Examples
edit
The most common examples are:
Name
Function
Alt. function
Natural variables
Entropy
S
=
1
T
U
+
P
T
V
−
∑
i
=
1
s
μ
i
T
N
i
{\displaystyle S={\frac {1}{T}}U+{\frac {P}{T}}V-\sum _{i=1}^{s}{\frac {\mu _{i}}{T}}N_{i}\,}
U
,
V
,
{
N
i
}
{\displaystyle ~~~~~U,V,\{N_{i}\}\,}
Massieu potential \ Helmholtz free entropy
Φ
=
S
−
1
T
U
{\displaystyle \Phi =S-{\frac {1}{T}}U}
=
−
A
T
{\displaystyle =-{\frac {A}{T}}}
1
T
,
V
,
{
N
i
}
{\displaystyle ~~~~~{\frac {1}{T}},V,\{N_{i}\}\,}
Planck potential \ Gibbs free entropy
Ξ
=
Φ
−
P
T
V
{\displaystyle \Xi =\Phi -{\frac {P}{T}}V}
=
−
G
T
{\displaystyle =-{\frac {G}{T}}}
1
T
,
P
T
,
{
N
i
}
{\displaystyle ~~~~~{\frac {1}{T}},{\frac {P}{T}},\{N_{i}\}\,}
where
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
ψ
{\displaystyle \psi }
, used by both Planck and Schrödinger . (Note that Gibbs used
ψ
{\displaystyle \psi }
to denote the free energy.) Free entropies where invented by French engineer François Massieu in 1869, and actually predate Gibbs's free energy (1875).
Dependence of the potentials on the natural variables
edit
Entropy
edit
S
=
S
(
U
,
V
,
{
N
i
}
)
{\displaystyle S=S(U,V,\{N_{i}\})}
By the definition of a total differential,
d
S
=
∂
S
∂
U
d
U
+
∂
S
∂
V
d
V
+
∑
i
=
1
s
∂
S
∂
N
i
d
N
i
.
{\displaystyle dS={\frac {\partial S}{\partial U}}dU+{\frac {\partial S}{\partial V}}dV+\sum _{i=1}^{s}{\frac {\partial S}{\partial N_{i}}}dN_{i}.}
From the equations of state ,
d
S
=
1
T
d
U
+
P
T
d
V
+
∑
i
=
1
s
(
−
μ
i
T
)
d
N
i
.
{\displaystyle dS={\frac {1}{T}}dU+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}.}
The differentials in the above equation are all of extensive variables , so they may be integrated to yield
S
=
U
T
+
P
V
T
+
∑
i
=
1
s
(
−
μ
i
N
T
)
.
{\displaystyle S={\frac {U}{T}}+{\frac {PV}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right).}
Massieu potential / Helmholtz free entropy
edit
Φ
=
S
−
U
T
{\displaystyle \Phi =S-{\frac {U}{T}}}
Φ
=
U
T
+
P
V
T
+
∑
i
=
1
s
(
−
μ
i
N
T
)
−
U
T
{\displaystyle \Phi ={\frac {U}{T}}+{\frac {PV}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right)-{\frac {U}{T}}}
Φ
=
P
V
T
+
∑
i
=
1
s
(
−
μ
i
N
T
)
{\displaystyle \Phi ={\frac {PV}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right)}
Starting over at the definition of
Φ
{\displaystyle \Phi }
and taking the total differential, we have via a Legendre transform (and the chain rule )
d
Φ
=
d
S
−
1
T
d
U
−
U
d
1
T
,
{\displaystyle d\Phi =dS-{\frac {1}{T}}dU-Ud{\frac {1}{T}},}
d
Φ
=
1
T
d
U
+
P
T
d
V
+
∑
i
=
1
s
(
−
μ
i
T
)
d
N
i
−
1
T
d
U
−
U
d
1
T
,
{\displaystyle d\Phi ={\frac {1}{T}}dU+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}-{\frac {1}{T}}dU-Ud{\frac {1}{T}},}
d
Φ
=
−
U
d
1
T
+
P
T
d
V
+
∑
i
=
1
s
(
−
μ
i
T
)
d
N
i
.
{\displaystyle d\Phi =-Ud{\frac {1}{T}}+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}.}
The above differentials are not all of extensive variables, so the equation may not be directly integrated. From
d
Φ
{\displaystyle d\Phi }
we see that
Φ
=
Φ
(
1
T
,
V
,
{
N
i
}
)
.
{\displaystyle \Phi =\Phi ({\frac {1}{T}},V,\{N_{i}\}).}
If reciprocal variables are not desired,[3] : 222
d
Φ
=
d
S
−
T
d
U
−
U
d
T
T
2
,
{\displaystyle d\Phi =dS-{\frac {TdU-UdT}{T^{2}}},}
d
Φ
=
d
S
−
1
T
d
U
+
U
T
2
d
T
,
{\displaystyle d\Phi =dS-{\frac {1}{T}}dU+{\frac {U}{T^{2}}}dT,}
d
Φ
=
1
T
d
U
+
P
T
d
V
+
∑
i
=
1
s
(
−
μ
i
T
)
d
N
i
−
1
T
d
U
+
U
T
2
d
T
,
{\displaystyle d\Phi ={\frac {1}{T}}dU+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}-{\frac {1}{T}}dU+{\frac {U}{T^{2}}}dT,}
d
Φ
=
U
T
2
d
T
+
P
T
d
V
+
∑
i
=
1
s
(
−
μ
i
T
)
d
N
i
,
{\displaystyle d\Phi ={\frac {U}{T^{2}}}dT+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i},}
Φ
=
Φ
(
T
,
V
,
{
N
i
}
)
.
