- Free entropy
A
thermodynamic "free entropy" is an entropicthermodynamic 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. Inmathematics , free entropy is the generalization of entropy defined infree 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::: is
entropy :: 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] :: is the Planck potential:: isinternal energy :: istemperature :: ispressure :: isvolume :: isHelmholtz free energy :: isGibbs free energy :: is number of particles (or number of moles) composing the "i"-th chemical component:: is thechemical potential of the "i"-th chemical component:: is the total number of components:: is the th componentsNote 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 , used by both
Planck andSchrödinger . (Note that Gibbs used 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
:
By the definition of a total differential,
:.
From the equations of state,
:.
The differentials in the above equation are all of extensive variables, so they may be integrated to yield
:.
Massieu potential Helmholtz free entropy
:::
Starting over at the definition of and taking the total differential, we have via a Legendre transform (and the
chain rule ):,:,:.
The above differentials are not all of extensive variables, so the equation may not be directly integrated. From we see that
:.
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:,:,:,:,:.
Planck potential Gibbs free entropy
:::
Starting over at the definition of and taking the total differential, we have via a Legendre transform (and the
chain rule ):::.
The above differentials are not all of extensive variables, so the equation may not be directly integrated. From we see that
:.
If reciprocal variables are not desired,rp|222
:,:,:,:,:.
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
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