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Bak-Tang-Wiesenfeld Sandpile Model

CSCI 577 - Grad Project Presentation
by

Blair Gemmer

on 30 September 2013

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Transcript of Bak-Tang-Wiesenfeld Sandpile Model

Bak-Tang-Wiesenfeld Sandpile Model
CSCI 577 - Graduate Project
Blair Gemmer
Introduction
1987 - Per Bak, Chao Tang, and Kurt Wiesenfeld (BTW) attempt to explain self-similarity and the naturally ubiquitous "1/f" noise based on what they called "self-organized criticality".
BTW wrote a paper on SOC which demonstrated their famous "sand pile model", and provided the method for propagating through a 2D cellular automaton representing a sand pile.
What is this 1/f noise?
BTW discovered that many things in nature exhibit power law behavior in their frequency.
where
and
is the number of interactions (topplings) in an avalanche. This is called the "size" of the avalanche.
is the distribution of the number of avalanches as a function of the avalanche size.
Method
Implemented 2D cellular automaton from BTW paper and other research.
2D matrix cells represented scalar slope of the sandpile. Each slope could hold one of 8 values:
at each column position (i,j)
The column is unstable if
The entire pile is unstable if any column is unstable.
if
then 4 units of slope are removed from column (i,j) and 1 unit of slope is added to each of the 4 neighboring columns:
Examples:
frequency vs. size of avalanches
frequency vs. magnitude of earthquakes
Results
Animation of lattice with colors from blue to red representing slope with the reddest being the closest to the critical point (angle of repose).
Demo
See the amazing automaton come to life!
Where is it found?
According to BTW:
Mountain formation, coastlines, hourglass, resistors, and even the luminosity of stars.
Plotted the results as an animation and a frequency distribution.
Mine is slightly different:
I implemented the "flow" model with "closed" boundaries.
Continuous flow of grains to a specified or random column.
No overflow, grains fall off edges.
Thus, self-organized criticality!
Critically stable state of sandpile (n=200),
Initialized with s(i,j) = 7
,
Full transcript