4 - Deterministic sandpiles

Summary

The models discussed in this chapter resemble some of the phenomenology of (naïve) sandpiles. As discussed earlier (Sec. 3.1), it is clear that the physics behind a real relaxing sandpile is much richer than can be captured by the following ‘sandpile models’. Yet, it would be unjust to count that as a shortcoming, because these models were never intended to describe all the physics of a sandpile. The situation is similar to that of the ‘Forest Fire Model’ which is only vaguely reminiscent of forest fires and was, explicitly, not intended to model them. The names of these models should not be taken literally, they merely serve as a sometimes humorous aide-memoire for their setup, similar to Thomson's Plum Pudding Model which is certainly not a model of a plum pudding.

In the following section, the iconic Bak-Tang-Wiesenfeld Model and its hugely important derivative, the Abelian Sandpile Model, are discussed in detail. This is followed by the ZHANG Model, which was intended as a continuous version of the BAK-TANG-WIESENFELD Model. Their common feature is a deterministic, rather than stochastic, relaxation rule. Although a lot of analytical and numerical progress has been made for all three models, their status quo, in particular to what extent they display true scale invariance, remains inconclusive.

The Bak-Tang-Wiesenfeld Model

The publication of the Bak-Tang-Wiesenfeld (BTW) Model (see Box 4.1) (Bak et al., 1987) marks the beginning of the entire field.