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. The species-specific Fermi sheet and Bose sheet expand the low-temperature branches.

Required background. Quantum statistics introduces occupations; the grand-canonical ensemble and partition functions explain the independent-mode sums.

Quadratic ideal gases

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
$n_0=N_0/V$	total bosonic ground-manifold 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 occupations, entropy, and grand potential are unchanged because $H-\mu\hat N$ is unchanged. The physical energy shifts by $C\langle\hat N\rangle$, and the fixed-$N$ Helmholtz free energy shifts by $CN$; it is not correct to leave every thermodynamic quantity 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

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

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

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

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

For $0<z<1$, define the convergent fugacity series

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

For real $s>0$, the integral below also defines the Fermi function for every $z>0$. For bosons it requires $0<z<1$, or $z=1$ with $s>1$. In particular, a degenerate Fermi gas with $z>1$ must not be evaluated using the divergent power series above:

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

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

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

For an ideal Bose condensate with $\epsilon_0=\mu=0$, a finite-degeneracy ground manifold 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 derives 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 $T\ll T_{\mathrm F}$,

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

The energy, heat capacity, entropy, and pressure are

$$\begin{aligned} \frac{U}{N} &= \frac35\epsilon_{\mathrm F} \left[ 1+ \frac{5\pi^2}{12} \left( \frac{T}{T_{\mathrm F}} \right)^2 +O[(T/T_{\mathrm F})^4] \right], \\ \frac{C_V}{Nk_{\mathrm B}} &= \frac{\pi^2}{2} \frac{T}{T_{\mathrm F}} + O[(T/T_{\mathrm F})^3], \\ \frac{S}{Nk_{\mathrm B}} &= \frac{\pi^2}{2} \frac{T}{T_{\mathrm F}} + O[(T/T_{\mathrm F})^3], \\ P &= \frac25n\epsilon_{\mathrm F} \left[ 1+ \frac{5\pi^2}{12} \left( \frac{T}{T_{\mathrm F}} \right)^2 +O[(T/T_{\mathrm F})^4] \right]. \end{aligned}$$

These coefficients are specific to a three-dimensional quadratic density of states at fixed density. The Sommerfeld Expansion page derives the general asymptotic method.

General Isotropic Power-Law Dispersion

For

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

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. For bosons require $z(\mathbf r)<1$ in the normal cloud; a condensed component is treated separately. The continuum potential minimum and the discrete trap ground energy should not be interchanged without the semiclassical approximation. 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.

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. 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.

3. 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 $d>2$.

4. 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.

References

• N. W. Ashcroft and N. D. Mermin, Solid State Physics, Harcourt (1976) — three-dimensional Fermi-gas state counting and low-temperature electron 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.
• NIST Digital Library of Mathematical Functions, §25.12, “Polylogarithms”, for series domains, integral representations and the Fermi/Bose index conventions.
• R. K. Pathria and P. D. Beale, Statistical Mechanics, 3rd ed., Elsevier (2011) — ideal quantum gases and ensemble thermodynamics.
• 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.