A pile of sand drives itself to its own critical point and stays there. That is the piece the chain was missing — not where the threshold is, but why a system should sit on it rather than pass through.
Bak, Tang and Wiesenfeld published Self-Organized Criticality: An Explanation of 1/f Noise in 1987, and the model in it is a pile of sand. Add one grain at a time. When a site holds four, it topples: four grains leave, one to each neighbour, and grains at the edge fall off. Repeat.
The claim was large and Bak made it larger in the book — How Nature Works, 1996 — where earthquakes, extinctions, forest fires, traffic and the economy are all the same pile. That claim is contested and this page does not endorse it. Sandpile experiments with real sand mostly do not show it; the earthquake case is strong, the biological cases much less so. OPEN CITED
The model is a different matter. It is exact, it is beautiful, and it says something this corpus needed and did not have.
Start with the theorem, because it is the one part of this chapter that is not statistics.
An unstable configuration stabilises to a unique final state, and the number of topplings is the same, whatever order the topplings are performed in.
Tested by stabilising the same random configuration twice, once last-in-first-out and once first-in-first-out: identical final grid and identical toppling count, every trial. Not close — equal. COMPUTED
This is worth sitting with. The intermediate history is wildly different: different sites topple, in different orders, at different moments. The endpoint does not care. A system whose path is arbitrary and whose destination is not — which is the property the dm³ chain claims for $G = U \circ F \circ K \circ C$ and rarely gets to verify on something exact.
Drive the pile — one grain at a time, always waiting for the avalanche to finish — and it walks to a stationary state and stays there.
N = 16 → 2.045
N = 32 → 2.084
N = 64 → 2.104
N = 100 → 2.109 (after 200 000 drops)
literature → 2.125 as N → ∞ COMPUTED
The N = 100 run initially returned 2.064, which would have broken the monotone trend and looked like a real finite-size effect. It was not: the burn-in was 18 000 drops for a 10 000-site lattice, and the pile had not reached stationarity. At 20 000 drops the density is 1.982; at 50 000 it is 2.106; at 200 000, 2.109. The first number measured the transient and would have been published as the answer. It is recorded here rather than silently replaced.
Nobody set the density. There is no knob. The pile is driven from outside by a process that knows nothing about criticality — drop a grain anywhere — and it arrives at one particular density and holds it.
And at that density the avalanches have no characteristic size. The distribution is a power law $P(s) \sim s^{-\tau}$ over every decade the lattice is large enough to show. A grain lands; usually nothing; sometimes four sites topple; occasionally forty thousand.
This page does not claim to have measured $\tau$. Four fits, over two lattice sizes and two windows, give $1.029$, $1.052$, $1.062$ and $1.063$ — agreeing with each other to $0.034$ and sitting, all four, about $0.15$ below the literature value near $1.2$. Mutually consistent and collectively wrong is the signature of a systematic error, here finite lattice size and a short window, and it is precisely the case where internal agreement tempts you to publish the number. Resolving it needs lattices and statistics far beyond what a chapter's verify script should run. OPENThis corpus's rule R10 is threshold, not scale: the fold happens at a point, the kind changes there, and nothing about it is gradual. Every chapter in this gallery so far has been an instance — the conic at $\beta = \alpha$, the plate at a degenerate eigenvalue, the top at $A = B = 2C$, the reaction at $d_c$.
Bak's pile is the apparent counterexample. Its avalanches are scale-free: no characteristic size, a power law, the opposite of a threshold.
They are not in conflict, and saying why is the point. R10 governs the control parameter: there is a critical value and the behaviour changes there. Self-organised criticality is what happens when the dynamics drives the control parameter to that value and pins it. You do not get scale-free response near the threshold. You get it at the threshold — and Bak's contribution is the mechanism by which a system arrives there with nobody tuning it.
Which is the piece the chain was missing. $K$ drives curvature toward $\kappa^*$; nothing in the corpus said why a system should sit at $\kappa^*$ rather than pass through it. SHOWN for the sandpile; OPEN for dm³.
| Operator | In this chapter | In dm³ |
|---|---|---|
| C | one grain, dropped anywhere — a driving that knows nothing | compression: the constraint applied blindly |
| K | the density climbing to 2.125 on its own | $\kappa \to \kappa^*$ — driven by the dynamics, not by a parameter |
| F | a toppling — four grains leave, and may set off any number more | the fold SHOWN |
| U | the unique stable configuration, independent of toppling order | Dhar's abelian property COMPUTED |
Every number on this page is produced by book7/ch-bak-verify.py. It records in its own closing block what it establishes and what it does not.
P. Bak, C. Tang and K. Wiesenfeld, “Self-organized criticality: an explanation of 1/f noise”, Phys. Rev. Lett. 59, 1987, 381–384.
D. Dhar, “Self-organized critical state of sandpile automaton models”, Phys. Rev. Lett. 64, 1990, 1613–1616 — the abelian property and the sandpile group.
P. Bak, How Nature Works: The Science of Self-Organized Criticality, Copernicus, 1996 — and the claims this page marks OPEN.
P. Grassberger and S. S. Manna, “Some more sandpiles”, J. Physique 51, 1990 — for the stationary density and the exponent estimates.
H. J. Jensen, Self-Organized Criticality, Cambridge, 1998.
J. Feder, “The evidence for self-organized criticality in sandpile dynamics”, Fractals 3, 1995 — on what real sand does and does not do.