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Critique of the cluster theory of critical point density fluctuations
Stillinger Jr., Frank H. Frisch, Harry L. 1961
Physica 27 75 l-752
LETTER TO THE EDITOR Critique of the cluster theory of critical point density fluctuations For two-dimensional lattice gases which possess a critical point (but where interactions are not necessarily restricted to nearest neighbours), we have investigated the form of the pair correlation function computed by a partial cluster diagram summation at the critical point. Diagrams taken into account are the entire set that may be constructed from series and parallel connections of the fundamental Mayer cluster bond. Since the result fails ever to agree in its asymptotic large distance form with the known exact result for the special case of nearest neighbor interactions in the square lattice i), the inevitable conclusion must be drawn that the highly connected “basic” s) or “prototype” 3) diagrams which were neglected, and which in any event are difficult to compute, are important in correctly determining fluctuation correlations in the critical region, at least in two dimensions. The first attempts to predict the nature of critical pair correlations in classical fluids, by Ornstein and Zernike 4), were based on assumption of independently variable Fourier components of microscopic density, and yielded a relation for G(r) = g@‘(y) -
1
(g(s) is the ordinary pair correlation function) of the form: VsG = ,csG, (1) where K is a constant which becomes zero at the critical point. In three dimensions, the appropriate spherically symmetric solution is well known to be an exponentially damped inverse distance function, which is perhaps qualitatively not unreasonable. Green 5) however, has shown that the same partial cluster summation technique as we have used, when applied to the three-dimensional continuum gas, casts considerable doubt on the validity of the Ornstein-Zernike theory. The inadequacy of eq. (1) is especially apparent in two dimensions for which the critical point solution is proportional to log r, which diverges at large distances in contradiction to the known general properties of g(s)(y). Kaufman and Onsager 1) have computed the persistence of order for pairs of sites along a row at the critical point of the two-dimensional square Ising net. In the language of the corresponding lattice gas theory, the asymptotic form of the pair correlation function (now defined only for vector separations + between lattice sites) is apparently isotropic e), and is given by: G(r) N C(r/a)-f G = exp{log(l
+ 2-*) -
(1 + y)/4 +S!l[~ log(l -
W2)
+ &I}
(2)
(y is Euler’s constant, and a is the nearest-neighbor distance). Van Leeuwen, Groeneveld and De Boer 2) have shown that the Mayer cluster theory of the pair correlation function leads to a coupled set of functional equations for this quantity which, for the lattice gas with pair interaction Q)(Y),may be exhibited in the form (/I = l/KT): g(2)(r12)= N(r12) =
Wl2)
exp[--/%h2) P~x(r13)
r3
= g(2Yr12) -
-
+
W12)
+
Wl2)l
[N(r32) +
x(r32)l
N(r12)
1
751 -
-
(3)
752
THE
The
density
CLUSTER
positively
THEORY
OF CRITICAL
p is the fraction
infinite
of lattice
for zero argument,
over all lattice sites.
POINT
sites filled
DENSITY
FLUCTUATIONS
by particles.
If a, is taken
to be
the sum in the second of eqs. (3) may be extended
E represents the sum of “basic” or “prototype” clusters connecting to factors G(ru) ; the associated diagrams
sites 1 and 2 which have bonds corresponding have
no articulation
be two
or more)
Green
points,
5) has pointed
to yield mation
diagrams
decay
lattice
We
and
is valid
incidentally
remark
Kaufman
or not
and Onsager,
(3) will preserve Thus
unlike
to speculate
particle
pair density
In order to preserve
approach;
for the square lattice that
fluctuations
versions
reasonable
the rate of decay
results in two interactions.
Ornstein-Zernike
three-dimensional
drop off as ~-a with result
of eqs.
of G however,
with nearest-neighbor
in the analogous
We
sites.
of the divergent
the Kaufman-Onsager
neighbors.
be 8 in the model of
that the approximate
yields qualitatively
over-correction
square, triangular),
to nearest
the value pe should rigorously
cluster theory
to be too great
two-
(4)
are restricted
the Ornstein-Zernike
of this apparant
critical
of g’s’ for the general
filled and unfilled
On account
E, of highly
the same approxi-
result:
between
is predicted
it is tempting
nature
that
point, and as is the area per site in the general
there is no assurance
this symmetry
the approximate
dimensions,
found
of lattice chosen (hexagonal,
interactions
that although
must
N 4(3r~z)*~~-+/,)-~/s
is independent
whether
point when the sum,
likewise
one site per unit cell, with
where pC is the value of p at the critical This expression
have
(there
themselves.
gas analogs of eqs. (3) may be analyzed
as to the asymptotic
gas, with
G(r) lattice.
and the field points
among
of g(2) at the critical
is disregarded.
may be investigated
dimensional
connected
out how the continuum
the asymptotic
connected
nodes, or subdiagrams,
are at least singly
theory,
case, the true
1 < a < 2 (G r e e n 5)).
