So what this means is that the randomness we observe in fluid flow cannot simply be a reflection of randomness that is inserted through the details of initial conditions. … Yet despite this, there is still apparent randomness in the overall behavior that is seen. … In the pictures on page 378 considerable randomness was already evident at the level of individual particles.

One aspect of the generation of randomness that we have noted several times in earlier chapters is that once significant randomness has been produced in a system, the overall properties of that system tend to become largely independent of the details of its initial conditions. … But whenever randomness is produced the overall patterns that are obtained look in the end almost indistinguishable. … So this means that if a system generates sufficient randomness, one can think of it as evolving towards a unique equilibrium whose properties are for practical purposes independent of its initial conditions.

But other numbers, even though they may be the result of simple mathematical operations, tend to have seemingly random forms. … Square roots yield repetitive sequences in this representation, but cube roots and higher roots yield seemingly random sequences.

But as the picture below shows, the detailed configuration of these cells changes rapidly in a seemingly random way. And just as in the other systems we have discussed, what then emerges on average from all these small-scale random changes is overall behavior that again seems in many ways smooth and continuous. … The boundary of the domain exhibits seemingly random fluctuations.

But at the speed used in the pictures on the next page , the array of eddies has begun to show random irregularities just like those associated with turbulence in real fluids. So where does this randomness come from? In the past couple of decades it has come to be widely believed that randomness in turbulent fluids must somehow be associated with

If a sequence is basically random but contains some short-range correlations then these will lead to smooth variations in the spectrum. And for example sequences that consist of random successions of specific blocks can yield any of the types of spectra shown below—and can sound variously like hisses, growls or gurgles. … But so far as I can Frequency spectra for long sequences obtained by concatenating blocks in random orders.

The position of this cell is chosen entirely at random, with the only constraint being that it should be adjacent to an existing cell in the cluster. … Unlike for the case of random walks, there is as yet no known way to make a rigorous mathematical analysis of this process. But just as for random walks, it appears once again that the details of the underlying rules for the system do not have much effect on the main features of the behavior that is seen.

[Generating] random networks One way to generate the connections for a "completely random" trivalent network with n nodes is just to apply a random permutation: RandomNetwork[n_?… Properties of random networks are discussed on page 963 . A convenient way to get somewhat random planar networks is from 2D Voronoi diagrams of the kind discussed on page 987 .

And with these types of initial conditions, systems like the one on the previous page always tend to exhibit increasing randomness. … It is clear that by starting with a simple state and then tracing backwards through the actual evolution of a reversible system one can find initial conditions that will lead to decreasing randomness. … The consequence of this is that no reasonable experiment can ever involve setting up the kind of initial conditions that will lead to decreases in randomness, and that therefore all practical experiments will tend to show only increases in randomness.

A fundamental fact known since the mid-1800s is that heat is a form of energy associated with the random microscopic motions of large numbers of atoms or other particles. … But as time goes on, the motion that occurs becomes progressively more random. … And in the second case, for example, the presence of a small fixed obstacle leads to rapid randomization in the arrangement of particles—very much like the randomization we saw in the one-dimensional cellular automaton that we discussed earlier in this section .