Kinetics of stochastically-gated diffusion-limited reactions and geometry of random walk trajectories

TL;DR

This paper investigates the kinetics of diffusion-limited reactions with stochastically-gated particles, providing exact solutions and bounds for various models, and linking survival probabilities to random walk trajectory functionals.

Contribution

It introduces four models of gated diffusion reactions on lattices, deriving exact solutions and bounds, and connects survival probabilities to random walk trajectory functionals.

Findings

Exact solution for a single immobile ungated target with mobile gated particles.

Rigorous bounds for survival probabilities in three other models.

Linking survival probabilities to generating functions of random walk functionals.

Abstract

In this paper we study the kinetics of diffusion-limited, pseudo-first-order A + B -> B reactions in situations in which the particles' intrinsic reactivities vary randomly in time. That is, we suppose that the particles are bearing "gates" which interchange randomly and independently of each other between two states - an active state, when the reaction may take place, and a blocked state, when the reaction is completly inhibited. We consider four different models, such that the A particle can be either mobile or immobile, gated or ungated, as well as ungated or gated B particles can be fixed at random positions or move randomly. All models are formulated on a $d$-dimensional regular lattice and we suppose that the mobile species perform independent, homogeneous, discrete-time lattice random walks. The model involving a single, immobile, ungated target A and a concentration of mobile, gated B particles is solved exactly. For the remaining three models we determine exactly, in form of rigorous lower and upper bounds, the large-N asymptotical behavior of the A particle survival probability. We also realize that for all four models studied here such a probalibity can be interpreted as the moment generating function of some functionals of random walk trajectories, such as, e.g., the number of self-intersections, the number of sites visited exactly a given number of times, "residence time" on a random array of lattice sites and etc. Our results thus apply to the asymptotical behavior of the corresponding generating functions which has not been known as yet.