From Newton to Mandelbrot: A Primer in Theoretical Physics

280 Pages·2017·9.07 MB·English

by  Dietrich Stauffer

Graduate Texts in Physics
Dietrich Stauffer
H. Eugene Stanley
Annick Lesne
From Newton
to Mandelbrot
A Primer in Theoretical Physics
Third Edition
Graduate Texts in Physics
Series editors
Kurt H. Becker, Polytechnic School of Engineering, Brooklyn, USA
Jean-Marc Di Meglio, Université Paris Diderot, Paris, France
Sadri Hassani, Illinois State University, Normal, USA
Bill Munro, NTT Basic Research Laboratories, Atsugi, Japan
Richard Needs, University of Cambridge, Cambridge, UK
William T. Rhodes, Florida Atlantic University, Boca Raton, USA
Susan Scott, Australian National University, Acton, Australia
H. Eugene Stanley, Boston University, Boston, USA
Martin Stutzmann, TU München, Garching, Germany
Andreas Wipf, Friedrich-Schiller-Universität Jena, Jena, Germany
Graduate Texts in Physics
GraduateTextsinPhysicspublishescorelearning/teachingmaterialforgraduate-and
advanced-levelundergraduatecoursesontopicsofcurrentandemergingfieldswithin
physics, both pure and applied. These textbooks serve students at the MS- or
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derivations,experimentsandapplications(asrelevant)fortheirmasteryandteaching,
respectively.Internationalinscopeandrelevance,thetextbookscorrespondtocourse
syllabisufficientlytoserveasrequiredreading.Theirdidacticstyle,comprehensive-
nessandcoverageoffundamentalmaterialalsomakethemsuitableasintroductions
orreferencesforscientistsentering,orrequiringtimelyknowledgeof,aresearchfield.
More information about this series at http://www.springer.com/series/8431
Dietrich Stauffer H. Eugene Stanley
(cid:129)
Annick Lesne
From Newton to Mandelbrot
A Primer in Theoretical Physics
Third Edition
123
Dietrich Stauffer Annick Lesne
Institute for Theoretical Physics Laboratoire PhysiqueThéorique dela
CologneUniversity MatièreCondensée(LPTMC)
Cologne UniversitéPierre etMarie Curie
Germany Paris
France
H.Eugene Stanley
Department ofPhysics—Center
forPolymer Studies
BostonUniversity
Boston, MT
USA
ISSN 1868-4513 ISSN 1868-4521 (electronic)
Graduate Textsin Physics
ISBN978-3-662-53683-4 ISBN978-3-662-53685-8 (eBook)
DOI 10.1007/978-3-662-53685-8
LibraryofCongressControlNumber:2016958466
©Springer-VerlagGmbHGermany1990,1996,2017
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Preface to the Third Edition
It’s very simple: This English edition is an updated translation of our Cours de
Physique, Springer, Paris 1999, which was an expanded French translation of
Stauffer & Stanley, From Newton to Mandelbrot, Springer, New York 1990 and
1996, which in turn was an expanded translation of Stauffer, Theoretische Physik,
Springer,Heidelberg1989(in German language).Thefirstexpansionwas Chap.5
on Fractals, the second expansion Chap. 6 on Dynamical Systems and Chaos. The
present version omits the diskettes added to the earlier version.
Chapters1–4containthestandardmaterialofcoursesintheoreticalphysicsand
are supposed to accompany lectures at the university; thus they are rather con-
densed. Theyaresupposedtofill1year ofteaching. Chapters5and6,incontrast,
are written less condensed since this material may not be part of standard lectures
andthuscouldbestudiedwithoutthehelpofauniversityteacher.Anappendixon
elementary particles lies somewhere in between: It could be a summary of a much
more detailed course, or studied without such a course.
