Physica A 173 (1991) 1-21 North-Holland
FIXED SCALE TRANSFORMATION APPLIED TO DIFFUSION LIMITED AGGREGATION AND DIELECTRIC BREAKDOWN MODEL IN THREE DIMENSIONS A. VESPIGNANI and L. PIETRONERO Dipartimento di Fisica, Universit~ di Roma "La Sapienza", Piazzale A. Moro 2, 00185 Roma, Italy Received 3 July 1990 Manuscript received in final form 25 September 1990 We extend the method of the fixed scale transformation (FST) to the case of fractal grow, th in three dimensions and apply it to diffusion limited aggregation and to the dielectric breakdown model for different values of the parameter 7/. The scheme is formally similar to the two-dimensional case with the following technical complications: (i) The basis conligurations for the fine graining process are five (instead of two) and consist of 2 × 2 cells. (ii) The treatment of the fluctuations of boundary conditions is far more complex ~lnd requires new schemes of approximations. In order to test the convergency of the theoretical results ~c consider three different schemes of increasing complexity. For DBM in three dimensions the computed values of the fractal dimension for 77 = 1. 2 and 3 result to bc in very good agreement with corresponding values obtained by computer simulations. These results provide an important test for the FST method as a new theoretical tool to stud,~ irreversible fractal growth.
I. Introduction
The fixed scale transformation (FST) is a new theoretical method, which appears particularly suitable for irreversible growth models that are intrinsically critical like diffusion limited aggregation (DLA) and the dielectric breakdown model (DBM) [1] (for a detailed description, see ref. [2]). The FST is completely different from the renormalization group in the sense that it exploits an additional invariance property with respect to the dynamical evolution at the same scale. This should also be combined with a scaleinvariance analysis of the growth process [3]. Originally it has been applied to D L A and DBM in two dimensions [1, 2]. After this we have studied the effect of empty configurations [4], the question of the scale invariance of the growth rules [3] and we have also made various applications to well known problems like percolation [5], invasion percolation [6] and lsing and Potts clusters [7]. 0378-4371/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
2
A. Vespignani, L. Pietronero / FST extended to fractal growth in 3D
For these problems the FST method allows to compute the fractal dimension within one percent from the exact result. In this paper we consider the application of the FST theory to DI, A and D B M in three dimensions. In principle this extension is conceptually straightforward and can be done along the lines of the two-dimensional calculations. In practice however, the three-dimensional case is substantially more complex for the following reasons: (i) In two dimensions one considers the intersection of the structure with a line perpendicular to the growth process and the resulting set can be analyzed by a fine (or coarse) graining process that contains only two configurations. In three dimensions the intersection is done with a plane and the basic configurations are five. (ii) In the explicit calculation of the FST matrix elements it is necessary to consider growth processes up to a relatively high order in order to achieve an accuracy comparable to the two dimensional calculation. The corresponding probability tree is therefore rather complex and it has been necessary to develop a computer algorithm for the automatic calculation of these matrix elements. (iii) The analysis of the boundary condition fluctuations is also more complex because of the same reasons. We consider three different schemes of calculation along a systematic line of increasing complexity in order to test the convergency of the method. Our best scheme of calculation gives the following results for DBM in three dimensions (in parenthesis we indicate the values obtained by simulations [8]: D = 2.49 (2.50) for 77= 1; D = 2.17 (2.13) for , 1 = 2 and D = 1.91 (1.89) for 77 = 3. A discussion about potential accuracy of the calculations and the possible limitations of the comparison with the simulations can be found in sections 5 and 6. These results provide an additional element in support of the validity of the FST theory for fractal growth processes. The paper is organized as follows: In section 2 we describe ~DLA and DBM in three dimensions and introduce the basic concepts for fie calculation. In section 3 we define the FST transformation for the three-dimensional case. In section 4 we discuss the scheme of the calculation for the matrix elements in the case of closed boundary conditions. In section 5 we generalize the approach to include the effect of the boundary condition fluctuations. In section 6 we obtain the final values for the fractal dimension and discuss these results.
