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 <f>n</f>-dimensional space (the order parameter space) of a function that also depends parametrically on the number <f>N</f> of degrees of freedom and on the inverse temperature <f>β</f>. 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 <f>n=1</f>. 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. [Copyright &y& Elsevier]