Ideal Bose Gas

One-Sentence Description

The ideal Bose gas is a noninteracting many-boson model whose occupations follow Bose–Einstein statistics and can accumulate macroscopically in a single mode.

Physical Setup

Choose one-particle modes and allow each bosonic mode to have occupation $0,1,2,\ldots$. The model omits interactions, so its thermodynamics is controlled by the one-particle spectrum and bosonic occupation counting.

Hilbert Space

The natural Hilbert space is bosonic Fock space over a one-particle Hilbert space $\mathcal h$:

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

For fixed particle number, one works in the symmetric $N$-particle sector.

Hamiltonian

In a diagonal mode basis,

$$H = \sum_{\alpha} \epsilon_\alpha a_\alpha^\dagger a_\alpha.$$

For a uniform continuum gas,

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

Parameters

Symbol	Meaning
$m$	particle mass
$n=N/V$	number density
$\mu$	chemical potential
$T$	temperature
$N_0$	occupation of the lowest mode

Solvability

The model is exactly solvable because the Hamiltonian is diagonal in bosonic occupation numbers. The Bose–Einstein occupation is

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

The denominator enforces $\mu$ below the lowest one-particle energy in the ideal gas.

Condensation Scale

For a uniform three-dimensional spinless ideal Bose gas, the condensation temperature is

$$k_BT_c = \frac{2\pi\hbar^2}{m} \left( \frac{n}{\zeta(3/2)} \right)^{2/3}.$$

Below $T_c$, the ideal-gas condensate fraction is

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

in the thermodynamic limit. Uniform one- and two-dimensional ideal Bose gases do not have the same finite-temperature condensation transition in the thermodynamic limit.

Key Observables

• Mode occupation numbers.
• Condensate occupation $N_0$.
• Thermal depletion in the ideal model.
• Density of states.
• Number fluctuations in different ensembles.

What It Teaches

The ideal Bose gas teaches Bose enhancement, the difference between statistics and interactions, the thermodynamic origin of ideal Bose–Einstein condensation, and why dimensionality matters.

It is a starting point for weakly interacting Bose gases, Bogoliubov theory, and lattice boson models.

Canonical Links

• Ideal Bose Gas Model Dossier
• Ideal Bose Gas
• Weakly Interacting Bose Gas Preview
• Bosonic Fock Space
• Bosonic Commutation Relations
• Bose–Einstein Statistics
• Bose–Einstein Distribution
• Mode Occupations

Variants

• uniform Bose gas;
• trapped Bose gas;
• lattice Bose gas;
• weakly interacting Bose gas;
• lower-dimensional Bose gases;
• quasiboson gases such as phonons or magnons.

Common Mistakes

• Treating ideal condensation as identical to superfluidity.
• Forgetting that interactions modify depletion and excitations.
• Ignoring dimensional restrictions on finite-temperature condensation.
• Letting the ideal-gas chemical potential exceed the lowest one-particle energy.

Quick Check

Why does the Bose–Einstein denominator contain a minus sign where the Fermi–Dirac formula contains a plus sign?

Solution

Bosonic modes can hold any nonnegative occupation number, so the grand-canonical sum is a geometric series. Fermionic modes have only occupations $0$ and $1$, giving a two-term sum and the plus sign in the Fermi–Dirac occupation.

References

• K. Huang, Statistical Mechanics, 2nd ed., Wiley, 1987.
• L. Pitaevskii and S. Stringari, Bose–Einstein Condensation, Oxford University Press, 2003.
• C. J. Pethick and H. Smith, Bose–Einstein Condensation in Dilute Gases, 2nd ed., Cambridge University Press, 2008.