Self-Organized Criticality (SOC) is a concept introduced by Per Bak, Chao Tang, and Kurt Wiesenfeld here at Brookhaven National Laboratory in 1987. I just returned from a conference on "Computational philosophy: lessons from simple models" celebrating the 20th anniversary of this model. Self-Organized Criticality refers to the tendency of large, slowly driven, dissipative systems to organize themselves to a critical (as in critical point of a 2nd order phase transition) stationary state. The dynamics in this state is intermittent with long periods of inactivity separated by well defined bursts of activity or avalanches. The criticality of this state is manifested as the absence of characteristic size or duration of these avalanches. In less fancy language: large avalanches occur rather often (there is no exponential decay of avalanche sizes, which would result in a characteristic avalanche size), and there is a variety of power laws characterizing various properties of the system. The paradigm model for this type of behavior is the sandpile cellular automaton also know as the Bak-Tang-Wiesenfeld (BTW) model. I wrote a JAVA applet which lets one play with this model by adding sand at any site or setting the height of a sand column at any site to any value, changing the system size, boundary conditions, etc.
Below I reproduce nice-looking snapshots of two SOC models: the BTW model (left) and the forest fire model (right), which was introduced in 1990 by Per Bak, Kan Chen, and Chao Tang .
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The identity state in the Bak-Tang-Wiesenfeld sandpile model on a 198×198 lattice. Heights of columns of sand are color coded as green-3,blue-2,red-1,gray-0. Adding this state to any recurrent configuration of the BTW sandpile model and letting it relax, in the end restores the initial configuration. Rules of the model are described here |
A snapshot of a self-sustained forest fire (yellow -fires, green -trees, black - empty spaces). Rules of the game are very simple: Trees grow on empty spaces with probability p; a fire burns down to an empty space in one time step but manages to ignite all nearest-neighbor trees. In order for this model to be a realistic model of forest fires, trees have to grow fast enough to replace their burned neighbors, while the fire is still burning :(. But the model is fun to watch anyhow. |
The above images were made by Mike Creutz who is working on high-energy physics here at Brookhaven. At his Xtoys page you can find more images , as well as neat X-windows-based interactive simulation programs you can play with. My JAVA applet of sandpile model is written under the influence of his xsand C-program for X-windows.
One of the original motivations of Self-Organized Criticality was to explain the ubiquitous appearance of the 1/f-noise and in non-equilibrium systems. In my recent paper with Chao Tang and Yi-Cheng Zhang we investigated the nature of long term configurational memory and the 1/f-noise in sandpile models defined on narrow stripes and driven by local addition of sand at one column. Interestingly, the underlying mechanism for the 1/f noise in this athermal non-equilibrium system is very similar to the mechanism traditionally used for the explanation of the 1/f-noise in voltage fluctuations in wires at thermal equilibrium. More details about this work as well as JAVA applets illustrating the main points of our paper with Chao Tang can be found here.
I also extensively studied SOC extremal models. These models, including the Bak-Sneppen model, Invasion percolation, Zaitsev model, etc., have a mechanism of organizing themselves to a scale-free stationary state, which is completely different from the conservation of sand in sandpile models. Their main feature is that activity always happens at the site, characterized by the global minimum (or maximum if you like it better) of some variable. As a result of this activity this variable is updated at the minimum site itself and its nearest neighbors according to some stochastic rule. The exact definition of this rule defines the universality class of the model. In these three papers you can find more about interesting properties of these models.
