Table Of ContentSCALEINVARIANCE
AND BEYOND
Les Houches Workshop, March 10-14, 1997
Editors
B. DUBRULLE, F. GRANER
and D. SORNETIE
EDP Sciences
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Centre de Physique des Houches
Books already published in this series
1 Porous Silicon Science and Technology
Jean-Claude VIAL and Jacques DERRIEN, Eds. 1995
2 Nonlinear Excitations in Biomolecules
Michel PEYRARD, Ed. 1995
3 Beyond Quasicrystals
Fran~oise AXEL and Denis ORAT lAS, Eds. 1995
4 Quantum Mechanical Simulation Methods for Studying Biological Systems
Dominique BICOUT and Martin FIELD, Eds. 1996
5 New Tools in Turbulence Modelling
Olivier METAIS and Joel FERZIOER, Eds. 1997
6 Catalysis by Metals
Albert Jean RENOUPREZ and Herve JOBIC, Eds. 1997
Book series coordinated by Michele LEDUC
Editors of "Scale Invariance and Beyond" (N0 7)
B. Dubrulle, Service d'Astrophysique, L'Orme des Merisiers, 709,
91191 Gif-sur-Yvette cedex, France
F. Graner, Laboratoire de Spectrometrie Physique, B.P. 87,
38402 Saint-Martin-d'Heres cedex, France
D. Somette, Laboratoire de Physique de la Matiere Condensle,
CNRS et Universitl de Nice -Sophia Antipolis, 06108 Nice, France
ISBN 978-3-540-64000-4 ISBN 978-3-662-09799-1 (eBook)
DOI 10.1007/978-3-662-09799-1
This wolk is subject to copyright. All rights are reserved, whether the whole or put of the material is
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laws of March Il, 1957 and September 9, 1965, respectively. Violations fall under the prosecution act
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@ Springer-Verlag Berlin Heidelberg 1997
Originally published by Springer-Verlag Berlin Heidelberg New York in 1997
AUTHORS
Ameodo A., Centre de Recherche Paul Pascal, avenue Schweitzer, 33600 Pessac,
France
Audit B., Centre de Recherche Paul Pascal, avenue Schweitzer, 33600 Pessac,
France
Bacry E., Centre de Mathematiques Appliquees, Ecole Polytechnique, 91128
Palaiseau, France
Biferale L., University of Rome "Tor Vergata", Department of Physics and INFN,
Via della Ricerca Scientifica 1, 00133 Rome, Italy
Bouchaud J.-P., Science & Finance, 109-111 rue Victor Hugo, 92532 Levallois,
France
Bouchaud J.-P., Service de Physique de l'Etat Condense, Centre d'Etudes de Saclay,
Orme des Merisiers, 91191 Gif-sur-Yvette cedex, France
Brechet Y., L.T.P.C.M., B.P. 75, I.N.P. Grenoble, 38402 Saint-Martin-d'Heres,
France
Butler S.L., Department of Physics, University of Toronto, Toronto, Ontario,
Canada, M5S 1A7
a
Castaing B., Centre de Recherches sur les Tres Basses Temperatures, Associe
l'Universite Joseph Fourier, CNRS, B.P. 166, 38042 Grenoble cedex 9,
France
Chatagny R., Group of Parallel Computing, CUI, University of Geneva, 24 rue du
General Dufour, 1211 Geneva 4, Switzerland
Chopard B., Group of Parallel Computing, CUI, University of Geneva, 24 rue du
General Dufour, 1211 Geneva 4, Switzerland
Cont R., Service de Physique de l'Etat Condense, Centre d'Etudes de Saclay, 91191
Gif-sur-Yvette, France, and, Science & Finance, 109-111 rue Victor Hugo,
92532 Levallois, France; and Laboratoire de Physique de la Matiere
Condensee, URA 190 du CNRS, Universite de Nice, 06108 Nice cedex 2,
France
Dahmen K.A., Department of Physics, Harvard University, Cambridge, MA 02138,
U.S.A.
