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Fractal Structures in Condensed Matter Physics

Fractal Structuresin Condensed Matter Physics
TSUNEYOSHI N AKAYAMA
Toyota Physical and Chemical Research Institute,
Nagakute, Japan
Article Outline
GlossaryDeﬁnition of the Subject
Introduction
Determining Fractal Dimensions
Polymer Chains in Solvents
Aggregates and Flocs
Aerogels
Dynamical Properties of Fractal Structures
Spectral Density of States and Spectral Dimensions
Future Directions
Bibliography
Glossary
Anomalous diﬀusion It is well known that the meansquare displacement hr2 (t)i of a diﬀusing particle ona uniform system is proportional to the time t suchFractal Structures in Condensed Matter Physics

as hr2 (t)i  t. This is called normal diﬀusion. Particles on fractal networks diﬀuse more slowly comparedwith the case of normal diﬀusion. This slow diﬀusion
called anomalous diﬀusion follows the relation given
by hr2 (t)i  t a , where the condition 0 < a < 1 always
holds.
Brownian motion Einstein published the important paper in 1905 opening the way to investigate the movement of small particles suspended in a stationary liquid, the so-called Brownian motion, which stimulated
J. Perrin in 1909 to pursue his experimental work conﬁrming the atomic nature of matter. The trail of a random walker provides an instructive example for understanding the meaning of random fractal structures.
Fractons Fractons, excitations on fractal elastic-networks, were named by S. Alexander and R. Orbach
in 1982. Fractons manifest not only static propertiesof fractal structures but also their dynamic properties. These modes show unique characteristics such as
strongly localized nature with the localization length ofthe order of wavelength.
Spectral density of states The spectral density of states ofordinary elastic networks are expressed by the Debye spectral density of states given by D(!)  ! d1 ,where d is the Euclidean dimensionality. The spectral density of states of fractal networks is given by
D(!)  ! d s 1 , where ds is called the spectral or fracton dimension of the system.
Spectral dimension This exponent characterizes the
spectral density of states for vibrational modes excited on fractal networks. The spectral dimension
constitutes the dynamic exponent of fractal networks
together with the conductivity exponent and the exponent of anomalous diﬀusion.
Definition of the Subject
The idea of fractals is based on self-similarity, which isa symmetry property of a system characterized by invariance under an isotropic scale-transformation on certain
length scales. The term scale-invariance has the implication that objects look the same on diﬀerent scales of observations. While the underlying concept of fractals is quite
simple, the concept is used for an extremely broad rangeof topics, providing a simple description of highly complex structures found in nature. The term fractal was ﬁrst
introduced by Benoit B. Mandelbrot in 1975, who gave
a deﬁnition on fractals in a simple manner “A fractal isa shape made of parts similar to the whole in some way”.
Thus far, the concept of fractals has been extensively used
to understand the behaviors of many complex systems orF

has been applied from physics, chemistry, and biology for
applied sciences and technological purposes. Examples offractal structures in condensed matter physics are numerous such as polymers, colloidal aggregations, porous media, rough surfaces, crystal growth, spin conﬁgurations ofdiluted magnets, and others. The critical phenomena ofphase transitions are another example where self-similarity plays a crucial role. Several books have been publishedon fractals and reviews concerned with special topics on
fractals have appeared.
Length, area, and volume are special cases of ordinary Euclidean measures. For example, length is the measure of a one-dimensional (1d) object, area the measure ofa two-dimensional (2d) object, and volume the measure ofa three-dimensional (3d) object. Let us employ a physicalquantity (observable) as the measure to deﬁne dimensions
for Euclidean systems, for example, a total mass M(r) ofa fractal object of the size r. For this, the following relation
should holdr / M(r)1/d ;(1)where d is the Euclidean dimensionality. Note that Euclidean spaces are the simplest scale-invariant systems.
We extend this idea to introduce dimensions for selfsimilar fractal structures. Consider a set of particles withunit mass m randomly distributed on a d-dimensional Euclidean space called the embedding space of the system.
Draw a sphere of radius r and denote the total mass of particles included in the sphere by M(r). Provided that the following relation holds in the meaning of statistical averagesuch asr / hM(r)i1/D f ;(2)

where h: : :i denotes the ensemble-average over diﬀerent
spheres of radius r, we call Df the similarity dimension. It
is necessary, of course, that Df is smaller than the embedding Euclidean dimension d. The deﬁnition of dimension
as a statistical quantity is quite useful to specify the characteristic of a self-similar object if we could choose a suitablemeasure.
There are many deﬁnitions to allocate dimensions.
Sometimes these take the same value as each other andsometimes not. The capacity dimension is based on the
coverage procedure. As an example, the length of a curved
line L is given by the product of the number N of straightline segment of length r needed to step along the curvefrom one end to the other such as L(r) D N(r)r. While,
the area S(r) or the volume V(r) of arbitrary objects canbe measured by covering it with squares or cubes of linear

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Fractal Structures in Condensed Matter Physics

size r. The identical relation,M(r) / N(r)rd(3)

should hold for the total mass M(r) as measure, for example. If this relation does not change as r ! 0, we have therelation N(r) / rd . We can extend the idea to deﬁne thedimensions of fractal structures such as
N(r) / rD f ;(4)from which the capacity dimension Df is given by
Df :D limln N(r)r!0 ln(1/r):(5)

The deﬁnition of Df can be rendered in the following implicit formlim N(r)r D f D const :

r!0

(6)

