title: Subdiffusion-reaction processes with A→B reactions versus subdiffusion-reaction processes with A+B→B reactions.
abstract: We consider the subdiffusion-reaction process with reactions of a type A+B→B (in which particles A are assumed to be mobile, whereas B are assumed to be static) in comparison to the subdiffusion-reaction process with A→B reactions which was studied by Sokolov, Schmidt, and Sagués [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→B reactions, particles B are distributed homogeneously), we show that subdiffusion-reaction equations describing these processes as well as their Green's functions are qualitatively different. The reason for this difference is as follows. In the case of the former reaction, particles A and B have to meet with some probability before the reaction occurs in contradiction with the case of the latter reaction. For the subdiffusion process with the A+B→B reactions we consider three models which differ in some details concerning a description of the reactions. We base the method considered in this paper on a random walk model in a system with both discrete time and discrete space variables. Then the system with discrete variables is transformed into a system with both continuous time and continuous space variables. Such a method seems to be convenient in analyzing subdiffusion-reaction processes with partially absorbing or partially reflecting walls. The reason is that within this method we can determine Green's 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 Green's 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 also derive equations related to the reaction parameters used in the considered models.
authors.affiliation: Institute of Physics, Jan Kochanowski University, ulica Świȩtokrzyska 15, 25-406 Kielce, Poland.
authors.forename: Tadeusz
authors.initials: T
authors.lastname: Kosztołowicz
authors.affiliation: Department of Radiological Informatics and Statistics, Medical University of Gdańsk, ulica Tuwima 15, 80-210 Gdańsk, Poland.
authors.forename: Katarzyna D
authors.initials: KD
authors.lastname: Lewandowska
mesh_terms: D004058:Diffusion; D008956:Models, Chemical; D009038:Motion; D013269:Stochastic Processes