On the theory of classical fluids

IL NUOVO CII~ENTO VOL. XVII f , N. 1 lo Ottobre 1960

On the Theory of Classical Fluids (').

L. VERL]~T

~Laboratoire de Physique Th~orique et Hautes E~ergies, Fa.cult~ des Scie~ces - Orsay

(ricevuto il 6 Maggio 1960)

R ~ s u m ~ . - - On montre que la sommation d 'nne large class de dia- grammes dans un d4veloppement du type ~, cluster,> permet l '6tablisse- ment d 'une 4quation int6grale pour la fonction de corr41ation dans les fluides classiques. Apr~s une approximat ion, cette 5quation se rSduit £ eelle de Born, Green et Kirkwood, et elle reprodui t plus exactcment que eelle-ci les coefficients du viriel. ()n donne les termes correctifs, de plus en plus c0mpliqu4s, qui permet tent de rendre exacte notre 6quation. On calcule enfin la fonction de corr41ation h trois corps, qui, introduite dans l '~qu~tion d 'Yvon-Born-Green, permet de cMcuter la fonetion de corr61ation k deux corps. On confirme ainsi lc calcul direct de cette fonction, et on montre que notre 4quation int4grale inclut des corrections

l ' app rox ima t ion de superposi t ion.

1 . - I n t r o d u c t i o n (**).

The p u r p o s e of th i s p a p e r is to d e t e r m i n e t h e r a d i a l d i s t r i b u t i o n f u n c t i o n

s(r) fo r ela, ss ica l f luids. K n o w i n g th i s func t ion , t h e t h e r m o d y n a m i c a l p rope r -

t i e s of h o m o g e n e o u s f luids in e q u i l i b r i u m can be deduced . I n p a r t i c u l a r

t h r o u g h t h e use of t h e v i r i a l t h e o r e m , t he e q u a t i o n of s t a t e is o b t a i n e d

(1) ~ = 1 - - - . o~ K T 6 T (r) ~ - r d r ,

(*) Supported in par t by the United States Air Force through the European Office, Air Research and Development Command.

(") At the moment when this paper was ready to be sent, Prof. Yvo~ kindly com- munica ted us prepr in ts of a w o r k due to Dr. E. M~ERO-', which is in many points s imilar to the present pa.per.

78 L. VERLET

where p is the pressure, ~o the densi ty (9 ~ 5V/V), K the Boltzmana~s con- s tant , T the absolute t empera tu re . We shall l imit ourselves to the case wehre

V(r) is a spherical ly symmet r i c potent ia l . The internal energy per particle, E, is given by the sum of the kinetic and

po ten t ia l energies

(2) E : 3 KT +

The theory of fluctuations enables us to establish the Ornstein-Zernicke relation (') which gives the isothermal compressibility through the equation

(3) K ~ \ ~ } ~ = ~ + (.~.(r) - - ~) d r .

The calculation of the correlat ion funct ion presented in this paper is based

on par t ia l summat ions of the cluster expansion (~) of the correlation function, which is expressed as a sum of ((cluster integrals ~) involving the funct ion

~o(Ir~-- rjl ) -- exp [ - - V ( i r ~ - - r~ i ) /KT] - - 1 ,

which approaches zero when Ir~--rb] tends to infinity. I n Section 2 we build the correlat ion funct ion by summing the te rms made

f rom simple chains of ~0 functions (see Fig. 2). Such an expansion has pre-

viously been considered (~); we repea t this calculat ion so ~s to introduce the

necessary notat ions. I n Section 3 we make an appl icat ion of the chain expansion scheme to the

equat ion of s ta te of p lasmas a t high tempera ture . When the asympto t ic form

of ~0 for large distances is used (4), Debye-Ht ickel laws (~) are obtained. I t will be Shown t h a t a b e t t e r approx imat ion for ~0 can improve the Debye-

Hiiekel laws for finite densities. I n the next paragraph , we introduce a generalized chain expansion in which

more and more chains are graf ted one over the other (see Yig. 6). A consistent

procedure for taking into account more and more complicated te rms will be established. The phi losophy involved in this approach is the same as 5fo-

(') L. S. ORNSTEIN and F. Z~RNIC~]~: Ams~e~dam Proc., 17, 793 (1914); F. ZER- ~'ICKE: Amsterdam Proc., 19, 1520 (1916); L. S. 0R.~'STEIN and F. ZERNICKE: Phys. Zeits., 19, 134 (1918); 26, 761 (1926).

(2) H. D. URSELL: Proc. Camb. Phil. Soc., 23, 685 (1927); J. E. MAYER and 3[. G. MAYER: Statistical Mechanics (New York, 1940); W. G. McMILLAN and J. E. MA~-]~R: Journ. Chem. Phys., 13, 276 (1945). See also ref. (a) and the excellent review paper of E. SAL~T]~R: A)~. Phys., 5, 183 (1958).

(a) j . E. MAYER and E. W. MONTROLL: Jour~. Chem. Phys., 9, 2 (1941). (4) j . E. MAYER: Journ. Chem. Phys., 18, 1426 (1956). (~) P. DEBYE and E. Hi)CKEL: Phys. Zeils., 24, 185 (1923).

ON THE THEORY OF CLASSICAL FLUIDS 7~

r i t e ' s (6), bu t we include some te rms which have been left out b y h i m , so tha t our results are different. A detailed comparison is made in the Appendix.

This approach will lead us to an integral equation (*) for the correlation

funct ion (Section 5). We shall establish the connection between the elements of our theory and some general relations holding for fluids, such as the Ornstein- Zernicke (1) and the Yvon (7) relations.

We then discuss the viria[ expansion obta ined f rom the integral equatiol~

of Section 5 and the generat ion of the t e rms of the cluster series which are

obta ined when this equat ion is solved by i terat ion, with the zero-densi ty corre lat ion funct ion as a s tart .

In Section 7, we show tha t the equat ion obtained when the Yvon-Born- Green set of integro-differential equations (s) is closed by the superposit ion

approximat ion (~) cart be reached by approx imat ing our integral equation. ~iVe show, in Section 8, how the correction te rms to the integral equat ion cart

be obtained. These correct ion t e rms involve more and more complicated integrals, bu t it is hoped tha t the simplest of these te rms can be computed. I t will be seen t ha t the effect of the correction te rms can be obtained, in prin-

ciple, f rom the exper imenta l de terminat ion of the correlation function.