Equation (5) brings out a key property of the Hausdorﬀ
dimension [10], where the product N(r)r D f remains ﬁnite
as r ! 0. If Df is altered even by an inﬁnitesimal amount,
this product will diverge either to zero or to inﬁnity. The
Hausdorﬀ dimension coincides with the capacity dimension for many fractal structures, although the Hausdorﬀ
dimension is deﬁned less than or equal to the capacity dimension. Hereafter, we refer to the capacity dimension or
the Hausdorﬀ dimension mentioned above as the fractal
dimension.
Introduction
Fractal structures are classiﬁed into two categories; deterministic fractals and random fractals. In condensed matter physics, we encounter many examples of random fractals. The most important characteristic of random fractals is the spatial and/or sample-to-sample ﬂuctuations intheir properties. We must discuss their characteristics byaveraging over a large ensemble. The nature of deterministic fractals can be easily understood from some examples. An instructive example is the Mandelbrot–Given fractal [12], which can be constructed by starting with a structure with eight line segments as shown in Fig. 1a (theﬁrst stage of the Mandelbrot–Given fractal). In the second stage, each line segment of the initial structure is replaced by the initial structure itself (Fig. 1b). This process is repeated indeﬁnitely. The Mandelbrot–Given fractal possesses an obvious dilatational symmetry, as seenfrom Fig. 1c, i. e., when we magnify a part of the structure,
the enlarged portion looks just like the original one. Let us
apply (5) to determine Df of the Mandelbrot–Given fractal. The Mandelbrot–Given fractal is composed of 8 partsof size 1/3, hence, N(1/3) D 8, N((1/3)2 ) D 82 , and so on.
We thus have a relation of the form N(r) / r ln3 8 , which
gives the fractal dimension Df D ln3 8 D 1:89278 : : :. The
Mandelbrot–Given fractal has many analogous featureswith percolation networks (see Sect. “Dynamical Properties of Fractal Structures”), a typical random fractal, suchthat the fractal dimension of a 2d percolation network is
Df D 91/48 D 1:895833: : :, which is very close to that of
the Mandelbrot–Given fractal.
The geometric characteristics of random fractals canbe understood by considering two extreme cases of random structures. Figure 2a represents the case in which
particles are randomly but homogeneously distributed ina d-dimensional box of size L, where d represents ordinary
Euclidean dimensionality of the embedding space. If we
divide this box into smaller boxes of size l, the mass density of the ith box isi (l) D

M i (l);ld(7)Fractal Structures in Condensed Matter Physics, Figure 1
Mandelbrot–Given fractal. a The initial structure with eight line segments, b the object obtained by replacing each line segment of
the initial structure by the initial structure itself (the second stage), and c the third stage of the Mandelbrot–Given fractal obtainedby replacing each line segment of the second-sage structure by the initial structureFractal Structures in Condensed Matter Physics

F

Fractal Structures in Condensed Matter Physics, Figure 2
a Homogeneous random structure in which particles are randomly but homogeneously distributed, and b the distribution functionsof local densities , where (l) is the average mass density independent of lwhere M i (l) represents the total mass (measure) insidebox i. Since this quantity depends on the box i, we plot the
distribution function P( ), from which curves like those in
Fig. 2b may be obtained for two box sizes l1 and (l2 < l2 ).
We see that the central peak position of the distributionfunction P( ) is the same for each case. This means thatthe average mass density yields(l) DhM i (l)i ild¯(l) D

becomes constant, indicating that hM i (l)i i / l d . The
above is equivalent to
D

m
ad

;

The other type of random structure is shown in Fig. 3a,
where particle positions are correlated with each otherand i (l) greatly ﬂuctuates from box to box, as shown in
Fig. 3b. The relation hM i (l)i i / l d may not hold at all forthis type of structure. Assuming the fractality for this system, namely, if the power law hM i (l)i i / lfD holds, the average mass density becomes(8)

where a is the average distance (characteristic lengthscale) between particles and the mass of a single particle.
This indicates that there exists a single length scale a characterizing the random system given in Fig. 2a.

hM i (l)i i/ l D f d ;ld

(9)where i (l) D 0 is excluded. In the case Df < d, ¯(l) depends on l and decreases with increasing l. Thus, there is
no characteristic length scale for the type of random structure shown in Fig. 3a. If (9) holds with Df < d, so thathM i (l)i i is proportional to lfD , the structure is said to be
fractal. It is important to note that there is no characteristic
length scale for the type of random fractal structure shownin Fig. 3b. Thus, we can extend the idea of self-similarity
not only for deterministic self-similar structures, but also

Fractal Structures in Condensed Matter Physics, Figure 3
a Correlated random fractal structure in which particles are randomly distributed, but correlated with each other, and b the distribution functions of local densities  with finite values, where the average mass densities depend on l

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Fractal Structures in Condensed Matter Physics

Fractal Structures in Condensed Matter Physics, Figure 4
a 2d site-percolation network and circles with different radii. b The power law relation holds between r and the number of particles
in the sphere of radius r, indicating the fractal dimension of the 2d network is Df D 1:89 : : : D 91/48

for random and disordered structures, the so-called random fractals, in the meaning of statistical average.
The percolation network made by putting particles orbonds on a lattice with the probability p is a typical example of random fractals. The theory of percolation was initiated in 1957 by S.R. Broadbent and J.M. Hammersley [5]in connection with the diﬀusion of gases through porous
media. Since their work, it has been widely accepted thatthe percolation theory describes a large number of physical and chemical phenomena such as gelation processes,
transport in amorphous materials, hopping conduction in
doped semiconductors, the quantum Hall eﬀect, and manyother applications. In addition, it forms the basis for studies of the ﬂow of liquids or gases through porous media.
Percolating networks thus serve as a model which helps
us to understand physical properties of complex fractal
structures.
For both deterministic and random fractals, it is remarkable that no characteristic length scale exists, andthis is a key feature of fractal structures. In other words,
fractals are deﬁned to be objects invariant under isotropic
scale transformations, i. e., uniform dilatation of the system in every spatial direction. In contrast, there exist systems which are invariant under anisotropic transformations. These are called self-aﬃne fractals.

