Simulations for trapping reactions with subdiffusive traps and subdiffusive particles

TL;DR

This paper investigates the survival probability of a subdiffusive particle amidst subdiffusive traps in one dimension, comparing asymptotic theoretical predictions with numerical simulations to understand their validity and limitations.

Contribution

The study provides a detailed comparison between asymptotic theoretical results and numerical simulations for reactions involving particles with different anomalous diffusion exponents.

Findings

Asymptotic theory matches simulations well in some parameter ranges.

In certain regimes, longer times are needed for simulations to reach asymptotic behavior.

For specific parameters, asymptotic behavior remains unconfirmed due to computational limitations.

Abstract

While there are many well-known and extensively tested results involving diffusion-limited binary reactions, reactions involving subdiffusive reactant species are far less understood. Subdiffusive motion is characterized by a mean square displacement $<x^2> \sim t^\gamma$ with $0<\gamma<1$. Recently we calculated the asymptotic survival probability $P(t)$ of a (sub)diffusive particle ($\gamma^\prime$) surrounded by (sub)diffusive traps ($\gamma$) in one dimension. These are among the few known results for reactions involving species characterized by different anomalous exponents. Our results were obtained by bounding, above and below, the exact survival probability by two other probabilities that are asymptotically identical (except when $\gamma^\prime=1$ and $0<\gamma<2/3$). Using this approach, we were not able to estimate the time of validity of the asymptotic result, nor the way in which the survival probability approaches this regime. Toward this goal, here we present a detailed comparison of the asymptotic results with numerical simulations. In some parameter ranges the asymptotic theory describes the simulation results very well even for relatively short times. However, in other regimes more time is required for the simulation results to approach asymptotic behavior, and we arrive at situations where we are not able to reach asymptotia within our computational means. This is regrettably the case for $\gamma^\prime=1$ and $0<\gamma<2/3$, where we are therefore not able to prove or disprove even conjectures about the asymptotic survival probability of the particle.