Sandpile · self-organized criticality

Every critical point so far (Ising, percolation) had to be FOUND by tuning a knob to one magic value. Can a system drive ITSELF to criticality with no tuned parameter at all?

▶ Run the simulationSee the measured result

Units: grains per site, heights 0..3 (17/8 exact; 25/8 in the 1..4 convention)

How the lab tests it

Run the Bak–Tang–Wiesenfeld sandpile: drop grains on random cells of an L×L grid; a cell that reaches 4 topples one grain to each neighbour, possibly cascading into an avalanche; grains that fall off the open boundary are lost. Drive grain by grain from empty, track the stationary mean density ⟨ρ⟩, and histogram the avalanche sizes s (total topplings) in log-bins.

What it checks

the pile self-organizing — with NO tuned knob — to a stationary critical slope (⟨ρ⟩ → 17/8 = 2.125, the exact Priezzhev value), and scale-free avalanches P(s) ∝ s^(−τ) over decades (the SOC signature; the fitted τ is an effective slope — the 2-D BTW distribution is famously multifractal, not one clean exponent). Grains are conserved exactly.

Abelian sandpile calculator — critical density, height statistics & how big a pile has to be

A pile that tunes itself. Every other critical point in this lab had a knob that someone turned to a magic value; drop sand at random and the pile finds its own, and the value it finds — a mean height of 17/8 grains per site — is exactly known. This page computes the things about that state which CAN be computed. The height-0 probability is one of them: Dhar's burning algorithm puts the recurrent configurations in bijection with spanning trees, the bonds of a uniform spanning tree form a determinantal process whose kernel is the lattice transfer current, and a 4×4 determinant of potential-kernel values lands on Majumdar and Dhar's 2/π² − 4/π³ = 0.073636230 to fourteen digits, with π arriving as an OUTPUT, 4/a(1,1), rather than as an input, and with the answer's own convergence residue printed beside it. The count is another: the same Brillouin zone, integrated logarithmically, says a large pile has e^(4G/π) = 3.209912 critical states per site where it has 4 stable ones, so Dhar's selection throws away 19.75% of configuration space per site — and that selection, nothing else, is what lifts the maximum-entropy pile's 3/2 to 17/8. What this page does NOT do is derive 17/8; that took from 1990 to 2011 and a loop-erased random walk, and nothing here assembles it. It is an editable field, used only to score, and what the page computes in its place is the number this lab actually measured: the L → ∞ intercept of five finite piles, fitted here from the published readings at 2.124499 ± 4.09e-4, one and a quarter of its own bars from 17/8 — where the same five numbers fitted WITHOUT the 1/L² term land 5.6 bars away, which is the estimator trap this world's oracle was built to avoid. The most useful thing here is probably the plainest: a pile 96 sites across reads 0.69% low, and to see 17/8 to one part in a thousand you need one about 845 across.