Fluctuation Effects in the Pair-Annihilation Process with L\'evy Dynamics

TL;DR

This paper studies how anomalous diffusion via Le9vy flights affects the decay dynamics in pair-annihilation processes, revealing universal behaviors and breakdowns of classical reaction laws near critical dimensions.

Contribution

It introduces a nonperturbative renormalization group analysis of Le9vy flight-driven reactions, highlighting universality and fluctuation effects near criticality.

Findings

Renormalized reaction rate approximates a universal law near critical dimension.

Law of mass action breaks down as criticality is approached.

Nonanalytic power law corrections become significant near critical dimension.

Abstract

We investigate the density decay in the pair-annihilation process A+A->0 in the case when the particles perform anomalous diffusion on a cubic lattice. The anomalous diffusion is realized via L\'evy flights, which are characterized by long-range jumps and lead to superdiffusive behavior. As a consequence, the critical dimension depends continuously on the control parameter of the L\'evy flight distribution. This instance is used to study the system close to the critical dimension by means of the nonperturbative renormalization group theory. Close to the critical dimension, the assumption of well-stirred reactants is violated by anticorrelations between the particles, and the law of mass action breaks down. The breakdown of the law of mass action is known to be caused by long-range fluctuations. We identify three interrelated consequences of these fluctuations. First, despite being a nonuniversal quantity and thus depending on the microscopic details, the renormalized reaction rate can be approximated by a universal law close to the critical dimension. The emergence of universality relies on the fact that long-range fluctuations suppress the influence of the underlying microscopic details. Second, as criticality is approached, the macroscopic reaction rate decreases such that the law of mass action loses its significance. And third, additional nonanalytic power law corrections complement the analytic law of mass action term. An increasing number of those corrections accumulate and give an essential contribution as the critical dimension is approached. We test our findings for two implementations of L\'evy flights that differ in the way they cross over to the normal diffusion.