  We address the problem of determining the stationary distribution of the
multi-allelic, neutral-evolution Wright-Fisher model in the diffusion limit. A
full solution to this problem for an arbitrary K x K mutation rate matrix
involves solving for the stationary solution of a forward Kolmogorov equation
over a (K - 1)-dimensional simplex, and remains intractable. In most practical
situations mutations rates are slow on the scale of the diffusion limit and the
solution is heavily concentrated on the corners and edges of the simplex. In
this paper we present a practical approximate solution for slow mutation rates
in the form of a set of line densities along the edges of the simplex. The
method of solution relies on parameterising the general non-reversible rate
matrix as the sum of a reversible part and a set of (K - 1)(K - 2)/2
independent terms corresponding to fluxes of probability along closed paths
around faces of the simplex. The solution is potentially a first step in
estimat- ing non-reversible evolutionary rate matrices from observed allele
frequency spectra.
