Sand-pile models

In the last chapter we considered the renormalization group method for treating large interactive systems. By assuming scale invariance a relatively small system could be scaled upward to a large interactive system. The approach is often applicable to systems that have critical point phenomena. In this chapter we consider the alternative approach to large interactive systems. This approach is called self-organized criticality. A system is said to be in a state of self-organized critically if it is maintained near a critical point (Bak et al., 1988). According to this concept a natural system is in a marginally stable state; when perturbed from this state it will evolve naturally back to the state of marginal stability. In the critical state there is no longer a natural length scale so that fractal statistics are applicable.

The simplest physical model for self-organized criticality is a sand pile. Consider a pile of sand on a circular table. Grains of sand are randomly dropped on the pile until the slope of the pile reaches the critical angle of repose. This is the maximum slope that a granular material can maintain without additional grains sliding down the slope. One hypothesis for the behavior of the sand pile would be that individual grains could be added until the slope is everywhere at an angle of repose. Additional grains would then simply slide down the slope. This is not what happens. The sand pile never reaches the hypothetical critical state.