The Minimal Reversible Coagulation-Fragmentation Process Having no Factorized Coagulation and Fragmentation Rates

The coagulation-fragmentation process (CFP) is a model description for the stochastic dynamics of a population of $N$ particles distributed into groups of different sizes that coagulate and fragment at some given rates. It arises in a variety of contexts. Coagulation and fragmentation rates whose ratio is of the form $a (i+j) / ( a(i) a(j) )$ are called factorized kernels, and provide a necessary condition for reversibility. We prove here that all reversible CFP's with $N \le 5$ particles have factorized kernels, and the smallest example of a reversible non factorized CFP is for $N=6$.