Fractional Fokker-Planck Equation Research Articles

Exact solutions of the fractional time‐derivative Fokker–Planck equation: A novel approach

In the present article, an approach to find the exact solution of the fractional Fokker–Planck equation (FFPE) is presented. It is based on transforming it to a system of first‐order partial differential equation via Hopf transformation, together with implementing the extended unified method. On the other hand, a theorem provides the reduction of the fractional derivatives to non‐autonomous ordinary derivative is given. Thus, the FFPE is reduced to non‐autonomous classical ones. Some explicit solutions of the classical, fractional time‐derivative Fokker–Planck equation are obtained. It is shown that the solution of the Fokker–Planck equation is bi‐Gaussian's, which was not found up to date. It is found that high friction coefficient plays a significant role in lowering the standard deviation. Further, it is found that the effect of the presence of the fractional derivative prevails that of the fractal derivative. Here, the most interesting result found is that mixed‐Gaussian solution is obtained. It is worthy to mention that the mixture of Gaussian's is a powerful tool in machine learning and also in the distribution of loads in networks. Further, varying the order of the fractional time derivatives results to slight effects in the probability distribution function. Also, it is shown that the mean and mean square of the velocity vary slowly.

Pencil-beam approximation of fractional Fokker-Planck

<p style='text-indent:20px;'>We consider the modeling of light beams propagating in highly forward-peaked turbulent media by fractional Fokker-Planck equations and their approximations by fractional Fermi pencil beam models. We obtain an error estimate in a 1-Wasserstein distance for the latter model showing that beam spreading is well captured by the Fermi pencil-beam approximation in the small diffusion limit.</p>

A preconditioned implicit difference scheme for semilinear two‐dimensional time–space fractional Fokker–Planck equations

AbstractTime–space fractional Fokker–Planck equations (TSFFPEs) are a class of very useful models for describing some practical phenomena in statistical physics. In the present article, we focus on the fast computation for semilinear two‐dimensional TSFFPEs. A stable implicit difference scheme with second‐order accuracy in time and space is derived. In order to accelerate the convergence rate of the scheme, a preconditioned strategy combining the Kronecker product splitting preconditioner and the generalized minimal residual method is suggested. Numerical experiments further confirm the theoretical accuracy of the difference scheme and computational advantage of the acceleration scheme.

Anomalous diffusion of molecules with rotating polar groups: The joint role played by inertia and dipole coupling in microwave and far-infrared absorption.

Budó's generalization [A. Budó, J. Chem. Phys. 17, 686 (1949)10.1063/1.1747370] of the Debye rotational diffusion model of dielectric relaxation of polar molecules to an assembly with internal interacting polar groups is extended to inertial anomalous diffusion. Thus, the theory can be applied both in the GHz and the THz regions, accounting for anomalous behavior as well as the necessary return to optical transparency at very high frequencies. The linking of both dispersion regions in a single model including anomalous effects is accomplished via a fractional Fokker-Planck equation in phase space based on the continuous time random walk ansatz. The latter is written via the Langevin equations for the stochastic dynamics of pairs of interacting heavy polar groups embedded in the frame of reference of a particular molecule or molecular dimer rotating about a space-fixed axis. The fractional Fokker-Planck equation is then converted to a three-term matrix differential recurrence equation for the statistical moments. This is solved in the frequency domain for the linear dielectric response using matrix continued fractions. Thus, one has the complex susceptibility χ(ω) for extensive ranges of damping, group dipole moment ratio, and friction. The susceptibility, as inferred from the small oscillation limit, inherently comprises a low frequency (GHz) band with width depending on the anomalous parameter and a far-infrared (THz) or Poley peak of resonant character with a comblike structure of harmonic peaks. This behavior is due to the double transcendental nature of the after-effect function.

