Derivation of the classical theory of correlations in fluids by means of functional differentiation

Physica 81A (1975) 47-71

DE’RIVATION IN FLUIDS

0 North-Holland

Publishing

OF THE CLASSICAL BY MEANS

Co.

THEORY

OF FUNCTIONAL

OF CORRELATIONS DIFFERENTIATION

D. J. GATES Mathematics

Department,

Parkville,

University

Victoria

3052,

of‘ Melbourne,

Australia

Received 21 March 1975

The k-particle, infinite-volume distribution functions Tik(r, , . . , Y~-~,7) and modified Ursell correlation functions 9 (rl , . . . , rk-l, y) of a classical system of particles with the two-body potential q(r) + yYK (yr) are considered. The limiting values of the functions 7ik(rl , . . , rk_l, y), (I -‘) ” Ok(sl/yr . . , sk- &, 7) in the limit y + 0 are calculated, under n,(s,~.....~~-~/~,./)and~ fairly weak conditions on q and K, by a method involving functional differentiation. These limiting functions are used to describe the molecular structure of the various states of the system both in the range of the potential q(r) and in the range of the potential yYK (yr). The direct correlation function C (r, y) is also considered and it is shown that for s # 0, lim,-O y-“C (s/y, y) = -,C?K(s), for all one-phase states, where j? is the reciprocal temperature. Special cases of our results confirm those of other authors, including the well-known results of Ornstein and Zernike.

1. Introduction In ref. 1 it was proved that the thermodynamic pressure and free-energy of a classical system of particles with the two-body potential u (r, Y) = q(v) + ~“K(yr),

density

(1.1)

tend to definite limits when y --) 0 (the “Van der Waals limit”), provided that q has a hard core and that both q and K satisfy fairly weak tempering conditions. Here ;g is a positive parameter, Y is the vector distance between a pair of particles, v is the number of dimensions, q(r) is called the “short-range” or “reference potential” and ~‘K(yv) is called the “long-range” or “Kac potential”. The limiting thermodynamic functions were expressed as variational principles. In the present paper, we use these results to examine the k-particle distribution function Ek (Ye, . . . , rk_ 1, 7) and k-particle Ursell correlation function z& (Ye, . . . , rk_ 1, y) for a system with the two-body potential (1.1). Some of the results were presented without proof in a letter 2. 47

48

D. J. GATES

The first results of this type were obtained by Uhlenbeck, Hemmer and Kac3) (UHK) for a one-dimensional system with q(r) = co for r < Ye, zero otherwise. and K(s) = &X e-‘, where r. and --x are positive they proved that, if the limiting thermodynamic phase transition

for densities

constants. functions

0 in, and only in. the interval

Among indicate

other things. a first-order

Q, < Q < c2, then

(1.2)

5,”(Y,

3

.

.

.

.

VI-I.?),

for

and

Q do1

CJ~02,

(1.3)

where i$ is the k-particle distribution function for a system (called the “reference system”) with two-body potential q(v). This result can be interpreted to mean that two phases of densities Q, and e2 coexist during the first-order transition (er < p d e2), but that otherwise the Kac potential does not affect the distribution ot groups of particles. Using a different method, Lebowitz and Penrose4) (LP) showed that (1.3) and (1.4) hold for k = 2, for any number of dimensions and for a certain class of functions K. Our first step (section 3) is to evaluate fii for all k = 2, 3, . . . , any number of dimensions and a wide class of functions K, by using a method like that of LP. Our formulae for fii apply even to crystalline phases where (1.3) and (1.4) do not hold. As pointed out by LP, E: describes the structure of the system only on the scale of the short-range potential q. We therefore call $ the “short-range distribution function”. To study the structure of the system on the scale of the Kac potential. i.e., over distances of order y-r, we calculate in section 4 the “long-range distribution functions”

Our simplest result is for a single fluid phase where we find iik = pk. Hence, in this case there is no correlation between particles on the scale of the Kac potential. Our third set of results is motivated by the expansions of the two-particle Ursell correlation function ii2 (v, 0, y) obtained by UHK3), Hemmer’), and Lebowitz. Stell and Baer6) (LSB), for gaseous (i.e., low-density) states. These expansions have the form [see eq. (4.10) of LSB] 21,(Y,0, y) =

27:Cr.e) +

y”

@) ) (’ d2fiii: y

f(yr,

Q) +

Q(y’“),

(1.6)

CLASSICAL

THEORY

where tii is the Ursell function

OF CORRELATIONS

of the reference

IN FLUIDS

49

system,

(1.7)

I?(P) =

sds

a:(p)

d2a0 (e)/f?e’,

=

e2;iip’s K(s)

(1.8)

(1.9)

a”(o) being the free-energy density of the reference system*. If, for a given p. the reference system is in a fluid phase we expect that ?i: (r, p) -+ p* as jr/ -+ co. Also we expect that $0) + 0 faster than ]rJ-y as r + co (i.e., it satisfies a “cluster property”‘), and hence, ~-‘hi (s/u) -+ 0 as y --+ 0. We therefore infer from (1.4) that y-“r& (s/u, Q, y) tends to a limit as y --) 0, and that

We show in section 5 that this indeed holds for s # 0 when the limiting thermodynamic functions indicate a single fluid phase. More generally, we show that ~7: is essentially the Green function of a simple integral operator. Similar considerations lead us to define

IA;(s,

)

.

.

.

)

sk) E lin~y-‘“-“Vii,(s,/y, Y-+0

. . . . sJy,y),

(1.11)

which we call “the weighted k-particle Ursell correlation function”. We calculate ~77explicitly and show in principle how to calculate fi: for higher values of k. Finally we consider the direct correlation function c (r, 7). The expansion of LSB [their equation

(5.18)] leads us to expect that

cw(s) = Jim y-“c (s/v, y) = -PK

(s).

(1.12)

y+O

We show that this indeed holds for all one-phase provided s # 0.

