Finite System-size Effects in Self-organized Criticality Systems

We explore upper limits for the largest avalanches or catastrophes in nonlinear energy dissipation systems governed by self-organized criticality. We generalize the idealized “straight” power-law size distribution and Pareto distribution functions in order to accommodate incomplete sampling, limited instrumental sensitivity, finite system-size effects, and “Black Swan” and “Dragon King” extreme events. Our findings are as follows. (i) Solar flares show no finite system-size limits up to L ≲ 200 Mm, but solar flare durations reveal an upper flare duration limit of ≲6 hr. (ii) Stellar flares observed with Kepler exhibit inertial ranges of E ≈ 10 34 –10 37 erg, finite system-size ranges of E ≈ 10 37 –10 38 erg, and extreme events at E ≈ (1–5) × 10 38 erg. (iii) The maximum flare energies of different spectral type stars (M, K, G, F, A, giants) reveal a positive correlation with the stellar radius, which indicates a finite system-size limit imposed by the stellar surface area. Fitting our finite system-size models to terrestrial data sets (earthquakes, wildfires, city sizes, blackouts, terrorism, words, surnames, web links) yields evidence (in half of the cases) for finite system-size limits and extreme events, which can be modeled with dual power-law size distributions.