Wolfram Science Group
Phoenix, AZ USA
Registered: Aug 2003
Sure, the most relevant sections to start thinking about it are the first few sections of chapter 9. Section 4 in particular deals with conserved quantities in NKS style systems, using CAs to illustrate. Rather than arising from difference relations as constants of integration, in NKS systems conservation is typically a built in feature of the evolution rule. In that case of Wolfram's network idea, any topological property left unchanged by the replacement rule used, would be an obvious potential source of such conserved quantities.
There are other potential connections. For instance, the prior assumption of micro randomness used to justify phase averages in statistical mechanics can be thought of as justified by the prevalence of class 3 behavior (and with it instrinsic randomness generation) in rules beyond a modest level of internal complexity.
One can throw the standard thermodynamics and statistical mechanics categories at CAs and see what it says about them. Wolfram did so in one of his early CA papers back in the 80s, "statistical mechanics of cellular automata". You can find that paper here -
There was also a fair amount of work later in the 80s and in the early 90s by others, some at Sante Fe in particular, making connections to dynamical systems theory, also looking at stochastic CAs and random fields in thermodynamic terms. Entropy measures are often used to characterize or filter CAs. Wuensche for example noticed that class 3 rules show maximal but typically unvarying entropy measure, simple rules low entropy measure, while 4s show high variance in the entropy, from middling values (corresponding to regions of stable periodic background e.g.) to high ones and back.
It is a reasonable area to work and plenty remains to be done.
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