My UW-Madison mathematics colleague Jordan Ellenberg has an interesting feature in Nautilus that describes a simple model capable of generating very complex behavior: “The Amazing, Autotuning Sandpile”. The dynamics of sand tumbling down when it reaches a critical point may help to inform us about the dynamics of social forces in the real world.
Real-world political phase transitions tend to happen not in neat sequences, but in sudden coordinated fits, like the Arab Spring, or the collapse of the Eastern Bloc. These reflect quiet periods punctuated by crises—like a sandpile. You can add grains of sand to the top of a sandpile for a while, to no apparent effect. Then, all at once, an avalanche sweeps sand down from the top in an irregular pattern, possibly setting off little sub-avalanches as it goes.
This analogy doesn’t necessarily get us anywhere. After all, real sand is hard to analyze, just like real politics. But here’s the miracle. A kind of abstraction of a sandheap, known as the “abelian sandpile model,” created by physicists Per Bak, Chao Tang, and Kurt Wiesenfeld in 1987, seems to capture some of the rich, chaotic features of real sandpiles, not to mention other complex systems from biology, physics, and social science—while remaining simple enough to study mathematically.
I think about sandpile models quite a lot. While Ellenberg focuses in this article upon the intricate fractal patterns that simple rules can generate, I find other issues more relevant, including the interaction of randomness with critical thresholds.