Physica A 168 (1990) 490-497
North-Holland
MASS MULTIFRACTALS
Tamfis V I C S E K
Institute for Technical Physics, Budapest, P.O. Box. 76, 1325 Hungary
In this paper the concept of mass multifractality is discussed and demonstrated on
deterministic and stochastic aggregation models of fractal growth. It is shown that in these
models the scaling of the density distribution of particles can be described in terms of an
infinite hierarchy of generalized dimensions. This behaviour can be observed on a length scale
which is much larger than the particle size, but is much smaller than the cluster size.
Numerical evidence is given that the distribution of mass within diffusion-limitedaggregates is
a multifractal.
1. Introduction
It has been demonstrated by numerous recent studies that far-from equilibrium growth p h e n o m e n a typically lead to fractal structures (see e.g., refs.
[1-3]). In this p a p e r we address the question whether the geometry of such
objects can be fully described in terms of ordinary self-similar fractals or, on
the contrary, the features of the given growth processes have their specific
impact on the structure. For example, one is interested in various scaling
properties of the mass distribution within the growth patterns other than the
isotropic power law decay characteristic for the simplest types of mathematical
fractals generated by recursion [4].
A much studied prototype of fractal growth is the diffusion-limited motion of
interfaces when the a d v a n c e m e n t of the surface of the evolving structure is
governed by a variable which in some approximations satisfies the Laplace
equation. The so-called diffusion-limited aggregation ( D L A ) model [5] has
been shown to capture the essential features of these processes and in the
following, when discussing the geometry of growing fractals in nature, we shall
use as an example the D L A model and concentrate on the density distribution
within such clusters.
Before presenting our related results for diffusion-limited aggregates (section
4), however, the concept of mass multifractality will be discussed and demonstrated on a deterministic model of growing clusters (section 2) and the method
of calculating the corresponding exponents will be described (section 3).
0378-4371/90/$03.50 © 1990- Elsevier Science Publishers B.V. (North-Holland)
T. Vicsek / Massmultifractals
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2. Mass (geometrical) multifractality
The formalism and the quantities describing mass multiffactals is given here
strictly for growing fractals with a well defined lower cutoff length scale (it is
assumed that the objects under study are clusters made of identical particles).
In the case of recursively generated finite fractals in the infinitely ramified limit
interesting conceptual questions arise, which are less unambiguous than in the
growing case and will be briefly discussed later.
The concept of mass (geometrircal) multifractality [6] is analogous to that of
the ordinary multifractal distributions with the important difference that we
assume the measure to be homogeneously spread over the growing cluster. In
this way it is only the number of particles (or the Lebesgue measure of the
structure) which contributes to the amount of measure within a given region on
the fractal. This is only one of the many possible measures one can define on a
fractal support, but since it corresponds to the mass distribution within the
fractal it is a natural choice from the point of view of a physicist.
In order to define the relevant quantities let us assume that the cluster, made
of particles of diameter a and having an extension L, is covered with a grid of
boxes of linear size l. It is required that 1 ~ L and a ~ I. According to our
experience, in order to see mass multifractality, L and l have to be at least two
orders of magnitude larger than 1 and a, respectively. This means that the
clusters must be extremely large.
T o characterize the mass distribution we determine M i which is the number
of particles in the ith box. Knowing the set of M i values one can calculate the
quantity N(M) which is the number of boxes with mass M. Assume that we
plot In N(M) versus In M/M o for various 1. If these histograms fall onto the
same universal (size independent) curve after rescaling both coordinates by a
factor ln(l/L) [7], the structure is a mass (geometrical) multifractal. From the
formalism for fractal measures [8-12] it follows that for multifractals there
exists an infinite hierarchy of generalized dimensions Dq, which for the mass
distribution are defined by
~ \Mo/
(1)
where the normalization factor M 0 is the total mass of the cluster.
Next we use a deterministic cluster model to demonstrate the above scaling
on the example of a mathematical fractal. Deterministic models based on
constructing a fractal using recursion represent a useful tool in studying fractal
growth because they allow for exact treatment of a number of properties [13].
