By F. Graner (auth.), B. Dubrulle, F. Graner, D. Sornette (eds.)
This e-book is a wonderful advent to the idea that of scale invariance, that's a turning out to be box of analysis with vast purposes. It describes the place and the way symmetry lower than scale transformation (and its a variety of kinds of partial breakdown) can be utilized to research ideas of an issue with no the necessity to explicitly clear up it. the 1st half offers descriptions of instruments and ideas; the second one is dedicated to contemporary makes an attempt to head past the invariance or symmetry breaking, to debate factors and effects, and to extract precious information regarding the approach. Examples are conscientiously labored out in fields as diversified as condensed topic physics, inhabitants dynamics, earthquake physics, turbulence, cosmology and finance.
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Extra info for Scale Invariance and Beyond: Les Houches Workshop, March 10–14, 1997
Scale symmetry breaking Like in any other symmetry, scale symmetry breaking can occur for a variety of reasons and can take different forms. Two specific aspects were discussed in the course of the school: the appearance of complex exponents generated by the breaking of the continuous scale symmetry into the discrete scale symmetry; the modifications of the laws induced by taking into account the presence of cut offs, or finite size effects. The first aspect is discussed in details by Sornette.
The hypothesis presented here is that the observed supercooling anomalies are caused by a critical point above which two metastable amorphous phases of ice, previously shown to be separated by a line of first order transitions, become indistinguishable. This model, which incorporated the two amorphous ice phases into our understanding of the liquid state of water, provides a more complete picture of the metastable and stable behavior of water. 2. From cosmology to earthquakes The presentations that follow discuss systems from natural sciences to social sciences.
Cont et al include finite size effects directly in the pdf of their Levy laws by truncating their range of application, so as to ensure a finite variance. Due to the truncation, the second moment exists and leads, in absence of severe correlations, to the slow convergence to the Gaussian law for large time intervals (months). This slow cross over from (truncated) Levy laws to a Gaussian law as the time scale increases reconcile the antagonist claims found in the lit~rature, and allows a better fit of the stock market price variations, even at small time scales (5 minutes).