3878 F Fractal Structures in Condensed Matter Physics

Fractal Structuresin Condensed Matter PhysicsTSUNEYOSHI NAKAYAMAToyota Physical and Chemical Research Institute,Nagakute, Japan

Article Outline

GlossaryDefinition of the SubjectIntroductionDetermining Fractal DimensionsPolymer Chains in SolventsAggregates and FlocsAerogelsDynamical Properties of Fractal StructuresSpectral Density of States and Spectral DimensionsFuture DirectionsBibliography

Glossary

Anomalous diffusion It is well known that the mean-square displacement hr2(t)i of a diffusing particle ona uniform system is proportional to the time t such

Fractal Structures in Condensed Matter Physics F 3879as hr2(t)i � t. This is called normal diffusion. Parti-cles on fractal networks diffuse more slowly comparedwith the case of normal diffusion. This slow diffusioncalled anomalous diffusion follows the relation givenby hr2(t)i � ta , where the condition 0 < a < 1 alwaysholds.

Brownian motion Einstein published the important pa-per in 1905 opening the way to investigate the move-ment of small particles suspended in a stationary liq-uid, the so-called Brownian motion, which stimulatedJ. Perrin in 1909 to pursue his experimental work con-firming the atomic nature of matter. The trail of a ran-dom walker provides an instructive example for un-derstanding the meaning of random fractal structures.

Fractons Fractons, excitations on fractal elastic-net-works, were named by S. Alexander and R. Orbachin 1982. Fractons manifest not only static propertiesof fractal structures but also their dynamic proper-ties. These modes show unique characteristics such asstrongly 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 De-bye spectral density of states given by D(!) � !d�1,where d is the Euclidean dimensionality. The spec-tral density of states of fractal networks is given byD(!) � !ds�1, where ds is called the spectral or frac-ton dimension of the system.

Spectral dimension This exponent characterizes thespectral density of states for vibrational modes ex-cited on fractal networks. The spectral dimensionconstitutes the dynamic exponent of fractal networkstogether with the conductivity exponent and the expo-nent 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 invari-ance under an isotropic scale-transformation on certainlength scales. The term scale-invariance has the implica-tion that objects look the same on different scales of obser-vations. While the underlying concept of fractals is quitesimple, the concept is used for an extremely broad rangeof topics, providing a simple description of highly com-plex structures found in nature. The term fractal was firstintroduced by Benoit B. Mandelbrot in 1975, who gavea 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 usedto understand the behaviors of many complex systems or

has been applied from physics, chemistry, and biology forapplied sciences and technological purposes. Examples offractal structures in condensed matter physics are numer-ous such as polymers, colloidal aggregations, porous me-dia, rough surfaces, crystal growth, spin configurations ofdiluted magnets, and others. The critical phenomena ofphase transitions are another example where self-similar-ity plays a crucial role. Several books have been publishedon fractals and reviews concerned with special topics onfractals have appeared.

Length, area, and volume are special cases of ordi-nary Euclidean measures. For example, length is the mea-sure 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 dimensionsfor Euclidean systems, for example, a total mass M(r) ofa fractal object of the size r. For this, the following relationshould hold

r / M(r)1/d ; (1)

where d is the Euclidean dimensionality. Note that Eu-clidean spaces are the simplest scale-invariant systems.We extend this idea to introduce dimensions for self-similar fractal structures. Consider a set of particles withunit massm randomly distributed on a d-dimensional Eu-clidean space called the embedding space of the system.Draw a sphere of radius r and denote the total mass of par-ticles included in the sphere byM(r). Provided that the fol-lowing relation holds in the meaning of statistical averagesuch as

r / hM(r)i1/Df ; (2)

where h: : :i denotes the ensemble-average over differentspheres of radius r, we call Df the similarity dimension. Itis necessary, of course, that Df is smaller than the embed-ding Euclidean dimension d. The definition of dimensionas a statistical quantity is quite useful to specify the charac-teristic 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 thecoverage procedure. As an example, the length of a curvedline L is given by the product of the number N of straight-line 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

3880 F Fractal Structures in Condensed Matter Physicssize r. The identical relation,

M(r) / N(r)rd (3)

should hold for the total massM(r) as measure, for exam-ple. 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�Df ; (4)

from which the capacity dimension Df is given by

Df :D limr!0

lnN(r)ln(1/r)

: (5)

The definition of Df can be rendered in the following im-plicit form

limr!0

N(r)rDf D const : (6)

Equation (5) brings out a key property of the Hausdorffdimension [10], where the product N(r)rDf remains finiteas r! 0. If Df is altered even by an infinitesimal amount,this product will diverge either to zero or to infinity. TheHausdorff dimension coincides with the capacity dimen-sion for many fractal structures, although the Hausdorffdimension is defined less than or equal to the capacity di-mension. Hereafter, we refer to the capacity dimension orthe Hausdorff dimension mentioned above as the fractaldimension.

