# Thermodynamics of the nonrelativistic free-electron Fermi gas in one, two,
and three dimensions from the degenerate to the nondegenerate temperature
regime

David C. Johnston  Ames Laboratory and Department of Physics and Astronomy,
Iowa State University, Ames, Iowa 50011, USA

###### Abstract

The thermodynamic properties of a nonrelativistic free-electron Fermi gas is
of fundamental interest in condensed matter physics. Properties previously
studied in three-dimensions (3D) in the low- and high-temperature limits
include the internal energy, heat capacity, zero-field magnetic spin
susceptibility, and pressure. Here we report solutions for the temperature
dependence spanning these two temperature regimes of the chemical potential,
internal energy, magnetic susceptibility, and the heat capacity at constant
volume in 1D, 2D, and 3D. Also calculated are the pressure, enthalpy, heat
capacity at constant pressure, isothermal compressibility, and thermal
expansion coefficient versus temperature in 2D and 3D. Of primary interest
here are the detailed dimension-dependent crossovers of these properties
between the degenerate and nondegenerate temperature regime, which are
graphically illustrated for each of the above properties.

##  I Introduction

A free-electron Fermi gas as defined here is a collection of a large number
$N$ of nonrelativistic noninteracting electrons with spin $S=1/2$ in zero
external potential. Thus the interaction of the electrons with a crystal
lattice of positive ions that would give rise to energy gaps in the band
structure and possibly superconductivity is ignored. The electron gas can be
in one dimension (1D), 2D, or 3D. The low- and high-temperature $T$ limits of
the thermal and magnetic properties of a 3D free-electron Fermi gas are well
known Sommerfeld1928 ; Ashcroft1976 ; Kittel1980 ; Ibach2002 ; Kittel2005 ;
Hook2010 ; Simon2013 ; Grosso2014 . Other aspects and consequences of the
Fermi statistics have also been considered McKelvey1964 ; Kapron1970 ;
Bludman1977 ; Arnaud1999 ; Balian1999 ; Mulin2003 ; Silbar2004 ; Durand2004 ;
Nattermann2005 ; Muller2015 ; Cetina1977 ; Sevilla2016 ; Li1998 ; Johnston2020
. In the limit of high $T$ the heat capacity at constant volume $C_{\rm V}$
and the pressure $p$ are the same as for a monatomic ideal gas and the
magnetic spin susceptibility $\chi$ is the same as for $N$ isolated electrons
following the Curie law $C/T$, where $C$ is the Curie constant for $S=1/2$. At
low $T$ in the degenerate regime, $C_{\rm V}$ is proportional to $T$, and
$\chi$ and $p$ saturate to constant values as $T\to 0$ K. Here expressions for
the chemical potential $\mu(T)$ are discussed Cowan2019  that allow the
crossovers of the dimension-dependent $\chi$, internal energy $U$ and $C_{\rm
V}$ in 1D, 2D, and 3D, and of $p$, enthalpy $H$, heat capacity at constant
pressure $C_{\rm p}$, isothermal bulk modulus $K_{T}$, and thermal expansion
coefficient $\alpha$ in 2D and 3D to be calculated between the low-$T$
degenerate regime and the high-$T$ nondegenerate regime. Illustratrative plots
of the $T$ dependences of these properties are provided.

There are many instances in which free-electron Fermi gas properties are
observed in real metals. An important example is the $T$-dependent $C_{\rm V}$
of metals at low $T$ compared to the Fermi temperature $T_{\rm F}=E_{\rm
F}/k_{\rm B}$ ($E_{\rm F}$ is the Fermi energy) which is proportional to $T$
irrespective of the dimensionality of the electron gas in the metal. However,
the proportionality constant $\gamma$, known as the Sommerfeld electronic heat
capacity coefficient, is proportional to the electronic density of states at
$E_{\rm F}$ which depends on the dimensionality of the metal for a given
electron concentration as discussed below. The free-electron model as
reflected in the theoretical value of $\gamma$ is often fairly close for a
particular metal to the measured value and hence is a good starting point for
interpreting the $\gamma$ value. Deviations occur due to features not included
in the free-electron Fermi-gas model such as the influences of the
interactions of the conduction electrons with the lattice (the electron-phonon
interaction) and of electron-electron interactions, such as described in Refs.
Sevilla2016 ; Johnston2020 .

In superconductors, the free-electron Fermi-gas model is not appropriate below
the superconducting transition temperature $T_{\rm c}$ even for simple metals
exhibiting so-called conventional superconductivity because within the BCS
theory Bardeen1957  the superconductivity arises from indirect attractive
interactions between the conduction electrons mediated by phonons. Here an
energy gap opens in the quasiparticle (electron/hole) excitation spectrum
below $T_{\rm c}$, so that the electronic heat capacity decreases faster than
linearly below $T_{\rm c}$ and approaches zero exponentially for $T\to 0$.

Interestingly, the free-electron Fermi-gas theory also sometimes applies to
nonmetals containing nuclei with nonzero spin, such as liquid 3He for which
the nuclei have spin $S=1/2$. Indeed, as shown in Fig. 3 below, the magnetic
susceptibility $\chi$ versus $T$ of liquid 3He is described well by 3D $S=1/2$
noninteracting-fermion theory all the way from the degenerate ($T\ll T_{\rm
F}$) to the nondegenerate ($T\gg T_{\rm F}$) temperature regime, and a fit to
the data yielded the value of $T_{\rm F}$ for this material. Additional
examples of 3D Fermi gases include the electron gas in white-dwarf stars and
the neutrons in neutron stars Kittel2005 , where the Fermi energies were
estimated. Important examples of 2D or quasi-2D electron gases occur in metal-
oxide-semiconductor field-effect transistors (MOSFETS) in which the Quantum
Hall Effect was discovered Klitzing1980  for which the 1985 Nobel Prize in
Physics was awarded Klitzing1985Nobel , as well as in other materials
MOSFETWiki .

Fundamental expressions and the notation used here are given in the Sec. II.
The equations from which $\mu(T)$ can be calculated for 1D, 2D, and 3D
electron Fermi gases are presented in Sec. III. The $\chi$ is obtained and
plotted versus $T$ from the degenerate to the nondegenerate regime for
electron Fermi gases in 1D, 2D, and 3D in Sec. IV, and the dimension-dependent
$U$ and $C_{\rm V}$ versus $T$ are calculated and plotted in Sec. V. The
pressure $p$, enthalpy $H$, heat capacity at constant pressure $C_{\rm p}$,
isothermal bulk modulus $K_{T}$, and thermal expansion coefficient $\alpha$ in
2D and 3D are derived and plotted from the degenerate to nondegenerate
temperature regime in Secs. VI–IX, respectively. Concluding remarks are given
in Sec. X.

Some of our results for the free-electron Fermi gas were already known from
either published or unpublished sources. The $\mu(T)$ in 1D, 2D, and 3D was
thoroughly investigated in Ref. Cowan2019 . The $U(T)$ and $C_{\rm V}(T)$ at
low $T$ for 1D, 2D, and 3D were obtained in Ref. Cetina1977 . Sommerfeld
expansions of the low-$T$ properties are routinely obtained for all three
dimensions in many sources. Calculations of $\mu(T)$, $p(T,V)$, and $C_{\rm
V}(T)$ in 1D, 2D, and 3D from the degenerate to the nondegenerate temperature
regime are available as of this writing in unpublished lecture notes
Muller2015 . Here we include our calculations and plots of these previously-
studied quantities for completeness. The calculations of $\mu(T)$ in 1D, 2D,
and 3D are required for the calculations of all other thermodynamic properties
versus dimensionality from the degenerate to the nondegenerate temperature
regime.

##  II  Fundamentals and Notation

The kinetic energy of an electron of mass $m$ is $E=p^{2}/2m$, where $p$ is
the magnitude of the momentum. Substituting the quantum-mechanical de Broglie
relation $p=\hbar k$ for a matter particle where $k$ is the magnitude of the
wave vector, one has

$E=\frac{\hbar^{2}k^{2}}{2m},$ (1)

which yields the differentials

$\displaystyle dk$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{\frac{2m}{\hbar^{2}}}\ \frac{dE}{\sqrt{E}}$
$\displaystyle k\,dk$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{2m}{\hbar^{2}}\right)\ dE$ (2)
$\displaystyle k^{2}\,dk$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{2m}{\hbar^{2}}\right)^{3/2}\sqrt{E}\ dE.$

For traveling waves, using periodic boundary conditions an allowed wave vector
has a volume of $2\pi/L$ in 1D, $(2\pi/L)^{2}=4\pi^{2}/A$ in 2D and
$(2\pi/L)^{3}=8\pi^{3}/V$ in 3D, where here $A$ is the area and $V$ is the
volume of the respective confining box Kittel2005 . Thus a line, circle, or
sphere of radius $k$ in k-space contains $N_{\rm W}$ traveling waves, given by

$\displaystyle N_{\rm W}$ $\displaystyle=$
$\displaystyle\frac{2k}{2\pi/L}=\frac{Lk}{\pi}\hskip 35.41269pt{\rm(1D)}$
$\displaystyle N_{\rm W}$ $\displaystyle=$ $\displaystyle\frac{\pi
k^{2}}{4\pi^{2}/A}=\frac{Ak^{2}}{4\pi}\hskip 25.29494pt{\rm(2D)}$ (3)
$\displaystyle N_{\rm W}$ $\displaystyle=$ $\displaystyle\frac{(4/3)\pi
k^{3}}{8\pi^{3}/V}=\frac{Vk^{3}}{6\pi^{2}}\hskip 14.45377pt{\rm(3D)},$

where the factor of two in the 1D case arises because $k_{x}=\pm k$. When one
includes the factor of two Zeeman degeneracy of the free electron, the density
of electron states versus energy is $D(E)=2\,dN_{\rm W}/dE$. Then Eqs. (3)
yield

$\displaystyle D(E)$ $\displaystyle=$
$\displaystyle\frac{L}{\pi}\sqrt{\frac{2m}{\hbar^{2}}}\
\frac{1}{\sqrt{E}}\hskip 39.02531pt{\rm(1D)}$ $\displaystyle D(E)$
$\displaystyle=$
$\displaystyle\frac{A}{2\pi}\left(\frac{2m}{\hbar^{2}}\right)\hskip
48.42076pt{\rm(2D)}$ (4) $\displaystyle D(E)$ $\displaystyle=$
$\displaystyle\frac{V}{2\pi^{2}}\left(\frac{2m}{\hbar^{2}}\right)^{3/2}\sqrt{E}\hskip
14.45377pt{\rm(3D)}.$

We consider an $n$-dimensional volume ($n=1,2,3$) containing $N$ electrons. At
temperature $T=0$, the $N$ electrons are placed into the energy states
$E=\hbar^{2}k^{2}/2m$ according to the Pauli Exlusion Principle with the
lowest energy states filled first. Here $m$ is the mass of the free electron.
The energy of the highest-filled state is the Fermi energy $E_{\rm F}$ and the
corresponding magnitude of the wave vector is the Fermi wave vector $k_{\rm
F}$. The number of filled states $N_{\rm S}$ up to an energy $E$ is given by
$N_{\rm S}=\int_{0}^{E}D(E)dE$. Setting $N_{\rm S}$ equal to the number $N$ of
electrons and the upper limit of the integral at $T=0$ to $E_{\rm F}$, one
obtains

$N=\int_{0}^{E_{\rm F}}D(E)dE.$ (5)

