User:EditingPencil/sandbox/ensembles

History

In the begening of the 20th century, four statistical ensembles were identified, namely, first adiabatic ensemble ie. microcanonical ensemble, and three isothermal ensembles which are canonical ensemble, grand canonical ensemble and Gibbs canonical ensemble.

Generalized ensemble theory

	Microcanonical	Canonical	Gibbs Canonical	Grand Canonical	Isoenthalpic–isobaric ensemble
Macrostate	N,V,E	N,V,T	N,P,T	mu,V,T	N,P,H
Probability of each microstate	$P(\mu _{i})=e^{-ST\beta }={\frac {1}{\Omega }}$	$P(\mu _{i})=e^{-ST\beta }={\frac {e^{-E\beta }}{Z}}$	$P(\mu _{i})=e^{-ST\beta }={\frac {e^{-H\beta }}{Z}}$	$P(\mu _{i})=e^{-ST\beta }={\frac {e^{-(E-\mu N)\beta }}{Z}}$
Normalization of probabilities	$\Omega =\sum _{i}1$	$Z=\sum _{i}e^{-{\frac {E(\mu _{i})}{K_{B}T}}}$	$Z=\sum _{i}e^{-{\frac {H(\mu _{i})}{K_{B}T}}}$	$Z=\sum _{i}e^{-{\frac {E(\mu _{i})-\mu N(\mu _{i})}{K_{B}T}}}=\sum _{i}e^{-{\frac {L(\mu _{i})}{K_{B}T}}}$
Entropy relation	$S=K_{B}\log(\Omega )$	$S=K_{B}\log({\mathcal {Z}})+K_{B}\beta E$	$S=K_{B}\log({\mathcal {Z}})+K_{B}\beta H$	$S=K_{B}\log({\mathcal {Z}})+K_{B}\beta (E-\mu N)$
Variable	$\Omega (N,V,E)=e^{\frac {S}{K_{B}}}$	${\mathcal {Z}}(N,V,T)=e^{-{\frac {F}{K_{B}T}}}$	${\mathcal {Z}}(N,P,T)=e^{-{\frac {G}{K_{B}T}}}$	${\mathcal {Z}}(\mu ,V,T)=e^{-{\frac {\Phi }{K_{B}T}}}$	${\mathcal {Z}}(N,P,H)$
Average relations		$-{\frac {\partial }{\partial \beta }}(\log {\mathcal {Z}})=\langle E\rangle$	$-{\frac {\partial }{\partial \beta }}(\log {\mathcal {Z}})=\langle H\rangle$	$-{\frac {\partial }{\partial \beta }}(\log {\mathcal {Z}})=\langle E\rangle -\mu \langle N\rangle$
Other relations	$P={\frac {1}{\beta }}{\frac {\partial \log(\Omega )}{\partial V}}$	$P={\frac {1}{\beta }}{\frac {\partial \log({\mathcal {Z}})}{\partial V}}$	$V=-{\frac {1}{\beta }}{\frac {\partial \log({\mathcal {Z}})}{\partial P}}$	$N={\frac {1}{\beta }}{\frac {\partial \log({\mathcal {Z}})}{\partial \mu }}$	$V=-{\frac {1}{\beta }}{\frac {\partial \log({\mathcal {Z}})}{\partial P}}$
	$\mu =-{\frac {1}{\beta }}{\frac {\partial \log(\Omega )}{\partial N}}$	$\mu =-{\frac {1}{\beta }}{\frac {\partial \log({\mathcal {Z}})}{\partial N}}$	$\mu =-{\frac {1}{\beta }}{\frac {\partial \log({\mathcal {Z}})}{\partial N}}$	$P=-{\frac {1}{\beta }}{\frac {\partial \log({\mathcal {Z}})}{\partial V}}$	$\mu =-{\frac {1}{\beta }}{\frac {\partial \log({\mathcal {Z}})}{\partial N}}$
	$\beta ={\frac {\partial \log(\Omega )}{\partial E}}$	$S=-{\frac {\partial F}{\partial T}}$	$S=-{\frac {\partial G}{\partial T}}$	$S=-{\frac {\partial \Phi }{\partial T}}$	$\beta ={\frac {\partial \log({\mathcal {Z}})}{\partial H}}$

about pseudo-ensembles

	Grand isobaric adiabatic ensemble	Grand isochoric adiabatic ensemble	Guggenheim ensemble
Macrostate	mu, P, R	mu, V, L	mu, P, T
Probability of each microstate			$P(\mu _{i})=e^{-ST\beta }={\frac {e^{-(E+PV-\mu N)\beta }}{Z}}$
Normalization of probabilities			$Z=\sum _{i}e^{-{\frac {E(\mu _{i})+PV(\mu _{i})-\mu N(\mu _{i})}{K_{B}T}}}=\sum _{i}e^{-{\frac {R(\mu _{i})}{K_{B}T}}}$
Entropy relation
Variable
Average relations
Other relations

https://www.google.co.in/books/edition/Molecular_Networking/ViDpEAAAQBAJ?hl=en&gbpv=1&pg=PT130&printsec=frontcover