Context :

Aims : 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 .

Methods : 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 energy dissipation rate f ( t ) occupies a fractal volume with dimension D _ { S } , ( 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 } , with S being the Eudlidean dimension .
We perform cellular automaton simulations in three dimensions ( S = 1 , 2 , 3 ) to test the theoretical model .

Results : The analytical model predicts the following statistical correlations : F \propto L ^ { D _ { S } } \propto T ^ { D _ { S } / 2 } for the flux , P \propto L ^ { S } \propto T ^ { S / 2 } for the peak energy dissipation rate , and E \propto FT \propto T ^ { 1 + D _ { S } / 2 } for the total dissipated energy ; The model 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 ) .
The cellular automaton simulations reproduce the predicted fractal dimensions , occurrence frequency distributions , and correlations within a satisfactory agreement within \approx 10 \% in all three dimensions .

Conclusions : 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 .