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	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	$\Xi=\Phi -\frac{P}{T} V$	$= - \frac{G}{T}$	$~~~~~\frac{1}{T},\frac{P}{T},\{N_i\}\,$

S

is entropy

Φ

is the Massieu potential^[1]^[2]

Ξ

is the Planck potential^[1]

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

μ 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 $ψ$ , used by both Planck and Schrödinger. (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).

1 Dependence of the potentials on the natural variables
2 References

[edit] Dependence of the potentials on the natural variables

[edit] 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,

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

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

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

[edit] Massieu potential \ Helmholtz free entropy

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

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

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

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

$d \Phi = d S - \frac {1} {T} dU - U d \frac {1} {T}$ ,

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

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

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

$\Phi = \Phi(\frac {1}{T},V,\{N_i\})$ .

If reciprocal variables are not desired,^[3]^:222

$d \Phi = d S - \frac {T d U - U d T} {T^2}$ ,

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

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

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

Φ = Φ(T, V,{N i})

[edit] Planck potential \ Gibbs free entropy

$\Xi = \Phi -\frac{P V}{T}$

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

$\Xi = \sum_{i=1}^s (- \frac{\mu_i N}{T})$

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

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

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

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

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

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

If reciprocal variables are not desired,^[3]^:222

$d \Xi = d \Phi - \frac{T (P d V + V d P) - P V d T}{T^2}$ ,

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

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

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

Ξ = Ξ(T, P,{N i})

[edit] References

^ ^a ^b Antoni Planes; Eduard Vives (2000-10-24). Entropic variables and Massieu-Planck functions. Entropic Formulation of Statistical Mechanics. Universitat de Barcelona. Retrieved on 2007-09-18.
^ T. Wada; A.M. Scarfone (12 2004). "Connections between Tsallis’ formalisms employing the standard linear average energy and ones employing the normalized q-average energy" (PDF). Physics Letters, Section A: General, Atomic and Solid State Physics 335 (5-6): 351–362. doi:10.1016/j.physleta.2004.12.054.
^ ^a ^b (1954) The Collected Papers of Peter J. W. Debye. Interscience Publishers, Inc..

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

Callen, Herbert B. (1985). Thermodynamics and an Introduction to Themostatistics, 2nd Ed., New York: John Wiley & Sons. ISBN 0-471-86256-8.

Categories: Thermodynamics | Thermodynamic entropy

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Free entropy

From Wikipedia, the free encyclopedia

Contents

[edit] Dependence of the potentials on the natural variables

[edit] Entropy

[edit] Massieu potential \ Helmholtz free entropy

[edit] Planck potential \ Gibbs free entropy

[edit] References

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