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Fixed scale transformation applied to cluster-cluster aggregation in two and three dimensions
Physica A 185 (1992) 202-210 North-Holland
Fixed scale transformation applied to cluster-cluster aggregation in two and three dimensions S. Sidoretti and A. Vespignani Dipartimento di Fisica, Universit~ "La Sapienza", P.le A. Moro 2, 1-00185 Rome, Italy
Recently it has been introduced a new theoretical framework named fixed scale transformation (FST), which appears particularly suitable to study the growth of fractal structures. This method allows the first analytical study of the process of cluster-cluster aggregation (CCA). The FST approach can in fact be generalized in a natural and relatively simple way to the case of CCA. Here we present detailed results for the analytical calculation of the fractal dimension of the aggregates. For CCA in two dimensions the computed value is D = 1.39 and in three dimensions is D = 1.9, to be compared with the simulation results that are respectively D = 1.45 and D = 1.8. Furthermore the approximation scheme of the FST can be implemented in a systematic way to estimate quantitatively higher order corrections and to study variation of the original model. This application is of particular relevance because CCA has eluded all the standard theoretical approach and in particular it cannot even be formulated from the point of view of renormalization group methods.
1. Introduction T h e fixed scale t r a n s f o r m a t i o n ( F S T ) m e t h o d [1] is a n e w t h e o r e t i c a l f r a m e w o r k t h a t allows to a d d r e s s t h e q u e s t i o n o f a s y s t e m a t i c i n v e s t i g a t i o n of e q u i l i b r i u m p r o p e r t i e s a n d a n a l y t i c a l c a l c u l a t i o n o f fractal d i m e n s i o n s for a wide range of irreversible growth models. In fact, t h e F S T has b e e n f o r m u l a t e d h a v i n g in m i n d t h e L a p l a c i a n i r r e v e r s i b l e g r o w t h m o d e l s ( D L A , D B M ) [2, 3]; h o w e v e r it can also b e a p p l i e d to t h e s t u d y o f t h e fractal p r o p e r t i e s o f e q u i l i b r i u m p r o b l e m s like p e r c o l a t i o n , i n v a s i o n p e r c o l a t i o n a n d Ising clusters, for which it gives v e r y g o o d results [4]. T h e F S T is a g e o m e t r i c a l a p p r o a c h to t h e s t u d y o f critical clusters l i n k i n g t h e g r o w t h r u l e of t h e m o d e l s to t h e f r a c t a l d i m e n s i o n a n d g e o m e t r i c a l s t r u c t u r e o f t h e a s y m p t o t i c a l l y g r o w n cluster. It is b a s e d o n b o t h t h e scale a n d t r a n s l a t i o n a l i n v a r i a n c e (at s a m e scale) p r o p e r t i e s o f t h e s e m o d e l s , a n d for this r e a s o n t h e F S T a p p r o a c h e l u d e s s o m e o f t h e p r o b l e m s affecting t h e r e a l s p a c e r e n o r m a l i z a t i o n g r o u p m e t h o d b a s e d o n t h e g r o w t h rules t h e m s e l v e s (an e x h a u s t i v e d i s c u s s i o n o f t h e F S T a p p r o a c h is d e v e l o p e d in this issue b y A . E r z a n [5]). 0378-4371/92/$05.00 © 1992- Elsevier Science Publishers B.V. All rights reserved
S. Sidoretti, A. Vespignani / Fixed scale transformation applied to CCA
203
Here we show how the FST method can be extended to study analytically the fractal features of the structure generated by the cluster-cluster aggregation (CCA) [6-9] process in two and three dimensions. The cluster-cluster aggregation model is one of the most studied problems of fractal growth because it describes a large amount of interesting physical phenomena in the field of colloids, polymer solutions and gels [6, 7]. From the theoretical point of view it has to be remarked that there is not a standard theory for the fractal features of cluster-cluster aggregates. In fact, the classical understanding of CCA is given by the mean field Smoluchowski equation [10] where the space dependence of all quantities is neglected so it cannot provide any informations about the geometrical characteristics of clusters. In addition, the fact that RG ideas are clearly inappropriate in such a case poses the CCA model as a major theoretical challenge. Within the FST scheme we will use the mass distribution, given by the mean field theory [6, 7, 11, 12], as input and taking into account the diffusive nature of the growth process (analogous to DLA), we will obtain analytically the fractal dimension of the aggregates forming in two or three dimension. Furthermore we are able to control quantitatively the approximation scheme used in the FST and to discuss its systematics.
