Multifractals and 1/ƒ Noise: Wild Self-Affinity in Physics (1963–1976)

448 Pages·1999·16.35 MB·English

by  Benoit B. Mandelbrot (auth.)

M ULTIFRACTALS and
1/f
NOISE
Springer-Verlag Berlin Heidelberg GmbH
SELECTED WORKS OF BENOIT B. MANDELBROT
REPRINTED, TRANSLATED, OR NEW
WITH ANNOTATIONS AND GUEST CONTRIBUTIONS
COMPANION TO THE FRACTAL GEOMETRY OF NATURE
Benoit B. Mandelbrot
MULTIFRACTALS and
1/f
NOISE
Wild Self-Affinity in
Physics (1963-1976)
SELECTA VOLUMEN
Includes contributions by
].M. Berger, ].-P. Kahane,
]. Peyri&re, and others
i
Springer
Benoit B. Mandelbrot
Mathematics Department
Yale University
New Haven, CT 06520-8283, USA
http://www.math.yale.edu/mandelbrot
and
IBM T.J. Watson Research Center
Yorktown Heights, NY 10598-0218, USA
Library ofCongress Cataloging-in-Publication Data
Mandelbrot, Benoit B.
Multifractals and 1ifnoise I Benoit B. Mandelbrot.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-4612-7434-6 (hardcover : alk. paper)
I. Multitractals. 2. Electronic noise. I. Title.
QA614.86.M27 1998
514' .742-dc21 98-3970
Printed on acid-free paper.
© 1999 Be110it B. Mandelbrot
Originally published by Springer-Verlag Berlin Heidelberg New York in 1999
Softcover reprint of the bardeover 1st edition 1999
All rights reserved. This work may not be translated or copied in whole or in part without the written
permission 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 inti:Jrmation storage and retricval, electronic adaptation, computer soft
ware, or by similar or dissimilar methodology now known or hereafter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the
former are not especially identified, is not to be taken as a sign that such names, as understood by thc
Trade Marksand Merchandise Marks Act, may accordingly be used freely by anyone.
Camera-ready copy provided by the author.
9 8 7 6 5 4 3 2 I
ISBN 978-1-4612-7434-6 ISBN 978-1-4612-2150-0 (eBook)
DOI 10.1007/978-1-4612-2150-0 SPIN 10679267
T o my wife, Aliette
I dedicate this intellectual fruit of mine,
her adoptive child
List of Chapters
In this List of Chapters, the sources given after the titles include (in parentheses)
the Ietter M followed by the year of publication and by the lower case Ietter that the
Bibliography uses to distinguish different texts published in the same year.
In the Bibliography, the items reproduced in this book and in Volumes N, H
and L of these Selecta are marked by a star followed by a chapter number, which in
some cases is incomplete or only tentative.
NP Preface (1998) .. . .. ... .. . . . .. .. ... .. .. .. .. ..... .. .. .. .. ... ..... .. ...... .. ..... .. .. . .. .. . .. ...... .. 1
I INTRODUCTIONS AND SHORT PIECES ............................ 17
N1 Panorama of grid-bound self-affine variability (1998) ............ 18
N2 Sketches of prehistory and history (1998) ................................. 62
N3 Scaling, invariants and fixed points (1998) ............................. 100
N4 Filtering and specifications of self-affinity (1998) .................. 111
NS Short excerpts (1964-1986) ......................................................... 117
II UNIFRACTAL ERRORS AND LEVY DUSTS ..................... 131
N6 New model for error dustering on telephone circuits
(Berger & M 1963). Additionaltests on dustering (1998) .... 132
N7 Self-similarity and conditional stationarity (M 1965c) .......... 166
viii
111 INTERMITTENT Vf NOISES AND
CONDITIONED RANDOM PROCESSES ........................... 207
NB 1/f noises and theinfrared catastrophe (M 1965b) ............. 208
N9 Co-indicator functions and related 1/f noises (M 1967i) .... 215
N10 Sporadic random functions and conditional spectra
self-similar examples and Iimits (M 1967b) ............................ 247
N11 Random sets of multiplicity for trigonometric series
(Kahane & M 1965) ...................................................................... 284
IV TURBULENCE AND MULTIFRACTAL MEASURES ....... 289
N12 Sporadic turbulence (M 1967k) .................................................. 290
N13 Intermittent free turbulence (M 1969b) .................................... 292
N14 Lognormal hypothesis and distribution of
energy dissipation in intermittent turbulence (M 1972j) ...... 294
N15 Intermittent turbulence in self-similar cascades:
divergence of high moments and
dimension of the carrier (M 1974f) ........................................... 317
N16 lterated random multiplications and invariance under
randomly weighted averaging (M 1974c) ................................ 358
N17 "On certain martingales of Benoit Mandelbrot"
Guest contribution (Kahane & Peyriere 1976) ........................ 372
N18 Intermittent turbulence and fractal dimension: the kurtosis
and the spectral exponent 5/3 + B (M 1976o) ...................... 389
