Free entropy

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A thermodynamic free entropy is an entropic thermodynamic 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. In mathematics, free entropy is the generalization of entropy defined in 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. The most common examples are:

Name Formula Natural variables
Entropy S\, ~~~~~U,V,\{N_i\}\,
Massieu potential / Helmholtz free entropy \Xi =S-\frac{1}{T} U = -A/T\, ~~~~~1/T,V,\{N_i\}\,
Planck potential / Gibbs free entropy \psi=S - \frac{1}{T} U -\frac{P}{T} V = -G/T\, ~~~~~1/T,P/T,\{N_i\}\,

where "U" is the internal energy, T = temperature, S = entropy, P = pressure, V = volume, "N" is the number of molecules of type i, "A" is the Helmholtz free energy, and "G" is the Gibbs free energy.

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

[edit] References

  • Massieu, M.F. (1869). "Compt. Rend." 69 (858): 1057.
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