Chemistry Reference
In-Depth Information
which is also described by the differential equation
∂
∂s
−
)
G
(
R
,
R
;
L
)=
δ
(
R
c
6
∇
r
+
U
(
R
)
δ
(
L
)
,
r
−
(2.101)
R
;
L
)isthe
Green function for the diffusion equation or propagator, and
δ
(
where
D
t
=
c/
6 is the translational diffusion coecient.
G
(
R
,
R
−
R
)
the Dirac delta function.
If the chain is worm-like, the bending of the chain will cost the energy
U
el
=
L
0
d
u
(
s
)
ds
2
ds,
ε
2
(2.102)
where
ε
is the bending constant,
u
is the tangent along the chain at contour
u
≡
length
s,
r
=
∂r/∂s
, and is a unit vector.
This elastic energy give rise to the Boltzmann factor of Equation 2.100,
therefore the path integral becomes the case of a worm-like chain,
˙
u
≡
∂
u
/∂s
,
U
;
L
)=
r
(
L
)=
R
r
(0)=
R
R
;
G
(
R
,
U
,
D
[
r
(
s
)]
D
[
u
(
s
)]
δ
ds
u
(
s
)
u
(
L
)=
U
L
×
R
−
u
(0)=
U
0
exp
u
2
(
s
)
L
ds
3
2
c
u
2
(
s
)+
βε
˙
×
−
(2.103)
2
0
where
R
=
L
0
u
(
s
)
ds
=
r
(
L
)
−
r
(0)
.
(2.104)
The unit vector
u
, enables Equation 2.103 to be rewritten as, by taking
D
[
R
(
s
)]
,
(
s
)] exp
u
2
(
s
)
,
U
;
L
)=
u
(
L
)=
U
u
(0)=
L
ds
βε
2
G
(
U
,
D
[
u
−
˙
(2.105)
U
0
where the uninteresting term exp(
−
3
L/
2
c
) is absorbed into the normaliza-
tion factor.