Introduction

Fractal structures are classified into two categories; deter-ministic fractals and random fractals. In condensed mat-ter physics, we encounter many examples of random frac-tals. The most important characteristic of random frac-tals is the spatial and/or sample-to-sample fluctuations in

Fractal Structures in Condensed Matter Physics, Figure 1Mandelbrot–Given fractal. a The initial structure with eight line segments, b the object obtained by replacing each line segment ofthe 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

their properties. We must discuss their characteristics byaveraging over a large ensemble. The nature of determin-istic fractals can be easily understood from some exam-ples. An instructive example is theMandelbrot–Given frac-tal [12], which can be constructed by starting with a struc-ture with eight line segments as shown in Fig. 1a (thefirst stage of the Mandelbrot–Given fractal). In the sec-ond stage, each line segment of the initial structure is re-placed by the initial structure itself (Fig. 1b). This pro-cess is repeated indefinitely. The Mandelbrot–Given frac-tal 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 usapply (5) to determine Df of the Mandelbrot–Given frac-tal. 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, whichgives the fractal dimension Df D ln3 8 D 1:89278 : : :. TheMandelbrot–Given fractal has many analogous featureswith percolation networks (see Sect. “Dynamical Proper-ties of Fractal Structures”), a typical random fractal, suchthat the fractal dimension of a 2d percolation network isDf D 91/48 D 1:895833: : :, which is very close to that ofthe Mandelbrot–Given fractal.

The geometric characteristics of random fractals canbe understood by considering two extreme cases of ran-dom structures. Figure 2a represents the case in whichparticles are randomly but homogeneously distributed ina d-dimensional box of size L, where d represents ordinaryEuclidean dimensionality of the embedding space. If wedivide this box into smaller boxes of size l, the mass den-sity of the ith box is

�i (l) DMi(l)l d

; (7)

Fractal Structures in Condensed Matter Physics F 3881

Fractal Structures in Condensed Matter Physics, Figure 2aHomogeneous 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 l

where Mi(l) represents the total mass (measure) insidebox i. Since this quantity depends on the box i, we plot thedistribution function P(�), from which curves like those inFig. 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) D hMi(l)iil d

becomes constant, indicating that hMi(l)ii / l d . Theabove is equivalent to

� D mad; (8)

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

Fractal Structures in Condensed Matter Physics, Figure 3a Correlated random fractal structure inwhich particles are randomly distributed, but correlatedwith each other, and b the distribu-tion functions of local densities�with finite values, where the averagemass densities depend on l

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 inFig. 3b. The relation hMi(l)ii / l d may not hold at all forthis type of structure. Assuming the fractality for this sys-tem, namely, if the power law hMi(l)ii / l Df holds, the av-erage mass density becomes

�̄(l) D hMi(l)iil d

/ l Df�d ; (9)

where �i(l) D 0 is excluded. In the case Df < d, �̄(l) de-pends on l and decreases with increasing l. Thus, there isno characteristic length scale for the type of random struc-ture shown in Fig. 3a. If (9) holds with Df < d, so thathMi(l)ii is proportional to l Df , the structure is said to befractal. It is important to note that there is no characteristiclength scale for the type of random fractal structure shownin Fig. 3b. Thus, we can extend the idea of self-similaritynot only for deterministic self-similar structures, but also

3882 F Fractal Structures in Condensed Matter Physics

Fractal Structures in Condensed Matter Physics, Figure 4a 2d site-percolation network and circles with different radii. b The power law relation holds between r and the number of particlesin 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 ran-dom 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 exam-ple of random fractals. The theory of percolation was ini-tiated in 1957 by S.R. Broadbent and J.M. Hammersley [5]in connection with the diffusion of gases through porousmedia. Since their work, it has been widely accepted thatthe percolation theory describes a large number of phys-ical and chemical phenomena such as gelation processes,transport in amorphous materials, hopping conduction indoped semiconductors, the quantumHall effect, andmanyother applications. In addition, it forms the basis for stud-ies of the flow of liquids or gases through porous media.Percolating networks thus serve as a model which helpsus to understand physical properties of complex fractalstructures.

For both deterministic and random fractals, it is re-markable 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 isotropicscale transformations, i. e., uniform dilatation of the sys-tem in every spatial direction. In contrast, there exist sys-tems which are invariant under anisotropic transforma-tions. These are called self-affine fractals.

