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Statistics of the occupation numbers

1. What is "occupation number"?

Imagine energy levels as "parking spots" for particles.

  • Pick one particular energy value \( \varepsilon \).
  • Let \( n_\varepsilon \) be the number of particles sitting in that energy level (0, 1, 2, 3, …).
  • We're usually interested in the average number of particles there: \(\langle n_\varepsilon \rangle\).

So the basic question is:

On average, how many particles are in a level of energy \( \varepsilon \)?

2. The main formula (one formula, three kinds of particles)

The book gives a single formula for the mean occupation number:

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

where

  • \(T\) = temperature
  • \(k\) = Boltzmann's constant
  • \(\mu\) = chemical potential (controls "how many particles" there are overall)

  • \(a\) distinguishes the kind of particles:

  • \(a = +1\): fermions (Fermi–Dirac statistics)

  • \(a = -1\): bosons (Bose–Einstein statistics)
  • \(a = 0\): classical particles (Maxwell–Boltzmann statistics)

So it's one master formula, and by choosing different \(a\), you get the three familiar distributions.

3. Behavior for different kinds of particles

Fermions (e.g. electrons, protons, neutrons) – \(a = +1\)

  • Pauli exclusion principle: each single-particle state can have at most one particle.
  • The formula guarantees \(0 \leq \langle n_\varepsilon \rangle \leq 1\).
  • In the graph, the fermion curve stays below 1 and flattens at that value.

Bosons (e.g. photons, helium-4 atoms at low T) – \(a = -1\)

  • No exclusion principle; any number of particles can pile into the same state.
  • As \(\mu\) approaches the lowest energy level \(\varepsilon_0\), the denominator \(e^{(\varepsilon-\mu)/kT} - 1\) can become very small, so \(\langle n_\varepsilon \rangle\) can become very large.
  • This is related to Bose–Einstein condensation: many bosons condense into the lowest energy state.

Classical particles (Maxwell–Boltzmann) – \(a = 0\)

If you set \(a=0\), the formula becomes

\[ \langle n_\varepsilon \rangle_{\text{MB}} = e^{(\mu - \varepsilon)/kT} \propto e^{-\varepsilon/kT}. \]
  • This is the usual classical exponential distribution.
  • No quantum restrictions; the occupation numbers are small and independent.

4. When do quantum and classical statistics become the same?

The key condition they emphasize is:

\[ e^{(\varepsilon - \mu)/kT} \gg 1 \]

("much larger than 1").

When this is true, adding \(+1\) or \(-1\) in the denominator doesn't matter:

\[ \frac{1}{e^{(\varepsilon-\mu)/kT} + a} \approx \frac{1}{e^{(\varepsilon-\mu)/kT}} \]

So fermions, bosons, and classical particles all have essentially the same mean occupation number.

This happens when:

  • the gas is dilute (low density), or
  • the energy level is high compared to \(\mu\), or
  • the temperature is high.

Physically: particles are so spread out that their quantum wavefunctions don't overlap much. Then the quantum effects (exclusion for fermions, bunching for bosons) become negligible → classical behavior.

From this condition they also deduce things like:

  • the "fugacity" \(z = e^{\mu/kT} \ll 1\),
  • and a standard low-density condition \(N \lambda^3 / V \ll 1\) (thermal wavelength \( \lambda \) is small compared to interparticle spacing).

5. Fluctuations of the occupation number

So far we looked at the average occupation \(\langle n_\varepsilon \rangle\). Now they ask:

How much does \(n_\varepsilon\) fluctuate around this mean?

They compute \(\langle n_\varepsilon^2 \rangle\) and then the variance

\[ \langle n_\varepsilon^2\rangle - \langle n_\varepsilon \rangle^2 \]

and show that the relative fluctuation is

\[ \frac{\langle n_\varepsilon^2 \rangle - \langle n_\varepsilon \rangle^2}{\langle n_\varepsilon \rangle^2} = \frac{1}{\langle n_\varepsilon \rangle} - a. \]

So:

  • Classical (\(a=0\)): \(\frac{\text{variance}}{\text{mean}^2} = \frac{1}{\langle n\rangle}\) → called normal fluctuations.

  • Fermions (\(a=+1\)): \(\frac{1}{\langle n\rangle} - 1\) → smaller than classical ("below normal") → fermions tend to avoid each other (anti-correlation).

  • Bosons (\(a=-1\)): \(\frac{1}{\langle n\rangle} + 1\) → larger than classical ("above normal") → bosons tend to bunch together (positive correlation).

This is another way of seeing the same physical idea:

  • Bosons like to sit together in the same state.
  • Fermions avoid sharing the same state.

6. The full probability \(p_\varepsilon(n)\)

Finally they look at the full probability that a level has exactly \(n\) particles, not just the average or variance.

Bose–Einstein case (bosons)

They show that

\[ p_\varepsilon(n)_{\text{BE}} \propto (z e^{-\beta \varepsilon})^n \]

and after normalization this becomes a geometric distribution in \(n\). Geometric distributions have large fluctuations → consistent with boson bunching.

Fermi–Dirac case (fermions)

Because of Pauli exclusion:

  • \(n\) can be only 0 or 1.

  • They find

  • \(p(0) = 1 - \langle n_\varepsilon \rangle\)

  • \(p(1) = \langle n_\varepsilon \rangle\).

Very restricted: again, fermions can't pile up in one state.

Maxwell–Boltzmann case (classical)

They get

\[ p_\varepsilon(n)_{\text{MB}} = \frac{(\langle n_\varepsilon \rangle)^n}{n!} e^{-\langle n_\varepsilon \rangle}, \]

which is a Poisson distribution.

  • For Poisson, the variance equals the mean.
  • This is what they call "normal" fluctuations and is typical of independent, random events (like counting radioactive decays).

7. Physical picture to keep in mind

  • Mean occupation: how full each "parking spot" (energy level) is on average.

  • Quantum vs classical:

  • If levels are very lightly populated (\(\langle n_\varepsilon \rangle \ll 1\)), all three types of statistics look the same → classical limit.

  • Differences show up when occupation gets large.
  • Bosons: can pile up; big fluctuations; bunching.
  • Fermions: at most one per state; smaller fluctuations; anti-bunching.
  • Classical particles: intermediate, "normal" fluctuations, Poisson statistics.