Chemistry Reference
In-Depth Information
random force superimposed, both representing the effect of the surrounding heat
bath on the nucleus.
A.
Fractional Diffusion
Here the objective is to provide a microscopic basis for the empirical stretched
exponential Eq.(81) of Bennett et al. [40]. Our particular hypothesis is [1] that is
may be explained via memory effects giving rise to fractional Brownian motion,
which preserves a few of the features of the continuous time random walk. In
order to achieve this via the Langevin equation, note that Lutz [2], following the
analysis of fractional Brownian motion given by Mandelbrot and van Ness, has
introduced a fractional Langevin equation for the translational motion of a free
Brownian particle, namely,
m
d
mβ
α
0
D
α
−
1
dt
v
(
t
)
+
v
(
t
)
=
λ
(
t
)
(136)
t
X
(
t
) is the velocity of the par-
ticle,
β
α
is the friction coefficient,
mβ
α
0
D
α
−
t
v
(
t
), and
λ
(
t
) are, respectively, the
generalized frictional and random forces with the properties
Here (as in the normal Langevin equation),
v
(
t
)
=
λ
(
t
)
=
λ
(
t
)
λ
(
t
)
=
mkTβ
α
(1
t
|
−
α
0
,
α
)
|
t
−
−
(
denotes the gamma function). The Riemann-Liouville fractional derivative [58]
is defined by
t
g
(
t
)
1
(
σ
)
0
D
−
σ
t
)
1
−
σ
dt
g
(
t
)
=
0
<σ<
1
(137)
t
(
t
−
0
has the form of a memory function so that Eq.(136) may be regarded as a gener-
alized Langevin equation [2, 15] describing subdiffusion.
The memory function
Kα
(
t
) is given (in accordance with the fluctuation dissi-
pation theorem [19, 54]) by the autocorrelation function
kT
λ
(0)
λ
(
t
)
1
K
α
(
t
)
=
(138)
and thus the fractional Langevin equation Eq.(136) can be written in the form [1],
t
m
dv
(
t
)
dt
t
)
v
(
t
)
dt
=
+
K
α
(
t
−
λ
(
t
)
(139)
0
If
α
equals unity, then Eq.(139) becomes the normal Langevin equation, in
which case the noise function becomes white, and its correlation function is a
delta function.