Determining Fractal Dimensions

There are severalmethods to determine fractal dimensionsDf of complex structures encountered in condensed mat-

ter physics. The following methods for obtaining the frac-tal dimension Df are known to be quite efficient.

Coverage Method

The idea of coverage in the definition of the capacity di-mension (see (5)) can be applied to obtain the fractal di-mension Df of material surfaces. An example is the frac-tality of rough surfaces or inner surfaces of porous media.The fractal nature is probed by changing the sizes of ad-sorbedmolecules on solid surfaces. Power laws are verifiedby plotting the total number of adsorbed molecules versustheir size r. The area of a surface can be estimated with theaid of molecules weakly adsorbed by van derWaals forces.Gas molecules are adsorbed on empty sites until the sur-face 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 : (10)

If the surface of the adsorbate is perfectly smooth, we ex-pect the measured area to be independent of the radius rof the probe molecules, which indicates the power law

N(r) / r�2 : (11)

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

Fractal Structures in Condensed Matter Physics F 3883surface with fractal dimension Df, (11) gives the relation

N(r) / r�Df : (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 rand count the number of boxes containing more than oneparticle, which we denote by N(r). From the definition ofthe capacity dimension (4), the number of particle

N(r) / r�Df : (13)

For homogeneous objects distributed in a d-dimensionalspace, 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 correla-tion function, which is the fundamental statistical quantityobserved by means of X-ray, light, and neutron scatteringexperiments. These techniques are available to bulk mate-rials (not surface), and is widely used in condensed matterphysics. Let �(r) be the number density of atoms at posi-tion r. The density-density correlation function G(r; r0) isdefined by

G(r; r0) D h�(r)�(r0)i ; (14)

where h: : :i denotes an ensemble average. This gives thecorrelation of the number-density fluctuation. Providedthat the distribution is isotropic, the correlation functionbecomes a function of only one variable, the radial dis-tance r D jr � r0j, which is defined in spherical coordi-nates. Because of the translational invariance of the systemon average, r0 can be fixed at the coordinate origin r0 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 probabilitythat 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) / rDf�d for a fractaldistribution, the correlation function becomes

G(r) / rDf�d ; (16)

whereDf and d are the fractal and the embeddedEuclideandimensions, respectively. This relation is often used di-rectly to determine Df for random fractal structures.

The scattering intensity in an actual experiment is pro-portional to the structure factor S(q), which is the Fouriertransform of the correlation function G(r). The structurefactor is calculated from (16) as

S(q) D 1V

Z

VG(r)ei q�rdr / q�Df (17)

where V is the volume of the system. Here dr is the d-di-mensional volume element. Using this relation, we can de-termine the fractal dimension Df from the data obtainedby scattering experiments.

When applying these methods to obtain the fractal di-mension 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. Fractalproperties should be observed only between these limits.

We describe in the succeeding Sections several exam-ples of fractal structures encountered in condensed matterphysics.

Polymer Chains in Solvents

Since the concept of fractal was coined by B.B. Mandelbrotin 1975, scientists reinterpreted random complex struc-tures found in condensed matter physics in terms of frac-tals. They found that a lot of objects are classified as fractalstructures. We show at first from polymer physics an in-structive example exhibiting the fractal structure. That isan early work by P.J. Flory in 1949 on the relationship be-tween the mean-square end-to-end distance of a polymerchain hr2i and the degree of polymerization N. Considera dilute solution of separate coils in a solvent, where thetotal length of a flexible polymer chain with a monomerlength a is Na. The simplest idealization views the poly-mer chain in analogy with a Brownianmotion of a randomwalker. The walk is made by a succession of N steps fromthe origin r D 0 to the end point r. According to the cen-tral limit theorem of the probability theory, the probabilityto find a walker at r after N steps (N 1) follows the dif-fusion equation and we have the expression for the proba-bility to find a particle after N steps at r

PN (r) D (2�Na2/3)�3/2 exp(�3r2/2Na2) ; (18)

where the prefactor arises from the normalization ofPN (r). The mean squared distance calculated from PN (r)becomes

hr2i DZ

r2PN (r)d3r D Na2 : (19)

Then, the mean-average end-to-end distance of a poly-mer chain yields R D hr2i1/2 D N1/2a. Since the number

3884 F Fractal Structures in Condensed Matter Physicsof polymerization N corresponds to the total mass M ofa polymer chain, the use of (19) leads to the relationsuch as M(R) � R2. The massM(R) can be considered asa measure of a polymer chain, the fractal dimension of thisideal chain as well as the trace of Brown motion becomesDf D 2 for any d-dimensional embedding space.

