Sand
Cellular Automata Modeling of Sand-Pile SOCs
Joseph Baker
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Physical systems of vast quantities of small, discrete particles exhibit fascinating self-organizing structures. When subjected to a
potential field, such as gravity or electrostatic charge, particles will sometimes fall into complex arrangements without any further
external influence. Such attractors occur in some form for an incredible variety of natural systems, which – seemingly regardless of
initial entropy and configuration – will organize themselves into precise, complex shapes. One example, which has been widely studied
since the late 1980s, is the sand-pile.
When poured from a local source, a collection of sand grains will form a smooth cone with a constant slope of
repose. This slope will remain effectively constant after it reaches a steep enough critical slope. If we further restrict this
pile to a finite grid or platform upon which to form, and disregard any grains which fall off of this grid, it’s obvious that
the pile will grow to a certain critical size and shape, and no further. However, these critical quantities are not some
fixed integer number – the number of grains, and mass of the system, will fluctuate around a critical value as
Figure 1: Idealized sand grain count, note 1/f noise. Credit Zehetner, Ref. in text (left); Author (right)
“avalanches” leave the grid, and sand is steadily added to the system. This state is called the critical state of the system.
In describing this model, it’s impossible not to point out a few truly remarkable details. After a moment’s
thought, it’s apparent that any arbitrary distribution pattern of sand on a finite grid will yield the same general critical
state, after an arbitrary time has passed. Next, we must consider the avalanches. The “size,” or number of grains
affected by these avalanches, scales according to a power-law distribution. (Tang, Wiesenfeld, & Bak, 27 July 1987).
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These self-reinforcing overflows occur most prominently once the system has reached its critical state, and are seen in
Figure 1 above (Zehetner, 2010) as the 1/f “noise” in the graph.
It’s clear that sand-piles are an example of a self-organizing criticality, or SOC. They’re a member of a
universality class which evolves to a very definite, yet only quasi-stable critical state, seemingly regardless of initial
conditions. Inasmuch, they’re an excellent model to study, if one wishes to understand SOC systems better. The
undisputed seminal work in this field of sand-squinting is that of Bak, Tang, and Wiesenfeld, originally published in the
Physics Review Letters (Tang, Wiesenfeld, & Bak, 27 July 1987). The three, who are frequently referred to as ‘BTW’ for
Mean-Field Theory
Sand Models
Cellular Automata Sand Models
1-D Models
2-D Models
Non-Square Cell
Automata
Square Cell Automata
Rectangular
Lattice
Directed
Non-Directed
Elliptical
Lattice
Finite
Quasi-Infinite
brevity’s sake, were interested in this 1/f noise seen in mass fluctuations, and devised a cellular automata system to
describe it. This method of modeling has remained essentially standard for sand-pile SOC theory ever since, and has
been applied in various dimensions and rule sets.
BTW’s initial model was simple. It described a row of N columns of an integer height, of which each column’s
height difference from its right-hand neighbor was measured as zn. Each “tick,” a grain of sand was added to a random
column of sand-towers. If zn on any column exceeded a certain value, the column would “tumble” one height integer to
its right neighboring column. This “half”-dimensional Directed Model gives rise to a simple “staircase” arrangement after
some time, with each additional grain tumbling all the way to the end of the row. However, when the same rule-set is
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generalized to two dimensions, all kinds of
interesting things happen. In one dimension, the staircase was extremely stable, as each action produced one reaction,
Figure 3: One-Dimensional Automata. Image credit Bak, Tang, Weisenfield 1987. Evolution to staircase (right). Image credit author
and noise could be perfectly transmitted through the system and out.
When we consider two dimensions, a noise signal from one sand grain will transmit its energy to multiple
neighboring grains. If those grains are also in a critical state, the addition of any energy will send them toppling,
continuing the cycle. Unlike in one dimension, where the critical slope was a “lossless” transmitter, zn,critical is now the
least stable position. Thus, the system will relax to a very different state, since high concentrations of zn cells create selfdestructive chain reactions. BTW summed this entire concept up beautifully in their 1987 publication: “The system will
become stable precisely at the point when the network of minimally stable clusters has been broken down to the level
where the noise signal [of grains being added] cannot be communicated through infinite distances. At this point there
will be no length scale, and, consequently, no time scale.”
It’s worth mentioning that BTW were not particularly interested in sand piles in their initial publication. While
they described their initial models with what they referred to as “sand language,” in order to provide a physical image
for the reader, they were keenly aware of the implications of their models to many self-organizing non-linear diffusion
systems, “describing the flow of electrons, or water, or light, etc.” (Tang, Wiesenfeld, & Bak, 27 July 1987, p. 38)
However, the sand pile is a very easy system to study in a laboratory, and thus many of the common characteristics of
SOCs have been analyzed by further study of the cellular automata models which describe sand’s accumulation, and the
comparison of these results against laboratory observations. (Lubeck, 2003)
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Directed models, in which cell-signal propagation is sent only one neighbor (in one direction) per dimension,
were quickly analyzed further and solved in an arbitrary number of dimensions (Dhar & Ramaswamy, Exactly Solved
Model of Self-Organized Critical Phenomena, 16 October 1989). However, the non-directed problem remains
incomplete, even in one-dimensional form. As with many other quasi-chaotic systems, it exhibits complex solutions
untraceably originating from complex conditions.