Non-Markovian Pitch-angle Scattering as the Origin of Particle Superdiffusion Parallel to the Magnetic Field

Abstract We develop a theoretical model for particle superdiffusive transport parallel to the average magnetic field, due to the pitch-angle scattering times having a non-Markovian, power-law probability distribution. We show that a non-Markovian Fokker–Planck equation can be derived, where the traditional time derivative is changed for a fractional time derivative. By solving the fractional Fokker–Planck equation, with the time-dependent part having solutions that are expressed by the Mittag-Leffler functions, it is found that an initial pitch-angle distribution slowly decays toward isotropy. This leads to a parallel velocity autocorrelation function that also has a slow power-law decay in time, thus implying superdiffusive transport in the direction parallel to the background magnetic field. In this framework, we derive for the first time the anomalous diffusion coefficient as a function of physical parameters like the background magnetic field, the resonant turbulence level, and the particle speed.

Nature of the nonequilibrium phase transition in the non-Markovian driven Dicke model.

The Dicke model famously exhibits a phase transition to a superradiant phase with a macroscopic population of photons and is realized in multiple settings in open quantum systems. In this paper, we study a variant of the Dicke model where the cavity mode is lossy due to the coupling to a Markovian environment while the atomic mode is coupled to a colored bath. We analytically investigate this model by inspecting its low-frequency behavior via the Schwinger-Keldysh field theory and carefully examine the nature of the corresponding superradiant phase transition. Integrating out the fast modes, we can identify a simple effective theory allowing us to derive analytical expressions for various critical exponents including the dynamical exponent. We find excellent agreement with previous numerical results when the non-Markovian bath is at zero temperature; however, contrary to these studies, our low-frequency approach reveals that the same exponents govern the critical behavior when the colored bath is at finite temperature unless the chemical potential is zero. Furthermore, we show that the superradiant phase transition is classical in nature, while it is genuinely nonequilibrium. We derive a fractional Langevin equation and conjecture the associated fractional Fokker-Planck equation that captures the system's long-time memory as well as its nonequilibrium behavior. Finally, we consider finite-size effects at the phase transition and identify the finite-size scaling exponents, unlocking a rich behavior in both statics and dynamics of the photonic and atomic observables.

Statistical solution to SDEs with $$\alpha $$-stable Lévy noise via deep neural network

The probability density function of the stochastic differential equations with $$\alpha $$ -stable Levy noise is determined by the deterministic fractional Fokker–Planck (FFP) equation with the Riesz fractional derivative. In this paper, to solve the FFP equation, a new algorithm which is an efficient numerical approach using the deep neural network is proposed in view of the mesh-dependence phenomenon in the traditional discrete scheme and the nonsmooth solution of the Monte Carlo method. Under the framework of the DL-FP algorithm, the “fractional centered derivative” approach is applied to approximate the Riesz fractional derivative of the output in the neural network, which is the major novelty of this approach. Numerical results are presented to demonstrate the accuracy of the approach. Comparing with the traditional discrete scheme and the Monte Carlo method, the proposed algorithm is mesh-less and smoother. Furthermore, the proposed technique performs well in dealing with the heavy-tail case through comparing with the analytical solution.

Anomalous reaction-diffusion equations for linear reactions.

Deriving evolution equations accounting for both anomalous diffusion and reactions is notoriously difficult, even in the simplest cases. In contrast to normal diffusion, reaction kinetics cannot be incorporated into evolution equations modeling subdiffusion by merely adding reaction terms to the equations describing spatial movement. A series of previous works derived fractional reaction-diffusion equations for the spatiotemporal evolution of particles undergoing subdiffusion in one space dimension with linear reactions between a finite number of discrete states. In this paper, we first give a short and elementary proof of these previous results. We then show how this argument gives the evolution equations for more general cases, including subdiffusion following any fractional Fokker-Planck equation in an arbitrary d-dimensional spatial domain with time-dependent reactions between infinitely many discrete states. In contrast to previous works which employed a variety of technical mathematical methods, our analysis reveals that the evolution equations follow from (1) the probabilistic independence of the stochastic spatial and discrete processes describing a single particle and (2) the linearity of the integro-differential operators describing spatial movement. We also apply our results to systems combining reactions with superdiffusion.

