Abelian Sandpile Model

In this talk I describe some rather elegant mathematical properties of a simple cellular automaton model for self organized criticality. I will discuss how a subset of states in this model form an Abelian group. Then I will show how to construct the non-trivial state which represents the identity for this group. The number of exact results known for this system suggests that it may ultimately be solvable. While the model discussed in this report was first presented to illustrate self organized criticality, this talk is not directly on that subject. Nevertheless, I begin with a brief summary of the concept. It is argued that strongly dissipative systems can drive themselves to a critical state. Unlike conventional critical phenomena, this should occur without any tuning of parameters to a critical value. The prototypical example of this phenomenon is a sandpile. If sand is slowly added to a heap on a table, the pile will evolve towards a critical slope. If it is too steep, a catastrophic avalanche will flatten it, and if it is too flat, the sand will gradually pile up to steepen the pile. Ultimately, the size of an avalanche produced by the random addition of an additional grain of sand will be unpredictable. The expected distribution of avalanche sizes is a power law.