Despre haos și teoria informației. Energy in natural systems exists on two levels: the macroscales, where everyday objects can be counted and measured, and the microscales, where countless atoms swim in random motion, unmeasurable except as an average entity, temperature. As Shaw noted, the total energy living in the microscales could outweigh the energy of the macroscales, but in classical systems this thermal motion was irrelevant—isolated and unusable. The scales do not communicate with one another. “One does not have to know the temperature to do a classical mechanics problem,” he said. It was Shaw’s view, however, that chaotic and near-chaotic systems bridged the gap between macroscales and microscales. Chaos was the creation of information. One could imagine water flowing past an obstruction. As every hydrodynamicist and white-water canoeist knows, if the water flows fast enough, it produces whorls downstream. At some speed, the whorls stay in place. At some higher speed, they move. An experimenter could choose a variety of methods for extracting data from such a system, with velocity probes and so forth, but why not try something simple: pick a point directly downstream from the obstruction and, at uniform time intervals, ask whether the whorl is to the right or the left. If the whorls are static, the data stream will look like this: left-left–left-left–left-left–left-left–left-left–left-left–left-left–left-left–left-left–left-left–. After a while, the observer starts to feel that new bits of data are failing to offer new information about the system. Or the whorls might be moving back and forth periodically: left-right–left-right–left-right–leftright– left-right–left-right–left-right–left-right–left-right–left-right–. Again, though at first the system seems one degree more interesting, it quickly ceases to offer any surprises. As the system becomes chaotic, however, strictly by virtue of its unpredictability, it generates a steady stream of information. Each new observation is a new bit. This is a problem for the experimenter trying to characterize the system completely. “He could never leave the room,” as Shaw said. “The flow would be a continuous source of information.” Where is this information coming from? The heat bath of the microscales, billions of molecules in their random thermodynamic dance. Just as turbulence transmits energy from large scales downward through chains of vortices to the dissipating small scales of viscosity, so information is transmitted back from the small scales to the large—at any rate, this was how Shaw and his colleagues began describing it. And the channel transmitting the information upward is the strange attractor, magnifying the initial randomness just as the Butterfly Effect magnifies small uncertainties into large-scale weather patterns. The question was how much. Shaw found—after unwittingly duplicating some of their work—that again Soviet scientists had been there first. A. N. Kolmogorov and Yasha Sinai had worked out some illuminating mathematics for the way a system’s “entropy per unit time” applies to the geometric pictures of surfaces stretching and folding in phase space. The conceptual core of the technique was a matter of drawing some arbitrarily small box around some set of initial conditions, as one might draw a small square on the side of a balloon, then calculating the effect of various expansions or twists on the box. It might stretch in one direction, for example, while remaining narrow in the other. The change in area corresponded to an introduction of uncertainty about the system’s past, a gain or loss of information. To the extent that information was just a fancy word for unpredictability, this conception merely matched the ideas that such scientists as Ruelle were developing. But the information theory framework allowed the Santa Cruz group to adopt a body of mathematical reasoning that had been well investigated by communications theorists. The problem of adding extrinsic noise to an otherwise deterministic system, for example, was new in dynamics but familiar enough in communications. The real appeal for these young scientists, however, was only partly the mathematics. When they spoke of systems generating information, they thought about the spontaneous generation of pattern in the world. “At the pinnacle of complicated dynamics are processes of biological evolution, or thought processes,” Packard said. “Intuitively there seems a clear sense in which these ultimately complicated systems are generating information. Billions of years ago there were just blobs of protoplasm; now billions of years later here we are. So information has been created and stored in our structure. In the development of one person’s mind from childhood, information is clearly not just accumulated but also generated—created from connections that were not there before.” - Chaos: Making a New Science, James Gleick
Posted on: Sat, 13 Dec 2014 18:51:44 +0000
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