For a given density of states function $D(E)$ one can solve for $E_{\rm F}$ as
a function of $N$ after doing the integral. Also, we have that $E_{\rm
F}=\hbar^{2}k_{\rm F}^{2}/(2m)$, so that $k_{\rm F}=\sqrt{2mE/\hbar^{2}}$.
Substituting the $D(E)$ functions in Eqs. (4) into (5) and doing the integrals
give

$\displaystyle E_{\rm F}$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{2m}\left(\frac{\pi N}{2L}\right)^{2},\hskip
14.45377ptk_{\rm F}=\frac{\pi N}{2L}\hskip 36.135pt({\rm 1D})$ $\displaystyle
E_{\rm F}$ $\displaystyle=$ $\displaystyle\frac{\hbar^{2}}{2m}\left(\frac{2\pi
N}{A}\right),\hskip 14.45377ptk_{\rm F}=\sqrt{\frac{2\pi N}{A}}\hskip
21.68121pt({\rm 2D})$ (6) $\displaystyle E_{\rm F}$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{2m}\left(\frac{3\pi^{2}N}{V}\right)^{2/3},\hskip
14.45377ptk_{\rm F}=\left(\frac{3\pi^{2}N}{V}\right)^{1/3},\hskip
7.22743pt({\rm 3D})$

where $L$ is the “volume” of the system in 1D, $A$ is the “volume” in 2D, and
$V$ is the volume in 3D. The Fermi velocity (speed) $v_{\rm F}$ is defined as
$v_{\rm F}=p_{\rm F}/m=\hbar k_{\rm F}/m$, where $p_{\rm F}=\hbar k_{\rm F}$
is the Fermi momentum of the electron, and hence $E_{\rm F}=\hbar^{2}k_{\rm
F}^{2}/2m$.

With the results in Eq. (6), one can determine the density of states at the
Fermi energy $D(E_{\rm F})$ by substituting the expressions for $E_{\rm F}$ in
Eqs. (6) for $E$ in Eqs. (4), yielding

$\displaystyle D(E_{\rm F})$ $\displaystyle=$
$\displaystyle\frac{2L}{\pi^{2}}\left(\frac{2m}{\hbar^{2}}\right)\left(\frac{L}{N}\right)\hskip
39.74872pt({\rm 1D})$ $\displaystyle D(E_{\rm F})$ $\displaystyle=$
$\displaystyle\frac{A}{2\pi}\left(\frac{2m}{\hbar^{2}}\right)\hskip
67.93398pt{\rm(2D)}$ (7) $\displaystyle D(E_{\rm F})$ $\displaystyle=$
$\displaystyle\frac{V}{2\pi^{2}}\left(\frac{2m}{\hbar^{2}}\right)\left(\frac{3\pi^{2}N}{V}\right)^{1/3}\hskip
7.22743pt{\rm(3D)}.$

Using Eqs. (6), one can write the $D(E)$ expressions (4) in terms of $E_{\rm
F}$ as

$\displaystyle D(E)$ $\displaystyle=$ $\displaystyle\frac{N}{2E_{\rm
F}}\sqrt{\frac{E_{\rm F}}{E}}\hskip 14.45377pt({\rm 1D})$ $\displaystyle D(E)$
$\displaystyle=$ $\displaystyle\frac{N}{E_{\rm F}}\hskip 44.80704pt({\rm 2D})$
(8) $\displaystyle D(E)$ $\displaystyle=$ $\displaystyle\frac{3N}{2E_{\rm
F}}\sqrt{\frac{E}{E_{\rm F}}}\hskip 14.45377pt({\rm 3D}).$

It is often more convenient to use expressions (8) for $D(E)$ instead of Eqs.
(4).

The probability that a state at energy $E$ of an electron gas is occupied at
temperature $T$ is given by the Fermi distribution function

$f(E,T)=\frac{1}{e^{(E-\mu)/k_{\rm B}T}+1},$ (9)

where $\mu$ is the chemical potential that in general depends on $T$. From Eq.
(9), the chemical potential is the energy at which the probability of the
state being occupied by an electron state is 1/2. One has $\mu(T\to 0)=E_{\rm
F}$, the Fermi energy.

In order to simplify notation and facilitate obtaining universal functions and
plots, we introduce dimensionless reduced variables

$\displaystyle\epsilon$ $\displaystyle\equiv$ $\displaystyle\frac{E}{E_{\rm
F}}$ $\displaystyle\bar{\mu}$ $\displaystyle\equiv$
$\displaystyle\frac{\mu}{E_{\rm F}}$ (10) $\displaystyle t$
$\displaystyle\equiv$ $\displaystyle\frac{k_{\rm B}T}{E_{\rm F}}.$

Then the Fermi function (9) becomes

$f(\epsilon,t)=\frac{1}{e^{(\epsilon-\bar{\mu})/t}+1},$ (11)

and the densities of states (8) become

$\displaystyle D(\epsilon)$ $\displaystyle=$ $\displaystyle\frac{N}{2E_{\rm
F}}\sqrt{\frac{1}{\epsilon}}\hskip 21.68121pt({\rm 1D})$ $\displaystyle
D(\epsilon)$ $\displaystyle=$ $\displaystyle\frac{N}{E_{\rm F}}\hskip
44.80704pt({\rm 2D})$ (12) $\displaystyle D(\epsilon)$ $\displaystyle=$
$\displaystyle\frac{3N}{2E_{\rm F}}\sqrt{\epsilon}\hskip 27.46295pt({\rm
3D}).$

We further define

$D(\epsilon)=\frac{N}{E_{\rm F}}{\tt D}(\epsilon),$

so the dimensionless reduced density of states ${\tt D}(\epsilon)$ is

$\displaystyle{\tt D}(\epsilon)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{\frac{1}{\epsilon}}\hskip 24.57153pt({\rm 1D})$
$\displaystyle{\tt D}(\epsilon)$ $\displaystyle=$ $\displaystyle 1\hskip
44.80704pt({\rm 2D})$ (13) $\displaystyle{\tt D}(\epsilon)$ $\displaystyle=$
$\displaystyle\frac{3}{2}\sqrt{\epsilon}\hskip 30.35326pt({\rm 3D}).$

##  III  Temperature Dependence of the Chemical Potential

###  III.1 Overall Temperature Dependence

In order to calculate the thermodynamic properties of a free-electron Fermi
gas over a wide $T$ range from the degenerate to the nondegenerate regimes,
one must first determine how the chemical potential $\mu$ in the Fermi
function (9) changes with $T$. The chemical potential is determined at each
$T$ by the requirement that the number $N$ of electrons in the Fermi gas be
equal to the number of occupied electron states:

$\displaystyle N$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}D(E)f(E,T)dE,$ $\displaystyle{\rm or}\ \ \ 1$
$\displaystyle=$ $\displaystyle\int_{0}^{\infty}{\tt
D}(\epsilon)f(\epsilon,t)d\epsilon,$ (14)

where $D(E)$ is the dimension-dependent density of states versus energy $E$
including the twofold Zeeman degeneracy of the electron, $\epsilon=E/E_{\rm
F}$ is the reduced energy defined in Eq. (10) where the Fermi energy $E_{\rm
F}$ is the chemical potential at $T=0$, ${\tt D}(\epsilon)$ is the reduced
density of states in Eq. (13), and $t$ is the reduced temperature $k_{\rm
B}T/E_{\rm F}$ where $k_{\rm B}$ is Boltzmann’s constant. The value of $E_{\rm
F}$ depends on the linear, areal, and volume density of electrons in 1D, 2D,
and 3D, respectively, as given in Eqs. (6).

To solve for the reduced chemical potential $\bar{\mu}(t)=\mu(t)/E_{\rm F}$ in
Eq. (11) using Eq. (14), we first find an expression for the definite integral
on the right-hand side of Eq. (14), which when substituted into Eq. (14)
yields an equation containing $\bar{\mu}$ and $t$ from which $\bar{\mu}(t)$
can be solved for. Using the ${\tt D}(\epsilon)$ expressions in Eqs. (13), one
can evaluate the integral in Eq. (14) for each of the three dimensions, and
Eq. (14) becomes

$\displaystyle-\frac{\sqrt{\pi t}}{2}\,{\rm
Li}_{1/2}\left(-e^{\bar{\mu}/t}\right)$ $\displaystyle=$ $\displaystyle
1\hskip 24.57153pt({\rm 1D})$ $\displaystyle
t\ln\left(1+e^{\bar{\mu}/t}\right)$ $\displaystyle=$ $\displaystyle 1\hskip
25.29494pt({\rm 2D})$ (15) $\displaystyle-\frac{3\sqrt{\pi}\,t^{3/2}}{4}\,{\rm
Li}_{3/2}\left(-e^{\bar{\mu}/t}\right)$ $\displaystyle=$ $\displaystyle
1\hskip 24.57153pt({\rm 3D}),$

where

${\rm Li}_{n}(x)\equiv\sum_{k=1}^{\infty}x^{k}/k^{n}$ (16)

is the polylogarithm function Mathematica ; PolylogWiki .

According to Eqs. (15), for the 2D case $\bar{\mu}(t)$ is given analytically
by

$\bar{\mu}(t)=t\ln(e^{1/t}-1)\hskip 36.135pt({\rm 2D}),$ (17)

as found previously McKelvey1964 . This immediately gives $\bar{\mu}(t\to
0)=1$ or $\mu(t\to 0)=E_{\rm F}$. For $t\gg 1$ one obtains
$\bar{\mu}(t)\approx-t\ln(t)$. For later use, Eq. (17) can be written

$1+e^{-\bar{\mu}/t}=\frac{1}{1-e^{-1/t}}\hskip 26.73953pt({\rm 2D}).$ (18)

In the 1D and 3D cases, Eqs. (15) are solved numerically for $\bar{\mu}(t)$.

Plots of $\bar{\mu}(t)$ for 1D, 2D, and 3D free-electron Fermi gases are shown
in Fig. 1, illustrating the crossover between the degenerate low-$T$ and
nondegenerate high-$T$ regime for each dimension. We note that the low-$T$
Sommerfeld expansion in 2D to any order in $T$ gives the $T$-independent value
$\mu=E_{\rm F}$ (Ref. Ashcroft1976 , p. 53), contrary to the $T$ dependence in
Fig. 1.

Figure 1: Chemical potential $\mu$ divided by the Fermi energy $E_{\rm F}$
($\equiv\bar{\mu}$) as a function of $k_{\rm B}T\equiv t$ for 1D, 2D, and 3D
free-electron Fermi gases. The analytic 2D $T$ dependence is given by Eq.
(17), and the 1D and 3D $T$ dependences were obtained by numerically solving
the respective equation in Eqs. (15).