* The function freduces of n (I), 0+) (see ref. 5).

to the Omstein-Zernike

states, including

crystalline

states.

formula for values of 9 near a critical point

50

D. 5. GATES

2. Assumptions and method Our method consists, essentially, of relating the limiting correlation various functional derivatives of the limiting free-energy density a(e,O+)

=

functions

linia(~,~>. g+O

to

(2.1)

Here a (g, y) is the free-energy density of a system with the two-body an external potential y (yx) and density Q, and is defined by

potential

( 1.1).

where Q is a cube of volume 15.1 and 2 (N, J2, y) is the partition particles in R, and is defined for N > 2 by

function

for A’

Z(N,Q,y)

E (l/N!.1”“)ldx, R

Hence A is the thermal

wavelength’)

and

In ref. 1 it was proved that if q and K satisfy fairly a (e, 0+) exists and is given by the variational formula a(@, 0+)

=

(2.3)

. ..Jd~~exp(-/3V.). u

weak conditions,

inf G(n), ns%(a)

where the functional

then

(2.5)

G is given by

s

G(n) = (l/V’l> jTdy {a” MY)I + h (Y) dy’ 4~') K(Y - Y')} >

(2.6)

the integral with respect to y’ being over all ov Y-dimensional space. Here +?(Q) is the class of functions that are: (i) bounded by 0 and eC (the maximum density permitted by q); (ii) Riemann-integrable over any bounded region of Y-dimensional space; (iii) periodic; and (iv) have space average Q, i.e.,

Wl);(

dyW

= e.

(2.7)

The region r (which depends on n) is the unit cell of IZ and has volume IT/, and U”(Q) was defined in section 1. To calculate Ei we first generalize (2.5) to include a k-particle short-range potential qk. Then, as exploited by Fisher’), the functional derivative of a (0, 0+) with

CLASSICAL THEORY OF CORRELATIONS

respect to qk is essentially derivative

of a (Q, 0+)

$.

Similarly,

we find that i$ is essentially

with respect to a k-particle

51

IN FLUIDS

generalized

the functional

Kac potential.

Also,

we find that 6f is essentially the kth functional derivative* of a (e, 0 +) with respect to an external potential. The form of the limiting correlation functions depends on the state of the system, i.e., on the local properties of a (Q, O+). For a given Q, the states can be broadly classified into one-phase states where g is not an interior point (but may be an end point) of a straight-line segment of a (Q, O+), and two-phase states where Q is an interior point of such a segment. Examples of one-phase states are: (i) one-phase jluid states, where a (Q, 0+) = a”(!?) + $02, and 01 = j ds K(s), whose existence was proved in refs. 4 and 12; and (ii) one-phase crystalline states whose existence was proved in ref. 12. Examples of two-phase states are: (iii) two-phase jluid states, where a (Q, 0 +) = CE [a”(o) + &xQ”] < a”(e) + &xQ~, (CEJ for any f is defined as the maximal convex function not exceeding f),and whose existence was proved in ref. 4; and (iv) two-phase crystalline states whose existence was also proved in ref. 12. These states can also be classified according to the existence and properties of a function TZ*(y, Q) E %?(p)which minimizes G(n), (y = O), i.e. for which the infimum in (2.5) is attained. We knowL2) that, for one-phase fluid states, n* = Qis the unique minimum. More generally, we shall adopt throughout this paper the following. Assumption. For a one-phase state, the infimum in (2.5) is attained for a function n* E V(c) which is unique apart from other minimal functions which differ from n* only by a rotation and/or a translation of its argument y, or on a negligible set in R’. (2.8) For a one-phase crystalline state we expect that all possible 12*‘sare periodic and differ only by the orientation of their unit cells; such functions would satisfy the assumption (2.8). This was verified for a one-dimensional cell model in ref. 10, where y was restricted to integral values, and it was found that

nzic(y, 0) =

Q+9

for

y

even,

e-9

for

y

odd,

e->

for

y

even,

@+3

for

y

odd.

(2.9)

or nZk(y, g) =

(2.10)

Here o+ and Q- are known

functions

of Q and 3 (Q+ + Q-) = Q.

* See Percus’) for other applications of such functional derivatives to classical fluids.

52

D. 5. GATES

For a two-phase

state. where !J~ < p < yZ say, it seems obvious

that the innmum

in (2.5) cannot be attained. For example, in a two-phase fluid state H* would have to indicate a proportion (c2 - c,).‘(o c z - 0,) by volume of uniform fluid of density 9, and a proportion (9 - I,,)/(?~ - 9,) of density p2, with a zero proportion of interface. But since 17” is periodic, it can only indicate a nonzero proportion of interface, so that no such function M”’is possible. As well as the above assumption, we shall assume that the infinite-volume correlation functions Sl, and Ck exist; no general proof of their existence has yet been found. We further assume the existence of their various limits (1.2), (1.3) and ( I .9) when “J + 0. Under all these assumptions, and others, which we mention as ti;ey arise, the limiting correlation functions can be evaluated. Although the! are not completely rigorous, our main results are stated formally as “propositions” for the sake of clarity.

3. Short-range correlation functions The k-particle (space-averaged) thermodynamic limit’)

Ek (y1 )....

distribution

functions

Ek are defined

by the

I’,,-,.$“/) (3.1)

where

with V,Vgiven by (2.4). To calculate the short-range correlations defined by (1.2) we generalize the result (2.5) to include a k-particle short-range potential i.q, (x, , . . . x,); where j. is a nonnegative parameter. That is, we consider a system with total potential energ)

We specify that qk (x1, . . . . xJ is symmetric in x1, . . . . xk. invariant simultaneous translation of x, , . . . , xk. and satisfies

qk

cx

, . . . . . x3

under

a,

if

Ix, - &I < To

for any

a f 6.

