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A power law for the correlation length close to the gas-liquid critical point
Physicu 63 (1973) 263-287
0 North-Holland Publishing CO.
A POWER LAW FOR THE CORRELATION CLOSE TO THE GAS-LIQUID
CRITICAL
LENGTH POINT
0. SEEBERG Institut fiir theoretische Physik der Universitdt Frankfurt/Main, Frankfurt am Main, Deu tschland
Received 15 March 1972
Synopsis The state of a system is characterized by occupation numbers of space cells. The correlation matrix G,, corresponds to a linear operator the eigenvalues of which give the “critical” part of the grand-canonical potential. This critical part is in the Omstein-Zemike case only a function of the correlation length A. By use of the thermodynamic definition of the isothermal compressibility ti and the 02 relation 3~z A2 a consistency differential equation for A is derived and solved for the critical isotherm. The solution turns out to be a power law: 1 N” [Apj -Z/5 ; or, again by use of the 02 relation: 3~z 1Ap1-4k
Within the framework of the theory of the scaling laws it is assumed that the critical thermodynamic quantities follow a power law close to the critical point. It is the aim of this paper to deduce such a power law for the correlation length using assumptions which lead to the Ornstein-Zernike theory. We consider a grand-canonical ensemble. The volume ~2 is divided into cells forming a cubic lattice. Thus we get a coarse-grained description of each system S by the set n of the occupation numbers Ni in the cells. In particular we get a coarsegrained correlation function Gij as follows: 1. Introduction,
The matrix (G,,) on the other hand corresponds to a linear self-adjoint operator and its eigenvalues and eigenvectors have the following physical meaning. The eigenvectors form a system of independently fluctuating observables while the eigenvalues give the fluctuations of these observables. From the translational invariance of the expectation values it follows that the eigenvectors correspond to plane waves while the eigenvalues are the Fourier components of the correlation function (appendix B). 263
264
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In section 2 we recall the Omstein-Zernike description. We write for the probability P[n]:
theory using the coarse-grai
where K is the grand-canonical potential. That means that the fluctuations assumed to be distributed according to a gaussian distribution. After smooth out the fluctuations (appendices A an4 B) we get by means of a Taylor expanz of the fluctuations an expression for the smooth ‘&direct correlation functi which corresponds to the operator K. Therefore the operator G_, is known. I the inverse of the operator K: G = @- X-l. Thus we arrive at the old Ornstt Zernike theory. But now the critical part of the partition function (or the gra canonical potential) can be calculated from the eigenvalues of the operato By “critical part” we mean the part due to long-ranged fluctuations. Now central idea is that only these critical fluctuations contribute to the crit behaviour. The isothermal compressibility, on the other hand, can be derj from the grand-canonical potential : K=
-P
(BP d Q
and 3G=
-2
02
(e> ap2
*
Thus x can be expressed as a function of the eigenvalues. But these eigenva: themselves are functions of x by means of the OZ theory, especially the Ornstt Zernike relation 3coc P. Thus we get a consistency relation in the form c differential equation. We solve this differential equation for /3 = PC* The SI tion is:
A = lAPI-2’5 llog
F(AP)~I~'~ G,
where F and G are constants. This solution is equivalent to a power il a lA/~]-‘l” when we use the definition of the critical exponents as given Griffith4). Of course it is possible to extend these considerations to the genl critical behaviour in the framework of the OZ theory. This will be done i forthcoming paper. 2. Ornstein-Zernike theory and Landau ansatz. We consider a grand-canon ensemble g which is characterized by the chemical potential p, 18= (kT)- l the volume Q. Let Q be a cubic box. This box is divided into cubic cells 2 equal volume Y = a3, so that the cells form a cubic lattice A. We considt system S E 9. A coarse-grained description of S is obtained from the set II of
CORRELATION
265
LENGTH CLOSE TO THE CRITICAL POINT
occupation numbers Nt, defined by
(2.1) We define the correlation matrix (G,,) by G, = <(K -) (4
(2 .2)
- <%>)>3
where the mean values are taken over the ensemble g. GtJ is related to the usual correlation function g by G ij =
z
Iz f
Ke)”
g
cc
f r2)
+
6 @I
-
r2)1
dr1
dr,
(2
f
l
3)
I
where g (u, , r2) = et”) (rl , r2) - (Q)~. For these definitions see ref. 1. G,, corresponds to a linear self-adjoint operator. This fact will be- used in the following analysis. Now let P[n] be the probability for n in 9. P[II] can always be written in the following form :
where K (p, /I, In) = -p (p, p) 52 is the grand-canonical thermodynamic potential. Eq. (2.4) is of course only a definition of K[n]. On the other hand, the grandcanonical probability P (x, IV) is given by: P (x, N) =
(N!h3N)-1exp [ -/?HN (x) + IV@ - Q&9].
(2.3
Thus we get:
where %S[?t]is the set of all configurations
exP c-m
2m7c
3/2N 1
( >
[rtl) = 7 Bh
which lead to the
Yg- exP .
Wi@~
j
exp
%mtl
E-PWdl
n. Thus we
QN. (2.6)
Now we write: j
exp c-BV
it3htl
ml QN = =P c--Bml)
j dqw x3h3
(2.7)
The integral on the 1.h.s. is of course the usual Boltzmann measure. We get:
sdqN = a3NN! (v
EMItI
N,!))-'.
(2 8) l
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266
By use of Stirling’s formula we get from (2.6) and (2.8) :
.
(2 9) l
Here we used that Ni + 1, which means that the Ni are macroscopic o bservables . Now V[n] itself must depend on & Therefore V[n] is not a true macroscopic observable, for such a variable must be uniquely determined by the vector n. We assume the existence of the most probable vector E: for all Furthermore
n * E.
(2.10)
we assume that P[n] can be written in the following form:
where SN 1 = Ni - Ni,
K,, =
d2K aNi aNj
[Iii
and c = exp(@K - @Km]). (2.11) is the first of our main assumptions. It is difficult to justify this assumption. Let us only emphasize that the occupation numbers Ni are macroscopic observables ; this is a crucial point. Furthermore (2.11) leads to the Ornstein-Zernike theory and our considerations are within the framework of this theory. It is therefore beyond the aim of this paper to discuss the validity of (2.11). n and i7 belong to the same ensemble g (p, /3,Q), which means that p, p, Q remain fixed. From (2.11) it follows now that (NJ
=
&,
G,, = (l//3
K,f
(2.12)
The proof can be found in appendix D. Now let us perform the smoothing S, defined in (A9). We get:
Now we replace &t (r2) on the r.h.s. of (2.13) by its Taylor approximation: (2.14) where R, = r20 - rl,.
CORRELATION
LENGTH
CLOSE TO THE CRITICAL
POINT
267
By use of the fact that K(R) is (almost) rotation invariant, we get after a short calculation :
(2.15)
= A I (8n)* (r) dr - (B/6) j grad* n(r) dr,
A
=jR(R)dR,
B
= j@‘?)R*dR.
The Taylor approximation (2.14) of course can only be valid if K(R) descreases sufficiently rapidly with increasing IRI and 6fi (r) is sufficiently smooth. Now R(R) corresponds to the usual direct correlation function and it is usually assumed that this function indeed fulfils the first condition, The second condition at least is fulfilled for long-range fluctuations, i.e., critical fluctuations. Eq. (2.15) of course corresponds to the well-known Landau ansatz2). Now let us interpret the quantities A, B. From (2.12) we get : (2.16) Using the fact that GiI, and Kkj are translation-invariant,
we get:
c Gk 1 Kkj = B-l* k
j
Now from (B7) and (B9) we get:
c
K(O)=j&R)dR=
&=A.
Rip*
Thus A = p-l
Furthermore
c
i,k
Glk
we have from (2.3): =
y31
Gik
kc
= Q(e) (1 + 4~ (e) s g(R) R2 dR) = Q(Q)~ x/?-l,
(2.17)
where 3~is the isothermal compressibility. We used the well-known compressibility integralI). Thus we have finally: A = ((@)’ m3)- l.
