Fermi Gas Formula Sheet

This sheet specializes the Quantum Gas Formula Sheet to ideal fermions. It collects the formulas most often needed for state counting, thermodynamics, response checks, and translation among one-, two-, and three-dimensional continuum conventions.

The derivations and physical interpretation remain in Fermi–Dirac Statistics, Ideal Fermi Gas, Degenerate Fermi Gas, Fermi Momentum and Fermi Energy, and Fermi Surface.

Default Scope

Unless stated otherwise:

• the particles are noninteracting fermions of mass $m$;
• the continuum is homogeneous and $d$ dimensional;
• $\epsilon_{\mathbf k}=\hbar^2k^2/(2m)$;
• $g$ counts equally populated internal components;
• $n=N/V$ is total density across those components;
• the thermodynamic limit follows periodic finite-volume state counting;
• $\epsilon=0$ is the one-particle band bottom;
• $\beta=1/(k_{\mathrm B}T)$ and $z=e^{\beta\mu}$;
• all density-of-states functions below include the factor $g$.

For lattice bands, traps, relativistic dispersions, spin imbalance, or interactions, keep the Fermi function but replace the spectrum, density of states, conserved quantities, and equation-of-state identities as required.

Core Distribution

The Fermi–Dirac occupation is

$$f(\epsilon) \equiv \frac{1}{e^{\beta(\epsilon-\mu)}+1} = \frac{1}{z^{-1}e^{\beta\epsilon}+1}.$$

It obeys

$$0 \le f(\epsilon) \le 1.$$

At zero temperature,

$$f(\epsilon) \longrightarrow \theta(\epsilon_{\mathrm F}-\epsilon),$$

with $\mu(0)=\epsilon_{\mathrm F}$ for the ideal continuum gas at fixed density.

The particle–hole identity around $\mu$ is

$$f(\mu+\delta) = 1-f(\mu-\delta).$$

Thermal Window

The derivative that localizes low-temperature corrections is

$$\begin{aligned} -\frac{\partial f}{\partial\epsilon} &= \beta f(\epsilon) \left[ 1-f(\epsilon) \right] \\ &= \frac{\beta}{ 4\cosh^2\!\left[ \beta(\epsilon-\mu)/2 \right] }. \end{aligned}$$

Its width is of order $k_{\mathrm B}T$, its maximum is $\beta/4$, and

$$\int_{-\infty}^{\infty} d\epsilon\, \left( -\frac{\partial f}{\partial\epsilon} \right) = 1.$$

For a spectrum bounded below by zero, extending the integral to $-\infty$ is accurate in the degenerate regime $\mu\gg k_{\mathrm B}T$.

Exact Independent-Mode Formulas

For one-particle modes $\alpha$,

$$H = \sum_\alpha \epsilon_\alpha n_\alpha, \qquad n_\alpha\in\{0,1\}.$$

The grand partition function and grand potential are

$$\begin{aligned} \Xi &= \prod_\alpha \left[ 1+e^{-\beta(\epsilon_\alpha-\mu)} \right], \\ \Omega &= -k_{\mathrm B}T \sum_\alpha \ln\left[ 1+e^{-\beta(\epsilon_\alpha-\mu)} \right]. \end{aligned}$$

The mean and variance of one mode are

$$\begin{aligned} \langle n_\alpha\rangle &= f(\epsilon_\alpha), \\ \operatorname{Var}(n_\alpha) &= f(\epsilon_\alpha) \left[ 1-f(\epsilon_\alpha) \right]. \end{aligned}$$

The one-mode entropy is

$$\frac{S_\alpha}{k_{\mathrm B}} = -f_\alpha\ln f_\alpha - (1-f_\alpha) \ln(1-f_\alpha).$$

These independent-mode fluctuation formulas are grand canonical. Fixing the total particle number correlates the occupations.

