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Is diffusion limited aggregation scale invariant?
PHYSICAIt\
Physica A 200 (1993) 165-170 North-Holland SDZ: 0378-4371(93)EO309-3
Is diffusion limited aggregation Antonio
scale invariant?
Coniglio
Dipartimento di Scienze Fisiche, Universitri di Napoli, Mostra d’oltremare 80125 Napoli, Italy
Pad. 19,
Recent work showing evidence for multiscaling in large two dimensional DLA clusters is reviewed. A connection between multiscaling and multifractality is also shown resulting in a weak localization of the growth sites (o-sites) in shells around the cluster seed.
One of the fundamental steps towards a complete theory in critical phenomena was the discovery of scaling laws. It was this discovery in fact which paved the way to the renormalization group. This theory is in fact based on the invariance of the thermodynamic quantities under an appropriate scale transformation. When the diffusion limited aggregation (DLA) model was proposed [l], it was natural to assume the same scaling laws valid in critical phenomena. In fact DLA shares with critical phenomena the common feature of a characteristic diverging length. However there are various differences from ordinary critical phenomena. For instance, DLA clusters show an interior frozen part and an external active periphery not present in other geometrical critical phenomena such as percolation. This feature of DLA is responsible for the poor performance of a real space renormalization group analysis, compared with analogous computations carried out in the percolation problem. In fact a different procedure for DLA in which the renormalization is limited to the mass in the frozen part, gives much better results [2]. The distinction between a frozen and a growth region was also stressed by Pietronero et al. [3], who have developed a theory of fractal growth which applies only to the frozen part. More recently it has been proposed that planar DLA does not have a simple self-similar geometry [4]. A more general concept of scale invariance was introduced [5] that may explain the various anomalies present in DLA and not in ordinary critical phenomena. This more general scale invariance leads to a multiscaling form for the density profile, which is the analog of the order parameter correlation function in critical phenomena. Defining the density profile as g(r, R) ddr = dM , Elsevier Science Publishers B.V.
166
A. Coniglio
I Is diffusion
limited aggregation
scale invariant?
where dM is the number of particles in the volume ddr at a distance r from the origin and R is the radius of gyration, the following multiscaling form was proposed [5] :
g(r, R) = ~-~+~(“~)c(r/R) ,
(1)
where C(x) is an amplitude that goes to zero rapidly for large x. Multiscaling contains a structure much richer than standard scaling. In the asymptotic regime, where r and R are both large with some fixed ratio x = r/R, the above form for the density profile predicts a different power law decay for each value of x and consequently a different local fractal dimensionality D(x) for each shell corresponding to that value of X. Specifically, moving from x = 0 toward increasingly larger values of x, D(X) varies from the fractal dimensionality D(0) of the inner frozen region to the fractal dimensionality of the outer shells in the growth region. The proposed behavior implies that DLA satisfies symmetries more complex than those found in ordinary critical phenomena. The density profile rescales by a factor which depends on x as opposed to standard scaling where D(x) = D independent of X. Knowledge of the symmetry of a problem is of fundamental importance in developing a theory. Then, if multiscaling is correct for DLA a complete theory should be based on such a multiscaling property, like in ordinary critical phenomena the renormalization group was based on standard scale invariance. The multiscaling form (1) has been tested for off lattice DLA up to lo5 particles [6] and more recently [7] for clusters of lo6 particles. D(x) was found to be roughly constant for small values of x, bending down at a value of x close to 1.5, with an overall behavior in agreement with the multiscaling picture. These result are in agreement with an independent calculation [8] recently done for off lattice DLA clusters up to one million particles. A single run of fifty million particles gave a rather inconclusive result. The plot of D(x) for the one million particles is given in fig. 1. To show the trend as a function of the number of particles the data for the two sets of measurements have been plotted. The unphysical bump, more pronounced in the set of lo5 particles, flattens out as the number of particles increases. In fact it can be shown [6] that asymptotically D(x) is a non-increasing function of x. It may be useful to characterize the behavior of D(X) for small values of x by an exponent p, D(x) -D(O)
Numerically
-
uxp.
it was found that p - 10 and a - l/2.
