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In-Depth Information
for
2 the micro-
scopic time scale diverges. If we multiply (
5.1
) by the velocity, average over the
fluctuations and assume that the velocity is statistically independent of the doorway
variable
μ>
2. However, with the power-law index in the interval 1
<μ<
)
ξ
=
V
(
0
)ξ (
t
0 we obtain
2
t
d
0
dt
t
)
0
(
t
)
dt
,
=−
0
ξ
(
t
−
(5.4)
where the velocity autocorrelation function is given by
)
≡
V
(
0
)
V
(
t
)
ξ
V
2
0
(
t
.
(5.5)
ξ
Inserting the hyperbolic autocorrelation function into (
5.4
) yields
μ
−
1
2
t
0
d
0
(
t
)
T
t
)
dt
,
=−
0
(
(5.6)
dt
T
+
t
−
t
t
which can be approximated by using
t
−
T
,
2
t
0
0
(
)
d
t
1
T
μ
−
1
t
)
dt
.
≈−
t
)
μ
−
1
0
(
(5.7)
dt
(
t
−
The evolution equation for the velocity autocorrelation function is simplified by
introducing a limiting procedure such that as
T
becomes infinitesimally small and
becomes infinitely large the indicated product remains finite:
T
μ
−
1
2
ϒ
=
lim
.
(5.8)
T
→
0
,
→∞
This limiting procedure is a generalization of a limit to an integral equation taken by Van
Hove and applied in this context by Grigolini
et al
.[
6
]. Adopting this ansatz reduces
(
5.7
) to the integro-differential equation
t
d
0
(
t
)
1
t
)
dt
.
=−
ϒ
t
)
μ
−
1
0
(
(5.9)
dt
(
t
−
0
Notice that in general the velocity autocorrelation function is related to the waiting-time
distribution density
ψ(
t
)
of the process under study as
t
0
ψ(
t
)
dt
0
(
t
)
=
1
−
(5.10)
and therefore, on taking the time derivative of (
5.10
), we obtain from (
5.9
)
t
1
t
)
dt
.
ψ(
t
)
=
ϒ
t
)
μ
−
1
0
(
(5.11)
(
t
−
0
In subsequent sections we show that one form of the fractional integral of interest to us
is given by
t
t
)
1
(α)
f
(
D
−
α
dt
[
f
(
t
)
]=
(5.12)
t
t
)
1
−
α
(
t
−
0
