Dear Colleagues,

Investigation of random processes in complex media has been attracting plenty of attention for years. Theoretical modeling of diffusion in heterogeneous and disordered media takes considerable part of these studies. Heterogeneous and disordered materials include various materials with defects, multi-scale amorphous composites, fractal and sparse structures, weighted graphs, and networks. Diffusion in such media with geometric constraints and random forces is often anomalous and is described by fractional calculus. Further development of the theoretical modeling of these random processes in a variety of realizations in physics, biology, social sciences, and finance is an essential part of modern studies, what we called complex systems.

New mathematical approaches shed light on many questions and also pose new ones. One such example is a random search process, whose systematic research stems from projects involving hunting for submarines, while the modern study of first-passage or hitting times covers a large area of search problems, from animal food foraging to molecular reactions and gene regulation. Moreover, random search processes in complex networks are important in order to understand animal food search strategies and improve web search engines, or to prolong or speed up survival times in first-encounter tasks.

Many of the aforementioned processes can be described by various random walk models,  as well as generalized (fractional) Fokker–Planck  and Langevin equations, which, in turn, may describe completely different problems with common features. In particular, a class of diffusion in the heterogeneous environment is closely connected to turbulent diffusion governed by inhomogeneous advection–diffusion equations, and also relates to the geometric Brownian motion, used to model stock prices.

The purpose of the Special Issue is to reflect current situation in fractional dynamics theory, and to collect various models for the description of anomalous diffusion and random walks in complex systems. We kindly invite researchers working in these fields to contribute with original research/review papers dedicated to theoretical modeling and applications.

Dr. Trifce Sandev
Guest Editor

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access quarterly journal published by MDPI.

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• anomalous diffusion and stochastic processes in complex systems
• diffusion and non-exponential relaxation in heterogeneous and disordered media
• diffusion in comb and fractal structures
• continuous time random walk
• random search processes and stochastic resetting
• fractional/generalized diffusion-wave equations and subordination
• fractional Brownian motion, Fokker–Planck equation, generalized Langevin equations
• fractional calculus and related special functions
• generalized geometric Brownian motion
• random walks on networks

1. Title: Dynamics Behind Stretched Exponential Behaviour

Authors: Katarzyna Gorska and Andrzej Horzela

2. Title: The Use of Fractals and Fractional Calculus in Physiology

Authors: Bruce West

3. Title: Quantum Recurrences in the Hilbert Space

Authors: Alexander Iomin

4. Title: Modeling of Phonon-Assisted Hopping in Fractal Nanosystems

Authors: Renat Sibatov and Alireza K. Golmankhaneh

5. Title: Diffusion in a Heterogenous Media and Sorption – Desorption Processes

Authors: A. Koltun, E. K. Lenzi, M. K. Lenzi, R. S. Zola

6. Title: Mean trapping time for an arbitrary site on a class of fractal scale-free trees

Authors: Long Gao, Junhao Peng, Chunming Tang

Abstract: In this paper, we study the discrete random walks on a class of fractal scale-free networks, the mean trapping time (MTT) for an arbitrary trap site is addressed analytically. Firstly, a method to label all the sites of the trees are presented. Then, a method, which bases on the connection between the Mean first-passage time and the effective resistances, to derive analytically the MTT for arbitrary trap site is also presented, some examples are also given to show the effectiveness of our method.
Finally, we compare the MTT for all the different trap sites of the tree, and find the site where the MTT achieves the minimum (or maximum) in the whole network.