Ideal Bose Gas

One-Sentence Description

The ideal Bose gas is a free many-body model in which bosonic occupation statistics, together with the infrared structure of a specified one-particle spectrum, determine equilibrium thermodynamics and whether an excited-state capacity limit permits macroscopic occupation of a lowest mode.

This dossier fixes the model data, solvability claim, important limits, observables, and validation targets. The Ideal Bose Gas teaching article owns the full box calculation, density-of-states derivation, and equation of state. Bose–Einstein Condensation owns the general condensate criterion and transition analysis.

Baseline Model Record

Unless a variant is named explicitly, this dossier uses the following model.

Field	Baseline choice
particles	one species of identical spinless bosons
space	a cubic box of side $L$ and volume $V=L^3$
boundaries	periodic in all three directions
one-particle Hamiltonian	$h=\mathbf p^2/(2m)$
ground energy	$\epsilon_{\mathbf0}=0$
many-body space	bosonic Fock space, or its fixed-$N$ sector
conserved charge	total particle number $N$
equilibrium controls	$T,V,\mu$ or $T,V,N$
thermodynamic limit	$N,V\to\infty$ at fixed density $n=N/V$
interactions	absent

This is not the same model as an ideal gas in a harmonic trap, an ideal lattice Bose gas, a multicomponent gas with separately conserved populations, or a gas of number-nonconserving quasiparticles. Those systems share mode factorization but not the same density of states, chemical-potential constraint, or condensation scale.

Degrees of Freedom and Hilbert Space

The one-particle Hilbert space is

$$\mathcal h = L^2(\mathbb T_L^3),$$

where $\mathbb T_L^3$ is the periodic cube. The bosonic Fock space is

$$\mathcal F_B(\mathcal h) = \bigoplus_{N=0}^{\infty} \operatorname{Sym}^N\mathcal h.$$

For fixed particle number, use only

$$\mathcal H_N = \operatorname{Sym}^N\mathcal h.$$

Momentum-mode operators obey

$$\begin{aligned} [a_{\mathbf k},a_{\mathbf q}^\dagger] &= \delta_{\mathbf k\mathbf q}, \\ [a_{\mathbf k},a_{\mathbf q}] &= 0, \\ [a_{\mathbf k}^\dagger,a_{\mathbf q}^\dagger] &= 0. \end{aligned}$$

Each occupation

$$n_{\mathbf k} = a_{\mathbf k}^\dagger a_{\mathbf k}$$

has eigenvalues $0,1,2,\ldots$. Bosonic exchange symmetry is already built into this occupation-number construction; there are no additional particle labels to symmetrize.

Hamiltonian

Periodic momenta and one-particle energies are

$$\mathbf k = \frac{2\pi}{L}\mathbf n, \qquad \mathbf n\in\mathbb Z^3,$$

and

$$\epsilon_{\mathbf k} = \frac{\hbar^2k^2}{2m}.$$

The many-body Hamiltonian and number operator are

$$\begin{aligned} H &= \sum_{\mathbf k} \epsilon_{\mathbf k}n_{\mathbf k}, \\ N &= \sum_{\mathbf k} n_{\mathbf k}. \end{aligned}$$

Grand-canonical calculations use

$$\begin{aligned} K &= H-\mu N \\ &= \sum_{\mathbf k} \left( \epsilon_{\mathbf k}-\mu \right) n_{\mathbf k}. \end{aligned}$$

At finite positive temperature, convergence requires

$$\mu < \epsilon_{\mathbf0}.$$

With the baseline energy zero, this becomes $\mu<0$. In the condensed thermodynamic limit, $\mu$ approaches zero from below; it does not cross above the lowest one-particle energy.

Energy-zero covariance

The absolute one-particle energy zero is conventional. Under

$$\epsilon_{\mathbf k}' = \epsilon_{\mathbf k}+C, \qquad \mu' = \mu+C,$$

one has

$$\begin{aligned} H' &= H+CN, \\ K' &= H'-\mu'N = K. \end{aligned}$$

Occupations and all grand-canonical probabilities are unchanged. A comparison that shifts $\epsilon_{\mathbf0}$ but not $\mu$ has changed the physical state rather than merely changing convention.