Density-of-States Workflow

Let $D(\epsilon)$ be the total one-particle density of states, including internal degeneracy and physical volume. Then

$$N = \int_0^\infty d\epsilon\, D(\epsilon)f(\epsilon).$$

At fixed $N$, this equation determines $\mu(T)$. The internal energy is

$$U = \int_0^\infty d\epsilon\, \epsilon D(\epsilon)f(\epsilon),$$

and the grand potential is

$$\begin{aligned} \Omega &= -k_{\mathrm B}T \int_0^\infty d\epsilon\, D(\epsilon) \mathcal L(\epsilon), \\ \mathcal L(\epsilon) &\equiv \ln\left[ 1+e^{-\beta(\epsilon-\mu)} \right]. \end{aligned}$$

The entropy is

$$\begin{aligned} S &= -k_{\mathrm B} \int_0^\infty d\epsilon\, D(\epsilon) \\ &\quad\times \left[ f\ln f + (1-f)\ln(1-f) \right]. \end{aligned}$$

These formulas remain useful for a noninteracting lattice band or trap after inserting the correct $D(\epsilon)$. Identities such as $U=dPV/2$ do not survive an arbitrary dispersion or confining potential.

Free-Continuum Density of States

For $\epsilon_{\mathbf k}=\hbar^2k^2/(2m)$,

$$\begin{aligned} D_d(\epsilon) &= \frac{gV}{\Gamma(d/2)} \left( \frac{m}{2\pi\hbar^2} \right)^{d/2} \\ &\quad\times \epsilon^{d/2-1}. \end{aligned}$$

In one, two, and three dimensions:

$$\begin{aligned} \frac{D_1(\epsilon)}{L} &= \frac{g}{\pi\hbar} \sqrt{ \frac{m}{2\epsilon} }, \\ \frac{D_2(\epsilon)}{A} &= \frac{gm}{2\pi\hbar^2}, \\ \frac{D_3(\epsilon)}{V} &= \frac{g}{4\pi^2} \left( \frac{2m}{\hbar^2} \right)^{3/2} \sqrt{\epsilon}. \end{aligned}$$

Here $L$, $A$, and $V$ denote one-, two-, and three-dimensional measures. The one-dimensional divergence at the band bottom is integrable in the particle-number integral away from singular limits.

Fermi-Sphere Counting

Let

$$\mathcal V_d \equiv \frac{\pi^{d/2}}{ \Gamma(d/2+1) }$$

be the volume of the unit $d$-ball. At $T=0$,

$$n = \frac{g\mathcal V_d}{(2\pi)^d} k_{\mathrm F}^d.$$

Equivalently,

Dimension	Density–momentum relation	Fermi momentum
$d=1$	$n=gk_{\mathrm F}/\pi$	$k_{\mathrm F}=\pi n/g$
$d=2$	$n=gk_{\mathrm F}^2/(4\pi)$	$k_{\mathrm F}=\sqrt{4\pi n/g}$
$d=3$	$n=gk_{\mathrm F}^3/(6\pi^2)$	$k_{\mathrm F}=(6\pi^2n/g)^{1/3}$

The associated scales are

$$\begin{aligned} \epsilon_{\mathrm F} &= \frac{\hbar^2k_{\mathrm F}^2}{2m}, \\ T_{\mathrm F} &= \frac{\epsilon_{\mathrm F}}{k_{\mathrm B}}, \\ v_{\mathrm F} &= \frac{\hbar k_{\mathrm F}}{m}, \\ \lambda_{\mathrm F} &= \frac{2\pi}{k_{\mathrm F}}. \end{aligned}$$

For unpolarized spin-$1/2$ fermions, $g=2$. For a polarized gas, count each component with its own density and Fermi momentum rather than inserting one common factor $g=2$.

Zero-Temperature Thermodynamics

For the quadratic continuum in $d$ dimensions,

$$\begin{aligned} \mu(0) &= \epsilon_{\mathrm F}, \\ \frac{U_0}{N} &= \frac{d}{d+2} \epsilon_{\mathrm F}, \\ P_0 &= \frac{2}{d+2} n\epsilon_{\mathrm F}. \end{aligned}$$

The scaling identity is

$$P = \frac{2}{d} \frac{U}{V}$$

at any temperature for this homogeneous quadratic ideal gas.

