Fractional master equations and fractal time random walks

A Generalized Kinetic Model of Fractional Order Transport Dynamics with Transit Time Heterogeneity in Microvascular Space

The aim of this study is to develop and validate a unifying kinetic model for microvascular transport by introducing an impulse response function that incorporates essential physiological parameters and integrates key features of existing models. This new methodology combines a one-compartment model of fractional order with a model that uses the gamma distribution to describe the distribution of capillary transit times. Central to this model are two primary parameters: [Formula: see text], representing the kurtosis of residue times, and [Formula: see text], signifying the width of the distribution of capillary transit times within a tissue voxel. To validate this proposed model, data from dynamic contrast-enhanced magnetic resonance imaging (DCI-MRI) were employed and the findings were compared with three existing models. Using the Akaike information criterion for model selection, the results demonstrate that the integrative model, especially at elevated blood flow rates, frequently offers superior fits in comparison to constrained models.

Random Walks on Comb-like Structures under Stochastic Resetting

We study the long-time dynamics of the mean squared displacement of a random walker moving on a comb structure under the effect of stochastic resetting. We consider that the walker's motion along the backbone is diffusive and it performs short jumps separated by random resting periods along fingers. We take into account two different types of resetting acting separately: global resetting from any point in the comb to the initial position and resetting from a finger to the corresponding backbone. We analyze the interplay between the waiting process and Markovian and non-Markovian resetting processes on the overall mean squared displacement. The Markovian resetting from the fingers is found to induce normal diffusion, thereby minimizing the trapping effect of fingers. In contrast, for non-Markovian local resetting, an interesting crossover with three different regimes emerges, with two of them subdiffusive and one of them diffusive. Thus, an interesting interplay between the exponents characterizing the waiting time distributions of the subdiffusive random walk and resetting takes place. As for global resetting, its effect is even more drastic as it precludes normal diffusion. Specifically, such a resetting can induce a constant asymptotic mean squared displacement in the Markovian case or two distinct regimes of subdiffusive motion in the non-Markovian case.

Time-fractional Caputo derivative versus other integrodifferential operators in generalized Fokker-Planck and generalized Langevin equations

Fractional diffusion and Fokker-Planck equations are widely used tools to describe anomalous diffusion in a large variety of complex systems. The equivalent formulations in terms of Caputo or Riemann-Liouville fractional derivatives can be derived as continuum limits of continuous-time random walks and are associated with the Mittag-Leffler relaxation of Fourier modes, interpolating between a short-time stretched exponential and a long-time inverse power-law scaling. More recently, a number of other integrodifferential operators have been proposed, including the Caputo-Fabrizio and Atangana-Baleanu forms. Moreover, the conformable derivative has been introduced. We study here the dynamics of the associated generalized Fokker-Planck equations from the perspective of the moments, the time-averaged mean-squared displacements, and the autocovariance functions. We also study generalized Langevin equations based on these generalized operators. The differences between the Fokker-Planck and Langevin equations with different integrodifferential operators are discussed and compared with the dynamic behavior of established models of scaled Brownian motion and fractional Brownian motion. We demonstrate that the integrodifferential operators with exponential and Mittag-Leffler kernels are not suitable to be introduced to Fokker-Planck and Langevin equations for the physically relevant diffusion scenarios discussed in our paper. The conformable and Caputo Langevin equations are unveiled to share similar properties with scaled and fractional Brownian motion, respectively.

Subdiffusion with particle immobilization process described by a differential equation with Riemann-Liouville-type fractional time derivative

An equation describing subdiffusion with possible immobilization of particles is derived by means of the continuous time random walk model. The equation contains a fractional time derivative of Riemann-Liouville type which is a differential-integral operator with the kernel defined by the Laplace transform; the kernel controls the immobilization process. We propose a method for calculating the inverse Laplace transform providing the kernel in the time domain. In the long time limit the subdiffusion-immobilization process reaches a stationary state in which the probability density of a particle distribution is an exponential function.

Subdiffusion equation with fractional Caputo time derivative with respect to another function in modeling transition from ordinary subdiffusion to superdiffusion

We use a subdiffusion equation with fractional Caputo time derivative with respect to another function g (g-subdiffusion equation) to describe a smooth transition from ordinary subdiffusion to superdiffusion. Ordinary subdiffusion is described by the equation with the "ordinary" fractional Caputo time derivative, superdiffusion is described by the equation with a fractional Riesz-type spatial derivative. We find the function g for which the solution (Green's function, GF) to the g-subdiffusion equation takes the form of GF for ordinary subdiffusion in the limit of small time and GF for superdiffusion in the limit of long time. To solve the g-subdiffusion equation we use the g-Laplace transform method. It is shown that the scaling properties of the GF for g-subdiffusion and the GF for superdiffusion are the same in the long time limit. We conclude that for a sufficiently long time the g-subdiffusion equation describes superdiffusion well, despite a different stochastic interpretation of the processes. Then, paradoxically, a subdiffusion equation with a fractional time derivative describes superdiffusion. The superdiffusive effect is achieved here not by making anomalously long jumps by a diffusing particle, but by greatly increasing the particle jump frequency which is derived by means of the g-continuous-time random walk model. The g-subdiffusion equation is shown to be quite general, it can be used in modeling of processes in which a kind of diffusion change continuously over time. In addition, some methods used in modeling of ordinary subdiffusion processes, such as the derivation of local boundary conditions at a thin partially permeable membrane, can be used to model g-subdiffusion processes, even if this process is interpreted as superdiffusion.

