Fractional Master Equation Research Articles

Population heterogeneity in the fractional master equation, ensemble self-reinforcement, and strong memory effects.

We formulate a fractional master equation in continuous time with random transition probabilities across the population of random walkers such that the effective underlying random walk exhibits ensemble self-reinforcement. The population heterogeneity generates a random walk with conditional transition probabilities that increase with the number of steps taken previously (self-reinforcement). Through this, we establish the connection between random walks with a heterogeneous ensemble and those with strong memory where the transition probability depends on the entire history of steps. We find the ensemble-averaged solution of the fractional master equation through subordination involving the fractional Poisson process counting the number of steps at a given time and the underlying discrete random walk with self-reinforcement. We also find the exact solution for the variance which exhibits superdiffusion even as the fractional exponent tends to 1.

Population heterogeneity in the fractional master equation, ensemble self-reinforcement, and strong memory effects.

We formulate a fractional master equationin continuous time with random transition probabilities across the population of random walkers such that the effective underlying random walk exhibits ensemble self-reinforcement. The population heterogeneity generates a random walk with conditional transition probabilities that increase with the number of steps taken previously (self-reinforcement). Through this, we establish the connection between random walks with a heterogeneous ensemble and those with strong memory where the transition probability depends on the entire history of steps. We find the ensemble-averaged solution of the fractional master equationthrough subordination involving the fractional Poisson process counting the number of steps at a given time and the underlying discrete random walk with self-reinforcement. We also find the exact solution for the variance which exhibits superdiffusion even as the fractional exponent tends to 1.

A phenomenological approach to anomalous transport in complex or disordered media

We aim to derive a phenomenological approach to link the theories of anomalous transport governed by fractional calculus and stochastic theory with the conductivity behavior governed by the semi-empirical conductivity formalism involving Debye, Cole–Cole, Cole–Davidson, and Havriliak–Negami type conductivity equations. We want to determine the anomalous transport processes in the amorphous semiconductors and insulators by developing a theoretical approach over some mathematical instruments and methods. In this paper, we obtain an analytical expression for the average behavior of conductivity in complex or disordered media via the fractional-stochastic differential equation, the Fourier–Laplace transform, some natural boundary initial conditions, and familiar physical relations. We start with the stochastic equation of motion called the Langevin equation, develop its equivalent master equation called the Klein–Kramers or Fokker–Planck equation, and consider the time-fractional generalization of the master equation. Once we derive the fractional master equation, we then determine the expressions for the mean value of the variables or observables through some calculations and conditions. Finally, we use these expressions in the current density relation to obtain the average conductivity behavior.

Variable-order fractional master equation and clustering of particles: non-uniform lysosome distribution.

In this paper, we formulate the space-dependent variable-order fractional master equation to model clustering of particles, organelles, inside living cells. We find its solution in the long-time limit describing non-uniform distribution due to a space-dependent fractional exponent. In the continuous space limit, the solution of this fractional master equation is found to be exactly the same as the space-dependent variable-order fractional diffusion equation. In addition, we show that the clustering of lysosomes, an essential organelle for healthy functioning of mammalian cells, exhibit space-dependent fractional exponents. Furthermore, we demonstrate that the non-uniform distribution of lysosomes in living cells is accurately described by the asymptotic solution of the space-dependent variable-order fractional master equation. Finally, Monte Carlo simulations of the fractional master equation validate our analytical solution. This article is part of the theme issue 'Transport phenomena in complex systems (part 1)'.

Non-Markovian process with variable memory functions

We present a treatment of the non-Markovian character of memory by incorporating different forms of Mittag-Leffler (ML) function, which generally arises in the solution of a fractional master equation, as different memory functions in the Generalized Kolmogorov-Feller Equation (GKFE). The cross-over from the short time (stretched exponential) to long time (inverse power law) approximations of the ML function incorporated in the GKFE is proven. We have found that the GKFE solutions are the same for negative exponential and upto first order expansion of the stretched exponential function for very small \(\tau \rightarrow 0\). A generalized integro-differential equation form of the GKFE along with an asymptotic case is provided.

