First-order canonical phase transition as origin of the non-equivalence of the microcanonical and canonical ensembles in the thermodynamic limit

By using a simple argument, the cause of the thermodynamic non-equivalence of the canonical and microcanonical ensembles is clarified to be the first-order canonical phase transition in the thermodynamic limit. There is ensemble equivalence only when the energy fluctuation is negligible with respect to the average energy. The energy fluctuation does not vanish during the first-order transition even in the thermodynamic limit, so that the ensemble non-equivalence cannot hold anymore. Moreover, similar results can also be derived for other dual ensembles.