Bose Gas Formula Sheet

This sheet specializes the Quantum Gas Formula Sheet to ideal bosons. The essential step is to separate a potentially macroscopic ground mode before replacing excited-state sums by continuum integrals.

Required background. Bose–Einstein statistics derives the mode distribution, and the ideal Bose gas explains thermal-cloud saturation.

Quadratic bosons and ground-state degeneracy

Unless stated otherwise:

• the particles are noninteracting bosons of mass $m$;
• the continuum is homogeneous and $d$ dimensional;
• $\epsilon_{\mathbf k}=\hbar^2k^2/(2m)$;
• the ground energy is shifted to $\epsilon_0=0$;
• $g$ counts internal components sharing one chemical potential and spectrum;
• $n=N/V$ is total density across those components;
• $\beta=1/(k_{\mathrm B}T)$ and $z=e^{\beta\mu}$;
• $0<z<1$ at finite volume;
• the thermodynamic limit follows periodic finite-volume state counting.

The factor $g$ is not mechanical: separately conserved components require separate number equations and chemical potentials, while a degenerate ground manifold can change how condensate occupation is distributed.

Core Distribution and Bound

The Bose–Einstein occupation is

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

Positivity and convergence require

$$\mu < \epsilon_0$$

for an ordinary finite-volume grand-canonical state. With $\epsilon_0=0$,

$$\mu<0, \qquad 0<z<1.$$

The thermodynamic condensation boundary is approached as

$$\mu\to0^-, \qquad z\to1^-.$$

For bosonic quasiparticles whose number is not conserved in equilibrium, the chemical potential is normally zero. Their oscillator frequencies must still be positive: an exactly zero-frequency unconstrained oscillator has a divergent partition function and needs separate treatment.

Exact Independent-Mode Formulas

For one-particle modes $\alpha$,

$$H = \sum_\alpha \epsilon_\alpha n_\alpha, \qquad n_\alpha=0,1,2,\ldots.$$

Define

$$x_\alpha \equiv e^{-\beta(\epsilon_\alpha-\mu)}.$$

The one-mode probability law is geometric:

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

The grand partition function and grand potential are

$$\begin{aligned} \Xi &= \prod_\alpha \frac{1}{1-x_\alpha}, \\ \Omega &= k_{\mathrm B}T \sum_\alpha \ln(1-x_\alpha). \end{aligned}$$

The mean and variance of one mode are

$$\begin{aligned} \langle n_\alpha\rangle &= \frac{x_\alpha}{1-x_\alpha}, \\ \operatorname{Var}(n_\alpha) &= \langle n_\alpha\rangle \left( 1+\langle n_\alpha\rangle \right). \end{aligned}$$

The one-mode entropy is

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

These fluctuation formulas apply to independent grand-canonical modes. A fixed total particle number correlates the occupations.

Separate the Ground Mode

Let $g_0$ be the finite number of degenerate ground modes, including internal components. The total mean occupation $N_0$ of this manifold is

$$N_0 = \frac{g_0z}{1-z}.$$

Therefore

$$\begin{aligned} z &= \frac{N_0}{N_0+g_0}, \\ \mu &= -k_{\mathrm B}T \ln\left( 1+\frac{g_0}{N_0} \right). \end{aligned}$$

For $N_0\gg g_0$,

$$\mu \simeq -\frac{g_0 k_{\mathrm B}T}{N_0}.$$

The exact number equation is

$$N = N_0 + \sum_{\alpha\notin\mathcal G_0} \frac{1}{z^{-1}e^{\beta\epsilon_\alpha}-1}.$$

Here $\mathcal G_0$ is the set of ground modes. For a single spatial minimum with $g$ degenerate internal components, $g_0=g$. Only the excited sum should be replaced by a smooth continuum integral near condensation. Absorbing the ground mode into the density of states erases the degree of freedom that becomes macroscopic.