0,

if

Ix,,-xJ>i

for any

u # b.

any

(3.4)

=

CLASSICAL

THEORY

OF CORRELATIONS

IN FLCIDS

53

and

Iqk6

1

3

. . . i

Xk)l d Al
otherwise,

(3.5)

.

where yc, (the hard-core diameter), ?, A and E are positive constants. The functions q and K are assumed to satisfy the conditions given in ref. 1. Then, by minor modifications to the proof used in ref. 1 to obtain (2.5), one can rigorously prove. Lemma

I. Under the stated conditions ing to (3.1) has the limit

a, (9. 0+)

= lim a, (0,~) i’- 0

where G, is defined

=

the free-energy

inf

density a, (9,~) correspond-

Gj,(k7),

(3.6)

nEV(Q)

to be

(1IO) d_v(a: bW1 + MY> J dy’ 4~‘) K(Y - Y’>>

(3.7)

and a&) is the free-energy density of a system with interaction >.qk. To apply the lemma we use the important functional relation (I/k!)jdr,,

. . . . dr,-,q,(x,x

R

+ YI, . . ..X + J’k-,)&@I,

potentials

q and

(see refs. 4 and 9) . . ..f’k-1.&Y)

(3.8) where R is the region where r. < 1~~- ~~1 < r for all a # b. In this integral, qk is independent of x because of the translational invariance of qk. Taking the Van der Waals limit of (3.8) we obtain, provided all the relevant limits and the derivative exist, (I,‘k!)Jdu,,...,

dt’,_,f&(X,X+Y

,,...,

X+i*,_,)fi;(i’,

,... >&i,c)

R

= $

a, (@, O+)lA=o*

(3.9)

The interchange of the limit y -+ 0 with the integration and differentiation in deriving (3.9) is justified on the left by Lebesgue’s bounded convergence theorem. On the right it is justified by the convexity of a, (p, y) as a function of 3,. We now obtain. Proposition

1. If ,n is a one-phase

4 (Yl~...,rk--l,d

state then, for almost

=(l/lr*(@)l)

ah

Y, ,

. . . , rk-, ,

j dyn,O[r,,...,uk-l,n~(y,e>],

In*

(3.10)

D. 5. GATES

54 where rz* (y, Q) is any minimal cell.

function

in %(c,) of G (n, cn), and P(v)

is its unit

We note firstly that (3.10) is independent of the choice of 17”because, by assumption (2.Q [r*(p)1 is independent of this choice, while, in the integral, r* transforms in the same way as IZ*. Secondly, let us consider a one-phase state Q of the modified system with free energy a,. By assumption (2.8) and lemma 1 we can write 01, (~3 O+)

=

G,(nT).

(3.11)

where nt (y, 9) is any minimal derivatives exist,

au,(@,0 +) al,

function

+ dGoG& ’

= aG, (6) i. = 0

a1

in W(Q) of G,. We deduce that, provided

dii

A=0

the

(3.12)

,i=O *

The second term on the right vanishes because Go is stationary with respect to variations of n in %Q) away from its minimal function nz. Assuming nz is continuous in A in a neighbourhood of?, = 0, then n: = a*. Then the first term on the right can be evaluated using (3.7), giving

(3.13)

Setting K = 0 in (3.8) gives (l/k!)

jdv,,

. . . . dr,_,q,(x,x

+ rl ,...,

x + ~._,)$(r

,,....

J._,,?)

R

(3.14)

and substituting L:aj.

this in (3.13) gives

(@*0 +)

an

A=0

= -

1

k!lr*l

dy du, ... dv,_, s I‘*

x qj( (x, x +

PI)

. .

.)

nk”[PI) . . . , n* (y,

Q)].

(3.15)

The integrations on the right side, being finite, can be interchanged. Equating the with the previous expression (3.9), and resulting expression for &, (Q, O+)/dill,=, noting that qk is arbitrary, we obtain proposition 1. For the special case of one-phase fluid states, where IZ* = Q, the proposition reduces to (1.3). For the one-dimensional cell model considered in ref. 10, one can

CLASSICAL THEORY OF CORRELATIONS

define distribution one-phase

functions

crystalline

?ii and ii: analogous

55

IN FLUIDS

to our present definitions.

For the

states of this model where (2.9) and (2.10) hold, proposition

1

reduces to ii: (rl , . . . . rk_l, p) =

$fi,O (r,,

....

r,_,p+)

+

$ii,O(r,,

....

,o-).

(3.16)

From (1.4), this is identical to the case of a two-phase fluid state consisting of equal proportions by volume of two fluids with densities Q+ and Q-. Hence, for the cell model, the function i$ does not distinguish between these two cases. The function iii does however [see (4.13)]. Our second main result is Proposition 2. If Q is a two-phase state for e1 < e < ez. where e1 and g, are onephase states, then for almost all rl, _. . , rk_ 1)

fii(...,

e)

=

I@-

c

i=ls2 where the definitions

@il

(@z - @I)

s

1 d.YGZ [...>II* (Y, pi)1> Ir*(@i)l r*te*)

are as in proposition

1.

We use the method of Fisher’). Suppose that a, (Q, 0+) for nl(A) < g < e,(A). By the definition of such a state

a,

(p, Of) = C

(3.17)

has a two-phase

I@- @iCn)l a?.[@iCn>, O+l.

state

(3.18)

i=1,2 @,(A) - Ql(A)

If da, (Q, 0 +)/de exists and is therefore a continuous function proof in the case k = 2), its values at ~~(2) and Q,(A) satisfy

O-t1 $_ aA (6 O+)LgicAj= aAk2Gh em -

of Q (see ref. 18 for a

4, k,GV,O+l QIG) ’

(3.19)

for i = 1, 2. Using this, we deduce from (3.18) that for ~~(0) < Q < ez(0), [where @i(O) F lim,+,

Pi(n)l, IQ- @do)1 z -a,

i=l. 2 @z(O) - @l(O) CY;i

kiC”>9 O+ll~=o~

(3.20)

If pi(A) is continuous in A in a neighbourhood of 3, = 0, we can replace e,(O) by ei in (3.20) where e1 and ez define the two-phase state of a (9, O+). The resulting equality together with (3.9) gives, for el < Q < ez and almost all rl, . . . , rk_ 1, ii; (. . . ,d=

‘e- @iIj$(..*,p) c (e2 - Ql)

i=l,Z

I

.

(3.21)

56

D. I. GATES

Combining this with proposition 1 gives proposition phase fluid states, where IZ*(y, ei) = pi, proposition

2. For the special case of two2 reduces to (1.4).

One can apply the methods of this section to determine the short-range behaviour of the Ursell correlation functions and the direct correlation function6). One finds in particular that, for one-phase fluid states, these functions also rend to their reference-system values, as in (I .3).

