“Clusters” in the Ising model, metastable states and essential singularity

ANNALS

OF PHYSICS

98, 390-417 (1976)

“Clusters”

in the king Model, Metastable Essential Singularity

States and

K. BINDER Fachrichtung II.1, Theoret. Physik, Universitat des Saarlandes, 66 Saarbrucken II, West Germany

ReceivedAugust 21, 1975 Various possibilities for the definition of “clusters” which are used in theories of critical phenomena and nucieation are discussed for the case of nearest neighbor Ising models. Using a two coordinate description in terms of a contour (of “size” s) around[reversed spins, it is shown that scaling assumptions for the cluster concentration p(d, s) imply that the critical behavior cannot be attributed to fully “ramified” clusters as suggested by Domb. Monte Carlo results for p(/, s) are also presented and are shown to be consistent with scaling. For large L’, a crossover to geometric behavior is found, and again interpreted in terms of scaling. Relating the “clusters” to fluctuations of a coarsegrained order parameter, the arguments of Andreev in favor of an essential singularity at the coexistence curve below the critical temperature are recovered. The stability limit of the metastable states, which can thus be defined in terms of dynamic considerations only, is obtained for the whole temperature range from computer simulations.

1. INTRODUCTION

Since the concept of “clusters” or “droplets” was introduced in the theory of dense gases [l-3], numerous attempts have been made to use this concept for the interpretation of static critical phenomena 13-141, dilute magnets and percolation [15-l 81, dynamic critical phenomena [ 1g-221, nucleation theory [23-271, phase separation [28-301, etc. Even compressible magnets [31], tricritical phenomena [32], and central peak phenomena at structural transitions [33] have been treated. Most of the above work relies heavily on unproven assumptions concerning the cluster properties, and part of it is clearly inappropriate [21, 32, 331. In view of this unsatisfactory situation, extensive computer studies have been carried out to test the various droplet models [34-391. Both the studies of clusters in the Ising model [27, 34-381 and in Lennard-Iones liquids [23, 391 are hampered by the fact that it is quite uncertain what is meant precisely by a “cluster.” In spite of these difficulties, the droplet model prediction l-5, 401 of an essential singularity at the coexistence curve attracted particular attention [3, 14, 24, 41-441 in view of the implications of this result for the properties of metastable states 390 Copyright All rights

0 1976 by Academic Press, Inc. of reproduction in any form reserved.

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391

[41-45]. Recently, Domb [3,43] suggested that a consideration of ramified clusters and cluster-cluster interactions [46] should remove the essential singularity from the coexistence line and allow for a spinodal line close to the critical point, in contrast to the model of Fisher [4-61. Since the Monte Carlo work for 3d-lattices also indicated much stronger deviations from the Fisher model [5] than earlier work in two dimensions [23,35], we attempt here to clarify these various approaches of cluster definition [20, 35, 471 and their consequences in a more systematic way than given previously. In Section 2, we introduce the “contour [48] picture” of clusters and formulate the appropriate homogeneity assumptions in the critical region. Apart from standard critical exponents [6], one new exponent enters into the description of the cluster concentration, which describes the degree of “compactness” of the clusters, and which was missing in the previous treatments [4-6, 11-13, 20, 471. The scaling structure of the Fisher model [5, 19, 201 follows only after a transformation of the “cluster coordinate” 1---f I’. It is shown that different “effective” cluster concentrations, n [, ncE enter into the critical contributions of magnetization and energy, as conjectured previously [20]. Reinterpretation of the transformed cluster coordinate I’ as the excess number of minus spins in a correlated volume of linear dimensions 6 = [,,(l - T/T,)-” yields the “fluctuation picture” [20, 471 of clusters (Section 3). Then both the Fisher cluster model [5] and the standard geometric model of classical nucleation theory [I, 231 turn out to be reasonable conjectures for two different limiting cases. In Section 4 computer simulation data, a short account of which was given earlier 1341, are presented in order to confirm this description. Section 5 discusses the consequences with respect to the essential singularity and the metastable states, while Section 6 summarizes the conclusions.

2. SCALING

THEORY FOR THE CLUSTER CONCENTRATION

In each state of the Ising system we may separate the minus spins from the plus spins by drawing contours around the minus spins [48] (Fig. 1). In two dimensions these contours are a set of nonintersecting closed polygons, in three dimensions polyhedra, and so forth. In the general case it may happen, however, that contours occur which extend throughout the whole system, i.e., a single contour contains a finite fraction of the minus spins in the system 1371. In order to avoid this “percolation problem” [49] we may introduce some additional prescription, which cuts the contours suitably into parts (as indicated by the wavy lines in Fig. I, where-only as an arbitrary example-it was required to cut the contour whenever flipping of a single spin would separate the contour into two contours). We now call the resulting entities “clusters” and denote the number of reversed spins within a cluster by G and the number of broken bonds at the

392

K.

BINDER

FIG. 1. Part of an Ising square lattice, where contours are drawn around “clusters” of reversed spins. Up spins are not shown. At positions marked by wavy lines an additional prescription is used, which cuts contours into parts which then touch each other, in order to safely avoid the occurence of percolating clusters of infinite size.

surface (contour) of the cluster by s. We stress the fact that clusters defined in this way may “touch” and that any contour may have several disconnected parts, i.e., there may be internal surfaces or holes within a cluster. It is possible that within the holes one again has smaller clusters [47], etc., as indicated in Fig. 1. At thermal equilibrium at a temperature T and field h = HpB we have Np(8, s) clusters characterized by the coordinates I, s at a lattice of N sites. In the case of nearest neighbor interactions 2J we have two exact relations for the average energy E and magnetization m, respectively [N+ co, m is normalized to unity for T-+ 0] m = 1 - 2 c ep’p(e, s) L,s

E= -J[4-2;Md]

-h[l

-2;W,s)].

Here q denotes the coordination number of the lattice. Equations (1) and (2) do not display the up-down symmetry of the king magnet which exists with respect to simultaneous change of the sign of the spins and the sign of the magnetic field. This symmetry can be borne out in a clear fashion, if we denote by Np-(e, s, H) the number of clusters of minus spins and by Np+(/, S, H) the number of clusters of plus spins, such that any spin must belong to one and only one cluster:

c QP+K 3, H> + P-V, s, f-01= 1, 8,s m = ~4p+(4 s, H) - P-V, s, fO1, 6.8

“CLUSTERS”

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q - 2 s[p+(Lp, S, H) + p-(t, s, If)] I - mh e,s = --J [q - 2 c sp-(C, s, H)] - mh. e.s In the last of these relations we have used the fact that each broken bond belongs at the same time to a “plus cluster” and a “minus cluster” (per definition), The above relations now exhibit all required symmetry properties, since p+g s, --H) = p-(C, s, H). If we now use the first of these relations to eliminate the sum over the ~‘(8, S, H) from the second relation, we immediately recover Eqs. (I), (2) by putting p-(T, s, H) == p(4, s). It is interesting to note that one may choose a definition of cluster coordinates where s counts the broken bonds at the external surface of a cluster only. Then one would have for the energy instead of the above relation

