Todd Rowland
Wolfram Research
Maryland
Registered: Oct 2003
Posts: 113 
Wasn't able to view the attachment (is it a special excel file?), so sorry about responding before seeing.
One thing about the rule being reversible is that one can take the last two steps, reverse their order and use them as an initial condition. The rules of this type are selfreversible so you can use 37R to make the reverse evolution of the original.
If the original had entropy increasing, the reversed version has entropy decreasing.
Because high entropy states outnumber low entropy states, when a rule has the behavior of changing entropy it has to increase entropy most of the time. (i.e. one can't break the theorems of information theory with this approach). So almost always one will see entropy follow the second law.
There are some technicalities about the right background for the infinite case, but for the finite case one is theoretically always going to cycle.
Last year I gave a talk on 37R at the NKS conference and there are some tools in those materials.
You can see pictures of typical evolutions from highly ordered states in the book, e.g., p.454. See also the discussion there about how this helps us understand the second law.
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