Scale Invariance and Beyond

Scale Invariance and Beyond B. Dubrulle, F. Graner and D. Somette · CNRS, CEAIDSMIDAPNWservice d'Astrophysique, L 'Orme des Merisiers, 709, 91191 Gif-sur-Yvette, France Laboratoire de Spectrometrie Physique, B.P. 87, 38402 Saint-Martin-d'Heres cedex, France Laboratoire de Physique de Ia Matiere Condensee, CNRS and Universite de Nice -Sophia Antipolis, Pare Valrose, 06108 Nice, France Department of Earth and Space Sciences and Institute of Geophysics and Planetary Physics, University of California, Los Angeles, California 90095-1567, U.S.A. 1. INTRODUCTION In physics, any symmetry is a bargain. It provides tools to analyze the solutions of a problem without need to explicitly solve the problem, or it can also suggest methods to simplify the problem. In real life, boundary conditions or even physical processes themselves break the initial beautiful symmetry. One has to deal with "approximate symmetry" and means to obtain useful informations from this broken symmetry have to be built. The topic of the school held in Les Houches in March 1997 is the illustration of these general ideas in the case of scale symmetry. Part of the school involved descriptions of tools or concepts used in scale invariant systems, and examples pertaining to them ("Scale Invariance ... "); the second half was devoted to recent attempts to go beyond the invariance or symmetry breaking, discuss causes and consequences, and extract useful informations about the system (" ... and beyond").