Institute for Nonlinear Science. Springer NewYork Berlin Heidelberg HongKong London Milan Paris Tokyo

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1 Institute for Nonlinear Science Springer NewYork Berlin Heidelberg HongKong London Milan Paris Tokyo

2 Institute für Nünlinear Science Henry D.I. Abarbanel Analysis ofchaotic Series (1996) Jordi Garcia-Ojalvo, Jose M. Sancho Noise in Spatially Extended Systems (1999) Leon Glass, Peter Hunter, Andrew McCullogh (Eds.) Theory 0/ Heart: Biomechanics, Biophysics, and Nonlinear Dynamic 0/ Cardiac Function (1991) Mark Millonas (Ed.) Fluctuations and Order: The New Synthesis (1996) Linda E. Reich! The Transition to Chaos in Conservative Classical Systems: Quantum Manifestations (1992) Bruce West, Mauro Bologna, Paolo Grigolini Physics 0/ Fractal Operators (2003)

3 Bruce J. West Mauro Bologna Paolo Grigolini Physics of Fractal Operators With 23 Illustrations Springer

4 Bruce J. West Department of the Anny U.S. Army Research Laboratory Army Research Office P.O. Box Research Triangle Park, NC USA Mauro Bologna Paolo Grigolini Department of Physics University of North Texas Denton, TX USA Editorial Board Institute for Nonlinear Science, University of California-San Diego Henry D.I. Abarbanel, Physics (Scripps Institution of Oceanography) Morteza Gharib, Applied Mechanics and Engineering Sciences Michael E. Gilpin, Biology Walter Heller, Economics Kat ja Lindenberg, Chemistry Manuel Rotenberg, Electrical and Computer Engineering John D. Simon, Chemistry Library of Congress Cataloging-in-Publication Data West, Bruce J. Physics of fractal operators / Bruce West, Mauro Bologna, Paolo Grigolini. p. cm. - (Institute for nonlinear seien ce) Includes bibliographie al references and index. ISBN ISBN (ebook) DOI / I. Fractional calculus. I. Bologna, Mauro. II. Grigolini, Paolo. III. Title. IV. Institute for nonlinear science (Springer-Verlag) QC20.7.F75 W I 5'5--dc Printed on acid-free paper Springer-Verlag New York, Inc. Softcover reprint ofthe hardcover 1st edition 2003 All rights reserved. This work may not be translated or copied in whole or in part without the written pennission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights I SPIN Typesetting: Pages created by the authors using a Springer TEX macro package. Springer-Verlag New York Berlin Heidelberg A member of Bertf'lsmannSpringer Science+Business Media GmbH

5 Preface In Chapter One we review the foundations of statistieal physies and fractal 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 memory and/or long-range spatial interactions. Since fractal functions are nondifferentiable, those phenomena described by such functions do not have differential equations of motion, but may have fractional-differential equations of motion. We argue that the traditional justification of statistieal mechanics 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 obtain fractional differential equations driven by stochastic processes. We explore the limitations for modeling complex phenomena using the analytic functions generally taught in mathematieal physics courses in Chapter Two. The need far less familiar functions, such as fractals, for modeling complex phenomena is discussed. The definitions of fractal functions, fractal dimensions, and statistieal fractals are reviewed. The generalized Weierstrass function (GWF) is argued to be a fractal function, to have self-similar scaling, and is used to motivate the definition of fractional integrals and derivatives. Some elementary concepts from the fractional calculus are introduced and the non-differentiable GWF is demonstrated to have fractional integrals and derivatives. In fact the fractional derivative (integral) of a regular function is shown to yield a fractal function. The extension of the familiar concepts from the traditional calculus, such as the chain rule for derivatives and the Leibniz rule for the derivative of products of functions, to the fractional calculus, is made in Chapter Three. We adopt a formalism for the fractional derivative operator that is different from that discussed in Chapter Two and consequently define generalizations of the exponential function, the trigonometric functions, and the hyperbolic trigonometrie functions. These new functions are very useful in generalizing such concepts as the Fourier transform for the analysis of nonanalytic functions. More important, we discuss the necessary properties for physical phenomena to be described by such functions. We briefly review Fourier analysis in Chapter Four at a level sufficient to understand the modeling of most linear physical phenomena. For example, we look at nondispersive wave propagation and ordinary diffusion. This discussion motivates the use of the generalized exponential function to define a generalization of the Fourier transform and the Dirac delta function. We use the Weyl fractional operators to suggest a generalization of