Even without a complete understanding of these cellular automata, we’ve found some interesting properties.
Various modifications to the rule set and precise workings of the BTW Model have been made, including the Abelian
Model, an elegant commutative model which has seen large amounts of study on its own. (Dhar, Self-Organized Critical
State of Sandpile Automata Models, 2 April 1990). The Abelian Model also describes proportionality exponents between
geometric parameters of the scale-less “fractal avalanche” occurrences very well. As the avalanches are self-similar
across scale, quantities such as their affected surface area, number of grains involved, and perimeter are related by a
logarithmic proportionality. By comparing the calculated proportionality exponents from real-world observation to
those of a model, it’s possible to quantify a model’s accuracy.
Recent work suggests that the Cartesian grid arrangement of a 2-D cellular automata may introduce effects
dissimilar to real-world conditions. Near the apex of the critical-state “cone,” cellular automata models create a four
sided pyramid, due to the non-trivial size of discrete cells in such small structures. These edges of the pyramid disrupt
the formation of avalanches near the peak, which would otherwise form on a smooth corner-less surface (Zehetner,
2010).
While no meaningful study has yet been conducted on non-Cartesian cellular automata models in this
application, a small computer game “Hexplode” is precisely this – a 24-cell hexagonal automata, which players attempt
to fill in their favor, “avalanching” their opponent off the grid. The game provided the author with several hours of
procrastination, and can be recommended as a simple demonstration of these “tumbling” cellular automata systems,
where cells spill their contents to neighbors upon achieving a set of parameters.
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Laboratory studies of sand as a SOC have also evolved in recent years. The 1/f noise log-log frequency
distribution observed in avalanches is known to be “heavy-tailed” just like many other real-world power law systems,
with a higher occurrence of large-scale events than should be seen from a perfect power law distribution. This disagrees
with most interpretations of the BTW model, and the 1996 “Oslo Rice Experiments” showed that rice piles, with their
elongated shape and low coefficient of friction, matched cellular automata models like BTW’s far closer than the original
piles of sand which the models were designed for. However, the models weren’t perfect, so new, more complicated
models were proposed to explain the rice piles (Pruessner & Jensen, 11 December 2003).
In the thirty years since the first study of Self-Organizing Criticalities in sand, we may not have counted the
number of grains of sand in the pile, but we’ve learned incredible things about that pile. As a simple intuitive system, the
sand-pile is perhaps the lab rat of Self-Organizing Criticalities. It’s led to the development of a whole new category of
cellular automata, and redefined how we think about Angle of Repose and avalanche prevention. Understanding of
“tumbling” cellular automata allows application to countless non-linear systems. It’s clear that we’re just beginning to
realize the scope of this field, and research is not slowing – over 2,000 journal articles and peer-reviewed papers
claiming relation to SOCs have been published since Bak, Tang, and Weisnefield’s studies, and more reach print daily.
Figure 4: Modified BTW Model cellular automata on a ~1000x1000 grid,
after 28 million steps. Colors represent Zn (gradient) value at that column.
Image credit: Wikipedia, Abelian Sandpile Model.
http://en.wikipedia.org/wiki/Abelian_sandpile_model
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WOW!! Awesome paper at every level. Grade = A.
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References
Dhar, D. (2 April 1990). Self-Organized Critical State of Sandpile Automata Models. Physical Review Letters, Vol 64, Num 14.
Dhar, D., & Ramaswamy, R. (16 October 1989). Exactly Solved Model of Self-Organized Critical Phenomena. Physics Review Letters,
Letter 63, 1659.
Lubeck, S. (2003). Mean-Field Theory for Self-Organized Critical Sandpile Models. Rehovot, Israel: Weizmann Institute of Science,
Dept. of Physics of Complex Systems.
Pruessner, G., & Jensen, H. J. (11 December 2003). Anisotropy and Universality: The Oslo Model, the Rice Pile Experiment, and the
Quenched Edwards-Wilkinson Equation. Physics Review Letters, Let. 91, 244303.
Tang, C., Wiesenfeld, K., & Bak, P. (27 July 1987). Self-Organized Criticality: An explanation of the 1/f Noise. Physics Review Letters,
Lett. 59, 381.
Zehetner, C. (2010). Gradient Based Bak-Tang-Wiesenfeld Sandpile Model. Unpublished Proceedings, Dept. of Applied Geology,
University of Freiburg.