A simple way of approximating the canonical partition functions in statistical mechanics

TL;DR

This paper introduces an accessible method using the Euler-MacLaurin formula for approximating canonical partition functions in statistical mechanics, suitable for undergraduate teaching, with two approaches of varying complexity.

Contribution

It presents a simple pedagogical approach with two methods for approximating partition functions, enhancing teaching tools in statistical mechanics courses.

Findings

First approach yields the first two terms of the expansion.

Second approach includes all correction terms.

Both methods are applied to translational, vibrational, and rotational partition functions.

Abstract

We propose a simple pedagogical way of introducing the Euler-MacLaurin summation formula in an undergraduate course on statistical mechanics. We put forward two alternative routes: the first one is the simplest and yields the first two terms of the expansion. The second one is somewhat more elaborate and takes into account all the correction terms. We apply both to the calculation of the simplest one-particle canonical partition functions for the translational, vibrational and rotational degrees of freedom.