The ground-mode grand potential is

$$\Omega_0 = g_0 k_{\mathrm B}T\ln(1-z).$$

Even when $N_0=O(V)$, $\Omega_0/V\to0$ in the thermodynamic limit. The ideal condensate therefore changes density without adding an extensive pressure contribution.

Free-Continuum Density of States

For the quadratic dispersion,

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

The thermal de Broglie wavelength is

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

Use

$$\sum_{\alpha\notin\mathcal G_0} F(\epsilon_\alpha) \longrightarrow \int_0^\infty d\epsilon\, D_d(\epsilon)F(\epsilon)$$

only when the excited spectrum is sufficiently dense on the relevant thermal scale.

Bose Functions

Define

$$g_s(z) \equiv \operatorname{Li}_s(z) = \sum_{\ell=1}^{\infty} \frac{z^\ell}{\ell^s}.$$

The integral representation below holds for real $s>0$ and $0<z<1$, or at $z=1$ for $s>1$. For nonpositive indices use the series at $z<1$ or analytic continuation, not this integral:

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

Useful identities are

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

For $z\ll1$,

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

The degeneracy factor $g$ and the Bose function $g_s(z)$ are different objects. The subscripted function should never be mistaken for the number of internal components.

Normal-Phase Master Formulas

Let

$$s \equiv \frac d2.$$

When the ground-mode density is negligible,

$$n = \frac{g}{\lambda_T^d} g_s(z).$$

The pressure, grand potential, and internal energy are

$$\begin{aligned} P &= \frac{gk_{\mathrm B}T}{\lambda_T^d} g_{s+1}(z), \\ \Omega &= -PV, \\ U &= sPV. \end{aligned}$$

The entropy per particle is

$$\frac{S}{Nk_{\mathrm B}} = (s+1) \frac{g_{s+1}(z)}{g_s(z)} - \ln z.$$

At fixed density, $z(T)$ follows from the number equation. Its logarithmic derivative is

$$\frac{d\ln z}{d\ln T} = -s \frac{g_s(z)}{g_{s-1}(z)}.$$

The normal-phase heat capacity at fixed $N$ and $V$ is

$$\frac{C_V}{Nk_{\mathrm B}} = s(s+1) \frac{g_{s+1}(z)}{g_s(z)} - s^2 \frac{g_s(z)}{g_{s-1}(z)}.$$

This formula assumes a homogeneous quadratic gas and a smooth thermodynamic limit. It should not be transplanted unchanged to a trap or lattice band.

Thermal-Cloud Saturation

At fixed $T$, the excited-state density increases with $z$ and reaches

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

when $z\to1^-$, provided $s>1$.

For a quadratic homogeneous gas, saturation therefore requires

$$d>2.$$

If $n>n_{\mathrm{th,max}}$, the excess density cannot enter the continuum thermal cloud. In the ideal thermodynamic limit it appears as $n_0=n-n_{\mathrm{th,max}}$.

Uniform Condensation Temperature

For $d>2$,

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

In three dimensions,

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

The condensate fraction below $T_c$ is

$$\frac{N_0}{N} = 1 - \left( \frac{T}{T_c} \right)^s, \qquad s=\frac d2.$$

These formulas assume a homogeneous ideal gas, fixed total density, a nondegenerate spatial ground mode apart from the stated internal structure, and a thermodynamic limit.