2. The model
The CCA model is described on a square lattice (in two or three dimensions) with periodic boundary conditions, where a fixed number of isolated particles is randomly inserted. Particles move following Brownian trajectories with probability of motion proportional to their mobility: p
= Cs ~ ,
(1)
where s is the mass of the aggregate and 3' is a parameter. When particles or clusters come to contact they join irreversibly. Clusters obtained by such a model have fractal dimension D ~ 1.45 in two dimensions and D ~ 1.8 in three dimensions, for 3' ~<0 [8, 9, 13, 14]. To develop an analytical approach to the geometrical properties of CCA aggregates, it is crucial to study the cluster mass distribution ns(t ). This last quantity gives the number of aggregates of mass s as a function of time, independently of the space's Euclidean dimensions the process develops in. To this extent it is necessary to distinguish clearly between some aspects of the ns(t ), due to finite size effects, and what we are interested in, i.e. the ns(t ) properties around the s value where at fixed time the aggregation is especially happening. So the interesting aspects can be recognized in simulations only by
204
S. Sidoretti, A. Vespignani / Fixed scale transformation applied to CCA
eliminating the influence of the first ones. The ns(t ) curves, obtained from simulations at different times [13, 14], show in the dependence on the y - e x p o n e n t a bell-shaped behaviour or that of a power law with exponent T<2: ns(t) ~ t-ws-~g(s/tZ) .
(2)
By the way, in both cases and for any value of y it has been shown that the e n v e l o p e of the ns(t ) curves at different times is a power law in s with exponent a = 2, that is the value predicted by Smoluchowski theory [6, 7, 11]. We emphasize that the tangent curve touches each ns(t) just in correspondence of the typical mass: 2
£-
E nsS Z n~s '
(3)
which identifies clusters m o r e favored to aggregation, so it individuates the p r o p e r t y we are interested in. A n o t h e r way to find this behaviour is to study C C A in the steady state regime, which is achieved by introducing a superior cut-off in the cluster's mass and a seed of k particles added each unitary time increment. In these conditions after a long enough time, the time dependence of ns(t ) disappears and the power law behaviour n,. ~ s-~ ,
(4)
with ~ = 2, is revealed [6, 7, 14]. The properties of n s are fundamental for our discussion as the key point that differentiates C C A from D L A because of the fact that the growth process can be due to incoming aggregates of different sizes. Since FST is based on the evaluation of matrix elements that hold for all scales, the n s distribution should be scale invariant and this is indeed the case. Given a generic length scale r. = 2"% (r 0 = 1 lattice unit) a basic element for the dynamical process is the relative probability K. that the next incoming particle has size r, with r. i < r < r. or r > r. (see next section). In the steady state we have s "~ r D
(5)
So we can evaluate rn
f P(r.)
=
rn
K" = p ( r " ) + P ( r > r")
n(r) r D ' d r 1 ce
f rn
= n(rl r p ' d r
1
1
-
(rr~_ll)-D(a-l)
(6)
205
s. Sidoretti, A. Vespignani / Fixed scale transformation applied to CCA
Since n ( r ) ~ r - D ~ is a power law K n results to be scale invariant. It is important to notice that the dependence of K on D will give rise to a strong non-linearity in the FST even at lowest orders.
3. Analytical calculation of the fractal dimension of CCAs in two and three dimensions
The starting point of the FST method is the identification of the elementary configurations involved in the fine or coarse graining process of the structure. In this respect it is convenient to consider the intersection of the structure with a line perpendicular to the local growth direction. In two dimension there are, as in D L A , two types of different configurations: type 1 consisting of an occupied site (black) and an empty one (white), and type 2 with both sites occupied. The statistics of the intersection set is given by the probability distribution (C1, C2) for the appearance of these two configurations. The probability distribution (C~, C2) can be simply related to the fractal dimension D of the structure by D = 1+
In (C l + 2C2) In 2
(7)
The fixed scale transformation is the iterative transformation that links the probability distribution for (C1, C2) of a given intersection with the distribution of another intersection at the same scale but translated in the growth direction. We then search for the fixed point that reflects the translational invariance of the structure. The matrix elements Mi, j of the FST correspond to the conditional probability that a configuration of type i is followed, in the growth direction, by a configuration of type j. These elements are calculated analytically through a graphical expansion that takes into account all the growth processes of the growing zone with a statistical weight given by the growth rule of the process considered (for a detailed description of the technical points see the work of A. Erzan in these proceedings [5]). For the explicit evaluation it is also necessary to define the appropriate boundary conditions and we have developed different schemes of increasing complexity to include self-consistently the fluctuations of boundary conditions. The problem is therefore reduced to the calculation of the FST matrix elements. Our discussion for the explicit calculation of the matrix elements will be based on a scheme similar to the one originally introduced for D L A [1, 5] and here we will focus mainly on the differences introduced by the CCA process. Considering the dynamics of the growth process at a generic scale of size r, one