Composite subdiffusion equation that describes transient subdiffusion

A composite subdiffusion equation with fractional Caputo time derivative with respect to another function g is used to describe a process of a continuous transition from subdiffusion with parameters α and D_{α} to subdiffusion with parameters β and D_{β}. The parameters are defined by the time evolution of the mean square displacement of diffusing particle σ^{2}(t)=2D_{i}t^{i}/Γ(1+i), i=α,β. The function g controls the process at intermediate times. The composite subdiffusion equation is more general than the ordinary fractional subdiffusion equation with constant parameters; it has potentially wide application in modeling diffusion processes with changing parameters.

An Analytical Approach for Fractional Hyperbolic Telegraph Equation Using Shehu Transform in One, Two and Three Dimensions

In the present research paper, an iterative approach named the iterative Shehu transform method is implemented to solve time-fractional hyperbolic telegraph equations in one, two, and three dimensions, respectively. These equations are the prominent ones in the field of physics and in some other significant problems. The efficacy and authenticity of the proposed method are tested using a comparison of approximated and exact results in graphical form. Both 2D and 3D plots are provided to affirm the compatibility of approximated-exact results. The iterative Shehu transform method is a reliable and efficient tool to provide approximated and exact results to a vast class of ODEs, PDEs, and fractional PDEs in a simplified way, without any discretization or linearization, and is free of errors. A convergence analysis is also provided in this research.

Rate equations, spatial moments, and concentration profiles for mobile-immobile models with power-law and mixed waiting time distributions

We present a framework for systems in which diffusion-advection transport of a tracer substance in a mobile zone is interrupted by trapping in an immobile zone. Our model unifies different model approaches based on distributed-order diffusion equations, exciton diffusion rate models, and random-walk models for multirate mobile-immobile mass transport. We study various forms for the trapping time dynamics and their effects on the tracer mass in the mobile zone. Moreover, we find the associated breakthrough curves, the tracer density at a fixed point in space as a function of time, and the mobile and immobile concentration profiles and the respective moments of the transport. Specifically, we derive explicit forms for the anomalous transport dynamics and an asymptotic power-law decay of the mobile mass for a Mittag-Leffler trapping time distribution. In our analysis we point out that even for exponential trapping time densities, transient anomalous transport is observed. Our results have direct applications in geophysical contexts, but also in biological, soft matter, and solid state systems.

A Survey of Some Recent Developments on Higher Transcendental Functions of Analytic Number Theory and Applied Mathematics

Often referred to as special functions or mathematical functions, the origin of many members of the remarkably vast family of higher transcendental functions can be traced back to such widespread areas as (for example) mathematical physics, analytic number theory and applied mathematical sciences. Here, in this survey-cum-expository review article, we aim at presenting a brief introductory overview and survey of some of the recent developments in the theory of several extensively studied higher transcendental functions and their potential applications. For further reading and researching by those who are interested in pursuing this subject, we have chosen to provide references to various useful monographs and textbooks on the theory and applications of higher transcendental functions. Some operators of fractional calculus, which are associated with higher transcendental functions, together with their applications, have also been considered. Many of the higher transcendental functions, especially those of the hypergeometric type, which we have investigated in this survey-cum-expository review article, are known to display a kind of symmetry in the sense that they remain invariant when the order of the numerator parameters or when the order of the denominator parameters is arbitrarily changed.

Stochastic interpretation of g-subdiffusion process

Recently, we considered the g-subdiffusion equation with a fractional Caputo time derivative with respect to another function g, T. Kosztołowicz et al. [Phys. Rev. E 104, 014118 (2021)2470-004510.1103/PhysRevE.104.014118]. This equation offers different possibilities for modeling diffusion such as a process in which a type of diffusion evolves continuously over time. However, the equation has not been derived from a stochastic model and the stochastic interpretation of g subdiffusion is still unknown. In this Letter, we show the stochastic foundations of this process. We derive the equation by means of a modified continuous time random walk model. An interpretation of the g-subdiffusion process is also discussed.

On Weighted (k, s)-Riemann-Liouville Fractional Operators and Solution of Fractional Kinetic Equation

In this article, we establish the weighted (k,s)-Riemann-Liouville fractional integral and differential operators. Some certain properties of the operators and the weighted generalized Laplace transform of the new operators are part of the paper. The article consists of Chebyshev-type inequalities involving a weighted fractional integral. We propose an integro-differential kinetic equation using the novel fractional operators and find its solution by applying weighted generalized Laplace transforms.

Subdiffusion equation with Caputo fractional derivative with respect to another function

We show an application of a subdiffusion equation with Caputo fractional time derivative with respect to another function g to describe subdiffusion in a medium having a structure evolving over time. In this case a continuous transition from subdiffusion to other type of diffusion may occur. The process can be interpreted as "ordinary" subdiffusion with fixed subdiffusion parameter (subdiffusion exponent) α in which timescale is changed by the function g. As an example, we consider the transition from "ordinary" subdiffusion to ultraslow diffusion. The g-subdiffusion process generates the additional aging process superimposed on the "standard" aging generated by "ordinary" subdiffusion. The aging process is analyzed using coefficient of relative aging of g-subdiffusion with respect to "ordinary" subdiffusion. The method of solving the g-subdiffusion equation is also presented.

