Single-site approximation for reaction-diffusion processes

Abstract

We consider the branching and annihilating random walk $A\to 2A$ and $2A\to 0$ with reaction rates $\sigma$ and $\lambda$, respectively, and hopping rate $D$, and study the phase diagram in the $(\lambda /D,\sigma /D)$ plane. According to standard mean-field theory, this system is in an active state for all $\sigma /D>0$, 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 $(\lambda /D,\sigma /D)$ 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.

15 pages, 2 figures, published version

References in corpus (1)

• Applications of Field-Theoretic Renormalization Group Methods to Reaction-Diffusion Problems