Poincaré recurrence and its implications for statistical physics and the arrow of time

Question

A very important theorem in mathematical physics is Poincaré’s recurrence theorem.

As you recall, this theorem states that given a dynamical system $(M , \phi , \mu)$ with $ \mu M < +\infty$, for every $A$ measurable the set of points of $A$ for which it exists some $N>0$ such that $\phi^Na \notin A$ for $n \geq N$ has measure zero.

This theorem applies, for example, to Hamiltonian dynamics, hence it has non trivial physical implications. For example a gas expanding in an empty room and then going back to the initial position.

What I would like to ask is: what are the physical implications of Poincaré theorem especially in statistical physics?

Is it possibile to use Poincaré’s recurrence in order to argue for a sort of circularity of time? In particular, can the evolution of the universe from its initial state be seen as dynamical system satisfying the hypotheses of Poincaré’s recurrence theorem?

I am writing it here on mathoverflow, because i) I see no difference between mathematics and physics ii) I am under the (possibly wrong) impression that not many physicists care much about Poincaré recurrence, while mathematicians do.

edit I have found this physics stackexchange quesiton of 8 years ago which asks one part of my question https://physics.stackexchange.com/questions/94122/is-poincare-recurrence-relevant-to-our-universe

Answer 1

Since the question is about physical implications of Poincaré recurrence one should take both quantum effects and gravitational effects into consideration. Quantum mechanics does not spoil the recurrences, any finite quantum mechanical system evolves quasi-periodically (Wikipedia has a simple proof).

Gravity provides more complications, which are not fully resolved because we lack a quantum theory of gravity. Black holes, once formed, grow if they are colder than the surrounding space, and thereby preempt the recurrence. See Gravity can significantly modify classical and quantum Poincare recurrence theorems.

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As an aside, the impression of the OP "that not many physicists care much about Poincaré recurrence" is not quite true. It is difficult to observe this effect in the laboratory, but it is an active topic of research. A recent publication, Recurrences in an isolated quantum many-body system, has received much attention (see this news item).

Answer 2

The Poincare recurrence (or, more general, the ergodic theorem that says that a system will, over time, evolve through essentially all microscopic states that are consistent with the total energy, particle number, etc.) seems like it should be precisely the thing needed to justify the microcanonical ensemble—and, from there, all of statistical mechanics. However, it does not work in practice. To explain why, I cannot do better than quote Kerson Huang's Statistical Mechanics (second edition, pages 90–91):

The time interval between two large fluctuations is called a Poincaré cycle. A crude estimate... shows that a Poincaré cylcle is of the order of $e^{N}$, where $N$ is the total number of molecules in the system. Since $N\approx10^{23}$, a Poincaré cycle is extremely long. In fact, it is essentially the same number, be it $10^{10^{23}}$ s or $10^{10^{23}}$ ages of the universe, (the age of the universe being a mere $10^{10}$ years.) Thus it has nothing to do with physics.

We mentioned the ergodic theorem..., but did not use it as a basis for the microcanonical ensemble, even though, on the surface, it seems to be the justification we need. The reason is that the existing proofs of the theorem all share a characteristic of the proof of the Poincaré theorem..., i.e.,an avoidance of dynamics. For this reason, they cannot provide the true relaxation time for a system to reach local equilibrium, (typically about $10^{-15}$ s for real systems,) but have a characteristic time scale of the order of the Poincaré cycle. For this reason, the ergodic theorem has so far been in interesting mathematical exercise irrelevant to physics.