21 +t 12(2+ 5I72 + 2T2V2 )] PI -31 Plql 2 q1 (2) (2) +ln T2 [X ( 3+p Vp +3Tp V )+n (V2+7TV2 +3Tr2 )]yx 3' Plql9 1 4m 1 ql P - pq -m' ql The system constants appearing in J 1 through A 9 are defined in table 4-1, which appears on page 60. It is unfortunate that things get so complicated so quicklyo In effect, we have expanded the collision integral operator in a power series related to the relative strength of internal interactions (i.e., a series similar to an inverse temperature expansion): c and L are the first order

57 terms in the expansion; A3 through, 9 constitute the second order term. It is apparent that it would be an extremely tedious task to obtain the third order terms of this expansion. However, one can make remarks concerning the general form of the higher order terms, even if analytic complexity precludes specific evaluation. The expansion of the integral term (4.15) may be written figuratively as: I[dT Op V;TV ) f (t-T) (4.47) [2 dt Op (P- Sp. qq -T 11 r~41 fo P^ -+ -4 ^AA 9kC319 - 3 - rdT T4 OPIOp p V V (' TV ) f (t(t) i ~T4E f (kdT T(2) Op Ek]L(Pl q;TV f (tATT) L2k3 1 1 ~ ql T 1,1 -Au-\ [ 2k-2JT Pi 2 where C\[\ are system constants for which M = 2k (cf. dis[2ck] cussion following 4.20o) Notice the manner in which the powers of T in the integrals increase directly with increasing powers of M. Terms in [k] the Op which do not contain derivatives re.spatial coordinates do not contain the time T explicitlyo Spatial terms in the operators appear only as powers of derivatives re _1; furthermore, every derivative term in gq is always coupled with a factor ( T /m)

58 It is to be noticed that the expansions for probability densities and correlation functions to be obtained in the following chapters will not be power series in the usual sense. There is no simple parameter available for an expansion; rather, expressions will contain the complicated parameters of the type defined by eqn. 4.20. Systems to which the analysis will be applied must be chosen judiciously. Actually, we wish to apply these calculations to assemblies for which interparticle forces are characterized by a hard repelling core plus short range attracting well ( cf. fig. 5.2, p.1, below.) In this case the system constants are roughly proportional to powers of ~-o, where go is the depth of the potential well. On the other hand, application of these expansions for studying plasmas will, in many cases, be quite unsatisfactory This is because, for the special case of a purely repulsive potential, the system constants are proportional to successively higher powers of the reciprocal of the appropriate coupling constant. For example, consider the system constants <V and 1 (cf. table 4.1), evaluated for a repulsive coulomb potential, 9(g +e/y. If one approximates g2 ( ) by ek-) i t ~ it is quite easily shown that y c,, Thus, rather than becoming progressively smaller, the system constants grow larger for high temperatures.

59 Even so, this does not necessarily preclude the employment of these expansions to study particular aspects of ionized media. For example, if one calculates the dielectric constant of a partially ionized gas, it is found that successive terms are actually related ( to a close approximation ) by cW< --- c where LC is the frequency of the incident radiation, and C is the density of electrons. HenCe, a useful expression is obtained at high frequencies, for which the expansion is strongly convergent. This point is examined in greater detail at the end of the next chapter. For systems with more complex force laws, it is quite difficult to determine the exact relationships between system constants of sucessively higher order. Suppose we make the following approximation for; 4) ( corresponding to an infinitely strong repulsive core). ric^ 1 z ~ $ YiS ( 4,48) Then, for example, ) - y %) AX / %)V where o is the depth of the attractive potential well. Thus in expansions for the slow neutron scattering cross-sections ( cf. chapter 6, below), where it is necessary to compare, e.g., terms proportional to t with those proportional to / Ar /1 k, one finds that the pertinent expansion parameter is effectively lo

60 Table 4-1: Definitions of system constants appearing in the collision integral expansion. (2) 3 (4xx) I 3 _ X 2I E gd3 g(c 2) i 2 4( Ii) I d g3 (&) D2 F ( [J 3 2 3 S2 g2 x x x X( () J d' g2(j[) a4 (0) (4xy) 3 x X{2x E Id3 g,2U)( T2 I - I d g () F (2F EyJ g(2) 0 J 292 63 ygi 23 x 2y a) X{ 432= 3Xg 2 ) (4xx) (d xyg ( 4xy) J2 X'1 dSad gC(E T) T492Fau2) ) Y(3 = 2 34 + 2 x!d3g2(4) 2F ( a2 F ( c) n = 3nY 3e) yd3 = 3 aF( 2 F() (see appendix C for proof (dlxx) (xyxc (4xy) { XY d 2) f2 (4 3x_) a xr = x Yr 2{l2 -.3(4xx) (4xy)2 e) 2X = 3X1 (.see appendix C for proof>o

CHAPTER V SOLUTION OF THE KINETIC EQUATION FOR f 1 ( xl;t|xk ); O1 10 MOMENTUM DENSITIES AND CORRELATION FUNCTIONS In the previous chapter, considerable effort was expended in obtaining an expansion for the collision integral which appears in the evolutionary equation for fl ( xl;t[ Xl). These expressions are a necessary preliminary for obtaining a solution of the kinetic equation. An iteration procedure for that purpose, consistent with the expansion of the collision integral, is developed in the following analysis. It is frequently fruitful to study the properties of dense gases or liquids at high temperatures by extension of theories appropriate to a study of dilute gases. It is somewhat in this spirit that we here attempt to obtain expansions for f,l ( xl;tlxo ) and related densities and correlation functionso Thus, let us search for an iteration solution to equation (4.1) ( the kinetic equation for fl ) having the form f ( x Itlx ) = f f[13 ] + fl a 1,1 10 1, 1f,1 fi] -oo.(5*1) [03 where fl,1 is the expression to be obtained were the assembly an ideal gas, and fl,, etc., represent successively higher order corrections due to the fact that there are 61

62 interactions between the members of the assembly. This scheme requires that the f[i] contain successively higher'1,1 powers of a small parameter characterizing the importance of the interactions; the parameters appropriate to this task are those already introduced in the previous chapter (cf. eqno 4.18 ff.) It might be said that this procedure resembles the Enskog solution to the Boltzmann equation** except, of course, the lather is an expansion about the equilibrium -ep1/2m density, f 2, whereas the zero'th order term (27fmkT) in our case is a time dependent six dimensional delta function in "-space, It is evident that the zero'th term of the solution for f is given byf0 mf1 PO f 0] = 0 ooo (5.2) 1at,11 ql 14 with the initial condition: f[0(xo)- 6 (x-x )oThe effects of collisions are absent from this equation. The first correction is obtained by solving the following kinetic equation: f:l -o f[] f[0( f ) o o 2 (5.3) a1fx + o Vql f U=0l( ) +2 (l =f = c d C )( 3+p V ) +ntV2 +TV2 ) 3 1 (tm 1 Pl Pl l Plql T 1,1 11] with initial condition fdii(xlo) E for all 1a, Similarly ** Enskog 36

63 the second order correction is provided by: f[23 + +q f[23 (5.4) t,11 + 2 1 1 i (f [ (J1]) (fOl with initial. condition f2( 0) and so forth. These kinetic equations may be solved by taking FourierLaplace transforms and then inverting the resulting expressions. The Laplace transform, taken with respect to t, has the following definition:? -St (.-st g [ G(t) ] |dt G(t) e (5.5) For our purposes, the Fourier transform is defined by 1 imd +ik q (5.6) 4[ g(q) 1 3/2 Lim d q g(ql e( co (56) k (27r) Q+W which, by the (Plancherel) Fourier Integral theorem converges in the mean to - +ik1 / Id q g(ql1 e It is quickly seen that, in accord with these definitions,:J<" 11 Ti;;T3/2 T V ir ) 1 c j01,1 e'"3 r/) (s-ik2..... m Inversion of the transform provides 6jlo1 PP1 P 4 _* -3 - (5.8) f, (xltlxl ) = 6(q6( - -L,~ i ^0 1qlLO 1o

64 which, of course, would be the only term in the expansion were the system an ideal gas. Similarly, for the first order correction, we obtain from 5.3: f I - -_ _1 ( f[ ] _]) - (f'1 (5.9) s 1 T T t oq 1 1 1 2p 1,2 1,1 (s-ik-P) -1 m The explicit evaluation of this term is made considerably easier by noticing that f[O( t-T ) f[O]( t t) 0o.......0. 0 (510) -T 1,1 1 1 so that (referring to 4.32, 4.40) (511) S\folooSo (5o11) (f11),0 2(f[~) - ct[ n(V2 +t V2 )+ (3+plVp )]fl (t) 1 2m P1q m 1 1,1 i.e., the integration in the collision integral becomes merely a simple integration of a polynomial in't Consider a typical calculation with 5.11 and 5.9. One has ~. tV2 f[o0(t) = 2' 6tf+(t) = V2 1 6(- ) PI 1, PI 11 P (S-ii)2 1P = (3(-ik/m)2 - 4(-ik/m)~V + V2. + ) (s-ikP1- (s-kip 13P 1 (st'P1~)2 1 1o m Tn Tn Therefore* m tV2 10 2 2p P2 6q(t) q - (s-ikl -)tV2 f (t) = t2 t62 6(t) -2tV 6 V 6? (s-ikp) 0 ^ 1,1 4 q1 13m 4K where T + 1 6q(t)V2 6P ) 2 Pi d 6(P11 ) =; 6(t) ES(1-q -,t) 00.....0 0 (5.12) t6 = rn+l)/snql = n-/sn+lo n = 01.2o... [nt1; = 0r(1)/+; n 0,1,2,....

65 In this way it is straight-forward determined that: f^3(1;t|2x ) = 00,0000 (5.13) = ct2( )[ q(t)V2 dt Vq &V tP+;6P2V 6q(t)] + P P 3m 1 1P + X) 36q -2t6P1 /q 6+6 (t) 6P] ) Proceding as before, one might calculate the explicit expressions for f 1] Unfortunately, the expressions are l,10 found to be very lengthy and complicated** On the other hand, many dynamical quantities may be obtained without having full ** That this is so is quickly demonstrated when investigating the contribution to f Z due only to the first two terms of the r.h.s. of 5.4.' From 4.40,4.32, and 5.13: c I FEf13) + 9(f~') = c2 Jdd, { 1+ ) x! ^ 1^ -2 q. m n ( 4 p x +t-)2 n [36q(tt-T)v26P(t-T_) + q ( 6q+6 9 m 36 m q1 1 P1 c2 [dT ( V2 + TV2 ] + 3+pof x 0o P Pq1 m 1M x(t-T) 6'[p6t(.Tt))V2 Cpp+(t-T)28c~ 22 pl Pi1 - ql 1 + hj3sP69(t) -2(tT)6pn q ++6 6 1] ) m 3 m 1 1 (footnote continued on bottom of next page..)

66 knowledge of f1 l(x; t [x ) Thus, rather than oppress the reader with a detailed accounting of the second term, f ] 1,1' we procede to calculate the dynamical quantities of primary interest by methods which circumvent the complete calculation of the p-space densityo The remainder of this chapter concerns the calculation of momentum space densities and associated expectation values and correlation functions. Specifically, we shall obtain: o(,l;t|li), the momentum timerelaxed density; <gl(t) 1El(O)>, which is the expected value of the momentum at time t, given the value of the momentum at t =0; < P(lt)~pl(O) > the momentum autocorrelation function; and, < Pl(O)P (tl)l(t2)Pl(t3) >, which is a fourx x x x time momentum correlation function to be related to the neutron scattering cross-sections in a later chaptero (footnote continued from previous page) c2t3 C n(2)V2 +x(2)3+ ])(n(2)[V2 6p3tt 6P + 6m M 1 4m 1 1 +3 t26PV2 6] + X~ ) [36q6P+ 6P P 20= qB 1 zp sa'% 22 1 n m (1 2 V + 4t (2)2 n[(2) 2 6P=4t 6qV 6P+ t2 6PV2 6q + 12 P ql PI ql P1 6m q1 + [ 366,P+6qp V 6P8 t6Pp Vq 6]) m 1 15m 1 It is apparent that these contributions to the inhomogeneous term of 5,4 are discouragingly lacking in simplicity. The corresponding contributions to fI due to these terms are found to be yet more involved. It is clear, however, that with a little work they may be obtained in the same manner as used to determine 5,13.

