Chaos In Physics

New developments in science that come under the general heading of "chaos" have been described as revolutionary and the basis for a genuinely new paradigm. It has brought about the realization that instead of understanding most of nature in principle, science has really addressed only the very restricted subset of phenomena that can be analyzed by simple methods. Chaos has had a broad impact in diverse disciplines, influencing mechanics, astronomy, solid-state physics, ecology, meteorology, and biology, as a few examples. For those interested in a popular book on the subject, Chaos: Making a New Science (Gleick 1987) is probably the best, with Fractals in Your Future (Lewis 1991) being particularly appropriate for teaching high-school level students. At a slightly more advanced level is Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise (Schroeder 1991).

Photos of the Mandelbrot Set (FRACTINT. PROGRAM), a classic example of chaos.

To understand the revolutionary nature of chaos requires some history. In the 1600s Newton developed classical mechanics to describe the motions of bodies in our solar system. In the 1800s Laplace extended the ideas of classical mechanics to suggest determinism — the motion of every body in the universe is completely determined. As a result, "Universal, immutable laws, absolute precision, and strict predictability were ideas that were habitually believed to characterize science." Newtonian physics assumed "that the initial data determine the future, unambiguously, uniquely, and forever ... [and] that the qualitative aspects of the motion are not too sensitive to the precise initial data." Newton's laws could be solved exactly for 2 bodies, such as the sun and the earth. For a system containing more than 2 bodies, such as the earth-moon-sun system, accurate approximations were used. More involved problems were — so was the common belief — only technically different from the special examples. Newtonian physics was a dramatic success in predicting the return of Halley's comet in 1757 after its observation in 1682, and in predicting a new planet (Neptune) based on the observed irregularities in the orbit of Uranus. "These ideas were so successful and suggestive that they stimulated the search for similar laws in all of physics (and most of science)" (Dresden 1992).
Unfortunately, "the general attitude toward classical physics was based on an uncritical, unanalyzed acceptance of the ideology of Newtonian physics." "It is not clear whether physicists and astronomers were aware of the tenuous mathematical basis of their (often unspoken) beliefs, but it is pretty clear that they didn't worry too much about it" (Dresden 1992). At the turn of the last century, Poincare in his The New Methods of Celestial Mechanics wondered about the stability of solar systems. He "discovered that with even the very smallest perturbation, some orbits behaved in an erratic, even chaotic way. This was true even for a closed system, that should be particularly amenable to analysis by classical mechanics. Quantum mechanics and the Heisenberg uncertainty principle yielded indeterminacy in another area a few years later, and Poincare's ideas were forgotten for a time. "Small wonder, since even Poincare had abandoned the ideas, saying, 'These things are so bizarre that I cannot bear to contemplate them'" (Briggs & Peat 1989). Poincare's ideas have only come to the forefront again in the last 15 years or so.
Now classical mechanics is being reevaluated. It has been found that "properties were inferred from the detailed examination of very few examples — maybe five or six altogether." "... all the general features attributed to classical mechanics are in general wrong. The exactly soluble examples are not generic; they are in fact quite atypical." "... chaotic behavior, contrary to earlier beliefs, was a rather general property and not a pathological feature of some contrived system," and even relatively simple systems can exhibit frightfully complex behavior (Dresden 1992).
For chaotic systems a slight imprecision results in indeterminism. This slight imprecision may result from uncertainty in the initial data, or from approximate calculations based on perturbations of an exact solution. For such systems, predictive errors develop exponentially with time and an initial, small imprecision eventually results in total loss of predictability (Davies 1990).
Examples of chaotic behavior are numerous. The famous Butterfly Effect suggests "that a butterfly stirring the air today in Peking can transform storm systems next month in New York" (Gleick 1987). Population dynamics of rabbits can be affected in unpredictable ways by small changes in food supply. Earthquakes, snow avalanches, and dinosaur extinctions have been studied using chaos methods. One simple example involved adding sand to a sand pile on a 4 cm plate, one grain at a time. Sand avalanches would occur after as few as one or as many as several thousand grains were added to the pile. The addition of a sand grain would cause an avalanche in unpredictable ways (Bak & Chen 1991).
Chaos theory brings attention to the fact that errors in describing the future based on present approximate data in classical mechanics develop exponentially with time. It is possible that errors in extrapolation from present earth-history data into the past also develop exponentially with time and therefore, less is known about the past than previously realized.

Benjamin L. Clausen

Selected References

• Bak, P. and K. Chen. 1991. Self-organized criticality. Scientific American 264(1):46-53.
• Briggs, J. and F. D. Peat 1989. Turbulent mirror. Harper & Row, New York, pp. 26-29.
• Davies, P. 1990. Chaos frees the universe. New Scientist 128(1737):48-51.
• Dresden, M. 1992. Chaos: a new scientific paradigm — or science by public relations? The Physics Teacher 30(1):10-14.
• Gleick, J. 1987. Chaos: making a new science. Viking Penguin. New York. 352 pp.
• Lewis, R.S. 1991. Fractals in your future. Media Magic, Nicasio, California. 265 pp.
• Schroeder, M. 1991. Fractals, chaos, power laws: minutes from an infinite paradise. Freeman, New York. 429 pp.