Critical correlations and the Yvon-Born-Green integral equation

PHYSICAL REVIEW A

VOLUME 25, NUMBER

JANUARY 1982

1

Critical correlations and the Yvon-Born-Green

integral equation

Moorad Alexanian~ Physics Department,

Montana State University, Bozeman, Montana 5971 7 (Received 11 May 1981)

It is suggested that the critical-point behavior of the Yvon-Born-Green integral equation for fluids for dimensions d &4 is described by a net correlation function which vanishes faster than any power of r as r 00, even though the isothermal compressibility E& diverges at the critical point.

~

Numerical solutions' of the Yvon-Born-Green (YBG) equation for the pair correlation function fluid give rise to g (r} for a square-well-potential values of the critical exponents y, P, and 5 in rath-

rise to the unphysical result that for d & 4 the net correlation function h(r) r T =g(r) r r —1 as oo with Dp &0. In what follows, we consider the possibility that h (r) r T van-

er good agreement with best experimental and theoretical estimates. Nonetheless, despite this success in obtaining critical exponents from a nonlinear integral equation linking directly the correlation function to the intermolecular potential a recent work on the critical-point exponent g gives

—

—

—d lng(r)= Qr

::

(r)

du

+p

s

du (s) g(s)[g( ds

i

r —s

~

-r

i

~

interested in the long-distance form of the YBG equation (1), which may be approximated by the nonlinear differential equation

dh

d

dr2 where h

—1

dh

r

dr

1

—up h

—1

h

2u2

u2

2

(2)

(r):—g (r) —1 and

u„(p, T) =

P

2"I

—+1

I (d/2+n/2+1)

2

xJ

~o

)

—1]ds,

du(s) ds

suines that at the critical point h (r) is of the form h(r) r r Do/r~ a— s r~oo with Do+0. It may

—

i

~

)

i

i

I

where p g(r)—(p(r)p(0)) — p5(r) and so g(r)~1 as 00, u (r)— P( r )/ks T with P(r) the intermolecular potential and p the molecular number density. If the potential u (r) has finite range Rp, as in the square-well-potential fluid, then the inhomogeneous term in (1) vanishes for r Rp and in the integral s & Ro. Since the exponent i) is de' +"' as r~ao, one is fined by h(r) r r C

r~

i

r~

ishes exponentially as 00. The Bogoliubov-Born-Green-Kirkwood- Yvon hierarchy for the pair correlation function g (r) becomes, with the aid of the Kirkwood superposition approximation, the YBG equation

—

Ir

i

~

r~

=Dp/r

g (s)s +"ds

(3)

=0, 2, 4, ... . It should be remarked that the long-distance form (2) of the YBG equation as-

for n

25

happen that Dp=0 in which case logarithmic terms may appear 3 as occurs for d =4. More precisely, in obtaining (2) from (1) for the asymptotic behavior of h (r) r zr, one assumes that ~

C

lim„„[r'h(r) r r ] '=0 for

some 0&@& 00. Now at the critical point the isothermal compressibility Kr diverges and since (2} gives that r for xr h(r)=D„e "'/r' 1, where — K (p, T}— (1 uo}/ui, one defines the critical point by ~(p, T) =0. It is interesting that the solution of the linear equation obtained from (2) by neglecting the quadratic term h and subject to the boundary condition h ( 00 ) =0 is h (r) =(r/ir)' Kzi2, (ar), where K„(z) is a modified Bessel function of the second kind. Notice that the eigenvalue (1 — up)/u2 is a functional of h (r). Now it is not absurd to suppose that a general solution of the nonlinear YBG Eq. (1) is given by an arbitrary linear superposition of solutions of the linearized equation valid for r 00 Eq. (2) without the h term. Note that this is a standard procedure in studies of the nonlinear Boltzmann equation, and there the assumption of the comi

:

»

~ —

572

1982

The American Physical Society

573

BRIEF REPORTS

25

studies of the classical Ornstein-Zernike theory of critical scattering. The representation (4) will admit a rather novel behavior of h (r) at the critical point which would eliminate the unphysical results of Ref. 3 for d (4, viz. , negative critical-point correlations for large r. Now the isothermal compressibility is given by

pleteness of the eigenfunctions of the linearized collision operator is correct for solutions with arbitrary initial states and at asymptotic times. Thus, we are lead to suppose the following most general representation for the net correlation function in terms of the solutions of the linearized equation obtained from (2): 1

—d/2

oo

[LJ(v AJ) 2(2~)'/2, g,

h(r)=

'Kdrq

dr&

~(r~A&)

+c.c. ],

ks TpKT

(4)

h(r)

l

l

l

l

T

T

j

—1) —I (d/2 „&2 d z

'

g ReLJ+

r' de

Re(LJ*AJ )

where the complex conjugate of z is denoted by z*. Thus, the divergence of ET occurs when a finite number of the AJ's vanish at the critical point, say, =1, 2. . .1. Hence, at the critical point and for

with Re~A~ &0, where lAJ & AJ+~ constant depending on p LJ is a finite (complex) conjugate. It the complex denotes and T, and c.c. in obtained previously was that (4) is interesting

for all

1+p— g

j

d&2

g

'Kdr2, (r~A~)+c. c. ] .