(2)
A. Coniglio
I Is diffusion limited aggregation scale invariant?
167
D(x) 2.0
1.0
1
I. 0.0
?
0.4
0.8
I.2
1.6
2.0
x
Fig. 1. Plot of D(x) versus n for (0) a set of 200 off-lattice DLA clusters of lo5 particles, a set of 66 off-lattice DLA clusters of lo6 particles. (After ref. [7].)
and (0)
The multiscaling form of the density profile g(r, R) implies a particular relation of mass versus radius involving a logarithmic prefactor. In fact from (1) by integrating g(r, R) over the volume we obtain the mass A4 as function of R. For large R then we find M-B(R)RD,
(3)
where D = D(0) is the fractal dimension of the frozen part and B(R) = (ln R)-D’p. The multiscaling form of the density profile (1) also implies multiscaling in the growth probability distribution P(r, M). This quantity was introduced by Plischke and Racz [9] and is defined in such a way that P(r, M) dr is the probability that the Mth particle added to the growing cluster sticks at a distance between r and r + dr from the origin. The multiscaling form on P(r, M) in turn implies that the fluctuation of the radius (width of growth region) approaches logarithmically the radius for large values of M, in agreement with numerical results [7,8]. The form of D(X) indicates that there is an inner region for x x0 for which the fractal dimension of the shells vary with continuity. Note that this second part is not a negligible surface effect as the width of the growing region scales as the radius. Moreover it is just the growth part where interesting phenomena linked to multifractality is manifested. Multiscaling and multifractality. Finally, we emphasize that in general multiscaling is a distinct phenomenon from multifractality [lo]. In the case of multifractality the existence of infinitely many exponents is the consequence of
168
A. Coniglio
I Is diffusion limited aggregation scale invariant?
the introduction of an appropriate measure on the fractal, and does not imply any condition on the symmetries#‘. For example in percolation there is multifractality with respect to the voltage distribution [12] of the random resistor network, nevertheless the pair connectedness function at criticality is invariant under standard scaling, and therefore does not obey multiscaling. This is the reason why standard renormalization group methods can be successfully applied to percolation. However, in the case of DLA the measure which gives multifractality is the growth probability {p,}, where pi is the probability that the perimeter site i will be the next to grow. The growth probabilities in turns characterizes the growth of the aggregate and therefore its geometrical structure. Thus multiscaling and multifractality in DLA may be linked together. In fact it was suggested [5] that multifractality combined with strong localization of the growth sites may imply multiscaling. In order to explore this connection Lee et al. [13] studied the joint distribution function N(a, X, M) where N((Y,x, M) da d_x is the number of perimeter sites with a-values in the range [(Y,CY+ da] and located in the annulus [x, x + dx] around the cluster seed. They showed that for large M the localization of the a-sites is peaked around a fixed value ,~(a) (fig. 2). Moreover they suggested on the basis of numerical results that the distribution N(a, x, M) obeys multiscaling and this is consistent with the multiscaling form of the density profile. Multiscaling in other systems. Multiscaling is manifested also in other models. In particular an exact solution [14] of the time dependent N = 00 GinzburgLandau model shows that the time dependent pair correlation function for a conserved order parameter exhibits a multiscaling form. Moreover, various models of invasion percolation have been studied numerically and the result of this analysis shows [15] that while ordinary invasion percolation does not exhibit a multiscaling form, the multiscaling form is found in the presence of a power law decay with the distance from the initial seed. In conclusion, we have given evidence that planar DLA clusters are invariant under a scaling transformation more complex than usually found in critical phenomena. Such scale invariance implies a multiscaling form (l), which is found not only in DLA but in other systems as well. I would like to thank the collaborators in this research: C. Amitrano, Meakin, J. Lee, S. Schwarzer, H.E. Stanley and M. Zannetti.
P.
