Entropy reversal in an infinite static universe?

[Suekdccia]

TL;DR Summary

Entropy reversal in an infinite static universe?

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 () 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 to one similar question on another physics-discussion site (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?

 

[Demystifier]

For the Poincare recurrence to exist, the essential requirement is that the phase space of all possible initial conditions has a finite volume. (In the quantum version, it translates to the finite dimension of the Hilbert space.) If there is a cosmological constant, then the universe has the horizon which makes it effectively finite, so the Poincare recurrence seems plausible. If there is no cosmological constant, and hence no horizon, the Poincare recurrence requires some other mechanism by which the universe has a finite volume. A reasonable possibility is that the universe is closed, i.e. that its spatial topology is a 3-dimensional sphere. I didn't watch the video, but I suspect that Susskind had such a closed universe in mind.