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