Fractional Klein-Kramers Research Articles

On the correlation between Kappa and Lévy stable distributions

This article investigates the correlation between the Kappa and Lévy distributions via two approaches of the Klein–Kramers equation. The first approach illustrates the velocity distribution functions via the solution of the fractional Klein–Kramers equation obtained using the Riesz fractional derivative. In contrast, the second approach shows the velocity distribution functions according to steady-state Kappa distribution, which arises from the solution of the Klein–Kramers equation with variable coefficients dependent on velocity. We find a unique and straightforward formula representing the relation between the Kappa exponent and the fractality index (Lévy stable index). The results indicate a viable probability distribution obtained from the fractional equations as an alternative to the Kappa distribution. Hence, our results may shed light on the stationary power-law distribution in non extensive statistics and introduce a new correlation between the order of the fractional derivative (α) and the nonthermal index (κ) of the distribution function. Our results also show exact matching with the probability distributions illustrated in the literature.

Super- and subdiffusive positions in fractional Klein–Kramers equations

The Ornstein–Uhlenbeck process is a Gaussian process with applications in various fields, originally as a model for the velocity of a Brownian particle subject to friction force. Its cumulative integral over time thus describes the position of the particle, which however remains overly regular, for instance, concerning Gaussianity and linear mean square displacement. Doubts cast on such over-regularity has led to two generalizations through time-expanding transformation: one with the velocity occasionally paused resulting in superdiffusive positions, the other one with the position trapped once in a while causing subdiffusive positions. The aim of the present work is to systematically derive and present various results to contrast those two similar yet different generalizations. In the framework of the Klein–Kramers equation, those two time-changing mechanisms make only a slight difference in what part of the operator the fractional derivative acts on, whereas by taking the probabilistic approach, we reveal that they are definitively distinct models through a variety of new findings in the theory of the super and subdiffusive positions. We expect our findings to provide an effective means for illustrative comparisons on the relevance of those modeling frameworks.

The local meshless collocation method for solving 2D fractional Klein-Kramers dynamics equation on irregular domains

PurposeThis study aims to propose a new numerical method for solving non-linear partial differential equations on irregular domains.Design/methodology/approachThe main aim of the current paper is to propose a local meshless collocation method to solve the two-dimensional Klein-Kramers equation with a fractional derivative in the Riemann-Liouville sense, in the time term. This equation describes the sub-diffusion in the presence of an external force field in phase space.FindingsFirst, the authors use two finite difference schemes to discrete temporal variables and then the radial basis function-differential quadrature method has been used to estimate the spatial direction. To discrete the time-variable, the authors use two different strategies with convergence orders and for 0 < γ < 1. Finally, some numerical examples have been presented to show the high accuracy and acceptable results of the proposed technique.Originality/valueThe proposed numerical technique is flexible for different computational domains.

Numerical evaluation of the fractional Klein–Kramers model arising in molecular dynamics

The time fractional Klein–Kramers model (TFKKM) is obtained by incorporating the subdiffusive mechanisms into the Klein–Kramers formalism. The TFKKM can efficiently express subdiffusion while an external force field is present in the phase space. The model describes the escape of a particle over a barrier and has a significant role in examining a variety of systems including slow (subdiffusion) dynamics. This paper describes a hybrid algorithm adopting the local radial basis functions based finite difference (LRBF–FD) for the numerical solution of the TFKKM. The time discretization is accomplished via the Grünwald-Letnikov formulation with second-order accuracy and the spatial derivatives are discretized by the LRBF–FD. The LRBF–FD is based on the local support domain that causes to a more sparse matrix and overcomes the ill-conditioning associated with the global collocation. The convergence and stability analysis of the time-discrete algorithm are deduced using the energy method. The feasibility and applicability of the LRBF–FD are demonstrated by using irregular domains. Numerical results are compared with analytical solution and with those obtained by other techniques to verify the accuracy and validity of the LRBF–FD.

