The Big Questions: Physics (20 page)

So, according to this interpretation of quantum theory at least, time emerged from a change in the nature of space. Hawking’s flexible, changeable universe is, on some levels, extremely attractive. It gives us participation in the universe, and does away with the problem of the pre-Big Bang state. But it is far from being universally accepted as the answer we have been looking for.

The important point to keep in mind is that no one understands how the quantum world really works. The interpretation you follow, whether it is Feynman’s sum over histories or Bohr’s rejection of an objective reality, is in many ways just a matter of taste. That’s why most physicists consider themselves to be adherents of the ‘shut up and calculate’ interpretation of quantum theory. In this pragmatic position, no one knows what it all means, but we do enjoy playing with it – and that can be enough.

The Cornell University physicist David Mermin put it best. The predictive power of the theory ‘is so beautiful and so powerful that it can, in itself, acquire the persuasive character of a complete explanation’. But it is not an explanation. So, while quantum theory says you can change the universe with a single glance, remember that we still see quantum theory through an obscuring fog of ignorance and incomprehension. It’s OK to believe you can make the moon appear. Just don’t try to convince anyone else of your powers.

The butterfly effect’s influence on weather, climate and the motions of the planets

It’s not a particularly new idea. You might even have grown up with the concept; it is written into a familiar children’s rhyme:

For want of a nail, the shoe was lost

For want of a shoe, the horse was lost

For want of a horse, the rider was lost

For want of a rider, the battle was lost

For want of a battle, the kingdom was lost

That is chaos theory, sometimes known as the ‘butterfly effect’. Can the lack of a single nail destabilize geopolitics? Can the flap of a butterfly’s wings result in a storm thousands of miles away? The answer is yes, and it happens all the time. Not necessarily in those exact ways, of course. The children’s rhyme is obviously a playful look at consequences. And even the butterfly was born in a throwaway comment.

Edward Lorenz, who initiated research in this field, was due to give a talk at the 1972 meeting of the American Association for the Advancement of Science, but had failed to provide a title. The meteorologist Philip Merilees, the session chair, eventually came up with something. In a paper nine years earlier, Lorenz had mentioned how a meteorologist had scorned chaos theory, pointing out that if it were correct, ‘one flap of a seagull’s wings
could change the course of weather forever’. Merilees evidently remembered this line, and concocted a variant that has entered popular culture like few other scientific concepts. The title of Lorenz’s talk was ‘Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?’

The official term for the butterfly effect is ‘sensitive dependence on initial conditions’. The basic idea is that most systems that change over time – whether they are natural, such as weather, or artificial, such as the numerical output from a computer program – will turn out very differently if even the tiniest adjustment is made to their starting point. This simple observation has such profound consequences that it has given rise to a whole new field of research.

‘Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?’

PHILIP MERILEES

The repercussions of chaos theory, as this field is known, have been felt across the whole of science. From the dynamics of the planets to the way epidemics spread through human populations, chaos theory’s influence is as wide-reaching as it is important. The entire universe is, it seems, in a state of chaos. This discovery would have come as a terrible shock to the grandly titled Pierre Simon, Marquis de Laplace. In the 18th century he had embraced the Newtonian revolution with relish. His book on the mechanisms of the universe, where he took Newton’s gravitational theory and used it to map out the movements of all the planets, was a masterpiece. A few years later, he boldly boasted of science’s power to tame every known phenomenon:

An intelligence which at a given instant knew all the forces acting in nature and the position of every object in the universe – if endowed with a brain sufficiently vast to make all necessary calculations – could describe with a single formula the motions of the largest astronomical bodies and those of the smallest atoms. To such an intelligence, nothing would be uncertain; the future, like the past, would be an open book.

Within a few decades of Laplace’s death, however, that vision had begun to unravel. It started in 1860, when the Scottish physicist James Clerk Maxwell discussed the amplification of small changes when considering what happens when molecules collide. Thirty years later, Henri Poincaré discovered that the mutual gravitational attraction of three moving objects displayed sensitive dependence on initial conditions. Then, in the 1920s, the Dutch engineer Balthasar van der Pol found chaos in the tones produced by a telephone handset connected to a vacuum tube. The electric current driving the tube occasionally triggered what we would recognize as feedback, and van der Pol was able to write down an equation to describe it.

‘To such an intelligence, nothing would be uncertain; the future, like the past, would be an open book.’

PIERRE SIMON, MARQUIS DE LAPLACE

Though this equation was enormously useful to engineers trying to build vacuum tubes into electronic systems such as broadcasting equipment, the whistling itself was no more than a nuisance. In fact, various mathematicians and engineers studied the phenomenon without noting anything particularly remarkable about it. Though the story of chaos theory has involved many players, it was Edward Lorenz who really brought it to life.

