(I am not sure if this is a mathematics or physics question so I am not sure where to post it. I am posting it here because the chief subject is an unreal universe that is purely a subject of mathematical theoretical analysis, but there are quite obvious and connections to the study of the real-life physics in our own universe.)
This is something I have wondered about. Conway's game of life provides a fictional universe where that it seems that many highly complex behaviors are possible (including what could arguably be considered to some extent a form of life, i.e. self-replicating information processing systems) but objects are also subject interestingly to extreme instability and rapid entropization, in that a small perturbation of an ordered system, such as the addition or removal of a single "cell" or filled grid square, will cause rapid disintegration of that system.
And this naturally invites comparisons with our own real-life universe: in this one, fortuitously, vanishingly small perturbations of systems do not generally lead to catastrophic entropy increases, but there is still a tendency toward increasing entropy in some way, and it seems thus that the CGoL universe could be thought of as perhaps in some regards having a "more aggressive" version of the 2nd law of thermodynamics.
However, in other regards, it appears to strongly violate the laws of thermodynamics that work in our universe: machines or life forms generally will run forever without any changes, and you can even create infinite streams of "matter" such as with a glider gun. It would seem that, in particular, energy is not conserved, and what we call as "perpetual motion" is possible in the CGoL universe but is not possible in our own.
But I wonder why this is, from a viewpoint of the mathematical structures of the two, in particular regarding the constraints on perpetual motion we know of in our universe. In particular, the typical retort as to why a perpetual motion machine is impossible is some variation on the first and second laws of thermodynamics (not sure what a "third law of thermodynamics" machine would be - I'd presume that's a machine that could refrigerate something to exactly absolute zero, then obtain 100% efficiency by using it as a cold bath). The typical reason it is asserted the first law is inviolate is Noether's theorem: dynamics is symmetric in temporal translation, meaning that, for a given configuration of particles, their future history does not depend on whether that configuration is created now or created (say) ten thousand years from now.
But Conway's universe also has this temporal translation symmetry property. The rules have no explicit time dependence. They are a discrete-time dynamical system (DTDS), sure, but the temporal symmetry group is maximal, so arbitrary time translations can be accessed (a counterexample would be a universe where that behavior is one way for even generation numbers and another for odd generation numbers), and thus you can propagate a quantity along a streamline in phase space, so I am not sure why that the same ways you could argue for Noether in our universe wouldn't still go through for the most part.
Is this correct? Does the temporal translation symmetry of Conway's universe give rise to a conserved quantity, that we might be able to call an "energy"? If so, how does or does not the ready and easy appearance of perpetual motion machines jive with its conservation? Moreover, could this also imply that if, say, hypothetically and someday a loophole were found in our universe that could permit what anyone else might call as perpetual motion, it would not necessarily imply a Noether temporal symmetry failure, but perhaps just a suitable redefinition or expansion of the idea of energy? And if not, why not?
By the way, here are some observations on what a possible "energy" function might have to look like.
- One interesting property of Conway patterns is that they can not only expand perpetually, as in the glider gun, but they can also disappear completely. If we are to assume a conserved energy functional for a particular pattern, it would stand to reason that any pattern that eventually dies completely must have energy equal to that of the vacuum (presumably, we could just set this to 0 conways).
- It would be nice if, at least under some suitable well-defined circumstances, energies are additive, i.e. if we put two patterns next to each other on a suitable grid and they do not interact with each other, the total energy in the grid should be equal to the sum of energies of the two patterns. Note that non-interactivity is vital: we could imagine a couple of patterns that, by themselves, have a positive energy, but when suitably composed even if not immediately overlapping, would result in a pattern that disappears completely, and thus the composition must have energy equal to the vacuum energy.
- Presumably, the smallest stable pattern, which is 3 cells in a horizontal or vertical bar (technically it's not "stable" in the strictest sense because it oscillates between horizontal and vertical orientation, but I'd call it stable because it never dies and moreover it maintains its shape), should have the least positive energy, but it's also possible this may contend with the square (4 cells) because, while it has one more cell, it doesn't move.
Does such an energy function exist? If so, but it is not unique, what further conditions could potentially single out a unique one? If it does not exist, which conditions should we relax and/or replace, and with what? And what would such energy functions suggest when applied to the situations that would seem like violations of conservation by the standards of the real-life universe, like the glider gun? Note that I'd also personally be inclined to think the above conditions are too restrictive in some ways because it's ostensibly still trying to assume or salvage a connection between energy and cell count and if anything we should take the existence of guns, patterns with changing cell count, and so forth as a big red flag suggesting "don't even bother" with that approach. However, then we need a different strategy for trying to find useful axioms.
(Note, clearly there's always a trivial energy function, given by assigning the same value to every pattern. That's also clearly not what we want, but I'm not sure how to eliminate it.)
It's also important to note the many differences between the symmetries of the Conway universe's dynamics and those of what we know so far about the dynamics of our own. For one, the Conway universe's symmetries are discrete, as I mentioned above. That may, alone, be enough to provide complete explaining power as to how that time translation can sit alongside infinite glider guns, but I am not sure. For another, some key symmetries that are present in ours are lacking: for one, rotational symmetry is not present except in the simple four-fold way. For another, boost symmetry is not present for any reasonable notion of boost - that is to say, the Conway universe admits a preferred, absolute rest frame: namely, the one in which a 2x2 square stays put. It seems to me any of these might frustrate or at least require a radical rethinking of what "energy" would have to mean. Perhaps these differences wholly and irrevocably sabotage the idea of energy?