Abelian Sandpile Model - Self-organized Criticality

Self-organized Criticality

The original interest behind the model stemmed from the fact that it is attracted to its critical state, at which point the correlation length of the system and the correlation time of the system go to infinity, without any fine tuning of a system parameter. This contrasts with earlier examples of critical phenomena, such as the phase transitions between solid and liquid, or liquid and gas, where the critical point can only be reached by precise tuning (e.g., of temperature). Hence, in the sandpile model we can say that the criticality is self-organized.

Once the sandpile model reaches its critical state there is no correlation between the system's response to a perturbation and the details of a perturbation. Generally this means that dropping another grain of sand onto the pile may cause nothing to happen, or it may cause the entire pile to collapse in a massive slide. The model also displays 1/ƒ noise, a feature common to many complex systems in nature.

This model only displays critical behaviour in two or more dimensions. The sandpile model can be expressed in 1D; however, instead of evolving to its critical state, the 1D sandpile model instead reaches a minimally stable state where every lattice site goes toward the critical slope.