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q and p
(
q
±
1
) =
1
/
2 with all other transition probabilities being zero, so that ( 4.6 )
reduces to
1
2 [
P n + 1 (
q
) =
P n (
q
+
1
) +
P n (
q
1
) ] .
(4.7)
The solution to the doubly discrete probability equation ( 4.7 )is[ 13 ]
N
!
1
2 N
(
2 ! (
2 !
P N (
m
) =
(4.8)
N
+
m
)/
N
m
)/
where m is the integer label of the lattice site and N is the total number of steps taken.
We leave it as a problem to prove that when N
m ( 4.8 ) is well approximated by a
Gaussian distribution.
This random-walk model can be generalized to the case where the second moment of
the diffusion variable is given by
2
t 2 H
Q
(
t
)
(4.9)
and H is the Hurst exponent confined to the interval 0
5
corresponds to the simple random-walk outline above, in which the probability of mov-
ing to the right or the left on a regular one-dimensional lattice is the same, resulting
in ordinary diffusion. The case H
H
1. The case H
=
0
.
5 corresponds to a random walk in which the
probability of a walker stepping in the direction of her previous step is greater than
that of reversing directions. The walk is called persistent and yields a mean-square dis-
placement that increases more rapidly in time than does the displacement in ordinary
diffusion, so the process is called superdiffusive . The case H
>
0
.
<
.
5 corresponds to a
random walk in which the probability of a walker stepping in the direction of her pre-
vious step is less than that of reversing directions. This walk is called anti-persistent
and yields a mean-square displacement that increases more slowly in time than does
the displacement in ordinary diffusion, so the process is subdiffusive . Thus, one kind
of anomalous diffusion has to do with memory in time for the random steps taken on
a regular lattice. This model has been used extensively in the interpretation of the fluc-
tuations in physiologic time series, as well as in physical and social phenomena [ 45 ].
Anomalous diffusion will be discussed in more detail in the following section.
The random-walk model discussed above assumes a spatial lattice upon which a
walker takes randomly directed steps in equal time intervals. Each step is completed
instantaneously and in the limit of a great many such steps the central limit theorem
(CLT) assures us that the resulting distribution of locations of the random walker is nor-
mal or Gaussian. However, there are a great many phenomena for which this kind of
diffusion assumption does not apply, either because the medium in which the walker
moves has structure, structure that might inhibit or facilitate the walker's steps, or
because the properties of the walker change over time. In either case a more general
kind of random walk is required, but before we turn our attention to that let us consider
an alternative limit of ( 4.8 ).
0
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