67 In chapter VI, consideration will be given to similar calculations of spatial densities and correlation functionso Proceding with this program, it is clear from the discussions of chapter III that the momentum time-relaxed density may be obtained from f i (xl; t I x1 ) by integrating over configurational coordinateso The probability density for finding particle #1 with momentum in the neighborhood of E1, at time t, given that at t 0 the particle had momentum E1' is defined byo o 3 3 (5 o14) 4p( |,t[p) E f d3qdS Ql flI;t lxlo)fJeq( ) (o o where feq (-l Il ) is the equilibrium probability density for finding the particle located at aq, given that it has momen-o turn p. Of course, in an isolated homogeneous system, 0 faq(ql pE) is independent of a1; in fact, it is given by f q(-l, loi a s, where Q volume of the system. Thus, o(tl.) f d3qld l1 fl(xltlxl)........ (5.14') Let us expand f in a series in direct correspondence with the iteration series for f,1 o Thus, [(t 0 (5.15) o( IE't =I I) -E -tl l k> oooooo o o o \zo

68 where [, [i] (l;t 1o) 1 Jd qld ql f ( tl) (5.16) 191 Xl.... In this way, using 5.16 and referring to 5.8, it is immediately found that the zero'th order (ideal gas) term for the momentum density is: [o] (5.17) a (, 9tt ILE ) 6( -Z a 0 oo e o o o o 1 10 1 10 Similarly, referring next to 5.13, because spatial integrations of the spatial gradients of delta functions which appear in 5013 provide zero boundary terms,** the first order correction is quickly seen to be: 2 (2) 2 +X(2 3+p (g, tl IL ct [n x +p v )]0 (2t| ) 1 92 P m P i 00000. (50 18) In principle, one might calculate the higher order corrections in this mannero Observe, however, that ~ could have been calculated directly from equation 5.3. If, in that equation, the configuration space integration is performed on both sides of the equations, one obtains: 111 r [ Cn = j1 dq dql 1(f 1) 211 ] (5.19) = c JdT (2)[3P P] 3n (2)V2 )6(pl ) o n 1 1 which clearly has the solution given by 5.18.## In the same way, the contributions to [2](p;tip ) are 1 10 ---- Ara 37 o 38 ** See, eogo, Arsac, Schwartz [0] ## Since, as initial condition for 5.19, we know that ~ (o) = 0 for all R1,g1o

69 determined directly from 5.4. For example, the contribution ~ fill +c~ (f~l~) a due to 1 (f2 1,1 is found to be: t t, (5 020) [2 c2 fdtl IdT + 3+r2 Pl (2 - 122 (P l im 2 2 2 4" )2..3p >2 2) c t [npV + _)(3+P ) 3 6(plPlo 24 1 m 1 The other parts of the inhomogeneous term of 5.4 have been found by this procedure and the results are tabulated on the next page,

70 Table 5-1: Second order terms of momentum density [21 Collision term Corresponding term of 4 (pl tlpl) 2t4 (4)1 2 (2 + 1 3 1,1 3 1 P P 1 1 Pi J4(fl, ) n [-10+5 2 +pc2tp22 (1+ X2 p 4 03 4ol 4x 2 2 41 2 [0^ 3C ) f ct4 X( )[ 15mV 5P + 5p 2 0+45m +P P)],12 WN 1 -- l 1v P 1Pl 24m B 1 L1 a 1P + x t2 + 12+ 10[ ] I + + J [ 4 2(4xx)[=10+5m VL +lb V 2 V ( 4x )] xy 2xx, T 1 1P 1 [0 1P + X 2y[32+Sp2V +3(p' )2] 2 + + X ( [18x) +9'V +(p +L vp p I ) ] + + t4 [ 33+20' + 3(' )2 24m (3) 3 1 P 3 P [03] (4xx) (4xy) j (f, ) ~ Ct ['T 72 + P (6V2 +3p V 2 )V 6 1,1 24m PI (xy x 1 1P 1 Pi0] +4 (-4V2 +6p ) ] + 4xy} p Pi [o + C2t4' 4) [ 5V2 + 1 V )2 JO ) si7 103,1 -c2t X 4 [(2)+[3 p ] Cx2)( ) + (2)2 ] [0 7 Pll (c P1 P P 24m 1 1

71 Table 5.1 (cont'd) E 2] Collision term.. Corresponding term of (PztlP ) 10Co] 4 (-4xx) (4xx) [01] p8( 1,) c(t. [ V2 V2 + y V2 (9+3pl ) ] + 2 —(3 P P l(3 m P 1 I + 3 l(43VlpVL -( Vp + C [0]+3PlVPl CG 9( [0] 2 4 r(2)V2 [ X(2)(3+p1 ) + n(2)v2 0] Q 9 (f 191 -c [ X (i P + VPi We are now in a position to notice that, referring to the kinetic equation for f,' had we earlier replaced the complete collision integral by the first correction term, i.e, had the kinetic equation for the full fl been approximated by Bf. + t *f -= cjdT [ X ( )(3+P ) ( n V2 2 )l ] fT ) It9 191 m 1 P1 P1 m P1q T1 91 0 then only the first term in table 5ol would have appeared as the contribution to ( I[2El;,l ). This would have been in error, for not only would we have overlooked the other terms [23 comprising - but, as is seen by this more rigorous calculation, the terms predicted by an approximate kinetic equation [ 2]2 )ontributions are exactly do not even appear; { (f) contrbutons are exactly 1,1'a~ }cotrbuton

72 cancelled by { (f 9 (f[~)} contributions!' It is cancelled by {~7(11 9 031 suspected that examination of the higher order terms of (pl;tPl ) would lead to similar conclusions. Thus, it is 1 10 to be construed that one must be very cautious when deducing information from an incomplete kinetic equation; truncation of a kinetic qtion fora probabiiit densit or distribution function. F, precludes determination of the functional form of F to hher order than the order of the truncation. It is now easy to calculate associated expressions for < n (t)[pl(0) >, the expected value of the momentum of a particle at t, given that the particle's initial momentum was -po By definition, + r 3 > oooooooo (5.21) < Pi (lotp ) (t) P [(O) > [d pi p lot|^) 1)loj Pl PjI l 1o It has already been demonstrated that the terms appearing in )(P;t|lo are of the form P1 6( ) 1 10 i p.1D 0 Appealing to the definition, 5.21, it is noticed that only those terms of 0 for which m > n-l contribute to < l (t) 1Pl(0) >. This simplifies matters considerably and, by a straightforward calculation, one obtains~ _+ 2 (2)-o = < Pl(t)|pl() > + ooooooo (5.22) ~1~1 *P10 2m10 +4!( xc2 (4) + c[(2x +4zx 5mx +10(o )pl Im (('X " ",) 2 ] + -X Pi Plo +o( t6, and terms of "sixth power of the forces" )C o

73 Next, let us use 5.22 to calculate the auto-correlation function, < pl(t) o Pl(0) >. The momentum auto-correlation function is defined by < P (t)~p (0) > d3p d3(p O.P )f (Pl(t),pl(0) ) I. 1 j lo 1 1 or, alternatively, (recognizing that the system of interest is an equilibrium assembly): 2 1i, ~ 1 o1 (2nrmkTT).- 0 2 However, it is clear that from the latter expression: o <i d3 — o (+ 2.... (5.24') <p (t) kPT(0)> =p < p (t)i p > p tO 32 ~,o ( -< e T) 3/2 o 1 o Thus, referring to 5.22, the following iteration is obtained for < pl(t)O~p1(-)> +. o+ 2 2 (2) (5.25) < P,(t) P1(0)> =<p > { 1 2 (2 2 + o eq0 22m c2t(4) +t4 C[5t [,) (4Xi 2 -, ) a1 -41( 3 m2 TiTI2 (3) 12T2m e_ -p2/2m In the above expression, <p2> eq 3 m/ = Id3p i <= POo o 3/ dPP3/2 the momentum auto-correlation function for an idea gas. It has already been noted by other investigators** that, as an approximation, the momentum auto-correlation function may be represented by an exponential function, viz., -+ p (=t)p ( > -At2/2 (5.26) < P(t)~p (0) > = 3n/B e 0000000oooo 39 40 **eog., Rice, De Gennes

74 with (2) (5.26') A C X oooooo m This expression is exact to o(t2j and the "second power of the forces." Furthermore, it satisfies the requirement that correlations vanish for long times, i.e., + r (5 27) Lim < p (t)~p (0) > =5 0.2 We should like to suggest here, however, that if one can seriously accept 5.26 as an approximation to 5.25, then the following observations are, at the least, of equal merit. (4) It is not unreasonable to approximate x (3) by,,r2 x. Certainly, in the high temperature limit where g2(*) Z e- and g3(z 3) —, 9 g3(s h ) g2(~Z)'f g2(Q3 g (S 2 gg (3), except for a small region of hyperspace for which 1 r [ < r*, r* being of the order of 2 3 a molecular radius. In addition, unless 1[E <r* and 2 3 | <r* a2(2) a2o( )3 0 so that integration over much of the exiluded'egion of hyperspace provides zero con(4) tribution to X(3), anyhow. Thus: x(3); 2 2 d33 -V(293 ['., ) (5.28) 3 3 nlgJt ofA5~28,'t _s __otied ta^t h _ In light of 5.28, it is noticed that the term of order

75 (ct2 x12)2 which actually occurs in < pl(t)~pl(0) > is the third term in a Taylor's series expansion of cos( cP)t2) / m rather than being the appropriate term of the series expansion of 5.26. For very dense systems, for which as a first approximation one may neglect terms of o(ctn) compared with terms o(c2tn), one obtains only the cosine term, viz., 12 0, 0... (5.29) < pl(tY) P(0) > cos A 2t This result is satisfying in that it demonstrates the existence of negative correlations, necessarily present in all dense systems. It is certainly unsatisfactory, however, in that the condition expressed by 5.27 is not fulfilled. Most likely, for moderately dense systems, the momentum auto-correlation function has a functional form intermediate between 5.26 and 5.29. For liquids, < p!(t)op (0) > prob_ ably has, in a rough way, the form of a damped oscillation (cf. fig 5-1.) However, 5.25 does not contain enough information to demonstrate this hypothesis explicitly. For example, suppose one tries to fit 5.25 to a damped cosine, e.g., Pl4t) Pl(0)> = cosA1/2 1+Bt < p(lt)~pl(O)>' cosA2t e oooo (5.30) where B c [n -5X +X +2X ] o00(5.30' 12m2 The expression given by 5.30 is exact through the fourth power of the forces. But, unfortunately, B is a positive quantity.

76 Thus, while (5.30) is an acceptable approximation to 5.25 for short times, it cannot be correct in general. The determination of the functional form of < pl(t)P (0) > for longer times requires an expansion to higher powers of the interaction parameters. re \' v^ \1jj~b 4Fig. 5-1: Momentum auto-correlation functions Are A and B actually positive constants, as has been asserted? The model two-particle potentials for which this expo.. sition is valid are typically of the form sketched in fig. 5.2; they are characterized by a steep repelling core plus attracting well, the latter falling to zero in a distance of the order of the molecular radius. In figure 5.2, the potentialanditsfirst two derivatives are plotted as a function of P, the intermolecular separation distance.

77 \l / \ / Fig. 5-2: Typical intermolecular potential Ive Fig. 5-3: Typical static pair correlation function

78 Consider the first of the constants, A, defined by (2) 5.26'. Using spherical coordinate, we can express x as.: x(s 2 d g[~ oi g) = (J 4dp p2 g2(p) [ a2+ ad ] j 2 W0 T 3 O ap P ap From figures 5.2 & 5.3 it is apparent that x(2) is positive. The positive portion of;20 is greater than the 0T negative; furthermore, g2(P) is greatest over the same range as is a2 positive. Similarly, the positive part of ap seems to exceed its negative contribution to the integral. This argument is most secure in the high temperature limit, where g2(5)'^ /\^ ~. Then, simple integration provides the relationship x =(2) where n2 jd3 g2() Fx() ) (see table 4-l,p.60 ). The latter is obviously a positive quantity. In general, (2 3 X = Id g2() a a B n, where * is a fictiax3x nx -6B (0) tious potential defined by g2() e (see fig. 5.3.) We see that * has essentially the same form as n(2) -(2) most likely is a good approximation of n( (2) Since x is positive, it follows that A is also a positive quantity. The second constant, B, defined by 5.30', may be analyzed in the same way. Let us first consider it in the high temperature limit. In this case, replacing g2(0) by ne Ea the following useful relationships between system constants (cf. table 4-1) may be obtained.

79 Previously, it has been shown that, in limit T-, (2) (2) e...o (5.31) X =an In addition, it is easy to show: (4) =n (4,xx); X(2.,2) = (4,xy).. (5.32a,b),(4,xx) = (4,xx); (4,xy) (xyx)=(4,xY)(533ab) = +2?4 4o5.33a,b) 3 2xx} (4xx) (4xx) {2xx} (4xy) (4xy) {2xx} 82p 2yy =aT 2x = B in; X2yy =BI xl; ^..(5.34a,b,c,) 3 {2x} (xy,xy) (4xy) X2 4=8ll -n {2xx> {2 xy 2 (4xy) (4xy) {xx} X )yy +2x 2xy} = 2 - 3 n =Xxx (5.34d) Not all of these relationships are needed here but, as they are useful in the discussion appearing in the next chapter, they are included at this time. Here, let 5.32a be used to obtain B [c }2x{x X}] e..e (5.35) B - [x dXXX+ 2XdX-Vf 12m Certainly, then, in the high temperature limit, because x{xx and x{ } are both positive constants (cf. table 4-1), B must be a positive quantity. The same conclusion is probably valid at lower temperatures. Before concluding this chapter, let brief consideration be given to the determination of multiple-time auto-correlation

80 functions(ioeo, time-relaxed correlation functions for more than two times.) Aside from satisfaction of our curiosity, these quantities are of interest because they may be related in certain approximations to neutron scattering cross-sectiont* (see chapter VI,) Specifically, consider the four-time auto-correlation function 00 (5 36) o ooo (5o 36) f 3 < pl(t )p1(t l-P(t2)p (t3) > = Jd poood3p3 p ooo3 ] (Dst iooo 23t3 ) x x x 2x 3 o 3 0x 3x i.e. the joint expectation of the xth component of the momentum of a particle at to,tl,t2,t3, (t3t is the joint probability density for finding a specified particle with momentum p at t =t P,...,po at t = to. In principle, a multiple time correlation function could be obtained by summing over the probability measure of all initial states consistent with the desired subsequent behavior of the system; ioeo, f(X t 0 ) J d (X 0 ~X r X 0o X (to)=X 00ooo0oX (t )=X no S s o s r Sr Or, one might integrate over all disregarded coordinates of a total system multiple-time correlation function, determined from solution of eqns. of motion, viZo, | oo dXnr f(X to;oo.Xn tr) f(Xsoto; 00 ox t dXoo o o d X 00Xt ) SI" r o r? no = dXnoooodXn f(X ) 6(Xn -e 1 LX_)oo6(X-e 6 o 0scA. hJman, et l0 0 **A. Rahman, et al41

81 Unfortunately, these two schemes are useless for actual calculation of reduced multiple time correlation functions, except in very special (and trivial) cases. What is required is an equation akin to the Liouville equation but, as yet, such an equation remains unknown. However, it is noticed that (Poto; pt3) = Po $,to 0 0 0P33 = (Pt3 IP2't2;Pl9tl;PoSto) xj(p2,t21pl'tl;Poto) x x (oPt t, Po'to X (PPo'to) If one now makes a Markoffian assumption, viz., that the probability of finding a particle with specific momentum at a later time is a function only of the momentum of the particle at t = 0, and not of the momentum of the particle at preceding times, then ooo OOO (5.37) < p!(to0)P(t1)P1(t2)P (t3) > jdp 3 P2 P3 ] Xt =.dpo ood [p p(3,t3- ) x 0 ~ XXXX X X Xt X(P St2-t1 I11)x(P1tl9tto IPO) It can be seen from the memory term of 3.18 that f (x;t I x1 ) is not strictly Markoffian (the kinetic equation involves integration over the previous history of the system.) However, it can be shown** that the system is Markoffian in momentum space for all except very short times (of the order of a'collision time'); furthermore, 5.37, being an expectation value, in some sense averages over the errors introduced ** — Prigogine 24