Lorenz had been a weather-watcher since childhood, and spent the Second World War as a weather forecaster for the Army Air Corps. Some years later, when working as a researcher at the Massachusetts Institute of Technology, Lorenz combined meteorology with mathematics and the relatively new science of computing. He built a processor that could model a simple version of the weather. And it was here that he discovered the butterfly effect.

As with many of the most important breakthroughs in science, it happened by accident. One afternoon in 1961 Lorenz was short of time and halfway through a weather simulation on his computer. Looking through a print out of where he wanted to
start, he punched in the numbers that would rerun the simulation from halfway onwards. It came out wrong – or at least wildly different from the original.

Alarmed by the apparent error, Lorenz checked what he had used as an input. He had, he noted, cut the numbers off after the third decimal place, assuming that the fine detail would make no difference. Where the computer had been using 0.506127, Lorenz had input only 0.506. It had made all the difference in the world. Lorenz had discovered the sensitive dependence on initial conditions: an unpredictability that arises through our limited knowledge.

Our rulers are not infinitely small, our actions are not infinitely smooth, our machines are not infinitely powerful. Thus every measurement we perform, and every computation carried out using those measurements, will have a small but finite error. Before Lorenz, that might have been considered to cause a problem as small as the error. But sensitive dependence on initial conditions means that, more often than not, the error will eventually be huge.

Everywhere we look we find chaos. The solar system, for instance, is chaotic because it involves the interaction of more than two bodies. As Henri Poincaré proved, while there are solutions to the equations that describe interactions between two bodies, add one more – or many more – and no exact solutions can be found. The mathematical equations describing the system simply cannot be solved.

With eight planets and a sun to consider, not to mention countless rocks, asteroids and comets, chaos reigns in the heavens. But a solar system that runs by chaos rather than clockwork does not mean we are in danger of colliding with another planet at any moment. Chaotic orbits are often ‘bounded’, moving in cycles that never quite repeat, but within a limited space, thus limiting the danger of collision.

This boundedness, where chaos operates within strict limits, has given rise to another icon of chaos: the ‘strange attractor’. Imagine a simple system that exhibits chaotic behaviour, something like a double pendulum, where two rigid rods are loosely joined and allowed to swing freely. The free movement of the double pendulum is similar to the movement of your leg from the hip, but with the knee able to bend in two directions. Until you have seen it with your own eyes, it is almost impossible to fathom the degree of unpredictability the double pendulum exhibits. It swings back and forth, with the endpoint of each oscillation seemingly at random.

Swinging freely from some arbitrary starting point, the bottom of the double pendulum traces out a pattern composed of loops that form definite shapes. Though it never makes the same loop twice, it also doesn’t deviate far from the established pattern. This ‘attraction’ to a particular form is what gave the strange attractor its name. Perhaps the most famous example is the butterfly-shaped Lorenz attractor. This is a map of the movement of a chaotic system in three dimensions. As the motion continues, the lines are denser and denser. But the trajectory never crosses itself, and never repeats.

A similar pattern emerges from a steel pendulum set in motion just above three magnets laid out as the corners of a triangle. Each magnet exerts a pull on the steel bob, and the pull varies as the bob moves in and out of the magnet’s field. The sum of all the competing pulls forces the pendulum into a
chaotic trajectory that is sensitive to the tiniest variation in its initial position or velocity. This is illustrated in its strange attractor.

The fact that these chaotic orbits are bounded does not mean they can’t have extreme consequences. This is demonstrated in the effects that the planets can have on each other. Though the orbits themselves do not stray too far from their expected paths, the chaotic motions might occasionally create a cataclysmic threat. Calculations show that a tiny kick to Saturn, from the particles that compose the solar wind, for example, can turn its orbit ‘aperiodic’. That means it will take a slightly different path each time it goes round the sun.

It’s a scary prospect, because it opens up the possibility that Jupiter, Saturn and the Sun will align at some moment. The combined gravitational pull of this trinity is enough to pull rocks out of the asteroid belt that lies between the orbits of Mars and Jupiter, and could unleash an asteroid storm. There have been claims that such an event preceded the asteroid impact that seems to have ended the age of the dinosaurs. If that is what happened, it would not be the first time that chaos has had an impact on biology – and it was certainly not the last. The butterfly effect rules biology as surely as it rules the swing of a double pendulum.

Chaos theory has had an enormous impact on the science of ecology. The idea of populations growing in good times and thinning out in the bad times has always been a strong part of biological thinking, but the rise of chaos theory and the butterfly effect created a Eureka moment. Before chaos came along, biologists of a mathematical bent would write down equations that approximated to the situations they were interested in. They might describe, for instance, how many squirrels were living in a square mile, how rich their food source was, how frequently they reproduced, and how many predators shared the territory.