Abelian Sandpile

Self-organized criticality: toppling rules, power-law avalanche distributions

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Avalanches
Abelian sandpile (Bak-Tang-Wiesenfeld 1987): if any cell has ≥4 grains, it topples — distributing one grain to each of 4 neighbors. Boundary cells lose grains. The system self-organizes to a critical state where avalanche sizes follow a power law s^{-τ}, τ≈1.1 (2D). The Abelian property: toppling order doesn't matter, final state is unique.