UK Nonlinear News, February 2004

Physics of Fractal Operators

By Bruce J West, Mauro Bologna and Paolo Grigolini

Reviewed by Henrik Jensen

354 pp, 23 figs., EUR 79.95, GBP 56.00, US $ 69.95 
ISBN 0-387-95554-2

Have you ever wondered about whether one can define differential derivative of non integer order and how useful these fractal derivatives would be? If the answer is yes this is the book to look at. The book is written by physicists with a pragmatic audience in mind. It contains a very thorough and clearly written discussion of the mathematical foundation as well as the applications to important and interesting mathematical and physical problems. All the topics are very main stream and of great general relevance.

The authors approach is guided by their observation that the calculus invented by Newton's and Leibniz's for smooth functions is simply not able to describe the non-analytic nature of many very important and common cases. The simplest and probably most wide spread of such examples is the trajectory of a Brownian particle or a random walker. The usual integer derivative is simply to be considered as a special case of a much larger, and more powerful, class of derivatives. It is curious to learn on page 1 that Leibniz himself in 1695 considered how to define the 1/2-power derivative of a monomial. Clearly, the non-integer derivatives are not just the latest fashion.

Although the authors open the book with the words: "This is not a text book", it could certainly serve the purpose as an excellent introductory text accompanying a course, say, on Dynamics either in Applied Mathematics or in Physics. The book is so carefully explain and so impressively comprehensive that it would be excellent an undergraduate as well as graduate course - one would, though, have to construct problems for the class since no the book contains no problems or questions at all. Obviously, the book is also of great relevance to the researcher who may need to become acquainted with Fractal Calculus in order to consider if Fractal Calculus is useful to whatever problem one is currently involved with in Statistical Mechanics, Complex Systems, Dynamical Systems etc.

The book starts out with two chapters in which the shortcomings of integer derivatives are discussed. Here, as in the rest of the book, physical and mathematical arguments are presented in parallel. To me this makes the reading more easy and enjoyable. Having realised the inadequacy of ordinary calculus we are now primed for the presentation of Fractal Calculus. This is done in the next three chapters where we learn about the elementary properties of fractal derivatives, generalised exponential and trigonometric functions. Fractal Fourier and Laplace transforms are introduced next in a reasonably self-contained way. A student won't really need to know much about either Fourier or Laplace transforms before hand. All the most important properties are introduced. Constantly many relevant and illuminating comments are given and each chapter contains a bibliography to the most essential papers and books.

The second half of the book contains the discussion of the applications of Fractal Calculus. There are three chapters of which the first is on Fractal randomness. The chapter focuses on random walks, stochastic time series and evolution equations (Fokker-Planck and Levy) for probability densities. The second application chapter is concerned with Fractal Rheology, i.e. a topic from materials science. Although this might sound a bit engineering like and therefore an unlikely arena for the demonstration of a new and advanced type of calculus, we are in fact here presented with concepts such as fractal memory, fractal viscoelasticity as well as path integrals. One is explained how fractal derivatives enables a quantitative description of stress relaxation experiments. The third of the application chapters discuss Gaussian processes which are non-Markovian due to long-time memory with a focus on fractal Brownian motion as introduced by Mandelbrot and van Ness. The last chapter is concerned with fractal propagation which leads to fractal eigenvalue problems and the fractal oscillator among other topics.

I am glad I got to know this book. I don't know yet whether fractal calculus will be of crucial importance to my own research in statistical mechanics and complex systems. But I got the feeling from this book that this might very well be the case. And if this happens, I now know exactly where to go for a highly readable and thorough introduction to the field. I think the book deserves to be present in mathematics and physics libraries. And I believe many interesting undergraduate and graduate projects in mathematics and its applications can start out from this book.

I have one little complaint. We all think of Springer as producing extremely high quality, and rather costly, books. The content of this book is first class, but I find it a bit surprising that the quality of the printing of the figures is so far below the quality of the printed text. Perhaps the book was produced entirely by the (in that case very careful) authors without the involvement of a copyeditor? I found one typo that indicates this: namely a "[?]" (on p. 249) instead of a reference number. This is of course what often happens when we use LaTex and is normally spotted by the copyeditor.

Nevertheless, get hold of the book for its exciting and well presented content. If you can't afford the book for yourself, then try to get it for a library near you.

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

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