I n the last paragraph, we calculate the three-body correlat ion funct ion: i t will be expressed as a sum of te rms involving the two-body correlation function. We show that , pu t t ing this three-body correlat ion funct ion in the Yvon-Born-Green equation, we obtain the two-body correlat ion funct ion de- r ived in the present paper.

2. - The c h a i n a p p r o x i m a t i o n .

The two-body correlat iou funct ion in classical mechanics is given by the expression

(4) s(1, 2) =

v2f~<l~ exp [ V(i, j)] K T ] d 3 " " d N

~ f i d l . . . d N

where we use the nota t ion i, j : I r ~ - - r j 1 .

(6) T. MORITA: Progr. Tbeor. Phys., 20, 920 (1958). See also E. MEERO~': Phys. Fluids, i, 246 (1958).

(') After this paper was written, we realized that this integral equation was con- tained, with different derivations, in the paper of Meeron quoted in ref. (~) and in a. paper by J. M. J. VAN L~uwE~-, J. GROENVELD and J. DE BOER: Physica, 25, 792 (1959).

(7) j . YvoN: Suppl. ~'uovo Cime.~to, 9, 144 (1958); Aclualit~s scienti]iques et indu- strielles (Paris, 1935), no. 203.

(8) M. BORN and H. S. GREEN: Proc. Roy. Sac. (Lm~don), A188, 10 (1946). (9) Z. G. KIRKWOOD: Jour~. Cke~t. Phys., 3, 300 (1935).

~0 L. VERLET

,(5)

As we said in the introduction, we define the quant i ty

J)] }o(i, j) = exp K T ]-- 1,

which tends to zero outside the range of interaction. I t has been shown (2) tha t (4) can be writ ten as follows:

V(I ,2)1 {6) s(1, 2) = exp ~ ] exp [~=,~ 9~vfi~v(1, 2)] .

There fl.~.(1, 2) is the << simple 1, 2 irreducible cluster integral ~) defined by the following symbolic formula, the meaning of which will appear below:

(7) fi~(1, 2) -~ ~ . d3 ... dN + 2 ~ 1-[ ~=o(i, ~).

For convenience we shall diagrammatically represent each ~o(i, j) appearing in (7) by a line joining the points i and ]. The products of all these ~o must be made. The sum sign indicates tha t one should sum over all the distin- guishable diagrams leading to the same integral which can be formed by joining the N + 2 points by }0-lines (the points 1 and 2 must not be linked together).

A diagram is considered as reducible:

a) if a par t of the diagram is disconnected;

2) if a par t of the diagram is connected with the rest through one point only;

-~-V ,vie

Fig. 1. - Redue- ible diagram.

3) if a part of the diagram can be separated from the

1 2 Fig. 2. - Chain

diagram.

rest by cutting a line joining point i and a line joining point 2 ; (the exponential in (6) reproduced this class of reducible diagrams).

To illustrate these prescriptions we have drawn a reducible diagram in Fig. 1. The A, B, C parts of the diagram violate prescriptions 1, 2, 3 respectively.

In this paragraph we shall consider the simple diagram of N + 2 points built by joining the points 1 and 2 to the 5 r other points by a simpte chain of ~o, which we shall call a << primary chain ~) as shown iu Fig. 2.

There are N! possible ways of forming this diagram and by (7) the corresponding ('luster integral is

(s) fl°(1, 2) = f d 3 ... d N + 2 ~:o(1, 3)~o(3, 4).. . ~o(N-~ 2, 2).

ON THE THEORY OF CL.a.S$ICAL FLUIDS 81

Introducing the Yourier transform of to

(9) ~o(k) = ~fexp [-- i k. r] to(r) d r ,

one has

(10) O~fl~(r) -- (2~s ° xp [i k-r] ~o~+~(k) dk.

Summing over all such diagrams, we introduce the function

(11) yo(r) = ~ ~fi~(r) - - ,~ 1 fexp [i k'r]Fo(k)dk,

with

Eo~(k) (12) G(k) - 1 - - E o ( k )

Using this approximation in (6), we then have

(13) So(r)= exp [-- ~--rT)l exp [yo(r)].

3. - An illustrative example: the equation of state of a high temperature plasma.

We shall consider the case of an electron plasma, at high temperature with a uniform background of ions.

We have ia this case for the electrons

(34) to(r) = e x p [ - - ~ J - - l ~

~vhere

e2 (]5) ~ _

KT"

We have plotted in Fig. 3 (curve (1)) (r/~)to(r). The calculation of the equation of state with the chain approximation has

:already been made by another author (4) who used for exp [-- a/r] its asymptotic expression for large values of r.

One has in this case

6 - I I N u o r o ( ' i m e ~ t o .

82 L. YERLET

(curve (2) of Fig. 3, represents (r/~)~Ao'(r)) which, with the help of equations (9), (12) and (11) gives

( [ r) (17) y~.(r) ~ 1 - - exp -- .

r

There h is the Debye length: h = (4~9 ) -~. This gives for large values of r

{2) - t ~ ' ( 3 )~ - - - - - - - - ' - ' - " - "

c/a

Fig. 3. - Curve (1)." (r/~)~o(r). Curve (2)." (r/e)~'~,(r). Curve (3): (r /e)~,(r) . Curve (4): (r /~)(s~, (r) - - l ) for ~ = 2h. Curve (5):

(r /~)(s(r) - - l ) for a=2h.

( l 8) .~'-~,(r) = 1 - - - - e×p -- ,

( r / ~ ) ( s ~ ( r ) - - l ) has been plot ted for h = 2~ in Fig. 3 (curve 4).

Using formulas (1) find (2), taking into account the uniform backgroun of positive ions, the Debye-Httckeld laws are obtained:

o K T " \ K T ] '

(20) K T -- 2 (,no)~ .