206
S. Sidoretti, A. Vespignani / Fixed scale transformation applied to CCA
has to distinguish between three possibilities. The first one is that the next incoming cluster has a size r > r,, the second possibility is that r = r, and the third is that r < r n. The latter case corresponds to the aggregation of small clusters with respect to the considered scale r,, and their contribution at small orders of the calculation can be neglected. The relevant contributions to the growth process are therefore given by the clusters of size r~ or larger. Their relative probability of occurrence can be expressed self-consistently as a function of (C 1, C2), using eqs. (6) and (7), as K , = K ( C I , C:) = 1 - 2 (1-t~)[l+ln(Cl+2C2)/ln 21
(8)
As an example we can consider the growth processes that are originated from a frozen configuration of type 1. With probability K the next cluster that comes in contact with our starting configuration is of size r n (the same as the coarse graining level considered). In this case the cluster is represented by a single site and it will occupy the site above the occupied one of the starting cell. This corresponds to the zero order growth process in the D L A scheme [1]. With probability 1 - K a larger cluster ( r > r , ) arrives. This gives rise to two possibilities as shown in fig. 1. The larger aggregate (encircled by a dashed line) can occupy a single site (above the starting cell) with probability C], or both sites with probability C e. This is because we assume that in the steady state regime the arriving aggregates have the same fractal dimension that we will compute in the end, characterized by the same distribution (CI, (?2). After the arrival of a large aggregate the probability tree is stopped because the eventual empty site above the starting cell is considered fully screened. At zero order we have therefore M,~ = K + ( 1 -
K)C 1
M]2=(1-
(9)
K)C 2 .
r---~
,-----:
•.,
i
el
a
.! i_...
• ....
:0', *
j
• ~
10 * ~ ....
• r
Oi
. . . . . . . .
0:
f
J
r..-~
:o
i.io
i..i r-..J
*i
L,.__J
(a)
(b)
Fig. 1. We consider the growth process conditional to the existence of a starting configuration of a certain type. In this case it is of type 1 (one filled and one e m p t y box). With probability 1 - K a large aggregate arrives (encircled by a dashed line) and this can lead to two possibilities. The occupation of a single site (a) or of both sites (b) above the starting configuration.
S. Sidoretti, A. Vespignani / Fixed scale transformation applied to CCA
207
In case a single particle has arrived (of size r = r n) the growth process should instead be continued. Again with probability 1 - K a large aggregate arrives but in this case it will stick to the tip of the growing cluster that is now one step higher. The probability that the incoming aggregate will penetrate to occupy the empty site above the starting cell is now of higher order and (for the m o m e n t ) we neglect it. With probability K the incoming aggregate has the size of a single cell and the process is analogous to D L A . It is possible to extend the construction of the probability tree to higher orders and the general expression for the matrix element at order l is [16] l-1
M,,(l) = (1 - K)C, + (1 - K)K + ~ K~+'(1 - K)MD~LA(n) n=l
+ K'+~MD~LA(I),
(10)
where MD1LA(n) corresponds
to the D L A matrix element of order n as discussed in refs. [1, 5]. It is important to notice that the incoming of large clusters produces a faster freezing effect, which is reflected in the fact that the matrix elements consist of power series in K and therefore the convergency is faster than in D L A . For this reason we can draw the general conclusion that DCcA
(11)
Actually in this expression we are making an approximation considering the starting cell fully screened as soon as a large cluster reaches the tip of the growing cluster. In fact the matrix elements are a combination of elements evaluated in different boundary conditions, so it is easy to see that our previous assumption is reasonable for closed boundary conditions, while for the open ones, because of the possibility of lateral growths, it should be necessary to introduce a correction that will raise the fractal dimension. Anyway, this correction gives a small contribution because at the lowest growth orders the screening effect is more relevant, and can be estimated to give an increase of about 2% of D. In analogy with the D L A and D B M models we have used in the explicit calculation of the matrix elements the first non-trivial scheme to include self-consistently the fluctuations of the boundary conditions. This scheme is called the o p e n - c l o s e d approximation and it is possible in this case to derive analytically [1] the fixed point distribution (C1, C2). In table I the results obtained for the fractal dimension up to third order of calculation are c o m p a r e d with the best simulation results actually available. In table I we
S. Sidoretti, A. Vespignani / Fixed scale transformation applied to CCA
208