ter physics. The following methods for obtaining the fractal dimension Df are known to be quite eﬃcient.
Coverage Method
The idea of coverage in the deﬁnition of the capacity dimension (see (5)) can be applied to obtain the fractal dimension Df of material surfaces. An example is the fractality of rough surfaces or inner surfaces of porous media.
The fractal nature is probed by changing the sizes of adsorbed molecules on solid surfaces. Power laws are veriﬁedby plotting the total number of adsorbed molecules versus
their size r. The area of a surface can be estimated with theaid of molecules weakly adsorbed by van der Waals forces.
Gas molecules are adsorbed on empty sites until the surface is uniformly covered with a layer one molecule thick.
Provided that the radius r of one adsorbed molecule andthe number of adsorbed molecules N(r) are known, thesurface area S obtained by molecules is given by
S(r) / N(r)r2 :If the surface of the adsorbate is perfectly smooth, we expect the measured area to be independent of the radius rof the probe molecules, which indicates the power law
N(r) / r2 :Determining Fractal Dimensions
There are several methods to determine fractal dimensions
Df of complex structures encountered in condensed mat-

(10)

(11)

However, if the surface of the adsorbate is rough or contains pores that are small compared with r, less of the surface area S is accessible with increasing size r. For a fractal

Fractal Structures in Condensed Matter Physics

surface with fractal dimension Df , (11) gives the relation
N(r) / rD f :(12)

Box-Counting Method
Consider as an example a set of particles distributed ina space. First, we divide the space into small boxes of size r
and count the number of boxes containing more than one
particle, which we denote by N(r). From the deﬁnition of
the capacity dimension (4), the number of particle
N(r) / rD f :(13)

For homogeneous objects distributed in a d-dimensional
space, the number of boxes of size r becomes, of course
N(r) / rd :Correlation Function
The fractal dimension Df can be obtained via the correlation function, which is the fundamental statistical quantity
observed by means of X-ray, light, and neutron scatteringexperiments. These techniques are available to bulk materials (not surface), and is widely used in condensed matterphysics. Let (r) be the number density of atoms at position r. The density-density correlation function G(r; r 0 ) isdeﬁned byG(r; r 0 ) D h (r) (r 0 )i ;(14)

where h: : :i denotes an ensemble average. This gives thecorrelation of the number-density ﬂuctuation. Provided
that the distribution is isotropic, the correlation function
becomes a function of only one variable, the radial distance r D jr  r0 j, which is deﬁned in spherical coordinates. Because of the translational invariance of the system
on average, r0 can be ﬁxed at the coordinate origin r 0 D 0.
We can write the correlation function as
G(r) D h (r) (0)i :(15)

The quantity h (r) (0)i is proportional to the probability
that a particle exists at a distance r from another particle.
This probability is proportional to the particle density (r)within a sphere of radius r. Since (r) / r D f d for a fractal
distribution, the correlation function becomesG(r) / r D f d ;(16)

where Df and d are the fractal and the embedded Euclidean
dimensions, respectively. This relation is often used directly to determine Df for random fractal structures.

F

The scattering intensity in an actual experiment is proportional to the structure factor S(q), which is the Fouriertransform of the correlation function G(r). The structure
factor is calculated from (16) as
Z
1
S(q) D
G(r)e i qr dr / qD f(17)
V V
where V is the volume of the system. Here dr is the d-dimensional volume element. Using this relation, we can determine the fractal dimension Df from the data obtainedby scattering experiments.
When applying these methods to obtain the fractal dimension Df , we need to take care over the following point.
Any fractal structures found in nature must have upperand lower length-limits for their fractality. There usuallyexists a crossover from homogeneous to fractal. Fractal
properties should be observed only between these limits.
We describe in the succeeding Sections several examples of fractal structures encountered in condensed matterphysics.
Polymer Chains in Solvents
Since the concept of fractal was coined by B.B. Mandelbrot
in 1975, scientists reinterpreted random complex structures found in condensed matter physics in terms of fractals. They found that a lot of objects are classiﬁed as fractal
structures. We show at ﬁrst from polymer physics an instructive example exhibiting the fractal structure. That is
an early work by P.J. Flory in 1949 on the relationship between the mean-square end-to-end distance of a polymer
chain hr2 i and the degree of polymerization N. Consider
a dilute solution of separate coils in a solvent, where thetotal length of a ﬂexible polymer chain with a monomer
length a is Na. The simplest idealization views the polymer chain in analogy with a Brownian motion of a random
walker. The walk is made by a succession of N steps fromthe origin r D 0 to the end point r. According to the central limit theorem of the probability theory, the probability
to ﬁnd a walker at r after N steps (N1) follows the diffusion equation and we have the expression for the probability to ﬁnd a particle after N steps at r
PN (r) D (2 Na 2/3)3/2 exp(3r2 /2Na2) ;(18)

where the prefactor arises from the normalization of
PN (r). The mean squared distance calculated from PN (r)becomesZ(19)
hr2 i D r2 PN (r)d3 r D Na2 :Then, the mean-average end-to-end distance of a polymer chain yields R D hr2 i1/2 D N 1/2 a. Since the number

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of polymerization N corresponds to the total mass M ofa polymer chain, the use of (19) leads to the relationsuch as M(R)  R2 . The mass M(R) can be considered asa measure of a polymer chain, the fractal dimension of thisideal chain as well as the trace of Brown motion becomes
Df D 2 for any d-dimensional embedding space.
The entropy of the idealized chain of the length
L D Na is obtained from (18) as
S(r) D S(0) 

3r2
;
2R2

(20)

from which the free energy Fel D U  T S is obtained as
Fel (r) D Fel(0) C

3k B Tr2
:
2R2

(21)

Here U is assumed to be independent of distinct conﬁgurations of polymer chains. This is an elastic energyof an ideal chain due to entropy where Fel decreases as
N ! large. P.J. Flory added the repulsive energy term
due to monomer-monomer interactions, the so-called excluded volume eﬀect. This has an analogy with self-avoiding
random walk. The contribution to the free energy is obtained by the virial expansion into the power series on theconcentration cint D N/r d . According to the mean ﬁeld
2 , we have the total
theory on the repulsive term Fint / cint
free-energy F such as
Fv(T)N 2
3r2
C
;
D
2
kB T
2Na
rd