{\displaystyle \Phi =\Phi (T,V,\{N_{i}\}).}
Planck potential / Gibbs free entropy
edit
Ξ
=
Φ
−
P
V
T
{\displaystyle \Xi =\Phi -{\frac {PV}{T}}}
Ξ
=
P
V
T
+
∑
i
=
1
s
(
−
μ
i
N
T
)
−
P
V
T
{\displaystyle \Xi ={\frac {PV}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right)-{\frac {PV}{T}}}
Ξ
=
∑
i
=
1
s
(
−
μ
i
N
T
)
{\displaystyle \Xi =\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right)}
Starting over at the definition of
Ξ
{\displaystyle \Xi }
and taking the total differential, we have via a Legendre transform (and the chain rule )
d
Ξ
=
d
Φ
−
P
T
d
V
−
V
d
P
T
{\displaystyle d\Xi =d\Phi -{\frac {P}{T}}dV-Vd{\frac {P}{T}}}
d
Ξ
=
−
U
d
2
T
+
P
T
d
V
+
∑
i
=
1
s
(
−
μ
i
T
)
d
N
i
−
P
T
d
V
−
V
d
P
T
{\displaystyle d\Xi =-Ud{\frac {2}{T}}+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}-{\frac {P}{T}}dV-Vd{\frac {P}{T}}}
d
Ξ
=
−
U
d
1
T
−
V
d
P
T
+
∑
i
=
1
s
(
−
μ
i
T
)
d
N
i
.
{\displaystyle d\Xi =-Ud{\frac {1}{T}}-Vd{\frac {P}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}.}
The above differentials are not all of extensive variables, so the equation may not be directly integrated. From
d
Ξ
{\displaystyle d\Xi }
we see that
Ξ
=
Ξ
(
1
T
,
P
T
,
{
N
i
}
)
.
{\displaystyle \Xi =\Xi \left({\frac {1}{T}},{\frac {P}{T}},\{N_{i}\}\right).}
If reciprocal variables are not desired,[3] : 222
d
Ξ
=
d
Φ
−
T
(
P
d
V
+
V
d
P
)
−
P
V
d
T
T
2
,
{\displaystyle d\Xi =d\Phi -{\frac {T(PdV+VdP)-PVdT}{T^{2}}},}
d
Ξ
=
d
Φ
−
P
T
d
V
−
V
T
d
P
+
P
V
T
2
d
T
,
{\displaystyle d\Xi =d\Phi -{\frac {P}{T}}dV-{\frac {V}{T}}dP+{\frac {PV}{T^{2}}}dT,}
d
Ξ
=
U
T
2
d
T
+
P
T
d
V
+
∑
i
=
1
s
(
−
μ
i
T
)
d
N
i
−
P
T
d
V
−
V
T
d
P
+
P
V
T
2
d
T
,
{\displaystyle d\Xi ={\frac {U}{T^{2}}}dT+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}-{\frac {P}{T}}dV-{\frac {V}{T}}dP+{\frac {PV}{T^{2}}}dT,}
d
Ξ
=
U
+
P
V
T
2
d
T
−
V
T
d
P
+
∑
i
=
1
s
(
−
μ
i
T
)
d
N
i
,
{\displaystyle d\Xi ={\frac {U+PV}{T^{2}}}dT-{\frac {V}{T}}dP+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i},}
Ξ
=
Ξ
(
T
,
P
,
{
N
i
}
)
.
{\displaystyle \Xi =\Xi (T,P,\{N_{i}\}).}
References
edit
^ a b Antoni Planes; Eduard Vives (2000-10-24). "Entropic variables and Massieu-Planck functions". Entropic Formulation of Statistical Mechanics . Universitat de Barcelona. Archived from the original on 2008-10-11. Retrieved 2007-09-18 .
^ T. Wada; A.M. Scarfone (December 2004). "Connections between Tsallis' formalisms employing the standard linear average energy and ones employing the normalized q-average energy". Physics Letters A . 335 (5–6): 351–362. arXiv :cond-mat/0410527 . Bibcode :2005PhLA..335..351W. doi :10.1016/j.physleta.2004.12.054. S2CID 17101164.
^ a b
The Collected Papers of Peter J. W. Debye . New York, New York: Interscience Publishers, Inc. 1954.
Bibliography
edit
Massieu, M.F. (1869). "Compt. Rend". 69 (858): 1057.
Callen, Herbert B. (1985). Thermodynamics and an Introduction to Thermostatistics (2nd ed.). New York: John Wiley & Sons. ISBN 0-471-86256-8 .