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# Free entropy

A thermodynamic free entropy is an entropic thermodynamic potential analogous to the free energy. Also know 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. In mathematics, free entropy is the generalization of entropy defined in 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. 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 $\Xi=\Phi -\frac{P}{T} V$ $= - \frac{G}{T}$ $~~~~~\frac{1}{T},\frac{P}{T},\{N_i\}\,$

 S is entropy Φ is the Massieu potential[1][2] Ξ is the Planck potential[1] U is internal energy T is temperature P is pressure V is volume A is Helmholtz free energy G is Gibbs free energy Ni is number of particles (or number of moles) composing the i-th chemical component μi is the chemical potential of the i-th chemical component s is the total number of components i is the ith components

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 ψ, used by both Planck and Schrodinger. (Note that Gibbs used ψ to denote the free energy.) Free entropies where invented by Massieu in 1869, and actually predate Gibb's free energy (1875).

## Dependence of the potentials on the natural variables

### Entropy

S = S(U,V,{Ni})

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 Φ 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 dΦ we see that

$\Phi = \Phi(\frac {1}{T},V,\{N_i\})$.

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

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

### 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 Ξ and taking the total differential, we have via a Legendre transform (and the chain rule)

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

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

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

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

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

## References

1. ^ a b Antoni Planes; Eduard Vives (2000-10-24). Entropic variables and Massieu-Planck functions. Entropic Formulation of Statistical Mechanics. Universitat de Barcelona. Retrieved on 2007-09-18.
2. ^ T. Wada; A.M. Scarfone (12 2004). "Connections between Tsallis’ formalisms employing the standard linear average energy and ones employing the normalized q-average energy" (PDF). Physics Letters, Section A: General, Atomic and Solid State Physics 335 (5-6): 351-362. doi:10.1016/j.physleta.2004.12.054. Retrieved on 2007-09-18.
3. ^ a b (1954) The Collected Papers of Peter J. W. Debye. Interscience Publishers, Inc..
• Massieu, M.F. (1869). "Compt. Rend." 69 (858): 1057.
• Callen, Herbert B. (1985). Thermodynamics and an Introduction to Themostatistics, 2nd Ed., New York: John Wiley & Sons. ISBN 0-471-86256-8.