Subdiffusion of Particles with a Nonlinear Interaction and Cell-Cell Adhesion

The main goal of this work is to propose a non-Markovian model for a subdiffusive transport of particles with nonlinear interaction that involves adhesion affects on escape rates from position x, in inhomogeneous media. In this case, the escape rates to be dependent on the particle density and also effected by the density at the neighbors as well as the chemotactic gradient. We systematically derive the subdiffusive fractional master equation. Considering the spatial continuum limit of the fractional master equation implies a stationary solution of the resulted fractional subdiffusive master equation. Finally, we analyze the role of adhesion in the resulted stationary density.

The entropy production paradox for fractional master equations

Time-fractional evolution equations for probability distributions provide a means to describe an important class of stochastic processes. Their solutions show features, which are essential in modeling a variety of phenomena in real world applications. One aspect, which has been observed in time-fractional diffusion equations, shows a surprising and unexpected behavior of the entropy production rate induced by these equations. The entropy production rate increases as one moves away from the fully irreversible case, corresponding to classical diffusion. This rate is analyzed for a new class of systems with state spaces that are finite and denumerable. We find that the entropy production paradox reemerges nonetheless, but in a new and unexpected form.

Between Waves and Diffusion: Paradoxical Entropy Production in an Exceptional Regime.

The entropy production rate is a well established measure for the extent of irreversibility in a process. For irreversible processes, one thus usually expects that the entropy production rate approaches zero in the reversible limit. Fractional diffusion equations provide a fascinating testbed for that intuition in that they build a bridge connecting the fully irreversible diffusion equation with the fully reversible wave equation by a one-parameter family of processes. The entropy production paradox describes the very non-intuitive increase of the entropy production rate as that bridge is passed from irreversible diffusion to reversible waves. This paradox has been established for time- and space-fractional diffusion equations on one-dimensional continuous space and for the Shannon, Tsallis and Renyi entropies. After a brief review of the known results, we generalize it to time-fractional diffusion on a finite chain of points described by a fractional master equation.

Birth–Death Stochastic Process of Optical Rogue Wave Described by Fractional Master Equation — Application of Confluent Heun Differential Equation —

The paper presents the birth–death stochastic process of an optical rogue wave with a long memory described by a fractional master equation. An exact analytic expression for the probability generating function is obtained with an integral representation of the confluent Heun function. This enables a full statistical analysis under any initial condition. It is demonstrated that the present mathematical approach can be utilized for the analysis of birth–death stochastic processes when the generating function can be described by a class of Heun differential equations.

Fractional Euler Limits and their Applications

Generalizations of the classical Euler formula to the setting of fractional calculus are discussed. Compound interest and fractional compound interest serve as motivation. Connections to fractional master equations are highlighted. An application to the Schlögl reactions with Mittag-Leffler waiting times is described.

Fractional calculus ties the microscopic and macroscopic scales of complex network dynamics

A two-state, master equation-based decision-making model has been shown to generate phase transitions, to be topologically complex, and to manifest temporal complexity through an inverse power-law probability distribution function in the switching times between the two critical states of consensus. These properties are entailed by the fundamental assumption that the network elements in the decision-making model imperfectly imitate one another. The process of subordination establishes that a single network element can be described by a fractional master equation whose analytic solution yields the observed inverse power-law probability distribution obtained by numerical integration of the two-state master equation to a high degree of accuracy.

Short note on the emergence of fractional kinetics

In the present Short Note an idea is proposed to explain the emergence and the observation of processes in complex media that are driven by fractional non-Markovian master equations. Particle trajectories are assumed to be solely Markovian and described by the Continuous Time Random Walk model. But, as a consequence of the complexity of the medium, each trajectory is supposed to scale in time according to a particular random timescale. The link from this framework to microscopic dynamics is discussed and the distribution of timescales is computed. In particular, when a stationary distribution is considered, the timescale distribution is uniquely determined as a function related to the fundamental solution of the space–time fractional diffusion equation. In contrast, when the non-stationary case is considered, the timescale distribution is no longer unique. Two distributions are here computed: one related to the M-Wright/Mainardi function, which is Green’s function of the time-fractional diffusion equation, and another related to the Mittag-Leffler function, which is the solution of the fractional-relaxation equation.

