# 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 $S={\frac }U+{\frac }V-\sum _^{\frac {\mu _}}N_\,$ $~~~~~U,V,\\}\,$ Massieu potential \ Helmholtz free entropy $\Phi =S-{\frac }U$ $=-{\frac }$ $~~~~~{\frac },V,\\}\,$ Planck potential \ Gibbs free entropy $\Xi =\Phi -{\frac }V$ $=-{\frac }$ $~~~~~{\frac },{\frac },\\}\,$ 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

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

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

From the equations of state,

$dS={\frac }dU+{\frac }dV+\sum _^(-{\frac {\mu _}})dN_$ .

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

$S={\frac }+{\frac }+\sum _^(-{\frac {\mu _N}})$ .

### Massieu potential / Helmholtz free entropy

$\Phi =S-{\frac }$ $\Phi ={\frac }+{\frac }+\sum _^(-{\frac {\mu _N}})-{\frac }$ $\Phi ={\frac }+\sum _^(-{\frac {\mu _N}})$ Starting over at the definition of $\Phi$ and taking the total differential, we have via a Legendre transform (and the chain rule)

$d\Phi =dS-{\frac }dU-Ud{\frac }$ ,
$d\Phi ={\frac }dU+{\frac }dV+\sum _^(-{\frac {\mu _}})dN_-{\frac }dU-Ud{\frac }$ ,
$d\Phi =-Ud{\frac }+{\frac }dV+\sum _^(-{\frac {\mu _}})dN_$ .

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

$\Phi =\Phi ({\frac },V,\\})$ .

If reciprocal variables are not desired,:222

$d\Phi =dS-{\frac }}$ ,
$d\Phi =dS-{\frac }dU+{\frac }}dT$ ,
$d\Phi ={\frac }dU+{\frac }dV+\sum _^(-{\frac {\mu _}})dN_-{\frac }dU+{\frac }}dT$ ,
$d\Phi ={\frac }}dT+{\frac }dV+\sum _^(-{\frac {\mu _}})dN_$ ,
$\Phi =\Phi (T,V,\\})$ .

### Planck potential / Gibbs free entropy

$\Xi =\Phi -{\frac }$ $\Xi ={\frac }+\sum _^(-{\frac {\mu _N}})-{\frac }$ $\Xi =\sum _^(-{\frac {\mu _N}})$ Starting over at the definition of $\Xi$ and taking the total differential, we have via a Legendre transform (and the chain rule)

$d\Xi =d\Phi -{\frac }dV-Vd{\frac }$ $d\Xi =-Ud{\frac }+{\frac }dV+\sum _^(-{\frac {\mu _}})dN_-{\frac }dV-Vd{\frac }$ $d\Xi =-Ud{\frac }-Vd{\frac }+\sum _^(-{\frac {\mu _}})dN_$ .

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

$\Xi =\Xi ({\frac },{\frac },\\})$ .

If reciprocal variables are not desired,:222

$d\Xi =d\Phi -{\frac }}$ ,
$d\Xi =d\Phi -{\frac }dV-{\frac }dP+{\frac }}dT$ ,
$d\Xi ={\frac }}dT+{\frac }dV+\sum _^(-{\frac {\mu _}})dN_-{\frac }dV-{\frac }dP+{\frac }}dT$ ,
$d\Xi ={\frac }}dT-{\frac }dP+\sum _^(-{\frac {\mu _}})dN_$ ,
$\Xi =\Xi (T,P,\\})$ .