Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 156

The entropy S associated with any density operator ρ having a trace equal to 1 is defined by relation (6) of Appendix VI:

(17)

The thermodynamic potential of the grand canonical ensemble is defined by the “grand potential” Φ, which can be expressed as a function of ρ by relation (Appendix VI, § 1-c-β):

(18)

Inserting (5) into (18), we see that the value of Φ at equilibrium, Φ_eq, can be directly obtained from the partition function Z:

(19)

We therefore have:

(20)

Consider now any density operator ρ and its associated function Φ obtained from (18). According to (5) and (20), we can write:

(21)

Inserting this result in (18) yields:

(22)

Now relation (16), used with ρ′ = ρ_eq, is written as:

(23)

Relation (22) thus implies that for any density operator ρ having a trace equal to 1, we have:

(24)

the equality occurring if, and only if, ρ = ρ_eq.

Relation (24) can be used to fix a variational principle: choosing a family of density operators ρ having a trace equal to 1, we try to identify in this family the operator that yields the lowest value for Φ. This operator will then be the optimal operator within this family. Furthermore, this operator yields an upper value for the grand potential, with an error of second order with respect to the error made on ρ.