book reviewdownloads.hindawi.com/journals/ddns/1998/326932.pdf · fractals. toapproachthe...

$: Book Reviewdownloads.hindawi.com/journals/ddns/1998/326932.pdf · fractals. Toapproachthe problemofcriticality in growth phenomena, multifractals are used as a tool. Multifractals$
Discrete Dynamics in Nature and Society, Vol. 2, pp. 73-75Reprints available directly from the publisherPhotocopying permitted by license only

(C) 1998 OPA (Overseas Publishers Association) N.V.Published by license under

the Gordon and Breach Science

Publishers imprint.Printed in India.

Book Review

Armin Bunde and Shlomo Havlin, (Fractals and Disordered Systems). Second Revised and EnlargedEdition, 408 pages with index and with 165 Figures and 10 Color Plates (Springer-Verlag, Berlin, 1996).

Shortly after Mandelbrot’s Fractal Geometry ofNature, there was explosive growth in fractalapplications to a variety of fields. Much like theoriginal development of "the calculus", fractalgeometry is a branch of mathematics inextricablytied to natural problems. It provides one of the firsttools for understanding complexity in both naturaland man-made settings. In a sense, it gives us a"calculus for complexity", allowing infinitelyjagged structures to be analyzed and interpreted.

The early stage in the growth of this new

discipline was largely phenomenological. A diver-sity of spatial and temporal behavior was examinedto see if scaling laws are obeyed. Fractal dimen-sions were ascribed to a range of structures. Thefractal dimension in a very loose sense replaces thederivative of calculus. Fractal curves, such as thejagged path of a lightning bolt, show scaleinvariance. That is, they show a similar jaggednessat any level of magnification. It is impossible totake the derivative of such a curve because thejaggedness persists at all scales. Regardless ofmagnification, a smooth region for drawing a

tangent does not exist. A consequence of this isthat fractal functions are non-analytic and are

governed by power law behavior. The fractaldimension appears in the exponent of these powerlaw expressions and provides a means of character-izing the degree of jaggedness of the function.

73

A very heady result in those early days of fractalswas the computer simulation by Witten andSanders of a diffusion limited aggregate (DLA).Here was a computer experiment that followed a

simple set of rules and produced a fractal structure.Not only that, it was the sort of image one sees in a

variety of physical phenomena ranging fromelectrochemical deposition to the growth of bacte-rial colonies. This early result suggested that suchcomputer experiments could lead from phenome-nology to a deeper understanding of fractals. Asthe present volume will attest to, this is anunfulfilled promise.

Fractals and Disordered Systems represents not

only the maturing of a field but also the inevitableconfronting of deep and difficult problems. The era

of phenomenology is over. It has been demon-strated that fractal structures pervade our lives.They exist virtually everywhere there is growth, beit in a plant such as cauliflower, in cracks andgrains of rock formations, in river basins or even inurban sprawl. As H.E. Stanley points out inChapter 1, the Metro of Paris has fractal-liketentacles. Interestingly, similar fractal dimensionsexist for a broad range of growth processes. Thefocus now centers on why fractals are so pervasivein nature. Is there a biological advantage to fractalgrowth patterns? If so, why do similar patternsappear in inorganic chemical systems? Why do

74 BOOK REVIEW

higher organizational structures have similar pat-terns? The goal now is to understand why suchdiverse structures are scale invariant. This questionruns throughout the edited volume.

Scale invariance has been familiar to physicistsand physical chemists for years. When a system isat a critical point between two phases, it will oftenshow scale invariance. For instance, at the criticalpoint between a gas and a liquid, the system willfluctuate wildly between the two states. Liquid-likeclusters of all sizes will be constantly forming andbreaking up. These clusters will exist on all lengthscales and are, therefore, considered to be scaleinvariant. They are said to have no "characteristiclength". This form of scale invariance has beenstudied extensively and there are mature and deeptheories that describe such phenomena.The problem facing the fractal field is that a

similar "criticality" occurs in growth patterns suchas DLA. The growth will hit a critical point wherescale invariant structures are formed. These areoften dendritic structures whose branches appearsimilar to magnified versions of branches of highergenerations. These patterns arise from frozen, non-

equilibrium structures and our level of under-standing of them is in its infancy. This is a centralchallenge revealed by Fractals and DisorderedSystems. Much of the book describes variousmathematical tools and approaches that brings us

to grips with this problem.The book begins with the chapter, Fractals and

Multifractals." The Interplay ofPhysics and Geome-try, by H. Eugene Stanley. This excellent chapterintroduces the basic concepts and definitions offractals. To approach the problem of criticality ingrowth phenomena, multifractals are used as a

tool. Multifractals are a generalization of thefractal concept to probability distributions. Themoments of a distribution of growth probabilitieswill often show power law scaling. One can thenassign a fractal dimension for each moment and,hence, the term multifractals. The multifractalspectrum provides a means of characterizing thedistribution. Stanley points out that the multi-fractal formalism provides parameters that are

mathematical analogues of thermodynamic quan-

tities. This allows the criticality of growth phenom-ena to be viewed in a similar context as phasetransitions.The multifractal approach of Chapter ties in

closely with the more mathematical treatments in

Chapter 4, Fractal Growth by Ammon Aharonyand Chapter 10 Exactly Self-similar Left-sidedMultifractals by Benoit B. Mandelbrot and CarlJ.G. Evertsz (with appendices by Rudolf H. Riediand Mandelbrot). These chapters focus on thenature of the multifractal spectrum of growthphenomena, especially DLA. They provide insight-ful views of the problems associated with theinterpretation of multifractal spectrum. They grap-ple with the question: Is growth multifractal? Thesechapters provide a deep and technical overview ofthe use of multifractals.