Symmetries and Conserved Quantities

Structure	Baseline statement	Qualification
global phase	$a_{\mathbf k}\mapsto e^{i\theta}a_{\mathbf k}$	generated by $N$
particle number	$[H,N]=0$	choose canonical or grand-canonical treatment explicitly
translation	total momentum is conserved	exact for periodic uniform geometry
cubic point group	the finite box preserves cubic rotations and reflections	full spatial rotations emerge only in the continuum thermodynamic description
time reversal	present in the absence of gauge fields or rotation	maps $\mathbf k$ to $-\mathbf k$
mode occupations	$[H,n_{\mathbf k}]=0$ for every $\mathbf k$	destroyed by generic interactions

The independently conserved mode occupations are the strongest signature of the free model. They make the equilibrium spectrum trivial to label, but they also mean the isolated model contains no collision mechanism that would dynamically establish a thermal distribution from a generic initial state.

Natural Scales and Control Parameters

The thermal de Broglie wavelength is

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

The principal dimensionless controls are

$$\begin{aligned} z &= e^{\beta\mu}, \\ \mathcal D &= n\lambda_T^3, \\ \eta_L &= \frac{L}{\lambda_T}. \end{aligned}$$

Here $z$ is fugacity, $\mathcal D$ is phase-space density for the spinless baseline, and $\eta_L$ measures finite-box resolution. The continuum thermodynamic formulas assume $\eta_L\gg1$ for the thermally occupied excited states, while the ground mode remains separate.

For the uniform three-dimensional baseline,

$$\mathcal D_c = \zeta\!\left(\frac32\right)$$

is the excited-state saturation value. Equivalently,

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

This $T_c$ is not portable to another geometry or dispersion without recomputing the one-particle density of states.

Exact Solution Status

An occupation configuration

$$\lvert\{n_{\mathbf k}\}\rangle$$

is an exact many-body eigenstate with

$$\begin{aligned} E &= \sum_{\mathbf k} \epsilon_{\mathbf k}n_{\mathbf k}, \\ N &= \sum_{\mathbf k} n_{\mathbf k}. \end{aligned}$$

For a finite set of modes, the grand partition function factorizes exactly:

$$\Xi = \prod_{\mathbf k} \frac{1} {1-e^{-\beta(\epsilon_{\mathbf k}-\mu)}}.$$

The mean occupation is

$$\overline n_{\mathbf k} = \frac{1} {e^{\beta(\epsilon_{\mathbf k}-\mu)}-1}.$$

The exactness claim is quantity-dependent.

Quantity	Status	Caveat
finite-volume energies and eigenstates	exact occupation-number solution	requires the declared one-particle spectrum
grand partition function	exact product over modes	converges only for $\mu<\epsilon_0$ at finite $T$
fixed-$N$ partition function	exactly defined and computable by coefficient extraction or recurrence	does not factor into independent grand-canonical mode sums
mode occupations	exact in equilibrium	ensemble must be stated
grand-canonical equal-time correlators	Gaussian and reducible by Wick contractions	fixed-$N$ states require number-conserving treatment
free dynamics	exact mode phases	no collisions or intrinsic equilibration
thermodynamic-limit formulas	exact leading limit for the stated geometry	finite-size corrections and order of limits remain separate
experimental condensate properties	not supplied by the ideal model alone	interactions, trapping, loss, and preparation matter

In the Heisenberg picture,

$$a_{\mathbf k}(t) = e^{-i\epsilon_{\mathbf k}t/\hbar} a_{\mathbf k}(0).$$

This solves free evolution, but it does not imply that a nonequilibrium occupation profile relaxes to a Bose–Einstein distribution. Thermal equilibrium is an ensemble assumption for the ideal Hamiltonian, not a consequence of ideal-gas collisions.