At the Fermi energy,

$$\frac{D_d(\epsilon_{\mathrm F})}{V} = \frac{dn}{2\epsilon_{\mathrm F}}.$$

The zero-temperature isothermal compressibility is

$$\kappa_T(0) = \frac{1}{n^2} \left( \frac{\partial n}{\partial\mu} \right)_{T=0} = \frac{d}{2n\epsilon_{\mathrm F}}.$$

Dimension-specific values are:

$d$	$U_0/N$	$P_0$	$\kappa_T(0)$
$1$	$\epsilon_{\mathrm F}/3$	$2n\epsilon_{\mathrm F}/3$	$1/(2n\epsilon_{\mathrm F})$
$2$	$\epsilon_{\mathrm F}/2$	$n\epsilon_{\mathrm F}/2$	$1/(n\epsilon_{\mathrm F})$
$3$	$3\epsilon_{\mathrm F}/5$	$2n\epsilon_{\mathrm F}/5$	$3/(2n\epsilon_{\mathrm F})$

The nonzero pressure at $T=0$ is a consequence of state filling under Pauli exclusion. It is not a two-body repulsive force.

Finite-Temperature Continuum Formulas

Define

$$\begin{aligned} f_s(z) &\equiv -\operatorname{Li}_s(-z) \\ &= \frac{1}{\Gamma(s)} \int_0^\infty dt\, \frac{t^{s-1}}{z^{-1}e^t+1}. \end{aligned}$$

For

$$\lambda_T = \sqrt{ \frac{2\pi\hbar^2}{mk_{\mathrm B}T} },$$

the number density, pressure, and energy are

$$\begin{aligned} n &= \frac{g}{\lambda_T^d} f_{d/2}(z), \\ P &= \frac{gk_{\mathrm B}T}{\lambda_T^d} f_{d/2+1}(z), \\ U &= \frac d2PV. \end{aligned}$$

The entropy density is

$$\begin{aligned} \frac{S}{k_{\mathrm B}V} &= \frac{g}{\lambda_T^d} \Bigg[ \left( \frac d2+1 \right) f_{d/2+1}(z) \\ &\qquad - \ln z\, f_{d/2}(z) \Bigg]. \end{aligned}$$

The number equation must be solved for $z$ or $\mu$ when density, rather than chemical potential, is fixed.

Degeneracy Diagnostics

Define

$$\vartheta \equiv \frac{T}{T_{\mathrm F}}$$

and the phase-space density per internal component

$$x \equiv \frac{n\lambda_T^d}{g}.$$

For the quadratic continuum,

$$x = \frac{1}{\Gamma(d/2+1)} \vartheta^{-d/2}.$$

Thus:

Regime	Equivalent diagnostics	Useful method
dilute classical	$x\ll1$, $z\ll1$, $T\gg T_{\mathrm F}$	fugacity or virial expansion
crossover	$x=O(1)$, $T=O(T_{\mathrm F})$	numerical Fermi integrals
degenerate	$x\gg1$, $z\gg1$, $T\ll T_{\mathrm F}$	Sommerfeld expansion

The condition “low temperature” is meaningful only relative to $T_{\mathrm F}$ and any other energy scales.

Sommerfeld Master Formula

For a sufficiently smooth function $\varphi(\epsilon)$ and $k_{\mathrm B}T\ll\mu$,

$$\begin{aligned} I_\varphi(T,\mu) &\equiv \int_0^\infty d\epsilon\, \varphi(\epsilon)f(\epsilon) \\ &= \int_0^\mu d\epsilon\, \varphi(\epsilon) \\ &\quad+ \frac{\pi^2}{6} (k_{\mathrm B}T)^2 \varphi'(\mu) \\ &\quad+ \frac{7\pi^4}{360} (k_{\mathrm B}T)^4 \varphi'''(\mu) + \cdots. \end{aligned}$$

The expansion assumes the lower spectral endpoint is far outside the thermal window and that $\varphi$ is smooth near $\mu$. Van Hove singularities, band edges, gaps, and rapidly varying matrix elements require separate treatment.

