The energy at any later time should be exactly equal to the initial energy, not merely equal on average. So why is the Boltzmann distribution subject to only a constraint on expected energy?
I mean, that’s just the distinction between microcanonical and canonical ensemble, no? In the canonical ensemble, there is some outside thermal bath with which the system can exchange energy. In the microcanonical one, there is indeed no exchange of energy, and you get exactly a distribution constrained on total energy.
I mean, that’s just the distinction between microcanonical and canonical ensemble, no? In the canonical ensemble, there is some outside thermal bath with which the system can exchange energy. In the microcanonical one, there is indeed no exchange of energy, and you get exactly a distribution constrained on total energy.