Thermodynamic Fingerprints

For the uniform three-dimensional continuum, separating the ground mode gives

$$n = \frac{N_0}{V} + \frac{1}{\lambda_T^3} \operatorname{Li}_{3/2}(z).$$

In the normal phase,

$$\begin{aligned} P &= \frac{k_{\mathrm B}T}{\lambda_T^3} \operatorname{Li}_{5/2}(z), \\ \frac{U}{V} &= \frac32 P. \end{aligned}$$

Below the ideal condensation scale, the thermodynamic-limit excited fraction is

$$\frac{N_{\mathrm{ex}}}{N} = \left( \frac{T}{T_c} \right)^{3/2},$$

so

$$\frac{N_0}{N} = 1- \left( \frac{T}{T_c} \right)^{3/2}.$$

These formulas are model fingerprints, not a second derivation. Their derivation, entropy, compressibility, and equation-of-state consequences are developed in the teaching article.

Infrared Criterion and Variants

Suppose the excited one-particle density of states near its minimum behaves as

$$\rho_{\mathrm{ex}}(\epsilon) \sim C\epsilon^{s-1}.$$

At saturation, the low-energy number integral behaves as

$$\int_0 d\epsilon\, \epsilon^{s-2}.$$

It is infrared finite only when

$$s>1.$$

For a uniform dispersion $\epsilon\propto k^p$ in $d$ dimensions,

$$s = \frac{d}{p}.$$

Thus a uniform quadratic gas has the ordinary finite-temperature capacity limit only for $d>2$. A harmonic trap has a different density-of-states exponent; a lattice has a band-dependent low-energy density of states. Bose–Einstein Condensation owns the general derivation and Quantum Gases in Traps owns trap state counting.

Important Limits

Limit	Controlled result	What must remain fixed
$z\ll1$	Maxwell–Boltzmann occupation plus exchange corrections	spectrum, $T$, and dilute phase-space density
$T\to0$ at fixed finite $N$	particles occupy a unique ground mode	ground-state degeneracy and ensemble
$V,N\to\infty$ at fixed $n$	sharp thermodynamic condensation boundary in the baseline model	order of limits and density
$L/\lambda_T\to\infty$	excited-mode sums approach continuum integrals	distinguished ground mode kept explicit
$d\le2$ with quadratic uniform dispersion	no ordinary finite-$T$ ideal-gas condensation	homogeneous thermodynamic limit
harmonic confinement	altered density of states and condensate-fraction power	trap frequencies and trap thermodynamic limit
weak repulsive interaction	ideal gas becomes the reference point for Bogoliubov theory	dilute parameter and scattering convention
lattice projection	free bosonic band model; interactions lead toward Bose–Hubbard physics	band, filling, and boundary conditions

The classical limit is not obtained merely by raising the temperature. The relevant requirement is small phase-space density, or equivalently small fugacity in the normal regime.

Finite Volume and Ensemble Audit

At finite $L$ and finite $N$:

• the spectrum is discrete;
• $\mu<\epsilon_0$ in the grand ensemble;
• all thermodynamic functions are smooth;
• there is no exact nonanalytic transition;
• canonical and grand-canonical condensate fluctuations can differ strongly.

With $\epsilon_0=0$, the exact grand-canonical ground-mode occupation is

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

Solving for $z$ gives

$$z = \frac{N_0}{N_0+1} < 1.$$

Therefore

$$\begin{aligned} \mu &= k_{\mathrm B}T\ln z \\ &= -k_{\mathrm B}T \ln\!\left( 1+\frac{1}{N_0} \right) \\ &\sim -\frac{k_{\mathrm B}T}{N_0} \qquad (N_0\gg1). \end{aligned}$$

The thermodynamic shorthand $z=1$ below $T_c$ means this limit has been taken after the ground mode was separated. Substituting $z=1$ into a finite-volume ground-mode geometric series would make that series diverge.

