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# A Statistical Fractal-Diffusive Avalanche Model of a Slowly-Driven Self-Organized Criticality System

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### Abstract

We develop a statistical analytical model that predicts the occurrence frequency distributions and parameter correlations of avalanches in nonlinear dissipative systems in the state of a slowly-driven self-organized criticality (SOC) system. This model, called the fractal-diffusive SOC model, is based on the following four assumptions: (i) The avalanche size $$L$$ grows as a diffusive random walk with time $$T$$, following $$L \propto T^{1/2}$$; (ii) The instantaneous energy dissipation rate $$f(t)$$ occupies a fractal volume with dimension $$D_S$$, which predicts the relationships $$F = f(t=T) \propto L^{D_S} \propto T^{D_S/2}$$, $$P \propto L^{S} \propto T^{S/2}$$ for the peak energy dissipation rate, and $$E \propto F T \propto T^{1+D_S/2}$$ for the total dissipated energy; (iii) The mean fractal dimension of avalanches in Euclidean space $$S=1,2,3$$ is $$D_S \approx (1+S)/2$$; and (iv) The occurrence frequency distributions $$N(x) \propto x^{-\alpha_x}$$ based on spatially uniform probabilities in a SOC system are given by $$N(L) \propto L^{-S}$$, which predicts powerlaw distributions for all parameters, with the slopes $$\alpha_T=(1+S)/2$$, $$\alpha_F=1+(S-1)/D_S$$, $$\alpha_P=2-1/S$$, and $$\alpha_E=1+(S-1)/(D_S+2)$$. We test the predicted fractal dimensions, occurrence frequency distributions, and correlations with numerical simulations of cellular automaton models in three dimensions $$S=1,2,3$$ and find satisfactory agreement within $$\approx 10%$$. One profound prediction of this universal SOC model is that the energy distribution has a powerlaw slope in the range of $$\alpha_E=1.40-1.67$$, and the peak energy distribution has a slope of $$\alpha_P=1.67$$ (for any fractal dimension $$D_S=1,...,3$$ in Euclidean space S=3), and thus predicts that the bulk energy is always contained in the largest events, which rules out significant nanoflare heating in the case of solar flares.

### Most cited references4

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### Self-organized criticality: An explanation of the 1/fnoise

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### Author and article information

###### Journal
20 December 2011
1112.4859
10.1051/0004-6361/201118237