Diffusion-limited reaction in one dimension: Paired and unpaired nucleation

We study the dynamics of diffusing particles in one space dimension with an nihilation on collision and nucleation (creation of particles) with constan t probability per unit time and length. The cases of nucleation of single p articles and nucleation in pairs are considered. A new method of analysis p ermits exact calculation of the steady-state density and its time evolution in terms of the three parameters describing the microscopic dynamics: the nucleation rate, the initial separation of nucleated pairs, and the diffusi vity of a particle. For paired nucleation at sufficiently small initial sep aration the nucleation rate is proportional to the square of the steady-sta te density. For unpaired nucleation, and for paired nucleation at sufficien tly large initial separation, the nucleation rate is proportional to the cu be of the steady-state density. (C) 2001 American Institute of Physics.