Equivalence of microcanonical and canonical ensemble

There are many partition functions, but how are their results related? If the usual formal derivation, e.g., of the canonical formalism from the microcanonical formalism, works as designed, all ensembles should give us the same thermodynamics; thermodynamic potentials corresponding to various partition functions are related mutually by Legendre transformations. Therefore, for ordinary particle-particle interactions (e.g., not long-ranged and with a sufficiently hard core), in the thermodynamic limit all the partition functions are expected to be equivalent in the sense that all give identical thermodynamic densities and fields. Here, let us demonstrate this ensemble equivalence of microcanonical and canonical ensembles.