Particular relationship between the partition function of an ensemble
The characteristic state function or Massieu's potential [ 1] statistical mechanics refers to a particular relationship between the partition function of an ensemble .
In particular, if the partition function P satisfies
P
=
exp
-->
(
− − -->
β β -->
Q
)
⇔ ⇔ -->
Q
=
− − -->
1
β β -->
ln
-->
(
P
)
{\displaystyle P=\exp(-\beta Q)\Leftrightarrow Q=-{\frac {1}{\beta }}\ln(P)}
P
=
exp
-->
(
+
β β -->
Q
)
⇔ ⇔ -->
Q
=
1
β β -->
ln
-->
(
P
)
{\displaystyle P=\exp(+\beta Q)\Leftrightarrow Q={\frac {1}{\beta }}\ln(P)}
in which Q is a thermodynamic quantity, then Q is known as the "characteristic state function" of the ensemble corresponding to "P". Beta refers to the thermodynamic beta .
Examples
The microcanonical ensemble satisfies
Ω Ω -->
(
U
,
V
,
N
)
=
e
β β -->
T
S
{\displaystyle \Omega (U,V,N)=e^{\beta TS}\;\,}
T
S
{\displaystyle TS}
The canonical ensemble satisfies
Z
(
T
,
V
,
N
)
=
e
− − -->
β β -->
A
{\displaystyle Z(T,V,N)=e^{-\beta A}\,\;}
Helmholtz free energy
A
{\displaystyle A}
The grand canonical ensemble satisfies
Z
(
T
,
V
,
μ μ -->
)
=
e
− − -->
β β -->
Φ Φ -->
{\displaystyle {\mathcal {Z}}(T,V,\mu )=e^{-\beta \Phi }\,\;}
Grand potential
Φ Φ -->
{\displaystyle \Phi }
The isothermal-isobaric ensemble satisfies
Δ Δ -->
(
N
,
T
,
P
)
=
e
− − -->
β β -->
G
{\displaystyle \Delta (N,T,P)=e^{-\beta G}\;\,}
Gibbs free energy
G
{\displaystyle G}
State functions are those which tell about the equilibrium state of a system
References
