On the derivation of the entropy of ideal quantum gases confined in a three-dimensional harmonic potential

Abstract

In this work, the entropy functions of ideal quantum gases in a three-dimensional harmonic trap are analytically calculated using temperature as an explicit variable. Afterward, the applicability of the analytical formulas is validated by comparison with the numerical calculation. The results illustrate that the obtained functions could be applied for the whole temperature regime with a maximum relative deviation of less than 7.5% in the vicinity of the critical temperature Tc in the case of Bose gases. Meanwhile, for Fermi gases, although the analytical formula fits well at very low- and high-temperature regimes, it cannot be applied at temperature in the range , where TF is the Fermi temperature. In addition, the consistency between our formulas and classical ones at significantly high temperatures is also discussed.

Numerical calculation of the dependence of chemical potential on temperature for ideal quantum gases in 1,2 and 3-dimensional spaces

“…This is obviously understood since the approximation in the analytical approach works well for low and high temperatures in the low-and hightemperature regime [33] due to the characteristics of the Sommerfeld and Taylor expansion, respectively. Also, the definition of low-and high-temperature regime in the case of Fermi gas cannot be clearly classified and has a relative meaning due to the lack of phase transition [25]. Especially in 3-D space, the absolute error between numerical and analytical results is less than 0.005 for the whole range of the considered reduced temperature, both in low-and hightemperature regimes.…”

“…The most common trap used in theoretical studies is parabolic trap or harmonic oscillator trap owing to its analytical solutions and many studies on thermodynamics properties of ultra-cold gases harmonically trapped have been conducted [16,17,18,19,20,21,22]. However, the explicit dependence of thermodynamic functions such as chemical potential, total energy, heat capacity and entropy of ideal quantum gases confined in a three-dimensional harmonic trap as a function of absolute temperature are fully studied in recent [23,24,25]. Besides, it is also essential to study the thermodynamics properties of ideal quantum gases in general.…”