Dubrulle B., CNRS, CEAIDSMIDAPNIA/Service d' Astrophysique, L'Orme des
Merisiers 709, 91191 Gif-sur-Yvette, France
IV
Graner F., CNRS, UMR 5588, Laboratoire de Spectrometrie Physique, Universite
Grenoble I - Joseph Fourier, B.P. 87, 38402 Saint-Martin-d'Heres cedex,
France
Harrington S.T., Center for Polymer Studies and Physics Department, Boston
University, Boston, MA 02215, U.S.A.
Henriksen R.N., Department of Physics, Queen's University at Kingston, Ontario,
K7L 3N6, Canada
Hilfer R., ICA-1, Universitlit Stuttgart, 70569 Stuttgart, Germany; and Institut filr
Physik, Universitlit Mainz, 55099 Mainz, Germany
Knopoff L., Department of Physics and Astronomy and Institute of Geophysics and
Planetary Physics, University of California, Los Angeles, U.S.A.
Lachieze-Rey M., C.E.-Saclay, 91191 Gif-sur-Yvette, France
Lebyodkin M., L.T.P.C.M., B.P. 75, I.N.P. Grenoble, 38402 Saint-Martin-d'Heres,
France
Manneville S., Centre de Recherche Paul Pascal, avenue Schweitzer, 33600 Pessac,
France
Marsili M., Institut de Physique Theorique, Universite de Fribourg, 1700 Fribourg,
Switzerland
Muzy J.F., Centre de Recherche Paul Pascal, avenue Schweitzer, 33600 Pessac,
France
Nottale L., DAEC, Observatoire de Paris-Meudon, CNRS and Universite Paris VII,
92195 Meudon cedex, France
Peltier W.R., Department of Physics, University of Toronto, Toronto, Ontario,
Canada, M5S 1A7
Perkovic 0., Laboratory of Atomic and Solid State Physics, Cornell University,
Ithaca, NY 14853-2501, U.S.A.
Pocheau A., IRPHE, Universite Aix-Marseille I & II, CNRS, Centre de Saint
Jerome, Case 252, 13397 Marseille, France
Poole P.H., Department of Applied Mathematics, University of Western Ontario,
London, Ontario N6A 5B7, Canada
Potters M., Science & Finance, 109-111 rue Victor Hugo, 92532 Levallois, France
Roux S.G., Centre de Recherche Paul Pascal, avenue Schweitzer, 33600 Pessac,
France
Sciortino F., Dipartimento di Fisica, Universita di Roma La Sapienza, 00185 Roma,
Italy
Sethna J.P., Laboratory of Atomic and Solid State Physics, Cornell University,
Ithaca, NY 14853-2501, U.S.A.
v
AUlHORS
Somette D., Laboratoire de Physique de Ia Matiere Condensee, CNRS and
Universite de Nice-Sophia Antipolis, Pare Valrose, 06108 Nice, France; and
Department of Earth and Space Sciences and Institute of Geophysics and
Planetary Physics, University of California, Los Angeles, California 90095-
1567, U.S.A.
Stanley H.E., Center for Polymer Studies and Physics Department, Boston
University, Boston, MA 02215, U.S.A.
Zajdenweber D., Department of Economics, Paris-X Nanterre University, Paris,
France
CONTENTS
FOREWORD
Scale Invariance and Beyond
by B. Dubrulle, F. Graner and D. Somette
1. Introduction .. .. .. ...... .. .. ......... ............ .. .. ... ........ .... .... ... ... .. ......... .. .. .. ...... .. . 1
2. Scale Invariance...................................................................................... 2
2.1. Concept............................................................................................ 2
2.1.1. A defmition of scale invariance............................................ 2
2.1.2. Open questions ..................................................................... 3
2.1.3. Analogy with translation invariance..................................... 3
2.1.4. Link with dimensional analysis............................................ 6
2.2. Tools to study scale in variance........................................................ 7
2.2.1. Wavelets............................................................................... 8
2.2.2. Fractional derivative............................................................. 8
2.2.3. Dimensional similarity ......................................................... 8
2.2.4. Stable laws............................................................................ 9
2.2.5. Computation 9f scaling exponents....................................... 10
2.3. Examples.......................................................................................... 10
2.3.1. Condensed matter systems................................................... 11
2.3.2. From cosmology to earthquakes........................................... 12
2.3.3. From biology to finance....................................................... 14
3. Beyond scale invariance ......................................................................... 16
3.1. From scale invariance to scale covariance....................................... 16
3.2. Scale symmetry breaking................................................................. 17
4. Conclusion.............................................................................................. 19
VIII CONTENTS
Scale lnvariance
Concept
LECTURE!
