Chemistry Reference
In-Depth Information
where we suppose that
t
approach zero (extremely small displacements
in infinitesimally short times) in such a way that [15]
and
2
D
γ
Gdt
2
D
=
=
lim
(107)
2
t
→
0
t
→
0
The diffusion coefficient
D
in Eq. (106) is obtained as follows. The change
in the phase
in an elapsed time interval
t
is
γX
t
0
G
(
t
)
dt
=
so that
γζ
−
1
t
0
λ
(
t
1
)
t
1
0
G
(
t
)
dt
dt
1
(
t
)
=
(108)
assuming that
(0)
0. This equation simply expresses the fact that the only way
the phase can change is via the equation of motion of
X
(
t
). In general, taking
account of inertia we would have
=
γ
t
0
X
(
t
1
)
t
1
0
G
(
t
)
dt
dt
1
(
t
)
=
(109)
In order to see how Eq. (108) leads to the correct result for the diffusion coeffi-
cient
D
, namely, Eq. (107), we consider (following Wang and Uhlenbeck [17])
the change
2
(
t
) in a small time
t
, then
γ
t
+
t
t
dt
1
t
1
0
λ
(
t
1
)
ζ
G
(
t
)
dt
=
(110)
so that
t
+
t
λ
(
t
1
)
dt
1
t
+
t
t
λ
(
t
2
)
dt
2
G
(
t
)
dt
2
γ
2
ζ
2
)
2
(
=
t
λ
(
t
1
)
λ
(
t
2
)
G
(
t
)
dt
2
dt
2
dt
1
γ
2
ζ
2
=
(111)
t