The entropy of the idealized chain of the lengthL D Na is obtained from (18) as

S(r) D S(0) � 3r2

2R2; (20)

from which the free energy Fel D U � TS is obtained as

Fel(r) D Fel(0)C3kBTr2

2R2: (21)

Here U is assumed to be independent of distinct con-figurations of polymer chains. This is an elastic energyof an ideal chain due to entropy where Fel decreases asN ! large. P.J. Flory added the repulsive energy termdue to monomer-monomer interactions, the so-called ex-cluded volume effect. This has an analogy with self-avoidingrandom walk. The contribution to the free energy is ob-tained by the virial expansion into the power series on theconcentration cint D N/rd . According to the mean fieldtheory on the repulsive term Fint / c2int, we have the totalfree-energy F such as

FkBT

D 3r2

2Na2C v(T)N

2

rd; (22)

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

M(R) / RdC23 : (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 D5/3 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 contain-ing small charged particles that are kept apart by Coulombrepulsion and kept afloat by Brownian motion. A changein the particle-particle interaction can be induced by vary-ing the chemical composition of the solution and in thismanner an aggregation process can be initiated. Aggre-gation processes are classified into two simple types: dif-fusion-limited aggregation (DLA) and diffusion-limited

cluster-cluster aggregation (DLCA), where a DLA is dueto the cluster-particle coalescence and a DLCA to the clus-ter-cluster flocculation. Inmost cases, actual aggregates in-volve a complex interplay between a variety of flocculationprocesses. The pioneering work was done by M.V. Smolu-chowski 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 distinctfrom the DLA process due to particle-cluster interaction.There are two distinct limiting regimes of the irreversiblecolloidal 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 fastprocess determined by the time for the clusters to en-counter 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 form-ing aggregates or flocs has been mainly due to computersimulations. 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 dis-tance r measured from the first seed particle is given byN(r) � r2:3. Though this relation surely exhibits the scal-ing form of (2), the applicability of this model for real sys-tems was doubted in later years. The researches on fractalaggregates has been developed from a simulation modelon DLA introduced by T.A. Witten and L.M. Sander in1981 [26] and on the DLCAmodel proposed by P. Meakinin 1983 [14] andM. Kolb et al. in 1983 [11], independently.The DLA has been used to describe diverse phenomenaforming fractal patterns such as electro-depositions, sur-face corrosions and dielectric breakdowns. In the simplestversion of the DLA model for irreversible colloidal ag-gregation, a particle is located at an initial site r D 0 asa seed for cluster formation. Another particle starts a ran-dom walk from a randomly chosen site in the sphericalshell of radius r with width dr( r) and center r D 0. Asa first step, a random walk is continued until the particlecontacts the seed. The cluster composed of two particlesis then formed. Note that the finite-size of particles is thevery reason of dendrite structures of DLA. This procedureis repeatedmany times, in each of which the radius r of thestarting spherical shell should be much larger than the gy-ration radius of the cluster. If the number of particles con-tained 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 aver-age. The fractality arises from the fact that the faster grow-ing parts of the cluster shield the other parts, which there-

Fractal Structures in Condensed Matter Physics F 3885

Fractal Structures in Condensed Matter Physics, Figure 5Simulated 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 arriv-ing randomwalker 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

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

illustrates a simulated result for a 2d DLA cluster obtainedby the procedure mentioned above. The number of parti-cles N inside a sphere of radius L ( the gyration radiusof the cluster) follows the scaling law given by

N / LDf ; (24)

where the fractal dimension takes a value of Df � 1:71 forthe 2d DLA cluster and Df � 2:5 for the 3d DLA clus-ter without an underlying lattice. Note that these fractaldimensions are sensitive to the embedding lattice struc-ture. The reason for this open structure is that a wander-ingmolecule will settle preferentially near one of the tips ofthe fractal, rather than inside a cluster. Thus, different siteshave different growth probabilities, which are high nearthe tips and decreasewith increasing depth inside a cluster.

One of the most extensively studied DLA processesis the growth of metallic forms by electrochemical depo-sition. The scaling properties of electrodeposited metalswere pointed out by R.M. Brady and R.C. Ball in 1984 forcopper electrodepositions. The confirmation of the frac-tality for zinc metal leaves was made by M. Matsushita etal. in 1984. In their experiments [13], zinc metal leaves aregrown two-dimensionally by electrodeposition. The struc-tures clearly recover the pattern obtained by computersimulations for the DLA model proposed by T.A. Wittenand L.M. Sander in 1981. Figure 6 shows a typical zinc

3886 F Fractal Structures in Condensed Matter Physicsdendrite that was deposited on the cathode in one of theseexperiments. The fractal dimensionality Df D 1:66˙ 0:33was obtained by computing the density-density correla-tion function G(r) for patterns grown at applied voltagesof less than 8V.

