Quantum Gas Formula Sheet

This sheet collects the formulas needed to move between discrete ideal-gas modes, continuum state counting, thermodynamic functions, and the principal Bose, Fermi, and dilute classical limits. It is a lookup and translation page, not the canonical derivation of quantum statistics.

The canonical explanations live in Quantum Statistics Overview, Bose–Einstein Statistics, Fermi–Dirac Statistics, Ideal Bose Gas, and Ideal Fermi Gas.

Default Scope

Unless a subsection says otherwise, the formulas assume:

• noninteracting identical particles;
• a homogeneous $d$-dimensional continuum;
• nonrelativistic dispersion $\epsilon_{\mathbf k}=\hbar^2k^2/(2m)$;
• physical volume $V$ and periodic boundary conditions before the continuum limit;
• $g$ independent internal components with the same mass and chemical potential;
• grand-canonical equilibrium at temperature $T$ and chemical potential $\mu$;
• a one-particle ground energy shifted to $\epsilon_0=0$;
• natural logarithms;
• thermal wavelength

$$\lambda_T \equiv \sqrt{ \frac{2\pi\hbar^2}{mk_{\mathrm B}T} }.$$

Finite systems should use exact mode sums before replacing them by integrals. Interactions, lattice bands, traps, relativistic dispersions, separately conserved species, and nonequilibrium occupations require modified formulas.

Symbols and Sign Convention

Symbol	Meaning
$\beta$	$1/(k_{\mathrm B}T)$
$z$	activity or fugacity $e^{\beta\mu}$ after setting $\epsilon_0=0$
$g$	internal degeneracy under the assumptions above
$\eta$	$+1$ for bosons and $-1$ for fermions
$\alpha$	complete one-particle mode label
$\epsilon_\alpha$	one-particle mode energy
$\bar n_\alpha$	mean occupation of mode $\alpha$
$D_d(\epsilon)$	total one-particle density of states in $d$ dimensions
$n=N/V$	total particle density
$n_0=N_0/V$	bosonic ground-mode density when separated
$\Phi_s^{(\eta)}(z)$	unified Bose/Fermi integral defined below

The chemical potential enters only through the invariant combination

$$z e^{-\beta\epsilon_\alpha} = e^{-\beta(\epsilon_\alpha-\mu)}.$$

If every one-particle energy is shifted by a constant $C$, shift $\mu$ by the same $C$. The occupation numbers and thermodynamics are then unchanged.

Exact Independent-Mode Formulas

For an ideal gas,

$$H = \sum_\alpha \epsilon_\alpha n_\alpha, \qquad \hat N = \sum_\alpha n_\alpha.$$

Define the one-mode activity

$$x_\alpha \equiv z e^{-\beta\epsilon_\alpha}.$$

For bosons, convergence requires $0\le x_\alpha<1$ for every mode. A fermionic mode allows any $x_\alpha\ge0$.

Partition functions and occupations

The one-mode and full grand partition functions are

$$\begin{aligned} \Xi_\alpha &= \left( 1-\eta x_\alpha \right)^{-\eta}, \\ \ln\Xi &= -\eta \sum_\alpha \ln\left( 1-\eta x_\alpha \right). \end{aligned}$$

The grand potential is

$$\Omega = \eta k_{\mathrm B}T \sum_\alpha \ln\left( 1-\eta x_\alpha \right).$$

The mean occupation is

$$\bar n_\alpha = \frac{1}{ z^{-1}e^{\beta\epsilon_\alpha}-\eta } = \frac{x_\alpha}{1-\eta x_\alpha}.$$

Thus

$$\begin{aligned} \bar n_\alpha^{\mathrm B} &= \frac{1}{z^{-1}e^{\beta\epsilon_\alpha}-1}, \\ \bar n_\alpha^{\mathrm F} &= \frac{1}{z^{-1}e^{\beta\epsilon_\alpha}+1}. \end{aligned}$$

These are mean mode occupations, not normalized probabilities over the mode label $\alpha$.

