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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). The script lets a finite open-boundary pile approach a stationary regime and plots the observed avalanche-size distribution. A power-law claim requires a fit comparison and finite-size analysis beyond the visual included here.

🧠 Idea

  1. Drop a grain of sand on a random cell.
  2. If any cell has ≥ 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 Possible and less frequent in typical runs
Size distribution Estimated from the run; model- and size-dependent
Tuning required Slow drive and open-boundary dissipation are built in

Why is this mind-blowing?

Many familiar phase transitions are studied by varying a control parameter through a critical region. The sandpile is a precise example in which slow driving and dissipation produce scale-rich cascades without tuning such a parameter to a single value. It motivated comparisons with many empirical systems, but their power laws and mechanisms must be established separately:

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

🖼 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 Ï„

The log-log plot and fitted slope are exploratory diagnostics. Establishing a power law requires testing alternative heavy-tailed distributions, a fitting range, goodness of fit, uncertainty, and finite-size effects.

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): Individual avalanches vary while aggregate distributions can be estimated with uncertainty.
  • Research connection: Critical regimes are candidates for studying information propagation and computation; this simulation does not show that intelligence is maximized there.

📚 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–¶ Run

cd simulation-models/emergent-dynamics/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