A perturbation analysis of the Wilemski–Fixman approximation for diffusion-controlled reactions

Abstract

Some time ago, Wilemski and Fixman suggested an approximate method for calculating reaction rates for diffusion‐controlled reactions. Their derivation contains a factorization assumption that makes it difficult to see how to derive higher order corrections systematically. In this paper, we assume that the reaction term can be regarded as small in a suitable sense, and develop a systematic perturbation analysis that yields the Wilemski–Fixman approximation in lowest order. This identification will be shown to imply that the Wilemski–Fixman approximation corresponds to a factorization of the resulting multiple integrals in a specific way. It will be shown that for a single localized reaction term (i.e., a delta function sink), the Wilemski–Fixman approximation leads to an exact expression for survival probability as a function of time, but the original factorization ansatz used by these authors is violated. We also develop a theory not making use of the restrictive assumption that the initial condition corresponds to an equilibrium density in the absence of reaction. Finally, we develop an exactly solvable model with a nonlocal reaction term against which the approximation can be tested.