Use of dirichlet distributions and orthogonal projection techniques for the fluctuation analysis of steady-state multivariate birth–death systems

Abstract

Approximate weak solutions of the Fokker–Planck equation represent a useful tool to analyze the equilibrium fluctuations of birth–death systems, as they provide a quantitative knowledge lying in between numerical simulations and exact analytic arguments. In this paper, we adapt the general mathematical formalism known as the Ritz–Galerkin method for partial differential equations to the Fokker–Planck equation with time-independent polynomial drift and diffusion coefficients on the simplex. Then, we show how the method works in two examples, namely the binary and multi-state voter models with zealots.