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But with an appropriate collection of tricks, it is in the end feasible to take almost any system of the type discussed here, and determine what pattern, if any, satisfies its constraint.
The issue with this mechanism, however, is that it can take a long time to get a given amount of good-quality randomness from it.
The answer, as illustrated on the next page , is that if there are enough particles, then the distribution one sees takes on a smooth and
If one takes some water and continuously increases its temperature, then for a while nothing much happens.
But if one just tries pushing balls together they almost always get stuck, and never take on anything like the arrangement shown.
The result is that in practice it is never possible to build perpetual motion machines that continually take energy in the form of heat—or randomized particle motions—and convert it into useful mechanical work.
What will the events associated with the second particle look like if one takes slices through the causal network so that the first particle appears to be at rest?
So if such processes can correspond to the evolution of systems like cellular automata, then it follows at least formally that differential equations should be able to do in finite time computations that would take a discrete system like a cellular automaton an infinite time to do.
The cellular automaton takes 2t 2 +t steps to emulate t steps of evolution in the Turing machine.
And so, for example, I suspect that it does not take a cellular automaton nearly as complicated as the one on page 767 for it to be an NP-complete problem to determine whether initial conditions exist that lead to particular behavior.