82 by assuming the Markoffian decomposition. In regard to the last point, it has been shown that for the calculation of many transport coefficients, rigorous expressions are obtained even after replacement of non-markoffian kinetic equations by approximate Markoffian equations.** Referring to 5.17, 5.18, the approximate expression (5.37) is given by: < Pl(to)Pl(t1)p1(t2)p(t3) > 0= O O (5.38) X X X X x x x x = Jd.od3 [Po oop ] ( 6(3- 2) (2 6( ) t( ) + + 6(3-P2)6(p^2l)C(tlt- t ) 2 ( ) ) + + dC(p 4p2c(t =t ) V"' t ) 3V )] (2 )6- ) +1 32i 2 P 1 P ( 2 )( ) +110 22 + ctgtt )2 -v2 3 V )]6(3-> 4 )(6(-?+ 3 I2 p ni 1 1 3 2 + o("fourth power in the forces") ) (t <t <t 3) Upon making the high temperature assumption, (2) x ()/ a straight-forward calculation provides < Pl(t )Pl(tt)P 2l(t3) > (5.39) X 0 x x X 3m2 (+c(2) [ 2t2 t +2t 2t2 t 2 23 - 3m2 exp=cx(2) t 2t t 2t +28t t7t22l t t +t 3_2m 3 3 34'"'e',g. Balescu 42

83 / is the value to be obtained were the assembly an ideal gas. We shall return to 5.39 at the end of the next chapter, when considering approximations to the slow neutron scattering cross-section. Finally, let us emphasize the appearance of secular terms in the expansions obtained for momentum densities and correlation functions. The presence of terms which increase with time is a basic deficiency of this exposition, resulting from the particular method chosen to solve the kinetic equation On the other hand, the expressions for densities and correlation functions thus obtained are useful in that they are exact for short times. As such, they serve as checks for any approximate solutions to the kinetic equation. In addition, in spite of these difficulties, the expansions obtained for momentum densities and correlation functions are often useful for the calculation of various transport coefficients. For example, it is known that the dielectric constant for an ionized assembly can be related to the Laplace transform of the momentum auto-correlation function 31,32. Upon assuming a linear relationship for the current produced by an external electric field of frequency A, the dielectric constant is found to be given by ) ^= U Uto gL A 0te P itet p^ t)> (5.43) **.. also here assuming a zero external magnetic field, Eqn. 5.40 can be modified for more general relationships between current and applied fields.

84 The summation is over all charge carriers of the ass mbly. Let us apply 5.40 to calculate the dielectric constant for a weakly ionized gas. Assuming that, due to the heavier mass of the ions, one need only consider the contribution to the current due to the electrons, we obtain'_p e, (5.41) In the above equation, Ce is the density of electrons, Y4e is the electron mass, and <~ 4~ is the momentum autocorrelation function for electrons. To obtain 5.41 we have also neglected electron-ion interactions. Neglecting, also, electron-electron interactions, from 5.25 and 5.41 one obtains: (rL I A'1' -;.+ < X j+ - + L..9)+ L -' (5.4,,), is the fraction of the atoms which are ionized..-u is defined similarly to previous definitions: beC i. <-.5> if5 (5.43) where Ve.k,~) is the interaction between electrons and neutral atoms; $Q( ) is probably well approximated by (5.44) CA2G <9 -7 e

85 In the next chapter corresponding expansions will be obtained for spatial time-relaxed densities. Secular terms also appear in those expansions. However, the expansions are not simple time expansions but are a sort of asymptotic expansion. In this form they are quite useful for calculating the scattering function; the secular terms in G (r,t) are not an impediment to taking the double Fourier transforms necessary for obtaining S(k,L3 ).

CHAPTER VI DETERMINATION OF G (r,t) As has been discussed in some detail in chapter 2, there is a close connection between the space-time correlation function, G(r,t), and the slow neutron scattering cross section. Primarily because of our interest in the latter, in this chapter attempts are made at calculation of Gs(r,t), the'self' part of the correlation function, which is related to the incoherent part of the cross section. As a continuation of the work of the preceding two chapters, G (r,t) (classical) is calculated through the'fourth power of the forces'; i.e., an expression is obtained for G which contains all terms for which M (as defined by 4.20) is _ 4. Corresponding expressions for the incoherent scattering function are then derived from the expressions for G (r,t). As is demonstrated, below, the cross section may be expressed as a sum of terms, each of which becomes successively more important as the temperature of the scattering system is lowered. The question of the calculation of the'distinct' part of the space-time correlation function, Gd, is deferred to chapter 8, below. Just as for the case of the momentum density discussed in the previous chapter, an expansion for G (r,t) may be found in direct correspondence with the expansion obtained for f (x; ti x! ) G (r,t) is defined as the conditional 1,1 -1l to s 86

87 probability density for finding a specified particle in the neighborhood of configuration phase point q + r at time t, given that at t = 0 the same particle was at 41. Since the systems to which this investigation pertain are isotropic, Gs does not depend on ql; furthermore, G (r,t) is a function S 0 5 only of the absolute value of r. Hence, 2 r r -8po /2m G (rt) Jdd3pj f( +t 1 4; l (6. 1) (2ifmkT)'2 i 1 lo 0oLo = T G]G,r t) -, S with 2 Elfrn 3P 3 -fBp /2Em +i] /+ G (rt = d d3p _ f (p,r;t lp 0) (6.2) CG] (rt) = 3Pld pf d3 / 2 e(r-)/pm 6 (jt) ) - (6.3) (2~mkT) 30 0 (mB/21ft2)3/2 e-marr2/2t2 which is the result for an ideal gas. The first order correction, i.e., that through the'second power of the forces', may be obtained from 5.13 G[O = fd d3P e japoT/2 fp1;tIlLo m) 00 (6.4) (2wmkT) }3 Integrating by parts re. p-: = ctJd pJd3p -/22 r WP[V2 stV2 t2/V26q]+ XV 1 ^.6q whc i theT resu PI for an ideal gas. = ct2jd3p0 eP-O2/2m2 S2 - 1 6(1-Qo-Pot/m) 2 ( 2 rkT) P 4 po 3 m P pi V% (fmkTl) $/ 1 -1

88 Again, integrating by parts: = t2 (( 2 2)C mbr2 3 ] ) G0(r,t) -t - _ B C! M a 3 1 - 1 m 2 4m t 3 s Finally: G rt) = ct4 ( 2) - X2) ) 2 G[0](rt) (65) G11jr~t) = Ct ~ (6 5) S 2m 4 3rr s. (2) (2) At high temperatures, n x/, so that 0- El] 4 (2) V2 G101 (6o5') lim G[Nrt) = ct4 X v2 G (r,t) 000 0 (65 S+ r s T-ow 24 m However, one could have obtained 6.5 directly from the kinetic equation for fl (5.3)o In general, the f are to be found from a kinetic equation of the form f] + lV f[] S(ql,Plot) with f (t=O) having fil f i S fl s1 i i s d.J1 1 S 1 J 1J. k the solution s-ip] t 1T PI jP'_ I j m m The OPij are derivative operators on El. Thence: G[i]~ fPd3p o /2 f43 p 2/2m-i GI ] dIV d d3p e 3^ ~ 1 k s mnkT) [s-ik~pl] [s-ikop m If one next integrates by parts re. 1,, he can immediately and easily perform the integration over o, yielding: r^i I f 3 "p2/2m.1 j-1 G^'3'jd3p e _ __1__ [i] 1 s G;~ 1 T^^T 2 k [i j I 3 om m1s-iko. ~2nmkT) Cs-f~In In

89 As an illustration, let us apply this procedure to a calculation leading to 6.5. It has previously been shown that afl m l f 11= 1oo.. (5 3) a m ~ q 1~1 ct(2)[V2 +t V2 2]+Xt3+Vp]) f[O (t) cPI, Plql p 1S1 f 1(Pl'q;0O) 0 Thus, [1] f I -p2/2m V-2 -.+ G = c d P. e (n -( vp)] + s 2 3-nmkTTf 3/2 is-ik2 1 m — m [s-ipj]3 (2) ++ + [2 ilp) ) 1 m [s-ikp]2 Cs-ik~P1 2 nm - -p 2/2m (2) 3+ + - + "l'"3 -S1 (r(2)t4v2 2 t pV ] }() d~P1 e n q 2M; V J)3/2 q.(q1-q~ - 3/2 PI —- V^2 q+ l2 t q]. - P- 8( m) (1 m 6 [0 C t (ln(2)- 1x(23) V2 G[ (rt) 9 as before.'m2 4 3 r s *2 r2 s In this way the complete calculation of f may be circumvented but the second correction to G may still be "'- 2 obtained. We start with the kinetic equation for f1 1: 22 i f f3 1 91-g fFiz(x;0o), o(5.4) 11) f + Ovq f 11 at A L q where 1 +49 are defined by 4.32, 4.37, 4.40-46. Corresponding to each of the. i will be a contribution to G[2 Consider, for example, the contribution due to (Wl+Q2)f[l] From 4.32 and 4.40: (fl +/.T)f~J 2 (.( (2)[vp2+ p 2 g (2)[3 + p ^ fl.^ ~.,(6.6) 1W2 -1 C!dT(1 [V +T V ]+X [3+p fl it-T') o

90 Thus, since 1] is present in the integral only as derivatives, except in: 5 ^ Wf 13 -= c2 dT^( [V +T(-ikV )+X2) [3+PlV ]e If (t-T) 121,1 P-1 p m Next, taking the Laplace transform and using the convolution theorem:'(1 2 3,1 Pc(2) CV2 r;1 - ikVpl 1 ],1 P [Ts' M] 2 n 1m1]+ X(2 3+Piv p m 1 1[s-ikp] 1,1 Finally, referring to 5.13: m -W c )f (6.7) 2 2 1, -2 - -:!<2) ((V2 1 ik/v )(-ik )-3 +1 V ) S' PlTs —) -P-T] 2 TPI[ - p1 ]3 -S-MPl si k ] [ Cs —-ik [ ]'m +(V2 1'ik )(3+ _2(-i(kpl) )) n(2) (([3+p V ] 1 )(3+ ) ]+ 3, )., m Now, utilize the procedure described on the bottom of page 2 6-4. For example, the contribution due to the x(2 term is (after integrating by parts twice) d=p2 2/2m -- 3 P 1 k 3; ) 61 id P1 3/2 e2 5m pP1, (i / (Lft62jm 1q 1 5- P l ) m 6( q-qo- Ip e(e L-32?rrk 3/2m5~

91 c t( t4 V2 V2 + - V2 ) G[3(rt) 180m r 360m s where, in the last step, use has been made of table 6-1 of Appendix E. In this way, it is determined that the total contribution to Gs2(r,t) due to (1 +) )f[j1 is G2Jdue t= c 24 (r3T2[19 t4 V2 V2] - G due to 2 8 280m2 r r (6.8) - X C[3 t4 V2 V2+1 t V2] + 0-_~. — ~-~2mB 2 r r m- r m 280m 8 40 mB r + [X 2 1 4 V2 V2+ 7 t V2]) G ](r,t) -;+ -2 r r r s m 90 m 18 m s As another example, consider the contribution due to t f13 *o [o] [o] J 3f[ 1 Again remembering thatt-T) [0 [ 1,1'1,1 and referring to 4.37, it it quickly seen that f = t3 (4) (15m + 5p + 5mP*vP + 2o ) f (t) 3Fa a, 6-'1 1 1,1 Thus, the contribution to G 2 is s 2 2m cxi' d p, l -/ +p2 1 ] 1 p(2"""T 13/2 --'p1'...] m m r -Bp2/2m (i 2+ -cx ) e t( + p )p ~V) 6(ql-q0-lt) so (2 fmkT)32 5!m 1 1 1 so that we finally have

92 G [2 (r,t) 00 = 00ooooooooooooo oooo (6o9) S due to. m 2 60 m-B2 6 m One procedes to calculate the contributions due to 4 — 9 t in the same way. However, rather than perform each calculation separately, since many of the terms in the collision integrals occur more than once, it is convenient to make a table (see appendix E, table II) in which are listed these terms and their corresponding contributions to GS] Then, after performing the T integration in the collision integrals, one can read off the contributions to G One finds: s 23 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (6. 1 0) m 2 36m a 3 mB s C(2 t 7................oooooooo (6.11) G [(T t) due totr (4xx) 4 4 n t t( V V2 V2 )G V) (rt)+ -~02 rr r s r m 2 i 22 6 mr + CX(3 t4(t4 V2 V2 + 7 t2 V2 [0] 3 2 82 28 r r OB'8(r,t)+ X { xt2 V2 G[0] 32 r s;2 2 20 m r s 2 X4 t (t4 2 )G (rt) m-2 ( 3 2 3 6 B2 r r r s m 2 36im a 180m-

93 G~2)rt) *00......... (6,12) (4xx 4 -S tt ( it4 V2 V2 + 1 t2V2) G rt) + m-~~2.We2 r 2 s 2m 1 42mSa 6 ma + c t4 (4XY) t22 ) G[ 3(rt) - 2m 60mn s 2 4) Bt4 ( t4 V2 v2+ t2 V2 )G[o(r,t) |2(3) - 2 r r 180 r s 2m 210m a a G23 t)...... (6.13) G (r,t) S due to, 7 2 x(2) tt (I4(2)7 t2V2 4 V2 V2] +.I-:I(+ ~ -Z4 2 r rIL 2m m 180in 90m a~ + r(2) [-t2 v - t4 V2 V2]) G [l(r,t) m r 22 r r s m 4OtmB 105min4 G2]2 (6.14) G (r~t) oo0~eoi S due toi c (4xx) ( 4 xx) + =ct4 B(( B11 -l --- ).2 Gc ^ ^ — $ 2 2 2r 192m 56 m2r Bt (4B ) - (43) 42 7vV) Gol (rt) 2mn (r m vv ~2] S (rtdue 0000000 (6.15) S ~~due to 000000 _ ~' ( A6-a t4 VI2V21+ ['Et4 V2V21 x - x - 2 rr *r- ~2 MI$~llf 22rr 2 ~ 168m - a3 192m B x G (rt)