A. Vespignani, L. Pietronero / FST extended to fractal growtl, 01 3 D
3
2. The dielectric breakdown model in three dimensions In analogy with the two-dimensional case [1, 2] the problem is defined on a cubic lattice on which the Laplace equation is solved, ~'z~(i, j, k) = 0 ,
(:.l)
with the boundary conditions
d~ (cluster) = O, (2.2) ~b (outer-electrode)= 1. One then defines the local field JVd~(i, ], k)J for all the bonds adjacent to the structure. From this the growth probability is defined a s
Iv l;
z.3)
P'= Z Iv,t,I, i
In order to define a growth direction it is convenient to consider the problcm between two infinite planes, in analogy with the cylindrical geometry of the two-dimensional case. Given a structure we intersect this with a plane perpendicular to the growth direction. For the two-dimensional case the intersection was done with a line and, in principle, the same could be done in three dimensions, but in such a case the problem of the boundary conditions would become extremely complex. So we believe it is more convenient in the three-dimensional case to make the intersection with a plane, despite the fact that this ~ves rise to complications of different nature. The intersection of a fractal structure of dimension D with a plane gives rise to a set of points with dimension D ' = D - 1.
(2.4)
Apart from peculiar cases eq. (2.4) holds in view of the law of codimension additivity. Note that in the case of self-affine structures as DLA seems to be eq. (2.4) has a restricted meaning but the method of the intersection is still perfectly consistent. We now analyze this set of points with a process of fine graining. This implies a box covering of the structure with squares of edge l characterized by a black dot if this square contains some point of the structure and an open dot otherwise. We further subdivide each of these boxes into four sub-boxes of edge 1/2 that again are characterized by black or white dots.
4
A. Vespignani, L. Pietronero I F S T extended to fractal growth in 3D
In the process of fine graining each box can be fragmented in five possible configurations as shown in fig. 1. Each of these configurations can have a multiplicity corresponding to non-equivalent rotations. For example the configuration of type 5 has a multiplicity equal to four. The probability that the process of fine graining gives rise to the configuration of type i is indicated as C~ (i = 1 , . . . , 5) and normalization requires E~__, C~ = 1. The average number of black sub-boxes generated in the fine graining process is
(n) =~.,n,C,=4C, + 3C 2 +
(2.5)
2 ( C 3 + C 4 ) + C5 ,
i
where n i indicates the number of black sub-boxes in the configuration of type i. In analogy to the two-dimensional case eq. (2.5) can be directly related to the fractal dimension of the entire structure by D=I+
In
(2.6)
which is therefore directly related to the probability distribution { Ci} (i = 1 , . . . , 5). Strictly speaking, with respect to the comparison with simulation, eq. (2.6) corresponds to the box counting dimension for growth between two infinite planes.
3. The fixed scale transformation for three dimensions
The fixed scale transformation (FST) is defined with respect to the dynamical evolution at the same scale. The idea is that the fixed point distribution for the occurrence probabilities of the various configurations {Ci} (i = 1 , . . . , 5) is invariant both with respect to scale transformation and with respect to the dynamical evolution. The renormalization group theory would exploit the first Q
C]
°
°°
°
o
C2
o
i°°1 i, °I "r3
o. °
o °
C4
-
C 5
Fig. 1. Basic configurations that appear in the process of fine graining or fragmentation.
A. Vespignani. L. Pietronero / FST extended to fractal growth in 3D
5
invariance while the FST is related to the second one and it appears more convenient for problems of irreversible fractal growth, in principle the FST should contain also the fully empty configuration, which cannot appeal in the fine graining process (eq. (2.5)). This problem has been studied in detail for the two-dimensional case and it gives rise, in general, to a high order correction [4]. Therefore for the present case we will neglect the empty configuration in the FST. In addition the FST should be complemented by a study of the scale invariance of the growth rules. This is a more subtle problem and requires further studies. For the present case we are going to use the growth dynamics at the minimal scale as for the two-dimensional case. The matrix elements of the FST, defined as M,. corres~nd to the probability that a configuration of type i is followed, in the growth direction, by a configuration of type j. This probability refers to the asymptotic [r~zcn structure, so its evaluation requires, in principle, an infinite n u m ~ r of gro~,,,~th processes. The FST transformation that describes the dynamical evolution at the same scale can be written as C
~
+!
M! I
" "
"
M.~I
t3.1) C
*l
Mi.~
---
M~5,
where k is the index of the iteration. As we have mentioned the matrix elements should include all the growth processes th~a lead trom configuration i to configuration j (on the following plane). Formaliy we can write M0 = ~ p(gr°wth processes i--~ j ) ,
(3.2)
Since these matrix elements will be used at all scales a finite number of groxx~th processes (repeated at all scales) in eq. (4.1) corresponds to a rather accurate description of the entire growth process. It is for this reason that, in general, a good convergency is obtained by considering only a relatively small number of processes in eq. (4.1). The fixed point {C~} of eq. (3.1) gives then the asymptotic probability distribution for the q,'e basic configurations. This distribution is of course the same for all scales and therefore it allows to determine the fractai dimension D via eq. (2.6). One should also notice the following normalization conditions that will be used in the calculations: 5
~ C, = 1, i=!
