Use of Dirichlet Distributions and Orthogonal Projection Techniques for the Fluctuation Analysis of Steady-State Multivariate Birth-Death Systems

TL;DR

This paper introduces a mathematical approach using Dirichlet distributions and orthogonal projection techniques to analyze fluctuations in steady-state multivariate birth-death systems, providing a bridge between simulations and exact solutions.

Contribution

It adapts the Ritz-Galerkin method to the Fokker-Planck equation with polynomial coefficients on the simplex for the first time in this context.

Findings

Successfully applied to binary and multi-state voter models with zealots

Provides approximate solutions that capture equilibrium fluctuations

Bridges numerical simulations and analytic methods

Abstract

Approximate weak solutions of the Fokker-Planck equation can 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 the present 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.