Statistical Mechanics - Physics at Oregon State University

6.3. MAXIMUM ENTROPY PRINCIPLE. 127
and at an extremum this should be zero for all variations ∆ρ, which gives us
the condition mentioned before. The changes in the quantity X are related to
changes in the operator ρ, and ultimately to changes in the density matrix Rij.
Suppose we only change one matrix element, hence only ∆Rmn is non-zero for
a specific value of m and n. This gives:
∆X = 〈n| δX
δρ
but we also have in terms of simple partial derivatives
which leads to
∂X
∆X =
〈n| δX
δρ
∂Rmn
|m〉 〈m| ∆ρ |n〉 (6.38)
〈m| ∆ρ |n〉 (6.39)
∂X
|m〉 =
∂Rmn
(6.40)
where we note that the order in which n and m appear is interchanged left and
right.
In our particular case we have
and hence
∂X
∂Rnm
X = −kB 〈i| ρ |j〉 〈j| log(ρ) |i〉 + λkB(
〈i| ρ |i〉 − 1) (6.41)
ij
∂ 〈j| log(ρ) |i〉
= −kB 〈m| log(ρ) |n〉 + λkBδnm − kB 〈i| ρ |j〉
ij
i
∂Rnm
(6.42)
It is the last term that causes problems in the calculations. This task can
be reformulated as follows. Suppose the operator ρ depends on some complex
variable x. Calculate Tr ρ . This can be done by using the definition of a
logarithm. We have
∂ρ
∂x
log(ρ) = τ ⇐ e τ = ρ (6.43)
We can relate derivatives of ρ to derivatives of τ. In general we have
and since the series converges uniformly, this gives
∂
∂x eτ(x) = ∂ 1
∂x n!
n
τ n (x) (6.44)
∂
∂x eτ(x) =
n
1 ∂
n! ∂x τ n (x) (6.45)