Table Of Content3
Volume 1 Fluctuations and Sensitivity in Nonequilibrium Systems
Editors: W. Horsthemke and D. K. Kondepudi
Volume 2 EXAFS and Near Edge Structure III
Editors: K.O.Hodgson, B.Hedman, and J.E.Penner-Hahn
Volume 3 Nonlinear Phenomena in Physics
Editor: F. Claro
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Nonlinear Phenomena
in Physics
Proceedings of the
1984 Latin American School of Physics,
Santiago, Chile, July 16-August 3, 1984
Editor: F. Claro
With 110 Figures
Springer-Verlag
Berlin Heidelberg New York Tokyo
Professor Francisco Claro, Ph. D.
Pontificia Universidad Catolica de Chile, Facultad de Fisica,
Casilla 114-D, Santiago, Chile
ISBN-13: 978-3-642-93291-5 e-ISBN-13: 978-3-642-93289-2
001 10 1007/978-3-642-93289-2
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Preface
It was almost four hundred years ago that Galileo wrote in Il Saggiatore that the
"Book of Nature is written in mathema ti ca 1 characters". Thi s sentence, i nspi red at
the dawn of physics has proved with the passage of time to contain a deep truth and
also a warning: in order to understand Nature, first we must learn to read mathema
tical characters. Indeed, writing physical law in such characters has proved not
as hard as unraveling the content of the resulting equations. In particular, the
lack of knowledge in the field of nonlinear mathematics has been a severe limita
tion in the past. Thus the solution to equations such as the Navier-Stokes equation
in fluid dynamics has remained elusive. The recent advent of fast computers and
some important analytical and numerical results in the study of bifurcations and
nonlinear waves have encouraged work both in theory and experiment involving non
linear phenomena. An explosive growth in the specialized literature penetrating
most research areas in physics in the last few years has ensued.
This book contains the most recent advances in nonlinear physics in various
fields including astrophysics, gravitation, particle physics, quantum optics, fluid
dynamics and the mathematics underlying the phenomena of chaos and nonlinear waves.
It presents a selection from the lectures delivered at the XXI '_atin American School
of Physics held in Santiago, Chile in July-August 1984 (EtAF'84). This School, at
tended by over 100 physicists from all parts of America, covered the subject matter
NONLINEAR PHENOMENA IN PHYSICS trying to balance information on the development of
new mathematical methods with the recent phenomenology most relevant to the subject.
The book reflects this spirit, together with the aim of publishing material that is
new and not contained in available review articles or books. Authors are renowned
specialists in each field,selected by the Organizing Committee of ElAF'84 consider
ing not only their expertise but also their ability to communicate. They were asked
to write an easy to read text that would be of use not only to specialists but also
to newcomers to the field and to advanced students. We feel these goals were achie
ved and that this volume will prove to be valuable and timely.
I would like to thank all those who made ElAF'84 possible. First, I owe special
gratitude to the Organizing Committee: Drs. M.C. Depassier, 1_. Gomberoff, D. Gott
lieb, F. lund, M. Orszag, H. Quintana, I. Schmidt, I. Schuller and E. Tirapegui,
who assisted me in all important decisions and many tasks. Then, my thanks to Ms.
Angela Bau who combined extreme efficiency and charm to make all practical matters
work, and Ms. tiliana Pineda for her patience and valuable assistance in preparing
the School and editing this book. Several colleagues and local students and friends
contri buted in vari ous ways to the success of ElAF '84 inc 1 udi ng L Arroyo, Z. Bar
ticevic, D.R. Hofstadter, J. Krause, M. I_oewe, I. Olivares, M. Pacheco, R. Ramirez,
S. Rios, R. Rojas and V. Tapia. Finally, the sponsors: Organization of American
States (O.A.S.), U.N.E.S.C.O., National Science Foundation, Committee on Science
and Technology in Developing Countries, Facultad de Ingenieria and Instituto de Mu
sica (Catholic University of Chile), International Centre for Theoretical Physics
in Trieste, Italy, C.E.R.N., European Southern Observatory (ESO), Service Culturel
et de Cooperation Scientifique et Technique of France, the British Council, Cerro
Tololo Interamerican Observatory, Argonne National '_aboratory U.S.A., CONICYT Chile,
C.N.P.Q. Brasil, the Chilean Academy of Sciences, Universidad F. Santa Maria publi
shing office, DERCO S.A. Chile, CORPORA S.A. Chile, and Centro de Perfecionamiento
del Magisterio, Chile. In the name of the Chilean Society of Physics, organizer of
EtAF'84, I would like to thank them all for their generosity.