###  III.2 Chemical Potential at High Temperatures

At temperatures sufficiently high that $\bar{\mu}$ in Fig. 1 becomes strongly
negative, the exponential factor $e^{-\bar{\mu}/t}$ in the denominator of the
Fermi function (11) becomes much larger than unity, which results in
$f(\epsilon,t)\ll 1$ for all electron energies $\epsilon>0$ of normal physical
interest. One can then ignore the additive factor of unity in the denominator
of the Fermi function, which then becomes

$f(\epsilon,t)=\frac{e^{-\epsilon/t}}{e^{-\bar{\mu}/t}}\equiv\frac{e^{-\epsilon/t}}{Z},$
(19)

which is just the Boltzmann probability distribution for a classical monatomic
ideal gas, where the partition function $Z$ is

$Z=e^{-\bar{\mu}(t)/t}\gg 1.$ (20)

Thus $e^{\bar{\mu}(t)/t}\ll 1$, and the Taylor-series expansion (16) of the
polylogarithm functions in Eqs. (15) about $x=0$ is

${\rm Li}_{n}(-x)=-x+\frac{x^{2}}{2^{n}}-\frac{x^{3}}{3^{n}}+\cdots,$ (21)

where in Eqs. (15) $x=e^{\bar{\mu}/t}\ll 1$ for the 1D and 3D cases. Keeping
only the first-order term in $x$, one has ${\rm Li}_{n}(-x)=-x$ for both
$n=1/2$ and 3/2 in 1D and 3D, respectively. In 2D, we set $e^{1/t}\approx
1+(1/t)$ in Eq. (17). Thus at high temperatures, Eqs. (15) yield

$\displaystyle\bar{\mu}$ $\displaystyle=$
$\displaystyle-\frac{t}{2}\ln\left(\frac{\pi t}{4}\right)\hskip
50.58878pt{\rm(1D,\ high}~{}t)$ $\displaystyle\bar{\mu}$ $\displaystyle=$
$\displaystyle-t\ln t\hskip 79.49744pt{\rm(2D,\ high}~{}t)$ (22)
$\displaystyle\bar{\mu}$ $\displaystyle=$
$\displaystyle-t\ln\left(\frac{3\sqrt{\pi}\,t^{3/2}}{4}\right)\hskip
27.46295pt{\rm(3D,\ high}~{}t).$

###  III.3 Fermi Function

At low temperatures $T\ll T_{\rm F}\equiv E_{\rm F}/k_{\rm B}$ where $T_{\rm
F}$ is the Fermi temperature, the Fermi function $f(\epsilon,t)$ is
independent of energy except near $E_{\rm F}$ where it decreases rapidly. This
feature suggests that to compute the temperature dependence of physical
properties at low temperatures, one should utilize expressions containing
$\partial f/\partial\epsilon$ which emphasizes the energy region around
$E_{\rm F}$. Using Eq. (11) for $f(\epsilon,t)$ gives

$\displaystyle\frac{\partial f(\epsilon,t)}{\partial\epsilon}$
$\displaystyle=$
$\displaystyle-\left(\frac{1}{2t}\right)\frac{1}{1+\cosh[(\epsilon-\bar{\mu})/t]}.$
(23)

The probability that one of the two Zeeman states of the electron within a
spatial orbital is occupied by an electron is $f$ and the probability that the
other one is unoccupied is $1-f$, so the probability that both events happen
at the same time is $f(1-f)$. Through a straightforward manipulation of
$f(\epsilon,t)$, one obtains

$f(1-f)=-t\frac{\partial f}{\partial\epsilon}.$ (24)

The low-temperature Sommerfeld expansion Ashcroft1976 ; WikiSommerfeld  of the
following integral $I(t)$ to order $t^{2}$ is

$I(t)=\int_{0}^{\infty}g(\epsilon)f(\epsilon,t)d\epsilon=\int_{0}^{\bar{\mu}}g(\epsilon)d\epsilon+\frac{dg(\epsilon)}{d\epsilon}\Big{|}_{\epsilon=\bar{\mu}}\frac{\pi^{2}t^{2}}{6},$
(25a) where here $g(\epsilon<0)=0$ and at low temperatures $\bar{\mu}\approx
1$. Let $G(\epsilon)$ be the indefinite integral $\displaystyle
G(\epsilon)=\int g(\epsilon)d\epsilon,\ \ {\rm or}\ \
g(\epsilon)=\frac{dG(\epsilon)}{d\epsilon}.$ (25b) Then an integration by
parts of the first integral in Eq. (25a) yields $\displaystyle I(t)$
$\displaystyle=$ $\displaystyle-\int_{0}^{\infty}G(\epsilon)\frac{\partial
f(\epsilon,t)}{\partial\epsilon}d\epsilon,$ (25c) $\displaystyle=$
$\displaystyle
G(\bar{\mu})+\frac{d^{2}G(\epsilon)}{d\epsilon^{2}}\bigg{|}_{\epsilon=\bar{\mu}}\frac{\pi^{2}t^{2}}{6}.$

Equations (25a) and (25c) are useful in different contexts.

###  III.4 Chemical Potential at Low Temperatures

The chemical potential is determined by Eq. (14). To evaluate the integral in
Eq. (14) in terms of the low-temperature expansion in Eq. (25a), the function
$g(\epsilon)={\tt D}(\epsilon)$, where ${\tt D}(\epsilon)$ is given in Eqs.
(13). Thus we have

$\displaystyle\int_{0}^{\bar{\mu}}{\tt D}(\epsilon)d\epsilon$ $\displaystyle=$
$\displaystyle\sqrt{\bar{\mu}},\ \ \ \ \frac{d{\tt
D}(\epsilon)}{d\epsilon}=-\frac{1}{4\bar{\mu}^{3/2}}\hskip 3.61371pt({\rm
1D})$ $\displaystyle\int_{0}^{\bar{\mu}}{\tt D}(\epsilon)d\epsilon$
$\displaystyle=$ $\displaystyle\bar{\mu},\ \ \ \ \ \ \ \frac{d{\tt
D}(\epsilon)}{d\epsilon}=0\hskip 30.35326pt({\rm 2D})$ (26)
$\displaystyle\int_{0}^{\bar{\mu}}{\tt D}(\epsilon)d\epsilon$ $\displaystyle=$
$\displaystyle\bar{\mu}^{3/2},\ \ \ \ \frac{d{\tt
D}(\epsilon)}{d\epsilon}=\frac{3}{4\sqrt{\bar{\mu}}}\hskip 10.84006pt({\rm
3D}).$

Inserting these results into Eq. (25a) gives

$\displaystyle\int_{0}^{\bar{\mu}}{\tt D}(\epsilon)f(\epsilon,t)d\epsilon$
$\displaystyle=$
$\displaystyle\sqrt{\bar{\mu}}-\frac{\pi^{2}t^{2}}{24\bar{\mu}^{3/2}}\hskip
3.61371pt({\rm 1D})$ $\displaystyle\int_{0}^{\bar{\mu}}{\tt
D}(\epsilon)f(\epsilon,t)d\epsilon$ $\displaystyle=$
$\displaystyle\bar{\mu}\hskip 56.3705pt({\rm 2D})$ (27)
$\displaystyle\int_{0}^{\bar{\mu}}{\tt D}(\epsilon)f(\epsilon,t)d\epsilon$
$\displaystyle=$
$\displaystyle\bar{\mu}^{3/2}+\frac{\pi^{2}t^{2}}{8\sqrt{\bar{\mu}}}\hskip
10.84006pt({\rm 3D}).$

####  III.4.1 One-Dimensional Fermi Gas

Setting the integral for 1D in Eq. (27) equal to unity according to Eq. (14)
gives

$\bar{\mu}^{2}-\bar{\mu}^{3/2}=\frac{\pi^{2}t^{2}}{24}.$ (28)

This is a 4th-order equation in $\bar{\mu}^{1/2}$. However, we take advantange
of the fact that $\bar{\mu}$ approaches unity at $t\to 0$. Thus we write

$\bar{\mu}(t)=1+\alpha(t)$ (29)

with $|\alpha(t)|\ll 1$ and use the Taylor-series expansion

$(1+\alpha)^{n}\approx 1+n\alpha$

to obtain from Eq. (28)

$(1+2\alpha)-\left(1+\frac{3}{2}\alpha\right)=\frac{\alpha}{2}=\frac{\pi^{2}t^{2}}{24}.$

Substituting $\alpha=\bar{\mu}-1$ gives

$\bar{\mu}=1+\frac{\pi^{2}t^{2}}{12}\hskip 14.45377pt({\rm 1D}).$ (30)

This increase in $\bar{\mu}$ with increasing $t$ at small $t$ agrees with the
1D plot in Fig. 1.

####  III.4.2 Two-Dimensional Fermi Gas

According to Eq. (17), for $t\ll 1$ one just obtains

$\displaystyle\bar{\mu}=1,$ (31)

indicating that there is no power-law Sommerfeld expansion for $\bar{\mu}(T)$
at low $T$, in agreement with Ref. Ashcroft1976 , p. 53.

####  III.4.3 Three-Dimensional Fermi Gas

Setting the integral for 3D in Eq. (27) equal to unity according to Eq. (14)
gives

$\bar{\mu}^{2}-\bar{\mu}^{1/2}=-\frac{\pi^{2}t^{2}}{8}.$ (32)

Then following the same steps as for the 1D case gives

$\bar{\mu}=1-\frac{\pi^{2}t^{2}}{12}\hskip 14.45377pt({\rm 3D}).$ (33)

This decrease in $\bar{\mu}$ with increasing $t$ is in agreement with the 3D
plot in Fig. 1 for small $t$. Interestingly, the temperature dependence of
$\bar{\mu}$ for 3D has the same magnitude but opposite sign as that in Eq.
(30) for 1D, and hence they are symmetrically disposed with respect to the
(nearly) temperature-independent behavior at low $t$ for the 2D case.

##  IV  Magnetic Spin Susceptibility

For a single free electron with spin $S=1/2$, the magnetic spin susceptibility
follows the Curie law

$\displaystyle\chi_{1}=\frac{C_{1}}{T}.$ (34a) The single-spin Curie constant
$C_{1}$ for spin $S=1/2$ is $\displaystyle C_{1}=\frac{g^{2}S(S+1)\mu_{\rm
B}^{2}}{3k_{\rm B}}=\frac{g^{2}\mu_{\rm B}^{2}}{4k_{\rm B}},$ (34b)

where $g\approx 2$ is the spectroscopic splitting factor of the electron
magnetic moment and $\mu_{\rm B}$ is the Bohr magneton. Here, with respect to
its spin, the term “free electron” means that the $z$-component of its
magnetic moment is free to fluctuate between its two spin magnetic quantum
numbers $m_{S}=\pm 1/2$. Such a spin is an “unpaired” spin. In a free-electron
gas, a spin is unpaired if another electron does not occupy the same orbital
with the other $m_{S}$ state, because according to the Pauli Exclusion
Principle the electron is then free to fluctuate back and forth between its
two $\mu_{z}=-gm_{S}\mu_{\rm B}$ $z$-component magnetic-moment values. The
probability that an electron occupies a spatial orbital with a particular spin
state ($\mu_{z}$ value) at energy $E$ and temperature $T$ is given by the
Fermi function $f(E,T)$. The probability that the other $\mu_{z}$ state is not
occupied is $1-f(E,T)$. Thus the probability $P_{\rm unpaired}(E,T)$ that an
electron is unpaired is given by

$P_{\rm unpaired}(E,T)=f(E,T)\,[1-f(E,T)].$

The magnetic susceptibility of the electron gas is then

$\chi=N_{\rm unpaired}\frac{C_{1}}{T}=N_{\rm unpaired}\frac{g^{2}\mu_{\rm
B}^{2}}{4k_{\rm B}T}.$ (35)

In dimensionless reduced temperature units $t=k_{\rm B}T/E_{\rm F}$ given in
Eqs. (10), Eq. (35) becomes

$\chi=N_{\rm unpaired}\frac{g^{2}\mu_{\rm B}^{2}}{4E_{\rm F}t}.$ (36)

The number of unpaired electrons at temperature $T$ is

$\displaystyle N_{\rm unpaired}(T)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}P_{\rm unpaired}(E,T)D(E)dE$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}f(E,T)[1-f(E,T)]D(E)dE.$

In dimensionless variables one has

$\frac{N_{\rm
unpaired}(t)}{N}=\int_{0}^{\infty}f(\epsilon,t)[1-f(\epsilon,t)]{\tt
D}(\epsilon)\,d\epsilon.$

Then Eq. (24) yields

$\displaystyle\frac{N_{\rm unpaired}(t)}{N}$ $\displaystyle=$
$\displaystyle-t\int_{0}^{\infty}{\tt D}(\epsilon)\frac{\partial
f(\epsilon,t)}{\partial\epsilon}\,d\epsilon$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{\infty}\frac{{\tt
D}(\epsilon)}{1+\cosh[(\epsilon-\bar{\mu})/t]}d\epsilon,$

where Eq. (23) was used to obtain the second equality.

