Utente:Grasso Luigi/sanbox1/Entropia libera

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.

Esempi

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 Template:Col-begin

$S$ 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).

Dipendenza dei potenziali dalle variabili naturali

Entropia

$S=S(U,V,\{N_{i}\})$

By the definition of a total differential,

$dS={\frac {\partial S}{\partial U}}dU+{\frac {\partial S}{\partial V}}dV+\sum _{i=1}^{s}{\frac {\partial S}{\partial N_{i}}}dN_{i}.$

From the equations of state,

$dS={\frac {1}{T}}dU+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}.$

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

$S={\frac {U}{T}}+{\frac {PV}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right).$

Potenziale di Massieu / entropia libera di Helmholtz

$\Phi =S-{\frac {U}{T}}$

$\Phi ={\frac {U}{T}}+{\frac {PV}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right)-{\frac {U}{T}}$

$\Phi ={\frac {PV}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right)$

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

$d\Phi =dS-{\frac {1}{T}}dU-Ud{\frac {1}{T}},$

$d\Phi ={\frac {1}{T}}dU+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}-{\frac {1}{T}}dU-Ud{\frac {1}{T}},$

$d\Phi =-Ud{\frac {1}{T}}+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}.$

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,222

$d\Phi =dS-{\frac {TdU-UdT}{T^{2}}},$

$d\Phi =dS-{\frac {1}{T}}dU+{\frac {U}{T^{2}}}dT,$

$d\Phi ={\frac {1}{T}}dU+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}-{\frac {1}{T}}dU+{\frac {U}{T^{2}}}dT,$

$d\Phi ={\frac {U}{T^{2}}}dT+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i},$

$\Phi =\Phi (T,V,\{N_{i}\}).$

Potenziale di Planck / Entropia libera di Gibbs

$\Xi =\Phi -{\frac {PV}{T}}$

$\Xi ={\frac {PV}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right)-{\frac {PV}{T}}$

$\Xi =\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right)$

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

$d\Xi =d\Phi -{\frac {P}{T}}dV-Vd{\frac {P}{T}}$

$d\Xi =-Ud{\frac {2}{T}}+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}-{\frac {P}{T}}dV-Vd{\frac {P}{T}}$

$d\Xi =-Ud{\frac {1}{T}}-Vd{\frac {P}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}.$

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,222

$d\Xi =d\Phi -{\frac {T(PdV+VdP)-PVdT}{T^{2}}},$

$d\Xi =d\Phi -{\frac {P}{T}}dV-{\frac {V}{T}}dP+{\frac {PV}{T^{2}}}dT,$

$d\Xi ={\frac {U}{T^{2}}}dT+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}-{\frac {P}{T}}dV-{\frac {V}{T}}dP+{\frac {PV}{T^{2}}}dT,$

$d\Xi ={\frac {U+PV}{T^{2}}}dT-{\frac {V}{T}}dP+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i},$

$\Xi =\Xi (T,P,\{N_{i}\}).$

Note

Bibliografia

• M.F. Massieu, Compt. Rend, vol. 69, n. 858, 1869, p. 1057.

• Herbert B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd, New York, John Wiley & Sons, 1985, ISBN 0-471-86256-8.