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Note that (
4.38
) has the same form as the transition probabilities (
4.33
) and therefore
satisfies our intuition about fractals, that being that the parts and the whole share a
common property. The scaling property of the Lévy distribution is obtained by scaling
the displacement
q
with a constant
λ
and the time
t
with a constant
ζ
to obtain
)
=
λ
−
μ
P
L
(
P
L
(λ
q
,ζ
t
q
,
t
)
(4.39)
/μ
. It should also be observed that (
4.39
) has the same scaling form as
fractional Brownian motion if the Lévy index
1
as long as
λ
=
ζ
μ
/
is the same as 1
H
. Thus, the self-affine
<μ<
scaling results from Lévy statistics with 0
2 or from Gaussian statistics with a
power-law spectrum having 0
The scaling relation is the same in both cases
and therefore cannot be used to distinguish between the two.
A further property of Lévy processes can be observed by taking the derivative of the
characteristic function (
4.36
) to obtain
.
5
<
H
≤
1
.
∂
P
(
k
,
t
)
|
μ
P
=−
γ
|
k
(
k
,
t
).
(4.40)
∂
t
In the case
2 the inverse Fourier transform of this equation yields a second deriva-
tive in the displacement. The resulting equation has the form of the FPE discussed in
the previous chapter. However, in general there is no explicit form for the inverse of
the Fourier transform of the Lévy characteristic function and consequently there is no
differential representation for the dynamics of the probability density. We shall subse-
quently see that the resulting equation for the Lévy probability density is a fractional
equation in the spatial variable.
The lack of a differential representation for Lévy statistics can perhaps be best inter-
preted using the random-walk picture. Unlike Brownian motion, for a Lévy process,
the random walk has steps at each point in time that can be of arbitrary length. Those
steps adjacent in time are not necessarily nearby in space, and the best we can do in
the continuum limit is obtain an integro-differential equation to describe the evolution
of the probability density. Consider a random flight defined by a jump distribution such
that the probability of taking a jump of unit size is 1
μ
=
/
a
, and the probability of taking a
jump a factor
b
larger is a factor 1
a
smaller. This argument is repeated again and again
as we did for the Weierstrass walk in the last section. As the flight progresses, the set
of sites visited consists of localized clusters of sites, interspersed by gaps, followed by
clusters of clusters over a larger spatial scale. This random flight generates a hierarchy
of clusters, the smallest having about
a
members in a cluster, the next largest being of
size
b
with approximately
a
2
members in the larger cluster and so on. The parameters
a
and
b
determine the number of subclusters in a cluster and the spatial scale size between
clusters, respectively. The fractal random flight is characterized by the parameter
/
ln
a
ln
b
,
μ
=
(4.41)
which is the fractal dimension of the set of points visited during the flight. It is worth
emphasizing that the inverse power-law index is given by the parameters that determine
how the spatial scales are separated in the random flight.