System of Time Fractional Models for COVID-19: Modeling, Analysis and Solutions

The emergence of the COVID-19 outbreak has caused a pandemic situation in over 210 countries. Controlling the spread of this disease has proven difficult despite several resources employed. Millions of hospitalizations and deaths have been observed, with thousands of cases occurring daily with many measures in place. Due to the complex nature of COVID-19, we proposed a system of time-fractional equations to better understand the transmission of the disease. Non-locality in the model has made fractional differential equations appropriate for modeling. Solving these types of models is computationally demanding. Our proposed generalized compartmental COVID-19 model incorporates effective contact rate, transition rate, quarantine rate, disease-induced death rate, natural death rate, natural recovery rate, and recovery rate of quarantine infected for a holistic study of the coronavirus disease. A detailed analysis of the proposed model is carried out, including the existence and uniqueness of solutions, local and global stability analysis of the disease-free equilibrium (symmetry), and sensitivity analysis. Furthermore, numerical solutions of the proposed model are obtained with the generalized Adam–Bashforth–Moulton method developed for the fractional-order model. Our analysis and solutions profile show that each of these incorporated parameters is very important in controlling the spread of COVID-19. Based on the results with different fractional-order, we observe that there seems to be a third or even fourth wave of the spike in cases of COVID-19, which is currently occurring in many countries.

Relation between generalized diffusion equations and subordination schemes

Generalized (non-Markovian) diffusion equations with different memory kernels and subordination schemes based on random time change in the Brownian diffusion process are popular mathematical tools for description of a variety of non-Fickian diffusion processes in physics, biology, and earth sciences. Some of such processes (notably, the fluid limits of continuous time random walks) allow for either kind of description, but other ones do not. In the present work we discuss the conditions under which a generalized diffusion equation does correspond to a subordination scheme, and the conditions under which a subordination scheme does possess the corresponding generalized diffusion equation. Moreover, we discuss examples of random processes for which only one, or both kinds of description are applicable.

Fractal Stochastic Processes on Thin Cantor-Like Sets

We review the basics of fractal calculus, define fractal Fourier transformation on thin Cantor-like sets and introduce fractal versions of Brownian motion and fractional Brownian motion. Fractional Brownian motion on thin Cantor-like sets is defined with the use of non-local fractal derivatives. The fractal Hurst exponent is suggested, and its relation with the order of non-local fractal derivatives is established. We relate the Gangal fractal derivative defined on a one-dimensional stochastic fractal to the fractional derivative after an averaging procedure over the ensemble of random realizations. That means the fractal derivative is the progenitor of the fractional derivative, which arises if we deal with a certain stochastic fractal.

Why Do Big Data and Machine Learning Entail the Fractional Dynamics?

Fractional-order calculus is about the differentiation and integration of non-integer orders. Fractional calculus (FC) is based on fractional-order thinking (FOT) and has been shown to help us to understand complex systems better, improve the processing of complex signals, enhance the control of complex systems, increase the performance of optimization, and even extend the enabling of the potential for creativity. In this article, the authors discuss the fractional dynamics, FOT and rich fractional stochastic models. First, the use of fractional dynamics in big data analytics for quantifying big data variability stemming from the generation of complex systems is justified. Second, we show why fractional dynamics is needed in machine learning and optimal randomness when asking: "is there a more optimal way to optimize?". Third, an optimal randomness case study for a stochastic configuration network (SCN) machine-learning method with heavy-tailed distributions is discussed. Finally, views on big data and (physics-informed) machine learning with fractional dynamics for future research are presented with concluding remarks.

Continuous time random walks under Markovian resetting

We investigate the effects of Markovian resetting events on continuous time random walks where the waiting times and the jump lengths are random variables distributed according to power-law probability density functions. We prove the existence of a nonequilibrium stationary state and finite mean first arrival time. However, the existence of an optimum reset rate is conditioned to a specific relationship between the exponents of both power-law tails. We also investigate the search efficiency by finding the optimal random walk which minimizes the mean first arrival time in terms of the reset rate, the distance of the initial position to the target, and the characteristic transport exponents.

A fractional order approach to modeling and simulations of the novel COVID-19

The novel coronavirus (SARS-CoV-2), or COVID-19, has emerged and spread at fast speed globally; the disease has become an unprecedented threat to public health worldwide. It is one of the greatest public health challenges in modern times, with no proven cure or vaccine. In this paper, our focus is on a fractional order approach to modeling and simulations of the novel COVID-19. We introduce a fractional type susceptible-exposed-infected-recovered (SEIR) model to gain insight into the ongoing pandemic. Our proposed model incorporates transmission rate, testing rates, and transition rate (from asymptomatic to symptomatic population groups) for a holistic study of the coronavirus disease. The impacts of these parameters on the dynamics of the solution profiles for the disease are simulated and discussed in detail. Furthermore, across all the different parameters, the effects of the fractional order derivative are also simulated and discussed in detail. Various simulations carried out enable us gain deep insights into the dynamics of the spread of COVID-19. The simulation results confirm that fractional calculus is an appropriate tool in modeling the spread of a complex infectious disease such as the novel COVID-19. In the absence of vaccine and treatment, our analysis strongly supports the significance reduction in the transmission rate as a valuable strategy to curb the spread of the virus. Our results suggest that tracing and moving testing up has an important benefit. It reduces the number of infected individuals in the general public and thereby reduces the spread of the pandemic. Once the infected individuals are identified and isolated, the interaction between susceptible and infected individuals diminishes and transmission reduces. Furthermore, aggressive testing is also highly recommended.

