Modified Representation of Canonical Average by Special Microscopic States for Classical Discrete Systems

TL;DR

This paper introduces a modified method for representing the canonical average in classical discrete systems using special microscopic states called projection states, improving accuracy and efficiency in predicting thermodynamic properties.

Contribution

It proposes a new representation of the canonical average based on transformation of moment matrices and Pade approximation, which enhances the accuracy over traditional methods.

Findings

Transformation of moment matrices improves representation accuracy.

Pade approximation offers better results except near singular points.

The method is validated through hypersphere integration.

Abstract

For substitutional crystalline solids typically referred to classical discrete system under constant composition, macroscopic structure in thermodynamically equilibrium state can be typically obtained through canonical average, where a set of microscopic structure dominantly contributing to the average should depend on temperature and many-body interaction through Boltzmann factor, exp(-bE). Despite these facts, our recent study reveals that based on configurational geometry, a few specially-selected microscopic structure (called projection state PS) independent of temperature and many-body interaction can reasonably characterize temperature dependence of macroscopic structure. Here we further modify representation of canonical average by using the same PSs, based on (i) transformation of multivariate 3-order moment matrix by one of the PS, and (ii) Pade approximation. We prove that the former can always results in better representation of canonical average than non-transformation one, confirmed by performing hypershere integration, while the latter approximation can provide better representaion except for e.g., inclusion of its own singular point within considered temperature, which can be known a priori.