Dynamics of social diversity

We introduce and solve analytically a model for the development of disparate social classes in a competitive population. Individuals advance their fitness by competing against those in lower classes, and in parallel, individuals decline due to inactivity. We find a phase transition from a homogeneous, single-class society to a hierarchical, multi-class society. In the former case, the population is uniformly poor. In the latter case, a finite-fraction condensate that consists of a static lower class remains. The rest of the population consists of an upwardly mobile middle class, on top of which lies a tiny upper class in the form of a thin boundary layer.