This is a good introduction to power law distributions.

Most natural events and outcomes follow Gaussian and Poisson distributions. They are made up of random, independent factors. But there are certain events which follow power-law distributions.

This is connected with phase transitions and critical points. When ice reaches a critical point it undergoes a phase transition to water. Likewise, the evaporation of water into gas. This is measured through the heat in the system. Self-organized Criticality is responsible for some, but not all, phase transitions. The Sand pile model most often used to illustrate the idea.

Take a cone-shaped sand pile. Drop one more grain of sand on top. What happens? Most of the time, very little. The grain may bounce and fall to the bottom of the pile. It may just land and sit at the top. Sometimes, the grain disturbs another grain on the pile.

Remember how the sandpile is shaped. It is made up of loose particles, interacting with each other and gravity. The pile is in stasis if you leave it alone. But when you start dropping more sand onto it, you change the shape. This disturbs the entire sand pile and pushes the pile further away from equilibrium. The sand pile is not longer sitting “right.”

Every so often, dropping one grain of sand causes an avalanche. That avalanche is a phase transition. The sand pile restructured itself after reaching the critical point. When the transition is over, the now larger sand pile is in a new state of stasis and retains its stable cone shape.

Sand pile avalanches follow a power law distribution and are an example of self-organized criticality.
Here’s an applet demonstrating the sand pile model.

Forest Fires are another example of SOC. The density of vegetation is a major factor in the severity distribution of forest fires. Simply put, if the trees are spaced out, fires will likely not spread as far. If they trees are clustered densely, fires will likely be more severe and widespread. Other factors like moisture levels and wind speed can be factored in.

This can be measured with Percolation models. The fire “percolates” from tree to tree. Here’s an explanation of percolation with another applet to demonstrate the forest fire model:

Percolation is the study of plausible paths between different points in a 2-dimensional or higher lattice network. Generally, there are two intermingled substances in the lattice (grid). If there exists a pathway from one side of the grid to the opposite side, we say that percolation has occurred.

In both cases, the events themselves are unpredictable, nonlinear, and seemingly random. But the distribution of events follows a precise and regular power law distribution.

Per Bak’s book, How Nature Works: The Science of Self-Organised Criticality gives a full explanation about how SOC produces avalanches, earthquakes, forest fires. and other natural events that are distributed by a power-law.(Reviewed here)

Self-Organized Systems have the following characteristics:

  • The system is open and dissipative, and its components are metastable.
  • The system organises itself in a critical state with avalanches of change at all sizes via which dissipation manifests itself. These avalanches are regular but not periodic.
  • The system is embedded in a single spatiotemporal fractal structure.
  • A critically self-organised system might become catastrophically unstable if it were manipulated and forced into certain optimal states which take it out of its self-organised state.

SOC appears in areas of evolution and the social sciences.