Equation of motion for the two-point probabilities

Let us study the next processes which take place on two different lattice sites I and m. The following processes are possible (Section 9.1.1)  

We want to introduce a diagrammatic description of these processes, equation (9.1.14). A process takes place on the site , whereas the symbol means that this site plays no role for the determination of the state of another site. Therefore the first two terms correspond to the cases (la) and (lb) and the third term represents the bimolecular step. The diagrams of type (la) and (lb) are of the form (9.1.17), the second term on the right hand side 

This equation is valid only for Z - m 1. The corresponding case when I and m are the nearest neighbours will be treated below. Let us first discuss the monomolecular step for Z - m 1. Using equation (9.1.60), we obtain 

Let us now treat the terms where a process on site I occurs without m being affected, but I and m are the nearest neighbours. For Z - m = 1 (the nearest neighbours) we change the expression for pP+ ) for 

Again we use equation (9.2.15). But now we get a non-complete mean-value because one site is in the state a-m- Therefore we introduce the definition 

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