Free entropy

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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.

The most common examples are:

Name	Function	Alt. function	Natural variables
Entropy	${\displaystyle S={\frac {1}{T}}U+{\frac {P}{T}}V-\sum _{i=1}^{s}{\frac {\mu _{i}}{T}}N_{i}\,}$		${\displaystyle ~~~~~U,V,\{N_{i}\}\,}$
Massieu potential \ Helmholtz free entropy	${\displaystyle \Phi =S-{\frac {1}{T}}U}$	${\displaystyle =-{\frac {A}{T}}}$	${\displaystyle ~~~~~{\frac {1}{T}},V,\{N_{i}\}\,}$
Planck potential \ Gibbs free entropy	${\displaystyle \Xi =\Phi -{\frac {P}{T}}V}$	${\displaystyle =-{\frac {G}{T}}}$	${\displaystyle ~~~~~{\frac {1}{T}},{\frac {P}{T}},\{N_{i}\}\,}$

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 ${\displaystyle \psi }$, used by both Planck and Schrödinger. (Note that Gibbs used ${\displaystyle \psi }$ to denote the free energy.) Free entropies where invented by French engineer François Massieu in 1869, and actually predate Gibbs's free energy (1875).

${\displaystyle S=S(U,V,\{N_{i}\})}$

By the definition of a total differential,

${\displaystyle dS={\frac {\partial S}{\partial U}}dU+{\frac {\partial S}{\partial V}}dV+\sum _{i=1}^{s}{\frac {\partial S}{\partial N_{i}}}dN_{i}.}$

From the equations of state,

${\displaystyle dS={\frac {1}{T}}dU+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}.}$

The differentials in the above equation are all of extensive variables, so they may be integrated to yield

${\displaystyle S={\frac {U}{T}}+{\frac {PV}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right)+{\textrm {constant}}.}$

${\displaystyle \Phi =S-{\frac {U}{T}}}$

${\displaystyle \Phi ={\frac {U}{T}}+{\frac {PV}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right)-{\frac {U}{T}}}$

${\displaystyle \Phi ={\frac {PV}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right)}$

Starting over at the definition of ${\displaystyle \Phi }$ and taking the total differential, we have via a Legendre transform (and the chain rule)

${\displaystyle d\Phi =dS-{\frac {1}{T}}dU-Ud{\frac {1}{T}},}$

${\displaystyle d\Phi ={\frac {1}{T}}dU+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}-{\frac {1}{T}}dU-Ud{\frac {1}{T}},}$

${\displaystyle d\Phi =-Ud{\frac {1}{T}}+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}.}$

The above differentials are not all of extensive variables, so the equation may not be directly integrated. From ${\displaystyle d\Phi }$ we see that

${\displaystyle \Phi =\Phi ({\frac {1}{T}},V,\{N_{i}\}).}$

If reciprocal variables are not desired,^[3]: 222

${\displaystyle d\Phi =dS-{\frac {TdU-UdT}{T^{2}}},}$

${\displaystyle d\Phi =dS-{\frac {1}{T}}dU+{\frac {U}{T^{2}}}dT,}$

${\displaystyle d\Phi ={\frac {1}{T}}dU+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}-{\frac {1}{T}}dU+{\frac {U}{T^{2}}}dT,}$

${\displaystyle d\Phi ={\frac {U}{T^{2}}}dT+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i},}$

${\displaystyle \Phi =\Phi (T,V,\{N_{i}\}).}$

${\displaystyle \Xi =\Phi -{\frac {PV}{T}}}$

${\displaystyle \Xi ={\frac {PV}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right)-{\frac {PV}{T}}}$

${\displaystyle \Xi =\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right)}$

Starting over at the definition of ${\displaystyle \Xi }$ and taking the total differential, we have via a Legendre transform (and the chain rule)

${\displaystyle d\Xi =d\Phi -{\frac {P}{T}}dV-Vd{\frac {P}{T}}}$

${\displaystyle d\Xi =-Ud{\frac {2}{T}}+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}-{\frac {P}{T}}dV-Vd{\frac {P}{T}}}$

${\displaystyle d\Xi =-Ud{\frac {1}{T}}-Vd{\frac {P}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}.}$

The above differentials are not all of extensive variables, so the equation may not be directly integrated. From ${\displaystyle d\Xi }$ we see that

${\displaystyle \Xi =\Xi \left({\frac {1}{T}},{\frac {P}{T}},\{N_{i}\}\right).}$

If reciprocal variables are not desired,^[3]: 222

${\displaystyle d\Xi =d\Phi -{\frac {T(PdV+VdP)-PVdT}{T^{2}}},}$

${\displaystyle d\Xi =d\Phi -{\frac {P}{T}}dV-{\frac {V}{T}}dP+{\frac {PV}{T^{2}}}dT,}$

${\displaystyle d\Xi ={\frac {U}{T^{2}}}dT+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}-{\frac {P}{T}}dV-{\frac {V}{T}}dP+{\frac {PV}{T^{2}}}dT,}$

${\displaystyle d\Xi ={\frac {U+PV}{T^{2}}}dT-{\frac {V}{T}}dP+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i},}$

${\displaystyle \Xi =\Xi (T,P,\{N_{i}\}).}$

• Massieu, M.F. (1869). "Compt. Rend". 69 (858): 1057. {{cite journal}}: Cite journal requires |journal= (help)

• Callen, Herbert B. (1985). Thermodynamics and an Introduction to Thermostatistics (2nd ed.). New York: John Wiley & Sons. ISBN 0-471-86256-8.