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2 Definition of Models

[2.3.1] Consider first the integral equation of motion for the CTRW-model [9, 10]. [2.3.2] The probability density pr,t obeys the integral equation


where λr denotes the probability for a displacement r in each single step, and ψt is the waiting time distribution giving the probability density for the time interval t between two consecutive steps. [2.3.3] The transition probabilities obey rλr=1, and the function Φt is the survival probability at the initial position which is related to the waiting time distribution through


[page 3, §0]   Fourier-Laplace transformation leads to the solution in Fourier-Laplace space given as [10]


where pk,u is the Fourier-Laplace transform of pr,t and similarly for ψ and λ.

[3.1.1] Two lattice models with different waiting time density will be considered. [3.1.2] In the first model the waiting time density is chosen as the one found in [1, 2]


where 0<α1,0<τ< is the characteristic time, and


is the generalized Mittag-Leffler function [14]. [3.1.3] In the second model the waiting time density is chosen as


where 0<α1, 0<τ< and c>0 is a suitable dimensional constant.

[3.2.1] The waiting time density ψ2t differs only little from ψ1t as shown graphically in Figure 1.

Figure 1: Waiting time densities ψ1t for model 1 and ψ2t for model 2 with α=0.8, τ=1 s and c=1 s-1.2.

[3.2.2] Note that both models have a long time tail of the form given in eq. (2), and the average waiting time 0tψitdt diverges.

[3.3.1] For both models the spatial transition probabilities are chosen as those for nearest-neighbour transitions (Polya walk) on a d-dimensional hypercubic lattice given as


where ej is the j-th unit basis vector generating the lattice, σ>0 is the lattice constant and δr,s=1 for r=s and δr,s=0 for rs.