Why the Mittag-Leffler Function Can Be Considered the Queen Function of the Fractional Calculus?

In this survey we stress the importance of the higher transcendental Mittag-Leffler function in the framework of the Fractional Calculus. We first start with the analytical properties of the classical Mittag-Leffler function as derived from being the solution of the simplest fractional differential equation governing relaxation processes. Through the sections of the text we plan to address the reader in this pathway towards the main applications of the Mittag-Leffler function that has induced us in the past to define it as the Queen Function of the Fractional Calculus. These applications concern some noteworthy stochastic processes and the time fractional diffusion-wave equation We expect that in the next future this function will gain more credit in the science of complex systems. Finally, in an appendix we sketch some historical aspects related to the author’s acquaintance with this function.

Biased Continuous-Time Random Walks with Mittag-Leffler Jumps

We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Poisson process as a strictly increasing walk with discrete Mittag-Leffler jumps time-changed with an independent (continuous-time) fractional Poisson process. We call this process ‘space-time Mittag-Leffler process’. We derive explicit formulae for the state probabilities which solve a Cauchy problem with a Kolmogorov-Feller (forward) difference-differential equation of general fractional type. We analyze a “well-scaled” diffusion limit and obtain a Cauchy problem with a space-time convolution equation involving Mittag-Leffler densities. We deduce in this limit the ‘state density kernel’ solving this Cauchy problem. It turns out that the diffusion limit exhibits connections to Prabhakar general fractional calculus. We also analyze in this way a generalization of the space-time Mittag-Leffler process. The approach of constructing good Laplacian generator functions has a large potential in applications of space-time generalizations of the Poisson process and in the field of continuous-time random walks on digraphs.

Reaction-diffusion and reaction-subdiffusion equations on arbitrarily evolving domains

Reaction-diffusion equations are widely used as the governing evolution equations for modeling many physical, chemical, and biological processes. Here we derive reaction-diffusion equations to model transport with reactions on a one-dimensional domain that is evolving. The model equations, which have been derived from generalized continuous time random walks, can incorporate complexities such as subdiffusive transport and inhomogeneous domain stretching and shrinking. Inhomogeneously growing domains are frequently encountered in biological phenomena involving stochastic transport, such as tumor growth and morphogen gradient formation. A method for constructing analytic expressions for short-time moments of the position of the particles is developed and moments calculated from this approach are shown to compare favorably with results from random walk simulations and numerical integration of the reaction transport equation. The results show the important role played by the initial condition. In particular, it strongly affects the time dependence of the moments in the short-time regime by introducing additional drift and diffusion terms. We also discuss how our reaction transport equation could be applied to study the spreading of a population on an evolving interface. From a more general perspective, our findings help to mitigate the scarcity of analytic results for reaction-diffusion problems in geometries displaying nonuniform growth. They are also expected to pave the way for further results, including the treatment of first-passage problems associated with encounter-controlled reactions in such domains.

Time Fractional Fisher-KPP and Fitzhugh-Nagumo Equations

A standard reaction-diffusion equation consists of two additive terms, a diffusion term and a reaction rate term. The latter term is obtained directly from a reaction rate equation which is itself derived from known reaction kinetics, together with modelling assumptions such as the law of mass action for well-mixed systems. In formulating a reaction-subdiffusion equation, it is not sufficient to know the reaction rate equation. It is also necessary to know details of the reaction kinetics, even in well-mixed systems where reactions are not diffusion limited. This is because, at a fundamental level, birth and death processes need to be dealt with differently in subdiffusive environments. While there has been some discussion of this in the published literature, few examples have been provided, and there are still very many papers being published with Caputo fractional time derivatives simply replacing first order time derivatives in reaction-diffusion equations. In this paper, we formulate clear examples of reaction-subdiffusion systems, based on; equal birth and death rate dynamics, Fisher-Kolmogorov, Petrovsky and Piskunov (Fisher-KPP) equation dynamics, and Fitzhugh-Nagumo equation dynamics. These examples illustrate how to incorporate considerations of reaction kinetics into fractional reaction-diffusion equations. We also show how the dynamics of a system with birth rates and death rates cancelling, in an otherwise subdiffusive environment, are governed by a mass-conserving tempered time fractional diffusion equation that is subdiffusive for short times but standard diffusion for long times.