We shall now make an approximation somawhat be t t e r thai1 (16), namely:

(21)

(rla)~o ~ has been plot ted in Fig. 3 (curve (3)) and one sees tha t it is fairly near to the exact curve (curve (1)). 5[oreover, the val idi ty of approximat ion (21} does not depend on the density. I t has been chosen so as to be able to perform analytically the Fourier t ransforms (9) and (11). We get in this approximat ion

(22) s(r) = e x p - - + exp -- + ~ / 1 - - 1 h -2

where

(23) A~= = ~ 1 (1 :]= ~ / 1 - 4h -~)

( r / ~ ) ( s ( r ) - 1) has been plot ted in Fig. 3 (curve (5)) for ~ = 2h and can be com- pared with the asymptot ic value obtained above (see Fig. 3, curve (4)). We

ON TIIE THEORY OF CLASSICAL FLUIDS 83

express the correction to the Debye-Hiickel law in the form

(24) P -- 1 1 1[ e~ '~+

E 3 (~o)~ (1 + fl). (25) K T - - 2 - K T

Comparing equations (24)-(25) with the Debye-Hiickel laws (19)-(20), we see tha t fi is a measure of the correction for non zero densities intro- 0.2 duced by the more exact t r ea tmen t fl ( ~ ) / tha t we have just m~de. fl is plot- 0.~ (21 U T, ted against ~/h in Fig. 4 (curve (1)). 0 " / ~ ' 1 ~ / - S L y - I t shows tha t the correction to the < L / Debye-Hfickel laws is small for -0.1 values of a smaller than the I)ebye length. - 0.2

The same conclusion was reached Fig. 4. - The correction factor against o:/h

by ABE (lO) who summed the chain- as given by the present theory (curve (1)) like terms in the cluster series for and by Abe's calculatidn (curve (2)).

the pressure and made an expansion in the density. Abe's result for /5 is plot ted in Fig. 4 (curve (2)). I t agrees with our result in the low density region.

4 . - The generalized chain expansion.

We shall now include more terms in the cluster expansion by a na tura l generalization of the chain diagrams: to a chain diagram built as above, where

o) b) c)

Fig. 5 . - Typical elements replacing ~0(i, i + 1).

between points i and

"-" (Fig. 5, a), we shall as sociate the diagrams of

2 type b (Yig. 5, b), where

Fig 6.- A primary chain in addition to the ~:o te rm constructed from ~ . ore or more chains s tar t

f rom i to go to i + ] , and the diagrams of type e (Fig. 5, e), where between i and i + 1 there is no $o bu t at least two chains start ing f rom i and going to i + l . We shall call such chains, grafted on the pr imary chain, secondary chains.

Le t us consider a diagram resulting from the operations described above on a prim,~ry chain of h r points Wig. 6). We shall sum all the diagrams which

(10) R. ABE: Proffr. Theor. Phys . , 22, 213 (1959).

84 L. VERLET

can be formed by considering all the terms of the type a, b and c, situated between i and i f l . Let Jf be the number of points of the initial diagram (including points i and i+l). The cluster integral corresponding to the diagram in which points i and i+l are joined by a to (diagram of type a) will be of the form

(26) p", =/d3 ... d ~ + 2 5,(i, i + 1) - .F,(l, 3) ... Ft(i - 1, i) Fl+l(i + ll i + 2 ) ... FN(Ar+ 2, 2 ) ,

where F, ... P, are cluster integrals resulting from the integration over the M - N variables belonging to the secondary chains. The factorial in defi- nition ( 7 ) disappears when the identity of the Jf points of the diagram is taken into account.

Let us add between i and i+l I chain5 of XI, AT2, ..., X I points each, thus forming a b-type diagranl between i and i+ l . We have altogether M=M+ +Nl+ ... N , points in the resulting diagram. The identity of the M, points leads to a factor M,! compensating the l/Xb! coming from (7). I t has to be taken into account that the permutation of the Z c b ~ i n s leads to identical diagrams, so that the cluster integral in thii caw i h written

The summation of all b-type diagrams included between i and i+l is thus taken into account by adding to the ,",(i, i+l) which comes in (26) the contribution

A similar reasoning yields for the contribution of the c-type diagrams included between i and i+l

Thus we see that we take into account a11 the secondary chains gra'fted on the primary chain between i and i f1 if v e replace the &(i, i+l) of the primary chamin by 5,(i, i+l), where

(30) $,(i, i + 1) = (E,(i, i + 1) + 1) esp i + l)] - 1 - %(i, i + 1).

ON T I I E T H E O R Y OF C L A S S I C A L F L U I D S 8 b

Tak ing (13) and (5) in to accoun t , we have

(31) ~(i , i + 1) = So(i, i 4- 1) - - 1 - - 7o(i, i -- 1) .

This r ep l acemen t can be on a ny line of the p r i m a r y chains.

Thus we are led to a a i m p r o v e d a p p r o x i m a t i o n for the

cor re la t ion f u n c t i o n : ~(r) is now cor~structed f r o m ~ chains

as So(r) was cons t ruc t ed in Sect ion 2 f rom $0 chains. We

can go to the n e x t order of a p p r o x i m a t i o n b y consider ing

d iagrams in which t he secondary chains are eoas t rue red

f r o m $~ ins tead of ~0 (see Fig. 7). I n the prim~try cha in

each ~0(r) is t h e n replaced b y

1 2

Fig. 7. - A primary chain constructed

from ~ .

(30') ~2(r) = ~o(r) + ~o(r)(exp [7~(r)] - - ]) 4- exp [y~(r)] - - 1 - - 7~(r) :

7~(r) represents here the sum of the chain cons t ruc t ed f rom ~ .

P roceed ing to h igher and higher degrees of complexi ty , we are thus led

to the following sys tem of equat ions , which define the i t e ra t ion p rocedure

f rom which the corre la t ion funct ion is obtained, val id for all n > 0:

(3] ')

(9') ~,~(k) ~Jfexp [ i k ' r l $ ( r ) d r ,

(~2')

-- exp i k. r] l~,(lc) , (11') y,,(r) (2~C,

(13') s,,(r) exp - - K T ] exp [7,,(r)] .

As we men t ioned in the in t roduct ion , ~[Ol~I$A ha s considered a deve lopment similar to ours, b u t some te rms inc luded here are missing in 3Ior i ta ' s expans ion (*).

A deta i led compar i son is m a d e in the Appendix .