Extreme statistics of anomalous subdiffusion following a fractional Fokker–Planck equation: subdiffusion is faster than normal diffusion

Anomalous subdiffusion characterizes transport in diverse physical systems and is especially prevalent inside biological cells. In cell biology, the prevailing model for chemical activation rates has recently changed from the first passage time (FPT) of a single searcher to the FPT of the fastest searcher out of many searchers to reach a target, which is called an extreme statistic or extreme FPT. In this paper, we investigate extreme statistics of searchers which move by anomalous subdiffusion. We model subdiffusion by a fractional Fokker–Planck equation involving the Riemann–Liouville fractional derivative. We prove an explicit and very general formula for every moment of subdiffusive extreme FPTs and approximate their full probability distribution. While the mean FPT of a single subdiffusive searcher is infinite, the fastest subdiffusive searcher out of many subdiffusive searchers typically has a finite mean FPT. In fact, we prove the counterintuitive result that extreme FPTs of subdiffusion are faster than extreme FPTs of normal diffusion. Mathematically, we employ a stochastic representation involving a random time change of a standard Ito drift–diffusion according to the trajectory of the first crossing time inverse of a Levy subordinator. A key step in our analysis is generalizing Varadhan’s formula from large deviation theory to the case of subdiffusion, which yields the short-time distribution of subdiffusion in terms of a certain geodesic distance.

Dispersive Transport Described by the Generalized Fick Law with Different Fractional Operators

The approach based on fractional advection–diffusion equations provides an effective and meaningful tool to describe the dispersive transport of charge carriers in disordered semiconductors. A fractional generalization of Fick’s law containing the Riemann–Liouville fractional derivative is related to the well-known fractional Fokker–Planck equation, and it is consistent with the universal characteristics of dispersive transport observed in the time-of-flight experiment (ToF). In the present paper, we consider the generalized Fick laws containing other forms of fractional time operators with singular and non-singular kernels and find out features of ToF transient currents that can indicate the presence of such fractional dynamics. Solutions of the corresponding fractional Fokker–Planck equations are expressed through solutions of integer-order equation in terms of an integral with the subordinating function. This representation is used to calculate the ToF transient current curves. The physical reasons leading to the considered fractional generalizations are elucidated and discussed.

Fractional-Order Genocchi–Petrov–Galerkin Method for Solving Time–Space Fractional Fokker–Planck Equations Arising from the Physical Phenomenon

In the current study, we present the fractional-order Genocchi–Petrov–Galerkin method to investigate the approximate solution of time–space fractional Fokker–Planck equations (FFPEs). In the proposed approach, due to fractional Genocchi functions (FGFs) and their operational matrices transform these equations into a system of algebraic equations. The operational matrices of fractional-order integration and derivative are obtained by utilizing Riemann–Liouville fractional integral operator and Caputo fractional derivative, respectively. The convergence analysis of the proposed technique is rigorously discussed. To illustrate the applicability of the error formulas, we present the process of obtaining results for some examples. These examples are carried out to confirm the effectiveness of the proposed method.

Numerical Scheme for the Fokker–Planck Equations Describing Anomalous Diffusions with Two Internal States

Recently, the fractional Fokker-Planck equations (FFPEs) with multiple internal states are built for the particles undergoing anomalous diffusion with different waiting time distributions for different internal states, which describe the distribution of positions of the particles [Xu and Deng, Math. Model. Nat. Phenom., $\mathbf{13}$, 10 (2018)]. In this paper, we first develop the Sobolev regularity of the FFPEs with two internal states, including the homogeneous problem with smooth and nonsmooth initial values and the inhomogeneous problem with vanishing initial value, and then we design the numerical scheme for the system of fractional partial differential equations based on the finite element method for the space derivatives and convolution quadrature for the time fractional derivatives. The optimal error estimates of the scheme under the above three different conditions are provided for both space semidiscrete and fully discrete schemes. Finally, one- and two-dimensional numerical experiments are performed to confirm our theoretical analysis and the predicted convergence order.

Information Geometric Investigation of Solutions to the Fractional Fokker–Planck Equation

A novel method for measuring distances between statistical states as represented by probability distribution functions (PDF) has been proposed, namely the information length. The information length enables the computation of the total number of statistically different states that a system evolves through in time. Anomalous transport can presumably be modeled fractional velocity derivatives and Langevin dynamics in a Fractional Fokker–Planck (FFP) approach. The numerical solutions or PDFs are found for varying degree of fractionality ( α ) of the stable Lévy distribution as solutions to the FFP equation. Specifically, the information length of time-dependent PDFs for a given fractional index α is computed.