(2.18)
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Now we turn to the interpretation of the quantity B. First of all we construct R from (2.15). Using the fact that x is an operator E a[R] and 81 is an arbitrary vector in 9, the unique solution turns out to be: (2.19)
The calculation can be found in appendix C. From (B7) and (B9) we therefore get, that the eigenvalues of the operator K are given by K(v) = A - 4x2~v2/6L2,
WED.
(2.20)
Thus we get from (2.12):
- q1 r2) = WC
=( -
z)+JZ1exp
A
(F(v,R)).
VED
(2.21)
The eigenvalues of G are of course given by G(v)
=
8-1
A
-
m!?? -I, 6L2
>
VED.
The r.h.s. of (2.15) must be positive. Hence A > 0 and B < 0, so that the eigenvalues are all positive. Now we use the invariance of the trace of a matrix in orthogonal transformations. We get
ii = CG 1
TrG.
(2.22)
Using the fact that all Gii are equal, we get
The sum is transformed into an integral by v (a/L) = 5. Thus we get: py3Gii
=
5
s
1
dC 1 + 47r2A2E2/a2 ’
D = {& I.&lG 3),
A2 = IBl/6A.
(2.23)
We choose two spheres KI of radius 4 and K2 of radius + 2’ with & c D c Kz
l
CORRELATION
LENGTH CLOSE TO THE CRITICAL POINT
269
Thus we get: ax < Gii, 1 - a arctan ax a > 3a2
Gil < -
@ PI
21i2 - A- arctan ax
Ax21/2 a
. >
Thus we get for the case A/a & 1, which is true close to the critical point :
where f < S < 2! Thus we have: (2.24) This is the desired interpretation meaning : Gii
wi -
=
Furthermore We have:
of the quantity IBI. Glf of course has a physical
wa2)*
we get a new expression for the correlation length defined in (2.23).
A = /3-l
(F G,,)-’ = p-ly3 (&Gik)-‘.
,
Inserting this into (2.23) we get: (2.25) Thus we get the following interpretation: A2 is the ratio of the sum of the nondiagonal elements to the diagonal element Gii of the matrix (G,,). Close to the critical point this ratio becomes very large. Furthermore it can be seen that A is independent of a only if Gii oc Q? Now it can be shown that the OrnsteinZernike assumption yields that the correlation function decreases cc r- l in the critical point5). We assume that this is also true in a neighbourhood. Then, indeed, we get : Gii K a’. On the other hand we get from (2.17) and (2.25): da5
a2
=
2q3Gii
2
= -16Ce)’ x In
(2.26)
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270
where IB’I = lBl/a3. (2.25) is the Omstein-Zernike relation. We assume that II?’ remains finite close to the critical point and furthermore that IB’I does not exhibil any critical behaviour. This case we call the Ornstein-Zernike case (O.Z. c.)_ 3. Calculation of the gmnd-cunonical potential. From (2.11) we get = 1
CCexp(-BK[n])
= exp Q9K(p, /9,Q) - PK [El) C exP ( - *8 C Kij 8Ni 8%) II
l
(3 1’, l
Therefore we get: K(p, p, In) = -QP(/3,
p) = K[fq - p-l
log s,
where
At first we calculate S. We use the spectral representation
of the operator K:
Thus we get:
The sum is transformed into an integral and we get:
=ndrlp, With Et fi”x, =q”andndXo an orthogonal matrix, we get:
which follows from the fact that (f/j iI
Thus we have:
or
(3 3
. P
\
l
Lcrr /
From (2.20) we therefore get: logs
A -
= -:clog$ VED
(
4X2BV2 --z-’
>
(3.4
CORRELATION
LENGTH CLOSE TO THE CRITICAL POINT
271
Here we used the fact that the eigenspace &, is at least of dimension 2. Now we transform the sum into an integral by (a/L) vi = & Thus we get: l
logs=
-*,3Jlog$(A-F)d&
(3.5)
I5 D is defined in (2.23). From (3.5) we get:
(3.6) Now let us put
Then we get for the integral on the r.h.s. of (3.6): J
=-
+Y 3
s
3
log(l
+
q2)dtl
D*
( > 5
1
where D* = (~1 lqll < d/a}. Again we choose two spheres & , Kz of radius x;l/a and 2%@2, respectively, with Kl c
D*
c
(3.7)
K2.
Hence there must be a sphere K* of radius a* with
1 log (1 + 7”) dq = j log (1 + 7’) dtl. D* K* Then : log
s =
-$y”
log
(z >
+ f
2
- + arctan $-
--log{$f[’ 1
2(a*)3
1 (+j++g($+) = 3.12
+
($)2]}+~n2:o*)’
W)
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272
Now, at least in the 0.2. case, log S remains finite in the critical region. On t other hand, log (@A/2x) diverges. Thus, we conclude that 746(~*)~ - l/a3 mt tend to zero. We choose therefore: a* = a (~/6)l/~,
(3
which is compatible
with (3.7) and we get from (3.2), (2.9), (3.8) and (3.9):
P(/Q)=(@)
p+$l~g~-fi-;log(,) >
( 1 2/Ya3
>I
1% 2 a2 ---
3 A2
7c 0-
213 +
6
$
arctan[$
(:)“‘I (3 .
Until now no explicit assumption has been made concerning cell length a. Let us investigate two different cases: I
l l
II . l
Ala =
D = const.
a = const.
the magnitude of t
(3. (3
l
Case I means that a is increased when il is increased. The conclusions in this cz can be derived with the following more physical argument. The critical phenome should only depend on the critical fluctuations, that is, only Auctuations w. lvij < EL/A are responsible for these phenomena. Note, that in our consideratic until now all fluctuations with /vi] < L/2a have been retained. F is a consta Furthermore, the determination of the eigenvalues of K [(Cl)] will only accurate for those 21which fulfil the condition a2v2/L2 < 1.
(3 .
Therefore one may retain only the critical fluctuations in calculating the grar canonical potential. More exactly, the contribution of the remaining fluctuatic can be considered as a constant with respect to the necessary differentiatic This idea leads to a differential equation the solutions of which are equivalent far as their analytical form is concerned to those of the differential equati corresponding to case I, Case I corresponds to the Kadanoff idea3) in the following sense. In order eliminate the cell length, which must cancel out, we put A/a = D. Then it tul out, that the results explicitly depend on the ratio D. But this fact must not taken too seriously. Let us assume, that (2.11) is valid for a fixed cell length
CORRELATION
LENGTH
CLOSE TO THE CRITICAL
POIl+iT
273
Let us increase this cell length by a factors S: a’ = sa. Then we get after some calculation, that the new probability P[~x] can bk written in the following form:
where e explicitly depends on s. Thus we are led to the following resolution of the above difficulty: There is a maximal value Dm, for which (2.11) is valid. For D < D, the results have to be changed as e depends on s. We decide to choose D = D,. D therefore has a physical meaning. Case II yields results, which have no physical sense. The treatment of this case can be found in appendix E. Of course there are still possibilities other than I and II, Now let us suppose for a moment, that all cells are enclosed within walls, so that no interaction occurs between different cells. In each cell there are (N)ym3 particles. Repeating our considerations, we find that the potential K in this case can be written in the following form: R = 1 Ki =
$
log g
p+$log~-
-(N)
> l
K cannot exhibit any critical behaviour, the same holds for the first term and therefore for the second one too. We assume for a moment, that iT = G. Then the second term is equal to K[B]. Thus we are led to the assumption that K[ti] cannot exhibit any critical behaviour. 4. Derivation of the consistertcy diflerential equation and its solution for a special case. The power law. Eq. (3.14) led us to the assumption that only log S is re-
sponsible for the critical phenomena. From (3.2), (3. lo), (3.11) we get: P(BtP)=<@>
(
3
I”.+$og,-fi-
1 -j- log (e>
2mx
Ph
>
-&log(g(l +E))+SF, where E = (6~~)~‘~D2 and +
Now we use the thermodynamical
relation
(4. 1)
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274
On the other hand we get from (2.26): x = 6A2((Q)’ jB’I)-k Furthermore A =
from (2.18) : D3
1
D3 1B’I
(e>’ m3 = (~)“d~
=
6P
l
Therefore we get
02 -II 2
6il2
2x=--*
I PI
au
On
the other hand we have: 2 aP a/J2
where :
p..=&+f+ H = 2F - log
H),
y
(1 + E)
x:
l
>
Thus we get by differentiation: 2
-0
aP2
D3 = p” = _
F
[6OlogA - 35 + 12 (H - logj9)]
2P + K[-15logA R4
+ 5 - 3(H-
lag/3)] . >
Now we put for brevity: &)
= &D3 [60 log il - 35 + 12 (N - log /3)],
g2@) = +D3 [-15logil
+ 5 - 3 (H - log@)].
