News, February 1999
Fractals, scaling and growth far from equilibrium
By P. Meakin
Reviewed by Miroslav M. Novak
Published by Cambridge University
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.
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
- 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.
- 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.
- 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.
- 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.
- 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
University Press for providing a copy of this book for
Miroslav M. Novak
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Last Updated: 5th January 1999.