Approach to Asymptotic Behaviour in the Dynamics of the Trapping Reaction

TL;DR

This paper analyzes the long-term survival probability of A particles in a one-dimensional trapping reaction with diffusing B particles, deriving asymptotic expressions for the probability as time approaches infinity.

Contribution

It introduces a novel formulation that eliminates B particles to derive the asymptotic behavior of the survival probability in the trapping reaction.

Findings

Derived asymptotic expression for Q(t) as t -> ∞

Identified the dependence of survival probability on particle densities and diffusion constants

Provided explicit formulas involving exponential decay with specific power laws

Abstract

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, $Q(t) \sim \exp[-(4/\sqrt{\pi})(\rho^2 D_Bt)^{1/2} - (C \rho^2 D_A t)^{1/3} + ...]$, where $\rho$ is the density of B particles and $C \propto D_A/D_B$ for $D_A/D_B << 1$.