Untangling Complex Systems, page 77
D = lim
[11.10]7
l→0
1
log
l
FIGURE 11.13 The black curve and the grey spot covered by a grid of squares.
7 The dimension D, determined by the Box Counting method for a fractal-like structure that we find in nature, is named as its Hausdorff dimension.
Chaos in Space
389
2.4
Slope: D = 1.2
Correlation coefficient r = 0.996
2.1
( l)) N
log( 1.8
1.5
−0.4
−0.2
0.0
0.2
log(1/ l)
FIGURE 11.14 Determination of the dimension D of the Britain coast by the Box Counting method.8
The jagged seacoast shapes are examples of statistical self-similar fractals because every por-
tion can be considered a reduced-scale image of the whole, from a statistical point of view. The
contoured path of a coastline fills more space than a line, but less space than a plane, and therefore
its fractal dimension will be included between 1 and 2. Highly corrugated coastlines fill more space
and have fractal dimensions close to 2. Smooth coastlines have dimensions closer to 1. It is pos-
sible to measure its dimension by using the Box Counting method. Before the advent of computers,
cartographers have encountered many problems in measuring the length of the coasts and borders.
For example, the determination of the length of Britain’s coast was really tough because it is par-
ticularly complex. The coastline measured on a large-scale map was approximately half the length
of the coastline measured on a detailed map. The closer cartographers looked, the more detailed
and longer the coastline became. If we apply the Box Counting method, we find that the dimension
of Britain’s coast is D = ( .
1 2 ± .
0 )
1 (see Figure 11.14).
The value of the fractal dimension D has some positive correlation with the irregularity or jag-
gedness of a frontier. In fact, the smoothest coast on the atlas is that of South Africa, and its Box
Counting Dimension has resulted in being D = 1.02 (Mandelbrot 1967). On the other hand, the land
frontier between Spain and Portugal, which is quite irregular, has D = 1.14. The rougher the border,
the larger the D, because it is not a simple line and it has space-filling characteristics in the plane.
The Box Counting method can be applied not only to complex structures projected in two dimen-
sions but also to objects observed in volumetric space. Instead of using squares, we must use cubes
that form lattices. The possible values of the fractal dimension are 0 ≤ D ≤ 1 for the fractal dusts;
1 ≤ D ≤ 2 for the fractal lines; 2 ≤ D ≤ 3 for the fractal surfaces, and 3 ≤ D ≤ 4 for the fractal volumes.
TRY EXERCISE 11.12
11.7 A METHOD FOR GENERATING FRACTAL-LIKE STRUCTURES IN THE LAB
An easy way for generating fractal-like structures in a laboratory is to use the Hele-Shaw cell
(Vicsek 1988). The Hele-Shaw cell is a pair of transparent, rigid plates separated by a small
gap, typically 0.5 mm high,9 with a hole in the center of the top plate (see Figure 11.15).
8 Mandelbrot (1967) calculated a dimension D = 1.25 for the coast of Great Britain. The data in Figure 11.14 refers to a Box Counting method applied to a Great Britain printed on A4 paper, and by using six distinct grids with the sides of the squares being 3, 2.5, 2, 1.5, 1, and 0.5 cm long, respectively (see also exercise 11.12). A reasonable estimate of the uncertainty is 10%. In fact, if we apply the same procedure to the grey speckle of Figure 11.13, we find that its Box Counting Dimension is D = ( .
1 8 ± .
0 2).
9 Henry Selby Hele-Shaw (1854–1941) was an English engineer, who studied the flow of water around a ship’s hull by using two parallel flat plates separated by an infinitesimally small gap. You can build your own Hele-Shaw cell by using 30 × 30 cm plates made of glass or clear plastic, like plexiglass.
390
Untangling Complex Systems
FIGURE 11.15 Picture of a Hele-Shaw cell.
A high-viscosity fluid, such as glycerin or a 4% aqueous solution of polyvinyl alcohol (PVA),
fills the space between the plates. For generating patterns that are visible easily, a colored
aqueous solution must be injected by a plastic syringe connected to the hole through a rubber
tubing.10
The aqueous solution displaces the viscous fluid generating beautiful structures characterized by
fingers of different shapes and sizes. The phenomenon of viscous fingering has been studied for a
long time because it is relevant in engineering. For instance, the petroleum industry has been trying
consistently to find ways of inhibiting viscous fingering because it limits oil recovery in a porous
media. Viscous fingering is also important for the study of pattern formation in non-equilibrium
conditions when the interplay between microscopic interfacial dynamics and external macroscopic
forces plays a crucial role (Ben-Jacob and Garik 1990). Figure 11.16 shows just a few examples of the uncountable morphologies that can be obtained in the Hele-Shaw cell.