Santiago October 1984 Francisco Claro
v
Contents
Part I Mathematical Methods and General
Dynamics for Golden Mean Rotation. By M.J. Fei genbaum ....•........•......... 2
1. I ntroduc ti on • . . . . • • . . . . . . . . . . . . . . • . . . . • . . . . . • • . . . . . . . . . . • . . . . . . . • . . • . . 2
2. Golden Mean Circle Maps .......•.......••..•..•........•....•.......... 3
3. Trajectory Scaling .......•.•.•.........•..•.••...•......•....•........ 5
4. The Golden Mean Power Spectrum ...•....•......•...•....•.....•.....•... 9
5. Concl usion ..•........••....•.....•................•................... 15
Nonlinear Phenomena in Dissipative Systems. By P. Coull et (Wi th 33 Fi gures) 16
1. Stability Analysis of Equilibria .......•...••••..............•...••... 16
2. Steady Non Equilibrium Behavior and Pattern Formation ..•..........•... 22
3. Time Dependent Periodic Phenomena ...............•......•.....•........ 36
4. Quasiperiodic Phenomena ............................................... 47
5. Chaoti c Phenomena • • . . . . • • . . . • • . . . . . . . • . • . . . • . . • . • . . . . . • . . . . . • • . . . . . . . . 53
References . . . . • . . . . . . . . • . . . . . . . • . . . . . . . . . . • . . . . . . . . • . . . . . • . . • . . . . . . . . . . . . 67
Statistical Mechanics of the sine-Gordon Field: Part I. By R. K. Bull ough •..•.•... 70
1. I ntroduc ti on . . . . . . • . . . . . . . • . . . . . . . . . . . . . . . . . . . • . . • . . . • . • . . . . . . . . . . • . . . 70
2. Elementary Aspects of Action-Angle Variables: Action-Angle Variables for
the Klein-Gordon Equation ..•...........•....•••......•......•..•...... 72
3. The Ferromagnetic CsNiF3; A Physical Example of the sine-Gordon Equation 76
4. Complete Integrability of the Classical sine-Gordon Equation .......... B2
5. Functional Integrals for the Partition Function Z ................•.... 91
6. The Partition Function for the l.inear Klein-Gordon Equation ....•...... 94
7. Transfer Integral Method for the Classical Partition Function of the
si ne-Gordon Eq uati on . . . . . . . . . . . . . . . . . . . • . . . . . • . . . . . . . . • • . . . . . . . . . . . . . . 96
Statistical Mechanics of the sine-Gordon Field: Part II
By R.K. Bullough, D.J. Pilling, and J. Timonen .....•...............•........ 103
B. Evaluation of the Partition Function Z for the sine-Gordon Field in Terms
of Action-Angle Variables: Construction of the Measure ......•...•..•.. 103
9. Floquet Theory for Periodic Problems ..••..............•...•........... 105
10. Partition Function for the sinh-Gordon Equation .••.•..••..•.....•.....• lOB
11. Sine-Gordon Statistical Mechanics Without Phonon-Phonon Phase-Shifts 110
12. Sine-Gordon Statistical Mechanics with Phonon-Phonon Phase-Shifts
I ncl uded ..••...................•........................•............. 113
13. Classical Fermions .•......••................................•......... 114
14. Unsol ved Probl ems .... , . . . . . • . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . •• . . . . 116
15. Conclusion .....••••......•...•......•.•.....•........•.•.............. 117
Appendices ..••...•..••••...•.•••. " ..... .•..•...•• ..•.... .•.•.... . .... .. .. lIB
References . . . . . . • . . . . • • . . . . . . . . • . • . . . • • . . . . . . . . • . . . . . . . . . . . . . . . . • . . . . . . . . . 126
Probabilistic Cellular Automata. By B.A. Huberman (With 1 Figure) .•• •••......•. 129
1. I ntroduc ti on . . . . • • . . . . . . . . • • • • . . . . . . • • • • . . . . • • • . . . . . . . . • . . . . . . . . . . . . . . 129