The fractality of uniformly sized gold-colloid aggre-gates according to the DLCA was experimentally demon-strated by D.A. Weitz in 1984 [25]. They used trans-mission-electron micrographs to determine the fractal di-mension of this systems to be Df D 1:75. They also per-formed quasi-elastic light-scattering experiments to inves-tigate the dynamic characteristics of DLCA of aqueousgold colloids. They confirmed the scaling behaviors for thedependence of the mean cluster size on both time and ini-tial concentration.

These works were performed consciously to examinethe fractality of aggregates. There had been earlier worksexhibiting the mass-size scaling relationship for actualaggregates. 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”, im-plying the fractality of aggregates. However, the data avail-able at that stage were not adequate and scattered. There-fore, this work did not provide decisive results on the frac-tal dimensions of aggregates. There were smarter experi-ments by N. Tambo and Y. Watanabe in 1967 [20], whichprecisely determined fractal dimensions of flocs formedin an aqueous solution. These were performed withoutbeing aware of the concept of fractals. Original workswere published in Japanese. The English versions of theseworks were published in 1979 [21].We discuss these worksbelow.

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 wa-ter and wastewater treatment. The papers by N. Tamboand Y. Watanabe pioneered the discussion on the so-called fractal approach to floc structures; they performedtheir own settling experiments to clarifying the size depen-dences of mass densities for clay-aluminum flocs by us-ing Stokes’ law ur / �(r)r2 where � is the differencebetween the densities of water and flocs taking so-smallvalues � � 0:01–0:001 g/cm3. Thus, the settling veloci-ties ur are very slow of the order of 0:001m/sec for flocs ofsizes r � 0:1mm, which enabled them to perform precisemeasurements. Since flocs are very fragile aggregates, theymade the settling experiments with special cautions onconvection and turbulence, and by careful and intensiveexperiments of flocculation conditions. They confirmed

from thousands of pieces of data the scaling relationshipbetween settling velocities ur and sizes of aggregates suchas ur / rb . 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 cwere found to take values from 1.25 to 1.00 depending onthe aluminum-ion concentration, showing that the frac-tal dimensions become Df D 1:75 to 2.00 with increas-ing aluminum-ion concentration. This is because the re-pulsive force between charged clay-particles is screened,and van der Waals attractive force dominates between thepair of particles. It is remarkable that these fractal dimen-sions Df show excellent agreement with those determinedfor actual DLCA and RLCA clusters in the 1980s by us-ing various experimental and computer simulation meth-ods. Thus, they had found that the size dependences ofmass densities of flocs are controlled by the aluminum-ion concentration dosed/suspended particle concentra-tion, which they named the ALT ratio. These correspond

Fractal Structures in CondensedMatter Physics, Figure 7Observed scaling relations between floc densities and their di-ameters where aluminum chloride is used as coagulants. Af-ter [21]

Fractal Structures in Condensed Matter Physics F 3887to the transition from DLCA (established now taking thevalue of Df � 1:78 from computer simulations) process tothe RLCA one (established at present from computer sim-ulations as Df � 2:11). The ALT ratio has since the publi-cation of the paper been used in practice as a criterion forthe coagulation to produce flocs with better settling prop-erties and less sludge volume.We show their experimentaldata in Fig. 7, which demonstrate clearly that flocs (aggre-gates) are fractal.

Aerogels

Silica aerogels are extremely light materials with porosi-ties as high as 98% and take fractal structures. The initialstep in the preparation is the hydrolysis of an alkoxysi-lane Si(OR)4, where R is CH3 or C2H5. The hydrolysisproduces silicon hydroxide Si(OH)4 groups which poly-condense into siloxane bonds –Si–O–Si–, and small par-ticles start to grow in the solution. These particles bindto each other by diffusion-limited cluster-cluster aggrega-tion (DLCA) (see Sect. “Aggregates and Flocs”) until even-tually they produce a disordered network filling the re-action volume. After suitable aging, if the solvent is ex-tracted above the critical point, the open porous structureof the network is preserved and decimeter-size monolithicblocks 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 detec-tors, supports for catalysis, or thermal insulators.

Silica aerogels possess two different length scales. Oneis 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 ho-mogeneous porous glass. Aerogels have a very low thermalconductivity, solid-like elasticity, and very large internalsurfaces.