Table I Values of the fractal dimension of the cluster-cluster aggregation model in two dimensions calculated analytically with the FST m e t h o d . T h e values reported correspond to two s c h e m e s of t r e a t m e n t of boundary conditions (see text) and to the evaluation up to fourth order in K of eq. (10). 2d
Closed
Open-closed
Simulations
2nd order in K 3rd order in K 4th order in K
1.29 1.32 1.32
1.31 1.37 1.39
1.45 +--0.05
report also the results obtained in a simpler scheme of approximation, the closed case [1]. It is worth while to r e m a r k the good convergence of the result as a function of the order of calculation and of the better treatment of the b o u n d a r y conditions problem. In addition, the result D = 1.39 obtained for the cluster-cluster model at third order can be directly c o m p a r e d with the D L A result at the same order D = 1.64. T h e approach can be naturally extended to three-dimensional C C A . Even in this case the ns(t ) is a power law with exponent a = 2 (see section 2). So from the theoretical point of view the development of the m e t h o d is the same as in two dimensions. Some complications occur in the evaluations of the matrix elements because higher Euclidean dimension implies more difficulties in taking into account the effect of the boundary conditions' fluctuations (actually this p r o b l e m is present in the same way in three dimensional D L A [15]. In table II we show the results obtained for the 3d C C A fractal dimension, obtained at different growth order in different boundary condition approxim a t i o n schemes. In table II the results for a more refined scheme of treatment of the b o u n d a r y conditions, called semi-open [15], are reported to show the convergence of the result with respect to an approximation scheme of increasingly complexity. We notice a m o r e rapid convergence towards a limit value of D c o m p a r e d to the D L A case and the relative influence of the introduction of the semi-open b o u n d a r y condition. The second r e m a r k is a consequence of the lower value of
Table II Values of the fractal dimension of the C C A model in three dimensions calculated analytically with the FST method. The values showed correspond to increasingly sophisticated s c h e m e s of approximation in the evaluation of the matrix elements. 3d
Closed
Open-closed
Semi-open
Simulations
3rd order in K 5th order in K
1.61 1.68
1.71 1.91
1.71 1.90
1.8 +- 0.1
s. Sidoretti, A. Vespignani / Fixed scale transformation applied to CCA
209
D (with respect to D L A ) [15], which gives m o r e relevance to open b o u n d a r y conditions. In particular the numerical estimate of D is expected to be sensible to i m p r o v e m e n t s of approximation schemes by the introduction of boundary conditions n e a r to open. For these reasons two-dimensional results can m o r e easily reach a better agreement with experimental data. By the way, even in three dimensions results are very good.
4. Conclusions We showed how the FST theoretical scheme allows to treat problems like C C A that are not yet satisfactory approached with the usual theoretical m e t h o d , as for example the renormalization group method. In addition the FST m e t h o d shows a good systematics in the refinement of the approximation schemes used and in the quantitative estimation of the correction due to higher order effects. For these reasons we obtain a good convergence of the results for the fractal dimension as a function of approximation schemes of increasing complexity and a good agreement with c o m p u t e r simulations. These results give rise to a deeper understanding of the C C A p h e n o m e n o n and allow to discuss in detail m a n y of the features appearing as puzzling in the simulations.
Acknowledgements Part of the work described here was carried out in collaboration with L. Pietronero, and we are very grateful to him. We also would like to thank A. E r z a n and P. Tartaglia for useful discussions and suggestions.
References [1] L. Pietronero, A. Erzan and C. Evertsz, Phys. Rev. Lett. 61 (1988) 861; Physica A 40 (1988) 5377. [2] T.A. Witten and L.M. Sander, Phys. Rev. Lett. 47 (1981) 1400. [3] L. Niemeyer, L. Pietronero and H.J. Wiesmann, Phys. Rev. Lett. 52 (1984) 1038. [4] L. Pietronero, in: Non-Linear Phenomena Related to Growth and Form, P. Pelce et al., eds. (Plenum, New York), in press. [5] A. Erzan, Physica A 185 (1992) 66, these Proceedings. [6] T. Vicsek, Fractal Growth Phenomena (World Scientific, Singapore, 1989). [7] R. Jullien and R. Botet, Aggregation and Fractal Aggregates (World Scientific, Singapore, 1987). [8] P. Meakin, Phys. Rev. Lett. 51 (1983) 1119.
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S. Sidoretti, A. Vespignani / Fixed scale transformation applied to CCA
[9] M. Kolb, R. Botet and R. Jullien, Phys. Rev. Lett. 51 (1983) 1123. [10] M.V. Smolukowski, Phys. Z. 17 (1916) 585. [11] F. Leyvraz, in: On Growth and Form, H.E. Stanley and N. Ostrowski, eds. (NijhofL Dordrecht, 1986) p. 136. [12] M.H. Ernst, in: Fractals in Physics, L. Pietronero and E. Tosatti, eds. (North-Holland, Amsterdam, 1986) p. 289. [13] T. Vicsek and F. Family, Phys. Rev. Lett. 52 (1984) 1669. [14] T. Vicsek, P. Meakin and F. Family, Phys. Rev. A 32 (1985) 1122. [15] A. Vespignani and L. Pietronero, Physica A 173 (1991) 1. [16] L. Pietronero, S. Sidoretti, P. Tartaglia and A. Vespignani, preprint.