Condensed-Phase Thermodynamics

Below $T_c$, set $z=1$ in the thermodynamic-limit thermal cloud. Then

$$\begin{aligned} P &= \frac{gk_{\mathrm B}T}{\lambda_T^d} \zeta(s+1), \\ U &= sPV. \end{aligned}$$

The entropy and heat capacity per total particle are

$$\begin{aligned} \frac{S}{Nk_{\mathrm B}} &= (s+1) \frac{\zeta(s+1)}{\zeta(s)} \left( \frac{T}{T_c} \right)^s, \\ \frac{C_V}{Nk_{\mathrm B}} &= s(s+1) \frac{\zeta(s+1)}{\zeta(s)} \left( \frac{T}{T_c} \right)^s. \end{aligned}$$

For the three-dimensional gas,

$$\begin{aligned} \frac{N_0}{N} &= 1- \left( \frac{T}{T_c} \right)^{3/2}, \\ \frac{C_V}{Nk_{\mathrm B}} &= \frac{15}{4} \frac{\zeta(5/2)}{\zeta(3/2)} \left( \frac{T}{T_c} \right)^{3/2}. \end{aligned}$$

At $T_c^-$, the last coefficient is approximately $1.926$. The ideal three-dimensional heat capacity is continuous at $T_c$, while its derivative is nonanalytic.

Three-Dimensional Lookup

For a homogeneous quadratic gas:

Quantity	Normal phase	Condensed phase
activity	$0<z<1$ from $n\lambda_T^3/g=g_{3/2}(z)$	$z\to1$
thermal density	$g\lambda_T^{-3}g_{3/2}(z)$	$g\lambda_T^{-3}\zeta(3/2)$
condensate density	negligible in thermodynamic normal phase	$n_0=n-g\lambda_T^{-3}\zeta(3/2)$
pressure	$gk_{\mathrm B}T\lambda_T^{-3}g_{5/2}(z)$	$gk_{\mathrm B}T\lambda_T^{-3}\zeta(5/2)$
energy	$3PV/2$	$3PV/2$
mode variance	$\bar n(1+\bar n)$	same mode law, with ground-mode caveat

Useful constants are

$$\begin{aligned} \zeta(3/2) &\simeq 2.612375, \\ \zeta(5/2) &\simeq 1.341487, \\ \zeta(3) &\simeq 1.202057. \end{aligned}$$

Compressibility and Number Response

In the normal phase,

$$\left( \frac{\partial n}{\partial\mu} \right)_T = \frac{\beta g}{\lambda_T^d} g_{s-1}(z).$$

The isothermal compressibility is

$$\kappa_T = \frac{\beta g}{n^2\lambda_T^d} g_{s-1}(z).$$

For $d=3$, $g_{1/2}(z)$ diverges as $z\to1^-$. The uniform ideal gas becomes infinitely compressible at the condensation boundary and throughout the condensed thermodynamic-limit phase, because added density can enter the zero-energy mode without changing $\mu$ or pressure.

For a uniform dilute three-dimensional repulsive gas at zero temperature with weak quantum depletion, repulsion supplies a restoring density cost; leading-order mean-field theory gives $\mu=g_{\mathrm{int}}n$ and $\kappa_T=1/(g_{\mathrm{int}}n^2)$ with interaction coupling $g_{\mathrm{int}}>0$. It is one reason the ideal condensate is not a complete model of a physical superfluid.

Ground-Mode Fluctuation Caveat

In the grand-canonical ideal gas,

$$\operatorname{Var}(N_0) = N_0(1+N_0/g_0).$$

If $N_0=O(N)$, then

$$\frac{ \sqrt{\operatorname{Var}(N_0)} }{N_0} \longrightarrow 1/\sqrt{g_0}.$$

For fixed $g_0$, this relative fluctuation remains nonzero; it equals one in the single-ground-mode case $g_0=1$. This behavior is often called the grand-canonical fluctuation problem or “grand-canonical catastrophe.” It reflects the combination of an exactly ideal zero mode and unconstrained particle exchange. Canonical fixed-$N$ calculations correlate $N_0$ with the thermal cloud and do not inherit this $O(N_0^2)$ variance unchanged.

Do not use this grand-canonical result as a universal claim about condensate fluctuations in finite traps or interacting gases.

Dilute Classical Limit

Let

$$x \equiv \frac{n\lambda_T^d}{g} \ll 1.$$

Then

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

The pressure is

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

The negative correction is an exchange-statistics effect, not an attractive two-body potential.

