Liouville's theorem applies to systems where the phase space volume is conserved, but it does not guarantee that Poincaré's recurrence theorem will hold in all cases. Poincaré's recurrence theorem requires that the system's energy and space remain constant, indicating that it is applicable primarily to microcanonical ensembles. In canonical ensembles, where energy is not conserved, Poincaré's theorem does not hold. Therefore, the relationship between Liouville's theorem and Poincaré's theorem is conditional on the conservation of energy. Understanding these distinctions is crucial for analyzing the behavior of dynamical systems over time.
