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We suggest a generalized definition of self-organized criticality (SOC)systems: SOC is a critical state of a nonlinear energy dissipation system thatis slowly and continuously driven towards a critical value of a system-wideinstability threshold, producing scale-free, fractal-diffusive, andintermittent avalanches with powerlaw-like size distributions. We develop herea macroscopic description of SOC systems that provides an equivalentdescription of the complex microscopic fine structure, in terms offractal-diffusive transport (FD-SOC). Quantitative values for the sizedistributions of SOC parameters (length scales $L$, time scales $T$, waitingtimes $\Delta t$, fluxes $F$, and energies $E$) are derived from firstprinciples, using the scale-free probability conjecture, $N(L) dL \proptoL^{-d}$, for Euclidean space dimension $d$. We apply this model toastrophysical SOC systems, such as lunar craters, the asteroid belt, Saturnring particles, magnetospheric substorms, radiation belt electrons, solarflares, stellar flares, pulsar glitches, soft gamma-ray repeaters, black-holeobjects, blazars, and cosmic rays. The FD-SOC model predicts correctly the sizedistributions of 8 out of these 12 astrophysical phenomena, and indicatesnon-standard scaling laws and measurement biases for the others.

A Macroscopic Description of a Generalized Self-Organized Criticality System: Astrophysical Applications