Let us consider the following example for the construction of an aggregate
T. Vicsek / Mass multifractals
492
o
Fig. 1. The first steps of the construction of a growing mass multifractal embedded into the d = 2
dimension.
type mass multifractal embedded into two dimensions (fig. 1). The rule is quite
simple: In the kth step the twice enlarged version of the configuration
corresponding to the (k - 1)th stage of growth is added to the four corners of
the already existing cluster. The mathematical fractal is obtained in the k--->
limit. To calculate the generalized dimensions defined by (1) for such a fractal
one can use the expression
5
rj(q-1)°qm iq =(1)q5(q-1)°q + 4( 4 )q( ~ ) (q-1)oq
= 1,
(2)
j=l
which follows from similarity arguments [7, 10]. Here the values rj = 5 and 5
express that the complete fractal is made of 5 parts, one of which is a scaled
down version of it by a factor 5, while the same factor for the other four parts
is equal to 5. On the other hand, the weights (mj) correspond to the fact that
the related masses are 1 and ~7 times less than the total mass.
The above equation can be solved for general q numerically, while in the
limits q---> ___ooit provides the explicit expressions D~ = In ~ / I n ~ = 1.579 and
D_~ = In 17/In 5 -----1.760. For -oo < q < ~ the corresponding Dq'S a r e between
the above limiting values.
For mathematical fractals with no lower cutoff length scale the situation is
rather delicate. In the formalism presented here for growing fractals all masses
are measured using the smallest unit, which we call particle. When this unit
becomes equal to zero (this is the case when the limiting set of finite
mathematical fractals is attained) it is less straightforward to provide a
physically plausible picture. Nevertheless, the above defined quantities should
exist in that limiting case as well. Furthermore, in the physically relevant
situations the condition of the existence of a smallest unit is always satisfied.
T. Vicsek / Mass multifractals
493
3. Generalized sand box method
To answer the question whether a stochastic fractal has a multifractal mass
distribution, one has to determine the generalized dimensions numerically.
When one tries to calculate Dq f r o m expression (1) for q < 0 the boxes which
contain a small number of particles (because they barely overlap with the
cluster) give an anomalously large contribution to the sum in the left-hand side
of (1) and, consequently, it is not possible to get reliable results for this case.
This problem can be solved by using the generalized sand box method [14],
which is based on studying the average of the masses M(R) (and their qth
moments) within boxes of size R with the centers of the boxes randomly
distributed on the fractal as is schematically shown in fig. 2. Thus, the question
is reduced to how (Mq(R)) scales with increasing R, where ( . . . ) denotes the
average over the centers. To use (1) in this context one first notes that the sum
in (1) corresponds to taking an average of the quantity (Mi/Mo) q-1 according
to the probability distribution Mi/M o. When the centers of the boxes are
chosen randomly the averaging is made over this distribution, and therefore
M(R)) q-1) ~ ( n )
((--~0
(ql,Dq .
(3)
Since in this method the boxes are centered on the fractal there will be no
boxes with too few particles in them and the scaling of the negative moments
can also be investigated.
Fig. 2. Schematic illustration of the generalized sand box method. In this approach the number of
particles within boxes of increasing size (R) is determined. The boxes are centered at randomly
chosen particles belonging to the cluster.
494
T. Vicsek / Mass multifractals
4. Mass multifractality of DLA clusters
The structure of diffusion-limited aggregates has been found to be much
more complex than that of the simplest isotropic fractals. In addition to lattice
effects [15], the density correlations in off-lattice aggregates have also been
shown to decay anisotropically [16]. Anomalous behaviour of the number of
boxes of size r containing a given number of particles was found in a study of
the mass distribution within diffusion-limited aggregates [17].
It is a natural idea to check whether the mass distribution in D L A clusters
can be described by a single exponent or it is a multifractal. The main
motivations to think that the latter case is more likely are the following: (i) A
"monofractal" distribution would be a rather special case of a more general,
possible distribution; (ii) more importantly, the density of particles in a cluster
is determined by the growth probability distribution, which is known to be a
multifractal [18-20].
