Reaction-Diffusion Processes on Scale-Free Networks

Abstract

Recently there has been considerable interest in the theoretical and numerical investigation on the reaction segregation phenomena in physics, chemistry, biology, and chemical engineering. It is well known that the binary reaction has been investigated numerically and experimentally on many phenomena. Ovchinnikov and Zeldovich have played an important role to develop a research of segregation phenomenon. The early time behavior for reaction-diffusion process investigated by Taitelbaum et al. shows that both the global reaction rate and the reaction front grow as t at very early times. Cornell et al. studied theoretically the generalized and more complicated nAþ mB ! C reactions under initially separated reactant conditions. Zumofen et al. studied the breakdown phenomena of Ovchinnikov–Zeldovich segregation in the Aþ B ! 0 reaction under Levy mixing which is responsible to anomalous diffusion. They found that the segregation disappear in d 1⁄4 3 dimensions for 0 and 1 < < 2. Yen et al. studied the early time scaling in the ternary reaction-diffusion system with initially separated reactants. On the other hand, the phenomena of small-world and scale-free networks are really shown to be different from the dynamical behavior of the regular lattice system. It is really of fundamental importance to discuss the numerical and analytical result of scale-free network models compared with that of regular and small-world network models. The bimolecular chemical reaction in scale-free networks was studied for the generation of the depletion zone and the segregation of the reactants, and it was found that the reaction-diffusion processes in scale-free networks are different in their nature compared to regular lattice models, due to the small diameter of networks and the existence of hubs. Similarly, Catanzaro et al. have found that the inverse particle density scales linearly as 1= ðtÞ t. From this result, the inverse particle density in an uncorrelated scale-free network is shown to cross over to a linear behavior. Gallos and Argyrakis have also discussed the reaction-diffusion process of two species on the scale-free network between the correlated and the uncorrelated configuration models, and they especially revealed that the two models are identical when 1⁄4 3:0. Our group has studied analytic and numerical model of Aþ Bþ C ! D based on the regular lattice and small world networks, and the global reaction rate has analytically been derived before and after the crossover in the system of Aþ Bþ C ! D. We obtained that the chemical reaction of three species of reactants occurs equivalently in both regular and small-world networks at early time regime, but the decay process on small-world network proceeds slightly faster than in the case of the regular network at long-time regime. The aim of this paper is to examine the result in the annihilation process of three species on Barabasi–Albert (BA) scale-free networks. Classically, the the particle density ðtÞ of the surviving particles in the reactions of Aþ A ! 0 and Aþ B ! 0 type on a d-dimensional regular lattice scales as