Chaos - James Gleick [76]
Somehow, in the process of choosing, the atoms of the metal must communicate information to one another. Kadanoff’s insight was that the communication can be most simply described in terms of scaling. In effect, he imagined dividing the metal into boxes. Each box communicates with its immediate neighbors. The way to describe that communication is the same as the way to describe the communication of any atom with its neighbors. Hence the usefulness of scaling: the best way to think of the metal is in terms of a fractal-like model, with boxes of all different sizes.
Much mathematical analysis, and much experience with real systems, was needed to establish the power of the scaling idea. Kadanoff felt that he had taken an unwieldy business and created a world of extreme beauty and self-containedness. Part of the beauty lay in its universality. Kadanoff’s idea gave a backbone to the most striking fact about critical phenomena, namely that these seemingly unrelated transitions—the boiling of liquids, the magnetizing of metals—all follow the same rules.
Then Wilson did the work that brought the whole theory together under the rubric of renormalization group theory, providing a powerful way of carrying out real calculations about real systems. Renormalization had entered physics in the 1940s as a part of quantum theory that made it possible to calculate interactions of electrons and photons. A problem with such calculations, as with the calculations Kadanoff and Wilson worried about, was that some items seemed to require treatment as infinite quantities, a messy and unpleasant business. Renormalizing the system, in ways devised by Richard Feynman, Julian Schwinger, Freeman Dyson, and other physicists, got rid of the infinities.
Only much later, in the 1960s, did Wilson dig down to the underlying basis for renormalization’s success. Like Kadanoff, he thought about scaling principles. Certain quantities, such as the mass of a particle, had always been considered fixed—as the mass of any object in everyday experience is fixed. The renormalization shortcut succeeded by acting as though a quantity like mass were not fixed at all. Such quantities seemed to float up or down depending on the scale from which they were viewed. It seemed absurd. Yet it was an exact analogue of what Benoit Mandelbrot was realizing about geometrical shapes and the coastline of England. Their length could not be measured independent of scale. There was a kind of relativity in which the position of the observer, near or far, on the beach or in a satellite, affected the measurement. As Mandelbrot, too, had seen, the variation across scales was not arbitrary; it followed rules. Variability in the standard measures of mass or length meant that a different sort of quantity was remaining fixed. In the case of fractals, it was the fractional dimension—a constant that could be calculated and used as a tool for further calculations. Allowing mass to vary depending on scale meant that mathematicians could recognize similarity across scales.
So for the hard work of calculation, Wilson’s renormalization group theory provided a different route into infinitely dense problems. Until then the only way to approach highly nonlinear problems was with a device called perturbation theory. For purposes of calculation, you assume that the nonlinear problem is reasonably close to some solvable, linear problem—just a small perturbation away. You solve the linear problem and perform a complicated bit of trickery with the leftover part, expanding it into what are called Feynman diagrams. The more accuracy you need, the more of these agonizing diagrams you must produce. With luck, your calculations converge toward a solution. Luck has a way of vanishing, however, whenever a problem is especially interesting. Feigenbaum, like every other young particle physicist in the 1960s, found himself doing endless Feynman diagrams. He was left with the conviction that perturbation theory was tedious, nonilluminating, and stupid. So he loved Wilson’s new renormalization group