Seeing Further (42 page)

Complexity, like crime, comes in organised and disorganised forms. The disorganised form goes by the name of
chaos
and has proven to be ubiquitous in Nature. The standard folklore about chaotic systems is that they are unpredictable. They lead to out-of-control dinosaur parks and frustrated meteorologists. However, it is important to appreciate the nature of chaotic systems more fully than the Hollywood headlines.

Classical (that is, non-quantum mechanical) chaotic systems are not in any sense intrinsically random or unpredictable. They merely possess extreme sensitivity to ignorance. As Maxwell was again the first to recognise in 1873, any initial uncertainty in our knowledge of a chaotic system’s state is rapidly amplified in time. This feature might make you think it hopeless even to try to use mathematics to describe a chaotic situation. We are never going to get the mathematical equations for weather prediction 100 per cent correct – there is too much going on – so we will always end up being inaccurate to some extent in our predictions. But although that type of inaccuracy can contribute to unpredictability, it is not in itself a fatal blow to predicting the future adequately. After all, small errors in the weather equations could turn out to have an increasingly insignificant effect on the forecast as time goes on. In practice, it is our inability to determine the weather everywhere at any given time with perfect accuracy that is the major problem. Our inevitable uncertainties about what is going on in between weather stations leaves scope for slightly different interpolations of the temperature and the wind motions in between their locations. Chaos means that those slight differences can produce very different forecasts about tomorrow’s weather.

An important feature of chaotic systems is that, although they become unpredictable when you try to determine the future from a particular uncertain starting value, there may be a particular stable statistical spread of outcomes after a long time, regardless of how you started out. The most important thing to appreciate about these stable statistical distributions of events is that they often have very stable and predictable average behaviours.
As a simple example, take a gas of moving molecules (their average speed of motion determines what we call the gas ‘temperature’) and think of the individual molecules as little balls. The motion of any single molecule is chaotic because each time it bounces off another molecule any uncertainty in its direction is amplified exponentially. This is something you can check for yourself by observing the collisions of marbles or snooker balls. In fact, the amplification in the angle of recoil,
$$
, in the successive (the
n
+1st and
nth)
collisions of two identical balls is well described by a rule:

where
d is
the average distance between collisions and
r
is the radius of the balls. Even the minimal initial uncertainty in θ
₀
allowed by Heisenberg’s uncertainty principle is increased to exceed θ = 360 degrees after only about 14 collisions. So you can then predict nothing about its trajectory.

The motions of gas molecules behave like a huge number of snooker balls bouncing off each other and the denser walls of their container. One knows from bitter experience that snooker exhibits sensitive dependence on initial conditions: a slight miscue of the cue-ball produces a big miss! Unlike the snooker balls, the molecules won’t slow down and stop. Their typical distance between collisions is about 200 times their radius. With this value
of d/r
the unpredictability grows 200-fold at each close molecular encounter. All the molecular motions are individually chaotic, just like the snooker balls, but we still have simple rules like Boyle’s Law governing the pressure
P,
volume
V,
and temperature
T–
the averaged properties
⁸
– of a confined gas of molecules:

The lesson of this simple example is that chaotic systems can have stable, predictable, long-term, average behaviours. However, it can be difficult to
predict when they will. The mathematical conditions that are sufficient to ensure it are often very difficult to prove. You usually just have to explore numerically to discover whether the computation of time averages converges in a nice way or not.
⁹

Considerable impetus was imparted to the study and understanding of this type of chaotic unpredictability and its influence on natural phenomena by theoretical biologists like Robert May (later to become the fifty-eighth President of the Royal Society in 2000) and George Oster, together with the mathematician James Yorke. They identified simple features displayed by wide classes of difference equation relating the
(n+1)st
to the
nth
state of a system as it made the transition from order to chaos.
¹⁰

Among complex outcomes of the laws of Nature, the most interesting are those that display forms of
organised complexity.
A selection of these are displayed in the diagram on the next page, in terms of their size, gauged by their information storage capacity, which is just how many binary digits are needed to specify them versus their ability to process information, which is simply how quickly they can change one list of numbers into another list.

As we proceed up the diagonal, increasing information storage capability grows hand in hand with the ability to transform that information into new forms. Organised complexity grows. Structures are typified by the presence of feedback, self-organisation and non-equilibrium behaviour. Mathematical scientists in many fields are searching for new types of ‘by-law’ or ‘principle’ which govern the existence and evolution of different varieties of complexity. These rules will be quite different from the ‘laws’ of the particle physicist. They will not be based upon symmetry and invariance, but upon principles of probability and information processing. Perhaps the second law of thermodynamics is as close as we have got to discovering one of this collection of general rules that govern the development of order and disorder.