General Dispersion and Critical Dimension

For

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

the low-energy density of states scales as

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

At $z=1$, the thermal-number integrand behaves as

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

It is integrable at zero energy only if

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

For a quadratic gas, $r=2$ and the criterion is $d>2$. A uniform ideal gas in one or two dimensions therefore has no finite-temperature thermodynamic Bose condensation transition. Finite size, confinement, altered dispersion, and interactions can produce crossovers or other kinds of order, but they do not invalidate the stated homogeneous ideal-gas result.

Harmonic-Trap Translation

For a three-dimensional anisotropic harmonic trap with

$$\bar\omega \equiv (\omega_x\omega_y\omega_z)^{1/3},$$

the semiclassical excited-state density of states is

$$D_{\mathrm{tr}}(\epsilon) = \frac{g\epsilon^2}{ 2(\hbar\bar\omega)^3 }.$$

Measure $\epsilon$ from the discrete trap ground energy. When $k_{\mathrm B}T$ is large compared with each $\hbar\omega_i$,

$$N_{\mathrm{ex}} = g \left( \frac{k_{\mathrm B}T}{\hbar\bar\omega} \right)^3 g_3(z).$$

The ideal trap condensation scale and condensate fraction are

$$\begin{aligned} T_c^{\mathrm{tr}} &= \frac{\hbar\bar\omega}{k_{\mathrm B}} \left[ \frac{N}{g\zeta(3)} \right]^{1/3}, \\ \frac{N_0}{N} &= 1- \left( \frac{T}{T_c^{\mathrm{tr}}} \right)^3. \end{aligned}$$

These are leading semiclassical ideal-trap formulas, with $N_0$ the total ground-manifold occupation. A sharp trap thermodynamic limit takes $N\to\infty$ and $\bar\omega\to0$ at fixed $N\bar\omega^3$ and fixed frequency ratios. Finite-size shifts, interactions, anisotropy beyond $\bar\omega$, and the experimental definition of the transition require the canonical Quantum Gases in Traps treatment.

Model-Boundary Translations

Lattice bosons

Use the actual band dispersion and Brillouin-zone density of states. A flat or bounded band can change low-energy counting, and interacting Bose–Hubbard systems are not described by independent Bose occupations of the bare lattice modes.

Weak interactions

Repulsive interactions generate a finite compressibility, alter the transition temperature, and replace free-particle low-energy excitations by collective modes. Use the Gross–Pitaevskii Equation and Bogoliubov Theory only within their stated dilute weak-coupling regimes.

One-dimensional gases

The Lieb–Liniger and Tonks–Girardeau limits are interacting one-dimensional models. Their correlations and thermodynamics cannot be obtained by inserting an effective chemical potential into the uniform ideal Bose formulas.

Condensation versus superfluidity

Macroscopic ground-mode occupation and superfluid response are distinct concepts. The uniform ideal Bose gas has condensation in $d>2$ but pathological compressibility and no interaction-generated sound mode.

Exercises

1. Ground-mode chemical potential

Starting from $N_0=g_0z/(1-z)$, solve for $\mu$ and find its large-$N_0$ behavior.

Solution

Rearranging gives

$$z = \frac{N_0}{N_0+g_0}.$$

Since $z=e^{\beta\mu}$,

$$\begin{aligned} \mu &= k_{\mathrm B}T \ln\left( \frac{N_0}{N_0+g_0} \right) \\ &= -k_{\mathrm B}T \ln\left( 1+\frac{g_0}{N_0} \right). \end{aligned}$$

For $N_0\gg g_0$, use $\ln(1+y)=y+O(y^2)$:

$$\mu = -\frac{g_0 k_{\mathrm B}T}{N_0} + O[k_{\mathrm B}T(g_0/N_0)^2].$$

Thus the finite-volume chemical potential remains negative and approaches the ground energy from below.

