Sandpile CA

Dribble sand from your hand. At first the sand piles into a shallow cone. After a point, single grains cause avalanches.

The sandpile is Per Bak's central example of self-organized criticality. Critical behavior is characterized by a dramatic and sudden change in macroscopic state. In normal systems, critical behavior is controlled by some external control variable. In self-organized critical systems, it emerges from interactions among the systems' countless components. In the sandpile, the size and frequency of avalanches is independent of any external controller -- for example, the slope of the pile is independent of the rate of sand addition -- and as in other self-organized critical phenomena, the magnitudes of events fall into a power-law distribution.

Here each cell contains some amount of sand, from one to eight units, inclusive. Lighter colored cells have more sand. When Go is on, one unit of sand is dropped on a random cell. If the cell is thus moved above the threshold four units of sand, that cell topples. When a cell topples, it loses four units and its four (von Neumann) neighbors each receive one unit. If any of these neighbors is moved above the threshold, it topples as well, possibly triggering toppling in turn. Cells that receive or lose sand are momentarily turned orange. Boundary cells are absorbing: sand that falls into a boundary cell is lost.

The histogram shows the frequency of several size classes of avalanche, on a log-log scale.

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Ted Wong, 2003