Equivalence of two parallel approaches to the cluster variation method: the multisite correlation functions method and the cluster effective fields method

We have analyzed relationship between two parallel approaches to the cluster variation method (CVM) as a theoretical tool for studying statistical properties of binary systems. The first approach to the method uses the multisite correlation functions as variational variables, while the other approach is the one that uses the cluster effective fields as variational variables. Using the Ising model description of physical systems studied, we have shown that the two approaches should produce identical final results, although they deal with quite different systems of nonlinear equations (which, in particular cases under study, must be solved numerically for a given temperature and chemical potential). To achieve identical final results, we show that it is necessary to introduce cluster fields for those clusters which appear to be subclusters of at least two different members of the Kikuchi cluster family. In addition, we demonstrate that variational variables of the two approaches generate two sets of cluster probabilities, whose intersection contains solution of the CVM approximation which corresponds to the thermodynamic equilibrium state. We also analyze the existence of the so-called consistency relations in both approaches to the CVM method, and, finally, we discuss the problem of convergency of numerical procedures that are used to analyze the low-temperature states of the model systems under study