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Lattice animal specific heats and the collapse of branched polymers
R. Dickman, W.C. Schieve
To cite this version:
R. Dickman, W.C. Schieve. Lattice animal specific heats and the collapse of branched polymers. Jour- nal de Physique, 1984, 45 (11), pp.1727-1730. �10.1051/jphys:0198400450110172700�. �jpa-00209914�
1727
Lattice animal specific heats and the collapse of branched polymers
R. Dickman (*) and W. C. Schieve
Center for Studies in Statistical Mechanics, The University of Texas at Austin, Austin, Texas 78712, U.S.A.
(Reçu le 4 avril 1984, accepté le 30 mai 1984)
Résumé. 2014 En utilisant une nouvelle forme de la méthode Monte Carlo, nous calculons la chaleur spécifique
d’un animal sur réseau. La chaleur spécifique présente un maximum au voisinage de la température d’effondrement obtenue par Derrida et Herrmann. Elle a aussi un maximum secondaire à basse temperature, liée, apparemment, à une transition rugueuse.
Abstract. 2014 The specific heat of one lattice animal is computed using a new Monte Carlo approach. The specific
heat exhibits a peak near the collapse transition temperature derived by Derrida and Herrmann. There is also
a secondary, low temperature peak which appears to be associated with a roughening transition.
J. Physique 45 (1984) 1727-1730 NOVEMBRE 1984,
Classification
Physics Abstracts
61.40K - 64.70
LE JOURNAL DE
PHYSIQUE
1. Introduction.
Connected clusters of occupied sites in a lattice, or
lattice animals, are of current interest in several contexts. Lattice animals have been studied in connec-
tion with clustering and nucleation [1-3], and in per- colation theory [4, 5]. Statistics and properties of
animals have been investigated by means of exact
enumeration [6], Monte Carlo methods [7-9, 16, 17],
momentum space renormalization group [10], and
the phenomenological renormalization group [11].
Recently, Derrida and Herrmann [12] used lattice animals as a model for the collapse transition in branched polymers. Their analysis indicates that at
a temperature To = 0.535 + 0.005, the structure of
square lattice animals changes from compact to ramified. If R is some measure of the length of an animal, (end-to-end distance, radius of gyration),
and if v is the exponent describing the asymptotic dependence of R on animal size, p, so that
then below To, v = 1/d in d dimensions, while for
T > To, v attains some higher value. (It has been
estimated [11] that v zr 0.64 for d = 2, above TO).
Derrida and Herrmann also presented evidence for the singular behaviour c - I T - To I-’ in the spe- cific heat, with a £x 0.48.
In this paper we present data on the specific heats
of animals which tends to support the conclusions of
Derrida and Herrmann. Thermal properties of lattice
animals may be ’derived from the partition function
where k is the number of nearest neighbour bonds and
up,k is the number of translationally distinct, p-particle,
k-bond animals. In the course of studies of lattice gas clusters [9], we have determined 6p,k for various p values in two dimensional lattices. The Monte Carlo
technique used to estimate 6 p,k is described in sec-
tion 2, and in section 3 we present our results on
specific heats.
2. Monte Carlo method for estimating the number of animals.
In order to compute thermal properties of a p-particle
animal directly from the partition function (2), the
coefficients up,k are required. For small sizes (p 9 in
the square lattice, p 7 in the triangle lattice), and
for maximally and nearly maximally bonded animals,
Qp,k may be determined by exact enumeration. In other cases we employed Monte Carlo methods to estimate Qn,k.
Most of the data was obtained using the following
scheme for sampling the configuration space of
p-particle animals. Given some configuration C.,
a trial configuration C’ is generated by displacing particle i, randomly chosen from the set { 2, ..., p 1,
from its position xi to x*, a randomly chosen nearest
or next nearest neighbour of xi. The trial configuration
is accepted with the usual Boltzmann-weighing :
if AE = E(C’) - E(C) 0, we take Cn + 1 = C’, but
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198400450110172700
1728
if AE > 0, Cn+ 1 = C’ only with probability e-fJåE"
and with probability 1 - e-¡JåE we take Cn+, 1 = Cn.