In elastic neutron scattering experiments, the scatter-ing differential cross-section measures the Fourier compo-nents of spatial fluctuations in the mass density. For aero-gels, the differential cross-section is the product of threefactors, and is expressed by

d�d˝D Af 2(q)S(q)C(q) C B : (25)

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

by B. The structure factor S(q) is proportional to the spatialFourier transform of the density-density correlation func-tion defined by (16), and is given by (17). Since the struc-ture 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, whileS(q) is independent of q for q� 1. The wavenumberregime 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 trans-fer (q� 1) 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.

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

3888 F Fractal Structures in Condensed Matter PhysicsThe results in Fig. 8 by R. Vacher et al. [22] are

from small-angle neutron scattering experiments on sil-ica aerogels. The various curves are labeled by the macro-scopic 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 pre-sented even in the particle regime q > 0:15Å-1 to em-phasize that the fits do not apply in the region, particu-larly for the denser samples. Remarkably, Df is indepen-dent 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�Df 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 trans-port on fractal support. For the diffusion of a particleon any d-dimensional ordinary Euclidean space, it is wellknown that the mean-square displacement hr2(t)i is pro-portional to the time such as hr2(t)i / t for any Euclideandimension 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 / t2/dw ; (26)

where dw is termed the exponent of anomalous diffusion.The exponents is expressed as dw D 2C � with a pos-itive � > 0 (see (31)), implying that the diffusion be-comes slower compared with the case of normal diffusion.This is because the inequality 2/dw < 1 always holds. Thisslow diffusions on fractal supports are called anomalousdiffusion.

The scaling relation between the length-scale andthe time-scale can be easily extended to the problem ofatomic vibrations of elastic fractal-networks. This is be-cause various types of equations governing dynamics canbe mapped onto the diffusion equation. This implies thatboth equations are governed by the same eigenvalue prob-lem, namely, the replacement of eigenvalues ! ! !2 be-tween the diffusion equation and the equation of atomicvibrations is justified. Thus, the basic properties of vibra-tions of fractal networks, such as the density of states,the dispersion relation and the localization/delocalizationproperty, can be derived from the same arguments fordiffusion on fractal networks. The dispersion relation be-tween the frequency ! and the wavelength �(!) is ob-tained from (26) by using the reciprocal relation t ! !�2(here the diffusion problem is mapped onto the vibrational

one) and hr2(t)i ! �(!)�2. Thus we obtain the disper-sion relation for vibrational excitations on fractal networkssuch as

! / �(!)dw/2 : (27)

If dw D 2, we have the ordinary dispersion relation! / �(!) for elastic waves excited on homogeneoussystems.

Consider the diffusion of a random walker on a perco-lating fractal network. How does hr2(t)i behave in the caseof fractal percolating networks? For this, P.G. deGennes 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. Ben-Avraham and S. Havlin in 1982 [3] investigated this prob-lem in terms of Monte Carlo simulations. The work byY. Gefen [9] triggered further developments in the dynam-ics of fractal systems, where the spectral (or fracton) di-mension ds is a key dimension for describing the dynamicsof fractal networks, in addition to the fractal dimensionDf.The fractal dimension Df characterizes how the geometri-cal distribution of a static structure depends on its lengthscale, whereas the spectral dimension ds plays a centralrole in characterizing dynamic quantities on fractal net-works. These dynamical properties are described in a uni-fied way by introducing a new dynamic exponent calledthe spectral or fracton dimension defined by

ds D2Dfdw

: (28)

The term fracton, coined by S. Alexander and R. Orbachin 1982 [1], denotes vibrational modes peculiar to frac-tal 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 ofanomalous diffusion dw by illustrating percolation frac-tal networks. The mean-square displacement hr2(t)i aftera sufficiently long time t should follow the anomalous dif-fusion described by (26). For a finite network with a size� , the mean-square distance at sufficiently large time be-comes hr2(t)i � �2, so we have the diffusion coefficient foranomalous diffusion from (26) such as

D / �2�dw : (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)

Fractal Structures in Condensed Matter Physics F 3889where 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 expo-nent for the correlation length defined by � / jp � pcj�� ,respectively. Comparing (29) and (30), we have the rela-tion between exponents such as

dw D 2Ct � ˇ�D 2C � : (31)

Due to the condition t > ˇ, and hence � > 0, implyingthat the diffusion becomes slow compared with the case ofnormal diffusion. This slow diffusion is called anomalousdiffusion.

