Free entropy

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


 * $$S$$ is entropy
 * $$\Phi$$ is the Massieu potential
 * $$\Xi$$ is the Planck potential
 * $$U$$ is internal energy


 * $$T$$ is temperature
 * $$P$$ is pressure
 * $$V$$ is volume
 * $$A$$ is Helmholtz free energy


 * $$G$$ is Gibbs free energy
 * $$N_i$$ is number of particles (or number of moles) composing the i-th chemical component
 * $$\mu_i$$ is the chemical potential of the i-th chemical component
 * $$s$$ is the total number of components
 * $$i$$ is the $$i$$th components.

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 François Massieu in 1869, and actually predate Gibbs's free energy (1875).

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,


 * $$d S = \frac{1}{T}dU+\frac{P}{T}dV + \sum_{i=1}^s \left(- \frac{\mu_i}{T}\right) d N_i .$$

The differentials in the above equation are all of extensive variables, so they may be integrated to yield


 * $$S = \frac{U}{T}+\frac{P V}{T} + \sum_{i=1}^s \left(- \frac{\mu_i N}{T}\right).$$

Massieu potential / Helmholtz free entropy

 * $$\Phi = S - \frac {U}{T}$$
 * $$\Phi = \frac{U}{T}+\frac{P V}{T} + \sum_{i=1}^s \left(- \frac{\mu_i N}{T}\right) - \frac {U}{T}$$
 * $$\Phi = \frac{P V}{T} + \sum_{i=1}^s \left(- \frac{\mu_i N}{T}\right)$$

Starting over at the definition of $$\Phi$$ and taking the total differential, we have via a Legendre transform (and the chain rule)


 * $$d \Phi = d S - \frac {1} {T} dU - U d \frac {1} {T} ,$$
 * $$d \Phi = \frac{1}{T}dU + \frac{P}{T}dV + \sum_{i=1}^s \left(- \frac{\mu_i}{T}\right) d N_i - \frac {1} {T} dU - U d \frac {1} {T},$$
 * $$d \Phi = - U d \frac {1} {T}+\frac{P}{T}dV + \sum_{i=1}^s \left(- \frac{\mu_i}{T}\right) d N_i.$$

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,


 * $$d \Phi = d S - \frac {T d U - U d T} {T^2} ,$$
 * $$d \Phi = d S - \frac {1} {T} d U + \frac {U} {T^2} d T ,$$
 * $$d \Phi = \frac{1}{T}dU + \frac{P}{T}dV + \sum_{i=1}^s \left(- \frac{\mu_i}{T}\right) d N_i - \frac {1} {T} d U + \frac {U} {T^2} d T,$$
 * $$d \Phi = \frac {U} {T^2} d T + \frac{P}{T}dV + \sum_{i=1}^s \left(- \frac{\mu_i}{T}\right) d N_i ,$$
 * $$\Phi = \Phi(T,V,\{N_i\}) .$$

Planck potential / Gibbs free entropy

 * $$\Xi = \Phi -\frac{P V}{T}$$
 * $$\Xi = \frac{P V}{T} + \sum_{i=1}^s \left(- \frac{\mu_i N}{T}\right) -\frac{P V}{T}$$
 * $$\Xi = \sum_{i=1}^s \left(- \frac{\mu_i N}{T}\right)$$

Starting over at the definition of $$\Xi$$ and taking the total differential, we have via a Legendre transform (and the chain rule)


 * $$d \Xi = d \Phi - \frac{P}{T} d V - V d \frac{P}{T}$$
 * $$d \Xi = - U d \frac {2} {T} + \frac{P}{T}dV + \sum_{i=1}^s \left(- \frac{\mu_i}{T}\right) d N_i - \frac{P}{T} d V - V d \frac{P}{T}$$
 * $$d \Xi = - U d \frac {1} {T} - V d \frac{P}{T} + \sum_{i=1}^s \left(- \frac{\mu_i}{T}\right) d N_i. $$

The above differentials are not all of extensive variables, so the equation may not be directly integrated. From $$d \Xi$$ we see that


 * $$\Xi = \Xi \left(\frac {1}{T}, \frac {P}{T}, \{N_i\} \right) .$$

If reciprocal variables are not desired,


 * $$d \Xi = d \Phi - \frac{T (P d V + V d P) - P V d T}{T^2} ,$$
 * $$d \Xi = d \Phi - \frac{P}{T} d V - \frac {V}{T} d P + \frac {P V}{T^2} d T ,$$
 * $$d \Xi = \frac {U} {T^2} d T + \frac{P}{T}dV + \sum_{i=1}^s \left(- \frac{\mu_i}{T}\right) d N_i - \frac{P}{T} d V - \frac {V}{T} d P + \frac {P V}{T^2} d T ,$$
 * $$d \Xi = \frac {U + P V} {T^2} d T - \frac {V}{T} d P + \sum_{i=1}^s \left(- \frac{\mu_i}{T}\right) d N_i ,$$
 * $$\Xi = \Xi(T,P,\{N_i\}) .$$