[LJ.(~A~)

(6)

Similarly for the Fourier transform

=, g t

h(k)

l

T

T

k J=1

h(k),

oo

ReL, + —,

g J=I+1

k'+AJ

+c.c.

(7)

I

On substituting (4) into the YBG Eq. (1), and conoo for T arbitrarily close to sidering the limit r T, we obtain

~

l=p,

(r s) 2s

du(s) ds

g(s)

l

z.

r ds =uo(p„T) .

any positive solution of (2) which satisfies h ( oo ) =0 must vanish. Here we find this feature

to be a consequence of our critical-point assumption for any d &0. It is important to remark that near the critical point and for k small

C

t

+ k'+A*. , k'+A.

h(k)=, Consequently, the condition uo(p„T, ) =1 is again valid even in this more general case and corresponds to a diverging KT, and so (8) serves to relate the critical parameters p, and T, . Since the (k) for single scattering is rescattering intensity =1+ph(k) &0 for I(k)/Io(k) to h(k) lated by k & 0, where Io(k) is the scattering intensity in the ReLJ & 0. All absence of correlation, then have concritical scattering works on previous sidered only the possibility g'. , ReLJ & 0; we now propose that the critical-point correlation of the — YBG equation satisfies , ReLJ 0 for Notice that KT still diverges at the criti0&d cal point but now h (r) r T does not vanish with

I

g.

&

g.

(4.

l

C

as r~ ~, instead,

it dea power-law decrease that only in particular Note creases exponentially. the first term in (6) vanishes identically. This somewhat similar to the observation that for d =1

which is large. Thus, g (r) is long-range and we still observe critical opalescence no matter how close we are to the critical point. Notice that if there is no singularity at k =0 in h(k) T C then l

r,

the above new critical-point behavior may be obtained from (4) for arbitrary d & 0. In order that h(r) r T be finite for 0&d &2 one must have, in

0, ReLJ — addition to the general condition g'. 1 d 2 —') .&2 =0 0 for &d Re(LJA~ the condition , =2. Therefore, and , Re(LJ lnAJ ) =0 for d the critical-point correlation becomes for d & 0 l

g.

&

g. 1

—d/2

„, g [Ln(~n)""

h«) r=r, = 2(2m. ) l

with Remi

A„& 0,

n

'Kd/2

—i

X(r~A„) +c.c. ] ... =1,2, , with Fourier coeffi-

BRIEF REPORTS

574

cients

h(k)

l

oo

r=r =

g

k z +A„

+cc.

(10)

Thus there is no vestige of the critical point neither in h(r) r r nor in h(k) z. whether in

r, as r~ oo

~

~

the form of a power-law decrease or a singularity at k =0 as k~O, respectively. Consequently, the critical part h, (r) of h (r) vanishes identically at the critical point and so the critical behavior is not of the usual type. Hence, in the absence of a singularity in h(k) r z. at k =0 and c since the usual definition of t), h(k) r r — k +" ~

~

as k ~0, is not valid here, we consider h(k) for k=O and T=T, and conclude that g=0. It is important to emphasize that, in general, this defini+" tion is not equivalent to h (r) r r

-r

I

25

the critical point and eliminates the unphysical long-distance behavior for h (r) found for d &4. Thus, (11) must have at least two solutions, one with a] — 0 corresponding to (8) and another with 0. Result (11) represents the simplest possible case. Of course, if h (r) r z. vanishes exponenC

a»

~

~

tially as r ao but with nonvanishing oscillations, that is, ~A, =a, + ib, with b, +0, then (11) has to be replaced by a somewhat more complicated condition. Also, in case there are several ~A„ =a„+ib„with equal real parts a„, then a more complicated condition may ensue. In any case, our result (11) is the simplest possible amongst this new characterization of a phase transition. Notice that (11) is a consequence of having to keep all the higher order derivatives of h (r) in {2), viz. , uq„h' " (r), since now h (r) r r vanishes

g„",

~

r~

but with a critical-point exponent g=0. We believe this new possible description of a critical point is related to the observation that "it is more reasonable to identify the critical-point behavior —0. Since with the limit ~~0 rather than with v= the nonclassical values of the critical exponents y, P, and 5 obtained from the YBG equation are definitely of interest, we conclude that the assumption of a power-law decrease in h (r) r r as