4. Long-range correlation functions To calculate the long-range distribution functions & defined by (1.5) we generalize the result (2.5) to include a k-particle generalized Kac potential i;f’“- ‘jV K,(yx,. . yxr), where 3. is a parameter. That is, we consider a system nith total potential energy

We specify that Kk (y, , . . ., yk) is symmetric in yl, . . . , y,, is invariant under any on any simultaneous translation of y, , . . . , y,, is bounded, is Riemann-integrable bounded region of the space of yr , . . . , yk, and satisfies for all yI , . . . _vr the condition IKk(yj. . . . . _Vk)l<

c,,rJ

IYO - Yll-y-6,

(4.2)

where C and (5 are positive constants. The condition (4.2) was also considered Fisher”). All the conditions ensure that the Riemann integral XI;

f

jds,

‘~._idSk-lKk(y,y

+ sr,...,y

+ S&l),

by

(4.3)

exists. is finite and is independent of y. The functions q and K are assumed to satisfy the conditions given in ref. 1. Then, by minor modifications to the proof used in ref. I to obtain (2.5). one can rigorously prove the following Lenma 2. Under the stated conditions ing to (4.1) has the limit &. (0. O+) E lim ui; (~1.;1) = ,-+O

inf

the free-energy

G”(n).

density un (,o, y) correspond-

(4.4)

flE%k(O)

where

G'(n)= G(M)+ (R/k!lTl) j dy,n (~1) j dyzn (yz) I’

...

j

dy,n (J'r) f6, (~1 >. . . , J./‘),

(4.5)

CLASSICAL THEORY OF CORRELATIONS

where G is defined by (2.6), and all integrals infinite except for the y1 integration.

57

IN FLUIDS

in the second term on the right are

One can combine and generalize lemmas 1 and 2 and obtain the free-energy density a (!?. 0 +) for a system whose potential energy is the sum of terms due to different many-body short-range potentials and many-body Kac potentials. In certain special cases the result simplifies by the method of ref. 12. For example, if we have m - 1 generalized Kac potentials, y’k-l)V& (yxr , . . . , yx,) for k = 2. . . . , m, which are all nonpositive, we find that a (I), 0+)

= CE

a”(e) + +“2e2 + $

The proof of this result is outlined Propvitiorz

m.

in the appendix.

3. If Q is a one-phase

.

(“3~~ + ... + -+

>

(4.6)

Our next main result is

state then, for almost

all s, , . . . , sk_ 1,

ii; (s, . ***,%-I,@) = U/l~*k”‘,ej,dyn* with the notation

of proposition

b’>d n* (Y + Sly d ... n* (Y +

sk-l,@),

(4.7)

1.

We note firstly that (4.7) like (3.10) is independent of the choice of n*. Secondly, we use the functional relation (3.8), which now takes the form (llk!)jdr,

~..duk_,y’k-“Y~k(y,y

x iik (Yl , . . . .

f y~r, . . ..y

+ yrk_r) (4.8)

Yk-,

Changing to the integration variables si = 7~~ and taking resulting equality gives, under our previous assumptions (1 jk!) j ds, ‘*‘ds,_,&(J’,J’ =

$

+ S1, . . . . y +

d (@,O+)I,=o.

Now consider a one-phase state e of the modified By assumption (2.8) and lemma 2 we can write

d (Q, 0 +) = GA@*“),

the limit y + 0 of the

sk-,)n~(s,,...,sk-,,@)

(4.9)

system with free energy ui (&O +).

(4.10)

D. J. GATES

58

where PZ*~(y, Q) is any minimal function (3.13), we here obtain from (4.5)

in V(p) of G”. By the argument

leading

to

....

dy,n* (yJ Kk (y, , . . . . yJ

= (lik!)jds, x (IjjZ’*l)

... ds,_,Kk(y. j dyne(y)

rP(y

y + ~1. . . . . y + sk_,) + s,) ... n”(y

+ So_,).

(4.11)

I’*

where the latter equality is obtained by setting y = y, , si = yi+, - yL for i=l . . , (k - l), and using the translational invariance of Kk. Comparing (4.9) with (4.1 I) we obtain. since Kk is arbitrary, the statement of proposition 3. An obvious consequence of the proposition is that I?; is periodic, with unit cell T*(Q), under a simultaneous translation of S, , . . . . sk_ I. For the special case of one-phase fluid states, where II’:: = 0, the proposition reduces

to (4.12)

Ek”(. . . . Q) = “k.

Thus, fir one-phase fluid states, the particles are statistically independent on the scale of the Kac potential. For the one-phase crystalline states of the cell model where (2.9) and (2.10) hold, we obtain, with minor modifications to the definitions, ii\ (s, p) = 4 [n”:(y) n* (y + s) + n* (y -t 1) n* (y + s + l)] Q+P- >

=i

3(&

for

+ &.

.s

odd,

for

s

(4.13) even,

which has the expected periodicity. A similar result can be obtained for 12;. and so on. For two-phase states we use a method like that used for proposition 2, and obtain an analogous result. In particular, for a two-phase fluid state where n L < ? < e2, we have the uniform distribution

a( ...1?) =

(--)d

+ (aL),p:.

The behaviour of the Ursell and direct correlation limit can also be calculated. One finds, in particular, phase fluid states.

