Free entropy

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Note: Conjugate variables in italics

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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

{{#invoke:see also|seealso}} The most common examples are:

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

where Template:Col-begin Template:Col-break

is entropy
is the Massieu potential[1][2]
is the Planck potential[1]
is internal energy

Template:Col-break

is temperature
is pressure
is volume
is Helmholtz free energy

Template:Col-break

is Gibbs free energy
is number of particles (or number of moles) composing the i-th chemical component
is the chemical potential of the i-th chemical component
is the total number of components
is the th components.

Template:Col-end

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 Francois Massieu in 1869, and actually predate Gibbs'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,[3]: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,[3]:222

,
,
,
,
.

References

  1. 1.0 1.1 Template:Cite web
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  3. 3.0 3.1 {{#invoke:citation/CS1|citation |CitationClass=book }}

Bibliography

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