What are self-organized criticality and the sandpile model?

Self-organized criticality (SOC) is a property of dynamical systems that have a critical point as an attractor. Such systems show the scale-invariant behavior typical of a phase transition's critical point, but they reach it by themselves rather than through precise tuning of a control parameter. The sandpile model, proposed by Per Bak, Chao Tang and Kurt Wiesenfeld in 1987, is a simple grid model that was the first system shown to exhibit SOC and remains the archetypal example. [1] [2]

Critical systems tend to show fractal geometry, heavy-tailed distributions and pink (1/f) noise. Freezing water is a standard example. Critical points are usually unstable, though: keeping water partly frozen requires active temperature control, and a small deviation pushes the system into one phase or the other. Yet many natural systems show signs of criticality. According to [greenteapress.com](https://greenteapress.com/complexity2/html/thinkcomplexity2009.html), this is the puzzle Bak, Tang and Wiesenfeld set out to answer, and SOC is their solution: from any initial condition the system moves toward a critical state and stays there without external control. [1]

[Wikipedia](https://en.wikipedia.org/wiki/Self-organized_criticality) adds that the key result of the 1987 paper was showing that complexity could emerge spontaneously from simple local interactions, robustly, without finely tuned details. It also notes that the paper followed earlier work by Jonathan Katz. [3]

The model is a two-dimensional cellular automaton in which each cell holds a number representing local slope or grain count. A cell that exceeds a threshold, usually set so that 3 is the stable limit, topples: it loses four units and each of its four neighbors gains one. Cells at the edge of the grid are kept at zero, so sand spills off. A toppling can push neighbors over the threshold, producing a chain reaction, or avalanche, that continues until every cell is stable. [1] [4]

Grains are then added one at a time at random places. Most additions cause no toppling, but occasionally one triggers an avalanche affecting a large part of the grid. Measured by duration or number of toppled cells, avalanche sizes follow a heavy-tailed, power-law distribution. Small events are common, large ones are rare but always possible, and the outcome of any single grain cannot be predicted without running the model. In 1990 Deepak Dhar showed the model is abelian: the final stable state does not depend on the order of topplings. [1] [2] [4]

One position is that it is not. According to [greenteapress.com](https://greenteapress.com/complexity2/html/thinkcomplexity2009.html), real sand is dense and not very sticky, so momentum affects avalanches. Real piles therefore show fewer very large and very small avalanches than the model predicts, and their distribution might not be heavy-tailed. [1]

The other position, attributed to Bak, is that this misses the point: the model is an abstraction, a simple example of a broad category of systems with many interacting elements. MIT course notes make a similar argument, saying the model's importance lies not in matching real sandpiles but in the apparent ubiquity of power-law phenomena in nature. The sources here do not say which view most specialists hold. [Wikipedia](https://en.wikipedia.org/wiki/Self-organized_criticality) cautions that, despite many identified examples, no known set of general characteristics guarantees that a system will display SOC. [1] [5] [3]