Free entropy explained

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

See also: List of thermodynamic properties. The most common examples are:

Name	Function	Alt. function	Natural variables
Entropy	S = (1)/(T) U + (P)/(T) V - \sum ^{s} _{i =1} (\mu _{i})/(T) N _{i}		~~~~~U,V,\{N _{i\}}
Massieu potential \ Helmholtz free entropy	\Phi = S - (1)/(T) U	= - (A)/(T)	~~~~~ (1)/(T) ,V,\{N _{i\}}
Planck potential \ Gibbs free entropy	\Xi = \Phi - (P)/(T) V	= - (G)/(T)	~~~~~ (1)/(, T) (P)/(T) ,\{N _{i\}}

where

S is entropy\Phi is the Massieu potential^{[1] [2]} \Xi is the Planck potential^[1] U is internal energyT is temperatureP is pressureV is volumeA is Helmholtz free energyG is Gibbs free energyN _{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 components is the total number of componentsi 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).

Dependence of the potentials on the natural variables

Entropy

S = S(U,V,\{N _{i\})}

By the definition of a total differential,

d S = (\partial S)/(\partial U) d U + (\partial S)/(\partial V) d V + \sum ^{s} _{i =1} (\partial S)/(\partial N _{i}) d N _{i.}

From the equations of state,

d S = (1)/(dU + T) (P)/(T) dV + \sum ^{s} _{i =1} \left( - (\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 = (U)/(+ T) (P V)/(T) + \sum ^{s} _{i =1} \left( - (\mu _{i} N)/(T) \right).

Massieu potential / Helmholtz free entropy

\Phi = S - (U)/(T) \Phi = (U)/(+ T) (P V)/(T) + \sum ^{s} _{i =1} \left( - (\mu _{i} N)/(T) \right) - (U)/(T) \Phi = (P V)/(T) + \sum ^{s} _{i =1} \left( - (\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 - (1)/(T) dU - U d (1)/(T) , d \Phi = (1)/(T) dU + (P)/(T) dV + \sum ^{s} _{i =1} \left( - (\mu _{i})/(T) \right) d N _{i} - (1)/(T) dU - U d (1)/(T) , d \Phi = - U d (1)/(+ T) (P)/(T) dV + \sum ^{s} _{i =1} \left( - (\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( (1)/(T) ,V, \{N _{i\})} .

If reciprocal variables are not desired,^[3]

d \Phi = d S - (T d U - U d T)/(T ^{2}) , d \Phi = d S - (1)/(T) d U + (U)/(T ^{2}) d T , d \Phi = (1)/(T) dU + (P)/(T) dV + \sum ^{s} _{i =1} \left( - (\mu _{i})/(T) \right) d N _{i} - (1)/(T) d U + (U)/(T ^{2}) d T, d \Phi = (U)/(T ^{2}) d T + (P)/(T) dV + \sum ^{s} _{i =1} \left( - (\mu _{i})/(T) \right) d N _{i} , \Phi = \Phi(T,V,\{N _{i\})} .

Planck potential / Gibbs free entropy

\Xi = \Phi - (P V)/(T) \Xi = (P V)/(T) + \sum ^{s} _{i =1} \left( - (\mu _{i} N)/(T) \right) - (P V)/(T) \Xi = \sum ^{s} _{i =1} \left( - (\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 - (P)/(T) d V - V d (P)/(T) d \Xi = - U d (2)/(T) + (P)/(T) dV + \sum ^{s} _{i =1} \left( - (\mu _{i})/(T) \right) d N _{i} - (P)/(T) d V - V d (P)/(T) d \Xi = - U d (1)/(T) - V d (P)/(T) + \sum ^{s} _{i =1} \left( - (\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( (1)/(T) , (P)/(T) , \{N _{i\}} \right) .

If reciprocal variables are not desired,^[3]

d \Xi = d \Phi - (T (P d V + V d P) - P V d T)/(T ^{2}) , d \Xi = d \Phi - (P)/(T) d V - (V)/(T) d P + (P V)/(T ^{2}) d T , d \Xi = (U)/(T ^{2}) d T + (P)/(T) dV + \sum ^{s} _{i =1} \left( - (\mu _{i})/(T) \right) d N _{i} - (P)/(T) d V - (V)/(T) d P + (P V)/(T ^{2}) d T , d \Xi = (U + P V)/(T ^{2}) d T - (V)/(T) d P + \sum ^{s} _{i =1} \left( - (\mu _{i})/(T) \right) d N _{i} , \Xi = \Xi(T,P,\{N _{i\})} .

References

1. Web site: Antoni Planes . Eduard Vives . 2000-10-24 . Universitat de Barcelona . Entropic variables and Massieu-Planck functions . 2007-09-18 . Entropic Formulation of Statistical Mechanics . 2008-10-11 . https://web.archive.org/web/20081011011717/http://www.ecm.ub.es/condensed/eduard/papers/massieu/node2.html . dead .
2. T. Wada . A.M. Scarfone . December 2004 . Connections between Tsallis' formalisms employing the standard linear average energy and ones employing the normalized q-average energy . Physics Letters A . 335 . 5–6 . 351–362 . 10.1016/j.physleta.2004.12.054 . cond-mat/0410527. 2005PhLA..335..351W . 17101164 .
3. Book: The Collected Papers of Peter J. W. Debye . Interscience Publishers, Inc. . New York, New York . 1954.

Bibliography

• M.F. . Massieu. 1869 . Compt. Rend . 69. 858. 1057 .
• Book: Callen, Herbert B. . Herbert Callen . 1985 . Thermodynamics and an Introduction to Thermostatistics . 2nd . John Wiley & Sons . New York . 0-471-86256-8 .