We can obtain either dense-branching morphologies (that consist of fingers of well-defined width
confined within a roughly circular envelope) like in (a) and (c) or dendritic morphologies like in (b),
(d), and (e). The splitting at the tip distinguishes the fingers from dendrites. The edge of the low- viscosity
fluid moves forward into the high-viscosity fluid because the pressure just behind the interface ( Pin)
is higher than that in the viscous fluid ( Pout) just in front of it. The speed at which the interface
advances depends on how steep this pressure gradient is ( Pin − Pout). When the interface randomly
develops bulges, the pressure gradients become steeper, rising to their highest values at the tips.
In fact, the excess pressure on the tip of a bulge is, according to the Laplace equation, ∆ P = 2γ / r, where γ is the surface tension and r is the curvature ray (see the sketch in Figure 11.17).
Therefore, a self-amplifying effect induces a faster growth of the bulges with respect to the other
parts of the interface. This positive feedback is counterbalanced by the surface tension that tends
to reduce the surface extension. When the viscous fluid has a limited solubility into the advancing
fluid, the negative effect exerted by the surface tension is relevant, and the pattern that sprouts, has
a dense-branching morphology, like in (a) and (c) of Figure 11.16. On the other hand, when the two fluids are perfectly miscible, like water and an aqueous polymer solution (for instance, 4% PVA made
10 The rubber tubing has an internal diameter of 2 mm.
Chaos in Space
391
(a)
(b)
(c)
(d)
(e)
FIGURE 11.16 Examples of dense-branching morphologies (a) and (c), obtained by using pure glycerin plus
a red-dye aqueous solution in (a), and red-colored glycerin plus air in (c). Examples of dendrites in (b), (d), and
(e) obtained by using 4% PVA reticulated by sodium borate plus a red-dye aqueous solution.
P
P
out
out
P ′ in
Viscous fluid
Pin
Pin
Inlet
Low-viscosity
propagating fluid
FIGURE 11.17 A circular interface between the viscous fluid and the low-viscosity propagating fluid having
a bulge generated by fluctuation.
392
Untangling Complex Systems
viscous by addition of a small amount of sodium borate11), the surface tension at the interphase is reduced at its minimum. Therefore, the bulges that sprout by fluctuations self-amplify easily. The
emerging patterns look like dendrites (like in b, d, and e of Figure 11.16).
TRY EXERCISE 11.13
11.8 DENDRITIC FRACTALS
Dendritic structures12 are spread throughout the animate and the inanimate worlds. The tree
branches, the plant roots, the branched projections of neurons, the alveoli in the lungs, the cell
colonies (for instance, cells of Bacillus subtilis, as shown by Matsushita et al. [1993]), the mineral
dendrites, the frost that forms on automobile windshields, dust, soot, and the dielectric breakdown
are just a few examples of dendritic shapes that we may find around us. 13 Dendritic structures (see
Figure 11.18) are statistically self-similar objects and have scale-invariant properties. For example, the density correlation function δ ( r), defined as the average density of units making up the object at
distance r from a point on the structure, is a measure of the average environment of a particle. δ ( r) is a scale-invariant function because it is a power law in r (Sander 1986):
−( − )
δ ( r) = kr dE d [11.11]
In [11.11], k is a constant, d is the Euclidean dimension of the space, and d is the fractal dimension.
E
δ ( r) of dendrites is a decreasing function of r: the average density decreases as the fractal becomes larger. For the determination of how the total mass M of the fractal scales with its radius of gyration
R, 14 we simply multiply equation [11.11], when r = R, by the volume RdE and we obtain: M R
KRd− dE RdE
( ) =
= KRd [11.12]
where K is another constant. For an ordinary curve, d = 1 because by doubling the length, the mass
doubles, too. For a disk, d = 2, and for a solid d = 3. For a dendrite projected on a plane, d is not an integer; it has been measured to be d ≈ 1
7
. .
