From Newton to Mandelbrot: A Primer in Theoretical Physics with Fractals for the Macintosh®

233 Pages·1996·13.418 MB·English

by  Professor Dr. Dietrich Stauffer, Professor Dr. H. Eugene Stanley (auth.)

From Newton to Mandelbrot
Dietrich Stauffer H. Eugene Stanley
From Newton
to Mandelbrot
A Primer in Theoretical Physics
with Fractals for the Macintosh®
Second Enlarged Edition
With 50 Figures, 16 Colored Plates,
47 Problems, 82 Short Questions with Answers
and a 3.5" Macintosh Diskette
, Springer
Professor Dr. Dietrich Stauffer
Institut fiir Theoretische Physik, Universităt Koln, Ziilpicher StraBe 77
D-50937 Koln, Germany
Professor Dr. H. Eugene Stanley
Center for Polymer Studies, Department of Physics,
590 Commonwealth Avenue, Boston, MA 02215, USA
Translator of Chapter 1-4: A. H. Armstrong
"Everglades", Brimpton Common, Reading, RG7 4RY Berks., UK
Library of Congress Cataloging-in-Publication Data
Stauffer, Dietrich. [Theoretische Physik, English] From Newton to Mandelbrot: a primer in theoretical
physics 1D ietrich Stauffer, H. Eugene Stanley; [translator of chapters 1-4,A. H. Armsstrong].-2nd enlarged
ed. p. cm. Indudes bibliographical references and index.
ISBN 3-540-59191-5 (softcover: alk. paper) 1. Mathematical physics. I. Stanley, H. Eugene (Harry Eugene).
1941-. II. Title. QC2o.S7413 1995 530.1'5-dc2o 95-31845 CIP
This book is an expanded version of the original German edition:
Theoretische Physik, 1. Aufl., by D. Stauffer © Springer-Verlag Berlin Heidelberg 1989
ISBN 978-3-540-59191-7 ISBN 978-3-642-86780-4 (eBook)
DOI 10.1007/978-3-642-86780-4
This work is subject to copyright. Ali rights are reserved, whether the whole or part of the material is
concerned, specifically the rights of translation, reprinting, reuse of illustrations, re citation, broadcast
ing, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this
publication or parts thereof is permitted only under the provisions of the German Copyright Law of
September 9, 1965, in its current version, and permission for use must always be obtained from
Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1990,1996
The use of general descriptive narnes, registered names, trademarks, etc. in this publication does not
imply, even in the absence of a specific statement, that such names are exempt from the relevant pro
tective laws and regulations and therefore free for general use.
Please note: Before using the programs in this book, please consult the technical manuals provided by
the manufacturer of the computer - and of any additional plug-in boards - to be used. The authors
and the publishers accept no legal responsibility for any darnage caused by improper use of the
instructions and programs contained herein. Although these programs have been tested with extreme
care, we can offer no formal guarantee that they will function correctly. The program on the enclosed
disc is under copyright-protection and may not be reproduced without written permission by
Springer-Verlag. One copy of the program may be made as a back-up, but ali further copies offend
copyright law.
Typesetting: Data conversion by Kurt Mattes, Heidelberg
Cover Design: Springer-Verlag, Design & Production
SPIN: 10500387 2156/3144-543210-Printed on acid-free paper
Preface
With increasing age, some authors gain more and more weight, scientifically and
gravitationally, and so do their books. Thus a new section on elementary particle
physics has been added. Its emphasis on computer simulation and phase transition
connects it with the end of the Statistical Physics chapter. In contrast to the first
four chapters, it does not lead to complicated exercises and may be more suited
to self-study; thus it is put into an appendix. The first four chapters, thought
to accompany a course or to summarize previous lectures, now also answer
the many questions at the end of each chapter; instructors may get solutions
of the more complicated problems by internet ([email protected]). For
the interested reader, we added to the four chapters recent literature references
wherever modern research aspects are touched upon in the text.
Some computer programs for the fractals in Chapter 5 are included in the
diskette that accompanies this book. More on the general subject of teaching
fractals can be found in the book Fractals in Science, edited by H. E. Stanley, E.
F. Taylor, and P. A. Trunfio (Springer, New York 1994, ISBN 0-387-94361-7 and
3-540-94361-7). The programs on the IDM diskette were constructed primarily
by S. V. Buldyrev, F. Caserta, A. Chandra, K. Shakhnovich, and E. F. Taylor
while those for the Macintosh diskette were written mainly by J. Blandey, S. V.
