Statistical Scale Symmetry Breaking
Many physical systems are characterized by fluctuating quantities over a wide range of scale. During this school, we met several examples: exchange rate in economics, velocity increments in turbulence, density fluctuations in cosmology, intensity of earthquakes, size of disordered domains in Ising models... In this lecture, I shall focus on the description of fluctuations in a scale invariant system, i.e. occuring in a system described by laws which keep the same shape at any scale. Mathematically, this property is referred to as “scale covariance” of the physical laws. Examples of such scale covariant laws are given in the lecture of Pocheau. It is now natural to wonder how this scale covariance is reflected in the random processes solutions of the laws. An easy answer, known by everybody, is when the random process keeps the symmetry, and is itself scale invariant. It is then characterized by power laws- the computation of the corresponding scaling exponents has become a whole industry in physics. This nice situation is however almost never reached in real systems: often, boundary conditions, finite size effects or forces spoil the system and induce a symmetry breaking. The solution is not scale invariant anymore, there are no exact power laws (see the lecture by Castaing and [3, 4, 5, 7]). This is a bit of a mess, then: if you do observe power laws, you have well defined tools and a well defined langage to describe to the guy next door your experiment. If he is interested and wants to compare his measurements to yours, he knows exactly what you are speaking about, and what quantities-namely exponents, and prefactors!- he should measure in his experiment, even though its experimental set up is quite different from yours. When scale symmetry is broken, what can you measure? What can you compare? What is likely to replace exponents and prefactors? The goal of this lecture is to present the tools and the “langage” which can be built to study the scale symmetry breaking.
KeywordsAnisotropy Covariance Helium Lost
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