Dendrites can be grouped in two sets, based on the process of their formation. There are den-
dritic structures that form by aggregation having “out-to-in” nature: particles come from outside
and join the growing cluster. But, there are also dendritic structures that grow from an “in-to-
out” dynamics, like those generated by viscous fingering or dielectric breakdown. Among the
“out-to-in” dendrites, many are generated by isotropic aggregation. Isotropic aggregation is the
process of suspended particles that move randomly and stick to one another to form connected
structures. It is different from the anisotropic aggregation that is a more constrained aggregation
where particles connect only in favored lattice directions. Basically, anisotropic aggregation is
crystallization and is responsible, for instance, of the dendritic six-branched structures of frost on
11 When Na B O is dissolved in water, it produces boric acid (H BO ) and borate ions (B(OH)−). If a small amount of 4
2 4
7
3
3
4
%
sodium borate solution is added to the 4% PVA, the borate ions bind to hydroxyl groups of PVA, promoting cross-linking among the macromolecules and increasing the viscosity of the polymer. The degree of cross-linking depends on the
amount of borate that is added (see the Figure in this note). The resulting viscous fluid is “Non-Newtonian” because its viscosity grows by increasing pressure.
OH OH OH OH OH
OH O
O
OH OH
B(OH) −
4
B
+ 4H2O
OH O
O
OH OH
OH OH OH OH OH
12 The word dendrite derives from the Greek δένδρον that means “tree.”
13 The ways the lichen spreads across rock, the malignant cells grow in tumors, the tea or coffee stains wet napkins, give rise to forms with ruffled, frond-like boundaries that radiate from a central focus. These boundaries are fractal (Ball 2009).
14 The radius of gyration is defined by R 2 = ( M ∑
)
( r
2
1
i − rc ) , where rc = (1 M ∑
)
r is the cluster’s center of mass.
i
i
i
Chaos in Space
393
FIGURE 11.18 Example of dendrite built by using the Diffusion Limited Aggregation model implemented in
the software NetLogo. (From Wilensky, U., and Rand, W., Introduction to Agent-Based Modeling: Modeling
Natural, Social and Engineered Complex Systems with NetLogo, MIT Press, Cambridge, MA, 2015.)
the windshield. The isotropic “out-to-in” aggregation is well described by the diffusion-limited
aggregation (DLA) model (Witten and Sander 1981, 1983). The initial condition is an immobile
particle (or “seed”) at the center of a disk. Then, at the boundary of the disk, a particle is released,
and it wanders by Brownian motion. Eventually, it hits the seed, and it sticks to it. Then, another
particle is released at the boundary, and it starts its random walk that stops when it hits the two-particles
cluster and sticks to it. This process repeats, again and … again. The possibility that the particles
rearrange after sticking, to find a more energetically favorable position, is excluded. The dendrite
in Figure 11.18 has been obtained by this model implemented in NetLogo (Wilensky and Rand
2015). It looks wispy and open because holes are formed and not filled up. In fact, a random
walker cannot wander down one of the channels in the cluster without getting stuck on the sides.
The growth of “in-to-out” dendritic shapes are triggered by fluctuations and are molded by two
forces: one negative and another positive. Fluctuations (due to the random motions of atoms and
molecules) on a propagating flat front may produce small bumps. The surface tension tends to sup-
press bumps smaller than a certain limit and smooths the interface. On the other hand, a positive
feedback process self-amplifies the small bumps producing larger bulges, and hence growing fin-
gers. In the case of viscous fingering, the positive feedback action is played by the stronger pressure
gradient that is present on top of a tip, as we learned in the previous paragraph (see Figure 11.17).
In the case of dielectric breakdown, which is the passage of a spark through an insulating material,
the electric field around the tips of a branching discharge is stronger than elsewhere. In the case of a
solidification process occurring in an undercooled liquid and initiated at a seed crystal, the positive
action is the temperature gradient that is steeper around the bulge than elsewhere; the latent heat
of solidification is shed around more rapidly at the tip of growing branches than on flat interfaces
(Langer 1980). When the positive self-amplifying action dominates, a dendritic shape is formed,
whereas when the negative force is overwhelming, a dense-branching morphology is produced.
