Models of hopping-controlled reactions with variable hopping range

We propose two classes of models for hopping-controlled reactions in which one of the reactants forms a random distribution of static traps and the hopping distances of the other reactants (random walkers) are independent random variables with a preassigned distribution. Specifically, in the discrete model, at each step the random walkers are allowed to make hops of all possible lengths of integer units up to a preassigned maximum value L, all with equal probability. In one of the continuous models, the hopping distances are Gaussian-distributed independent random variables with a mean L. In the other continuous model, the distribution of the hopping distances r follow an exponential distribution, namely, exp(-‖r‖/L). We predict the L dependence as well as the time dependence of the reaction rates (the decay of the particle density as a function of time) for these models analytically. We also verify some of these predictions by Monte Carlo computer simulations.