On the Chemical Potential of Ideal Fermi and Bose Gases

Abstract

Knowledge of the chemical potential is essential in application of the Fermi-Dirac and the Bose-Einstein distribution functions for the calculation of properties of quantum gases. We give expressions for the chemical potential of ideal Fermi and Bose gases in 1, 2 and 3 dimensions in terms of inverse polylogarithm functions. We provide Mathematica functions for these chemical potentials together with low-and hightemperature series expansions. In the 3d Bose case we give also expansions about T B. The Mathematica routines for the series allow calculation to arbitrary order.

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

“…We proceed to present our numerical results for three cases of ideal Maxwell, Fermi, and Bose gases. The comparison with the analytical work in Ref [33] is thoroughly considered to validate our numerical results. Vice versa, our numerical results can be used as a benchmark to investigate the applicability of the approximately analytical formulae in each temperature regime.…”

“…Among the thermodynamics functions, the chemical potential plays a significant role in deriving the others. The most comprehensive works on chemical potential in 1-, 2-, and 3dimension spaces for the ideal quantum and classical gases were presented in [32,33]. However, to our understanding, an accurately numerical consideration for the chemical potential in ddimensional space for the ideal quantum and classical gases has not been done yet.…”

“…However, to our understanding, an accurately numerical consideration for the chemical potential in ddimensional space for the ideal quantum and classical gases has not been done yet. Indeed, Cowan also presented the analytical results which express the dependence of the chemical potential of ideal quantum and classical gases in d-dimensional space by using a built-in function in Mathematica to expand mathematical series [33]. However, apparently this approach cannot be taken into account as true numerical calculations.…”

“…Hence, the applicability of the analytical formulae was not clearly clarified in Ref. [33] and is deserved a thorough study using a purely numerical approach in addition to the above-mentioned knowlegde gaps. It is noted that the current work is vital for the sake of numerically and analytically calculating the other thermodynamics functions of ideal quantum and classical gases in d-dimensional space.…”

“…In this work, we numerically calculate the chemical potential for all cases stated in [33] using the numerical scheme given in Section 2. In Section 3, we conduct comparisons and draw conclusions regarding the application range of the analytical functions in [33]. Section 4 concludes the paper.…”