Fermi-Dirac statistics

Fermi-Dirac statistics is applicable to a system of $𝑁$ identical and indistinguishable 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 $| 𝑖 ⟩$ , such that all occupation numbers add to $𝑁 = \sum 𝑖 𝐾 𝑖$ ,

⃗ 𝐊 \in F D = {⃗ 𝐊 \in {0, 1} 𝑁 : \sum 𝑖 𝐾 𝑖 = 𝑁}

and the energy of such a microstate is given by

𝐸 ⃗ 𝐊 = \sum 𝑖 𝐾 𝑖 𝜀 𝑖 = ⃗ 𝐊 \cdot ⃗ 𝜀

Hence the canonical partition function is given by

𝑍 = \sum ⃗ 𝐊 \in F D e x p (- ⃗ 𝐊 \cdot ⃗ 𝜀 𝑘 𝐵 𝑇)

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