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

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 are not constant but vary randomly in time. That is, we suppose that the particles are bearing "gates" which fluctuate in time, randomly and independently of each other, between two states-an active state, when the reaction may take place between A and B particles appearing in close contact, and a blocked state, when the reaction is completely inhibited. We focus here on two customary limiting cases of pseudo-first-order reactions-the so-called target annihilation and the Rosenstock trapping model-and consider four different particular models, such that the A particle can be either mobile or immobile or gated or ungated, and 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 the form of rigorous lower and upper bounds showing the same N dependence, the large-N asymptotical behavior of the probability that the A particle survives until the Nth step. We also realize that for all four models studied here the A particle survival probability 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, the "residence time" on a random array of lattice sites, etc. Our results thus apply to the asymptotic behavior of corresponding generating functions which are not known as yet.