Free entropy

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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 transform of the entropy. The different potentials correspond to different constraints to which the system may be subjected.

Examples

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\}\,

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 \psi, used by both Planck and Schrödinger. (Note that Gibbs used \psi to denote the free energy.) Free entropies where invented by French engineer Francois Massieu in 1869, and actually predate Gibbs'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,[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,
\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 {2} {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,[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,
\Xi = \Xi(T,P,\{N_i\}).

References

  1. 1.0 1.1 Antoni Planes; Eduard Vives (2000-10-24). "Entropic variables and Massieu-Planck functions". Entropic Formulation of Statistical Mechanics. Universitat de Barcelona. Retrieved 2007-09-18.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
  2. Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).
  3. 3.0 3.1 The Collected Papers of Peter J. W. Debye. New York, New York: Interscience Publishers, Inc. 1954.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>

Bibliography

  • Massieu, M.F. (1869). "Compt. Rend". 69 (858): 1057.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
  • Callen, Herbert B. (1985). Thermodynamics and an Introduction to Thermostatistics (2nd ed.). New York: John Wiley & Sons. ISBN 0-471-86256-8.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>