Lattice Kinetics of Diffusion-Limited Coalescence and Annihilation with   Sources
  We study the 1D kinetics of diffusion-limited coalescence and annihilation with back reactions and different kinds of particle input. By considering the changes in occupation and parity of a given interval, we derive sets of hierarchical equations from which exact expressions for the lattice coverage and the particle concentration can be obtained. We compare the mean-field approximation and the continuum approximation to the exact solutions and we discuss their regime of validity. 
E. Abad, T. Masser, D. ben-Avraham