Front propagation inA+B→2Areaction under subdiffusion

Abstract

We consider an irreversible autocatalytic conversion reaction A + B → 2A under subdiffusion described by continuous-time random walks. The reactants' transformations take place independently of their motion and are described by constant rates. The analog of this reaction in the case of normal diffusion is described by the Fisher-Kolmogorov-Petrovskii-Piskunov equation leading to the existence of a nonzero minimal front propagation velocity, which is really attained by the front in its stable motion. We show that for subdiffusion, this minimal propagation velocity is zero, which suggests propagation failure.

Fronts in anomalous diffusion–reaction systems

“…Nonlinear generalizations of equation (3.8) have been suggested by Froemberg et al [60] and Fedotov [61]. In the case of nonlinear dependence of reaction rates on reactant densities, it is typically impossible to write a closed system of equations for concentrations u i (x, t).…”

“…Surprisingly, solutions with a constant velocity have been found also in models without 'rejuvenation' [29,60,61]. The minimum front velocity can be either finite [61] (for reaction A + B → B + 2A) or zero [60] (for reaction A + B → 2A).…”

“…If the waiting time distribution for jumps is determined by the probability distribution of fluctuations in the environment that make the molecule release possible, it is natural to assume that the age of the molecule, understood as the argument of the waiting time pdf, is not changed owing to the participation of the molecule in the reaction [24,30]. In that case, the 'age structure' of the population of molecules depends on time and has a tendency of 'ageing', which leads to the decrease in the jump probability and hence slowing down of all the processes, including the front propagation ('propagation failure') [33,60].…”

Mathematical Modelling of Subdiffusion-reaction Systems

“…In the framework of system (3.12), (3.13), the "age structure" of the population of molecules depends on time and has a tendency of "aging", which leads to the decrease of the jump probability and hence slowing down of all the processes, including the front propagation ("propagation failure") [46], [30]. Specifically, if the jump rates of all the components coincide, W i (τ ) = W (τ ) for all i, in the large scale limit the problem is reduced to a normal diffusion-reaction system with the matrix of the diffusion coefficients decaying like t γ−1 [34].…”

“…Froemberg et al [46] have considered the A + B → 2A reaction under subdiffusion characterized by a waiting time pdf, w(t). An integro-differential equation,…”