Field Theoretic Approach to Long Range Reactions

Most chemical reactions occur through a local mechanism, where reaction occurs at a finite rate only when reactants are closer than a capture radius. In some unusual cases, however, the reaction can proceed by a longrange mechanism. Exciton decay by a multipolar interaction is one such example. In these long-range cases, the rate of reaction between two reactants is distance dependent: w(r) = γ/r for two reactants separated by a distance r. Here the strength of the interaction is given by γ = R 0 /τ , where R0 is the Forster radius, and τ is the decay lifetime. The case of long-range reaction has received less theoretical attention than has the more common case of local reaction. The one-species, A + A → ∅ reaction between immobile reactants without adsorption has been investigated by placing bounds on a mean-field treatment [1]. An interesting power-law decay of the concentration was found in d spatial dimensions at long times: c(t) ∼ at−d/n as t → ∞ for n > d. An extension to the A+B → ∅ case gave the scaling c(t) ∼ at−d/(2n−d) as t → ∞ for n > d. The single-species reaction, A + A → ∅ reaction between immobile reactants in the presence of adsorption has been investigated with a similar treatment [2]. A non-trivial scaling of the concentration with the adsorption rate J was found: c(J) ∼ aJ as J → 0, n > d. In principle, however, these mean-field results could be modified by renormalization effects in low spatial dimensions. In this paper, we apply the renormalization group approach to reaction kinetics to derive the asymptotically exact behavior of these long range reactions. There are eight distinct physical cases, considering all possibilities of oneor two-species reactions, with or without adsorption, and mobile or immobile reactants. Our paper is orga-