Thursday, 1 September 2011

random thoughts

Here’s an interesting experiment that you might like to try: ask someone to choose a random number between one and six. Ask them again. And again, six times in all. This is not a scientifically verified statement, but it is very likely that your subject will select six different numbers. Now, it is possible for a random selection to give precisely this result, but it isn’t likely. There are 46,656 possible ways of choosing six numbers from a pool of six, only 720 of which make up a set of six different numbers.

Many years ago, I conducted a similar experiment to that outlined above with a friend. I would roll a single die, behind a screen, and note the result. He would guess the result and write it down. We did this 36 times. When we compared notes, it turned out that my friend had been correct 19 times. Nothing remarkable in that, you might think, but this result is considerably above chance expectations, which would be six correct answers out of 36. But the most interesting part of the result was a sequence in which my friend had written ‘one’ five times in a row. He was correct each time.

This highlights one of the characteristics of randomness: that there are likely to be clusters rather than a roughly even distribution. The next time you see a flock of sheep in a field, if there are no external distractions and the sheep have been grazing for some time, there will be a noticeable concentration of animals in some parts of the field, while other parts will be almost empty. If we assume that the quality of grazing is uniform across the field, the distribution will be random.

The random distribution of sheep in a field.

Now consider what is not random. Chaos theory is, philosophically, one of the most intriguing theories in modern science, and about ten years ago, when my full-time occupation was editing academic textbooks, it was ‘flavour of the month’ and everyone had something to say on the subject. Most of it was utter drivel, usually because the author simply didn’t understand the theory, but one example stands out as something worse: it was wrong. I was unable to persuade the author of An Introduction to Global Environmental Issues that his definition of chaos theory in the glossary was fundamentally incorrect, precisely because it included the word ‘randomness’:
chaos theory — theory explaining phenomena as being a consequence of inherent randomness in a system. There are mathematical models to simulate chaotic systems.
In the same glossary, chaos was defined as ‘unpredictable or random processes and their consequences’, which is also incorrect.

I had the melancholy satisfaction, later that same year, of watching the Royal Institution Christmas Lectures on TV. These lectures, aimed at children, are given each year by a scientist who is a leader in their particular field. That year, they were delivered by a mathematician, who tackled the subject of chaos theory head on. He was emphatic that there is nothing random about chaos. It is merely predictably unpredictable. I wondered whether the errant author had been watching, and if so, whether he took on board what he was hearing.

In fact, chaos theory is best explained with reference to the following hypothetical scenario, which in its initial state is a typical example of a linear system. Imagine two bodies that interact through mutual gravitational attraction—a single planet orbiting a star, for example. In this system, any slight change in the velocity of the planet will lead to only a slight change in its orbit at any time in the future, even millions of years later. Such stability means that the position of the planet can be predicted millions of years into the future, the only constraints on the accuracy of prediction being the accuracy of the initial measurements.

However, such simple, linear systems are rare. They are merely approximations of nature or abstractions from it, but the hypothetical model can be made rather more realistic by the addition of a third body—a comet that is also orbiting the star. The comet will be influenced by the planet’s gravity, in the same way that comets in our own solar system are influenced by the planets, especially Jupiter.

We now have a nonlinear system: the planet will orbit the star for ever, but after a finite number of orbits the comet will inevitably approach the planet so closely that it will be thrown out of the system. This system is nonlinear because any tiny change in the comet’s calculated velocity results in changes in its predicted position that grow exponentially with time. Thus, if there is a minuscule mistake in measuring the velocity today, within a relatively short time, a few orbits, it will be impossible to predict even roughly the comet’s position or whether it will have been expelled from the system entirely.

To illustrate this situation, a computer simulation was devised some years ago to predict the number of orbits that such a comet would make before being expelled from the system. This model included only the Sun and Jupiter, and the accuracy with which the orbit of the comet was calculated was varied. If all velocities were calculated to a precision of one part in a million, the model predicted that the comet would stick around for 757 orbits. When the accuracy was improved to one part in ten million, the prediction was 38 orbits; one part in one hundred million, 235 orbits; and so on down to one part in ten thousand trillion, when the comet was predicted to disappear after 17 orbits. There was absolutely no tendency for the predictions to approach a single solution with increasing accuracy—increases in the accuracy of the initial measurements had no predictable effect.

Without absolute, infinite knowledge of the comet’s velocity and infinite precision in calculation, its orbit is simply unpredictable. Unexpectedly, this is not an effect of chance. At all points, the orbit is under the direct control of gravitation, and the noted unpredictability arises as a direct result of the instability of the three-body interaction. Instability and nonlinearity are thus defining characteristics of chaos.

In the real world, the most obvious manifestation of chaos is in forecasting future weather. Modern-day weather forecasts are derived from computer simulations, and it is now standard practice to run not one but hundreds of such simulations. In most of these, the results are broadly similar up to five days in advance, but beyond five days this similarity rapidly disappears, and a wide range of different results appear, hence the pointlessness of offering a weather forecast more than five days in advance.

If you’ve been unable to follow this explanation of chaos theory, then the ‘classic’ explanation—that a butterfly flapping its wings in the Amazonian jungle could cause a hurricane in the Atlantic Ocean—is unlikely to be any more helpful. It’s true, of course, but unless it’s explained to you, the point of this analogy—that tiny changes in initial conditions can have massive long-term effects—will be lost, especially when you bear in mind that the opposite scenario, that of a butterfly failing to flap its wings because it has just been eaten by a passing iguana, can have equally devastating consequences.

3 comments:

  1. Excellent, wonderful example with the comet. Chaos Theory seemed to work with my mind for some reason, but it's extremely hard to explain it to others! Definitely not a *random* thing though!

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  2. Dennis, I find this topic so fascinating. It seems that many people confuse chaos with randomness. I also think many people don't want to think about what chaos is at all! It's gotten such a bad rap. The examples you use to illustrate chaos theory here are wonderfully intuitive. Thanks for sharing this with me! Kris

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  3. You’re absolutely right Kris, people do confuse chaos with randomness, which is why I started this short essay with an explanation of randomness. It grieves me to this day that I was unable to persuade the author of An Introduction to Global Environmental Issues that he was wrong, because this was a major recommended textbook for third-year university students! I’d like to think that he might read this explanation one day.

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