TRY EXERCISE 11.14
394
Untangling Complex Systems
11.9 MULTIFRACTALS
Look at the dendrites shown in Figures 11.16 and 11.18. You may notice that the density of points on the periphery of the dendrites is much lower than that in the cores of them. A fractal structure, like
that of a dendrite, has a scale symmetry that is different from place to place. If we characterize a den-
drite by determining just one dimension, we give an approximate description of its features. As far
as such a complex fractal structure is concerned, it is more appropriate to define a distribution function
that describes how the dimension varies in it. Such structure is called multifractal. A convenient
way for quantifying the local variations in scale symmetry of a multifractal is the determination
of its pointwise fractal dimensions. In the fractal, F, we select a point xi and we consider a ball
of radius r centered in xi. We count the number of points that are inside the ball, N ( r), for r → 0.
i
We, then, define α = lim( lnNi( r) / ln( r)). We repeat the determination for many other points of the r→0
structure. Let Sα the subset of F consisting of all the points with a pointwise dimension α. If α is a typical scaling factor in F, Sα will be a large subset; on the other hand, if α is unusual, Sα will be small. Each set Sα is itself a fractal, so it is reasonable to measure its size by its fractal dimension,
f (α). f (α) is called the multifractal spectrum of F (Halsey et al. 1986). Therefore, a multifractal is an interwoven set of fractals of different dimensions α having f (α) as relative weights. The function f (α) is bell-shaped as shown in Figure 11.19. The maximum value of f (α) is the dimension calculated by the box counting method.
Multifractal structures arise in many nonlinear events and not just in DLA, viscous fingering
and dielectric breakdown. Multifractals may be encountered in both fully-developed turbulences
(Paladin and Vulpiani 1987) and fluid flows within random porous media (Stanley and Meakin
1988) and fractures in disordered solids (Coniglio et al. 1989) and chaotic attractors (Paladin and
Vulpiani 1987). The concept of multifractality is also used to describe and interpret complex images
and time series.
11.9.1 analysis of The comPlex images
Images that are discontinuous and fragmented are challenging to interpret. Such images may be
analyzed by using the multifractals theory. A grayscale image can be described by the distribution
of the intensity: bright areas have high densities, whereas dark areas have low densities. The local
dimension α describes the pointwise regularity of the objects by considering the distribution of
[11.10]7
l→0
1
log
l
FIGURE 11.13 The black curve and the grey spot covered by a grid of squares.
7 The dimension D, determined by the Box Counting method for a fractal-like structure that we find in nature, is named as its Hausdorff dimension.
Chaos in Space
389
2.4
Slope: D = 1.2
Correlation coefficient r = 0.996
2.1
( l)) N
log( 1.8
1.5
−0.4
−0.2
0.0
0.2
log(1/ l)
FIGURE 11.14 Determination of the dimension D of the Britain coast by the Box Counting method.8
The jagged seacoast shapes are examples of statistical self-similar fractals because every por-
tion can be considered a reduced-scale image of the whole, from a statistical point of view. The
contoured path of a coastline fills more space than a line, but less space than a plane, and therefore
its fractal dimension will be included between 1 and 2. Highly corrugated coastlines fill more space
and have fractal dimensions close to 2. Smooth coastlines have dimensions closer to 1. It is pos-
sible to measure its dimension by using the Box Counting method. Before the advent of computers,
cartographers have encountered many problems in measuring the length of the coasts and borders.
For example, the determination of the length of Britain’s coast was really tough because it is par-
ticularly complex. The coastline measured on a large-scale map was approximately half the length
of the coastline measured on a detailed map. The closer cartographers looked, the more detailed
and longer the coastline became. If we apply the Box Counting method, we find that the dimension
of Britain’s coast is D = ( .
1 2 ± .
0 )
1 (see Figure 11.14).
The value of the fractal dimension D has some positive correlation with the irregularity or jag-
gedness of a frontier. In fact, the smoothest coast on the atlas is that of South Africa, and its Box
Counting Dimension has resulted in being D = 1.02 (Mandelbrot 1967). On the other hand, the land
frontier between Spain and Portugal, which is quite irregular, has D = 1.14. The rougher the border,
the larger the D, because it is not a simple line and it has space-filling characteristics in the plane.
