Biomedical Engineering Reference
In-Depth Information
that coherences between states with different particle number are decoupled from
the populations and vanish exactly in the stationary limit:
i
h
(
N
E
)
ρ
N
EE
ρ
˙
EE
=
−
E
−
2
P
NE
d
i
σ
Γ
χσ
d
j
σ
1
i
π
−
χσ
ij
F
f
χ
(
(
F
−
H
sys
)
−
p
χ
(
F
−
H
sys
)+
F
−
H
sys
)
ij
d
i
σ
i
π
d
j
σ
Γ
χσ
f
χ
(
N
FE
+
(
H
sys
−
F
)
−
p
χ
(
H
sys
−
F
)+
H
sys
−
F
)
ρ
ij
d
i
σ
Γ
χσ
d
j
σ
1
2
ρ
i
π
−
χσ
ij
F
N
EF
f
χ
(
(
F
−
H
sys
)
+
p
χ
(
F
−
H
sys
)+
F
−
H
sys
)
ij
d
i
σ
i
π
d
j
σ
Γ
χσ
f
χ
(
+
(
H
sys
−
F
)
+
p
χ
(
H
sys
−
F
)+
H
sys
−
F
)
P
NE
ij
2
P
NE
d
i
σ
ρ
1
i
π
d
j
σ
Γ
χσ
+
χσ
ij
FF
N
−
1
E
−
F
)
E
−
F
)+
f
χ
(
E
−
F
)
(
+
p
χ
(
FF
ij
i
π
d
i
σ
ρ
N
−
1
d
j
σ
Γ
χσ
f
χ
(
+
(
E
−
F
)
−
p
χ
(
E
−
F
)+
E
−
F
)
FF
ij
i
π
d
i
σ
Γ
χσ
N
+
1
F
−
E
)
F
−
E
)+
f
χ
(
F
−
E
)
+
d
j
σ
ρ
(
+
p
χ
(
FF
ij
i
π
d
i
σ
Γ
χσ
N
+
1
f
χ
(
+
d
j
σ
ρ
(
F
−
E
)+
−
p
χ
(
F
−
E
)+
F
−
E
)
P
NE
.
(7.61)
FF
ij
Equation (
7.5
) represents a special case of Eq. (
7.61
) in which all energy spacings
between states with the same particle number are either zero or much larger than
the level broadening
h
. The problem of a master equation in presence of quasi-
degenerate states in order to study transport through molecules has been addressed
in the work of Schultz et al. [
43
]. The authors claim in their work that the singular
coupling limit should be used in order to derive an equation for the density matrix in
presence of quasi-degenerate states. Equation (
7.61
) is derived in the weak coupling
limit and bridges all the regimes as illustrated by Figs.
7.18
-
7.20
.
The current operators associated with the master equation just presented read:
Γ
1
2
NEF
ij
σ
P
NE
d
i
σ
Γ
I
χ
=
d
j
σ
i
π
χσ
ij
f
χ
(
(
E
−
H
sys
)
+
p
χ
(
E
−
H
sys
)+
E
−
H
sys
)
d
j
σ
i
π
d
i
σ
Γ
χσ
f
χ
(
+
(
F
−
H
sys
)
−
p
χ
(
F
−
H
sys
)+
F
−
H
sys
)
ij
