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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.

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## 1. Examples

The most common examples are:

 Name Function Alt. function Natural variables Entropy $\displaystyle{ S = \frac {1}{T} U + \frac {P}{T} V - \sum_{i=1}^s \frac {\mu_i}{T} N_i \, }$ $\displaystyle{ ~~~~~U,V,\{N_i\}\, }$ 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 where invented by French engineer François Massieu in 1869, and actually predate Gibbs's free energy (1875).

## 2. Dependence of the Potentials on the Natural Variables

### 2.1. Entropy

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

By the definition of a total differential,

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

From the equations of state,

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

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

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

### 2.2. Massieu Potential / Helmholtz Free Entropy

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

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

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

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

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

### 2.3. Planck Potential / Gibbs Free Entropy

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

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

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

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

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

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

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

### References

1. Antoni Planes; Eduard Vives (2000-10-24). "Entropic variables and Massieu-Planck functions". Entropic Formulation of Statistical Mechanics. Universitat de Barcelona. http://www.ecm.ub.es/condensed/eduard/papers/massieu/node2.html.
2. T. Wada; A.M. Scarfone (December 2004). "Connections between Tsallis' formalisms employing the standard linear average energy and ones employing the normalized q-average energy". Physics Letters A 335 (5–6): 351–362. doi:10.1016/j.physleta.2004.12.054. Bibcode: 2005PhLA..335..351W.  https://dx.doi.org/10.1016%2Fj.physleta.2004.12.054
3. The Collected Papers of Peter J. W. Debye. New York, New York: Interscience Publishers, Inc.. 1954.
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