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Approximate weak solutions of the Fokker-Planck equation can represent auseful tool to analyze the equilibrium fluctuations of birth-death systems, asthey provide a quantitative knowledge lying in between numerical simulationsand exact analytic arguments. In the present paper, we adapt the generalmathematical formalism known as the Ritz-Galerkin method for partialdifferential equations to the Fokker-Planck equation with time-independentpolynomial drift and diffusion coefficients on the simplex. Then, we show howthe method works in two examples, namely the binary and multi-state votermodels with zealots.

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