For $t\ll 1$, we write Eq. (IV) as

$\displaystyle\lim_{t\to 0}\frac{N_{\rm unpaired}(t)}{N}$ $\displaystyle=$
$\displaystyle t\int_{0}^{\infty}{\tt
D}(\epsilon)\,\delta(\epsilon-1)\,d\epsilon$ (38) $\displaystyle=$
$\displaystyle t\,{\tt D}(\epsilon=1),$

where $\delta(x)$ is the Dirac delta function. Switching back to conventional
quantities using the conversion expressions in Sec. II gives

$\lim_{T\to 0}N_{\rm unpaired}=k_{\rm B}T\,D(E_{\rm F}).$

Then using Eq. (35) one obtains the well-known result for the Pauli spin
susceptibility

$\displaystyle\chi^{\rm Pauli}$ $\displaystyle\equiv$
$\displaystyle\chi(t=0)=[k_{\rm B}TD(E_{\rm F})]\frac{g^{2}\mu_{\rm
B}^{2}}{4k_{\rm B}T}$ (39) $\displaystyle=$ $\displaystyle\frac{g^{2}\mu_{\rm
B}^{2}}{4}D(E_{\rm F})=\frac{Ng^{2}\mu_{\rm B}^{2}}{4E_{\rm F}}{\tt D}(1).$

Then Eqs. (13) yield

$\displaystyle\chi(0)$ $\displaystyle=$ $\displaystyle\frac{Ng^{2}\mu_{\rm
B}^{2}}{8E_{\rm F}}\quad({\rm 1D})$ (40a) $\displaystyle=$
$\displaystyle\frac{Ng^{2}\mu_{\rm B}^{2}}{4E_{\rm F}}\quad({\rm 2D})$ (40b)
$\displaystyle=$ $\displaystyle\frac{3Ng^{2}\mu_{\rm B}^{2}}{8E_{\rm
F}}\quad({\rm 3D}).$ (40c)

Figure 2: (a) Normalized magnetic spin susceptibility $\chi(t)/\chi(0)$ versus
reduced temperature $t=k_{\rm B}T/E_{\rm F}$ for $S=1/2$ Fermi gases in 1D, 2D
and 3D, where $\chi(0)$ is the respective zero-temperature susceptibility. (b)
Inverse normalized susceptibility versus $t$. At high finite temperatures, the
susceptibilities in 1D and 3D approach a Curie-law behavior $C/T$ whereas the
data in 2D approach Curie-Weiss behavior $C/(T-\theta_{\rm 2D})$. The negative
$\theta_{\rm 2D}$ value arises from the Fermi statistics rather than from
electron interactions. Figure 3: Nuclear magnetic susceptibility data
normalized by the value at $T\to 0$, $\chi(T)/\chi(0)$, obtained from NMR
measurements versus temperature $T$ for liquid 3He under a pressure of 20.5
atm. (filled circles), where the data point at $T=0$ is extrapolated
Thomson1962 . A fit of the data by Eq. (41) is shown by the solid curve where
the fitted Fermi temperature is 0.30 K.

Using Eqs. (35), (IV), and (39), the normalized dimension-dependent
susceptibility in reduced variables is given by

$\displaystyle\frac{\chi(t)}{\chi(0)}=\frac{1}{2t\,{\tt
D}(1)}\int_{0}^{\infty}\frac{{\tt
D}(\epsilon)}{1+\cosh[(\epsilon-\bar{\mu})/t]}d\epsilon,$ (41)

where the dimension-dependent reduced densities of states ${\tt D}(\epsilon)$
are given in Eq. (13). Figure 2(a) shows plots of $\chi(t)/\chi(0)$ for 1D,
2D, and 3D $S=1/2$ Fermi gases from $t=0$ to 3. The dimensionality of the
Fermi gas is seen to significantly affect the results. The inverse normalized
susceptibilities are shown in Fig. 2(b) which will be discussed in Sec. IV.2.

Figure 3 shows our fit to the measured Thomson1962  spin susceptibility of the
$S=1/2$ nuclei in liquid 3He under a pressure of 20.5 atm which illustrates
the crossover from the degenerate regime at 0.045 K to the nondegenerate
regime at 1.0 K. The single-parameter fit yielded the Fermi temperature
$T_{\rm F}=0.30$ K.

In an applied magnetic field $H$, the Zeeman energy levels of the 3He nuclei
with azimuthal quantum number $m_{S}=\pm 1/2$ are split by the $H$ according
to $\Delta E=\gamma H$, where $\gamma$ is the nuclear gyromagnetic ratio. For
the fixed angular frequency $\omega$ of the rf magnetic field applied to the
sample, resonance occurs when the quantum of energy of the field $\hbar\omega$
equals $\Delta E$ ($\hbar$ is Planck’s constant divided by 2$\pi$), at which
frequency power is absorbed from the rf source. Plotting the absorbed power
versus $H$ at a given $T$ yields the rf absorption line which has a certain
linewidth. The integral of the absorption line versus $H$ is proportional to
the magnetic susceptibility of the sample at the given $T$. Then normalizing
the area of the absorption line to that in the “high”-$T$ Curie-law region at
1 K yields the $\chi(T)/\chi(T=0)$ data plotted in Fig. 3.

###  IV.1 Low-Temperature Behavior

In order to obtain the lowest-order power-law temperature dependence of the
magnetic susceptibility at low temperatures in 1D and 3D, we note that the
integral in Eq. (IV) has the form of the Sommerfeld expansion in Eq. (25c)
with $G(\epsilon)={\tt D}(\epsilon)$. Therefore in 1D and 3D we use Eqs. (25c)
and (IV) to obtain

$\displaystyle\frac{N_{\rm unpaired}(t)}{N}=t\left[{\tt
D}(\bar{\mu})+\frac{d^{2}{\tt
D}(\epsilon)}{d\epsilon^{2}}\Big{|}_{\bar{\mu}}\frac{\pi^{2}t^{2}}{6}\right].$
(42)

Then from Eq. (36) the spin susceptibility is

$\chi=\frac{Ng^{2}\mu_{\rm B}^{2}}{4E_{\rm F}}\left[{\tt
D}(\bar{\mu})+\frac{d^{2}{\tt
D}(\epsilon)}{d\epsilon^{2}}\Big{|}_{\bar{\mu}}\frac{\pi^{2}t^{2}}{6}\right].$
(43)

In 2D we obtain an analytic result for all $T$ below.

####  IV.1.1 1D Electron Gas

In one dimension, from Eqs. (13) we have

${\tt D}(\bar{\mu})=\frac{1}{2}\bar{\mu}^{-1/2}$ (44)

and

$\displaystyle\frac{d^{2}{\tt
D}(\epsilon)}{d\epsilon^{2}}\Big{|}_{\bar{\mu}}=\frac{3}{8}\bar{\mu}^{-5/2}.$
(45)

The 1D $t$ dependence of $\bar{\mu}$ at low temperatures is given by Eq. (30).
Then using the Taylor-series expansion $(1+x)^{n}\approx 1+nx$ gives, to order
$t^{2}$,

$\displaystyle{\tt D}(\bar{\mu})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(1-\frac{\pi^{2}t^{2}}{24}\right).$ (46a)
$\displaystyle\frac{d^{2}{\tt
D}(\epsilon)}{d\epsilon^{2}}\Big{|}_{\bar{\mu}}t^{2}$ $\displaystyle=$
$\displaystyle\frac{3}{8}t^{2}.$ (46b)

Inserting these results into Eq. (43) using Eqs. (39) and (13) gives

$\chi(t)=\chi_{\rm 1D}(0)\left(1+\frac{\pi^{2}t^{2}}{12}\right).$ (47)

####  IV.1.2 3D Electron Gas

In three dimensions, from Eqs. (13) one has

${\tt D}(\bar{\mu})=\frac{3}{2}\sqrt{\bar{\mu}}.$ (48)

We then obtain

$\frac{d^{2}{\tt
D}(\epsilon)}{d\epsilon^{2}}\Big{|}_{\bar{\mu}}=-\frac{3}{8}\bar{\mu}^{-3/2}.$

The reduced 3D chemical potential at low temperatures is given by Eq. (33).
Again using the Taylor series expansion $(1+x)^{n}\approx 1+nx$ for $x\ll 1$
gives, to order $t^{2}$,

$\displaystyle{\tt D}(\bar{\mu})$ $\displaystyle=$
$\displaystyle\frac{3}{2}\left(1-\frac{\pi^{2}t^{2}}{24}\right).$ (49)
$\displaystyle\frac{d^{2}{\tt
D}(\epsilon)}{d\epsilon^{2}}\Big{|}_{\bar{\mu}}t^{2}$ $\displaystyle=$
$\displaystyle-\frac{3}{8}t^{2}.$ (50)

Inserting these results into Eq. (43) using Eqs. (39) and (13) gives, to order
$t^{2}$,

$\chi(t)=\chi_{\rm 3D}(0)\left(1-\frac{\pi^{2}t^{2}}{12}\right).$ (51)

Thus the 1D and 3D temperature behaviors in Eqs. (47) and (51), respectively,
have the same temperature dependence magnitude but with opposite signs at low
temperatures. These behaviors agree with the respective plots in Fig. 2(a) at
low temperatures.

####  IV.1.3  2D Electron Gas

For an electron gas in two dimensions an analytic expression for the spin
susceptibility can be obtained. The 2D case is special because the density of
states is independent of the energy of the $S=1/2$ fermion, as shown in Eqs.
(4) and (13). Therefore Eq. (IV) becomes

$\displaystyle\frac{N_{\rm unpaired}(T)}{N}$ $\displaystyle=$
$\displaystyle-t{\tt D}(\epsilon)\int_{0}^{\infty}df(\epsilon)$
$\displaystyle=$ $\displaystyle-t{\tt
D}(\epsilon)[f(\epsilon=\infty)-f(\epsilon=0)].$

Since $f(\epsilon,t)=1/[e^{(\epsilon-\bar{\mu})/t}+1]$, one has

$f(\epsilon=\infty)=0$

and

$f(\epsilon=0)=\frac{1}{e^{-\bar{\mu}/t}+1}=1-e^{-1/t},$

where the latter equality was given in Eq. (18). Then Eq.
(LABEL:Eq:Nunpaired2D) becomes

$\frac{N_{\rm unpaired}(T)}{N}=t{\tt D}(\epsilon)\left(1-e^{-1/t}\right).$
(53)

Using ${\tt D}(\epsilon)=1$ and $t=k_{\rm B}T/E_{\rm F}$ from Sec. II gives

$N_{\rm unpaired}(T)=N\frac{k_{\rm B}T}{E_{\rm F}}\left(1-e^{-E_{\rm F}/k_{\rm
B}T}\right).$ (54)

Finally, substituting this into Eq. (35) gives

$\chi=\frac{Ng^{2}\mu_{\rm B}^{2}}{4E_{\rm F}}\left(1-e^{-E_{\rm F}/k_{\rm
B}T}\right),$ (55a) where using Eqs. (12) one obtains
$\chi(T=0)=\frac{Ng^{2}\mu_{\rm B}^{2}}{4E_{\rm F}}=\frac{g^{2}}{4}\mu_{\rm
B}^{2}D(E_{\rm F}),$ (55b)

as in the general Eq. (39). The $\chi(t)/\chi(0)$ from Eqs. (55) is plotted
versus $t$ in Fig. 2(a) along with the 1D and 3D results.

