But even in cases where the behavior obtained with a particular random initial condition is very complicated the structure of the attractor is almost invariably quite simple.

Note (c) for Chaos Theory and Randomness from Initial Conditions…In 1860 James Clerk Maxwell discussed how collisions between hard sphere molecules could lead to progressive amplification of small changes and yield microscopic randomness in gases. … Work by Robert Shaw in the late 1970s clarified connections between information content of initial conditions and apparent randomness of behavior. … So in 1985 when I raised the possibility that intrinsic randomness might instead be a key phenomenon this was greeted with much hostility by some younger proponents of chaos theory.

Note (d) for Chaos Theory and Randomness from Initial Conditions…Spinning and tossing [as sources of randomness] Starting with speed v , the speed of the ball at time t is simply v - a t , where a is the deceleration produced by friction.

With typical random initial conditions the most common structures to occur are: The next most common moving structure is the so-called "spaceship": The complete set of structures with less than 8 black cells that remain unchanged at every step in the evolution are: More complicated repetitive and moving structures are shown in the pictures below.

But it turns out that if one picks a number at random subject only to the constraint that its size be in a certain range, then it is overwhelmingly likely that the number one gets will have a digit sequence that is essentially random. … But as I have discussed above, the only randomness that can actually come out of such a system is randomness that was explicitly put in through the details of its initial conditions. … For what it says is that the randomness we see somehow comes from randomness that is already present—but it does not explain where that randomness comes from.

And in particular what this means is that randomness which comes from the mechanism of intrinsic randomness generation discussed in the previous section is able to make systems with discrete components behave in seemingly continuous ways. … There is no randomness in the rules or the initial conditions for this system. But through the mechanism of intrinsic randomness generation, the behavior of the system exhibits considerable randomness.

continuous form, and shows no trace of the underlying discreteness of the system; the randomness has in a sense successfully washed out essentially all the microscopic details of the system. … The distribution of positions by reached particles that follow random walks. The top left shows three individual examples of random walks, in which each particle randomly moves one position to the left or right.

effect on overall patterns of flow, one cannot realistically attribute the large-scale randomness that one sees in a turbulent fluid to randomness that exists at the level of individual particles. Instead, what seems to be happening is that intrinsic randomness generation occurs directly at the level of large-scale fluid motion. … And although many details are different from what one sees in real fluids, the overall mixture of regularity and randomness is strikingly similar.

So are there mobile automata in which the motion of the active cell is also seemingly random? … A mobile automaton that yields a pattern with seemingly random features. … But when viewed in compressed form, as on the left, the overall pattern of colors seems in many respects random.

particles in physics one must consider structures on much more complicated and random backgrounds. … Yet at first one might think that such randomness would inevitably disrupt any kind of definite persistent structure. … It turns out that here again it is possible to get definite structures that persist even in the presence of randomness.