Equivalence between microcanonical methods for lattice models

The development of reliable methods for estimating microcanonical averages constitutes an important issue in statistical mechanics. One possibility consists of calculating a given microcanonical quantity by means of typical relations in the grand-canonical ensemble. But given that distinct ensembles are equivalent only at the thermodynamic limit, a natural question is if finite size effects would prevent such procedure. In this work we investigate thoroughly this query in different systems yielding first and second order phase transitions. Our study is carried out from the direct comparison with the thermodynamic relation $(\frac{\partial s}{\partial e})$, where the entropy is obtained from the density of states. A systematic analysis for finite sizes is undertaken. We find that, although results become inequivalent for extreme low system sizes, the equivalence holds true for rather small $L$'s. Therefore direct, simple (when compared with other well established approaches) and very precise microcanonical quantities can be obtained from the proposed method.