Stanley’s introductory chapter is followed bytwo fine chapters on percolation, simply entitledPercolation I by Armin Bunde and Shlomo Havlinand Percolation H by Shlomo Havlin and ArminBunde. Percolation models have been used to

describe such diverse phenomena as epidemics,forest fires and dielectric breakdown. Percolationhas considerable theoretical appeal because it

represents a "geometric phase transition". Addi-

tionally, it is an area where significant progress hasbeen made. Again, scale invariance appears fromcomputer models based on a simple set of rules.These two comprehensive and very readablechapters consecutively cover static and dynamicphenomena. These chapters also serve as introduc-tion to material that appears throughout the book.The first four chapters comprise more than half ofthe book and lay a firm basis for later more specificchapters. They present a coherent overview of the

theory of scale invariant structures and relate themto the underlying physics of disordered systems.The following four chapters move from theory

and computer simulations to a closer tie with theexperimental world. Fractures by Hans J. Herrmannis a self-contained chapter of great technologicalimportance. Herrmann details the development ofmodels of fracture in solids, starting with a

phenomenological description of elasticity andproceeding to lattice models. It is seen that certain

BOOK REVIEW 75

fracture phenomena can be realistically modeledwith lattice approaches. It is also seen how suchmodels give rise to fractal structures. Fracturecan be considered a growth phenomenon andas such has inherent fractal and multifractalcharacteristics.

Chapter 6 (Transport Across Irregular Interfaces."Fractal Electrodes, Membranes and Catalysts byBernard Sapoval) and Chapter 7 (Fractal Surfacesand Interfaces Jean-Frangois Gouyet, Michel Ros-so and Bernard Sapoval) are another pair ofcomplementary chapters. Unlike the percolationchapters, these chapters start with dynamics andthen move to structure. Chapter 6 is concernedwith transport processes across irregular surfaces.Biological examples of these are fluid exchange inthe root system of a tree and the exchange ofoxygen in the lung. A technologically importantexample is the optimization of the power of a carbattery by using a porous electrode. These pro-blems can be viewed as the "irrigation" ofstructures of large surface area and low volume.The chapter demonstrates that a wide range ofthese phenomena can be treated by a generalformulation of Laplacian transfer across irregularsurfaces. This problem is not a fractal one per se.

Yet a fractal interface provides a mathematicalconvenience that actually makes the problem moretractable. One can then use these results todistinguish phenomena due to the fractal natureof the interface from general disorder phenomena.

Fractals appear again as a useful tool inChapter 7. This chapter deals with the descriptionand properties of surfaces and interfaces, a topic ofconsiderable recent attention. Additionally, thenatural question of what creates specific interfacegeometries is asked. Interfaces reflect a competitionbetween forces such as surface tension that tend tosmooth them with forces that drive the system todisorder. The fractal formalism offers an obvioustool in handling objects of such complexity. Again,it is seen that for structures that are far fromequilibrium, simple physical processes such as

diffusion or aggregation give rise to fractal struc-tures. A variety of physical examples of suchbehavior are discussed. It is speculated that the

combination of random processes and non-equilib-rium conditions is ubiquitous in fractal formation.

Chapter 8, Fractals and Experiments by JorgenK. Kjems picks up themes of previous threechapters. This is an easy reading, qualitativechapter. It describes how you make fractals in thelaboratory under controlled conditions, how youmeasure their fractal dimension and what physicalproperties they have. It covers a number ofdifferent experimental scenarios that are physicalrealizations of the theoretical models, such as

DLA, discussed in earlier chapters. There is a niceexposition of the use of scattering techniques todetermine the fractal dimension. Fractality hasstrong implications for the mechanical, vibrationaland thermal properties of a structure. This chaptercollects a variety of results from the literature,providing a useful survey.

In 1969 Stuart Kauffman proposed a model todescribe cell differentiation in developing organ-isms. This model has grown into a whole class ofmodels known as cellular automata. This is thetopic of Chapter 9, Cellular Automata by DietrichStauffer. As in percolation, these models also showgeometric phase transitions. They also showhidden fractal and multifractal behavior. This is a

very brief chapter that may not be as tractable tothe layman. Nevertheless, there are a range ofinteresting biological applications that enhance theappeal of the subject.

This book brings together leaders in the field in acomprehensive survey of recent developments. It is

surprisingly cohesive for a multi-author work andthe editors should be given credit for maintainingthe uniformity of presentation. This is a sophisti-cated volume and not all of the chapters will appealto the layman. My main criticism of the text is thatit is not substantially different from the firstedition. Regardless, this volume is an excellententree to fractals and disordered systems andcaptures the excitement of the field.

T. GREGORY DEWEYDepartment of Chemistry and Biochemistry,

University of Denver, Denver, CO 80208, USATel.: 303-871-3100. Fax: 303-871-2254.

E-mail: [email protected].

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