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associated with the mass distribution, is equal to the Euclidean dimension E of the
physical space, for example, in three-dimensional space d
3.
Suppose that on closer inspection of Figure 2.15 the mass points are seen to be not
uniformly distributed in space, but instead clumped in distinct spheres of size R
=
E
=
/
b each
/
having a mass that is 1
a smaller than the total mass. Here a and b are independent
parameters each greater than unity. Thus, what had initially appeared as a huge globe
filled uniformly with sand (mass points) turns out to resemble one filled with basket-
balls, each of the basketballs being filled uniformly with sand. We now examine one
of these smaller spheres (basketballs) and find that instead of the mass points being
uniformly distributed in this reduced region the sphere consists of still smaller spheres,
each of radius R
a 2 smaller than the total mass. Now
again the image changes so that the basketballs appear to be filled with baseballs, and
each baseball is uniformly filled with sand. It is now assumed that this procedure of
constructing spheres within spheres can be telescoped indefinitely to obtain
b 2 and each having a mass 1
/
/
a n M R
b n
M
(
R
) =
lim
.
(2.28)
n
→∞
Relation ( 2.28 ) is one of those equations that looks formidable, but is actually easy to
solve when looked at physically. Physically the mass of this object must be finite, and
one way to accomplish this is by requiring conditions on the parameters that yield a
finite value for the total mass in the limit of n becoming infinitely large. Recall that the
mass increases as the d th power of the radius so that
a n R
b n
d
M
(
R
)
lim
n
.
(2.29)
→∞
Thus, the total mass is finite if the parameters satisfy the relation
a
b d
n
=
1
(2.30)
independently of n and therefore the dimension d must satisfy the relation
ln a
ln b .
d
=
(2.31)
The index of the power-law distribution of mass points within the volume can therefore
be distinct from the Euclidean dimension of the space in which the mass is embedded,
that is, in general d
D , the fractal dimension.
The fractal idea also spills over from the geometry in Figure 2.15 into the realm of
statistics. We do not go into detail here to explain fractal statistics, but observe that frac-
tional exponents arise in the analysis of time series stemming from complex dynamic
webs, and such time series and the webs that generate them are among the main top-
ics of this topic. The fractional index, either in time or in displacements, is the basis
of interpretation of these time-series data and of the associated probability distribu-
tion functions in terms of the fractal dimension. Although the arguments given above
are geometric, they apply with equal validity to the trace of time series and therefore
to the time series themselves, often describing processes that have a temporal fractal
=
E
=
3. Consequently, d
=
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