Fermi–Dirac statistics

Remarks

The formula follows from the grand partition function of one state, $1 + e^{-\beta(\varepsilon - \mu)}$, which has one term for each allowed occupation. The mean occupation is $e^{-\beta(\varepsilon - \mu)} / (1 + e^{-\beta(\varepsilon - \mu)})$, which equals the formula above.

Example: at $\varepsilon = \mu$, $\bar n = 1/(1 + 1) = 1/2$. For every $\varepsilon$, $0 < \bar n(\varepsilon) < 1$, consistent with at most one particle per state. As $T \to 0$, $\bar n(\varepsilon)$ tends to $1$ for $\varepsilon < \mu$ and to $0$ for $\varepsilon > \mu$.

Non-example: Maxwell–Boltzmann statistics gives $\bar n = e^{-\beta(\varepsilon - \mu)}$, which exceeds $1$ when $\varepsilon < \mu$, so it violates the exclusion principle. The two agree up to small relative error when $\beta(\varepsilon - \mu) \gg 1$.

Integrals of $f$ against powers of $\varepsilon$ are Fermi–Dirac integrals.