Relation [of powers] to substitution systems Despite the uniform distribution result in the note above, the sequence Floor[(n + 1) h] - Floor[n h] is definitely not completely random, and can in fact be generated by a sequence of substitution rules.

And with random initial conditions, I found more complicated behavior.

If one ignores the first component of the spectrum the remainder is flat for a constant sequence, or for a random sequence in the limit of infinite length.

Typically the network topology of a foam continually rearranges itself through cascades of seemingly random T1 processes (rule (b) from page 511 ), with regions that reach zero size disappearing through T2 processes (reversed rule (a)).

In the continued fraction for a randomly chosen number, the probability to find a term of size s is Log[2, (1 + 1/s)/(1 + 1/(s + 1))] , so that the probability of getting a 1 is about 41.50%, and the probability of getting a large term falls off like 1/s 2 . … Note that although the usual continued fraction for π looks quite random, modified forms such as 4/(Fold[(#2/#1 + 2)&, 2, Reverse[Range[1, n, 2] 2 ]] - 1) can be very regular. … In analogy to digits in a concatenation sequence the terms in the sequence Flatten[Table[Rest[ContinuedFraction[a/b]], {b, 2, n}, {a, b - 1}]] are known to occur with the same frequencies as they would in the continued fraction representation for a randomly chosen number.

But in many frequency bands one hears instead either very regular or seemingly quite random signals. … When there are seemingly random signals some arise say from transmission of analog video (though this typically has very rigid overall structure associated with successive lines and frames), but most are now associated with digital data.

Intrinsic randomness generation always tends to lead to a certain uniformity in networks. … And in fact many kinds of causal networks—say associated with early randomly connected space networks—will inevitably yield common ancestors for distant parts of the universe.

But even with m = {1, 0, 1, 0}/2 the method generates fairly random patterns, as in the second row below.

(This sequence is qualitatively not unlike the initiation of randomness in turbulent fluid flow and many other systems.)

If we then assume perfect underlying randomness, the density at a particular position must be given in terms of the densities at neighboring positions on the previous step by f[x, t + dt]  p 1 f[x - dx, t] + p 2 f[x, t] + p 3 f[x + dx, t] Density conservation implies that p 1 + p 2 + p 3  1 , while left-right symmetry implies p 1  p 3 .