Information Technology Reference
In-Depth Information
2
1
0
-1
-2
-3
0.5
1
1.5
2
2.5
Figure 2.9.
Summation of main stream-length distributions for the Mississippi River. Both axes are
logarithmic, the unit of length is km and the vertical axis is the probability density with units of
km
−
1
. Distributions of lengths at the
n
th generation
l
n
orders
n
=
3 (circles),
n
=
4 (squares)
and
n
=
5 (triangles) are shown. The distributions sum together to give an inverse power-law tail
(stars). The power-law distribution is vertically offset by a factor of ten to enhance its visibility,
and is the summation of the distributions below as well as the distributions for order
n
=
6main
stream lengths [
18
]. Reproduced with permission.
statistical behavior, having a finite mean and variance, to inverse power-law behavior in
which the second moment diverges and in some cases even the mean diverges.
Let
g
(ξ)
x
be amplified such that in addition to the mean value
it also contains a
new mean value
N
x
and that this new value occurs with probability
p
. In this way the
probability of the dimensionless random variable being within an interval
(ξ,ξ
+
d
ξ)
changes:
g
(ξ)
d
ξ
→
g
(ξ/
N
)
d
ξ/
N
.
In the second stage of amplification, which we assume occurs with probability
p
2
,the
mean value becomes
N
2
x
. The new distribution function
G
(ξ)
that allows for the
possibility of continuing levels of amplification is
p
2
N
2
g
p
N
g
N
2
G
(ξ)
=
g
(ξ)
+
(ξ/
N
)
+
(ξ/
)
+···
.
(2.7)
The infinite series (
2.7
) may be reorganized to obtain
p
N
G
G
(ξ)
=
g
(ξ)
+
(ξ/
N
),
(2.8)
which is an implicit expression for the infinite series known as a renormalization-group
relation. We note that (
2.8
) was first obtained using a different method by Novikov [
54
]
in his study of intermittency in turbulent fluid flow [
38
] and later independently using
the present method by Montroll and Shlesinger [
50
]. We also find this relation useful in
the study of a number of biomedical phenomena such as the analysis of the architecture
of the mammalian lung and the cardiac conduction network, both of which we discuss
later [
94
].
