Fractional Dynamics: Theory and Applications

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Complexity".

Deadline for manuscript submissions: closed (22 January 2022) | Viewed by 23157

Special Issue Editor

1. Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia
2. Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University, Arhimedova 3, 1000 Skopje, Macedonia
3. Institute of Physics & Astronomy, University of Potsdam, D-14776 Potsdam, Germany
Interests: statistical mechanics; mathematical physics; stochastic processes; anomalous diffusion; fractional calculus
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

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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Keywords

  • 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

Published Papers (11 papers)

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Editorial

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3 pages, 180 KiB  
Editorial
Editorial for Special Issue “Fractional Dynamics: Theory and Applications”
by Trifce Sandev
Fractal Fract. 2022, 6(11), 668; https://doi.org/10.3390/fractalfract6110668 - 11 Nov 2022
Viewed by 983
Abstract
The investigation of fluctuations and random processes in complex systems and random environments has been attracting much attention for years [...] Full article
(This article belongs to the Special Issue Fractional Dynamics: Theory and Applications)

Research

Jump to: Editorial

23 pages, 883 KiB  
Article
Asymmetric Lévy Flights Are More Efficient in Random Search
by Amin Padash, Trifce Sandev, Holger Kantz, Ralf Metzler and Aleksei V. Chechkin
Fractal Fract. 2022, 6(5), 260; https://doi.org/10.3390/fractalfract6050260 - 8 May 2022
Cited by 7 | Viewed by 2378
Abstract
We study the first-arrival (first-hitting) dynamics and efficiency of a one-dimensional random search model performing asymmetric Lévy flights by leveraging the Fokker–Planck equation with a δ-sink and an asymmetric space-fractional derivative operator with stable index α and asymmetry (skewness) parameter β. [...] Read more.
We study the first-arrival (first-hitting) dynamics and efficiency of a one-dimensional random search model performing asymmetric Lévy flights by leveraging the Fokker–Planck equation with a δ-sink and an asymmetric space-fractional derivative operator with stable index α and asymmetry (skewness) parameter β. We find exact analytical results for the probability density of first-arrival times and the search efficiency, and we analyse their behaviour within the limits of short and long times. We find that when the starting point of the searcher is to the right of the target, random search by Brownian motion is more efficient than Lévy flights with β0 (with a rightward bias) for short initial distances, while for β>0 (with a leftward bias) Lévy flights with α1 are more efficient. When increasing the initial distance of the searcher to the target, Lévy flight search (except for α=1 with β=0) is more efficient than the Brownian search. Moreover, the asymmetry in jumps leads to essentially higher efficiency of the Lévy search compared to symmetric Lévy flights at both short and long distances, and the effect is more pronounced for stable indices α close to unity. Full article
(This article belongs to the Special Issue Fractional Dynamics: Theory and Applications)
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14 pages, 2799 KiB  
Article
Modelling of Electron and Thermal Transport in Quasi-Fractal Carbon Nitride Nanoribbons
by Renat T. Sibatov, Alireza Khalili Golmankhaneh, Ruslan M. Meftakhutdinov, Ekaterina V. Morozova and Diana A. Timkaeva
Fractal Fract. 2022, 6(2), 115; https://doi.org/10.3390/fractalfract6020115 - 15 Feb 2022
Cited by 3 | Viewed by 2066
Abstract
In this work, using calculations based on the density functional theory, molecular dynamics, non-equilibrium Green functions method, and Monte Carlo simulation, we study electronic and phonon transport in a device based on quasi-fractal carbon nitride nanoribbons with Sierpinski triangle blocks. Modifications of electronic [...] Read more.
In this work, using calculations based on the density functional theory, molecular dynamics, non-equilibrium Green functions method, and Monte Carlo simulation, we study electronic and phonon transport in a device based on quasi-fractal carbon nitride nanoribbons with Sierpinski triangle blocks. Modifications of electronic and thermal conductance with increase in generation g of quasi-fractal segments are estimated. Introducing energetic disorder, we study hopping electron transport in the quasi-fractal nanoribbons by Monte Carlo simulation of a biased random walk with generalized Miller–Abrahams transfer rates. Calculated time dependencies of the mean square displacement bear evidence of transient anomalous diffusion. Variations of anomalous drift-diffusion parameters with localization radius, temperature, electric field intensity, and energy disorder level are estimated. The hopping in quasi-fractal nanoribbons can serve as an explicit physical implementation of the generalized comb model. Full article
