Entropy reversal in an infinite static universe?

Question

As far as I know, entropy could be reversed by the Poincaré recurrence theorem if it had a finite horizon given by some amount of vacuum energy causing an accelerating expansion.

However, I found this lecture by Leonard Susskind (https://youtu.be/n7eW-xPEvoQ?t=2324) where he tells a way through which the vacuum could decay into a vacuum state with no energy and therefore no expansion would occur. In this case, we would have a static universe. However, he says that in this case no recurrence would take place.

But, in this answer I got to one of my questions (Could any new structures be formed after the heat death of the universe?), it says that in a static universe with no accelerated expansion (and therefore no cosmological constant) the Poincarré recurrence theorem would hold. And also, I understand that in a non-accelerating expanding universe there would be no maximal entropy reached so the recurrence should occur.

So, what am I missing here?

Answer 1

Let me tell you my understanding of the issue. I caution that it might not sound very orthodox to some. But here is the deal. Poincare's recurrence theorem is a mathematically strict statement in Mechanics. On the other hand Second Law of Thermodynamics is an independent, standalone statement, which is more fundamental than Classical Mechanics. As such, one should not expect the latter to be derivable from the former.

In fact, it must be these kinds of thoughts that drove Boltzmann crazy. To remind, he was attempting to do precisely that: to derive the Second Law from Mechanics. And then, when Poincare came up with his theorem, with which no one could argue, some have thought that if entropy is reducible to pure mechanics, in some corners of the Universe the entropy must be decreasing. But since all existing astronomical evidence is attesting to the contrary, this became an important argument against purely mechanical nature of entropy. In the narrow sense this has proved Boltzmann wrong (not that entropy has to do with disorder, but that H-theorem is a actual theorem in mathematical sense), and ultimately drove him into depression with its infamous and sad ending.

To summarize: entropy will not go down under any circumstance. This is a postulate that summarizes the totality of empirical evidence collected by humankind so far (just like the First Law of Thermodynamics). And the fact that Poincare theorem says otherwise, just means that Classical Mechanics, upon which it is based, does not capture the entirety of phenomena in our Universe.