The Box Counting method can be applied not only to complex structures projected in two dimen-
sions but also to objects observed in volumetric space. Instead of using squares, we must use cubes
that form lattices. The possible values of the fractal dimension are 0 ≤ D ≤ 1 for the fractal dusts;
1 ≤ D ≤ 2 for the fractal lines; 2 ≤ D ≤ 3 for the fractal surfaces, and 3 ≤ D ≤ 4 for the fractal volumes.
TRY EXERCISE 11.12
11.7 A METHOD FOR GENERATING FRACTAL-LIKE STRUCTURES IN THE LAB
An easy way for generating fractal-like structures in a laboratory is to use the Hele-Shaw cell
(Vicsek 1988). The Hele-Shaw cell is a pair of transparent, rigid plates separated by a small
gap, typically 0.5 mm high,9 with a hole in the center of the top plate (see Figure 11.15).
8 Mandelbrot (1967) calculated a dimension D = 1.25 for the coast of Great Britain. The data in Figure 11.14 refers to a Box Counting method applied to a Great Britain printed on A4 paper, and by using six distinct grids with the sides of the squares being 3, 2.5, 2, 1.5, 1, and 0.5 cm long, respectively (see also exercise 11.12). A reasonable estimate of the uncertainty is 10%. In fact, if we apply the same procedure to the grey speckle of Figure 11.13, we find that its Box Counting Dimension is D = ( .
1 8 ± .
0 2).
9 Henry Selby Hele-Shaw (1854–1941) was an English engineer, who studied the flow of water around a ship’s hull by using two parallel flat plates separated by an infinitesimally small gap. You can build your own Hele-Shaw cell by using 30 × 30 cm plates made of glass or clear plastic, like plexiglass.
390
Untangling Complex Systems
FIGURE 11.15 Picture of a Hele-Shaw cell.
A high-viscosity fluid, such as glycerin or a 4% aqueous solution of polyvinyl alcohol (PVA),
fills the space between the plates. For generating patterns that are visible easily, a colored
aqueous solution must be injected by a plastic syringe connected to the hole through a rubber
tubing.10
The aqueous solution displaces the viscous fluid generating beautiful structures characterized by
fingers of different shapes and sizes. The phenomenon of viscous fingering has been studied for a
long time because it is relevant in engineering. For instance, the petroleum industry has been trying
consistently to find ways of inhibiting viscous fingering because it limits oil recovery in a porous
media. Viscous fingering is also important for the study of pattern formation in non-equilibrium
conditions when the interplay between microscopic interfacial dynamics and external macroscopic
forces plays a crucial role (Ben-Jacob and Garik 1990). Figure 11.16 shows just a few examples of the uncountable morphologies that can be obtained in the Hele-Shaw cell.
We can obtain either dense-branching morphologies (that consist of fingers of well-defined width
confined within a roughly circular envelope) like in (a) and (c) or dendritic morphologies like in (b),
(d), and (e). The splitting at the tip distinguishes the fingers from dendrites. The edge of the low- viscosity
fluid moves forward into the high-viscosity fluid because the pressure just behind the interface ( Pin)
is higher than that in the viscous fluid ( Pout) just in front of it. The speed at which the interface
advances depends on how steep this pressure gradient is ( Pin − Pout). When the interface randomly
develops bulges, the pressure gradients become steeper, rising to their highest values at the tips.
In fact, the excess pressure on the tip of a bulge is, according to the Laplace equation, ∆ P = 2γ / r, where γ is the surface tension and r is the curvature ray (see the sketch in Figure 11.17).
Therefore, a self-amplifying effect induces a faster growth of the bulges with respect to the other
parts of the interface. This positive feedback is counterbalanced by the surface tension that tends
to reduce the surface extension. When the viscous fluid has a limited solubility into the advancing
fluid, the negative effect exerted by the surface tension is relevant, and the pattern that sprouts, has
a dense-branching morphology, like in (a) and (c) of Figure 11.16. On the other hand, when the two fluids are perfectly miscible, like water and an aqueous polymer solution (for instance, 4% PVA made
10 The rubber tubing has an internal diameter of 2 mm.
Chaos in Space
391
(a)
(b)
(c)
(d)
(e)
FIGURE 11.16 Examples of dense-branching morphologies (a) and (c), obtained by using pure glycerin plus
a red-dye aqueous solution in (a), and red-colored glycerin plus air in (c). Examples of dendrites in (b), (d), and
(e) obtained by using 4% PVA reticulated by sodium borate plus a red-dye aqueous solution.
