Chemistry Reference
In-Depth Information
If we now average this random variable over its realizations and assume a
Maxwellian distribution of velocities for the initial velocity
v
0
, so that it is no
longer a point value but has the Gaussian distribution wih
σ
2
=
kT /m
,wefind
t
t
1
e
−
β
(
t
−
t
)
1
e
−
β
(
t
−
t
)
kT
mβ
2
(1
1
(
mβ
)
2
X
2
(
t
)
e
−
βt
)
2
=
−
+
−
−
0
0
λ
(
t
)
λ
(
t
)
dt
dt
t
t
t
)
1
e
−
β
(
t
−
t
)
kT
mβ
2
(1
2
ζkT
(
mβ
)
2
e
−
βt
)
2
δ
(
t
−
=
−
+
−
0
0
1
e
−
β
(
t
−
t
)
dt
dt
−
t
1
e
−
β
(
t
−
t
)
2
dt
kT
mβ
2
(1
2
kT
(
mβ
)
e
−
βt
)
2
=
−
+
−
0
Examining the second term,
t
t
1
e
−
β
(
t
−
t
)
2
dt
=
1
e
−
2
βt
e
2
βt
dt
2
kT
(
mβ
)
2
kT
(
mβ
)
2
e
−
βt
e
βt
−
−
+
0
0
t
2
β
e
−
2
βt
2
β
−
−
2
kT
mβ
2
β
+
1
2
β
e
−
βt
1
=
−
+
Reintroducing the first term leads to
mβ
2
1
e
−
2
βt
kT
X
2
(
t
)
2
e
−
βt
e
−
2
βt
4
e
−
βt
=
−
+
+
2
βt
−
3
+
−
Thus we have the Ornstein-Uhlenbeck [15, 22] result exactly for the mean-
square displacement of a Brownian particle including the inertia.
X
2
(
t
)
=
mβ
2
βt
e
−
βt
2
kT
−
+
1
(122)
Einstein concluded since the inertial time
β
−
1
is on the order of 10
−
7
s for
Brownian particles [15] that the inertial effects could be ignored (cf. Eq. 3.1.11 of
[15]). We remark that the characteristic function (corresponding of course to the
intermediate scattering function [54]) of
X
(
t
) is simply
exp
e
−
βt
κ
2
kT
β
2
m
βt
F
x
(
κ, t
)
=
−
−
1
+
(123)
where
κ
denotes the wave number. Clearly, this equation is characterized by the
times (
κ
2
D
)
−
1
and
β
−
1
and its double transcendental nature gives rise to an infinity
of exponential decay modes [15] similar to those encountered in the dielectric
relaxation of a fixed-axis rotator. Moreover, for small inertial effects, Eq.(123) is
