The Poincaré recurrence theorem states that, for a bound phase space, the system will return to a state very close to the initial conditions, in some finite time $\tau$.

For example, let's say I have an atom in a $E \propto k^2$ dispersion relation parabola, but its total energy makes only the three lowest states accessible.
The "frequencies" in the system are therefore (normalised to some unit) $1, 4,$ and $9$.

I can find the least common multiple $1\cdot 4\cdot 9 = 36$ to the "frequency" after which I recover the initial state, since $36$ contains an integer multiple of $1$, of $4$ and of $9$.

What if I had irrational numbers though? (or in general incommensurate ratios)
So like $1$ and $\sqrt{2}$. The least common multiple would be $\sqrt{2}$, after such time I have not gone through an integer multiple of the $1$ frequency... so I cannot be back in the initial state.

Can I still find a Poincaré time/frequency for incommensurate frequencies?

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    $\begingroup$ Well, you answered the question yourself in the first sentence. The theorem only says that you'll get to a state close to the initial one. $\endgroup$ – knzhou Jan 14 at 14:17

What if I had irrational numbers though? [...] I cannot be back in the initial state.

As knzhou already commented, the theorem states you should be able to get arbitrarily close to the initial state, but not necessarily exactly on top of it.

So, in the situation you describe, you'd need better and better rational approximations of the irrational number, the closer you'd require the system to get to its initial state.


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