Buldyrev, T. Mekonen, R. L. Selinger, P. Trunfio, and B. Volbright. We thank
these individuals for their contribution, and also thank H. Rollnik, F.-W. Eicke,
F. W. Hehl, E. W. Mielke, and J. Potvin for their help with the additions to the
book. We hope readers who note further imperfections, or in any way wish to
make constructive suggestions, will communicate their thoughts to the authors.
Cologne, Germany Dietrich Stauffer
Boston, Massachusetts H. Eugene Stanley
June 1995
Preface to the First Edition
This is not a book for theoretical physicists. Rather it is addressed to profession
als from other disciplines, as well as to physics students who may wish to have
in one slim volume a concise survey of the four traditional branches of theo
retical physics. We have added a fifth chapter, which emphasizes the possible
connections between basic physics and geometry. Thus we start with classical
mechanics, where Isaac Newton was the dominating force, and end with frac
tal concepts, pioneered by Benoit Mandelbrot. Just as reading a review article
should not replace the study of original research publications, so also perusing the
present short volume should not replace systematic study of more comprehensive
texts for those wishing a firmer grounding in theoretical physics.
The opening paragraphs of Chapter 5 benefitted from input by B. Jorgensen.
We wish to thank G. Daccord for providing us with Plates 7 and 8, F. Family
for Plates 1 and 15, A.D. Fowler for Plate 3, R. Lenormand for Plate 11, P.
Meakin for Plate 14 as well as the cover illustration, J. Nittmann for Plate 13,
U. Oxaal for Plate 10, A. Skjeltorp for Plates 4, 9 and 16, K.R. Sreenivasan for
Plate 5, R.H.R. Stanley for Plate 2, and P. Trunfio for Plates 6 and 12. We also
thank A. Armstrong, A. Coniglio, J. Hajdu, F.W. Hehl, K.W. Kehr, 1. Kertesz,
A. Margolina. R. Selinger, P. Trunfio, and D.E. Wolf as well as many students
- particularly L. Jaeger - who offered their feedback at appropriate occasions
and A. Armstrong for translating Chapters 1-4 from the original German edition
published by Springer.
JUlich and Boston D. Stauffer
July 1990 H.E. Stanley
Contents
1. Mechanics ............................................ . 1
1.1 Point Mechanics ................................... . 1
1.1.1 Basic Concepts of Mechanics and Kinematics .... . 1
1.1.2 Newton's Law of Motion ..................... . 3
1.1.3 Simple Applications of Newton's Law .......... . 5
1.1.4 Hannonic Oscillator in One Dimension ......... . 12
1.2 Mechanics of Point Mass Systems .................... . 15
1.2.1 The Ten Laws of Conservation ................ . 16
1.2.2 The Two-body Problem ...................... . 17
1.2.3 Constraining Forces and d' Alembert's Principle .. . 18
1.3 Analytical Mechanics ............................... . 22
1.3.1 The Lagrange Function ...................... . 22
1.3.2 The Hamilton Function ...................... . 24
1.3.3 Hannonic Approximation for Small Oscillations .. . 26
1.4 Mechanics of Rigid Bodies .......................... . 30
1.4.1 Kinematics and Inertia Tensor ................. . 31
1.4.2 Equations of Motion ......................... . 35
1.5 Continuum Mechanics .............................. . 40
1.5.1 Basic Concepts ............................. . 40
1.5.2 Stress, Strain and Hooke's Law ................ . 44
1.5.3 Waves in Isotropic Continua .................. . 46
1.5.4 Hydrodynamics ............................. . 48
Questions 54
Problems 55
2. Electricity and Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.1 Vacuum Electrodynamics ............................. 59
2.1.1 Steady Electric and Magnetic Fields ............. 59
2.1.2 Maxwell's Equations and Vector Potential ........ 63
2.1.3 Energy Density of the Field .................... 65
2.1.4 Electromagnetic Waves ....................... 66
2.1.5 Fourier Transformation ........................ 67
2.1.6 Inhomogeneous Wave Equation ................. 68
2.1.7 Applications ................................ 69
X Contents
2.2 Electrodynamics in Matter ........................... 73
2.2.1 Maxwell's Equations in Matter ................. 73
2.2.2 Properties of Matter .......................... 74
2.2.3 Wave Equation in Matter ...................... 76
2.2.4 Electrostatics at Surfaces ...................... 77
2.3 Theory of Relativity ................................ 80
2.3.1 Lorentz Transfonnation ....................... 80
2.3.2 Relativistic Electrodynamics ................... 83
2.3.3 Energy, Mass and Momentum .................. 85
Questions .............................................. 87
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3. Quantum Meehanies .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.1 Basic Concepts ..................................... 89
