Chaos - James Gleick [144]
Somehow, after all, as the universe ebbs toward its final equilibrium in the featureless heat bath of maximum entropy, it manages to create interesting structures. Thoughtful physicists concerned with the workings of thermodynamics realize how disturbing is the question of, as one put it, “how a purposeless flow of energy can wash life and consciousness into the world.” Compounding the trouble is the slippery notion of entropy, reasonably well-defined for thermodynamic purposes in terms of heat and temperature, but devilishly hard to pin down as a measure of disorder. Physicists have trouble enough measuring the degree of order in water, forming crystalline structures in the transition to ice, energy bleeding away all the while. But thermodynamic entropy fails miserably as a measure of the changing degree of form and formlessness in the creation of amino acids, of microorganisms, of self-reproducing plants and animals, of complex information systems like the brain. Certainly these evolving islands of order must obey the Second Law. The important laws, the creative laws, lie elsewhere.
Nature forms patterns. Some are orderly in space but disorderly in time, others orderly in time but disorderly in space. Some patterns are fractal, exhibiting structures self-similar in scale. Others give rise to steady states or oscillating ones. Pattern formation has 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. As water freezes, crystals send out tips; the tips grow, their boundaries becoming unstable, and new tips shoot out from the sides. Snowflakes obey mathematical laws of surprising subtlety, and it was impossible to predict precisely how fast a tip would grow, how narrow it would be, or how often it would branch. Generations of scientists sketched and cataloged the variegated patterns: plates and columns, crystals and polycrystals, needles and dendrites. The treatises treated crystal formation as a classification matter, for lack of a better approach.
Growth of such tips, dendrites, is now known as a highly nonlinear unstable free boundary problem, meaning that models need to track a complex, wiggly boundary as it changes dynamically. When solidification proceeds from outside to inside, as in an ice tray, the boundary generally remains stable and smooth, its speed controlled by the ability of the walls to draw away the heat. But when a crystal solidifies outward from an initial seed—as a snowflake does, grabbing water molecules while it falls through the moisture-laden air—the process becomes unstable. Any bit of boundary that gets out ahead of its neighbors gains an advantage in picking up new water molecules and therefore grows that much faster—the “lightning-rod effect.” New branches form, and then subbranches.
One difficulty was in deciding which of the many physical forces involved are important and which can safely be ignored. Most important, as scientists have long known, is the diffusion of the heat released when water freezes. But the physics of heat diffusion cannot completely explain the patterns researchers observe when they look at snowflakes under microscopes or grow them in the