# Free entropy

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.

## Examples

The most common examples are:

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

### Entropy

$S=S(U,V,\{N_{i}\})$ By the definition of a total differential,

$dS={\frac {\partial S}{\partial U}}dU+{\frac {\partial S}{\partial V}}dV+\sum _{i=1}^{s}{\frac {\partial S}{\partial N_{i}}}dN_{i}.$ From the equations of state,

$dS={\frac {1}{T}}dU+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}.$ The differentials in the above equation are all of extensive variables, so they may be integrated to yield

$S={\frac {U}{T}}+{\frac {PV}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right).$ ### Massieu potential / Helmholtz free entropy

$\Phi =S-{\frac {U}{T}}$ $\Phi ={\frac {U}{T}}+{\frac {PV}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right)-{\frac {U}{T}}$ $\Phi ={\frac {PV}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right)$ Starting over at the definition of $\Phi$ and taking the total differential, we have via a Legendre transform (and the chain rule)

$d\Phi =dS-{\frac {1}{T}}dU-Ud{\frac {1}{T}},$ $d\Phi ={\frac {1}{T}}dU+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}-{\frac {1}{T}}dU-Ud{\frac {1}{T}},$ $d\Phi =-Ud{\frac {1}{T}}+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}.$ The above differentials are not all of extensive variables, so the equation may not be directly integrated. From $d\Phi$ we see that

$\Phi =\Phi ({\frac {1}{T}},V,\{N_{i}\}).$ If reciprocal variables are not desired,: 222

$d\Phi =dS-{\frac {TdU-UdT}{T^{2}}},$ $d\Phi =dS-{\frac {1}{T}}dU+{\frac {U}{T^{2}}}dT,$ $d\Phi ={\frac {1}{T}}dU+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}-{\frac {1}{T}}dU+{\frac {U}{T^{2}}}dT,$ $d\Phi ={\frac {U}{T^{2}}}dT+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i},$ $\Phi =\Phi (T,V,\{N_{i}\}).$ ### Planck potential / Gibbs free entropy

$\Xi =\Phi -{\frac {PV}{T}}$ $\Xi ={\frac {PV}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right)-{\frac {PV}{T}}$ $\Xi =\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right)$ Starting over at the definition of $\Xi$ and taking the total differential, we have via a Legendre transform (and the chain rule)

$d\Xi =d\Phi -{\frac {P}{T}}dV-Vd{\frac {P}{T}}$ $d\Xi =-Ud{\frac {2}{T}}+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}-{\frac {P}{T}}dV-Vd{\frac {P}{T}}$ $d\Xi =-Ud{\frac {1}{T}}-Vd{\frac {P}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}.$ The above differentials are not all of extensive variables, so the equation may not be directly integrated. From $d\Xi$ we see that

$\Xi =\Xi \left({\frac {1}{T}},{\frac {P}{T}},\{N_{i}\}\right).$ If reciprocal variables are not desired,: 222

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