Fractional Diffusion to a Cantor Set in 2D

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.

Quantum dynamics and relaxation in comb turbulent diffusion

“…Then, in the backbone's transport equation, the Lévy-Smirnov distribution P y (y 0 , t) plays a role of a strength of a δ sourcesink term at x = x 0 . This situation differs from both scenarios considered above and has been discussed for anomalous diffusion and random search in two and three dimensional combs [30,31].…”

General characteristics of a fractal Sturm–Liouville problem

“…and they are often too irregular to have any smooth differentiable structure defined on them, which results in delivering the methods and techniques of ordinary calculus powerless and inapplicable. Some approaches have been developed to deal with this inapplicability by means of fractional derivatives [6,10,21,23,24,34,38,42], fractional spaces [1,7,20,35], harmonic analysis [2,5,8,22,25,37,39], and measure theory, non-standard methods, and stochastic process [3, 29-31, 36, 40]. Yet, there has still been a gap in the literature on how to develop an appropriate calculus mainly related to the disconnected fractal subsets of R .…”