A part of a semi-Markovian chain with only nearest neighbor transitions and directional WT-PDFs, and . A way to simulate such a process is to first draw a random number out of a uniform density that determines the propagation direction according to the transition probabilities, and then to draw a random time out of the relevant WT-PDF.
A circular three-state chain. The system is characterized by the waiting time PDFs, here denoted by , where and .
A two-state semi-Markovian chain characterized by the WT-PDFs and .
as a function of , for and . The exact expression in Eq. (16) (dotted curve), the Gaussian approximation (dashed curve), and the rectangular approximation are shown.
The Green’s function in Eq. (18), the expression in Eq. (19) that uses the rectangular approximation, and packets of path PDFs (the size of each packet is shown at the base of the arrow pointing on the packet). Panel (a) shows, on a linear-log scale, that the approximation of a packet of path PDFs is valid only at the maximum of the packet, where panel (b) emphasizes that the packet width increases at large times (the width of the packet scales as ). Here, and with appropriate arbitrary units.
A three-state chain. The system is characterized by the waiting time PDFs , , , and .
A four-state chain. The system is characterized by the waiting time PDFs , , , , , and .
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