5 - Dissipative models

Summary

Dissipation is a major theme in SOC for several reasons. Like every relaxation process, avalanching in a sandpile generally can be seen as a form of dissipation, quite literally so for the sand grains that dissipate potential energy in the BTW Model. In that sense, sandpile models are inherently dissipative. Yet, their dynamics can be expressed in terms of variables, which are conserved under the bulk dynamics (also ‘local’ dynamics), such as the number of slope units in the Abelian BTW Model.

The models described in the present chapter, however, go a step further by obeying dynamical rules without local bulk conservation (also ‘local’ conservation), although all models develop towards a stationary state even in the non-conserved variable, i.e. overall there is asymptotic conservation on average. The observation of scale-free dissipation in turbulence triggered the development of the Forest Fire Model (Sec. 5.1), i.e. it was explicitly designed in a dissipative fashion. The situation is somewhat similar for the OFC Model (Sec. 5.3) which incorporates a dissipation parameter α, whereas in the BS Model (Sec. 5.4) dissipation is a necessary by-product. Both the OFC Model and the BS Model are examples of models driven by extremal dynamics, which consists of identifying the ‘weakest link’ among all sites and starting relaxation from there.

In the light of Hwa and Kardar's (1989a) work (Sec. 9.2.2), which suggested that scale invariant phenomena arise naturally, even generically in the presence of bulk conservation, the existence of non-conservative SOC models is particularly important.