Thus we get the following differential equation from (4.5):
CORRELATION
LENGTH
CLOSE TO THE CRI’TICAL POINT
275
Before solving this differential equation let us discuss the results which follow from the “physical idea” mentioned above. We have 1 log s. P ct = PQ Instead of (3.4) we now get log S” = -fClog$(A-F), VEP
where
Again we transform the sum into an integral by lv@
= 61.
We get: log S” = -3
(0 (1 -t 47c2p253 de Q +
g b
=-s(p)” a3
(-flog: -
+c* , >
c* = J log (I + 4x2F2t2)de. 115 With (2_18) we therefore get: *
Per = --
F3
w
(
log
w +c*
@
12za2a3
~$~-log~+*+log~~].
> (4.7)
Now the comparison between (4.7) and (4.4) yields that PC*,and pCr are equivalent as far as their analytical form is concerned. Now we only use that analytical form in the following calculations. Therefore the analytical form of the solutions cannot be altered. Let us now solve the differential eqbation (4.6). At tist glance this equation seems to be highly nonlinear. But this difficulty can be resolved very easily. We choose A = x to be the independent variable, while it’ = z is now the
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276 unknown function. We get:
where zfl) = dz/dx. Let us substitute y = z2. Thus we get:
(4 8) l
This is an ordinary linear differential equation which can easily be solved. g, has no zeros in a sufficiently small neighbourhood of the critical point. Thus, defining
a1 (4 ml =xg2
we
0
12/3x6
ecx1 =
’
P’I
g2(x)p
get
(4 9) l
The solution is:
where 2 is an integration constant. After some calculation we get: 2g 1 dx = -810gx s
+ 2log[logx
+ 3(H-
(4.10)
log/?) - $1.
xg2
Now we calculate the integral J:
=
s
12’
x2
PI
v = -51B'p3
8P = -5p703
[lug x + 3 (H - lug @) - +I” dx
g2
s
[logx
+ $(H-
(-_i
log@ -
+]x-“dx
(log x + 1) + ; r+ - +(H - log@)]
x
>
l
CORRELATION
LENGTH CLOSE TO THE CRITICAL POINT
277
In this way we get:
(4.11)
Now we consider the coordinate system (p, 18). Let us consider the point C;, /3) on the line /3 = const, where ;3.= x has its maximum xM. At this point there must be dA 2 = -=o,
therefore
y = 0
also.
dP
Thus we get: Y
SPx8
= 5 /B’I D3
1 + x
[logx + 3(H-
log@) - $]-2
log Exexp (3 (H - log 8) + 31
(4.12)
l
>
From (4.12) it is seen that indeed y > 0, as it should be. This point is taken up in appendix E, where case II is considered. In particular we choose the line 16= PC. The maximal value of x = 2 therefore is infinite. Thus we get: Y =
5;;;;;3 Ilog [xexP 6 w - 1ogP)_!_ log Ix exP (3 (H x
x
-
1% B) +
3)l
+)I}-” (4.13)
l
Now we have y=
da
2
(> -
l
dP
Thus dx -=
+ (log {x exp 13 (H - log/D +
diu
1% (x =P L-k(H - m/9
-
31D3 SII ’
(4.14)
278
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where q = _+l, so that we get:
{x =l(i,;tD3)+S* w-
log “P I-&cH - log8) - +I> x7’2(log (x exP f-k log 8) + 3w
dx
co
(log {xexp[+cH - log b) + +I} - logd
dx .
Now we put (4.15) Thus we get:
ilru =11
-
5’2(log AX)%
-(= 1*{ 500
1 - 0 [(* logA1x)*]}]
(4.16) @ is the error function. Now we use the following estimation: $ [ 1 - Q(x)] N exp ( - x2)/2&
(x > I),
from which we obtain: (4.17) We consider only the leading term (Aa & 1) and get: (4.18) Let us reverse this expression. Putting 125 IB’I D3 (A~)~/32/9&
= y;
(a A)-s = x,
CORRELATION
279
LENGTH CLOSE TO THE CRITICAL POINT
(4.19)
Y = x llog xl.
Let x(y) be the solution of (4.19). From log y = log x + log Ilog xl, lim,,o kg X(Y)/l% Yl = 1. Therefore we write the solution in the furm:
we
get
x = Y IlWYY dY). Inserting this into eq. (4.19X we get: 1 =
y(y) Ilog xl lb
YI-l*
Thus we get: lim,,* v(y) = 1. Therefore we have l
lim
X(Y)
Y+O
-
Y llogYl-l
=
0 3
Y bgYrl
or x(y) N y Ilog yI - l, leading to :
(614 -5 = 125 1B’ID3 (AP)~ N
32p G5
125 IB’I D3 (AP)~
-I f
w cm5
2 -= lApl-2/5 jlog 8(Ap)zj1/5 G,
(4.20)
Thus we have obtained a power law: it oc IApld215 for the correlation length 2 cluse to the critical point along the critical isotherm b = PC. We shall investigate the general critical behaviour by solving the differential equation (4.6) in a forthcoming paper. The difficulty which hereby arises is that for lines @+ PCthe initial values are not known. We therefore must introduce some additional hypotheses. In the case b = PC we have only used that il must diverge, when we go to the critical point PC = (p,, PC). From (4.20) we get by use of the Ornstein-Zernike relation (2.26): 3c = IApl-4/5. Acknowledgements. I wish to thank Professor M.Wertheim deaux fur many helpful discussions.
and Dr,D.Be-
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280
APPENDIX
A
Smoothing. Theorem(A1). LetA beanisomorphismfromavectorspaceeof dimension y3 into a vector space 8. In both spaces a product is defined. Let A have the following property :
(Aa, Ab) = (a, b) .
(AZ)
(I) Then an isomorphism H from the algebra a[CJ of the linear operators in G into the algebra a[%] of the linear operators in 8 is induced by: ALa = H(L) Aa,
where u E e, L E a[G]. The proof can be omitted. (II) Let L be a self-adjoint operator E a[eJ. The eigenvectors {nr> form a coiplete orthonormal system in 6 Then H(L) is self-adjoint, its eigenfunctions are given by {Aq} and the eigenvalues & are left invariant. Proof: 1. (Aa, H(L) Ab) = (Aa, ALb) = (a, Lb) = (La, b) = (ALa, Ab) = (H(L) Aa, Ab). 2. Ri = (nil Ln,) = (An,, ALq)
= (An,, H(L) An,).
let (G,,) be the correlation matrix defined in (2.3). (G,,) corresponds to a linear self-adjoint operator G in a vector space G of dimension y3 = QaB3. Its eigenvalues are positive. This follows from the use of the grand-canonical ensemble g. In G we introduce the usual product: Now
(W
(a, b) = 1 aibi.
