An Urn Model for Population Mixing and the Phases Within

Abstract

We introduce a generalization of the Ehrenfest urn model in which samples of independent identically distributed sizes with a general generating discrete distribution (the generator) are taken out of an urn containing white and blue balls (n in total). Each ball in the sample is repainted with the opposite color and the sample is replaced in the urn. We study the phases in the gradual change from the initial condition to the steady state for numerous cases where such a steady state exists. We look at the status of the urn after a number of draws. We identify a concept of linearity based on a combination of the generator and the number of draws, below which we consider the case to be sublinear and above which the case is superlinear. In a properly defined upper subphase of the sublinear phase the number of white balls is asymptotically normally distributed, with parameters that are influenced by the initial conditions and the generator. In the linear phase a different normal distribution applies, in which the influence of the initial conditions and the generator are attenuated. At the superlinear phase the mix is nearly perfect, with a nearly symmetrical normal distribution in which the effect of the initial conditions and the generator is obliterated. We give interpretations for how the results in different phases conjoin at the “seam lines.” The Gaussian results are obtained via martingale theory. We give a few concrete examples.