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Chaos Theory and Randomness from Initial Conditions…perfect randomness. … For to know in any real way why the motion of the boat is random, we must inevitably ask more about the randomness of the ocean surface. … But there is still randomness in the initial conditions, and the point is that as the system evolves, it samples more and more of this randomness, and as a result produces behavior that is correspondingly random.
The Intrinsic Generation of Randomness…investigate what happens if at every step one randomly perturbs the gray level of each cell by a small amount. … But the crucial point is that even when the behavior involves intrinsic randomness generation, perturbations of at least some size can still be tolerated.
Chaos Theory and Randomness from Initial Conditions…And in the first case on the facing page , it so happens that the sequence of digits for each of the initial points shown is indeed quite random, so the behavior we see is correspondingly random. … But in order for the behavior of the system to be random, it is necessary in addition that the sequence of digits be random. And indeed, the whole idea of the mechanism for randomness in this section is precisely that any randomness we see must come from randomness in the initial conditions for the system we are looking at.
Chaos Theory and Randomness from Initial Conditions…the bumps, and if there is randomness in these bumps it leads to corresponding randomness in the motion of the car. … But where does this randomness come from? … It is such sensitivity to randomness in the initial conditions that makes processes such as rolling dice or tossing coins yield seemingly random output.
There is considerable randomness in the patterns produced in each case. … So just how complex can the behavior of a cellular automaton that starts from random initial conditions be? … The initial conditions used are again completely random.
Chaos Theory and Randomness from Initial Conditions…And indeed on successive trials there will usually be sufficiently large random variations in the initial speed that the outcomes will seem completely random. … And in each case the basic mechanism that leads to the randomness we see is a sensitive dependence on randomness that is present in the typical initial conditions that are provided. Without randomness in the initial conditions, however, there is no randomness in the output from these systems.
My purpose in this chapter is to go to the opposite extreme, and to consider completely random initial conditions, in which, for example, every cell is chosen to be black or white at random. One might think that starting from such randomness no order would ever emerge. But in fact what we will find in this chapter is that many systems spontaneously tend to organize themselves, so that even with completely random initial conditions they end up producing behavior that has many features that are not at all random.
Note (a) for Randomness in Class 3 Systems…Probabilistic estimates [of cellular automaton properties] One way to get estimates for density and other properties of class 3 cellular automata is to make the assumption that the color of each cell at each step is completely random.
Starting from random initial conditions, these cellular automata again quickly settle down to stable states. … A cellular automaton that evolves to a simple uniform state when started from any random initial condition. … Four more examples of cellular automata that evolve from random initial conditions to completely uniform states.
Chaos Theory and Randomness from Initial Conditions…And indeed to assume that it does is effectively just to ignore the fundamental question of where randomness in nature comes from. … And often such effects will tend to introduce new randomness from the environment. So the idea that randomness comes purely from initial conditions can be realistic only for a fairly small number of steps; randomness which is seen after that must therefore typically be attributed to other mechanisms.
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