(4.14)

function?) in the long-range that they all vanish for one-

CLASSICAL

THEORY

OF CORRELATIONS

IN FLUIDS

59

5. Density function in an external field We consider the long-range (or large-scale) behaviour of the one-particle distribution function or density function n, in the Van der Waals limit and in the grandcanonical formulation. This function is not expected to have a thermodynamic limit except in the presence of a bounded, periodic external field, Y(X) say, for the reasons given by Fisher’). Also it is more convenient to avoid applying the thermodynamic limit to the definition (3.2) of nr. Instead, we define the infinite-volume density

function

f-7

n’, directly

by the relation

dx Y’(X) n; (Y; x) = - $

p (Y + jlY’)l,=,

,

(5.1)

s R

for all bounded, periodic functions Y’ that have the same unit cell R as Y. Here p(Y) is the pressure of the system in field Yin the grand-canonical formulation. As proved in ref. 1, the function p(Y) exists. Also, we expect (Ruelle17) gives a proof for lattice systems) that the derivative on the right side of (5.1) exists for a large class of states (the one-phase states). We regard n; as existing, when there is a unique function n; (ignoring sets of zero measure) which satisfies (5.1). It follows that 12; exists for all one-phase states. The definition (5.1) is motivated by the observation that the finite volume density function, defined by (3.2), satisfies a similar equation (see Lebowitz and Percus14)). We define the large-scale density nk by

12:(y; Y) = lim ni,

(5.2)

where Y?(x) = y (yx) and y is a bounded, periodic function. This field has the property that its period tends to infinity at the same rate as the range of the Kac potential when y + 0. To find nk we need Lemmu 3. For a system with total potential energy V, + CIGaSN y (yx,), where V, is given by (2.4) and y is bounded, periodic, and Riemann-integrable on any bounded region of v-dimensional space, the pressure p (y; y) has the Van der Waals limit p(y;O+)

= limP(Y7;y) y+O

= supF(y;n),

(5.3)

n 6.24

where

WY; n) =J\;clilDl)

j dyn (Y> [iu- ~(~11- G(n).

D

(5.4)

Here G(n) is defined by (2.6), the limit D + 03 is taken over an ascending sequence of cubes, and .% is the class of functions n(y) which are Riemann-integrable on bounded regions, periodic and positive.

D. J. GATES

60

The proof of this result was given in ref. I for the case where y has a cubic unit cell. The proof can be extended to include cells that are parallelepipeds of any shape. We suppose that the external field y has the property of removing the arbitrariness of the maxima1 function n* (y; y) of F (y; n) under rotations and translations. Such a y is usually referred to as a “symmetry breaking field”. We take y to have the same unit cell J as IZ* (0; y). It follows from (5.4) that n* (y; y) has the same unit cell. One of our basic results is Proposition

4. For all one-phase

for almost

all y> where I? (y; y) is the unique

states jr? is unique

and is given by

maxima1 function

of F (y; n).

We replace Y and !P’ by !PY(x) = y (yx) and Y;(x) = y’ (yx), respectively, in (5.1), where y and y’ have unit cell J; set y = yx on the left side, take the limit y -+ 0 of both sides and use (5.3), which gives

dw+W&(y;y) =

--$-I’(y. +Ayr;o+)l;,=,.

(5.6)

3 /

assuming that /z’; and the right side exist. In obtaining (5.6), the interchange of the limit y -+ 0 with the derivative is justified by the convexity in Itiofp (y + Ay’; 0+) which is easily proved. The interchange of the limit y -+ 0 with the space average is justified because of the periodicity of I$ and the expected periodicity of Now, we have from (5.3) and for one-phase states

ny.

/J (Y + AY’ ; o+)

= F[y

+ iy’ ; C* (y + iy’;

. ..)I.

(5.7)

Hence we have

$P (Y +1LY'i OfLo n* (y

+ ly’; ...)]l/.=o

+ +

[?/I + 3.y’; II* (y:

)]lizO.

(5.8)

Since E‘(y :n) is stationary with respect to variations of n in %! away from tz* (y ; J), it follows that the first term on the right side of (5.8) vanishes. The remaining term can be found from (5.4), which gives

(5.9)

CLASSICAL

THEORY

OF CORRELATIONS

IN FLUIDS

Substituting this in (5.6), and noting that the resulting equation class of functions y’, we obtain the statement of proposition 4. Proposition

5. (Van Kampen’s

PO k* (Y;

equation).

For one-phase

holds for a large

states

~11 + ~0) + s WK (Y - y’) II* (Y; y’) = 0.

where ,LL”(Q)= &z” (~)/a?, which is the chemical Also n: (y, y) satisfies the same equation.

potential

61

of the reference

(5.10) system.

This proposition was proved in ref. 13 for the case where n* is periodic. The general proof seems difficult, but (5.10) simply states that the functional derivative of F (y; n) is zero at its maximum, as one would expect. The function ny satisfies (5.10) because of proposition 4. If y(y) = 0 for all y, eq. (5.10) still holds for all minimal functions II* [see assumption (2.8)] but nk does not exist. On the other hand, suppose we replace y by Ay everywhere in (5.10) and let 1, --f 0. This, together with proposition 4, leads to the conclusion that the function fz* (J; y) = lim nT (3,~; y)

(5.11)

R-t0

and the similarly defined function nk (J; y) both exist, are equal to each other and to the particular minimal function in zero field that has unit cell J; and that both satisfy (5.10) with y = 0. (We use this later in proposition 9.)

6. Modified Ursell functions in an external field In this section we consider the weighted, modified Ursell correlation function cl,” [see (l.ll)]. Our method requires that we use Ursell functions for systems in external fields: we calculate the zero-field functions in the next section. The infinite-volume, k-particle, modified Ursell correlation function C;, (Y; X,. . . . . x,), in a periodic external potential Y(X), can be defined by the recurrence relation

= ---

I

au, (Y + iY’;x,,

. . . . x,)l&=o,

/3 i%

for k =: 1, 2, . . . , where the equality is required to hold for all bounded, functions P(x) with the same unit cell as Y’(x), and where u, (!P;x)

= n; (!P;x),

(6.1) periodic

(6.2)

62

D. J. GATES

with fz’, defined by (5.1). The discussion of (5.1), in the previous section. implies here that 0;: exists for all one-phase states. It is more u~ual’~) to define these Ursell functions in terms of the distribution functions 11~: for example. iJ2 is usually defined to be

(where the dependence on ‘V and p is omitted). Lebowitz and Percus14) showed, fol finite volumes. that with the usual definitions, the U,‘s satisfy equations like (6.1). Because of the difficulty of taking the thermodynamic limit of their equations, we prefer to adopt (6.1) as our definition. The weighted. modified Ursell functions in the Van der Waals limit are defined by Ci,“(w;y,

,...,.

vn) = lirny”-“‘“Ci,(‘Y.;4,,~/ .,-,O

,....

yr/lf’,;,J),

(6.4)

where ‘Y,(x) = ~1(yx) and U, (Y’; X, , . . . , xk, 7) is the Ursell function corresponding to the two-body potential ( 1.l). [Nore that U,” certainly does not exist for twophase states, because U,<(Y/; .vil, ..,x,)++ 0 as (xi - xj( --f c/? for such states.] The first of these functions is just the long-range density considered in section 5: c/p

(y;y)

=

t7k (y:y).