Fixed-Density Low-Temperature Results

For the homogeneous quadratic gas, let $\vartheta=T/T_{\mathrm F}\ll1$. At fixed density,

$$\frac{\mu(T)}{\epsilon_{\mathrm F}} = 1 - \frac{\pi^2}{12} (d-2)\vartheta^2 + O(\vartheta^4).$$

The energy and pressure are

$$\begin{aligned} \mathcal R_d(\vartheta) &\equiv 1+ \frac{\pi^2(d+2)}{12} \vartheta^2 + O(\vartheta^4), \\ \frac{U}{N} &= \frac{d}{d+2} \epsilon_{\mathrm F} \mathcal R_d(\vartheta), \\ P &= \frac{2}{d+2} n\epsilon_{\mathrm F} \mathcal R_d(\vartheta). \end{aligned}$$

The heat capacity and entropy are

$$\begin{aligned} \frac{C_V}{Nk_{\mathrm B}} &= \frac{\pi^2d}{6} \vartheta + O(\vartheta^3), \\ \frac{S}{Nk_{\mathrm B}} &= \frac{\pi^2d}{6} \vartheta + O(\vartheta^3). \end{aligned}$$

For $d=3$, these reduce to

$$\begin{aligned} \mu(T) &= \epsilon_{\mathrm F} \left( 1- \frac{\pi^2}{12}\vartheta^2 \right), \\ \frac{C_V}{Nk_{\mathrm B}} &= \frac{\pi^2}{2}\vartheta. \end{aligned}$$

In $d=2$, the free-particle density of states is constant. The algebraic $T^2$ shift of $\mu$ vanishes; the leading fixed-density correction is exponentially small, as shown below.

Exact Two-Dimensional Chemical Potential

For a two-dimensional quadratic gas,

$$\frac{D_2}{A} = \frac{gm}{2\pi\hbar^2} \equiv \mathcal D_0.$$

The number equation is

$$n = \mathcal D_0 k_{\mathrm B}T \ln(1+z).$$

Since $\epsilon_{\mathrm F}=n/\mathcal D_0$,

$$\begin{aligned} \mu(T) &= k_{\mathrm B}T \ln\left( e^{\epsilon_{\mathrm F}/(k_{\mathrm B}T)}-1 \right) \\ &= \epsilon_{\mathrm F} + k_{\mathrm B}T \ln\left( 1-e^{-\epsilon_{\mathrm F}/(k_{\mathrm B}T)} \right). \end{aligned}$$

Therefore $\mu-\epsilon_{\mathrm F}$ is exponentially small when $T\ll T_{\mathrm F}$. A finite band edge or energy-dependent density of states changes this special result.

Number Response and Fluctuations

The number derivative is

$$\begin{aligned} \left( \frac{\partial N}{\partial\mu} \right)_T &= \beta \int_0^\infty d\epsilon\, D(\epsilon) f(\epsilon) \\ &\quad\times \left[ 1-f(\epsilon) \right]. \end{aligned}$$

In the grand-canonical ensemble,

$$\operatorname{Var}(N) = k_{\mathrm B}T \left( \frac{\partial N}{\partial\mu} \right)_T.$$

At low temperature,

$$\begin{aligned} \left( \frac{\partial N}{\partial\mu} \right)_T &\simeq D(\epsilon_{\mathrm F}), \\ \operatorname{Var}(N) &\simeq k_{\mathrm B}T D(\epsilon_{\mathrm F}). \end{aligned}$$

For the quadratic continuum,

$$\frac{\operatorname{Var}(N)}{N} \simeq \frac d2 \frac{T}{T_{\mathrm F}}.$$

This is a variance-to-mean ratio, not the squared relative fluctuation. The relative root-mean-square fluctuation still vanishes with increasing $N$ at fixed thermodynamic conditions.

The isothermal compressibility is

$$\kappa_T = \frac{1}{n^2} \left( \frac{\partial n}{\partial\mu} \right)_T.$$

Dilute Classical Limit

For

$$x = \frac{n\lambda_T^d}{g} \ll 1,$$

the activity and chemical potential are

$$\begin{aligned} z &= x + \frac{x^2}{2^{d/2}} + O(x^3), \\ \frac{\mu}{k_{\mathrm B}T} &= \ln x + \frac{x}{2^{d/2}} + O(x^2). \end{aligned}$$

The pressure is

$$\frac{P}{nk_{\mathrm B}T} = 1 + \frac{x}{2^{d/2+1}} + O(x^2).$$

The positive correction is the leading ideal exchange effect for fermions. It is not evidence for a microscopic repulsive potential.

