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

Abstract

We consider the subdiffusion-reaction process with reactions of a type A+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 B reactions 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 between particles which continue to exist is taken into account. Although in both processes a probability of the vanishing of particle A due to a reaction is independent of both time and space variables (assuming that in the system with the A+B\arrow B reactions, particles B are distributed homogeneously) we show that subdiffusion-reaction equations describing these processes as well as their Greens' functions are qualitatively different. For the subdiffusion process with the A+B\arrow B reactions we consider three models. We base the method considered in this paper on a random walk model in a system with both discrete time and space variables. Then, the system with discrete variables is transformed into a system with continuous variables. Such a method seems to be convenient in analysing subdiffusion-reaction processes with partially absorbing or partially reflecting walls. The reason is that within this method we can determine Greens' functions without a necessity of solving a fractional differential subdiffusion-reaction equation with boundary conditions at the walls. As an example we use the model to find the Greens' functions for a subdiffusive-reaction system (with the reactions mentioned above), which is bounded by a partially absorbing wall. This example shows how the model can be used to analyze the subdiffusion-reaction process in a system with partially absorbing or reflecting thin membranes. Employing a simple phenomenological model, we derive equations related to the reaction parameters used in the considered models.