A more accurate analysis of Bose-Einstein condensation in harmonic traps

TL;DR

This paper analytically calculates the internal energy and specific heat of non-interacting bosons in a harmonic trap, revealing a sharp phase transition and confirming previous numerical and approximate results.

Contribution

It provides an exact analytical approach using Euler-Maclaurin summation to study Bose-Einstein condensation in harmonic traps, improving understanding of thermodynamic properties.

Findings

Identification of a sharp λ-like peak in specific heat at transition

Full agreement with numerical calculations across temperature ranges

Exponential suppression of excitations at very low temperatures

Abstract

Using the Euler-Maclaurin summation we calculate analytically the internal energy for non-interacting bosons confined within a harmonic oscillator potential. The specific heat shows a sharp $\lambda$-like peak indicating a condensation into the ground state at a well-defined transition temperature. Full agreement is obtained with direct numerical calculation of the same quantities. When the number of trapped particles is very large and at temperatures near and above the transition temperature, the results also agree with previous approximate calculations. At extremely low temperatures both the specific heat and the number of particles excited from the condensate are exponentially suppressed.