On Thursday, April 25, 2002, at 07:45 AM, Major Variola (ret) wrote: > At 09:42 AM 4/23/02 -0700, Tim May wrote: >> >> And even if the world were Newtonian, in a classical billiard ball >> sense, with Planck's constant precisely equal to zero, predictability > is >> a chimera. Consider a game of billiards, with perfectly spherical >> billiard balls, a perfectly flat table, etc. Trajectories depend on >> angles to a precision that keeps going deeper and deeper into the >> decimals. For example, predicting the table state after, say, 3 > seconds, >> might require knowing positions, speeds, and angles (in other words, > the >> vectors) to a precision of one part in a thousand. Doable, one might > say. > > 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. 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_. (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.) > > You may have meant to imply this --if spherical balls > hit other balls the uncertainty is similarly magnified-- but its worth > noting the difference in predictability between flat and curved-wall > abstract billiards. And it's also useful for people to understand, deeply, that the divergences may "enter" at the fourth decimal place, or the sixth, or the twelfth. But they enter, and enter quickly, and the cascade of divergences doesn't much involve issues of flat vs. curved walls. The divergences are at a much more profound level. 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. 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. But the point here is that the positions and veocities of the billard balls are "unpredictable" after some time t, where t is probably on the order of tens of seconds, even if Planck's constant were zero and there were no quantum effects whatsoever. --Tim May "To those who scare peace-loving people with phantoms of lost liberty, my message is this: Your tactics only aid terrorists." --John Ashcroft, U.S. Attorney General
