Solutions to the Smoluchowski equation for problems involving the anisotropic diffusion or absorption of a particle

The escape probabilities or the recombination probabilities in diffusion-controlled reactions are obtained for some special systems which are not centrosymmetric. Specifically, we calculate (1) the escape probability of an electron from N fixed cations, (2) the recombination probability of a pair of radicals in an anisotropic medium, (3) the recombination probability of a radical with each of two statics sinks, and obtain (4) approximate expressions for the recombination probability of a radical with N static sinks in particular configurations. As a by-product, we obtain the bulk reaction rate for each system. Of special note is our result for the first system listed above. With the proviso that the cation radii are vanishingly small, we find the escape probability to be equal to the reciprocal of the Boltzmann factor for the system. This result factors into a product of isolated geminate ion pair escape probabilities. No competitive effects due to the geometric distribution of the cations appear.