Space-Time Inversion of Stochastic Dynamics

This article introduces the concept of space-time inversion of stochastic Langevin equations as a way of transforming the parametrization of the dynamics from time to a monotonically varying spatial coordinate. A typical physical problem in which this approach can be fruitfully used is the analysis of solute dispersion in long straight tubes (Taylor-Aris dispersion), where the time-parametrization of the dynamics is recast in that of the axial coordinate. This allows the connection between the analysis of the forward (in time) evolution of the process and that of its exit-time statistics. The derivation of the Fokker-Planck equation for the inverted dynamics requires attention: it can be deduced using a mollified approach of the Wiener perturbations “a-la Wong-Zakai” by considering a sequence of almost everywhere smooth stochastic processes (in the present case, Poisson-Kac processes), converging to the Wiener processes in some limit (the Kac limit). The mathematical interpretation of the resulting Fokker-Planck equation can be obtained by introducing a new way of considering the stochastic integrals over the increments of a Wiener process, referred to as stochastic Stjelties integrals of mixed order. Several examples ranging from stochastic thermodynamics and fractal-time models are also analyzed.

Quenched and annealed disorder mechanisms in comb models with fractional operators

Recent experimental findings on anomalous diffusion have demanded novel models that combine annealed (temporal) and quenched (spatial or static) disorder mechanisms. The comb model is a simplified description of diffusion on percolation clusters, where the comblike structure mimics quenched disorder mechanisms and yields a subdiffusive regime. Here we extend the comb model to simultaneously account for quenched and annealed disorder mechanisms. To do so, we replace usual derivatives in the comb diffusion equation by different fractional time-derivative operators and the conventional comblike structure by a generalized fractal structure. Our hybrid comb models thus represent a diffusion where different comblike structures describe different quenched disorder mechanisms, and the fractional operators account for various annealed disorder mechanisms. We find exact solutions for the diffusion propagator and mean square displacement in terms of different memory kernels used for defining the fractional operators. Among other findings, we show that these models describe crossovers from subdiffusion to Brownian or confined diffusions, situations emerging in empirical results. These results reveal the critical role of interactions between geometrical restrictions and memory effects on modeling anomalous diffusion.

Analogues to Lie Method and Noether’s Theorem in Fractal Calculus

In this manuscript, we study symmetries of fractal differential equations. We show that using symmetry properties, one of the solutions can map to another solution. We obtain canonical coordinate systems for differential equations on fractal sets, which makes them simpler to solve. An analogue for Noether’s Theorem on fractal sets is given, and a corresponding conservative quantity is suggested. Several examples are solved to illustrate the results.

Fractional electron transfer kinetics and a quantum breaking of ergodicity

The dissipative curve-crossing problem provides a paradigm for electron-transfer (ET) processes in condensed media. It establishes the simplest conceptual test bed to study the influence of the medium's dynamics on ET kinetics both on the ensemble level, and on the level of single particles. Single electron description is particularly important for nanoscaled systems like proteins, or molecular wires. Especially insightful is this framework in the semiclassical limit, where the environment can be treated classically, and an exact analytical treatment becomes feasible. Slow medium's dynamics is capable of enslaving ET and bringing it on the ensemble level from a quantum regime of nonadiabatic tunneling to the classical adiabatic regime, where electrons follow the nuclei rearrangements. This classical adiabatic textbook picture contradicts, however, in a very spectacular fashion to the statistics of single electron transitions, even in the Debye, memoryless media, also named Ohmic in the parlance of the famed spin-boson model. On the single particle level, ET always remains quantum, and this was named a quantum breaking of ergodicity in the adiabatic ET regime. What happens in the case of subdiffusive, fractional, or sub-Ohmic medium's dynamics, which is featured by power-law decaying dynamical memory effects typical, e.g., for protein macromolecules, and other viscoelastic media? Such a memory is vividly manifested by anomalous Cole-Cole dielectric response in such media. We address this question based both on accurate numerics and analytical theory. The ensemble theory remarkably agrees with the numerical dynamics of electronic populations, revealing a power-law relaxation tail even in a profoundly nonadiabatic electron transfer regime. In other words, ET in such media should typically display fractional kinetics. However, a profound difference with the numerically accurate results occurs for the distribution of residence times in the electronic states, both on the ensemble level and the level of single trajectories. Ergodicity is broken dynamically even in a more spectacular way than in the memoryless case. Our results question the applicability of all the existing and widely accepted ensemble theories of electron transfer in fractional, sub-Ohmic environments, on the level of single molecules, and provide a real challenge to face, both for theorists and experimentalists.

Recent Developments in Dispersive Kinetics

In general, chemical reactions proceeding on time scales comparable to, or shorter than, those of internal rearrangements in a reaction system renewing the environment of the reactants (mixing), are dispersive. For dispersive kinetics, as for dispersive transport and dispersive relaxation, many time scales coexist. The rate coefficients for dispersive processes depend on time. For a time-dependent specific reaction rate, using the concept of energy profile along the reaction path, one finds the potential energy barrier separating reactants from products to evolve in time during the course of reaction. The evolution of the energy barrier during the course of reaction is described in terms of energy distribution functions related directly to the distribution function of logarithms of lifetimes calculable from kinetic equations with a time-dependent specific reaction rate. This phenomenological approach is compared with that in which the kinetic equations with time-dependent specific reaction rates are interpreted in terms of the superposition of classical reaction patterns. Special attention is paid to renor-malization of rate coefficients following from the stochastic theory of renewals (structural relaxation) in the reaction system. This phenomenological approach to kinetics is taken as a convenient basis to present a number of comprehensive models of dispersive kinetics developed in the 1990s and to discuss some recently published experimental data to show what one derives directly from experimental data and what the detailed mechanistic models have to account for to be acceptable.