Now we consider the vector space % of the step functions defined on JI 8 = {fl
y19 r2 E zt Tf(r,)
=
f0
i} .
for all
WI
In % a product is defined by:
W) Now we construct an isomorphism A from @ into 8 by:
for
FEZ,,
0
otherwise
l
CORRELATION
LENGTH CLOSE TO THE CRITICAL POINT
281
Therefore we have:
(a, b) = (& Air), and the presuppositions of theorem (Al) are fulfilled. The eigenfunctions of any self-adjoint operator K are of course step functions. This is inconvenient, because they are not differentiable. Therefore we construct a third vector space 32by
where
m
Zi(r) = B[e,] (v) = C exp VED
Here Xt is the centre of the cell Zr and D is defined by:
D = {VI
2n-k 1 =r}*
141G n,
In 9 again the product (A5) is introduced. we have
&has the dimension dim R = y3 and
where we have used that C exp V
(-9
(v,&
-
XJ)YB3 = &km
Thus the mapping B defined in (A7) is again an isomorphism which fulfils the presuppositions of (Al). The eigenfunctions now are differentiable. Therefore we call the mapping
the smoothing operator. Let us, for convenience, compile our notation
Let K be an operator E a[G]. Then the corresponding operator &(K) E am] is denoted by g, while the corresponding operator H2 [HI(K)] is denoted by R.
282
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I? and E correspond to integral operators :
APPENDIX
B
Consdruction of the eigenvectors and eigenvalues of the operafor G. The function G Is given by [see (All)]:
At first we expand e into a Fourier series:
After a short calculation we get:
e(fl7PI = c exp
-$(u,x,,
- F(p.&)
m
G,,y-3. >
Now Gij can only depend on the difference vector Rij = Xi - Xj, because the expectation values ‘of the observables xnust be invariant under the subgroup T, of the translation group T. T, contains all translations corresponding to a vector of the form t = {m, a, m2a, m3a),where the mi are integers. Thus we get:
G = ym3 C exp t
-F
(U + p,Xi)
)
C mi
eXp
(~(P3Rijl)
Gil,
w
(
where mi contains all vectors Rij with - Rij + Xi E Q for fixed Xi* NOW we make use of the following assumption about T: For physical results the sum
can be calculated, as if it were independent of Xi. This is true in the thermodynamic limit. With (A8) we then get:
CORRELATION
283
LENGTH CLOSE TO THE CRITICAL POINT
Inserting this into (B2) yields:
(7 =
c exp
F
(tl, r1 - rz)
(
Q-W(v), >
where K(v) =~@Z)exp
(+,R))dR
m Thus we get by use of the properties of Glj:
tt, rl 1 sin -2x (tr, r2) Q-lK(fl)*
+ sin +(
L
>
Let us define the set E by E
=
{VI
vu3
>
0)
u
{tq
v3
=
0,
v2
>
o>
u
{VI
113 =
212 =
0,211
>
0). @8)
Thus we can write: CC
2
c( -
lJEE
cos -;(
v, PI1 cos $(
v, r2 )
Q
+ sin $(
2f,pl ) sin -27T (v, r2) K(u) + Q-%
L
(0).
(B9)
>
Now the functions
P V
=
2
*
(> -
cos
I2
-
;(
v,
r
)
2 ,
&
=
-
0Q
*
sin +(
VJ ) 3
with tl E E, and g, = Q-’ together form a complete orthonormal system in 9. Thus we conclude that these functions are the eigenfunctions of the integral operator E, while the Fourier components of K are the eigenvalues. With the theorem (Al) we therefore have found the eigenvalues of G, if the Fourier components of (? are known. Furthermore it can easily be seen, that the eigenspace & is at least of dimension 2 and it is representation space of the group Tae
0. SEEBERG
284
APPENDIX
C
Calculation of the direct correlation function. Our starting point is the following equation : .
Now 6~ E R Therefore the x.h.s. becomes: R = A
c (6N,)2
-
z C &Vi 6
F
exp
Slv, C VED
(*L
ym3.
(v, Xk - Xi))
The 1.h.s. becomes :
Thus we get: 4X2V2 exp x 6 DEb L2
2xi (#,X, - Xi) y-3. L >
Kik = Aa,, - B
With (Al 1) we therefore get:
&h, rz) = =
C .Lk
Kikzi
Crl)
e”k(r2)
A6ik - ’
6
4;n;*v2 C exp #ED
1
ym3
L2
>
With (A8) we finally have: 2xi
ry
K( r1 f y2)
=
-(v,Y, AC
-v,)
APPENDIX
A-
B 4x2v2
--. 6
L2
D
Proof of eqs. (2.12). Our starting point is (2.1 I). We use the spectral representation of the operator K:
CORRELATION
LBNGTH CLOSE TO THE CRITICAL POINT
285
Thus:
Thus we get: c=JJ
V
(E$*,>
W)
[see (3.3)]. Now let us put: clff” Slv, = vu. Therefore we get:
This is the first part of the proof. Now let us calculate:
As above, we get:
Here we used that (f;3 is an orthogonal matrix. With (Dl) ‘we get:
= CfiyK-‘(&3-l
(w
=@-'KG'.
Q
This is the second part of the proof.
APPENDIX
E
Case II. Case II is defined by condition (3.12). Instead of calculating the critical part of the grand-canonical potential we calculate directly its derivatives. Only these derivatives are needed in the consistency differential equation. Our starting point is (3.4). Now we have:
pm= (l/@)
log s.
Let us therefore differentiate eq. (3.4) with respect to p. Then we get:
0. SEEBERG
286
Again we transform the sum into an integral by putting (a/L) z+ = & and we get:
aP
a log s -=c
-3r
49 6 2 aB
--dA
3 SK
ap
6~2
ap
A--
5
4n2Bf2 6a2
>Ide
Now in the 02 case, B dues nut exhibit any critical behaviour. neglect the term ccdB/@ and get:
’
Thus we can
From (2.23) we thus get:
Therefore we obtain with (2.24), (2.23): -a i0g
s
aA 3a2e3 = -- 1 --=
aP
2 ap
7TI4
a
-1
--- QS a -.1 4~~ ap ( a2 >
(W
Thus we have: --- 6
aP Cr -=
(>
4ag ap 22
aP
l
Therefore we get by the use of (4.3) : 6A2 -=--_ I IS I
(E3)
Thus we have: a2 =-
c
3(i1’)2 ---( a4
P I3 J
I
where
c
26 P’l > = 24axj9
0
l
As in section 4, we choose it = x as the independent variable, while 1’ = z now is the unknown function. Therefore we get with
CORRELATION
LENGTH
CLOSE TO THE CRITICAL
POINT
287
The solution of this equation is: y = x6
D + -
2 c
log x , >
where D is an integration constant. Now we consider the line /3 = const. We assume, that close to the critical point il should possess a maximum as function ofp along the line /3 = const. At this point therefore we must have: dn/dp = z = 0, and thus y = 0. Therefore we have: Y -_ x6
2 --logx~+
(
C
2 -1ogx c
l
>
Now y must be positive by definition. Therefore: 2 -1ogx C
2 - -logx, c
> 0.
But then xM is a minimum. This is in contradiction
to our physical assumption.
REFERENCES
1) Master, 2) 3 4) 5)
A., Statistische Thermodynamik Kondensierter Phasen P.H. XIII, Springer (Berlin, 1962). Landau, L. D. and Lifschitz, E. M., Lehrbuch der Theoretischen Physik V, Akademie-VerIag (Berlin, 1966). Kadanoff, L. P., Gijtze, W., Hamblen, D,, Hecht, R., Lewis, E. A. S., PaIciauscas, V. V., Rayl, M. and Swift, J., Rev. mod. Phys. 39 (1967) 395. Griffith, R.B., J. them. Phys. 43 (1965) 1958. Wertheim, M. and Hynne, F., Integral Equations in the Classical Theory of Fluids, Kemisk Laboratorium III, H.C. &sted Institutet.
