Evolution of the Correlation Function for a Class of Processes involving Non Local Self - Replication

Abstract

A large class of evolutionary processes can be modeled by a rule which involves self-replication of some physical quantity with a non local rescaling. I show that a class of such models are exactly solvable -- in the discrete as well as continuum limit -- and can represent several physical situations as varied 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 evolve unstably as $t \to \infty$ for generic initial conditions. They can however exhibit (unstable) power law correlation function in the continuum limit, for an intermediate range of times and length scales.