94 Upon adding these contributions (6.8 - 6.15) one obtains for the total second order correction to Gs(r,t): GC2(r,t) =(6.16) = c t 2r22 [t VV + X o) V2V2]) G[ t 1-czB 3-) s (r,t) 2m m2 rn B m 21OrnB 2 4 ~(^ -x 2 +2 ([2(4 +) 1 9T(.4)+ (4)] 4 v22 + -2 3 (3) 840 36 fr 2m 6 ( 3 3)' -8 ( m —3r +[-BT(4)+13X( 4 ] 2V 2)GEt).3) -~ 3 M r s (4xx) (4xx) {?K]} (4xx) (4xx) 4 -x ~x rrl ~9~X f3\Ya]tm V2V2 + -2m 18 60- 36 24 92 r (4xx) (4xx) {2xx} {2xy (4xy) (4xx) +[_% -x + +3x +' - 2 66 60 20 60 6 x t2V2 )G[0] tM —r s (r,t) At high temperatures, where g e, one may make use of the relationships 5.30 - 5.33, also noting that B2p(4) (3) 3 X(4 ) xU2, so that: 0 0 0 0 OOO O0 (6 17) G[2(r, t) T — = c 24 X2(23 t V2 V2 + 1 t V2])G (rt) + -'I2 6-x-' r2r r 1 2m 630rnm 15 mTB (4xx) (4xx) +ct (C +n 2 ] t 2 2m2 90 1728 -- (4xx) (4xx) {2xy (4xy) +[13 n _B2 + 3X T+ ] x 60 60 20 60 t 2V2 ) G~[0r,t) ma r s Am AC On the other hand, as a result of equations 6~3, 6.5, and 6.16, the general form of an expansion for G (r,t) appears: s

95 (6.18) G (rt) = It2 D[n ] G[O](Rt) 0000000 S R S where 1/2 [0 3 R/2 / 2 (619) R E(mg) r; G (R,t) = (m/2nt2)3/ e.61 -M s The D[] are purely differential operators in R and are not -n] explicit functions of t. The differential order of D[n] is < 2n From 6.18 one can determine that the "intermediate scattering function" (classical) is of the form _ -t2k2/2mB -o (6620) - (k,t) = e I2n t n tk l/2) -1 Io (ma) where the n( tk ) are polynomials in tk/(m3)1/2 TCm. )l/2 n corresponding to thbe D and contain terms of even power of tk/(mg)1/2 less or equal to than 2n. For example wCn]' t2 k2/mS corresponds to the differential operaEn] tor D R. R It is now easy to obtain an inverse temperature expansion for the classical approximation to the incoherent scattering function pertinent to slow-neutron scattering crosssection. We have: owt " -ik3 r.o.,...ooo (6,21) Snc (k,w)E 1 t e d3r e GC(rt) incoS- S 2r -i i"t - 0 1 -2[i(t+k2t 2] ) _=1 dt: e 00nckt)= dt e B- Ct2nw[n]( tk ~,*0,00

96 so that, from 6.3, 6.5, and 6.19: ( O -(iwt+k2t2) 00000000000 (6.22) ~C1i~~~ 2 S. (k,w) =1 Jdt e C[ 1 ct2(Bn- x1() 4 o( ~001 nc 2 _. 2mn 4 3 mB~2 k T2 = 1 e 1 - ck (T+imw (T - ) +oo00 21 - c 2m B 4 3 mSW2 m282 4)(ayIL) 0 = mBI e- ~ [ 1- cB (3-1Omgo2 + m282 4)( 8- X) +00] -2nk" 2k k 4 3 Thus, in the high temperature limit: maW2 0 000 00 00 0 00 oo ( 6 22' ) Si (k,) = m e 2k [ 1+ cB (3-0lm2+mn2w'4 )X(2) + n2,k 224k k + terms in the"fourth power of the forces" +0000] However, there yet remains the task of correcting 6.22 so as to take into account the basic quantum nature of neutron scattering processes. Rosenbaum* has investigated quantum corrections to the classical scattering function and has found%-o Bli/2 8Fi2k2/8m c (6 23) S. (k,w) =e e [ sCl (ko) + inc inc +ctn22 X(2) - e 2 (1-2 + 2k2S) + o('44)+o 48 82rmk' k 8m 43 J*Rosenbaum, thesis - especially eqno 5.45;% Rewritten to our notation.

97 Thus, using 6.22', one can express the exact scattering function at high temperature as: hw/2 - Bi2k2/8m -mB32/2k2 (6,24) S (k,w) e e e x 0000 inc 2rk x [ 1 + cx (2)(3- lOmBn2 + m282w4) A (k, w,) + + terms of o(,se 4) 00ooo ] In the above equation, A.(kw,8) describes the quantum correction to the second term in the scattering function and would be unity if no correction were necessary. Explicitly, 1 -w2mB + h2k2 0 0 0 ( 625) A (k,a ) E 1 + Bh2k2 2 8m 2 2m 3- 1OmBw+m aw Rosenbaum's analysis has not been explicitly developed beyond the statement given by 6.23, above. But, it can be surmised that the exact scattering function can probably be expressed as a sum of terms, each proportional to one of the system constants which we have already defined; i.e., a-hw/2 -Bh2k2/8m Sinc(k,) = e e [ Sc (k,1) + a (ksw)X(2) + inc 1 + a (k,i,)X(4) + 0, etc, ] The same system constants which demonstrate the deviation of Sinc (k^) from that for an ideal gas are also intimately involved in expressing the intrinsic quantum nature of the

98 scattering process (cfo, 6o24, 6o25)o In general, one cannot separate the "quantum effects" in the cross-section from contributions due to interactions between particleso On the other hand, referring to 6025, it is seen that for small momentum and energy transfer, A2(kwB) 1 Thus, under these conditions, quantum corrections to the scattering function are much less significant than the normal corrections due to interactions between particles (at least, to an order of the "fourth power of the forces.") Of course, the fundamental quantum correction manifest in the B 3w/2 -fh2k2/8m multiplicative factors, e e, remains; however, the quantum interference effects demonstrated by A (kwSB) 2 may be neglected if ~B2k2 << 1 If this require2m ment is met, the A-C-R-Z prescription7/o, viz, aw/2 _-52k2/am, ( 6 26) S. (k,w) - e e S (k)w) inc inc is probably valido For high momentum and energy transfer quantum interference terms are comparable to interaction terms and 6.26 must be employed with care. Equations 6.22-6.24 suggest that it may be feasible to measure the temperature dependence and magnitudes of the basic system constants [ Sn r- x ] 4 3 The measured values could eventually serve as a check on values yet to be determined by direct calculation from postulated interparticle forceso In addition, though, many physical phenomena are probably adequately described %7oo Aamodt, Case, Rosenbaum, Zweifel 8

99 by theories for which the basic parameters are the system constants, rather than the forces, so that there is sufficient reason to obtain information about these constants for their own sake. In measuring the M= 2 system constants, as demonstrated by eqns. 6.22 & 6.22', it would be desirable to scatter neutrons from a high pressure gas at very high temperature, studying neutrons which have experienced a low net momentum and energy transfer with the bombarded system. Under these circumstances, the second order term in 6.24 is a small correction to the zero'th, and the fourth and higher terms may be neglected. Quantum effects would not be too important, but they could be calculated, One would measure Sn (k9w) inc at constant k and w, as a function of TtB Of course, 6.24 is related to the incoherent part of the scattering cross-section, only. Hence, these last remarks are conditional upon the possibility of determining the temperature dependence of the coherent part of the crosssection also. This point shall be considered in chapter 8, where determination of Gd(r,t) is discussed. At this time, let us return to consideration of (classical) Gs(r,t). From equations 6.3 & 6.5, oOO.oooOoo (6.27) GS(rt) = E 1 + c(nt)+o(4) 0 GOn (r t) sG(r~'t) 3= 1, + ct4 ( n y ) V + oS()oooo G (rft) 2m 4 3B S

100 But, 1m nr2/2t2 G[O](r,t) = (mB/2=t2)3/2 e,oo (6,28) s (t2/2mB) V2 ** r 6() [ 1 + t2 72 + t4 v2 V2 + I000 6(r) 2m r m-7B2 r r As a consequence, one might wish to represent G (r,t) for short times as: 3/2 -mr2/2Wd(t) (6.29) G (r,t) (mB/21W2(t) )3/2 e mr/2o (6.29) S with W (t) = t2[ 1 + t2 ( Snr- xt) ] 0ooo0 (6.30) m 4 3 rJ t2[ 1 - ct2 X ] 12m The higher the temperature, and/or the lower the density, the better is this representation. For consistency, let Gs(r,t) be calculated according to the'Gaussian approximation". " If an ansatz is made to impose a Gaussian form on Gs(r,t), then it has been shown that -r2/2w2(t) G (r,t) - [ 2rw2(t) ] 3/2 er/2(t) (6.31) S 3____ r2/2t2 **One notices that the Fourier transforms of.(m-22 e and (t2/2mR) V2 _+ t/2 r 6(r) are identical, so that the functions themselves must be equal. (3[@ 10 9 See, e.g., Vineyard, Schofield

101 where t w2(t) - 2 fdt (t-t') <v(o)'V(t')> (6 32) 3 Reference to eqn. 5.25 demonstrates w (t) t2 - 2 [ 2)t X +,o(4) +..Q (6 33) ma 12m co (6.33) rnm 12m i.e., for short times, the Gaussian approximation provides the correct expression for Gs(r,t).** This result is well known; it is also known that the Gaussian approximation does not hold for intermediate times. Here, however, it is explicitly demonstrated that the higher the temperature, the better is the approximation and the longer will be the interval for which it is valid. 41 Rahman et al. have indicated how the Gaussian approximation may be improved upon, and have suggested C ca. (6.34) t9o Cal(k,t) = exp 1(-k2)1 y (t) inc n 5=i where y1(t) E dt1 jdt2 Vk(t2) vk(tl)> Oooo (6o35) t tL t Y2(t) = t1 dt2 d0 <4 (t4)oooV (t )> 2 d t 2 4 k 4 ~ 2 1 and so forth,. Calculating y1 and Y from 5.25 and 5.39, 6.34 provides *t*Assuming; of course, that the correct expansion for <.is used, as per 5.25.

102 (at high temperature): \cl -k2[t2 (1-ct2x(2)) oo] +k4[ct6 X(2)+oo] -k6y inc (k't) = e 2m 12m e I8mr 2T e 6oo 1- k2t2+ ct4x(2)k2+ k4t4 k4ct6x()l + 1 ] + 2m- 24mLB 8m-l2 m ~ z 48 288 + terms o(t6E0), o(t8e2), o(E4), etcoooooooo 0 0 0 0 0 (6o 36) On the other hand, from the exact calculation, 6.20 and 6,5, it is found that -k2t 2/2mB o 2o ( 6.37) c'(kt) =e [ c (2) + o(4).o]'inc 24m 7 1 - k2t2 + ct4X(2)k2 + k4t4 - k4ct6(2)[1 + 2mB 24m'L 8m-jB 3 mj' 48 + terms o(t6c0), o(t82), o(c4), etc OO... Comparing 6.37 with 6.36, it is seen that the k4t6X(2) terms do not agree; the imposition of the Markoffian assumption in calculating the four-time momentum correlation function is probably the source of error.

CHAPTER VII BOGOLIUBOV FORMULATION; EXPANSIONS IN DENSITY In the previous chapters, an investigation has been performed to obtain expansions for conditional probability densities and associated time-relaxed correlation functions. As it turns out, however, the resulting expressions demonstrate, only, the successively higher order contributions of the interparticle forces; it is not clear from the preceding work how one might also arrange the terms of the expansions to observe effects due primarily to changes in the average density of particles. In the work that follows, below, a kinetic equation is derived for f,1 (x;tlxl ) in which the terms of the collislon integral are arranged not only according to increasing powers of the interaction between particles, but in powers of the average density, as well. As such, this work 3 follows closely that of Bogoliubov, who studied the more usual case of a system which is totally symmetric under interchange of particles.** Bogoliubov considered models with repulsive interactions only, searching for an asymptotic kinetic equation for fl, of the form: af t(l;tlX ) = cn A (t) 0 0 o 0 (7.) ~*The system of interest here is not, because attention is focussed on specific members of the assembly. 103

104 ** Ao was found to be the streaming operator; Al is closely related to binary collisions; A2, ternary collisions, etc. The denser the system, the more important are the effects of multiple collisions. In the following derivation the more usual systems, in which attractive forces are also important, are considered. Hence, also needed is a parameter with which to characterize the effects of the attractive forces. The van der Waals theory for the equilibrium equation of state44 suggests choosing cc for this purpose. c is designated as a coupling parameter for the attractive forces which appear in the Hamiltonian, measuring the importance of the attractive potential energy relative to the average kinetic energy of the particles. It is typically proportional to the depth of the potential well, divided by kT. Hence, rather than in Eq. (7.1), our asymptotic collision integrals will appear as coefficients of a double expansion, in c and ec. Bogoliubov makes yet another important assumption, which is the essence of his approach to the problem. He assumes that after a very short time, of the order of the time during which two particles are close enough to interact (i.e., the "collision time"), all correlations between particles are destroyed. From that time onward, the tempral dependence of the densities, fs s(xl,.., x;t), for s > 1, is assumed to be only implicit: the higher order ** Specifically, -Pl' 1ql fll(xl7t)

105 densities are time independent functionals of fl, (xl;t) and depend on the time only through the change of fl,1. with time, Bogoliubov considers dilute, non-crystalline systems characterized by purely repelling forces. Hence, his assumption is plausible because, by virtue of the latter ch-racteristic, if two particles were to interact they would always move a'way from each other. After a short time it would be impossible to specify, with anything more than a very small probability, which molecules might have interacted to give rise to the dynamical state in which the particles are found. Hence, higher order correlations are being destroyed too quickly to be of importance in the gross evolution of the distribution functions. As an extension of this argument, it is assumed. that af, is also dependent on the time in this way. at, In the case of systems with attractive forces it is harder to find a plausible argument to justify the assumption. The great difficulty here is that, due to the forces of attraction, particles can be trapped in each others' force fields and correlations would persist for longer periods of time. However, if the systems are not too dense, the frequency of occurrence of these arrangements will be very low, In accord with these remarks, perform the following expansions:

