An ideal gas in other quantum-mechanical ensembles

1. Describing a gas by "occupation numbers"

We have an ideal gas of indistinguishable particles that can occupy single-particle energy levels \( \varepsilon \).

Let \( n_\varepsilon \) = how many particles sit in level \( \varepsilon \).
The total energy of the gas is then

\[ E = \sum_{\varepsilon} n_\varepsilon \, \varepsilon \]

The total number of particles is fixed:

\[ \sum_{\varepsilon} n_\varepsilon = N \]

So a macrostate of the gas is completely described by the set \( \{n_\varepsilon\} \): how many particles occupy each level.

2. Canonical partition function in terms of occupation numbers

For a system with fixed \( N, V, T \), the canonical partition function is

\[ Q_N(V,T) = \sum_{E} e^{-\beta E}, \quad \beta = 1/kT. \]

Using \( E = \sum_\varepsilon n_\varepsilon \varepsilon \), we can rewrite this as a sum over all allowed sets \( \{n_\varepsilon\} \):

\[ Q_N(V,T) = \sum_{\{n_\varepsilon\}}' g_{\{n_\varepsilon\}} e^{-\beta \sum_\varepsilon n_\varepsilon \varepsilon}. \]

The prime \( \sum' \) means we only sum over sets that satisfy \( \sum_\varepsilon n_\varepsilon = N \).
\( g_{\{n_\varepsilon\}} \) is a weight factor: it counts how many different microstates correspond to that same pattern of occupations.

The form of \( g_{\{n_\varepsilon\}} \) depends on the statistics:

Bosons (Bose–Einstein): many particles can occupy the same state. \( \Rightarrow g_{\text{BE}} = 1 \) (no restriction beyond the total \( N \)).
Fermions (Fermi–Dirac): at most one particle per single-particle state (Pauli principle).

\[ g_{\text{FD}} = \begin{cases} 1 & \text{if all } n_\varepsilon = 0 \text{ or } 1,\\ 0 & \text{otherwise.} \end{cases} \]

Classical Maxwell–Boltzmann (MB): particles are distinguishable in the counting, which gives

\[ g_{\text{MB}} = \prod_\varepsilon \dfrac{1}{n_\varepsilon!}. \]

So: same general formula for \( Q_N \), just three different weight factors.

3. Maxwell–Boltzmann case: getting the usual ideal-gas result

Plugging the MB weight into \( Q_N \) and using some combinatorics (multinomial theorem), you end up with

\[ Q_N(V,T) = \frac{1}{N!} \left[ Q_1(V,T) \right]^N, \]

where \( Q_1 \) is the one-particle partition function:

\[ Q_1(V,T) = \sum_\varepsilon e^{-\beta \varepsilon}. \]

For a free ideal gas, computing \( Q_1 \) (by replacing the sum over energy levels with an integral over momentum) gives

\[ Q_1 = \frac{V}{\lambda^3}, \quad \lambda = \frac{h}{\sqrt{2\pi m kT}} \]

(\( \lambda \) is the thermal de Broglie wavelength).

So finally:

\[ Q_N = \frac{1}{N!} \left(\frac{V}{\lambda^3}\right)^N. \]

That's the standard canonical partition function of a classical ideal gas.

4. Grand partition function and the \( q \)-potential

Now we change ensemble: instead of fixing \( N \), we allow \( N \) to fluctuate and introduce the fugacity

\[ z = e^{\beta \mu} \]

(\( \mu \) = chemical potential).

The grand partition function is

\[ \mathcal{Q}(z,V,T) = \sum_{N=0}^{\infty} z^N Q_N(V,T). \]

For the MB gas you plug in the \( Q_N \) we just found and get

\[ \mathcal{Q}(z,V,T) = \exp\!\left( \frac{zV}{\lambda^3} \right). \]

They define the \( q \)-potential as

\[ q(z,V,T) \equiv \ln \mathcal{Q}(z,V,T) = \frac{PV}{kT}, \]

so all thermodynamic quantities can be obtained by differentiating \( q \).

5. Doing the same for Bose–Einstein and Fermi–Dirac

Instead of trying to compute \( Q_N \) directly (which is messy for BE/FD), they go straight for the grand partition function:

\[ \mathcal{Q}(z,V,T) = \sum_{N=0}^\infty z^N \sum_{\{n_\varepsilon\}}' e^{-\beta \sum_\varepsilon n_\varepsilon \varepsilon}. \]

They then:

Swap the order of summations.
Notice that the constraint on total \( N \) disappears thanks to the sum over all \( N \).
The result factorizes into a product over independent single-particle levels.