(2) for the near-neighbor
square
lattice, it is easy to demonstrate that E(r) necessarily is proportional to Y-) for large r. Therefore, in spite of the fact that the set of clusters contributing to E is highly crosslinked, the resulting sum decays slowly to zero near the critical point. Furthermore, Eq. (2) suffices to demonstrate that many (though not all) of the clusters in E individually
diverge
The conclusion cooperative
at the critical
behavior
a proper
description
therefore
appears
is that highly connected of at least
fundamentally
to seek a renormalization vergent
result
point.
to be drawn for the cluster theoretic the critical necessary,
transformation
under conditions
approach
to systems exhibiting
clusters must be considered point,
at least
for
phase.
two-dimensional
of the terms in the
of cooperative
in obtaining
if not the condensed
E series to provide a con-
transition. FRANK
H.
HARRY
L. FRISCH
STILLINGER,
JR.
Bell Telephone Laboratories, Incorporated Murray Hill, New Jersey, U.S.A. Received
:
It
systems,
5-4-6 1 REFERENCES
1) Kaufman, I3. and Onsager, L., Phys. Rev. 73 (1949) 1244. 2) Van Leeuwen, J. M. J., Groeneveld, J. md De Boer, J., Physica 25 (1959) 792. 3) Meeron, B., J. math. Phys. 1 (1960) 192. 4) Ornstein, L. S. and Zernike, F., Proc. Acad. Sci. Amsterdam 17 (1914) 793. 5) Green, M. S., J. them. Phys. 33 (1960) 1403. 6) Domb, C., Adv. Phys. 3 (1960) 199-201.
Physica 27 75 l-752
LETTER TO THE EDITOR Critique of the cluster theory of critical point density fluctuations For two-dimensional lattice gases which possess a critical point (but where interactions are not necessarily restricted to nearest neighbours), we have investigated the form of the pair correlation function computed by a partial cluster diagram summation at the critical point. Diagrams taken into account are the entire set that may be constructed from series and parallel connections of the fundamental Mayer cluster bond. Since the result fails ever to agree in its asymptotic large distance form with the known exact result for the special case of nearest neighbor interactions in the square lattice i), the inevitable conclusion must be drawn that the highly connected “basic” s) or “prototype” 3) diagrams which were neglected, and which in any event are difficult to compute, are important in correctly determining fluctuation correlations in the critical region, at least in two dimensions. The first attempts to predict the nature of critical pair correlations in classical fluids, by Ornstein and Zernike 4), were based on assumption of independently variable Fourier components of microscopic density, and yielded a relation for G(r) = g@‘(y) -
1
(g(s) is the ordinary pair correlation function) of the form: VsG = ,csG, (1) where K is a constant which becomes zero at the critical point. In three dimensions, the appropriate spherically symmetric solution is well known to be an exponentially damped inverse distance function, which is perhaps qualitatively not unreasonable. Green 5) however, has shown that the same partial cluster summation technique as we have used, when applied to the three-dimensional continuum gas, casts considerable doubt on the validity of the Ornstein-Zernike theory. The inadequacy of eq. (1) is especially apparent in two dimensions for which the critical point solution is proportional to log r, which diverges at large distances in contradiction to the known general properties of g(s)(y). Kaufman and Onsager 1) have computed the persistence of order for pairs of sites along a row at the critical point of the two-dimensional square Ising net. In the language of the corresponding lattice gas theory, the asymptotic form of the pair correlation function (now defined only for vector separations + between lattice sites) is apparently isotropic e), and is given by: G(r) N C(r/a)-f G = exp{log(l
+ 2-*) -
(1 + y)/4 +S!l[~ log(l -
W2)
+ &I}
(2)
(y is Euler’s constant, and a is the nearest-neighbor distance). Van Leeuwen, Groeneveld and De Boer 2) have shown that the Mayer cluster theory of the pair correlation function leads to a coupled set of functional equations for this quantity which, for the lattice gas with pair interaction Q)(Y),may be exhibited in the form (/I = l/KT): g(2)(r12)= N(r12) =
Wl2)
exp[--/%h2) P~x(r13)
r3
= g(2Yr12) -
-
+
W12)
+
Wl2)l
[N(r32) +
x(r32)l
N(r12)
1
751 -
-
(3)
752
THE
The
density
CLUSTER
positively
THEORY
OF CRITICAL
p is the fraction
infinite
of lattice
for zero argument,
over all lattice sites.
