Application of Mayer’s activity expansions to the Ising problem

Abstract

Recent successes in approximate evaluations of high-order cluster integrals have greatly advanced the quantitative applicability of Mayer’s cluster expansion for various statistical models of matter and even real substances. In this paper, the thermodynamics of the Ising model is represented in a general form of the virial series in powers of activity, which is not restricted by any special geometry or the simplification of nearest-neighbor interactions. In order to make the presented theoretical expressions applicable in quantitative studies, a number of techniques are considered, which approximate the unlimited (almost infinite) set of reducible cluster integrals, bn, based on only a few number of irreducible integrals, βk. Namely, such short βk sets are defined here for a number of magnetic models of various geometry, dimensionality, as well as interaction ranges (not limited by the nearest coordination sphere), and the calculations performed for these models in subcritical regimes (below the Curie point) indicate the adequacy of the proposed approach in general and good agreement with the known exact solution in a particular case of the two-dimensional square model with nearest-neighbor interactions. In addition, the quantitative similarity of subcritical magnetization curves is observed for all the considered models at the same values of a certain reduced temperature and the nature of this similarity may deserve further attentive studying.