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

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 $L\propto T^{1\mathrm{/}2}$; (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 $E\propto FT\propto T^{1+D_S\mathrm{/}2}$ for the total dissipated energy; (iii) The mean fractal dimension of avalanches in Euclidean space $S=1,2,3$ is $D_S\approx (1+S)\mathrm{/}2$; 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 ${\alpha }_T=(1+S)\mathrm{/}2$, ${\alpha }_F=1+(S-1)\mathrm{/}{D}_S$, ${\alpha }_P=2-1\mathrm{/}S$, and ${\alpha }_E=1+(S-1)\mathrm{/}(D_S+2)$. 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 $D_S=1,\mathrm{.}\mathrm{.}\mathrm{.},3$ 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.