Read The Big Questions: Physics Online
Authors: Michael Brooks
The Big Questions: Physics (21 page)
From that, they could work out how that population of squirrels would grow and diminish over time. But any time the equations gave results that seemed to be going out of control, the mathematical biologists would ‘reset’ the scenario, assuming that there was some instability in the system that needed to be reined in. With the advent of chaos theory, it became clear that the wild changes could quite easily be a natural part of the system.
Imagine, for instance, a population of squirrels with no predators. If, on average, each adult produces less than one offspring per generation, the population will dwindle to zero. If the number of offspring is between one and three, there is some stability. If the average number of offspring per generation is more than three, however, things get strange.
A tendency to ‘boom and bust’ appears in the population statistics. It is, essentially, the same as that whistling valve tube in the telephone line mentioned earlier: a process of feedback creates wild oscillations. Chaos theorists call it a ‘bifurcation’. It means that the population is uniquely sensitive to the number of offspring per generation. In one year, the population will boom, but in the next it busts. There’s nothing in between. Then the bifurcation splits again and again, and things eventually start to look random: there is no apparent pattern. But only for a while: as time goes on, the number of offspring increases again, and, out of nowhere, another bifurcation comes into play.
Such complexity is everywhere in the natural world, and understanding it can save lives. The hit and miss, up and down, boom and bust occurs in epidemics of diseases like AIDS, measles and polio, for instance. Because the number of cases follows a chaotic trajectory, it is sensitive to a knock from something like an inoculation programme. But that knock doesn’t always wipe out the disease; instead, the numbers can be thrown into an unstable regime – around a bifurcation region, for instance. This means that short-term figures for the disease might rise, suggesting that a programme of inoculation has failed. Awareness of chaos allows medical researchers to see beyond the initial response, and allow for the chaotic response, mapping what is hopefully a downward trajectory over the long term.
FRACTAL PATTERNS
Here’s a simple question: how long is the coastline of Britain? Look it up in a few encyclopedias, and you’ll get wildly differing answers – the difference can be as much as 20 per cent. That’s because the most straightforward answer anyone can give is that it depends on the size of your ruler. Given a smaller ruler to work with, there will always be smaller features to measure. Coastlines, like myriad features of the natural world, are self-similar, or ‘fractal’.
As a result, you can always measure a fractal coastline more closely, and add to the total length you already have. As you zoom in on a fractal structure, the essence of your view does not change. At each size-scale, the same patterns are repeated. Given a picture showing nothing but desert sand dunes or ocean swell, for instance, you cannot tell whether you are looking at a few square kilometres or a few square centimetres. It’s the same with coastlines.
Creating a fractal is just a matter of drawing simple but repeated shapes. The Koch snowflake, for instance, is composed of triangles added to the centre of each side of an existing triangle. The added triangles have side lengths one-third of the side on which they rest. After a few iterations, the result is an astonishingly detailed pattern.
This intricacy is the hallmark of fractals, and it is the point at which they relate to chaos theory. Chaos theory says that generating a perfectly accurate picture depends on starting with perfectly accurate information – which is impossible. In a chaotic system, any inaccuracies in the
information are grotesquely amplified to give a badly distorted picture. Fractals are something like chaos theory turned on its head: the accuracy of the information you gain from a picture depends on how closely you are looking at it – and you can never get close enough to get perfectly accurate information.
As a result, the manifestations of fractal behaviour intrigue and infuriate in equal measure. Researchers have been intrigued to find fractal structures outside of the natural world. Financial data – records of stock-market trades, for instance – often take a fractal form, suggesting that their detailed structure arises from absurdly simple rules. But inferring useful information about those rules is absurdly and frustratingly difficult.
The most famous fractal structure is the Mandelbrot set. First created by French mathematician Benoît Mandelbrot, it is defined by a relatively simple equation, but forms a complex collage of balloons, thorns, spirals and jets, all of which contain similar structures. Its aesthetic appeal is almost unparalleled in mathematics, but what does it mean?
That is still not clear. Researchers have suggested that the fractal structure of financial markets may mean that markets are governed by simple rules and are thus far easier to analyse (and thus, perhaps, predict) than one might imagine. The ubiquity of fractal structures in natural and artificial systems, and in human culture – many common musical patterns, and Jackson Pollock’s art, for example – has led Mandelbrot to claim that they are a key to unlocking deep secrets of the universe. As yet, however, the study of fractals has not produced any insights we can truly call significant.
An understanding of biological chaos and the butterfly effect is saving lives around the world in a more immediate way too. Your heart beats because of co-ordinated pulses of electricity that work through the cells in a kind of wave, causing the muscle to contract in specific ways and at specific times. When this breaks down, an ‘arrhythmia’ occurs. Heart arrhythmias kill hundreds of thousands of people each year: for myriad reasons, the heart can stop beating normally – or, indeed, at all. Often the muscles are all contracting randomly, and the heart is no longer a pump, but a seething, pulsating mess of tissue. It’s a chaotic system – and one where a good kick can take the chaos away.