E = --J 14-

2 z sIp+(t, s, H) + p-(6, s, H)]\

= --J

q - 2 c s[p-(f, s, --H) + p-(6 s, H)]j 1 CA

= -.I

q - 2 1 sp&, s)/ - mh, 1 cf.8

- mh - mh

with pE(C, S) = p-(t, s, H) + p-(t, S, --H). This latter definition is preferable for the following reason. For very large clusters (C-t co), “local equilibrium” will be established within each cluster, i.e., at nonzero temperature the total number of clusters of the another sign within this cluster also must be proportional to ?, and hence the number of broken bonds of such clusters would be trivially proportional to the cluster size. In the following we will disregard the difference between p(& s) and PE(/, s), however, since PE(8, s) must satisfy the same scaling assumption, if p(t, s) satisfies scaling [Eq. (4) below]. If we had not taken precautions against the percolation problem, it would be conceivable that important contributions to Eqs. (I), (2) come from C = 03, i.e., a single infinite network, as found in the computer simulations of MiillerKrumbhaar [36, 371. If near T, , m and E satisfy static scaling hypotheses

[E==zI 1 - T/T, II Fl m = 8&h&*),

E = EC + +“z;(h@),

(3)

394

K.

BINDER

with the usual scaling relations between critical exponents 2 - CII= /% + /3 = y + 2/3 = dv it is reasonable to assume that p(~?,s) is also represented in terms of a generalized homogeneous function: p(c, s) = e-ry(se-z, he: &),

(4)

where 70, x > 0, y > 0, z > 0 are four scaling powers characteristic for a function of four variables I, s, h, and E. It must be stressed that Eq. (4) will not be true for arbitrary prescriptions of cutting the contours into parts, but it seems reasonable that a suitable prescription, i.e., a suitable cluster definition can be found, such that Eq. (4) holds close to the critical point. Here we are not concerned with the problem of giving this prescription explicitly, since at low temperatures it seems to be rather unimportant, but rather we exploit the consequences of Eq. (4). In the scaling limit h -+ 0, E-+ 0, hi-@ = finite, we must also have G+ 03, h&! = finite, EL+= finite. The derivatives of m and E with respect to h and E diverge in this limit. Since p is finite at all values of its argument, it is seen from Eqs. (l), (2) that critical divergencies can be obtained only from the fact that the 8, s summations diverge. Hence we may replace the sums by integrals with negligible error for the singular parts, e.g., (5) Here jW denotes the derivative of p with respect to the argument geometric reasons we have in d dimensions

&, and for

(6)

where c1, c2 are constants of order unity. Clusters with s close to smin we denote as “compact” (for instance, the clusters of classical nucleation theory [23]), while clusters with s close to smax are “ramified” [3, 431 (for instance, clusters on a Bethe lattice). The additional cutting prescriptions may have the consequence that fully ramified clusters (i.e., with c1 = q - 2) do not occur. But in spite of the touching of clusters we must have cz > 0. Then Eq. (5) is rewritten (7) As a first possibility

we assume 1 - l/d < x < 1 and that the t-integration

is

“CLUSTERS”

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convergent both for t--f 0 and t -+ co (other possibilities appendix). Then we may introduce a cluster concentration

are investigated in the n by [T = TV - X]

In the integration limits 0 is replaced by c2 if x = 1 - l/d, and 00 is replaced by cp if x = 1. Combining Eqs. (7) and (8) it is found that asymptotically for h+O,c+O

02 &n/i%

cc

.c0

d/ P-~+%(h~~,

(9)

El’),

which yields for h = 0, changing the integration variable C to II = G;?$ amI& r E-(?-T+z)“, p = (T - 2)/z, where a comparison with Eq. (3) was made. Similarly am/ah x h-(2-W/~,

(10)

one finds for E = 0:

r/s = (r - 2)/J>,

ps = y/z.

(I 1)

If one calculates am/ah and puts afterwards h = 0, one finds instead am/ah CCE-r2-~+v)i~, y = (2 - 7 f y)/z.

(12)

Note that Eqs. (lo)-(12) satisfy the scaling relation y = p(S - 1). The droplet model exponents T, y, z are not uniquely determined in terms of critical exponents (one of then could still be chosen arbitrarily). Next we consider the specific heat (13)

which can be written in terms of “energy clusters” ntE

as

The “energy clusters” ncE have been introduced previously [20] on an ad hoc basis. The difference between ne and ndEmay be related to the average size 5 of the contour

396

K. BINDER

CIusters with large “surface” s are more important Eq. (15) we immediately derive aE

-$-g = c-e,

for neE than for Q. From

CX= 1--7+2+x z

(17)

From the equality of cross derivatives of the free energy a2F PC ac ah

a2F ahae’

am X”ah

aE

(18)

we find, on the other hand, E

ah

K

&l

’

p-1=

l--7+y+x

z

which implies that the scaling relation 2 - CII= p(S + 1) is fulfilled can be related to the exponents z and y x=z-y+1=1-z(/36-I).

(19)

and that x W)

Hence we find that the scaling structure of the cluster concentration is determined by bulk critical exponents and one new exponent, for instance x, which describes the average size of the contour as 8 + co [Eq. (16)] at E = 0, h = 0. We note that the case x = 1, where the critical behavior would be dominated by ramified clusters, would imply z = y, i.e., [Eq. (1 l)] PS = 1. Hence we conclude that the suggestion of Domb [3, 431 that the critical behavior can be attributed to ramified clusters is inconsistent with scaling [Eq. (4)], and we believe this suggestion is incorrect. This conclusion remains true as well in the cases considered in the appendix, where the main contributions to magnetization clusters (q) and energy clusters (nf) come from different regimes in the interval rmi, < s < s,,, . In these cases Eq. (20) does not hold, however, and has to be replaced by Eq. (A.6) or (A. IO), respectively. We summarize these cases as follows. (1) If the dominant contributions to both magnetization clusters and energy clusters are due to compact clusters, Eqs. (A.4), (A.5) are valid. If the compact clusters dominate the magnetization clusters only, we have instead

i.e., now two independent exponents x, z are necessary to characterize the scaling of the cluster distribution. Note, however, that x does not enter the thermodynamic functions.

