# 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 transform of the entropy. The different potentials correspond to different constraints to which the system may be subjected.

## Examples

${\displaystyle S}$ is entropy
${\displaystyle \Phi }$ is the Massieu potential[1][2]
${\displaystyle \Xi }$ is the Planck potential[1]
${\displaystyle U}$ is internal energy
${\displaystyle T}$ is temperature
${\displaystyle P}$ is pressure
${\displaystyle V}$ is volume
${\displaystyle A}$ is Helmholtz free energy
${\displaystyle G}$ is Gibbs free energy
${\displaystyle N_{i}}$ is number of particles (or number of moles) composing the i-th chemical component
${\displaystyle \mu _{i}}$ is the chemical potential of the i-th chemical component
${\displaystyle s}$ is the total number of components
${\displaystyle i}$ is the ${\displaystyle i}$th components.

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 Francois Massieu in 1869, and actually predate Gibbs's free energy (1875).

## Dependence of the potentials on the natural variables

### Entropy

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

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,

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

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

${\displaystyle S={\frac {U}{T}}+{\frac {pV}{T}}+\sum _{i=1}^{s}(-{\frac {\mu _{i}N}{T}})}$.

### Massieu potential / Helmholtz free entropy

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

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

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

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

${\displaystyle d\Phi =dS-{\frac {TdU-UdT}{T^{2}}}}$,
${\displaystyle d\Phi =dS-{\frac {1}{T}}dU+{\frac {U}{T^{2}}}dT}$,
${\displaystyle d\Phi ={\frac {1}{T}}dU+{\frac {P}{T}}dV+\sum _{i=1}^{s}(-{\frac {\mu _{i}}{T}})dN_{i}-{\frac {1}{T}}dU+{\frac {U}{T^{2}}}dT}$,
${\displaystyle d\Phi ={\frac {U}{T^{2}}}dT+{\frac {P}{T}}dV+\sum _{i=1}^{s}(-{\frac {\mu _{i}}{T}})dN_{i}}$,
${\displaystyle \Phi =\Phi (T,V,\{N_{i}\})}$.

### Planck potential / Gibbs free entropy

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

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

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

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

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

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

${\displaystyle d\Xi =d\Phi -{\frac {T(PdV+VdP)-PVdT}{T^{2}}}}$,
${\displaystyle d\Xi =d\Phi -{\frac {P}{T}}dV-{\frac {V}{T}}dP+{\frac {PV}{T^{2}}}dT}$,
${\displaystyle d\Xi ={\frac {U}{T^{2}}}dT+{\frac {P}{T}}dV+\sum _{i=1}^{s}(-{\frac {\mu _{i}}{T}})dN_{i}-{\frac {P}{T}}dV-{\frac {V}{T}}dP+{\frac {PV}{T^{2}}}dT}$,
${\displaystyle d\Xi ={\frac {U+PV}{T^{2}}}dT-{\frac {V}{T}}dP+\sum _{i=1}^{s}(-{\frac {\mu _{i}}{T}})dN_{i}}$,
${\displaystyle \Xi =\Xi (T,P,\{N_{i}\})}$.

## References

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## Bibliography

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