Applications of statistical mechanics to natural hazards and landforms

The concept of self-organized criticality was introduced to explain the behavior of the cellular-automata sandpile model. A variety of multiple slider-block and forest-fire models have been introduced which are also said to exhibit self-organized critical behavior. It has been argued that earthquakes, landslides, forest-fires, and extinctions are examples of self-organized criticality in nature. The basic forest-fire model is particularly interesting in terms of its relation to the critical-point behavior of the site-percolation model. In the basic forest-fire model trees are randomly planted on a grid of points, periodically sparks are randomly dropped on the grid and if a spark drops on a tree that tree and the adjacent trees burn in a model fire. In the forest-fire model there is an inverse cascade of trees from small clusters to large clusters, trees are lost primarily from model fires that destroy the largest clusters. This quasi-steady-state cascade gives a power-law frequency–area distribution for both clusters of trees and smaller fires. The site-percolation model is equivalent to the forest-fire model without fires. In this case there is a transient cascade of trees from small to large clusters and a power-law distribution is found only at a critical density of trees. The earth's topography is an example of both statistically self-similar and self-affine fractals. Landforms are also associated with drainage networks, which are statistical fractal trees. A universal feature of drainage networks and other growth networks is side branching. Deterministic space-filling networks with side-branching symmetries are illustrated. It is shown that naturally occurring drainage networks have symmetries similar to diffusion-limited aggregation clusters.