Fixed scale transformation applied to cluster-cluster aggregation in two and three dimensions S. Sidoretti and A. Vespignani Dipartimento di Fisica, Universit~ "La Sapienza", P.le A. Moro 2, 1-00185 Rome, Italy
Recently it has been introduced a new theoretical framework named fixed scale transformation (FST), which appears particularly suitable to study the growth of fractal structures. This method allows the first analytical study of the process of cluster-cluster aggregation (CCA). The FST approach can in fact be generalized in a natural and relatively simple way to the case of CCA. Here we present detailed results for the analytical calculation of the fractal dimension of the aggregates. For CCA in two dimensions the computed value is D = 1.39 and in three dimensions is D = 1.9, to be compared with the simulation results that are respectively D = 1.45 and D = 1.8. Furthermore the approximation scheme of the FST can be implemented in a systematic way to estimate quantitatively higher order corrections and to study variation of the original model. This application is of particular relevance because CCA has eluded all the standard theoretical approach and in particular it cannot even be formulated from the point of view of renormalization group methods.
1. Introduction T h e fixed scale t r a n s f o r m a t i o n ( F S T ) m e t h o d [1] is a n e w t h e o r e t i c a l f r a m e w o r k t h a t allows to a d d r e s s t h e q u e s t i o n o f a s y s t e m a t i c i n v e s t i g a t i o n of e q u i l i b r i u m p r o p e r t i e s a n d a n a l y t i c a l c a l c u l a t i o n o f fractal d i m e n s i o n s for a wide range of irreversible growth models. In fact, t h e F S T has b e e n f o r m u l a t e d h a v i n g in m i n d t h e L a p l a c i a n i r r e v e r s i b l e g r o w t h m o d e l s ( D L A , D B M ) [2, 3]; h o w e v e r it can also b e a p p l i e d to t h e s t u d y o f t h e fractal p r o p e r t i e s o f e q u i l i b r i u m p r o b l e m s like p e r c o l a t i o n , i n v a s i o n p e r c o l a t i o n a n d Ising clusters, for which it gives v e r y g o o d results [4]. T h e F S T is a g e o m e t r i c a l a p p r o a c h to t h e s t u d y o f critical clusters l i n k i n g t h e g r o w t h r u l e of t h e m o d e l s to t h e f r a c t a l d i m e n s i o n a n d g e o m e t r i c a l s t r u c t u r e o f t h e a s y m p t o t i c a l l y g r o w n cluster. It is b a s e d o n b o t h t h e scale a n d t r a n s l a t i o n a l i n v a r i a n c e (at s a m e scale) p r o p e r t i e s o f t h e s e m o d e l s , a n d for this r e a s o n t h e F S T a p p r o a c h e l u d e s s o m e o f t h e p r o b l e m s affecting t h e r e a l s p a c e r e n o r m a l i z a t i o n g r o u p m e t h o d b a s e d o n t h e g r o w t h rules t h e m s e l v e s (an e x h a u s t i v e d i s c u s s i o n o f t h e F S T a p p r o a c h is d e v e l o p e d in this issue b y A . E r z a n [5]). 0378-4371/92/$05.00 © 1992- Elsevier Science Publishers B.V. All rights reserved
S. Sidoretti, A. Vespignani / Fixed scale transformation applied to CCA
203
Here we show how the FST method can be extended to study analytically the fractal features of the structure generated by the cluster-cluster aggregation (CCA) [6-9] process in two and three dimensions. The cluster-cluster aggregation model is one of the most studied problems of fractal growth because it describes a large amount of interesting physical phenomena in the field of colloids, polymer solutions and gels [6, 7]. From the theoretical point of view it has to be remarked that there is not a standard theory for the fractal features of cluster-cluster aggregates. In fact, the classical understanding of CCA is given by the mean field Smoluchowski equation [10] where the space dependence of all quantities is neglected so it cannot provide any informations about the geometrical characteristics of clusters. In addition, the fact that RG ideas are clearly inappropriate in such a case poses the CCA model as a major theoretical challenge. Within the FST scheme we will use the mass distribution, given by the mean field theory [6, 7, 11, 12], as input and taking into account the diffusive nature of the growth process (analogous to DLA), we will obtain analytically the fractal dimension of the aggregates forming in two or three dimension. Furthermore we are able to control quantitatively the approximation scheme used in the FST and to discuss its systematics.
