The diffusion-limited coalescence model, A+AA, can be treated exactly in one dimension. The process is unexpectedly rich, displaying self-critical ordering in a nonequilibrium system, a kinetic phase transition, and a lattice version of Fisher waves. Thus, in spite of its simplicity it sheds light on many important aspects of anomalous kinetics. It also serves as a benchmark test for approximation methods and simulation algorithms. The coalescence model will concern us throughout the remainder of the book. Here we introduce the model and explain the technique which allows its exact analysis.

The one-species coalescence model

Our basic model is a lattice realization of the one-dimensional coalescence process A + A → A. The exact analysis can also be extended to the reversible process, A → A + A, as well as to the input of A particles. The system is defined on a one-dimensional lattice of lattice spacing Δx. Each site may be either occupied by an A particle or empty. The full process consists of the following dynamic rules.

Diffusion. Particles hop randomly to the nearest lattice site with a hopping rate 2D/(Δx)2. The hopping is symmetric, with rate D/(Δx)2 to the right and D/(Δx)2 to the left. At long times this yields normal diffusion, with diffusion coefficient D.

Birth. A particle gives birth to another at an adjacent site, at rate ν/Δx. This means a rate of ν/(2Δx) for birth on each side of the original particle. Notice that, while ν is a constant (with units of velocity), the rate ν/Δx diverges in the continuum limit of Δx → 0.