Single-site approximation for reaction-diffusion processes

TL;DR

This paper demonstrates that a simple single-site approximation can accurately reproduce the phase diagram of a reaction-diffusion process involving branching and annihilation, aligning with nonperturbative predictions across all dimensions.

Contribution

The authors introduce a single-site approximation method that effectively captures the nonperturbative phase diagram of reaction-diffusion processes for all dimensions.

Findings

Single-site approximation reproduces the phase diagram accurately.

Method aligns with nonperturbative predictions.

Approach is potentially applicable to other absorbing state transitions.

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.