  We suggest a generalized definition of self-organized criticality (SOC)
systems: SOC is a critical state of a nonlinear energy dissipation system that
is slowly and continuously driven towards a critical value of a system-wide
instability threshold, producing scale-free, fractal-diffusive, and
intermittent avalanches with powerlaw-like size distributions. We develop here
a macroscopic description of SOC systems that provides an equivalent
description of the complex microscopic fine structure, in terms of
fractal-diffusive transport (FD-SOC). Quantitative values for the size
distributions of SOC parameters (length scales $L$, time scales $T$, waiting
times $\Delta t$, fluxes $F$, and energies $E$) are derived from first
principles, using the scale-free probability conjecture, $N(L) dL \propto
L^{-d}$, for Euclidean space dimension $d$. We apply this model to
astrophysical SOC systems, such as lunar craters, the asteroid belt, Saturn
ring particles, magnetospheric substorms, radiation belt electrons, solar
flares, stellar flares, pulsar glitches, soft gamma-ray repeaters, black-hole
objects, blazars, and cosmic rays. The FD-SOC model predicts correctly the size
distributions of 8 out of these 12 astrophysical phenomena, and indicates
non-standard scaling laws and measurement biases for the others.
