Average energy approximation of the ideal Bose-Einstein gas and condensate

TL;DR

This paper introduces an average energy approximation for the ideal Bose-Einstein gas, deriving thermodynamic properties and phase behavior, including Bose-Einstein condensation, with results consistent with traditional probabilistic approaches.

Contribution

It presents a simplified, average energy-based method to analyze the thermodynamics and phase transition of an ideal Bose-Einstein gas, aligning with established results.

Findings

Entropy expression recovers Sakur-Tetrode entropy in classical limit

Identifies Bose-Einstein condensate and two-phase region

Results match standard probabilistic derivations

Abstract

If the N bosons that compose an ideal Bose-Einstein gas with energy E and volume V are each assumed to have the average energy of the system E/N, the entropy is easily expressed in terms of the number of bosons N and the number of single-particle microstates n they can occupy. Because the entropy derived is a function of only N and n, and the latter is a function of the extensive variables, E, V, and N, this entropy describes all that can be known of the thermodynamics of this system. In particular, the entropy very simply recovers the Sakur-Tetrode entropy in the classical limit and at sufficiently low temperature describes an unstable system. A thermodynamic stability analysis recovers the Bose-Einstein condensate and a two-phase region. Apart from numerical factors of order one, results are identical with those derived via standard, probabilistic methods.