Bak's Sandpile
Sandpile is a cellular automaton. A sand grain is added to a randomly selected cell. Grains stack one upon another. If a cell contains four or more grains those grains will topple at the next timestep, transferring one grain to each of the 4 neighboring cells. This transfer may cause one or more of the neighbors to topple. Avalanches are allowed to run to completion before more sand is added to the pile.
Sandpile exhibits self-organized criticality. Small perturbations to the system (the random addition of a single sand grain) may cause large or small avalanches. Watch the graph of avalanche frequency versus avalanche size evolve, as the number of avalanches grows. The data will tend to fall on a line with a negative slope. Because this is a double log graph, frequency and size really follow an inverse power law; small avalanches are much more frequent than large avalanches. The slope of the line equals the exponent in the power law.
The power law relation between avalanche frequency and avalanche size is a macroscopic feature that emerges from local interactions among cells. Nothing in the rules specifies the relationship should be a power law. In nature, snowmelt- and earthquake-induced landslides follow inverse power laws with exponents of 2.5 ± 0.5.
The game's speed can be changed from the pull-down menu. To watch individual avalanches evolve choose 1 Timestep. To generate the frequency plot quickly, select 100 Avalanches.
Bak, P., 1996, How Nature Works: The Science of Self-Organized Criticality, Copernicus Press, 212 pp.
White cells contain no grains while green cells contain four or more grains. Red cells contain fewer than 4 grains because the grains toppled (and were redistributed) at least once in the current avalanche.