Introduction

The present chapter is about models. All the models are inspired by some physical system: sandpiles, earthquakes, magnetic vortex motion, forest fires, interface growth, or biological evolution. The models are defined in terms of a dynamical variable – for example, the local slope of the sand heap or the stress in the earthquake fault. The dynamical variable or field is updated in every time step according to some algorithm. The choice of the updating algorithm is, to some degree, arbitrary. The criteria for choosing the relevant definitions are, for the most part, simplicity and intuition. Statistical mechanicians have this overall belief that complexity arises from simplicity: that the intricate and complex behavior found in many systems is due to the large number of degrees of freedom, rather than caused by some very complicated behavior of the individual degrees of freedom. All the models described in this chapter are formulated according to this paradigm. In addition, they are all designed with an eye toward numerical ease and efficiency when simulated on computers.

For ease of presentation, for each model we shall first simply present the definition of the model and the conclusions derived from numerical studies thereof. Since the aim is to understand the world around us, we shall use separate sections to discuss the relevance and shortcomings of each model. As always when building models of Nature, one proceeds through steps of increasing refinement, and so also for models of SOC systems.