exponentially as ~ rather than as a power. Finally, exact results establish the absence of a phase transition in one dimensional fluids with shortrange forces. However, our critical-point conditions for d =1, 0 and ReLJ — , Re(LJ /~A~ ) =0, do not allow us to exclude, in general, a diverging K~, that is, =1,2, ... 1. For , Re(LJ/AJ) ~oo as AJ~O, instance, the simplest case with I =1 and A real with L ~ ~ A '~'+" as A where 0 & e & 1. Thus, ReL, and Re(L /~A, ) but (6 —1)/2 Kz- ~ A op as A ] note that y=&(1 —e). Thus, for this case limr r h(r)dr h(r) r r dr & ao, and

its consequent negative values in the critical scattering intensity as k~O for 2 &d &~ is questionable for d &4. Suppose the critical point is described by a diverging Kz but with an h (r) r r which vanishes exponentially as r~ao, that is, ~A& =—a& ~0 and Re~A„&a~ for n & 2 in (9), then from the YBG Eq. (1), we have that

so the integral does not converge uniformly for T & T, . Hence, our representation (4) is quite general and thus allows for phase transitions for d = 1 presumably corresponding to long-range— forces. In fact, on multiplying (1) by r and integrating over all space one obtains for 7' limr ling(r) =lim, or lng(r)=0 and r (du/dr)dr & oo that

~

r~

C

as oo and our present study further illustrates this difference. Therefore, the critical-point divergence of K& has been shown to be possible even in the absence of a singularity in h(k) r r as k~0, ~

"

r~

oo

—with

~

C

~

1=

pc

/2

I (d/2+1)

R d J s ~0

X

=

J

du(s) ds

2

&

j

~

~0

~0—

~

~

y

~

f

&f

I (d /2+ 1)Id/z(a (a)s) d/2

&s)

~0

~

—

~

f

~

r f In[1+h(r)]dr= — d f

7.

&

~0,

1

g($)

du (r) dr

dr

+(k~ TpK~ —1)uo(p, T)/p .

ds

where I„(z) i (iz) is a Bessel function with purely imaginary argument. Notice that is a monotonically increasing function Id/z(x)/x of x for x 0 and approaches [2 l (d/2+1)] as x~O. Therefore, a nonzero value for a& will represent a departure from a power-law decrease at

)

g.

(12)

:

Since ln(1+h) &h for that

1+h &0, we

have from (12)

I„—f r dudr dr & d (ksTpKr —1)[1—uo(p, T)] . p

(13)

If I„&0, then

575

BRIEF REPORTS

25 u p(p,

T) & 1 —[dka T (Kr

K— r )]

~

—Kr)]

I„~

lim

KT~ ~

I

—

*On leave from Departamento de Fisica, Centro de Investigacion y Estudios Avanzados, Apartado Postal

14-740, Mexico 14, D.F., Mexico. Green, K. D. Luks, and J. J. Kozak, Phys. Rev. Lett. 42, 985 (1979). 2K. A. Green, K. D. Luks, E. Lee, and J. J. Kozak, Phys. Rev. A 21, 356 (1980). G. L. Jones, J. J. Kozak, E. Lee, S. Fishman, and M.

'K. A.

we expect

I„/— d—

where h, represents the critical part of h (r) in the representation (4), h, is given by the sum of terms with vanishing Az s at the critical point. Unfortunately, we have not been able to determine

„(1

lims

T(Ky'

I [h, —ln(1+h, )]dr=kaT, K

~I„~.

~1 as Kr~ oo.

for 1lk+Tp=K& &Kq & ~; however, if 1„&0, then uo(p, T) may exceed the value unity for Kr & Kr & a& since u p(p, T) [dka

(1+

'

(h

Nevertheless,

Therefore, if'

u— p)K&

2v)y.

where K,—

we do know that up then 1—

Consequently,

up-g,

—oo &K & oo

since

by (12)

—h, ) r r dr=C&0, ~

I

the value of the constant C; however, if C =0, then limz ~ —+ (g) „h, =0; thus proving the critical-point behavior proposed in the text for the YBG integral equation.

E. Fisher, Phys. Rev. I.ett. 46, 795 (1981). See, for instance, T. L. Hill, Statistical Mechanics (McGraw-Hill, New York, 1956), p. 206. M. Alexanian, Phys. Lett. A 74, 1 (1979); M. Alexanian and B. Grinstein, Phys. Lett. A 78, 209 (1980); and references therein. M. Alexanian, Phys. Rev. A 12, 1609 (1975). 7M. E. Fisher, J. Math. Phys. 5, 944 (1964).