Fractional Discrete Processes: Compound and Mixed Poisson Representations

We consider two fractional versions of a family of nonnegative integer-valued processes. We prove that their probability mass functions solve fractional Kolmogorov forward equations, and we show the overdispersion of these processes. As particular examples in this family, we can define fractional versions of some processes in the literature as the Pólya-Aeppli process, the Poisson inverse Gaussian process, and the negative binomial process. We also define and study some more general fractional versions with two fractional parameters.

State-Dependent Fractional Point Processes

In this paper we analyse the fractional Poisson process where the state probabilities pkνk(t), t ≥ 0, are governed by time-fractional equations of order 0 < νk ≤ 1 depending on the number k of events that have occurred up to time t. We are able to obtain explicitly the Laplace transform of pkνk(t) and various representations of state probabilities. We show that the Poisson process with intermediate waiting times depending on νk differs from that constructed from the fractional state equations (in the case of νk = ν, for all k, they coincide with the time-fractional Poisson process). We also introduce a different form of fractional state-dependent Poisson process as a weighted sum of homogeneous Poisson processes. Finally, we consider the fractional birth process governed by equations with state-dependent fractionality.

Generalized fractional diffusion equations for subdiffusion in arbitrarily growing domains

The ubiquity of subdiffusive transport in physical and biological systems has led to intensive efforts to provide robust theoretical models for this phenomena. These models often involve fractional derivatives. The important physical extension of this work to processes occurring in growing materials has proven highly nontrivial. Here we derive evolution equations for modeling subdiffusive transport in a growing medium. The derivation is based on a continuous-time random walk. The concise formulation of these evolution equations requires the introduction of a new, comoving, fractional derivative. The implementation of the evolution equation is illustrated with a simple model of subdiffusing proteins in a growing membrane.

A Fractional-Order Infectivity and Recovery SIR Model

The introduction of fractional-order derivatives to epidemiological compartment models, such as SIR models, has attracted much attention. When this introduction is done in an ad hoc manner, it is difficult to reconcile parameters in the resulting fractional-order equations with the dynamics of individuals. This issue is circumvented by deriving fractional-order models from an underlying stochastic process. Here, we derive a fractional-order infectivity and recovery Susceptible Infectious Recovered (SIR) model from the stochastic process of a continuous-time random walk (CTRW) that incorporates a time-since-infection dependence on both the infectivity and the recovery of the population. By considering a power-law dependence in the infectivity and recovery, fractional-order derivatives appear in the generalised master equations that govern the evolution of the SIR populations. Under the appropriate limits, this fractional-order infectivity and recovery model reduces to both the standard SIR model and the fractional recovery SIR model.

Fractional Transport of Photons in Deterministic Aperiodic Structures

The propagation of optical pulses through primary types of deterministic aperiodic structures is numerically studied in time domain using the rigorous transfer matrix method in combination with analytical fractional transport models. We demonstrate tunable anomalous photon transport, including the elusive logarithmic Sinai sub-diffusion in photonic systems for the first time. Our results are in excellent agreement with the scaling theory of transport in aperiodic media with fractal spectra, and additionally demonstrate logarithmic sub-diffusion in the presence of multifractality. Moreover, we establish a fruitful connection between tunable photon diffusion and fractional dynamics, which provides analytical insights into the asymptotic transport regime of optical media with deterministic aperiodic order. The demonstration of tunable sub-diffusion and logarithmic photon transport in deterministic aperiodic structures can open novel and fascinating scenarios for the engineering of wave propagation and light-matter interaction phenomena beyond the conventional diffusive regime.

A Fractional Order Recovery SIR Model from a Stochastic Process

Over the past several decades, there has been a proliferation of epidemiological models with ordinary derivatives replaced by fractional derivatives in an ad hoc manner. These models may be mathematically interesting, but their relevance is uncertain. Here we develop an SIR model for an epidemic, including vital dynamics, from an underlying stochastic process. We show how fractional differential operators arise naturally in these models whenever the recovery time from the disease is power-law distributed. This can provide a model for a chronic disease process where individuals who are infected for a long time are unlikely to recover. The fractional order recovery model is shown to be consistent with the Kermack-McKendrick age-structured SIR model, and it reduces to the Hethcote-Tudor integral equation SIR model. The derivation from a stochastic process is extended to discrete time, providing a stable numerical method for solving the model equations. We have carried out simulations of the fractional order recovery model showing convergence to equilibrium states. The number of infecteds in the endemic equilibrium state increases as the fractional order of the derivative tends to zero.

State-Dependent Fractional Point Processes

In this paper we analyse the fractional Poisson process where the state probabilitiespkνk(t),t≥ 0, are governed by time-fractional equations of order 0 &lt; νk≤ 1 depending on the numberkof events that have occurred up to timet. We are able to obtain explicitly the Laplace transform ofpkνk(t) and various representations of state probabilities. We show that the Poisson process with intermediate waiting times depending on νkdiffers from that constructed from the fractional state equations (in the case of νk= ν, for allk, they coincide with the time-fractional Poisson process). We also introduce a different form of fractional state-dependent Poisson process as a weighted sum of homogeneous Poisson processes. Finally, we consider the fractional birth process governed by equations with state-dependent fractionality.