Positivity preserving schemes for the fractional Klein-Kramers equation with boundaries

The fractional Klein-Kramers equation describes the process of subdiffusion in the presence of an external force field in phase space and incorporates a fractional operator in time of order α, 0 < α < 1. We present a family of finite volume schemes for the fractional Klein-Kramers equation, that includes first or second-order schemes in phase space, and implicit or explicit schemes in time with an order of accuracy that can change between α and 2−α. It is proved, for the open domain, that the schemes satisfy the positivity preserving property. The positivity preserving property for the explicit schemes imposes a strong condition in the relation between time step, space step and phase step, for small values of α, highlighting the advantage of using implicit schemes in these cases. For a bounded domain in space, two types of boundary conditions are considered, absorbing boundary conditions and reflecting boundary conditions. The inclusion of boundary conditions leads to some technical complications that require changes in the schemes near the boundary. The positivity preserving property holds for the new formulation and the overall accuracy is ensured with the use of non-uniform meshes. Numerical tests are presented in the end to show the convergence of the finite volume schemes.

Fractional diffusion equation for an n-dimensional correlated Lévy walk.

Lévy walks define a fundamental concept in random walk theory that allows one to model diffusive spreading faster than Brownian motion. They have many applications across different disciplines. However, so far the derivation of a diffusion equation for an n-dimensional correlated Lévy walk remained elusive. Starting from a fractional Klein-Kramers equation here we use a moment method combined with a Cattaneo approximation to derive a fractional diffusion equation for superdiffusive short-range auto-correlated Lévy walks in the large time limit, and we solve it. Our derivation discloses different dynamical mechanisms leading to correlated Lévy walk diffusion in terms of quantities that can be measured experimentally.

Generalized Klein–Kramers equation: solution and application

A generalized Klein–Kramers equation based on the continuous time random walk model is investigated. The equation generalizes the ordinary and fractional Klein–Kramers equations. Analytic solutions for the probability density and first two moments (for the force-free case) are obtained, and their dynamic behaviors are investigated in detail. The model is used to describe the cell migration of two migrating transformed renal epithelial Madin–Darby canine kidney (MDCK-F) cell strains: wild-type (NHE+) and NHE-deficient (NHE−) cells. Our theoretical predictions are in good agreement with experimental work in the paper (Dieterich et al 2008 Proc. Nat. Acad. Sci. USA 105 459).

Generalized Klein-Kramers equations

A generalized Klein-Kramers equation for a particle interacting with an external field is proposed. The equation generalizes the fractional Klein-Kramers equation introduced by Barkai and Silbey [J. Phys. Chem. B 104, 3866 (2000)]. Besides, the generalized Klein-Kramers equation can also recover the integro-differential Klein-Kramers equation for continuous-time random walk; this means that it can describe the subdiffusive and superdiffusive regimes in the long-time limit. Moreover, analytic solutions for first two moments both in velocity and displacement (for force-free case) are obtained, and their dynamic behaviors are investigated.

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).

A finite difference approach for the initial-boundary value problem of the fractional Klein-Kramers equation in phase space

Abstract Considering the features of the fractional Klein-Kramers equation (FKKE) in phase space, only the unilateral boundary condition in position direction is needed, which is different from the bilateral boundary conditions in [Cartling B., Kinetics of activated processes from nonstationary solutions of the Fokker-Planck equation for a bistable potential, J. Chem. Phys., 1987, 87(5), 2638–2648] and [Deng W., Li C., Finite difference methods and their physical constrains for the fractional Klein-Kramers equation, Numer. Methods Partial Differential Equations, 2011, 27(6), 1561–1583]. In the paper, a finite difference scheme is constructed, where temporal fractional derivatives are approximated using L1 discretization. The advantages of the scheme are: for every temporal level it can be dealt with from one side to the other one in position direction, and for any fixed position only a tri-diagonal system of linear algebraic equations needs to be solved. The computational amount reduces compared with the ADI scheme in [Cartling B., Kinetics of activated processes from nonstationary solutions of the Fokker-Planck equation for a bistable potential, J. Chem. Phys., 1987, 87(5), 2638–2648] and the five-point scheme in [Deng W., Li C., Finite difference methods and their physical constrains for the fractional Klein-Kramers equation, Numer. Methods Partial Differential Equations, 2011, 27(6), 1561–1583]. The stability and convergence are proved and two examples are included to show the accuracy and effectiveness of the method.

