Self-organised criticality

In the rest of this chapter, we will discuss briefly the theoretical ideas and the models employed for the study of failure of disordered solids, and other dynamical systems. In particular, we give a very brief summary of the percolation theory and the models (both lattice and continuum). The various lattice statistical exponents and the (fractal) dimensions are introduced here. We then give brief introduction to the concept of stress concentration around a sharp edge of a void or impurity cluster in a stressed solid. The concept is then extended to derive the extreme statistics of failure of randomly disordered solids. Here, we also discuss the competition between the percolation and the extreme statistics in determining the breakdown statistics of disordered solids. Finally, we discuss the self-organised criticality and some models showing such critical behaviour. 

We discuss the various dynamical models of earthquake-like failures in Chapter 4. Specifically, the properties of the Burridge-Knopoff stick-slip model (Burridge and Knopoff 1967) and of the self-organised criticality models, the Guttenberg-Richter type power laws, for the frequency distribution of earthquakes in these models are discussed here. 

Recently, some very simple cellular automata models of such randomly driven dissipative systems have been developed and have been studied extensively. It has been shown that the dynamics of such models leads to a critical state characterised by power laws induced by stochastically developed self-similarities in the system. One such popular model, known as the BTW model, introduced by Bak et al (1987,1988), attempts to capture the avalanche dynamics of a sandpile where the sand grains are being added to the pile at a constant rate. The model has been studied extensively, both numerically and analytically, and the existence of the self-organised criticality in the model has been established. 

Dynamic annealed impurity and self-organised criticality in fracture... 

Self-organised criticality and cellular automata models of earthquakes... 

Response of sandpile models to weak pulses and precursors of self-organised criticality and earthquakes... 

Obviously, it would be quite useful to know the precursors of such critical states, and be able to predict imminent catastrophes. It has been suggested recently (by Acharyya and Chakrabarti 1996a,6) that looking at the growth of the responses to the appropriate local and weak pulsed perturbations in some models, the approach to the self-organised critical state can be studied and its appearance can be predicted. This has been demonstrated, in particular, in the BTW critical height model (introduced in Section 1.2.3 of the first chapter, and also discussed in the previous section). It has been... 

P. Bak How Nature Works. The Science of Self-Organised Criticality (Oxford University Press., 1996)... 

Optimal paths in the presence of strong disorder is found to belong to the universality class of the shortest paths on self-organised critical clusters in invasion percolation with and without trapping [38]. The length L of the shortest paths on percolation clusters scales with the end-to-end distance R as... 

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