Ensemble equivalence for non-Boltzmannian distributions

We discuss the possibility of using generalized canonical distributions, i.e., using other factors than exp(-βE), in order to compute the equilibrium properties of physical systems. It will be shown that some other choices can, in certain cases, lead to a simpler calculation of those properties. The corresponding equivalence between the canonical and the generalized canonical distributions is derived using well-known principles of statistical mechanics and we exemplify the method by deriving in a simple way the equilibrium properties of the long-range Ising model.