2. The model
The CCA model is described on a square lattice (in two or three dimensions) with periodic boundary conditions, where a fixed number of isolated particles is randomly inserted. Particles move following Brownian trajectories with probability of motion proportional to their mobility: p
= Cs ~ ,
(1)
where s is the mass of the aggregate and 3' is a parameter. When particles or clusters come to contact they join irreversibly. Clusters obtained by such a model have fractal dimension D ~ 1.45 in two dimensions and D ~ 1.8 in three dimensions, for 3' ~<0 [8, 9, 13, 14]. To develop an analytical approach to the geometrical properties of CCA aggregates, it is crucial to study the cluster mass distribution ns(t ). This last quantity gives the number of aggregates of mass s as a function of time, independently of the space's Euclidean dimensions the process develops in. To this extent it is necessary to distinguish clearly between some aspects of the ns(t ), due to finite size effects, and what we are interested in, i.e. the ns(t ) properties around the s value where at fixed time the aggregation is especially happening. So the interesting aspects can be recognized in simulations only by
204
S. Sidoretti, A. Vespignani / Fixed scale transformation applied to CCA
eliminating the influence of the first ones. The ns(t ) curves, obtained from simulations at different times [13, 14], show in the dependence on the y - e x p o n e n t a bell-shaped behaviour or that of a power law with exponent T<2: ns(t) ~ t-ws-~g(s/tZ) .
(2)
By the way, in both cases and for any value of y it has been shown that the e n v e l o p e of the ns(t ) curves at different times is a power law in s with exponent a = 2, that is the value predicted by Smoluchowski theory [6, 7, 11]. We emphasize that the tangent curve touches each ns(t) just in correspondence of the typical mass: 2
£-
E nsS Z n~s '
(3)
which identifies clusters m o r e favored to aggregation, so it individuates the p r o p e r t y we are interested in. A n o t h e r way to find this behaviour is to study C C A in the steady state regime, which is achieved by introducing a superior cut-off in the cluster's mass and a seed of k particles added each unitary time increment. In these conditions after a long enough time, the time dependence of ns(t ) disappears and the power law behaviour n,. ~ s-~ ,
(4)
with ~ = 2, is revealed [6, 7, 14]. The properties of n s are fundamental for our discussion as the key point that differentiates C C A from D L A because of the fact that the growth process can be due to incoming aggregates of different sizes. Since FST is based on the evaluation of matrix elements that hold for all scales, the n s distribution should be scale invariant and this is indeed the case. Given a generic length scale r. = 2"% (r 0 = 1 lattice unit) a basic element for the dynamical process is the relative probability K. that the next incoming particle has size r, with r. i < r < r. or r > r. (see next section). In the steady state we have s "~ r D
(5)
So we can evaluate rn
f P(r.)
=
rn
K" = p ( r " ) + P ( r > r")
n(r) r D ' d r 1 ce
f rn
= n(rl r p ' d r
1
1
-
(rr~_ll)-D(a-l)
(6)
205
s. Sidoretti, A. Vespignani / Fixed scale transformation applied to CCA
Since n ( r ) ~ r - D ~ is a power law K n results to be scale invariant. It is important to notice that the dependence of K on D will give rise to a strong non-linearity in the FST even at lowest orders.
3. Analytical calculation of the fractal dimension of CCAs in two and three dimensions
The starting point of the FST method is the identification of the elementary configurations involved in the fine or coarse graining process of the structure. In this respect it is convenient to consider the intersection of the structure with a line perpendicular to the local growth direction. In two dimension there are, as in D L A , two types of different configurations: type 1 consisting of an occupied site (black) and an empty one (white), and type 2 with both sites occupied. The statistics of the intersection set is given by the probability distribution (C1, C2) for the appearance of these two configurations. The probability distribution (C~, C2) can be simply related to the fractal dimension D of the structure by D = 1+
In (C l + 2C2) In 2
(7)
The fixed scale transformation is the iterative transformation that links the probability distribution for (C1, C2) of a given intersection with the distribution of another intersection at the same scale but translated in the growth direction. We then search for the fixed point that reflects the translational invariance of the structure. The matrix elements Mi, j of the FST correspond to the conditional probability that a configuration of type i is followed, in the growth direction, by a configuration of type j. These elements are calculated analytically through a graphical expansion that takes into account all the growth processes of the growing zone with a statistical weight given by the growth rule of the process considered (for a detailed description of the technical points see the work of A. Erzan in these proceedings [5]). For the explicit evaluation it is also necessary to define the appropriate boundary conditions and we have developed different schemes of increasing complexity to include self-consistently the fluctuations of boundary conditions. The problem is therefore reduced to the calculation of the FST matrix elements. Our discussion for the explicit calculation of the matrix elements will be based on a scheme similar to the one originally introduced for D L A [1, 5] and here we will focus mainly on the differences introduced by the CCA process. Considering the dynamics of the growth process at a generic scale of size r, one
