Abstract
A random walk on a two dimensional square in R2 space with a hidden absorbing fractal set Fμ is considered. This search-like problem is treated in the framework of a diffusion–reaction equation, when an absorbing term is included inside a Fokker–Planck equation as a reaction term. This macroscopic approach for the 2D transport in the R2 space corresponds to the comb geometry, when the random walk consists of 1D movements in the x and y directions, respectively, as a direct-Cartesian product of the 1D movements. The main value in task is the first arrival time distribution (FATD) to sink points of the fractal set, where travelling particles are absorbed. Analytical expression for the FATD is obtained in the subdiffusive regime for both the fractal set of sinks and for a single sink.
Highlights
In this paper, we study a 2D continuous subdiffusive transport in the presence of an absorbing fractal set
A random walk on a two dimensional square in R2 space with a hidden absorbing fractal set Fμ is considered. This search-like problem is treated in the framework of a diffusion–reaction equation, when an absorbing term is included inside a Fokker–Planck equation as a reaction term
This macroscopic approach for the 2D transport in the R2 space corresponds to the comb geometry, when the random walk consists of 1D movements in the x and y directions, respectively, as a direct-Cartesian product of the 1D movements
Summary
We study a 2D continuous subdiffusive transport in the presence of an absorbing fractal set. An intermittent search process on a dendrite tree has been considered [11], while a relation between the comb model and a spine dendrite structure has been established as well [12,13,14,15] with further explorations of anomalous diffusion inside spine dendrites [16,17,18] Another important application relates to fabrication of fractal absorbers in metamaterials applied for antenna technology (see, e.g., [19] and references therein). Taking into account that the fractal set in continuous space is not a frozen-fixed structure, the limit μ → 0 is performed, and the FATD for a single sink is obtained in the form of the Fox H-functions.
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