(3.3)
6
A. Vespignani, L. Pietronero / F S T extended to fractal growth in 3D
Mij =
1.
(3.4)
J
The basic proble.m is therefore the determination of the matrix elements Mij. As we are going tr~ scc in the following, these matrix elements will only be scalars in the case of fixed boundary conditions. If one includes instead the fluctuations of boundary conditions the matrix elements will be themselves functions of the probabilities { Ci}. This will give rise therefore to a non-linear iteration for the FST (eq. (3.1)), which makes the problem more complex but also more accurate.
4. The matrix e l e m e n t s Mij for closed boundary conditions
In this section we describe a scheme for the explicit calculation of the matrix elements Mij in the case of closed (or periodic) boundary conditions. In the two-dimensional case we consider explicitly the growth processes defined in the column above the two sites referring to the starting configuration. Growth processes outside this column are described only in terms of boundary conditions. One may notice, in this respect, that the study of the effect of empty configurations ,~n the FST provides also a test for the validity of this scheme. In the three-dimensional case we consider therefore the growth processes in the three-dimensional column above the starting (2 × 2) configuration of four sites defined by the process of fine graining. Each plane of this column has four sites as shown in fig. 2a and a site is defined by three coordinates (i, j, k), where k refers to the plane and i, j define the position within the plane as shown in fig. 2b. The plane with k = 1 contains the starting configuration, which is considered as frozen in the sense that no growth can occur within this configuration. This is due to the fact that we consider growth conditional to the existence of a given starting configuration. Growth processes are confined to
J°J°_7 .~-ojo f
a)
1 ,2
2,2
1,1
2,1
b)
Fig. 2. (a) Example of the three-dimensional column on which the growth processes are considered. (b) Labelling of the sites within a given plane.
A. Vespignani, L. Pietronero I F S T extended to fractai growth in I D
7
the sites of this column and the growth probability is normalized accordingly. The probability of a growth process is given by
Iv, , kl
(4.!)
P'J~ = Z IV,~,~,! ' Ok
and it implies the solution of the Laplace equation for each configuration. This requires the definition of the boundary conditions. Here we will adopt ~ r i ~ i c boundary conditions that correspond essentially to the case of closed ~ u n d a r y conditions. This implies that our column is surrounded on its four sites by replicas of the same configurations. Later on we are going to imp~ve the method by considering also the possibility of open ~ u n d a ~ ~nditions. As an example we can consider the growth processes corres~nding to a starting configuration of type 3 as shown in fig. 3. The first growth process is trivial because it will necessarily occur above one of the starting occupied sites with equal probability. The corresponding configurations are equivalent by symmetry, so this process is already included in the starting configuration. ~ e
P
. .
joj,=
j
Jo.Jo
J
J o J , = J
f o J o J a)
b)
c)
Fig. 3. Example of the growth processes starting with a configuration of type 3 (see fig. 1 ). The first growth process is trivial and it is already added in the starting configurations. The second growth process requires an analysis of the various configurations and the calculation of the corresponding potentials.
8
A. Vespignani, L. Pietronero I FST extended to fractal growth in 3D
total growth probability leading to the occupation of a given site will be denoted by P,(i, j, k) where (i, j, k) are the coordinates of the site and n refers to t h e n u m b e r 3f particles added. In the counting of n in general we will not consider the first (trivial) process. This probability of site occupation may include more than one bond leading to the same site. For example in the case of fig. 3 the first order growth probabilities are ~b(1, 1, 3) P,(1, 2, 3) = th(1, 1 , 3 ) + 2th(2, 1,2) + 3~b(1, 2, 2) '
(4.2)
3~b(1, 2, 2) P,(1, 2, 2) = ~b(1, 1 , 3 ) + 2~b(2, 1,2) + 3~b(1, 2, 2) '
(4.3)
P,(2, 1 , 2 ) = 1 - P,(1, 1 , 3 ) - P,(1, 2, 2).
(4.4)
Therefore at first order a cell of type 3 has a probability P ~ ( 1 , 2 , 2 ) + P1(2, 1, 2) to be followed, in the growth direction, by a cell of the same type (also rotated). This gives M33 = P,(1, 2, 2) + P,(2, 1 , 2 ) ,
(4.5)
in addition M35 = P,(1, I, 3)
(4.6)
M31 = M32-- M34 = 0.
