Statistical Mechanics - Physics at Oregon State University

130 CHAPTER 6. DENSITY MATRIX FORMALISM.
Now we can interchange factors and we find at the extremum that
or
∂
∂x eτ(x) =
∞
n=1
1
(λ − 1)n−1
(n − 1)!
∂ρ ∂ log(ρ)
= ρ
∂x ∂x
Using x = ρij we get at the extremum
∂ρ ∂ log(ρ)
= ρ
∂ρij
∂ρij
∂τ
= e
∂x
λ−1
∂τ
∂x
and we can now take the element nm of this operator equation to get
∂ρmn ∂ log(ρmn)
= ρ
∂ρij
∂ρij
(6.65)
(6.66)
(6.67)
(6.68)
The left side is easy, and equal to δmiδnj. Hence at the extremum we have
2 ∂ X
= −kBe 1−λ δmiδnj
(6.69)
∂ρij∂ρnm
Suppose we make an arbitrary small change in the density matrix around
the extremum. In second order we have
∆X =
2 ∂ X
or
ijmn
∂ρij∂ρnm
1−λ
∆ρij∆ρnm = −kBe
∆X = −kBe
ijmn
1−λ
∆ρij∆ρji
ij
δmiδnj∆ρij∆ρnm (6.70)
(6.71)
Now we use the fact that the density matrix has to remain Hermitian, and hence
∆ρ∗ ij = ∆ρji to get
1−λ
|∆ρij| 2
(6.72)
∆X = −kBe
which shows that the extremum is a maximum, indeed.
Why bother?
The whole approach sketched above seems very complicated, and for the
simple microcanonical ensemble (states with given U, V, N) it is indeed. The
advantage is that it is easier to generalize for more complex systems. That is
where we gain in this formalism. The only thing we have shown above is that
ij