106 afl (X;tlx ) A(f ) = cm(c)n A (x;f ) 72) at 11 m, n 1,1 and, for s 2, f (X;tl1 o)= Ticm(ec)n fS(xS;fll( X;t SOS S 0| N 1 L C)C S, 5;f^t191(-lo 0.0t ().3) As has already been discussed in Chapter 2 (and, also, Appendix 7-2) the Liouville equation may be transformed into a set of coupled differential equations (the BBGKY hierarchy) which provide the temporal evolution of the reduced probability density functions for specified sets of particles in the assembly. The hierarchy may be written in the following form (having singled out for attention only one particle of the assembly) af (X;tlx ) + [ H - O ] f w S -lIo S s s s, - - ~8~ f + c ds l,[ r r s)H~l 7l B s s | Is+1 AS s s s+l,s+l 1 < s << n The operators appearing in the equation have the following definitions: **At the end of thischapterin preparation for studying G (r,t), which is related to a probability density conditiona upon the initial positions of two particles, a different form of the hierarchy is employed (see, also, Appendix 7.2).:7Relating to the notation of the preceding chapters, it is seen that Lo = -H and L or Oa (see next page~) n 1 n n

107.4mi i i P ] ) j hi 1 r $F>~J $ rS _ i E.ijLa Pi P. 1 1 sents respectively, the repulsive and attractive terms. C is a positive constant; fa is the absolute value of the attractive part of the potential. As n the earlier parts of this study t s being saj % -~i( VV Vp. fGL ~ s,j j. Pi- B- assumed that the two pnteracticle interaction poteartices s adequately harepres beented broke two-bodyerms, potentials eac whih consistsents respectively throng repulsive core and weak short-range attractive terms. is a positive constant; is the absolveragute valudensity of the assembly. attractive part of the potential. As in the earlier parts of this study, it is being assumed that the interaction between particles is adequately irepresented by two-body, additive, potentials each consisting of a strong repulsive core and. weak short-range attractive "tail". c is the average density of the assembly. Placing 7.2 and 7.3 into the BBGKY hierarchy (7.4) with s = 1, and equating coefficients of powers of the expansion parameters, one finds: 00000o0o (705) a) — H1 fl(Xl;t); b) Am = njdx2[f n-f2 tm,n 2[ lf2 92 1]9 for m,n 1; c) Am dx2 (2) 2,2; d) A0 n Jdx2ax2) 2 2

108 Also, for s > 2: 4f ix;f1x = f = f 6f A(x1;f Ix,) — s.~ s 1,1 30 j's's ^.1 9s's I* 1.1, <0 = cm(cE)n A,m;fl )6f 1,1 Define Dmn as the operator which takes the partial derivative with respect to time according to the chain rule, and af then replaces 1 by Am n In other words, D - ( 6 af ) o.ooo (707) m,n 6f-1 0,ll f - A 1,1 r',1 mn Hence, ooP af = c (ec)" D f (Xs of., 00 00 (7.8) at,s m~n s,s s In this way, using Equations (7.8) and (7.3) in the BBGKY hierarchy for s > 2, and then equating coefficients of powers of the expansion parameters, the following equation is derived. D fO n + ( H Or) fsn(X;fl,lx ) - fmi' D fmS-,n. D fm - D f -? p:o sis 0,q SS'pq s,s.a fl, nlJ^i0 ) rfn m-ln, a m n-l;.. (.79) 0a fmtl, n- 1 +r [ s f - S J s+J s s+ls+ s s+l1,s+1

109 Equations (7.9) and (7.5) are the basis for obtaining the asymptotic evolutionary equations which we are seeking. However, before we can proceed it is necessary to impose some boundary conditions on the fss To this end, in the spirit of Bogoliubov, assume that: Tet 3x -T ll (7.s 10) -+fl(x) f(xi) 0 S) t (Z (s) is the reduced "kinetic operator" for an s-body reduced dynamical problem which involves only the repulsive part of the forces. The definition of this operator is that when acting on the coordinates of a particle at time zero, it transforms the phase space position of the particle to that which it would have at time t were it acted upon by only the repulsive part of the forces of the other particles. Hence, () A(X ) = A(X ) where A(X ) is the solution t So S S of A(X ) = [ H - 0 ] A(X ) at s s s A(X ) being any arbitrary function of the coordinates so -(Hs-Or )t of the s-dimensional phase space, i.e.,. = e o Condition 7.10 corresponds, in a physical sense, to the weakening of correlations when the interacting particles move far away, in space-time, from the region of interaction. The full meaning of the assertion is far from clear, but, as 22 Uhlenbeck says, "it works." Hollinger has indicated * -18 Uhlenbeck p. 47.

110 that this assumption is equivalent to the Strosszahlansatz, implying the lack of correlations before a collision takes place. In any event, (7.10) implies, after equating coefficients of powers of the expansion parameters: liraM (s) lf l,(1) lim g TfS S(Xs; A +T ) =, T-O-w -T s: (s (+T $, < 1)...... (7.11a) = lim ($) ) fl gXl +T,i T-w'-T T +1 and. lim 9r(s)fm'n (lX (x 0 T — /-T S S S S+T 1 ) ) for m or n * 0 7.11b) It is now possible to obtain explicit representations for the Amn which make up the collision terms in the evolutionary equation for fll In Equation (7.9), let m = n = 0. In this case, the equation reads: D fo;f 1,) + [ Hs r8 ] fo ~( fl ) = 0 0 $ 1$ S S $ IS I ooo.. (7.12) However, this equation, which holds for fl,, is equally valid for $ (1)f1.' Noting that, +T D f~~~( * f f - (f af ^ Do0ofs0 (; +T S C OTtll f -+A1 1,1 ot Aop.701 =ref.705a = Hl f, 6f~0 T~r 1.1 -.s,1 H iT so that, because w1) e, we can write D f~o;'.... (7.13. ),o 0 SS1, ( lbdT)

1ll Equation (7.12) becomes: ( d/dT - t H ] ) f (; f 0'l I) =........... (7.14) This equation has the formal solution, r fo/01 f. (s ) f(,o) (;O )/ O foSos1(; l 1, (7.15) Multiplying 7.15 by _r(s) the equation yields: T f0o0(; ) = ts) fo,o(; r1) fOS(;fl) =l fs SS;+T fl,l).. (15a) Since the left hand side of the above equation does not depend on T, the right hand side may be evaluated at any value of T which is convenient. For this purpose, let +Tso that in virtue of Equation (7.11a):, - f1 1) 1 +Tfl, xl;l * O,n How about expressions for f? From (7.9)::,S D fo~n + [ H -Or] fon = 0,0 SS S S SS ~"0000000000(7.17) apVsf -5 fS+j1+1 - ID f = -afJ.Pnsf s 4,s' -a na o,n-f S ss s+1 s s+l o,q s,s In general, for equations of the form D fO nf;fl ) + [ H _-r ] fon (;fl) (7.18) 090 SOS 11 S S SOS 1 01 o it is noticed that, upon letting fl 1 + fl and by virtue of d H-OrS )f on( + f l) ( d -[H -Or])(~ s-f~ 9(,s f ) - Td s ST S S I;SS *+T 1, - S S )(; S O,1) ti T A A-It.lt

112 that (7.18) becomes r N \ A (s) dT [ s f0,s f) -%1;sf ) ~ A +T [ A-T S9S Sr s This has the solution: TrT ( TSOS', s,,1) -T' S TV 0 Using (7.11b), we obtain finally: o S~ins + fdT r) ~ 000000 (7.19) 0 fS S if;f1 + JdT? fS J8+T w 1 0 0 0 0 0 0 (7 19) In this way, Equation (7.17) has the solution: fon(;f ) = s$S,1 o0 r( ( 0 on-q a S n o o S a 5o (7.20) f f )f + 0 f 0 OqSS Il S S + Six dx:, +4 f ) dT In particular, for s = 2, n = 1: ss t 2,2 fl,1) = *fc ^^ ^.^i.!'a0 00000000 (7.20a) 2,2 + J ~;9f2)d~ -S (lf2 2(0fs f2 32f3 0 (7.21) = (*fl ) -=) and, in particular: f2 2 (x1x2;fl ) ooooooooo a(7.21a) r~6s) -M DiP, o.,, ))d, 0 d ( 9 ~22 lx 2 ( 2) f~;~(;; f ) ) d. Finally, using Equations (7.2), (7-5), (7.20), and (7.21), the kinetic equation can be written as:

113 fl + P-l" v I.f c I (X;t) + c2I ( t) + 1,I m2 1 f,1 =- -~l;}+c I Xl t) +.... (7.22) + o(c ) +...... where, r rrr (;t)= Qjdx [Dl(X )-Xe:a(x )]X 2 f 1..0.... (7.23) I^(xj;t) QJdx2E(X2)-C (xEO2)J x * *...... (7.24) Equation (22) might now be solved for Of course, these may be illusory hopes, for it is noticed that successively highone would proceed were heollision integrals contain obtaining higher order terms. Equation (7.22) might now be solved for f!,_(Xl;tlx|l ). Of course, these may be illusory hopes, for it is noticed that successively higher order collision integrals contain within them solutions to many-body mechanical problems of correspondingly higher order. The latter, themselves, have unfortunately never been solved except in the most trivial cases. Truly, this is a serious limitation to the scheme, and it is one reason why the methods used earlier in this dissertation have been preferred. However, let us extend this investigation in preparation for work which follows in the next chapter. Suppose, now, that instead of focussing attention, initially, on only one particle, we specify the initial positions in phase space of two of the particles. In this case, the proper form of the BBGKY hierarchy is not (7.4), but rather (see AppF):

114 af l(Xl;t X' ) + H (x) f l(xl;t ) 31, lo 0 1 1 1, 00.0000 (7.25a) - dx [r c a If (X x;tx )O + = 2 1J2- 12,2 1 2 1X2; 20 +(n-2) Idx3[~ r co aIf af (2;t [Xl,x) + H (x ) f l(;t[ ) =.o (7.25b) t 1 2 20 1 2 1, 2 = Id~l[l^ l1 lr a I f 12 192 l f2,2(xl x2;tlxl1,X2 )+ +(n-2) dx3[C 3 - a3]f2,(xc a;t i 3 293 2,3]f2,(x2 x2t3) af + [C H - ]fss =.. (725c) S S00SS0 (725C00000) = -Eaf + (n-s) Jdxs+l[r -a] fl,+ (s 22; (1,2)e{s} ) Now, let us make the assumption, analogous to (7.3), that fs, (X; tlxl, x 2 )is not an explicit function of the time; rather, is functionally dependent on both fl, (x; t1 i, x2 ) and f,1 (x2; t xl x ) and changes 0 ** 0 20 with time as the latter change. Thus, one performs an expansion: fs sX;tlx x) =....o..... (7.26) - Cm(EC) f'n(Xs; fl,(X1), f ll(X2) ) *V*fO: 2..but, we do not assume explicit dependence upon a two particle density. As such, this assumption might be regarded as a "superposition approximation."

115 Also assume: af (xtlX x) = C A (;f(xl) ) (7.27a) at.L 9 1 -lo 72om,n ~ 0 f 0 0 lf(X;tl|X s = Ad ~C ( EC) Am V ((c )... ~ ~ (7.27b) iat'x 10 "2O oo mTn 1 ooo' Inserting (7.27a) into (7.25a) and equating coefficients of expansion parameters, one obtains exactly the same set of functions for the Am n(xl; fl 1 (l) as obtained previously (Eq. (7.5), above). Also, from (7.27b) and (7.25b), a similar set is obtained for A(x^; fl 1(x2)) Continuing as before, one obtains the same equations as (7.8) and (7.9), except that, now, Dm n must be interpreted as follows: D = [ a o af (x,) + 6 of (x) ] *f(lX) at l, (X2 o oo (7.28) with: %f l(xl) +Am (X);,f Ap(x2) The boundary conditions (7.10) and (7.11) are unchanged. Similarly, with minor modification, Eq. (7.15a,7.20,7.21) follow, except that one must keep in mind the new definition of Dm n. For illustration, let us now write down some explicit terms of the expansion for f,22. Immediately, one obtains: f 09(xx,;t12_, i l _2o' ^(-1>;~i^X( X2 fixl;tl/Xofl >;tll\2,)o7.29) V-~O )00 ~=I*X.q")( ). I VX*+ However, to do so, one neglects the first integrals occurring on the right hand sides of Equations (7.25a,b) as compared with the second.

116 To obtain the terms providing the first order density correction we must calculate f10 + ff0'1 as provided by,,2 92 2 by (7.20a) and (7.21a). To this end, consider [D1 ODo, 1] f2 (X fl 1 fx ) 6f (,, x ), )((X ) 1(X) +,2 1,1 1 91 2 at +;],g2(s~;) c atf,,X2)l ~ ^f01l^;,, Cx\ (' cf )-*A (x.fx )) +.t 90 1 0 1, it + CA But, af.1'l(x _ 0,1 at - f2,2 (; fl,l(x), lX2) X ) 2 (x lX2 fl l(Xl)fl 19('X2 *fl(([zXlX 2,'t' x)]2,~l)) Xlt{ f) x f!,,X22o20 xf1 ([ x9x ) (x2 )] x2)) x2;tlx, x2 ) (7.31) Thu s, by-o (['4oo *'^*n ) 1 -o 0 *. b) Thus, by (.E.4b) and (E,4c) of Appendix E: 0ooooo (7o32) /'^',l Cx)): 6= (Zi -x 1)/Q t ) wh er 2 e +.f11 X fx ) ) 66 1 lX+t 2 fi2;tl2o where 1Z~ [t(X1x2 (X1 ] X1 2 [,g-:'X " x2)..(x2.. (7...)

