Diffusion-assisted long-range reactions in confined systems: Projection operator approach

The diffusion-assisted long-range reversible reaction equation is solved fo r the pair survival probability using a projection operator method in terms of the diffusion propagator in the absence of reaction. For a localized (d elta function) reaction sink, the well-known analytical solution is immedia tely reproduced from the operator expression. It is emphasized that the mea n reaction time approach, often used to approximate the overall reaction ra te, is not adequate for a nonequilibrium initial condition. The general ope rator solution for a delocalized sink is shown to reduce to a closed matrix form, provided the propagator has a discrete spectrum of eigenmodes. The m atrix solution is exact and applies for an arbitrary functional form and st rength of the reaction sink. Although matrices of infinite dimensions are i nvolved, they can be truncated at a certain finite dimension to attain any prescribed precision. Convergence of the truncated matrix solution is fast and often only a few of the lowest eigenmodes are sufficient to obtain quan titatively reasonable results. Several long-range reaction models are analy zed in detail revealing the breakdown of the widely used closure approximat ion obtained as a first-order Pade approximation of the operator solution. (C) 1999 American Institute of Physics. [S0021-9606(99)50716-2].