Nonlinear subdiffusive fractional equations and the aggregation phenomenon

In this article we address the problem of the nonlinear interaction of subdiffusive particles. We introduce the random walk model in which statistical characteristics of a random walker such as escape rate and jump distribution depend on the mean density of particles. We derive a set of nonlinear subdiffusive fractional master equations and consider their diffusion approximations. We show that these equations describe the transition from an intermediate subdiffusive regime to asymptotically normal advection-diffusion transport regime. This transition is governed by nonlinear tempering parameter that generalizes the standard linear tempering. We illustrate the general results through the use of the examples from cell and population biology. We find that a nonuniform anomalous exponent has a strong influence on the aggregation phenomenon.

On a Fractional Master Equation and a Fractional Diffusion Equation

In this paper , we derive the solutions of fractional master equation defined by (2.1) and fractional diffusion equation defined by (3.3). The method followed in deriving the solution is that of Laplace and Fourier transforms. The solutions are obtained in a neat and compact forms in terms of the generalized Mittag -Leffler function and Fox' H-function. The results established are of general character and include some known results, as special cases.

Random death process for the regularization of subdiffusive fractional equations

The description of subdiffusive transport in complex media by fractional equations with a constant anomalous exponent is not robust where the stationary distribution is concerned. The Gibbs-Boltzmann distribution is radically changed by even small spatial perturbations to the anomalous exponent [S. Fedotov and S. Falconer, Phys. Rev. E 85, 031132 (2012)]. To rectify this problem we propose the inclusion of the random death process in the random walk scheme, which is quite natural for biological applications including morphogen gradient formation. From this, we arrive at the modified fractional master equation and analyze its asymptotic behavior, both analytically and by Monte Carlo simulation. We show that this equation is structurally stable against spatial variations of the anomalous exponent. We find that the stationary flux of the particles has a Markovian form with rate functions depending on the anomalous rate functions, the death rate, and the anomalous exponent. Additionally, in the continuous limit we arrive at an advection-diffusion equation where advection and diffusion coefficients depend on both the death rate and anomalous exponent.

Generalized Fractional Master Equation for Self-Similar Stochastic Processes Modelling Anomalous Diffusion

The Master Equation approach to model anomalous diffusion is considered. Anomalous diffusion in complex media can be described as the result of a superposition mechanism reflecting inhomogeneity and nonstationarity properties of the medium. For instance, when this superposition is applied to the time-fractional diffusion process, the resulting Master Equation emerges to be the governing equation of the Erdélyi-Kober fractional diffusion, that describes the evolution of the marginal distribution of the so-called generalized grey Brownian motion. This motion is a parametric class of stochastic processes that provides models for both fast and slow anomalous diffusion: it is made up of self-similar processes with stationary increments and depends on two real parameters. The class includes the fractional Brownian motion, the time-fractional diffusion stochastic processes, and the standard Brownian motion. In this framework, the M-Wright function (known also as Mainardi function) emerges as a natural generalization of the Gaussian distribution, recovering the same key role of the Gaussian density for the standard and the fractional Brownian motion.