The leading classical entropy per particle is

$$\frac{S}{Nk_{\mathrm B}} = \frac d2+1 - \ln\left( \frac{n\lambda_T^d}{g} \right) + O(x).$$

Complete-Degeneracy Table

For the quadratic continuum at $T=0$:

Quantity	General $d$	Three dimensions
density	$g\mathcal V_d k_{\mathrm F}^d/(2\pi)^d$	$gk_{\mathrm F}^3/(6\pi^2)$
Fermi energy	$\hbar^2k_{\mathrm F}^2/(2m)$	same
energy per particle	$d\epsilon_{\mathrm F}/(d+2)$	$3\epsilon_{\mathrm F}/5$
pressure	$2n\epsilon_{\mathrm F}/(d+2)$	$2n\epsilon_{\mathrm F}/5$
DOS per volume at $\epsilon_{\mathrm F}$	$dn/(2\epsilon_{\mathrm F})$	$3n/(2\epsilon_{\mathrm F})$
compressibility	$d/(2n\epsilon_{\mathrm F})$	$3/(2n\epsilon_{\mathrm F})$

Model-Boundary Translations

Lattice bands

Use the band dispersion $\epsilon_{a\mathbf k}$ and Brillouin-zone sum:

$$N = \sum_{a,\mathbf k} f(\epsilon_{a\mathbf k}).$$

The Fermi surface is defined by $\epsilon_{a\mathbf k}=\mu$ at low temperature. The free-space formulas for $k_{\mathrm F}$, pressure, and $D_d(\epsilon)$ generally do not apply.

Traps

Replace the homogeneous density of states by the trap density of states or use a controlled local-density approximation. The symbol $V$ should not be interpreted as a uniform box volume when no such volume exists.

Relativistic fermions

Replace the dispersion and state-counting measure consistently. In the ultrarelativistic limit, $P=U/(dV)$ rather than $P=2U/(dV)$.

Interacting fermions

The exact momentum occupation need not be a Fermi–Dirac function of the bare one-particle energy. Hartree–Fock theory, quasiparticle theory, and Fermi-liquid theory introduce effective spectra and renormalized response coefficients. State which spectrum enters $f$.

Fast Consistency Checks

• $0\le f\le1$ for every mode.
• $D(\epsilon)$ has dimensions of inverse energy.
• $D(\epsilon_{\mathrm F})/V=dn/(2\epsilon_{\mathrm F})$ for a quadratic continuum.
• $T\to0$ gives a filled Fermi ball, not zero kinetic energy.
• $T\gg T_{\mathrm F}$ gives $z\ll1$ and the Maxwell–Boltzmann law.
• $C_V$ and $S$ vanish linearly as $T\to0$ in the ideal continuum.
• $g=2$ gives $k_{\mathrm F}=(3\pi^2n)^{1/3}$ in three dimensions.
• A fixed-density low-temperature calculation must include the shift of $\mu(T)$.
• Any quoted pressure identity must match the dispersion and geometry.

Common Mistakes

• Using total density $n$ with a one-component formula while also setting $g=2$.
• Calling $f(\epsilon)$ a probability density over energy without multiplying by $D(\epsilon)$ and normalizing.
• Replacing the finite-temperature chemical potential by $\epsilon_{\mathrm F}$ at all temperatures.
• Assigning the classical heat capacity $dNk_{\mathrm B}/2$ to a degenerate gas.
• Applying the three-dimensional square-root density of states in one or two dimensions.
• Treating degeneracy pressure as a new pair force.
• Using a Sommerfeld expansion at a band edge or singular density of states.
• Confusing grand-canonical number fluctuations with fixed-$N$ fluctuations.
• Applying free-space $k_{\mathrm F}$ formulas to a lattice Fermi surface.
• Using nonrelativistic pressure formulas when $\epsilon_{\mathrm F}$ is comparable to the rest energy.