POINT
sites filled
DENSITY
FLUCTUATIONS
by particles.
If a, is taken
to be
the sum in the second of eqs. (3) may be extended
E represents the sum of “basic” or “prototype” clusters connecting to factors G(ru) ; the associated diagrams
sites 1 and 2 which have bonds corresponding have
no articulation
be two
or more)
Green
points,
5) has pointed
to yield mation
diagrams
decay
lattice
We
and
is valid
incidentally
remark
Kaufman
or not
and Onsager,
(3) will preserve Thus
unlike
to speculate
particle
pair density
In order to preserve
approach;
for the square lattice that
fluctuations
versions
reasonable
the rate of decay
results in two interactions.
Ornstein-Zernike
three-dimensional
drop off as ~-a with result
of eqs.
of G however,
with nearest-neighbor
in the analogous
We
sites.
of the divergent
the Kaufman-Onsager
neighbors.
be 8 in the model of
that the approximate
yields qualitatively
over-correction
square, triangular),
to nearest
the value pe should rigorously
cluster theory
to be too great
two-
(4)
are restricted
the Ornstein-Zernike
of this apparant
critical
of g’s’ for the general
filled and unfilled
On account
E, of highly
the same approxi-
result:
between
is predicted
it is tempting
nature
that
point, and as is the area per site in the general
there is no assurance
this symmetry
the approximate
dimensions,
found
of lattice chosen (hexagonal,
interactions
that although
must
N 4(3r~z)*~~-+/,)-~/s
is independent
whether
point when the sum,
likewise
one site per unit cell, with
where pC is the value of p at the critical This expression
have
(there
themselves.
gas analogs of eqs. (3) may be analyzed
as to the asymptotic
gas, with
G(r) lattice.
and the field points
among
of g(2) at the critical
is disregarded.
may be investigated
dimensional
connected
out how the continuum
the asymptotic
connected
nodes, or subdiagrams,
are at least singly
theory,
case, the true
1 < a < 2 (G r e e n 5)).
(2) for the near-neighbor
square
lattice, it is easy to demonstrate that E(r) necessarily is proportional to Y-) for large r. Therefore, in spite of the fact that the set of clusters contributing to E is highly crosslinked, the resulting sum decays slowly to zero near the critical point. Furthermore, Eq. (2) suffices to demonstrate that many (though not all) of the clusters in E individually
diverge
The conclusion cooperative
at the critical
behavior
a proper
description
therefore
appears
is that highly connected of at least
fundamentally
to seek a renormalization vergent
result
point.
to be drawn for the cluster theoretic the critical necessary,
transformation
under conditions
approach
to systems exhibiting
clusters must be considered point,
at least
for
phase.
two-dimensional
of the terms in the
of cooperative
in obtaining
if not the condensed
E series to provide a con-
transition. FRANK
H.
HARRY
L. FRISCH
STILLINGER,
JR.
Bell Telephone Laboratories, Incorporated Murray Hill, New Jersey, U.S.A. Received
:
It
systems,
5-4-6 1 REFERENCES
1) Kaufman, I3. and Onsager, L., Phys. Rev. 73 (1949) 1244. 2) Van Leeuwen, J. M. J., Groeneveld, J. md De Boer, J., Physica 25 (1959) 792. 3) Meeron, B., J. math. Phys. 1 (1960) 192. 4) Ornstein, L. S. and Zernike, F., Proc. Acad. Sci. Amsterdam 17 (1914) 793. 5) Green, M. S., J. them. Phys. 33 (1960) 1403. 6) Domb, C., Adv. Phys. 3 (1960) 199-201.