The defining characteristic of the structures in the diagram below is that they are more than the sum of their parts. They are what they are, they display the behaviour that they do, not because they are made of atoms or molecules (which they all are), but because of the way in which their constituents are organised. It is the circuit diagram of the neutral network that is the root of its complex behaviour. The laws of electromagnetism alone are insufficient to explain the working of a brain. We need to know how it is wired up and its circuits inter-connected. No theory of everything
that the particle physicists supply us with is likely to shed any light upon the complex workings of the human brain or a turbulent waterfall.

The advent of small, inexpensive, powerful computers with good interactive graphics has enabled large, complex, and disordered situations to be studied observationally – by looking at a computer monitor. Experimental mathematics is a new tool. A computer can be programmed to simulate the evolution of complicated systems, and their long-term behaviour observed, studied, modified and replayed. By these means, the study of chaos and complexity has become a multidisciplinary subculture within science. The study of the traditional, exactly soluble problems of science has been augmented by a growing appreciation of the vast complexity expected in situations where many competing influences are at work. Prime candidates are provided by systems that evolve in their environment by natural selection, and, in so doing, modify those environments in complicated ways.

As our intuition about the nuances of chaotic behaviour has matured by exposure to natural examples, novelties have emerged that give important hints about how disorder often develops from regularity. Chaos and order have been found to coexist in a curious symbiosis. Imagine a very large egg-timer in which sand is falling, grain by grain, to create a growing sand pile. The pile evolves under the force of gravity in an erratic manner. Sandfalls of all sizes occur, and their effect is to maintain the overall gradient of the sand pile in a temporary equilibrium, always just on the verge of collapse. The pile steadily steepens until it reaches a particular slope and then gets no steeper. This self-sustaining process was dubbed ‘self-organising criticality’ by its discoverers, Per Bak, Chao Tang and Kurt Wiesenfeld, in 1987. The adjective ‘self-organising’ captures the way in which the chaotically falling grains seem to arrange themselves into an orderly pile. The title ‘criticality’ reflects the precarious state of the pile at any time. It is always about to
experience an avalanche of some size or another. The sequence of events that maintains its state of large-scale order is a slow local build-up of sand somewhere on the slope, then a sudden avalanche, followed by another slow build-up, a sudden avalanche, and so on. At first, the infalling grains affect a small area of the pile, but gradually their avalanching effects increase to span the dimension of the entire pile, as they must if they are to organise it.

At a microscopic level, the fall of sand is chaotic, yet the result in the presence of a force like gravity is large-scale organisation. If there is nothing peculiar about the sand,
¹¹
that renders avalanches of one size more probable than all others, then the frequency with which avalanches occur is proportional to some mathematical power of their size (the avalanches are said to be ‘scale-free’ processes). There are many natural systems – like earthquakes – and man-made ones – like some stock market crashes – where a concatenation of local processes combine to maintain a semblance of equilibrium in this way. Order develops on a large scale through the combination of many independent chaotic small-scale events that hover on the brink of instability. Complex adaptive systems thrive in the hinterland between the inflexibilities of determinism and the vagaries of chaos. There, they get the best of both worlds: out of chaos springs a wealth of alternatives for natural selection to sift; while the rudder of determinism sets a clear average course towards islands of stability.

Originally, its discoverers hoped that the way in which the sandpile organised itself might be a paradigm for the development of all types of organised complexity. This was too optimistic. But it does provide clues as to how many types of complexity organise themselves. The avalanches of sand can represent extinctions of species in an ecological balance, traffic flow on a motorway, the bankruptcies of businesses in an economic system, earthquakes or volcanic eruptions in a model of the pressure equilibrium of the Earth’s crust, and even the formation of ox-bow lakes by a meandering river. Bends in the river make the flow faster there, which erodes the bank, leading to an ox-bow lake forming. After the lake forms, the river is left a
little straighter. This process of gradual build-up of curvature followed by sudden ox-bow formation and straightening is how a river on a flat plain ‘organises’ its meandering shape.

It seems rather remarkable that all these completely different problems should behave like a tumbling pile of sand. A picture of Richard Solé’s, showing a dog being taken for a bumpy walk, reveals the connection.
¹²
If we have a situation where a force is acting – for the sand pile it is gravity, for the dog it is the elasticity of its leash – and there are many possible equilibrium states (valleys for the dog, stable local hills for the sand), then we can see what happens as the leash is pulled. The dog moves slowly uphill and then is pulled swiftly across the peak to the next valley, begins slowly climbing again, and then jumps across. This staccato movement of slow build-up and sudden jump, time and again, is what characterises the sandpile with its gradual buildup of sand followed by an avalanche. We can see from the picture that it will be the general pattern of behaviour in any system with very simple ingredients.