The discussion centers on the relationship between Liouville's theorem and Poincaré's recurrence theorem in statistical mechanics. It is established that Poincaré's recurrence theorem applies to systems where both energy and spatial configuration remain constant, specifically in microcanonical ensembles. However, in canonical ensembles where energy is not conserved, Poincaré's recurrence theorem does not hold. This distinction is crucial for understanding the behavior of dynamical systems over time.
PREREQUISITESThis discussion is beneficial for physicists, mathematicians, and students of statistical mechanics who are exploring the foundational theorems that govern dynamical systems and entropy. It is particularly relevant for those studying thermodynamics and chaos theory.
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.Heirot said: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?