グランドカノニカル分布 (Grand canonical ensemble)

基本的性質

⟨N⟩=∑iNipi=1Ξ(β,μ)∑iNie−β(Ei−μNi)=1β∂∂μlnΞ(β,μ)\begin{aligned} \langle N \rangle & = \sum_{i} N_i p_i \\ & = \frac{1}{\Xi \left( \beta ,\mu \right)} \sum_i N_i e^{-\beta \left( E_i -\mu N_i \right)} \\ & = \frac{1}{\beta } \frac{\partial }{\partial \mu } \ln \Xi \left( \beta ,\mu \right) \\ \end{aligned}