BTW Sandpile — Self-Organized Criticality

Self-organized criticality (SOC) was introduced by Bak, Tang, and Wiesenfeld (1987) using the abelian sandpile. Grains are added at random sites. When a cell reaches threshold z ≥ 4, it topples: loses 4 grains, each neighbor gains 1 (grains leave at boundaries). The system self-tunes to a critical state with power-law avalanche size distributions P(s) ~ s^−τ, τ≈1.29 in 2D. No fine-tuning is needed — the criticality is self-organized. The fractal structure emerges from the long-range correlations at criticality. SOC has been proposed as a mechanism for 1/f noise, earthquake statistics (Gutenberg-Richter law), extinction cascades, and forest fires.