P
P
out
out
P ′ in
Viscous fluid
Pin
Pin
Inlet
Low-viscosity
propagating fluid
FIGURE 11.17 A circular interface between the viscous fluid and the low-viscosity propagating fluid having
a bulge generated by fluctuation.
392
Untangling Complex Systems
viscous by addition of a small amount of sodium borate11), the surface tension at the interphase is reduced at its minimum. Therefore, the bulges that sprout by fluctuations self-amplify easily. The
emerging patterns look like dendrites (like in b, d, and e of Figure 11.16).
TRY EXERCISE 11.13
11.8 DENDRITIC FRACTALS
Dendritic structures12 are spread throughout the animate and the inanimate worlds. The tree
branches, the plant roots, the branched projections of neurons, the alveoli in the lungs, the cell
colonies (for instance, cells of Bacillus subtilis, as shown by Matsushita et al. [1993]), the mineral
dendrites, the frost that forms on automobile windshields, dust, soot, and the dielectric breakdown
are just a few examples of dendritic shapes that we may find around us. 13 Dendritic structures (see
Figure 11.18) are statistically self-similar objects and have scale-invariant properties. For example, the density correlation function δ ( r), defined as the average density of units making up the object at
distance r from a point on the structure, is a measure of the average environment of a particle. δ ( r) is a scale-invariant function because it is a power law in r (Sander 1986):
−( − )
δ ( r) = kr dE d [11.11]
In [11.11], k is a constant, d is the Euclidean dimension of the space, and d is the fractal dimension.
E
δ ( r) of dendrites is a decreasing function of r: the average density decreases as the fractal becomes larger. For the determination of how the total mass M of the fractal scales with its radius of gyration
R, 14 we simply multiply equation [11.11], when r = R, by the volume RdE and we obtain: M R
KRd− dE RdE
( ) =
= KRd [11.12]
where K is another constant. For an ordinary curve, d = 1 because by doubling the length, the mass
doubles, too. For a disk, d = 2, and for a solid d = 3. For a dendrite projected on a plane, d is not an integer; it has been measured to be d ≈ 1
7
. .
Dendrites can be grouped in two sets, based on the process of their formation. There are den-
dritic structures that form by aggregation having “out-to-in” nature: particles come from outside
and join the growing cluster. But, there are also dendritic structures that grow from an “in-to-
out” dynamics, like those generated by viscous fingering or dielectric breakdown. Among the
“out-to-in” dendrites, many are generated by isotropic aggregation. Isotropic aggregation is the
process of suspended particles that move randomly and stick to one another to form connected
structures. It is different from the anisotropic aggregation that is a more constrained aggregation
where particles connect only in favored lattice directions. Basically, anisotropic aggregation is
crystallization and is responsible, for instance, of the dendritic six-branched structures of frost on
11 When Na B O is dissolved in water, it produces boric acid (H BO ) and borate ions (B(OH)−). If a small amount of 4
2 4
7
3
3
4
%
sodium borate solution is added to the 4% PVA, the borate ions bind to hydroxyl groups of PVA, promoting cross-linking among the macromolecules and increasing the viscosity of the polymer. The degree of cross-linking depends on the
amount of borate that is added (see the Figure in this note). The resulting viscous fluid is “Non-Newtonian” because its viscosity grows by increasing pressure.
OH OH OH OH OH
OH O
O
OH OH
B(OH) −
4
B
+ 4H2O
OH O
O
OH OH
OH OH OH OH OH
12 The word dendrite derives from the Greek δένδρον that means “tree.”
13 The ways the lichen spreads across rock, the malignant cells grow in tumors, the tea or coffee stains wet napkins, give rise to forms with ruffled, frond-like boundaries that radiate from a central focus. These boundaries are fractal (Ball 2009).
