Chemistry Reference
In-Depth Information
since we have set
=
0at
t
=
0. Next, we have for a step-field gradient
2
Dγ
2
G
2
t
0
t
dt
1
t
0
dt
2
dt
=
2
3
Dγ
2
G
2
t
3
σ
2
=
(118)
0
which obviously yields the result [see Eq. (102)] obtained from the Langevin
equation.
Equation
(119)
is
simply
a
special
case
of
a
Lemma
due
to
Chandrasekhar [115].
C.
Phase Diffusion Including the Inertia
The analysis given above ignores the inertia of the Brownian particles. If the
inertial effects are included, the translational process,
X
(
t
), now possesses
two
characteristic times, as discussed in Section II.A. One characterizing the slow
diffusion associated with the noninertial motion that we have already analyzed.
The other is the correlation time,
τ
v
=
m/ζ
, of the velocity correlation function. It
is of interest to include these in the phase diffusion and therefore we show how the
calculation just outlined using the noninertial Langevin equation, Eq. (92), may
be extended for a free particle of mass
m
as in [1].
For the inertial motion of a Brownian particle, an explicit formula for the dis-
placement
X
(
t
) is available from the Ornstein-Uhlenbeck theory [15, 22]. This
theory is simply the Einstein or Langevin theory with the inertia of the particles
included. We again start by writing again the full Langevin equation in phase space,
mv
(
t
)
=−
ζv
+
λ
(
t
)
It is assumed that the particle starts off at a definite phase point (
x
0
,v
0
) such
that the state vector becomes,
t
β
1
e
−
βt
+
1
e
−
β
(
t
−
t
)
λ
(
t
)
dt
v
0
1
mβ
X
(
t
)
=
x
0
+
−
−
(119)
0
where
β
=
ζ/m
. It follows that:
t
1
m
e
−
β
(
t
−
t
)
)
λ
(
t
)
dt
X
(
t
)
v
0
e
−
βt
v
(
t
)
=
=
+
(1
−
(120)
0
As far as our calculation is concerned the instantaneous displacement Eq.(120)
consists of a deterministic term and a random term. Both
x
0
and
t
0
may be set to
zero without loss of generality, so therefore on squaring
β
2
1
v
0
e
−
βt
2
X
2
(
t
)
=
−
(
mβ
)
2
t
t
1
e
−
β
(
t
−
t
)
1
e
−
β
(
t
−
t
)
λ
(
t
)
λ
(
t
)
dt
dt
+
1
+
−
−
CC
0
0
(121)
