[[Statistical thermodynamics MOC]]
# Fermi-Dirac statistics

**Fermi-Dirac statistics** is applicable to a system of $N$ identical and indistinguishable [[Fermion|fermions]], i.e. particles obeying the [[Pauli exclusion principle]] and thus unable to occupy the same states.
A unique microstate is therefore labelled by every combination of occupation numbers $K_i \in \{ 0,1 \}$ for each particle state $\ket{i}$,
such that all occupation numbers add to $N = \sum_{i} K_{i}$,
$$
\begin{align*}
\vab K \in \mathrm{FD} = \left\{  \vab K \in \{ 0,1 \}^N : \sum_{i} K_{i} = N  \right\}
\end{align*}
$$
and the energy of such a microstate is given by
$$
\begin{align*}
E_{\vab K} = \sum_{i} K_{i} \varepsilon_{i} = \vab K \cdot \vab \varepsilon
\end{align*}
$$
Hence the [[Canonical ensemble|canonical partition function]] is given by
$$
\begin{align*}
Z = \sum_{\vab K \in \mathrm{FD}} \exp\left( \frac{-\vab K \cdot \vab \varepsilon}{k_{B}T} \right)
\end{align*}
$$

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