117 Thus, (7.30) becomes: D[ D, + eDo,1 3] f92f 0 ) ED 2,2'. i. Ti,) (Z3 -x1) Jdx3j(ztx: X 1 1.+T 1491 2 20 +l (xl) dx3[ x x )_Ea] f (xX,0.f).)) 2lX3 1 f2 22(XX3;f l (Xl)~)2 1 00000,000 (7o 34) Proceeding in a similar way in the calculation of the other terms appearing in (7.24), it is found that the density correction of first order is given by: f 10;f (x 00 + ef 01(X S 00. 0 - 2,2 2 1,1 1 2,2 2 o o o oooo (7 35) J6(-T Z+T(Xl (2t1X2) X - O 40 a I fo (oo | 6(Z )f;tx) |d [f (xZ3)]f 2^x;x.(xx;f ) + +z1( _xfl l Z2;tlX20 )d~3 13x 29 29 3 1s1 -SQjdx3[(+2 (x X2xX) f33 (X iX.3;fll(Xl).f 1, );1+ 0oO ~ 2 j 2,,tX, T, x,j T? C ( Xl~fl 1( 2;tX * X2), oetch Same as above, only with ia and rd+a replaced by 1r and drespectivelyo.

118 It is interesting to note that upon assuming that particles move independently of each other, (7.35) is identically zero. Thus, according to the approximations already described, a density expansion of f2,2(xl, x2; tIx1, x ) is provided 0o by f2 2(X2 f2 c2)...... 2 (7.36) with f2,2 and f1. given by (7.29) and (7.35), respectively. Clearly, however, things are quite labyrinthine. But, some useful information is obtained from this work; further reference to these equations is made at the end of the next chapter.

CHAPTER VIII CALCULATIONS OF Gd (r,t) In the preceding chapters, primary interest has been the derivation and solution of equations providing the evolution of the one particle conditional density, fli,(x1; tlxl ). Unfortunately, it has been seen that the required analysis is discouragingly complicated. Yet, even greater complication is involved in the calculation of a two particle density, f2 2 (xl1,x2;tlx1,x2 ). Properly, one would start with Equation (3.23), with s = 2: af 2(X2tix +( t v (v - )) f + t T(l-<)L ^(8 1) 0= jd L e 1-q2 qLf 2,2 ) The collision integral would. be expanded using methods similar to those used in Chapter 4; iteration solutions would be subsequently obtained. as has been done in Chapter 5. In this way, the zerolth term in the expansion would be taken as the solution of /P + + [~](x;tI -) 0 (a/at +?Vq + Vq l0 0(V.C-v2 IJim 1 12 P P2 f2,2 2 (8.2) with 2,2 CX 2;i 2) = 6(1X ) 6( 2- ) i.e., to zero'th order, one considers two particles to be moving independently of the other particles of the assembly; however, they are influenced by their mutual force field. 119

120 Thus, the first step in the program would be the solution of (8.2). But, difficulties are immediately apparent because, in general, such solution remains unknown. Suppose, however, we wish to calculate Gd(r,t), defined by Gd (rt) 2 o2 o ooo (8o3) f (X;tP) =C dpDP2dp dp2dqdq g2(q4qo ) e - 10 2 (X;tx ) j ld 2o _1 21 20 2 20 2 22 20 where g2 (1n,'20 ) is defined as in Equation (4.19). In this case it is not unreasonable to resort to a form of'superposition' approximation, viz., assume f (X;t I >) f (;tjx ) f ( xtjx) 0(oxo;to (8.4) 2,2 2 20 Ml'1o i12'20!,1 1,1 At least for short times, it is to be expected that Gd(r,t) will not be much affected by such a simplification; close orbits, i.e., those for which (8.4) is not a good approximation, contribute relatively little to the calculation of (8.3) because of the presence of the weighting factor, g2(ql1, q2 ), appearing in the integral. O O Hence, by (5.8), (5.13) and (8.4): f 2(X2;tlx2 ) _ (S)ql 62(t)6Pl16P2 + OO. (805) +ct2n(2)('6~1) 6.t& S+V2, 6 6 6/ V6 *06,p6 6 ~ () 2 +? x ) ) 6q7 ^^(2 F m i a + 7B ^ - PSS^2+9t )if 66 ) +ct2X 2) 61 (6-671- 2 6F'?6,~ - 6 ~652 +t ( PA+P+ ) 6W - ) 2 3m P, P~ + terms o(c4) cooe7o Pl Cf. Chapter 2, above.

121 Consider, e.g., the zero'th term of Gd, defined by: E0a f - (p2+p2 ) Gd(rt) =c ld dp2dx ddq20 g(qlq2 )e Zm o doql(t)d2(t6PlP2 J lo 2o 1 2dq2o 2 1 q2o (2rmkT-)-mB(qo-q) 2/2t 2 =c dq2 g (qq2o )1mS 13/2 e 2 (8.6) c [ t2] e 0 o'rt mBr2 t2V2 But, as before, notice that [m 2 e 2t e2m 5r) Thus, [03 -(t2/2mB)v2 Gd(rt) = c jdq2 (e (q -(t2/2m) V2 = c e q2 g2(qq) =. (8') -(t2/2m ) 2 e r ge (r) g2 Next, consider E1] 2 2 -f (2P2 ) G(r,t) c t dpdp dxdpd 2m 1 2 g2(ql2 ) x d 1- 1o2o1220 2dPd2 2 Om(2 ( 2mkT) - o X(2) t[6 qlt)^ 2(t) (V2 +V2 )6 P- P2 1~t ~ P1 P2 0 o o G e eo o o eo 0 ] 1 + Pl(2) P q2 + (2) 66 (0t)6 26 (t) -.......C0. ] 0 m ( see eqn. 8.5, above, for detail ). o0o.o.. e.. o * o a e ( 7)

122 Integrate by parts re. p1 and P2 to obtain: [1 2t -(P+P2 )/2m x I V2~ V2 ) + Xp )3 ] V2 + V [4 (Plo P20 3m l P +P2 ) t 2 2 6(P+P2 )/2 = [Ct qdq dp dp e Jt 0 g (q q ) x 7. 2 2 j2, 9q2. 2) V2 + 1 V2 (2 2t-+ + + 6ql2t 6q2 XE n(2) - 12 p p2 )/P +P2 )-6] t) (t) 4m 3m Pl (t) 3 (t Further reduction yields, = cd(rt 2 ) x= cfq4 q - f ( )(V 2+v2 x Gd20t 2 4 3 0 g2( 2qmkT 2 )1 2 [O] [03 2 ^03 - ~ s 2 2 But Jdql V2 1Gs(qf1loields zero boundary terms; thus, l (t) 4 n ((t2/2m() 2( 24 C -(2) (2) 2 rlq? C1q qV2 Gdrt) = ct 2n DEn)3 e R g ( t /(m)1/2) [ 0 R 2 (8.8) where the DR are identical to those defined previously in the study of GS (cf., Eq. (6.18). In fact, because jiG i cI ) - ~ 0for all j 1, upon reflection it is seen that (8.8) is true for all terms in the series.

123 Thus, re-writing (6.18) and combining with (8.8), the following result is obtained for the complete G(r,t): C[ R +V2 G C(r,t) = t2n [n] e R GCl.( Rt ( 1/2; 0 )(8. where G(r,0) = 6(r) + c g (r) 2............. (8.10) The expression given by (8.9) is formally identical to,,,,10 Vineyard's convolution approximation.n Thus, it is seen that the basis for the latter is to be found in the assumption of superposition, (8.4). One may now calculate the coherent scattering function. A calculation similar to the one leading to (6.22) provides: clo -mBa2/2k2 S (k,)) ='\WM; e [ 1 + c g2(k, ) ] 2 coh. N 2irk 2 [ l+cg3 (3-10mSo2+m22w4[ 4] - X ] +o( 4)o..o] (8.11) 2kL k 4 3 Hence, a propos the discussion on p. 6-13 concerning the measurement of system constants, it is now apparent how a proper measurement of the scattering cross sections will provide the desired information. For fixed momentum and energy transfer: d2 oh Sco (kw) + oA n S. (km) 3 dQ — oooo (8o12)

124 -mBw2/2k2 X' Xcoh ( 1l+ Cg2 (ks ) ) + inc ] x u-rk 2 ~ coh 2nc x 1 + cB (3-10moB2+ m282w4)( Br- X') + oooo 2k2 k k 4 3 2 (k, ) is the Fourier transform of g2(r), and can be 45,46,47 determined from x-ray scattering data. Of course, (8.12) is incomplete in that quantum effects are not included. Unfortunately, the counterpart of 43 Rosenbaum's analysis for the incoherent scattering function is not available and the quantum corrections to the coherent scattering function are uncertain. It has x **been suggested that the basic correction is that previously noted, viz., 3hw/2 -S2k2/8m S(kw)= e e fi2) + (k,)- e e- SC l(k,w) + oC) + to] ~ ~~ (8.13) However, a term by term analysis for Scoh. is not available (see (6.23) and ff. ). But, it is suspected that, as before, for small momentum and energy changes, quantum effects are small when compared with contributions to the scattering functions due to forces between particles. Can anything more be said about the approximation made in determining Gd(r,t), i.e., about the'superposition approximation', (8.4)? Let us, for the moment, refer to the ** 11 Singwi, K.S., and Sjolander, A.

125 zero'th term in the density expansion obtained in the previous chapter: 0o..o. (7.29) f2,2(x2;t x20) a (=l 2) ( ( f' f111;tlx )f'(X;[x ) 2 20;t lox;o > sw 20 *OQ +~,l The action of the kinetic operators is made clear if one notes that, acting upon any arbitrary function of Xs: e ne) ooo (x) F(X;t) = (8.14) +1 1 s 0 =F(q + TPi; oo;q+iTP;t) Hence, 1 m 1 s)(X )(x) o0o000 x ) F(X;t) = — ~ S S S.. 0 00 (8.15) $s 1t 1 1<3 s i ( +T;00000 q T 000000 L M$ m Note that (s (Xs) =im p i is the momentum which the ith particle has before the occurrence of the interaction which results in the configuration {X } at time t. S In other words, P. (x )is the constant momentum which the ith particle has when it is infinitely far removed from the other particles before collision; it is compatible with the state X. at t, which is obtained through the reduced s-body mechanical interaction. Similarly, Q (Xs) Elm qim + Pi )is the position at T-Soo s_% i 1 which the i particle would have been found at time t if +(s) it had proceeded from infinity with momentum P. (X )but 1 S that as a result of the s-body interaction it is at gi, instead.

126 Referring to the two particle density described by (7.36), for almost all pairs of particles: l(X2) q; P1(X2) = P Q2(2 = q2; and,P2(X2) =P2 When this is the case, terms in the density expansion of order higher than the zero th are zero. Thus, for most pairs, f2,2(X2;t X2 ) = f ( t ) f (X2;t ) 2 2 2_ -CL 1,1 -2 o 22

CHAPTER IX SUMMARY AND COMMENTARY The main object of this thesis has been the calculation of time-relaxed probability densities and associated correlation functions for particles moving through an equilibrium assembly of like particles. First, an integro-differential equation (318 ) describing the temporal evolution of the probability densities was obtained from the Liouville equation. The equation is generally valid for large systems, providing there are no external fields acting on the assembly. It was assumed that the interaction between particles can be represented by additive central force potentials, and that the equilibrium probability density for the total assembly is a canonical distribution. No other assumptions were required for the derivation of the kinetic equation. In contrast to the BBGKY hierarchy, the derived equation contains all the information necessary for its solution; i.e., the kfnetic equation for a reduced probability density does not contain integrals over unknown higher order densities. Particular attention was given the kinetic equation for the one-particle density. By manipulating the collision integral appearing in that equation, the latter was put in the form M Cj]3 aatf-Cx.;tlx,~ ) + plv, A f = I (. 127 - 127

128 where the successively demonstrate higher order perturbations to the evolution of fl,1 due to the internal potential energy of the assembly. These terms may be written, figuratively, in the following functional form: co t (2k-2) [k]. I = m dT T- Op (pl;Vp; TV f lt-T) k=t [2kl o (2k-2)! 1 l 11 oo92 It is apparent that the collision integral contains a "memory", i.e., it is necessary to integrate over the past [k] history of fl l. A, is the streaming operator; op (p,v;TV ) 1 "^ 1 -T 0Pi- P ql are mixed differential operators in Pi and ql The constants'^ [2kjare of the form: [2k] j2 2kg2k 2k), a'(,o 3 — ( 2 2k where 0 is the interparticle potential and g2k(2g 0000og ) is related to the 2k-particle equilibrium correlation function (see 4.18, and 4.19). It is noticed that these constants are equilibrium constants, i.e., derivatives of interparticle potentials averaged over equilibrium static correlation functions. Regarding spatial integration over interparticle potentials, there is no need to introduce any arbitrary cut-off procedures into this analysis. At small interparticle distances, g2k( 52'..e..'o2k ) decreases to zero faster than do the derivatives of the potentials diverge. On the other hand, *'*o0owith the condition, I ]m.. = 2k A: i"

129 although for large values of ( E L g 1. 2 00o00, 2k 2k1 the derivatives of the potential fall off to zero fast enough for the integral to converge. The latter is true even for potentials of the form (P0)' l/p, because we are integrating over derivatives of the potential rather than the potential itself (cf. 9.3). Collision integral terms through the "fourth power of the forces" have been obtained explicitly. The kinetic equation has been solved. by an iteration procedure: [03 [1] [2] f f + f +f + O(E6) 1,1 1,1 1,1 3~1000 9I. 000000 (9.4 ) where f1, would be the only term in the expansion were the system an ideal gas. f11 is a first correction to account for the presence of forces between particles and is proportional to system constants of the type ^VY4 (cf0 9o3); [123 f1, is a second correction and is proportional to system constants of the type'\ f\ etc. The time-relaxed momentum density is obtained from the full density, (9g4), by integrating over spatial coordinates. Some related correlation functions are also investigated, Most interesting, however, is the demonstration by an explicit example that calculations based upon a truncated kinetic equation (i.e., an equation similar to (9.1), but one which contains only a finite set of [JJ ) will be *"For ( lO.

