Untangling Complex Systems, page 78
intensity, and it must be calculated for each point of the image. Then, the image can be partitioned
into subsets consisting of points where the local dimension is α. For a two-dimensional image,
points with α ≈ 2 are points where the intensity is regular. Points with α ≠ 2 lie in regions where
there are singularities. If α is much larger or much smaller than 2, the region is characterized by
discontinuities of the intensity. The value of f( α) gives the global information of the image. Points in a homogeneous region are abundant; therefore, they have a significant value of f( α). Edge points
f( α)
dbox
α
FIGURE 11.19 Shape of a multifractal spectrum f (α).
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395
have smaller f( α) values because they are rare. If many “edge” points, having α ≠ 2, are present, we are in the presence of a homogeneous textured image.
Multifractal analysis has revealed itself as a powerful tool in the analysis of complex medical
images for distinguishing healthy and pathological tissues (Lopes and Betrouni 2009); in geology
for exploring geochemical data that have rugged and singular spatial variability (Zuo and Wang
2016). Multifractals may also be useful for predicting earthquakes (Harte 2001). The intuitive moti-
vation is that earthquakes derive from fractal fractures of the terrestrial crust. Major fractures occur
along major faults, the most dramatic being the tectonic plate boundaries. Within major fault sys-
tems, there are smaller faults that branch off and form a self-similar hierarchy of networks. These
networks are often characterized by multifractal dimensions.
11.9.2 analysis of The comPlex Time series
Most of the time series that we encounter in the economy, biology, geology, and engineering (Tang
et al. 2015) display many singularities, which appear as step-like or cusp-like features. Such singulari-
ties may be treated as fractals having different dimensions α and weights f( α). For instance, it has been demonstrated that heart rate fluctuations of healthy individuals are multifractal, whereas congestive
heart failure, which is a life-threatening condition, leads to a loss of multifractality (Ivanov et al. 1999).
In economy, the classical financial theories conceive stock prices as moving according to a random
walk. Information digested by the market is the motor of the random movement of the prices. Since
traditional finance theory assumes that the arrival of news is random, and no one knows if it will be
good or bad news, then, the prices move randomly. However, because economies of the industrialized
countries tend to grow, on average, economists model the randomness of news as having a slight bias
toward good news. This model is referred to as the theory of random walk with drift. This theory can-
not account for the sudden substantial price changes and spikes that are frequent in specific periods.
According to Mandelbrot and Hundson (2004), markets are like roiling seas. Multifractals can describe
their turbulence. Multifractal simulations of the market activity provide estimates of the probability of
what the market might do and allow one to prepare for inevitable changes.
11.10 DIFFUSION IN FRACTALS
The diffusion exhibits a peculiar behavior when it occurs in fractals rather than in Euclidean spaces.
In Euclidean spaces, the root-mean-square distance R 2 1 2 traveled by a random walker is propor-
tional to ( Dt)1 2, where D is the diffusion coefficient (remind exercises 3.10 and 3.11). In a fractal, a random walker travels shorter distances after the same time interval because the path is winding,
having many looped and branched routes. Therefore, the root mean square distance is
1
1
R 2 2 ∝ ( Dt dw
) [11.13]
wherein dw > 2 (Havlin and Ben-Avraham 1987). Equation [11.13] means that the Brownian motion
on a fractal substrate, forcing the walker into many detours, is delayed: ( Dt)1 dw
( Dt)1 2
<
.
The probability of finding a random walker at position r in Euclidean space d , departing from
E
r = 0, and after time t is given by the Gaussian distribution (Rothschild 1998)
r 2
−
1
p( r, t) =
e 4 Dt [11.14]
dE
(4π Dt) 2
The Gaussian distribution [11.14] has zero mean and standard deviation σ = 2 Dt . The probability
of returning to the starting point p( ,
0 t) is proportional to (4
)− 2
π Dt dE . The smaller p( ,
0 t) and the
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Untangling Complex Systems
wider the random walker’s excursion. Hence, the reciprocal [ (
p ,
0 t)] 1
− is a measure of the spatial
excursion taken by the random walker. If the random walk occurs in a fractal having the inter-
nal dimension d , the spatial excursion will be proportional to ( t) dfr 2. But such excursion can be fr
expressed as the root mean square distance traveled in the fractal having dimension d:
d fr
d
d
[ (
p , t)]− ∝ ( t)
∝ R
∝ ( t ) dw
0
1
2
2 2
[11.15]
By comparing the time exponents in equation [11.15], we obtain the relationship between the expo-
nents d, d , and d :
w
fr
d
d fr = 2 [11.16]
dw
d is called the fractal dimension of the path of the anomalous random walk (Havlin and Ben-
w
Avraham 1987), whereas d is called fracton. d is also called spectral dimension of the fractal
fr
fr
having dimension d (Alexander and Orbach 1982) because the time-dependent probability of a ran-
dom walker to return to a certain, previously visited site (i.e., [ (
p , t)]− ∝ ( t) dfr
0
1
2), has a frequency-
dependent counterpart, which is the density of vibrational states of the low-frequency phonons
ρ ν
( ) ~ν dfr−1 [11.17]
The dynamics of acoustic phonons, which are longitudinal displacement modes, conceptually equiv-
alent to translational particle diffusion, are probed by vibrational spectroscopy using wavelengths
of the order of mm (i.e., wavenumbers of a few cm−1). If we use too long or too short wavelengths,
we cannot detect the details of the connectivity of a fractal. It is like trying to characterize a fractal
surface of an adsorbent material by exploiting molecules whose linear sizes are not comparable to
the irregularities that we want to probe.
