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

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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 simplifying assumptions about the evolution of the spatial structure of the systems. It is shown that, although the asymptotic behavior of the annihilating systems does depend on the structure of the initial configuration, the deterministic approximations are not able to detect this dependence, and the approximation is poor when the particles are initially highly clustered.

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