Cross-Links

• Quantum Gas Formula Sheet
• Symbols and Conventions
• Fermi–Dirac Distribution Formula Card
• Fermi–Dirac Statistics
• Ideal Fermi Gas
• Degenerate Fermi Gas
• Fermi Momentum and Fermi Energy
• Fermi Surface
• Sommerfeld Expansion
• Maxwell–Boltzmann Limit
• Low-Dimensional Quantum Gases
• Quantum Gases in Traps
• Hartree–Fock Approximation
• Fermi-Liquid Theory Preview

References

• R. K. Pathria and P. D. Beale, Statistical Mechanics, 3rd ed., Elsevier (2011) — ideal Fermi gases and thermodynamic ensembles.
• K. Huang, Statistical Mechanics, 2nd ed., Wiley (1987) — density-of-states integrals, virial expansion, and degenerate limits.
• L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1, 3rd ed., Butterworth–Heinemann (1980) — degenerate Fermi thermodynamics and low-temperature expansions.
• N. W. Ashcroft and N. D. Mermin, Solid State Physics, Harcourt (1976) — electron-gas state counting, Fermi surfaces, and low-temperature response.
• G. D. Mahan, Many-Particle Physics, 3rd ed., Springer (2000) — Fermi functions, response, and interacting extensions.
• P. Coleman, Introduction to Many-Body Physics, Cambridge University Press (2015) — Fermi seas, quasiparticles, and many-body conventions.

Exercises

1. Count states in arbitrary dimension

Starting from periodic momentum-state density $V/(2\pi)^d$, derive

$$n = \frac{g\mathcal V_d}{(2\pi)^d} k_{\mathrm F}^d.$$

Solution

At zero temperature, all states inside the Fermi ball are occupied. Therefore

$$\begin{aligned} N &= gV \int_{k<k_{\mathrm F}} \frac{d^dk}{(2\pi)^d} \\ &= gV \frac{\mathcal V_d k_{\mathrm F}^d}{ (2\pi)^d }. \end{aligned}$$

Dividing by $V$ gives the result. The factor $g$ counts distinct internal states at each momentum, while $\mathcal V_d k_{\mathrm F}^d$ is geometric momentum-space volume.

2. Density of states at the Fermi level

Use $N\propto\epsilon_{\mathrm F}^{d/2}$ to show that

$$\frac{D_d(\epsilon_{\mathrm F})}{V} = \frac{dn}{2\epsilon_{\mathrm F}}$$

and derive the zero-temperature compressibility.

Solution

At zero temperature,

$$N = \int_0^{\epsilon_{\mathrm F}} d\epsilon\, D_d(\epsilon).$$

Because $D_d(\epsilon)\propto\epsilon^{d/2-1}$,

$$N = \frac{2}{d} D_d(\epsilon_{\mathrm F}) \epsilon_{\mathrm F}.$$

Divide by $V$ and rearrange to obtain

$$\frac{D_d(\epsilon_{\mathrm F})}{V} = \frac{dn}{2\epsilon_{\mathrm F}}.$$

Since $\mu(0)=\epsilon_{\mathrm F}$,

$$\begin{aligned} \kappa_T(0) &= \frac{1}{n^2} \left( \frac{\partial n}{\partial\mu} \right)_{T=0} \\ &= \frac{d}{2n\epsilon_{\mathrm F}}. \end{aligned}$$

3. Low-temperature chemical potential in dimension d

For $D_d(\epsilon)=C\epsilon^{d/2-1}$, use the Sommerfeld expansion at fixed $N$ to derive the leading shift of $\mu(T)$.

