Self-organized Criticality: Sandpiles and Flux Lines

1. Self-Organized Criticality 1.1. Introduction There is an abundance of the scale invariant phenomena in nature, e.g. fractals, earthquakes, 1=f noise, uctuation of the stock index, etc. Scale invariance means that there are many scales (or equivalently, no typical scale) in the system. One asks why and how so many diierent scales emerge naturally in a physical system. In 1987, Bak, Tang, and Wiesenfeld proposed the notion of Self-Organized Criticality (SOC) 1 as a mechanism for some of the scale invariant phenomena. The idea is best illustrated by thinking of a model sandpile. Imaging that we build a sandpile from scratch by adding grains of sand slowly and randomly. At the beginning, the pile is rather at. The added grain usually nestles where it is sprinkled, or it may cause a small local rearrangement of sand grains. As the pile grows we sometimes see larger and larger local rearrangements or avalanches. Eventually the pile will reach a statistically stationary state where its slope has a \critical" value. If the slope of the pile is smaller than the critical value, addition of sand grains will increase the slope. On the other hand, if the slope of the pile is larger than the critical value, addition of sand grains will trigger large avalanches which will bring the slope back to the critical value. So there is a feedback mechanism in the dynamics to keep the system at a critical point. At this critical state, the addition of sand grains can have many diierent consequences: it may simply nestle where it is added, or it may cause a small avalanche, or it may on occasion trigger a very large avalanche. At this point, there is no typical scale for the avalanches or uctuations: they can have any sizes up to the size of the system. It has been demonstrated in some simple models that indeed the system will be driven into a critical state where the distribution of uctuations has a power-law form or a multifractal form. The physics generating small avalanches is the same as that generating the big ones. The big avalanches are nothing special but part of the critical uctuations. Scale-invariant uctuations are common in equilibrium critical phenomena where a critical state can be reached by ne-tuning certain parameters. While for certain nonequilibrium dynamical systems, the dynamical feedback mechanism mentioned above tunes the system to be at a critical point. …