Fractional Reaction-Diffusion Equation

TL;DR

This paper derives a fractional reaction-diffusion equation incorporating memory effects in both diffusion and reaction terms from a continuous time random walk model, and applies it to geminate recombination, revealing reactivity dependence.

Contribution

It introduces a novel fractional reaction-diffusion equation with a memory-dependent reaction term, advancing the modeling of dispersive transport and reactions.

Findings

Recombination depends on intrinsic reaction rate.

Memory effects influence the reaction and diffusion processes.

Numerical simulations confirm reactivity dependence.

Abstract

A fractional reaction-diffusion equation is derived from a continuous time random walk model when the transport is dispersive. The exit from the encounter distance, which is described by the algebraic waiting time distribution of jump motion, interferes with the reaction at the encounter distance. Therefore, the reaction term has a memory effect. The derived equation is applied to the geminate recombination problem. The recombination is shown to depend on the intrinsic reaction rate, in contrast with the results of Sung et al. [J. Chem. Phys. {\bf 116}, 2338 (2002)], which were obtained from the fractional reaction-diffusion equation where the diffusion term has a memory effect but the reaction term does not. The reactivity dependence of the recombination probability is confirmed by numerical simulations.