Fluctuation Effects in the Pair Annihilation Process with Levy Dynamics

Reaction diffusion models provide a plethora of intensively studied classical nonequilibrium many body systems. One example is the diffusion limited pair annihilation process $A+A\to 0$, where the reactants diffuse in space and annihilate on contact. Inspired by the fact that many phenomena observed in nature exhibit superdiffusive behavior, we investigate the pair annihilation process in the case where the particles perform superdiffusion, realized by Levy flights. As a consequence, the critical dimension depends continuously on the control parameter of the Levy flight distribution. This instance is used to study the density decay in the pair annihilation process close to the critical dimension by means of the non-perturbative renormalization group theory. Close to the critical dimension, long-range fluctuations cause the law of mass action to break down. One crucial consequence of these fluctuations is that the law of mass action is complemented by additional non-analytic correction terms above the critical dimension. An increasing number of those corrections accumulate and give an essential contribution as the critical dimension is approached.