Drop grains one at a time. Avalanches follow a power law — the system self-tunes to criticality.
20
Total grains: 0
Last avalanche: 0 topplings
Max avalanche: 0
Avalanches: 0
Bak-Tang-Wiesenfeld Sandpile (1987): each cell holds up to 3 sand grains. When a cell reaches 4, it topples — sending one grain to each of its four neighbors (boundary grains fall off). A single dropped grain can trigger a cascade of topplings (avalanche) of any size.
Self-organized criticality (SOC): without tuning any parameters, the system spontaneously evolves to a critical state where avalanche sizes follow a power law P(s) ~ s^(−τ) with τ ≈ 1.0–1.4. This is unlike equilibrium phase transitions, which require fine-tuning to reach criticality.
SOC has been proposed as a mechanism for 1/f noise, earthquake Gutenberg-Richter law, neural avalanches in the brain, stock market fluctuations, and species extinction events. The sandpile is Abelian — the final configuration is independent of the order in which topplings occur (Dhar 1990).