Information Technology Reference
In-Depth Information
Implementing the assumption that the initial condition and the random force fluctu-
ations are statistically independent of one another yields for the second moment of the
web's dynamic response
Q
t
dt
1
t
0
2
2
(
t
)
ξ
=
dt
2
ξ(
t
1
)ξ(
t
2
)
ξ
+
Q
(
0
)
.
(3.66)
0
The integral in (
3.66
) can be simplified by assuming that the
-process is stationary in
time; that is, its moments are independent of the origin of time and consequently
ξ
ξ(
t
1
)ξ(
t
2
)
ξ
=
ξ(
t
1
−
t
2
)ξ
(
)
ξ
.
0
(3.67)
Inserting (
3.67
)into(
3.66
) and taking the time derivative leads to the differential
equation of motion for the second moment
2
t
0
2
d
Q
(
t
)
ξ
dt
ξ(
t
)ξ(
=
0
)
ξ
.
(3.68)
dt
Clearly the second moment of the web response appearing on the lhs of (
3.68
)mustbe
connected with the long-time diffusional regime described by
Q
2
Ct
2
α
,
(
t
)
ξ
=
(3.69)
where
C
is a constant. It is evident that the physical bounds on the possible values of
the scaling exponent in (
3.69
)aregivenby0
0 defines the case of local-
ization, which is the lower limit of any diffusion process. The value
≤
α
≤
1;
α
=
α
=
1 refers to the
case of many uncorrelated deterministic trajectories with
Q
(
t
)
−
Q
(
0
)
being linearly
proportional to time for each of them. The constraint
1 is a consequence of the fact
that a classical diffusion process cannot spread faster than a collection of deterministic
ballistic trajectories!
It is worth pointing out that turbulent diffusion given by the motion of a passive scalar,
such as smoke, in a turbulent flow field, such as wind a few meters above the ground,
leads to a mean-square separation of the smoke particles that increases faster than bal-
listic; that is,
α
≤
1. In fact, Richardson [
41
] observed the width of smoke plumes from
chimneys increasing with a power-law index slightly larger than
α>
Such behavior
occurs because the flow field drives the diffusing scalar particles in a manner having to
do with the scaling of vortices in turbulence. The first scaling model of fluid turbulence
was that proposed by Kolmogorov [
27
] in which eddies are nested within one another
with the spatially larger eddies rotating faster than the smaller ones. Consequently, when
a local eddy pulls the scalar particles apart the effect is accelerated because an increas-
ing separation distance produces a larger relative velocity. This shearing effect results
in the growth of the mean-square separation of the passive scalar particles becoming
faster than ballistic; it is a driven process. In this way the dynamics of the environment
directly influences the measured properties of the stochastic web variable.
Finally, the condition
α
=
1
.
5
.
5 is obtained for simple diffusion, where the variance
increases linearly with time. We dwell on these physical models here because they are
useful in interpreting phenomena outside the physical sciences that satisfy analogous
α
=
0
.
