# Re-examining the Statistical Mechanics of an Interacting Bose Gas

## TL;DR

This paper critically re-examines the statistical mechanics of interacting Bose gases, clarifies the canonical versus grand-canonical treatments, and proposes a new framework that allows the number of bosons to vary with temperature.

## Contribution

It identifies the misinterpretation in previous treatments as a canonical approach and develops a canonical and grand-canonical formulation where boson number varies with temperature.

## Key findings

- The grand-canonical treatment in previous theories is effectively canonical.
- Discontinuity in condensate density at T_c is due to an inappropriate extension of Bogoliubov's prescription.
- A new canonical and grand-canonical framework for interacting bosons is proposed.

## Abstract

We re-examine the way in which Bogoliubov's theory of a dilute Bose gas at $T=0$ has been extended to describe the statistical mechanics of interacting bosons at finite temperature. We show explicitly that the field-theoretic calculation of the grand partition function in this formulation amounts to a canonical trace over the eigenfunctions of the Bogoliubov Hamiltonian at fixed total number of bosons $N$, and that the additional trace over $N$ that is required in the grand-canonical formalism is never carried out. This implies that what usually passes as the grand-canonical treatment of the Bogoliubov Hamiltonian is not quite grand-canonical, and is in fact a canonical one. We also show that the discontinuity in the condensate density predicted by previous formulations of this theory as the temperature $T$ goes past the critical transition temperature $T_c$ is a direct consequence of an inappropriate generalization of the Bogoliubov prescription to finite temperatures, and that this discontinuity disappears when this prescription is either used as a zero temperature approximation or avoided altogether. Armed with the above findings, we reformulate the statistical mechanics of interacting bosons in the canonical ensemble and derive the thermodynamics of the system. We then show how the canonical treatment can be used to setup a truly grand-canonical description of the statistical mechanics of a weakly interacting Bose gas where the average number of bosons in the system varies with temperature, unlike existing formulations where the total number of bosons $N$ is taken to be a constant that does not depend on $T$. Consequences on the physics of interacting bosons are briefly discussed.