Subdiffusion--reaction process with $A\longrightarrow B$ reactions versus subdiffusion--reaction process with $A+B\longrightarrow B$ reactions

摘要

We consider the subdiffusion-reaction process with reactions of a typeA+B\arrow B (in which particles A are assumed to be mobile whereas B - static)in comparison to the subdiffusion-reaction process with A\rightarrow Breactions which was studied by I.M. Sokolov et al. in Phys. Rev. E 73, 031102(2006). In both processes a rule that reactions can only occur betweenparticles which continue to exist is taken into account. Although in bothprocesses a probability of the vanishing of particle A due to a reaction isindependent of both time and space variables (assuming that in the system withthe A+B\arrow B reactions, particles B are distributed homogeneously) we showthat subdiffusion-reaction equations describing these processes as well astheir Greens' functions are qualitatively different. For the subdiffusionprocess with the A+B\arrow B reactions we consider three models. We base themethod considered in this paper on a random walk model in a system with bothdiscrete time and space variables. Then, the system with discrete variables istransformed into a system with continuous variables. Such a method seems to beconvenient in analysing subdiffusion-reaction processes with partiallyabsorbing or partially reflecting walls. The reason is that within this methodwe can determine Greens' functions without a necessity of solving a fractionaldifferential subdiffusion-reaction equation with boundary conditions at thewalls. As an example we use the model to find the Greens' functions for asubdiffusive-reaction system (with the reactions mentioned above), which isbounded by a partially absorbing wall. This example shows how the model can beused to analyze the subdiffusion-reaction process in a system with partiallyabsorbing or reflecting thin membranes. Employing a simple phenomenologicalmodel, we derive equations related to the reaction parameters used in theconsidered models.