“CLUSTERS"

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(2) If neither compact nor ramified clusters dominate, the last inequality of Eq. (21) is replaced by an equality [Eq. (20)]. (3) If neither compact nor ramified clusters dominate the magnetization clusters, but ramified clusters dominate the energy clusters, we have again two independent exponents z, x, but instead of Eq. (21) we have now s

x

fl-zw-1)

1 9

O
1 - z(/3S -

qp-I---

I) c: X
(4) The case that ramified clusters dominate both energy and magnetization clusters would imply PS = 1 and is thus not of physical interest. 3. RELATION

TO STANDARD

CLUSTER MODELS

Since most of the theories on cluster distributions do not consider the coordinate s explicitly, it is convenient to rewrite our results as follows

ne YEI=pP+(Ylw@p,

&Y/y,

n/E

=

[-Y+(Y/~)lfiE(h/~,

&/OS),

(25)

and in all cases the exponent y has to be within the limits 0 < y < /3S/[d(pS - I)]. Note that in usual cluster models [5, 131 the exponent y is put equal to unity. Obviously this choice is not excluded for d = 2 (y < 15/14 if ,6 = l/g, 6 = 15 is used [6]), but it is excluded both for d = 3 (from [6], /3 = 5/16, 6 = 5 one finds y < 25127) and for d = 4 (from p = l/2, 6 = 3 one finds y < 314). We stress the fact, however, that it is possible to obtain precisely the scaling structure of the Fisher [5] model if one transforms from the coordinate C to a generalized cluster coordinate L’ by writing e’ = PY and requiring the relations to satisfy the sum rules n& dG= n&’

dt’,

nLEP dt = I$/~“’ d/‘,

(26)

which gives w’ = l//?S and ,ld’ zzzpc2+wjid,(hf’,

&lP),

$7

=

p2+v)@(/f,

,pPy.

(27)

Thus if one defines [20] a cluster coordinate 8’ by the requirement that a dependence 595/9W8

398

K.

BINDER

on h occurs in the cluster concentration in the combination hl’ only, Eq. (27) is the most general description. It remains to clarify the physical meaning of this generalized coordinate, however. This is done conveniently considering [20,47] the average number zcontributing to the divergence of the susceptibility as E- 0, h = 0 (consideration of any other diverging quantity would give the same result). This yields

i.e., the typical number of reversed spins in a cluster should increase as ,-flSiv as E - 0. On the other hand, it is well known that close to T, all states of the system can be represented in terms of a coarse grained order parameter [50], which has fluctuations on a length scale of the correlation distance t and whose average value is proportional l . This fact is consistent with our cluster description only, if we require that [ is a typical linear dimension of the “rather compact” regions of our typical clusters (cf. Fig. l), since the regions of the lattice where the cluster is less compact (and thus roughly equal number of spins point up and down) produce this average background magnetization (m) and give little contribution to the correlation function. A cluster with P reversed spins will thus produce a deviation of the coarse grained local order parameter k(r) in a volume region V, . Measuring lengths in units of the lattice spacing, we must have V, ,( constant G, of course, where the constant is of order unity. It is then required, that the typical volume Vl of these rather compact regions of the clusters should be of the order P CCF-~” = E+-~). But then the total number of reversed spins t in a typical cluster must be at least of the same order of magnitude k z P,

p/y

2 2 - 01,

y e 1 + 1/s.

(29) It is appealing but not necessary to choose the prescriptions for the cutting of the contours such that V, = constant L, and then Eq. (29) also holds as an quality. Then we would have y = l/(1 + l/S) and 8’ cc &’l / c1+1/sj. Before discussing this possibility in detail, let us assume that we have a more general relation which holds asymptotically for large L’, V, cc /c, [ < 1. We now construct a relation between the exponents y and {. For this purpose we consider large clusters which give rise to “domains” of the other phase with volume V, > 4”. Such domains may occur in thermal equilibrium by thermal fluctuations and should thus be contained in our description [Eqs. (4), (25)] as a limiting case for 8 --f co. Since their cost in free energy Fd is very large and thus these fluctuations will be extremely seldom, we may estimate their probability by writing down the Boltzmann factor: nd a exp[--FJkBT].

(30)

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On the other hand, the excess free energy Ft of a macroscopic domain of volume bi and order parameter difference 2m cc & is given by [23, 271 F/

=

vLeRha’

+

~;-l/d~B-a-vb

f

.

(31)

Here a’, b’ are constants and the appropriate critical exponent of the surface free energy has been used [51]. Now it is easy to see that Eqs. (25) and (30) are consistent with each other only if V, cc 4ac1i-1is),i.e., < = ~(1 + l/S). Now we discuss a physical interpretation for the coordinate /’ CC/‘I. This coordinate /’ is interpreted as the total excess magnetization produced by the ! reserved spins within the volume V!, and we have P/L’, cc (/‘)-I/“. This interpretation is consistent with the fact that the excess magnetization per spin (i.e.. the difference between m(r) and WI) in a correlated volume LJ” should have the order of magnitude 8 as well S/j7[ = (&U/S E (E-Rs/Y)-” 6 = CR.

(32)

The fact that the scaled cluster concentration depends on the product ht’ only also becomes an obvious physical explanation: In the free energy of a cluster we must have a contribution proportional to the product of the average magnetization of a cluster (P) times the magnetic field h. In addition to this “bulk energy” of a cluster, it is tempting to associate the other variable .~(P,)llfl~ with the surface free energy of a cluster: Then below T, for large enough G the excess free energy associated with a cluster is [a, b are constants] F/ = Fcbulk + Fdsurface + ... = ah/’ + br(l’)1:B6 + ... .

since even for h = 0 the cost in energy due to broken bonds in the surface region of the cluster, where it is less compact, should give a contribution. If we would have Vc = /, the surface area of V, would be c.cCs[cf. Eq. (16)], and one would expect Fc,,,,,, to be proportional to the product of the effective surface area P = (8’p”+1/6) and the effective order parameter difference %/Ve = /‘-Us. which gives

(~‘)2(1+1/6)-1!s

E (e’)[l-!/Cl-liSS)lCl+Ijs,-ljs

=

([‘)1/66,

if we

use

y =

1 +

[/a;

in

this case the interpretation given for the second term of Eq. (33) is quite obvious. In the general case [where y < 1 + l/6 and where it is unclear how many of the lL broken bonds are in the surface region of V,] it is interesting to note that the second terms in Eqs. (31) and (33) have the same order of magnitude at V! g [(f [cf. Eq. (32)]: bE[fd@]W ($td)l-l/d

E2--ar--ub'

z be[e-(~~+6)EB]1/6~= b, =

E-(d-l)v~-v~2-ab’

=

E-dv62-ub’

and also the bulk energy terms in Eq. (31) and (33) “match”

(3‘td) _

b’

9

(,34b)

for arbitrary values

400

K.

