# On the Chemical Potential of Ideal Fermi and Bose Gases

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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 \(T_{{{{\mathrm {B}}}}}\). The *Mathematica* routines for the series allow calculation to arbitrary order.

## Keywords

Quantum gases Chemical potential Mathematica## 1 Introduction

The properties of ideal Fermi and Bose gases are the starting points for the understanding of the low-temperature behaviour of a broad range of physical systems, including electrons in metals [1, 2], the helium liquids [3, 4] and systems of trapped gases [5]. Associated with these is interest in systems of lower dimensionality including graphene [6], helium films [7] and ultra-cold atoms in quasi-1d and quasi-2d traps [8].

The properties of ideal quantum gases are expressed, conveniently and succinctly in terms of the so-called Fermi–Dirac and Bose–Einstein functions. And essential to this is knowledge of the chemical potential [9].

Traditionally Fermi–Dirac and Bose–Einstein functions have been found from tables and power series expansions. The 1938 paper by McDougall and Stoner [10] gave extensive tables for fermions and there was a discussion of the corresponding functions for bosons by London [11], with series expansions given by Robinson [12] and generalized by Clunie [13]. A consolidated treatment of these functions is found in Pathria [14].

The problem with the McDougal–Stoner tables and the above treatments was that the functions were given for different values of \(\mu /kT\) which made for difficulties finding the chemical potential \(\mu \). In 1974 Betts [15] published tables of (reduced) chemical potential as a function of (reduced) temperature for 3d fermions and bosons; at that time, it was a heroic achievement of minicomputer programming. Ebner and Fu [16] gave useful series expressions for 3d fermions and analytic expressions for 2d fermions. Also they produced extensive tables [17], sadly now unavailable.

A further provision of series expansions for fermions was by Hore and Frankel [18] who gave expressions for the *intermediate* quantum region. These series were expanded about \(z=1\) where *z* is the fugacity. Expansions in the intermediate region for both bosons and fermions have been given by Sotnikov et al. [19].

The advent of symbolic mathematics software such as *Mathematica* has revolutionized the way Fermi–Dirac and Bose–Einstein calculations may be carried out. Such software can perform series expansions, reversion of series, symbolic integration, numerical integration to arbitrary precision and general symbolic manipulation. Moreover *Mathematica* “knows of” a large number of the “special functions” and their properties. Also it has inverse function support.

The Fermi–Dirac and the Bose–Einstein integrals may both be expressed in terms of the polylogarithm functions, once the chemical potential is known. In this paper we provide *Mathematica* functions to obtain the chemical potential. From these it is then straightforward to evaluate properties of Bose and Fermi gases. For reference we include the classical (Maxwell) gas. Of course at low temperatures such a gas is un-physical; it violates the third law of thermodynamics. But at high temperatures the way that the Bose and Fermi gases deviate from the classical is instructive.

## 2 General Methodology

### 2.1 The Polylogarithm Functions

*a*is understood to be \(+1\) for fermions and \(-1\) for bosons. Classical particles also can be accommodated by taking \(a=0\). We may refer to such particles as “maxwellons”.

The distribution function involves the chemical potential \(\mu \), and so, this must be known before any later calculations. A key result of this paper is the provision of *Mathematica* functions for evaluation of the chemical potential for free fermion and boson gases in one, two and three dimensions.

### 2.2 Common Energy Scale

*is*appropriate in the discussion of maxwellons and bosons. In this spirit we shall introduce a characteristic wave vector, which we call the

*quantum*wave vector \(k_{{\mathrm {q}}}\), and relate this to a

*quantum*energy \(\varepsilon _{{\mathrm {q}}}=\hbar ^2k_{{\mathrm {q}}}^2/2m\). This would be the Fermi energy in the case of fermions. In 1, 2 and 3d this is

### 2.3 Density of States

*d*dimensions

## 3 Chemical Potential and Fugacity

*N*so that

*N*cancels and Eq. (12) leads to an expression for the quantum energy

^{1}

*Mathematica*routines for these functions (Fermi and Bose) are given in Appendix A.

## 4 Functions and Their Series Expressions

Details of the calculation of the low-temperature and high-temperature series are given in the appendices. Associated *Mathematica* Notebooks are included in Supplementary Information, allowing evaluation of these series to arbitrary order.

A key point about the *Mathematica* calculations is that while *Mathematica* provides the InverseFunction command used in the chemical potential and fugacity expressions, the InverseFunction provision does not extend to symbolic series calculations. For this reason we have to perform the series expansion first and then obtain the inverse by reversing the power series.

In this section below we summarize the results of these calculations. We shall use the *Mathematica* functions of Appendix A to create plots of the chemical potential and the series approximations given. A *Mathematica* notebook creating the plots below is given in Supplementary Information MOESM1_ESM.nb.

### 4.1 Maxwellons in 1d

### 4.2 Fermions in 1d

*Chemical potential—low-temperature series*

*Chemical potential—high-temperature series*

*Fugacity—low-temperature series.*Since \(z_{{\mathrm {F1}}}(\tau )\) diverges as \(\tau \rightarrow 0\) there is no simple low-temperature power series. But the low-temperature behaviour can be expressed as

*Fugacity—high-temperature series*

### 4.3 Bosons in 1d

*Chemical potential—low-temperature series*

*Chemical potential—high-temperature series*

*Fugacity—low-temperature series*

*Fugacity—high-temperature series*

### 4.4 Maxwellons in 2d

### 4.5 Fermions in 2d

*Chemical potential—low-temperature series*There is no series in ascending powers of \(\tau \), but the following series of exponentials follows from Eq. (43) at low temperatures.