The grand-canonical mode variance,

$$\operatorname{Var}(n_{\mathbf k}) = \overline n_{\mathbf k} \left( 1+\overline n_{\mathbf k} \right),$$

produces order-one relative fluctuations for a macroscopically occupied ideal ground mode. That result is ensemble- and model-specific; it is not a universal prediction for an interacting fixed-$N$ condensate.

Typical Observables

Observable	Ideal-model form or route	Interpretation caveat
momentum occupation	$\overline n_{\mathbf k}$	depends on $\mu$, $T$, and energy-zero convention together
ground-mode occupation	$N_0$	macroscopic occupation is not by itself superfluidity
pressure and energy	polylogarithm formulas in the continuum baseline	geometry and density of states matter
one-body density matrix	Fourier transform of momentum occupations	Penrose–Onsager eigenvalues give the basis-independent criterion
density correlations	Wick contractions in the grand Gaussian state	canonical number projection changes fluctuation relations
compressibility	derivative of the number equation	ideal condensed gas is pathologically soft
free time correlations	mode phases $e^{-i\epsilon_{\mathbf k}t/\hbar}$	no damping or collision-induced linewidth

The model describes Bose enhancement, exchange-statistical pressure corrections, excited-state saturation, and ideal condensation. It does not establish a finite sound speed, vortex energetics, collision rates, hydrodynamic equilibration, or a nonzero Landau critical velocity.

Minimal Worked Example

Consider two identical bosons restricted to a nondegenerate ground mode of energy $0$ and one excited mode of energy $\Delta>0$. Work in the fixed-$N=2$ sector. The three occupation states are

$$\lvert2,0\rangle, \qquad \lvert1,1\rangle, \qquad \lvert0,2\rangle,$$

with energies $0,\Delta,2\Delta$. Defining

$$q = e^{-\beta\Delta},$$

the canonical partition function is

$$Z_2 = 1+q+q^2.$$

The mean excited-mode occupation is

$$\langle n_1\rangle = \frac{q+2q^2} {1+q+q^2}.$$

Hence

$$\begin{aligned} T\to0 &: \langle n_1\rangle\to0, \\ T\to\infty &: \langle n_1\rangle\to1. \end{aligned}$$

The high-temperature limit weights the three bosonic Fock configurations equally. There is no extra multiplicity for “which particle” is excited because the particles are identical. This finite model shows Bose occupation counting but has no phase transition.

Numerical Benchmark

The canonical numerical contract is MB-B006: Ideal Bose Gas.

Its reference fingerprints include

$$\begin{aligned} \zeta(3/2) &= 2.612375348685488\ldots, \\ \zeta(5/2) &= 1.341487257250917\ldots. \end{aligned}$$

For the normal-state target

$$n\lambda_T^3=1,$$

the physical fugacity root is

$$z = 0.698614359135065.$$

At

$$\frac{T}{T_c} = \frac12,$$

the thermodynamic condensate fraction is

$$\frac{N_0}{N} = 0.646446609406726.$$

A valid implementation must:

1. bracket the normal-state root inside $0<z<1$;
2. report the number-equation residual separately from quadrature error;
3. reproduce the zeta values independently of the root solver;
4. separate the ground mode below $T_c$ rather than seeking an unphysical $z>1$;
5. distinguish the thermodynamic $z=1$ prescription from a finite-box calculation;
6. refine arithmetic or quadrature until the reported digits stabilize.

The notebook name ideal_bose_gas_bec.ipynb is currently a planned artifact, not a committed executable. Its release status and promotion requirements belong to Reproducible Notebooks.

Variants and Handoffs

Internal components

An internal degeneracy factor $g$ multiplies the normal-state density of states only when the components have matching spectra and chemical potentials. If component populations are separately conserved or the ground level is degenerate, condensate allocation requires additional state data.

Harmonic traps

Confinement changes the low-energy density of states and the thermodynamic limit. The uniform-box power $3/2$ for the condensate fraction is not the three-dimensional harmonic-trap power. Use Quantum Gases in Traps for the canonical trapped treatment.

