Chemistry Reference
In-Depth Information
lying horizontally, and
W
(
t
) is the diagonal matrix with the elements
e
−
μ
t
.
In
fact,
E
−
1
E
·
W
(
t
)
·
=
exp (
t
)
(85)
The asymptotic value of
ρ
(
t
) is described by the zero eigenvalue
1
=
0,
ρ
(
∞
)
=
R
1
(
L
1
·
ρ
(0))
(86)
eq
eq
Here
ρ
(
∞
)
=
ρ
should be satisfied, where
ρ
follows from Eq. (54), and this
result should be independent of
ρ
(0)
.
Thus one concludes that
exp
δ
α
(
a
)
β
(
a
)
1
Z
s
ε
α
(
a
)
k
B
T
L
1
a
=
δ
α
(
a
)
β
(
a
)
,
=
−
(87)
1
a
where
α
(
a
) and
β
(
a
) are given by Eq. (79). This means that
L
1
is related to the
normalization of the DM
,
while
R
1
contains the information about the equilibrium
state. Indeed, one obtains
L
1
·
ρ
(0)
=
L
1
a
ρ
a
(0)
=
δ
αβ
ρ
αβ
(0)
=
ρ
αα
(0)
=
1
(88)
a
α
αβ
eq
,
as it should be. Note that
R
1
and
L
1
satisfy the orthonor-
mality condition in Eq. (82),
and
ρ
(
∞
)
=
R
1
=
ρ
exp
δ
α
(
a
)
β
(
a
)
=
exp
δ
αβ
=
1
Z
s
ε
α
(
a
)
k
B
T
1
Z
s
ε
α
k
B
T
L
1
a
R
1
a
=
−
−
1
a
a
αβ
(89)
The time dependence of any physical quantity
A
is given by Eq. (2) that can be
rewritten in the form
A
(
t
)
=
A
βα
ρ
αβ
(
t
)
=
A
a
ρ
a
(
t
)
(90)
αβ
a
where
βα
≡
β
A
α
A
a
≡
A
β
(
a
)
α
(
a
)
,
(91)
Writing Eq. (90) in the vector form as
A
(
t
)
=
A
·
ρ
(
t
) one obtains
A
R
μ
e
−
μ
t
L
μ
·
ρ
(0)
A
(
t
)
=
·
(92)
μ
or, in the fully vectorized form,
E
−
1
A
(
t
)
=
A
·
E
·
W
(
t
)
·
·
ρ
(0)
(93)
