Wolfram Science Group
Phoenix, AZ USA
Registered: Aug 2003
(1) Well it can't "terminate", though it can repeat. Since it is reversible, if it is "alive" in the direction that gets it to step n, it is "alive" in the direction that keeps going beyond step n. So it doesn't terminate. But it can "close", by revisiting a pair of successive steps that it has hit before. Then it necessarily goes around in a cycle, because it depends only on what happened on the previous 2 steps.
You can easily get it to repeat from a small initial condition, especially one with lots of divisors to allow subcycles. This follows from the "class 2 behavior in systems of limited size" argument, with the only wrinkle being that now one has two time steps rather than one that have to be the same, to have the same future. E.g. with a grid only 6 cells wide and initial condition
- for the previous step and "start" step, you get a cycle with period 24 steps. You can probably find special initials that close as fast as divisor related subcycles allow, if you search for and use the best (few) of the 2^12 initial conditions.
But if you e.g. take a random initial condition 137 cells wide - prime, so there aren't any subcycles to hit randomly - then you have something like 2^274 possible states. You can check for a cycle by scanning for two successive lines that have occurred before. From a random initial, I get none out to 100,000 steps. A million steps runs me out of memory without finding a repetition.
On (2), you can have a trillion cycles within the Planck time and it won't make any appreciable difference, unless the system is limited in width, or you specifically engineer a special initial that cycles. The combinatorial explosion is exponential in the system size. And typically when we are talking about second law processes we are talking about mole numbers of elements or something like it. Do whatever you like on the time side and you won't get within spitting distance... You are only going to see this sort of system "close" for special initial conditions or very limited system sizes.
There are simply vastly more possible arrangements than there are elements - that is the underlying driver. In realistic times, you are only going to visit a tiny subset of those possible arrangements. Instantiated reality is measure zero in overall possibility space, so you can be on transients "forever" - that is the moral. The question is therefore what typical transients look like, not what infinite time behavior looks like. (As Wolfram is fond of pointing out, the universe to date is a transient; ergo dynamics matter.)
I hope this helps.
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