Studies in nonlinear stochastic processes. IV. A comparison of statistical linearization, diagrammatic expansion, and projection operator methods

Abstract

We compare the methods of statistical linearization, perturbation expansions, and projection operators for the approximate solution of nonlinear multimode stochastic equations. The model equations we choose for this comparison are coupled, nonlinear, first-order, one-dimensional complex mode rate equations. We show that the method of statistical linearization is completely equivalent to the neglect of certain well-defined diagrams in the perturbation expansion resulting in the first Kraichnan-Wyld approximation, and to the retention of only Markovian terms in the projection operator method, i.e., those terms that are local in time.