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:

::$S$ is entropy::$Phi$ is the Massieu potentialcite web |author=Antoni Planes |coauthors=Eduard Vives |date=2000-10-24 |publisher=Universitat de Barcelona |url=http://www.ecm.ub.es/condensed/eduard/papers/massieu/node2.html |title=Entropic variables and Massieu-Planck functions |accessdate=2007-09-18 |work=Entropic Formulation of Statistical Mechanics ] [cite journal |author=T. Wada |coauthors=A.M. Scarfone |year=2004 |month=12 |title=Connections between Tsallis’ formalisms employing the standard linear average energy and ones employing the normalized q-average energy |journal=Physics Letters, Section A: General, Atomic and Solid State Physics |volume=335 |issue=5-6 |pages=351–362 |doi=10.1016/j.physleta.2004.12.054 |url=http://arxiv.org/abs/cond-mat/0410527v1 |accessdate=2007-09-18] ::$Xi$ is the Planck potential::$U$ is internal energy::$T$ is temperature::$P$ is pressure::$V$ is volume::$A$ is Helmholtz free energy::$G$ is Gibbs free energy::$N\_i$ is number of particles (or number of moles) composing the "i"-th chemical component::$mu\_i$ is the chemical potential of the "i"-th chemical component::$s$ is the total number of components::$i$ is the $i$^th 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 $psi$, used by both Planck and Schrödinger. (Note that Gibbs used $psi$ 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

:$S\; =\; S(U,V,\{N\_i\})$

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,cite book
title=The Collected Papers of Peter J. W. Debye
publisher=Interscience Publishers, Inc.
place=New York, New York
year=1954] rp|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)

:$d\; Xi\; =\; d\; Phi\; -\; frac\{P\}\{T\}\; d\; V\; -\; V\; d\; frac\{P\}\{T\}$:$d\; Xi\; =\; -\; U\; d\; frac\; \{1\}\; \{T\}\; +\; frac\{P\}\{T\}dV\; +\; sum\_\{i=1\}^s\; (-\; frac\{mu\_i\}\{T\})\; d\; N\_i\; -\; frac\{P\}\{T\}\; d\; V\; -\; V\; d\; frac\{P\}\{T\}$:$d\; Xi\; =\; -\; U\; d\; frac\; \{1\}\; \{T\}\; -\; V\; d\; frac\{P\}\{T\}\; +\; sum\_\{i=1\}^s\; (-\; frac\{mu\_i\}\{T\})\; d\; N\_i$.

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

:$Xi\; =\; Xi(frac\; \{1\}\{T\},frac\; \{P\}\{T\},\{N\_i\})$.

If reciprocal variables are not desired,rp|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

*cite journal
first =M.F. |last = Massieu|year=1869 |title= Compt. Rend.
volume=69
issue= 858
pages= 1057

*cite book
first = Herbert B. | last = Callen | authorlink = Herbert Callen | year = 1985
title = Thermodynamics and an Introduction to Themostatistics | edition = 2nd Ed.
publisher = John Wiley & Sons | location = New York | id = ISBN 0-471-86256-8