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has to distinguish between three possibilities. The first one is that the next incoming cluster has a size r > r,, the second possibility is that r = r, and the third is that r < r n. The latter case corresponds to the aggregation of small clusters with respect to the considered scale r,, and their contribution at small orders of the calculation can be neglected. The relevant contributions to the growth process are therefore given by the clusters of size r~ or larger. Their relative probability of occurrence can be expressed self-consistently as a function of (C 1, C2), using eqs. (6) and (7), as K , = K ( C I , C:) = 1 - 2 (1-t~)[l+ln(Cl+2C2)/ln 21
(8)
As an example we can consider the growth processes that are originated from a frozen configuration of type 1. With probability K the next cluster that comes in contact with our starting configuration is of size r n (the same as the coarse graining level considered). In this case the cluster is represented by a single site and it will occupy the site above the occupied one of the starting cell. This corresponds to the zero order growth process in the D L A scheme [1]. With probability 1 - K a larger cluster ( r > r , ) arrives. This gives rise to two possibilities as shown in fig. 1. The larger aggregate (encircled by a dashed line) can occupy a single site (above the starting cell) with probability C], or both sites with probability C e. This is because we assume that in the steady state regime the arriving aggregates have the same fractal dimension that we will compute in the end, characterized by the same distribution (CI, (?2). After the arrival of a large aggregate the probability tree is stopped because the eventual empty site above the starting cell is considered fully screened. At zero order we have therefore M,~ = K + ( 1 -
K)C 1
M]2=(1-
(9)
K)C 2 .
r---~
,-----:
•.,
i
el
a
.! i_...
• ....
:0', *
j
• ~
10 * ~ ....
• r
Oi
. . . . . . . .
0:
f
J
r..-~
:o
i.io
i..i r-..J
*i
L,.__J
(a)
(b)
Fig. 1. We consider the growth process conditional to the existence of a starting configuration of a certain type. In this case it is of type 1 (one filled and one e m p t y box). With probability 1 - K a large aggregate arrives (encircled by a dashed line) and this can lead to two possibilities. The occupation of a single site (a) or of both sites (b) above the starting configuration.
S. Sidoretti, A. Vespignani / Fixed scale transformation applied to CCA
207
In case a single particle has arrived (of size r = r n) the growth process should instead be continued. Again with probability 1 - K a large aggregate arrives but in this case it will stick to the tip of the growing cluster that is now one step higher. The probability that the incoming aggregate will penetrate to occupy the empty site above the starting cell is now of higher order and (for the m o m e n t ) we neglect it. With probability K the incoming aggregate has the size of a single cell and the process is analogous to D L A . It is possible to extend the construction of the probability tree to higher orders and the general expression for the matrix element at order l is [16] l-1
M,,(l) = (1 - K)C, + (1 - K)K + ~ K~+'(1 - K)MD~LA(n) n=l
+ K'+~MD~LA(I),
(10)
where MD1LA(n) corresponds
to the D L A matrix element of order n as discussed in refs. [1, 5]. It is important to notice that the incoming of large clusters produces a faster freezing effect, which is reflected in the fact that the matrix elements consist of power series in K and therefore the convergency is faster than in D L A . For this reason we can draw the general conclusion that DCcA
(11)
Actually in this expression we are making an approximation considering the starting cell fully screened as soon as a large cluster reaches the tip of the growing cluster. In fact the matrix elements are a combination of elements evaluated in different boundary conditions, so it is easy to see that our previous assumption is reasonable for closed boundary conditions, while for the open ones, because of the possibility of lateral growths, it should be necessary to introduce a correction that will raise the fractal dimension. Anyway, this correction gives a small contribution because at the lowest growth orders the screening effect is more relevant, and can be estimated to give an increase of about 2% of D. In analogy with the D L A and D B M models we have used in the explicit calculation of the matrix elements the first non-trivial scheme to include self-consistently the fluctuations of the boundary conditions. This scheme is called the o p e n - c l o s e d approximation and it is possible in this case to derive analytically [1] the fixed point distribution (C1, C2). In table I the results obtained for the fractal dimension up to third order of calculation are c o m p a r e d with the best simulation results actually available. In table I we
S. Sidoretti, A. Vespignani / Fixed scale transformation applied to CCA