Mode probabilities and fluctuations

For one bosonic mode,

$$\begin{aligned} P_\alpha(n) &= (1-x_\alpha)x_\alpha^n, \\ n &= 0,1,2,\ldots. \end{aligned}$$

whereas a fermionic mode has

$$\begin{aligned} P_\alpha(0) &= \frac{1}{1+x_\alpha}, \\ P_\alpha(1) &= \frac{x_\alpha}{1+x_\alpha}. \end{aligned}$$

The grand-canonical mode variance is

$$\operatorname{Var}(n_\alpha) = \bar n_\alpha \left( 1+\eta\bar n_\alpha \right).$$

Bosonic fluctuations are enhanced and fermionic fluctuations are suppressed relative to a Poisson variable with the same mean. Fixed-total-number constraints correlate modes and invalidate a direct sum of independent grand-canonical variances.

The entropy of one independent mode is

$$\begin{aligned} \frac{S_\alpha}{k_{\mathrm B}} &= \frac{1+\eta\bar n_\alpha}{\eta} \ln\left( 1+\eta\bar n_\alpha \right) \\ &\quad - \bar n_\alpha \ln\bar n_\alpha. \end{aligned}$$

Substituting $\eta=+1$ gives the bosonic expression; substituting $\eta=-1$ gives the binary fermionic entropy.

From Mode Sums to Integrals

For a periodic homogeneous continuum,

$$\sum_{\mathbf k,\sigma} \longrightarrow gV \int \frac{d^dk}{(2\pi)^d}.$$

For the quadratic dispersion, the density of states is

$$\begin{aligned} D_d(\epsilon) &= \frac{gV}{\Gamma(d/2)} \left( \frac{m}{2\pi\hbar^2} \right)^{d/2} \\ &\quad\times \epsilon^{d/2-1}, \qquad \epsilon>0. \end{aligned}$$

It is normalized so that

$$\sum_\alpha F(\epsilon_\alpha) \longrightarrow \int_0^\infty d\epsilon\, D_d(\epsilon)F(\epsilon).$$

In three dimensions,

$$D_3(\epsilon) = \frac{gV}{4\pi^2} \left( \frac{2m}{\hbar^2} \right)^{3/2} \sqrt{\epsilon}.$$

The continuum replacement is controlled only when the relevant thermal or Fermi energy window contains many discrete levels. Near a bosonic ground mode, separate that mode before making the replacement.

Bose and Fermi Integrals

Define the unified function

$$\Phi_s^{(\eta)}(z) \equiv \sum_{\ell=1}^{\infty} \frac{ \eta^{\ell-1}z^\ell }{\ell^s}.$$

For the convergent integral representation,

$$\Phi_s^{(\eta)}(z) = \frac{1}{\Gamma(s)} \int_0^\infty dt\, \frac{t^{s-1}}{z^{-1}e^t-\eta}.$$

The standard names are

$$\begin{aligned} g_s(z) &\equiv \Phi_s^{(+1)}(z) = \operatorname{Li}_s(z), \\ f_s(z) &\equiv \Phi_s^{(-1)}(z) = -\operatorname{Li}_s(-z). \end{aligned}$$

Some references shift the index of “Fermi integrals” by one. The polylogarithm form removes that ambiguity.

Useful identities are

$$\begin{aligned} z\frac{d}{dz} \Phi_s^{(\eta)}(z) &= \Phi_{s-1}^{(\eta)}(z), \\ g_s(1) &= \zeta(s), \qquad s>1. \end{aligned}$$

For small $z$,

$$\Phi_s^{(\eta)}(z) = z + \eta\frac{z^2}{2^s} + O(z^3).$$

At the Bose boundary $z\to1^-$, $g_s(1)$ is finite only for $s>1$. This convergence condition controls ideal-gas condensation in the thermodynamic limit.

Uniform-Gas Master Formulas

For a homogeneous quadratic gas in $d$ dimensions, the thermal-mode density is

$$n_{\mathrm{th}} = \frac{g}{\lambda_T^d} \Phi_{d/2}^{(\eta)}(z).$$

The total density is

$$n = \begin{cases} n_0+ \dfrac{g}{\lambda_T^d} g_{d/2}(z), &\text{bosons}, \\[6pt] \dfrac{g}{\lambda_T^d} f_{d/2}(z), &\text{fermions}. \end{cases}$$

The pressure and grand potential are

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

For a quadratic dispersion with no additive rest energy,

$$U = \frac{d}{2}PV.$$

In the normal phase, the entropy density is

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

For an ideal Bose condensate with $\epsilon_0=\mu=0$, the ground mode carries no extensive energy, pressure, or entropy; the thermal cloud supplies the expressions above with $z=1$.

