At 10:18 AM 4/25/02 -0700, Tim May wrote: >On Thursday, April 25, 2002, at 07:45 AM, Major Variola (ret) wrote: >> Predictability gets much worse if one of the walls of a pool-table is
>> curved, >> then the uncertainty in a perfectly-round ball's momentum is >> magnified after reflection, compared to a pool-table of 3 or more >> flat walls. > >Yes, of course. There are many sources of divergence, and curved walls >certainly add divergence. But, as you acknowledge, the curvature of the >spherical balls is a source. In fact, the radius of curvature of a ball >is much smaller than that of curved side walls, so of course they are >huge sources of divergence. Yep. I misremembered the model. The importance of curved walls is only in abstract-billiards with point-balls. Otherwise, analytically, you get into fixed sequences with flat walls. Pong-like games can demonstrate this. Its all about, as you said, the effect of divergence on initial uncertainty. >And it's important for people not to think that the curved walls or >curved balls are important to the phenomenon: if all surfaces were >nominally flat (say, to a sixteenth wavelength, about the best a >telescope mirror is ground to), the divergences would _still_ occur. >Tiny alternations in temperature would affect dimensions, friction, >speeds, and hence would alter arrival times. At some point, objects in >one history would bounce and in another history would miss...the changes >at this point are _huge_. These are unavoidable *empirical* sources of uncertainty; the above gedankenbilliards & divergence addresses *analytic* magnification of uncertainty. >(I actually did a project at Intel which exploited this. I devised a >scheme whereby "known good" microprocessors would be imaged by an >electron microcope, state by state (using "beam blanking" to only >illuminate the chip during a specific state), and then would be >digitally subtracted or otherwise compared to the internal states of >chips having some speed problem, or some voltage problem, or just plain >failing. By examining the time evolution of divergent states, especially >by running the history backwards, we could pinpoint the "first >divergence," the first state where voltage levels differed noticeably. >This was usually a place where a design needed to be tweaked, or where a >marginality was present, or a flaw, etc. This machine, the Dynamic Fault >Imager, was used to get speeds up on the 80286 and later processors.) Much more finesse than your 'cutting the crap out of packaging materials' trick... >BTW, someone was speculating about "history healing itself" (this is my >term for it, a familiar trope in SF). A look at the billiard ball model >will show how this cannot conceivably happen: as soon as one ball >"misses" another, all of the later trajectories are radically different. >If this is not immediately clear, spend several minutes drawing pictures. "Time-machine inventor reports that his stepping on a butterfly in Brazil averted a Chinese cyclone..." >If the writer was talking about "conserved quantities" (I think he >mentioned vapors in an elevator, for example), this is not at all what >we are talking about. Yes, the total energy of the billiard balls will >remain roughly the same in both histories, though divergences will occur >even in total energy, just more slowly. Yes, the earth's overall climate >is not dramatically affected by butterflies. To track one (odor) molecule's position into the future, you'd need exponentially more information (about the other molecules it interacts with) the farther you want to predict in time. ... Jim Bell was just doing physics-of-computation experiments in IRS offices..
