Physics of Fractal Operators

Bruce West, Mauro Bologna, Paolo Grigolini

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English

Springer-Verlag New York Inc.
14 January 2003

Fractal geometry; Analytic topology

Series: Institute for Nonlinear Science

This text describes how fractal phenomena, both deterministic and random, change over time, using the fractional calculus.

The intent is to identify those characteristics of complex physical phenomena that require fractional derivatives or fractional integrals to describe how the process changes over time.

The discussion emphasizes the properties of physical phenomena whose evolution is best described using the fractional calculus, such as systems with long-range spatial interactions or long-time memory. In many cases, classic analytic function theory cannot serve for modeling complex phenomena; ""Fractal Operators"" shows how classes of less familiar functions, such as fractals, can serve as useful models in such cases.

Because fractal functions, such as the Weierstrass function (long known not to have a derivative), do in fact have fractional derivatives, they can be cast as solutions to fractional differential equations.

The traditional techniques for solving differential equations, including Fourier and Laplace transforms as well as Green's functions, can be generalized to fractional derivatives. Fractal Operators addresses a general strategy for understanding wave propagation through random media, the nonlinear response of complex materials, and the fluctuations of various forms of transport in heterogeneous materials. This strategy builds on traditional approaches and explains why the historical techniques fail as phenomena become more and more complicated.

By: Bruce West, Mauro Bologna, Paolo Grigolini
Imprint: Springer-Verlag New York Inc.
Country of Publication: United States
Edition: 2003 ed.
Dimensions: Height: 235mm, Width: 155mm, Spine: 22mm
Weight: 747g
ISBN: 9780387955544
ISBN 10: 0387955542
Series: Institute for Nonlinear Science
Pages: 354
Publication Date: 14 January 2003
Audience: College/higher education , Professional and scholarly , Professional & Vocational , A / AS level , Further / Higher Education
Format: Hardback
Publisher's Status: Active

1 Non-differentiable processes.- 1.1 Classical mechanics.- 1.2 Langevin equation.- 1.3 Comments on the physics of the fractional calculus.- 1.4 Commentary.- 2 Failure of traditional models.- 2.1 Fractals; geometric and otherwise.- 2.2 Generalized Weierstrass function.- 2.3 Fractional operators.- 2.4 Intervals of the generalized Weierstrass function.- 2.5 Commentary.- 3 Fractional dynamics.- 3.1 Elementary properties of fractional derivatives.- 3.2 The generalized exponential functions.- 3.3 Parametric derivatives.- 3.4 Commentary.- 4 Fractional Fourier transforms.- 4.1 A brief review of Fourier analysis.- 4.2 Linear fields.- 4.3 Fourier transforms in the fractional calculus.- 4.4 Generalized Fourier transform.- 4.5 Commentary.- 5 Fractional Laplace transforms.- 5.1 Solving differential equations.- 5.2 Generalized exponentials.- 5.3 Fractional Green’s functions.- 5.4 Commentary.- 6 Fractional randomness.- 6.1 Ordinary random walk.- 6.2 Continuous-time random walk.- 6.3 Fractional random walks.- 6.4 Fractal stochastic time series.- 6.5 Evolution of probability densities.- 6.6 Langevin equation with Lévy statistics.- 6.7 Commentary.- 7 Fractional Rheology.- 7.1 History and definitions.- 7.2 Fractional relaxation.- 7.3 Path integrals.- 7.4 Commentary.- 8 Fractional stochastics.- 8.1 Fractional stochastic equations.- 8.2 Memory kernels.- 8.3 The continuous master equation.- 8.4 Back to Langevin.- 9 The ant in the gurge metaphor.- 9.1 Lévy statistics and renormalization.:.- 9.2 An ad hoc derivation.- 9.3 Fractional eigenvalue equation.- 9.4 Fractional stochastic oscillator.- 9.5 Fractional propagation-transport equation.- 9.6 Commentary.- 10 Appendix.- 10.1 Special functions.- 10.2 Fractional derivatives.- 10.3 Mellin transforms.

Reviews for Physics of Fractal Operators

"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’sabundance 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)"