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

The most common examples are:where

1. is entropy

2. is the Massieu potential

3. is the Planck potential

4. is internal energy

5. is temperature

6. is pressure

7. is volume

8. is Helmholtz free energy

9. is Gibbs free energy

10. is number of particles (or number of moles) composing the i-th chemical component

11. is the chemical potential of the i-th chemical component

12. is the total number of components

13. th components.

, used by both Planck and Schrödinger to denote the free energy.) Free entropies were invented by French engineer François 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

and taking the total differential, we have via a Legendre transform (and the chain rule)

we see that

If reciprocal variables are not desired,

Planck potential / Gibbs free entropy

and taking the total differential, we have via a Legendre transform (and the chain rule)

we see that

If reciprocal variables are not desired,

References

Bibliography