Diffusion-controlled reactions in one dimension: Exact solutions and deterministic approximations

Abstract

One-dimensional systems of annihilating and coalescing random walks and Brownian motions with arbitrary initial configurations of particles are studied. Exact results are derived for site-occupancy probabilities (densities), local fluctuations in these probabilities, and distributions of nearest-neighbor distances. Two types of initial configuration are investigated in detail: the homogeneous Poisson process and a one-parameter family of pairwise clustered locations. The systems studied can be regarded as simple models of diffusion-controlled chemical reactions and hence the exact results derived here can be compared with the predictions of traditional, deterministic models that are based on..