Springer-Verlag 2000

Pages: 434

ISBN: 3-540-67462-4 (Hardcover).

The title alone is enough to make one realise that this is a very ambitious book. As if it were not enough to try to look at such a range of interesting mathematical and physical concepts, the author develops much of the mathematical and statistical machinery used by researchers in probability, statistical physics, stochastic processes and applied mathematics and the author is eager to make good use of the machinery throughout the book.

Topics covered include Gaussian and Levy Power Laws; Fractals and Multifractals; Statistical Mechanics; Critical Phenomena; Percolation and Directed Percolation and Self-Organised Criticality and Randomness in a variety of guises.

Personally, I have been waiting for a book that would try to work through the connections between these many, clearly related topics. I was a little disappointed that some of the topics, such as the Ising model were not illustrated by fully worked examples which would really enhance the understanding of those of us who are slower students. Additionally, some simple lines of code which could illustrate the fractals, the self-organised criticality, the randomness, etc. would have been a great aid. While I was able to improve my knowledge of the mathematical connections, it was an arduous path and I was only able to consult a small number of the extraordinary 832 (count them if you dare!) references to texts and original literature.

It is interesting to see yet another account of fractals in nature which seeks ways to justify the description of natural objects as fractals, but cannot offer a realistic mechanism for the way these objects arise. Perhaps the most well known example of this is in botany where fully grown plants may be described by fractals, but this gives no insight into how they actually grew to look thus. The end result of rivers and mountains are similarly described by fractals with no insight into the processes that created them. Perhaps Diffusion Limited Aggregation (DLA) is the answer, but the allusions are still not entirely convincing. Similarly, the seemingly ubiquitous power laws in nature are discussed with reference to several possible mechanisms, plausible in selected cases and leading to a description of self-organised criticality (SOC). In some ways, chapter 15 (on SOC) is the most valuable part of the book, as it trys to "cut through the crap" that is the hallmark of so many descriptions of SOC and it also trys to clearly link the ideas with mechanisms, physical descriptions and a mathematical framework. The extent to which this is successful depends upon the point of view of the reader, as there are many who would argue that our obsession with fitting power laws, etc. is the main reason that they seem to be so ubiquitous in nature.

Complex systems are of course fiendishly difficult to characterise and require many of the techniques as discussed in this book, as well as some that are yet to be invented. Geological and Atmospheric science are two areas where there is an increasing use of these techniques in order to try to describe observations and especially when attempting to use models in any predictive sense, such as the location of unknown ore deposits. Given the obvious economic and scientific interest, it really is valuable to examine all approaches to modeling and understanding complex systems and this book (despite its shortcomings) is to be commended for its contribution.

For the right combination of instructor and graduate student, this book would make an excessively diverting and stimulating course and could well lead on to research topics of real interest. Your library should have a copy, but it is not for the faint hearted.

A listing of books reviewed in `UK Nonlinear News`
is available.

`UK Nonlinear News` thanks
Springer Verlag for
providing a review copy of this book.

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Last Updated: 21st February 2001.

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