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⚡ Self-Organized Criticality – Bak's Sandpile

This simulation implements the Bak-Tang-Wiesenfeld sandpile (1987), the canonical model of self-organized criticality (SOC): a system that drives itself to a critical state where perturbations trigger avalanches of all sizes, following a power-law distribution.

🧠 Idea

  1. Drop a grain of sand on a random cell.
  2. If any cell has â‰Ĩ 4 grains, it topples:
  3. loses 4 grains
  4. each of its 4 neighbours gains 1 grain
  5. Toppling can cascade → avalanches
  6. Grains at the boundary are lost (open boundaries = dissipation).

After a transient phase, the system reaches a critical state where:

Property Value
Small avalanches Very frequent
Large avalanches Rare but inevitable
Size distribution P(s) ~ s^(−Ī„), Ī„ ≈ 1.1 – 1.3
Tuning required None – criticality is self-organized

Why is this mind-blowing?

Most phase transitions require precise parameter tuning (temperature at exactly the Curie point, coupling at exactly K_c). The sandpile tunes itself to criticality – no external hand needed. This is why power laws appear everywhere:

  • Earthquakes (Gutenberg-Richter law)
  • Forest fires (fire size distribution)
  • Neural avalanches (Beggs & Plenz, 2003)
  • Financial crashes (Mandelbrot)
  • Extinction events (punctuated equilibrium)

đŸ–ŧ Visualisation

Two-panel display:

Panel Content
Left Current sandpile height map (sand-coloured heatmap)
Right Log-log plot of avalanche size distribution with fitted power-law exponent Ī„

The power law becomes visible after a few thousand grains, and gets cleaner with more data. The fitted Ī„ is displayed as a dashed line.

Press ESC to exit.

🔗 Connection to System Intelligence

  • Regulation (R): The average height self-regulates near 2.0 – too much sand → large avalanches dissipate it
  • Predictive Power (P): While individual avalanches are unpredictable, the distribution is perfectly lawful
  • The deep point: In SOC, the system is at the boundary between order and chaos – where many theorists believe computation and intelligence are maximised

📚 References

  • Bak, P., Tang, C. & Wiesenfeld, K. (1987). Self-organized criticality: An explanation of 1/f noise. Physical Review Letters.
  • Bak, P. (1996). How Nature Works: The Science of Self-Organized Criticality. Copernicus.
  • Beggs, J. M. & Plenz, D. (2003). Neuronal avalanches in neocortical circuits. Journal of Neuroscience.

â–ļ Run

cd simulation-models/self-organized-criticality
python3 sandpile.py

Experiment ideas

  • Increase NUM_GRAINS to 200000 for a cleaner power law
  • Try GRID_SIZE = 128 for larger avalanches (slower)
  • Drop grains only in the centre: r = c = GRID_SIZE // 2 → beautiful symmetric patterns