Distributed-order diffusion equations and multifractality: Models and solutions

We study distributed-order time fractional diffusion equations characterized by multifractal memory kernels, in contrast to the simple power-law kernel of common time fractional diffusion equations. Based on the physical approach to anomalous diffusion provided by the seminal Scher-Montroll-Weiss continuous time random walk, we analyze both natural and modified-form distributed-order time fractional diffusion equations and compare the two approaches. The mean squared displacement is obtained and its limiting behavior analyzed. We derive the connection between the Wiener process, described by the conventional Langevin equation and the dynamics encoded by the distributed-order time fractional diffusion equation in terms of a generalized subordination of time. A detailed analysis of the multifractal properties of distributed-order diffusion equations is provided.

Consequences of animal interactions on their dynamics: emergence of home ranges and territoriality

Animal spacing has important implications for population abundance, species demography and the environment. Mechanisms underlying spatial segregation have their roots in the characteristics of the animals, their mutual interaction and their response, collective as well as individual, to environmental variables. This review describes how the combination of these factors shapes the patterns we observe and presents a practical, usable framework for the analysis of movement data in confined spaces. The basis of the framework is the theory of interacting random walks and the mathematical description of out-of-equilibrium systems. Although our focus is on modelling and interpreting animal home ranges and territories in vertebrates, we believe further studies on invertebrates may also help to answer questions and resolve unanswered puzzles that are still inaccessible to experimental investigation in vertebrate species.

A directed continuous time random walk model with jump length depending on waiting time

In continuum one-dimensional space, a coupled directed continuous time random walk model is proposed, where the random walker jumps toward one direction and the waiting time between jumps affects the subsequent jump. In the proposed model, the Laplace-Laplace transform of the probability density function P(x, t) of finding the walker at position x at time t is completely determined by the Laplace transform of the probability density function φ(t) of the waiting time. In terms of the probability density function of the waiting time in the Laplace domain, the limit distribution of the random process and the corresponding evolving equations are derived.

Fractional Discrete Processes: Compound and Mixed Poisson Representations

We consider two fractional versions of a family of nonnegative integer-valued processes. We prove that their probability mass functions solve fractional Kolmogorov forward equations, and we show the overdispersion of these processes. As particular examples in this family, we can define fractional versions of some processes in the literature as the Pólya-Aeppli process, the Poisson inverse Gaussian process, and the negative binomial process. We also define and study some more general fractional versions with two fractional parameters.

Nonlinear degradation-enhanced transport of morphogens performing subdiffusion

We study a morphogen gradient formation under nonlinear degradation and subdiffusive transport. In the long-time limit, we obtain the nonlinear effect of degradation-enhanced diffusion, resulting from the interaction of non-Markovian subdiffusive transport with a nonlinear reaction. We find the stationary profile of power-law type, which has implications for robustness, with the shape of the profile being controlled by the anomalous exponent. Far away from the source of morphogens, any changes in the rate of production are not felt.

Continuous-time random walks on networks with vertex- and time-dependent forcing

We have investigated the transport of particles moving as random walks on the vertices of a network, subject to vertex- and time-dependent forcing. We have derived the generalized master equations for this transport using continuous time random walks, characterized by jump and waiting time densities, as the underlying stochastic process. The forcing is incorporated through a vertex- and time-dependent bias in the jump densities governing the random walking particles. As a particular case, we consider particle forcing proportional to the concentration of particles on adjacent vertices, analogous to self-chemotactic attraction in a spatial continuum. Our algebraic and numerical studies of this system reveal an interesting pair-aggregation pattern formation in which the steady state is composed of a high concentration of particles on a small number of isolated pairs of adjacent vertices. The steady states do not exhibit this pair aggregation if the transport is random on the vertices, i.e., without forcing. The manifestation of pair aggregation on a transport network may thus be a signature of self-chemotactic-like forcing.

Anomalous transport in the crowded world of biological cells

A ubiquitous observation in cell biology is that the diffusive motion of macromolecules and organelles is anomalous, and a description simply based on the conventional diffusion equation with diffusion constants measured in dilute solution fails. This is commonly attributed to macromolecular crowding in the interior of cells and in cellular membranes, summarizing their densely packed and heterogeneous structures. The most familiar phenomenon is a sublinear, power-law increase of the mean-square displacement (MSD) as a function of the lag time, but there are other manifestations like strongly reduced and time-dependent diffusion coefficients, persistent correlations in time, non-Gaussian distributions of spatial displacements, heterogeneous diffusion and a fraction of immobile particles. After a general introduction to the statistical description of slow, anomalous transport, we summarize some widely used theoretical models: Gaussian models like fractional Brownian motion and Langevin equations for visco-elastic media, the continuous-time random walk model, and the Lorentz model describing obstructed transport in a heterogeneous environment. Particular emphasis is put on the spatio-temporal properties of the transport in terms of two-point correlation functions, dynamic scaling behaviour, and how the models are distinguished by their propagators even if the MSDs are identical. Then, we review the theory underlying commonly applied experimental techniques in the presence of anomalous transport like single-particle tracking, fluorescence correlation spectroscopy (FCS) and fluorescence recovery after photobleaching (FRAP). We report on the large body of recent experimental evidence for anomalous transport in crowded biological media: in cyto- and nucleoplasm as well as in cellular membranes, complemented by in vitro experiments where a variety of model systems mimic physiological crowding conditions. Finally, computer simulations are discussed which play an important role in testing the theoretical models and corroborating the experimental findings. The review is completed by a synthesis of the theoretical and experimental progress identifying open questions for future investigation.

