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Table 3
Theoretically predicted and numerically simulated correlations between the length scale L, time duration T, peak energy dissipation rate P, and total energy E for cellular automaton models with Euclidean dimension S = 1,2,3 in the state of self-organized criticality.
| Reference | S = 1 | S = 2 | S = 3 | |
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| Fractal Dimension: DS | ||||
| Diffusive scaling of time with length: L ∝ TβTL = T1/2 | ||||
| Theory: βTL | 0.50 | 0.50 | 0.50 | |
| Slopes (N = 256,64,24) | 0.68 ± 0.25 | 0.49 ± 0.21 | ||
| Slopes (N = 128,32,16) | 0.76 ± 0.13 | 0.48 ± 0.21 | ||
| Regression (N = 256,64,24) | 0.53 ± 0.15 | 0.65 ± 0.02 | 0.53 ± 0.02 | |
| Regression (N = 128,32,16) | 0.65 ± 0.13 | 0.61 ± 0.01 | 0.53 ± 0.02 | |
| Correlation of power with duration: P ∝ TβTP | ||||
| Theory: βTP | 0.50 | 1.00 | 1.50 | |
| Slopes (N = 256,64,24) | 0.91 ± 0.19 | 0.79 ± 0.24 | ||
| Slopes (N = 128,32,16) | 0.99 ± 0.12 | 0.80 ± 0.22 | ||
| Regression (N = 256,64,24) | 0.67 ± 0.20 | 1.04 ± 0.11 | 1.01 ± 0.12 | |
| Regression (N = 128,32,16) | 0.73 ± 0.14 | 1.01 ± 0.08 | 1.02 ± 0.14 | |
| Correlation of energy with duration: E ∝ TβTE | ||||
| Theory: βTE | 1.50 | 1.75 | 2.00 | |
| Slopes (N = 256,64,24) | 1.61 ± 0.08 | 1.51 ± 0.20 | ||
| Slopes (N = 128,32,16) | 1.73 ± 0.10 | 1.48 ± 0.19 | ||
| Regression (N = 256,64,24) | 1.56 ± 0.09 | 1.80 ± 0.05 | 1.75 ± 0.06 | |
| Regression (N = 128,32,16) | 1.65 ± 0.07 | 1.82 ± 0.04 | 1.75 ± 0.06 | |
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