A procedure for high-accuracy numerical derivation of the thermodynamic properties of ideal Bose gases

Abstract

The thermodynamic functions of ideal Bose gases are important in fundamental physics and have been widely studied via both analytical and numerical methods. Studying these functions is important for undergraduates as it is the first step for exploring ultracold physics and quantum phenomena. However, the accuracy of the calculations is usually limited owing to the discontinuity of the functions at critical points. In this study, we present an analytical procedure for deriving the chemical potential, the total energy, and the heat capacity as functions of the absolute temperature. The approximated analytical results are compared with values obtained using the numerical method to evaluate their accuracy and to determine the applicable range of the procedure. Moreover, a correction for the chemical potential calculated around the critical temperature was proposed to reduce the deviation between the approximated and numerical approaches.

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

“…Owing to the phase transition at a critical temperature T c of bosons (the BEC), the thermodynamic properties are treated with two separate phases, condensed and normal phases [14,16,23]. The entropy S, therefore, is also proceeded.…”

“…Thermodynamic quantities such as the chemical potential, the total energy, and the heat capacity of trapped quantum gases have been extensively studied by several authors [14][15][16][17][18][19][20][21][22][23]. However, the entropy, which plays a crucial role in the interpretation of phase transition in physics, has not been mentioned in detail, especially for Fermi gas where other functions are usually considered [15,19,21].…”

“…However, the entropy, which plays a crucial role in the interpretation of phase transition in physics, has not been mentioned in detail, especially for Fermi gas where other functions are usually considered [15,19,21]. In comparison to the entropy of harmonically trapped Fermi gases, that of Bose gases receives less concern [14,16,18,22,23]. Grether et al have recently examined the properties of entropy in such a system; however, they expressed the results in terms of polylogarithm functions of chemical potential [17].…”

Bose-Einstein statistics for a finite number of particles

“…In these experiments the number of particles ranged in the order of thousands [4] to millions [6]. This sprung interest in studies of BE statistics that do not rely on the thermodynamic limit (number of particles N → ∞) [7][8][9][10][11][12][13][14]. It is important to notice that the non-analytical behaviour mentioned previously is a consequence of the thermodynamic limit applied to BE statistics and therefore shouldn't be expected for a finite number of particles.…”

“…This is fundamental for the study of BE statistics for a finite number of particles since the thermodynamical quantities that identify the phase transition -fraction of particles in the ground state and specific heat -can be written exactly in terms of β and ξ, as we will see in the present article. Most of the investigations done in BE statistics for finite number of particles [7][8][9][10][11][12] is restricted to harmonically trapped gases, η = D − 1. Among the investigations that went beyond the study of harmonically trapped gases, it is important to mention the work of Jaouadi et.…”