0 North-Holland Publishing CO.
A POWER LAW FOR THE CORRELATION CLOSE TO THE GAS-LIQUID
CRITICAL
LENGTH POINT
0. SEEBERG Institut fiir theoretische Physik der Universitdt Frankfurt/Main, Frankfurt am Main, Deu tschland
Received 15 March 1972
Synopsis The state of a system is characterized by occupation numbers of space cells. The correlation matrix G,, corresponds to a linear operator the eigenvalues of which give the “critical” part of the grand-canonical potential. This critical part is in the Omstein-Zemike case only a function of the correlation length A. By use of the thermodynamic definition of the isothermal compressibility ti and the 02 relation 3~z A2 a consistency differential equation for A is derived and solved for the critical isotherm. The solution turns out to be a power law: 1 N” [Apj -Z/5 ; or, again by use of the 02 relation: 3~z 1Ap1-4k
Within the framework of the theory of the scaling laws it is assumed that the critical thermodynamic quantities follow a power law close to the critical point. It is the aim of this paper to deduce such a power law for the correlation length using assumptions which lead to the Ornstein-Zernike theory. We consider a grand-canonical ensemble. The volume ~2 is divided into cells forming a cubic lattice. Thus we get a coarse-grained description of each system S by the set n of the occupation numbers Ni in the cells. In particular we get a coarsegrained correlation function Gij as follows: 1. Introduction,
The matrix (G,,) on the other hand corresponds to a linear self-adjoint operator and its eigenvalues and eigenvectors have the following physical meaning. The eigenvectors form a system of independently fluctuating observables while the eigenvalues give the fluctuations of these observables. From the translational invariance of the expectation values it follows that the eigenvectors correspond to plane waves while the eigenvalues are the Fourier components of the correlation function (appendix B). 263
264
0. SEEBERG
In section 2 we recall the Omstein-Zernike description. We write for the probability P[n]:
theory using the coarse-grai
where K is the grand-canonical potential. That means that the fluctuations assumed to be distributed according to a gaussian distribution. After smooth out the fluctuations (appendices A an4 B) we get by means of a Taylor expanz of the fluctuations an expression for the smooth ‘&direct correlation functi which corresponds to the operator K. Therefore the operator G_, is known. I the inverse of the operator K: G = @- X-l. Thus we arrive at the old Ornstt Zernike theory. But now the critical part of the partition function (or the gra canonical potential) can be calculated from the eigenvalues of the operato By “critical part” we mean the part due to long-ranged fluctuations. Now central idea is that only these critical fluctuations contribute to the crit behaviour. The isothermal compressibility, on the other hand, can be derj from the grand-canonical potential : K=
-P
(BP d Q
and 3G=
-2
02
(e> ap2
*
Thus x can be expressed as a function of the eigenvalues. But these eigenva: themselves are functions of x by means of the OZ theory, especially the Ornstt Zernike relation 3coc P. Thus we get a consistency relation in the form c differential equation. We solve this differential equation for /3 = PC* The SI tion is:
A = lAPI-2’5 llog
F(AP)~I~'~ G,
where F and G are constants. This solution is equivalent to a power il a lA/~]-‘l” when we use the definition of the critical exponents as given Griffith4). Of course it is possible to extend these considerations to the genl critical behaviour in the framework of the OZ theory. This will be done i forthcoming paper. 2. Ornstein-Zernike theory and Landau ansatz. We consider a grand-canon ensemble g which is characterized by the chemical potential p, 18= (kT)- l the volume Q. Let Q be a cubic box. This box is divided into cubic cells 2 equal volume Y = a3, so that the cells form a cubic lattice A. We considt system S E 9. A coarse-grained description of S is obtained from the set II of
CORRELATION
265
LENGTH CLOSE TO THE CRITICAL POINT
occupation numbers Nt, defined by
(2.1) We define the correlation matrix (G,,) by G, = <(K -
(2 .2)
- <%>)>3
where the mean values are taken over the ensemble g. GtJ is related to the usual correlation function g by G ij =
z
Iz f
Ke)”
g
cc
f r2)
+
6 @I
-
r2)1
dr1
dr,
(2
f
l
3)
I
where g (u, , r2) = et”) (rl , r2) - (Q)~. For these definitions see ref. 1. G,, corresponds to a linear self-adjoint operator. This fact will be- used in the following analysis. Now let P[n] be the probability for n in 9. P[II] can always be written in the following form :
where K (p, /I, In) = -p (p, p) 52 is the grand-canonical thermodynamic potential. Eq. (2.4) is of course only a definition of K[n]. On the other hand, the grandcanonical probability P (x, IV) is given by: P (x, N) =
(N!h3N)-1exp [ -/?HN (x) + IV@ - Q&9].
(2.3
Thus we get:
where %S[?t]is the set of all configurations
exP c-m
2m7c
3/2N 1
( >
[rtl) = 7 Bh
which lead to the
Yg- exP .
Wi@~
j
exp
%mtl
E-PWdl
n. Thus we
QN. (2.6)
Now we write: j
exp c-BV
it3htl
ml QN = =P c--Bml)
j dqw x3h3
(2.7)
The integral on the 1.h.s. is of course the usual Boltzmann measure. We get:
sdqN = a3NN! (v
EMItI
N,!))-'.
(2 8) l
0. SEEBERG
266
By use of Stirling’s formula we get from (2.6) and (2.8) :
.
(2 9) l
Here we used that Ni + 1, which means that the Ni are macroscopic o bservables . Now V[n] itself must depend on & Therefore V[n] is not a true macroscopic observable, for such a variable must be uniquely determined by the vector n. We assume the existence of the most probable vector E: for all Furthermore
n * E.
(2.10)
we assume that P[n] can be written in the following form:
where SN 1 = Ni - Ni,
K,, =
d2K aNi aNj
[Iii
and c = exp(@K - @Km]). (2.11) is the first of our main assumptions. It is difficult to justify this assumption. Let us only emphasize that the occupation numbers Ni are macroscopic observables ; this is a crucial point. Furthermore (2.11) leads to the Ornstein-Zernike theory and our considerations are within the framework of this theory. It is therefore beyond the aim of this paper to discuss the validity of (2.11). n and i7 belong to the same ensemble g (p, /3,Q), which means that p, p, Q remain fixed. From (2.11) it follows now that (NJ
=
&,
G,, = (l//3
K,f
(2.12)
The proof can be found in appendix D. Now let us perform the smoothing S, defined in (A9). We get:
Now we replace &t (r2) on the r.h.s. of (2.13) by its Taylor approximation: (2.14) where R, = r20 - rl,.
CORRELATION
LENGTH
CLOSE TO THE CRITICAL
POINT
267
By use of the fact that K(R) is (almost) rotation invariant, we get after a short calculation :
(2.15)
= A I (8n)* (r) dr - (B/6) j grad* n(r) dr,
A
=jR(R)dR,
B
= j@‘?)R*dR.
The Taylor approximation (2.14) of course can only be valid if K(R) descreases sufficiently rapidly with increasing IRI and 6fi (r) is sufficiently smooth. Now R(R) corresponds to the usual direct correlation function and it is usually assumed that this function indeed fulfils the first condition, The second condition at least is fulfilled for long-range fluctuations, i.e., critical fluctuations. Eq. (2.15) of course corresponds to the well-known Landau ansatz2). Now let us interpret the quantities A, B. From (2.12) we get : (2.16) Using the fact that GiI, and Kkj are translation-invariant,
we get:
c Gk 1 Kkj = B-l* k
j
Now from (B7) and (B9) we get:
c
K(O)=j&R)dR=
&=A.
Rip*
Thus A = p-l
Furthermore
c
i,k
Glk
we have from (2.3): =
y31
Gik
kc
= Q(e) (1 + 4~ (e) s g(R) R2 dR) = Q(Q)~ x/?-l,
(2.17)
where 3~is the isothermal compressibility. We used the well-known compressibility integralI). Thus we have finally: A = ((@)’ m3)- l.
