Can a butterfly stirring the air in Beijing today transform storms in New York next month? Watch two bits of foam flowing side by side at the bottom of a waterfall. What can you guess about how close they were at the top? Nothing. As far as theoretical physics is concerned, God might just as well have taken all those water molecules under the table and shuffled them personally. Traditionally, when physicists saw complex results, they looked for complex causes
Now all that has changed. In the last 20 years, scientists have created an alternative set of ideas: Simple systems give rise to complex behavior. Complex systems give rise to simple behavior. More important, the laws of complexity hold universally, caring not at all for the details of a system's constituent atoms. And "chaos"--the obstinate element of disorder within order, of variation where predictability was expected--has become a shorthand name for a fast-growing movement that is reshaping the fabric of the scientific establishment in the United States, Europe, and Japan.
Where chaos begins, classical science stops. For as long as the world has had physicists inquiring into the laws of nature, it has suffered a special ignorance about disorder: in the atmosphere, in the turbulent sea, in the fluctuations of wildlife populations, in the oscillations of the heart and brain. The irregular side of nature, the discontinuous and erratic side--these have been puzzles to science, or worse, monstrosities.
But just over a decade ago, a few scientists in the United States and Europe began to find a way through disorder. They were mathematicians, physicists, biologists, chemists--all seeking connections between different kinds of irregularity. Physiologists found a surprising order in the chaos that develops in the human heart. Ecologists explored the rise and fall of gypsy moth populations. Economists dug out old stock price data and tried a new kind of analysis. The insights that emerged led them to parallels in the natural world--the shapes of clouds, the paths of lightning, the microscopic intertwining of blood vessels, the galactic clustering of stars.
Such puzzles and monstrosities are the domain of the revolution in chaos--a revolution not of technology, like the laser revolution or the computer revolution, but a revolution of ideas. This revolution began with a set of ideas having to do with disorder in nature: from turbulence in fluids, to the erratic flows of epidemics, to the arrhythmic writhing of a human heart in the moments before death. It has continued with an even broader set of ideas that might be better classified under the rubric of complexity.
It is a revolution in every sense that that word can be applied to science. It has been sudden. It has radically transformed the way scientists think about their universe, the way they perceive the complexity that always exists in their slices of nature. It has brought a good deal of turbulence in the bureaucracy of science--from hurt feelings and rejected papers in the early days, to new, expensive, interdisciplinary institutes in these days of trendy success.
Those concerned about education argue that chaos should be--and can be--part of the earliest education of scientists and nonscientists. One prominent biologist, Robert May of Oxford University, has called on educators to give every student a pocket calculator and show them how to play with a simple chaotic equation. He contends that it would change the way they think about everything from the theory of business cycles to the propagation of rumors.
Where does complexity come from: complex behavior or complex organization? This has been an odd question, in a way. Scientists were not used to thinking about complexity as a thing with a life of its own. But the question of how complexity is born and how it evolves is one that this new science of chaos is beginning to crack. Those who care about the behavior of groups of people--the behavior that drives the stock market, and the behavior that shapes the organization of large corporations--are beginning to pay attention to chaos.
As recently as two decades ago, most practicing scientists shared a few unspoken beliefs. For example, simple systems behave in simple ways. A mechanical contraption like a pendulum, a small electrical circuit, an idealized population of fish in a pond--as long as these systems could be reduced to a few perfectly understood, perfectly deterministic laws, their long-term behavior would be stable and predictable. Another belief: Complex behavior seemed to imply complex causes. A mechanical device, an electrical circuit, a wildlife population, a fluid flow, a biological organ, a particle beam, an atmospheric storm, a national economy--systems that were visibly unstable, unpredictable, or out of control must, scientists believed, either be governed by a multitude of independent components or subject to random external influences.
The first chaos theorists, the scientists who set the discipline in motion, shared certain sensibilities. They had an eye for pattern, especially pattern that appeared on different scales at the same time. They had a taste for randomness and complexity, for jagged edges and sudden leaps. Believers in chaos--and they sometimes call themselves believers, or converts, or evangelists--speculate about determinism and free will, about evolution, about the nature of conscious intelligence. They feel that they are turning back a trend in science toward reductionism, the analysis of systems in terms of their constituent parts: quarks, chromosomes, or neurons. They believe that they are looking for the whole.
