Exact solution for probabilities of complex particles

In principle, this time-evolution equation can be solved subject to the given initial condition psc (0) 8 (s — so) 8 (c — cy) to obtain ps c (t) uniquely for all 

To illustrate how to proceed using the cumulant generating functions, the well-known two-compartment model and the enzymatic reaction will be presented as examples of linear and nonlinear systems, respectively. In these examples, there are two interacting populations (m = 2) and the cumulant generating function is 

Initial conditions for the compartmental model and the enzymatic reaction were set to nq = [100 50], and so = 100, eo = 50, and Co = 0, respectively. These values are higher than those used previously and they are more likely to resemble experimental reality. 

Two-Compartment Model The model assumptions in (9.45) were substituted directly into operator equation (9.39), which was transformed via (9.37) to yield 

Upon substituting the series expansion (9.48) into the previous equation and equating coefficients of 0i and 02, one has the following differential equations 

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