Bose-Einstein statistics

Bose-Einstein statistics is applicable to a system of $𝑁$ identical and indistinguishable bosons, i.e. particles not obeying the Pauli exclusion principle and thus able to occupy the same states. A unique microstate is therefore labelled by every combination of occupation numbers $𝐾 𝑖$ for each particle state $| 𝑖 ⟩$ , such that all occupation numbers add to $𝑁 = \sum 𝑖 𝐾 𝑖$

⃗ 𝐊 \in B E = {⃗ 𝐊 \in (ℕ 0) 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 B E e x p (- ⃗ 𝐊 \cdot ⃗ 𝜀 𝑘 𝐵 𝑇)

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