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limit distributions having properties very different from normal statistics, which were
subsequently referred to as Lévy processes and Lévy distributions. In the present context
the random-walker nomenclature is replaced by that of the random flyer to indicate that
the moving entity does not touch all the intermediate lattice points but flies above them
to touch down some distance from launch. The Lévy flight produces the kind of cluster-
ing so prominent in Figure
4.5
. This clustering arises from the separation of time scales
as discussed in the previous section and to which we return below. It is worth pointing
out here that asymptotically the Lévy distribution has an inverse power-law form and
therefore is often closely related to phenomena having those statistical properties. Let us
now consider what happens when the steps in the random flight can become arbitrarily
long, resulting in the divergence of the second moment.
For simplicity consider the transition probability used in the last section to have the
inverse power-law form
C
p
(
q
)
=
|
μ
+
1
,
0
<μ<
2
,
(4.33)
|
q
where
C
is the normalization constant. The characteristic function of the transition
probability is given by the Fourier transform to be
)
=
FT
p
k
≈
|
μ
p
(
k
(
q
)
;
1
−
c
|
k
(4.34)
for small
k
, whose inverse Fourier transform gives the probability per unit time of
making long flights. The probability density for
N
flights is an
N
-fold convolution of
the single-flight transition probabilities, which implies that the characteristic function
for the
N
-flight process is given by the product of the characteristic functions for the
single-flight process
N
times,
)
=
)
N
≈
1
|
μ
N
exp
−
|
μ
.
P
N
(
k
p
(
k
−
c
|
k
≈
Nc
|
k
(4.35)
The exponential approximation used here can be made rigorous, but that does not con-
tribute to the argument and therefore is not presented. In the continuum limit where we
replace the total number of jumps
N
by the continuum time
t
, we replace (
4.35
) with
exp
−
γ
|
μ
P
(
k
,
t
)
=
t
|
k
(4.36)
and the probability density is given by the inverse Fourier transform
∞
exp
−
|
μ
dk
1
2
P
L
(
q
,
t
)
=
ikq
−
γ
t
|
k
;
0
<μ
≤
2
.
(4.37)
π
−∞
This is the equation for the centrosymmetric Lévy distribution in one dimension and
consequently we use the subscript indicated.
Montroll and West [
33
] showed that for large values of the random variable the Lévy
distribution becomes an infinite algebraic series. The lowest-order term in this series for
large values of the variate is the inverse power law
1
P
L
(
,
)
∝
|
μ
+
1
;
<μ
≤
.
lim
q
q
t
0
2
(4.38)
|
q
→∞
