4.9: Summary of Major Ensembles
- Page ID
- 18900
boundary | variables | probability of microstate | p.f. | master function | |
microcanonical | adiabatic (no-skid) | E, V, N | \( \frac{d \Gamma}{N ! h_{0}^{3 N}} \frac{1}{\Omega} \text { or } 0\) | Ω | S(E, V, N) = k_{B} 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) = −k_{B}T lnZ |
grand canonical | heat bath, with holes | T, V,
μ |
\( \frac{d \Gamma_{N} e^{-\beta H\left(\Gamma_{N}\right)-\alpha N}}{N ! h_{0}^{3 N}} \frac{1}{\Xi}\) | Ξ | Π(T, V, µ) = −k_{B}T ln Ξ |
In all cases, the partition function (p.f. in the above table) is the normalization factor
\[\mathrm{p.f.}=\sum_{\text { microstates }} \text { unnormalized probability. }\]