(') Note added in proo]. - Ir~ an article which appeared recently (Progr. Theor. Phys.,. 23, 829 (1960)) MORIT* has added to his preeeeding expansion the missing terms. He thus obtains the eqs. (37-38) of the present paper: moreover, the same author has given the expression of the correction terms to the integral equation as in the preprint of Dr. M~ERO_~ and as irL the present paper (Progr. Theor. Phys., 28, 385 (1960)).

8 6 L . V E R L E T

5. - The integral equation for the generalized chain expansion.

Let us assume tha t the i terat ion process converges: in tile limit of n going to infinity (31') becomes

(32) ~(r) = g ( r ) - - 7 ( r ) ,

where we h~ve put

(33) g ( r ) = s ( r ) - - 1 .

We note that. g(r) approaches zero when r tends to infini D. Taking the Fourier t ransform of (32)~ one has

(34) G(k) :== 9 ;exp [ i k . r ] g ( r ) d r :- Y.(k) T l ' (k) ,

which by (12') become.~

(35) a(k) = .~(k)/(1 ---~tk)) ,

This equation~ with (12') a~d (11.') leads to

(36) 7(1, 2) --- ~2f(13g(l, 3)~(2, 3) m

Thus the correlation function i~ defined through the two coupled equations

(37) s(r) exp -- K T ] exp [7(r) l,

@r' ( , - { : _ - - - v ( , - ' ) ) • (3S) y(r)

This last equation can be put in the following' form, convenient for a numerical ea, lculation,

co

2~ o f~ (39) 7(r) = r ~ " 0

where

(4o)

a,. ' (.,.(,.') - 1 - ~ ( , . ' ) ) ( ¢ , ( , . + ,.') - ¢ ( i , - r' i ) ) ,

r

~(r) =/(,,,'(r') - - 1)r' d r ' .

0

ON TH E TI IEORY OF CLASSICAL F L U I D S 87

The set of equations (37)-(38) is equivalent to one non-linear integral equa- t ion for y(r) or s(r). 7(r) is certainly a function more regular tham is s(r), and it will be be t te r practically to solve for y(r). The elimination of y(r) will however be useful in the following and we shall perform it now.

F rom (35) one has

(41) S(k) G(k)/(1 '-- (I(I~)) .

Thus we h~ve, using (36) and (37),

(42) .ql, 2) -- g(1, 2) -- 1 =

-- e x p [ - - KT -- (1, 3)g(3, 2 ) d 3 - - 2 (1,3)g(3, 4)g(4, 2) d3 d 4 +

+ . . .+o ~(- 1)"+If q(1, 3)g(3, 4).. . g ( N + 2, 2)(13 ... d N + 2 + ""l"

Equat io~ (35) can be wri t ten in co-ordinate space

(43) g(r) =- $(r) -- 9/g(]r-- r' i )~ ( r ' )d r ' .

We see tha t this relat ion is identical with a quite general relation holding for fluids which h~s been derived by O~sT~I~" and ZE~NICK~ (~). The present theory enables us, using (32), to calculate }(r) which is interpreted in ref. (11) as the direct interaot ion between two par tMes.

This funct ion is also of interest in the calculation of the quant i ty (3p/3~)~ which goes to zero at the critical point. I t is obtuined from relation (3)

(44> 1 (cP/ 1 1/( l ' G(0)) KT \ ~ ] r 1 + o fg ( r )d r --

~nd, using (34),

(45) I ('Q: t - ~f~ KT~o/r - - l - - S ( 0 ) : : 1 - - (r) d r .

Yvo~ " has derived (17) a relation similar to (45), and the functior~ ~(r) is apar t from a trivia.1 multiplicative constant identical with his function ttldr). Yvo~ has shown, by considering the first terms of the cluster expansion of ~(r) tha t its range is of the order of the range e of the potential, even in the neighborhood of the critical point. We can show more generally tha t this is indeed the case:

(n) See ref. (1) and L. (40LDSTE1N: Phys. l:er., 84, 466 (1951).

outside the range of the potential, one has

Tr(r) rn 0 for r 2 ~ .

We know from experiment ( I 2 ) that evcn at the critical point s (r ) is not very different from one, so that we can suppose that in t'his case y(r) , although of long range, is appreciably smaller than one. 11-e have

So that, for r 2 8, ( ( r ) is an order of magnitude smaller than g ( ~ ) . At the critical point, one has

so that 7 a(0) = 1. .

If, for example, we choose %(k) of the form

(48) E(k) = pZ/(p" P )

which gives for &r)

(for small k ) ,

with p = 1/& then we have

E(k) g(r) = -

L C ~ exp[i k . r ] d r d k = - for r 2 E . 2 n 2 p

This function is the long range function expected at the critical point (I1).

6. - Relation with the virial coefficients.

We shall see now how the virial coefficients are reproduced by the int.egral equation (37)-(38).

We use first the iteration procedure of Sect8ion 3. When only primarr chains

( I 2 ) A. EISENSTEIN and N. S. GI~GRICII : P h y s . Rev., 62, 261 (1942).

ON T I t E T t t E O R Y OF CLAS,~ICAL :FLUIDS 89 '

are considered, we get the first three virial coefficients and pieces of all t h e others. In part icular there is a contribution to the four th virial coefficients, represented by the two diagrams (a) of Fig'. 8. The first of these diagrams comes from the single chain of two points, the second te rm is the reducible diagram which appears in the expansion of the exponential in eq. (6), the corresponding irreducible diagram being a chain with one point. When secondary chains are considered, there comes a new contr ibut ion to all the virial

a) b) c)

Fig. 8. - Diagrams contributing to the fourth virial coefficients. Here the dotted lines represent ~f)+l,

the solid line, ~0.

coefficients of an order higher than the third. The contr ibut ion to the four th virial coefficients is represented by diagrams (b) of Fig. 8. Tile next order of complication of the chains will modify the virial coefficients s tart ing from the fifth one. I t is clear tha t the diagram (c) of Fig'. 8, which is also a contribution to the four th virial coefficient, will not appear. I t is quite evident tha t a.n integral equation of the type (37)-(38), involving only 3 points, is unable to generate terms like the one of diagram (e).