Solution

Let $a=d/2-1$. The number equation is

$$\begin{aligned} N &= C \frac{\mu^{a+1}}{a+1} \\ &\quad+ \frac{\pi^2}{6} (k_{\mathrm B}T)^2 Ca\mu^{a-1} \\ &\quad+ O(T^4). \end{aligned}$$

Write $\mu=\epsilon_{\mathrm F}+\delta\mu$ and compare with the zero-temperature value. To order $T^2$,

$$C\epsilon_{\mathrm F}^{a} \delta\mu + \frac{\pi^2}{6} (k_{\mathrm B}T)^2 Ca\epsilon_{\mathrm F}^{a-1} = 0.$$

Hence

$$\frac{\delta\mu}{\epsilon_{\mathrm F}} = -\frac{\pi^2}{6} a \left( \frac{T}{T_{\mathrm F}} \right)^2.$$

Using $a=(d-2)/2$ gives

$$\begin{aligned} \frac{\mu(T)}{\epsilon_{\mathrm F}} &= 1 - \frac{\pi^2}{12} (d-2) \\ &\quad\times \left( \frac{T}{T_{\mathrm F}} \right)^2 + O\!\left[ \left( \frac{T}{T_{\mathrm F}} \right)^4 \right]. \end{aligned}$$

4. Exact two-dimensional chemical potential

Starting from the constant two-dimensional density of states, derive the exact fixed-density expression for $\mu(T)$.

Solution

With $D_2/A=\mathcal D_0$,

$$\begin{aligned} n &= \mathcal D_0 \int_0^\infty d\epsilon\, \frac{1}{e^{\beta(\epsilon-\mu)}+1} \\ &= \mathcal D_0 k_{\mathrm B}T \ln(1+e^{\beta\mu}). \end{aligned}$$

Because $n=\mathcal D_0\epsilon_{\mathrm F}$,

$$e^{\beta\mu} = e^{\beta\epsilon_{\mathrm F}}-1.$$

Taking the logarithm gives

$$\mu(T) = k_{\mathrm B}T \ln\left( e^{\epsilon_{\mathrm F}/(k_{\mathrm B}T)}-1 \right).$$

Factoring out the large exponential yields the equivalent form

$$\mu(T) = \epsilon_{\mathrm F} + k_{\mathrm B}T \ln\left( 1-e^{-\epsilon_{\mathrm F}/(k_{\mathrm B}T)} \right).$$

5. Grand-canonical number variance

Show that a low-temperature quadratic gas satisfies

$$\frac{\operatorname{Var}(N)}{N} \simeq \frac d2 \frac{T}{T_{\mathrm F}}.$$

Solution

The fluctuation relation gives

$$\operatorname{Var}(N) = k_{\mathrm B}T \left( \frac{\partial N}{\partial\mu} \right)_T.$$

At low temperature, the derivative is localized at the Fermi energy:

$$\left( \frac{\partial N}{\partial\mu} \right)_T \simeq D_d(\epsilon_{\mathrm F}).$$

Using

$$D_d(\epsilon_{\mathrm F}) = \frac{dN}{2\epsilon_{\mathrm F}},$$

one obtains

$$\frac{\operatorname{Var}(N)}{N} \simeq \frac d2 \frac{k_{\mathrm B}T}{\epsilon_{\mathrm F}} = \frac d2 \frac{T}{T_{\mathrm F}}.$$

This result is grand canonical; it is identically zero for an exactly fixed total $N$.

6. Leading classical correction

Let $x=n\lambda_T^d/g\ll1$. Derive the first Fermi correction to the classical pressure and explain its sign.

Solution

The Fermi integrals have expansions

$$\begin{aligned} x &= z- \frac{z^2}{2^{d/2}} + O(z^3), \\ \frac{P\lambda_T^d}{ gk_{\mathrm B}T } &= z- \frac{z^2}{2^{d/2+1}} + O(z^3). \end{aligned}$$

Inverting the first series gives

$$z = x+ \frac{x^2}{2^{d/2}} + O(x^3).$$

Substitution into the pressure yields

$$\frac{P\lambda_T^d}{ gk_{\mathrm B}T } = x+ \frac{x^2}{2^{d/2+1}} + O(x^3).$$

Dividing by $x$ gives

$$\frac{P}{nk_{\mathrm B}T} = 1+ \frac{x}{2^{d/2+1}} + O(x^2).$$

The correction is positive because Pauli exclusion suppresses multiple occupancy of the same one-particle mode. It is statistical rather than a pairwise repulsion.