So the two-dimensional reversible cellular automaton Critters (which you can simulate online at https://dmishin.github.io/js-revca/index.html#) on the Torus does not seem to follow the second law of thermodynamics. For example, if a $10\times 10$ block in the torus is filled with random data and the cells outside of this $10\times 10$ block are all zero, then there is a good chance that after several million generations on a $256\times 256$ grid, one can see the cellular automaton return to its original state. Why is the Poincare recurrence time for the reversible cellular automata Critters so small in this case? The rule Critters is universal for reversible computation, and it exhibits chaotic behavior that one would expect in a reversible cellular automaton. Therefore, one would expect that after running the cellular automaton for enough generations, the entropy would constantly increase and hence one would need to wait a very long time (much longer than several million generations) to observe the Poincare recurrence. Is there any explanation for this phenomenon? Are there other chaotic, reversible, universal cellular automata of dimension 2 that exhibit this phenomenon?
Edit: Please see the comments for why this does not address the question, however, I think I should not delete it in order to preserve the discussion...
Not an expert, but my impression is that reversible computation does not by itself result in a change in entropy: the amount of information in the system has not changed, only the format in which it is stored. In other words such processes obey the second law with only a weak inequality, not a strict one. To get a strict increase in entropy, one has to erase or destroy information, i.e. irreversible computation. I don't see why the rest of the details in your question -- two dimensions, cellular automata, chaos, recurrence -- would affect this.
Happy to be corrected by any experts, however.