(22)

where v(T) is the excluded volume parameter. We can obtain a minimum of F(r) at r D R by diﬀerentiating F(r)with respect to r such that
dC2

M(R) / R 3 :(23)

Here the number of polymerization N corresponds to thetotal mass M(R) of a polymer chain. Thus, we have thefractal dimension Df D (d C 2)/3, in particular, Df D
5/3 D 1:666 : : : for a polymer chain in a solvent.
Aggregates and Flocs
The structures of a wide variety of ﬂocculated colloids insuspension (called aggregates or ﬂocs) can be described interms of fractals. A colloidal suspension is a ﬂuid containing small charged particles that are kept apart by Coulomb
repulsion and kept aﬂoat by Brownian motion. A change
in the particle-particle interaction can be induced by varying the chemical composition of the solution and in thismanner an aggregation process can be initiated. Aggregation processes are classiﬁed into two simple types: diffusion-limited aggregation (DLA) and diﬀusion-limited

cluster-cluster aggregation (DLCA), where a DLA is dueto the cluster-particle coalescence and a DLCA to the cluster-cluster ﬂocculation. In most cases, actual aggregates involve a complex interplay between a variety of ﬂocculation
processes. The pioneering work was done by M.V. Smoluchowski in 1906, who formulated a kinetic theory forthe irreversible aggregation of particles into clusters andfurther clusters combining with clusters. The inclusionof cluster-cluster aggregation makes this process distinct
from the DLA process due to particle-cluster interaction.
There are two distinct limiting regimes of the irreversible
colloidal aggregation process: the diﬀusion-limited CCA(DLCA) in dilute solutions and the reaction-limited CCA(RLCA) in dense solutions. The DLCA is due to the fast
process determined by the time for the clusters to encounter each other by diﬀusion, and the RLCA is due tothe slow process since the cluster-cluster repulsion has todominate thermal activation.
Much of our understanding on the mechanism forming aggregates or ﬂocs has been mainly due to computer
simulations. The ﬁrst simulation was carried out by Voldin 1963 [23], who used the ballistic aggregation modeland found that the number of particles N(r) within a distance r measured from the ﬁrst seed particle is given byN(r)  r2:3 . Though this relation surely exhibits the scaling form of (2), the applicability of this model for real systems was doubted in later years. The researches on fractal
aggregates has been developed from a simulation modelon DLA introduced by T.A. Witten and L.M. Sander in1981 [26] and on the DLCA model proposed by P. Meakinin 1983 [14] and M. Kolb et al. in 1983 [11], independently.
The DLA has been used to describe diverse phenomena
forming fractal patterns such as electro-depositions, surface corrosions and dielectric breakdowns. In the simplest
version of the DLA model for irreversible colloidal aggregation, a particle is located at an initial site r D 0 asa seed for cluster formation. Another particle starts a random walk from a randomly chosen site in the sphericalshell of radius r with width dr( r) and center r D 0. As
a ﬁrst step, a random walk is continued until the particle
contacts the seed. The cluster composed of two particles
is then formed. Note that the ﬁnite-size of particles is thevery reason of dendrite structures of DLA. This procedure
is repeated many times, in each of which the radius r of thestarting spherical shell should be much larger than the gyration radius of the cluster. If the number of particles contained in the DLA cluster is huge (typically 104  108 ), thecluster generated by this process is highly branched, andforms fractal structures in the meaning of statistical average. The fractality arises from the fact that the faster growing parts of the cluster shield the other parts, which there-Fractal Structures in Condensed Matter Physics

F

illustrates a simulated result for a 2d DLA cluster obtainedby the procedure mentioned above. The number of particles N inside a sphere of radius L ( the gyration radiusof the cluster) follows the scaling law given by
N / L Df ;Fractal Structures in Condensed Matter Physics, Figure 5
Simulated results of a 2d diffusion-limited aggregation (DLA).
The number of particles contained in this DLA cluster is 104

fore become less accessible to incoming particles. An arriving random walker is far more likely to attach to one of thetips of the cluster. Thus, the essence of the fractal-patternformation arises surely from nonlinear process. Figure 5(24)

where the fractal dimension takes a value of Df  1:71 forthe 2d DLA cluster and Df  2:5 for the 3d DLA cluster without an underlying lattice. Note that these fractal
dimensions are sensitive to the embedding lattice structure. The reason for this open structure is that a wandering molecule will settle preferentially near one of the tips ofthe fractal, rather than inside a cluster. Thus, diﬀerent sites
have diﬀerent growth probabilities, which are high near
the tips and decrease with increasing depth inside a cluster.
One of the most extensively studied DLA processes
is the growth of metallic forms by electrochemical deposition. The scaling properties of electrodeposited metals
were pointed out by R.M. Brady and R.C. Ball in 1984 forcopper electrodepositions. The conﬁrmation of the fractality for zinc metal leaves was made by M. Matsushita et
al. in 1984. In their experiments [13], zinc metal leaves aregrown two-dimensionally by electrodeposition. The structures clearly recover the pattern obtained by computer
simulations for the DLA model proposed by T.A. Witten
and L.M. Sander in 1981. Figure 6 shows a typical zinc

Fractal Structures in Condensed Matter Physics, Figure 6
The fractal structures of zinc metal leaves grown by electrodeposition. Photographs a–d were taken 3,5,9, and 15 min after initiating
the electrolysis, respectively. After [13]