*Chemical potential—high-temperature series*

*Fugacity—low temperatures.*Since \(z_{{\mathrm {F2}}}(\tau )\) diverges as \(\tau \rightarrow 0\) there is no simple low-temperature power series. But in the spirit of Eq. (45) and writing \(z_{{\mathrm {F2}}}(\tau )\) as

*Fugacity—high-temperature series*

### 4.6 Bosons in 2d

*Chemical potential—low-temperature series*There is no series in ascending powers of \(\tau \), but the following series of exponentials follows from Eq. (52) at low temperatures.

*Chemical potential—high-temperature series*

*Fugacity—low temperatures*There is no low-temperature series for the fugacity, but we may regard the expression for \(z_{{\mathrm {B2}}}(\tau )\), Eq. (53), as a terminating low-temperature expansion:

*Fugacity—high-temperature series*

### 4.7 Maxwellons in 3d

### 4.8 Fermions in 3d

*Chemical potential—low-temperature series*

*Chemical potential—high-temperature series*

*Fugacity—low-temperature series*Since \(z_{{\mathrm {F3}}}\) diverges as \(\tau \rightarrow 0\) there is no simple low-temperature power series. But the low-temperature behaviour can be expressed as

*Fugacity—high-temperature series*

### 4.9 Bosons in 3d

In three dimensions bosons can undergo Bose–Einstein condensation. When this happens the chemical potential will be zero and the fugacity will be unity. We denote the reduced Bose temperature by \(\tau _{{{{\mathrm {B}}}}}\).

*z*when \(\tau >\tau _{{{{\mathrm {B}}}}}\). Then the reduced chemical potential and the fugacity are given by

*Chemical potential—low-temperature series*

*Chemical potential—high-temperature series*

*Fugacity—low-temperature series*

*Fugacity—high-temperature series*

## 5 Chemical Potential Plots in 1, 2 and 3d

*Mathematica*functions to create plots of the chemical potential in one-, two- and three dimensions. These will contrast the differences and similarities of the Fermi, Bose and Maxwell cases.

### 5.1 Three Dimensions

At low temperatures the fermion \(\mu ^*(\tau )\rightarrow 1\) as \(\tau \rightarrow 0\) (\(\mu (T)\rightarrow \varepsilon _{{\mathrm {F}}}\) as \(T\rightarrow 0\)).

Upon cooling, bosons in 3d undergo BEC at the Bose temperature \(\tau _{{{{\mathrm {B}}}}}\) where the chemical potential goes to zero. Just above the Bose temperature \(\mu ^*(\tau )\) increases quadratically in \((\tau -\tau _{{{{\mathrm {B}}}}})\), while below the Bose temperature \(\mu ^*(\tau )\) is identically zero. There is a discontinuity in the second derivative of \(\mu ^*(\tau )\) at \(\tau =\tau _{{{{\mathrm {B}}}}}\). We shall in the following sections see that there is no BEC for \(d<3\); this is an example of the Mermin–Wagner theorem [25]. Then there is macroscopic occupation only at \(T=0\).

The maxwellon chemical potential increases from zero as the temperature increases from zero, with a nonzero slope; this is a violation of the third law of thermodynamics, which requires \(\partial \mu /\partial T\rightarrow 0\) as \(T\rightarrow 0\).

### 5.2 Two Dimensions

*towards*the classical Maxwell behaviour

At low temperatures the Bose chemical potential goes to zero. There is no BEC, so \(\mu \) is zero only *at* zero temperature. But the low-temperature \(\mu \) is “very flat” (see Fig. 8 right plot); it is as if the 2d bosons are “trying to” condense. We shall see in Sect. 5.3 that there is certainly no BEC in 1d. But here, in 2d, we might say that BEC is “marginal”.

The low-temperature Fermi chemical potential is similar to the Bose case, just shifted up by \(\varepsilon _{{\mathrm {q}}}\).

### 5.3 One Dimension

The temperature dependence of the 1d chemical potential is shown in Fig. 9.

*fractional*deviation \((\mu _{{\mathrm {F,B}}}^*(\tau )-\mu _{{\mathrm {M}}}^*(\tau ))/\mu _{{\mathrm {M}}}^*(\tau )\) does go to zero as \(\tau \rightarrow \infty \).

The low-temperature Bose chemical potential goes to zero as \(\tau ^2\). There is not even a hint of BEC; the macroscopic occupation of the ground state occurs only at \(T=0\).

The low-temperature Fermi chemical potential has an interesting form. As the temperature increases from zero \(\mu \) increases a little before turning over and decreasing. This is not a violation of the third law since \(\partial \mu /\partial T \rightarrow 0\) as \(T\rightarrow 0\).

The Maxwell chemical potential increases from zero with nonzero slope as the temperature increases from zero, in violation of the third law of thermodynamics.

## Footnotes

## Notes

### Acknowledgements

The author is grateful to Bob Swarup; aspects of this work, particularly the role of dimensionality, were discussed in his MSc thesis [29]. Special thanks go to Bill Mullin. He critiqued an early version of the manuscript, giving invaluable advice. He also made helpful suggestions on the presentation.

## Supplementary material

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