Figure 4: Comparison of the numerically-exact calculations of the normalized
magnetic susceptibility $\chi(T)/\chi(0)$ in 1D, 2D, and 3D (solid curves)
with the predictions of Sommerfeld expansions (dashed curves) at low
temperatures relative to the Fermi temperature $E_{\rm F}/k_{\rm B}$.

A Sommerfeld expansion of the low-$T$ magnetic susceptibility behavior in 2D
to any order in $t$ gives $\chi(t)/\chi(0)=1$ Ashcroft1976 . Comparisons of
the numerically-exact predictions of the normalized susceptibilities
$\chi(t)/\chi(0)$ in 1D, 2D, and 3D in Fig. 2 at low temperatures with the
Sommerfeld expansion predictions are shown in Fig. 4. The agreement of these
expansions with the exact results is seen to be limited to the temperature
range $t\lesssim 0.1$.

###  IV.2  High-Temperature Behavior

According to Eq. (19), at very high temperatures the Fermi function becomes
the Boltzmann distribution

$f(\epsilon,t)=\frac{e^{-\epsilon/t}}{Z},$

where $Z(t)=e^{-\bar{\mu}/t}$ is the partition function. Then

$\frac{\partial
f(\epsilon,t)}{\partial\epsilon}=-\frac{1}{t}\,\frac{e^{-\epsilon/t}}{Z}=-\frac{1}{t}f(\epsilon,t),$
(56)

and using Eq. (14), Eq. (IV) becomes

$\frac{N_{\rm unpaired}(t)}{N}=\int_{0}^{\infty}{\tt
D}(\epsilon)f(\epsilon,t)\,d\epsilon=1.$ (57)

Thus for $k_{\rm B}T\gg E_{\rm F}$, the number of unpaired electrons is equal
to the total number $N$ of electrons in the system, and the spin
susceptibility is just that of $N$ free spins-1/2,

$\displaystyle\chi=\frac{NC_{1}}{T}=\frac{Ng^{2}\mu_{\rm B}^{2}}{4k_{\rm
B}T}=\frac{Ng^{2}\mu_{\rm B}^{2}}{4E_{\rm F}t},$ (58)

which is the same for 1D, 2D, and 3D free-electron gases, where $C_{1}$ is the
single-electron Curie constant in Eq. (34b).

It is of interest to calculate the spin susceptibility in 2D at high
temperatures, but not in the limit $T\to\infty$, by keeping one additional
term in the Taylor-series expansion of the exponential, i.e.,
$e^{x}=1+x+x^{2}/2$. Then Eq. (55a) becomes

$\displaystyle\chi$ $\displaystyle=$ $\displaystyle\frac{Ng^{2}\mu_{\rm
B}^{2}}{4E_{\rm F}}\left[\frac{E_{\rm F}}{k_{\rm
B}T}-\frac{1}{2}\left(\frac{E_{\rm F}}{k_{\rm B}T}\right)^{2}\right]$
$\displaystyle=$ $\displaystyle\frac{Ng^{2}\mu_{\rm B}^{2}}{4k_{\rm
B}T}\left(1-\frac{1}{2}\frac{E_{\rm F}}{k_{\rm B}T}\right).$

Using the Taylor-series expansion $(1+x)^{n}\approx 1+nx$ for small $x$ which
gives $1-x\approx 1/(1+x)$ with $x=E_{\rm F}/(2k_{\rm B}T$), Eq. (IV.2) can be
written

$\displaystyle\chi$ $\displaystyle=$ $\displaystyle\frac{Ng^{2}\mu_{\rm
B}^{2}}{4k_{\rm B}T\left(1+\frac{E_{\rm F}}{2k_{\rm
B}T}\right)}=\frac{Ng^{2}\mu_{\rm B}^{2}}{4k_{\rm B}\left(T+\frac{E_{\rm
F}}{2k_{\rm B}}\right)}.$ (60)

This has the form of the Curie-Weiss law

$\chi=\frac{C}{T-\theta_{\rm 2D}},$ (61)

where $C=Ng^{2}\mu_{\rm B}^{2}/(4k_{\rm B})$ is the Curie constant of the
$S=1/2$ Fermi gas and the Weiss temperature is

$\displaystyle\theta_{\rm 2D}=-E_{\rm F}/2k_{\rm B}.$ (62)

In reduced units one has

$\displaystyle\bar{\theta}_{\rm 2D}\equiv\frac{k_{\rm B}\theta_{\rm
2D}}{E_{\rm F}}=-\frac{1}{2}.$ (63)

The negative value agrees with expectation from Fig. 2(b) obtained by
extrapolating the nearly linear $\chi(0)/\chi(t)$ data for $t>1$ to the
horizontal axis. In contrast to the usual interpretation of the Weiss
temperature, the Weiss temperature does not arise from interactions between
the spins but rather from the temperature-dependent Fermi statistics.

Figure 5: Approach of the normalized susceptibilities $\chi(t)t/\chi(0)$ in 1D
and 3D and $\chi(t)(t-\bar{\theta}_{\rm 2D})/\chi(0)$ in 2D to the high-$t$
Curie-law limit, with $\bar{\theta}_{\rm 2D}=-1/2$. In 1D, 2D, and 3D, panels
(a–c) respectively show that $\chi(t)t$ approaches the asymptotic high-$t$
limits of 2, 1, and 2/3 logarithmically in $t$, linearly in $t$, and inversely
in $t$, respectively.

It is also of interest to examine the manner in which the 1D, 2D, and 3D
susceptibilities asymptote to the Curie-law limit at high temperatures. In
Figs. 5(a–c) we plot $\frac{\chi(t)t}{\chi(0)}$ versus $\log_{10}(t)$ for 1D,
$\chi(t)(t-\bar{\theta}_{\rm 2D})/\chi(0)$ versus $t$ for 2D, and
$\frac{\chi(t)t}{\chi(0)}$ versus $1/t$ for 3D. The choice for the abscissa
label for 2D was motivated by the Curie-Weiss behavior in Eq. (61). In 1D, the
approach of $\chi T$ to its high-$T$ limit was extremely slow, which was found
to be logarithmic. In 3D, the approach of $\chi T$ to its high-$T$ limit was
determined by trial and error to vary as $1/T$. The high-$T$ limits are seen
from Fig. 5 to be

$\displaystyle\lim_{t\to\infty}\frac{\chi(t)t}{\chi(0)}$ $\displaystyle=$
$\displaystyle 2\quad{\rm(1D)}$ (64a) $\displaystyle=$ $\displaystyle
1\quad{\rm(2D)}$ (64b) $\displaystyle=$
$\displaystyle\frac{2}{3}\quad{\rm(3D)}.$ (64c)

Using Eqs. (40) and (58), these results confirm that at high $T$, the
susceptibility in each dimension is just that of $N$ isolated spins.

##  V  Internal Energy and Heat Capacity at Constant Volume

The heat capacity $C_{\rm V}$ at constant volume $V$ is defined as

$C_{\rm V}=\left(\frac{dE_{\rm ave}}{dT}\right)_{V},$ (65)

where the thermal average (internal) energy per electron is

$E_{\rm ave}(T)=\int_{0}^{\infty}E\,f(E,T)\,D(E)\,dE.$ (66)

At $T=0$, the Fermi function $f(E,T=0)$ is a step function that is equal to 1
below $E_{\rm F}$ and is equal to 0 above $E_{\rm F}$. Therefore

$E_{\rm ave}(T=0)=\int_{0}^{E_{\rm F}}E\,D(E)\,dE.$ (67)

Using the density of states $D(E)$ functions in Eqs. (8), one obtains the
ground-state energy as

$\displaystyle E(T=0)$ $\displaystyle=$ $\displaystyle\frac{1}{3}NE_{\rm
F}\hskip 14.45377pt({\rm 1D})$ $\displaystyle E(T=0)$ $\displaystyle=$
$\displaystyle\frac{1}{2}NE_{\rm F}\hskip 14.45377pt({\rm 2D})$ (68)
$\displaystyle E(T=0)$ $\displaystyle=$ $\displaystyle\frac{3}{5}NE_{\rm
F}\hskip 14.45377pt({\rm 3D}).$

The increase in $E_{\rm ave}(T=0)$ with increasing dimensionality follows
because the density of states decreases with energy $E$ in 1D, is constant in
2D and increases with $E$ in 3D.

Figure 6: Normalized average reduced energy per electron $\epsilon_{\rm
ave}=E_{\rm ave}/NE_{\rm F}$ versus reduced temperature $t=k_{\rm B}T/E_{\rm
F}$ for free-electron Fermi gases in 1D, 2D, and 3D calculated using Eqs.
(70).

By rewriting Eq. (67) in terms of the dimensionless variables in Sec. II, one
obtains the following expression for the reduced average energy per electron
versus reduced temperature

$\epsilon_{\rm ave}(t)=\int_{0}^{\infty}\epsilon\,f(\epsilon,t)\,{\tt
D}(\epsilon)\,d\epsilon.$ (69)

Then using the ${\tt D}(\epsilon)$ functions in Eqs. (13), we obtain

$\displaystyle\epsilon_{\rm ave}(t)$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{\pi}}{2}\,t^{3/2}\,{\rm
Li}_{3/2}\left(-e^{\bar{\mu}/t}\right)\hskip 10.84006pt({\rm 1D})$
$\displaystyle\epsilon_{\rm ave}(t)$ $\displaystyle=$
$\displaystyle-t^{2}\,{\rm Li}_{2}\left(-e^{\bar{\mu}/t}\right)\hskip
46.97505pt({\rm 2D})$ (70) $\displaystyle\epsilon_{\rm ave}(t)$
$\displaystyle=$ $\displaystyle-\frac{9\sqrt{\pi}}{8}\,t^{5/2}\,{\rm
Li}_{5/2}\left(-e^{\bar{\mu}/t}\right)\hskip 7.22743pt({\rm 3D}),$

where the polylogarithm function Li${}_{n}(x)$ was introduced in Eq. (16). In
order to plot and/or otherwise utilize these functions, one must first insert
the reduced chemical potential $\bar{\mu}(t)$ determined from Eqs. (15). The
$\epsilon_{\rm ave}(t)$ functions are plotted in Fig. 6 in 1D, 2D, and 3D. The
data in Fig. 6 for all three dimensions are approximately linear in $T$ for
$t\gtrsim 1$, which means that the constant-volume heat capacity $C_{\rm
V}(T)=dE_{\rm ave}/dT$ is approximately constant above this temperature. This
is confirmed below.

