Rate equation of the A+A→A reaction with probability of reaction and diffusion

We study the coagulation reaction A+A→A with probability p of reaction in a one-dimensional lattice. We show that dρ/dt=-pΓ₁, where ρ(t) is the density of particles and Γ₁(t) is the density of nearest-neighbor occupied sites. From the analysis of the crossover between the reaction-controlled-reaction and the diffusion-controlled-reaction regimes an approximate scaling ansatz for Γ₁ and ρ for different values of p is proposed. Using this scaling, the collapse of Monte Carlo data improves when p decreases. From this scaling an analytical approximation of the density is found which agrees well with the numerical results of ρ(t) for different values of p.