A thermodynamic free entropy is an entropic thermodynamic potential analogous to the free energy. Also known as a Massieu, Planck, or Massieu–Planck potentials (or functions), or (rarely) free information. In statistical mechanics, free entropies frequently appear as the logarithm of a partition function. The Onsager reciprocal relations in particular, are developed in terms of entropic potentials. In mathematics, free entropy means something quite different: it is a generalization of entropy defined in the subject of free probability.
A free entropy is generated by a Legendre transformation of the entropy. The different potentials correspond to different constraints to which the system may be subjected.
See also: List of thermodynamic properties. The most common examples are:
| Name | Function | Alt. function | Natural variables | |||||||||||||||
| Entropy | dS=
dU+
dV-
dNi | ~~~~~U,V,\{Ni\} | ||||||||||||||||
| Massieu potential \ Helmholtz free entropy | \Phi=S-
U | =-
|
,V,\{Ni\} | |||||||||||||||
| Planck potential \ Gibbs free entropy | \Xi=\Phi-
V | =-
|
,\{Ni\} |
where
S
\Phi
\Xi
U
T
P
V
A
G
Ni
\mui
s
i
i
Note that the use of the terms "Massieu" and "Planck" for explicit Massieu-Planck potentials are somewhat obscure and ambiguous. In particular "Planck potential" has alternative meanings. The most standard notation for an entropic potential is
\psi
\psi
S=S(U,V,\{Ni\})
By the definition of a total differential,
dS=
| \partialS | |
| \partialU |
dU+
| \partialS | |
| \partialV |
dV+
| s | |
| \sum | |
| i=1 |
| \partialS | |
| \partialNi |
dNi.
From the equations of state,
dS=
| 1 | dU+ | |
| T |
| P | |
| T |
dV+
| s | |
| \sum | |
| i=1 |
\left(-
| \mui | |
| T |
\right)dNi.
The differentials in the above equation are all of extensive variables, so they may be integrated to yield
S=
| U | + | |
| T |
| PV | |
| T |
+
| s | |
| \sum | |
| i=1 |
\left(-
| \muiN | |
| T |
\right)+rm{constant}.
\Phi=S-
| U | |
| T |
\Phi=
| U | + | |
| T |
| PV | |
| T |
+
| s | |
| \sum | |
| i=1 |
\left(-
| \muiN | |
| T |
\right)-
| U | |
| T |
\Phi=
| PV | |
| T |
+
| s | |
| \sum | |
| i=1 |
\left(-
| \muiN | |
| T |
\right)
Starting over at the definition of
\Phi
d\Phi=dS-
| 1 | |
| T |
dU-Ud
| 1 | |
| T |
,
d\Phi=
| 1 | |
| T |
dU+
| P | |
| T |
dV+
| s | |
| \sum | |
| i=1 |
\left(-
| \mui | |
| T |
\right)dNi-
| 1 | |
| T |
dU-Ud
| 1 | |
| T |
,
d\Phi=-Ud
| 1 | + | |
| T |
| P | |
| T |
dV+
| s | |
| \sum | |
| i=1 |
\left(-
| \mui | |
| T |
\right)dNi.
The above differentials are not all of extensive variables, so the equation may not be directly integrated. From
d\Phi
\Phi=\Phi(
| 1 | |
| T |
,V,\{Ni\}).
If reciprocal variables are not desired,[3]
d\Phi=dS-
| TdU-UdT | |
| T2 |
,
d\Phi=dS-
| 1 | |
| T |
dU+
| U | |
| T2 |
dT,
d\Phi=
| 1 | |
| T |
dU+
| P | |
| T |
dV+
| s | |
| \sum | |
| i=1 |
\left(-
| \mui | |
| T |
\right)dNi-
| 1 | |
| T |
dU+
| U | |
| T2 |
dT,
d\Phi=
| U | |
| T2 |
dT+
| P | |
| T |
dV+
| s | |
| \sum | |
| i=1 |
\left(-
| \mui | |
| T |
\right)dNi,
\Phi=\Phi(T,V,\{Ni\}).
\Xi=\Phi-
| PV | |
| T |
\Xi=
| PV | |
| T |
+
| s | |
| \sum | |
| i=1 |
\left(-
| \muiN | |
| T |
\right)-
| PV | |
| T |
\Xi=
| s | |
| \sum | |
| i=1 |
\left(-
| \muiN | |
| T |
\right)
Starting over at the definition of
\Xi
d\Xi=d\Phi-
| P | |
| T |
dV-Vd
| P | |
| T |
d\Xi=-Ud
| 2 | |
| T |
+
| P | |
| T |
dV+
| s | |
| \sum | |
| i=1 |
\left(-
| \mui | |
| T |
\right)dNi-
| P | |
| T |
dV-Vd
| P | |
| T |
d\Xi=-Ud
| 1 | |
| T |
-Vd
| P | |
| T |
+
| s | |
| \sum | |
| i=1 |
\left(-
| \mui | |
| T |
\right)dNi.
The above differentials are not all of extensive variables, so the equation may not be directly integrated. From
d\Xi
\Xi=\Xi\left(
| 1 | |
| T |
,
| P | |
| T |
,\{Ni\}\right).
If reciprocal variables are not desired,[3]
d\Xi=d\Phi-
| T(PdV+VdP)-PVdT | |
| T2 |
,
d\Xi=d\Phi-
| P | |
| T |
dV-
| V | |
| T |
dP+
| PV | |
| T2 |
dT,
d\Xi=
| U | |
| T2 |
dT+
| P | |
| T |
dV+
| s | |
| \sum | |
| i=1 |
\left(-
| \mui | |
| T |
\right)dNi-
| P | |
| T |
dV-
| V | |
| T |
dP+
| PV | |
| T2 |
dT,
d\Xi=
| U+PV | |
| T2 |
dT-
| V | |
| T |
dP+
| s | |
| \sum | |
| i=1 |
\left(-
| \mui | |
| T |
\right)dNi,
\Xi=\Xi(T,P,\{Ni\}).
