Evolution of the correlation function for a class of processes involving nonlocal self-replication

Abstract

A large class of evolutionary processes can be modeled by a rule that involves self-replication of some physical quantity with a nonlocal rescaling. We show that a class of such models is exactly solvable in the discrete as well as the continuum limit and can represent several physical situations, as varied as from the formation of galaxies in some cosmological models to growth of bacterial cultures. This class of models, in general, has no steady state solution and evolves unstably as t → ∞ for generic initial conditions. The models can, however, exhibit an (unstable) power-law correlation function in the continuum limit, for an intermediate range of times and length scales.