# 3 Results

[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

D0+α,1pr,t=∑r′wr-r′pr′,t | | (10) |

with intitial condition

and fractional transition rates

Here the fractional time derivative
D0+α,1 of order α and type 1
in eq. (10) is defined as [15]

D0+α,1pr,t=1Γ1-α∫0tt-t′-α∂∂tpr,t′dt′ | | (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 ψ1t

and the Fourier transform of λr, the so called
structure function

λk=1d∑j=1dcosσkj, | | (15) |

into eq. (5).
[5.1.2] This gives

p1k,u=1uτuα1+τuα-λk=uα-1uα-wk | | (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 ψ1t 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 σ,τ→0 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

leads for all fixed k,u to

p1¯k,u=limτ→0σ→0p1k,u=uα-1uα+Cαk2. | | (18) |

[5.2.3] Here the expansion cosx=1-x2/2+x4/24-… has been used.
[5.2.4] Therefore the solution of the first model with
waiting time density ψ1t converges in the continuum limit
to the solution of the fractional diffusion equation

D0+α,1p1¯r,t=CαΔp1¯r,t | | (19) |

with initial condition analogous to eq. (11).

[5.3.1] Consider now the second model with waiting time density ψ2t
given by eq. (8).
[5.3.2] In this case

ψ2u=12+2cτ2uα+12+2τu | | (20) |

and

p2k,u | =1u1-λk-1ψ2u1-ψ2u-1 | |

| =1u1-λk-12+τu+cτ2uατu+cτ2uα+2cτ3uα+1-1 | | (21) |

| =1u1+1ταuασ2k22d-σ4k424d+…2+τu+cτ2uατu1-α+cτ2-α+2cτ3-αu-1. | |

From this follows

p2¯k,u=limτ→0σ→0p2k,u=0 | | (22) |

showing that the continuum limit as in eq. (17) with
finite Cα
does not give rise to the propagator of fractional diffusion.
[5.3.3] On the other hand the conventional continuum limit with
C1=limτ→0σ→0σ2/2dτ
exists and yields

p2¯k,u=limτ→0σ→0p2k,u=1u+C1k2. | | (23) |

the Gaussian propagator of ordinary diffusion with diffusion constant
C1.