The most passionate advocates of the new science go so far as to say that 20th-century science will be remembered for just three things: relativity, quantum mechanics, and chaos. Chaos, they contend, has become the century's third great revolution in the physical sciences. Like those revolutions before it, chaos cuts away at the tenets of Newton's physics. As one physicist puts it: "Relativity eliminated the Newtonian illusion of absolute space and time; quantum theory eliminated the Newtonian dream of a controllable measurement process; and chaos eliminates the Laplacian fantasy of deterministic predictability."
Of the three, the revolution in chaos applies to the universe we see and touch, to objects at human scale. Everyday experience and real pictures of the world have become legitimate targets for inquiry. There has long been a feeling, not always expressed openly, that theoretical physics has strayed far from human intuition about the world. Whether this will prove to be fruitful heresy or just plain heresy, no one knows. But some of those who thought physics might be working its way into a corner now look to chaos as a way out.
The modern study of chaos began with the creeping realization in the 1960's that quite simple mathematical equations could model systems every bit as violent as a waterfall. Tiny differences in input could quickly become overwhelming differences in output--a phenomenon given the name "sensitive dependence on initial conditions." In weather, for example, this translates into what is known as the Butterfly Effect--the notion that a butterfly stirring the air today in Beijing can transform storm systems next month in New York.
In 1961, Edward Lorenz, a theoretical meteorologist at the Massachusetts Institute of Technology, developed a simulated weather model in his new electronic computer, based on 12 numerical rules--equations that expressed the relationships between temperature and pressure, pressure and wind speed, and so forth. Lorenz understood that he was putting into practice the laws of Newton. Thanks to the determinism of physical law, further intervention would then be unnecessary. Those who made such models took for granted that, from present to future, the laws of motion provide a bridge of mathematical certainty. Understand the laws and you understand the universe. But there was always one small compromise, so small that working scientists usually forgot it was there, lurking in a corner of their philosophies like an unpaid bill: Measurement could never be perfect. Scientists marching under Newton's banner actually waved another flag that said something like this: Given an approximate knowledge of a system's initial conditions and an understanding of natural law, one can calculate the approximate behavior of the system. This assumption lay at the heart of science. As one theoretician liked to tell his students: "There's a convergence in the way things work, and arbitrarily small influences don't blow up to have arbitrarily large effects."
At first, Lorenz's printouts seemed to behave in those recognizable ways. They matched his cherished intuition about the weather, his sense that it repeated itself, displaying familiar patterns over time, pressure rising and falling, the air stream swinging north and south. But the repetitions were never exact. There were patterns, with disturbances. An orderly disorder.
One day in the winter of 1961, wanting to examine one sequence at greater length, Lorenz took a shortcut. Instead of starting the whole run over, he started midway through. To give the machine its initial conditions, he typed the numbers straight from the earlier printout. Then he walked down the hall to get away from the noise and drink a cup of coffee.
The new run should have exactly duplicated the old. Yet as he stared at the new printout, Lorenz saw his weather diverging so rapidly from the pattern of the last run that, within just a few simulated "months," all resemblance had disappeared. At first he suspected a malfunction. But suddenly he realized the truth. In the computer's memory, six decimal places were stored. On the printout, to save space, just three appeared. Lorenz had entered the shorter, rounded-off numbers, assuming that the difference--one part in a thousand--was inconsequential.
A small numerical error was like a small puff of wind--surely the small puffs faded or canceled each other out before they could change important, large-scale features of the weather. Yet in Lorenz's system of equations, small error proved catastrophic. For reasons of mathematical intuition that his colleagues would understand only later, Lorenz felt a jolt. The practical import could be staggering. The Butterfly Effect meant that long-term weather forecasting would be impossible.
But Lorenz saw beyond the randomness embedded in his weather model. He saw a fine geometric structure, order masquerading as randomness. He turned his attention more and more to the mathematics of systems that never found a steady state, that almost repeated themselves but never quite succeeded, trying to find simple equations that would produce the aperiodicity he was seeking. At first his computer tended to lock into repetitive cycles. But he finally succeeded when he put in an equation that varied the amount of heating from east to west, corresponding to the real-world variation between the way the sun warms the western coast of Europe and the way it warms the Atlantic Ocean. The repetition disappeared.