We now examine the diagrams which are generated when the integral equation (37)-(38) is solved by iten~tion, s tar t ing from the zero-density correlat ion funct ion

I-- V/1 KTJ

~rom (25), we have, as a first approximat ion,

(5]) y~(], 2) = o f Q(1, 3)~,,(2, 3 ) d 3 ,

which is symbolized by the diagram: Equat ion (24) gives sa, which we can writ(; schematic~dly as

<> $I (1,2) . . . . . + ~ + ~ + ~ "I-''.

We see tha t the first three virial coefficients are obtained as well as one, t e rm of the four th virial coefficient.

In the nex t order, we have, using (25) agMn,

} / 2 ( 1 , 2 ) = /x. + Pl . ~q + ~

1 1

9 0 I,. V E R L E T

We now have all the contributions to the four th virial coefficient except~

naturally, the t e rm represented by diagram (c) of Fig. 8. We note tha t with this i terat ion procedure the chain of n points appears only at the n-th i teration. Thus the terms of the cluster series are summed in an order quite different f rom tha t in Section 4. Of course, the same diagrams are ul t imately obtained.

7. - C o n n e c t i o n w i t h the B o r n - G r e e n - K i r k w o o d e q u a t i o n .

We shall show now tha t the integral equation (37)-(38) contains the equa- t ion obtained when the system of integro-differerttial equations due to Yvon, Born and Green (s) is closed by the superposition approximation (%

Let us first briefly rederive the Yvon-Born-Grecn equations: t~king the gradient of (4) with respect of (.o-ordinate r~, we get. the following exact

equation:

(5 ̀ ) ) V~s(1, 2) = X(1, 2),s'(1, 2} + 91\Y(1 , 3)s(1, 2, 3 ) d 3 ,

where we h:~ve put

(53) X ( i , 2) = - - V~v(1 , 2 ) / K T .

The 3-particle correlation fmwtion is defined by

(5~) ~(1, 2, 3) = V aJ ~~2 [ KT I d4 ... (iN

d l ... d N

We shall now use the superposition approximatiou

(55) ~'(I, 2, 3) : s'(l, 2) x(l, 3) ,s'(2, 3) .

This permits us to express (52~ as an il~tegro-differential equation for the

correlation function. To make tile comparison with the integral equation (37)-(38), we shall

make a trivial modification of eq. (52); for this we use the relation

(56) f s ( ] , 2) X(1, 2)d2 ~ 0 ,

O N T I l E T l l ] ~ ; O R Y O1,' C L A S S I C A L I " I A ' I I ) S 9 ] .

which is obtaiited by notitlg th'~t~ for a homogeneous fluid we have

(57) V f g exp [--V~TJ)J d2 ,.. d N 0 .

K T I d l ... (IN

This leads to the equation

. f x ( 1 3) s(1, 3) [.~(2, 3) -- 1 ]d3 (58) V~s(1, 2) = X(1, 2)s(1, 2) + os(1, 2)f '

This equation has been extensively applied to the theory of liquid state, in particular by KIg_~woo~ and his collaborators (~'~). For the sake of brevi ty we shall (;all this equation the BGK equation. We shall derive this equation as an approximation of the integral equat ion (37)-(38) obtuined from the generalized chain approach. For this, we take the gradient of (37) with respect to co-ordim:te 1

(59) V~s(l, 2) X(1, 2) s(l, 2) - - V~ ;(1, 2) , (1 , 2 ) .

Taking the gradient of (38), we obtaill

(6o) /,

V,y(t , 2) - / j (W~s(1, 3) -- V~?,(1, 3))(s(2, 3 ) - l)(13.

Combining this equation with the preceeding one, we obtain

(ol) Vx.s'(], 2) = X(I , 2)s(1, 2) ~- ,2s(1, ' ) )fX(1, 3)s(1, 3)(s(2, 3) 1) d3 +

+ 3) - (.,.(,_,, 3) - 3 ) d 3 .

This equation is identical with the BGK equation {58) except for tile l~st term which vanishes only if we replace i~l it the correlation function by its value for large distances.

The relation between the BGK equation and the integral equation (37)-(38) can be seen in another way: we c'm, from the BGK equation (58), using deK-

(,a) j . G. KIRKWOOD, V. A. I,EWINSON and 1;. J. ALDER: Journ. Chem. l'h9,~., 20, 929 (1952).

92 L. ¥ERLET

nition (33), calculate the quan t i ty

g(1, 3) + o.s'(l, 3 ) ;X(1 , 4) s(1, ~) g(4, 3) (62) X(1, 3) 8(]~ 3) V~ d4 . . ]

Subst i tut ing in (58), we get a te rm ~2 in which the quant i ty X(1, 4)s(1, 4) appears. Subst i tut ing (62) ~gMn and again so as to get rid of this type of term, we obtain,

( 21@ (63) V~s(1, 2) = s(l, 2)V1 \ KT + (1, 3)g(2, 3 ) d 3 - -

(,~ fg(1, 3)g(3, 4)g(4, 2 ) d 3 4 4 + ~o3(g(l, 3)g(3, ~)g(4, 5)g(5, 2)d3 d4 d 5 - . . . ) - . 2 /

9':s(1, 2)[V~g(], 3)g(1, 4)g(3, 4)g(4, 2) d3 d4 ÷ I /

93 s(1, 2)/V~ g(l, 3) g(l, 4) g(3, 4) g(4, 5) g(5, 2) + d3 d4 d5 . J

Using (41) and (42), this equati on can be put in the compact form

(6t) V 1 8 ( ] , 2 ) : ,~'(] ? 2 ) V I ( V(1, 2) )

KT + 7(1' 2) - -

--o2s(1, 2)fV~g(1, 3).(/(1, 4)g(3, 4)~(4, 2 ) d 3 d 4 .

If we take into account the first term only, we get, after integration, .% correlation function identical with ours (eq. (37)). By replacing in the second term the ~ and the g's by ~0's, we see that it cannot be interpreted as a term of the cluster expansion. Apar t from the terns of type a (Fig. 8), which cannot be obtained from an integral equation involving only three points, the fourth virial coefficient is correctly obtained from the first te rm of (54). The second term must be considered as a spurious term due to the superposition approximation. In fact, we shM1 see in Section 9, tha t a more correct three-body correlation function brings in counter terms which automatically cancel the spurious terms of equations (64) and (65).