2. Automa ta Dynami c s . . . . . . . . . . • . . . . . . . . . • • . . • . • • • . . . . . . . . . . . . . . . . . . . . . . . . 130
3. The Stick-Slip Behavior of Faults ..........••.•.......•.....•......... 133
References . . . • • .. . . . . .. • . . . . . . . • . . . . . . . • •. . . . . . • • • . . • . •. . . • • . . . . . . •. . . . . . 136
VII
Part II Quantum Optics
Chaos, Generalized Multistability and Low Frequency Spectra in Quantum Optics
By F. T. Arecchi (With 18 Fi gures) . . . . . . . . • . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 140
1. Introduction: Order and Chaos in Quantum Optics ....................... 140
2. Physics of Stimulated Emission Processes .............................. 145
3. The Maxwell-Bloch Equations ........................................... 150
4. Onset of Chaos, Generalized Multistability and tow Frequency Spectra in
Quantum Optic s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
From Optical Bistability to Chaos. By H.M. Nussenzveig (With 9 Figures) ........ 162
1. I ntroduc ti on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
2. The Ring Cavity ....................................................... 163
3. The Maxwell -Bloch Equati ons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
4. The Linear Approximation ...... ........................................ 171
5. Opti cal Bi stabil ity ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6. Semiclassical Theory: Steady State .................................... 176
7. Di scussi on of the State Equati on ...................................... 182
8. Linear Stability Analysis ............................................. 187
9. Dispersive Optical Bistability ........................................ 195
10. Quantum Theory of Optical Bistabi1ity ................................. 198
11. Spec trum of the Transmi tted ti ght . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 204
12. Photon Stati stics .,. . . . . . . . . . . .. .. . . . . . . . . . . .. . . . .. . ... . . . .. .. . . . . . . . . 207
13. Si de Mode I nstabil ities . . . . . . . .. . . . . . . .. . . . . .. .. .. . . . . . . . . . .. . . . .. . . .. 210
14. Mu1tistabi1ity ........................................................ 215
15. Ikeda Instability. Appraoch to Chaos .................................. 220
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
Part III Fluids
Recent Developments in Rayleigh-Benard Convection and I nterface Dynamics
By A. Li bchaber (With 6 Figures) ............................................ 228
1. The Ray1 ei gh-Benard I nstabi 1 ity ..............................•........ 228
2. Fluid-Fluid Interface, the Saffman-Taylor Problem ..................... 231
3. Solid-liquid Interface, from Cells to Dendrites ....................... 234
References ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
Part I V Astrophysics and General Relativity
Nonlinear Problems in Stellar Dynamics. By G. Contopoulos (With 8 Fi gures) .... 238
1. Introduction ........................._ ................................. 238
2. Stellar Motions on the Plane of Symmetry of a Spiral Galaxy ........... 239
3. Stellar Motions on the Meridian Plane of an Axisymmetric Galaxy ....... 243
4. Transition from Ordered to Stochastic Motion .......................... 244
5. Bifurcations and Gaps ................................................. 247
6. Three-Dimensional Systems ............................................. 249
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
Solitons in General Relativity and Supergravity. By G.W. Gibbons ............ 255
