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4.9: Summary of Major Ensembles

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  • boundary variables probability of microstate p.f. master function
    microcanonical adiabatic (no-skid) E, V, N \( \frac{d \Gamma}{N ! h_{0}^{3 N}} \frac{1}{\Omega} \text { or } 0\) S(E, V, N) = kB ln Ω
    canonical heat bath T, V, N \( \frac{d \Gamma e^{-\beta H(\Gamma)}}{N ! h_{0}^{3 N}} \frac{1}{Z}\) Z F(T, V, N) = −kBT lnZ
    grand canonical heat bath, with holes T, V,


    \( \frac{d \Gamma_{N} e^{-\beta H\left(\Gamma_{N}\right)-\alpha N}}{N ! h_{0}^{3 N}} \frac{1}{\Xi}\) Ξ Π(T, V, µ) = −kBT ln Ξ

    In all cases, the partition function (p.f. in the above table) is the normalization factor

    \[\mathrm{p.f.}=\sum_{\text { microstates }} \text { unnormalized probability. }\]

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