(2.18)
268
0. SEEBERG
Now we turn to the interpretation of the quantity B. First of all we construct R from (2.15). Using the fact that x is an operator E a[R] and 81 is an arbitrary vector in 9, the unique solution turns out to be: (2.19)
The calculation can be found in appendix C. From (B7) and (B9) we therefore get, that the eigenvalues of the operator K are given by K(v) = A - 4x2~v2/6L2,
WED.
(2.20)
Thus we get from (2.12):
- q1 r2) = WC
=( -
z)+JZ1exp
A
(F(v,R)).
VED
(2.21)
The eigenvalues of G are of course given by G(v)
=
8-1
A
-
m!?? -I, 6L2
>
VED.
The r.h.s. of (2.15) must be positive. Hence A > 0 and B < 0, so that the eigenvalues are all positive. Now we use the invariance of the trace of a matrix in orthogonal transformations. We get
ii = CG 1
TrG.
(2.22)
Using the fact that all Gii are equal, we get
The sum is transformed into an integral by v (a/L) = 5. Thus we get: py3Gii
=
5
s
1
dC 1 + 47r2A2E2/a2 ’
D = {& I.&lG 3),
A2 = IBl/6A.
(2.23)
We choose two spheres KI of radius 4 and K2 of radius + 2’ with & c D c Kz
l
CORRELATION
LENGTH CLOSE TO THE CRITICAL POINT
269
Thus we get: ax < Gii, 1 - a arctan ax a > 3a2
Gil < -
@ PI
21i2 - A- arctan ax
Ax21/2 a
. >
Thus we get for the case A/a & 1, which is true close to the critical point :
where f < S < 2! Thus we have: (2.24) This is the desired interpretation meaning : Gii
wi -
=
Furthermore We have:
of the quantity IBI. Glf of course has a physical
wa2)*
we get a new expression for the correlation length defined in (2.23).
A = /3-l
(F G,,)-’ = p-ly3 (&Gik)-‘.
,
Inserting this into (2.23) we get: (2.25) Thus we get the following interpretation: A2 is the ratio of the sum of the nondiagonal elements to the diagonal element Gii of the matrix (G,,). Close to the critical point this ratio becomes very large. Furthermore it can be seen that A is independent of a only if Gii oc Q? Now it can be shown that the OrnsteinZernike assumption yields that the correlation function decreases cc r- l in the critical point5). We assume that this is also true in a neighbourhood. Then, indeed, we get : Gii K a’. On the other hand we get from (2.17) and (2.25): da5
a2
=
2q3Gii
2
= -16Ce)’ x In
(2.26)
0. SEEBERG
270
where IB’I = lBl/a3. (2.25) is the Omstein-Zernike relation. We assume that II?’ remains finite close to the critical point and furthermore that IB’I does not exhibil any critical behaviour. This case we call the Ornstein-Zernike case (O.Z. c.)_ 3. Calculation of the gmnd-cunonical potential. From (2.11) we get = 1
CCexp(-BK[n])
= exp Q9K(p, /9,Q) - PK [El) C exP ( - *8 C Kij 8Ni 8%) II
l
(3 1’, l
Therefore we get: K(p, p, In) = -QP(/3,
p) = K[fq - p-l
log s,
where
At first we calculate S. We use the spectral representation
of the operator K:
Thus we get:
The sum is transformed into an integral and we get:
=ndrlp, With Et fi”x, =q”andndXo an orthogonal matrix, we get:
which follows from the fact that (f/j iI
Thus we have:
or
(3 3
. P
\
l
Lcrr /
From (2.20) we therefore get: logs
A -
= -:clog$ VED
(
4X2BV2 --z-’
>
(3.4
CORRELATION
LENGTH CLOSE TO THE CRITICAL POINT
271
Here we used the fact that the eigenspace &, is at least of dimension 2. Now we transform the sum into an integral by (a/L) vi = & Thus we get: l
logs=
-*,3Jlog$(A-F)d&
(3.5)
I5 D is defined in (2.23). From (3.5) we get:
(3.6) Now let us put
Then we get for the integral on the r.h.s. of (3.6): J
=-
+Y 3
s
3
log(l
+
q2)dtl
D*
( > 5
1
where D* = (~1 lqll < d/a}. Again we choose two spheres & , Kz of radius x;l/a and 2%@2, respectively, with Kl c
D*
c
(3.7)
K2.
Hence there must be a sphere K* of radius a* with
1 log (1 + 7”) dq = j log (1 + 7’) dtl. D* K* Then : log
s =
-$y”
log
(z >
+ f
2
- + arctan $-
--log{$f[’ 1
2(a*)3
1 (+j++g($+) = 3.12
+
($)2]}+~n2:o*)’
W)
0. SEEBERG
272
Now, at least in the 0.2. case, log S remains finite in the critical region. On t other hand, log (@A/2x) diverges. Thus, we conclude that 746(~*)~ - l/a3 mt tend to zero. We choose therefore: a* = a (~/6)l/~,
(3
which is compatible
with (3.7) and we get from (3.2), (2.9), (3.8) and (3.9):
P(/Q)=(@)
p+$l~g~-fi-;log(,) >
( 1 2/Ya3
>I
1% 2 a2 ---
3 A2
7c 0-
213 +
6
$
arctan[$
(:)“‘I (3 .
Until now no explicit assumption has been made concerning cell length a. Let us investigate two different cases: I
l l
II . l
Ala =
D = const.
a = const.
the magnitude of t
(3. (3
l
Case I means that a is increased when il is increased. The conclusions in this cz can be derived with the following more physical argument. The critical phenome should only depend on the critical fluctuations, that is, only Auctuations w. lvij < EL/A are responsible for these phenomena. Note, that in our consideratic until now all fluctuations with /vi] < L/2a have been retained. F is a consta Furthermore, the determination of the eigenvalues of K [(Cl)] will only accurate for those 21which fulfil the condition a2v2/L2 < 1.
(3 .
Therefore one may retain only the critical fluctuations in calculating the grar canonical potential. More exactly, the contribution of the remaining fluctuatic can be considered as a constant with respect to the necessary differentiatic This idea leads to a differential equation the solutions of which are equivalent far as their analytical form is concerned to those of the differential equati corresponding to case I, Case I corresponds to the Kadanoff idea3) in the following sense. In order eliminate the cell length, which must cancel out, we put A/a = D. Then it tul out, that the results explicitly depend on the ratio D. But this fact must not taken too seriously. Let us assume, that (2.11) is valid for a fixed cell length
CORRELATION
LENGTH
CLOSE TO THE CRITICAL
POIl+iT
273
Let us increase this cell length by a factors S: a’ = sa. Then we get after some calculation, that the new probability P[~x] can bk written in the following form:
where e explicitly depends on s. Thus we are led to the following resolution of the above difficulty: There is a maximal value Dm, for which (2.11) is valid. For D < D, the results have to be changed as e depends on s. We decide to choose D = D,. D therefore has a physical meaning. Case II yields results, which have no physical sense. The treatment of this case can be found in appendix E. Of course there are still possibilities other than I and II, Now let us suppose for a moment, that all cells are enclosed within walls, so that no interaction occurs between different cells. In each cell there are (N)ym3 particles. Repeating our considerations, we find that the potential K in this case can be written in the following form: R = 1 Ki =
$
log g
p+$log~-
-(N)
> l
K cannot exhibit any critical behaviour, the same holds for the first term and therefore for the second one too. We assume for a moment, that iT = G. Then the second term is equal to K[B]. Thus we are led to the assumption that K[ti] cannot exhibit any critical behaviour. 4. Derivation of the consistertcy diflerential equation and its solution for a special case. The power law. Eq. (3.14) led us to the assumption that only log S is re-
sponsible for the critical phenomena. From (3.2), (3. lo), (3.11) we get: P(BtP)=<@>
(
3
I”.+$og,-fi-
1 -j- log (e>
2mx
Ph
>
-&log(g(l +E))+SF, where E = (6~~)~‘~D2 and +
Now we use the thermodynamical
relation
(4. 1)
0, SEEBERG
274
On the other hand we get from (2.26): x = 6A2((Q)’ jB’I)-k Furthermore A =
from (2.18) : D3
1
D3 1B’I
(e>’ m3 = (~)“d~
=
6P
l
Therefore we get
02 -II 2
6il2
2x=--*
I PI
au
On
the other hand we have: 2 aP a/J2
where :
p..=&+f+ H = 2F - log
H),
y
(1 + E)
x:
l
>
Thus we get by differentiation: 2
-0
aP2
D3 = p” = _
F
[6OlogA - 35 + 12 (H - logj9)]
2P + K[-15logA R4
+ 5 - 3(H-
lag/3)] . >
Now we put for brevity: &)
= &D3 [60 log il - 35 + 12 (N - log /3)],
g2@) = +D3 [-15logil
+ 5 - 3 (H - log@)].