An efficient computational technique for local fractional Fokker Planck equation

The key aim of the present study is to compute the solution of local fractional Fokker Planck equation (LFFPE) on the Cantor set. We perform a comparison between the reduced differential transform method (RDTM) and local fractional series expansion method (LFSEM) employed to the LFFPE. The operators are considered in the local nature. The outcomes demonstrate the important characteristic of the two techniques which are very successful and simple to solve the differential equations having fractional derivative operator of local nature.

An efficient high-order numerical algorithm for the time fractional Fokker–Planck equations

In this paper, a high-order numerical algorithm is derived for a class of time fractional Fokker–Planck equations. To avoid discrete approximation of convection term, we first transform the original equation into an equivalent form, then the spatial derivative and time Riemann–Liouville derivative are approximated by a fourth-order compact formula and a second-order midpoint formula, respectively. The stability and convergence of the developed difference scheme is studied by the energy method. Finally, numerical experiments are provided to demonstrate the effectiveness of the algorithm.

Anomalous diffusion of a dipole interacting with its surroundings.

A fractional Fokker-Planck equation based on the continuous time random walk Ansatz is written via the Langevin equations for the dynamics of a dipole interacting with its surroundings, as represented by a cage of dipolar molecules. This equation is solved in the frequency domain using matrix continued fractions, thus yielding the linear dielectric response for extensive ranges of damping, dipole moment ratio, and cage-dipole inertia ratio, and hence the complex susceptibility. The latter comprises a low frequency band with width depending on the anomalous parameter and a far infrared (THz) band with a comb-like structure of peaks. Several physical consequences of the model relevant to anomalous diffusion in the presence of interactions are discussed. The entire calculation may be regarded as an extension of the cage model interpretation of the dynamics of polar molecules to anomalous diffusion, taking into account inertial effects.

Fractional Hawkes processes

Hawkes processes have a self-excitation mechanism used for modeling the clustering of events observed in natural or social phenomena. In the first part of this article, we find the forward differential equations ruling the probability density function and the Laplace’s transform of the intensity of a Hawkes process, with an exponential decaying kernel. In the second part, we study the properties of the fractional version of this process. The fractional Hawkes process is obtained by subordinating the point process with the inverse of a α-stable Lévy process. This process is not Markov but the probability density function of its intensity is solution of a fractional Fokker–Planck equation. Finally, we find closed form expressions for moments and autocovariance of the fractional intensity.

An efficient analytical approach for solving fractional Fokker-Planck equations

An efficient analytical approach for solving fractional Fokker-Planck equations

Anomalous decay of linear and quadratic states

A study about subdiffusive escape to an absorbing barrier in linear and quadratic potentials is presented. A comparison is made between the mechanisms of the generalized Langevin equation (GLE) and the fractional Fokker–Planck equation (FFPE), in terms of the decay properties of the survival probability in a given region of interest. We find that the asymptotic decay of the survival probability for the case of FFPE is slower (power-law) as compared to that of the GLE (stretched-exponential or exponential). This observation is independent of the details of the underlying potential in which the motion is taking place. As a consequence, the qualitative difference in the escape behavior can be employed as an experimental test to distinguish between the two mechanisms of subdiffusion. This is particularly important in situations where individual trajectories are unavailable.

Anomalous heat diffusion from fractional Fokker–Planck equation

Anomalous heat diffusion, which is commonly characterized by the nonlinear growth of mean square of displacement (MSD), |Δx|2∼tβ (0<β≤2), is usually paired with a length-dependence of effective thermal conductivity κeff, namely, κeff∼Lα with L the system length. In this work, a generic time- and length-dependence of κeff is obtained based on the fractional Fokker–Planck equation (FFPE) with orders γ,μ∈R2, namely, κeff∝tγ−1L−μ. Two existing paradigmatic results, κeff∝tβ−1 and κeff∝L2−2∕β, are first unified in our work, which reflect memory effects and nonlocality in energy fluctuations, respectively. We formulate the effective thermal conductivity in terms of entropy generation, which does not rely on the local-equilibrium hypothesis.