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Billiard table with convexobstacle
A deterministic chaotic system has, roughly speaking, two salientfeatures: (i) the evolution of the system over a long time periodeffectively mimics a random or stochastic process—it lackspredictability or computability in some appropriate sense; (ii) twosystems with nearly identical initial states will have radicallydivergent future developments, within a finite (and typically, short)timespan. We will use “randomness” to denote the firstfeature, and “sensitive dependence on initial conditions”(SDIC) for the latter. Definitions of chaos may focus on either orboth of these properties; Batterman (1993) argues that only (ii)provides an appropriate basis for defining chaotic systems.
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If quantum theories were unquestionably indeterministic, anddeterministic theories guaranteed repeatability of a strong form,there could conceivably be further experimental input on the questionof determinism's truth or falsity. Unfortunately, the existence of casts strong doubt on the former point, while chaos theorycasts strong doubt on the latter. More will be said about each ofthese complications below.
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A simple and very important example of a chaotic system in bothrandomness and SDIC terms is the Newtonian dynamics of a pool tablewith a convex obstacle (or obstacles) (Sinai 1970 and others). See .
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The usual idealizing assumptions are made: no friction, perfectlyelastic collisions, no outside influences. The ball's trajectory isdetermined by its initial position and direction of motion. If weimagine a slightly different initial direction, thetrajectory will at first be only slightly different. And collisionswith the straight walls will not tend to increase very rapidly thedifference between trajectories. But collisions with the convex objectwill have the effect of amplifying the differences. Afterseveral collisions with the convex body or bodies, trajectories thatstarted out very close to one another will have become wildlydifferent—SDIC.
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Even if the first hurdle can be overcome, the second, namelyestablishing precisely what the actual laws are, may seem dauntingindeed. In a sense, what we are asking for is precisely what19th and 20th century physicists sometimes setas their goal: the Final Theory of Everything. But perhaps, as Newtonsaid of establishing the solar system's absolute motion, “thething is not altogether desperate.” Many physicists in the past60 years or so have been convinced of determinism's falsity, becausethey were convinced that (a) whatever the Final Theory is, it will besome recognizable variant of the family of quantum mechanicaltheories; and (b) all quantum mechanical theories arenon-deterministic. Both (a) and (b) are highly debatable, but thepoint is that one can see how arguments in favor of these positionsmight be mounted. The same was true in the 19th century,when theorists might have argued that (a) whatever the Final Theoryis, it will involve only continuous fluids and solids governed bypartial differential equations; and (b) all such theories aredeterministic. (Here, (b) is almost certainly false; see Earman(1986),ch. XI). Even if we now are not, we may in future be in aposition to mount a credible argument for or against determinism onthe grounds of features we think we know the Final Theory musthave.