(6.5)

To evaluate the higher-order weighted Ursell functions we return to (5.1), replace Y’(x) and Y”(x) by Y’,,(x) = w (yx) and y;(x) = v’ (yx), respectively, in this equation, set yi = yxi for i x I. 2. . . lc + 1. multiply both sides by 7~(~-~)‘. and take the limit y --f 0. This gives jdY,+,YW,+,)

UhL!YLh. ....!‘ii+1) (6.6)

for one-phase states, assuming that the limits and the derivative exist. In obtaining (6.6) we expect that the interchange of the limit y + 0 with: (i) the integration. on rhe left. is justified because I!/~?+, is expected to satisfy a cluster property’) for onephase states; and with (ii) the derivative, on the right, because ;~(l-~)“Di~ is expected to tend to U,” uniformly in i. for small enough I., for one-phase states. To evaluate Uf we set 1~ = I in (6.6) and. using (6.2) and proposition 4, replace UT by y1*on the right side. To calculate this right side, we replace v by y + 1.~)’in Van Kampen’s equation (5. IO) and differentiate with respect to A. This gives

CLASSICAL THEORY OF CORRELATIONS

where the operator H, f(y) and a&)

= Pa:

H, is defined for any functionf(y)

j dy2y’ (~2) H,$+’

Since y’ is arbitrary,

(6.8)

(Y; ~1, YJ

1 gives (6.9)

= Y’(YJ.

we deduce

H,$‘,W (Y: Y,Y,~)

(6.10)

= 6 (Y, - ~2).

6. For all one-phase

~my;Y,,Y,)

(6.7) with (6.6) for k

Combining

63

by

b*Wl .fb) + ioj WfW Kb - ~‘1

= Pa0 (e)/+‘.

Proposition

IN FLUIDS

states of p (y; p, 0 +), (6.11)

= ~[fl*“(y;...),Y,,Y,l,

for almost all y1 and y2, where 9 (IZ’~,y1 , yJ is the Green function H Y’

of the operator

We have chosen, for later convenience (see proposition 10) to regard H, and 9 as functionals of n*. Eq. (6.11) confirms that U,” is, as expected, symmetric in yI and y, because H,, being a symmetric Hilbert-space operator, has a symmetric Green function. We shall deduce some more familiar results from proposition 6 in the next section. To evaluate U,” we set k = 2 in (6.6). To calculate the right side we replace y by y + 1.y’ in (6.10) and differentiate both sides with respect to ii. This gives

h1

$ +

U,”(Y + 1~':

PC

(Yi Yl> Y2)

Y, ~dA=o 3

4 b*

Y1)l5

(y;

’/

where a:(~) 3 a3a0 (e)/ap3. Combining the arbitrariness of y’, gives J-I,IU3w(y;~1,~2,~3)

= -Bai

n* cy + 1.y’; Y,)l,=o

= 0,

(6.12)

this with (6.6) for k = 1 and 2, and using

b”(y;ydl

U~W(Y;Y~~YJ

U~~(Y;Y~,Y~).

(6.13) Multiplying proposition

both sides of this by U,” (y; y, , y), integrating 6, we obtain

Proposition

7. For all one-phase

U,~(Y,,Y,>Y,)

= -b’s

dyd

over all y, and using

states of p (y; 0 +). b*(y)1

U,“(Y>Y,)

U,“(Y,Y,)

U,“(Y,Y,),

(6.14)

D. J. GATES

63 for almost shown).

all

Y,

.

yr and Y3 (where dependence

oi functions

on y and ,u is not

Clearly r/y is symmetric in Y, , y, and Y3 as expected. An equation similar to (6.14) was used as a “superposition approximation” by Percus’). To determine U,” one sets k = 3 in (6.6) and calculates the right side by functional differentiation of (6.13). One can calculate UT to any order by repeating this process. Consider now the “modified direct correlation function” C (ku; X, , .x2) which can be definedL4) by JdxC(!&‘;x,x,)

Uz (Y’:x.x,)

We define the weighted

= -n(~,

limit of this function

(6.15)

-x2). by

(6.16)

C” (yj:~‘, .Y2) = lim y-“C(~~;Y,,‘y,Y,is,). p+O

To evaluate Cw, we change to y variables in (6. I5), divide both sides by y’ and let 1’ + 0. Assuming that the limits exist. and that C/F satisfies a suitable cluster property. we obtain

jdYCW(ly;Y,.Y,)

u%&Y~.yz)

= -h(Y,

(6.17)

-VI).

Comparing this with (6.10) shows that - Cw is just the kernel corresponding operator H,. This leads to the fo!lowing. Proposition

8. For all one-phase

cw(YlY,,Y,) for almost

= -W(y,

to the

states

- Y2) -

pa;In* (y;y,)l2j

(y, - y2),

(6.18)

all Y, and y2.

We deduce that Cw(~;~7,,~,)

= -W(Y,

-

~2).

which is independmt of 11)and 9. Other refs. 6 and 14 [see also (1.12)].

for

Yl + Yz.

formulations

of this result

(6.19) are given

in

7. Modified Ursell functions in zero field Although the results of the previous section are instructive, we are primarily interested in correlation functions in the absence of external fields. Since the Ursell functions Uk are not themselves well defined in this case, we consider instead their

CLASSICAL

THEORY

OF CORRELATIONS

65

IN FLUIDS

space averages

0, (r L, . . . . Y~-~)

dxlim U,(1,Y;x,x = lim -JL IDl-+rn IDI s 1-0

+ y1 ,...,

x + Y~-~),

(7.1)

D

with 11 and Y as before. As indicated by the notation, we assume that u,+ is independent of !J? This is because the various limits lim,,, U, (LY/: x1, . . . , x,) for various Y’s are expected to differ only by simultaneous rotations and/or translztions of x, , . . . , xk. It would, perhaps, be more natural to define uk like fik in (3.1). Probably, the two definitions are equivalent, but the latter is more difficult to deal with. The weighted functions are defined by

q
s~_~) E lim~“-‘i)VU1;

(s,/v,

. . . . sl,_,:‘y, y),

(7.2)

y-0

to the two-body potential (1.1). To calwhere i?, (F~, . . . , rk_ 1, 7)) corresponds culate Dkw, we substitute (7.1) in (7.2), replace !P by u’,, set y = Y_Xand use (6.4). This gives, for one-phase states,

c’,” (s I,...,sk_,)

=

lim IDI+m

1

dylim

IDI

s

U~(?.y;y,y

+

sl,...,y

,X+0

+

sk_,).