Lecture 1: Black Hole Solitons in Ungauged Extended Supergravity ......... 255
1. The Sol i ton Concept for Gravity ....................................... 255
2. Absence of Solitons Without Horizons .................................. 256
3. Black Holes and the Breakdown of Supersymmetry at Finite Temperature 258
4. Extended Supersymmetry: Central Charges and Multiplet Shortening ...... 262
5. Black Holes in N=2 Supergravity: Charge Without Charge ................ 264
6. The Bogomo 1 ny I nequa 1 ity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
VIII
7. The Zero Modes . • • . . . • . . . . . • • • . . . . • • . • . • . . . . . • • . . • . • . . • . . • . . . . . • . . . . . . • 267
8. Conclusion ..•.•.•...••....•...•.••..•.....••....•....••.•.....•.....•• 268
References .. • . . • . • . • • . . . . . . . . • . •• . .. . . .• . . • . •. . . . . •• . . . . . . •• . . . . . . . . . • • •• 268
Lecture 2: Supersymmetric Monopoles in Kaluza-Klein Theory •.••••.••.•••• : 270
1. Kaluza-Klein Theory .•••.••••.•....••...••.•.••••.•..•••••....••..•..•. 270
2. Gross-Perry-Sorkin Monopol es .•.••....• .•...•••.•.. .... •••. .. .•••. ..... 271
3. N = 8 Supergravity in 5-Dimensions ...••.•.•........•..•.•.•..••...••.. 273
4. Supersymmetries and Zero Models .•..•....•.•..•.•..........••......•.•• 275
5. Electric Magnetic Duality .•••.•••.•••..••••.••••..•.•..••••••.••..•..• 276
6. Monopol e-pyrgon Dual ity .••••••••.•...•...••.•••••••..•..•..•••••....••• 278
References . • . . . • . • . . . . • • • • . . . • • • • • . • . . . . . . . . . . . . . . . • . . . • • • • . . . . . . • • . • . • • . 279
The Dynamics of Spacetime Curvature: Nonlinear Aspects
By K. S. Thorne (Wi th 6 Fi gures) . • • • . . . . . . . . •• . • . • . • . . . . . . . • • . . . . • • . . • • . • . • • 280
1. I ntroducti on • • •• .. . • .• . . . • • • . . • . .• .•. . •• • . • . • . . . •. . . •• . • • . . . . • • ••. . . .• 280
2. Gravitational Waves .•....•.•.•......•••.••.....•..•.......•••.••..•••• 281
3. Geons . • . • • • • . • • • . . • . . . • . . . • . . . • . . . . . . . • . • . • . . . . . • • • . . . • . . . . • • • • • . . • • . . 283
4. Black Holes ...•.....•.••.••.•.•..••...•.•.......•.•...•.•..•.•....•••. 284
5. White Holes .•••.•..••...•..•••••..•••••••.•.•••••.•.•••••..•...••••••• 285
6. Wormholes ...•......•....•.••••..••.....•.•..•.•.•......•...•••....•.•. 286
7. Singularities .•....•.•...•.•...........•....•...•....•.•.•.•.•.•...••. 288
8. Techniques of Future Research: Soliton Theory and Numerical Relativity. 290
References . • . • . . • . • . • . . • • . . . . • • • • • • . . . . . . • . . . • . . . . . • . . • . • . • . • • . . . • • . . . . • . 290
Part V High Energy Physics
Unification, Supersymmetry and Cosmology. By J. E1Hs ......•.•••.••.......•. 294
1. The Standard Model . . . •• . . . • • • . . • • . . . .. . • • .. . . . . . •. • •• • • • • .. . . • .•. • . • • • 294
2. From the Standard Model to Grand Unification ...••••••.•••........•.••. 299
3. Simple GUT Models ..••.•...•.•.•••.•................•...•.•.•.•••.•..•• 303
4. New Interactions in GUTs ..••.•..•.•.•..•.•..•.•.•...••••..••...•..••.. 308
5. The Hierarchy Problem •.••••••.••••.....•........•..•.•..•.....••.•.••• 313
6. Supersymmetri c GUTs . • .•. . • . •• • . . . •• •• •• • • . • .• •••. • •• . ••• . •• • •••• . . ••. . 317
7. Supersymmetry at Low Energies •••••.•...•.........•••...........•..•..• 322
8. Supersymmetry and CERN Collider Data ..•...•......•••.••••...•.•..••.•• 327
9. GUTs and Cosmology .•••••.••••.•••.....•••.•...••.•••.••...•••.••..•••• 332
