Trapping Reactions with Randomly Moving Traps: Exact Asymptotic Results for Compact Exploration

TL;DR

This paper demonstrates that in systems with compactly exploring traps, the survival probability of a diffusing particle becomes independent of its diffusion coefficient in the long-time limit, extending previous 1D results to higher dimensions.

Contribution

The study generalizes the asymptotic behavior of survival probabilities to arbitrary dimensions and fractal systems with compact exploration, revealing conditions for independence from particle diffusivity.

Findings

Survival probability is independent of particle diffusion coefficient in the asymptotic limit for compact exploration.

The results extend to systems of arbitrary dimension, including fractal geometries.

In the marginal case of 2D, the decay form is determined up to a numerical factor.

Abstract

In a recent Letter Bray and Blythe have shown that the survival probability P(t) of an A particle diffusing with a diffusion coefficient D_A in a 1D system with diffusive traps B is independent of D_A in the asymptotic limit t \to \infty and coincides with the survival probability of an immobile target in the presence of diffusive traps. Here we show that this remarkable behavior has a more general range of validity and holds for systems of an arbitrary dimension d, integer or fractal, provided that the traps are "compactly exploring" the space, i.e. the "fractal" dimension dw of traps' trajectories is greater than d. For the marginal case when dw = d, as exemplified here by conventional diffusion in 2D systems, the decay form is determined up to a numerical factor in the characteristic decay time.