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Approach to Asymptotic Behaviour in the Dynamics of the Trapping Reaction

We consider the trapping reaction A + B -> B in space dimension d=1, where the A and B particles have diffusion constants D_A, D_B respectively. We calculate the probability, Q(t), that a given A particle has not yet reacted at time t. Exploiting a recent formulation in which the B particles are eliminated from the problem we find, for t -> \infty, $Q(t) \sim \exp[-(4/\sqrtπ)(ρ^2 D_Bt)^{1/2} - (C ρ^2 D_A t)^{1/3} + ...]$, where $ρ$ is the density of B particles and $C \propto D_A/D_B$ for $D_A/D_B << 1$.