# Poincaré recurrence theorem with irrational frequencies?

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?

• 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. – knzhou Jan 14 at 14:17