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a quantity known as the relative dispersion in the biomedical literature [
2
]. In the next
section we determine the consequences of having a waiting-time distribution with a
diverging average sojourn time.
5.4.3
Infinite moments
The finite moments of the random-walk process in the CTRW formalism have been
shown to be a direct consequence of the waiting-time distribution having finite
moments. This is made explicit in the Taylor expansion of the Laplace transform given
in (
5.141
). However, there are examples in the physical, social and life sciences where
the central moments diverge. This divergence of the moments in space and/or time is
a consequence of non-locality in the corresponding distribution functions. In time this
non-local effect is called memory and is the influence of past behavior on future activ-
ity. In particular, when the first moment of the distribution diverges the memory extends
over a very long time compared with that of an exponential relaxation, which is to say
that there is no characteristic relaxation time for the statistical process.
This observation is not particularly profound when we examine how things work in
the real world. Pretty much everything that happens is dependent on history; whether
being late for work is overlooked as an anomaly or is considered just one more incident
in a train of irresponsible actions depends on your work record; similarly, whether your
wife treats your forgetting an anniversary as the foible of an absent-minded but loving
husband depends on the warmth of the day-to-day exchanges. But, as much as we rec-
ognize this dependence of our personal lives on history, it has proven to be too difficult
to include it formally in our mathematical models. It is not the case that we have never
been able to do it, but that when there is success it is considered extremely significant.
In an analogous way the non-local effect in space implies that what is occurring at
one point in space is not merely dependent on what is happening in the immediate
neighborhood of that point, but depends on what is happening in very distant spatial
regions as well. Here we have the spatial analogue of memory, spatial anisotropy. In
our private lives we know that the time it takes to get from point A to point B depends
on the path taken. In society these paths are laid out for us. We do not drive the wrong
way on a road or go “up the down staircase.” We learn very young how to traverse
the physical as well as the psychological paths to minimize resistance. Zipf thought
that this was a universal property of human behavior resulting in the inverse power-law
distribution. We cannot change our direction of motion when we wish, only when we
are allowed to do so. The constraints on us depend not only on where we are, but also
on how where we are is connected to where we want to go. This connectedness imposes
non-local constraints on our movements.
These structural constraints are included in modeling the dynamics through the
waiting-time and step-length distributions. When the random walker is moving on a
desert plain these functions have a slow decay. However, when the walker is in an
urban environment or in other contexts having rich structure these functions may decay
much more rapidly. Therefore we extend the preceding analysis to cases where the
