Chaos Bound by N. Katherine Hayles

Author:N. Katherine Hayles
Language: eng
Format: epub
Publisher: Cornell University Press
Published: 2017-09-23T00:00:00+00:00

Chaos and Symmetry

The mathematics Feigenbaum used to reveal recursive symmetries was developed by Kenneth Wilson, winner of the 1982 Nobel Prize in physics. To understand Wilson’s approach, consider what happens when a flow becomes turbulent (Wilson, 1983). Often microscopic fluctuations within a flowing liquid cancel each other out, as when a river flows smoothly between its banks. In this case each water molecule follows much the same path as the one before it, so that molecules starting close together continue to be close. Sometimes, however, microscopic fluctuations persist and are magnified up to the macroscopic level, causing eddies and backwaters to form. Then molecules that began close together may quickly separate, and molecules that were far apart may come close together. As a result it becomes extremely difficult to calculate how the flow will evolve. “Theorists have difficulties with these problems,” Wilson explains, “because they involve very many coupled degrees of freedom. It takes many variables to characterize a turbulent flow or the state of a fluid near the critical point” (p. 583). Indeed, the mathematics of turbulence is so complex that even the new supercomputers are inadequate to handle it. Computation times become unreasonably long after only three or four variables are considered, whereas dozens are necessary to create a model that can simulate turbulence accurately. Hence the importance of being able to solve these problems analytically.

Wilson’s contribution was to devise a method that would sometimes yield analytical solutions. The essence of his approach was to shift from following individual molecules to looking for symmetries between different scales. In turbulent flow, for example, large swirls of water have smaller swirls within which are still smaller swirls…. To model these recursive symmetries, Wilson used renormalization groups. Renormalization had first emerged as a technique in quantum mechanics; physicists used it to get rid of infinite quantities when they appeared in the equations. Originally the only justification for the procedure was that it made the answers come out right. If you think this sounds suspiciously aribitrary, you are not alone. For years virtually all mathematicians and some physicists regarded renormalization as no more than hand-waving. But Wilson saw its deeper implications.

He knew that in renormalization certain quantities regarded as fixed, such as particle mass, are treated as if they were variable. He realized that there is a sense in which this is a profound truth rather than an arbitrary procedure. For example, we tend to think of a golf ball as a smooth sphere. But to a mosquito it would appear as a pocked irregular surface, and to a bacterium as the Wilson Alps. Renormalization implied that the choice of ruler used to measure physical properties affected the answer. At the same time, it revealed that there was something else—something not normally considered—that remained constant over many measurement scales. This was the scaling factor. By combining the renormalization process with the idea of a mathematical group, Wilson arrived at a method whereby this factor could be defined and calculated.

“Group,” as it is used in mathematics, denotes a set of objects that is invariant under symmetry operations.

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