8. - The correct ion t erms .

We now want to im:lude in our approximation scheme the more complicated terms which have been left out of the generalized chain expansion of Section 3. We shall first sum all the combinations of terms whose proto type

ON T H E THEOR, Y OF C L A S S I C A L F L U I D S 93

is diagram v of Fig. 8 with the terms already considered, forming diagrams of the type shown in Fig. 9.

For this pmoose we shall consider the i terat ion scheme defined by the following equations :

(65) 7~, = @~(1, 3)~.(3, 2) da + @ . ( ~ , 3)~.~(,, 4) ~.(~, 2)dad4 +

+ ~'~f~.(1, 3)~n(3, 4)~,,(4, 5)~.(5, 2 )d3d4d5 + . . . - -

-- ~(1, 2) @ ),~(1, 2) = @o(1, 3) ~o(3, 2) d3,

03 7 (66) 7~ = 2 Zn-~(1, 3) }~-,(1, 4) ~,~_,(2, 3) ,~n_1(2, ~)/~n-~(3, 4) d3 d4,

( 6 7 ) ~ ( r ) = 8n_i(r ) - - 1 - - ~(nl)- l(r) ,

(68) exp [y~( r )_ y~)(r)]. [ V(r)]

s , ( r ) = e x p l-- L

We star t with

(69) y(~(¢)=0, s_~(r]=~o(r), 2_t(r)=$o(r).

1 2 Fig. 9. - A typical diagram obtained from the first correction term tothe integral eqs. (37)-(38). The solid lines represent ~o's.

After the first i terat ion we obtain for ~)(r) the same result as in Section 3: y~l)(r) is the sum of chains of ~o. 7~ ~) is represented by diagram v of Fig. 8. We notice t ha t at this stage all the four first virial coefficients have been obtained except for the terms represented by diagram b of Fig. 8.

We then proceed as in Section 3 to build ~1, which we represent graphi-

eally as

(70) + . . .

9 ~ L. V E R L E T

The last t e rm corresponds to - - y l ' ( r ) in fornmla (67). I t has to be sub-

t rac ted as was the case in Section 3, so tha t the chain diagrams arc not counted

twice. I t is clear tha t a similar t e rm has not to be subt rac ted for y:a>(r). y(~l)(r) is the sum of the chains constructed from ~. y(~)(r) is given symbol-

ically by the equatiml (71):

1 (7,) r':' c~= ~ R ,

where the wiggly lines of the diagrams represent the sum of $o chains as given by )..(r) of eq. (65).

Going to the nex t approximation, ~ will have the same general shape as #~ of eq. (70), bu t the chains will be now constructed from ~ , and the side of the crossed diagrams will now be made from chains constructed from ~o, a~ given by 2o(r) of eq. (65).

After an infinite number of i terat ions have been made, we obtain, comparing (67) and (65),

(72) ~t(r) = s(r) - - 1 = g(r) .

The correlation funct ion is now defined by the set of equations

(7a)

(74)

(75)

V(r)] s(r) = g(r) ÷ 1 = exp -- K T J exp [gin(r) + y(~)(r)],

2) _- i f ( u < a) - 3)) g(3, 2) a n ,

~o~fq y('-)(1, 2) = ~ (1, 3)g( , , 4)g(2, 3)g(2, 4)g(3, 4 ) d a d 4 .

There is some arbitrariness about the way to introduce the correction terms° :For instance we might have wr i t ten instead of eq. (66)

(66') = 2 J ~-=' ' 2)an_~( , , 4)a._o(e, 4)a~_~/2, a)~.~_2(a, 4)dad4,

with in addit ion of conditions (69) the prescription

yn-2(r) = 0 .

We star t the i terat ion procedure as above. ~*e have af ter thef i rs t i terat ion the same result as in Section 3. After the second i tera t ion we have for $., the same type of diagram as those given by (70) except tha t the chains in it ar~ now constructed f rom ~1. Taking again the example of the four th virial coef-

ON T h E TIIEORY OF C L A S S I C A L FLUIDS 9 5

ficients, we see tha t the first i terat ion gives the diagram (a) of Fig. 8, when the second i tera t ion gives also diagrams (b) and (e). Ultimately, however,

the same diagrams are summed and the same limiting equations are obtained. The next correction terms forming y'8)(r) will be defined as those which

cannot be reduced to terms included in y~2)(r) or in y~)(r), bu t which can be so reduced by cut t ing a link somewhere in the representa t ive diagram.

We write graphically the formula for y~)(r). The lines in the diagrams~ af ter an infinite number of i terations have been made, represent g-functions. These terms generalize the te rm given by y~)(r) in equat ion (75). We separate the contr ibut ion of 5-point and 6-point diagrams, which are of order o 3 and ~o ¢ respectively:

(76) Y~5) (r) = +

cs) ( r ) = + +

The 5-point diagrams of a higher degree of complication will also be useful in the next paragraph. They are

(77)

The integrals corresponding to y~), y(4).., are obviously very difficult t o calculate. I t may be hoped, however, that, given the great number of links they involve, they are small.

We shall put

(78) y°(r) =: 7(1)(r) ;

?~(r) represents now the sum of the chains constructed from completely re- normahzed ~'s.

The sum of the rest of the contr ibution to 7(r) will be given by

(79) 7"(r) = ~ y"'(r).

The correlation funct ion is now given by

(80) [ V(r)]

s(r) = exp [- - K T J exp [y ' ( r) q- y~(r)] .

9ti L. VERLET

We easily see tha t the Ornstein-Zernicke relation (43) is still valid in the present case, with ~(r) defined by the equation

(67') ~(r) = g ( r ) - - y q r ) .

The Yvon relation (34) is also maintained, but the demonstration of the smallness of ~(r) outside the range of the potential now requires tha t yR(r) should be for large distances an order of magnitude smaller than y~(r), which is probably true, so tha t the essential of the demonstration of Section 5 remains.

