UK Nonlinear News, February 1999

Fractals, scaling and growth far from equilibrium

By P. Meakin

Reviewed by Miroslav M. Novak

Published by Cambridge University Press, 1998,
ISBN 0-521-45253-8, GBP 75.00

Attempts to understand macroscopic objects in terms of their microscopic constituents are fraught with problems. It suffices to recall the difficulties encountered when dealing with simple molecules containing a few atoms. As a result, another approach has been developed in the last two decades, based upon ideas originating in statistical physics. A realisation that a scale independent symmetry approach can describe many processes and systems, has dramatically improved our understanding, without the requirement to consider microscopic detail. These new ideas have facilitated the development of analytical tools, that make it possible to quantify a broad range of phenomena.

This monograph provides an extensive exposition of the fundamental aspect of scaling, pattern formation far-from-equilibrium and fractals. It covers a wide range of natural phenomena, but primarily concentrates on the diffusion-limited processes and the growth of rough surfaces. The strength of the book rests with application of and simulation techniques with fractals and scaling. The reader will find numerous descriptions of experiments, including their detailed discussion.

There are five chapters with two appendices, arranged in the following order

  1. Pattern Formation Far From Equilibrium. In general, under close-to equilibrium conditions, the formation of simple Euclidean shapes is prevalent and the ideas of equilibrium thermodynamics apply. However, non-equilibrium processes are responsible for the majority of natural shapes. Although by far the most applications relate to physical and life sciences, there are many examples of human activity (railroad network, growth of cities) alluded to that that exhibit fractal distribution. A large part of this section considers various growth processes and the resultant structures.
  2. Fractals and Scaling. There are many different types of fractals. This chapter describes some of the most common ones and considers several ways of quantifying them through their dimension. The ideas behind self-similarity, self-affinity and lacunarity are introduced, as are the ways of calculating various dimensionalities. In nature, the finite size of the structure introduces limitations on the idealized conditions, and interpretation of the results must take this effect into account. In the simplest scaling model, the underlying power law is described by a fixed index. More generally, a function is needed to describe the multiscaling behaviour, leading to multifractals.
  3. Growth Models. The growth paradigms (Eden model, diffusion-limited aggregation, ballistic deposition and percolation) form the basis for the study of the non-equilibrium growth. Particular emphasis is placed on a DLA model, which is still not fully understood from the theoretical standpoint. Some theoretical methods dealing with growth are also examined in this section.
  4. Experimental Studies. An exhaustive survey of experiments attempting to quantify the geometry of growth patterns can be found here. The DLA and percolation processes are central to this investigation. The chapter provides excellent overview of the latest work in this field.
  5. The Growth of Surface and Interfaces. Many systems can be described by uniform regions that are separated from other regions by a boundary, along which some of the properties change abruptly. This chapter describes some of the theoretical and computational advances in surface growth phenomena. It also contains an experimental section dealing with directed growth.

The linear stability of the moving boundary equations is described in the first appendix, whilst the characterization of multifractal set and its application to growth are dealt with in the Appendix 2.

The mathematical treatment of the book is carefully presented, emphasizing the results, rather than intricate mathematical detail. It is suitable for graduate students, with background in physical sciences. On the whole, the book provides readable treatment of the basics of fractal geometry and scaling, to facilitate quantification of the structures grown under the non-equilibrium conditions. An extensive bibliography of over 1300 references makes this monograph a solid contribution in the field.

UK Nonlinear News thanks Cambridge University Press for providing a copy of this book for review.

Miroslav M. Novak (

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