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dendrite that was deposited on the cathode in one of theseexperiments. The fractal dimensionality Df D 1:66 ˙ 0:33
was obtained by computing the density-density correlation function G(r) for patterns grown at applied voltagesof less than 8V.
The fractality of uniformly sized gold-colloid aggregates according to the DLCA was experimentally demonstrated by D.A. Weitz in 1984 [25]. They used transmission-electron micrographs to determine the fractal dimension of this systems to be Df D 1:75. They also performed quasi-elastic light-scattering experiments to investigate the dynamic characteristics of DLCA of aqueousgold colloids. They conﬁrmed the scaling behaviors for the
dependence of the mean cluster size on both time and initial concentration.
These works were performed consciously to examine
the fractality of aggregates. There had been earlier works
exhibiting the mass-size scaling relationship for actual
aggregates. J.M. Beeckmans [2] pointed out in 1963 thepower law behaviors by analyzing the data for aerosol andprecipitated smokes in the literature (1922–1961). He usedin his paper the term “aggregates-within-aggregates”, implying the fractality of aggregates. However, the data available at that stage were not adequate and scattered. Therefore, this work did not provide decisive results on the fractal dimensions of aggregates. There were smarter experiments by N. Tambo and Y. Watanabe in 1967 [20], which
precisely determined fractal dimensions of ﬂocs formed
in an aqueous solution. These were performed without
being aware of the concept of fractals. Original works
were published in Japanese. The English versions of theseworks were published in 1979 [21]. We discuss these works
below.
Flocs generated in aqueous solutions have been thesubject of numerous studies ranging from basic to appliedsciences. In particular, the settling process of ﬂocs formedin water incorporating kaolin colloids is relevant to water and wastewater treatment. The papers by N. Tamboand Y. Watanabe pioneered the discussion on the socalled fractal approach to ﬂoc structures; they performed
their own settling experiments to clarifying the size dependences of mass densities for clay-aluminum ﬂocs by using Stokes’ law ur /(r)r2 whereis the diﬀerence
between the densities of water and ﬂocs taking so-smallvalues 0:01–0:001 g/cm3 . Thus, the settling velocities ur are very slow of the order of 0:001 m/sec for ﬂocs ofsizes r  0:1 mm, which enabled them to perform precise
measurements. Since ﬂocs are very fragile aggregates, they
made the settling experiments with special cautions onconvection and turbulence, and by careful and intensive
experiments of ﬂocculation conditions. They conﬁrmed

from thousands of pieces of data the scaling relationshipbetween settling velocities ur and sizes of aggregates suchas ur / r b . From the analysis of these data, they foundthe scaling relation between eﬀective mass densities andsizes of ﬂocs such as(r) / rc , where the exponents c
were found to take values from 1.25 to 1.00 depending on
the aluminum-ion concentration, showing that the fractal dimensions become Df D 1:75 to 2.00 with increasing aluminum-ion concentration. This is because the repulsive force between charged clay-particles is screened,and van der Waals attractive force dominates between thepair of particles. It is remarkable that these fractal dimensions Df show excellent agreement with those determined
for actual DLCA and RLCA clusters in the 1980s by using various experimental and computer simulation methods. Thus, they had found that the size dependences ofmass densities of ﬂocs are controlled by the aluminumion concentration dosed/suspended particle concentration, which they named the ALT ratio. These correspond

Fractal Structures in Condensed Matter Physics, Figure 7
Observed scaling relations between floc densities and their diameters where aluminum chloride is used as coagulants. After [21]Fractal Structures in Condensed Matter Physics

to the transition from DLCA (established now taking thevalue of Df  1:78 from computer simulations) process to
the RLCA one (established at present from computer simulations as Df  2:11). The ALT ratio has since the publication of the paper been used in practice as a criterion forthe coagulation to produce ﬂocs with better settling properties and less sludge volume. We show their experimental
data in Fig. 7, which demonstrate clearly that ﬂocs (aggregates) are fractal.
Aerogels
Silica aerogels are extremely light materials with porosities as high as 98% and take fractal structures. The initial
step in the preparation is the hydrolysis of an alkoxysilane Si(OR)4 , where R is CH3 or C2 H5 . The hydrolysis
produces silicon hydroxide Si(OH)4 groups which polycondense into siloxane bonds –Si–O–Si–, and small particles start to grow in the solution. These particles bindto each other by diﬀusion-limited cluster-cluster aggregation (DLCA) (see Sect. “Aggregates and Flocs”) until eventually they produce a disordered network ﬁlling the reaction volume. After suitable aging, if the solvent is extracted above the critical point, the open porous structureof the network is preserved and decimeter-size monolithic
blocks with a range of densities from 50 to 500 kg/m3 canbe obtained. As a consequence, aerogels exhibit unusualphysical properties, making them suitable for a number ofpractical applications, such as Cerenkov radiation detectors, supports for catalysis, or thermal insulators.
Silica aerogels possess two diﬀerent length scales. One
is the radius r of primary particles. The other length is thecorrelation length of the gel. At intermediate length scales,
lying between these two length scales, the clusters possessa fractal structure and at larger length scales the gel is a homogeneous porous glass. Aerogels have a very low thermal
conductivity, solid-like elasticity, and very large internalsurfaces.
In elastic neutron scattering experiments, the scattering diﬀerential cross-section measures the Fourier components of spatial ﬂuctuations in the mass density. For aerogels, the diﬀerential cross-section is the product of threefactors, and is expressed byd
D Af 2 (q)S(q)C(q) C B :d˝

Fby B. The structure factor S(q) is proportional to the spatial
Fourier transform of the density-density correlation function deﬁned by (16), and is given by (17). Since the structure of the aerogel is fractal up to the correlation length
 of the system and homogeneous for larger scales, thecorrelation function G(r) is expressed by (25) for r
and G(r) D Const. for r
. Corresponding to this, thestructure factor S(q) is given by (17) for q
1, while
S(q) is independent of q for q
1. The wavenumber
regime for which S(q) becomes a constant is called theGuinier regime. The value of Df can be deduced from theslope of the observed intensity versus momentum transfer (q1) in a double logarithmic plot. For very large q,
there exists a regime called the Porod regime in which thescattering intensity is proportional to q4 .

