Wednesday, March 12, 2008

Chaos is weakly constrained

You can have chaos with as few as two degrees of freedom in a closed system, and with only one if the system is "pumped" from the outside. But much of the chaos around us is a result of systems with many degrees of freedom. Such systems, whether open or closed, are often analyzed with statistical methods: statistical physics, or statistical mechanics, from which we get statistical distributions, averages, variances, and all the other paraphenalia. In physics, the distributions and their moments (the generalized weights of the distribution - the mean and variance are the first two) are fixed by general conservation laws, such as the conservation of the number of bodies (or particles) and of the energy in the system.

If we look at each changing degree of freedom in detail, such conservation laws tend to reduce the overall chaos in the system. The more conservation laws, the less chaos. Each conservation law is related to a symmetry of the system: the dynamics of an isolated system is invariant in time for example and leads to a conserved total energy; if the identity of its constituent parts is unchanging, the number of particles is conserved too. Non-chaotic systems - the standard ones in physics being a single planet orbiting a star (or one electron orbiting a nucleus) and a simple harmonic oscillator - have so much symmetry that their motion is non-chaotic and can actually be solved for exactly with pencil and paper (in closed form). This is an unusual property in general; most systems cannot be so solved, most are nonlinear, and many are chaotic. Chaos is the norm; nonchaotic motion the exception. A few conservation laws still hold rigidly, but they are far too few to place much constraint on the motion of a system of many degrees of freedom.

Consider a modest volume of air in the room, a mole, with about 1024 molecules (1 followed by 24 zeros). The constraints imposed by conserving their total energy, total volume, and total number place three restrictions on the degrees of freedom of the gas: six times the number of molecules. It's "six" because each molecule can translate in three separate directions and rotate its orientation in three independent ways.

If we don't follow each degree of freedom in detail and instead fall back on a statistical description, the conservation laws announce themselves in a different way. For each conserved total quantity (energy, number, and volume, say), a statistical distribution results whose properties are fixed by that total. Energy is the best-known; its distribution is controlled by a parameter equivalent to temperature. Furthermore, a not-totally-strict, but very restrictive, type of "equal distribution" of energy among the individual particles results (equipartition). It's very unlikely that the particles will share the fixed total energy in any way other than sharing it equally. Similarly for fixed total volume (giving rise to pressure, as particles bounce off the walls that constrain them to remain in that volume) and fixed total number (giving rise to chemical potential, the "energy cost" of introducing or removing one particle from the system). These laws are the reason, for example, why it is overwhelmingly unlikely that the air in the room I am sitting in right now will find itself all crammed in one corner, leaving the rest of the space a vacuum. And that's a good thing.

But we can measure an infinity of other properties about the system, and, in general, the answers will not be constrained by a conservation law. In that case, the particles aren't sharing a fixed total of something. The resulting patterns of some such variable, in general, are chaotic and generally exhibit a wide range of possible "sharing schemes." The statistical distributions that result will look nothing like the "law of large numbers" or equipartition world just outlined for the conserved case. Instead, they will be distributed according to a fractal, or scaling, law, one that is self-similar on many scales. For a truly chaotic system, after waiting an infinite amount of time, the distribution would be a perfect fractal, looking like itself no matter if you zoom in or zoom out. It's utterly different from the sharply-defined world of averages and small deviations defined by conservation laws.

We'll meet this distinction - a world of averages versus of a world of scaling - again, when we consider the statistical nature of chaos and take a look at a remarkable new book.

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