(7.3)

D

In obtaining this, we expect that the interchange of the limit 1/ --f 0 with (i) the space average is justified because of the periodicity of the integrand; and with (ii) the limit ;1 + 0 because, as in (6.6), ,I(~-~)~ Uk is expected to tend to U,” uniformly in 1 for small enough A. Since y(y) and n* (y; y) both have unit cell J, we can show that

LJ,“(?.y;y, + b,y,

y, + 6) = u,“(r$;y

+ b ,...,

,,...r

yk),

for any vector b joining two lattice points of the lattice with unit cell k = 2, this follows from proposition 6 and the definition of 9; for k from proposition 7. and so on. Now let us regard UT (y: yt, . . . , y,, tional UT (~2~‘;y , . . . . , yk) of /P, (see proposition 7). Then (7.3) can

UT (s I.

.

..>S.-,> =

WlJI>jdyU,W [a*;Y,J’

+

SI, . . . ..I’

+

(7.4) J. In the case = 3 it follows ,u) as a funcbe written

Sk-l]r

(7.5)

J

where II” (J; y) is defined by (5.16). Here, the transferring of the limit il -+ 0 inside the function U,” IS justified for k = 2 by replacing y by Ay in (6.10) and letting 2 --f 0; for k = 3 by doing the same to (6.13), and so on. Although the right side of (7.5) is independent of y, it is not manifestly so because of the appearance of the unit cell J of y. However, by the argument conclud-

D. J. GATES

66

ing section 5, /li’ (J; ~7) is just the particular zero-field maximal function, those allowed by assumption (2.8) that has unit cell J. By considering (6.10), that if a given II’) is rep!aced by another M” diffe;.in g by a given rotation translation ol‘~‘. then Up (~1: y, . y2) changes only by the same operation on

among we see and or y, and

yr simultaneously. But this makes no difference to the space average of Up in (7.5). By considering (6.13). WC:t-each the same conclusion about Uw. and so on for a!! /,. Consequently we can replace II* (J; y) and J in (7.5) by any zero-field minimal function n*(y) and its ~:nit cell I"" respectively. which gives Propositim 9. For ail one-phase

u/y(s, for almost

,

..

Sk_

all s, .

/!

= (I q/y

states

J dy L:; j/l*:: y.

, .I

J’

i

s,

. 4‘

+

Sk-,

).

(7.6)

s\>_i

This proposition enables us to calculate 0,” for any /Cby using the method of the previous section for obtaining UT. The most important special case of proposition 9 is Proposition IO. For all one-phase

states (7.7)

for almost

a!1 s. where ?I is the Green

function

of H, defined

by (6.8).

This follows directly from propositions 6 and 9. In the case of the cell model where (2.9) and (2.10) apply, one can calculate U,” explicitly lo). It decays exponentially with a superimposed periodicity. Proposition 10 has the following Corollary. For one-phase

states

(7.8)

This is an analogue of the well-known “compressibility formula”i4). To prove (7.8). we set y = 0 in Van Kampen’s equation (5.13) and differentiate with respect to ,u, which gives

(7.9)

CLASSICAL

Multiplying 1

j-

OF CORRELATIONS

by 9 (y, y’) and integrating

a ;G

THEORY

n*(y'> = j

Differentiating

IN FLUIDS

over all y gives (7.10)

dy9 (y. y’).

the formula

p

(p.

67

0+)

= F(0; n*), (7.11)

10 and (7.11) gives Hence, Iintegrating (7.10) over all y’ in I’*’ and using proposition (7.8). The same result could perhaps be obtained by taking the Van der Waals limit of the compressibility formula for p (,u, 7). For the special case of one-phase fluid states (see section 2) where

o+>= ZP CPU, 0+)1+c,

n*(y) = @(p, one can obtain

@j((.y1,y2)

U,” or 9 from (6.10) by Fourier

=

transform.

This gives

_!_idpexp~~‘(y~-y2)1,

(7.12)

az(e>+ a4 where k(p) is defined

by (1.8). Substituting

Pvoposii+ionII. For one-phase

then in (7.7) gives.

fluid states (7.13)

for almost

all s.

A more revealing

way of writing

this result is (7.14)

where f is defined by (1.7) and i?p is the space-averaged, modified Ursell function of the reference system. To obtain (7.14) from (7.13) we simply use the compressibility formula14) j dyui (v, Q) = [Da: (Q)]-‘. The first term on the right side of (7.14) can be regarded as the contribution to 0: due to the short-range potential q(v), and the second term as the contribution due to the Kac potential YAK. For s # 0, only the second term remains, giving the grand-canonical formulation of (1.10).

D. J. GATES

68

Eq. (7.4) illustrates that the order in which some of our limit operations taken is important. Since f(s, 0) is continuous in s near s = 0, unlike u:(s), deduce that hm lim y-l’rsiz (s/y, y) = lim u,“(s) s- 0 j,+O 7-O

are we

(7.15)

= ,f(o. p).

On the other hand lim lim r-POz 7-o r-0

(s/v, y) = lim j,-y(-yz) ;’- 0

= -_x.

The original (unmodified) function of Ursell of uz, for one-phase states of p (p, y), by

(7.16)

U2 (v. y) may be definedg) in terms

(7.17)

c2 (v, y) = u2 (I’, y) - 0 (/& y) O(r). The corresponding u;(s)

weighted

function

= O,w(s) - 0 (/L, 0+)

Uy, defined

like (7.2)

is therefore

given by (7.18)

n(s).

