The Poincarè recurrence theorem doesn’t imply that. It doesn’t imply the system is ergodic, and it only applies to “almost all” states (the exceptions are guaranteed to have measure zero, but then again, so is the set of all numbers anyone will ever specifically think about). In any case, the entropy doesn’t change at all because it’s a property of an abstraction.
An ideal gas in a box is an egodic system. The Poincarè recurrence theorem states that a volume preserving dynamical system (i.e. any conservative system in classical physics) returns infinitely often in any neighbourhood (as small as you want) of any point of the phase space.
The Poincarè recurrence theorem doesn’t imply that. It doesn’t imply the system is ergodic, and it only applies to “almost all” states (the exceptions are guaranteed to have measure zero, but then again, so is the set of all numbers anyone will ever specifically think about). In any case, the entropy doesn’t change at all because it’s a property of an abstraction.
An ideal gas in a box is an egodic system. The Poincarè recurrence theorem states that a volume preserving dynamical system (i.e. any conservative system in classical physics) returns infinitely often in any neighbourhood (as small as you want) of any point of the phase space.