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
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
from the box. As the lowering wall moves slowly enough, we can expect
sand tumble out of sporadically forming distinct avalanche events
of a continuous out flow. This gives us a very simple avalanche system.
We can characterize each avalanche of the system by the dimensions
the avalanche cluster in the three spatial directions, the
direction parallel to the lowering wall which we call width (w),
direction perpendicular to the wall which we call length (l) and
vertical height direction which we call depth (d), as well as
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
in your browser, you can use this version.)
avagl: an OpenGL program
written in C++.
Building this program requires
working installation of gtkmm-1.2.xx and
Numerical methods and results
In the study of sandbox models, we use Monte Carlo simulation to
perform the avalanche dynamics and measure various distribution
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
This mapping allows us to gain understanding of avalanche systems in a
Directed percolation roughening transition
For the discrete height version of the sandbox model described above,
are only two possible height values a toppling site can settle into. It
be just the same height as the minimum of it's supporting site or one
more than that. We call them the minimum stable height and maximum
height for the toppling site correspondingly. We can introduce a
parameter which represents the probability that the site settles into
maximum stable height instead of the minimum one.
In the model, an interesting deepening transition arises as we vary
cohesion parameter from 0 to 1 at the critical value p_c. Bellow the
point, the avalanches are in a flat phase where each involves only a
surface layers of the materials. Above the transition point, the
penetrates into the surface and have their depths scales with their
The nature of this transition is related to the directed percolation
transition which describes epidemic propagation.
From interface picture of the sandbox model, the deepening transition
related to the directed percolation roughening transition of the
growth which have received several studies in recent year. However, a
hierarchical directed percolation structure prevents a exact
of the scaling behavior of the interface roughness at the transition
The fractal structure of the avalanche clusters at the transition point
breaks a hyper-scaling relation of the scaling exponent which is derived
the compactness of avalanche clusters and holds in both the shallow and
phase of the avalanches.
You can find the slides for my talk, An
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,
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,
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
out the paper, Cohesion-induced
transition of avalanches at Phys.
Rev. E 66, 061304 (2002).