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equations. The deviation from the classical diffusion growth of the mean-square separa-
tion is particularly interesting because it may provide a probe into the dynamics of the
environment in which the web of interest is embedded. But then again life is not always
so straightforward and deviations from simple diffusion may also be caused by internal
dynamics of the web rather than from the influence of the environment on the web.
One probe of the web's dynamics can be obtained by substituting the definition of
the autocorrelation function given by (
3.62
)into(
3.68
), and thereby obtaining the time-
dependent diffusion coefficient
t
0
ξ
(
2
d
Q
(
t
)
ξ
2
t
)
dt
.
2
D
(
t
)
=
=
2
ξ
ξ
(3.70)
dt
The deviation of the scaling index from the conventional diffusion value of
5 can
now be explained in terms of the long-term memory in the random force. This memory
may be a consequence of internal web dynamics not explicitly included in the equations
of motion or it can be produced by a modulation of the web dynamics by the motion
of the environment. Part of the challenge is to determine the source of the memory
from the context of the phenomenon being investigated. Inserting the inverse power-
law form of the autocorrelation function given by (
3.65
)into(
3.70
) and taking another
time derivative leads to the long-time prediction
d
2
α
=
0
.
2
Q
(
t
)
ξ
1
t
β
.
Ct
2
α
−
2
=
2
α(
2
α
−
1
)
≈±
(3.71)
dt
2
The equalities in (
3.71
) are a consequence of having assumed that the long-time limit
of the autocorrelation function is dominated by the inverse power law shown in (
3.65
).
A realization of the positive sign in (
3.71
) is depicted by the solid line in Figure
3.3
,
whereas a realization of the negative sign is depicted by the dashed line in that figure.
Note that the dashed curve goes negative near a lag time of 24 and then asymptotically
approaches zero correlation from below.
The relation between the two exponents, the one for anomalous diffusion and the
other for the random force autocorrelation function, is obtained using (
3.71
)tobe
α
=
1
−
β/
2
,
(3.72)
1.0
0.8
0.6
0.4
0.2
0.0
0
50
100
150
Time (arb.
units)
The solid curve is the positive inverse power-law autocorrelation function for the random force
and remains positive for all time, approaching zero asymptotically from above. The dashed
curve has the negative autocorrelation function at long times and approaches zero asymptotically
from below [
32
]. Reproduced with permission.
Figure 3.3.