(This article belongs to the Special Issue Fractional Dynamics: Theory and Applications)
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23 pages, 874 KiB  
Article
Tuning of the Dielectric Relaxation and Complex Susceptibility in a System of Polar Molecules: A Generalised Model Based on Rotational Diffusion with Resetting
by Irina Petreska, Ljupco Pejov, Trifce Sandev, Ljupco Kocarev and Ralf Metzler
Fractal Fract. 2022, 6(2), 88; https://doi.org/10.3390/fractalfract6020088 - 5 Feb 2022
Cited by 5 | Viewed by 1839
Abstract
The application of the fractional calculus in the mathematical modelling of relaxation processes in complex heterogeneous media has attracted a considerable amount of interest lately. The reason for this is the successful implementation of fractional stochastic and kinetic equations in the studies of [...] Read more.
The application of the fractional calculus in the mathematical modelling of relaxation processes in complex heterogeneous media has attracted a considerable amount of interest lately. The reason for this is the successful implementation of fractional stochastic and kinetic equations in the studies of non-Debye relaxation. In this work, we consider the rotational diffusion equation with a generalised memory kernel in the context of dielectric relaxation processes in a medium composed of polar molecules. We give an overview of existing models on non-exponential relaxation and introduce an exponential resetting dynamic in the corresponding process. The autocorrelation function and complex susceptibility are analysed in detail. We show that stochastic resetting leads to a saturation of the autocorrelation function to a constant value, in contrast to the case without resetting, for which it decays to zero. The behaviour of the autocorrelation function, as well as the complex susceptibility in the presence of resetting, confirms that the dielectric relaxation dynamics can be tuned by an appropriate choice of the resetting rate. The presented results are general and flexible, and they will be of interest for the theoretical description of non-trivial relaxation dynamics in heterogeneous systems composed of polar molecules. Full article
(This article belongs to the Special Issue Fractional Dynamics: Theory and Applications)
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28 pages, 831 KiB  
Article
A Novel Analytical Formula for the Discounted Moments of the ECIR Process and Interest Rate Swaps Pricing
by Ratinan Boonklurb, Ampol Duangpan, Udomsak Rakwongwan and Phiraphat Sutthimat
Fractal Fract. 2022, 6(2), 58; https://doi.org/10.3390/fractalfract6020058 - 24 Jan 2022
Cited by 8 | Viewed by 2303
Abstract
This paper presents an explicit formula of conditional expectation for a product of polynomial functions and the discounted characteristic function based on the Cox–Ingersoll–Ross (CIR) process. We also propose an analytical formula as well as a very efficient and accurate approach, based on [...] Read more.
This paper presents an explicit formula of conditional expectation for a product of polynomial functions and the discounted characteristic function based on the Cox–Ingersoll–Ross (CIR) process. We also propose an analytical formula as well as a very efficient and accurate approach, based on the finite integration method with shifted Chebyshev polynomial, to evaluate this expectation under the Extended CIR (ECIR) process. The formulas are derived by solving the equivalent partial differential equations obtained by utilizing the Feynman–Kac representation. In addition, we extend our results to derive an analytical formula of conditional expectation of a product of mixed polynomial functions and the discounted characteristic function. The accuracy and efficiency of the proposed scheme are also numerically shown for various modeling parameters by comparing them with those obtained from Monte Carlo simulations. In addition, to illustrate applications of the obtained formulas in finance, analytical pricing formulas for arrears and vanilla interest rate swaps under the ECIR process are derived. The pricing formulas become explicit under the CIR process. Finally, the fractional ECIR process is also studied as an extended case of our main results. Full article
(This article belongs to the Special Issue Fractional Dynamics: Theory and Applications)
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20 pages, 495 KiB  
Article
Non-Debye Relaxations: The Ups and Downs of the Stretched Exponential vs. Mittag–Leffler’s Matchings
by Katarzyna Górska, Andrzej Horzela and Karol A. Penson
Fractal Fract. 2021, 5(4), 265; https://doi.org/10.3390/fractalfract5040265 - 7 Dec 2021
