Bose–Einstein and Fermi–Dirac Statistics

At finite temperature a gas of identical particles distributes itself across energy levels. The exchange symmetry that classified particles into bosons and fermions also dictates how many of them can sit in each level, and that gives two distinct statistical laws — completely different from the classical Maxwell–Boltzmann result.

Counting states correctly

In classical statistics, particles are distinguishable, so swapping two of them between levels counts as a new arrangement. For identical quantum particles it does not: the same set of occupation numbers is one and only one state. Bosons and fermions then differ in which occupation numbers are allowed:

Bosons: any number of particles may share a single-particle state. Occupation $n = 0, 1, 2, \ldots$
Fermions: at most one particle per single-particle state (Pauli). Occupation $n = 0$ or $1$ .

These two counting rules, fed through the grand-canonical ensemble, produce the two famous distributions.

The two distributions

The average occupation of a single-particle level of energy $\varepsilon$ , at temperature $T$ and chemical potential $\mu$ , is

\bar{n}_\text{BE}(\varepsilon) = \frac{1}{e^{(\varepsilon - \mu)/k_B T} - 1} \qquad\text{(Bose–Einstein)},

\bar{n}_\text{FD}(\varepsilon) = \frac{1}{e^{(\varepsilon - \mu)/k_B T} + 1} \qquad\text{(Fermi–Dirac)}.

The only difference is the sign in the denominator: $-1$ for bosons, $+1$ for fermions. In the limit $e^{(\varepsilon-\mu)/k_B T} \gg 1$ (high temperature or low density), both reduce to the classical Maxwell–Boltzmann form $\bar{n} \approx e^{-(\varepsilon-\mu)/k_B T}$ , where quantum statistics become irrelevant because occupations are tiny.

Reading the physics off the sign

The $+1$ in Fermi–Dirac guarantees $\bar{n}_\text{FD} \le 1$ for all energies — the Pauli principle written into thermodynamics. As $T \to 0$ , every level below $\mu$ (the Fermi energy $E_F$ ) is filled exactly once and every level above is empty: a sharp Fermi sea. This is why metals conduct, why electrons in a white dwarf resist compression, and why the heat capacity of a metal's electrons is small.

The $-1$ in Bose–Einstein allows $\bar{n}_\text{BE}$ to grow without bound as $\varepsilon \to \mu$ . Cool a boson gas enough and a macroscopic fraction of the particles crashes into the single lowest state — Bose–Einstein condensation, first achieved with dilute atomic gases in 1995 and visible in superfluidity and lasing.

A small example

Two identical particles, two levels of energies $0$ and $\varepsilon$ . Count the distinct two-particle states:

Distinguishable: 4 states (each particle independently in level 0 or $\varepsilon$ ).
Bosons: 3 states — both in level 0; both in level $\varepsilon$ ; one in each. Swapping particles in the "one in each" case is not a new state.
Fermions: 1 state — one particle per level (both-in-one-level is forbidden by exclusion).

The same two levels host different numbers of states depending on the statistics, which is exactly why the thermal averages differ.

The takeaway

Exchange symmetry, not any new force, gives identical particles their thermal behaviour: fermions obey Fermi–Dirac statistics and fill states one at a time up to the Fermi energy, while bosons obey Bose–Einstein statistics and can pile into a single state. Both collapse to the classical Maxwell–Boltzmann law when occupations are sparse. These distributions underpin solid-state physics, astrophysics, and the entire field of ultracold atoms.