Front propagation in a one-dimensional autocatalytic reaction-subdiffusion system

Abstract

We study numerically the autocatalytic irreversible reaction A+B-->2A on a one-dimensional lattice for the case of subdiffusive reactants performing symmetric continuous-time random walks with the power-law waiting time density function psi(t) proportional, t(-1-alpha) with 0<alpha<1 in a large range of other relevant parameters. We show that the propagation failure previously found by the same authors [Phys. Rev. E 78, 011128 (2008)] essentially corresponds to a propagation of a front of a stable form at a velocity which decays with time.

Nonuniversality and the role of tails in reaction-subdiffusion fronts

“…In our case II, studied previously by Sokolov et al ͓17,14,21͔, the long tail of the distribution ͑1͒ is the responsible for the propagation failure, as more particles get gradually trapped in the tail. This results in a frozen state in which all particles remain waiting and so front propagation is not possible.…”

Mathematical Modelling of Subdiffusion-reaction Systems

“…The consequences of both assumptions are discussed in [29]. The microscopic simulations are in favor of the latter assumption [30], but generally the validity of both assumptions can depend on the particular physical problem.…”

“…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].…”

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

“…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.…”