14 The radius of gyration is defined by R 2 = ( M ∑
)
( r
2
1
i − rc ) , where rc = (1 M ∑
)
r is the cluster’s center of mass.
i
i
i
Chaos in Space
393
FIGURE 11.18 Example of dendrite built by using the Diffusion Limited Aggregation model implemented in
the software NetLogo. (From Wilensky, U., and Rand, W., Introduction to Agent-Based Modeling: Modeling
Natural, Social and Engineered Complex Systems with NetLogo, MIT Press, Cambridge, MA, 2015.)
the windshield. The isotropic “out-to-in” aggregation is well described by the diffusion-limited
aggregation (DLA) model (Witten and Sander 1981, 1983). The initial condition is an immobile
particle (or “seed”) at the center of a disk. Then, at the boundary of the disk, a particle is released,
and it wanders by Brownian motion. Eventually, it hits the seed, and it sticks to it. Then, another
particle is released at the boundary, and it starts its random walk that stops when it hits the two-particles
cluster and sticks to it. This process repeats, again and … again. The possibility that the particles
rearrange after sticking, to find a more energetically favorable position, is excluded. The dendrite
in Figure 11.18 has been obtained by this model implemented in NetLogo (Wilensky and Rand
2015). It looks wispy and open because holes are formed and not filled up. In fact, a random
walker cannot wander down one of the channels in the cluster without getting stuck on the sides.
The growth of “in-to-out” dendritic shapes are triggered by fluctuations and are molded by two
forces: one negative and another positive. Fluctuations (due to the random motions of atoms and
molecules) on a propagating flat front may produce small bumps. The surface tension tends to sup-
press bumps smaller than a certain limit and smooths the interface. On the other hand, a positive
feedback process self-amplifies the small bumps producing larger bulges, and hence growing fin-
gers. In the case of viscous fingering, the positive feedback action is played by the stronger pressure
gradient that is present on top of a tip, as we learned in the previous paragraph (see Figure 11.17).
In the case of dielectric breakdown, which is the passage of a spark through an insulating material,
the electric field around the tips of a branching discharge is stronger than elsewhere. In the case of a
solidification process occurring in an undercooled liquid and initiated at a seed crystal, the positive
action is the temperature gradient that is steeper around the bulge than elsewhere; the latent heat
of solidification is shed around more rapidly at the tip of growing branches than on flat interfaces
(Langer 1980). When the positive self-amplifying action dominates, a dendritic shape is formed,
whereas when the negative force is overwhelming, a dense-branching morphology is produced.
TRY EXERCISE 11.14
394
Untangling Complex Systems
11.9 MULTIFRACTALS
Look at the dendrites shown in Figures 11.16 and 11.18. You may notice that the density of points on the periphery of the dendrites is much lower than that in the cores of them. A fractal structure, like
that of a dendrite, has a scale symmetry that is different from place to place. If we characterize a den-
drite by determining just one dimension, we give an approximate description of its features. As far
as such a complex fractal structure is concerned, it is more appropriate to define a distribution function
that describes how the dimension varies in it. Such structure is called multifractal. A convenient
way for quantifying the local variations in scale symmetry of a multifractal is the determination
of its pointwise fractal dimensions. In the fractal, F, we select a point xi and we consider a ball
of radius r centered in xi. We count the number of points that are inside the ball, N ( r), for r → 0.
i
We, then, define α = lim( lnNi( r) / ln( r)). We repeat the determination for many other points of the r→0
structure. Let Sα the subset of F consisting of all the points with a pointwise dimension α. If α is a typical scaling factor in F, Sα will be a large subset; on the other hand, if α is unusual, Sα will be small. Each set Sα is itself a fractal, so it is reasonable to measure its size by its fractal dimension,
f (α). f (α) is called the multifractal spectrum of F (Halsey et al. 1986). Therefore, a multifractal is an interwoven set of fractals of different dimensions α having f (α) as relative weights. The function f (α) is bell-shaped as shown in Figure 11.19. The maximum value of f (α) is the dimension calculated by the box counting method.
Multifractal structures arise in many nonlinear events and not just in DLA, viscous fingering
and dielectric breakdown. Multifractals may be encountered in both fully-developed turbulences
(Paladin and Vulpiani 1987) and fluid flows within random porous media (Stanley and Meakin
1988) and fractures in disordered solids (Coniglio et al. 1989) and chaotic attractors (Paladin and
Vulpiani 1987). The concept of multifractality is also used to describe and interpret complex images
and time series.
11.9.1 analysis of The comPlex images
Images that are discontinuous and fragmented are challenging to interpret. Such images may be
analyzed by using the multifractals theory. A grayscale image can be described by the distribution
of the intensity: bright areas have high densities, whereas dark areas have low densities. The local
dimension α describes the pointwise regularity of the objects by considering the distribution of