Scale Invariance Without Mechanism?
by F. Graner
1. Introduction ....... ........... .. .. .. .. .. .. ...... .. .... ..... .. .. .. .... .... ...... .. .. ............. ... .. ... 25
2. Magic in the solar system?...................................................................... 25
2.1. A « log versus rank » plot................................................................ 26
2.2. Explanations of this geometrical progression ....... .. .. .... ..... ... .. .. ... .. .. 27
2.3. Comments........................................................................................ 28
3. Hand-waving symmetry argument.......................................................... 28
3.1. Generic equations............................................................................ 28
3.2. Generic solutions ............................................................................. 29
3.3. Comments........................................................................................ 30
4. Example: the flat, cold disk .................................................................... 30
4.1. Symmetry of the equations .............................................................. 31
4.2. Linear stability analysis ................................................................... 31
4.3. Unstable modes................................................................................ 32
5. Discussion............................................................................................... 33
5 .1. From continuous to discrete scale in variance .. ....... ... .. .. ......... ......... 33
5.2. Warnings.......................................................................................... 34
6. Summary................................................................................................. 35
IX
Tools
LECTURE2
Scale Invariance and Beyond: What Can We Learn from
Wavelet Analysis?
by A. Ameodo, B. Audit, E. Bacry, S. Manneville, J.F. Muzy
and S.G. Roux
1. From global to local characterization of the regularity
of fractal functions.................................................................................. 37
2. Statistical analysis of the regularity of fractal functions:
the multifractal formalism...................................................................... 38
3. A thermodynamics of fractal signals based on wavelet analysis............ 39
3.1. Singularity detection and processing with wavelets........................ 39
3.2. The wavelet transform modulus maxima method............................ 40
3.3. The structure function approach versus the WTMM method.......... 41
4. Numerical and experimental applications of the WTMM method......... 41
. 4.1. Fully developed turbulence.............................................................. 42
4.2. DNA sequences................................................................................ 45
5. Further developments and perspectives.................................................. 48
LECTURE3
Fractional Derivatives in Static and Dynamic Scaling
by R. Hilfer
1. Introduction ................................................................. ........................... 53
2. Fractional derivatives.............................................................................. 54
3. Scaling in thermodynamics..................................................................... 56
4. Scaling in macroscopic dynamics........................................................... 58
X CONTENTS
LECTURE4
Multi-Dimensional Self-Similarity, and Self-Gravitating
N-Body Systems
by R.N. Henriksen
1. Introduction ... ... . .. ........... .. ... ........... ............. ....... .. ............... .... ... .. .......... 63
2. CH prescription for multi-dimensional self-similarity............................ 65
3. Rescaling symmetries inN-body self-gravitating systems..................... 68
3.1. Stationary example.......................................................................... 68
3.2. Asymptotic self-similarity: speculations.......................................... 69
LECTURES
Scaling in Stock Market Data: Stable Laws and Beyond
by R. Cont, M. Potters and J.-Ph. Bouchaud
1. Introduction ............................................................................................ 75
2. Statistical description of market data...................................................... 76
3. Scale invariance and stable laws............................................................. 77
4. Beyond scale invariance: truncated Levy flights.................................... 78
5. Correlation and dependence.................................................................... 81
6. Turbulence and finance........................................................................... 82
7. Conclusion ....... .. ..... ......... ... ......... .. .. .. .. ..... .... .. . ........... .. .. .. .. ..... .. ............. 83
Examples
Condensed Matter Systems
LECTURE6
Hysteresis, Avalanches, and Barkhausen Noise
by J.P. Sethna, 0. Perkovic and K.A. Dahmen
1. Introduction .... ... ................ .... .. ..... .. .. .. .......... ....................... ..... ..... .. ..... .. 87
2. The model............................................................................................... 89
