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.

Examples

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The most common examples are:

Name Function Alt. function Natural variables
Entropy    
Massieu potential \ Helmholtz free entropy      
Planck potential \ Gibbs free entropy      

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

Dependence of the potentials on the natural variables

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Entropy

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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

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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,[3]: 222 

 
 
 
 
 

Planck potential / Gibbs free entropy

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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,[3]: 222 

 
 
 
 
 

References

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  1. ^ a b Antoni Planes; Eduard Vives (2000-10-24). "Entropic variables and Massieu-Planck functions". Entropic Formulation of Statistical Mechanics. Universitat de Barcelona. Archived from the original on 2008-10-11. Retrieved 2007-09-18.
  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. arXiv:cond-mat/0410527. Bibcode:2005PhLA..335..351W. doi:10.1016/j.physleta.2004.12.054. S2CID 17101164.
  3. ^ a b The Collected Papers of Peter J. W. Debye. New York, New York: Interscience Publishers, Inc. 1954.

Bibliography

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  • Massieu, M.F. (1869). "Compt. Rend". 69 (858): 1057. {{cite journal}}: Cite journal requires |journal= (help)