Sandpile models

Chapter 8 describes a number of generalized CA models, including reversible CA, coupled-map lattices, quantum CA, reaction-diffusion models, immunologically motivated CA models, random Boolean networks, sandpile models (in the context of self-organized criticality), structurally dynamic CA (in which the temporal evolution of the value of individual sites of a lattice are dynamically linked to an evolving lattice structure), and simple CA models of combat. 

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

V. Frette. Sandpile models with dynamically varying critical slopes, Phys. Rev. Lett. 70, 2762-2765 (1993). 

Hierarchical Structures Huberman and Kerzberg [huber85c] show that 1// noise can result from certain hierarchical structures, the basic idea being that diffusion between different levels of the hierarchy yields a hierarchy of time scales. Since the hierarchical dynamics approach appears to be (on the surface, least) very different from the sandpile CA model, it is an intriguing challenge to see if the two approaches are related on a more fundamental level. 

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. 

J. Duran, T. Mazozi, E. Clement, and J. Rajchenbach. Decompaction modes of a two-dimensional "sandpile" under vibration Model and experiments. 

The OSL model was constructed in order to explain curious observations reporting that the maximum pressure P in a sandpile was not necessarily directly below the pile s peak but, rather, could occur on a ring of nonzero radius [49-52] (see also Savage [53]). In some cases, the pressure at the base was actually reported to have a local minimum under the peak, the so-called stress dip phenomenon. The 2D OSL model has a Janssen-like constitutive relation of the form (Jxx = ( zz + where z is the vertical and x is the horizontal direction. When coupled with the constraint of stress balance, this leads to the proposal that (static) stresses within a granular material satisfy a hyperbolic PDF in the spatial variables, x and z. Bouchaud et al. then showed that this model could predict a stress dip. Savage [53] argued that soil mechanics models [14] can also account for a stress dip. Elasto-plastic soil mechanics models [ 14] are elastic below yield and are described in this case by elliptic equations (above yield, they are characterized by hyperbolic equations). Hence, the OSL and soil mechanics approaches are inherently different types of models. 

There has been considerable interplay between experimental tests, stimulated for instance by the Q-model and the OSL model, and the development of new models in response to experiments. Both experiments and numerical simulations [54,55, 59,60,81] have shown that the existence of non-isotropic textures due to different deposition procedures of sandpiles or other packing procedures can determine the way forces are transmitted and produce different stress distributions. 

M. Paczuski and S. Boettcher. Universality in sandpiles, interface depinning, and earthquake models, Phys. Rev. Lett. 77, 111-114 (1996). 

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