Statistics of a Population with Creation and Recombination Dependent on Existing Numbers

Abstract

Studies of fluctuations in gas density are commonly based on a Poisson distribution of probability of the number present. It is pointed out that this applies only to a population in which the rate of creation is constant and the rate of loss proportional to the first power of the number present. Populations liable to loss by recombination are likely to have a rate of loss proportional to the square of the number present; and the distribution function is then related either to a Bessel function or to a Laguerre polynomial, according as the rate of creation is either constant or dependent on the complement of the number already present. The variance of the Bessel type of distribution is half that of a corresponding Poisson distribution.