Finally, we note that , supposing V(r) and s(r) known, it is possible to deduce y~(r)q-yR(r), a t least outside the repulsive region of the potential. On the other hand, equation (74) can be wri t ten after a Fourier t ransformation

G~(k) (743 ]'°(k) = ~ :~ G(k) '

y°(r) can therefore be obtained equally well from the knowledge of the correlation function. If the experimental determination of this functions were precise enough, i t should be possible to measure the effects of the correction terms introduced in this paragraph.

9. - The t h r e e - b o d y corre la t ion funct ion .

In this paragraph we shall give the first corrections to the superposition approximation (14). We shall show that , when the corrected 3-body correlation function is introduced in the Yvon-]3orn-Green equation (52) the correlation function calculated in the preceeding paragraph is obtained and the spurious t e rms of the BGK equation disappear.

The cluster expansion for the three-body correlation function is given in the paper of SALPETEg quoted in ref. (~). The three-body correlation function is defined as

(81) g(1, 2, 3) = s(1, 2)g(2, 3)g(3, 2) exp [ i Osfls(1, 2, 3)] . ~Vffil

The fl~.(1, 2, 3) are the exact generalization of the two-body irreducible cluster integrals: an irreducible diagram is defined by the same prescription as in Section 2, except for the last prescription which now reads: 3') a diagram is

(14) Various authors have investigated this problem by considering the first few -terms of the expansion of (81): Yvo~: private communication; R. ABe: Progr. Theor. Phys., 21, 421 (1959).

ON THE TI IEORY OF CLASSICAL F L U I D S 97

reducible if a pa r t of it can be separa ted f rom the rest by cut t ing lines joilting

points 1, 2 and 3. The fi~(1, 2, 3) are given by a formula similar to formula (7). of Section 2.

The first t e rm of the cluster expansion leads to the superposi t ion approximat ion (we note tha t in (81) the exact two-body correlat ion funct ion ap-

pears) . The first correct ion t e rm wilt / N

3 4 be represented by the d iagram shown in Fig. 10. In this diagranl the dot ted 3 ,~7.~, 3,

2 . . . . . l lines represent s functions. I n the 1

Fig. 10. - The first usual cluster expansion the solid Fig. l l . - Typical correction to the lines of the d iagram of Fig. 10 would diagrams included superposition ap- represent ~,, functions. I t is clear in the first correc- proximation. The t ha t by the eonsideration.~ of the pre- tion to the super- dotted and solid position as described lines represent s and ceding paragraph these lines should byFig, lO. Herethe g tunetions respe- be now replaeed by g functions, solid lines repre-

ctively. Once this replacement has been sent ~o functions. made, a large ('lass of t e rms of (81)

is summed. For instance diagrams such as the diagrams shown in Fig. 11 are

included, when equations (79)-(80) are used to define g. The t e rm N = 2 in the exponent of (81) will give rise to the irreducible

d iagrams shown in Fig. 12, which are the p ro to type of the 92 correction to ~o the superposi t ion approximat ion .

Fig. 12. - The second correction to the superposition approximation. and solid lines represent s and g functions rcspeetively.

+ the two d iagrams obta ined ~-hen pe lmu t ing ]~ 2 ar_d 3

+ the two dia, o'r~ms obtMned when pe rmut ing 1, 2 and 3

The dotted

There again the dot ted lines represent s functions, the solid fines, g functions when the summat ion of t e rms considered in the preceeding paragraph has been made.

We ~hall ~how now tha t the introduct ion of the corrected three-body correlation funct ion in the Yvon-Born-Green equat ion (52) leads to equations (79)-(80) for the two-body correlat ion funetic.n.

We first consider a three-body correlat ion function including the first correct ion to the superposit ion approximat ion, which is, wri t ing the expression

7 - l l N ~ ¢ o r o ( ' i m e M o .

98 L, V E R L E T

represented by the diagram of Fig. 10,

(82) s~(1, 2, 3 ) = s(1, 2)s(1, 3)s(2, 3)(1-~ o fg(1,4)g(2,4)g(3,4)d4).

We introduce this function in the Yvon-Born-Green equation (52). This replacement taking (56) into account, yields

(s3) V~s(1, 2) = X(1, 2) s(1, 2) + os(1, 2)fx(1, 3)s(1, 3)g(2, 3)d3 ~-

÷ ~o2s(1, 2) fx(1 , 3)s(1, 3)s~2, 3)g(1, 4)g(2, 4) g(3, 4 ) d 3 d 4 .

Eliminating the terms containing the potential, as was done in Section 6, and neglecting the terms of an order in ~ higher than the second, we obtain the equation

(84) V,s(1, 2) = X(1, 2)s(1, 2) ÷ ~os(1, 2) Vljg(1, 3)g(2, 3)d3 --

-- ~s(1, 2) Vlfg(1, 3)g(3, 4) g(4, 2) d 3 d 4 - -

- - ~2 s(1, 2)fVlg(1, 3)g(1, 4)g(3, 4)g(4, 2)d3d4 +

_~ ~2 8(1, 2)fVl g(l, 3)g(1, 4)g(3, 4)g(4, 2)d3 d4 T

4~,~°~ s(1, 2) v,fg(1, 3) g(1, 4)g(2, 3) g(2, 4)g(3, 4) d3 d4

The second and third terms are the beginning of the expansion of ~s(1,2)V17~(l,2) aS can be seen by a comparison with equations (63)-(64). The fourth term is the first spurious term appearing in the BGK equation (see eq. (63)). I t is cancelled by the fifth term. The last term is equal to os(1, 2)VlV~2'(1, 2).

Equation (84) is thus equivalent to the equations (79)-(80) up to the order ~2. Using the next correction to the superposition approximation which is

represented diagramatically in Fig. 12, we have pushed the calculation up to the order ¢3. We do not give the details of the caclulation as it is too long to be reproduced and quite straightforward. We only quote the results:

1) y0(1, 2) is obtained up to the order ~o 3.

2) All spurious terms are eliminated up to the order ~3.

ON TIIE THEORY OF CLASSICAL F L U I D S 99

3) All the ~o a contributions to yR(1, 2), given by equations (76)-(77) are obtained with the right coefficients.