(25)

Here A is a coeﬃcient proportional to the particle concentration and f (q) is the primary-particle form factor.
The structure factor S(q) describes the correlation between
particles in a cluster and C(q) accounts for cluster-cluster correlations. The incoherent background is expressed

Fractal Structures in Condensed Matter Physics, Figure 8
Scattered intensities for eight neutrally reacted samples. Curves
are labeled with  in kg/m3 . After [22]

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Fractal Structures in Condensed Matter Physics

The results in Fig. 8 by R. Vacher et al. [22] arefrom small-angle neutron scattering experiments on silica aerogels. The various curves are labeled by the macroscopic density of the corresponding sample in Fig. 8.
For example, 95 refers to a neutrally reacted sample with
D 95 kg/m3 . Solid lines represent best ﬁts. They are presented even in the particle regime q > 0:15 Å-1 to emphasize that the ﬁts do not apply in the region, particularly for the denser samples. Remarkably, Df is independent of sample density to within experimental accuracy:
Df D 2:40 ˙ 0:03 for samples 95 to 360. The departureof S(q) from the qD f dependence at large q indicates thepresence of particles with gyration radii of a few Å.
Dynamical Properties of Fractal Structures
The dynamics of fractal objects is deeply related to thetime-scale problems such as diﬀusion, vibration and transport on fractal support. For the diﬀusion of a particle
on any d-dimensional ordinary Euclidean space, it is wellknown that the mean-square displacement hr2 (t)i is proportional to the time such as hr2 (t)i / t for any Euclidean
dimension d (see also (19)). This is called normal diﬀusion.
While, on fractal supports, a particle more slowly diﬀuses,
and the mean-square displacement follows the power law
hr2 (t)i / t 2/d w ;(26)

where dw is termed the exponent of anomalous diﬀusion.
The exponents is expressed as dw D 2 C  with a positive  > 0 (see (31)), implying that the diﬀusion becomes slower compared with the case of normal diﬀusion.
This is because the inequality 2/dw < 1 always holds. This
slow diﬀusions on fractal supports are called anomalous
diﬀusion.
The scaling relation between the length-scale andthe time-scale can be easily extended to the problem of
atomic vibrations of elastic fractal-networks. This is because various types of equations governing dynamics canbe mapped onto the diﬀusion equation. This implies thatboth equations are governed by the same eigenvalue problem, namely, the replacement of eigenvalues ! ! ! 2 between the diﬀusion equation and the equation of atomic
vibrations is justiﬁed. Thus, the basic properties of vibrations of fractal networks, such as the density of states,
the dispersion relation and the localization/delocalization
property, can be derived from the same arguments fordiﬀusion on fractal networks. The dispersion relation between the frequency ! and the wavelength (!) is obtained from (26) by using the reciprocal relation t ! ! 2
(here the diﬀusion problem is mapped onto the vibrationalone) and hr2 (t)i ! (!)2 . Thus we obtain the dispersion relation for vibrational excitations on fractal networks
such as! / (!)d w /2 :(27)

If dw D 2, we have the ordinary dispersion relation! / (!) for elastic waves excited on homogeneous
systems.
Consider the diﬀusion of a random walker on a percolating fractal network. How does hr2 (t)i behave in the caseof fractal percolating networks? For this, P.G. de Gennes in1976 [7] posed the problem called an ant in the labyrinth.
Y. Gefen et al. in 1983 [9] gave a fundamental descriptionof this problem in terms of a scaling argument. D. BenAvraham and S. Havlin in 1982 [3] investigated this problem in terms of Monte Carlo simulations. The work by
Y. Gefen [9] triggered further developments in the dynamics of fractal systems, where the spectral (or fracton) dimension ds is a key dimension for describing the dynamics
of fractal networks, in addition to the fractal dimension Df .
The fractal dimension Df characterizes how the geometrical distribution of a static structure depends on its lengthscale, whereas the spectral dimension ds plays a central
role in characterizing dynamic quantities on fractal networks. These dynamical properties are described in a uniﬁed way by introducing a new dynamic exponent called
the spectral or fracton dimension deﬁned by
ds D

2Df
:
dw

(28)

The term fracton, coined by S. Alexander and R. Orbach
in 1982 [1], denotes vibrational modes peculiar to fractal structures. The characteristics of fracton modes covera rich variety of physical implications. These modes arestrongly localized in space and their localization length isof the order of their wavelengths.
We give below the explicit form of the exponent of
anomalous diﬀusion dw by illustrating percolation fractal networks. The mean-square displacement hr2 (t)i after
a suﬃciently long time t should follow the anomalous diffusion described by (26). For a ﬁnite network with a size
, the mean-square distance at suﬃciently large time becomes hr2 (t)i   2 , so we have the diﬀusion coeﬃcient for
anomalous diﬀusion from (26) such as
D /  2d w :(29)

For percolating networks, the diﬀusion constant D in thevicinity of the critical percolation density pc behaves
D / (p  pc ) tˇ /  (tˇ )/ ;

(30)

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Fractal Structures in Condensed Matter Physics

where t is called the conductivity exponent deﬁned bydc  (p  pc ) t , ˇ the exponent for the percolation orderparameter deﬁned by S(p) / (p  pc )ˇ , and  the exponent for the correlation length deﬁned by  / jp  pc j ,respectively. Comparing (29) and (30), we have the relation between exponents such as
dw D 2 Ctˇ
D2C :
D( !) /
(31)

Due to the condition t > ˇ, and hence  > 0, implyingthat the diﬀusion becomes slow compared with the case of
normal diﬀusion. This slow diﬀusion is called anomalous
diﬀusion.
Spectral Density of States and Spectral Dimensions
The spectral density of states of atomic vibrations is themost fundamental quantity describing the dynamic properties of homogeneous or fractal systems such as speciﬁcheats, heat transport, scattering of waves and others. The
simplest derivation of the spectral density of states (abbreviated, SDOS) of a homogeneous elastic system is given
below. The density of states at ! is deﬁned as the numberof modes per particle, which is expressed byD( !) D

1;
!Ld

SDOS for fractal networks in the same line with this treatment. Consider the SDOS of a fractal structure of size Lwith fractal dimension Df . The density of states per particle at the lowest frequency ! for this system is, as in thecase of (32), written as
1
:
LfD !

