Energy exponents of avalanches and Hausdorff dimensions of collapse patterns

Casals, Blai and Salje, Ekhard K. H. (2021) Energy exponents of avalanches and Hausdorff dimensions of collapse patterns. Physical Review E, 104 (5). ISSN 2470-0045 DOI https://doi.org/10.1103/physreve.104.054138

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Abstract

A simple numerical model to simulate athermal avalanches is presented. The model is inspired by the “porous collapse” process where the compression of porous materials generates collapse cascades, leading to power law distributed avalanches. The energy (E), amplitude (Amax), and size (S) exponents are derived by computer simulation in two approximations. Time-dependent “jerk” spectra are calculated in a single avalanche model where each avalanche is simulated separately from other avalanches. The average avalanche profile is parabolic, the scaling between energy and amplitude follows E∼A2max, and the energy exponent is ε=1.33. Adding a general noise term in a continuous event model generates infinite avalanche sequences which allow the evaluation of waiting time distributions and pattern formation. We find the validity of the Omori law and the same exponents as in the single avalanche model. We then add spatial correlations by stipulating the ratio G/N between growth processes G (linked to a previous event location) and nucleation processes N (with new, randomly chosen nucleation sites). We found, in good approximation, a power law correlation between the energy exponent ε and the Hausdorff dimension HD of the resulting collapse pattern HD−1∼ɛ−3. The evolving patterns depend strongly on G/N with the distribution of collapse sites equally power law distributed. Its exponent ɛtopo would be linked to the dynamical exponent ε if each collapse carried an energy equivalent to the size of the collapse. A complex correlation between ɛ, ɛtopo, and HD emerges, depending strongly on the relative occupancy of the collapse sites in the simulation box.

Item Type:	Article
Uncontrolled Keywords:	2021AREP, IA77
Subjects:	03 - Mineral Sciences
Divisions:	03 - Mineral Sciences 08 - Green Open Access
Journal or Publication Title:	Physical Review E
Volume:	104
Identification Number:	https://doi.org/10.1103/physreve.104.054138
Depositing User:	Sarah Humbert
Date Deposited:	10 Jan 2022 18:26
Last Modified:	10 Jan 2022 18:26
URI:	http://eprints.esc.cam.ac.uk/id/eprint/6041

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