Self-Organized Criticality – Sandpile

Bak-Tang-Wiesenfeld Sandpile Model (1987)

The Abelian sandpile model is the canonical example of self-organized criticality (SOC). Add one grain to a random cell. If height ≥ 4, topple: the cell loses 4 grains, each neighbor gains 1. This cascade (avalanche) continues until stable.

Toppling rule: z(i,j) → z(i,j)−4 if z≥4; each neighbor z(i±1,j±1) → z+1

Avalanche size distribution: P(s) ~ s^(−τ) τ ≈ 1.22 (2D)

The system self-organizes to a critical state without tuning any parameter. Avalanches follow a power law spanning all sizes — no characteristic scale. The log-log plot shows the avalanche size distribution accumulating over time.

SOC appears in earthquake (Gutenberg-Richter law), forest fires, neuronal avalanches, and financial market crashes.