Cited by 6 | Viewed by 2044
Abstract
Experimental data collected to provide us with information on the course of dielectric relaxation phenomena are obtained according to two distinct schemes: one can measure either the time decay of depolarization current or use methods of the broadband dielectric spectroscopy. Both sets of [...] Read more.
Experimental data collected to provide us with information on the course of dielectric relaxation phenomena are obtained according to two distinct schemes: one can measure either the time decay of depolarization current or use methods of the broadband dielectric spectroscopy. Both sets of data are usually fitted by time or frequency dependent functions which, in turn, may be analytically transformed among themselves using the Laplace transform. This leads to the question on comparability of results obtained using just mentioned experimental procedures. If we would like to do that in the time domain we have to go beyond widely accepted Kohlrausch–Williams–Watts approximation and become acquainted with description using the Mittag–Leffler functions. To convince the reader that the latter is not difficult to understand we propose to look at the problem from the point of view of objects which appear in the stochastic processes approach to relaxation. These are the characteristic exponents which are read out from the standard non-Debye frequency dependent patterns. Characteristic functions appear to be expressed in terms of elementary functions whose asymptotics is simple. This opens new possibility to compare behavior of functions used to describe non-Debye relaxations. It turnes out that the use of Mittag-Leffler function proves very convenient for such a comparison. Full article
(This article belongs to the Special Issue Fractional Dynamics: Theory and Applications)
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19 pages, 645 KiB  
Article
Optimizing the First-Passage Process on a Class of Fractal Scale-Free Trees
by Long Gao, Junhao Peng and Chunming Tang
Fractal Fract. 2021, 5(4), 184; https://doi.org/10.3390/fractalfract5040184 - 25 Oct 2021
Cited by 6 | Viewed by 1527
Abstract
First-passage processes on fractals are of particular importance since fractals are ubiquitous in nature, and first-passage processes are fundamental dynamic processes that have wide applications. The global mean first-passage time (GMFPT), which is the expected time for a walker (or a particle) to [...] Read more.
First-passage processes on fractals are of particular importance since fractals are ubiquitous in nature, and first-passage processes are fundamental dynamic processes that have wide applications. The global mean first-passage time (GMFPT), which is the expected time for a walker (or a particle) to first reach the given target site while the probability distribution for the position of target site is uniform, is a useful indicator for the transport efficiency of the whole network. The smaller the GMFPT, the faster the mass is transported on the network. In this work, we consider the first-passage process on a class of fractal scale-free trees (FSTs), aiming at speeding up the first-passage process on the FSTs. Firstly, we analyze the global mean first-passage time (GMFPT) for unbiased random walks on the FSTs. Then we introduce proper weight, dominated by a parameter w (w > 0), to each edge of the FSTs and construct a biased random walks strategy based on these weights. Next, we analytically evaluated the GMFPT for biased random walks on the FSTs. The exact results of the GMFPT for unbiased and biased random walks on the FSTs are both obtained. Finally, we view the GMFPT as a function of parameter w and find the point where the GMFPT achieves its minimum. The exact result is obtained and a way to optimize and speed up the first-passage process on the FSTs is presented. Full article
(This article belongs to the Special Issue Fractional Dynamics: Theory and Applications)
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12 pages, 1481 KiB  
Article
Diffusion in Heterogenous Media and Sorption—Desorption Processes
by Ana Paula S. Koltun, Ervin Kaminski Lenzi, Marcelo Kaminski Lenzi and Rafael Soares Zola
Fractal Fract. 2021, 5(4), 183; https://doi.org/10.3390/fractalfract5040183 - 25 Oct 2021
Cited by 5 | Viewed by 1527
Abstract
We investigate particle diffusion in a heterogeneous medium limited by a surface where sorption–desorption processes are governed by a kinetic equation. We consider that the dynamics of the particles present in the medium are governed by a diffusion equation with a spatial dependence [...] Read more.
