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# 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,1⁢p⁢r,t=∑r′w⁢r-r′⁢p⁢r′,t (10)

with intitial condition

 p⁢r,0=δr,0 (11)

and fractional transition rates

 w⁢r=λ⁢r-1τα. (12)

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

 D0+α,1⁢p⁢r,t=1Γ⁢1-α⁢∫0tt-t′-α⁢∂∂⁡t⁢p⁢r,t′⁢d⁢t′ (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

 ψ1⁢u=11+τ⁢uα (14)

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

 λ⁢k=1d⁢∑j=1dcos⁡σ⁢kj, (15)

into eq. (5). [5.1.2] This gives

 p1⁢k,u=1u⁢τ⁢uα1+τ⁢uα-λ⁢k=uα-1uα-w⁢k (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,τ0 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

 Cα=limτ→0σ→0⁡σ2⁢d⁢τα (17)

leads for all fixed k,u to

 p1¯⁢k,u=limτ→0σ→0⁡p1⁢k,u=uα-1uα+Cα⁢k2. (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,1⁢p1¯⁢r,t=Cα⁢Δ⁢p1¯⁢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

 ψ2⁢u=12+2⁢c⁢τ2⁢uα+12+2⁢τ⁢u (20)

and

 p2⁢k,u =1u⁢1-λ⁢k-1⁢ψ2⁢u1-ψ2⁢u-1 =1u⁢1-λ⁢k-1⁢2+τ⁢u+c⁢τ2⁢uατ⁢u+c⁢τ2⁢uα+2⁢c⁢τ3⁢uα+1-1 (21) =1u⁢1+1τα⁢uα⁢σ2⁢k22⁢d-σ4⁢k424⁢d+…⁢2+τ⁢u+c⁢τ2⁢uατ⁢u1-α+c⁢τ2-α+2⁢c⁢τ3-α⁢u-1.

From this follows

 p2¯⁢k,u=limτ→0σ→0⁡p2⁢k,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=limτ0σ0σ2/2dτ exists and yields

 p2¯⁢k,u=limτ→0σ→0⁡p2⁢k,u=1u+C1⁢k2. (23)

the Gaussian propagator of ordinary diffusion with diffusion constant C1.