The energy E(C) is - k(C), where k(C) is the number of bonds in C, unless C has an overlap (two particles
occupy the same site), or C is disconnected. Discon- nected and overlap configurations are assigned energy + oo. (Thus if x* is occupied, or if moving particle i to
x* destroys connectivity, then Cn+ 1 = CJ.
The Monte Carlo process described above is a Markov chain whose state space is the set of p-particle
animal configurations. Each translationally distinct
animal is represented exactly p times, since particle 1
is fixed at the origin. The inclusion of next nearest as well as nearest neighbour moves ensures that it is
possible to get from any given animal configuration
to any other one (the Markov chain is irreducible).
Suppose that C and C’ are mutually accessible confi-
gurations, i.e., subjecting one of the particles in C
to a nearest or next nearest neighbour displacement yields C’. The probability of going from C to C’ is
where q’ is the number of nearest and next nearest
neighbours. We see that the transition probability
satisfies detailed balance
which implies that the invariant distribution of the Markov chain is P(C) oc e-¡JE(C). Thus the probability
of realizing a k-bound configuration obeys
and an unbiased estimator of the ratio up,klap,k’ is
e¡J(k’-k) Nk/Nk-, where Nk is the number of realizations of k-bond animals. Using the results of Sykes et al. [6]
for the asymptotic growth of the total number of p-
particle animals, we are then able to derive estimates of individual ap,k values.
In studies of larger animals, it was found that relaxation from the initial configuration is greatly
enhanced if o non-local » transitions are permitted [13].
To generate the trial configuration C’ from C,,, par-
ticle i (chosen at random from the set { 2, ..., p 1, as before), is moved to x*, which now may lie anywhere
on the cluster perimeter : an integer j is chosen at random from { 1, ..., p }, and x* is a randomly chosen
nearest neighbour of xj. (If x* is occupied, if x* = xi,
or if C’ is disconnected, then Cn+ 1 = C,,.) The proba- bility that a site y is chosen to be x* is v(y ; Cn)/qp, where q is the coordination number and v(y, C) is the
number of occupied nearest neighbours of y in C.
In passing from configuration Cn to C’, the change
in the number of bonds is
If E(C’) cc, we take Cn+ 1 = C’ with probability
r = v(x,; C’)/v(x*; Cn) if Ak > 0, and with probabi- lity efJåk if Ak 0. (Cn+ 1 = Cn with probability
1 - r, (Ak a 0), or 1 - efJåk (Ak 0)). The transition
probability for configurations related by a single par- ticle displacement is now
which again satisfies detailed balance.
Stauffer [16], and Peters et al. [17] used a Monte
Carlo method to study « random animals » (animals
at T = oo). In this process a trial configuration is generated by moving a randomly selected particle to
any randomly selected perimeter site (any unoccupied
site with at least one occupied nearest neighbour), and
the trial configuration is accepted so long as it is
connected. As noted by Stauffer, this method suffers from a small bias. The source of the bias is that the number of allowed transitions (i.e. the number of
perimeter sites) is not constant, and so the transition
probabilities do not satisfy detailed balance.
For larger animals we sample configurations at
several temperatures, in order to obtain precise
estimates of Qn,k over the entire range of k values.
Consistency tests of estimates derived at different temperatures permit one to check for adequate relaxa-
tion from the initial configuration. For p = 64
(square lattice), five temperatures - e-P = 0.5, 0.25, 0.135, 0.075, and 0.05 - were used. A number of runs were made, for a total of 20 to 40 million events at each temperature. (At the start of each run, 1 or
2 million events were allowed for relaxation.) Uncer-
tainties in Qp.k estimates (computed from variances of the data averaged in blocks of 1 million events) do not
exceed 5 %. As might be expected, for smaller sizes this level of precision can be achieved with smaller numbers of events and fewer different temperatures.
Our Monte Carlo methods reproduce the exact
up,k ratios for p = 9 (square lattice) to within uncer- tainty, with an accuracy of 0.5 %.
For p = 100, runs were made at T = 0.3, 0.434, 0.5, 0.55, and 0.6, so as to provide high precision (uncertainty 0.4 %) estimates of the specific heat
near To.
In addition to the methods described above, the
random growth scheme described in [9] was used to
derive upk estimates for some of the smaller sizes.