We investigate particle diffusion in a heterogeneous medium limited by a surface where sorption–desorption processes are governed by a kinetic equation. We consider that the dynamics of the particles present in the medium are governed by a diffusion equation with a spatial dependence on the diffusion coefficient, i.e., K(x) = D|x|η, with −1 < η and D = const, respectively. This system is analyzed in a semi-infinity region, i.e., the system is defined in the interval [0,∞) for an arbitrary initial condition. The solutions are obtained and display anomalous spreading, that is, the dynamics may be viewed as anomalous diffusion, which in turn is related, and hence, the model can be directly applied to several complex systems ranging from biological fluids to electrolytic cells. Full article
(This article belongs to the Special Issue Fractional Dynamics: Theory and Applications)
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14 pages, 344 KiB  
Article
Quantum Walks in Hilbert Space of Lévy Matrices: Recurrences and Revivals
by Alexander Iomin
Fractal Fract. 2021, 5(4), 171; https://doi.org/10.3390/fractalfract5040171 - 18 Oct 2021
Cited by 1 | Viewed by 1205
Abstract
The quantum evolution of wave functions controlled by the spectrum of Lévy random matrices is considered. An analytical treatment of quantum recurrences and revivals in the Hilbert space is performed in the framework of a theory of almost periodic functions. It is shown [...] Read more.
The quantum evolution of wave functions controlled by the spectrum of Lévy random matrices is considered. An analytical treatment of quantum recurrences and revivals in the Hilbert space is performed in the framework of a theory of almost periodic functions. It is shown that the statistics of quantum recurrences in the Hilbert space of quantum systems is sensitive to the statistics of the corresponding quantum spectrum. In particular, it is shown that both the Poisson energy level statistics and the Brody distribution correspond to the power law of the quantum recurrences, while the Wigner–Dyson and Lévy–Smirnov statistics of the energy spectra are responsible for the exponential statistics of the quantum returns of the wave function. Full article
(This article belongs to the Special Issue Fractional Dynamics: Theory and Applications)
14 pages, 338 KiB  
Article
An Inverse Source Problem for the Generalized Subdiffusion Equation with Nonclassical Boundary Conditions
by Emilia Bazhlekova
Fractal Fract. 2021, 5(3), 63; https://doi.org/10.3390/fractalfract5030063 - 30 Jun 2021
Cited by 4 | Viewed by 1762
Abstract
An initial-boundary-value problem is considered for the one-dimensional diffusion equation with a general convolutional derivative in time and nonclassical boundary conditions. We are concerned with the inverse source problem of recovery of a space-dependent source term from given final time data. Generalized eigenfunction [...] Read more.
An initial-boundary-value problem is considered for the one-dimensional diffusion equation with a general convolutional derivative in time and nonclassical boundary conditions. We are concerned with the inverse source problem of recovery of a space-dependent source term from given final time data. Generalized eigenfunction expansions are used with respect to a biorthogonal pair of bases. Existence, uniqueness and stability estimates in Sobolev spaces are established. Full article
(This article belongs to the Special Issue Fractional Dynamics: Theory and Applications)
11 pages, 556 KiB  
Article
Subdiffusive Reaction Model of Molecular Species in Liquid Layers: Fractional Reaction-Telegraph Approach
by Ashraf M. Tawfik and Mohamed Mokhtar Hefny
Fractal Fract. 2021, 5(2), 51; https://doi.org/10.3390/fractalfract5020051 - 3 Jun 2021
Cited by 5 | Viewed by 2099
Abstract
In recent years, different experimental works with molecular simulation techniques have been developed to study the transport of plasma-generated reactive species in liquid layers. Here, we improve the classical transport model that describes the molecular species movement in liquid layers via considering the [...] Read more.
In recent years, different experimental works with molecular simulation techniques have been developed to study the transport of plasma-generated reactive species in liquid layers. Here, we improve the classical transport model that describes the molecular species movement in liquid layers via considering the fractional reaction–telegraph equation. We have considered the fractional equation to describe a non-Brownian motion of molecular species in a liquid layer, which have different diffusivities. The analytical solution of the fractional reaction–telegraph equation, which is defined in terms of the Caputo fractional derivative, is obtained by using the Laplace–Fourier technique. The profiles of species density with the mean square displacement are discussed in each case for different values of the time-fractional order and relaxation time. Full article
(This article belongs to the Special Issue Fractional Dynamics: Theory and Applications)
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