Sandbox Model

Sandbox model is a directed avalanche system that exhibits self-organized criticality. Because of the direct nature of its toppling rules, the dynamics of a sandbox system can be mapping to a fluctuating interface in one lower dimension. This underlying interface growth dynamics of sandbox system belongs to the Kardar-Parisi-Zhang (KPZ) universality class. Through this mapping the exponents that characterize the scaling behavior of avalanche distribution functions can be derived from the known KPZ critical exponents.

Physical system

Consider a cubic box filled with sand. One of the retaining wall of the box can be lowered in an arbitrarily slow rate so to unload the sand from the box. As the lowering wall moves slowly enough, we can expect the sand tumble out of sporadically forming distinct avalanche events instead of a continuous out flow. This gives us a very simple avalanche system.

We can characterize each avalanche of the system by the dimensions of the avalanche cluster in the three spatial directions, the horizontal direction parallel to the lowering wall which we call width (w), the direction perpendicular to the wall which we call length (l) and the vertical height direction which we call depth (d), as well as the total amount of sand got displaced which we call mass (m).

A very interesting observation is that when we plot the frequency of the avalanches for a given kind of sizes versus the size itself on a double log scale, we find the resulting curve is a straight line. This means the distribution follows power law decay and we can't define a typical size for the avalanches.

Model formulation (If you do not have MathML support in your browser, you can use this version.)

Visualizations

Interactive program

avagl: an OpenGL program written in C++.
Building this program requires working installation of gtkmm-1.2.xx and gtkglarea-1.1.xx.

Numerical methods and results

In the study of sandbox models, we use Monte Carlo simulation to perform the avalanche dynamics and measure various distribution functions of avalanche sizes.

Mapping to interface growth

The directness and locality of the dynamic rules of the sandbox allows us to reinterpret the sandbox dynamics in terms of interface growth dynamics in one lower dimension. Interface growth dynamics has been studied for a much longer time and our understanding is in a much better shape. This mapping allows us to gain understanding of avalanche systems in a cost effective way.

Directed percolation roughening transition

For the discrete height version of the sandbox model described above, there are only two possible height values a toppling site can settle into. It can be just the same height as the minimum of it's supporting site or one unit more than that. We call them the minimum stable height and maximum stable height for the toppling site correspondingly. We can introduce a cohesion parameter which represents the probability that the site settles into the maximum stable height instead of the minimum one.

In the model, an interesting deepening transition arises as we vary this cohesion parameter from 0 to 1 at the critical value p_c. Bellow the transition point, the avalanches are in a flat phase where each involves only a few surface layers of the materials. Above the transition point, the avalanches penetrates into the surface and have their depths scales with their sizes. The nature of this transition is related to the directed percolation phase transition which describes epidemic propagation.

From interface picture of the sandbox model, the deepening transition is related to the directed percolation roughening transition of the interface growth which have received several studies in recent year. However, a novel hierarchical directed percolation structure prevents a exact characterization of the scaling behavior of the interface roughness at the transition point. The fractal structure of the avalanche clusters at the transition point also breaks a hyper-scaling relation of the scaling exponent which is derived from the compactness of avalanche clusters and holds in both the shallow and deep phase of the avalanches.

References

You can find the slides for my talk, An Interface View of a Directed Avalanche System, in 2001 APS March meeting.

Also, a related paper, Interface view of directed sandpile dynamics, is published at Phys. Rev. E 65, 031309 (2002).

A longer paper, Directed avalanche processes with underlying interface dynamics, which resolved some issues raised in the short paper is published at Phys. Rev. E 66, 011306 (2002).

Also, another talk given in condensed matter physics journal club on 2002-02-06, Interface dynamics prospect of avalanche systems.

For more details for the deepening transition of avalanches, you can check out the paper, Cohesion-induced deepening transition of avalanches at Phys. Rev. E 66, 061304 (2002).

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