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Free entropy

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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.

Contents

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.

Examples

The most common examples are:

where

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 French engineer Francois 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 = ∂ S ∂ U d U + ∂ S ∂ V d V + ∑ i = 1 s ∂ S ∂ N i d N i .

From the equations of state,

d S = 1 T d U + P T d V + ∑ i = 1 s ( − μ i T ) 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 + ∑ i = 1 s ( − μ i N T ) .

Massieu potential / Helmholtz free entropy

Φ = S − U T Φ = U T + P V T + ∑ i = 1 s ( − μ i N T ) − U T Φ = P V T + ∑ i = 1 s ( − μ 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 Φ = d S − 1 T d U − U d 1 T , d Φ = 1 T d U + P T d V + ∑ i = 1 s ( − μ i T ) d N i − 1 T d U − U d 1 T , d Φ = − U d 1 T + P T d V + ∑ i = 1 s ( − μ 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

Φ = Φ ( 1 T , V , { N i } ) .

If reciprocal variables are not desired,

d Φ = d S − T d U − U d T T 2 , d Φ = d S − 1 T d U + U T 2 d T , d Φ = 1 T d U + P T d V + ∑ i = 1 s ( − μ i T ) d N i − 1 T d U + U T 2 d T , d Φ = U T 2 d T + P T d V + ∑ i = 1 s ( − μ i T ) d N i , Φ = Φ ( T , V , { N i } ) .

Planck potential / Gibbs free entropy

Ξ = Φ − P V T Ξ = P V T + ∑ i = 1 s ( − μ i N T ) − P V T Ξ = ∑ i = 1 s ( − μ 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 Ξ = d Φ − P T d V − V d P T d Ξ = − U d 2 T + P T d V + ∑ i = 1 s ( − μ i T ) d N i − P T d V − V d P T d Ξ = − U d 1 T − V d P T + ∑ i = 1 s ( − μ 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

Ξ = Ξ ( 1 T , P T , { N i } ) .

If reciprocal variables are not desired,

d Ξ = d Φ − T ( P d V + V d P ) − P V d T T 2 , d Ξ = d Φ − P T d V − V T d P + P V T 2 d T , d Ξ = U T 2 d T + P T d V + ∑ i = 1 s ( − μ i T ) d N i − P T d V − V T d P + P V T 2 d T , d Ξ = U + P V T 2 d T − V T d P + ∑ i = 1 s ( − μ i T ) d N i , Ξ = Ξ ( T , P , { N i } ) .

References

Free entropy Wikipedia
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