Drop grains of sand onto a grid. When a cell reaches the toppling threshold it gives one grain to each neighbour, which may push those neighbours over in turn. The Bak-Tang-Wiesenfeld model is the original example of self-organized criticality, producing scale-free avalanches and fractal patterns.
How it works: Start drops grains one at a time at the chosen site. Each site holds grains below the threshold at rest, drawn as increasing brightness; a site that reaches the threshold topples and distributes one grain to each of its neighbours. Grains that topple off the edge are lost, and that open boundary is what lets the pile reach a steady state rather than filling forever. The graphs average over all avalanches and demonstrate convergence to the critical density as the simulation progresses.
How it works
Start drops grains one at a time at the chosen site. Each site
holds grains below the toppling threshold at rest, drawn as increasing
brightness: an empty site is nearly black and a site one grain short of toppling
is full orange. A site that reaches the threshold gives one grain to each of its
neighbours, which can push them over in turn. Grains that topple off the edge are
lost, and that leak is what lets the pile reach a steady state.
Relaxation is shown as parallel sweeps: every unstable site topples at once, then
the next sweep runs, so the avalanche front travels outward a ring at a time and
the sweep count gives the avalanche duration. The resting configuration reached
is independent of the order in which topples are performed, which is the abelian
property the model is named for[3].
Variants:
Ruleset selects the neighbourhood. The classic rule topples to
the 4 edge-neighbours at a threshold of 4; the Moore variant feeds the 4
diagonals as well, and the threshold rises to 8 with it, because a toppling site
gives exactly one grain to each neighbour it has. A higher threshold means taller
resting piles: heights run 0–3 under the 4-rule and 0–7 under the
8-rule, and the critical density roughly doubles, from $\rho_c \simeq 2.125$ to a
measured 4.45.
Drop site is the centre or a uniformly random site.
Centre-dropping builds the fractal target pattern; random dropping gives cleaner
avalanche statistics, since every site takes a turn as the seed.
Initial state sets the lattice empty, half full, maximally full,
or random. The pile reaches the same critical density from all four, which is the
self-organization: nothing here is tuned to a critical point.
Graphs:
(a) avalanche-size distribution N(s), the
log-binned frequency of avalanches of s topples on log-log axes. A straight line
means avalanches have no characteristic size, so a single grain can set off
anything from nothing to a system-spanning cascade. The cutoff at large s is set
by the lattice size. (b) mean height $\langle h \rangle$ against
avalanche number, the self-organization itself. The dotted line is the critical
density for the active ruleset. The curve climbs to it from an empty start or
falls to it from a full one, then stays there.
Grassberger, P. & Manna, S. S. (1990).
Some more sandpiles.
J. Phys. France 51, 1077-1098.source of the critical density 2.125 for the 4-neighbour rule
