Two species coagulation approach to consensus by group level interactions

We explore the self-organization dynamics of a set of entities by considering the interactions that affect the different subgroups conforming the whole. To this end, we employ the widespread example of coagulation kinetics, and characterize which interaction types lead to consensus formation and which do not, as well as the corresponding different macroscopic patterns. The crucial technical point is extending the usual one species coagulation dynamics to the two species one. This is achieved by means of introducing explicitly solvable kernels which have a clear physical meaning. The corresponding solutions are calculated in the long time limit, in which consensus may or may not be reached. The lack of consensus is characterized by means of scaling limits of the solutions. The possible applications of our results to some topics in which consensus reaching is fundamental, like collective animal motion and opinion spreading dynamics, are also outlined.