3.1.1 Introduction ................................. 89
3.1.2 Mathematical Foundations ..................... 90
3.1.3 Basic Axioms of Quantum Theory .............. 91
3.1.4 Operators ................................... 94
3.1.5 Heisenberg's Uncertainty Principle .............. 95
3.2 Schrodinger's Equation .............................. 96
3.2.1 The Basic Equation .......................... 96
3.2.2 Penetration ........................ . . . . . . . . . 98
3.2.3 Thnnel Effect ............................... 99
3.2.4 Quasi-classical WKB Approximation ............ 100
3.2.5 Free and Bound States in the Potential Well ...... 101
3.2.6 Hannonic Oscillators ......................... 102
3.3 Angular Momentum and the Structure of the Atom ....... 105
3.3.1 Angular Momentum Operator .................. 105
3.3.2 Eigenfunctions of L2 and Lz •.•...••.•.•••.•••. 106
3.3.3 Hydrogen Atom ............................. 107
3.3.4 Atomic Structure and the Periodic System ........ 110
3.3.5 Indistinguishability ........................... 111
3.3.6 Exchange Reactions and Homopolar Binding ...... 112
3.4 Perturbation Theory and Scattering ..................... 115
3.4.1 Steady Perturbation Theory .................... 115
3.4.2 Unsteady Perturbation Theory .................. 116
3.4.3 Scattering and Born's First Approximation ........ 118
Questions ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Problems . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . .. . . . . . . . . 120
4. Statistieal Physies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.1 Probability and Entropy .............................. 123
4.1.1 Canonical Distribution ........................ 123
4.1.2 Entropy, Axioms and Free Energy .............. 126
Contents XI
4.2 Thermodynamics of the Equilibrium ................... . 129
4.2.1 Energy and Other Thermodynamic Potentials ..... . 129
4.2.2 Thermodynamic Relations .................... . 131
4.2.3 Alternatives to the Canonical Probability Distribution 133
4.2.4 Efficiency and the Carnot Cycle ............... . 135
4.2.5 Phase Equilibrium and the Clausius-Clapeyron
Equation .................................. . 136
4.2.6 Mass Action Law for Gases ................... . 139
4.2.7 The Laws of Henry, Raoult and van't Hoff ...... . 140
4.2.8 Joule-Thomson Effect ........................ . 142
4.3 Statistical Mechanics of Ideal and Real Systems ......... . 143
4.3.1 Fermi and Bose Distributions ................. . 143
4.3.2 Classical Limiting Case f3J1. -t -00 ............ . 145
4.3.3 Classical Equidistribution Law ................ . 147
4.3.4 Ideal Fermi-gas at Low Temperatures f3J1. -t +00 .. 148
4.3.5 Ideal Bose-gas at Low Temperatures f3J1. -t 0 .... . 149
4.3.6 Vibrations ................................. . 152
4.3.7 Virial Expansion for Real Gases ............... . 153
4.3.8 Van der Waals' Equation ..................... . 154
4.3.9 Magnetism of Localised Spins ................. . 156
4.3.10 Scaling Theory ............................. . 160
Questions 162
Problems 162
5. Fractals in Theoretical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.1 Non-random Fractals ................................ 166
5.2 Random Fractals: The Unbiased Random Walk ........... 168
5.3 'A Single Length' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
5.3.1 The Concept of a Characteristic Length .......... 170
5.3.2 Higher Dimensions ........................... 170
5.3.3 Additional Lengths that Scale with..fi ........... 171
5.4 Functional Equations and Scaling: One Variable .......... 172
5.5 Fractal Dimension of the Unbiased Random Walk ........ 172
5.6 Universality Classes and Active Parameters .............. 173
5.6.1 Biased Random Walk ......................... 173
5.6.2 Scaling of the Characteristic Length ............. 174
5.7 Functional Equations and Scaling: Two Variables ......... 175
5.8 Fractals and the Critical Dimension .................... 177
5.9 Fractal Aggregates .................................. 182
5.10 Fractals in Nature ................................... 185
Appendix: Elementary Particles ... . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
A.1 Basic Facts and Other Lies ........................... 191
A.I.l Particles.................................... 191
A.1.2 Forces ..................................... 194
XII Contents
A.2 Quantum Field Theory of Elementary Particles ........... 196
A.2.1 Quantum and Thermal Fluctuations .............. 196
A.2.2 Simulations at T = 0 .......................... 198
A.2.3 Simulations in the TeraKelvin Region ............ 199
Questions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
Answers to Questions .................... ,.................. 201
Further Reading ............................................ 206
Name and Subjeet Index ..................................... 207