Microcanonical
Canonical
Gibbs Canonical
Grand Canonical
Isoenthalpic–isobaric ensemble[1]
Macrostate
N,V,E
N,V,T
N,P,T
mu,V,T
N,P,H
Probability of each microstate
P
(
μ
i
)
=
e
−
S
T
β
=
1
Ω
{\displaystyle P(\mu _{i})=e^{-ST\beta }={\frac {1}{\Omega }}}
P
(
μ
i
)
=
e
−
S
T
β
=
e
−
E
β
Z
{\displaystyle P(\mu _{i})=e^{-ST\beta }={\frac {e^{-E\beta }}{Z}}}
P
(
μ
i
)
=
e
−
S
T
β
=
e
−
H
β
Z
{\displaystyle P(\mu _{i})=e^{-ST\beta }={\frac {e^{-H\beta }}{Z}}}
P
(
μ
i
)
=
e
−
S
T
β
=
e
−
(
E
−
μ
N
)
β
Z
{\displaystyle P(\mu _{i})=e^{-ST\beta }={\frac {e^{-(E-\mu N)\beta }}{Z}}}
Normalization of probabilities
Ω
=
∑
i
1
{\displaystyle \Omega =\sum _{i}1}
Z
=
∑
i
e
−
E
(
μ
i
)
K
B
T
{\displaystyle Z=\sum _{i}e^{-{\frac {E(\mu _{i})}{K_{B}T}}}}
Z
=
∑
i
e
−
H
(
μ
i
)
K
B
T
{\displaystyle Z=\sum _{i}e^{-{\frac {H(\mu _{i})}{K_{B}T}}}}
Z
=
∑
i
e
−
E
(
μ
i
)
−
μ
N
(
μ
i
)
K
B
T
=
∑
i
e
−
L
(
μ
i
)
K
B
T
{\displaystyle 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
(
Ω
)
{\displaystyle S=K_{B}\log(\Omega )}
S
=
K
B
log
(
Z
)
+
K
B
β
E
{\displaystyle S=K_{B}\log({\mathcal {Z}})+K_{B}\beta E}
S
=
K
B
log
(
Z
)
+
K
B
β
H
{\displaystyle S=K_{B}\log({\mathcal {Z}})+K_{B}\beta H}
S
=
K
B
log
(
Z
)
+
K
B
β
(
E
−
μ
N
)
{\displaystyle S=K_{B}\log({\mathcal {Z}})+K_{B}\beta (E-\mu N)}
Variable
Ω
(
N
,
V
,
E
)
=
e
S
K
B
{\displaystyle \Omega (N,V,E)=e^{\frac {S}{K_{B}}}}
Z
(
N
,
V
,
T
)
=
e
−
F
K
B
T
{\displaystyle {\mathcal {Z}}(N,V,T)=e^{-{\frac {F}{K_{B}T}}}}
Z
(
N
,
P
,
T
)
=
e
−
G
K
B
T
{\displaystyle {\mathcal {Z}}(N,P,T)=e^{-{\frac {G}{K_{B}T}}}}
Z
(
μ
,
V
,
T
)
=
e
−
Φ
K
B
T
{\displaystyle {\mathcal {Z}}(\mu ,V,T)=e^{-{\frac {\Phi }{K_{B}T}}}}
Z
(
N
,
P
,
H
)
{\displaystyle {\mathcal {Z}}(N,P,H)}
Average relations
−
∂
∂
β
(
log
Z
)
=
⟨
E
⟩
{\displaystyle -{\frac {\partial }{\partial \beta }}(\log {\mathcal {Z}})=\langle E\rangle }
−
∂
∂
β
(
log
Z
)
=
⟨
H
⟩
{\displaystyle -{\frac {\partial }{\partial \beta }}(\log {\mathcal {Z}})=\langle H\rangle }
−
∂
∂
β
(
log
Z
)
=
⟨
E
⟩
−
μ
⟨
N
⟩
{\displaystyle -{\frac {\partial }{\partial \beta }}(\log {\mathcal {Z}})=\langle E\rangle -\mu \langle N\rangle }
Other relations
P
=
1
β
∂
log
(
Ω
)
∂
V
{\displaystyle P={\frac {1}{\beta }}{\frac {\partial \log(\Omega )}{\partial V}}}
P
=
1
β
∂
log
(
Z
)
∂
V
{\displaystyle P={\frac {1}{\beta }}{\frac {\partial \log({\mathcal {Z}})}{\partial V}}}
V
=
−
1
β
∂
log
(
Z
)
∂
P
{\displaystyle V=-{\frac {1}{\beta }}{\frac {\partial \log({\mathcal {Z}})}{\partial P}}}
N
=
1
β
∂
log
(
Z
)
∂
μ
{\displaystyle N={\frac {1}{\beta }}{\frac {\partial \log({\mathcal {Z}})}{\partial \mu }}}
V
=
−
1
β
∂
log
(
Z
)
∂
P
{\displaystyle V=-{\frac {1}{\beta }}{\frac {\partial \log({\mathcal {Z}})}{\partial P}}}
μ
=
−
1
β
∂
log
(
Ω
)
∂
N
{\displaystyle \mu =-{\frac {1}{\beta }}{\frac {\partial \log(\Omega )}{\partial N}}}
μ
=
−
1
β
∂
log
(
Z
)
∂
N
{\displaystyle \mu =-{\frac {1}{\beta }}{\frac {\partial \log({\mathcal {Z}})}{\partial N}}}
μ
=
−
1
β
∂
log
(
Z
)
∂
N
{\displaystyle \mu =-{\frac {1}{\beta }}{\frac {\partial \log({\mathcal {Z}})}{\partial N}}}
P
=
−
1
β
∂
log
(
Z
)
∂
V
{\displaystyle P=-{\frac {1}{\beta }}{\frac {\partial \log({\mathcal {Z}})}{\partial V}}}
μ
=
−
1
β
∂
log
(
Z
)
∂
N
{\displaystyle \mu =-{\frac {1}{\beta }}{\frac {\partial \log({\mathcal {Z}})}{\partial N}}}
β
=
∂
log
(
Ω
)
∂
E
{\displaystyle \beta ={\frac {\partial \log(\Omega )}{\partial E}}}
S
=
−
∂
F
∂
T
{\displaystyle S=-{\frac {\partial F}{\partial T}}}
S
=
−
∂
G
∂
T
{\displaystyle S=-{\frac {\partial G}{\partial T}}}
S
=
−
∂
Φ
∂
T
{\displaystyle S=-{\frac {\partial \Phi }{\partial T}}}
β
=
∂
log
(
Z
)
∂
H
{\displaystyle \beta ={\frac {\partial \log({\mathcal {Z}})}{\partial H}}}
