BTW Sandpile and Self-Organized Criticality

What you are seeing: the Bak- sandpile model on a 32 x 32 lattice. Drop a single grain at a random site; if any site reaches height 4, it topples (sends 1 grain to each neighbor). Cascades of toppling form avalanches. After equilibration the system sits at a critical state where avalanche sizes are power-law distributed, $P(s) \sim s^{-\tau}$ with $\tau \approx 1.21$ in 2D.

The lattice on the left shows current heights (0 = darkest, 3 = brightest). The right panel plots the avalanche-size histogram on log-log scale. After enough drops the histogram should display a clear power-law tail. The system "self-organizes" to the critical state without any external tuning of parameters: the only ingredients are drive (grain drop), threshold (height = 4), and dissipation (grains fall off the edge).

Figure 1. BTW sandpile. Method: drive-threshold-dissipation rule on 32 x 32 lattice.

speed3

z_c4

WHAT TO TRY

Drop grains and watch the pile self-organize: it climbs to the critical slope on its own and then sits there, no tuning needed, the defining trick of self-organized criticality.
A single added grain may do nothing or trigger an avalanche that spans the lattice: the size distribution is a power law with no characteristic scale.
Lower the toppling threshold z_c and the pile destabilizes sooner, yet the same scale-free cascades appear, the model Bak proposed for earthquakes, forest fires and 1/f noise.