Dimension d is related to the morphology of a fractal, whereas d and d are related to its topol-fr
w
ogy. The fracton includes both the static attributes of the medium, characterized by the fractal
dimension d and the dynamic aspects of the motion of the random walker, characterized by d .
w
For the whole class of random fractals (such as the dendritic DLA) embedded in Euclidean two- or
three-dimensions, dw > 2. Hence, based on equation [11.16], it is dfr < d < dE, and dfr is always ≈ 4 3
/
(Alexander and Orbach 1982).
The diffusion in a linear polymer corresponds to a self-avoiding random walk, and it has d fr = 1,
dw = 2 d, no matter what its fractal dimensionality is. For a branched polymer, d fr ≠ 1 . When the density of bridges is high enough, d fr equals d because dw = 2, independently of d (Helman et al. 1984).
TRY EXERCISE 11.15
11.11 CHEMICAL REACTIONS ON FRACTALS AND
FRACTAL-LIKE KINETICS IN CELLS
Usually, we assume that the chemical reactions occur in dilute solutions that are spatially homo-
geneous. However, when diffusion-limited reactions occur on fractals or dimensionally restricted
architectures, 15 the elementary chemical kinetics are quite different from those valid in traditional systems. In fact, in unconventional media, it has been found that, both phenomenologically and
theoretically, the rate law of either a homodimeric ( A + A → P) or heterodimeric ( A + B → P)
15 Examples of dimensionally restricted architectures are those encountered within a cell where chemical transformations may be confined to two-dimensional membranes or one-dimensional channels.
Chaos in Space
397
diffusion-limited reaction exhibits a characteristic reduction of the rate constant with time
(Kopelman 1988):
k k t h
=
−
0
[11.18]
In [11.18], k is the instantaneous rate coefficient, 16 t ≥ 1, and 0 ≤ h < 1. For a homodimeric reaction of the type
A + A → P [11.19]
the differential equation describing its anomalous evolution looks like
d[ A]
−
=
−
k
2
0 t h[ A] [11.20]
dt
After separating the variables and integrating between ( t = 0;[ A] = [ A
t;[ ]
A
0
] ) and (
), we obtain
1
1
k
0
1
( − )
t h
−
=
[11.21]
[ A
−
0
]
[ ]
A
( h 1)
When the homodimeric reaction occurs in a homogeneous medium or systems made homogenous
by effective stirring, h = 0, the equation [11.21] becomes the traditional integrated kinetic law with k as a constant.
Equation [11.20], having the rate coefficient dependent on time, is equivalent to a time-invariant
rate law with an increased kinetic order, n. In fact, from [11.21], if we assume that [ ]
A 1
−0 is negligible
respect to [ ]
A −1, we achieve
1
1 h −1 h
t ≈
−
[11.22]
k 0[ A]
The insertion of [11.22] into [11.20] yields
h
d[ A]
k 0
h
−
1 h
2+
−
= k
1
0
−
[ A
h
]
[11.23]
dt
1− h
It is evident that the reaction order n = 2 + ( h 1
( − h)) with respect to A exceeds 2. As the reaction
becomes increasingly diffusion-limited or dimensionally restricted, h increases, the rate constant
decreases more quickly with time (see equation [11.20]), and the kinetic order in the time-invariant
rate law [11.23] grows well beyond the molecularity of the reaction (Savageau 1995). When the
homodimeric reaction [11.19] takes place in a random fractal, embedded in a Euclidean two- or
three-dimensional space,
d
h
fr
= 1−
[11.24]
2
Introducing [11.24] in equation [11.23], we obtain
+
d A
2
1
− ∝ A dfr
[ ]
[11.25]
dt
16 The term “coefficient” rather than “constant” must be used when k depends on time.
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Untangling Complex Systems
Since d fr ≈ 4 / 3 for random fractals in two or three dimensions, the reaction order respect to A
results in n ≈ 2 5
. (Kopelman 1988). The reaction [11.19] may also proceed on catalytic islands
distributed on non-catalytic supports. Whether such “catalytic dust” is strictly fractal or is just
made of monodisperse islands, the result is anomalous fractal-like reaction kinetics. In particular,
for fractal dust, with 0 < d fr < 1, the reaction order with respect to A might assume large values:
3 < n < ∞.
