On the evaluation of the internal energy and chemical potential of an ideal fermion system

Abstract

The internal energy and chemical potential of a noninteracting Fermi gas has been evaluated in three ways: numerically (applicable to arbitrary temperatures), through an asymptotic power series expansion (suitable for low temperatures), and by means of standard Padé approximants (appropriate for low and moderate temperatures). The main conclusion of this work is that the Padé approximant, which is constructed from the asymptotic series employing only minimal labor, is far superior to the truncated series expansion.

Chemical potential and internal energy of the noninteracting Fermi gas in fractional-dimensional space

“…This method has been described by Baker [9]. As pointed out earlier, this method has been applied to truncated series of μ/E F and U/U 0 in 3D containing terms up to the 8th power of T [5] …”

“…In this technique, the truncated series is expressed as a ratio of the two polynomials of finite sizes so that the ratio is convergent. Using this method, the truncated series of μ and U valid in the low-temperature and high-density regimes are extended to intermediate-temperature and low-density regimes [5].…”

“…Furthermore, the Sommerfeld method can be easily extended to calculate several ground state properties of the free fermion gas using the thermodynamic relations [2]. Both μ and U in 3D were calculated up to order T 6 by Kiess [4] and up to order T 8 by Aguilera-Navarro et al [5]. The only limitation of the Sommerfeld method is that it cannot be used for two-dimensional (2D) systems.…”