Physics of Fractal Operators (Institute for Nonlinear Science)

In Chapter One we review the foundations of statistieal physies and frac­ tal functions. Our purpose is to demonstrate the limitations of Hamilton’s equations of motion for providing a dynamical basis for the statistics of complex phenomena. The fractal functions are intended as possible models of certain complex phenomena; physical.systems that have long-time mem­ ory and/or long-range spatial interactions. Since fractal functions are non­ differentiable, those phenomena described by such functions do not have dif­ ferential equations of motion, but may have fractional-differential equations of motion. We argue that the traditional justification of statistieal mechan­ ics relies on aseparation between microscopic and macroscopie time scales. When this separation exists traditional statistieal physics results. When the microscopic time scales diverge and overlap with the macroscopie time scales, classieal statistieal mechanics is not applicable to the phenomenon described. In fact, it is shown that rather than the stochastic differential equations of Langevin describing such things as Brownian motion, we ob­ tain fractional differential equations driven by stochastic processes.

From the reviews:

"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...

"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."

- UK Nonlinear News

"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 mainstream and of great general relevance. ... Obviously, the book is also of great relevance to the researcher who may need to become acquainted with Fractal Calculus ... . I am glad I got to know this book." (Henrik Jensen, UK Nonlinear News, February, 2004)

"Physics of Fractal Operators ... is a timely introduction that discusses the basics of fractional calculus. ... Physics of Fractal Operators, which actively promotes the use of fractional calculus in physics, may help teachers develop an appropriate curriculum. ... the book’s abundance of material makes it very useful to researchers working in the field of complex systems and stochastic processes. It should help those who want to teach fractional calculus and it will definitely motivate those who want to learn ... ." (Igor M. Sokolov, Physics Today, December, 2003)

"The main merit of this well-written book is that it brings out rather clearly the relevance of the fractional calculus leading to the fractal operators and fractal functions. ... Each chapter contains an extensive list of relevant references. ... The overall style of presentation of the material covered in this book makes it rather useful for physicists and applied mathematicians carrying out a self-study of the fractal calculus and its applications." (Suresh V. Lawande, Mathematical Reviews, 2004 h)

"‘Physics of Fractal Operators’ is one of the great ideas books of our time. It may well become one of the most influential books with the paradigm of using fractional calculus to describe systems with emerging and evolving fractal complexities becoming widely used across the sciences. This important book should be mandatory reading for all PhD students in physics, and it should be at the side of all scientists working with fractals and complexity." (B I Henry, The Physicist, Vol. 40 (5), 2003)

"This book introduces the reader to the interesting mathematical notion of fractal operators and its usefulness to physics. ... a comprehensive, well written introduction to the subject ... useful to researchers and teachers alike. It is indeed targeted towards a wide, non specialist audience and provides the mathematical basis of fractional calculus ... . This book offers a lot of high-quality material to learn from and was definitely a very interesting and enjoyable read for me." (Yves Caudano, Physicalia, Vol. 28 (4-6), 2006)