Diffusion-controlled reactions with non-Markovian binding/unbinding kinetics

TL;DR

This paper develops a theoretical framework for reversible diffusion-controlled reactions incorporating non-Markovian binding and unbinding kinetics, revealing how heavy-tailed distributions influence long-term reaction behavior and boundary conditions.

Contribution

It introduces a generalized model for reversible reactions with non-exponential waiting times, extending classical Markovian theories to include memory effects and anomalous kinetics.

Findings

Spectral expansions for propagator and flux derived

Heavy-tailed distributions can cause anomalous reaction dynamics

Distinctions between different types of reactivity are clarified

Abstract

We develop a theory of reversible diffusion-controlled reactions with generalized binding/unbinding kinetics. In this framework, a diffusing particle can bind to the reactive substrate after a random number of arrivals onto it, with a given threshold distribution. The particle remains bound to the substrate for a random waiting time drawn from another given distribution and then resumes its bulk diffusion until the next binding, and so on. When both distributions are exponential, one retrieves the conventional first-order forward and backward reactions whose reversible kinetics is described by generalized Collins-Kimball's (or back-reaction) boundary condition. In turn, if either of distributions is not exponential, one deals with generalized (non-Markovian) binding or unbinding kinetics (or both). Combining renewal technique with the encounter-based approach, we derive spectral expansions for the propagator, the concentration of particles, and the diffusive flux on the substrate. We study their long-time behavior and reveal how anomalous rarity of binding or unbinding events due to heavy tails of the threshold and waiting time distributions may affect such reversible diffusion-controlled reactions. Distinctions between time-dependent reactivity, encounter-dependent reactivity, and a convolution-type Robin boundary condition with a memory kernel are elucidated.