Medics have long known that a jolt of electricity can set this problem right again, but you can’t just put any old jolt into a human heart. To set the rhythm right again requires an understanding of its chaotic dynamics. The heart is, effectively, an oscillator like a pendulum. And when you know how a chaotic pendulum can be controlled, you can also design a defibrillator that works much better than the ones designed by trial and error. The main area where the butterfly effect has been put to work, though, is exactly where it started: the weather.
Meteorologists like to run hugely complex simulations of the Earth’s weather systems on massive supercomputers. The simulations are based on the laws of physics, and model things such as how ocean and air currents move around the globe. Before the butterfly era, we might have thought that a suitably powerful simulation would predict the weather weeks, months or maybe even years in advance. Chaos tells us this is just not possible.
The trouble is, the physics of the models is approximate, and the data used to set up the simulation even more so. The weather stations used to gather information are scattered over the Earth, with large gaps between them: we don’t have the information
from the places between the weather stations. Scientists now know that, within just a few days, these sources of error are enough to set the weather models off on a trajectory that will bear no relation to what actually happens to the weather. A butterfly flapping its wings somewhere between the weather stations could cause a storm that no one saw coming.
Of course, the meteorologists reset their models every time they get new data in. They also run ‘ensemble’ forecasts, where they put slightly different initial conditions into the model and look at how much the outcome varies. That allows them to create an averaged forecast that is likely to be more accurate than any one prediction. It also allows a measure of the reliability of their forecast. It’s not enough to have a prediction – it’s better to have an idea of how much you should trust that prediction.
Ironically, the longest-term predictions come out OK: the science of climate prediction is not so sensitive to initial conditions as are short-term weather predictions. That’s essentially because climate prediction deals more in generalities than specifics. The flap of a butterfly’s wings might cause a storm in Texas, but another flap might calm a storm that was already blowing up. Over the 30-year average that constitutes a climate analysis, the number of storms evens out, and each butterfly becomes irrelevant.
Lorenz used the equations of chaos theory to show this. When you look at a strange attractor, you see a particular shape. Applied to climate science, the shape you see indicates the future climate. The flowing line that moves around unpredictably, gradually creating the shape, is like noise on the signal; it is not the parameter of interest. That means running climate simulations, however chaotic their predictions might be over the short term, reveals a reliable broad-brush picture of what is coming. So, does chaos theory spell disaster? Quite the opposite: that flapping butterfly has been instrumental in warning us about the greatest threat faced by humanity: runaway climate change induced by human activity. Chaos is not always a problem.
A strange kind of wave, and an even stranger kind of particle
‘What is poetry? Why, Sir, it is much easier to say what it is not. We all know what light is, but it is not easy to tell what it is.’ Samuel Johnson thought this a convincing justification for the difficulties of defining poetry. Unfortunately, the idea that we all know what light is has one fundamental flaw. It’s not clear that we do.
When Johnson wrote those words in 18th-century England, Isaac Newton’s view of light as particles or ‘corpuscles’ of energy reigned supreme. Within 20 years of Johnson’s death, Thomas Young had ‘proved’ light was a wave, not a particle. A century later, Albert Einstein showed light to be, once again, particles. Now we have to think of it as both – or neither. Light, the universal metaphor for understanding and revelation, is astonishingly opaque.
One thing about light is certain: it is essential to our existence. Without light from the sun, plants would not be able to use photosynthesis to harvest energy and grow, and we would have nothing to eat. Humans deprived of light suffer depression – researchers who kept rats in the dark for six weeks watched their brain cells die for lack of light. Insufficient exposure to direct sunlight creates skeletal problems such as rickets. Whatever light is, we need it.
This was recognized by ancient civilizations. The Neolithic monument at Stonehenge is, it seems, a temple to the light-giving sun. The Egyptians worshipped Ra, the sun god, as the giver of life. The first people to attempt a definition of light, though, were the ancient Greeks who were a little more circumspect than the Egyptians: to them, light was not something to be revered, but a by-product of fire, one of the four fundamental elements that made up the universe.
There were various Greek ideas about the nature of light and vision. Euclid’s was the most developed. Light from an object mixed with the light from the eye, he said, but a person could only see the object when the eye’s fire reflected directly back from the object. It was only close to the modern scientific view in that it had light travelling in straight lines, however. And we had to wait for nearly two millennia before anyone attempted to move our understanding of light forward. That progress was kicked off by a Frenchman, René Descartes, early in the 17th century.
Not that Descartes’s contribution lasted. His idea was that space is filled with an invisible fluid that he called the ‘plenum’. The plenum, Descartes said, has a ‘tendency to motion’ such that a candle creates a pressure in the plenum in much the same way as a drum creates sound waves in the air. This pressure is passed on to the eyeball and manifests as light. Almost as soon as he started thinking about it, Isaac Newton debunked the idea.
If light is just pressure of the plenum on the eye, Newton argued, then breaking into a run on a dark night should flood the world with light. Newton was a big fan of the emerging idea of the atom – that, on the smallest scales, everything can be divided into component parts. Light, he argued, should be no different. It was, he suggested, composed of atomic elements that Newton referred to as ‘corpuscles’.