#’ A different case is when multifractality is obtained with respect to the mass distribution as found in [ll]. In this case multifractality is directly related to a more complex geometry of the aggregate and is closer to multiscaling.
A. Coniglio
I Is diffusion
limited aggregation
scale invariant?
169
Fig. 2. Location of LYsites from 18 off DLA clusters of M = 20 000. In (a) we have 1.5 <(I < 1.9, in (b) 2.8 <(Y < 3.0, and in (c) (Y> 6. (After ref. [13].)
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo]
T.A. Witten and L.M. Sander, Phys. Rev. Lett. 47 (1981) 1400. C. Amitrano, A. Coniglio and F. di Liberto, J. Phys. A 21 (1988) L201. L. Pietronero, A. Erzan and C. Evertsz, Phys. Rev. Lett. 61 (1988) 861. B.B. Mandelbrot, Physica A 191 (1992) 95. A. Coniglio and M. Zannetti, Physica A 163 (1990) 325. C. Amitrano, A. Coniglio, P. Meakin and M. Zannetti, Phys. Rev. B 44 (1991) 4974. C. Amitrano, A. Coniglio, P. Meakin and M. Zannetti, submitted to Fractals (1993). P. Ossadnik, Physica A 195 (1993) 319. M. Plischke and Z. Racz, Phys. Rev. Lett. 53 (1984) 415. C. Amitrano, A. Coniglio and F. di Liberto, Phys. Rev. Lett. 57 (1987) 1016;
170
A. Coniglio
/ Is diffusion limited aggregation scale invariant?
T.C. Halsey, M.H. Jensen, L.P Kadanoff, I. Procaccia and B.I. Shraiman, Phys. Rev. A 33 (1986) 1141. 111 T. Vicsek, F. Family and P. Meakin Europhys. Lett. 12 (1990) 217. 121 L. de Arcangelis, S. Redner and A. Coniglio, Phys. Rev. B 31 (1985) 4725. 131 J. Lee, S. Schwarzer, A. Coniglio and H.E. Stanley, Phys. Rev. E (1993), to appear. 141 A. Coniglio and M. Zannetti, Europhys. Lett. 10 (1989) 575. 151 L. de Arcangelis and H.L. Hermann, J. Phys. A 23 (1990) L923.
Physica A 200 (1993) 165-170 North-Holland SDZ: 0378-4371(93)EO309-3
Is diffusion limited aggregation Antonio
scale invariant?
Coniglio
Dipartimento di Scienze Fisiche, Universitri di Napoli, Mostra d’oltremare 80125 Napoli, Italy
Pad. 19,
Recent work showing evidence for multiscaling in large two dimensional DLA clusters is reviewed. A connection between multiscaling and multifractality is also shown resulting in a weak localization of the growth sites (o-sites) in shells around the cluster seed.
One of the fundamental steps towards a complete theory in critical phenomena was the discovery of scaling laws. It was this discovery in fact which paved the way to the renormalization group. This theory is in fact based on the invariance of the thermodynamic quantities under an appropriate scale transformation. When the diffusion limited aggregation (DLA) model was proposed [l], it was natural to assume the same scaling laws valid in critical phenomena. In fact DLA shares with critical phenomena the common feature of a characteristic diverging length. However there are various differences from ordinary critical phenomena. For instance, DLA clusters show an interior frozen part and an external active periphery not present in other geometrical critical phenomena such as percolation. This feature of DLA is responsible for the poor performance of a real space renormalization group analysis, compared with analogous computations carried out in the percolation problem. In fact a different procedure for DLA in which the renormalization is limited to the mass in the frozen part, gives much better results [2]. The distinction between a frozen and a growth region was also stressed by Pietronero et al. [3], who have developed a theory of fractal growth which applies only to the frozen part. More recently it has been proposed that planar DLA does not have a simple self-similar geometry [4]. A more general concept of scale invariance was introduced [5] that may explain the various anomalies present in DLA and not in ordinary critical phenomena. This more general scale invariance leads to a multiscaling form for the density profile, which is the analog of the order parameter correlation function in critical phenomena. Defining the density profile as g(r, R) ddr = dM , Elsevier Science Publishers B.V.