TRY EXERCISE 11.16
The relevant role that a fractal structure may have on the chemical kinetics is also proved in the
nucleus of a eukaryotic cell. In the nucleus, the chromatin is a long polymer (each human cell
contains about two meters of DNA) that fills the available volume with a compact polymorphic
structure on which the transcription, replication, recombination, and repair of the genome occur.
The three-dimensional architecture of chromatin is crucial to the functioning of a cell. Chromatin
is a hierarchical structure spanning four spatial scales: from few nanometers to tens of microm-
eters. In fact, it is constituted by DNA (≈2 nm tick); nucleosomes (i.e., segments of DNA, each
one wrapping eight histone proteins and forming a complex ≈10 nm long); chromatins that are
fibers ≈30 nm high, generated by the nucleosomes that fold up; higher order chromatin loops
and coils ≈300 nm in length, and, finally, ≈1.4 μ m chromosome territories contained in the
nucleus ≈20 μm large. Chromatin is actually a fractal (Bancaud et al. 2012). Some areas of the
nucleus contain heterochromatin made of DNA packed tightly around histones. Other nuclear
areas contain euchromatin that is DNA loosely packed. Usually, genes in euchromatin are active,
whereas those in heterochromatin are inactive. The different activity is related to the different
fractal structure (Bancaud et al. 2009). In fact, euchromatin seems to have a higher fractal dimen-
sion ( d ≈ 2.6), exposing a broader and rougher surface to the proteins scanning for their target
sequences. Heterochromatin has a smaller fractal dimension ( d ≈ 2.2) because it is flatter and
smoother and has a less extended surface. The more accessible DNA surfaces in euchromatin can
be scanned more efficiently by nuclear factors favoring active transcription than in the hetero-
chromatin. These conclusions have been drawn by collecting the time evolution of luminescent
signals emitted by fluorescent proteins and markers (Bancaud et al. 2009). The decays of light-
emitted intensities ( I( t)) coming from fractal structures can be fitted by the stretched exponential
(or Kohlrausch or KWW) function:
t β
−
I( t) = I
τ
0 e [11.26]
where 0 < β ≤ 1, and τ is a parameter having, of course, the dimension of time (Berberan-Santos
et al. 2005). The stretched exponential function [11.26] is the integrated form of the first order dif-
ferential equation
dN
= − k( t) N [11.27]
dt
where N is the concentration of the excited luminophores, and k( t) is the time-dependent rate coefficient. When k is time-independent, we obtain the traditional exponential decay function [11.26] with
β = 1. The stretched exponential function is used to describe the luminescence decay collected for
any inhomogeneous, disordered medium in alternative to the maximum entropy method presented
in Appendix B. Moreover, it is suitable to fit the transient signal originated by Resonance Energy
Chaos in Space
399
Transfer (RET) phenomenon based on the electrostatic interaction between dipoles and/or multi-
poles. In the case of RET, β = d/ s, where d is the fractal or Euclidean dimension and s represents a parameter whose value depends on the RET mechanism. In fact, s is equal to 6, 8, or 10 for dipole-dipole, dipole-quadrupole, and quadrupole-quadrupole interactions, respectively.
11.12 POWER LAWS OR STRETCHED EXPONENTIAL FUNCTIONS?
If we look back at the equations appearing in this chapter, we might notice that many of them
are power laws. For example, the equation describing the relationship between the mass of a
dendritic fractal and its radius of gyration (equation [11.12]), or the relationship between the
kinetic constant of a dimeric elementary reaction and time (equation [11.18]). Actually, power
laws are widespread in science (West 2017): they may be encountered in many disciplines. In
biology, the basal metabolic rate of an organism ( R ), the rate at which its cells convert nutrients
E
to energy, scales with the organism’s body mass ( M) according to the following empirical power
law (Schmidt-Nielsen 1984):
3
R
4
E ∝ M [11.28]
Moreover, the shape of a growing biological organism is ruled by the empirical power law
y xb
∝ [11.29]
where x is the size of one part of the organism, whereas y is the size of another part, and b is a constant (Huxley 1932). The power law [11.29] describes the allometric morphogenesis, which is
the development of form or pattern by differential growth among the component parts. It has been
confirmed many times in biological context, but it is not confined only to organisms (Savageau
1979). Examples of other phenomena that obey power laws are the intensity of solar flares, the
