A demonstration of self-organised criticality

I do not want to overstress the concept of self-organised criticality (SOC), but it is something I will be returning to from time to time.

The following program shows SOC with an example from Per Bak's How Nature Works. The square at top left is a simulated sandpile. When you press the Step button, a 'sand grain' is added to a random position on the pile. If the height at a given position is 4 grains or more, the height is decreased by 4 and 1 grain is added to each of the neighbouring positions, to the north, east, west and south. If the position is at the edge, the grains 'fall off the side'. The height at a given position is shown by shades of grey, with white for 0 grains and black for 3 grains. A field shows the mean height across the whole sandpile.

At first, grains appear in scattered positions, but eventually the pile has grown to the extent that 'avalanches' can occur, whereby the grains 'toppling' from one position cause other positions to become too high and they topple in turn. A field shows the size of each avalanche, in terms of the number of positions where there is a change in height (minimum 1).

As the simulation proceeds, the number of avalanches of different sizes is recorded and eventually a log-log graph is plotted, i.e. a plot of log(N_S) against log(S), where S is the size of avalanche and N_S the number of avalanches of that size. The tell-tale signature of SOC is that such a graph should give a straight line.

Pressing the Step button repeatedly is time-consuming and labour-intensive. Instead, you can press the Run button, which keeps stepping automatically. To pause the simulation, press the Run button again. After the graph has been plotted, the simulation goes into Pause mode, but you can start it again by pressing the Run button.

A field shows the number of events to collect before plotting the graph. This is initially set to 10,000, but you may change it to any value. If the simulation is in Run mode, changing the value does not take effect until after it has gone back to Pause mode.

I may be improving this program in due course.

Self-organised criticality and uniformitarianism

Per Bak, discoverer of self-organised criticality, stated that his researches 'disproved' uniformitarianism. In reality, they confirmed it at a deeper level.

IBM sandpile experiment, demonstrating self-organised criticality

Bak misunderstood uniformitarianism as meaning that change is always gradual. However, the deeper meaning of uniformitarianism is that extraordinary effects do not have to have extraordinary causes.

In his famous sandpile experiment, Bak showed that a uniform input (sand falling onto a pile one grain at a time) could produce effects (avalanches of sand falling off the pile) on all scales. Small slippages occurred frequently, medium slippages less often, and large slippages very rarely (once the pile had 'self-organised' to the 'critical' state, that is). Contrary to Bak's view, this is a validation of the uniformitarian principle since it shows that large and unusual phenomena (big avalanches) can arise from the same processes as produce ordinary ones (the continual small avalanches).

Self-organised criticality shows how something like the extinction of the dinosaurs or the fall of the Roman empire need not require some special factor such as a meteorite impact or a barbarian invasion. It may come about through the same processes as lie behind smaller effects, such as the extinction of individual species or everyday fluctations in people's political and economic fortunes.

As used on this website, uniformitarianism means precisely this. History, despite its enormous variety, is to be understood not as a series of chance events with ad hoc causes, but as the expression of eternal and constant principles.

Uniformitarianism and the principle of mediocrity

Two philosophical underpinnings of the theory of history and society presented here are uniformitarianism and the principle of mediocrity. Uniformitarianism means attempting to explain apparently disparate phenomena in terms of uniform causes, while the principle of mediocrity means viewing our own position and perspective as normal rather than central.

Charles Lyell is credited with introducing the concept of uniformitarianism in the study of geology. He meant the assumption that the processes we observe today are those that operated in the past. The point was that, rather than imagining special eventualities--such as Noah-like floods, thunderbolts or divine creation--to explain features of the landscape, we should understand them as arising through the normal processes--such as erosion and sedimentation--we can see occurring in front of our eyes. An important consequence, considering the time it would take to erode the Grand Canyon or pile up Mount Everest, was a recognition of the great age of the earth. It is said that Lyell's ideas influenced Charles Darwin, both giving him the idea that species could emerge through slow-acting selection pressures and supplying the vast spans of time that his theory of evolution would need.

Uniformitarianism can be seen as a default assumption of science. Although we have sampled an infinitesimally tiny portion of space, we assume that the laws of physics we observe operating here on our planet also apply everywhere throughout the universe. Astronomers interpret what they see in the heavens by applying familiar principles of science, rather than by postulating novel principles in an ad hoc manner - e.g. explaining the forms of galaxies in terms of gravitational and electro-magnetic forces, rather than in terms of special 'galactic forces'.

Related to this is the principle of mediocrity. It says that we should assume we are in a typical rather than a privileged position with respect to the phenomena we observe. The history of cosmology has been a gradual unveiling of the principle of mediocrity. Early humans thought the earth was at the centre of the universe and that everything revolved around it. Now, not only do we know that the earth is just one of many planets orbiting the sun, but the sun itself has been revealed as an ordinary star in a humdrum part of the Milky Way, which is one of countless similar galaxies spread through the universe.

Neither uniformitarianism nor the principle of mediocrity should be applied in a dogmatic manner. It is rather that they are the most sensible default assumptions, in the absence of evidence to the contrary. In a later post, I will discuss how criticism of uniformitarianism arising from one theory of complex systems reflects a misreading of the concept.

Uniformitarianism does not require rigid insistence that systems' behaviour and properties never change. Astrophysicists entertain the possibility of secular variation in physical laws, and since there are, for example, anomalies in the rotation of galaxies, it is possible that there may be physical laws we have not yet discovered. Nevertheless, the point is that we should seek common explanations that can account for dramatic, large-scale phenomena alongside modest, everyday, local phenomena, before we resort to invoking special, unusual forces. In this respect, it is an example of Ockham's razor.

Similarly, the principle of mediocrity is a point to start from rather than something that must inevitably be true. Suppose you were asked to pick a ball from a bag and it turned out to be blue. It might be that you happened to pick out the single blue ball from a bag full of red balls, but there is no real reason to think that. If you had to predict the colour of a second ball from the same bag, your best bet would be to assume the principle of mediocrity and say blue, even though that is far from guaranteed. This is an artificial situation, but in science the principle of mediocrity is usually a reliable guide.

Applied to the study of history and society these concepts lead to the following assumptions on which the theory is built:

1. Human beings have been much the same at all times and in all places.
2. We do not live at a special time in history, nor does our country/society have a special place in the world.