Equivalence between microcanonical methods for lattice models

TL;DR

This paper investigates whether microcanonical averages can be accurately estimated using grand-canonical relations in finite lattice systems, finding that the methods are effective even for small system sizes.

Contribution

The study demonstrates the equivalence of microcanonical and grand-canonical methods for finite systems across different phase transitions, providing a simple and precise approach.

Findings

Equivalence holds for small system sizes.

Microcanonical quantities can be accurately estimated from grand-canonical relations.

Method is effective for systems with first and second order phase transitions.

Abstract

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.