Assuming that the dispersion relation foring to (33) is(36)! correspond-! / Lz ;(37)

we can eliminate L from (36) and obtainD( !) /

! D f /z1 :(38)

The exponent z of the dispersion relation (37) is evaluatedfrom the exponent of anomalous diﬀusion dw . Considering the mapping correspondence between diﬀusion andatomic vibrations, we can replace hr2 (t)i and t by L2 and1/ ! 2 , respectively. Equation (26) can then be read asL/! 2/d w :(39)

The comparison of (28),(37) and (39) leads to(32)
zD

where ! is the frequency interval between adjacenteigenfrequencies close to ! and L is the linear size of thesystem. In the lowest frequency region, ! is the lowest eigenfrequency which depends on the size L. The relation between the frequency ! and L is obtained fromthe well-known linear dispersion relationship ! D ( k,where ( is the velocity of phonons (quantized elastic
waves) such that

Df
dw
:
D
2
ds

(40)

Since the system has a scale-invariant fractal (self-similar)
structure !, can be replaced by an arbitrary frequency !.
Hence, from (38) and (40) the SDOS for fractal networks
is found to beD(!) / ! d s 1 ;(41)

and the dispersion relation (39) becomes
!D

2(

1
/ :
L

(33)

The substitution of (33) into (32) yields
D( !) /

! d1 :

(34)

Since this relation holds for any length scale L due tothe scale-invariance property of homogeneous systems, we
can replace the frequency ! by an arbitrary !. Therefore, we obtain the conventional Debye density of states asD(!) / ! d1 :(35)

It should be noted that this derivation is based on the scale
invariance of the system, suggesting that we can derive the! / L(!)D f /d s :(42)

For percolating networks, the spectral dimension is obtained from (40)
ds D

2Df
2( Df
D
:
2C
2( C   ˇ(43)This exponent ds is called the fracton dimension after
S. Alexander and R. Orbach [1] or the spectral dimension after R. Rammal and G. Toulouse [17], hereafter we
use the term spectral dimension for ds . S. Alexander and
R. Orbach [1] estimated the values of ds for percolating networks on d-dimensional Euclidean lattices fromthe known values of the exponents Df ; (;  and ˇ. They

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Fractal Structures in Condensed Matter Physics

pointed out that, while these exponents depend largely
on d, the spectral dimension (fracton) dimension ds doesnot.
The spectral dimension ds can be obtained from thevalue of the conductivity exponent t or vice versa. In thecase of percolating networks, the conductivity exponent t
is related to ds through (43), which means that the conductivity dc  (p  pc ) t is also characterized by the spectral dimension ds . In this sense, the spectral dimension ds
is an intrinsic exponent related to the dynamics of fractal
systems. We can determine the precise values of ds fromthe numerical calculations of the spectral density of statesof percolation fractal networks.
The fracton SDOS for 2d, 3d, and 4d bond percolation networks at the percolation threshold p D pc are
given in Fig. 9a and b, which were calculated by K. Yakubo
and T. Nakayama in 1989. These were obtained by largescale computer simulations [27]. At p D pc , the correlation length diverges as  / jp  pc j and the network
has a fractal structure at any length scale. Therefore, fracton SDOS should be recovered in the wide frequency range!L!
!D , where !D is the Debye cutoﬀ frequency
and !L is the lower cutoﬀ determined by the system size.
The SDOSs and the integrated SDOSs per atom are shownby the ﬁlled squares for a 2d bond percolation (abbreviated, BP) network at pc D 0:5. The lowest frequency !Lis quite small (!  105 for the 2d systems) as seen fromthe results in Fig. 9 because of the large sizes of the systems.

The spectral dimension ds is obtained as ds D 1:33 ˙ 0:11
from Fig. 9a, whereas data in Fig. 9b give the more precise value ds D 1:325 ˙ 0:002. The SDOS and the integrated SDOS for 3d BP networks at pc D 0:249 are given
in Fig. 9a and b by the ﬁlled triangles (middle). The spectral dimension ds is obtained as ds D 1:31 ˙ 0:02 from
Fig. 9a and ds D 1:317 ˙ 0:003 from Fig. 9b. The SDOS
and the integrated SDOS of 4d BP networks at pc D 0:160.
A typical mode pattern of a fracton on a 2d percolation network is shown in Fig. 10a, where the eigenmode
belongs to the angular frequency ! D 0:04997. To bringout the details more clearly, Fig. 10b by K. Yakubo and
T. Nakayama [28] shows cross-sections of this fractonmode along the line drawn in Fig. 10a. Filled and open
circles represent occupied and vacant sites in the percolation network, respectively. We see that the fracton core(the largest amplitude) possesses very clear boundaries forthe edges of the excitation, with an almost step-like character and a long tail in the direction of the weak segments.
It should be noted that displacements of atoms in deadends (weakly connected portions in the percolation network) move in phase, and fall oﬀ sharply at their edges.
The spectral dimension can be obtained exactly for deterministic fractals. In the case of the d-dimensional Sierpinski gasket, the spectral dimension is given by [17]ds D

2 log(d C 1):log(d C 3)

Fractal Structures in Condensed Matter Physics, Figure 9
a Spectral densities of states (SDOS) per atom for 2d, 3d, and 4d BP networks at p D pc . The angular frequency ! is defined with massunits m D 1 and force constant Kij D 1. The networks are formed on 1100  1100 (2d), 100  100  100 (3d), and 30  30  30  30(4d) lattices with periodic boundary conditions, respectively. b Integrated densities of states for the same

Fractal Structures in Condensed Matter Physics

F

Fractal Structures in Condensed Matter Physics, Figure 10
a Typical fracton mode (! D 0:04997) on a 2d network. Bright region represents the large amplitude portion of the mode. b Crosssection of the fracton mode shown in a along the white line. The four figures are snapshots at different times. After [28]We see from this that the upper bound for a Sierpinski gasket is ds D 2 as d ! 1. The spectral dimension for the
Mandelbrot–Given fractal depicted is also calculated analytically as
ds D

