Canonical and microcanonical partition functions in long-range systems with order parameter space of arbitrary dimension

We consider systems in which the canonical partition function can be expressed as the integral in an n-dimensional space (the order parameter space) of a function that also depends parametrically on the number N of degrees of freedom and on the inverse temperature β. We show how to compute, together with the canonical entropy and specific heat, also the corresponding microcanonical quantities, generalizing in this way some results already in the literature for the case n=1. From the expressions that are obtained it is possible to derive the necessary conditions for the equivalence of the canonical and of the microcanonical ensembles. We finally study a simple model, with a two-dimensional order parameter space, in which ensemble inequivalence is realized.