Complexity of representations of coefficients of power series in classical statistical mechanics. Their classification and complexity criteria

TL;DR

This paper introduces a classification and criteria for the complexity of representations of power series coefficients in classical statistical mechanics, aiming to simplify their estimation and comparison.

Contribution

It formulates a new classification of coefficient representations and develops criteria to compare their complexity, demonstrated through examples involving virial coefficients.

Findings

Ree-Hoover representations are compared with graph-based representations.

New criteria effectively distinguish between different representation complexities.

Results are summarized in tables with detailed comments.

Abstract

It is declared that the aim of simplifying representations of coefficients of power series of classical statistical mechanics is to simplify a process of obtaining estimates of the coefficients using their simplified representations. The aim of the article is: to formulate criteria for the complexity (from the above point of view) of these representations and to demonstrate their application by examples of comparing Ree-Hoover representations of virial coefficients and such representations of power series coefficients that are based on the conception of the frame classification of labeled graphs. To solve these problems, mathematical notions were introduced (such as a base product, a base integral, a base linear combination of integrals, a base linear combination of integrals with coefficients of negligible complexity, a base set of base linear combinations of integrals with coefficients of negligible complexity); and a classification of representations of coefficients of power series of classical statistical mechanics is proposed. In this classification the class of base linear combinations of integrals with coefficients of negligible complexity is the most important class. It includes the most well-known representations of the coefficients of power series of classical statistical mechanics. Three criteria are formulated to estimate the comparative complexity of base linear combinations of integrals with coefficients of negligible complexity and their extensions to the totality of base sets of base linear combinations of integrals with coefficients of negligible complexity are constructed. The application of all the constructed criteria is demonstrated by examples of comparing with each other of the above power series coefficients representations. The obtained results are presented in the tables and commented.