Trifce Sandev

Recent developments in modeling of different physical and biological systems undoubtedly confirmed that fractional calculus is not just an exotic mathematical theory, as it might appeared at the dawn of its emergence. The present work aims to demonstrate this through physical examples of diffusion and quantum motion in two-dimensional comb-like structures, leading to fractional calculus description of the corresponding processes. Comb geometry is one of the most simple paradigm of a two-dimensional structure, where anomalous diffusion can be realized. The comb model, for the first time, has been applied to investigate anomalous diffusion in low-dimensional percolation clusters, and nowadays it has many applications in description of transport properties in porous dielectrics and low dimensional composites, turbulent diffusion on a comb, tracer dynamics in subsurface aquifers, and anomalous diffusion in spiny dendrites, to name but a few. In this presentation we will show how the time fractional diffusion and Schrödinger equations can be derived by projection of the two dimensional comb dynamics in the one-dimensional configuration space. The anomalous character of the corresponding process can be described by the fact that the particle which moves along the main backbone channel can get trapped in the fingers of the comb. We will also show how the anomalous diffusion along the main backbone depends on the fractal structure of the comb, resulting in natural appearance of fractional derivatives in the corresponding equations for the probability density function. The connection between the anomalous diffusion exponent and the fractal dimension of the comb/mesh structures has been recently experimentally observed in the anomalous transport through porous structurally inhomogeneous media.