BINDER

of the scaled field /~--8~, due to Eq. (32). Equation (34) implies that FZ/kBT is a constant of order unity in the region where fluctuations of size 5 are involved. AS long as this constant exceeds unity considerably, i.e., for 1 < ,(/‘)l/sS Q ~0, it seems reasonable to assume that n is proportional to exp(-Ft/k,T) [Eq. (30)], and combining this assumption with Eq. (27) one finds [20] ~8’ = constant(e’)-‘2+1/s) exp[-ahe’/k,T

- b~(t”)l/B~/k,T],

(35)

which is the droplet model of Fisher [5]. While this “derivation” certainly does not have the rigor of the treatment given in Section 2, it is interesting to note that some of the criticism [3] pertinent to the rather different original derivation [5] (where no difference between P and e’ was allowed for) does not apply to the present one. If Eq. (35) or similar other formulas due to Reatto and Rastelli [13] are used for arbitrary values of ,(8’)l/fls and he’, one obtains relations between the various prefactors (a’, b’, etc.) and critical amplitudes. Obviously this procedure is not expected to be accurate, since Eq. (35) is valid for F/kBT > I only, where .~(C,)lj@ > 1 at the coexistence curve, and Eq. (35) should not be used for ,(L’)l/@* < 1. The use of this model Eq. (35) with prefactors determined in this way for applications in nucleation theory [25-271 is thus of doubtful validity, although this procedure works surprisingly well in the case of the two-dimensional kinetic Tsing model [27, 41, 521. Furthermore, we stress the observation that the crossover from the regime of critical fluctuations [as described, e.g., in Eqs. (35), (33)] to the regime of macroscopic domains [as described by Eqs. (30), (31)] for ,(d’)liO” -+ co is irrelevant for critical amplitudes, since the contributions of the geometric regime are exponentially small. This crossover is important for questions like nucleation close to the coexistence curve, or the essential singularity, however. For the sake of clarity we summarize the main concepts introduced in this section. A cluster with P reversed spins (i.e., C measures the deviation of the magnetization from Z(r) = 1) gives rise to an excess magnetization of el cc eV (i.e., t’ measures the deviation of the magnetization from m(r) = m) within a volume region V cc tP< with 5 = ~(1 + l/S). The latter relation follows from with the classical free energy of large scaling, i.e., a “matching condition” “droplets” [Eq. (30)]. Are there arguments to determine 5 or y uniquely. Let us require that the total number of broken bonds in a large domain Vc be proportional to V, itself, in order to have agreement with the bulk internal energy E as well. Since S = @ and x = 1 - ~(1 - I//%) [Eqs. (16), (20)], the relation 5 = x would yield

Y+P

y=y+p+l--cu.’

1 Z=r+i3+1-CX’

2-a X=r+/I+l--“l

(364

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401

For d = 2, 3,4 this would result in x = 16123 = 0.69, lo/13 = 0.77, and 415 = 0.8, respectively. Of course, this requirement is not absolutely necessary, since one may allow for smaller clusters occurring within a large cluster: The energy contribution of the volume region filled by the large cluster may then be mainly due to the smaller clusters. The extremum possibility in favor of this case is [cf. Eq. (29)] y = [l + l/S]-l,

z = (2 - cd-l,

x = (1 + /3)/(2 - a).

(36b)

For d = 2, 3,4 this would result in x = 9116 m 0.56, 7110 = 0.7, 314 = 0.75, respectively. Available computer simulations [34] are more consistent with Eq. (36b), but more detailed results would be desirable.

4. COMPARISON

WITH COMPUTER

SIMULATIONS

While computer simulations are available where IZ was obtained from direct counting of clusters in the generated Monte Carlo configurations [35-381, a determination of the full distribution p(L, s) would require prohibitively large amounts of computing time. It is possible to study the reduced distribution P,(s)

by generating a Markov chain through a subspace of the configuration space. which contains a cluster with /spins. This is done [34] by starting from an arbitrarily chosen configuration of a system of N = Nld sites on a square (or simple cubic) lattice, with a cluster of (= [Id (L’, being an integer) reversed spins in it, where N1 > PI . Then transitions are generated by exchanging pairs of spins subject to the restriction that the number of spins P within the cluster is conserved, and the transition probability is chosen such that it obeys a detailed balance condition with the canonic ensemble distribution [53]. In the actual simulation two approximations were necessary for the sake of simplicity: (i) No prescription for cutting the contours into parts was used. Since iV1 =Z 3C, was chosen and T < 0.9 T, , difficulties with the percolation of clusters did not yet arise. (ii) Instead of recording the total number of broken bonds only s’ was recorded, the surface area of the outer contour of a cluster [34]. For not too large C it is quite improbable that cluster contain holes (Fig. l), and indeed very few were found. Then P!(s) and P!(s’) should be very similar, if P/(Y) is defined in analogy with Eq. (36). Figures 2-4 give numerical results for Pe(s’) at two dimensions for these temperatures. These distributions tend strongly to zero for s’ -+ s,in , indicating

402

K. BINDER

that the case considered in Eqs. (A.4) (AS) is not realized. For large 6’ these distributions always have a rather broad tail for s’ > 3, even at rather low temperatures. This property can be understood from the fact that the interface between two coexisting phases is “rough” at arbitrarily low temperatures in the 2d-Ising model [54]. But P&‘) tends strongly to zero long before sdaX (W 2/) is reached.

FIG. 2. Reduced distribution P&‘) versus s’ for the square Ising lattice at T/Tc = 0.44. Parameter of the curves is 8. Arrows denote the position of &,,, .

FIG. 3. Reduced distribution P&‘) versus s’ for the square Ising lattice at T/T, = 0.7. Parameter of the curves is 8. Arrows denote the position of &,,, .

“CLUSTERS”

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MODEL

403

P,(S'l 43 f

FIG. 4. Reduced distribution I’&‘) versus s’ for the square Ising lattice at T/rC - 0.9. Parameter of the curves is /. Arrows denote the position of &,,, .

Hence it is plausible to assume that none of the cases considered in the appendix is realized in the Zd-Ising model, and both magnetization and energy clusters are dominated by s oc P with x somewhat larger than 1 - l/d. Figures 5 and 6 give numerical results for P(d) at three dimensions for two temperatures. Again these distributions tend strongly to zero for s’ + .smrn~ but now S’ is relatively closer to &,, (Z 4P), and the distributions are much broader

0

I

FIG 5. Reduced distribution PC($) versus s’ for the simple cubic king lattice at 7;:T, -A 0.7. Parameter of the curves is l. Arrows denote the position of S& . The curve on the right side refers to L = 27.

404

K.

BINDER

.

