Self-Organized Criticality

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The BTW sandpile model (Bak-Tang-Wiesenfeld 1987): add grains randomly; when a cell exceeds threshold Z_c, it topples — distributing grains to neighbors, possibly triggering an avalanche. The system self-organizes to a critical state without tuning parameters. Avalanche sizes follow a power law P(s) ~ s−τ (τ≈1.2 in 2D), visible in the histogram as a straight line on log-log scale. This is the canonical example of SOC: criticality as an attractor of the dynamics, not a coincidence.