Canonical partition functions: ideal quantum gases, interacting classical gases, and interacting quantum gases

TL;DR

This paper introduces a mathematical approach to calculate canonical partition functions for ideal and interacting gases using symmetric functions and Bell polynomials, enabling precise computation of virial coefficients.

Contribution

It presents a novel method leveraging symmetric functions and Bell polynomials to compute canonical partition functions for various gases, including interacting systems.

Findings

Exact canonical partition functions for ideal quantum gases derived.

Virial coefficients computed directly from canonical partition functions.

Method applicable to both classical and quantum interacting gases.

Abstract

In statistical mechanics, for a system with fixed number of particles, e.g., a finite-size system, strictly speaking, the thermodynamic quantity needs to be calculated in the canonical ensemble. Nevertheless, the calculation of the canonical partition function is difficult.\textbf{ }In this paper, based on the mathematical theory of the symmetric function, we suggest a method for the calculation of the canonical partition function of\ ideal quantum gases, including ideal Bose, Fermi, and Gentile gases. Moreover, we express the canonical partition functions of interacting classical and quantum gases given by the classical and quantum cluster expansion methods in terms of the Bell polynomial in mathematics. The virial coefficients of ideal Bose, Fermi, and Gentile gases is calculated from the exact canonical partition function. The virial coefficients of interacting classical and quantum gases is calculated from the canonical partition function by using the expansion of the Bell polynomial, rather than calculated from the grand canonical potential.