166
A. Coniglio
I Is diffusion
limited aggregation
scale invariant?
where dM is the number of particles in the volume ddr at a distance r from the origin and R is the radius of gyration, the following multiscaling form was proposed [5] :
g(r, R) = ~-~+~(“~)c(r/R) ,
(1)
where C(x) is an amplitude that goes to zero rapidly for large x. Multiscaling contains a structure much richer than standard scaling. In the asymptotic regime, where r and R are both large with some fixed ratio x = r/R, the above form for the density profile predicts a different power law decay for each value of x and consequently a different local fractal dimensionality D(x) for each shell corresponding to that value of X. Specifically, moving from x = 0 toward increasingly larger values of x, D(X) varies from the fractal dimensionality D(0) of the inner frozen region to the fractal dimensionality of the outer shells in the growth region. The proposed behavior implies that DLA satisfies symmetries more complex than those found in ordinary critical phenomena. The density profile rescales by a factor which depends on x as opposed to standard scaling where D(x) = D independent of X. Knowledge of the symmetry of a problem is of fundamental importance in developing a theory. Then, if multiscaling is correct for DLA a complete theory should be based on such a multiscaling property, like in ordinary critical phenomena the renormalization group was based on standard scale invariance. The multiscaling form (1) has been tested for off lattice DLA up to lo5 particles [6] and more recently [7] for clusters of lo6 particles. D(x) was found to be roughly constant for small values of x, bending down at a value of x close to 1.5, with an overall behavior in agreement with the multiscaling picture. These result are in agreement with an independent calculation [8] recently done for off lattice DLA clusters up to one million particles. A single run of fifty million particles gave a rather inconclusive result. The plot of D(x) for the one million particles is given in fig. 1. To show the trend as a function of the number of particles the data for the two sets of measurements have been plotted. The unphysical bump, more pronounced in the set of lo5 particles, flattens out as the number of particles increases. In fact it can be shown [6] that asymptotically D(x) is a non-increasing function of x. It may be useful to characterize the behavior of D(X) for small values of x by an exponent p, D(x) -D(O)
Numerically
-
uxp.
it was found that p - 10 and a - l/2.
(2)
A. Coniglio
I Is diffusion limited aggregation scale invariant?
167
D(x) 2.0
1.0
1
I. 0.0
?
0.4
0.8
I.2
1.6
2.0
x
Fig. 1. Plot of D(x) versus n for (0) a set of 200 off-lattice DLA clusters of lo5 particles, a set of 66 off-lattice DLA clusters of lo6 particles. (After ref. [7].)
and (0)
The multiscaling form of the density profile g(r, R) implies a particular relation of mass versus radius involving a logarithmic prefactor. In fact from (1) by integrating g(r, R) over the volume we obtain the mass A4 as function of R. For large R then we find M-B(R)RD,
(3)
where D = D(0) is the fractal dimension of the frozen part and B(R) = (ln R)-D’p. The multiscaling form of the density profile (1) also implies multiscaling in the growth probability distribution P(r, M). This quantity was introduced by Plischke and Racz [9] and is defined in such a way that P(r, M) dr is the probability that the Mth particle added to the growing cluster sticks at a distance between r and r + dr from the origin. The multiscaling form on P(r, M) in turn implies that the fluctuation of the radius (width of growth region) approaches logarithmically the radius for large values of M, in agreement with numerical results [7,8]. The form of D(X) indicates that there is an inner region for x
168
A. Coniglio
I Is diffusion limited aggregation scale invariant?