N19 Fractal dimension, dispersion, and singularities of
fluid motion (M 1976c) ............................................................... 416
Cumulative Bibliography ........................................................... 419
Index ............................................................................................... 432
NP
Preface
CERTAIN NOISES, MANY ASPECTS OF TURBULENCE, and nearly
every aspect of finance, exhibit a level of temporal and spatial variability
whose "wildness" vividly impressed itself upon me in the early nineteen
sixties. I soon concluded that those phenomena cannot be described- and
much less handled, understood, or explained - by simply adapting or
extending slightly the statistical techniques that had sufficed in earlier
physics.
At an International Congress of Philosophy of Science (Jerusalem,
1964), I argued that successful existing statistical models represented a first
stage of indeterminism in science. As I saw it, the study of finance and
turbulence could not move forward without the recognition that those
phenomena must involve a new second stage of indeterminism. Alta
gether new mathematical tools were needed.
A related need was to acknowledge that all possible questions one
might raise must be sorted out in two categories. Same questions are
"well-posed," and can expect an answer. Other questions are "ill posed,"
and - at least for the time being - are not even worth raising. Same
questions that are well-posed in first-stage indeterminism should be
expected tobe ill-posed in the second stage.
This sharp distinction echoes one that Hadamard raised in mechanics.
In my work, it first arose in finance but soon became essential elsewhere.
In the context of climatology, this underlying idea is becoming widely
known to everyone therefore deserves mention, even though the topic is
pursued in M1998H rather than in this book. The question is that of
anthropogenic global warming. If climate follows the (implicit) assumption
that its fluctuations belang to first-stage indeterminism, it is legitimate to
2 PREFACE 0 0 NP
view those fluctuations as varying around a durable equilibrium level. If
so, to evaluate Man's influence on climate may be a well-posed problem.
If second-stage indeterminism prevails, the search for equilibrium might,
to the contrary, be an ill-posed problern and may have tobe rephrased.
The early technical papers in which I reported on all those issues are
being reprinted in several books, of which this is the second to appear.
Those books are largely independent of one another. They also contain
substantial additional material, but will be referred to by the old-fashioned
Latin word for "Selected Papers," which is Selecta.
Having met Edward Lorenz around the timeofthat 1964 Congress in
Jerusalem, I was privileged to follow the development of the theory of
deterministic chaos from the beginning of its flowering. In due time, I con
tributed to it by discovering the Mandelbrot set. However, this work does
not concern the theory of chaos, but- to use a good old-fashioned term
the theory of chances. Incidentally, an unintended but amusing loose
parallelism exists between what I called in 1964 a second stage of
indeterminism and chaos viewed as a second stage of determinism. Both
push back the "edge of messiness," beyond which simple descriptions are
not yet available.
My talk at the above-mentioned 1964 Jerusalem congress was not actu
ally printed until M1987r. This first reference is an opportunity to draw
attention to the style adopted throughout this book. The principle is
explained on the first page of the bibliography and can be illustrated by a
few examples. The Selecta books include M1997E, Ml998H and Ml998L,
as well as Ml997FE. The present book is denoted in other writings as
Ml998N. My book-length Essays on fractals are denoted as Ml9750,
Ml977F and Ml982F, the third one being also denoted as FGN.
Another idiosyncratic usage is this: instead of writing "the model pre
sented in M1972j{N14}," this book will often write "the M1972 model."
T
o continue in more specific terms, this book's subtitle lists two broad
themes. Both are essential to the study of the substantive topic and tool
listed in the title, and to many of my other contributions to science.
The theme of wild versus mild or slow variability amplifies and
deepens my 1964 theme of successive "stages of indeterminism." It has
become easiest to express by a metaphor that begins in physics. As is
well-known, mechanics relies on a unique set of laws of great generality.
However, there are several quite distinct states of matter. Best known are
gases, liquids, and solids. It is equally well-known that probability theory