The normal-phase number susceptibility is

$$\left( \frac{\partial n}{\partial\mu} \right)_T = \frac{\beta g}{\lambda_T^d} \Phi_{d/2-1}^{(\eta)}(z).$$

This derivative can diverge at an ideal Bose boundary. Interactions qualitatively alter that response.

Three-Dimensional Lookup

For $d=3$ and $\epsilon_0=0$:

Quantity	Bosons	Fermions
mode occupation	$[z^{-1}e^{\beta\epsilon}-1]^{-1}$	$[z^{-1}e^{\beta\epsilon}+1]^{-1}$
thermal density	$g\lambda_T^{-3}g_{3/2}(z)$	$g\lambda_T^{-3}f_{3/2}(z)$
pressure	$gk_{\mathrm B}T\lambda_T^{-3}g_{5/2}(z)$	$gk_{\mathrm B}T\lambda_T^{-3}f_{5/2}(z)$
internal energy	$3PV/2$	$3PV/2$
mode variance	$\bar n(1+\bar n)$	$\bar n(1-\bar n)$
activity range	$0<z\le1$	$0<z<\infty$

For bosons, the total density is $n=n_0+n_{\mathrm{th}}$. The table’s thermal-density entry does not include a separately occupied ground mode.

Numerical constants used often are

$$\begin{aligned} \zeta(3/2) &\simeq 2.612375, \\ \zeta(5/2) &\simeq 1.341487. \end{aligned}$$

Maxwell–Boltzmann Regime

Quantum occupations share the dilute asymptotic form

$$\bar n_\alpha \simeq z e^{-\beta\epsilon_\alpha}$$

when $x_\alpha=z e^{-\beta\epsilon_\alpha}\ll1$ for every appreciably occupied mode.

For the homogeneous gas, define phase-space density per internal state

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

The dilute criterion is $x\ll1$. Then

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

and the leading ideal exchange correction is

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

Thus Bose statistics lowers and Fermi statistics raises the pressure relative to the classical ideal-gas value at fixed $n$ and $T$. These signs arise from exchange statistics, not from an attractive or repulsive interparticle potential.

In three dimensions,

$$\frac{P}{nk_{\mathrm B}T} = 1 - \eta \frac{n\lambda_T^3}{2^{5/2}g} + O\!\left[ \left( \frac{n\lambda_T^3}{g} \right)^2 \right].$$

The Maxwell–Boltzmann Limit page owns normalized one-particle distributions, the Gibbs factor, classical thermodynamics, and quantitative accuracy tests.

Bose Chemical-Potential Bound

For bosons, every mode requires

$$\mu < \epsilon_\alpha$$

at finite volume in an ordinary grand-canonical state. Hence

$$\mu < \epsilon_0.$$

With $\epsilon_0=0$, this is $z<1$. In the thermodynamic condensation limit, $z\to1^-$ and the ground mode must be separated from the continuum integral.

For fermions there is no analogous convergence bound because each mode has only occupations zero and one.

For particles whose number is not conserved in equilibrium, such as photons in a blackbody cavity, the equilibrium chemical potential is normally fixed to zero rather than determined by a number equation.

Uniform Ideal Bose Condensation

For a quadratic homogeneous gas, the maximum thermal density at $z=1$ is

$$n_{\mathrm{th,max}} = \frac{g}{\lambda_T^d} \zeta(d/2).$$

It is finite only for $d>2$. Under the declared ideal-gas assumptions,

$$T_c = \frac{2\pi\hbar^2}{mk_{\mathrm B}} \left[ \frac{n}{g\zeta(d/2)} \right]^{2/d}, \qquad d>2.$$

Below $T_c$,

$$\frac{N_0}{N} = 1 - \left( \frac{T}{T_c} \right)^{d/2}.$$

In three dimensions,

$$T_c = \frac{2\pi\hbar^2}{mk_{\mathrm B}} \left[ \frac{n}{g\zeta(3/2)} \right]^{2/3}.$$

These are thermodynamic-limit ideal-gas results. A finite system has a crossover rather than a nonanalytic phase transition. Ground-state degeneracy, separately conserved internal populations, trapping, and interactions modify the interpretation and sometimes the formulas. See Bose–Einstein Condensation.

