Single-site approximation for reaction-diffusion processes

We consider the branching and annihilating random walk $A\to 2A$ and $2A\to 0$withreactionrates$\sigma$and$\lambda$,respectively,andhoppingrate$D$,andstudythephasediagraminthe$(\lambda/D,\sigma/D)$ plane. According to standard mean-field theory, this system is in an active state for all $\sigma \mathrm{/}D>0$, 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 $(\lambda \mathrm{/}D,\sigma \mathrm{/}D)$ 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.