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

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Abstract
The diffusion-assisted long-range reversible reaction equation is solved for the pair survival probability using a projection operator method in terms of the diffusion propagator in the absence of reaction. For a localized (delta function) reaction sink, the well-known analytical solution is immediately reproduced from the operator expression. It is emphasized that the mean reaction time approach, often used to approximate the overall reaction rate, is not adequate for a nonequilibrium initial condition. The general operator solution for a delocalized sink is shown to reduce to a closed matrix form, provided the propagator has a discrete spectrum of eigenmodes. The matrix solution is exact and applies for an arbitrary functional form and strength of the reaction sink. Although matrices of infinite dimensions are involved, 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 quantitatively reasonable results. Several long-range reaction models are analyzed in detail revealing the breakdown of the widely used closure approximation obtained as a first-order Padé approximation of the operator solution.