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5.4.2
Continuous-time random walks (CTRWs)
One of the most useful generalizations of the random-walk model was made by Mon-
troll and Weiss [
19
] in 1965, when they included random intervals between successive
steps in the walking process to account for local structure in the environment. In this
way the time interval between successive steps
t
n
becomes a random vari-
able. This generalization is referred to as the continuous-time random-walk (CTRW)
model since the walker is allowed to step at a continuum of times and the time between
successive steps is considered to be a second random variable, the vector value of the
step being the first random variable. The CTRW has been used to model a number of
complicated statistical phenomena, from the microscopic structure of individual lattice
sites for heterogeneous media to the stickiness of stability islands in chaotic dynamical
networks [
33
]. In our continuing discussion of random walks we remain focused on the
dynamical variable.
The CTRW explicitly assumes that the sequence of time differences {
τ
n
=
t
n
+
1
−
τ
n
} constitutes
a set of independent identically distributed random variables. The probability that the
random variable
τ
,
+
ψ(
)
ψ(
)
is
the now-familiar waiting-time distribution function. The name reflects the fact that the
walker waits for a given time at a lattice site before stepping to the next site. The length
of the sojourn is characteristic of the structure of the medium, as we discussed in the
previous chapter on memory and fractional dynamical equations, where the waiting-
time distribution was found to be a renewal process.
The waiting-time distribution is also used to define the probability that a step has not
been taken in the time interval (0
lies in an interval {
t
t
dt
} is given by
t
dt
, where
t
is the probability that
a walker is at the lattice site
q
at a time
t
immediately after a step has been taken, and
P
,
t
). Thus, if the function
W
(
q
,
t
)
is the probability of a walker being at
q
at time
t
, then in terms of the survival
probability
(
q
,
t
)
t
0
(
t
)
t
)
dt
.
P
(
q
,
t
)
=
δ
q
,
0
0
(
t
)
+
t
−
W
(
q
,
(5.124)
If a walker arrives at
q
at time
t
and remains there for a time (
t
t
), or the walker has
−
not moved from the origin of the lattice,
P
(
q
,
t
)
is the average over all arrival times with
t
<
0
can
be determined from the transition probabilities. It should be pointed out that the function
P
<
t
. Note that the waiting time at the origin might be different, so that
0
(
t
)
is a probability when
q
is a lattice site and a probability density when
q
is a point
in the spatial continuum.
The probability function
W
(
q
,
t
)
(
q
,
t
)
itself satisfies the recurrence relation
t
q
,
t
)
q
,
t
)
dt
,
W
(
q
,
t
)
=
p
0
(
q
,
t
)
+
p
(
q
−
t
−
W
(
(5.125)
0
q
where
p
dt
is the probability that the time between two successive steps is
between
t
and
t
(
q
,
t
)
+
dt
, and the step length extends across
q
lattice points. The transition
