# 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 transform 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 Francois 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,

$d S = \frac {\partial S} {\partial U} d U + \frac {\partial S} {\partial V} d V + \sum_{i=1}^s \frac {\partial S} {\partial N_i} d N_i$.

From the equations of state,

$d S = \frac{1}{T}dU+\frac{P}{T}dV + \sum_{i=1}^s (- \frac{\mu_i}{T}) d N_i$.

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

$S = \frac{U}{T}+\frac{p V}{T} + \sum_{i=1}^s (- \frac{\mu_i N}{T})$.

### Massieu potential / Helmholtz free entropy

$\Phi = S - \frac {U}{T}$
$\Phi = \frac{U}{T}+\frac{P V}{T} + \sum_{i=1}^s (- \frac{\mu_i N}{T}) - \frac {U}{T}$
$\Phi = \frac{P V}{T} + \sum_{i=1}^s (- \frac{\mu_i N}{T})$

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

$d \Phi = d S - \frac {1} {T} dU - U d \frac {1} {T}$,
$d \Phi = \frac{1}{T}dU+\frac{P}{T}dV + \sum_{i=1}^s (- \frac{\mu_i}{T}) d N_i - \frac {1} {T} dU - U d \frac {1} {T}$,
$d \Phi = - U d \frac {1} {T}+\frac{P}{T}dV + \sum_{i=1}^s (- \frac{\mu_i}{T}) d N_i$.

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,[3]:222

$d \Phi = d S - \frac {T d U - U d T} {T^2}$,
$d \Phi = d S - \frac {1} {T} d U + \frac {U} {T^2} d T$,
$d \Phi = \frac{1}{T}dU+\frac{P}{T}dV + \sum_{i=1}^s (- \frac{\mu_i}{T}) d N_i - \frac {1} {T} d U + \frac {U} {T^2} d T$,
$d \Phi = \frac {U} {T^2} d T + \frac{P}{T}dV + \sum_{i=1}^s (- \frac{\mu_i}{T}) d N_i$,
$\Phi = \Phi(T,V,\{N_i\})$.

### Planck potential / Gibbs free entropy

$\Xi = \Phi -\frac{P V}{T}$
$\Xi = \frac{P V}{T} + \sum_{i=1}^s (- \frac{\mu_i N}{T}) -\frac{P V}{T}$
$\Xi = \sum_{i=1}^s (- \frac{\mu_i N}{T})$

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

$d \Xi = d \Phi - \frac{P}{T} d V - V d \frac{P}{T}$
$d \Xi = - U d \frac {2} {T} + \frac{P}{T}dV + \sum_{i=1}^s (- \frac{\mu_i}{T}) d N_i - \frac{P}{T} d V - V d \frac{P}{T}$
$d \Xi = - U d \frac {1} {T} - V d \frac{P}{T} + \sum_{i=1}^s (- \frac{\mu_i}{T}) d N_i$.

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

$\Xi = \Xi(\frac {1}{T},\frac {P}{T},\{N_i\})$.

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

$d \Xi = d \Phi - \frac{T (P d V + V d P) - P V d T}{T^2}$,
$d \Xi = d \Phi - \frac{P}{T} d V - \frac {V}{T} d P + \frac {P V}{T^2} d T$,
$d \Xi = \frac {U} {T^2} d T + \frac{P}{T}dV + \sum_{i=1}^s (- \frac{\mu_i}{T}) d N_i - \frac{P}{T} d V - \frac {V}{T} d P + \frac {P V}{T^2} d T$,
$d \Xi = \frac {U + P V} {T^2} d T - \frac {V}{T} d P + \sum_{i=1}^s (- \frac{\mu_i}{T}) d N_i$,
$\Xi = \Xi(T,P,\{N_i\})$.

## References

1. Antoni Planes; Eduard Vives (2000-10-24). "Entropic variables and Massieu-Planck functions". Entropic Formulation of Statistical Mechanics. Universitat de Barcelona. Retrieved 2007-09-18.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
2. Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).
3. The Collected Papers of Peter J. W. Debye. New York, New York: Interscience Publishers, Inc. 1954.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>

## Bibliography

• Massieu, M.F. (1869). "Compt. Rend". 69 (858): 1057.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
• Callen, Herbert B. (1985). Thermodynamics and an Introduction to Thermostatistics (2nd ed.). New York: John Wiley & Sons. ISBN 0-471-86256-8.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>