Multigroup criticality condition for a space-independent multiplying assembly via the Chapman--Kolmogorov equation

We consider the multigroup time-dependent probability distribution of neutrons in a space-independent, source-free multiplying assembly. The first-order partial differential equation for the probability generating function can be derived from the forward Chapman--Kolmogorov equations of the stochastic process, and solved: in the principle: by the method of characteristics. It is well known that the asymptotic extinction probability is related to the singular point of the system of characteristic equations. An analysis of the location of this point in the hyperspace of dummy variables associated with neutrons of various velocity groups leads to a multigroup criticality condition for the assembly