On the Continuum Time Limit of Reaction-Diffusion Systems

TL;DR

This paper investigates the continuous time limit of reaction-diffusion systems, specifically the parity conserving branching-annihilating random walk, revealing non-trivial scaling behaviors at criticality that differ from discrete time formulations.

Contribution

It demonstrates that the continuous time limit of the pc-BARW model exhibits non-trivial scaling of branching and hopping probabilities, a phenomenon not observed in some other reaction-diffusion systems.

Findings

Continuous time limit involves different scaling powers for branching and hopping probabilities.

The phenomenon is likely generic and not restricted to the 1D pc-BARW model.

The critical behavior differs significantly between discrete and continuous time formulations.

Abstract

The parity conserving branching-annihilating random walk (pc-BARW) model is a reaction-diffusion system on a lattice where particles can branch into $m$ offsprings with even $m$ and hop to neighboring sites. If two or more particles land on the same site, they immediately annihilate pairwise. In this way the number of particles is preserved modulo two. It is well known that the pc-BARW with $m=2$ in 1 spatial dimension has no phase transition (it is always subcritical), if the hopping is described by a continuous time random walk. In contrast, the $m=2$ 1-d pc-BARW has a phase transition when formulated in discrete time, but we show that the continuous time limit is non-trivial: When the time step $\delta t\to 0$, the branching and hopping probabilities at the critical point scale with different powers of $\delta t$. These powers are different for different microscopic realizations. Although this phenomenon is not observed in some other reaction-diffusion systems like, e.g. the contact process, we argue that it should be generic and not restricted to the 1-d pc-BARW model.