The pile
Grains dropped one at a time onto a table. Nothing tunes it, nothing balances it, and nobody is holding it at any particular state. It walks to the edge of its own stability and stays there, and once it is there the next grain may do nothing at all 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 attached that is still doing work: that a system with many parts and a slow drive will move by itself to a critical point, without anybody setting a parameter to a special value, and that once there its behaviour has no characteristic size.
The model is almost nothing. A grid of cells, each holding a small number of grains. Drop a grain on a cell. If a cell ever holds four or more, it topples: it gives one grain to each of its four neighbours and keeps the rest. Toppling can push a neighbour over its own limit, so a single grain can start a chain that runs 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 the record: every avalanche so far, sorted by size, drawn 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, and the straighter it gets the more grains you drop.
The pile
Avalanche sizes, both axes logarithmic
02What the straight line means
A straight line on a plot where both axes are logarithmic says that the number of avalanches of a given size falls as some fixed power of that size. Halving the frequency requires multiplying the size by a constant factor, whatever size you started from. That is what it means for a distribution to have no characteristic scale: there is no typical avalanche, and asking for the average size of the next one is asking a question the distribution does not answer usefully.
number of avalanches of size s ∝ s to the power of minus τ The exponent printed under the plot is fitted by least squares to the logarithmically binned counts, over the bins that have enough avalanches in them to be worth fitting. It is a crude estimate on a small table and it should be read as an indication that the line is straight rather than as a measurement of τ.
Notice how the plot fills in. The small avalanches arrive immediately and in enormous numbers. The large ones arrive rarely, and the largest one you have seen keeps being overtaken. That is not the record settling down towards a true maximum. 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 in the readout. It climbs while the pile is filling, then stops climbing and stays put. From that point on, whatever you do to the pile, it returns to roughly the same average height. Grains going in are balanced by grains falling off the edges, and the balance is maintained by the avalanches themselves.
This is what makes the word organised do work in the phrase. The pile is not held at criticality by anybody. Add grains faster and it is still there. Start it from an empty table and it walks there. Start it from an overloaded table and it collapses back to it. The critical state is where the dynamics take it, 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 the forecasting point. In a state like this the size of the next event is genuinely unforecastable, and not because anybody is ignorant. The distribution is known exactly here, since I wrote it, and knowing it does not help: it tells you the odds over a run and says nothing at all about the next grain. There is a habit of treating unpredictability as a shortfall of data or of model quality, on the assumption that a better instrument would resolve it. Some unpredictability is a property of the system and survives any instrument, and a research programme that wants to talk about what can and cannot be forecast has to be able to tell the two apart.
The second is about responsibility. When a large avalanche happens, the grain that started it is indistinguishable from the thousands of grains that started nothing. It was not a special grain. It did not fall harder. Looking for what was different about it will find nothing, because nothing was, and the difference lies entirely in the state the pile had already reached. Every institution that reviews a disaster by looking for the last action before it 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 not sandReal granular piles were tested against this model and behave less cleanly than it does, for reasons to do with inertia and friction that the model has no way of representing. The name is a picture and not a claim about sand.
- The table is smallEvery power law here is cut off by the size of the grid, which is why the right hand end of the plot bends down. The exponent fitted from a few thousand avalanches on a table this size is a rough indication and nothing more.
- The transient is includedEvery avalanche is recorded, including those from the early period when the pile was still filling up and had not 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 is run to completion before the next grain is dropped. That separation of timescales is an assumption of the model, and 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 on distinguishing a power law from other heavy tailed distributions is not kind to eyeball fits. This page shows a mechanism that produces one, not a method for detecting one.
- An analogy is not a findingThe two paragraphs above about forecasting and about responsibility are arguments by analogy. This model does not establish anything 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.