Zero-Temperature Fermi Gas

At $T=0$,

$$\bar n_{\mathbf k\sigma} = \theta(k_{\mathrm F}-k).$$

The volume of the occupied $d$-dimensional Fermi ball gives

$$n = \frac{g}{(2\pi)^d} \frac{\pi^{d/2}}{ \Gamma(d/2+1) } k_{\mathrm F}^d.$$

The Fermi scales are

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

For a quadratic dispersion in $d$ dimensions,

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

In three dimensions,

$$\begin{aligned} k_{\mathrm F} &= \left( \frac{6\pi^2n}{g} \right)^{1/3}, \\ \frac{U_0}{N} &= \frac35\epsilon_{\mathrm F}, \\ P_0 &= \frac25n\epsilon_{\mathrm F}. \end{aligned}$$

For spin-$1/2$ particles with $g=2$, the familiar relation is $k_{\mathrm F}=(3\pi^2n)^{1/3}$.

Low-Temperature Fermi Gas

For a three-dimensional quadratic gas at fixed density and $T\ll T_{\mathrm F}$,

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

The energy, heat capacity, entropy, and pressure are

$$\begin{aligned} \frac{U}{N} &= \frac35\epsilon_{\mathrm F} \left[ 1+ \frac{5\pi^2}{12} \left( \frac{T}{T_{\mathrm F}} \right)^2 \right], \\ \frac{C_V}{Nk_{\mathrm B}} &= \frac{\pi^2}{2} \frac{T}{T_{\mathrm F}} + O(T^3), \\ \frac{S}{Nk_{\mathrm B}} &= \frac{\pi^2}{2} \frac{T}{T_{\mathrm F}} + O(T^3), \\ P &= \frac25n\epsilon_{\mathrm F} \left[ 1+ \frac{5\pi^2}{12} \left( \frac{T}{T_{\mathrm F}} \right)^2 \right]. \end{aligned}$$

The displayed $O(T^4)$ and $O(T^3)$ abbreviations mean powers of the dimensionless ratio $T/T_{\mathrm F}$. These coefficients are specific to a three-dimensional quadratic density of states at fixed density. The Sommerfeld Expansion page owns the general asymptotic method.

General Isotropic Power-Law Dispersion

For

$$\epsilon_{\mathbf k} = A k^r, \qquad A>0,$$

the continuum density of states is

$$\frac{D_{d,r}(\epsilon)}{V} = \frac{gS_{d-1}}{ (2\pi)^d r A^{d/r} } \epsilon^{d/r-1},$$

where

$$S_{d-1} = \frac{2\pi^{d/2}}{\Gamma(d/2)}$$

is the area of the unit $(d-1)$-sphere.

For bosons at $z=1$, the low-energy number integral is finite only when

$$\frac{d}{r} > 1.$$

The quadratic result follows from $r=2$. This criterion concerns a homogeneous ideal continuum and does not by itself decide condensation in a trap, lattice band, finite system, or interacting fluid.

Slowly Varying External Potentials

In a semiclassical local-density approximation, define

$$z(\mathbf r) = z e^{-\beta V_{\mathrm{ext}}(\mathbf r)}.$$

The local thermal density is then

$$n_{\mathrm{th}}(\mathbf r) = \frac{g}{\lambda_T^d} \Phi_{d/2}^{(\eta)} \left[ z(\mathbf r) \right].$$

Integrating over position gives the total thermal number. This approximation requires the potential to vary slowly on the relevant microscopic and thermal length scales. Harmonic traps change the effective density of states and therefore the condensation exponent; use Quantum Gases in Traps for the canonical treatment.

Finite-Size and Ensemble Cautions

1. Keep exact sums when the level spacing is resolved. Replace $\sum_\alpha$ by an integral only after comparing the level spacing with $k_{\mathrm B}T$, $\epsilon_{\mathrm F}$, or the probe resolution.
2. Separate a bosonic ground mode. A continuum density of states can erase the mode whose occupation becomes macroscopic.
3. Do not infer a finite-size singularity. Exact finite partition functions are analytic away from convergence boundaries.
4. Distinguish grand-canonical and canonical fluctuations. Independent-mode variances apply to the grand-canonical ideal gas; a fixed total number introduces correlations.
5. Declare internal constraints. A factor $g$ is valid only when the components share the stated chemical potential and spectrum.
6. Preserve energy-zero covariance. Shift $\epsilon_\alpha$ and $\mu$ together.