Thus we get the following differential equation from (4.5):
CORRELATION
LENGTH
CLOSE TO THE CRI’TICAL POINT
275
Before solving this differential equation let us discuss the results which follow from the “physical idea” mentioned above. We have 1 log s. P ct = PQ Instead of (3.4) we now get log S” = -fClog$(A-F), VEP
where
Again we transform the sum into an integral by lv@
= 61.
We get: log S” = -3
(0 (1 -t 47c2p253 de Q +
g b
=-s(p)” a3
(-flog: -
+c* , >
c* = J log (I + 4x2F2t2)de. 115 With (2_18) we therefore get: *
Per = --
F3
w
(
log
w +c*
@
12za2a3
~$~-log~+*+log~~].
> (4.7)
Now the comparison between (4.7) and (4.4) yields that PC*,and pCr are equivalent as far as their analytical form is concerned. Now we only use that analytical form in the following calculations. Therefore the analytical form of the solutions cannot be altered. Let us now solve the differential eqbation (4.6). At tist glance this equation seems to be highly nonlinear. But this difficulty can be resolved very easily. We choose A = x to be the independent variable, while it’ = z is now the
0. SEEBERG
276 unknown function. We get:
where zfl) = dz/dx. Let us substitute y = z2. Thus we get:
(4 8) l
This is an ordinary linear differential equation which can easily be solved. g, has no zeros in a sufficiently small neighbourhood of the critical point. Thus, defining
a1 (4 ml =xg2
we
0
12/3x6
ecx1 =
’
P’I
g2(x)p
get
(4 9) l
The solution is:
where 2 is an integration constant. After some calculation we get: 2g 1 dx = -810gx s
+ 2log[logx
+ 3(H-
(4.10)
log/?) - $1.
xg2
Now we calculate the integral J:
=
s
12’
x2
PI
v = -51B'p3
8P = -5p703
[lug x + 3 (H - lug @) - +I” dx
g2
s
[logx
+ $(H-
(-_i
log@ -
+]x-“dx
(log x + 1) + ; r+ - +(H - log@)]
x
>
l
CORRELATION
LENGTH CLOSE TO THE CRITICAL POINT
277
In this way we get:
(4.11)
Now we consider the coordinate system (p, 18). Let us consider the point C;, /3) on the line /3 = const, where ;3.= x has its maximum xM. At this point there must be dA 2 = -=o,
therefore
y = 0
also.
dP
Thus we get: Y
SPx8
= 5 /B’I D3
1 + x
[logx + 3(H-
log@) - $]-2
log Exexp (3 (H - log 8) + 31
(4.12)
l
>
From (4.12) it is seen that indeed y > 0, as it should be. This point is taken up in appendix E, where case II is considered. In particular we choose the line 16= PC. The maximal value of x = 2 therefore is infinite. Thus we get: Y =
5;;;;;3 Ilog [xexP 6 w - 1ogP)_!_ log Ix exP (3 (H x
x
-
1% B) +
3)l
+)I}-” (4.13)
l
Now we have y=
da
2
(> -
l
dP
Thus dx -=
+ (log {x exp 13 (H - log/D +
diu
1% (x =P L-k(H - m/9
-
31D3 SII ’
(4.14)
278
0. SEEBERG
where q = _+l, so that we get:
{x =l(i,;tD3)+S* w-
log “P I-&cH - log8) - +I> x7’2(log (x exP f-k log 8) + 3w
dx
co
(log {xexp[+cH - log b) + +I} - logd
dx .
Now we put (4.15) Thus we get:
ilru =11
-
5’2(log AX)%
-(= 1*{ 500
1 - 0 [(* logA1x)*]}]
(4.16) @ is the error function. Now we use the following estimation: $ [ 1 - Q(x)] N exp ( - x2)/2&
(x > I),
from which we obtain: (4.17) We consider only the leading term (Aa & 1) and get: (4.18) Let us reverse this expression. Putting 125 IB’I D3 (A~)~/32/9&
= y;
(a A)-s = x,
CORRELATION
279
LENGTH CLOSE TO THE CRITICAL POINT
(4.19)
Y = x llog xl.
Let x(y) be the solution of (4.19). From log y = log x + log Ilog xl, lim,,o kg X(Y)/l% Yl = 1. Therefore we write the solution in the furm:
we
get
x = Y IlWYY dY). Inserting this into eq. (4.19X we get: 1 =
y(y) Ilog xl lb
YI-l*
Thus we get: lim,,* v(y) = 1. Therefore we have l
lim
X(Y)
Y+O
-
Y llogYl-l
=
0 3
Y bgYrl
or x(y) N y Ilog yI - l, leading to :
(614 -5 = 125 1B’ID3 (AP)~ N
32p G5
125 IB’I D3 (AP)~
-I f
w cm5
2 -= lApl-2/5 jlog 8(Ap)zj1/5 G,
(4.20)
Thus we have obtained a power law: it oc IApld215 for the correlation length 2 cluse to the critical point along the critical isotherm b = PC. We shall investigate the general critical behaviour by solving the differential equation (4.6) in a forthcoming paper. The difficulty which hereby arises is that for lines @+ PCthe initial values are not known. We therefore must introduce some additional hypotheses. In the case b = PC we have only used that il must diverge, when we go to the critical point PC = (p,, PC). From (4.20) we get by use of the Ornstein-Zernike relation (2.26): 3c = IApl-4/5. Acknowledgements. I wish to thank Professor M.Wertheim deaux fur many helpful discussions.
and Dr,D.Be-
0. SEEBERG
280
APPENDIX
A
Smoothing. Theorem(A1). LetA beanisomorphismfromavectorspaceeof dimension y3 into a vector space 8. In both spaces a product is defined. Let A have the following property :
(Aa, Ab) = (a, b) .
(AZ)
(I) Then an isomorphism H from the algebra a[CJ of the linear operators in G into the algebra a[%] of the linear operators in 8 is induced by: ALa = H(L) Aa,
where u E e, L E a[G]. The proof can be omitted. (II) Let L be a self-adjoint operator E a[eJ. The eigenvectors {nr> form a coiplete orthonormal system in 6 Then H(L) is self-adjoint, its eigenfunctions are given by {Aq} and the eigenvalues & are left invariant. Proof: 1. (Aa, H(L) Ab) = (Aa, ALb) = (a, Lb) = (La, b) = (ALa, Ab) = (H(L) Aa, Ab). 2. Ri = (nil Ln,) = (An,, ALq)
= (An,, H(L) An,).
let (G,,) be the correlation matrix defined in (2.3). (G,,) corresponds to a linear self-adjoint operator G in a vector space G of dimension y3 = QaB3. Its eigenvalues are positive. This follows from the use of the grand-canonical ensemble g. In G we introduce the usual product: Now
(W
(a, b) = 1 aibi.
