Annihilations of stationary particles on a lattice

The time evolution of a system of particles which initially occupy points on a lattice and subsequently undergo pairwise annihilation, is investigated with the help of the Master equation. An exact analysis is given for a onedimensional lattice of finite as well as infinite extent. Hierarchy equations are derived with the help of a generating function technique, and are solved to give expressions for the number of particles, pairs, and $n$-tuples for arbitrary time. Computer calculations for the number of singletons left behind are given for lattices of various sizes and shown to be in complete agreement with the analytical results. The motivation of this study comes primarily from pycnonuclear reactions in the interiors of dense stars and secondarily from recombination of free radicals on surfaces and annihilation of Frenkel excitons in molecular crystals.