FIG. 6. Reduced distribution P&‘) versus s’ for the simple cubic Ising lattice at T/Tc = 0.9. Parameter of the curves is 8. Arrows denote the position of Cl,, .

than the corresponding distributions for d = 2 and similar values of 8. If one compares distributions with similar S, however, then there is no qualitative difference. Consideration of the distribution s’P{(s’) shows that its maximum occurs nearly at the same position as the maximum of the distribution PG(s’) itself. Since also the distribution s’~P~(s’) is quickly decreasing after this maximum value has been reached, we conclude that both energy and magnetization clusters are dominated from the same parts of the contour size distribution. These distributions are not taken in the asymptotic regime d-t co, E-P 0, and exhibit also considerable scatter, and thus a direct verification of the scaling ideas of the previous sections is difficult. Thus we consider here only the scaling of S’, where we expect on the basis of Eqs. (8), (14) (16), and (20) and y = 1 + l/6 s =

d(l+l/o,/ll+s,f(,2-ae))

x

{I-l/d+-l-v,

E2--ae-

co

(38)

invoking also the ideas concerning the crossover to geometric behavior developed in Section 3. We have /?S - 1 - v < 0, as expected. Hence the data for S’ obtained [34] from the distributions in Figs. 2-6 are replotted in scaled form, as suggested by Eq. (38), in Fig. 7. Of course, we cannot really expect that data for G < 10 or E as large as 0.6 fit to a scaling representation. In fact, while the data for E = 0.1 and 0.3 fall fairly close to each other in that plot, strong deviations occur for E M 0.6. In three dimensions the deviations are much larger than they are in two dimensions. This observation may be attributed to the fact that E = 0.6 is below the roughening temperature T, of the interface [55]; it is suspected that the behavior of large clusters changes significantly at T, , below TR the clusters are probably much more compact. At d = 2, TR = 0, however [54], and thus the

“CLUSTERS"

IN THE ISING MODEL

405

FIG. 7. Log-log plot of the scaled average surface area s’/@+~/@/~~+~’ versus &@+W in three dimensions (upper part) and two dimensions (lower part).

deviations from the scaling regime are less drastic throughout the low temperature regime. But particularly for d = 2, the crossover to geometric behavior is clearly seen, as indicated by the straight lines which have the corresponding theoretical slopes. Clearly, additional data for both smaller E and larger ( would be desirable but due to the need of going both to larger systems and larger Markov chains considerably more amount of computing time would be required [53].

5. METASTABLE

STATES AND ESSENTIAL

SINGULARITY

It has been proven [56] rigorously that all derivatives of the free energy exist in the Ising model at the coexistence curve as h --) O+. If in this limit at the coexistence curve a singularity occurs at all, it can be an essential singularity only, where

i.e., expanding F as a power series in h the radius of convergence would be zero. In the framework of Eqs. (1) and (4) one has to consider the behavior of the derivative (401

where P) is the kth derivative of n” with respect to the argument hb, as k - m.

406

K. BINDER

If the upper limit of the integration in Eq. (40) were replaced by some cutoff L, one would obtain the following approximation for 3m/i3hk jhSOwhich becomes asymptotically exact as k -+ 03 aktn

ahx’ h-0

since n”(k)(0, &Ps) get

I w ?i’“‘(O, cL”/BS) kL’ _ 1 _ J,,s Lb-”

JS)

(41)

must be a bounded function of its argument. Then one would

for any finite L. Thus for a discussion of the possible essential singularity, only the asymptotic behavior of 3”) (0, X+ co) is relevant. Calculating the free energy of the king system, it is legitimate to do an averaging over regious of size L > td, as an intermediate step [50] to obtain a coarse-grained free energy Fe, . The total free energy of the system is then FCBplus a correction due to fluctuations of size larger than L. From the above argument it follows that we have to estimate the contribution of these fluctuations, in order to elucidate

FIG. 8. Various fluctuations of a coarse grained order parameter m(z), shown schematically as functions of one spatial coordinate z.

“CLUSTERS”

IN

THE

KING

407

MODEL

the question of the essential singularity. This is possible since we have chosen L1’ll much larger than the correlation length of fluctuations. Possible remaining fluctuations are shown schematically in Fig. 8, where m is the magnetization which would be obtained from Fcg . These fluctuations have to be represented by the clusters with / > L. The fluctuation in Fig. 8a is of the domain-type. In the limit L/t” -+ a, m -+ 0 we also have am/m ---f0 in the fluctuations considered in Figs. 8b and 8c; in this limit these latter fluctuations should become symmetric with respect to m, and thus their contribution to the magnetization clusters nt should asymptotically cancel each other. Note that the fluctuations of Figs. 8b and 8c are qualitatively not different from fluctuations in the paramagnetic region or at the critical isotherm, where we expect no essential singularity to occur. Thus only the contribution of domain-like fluctuations matter. The concentration of these fluctuations is made arbitrarily small if L/td is made only large enough. Hence any interaction contribution of these fluctuations is safely neglected, as can be seen from a treatment similar to [46], and the correction to the free energy is given by the total excess free energy Ft of a fluctuation multiplied by its concentration, which must be proportional to exp[-Fe/ks7’] in this limit [cf. Eqs. (30), (31), from which we also find (9%/W) (0, &JBG) = (--a’@VJk,T)“: fi (0, &/Ss)], and hence we have from Eq. (40)

Here we made explicit use of the fact that the fluctuation in Fig. 8a can have only a surface contribution at the coexistence curve, and note further that c > 0. For k -+ cc the first contribution on the right-hand side of Eq. (41) is negligible in comparison with the second, and there the lower limit of integration may be put equal to zero. Then one finds

= constant x li+li (k + 1)

T((d/d - l)(k - 1 -- J’ - l/d) - 1) T((d/d - l)(k - 4” - l/d) - 1)

= constant x lim k--lltd--l) = 0 k+m

y(1 -i- l/S)

I.

Thus one expects an essential singularity for all d (except d S; 1). The same argument implies that there should be no essential singularity if the free energy of the fluctuations is asymptotically proportional to their volume, replacing cY1--lirl by

408

K.

BINDER

CG in Eq. (42). These findings disagree with the suggestion of Domb [3, 431 that contributions of ramified clusters should remove the essential singularity in favor of a spinodal line, at least close to T, . Since our scaling arguments are restricted to the critical region, the case of low temperatures requires separate discussion. For T+ 0, however, the rigorous analysis of Capocaccia et al. [42] implies the existence of an essential singularity as well. These authors show that for small T and negative h metastable states can be defined in the Ising square lattice only, if certain large enough clusters are excluded from the partition function. The resulting metastable states are to some extent nonunique, of course, since there is considerable arbitrariness in this exclusion of clusters from the partition function. In the case of an essential singularity a unique continuation of the free energy into the metastable region cannot be given [24]. It has been suggested earlier [41, 571 that metastable states have to be defined on the basis of dynamic criteria in such a case. For instance, one may require that the lifetime of the metastable state should exceed the thermal equilibrium relaxation time at the coexistence curve by at least a factor of 102, in order that one can speak of a well-defined metastable “state” (i.e., whose properties do not change appreciably during a reasonable period of observation). Enhancing this (arbitrary) factor by some orders of magnitude would not change the resulting stability limit h* too much, since the lifetime varies exponentially fast with h in this regime. Results for the metastable states of the square Ising lattice were obtained in the critical region, and did not exhibit any indication of a spinodal singularity [41]. Here we extend this approach to lower temperatures. Again we find that homogeneous nucleation limits the lifetime of metastable states in the Glauber [58] kinetic Ising model at such values of h already where am/ah is only slightly enhanced over its value at the coexistence curve. The resulting stability limit is shown as a function of temperature in Fig. 9, together with the molecular field result for the spinodal line [41] and the rigorous bounds of Capocaccia et al. [42]. The data agree quite well with an extrapolation based on the Schofield [59] linear model equation of state (for details see [41]) but this coincidence is probably accidental. For very low temperatures all these predictions result in stability limits of the same order of magnetude. The stability limits of Capocaccia et al. [42] were taken from their rigorous bounds for the “escape rate” per unit volume ueSc , which is proportional to the initial slope of the magnetization as a function of time after the field was put on its negative value. Their result can be written as

if the initial

state does not contain any clusters with c2 (or more) reversed spins.