208
Table I Values of the fractal dimension of the cluster-cluster aggregation model in two dimensions calculated analytically with the FST m e t h o d . T h e values reported correspond to two s c h e m e s of t r e a t m e n t of boundary conditions (see text) and to the evaluation up to fourth order in K of eq. (10). 2d
Closed
Open-closed
Simulations
2nd order in K 3rd order in K 4th order in K
1.29 1.32 1.32
1.31 1.37 1.39
1.45 +--0.05
report also the results obtained in a simpler scheme of approximation, the closed case [1]. It is worth while to r e m a r k the good convergence of the result as a function of the order of calculation and of the better treatment of the b o u n d a r y conditions problem. In addition, the result D = 1.39 obtained for the cluster-cluster model at third order can be directly c o m p a r e d with the D L A result at the same order D = 1.64. T h e approach can be naturally extended to three-dimensional C C A . Even in this case the ns(t ) is a power law with exponent a = 2 (see section 2). So from the theoretical point of view the development of the m e t h o d is the same as in two dimensions. Some complications occur in the evaluations of the matrix elements because higher Euclidean dimension implies more difficulties in taking into account the effect of the boundary conditions' fluctuations (actually this p r o b l e m is present in the same way in three dimensional D L A [15]. In table II we show the results obtained for the 3d C C A fractal dimension, obtained at different growth order in different boundary condition approxim a t i o n schemes. In table II the results for a more refined scheme of treatment of the b o u n d a r y conditions, called semi-open [15], are reported to show the convergence of the result with respect to an approximation scheme of increasingly complexity. We notice a m o r e rapid convergence towards a limit value of D c o m p a r e d to the D L A case and the relative influence of the introduction of the semi-open b o u n d a r y condition. The second r e m a r k is a consequence of the lower value of
Table II Values of the fractal dimension of the C C A model in three dimensions calculated analytically with the FST method. The values showed correspond to increasingly sophisticated s c h e m e s of approximation in the evaluation of the matrix elements. 3d
Closed
Open-closed
Semi-open
Simulations
3rd order in K 5th order in K
1.61 1.68
1.71 1.91
1.71 1.90
1.8 +- 0.1
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209
D (with respect to D L A ) [15], which gives m o r e relevance to open b o u n d a r y conditions. In particular the numerical estimate of D is expected to be sensible to i m p r o v e m e n t s of approximation schemes by the introduction of boundary conditions n e a r to open. For these reasons two-dimensional results can m o r e easily reach a better agreement with experimental data. By the way, even in three dimensions results are very good.
4. Conclusions We showed how the FST theoretical scheme allows to treat problems like C C A that are not yet satisfactory approached with the usual theoretical m e t h o d , as for example the renormalization group method. In addition the FST m e t h o d shows a good systematics in the refinement of the approximation schemes used and in the quantitative estimation of the correction due to higher order effects. For these reasons we obtain a good convergence of the results for the fractal dimension as a function of approximation schemes of increasing complexity and a good agreement with c o m p u t e r simulations. These results give rise to a deeper understanding of the C C A p h e n o m e n o n and allow to discuss in detail m a n y of the features appearing as puzzling in the simulations.
Acknowledgements Part of the work described here was carried out in collaboration with L. Pietronero, and we are very grateful to him. We also would like to thank A. E r z a n and P. Tartaglia for useful discussions and suggestions.
References [1] L. Pietronero, A. Erzan and C. Evertsz, Phys. Rev. Lett. 61 (1988) 861; Physica A 40 (1988) 5377. [2] T.A. Witten and L.M. Sander, Phys. Rev. Lett. 47 (1981) 1400. [3] L. Niemeyer, L. Pietronero and H.J. Wiesmann, Phys. Rev. Lett. 52 (1984) 1038. [4] L. Pietronero, in: Non-Linear Phenomena Related to Growth and Form, P. Pelce et al., eds. (Plenum, New York), in press. [5] A. Erzan, Physica A 185 (1992) 66, these Proceedings. [6] T. Vicsek, Fractal Growth Phenomena (World Scientific, Singapore, 1989). [7] R. Jullien and R. Botet, Aggregation and Fractal Aggregates (World Scientific, Singapore, 1987). [8] P. Meakin, Phys. Rev. Lett. 51 (1983) 1119.
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[9] M. Kolb, R. Botet and R. Jullien, Phys. Rev. Lett. 51 (1983) 1123. [10] M.V. Smolukowski, Phys. Z. 17 (1916) 585. [11] F. Leyvraz, in: On Growth and Form, H.E. Stanley and N. Ostrowski, eds. (NijhofL Dordrecht, 1986) p. 136. [12] M.H. Ernst, in: Fractals in Physics, L. Pietronero and E. Tosatti, eds. (North-Holland, Amsterdam, 1986) p. 289. [13] T. Vicsek and F. Family, Phys. Rev. Lett. 52 (1984) 1669. [14] T. Vicsek, P. Meakin and F. Family, Phys. Rev. A 32 (1985) 1122. [15] A. Vespignani and L. Pietronero, Physica A 173 (1991) 1. [16] L. Pietronero, S. Sidoretti, P. Tartaglia and A. Vespignani, preprint.