Single-site approximation for reaction-diffusion processes

We consider the branching and annihilating random walk A → 2A and 2A → 0 with reaction rates σ and λ, respectively, and hopping rate D, and study the phase diagram in the (λ/D, σ/D) plane. According to standard mean-field theory, this system is in an active state for all σ/D > 0, and perturbative renormalization suggests that this mean-field result is valid for d > 2; however, nonperturbative renormalization predicts that for all d there is a phase transition line to an absorbing state in the (λ/D, σ/D) plane. We show here that a simple single-site approximation reproduces with minimal effort the nonperturbative phase diagram both qualitatively and quantitatively for all dimensions d > 2. We expect the approach to be useful for other reaction-diffusion processes involving absorbing state transitions.