Spectral Density of States and Spectral Dimensions

The spectral density of states of atomic vibrations is themost fundamental quantity describing the dynamic prop-erties of homogeneous or fractal systems such as specificheats, heat transport, scattering of waves and others. Thesimplest derivation of the spectral density of states (abbre-viated, SDOS) of a homogeneous elastic system is givenbelow. The density of states at ! is defined as the numberof modes per particle, which is expressed by

D(!) D 1

!Ld

; (32)

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

! 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, wecan replace the frequency ! by an arbitrary !. There-fore, we obtain the conventional Debye density of states as

D(!) / !d�1 : (35)

It should be noted that this derivation is based on the scaleinvariance of the system, suggesting that we can derive the

SDOS for fractal networks in the same line with this treat-ment. Consider the SDOS of a fractal structure of size Lwith fractal dimension Df. The density of states per parti-cle at the lowest frequency! for this system is, as in thecase of (32), written as

D(!) / 1LDf !

: (36)

Assuming that the dispersion relation for! correspond-ing to (33) is

! / L�z ; (37)

we can eliminate L from (36) and obtain

D(!) / !Df/z�1 : (38)

The exponent z of the dispersion relation (37) is evaluatedfrom the exponent of anomalous diffusion dw. Consider-ing 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 as

L / !�2/dw : (39)

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

z D dw2D Df

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 networksis found to be

D(!) / !ds�1 ; (41)

and the dispersion relation (39) becomes

! / L(!)�Df/ds : (42)

For percolating networks, the spectral dimension is ob-tained from (40)

ds D2Df2C � D

2(Df2( C � � ˇ : (43)

This exponent ds is called the fracton dimension afterS. Alexander and R. Orbach [1] or the spectral dimen-sion after R. Rammal and G. Toulouse [17], hereafter weuse the term spectral dimension for ds. S. Alexander andR. Orbach [1] estimated the values of ds for percolat-ing networks on d-dimensional Euclidean lattices fromthe known values of the exponents Df; (; � and ˇ. They

3890 F Fractal Structures in Condensed Matter Physicspointed out that, while these exponents depend largelyon 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 tis related to ds through (43), which means that the con-ductivity �dc � (p � pc)t is also characterized by the spec-tral dimension ds. In this sense, the spectral dimension dsis an intrinsic exponent related to the dynamics of fractalsystems. 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 perco-lation networks at the percolation threshold p D pc aregiven in Fig. 9a and b, which were calculated by K. Yakuboand T. Nakayama in 1989. These were obtained by large-scale computer simulations [27]. At p D pc, the correla-tion length diverges as � / jp � pcj�� and the networkhas a fractal structure at any length scale. Therefore, frac-ton SDOS should be recovered in the wide frequency range!L ! !D, where !D is the Debye cutoff frequencyand !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 (abbrevi-ated, 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.

Fractal Structures in Condensed Matter Physics, Figure 9a Spectral densities of states (SDOS) per atom for 2d, 3d, and4d BP networks at p D pc. The angular frequency! is definedwithmassunitsmD 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

The spectral dimension ds is obtained as ds D 1:33˙ 0:11from Fig. 9a, whereas data in Fig. 9b give the more pre-cise value ds D 1:325˙ 0:002. The SDOS and the inte-grated SDOS for 3d BP networks at pc D 0:249 are givenin Fig. 9a and b by the filled triangles (middle). The spec-tral dimension ds is obtained as ds D 1:31˙ 0:02 fromFig. 9a and ds D 1:317˙ 0:003 from Fig. 9b. The SDOSand the integrated SDOS of 4d BP networks at pc D 0:160.

A typical mode pattern of a fracton on a 2d percola-tion network is shown in Fig. 10a, where the eigenmodebelongs to the angular frequency ! D 0:04997. To bringout the details more clearly, Fig. 10b by K. Yakubo andT. Nakayama [28] shows cross-sections of this fractonmode along the line drawn in Fig. 10a. Filled and opencircles represent occupied and vacant sites in the perco-lation 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 char-acter 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 net-work) move in phase, and fall off sharply at their edges.

The spectral dimension can be obtained exactly for de-terministic fractals. In the case of the d-dimensional Sier-pinski gasket, the spectral dimension is given by [17]

ds D2 log(d C 1)log(d C 3) :

Fractal Structures in Condensed Matter Physics F 3891

Fractal Structures in Condensed Matter Physics, Figure 10a Typical fracton mode (! D 0:04997) on a 2d network. Bright region represents the large amplitude portion of the mode. b Cross-section of the fracton mode shown in a along thewhite line. The four figures are snapshots at different times. After [28]

We see from this that the upper bound for a Sierpinski gas-ket is ds D 2 as d !1. The spectral dimension for theMandelbrot–Given fractal depicted is also calculated ana-lytically as

ds D2 log 8log 22

D 1:345 : : : :

