THOMPSON's METHOD APPLIED TO THE BROWNIAN AND NON-BROWNIAN DIFFUSION-CONTROLLED REACTIONS OF TYPE kA → lA (l < k)

Abstract

The diffusion-controlled reactions of type kA → lA, with l < k, including the case of the annihilation reaction kA → 0, are studied by using Thompson's method, both in the case of Brownian (γ = 2) as in the case of anomalous (γ ≠ 2) diffusion conditions. These reactions are known to be strongly dependent on fluctuations below some critical dimension dc. We find dc = γ/(k-1) and the asymptotic behavior of density [Formula: see text]. At d = dc, the density goes as <∊> ~ ( ln t/t)1/k-1. For γ = 2, the scaling results obtained here are in agreement with the renormalization group calculations of Lee15 and a more recent work of Oliveira.8 We go further by also studying the case of an external homogeneous source (h) of single species particles A in the case of the stationary regime. Then we obtain the critical exponent δ in [Formula: see text], the dynamical exponent for the relaxation time Δ′ in (τh ~ hΔ′) and the exponent for the concentration decay ξ in (∊ ~ τ- ξ), with all these quantities evaluated in the limit h → 0. We get relations of scaling among critical indexes, and, in the special case of γ = 2, we recover those results previously obtained by Rácz.30 Thompson's method is a simple alternative way to the renormalization group scheme and has been shown to be a good description for the long-time (long wave-length) regime.