“CLUSTERS“

409

IN THE ISING MODEL

FIG. 9. Stability limit of metastable states according to various predictions (cf. text). The crosses denote Monte Carlo results of Binder and Miiller-Krumbhaar [41]; the points denote present Monte Carlo results.

Note also that MaxiF,) = 1 and Min{F,} = 36. While this inequality does not give any useful results in the thermodynamic limit, where clusters of any size occur, however small their concentration is, this inequality is useful for finite systems at low temperatures, where the appropriate c is found from putting NnL == 1 with ( = c2. The most favorable possibility, c = I, results in the two bounds shown for H* in Fig. 9 by interpreting Eq. (44) as two equalities and putting weSc* I. Of course the approach of Capocaccia et al. [42] is subject to severe criticism also, apart from the fact that it becomes meaningless at higher temperatures since for c > 1 the two bounds differ by many orders of magnitude. First of all we note that mesCcharacterizes the relaxation from the initial state towards the metastable state rather than the relaxation from the metastable to the stable state. Thus w& may underestimate the lifetime of the metastable state significantly. This happens in stochastic molecular field models, where UJ& is finite but the lifetime of the metastable state is infinite for 1H 1 < ! H* i. In the nearest neighbor kinetic lsing model it can be shown [41] that for T----f T, West cc E-B 1h I,

h ---f O-

(45)

while nucleation theory implies an exponential variation [41, 271 (lifetime)-l cc &‘r exp{ -constant[c-fi8 I h i]-‘“-l)),

h 4 0-,

(46)

where L$.L~is the exponent of critical slowing down [60]. Finally, we comment on the possibility of interpreting experimental data on metastable states in terms of a “pseudospinodal curve” [61]. There one fits data on the equation of state in the stable or metastable region to a function which has a singularity in the unstable region. This pseudospinodal singularity is there-

410

K. BINDER

fore not a phenomenon accessible to direct observation but rather an artificial way of parameterizing the equation of state in a region which is rather remote from this singularity. If such a fit is possible, this therefore does nor constitute any evidence for a singularity at the actual stability limit, which has nothing to do with this “pseudospinodal.” Figures 10 and 11 may be taken as some evidence for this point. In Fig. 10 data on the magnetization of metastable states and its fluctuation (denoted by

FIG. 10. Inverse relaxation time +a-I (left) and magnetization m and its fluctuation ksTx (right) of metastable states plotted as a function of the field. The points are Monte Carlo results for a square 55 x 55 lattice with periodic boundary conditions and nearest neighbor interactions at J/kBT = 0.6.

FIG. 11. Variation of the magnetization (coarse-grained over a time-interval of 40 Monte Carlo steps/spin) with time for a field in the vicinity of the pseudospinodal (top). The pseudospinodal H* is determined by fitting the data on ksTx (Fig. 10) on a log-log plot versus (H - H*) to a straight line with slope l/2 (bottom).

“CLUSTERS”

IN

THE

ISING

MODEL

411

k,~~) are shown as a function of the magnetic field. At more negative fields the magnetization is distinctly relaxing (this distinct relaxation during an observation time of about 103 Monte Carlo steps/spin was observed first at a field -0.1, as indicated by the arrow), and there the relaxation time rR pBHIkBT-needed for a magnetization reversal is given. An interpretation in terms of a pseudospinodal at H* would imply that k,TX oc (H - H*)-“‘. Figure 11, lower part, shows that the data can in fact be represented in this form, and the estimate for H* obtained is paH*/kBT = -0.11 + 0.002. This estimate is included also in Fig. 10. We find, however, that one cannot come closer to this “pseudospinodal” than 1H/H* - 11 * 15 %: If one went closer, the magnetization is so strongly relaxing due to homogeneous nucleation that any meaningful observation of a static quantity clearly would be impossible. This fact is illustrated on the upper part of Fig. Il. In conclusion of this section, we stress the fact that all metastable states m can be defined only within some uncertainty Sm (with 6m/m ---f 0 as H---z Om-)[41, 571: Since the lifetime of a metastable state is finite, one necessarily has some remnant of the relaxation into the metastable state (following the change of variables which servesto bring the system from a stable state into this metastable state) during the whole period of observation. This relaxation continuously goes over into the relaxation which leads out of this metastable state towards the truly stable state. Hence any metastable state found experimentally has some intrinsic time-dependence, and the “static properties” of a metastable state depend to some extent on the way in which this state is prepared. A unique characterization of “metastability” is thus provided in terms of dynamic “nonequilibrium relaxation functions” only [57, 411. Close to the coexistence curve this intrinsic time-dependence is negligible for all practical purposes [cf. Eq. (46)]. This is not true in the vicinity of the actual stability limit, however: The location of the latter necessarily dependson the time of observation, although for practical purposes this dependence is also small. Thus the actual stability limit is not a precisely defined point (H*), but rather a region (H* k SH), although SH may be very small in practical cases.A spinodal singularity could occur at a precisely defined point only and not in a whole region. and therefore it cannot be observed in principle if the lifetime of metastable states is finite. Therefore, even if one did not have an essentialsingularity at the coexistence curve and could give a precisely defined analytic continuation of the equation of state in the stable region up to some spinodal curve, as suggestedby Domb [3, 431, the physical significance of this continuation would be obscure. In fact, the ideal Bose gasin three dimensionsprovides a nice example for this observation [62]: The free energy of the analytic continuation for densities between the coexistence curve is lower than the free energy at the coexistence curve itself [63], contrary to the metastability concept. A well-defined spinodal is rather found as a charac-

412

K. BINDER

teristic of a coarse-grained free energy, which is rather different from the true free energy, and not accessible to any direct observation but rather a valuable aid in intermediate steps of calculations of dynamic relaxation, as pointed out convincingly by Langer [64].

6. CONCLUSIONS

We conclude by summarizing

the main statements of this investigation.

