Abstract
We study fluctuation effects in the two-species reaction-diffusion systemA+B→ Ø andA+A→ (Ø,A). In contrast to the usually assumed ordinary short-range diffusion spreading of the reactants we consider anomalous diffusion due to microscopic long-range hops. In order to describe the latter, we employ the Lévy stochastic ensemble. The probability distribution for the Lévy flights decays inddimensions with the distanceraccording to a power-lawr−d−σ. For anomalous diffusion (including Lévy flights) the critical dimensiondc=σdepends on the control parameterσ, 0<σ ≤ 2. The model is studied in terms of the field theoretic approach based on the Feynman diagrammatic technique and perturbative renormalization group method. We demonstrate the ideas behind theBparticle density calculation.
Highlights
IntroductionGenuine reaction-diffusion models describe a multitude of phenomena in various disciplines, from population dynamics in ecology, competition of bacterial colonies in microbiology, dynamics of magnetic monopoles in the early universe in cosmology, to the stock market in economy, opinion exchange in sociology, etc [1]
Genuine reaction-diffusion models describe a multitude of phenomena in various disciplines, from population dynamics in ecology, competition of bacterial colonies in microbiology, dynamics of magnetic monopoles in the early universe in cosmology, to the stock market in economy, opinion exchange in sociology, etc [1].We consider a system consisting of two particle species A and B, with the corresponding diffusion constants DnA and DnB
In order to calculate the large-time behavior of the B particle density below the critical dimension, it is advantageous to employ the field theoretic approach followed by the perturbative renormalization group formalism
Summary
Genuine reaction-diffusion models describe a multitude of phenomena in various disciplines, from population dynamics in ecology, competition of bacterial colonies in microbiology, dynamics of magnetic monopoles in the early universe in cosmology, to the stock market in economy, opinion exchange in sociology, etc [1]. Where a and b denote the densities of corresponding reacting particles A and B, parameters λ and λ are the reaction rates. For a space dimension d larger than the upper critical dimension dc (dc = 2 for the ordinary diffusion) the result for particle density A decreases with time as a ∼ 1/(λt) [3,4,5]. The analysis of the canonical dimensions shows that for the case σ < 2 the ordinary diffusion terms ∝ ∇2 are infrared irrelevant with respect to the anomalous diffusion terms ∝ ∇σ. In order to calculate the large-time behavior of the B particle density below the critical dimension, it is advantageous to employ the field theoretic approach followed by the perturbative renormalization group formalism. We concentrate on explaining how the density of B particles can be calculated
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