The random growth scheme also proved useful for
generating initial configurations in the studies des- cribed above.
3. Lattice animal specific heats.
We have computed the specific heat per particle in
a p particle animal
where ( kn ) = z; 1 L kn (J p,k ek, T. The specific heat
k
of square lattice animals is plotted as a function of temperature in figure 1. Figures 2 and 3 are similar plots for the triangle and FCC lattices (in the latter
case only data forp = 19 is available). The uncertainty
in the specific heat does not exceed 1 %.
As is evident from the figures, the specific heat of larger animals exhibits a strong, fairly broad peak.
With increasing size, the temperature, Tm, at which
the specific heat attains its maximum increases, and
the peak grows higher and sharper. In the square lattice, for p = 100, Tm = 0.542 ± 0.005. In figure 4
we plot Tm versus p-0 for square lattice animals, using
the crossover exponent value 0 = 0.657 [12]. While
the data does not fall on a straight line, as one would expect on the basis of finite size scaling theory [18],
it is clear that the Monte Carlo results are consistent with the Derrida-Herrmann estimate, T 8 = 0.535
± 0.005. For the sizes studied, the rounding of the specific heat peak is too severe to permit estimation of
Fig. 1. - Specific heat against temperature for square lattice animals.
Fig. 2. - Specific heat against temperature for triangle
lattice animals.
Fig. 3. - Specific heat against temperature for a p = 19 animal in the FCC lattice.
the exponent a. It appears that data for larger p values
is needed before critical exponents can be determined
through a finite size scaling analysis.
In finite Ising lattices with free boundaries [14, 15]
a similar dependence of the specific heat on size is
observed, i.e., the peak grows sharper and higher, and approaches the critical temperature from below, with increasing lattice size. In this regard it is interesting to
note that for square lattice animals, To is close to, the
1730
Fig. 4. - Variation of Tm with size for square lattice animals. The arrow indicates the Derrida-Herrmann esti- mate for Te.
lattice gas critical temperature T,, 0.567. Turning
to the triangle lattice, we find Tm = 0.90 + 0.01 for
p = 61, while for this lattice Tc 0.912. The peak in
the FCC p = 19 specific heat is at T = 1.09, well
below the critical temperature of 2.448 8, but an animal of this size in the FCC lattice is entirely sur-
face, and so no similarity with bulk properties is to be expected.
A surprising feature of the specific heat plots is the subsidiary peak at low temperature, which is, of
course, not expected on the basis of bulk lattice gas or
animal properties. With increasing size, the peak
shifts to lower temperatures, is reduced in magnitude,
and becomes somewhat sharper. In the temperature
range (T 0.2) where the peak is observed, small and
medium sized animals (p 100) tend not to have any vacant sites within their perimeter. The predominant
type of excitation in this temperature range is a
rearrangement of the surface. Thus the subsidiary peak may correspond to a roughening transition.
Since roughening occurs at T = 0 in two dimensional systems, we would then expect the temperature T,
where the peaks occurs to approach zero ass -+ oo.
The interpretation of the subsidiary peak as a sur-
face effect is also supported by the observations : (1)
In the square lattice, the height cs of the peak obeys p1/2 c. = const. (2) In the square lattice, when
p =A n2, a distinct peak is absent, although there is
still a pronounced shoulder in the low temperature specific heat. When p 0 n2, the maximally bonded
state is degenerate and has steps on the surface, so that, in effect, the surface is already somewhat disor- dered at T = 0. In the square lattice the product Tg log p is nearly independent of p. For the triangle
lattice, T, log p is also roughly constant, but pI/2 C, increases monotonically with p.
For small animals (p = 9 in the square lattice, p = 7 in the triangle lattice), there is only a single, low temperature peak in the specific heat. As p increases,
the peak at higher temperature, associated with the
collapse transition, gradually develops, while the original peak becomes weaker and shifts to lower temperature.
4. Summary.
We have presented data on the specific heats of lattice animals. In two dimensions, the animal collapse tem- perature appears to be quite close to the lattice gas critical temperature. The data for the square lattice is in agreement with the Derrida-Herrmann estimate
Te = 0.535. However, to estimate critical exponents it will be necessary to study larger animals. Finally,
we have pointed out what appears to be a surface
roughening effect in animals.
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