Self-Organized Criticality Theory

Power law phenomena exist extensively in nature. For example, Pareto's law in economics: a minority's income is far greater than that of the majority. The probability that individual income X is not less than a specific value x has a simple inverse relationship with the constant power of x: P[X>=k] ~ x^(-k). Common examples also include the distribution of earthquake magnitudes (Gutenberg-Richter law), the distribution of lunar crater diameters, the distribution of sizes of interplanetary debris, the distribution of solar flare intensities, the distribution of computer file sizes, the distribution of war scales, the distribution of word frequencies in human language, and more (Hu Haibo and Wang Lin, 2005).

Statistical physics has discovered that a closed equilibrium system can display complex behavior characterized by power laws, but only under very special circumstances, namely at the critical point separating two phases. And the critical state needs to be reached through precise adjustment of parameters such as temperature and pressure. Are the power law relationships widely occurring in nature also related to critical states?

In 1987, Bak and others proposed Self-Organized Criticality (SOC, Self-Organized Critical) theory to explain this. Self-organized criticality theory holds that large systems with multiple interacting elements can spontaneously evolve toward a critical state. At this self-organized critical state, a small event can lead to a large event or even a sudden change. Self-organized criticality theory is a new way of observing nature. Its basic stance is that nature is always in a state of continuous non-equilibrium. Due to interactions among elements within the system, they can organize into a critically stable state, that is, the critical state (Bak et al., 1987).

When self-organized critical systems evolve to complex critical states, they are not intervened by any external forces, and they experience a very long transient period. Bak used the sandpile model to demonstrate the self-organized critical process. Imagine continuously adding sand to a sandpile on a table. Initially the sandpile is flat and low, sand avalanches rarely occur. As the sandpile's height increases, its slope also increases. At a certain point, the sandpile's slope reaches a critical value. At this time, adding a single grain of sand may cause avalanches of different sizes. At this point, the entire sandpile is in a self-organized critical state, and the size of sand avalanches follows a power law relationship with their frequency. Bak also used self-organized criticality theory to infer the evolution of society. "If this is the true scenario of the real world, then we must accept the perspectives of biology, history, and economics: instability and great disasters are inevitable. Because the results of those specific unimportant events in the past are contingent, we must also abandon the view of detailed long-term determinism or predictability. Economically, from a selfish perspective, all we can do is transfer disasters to neighboring states." "Self-organized criticality can be seen as the theoretical criterion for catastrophism." (Bak, 1996).