Subdiffusive master equation with space-dependent anomalous exponent and structural instability

We derive the fractional master equation with space-dependent anomalous exponent. We analyze the asymptotic behavior of the corresponding lattice model both analytically and by Monte Carlo simulation. We show that the subdiffusive fractional equations with constant anomalous exponent μ in a bounded domain [0,L] are not structurally stable with respect to the nonhomogeneous variations of parameter μ. In particular, the Gibbs-Boltzmann distribution is no longer the stationary solution of the fractional Fokker-Planck equation whatever the space variation of the exponent might be. We analyze the random distribution of μ in space and find that in the long-time limit, the probability distribution is highly intermediate in space and the behavior is completely dominated by very unlikely events. We show that subdiffusive fractional equations with the nonuniform random distribution of anomalous exponent is an illustration of a "Black Swan," the low probability event of the small value of the anomalous exponent that completely dominates the long-time behavior of subdiffusive systems.

On a Fractional Master Equation

A fractional order time‐independent form of the wave equation or diffusion equation in two dimensions is obtained from the standard time‐independent form of the wave equation or diffusion equation in two‐dimensions by replacing the integer order partial derivatives by fractional Riesz‐Feller derivative and Caputo derivative of order α, β, 1 < ℜ(α) ≤ 2 and 1 < ℜ(β) ≤ 2 respectively. In this paper, we derive an analytic solution for the fractional time‐independent form of the wave equation or diffusion equation in two dimensions in terms of the Mittag‐Leffler function. The solutions to the fractional Poisson and the Laplace equations of the same kind are obtained, again represented by means of the Mittag‐Leffler function. In all three cases, the solutions are represented also in terms of Fox′s H‐function.

Fractional and operational calculus with generalized fractional derivative operators and Mittag–Leffler type functions

In this paper, we study a certain family of generalized Riemann–Liouville fractional derivative operators of order α and type β, which were introduced and investigated in several earlier works [R. Hilfer (ed.), Applications of Fractional Calculus in Physics, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 2000; R. Hilfer, Fractional time evolution, in Applications of Fractional Calculus in Physics, R. Hilfer, ed., World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 2000, pp. 87–130; R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials, J. Chem. Phys. 284 (2002), pp. 399–408; R. Hilfer, Threefold introduction to fractional derivatives, in Anomalous Transport: Foundations and Applications, R. Klages, G. Radons, and I.M. Sokolov, eds., Wiley-VCH Verlag, Weinheim, 2008, pp. 17–73; R. Hilfer and L. Anton, Fractional master equations and fractal time random walks, Phys. Rev. E 51 (1995), pp. R848–R851; R. Hilfer, Y. Luchko, and Ž. Tomovski, Operational method for solution of the fractional differential equations with the generalized Riemann-Liouville fractional derivatives, Fract. Calc. Appl. Anal. 12 (2009), pp. 299–318; F. Mainardi and R. Gorenflo, Time-fractional derivatives in relaxation processes: A tutorial survey, Fract. Calc. Appl. Anal. 10 (2007), pp. 269–308; T. Sandev and Ž. Tomovski, General time fractional wave equation for a vibrating string, J. Phys. A Math. Theor. 43 (2010), 055204; H.M. Srivastava and Ž. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel, Appl. Math. Comput. 211 (2009), pp. 198–210]. In particular, we derive various compositional properties, which are associated with Mittag–Leffler functions and Hardy-type inequalities for the generalized fractional derivative operator . Furthermore, by using the Laplace transformation methods, we provide solutions of many different classes of fractional differential equations with constant and variable coefficients and some general Volterra-type differintegral equations in the space of Lebesgue integrable functions. Particular cases of these general solutions and a brief discussion about some recently investigated fractional kinetic equations are also given.

An analysis of dielectric relaxation using the fractional master equation of the stochastic Ising model

In this paper, we begin introducing some basic definitions and mathematical preliminaries of the fractional calculus theory. By using the fractional calculus technique (that is, calculus of derivatives and integrals of any arbitrary real or complex order) a solution of the fractional master equation derived from the stochastic Ising model of Glauber has been obtained and the result is applied to an analysis of the dielectric relaxation processes. From the solution of the equation, the Cole–Cole dispersion relation, KWW (Kohlrausch–William–Watts) equation and algebraic decay relaxation functions are obtained easily. Then these functions are compared with Bozdemir's earlier analysis of the stochastic Ising model.