Lattices

For ideal bosons in a band,

$$H = \sum_{\mathbf k} \epsilon_{\mathrm{band}}(\mathbf k) n_{\mathbf k}.$$

Mode factorization survives, but bandwidth, band minima, boundary twists, and van Hove structure alter thermodynamics. Adding onsite repulsion produces the Bose–Hubbard Model, which is a different interacting model.

Weak interactions

The contact-interaction term

$$\frac{g}{2} \int d^3x\, \psi^\dagger\psi^\dagger\psi\psi$$

destroys independent conservation of every momentum occupation. Weakly Interacting Bose Gas Preview owns the dilute control parameter, sound, depletion, and the bridge to Bogoliubov quasiparticles.

Number-nonconserving quasiparticles

Equilibrium photons and phonons usually have $\mu=0$ because their number is not conserved. Their Bose occupation law does not make them the same fixed-particle-number gas used for the baseline condensation calculation.

Canonical Boundaries

This dossier owns:

• the convention-complete baseline record;
• the energy-zero and chemical-potential audit;
• the quantity-specific exactness statement;
• the comparison of finite-volume and thermodynamic claims;
• the model-variant and benchmark handoffs.

It does not rederive:

• Bose occupation statistics, owned by Bose–Einstein Statistics;
• uniform thermodynamic formulas, owned by Ideal Bose Gas;
• the general condensate criterion and transition, owned by Bose–Einstein Condensation;
• trap and low-dimensional state counting, owned by Quantum Gases in Traps and Low-Dimensional Quantum Gases;
• interacting condensate theory, owned by Weakly Interacting Bose Gas Preview and Bogoliubov Theory;
• species, apparatus, and experimental chronology, owned by the experimental and AMO volumes.

Common Mistakes

• Writing “ideal Bose gas” without specifying the one-particle spectrum, geometry, and ensemble.
• Treating the grand Hamiltonian $K=H-\mu N$ as though it were the isolated Hamiltonian $H$.
• Shifting one-particle energies without shifting the chemical potential.
• Letting $\mu$ exceed the lowest one-particle energy.
• Replacing the entire finite spectrum by a continuum integral and losing the ground mode.
• Setting $z=1$ in a finite-volume ground-mode geometric series.
• Calling a finite-size crossover an exact phase transition.
• Using the uniform three-dimensional critical temperature for a trap or lattice.
• Inserting an internal degeneracy factor without specifying population constraints.
• Calling a fixed-$N$ thermal state a product of independent grand-canonical mode states.
• Assuming exact free evolution implies collision-driven thermalization.
• Identifying condensate fraction with superfluid fraction.
• Inferring sound, stiffness, or vortex stability from the noninteracting model.
• Treating the anomalous grand-canonical condensate fluctuation as ensemble independent.
• Using an unconstrained numerical root finder that can return $z>1$.

Exercises

1. Energy-zero covariance. Show directly that the mean occupation is unchanged by

$$\epsilon_\alpha\mapsto\epsilon_\alpha+C, \qquad \mu\mapsto\mu+C.$$

What goes wrong if only $\epsilon_\alpha$ is shifted?

Solution

The occupation depends only on the difference:

$$\overline n_\alpha = \frac{1} {e^{\beta(\epsilon_\alpha-\mu)}-1}.$$

After shifting both quantities,

$$\epsilon_\alpha' -\mu' = \epsilon_\alpha-\mu,$$

so every occupation and grand-canonical weight is unchanged. If only $\epsilon_\alpha$ is shifted, then $\epsilon_\alpha-\mu$ changes by $C$; the fugacity relative to the spectrum changes, so this is a different equilibrium state.

2. Finite-box chemical potential. Starting from $N_0=z/(1-z)$ with $\epsilon_0=0$, derive the large-$N_0$ behavior of $\mu$. Explain why it approaches zero from below.