The $C_{\rm V}(t)$ can be determined by differentiating Eq. (69) with respect
to $t$, which gives $C_{\rm V}(t)$ per mole of electrons as

$C_{\rm V}(t)=R\frac{d\epsilon_{\rm
ave}(t)}{dt}=R\int_{0}^{\infty}\epsilon\,\frac{df(\epsilon,t)}{dt}\,{\tt
D}(\epsilon)\,d\epsilon,$ (71)

where $R=N_{\rm A}k_{\rm B}$ is the molar gas constant and $N_{\rm A}$ is
Avogadro’s number. Keeping in mind that $\bar{\mu}$ is a function of $t$, the
total derivative $df/dt$ is obtained as

$\displaystyle\frac{df(\epsilon,t)}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{t^{2}}\frac{e^{(\epsilon-\bar{\mu})/t}}{\left[e^{(\epsilon-\bar{\mu})/t}+1\right]^{2}}\left[\epsilon-\bar{\mu}+t\frac{d\bar{\mu}(t)}{dt}\right].$
(72)

The temperature derivative $d\bar{\mu}(t)/dt$ is found by taking the total $t$
derivative of Eq. (14) and then solving for $d\bar{\mu}/dt$, which gives

$\frac{d\bar{\mu}}{dt}(t)=-\frac{1}{t}\frac{\int_{0}^{\infty}\frac{{\tt
D}(\epsilon)(\epsilon-\bar{\mu})e^{(\epsilon-\bar{\mu})/t}}{\left[e^{(\epsilon-\bar{\mu})/t}+1\right]^{2}}d\epsilon}{\int_{0}^{\infty}\frac{{\tt
D}(\epsilon)e^{(\epsilon-\bar{\mu})/t}}{\left[e^{(\epsilon-\bar{\mu})/t}+1\right]^{2}}d\epsilon}.$
(73)

Inserting Eq. (73) into (72) and the result into (71) gives

$\frac{C_{\rm V\,1D}}{R}=\frac{\sqrt{\pi t}}{8}\bigg{\\{}\frac{[{\rm
Li}_{1/2}(-e^{\bar{\mu}(t)/t})]^{2}}{{\rm
Li}_{-1/2}(-e^{\bar{\mu}(t)/t})}-3{\rm
Li}_{3/2}\left(-e^{\bar{\mu}(t)/t}\right)\bigg{\\}},$ (74a)
$\displaystyle\frac{C_{\rm V\,2D}}{R}$ $\displaystyle=$
$\displaystyle-t\left(e^{-\mu(t)/t}+1\right)\left[\ln\left(1+e^{\mu(t)/t}\right)\right]^{2}$
(74b) $\displaystyle-2t{\rm Li}_{2}\left(-e^{\mu(t)/t}\right),$
$\displaystyle\frac{C_{\rm V\,3D}}{R}$ $\displaystyle=$
$\displaystyle\frac{9\sqrt{\pi}t^{3/2}}{16}\bigg{\\{}\frac{3\left[{\rm
Li}_{3/2}\left(-e^{\mu(t)/t}\right)\right]^{2}}{{\rm
Li}_{1/2}\left(-e^{\mu(t)/t}\right)}$ $\displaystyle-5{\rm
Li}_{5/2}\left(-e^{\bar{\mu}(t)/t}\right)\bigg{\\}}.$

Figure 7: (a) Reduced molar heat capacity at constant volume $C_{\rm V}/R$
versus reduced temperature $t=k_{\rm B}T/E_{\rm F}$ for free-electron gases in
1D, 2D, and 3D. The horizontal dashed lines are the predictions $C_{\rm
V}/R=d/2$ for the ideal monatomic gas in 1D, 2D, and 3D, respectively, where
$d$ is the dimensionality. $C_{\rm V}(T)/R$ for the 1D case asymptotically
decreases with increasing temperature towards the value $C_{\rm V}/R=1/2$, in
contrast to the monotonic asymptotic increases in the 2D and 3D cases. (b)
Expanded plots of the low-$t$ regions of (a), where $C_{\rm V}\propto T$ at
the lowest $T$ for all three dimensions as described by Eq. (79a).

Using Eqs. (74), $C_{\rm V}(t)$ was calculated for 1D, 2D and 3D $S=1/2$ Fermi
gases which are plotted in Fig. 7(a). At high temperatures, $C_{\rm V}(t)$
approaches the temperature-independent value $C_{\rm V}=Rd/2$ obtained from
the equipartition theorem for a classical monatomic ideal gas in the
respective dimension $d$ where $R$ is the molar gas constant. Expanded plots
of the low-temperature region are shown in Fig. 7(b), where $C_{\rm V}\propto
T$ at the lowest temperatures $t\lesssim 0.05$ for all three dimensions, as
discussed next.

Using the ${\tt D}(\epsilon)$ functions in Eqs. (13), the function
$g(\epsilon)=\epsilon{\tt D}(\epsilon)$ in Eq. (25a) is given by

$\displaystyle g(\epsilon)$ $\displaystyle=$
$\displaystyle\frac{\epsilon^{1/2}}{2}\hskip 36.135pt({\rm 1D})$
$\displaystyle g(\epsilon)$ $\displaystyle=$ $\displaystyle\epsilon\hskip
51.31218pt({\rm 2D})$ (75) $\displaystyle g(\epsilon)$ $\displaystyle=$
$\displaystyle\frac{3\,\epsilon^{3/2}}{2}\hskip 29.63095pt({\rm 3D}).$

Substituting these $g(\epsilon)$ functions into Eq. (25a) gives

$\displaystyle\epsilon_{\rm ave}(t)$ $\displaystyle=$
$\displaystyle\frac{\bar{\mu}^{3/2}}{3}+\bar{\mu}^{-1/2}\,\frac{\pi^{2}t^{2}}{24}\hskip
18.06749pt({\rm 1D})$ $\displaystyle\epsilon_{\rm ave}(t)$ $\displaystyle=$
$\displaystyle\frac{\bar{\mu}^{2}}{2}+\frac{\pi^{2}t^{2}}{6}\hskip
53.4802pt({\rm 2D})$ (76) $\displaystyle\epsilon_{\rm ave}(t)$
$\displaystyle=$
$\displaystyle\frac{3\bar{\mu}^{5/2}}{5}+\bar{\mu}^{1/2}\,\frac{9\pi^{2}t^{2}}{24}\hskip
15.17719pt({\rm 3D}).$

Inserting the $\bar{\mu}(t)$ dependences in Eqs. (30), (31), and (33) into
these expressions, respectively, and using the Taylor-series expansion
$(1+x)^{n}\approx 1+nx$ gives, to order $t^{2}$,

$\displaystyle\epsilon_{\rm ave}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{3}+\frac{\pi^{2}t^{2}}{12}\hskip 18.06749pt({\rm 1D})$
$\displaystyle\epsilon_{\rm ave}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}+\frac{\pi^{2}t^{2}}{6}\hskip 18.06749pt({\rm 2D})$
(77) $\displaystyle\epsilon_{\rm ave}(t)$ $\displaystyle=$
$\displaystyle\frac{3}{5}+\frac{\pi^{2}t^{2}}{4}\hskip 18.06749pt({\rm 3D}).$

The values of zero-temperature energy per particle, $\epsilon_{\rm
ave}(t=0)=E_{\rm ave}(T=0)/NE_{\rm F}$, agree with the respective values in
Eqs. (68). Finally, using Eq. (71) we obtain the low-temperature molar heat
capacity expansions

$\displaystyle C_{\rm V}$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}R}{6}t\hskip 18.06749pt({\rm 1D})$ $\displaystyle
C_{\rm V}$ $\displaystyle=$ $\displaystyle\frac{\pi^{2}R}{3}t\hskip
18.06749pt({\rm 2D})$ (78) $\displaystyle C_{\rm V}$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}R}{2}t\hskip 18.06749pt({\rm 3D}).$

In terms of the respective conventional densities of states at the Fermi
energy $D(E_{\rm F})$ in Eqs. (8), these equations can be written for all
three dimensions as

$C_{\rm V}=\frac{\pi^{2}k_{\rm B}^{2}}{3}D(E_{\rm F})T\equiv\gamma T,$ (79a)
where the Sommerfeld coefficient (electronic heat capacity coefficient
$\gamma$) is given by the well-known result $\gamma=\frac{\pi^{2}k_{\rm
B}^{2}}{3}D(E_{\rm F}).$ (79b)

From Fig. 7(b), the linear temperature dependence of $C_{\rm V}$ is seen to be
followed for temperatures $T\lesssim 0.05\,T_{\rm F}$.

##  VI  Pressure in Two and Three Dimensions

For a monatomic ideal gas, the equation of state is $pV=Nk_{\rm B}T$, where
$p$ is the pressure and $V$ is the volume in 3D and the area in 2D. From the
Equipartition Theorem the internal energy is $U=Nk_{\rm B}T$ in 2D and
$U=3Nk_{\rm B}T/2$ in 3D. Thus one has

$\displaystyle pV(T)=\frac{2}{d}U(T),$ (80)

where $d$ is the dimensionality of the gas. The same expression is valid at
each temperature for the 2D and 3D quantum $S=1/2$ Fermi gases, respectively
Mancarella2014 . In reduced units Eq. (80) becomes

$\displaystyle\frac{pV}{NE_{\rm F}}(t)$ $\displaystyle=$
$\displaystyle\frac{2}{d}\epsilon_{\rm ave}(t).$ (81)

Using the second and third of Eqs. (70), we obtain

$\displaystyle\frac{pV}{NE_{\rm F}}$ $\displaystyle=$ $\displaystyle-t^{2}{\rm
Li}_{2}(-e^{\bar{\mu}(t)/t})\quad({\rm 2D}),$ (82a)
$\displaystyle\frac{pV}{NE_{\rm F}}$ $\displaystyle=$
$\displaystyle-\frac{3\sqrt{\pi}}{4}\,t^{5/2}\,{\rm
Li}_{5/2}(-e^{\bar{\mu}(t)/t})\quad({\rm 3D}),$ (82b)

where $\bar{\mu}(t)$ is calculated by solving the second and third of Eqs.
(15), respectively. The compression factors obtained from Eqs. (82) are

$\displaystyle\frac{pV}{Nk_{\rm B}T}$ $\displaystyle=$ $\displaystyle-t{\rm
Li}_{2}(-e^{\bar{\mu}(t)/t})\ \ ({\rm 2D}),$ (83a)
$\displaystyle\frac{pV}{Nk_{\rm B}T}$ $\displaystyle=$
$\displaystyle-\frac{3\sqrt{\pi}}{4}\,t^{3/2}\,{\rm
Li}_{5/2}(-e^{\bar{\mu}(t)/t})\ {(\rm 3D)}.$ (83b)

Figure 8: (a) Reduced pressure $pV/NE_{\rm F}$ versus reduced temperature
$t=k_{\rm B}T/E_{\rm F}$ for a free-electron Fermi gas in three dimensions.
(b) Compression factor $pV/Nk_{\rm B}T$ versus $t$ for the same gas. For 2D,
$V$ is the area of the gas.

The scaled pressures $pV/NE_{\rm F}$ versus reduced temperature $t=k_{\rm
B}T/E_{\rm F}$ computed for 2D and 3D electron Fermi gases using Eqs. (82) are
plotted in Fig. 8(a) Wiki . As seen in the figure, the reduced degeneracy
pressure is $pV/NE_{\rm F}=0.4$ at $t\to 0$ in 3D and 0.5 in 2D. The pressure
increases with increasing temperature, approaching a classical proportional
behavior at high temperatures. This behavior is examined in more detail in
Fig. 8(b), where the compression factors $pV/Nk_{\rm B}T$ are plotted versus
$t$ using Eqs. (83). At low $t$, this quantity diverges as $1/t$ because $p$
attains a constant value for $t\to 0$ as shown in Fig. 8(a). At high $t$, the
compression factors approach unity since the high-$t$ limit corresponds to the
ideal-gas law.

