Drop grains of sand one at a time onto a square table. A cell that holds four grains topples: it sends
one grain to each neighbour. Most drops cause nothing; some set off a chain of topplings, an avalanche. Over many
random drops the avalanche sizes fall off as a rough power law. Nobody sets that law by hand. The model is Bak,
Tang and Wiesenfeld's from 1987.
What the numbers mean
A cell topples when it holds four or more grains. In this model one toppling takes four grains away from the cell
and gives one to each of its four neighbours. Grains pushed off the edge are gone. The size of an avalanche
is the number of topplings it needed before every cell held three or fewer.
The final pile does not depend on the order in which the unstable cells topple. That is the abelian property,
which is why a fixed seed gives one definite answer. The number of topplings does not depend on the order either.
The exponent is the slope, with the sign flipped, of a straight line fitted to the log-log histogram. Sizes are
binned by powers of two, and bins holding fewer than five avalanches are left out. Only drops made after the warm-up
(20 × side² grains) are recorded, because a table that is still filling does not show the settled behaviour.
The fitted value is a rough estimate from a few decades of sizes, not a precise constant. Published estimates for
this two-dimensional model are usually quoted near 1.2 to 1.3; that figure is from memory and was not checked against
a source here.