Free entropy

The most common examples are:

Name	Function	Alt. function	Natural variables
Entropy	${\displaystyle dS={\frac {1}{T}}dU+{\frac {P}{T}}dV-\sum _{i=1}^{s}{\frac {\mu _{i}}{T}}dN_{i}\,}$		${\displaystyle ~~~~~U,V,\{N_{i}\}\,}$
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 were invented by French engineer François Massieu in 1869, and actually predate Gibbs's free energy (1875).