(4.7)
and
The probability distribution at second order corresponds to the addition of an extra particle at the end configurations a, b, c of fig. 3. This gives for example M33 = PI(1, 1, 3)[P2(1, 2, 2) + P2(2, 1,2)] + P,(2, 1 , 2 ) [ 1 - P2(2, 2, 2 ) - P2(1, 2, 2)1 + PI(1, 2, 2 ) [ 1 - Pzb(2, 2, 2 ) - P2(2, 1, 2)1,
(4.8)
where the index a, b or c refers to the three configurations of fig. 3. The same should be done also for the other matrix elements. In principle one should consider higher and higher orders until the empty sites in the configuration above the starting one become frozen, in the sense that the corresponding grewth probability is virtually zero. In practice one can
A. Vespignani, L. Pietronero / F S T extended to fractal growth in 3 D
t,)
/ /
l)l tl, 2, 2)
!' I (2, 1, 2)
/',,,
I)2 (2, I, 2)
1'2 (2, 2, 2)
1'2(I, 2, 2)
1'2 (2, 2, 2)
1'~(2, 2, 2)
P3(2. l, 2)
1
l ) ~ . 2.2)
l'~(I. 2.2)
1' I f2, 2.2)
1',(1...,~ 2)
i'z(2, l...~')
/
Fig. 4. Scheme of the probability tree corresponding to the calculation of M~ at third order. This implies the addition of three particles in addition to the first, trivial, growth process. Th.e probabilities of different branches are different even if they have the same indices, in view of the different boundary conditions.
observe that in t w o dimensions a relatively small order provides already a good convergency in view of the screening properties of the Laplace equation. In three dimension the situation is more complex because one has to go at least up to third order to have a non-zero probability for the configuration of type 1. To get a good convergency in addition one has to go much higher than third order. In fig. 4 we show the pJ'obability tree for the matrix element M!1 up to third order. This is one of the simplest cases that shows the difficulty of going to higher orders with a direct analysis. We have therefore constructed a computer program that generates all the relevant configurations and computes automatically the matrix elements. In this way it is possible to compute the
A. Vespignani, L. Pietronero / FST extended to fractal growth in 3D
10
matrix elements up to the sixth order, which roughly corresponds to the third order of the two-dimensional case. This is due to the fact that the cells for three dimensions contain twice the sites as those for two dimensions. In order to check the degree of convergency as a function of the order we have computed a few matrix elements up to the eighth order. As an example in fig. 5 we show the behavior of M31 at various orders. One can see that the sixth order has already a good degree of convergency. Finally we should mention that for the matrix elements of type M2j the first growth process is not symmetric. In fact, contrary to all other cases, if one starts with a configuration of type 2 (fig. 6) one can see that the growth probability corresponding to the occupation of the site (1, 1, 2) is different than that corresponding to the sites (2, 1, 2) or (1,2, 2). In this case therefore we have to consider also a non-trivial zero order growth process. As shown in fig. 6 we have
M2j = [P o(2, 1, 2) + Po(1, 2, 2)]M2b.j + Po(1, 1, 2) M2~.j,
(4.9)
where P0(i, 1, k) refers to the probability to occupy a site in the empty plane k=2.
-'
'
I
. . . .
I
. . . .
I
'~-'
'
1
. . . .
I
. . . .
I
'~-
0.08
J 0.07
cO
v
0.06
t 0.05
,,,I,,,,I,, 3
4
5 6 Order of growth
,,I,,,,I,, 7
8
Fig. 5. Behavior of a typical matrix element (Ma~) as a function of the order of the calculation for the case of closed (periodic) boundary conditions.
A. Vespignani, L. Pietronero / FST extended to fractal growth in 3D
11
We have now all the elements to compute the matrix elements up to the sixth order. The matrix corresponding to eq. (3.1) is in this case given by 0.114 0.317 0.302 0.066 0.201
t
0.095 0.292 0.327 0.052 0.234
0.077 0.257 0.387 0.0 0.278
0.085 0.272 0.283 0.092 0.268
0.068\ 0.233| 0.377 / . 00 / 01322
(4.10)
By inserting these values into the fixed scale transformation (eq. (3.1)) we can compute the corresponding fixed point. This corresponds to the follo~ng asymptotic probability distribution for the five elementary configurations: C ! =0.083,
C., = 0 . 2 6 5 ,
C3 = 0 . 3 5 9 ,
(7-.4=0.021 ,
C~ = 0.272. (4.1!)
From these we obtain, via eqs. (2.5) and (2.6), ( n ) = 2.158,
(4.12)
D=2.11.
