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-orde r A + B-->B reactions in situations in which the particles' intrinsic react ivities are not constant but vary randomly in time. That is, we suppose tha t the particles are bearing "gates" which fluctuate in time, randomly and i ndependently of each other, between two states-an active state, when the re action may take place between A and B particles appearing in close contact, and a blocked state, when the reaction is completely inhibited. We focus h ere on two customary limiting cases of pseudo-first-order reactions-the so- called target annihilation and the Rosenstock trapping model-and consider f our different particular models, such that the A particle can be either mob ile or immobile or gated or ungated, and ungated or gated B particles can b e fixed at random positions or move randomly. All models are formulated on a d-dimensional regular lattice, and we suppose that the mobile species per form independent, homogeneous, discrete-time lattice random walks. The mode l involving a single, immobile, ungated target A and a concentration of mob ile, 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 int erpreted as the moment generating function of some functionals of random wa lk trajectories, such as, e.g., the number of self-intersections, the numbe r 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 asympt otic behavior of corresponding generating functions which are not known as yet.