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In-Depth Information
It is interesting to note that this argument is similar to one presented in association
with the data depicted in Figure
2.9
. In that case the contributions from the increas-
ing orders of branching within the web contribute to the total distribution of lengths
with decreasing amplitudes. Consider the situation in which the dimensionless variable
becomes very large,
ξ
→∞
(
ξ
)
, and the basic function
g
asymptotically goes to zero
(ξ),
faster than does
G
such that
(ξ)
G
g
(ξ)
=
.
lim
ξ
→∞
0
(2.9)
In this asymptotic domain (
2.8
) can be approximated by
p
N
G
G
(ξ)
≈
(ξ/
N
).
(2.10)
Consequently, the asymptotic form of the distribution is determined by this simple
scaling relation in the domain where (
2.9
)isvalid
.
Equation (
2.10
) must provide the
non-analytic part of the distribution since
g
is analytic by assumption, that is, the
inverse power law is independent of the form of the basic distribution.
The solution to (
2.10
) is readily obtained by assuming a solution to the scaling
equation of the form
(ξ)
A
(ξ)
ξ
μ
.
G
(ξ)
=
(2.11)
Note that this is how we solve differential equations. We assume the form of the solution
with the appropriate number of constants to fit the initial and/or boundary conditions.
Then we insert the assumed form of the solution (
2.11
) into the “dynamical” equation
(
2.10
) to yield the relations for the amplitude
A
(ξ)
=
A
(ξ/
N
)
(2.12)
and the power-law index
N
μ
ξ
μ
.
1
ξ
μ
=
p
N
(2.13)
The benign-looking (
2.12
) implies that the amplitude function is periodic in the loga-
rithm of
with period ln
N
and consequently the general function can be expressed as
the infinite but discrete Fourier series
ξ
A
n
exp
i
2
∞
π
n
ln
ξ
A
(ξ)
=
.
(2.14)
ln
N
n
=−∞
The coefficients
A
n
are empirically determined by the problem of interest and very often
only the
n
0 term contributes, that being a constant amplitude. The equation for the
power-law index yields
=
ln
(
1
/
p
)
μ
=
1
+
(2.15)
ln
N
