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

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

• To me this question - asking for the physical interpretation of a mathematical result - is more physics than math; but anyways I have routinely seen ergodicity (including Poincaré’s work) contrasted with KAM theory (which would say something "opposite": that the system retains some knowledge of its initial conditions). For example I believe Cédric Villani has given many talks with this theme. But probably others who are experts in mathematical physics can say more. Aug 28, 2022 at 17:25
• thanks for the comment. I see your point, but i believe interpretations of mathematical physics theorems should be seen as part of mathematical physics, hence of mathematics. Hence an answer to this question would be of interest for a mathematical audience. But someone already downvoted this question, so maybe people share your view. Aug 28, 2022 at 17:59
• The Poincare recurrence time for a macroscopic gas is on the order of something like $2^{10^{23}}$, a completely unphysical number that physicists don't care about, and much larger than the expected lifetime of the universe. It's like arguing that the central limit theorem can technically fail with some tiny probability for a large but finite number of independent samples; that's true but with sufficiently many observations it's vanishingly unlikely. Aug 28, 2022 at 18:31
• @QiaochuYuan I don’t think it’s the same as for the central limit theorem. To me what you say is just that we cannot ever experience nor observe the example of the gas, which of course I agree with. But the rest of what you say is quite speculative: I may be wrong but i don’t think in physics there is a clear consensus about the “expected lifetime” of the universe, let alone what would it mean for it to “end”, and it does not exclude a recurrence phenomen. Aug 28, 2022 at 19:11
• It does not really answer the question, but here is what Arnold had to say, in "Mathematical methods of classical mechanics": "The following prediction is a paradoxical conclusion from the theorems of Poincare and Liouville: if you open a partition separating a chamber containing gas and a chamber with a vacuum, then after a while the gas molecules will again collect in the first chamber[...] The resolution of the paradox lies in the fact that 'a while' may be longer than the duration of the solar system's existence."
– user164898
Aug 28, 2022 at 19:31

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.

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).

• this is a great answer! thank you. Especially the first paper is very close to what I was asking. And I am happy to hear that my impression was wrong and that this topic is still one of active research for physicists Aug 29, 2022 at 8:53
• Do black holes only grow if they are colder than the surrounding space? What does this mean exactly? Aug 29, 2022 at 19:13
• @bananenheld --- Following Hawking, one can assign to a black hole a radiation temperature, inversely proportional to the black hole's mass. The cosmic background radiation (CMB) has a temperature of 2.7 K. A stellar-mass black hole starts out colder than the CMB, it will absorb CMB radiation, become heavier and then even colder. Converseluy, f the black hole starts out hotter than the CMB, it will emit radiation, become lighter and then hotter, radiating faster until it disappears. Aug 29, 2022 at 20:17

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.