We calculated the Dqspectrum using five off-lattice D L A clusters generated
in two dimensions each consisting of one million particles [21]. When applying
the above-described generalized sand box method we randomly picked 5 × 10 4
particles from the central 3000 × 3000 lattice unit region of the clusters and
determined the number of particles M(R) within the square shaped boxes of
size R centered at the selected particles. Then we calculated ( [ M ( R ) / M o ] q - 1 ) ,
the average of the ( q - 1)th m o m e n t of the normalized number of particles as
a function of the box size, and plotted its logarithm divided by (q - 1) versus
In(R/L) for various q values.
Fig. 3 shows a typical set of results, where the scaling of the moments of the
box masses is demonstrated for q = - 1 0 , q = 0 and q = 10. It is clear from this
figure that the various moments scale differently, although the slopes of the
straight lines (corresponding to t h e Dq'S) fitted to the data are rather similar.
However, one does not expect them to be very different, since mass multifractals embedded into two dimensions must satisfy the condition 1 ~< Dq <~2.
Consequently, the structure of D L A is described by an infinite set of mass
exponents. The overall dependence o f Dq on q is consistent with the expected
behavior, namely, it exhibits a monotonic decay with growing q.
5. Discussion
In section 2 it was demonstrated that recursive growing fractals may have a
multifractal mass distribution. To see whether this situation can occur in
physically motivated models we calculated the generalized dimensions corre-
495
T. Vicsek / Mass multifractals
-3'
-4
-5
q=lO
6.
7
-6
-7
N
-8i
10
-5
-4
-3
-2
-1
in(R/L)
Fig. 3. Estimation of Dq, the generalized dimension associated with the mass distribution, for
q = - 1 0 , 0 and 10 from the slopes of the plots in([Mo/M(R)] l - q ) / ( q - 1) versus l n ( R / L ) . These
data were obtained for two off-lattice D L A clusters of one million particles each, and for 50000
randomly selected sand box centers.
sponding to the distribution of mass in D L A clusters and found a hierarchy of
exponents.
To test our results for D L A [21], we calculated the Dq spectrum for two
other kinds of models for which the mass distribution is well understood.
(i) In the first simulation we determined the generalized dimensions for
randomly distributed particles in a 3000 x 3000 cell. We found that after a
transient for small sizes the box masses scale with essentially the same
exponent D = 2 for different q values.
(ii) The second simulation was m a d e to test whether an apparent multifractal mass distribution is obtained for an object which is known to be described
by a single fractal dimension. We carried out the calculations for the Sierpinski
gasket, which is a simple deterministic fractal having a unique fractal dimension. The results are given in fig. 4, where it is demonstrated that the slopes of
the data sets (corresponding t o Dq) are indistinguishable, indicating that in
contrast to D L A clusters our analysis in this case detects a monofractal
distribution of mass.
We conclude that neither randomness nor a purely monofractal behavior
seems to be responsible for our result for the mass distribution in D L A
clusters.
An interesting consequence of fig. 3 is that Dq= 2 # Dq=0, where Oq=o is the
496
T. Vicsek / Mass multifractals
-2
-4'
"~
7
-6'
-8'
slope--1.595
-10
-12
-7
-6
-5
-4
•
q = -10
•
q=0
•
q=10
-3
-2
-1
0
tn(R/L)
Fig. 4. The same as fig. 3, but for the Sierpinski gasket, which is known to be described by a single
fractal dimension. This figure demonstrates that a monofractal distribution of particles does not
lead to the multifractal mass distribution we observed for DLA clusters.
true fractal dimension D of the aggregates. This is significant because the
standard methods for the determination of the fractal dimension (such as the
radius of gyration method or the calculation of the density-density correlation
function) yield O q= 2.
Acknowledgements
Parts of the research presented here were carried out in collaboration with F.
Family, P. Meakin and T. T61. Helpful discussions with C. Evertsz and J. Feder
are also acknowledged.
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T. Vicsek / Mass multifractals
497
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