Thermodynamic characteristics of ideal quantum gases in harmonic potentials within exact and semiclassical approaches

TL;DR

This paper compares exact and semiclassical methods to analyze thermodynamic properties of trapped ideal Bose and Fermi gases, revealing finite-size effects and improving critical temperature estimates.

Contribution

It introduces a more accurate approach for finite particle corrections to the critical temperature and highlights deviations from semiclassical predictions.

Findings

Finite-size effects significantly alter Fermi energy calculations for small particle numbers.

Bose gases show softened phase transition features compared to semiclassical predictions.

The new methodology improves the accuracy of critical temperature estimates for finite systems.

Abstract

We theoretically examine equilibrium properties of the harmonically trapped ideal Bose and Fermi gases in the quantum degeneracy regime. We analyze thermodynamic characteristics of gases with a finite number of atoms by means of the known semiclassical approach and perform comparison with exact numerical results. For a Fermi gas, we demonstrate deviations in the Fermi energy values originating from a discrete level structure and show that these are observable only for a small number of particles. For a Bose gas, we observe characteristic softening of phase transition features, which contrasts to the semiclassical predictions and related approximations. We provide a more accurate methodology of determining corrections to the critical temperature due to finite number of particles.