He continued to seek simpler and simpler models and to examine other dynamic fluid systems. His models turned out to have exact analogues in real systems. For example, his equations precisely described an old-fashioned electrical dynamo, where current flows through a disc that rotates through a magnetic field. Under certain conditions, the dynamo can reverse itself. Such behavior, scientists later suggested, might provide an explanation for another peculiar reversing phenomenon: the earthly magnetic field, or "geodynamo," that is known to have flopped many times during the earth's history, at intervals that seem erratic and inexplicable. Another system precisely described by Lorenz's equations is a simple-looking water wheel--simple, yet capable of stunningly chaotic reversals of direction.
As Lorenz's ideas spread and others arrived at similar kinds of conclusions, many scientists felt an intellectual excitement that comes with the truly new. To the Nobel Prize-winning physicist Freeman Dyson at the Institute for Advanced Study in Princeton, the news of chaos came "like an electric shock" in the 1970's. Physicists began to face what many believed was a deficiency in their education about even such simple systems as the pendulum.
The pendulum was the classical model of measurable regularity. Galileo believed that a pendulum of a given length not only keeps precise time but keeps the same time no matter how wide or narrow the angle of its swing. He made his claim in terms of experimentation, but the theory made it convincing--so much so that it is still taught as gospel in most high school physics classes. But it is wrong. The regularity Galileo saw is only an approximation. The changing angle of the bob's motion creates a slight nonlinearity in the equations.
Consider a playground swing. It accelerates on its way down, decelerates on its way up, all the while losing a bit of speed to friction. It gets a regular push--say, from some clockwork machine. All our intuition tells us that, no matter where the swing might start, the motion will eventually settle down to a regular back-and forth pattern, with the swing coming to the same height each time. That can happen. Yet odd as it seems, the motion can also turn erratic, first high, then low, never settling down to a steady state and never exactly repeating a pattern of swings that came before. The surprising behavior comes from a nonlinear twist in the flow of energy in and out of this simple oscillator. The swing is damped and it is driven: damped because friction is trying to bring it to a halt; driven because it is getting periodic pushes. Even when a damped, driven system is at equilibrium, it is not at equilibrium--and the world is full of such systems, beginning with the weather: damped by the friction of moving air and water, and by the dissipation of heat to outer space; and driven by the constant push of the sun's energy.
As chaos began to unite the study of different systems, pendulum dynamics broadened to cover high technologies from lasers to superconductors. Some chemical reactions displayed pendulum-like behavior, as did the beating heart. In all these phenomena, nature forms patterns. Some are orderly in space but disorderly in time, others orderly in time but disorderly in space. Some patterns are fractal, highly irregular and fragmented, exhibiting structures self-similar in scale. Self- similarity is symmetry across scale. It implies recursion, pattern inside of pattern, like an aerial picture of a coastline or a snowflake shape. Others give rise to steady states or oscillating ones. Pattern formation has now become a branch of physics and of materials science, allowing scientists to model the aggregation of particles into clusters, the fractured spread of electrical discharges, and the growth of crystals in ice and metal alloys.
The dynamics seem so basic--shapes changing in space and time-yet only now are the tools available to understand them. It is a fair question now to ask a physicist, "Why are all snowflakes different?"
Ice crystals form in the turbulent air with a famous blending of symmetry and chance, the special beauty of six-fold indeterminacy, obeying mathematical laws of surprising subtlety, making it formerly impossible to predict how fast a flake tip would grow, how narrow it would be, or how often it would branch. When a crystal solidifies outward from an initial seed--as a snowflake does, grabbing water molecules as it falls--the process becomes unstable.
The physics of heat diffusion cannot completely explain the patterns. But recently scientists worked out a way to incorporate another process--surface tension. The heart of the new snowflake model is the essence of chaos: a delicate balance between forces of stability and forces of instability, a powerful interplay of forces on atomic scales and forces on everyday scales.
Because the laws of growth are purely deterministic, snowflakes maintain a near-perfect symmetry. But the nature of the turbulent air is such that any pair of snowflakes will travel very different paths. The final flake records the history of all the changing weather conditions it has experienced, and the combinations may as well be infinite.
Classical physics could complete its mission without answering some of the most fundamental questions about nature. How does life begin? What is turbulence? And above all, in a universe ruled by entropy, drawing inexorably toward greater and greater disorder, how does order arise? Only a new kind of science could begin to address those questions, to cross the vast gulf between knowledge of what one thing does--one water molecule, one neuron--and what millions of them do.
"God plays dice with the universe," says one chaos specialist, the physicist Joseph Ford, in answer to Einstein's famous question. "But they're loaded dice. And the main object of physics now is to find out by what rules were they loaded and how we can use them for our own ends.
