Diffusion-limited reaction in one dimension

Reaction rates that are controlled by collisions between diffusing particles depend on the distribution of distances between particles as well as on the concentration of particles. Classical treatment of reaction rates thus produces a hierarchy of particle correlation functions. However, when collision between particles results in annihilation, it is possible to sidestep this hierarchy and find an exact solution for the mean number of particles per unit length as a function of time. This article is concerned with the case, in one space dimension, where new particles are born or “nucleated”, two at a time, at random times and positions, with a fixed distance bb between the particles at birth. We review an exact method for calculating the mean lifetime of a particle and an exact rate equation in terms of the correlation function. In addition, the distribution of particle lifetimes is calculated under a “constant-killing-rate” approximation that compares favourably with the results of numerical experiments.