Cologne, Germany Dietrich Stauffer
Boston, Massachusetts H. Eugene Stanley
Paris, France Annick Lesne
December 2015
v
Preface to the Second Edition
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
connectsitwiththeendoftheStatisticalPhysicschapter.Incontrasttothefirstfour
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
accompanyacourseortosummarizepreviouslectures,nowalsoanswerthemany
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.
SomecomputerprogramsforthefractalsinChap.5areincludedinthediskette
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,NewYork1994,ISBN0-387-94361-7and3-540-94361-7).
The programs on the IBM diskette were constructed primarily by S.V. Buldyrev,
F. Caserta, A. Chandra, K. Shakhnovich, and E.F. Taylor while those for the
MacintoshdiskettewerewrittenmainlybyJ.Blandey,S.V.Buldyrev,T.Mekonen,
R.L. Selinger, P. Trunfio, and B. Volbright. We thank these individuals for their
contribution,andalsothankH.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
vii
Preface to the First Edition
This is nota book for theoretical physicists. Rather it is addressedto professionals
fromotherdisciplines,aswellastophysicsstudentswhomaywishtohaveinone
slimvolumeaconcisesurveyofthefourtraditionalbranchesoftheoreticalphysics.
Wehaveaddedafifthchapter,whichemphasizesthepossibleconnectionsbetween
basic physics and geometry. Thus we start with classical mechanics, where Isaac
Newton was the dominating force, and end with fractal concepts, pioneered by
Benoit Mandelbrot.Just asreadingareviewarticle shouldnotreplacethestudyof
originalresearchpublications,soalsoperusingthepresentshortvolumeshouldnot
replace systematic study of more comprehensive texts for those wishing a firmer
grounding in theoretical physics.
The opening paragraphs of Chap. 5 benefitted from input by B. Jorgensen. We
wishtothankG.DaccordforprovidinguswithPlates7and8,F.FamilyforPlates
1and15,A.D.FowlerforPlate3,R.LenormandforPlate11,P.MeakinforPlate
14aswellasthecoverillustration,J.NittmannforPlate13,U.OxaalforPlate10,
A.SkjeltorpforPlates4,9and16,K.R.SreenivasanforPlate5,R.H.R.Stanleyfor
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, J. 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
fortranslatingChaps.1–4fromtheoriginalGermaneditionpublishedbySpringer.
Jülich and Boston D. Stauffer
July 1990 H.E. Stanley
ix
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.. .... .... .... 6
1.1.4 Harmonic Oscillator in One Dimension . .... .... .... 13
1.2 Mechanics of Point Mass Systems.... .... .... .... .... .... 17
1.2.1 The Ten Laws of Conservation.... .... .... .... .... 17
1.2.2 The Two-Body Problem..... .... .... .... .... .... 19
1.2.3 Constraining Forces and d’Alembert’s Principle... .... 20
1.3 Analytical Mechanics. .... .... ..... .... .... .... .... .... 24
1.3.1 The Lagrange Function. ..... .... .... .... .... .... 24
1.3.2 The Hamilton Function. ..... .... .... .... .... .... 26
1.3.3 Harmonic Approximation for Small Oscillations .. .... 28
1.4 Mechanics of Rigid Bodies .... ..... .... .... .... .... .... 33
1.4.1 Kinematics and Inertia Tensor .... .... .... .... .... 33
1.4.2 Equations of Motion... ..... .... .... .... .... .... 36
1.5 Continuum Mechanics.... .... ..... .... .... .... .... .... 42
1.5.1 Basic Concepts... .... ..... .... .... .... .... .... 42
1.5.2 Stress, Strain and Hooke’s Law ... .... .... .... .... 47
