[1.3.3.1] Let us start by recalling briefly the general theory of continuous time random walks [5, 7, 8]. [1.3.3.2] The basic equation of motion is the continous time random walk (CTRW) integral equation [16]
![]() |
(2.1) |
[page 2, §0]
describing a random walk in continous time without correlation
between its spatial and temporal behaviour.
[2.1.0.1] Here, as in (1.2),
denotes the probability density to find the diffusing
entity at the position
at time
if it started from
the origin
at time
.
[2.1.0.2]
is the probability
for a displacement
in each single step, and
is the
waiting time distribution giving the probability density
for the time interval
between two consecutive steps.
[2.1.0.3] The transition probabilities obey
.
[2.1.0.4] The function
is the survival probability at the initial
position which is related to the waiting time distribution through
![]() |
(2.2) |
[2.1.0.5] The objective of this paper which was defined in the introduction is to show that the fractional master equation (1.2) is a special case of the CTRW-equation (2.1), and to find the appropriate waiting time density.
[2.1.1.1] The translation invariant form of the transition probabilities in (2.1) allows a solution through Fourier-Laplace transformation. [2.1.1.2] Let
![]() |
(2.3) |
denote the Laplace transform of and
![]() |
(2.4) |
the Fourier transform of , which is also called
the structure function of the random walk [5].
[2.1.1.3] Then the
Fourier Laplace transform
of the solution to (2.1)
is given as [5, 7, 8, 16]
![]() |
(2.5) |
where is the Laplace transform of the survival probability.
[2.1.2.1] Similarly the fractional master equation (1.2) can be solved in Fourier-Laplace space with the result
![]() |
(2.6) |
where is the Fourier transform of the kernel
in (1.2).
[2.1.2.2] Eliminating
between (2.5) and
(2.6) gives the result
![]() |
(2.7) |
where is a constant.
[2.1.2.3] The last equality obtains because the
left hand side of the first equality is
-independent while
the right hand side is independent of
.
[2.1.3.1] From (2.7) it is seen that the fractional
master equation characterized by the kernel and the
order
corresponds to a special class of space time
decoupled continuous time random walks characterized by
and
.
[2.1.3.2] This correspondence is given precisely as
![]() |
(2.8) |
and
![]() |
(2.9) |
with the same constant appearing in both equations.
[2.2.0.1] Not unexpectedly the correspondence defines the waiting
time distribution uniquely up to a constant while the
structure function is related to the Fourier transform
of the transition rates.
[2.2.1.1] To invert the Laplace transformation in (2.8)
and exhibit the form of the waiting time density
in the time domain it is convenient to introduce the
Mellin transformation
![]() |
(2.10) |
for a function .
[2.2.1.2] The Mellin transformed waiting time
density is obtained as
![]() |
(2.11) |
where denotes the Gamma function.
[2.2.1.3] To obtain (2.11)
from (2.8) the relation between Laplace and Mellin transforms
![]() |
(2.12) |
the special result
![]() |
(2.13) |
and the general relation
![]() |
(2.14) |
valid for have been employed.
[2.2.1.4] Using the definition of the general
-function given in the
appendix one obtains the result
![]() |
(2.15) |
which may be rewritten as
![]() |
(2.16) |
with the help of general relations for -functions [24].
[2.2.1.5] The dependence on the parameters
and
has been indicated
explicitly.
[2.2.1.6] From the series expansion of
-functions given in the appendix
one finds
[page 3, §0]
![]() |
(2.17) |
showing that behaves as
![]() |
(2.18) |
for small .
[3.1.0.1] Because
the waiting time density
is singular at the origin.
[3.1.0.2] The series representation (2.17)
shows that the waiting time density is a natural generalization of an
exponential waiting time density to which it reduces for
,
i.e.
.
[3.1.0.3] The series in (2.17) is recognized as the generalized
Mittag-Leffler function
[25], and
may thus be written alternatively as
![]() |
(2.19) |
[3.1.0.4] Of course the result (2.17) can also be obtained more directly, but we have presented here a method using Mellin transforms because it remains applicable in cases where a direct inversion fails [23]. [3.1.0.5] The asymptotic expansion of the Mittag-Leffler function for large argument [25] yields
![]() |
(2.20) |
for large and
.
[3.1.0.6] This result shows that the waiting time
distribution has an algebraic tail of the kind usually considered
in the theory of random walks [12, 13, 14, 15, 16].