Reactions governed by a binomial redistribution process—The ehrenfest urn problem

A distributive process of the binomial type in a one-dimensional discrete space with an absorbing barrier is studied. A simple expression for the particle number Σ(t) is derived. The analysis is based on recursion relationships and sum rules for the underlying eigenvectors, the Krawtchouk polynomials. The first passage time is determined, and the validity of the passage time approximation to Σ(t) tested. The continuous limit, corresponding to the diffusion and reaction of a harmonically bound particle, is briefly described.