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Journal of Low Temperature Physics

On the Chemical Potential of Ideal Fermi and Bose Gases

Brian P. Cowan

Published Sep 12, 2019• 16 citations
Publisher

Abstract

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 high-temperature series expansions. In the 3d Bose case we give also expansions about $$T_{{{{\mathrm {B}}}}}$$ T B . The Mathematica routines for the series allow calculation to arbitrary order.

DOI
10.1007/s10909-019-02228-0

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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 high-temperature series expansions. In the 3d Bose case we give also expansions about TBT_{{{{\mathrm {B}}}}}TB​ T B . The Mathematica routines for the series allow calculation to arbitrary order.
Calculations of chemical potentials for ideal monatomic gases with Bose-Einstein and Fermi-Dirac statistics as functions of temperature, across the temperature region that is typical for the collective quantum degeneracy effect, are presented. Numerical calculations are performed without any additional approximations, and explicit dependences of the chemical potentials on temperature are constructed at a fixed density of gas particles. Approximate polynomial dependences of chemical potentials on temperature are obtained that allow for the results to be used in further studies without re-applying the involved numerical methods. The ease of using the obtained representations is demonstrated on examples of deformation of distribution for a population of energy states at low temperatures, and on the impact of quantum statistics (exchange interaction) on the equations of state for ideal gases and some of the thermodynamic properties thereof. The results of this study essentially unify two opposite limiting cases in an intermediate region that are used to describe the equilibrium states of ideal gases, which are well known from university courses on statistical physics, thus adding value from an educational point of view.
The chemical potential and fluctuations in number of particles in a D-dimensional free Fermi gas at low temperatures are obtained by means of polylogarithms. This idea is extended to show that the density of any ideal gas, whether Fermi, Bose, or classical, can be expressed in polylogarithms. The densities of different statistics correspond to different domains of polylogarithms in such a way that there emerges a unifying picture. The density of the classical ideal gas represents a fixed point of polylogarithms. Inequalities for polylogarithms are used to provide a precise bound on errors in the fermion chemical potential at low temperatures.
Abstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussed
In this work we revisit the concept of chemical potential μμμ in both classical and quantum gases from a perspective of Equilibrium Statistical Mechanics (ESM). Two new results regarding the equation of state μ=μ(n,T)μ=μ(n,T)μ=μ(n,T), where nnn is the particle density and TTT the absolute temperature, are given for the classical interacting gas and for the weakly-interacting quantum Bose gas. In order to make this review self-contained and adequate for a general reader we provide all the basic elements in an advanced-undergraduate or graduate statistical mechanics course required to follow all the calculations. We start by presenting a calculation of μ(n,T)μ(n,T)μ(n,T) for the classical ideal gas in the canonical ensemble. After this, we consider the interactions between particles and compute the effects of them on μ(n,T)μ(n,T)μ(n,T) for the van der Waals gas. For quantum gases we present an alternative approach to calculate the Bose-Einstein (BE) and Fermi-Dirac (FD) statistics. We show that this scheme can be straightforwardly generalized to determine what we have called Intermediate Quantum Statistics (IQS) which deal with ideal quantum systems where a single-particle energy can be occupied by at most jjj particles with 0⩽j⩽N0 \leqslant j \leqslant N0⩽j⩽N with NNN the total number of particles. In the final part we address general considerations that underlie the theory of weakly interacting quantum gases. In the case of the weakly interacting Bose gas, we focus our attention to the equation of state μ=μ(n,T)μ=μ(n,T)μ=μ(n,T) in the Hartree-Fock mean-field approximation (HF) and the implications of such results in the elucidation of the order of the phase transitions involved in the BEC phase for non-ideal Bose gases.
In this paper, the chemical potential of two-dimensional (2D) and quasi-one-dimensional (Q1D) multisubband charged Fermi gases is evaluated. We start with a rather general formula for the thermodynamic potential of an ideal quantum statistical system with arbitrary occupation-number to calculate, as a particular case, the chemical potential of the multisubband 2D Fermi gas described by a quadratic energy spectrum. The chemical potential is also studied in the case of a low-dimensional Fermi gas in the presence of a quantizing magnetic field. The same approach is used to study the chemical potential of a multisubband Q1D Fermi gas. The influence of temperature is considered. We found interesting analytical results for special limits as well as numerical results that may be important for the description of both the thermodynamic and kinetic properties of Fermi systems in low dimensionality.