For each level \( \varepsilon \), you have a local sum over allowed \( n_\varepsilon \):

Bosons: \( n_\varepsilon = 0,1,2,\dots \) \( \Rightarrow \sum_{n=0}^{\infty} (ze^{-\beta\varepsilon})^n = 1/(1 - ze^{-\beta\varepsilon}) \).
Fermions: \( n_\varepsilon = 0,1 \) \( \Rightarrow \sum_{n=0}^{1} (ze^{-\beta\varepsilon})^n = 1 + ze^{-\beta\varepsilon} \).

So

\[ \mathcal{Q}(z,V,T) = \begin{cases} \displaystyle \prod_\varepsilon \frac{1}{1 - ze^{-\beta\varepsilon}} & \text{(Bosons)}\\[0.8em] \displaystyle \prod_\varepsilon \bigl(1 + ze^{-\beta\varepsilon}\bigr) & \text{(Fermions)} \end{cases} \]

Then they take logs to get \( q = \ln \mathcal{Q} \), which becomes a sum over levels.

To write all three cases in one formula, they introduce a parameter \( a \):

Bosons: \( a = -1 \)
Fermions: \( a = +1 \)
Classical: \( a \to 0 \)

Then

\[ q(z,V,T) = \frac{PV}{kT} = \frac{1}{a} \sum_\varepsilon \ln \bigl(1 + a z e^{-\beta \varepsilon}\bigr). \]

(If you plug \( a=-1 \) or \( +1 \) you get the BE/FD expressions; the MB limit is obtained as \( a\to 0 \).)

6. Getting average particle number and energy

Thermodynamics in the grand ensemble says:

Mean particle number:

\[ \bar N = z \left(\frac{\partial q}{\partial z}\right)_{V,T} \]

Mean energy:

\[ \bar E = - \left(\frac{\partial q}{\partial \beta}\right)_{z,V}. \]

If you take those derivatives using the expression for \( q \), you get

\[ \bar N = \sum_\varepsilon \frac{1}{z^{-1} e^{\beta\varepsilon} + a}, \]

\[ \bar E = \sum_\varepsilon \frac{\varepsilon}{z^{-1} e^{\beta\varepsilon} + a}. \]

Look at \( \bar N \): it's a sum over levels. That means the average occupation number of a single level \( \varepsilon \) is

\[ \boxed{\langle n_\varepsilon \rangle = \frac{1}{z^{-1} e^{\beta\varepsilon} + a}}. \]

They also show you can get this more directly by differentiating \( q \) with respect to \( \varepsilon \) itself, but the result is the same.

7. Writing it in terms of \( \varepsilon - \mu \)

Using \( z = e^{\beta\mu} \), we can rewrite

\[ \langle n_\varepsilon \rangle = \frac{1}{e^{(\varepsilon - \mu)/kT} + a}. \]

This is the compact final formula in section 6.3:

Fermi–Dirac (fermions, \( a=+1 \))

\[ \langle n_\varepsilon \rangle = \frac{1}{e^{(\varepsilon - \mu)/kT} + 1} \]

Occupation never exceeds 1.

Bose–Einstein (bosons, \( a=-1 \))

\[ \langle n_\varepsilon \rangle = \frac{1}{e^{(\varepsilon - \mu)/kT} - 1} \]

Occupation can become large, which leads to Bose–Einstein condensation when \( \mu \) approaches the lowest energy level.

Maxwell–Boltzmann limit (classical gas, \( a \to 0 \)) Expand denominator for small \( a \): \( \langle n_\varepsilon \rangle \approx z e^{-\beta\varepsilon} \), which is the usual Boltzmann factor.

Super-short summary

Describe a many-particle state by how many particles sit in each single-particle level \( \{n_\varepsilon\} \).
Write the partition functions as sums over these occupation numbers, with a weight \( g \) that encodes the statistics (BE/FD/MB).
In the grand ensemble, levels become independent and the grand partition function factorizes into simple sums over \( n_\varepsilon \).
Taking logs and derivatives of the grand partition function gives you:
The pressure,
The average number of particles,
The mean energy,
And particularly the mean occupation number of each level:

\[ \langle n_\varepsilon \rangle = \frac{1}{e^{(\varepsilon - \mu)/kT} + a}, \]

which is the universal formula covering Bose, Fermi, and classical statistics.