Solution

Solving for $z$ gives

$$z = \frac{N_0}{N_0+1}.$$

Therefore

$$\mu = -k_{\mathrm B}T \ln\!\left( 1+\frac{1}{N_0} \right).$$

Using $\ln(1+x)=x+O(x^2)$,

$$\mu = -\frac{k_{\mathrm B}T}{N_0} + O\!\left( \frac{k_{\mathrm B}T}{N_0^2} \right).$$

For every finite $N_0$, $z<1$ and $\mu<0$. Zero is reached only as the occupation becomes macroscopic in the relevant thermodynamic sequence.

3. General infrared test. Let $\epsilon\propto k^p$ in $d$ uniform spatial dimensions. Determine when the saturated excited-state number is infrared finite.

Solution

The low-energy density of states scales as

$$\rho(\epsilon) \propto \epsilon^{d/p-1}.$$

At saturation,

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

The number integrand is therefore proportional to

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

Its integral converges at zero when

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

or

$$d>p.$$

For quadratic dispersion, $p=2$, so the uniform finite-temperature capacity limit requires $d>2$.

4. Two-mode canonical check. For the two-boson, two-mode example, compute the mean energy and its low- and high-temperature limits.

Solution

Since $E=\Delta n_1$,

$$\langle E\rangle = \Delta \frac{q+2q^2} {1+q+q^2}.$$

As $T\to0$, $q\to0$ and $\langle E\rangle\to0$. As $T\to\infty$, $q\to1$ and

$$\langle E\rangle \longrightarrow \Delta.$$

The latter is the average of the three equally weighted energies $0,\Delta,2\Delta$.

5. Uniqueness of the benchmark root. Show that the normal-state equation

$$\operatorname{Li}_{3/2}(z)=1$$

has at most one root in $0<z<1$. Why is bracketing preferable to an unconstrained Newton iteration?

Solution

For $0<z<1$,

$$\frac{d}{dz} \operatorname{Li}_{3/2}(z) = \frac{ \operatorname{Li}_{1/2}(z) }{z} > 0.$$

Thus $\operatorname{Li}_{3/2}(z)$ is strictly increasing and can cross the target only once. It rises from $0$ to $\zeta(3/2)>1$, so a root exists and is unique.

A bracketed method preserves the physical interval and the existence guarantee. An unconstrained Newton step can leave $0<z<1$, encounter an unphysical branch, or produce $z>1$ even when the initial residual is small.

References

1. R. K. Pathria and P. D. Beale, Statistical Mechanics, 3rd ed., Elsevier (2011).
2. K. Huang, Statistical Mechanics, 2nd ed., Wiley (1987).
3. M. Kardar, Statistical Physics of Particles, Cambridge University Press (2007).
4. A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover (2003).
5. C. J. Pethick and H. Smith, Bose–Einstein Condensation in Dilute Gases, 2nd ed., Cambridge University Press (2008).
6. L. Pitaevskii and S. Stringari, Bose–Einstein Condensation and Superfluidity, Oxford University Press (2016).
7. O. Penrose and L. Onsager, “Bose–Einstein Condensation and Liquid Helium”, Physical Review 104, 576–584 (1956).
8. V. Bagnato and D. Kleppner, “Bose–Einstein Condensation in Low-Dimensional Traps”, Physical Review A 44, 7439–7441 (1991).
9. F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose–Einstein Condensation in Trapped Gases”, Reviews of Modern Physics 71, 463–512 (1999).
10. I. Bloch, J. Dalibard, and W. Zwerger, “Many-Body Physics with Ultracold Gases”, Reviews of Modern Physics 80, 885–964 (2008).

Cross-Links

• Model Encyclopedia Overview for the shared dossier and exactness conventions.
• Ideal Fermi Gas Model Dossier for the fermionic comparison and Pauli-filled baseline.
• Ideal Bose Gas for the full thermodynamic derivation.
• Bose–Einstein Condensation for the basis-independent criterion and transition physics.
• Bose–Einstein Distribution for quick formula lookup.
• Ideal Bose Gas Model Card for the short Reference locator.
• MB-B006 Benchmark for the stable numerical contract.
• Reproducible Notebooks for the planned artifact and promotion gates.