130 incorrect if carried to an order of expansion parameters (the XY ) greater than the order of the approximation [2k] made for the collision integral. The latter would be a trite observation were it not that much effort is expended on the solution of kinetic equations which are similarly only approximations. Both the Boltzmann gas equation and the Vlasov equation are of 3,21 the latter category.' There is undoubtedly some point to investigating whether intrinsic limitations exist regarding the information to be obtained from solution of such equations. Like the momentum density, an expansion for Gs(r,t) is found in direct correspondence with the expansion obtained for f,l1 (xl;tlxo ). As a consequence of the prior ordering of the terms appearing in fl,1, Gs(r,t) is expressed as a series of terms which are successively less important at high temperatures. The expansion for the latter may be written as: G (r,t) = a t2n D[n] G0](R,t) 00000 (9.5) S n R s or, alternatively, 1 2 t2n [n] 7 R G (rt) = a t DR] e 6(R) 0oo oo0 (9o5') s n R [03 _R2/ + 1/29 [01! -R/2 In] where R = (mB/t2) r, G (R,t) = (mS/2Tt2)U e and the D are purely differential operators R are purely differential operators in R (i.e., they are not

131 explicity functions of t). Here, too, terms in the expansion are obtained to the "fourth power of the forces" (i.e., the second order correction to an ideal gas calculation.) The expansions for Gs(r,t) have been obtained in a form demonstrating the temperature dependence of f, l(l;txl ). They are, roughly, expansions in the ratio of potential to kinetic energy of the assembly. On the other hand, they are not expansions in the time, per se. By taking Fourier transforms of Gs, first with respect to the spatial variable and then also the time variable, one can obtain the classical approximations to, respectively, the "intermediate scattering function" and the "scattering function" for incoherent scattering of slow neutrons. For these, one obtains series in direct correspondence with the expansions for Gs(r,t). Here, too, the temperature dependence of these quantities is of particular interest. Whereas it is true that various similar expansions have already been obtained for these transformed quantities, 930 it is believed that the temperature dependence of the latter is more surely and explicitly demonstrated by this exposition. An additional benefit of proceeding as we have is that, because the functional form of Gs(r,t) has been demonstrated by our calculations, the task of deducing meaningful information about Gs(r,t) from the transforms is probably simplified Of course, if one is seriously interested in calculating scattering cross sections, he must find some way of relating

132 the classical scattering functions to the correct scattering functions which are defined within the context of quantum mechanics. Some of the schemes which have been proposed for this purpose were examined in Chapter six. It was concluded that they are probably valid if applied to cross-section calculations for low momentum and energy transfer, but they must otherwise be applied with care. The equation giving the correct scattering function is typically of the form: 000.0. (9,6) 3hw/2 -tfi2k2/8m Snc (k,w) = e (Scl (k,a,8)+ h2A2(k,,6)+ h4A (k.w,6) +,,) iL~~nci nc 2 4 where the {Ai} are functions of k, w and also the system constants given by (9.3). But, Scoh (kw) has also been obtained as an expansion in system constants. Comparison of all terms appearing in (9.6) which are proportional to system constants of type r[2s1shows that fi2A2(k,w,) can be neglected when compared with S?,c (kw, a) only if lnCo k and ware small; Eq. (9.6) is correct for all k and w, but to apply it in general, one should retain all terms in the expansion in t2 consistent with the expansion for Sc (k,w) inc The approximation, Btw/2 -_fh2k2/8m SinC(kw) - e e Sic (k,w) ~00 (9 7) is correct only for k, w small or B1 very large. If Sinc(k,'>,) is known to a given order in system constants,

133 then to obtain the exact cross section in a meaningful way, one must also know a large number of the A.} Is there any point to obtaining G(r,t) if one is interested only in a calculation of the cross section? The latter is defined in terms of the scattering function, whereas G(r,t) has been introduced primarily to aid in relating the scattering process to the internal motion of the scattering system. To obtain the expansions obtained in this thesis, it might be easier to calculate the scattering functions from their basic definitions, rather than from G(r,t) Suppose, on the other hand, that one chooses a model Hamiltonian for the scattering system for the purpose of simplifying the calculations, Even if in this case the scattering function is calculated directly from its quantum mechanical definition, it must be acknowledged that classical considerations enter into the calculation. The choice of the model Hamiltonian is almost always determined from some prior knowledge about the gross motions of particles in such model systems and, in fact, the arguments used to justify such Hamiltonians are usually classical concepts. For example, one refers to a particle moving within a cluster of particles or, a particle trapped in a potential well or, a particle diffusing according to the Langevin equation, and so forth. Remember that here, however, G(r,t) was of primary interest; the cross section calculation was ancillary.

134 In addition to examination of expansions in powers of the forces, investigation was made of the derivation of equations pertinent to density expansions of fl, 1 Bogoliubov type kinetic equations were derived for fl which, in contrast 1,1 to earlier derivations, are valid for systems for which interparticle forces contain attractive as well as repelling terms. Finally, Gd(r,t) was derived from a superposition approximation for the two-particle probability density, f2 2(Xl, X;tlxl,x ) It was pointed out that this 2,2 1 1 approximation is equivalent to Vineyard's "convolution approximation for G(r,t). Certainly, there are still some incomplete aspects of this research. For example, an H-theorem has not yet been established for the kinetic equations derived in chapter 3. The proof of such a theorem would be of great importance because of the general applicability of these equations. Then, too, there is the question of the secular terms appearing in the expressions obtained for the solutions to the kinetic equation ( i.e., the time-relaxed densities. ) These expansions have been useful for the calculation of certain quantities pertinent to this investigation. Furthermore, they are the exact expressions for short times that any approximate solution must satisfy. However, they are clearly unsatisfactory for extension over the entire time scale, and other procedures should be investigated for solution of the kinetic equations. In this regard, it might be fruitful

135 to employ methods recently introduced by Frieman and his collaborators. Also, it might be interesting to investigate the derivation of the equations of hydrodynamics after starting with similar kinetic equations for a more general one-particle distribution. Whereas we have been particularly interested, here, in studying the temporal evolution of probability densities for particles moving through an equilibrium assembly, analogous kinetic equations for distribution functions could be obtained for many other non-equilibrium problems. In fact, one can even obtain kinetic equations to provide timerelaxed correlation functions directly. Another task yet to be performed is a more complete investigation of the "system constants" appearing in these expansions. Can prescriptions be devised for their calculation similar to the diagram theories invented for the evaluation of equilibrium distribution functions? Let us conclude by pointing out that these calculations are, in a broad sense, only preliminary. The equilibrium and quasi-equilibrium (stationary state) properties of matter have ** For example, for the momentum auto-correlation function, (t) (0) it)' t) where e- ~~,o~ 1-ko - e %fzt - <L^<tt) + e L (-Q)L <CKiJTUa, where;- tat ^ C -; and, D A

136 been well studied, and we are cognizant of the great successes which have been achieved in these areas. Even more exciting, though, is the thought that recent investigation into nonequilibrium problems is just the beginning of a similarly fruitful period of investigation into a more extensive but less well understood class of natural phenomena.

APPENDIX A NOTES ON PROBABILITY THEORY"' 1. THE NOTION OF RANDOM VARIABLES, DENSITIES, AND DISTRIBUTIONS Consider a collection of entities ("sample space"), S, complete in the sense that all possible varieties of any pertinent property or set of properties possessed by the members of the collection are represented by some subset of S (albeit a set of measure zero, if necessary). A. 1 Definition: A property X( ), is a "function defined on the space S "if to every member s of S there is a real number, denoted by X(s), which is called the value of the function X at s e.g., Consider a set of cows, C, and consider the pertinent property to be the color of the eyes of the individual cows. If, for example, the number 1 is assigned to the property: blue eyes, 2 47: red eyes, 3 4= green eyes, etc., then eye-color (E(-)) is a function defined on the space C. A. 2 Definition: A property, X(~), is said to be a random variable if it is a real valued function defined on a sample space on whose subsets a probability function,/4p (o), has been defined. The probability function (measure) is an a priori'weight' assigned to the members of S. This notion will become more familiar, below. Ref: Parzan49, Feller,50 Doob'51 This presentation is similar to the discussion found in Parzan's book, and is included as an aid for those readers who might be unfamiliar with some of the terminology used in the main text, above. 137

138 A. 3 Definition: The probability function Pr X() of a random variable X is a set function defined for every set of real numbers. Its value is the "probability that X is in B". The probability measure of X, Prx('), is obtained from a probability function Ap (.), defined on the sample space, by summing up the measure of all subsets of S such that X(s) = B. i.e., e.g., If we assigne equal measure/i (e), to each cow in our sample space, C. then PrE(') is the probability measure for a given eye-color = /# (s: E = given eyecolor) = NE/N, where NE = the number of cows having a given eyecolor, N = the total number of members of the sample space. However, the choice of the basic probability measure is not unique. It is an a priori assignment, postulated for any given problem in accord with the relevant Logic. For example, with reference to the sample space C, one might have chosen the probability measure to be proportional to the weight of the cows involved. In this case, PrE(.), also, would be proportional to the weight of the cows having a given eyecolor. A.4 Axioms: For a measure to be a probability measure, it must satisfy the following axioms of probability theory: Ax. 1: /lpr (s)O0 for every subset s6SS. (As a consequence, Prx(. )70 for every event.) Ax. 2: \ aul, S) s l ( -T\ & Pai - ) Ax. 3: For subsets S1 and S of S: Isk1%v \ SiS

139 A.5 Definition: A random variable X(.) is discrete if Pr (.) is>O for at most a denumerably infinite number of points, x, and if i p)='1. px(x) is the probability that the observed value of the random variable X will be equal to x. A. 6 Definition: The values x of a discrete r.v. for which PrX(x)> 0 are called probability mass points. A. 7 Definition: A random variable is continuous if it possesses no probability mass points. For a continuous r.v., the question, "what is the probability that X = x? " has no meaning. Rather, one must ask: "What is the probability that X is within the range B, i.e., what is P,(B)? " Prv(B) can be obtained from a function fx(x) by integrating the latter (in the Lebesgue sense) over all points x~B, i.e., (Note: i. \ p ), A.8 Definition: fx(x), defined in the above manner, is called the probability density function of the random variable X. Note that fx(x) is never negative. e.g, An example of a probability density function is the normalized neutron density, f(E) = n(E)/n (such that f,, (c(Q&( is the prob. that the energy of a neutron, chosen at random from the assembly, is in the range E'.)

140 A. 9 Definition: The distribution function of a random variable, X, denoted by FX(.) is defined by - <,0, 0x1- $) (In the case of a discrete r.vo, Up.^x* ) - But, if integration is taken in the Lebesgue sense, then where, for the discrete case: Note that FX(x) is continuous if X is a continuous random variable but it possesses step distontinuities if X is a discrete r.v.) e.g., If the probability that X = x is 1, then 0 if x< x 1 if x>x o Also, since fx) = t F(x), thus fx(x) = (x-x) It is seen that F (x) is a never decreasing function of x; X - loo F (- oo) = 0; F (+ oo) = 1; f(x) dx = 1. 2. MULTIVARIATE DISTRIBUTIONS AND DENSITIES, JOINTLY DISTRIBUTED r.v.'s A. 10 Definition: Two or more random variables can be defined simultaneously on the same sample space. In such a case, the random variables are said to be "jointly distributed". Consider, for example, the particular case of two random variables, X and X2. In analogy to the previous discussion, one has 1 2

141 Pr 1, X (B) =Pr / (s in S: VX(s), X2(s)lt B) for the joint probability function. It is the cumulative measure of all points s S such that the two dimensional phase point, (Xl(s), X (s)) associated with each s is within the set B. A. 11 Definition: The joint distribution function of the random variables X1 and X2 is defined by: Fx, x I(xl, X2) = P\ s: (X1 < xl X2 < x XI.P1 _ Pr _ = Pr (X, < x1 X2< x2) Notice that lim F (Xx 0;similarly v. - 0.-.X 1,X2 Also,' 2 A. 12 Definition: Similarly, V (,,y) - (Xr is the marginal distribution function of the random variable X2 corresponding to the joint distribution FX1 XZ. (A similar definition holds for L V- -V nX), VX2' ^ Io^ ) 0 A. 13 Definition: In the case of continuous random variables, Pr x x(B) fx, (xl, X) dxl dx2 B where fX X2(X1,X2) - "joint probability density" Note that f 2 > for all (x1,x2), and that fx (x(,x ) 1 - X- - 1 9 22 ~ --- F X XZ,X ). 1'l^ l' 2 z2

142 A. 14 Definition: The marginal probability density function of the random variable X1 corresponding to the joint distribution FX X is CPO 1 2 -oO It is seen that -. I>. o 5= c Remarks similar to these for discrete r.v.'s follow in an obvious way. Also, generalizations to multivariate distributions of dimension greater than 2 are easy and shall not be mentioned here. 3. INDEPENDENT RANDOM VARIABLES Let X1 and X2 be jointly distributed, with individual (marginal) densities FX and FX, respectively, and joint distribution FX1 X. A. 15 Definition: X and X2 are independent if PrS Xle B1, X2 B2I = Pr X1E B1 x PrX2E BZ2 i.e., if there is no dependence between the values taken on jointly by X1 and X2. As a consequence, for independent random variables, e.go, The space occupancies of the particles of an ideal gas are mutually independepe nt. Indeed, we have (x).^) (.) for the ideal gas. -

143 4. CONDITIONAL PROBABILITIES AND CONDITIONAL DENSITIES, ETC o A. 16 Definition: For discrete random variables, given two properties, A & B, defined on the same probability space, the conditional probability of property A, given the property B, is defined as: ZvS A\ts\- \' t\W'm/WAwl,';P R> O For continuous r.v.o s, the above definition is going to lead to difficulty since, if we try to define Pr\A\X1 - Pr A, //PrXj, trouble arises because Pr X = x= 0. Instead: A. 17 Definition: The conditional probability distribution, P )A\X = xt, for continuous variables, is defined by: 5gt(LS:YSo QS X- e Y.,(S)C ss Pi (< 6 j^p^ X\;r'^C,^ ^ \ I Ais&'- S fpw^t -'U')A l tX ) 3N Note that V,%= \ \\X:^X\ 0. Furthermore, if A and X are independent, VPAa\ p4, - \ and the above equation is trivially satisfied A. 18 Definition: The conditional distribution function of Y, given X, is defined by: Fy x(y/x) - PrlYoy X = xl. Consequently, th^X~-y La=^ (y(^) \,,;t ^ ^\sh(k t).

