Poincaré recurrence applicability condition?

  • #1
This is how Wikipedia summarizes the Poincaré Recurrence Theorem:
In mathematics, the Poincaré recurrence theorem states that certain systems will, after a sufficiently long time, return to a state very close to the initial state. The Poincaré recurrence time is the length of time elapsed until the recurrence. The result applies to physical systems in which energy is conserved.

This is wrong, isn't it? Don't you need to ensure the phase space is bounded, and isn't conservation of energy an insufficient justification for that? Like, imagine throwing two baseballs away from each other into infinite space at escape velocity or higher; surely energy is conserved, yet they'll never come back together.

That seems incredibly basic, so I apologize if I'm asking something really stupid here, but please check me on this.


Elsewhere, I've seen the theorem presented like this:
if the system has a fixed total energy that restricts its dynamics to bounded subsets of its phase space, the system will eventually return as closely as you like to any given initial set of molecular positions and velocities.


My question is this: How do you know if the dynamics are restricted to a bounded subset of phase space or not? What condition establishes that fact?
 

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  • #2
haruspex
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The piece you quote says "certain systems". If you read further down the Wikipedia article it brings in invariant phase space volume.
 

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