Asymptotic front behavior in anA+B→2Areaction under subdiffusion

Abstract

We discuss the front propagation in the A + B → 2A reaction under subdiffusion which is described by continuous time random walks with a heavy-tailed power law waiting time probability density function. Using a crossover argument, we discuss the two scaling regimes of the front

Fronts in anomalous diffusion–reaction systems

“…In the model of Froemberg et al [60], the produced particles inherit the age of B-particles, and their mobility decreases with time. In the latter case, another kind of front solution, with velocity decreasing as t −α , α = (1 − γ )/2 has been found both by means of continuous reaction-subdiffusion equations [64] and in probabilistic numerical simulations [32,33]. One more law of the front motion, c ∼ t −2α , which takes place at small concentrations of components (in the fluctuation-dominated region), has been found in simulations [33] and explained in Froemberg et al [64].…”

Application of hyperbolic scaling for calculation of reaction-subdiffusion front propagation

“…[11]. According to the arguments, for short times the behavior of the velocity v(H) must be similar to that in subdiffusion (it does not feel the cutoff), whereas for long times the behavior is the classical one with a constant minimal velocity, and there has to be a crossover (no jump!)…”

“…Introduction.-The problem of front propagation in reaction-transport equations is attracting much attention that is related to the considerable progress in our understanding of this phenomenon via the generalization of the standard reaction-diffusion scheme in the framework of the Fisher-Kolmogorov-Petrovskii-Piskunov (FKPP) equation for fractional reaction-subdiffusion systems [1][2][3][4][5][6][7][8][9][10][11]. The description of reactions under subdiffusion is relevant to strongly inhomogeneous environments, in porous media such as certain geological formations or gels, in crowded cell interiors, and so on.…”

“…To determine the crossover time, it is plausible to argue with the amount of performed steps, as a measure of mobility, which for the normal regime is n D (t) ∝ t/t 1−α T α and in the subdiffusive regime reads n SD (t) ∝ (t/t 0 ) α , equating them, one finds v(t < t cr ) ∼ t obtained in Refs. [10,11]. An important point when considering the relaxation in the framework of hyperbolic scaling is that the process of relaxation for a subdiffusive front can be treated for the finite energy H = 2k in the framework of the Markovian case.…”

“…[11]. Another specific property of the method is an effective linearization of the generalized FKPP Eq.…”