Now we consider the vector space % of the step functions defined on JI 8 = {fl
y19 r2 E zt Tf(r,)
=
f0
i} .
for all
WI
In % a product is defined by:
W) Now we construct an isomorphism A from @ into 8 by:
for
FEZ,,
0
otherwise
l
CORRELATION
LENGTH CLOSE TO THE CRITICAL POINT
281
Therefore we have:
(a, b) = (& Air), and the presuppositions of theorem (Al) are fulfilled. The eigenfunctions of any self-adjoint operator K are of course step functions. This is inconvenient, because they are not differentiable. Therefore we construct a third vector space 32by
where
m
Zi(r) = B[e,] (v) = C exp VED
Here Xt is the centre of the cell Zr and D is defined by:
D = {VI
2n-k 1 =r}*
141G n,
In 9 again the product (A5) is introduced. we have
&has the dimension dim R = y3 and
where we have used that C exp V
(-9
(v,&
-
XJ)YB3 = &km
Thus the mapping B defined in (A7) is again an isomorphism which fulfils the presuppositions of (Al). The eigenfunctions now are differentiable. Therefore we call the mapping
the smoothing operator. Let us, for convenience, compile our notation
Let K be an operator E a[G]. Then the corresponding operator &(K) E am] is denoted by g, while the corresponding operator H2 [HI(K)] is denoted by R.
282
0, SEEl3ERG
I? and E correspond to integral operators :
APPENDIX
B
Consdruction of the eigenvectors and eigenvalues of the operafor G. The function G Is given by [see (All)]:
At first we expand e into a Fourier series:
After a short calculation we get:
e(fl7PI = c exp
-$(u,x,,
- F(p.&)
m
G,,y-3. >
Now Gij can only depend on the difference vector Rij = Xi - Xj, because the expectation values ‘of the observables xnust be invariant under the subgroup T, of the translation group T. T, contains all translations corresponding to a vector of the form t = {m, a, m2a, m3a),where the mi are integers. Thus we get:
G = ym3 C exp t
-F
(U + p,Xi)
)
C mi
eXp
(~(P3Rijl)
Gil,
w
(
where mi contains all vectors Rij with - Rij + Xi E Q for fixed Xi* NOW we make use of the following assumption about T: For physical results the sum
can be calculated, as if it were independent of Xi. This is true in the thermodynamic limit. With (A8) we then get:
CORRELATION
283
LENGTH CLOSE TO THE CRITICAL POINT
Inserting this into (B2) yields:
(7 =
c exp
F
(tl, r1 - rz)
(
Q-W(v), >
where K(v) =~@Z)exp
(+,R))dR
m Thus we get by use of the properties of Glj:
tt, rl 1 sin -2x (tr, r2) Q-lK(fl)*
+ sin +(
L
>
Let us define the set E by E
=
{VI
vu3
>
0)
u
{tq
v3
=
0,
v2
>
o>
u
{VI
113 =
212 =
0,211
>
0). @8)
Thus we can write: CC
2
c( -
lJEE
cos -;(
v, PI1 cos $(
v, r2 )
Q
+ sin $(
2f,pl ) sin -27T (v, r2) K(u) + Q-%
L
(0).
(B9)
>
Now the functions
P V
=
2
*
(> -
cos
I2
-
;(
v,
r
)
2 ,
&
=
-
0Q
*
sin +(
VJ ) 3
with tl E E, and g, = Q-’ together form a complete orthonormal system in 9. Thus we conclude that these functions are the eigenfunctions of the integral operator E, while the Fourier components of K are the eigenvalues. With the theorem (Al) we therefore have found the eigenvalues of G, if the Fourier components of (? are known. Furthermore it can easily be seen, that the eigenspace & is at least of dimension 2 and it is representation space of the group Tae
0. SEEBERG
284
APPENDIX
C
Calculation of the direct correlation function. Our starting point is the following equation : .
Now 6~ E R Therefore the x.h.s. becomes: R = A
c (6N,)2
-
z C &Vi 6
F
exp
Slv, C VED
(*L
ym3.
(v, Xk - Xi))
The 1.h.s. becomes :
Thus we get: 4X2V2 exp x 6 DEb L2
2xi (#,X, - Xi) y-3. L >
Kik = Aa,, - B
With (Al 1) we therefore get:
&h, rz) = =
C .Lk
Kikzi
Crl)
e”k(r2)
A6ik - ’
6
4;n;*v2 C exp #ED
1
ym3
L2
>
With (A8) we finally have: 2xi
ry
K( r1 f y2)
=
-(v,Y, AC
-v,)
APPENDIX
A-
B 4x2v2
--. 6
L2
D
Proof of eqs. (2.12). Our starting point is (2.1 I). We use the spectral representation of the operator K:
CORRELATION
LBNGTH CLOSE TO THE CRITICAL POINT
285
Thus:
Thus we get: c=JJ
V
(E$*,>
W)
[see (3.3)]. Now let us put: clff” Slv, = vu. Therefore we get:
This is the first part of the proof. Now let us calculate:
As above, we get:
Here we used that (f;3 is an orthogonal matrix. With (Dl) ‘we get:
= CfiyK-‘(&3-l
(w
=@-'KG'.
Q
This is the second part of the proof.
APPENDIX
E
Case II. Case II is defined by condition (3.12). Instead of calculating the critical part of the grand-canonical potential we calculate directly its derivatives. Only these derivatives are needed in the consistency differential equation. Our starting point is (3.4). Now we have:
pm= (l/@)
log s.
Let us therefore differentiate eq. (3.4) with respect to p. Then we get:
0. SEEBERG
286
Again we transform the sum into an integral by putting (a/L) z+ = & and we get:
aP
a log s -=c
-3r
49 6 2 aB
--dA
3 SK
ap
6~2
ap
A--
5
4n2Bf2 6a2
>Ide
Now in the 02 case, B dues nut exhibit any critical behaviour. neglect the term ccdB/@ and get:
’
Thus we can
From (2.23) we thus get:
Therefore we obtain with (2.24), (2.23): -a i0g
s
aA 3a2e3 = -- 1 --=
aP
2 ap
7TI4
a
-1
--- QS a -.1 4~~ ap ( a2 >
(W
Thus we have: --- 6
aP Cr -=
(>
4ag ap 22
aP
l
Therefore we get by the use of (4.3) : 6A2 -=--_ I IS I
(E3)
Thus we have: a2 =-
c
3(i1’)2 ---( a4
P I3 J
I
where
c
26 P’l > = 24axj9
0
l
As in section 4, we choose it = x as the independent variable, while 1’ = z now is the unknown function. Therefore we get with
CORRELATION
LENGTH
CLOSE TO THE CRITICAL
POINT
287
The solution of this equation is: y = x6
D + -
2 c
log x , >
where D is an integration constant. Now we consider the line /3 = const. We assume, that close to the critical point il should possess a maximum as function ofp along the line /3 = const. At this point therefore we must have: dn/dp = z = 0, and thus y = 0. Therefore we have: Y -_ x6
2 --logx~+
(
C
2 -1ogx c
l
>
Now y must be positive by definition. Therefore: 2 -1ogx C
2 - -logx, c
> 0.
But then xM is a minimum. This is in contradiction
to our physical assumption.
REFERENCES
1) Master, 2) 3 4) 5)
A., Statistische Thermodynamik Kondensierter Phasen P.H. XIII, Springer (Berlin, 1962). Landau, L. D. and Lifschitz, E. M., Lehrbuch der Theoretischen Physik V, Akademie-VerIag (Berlin, 1966). Kadanoff, L. P., Gijtze, W., Hamblen, D,, Hecht, R., Lewis, E. A. S., PaIciauscas, V. V., Rayl, M. and Swift, J., Rev. mod. Phys. 39 (1967) 395. Griffith, R.B., J. them. Phys. 43 (1965) 1958. Wertheim, M. and Hynne, F., Integral Equations in the Classical Theory of Fluids, Kemisk Laboratorium III, H.C. &sted Institutet.