We have still to show that the introduction of the corrected three-body correlation function (82) in the Born and Green equation leading to equation (83), added to the requirement that the resulting integrodifferential equation be a three-point equation, leads to the equation (37)-(38). In other words~ we shall perform the elimination of the spurious terms of the BGK equation at all orders in ~o. To do this, we replace in (83), s(2, 4) by its asymptotic value which is unity. Thus we o'et

(85)

where

(86)

V~s(1, 2) X(1, 2) s(1, 2) + (,.~(1, 2)fF(1, 3) g(3, 2)d3,

F(1, 3) = X(1, 3)s(1, 3) -- of X(1, 4).~'(1, 4)8(4, 3)g(1, 3) d4.

In this last equation, we eliminate the first term with the help of equation (83). This leads to the equation

(87) F(1, 3) --- Vlg(l~ 3) o(l÷g(1,3))fx(1,4)s(1,4)g(4,3)d4-- -- o2s(1, 3)/X(l: 5)s(1, 5)g(1, t) g(5: 4)g(4, 3)d5d4 d-

+ og(1 3)fx(a: 4)8(1, l)g(4, 3)d4.

We see that in this equation the fourth term is cancelled by the second part of the second term. We shall replace in the third term s(1, 3) by i t s asymptotic value so as to obtain a three-point integral equation.

We then obtain for F(1, 3) the integral equation

(88) ~fF(1, 4) g(4, 3)d4. F(I: 3) = Vlg(1, 3) -- o

This equation combined with equation (86) leads to equation (61) which is equivalent to the integral equation (37)-(38). The elimination of all the spurious terms of the BGK equation has therefore been achieved and a value of y'(r) correct to all orders in o is obtained. We conclude that equation (37)-(38) includes the main part of the corrected three-body correlation function (82) and enables us to go one step beyond the superposition approximation used in the BGK equation. ~oreover, we have seen that the precision can still be improved by the use of a more correct three-body correlation function which leads to the correction terms described in Section 8.

1(10 L. V~;RLET

10. - Conclusion.

In this paper we have established an integral for the three-body corre-

lat ion funct ion in classical fluids. This equat ion contains a main equat ion

which is vmT simple and correct ion terms. The solution of the main equat ion

appears at first r a the r easy, but it should be noted t ha t i t is a highly non- l inear e~tuation. I n fact , we have t r ied to solve it iteratively~ bo th by using the

i te ra t ion procedure described in Section 3 and by a simple i te ra t ive process s tar t ing f rom the zero-densi ty correlat ion funct ion as in Section 6. In the low

density, convergence is a t ta ined and bo th methods lead to the same result ; bu t the two approaches fail to converge in the high densi ty region. I t will

t hen be necessary to use a non- i tera t ive method to solve this equation. This

computa t ion is present ly in progress as well as the calculation of the first cor-

rections to the integral equation.

We wish to acknowledge some interest ing discussions with Prof. J . Y v o x

Dr. P. DE GEN>'ES and Dr. D. E. 3[cC~.~BEe.

A P P E N D I X

Morita's expansion for the two-body correlation function.

~ o ~ r r . t (6) has de te rmined the 2-body correlat ion funct ion wi th the help of an i tera t ion scheme resembling ottrs: the correlat ion funct ion is defined by eq. (9'), (11'), (12')~ (13'), bu t the equat ion (31') is replaced in his work b y the equat ion

(A.1) ~ ( r ) = (~,_~{,)~ • -- 1) exp ,[_y,,M *t""J __ 7{- 2(r)] 1 - - (7~_,(r) - - y~_2(r)) , M

where the superscr ipt refers to the quant i t ies calculated by 5IORITA. A direct comparison with the equat ion (31') m a ) be made if we el iminate

~_~(r) by a repeated appl icat ion of (A.1). We thus obtain

(A.2) ~ ( r ) = {(~0(r) ~- 1) exp [7~_~{r)] 1 - - 7 M t(r)}--

iffin

ON T I I E T H E O R Y OF CLASSICAL F L U I D S I01

The t e r m b e t w e e n b r a c k e t s is i d e n t i c a l w i t h ~ ( r ) of (31'). F r o m th is t e r m is s u b t r a c t e d a sum of t e r m s which does no t v a n i s h when n t e n d s to inf i l l i ty . so t h a t (A.2) is f u n d a m e n t a l l y d i f f e ren t f rom (31').

W e can i l l u s t r a t e th i s d i f ference b y c a l c u l a t i n g M ~2 (r) ( ~ ( r ) is i d e n t i c a l w i t h ~:~(r))

(A.3) ~ ( r ) = ~ ( r ) - - yo(r)(exp [)~(r) - - yo(r)] - - 1) .

q7 Fig. 13. - A typical term subtracted from ~2(r) in 5[orita 's ex-

pansion scheme.

W e see t h a t f r o m ~ ( r ) a r e s u b t r a c t e d t e r m s such as r e p r e s e n t e d in F ig . 13, in which t h e r e is one cha in con- s t r u c t e d f r o m ~0's a n d t h e chMns c o n s t r u c t e d f rom $1's, wh ich ~re n o t i d e n t i c a l w i t h t h e p r i m a r y cha ins . The re does n o t seem to be a n y r e a s o n w h y such t e r m s shou ld be e x c l u d e d f rom the e x p a n s i o n scheme.

R I A S ~ V N T O (*)

Si mostra c h e l a somma d i u n a larga classe di diagrammi in uno sviluppo del t ipo <( cluster ~> permette di scrivere una equazione integrale per la funzione di correlazione nei fluidi classici. Con una approssimazione, questa equazione si riduce a queUa 4i Born, Green e Kirkwood e riproduce pifl esa t tamente di quella i coefficienti del viriale. Si danno i termini corrett ivi , sempre pi~ complicati , che permettono di rendere esa t ta la nostra equazione. Si calcola infine la funzione di correlazione a tre corpi, che, intro- dot ta nell 'equazione di Yvon-Born-Green, permet te di calcolare la funzione di correlazione a due corpi. Si conferma cosl il calcolo d i re t to di questa funzione, e si mostra che la nostra equazione integrale inclnde de|le eorrezioni all 'approssimazione di sovrap- posizione.

(*) T r a d u z i o n e a c u r a d e l l a R e d a z i o n e .

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On the theory of classical fluids