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So even though the total of the energy of all particles remains the same, the distribution of this energy becomes progressively more random, just as the usual Second Law implies.
With black and white interpreted as True and False , the forms of operators shown here correspond respectively to And , Equal , Implies and Nand .
But the Principle of Computational Equivalence implies that in the end essentially any method of perception and analysis that can actually be implemented in our universe must have a certain computational equivalence, and must therefore at least in some respects come to the same absolute conclusions.
(Euler's theorem implies that at any stage there must always be exactly 12 pentagonal faces.)
But such equilibrium implies uniformity, and from this it follows that the initial conditions for the gravitational forces in the universe must have been highly regular, resulting in simple overall expansion, rather than random expansion in some places and contraction in others.
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 .
Note also that as discussed in the main text having maximal entropy does not by any means imply perfect randomness.
For it implies that even at the smallest scales the laws of physics will show the same computational sophistication that we normally associate with intelligence.
Implicational calculus
With k = 2 the operator can be either 2 or 11 ( Implies ), with k = 3 {2694, 9337, 15980} , and with k = 4 any of 16 possibilities.
If k is irrational then equidistribution of Mod[Binomial[t, x], k] implies that all possible values eventually appear; the corresponding patterns seem fairly irregular, as shown below.