Fractional Klein–Kramers dynamics for subdiffusion and Itô formula

Subdiffusion in the presence of an external force field has been recently described in phasespace by the fractional Klein–Kramers equation. In this paper using a subordinationmethod, we identify a two-dimensional stochastic process (position, velocity) whoseprobability density function is a solution of the fractional Klein–Kramers equation. Thestructure of this process agrees with the two-stage scenario underlying the anomalousdiffusion mechanism, in which trapping events are superimposed on the Langevindynamics. Applying an extension of the celebrated Itô formula for subdiffusion we foundthat the velocity process can be represented explicitly by a corresponding fractionalOrnstein–Uhlenbeck process. A basic feature arising in the context of this stochasticrepresentation is the random change of time of the system made by subordination. For theposition and velocity processes we present a computer visualization of their sample pathsand we derive an explicit expression for the two-point correlation function of thevelocity process. The obtained stochastic representation is crucial in constructing analgorithm to simulate sample paths of the anomalous diffusion, which in turnallows us to detect and examine many relevant properties of the system underconsideration.

Finite difference methods and their physical constraints for the fractional klein-kramers equation

Incorporating subdiffusive mechanisms into the Klein-Kramers formalism leads to the fractional Klein-Kramers equation. Then, the equation can effectively describe subdiffusion in the presence of an external force field in the phase space. This article presents the finite difference methods for numerically solving the fractional Klein-Kramers equation and does the detailed stability and error analyses. The stability condition, mvβ ≤ 16, shows the ratio between the kinetic energy of the particle and the temperature of the fluid can not be too large, which well agrees with the physical property of the subdiffusive particle, we call it “physical constraint.” The numerical examples are provided to verify the theoretical results on rate of convergence. Moreover, we simulate the fractional Klein-Kramers dynamics and the simulation results further confirm the effectiveness of our numerical schemes. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1561–1583, 2010

Spectral density of fluctuations in fractional bistable Klein-Kramers systems

We investigate the stationary spectral density of fractional bistable Klein-Kramers systems. First, we deduce a dissipation-fluctuation relation between the stationary spectral density at thermal equilibrium and the linear response of the system to an applied perturbation. Second, we describe how to obtain the linear dynamic susceptibility from the method of moments, and thus we derive the fluctuating spectral density from the dissipation-fluctuation relation. Finally, we exhibit the structure of this fluctuating spectral distribution and explore the effect of the subdiffusion on it. Compared with the standard bistable Klein-Kramers systems, our observation on the spectral distribution in fractional systems reveals that the subdiffusion weakens the oscillatory components of the intrawell oscillation and the above-barrier motion. This phenomenon should reflect a fact that the particles tend to stand still in separate wells in subdiffusive processes.

Anomalous dynamics of cell migration

Cell movement--for example, during embryogenesis or tumor metastasis--is a complex dynamical process resulting from an intricate interplay of multiple components of the cellular migration machinery. At first sight, the paths of migrating cells resemble those of thermally driven Brownian particles. However, cell migration is an active biological process putting a characterization in terms of normal Brownian motion into question. By analyzing the trajectories of wild-type and mutated epithelial (transformed Madin-Darby canine kidney) cells, we show experimentally that anomalous dynamics characterizes cell migration. A superdiffusive increase of the mean squared displacement, non-Gaussian spatial probability distributions, and power-law decays of the velocity autocorrelations is the basis for this interpretation. Almost all results can be explained with a fractional Klein-Kramers equation allowing the quantitative classification of cell migration by a few parameters. Thereby, it discloses the influence and relative importance of individual components of the cellular migration apparatus to the behavior of the cell as a whole.

Numerical approach to the fractional Klein-Kramers equation

Subdiffusion in the presence of an external force field can be described in phase space by the fractional Klein-Kramers equation. In this paper, we explore the stochastic structure of this equation. Using a subordination method, we define a random process whose probability density function is a solution of the fractional Klein-Kramers equation. The structure of the introduced process agrees with the two-stage scenario underlying the anomalous diffusion mechanism, in which trapping events are superimposed on the Langevin dynamics. We develop an efficient computer algorithm for visualization of fractional Klein-Kramers dynamics and present some simulation results based on Monte Carlo techniques.

