The pile
Grains dropped one at a time onto a table. Nothing tunes it and nobody holds it at any particular state. It walks to the edge of its own stability and stays there, where the next grain may do nothing or may bring down a quarter of the pile, and no amount of watching will tell you which.
Bak, Tang and Wiesenfeld published this in 1987 with a claim that is still doing work: a system with many parts and a slow drive moves by itself to a critical point, without anybody setting a parameter, and once there its behaviour has no characteristic size.
A grid of cells, each holding a few grains. Drop a grain on a cell. If a cell ever holds four or more, it topples: one grain to each of its four neighbours. A toppling can push a neighbour over its own limit, so one grain can start a chain across the table. Grains that fall off the edge are gone.
That is the whole of it. There is no parameter to tune, and that is the point.
01The table and the record
On the left is the pile, shaded by how many grains each cell holds, with the cells that toppled in the last avalanche marked. On the right is every avalanche so far, sorted by size, on a plot where both axes are logarithmic. A distribution with a characteristic size would show a hump. A power law shows a straight line.
The pile
Avalanche sizes, both axes logarithmic
02What the straight line means
A straight line on a log log plot says that the number of avalanches of a given size falls as a fixed power of that size: halving the frequency means multiplying the size by a constant factor, whatever size you started from. There is no typical avalanche, and the average size of the next one is a question the distribution does not answer usefully.
number of avalanches of size s ∝ s to the power of minus τ The exponent under the plot is fitted by least squares to the logarithmically binned counts, over the bins with enough avalanches to fit. On a small table it says the line is straight; it is no measurement of τ.
The small avalanches arrive at once and in enormous numbers. The large ones arrive rarely, and the largest you have seen keeps being overtaken. There is no true maximum here except the size of the table, and on a larger table the largest event would simply be larger.
03Nothing is being tuned
Watch the average number of grains per cell. It climbs while the pile is filling, then stays put. From then on, whatever you do to the pile, it returns to roughly the same height: grains going in are balanced by grains falling off the edges, and the avalanches themselves keep the balance.
This is what makes the word organised do work. The pile is not held at criticality by anybody. Add grains faster and it is still there. Start from an empty table and it walks there; start from an overloaded one and it collapses back to it. The critical state is where the dynamics go, which is why the effect turns up in systems nobody is administering.
04Why this belongs to SIMEA
Two reasons, and the second is the one I care about.
The first is forecasting. In a state like this the size of the next event is unforecastable, and ignorance has nothing to do with it: the distribution is known exactly here, since I wrote it, and it gives the odds over a run and nothing about the next grain. Some unpredictability is a property of the system and survives any instrument, and a research project about what can be forecast has to tell the two apart.
The second is responsibility. The grain that starts a large avalanche is indistinguishable from the thousands that started nothing. It was an ordinary grain; the difference lies entirely in the state the pile had already reached. Every review that looks for the last action before a disaster is doing this, and the pile is the cleanest demonstration I know that the last action can be innocent and the outcome still enormous.
05Where the model stops
Assumptions and limits
- It is no sandReal granular piles behave less cleanly than this model, for reasons of inertia and friction it cannot represent. The name is a picture and says nothing about sand.
- The table is smallEvery power law here is cut off by the grid, which is why the right hand end of the plot bends down. An exponent fitted from a few thousand avalanches on a table this size is a rough indication.
- The transient is includedEvery avalanche is recorded, including those from before the pile reached its critical state. Press reset and drop a few thousand grains before reading the plot seriously.
- The drive is separated from the responseEach avalanche runs to completion before the next grain drops. A system driven faster than it can relax would behave differently.
- Power laws are easy to see and hard to establishA straight stretch on a log log plot is weak evidence on its own, and the statistical literature is unkind to eyeball fits. This page shows a mechanism that produces one; it is no method for detecting one.
- An analogy is no findingThe paragraphs on forecasting and responsibility are arguments by analogy. This model establishes nothing about markets, institutions or disasters, and no number from this page belongs in a claim about them.
06Source
- The paper that introduced the model and the idea of self organised criticality: dynamical systems with spatial degrees of freedom naturally evolve into a self organised critical state, which the authors connect to flicker noise and to the origin of fractal structures. Per Bak, Chao Tang and Kurt Wiesenfeld, Self-organized criticality: An explanation of the 1/f noise, Physical Review Letters 59(4), 27 July 1987, page 381. journals.aps.org/prl/abstract/10.1103/PhysRevLett.59.381 Checked 2026 08 22.