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.

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.

- discrete height model with control parameter(Java)
- continuous height model running surface + cross section(Java)
- animated GIF files for two different samplings

Building this program requires working installation of gtkmm-1.2.xx and gtkglarea-1.1.xx.

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.

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).

Home