Experiment 07
Sandpile: self-organized criticality
Drop grains one at a time: the pile finds its own critical point, where an avalanche can be tiny or sweep across the whole board — with no characteristic scale.
What is the sandpile?
On a grid, each cell holds a stack of grains. As soon as a cell reaches 4 grains, it topples: it gives one grain to each of its 4 neighbours. Grains that fall off the edge of the board are lost.
One toppling can trigger others: that's an avalanche. All unstable cells are relaxed until things settle — then the next grain is added.
Why this is emergence — a power law, with no tuning at all
Without ever touching a parameter, the pile settles around a critical mean height (~2.1). At that point, avalanche sizes follow a power law: there is no "typical" avalanche, only events at every scale.
This is self-organized criticality (Bak, Tang, Wiesenfeld, 1987) — the same principle invoked for earthquakes, forest fires, extinctions, and market crashes: systems that tune themselves to the edge of chaos.
Pilot
Pilot the pile live
Understanding the vital signs
Four measurements that track self-organization toward the critical state.
- Avalanche size
- Total number of topplings in the last avalanche.
- Avalanche area
- Number of distinct cells touched by the last avalanche.
- Mean height
- Mean height of the pile; converges to ~2.1 at the critical state.
- Critical fraction
- Fraction of cells at 3 grains — "on the edge" of toppling.