This value is close to those for percolating networks men-tioned above, in addition to the fact that the fractal dimen-sion ds D log 8/ log 3 of the Mandelbrot–Given fractal isclose to Df D 91/48 for 2d percolating networks and thatthe 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] observedfracton 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 physicshas developed by making efforts to elucidate the physi-cal mechanisms of smaller and smaller structures such asmolecules, atoms, and elementary particles. An example in

condensed matter physics is the band theory of electronsin solids. Energy spectra of electrons can be obtained byincorporating group theory based on the translational androtational symmetry of the systems. The use of this math-ematical tool greatly simplifies the treatment of systemscomposed of 1022 atoms. If the energy spectrum of a unitcell molecule is solved, the whole energy spectrum of thesolid can be computed by applying the group theory. Inthis context, the problem of an ordered solid is reducedto that of a unit cell. Weakly disordered systems can behandled by regarding impurities as a small perturbation tothe corresponding ordered systems. However, a differentapproach is required for elucidating the physical proper-ties of strongly disordered/complex systems with correla-tions, or of medium-scale objects, for which it is difficultto find an easily identifiable small parameter that would al-low a perturbative analysis. For such systems, the conceptof fractals plays an important role in building pictures ofthe realm of nature.

Our established knowledge on fractals is mainly due toexperimental observations or computer simulations. Theresearches are at the phenomenological level stage, not atthe intrinsic level, except for a few examples. Concerningfuture directions of the researches on fractals in condensed

3892 F Fractal Structures in Condensed Matter Physicsmatter 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 ba-sic aspect such as the problem “Why are there numerousexamples showing fractal structures in nature/condensedmatter?” However, this type of question is hard. The ki-netic growth-mechanisms of fractal systems have a richvariety of applications from the basic to applied sciencesand attract much attention as one of the important sub-jects in non-equilibrium statistical physics and nonlin-ear physics. Network formation in society is one examplewhere the kinetic growth is relevant. However, many as-pects related to the mechanisms of network formations re-main puzzling because arguments are at the phenomeno-logical stage. If we compare with researches on Brown-ian 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 no-table that the DLA is a stochastic version of the Hele–Shawproblem, the flow in composite fluids with high and lowviscosities: the particles diffuse in the DLA, while the fluidpressure diffuses in Hele–Shaw flow [19]. These are deeplyrelated to each other and involve many open questions forbasic physics and mathematical physics.

Concerning the opposite direction, one of the impor-tant issues in fractal research is to explore practical usesof fractal structures. In fact, the characteristics of fractalsare applied to many cases such as the formation of tailor-made nano-scale fractal structures, fractal-shaped anten-nae with much reduced sizes compared with those of ordi-nary antennae, and fractal molecules sensitive to frequen-cies in the infrared region of light.

Deep insights into fractal physics in condensed matterwill open the door to new sciences and its application totechnologies in the near future.

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Fractals andWavelets:What CanWe Learn on Transcriptionand Replication fromWavelet-BasedMultifractal Analysis of DNASequences?

ALAIN ARNEODO1, BENJAMIN AUDIT1,EDWARD-BENEDICT BRODIE OF BRODIE1,SAMUEL NICOLAY2, MARIE TOUCHON3,5,YVES D’AUBENTON-CARAFA4, MAXIME HUVET4,CLAUDE THERMES41 Laboratoire Joliot–Curie and Laboratoire de Physique,ENS-Lyon CNRS, Lyon Cedex, France

2 Institut de Mathématique, Université de Liège,Liège, Belgium

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 Pierreet Marie Curie, Paris, France

Article Outline

GlossaryDefinition of the SubjectIntroductionA Wavelet-BasedMultifractal Formalism:

The Wavelet Transform Modulus Maxima MethodBifractality of Human DNA Strand-Asymmetry Profiles

Results from TranscriptionFrom the Detection of Relication Origins Using

the Wavelet Transform Microscope to the Modelingof Replication in Mammalian Genomes

A Wavelet-BasedMethodology to DisentangleTranscription- and Replication-Associated StrandAsymmetries Reveals a Remarkable Gene Organizationin the Human Genome

Future DirectionsAcknowledgmentsBibliography

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 proper-ties that can either fluctuate from point to point (mul-tifractal) or be homogeneous (monofractal). Mathe-matically, these properties should hold over all scales.However, in the real world, there are necessarily lowerand 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. TheWT has been early recognized as a mathematical mi-croscope that is well adapted to characterize the scale-invariance properties of fractal objects and to revealthe hierarchy that governs the spatial distribution ofthe singularities of multifractal measures and func-tions. More specifically, the WT is a space-scale analy-sis which consists in expanding signals in terms ofwavelets that are constructed from a single function,the analyzing wavelet, by means of translations anddilations.