Face-to-diagonal reduction of Kramers-Wannier approximation for cubic lattice particle systems with nearest neighbour exclusion

TL;DR

This paper develops a face-to-diagonal reduction of the Kramers-Wannier approximation within the Cluster Variation Method for cubic lattice particle systems with nearest neighbour exclusion, enabling more efficient thermodynamic modeling of complex disordered media.

Contribution

It introduces a novel face-to-diagonal reduction of the Kramers-Wannier entropy approximation for NNE systems, extending the applicability of CVM to more complex configurations.

Findings

The reduction simplifies entropy calculations for face clusters.

Application potential for modeling disordered materials like liquid silicates.

Discussion of numerical implementation challenges and solutions.

Abstract

The paper is concerned with interacting particle systems on the simple cubic lattice obeying the nearest neighbour exclusion (NNE). This constraint forbids any two neighbouring sites of the lattice to be simultaneously occupied, thus reducing the set of admissible configurations for the cubic cell and its subclusters such as edges and faces. This reduction extends applicability of Kikuchi's Cluster Variation Method (CVM) with higher-order clusters to systems with complex site configurations and short-range ordering, which would be impractical beyond the NNE framework because of the "curse of dimensionality". For edges of the cubic cell, which are the operational clusters of the Bethe-Peierls entropy approximation in the CVM hierarchy, the edge-to-site reduction of the entropy cumulants was studied previously. In extending the earlier results, we develop a face-to-diagonal reduction of the Kramers-Wannier entropy approximation of the CVM in the NNE setting. We also outline an application of the resulting approximation to thermodynamic modeling of disordered condensed media, such as liquid silicates, and discuss combinatorial and numerical aspects of the implementation of this approach.