[page 4, §1]
[4.1.1] It follows from the general results in Ref. [1] that
the first model defined by eqs. (6) and
(9) is equivalent to
the fractional master equation
(10) |
with intitial condition
(11) |
and fractional transition rates
(12) |
Here the fractional time derivative of order and type in eq. (10) is defined as [15]
(13) |
thereby giving a more precise meaning to the symbolic notation in eq. (1).
[page 5, §1] [5.1.1] The result is obtained from inserting the Laplace transform of
(14) |
and the Fourier transform of , the so called structure function
(15) |
into eq. (5). [5.1.2] This gives
(16) |
where the Fourier transform of eq. (12) was used in the last equality and the subscript refers to the first model. [5.1.3] Equation (16) equals the result obtained from Fourier-Laplace transformation of the fractional Cauchy problem defined by equations (10) and (11). [5.1.4] Hence a CTRW-model with and the fractional master equation describe the same random walk process in the sense that their fundamental solutions are the same.
[5.2.1] The continuum limit was the background and motivation for the discussion in Ref. [2]. [5.2.2] It follows from eq. (1.9) in Ref. [2] by virtue of the continuity theorem [16] for characteristic functions that for the first model the continuum limit with
(17) |
leads for all fixed to
(18) |
[5.2.3] Here the expansion has been used. [5.2.4] Therefore the solution of the first model with waiting time density converges in the continuum limit to the solution of the fractional diffusion equation
(19) |
with initial condition analogous to eq. (11).
[5.3.1] Consider now the second model with waiting time density given by eq. (8). [5.3.2] In this case
(20) |
and
(21) | |||
From this follows
(22) |
showing that the continuum limit as in eq. (17) with finite does not give rise to the propagator of fractional diffusion. [5.3.3] On the other hand the conventional continuum limit with exists and yields
(23) |
the Gaussian propagator of ordinary diffusion with diffusion constant .