2. Condensate fraction and heat capacity

For a homogeneous quadratic gas with $d>2$, derive the condensate fraction and the condensed-phase heat capacity at fixed $N$.

Solution

Below $T_c$, $z=1$ and

$$N_{\mathrm{ex}}(T) = \frac{gV}{\lambda_T^d} \zeta(s), \qquad s=\frac d2.$$

Because $\lambda_T^{-d}\propto T^s$ and $N=N_{\mathrm{ex}}(T_c)$,

$$\frac{N_{\mathrm{ex}}(T)}{N} = \left( \frac{T}{T_c} \right)^s.$$

Therefore

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

The thermal energy is

$$U = s \frac{gVk_{\mathrm B}T}{\lambda_T^d} \zeta(s+1),$$

so $U\propto T^{s+1}$. Differentiating and eliminating $gV/\lambda_{T_c}^d$ with the critical-number equation gives

$$\frac{C_V}{Nk_{\mathrm B}} = s(s+1) \frac{\zeta(s+1)}{\zeta(s)} \left( \frac{T}{T_c} \right)^s.$$

3. Grand-canonical ground-mode fluctuation

Find the relative root-mean-square fluctuation of the total population of $g_0$ independent degenerate ground modes as $N_0$ becomes macroscopic.

Solution

Adding the variances of $g_0$ independent geometric modes, each with mean $N_0/g_0$, gives

$$\operatorname{Var}(N_0) = N_0(1+N_0/g_0).$$

Hence

$$\begin{aligned} \frac{\sqrt{\operatorname{Var}(N_0)}}{N_0} &= \sqrt{ \frac{1}{g_0}+\frac{1}{N_0} } \\ &\longrightarrow \frac1{\sqrt{g_0}} \qquad (N_0\to\infty). \end{aligned}$$

The result is specific to the grand-canonical ideal zero mode. A fixed total number or interactions change the fluctuation problem.

4. Harmonic-trap exponent

Use the three-dimensional harmonic-trap density of states to derive the ideal condensate fraction exponent.

Solution

At $z=1$,

$$N_{\mathrm{ex}} = g \left( \frac{k_{\mathrm B}T}{\hbar\bar\omega} \right)^3 \zeta(3).$$

At the trap condensation temperature,

$$N = g \left( \frac{k_{\mathrm B}T_c^{\mathrm{tr}}}{ \hbar\bar\omega } \right)^3 \zeta(3).$$

Taking the ratio gives

$$\frac{N_{\mathrm{ex}}(T)}{N} = \left( \frac{T}{T_c^{\mathrm{tr}}} \right)^3.$$

Therefore

$$\frac{N_0}{N} = 1- \left( \frac{T}{T_c^{\mathrm{tr}}} \right)^3.$$

The exponent $3$ comes from the harmonic-trap density of states and differs from the homogeneous three-dimensional exponent $3/2$.

References

• K. Huang, Statistical Mechanics, 2nd ed., Wiley (1987) — Bose functions, condensation, and fluctuations.
• W. Ketterle and N. J. van Druten, “Bose–Einstein condensation of a finite number of particles trapped in one or three dimensions,” Physical Review A 54, 656–660 (1996) — finite trapped-gas state counting and dimensional crossover.
• L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1, 3rd ed., Butterworth–Heinemann (1980) — ideal quantum gases and thermodynamic limits.
• R. K. Pathria and P. D. Beale, Statistical Mechanics, 3rd ed., Elsevier (2011) — ideal Bose gases and ensemble thermodynamics.
• C. J. Pethick and H. Smith, Bose–Einstein Condensation in Dilute Gases, 2nd ed., Cambridge University Press (2008) — uniform and trapped gases and interaction boundaries.
• L. Pitaevskii and S. Stringari, Bose–Einstein Condensation and Superfluidity, Oxford University Press (2016) — finite-temperature Bose gases, condensates, and collective behavior.