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THE SAND PILE MODEL AND SELF ORGANISED
CRITICALITY
In 1987,
Bak, Tang and Wiesenfeld simulated the behaviour of a sand pile which builds
up when sand is dropped on a table. Sand might fall off the sides of the
table. If a sand grain falls at a position where the slope of the pile is
large, it will slide down, and eventually cause sand grains of the existing
pile to slide down as well. Bak and
co-workers modelled the sand pile as a regular array of columns consisting of
cubic sand grains. Addition of new grains is simply performed by selecting a
column at random and increasing its height by one. If the column then exceeds
its neighbours in height more than some threshold, it will ``collapse'': it
will loose some grains which are distributed evenly over its nearest
neighbours. As this collapse alters the height differences involving those
neighbours, there is the possibility that they collapse in turn. A cascade
process sets in until all height differences are below the threshold. The
size of such an avalanche is defined as the number of sand grains sliding as
a result of a single grain of sand being added to the pile. What is so
interesting about the sand pile model? It turns out that the sides of the
sand pile acquire a specific slope, which is such that the distribution of
avalanches as function of size scales as a power law. Power laws indicate the
absence of scale and indeed avalanches on all scales are sustained for the
equilibrium slope. If the slope is changed artificially from its equilibrium
value, the distribution is no longer a power law, but it will have an
intrinsic scale (e.g. exponential). Power laws and absence of scale are the
signature of a system being critical. Because the sand pile tends to adjust
the slope of its sides until the power law scaling sets in, the criticality
is called ``self-organised''. The field of self-organised criticality has
become very popular since the discovery of this phenomenon by the authors
mentioned. Self-organised
criticality has been found in a variety of phenomena such as earthquakes,
volcanic activity, the game of life, landscape formation and stock markets. My sand pile
simulation shows a quadrant of a sand pile. It can be considered as a sand
pile on a square horizontal table with vertical planes mounted at two
adjacent sides, such that the sand heaps up against these planes and can
slide off the table only at the remaining two sides. The simulation draws the
sand pile in a perspective view each time a certain number of grains has been
added. This number is called ``DrawInterval". The pile is shown in
perspective - a color coding is used which indicates the height. Avalanches
exceeding some threshold in size are shown in red (note that this threshold
is different from the height difference threshold beyond which a column
collapses, which is taken to be 4 in the simulation). You can now
go to the simulation.
Have fun! |