##  VII Enthalpy and Heat Capacity at Constant Pressure

Figure 9: Reduced enthalpy $H/NE_{\rm F}$ versus reduced temperature $t=k_{\rm
B}T/E_{\rm F}$ for a $S-1/2$ Fermi gas in 2D and 3D.

Here only the 2D and 3D cases are relevant. The enthalpy $H$ is defined as

$\displaystyle H=U+pV.$ (84)

In reduced variables, this expression becomes

$\displaystyle\frac{H}{NE_{\rm F}}(t)=\epsilon_{\rm ave}(t)+\frac{pV}{NE_{\rm
F}}(t),$ (85)

where the temperature dependences of the first and second term are given in
Eqs. (70) and (82) and plotted for 2D and 3D in Figs. 6 and 8(a),
respectively. Plots of the reduced enthalpy versus reduced temperature in 2D
and 3D are shown in Fig. 9.

The heat capacity at constant pressure is given by $C_{\rm p}=(\partial
H/\partial T)$. This yields the molar heat capacity as

$\displaystyle\frac{C_{\rm p}}{R}=\frac{\partial[H/(NE_{\rm F})]}{\partial
t}.$ (86)

Then using Eq. (85) one obtains

$\displaystyle\frac{C_{\rm p}(t)}{R}=\frac{C_{\rm
V}(t)}{R}+\frac{\partial}{\partial t}\left(\frac{pV}{NE_{\rm F}}(t)\right),$
(87)

where the first and second terms are calculated from Eqs. (74) and Eqs. (82),
respectively. At all temperatures we find the exact results

$\displaystyle\frac{C_{\rm p}(t)}{C_{\rm V}(t)}$ $\displaystyle=$
$\displaystyle 2\qquad(2{\rm D}),$ (88a) $\displaystyle\frac{C_{\rm
p}(t)}{C_{\rm V}(t)}$ $\displaystyle=$ $\displaystyle\frac{5}{3}\qquad(3{\rm
D}),$ (88b)

where $C_{\rm V}(t)$ was calculated for both dimensions in Sec. V and plotted
in 1D, 2D, and 3D in Fig. 7. Interestingly, the results in Eqs. (88) are the
same as for a monatomic ideal gas in 2D and 3D, respectively.

##  VIII  Isothermal Bulk Modulus

The isothermal bulk modulus $K_{T}$ is defined as

$\displaystyle K_{T}=-V\left(\frac{dp}{dV}\right)_{T},$ (89)

with dimensions of pressure. Figure 8(b) plotted $pV/Nk_{\rm B}T$ versus
$t=k_{\rm B}T/E_{\rm F}$. Thus at fixed $T$ one has

$\displaystyle\frac{pV}{Nk_{\rm B}T}$ $\displaystyle=$ $\displaystyle f(t)\
({\rm a\ number}),$ (90a) $\displaystyle p$ $\displaystyle=$
$\displaystyle\frac{Nk_{\rm B}T}{V}f(t),$ (90b)
$\displaystyle\left(\frac{\partial p}{\partial V}\right)_{T}$ $\displaystyle=$
$\displaystyle-\frac{p}{V}.$ (90c)

Hence using Eq. (89) one obtains for 2D and 3D free-electron Fermi gases the
same expression

$\displaystyle K_{T}(t)$ $\displaystyle=$ $\displaystyle p(t),$ (91a)
$\displaystyle\frac{K_{T}V}{NE_{\rm F}}$ $\displaystyle=$
$\displaystyle\frac{pV}{NE_{\rm F}}.$ (91b)

Equation (91a) is the same as for a monatomic ideal gas in either 2D or 3D.
The reduced pressure $pV/NE_{\rm F}$ was plotted versus $t$ for 2D and 3D
free-electron Fermi gases in Fig. 8(a).

##  IX  Thermal Expansion Coefficient

Figure 10: (a) Reduced thermal expansion coefficient $\alpha E_{\rm F}/k_{\rm
B}$ versus reduced temperature $t=k_{\rm B}T/E_{\rm F}$ for a free-electron
Fermi gas in 2D and 3D. (b) $\alpha T$ versus $t$. The data approach unity at
high temperatures, which is the property of a monatomic ideal gas.

The thermal expansion coefficient $\alpha$ is defined as

$\displaystyle\alpha=\frac{1}{V}\left(\frac{\partial V}{\partial
T}\right)_{p}.$ (92)

A useful Maxwell relation is

$\displaystyle\alpha=\frac{1}{K_{T}}\left(\frac{\partial p}{\partial
T}\right)_{V},$ (93)

where for ideal gases and $S=1/2$ Fermi gases $K_{T}=p$ from Eq. (91a).
Therefore

$\displaystyle\alpha=\frac{1}{p}\left(\frac{\partial p}{\partial
T}\right)_{V}.$ (94)

The reduced pressure and temperature are

$\displaystyle\bar{p}=\frac{pV}{NE_{\rm F}},\qquad t=\frac{k_{\rm B}T}{NE_{\rm
F}},$ (95)

so the reduced thermal expansion coefficient is

$\displaystyle\frac{\alpha E_{\rm F}}{k_{\rm
B}}=\frac{1}{\bar{p}}\left(\frac{\partial\bar{p}}{\partial t}\right)_{V},$
(96)

where $\bar{p}(t)$ was calculated in Sec. VI. In terms of the $T$-dependent
chemical potential, we obtain

$\displaystyle\frac{\alpha E_{\rm F}}{k_{\rm
B}}=\frac{1}{2t}\left\\{5-\frac{3\left[{\rm
Li}_{3/2}(-e^{\bar{\mu}(t)/t})\right]^{2}}{{\rm
Li}_{1/2}(-e^{\bar{\mu}(t)/t}){\rm Li}_{5/2}(-e^{\bar{\mu}(t)/t})}\right\\}.$
(97)

Figure 10(a) shows plots of $\alpha E_{\rm F}/k_{\rm B}$ versus $t$ for 2D and
3D $S=1/2$ Fermi gases, which show that $\alpha(t=0)=0$. The curves show peaks
at $t\approx 0.3$, followed by a smooth decrease at higher temperatures. To
examine this decrease quantitatively, $\alpha T$ versus $t$ is plotted for the
two dimensions in Fig. 10(b). The data for both dimensions asymptotically
approach unity, which is the value for the ideal gas since $\alpha=1/T$ for
that gas.

##  X  Concluding Remarks

The properties calculated versus dimension and temperature in this paper
include the magnetic spin susceptibility, internal energy, heat capacity at
constant volume, pressure, enthalpy, heat capacity at constant pressure,
isothermal bulk modulus, and thermal-expansion coefficient. The crossovers of
these properties between the low-temperature degenerate regime and the high-
temperature nondegenerate regime were elucidated and analyzed. The high-
temperature limit of the magnetic susceptibility is that of isolated
spins-1/2, and those of the other thermodynamic quantities correspond to the
respective properties of the monatomic ideal gas.

The expressions for three of the thermodynamic properties have forms analogous
to those for the monatomic ideal gas. The temperature-independent ratio of
heat capacities $C_{\rm p}/C_{\rm V}$ for the free-electron Fermi gas in 2D
and 3D in Eqs. (88), the temperature-dependent expressions for the pressure
times the volume in terms of the internal energy for the respective dimension,
and in each of 2 and 3 dimensions the expression (91a) for the isothermal bulk
modulus for both 2D and 3D electron Fermi gases are the same as for the
monatomic ideal gas.

Usually the theory for the nonrelativistic free-electron Fermi gas treated
here is only of practical interest at temperatures low compared to the Fermi
temperature, because the latter temperature for most metals is very high
compared to the maximum accessible measurement temperature. However one can
envision metals with low Fermi temperatures for which the present results
might be useful. Of particular interest might be semimetals with low carrier
concentrations. The challenge would be to separate the properties of the Fermi
gas from those of the lattice. However, the magnetic properties of a Fermi gas
might be measurable from the degenerate to the nondegenerate regime if the
purity of the nonmagnetic host was sufficient. A special case is the $S=1/2$
nuclear magnetic susceptibility of liquid 3He, for which the experimental data
from the literature for the magnetic susceptibility versus temperature at a
pressure of 20.5 atm in Fig. 3 were fitted well by the theory from a
temperature of 0.045 K in the degenerate temperature regime to 1 K in the
nondegenerate temperature regime, yielding a Fermi temperature of 0.30 K.

###### Acknowledgements.

I am grateful to H. Godfrin, R. J. McQueeney, and D. Vaknin for helpful
comments. This research was partially supported by the U.S. Department of
Energy, Office of Basic Energy Sciences, Division of Materials Sciences and
Engineering. Ames Laboratory is operated for the U.S. Department of Energy by
Iowa State University under Contract No. DE-AC02-07CH11358.

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# Validation and Implementation of ILBFS

Fred Metaneal Grabovski, Lior Yasur

###### Abstract

Recursive Best-First Search (RBFS) is a heuristic search algorithm known for
its efficient memory usage compared to traditional best-first search methods
like A*. Despite its theoretical advantages, RBFS is complex and difficult to
teach and to implement, limiting its widespread adoption. To address these
challenges, Iterative Linear Best-First Search (ILBFS) was introduced as a
simpler, more intuitive alternative while maintaining the linear space
complexity of RBFS. In this paper, we present the first implementation of
ILBFS, validate its memory usage and node expansion order claims, and explore
critical aspects of its implementation, such as tie-breaking and node deletion
mechanisms. Our findings demonstrate that ILBFS can serve as an effective
stepping stone for researchers and practitioners looking to use memory
efficient best-first search methods, facilitating the adoption of RBFS-like
algorithms.

## Introduction

Recursive Best-First Search (RBFS) is a heuristic search algorithm designed to
optimize memory usage while maintaining the benefits of best-first search
methods (?). Developed as an alternative to A* for large state spaces, RBFS
performs a depth-first search while keeping track of the best alternative path
in its recursive call stack. This allows RBFS to use linear space, a
significant improvement over the memory-intensive A* algorithm. However, RBFS
is known for its complexity and the difficulty in understanding and
implementing it, which has limited its widespread adoption despite its
theoretical advantages.

To address the pedagogical challenges of RBFS, Iterative Linear Best-First
Search (ILBFS)(?) was introduced. In (?), the author first formalizes the
collapse and restore macros, that manage memory by temporarily removing
subtrees from memory and later restoring them when needed. The author then
uses these macros to introduce ILBFS. The use of collapse and restore ensures
that ILBFS maintains a linear space complexity similar to RBFS. The key
advantage of ILBFS is its simpler, more intuitive algorithm, making it easier
to teach and understand compared to the recursive nature of RBFS. This
accessibility aims to encourage more researchers and practitioners to adopt
RBFS-like algorithms by providing a stepping stone through ILBFS.

In this work, we implement ILBFS for the first time and validate its claims of
equivalence in memory usage and node expansion order compared to RBFS. We also
touch on certain points of implementation such as tie breaking and node
deletion which are crucial in implementing ILBFS.

### Contributions

In this work, we present the following contributions:

The first implementation of ILBFS. A validation of ILBFS’s claims of
equivalence in memory usage and node expansion order compared to RBFS. An
exploration of critical implementation details such as tie-breaking and node
deletion mechanisms. An empirical analysis demonstrating the performance and
efficiency of ILBFS.