Dimensional and Limiting Checks

Use these fast tests:

• $\lambda_T$ has dimensions of length.
• $D_d(\epsilon)$ has dimensions of inverse energy.
• $g/\lambda_T^d$ has dimensions of number density.
• $k_{\mathrm B}T/\lambda_T^d$ has dimensions of pressure or energy density.
• $z$ and $\Phi_s^{(\eta)}(z)$ are dimensionless.
• $z\to0$ recovers Maxwell–Boltzmann occupations.
• $T\to0$ gives a fermionic step and, for a condensed ideal Bose gas, a vanishing thermal fraction.
• $g=2$ in the three-dimensional Fermi formula gives $n=k_{\mathrm F}^3/(3\pi^2)$.
• $d=3$ in $U=(d/2)PV$ gives $U=3PV/2$.

Common Mistakes

• Treating a mean occupation $\bar n_\alpha$ as a probability distribution over modes.
• Inserting $g$ without checking whether internal populations interconvert or have separate chemical potentials.
• Using $z=1$ for conserved massive particles outside the bosonic condensation boundary.
• Applying the Bose continuum integral to the ground mode.
• Using the free-space density of states for a trap or lattice band.
• Forgetting the factor of $g$ in phase-space density and Fermi momentum.
• Calling the Bose or Fermi virial correction an interaction energy.
• Applying low-temperature Fermi coefficients at fixed $\mu$ when the quoted result assumes fixed $N$.
• Using $T\ll T_{\mathrm F}$ as a criterion for a Bose gas.
• Assuming that an ideal Bose condensate is automatically a superfluid.
• Using a thermodynamic-limit transition formula for a small discrete spectrum without a finite-size analysis.

Cross-Links

• Symbols and Conventions
• Ensemble Formula Sheet
• Fermi Gas Formula Sheet
• Bose Gas Formula Sheet
• Quantum Statistics Overview
• Bose–Einstein Statistics
• Fermi–Dirac Statistics
• Maxwell–Boltzmann Limit
• Ideal Bose Gas
• Ideal Fermi Gas
• Bose–Einstein Condensation
• Fermi Momentum and Fermi Energy
• Sommerfeld Expansion
• Quantum Gases in Traps
• Low-Dimensional Quantum Gases
• Bose–Einstein Distribution Formula Card
• Fermi–Dirac Distribution Formula Card

References

• R. K. Pathria and P. D. Beale, Statistical Mechanics, 3rd ed., Elsevier (2011) — ideal quantum gases and ensemble thermodynamics.
• K. Huang, Statistical Mechanics, 2nd ed., Wiley (1987) — density of states, virial expansion, and ideal Bose and Fermi gases.
• L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1, 3rd ed., Butterworth–Heinemann (1980) — quantum distributions and degenerate gases.
• C. J. Pethick and H. Smith, Bose–Einstein Condensation in Dilute Gases, 2nd ed., Cambridge University Press (2008) — ideal-gas condensation and trapped-gas conventions.
• L. Pitaevskii and S. Stringari, Bose–Einstein Condensation and Superfluidity, Oxford University Press (2016) — Bose gases, finite temperature, and interaction boundaries.
• N. W. Ashcroft and N. D. Mermin, Solid State Physics, Harcourt (1976) — three-dimensional Fermi-gas state counting and low-temperature electron thermodynamics.

Exercises

1. Unified one-mode variance

Starting from

$$\Xi_\alpha = \left( 1-\eta x_\alpha \right)^{-\eta},$$

derive the mean occupation and variance using derivatives with respect to $\ln x_\alpha$.

