Deriving the Quantum Distributions from the Grand Ensemble

The previous lesson stated the Bose-Einstein and Fermi-Dirac laws and read off their consequences, but the normalization constant $\alpha$ was left as a quantity fixed after the fact by a particle-number integral. The grand canonical ensemble removes that awkwardness. It fixes the temperature and the chemical potential rather than the particle number, and it makes the single-particle modes of an ideal quantum gas statistically independent. The whole derivation is then a sum over the occupancy of one mode, done twice: a geometric series for bosons, a two-term sum for fermions. The constant $\alpha$ of the survey lesson becomes $-\beta \mu$, an equilibrium property of the reservoir rather than a fitting parameter.

The occupation-number representation

An ideal gas has no interactions, so its exact energy eigenstates are products of single-particle states. Label the single-particle levels by an index $k$ with energies ${\epsilon }_k$. A microstate of the whole gas is fixed by stating how many particles occupy each level: the occupation numbers $\{n_k\}$. The total particle number and energy are

The exchange symmetry of the previous lesson restricts the allowed values of each $n_k$:

• Bosons. Any number of identical bosons may share a level, so $n_k\in \{0,1,2,\dots \}$ without bound.
• Fermions. The exclusion principle permits at most one fermion per state, so $n_k\in \{0,1\}$.

In the canonical ensemble the constraint ${\sum }_k{n}_k=N$ couples the levels: the occupancies are not free to vary independently, because raising one $n_k$ forces another down to keep $N$ fixed. That coupling is what makes the fixed-$N$ sum hard. Releasing $N$ decouples the levels.

One mode as an independent subsystem

Fix the temperature through $\beta =1/k_{B}T$ and the chemical potential $\mu$. The grand partition function sums the Gibbs weight $e^{-\beta (E-\mu N)}$ over every microstate. Writing $E-\mu N={\sum }_k({\epsilon }_k-\mu )n_k$ turns the weight into a product over levels, and because the occupancies now range independently, the sum over microstates factorizes:

Each factor ${\Xi }_k$ is the grand partition function of a single level treated as its own subsystem in contact with the reservoir. The grand potential is a sum of one-level contributions,

and every thermodynamic quantity of the ideal quantum gas reduces to a single-level calculation summed over $k$.¹

Summing the two series

The two allowed ranges give two elementary sums.

For bosons the sum is a geometric series in the ratio $e^{-x}$. It converges only if $e^{-x}<1$, i.e. $\mu <\epsilon$; applied to the lowest level this requires the chemical potential to lie below the ground-state energy, a condition that returns with force in Bose-Einstein condensation. The sum is

For fermions only $n=0$ and $n=1$ contribute, so the sum has two terms:

Mean occupation from a single derivative

The mean occupation of a level is the reservoir-weighted average of $n_k$. It comes directly from ${\Xi }_k$ by differentiating with respect to the level energy, since each factor of $e^{-\beta {\epsilon }_k{n}_k}$ carries an $n_k$ down:

Applying this to each closed-form ${\Xi }_k$ gives the two quantum distributions with no free constants left.

The constant $\alpha$ of the survey lesson is now identified: writing $e^{\alpha }=e^{-\beta \mu }$ reproduces the earlier form $1/(e^{\alpha }{e}^{\epsilon /k_{B}T}\mp 1)$. What was a normalization constant is the chemical potential of the reservoir, fixed by thermodynamics rather than by a counting integral. The fermion occupation is bounded, $0\le ⟨n⟩\le 1$, and equals $\frac{1}{2}$ at $\epsilon =\mu$; the boson occupation is unbounded and diverges as $\epsilon \to {\mu }^{+}$.

The Maxwell-Boltzmann limit

Both distributions collapse to a common form when every level is nearly empty. That happens when $e^{\beta (\epsilon -\mu )}\gg 1$ for all occupied levels, which the lowest level makes the condition $e^{-\beta \mu }\gg 1$, i.e. a large negative $\mu$. In that regime the $\mp 1$ is negligible beside the exponential and

The occupancy is the Boltzmann factor scaled by the fugacity $z=e^{\beta \mu }$, and the distinction between the two statistics disappears: when levels are almost never doubly occupied, whether double occupancy is enhanced, forbidden, or neutral cannot matter. This is the microscopic content of the classical-validity criterion of the survey lesson, and the next lesson develops it into the quantum concentration.³

Occupation-number fluctuations

The grand ensemble delivers the variance of the occupation as easily as the mean, through a second derivative of $\ln {\Xi }_k$. Since $⟨n_k⟩$ is itself ${\beta }^{-1}{\partial }_{\mu }\ln {\Xi }_k$, one more $\mu$-derivative gives

with the lower sign for bosons and the upper for fermions. The identity ${\partial }_{\mu }⟨n⟩=\beta ⟨n⟩(1\mp ⟨n⟩)$ follows from $e^x=(1\mp ⟨n⟩)/⟨n⟩$. The three statistics separate cleanly:

• Bosons have $⟨\Delta n^2⟩=⟨n⟩(1+⟨n⟩)$, larger than the Poisson value $⟨n⟩$. The excess $⟨n{⟩}^2$ is bunching: bosons are found together more often than independent particles would be.
• Fermions have $⟨\Delta n^2⟩=⟨n⟩(1-⟨n⟩)$, smaller than Poisson and vanishing at both $⟨n⟩=0$ and $⟨n⟩=1$. A filled or empty level does not fluctuate. This suppression is anti-bunching.
• The Maxwell-Boltzmann limit $⟨n⟩\ll 1$ sends both to the Poisson variance $⟨\Delta n^2⟩\approx ⟨n⟩$, where the $⟨n{⟩}^2$ correction is negligible.

The variance identities are the seed of the correlation properties of quantum light and matter: boson bunching underlies the Hanbury Brown-Twiss effect and the photon statistics of thermal light, and fermion anti-bunching underlies the Pauli-suppressed shot noise of a degenerate electron gas.⁴

Summary

• In the occupation-number representation an ideal quantum gas is a set of single-particle levels, each holding an occupation $n_k$; the total energy and number are ${\sum }_k{n}_k{\epsilon }_k$ and ${\sum }_k{n}_k$.
• Fixing $T$ and $\mu$ decouples the levels: $\Xi ={\prod }_k{\Xi }_k$ with ${\Xi }_k={\sum }_n{e}^{-\beta ({\epsilon }_k-\mu )n}$. Bosons sum a geometric series ($1/(1-e^{-x})$), fermions sum two terms ($1+e^{-x}$).
• One derivative gives $⟨n⟩=1/(e^{\beta (\epsilon -\mu )}\mp 1)$, and the survey-lesson constant is $e^{\alpha }=e^{-\beta \mu }$.
• When $e^{\beta (\epsilon -\mu )}\gg 1$ both reduce to $⟨n⟩\approx z{e}^{-\beta \epsilon }$, the Maxwell-Boltzmann limit.
• The occupation variance is $⟨n⟩(1\pm ⟨n⟩)$: boson bunching above Poisson, fermion anti-bunching below it and zero at full occupancy.