## Implementation Notes

While ILBFS can be considered easier both to teach and understand,
implementing it presents several unique challenges compared to RBFS. While
both algorithms optimize memory usage and maintain efficient search
properties, ILBFS requires specific considerations to ensure it functions
correctly and efficiently in an iterative framework.

In this section, we discuss the general principles and considerations involved
in implementing ILBFS. These include the importance of maintaining correct
tie-breaking mechanisms to avoid infinite loops, strategies for efficient node
deletion from the open list to preserve linear memory usage, and overall
algorithmic efficiency.

By addressing these implementation details, we aim to provide a robust
framework for ILBFS that mirrors the theoretical efficiency and correctness of
RBFS while being easier to understand, in the hopes that it will increase the
practical accessibility of RBFS-like algorithms. The following subsections go
into specific aspects of these considerations, providing detailed insights and
solutions for effective ILBFS implementation.

### Importance of Tie Breaking

Algorithm 1 Low-level ILBFS

1:Input: Root $R$

2:Insert $R$ into OPEN and TREE

3:oldbest = NULL

4:while OPEN not empty do

5: best = extractmin(OPEN)

6: if goal(best) then

7: exit

8: end if

9: while oldbest $\neq$ best.parent do

10: oldbest.val $\leftarrow$ min(values of oldbest children)

11: Insert oldbest to OPEN

12: Delete all children of oldbest from OPEN and TREE

13: oldbest $\leftarrow$ oldbest.parent

14: end while

15: for all child $C$ of best do

16: $F(C)\leftarrow f(C)$

17: if $F(best)>f(best)$ and $F(best)>F(C)$ then

18: $F(C)\leftarrow F(best)$

19: end if

20: Insert $C$ to OPEN and TREE

21: end for

22: oldbest $\leftarrow$ best

23:end while

The importance of tie-breaking in search algorithms cannot be overstated, as
it can significantly impact their runtime. In the case of ILBFS, correct tie-
breaking during the extractmin operation (row 5 in Algorithm 1) is even more
critical. Due to the collapse macro, incorrect tie-breaking can lead the into
expanding nodes in a different manner than RBFS and might even result in an
infinite loop.

The algorithm collapses a subtree only when best\- the node that is currently
extracted from open, is not a child of oldbest\- the previously extracted
node, indicating the existence of a node with a smaller F value in a different
subtree than the subtree where oldbest is located. However, in cases where the
F value of oldbest’s children is equal to the F value of other nodes, tie-
breaking becomes crucial.

Consider the following scenario: A and B are two subtrees, and ”best”
currently resides in subtree A with an F value of 20. Assume subtree B is
collapsed with an F value of 20, meaning the root of B exists in the open list
with an F value of 20. When we expand ”best” in subtree A, and the child with
the lowest F value is also 20 (Figure 1 (a)), incorrect tie-breaking can
occur. If the tie-breaking mechanism does not prioritize this child of A, the
algorithm may move ”best” to the root of B, causing subtree A to collapse. If
expanding B again results in a situation where the lowest F value of the
frontier is 20 (Figure 1 (b)), the algorithm would move ”best” back to the
root of A and collapse B, leading to an infinite loop.

The unsurprising solution is a tie-breaking strategy that is mentioned briefly
in (?), as part of the explanation of the restore macro, but as shown here is
a crucial part of the algorithm. The solution is to tie-break nodes in a
depth-first order, meaning that for nodes with equal F value, the node that
was put into open first will be expanded first. This approach prevents
unnecessary switching between subtrees and employs the collapse macro only
when absolutely necessary, i.e., when the F value of the other subtree is
lower than the lowest F value of the frontier of the currently explored
subtree. In our implementation, we established an incremental variable called
creation time, where each node is assigned the current creation time as
attribute. This attribute is used to tie break nodes with equal F.
Alternatively, one could use an epsilon ($\epsilon$) to relax the tie-breaking
condition, as suggested by (?). This technique, known as RBFSCR, allows the
search to persevere longer in each subtree before backtracking, which also
prevents the aforementioned infinite loop. As this issue does not arise in
RBFS due to it’s recursive DFS nature, in ILBFS it needs to be specified as
part of the ILBFS algorithm.

Figure 1: Example scenario where the ILBFS algorithm could result in an
infinite loop for certain tie-breakings. The algorithm could potentially
switch back and forth between a and b (left and right).

### Removal from the Open List

In order for ILBFS to maintain linear memory usage, the algorithm must
explicitly delete nodes during the collapse macro. Unlike RBFS, which
inherently manages node storage through the recursion stack, ILBFS requires
the explicit removal of nodes from the open list. The open list is typically
implemented as a min-heap, which allows for efficient extraction of the
minimum element with a time complexity of $O(1)$. However, removing specific
nodes from a heap without disrupting its order requires careful handling and
additional implementation decisions.

In this section, we present two methods for managing node deletion from the
open list in ILBFS: the heapify method and the sift down method. We compare
these approaches to the node management in RBFS to evaluate their impact on
runtime and overall algorithm efficiency. The results of this comparison are
detailed in section Results and Discussion.

### Heapify

The heapify method is a straightforward approach to maintaining the heap
property after removing nodes. In this method, when nodes are removed from the
open list during the collapse macro, we simply remove them from the array of
the open list without regard for the heap’s structure. This operation has a
time complexity of O(n) where n is the number of elements in the array. After
all necessary removals are complete, we re-order the heap using heapify. The
heapify operation reconstructs the heap from the bottom up, ensuring that the
heap property is maintained. This operation has a time complexity of O(n),
where n is the number of elements in the heap. While this method is simple to
implement, it can be inefficient for large heaps, as it requires rebuilding
the entire heap structure after each set of removals. The time complexity for
removing k nodes and then heapifying is O(kn + n), or O(kn) in the worst case
where k is on the order of n. We also note that since the collapse macro does
not require the heap to be sorted, we are able to heapify the heap just once
at the end of each collapse.

### Sift Down

The sift down method offers a more efficient approach to maintaining the heap
property when removing nodes. In this method, when a node is removed, we
replace it with the last element in the heap and then perform a sift-down
operation to restore the heap property. The process works as follows:

  1. 1.

Find the index of the node to be removed (O(n) operation).

  2. 2.

Replace the node at this index with the last element in the heap.

  3. 3.

Remove the last element from the heap.

  4. 4.

Perform a sift-up operation from the replaced node’s position to ensure it’s
not smaller than its parent.

  5. 5.

Perform a sift-down operation from the same position to ensure it’s not larger
than its children.

The sift-up and sift-down operations each have a time complexity of O(log n)
in the worst case, where n is the number of elements in the heap. This method
is more efficient than the heapify method, especially for large heaps, as it
only needs to fix the heap structure along a single path from the removed node
to a leaf, rather than rebuilding the entire heap. The time complexity for
removing k nodes using this method is O(k(n + log n)), or O(kn) in the worst
case, which is better than the heapify method when k is significantly smaller
than n. However, since we need the location of the removed removed node to fix
the heap, we must fix the heap after each deletion, instead of once per
collapse like in heapify. We note that the difference between the method is
only a constant difference and not a order of magnitude difference, but it
could potentially be meaningful in larger.

Our code for RBFS is heavily based on
111https://github.com/NiloofarShahbaz/8-puzzle-search-implementation.

The entire codebase for our experiment can be found at
222https://github.com/FreddieMG/Validation-and-Implementation-of-ILBFS.

## Results and Discussion

In this study, we evaluated the performance of RBFS and ILBFS algorithms using
the 8-tile problem. We generated random problem instances with optimal
solution lengths ranging from 2 to 22 moves to ensure a comprehensive
evaluation across varying complexities.

For each solution length, we created 10 problem instances, ran the algorithms
on each of them to gather sufficient data for analysis and averaged them. We
measured both the runtime and peak memory usage for each algorithm variant,
specifically RBFS, ILBFS with sift down, and ILBFS with heapify.

We also verified during the experiment that ILBFS indeed expands nodes in the
same order as RBFS, provided the correct tie breaking as explained in Section
Importance of Tie Breaking

The results are illustrated in Figures 2 and Table 1.

Table 1: Runtime Comparison of RBFS and ILBFS Algorithms Input Size | RBFS Runtime (s) | ILBFS Sift Down Runtime (s) | ILBFS Heapify Runtime (s)
---|---|---|---
2 | $1.58\times 10^{-4}$ | $1.21\times 10^{-4}$ | $1.09\times 10^{-4}$
4 | $3.16\times 10^{-4}$ | $3.12\times 10^{-4}$ | $2.83\times 10^{-4}$
6 | $4.35\times 10^{-4}$ | $4.90\times 10^{-4}$ | $4.71\times 10^{-4}$
8 | $5.90\times 10^{-4}$ | $7.27\times 10^{-4}$ | $7.09\times 10^{-4}$
10 | $1.40\times 10^{-3}$ | $1.98\times 10^{-3}$ | $1.89\times 10^{-3}$
12 | $6.85\times 10^{-3}$ | $1.28\times 10^{-2}$ | $1.25\times 10^{-2}$
14 | $2.23\times 10^{-3}$ | $3.67\times 10^{-3}$ | $3.53\times 10^{-3}$
16 | $3.16\times 10^{-2}$ | $7.13\times 10^{-2}$ | $7.04\times 10^{-2}$
18 | $4.58\times 10^{-1}$ | $1.18$ | $1.18$
20 | $6.43\times 10^{-1}$ | $1.92$ | $1.76$
22 | $8.69$ | $22.89$ | $22.69$

Unsurprisingly, for both implementations of ILBFS, the runtime is up to 3
times longer than RBFS’s, due to the handling and fixing of the open list
after we remove nodes. The different approaches provided in this study,
heapify and sift down, resulted in near identical performance.

Figure 2: Comparison of Average Memory Usage by Solution Length for RBFS and
ILBFS Variants.

The memory usage analysis of both RBFS and ILBFS reveals that both algorithms
manage to solve the 8-tile problem with linear memory. This efficiency is due
to the nature of each algorithm’s approach to memory management. ILBFS
maintains both an open list and a tree variable, which occasionally results in
slight differences in memory consumption compared to RBFS. In contrast, RBFS
relies on the call stack to handle open frames, using recursion to manage the
memory. This means that RBFS efficiently uses the stack space but can
encounter limitations due to stack depth in larger problems. Overall, both
methods demonstrate effective memory management, with ILBFS offering a more
explicit handling of search states.

In all, both algorithms manage to solve best-first search problems using
linear memory while maintaining best-first search and using exactly the same
node expansion order. Although ILBFS is slower than RBFS, it offers a
pedagogical advantage by being more straightforward and intuitive. This makes
ILBFS a valuable learning tool for understanding the principles of best-first
search algorithms, without giving up on the advantages of RBFS.

## References

  * [Korf(1993)] Richard E Korf.  Linear-space best-first search.  _Artificial intelligence_ , 62(1):41–78, 1993.
  * [Felner(2015)] Ariel Felner.  Position paper: The collapse macro in best-first search algorithms and an iterative variant of rbfs.  In _Proceedings of the International Symposium on Combinatorial Search_ , volume 6, pages 28–34, 2015.
  * [Hatem et al.(2015)Hatem, Kiesel, and Ruml] Matthew Hatem, Scott Kiesel, and Wheeler Ruml.  Recursive best-first search with bounded overhead.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 29, 2015.