(i) Describing clusters of 8 reversed spins in terms of certain contours of size s, the scaling assumption for the probability distribution p(/, S) involves four exponents x, y, z, T, [Eq. (4)]. The exact sum rules [Eqs. (I), (2)J and the scaling of thermodynamic functions relate these exponents to critical exponents, apart from one new exponent y [Eq. (25)]. Integrating over the contour size distribution, it is found that two different cluster concentrations ne and neEmust be distinguished, where nd describes contributions to the magnetization and neE contributions to the energy [Eqs. (8), (14), (25)]. If both cluster concentrations are dominated by the same region in the contour size distribution, y is related to the exponent x of the typical contour size [Eq. (16)] by the relation x = 1 - v( 1 - I/@) [Eqs. (1 I), (20)]. If instead the dominating contributions to nt and ncE come from different regions in the contour size distribution ~(8, s), this equality is replaced by weaker inequalities [Eqs. (21), (22)], but the then undetermined exponent x does not enter thermodynamic quantities [Eq. (25) 1. (ii) The typical clusters cannot be “ramified” (which would mean x = l), as suggested by Domb [3, 431, if scaling assumptions hold for p(C, s). (iii) According to the usual derivation of the cluster model of Fisher [5] and related models y = 1, which is incorrect in general, violating geometric constraints for x. However, the Fisher model (or more generally speaking, its scaling structure [20]) is recovered as a valid description if its cluster coordinate is identified as a generalized cluster coordinate 6’ CC8~. (iv) The question of the essential singularity at the coexistence curve is traced back to the asymptotic behavior of the cluster concentration n as &‘YJBs--f co. Thus it is argued that the validity of the Fisher [5] and related [13] cluster models is irrelevant for this question, since in this asymptotic region the clusters have to be related to macroscopic domains, which are described in terms of the classical Becker-Doring-Frenkel droplet picture [ 1, 231. The requirement that the crossover of the cluster properties from the critical regime to this geometric regime is also consistent with scaling yields the result y < l/(1 + l/S). We find that an essential singularity occurs at the coexistence curve for all d, contrary to suggestions of Domb [3, 431 but in agreement with Fisher [5], Andreev [40], and Langer [24].

“CLUSTERS”

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ISING

MODEL

413

(v) The generalized cluster coordinate P = PJ can be interpreted physically as the excess magnetization of a cluster [47], since then the typical cluster “volume” becomes equal to the correlation volume 5”. For clusters of this size the excess magnetization is of order 8, and the excess surface free energy is of order kBT. This interpretation of clusters as some representation of typical fluctuations has been proposed earlier on heuristic grounds 120, 471 and is put by the present calculations on a more coherent basis. (vi) The cluster model of Fisher [5] is rederived as an interpolation between the asymptotic regime &/ss --t cc, where fluctuations require a formation energy much larger than kBT and the classical droplet model [l, 231 is valid. and the regime where fluctuations require less than kBT (and where ,/ajas < I). While this treatment justifies the use of this model in nucleation theory [25-271, where large E&‘/@ are involved, it shows at the same time that there is little reason to assume that this model is valid for ee’JIBs < 1, and hence a determination of its prefactors in terms of critical amplitudes [9, 251 is inaccurate, in general. since there an integration over all values of the scaling variable EP~/~~is involved. (vii) The fact that the Monte Carlo result of Stoll et al. [35] for nf and d = 2 where consistent with the Fisher [5] cluster model taking y = 1 is understood from the fact that due to the large value of 6 the deviation of L’ from (is insignificant for the range of /values studied. In three dimensions the agreement with the Fisher model and y = 1 is in fact much less convincing [36]. It is difficult to test our more general scaling theory, since the data are restricted to rather low temperatures, in order to avoid the percolation of clusters, which occurs in the SClattice for about [37] T w 0.93 T, . In our theory it is assumed that the percolation difficulties are avoided by suitable additional prescriptions by which otherwise percolating contours are cut into parts, but an explicit formulation of such prescriptions suitabIe for computer studies remains to be given! (viii) The present Monte Carlo results on p(/, s) for T < 0.9 T, show that fully ramified clusters are unimportant and suggest that the dominant contributions to both energy and magnetization clusters nGE, nd come from the same region of the contour size distribution, Some indications of a crossover to the geometric regime are clearly seen. The data are not close enough to T, to show convincingly that scaling assumptions actually hold, but the observed deviations from scaling have the expected order of magnitude. (ix) For d = 3 some indications have been observed that p((; s) changes drastically in the vicinity of the interface roughening temperature [55] T, for large /. Hence a study of ~(6, s) by low temperature series expansion techniques as suggested by Domb [3, 431 will probably rather reflect this singularity than the behavior of p(P, s) in the critical region. 595/98/z-9

414

K. BINDER

(x) Monte Carlo studies of metastable states in the kinetic two-dimensional Ising model yield a very smooth variation of the stability limit with temperature, and no indication of a spinodal line. While at T -+ 0 our stability limit falls within the bounds obtained from the escape rate considerations of Capocaccia et al. [42], we present arguments that the usefulness of their treatment is very restricted.

APPENDIX:

CLUSTER SCALING FOR THE CASE WHERE CONTOUR SIZE DISTRIBUTION Is PEAKED AT A BOUNDARY

The step from Eq. (7) to Eq. (8) is unjustified, if the dominant contributions to the integral come from the vicinity of the boundaries of the integration interval [i.e., from t -+ 0 or t + cc instead of t = finite and nonzero, as supposed in Eq. (S)]. As first case we assume now

with & > 2. Then we find cd ds f$c)(s&-“, j#, s,zp-lld

#)

ot ,,-(l-l’d--2)‘~‘-“fi(~~Y,

&),

(A.3

i.e., Eqs. (8)-(12) are still valid if we redefine the exponent 7 as T = 70 - x + (& - I)[1 - (l/d) - x]. Similarly, we find from Eqs. (13) and (A.l) that 63)

which yields 01= [l - 70 + z - 2x - (1 - l/d - x)(#Q - 21/z instead of Eq. (17). However, using the Maxwell relation, Eq. (18), it is easily seen that the scaling relation 2 - CY. = p(S + 1) still holds, but Eq. (20) is replaced by 1 - l/d = Z -2

+ 1,

z = l/[&36

- I)],

1’ = j%/[d(@

-

l)].

(A.4)

where Eq. (11) was used, and Eq. (16) is replaced by 3 cc p--lid

>

(A.5)

as expected. In this case the exponent x is unimportant for nd and nf, and 7, y, and z are determined uniquely in terms of critical exponents. If Eq. (A.l) holds with 1 < & < 2, Eq. (A.2) still holds but the dominant contributions to the energy clusters do not come from t -+ 0, due to an extra

“CLUSTERS”

IN THE ISING MODEL

315

factor t in their definition [Eq. (14)]. Then Eqs. (16), (17) remain valid, and Eq. ( 18) yields now 3’ + 9 + (1 - I/d-

x)(&

- 1) x z T I,

(A.6)

which again implies 2 - 131 = /3(8 + 1). Finally, we note that the boundary cases q$ = 1 or f$r .= 2 would lead to power laws modified by logarithmic corrections. As second case we consider p( t, hP, a“) cc t”‘j&“,

et*),

t--t a.

