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Monday, June 26, 2017

Cellular Automata Laboratory: Bak-Tang-Wiesenfeld Sandpile Model

A new rule in Cellular Automata Laboratory (CelLab) implements the Bak-Tang-Wiesenfeld sandpile model [BakTang&Wiesenfeld87]. In each generation, a single grain of sand falls on the cell at the center of the map. When the pile of sand in any cell reaches a height of four grains, it becomes unstable and topples, with the four grains it contains distributed to its four von Neumann neighbors. If this process results in one of more of the neighbors containing four grains, they in turn topple and the process continues until no cell contains four grains. This was the first model discovered which exhibits the property of self-organized criticality. The system exhibits avalanches whose size follows a power law: many small, local events, and a few rare large ones.

Color is used to represent the number of grains in each cell: grey for none, blue for 1, yellow for 2, and red for 3. Since a cell with four grains immediately topples, no cell can contain more than three grains. As the pile grows, you will see how the addition of a single grain can cause cascades of all sizes. While you might expect a smoothly growing structure, in fact the depth of the sand in the pile exhibits a complex fractal pattern that emerges as the pile grows. The edges of the map consume any grains which reach them: they limit the growth of the pile.

If you're patient and have a high-resolution screen, try running Sand in the double-wide simulator—it will produce intricate mandala patterns. The Sand rule is entirely implemented within the sand user evaluator. This is an interesting mathematical model which has proved useful in analyzing emergent processes in a variety of fields. It does not, however, accurately model the behavior of actual piles of sand.

Run the Sandpile simulation in CelLab
Run the Sandpile simulation (double-wide)

Posted at June 26, 2017 13:03