the introduction of an appropriate measure on the fractal, and does not imply any condition on the symmetries#‘. For example in percolation there is multifractality with respect to the voltage distribution [12] of the random resistor network, nevertheless the pair connectedness function at criticality is invariant under standard scaling, and therefore does not obey multiscaling. This is the reason why standard renormalization group methods can be successfully applied to percolation. However, in the case of DLA the measure which gives multifractality is the growth probability {p,}, where pi is the probability that the perimeter site i will be the next to grow. The growth probabilities in turns characterizes the growth of the aggregate and therefore its geometrical structure. Thus multiscaling and multifractality in DLA may be linked together. In fact it was suggested [5] that multifractality combined with strong localization of the growth sites may imply multiscaling. In order to explore this connection Lee et al. [13] studied the joint distribution function N(a, X, M) where N((Y,x, M) da d_x is the number of perimeter sites with a-values in the range [(Y,CY+ da] and located in the annulus [x, x + dx] around the cluster seed. They showed that for large M the localization of the a-sites is peaked around a fixed value ,~(a) (fig. 2). Moreover they suggested on the basis of numerical results that the distribution N(a, x, M) obeys multiscaling and this is consistent with the multiscaling form of the density profile. Multiscaling in other systems. Multiscaling is manifested also in other models. In particular an exact solution [14] of the time dependent N = 00 GinzburgLandau model shows that the time dependent pair correlation function for a conserved order parameter exhibits a multiscaling form. Moreover, various models of invasion percolation have been studied numerically and the result of this analysis shows [15] that while ordinary invasion percolation does not exhibit a multiscaling form, the multiscaling form is found in the presence of a power law decay with the distance from the initial seed. In conclusion, we have given evidence that planar DLA clusters are invariant under a scaling transformation more complex than usually found in critical phenomena. Such scale invariance implies a multiscaling form (l), which is found not only in DLA but in other systems as well. I would like to thank the collaborators in this research: C. Amitrano, Meakin, J. Lee, S. Schwarzer, H.E. Stanley and M. Zannetti.
P.
#’ A different case is when multifractality is obtained with respect to the mass distribution as found in [ll]. In this case multifractality is directly related to a more complex geometry of the aggregate and is closer to multiscaling.
A. Coniglio
I Is diffusion
limited aggregation
scale invariant?
169
Fig. 2. Location of LYsites from 18 off DLA clusters of M = 20 000. In (a) we have 1.5 <(I < 1.9, in (b) 2.8 <(Y < 3.0, and in (c) (Y> 6. (After ref. [13].)
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo]
T.A. Witten and L.M. Sander, Phys. Rev. Lett. 47 (1981) 1400. C. Amitrano, A. Coniglio and F. di Liberto, J. Phys. A 21 (1988) L201. L. Pietronero, A. Erzan and C. Evertsz, Phys. Rev. Lett. 61 (1988) 861. B.B. Mandelbrot, Physica A 191 (1992) 95. A. Coniglio and M. Zannetti, Physica A 163 (1990) 325. C. Amitrano, A. Coniglio, P. Meakin and M. Zannetti, Phys. Rev. B 44 (1991) 4974. C. Amitrano, A. Coniglio, P. Meakin and M. Zannetti, submitted to Fractals (1993). P. Ossadnik, Physica A 195 (1993) 319. M. Plischke and Z. Racz, Phys. Rev. Lett. 53 (1984) 415. C. Amitrano, A. Coniglio and F. di Liberto, Phys. Rev. Lett. 57 (1987) 1016;
170
A. Coniglio
/ Is diffusion limited aggregation scale invariant?
T.C. Halsey, M.H. Jensen, L.P Kadanoff, I. Procaccia and B.I. Shraiman, Phys. Rev. A 33 (1986) 1141. 111 T. Vicsek, F. Family and P. Meakin Europhys. Lett. 12 (1990) 217. 121 L. de Arcangelis, S. Redner and A. Coniglio, Phys. Rev. B 31 (1985) 4725. 131 J. Lee, S. Schwarzer, A. Coniglio and H.E. Stanley, Phys. Rev. E (1993), to appear. 141 A. Coniglio and M. Zannetti, Europhys. Lett. 10 (1989) 575. 151 L. de Arcangelis and H.L. Hermann, J. Phys. A 23 (1990) L923.