10. Inflation and Supersymmetric Cosmology •.•.•....•.••.•.•.•.••.•••....•• 336
References • . . • • • • • • . . . • . • . . . . • . • . • . . • . . . • • . . . . • . • . . • • . . . • . . . . . . . . . • • . . . • . . 340
Lectures on Quark Flavor Mixing in the Standard Model
By E. de Rafael (With 29Figures) •.•.•••••....•.•.••.•••••.•••..•••••.•.••••• 344
1. Overview of the Standard Model ...•••••..•.•....•..••.••.•....••..•.•.• 344
2. Phenomenological Determination of the Cabibbo-Kobayashi-Maskawa Matrix
El ements . . • . . .• . •• . • . . . . . • . • . . . . . . . . . . . . . . . • . . . . • . . . . • . • • . . • • • • . . • . . •• 360
3. The ~s = 1 Effective Hamiltonian for Non-Leptonic Decays in the Standard
Model •• .. ..•.•..•.. . .... .....•. .. ....••..••.. ..•.•..••.••• ...••. ..•••. 374
4. Violation of CP-Invariance in the St~ndard Model •..••..••.•••••.•••.•• 405
References . • • • . . • • • . . . • . . • • • . . • • . . . . • • • • • • • • • • . . . • • . . . . • • • • • • • . • • • • . . • . . . 434
Index of Contributors .•••......•••.••.......••.•••...••...•........•••..••.. 441
IX
Part I Mathematical Methods and General
Dynamics for Golden Mean Rotation
Mitchell J. Feigenbaum
Cornell University, Physics Department, Ithaca, NY 14853, USA
Trajectory scalings for golden mean quasiperiodic motion are determined from its
renormalization group. The power spectrum is then computed in an accurate
approximation.
I. Introduction
This paper, of a partially pedagogic nature, assumes some familiarity with the
renormalization group methods for deterministic systems. Indeed, it represents a
(not entirely trivial) reworking of the tools developed for period doubling in a
somewhat more complex setting. Background material will be mentioned in allusion.
Basically, we have the following situation. A high magnification of
appropriately high iterates of a mapping on the circle near a special point is an
arbitrarily accurate approximation to a fixed point (function) of a transformation
(the renormalization group transformation) on a space of (actually ~airs of)
functions. Emphatically, high iterates of the map everywhere on the circle are
not well approximated. The power spectrum (as an example of a general dynamical
object of study) is a transform of the entire motion, and so the question arises
as to how knowledge about some particular point can suffice to determine such an
object. Because (modulo a proviso to follow) indeed it is possible to accomplish
just this. The proviso is that a number of measurements must be provided as
initial data, and the better the desired result, the more such measurements must
be provided.
Rather than a limitation that could be removed by a better method, this point
of formulating a "new" initial data problem represents a significant insight into
what are reasonable physical questions, and what should be the significant
fundamental "variables" of a physical situation characterized by a large degree_of
irregularity and hence embracing large amounts of information. Thus, a "better"
answer can probably require calculation approaching, if not identical to, the
actual simulation of the precise process, and so a computation esthetically
pleasing to only a computer. That is, it offers no "physical" insight and is
"understanding" insofar as the witnessing of an event can be equated to its
understanding.
The method of solution presented here is that of the "trajectory scaling
function" originally implemented for the determination of the dynamical properties
2
