Renormalisation for Reaction-Diffusion Systems with Non-Local Interactions

TL;DR

This paper investigates how non-local interactions in reaction-diffusion systems influence divergences and renormalisation, revealing universal behaviour and providing new methods to solve related equations.

Contribution

It demonstrates that non-local interactions can regulate ultra-violet divergences and introduces a space-time-field rescaling approach to solve Callan-Symanzik equations.

Findings

Non-local interactions regulate ultra-violet divergences.

Infra-red divergences persist in asymptotic regimes.

Renormalisation group preserves action structure and simplifies solutions.

Abstract

Models of reaction diffusion processes usually employ discrete lattice models with particles interacting at the same site, resulting in localized reactions in the continuum limit. Here, various non-local interactions are considered, and two features reported. Firstly, it is shown that sufficiently non-local interactions will regulate ultra-violet divergences that perturbative methods with local interactions produce. However, in asymptotic regimes, infra-red divergences persist and ultra-violet divergences can reappear. Renormalisation methods are shown to report the same universal behaviour as local interactions at critical points. Secondly, the renormalisation group can be interpreted as a space-time-field rescaling that preserves action structure. This can be used to extract solutions to Callan-Symanzik equations directly without having to solve (or construct) the equation. These observations are exemplified for two paradigm models; annihilation $A_p+A_q\rightarrow \phi$, and this process paired with branching, birth and death.