6 vi PREFACE Fourier transforms and from this generalization construct fractional wave equations and fractional diffusion equations. In Chapter Five we briefly review Laplace transforms at a level suflicient to understand how to solve systems of linear differential equations, whether ordinary or fractional. We discuss how to use the generalized exponential functions in the Laplace transform formalism to find unfamiliar inverse Laplace transforms. We explore how to solve an initial value problem using Laplace and Mellin transforms and relate the results to viscoelastic phenomena. In addition we introduce fractional Green's functions as an alternate method for solving such equations. The discussions in Chapter Six focus on ways to generate a stochastic process with long-term memory, that is, a random process with an inverse power-law correlation function. One way to model the non-local nature of the memory is by using the properties of fractional derivatives in time. We review ordinary random walks, continuous time random walks, and finally fractional random walks. We also investigate the properties of time series that are generated by complex phenomena and show that the inverse power-law memory is the signature of statistical fractal processes. This is shown to lead to Levy stable processes. We review some of the essential elements of c1assical rheology in Chapter Seven, in order to show that ordinary and partial differential equations are insuflicient to model the stress relaxation in viscoelastic materials. The fractional calculus is used to construct a general theory of rheology and the predictions compare favorably with experiments. In Chapter Eight the fractional calculus is used to obtain insight into the stochastic dynamical equations considered earlier. The exact solutions to these equations are obtained using the techniques of the fractional calculus developed in these lectures. The central moments are calculated exactly using the solutions to the fractional dynamical stochastic equation. We also discuss the kinds of physical systems that could give rise to such behavior. Wave propagation and transport in heterogeneous media, such as spatial fractals, are discussed in Chapter Nine. We find that the eigenfunction expansion technique for solving partial differential equations is applicable to these propagation-transport fractional equations describing fractional wave propagation and anomalous diffusion. We demonstrate that these exact solutions share a number of properties with familiar physical systems, such as relaxation to a Boltzmann equilibrium. A number of fractional derivatives and fractional integrals are briefly discussed in Chapter Ten, which is actually an Appendix. The special functions from mathematical physics, such as the Bessel function, the Hankel function, and so on, are shown to generalize and to have generators in terms of fractiönal derivatives. In addition, the details of some of the techniques used to solve differential equations in the text, inc1uding Fox functions and Mellin transforms, are included in this chapter.

7 Contents Preface... v 1 Non-differentiable proeesses Classieal mechanics Euler-Lagrange equations of motion Hamilton's equations of motion Langevin equation Hamiltonian model Stoehastie ealeulus Comments on the physies of the fraetional ealculus Markov approximation Inverse power-law memory Commentary Failure of traditional models Fraetalsj geometrie and otherwise Fraetal dimension Generalized Weierstrass flinction Fraetional operators Series representations of fractional operators Riemann-Liouville fraetional operators Some elementary fractional derivatives and integrals Intervals of the generalized Weierstrass function Fraetional integral of the IGWF Fraetional derivative of the IGWF What about the GWF? Conclusions about fractal funetions Commentary Fraetional dynamies Elementary properties of fraetional derivatives Constant funetions The generalized Leibniz rule The generalized exponential funetions Generalized trigonometrie funetions Parametrie derivatives Commentary Fraetional Fourier transforrns A brief review of Fourier analysis Linear fields Inhomogeneous linear wave fields Linear diffusive fields Fourier transforrns in the fraetional ealeulus Fourier transforrns of fractional derivatives Weyl's fraetional operators The Dirae delta function Generalized Fourier transform Examples of generalized transforms GeneFalized eonvolutions

8 viii CONTENTS 4.5 Commentary Fractional Laplace transforms Solving differential equations Extension of fractional powers Generalized exponentials Solutions to differential equations Incomplete symmetry Integral properties of the generalized exponential..., Fractional Green's functions Commentary Fractional randomness Ordinary random walk... ~ Continuous-time random walk Fractional random walks Inverse power-iaw spectra Fractional Brownian motion Fractal stochastic time series Colored noise Box dimension Power-law correlations Evolution of probability densities The Fokker-Planck equation The Levy evolution equation Langevin equation with Levy statistics Commentary Fractional Rheology History and definitions Complex moduli The standard model Fractional memory Fractional relaxation Using Fox functions A fractional theory of viscoelasticity Path integrals Wiener path integral Levy path integral Commentary Fractional stochastics Fractional stochastic equations A simpler equation Non-Markovian statistics Memory kernels The continuous master equation Back to Langevin InhomogeneoUs fractional solution Velocity autocorrelation function

9 CONTENTS ix Fractional mean-square displacement Cornmentary The ant in the gurge metaphor Levy statistics and renormalization An ad hoc derivation Fractional eigenvalue equation Fractional stochastic oscillator Fractional propagation-transport equation Cornmentary Appendix Special functions Jacobi and Laguerre Polynomials Applications of the generalized functions Bessel functions Fraetional derivatives j2-fraetional derivative Generalized trigonometrie funetions Miseellaneous integrals Complex integrals Mellin transforms Fox funetions Index 350