(4.13)
This result corresponds to the sixth order calculation for the case of closed boundary conditions. It is rather different (lower) than the fractal dimension obtained by computer simulations D = 2.5 but this is not surprising since the closed boundary conditions enhance the screening effect of the Laplace equation and therefore they give a low value for D. In the next section we are going to see that an improvement of the boundary condition treatment will also improve the numerical results.
/,,~/oe/o
/
/ o / o / ~ / o /
/
/ o/o / o / o /
/
/
/ e / o / o / o /
/
j
o f
o f
/ Q / o /
/
[
!
b
a
Fig. 6. Representation of the three possible positions of the first growth site with a starting configuration of type 2. The two cases (b) are , o t equivalent to the case (a). so the zero order growth process in this case is non-trivial.
12
A. Vespignani, L. Pietronero / FST extended to fractal growth in 3 D
5. Inclusion of boundary condition fluctuations
In this section we generalize the treatment by allowing the growth process to have different boundary conditions each weighted by its probability of occurrence. This corresponds to a generalization of the scheme of calculation along a systematic line of increasing complexity. The situation is in this respect substantially more complex than the two-dimensional case [1, 2, 9] because there are four sides to consider as shown in fig. 7 in which the central cell corresponds to the 2 x 2 cells we consider as the starting configurations for growth. In principle one may also consider the diagonal sites but these are only indirectly linked to the growing structure, so we will neglect this effect. Our objective is now to express the probability distribution of the various boundary conditions as a function of the probability distribution { C~} ( i 1 , . . . , 5) for the elementary configurations. To this purpose it is convenient to introduce the minimal scale b corresponding to the size of the smallest structure considered and define a series of length scales A,, = 2"b (n = 0,1,2,...). A given box of size b becomes grouped, at the level of boxing 2b, in one of the four possible ways shown in fig. 8. We call this process boxing and the probability for each of the four boxing processes is ~ because the positions of the boxes in a coarse graining proces~ are uncorrelated with the structure. These four possible process will be indicated as left-up or right-up boxing and as left-down or right-down boxing (lu, ru, ld, rd). We begin this analysis by considering that a black box is neighboured on its right and left side by a black box without considering what is the situation in the other two neighbouring boxes. We start therefore with a black box of size b and consider the first boxing process and suppose that this is of type (ru) (fig.
8). The probability to have a black box on the right, conditional to the fact that there is already a black site in a box of size 2b, is given by P1 =
e,,
,
(5.1)
X
I× X Fig. 7. The type of neighbouring cells on the sides of the starting 2 x 2 cell define the boundary conditions for the growth process.
A. Vespignani, L. Pietronero / FST extended to fractai growth in 3D
13
LEFT-UP BOXING
RIGHT-UP BOXING
RIGt4T-DOWN BOXING
I. EF'I-DOWN BOXING
Fig. 8. An occupied box of size b can be grouped in four pogsible ways within a box of s~e 2b.
t,,, =
+
3
+ -',( c , + c , ) + c , .
(5.2)
In order to consider the probability to have a black box also on the left side we have to go to the next level of coarse graining, for which there are again four possibilities of boxing. If the next boxing is of type (lu) or (ld). the probability that the box of size 2b, on the left of the box of size 2b that contains the starting black box, is black is P~. This should then be multiplied by Pt,, that is the probability that within the (2b) left box there is a black box of size b adjacent to the left side of the starting box of size b. So the final probability for a black box to have black boxes on the left and on the fight is ~P~P0. If instead the second boxing is (ru) or (rd), we have to go to a still higher level of coarse graining and so on. Iterating this procedure we obtain the following probability tree:
t,.u~. ~ <
.(lu) + (ld): ~P~Po tl,,~ .
p2 p2 tl,-i~. ~. t - ,
(ru) + (rd): ~
(lu) + (ld) . _~PiPo 2 3 (ru) + (rd): ½
< (ru) + (rd)"
1
(5.3)
where, for example, (lu) + (ld) corresponds to the sum of the probabilities that the boxing is (lu) or (ld). The tree shown in eq. (5.3) corresponds to a starting box of type (ru). One can see that, starting with a different type of boxing, a
A. Vespignani, L. Pietronero / FST extended to fractal growth in 3D
14
similar tree is generated. By resumming all the various cases the final tree is 1
2 PIPo
(ru) + (rd) + (lu) + (ld): , < . , 1
, PtPo 2 2 f
~
:< 2
~1 P~2 Po3 !