It follows that z?y is given by the right side of (7.10) with i?i,” replaced by i$. the unmodified Ursell function of the reference system. An expression for the function i?,” can be obtained by combining propositions 8 and 10. In the special case of one-phase fluid states where n* = e (p, O+). this espression can be simplified by taking Fourier transforms and using eq. (7.12). This gives Proposition

i7y (s,

12. For one-phase , s2) =

--(a::/~‘, X

for almost

fluid states

.rs dP,

exp [2zi (p, - SI + P2 - sdl

dP2

k4

+

@PI)1

k4 + RP,)l

+

P2f

of CE~and ai is 0 (,u, O+).

It is probable that the right side of (7.19) would be 03 (r, , r2, y) were expanded in a form like (1.6). A similar obtained in ref. 6 as a term in the expansion of a (Q, y) in We can define cw in terms of Cw by a formula like (7.1). to proposition 9 can be applied (perhaps more dubiously)

c”(s) = -PK (s) - P [S(s>/ll’*ll j dya; I’*

(I. 12), and simplifies

+ R(P,

(7.19)

all s, and s2. where the argument

which confirms

[4

obtained as a term if looking expression was powers of y. The arguments leading to cW. This gives (7.20)

[n*(y)],

in an obvious

way for one-phase

fluid states.

CLASSICAL

THEORY

OF CORRELATIONS

IN FLUIDS

69

8. Discussion Our results are summarized by the propositions and corollaries. They show how to sensibly define and to calculate correlation functions in the Van der Waals limit. Special cases of our results are equivalent to those of other authors, including the well known results of Ornstein and Zernike. Our results apply also to lattice gases. By simple changes of notation, one can obtain corresponding results for the spin correlation functions in the Ising model of magnetic systems. As pointed out by Lebowitz and Penrose4), the result (1.3), which gives fii for a one-phase fluid state of a (Q, O-t), presents a certain paradox. Suppose K is positive-definite [i.e. &) > 0 for all ,p]: then, as shown4), a (Q, 0+) = a’(o) + +3,02 where n = s dsK (s) > 0. Suppose also that U”(Q) has a first-order phase transition for Q, < Q < cb: then $ has a two-phase form [like (1.4), see ref. 81, but a (9, O+) clearly does not and hence, from (4.12) neither does Ek. This means that the system has a two-phase structure in the short range [i.e. the range of q(r)], but a one-phase structure in the long range (i.e., the range y-l of the Kac potential). To understand the cause of this structure, we consider the surface energy 0 due to the imposition of a Kac potential on a reference system in a two-phase state. A rough estimate of g is14): (positive

constant)

x (--a)

x (surface

of two-phase

interface),

which must be smallest for the most probable structure of the system. For II < 0, we see that cr is small if the surface area is small: hence the separation of phases persists in the long range and is indicated by a (e, O+) and Ek. But for 1 > 0, a large surface area gives a small C: hence the phases tend to be mixed up by the Kac potential and there is no separation of phases in the long range. This resolves the parad ox. The most important unsolved problem is the justification of assumption (2.8) concerning the existence and uniqueness of the extremal functions IZ*. It should be possible to prove the existence of n* (y; y), [i.e., the attainment of the supremum in (5.3)], using the fact’) that F(y; n) is upper semicontinuous in the space L, It is thus only necessary to prove that 9f?has a compact subset whose image under F contains the desired supremum as a limit point. Our other assumptions throughout the paper may be more difficult to justify: in particular, the existence of the functions Ei, ti: and i?,“. (The existence of E, and or for y # 0 has not yet been proved.) Physically, one can understand why such limirs might be difficult to establish: for instance, if E, (u, y) has oscillations in P. then iii (s/v, y) has oscillations in s which may become more rapid as ;I -+ 0. Possibly, this difficulty can be overcome by using correlation functions which are locally space-averaged over a region cc)which is small compared to y-l. For example, one can define a longrange density by

D. J. GATES

70

where PJ is a cube of volume

/w/ centred

at P = 0, and /zI is the usual one-particle

distribution function for volume M in the grand forma!ism. If K = 0 but y # 0. we can prove rigorously”). under fairly weak conditions on y’. that fiy exists. and saiisiics

This is just the standard condition for hydrostatic case of Van Kampen’s equation (5.10).

equilibrium,

and is a special

~lrc i hu der Waais equation for mcmy-body attractire itzteractiom We outline here the derivation of the generalized Van der Waals equation (4.6) for a system with nl - 1 generalized Kac potentials, $-l)\‘Kk (yx, , . ..) pk) for /< E -. 7 . . . . ~1 - 1, which are all nonpositive. We deduce this result from the variational principle (lemma 2), which takes the form U(C). C+)

=

inf G(12. a), IIs%

(A.1)

lvherc G (17. x)

We

deduce

z (l//r’l)j

dya’ [n(y)]

(4.6) by obtaining

upper and lower bounds

on CI(0, O+). using (A.1).

The latter gives, since Q E g(e), a (,o, 0 +) < G (Q, m) = Q’(P) + Eli=, Since 11(I, 0+) is convex we deduce that

(l/k!)

a,ok.

nI

a (“. 0) < CE

a’(o) + C (I/k!) k=2

a,~”

(A.3)

. >

To obtain an upper bound we use the famous inequality and arithmetic means of positive numbers. This gives

relating

the geometric

(A.4) Substituting

in (A.2) gives, since Kk 9 0,

G (17, XI) > (Iljl’l)

1 dy r

(1’ [/Z(y)] + f (l/k!) k=2

%,$?(Jj)’

* >

(A.5)

CLASSICAL

Lemma

THEORY

OF CORRELATIONS

IN FLUIDS

71

7 of ref. 12 states that, for any functionf(?).

irlf U/l~l> _f4vfW)l

nt%(g)

r

(A.61

= CEf(n).

provided that the left side exists. Applying this to (A. 1) and (A.5) yields the reverse of the inequality (A.3). Combining the two yields the desired result ,?I

a(p, O+)

= CE

a”(9) +

c (I/k!)a,$

h=Z

.

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