PMID: 11909200Mar 23, 2002Paper

Log-periodic route to fractal functions

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
S Gluzman, D Sornette

Abstract

Log-periodic oscillations have been found to decorate the usual power-law behavior found to describe the approach to a critical point, when the continuous scale-invariance symmetry is partially broken into a discrete-scale invariance symmetry. For Ising or Potts spins with ferromagnetic interactions on hierarchical systems, the relative magnitude of the log-periodic corrections are usually very small, of order 10(-5). In growth processes [diffusion limited aggregation (DLA)], rupture, earthquake, and financial crashes, log-periodic oscillations with amplitudes of the order of 10% have been reported. We suggest a "technical" explanation for this 4 order-of-magnitude difference based on the property of the "regular function" g(x) embodying the effect of the microscopic degrees of freedom summed over in a renormalization group (RG) approach F(x)=g(x)+mu(-1)F(gamma x) of an observable F as a function of a control parameter x. For systems for which the RG equation has not been derived, the previous equation can be understood as a Jackson q integral, which is the natural tool for describing discrete-scale invariance. We classify the "Weierstrass-type" solutions of the RG into two classes characterized by the amplitudes A(n) of the po...Continue Reading

References

Sep 29, 1999·Proceedings of the National Academy of Sciences of the United States of America·R MetzlerJ Jortner
Dec 2, 2000·Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics·B DavidovitchI Procaccia
Apr 24, 2002·Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics·G Paul
Jun 1, 1981·Proceedings of the National Academy of Sciences of the United States of America·B D HughesE W Montroll

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Citations

Jan 15, 2016·Journal of the Royal Society, Interface·Amir A ToorMasoud H Manjili
Jan 20, 2018·Physical Review. E·R M B DinizJ M de Araújo
Sep 19, 2020·Physical Review. E·Ehoud Pazy
Apr 7, 2010·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·A S FerreiraMarco Antonio Alves da Silva
Jan 15, 2011·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·M Ghasemi Nezhadhaghighi, M A Rajabpour
Jul 13, 2004·Physical Review Letters·Hugo L D de S Cavalcante, J R Rios Leite
Jan 7, 2003·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·William J Reed, Barry D Hughes

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