(5.4)
This tree can be resummed and the result is oc
e P0
P * = P(Ar= b; At = b ) = ½P,Po ,~=0(½P0)"- 2 - Vo '
(5.5)
where A r -" b (A t = b) implies that on the right (left) our starting black box is immediately followed by another black box at a distance corresponding to the minimal scale b. In a similar but more complex way one can compute the probability P ÷ that a black box is surrounded on all four sites by black boxes. The detailed calculation is reported in appendix A and the result is e ÷-
2
2e,e0e
3
4 - p2 + ( 2 - P,,)(4- p2)
(5.6)
having defined 1C2+
pN = 4
C
i
eo
(5.7)
In order to include different types of boundary conditions in the calculation of the fractal dimension it is necessary to generalize the structure of the fixed scale transformation. We characterize a given type of boundary condition by the four variables (A r, At, Au, Ad) , which define the distance for various directions (right, left, up, down) at which a new occupied site can be found. The case of the previous section corresponds to the situation in which A = 0 in all directions, ha the more general case the matrix elements Mij become the weighted average of the matrix elements referring to different boundary conditions. Formally we can express this as
M~j= ~, P({A}) {a}
Mij(ih}),
(5.8)
where {A} refers to a particular configuration of the four A's and the sum extends in principle over all possible configurations. We have seen that the
A. Vespignani, L. Pietronero I FST extended to fractal growth in 3D
15
probability for a given configuration can be expressed in terms of the { C,} ( i = 1 , . . . ,5). Therefore the fixed scale transformation is generalized as follows:
C
~
=
(5.9)
P(IA};
C' +
\M,si{ A})
Mss({ A}
C~/
and it becomes a non-linear transformation. In principle its ~mplete determination would imply considering all possible types of boundaD" conditions. In practice, however, one can define suitable t[uncation ~h emes by limiting the sum to a few, particularly relevant, types of boundaD, conditions like the open-closed approximation for the two-dimensional case. For the three-dimensional case we have devised two schemes of approximation that include two or three different types of boundaD, conditions and the
corresponding matrix elements. In the first scheme the two types of boundary, conditions correspond Io A = 0 or A = ~: for all the four sites. This gives
M# = P cl M#cl + P
op
M#,,p ,
(~.10)
where the indices "'cl" or "op" refer to closed (A = 0 ) or open ( a = ~) boundary conditions. The probability P¢~ refers to the situation in which a black box is completely surrounded by four black boxes, however it is not identical to P+ of eq. (5.6). This is due to the fact that now we have to interpret the starting black box as the 2 x 2 cell to be used in the growth process. So we have to specify in greater detail the configurations of the neighbouring 2 x 2 cells. In order to consider the configuration as closed we require that for each side there must be at least one (or two) black subcells adjacent to the central 2 x 2 cell. This probabifity for each 2 × 2 neighbouring cell is Pc,, = C2 + C~ + 3zC 3 + ~C 5 + C 4 .
(5.11)
We have therefore eel =
p
+ (pco)4 =
:)2 p 2 N--0
2 p21p3pN
)
4 - p2 + ( 2 - P o ) ( 4 - P~) (Pc,,
2)
)4
•
(5.1
16
A. Vespignani, L. Pietronero / F S T extended to fractal growth in 3D ci
For the matrix elements we have that M ij corresponds to those computed in op the previous section while for M 0 a new calculation is necessary. It is easy to see that this calculation (carried to large enough order) should overestimate the fractal dimension. In fact all configurations that are not completely closed, are considered as completely open, and therefore the screening effect for these configurations is lowered by our scheme. In the second scheme we try to improve on this point by including also a third configuration, intermediate between the previous two. This third configuration has closed boundary conditions on at least two opposite sides but not on more than three. The point is that one open side is not very different from two open sides with respect to the screening properties of the Laplace equation. In this case we have ci + P op2 M~j op2 Mq = P cl M~j
+
p°P4M~P4,
(5.13)
where pop2 corresponds to the new added type of boundary condition while pop4 corresponds to a configuration that is open in all four sides. In this case pop4 =
1 - pop2 _ pc1.
(5.14)
Considering that P* (eq. (5.5)) gives the probability that a black box is surrounded by two black boxes (on opposite directions) independent of the occupation of the other two neighbouring boxes, the probability that at least one side but not more than two is open is given by P*(Pco) 2 - Pc1. We have therefore
poo: = 2[P*(e¢o): - e d ] ,
(5.15)
where the factor of 2 is due to the fact that we have two ways to fix the pair of opposite black boxes. This expression, together with eqs. (5.12) and (5.14), gives all the three probabilities that appear in eq. (5.13). For each of these two schemes of fluctuating boundary conditions we can now compute the fractal dimension. This is described in the next section.
