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4.9: Summary of Major Ensembles

Last updated

Sep 10, 2020
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- 4.8: The Grand Canonical Ensemble in the Thermodynamic Limit
- 4.10: Quantal Statistical Mechanics

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	boundary	variables	probability of microstate	p.f.	master function
microcanonical	adiabatic (no-skid)	E, V, N	$\frac{d \Gamma}{N ! h_{0}^{3 N}} \frac{1}{\Omega} \text { or } 0$	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_BT 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_BT 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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