1.5.3 Waves in Isotropic Continua.. .... .... .... .... .... 49
1.5.4 Hydrodynamics... .... ..... .... .... .... .... .... 50
2 Electricity and Magnetism. .... .... ..... .... .... .... .... .... 61
2.1 Vacuum Electrodynamics.. .... ..... .... .... .... .... .... 61
2.1.1 Steady Electric and Magnetic Fields.... .... .... .... 61
2.1.2 Maxwell’s Equations and Vector Potential... .... .... 66
2.1.3 Energy Density of the Field .. .... .... .... .... .... 68
2.1.4 Electromagnetic Waves. ..... .... .... .... .... .... 68
2.1.5 Fourier Transformation. ..... .... .... .... .... .... 69
2.1.6 Inhomogeneous Wave Equation ... .... .... .... .... 70
xi
xii Contents
2.1.7 Applications . .... .... ..... .... .... .... .... .... 71
2.2 Electrodynamics in Matter. .... ..... .... .... .... .... .... 76
2.2.1 Maxwell’s Equations in Matter.... .... .... .... .... 76
2.2.2 Properties of Matter ... ..... .... .... .... .... .... 77
2.2.3 Wave Equation in Matter .... .... .... .... .... .... 79
2.2.4 Electrostatics at Surfaces..... .... .... .... .... .... 80
2.3 Theory of Relativity.. .... .... ..... .... .... .... .... .... 83
2.3.1 Lorentz Transformation. ..... .... .... .... .... .... 84
2.3.2 Relativistic Electrodynamics.. .... .... .... .... .... 87
2.3.3 Energy, Mass and Momentum .... .... .... .... .... 89
3 Quantum Mechanics.. .... .... .... ..... .... .... .... .... .... 93
3.1 Basic Concepts . .... .... .... ..... .... .... .... .... .... 93
3.1.1 Introduction . .... .... ..... .... .... .... .... .... 93
3.1.2 Mathematical Foundations ... .... .... .... .... .... 94
3.1.3 Basic Axioms of Quantum Theory. .... .... .... .... 96
3.1.4 Operators ... .... .... ..... .... .... .... .... .... 98
3.1.5 Heisenberg’s Uncertainty Principle. .... .... .... .... 100
3.2 Schrödinger’s Equation ... .... ..... .... .... .... .... .... 101
3.2.1 The Basic Equation.... ..... .... .... .... .... .... 101
3.2.2 Penetration .. .... .... ..... .... .... .... .... .... 102
3.2.3 Tunnel Effect .... .... ..... .... .... .... .... .... 104
3.2.4 Quasi-classical WKB Approximation ... .... .... .... 105
3.2.5 Free and Bound States in the Potential Well . .... .... 106
3.2.6 Harmonic Oscillators .. ..... .... .... .... .... .... 107
3.3 Angular Momentum and the Structure of the Atom... .... .... 110
3.3.1 Angular Momentum Operator. .... .... .... .... .... 110
3.3.2 Eigenfunctions of L2 and Lz.. .... .... .... .... .... 111
3.3.3 Hydrogen Atom .. .... ..... .... .... .... .... .... 112
3.3.4 Atomic Structure and the Periodic System ... .... .... 115
3.3.5 Indistinguishability.... ..... .... .... .... .... .... 116
3.3.6 Exchange Reactions and Homopolar Binding. .... .... 118
3.4 Perturbation Theory and Scattering ... .... .... .... .... .... 120
3.4.1 Steady Perturbation Theory... .... .... .... .... .... 120
3.4.2 Unsteady Perturbation Theory. .... .... .... .... .... 122
3.4.3 Scattering and Born’s First Approximation... .... .... 124
4 Statistical Physics .... .... .... .... ..... .... .... .... .... .... 129
4.1 Probability and Entropy... .... ..... .... .... .... .... .... 129
4.1.1 Canonical Distribution . ..... .... .... .... .... .... 129
4.1.2 Entropy, Axioms and Free Energy. .... .... .... .... 132
4.2 Thermodynamics of the Equilibrium .. .... .... .... .... .... 135
4.2.1 Energy and Other Thermodynamic Potentials. .... .... 135
4.2.2 Thermodynamic Relations ... .... .... .... .... .... 138