6. Results and conclusions
The calculation of section 4 refers to closed boundary conditions, which we will identify as A = 0. For the two schemes defined in section 5 in which the
A. Vespignani, L. Pietronero / FST extended to fractai growth in 3D
17
boundary conditions are allowed to fluctuate, we will use the notation "scheme 2" or "scheme 3" depending on the number of boundary conditions included. These three different schemes of calculation correspond to a systematic line of increasing complexity and can provide an idea of the convergency properties of the method. el The matrix elements M ij are the same as those of section 4. So for scheme 2 we have only to compute M ijop that correspond to an isolated growing structure, in the sense that A = ~ on all four sides. In practice the calculation is performed for periodic boundary conditions in which A = 10b, where b is the lattice spacing. One can check that for growth processes of relatively low order this is a good representation of the situation of open boundary conditions (b.c.). We compute the matrix elements up to 6th order in the growth process and determine the fixed point of the non-linear FST by numerical iteration. The fixed point distribution for the elementary configurations is C i =0.290,
C:=0.396,
C3=0.212,
C~=0.031,
C 5 =0.071,
(o.1) and the corresponding fractal dimension is D = 2.54
(6.2)
(scheme 2, 6th order).
By comparing this value with that corresponding only to closed b.c. (D = 2. I 1, eq. (4.13)) and to the value obtained from numerical simulations (D = 2.5) one can appreciate the importance of the fluctuations of boundary conditions. For scheme 3 the new matrix elements to be computed are those of type MOp2 There are computed with two (opposite) sides closed and the other two ij open. There may be ambiguities in the specific definition of the site occupation as shown in fig. 9 for the case of a starting configuration of type 3. In these cases we compute the matrix elements by averaging over the different configu-
0
0
[o[o
0
0
0
°
0
o o
0
•
0
°
0
0
0
0
i
o
~o o o o o o a)
°
® :°
°
°
b)
®
L
Fig. 9. Two possible growth scenarios corresponding to a starting configuration of type 3 in the case of intermediate boundary conditions. In these cases the calculation of the matrix elements (M~ p) is performed by averaging over these different scenarios.
A. Vespignani, L. Pietronero / FST extended to fractal growth in 3D
18
rations with equal weight The fixed point is in this case given by C 1=0•257,
C2=0.386,
C3 = 0 . 2 3 6 ,
Ca = 0 . 0 3 2 ,
C5 = 0 . 0 8 9 , (6.3)
and the corresponding fractal dimension is D = 2.49
(scheme 3, 6th order).
(6.4)
As expected the value of D is lower with respect to scheme 2 and it is in excellent agreement with the numerical result (D -----2.5). Concerning the comparison with the results of the simulations some caution is however necessary. Strictly speaking the FST method refers to the properties of growth between two infinite lines or planes (in 3D) and its outcome corresponds to the box counting dimension. For this geometry there should be no problem of lattice anisotropy as discussed in ref. [10]. The problem of the lattice anisotropy arises from the fact that, in two dimensions and in the radial geometry, the clusters seem to take asymptotically a deterministic shape with elongated branches• From the point of view of the gyration radius this effect may appear to change the value of the fractal dimension but this is, in some sense, a misleading conclusion. The important feature with respect to the box counting fractal dimension is what is left after the growing front passes through a certain region and not the eventual instability of the shape of the growing front. From this point of view the lattice anisotropy is not such a relevant problem and in particular, in three dimensions, this effect appears to be essentially negligible• Our comparison with simulations is done with the best 3D data available that correspond to off lattice growth for radial geometry and they should represent a fair comparison to the theory. Until now we always discussed the case of DBM witi~ rl = 1 equivalent to DLA. However our calculations can be easily generalized to the case 71~ 1 by using a different expression for the growth probability,
p_
Iv 17
'
Y, Iv l"i
•
(6.5/
i
We have considered explicitly the cases 71= 2 and r / = 3 for which we have repeated the entire calculation in the three schemes considered. All results are reported in table I. Concerning the convergency of the results one has to consider two different
A. Vespignani, L. Pietronero / FST extended to fractal growth in 3D
19
Table I Values of the fractal dimension D of the DBM model in three dimensions computed with the fixed scale transformation theory. The values reported correspond to three different values of the parameter ~ and to increasingly more sophisticated schemes of calculations. For comparison we also list the values of D obtained by computer simulations (from ref. [8]).