Inertial effects in the fractional translational diffusion of a Brownian particle in a double-well potential

The anomalous translational diffusion including inertial effects of nonlinear Brownian oscillators in a double well potential V(x)=ax{2}/2+bx{4}/4 is considered. An exact solution of the fractional Klein-Kramers (Fokker-Planck) equation is obtained allowing one to calculate via matrix continued fractions the positional autocorrelation function and dynamic susceptibility describing the position response to a small external field. The result is a generalization of the solution for the normal Brownian motion in a double well potential to fractional dynamics (giving rise to anomalous diffusion).

Inertial effects in anomalous dielectric relaxation

The linear dielectric response of an assembly of symmetric top dipole molecules (each of which is free to rotate in space) is evaluated in the context of fractional dynamics. The infinite hierarchy of differential-recurrence relations for the relaxation functions appropriate to the dielectric response is derived by using the underlying inertial fractional Fokker–Planck (fractional Klein–Kramers) equation. On solving this hierarchy in terms of continued fractions (as in normal rotational diffusion), the complex dynamic susceptibility is calculated for typical values of the model parameters for the particular case of linear molecules. It is shown that the model can reproduce nonexponential anomalous dielectric relaxation behaviour at low frequencies ( ωτ≤1, where τ is the Debye relaxation time) and the inclusion of inertial effects ensures that optical transparency is regained at very high frequencies (in the far infrared region) so that Gordon's sum rule for integral dipolar absorption is satisfied.

Anomalous dielectric relaxation in a double-well potential

The fractional Klein–Kramers equation describing the fractal time dynamics of an assembly of fixed axis dipoles rotating in a double-well cosine potential representing the internal field due to neighboring molecules is solved using matrix continued fractions. The solution includes both inertial and strong internal field effects which in combination produce a strong resonance peak (Poley absorption) at high frequencies due to librations of the dipoles in the potential, an anomalous relaxation band at low-frequencies mainly arising from overbarrier relaxation and a weaker relaxation band at mid frequencies due to the fast intrawell modes. The high frequency behavior is controlled by the inertia of a dipole, so that the Gordon sum rule for dipolar absorption is satisfied ensuring a return to optical transparency at very high frequencies. Application of the model to the broad-band (0-THz) dielectric loss spectrum of a dilute solution of the probe dipolar molecule CH 2Cl 2 in glassy decalin is demonstrated.

Inertial effects in anomalous dielectric relaxation of symmetrical top molecules.

The linear dielectric response of an assembly of noninteracting symmetrical top molecules (each of which is free to rotate in space) is evaluated in the context of fractional dynamics. The infinite hierarchy of differential-recurrence relations for the relaxation functions appropriate to the dielectric response is derived by using the underlying inertial fractional Klein-Kramers equation. On solving this hierarchy in terms of matrix continued fractions (as in the normal rotational diffusion), the complex dynamic susceptibility is obtained and is calculated for typical values of the model parameters. For the limiting case of spherical top molecules, the solution is obtained in terms of an ordinary continued fraction. It is shown that the model can reproduce nonexponential anomalous dielectric relaxation behavior at low frequencies (omegatau<or=1, where tau is the Debye relaxation time) and the inclusion of inertial effects ensures that optical transparency is regained at very high frequencies (in the far infrared region) so that Gordon's sum rule for integral dipolar absorption is satisfied.

Anomalous diffusion and dielectric relaxation in anN-fold cosine potential

The fractional Klein-Kramers (Fokker-Planck) equation describing the fractal time dynamics of an assembly of fixed axis dipoles rotating in an N-fold cosine potential representing the internal field due to neighboring molecules is solved using matrix continued fractions. The result can be considered as a generalization of the solution for the normal Brownian motion in a cosine periodic potential to fractional dynamics (giving rise to anomalous diffusion) and also represents a generalization of Fröhlich's model of relaxation over a potential barrier. The solution includes both inertial and strong internal field effects, which in combination produce a strong resonance peak (Poley absorption) at high frequencies due to librations of the dipoles in the potential, an anomalous relaxation band at low frequencies mainly arising from overbarrier relaxation, and a weaker relaxation band at midfrequencies due to the fast intrawell modes. The high-frequency behavior is controlled by the inertia of the dipole, so that the Gordon sum rule for dipolar absorption is satisfied, ensuring a return to optical transparency at very high frequencies. Application of the model to the broadband (0-THz) dielectric loss spectrum of a dilute solution of the probe dipolar molecule CH2Cl2 in glassy decalin is demonstrated.