Do Liouville's and Poincaré's theorems both hold in the canonical ensemble?

[Heirot]

If we have a system for which the Liouville's tm holds, can we automaticly say the Poincare's recurrence tm also holds? Presumably this is true in microcanonical ansable, but how about canonical, where the energy isn't constant?

 

[Andrew Mason]

If we have a system for which the Liouville's tm holds, can we automaticly say the Poincare's recurrence tm also holds? Presumably this is true in microcanonical ansable, but how about canonical, where the energy isn't constant?

Poincaré's recurrence theorem holds for any statistical system in which system's energy and the system's space do not change. It says, essentially, that such a system will return to a microstate that is to within an arbitrarily close approximation of its original microstate, if it is given enough time. So if energy is not conserved in the system, Poincaré's recurrence theorem does not hold.

AM