Anomalous diffusion and multifractional Brownian motion: simulating molecular crowding and physical obstacles in systems biology

There have been many recent studies from both experimental and simulation perspectives in order to understand the effects of spatial crowding in molecular biology. These effects manifest themselves in protein organisation on the plasma membrane, on chemical signalling within the cell and in gene regulation. Simulations are usually done with lattice- or meshless-based random walks but insights can also be gained through the computation of the underlying probability density functions of these stochastic processes. Until recently much of the focus had been on continuous time random walks, but some very recent work has suggested that fractional Brownian motion may be a good descriptor of spatial crowding effects in some cases. The study compares both fractional Brownian motion and continuous time random walks and highlights how well they can represent different types of spatial crowding and physical obstacles. Simulated spatial data, mimicking experimental data, was first generated by using the package Smoldyn. We then attempted to characterise this data through continuous time anomalously diffusing random walks and multifractional Brownian motion (MFBM) by obtaining MFBM paths that match the statistical properties of our sample data. Although diffusion around immovable obstacles can be reasonably characterised by a single Hurst exponent, we find that diffusion in a crowded environment seems to exhibit multifractional properties in the form of a different short- and long-time behaviour.

Fractional diffusion in a periodic potential: Overdamped and inertia corrected solutions for the spectrum of the velocity correlation function

Anomalous diffusion of a particle in a cosine periodic potential is treated using fractional diffusion equations in both phase and configuration space. Exact solutions of two distinct forms of the fractional Klein-Kramers (Fokker-Planck) equation for the distribution function in phase space are obtained via matrix continued fractions yielding the average velocity, the velocity autocorrelation function, its spectrum, etc. In the overdamped limit, the results yielded by both equations agree with those from a fractional probability density diffusion equation in configuration space. A simple analytic solution for the spectrum of the velocity correlation function is also given using the effective eigenvalue approximation. The results represent generalizations of the conventional solutions for the normal diffusion of a Brownian particle in a cosine potential to fractional dynamics (giving rise to anomalous diffusion).

Analytical Solution for the Differential Equation Containing Generalized Fractional Derivative Operators and Mittag-Leffler-Type Function

We discuss and derive the analytical solution for the fractional partial differential equation with generalized Riemann-Liouville fractional operator of order and . Here, we derive the solution of the given differential equation with the help of Laplace and Hankel transform in terms of Fox's -function as well as in terms of Fox-Wright function .

Linking animal movement to site fidelity

Site fidelity, the recurrent visit of an animal to a previously occupied area is a wide-spread behavior in the animal kingdom. The relevance of site fidelity to territoriality, successful breeding, social associations, optimal foraging and other ecological processes, demands accurate quantification. Here we generalize previous theory that connects site fidelity patterns to random walk parameters within the framework of the space-time fractional diffusion equation. In particular, we describe the site fidelity function in terms of animal movement characteristics via the Lévy exponent, which controls the step-length distribution of the random steps at each turning point, and the waiting time exponent that controls for how long an animal awaits before actually moving. The analytical results obtained will provide a rigorous benchmark for empirically driven studies of animal site fidelity.

A statistical mechanical theory of proton transport kinetics in hydrogen-bonded networks based on population correlation functions with applications to acids and bases

Extraction of relaxation times, lifetimes, and rates associated with the transport of topological charge defects in hydrogen-bonded networks from molecular dynamics simulations is a challenge because proton transfer reactions continually change the identity of the defect core. In this paper, we present a statistical mechanical theory that allows these quantities to be computed in an unbiased manner. The theory employs a set of suitably defined indicator or population functions for locating a defect structure and their associated correlation functions. These functions are then used to develop a chemical master equation framework from which the rates and lifetimes can be determined. Furthermore, we develop an integral equation formalism for connecting various types of population correlation functions and derive an iterative solution to the equation, which is given a graphical interpretation. The chemical master equation framework is applied to the problems of both hydronium and hydroxide transport in bulk water. For each case it is shown that the theory establishes direct links between the defect's dominant solvation structures, the kinetics of charge transfer, and the mechanism of structural diffusion. A detailed analysis is presented for aqueous hydroxide, examining both reorientational time scales and relaxation of the rotational anisotropy, which is correlated with recent experimental results for these quantities. Finally, for OH(-)(aq) it is demonstrated that the "dynamical hypercoordination mechanism" is consistent with available experimental data while other mechanistic proposals are shown to fail. As a means of going beyond the linear rate theory valid from short up to intermediate time scales, a fractional kinetic model is introduced in the Appendix in order to describe the nonexponential long-time behavior of time-correlation functions. Within the mathematical framework of fractional calculus the power law decay ∼t(-σ), where σ is a parameter of the model and depends on the dimensionality of the system, is obtained from Mittag-Leffler functions due to their long-time asymptotics, whereas (stretched) exponential behavior is found for short times.