Evidence of Discrete Scale Invariance in DLA and Time-to-Failure by Canonical Averaging

A. Johansen and D. Sornette ()
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A. Johansen: CATS, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark
D. Sornette: Institute of Geophysics and Planetary Physics and Department of Earth and Space Sciences, UCLA, Los Angeles, CA 90095-1567, USA;

International Journal of Modern Physics C (IJMPC), 1998, vol. 09, issue 03, 433-447

Abstract: Discrete scale invariance, which corresponds to a partial breaking of the scaling symmetry, is reflected in the existence of a hierarchy of characteristic scalesl0,l0λ,l0λ2,…,where λ is a preferred scaling ratio andl0a microscopic cut-off. Signatures of discrete scale invariance have recently been found in a variety of systems ranging from rupture, earthquakes, Laplacian growth phenomena, "animals" in percolation to financial market crashes. We believe it to be a quite general, albeit subtle phenomenon. Indeed, the practical problem in uncovering an underlying discrete scale invariance is that standard ensemble averaging procedures destroy it as if it was pure noise. This is due to the fact, that while λ only depends on the underlying physics,l0on the contrary is realization-dependent. Here, we adapt and implement a novel so-called "canonical" averaging scheme which re-sets thel0of different realizations to approximately the same value. The method is based on the determination of a realization-dependent effective critical point obtained from, e.g., a maximum susceptibility criterion. We demonstrate the method on diffusion limited aggregation and a model of rupture.

Keywords: Scaling; Discrete Scale Invariance; Complex Dimensions; Log-Periodicity; Diffusion Limited Aggregation; Rupture; Critical; Prediction; Power Laws; Averaging; Disorder (search for similar items in EconPapers)
Date: 1998
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DOI: 10.1142/S0129183198000339

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