FST scheme
1
2
3
A=0 6th order
2.11
1.49
1.19
scheme 2 6th order
2.54
2.21
1.92
scheme 3 6th order
2.49
2.17
1.91
numerical simulations (off lattice)
2.5
2.13
1.89
processes. The first one refers to the convergency with respect to the order of the calculation within a given scheme of the t, oundary condition treatment. In this respect this convergency is faster for larger values of 77 in view of the enhanced screening effect of the Laplace equation. A further convergency problem can be defined with respect to increasingly better schemes for the treatment of the boundary conditions once the order is extrapolated to infinity. A proper extrapolation procedure should include both effects and it is hard to implement in practice. In general, however, our results (non extrapolated) are within one percent from the numerical ones. The study of three-dimensional D L A and DBM growth processes provides therefore an important test for the validity of the FST method as a new theoretical tool to study irreversible fractal growth.
Appendix A In this appendix we show the explicit calculation of the probability P+ to have a black box surrounded on all four sites by black boxes. Let P0 and Pt be the same probability of section (5) and call PN the probability of having two black sites in a 2 × 2 box given that another black site is present.
20
A. Vespignani, L. Pietronero / FST extended to fractal growth in 3 D
One can easily show: ion =
1C, 4 _ + Cl Po "
(A.1)
As in section 5, let us start with a black box of size b and suppose that the first boxing process is of (ru) type. The probability to have two black boxes adjacent to the given starting box is ~ PNIn order to calculate the probability of a black box also on the two remaining sides we have to consider the next level of coarse-graining. If the next boxing is of type (ld) the final probability is -~PNPo. 1 2 On the contrary, if the second boxing is of (lu) type the probability of a third black box adjacent to the starting box is ~ PNP0, and we have to consider the next order of boxing. If the third boxing is of (ld) or (rd) type the probability to have also a fourth box adjacent to the starting site is ¼p~p2. In the opposite case another level of coarse-graining must be considered. By iterating this procedure we obtain the following probability tree: (ld)" ~1PNPo2 (rd) + (ld)" ½p, p2
(ru)- 41PN -
! 3 (rd) + (ld)" ~P,P,,
(lu)" ~P,P,, (ru) + (lu)" ½
(ru) + (lu): ½"-" (A.2) For symmetry if the second boxing is of (rd) type the generated tree is equivalent to the last, so that (ld)" -~PNP~ (ru). pN
. 12 pip~ - - (rd) + (lu)" ½P, Po < ,
<, \
• 12
Pl p3
: ~-"_
(A.3)
The branch of the tree relative to the (rd) + (lu) boxing can be resummed and the result is 7~
1
7.
.3
P,Po >'~ ( ~Po)" . n ={I
(A.4)
A. Vespignani, L. Pietronero / FST extended to fractal growth in 3D
21
Finally we have to consider a second boxing of (ru) type. The tree generated in this case is the same as in (A.3) except for a factor ~ P~. By repeating this consideration we can write the complete tree as
.Od)
"
I 2 3 ~ (~'Po)" (ru)" ~ PN ~(lu) +(rd) "-~PiPo
\
n=O
(ru)" ~ p2
~
(lu)+ (rd): ~P~P~~
( ~P,,)"
n~O
(ru): ~,e~'" (A.5)
In the case of different starting boxing process a similar tree is generated. By resumming all the cases we obtain (A.6) n=O
-0
from which the final expression used in section 5 is recovered.
References [ll [2] [3] [4] [5] [6] [7] [8]
L. Pietronero, A. Erzan and C. Evertsz, Phys. Rev. Lett. 61 (1988) 861. L. Pietronero, A. Erzan and C. Evertsz, Physica A 151 (1988) 207. R. De Angelis, M. Marsili, L. Pietronero and A. Vespignani, to be published. A. Vespignani and L. Pietronero, Physica A 168 (1990) 723. L. Pietronero and A. Stella, Physica A 170 (1990) 64. L. Pietronero and W.R. Schneider, Physica A 170 (1990) 81. A. Erzan and L. Pietronero, J. Phys. A (199I), in press. S. Tolman and P. Meakin, Physica A 158 (1989) 801. P. Meakin and S. Tolman, in: Fractals" Physical Origin and Properties. L. Pietronero, ed (Plenum, New York, London, 1989). ~,,~,. A -,,, A n Siebesma, Phys. ~^'" ~'~ (1989) 5377. [91 R.R. Tremblay and rLr. C. Evertsz, Phys. Rev. A 41 (1990) 1830. [lO1