(A.7)

with c$?> - 2. If & > -1, Eq. (A.2) is replaced by .CIC & j ‘qp, I - e,2/I--l.d

/!L”, &)

E p-i-(+!) ‘l--“‘!f2(/l/‘: &f),

and Eqs. (K-(12) are still valid if T is reinterpreted as T =m7,)- s Similarly, we find from Eqs. (13) and (A.7) that

(A.8) (I#~- I)(1 - s).

(A.3 which yields :z -= [I - 7” $ 2s -. z t (2 i- &)(l - x)1/z instead of Eq. (17). Equation (18) now yields J = z, which implies either a breakdown of the scaling assumption Eq. (4) or ,3S = 1, which is not true in general. The case-2 -=c& < - 1 cannot be ruled out, however, since then the dominant contributions to the magnetization clusters nt come from finite t, while the dominant contributions to the energy clusters come from t --f cc. In this case Eqs. (8)-(12) remain valid with T = 7O-- x, while Eq. (15) has to be replaced by Eq. (A.9). From Eq. (18) we then find j’ + x J- (2 + q$)(I - x) = z L- 1, (A.10) and Eq. (16) has to be replaced by s *

~r+(l-xH2+d2)

the boundary cases CJ$= -2 result. III

=

p+1-u

or -1,

=

p4-1.

(A.1 1)

logarithmic corrections would again

ACKNOWLEDGMENTS It is a pleasure to thank H. Muller-Krumbhaar and C. Domb for stimulating correspondence and discussions and information on their work prior to publication. Thanks are due to M. E. Fisher, G. Gallavotti, A. J. Guttmann, J. L. Lebowitz, and D. Stauffer for interesting discussions or correspondence, and to the latter for a careful reading of the manuscript.

416

K.

BINDER

REFERENCES 1. R. BECKER AND W. DBRING, Ann. Physik 24 (1935), 719. 2. J. E. MAYER, 1. Chem. Phys. 5 (1937), 67. 3. A beautiful review of the historical development and more complete references may be found in C. DOMB, “Metastability and Spinodals in the Lattice Gas Model,” preprint. 4. J. W. E~SAM AND M. E. FISHER, J. Chem. Phys. 38 (1963), 802. 5. M. E. FISHER, Physics 3 (1967), 255. The prediction of the essential singularity published in this paper was actually presented first by M. E. FISHER, IUPAP Conference on Statistical Mechanics, Boston, 1962 (unpublished), as cited by S. KATSURA, Adu. Phys. 12 (1963), 416. 6. M. E. FISHER, in ‘Critical Phenomena” (M. S. Green, Ed.), p. 1, Academic Press, New York, 1971. 7. M. E. FISCHER AND B. U. FELDERHOF, Ann. Phys. (N.Y.) 58 (1970), 176, 217, 268, 281. 8. C. S. KIANG, Phys. Rev. Lett. 24 (1970), 47. 9. C. S. KIANG AND D. STAUFFER, Z. Phys. 235 (1970), 130. 10. W. RATHJEN, D. STAUFFER, AND C. S. KIANG, Phys. Lett. 40A (1972), 345. 11. L. REATTO, in “Critical Phenomena” (M. S. Green, Ed.), p. 265, Academic Press, New York, 1971. 12. L. REATTO, Phys. Left. 33A (1970), 519. 13. L. REATTO AND E. RASTELLI, J. Phys. C5 (1972), 2785. 14. C. DOMB, J. Phys. C6 (1973), 39. 15. P. G. DE GENNES, P. LAFORE, AND J. P. MILLOT, J. Phys. Chem. Sol. 11 (1959), 105. 16. J. W. ESSAM AND K. M. GWILYM, J. Phys. C4 (1971), L 228. 17. C. DOMB, J. Phys. C7 (1974) 2677. 18. D. STAUFFER, J. Phys. C8 (1975), L 172; Phys. Rev. Left. 35 (1975), 394; Z. Phys. B22 (1975), 161. 19. K. BINDER, D. STAUFFER, AND H. MULLER-KRUMBHAAR, Phys. Rev. BlO (1974), 3853. 20. K. BINDER, D. STAUFFER, AND H. MILLER-KRUMBHAAR, Phys. Rev. B12 (1975), 5261. 21. B. J. ACKERSON, C. M. SORENSEN, R. C. MOCKLER, AND W. J. O’SULLIVAN, Phys. Rev. Lett. 34 (1975), 1371. 22. G. J. ADRIAENSSENS, Phys. Rev. 812 (1975), 5116. 23. For reviews, see, e.g. A. C. ZETTLEMOYER (Ed.), “Nucleation,” Dekker, New York, 1969; “Nucleation II,” to appear. 24. J. S. LANGER, Ann. Phys. (N.Y.) 41 (1967), 108. 25. C. S. KIANG, D. STAUFFER, G. H. WALKER, 0. P. PURI, J. D. WISE, JR., AND E. M. PATTERSON, J. Atm. Sci. 28 (1971), 1112. 26. A. EGGINGTON, C. S. KIANG, D. STAUFFER, AND G. H. WALKER, Phys. Rev. Lett. 26 (1971), 820. 27. K. BINDER, in “Physics of Nonequilibrium Systems” (T. Riste, Ed.), p. 53, Plenum Press, New York, 1976. 28. I. M. LIFSHITZ AND V. V. SLYOZOV, J. Phys. Chem. Solids 19 (1961), 35. 29. K. BINDER AND D. STAUFFER, Phys. Rev. Lett. 33 (1974), 1006, and to appear. 30. J. MARRO, Thesis, Yeshiva University, New York, 1975, unpublished. 31. H. WAGNER, Phys. Rev. Lett. 25 (1970), 31, 261. 32. L. REATTO, Phys. Rev. B5 (1972), 204; D. STAUFFER, Phys. Rev. B6 (1972), 1839. 33. J. KRUMHANSL AND R. SCHRIEFFER, Phys. Rev. Bll (1975), 3535, and references contained therein. 34. K. BINDER AND D. STAUFFER, J. Statist. Phys. 6 (1972), 49. 35. E. STOLL, K. BINDER, AND T. SCHNEIDER, Phys. Rev. B6 (1972), 2777. 36. H. MILLER-KRUMBHAAR, Phys. Lett. 48A (1974), 459.

“CLUSTERS” 37. H. MILLER-KRUMBHAAR, 38. H. MijL~fi~-KRUMBHAAR

39. 40. 41. 42. 43. 44. 45.

46. 47.

48. 49. 50. 51. 52. 53.

54. 55.

56. 57.

58. 59. 60.

61. 62. 63. 64.

IN

THE

ISING

MODEL

417

Lat. 50A (1974), 27. AND E. STOLL, “Cluster Statistics in the Lattice Gas Model,”

Phys.

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