Solution

The mean is

$$\begin{aligned} \bar n_\alpha &= x_\alpha \frac{\partial}{\partial x_\alpha} \ln\Xi_\alpha \\ &= \frac{x_\alpha}{1-\eta x_\alpha}. \end{aligned}$$

The second cumulant is

$$\begin{aligned} \operatorname{Var}(n_\alpha) &= x_\alpha \frac{\partial\bar n_\alpha}{ \partial x_\alpha } \\ &= \frac{x_\alpha}{ (1-\eta x_\alpha)^2 }. \end{aligned}$$

Since $\bar n_\alpha=x_\alpha/(1-\eta x_\alpha)$,

$$\operatorname{Var}(n_\alpha) = \bar n_\alpha \left( 1+\eta\bar n_\alpha \right).$$

2. Density integral and thermal wavelength

Use the quadratic density of states to show that the thermal-mode density is

$$n_{\mathrm{th}} = \frac{g}{\lambda_T^d} \Phi_{d/2}^{(\eta)}(z).$$

Solution

Start from

$$n_{\mathrm{th}} = \frac{1}{V} \int_0^\infty d\epsilon\, \frac{D_d(\epsilon)}{ z^{-1}e^{\beta\epsilon}-\eta }.$$

Substitute the density of states and set $t=\beta\epsilon$:

$$\begin{aligned} n_{\mathrm{th}} &= \frac{g}{\Gamma(d/2)} \left( \frac{m}{2\pi\hbar^2\beta} \right)^{d/2} \\ &\quad\times \int_0^\infty dt\, \frac{t^{d/2-1}}{z^{-1}e^t-\eta}. \end{aligned}$$

The integral is $\Gamma(d/2)\Phi_{d/2}^{(\eta)}(z)$, and

$$\left( \frac{m}{2\pi\hbar^2\beta} \right)^{d/2} = \lambda_T^{-d}.$$

Combining the factors gives the result.

3. Leading exchange correction

Let $x=n\lambda_T^d/g$. Derive

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

Solution

The small-$z$ expansions give

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

Series inversion gives

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

Substitution into the pressure series yields

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

Divide by $x=n\lambda_T^d/g$. Bosons have $\eta=+1$ and therefore a negative correction; fermions have $\eta=-1$ and a positive correction.

4. Condensation criterion for a power-law dispersion

For $\epsilon_{\mathbf k}=Ak^r$, determine when the bosonic thermal density at $z=1$ is finite at low energy.

Solution

The density of states scales as

$$D_{d,r}(\epsilon) \propto \epsilon^{d/r-1}.$$

At $z=1$ and small $\epsilon$,

$$\frac{1}{e^{\beta\epsilon}-1} \sim \frac{1}{\beta\epsilon}.$$

The number integrand therefore behaves as

$$\epsilon^{d/r-2}.$$

The integral at the lower endpoint is finite when

$$\frac{d}{r}-2 > -1,$$

or

$$\frac{d}{r}>1.$$

For a quadratic gas, $r=2$, so a homogeneous ideal-gas condensation transition requires $d>2$.

5. Three-dimensional Fermi-sphere count

Derive the relation between density and Fermi momentum for a three-dimensional gas with internal degeneracy $g$, then specialize to spin-$1/2$ particles.

Solution

At zero temperature, each internal component occupies a momentum-space sphere of radius $k_{\mathrm F}$. Therefore

$$\begin{aligned} n &= g \int_{k<k_{\mathrm F}} \frac{d^3k}{(2\pi)^3} \\ &= \frac{g}{(2\pi)^3} \frac{4\pi k_{\mathrm F}^3}{3} \\ &= \frac{gk_{\mathrm F}^3}{6\pi^2}. \end{aligned}$$

Hence

$$k_{\mathrm F} = \left( \frac{6\pi^2n}{g} \right)^{1/3}.$$

For spin-$1/2$ particles with both spin states populated, $g=2$ and

$$k_{\mathrm F} = \left( 3\pi^2n \right)^{1/3}.$$

6. Local-density classical limit

In an external potential, show that the local-density formula reduces to the barometric form in the dilute regime.

Solution

The local activity is

$$z(\mathbf r) = z e^{-\beta V_{\mathrm{ext}}(\mathbf r)}.$$

For $z(\mathbf r)\ll1$,

$$\Phi_{d/2}^{(\eta)} \left[z(\mathbf r)\right] \simeq z(\mathbf r)$$

for either statistics. Thus

$$n(\mathbf r) \simeq \frac{g z}{\lambda_T^d} e^{-\beta V_{\mathrm{ext}}(\mathbf r)}.$$

Writing $n_\star=gz/\lambda_T^d$ gives

$$n(\mathbf r) = n_\star e^{-\beta V_{\mathrm{ext}}(\mathbf r)},$$

the barometric or Boltzmann profile. Quantum-statistical differences first enter at higher powers of the local activity.