6 - Stochastic sandpiles

Summary

Both of the following models incorporate a form of stochasticity in the relaxation mechanism. In the MANNA Model particles topple to sites randomly chosen among nearest neighbours and in the OSLO Model the local critical slopes are chosen at random. They both display robust scaling and belong to the same enormous universality class, which contains two large classes of ordinary (tuned), non-equilibrium critical phenomena: directed percolation with conserved field (C-DP) and the quenched Edwards-Wilkinson equation (qEW equation). The former is paradigmatically represented by the MANNA Model, the latter by the OSLO Model.

Both models are generally considered to be Abelian, even when they strictly are not. In their original versions, the relaxation of the MANNA Model is non-Abelian and so is the driving in the OSLO Model. This can be perceived as a shortcoming, not only because the BTW Model has been understood in much greater detail by studying its Abelian variant, but also because of the simplification of their implementation, as the final configurations become independent of the order of updates. Nowadays, the MANNA Model and, where the issue arises, the OSLO Model are studied in their Abelian variant. The MANNA Model is currently probably the most intensely studied model of SOC.

In their Abelian form, both models can be described in terms of stochastic equations of motion (Sec. 6.2.1.3 and Sec. 6.3.4.2). These look very different for the two models.