2 log 8
D 1:345 : : : :log 22This value is close to those for percolating networks mentioned above, in addition to the fact that the fractal dimension ds D log 8/ log 3 of the Mandelbrot–Given fractal isclose to Df D 91/48 for 2d percolating networks and that
the Mandelbrot–Given fractal has a structure with nodes,
links, and blobs as in the case of percolating networks.
For real systems, E. Courtens et al. in 1988 [6] observed
fracton excitations in aerogels by means of inelastic lightscattering.
Future Directions
The signiﬁcance of fractal researches in sciences is that thevery idea of fractals opposes reductionism. Modern physics
has developed by making eﬀorts to elucidate the physical mechanisms of smaller and smaller structures such as
molecules, atoms, and elementary particles. An example incondensed matter physics is the band theory of electrons
in solids. Energy spectra of electrons can be obtained byincorporating group theory based on the translational androtational symmetry of the systems. The use of this mathematical tool greatly simpliﬁes the treatment of systems
composed of 1022 atoms. If the energy spectrum of a unit
cell molecule is solved, the whole energy spectrum of thesolid can be computed by applying the group theory. In
this context, the problem of an ordered solid is reduced
to that of a unit cell. Weakly disordered systems can behandled by regarding impurities as a small perturbation to
the corresponding ordered systems. However, a diﬀerent
approach is required for elucidating the physical properties of strongly disordered/complex systems with correlations, or of medium-scale objects, for which it is diﬃcult
to ﬁnd an easily identiﬁable small parameter that would allow a perturbative analysis. For such systems, the conceptof fractals plays an important role in building pictures of
the realm of nature.
Our established knowledge on fractals is mainly due toexperimental observations or computer simulations. The
researches are at the phenomenological level stage, not at
the intrinsic level, except for a few examples. Concerning
future directions of the researches on fractals in condensed

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matter physics apart from such a question as “What kindsof fractal structures are involved in condensed matter?”,
we should consider two directions: one is the very basic aspect such as the problem “Why are there numerous
examples showing fractal structures in nature/condensed
matter?” However, this type of question is hard. The kinetic growth-mechanisms of fractal systems have a richvariety of applications from the basic to applied sciences
and attract much attention as one of the important subjects in non-equilibrium statistical physics and nonlinear physics. Network formation in society is one example
where the kinetic growth is relevant. However, many aspects related to the mechanisms of network formations remain puzzling because arguments are at the phenomenological stage. If we compare with researches on Brownian motion as an example, the DLA researches need toadvance to the stage of Einstein’s intrinsic theory [8], orthat of Smoluchowski [18] and H. Nyquist [15]. It is notable that the DLA is a stochastic version of the Hele–Shaw
problem, the ﬂow in composite ﬂuids with high and low
viscosities: the particles diﬀuse in the DLA, while the ﬂuid
pressure diﬀuses in Hele–Shaw ﬂow [19]. These are deeply
related to each other and involve many open questions forbasic physics and mathematical physics.
Concerning the opposite direction, one of the important issues in fractal research is to explore practical usesof fractal structures. In fact, the characteristics of fractals
are applied to many cases such as the formation of tailormade nano-scale fractal structures, fractal-shaped antennae with much reduced sizes compared with those of ordinary antennae, and fractal molecules sensitive to frequencies in the infrared region of light.
Deep insights into fractal physics in condensed matter
will open the door to new sciences and its application totechnologies in the near future.
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Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . ?

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Fractals and Wavelets:
What Can We Learn on Transcription
and Replication from Wavelet-BasedMultifractal Analysis of DNA
Sequences?
ALAIN ARNEODO1 , BENJAMIN AUDIT1 ,EDWARD-BENEDICT BRODIE OF BRODIE1 ,SAMUEL N ICOLAY2, MARIE TOUCHON3,5 ,YVES D’AUBENTON-CARAFA 4 , MAXIME HUVET4 ,CLAUDE THERMES4
1 Laboratoire Joliot–Curie and Laboratoire de Physique,
ENS-Lyon CNRS, Lyon Cedex, France
2 Institut de Mathématique, Université de Liège,
Liège, Belgium

F

3 Génétique des Génomes Bactériens, Institut Pasteur,

CNRS, Paris, France
4 Centre de Génétique Moléculaire, CNRS,

Gif-sur-Yvette, France
5 Atelier de Bioinformatique, Université Pierre

et Marie Curie, Paris, France
Article Outline
GlossaryDeﬁnition of the Subject
Introduction
A Wavelet-Based Multifractal Formalism:
The Wavelet Transform Modulus Maxima Method
Bifractality of Human DNA Strand-Asymmetry ProﬁlesResults from Transcription
From the Detection of Relication Origins Using
the Wavelet Transform Microscope to the Modelingof Replication in Mammalian Genomes
A Wavelet-Based Methodology to Disentangle
Transcription- and Replication-Associated Strand
Asymmetries Reveals a Remarkable Gene Organizationin the Human Genome
Future Directions
Acknowledgments
Bibliography
Glossary
Fractal Fractals are complex mathematical objects thatare invariant with respect to dilations (self-similarity)and therefore do not possess a characteristic lengthscale. Fractal objects display scale-invariance properties that can either ﬂuctuate from point to point (multifractal) or be homogeneous (monofractal). Mathematically, these properties should hold over all scales.
However, in the real world, there are necessarily lower
and upper bounds over which self-similarity applies.
Wavelet transform The continuous wavelet transform(WT) is a mathematical technique introduced in theearly 1980s to perform time-frequency analysis. The
WT has been early recognized as a mathematical microscope that is well adapted to characterize the scaleinvariance properties of fractal objects and to reveal
the hierarchy that governs the spatial distribution of
the singularities of multifractal measures and functions. More speciﬁcally, the WT is a space-scale analysis which consists in expanding signals in terms ofwavelets that are constructed from a single function,
the analyzing wavelet, by means of translations anddilations.

3893