144 A. 19 Definition: The conditional probability-density function of the random variable Y, given the random variable X, is defined by: fy x(ylx) F y\X(xly)~ We are thus led to the important relation: Ylx(yx) = X, y(xy)/ X(X) Proof: In light of A. 19: > IC<,-\ [\9c\.X). Next, differentiating re y: re Ye )"I v I( = b\% tst5 ~ t v Also consider: Proof: The "conditional probability" of realizing an event A, given that Y = y', Z = z', is defined by the following relationship. ^\t+Xt l-X~i% -- \\ S^(A\\ —5 \l''l t 1'\\) —'J-'T=s3'v ty~t, y-~^^z% ^V But, if B is such thatN-( oo. o, then?.A s.It.kY,. f t -y -,\S~a.Yt'fe where B' = range of Y. Thus, Now, suppose that B is the ange d A i the even Now, suppose that B1 is the range and A is the event within the range A = 5- 0 \o ~ We thus find the following relationship for probability distribution functions:.c ~ ~, \ 3 -co-D..'

145 If we next take derivatives re x, y t, AA' where f ^ 6 — % [llas, - conditional probability density. Hence, since fX y(x,y) = fX y(x y) ~ fy(y) $x,, y (s) =Z k @ Ay ) tX \ (\. &. 5. EXPECTATIONS Given a function of the random variable x: g(x). Ao 20 Definition: The expected value of g(x), with regard to the distribution F (x) is defined as: Alternatively 4ao 4a Similarly, if h(x,y) is a function of the random variables X,Y: iL>, =- ~\ \^( +- - \A &\ ^ l. = \\ \V^ ^U>)t~s,%L, _oo Notice, if h(x,y) = hl(x) ~ h2(y), and if x and y are independent: 4h(x,y)> =<hl >. < h2

APPENDIX B REDUCTION OF COLLISION INTEGRALS As noted in the main text, upon expanding the collision integral according to 4.15, one finds it convenient to write: 5\54 -51 9t\ - \t e \-.:1 2.Ac- (B. 1) according to the definitions: as -e. Q\^\ t1 6^ 2 LOLo Si)- (B. 2a) \I iiQ\ 9\^ L\<e 1 Z Loll Ln -rmll~~ — 2 (B.2b) 3^ _ f.1 (U, f- (B.2c) H 2t &toKt, Vt L4 -- (B. Zd) <^'1 -f (' t tul 1 - t\aKa We 1Xe~t~t\:t,'5 _ (B.2e) In addition, expand 146 Ar

147 t 1 g~t +tt (S)\_tt,2 L \_t$ t..._-. _ (B. 4b) C.tt tt L, LI U -.) _.j - (B 4c),, combines with one of the terms of 1 to provide the'basic' o are shnc' collision integral term Q (cf., 4.32). A, 4- 4, are shown to yield,, defined by 4.37. The other of the nine terms (,...., ~q ), defined by equations (4.32), (4.37),(4.40 - 4.46), are obtained in the same way. The following is an abbreviated exposition of the procedure by which these terms are obtained. One expands,.~c1 o —T- i4% -s l __o AeJ W ^ -'' ^"' _ (B.3 5) with (B. 6a)'Vl l, (&,) (B. 6b) (B.6c) etc. It is then found that (cf., Eq. (4.40), (4,41) a, t ~Z. (B.7)

148 7r - t1 2 \/- (B.8) Similarly, expand A7> "5 -'-IL" -+^~ -\- 91-k-(B 9) where \,9'. la, k - f - ~~~~~~~~~~~ \.... U o (B. 1Ob) <~> \ _ ^ t <C (^ t-tqA <. L 0. > l- - - - (B. lOcb) u B. 1(B.10d) Then, (cf., (4.42) _) ^\x o,l \'N1 t\ $6 3'& y U ^ ^, - - _, (B.11) t) where 9', is obtained from (B. 2c) by an obvious definition. c'1 \3AW1 o~0 o~ - tand are obtained from the expansion for viz. - 1s \. t ^ ^^ + -- (B. 12) where Ko 1,, K1 L TJ3~, B1,)

149 (E5-,o \. rL.~ tL 0 Li\ L L'j^ SIt^ $Lil~oSQ!x.)L1 ^l K~ (B. 13b) ^''^ 0. etc. For 9,(f,4.43) we have: n c 1 - ^ 1 (B.14) b - 3- 7,> + z - - - U ^1 "M l41 34 where % is obtained from an expansion of B.2d. " and;. follow from an expansion of'y, defined by: 4 1 -C a' 11 (Tf.'o)L <Lt \,a o^ (1- l LIDLI kC,) (B. 15) Similarly, defining (cfo B. lOb) >fr~ ~~ 1 \ ( 3 o ( (B. 16) we have f l io 1t1 h - - (B. 17) Finally, kj% o~- o oo41' q (B.18) QCq oo -.c'''t4 L(B.19)1 -_- - - - - _ (B. 19)

150 with i3,0 - uI tE -BI; i~~9 l~g~it ",? $>\\, (B.20) QUU1 01 ul_ 2i4, 7, L 4^ ^' LZ (..,t-,, S'-''', (B. 21) W^^013 ~p l -. \eO\ 1 -^^ *^ ^.,, I I, ^Q1_.. (B. 22) All terms to i(E) are included in i'7.. o Let us conclude by performing as an example, one of the partial integrations somewhat more obscure than those provided in the text (cf. Eq. 4.38 ff. Consider, Z, t~ 94 t'-_i) c1,.l^J ^^^ which is to be compared with Notice that W ~'; where -\ so that /t3'*"'91- 9- L\i -t^ L T2 i L-,L, r\ K-LIl\ —-) Integrate by parts over and thens:'4?-^.^A'^'^^- A'. t^

151 Next, integrate by parts over 1', Q ~* 2 L ^ - r L + -?'% R — e ~L.,:, q:-,,,X <x'\ ~:, % (B.18). 0 L. - C r Lq",-.. L~ -~*,; i-V, -* oThus, we define one of the'i' llii'' integrals +, Z T i.ik', -,(C e. 2 t-^ OE^ C\,f^-'i

APPENDIX C RELATIONSHIPS BETWEEN SYSTEM CONSTANTS We shall prove that K -,; it is expeditious to evaluate these coefficients in spherical coordinates. Thus, X E\& %4) V i - _.W f \g 52 1' t (C. 1) I' a + Hence, since e SA.AA' c a - u ( <S ~-,;a'pO-n Next, performing the integration over5and(Q one obtains ^^<) a \3ys),t'oA(C. 1') 152aPp-~~ c~p~l Ssc~C co~

153 In the same way, one shows: It is quite straightforward to show the validity of the other relationships stated on page 4-20. Their proofs are essentially no different from the one here described.

APPENDIX D CALCULATIONAL AIDS TO CHAPTER 6 Table D-lg Some Useful Integrals Proof eNi~i,: ^/'51~~ <_a c,1 i ^ e R,,_: -s _ t\r i G ( 6.t- AS)-'C > + 1 _ A, _s -t.10,, \ *Proof: 154 \~ ---— a6A'S Al

155 (2) Table D-2: Contributions to Gs (r,t) Term in Collsn Corresponding Contribution Integral to G (r,t) totU~~~LS~S r#A $0 s o -^ \r^ P (O tvAC f] 1~k~ c, ~A 4% _ ~ ~ ~ ~ ~ ~~~~~~~v i^:;S ty it _ c ____~_. — 1 — -L ~~C —C*~I I _____ -__-_ ___ -__ ____I — ^v ^^^g^^^ QC ^^^ t'^ f.^" t' ^>'1-v?^ -'' ^'>.Z^.__^r5 ~s ^ ^o^ i5 ( t,^ ~~ ~~Qt~'c"^ 02^ 0 ^~~~~~~c -w 3 l:

156 Table D-2 (Cont.) Term in Collsn Corresponding Contribution Integral i to Gs (r,t) i- ^ t ra,. iQv V,5 I ^- t' s7 \ ^ I r WI _ it~,,.%,. -l, I'\~ I~ -,,,,,-,< 1A,.-, /\ -t i,^ 1G L5 %0O WIe Y^N'R. t1 ~~ V^ M Vol.- 7 t - th's40 Xs'%' -, t'.. ~~ 2, ~~3 1 U1SJoS

APPENDIX E THE FUNCTIONAL CALCULUS In the derivation of kinetic equations appearing in Chapter 7, notions of the functional calculus play an important role. For this reason it is probably helpful to consider some aspects of functionals and their derivatives. 1. "Z is a functional of the function x(t) in the interval (a,b) when it depends on all the values taken on by x(t) when t varies in the interval v., (a,b)." Thus, the functional is a generalization of the concept of a function of many variables. Ordinarily, one writes a function of the parameters o0,, - in the following form: Z(ot,,..... Changing the notation, Z can also be written as: Z(u, u,,..,). Now, suppose the set., % % is a continuous set of infinitely many points in the interval (a,b). The set of variables lu, ~ f,... } can be represented by the function u(t), te (a, b). In this way, Z is a functional of u(t), and is written as Z( (k]), Similarly, one can consider quantities which are functionals of more than one function, say, u(t) and v(t) and which are, also, functions of parameters y1, y2,.. Such a functional might be written in the following form: F(yl, y. ~ u(t), v(t))) Volterra, 52 Chapter 1, Section 1. 157

158 2. With regard to derivatives, too, the theory of the calculus of functionals is a direct generalization of the theory of the calculus of functions of many independent variables. For ordinary functions, the partial derivative - is related to the total differential through )u{ 3a+ $ ~I SU &;. In the theory of functionals, the summation becomes the integral, IL %W) C% - 16? ~% and leads to the following definition:* F [UJl; ] D defined as the first derivative of the functional F with respect to the function u(t) at the point, is determined by the following relationship: r r -. Susl) Another definition, involving somewhat more rigor, is stated as follows. In the neighborhood of a value xO, let y(x) be given a continuous increment (x) which does not change sign, and such that \ 4(}\ (, for x -h< x < x +h and 9(x) = 0 for x otherwise. Form the limit o - -- o G b 6 where FF = Fly(x) + (x)] - F[y(x)] and 6 = e(x) dx. If M and if LV approaches (uniformly) a limit ase and h approach zero in an arbitrary manner, then the limit is defined as the functional derivative of F y(x)] at the point xo: F'[y(x); x o. Hence, the definition is analogous to the definition of the ordinary partial derivative: * ibid., chapt.I, sec. II.

159 3y; \ryo h iS0 - We note that F' [y(x);x ] is a functional of y(x) and x 3. In general, by applying successively the rule for to the functionals G = -, etc., one can show, as a logical consequence, that the appropriate defining relationship for the nth functional derivative of F is: bb \ 4. Let us now make use of the definition of the functional derivative to derive some relations which shall be useful in the ensuing discussion. a) Suppose F [u] = a(x) u(x) dx. The\)/F = F [u(,x+ C\( x)] - -F[ u ()] = a (x) \(x) dx + tQ ) so that the definition,'~_ o'IE ~ F AS)~ ~ implies: -..) b) Suppose F \uj = u(y). Therefore,.L*~\i]Cl q'Y\)i\- x\I. Hence, i; _ _

160 c) Product Differential Rule for Functionals: Suppose F = Aju(x)l * B [u(x)j, where A and B are both regular homogeneous functionals of degree 1 *, viz., A 1u(x) = =\a(x) u(x) dx and B u(x)= S b(y) u(y) dy. In this case, V t -- V \ i1 - - Lt \<-)cY1A(u$<44,<^ Y (uE l~ ~- \\^W^{^\ ^^b, CO4ab(^ ^U^Q) k +~C&t) _. ___ k 0W r Qu 6z d) Suppose F.u(x)l is a regular homogeneous functional of n degree n, i.e., Fn ux= (x)1 ia(xl,..,xn) u(x1)...u(xn) dxl..dx n degee, n i n n n 1 n and, in addition, the kernel a (x1,....,x) is symmetric with respect to permutation of its n variables. Then, ^, - s1U+^1-ts l * ~\..\ax3.Y \.(,^V.w4+ A functional of the form:; ( r\, b, is called a regular homogeneous functional of degree n.

161 Hence, due to the symmetric property of the kernel, a, the first functional derivative is ae i,) c Repeated differentiation yields, for the sth functional derivative: i S, Ah \ \^l*(*^ v''**^ ^s'' ut&. ~c

APPENDIX F THE BBGKY HIERARCHY By successive integration over the coordinates of the particles of the assembly, the Liouville equation can be transformed into a hierarchy of coupled integro-differential equations, each of which provides the temporal evolution of the s particle probability density in terms of the s + 1st density. For an isolated system for which the internal potential energy can be considered to be the sum of two-body interactions, the Liouville equation may be written as: e>ii,^ an * ^ A _,, (F.1) where %" -\ t/ is the Hamiltonian of motion and -,* is the internal potential energy of the assembly. Let the n-particle density be designated as t,_~ ~ lo, as a reminder that the system is symmetric under interchange of designation of any pair of particles, except a pair in which one of the members is particle nurmber 1. Integrate the Liouville equation over the coordinates x +l....,x - -s+ 1 -n One obtains: In other words, all particles of the assembly are assumed to be of the same type; however, by focusing attention on particle #1, the initial probability,\,('L, 0;O o is not symmetric under interchange of the variable x1 with any other x%. On the other hand, ~t'_.^(v, oio% is invariant under interchange of the designation of all other particles; %,a;"S'\ will therefore have a similar symmetry. 162

163 Xt s-,_ - ^ (F. 2) where,t\'L\) is defined by Eq. (3.2) of the text. To derive Eq. (F. 2) we have noted that \\& V Vi;;, ^ = O and that, also, \XakY C\! ^ Now, because of the symmetry properties of f ( X; tx n,n — n x-1, the previous equation yields the hierarchy: _ -^\kA i<-Y s^ -** 5,<+ $e~s1.,(F.3) On the other hand, suppose we had specified the initial positions of two of the particles. Then, in contrast to the usual form of the equations of the hierarchy (F. 3), the following set would be obtained: i 8 (X&'1 \Us,>^ 1 ^nn~71 V^. ^, 3l~ t\XowLo)X-)C 1.h, 0 (1t ('t \ t; oo + *-0 - (F. 4a) It^ 3,0 ")8^ 2) ^^, 1 33t,\' t u ~~tL ~~zr)V~

164 t;F /- 4b ]\L3 t -,3 W.^ - < ^'L, (F;\' }1~LIVSLS. anal, yses^. Both sets, (F. 3) and (F.4) are pertinent to the main text of this analysis.

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